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PhysRevB.79.115102.pdf	Disproportionation and electronic phase separation in parent manganite LaMnO 3 A. S. Moskvin Ural State University, 620083 Ekaterinburg, Russia /H20849Received 22 September 2008; revised manuscript received 7 December 2008; published 4 March 2009 /H20850 Nominally pure undoped parent manganite LaMnO 3exhibits a puzzling behavior inconsistent with a simple picture of an A-type antiferromagnetic insulator /H20849A-AFI /H20850with a cooperative Jahn-Teller ordering. We do assign its anomalous properties to charge transfer /H20849CT/H20850instabilities and competition between insulating A-AFI phase and metalliclike dynamically disproportionated phase formally separated by a ﬁrst-order phase transition atT disp=TJT/H11015750 K. The unconventional high-temperature phase is addressed to be a speciﬁc electron-hole /H20849EH/H20850Bose liquid /H20849EHBL /H20850rather than a simple “chemically” disproportionated La /H20849Mn2+Mn4+/H20850O3phase. The phase does nucleate as a result of the CT instability and evolves from the self-trapped CT excitons or speciﬁcEH dimers, which seem to be a precursor of both insulating and metalliclike ferromagnetic phases observed inmanganites. We arrive at highly frustrated system of triplet /H20849e g2/H208503A2gbosons moving in a lattice formed by hole Mn4+centers. Starting with different experimental data we have reproduced a typical temperature dependence of the volume fraction of high-temperature mixed-valence EHBL phase. We argue that a slight nonisovalentsubstitution, photoirradiation, external pressure, or magnetic ﬁeld gives rise to an electronic phase separationwith a nucleation or an overgrowth of EH droplets. Such a scenario provides a comprehensive explanation ofnumerous puzzling properties observed in parent and nonisovalently doped manganite LaMnO 3including an intriguing manifestation of superconducting ﬂuctuations. DOI: 10.1103/PhysRevB.79.115102 PACS number /H20849s/H20850: 71.30. /H11001h, 75.47.Lx, 71.35. /H11002y I. INTRODUCTION Perovskite manganites RMnO 3/H20849R=rare earth or yttrium /H20850 manifest many extraordinary physical properties. UndopedTbMnO 3and DyMnO 3reveal multiferroic behavior.1Under nonisovalent substitution all the orthorhombic manganitesreveal an insulator-to-metal /H20849IM/H20850transition and colossal magnetoresistance /H20849CMR /H20850effect which are currently ex- plained in terms of an electronic phase separation /H20849EPS /H20850trig- gered by a hole doping. Overview of the current state of theart with theoretical and experimental situation in dopedCMR manganites R 1−xSr/H20849Ca/H20850xMnO 3can be found in many review articles.2–6 However, even nominally pure undoped stoichiometric parent manganite LaMnO 3does exhibit a puzzling behavior inconsistent with a simple picture of an A-type antiferromag- netic insulator /H20849A-AFI /H20850which it is usually assigned to.2–6 First it concerns anomalous transport properties,7–9photoin- duced /H20849PI/H20850absorption,10pressure-induced effects,11dielectric anomalies,12and the high ﬁeld-induced IM transition.13Be- low, in the paper we demonstrate that the unconventionalbehavior of parent manganite LaMnO 3can be explained to be a result of an electronic phase separation inherent even fornominally pure stoichiometric manganite with a coexistenceof conventional A-AFI phase and unconventional electron-hole /H20849EH/H20850Bose liquid /H20849EHBL /H20850which nucleation is a result of a charge transfer /H20849CT/H20850instability of A-AFI phase. In a sense, hereafter we report a comprehensive elaboration of aso-called “disproportionation” scenario in manganites whichwas addressed earlier by many authors; however, by now itwas not properly developed. The paper is organized as follows. In Sec. IIwe discuss an unconventional ﬁrst-order phase transition in parent manga-nite LaMnO 3and argue that it should be addressed to be a disproportionation rather than a Jahn-Teller /H20849JT/H20850phase tran-sition. Then we show that the resonant x-ray scattering data can be used to reconstruct a “phase diagram” which shows atentative temperature dependence of the volume fraction oftwo competing phases for parent LaMnO 3. The electron- lattice relaxation effects and the self-trapping of the CT ex-citons with nucleation of electron-hole droplets are consid-ered in Sec. III. In Sec. IVwe describe the details of the charge and spin structure of electron-hole dimers to be themain building blocks of the EHBL phase in a parent manga-nite. The effective Hamiltonian of the EHBL phase equiva-lent to a triplet boson double-exchange /H20849DE/H20850model is ad- dressed in Sec. V. Numerous optical, magnetic, and other manifestations of the EH dimers and EH droplets in parentand low-hole-doped manganites are considered in Sec. VI. Short comments on the hole doping effects are made in Sec.VII. Short conclusions are presented in Sec. VIII. II. EXPERIMENTAL SIGNATURES OF DISPROPORTIONATION AND ELECTRONIC PHASE SEPARATION IN PARENT MANGANITE LaMnO 3 A. Unconventional ﬁrst-order phase transition in LaMnO 3 Measurements on single crystals of the high-temperature transport and magnetic properties,7–9,14resonant x-ray scattering,15,16and neutron-diffraction17studies of the RMnO 3family point to a ﬁrst-order electronic phase transi- tion at T=TJT/H20849TJT/H11015750 K in LaMnO 3/H20850from the low- temperature orbitally ordered /H20849OO /H20850antiferromagnetic insu- lating phase /H20849O/H11032orthorhombic Pbnm /H20850, with a cooperative Jahn-Teller ordering of the occupied orbitals of the MnO 6 octahedra to a high-temperature charge and orbitally disor-dered phase /H20849O orthorhombic or “pseudocubic” Pbnm /H20850.I ti s worth noting that the “ﬁrst orderness” is rather unexpectedpoint for the cooperative Jahn-Teller ordering as a commonPHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 1098-0121/2009/79 /H2084911/H20850/115102 /H2084919/H20850 ©2009 The American Physical Society 115102-1viewpoint implies that it is to be a second-order “order- disorder-type” phase transition. According to the conven-tional model of the ﬁrst-order phase transitions, there are two characteristic temperatures, T 1/H11569/H11021TJTand T2/H11569/H11022TJT/H20849“super- cooling” and “superheating” spinodals, respectively /H20850, which determine the temperature range of the coexistence of bothphases. Both temperatures are hardly deﬁned for parent man-ganites. A change in slope of the temperature dependence of the thermoelectric power at T 1/H11569/H11015600 K in LaMnO 3/H20849Refs. 9 and14/H20850is considered to be due to nucleation of an orbitally disordered phase on heating or homogeneous nucleation ofthe low- TOO phase on cooling. The volume fraction of charge and orbitally disordered phase monotonically grows with increasing temperature in the interval T 1/H11569/H11021T/H11021TJTbut increases discontinuously on heating across TJT. The low- T OO phase looses stability only at T2/H11569/H11022TJT. Weak diffuse x-ray scattering consistent with orbital ﬂuctuations was ob-served in LaMnO 3with the intensity falling gradually with increasing temperature and disappearing above T2/H11569 /H110111000 K concomitant with the suppression of the octahe- dral tilt ordering and a structural transition to a rhombohedralphase. 16 The x-ray diffraction data18for LaMnO 3have revealed a coexistence of two orthorhombic Pbnm phases O /H11032a n dOi n a wide temperature range both below and above TJT.I t means that a sizable volume fraction of large /H20849/H110111000 Å /H20850 domains of low- /H20849high- /H20850temperature phase survives between TJTandT2/H11569/H20849T1/H11569andTJT/H20850, respectively. However, it does not prevent the nanoscopic size droplets to survive outside thistemperature range. Furthermore, the neutron-diffraction mea-surements /H20849T/H11021300 K /H20850for several samples of nominal com- position LaMnO 3after different heat treatments seemingly provoking the nucleation of a high-temperature phase19have revealed a coexistence of bare orthorhombic A-AFI phasewith another orthorhombic and rhombohedral ferromagneticphases with a considerably /H20849/H110112%/H20850smaller unit-cell volume and ordering temperatures T Cnear TN. Puzzlingly, this coex- istence spreads out over all temperature range studied fromroom temperature up to 10 K. Similar effects have been ob-served in a complex /H20849ac initial magnetic susceptibility, mag- netization, magnetoresistance, and neutron-diffraction /H20850study /H20849T/H11021300 K /H20850of slightly nonstoichiometric LaMnO 3+/H9254 system.20Interestingly that all over the ferromagnetic phases the thermal factors of oxygen atoms present an excess /H9004B /H110110.3–0.5 Å2as compared with antiferromagnetic A-AFI phase that points to a speciﬁc role of dynamic lattice effects. Even in the absence of chemical doping, LaMnO 3shows the ability to accommodate a so-called “oxidative nonstoichi-ometry,” which also involves the partial oxidation of someMn 3+to Mn4+which smaller size leads to an increase in the tolerance factor, thus stabilizing the perovskite structure.21 The manganite crystals grown by the ﬂoating zone methodseem to preserve well-developed traces of the high-temperature phase. Interestingly, that the LaMnO 3crystals do not tolerate repeated excursions to high temperatures, 800 K,before changing their properties. Such an anomalousmemory effect with an overall loss of long-range orbital or-der in one sample of the LaMnO 3after extended cycling above 1000 K and cooling back to room temperature wasobserved by Zimmermann et al. 16It is worth mentioning thatthe characteristic temperatures T1/H11569,TJT, and T2/H11569for the phase transition are believed to depend on the initial content ofMn 4+/H20849Ref. 17/H20850: the sample used in Ref. 22gave TJT=600 K and T2/H11569=800 K, suggesting the presence of a non-negligible amount of Mn4+that reduces the temperatures of the phase transition. All these data evidence an existenceof electronic phase separation inherent for parent stoichio-metric LaMnO 3with the phase volume fraction sensitive to sample stoichiometry, prehistory, and morphology. B. Disproportionation rather than the JT nature of the phase transition in parent LaMnO 3 The electronic state in the high-temperature O orthorhom- bic phase of parent LaMnO 3remains poorly understood. The transport measurements9/H20851resistivity /H9267/H20849T/H20850and thermoelectric power /H9251/H20849T/H20850; see, also Refs. 7,8,23, and 24/H20852were interpreted by the authors as a striking evidence of the R/H20849Mn2+Mn4+/H20850O3 disproportionation rather than a simple orbitally disordered RMn3+O3character of the high-temperature phase. Let us shortly overview the argumentation by Zhou andGoodenough. 9Thermoelectric power reveals an irreversible change from /H9251/H20849300 K /H20850=−600 /H9262V/K to about 550 /H9262V/K on thermal cycling to 1100 K with a nearly zero value atT/H11022T JT. Small-polaron conduction by a single charge carrier would give a temperature-independent thermoelectric powerdominated by the statistical term /H9251=− /H20849k/e/H20850ln/H20851/H208491−c/H20850/c/H20852, /H208491/H20850 where cis the fraction of Mn sites occupied by a charge carrier and the spin degree of freedom is lifted by the strongintra-atomic exchange. Near stoichiometry, two types ofcharge carriers may be present but with only one dominatingat room temperature to give a large negative or large positive /H9251/H20849300 K /H20850for a small value of c. From Eq. /H208491/H20850value of /H9251/H20849300 K /H20850/H11015/H11006600/H9262V/K in the virgin crystal reﬂects a small fraction /H20849c/H110150.001 /H20850of a imbalance between electron- like and holelike mobile/immobile charges. An abrupt dropin /H9251/H20849T/H20850and/H9267/H20849T/H20850atTJTto a nearly temperature-independent and a nearly zero value for T/H11022TJTwith a reversible behavior of both quantities agrees with a phase transition to a fullydisproportionated Mn 2++Mn4+or, more precisely, to an electron-hole liquid phase25–27with a two-particle transport andceff=0.5. However, the system retains a rather high value of resistivity, that is, the EH liquid phase manifests a “poor”metal behavior. Strictly speaking, the disproportionationphase transition at T=T disp=TJTis governed ﬁrst by a charge order rather than the orbital order parameter. In other words,the Jahn-Teller ordering at T=T JTonly accompanies the charge ordering at T=Tdisp=TJT; hence a simpliﬁed Jahn- Teller picture does misinterpret a true sense of the phenom-enon. In contrast with the high-temperature measurements car- ried out in a vacuum of 10 −3torr,9the transport measure- ments performed in air28evidenced another evolution of /H9251/H20849T/H20850/H20849see Fig. 1/H20850. On heating the thermoelectric power starts from large but positive values and on cooling from T/H11022TJT/H9251/H20849T/H20850does not return to its original value because the sample, according to authors,28becomes slightly /H20849/H110111%/H20850oxi-A. S. MOSKVIN PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-2dized. A simple comparison of the two data sets9,28points to an unconventional behavior of parent manganite on crossing the “supercooling spinodal” temperature T1/H11569. The system can memorize a high-temperature phase up to temperatures be-low 300 K. The role of a slight oxidation seemingly reducesto be an additional regulative factor governing the A-AFI/EHBL phase volume fraction. Strong and irreversible temperature dependence of /H9251/H20849T/H20850 and/H9267/H20849T/H20850atT/H11021T1/H11569agrees with a scenario of a well-developed electronic phase separation with a puzzling electron-holesymmetry and a strong sensitivity of transport propertiesboth to sample morphology and quality. The magnitude ofthe resistivity and character of irreversibility agrees with apoor metal like conductivity of high-temperature phase and points to a considerable volume fraction of this phase tosurvive up to room temperature. Resistivity of differentsamples of the nominally same composition can differ byorders of magnitude. Interestingly that these data point to apossibility of colossal, up to 6 orders of magnitude, varia-tions in resistivity in parent LaMnO 3at a constant tempera- ture well below TJTonly due to the variation in its A-AFI/ EHBL volume fraction composition which can be realized bythe temperature change, pressure, isotopic substitution, appli- cation of external magnetic/electric ﬁeld, and photoirradia-tion. This behavior can hardly be directly related with thecolossal magnetoresistivity observed for the hole doped man-ganites; however, this phase can be an important participantof electronic transformations in manganites. Below T=T 1/H11569/H20849T1/H11569/H11015600 K in LaMnO 3/H20850or the temperature of the homogeneous nucleation of the low- TOO phase, the high-temperature mixed-valence EH phase loses stability,however, it survives due to various charge inhomogeneitiesforming EH droplets pinned by statically ﬂuctuating electricﬁelds. C. Temperature dependence of the EHBL volume fraction By now we have no information about how both phases share the volume fraction on cooling from high temperatures.Clearly, such an information depends strongly on the tech-niques used. For instance, both long-lived static domains andshort-lived dynamic ﬂuctuations of either phase contribute tooptical response, while only the large static domains are seenin conventional x-ray or neutron scattering measurements.Fortunately, the resonant x-ray scattering data 15,16can be used to reconstruct the tentative T-fEHphase diagram of a manganite with fEHbeing a volume fraction of EH droplets. Indeed, the intensity of this scattering depends on the size ofthe splitting /H9004of the Mn 4 plevels, induced by the orbital ordering of Mn 3 de gstates, hence is nonzero only for orbit- ally ordered Mn3+ions in distorted MnO 6octahedra. The ﬁrst-order nature of the cooperative JT phase transition inLaMnO 3/H20849Ref. 9/H20850implies that the local orbital order param- eter such as /H9004in Ref. 15remains nearly constant below the transition temperature;29hence the temperature behavior of resonant x-ray scattering intensity has to reﬂect the tempera-ture change in the net /H20849static+dynamic /H20850OO phase volume fraction rather than /H9004/H20849T/H20850effect. This suggestion agrees with the neutron-diffraction studies by Rodríguez-Carvajal et al., 17evidencing no visible effect of the antiferromagnetic spin ordering at T=TN/H11015140 K on the OO parameter, while the x-ray scattering intensity dramatically /H20849up to 40% /H20850falls upon heating above TN.15Overall, the temperature depen- dence of the resonant x-ray scattering intensity in LaMnO 3 shows up an unusual behavior with an arrest or even clearhole between 300 and 500 K, a sharp downfall above T=T 1/H11569/H11015600 K, and vanishing right after T=TJT/H11015750 K. Thus, the x-ray data15,16can be used to ﬁnd the temperature behavior of the resultant static and dynamic EH droplet vol-ume fraction in the sample. In Fig. 1we have reproduced experimental data from Refs. 15and16renormalized and transformed into a relative volume fraction of a “non-OO” phase which is supposed tobe an EH droplet phase. The renormalization implied thelow-temperature 75% volume fraction of the OO phase. Dif-ferent ﬁlling /H20849from top to bottom /H20850points to an A-AFI phase, orbital ﬂuctuation phase near T JT, and dynamic and static EH droplet phase. Despite the overall fall of the EH droplet vol-ume fraction on cooling from T JT, we expect some intervals of the re-entrant behavior due to a subtle competition of twophases. It is clear that any ordering does lower the free en-T= Tdisp JTO orthorhombic JT ordering A-type AFM’ T( K )fEH TN01.0 dynamic static 200 400 600 800EH droplets 1000 T1T21000 300 400 500 600 700 800 900-400-2000200400600 T (K)-600Seebeck ( V/K)/CID1Resistivity ( cm) /CID3 102 10-210-1103104 100101105 Seebeck ResistivityLaMnO3 O orthorhombic FM-EHBL TC Tg? FIG. 1. /H20849Color online /H20850Top panel: temperature dependence of thermoelectric power and resistivity in parent manganite LaMnO 3 /H20849reproduced from Refs. 9and28/H20850. Bottom panel: schematic T-fEH phase diagram of a parent perovskite manganite, fEHbeing the vol- ume fraction of mixed-valence phase. Small and large circles showup experimental data from Refs. 15and16transformed into a re- sultant volume fraction of a non-OO phase supposed to be a systemof static and dynamic EH droplets. Different ﬁlling /H20849from top to bottom /H20850points to an A-AFI phase, orbital ﬂuctuation phase near T JT, and dynamic and static EH droplet phase. Note a difference in TJTvalues in Refs. 15and16and Ref. 9.DISPROPORTIONATION AND ELECTRONIC PHASE … PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-3ergy of the phase thus resulting in a rise of its volume frac- tion. Taking into account experimental data from Ref. 19 pointing to close temperatures of AFI and ferromagnetic in-sulator /H20849FI/H20850orderings in competing phases /H20849T NandTC, re- spectively /H20850, we may assign a signature of a re-entrant behav- ior at 400–600 K to a glasslike transition within the EHliquid near T=T g/H11011400 K. Surely, we are aware that the picture shown in Fig. 1is not a real phase diagram; however, it is very instructive for a qualitative understanding of a com-plex phase competition in parent manganite. Concluding the section, we should once more emphasize a dramatic charge instability of parent manganite LaMnO 3 with extreme sensitivity to different external factors, samplestoichiometry, and prehistory. In this connection, it is worthnoting that highly stoichiometric LaMnO 3samples measured by Subías et al.30did not show noticeable temperature de- pendence of the resonant intensity for the /H208493,0,0 /H20850reﬂection from 10 to 300 K, in contrast with the data by Murakami et al.15Further work at an even higher temperature range and for different samples seems to be necessary in order to dis-tinctly reveal and examine the phase-separated state in a par-ent manganite. III. ELECTRON-LATTICE RELAXATION AND NUCLEATION OF EH DROPLETS IN A PARENT MANGANITE A. Electron-lattice relaxation and self-trapping of CT excitons At ﬁrst glance the disproportionation in manganese com- pounds is hardly possible since manganese atom does notmanifest a valence-skipping phenomenon as, e.g., bismuthatom which can be found as Bi 3+or Bi5+, but not Bi4+, with a generic bismuth oxide BaBiO 3to be a well-known example of a charge disproportionated system. Strictly speaking,sometimes manganese reveals a valence preference, e.g.,while both Mn 2+and Mn4+are observed in MgO:Mn and CaO:Mn, the Mn3+center is missing.31Furthermore, the d4 conﬁguration of Mn3+ion is argued32to be a missing oxida- tion state due to the large exchange-correlation energy gainthat stabilizes the d 5electronic conﬁguration thus resulting in the charge disproportionation or dynamical charge ﬂuctua-tiond 4+d4→d3+d5. The reason for valence skipping or valence preference observed for many elements still remains a mystery. Re-cently, Harrison 33argued that most likely traditional lattice relaxation effects, rather than any intra-atomic mechanisms/H20849speciﬁc behavior of ionization energies, stability of closed shells, and strong screening of the high-charged states /H20850, are a driving force for disproportionation with formation of so-called “negative- U” centers. Anyhow the disproportionation in an insulator signals a well-developed CT instability. What is a microscopic originof the CT instability in parent manganites? The dispropor-tionation reaction can be considered to be a ﬁnal stage of aself-trapping of the d-dCT excitons /H20849Mott-Hubbard exci- tons /H20850that determine the main low-energy CT band peaked near 2 eV in LaMnO 3.34Indeed, these two-center excitations due to a charge transfer between two MnO 6octahedra may be considered as quanta of the disproportionation reaction,MnO69−+ MnO69−→MnO68−+ MnO610−, /H208492/H20850 with the creation of electron MnO610−and hole MnO48−cen- ters. Within a simplest model26the former corresponds to a nominal 3 d5/H20849Mn2+/H20850conﬁguration, while the latter does to the 3 d3/H20849Mn4+/H20850one. The minimal energy cost of the optically excited dispro- portionation or electron-hole formation in insulating manga-nites is 2.0–2.5 eV. 34However, the question arises: what is the energy cost for the thermal excitation of such a localdisproportionation or effective correlation energy U? The an- swer implies ﬁrst of all the knowledge of relaxation energyor the energy gain due to the lattice polarization by the lo-calized charges. The full polarization energy Rincludes the cumulative effect of electronic and ionic terms related with the displacement of electron shells and ionic cores,respectively. 35The former term Roptis due to the nonretarded effect of the electronic polarization by the momentarily lo-calized electron-hole pair given the ionic cores ﬁxed at theirperfect crystal positions. Such a situation is typical for latticeresponse accompanying the Franck-Condon transitions /H20849op- tical excitation and photoionization /H20850. On the other hand, all the long-lived excitations, i.e., all the intrinsic thermally ac-tivated states and the extrinsic particles produced as a resultof doping, injection, or optical pumping, should be regardedas stationary states of a system with a deformed lattice struc-ture. The lattice relaxation energies, − /H9004R th, associated with the hole/electron localization in 3 doxides are particularly large. For instance, in LaMnO 3the optical /H20849nonrelaxed /H20850energies of the creation of the hole on Mn and O sites are 2.6 and 4.9 eV, respectively, while − /H9004RthMn=0.7–0.8 and − /H9004RthO=2.4 eV.36 In other words, the electronic hole is marginally more stable at the Mn site than at the O site in the LaMnO 3lattice; however, both possibilities should be treated seriously. Shell-model estimations36yield for the energy of the op- tically excited disproportionation /H208492/H20850or electron-hole forma- tion in parent manganite LaMnO 3:Eopt/H110153.7 eV, while the respective thermal relaxation energy is estimated as−/H9004R th/H110151.0 eV. Despite the estimations imply the noninter- acting electron and hole centers these are believed to providea sound background for any reasonable models of self-trapped d-dCT excitons. Thorough calculation of the local- ization energy for electron-hole dimers remains a challeng-ing task for future studies. It is worth noting that despite theirvery large several eV magnitudes, the relaxation effects arenot incorporated into current theoretical models of mangan-ites. Figure 2illustrates two possible ways the electron-lattice polarization governs the CT exciton evolution. Shown arethe adiabatic potentials /H20849APs /H20850for the two-center ground-state /H20849GS/H20850M 0-M0conﬁguration and excited M/H11006-M/H11007CT or dis- proportionated conﬁguration. The Qcoordinate is related with a lattice degree of freedom. For lower branch of AP inthe system we have either a single minimum point for the GSconﬁguration /H20851Fig.2/H20849a/H20850/H20852or a two-well structure with an ad- ditional local minimum point /H20851Fig.2/H20849b/H20850/H20852associated with the self-trapped CT exciton. This “bistability” effect is of pri-mary importance for our analysis. Indeed, these two minimaA. S. MOSKVIN PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-4are related with two /H20849meta /H20850stable charge states with and without CT, respectively, which form two candidates tostruggle for a ground state. It is worth noting that the self-trapped CT exciton may be described as a conﬁguration withnegative disproportionation energy U. Thus one concludes that all the systems such as manganites may be divided intotwo classes: CT stable systems with the only lower AP branch minimum for a certain charge conﬁguration, andbistable, or CT unstable systems with two lower AP branch minima for two local charge conﬁgurations, one of which isassociated with the self-trapped CT excitons resulting fromself-consistent charge transfer and electron-lattice relaxation.Such excitons are often related with the appearance of thenegative- Ueffect. It means that the three types of MnO 6 centers MnO48,9,10−should be considered in manganites on equal footing.26,27 Above we have presented a generalized disproportion- ation scenario for parent manganites in which an unconven-tional phase state with a 2Mn 3+→Mn2++Mn4+dispropor- tionation nominally within manganese subsystem evolvesfrom EH dimers or self-trapped d-dCT excitons. However, such a scenario in parent manganites would compete withanother “asymmetric” disproportionation scenario, Mn 3++O2−→Mn2++O1−, /H208493/H20850 which evolves from a self-trapping of low-energy p-dCT excitons. Indeed, we should make a remarkable observation,which to the best of our knowledge has not been previouslyreported for these materials, that is, the famous “manganite”2 eV absorption band has a composite structure being a su-perposition of a rather broad and intensive CT d-dband and several narrow and relatively weak CT p-dbands. 34,37A dual nature of the dielectric gap in nominally stoichiometric par-ent perovskite manganites RMnO 3, being formed by a super- position of forbidden or weak dipole allowed p-dCT transi- tions and intersite d-dCT transitions, means that these should rather be sorted neither into the CT insulator nor theMott-Hubbard insulator in the Zaanen-Sawatzky-Allen 38 scheme. A detailed analysis of the CT p-dtransitions in LaMnO 3has been performed by the present author in Ref. 37. Among the ﬁrst p-dcandidates for a self-trapping we should point to the low-energy CT state /H20851/H20849t2g34A2g;eg23A2g;6A1g/H20850;t/H60181g/H208525,7T1gin MnO69−octahedron which arises as a result of the O 2 pelectron transfer from the highest in energy nonbonding t1gorbital to the egmanganese orbital. Simplest view of this exciton implies the oxygen t1g hole rotating around nominally Mn2+ion with ferro- /H208497T1g/H20850orantiferro- /H208495T1g/H20850ordering. It has a number of unconventional properties. First, orbitally degenerated ground T1gstate im- plies a nonquenched orbital moment and strong magneticanisotropy. May be more important to say that we deal witha Jahn-Teller center unstable with regard to local distortions. Second, we expect a high-spin S=3 ground state 7T1gbe- cause of usually ferromagnetic p-dexchange coupling. Oxy- gen holes can form the so-called O−bound small polarons.39 Shell-model estimations36yield for the energy of optically excited asymmetric disproportionation /H208493/H20850in parent manga- nite LaMnO 3:Eopt/H110154.75 eV, while the respective thermal relaxation energy is estimated as − /H9004Rth/H110151.25 eV. However, these qualitative estimations do not concern a number ofimportant points such as p-dandp-pcovalencies, and a par- tial delocalization of oxygen holes. A sharp electron-hole asymmetry and a rather big S=3 ground-state spin value most likely exclude the self-trapped p-dCT excitons as candidates to form a high-temperature T/H11022T JTphase of parent manganite. However, the “danger- ous” closeness to the ground state makes them to be thepotential participants of any perturbations taking place forparent manganites. B. Nucleation of EH droplets in a parent manganite The AP bistability in CT unstable insulators points to tempting perspectives of their evolution under either externalimpact. Metastable CT excitons in the CT unstable M 0phase or EH dimers present candidate “relaxed excited states” tostruggle for stability with ground state and the natural nucle-ation centers for electron-hole liquid phase. What way theCT unstable M 0phase can be transformed into novel phase? It seems likely that such a phase transition could be realizeddue to a mechanism familiar to semiconductors with ﬁlledbands such as Ge and Si where given certain conditions oneobserves a formation of metallic EH liquid as a result of theexciton decay. 40However, the system of strongly correlated electron M−and hole M+centers appears to be equivalent to an electron-hole Bose liquid in contrast with the electron-hole Fermi liquid in conventional semiconductors. The Mott-Wannier excitons in the latter wide-band systems dissociateeasily producing two-component electron-hole gas orplasma, 40while small CT excitons both free and self-trapped are likely to be stable with regard to the EH dissociation. Atthe same time, the two-center CT excitons have a very largeﬂuctuating electrical dipole moment /H20841d/H20841/H110112eR MMand can be involved into attractive electrostatic dipole-dipole interac-tion. Namely, this is believed to be important incentive to theproliferation of excitons and its clusterization. The CT exci-tons are proved to attract each other and form moleculescalled biexcitons, and more complex clusters, or excitonicstrings, where the individuality of the separate exciton islikely to be lost. Moreover, one may assume that like thesemiconductors with indirect band gap structure, it is ener-getically favorable for the system to separate into a low den-sity exciton phase coexisting with the microregions of a highdensity two-component phase composed of electron M −and hole M+centers or EH droplets. Indeed, the excitons may be considered to be well deﬁned entities only at small content,Q QU<0 U>0 U>0 self-trapped CT excitonCT exciton b) a)Mn -Mn3+ 3+Mn -Mn3+ 3+Mn -Mn2+ 4+ FIG. 2. /H20849Color online /H20850Simple illustration of the electron-lattice polarization effects for CT excitons /H20849see text for details /H20850.DISPROPORTIONATION AND ELECTRONIC PHASE … PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-5whereas at large densities their coupling is screened and their overlap becomes so considerable that they loose individual-ity and we come to the system of electron M −and hole M+ centers, which form a metalliclike electron-hole Bose liquid with a main two-particle transport mechanism.27An increase in injected excitons in this case merely increases the size ofthe EH droplets, without changing the free exciton density. An EH droplet seems to have no distinct boundary, most likely it looks like a core with more or less stable electronand hole centers surrounded by a cloud of metastable CTexcitons. Homogeneous nucleation implies the spontaneous formation of EH droplets due to the thermodynamic ﬂuctua-tions in exciton gas. Generally speaking, such a state with anonzero volume fraction of EH droplets and the spontaneousbreaking of translational symmetry can be stable in nomi-nally pure insulating crystal. However, the level of intrinsicnonstoichiometry in 3 doxides is signiﬁcant /H20849one charged defect every 100–1000 molecular units is common /H20850. The charged defect produces random electric ﬁeld, which can bevery large /H20849up to 10 8Vc m−1/H20850thus promoting the condensa- tion of CT excitons and the inhomogeneous nucleation of EH droplets. Deviation from the neutrality implies the existence of ad- ditional electron or hole centers that can be the natural cen-ters for the inhomogeneous nucleation of the EH droplets.Such droplets are believed to provide a more effectivescreening of the electrostatic repulsion for additionalelectron/hole centers than the parent insulating phase. As aresult, the electron/hole injection to the insulating M 0phase due to a nonisovalent substitution as in La 1−xSrxMnO 3or change in stoihiometry as in La xMnO 3, LaMnO 3−/H9254, or ﬁeld effect is believed to shift the phase equilibrium from theinsulating state to the unconventional electron-hole Bose liq-uid or in other words induce the insulator-to-EHBL phasetransition. This process results in a relative increase in theenergy of the parent phase and creates proper conditions forits competing with other phases capable to provide an effec-tive screening of the charge inhomogeneity potential. Thestrongly degenerate system of electron and hole centers inEH droplet is one of the most preferable ones for this pur-pose. At the beginning /H20849nucleation regime /H20850an EH droplet nucleates as a nanoscopic cluster composed of several num-bers of neighboring electron and hole centers pinned by dis-order potential. It is clear that such a situation does not ex-clude the self-doping with the formation of a self-organizedcollective charge-inhomogeneous state in systems which arenear the charge instability. EH droplets can manifest itself remarkably in various properties of the 3 doxides even at small volume fraction or in a “pseudoimpurity regime.” Insulators in this regimeshould be considered as phase inhomogeneous systems with,in general, thermoactivated mobility of the interphase bound-aries. On the one hand, main features of this pseudoimpurityregime would be determined by the partial intrinsic contri-butions of the appropriate phase components with possiblelimitations imposed by the ﬁnite size effects. On the otherhand, the real properties will be determined by the peculiargeometrical factors such as a volume fraction, the averagesize of droplets and its dispersion, the shape and possibletexture of the droplets, and the geometrical relaxation rates.These factors are tightly coupled, especially near phase tran- sitions for either phase /H20849long-range antiferromagnetic order- ing for the parent phase, the charge ordering, and other phasetransformations for the EH droplets /H20850accompanied by the variation in a relative volume fraction. Numerous examples of the unconventional behavior of the 3 doxides in the pseudoimpurity regime could be easily explained with taking into account the interphase boundaryeffects /H20849coercitivity, mobility threshold, non-Ohmic conduc- tivity, oscillations, relaxation, etc. /H20850and corresponding char- acteristic quantities. Under increasing doping the pseudoim-purity regime with a relatively small volume fraction of EHdroplets /H20849nanoscopic phase separation /H20850can gradually trans- form into a macro /H20849chemical /H20850“phase-separation regime” with a sizable volume fraction of EH droplets and ﬁnally to an-other EH liquid phase. IV . ELECTRON-HOLE DIMERS IN PARENT MANGANITE A. EH dimers: Physical versus chemical view Parent manganites are believed to be unconventional sys- tems which are unstable with regard to a self-trapping of thelow-energy charge transfer excitons which are precursors ofnucleation of the EH Bose liquid. Hereafter we should em-phasize once more that a view of the self-trapped CT excitonto be a Mn 2+-Mn4+pair is typical for a chemical view of disproportionation and is strongly oversimpliﬁed. Actuallywe deal with an EH dimer to be a dynamically charge ﬂuc- tuating system of coupled electron MnO 610−and hole MnO48− centers having been glued in a lattice due to a strong electron-lattice polarization effects. In other words, weshould proceed with a rather complex physical view of dis- proportionation phenomena which ﬁrst implies a charge ex-change reaction, Mn 2++M n4+↔Mn4++M n2+, /H208494/H20850 governed by a two-particle charge transfer integral, tB=/H20855Mn2+Mn4+/H20841HˆB/H20841Mn4+Mn2+/H20856, /H208495/H20850 where HˆBis an effective two-particle /H20849bosonic /H20850transfer Hamiltonian, and we assume a parallel orientation of all thespins. As a result of this quantum process the bare ionicstates with site-centered charge order and the same bare en-ergy E 0transform into two EH-dimer states with an indeﬁ- nite valence and bond-centered charge order, /H20841/H11006/H20856=1 /H208812/H20849/H20841Mn2+Mn4+/H20856/H11006/H20841Mn4+Mn2+/H20856/H20850 /H20849 6/H20850 with the energies E/H11006=E0/H11006tB. In other words, the exchange reaction restores the bare charge symmetry. In both /H20841/H11006/H20856 states the site manganese valence is indeﬁnite with quantumﬂuctuations between +2 and +4, however, with a mean value+3. Interestingly that, in contrast with the ionic states, theEH-dimer states /H20841/H11006/H20856have both a distinct electron/hole and an inversion symmetry, even parity /H20849s-type symmetry /H20850for /H20841+/H20856and odd parity /H20849p-type symmetry /H20850for /H20841−/H20856states, respec- tively. Both states are coupled by a large electric-dipole ma-trix element,A. S. MOSKVIN PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-6/H20855+/H20841dˆ/H20841−/H20856=2eRMnMn , /H208497/H20850 where RMnMn is a Mn-Mn separation. The two-particle trans- port Mn2+-Mn4+→Mn4+-Mn2+can be realized through two successive one-particle processes with the eg-electron trans- fer as follows: Mn2++M n4+→eg Mn3++M n3+→eg Mn4++M n2+. Hence the two-particle transfer integral tBcan be evaluated as follows: tB=−teg2/U, /H208498/H20850 where tegis one-particle transfer integral for egelectron and Uis a mean transfer energy. It means that the two-particle bosonic transfer integral can be directly coupled with the kinetic egcontribution Jkinegto Heisenberg exchange integral. Both tBandJkinegare determined by the second-order one- particle transfer mechanism. It should be noted that negativesign of the two-particle CT integral t Bpoints to the energy stabilization of the s-type EH-dimer state /H20841+/H20856. Second, one should emphasize once more that the stabili- zation of EH dimers is provided by a strong electron-latticeeffect with a striking intermediate oxygen atom polarizationand displacement concomitant with charge exchange. In asense, the EH dimer may be addressed to be a bosonic coun-terpart of the Zener Mn 4+-Mn3+polaron.41It is no wonder that even in a generic disproportionated system BaBiO 3in- stead of simple checkerboard charge ordering of Bi3+and Bi5+ions we arrive at charge-density wave /H20849CDW /H20850state with the alteration of expanded Bi/H208494−/H9267/H20850+O6and compressed Bi/H208494+/H9267/H20850+O6octahedra with 0 /H11021/H9267/H112701.42Enormously large val- ues of oxygen thermal parameters in BaBiO 3/H20849Ref. 43/H20850evi- dence a great importance of dynamical oxygen breathingmodes providing some sort of a “disproportionation glue.”Sharp rise of the oxygen thermal parameter in the high-temperature O phase of LaMnO 3/H20849Ref. 17/H20850or in several “competing” phases found by Huang et al.19as compared with the bare AFI phase is believed to be a clear signature ofthe manganese disproportionation. The formation of EH dimers seems to be a more complex process than it is assumed in simpliﬁed approaches such asPeierls-Hubbard model /H20849see, e.g., Ref. 44/H20850or Rice-Sneddon model. 45As a rule, these focus on the breathing mode for the intermediate oxygen ion and neglect strong effects of theoverall electron-lattice relaxation. The EH dimer can beviewed as a Jahn-Teller center /H20849JT polaron /H20850with the energy spectrum perturbed by strong electron-lattice effects. Thuswe see that a simple chemical view of the disproportionationshould be actually replaced by a more realistic physical viewthat implies a quantum and dynamical nature of the dispro- portionation reaction. B. EH dimers: Spin structure Let us apply to spin degrees of freedom which are of great importance for magnetic properties both of isolated EHdimer and of the EHBL phase that evolves from the EHdimers. The net spin of the EH dimer is S=S 1+S2, whereS1/H20849S1=5 /2/H20850andS2/H20849S1=3 /2/H20850are spins of Mn2+and Mn4+ ions, respectively. In nonrelativistic approximation the spin structure of the EH dimer will be determined by isotropicHeisenberg exchange coupling, V ex=J/H20849Sˆ1·Sˆ2/H20850, /H208499/H20850 with Jbeing an exchange integral, and two-particle charge transfer characterized by a respective transfer integral whichdepends on spin states as follows: /H208835 23 2;SM/H20879HˆB/H208793 25 2;SM/H20884=1 20S/H20849S+1/H20850tB, /H2084910/H20850 where tBis a spinless transfer integral. Making use of this relation we can introduce an effective spin-operator form forthe boson transfer as follows: Hˆ Beff=tB 20/H208512/H20849Sˆ1·Sˆ2/H20850+S1/H20849S1+1/H20850+S2/H20849S2+1/H20850/H20852, /H2084911/H20850 which can be a very instructive tool both for qualitative and quantitative analyses of boson transfer effects, in particular,the temperature effects. For instance, the expression points toa strong, almost twofold, suppression of effective transferintegral in paramagnetic phase as compared with its maximalvalue for a ferromagnetic ordering. Both conventional Heisenberg exchange coupling and un- conventional two-particle bosonic transfer or bosonic doubleexchange can be easily diagonalized in the net spin Srepre- sentation so that for the energy we arrive at E S=J 2/H20875S/H20849S+1/H20850−25 2/H20876/H110061 20S/H20849S+1/H20850tB, /H2084912/H20850 where /H11006corresponds to two quantum superpositions /H20841/H11006/H20856 written in a spin representation as follows: /H20841SM /H20856/H11006=1 /H208812/H20873/H208795 23 2;SM/H20884/H11006/H208793 25 2;SM/H20884/H20874, /H2084913/H20850 with s- and p-type symmetries, respectively. It is worth not- ing that the bosonic double-exchange contribution formallycorresponds to ferromagnetic exchange coupling with J B=−1 10/H20841tB/H20841. We see that the cumulative effect of the Heisenberg ex- change and the bosonic double-exchange results in a stabili-zation of the S=4 high-spin /H20849ferromagnetic /H20850state of the EH dimer provided /H20841t B/H20841/H1102210Jand the S=1 low-spin /H20849ferrimag- netic /H20850state otherwise. Spin states with intermediate Svalues, S=2,3, correspond to a classical noncollinear ordering. To estimate both quantities tBandJwe can address the results of a comprehensive analysis of different exchangeparameters in perovskites RFeO 3,RCrO 3, and RFe1−xCrxO3 with Fe3+and Cr3+ions46isoelectronic with Mn2+and Mn4+, respectively. For the superexchange geometry typical forLaMnO 3/H20849Ref. 21/H20850with the Mn-O-Mn bond angle /H9258/H11015155° the authors have found J=J/H20849d5−d3/H20850= +7.2 K while for J/H20849egeg/H20850/H11015−tB=295.6 K. In other words, for a net effective exchange integral we come to a rather large value:J eff=J−0.1 /H20841tB/H20841/H1101522.4 K. Despite the antiferromagnetic sign of the Heisenberg superexchange integral these data unam-DISPROPORTIONATION AND ELECTRONIC PHASE … PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-7biguously point to a dominant ferromagnetic contribution of the bosonic double-exchange mechanism. It is worth noting that the authors46have predicted the sign change in the superexchange integral in the d5-O2−-d3 system Fe3+-O2−-Cr3+in perovskite lattice from the antifer- romagnetic to ferromagnetic one on crossing the superex-change bonding angle /H9258/H11015162°. Interestingly that the param- eterJ/H20849egeg/H20850/H11015−tBis shown to rapidly fall with the decrease in the bond angle /H9258in contrast with J=J/H20849d5−d3/H20850which re- veals a rapid rise with /H9258. For the bond angle /H9258=143° typical for the heavy rare-earth manganites RMnO 3 /H20849R=Dy,Ho,Y,Er /H20850/H20849Ref. 21/H20850the relation between tB/H11015−153.8 K and J=J/H20849d5−d3/H20850/H1101514.4 K /H20849Ref. 46/H20850ap- proaches to the critical one, /H20841tB/H20841=10J, evidencing a destabi- lization of the ferromagnetic state for the EH dimers. In otherwords, the structural factor plays a signiﬁcant role for stabi-lization of one or another spin state of the EH dimers. Spinstructure of the EH dimer given antiferromagnetic sign ofexchange integral J/H110220 and /H20841t B/H20841=20Jis shown in Fig. 3.W e see a dramatic competition of two opposite trends, governedby one- and two-particle transports. EH dimers can manifest typical superparamagnetic behav- ior with large values of the effective spin magnetic momentup to /H9262eff/H110159/H9262B. Both bare Mn2+and Mn4+constituents of the EH dimer are s-type ions; i.e., these have an orbitally nondegenerated ground state that predetermines a rathersmall spin anisotropy. Local magnetic ﬁelds on the manganese nuclei in both bond-centered /H20841SM /H20856 /H11006states of the EH dimer are the same and determined as follows: Hn=1 2/H20875S/H20849S+1/H20850+5 2S/H20849S+1/H20850A2+S/H20849S+1/H20850−5 2S/H20849S+1/H20850A4/H20876/H20855S/H20856, /H2084914/H20850 where A2andA4are hyperﬁne constants for Mn2+and Mn4+, respectively, and we neglect the effects of transferred andsupertransferred hyperﬁne interactions. Starting with typical for Mn 2+and Mn4+values of5 2A2=600 MHz and 3 2A4=300 MHz, respectively, we arrive at maximal values of 55Mn nuclear magnetic resonance /H20849NMR /H20850frequencies for S=4, 3, 2, and 1 spin states of the EH dimer to be 450, 342.5,237, and 135 MHz, respectively. The55Mn NMR frequencies for bare Mn4+,3+,2+ions in LaMnO 3/H20849Refs. 47–49/H20850and theo- retical predictions for the EH dimer in different spin statesare shown in Fig. 4. Comparing these values with two bare frequencies we see that 55Mn NMR can be a useful tool to study the EH dimers in a wide range from bond-centered to site-centered states. Experimental55Mn NMR signal for slightly nonstoichiometric LaMnO 3/H20849Ref. 50/H20850is shown in Fig.4by ﬁlling /H20849see Sec. VIfor discussion /H20850. Concluding the section we should point to unconventional magnetoelectric properties of the EH dimer. Indeed, the two-particle bosonic transport and respective kinetic contributionto stabilization of the ferromagnetic ordering can be sup- pressed by a relatively small electric ﬁeld that makes the EHdimer to be a promising magnetoelectric cell especially forthe heavy rare-earth manganites RMnO 3/H20849R=Dy,Ho,Y,Er /H20850 with supposedly a ferroantiferroinstability. In addition, it isworth noting a strong anisotropy of the dimer’s electric po-larizability. In an external electric ﬁeld the EH dimers tend toalign along the ﬁeld. C. EH-dimer dynamics: Immobile and mobile dimers Above we addressed the internal electron-hole motion in a localized immobile EH dimer resulting in an s-psplitting. However, the EH dimer can move in three-dimensional /H208493D/H20850 lattice thus developing new translational and rotationalmodes. For simplicity, hereafter we address an ideal cubicperovskite lattice where the main modes are rotations of thehole /H20849electron /H20850around the electron /H20849hole /H20850by 90° and 180° and axial translations. It is interesting to note that the 90° and180° rotations of the hole /H20849electron /H20850around the electron /H20849hole /H20850correspond to the next-nearest-neighbor /H20849NNN /H20850and next-next-nearest-neighbor /H20849NNNN /H20850hoppings of the hole /H20849electron /H20850MnO 68−/H20849MnO610−/H20850center in the lattice formed by the MnO69−centers. We can introduce a set of transfer param- eters to describe the dimer dynamics ts=−tp/H110151 2/H20849tNNNNe+tNNNNh/H20850, tsp=−tps/H110151 2/H20849tNNNNe−tNNNNh/H20850 for the collinear exciton motion andS=4 S=3S=4 S=3 S=2 S=2S=1S=1Mn -2+Mn4+Mn -4+Mn2+P P Two-particle transferS=1S=2S=3 S=4Exchange coupling (one-particle transfer) Even-parity states s-typeOdd-parity p states-type FIG. 3. /H20849Color online /H20850Spin structure of the self-trapped CT exciton or EH dimer with a step-by-step inclusion of one- and two-particle charge transfers. Arrows point to electric-dipole moment forbare site-centered dimer conﬁgurations.55Mn NMR frequencies for EH-dimer 100 200 300 400 500 600Mn2+Mn4+S=1 S=4 S=3 S=2 NMR fre quencies (MHz )Mn3+ FIG. 4. /H20849Color online /H2085055Mn NMR frequencies for bare Mn4+,3+,2+ions in LaMnO 3/H20849Refs. 47–49/H20850and theoretical predic- tions for the EH dimer in different spin states. Shown by ﬁlling is a 55Mn NMR signal for slightly nonstoichiometric LaMnO 3repro- duced from Ref. 50.A. S. MOSKVIN PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-8tsxy=−tpxy/H110151 2/H20849tNNNe+tNNNh/H20850, tspxy=tpsxy/H110151 2/H20849tNNNe−tNNNh/H20850, corresponding to a 90° rotation /H20849x→ymotion /H20850of the exci- ton. All these parameters have a rather clear physical sense.The electron /H20849hole /H20850transfer integrals for collinear exciton transfer t NNNNe,hare believed to be smaller than tNNNe,hintegrals for rectangular transfer. In other words, the two-centerdimers prefer to move “crablike” rather than in the usualcollinear mode. This implies a large difference for the dimerdispersion in /H20851100 /H20852and /H20851110 /H20852directions. The motion of the EH dimer in the bare LaMnO 3lattice with the orbital order of the Jahn-Teller Mn3+ions bears an activation character with an activation energy /H9004E=1 2/H9004JT, where /H9004JTis the Jahn-Teller splitting of the eglevels in Mn3+ ions. Thus one may conclude that the EH-dimer energy band in the bare LaMnO 3lattice would be composed of the low- energy subband of immobile localized EH dimers or spdou- blet with the energy separation of 2 /H20841tB/H20841and the high-energy subband of mobile EH dimers shifted by1 2/H9004JTwith the band- width W/H110116tNNN, where tNNN is an effective next-nearest- neighbor eg−egtransfer integral in Mn3+-Mn3+pair. Sche- matically the spectrum is shown in Fig. 5. An optical portrait of the EH-dimer bands is composed of a rather narrow low-energy line due to electrodipole CT s-ptransition for immo- bile dimers peaked at E sp=2/H20841tB/H20841and a relatively broad high- energy line due to electrodipole photoinduced dimer transport peaked at E/H110151 2/H9004JT+/H20841tB/H20841. To estimate these energies one might use our aforementioned estimates for /H20841tB/H20841 /H110150.03 eV and reasonable estimates of the Jahn-Teller split- ting/H9004JT/H110150.7 eV /H20849see, e.g., Ref. 34/H20850. Thus we predict a two-peak structure of the EH-dimer optical response with anarrow line at /H110110.06 eV and a broad line at /H110110.4 eV. Our estimate of the sp-separation E sp=2/H20841tB/H20841does not account for the Jahn-Teller polaronic effects in the EH dimer that canresult in its strong increase. It is worth noting that the activation character for the mo- tion of the EH dimer in parent manganite lattice implies thesame feature for the generic 2 eV d-dCT exciton resulting in its weak dispersion. Indeed, the resonant inelastic x-ray scat-tering /H20849RIXS /H20850experiments on parent manganite LaMnO 3by Inami et al.51found the energy dispersion of the 2.0–2.5 eV peak to be less than a few hundred meV. D. EH dimers: EH dissociation and recombination. The EH-dimer dissociation or uncoupling energy may be estimated to be on the order of 1.0–1.5 eV. The EH couplingwithin the dimer is determined by a cumulative effect ofelectrostatic attraction and local lattice relaxation /H20849reorgani- zation /H20850energy. The EH recombination in the EH dimer resembles an in- verse disproportionation reaction, MnO 68−+ MnO610−→MnO69−+ MnO69−. /H2084915/H20850 The inverse counterpart of 2 eV d-dCT transition in the bare parent manganite is expected to have nearly the same energy.CT transition /H2084915/H20850in EH dimer can be induced only in E /H20648RMnMn polarization. However, this CT transition can be hardly photoinduced from the ground s-type state of the EH dimer in contrast with the p-type state due to selection rules for electrodipole transitions. It means that at least at ratherlow temperatures kT/H112702/H20841t B/H20841the EH recombination band would be invisible; that is, the optical response of EH dimerswould be reduced to two aforementioned low-energy bandsthat are developed within the energy gap of the bare parentmanganite. In addition, we should point to different p-dCT transitions within electron MnO 610−and hole MnO68−centers with the onset energy near 3 eV. It is worth noting that theoverall optical response of the EH dimers in weakly distortedperovskite lattice is expected to be nearly isotropic at vari-ance with the CT response of parent LaMnO 3in its bare A-AFI phase.34 V . ELECTRON-HOLE BOSE LIQUID: THE TRIPLET BOSON DOUBLE-EXCHANGE MODEL A. Effective Hamiltonian To describe the electron-hole Bose liquid /H20849EHBL /H20850phase that evolves from EH dimers we restrict ourselves with or- bital singlets6A1gand4A2gfor the electron MnO610−and hole MnO68−centers, respectively. Speciﬁc electron conﬁgurations of these centers, t2g3;4A2geg2;3A2g:6A1gandt2g3;4A2g, respec- tively, enable us to consider the electron center MnO610−to be composed of the hole MnO68−center and a two-electron eg2;3A2gconﬁguration which can be viewed as a composite triplet boson. In the absence of the external magnetic ﬁeldthe effective Hamiltonian of the electron-hole Bose liquidtakes the form of the Hamiltonian of the quantum latticeBose gas of the triplet bosons with an exchange coupling, Hˆ=Hˆ QLBG +Hˆex=/H20858 i/HS11005j,mtB/H20849ij/H20850Bˆ im†Bˆjm+/H20858 i/H11022jVijninj−/H9262/H20858 ini +/H20858 i/H11022jJijhh/H20849Sˆi·Sˆj/H20850+/H20858 i/HS11005jJijhb/H20849sˆi·Sˆj/H20850+/H20858 i/H11022jJijbb/H20849sˆi·sˆj/H20850 +/H20858 iJiihb/H20849sˆi·Sˆj/H20850. /H2084916/H208502\|t \|B6\|t \|nnn /CID2JT Immobile EH-dimersMobile EH-dimers s-p- FIG. 5. /H20849Color online /H20850Schematic energy spectrum of immobile /H20849localized /H20850and mobile EH dimers. Bold arrows point to allowed electrodipole transitions.DISPROPORTIONATION AND ELECTRONIC PHASE … PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-9Here Bˆ im†denotes the S=1 boson creation operator with a spin projection mat the site iandBˆimis a corresponding annihilation operator. The boson number operator nˆim =Bˆ im†Bˆimatisite due to the condition of the on-site inﬁnitely large repulsion Vii→+/H11009/H20849hardcore boson /H20850can take values 0 or 1. The ﬁrst term in Eq. /H2084916/H20850corresponds to the kinetic en- ergy of the bosons; tB/H20849ij/H20850is the transfer integral. The second one reﬂects the effective repulsion /H20849Vij/H110220/H20850of the bosons located on the neighboring sites. The chemical potential /H9262is introduced to ﬁx the boson concentration: n=1 N/H20858i/H20855nˆi/H20856. For EHBL phase in parent manganite we arrive at the same num- ber of electron and hole centers, that is, to n=1 2. The remain- ing terms in Eq. /H2084916/H20850represent the Heisenberg exchange in- teraction between the spins of the hole centers /H20849term with Jhh/H20850, spins of the hole centers and the neighbor boson spins /H20849term with Jhb/H20850, boson spins /H20849term with Jbb/H20850, and the very last term in Eq. /H2084916/H20850stands for the intracenter Hund exchange between the boson spin and the spin of the hole center. In order to account for the Hund rule one should consider Jiihbto be inﬁnitely large ferromagnetic. Generally speaking, thismodel Hamiltonian describes the system that can be consid-ered as a Bose analog of the one orbital double-exchange model system. 2 Aforementioned estimates for different superexchange couplings given the bond geometry typical for LaMnO 3pre- dict antiferromagnetic coupling of the nearest-neighbor /H20849NN /H20850 hole centers /H20849Jhh/H110220/H20850, antiferromagnetic coupling of the two nearest-neighbor bosons /H20849Jbb/H110220/H20850, and ferromagnetic cou- pling of the boson and the nearest-neighbor hole centers/H20849J hb/H110210/H20850. In other words, we arrive at highly frustrated sys- tem of triplet bosons moving in a lattice formed by holecenters when the hole centers tend to order G-type antiferro- magnetically; the triplet bosons tend to order ferromagneti-cally both with respect to its own site and its nearest neigh-bors. Furthermore, nearest-neighboring bosons stronglyprefer an antiferromagnetic ordering. Lastly, the boson trans-port prefers an overall ferromagnetic ordering. B. Implications for phase states and phase diagram By now we have no comprehensive analysis of phase states and phase diagram for the generalized triplet bosondouble-exchange model. The tentative analysis of the modelin framework of a mean-ﬁeld approximation /H20849MFA /H20850/H20849Ref. 52/H20850allows us to predict a very rich phase diagram even at half-ﬁlling /H20849n= 1 2/H20850with a rather conventional diagonal long- range order /H20849DLRO /H20850with ferromagnetic insulating or ferro- magnetic metallic /H20849FM /H20850phase and unconventional off- diagonal long-range order /H20849ODLRO /H20850with a coexistence of superﬂuidity of triplet bosons and ferromagnetic ordering.However, it is unlikely that the MFA approach can provide arelevant description of such a complex system. Some impli-cations may be formulated from the comparison with famil-iar double-exchange model, 2singlet boson Hubbard model /H20849see, e.g., Ref. 53/H20850, and with generic bismuthate oxide BaBiO 3as a well documented disproportionated system which can be described as a 3D system of the spin-singletlocal bosons.If the boson transfer is excluded we arrive at a spin sys- tem resembling that of mixed orthoferrite-orthochromite LaFe 1−xCrxO3/H20851nB=1 2/H208491−x/H20850/H20852which is a G-type antiferromag- net all over the dilution range 0 /H11021x/H110211 with TN’s shifting from TN=740 K for LaFeO 3toTN=140 K for LaCrO 3.54 However, at variance with a monovalent /H20849Fe3+-Cr3+/H20850 orthoferrite-orthochromite the Mn2+-Mn4+charge system in the EHBL phase would reveal a trend to a charge ordering, e.g., of a simple checkerboard Gtype in LaMnO 3/H20849nB=1 2/H20850.I t is worth noting that the naively expected large values of aboson-boson repulsion V ijwould result in a large tempera- tureTCOof the charge ordering well beyond room tempera- ture. However, the manganites must have a large dielectricfunction and a strong screening of the repulsion; hence mod-erate values of V NNandTCO’s predicted. However, such a scenario breaks when the boson trans- port is at work. It does suppress both types of charge andspin ordering and we arrive most likely at an inhomogeneoussystem with a glasslike behavior of charge and spin sub-systems, which does or does not reveal a long-range ferro-magnetic order at low temperatures. A question remains:whether the EHBL Hamiltonian /H2084916/H20850can lead to uniform solutions beyond MFA? According to experimental data 19the phases in LaMnO 3, which we relate with EHBL, exhibit a long-range ferromag-netic order below T C/H11015140 K, however, with rather small values of a mean magnetic moment, which agrees with a spininhomogeneity. It is worth noting that the glass scenario im-plies a speciﬁc “freezing” temperature T gto be a remnant of the MFA critical temperature. Such a temperature should berevealed in physical properties of the system. With a deviation from half-ﬁlling to n B/H110211 2the local trip- let bosons gain in freedom to move and improve their kineticenergy. On the other hand it is accompanied by a sharp de-crease in the number of the boson-boson pairs with the moststrong e g-egantiferromagnetic coupling. In other words, a FM phase becomes a main candidate to a ground state. Interestingly, that an intent reader can note that here we describe main features of phase diagrams typical for holedoped manganites such as La 1−xCaxMnO 3. Indeed, this re- semblance seems not to be accidental one and points to aprofound role of the EHBL phase in unconventional proper-ties of doped manganites as well. One of the most intriguing and challenging issues is re- lated with the probable superﬂuidity of the triplet localbosons. Indeed, the boson transfer integral t Bdeﬁnes a maxi- mal temperature Tmax/H11015tBof the onset of local superconduct- ing ﬂuctuations in the hardcore boson systems.55Our estima- tions point to Tmax/H11015300–700 K, where the lower bound is taken from theoretical estimations, while the upper bound isderived from optical data on the 0.1 eV spectral feature.However, these high values of T maxdo not give rise to opti- mistic expectations regarding the high- Tcbulk superconduc- tivity in the EHBL phase of parent manganites ﬁrst becauseof a spin frustration. Nevertheless, despite the fact that theemergence of a bulk superconductivity in a highly frustratedmulticomponent EHBL phase seems to be a very uncommonphenomenon, the well-developed local superconducting ﬂuc-tuations can strongly inﬂuence the transport as well as otherphysical properties. A detailed analysis of the bosonicA. S. MOSKVIN PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-10double-exchange model, in particular, of the off-diagonal su- perconducting order with the superﬂuidity of the triplet localbosons remains to be a challenging issue for future studies. Itis worth noting that the electron-lattice coupling can bestrongly involved into the buildup of the electronic structureof the bosonic double-exchange model, in particular,strengthening the EH-dimer ﬂuctuations. VI. EXPERIMENTAL MANIFESTATION OF EH DROPLETS IN PARENT AND LOW-HOLE-DOPED MANGANITES Above, in Sec. IIwe addressed some experimental data that somehow pointed to a disproportionation scenario andhave been used to start with a detailed analysis of the EHBLphase. Hereafter, we address different new experimental datathat support our scenario in some details. A. Optical response of electronically phase-separated manganites The CT unstable systems will be characterized by a well- developed volume fraction of the short- and long-lived CTexcitons or the EH droplets that can give rise to a speciﬁcoptical response in a wide spectral range due to different p-d andd-dCT transitions. First, these are the low-energy intra- center CT transitions and high-energy inverse d-dCT transi- tions, or EH recombination process in EH dimers and/ornanoscopic EH centers, and different high-energy CT transi-tions in electron and hole centers. It is worth noting that,strictly speaking, the optical measurements should alwaysdisplay a larger volume fraction of EH droplets as comparedwith static or quasistatic measurements because these “see”short-lived droplets as well. What are the main optical sig-natures of the CT instability? A simpliﬁed picture implies thespectral weight transfer from the bare CT band to the CT gapwith an appearance of the midgap bands and smearing of thefundamental absorption edge. Such a transformation of theoptical response is shown schematically in Fig. 6. The trans- ferred spectral weight can be easily revealed in the spectralwindow of the bare insulator to be a direct indicator of theCT instability. It is worth noting that the fragile “matrix-droplet ” structure of the parent manganites makes the opti-cal response to be very sensitive to such factors as tempera-ture, sample shape /H20849bulk crystal, thin ﬁlm /H20850and quality, and external magnetic ﬁeld, which can explain some inconsisten-cies observed by different authors /H20849see, e.g., Refs. 34and 56–58/H20850. Great care is needed if one wants to separate off the volume fraction effects to obtain the temperature behavior ofspectral weight for certain band and compare the results withthose observed by different groups on different samples.Charge transfer instability and the CT exciton self-trappingin nominally pure manganites are indeed supported by thestudies of their optical response. Anisotropic optical conductivity spectra for a detwinned single crystal of LaMnO 3, which undergoes the orbital ordering below TJT/H11015780 K, have been derived from the reﬂectivity spectra investigated by Tobe et al.56over a wide temperature range, 10 K /H11021T/H11021800 K /H20849see Fig. 7/H20850.A stemperature is increased, the EH dimers generating d-dCT transition peaked around 2 eV show a dramatic loss of spec-tral weight with its partial transfer to the low energies. Si-multaneously one observes a suppression of optical aniso-tropy. Above T JT, the gap feature becomes obscure and the anisotropy disappears completely. Such a behavior of the 2eV band can be hardly explained by the effect of spinﬂuctuations, 34most likely it points to a shrinking of the A-AFI phase volume fraction with approaching to Tdisp =TJTand phase transition to an unconventional metalliclike phase. However, the optical conductivity does not reveal anysignatures of Drude peak, which together with a rather largeresistivity 9points to an unusual charge transport. Main features of the optical response56agree with predic- tions followed from the EPS phase diagram and isotropiccharacter of the optical response of EH droplets. However,the reﬂectivity data did not reveal any midgap structureswhich observation and identiﬁcation needs usually in directabsorption/transmission measurements. The most detailedstudies of spectral, temperature, and doping behavior of themidgap bands were performed in Refs. 58–62. All the man- ganites investigated, both parent and hole/electron doped,show up two speciﬁc low-energy optical features peakednear 0.10–0.15 eV /H208490.1 eV band /H20850and 0.3–0.6 eV /H208490.5 eV band /H20850. Results of the ellipsometric and direct absorption measurements for a single-crystalline parent LaMnO 3sample are shown in Fig. 8; these directly reveal both 0.1 and 0.5 eVPhoton energyOptical portrait of CT instability Absorption Intra-center transitions EH recombination FIG. 6. /H20849Color online /H20850Optical response /H20849schematically /H20850of the self-trapped CT excitons and EH droplets /H20849dotted curves /H20850. Arrows point to a spectral weight transfer from the bare CT band to the CTgap with an appearance of the midgap bands and/or smearing of thefundamental absorption edge. Optical conductivity (10 cm )3-1 -1/CID2 4012 1 2 3 0T= 10 K =300 K=700 K=800 KLaMnO3 Photon energy (eV) FIG. 7. /H20849Color online /H20850Temperature dependence of optical con- ductivity of parent LaMnO 3forE/H20648ab /H20849reproduced from Ref. 56/H20850.DISPROPORTIONATION AND ELECTRONIC PHASE … PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-11features in the spectral window of the bare matrix.58These two bands can be naturally attributed to the CT transitionswithin the immobile EH dimers and to the dimer transportactivating transitions, respectively. Respective energies agreewith theoretical predictions, although more accurate value of0.15 eV for the “0.1 eV” peak points most likely to an es-sential electron-lattice effect. The 0.5 eV band in LaMnO 3was revealed by photoin- duced absorption spectroscopy under light excitations withthe photon energy near 2.4 eV that provides optimal condi-tions for the EH-pair creation. Photoinduced absorption wasobserved 10with a strong broad midinfrared peak centered at /H110115000 cm−1/H110150.62 eV. Since the laser photoexcitation and measurement are pseudocontinuous, the photoexcited EH-pair lifetimes need to be quite long for any signiﬁcant pho-toexcited EH-pair density to build up. It means that the lat-tice is arranged in the appropriate relaxed state. The origin ofthe photoinduced /H20849PI/H20850absorption peak was attributed 10to the photon-assisted hopping of anti-Jahn-Teller polarons formedby photoexcited charge carriers. This interpretation wasbased on the assumption of primary p-dCT transition in- duced by excitation light with the energy h /H9263=2.41 eV. However, the d-dCT transition nature of 2 eV absorption band in LaMnO 3/H20849Ref. 34/H20850unambiguously points to the EH dimers to be main contributor to PI absorption peak. In sucha case, the PI absorption peak energy /H20849/H110110.6 eV /H20850may be attributed to the energy of the photon-assisted hopping of therelaxed EH dimers /H20849see Fig. 5/H20850and can be used as an esti- mate of the Jahn-Teller energy /H9004 JT. Similarly, so-called midgap features in nominally pure manganites were directly or indirectly observed by many au-thors. Furthermore, it seems that some authors did not reportthe optical data below 1.5 eV to avoid the problems withthese odd features. Observation of the MIR features agreeswith the scenario of well-developed intrinsic electronic inho-mogeneity inherent to nominally stoichiometric insulating manganites and composed of volume fraction of conceivablyEH droplet phase. Finally, it is instructive to compare the midgap absorption spectrum of parent manganite with IR optical spectra ofchemically doped compounds to see whether the nonisova-lent substitution stimulates the condensation of EH pairs andrespective rise of the EH droplet volume fraction. Indeed,Okimoto et al. 63observed in La 0.9Sr0.1MnO 3a broad absorp- tion peaked around 0.5 eV which is absent at room tempera-ture and increases in intensity with decreasing temperature.In addition, the absorption feature reported also shifts tolower energy as doping is increased, in agreement with PImeasurements. 10A midgap state with a similar peak energy and similar doping dependence was also observed at roomtemperature by Jung et al. 64in La 1−xCaxMnO 3. Thus we see that the strong and broad midinfrared optical feature peaked near 0.5 eV and observed in all the perovskitemanganites studied can be surely attributed to the opticalresponse of isolated EH dimers or small EH droplets edgedby the JT Mn 3+centers, more precisely, to an optical activa- tion of the dimer transport in such a surroundings. The peakenergy may be used to estimate the Jahn-Teller splitting fore glevels in Mn3+centers and its variation under different conditions. B. Lattice effects in parent LaMnO 3 The unusual abrupt unit-cell volume contraction by 0.36% has been observed by Chatterji et al.65in LaMnO 3atTJT. The high-temperature phase just above TJThas less volume than the low-temperature phase. The local structure of stoichiometric LaMnO 3across the Jahn-Teller transition at TJTwas studied by means of ex- tended x-ray absorption ﬁne structure /H20849EXAFS /H20850at Mn K edge66and high real space resolution atomic pair distribution function /H20849PDF /H20850analysis.67Both techniques reveal two differ- ent Mn-O separations, 1.92 Å /H208491.94 Å /H20850and 2.13 Å /H208492.16 Å /H20850, distributed with intensity 2:1, respectively. Com- paring these separations with room-temperature neutron-diffraction data 21/H208491.907, 1.968, and 2.178 Å /H20850both groups point to a persistence of the JT distortions of MnO 6octahe- dra on crossing TJT. However, both this result and that of Chatterji et al.65most likely point to a transition to Mn-O separations speciﬁc for EH dimers or nearest-neighbor elec- tron MnO610−/H20849Mn2+/H20850and hole MnO610−/H20849Mn4+/H20850centers coupled by fast electron exchange. In any case the picture is that inthe high-temperature O phase the local distortions of theMn-O separations are dynamical in character similar to thosein BaBiO 3. A signature of that is an excess increase in the thermal factors of oxygen atoms in going from O /H11032to the O phase.17The observed Raman spectra for undoped LaMnO 3 crystal at ambient pressure and room temperature reveal anumber of additional lines, in particular, strong /H20849A 1g+B2g/H20850 mode 675 cm−1, which are also have been observed in the spectra of doped materials and may be attributed to dropletsof EHBL phase. 68 Strong variation in the LaMnO 3Raman spectra, both of intensity and energy shift with increasing laser power,690 12 3 4 5 E(eV)/CID1/CID34.0 2.03.0 1.0 00.1 0.5 0.4 0.6 0.7 0.8 0.3 0.2 0.0 E (eV)K (cm )-1 298 K 80 K 200 0100300400500600 LaMnO3 LaMnO3La Sr MnO0.93 0.07 3 /CID4T (a.u.)E (eV) 0.5 1.0 FIG. 8. /H20849Color online /H20850Imaginary part of the dielectric function /H9255abin LaMnO 3/H20849solid triangles /H20850and La 0.93Sr0.07MnO 3/H20849open circles /H20850 /H20849Ref. 58/H20850. Low-energy part of the spectrum is a guide for eyes from the infrared absorption data /H20849see right-hand inset /H20850. Right-hand inset: infrared absorption for parent LaMnO 3at 80 and 298 K /H20849reproduced from Ref. 58/H20850. Left-hand inset: photoinduced transmittance of par- ent LaMnO 3atT=25 K /H20849reproduced from Ref. 10/H20850.A. S. MOSKVIN PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-12could be related to the photoinduced nucleation and the vol- ume expansion of the EH Bose liquid. Surely, laser annealingcan simply increase the temperature thus resulting in anA-AFI/EHBL volume fraction redistribution. The strongvariations in the LaMnO 3Raman spectra on the excitation laser power provide evidence for a structural instability thatmay result in a laser-irradiation-induced structural phasetransition. It is worth noting a strong resonant character ofthe excitation of the Raman specta 70that points to a need in more extensive studies focused on the search of the EH drop-let response. The intrinsic electronic phase separation inherent for nominally undoped stoichiometric LaMnO 3manifests itself in remarkable variations in x-ray diffraction pattern, opticalreﬂectivity and Raman spectra, and resistivity under pres-sures up to 40 GPa. 11The pressure-induced variations in Ra- man spectra, in particular, a blueshift and the intensity loss ofthe in-phase O2 stretching B 2gmode with a concomitant emergence of a peak at /H1101145 cm−1higher in energy evi- denced some kind of electronic phase separation with a steeprise of the volume fraction of the domains of a phase withinthe parent A-AFI phase /H20849“sluggish” transition 11/H20850. Evolution of phase was accompanied by a dramatic change in reﬂec-tance which resembles that of LaMnO 3at ambient pressure on heating from low temperatures to T/H11022TJT.56Furthermore, the system exhibited an anomalously strong pressure-induced fall of the room-temperature resistivity by 3 ordersof magnitude in the range of 0–30 GPa with an IM transitionat 32 GPa. An overall fall of resistivity in the range of 0–32GPa amounts to 5 orders of magnitude. However, the systemretains a rather high resistance, exhibiting a “poor” metallicbehavior typical for EHBL phase. It is worth noting that athigh pressures /H1102230 GPa the resistivity does not reveal siz- able temperature dependence between 80 and 300 K simi-larly to the high-temperature T/H11022T JTbehavior of LaMnO 3at ambient pressure /H20849see Ref. 9and Fig. 1/H20850. Overall these data provide a very strong support for our scenario of the A-AFI/EHBL electronic phase separation in parent manganite takingplace without any hole/electron doping. The effect of the O 16→O18isotope substitution on the IM transition and optical response71can be easily explained as a result of an energy stabilization of the parent A-type antifer- romagnetic phase as compared with the EH Bose liquid. Thepercolation mechanism of the isotope effect in manganites isconsidered in Ref. 71. C. Magnetic and resonance properties of EHBL phase in LaMnO 3 What about the magnetic properties of the phase? In the framework of our scenario the EH Bose liquid in LaMnO 3 evolves from the EH dimers which are peculiar magneticcenters with intrinsic spin structure and with enormouslylarge magnetic moments in their ground ferromagnetic state.However, the EH dimers exist as well deﬁned entities only atvery initial stage of the EHBL evolution. Within well-developed EH Bose liquid we deal with a strong overlap ofEH dimers when these lose individuality. A tentative analysisof the EH liquid phase in parent manganites 26shows that itmay be addressed to be a triplet bosonic analog of a simple fermionic double-exchange model with a well-developedtrend to a ferromagnetic ordering. It is interesting that bothmodels have much in common that hinders their discerning.In both cases the net magnetic moment of calcium/H20849strontium /H20850-doped manganite La 1−xCa/H20849Sr/H20850xMnO 3saturates to the full ferromagnetic value /H11015/H208494−x/H20850/H9262B/f.u. Well developed ferromagnetic ﬂuctuations within EHBL phase in LaMnO 3 have been observed in high-temperature susceptibility mea-surements by Zhou and Goodenough 9which measured the temperature dependence of paramagnetic susceptibility bothbelow and above T JT. They observed a change from an an- isotropic antiferromagnetism to an isotropic ferromagnetismcrossing T JTaccompanied by an abrupt rise of magnetic sus- ceptibility. These data point to an energy stabilization of theEH Bose liquid in an external magnetic ﬁeld as comparedwith a parent A-type antiferromagnetic phase. The dc magnetic susceptibility shows two distinct regimes 72,73for LaMnO 3, above and below TJT. For T/H11022TJT, /H9273dc/H20849T/H20850follows a Curie-Weiss /H20849CW /H20850law,/H9273dc/H20849T/H20850=C//H20849T−/H9008/H20850, with C=3.4 emu K /mol /H20849/H9262eff/H110155.22/H9262B/H20850and/H9008/H11015200 K. ForT/H11021TJTthe behavior of magnetic susceptibility strongly depends on the samples studied. Zhou and Goodenough9ob- served an abrupt fall in the Weiss constant on crossing TJT from large ferromagnetic to a small antiferromagnetic /H9008 /H1101550 K, while Causa and co-workers72,73found that the Curie-Weiss behavior of /H9273dc/H20849T/H20850is recovered only near room temperature with a reduced antiferromagnetic /H9008/H1101575 K. In- terestingly that instead of a natural suggestion of an elec-tronic phase-separated state below T JTwith a coexistence of low- and high-temperature phases and steep change in effec-tive/H9008, the authors 72,73explained their data as a manifesta- tion of dramatic changes in exchange parameters induced bycrystal distortions. They refer to theoretical calculations 74 which show that Jabin parent manganites is FM and de- creases with the JT distortion while Jcchanges from FM in the pseudocubic O phase to AFM in the O /H11032phase. However, the aforementioned estimations46based on the experimental data for isostructural orthoferrites, orthochromites, andmixed orthoferrites chromites point to a more reasonable an-tiferromagnetic orbitally averaged exchange coupling of twoMn 3+ions with bond geometry typical for LaMnO 3:J /H1101512.6 K. Magnetic measurements for low-hole-doped LaMnO 3 samples75–79reveal a coexistence of antiferromagnetic matrix with ferromagnetic clusters or spin-glass behavior, accompa-nied by magnetic hysteresis phenomena. Anomalous magni-tudes of the effective magnetic moment per manganese ionthat considerably exceed expected theoretical values, up to /H9262eff/H110156/H9262Bin La 0.9Sr0.1MnO 3/H20849Ref. 76/H20850, were explained to be an evidence of a disproportionation 2Mn3+→Mn4++Mn2+ /H20849Ref. 75/H20850or a superparamagnetic behavior of ferromagnetic clusters.76As a whole, magnetic measurements for nearly stoichiometric LaMnO 3support the disproportionation sce- nario. The electronic spin resonance /H20849ESR /H20850spectrum of LaMnO 3in a wide temperature range above TNand up to temperature /H11011800 K above TJTshows a single Lorentzian line with g/H110111.98–2.00 and /H9004H/H110112400 Gauss at room temperature.72,80In common, the spectrum intensity followsDISPROPORTIONATION AND ELECTRONIC PHASE … PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-13the dc susceptibility; however, the consistent interpretation of the origin of ESR signal, especially in O pseudocubicphase, is still lacking. Two different electronic phases aredocumented by ESR measurements in slightly La-deﬁcientLa 0.99MnO 3.79Further experimental ESR studies have to be carried out to clarify the issue. The55Mn nuclear magnetic resonance /H20849NMR /H20850data sup- port most likely the EHBL scenario. Indeed, the zero-ﬁeld 55Mn NMR spectrum in a nominally undoped LaMnO 3con- sists of a sharp central peak at 350 MHz due to bare Mn3+O69−centers and two minority signals at approximately 310 and 385 MHz,48which can be assigned to a localized hole MnO68−/H20849=Mn4+/H20850center and EH dimers with a fast bosonic exchange, respectively. Evolution of such a picture with Ca /H20849Sr/H20850doping can easily explain a complex55Mn NMR line shape in La 1−xCa/H20849Sr/H20850xMnO 3samples.48,77It is worth not- ing that Tomka et al.47observed three55Mn NMR signals in a hole doped PrMnO 3around 310, 400, and 590 MHz, which can be attributed to localized hole MnO68−and electron MnO610−centers /H20849narrow resonances around 310 and 590 MHz, respectively /H20850and to EH droplets with a fast bosonic exchange /H20849broad resonance around 400 MHz /H20850. It is worth noting that the55Mn NMR line shape in La1−xCa/H20849Sr/H20850xMnO 3samples48,77with a most part of intensity shifted to a very broad line in the range of 350–450MHz can hardly be explained in framework of a so-calleddouble-exchange /H20849DE/H20850line 48with a frequency fDE =1 2/H20851f/H20849Mn3+/H20850+f/H20849Mn4+/H20850/H20852derived from that typical for Mn3+/H20849350 MHz /H20850and Mn4+/H20849310 MHz /H20850. Our scenario with a broad line centered with more or less redshiftfrom a frequency speciﬁc for a high-spin state of the EH dimer: f EH=1 2/H20851f/H20849Mn2+/H20850+f/H20849Mn4+/H20850/H20852/H11015450 MHz with f/H20849Mn2+/H20850/H11015590 MHz and f/H20849Mn4+/H20850/H11015310 MHz is believed to be more appropriate one. It is worth noting that the55Mn NMR response of EH dimers can shed some light on several 55Mn NMR puzzles, in particular, observation of the low- temperature /H208494.2 K /H20850low-frequency NMR lines at 260 MHz in one of nominally undoped LaMnO 3samples81and even at 100 MHz in a more complex manganite /H20849BiCa /H20850MnO 3.49In both cases we deal seemingly with a some sort of a stabili-zation of low-spin states for EH dimers, for instance, due tothe Mn-O-Mn bond geometry distortions resulting in an an-tiferromagnetic Mn 2+-O-Mn4+superexchange. The55Mn NMR spectra of slightly nonstoichiometric LaMnO 3/H20849Ref. 50/H20850may be viewed as the most striking evi- dence of the EH-dimer response in a spin inhomogeneousglasslike state. A simple comparison of experimental spectrawith theoretical predictions for EH dimers /H20849see Fig. 4/H20850shows a clear manifestation of the S=4,3,2 spin multiplets of the EH dimers with the mixing effects due to a spin noncol-linearity. Magnetic and transport properties of a single-crystalline parent undoped manganite LaMnO 3have been studied re- cently under ultrahigh mega-Gauss magnetic ﬁeld at heliumtemperatures. 13In accordance with theoretical predictions82a sharp magnetic spin-ﬂip transition was observed at about 70T without visible transport anomalies. On further rising themagnetic ﬁeld the authors observed unusual magnetoinducedIM transition at H IM/H11011220 T that is considerably above theﬁeld of the magnetic saturation of the A-AFI phase. Large values of the p-dord-dcharge transfer energies in bare A-AFI phase of parent manganites /H20851/H110112 eV in LaMnO 3/H20849Ref. 34/H20850/H20852make the energy difference between the A-AFI ground state and any metallic phase seemingly too large to be over-come even for magnetic ﬁelds as large as hundreds of tesla.Zeeman energy associated with such a ﬁeld is clearly morethan 1 order of magnitude smaller than the charge reorderingenergy. Thus we see that a puzzling ﬁeld-driven IM transi-tion cannot be explained within a standard scenario implyingthe parent manganite LaMnO 3to be a uniform system of the Jahn-Teller Mn3+centers with an A-type antiferromagnetic order and needs a revisit of our view on the stability of itsground state. However, our scenario can easily explain thepuzzling ﬁeld-driven IM transition in perovskite manganiteLaMnO 3/H20849Ref. 13/H20850to be a result of a percolative transition in an inhomogeneous phase-separated A-AFI/EHBL state. Thevolume fraction of the ferromagnetic EHBL phase grows inan applied magnetic ﬁeld, and at a sufﬁciently high ﬁeld thisfraction reaches its percolation threshold to give the IM tran-sition. It is clear that a relatively small zero-ﬁeld volumefraction of ferromagnetic EHBL phase in the parent manga-nite has required large magnetic ﬁeld to induce the IM tran-sition. D. Dielectric anomalies in LaMnO 3 The broadband dielectric spectroscopy helps in character- izing the phase states and transitions in Mott insulator.Above we pointed to anomalous electric polarizability of theEH dimers and EH droplets that would result in dielectricanomalies in the EHBL phase and the phase-separated stateof LaMnO 3. Indeed, such anomalies were reported recently both for polycrystalline and single-crystalline samples ofparent LaMnO 3. First of all, one should note relatively high static dielectric constant in LaMnO 3atT=0 /H20849/H92550/H1101118–20 /H20850 approaching to values typical for genuine multiferroic sys-tems /H20849/H9255 0/H1101525/H20850, whereas for the conventional nonpolar sys- tems, /H92550varies within 1–5. The entire /H9255/H11032/H20849/H9275,T/H20850−Tpattern across 77–900 T has two prominent features: /H20849i/H20850near TNand /H20849ii/H20850near TJTto be essential signatures of puzzlingly unex- pected multiferroicity. Far below TN,/H9255/H11032/H20849/H9275,T/H20850is nearly tem- perature and frequency independent, as expected. Followingthe anomaly at T N,/H9255/H11032/H20849/H9275,T/H20850rises with Tby 5 orders of mag- nitude near TJT. Finally, /H9255/H11032becomes nearly temperature in- dependent beyond TJT. The P-Eloop does not signify any ferroelectric order yet the time-dependence plot resemblesthe “domain-switching-like” pattern. The ﬁnite loop area sig-niﬁes the presence of irreversible local domain ﬂuctuations.From these results, it appears that the intrinsic electrical po-larization probably develops locally with no global ferroelec-tric order. The nature of the anomaly at T JTvaries with the increase in Mn4+concentration following a certain trend— from a sharp upward feature to a smeared plateau and then adownward feature to ﬁnally a rather broader downward peak. The observation of an intrinsic dielectric response in glo- bally centrosymmetric LaMnO 3, where no ferroelectric order is possible due to the absence of off-center distortion inMnO 6octahedra, cannot be explained in frames of the con-A. S. MOSKVIN PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-14ventional uniform antiferromagnetic insulating A-AFI sce- nario and agrees with the electronic A-AFI/EHBL phase-separated state with a coexistence of nonpolar A-AFI phaseand highly polarizable EHBL phase. E. Comment on the experimental nonobservance of the EHBL phase in LaMnO 3 By now there has been no systematic exploration of exact valence and spin state of Mn in perovskite manganites. Us-ing electron paramagnetic resonance /H20849EPR /H20850measurements Oseroff et al. 80suggested that below 600 K in LaMnO 3there are no isolated Mn atoms with valences of +2, +3, and +4;however they argued that EPR signals are consistent with acomplex magnetic entity composed of Mn ions of differentvalences. Park et al. 83attempted to support the Mn3+/Mn4+model based on the Mn 2 px-ray photoelectron spectroscopy /H20849XPES /H20850and O 1 sabsorption. However, the signiﬁcant dis- crepancy between the weighted Mn3+/Mn4+spectrum and the experimental one for given xsuggests a more complex doping effect. Subias et al.84examined the valence state of Mn utilizing Mn K-edge x-ray absorption near edge spectra /H20849XANES /H20850; however, a large discrepancy is found between experimental spectra given intermediate doping and appro-priate superposition of the end members. The valence state of Mn in Ca-doped LaMnO 3was stud- ied by high-resolution Mn K/H9252emission spectroscopy by Ty- sonet al.85No evidence for Mn2+was claimed at any x values seemingly ruling out proposals regarding the Mn3+ disproportionation. However, this conclusion seems to be ab- solutely unreasonable one. Indeed, electron center MnO610− can be found in two conﬁgurations with formal Mn valences Mn2+and Mn1+/H20849not simple Mn2+/H20850. In its turn, the hole center MnO68−can be found in two conﬁgurations with formal Mn valences Mn4+and Mn3+/H20849not simple Mn4+/H20850. Furthermore, even the bare center MnO69−can be found in two conﬁgura- tions with formal Mn valences Mn3+and Mn2+/H20849not simple Mn3+/H20850. So, within the model the Mn K/H9252emission spectrum for the Ca-doped LaMnO 3has to be a superposition of ap- propriately weighted Mn1+,M n2+,M n3+, and Mn4+contribu- tions /H20849not simple Mn4+and Mn3+, as one assumes in Ref. 85/H20850. Unfortunately, we do not know the Mn K/H9252emission spectra for the oxide compounds with Mn1+ions; however a close inspection of the Mn K/H9252emission spectra for the series of Mn oxide compounds with Mn valence varying from 2+to 7+ /H20849Fig. 2 in Ref. 85/H20850allows us to uncover a rather clear dependence on valence and indicates a possibility to explainthe experimental spectrum for Ca-doped LaMnO 3/H20851Fig.4/H20849a/H20850/H20852 as a superposition of appropriately weighted Mn1+,M n2+, Mn3+, and Mn4+contributions. Later86it has been shown that MnL-edge absorption rather than that of Kedge is com- pletely dominated by Mn 3 dstates and, hence, is an excel- lent indicator of Mn oxidation state and coordination. Inter-estingly that the results of the x-ray absorption and emissionspectroscopy in vicinity of the Mn L 23edge87provide a strik- ing evidence of a coexistence of Mn3+and Mn2+valence states in a single-crystalline LaMnO 3. This set of conﬂicting data together with a number of additional data88suggests the need for an in-depth explora-tion of the Mn-valence problem in this perovskite system. However, one might say, the doped manganites are not onlysystems with mixed valence but systems with indeﬁnite va-lence, where we cannot, strictly speaking, unambiguouslydistinguish Mn species with either distinct valence state. It seems, by now, that there are no techniques capable of direct and unambiguous detection of electron-hole Bose liq-uid. However, we do not see any sound objections againstsuch a scenario that is shown to explain a main body ofexperimental data. VII. HOLE DOPING OF PARENT MANGANITE Evolution of the electronic structure of nominally insulat- ing 3 doxides under a nonisovalent substitution as in La1−x3+Srx2+MnO 3remains one of the challenging problems in physics of strong correlations. A conventional model ap-proach focuses on a hole doping and implies a change in the/H20849quasi /H20850particle occupation in the valence band or a hole lo- calization in either cation 3 dorbital or anion O 2 porbital or in a proper hybridized molecular orbital. However, in the 3 d oxides unstable with regard to a charge transfer such as par-ent manganites one should expect just another scenario whenthe nonisovalent substituents do form the nucleation centersfor the EH droplets thus provoking the ﬁrst-order phase tran-sition into an EH disproportionated phase with a proper de-viation from a half-ﬁlling. Conventional double-exchange model implies the manga- nese location of the doped hole and its motion in the latticeformed by nominal parent manganite. 2However, by now there are very strong hints at oxygen location of doped holes.One might point to several exciting experimental results sup-porting the oxygen nature of holes in manganites. The ﬁrst isa direct observation of the O 2 pholes in the O 1 sx-ray absorption spectroscopy measurements. 89Second, Tyson et al.85in their Mn K/H9252emission spectra studies of the Ca- doped LaMnO 3observed an “arrested” Mn-valence response to the doping in the x/H110210.3 range, also consistent with cre- ation of predominantly oxygen holes. Third, Galakhov et al.90reported Mn 3 sx-ray emission spectra in mixed-valence manganites and showed that the change in the Mn formalvalency from 3 to 3.3 is not accompanied by any decrease inthe Mn 3 ssplitting. They proposed that this effect can be explained by the appearance in the ground-state conﬁgura-tion of holes in the O 2 pstates. The oxygen location of the doped holes is partially supported by observation of anoma-lously large magnitude of saturated magnetic moments inferromagnetic state for different doped manganites. 75,76 Two oxygen-hole scenarios are possible. The ﬁrst implies the hole doping directly to bare A-AFI phase of parent man-ganite. Given light doping we arrive at the hole trapping inpotential wells created by the substituents such as Ca 2+,S r2+, or cation vacancies. This gives rise to evolution of hole-richorbitally disordered ferromagnetic phase. The volume frac-tion of this phase increases with x, and ferromagnetic order- ing within this phase introduces spin-glass behavior wherethe ferromagnetic phase does not percolate in zero magneticﬁeld H=0; but growth of the ferromagnetic phase to beyond percolation in a modest ﬁeld can convert the spin glass to aDISPROPORTIONATION AND ELECTRONIC PHASE … PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-15bulk ferromagnetic insulator. On further increasing the hole doping the ferromagnetic metallic ground state is obtainedwith itinerant oxygen holes and degenerate e gorbitals of Mn3+ions. In second scenario one proposes that doped holes trigger the phase transition to an “asymmetrically” disproportion-ated phase with nominal non-JT Mn 2+ions and oxygen holes that can form a band of itinerant carriers. This scenario im-plies that the doped holes simply change a hole band ﬁlling. Both scenario imply an unconventional system with two, Mn 3 dand O 2 p, unﬁlled shells. One should note that de- spite a wide-spread opinion the correlation effects for theoxygen holes can be rather strong. These could provide acoexistence of the two /H20849manganese and oxygen /H20850nonﬁlled bands. Such a p-dmodel with ferromagnetic p-dcoupling imme- diately explains many unconventional properties of the holedoped manganites. First of all, at low-hole content we deal with hole localization in impurity potential. Then, given fur-ther hole doping a percolation threshold occurs accompaniedby insulator-anionic oxygen metal phase transition and fer-romagnetic ordering both in oxygen and Mn sublattices dueto a strong ferromagnetic Heisenberg pdexchange. However, it should be noted that ferromagnetic sign of pdexchange is characteristic of nonbonding panddorbitals. The oxygen-hole doping results in a strong spectral weight transfer from the intense O 2 p-Mn 3 dCT transition bands to the O 2 pband developed. The Mn 3+d-dtransitions will gradually shift to the low energies due to a partial O 2 p hole screening of the crystalline ﬁeld. In a whole, opticaldata do not disprove the oxygen-hole scenario. Despite many controversial opinions regarding the elec- tronic structure of doped holes the current description ofcomplex phase diagrams for doped manganites implies awell-developed phase separation with coexistence of bare an-tiferromagnetic and several ferromagnetic phases. 2,91What is the role played by the EHBL phase inherent for parent man-ganites? Hole doping of parent manganite is produced by a nonis- ovalent substitution as in La 1−xSrxMnO 3or by an oxygen nonstoichiometry. The Sr2+and Ca2+substituents form effec- tive trapping centers for the EH dimers and the nucleationcenters for the EH Bose liquid. At a critical substituent con-centration x c/H110150.16 one arrives at a percolation threshold3 when the conditions for an itinerant particle hopping do emerge. Holes are doped into EH Bose liquid of parentLaMnO 3similar to generic BaBiO 3system only pairwise, transforming formally electron MnO610−center to hole MnO68−center. Similarly to BaBiO 3doped hole centers form local composite bosons which shift the system from half-ﬁlling /H20849n B=1 /2/H20850. It seems the EHBL phase addressed above appears to be an important precursor for a ferromagnetic metallic phaseresponsible for colossal magnetoresistance observed indoped manganites. Existence of such an intermediate “poormetallic” phase seems to be essential for a transformation ofbare insulating A-AFI phase to a “good-metallic” phase un-der hole doping. Low-energy CT excitations typical forEHBL phase and well exhibited in optical response /H20849see Figs. 7and8/H20850give rise to a signiﬁcant screening of electrostaticinteractions and to a suppression of localization trend for doped charge carriers with their escape out of charge trapsand the evolution of itineracy. This trend is well illustrated inFig. 8, where the dielectric function /H9255 2is shown both for parent and slightly hole doped LaMnO 3. We see a clear red- shift both for low-energy /H208492e V /H20850d-dCT band and high- energy /H208494.5 eV /H20850p-dCT band with a rise of intensity for both bands, particularly sharp for the 2 eV band. All these effectsevidence the lowering of effective values for the chargetransfer energies, which is a clear trend to “metallicity.” One of the intriguing issues is related with seemingly masked superconducting ﬂuctuations in doped manganitesand its relation to colossal magnetoresistance. Indeed, dopedmanganites reveal many properties typical for superconduct-ing materials or, rather, unconventional superconductors suchas cuprates. Kim 92proposed the frustrated p-wave pairing superconducting state similar to the A1state in superﬂuid He-3 to explain the CMR, the sharp drop of resistivity, thesteep jump of speciﬁc heat, and the gap opening in tunnelingof manganese oxides. In this scenario, colossal magnetore-sistance /H20849CMR /H20850is naturally explained by the superconduct- ing ﬂuctuation with increasing magnetic ﬁelds. This idea isclosely related to the observation of anomalous proximityeffect between superconducting YBaCuO and a manganeseoxide, La 1−xCaxMnO 3or La 1−xSrxMnO 3,93and also the concept of local superconductivity manifested by dopedmanganites. 94 VIII. CONCLUSION To summarize, we do assign anomalous properties of par- ent manganite LaMnO 3to charge transfer instabilities and competition between insulating A-AFM phase and metallic-like dynamically disproportionated phase formally separatedby a ﬁrst-order phase transition at T disp=TJT/H11015750 K. We report a comprehensive elaboration of a so-called dispropor-tionation scenario in manganites which was addressed earlierby many authors; however, by now it was not properly de-veloped. The unconventional high-temperature phase is ad-dressed to be a speciﬁc electron-hole Bose liquid rather thana simple “chemically” disproportionated R/H20849Mn 2+Mn4+/H20850O3 phase. We arrive at highly frustrated system of triplet /H20849eg2/H208503A2gbosons moving in a lattice formed by hole Mn4+ centers when the latter tend to order G-type antiferromag- netically and the triplet bosons tend to order ferromagneti-cally both with respect to its own site and its nearest neigh-bors, nearest neighboring bosons strongly prefer anantiferromagnetic ordering. Lastly, the boson transport pre-fers an overall ferromagnetic ordering. Starting with different experimental data we have repro- duced a typical temperature dependence of the volume frac-tion of the high-temperature mixed-valence EHBL phase.New phase nucleates as a result of the CT instability andevolves from the self-trapped CT excitons or speciﬁc EHdimers, which seem to be a precursor of both insulating andmetalliclike ferromagnetic phases observed in manganites.We present a detailed analysis of electronic structure, energyspectrum, optical, magnetic, and resonance properties of EHdimers. We argue that a slight nonisovalent substitution, pho-A. S. MOSKVIN PHYSICAL REVIEW B 79, 115102 /H208492009 /H20850 115102-16toirradiation, external pressure, or magnetic ﬁeld gives rise to an electronic phase separation with a nucleation or anovergrowth of EH droplets. Such a scenario provides a com-prehensive explanation of numerous puzzling properties ob-served in parent and nonisovalently doped manganiteLaMnO 3including an intriguing manifestation of supercon- ducting ﬂuctuations. We argue that the unusual55Mn NMR spectra of nonis- ovalently doped manganites LaMnO 3may be addressed to be a clear signature of a quantum disproportionation and forma-tion of EH dimers. Given the complex phase-separation dia-gram of this class of materials, the study of the nominally stoichiometric parent compound could give a deep insightinto the physics governing the doped version of these man- ganese oxides. It would be important to verify the expecta-tions of EHBL scenario by more extensive and goaledstudies. ACKNOWLEDGMENTS I thank N. N. Loshkareva, Yu. P. Sukhorukov, K. N. Mikhalev, Yu. B. Kudasov, and V. V. Platonov for stimulatingand helpful discussions. The work was supported by RFBRunder Grants No. 06-02-17242, No. 07-02-96047, and No.08-02-00633. 1T. 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PhysRevB.85.045104.pdf	PHYSICAL REVIEW B 85, 045104 (2012) Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors Shinsei Ryu,1Joel E. Moore,1,2and Andreas W. W. Ludwig3 1Department of Physics, University of California, Berkeley, California 94720, USA 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3Department of Physics, University of California, Santa Barbara, California 93106, USA (Received 30 December 2010; revised manuscript received 25 May 2011; published 5 January 2012) One of the deﬁning properties of the conventional three-dimensional (“ Z2” or “spin-orbit”) topological insulator is its characteristic magnetoelectric effect, as described by axion electrodynamics. In this paper,we discuss an analog of such a magnetoelectric effect in the thermal (or gravitational) and magnetic dipoleresponses in all symmetry classes that admit topologically nontrivial insulators or superconductors to exist inthree dimensions. In particular, for topological superconductors (or superﬂuids) with time-reversal symmetry,which lack SU(2) spin rotation symmetry (e.g., due to spin-orbit interactions), such as the B phase of 3He, the thermal response is the only probe that can detect the nontrivial topological character through transport.We show that, for such topological superconductors, applying a temperature gradient produces a thermal-(or mass-) surface current perpendicular to the thermal gradient. Such charge, thermal, or magnetic dipoleresponses provide a deﬁnition of topological insulators and superconductors beyond the single-particle picture.Moreover, we ﬁnd, for a signiﬁcant part of the “tenfold” list of topological insulators found in previous workin the absence of interactions, that in general dimensions, the effective ﬁeld theory describing the space-timeresponses is governed by a ﬁeld theory anomaly. Since anomalies are known to be insensitive to whether theunderlying fermions are interacting, this shows that the classiﬁcation of these topological insulators is robustto adiabatic deformations by interparticle interactions in general dimensionality. In particular, this applies tosymmetry classes DIII, CI, and AIII in three spatial dimensions, and to symmetry classes D and C in two spatialdimensions. DOI: 10.1103/PhysRevB.85.045104 PACS number(s): 72 .10.−d, 73.21.−b, 73.50.Fq I. INTRODUCTION The considerable recent progress in understanding topolog- ical insulating phases in three dimensions was initiated by stud-ies of single-particle Hamiltonians describing electrons with time-reversal invariance. 1–5In both two and three dimensions, time-reversal invariant Fermi systems that have topologicalinvariants of Z 2type are known to exist: insulators can be classiﬁed as “ordinary” or “topological” by band-structure integrals similar to the integer-valued integrals that appear inthe integer quantum Hall effect. 6,7These invariants survive when disorder is added to the system. In fact, stability to disorder is one of the deﬁning properties of topological insulating phases (and also topological superconductors). Thecomplete classiﬁcation of topological insulators and topolog- ical superconductors in any dimension has been obtained in Refs. 8and 9, and in every dimension, ﬁve of the ten Altland-Zirnbauer symmetry classes 11,12of single-particle Hamiltonians (including some describing the Bogoliubov quasiparticles of superconductors or superﬂuids, rather thanordinary electrons) contain topological insulating phases with topologically protected gapless surface states. An important question is, how can these various phases be deﬁned in terms of a physical response function? Asidefrom aiding in experimental detection, such deﬁnitions alsoindicate that the phase is well-deﬁned in the presence ofinteractions. The best studied example is the conventionalthree-dimensional (“ Z 2” or “spin-orbit”) topological insulator with no symmetries beyond time-reversal, which has beenrecently observed in various materials, including Bi xSb1−xalloys,13Bi2Se3, and Bi 2Te3.14–17Such materials support a quantized magnetoelectric response generated by the orbitalmotion of the electrons, i.e., the phase can be deﬁned bythe response of the bulk polarization to an applied magneticﬁeld. 18,19The possibility of such a bulk response was discussed some time ago as a condensed-matter realization of “axionelectrodynamics.” 20 The ﬁrst goal of this paper is to ﬁnd, for all three- dimensional topological insulators and superconductors, thecorresponding responses that result from the coupling of thetheory to gauge and gravitational 21ﬁelds. The second goal of this paper is to understand to what extent the classiﬁca-tion scheme found previously for topological insulators ofnoninteracting fermions can be stable to fermion interactions.This addresses the question of whether certain topologicalinsulators that describe distinct topological phases in theabsence of fermion interactions (connected only by quantumphase transitions at which the bulk gap closes) can beadiabatically deformed into each other when interactions areincluded (without closing the bulk gap). We ﬁnd that thiscannot happen, e.g., in symmetry classes DIII, CI, and AIIIin three spatial dimensions, and in symmetry classes D andC in two dimensions. More generally, in the ﬁnal (moretechnical) section of this paper, we provide an answer to thisquestion in general dimensionalities for a signiﬁcant part ofthe list of topological insulators (superconductors) within the“tenfold” classiﬁcation scheme, obtained for noninteractingparticles. 8–10,22In particular, we relate the topological fea- tures of these topological insulators to the appearance ofa topological term in the effective ﬁeld theory describing 045104-1 1098-0121/2012/85(4)/045104(15) ©2012 American Physical SocietySHINSEI RYU, JOEL E. MOORE, AND ANDREAS W. W. LUDWIG PHYSICAL REVIEW B 85, 045104 (2012) TABLE I. Electromagnetic and gravitational (thermal) responses for ﬁve out of ten Altland-Zirnbauer symmetry classes (AII, CI, CII, DIII, and AIII). The assumptions made in the ﬁrst four classes are that U(1) conserved currents arise from electrical charge and thatSU(2) conserved currents arise from spin. In class AIII (as indicated by asterisks), the U(1) conservation law may arise either from charge or one component of spin. Symmetry Charge Gravitational Dipole AII√√ CI√√ CII√√ DIII√ AIII∗√∗ space-time-dependent responses. Alternately, we relate these topological terms to what are known as “anomalies” appearingin the theories describing the responses. Since the “anomalies”are known to be insensitive to whether the underlyingfermions are interacting or not, our so-obtained descriptionof the topological features demonstrates the insensitivity ofthese topological insulators (superconductors) to adiabaticdeformations by interactions. The general picture emerging from the results presented in this paper is that the topological insulators (supercon-ductors) appearing in the “ten-fold list” can be viewed asgeneralizations of the d=2 Integer Quantum Hall Effect to systems in different dimensions dand with different (“anti-unitary”) symmetry properties. 8While the “ten-fold classiﬁcation scheme” was originally established in Refs. 8 and 9for noninteracting fermions, the characterization in terms of anomalies implies that this extends also to all thoseinteracting systems which can be adiabatically connected to noninteracting topological insulators (superconductors)without closing the bulk gap. (This may include fairlystrong interactions, albeit typically not expected to exceedthe noninteracting bulk gap.) One may expect that to any ofthe topological insulators (superconductors) in the “ten-foldlist” (viewed as generalizations of the Integer Quantum HallEffect) corresponds a set of “fractional” topological insulators(superconductors) notadiabatically connected to a noninter- acting one, in analogy to the case of the two-dimensionalQuantum Hall Effect. This includes, e.g., a recently proposedthree-dimensional “fractional” topological insulator Ref. 23. One expects a description in terms of anomalies to carryover to all such systems and to play a role in a (future)perhaps comprehensive characterization of such “fractional”topological insulators (superconductors). In the present paper,however, we focus on those interacting topological insulators(superconductors) which can be adiabatically connected to anoninteracting system of fermions. Let us focus now on the topological insulators (supercon- ductors) in d=3 spatial dimension (see also Table I). From a conceptual point of view, it is the surface responses that are simplest to describe, and they are quantized (but they maynot necessarily be the most easily accessible experimentally;therefore, we also discuss the bulk responses further below).Charge surface response . This is, in particular, relevant for the (“ Z 2” or “spin-orbit”) topological insulator, which is time-reversal-invariant. Upon subjecting its surface to aweak time-reversal symmetry-breaking perturbation (in thezero-temperature limit), the surface turns into a quantum Hallinsulator whose electrical surface Hall conductance takes onthe quantized value 24 σxy/(e2/h)=n 2(1) (a multiple of half the conductance quantum) as the strength of the symmetry-breaking perturbation is reduced to zero (alwaysat zero temperature). Here, n=0 and 1 for the “ Z 2” (or “spin- orbit”) topological insulator18,24(in the so-called symmetry class AII), in the topologically trivial and nontrivial phase,respectively. While the surface of Z 2topological insulators in class AII may exhibit any odd (even) number Dirac cones inthe topologically nontrivial (trivial) phase at the microscopiclevel, only the odd-even parity, n=1 and 0 of that number, is topologically protected. For the less familiar topologicalinsulator in symmetry class AIII a relation analogous to Eq. ( 1) applies. 22,26,45 Spin surface response . Analogous effects are known29for the time-reversal-invariant topological (spin-singlet) super-conductor in symmetry class CI in d=3 spatial dimension. Subjecting its surface, as above, to a weak time-reversalsymmetry-breaking perturbation (in the zero-temperaturelimit), the surface turns into what is known as the “spinquantum Hall insulator.” 27,28Due to spin-singlet pairing, this superconductor has SU(2) Pauli-spin rotation symmetry,which permits the deﬁnition of the “surface spin conductivity.”In particular, 27a gradient of magnetic ﬁeld within the surface (say in the zdirection of spin space) leads to a spin current perpendicular to the gradient (and within the surface). Thisdeﬁnes the “surface spin-Hall conductance,” which, similar toEq. ( 1), takes on the quantized value σ (spin) xy/slashbigg(¯h/2)2 h=n 2(2) [n-times half the “spin-conductance quantum”(¯h/2)2 h]a st h e time-reversal symmetry-breaking perturbation is reduced tozero. 8,29,45 Thermal surface response . As we show in Sec. III B of this paper, an analogous effect occurs for the thermalresponse at the surface of the time-reversal-invariant topo-logical superconductor in symmetry class DIII in d=3 spatial dimensions: subjecting its surface, as above, to aweak time-reversal symmetry-breaking perturbation (in thelow-temperature limit), a temperature gradient within thesurface leads to a heat (energy) current in the perpendiculardirection in the surface. The so-deﬁned surface thermal Hallconductance σ T xy(when divided by temperature) tends, similar to Eqs. ( 1) and ( 2), in the zero-temperature limit to a quantized value /parenleftbig σT xy/T/parenrightbig/slashbigg(πkB)2 3h=±c/2,where c=n/2( 3 ) 045104-2ELECTROMAGNETIC AND GRA VITATIONAL RESPONSES ... PHYSICAL REVIEW B 85, 045104 (2012) as the symmetry-breaking perturbation is reduced to zero.8,22,45 [c×(πkB)2 3his the thermal conductance for a Majorana fermion whenc=1/2 (its central charge).] If we start out with a noninteracting topological insulator, one can explicitly compute the theory describing variousspace-time-dependent responses. [For the thermal responsesof the DIII topological superconductor in d=3 spatial dimension, this is done in Sec. III B of this paper. For the SU(2) spin responses of the topological singlet superconductorin symmetry class CI this was done in Ref. 29. For a signiﬁcant part of the list of all topological insulators (superconductors),this is done more generally in Sec. Vof this paper for all dimensionalities.] Due to the fact that the underlyinginsulators are topological, the ﬁeld theories for the responsesturn out to be described by what are called anomalies. Theanomalies turn out to provide 63an alternative characterization of topological insulators (superconductors) [except in certainone-dimensional cases 53]. The charge, spin, and thermal surface responses discussed above are consequences of suchanomalies. 30Anomalies are known to be insensitive to the presence or absence of interactions. They are thus independentof the strength of the interactions and can only change whena bulk quantum phase transition is crossed (at which the bulkgap closes). While these surface responses are quantized and theo- retically useful in that they permit one to understand thestability of the topological insulator (superconductor) phasesto interactions (for the cases discussed above, and in Sec. V for general dimensionalities), they may not all be directlyaccessible experimentally. Therefore, we discuss below alsothe various bulk responses. The bulk responses that we ﬁnd are of three types: charge response, previously shown to lead to a quantized E·Bterm in the ordinary Z 2topological insulator (“axion electrodynamics”);18–20gravitational response, when energy ﬂows lead to an analog of this term for gravitational ﬁelds,leading to a Lense-Thirring frame-dragging effect 31when a temperature gradient is applied; and magnetic dipole response, when a magnetic dipole current induced by an applied perturbation leads to an electrical ﬁeld. A single phase may show more than one of these effects; for example, a phasewith a conserved SU(2) spin current can show a non-Abelianresponse of this type in the presence of an SU(2) gauge ﬁeldcoupling to this current, but will also show a magnetic dipoleresponse via its coupling to ordinary U(1) electromagnetism.We obtain these possible responses for each of the ﬁvesymmetry classes in three dimensions supporting topologicalphases. 8,9As in the classiﬁcation in Ref. 8, the approach we take is based upon the surfaces of these topologicalphases; these surfaces carry currents leading to new termsin the effective action of gravitational and electromagneticﬁelds. Our results for the various symmetry classes withtopological invariants in three dimensions are summarized inTable I. These bulk responses are “topological” to varying degrees. The charge response is topological both in its spatial depen-dence and as a term of the effective action: quantization ofthe response is tied to quantization of the electrical charge andthe Dirac quantization condition. The gravitational responseis topological in terms of the spatial dependence, but its coefﬁcient is related to the mass or energy of the underlyingparticles and hence not quantized to the same degree asthe charge response. The magnetic dipole response is nottopological in the sense of being metric-independent, but itdoes arise from sample boundaries in the same way as theother responses. This paper is organized as follows: We begin in Sec. II by reviewing the axion electromagnetism for the three-dimensional topological insulators in the spin-orbit symmetryclass (symmetry class AII). In Sec. III, the thermal response of three-dimensional time-reversal invariant topological su-perconductors (such as the B-phase of 3He) is discussed by exploiting a close analogy of electromagnetism and gravityin Newtonian approximation. In Sec. IV, the dipole response is discussed for three-dimensional topological phases when atleast one component of spin is conserved. All these responseswill be discussed from a much broader perspective in Sec. V in terms of anomalies of various kinds (chiral anomaly, gaugeanomaly, gravitational anomaly), and the descent relationpertaining to these anomalies. We conclude in Sec. VI. II. CHARGE RESPONSES For an explicit example, consider a cylinder of a topological insulator with surface Hall conductance ±e2/(2h), deﬁned with reference to the outward normal (see Fig. 1). (Below, we choose a plus sign for the surface Hall conductance bysubjecting the surface to a weak external time-reversal sym-metry source.) The motivation for considering this examplein some detail is that it will lead to a direct interpretation ofthe corresponding gravitational response below. The currentresponse to an applied electrical ﬁeld along the cylinder axisis (see Fig. 1) j=j θˆθ,where jθ=e2 2hEz. (4) Now the magnetic ﬁeld induced by this current follows from one of Maxwell’s equations, ∇×B=4π cj, (5) which leads to the magnetostatic equation B(x)=1 c/integraldisplay j(x/prime)×(x−x/prime) \|x−x/prime\|3d3x/prime. (6) The result for a thin cylinder is that the magnetic ﬁeld at the cylinder axis, well away from the cylinder ends, is given byB=B zˆzwith Bz=1 c/integraldisplay∞ −∞r(2πr)jθ (r2+a2)3/2da=4π cjθ=2πe2Ez hc.(7) This magnitude follows from minimizing the magnetic energy, HB=B2 8π−e2 2hcE·B, (8) 045104-3SHINSEI RYU, JOEL E. MOORE, AND ANDREAS W. W. LUDWIG PHYSICAL REVIEW B 85, 045104 (2012) ___ _ _ __ _ ++++ +++ +(b) (a) FIG. 1. Electric and thermal response of topological insulators, and thermal response of topological triplet superconductors, in a cylindrical geometry. (a) Electric ( j)o rt h e r m a l( jT) current driven by applied electric ﬁeld ( E) or thermal gradient ( ∇T/T ). (b) A response dual to (a) where an applied magnetic ﬁeld in the zdirection induces charge polarization. which follows from the Maxwell Lagrangian supplemented with the θterm (axion term) Lθ=θe2 2πhcE·B=θe2 16πhc/epsilon1μνρλFμνFρλ (9) for the coupling θ=−π. (The negative sign in this equation is picked out by the choice of the direction of the current ﬂowaround the cylinder.) To understand the dual response (see Fig. 1), which is an electrical ﬁeld induced by an applied magnetic ﬁeld, one needsto include the ends of the cylinder. Applying a magnetic ﬁeldnormal to a Hall layer increases or decreases the charge densitydepending on the direction of the ﬁeld, as is required for thecharge continuity equation to follow from Maxwell’s equation ∂B ∂t+∇×E=0. (10) Hence an applied magnetic ﬁeld induces an electrical polar- ization along the interior of the cylinder. We now turn to agravitational version of the above physics, generated by energyﬂows from surface thermal Hall layers. III. GRA VITATIONAL RESPONSES A. Gravitoelectromagnetism Our approach will be to start from the energy ﬂow at surfaces of a topological phase, which is the microscopicsource of the gravitational response. The importance of thisresponse is that it is the only one that exists in the importantsymmetry class DIII, which includes superﬂuid 3He. We use this phase as an explicit example in the following. The surfaceMajorana mode that exists in this phase does not carry charge,but it does carry heat, leading to a thermal Hall effect. Hencea temperature gradient applied to a cylinder leads to an energy ﬂow perpendicular to the applied gradient, j T θ=σT xy(−∂zT)=c−2TσT xyEg,z, (11) where for future use we have treated temperature as a scalar potential generating a ﬁeld Eg=−c2(∇T)/Twith units of acceleration. The physical meaning of this scalar potentialwas worked out by Luttinger in his derivation of the thermaltransport coefﬁcients: 32in a near-equilibrium system, the effect of a thermal gradient is equivalent to that obtained froma gravitational potential ψsuch that ∇ψ=∇T T, (12) where ψis the gravitational potential energy per mass, divided byc2. This rotational energy ﬂow couples to the gravitational ﬁeld at the ﬁrst post-Newtonian approximation (i.e., the couplingis down by a factor v/c compared to the static gravitational effect present in the absence of the applied gradient). Becausetemperature couples to the local energy density in the sameway as an applied gravitational potential, as used by Luttingerin his derivation of the thermal Kubo formula, 32we can view this effect similarly to the charge response above, asa gravitational “magnetic” ﬁeld resulting from the energy ﬂowthat was induced by a gravitational “electric” ﬁeld reﬂectingthe temperature gradient. This analogy can be made precise in the near-Newtonian limit using the gravitoelectromagnetic equations 33that apply to a near-Minkowski metric. The relevant equation is that amass current induces a gravitomagnetic ﬁeld B g, deﬁned more precisely below, via the equation ∇×Bg=−4πGjm c. (13) Here jmis the (three-dimensional) mass current density, satisfying jm=jT/c2, andGis the effective Newton constant of the material. The negative sign in this equation compared tothe corresponding Maxwell’s equation is physically signiﬁcantand results from the difference that equal masses attract,while equal charges repel. The ﬁeld E g, like Bg, has units of acceleration, and the gravitational force on a test particle ofsmall mass m testis F=mtest/parenleftbigg Eg+2v c×Bg/parenrightbigg , (14) where vis the particle velocity. The factor of 2 here results from the spin-2 nature of the gravitational ﬁeld. Now, by the same steps as above, there is an induced ﬁeld along the cylinder axis, Bg=4πGjT θ c3=4πG c3TσT xyEgz c2. (15) Since σT xyhas the units k2 BT/h of a two-dimensional thermal conductivity, the ratio between BgandEgis of the form G(energy2)/(hc5), which is dimensionless (the gravitational analog of the ﬁne-structure constant that appears in the chargecase). The gravitomagnetic ﬁeld then has exactly the same spatial dependence as the magnetic ﬁeld in the axion case computed 045104-4ELECTROMAGNETIC AND GRA VITATIONAL RESPONSES ... PHYSICAL REVIEW B 85, 045104 (2012) above. In particular, it is topological (e.g., the ﬁeld at the cylinder axis does not fall off as the cylinder radius becomeslarger) and scales with the energy ﬂow, which in turn scalesquadratically with the mass of the underlying particles. B. Gravitational instanton term We now discuss the gravitational response in topological insulators and superconductors from a more formal point ofview. When discussing electromagnetic responses in topo-logical insulators, we can couple electrons to an external(background) U(1) gauge ﬁeld. The θterm in the effective action for the gauge ﬁeld then results by integrating overthe gapped electrons. To discuss gravitational and thermalresponses, we can take a similar approach: we can introduce anexternal gravitational ﬁeld that couples to fermions (electronsfor topological insulators, and fermionic Bogoliubov quasipar-ticles for topological superconductors). By integrating over thegapped fermions, we obtain an effective gravitational action.The derivation of the effective action proceeds in a way quiteparallel to that of the U(1) case: Indeed, both of them arerelated to a chiral anomaly, as we will see below. For topological insulators or superconductors deﬁned on a lattice, it is not obvious how to couple fermions to gravity in away fully invariant under general coordinate transformations.Also, there is of course no Lorentz symmetry on a lattice. Yet,energy and momentum are conserved, and one can think ofintroducing an external ﬁeld that couples to these conservedquantities. The gravitoelectromagnetic approach discussed inthe previous subsection is based on a particular background(ﬂat Minkowski metric), and is an approximation of the fullEinstein gravity in the limit where the mass ﬂows are small insome particular reference frame deﬁned by the system with nothermal perturbation. However, all topological insulators (superconductors) are known 22to possess a representative in the same topological phase, which is described by a Dirac Hamiltonian. Fermionswhose dynamics is described by a Dirac Hamiltonian cannaturally be coupled to a gravitational background ﬁeld. (Thetheory is fully Lorentz invariant, and the coupling to gravityis fully invariant under general coordinate transformations,and can be described in terms of the spin connection.) Forthis reason, we provide (below) a derivation of the effectiveaction in terms of the Dirac representative of the topologicalphases. The topological features of the effective action forthe gravitational responses are expected to be independent ofthe choice of representative in the topological class, and thusto have a much more general applicability. Physically, suchgravitational responses describe thermal response functions. 32 We thus consider the following single 4 ×4 continuum Dirac model: H=/integraldisplay d3xψ†(−i∂·α+mβ)ψ, (16) where ψ†andψrepresent creation and annihilation operator of complex fermions, respectively, and α=σ1⊗σand β=σ3⊗σ0are the Dirac matrices ( σ0,1,2,3are standard Pauli matrices). (In this subsection, we use natural units, c=¯h=1, and set the Fermi velocity to be 1 for simplicity.) Fortopological superconductors, we need to use real (Majorana) fermions instead of complex fermions. We assume the Dirac model is in a topologically nontrivial phase for m> 0 while it is in a trivial phase for m< 0: While this does not look apparent from the action in the continuumlimit, when the Dirac model is derived from an appropriatelattice model, the sign of the mass does determine the natureof the phase. In the presence of a gravitational background,the fermionic action is given by 34 S[m,¯ψ,ψ,e ]=/integraldisplay d4x√gL, (17) L=¯ψeaμiγa/parenleftbigg ∂μ−i 2ωμab/Sigma1ab/parenrightbigg ψ−m¯ψψ, where μ,ν,... =0,1,2,3 is the space-time index, and a,b,... =0,1,2,3 is the ﬂat index; eaμis the vielbein, and ωμabis the spin connection; /Sigma1ab=[γa,γb]/(4i). (See Ref. 35 for our conventions of metric, vielbein, spin connection,etc.) The effective gravitational action W eff[m,e]f o rt h e gravitational ﬁeld is then obtained from the fermionic pathintegral e iWeff[m,e]=/integraldisplay D[¯ψ,ψ ]eiS[m,¯ψ,ψ,e ]. (18) A key observation is that the continuum Hamiltonian H enjoys a continuous chiral symmetry: we can ﬂip the sign ofmass, in a continuous fashion, by the following chiral rotation: ψ→ψ=e iφγ 5/2ψ/prime,ψ†→ψ†=ψ†/primee−iφγ 5/2,(19) under which ¯ψ(i∂μγμ−m)ψ=¯ψ/prime[i∂μγμ−m/prime(φ)]ψ/prime, (20) m/prime(φ)=meiφγ 5=m[cosφ+iγ5sinφ], so that m/prime(φ=0)=mandm/prime(φ=π)=−m. Since mcan continuously be rotated into −m, one would think, naively, Weff[m,e]=Weff[−m,e]. This naive expectation is, however, not true because of chiral anomaly. The chiral transformationthat rotates mcontinuously costs the Jacobian Jof the path integral measure, D[¯ψ,ψ ]=JD[¯ψ /prime,ψ/prime]. (21) The chiral anomaly (the chiral Jacobian J) is responsible for theθterm. The Jacobian Jcan be computed explicitly by the Fujikawa method,36with the result Wθ eff:=− lnJ =θ1 2/bracketleftbigg1 2×384π2/integraldisplay d4x√g/epsilon1cdefRa bcdRb aef/bracketrightbigg (22) when m> 0 while Wθ eff=0 when m< 0. The expression in square brackets is the so-called Dirac genus (see Sec. V below for details), which is equal,34by the Atiyah-Singer index theorem, to the index of the Dirac operator in the curvedbackground. The multiplicative prefactor 1 /2 arises because of the Majorana nature of the Bogoliubov quasiparticles. Theindex in square brackets is in fact an even integer (by Rochlin’stheorem 39). Therefore, (1 /2) of that expression, i.e., half the index, is an integer. Thus the gravitational effective action Wθ eff in Eq. ( 22) equals θtimes an integer, i.e., it is a so-called θterm. 045104-5SHINSEI RYU, JOEL E. MOORE, AND ANDREAS W. W. LUDWIG PHYSICAL REVIEW B 85, 045104 (2012) As we rotate the angle θ=φ,E q .( 20), from zero to 2 π,t h e partition function winds an integer number of times around theorigin in the complex plane. This winding number measuresthe integer 8,10,22of the topological insulator (superconductor). See also Ref. 63. [This winding number is ultimately related to a property of the underlying massless theory. See, e.g., Eq. ( 46) and its generalizations.] Now, since θ→−θunder time reversal, the θangle is ﬁxed by time-reversal symmetry and periodicity to either θ=0o rθ=π. The former corresponds to a topologically trivial state, and θ=πto the topologically nontrivial state. [For a similar discussion on the derivationof the θterm, i.e., the E·Bterm, for the electromagnetic response, see Ref. 26, and for the non-Abelian SU(2) response, see Ref. 29.] Note that if instead we consider complex (Dirac) fermions in the background gravity ﬁeld, the theta angle θis an integer multiple of 2 π, but not of πas in the Majorana case. The part of the effective action that is not related to the Fujikawa Jacobian takes the form of the Einstein-Hilbertaction W EH=(16πG)−1/integraltext d4x√gR, where Gis the effective Newton constant in the bulk of the topological insula-tor (superconductor). The gravitoelectromagnetism equationsmentioned above can be derived from the effective action bytaking the Newtonian limit (near Minkowski limit). To make the connection with the existence of topologically protected surface modes, we note that when there are bound-aries (say) in the x 3direction at x3=L+and at x3=L−,t h e gravitational instanton term Wθ eff, at the nontrivial time-reversal invariant value θ=πof the angle θ, can be written in terms of the gravitational Chern-Simons terms at the boundaries, Wθ eff=ICS\|x3=L+−ICS\|x3=L−, (23) where ( i,j,k=0,1,2) ICS=1 21 4πc 24/integraldisplay d3x/epsilon1ijktr/parenleftbigg ωi∂jωk+2 3ωiωjωk/parenrightbigg (24) withc=1/2. This kind of relationship between the θ-term and the Chern-Simons type term in one lower dimension isa special case of the so-called descent relation and will bediscussed further in Sec. V. This value of the coefﬁcient of the gravitational Chern-Simons term is one-half of thecanonical value (1 /4π)×(c/24) with c=1/2. As before, for fermions with a reality condition (Majorana fermions),the canonical value of the coefﬁcient of the gravitationalChern-Simons term corresponds to c=1/2, as opposed to c=1 for fermions without a reality condition. As discussed by V olovik 37and Read and Green38in the context of the two-dimensional chiral p-wave superconductor, the coefﬁcient of the gravitational Chern-Simons term is directly relatedto the thermal Hall conductivity, which in our case is carriedby the topologically protected surface modes. 40[See Eq. ( 3) of the Introduction.] IV . DIPOLE RESPONSES A. Topological singlet superconductor (class CI) and spin chiral topological insulator (class CII) The last response we consider can be measured in systems with a conserved spin or magnetic dipole current. Amongthe ﬁve symmetry classes that admit a topological phase inthree-spatial dimensions, we thus focus on topological singlet superconductors in symmetry class CI (possessing time-reversal and spin rotation invariance), and also on topologicalinsulators in symmetry class CII (possessing time-reversal butwithout spin rotation invariance) (see Table I). Simple lattice models of the three-dimensional topological singlet superconductor in symmetry class CI were discussedpreviously on the diamond lattice 29and on the cubic lattice,26 for which, in the presence of a boundary (surface), there is astable and nonlocalizing Andreev bound state. Similar to thequantized E·Bterm for the charge response in the topological insulator, the response of topological singlet superconductorsto a ﬁctitious external SU(2) gauge ﬁeld (a “spin” gauge ﬁeld,which couples to conserved spin current) is described by theθterm at θ=πin the (3 +1)-dimensional SU(2) Yang-Mills theory. 29Theθterm predicts the surface quantum Hall effect for spin transport (the spin quantum Hall effect), as alreadymentioned in the Introduction (Sec. I). To detect such a quantum Hall effect for the SU(2) symmetric spin current requires a ﬁctitious external spingauge ﬁeld, and hence one would think it cannot be detectedexperimentally. Nevertheless, we discuss in this section thatthe electromagnetic response carried by the dipole moment ofthe spin current can be measurable. (See Ref. 41for a similar discussion on the dipole response in a 3He-Asuperﬂuid thin ﬁlm or two-dimensional p-wave paired states.) The topological insulator in symmetry class CII (called a “spin chiral topological insulator” in Ref. 26) is in many ways analogous to the more familiar quantum spin Hall effect intwo spatial dimensions, but requires the chiral symmetry inaddition to time-reversal symmetry. (For a lattice model of theZ 2topological insulator in symmetry class CII, see Ref. 26.) Just as an intuitive understanding of the quantum spin Halleffect can be obtained by starting from two decoupled and inde-pendent quantum Hall systems with opposite chirality for eachspin and then gluing them together, this spin chiral topologicalinsulator can be obtained by considering two independenttopological insulators in symmetry class AIII. More generalquantum spin Hall states or spin chiral topological insulatorscan then be obtained by destroying the S zconservation by mix- ing spin-up and -down components. The dipole response forclass CII topological insulators, which we will describe below,assumes that a U(1) part of the SU(2) spin rotation symmetry isconserved (i.e., one component of spin is conserved). However,even when there is no such symmetry, if mixing between twospecies is weak, we can still have such a dipole response. B. Magnetic dipole responses The spin current response at the surface of such a system to an applied magnetic ﬁeld Bvia the Zeeman effect can be written as ja i=α/epsilon1ijk(∂jθ)∂kBa, (25) where αis some constant. Here we have introduced a scalar ﬁeldθ(“axion” ﬁeld),29by analogy with the local electro- magnetic polarizability of the (AII, spin-orbit) topologicalinsulator, to describe the spatial location of the dipole current,which as before is a surface property. Here j a irepresents the ath component of a magnetic dipole current of dipoles in 045104-6ELECTROMAGNETIC AND GRA VITATIONAL RESPONSES ... PHYSICAL REVIEW B 85, 045104 (2012) spatial direction i. Such a current can generate two types of static electromagnetic responses: a dipole density through the continuity equation ∂ija i+∂tna=0, (26) and an electrical ﬁeld through the equation (∇×E)i=/epsilon1ijk∂jEk=μ 4π∂aja i, (27) where μis the permeability of the material of interest. (One could alternately have a time-varying magnetic ﬁeld, just asa current density can produce either a constant magnetic ﬁeldor a time-varying electrical ﬁeld.) The second response maybe unfamiliar but can be derived from elementary principles;see Ref. 42for a discussion of how it can be measured experimentally. Start from a dipole ﬁeld in the laboratoryframe. Take one copy with the dipoles pointing along somedirection ˆnand boost that along v, and take another copy with the dipoles pointing along −ˆnand boost that along −v.F o r a dipole density n a, this leads, in the comoving frame, to the ﬁeldBa=(μ/4π)na, and hence ∇·B=μ 4π∂ana. (28) Using the nonrelativistic Lorentz transformation law E→γ(E+v×B) (29) withγ/similarequal1 leads to Eq. ( 27), with ja i=vina. Now we consider these responses for the surface spin current of a three-dimensional topological singlet supercon-ductor. The spin Hall current is always divergence-free bycommutation of derivatives, ∂ ija i=α/epsilon1ijk∂i(∂jθ∂kBa)=0, (30) since whichever term the ∂iacts on gives zero. However, the electromagnetic response can be nonzero: /epsilon1ijk∂jEk=μ 4π∂aja i=μα 4π∂a(/epsilon1lmn∂mθ∂nBa). (31) There are two parts to this: one “monopole” part is only nonzero if ∂aBa/negationslash=0,and we therefore neglect it. There is also a term μα 4π/epsilon1lmn(∂a∂mθ)∂nBa. (32) C. Example As an example, we compute this response for the case of a surface of a topological singlet superconductor, where the thetaangleθvaries as a function of the distance from the surface (Fig. 2). For the response to be nonzero, we need a=m=z, so the response is to the zcomponent of the magnetic ﬁeld. We get, up to a possible sign, (∇×E) x=−αμ 4π∂2 zθ∂yBz,(∇×E)y=αμ 4π∂2 zθ∂xBz. (33) For the case in which θis ﬁrst constant, then changes linearly inzwithin a surface surface layer, and is then constant again outside this layer (Fig. 2), this response will occur entirely at the top and bottom surfaces of the region of linear change.topological superconductor FIG. 2. Surface of a spin chiral topological insulator (class CII) or topological singlet superconductor (class CI). As an example relevant to possible experiments, we compute this response for the magnetic ﬁeld produced by a magneticmonopole ﬁeld of strength q m(i.e., from one end of a long magnetic dipole), suspended a distance z0above a spin Hall surface layer where θchanges linearly across a thickness d. This surface layer gives two surfaces with (∇×E)x=jm x=∓β∂yBz,(∇×E)y=jm y=±β∂xBz, (34) where β=(αμ)/(4π)π/d. At the top layer, the zcomponent of magnetic ﬁeld is, in cylindrical coordinates, Bz=qmz0/parenleftbig r2+z2 0/parenrightbig3/2, (35) which leads to a surface magnetic current of magnitude, jm θ=3βqmz0r /parenleftbig r2+z2 0/parenrightbig5/2, (36) at the top surface. Since E(r)=/integraldisplay d3r/prime(r−r/prime)×j(r/prime) \|r−r/prime\|2, (37) we obtain that the electrical ﬁeld from the top surface, at a height z1above the top surface (and directly above or below the original monopole), is Ez(z1)=/integraldisplay∞ 0(2πr)dr3βqmz0r /parenleftbig r2+z02/parenrightbig5/2r r2+z12.(38) Evaluating this at the original height z0gives Ez(z0)=(6πβq mz0)2 15z04=4πβq m 5z03. (39) Comparing this to the case of an image charge above a metal, we see that the electrical ﬁeld falls off by one more power ofheight. From the above, the dipole currents are localized tothe top and bottom surfaces of the region where θchanges. The bottom surface contributes with an opposite sign and withz→z+d, so we obtain E z(z0)=4πβq m 5/bracketleftbig z0−3−(z0+d)−3/bracketrightbig , (40) so that for d/lessmuchz0the electric ﬁeld falls off as the fourth power of distance. 045104-7SHINSEI RYU, JOEL E. MOORE, AND ANDREAS W. W. LUDWIG PHYSICAL REVIEW B 85, 045104 (2012) We can understand the scaling of the result by noting that qm divided by length cubed has units of magnetic ﬁeld per length; multiplying by βconverts this to a two-dimensional magnetic charge current density, which has the same units as an electricﬁeld. While the dipole response originates in a topologicalphase, it is not itself “topological” but depends sensitively onthe geometry used to probe it. V . TOPOLOGICAL FIELD THEORIES FOR SPACE-TIME-DEPENDENT RESPONSES IN TOPOLOGICAL INSULATORS AND SUPERCONDUCTORS IN GENERAL DIMENSIONS FROM ANOMALIES The previous sections of this paper complete the list of the (topological) ﬁeld theories describing the space-time-dependent responses of all topological insulators andsuperconductors in three spatial dimensions (3 +1 space-time dimensions). In this section, we will describe, more generally,the (topological) ﬁeld theories for such responses in generaldimensions. Most importantly, the main result obtained in thissection is a general connection between the appearance of suchtopological terms in the ﬁeld theories for the responses and theappearance of what are called anomalies 43for the ﬁeld theories in those space-time dimensions in which topological insulators(superconductors) appear. In fact, we may ask if the existenceof a particular type of anomaly in a given dimension allowsus to predict the existence of a topological insulator (super-conductor) of the “tenfold” classiﬁcation in that dimension.The answer to this question is afﬁrmative. As we demonstratebelow, a large part of the “tenfold” classiﬁcation can be derivedfrom the existence of the known anomalies in correspondingquantum ﬁeld theories in space-time. This can then be thoughtof as yet another derivation of the “tenfold” classiﬁcation, in addition to the previously known derivations such as that based on Anderson localization at the sample boundaries, 8 and K-theory9(as well as a later point of view based on D- branes46,47). Moreover, and most importantly, the appearance of an anomaly is a statement about the respective quantumﬁeld theory (of space-time linear responses) independent ofthe assumption of the absence of interparticle interactions.Thus, anomalies provide a description of topological insulators(superconductors) in the context of interacting systems. A. Topological insulators (superconductors) in the two complex symmetry classes A and AIII from anomalies in the gauge ﬁeld action 1. The integer quantum Hall effect (class A) Let us begin by describing the topological ﬁeld theories describing the space-time-dependent responses of the two“complex” symmetry classes, classes A and AIII in theCartan (Altland-Zirnbauer) classiﬁcation. 8,10,22This includes the most familiar example, namely the integer quantum Hallinsulator (IQH), belonging to symmetry class A. In bothsymmetry classes, A and AIII, there has to exist a conservedU(1) charge (particle number). This is the electromagneticcharge, since these symmetry classes can be realized asnormal electronic systems (as opposed to superconductingquasiparticle systems). 48Therefore, we can minimally couplethese topological insulators to an external U(1) gauge ﬁeld. The ﬁeld theory describing the space-time-dependent linearresponses of the topological insulator can then be obtainedby integrating out the gapped fermions. The fact that theunderlying insulator is topological is reﬂected in the factthat the effective action for the external U(1) gauge ﬁeld,describing the electromagnetic linear responses, contains aterm of “topological origin,” such as, e.g., a Chern-Simonsor aθterm, or corresponding higher-dimensional analogs of these terms (see below for more details). In turn, the presence of terms of topological origin in the so-obtained effective action for the external U(1) gauge ﬁeldis closely related to the presence of a so-called anomaly.To see how an anomaly for the theory of the external U(1)gauge ﬁeld can actually predict the presence of a topologicalphase, let us consider ﬁrst, as the simplest example, the IQHinsulator in d=2 spatial dimensions—symmetry class A. (The space-time dimension is thus D=2+1.) In fact, let us ﬁrst focus attention on the theory of the sample boundary(the edge state), which has d=1 spatial dimensions. It is known (see below) that the effective theory for the linearresponses of the U(1) gauge ﬁeld in D=1+1 space-time dimensions (i.e., of the edge state) can have what is calleda “gauge anomaly” since the space-time dimension Dis even. 33,36The presence of this anomaly simply means that U(1) charge conservation is spoiled by quantum mechanics.In the condensed-matter setting of the IQH insulator, themeaning of this anomaly is that the system (i.e., the edge)inD=1+1 space-time dimensions, exhibiting the anomaly, does not exist in isolation, but is necessarily realized as theboundary of a topological insulator in one dimension higher. Inthis case, the breakdown of the conservation law of U(1) chargeconservation at the boundary simply means that the current“leaks” into the bulk. Thus, in the condensed-matter setting,the presence of the anomaly in the theory at the boundary isnot something abnormal, but it is a physical effect: it is theinteger quantum Hall effect. As we will discuss brieﬂy below,the same reasoning applies to all even space-time dimension,D=2k. Consequently, we see that the presence of a U(1) gauge anomaly predicts the presence of a topological insulatorin one dimension higher. That is, this predicts the presenceof a topological insulator in symmetry class A in D=2k+1 space-time dimensions, in agreement with the “tenfold” wayclassiﬁcation. 2. Three-dimensional insulator (superconductor) in symmetry class AIII Let us now consider the topological insulator (supercon- ductor) in the other complex symmetry class, class AIII, ind=3 spatial dimensions. Again, the space-time dimension D=3+1=4 is even. It is known (see below) that in all even space-time dimensions, the effective action for thespace-time-dependent U(1) gauge ﬁeld may also possess adifferent anomaly [in contrast to the discussion in the precedingsubsection], often referred to as the “chiral (or axial) anomalyin a background U(1) gauge ﬁeld.” 34The meaning of such an anomaly can be explained using Eq. ( 46) below: the so-called axial (or chiral) U(1) current Jμ 5(x)i snotconserved in the presence of a background U(1) gauge ﬁeld, i.e., DμJμ 5(x)/negationslash=0, 045104-8ELECTROMAGNETIC AND GRA VITATIONAL RESPONSES ... PHYSICAL REVIEW B 85, 045104 (2012) where Dμdenotes the covariant derivative in the presence of a background gauge ﬁeld. In the simplest case of a single copyof a massive Dirac fermion (mass m), this covariant derivative of the current is given by Eq. ( 46) below. As displayed in this equation, there are two sources of the lack of conservation: (i)a ﬁnite mass m/negationslash=0 and (ii) the extra “anomaly” term A 2n+2 (to be discussed in more detail below), which represents the breaking of the conservation of Jμ 5by quantum effects.50Now, as discussed in Ref. 26, the presence of a “chiral (or axial) anomaly in a background U(1) gauge ﬁeld” implies directlythe possibility of having a nonvanishing θterm when deriving the effective action for the external U(1) gauge ﬁeld. 51(The θangle is ﬁxed22toθ=πby a discrete symmetry, which is the chiral symmetry for symmetry class AIII.) Thus, the pres-ence of a “chiral (or axial) anomaly in a background U(1) gaugeﬁeld” in D=2kspace-time dimensions signals the existence of a topological insulator in this space-time dimension throughthe appearance of a θterm in the (topological) ﬁeld theory for the linear responses. 3. Anomaly polynomials and descent relation Observe that above we have used anomalies of two kinds , and we used them in two different ways : (i) In case 1. there was an anomaly in the theory of the responses at the boundary [which had D=(d−1)+1 space- time dimensions]. In this case the anomalous theory (i.e., theone at the boundary) was gapless (critical); we refer to thissituation as a gauge anomaly [i.e., nonconservation of the U(1)charge in question]. The presence of this anomaly implied theexistence of a topological insulator in one dimension higher,i.e., in D /prime=d+1 space-time dimensions. The responses of this topological insulator are described by an effective Chern-Simons action for the U(1) gauge ﬁeld in D /prime=d+1 space- time dimensions. [See also Eq. ( 42).] (ii) In case 2. there existed an anomaly in the massive bulk theory in D=d+1 space-time dimensions. This was a chiral anomaly [referring to the violation of the conservationof the global axial U(1) current Jμ 5] in the background of a nonvanishing U(1) background gauge ﬁeld. There are important relationships between the following different anomalies: (i) the U(1) gauge anomaly in D=2n, (ii) the Chern-Simons term (i.e., parity anomaly) in D=2n+ 1,and (iii) chiral anomaly in the presence of a background gauge ﬁeld in D=2n+2, which can be summarized, in terms of the so-called descent relation of the “anomaly polynomial.”34Let us now explain this relation. As mentioned above, it is known that in even space-time dimensions D=2n, there is a U(1) gauge anomaly. If there is a gauge anomaly, the (Euclidean) effective action ln Z[A] in the presence of the gauge ﬁeld Ais not invariant under a gauge transformation A→A+v. Thus we can write δvlnZ[A]=2πi/integraldisplay M2n/Omega1(1) 2n(v,A,F), (41) where the variation δvis the gauge transformation in question, and/Omega1(1) 2nis a 2n-form built from the connection 1-form, A= Aμdxμ, its ﬁeld-strength 2-form, F=(1/2)Fμνdxμdxν, and the variation v=vμdxμof the gauge ﬁeld. [By deﬁnition, /Omega1(1) 2nis linear in v. The integral is taken over the physicalD=2n-dimensional (Euclidean) space-time M2n.] Now, the descent relation tells us that /Omega1(1) 2ncan be derived from the so-called anomaly polynomial /Omega12n+2(F), which is a (2 n+2)- form built from the curvature 2-form F, with the aid of yet another (2 n+1)-form /Omega1(0) 2n+1,b y /Omega12n+2=d/Omega1(0) 2n+1,δ v/Omega1(0) 2n+1=d/Omega1(1) 2n. (42) That is, /Omega12n+2is closed, and gauge invariant, and hence can be written as a polynomial in F.H e r e /Omega1(0) 2n+1(A,F) is its corresponding Chern-Simons form. There is a simple closed-form expression for the anomaly polynomial /Omega12n+2that is given by /Omega1D(F)=ch(F)\|D. (43) Let us explain the notation: ch( F) is the following power series (“characteristic class”) constructed from the ﬁeld-strength 2-formF, and is given by ch(F)=r+i 2πtrF−1 2(2π)2trF2+···. (44) This expression is written for the general case of a gauge ﬁeld transforming in an r-dimensional irreducible representation of a (possibly non-Abelian) gauge group, where tr denotes thetrace in this representation. Observe that ch( F) consists of a sum of different p-forms with different pwhere p=even. The notation ···\| Din Eq. ( 43) means we extract a D-form from ch( F). While up to this point the differential forms /Omega1(0) 2n+1and /Omega12n+2appear to have been introduced solely to express theD=2n-dimensional gauge anomaly in terms of other objects, they themselves are known to be related to other types of anomalies: the Chern-Simons form /Omega1(0) 2n+1represents an anomaly in a discrete symmetry (parity or charge-conjugationsymmetry, depending on dimensionality) discussed in moredetail in Sec. VA4 below, and /Omega1 2n+2represents34the chiral anomaly in the presence of a background gauge ﬁeld, discussedin Sec. VA2 above. The integral of /Omega1 2n+2overD=(2n+2)- dimensional space-time, on the other hand, represents the θ term (see also Sec. VA5 below). 4. The Chern-Simons term The integral of /Omega1(0) 2n+1(A,F) over D=(2n+1)- dimensional space-time is the Chern-Simons-type action forthe gauge ﬁeld A, and represents, as already mentioned, an anomaly in a discrete symmetry: the parity or charge-conjugation anomaly. In turn, the presence of such a Chern-Simons term in the effective (bulk) action for the gauge ﬁeld AinD=(2n+1)- dimensional space-time signals the presence of a topologicalphase: when there is a boundary in the system, the integralof the Chern-Simons term is not invariant on its own; rather,upon making use of the descent relation Eq. ( 42), one obtains δ v/integraldisplay M2n+1/Omega1(0) 2n+1=/integraldisplay M2n+1d/Omega1(1) 2n=/integraldisplay ∂M 2n+1/Omega1(1) 2n. (45) This is something we are familiar with from the physics of the quantum Hall effect: the presence of the boundary term/integraltext ∂M 2n+1/Omega1(1) 2nappearing on the right-hand side of Eq. ( 45) signals 045104-9SHINSEI RYU, JOEL E. MOORE, AND ANDREAS W. W. LUDWIG PHYSICAL REVIEW B 85, 045104 (2012) the presence of an edge mode. In turn, as we have seen in Sec. VA1 , the gauge anomaly in D=(2n)-dimensional space-time, which is represented by the integral over /Omega1(1) 2n, itself signals the presence of a topological phase in D=2n+1 space-time dimensions, i.e., in one dimension higher. 5. The θterm The integral of the anomaly polynomial /Omega12n+2overD= (2n+2)-dimensional space-time is the θterm and represents a chiral anomaly in the presence of a background gauge ﬁeld(discussed in Sec. VA2 above). Again, to be more explicit, in the presence of such an axial anomaly, the axial currentJ μ 5(x) [which in the present case is an axial U(1) current] is not conserved: DμJμ 5(x)/negationslash=0, where Dμis the covariant derivative in the presence of the gauge ﬁeld. For a single copyof a massive Dirac fermion, it is given by D μJμ 5(x)=2im¯ψγ 2n+1ψ+2iA2n+2(x), (46) where the ﬁrst term represents the explicit breaking of the chiral symmetry by the mass term, whereas the second termrepresents the breaking of the chiral symmetry by quantumeffects. A 2n+2quantifying the breaking of the axial current conservation by an anomaly is essentially identical to /Omega12n+2, and given by removing all dxμthat appear in the differential form/Omega12n+2. Just as was the case for the Chern-Simons term, the presence of such a θterm in the effective action for the gauge ﬁeld signals the presence of a topological phase. In particular, thedescent relation tells us that/integraldisplay M2n+2/Omega12n+2=/integraldisplay M2n+2d/Omega1(0) 2n+1=/integraldisplay ∂M 2n+2/Omega1(0) 2n+1. (47) This is, again, something we are familiar with from the physics of the three-dimensional topological insulator in class AIII,which is described by the θterm (the axion term). In the presence of a boundary ∂M 2n+2, such a topological state supports boundary degrees of freedom, as signaled by the boundary term/integraltext ∂M 2n+1/Omega1(0) 2n+1, which is a Chern-Simons term.52 Let us summarize: to derive the existence of topological phases in symmetry class A and AIII, we start from theanomaly polynomial /Omega1 2n+2. Then the terms/integraltext M2n+2/Omega12n+2and/integraltext M2n+1/Omega1(0) 2n+1are the effective actions for the (topological) ﬁeld theory of the space-time linear responses for the gauge ﬁeld for the topological phases in class AIII ( D=2n+2) and A (D=2n+1), respectively. B. Topological insulators (superconductors) in the remaining eight “real” symmetry classes from gravitational and mixed anomalies 1. Gravitational anomaly and axial anomaly in the presence of background gravity For the remaining eight “real” of the ten symmetry classes, having a conserved U(1) quantity is less trivial. Classes AI,AII, and CII are naturally realized as a normal (as opposedto superconducting) electronic system, and thus for thesethere is a natural notion of a conserved U(1) quantity (theelectrical charge). One realization of the BDI symmetry class,which is only part 53of the entire symmetry class, can also beconsidered to have a conserved U(1) quantity, and we consider this realization in this subsection. On the other hand, classesD, DIII, C, and CI are naturally realized as BdG systems.While for classes C and CI, SU(2) spin is conserved [so aconserved U(1) charge exists], for classes D and DIII, there isno conserved U(1) quantity at all. Since for the latter four of eight real symmetry classes (D, DIII, C, CI) we cannot rely on a conserved U(1) quantity todescribe these topological phases, it is not possible to couplethese systems minimally to a U(1) gauge ﬁeld. However,it is natural to consider a coupling of these topologicalphases to gravity. Let us focus ﬁrst on topological insulators(superconductors) with an integer topological charge, Z,b u t not on those with a binary topological charge, Z 2. For now we also do not consider topological insulators or superconductorsw i t ha2 Zcharge. An analog of the U(1) gauge anomaly, which we have described in Sec. VA1 at the boundary (of space-time di- mension D=2n) of topological phases in symmetry class A, is the gravitational anomaly. It corresponds to the breakdownof energy-momentum conservation, and when it happens, itmust be realized in a system that represents the boundary ofa topological phase in one dimension higher [in analogy tothe case of a U(1) gauge anomaly, Sec. VA1 ]. We refer to this anomaly also as a “purely gravitational anomaly.” In thefollowing, we will show that one can predict the appearance ofthe topological phases in symmetry classes D, C, DIII, CI [i.e.,those without conserved U(1) charge] from the presence of apurely gravitational anomaly that appears in the ﬁeld theoryfor the gravitational (or thermal 32) responses. Finally, we will need to discuss the still remaining sym- metry classes AI, BDI, AII, and CII. Topological insulators(superconductors) in these symmetry classes can be coupled toboth a U(1) gauge ﬁeld 54as well as a gravitational background. We will show that the ﬁeld theories for the space-time-dependent linear responses for these topological insulatorspossess a so-called mixed anomaly. Indeed, we will show thatthe appearance of a mixed gravitational and electromagneticaxial anomaly signals the existence of topological phases inthese symmetry classes. 2. Topological insulators (superconductors) in symmetry classes D, C, DIII, and CI from the purely gravitational anomaly As mentioned earlier in this paper, each topological insulator (in any dimension) has a Dirac Hamiltonianrepresentative. 22We can consider the coupling of this Dirac theory to a space-time-dependent gravitational background.Upon integrating out the massive fermions, we obtain aneffective gravitational action in Dspace-time dimensions. If there is a gravitational anomaly, the (Euclidean) effectiveaction ln Z[e,ω] in the presence of the gravitational background is not invariant under a general coordinatetransformation x μ→xμ+/epsilon1μ, where eis the vielbein and ω is the spin-connection 1-form. That is, δvlnZ[e,ω]=2πi/integraldisplay MD/Omega1(1) D(v,ω,R), (48) where δvrepresents an inﬁnitesimal SO( D) rotation, under which ω, the spin-connection 1-form ω, is transformed as ω→ 045104-10ELECTROMAGNETIC AND GRA VITATIONAL RESPONSES ... PHYSICAL REVIEW B 85, 045104 (2012) ω+v;/Omega1(1) D(v,ω,R)i sa D-form related to the gravitational anomaly. In complete analogy to the case of the gauge anomaly discussed above, /Omega1(1) D(v,ω,R) can be derived from a corre- sponding anomaly polynomial /Omega1D+2(R)[ s e eE q s .( 54) and (55) below] through its Chern-Simons form /Omega1(0) D+1(ω,R), by using a descent relation that takes a form identical to Eq. ( 42). Thus, once the existence of the (purely) gravitational anomalyis known for a given dimension D, it predicts the presence of topological phases in D+1 andD+2 dimensions, using the same logic as in the gauge ﬁeld case above. Now, according to Ref. 55, a purely gravitational anomaly can exist in D=4k+2(d=4k+1). (49) Thus, breakdown of energy-momentum conservation due to quantum effects can occur in these dimensions. As in the caseof symmetry class A, discussed above, we take this as evidencefor the existence of a topological bulk in one dimension higher,i.e., in space-time dimensions D=4k+3(d=4k+2). (50) This thus predicts the appearance of topological phases in class D ( d=2),class C ( d=6), (51) as well as all the other higher-dimensional topological phases that we can obtain from these by Bott periodicity. (These arecolored red in Table 2.) On the other hand, there is an analog of the “axial anomaly in the presence of a background gauge ﬁeld,” which wediscussed in Sec. VA2 in the context of symmetry class AIII in D=2nspace-time dimensions. This analog is the “axial anomaly in the presence of a background gravitationalﬁeld.” If only a background gravitational ﬁeld is present, thisanomaly exists in space-time dimensions D=4k(d=4k−1). (52)This covers symmetry classes class DIII ( d=3),class CI ( d=7), (53) as well as all higher-dimensional topological phases that we can obtain from these by Bott periodicity. (These are coloredblue in Table II.) The anomaly polynomial related to the gravitational anoma- lies is known explicitly. It can be written as /Omega1 D=4k=ˆA(R)\|D, (54) where ˆA(R) is the so-called Dirac genus given by36 ˆA(R)=1+1 (4π)21 12trR2 +1 (4π)2/bracketleftbigg1 288(trR2)2+1 360trR4/bracketrightbigg +···.(55) HereRis theD×Dmatrix of 2-forms, Rμν:=1 2Rαβμνdxαdxβ, (56) where Rαβμνis the usual Riemann curvature tensor, and the trace refers to the D×Dmatrix structure. This deﬁnes, by the descent relation [which takes a form identical to Eq. ( 42)], the differential forms /Omega1(0) 4k−1and/Omega1(1) 4k−2. As before, the notation ˆA(R)\|Dextracts a D-form from ˆA(R). It is obvious from ( 55) that the anomaly polynomial exists only for D=4kbecause Eq. ( 55) is a function of R2. [Note that the descent relation Eq. ( 42) then implies the existence of a purely gravitational anomaly /Omega1(1) 4k+2(R)i nD=4k+2 space-time dimensions, in agreement with Ref. 55.] 3. Topological insulators (superconductors) in symmetry classes AI, BDI, AII, and CII from the mixed anomaly Before proceeding, let us brieﬂy summarize the previous subsection: by considering various anomalies related to grav-ity, we can predict the integer topological phases in the BdGsymmetry classes D, DIII, C, and CI. (As mentioned above, TABLE II. Topological insulators (superconductors) with an integer ( Z) classiﬁcation, (a) in the complex symmetry classes, predicted from the chiral U(1) anomaly, and (b) in the real symmetry classes, predicted from the gravitational anomaly (red), the chiral anomaly in thepresence of background gravity (magenta), the mixed anomaly under gauge and coordinate transformations (blue) and the chiral anomaly in the presence of both background gravity and U(1) gauge ﬁeld (green). Cartan \d 0 1 2 3 4 5 6 7 8 9 10 11 ··· A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 ··· AIII Z 0 0 Z 0 Z 0 Z 0 Z 0 Z ··· AI Z 2Z 0 0 00 Z2 Z2 Z 000 ··· BDI Z2 Z 000 2 Z 0 Z2 Z2 Z 00 ··· D Z2 Z2 Z 000 2 Z 0 Z2 Z2 Z 0 ··· DIII Z2 Z2 Z 00 0 2 Z 0 Z2 Z2 Z ··· AII 0 2Z0 0 0 00 00Z2 Z2 Z 00 0 2 Z 0 Z2 Z2 ··· CII 2 Z 0 Z2 Z2 Z 00 0 2 Z 0 Z2 ··· C2 Z 0 Z2 Z2 Z 00 0 2 Z 0 ··· CI 2 Z 0 Z2 Z2 Z 000 2 Z ··· 045104-11SHINSEI RYU, JOEL E. MOORE, AND ANDREAS W. W. LUDWIG PHYSICAL REVIEW B 85, 045104 (2012) for the moment we do not consider topological phases with Z2or 2Ztopological charges.) On the other hand, we have so far not covered the description of topological insulators insymmetry classes AI, BDI, AII, and CII in terms of anomalies. So far, we have considered for the “real” symmetry classes only those anomalies that involve solely gravity. Since the(gapped) topological insulators in symmetry classes AI, BDI,AII, and CII, also possess a conserved U(1) charge, 54we can couple those to both a U(1) gauge ﬁeld as well as a gravitationalbackground. Therefore, it is natural to consider an anomalythat occurs in the presence of both a background gauge and abackground gravitational ﬁeld. As it turns out, even in the presence of both gauge and gravitational ﬁelds, the structure of the anomaly issimilar to the one discussed so far: the noninvariance of theeffective action under a gauge transformation or coordinatetransformation can be expressed as δ vlnZ[A,e,ω ]=2πi/integraldisplay MD/Omega1(1) D(v,A,ω,F,R), (57) where /Omega1(1) D(v,A,ω,F,R) can be derived from an associated anomaly polynomial, which reads34,36 /Omega1D(R,F)=/bracketleftbig ch(F)ˆA(R)/bracketrightbig \|D. (58) As the right-hand side is given simply by the product of the anomaly polynomials for a gauge ﬁeld [Eq. ( 44)] and gravity [Eq. ( 55)], by switching off either RorF, we recover the results discussed in the previous subsections: for all evenspace-time dimensions D=d+1=2k(k=1,2,...)w e obtain a nonvanishing anomaly polynomial /Omega1 D(R=0,F)= /Omega1D(F), which we have already used to predict topological insulators or superconductors in class A ( D=2k+1) and AIII (D=2k). For space-time dimensions D=d+1=4k (k=1,2,...) we obtain a nonvanishing anomaly polynomial /Omega1D(R,F=0)=/Omega1D(R), which we have already used to predict topological insulators or superconductors in class DIII(D=4+8k) and CI ( D=8+8k). On the other hand, while the anomaly polynomial /Omega1 D(R,F=0)=/Omega1D(R) vanishes in D=4k+2 dimensions, the one obtained from Eq. ( 58), namely /Omega1D(R,F), is nonvanishing in these dimensions. As before, the anomaly polynomial itself is related to a “chiral anomaly in the presence of both gauge ﬁeld andgravity” of the massive bulk system in D=4k+2 space-time dimensions, D μJμ 5(x)=2im¯ψγD−1ψ+2iAD(x), where AD(x) is given in terms of /Omega1D(R,F). For this reason, one predicts an additional topological insulator (superconductor)in these space-time dimensions (besides the one of Sec. VA2 ). Therefore, one predicts the occurrence of topological phasesin spatial dimensions d=9(d=1) and d=5, class BDI [ d=9(d=1)],class CII ( d=5),(59) as well as of all higher-dimensional topological phases that we can obtain from these by Bott periodicity. 56(These are colored green in Table II.) Indeed, for classes BDI and CII, we can realize these symmetry classes as a normal (i.e.,not superconducting) system, and hence they have a naturalU(1) charge. 54The effective topological ﬁeld theory for the space-time-dependent linear [electrical and gravitational(thermal)] responses possesses a term of topological origin of the form/integraltext /Omega1D(R,F), where D=4k+2. Moreover, it turns out that a descent relation that is identical in form to Eq. ( 42) also holds for the “mixed” anomaly polynomial deﬁned in Eq. ( 58). Therefore, the space-time integral of the Chern-Simons form /Omega1(0) 4k+1of/Omega14k+2, which is obtained from /Omega14k+2by using the descent relation, d/Omega1(0) 4k+1= /Omega14k+2, describes the term of topological origin in the effective action for the linear responses in D=4k+1 space-time dimensions. This corresponds to a “mixed anomaly” /Omega1(1) 4kin the corresponding boundary theory in 4 kspace-time dimensions. For this reason, one predicts the occurrence of additionaltopological insulators in spatial dimensionalities d=0 and 4 (besides the ones in Sec. VA1 ), for the two symmetry classes class AI ( d=0),class AII ( d=4), (60) as well as for all their higher-dimensional equivalents obtained from the Bott periodicity (These are colored magenta inTable II.) 4. Atiyah-Singer index theorem For all the symmetry classes with chiral symmetry ,t h e Hamiltonian can be brought into block off-diagonal form.8 Above, we have discussed all symmetry classes of this formthat possess topological insulators with a Zclassiﬁcation (i.e., AIII in D=2n,D I I Ii n D=4+8k,C Ii n D=8+8k, CII in D=6+8k,B D Ii n D=10+8k). A Dirac Hamil- tonian Hwith chiral symmetry possesses an index, and the Atiyah-Singer index theorem 34relates the integral of the anomaly polynomial discussed above to this index throughthe formula index(H)=/integraldisplay MD/Omega1D(R,F), (61) where /Omega1D(R,F) is the most general anomaly polynomial, as deﬁned in Eq. ( 58) above. Here, the Dirac Hamiltonian H refers to the Hamiltonian in a gravitational background and abackground (Abelian or non-Abelian) gauge ﬁeld. The indexindex(H) is by deﬁnition an integer. We note that it is because of this theorem that the space-time integral of the anomalypolynomial represents a θterm for the theory of the space- time-dependent linear gauge and gravitational responses, andthat the θterms only occur for symmetry classes possessing a chiral symmetry. 5. Global gravitational anomalies The discussion that we have presented so far for the connection between anomalies and topological insulators andsuperconductors in “the primary series” (those located inthe diagonal of the Periodic Table and characterized by aninteger topological invariant) can be extended to some of the“ﬁrst and second descendants” (the topological insulators andsuperconductors in the same symmetry class, but in one andtwo dimensions less than the one with a Zinvariant; these are each characterized by a Z 2invariant). We propose that for these we need to use so-called global anomalies, instead ofthe so-called perturbative anomalies that we have made use ofin this section. Such anomalies do not affect inﬁnitesimal, butrather large (of order 1) symmetry transformations. 045104-12ELECTROMAGNETIC AND GRA VITATIONAL RESPONSES ... PHYSICAL REVIEW B 85, 045104 (2012) It was found in Ref. 55that global gravitational anomalies can exist, given certain assumptions are satisﬁed, (i) in D= 8k, (ii) in D=8k+1, and (iii) in D=4k+2 space-time dimensions. If so, then following the same reasoning as above,the presence of these anomalies would indicate the existence ofa topological insulator in one dimension higher (of which theanomalous system is the boundary). This would then indicatethe existence of topological insulators (superconductors) inspace-time dimensions (i) D=8k+1, (ii) D=8k+2, and (iii)D=4k+3 [corresponding to spatial dimensions (i) d= 8k, (ii)d=8k+1, and (iii) d=4k+2]. Indeed, there exist Z 2topological insulators in these dimensions (Table II). More precisely, there exist twoZ2topological insulators in these dimensions, and at this point we have not yet explored indetail which of the two (or if both) could be related to thisglobal gravitational anomaly. Moreover, we note that therealso exist other (i.e., not gravitational) global anomalies, andwe propose that the other, as yet not yet covered, Z 2topological insulators can be obtained from considering these other globalanomalies. We end by mentioning that the notions presented in this section (Sec. V) may also be further supported by the connection with the tenfold classiﬁcation of D-branes: 46,47 In the D-brane realizations of topological insulators and superconductors, massive fermion spectra arise as open stringexcitations connecting two D-branes, which are in one-to-onecorrespondence with the Dirac representative of the tenfoldclassiﬁcation of topological insulators and superconductors,and come quite naturally with gauge interactions. The Wess-Zumino term of the D-branes gives rise to a gauge ﬁeld theoryof topological nature, such as those with the Chern-Simonsterm or the θterm in various dimensions. VI. CONCLUSIONS There are various important future research directions in the ﬁeld of topological insulators and superconductors. Letus mention two here. One is the search for experimentalrealizations of the topological singlet and triplet supercon-ductors in three spatial dimensions, besides the B phase ofthe 3He superﬂuid. Given how fast experimental realizations of the quantum spin Hall effect in two spatial dimensionsand the Z 2topological insulators in three dimensions have been found, one may perhaps anticipate a similar develop-ment for these three-dimensional topological superconductingphases. Notably, Cu xBi2Se3, which arises from the familiar three-dimensional topological insulators Bi 2Se3, was found to be superconducting at 3.8 K.57Subsequent theoretical work proposed that this superconducting phase should be atopological superconductor. 58The various linear responses discussed in this paper, as summarized in Table I, may become helpful in the search for, and identiﬁcation of, such varioustopological phases. Another important issue is to complete the study of the effect of interactions for the symmetry classes so far notyet included in the discussion given in Sec. V. (These include, in general dimensionalities, the topological insulators(superconductors) with a 2 Zclassiﬁcation, as well as the majority of those with a Z 2classiﬁcation.) Moreover, this includes the case of symmetry class BDI in d=1 spatial dimension (recall also Refs. 53and56), discussed in the work of Refs. 59and 61. Further important outstanding questions concern possible topological phases (besides superconductors)which may arise from interactions rather than from bandeffects. How can one describe “fractional” versions of thetopological insulators (superconductors), 23and how can one classify bosonic systems such as, e.g., spin systems?62Clearly, to address any of these interaction-dominated issues, onecannot rely on a topological invariant deﬁned in terms ofsingle-particle Bloch wave functions. Rather, a deﬁnition oftopological quantum states of matter in terms of responses tophysical probes is necessary. In this paper, we have developeda description of this type for all topological insulators in threespatial dimensions, and for a signiﬁcant part of the topologicalinsulators in general dimensions. From a conceptual point ofview, the gravitational responses are the most fundamentalones in that they apply to all topological insulators. Owingto Luttinger’s derivation 32of the thermal Kubo formula, these correspond physically to thermal response functions. ACKNOWLEDGMENTS We thank Taylor Hughes, Charles Kane, Alexei Kitaev, Shunji Matsuura, Xiao-Liang Qi, Tadashi Takayanagi, AshvinVishwanath, and Shou-Cheng Zhang for useful discussions.S.R. thanks the Center for Condensed Matter Theory at theUniversity of California, Berkeley for its support. J.E.M.acknowledges support from NSF Grant No. DMR-0804413.This work was supported, in part, by the NSF under Grant No.DMR-0706140 (A.W.W.L.). 1C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). 2L. Fu, C. L. Kane, and E. J. Mele, P h y s .R e v .L e t t . 98, 106803 (2007). 3L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007). 4A. M. Essin and J. E. Moore, Phys. Rev. B 76, 165307 (2007). 5J. E. Moore and L. Balents, P h y s .R e v .B 75, 121306 (2007). 6D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, P h y s .R e v .L e t t . 49, 405 (1982). 7M. 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Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y . S. Hor, R. J. Cava, and M. Z. Hasan, Nature Phys. 5, 398 (2009). 16D. Hsieh, Y . Xia, D. Qian, L. Wray, J. H. Dil, F. Meier,J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V .Fedorov, H. Lin, A. Bansil, D. Grauer, Y . S. Hor, R. J. Cava, andM. Z. Hasan, Nature (London) 460, 1101 (2009). 17Y . L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang,I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science 325, 178 (2009). 18X.-L. Qi, T. L. Hughes, and S.-C. Zhang, P h y s .R e v .B 78, 195424 (2008). 19A. M. Essin, J. E. Moore, and D. Vanderbilt, P h y s .R e v .L e t t . 102, 146805 (2009). 20F. Wilczek, P h y s .R e v .L e t t . 58, 1799 (1987). 21As will be explained below, we use Luttinger’s derivation32of the thermal Kubo formula to relate the (more abstract) gravitationalresponses to physical thermal transport coefﬁcients. 22S. Ryu, A. Schnyder, A. Furusaki, and A. W. W. Ludwig,New J. Phys. 12, 065010 (2010). 23See, for example, J. Maciejko, X.-L. Qi, A. Karch, and S.-C. Zhang, Phys. Rev. Lett. 105, 246809 (2010); B. Swingle, M. Barkeshli, J. McGreevy, and T. Senthil, P h y s .R e v .B 83, 195139 (2011). 24This effect has been discussed in the earlier literature in Ref. 25 [see Eq. (30) of this reference]. 25A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein,P h y s .R e v .B 50, 7526 (1994). 26P. Hosur, S. Ryu, and A. Vishwanath, Phys. Rev. B 81, 045120 (2010). 27T. Senthil, J. B. Marston, and M. P. A. Fisher, P h y s .R e v .B 60, 4245 (1999); I. A. Gruzberg, A. W. W. Ludwig, and N. Read, Phys. Rev. Lett. 82, 4524 (1999). 28Not to be confused with the quantum Spin Hall effect (QSHE). 29A. P. Schnyder, S. Ryu, and A. W. W. Ludwig, Phys. Rev. Lett. 102, 196804 (2009). 30The theory of the space-time-dependent responses at the surface isthe Chern-Simons theory (see Sec. Vfor more details). The charge, spin, and thermal surface conductivities (in natural units) are thecoupling constants of the Chern-Simons terms. 31H. Thirring and J. Lense, Phys. Z 19, 156 (1918) [Gen. Relativ. Gravitation 16, 727 (1984)]. 32J. M. Luttinger, Phys. Rev. 135, A1505 (1964). 33S. J. Clark and R. W. Tucker, Class. Quantum Grav. 17, 4125 (2000). 34See, e.g., M. Nakahara, Geometry, Topology and Physics (Institute of Physics, Bristol, 1998). 35We hereby collect our conventions for metric, vielbein, spinconnection, etc. We start from the metric and its inverse, g μν,g ρσ,with gμνgνρ=δμ ρ. (62) The components of the Levi-Civita connection are /Gamma1μ νρ=1 2gμα(∂νgρα+∂ρgνα−∂αgνρ). (63)The vielbein eaμandea μdiagonalizes the metric, and is deﬁned by gμνeaμebν=ηab,ηabea μeb ν=gμν. (64) Here, ηabis a ﬂat (Minkowski) metric, and we use Greek indices μ, ν,... for coordinates of the manifold, and Roman indices a, b,... for the ﬂat coordinates at some point x0of the manifold; they are raised and lowered by gμν,gμνandηab,ηab, respectively. Since the vielbein eaμtransforms as a covariant vector under general coordinate transformation, it is convenient to introduce a one-form ea=ea μdxμ. (65) The spin connection is ωμa b=ea α/bracketleftBig ∂μebα+/Gamma1α μβebβ/bracketrightBig . (66) This can be written in terms of a covariant vector ebμ, which is the bth eigenvector of the metric, by using the covariant derivative with respect to the Levi-Civita connection /Gamma1μνρas ωμa b=ea α∇μebα. (67) We deﬁne the connection one-form by ωa b=ωμa bdxμ. (68) The curvature tensor is Rμ ναβ=∂α/Gamma1μ νβ−∂β/Gamma1μ να+/Gamma1μ σα/Gamma1σ νβ−/Gamma1μ σβ/Gamma1σ να, (69) Rμα=gνβRμναβ. The curvature tensor can also be constructed from the spin connection: Ra b=dωa b+ωa c∧ωc b=Rμνc bdxμdxν, (70) where Rμνcb=Rμνρλecρebλ. 36K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies (Oxford University Press, Oxford, 2004). 37G. E. V olovik, JETP Lett. 51, 125 (1990). 38N. Read and Dmitry Green, Phys. Rev. B 61, 10267 (2000). 39See, e.g., M. H. Freedman and R. Kirby, Proceedings of the Sym- posium on Pure Mathematics (Stanford University Press, Stanford, CA, 1976), Pt. 2, pp. 85–97; Proceedings of the Symposium on Pure Mathematics, XXXII (American Mathematical Society, Providence, RI, 1978). 40The quantity cdenotes the conformal central charge of the confor- mal ﬁeld theory describing the (topologically protected) chiral edgemodes that would appear at a spatial (1 +1)-dimensional boundary. 41J. Goryo, M. Kohmoto, and Y .-S. Wu, P h y s .R e v .B 77, 144504 (2008). 42F. Meier and D. Loss, Phys. Rev. Lett. 90, 167204 (2003). 43We will give a brief explanation of the relevant concepts below. See also Refs. 33,35,43, and 54. 44L. Alvarez-Gaum ´e and P. Ginsparg, Ann. Phys. (NY) 161, 423 (1985). 45Speciﬁc details of how the surface of a topological insulator(superconductor) is gapped may inﬂuence the speciﬁc responsesresulting at the surface, which thus may not depend solely on thesymmetry class of the bulk. A discussion of such effects has beengiven recently for the thermal case (our Eq. ( 3)) in Z. Wang, X.-L. Qi, and S.-C. Zhang, e-print arXiv:1011.0586 . 46S. Ryu and T. Takayanagi, Phys. Lett. B 693, 175 (2010). 045104-14ELECTROMAGNETIC AND GRA VITATIONAL RESPONSES ... PHYSICAL REVIEW B 85, 045104 (2012) 47S. Ryu and T. Takayanagi, Phys. Rev. D 82, 086014 (2010). 48It is known49that symmetry class AIII can also be realized as a quasiparticle system within a (spinful) superconducting groundstate, which conserves one component (say the S zcomponent) of Pauli spin. In this case, the U(1) charge associated with theconservation of S zcan be used in lieu of the conserved particle number of a normal (not superconducting) system in class AIII. 49M. S. Foster and A. W. W. Ludwig, P h y s .R e v .B 77, 165108 (2008). 50As explained in the paragraph below Eq. ( 46), the quantity A2n+2(x) appearing in this equation is given by Eqs. ( 42)a n d( 43)b e l o w , and vanishes in the absence of the electromagnetic ﬁeld strengthF μν. Therefore, this anomaly is called “chiral (axial) anomaly in a background U(1) gauge ﬁeld.” 51The argument is essentially the same as that presented in Sec. III B . As explained below, the calculations performed in this subsectionamount to a derivation of what we will call below a “chiral U(1) gauge anomaly in the presence of a background gravitational ﬁeld.” 52The appearance of this term for the non-Abelian gauge groupSU(2) was ﬁrst pointed out in the context of topological insulators(superconductors) in Ref. 29for the spin-singlet topological superconductor in symmetry class CI in d=3 spatial dimensions. 53For class BDI there exist two distinct physical realizations, one as (“spinless” time-reversal invariant) superconductors and one as nor-mal (nonsuperconducting) electronic systems. Without consideringinteractions, there is basically no difference between the two, exceptthat the number of species of Majorana fermions is even in the lattercase, where a pair of Majorana fermions is thought to be combinedinto a complex fermion, carrying a U(1) charge, or particle number.The discussion of anomalies, considered in the current sectionof this paper, is aimed at the discussion of interacting theories(as explained, e.g., in the Introduction). Now, when inclusion ofinteractions is considered, the two above-mentioned realizationsof symmetry class BDI behave very differently. Obviously, in thelatter (normal, nonsuperconducting) realization, the interactions areto respect the U(1) symmetry, whereas in the former (supercon-ducting) realization, there is no such constraint on the form of the interactions. In this subsection, we will consider solely thelatter realization. In this case, there is thus always a conserved U(1)quantity. The former (superconducting) case was discussed recentlyin Refs. 59–61. (For similar methods applied to a gapped spin chain, see, e.g., Ref. 62.) At present, we do not have an understanding of that case in terms of anomalies. We hope to be able to address thiscase in future work. 54For symmetry class BDI, recall the comment in Ref. 53. 55L. Alvarez-Gaum ´e and E. Witten, Nucl. Phys. B 234, 269 (1983). 56For class BDI in d=1 spatial dimensions, the mixed anomaly polynomial in the corresponding space-time dimensionality D=2 is simply equal to the anomaly polynomial for the U(1) gaugeanomaly, discussed above (describing the ﬁeld strength in D=2). The lowest spatial dimension d=1 behaves thus differently from all other dimensions d=8k+1(k/greaterorequalslant1) related by Bott periodicity, in which an independent mixed anomaly polynomial exists. For this reason, we have denoted d=1 in parentheses. 57Y . S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong, and R. J. Cava,Phys. Rev. Lett. 104, 057001 (2010). 58L. Fu and E. Berg, P h y s .R e v .L e t t . 105, 097001 (2010). 59L. Fidkowski and A. Kitaev, Phys. Rev. B 81, 134509 (2010). 60L. Fidkowski and A. Kitaev, Phys. Rev. B 83, 075103 (2011). 61A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B 83, 075102 (2011). 62Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009); X. Chen, Z.-C. Gu, and X.-G. Wen, ibid. 83, 035107 (2011); F. Pollmann, E .B e r g ,A .M .T u r n e r ,a n dM .O s h i k a w a ,e - p r i n t arXiv:0909.4059 . 63For topological insulators (superconductors) in symmetry classes with chiral symmetry in even space-time dimensions with an integerclassiﬁcation, this integer corresponds to the winding number ofthe partition function as a function of the angle of an “axial” U(1)rotation (as in Sec. III B ). Integers of the classiﬁcation in odd space- time dimensions are represented by the coefﬁcients of generalizedChern-Simons terms (see Sec. Vfor more details). 045104-15
PhysRevB.81.014401.pdf	Phase separation in the CoO 2layer observed in thermoelectric layered cobalt dioxides Tsuyoshi Takami,1,Hiroshi Nanba,1Yasuhide Umeshima,1Masayuki Itoh,1Hiroshi Nozaki,2Hiroshi Itahara,2and Jun Sugiyama2 1Department of Physics, Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan 2Toyota Central Research and Development Laboratories, Inc., Nagakute, Aichi 480-1192, Japan /H20849Received 5 June 2009; revised manuscript received 12 October 2009; published 4 January 2010 /H20850 59Co nuclear magnetic resonance /H20849NMR /H20850measurements have been performed to study the local magnetic properties of the misﬁt layered cobalt dioxides /H20849MLCO’s /H20850with the CoO 2and rock-salt layers, /H20851Ca2CoO 3/H208520.62CoO 2/H20849/H11013Ca3Co3.92O9.34/H20850and Ca 3Co3.92O9.34−/H9254with oxygen nonstoichiometry. The59Co NMR spectrum consists of mainly ﬁve lines at 4.2 K at which the samples are in a magnetically ordered state. Amongthe ﬁve NMR lines for Ca 3Co3.92O9.34, three lines at higher frequencies /H20849f’s/H20850satisfy the resonance condition with two branches indicating the presence of antiferromagnetic internal ﬁelds /H20849Hint’s/H20850. The other two lines exhibit one branch, and one of the two has a nonzero Hintunder zero external ﬁeld /H20849ZF/H20850, which signiﬁes the existence of ferromagnetic /H20849FM/H20850Hint’s. The other has a zero Hintunder ZF. By taking account of both the valence state of the Co ions in each layer and the lattice modulation due to the misﬁt between the CoO 2layer and the rock-salt layer, the NMR spectra at higher f’s are attributed to the Co in the rock-salt layer, whereas those at lower f’s to the Co in the CoO 2layer. Furthermore, a spin-density wave order appears to coexist with a FM order in the CoO 2layer for MLCO’s. The magnetic and transport properties of these materials are discussed in terms of a separation between two phases. DOI: 10.1103/PhysRevB.81.014401 PACS number /H20849s/H20850: 72.15.Jf, 75.50.Gg, 75.30.Fv I. INTRODUCTION Since the discovery1of the coexistence of a large ther- mopower and a low electrical resistivity in NaCo 2O4, which has a close-packed two-dimensional /H208492D/H20850CoO 2array, exten- sive investigations of other cobalt oxides have been under-taken in a search for practical materials for thermoelectricconversion. Furthermore, the sodium content xin Na xCoO 2 can be varied over a wide range, and this system has been reported to show various magnetic and electrical propertieswith changing xand/or T, such as superconductivity for wa- ter intercalated Na 0.35CoO 2,2a charge ordered state for Na0.5CoO 2,3and a spin-density wave /H20849SDW /H20850ordered state for Na xCoO 2with x/H113500.75.4These experimental ﬁndings have also drawn much interest in the inter-relationship be-tween dimensionality and physical/transport propertiesamong the cobalt oxides. For instance, perovskite-type R 1−xSrxCoO 3/H20849R=La, Pr, Nd, and Sm /H20850with x=0.05–0.1 could be potential thermo- electric materials at around room temperature.5–7Their struc- ture consists of corner-sharing CoO 6octahedra forming a three-dimensional /H208493D/H20850network. However, none of the Co- oxide perovskites can be used at high Tbecause their ther- mopower decreases rapidly above /H11015500 K due to a spin- state transition and/or a metal-insulator transition. In contrast to the 3D system, the quasi-one-dimensional /H20849Q1D /H20850cobalt oxides, An+2Con+1O3n+3/H20849A: alkaline-earth metal, n=1–5 and /H11009/H20850, in which each 1D CoO 3chain is sur- rounded by six equally spaced chains forming a triangularlattice in the abplane, exhibit no spin-state transition at least between 2–600 K. 8–10The unusual magnetic properties, such as a partially disordered antiferromagnetic state, were foundin Ca 3Co2O6/H20849A=Ca, n=1/H20850,11,12which is a 2D antiferro- magnet with ferromagnetic /H20849FM/H20850Ising-spin chains, and the magnetic phase diagram with various nhas been proposedfrom positive muon spin rotation and relaxation /H20849/H9262+SR/H20850and magnetization measurements.8–10,13Partially due to the lack of a spin-state transition and their chemical stability, at leastup to 1300 K, the Q1D cobalt oxides with n=1 and 2 have been suggested to be potential candidates for thermoelectricmaterials at /H110151300 K. 14,15 For the Q1D and 3D systems, the dimensionless ﬁgure of merit ZT=S2T//H9267/H9260, which is related to the efﬁciency and per- formance of thermoelectric power generation or cooling, isstill not high enough for practical application; further inves-tigations to improve their thermoelectric properties areneeded as far as we know. Here, S, /H9267,/H9260, and Tare ther- mopower, electrical resistivity, thermal conductivity, and ab-solute temperature, respectively. On the other hand, misﬁtlayered cobalt dioxides /H20849MLCO’s /H20850,/H20851Ca 2CoO 3/H208520.62CoO 2and /H20851Ca2Co1.3Cu0.7O4/H208520.62CoO 2, have attracted considerable at- tention because of their large S, low /H9267, and low /H9260,a si nt h e case of Na xCoO 2. In particular, MLCO’s exhibit excellent thermoelectric performance at high Tcompared to NaxCoO 2,16,17since MLCO’s are more stable at high Tthan NaxCoO 2. Structurally, MLCO’s share common components, CoO 2and rock-salt layers. The CoO 2layer consists of a 2D triangular lattice of edge-sharing CoO 6octahedra in the ab plane. In the rock-salt layer, on the other hand, cations andthe O 2−ions make a rock-salt lattice. Triple and quadruple subsystems form the rock-salt layer in /H20851Ca2CoO 3/H208520.62CoO 2 and /H20851Ca2Co1.3Cu0.7O4/H208520.62CoO 2, respectively. The overall crystal structure of these materials consists of alternating lay-ers of the CoO 2and rock-salt layers stacked along the caxis. In addition, there is a misﬁt between the two layers along thebaxis, i.e., the spatial period along the baxis of the CoO 2 layer is incommensurate with that of the rock-salt layer. Motivated by the geometrical frustration in the CoO 2 layer for /H20851Ca2CoO 3/H208520.62CoO 2and /H20851Ca2Co1.3Cu0.7O4/H208520.62CoO 2 with 2D triangular lattices, the magnetic nature of these com- pounds has also been actively studied. /H9262+SR and magnetiza-PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 1098-0121/2010/81 /H208491/H20850/014401 /H2084912/H20850 ©2010 The American Physical Society 014401-1tion experiments on /H20851Ca2CoO 3/H208520.62CoO 2indicated the exis- tence of a short-range order of an incommensurate /H20849IC/H20850SDW state below /H11015100 K; a long-range IC-SDW order was completed below /H1101530 K.18,19/H9267increases drastically upon cooling particularly below 100 K.16With a further de- crease in T, the ferrimagnetic /H20849FR/H20850transition was reported to take place at /H1101519 K.18,19Also, /H20851Ca2Co1.3Cu0.7O4/H208520.62CoO 2 exhibits similar magnetic transitions, i.e., a SDW state and a magnetically ordered state, but the onset of the transitionT’s to the ordered states are higher than those for /H20851Ca 2CoO 3/H208520.62CoO 2. That is, a transition to a short-range or- der of the IC-SDW state at /H11015180 K with decreasing Twas found; then the long-range order and a 3D antiferromagnetic/H20849AF/H20850/H20849or FR /H20850order appeared below /H11015140 and /H1101585 K, respectively. 20Quite recently, we have performed59Co nuclear magnetic resonance /H20849NMR /H20850measurements on the lat- ter compound, and the observed59Co NMR spectra with varying Twere in agreement with the phase diagram.21 In the lattice of MLCO’s, there are at least two Co sites, namely, one is in the CoO 2layer and the other is in the rock-salt layer. This is partially, to our knowledge, the pre-dominant reason for the complex magnetic properties of theMLCO’s. In particular, the charge-carrier transport of theMLCO’s is restricted mainly to the CoO 2layer, which means that the transport properties are mostly governed by electronsin this layer. Interestingly, the degeneracy of spins and orbit-als of the 3 delectrons of the Co ions has been theoretically pointed out to be important for enhancing S. 22However, the local magnetic properties in each layer of MLCO’s have notbeen fully established in contrast to Na xCoO 2. This situation is partially due to the complex crystal structure and the dif-ﬁculty in controlling widely the carrier density in the CoO 2 layer by changing the amount of cations. Recently, it hasbeen reported that the transport properties of an MLCO alsodepend on their oxygen deﬁciency /H20849 /H9254/H20850,23as well as on xfor NaxCoO 2. Therefore, a systematic study with changing /H9254and the number of the rock-salt layers could be one way to ad-dress this issue. In this paper, in order to clarify the local magnetism in each layer and understand the mechanism of theexcellent thermoelectric properties of the MLCO’s, we have performed 59Co NMR measurements on /H20851Ca2CoO 3/H208520.62CoO 2 /H20849/H11013Ca3Co3.92O9.34/H20850and Ca 3Co3.92O9.34−/H9254with/H9254=0.34, to- gether with a c-axis-aligned sample of Ca 3Co3.92O9.34−/H9254 with/H9254=0.24. We report the NMR results on the three samples with different oxygen contents in detail and com-pare the results with those on the other MLCO,/H20851Ca 2Co1.3Cu0.7O4/H208520.62CoO 2. II. EXPERIMENT The polycrystalline samples of Ca 3Co3.92O9.34 and Ca3Co3.92O9.34−/H9254used in this study were prepared by solid- state reaction. A mixture of the starting materials, CaCO 3and Co3O4powders, was pressed into pellets and calcined at 900 °C for 20 h in an O 2ﬂow. After regrinding, the powders were pelletized and calcined again under the same condi-tions. This process was repeated several times in order toobtain well-crystallized single-phase samples. The deoxy-genation was carried out in pure N 2gas with high purity /H2084999.9998% /H20850according to Ref. 23. The c-axis-aligned Ca3Co3.92O9.34−/H9254sample was synthesized by a reactive tem- plated grain growth technique at Toyota Central Researchand Development Laboratories. Inc. 24Diffraction peaks only from /H2084900l/H20850planes were observed for this sample. The Lot- gering factor was estimated to be over 0.95 using the x-raydiffraction intensity, indicative of a strong c-axis orientation. Further detailed preparation and characterization of thissample have been already published elsewhere. 25The oxy- gen contents in the deoxygenated sample and thec-axis-aligned sample were chemically determined by iodo- metric titration and were found to be 9 and 9.1, respectively. X-ray diffraction measurements were carried out with CuK /H9251radiation to conﬁrm the phase purity. All the x-ray diffraction peaks of the MLCO’s studied in this work wereindexed by a monoclinic unit cell consistent with theliterature, 17,26indicating that these samples are single phase. Thec-axis length increased with /H9254, while the change in the a-axis length was quite small. The two b-axis lengths exhib- ited an opposite trend with increasing /H9254, i.e., the b1-axis length for the CoO 2layer increased, whereas the b2-axis length for the rock-salt layer decreased. These results areconsistent with the previous study. 23We have further tested the phase purity by NMR measurements, which are moresensitive compared to x-ray diffraction measurements. Impu-rity phases, such as Co 3O4and Ca 3Co2O6, were not observed in the NMR spectrum. These results suggest that we success-fully obtained MLCO’s of high purity. NMR measurementswere performed using a coherent pulsed spectrometer and asuperconducting magnet with a constant ﬁeld of H =6.1065 T. 59Co NMR spectra in the ﬁeld were obtained after Fourier transformation of spin-echo signals collected atsome frequencies /H20849f’s/H20850.F-swept NMR spectra under zero external ﬁeld /H20849ZF/H20850were also taken point by point of f. III. RESULTS A. Randomly oriented polycrystalline Ca 3Co3.92O9.34 In a magnetically ordered state, in general, the nuclei are subjected to an internal ﬁeld /H20849Hint/H20850due to the spontaneous magnetic moments. Consequently, an NMR spectrum can be detected even under ZF. The f-swept59Co NMR spectrum at 4.2 K under ZF was measured in the wide frange up to 300 MHz, which is shown in Fig. 1/H20849a/H20850. Following the general trend, we observed a59Co NMR spectrum with several com- ponents in the FR state under ZF. This result clearly demon-strates the existence of nonequivalent Co sites with differentH int’s. Note here that the NMR lines corresponding to small Hint’s are distributed near 0 MHz in the ZF-NMR spectrum. Therefore, although it is difﬁcult to detect these NMR linesin this measurement condition, the presence of the two com-ponents, S1 and S2, is conﬁrmed by other measurement con-ditions in later, i.e., the Tdependence of the NMR spectrum under an external ﬁeld Hand the Hdependence of the reso- nance fas discussed below. By taking account of both the valence state of Co in each layer and the lattice modulationdue to the misﬁt, 16,26we have concluded that the NMR spec- tra for S3–S5 are attributed to the Co in the rock-salt layer,TAKAMI et al. PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-2whereas those for S1 and S2 belong to the Co in the CoO 2 layer.21The spin quantum number for the Co in the rock-salt layer has been claimed to be about six times larger than thatfor the CoO 2layer from neutron powder diffraction and magnetic-susceptibility measurements,27which is consistent with relatively large Hint’s for S3–S5. Furthermore, the x-ray diffraction data reveal three different Co-O bond lengths inthe rock-salt layer, 28which is compatible with our suggestion that three signals, S3–S5, come from the Co in the rock-saltlayer. On the other hand, for /H20851Ca 2Co1.3Cu0.7O4/H208520.62CoO 2, a com- plex NMR spectrum at higher f’s was observed in a wider f range compared to that for the present Ca 3Co3.92O9.34.21Con- sidering the random distribution of Co and Cu in the rock-salt layer and/or quadruple-layered blocks, magnetic envi-ronments for the Co nuclei in the rock-salt layer are naturallyexpected to be complex, resulting in a wide distribution ofH int’s. Hence, we have postulated that the NMR spectra at higher f’s are assigned as signals from the Co in the rock-salt layer and the other spectra with a broad peak located from/H110150t o /H1101525 MHz are assigned as signals from the Co in the CoO 2layer similar to Ca 3Co3.92O9.34.21 In order to detect clearly the NMR spectra that locate at f/H1134920 MHz under ZF for Ca 3Co3.92O9.34, we measured the 59Co NMR spectra under 6.1065 T. Figure 1/H20849b/H20850shows the f-swept59Co NMR spectrum at 4.2 K. Although the spec- trum exhibits a broad peak around 75 MHz, the spectrumshape is well explained by the two signals, i.e., S1 and S2,and additional signals from the Cu coil. Two signals detectedunder Hare also observed for Na xCoO 2that has the CoO 2 layer,29which also implies that they come from the Co in the CoO 2layer.Since the nucleus in a magnetically ordered state would experience a local magnetic ﬁeld, the resonance frequency f/H20849fr/H20850is expressed by /H9275r=2/H9266fr=/H9253/H20881H2+Hint2+2/H20841H/H20841/H20841Hint/H20841cos/H9258, /H208491/H20850 where /H9253,H,Hint, and/H9258are the nuclear gyromagnetic ratio, the external ﬁeld, the internal ﬁeld, and the angle between H andHint, respectively. A straightforward calculation of the above equation with subsequent insertion of /H9258=0°, 180°, and 90° leads to simple equations /H9275r,/H9258=0°=/H9253/H20841H+Hint/H20841, /H208492/H20850 /H9275r,/H9258=180°=/H9253/H20841H−Hint/H20841, /H208493/H20850 /H9275r,/H9258=90°=/H9253/H20881H2+Hint2. /H208494/H20850 Therefore, the magnitude and the direction of Hintcan be determined by measuring fras a function of H. TheH-swept59Co NMR spectra at 4.2 K taken at several f’s for the randomly oriented Ca 3Co3.92O9.34powder are shown in Fig. 2together with the calculated AF powder pat- terns. It is well known that the NMR spectrum for randompowders in the FM ordered state is different from that in theAF ordered state. When FM H int’s are formed, the NMR spectrum is observed at H0/H11006Hint, where H0is the resonance0 100 200 300(a) CoO 2layerrocksalt-type layer S1 S2S3S4 S5 (b)Frequency (MHz)59Co Spin-Echo Amplitud e (arb. units)Ca3Co3.92O9.34,T=4 . 2K S1S2 60 70 8059Co Spin-Echo Amplitude (arb. units) Frequency (MHz)Ca3Co3.92O9.34 H= 6.1065 T T=4 . 2K 63Cu65Cu FIG. 1. Frequency-swept59Co NMR spectra for Ca 3Co3.92O9.34 at 4.2 K under /H20849a/H20850ZF/H20849Ref. 21/H20850and /H20849b/H208506.1065 T. S1–S5 represent the peak positions of the NMR spectra. The solid line in Fig. 1/H20849a/H20850is a guide to the eyes. The peaks observed at 68.91 and 73.82 MHz inFig. 1/H20849b/H20850are the 63Cu and65Cu NMR signals in an NMR coil, respectively.0 2 4 6 8 10Ca3Co3.92O9.34 H(T)59Co Spin-Echo Amplitude (arb. units) T=4 . 2K85 MHz95 MHz100 MHz120 MHz130 MHz150 MHz 85 MHz 6 7 8 H(T)59Co Spin-Echo Amplitude (arb. units)S3 S4S5 S1S2 65Cu63Cu19F1H FIG. 2. /H20849Color online /H20850Field-swept59Co NMR spectra for Ca3Co3.92O9.34at 4.2 K taken at various frequencies together with the calculated AF powder patterns. Two sharp peaks observed atlower H’s and higher H’s are the 1H and19F NMR signals, respec- tively, and they are caused by cellophane and polytetraﬂuoroethyl-ene tapes. For instance, the former distributes at 1.996 T and thelatter at 2.212 T in the data taken at 85 MHz. In the inset, H-swept 59Co NMR spectrum above 5.5 T taken at 85 MHz is displayed as an expanded scale. The peaks observed at 7.532 and 7.031 T are the 63Cu and65Cu NMR signals in an NMR coil, respectively.PHASE SEPARATION IN THE CoO 2LAYER … PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-3ﬁeld at Knight shift K=0, and the sign of Hintdepends on its direction. This is because FM moments are rotated easily tothe direction of H. On the other hand, when H intis AF, the NMR spectrum has a peak and a step at H0−Hint/H20849/H9258=180° /H20850 andH0+Hint/H20849/H9258=0°/H20850, respectively, and distributes between these ﬁelds. For the NMR spectra arising from AF Hint’s, the positions at higher f’s were determined as the step position, whereas those at lower f’s were taken as the peak position /H20849see Fig. 2/H20850. Note here that a new broad peak at intermediate H’s is attributable to the increase in a rotation of AF mo- ments by Howing to the decrease in the anisotropic and molecular ﬁelds. However, the experimental results did notagree completely with the calculated AF powder patterns,which is probably because a simple AF order is not formeddue to a complex crystal structure, e.g., a misﬁt between twolayers. Therefore, the error bars are added in Fig. 3. The NMR spectrum taken at 85 MHz at the Hrange displayed in the inset of Fig. 2did not show the powder pattern expected for AF H int’s. Also, this powder pattern was not observed, even when fwas decreased down to 15 MHz. Therefore, the positions of the NMR spectra showing these behaviors weredetermined as the peak positions. Figure 3shows the f ras a function of Hat 4.2 K for the same sample. It is found that there are ﬁve components, S1–S5, for the NMR spectrum at 4.2 K under ZF. Both thenumber and their values under ZF accord with those ob-served in the ZF-NMR spectrum shown in Fig. 1/H20849a/H20850. The values of H int’s under ZF are estimated as 0 T for S1, 1.5 T for S2, 9.3 T for S3, 12.7 T for S4, and 14.0 T for S5. Threeof them, S3–S5, are found to agree with the two resonanceconditions; that is, f rincreases /H20849decreases /H20850linearly with H, i.e., satisﬁes “two branches.” On the contrary, frfor S1 andS2 increases linearly with H, i.e., satisﬁes “one branch.” Fur- thermore, the Hdependence of fris well explained by Eq. /H208492/H20850 or/H208493/H20850. Note that the slope of the solid lines in Fig. 3is described based on the nuclear gyromagnetic ratio of59Co, i.e., 2/H9266/H1100310.054 MHz /T. These results demonstrate that AF Hint’s are formed in the rock-salt layer, whereas FM Hint’s are done partially in the CoO 2layer. Also, it is reasonable to conclude that the values of fr’s for S1 and S2 at 6.1065 T predicted from the frversus Hlines at 4.2 K coincide with those observed in the59Co NMR fspectrum shown in Fig. 1/H20849b/H20850. Hintat 0 K was reported to be independent of the substi- tution elements, the amount of the replaced elements, and thenumber of the rock-salt layers from /H9262+SR experiments, which suggests that the IC-SDW ordered state exists in theCoO 2layer.20Furthermore, since the transport properties are mainly determined by the electronic states in the CoO 2layer, information on the local magnetic properties of this layer iscritical in order to understand the physics behind the excel-lent thermoelectric properties of the MLCO’s. Figure 4 shows the f-swept 59Co NMR spectra for S1 and S2, which correspond to the signals from the CoO 2layer of Ca3Co3.92O9.34, measured under 6.1065 T at various T’s. The NMR spectrum was clearly found to consist of two compo-nents below T m1and they have asymmetric shape above Tm1. The physical meaning and the origin of Tm1are explained below. S2, whose intensity is larger than that of S1, is ob-served over the whole Trange measured, which means that the two sites are in different proportions. The incommensu-S1S2S3S4S5 2 4 6 8 10100200 0Ca3Co3.92O9.34 H(T)Frequency (MHz)T=4 . 2K FIG. 3. /H20849Color online /H20850Hdependence of the resonance frequency for Ca 3Co3.92O9.34at 4.2 K. The solid lines in the ﬁgure are the results of ﬁtting Eqs. /H208492/H20850and /H208493/H20850to the data, and their slope is the nuclear gyromagnetic ratio of59Co, i.e., 2 /H9266/H1100310.054 MHz /T. S1–S5 represent the peak positions of the NMR spectra and corre-spond to those in Fig. 1.60 70 80 9059Co Spin-Echo Amplitude (arb. units) Frequenc y(MHz)Ca3Co3.92O9.34 H= 6.1065 T 300 K240 K200 K160 K120 K100 K80 K60 K50 K40 K35 K30 K27 K23 K17 K4.2 K 280 KPMFRS1 S263Cu 65Cu TFR Tm1×50 Tm2 FIG. 4. Frequency-swept59Co NMR spectra for Ca 3Co3.92O9.34 measured under 6.1065 T at various T’s. PM and FR denote the paramagnetic phase and the ferrimagnetic phase, respectively. Theinverted triangles in the ﬁgure represent the peak positions of thespectra, and S1 and S2 correspond to those in Fig. 1/H20849b/H20850.T m1,Tm2, andTFRare the characteristic T’s/H20849see Fig. 5and text /H20850. The peaks observed at 68.91 and 73.82 MHz are the63Cu and65Cu NMR signals in an NMR coil, respectively. The spin-echo amplitude at4.2 K is ampliﬁed by 50 times.TAKAMI et al. PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-4rability of the nearby rock-salt layers strongly distributes the Co electric ﬁeld gradient /H20849EFG /H20850at the Co site in the CoO 2layer, which may make the quadrupolar structure of the NMR spectrum very ambiguous compared to thatobserved for Na xCoO 2. In order to resolve the NMR spectrum with asymmetric shape and elucidate the origin of its asymmetric shape, we measured the59Co NMR spec- tra for the c-axis-aligned Ca 3Co3.92O9.1sample under the same condition; the result is explained in detail in Sec. III B. Furthermore, the peak position for S1 was almost Tindepen- dent, whereas that for S2 shifted toward a lower fwith increasing T, particularly below Tm1, due to the decrease inHintwith T. A similar behavior was also observed for /H20851Ca2Co1.3Cu0.7O4/H208520.62CoO 2.21On the other hand, the NMR signals, corresponding to S3–S5, were not observed in thismeasurement condition, probably because the nuclei spin-spin relaxation time /H20849T 2/H20850is too short to be observable due to the magnetic interaction between the Co ions in the rock-saltlayer. In order to clarify the changes in the S1 and S2 signals with varying T, the Tdependences of the 59Co Knight shift K andHintfor Ca 3Co3.92O9.34are plotted in Fig. 5/H20849a/H20850. Here, we deﬁne KasK=/H20849fr−f0/H20850/f0, where f0=/H9253H/2/H9266with/H9253=2/H9266 /H1100310.054 MHz /T and H=6.1065 T. Kfor S1 /H20849KS1/H20850was about 1.8%, while KS2showed a Tdependence. KS2above /H11015100 K, expressed as Tm1, obeyed the Curie-Weiss law, KS2=1.84+50.6 //H20849T−35.7 /H20850%/H20851a solid curve in Fig. 5/H20849a/H20850/H20852. The fairly good ﬁt and the positive Weiss temperature of 35.7 Kwith quite small error indicate a FM interaction between the Co ions at the Co sites that are responsible for S2. This resultis consistent with the conclusion derived from the relation-ship between f randH. The KS2/H20849T/H20850curve exhibited a plateau at/H1101540 K. Also, this curve suggests the presence of Hintand theHint/H20849T/H20850curve increased signiﬁcantly below 23 K /H20849TFR/H20850, below which the FR order appears, as is clearly seen in theinset of Fig. 5/H20849a/H20850. Here, H int=2/H9266/H20841fr−f0/H20841//H9253. Similar Tdepen- dence of KS1,KS2, and Hinthas also been observed for /H20851Ca2Co1.3Cu0.7O4/H208520.62CoO 2.21 Figure 5/H20849b/H20850shows the Tdependence of the half-width at half maximum /H20849Whwhm/H20850of the59Co NMR spectra measured under 6.1065 T for Ca 3Co3.92O9.34.Whwhmis known to de- pend on the ﬁeld inhomogeneities arising from the variationin the demagnetizing ﬁeld within a given particle and be-tween different particles, the nuclear-nuclear dipolar interac-tion, and the time-dependent electron-nuclear magnetic inter-action. Basically, W hwhmfor this compound plotted in Fig. 5/H20849b/H20850is determined by ﬁtting the NMR spectrum with a com- bination of two Gaussian functions. Whwhm’s for S1 and S2 /H20849WS1hwhmandWS2hwhm/H20850are found to increase with decreasing T. In particular, the WS2hwhm/H20849T/H20850curve changes its slope at Tm1 andTm2; that is, the slope becomes steeper with decreasing T. Note that it was difﬁcult to estimate WS1hwhmfor every T point due to the weak intensity of the S1 signal. Figure 5/H20849c/H20850shows the Tdependence of the integrated in- tensity Ifor S2 /H20849IS2/H20850for Ca 3Co3.92O9.34. In the paramagnetic /H20849PM/H20850phase, the change in Iwith varying Twas small. Upon cooling, Iincreased gradually and exhibited a peak; then decreased and ﬁnally increased again below Tm2. The changes in IatTm1andTm2with varying Tare likely to correlate with the Tvariations in Kand/or Whwhm. However, although T2, which determines IS2, would be very short, par- ticularly below /H1101560 K due to the magnetic order, the T dependences of the NMR parameters /H20849KS2,WS2hwhm, and IS2/H20850 are still not fully explained at present. B.c-axis-aligned Ca 3Co3.92O9.1 The59Co NMR spectrum in the FR state for the c-axis-aligned sample of Ca 3Co3.92O9.1under ZF is shown in Fig. 6/H20849a/H20850. As is clear from this ﬁgure, the spectrum at f /H1135040 MHz consisted of mainly three components. In the 59Co NMR measurements under 6.1065 T at 4.2 K, we also observed two sets of59Co NMR spectra, as in the case of Ca3Co3.92O9.34 /H20851see Fig. 6/H20849b/H20850/H20852. The S1 and S2 signals ob- served under 6.1065 T are located at f/H1134940 MHz under ZF. These results also demonstrate the existence of the ﬁve non-equivalent Co sites. Because the crystal structure ofCa 3Co3.92O9.34−/H9254does not change signiﬁcantly with /H9254, the NMR lines at higher f’s are assigned as signals from the Co in the rock-salt layer and the others are assigned as signalsfrom the Co in the CoO 2layer. TheH-swept59Co NMR spectra at 4.2 K taken at various f’s were obtained for the c-axis-aligned Ca 3Co3.92O9.1 sample for H/H20648thecaxis. We observed the59Co NMR spec- trum with a few components at each f. Fundamentally, the position of the signals S1–S5 was determined as the peakposition. The NMR spectrum for S3 was broad, which is dueTm2 S1 S2TFR S2Tm1S1S2TFR Tm1 Tm2TFRTm2Tm1 0 100 200 300(b) T(K)I(arb. units)0246810 (c)K(%) 100101Whwhm(MHz)(a) 5101520250.51.01.5 0Hint(T) T(K) FIG. 5. /H20849Color online /H20850Tdependences of /H20849a/H20850the Knight shift, /H20849b/H20850 the half-width at half maximum, and /H20849c/H20850the integrated intensity of the59Co NMR spectra for Ca 3Co3.92O9.34. The inset shows the T dependence of the internal ﬁeld and the solid curve is a guide to theeyes. T m1,Tm2, and TFRare the characteristic T’s/H20849see text /H20850. The solid curves in the main panel are guides to the eyes except for theresult of the Curie-Weiss ﬁtting in Fig. 5/H20849a/H20850.PHASE SEPARATION IN THE CoO 2LAYER … PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-5to the electric quadrupole interaction. The quadrupolar fre- quency /H20849/H9263Q/H20850for S3 was found to be /H110153 MHz. As for the S3 signal, we plotted the positions of central lines in Fig. 7.fr/H20648 for this sample is plotted as a function of Hin Fig. 7. The values of Hint’s under ZF are estimated as 0, 1.0, 5.9, 12.4, and 15.9 T for S1, S2, S3, S4, and S5, respectively. Theresonance conditions of f r/H20648for S1–S5 were the same as those for Ca 3Co3.92O9.34. We also measured the H-swept59Co NMR spectra at 4.2 K and various f’s for H/H11036thecaxis. The NMR spectrum with a few components was observed at each findependent of the direction of H. Although the NMR spectra measured forH/H11036thecaxis were broad compared to those for H/H20648thec axis, we roughly determined the peak positions, as in thecase of H /H20648thecaxis. As displayed in Fig. 8, the Hdepen- dence of fr/H11036followed Eq. /H208494/H20850. The values of Hint’s under ZF forH/H11036thecaxis are naturally the same as those for H/H20648the caxis. By taking advantage of the orientation, the conclusion that the direction of Hintis along the caxis can be derived from the results plotted in Figs. 7and8. Figures 9and10show the Tdependence of the f-swept 59Co NMR spectra for S1 and S2, which correspond to the signals from the CoO 2layer of the c-axis-aligned Ca3Co3.92O9.1sample, measured under 6.1065 T for H/H20648thec axis and H/H11036thecaxis, respectively. The59Co NMR spec- trum at 120 K is displayed in the inset as an expanded scaletogether with that of the randomly oriented Ca 3Co3.92O9.34 sample. The clear peak structure attests the high quality of the sample. The I=7 /2 nuclear spin of59Co senses the mag- netic properties of the Co site and couples through its nuclearquadrupole moment to the EFG tensor created by its chargeenvironment. The 59Co NMR spectrum for H/H20648thecaxis is the most typical one for the two sites for which the caxis is the principal axis of the EFG. This result indicates that theNMR spectrum observed under 6.1065 T for these samplesconsists of the signals from S1 and S2 even in the PM phase,although the NMR spectrum for S2 overlapped that for S1 at65Cu 63Cu0 100 200 300(a) S3 S4S5 (b)Frequency (MHz)59Co Spin-Echo Amplitud e (arb. units)Ca3Co3.92O9.1,T=4 . 2K S1S2 60 70 8059Co Spin-Echo Amplitude (arb. units) Frequenc y(MHz)Ca3Co3.92O9.1 H= 6.1065 T H//caxis T=4 . 2Krocksalt-type layer FIG. 6. Frequency-swept59Co NMR spectra for the c-axis-aligned sample of Ca 3Co3.92O9.1at 4.2 K under /H20849a/H20850ZF and /H20849b/H208506.1065 T for H/H20648thecaxis. S1–S5 represent the peak positions of the NMR spectra. The solid line in Fig. 6/H20849a/H20850is a guide to the eyes. The peaks observed at 68.91 and 73.82 MHz in Fig. 6/H20849b/H20850are the63Cu and65Cu NMR signals in an NMR coil, respectively. The NMR line at 77.8 MHz is from a radio FM broadcast.S1S2S3S4S5 2 4 6 850100150 0Ca3Co3.92O9.1 H(T)Frequency (MHz ) T=4 . 2K , H//caxis FIG. 7. /H20849Color online /H20850Hdependence of the resonance frequency for the c-axis-aligned sample of Ca 3Co3.92O9.1at 4.2 K for H/H20648thec axis. The solid lines in the ﬁgure are the results of ﬁtting Eqs. /H208492/H20850 and /H208493/H20850to the data, and their slope is the nuclear gyromagnetic ratio of59Co, i.e., 2 /H9266/H1100310.054 MHz /T. S1–S5 correspond to those in Fig.6. S1S2S3S4S5 2 4 6 850100150 0Ca3Co3.92O9.1 H(T)Frequency (MHz ) T=4 . 2K , H⊥caxis FIG. 8. /H20849Color online /H20850Hdependence of the resonance frequency for the c-axis-aligned sample of Ca 3Co3.92O9.1at 4.2 K for H/H11036the caxis. The solid curves in the ﬁgure are the results of ﬁtting Eq. /H208494/H20850 to the data. S1–S5 correspond to those in Fig. 6.TAKAMI et al. PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-6around room temperature. /H9263Qand the asymmetric parameter /H9257for S1 are evaluated to be /H110151 MHz and 0.20, respectively. The59Co NMR funder ZF generally depends on both /H9263Q and/H9257, and their estimated values rule out the possibility that the NMR spectrum under 6.1065 T consisting of two signalscomes from two components among S3–S5 assuming thatthe charge distribution around cobalt nucleus remains unal-tered with varying Tbecause of no structural phase transi- tion. As Tis lowered, the quadrupole singularities spread and Lorentzian NMR spectra were observed. On the other hand,the crystallites are almost random for H/H11036thecaxis, result- ing in powder spectra /H20849see Fig. 10/H20850. Hence, although the splitting due to the electric quadrupole interaction was am-biguous compared to that for H /H20648thecaxis, the spectrum consisting of the S1 and S2 signals was observed. Figures 11and12show the Tdependences of K,Hint, and Ifor the c-axis-aligned sample of Ca 3Co3.92O9.1forH/H20648thec axis and H/H11036thecaxis, respectively. The KS1measured in both conditions was almost Tindependent /H20849/H110153.5%/H20850, but KS2 was dependent on T. The Tdependence of KS2, particularly in the Trange above Tm1, was ﬁtted by a Curie-Weiss for-mula, KS2=1.77+46.6 //H20849T−41.5 /H20850%, which is shown in Fig. 11/H20849a/H20850as the solid curve. Hintfor S2 was found to be /H110151T a t 4.2 K and decreased drastically upon heating to TFR. As can be seen from Fig. 12/H20849b/H20850, although the change in the Iversus Tcurve for H/H11036thecaxis was less clear than that for H/H20648the caxis, an increase in Ibelow Tm2was commonly observed. TheTvariations in K,Hint, and Icorresponding to S2 for H/H20648 thecaxis seem to exhibit changes at Tm1,Tm2, and TFRas in the case of Ca 3Co3.92O9.34. Furthermore, the anisotropy of Kwas smaller than that of the magnetic susceptibility /H9273. For instance, the ratio /H9273c//H9273ab has been reported to be about 2 at 100 K for thec-axis-aligned /H20851Ca 2CoO 3−/H9254/H208520.62CoO 2sample prepared by ap- plying magnetic alignment in which /H9273cand/H9273abare the mag- netic susceptibility when His applied parallel to the caxis and the abplane, respectively.30The small anisotropy of K implies that the macroscopic magnetism of the MLCO’s witha triple subsystem is dominated by the local magnetic prop-erties coming from the Co in the rock-salt layer. However,only AF H int’s are formed in the rock-salt layer. Therefore, the magnetic interaction in the CoO 2layer is not negligibly weak to stabilize the FR state. Furthermore, TFRdepended on the oxygen content and decreased with increasing /H9254/H20851see the insets in Figs. 5/H20849a/H20850and11/H20849a/H20850/H20852. This behavior is probably due to the smaller concentration of holes in the Co4+/Co3+ couple.PMS2 S163Cu 65Cu TFR Tm2 Tm1×10 ×3 S2 S1S1S2 ▼ ▼ 60 70 80 9059Co Spin-Echo Amplitude (arb. units) Frequenc y(MHz)Ca3Co3.92O9.1 300 K240 K210 K180 K150 K120 K100 K80 K60 K50 K40 K28 K22 K12 K7K4.2 K 270 KH=6 . 1 0 6 5T H//caxis 60 62 64 66 Frequency (MHz)59Co Spin-Echo Amplitude (arb. units)(a) (b)Ca3Co3.92O9.34 T= 120 K Ca3Co3.92O9.1 T=1 2 0KFR FIG. 9. /H20849Color online /H20850Frequency-swept59Co NMR spectra for thec-axis-aligned sample of Ca 3Co3.92O9.1measured under 6.1065 T at various T’s for H/H20648thecaxis. The inverted triangles in the ﬁgure represent the peak positions of the central lines split by theelectric quadrupole interaction. PM, FR, T m1,Tm2, and TFRhave the same meaning as those in Fig. 4/H20849see Fig. 11and text /H20850. The peaks observed at 68.91 and 73.82 MHz are the63Cu and65Cu NMR signals in an NMR coil, respectively. The sharp line at 77.8 MHz isfrom a radio FM broadcast. The spin-echo amplitudes at 4.2 and 7K are ampliﬁed by 10 times and 3 times, respectively. The insetshows f-swept 59Co NMR spectra for /H20849a/H20850the randomly oriented polycrystalline Ca 3Co3.92O9.34sample and /H20849b/H20850thec-axis-aligned Ca3Co3.92O9.1sample measured under 6.1065 T at 120 K. In the latter compound, His applied parallel to the caxis. The arrows denote the59Co NMR lines split by the electric quadrupole interaction.PM65Cu63CuS1S2 TFR Tm2 Tm1×25 ×10 ×5FR 60 70 80 9059Co Spin-Echo Amplitude (arb. units) Frequenc y(MHz)Ca3Co3.92O9.1 H= 6.1065 T, H⊥caxis 300 K240 K180 K150 K120 K100 K80 K60 K40 K30 K26 K22 K17 K10 K7K4.2 K FIG. 10. Frequency-swept59Co NMR spectra for the c-axis-aligned sample of Ca 3Co3.92O9.1measured under 6.1065 T at various T’s for H/H11036thecaxis. The inverted triangles in the ﬁgure represent the peak positions of the spectra. PM, FR, Tm1,Tm2, and TFRhave the same meaning as those in Fig. 4/H20849see Fig. 12and text /H20850. The peaks observed at 68.91 and 73.82 MHz are the63Cu and65Cu NMR signals in an NMR coil, respectively. The sharp line at 77.8MHz is from a radio FM broadcast. The spin-echo amplitudes at4.2, 7, and 10 K are ampliﬁed by 25 times, 10 times, and 5 times,respectively.PHASE SEPARATION IN THE CoO 2LAYER … PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-7C. Randomly oriented polycrystalline Ca 3Co3.92O9with large oxygen vacancy The existence of ﬁve nonequivalent Co sites with differ- entHint’s at 4.2 K was also conﬁrmed by the59Co NMRmeasurements under ZF and 6.1065 T /H20849see Fig. 13/H20850,a si nt h e cases of Ca 3Co3.92O9.34and Ca 3Co3.92O9.1. Also, the intensity ratio of the S1 and S2 signals was found to depend stronglyon the oxygen content in the MLCO’s with a triple sub-system in comparison with Figs. 1/H20849b/H20850,6/H20849b/H20850, and 13/H20849b/H20850.I n other words, the relative intensity of the S1 signal increasedwith decreasing oxygen content. Figure 14shows the f-swept 59Co NMR spectra measured under 6.1065 T at various T’s for S1 and S2, which corre- spond to the signals from the CoO 2layer of Ca 3Co3.92O9. The quadrupole-broadened NMR spectrum consists of twocomponents, as in the cases of Ca 3Co3.92O9.34,C a 3Co3.92O9.1, and /H20851Ca2Co1.3Cu0.7O4/H208520.62CoO 2. Here, it is worth emphasiz- ing that the NMR spectrum was governed by the componentcorresponding to S1 whose Kexhibited almost T-independent behavior, which is an opposite trend com- pared to Ca 3Co3.92O9.34with almost no oxygen vacancy. This result indicates that the dominant interaction affecting thelocal magnetism in the CoO 2layer at lower T’s in the ML- CO’s with a triple subsystem depends strongly on the oxygencontent, i.e., the carrier concentration. Figure 15/H20849a/H20850shows the Tdependence of W S1hwhmof the NMR spectra measured under 6.1065 T for Ca 3Co3.92O9. Whwhmof the NMR spectrum for this material was analyzed by ﬁtting a single Gaussian function to the data because of adominant contribution of S1 to the NMR spectrum as alreadymentioned above. W hwhmfor S1 increased below Tm1and exhibited a plateau at /H1101550 K. When further cooled, WS1hwhm increased again below /H11015Tm2. Two characteristic tempera- tures Tm1andTm2below which WS1hwhmincreased were found to correlate with the Tat which IS1showed the peculiar changes as in the case of Ca 3Co3.92O9.34/H20851see Fig. 15/H20849b/H20850/H20852.TFR Tm1 Tm1 Tm20510K(%)Ca3Co3.92O9.1 H= 6.1065 T H//caxis S1 S2 (b) 0 100 200 300 T(K)I(arb. units)Ca3Co3.92O9.1 H= 6.1065 T H//caxis(a) S25101520250.51.01.5 0 T(K)Hint(T) FIG. 11. /H20849Color online /H20850Tdependences of /H20849a/H20850the59Co Knight shift and /H20849b/H20850the integrated intensity for the c-axis-aligned sample of Ca3Co3.92O9.1forH/H20648thecaxis. The inset shows the Tdependence of the internal ﬁeld. The solid curve in the main panel of Fig. 11/H20849a/H20850 shows the result of the Curie-Weiss ﬁtting and the other curves areguides to the eyes. T m1,Tm2, and TFRhave the same meaning as those in Fig. 4. Tm20246810K(%)Ca3Co3.92O9.1 H=6 . 1 0 6 5T H⊥caxis S1 S2 Ca3Co3.92O9.1 H=6 . 1 0 6 5T H⊥caxis 0 100 200 300 T(K)(a) (b)I(arb. units)S2 FIG. 12. /H20849Color online /H20850Tdependences of /H20849a/H20850the59Co Knight shift and /H20849b/H20850the integrated intensity for the c-axis-aligned sample of Ca3Co3.92O9.1forH/H11036thecaxis. The solid curve in Fig. 12/H20849b/H20850is a guide to the eyes. Tm2is the temperature below which Iincreased rapidly.65Cu 63Cu0 100 200 300(a) S3S4S5 (b)Frequency (MHz)59Co Spin-Echo Amplitud e (arb. units)Ca3Co3.92O9,T=4 . 2K S1 S2 60 70 8059Co Spin-Echo Amplitude (arb. units) Frequency (MHz)Ca3Co3.92O9 H= 6.1065 T T=4 . 2Krocksalt-type layer FIG. 13. Frequency-swept59Co NMR spectra for Ca 3Co3.92O9 at 4.2 K under /H20849a/H20850ZF and /H20849b/H208506.1065 T. S1–S5 represent the peak positions of the NMR spectra. The solid line in Fig. 13/H20849a/H20850is a guide to the eyes. The peaks observed at 68.91 and 73.82 MHz in Fig.13/H20849b/H20850are the 63Cu and65Cu NMR signals in an NMR coil, respectively.TAKAMI et al. PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-8IV. DISCUSSION A. Origin of the magnetism By a systematic study of59Co NMR measurements for the MLCO’s, the59Co NMR spectrum coming from the Co in the CoO 2layer was found to consist of mainly two lines. Oneof them, S1, has a zero Hintunder ZF and the other, S2, has a nonzero Hintunder ZF /H20849FMHint’s/H20850. This behavior is uncon- ventional because two of the Hint’s exist simultaneously in a single layer even consisting of one crystallographicallyequivalent Co site. There may be a few scenarios to explainthese experimental ﬁndings. One is that there are two non-equivalent sites in a single uniform phase, wherein two dif-ferent electronic states around equivalent cobalt nuclei exist,for example, due to a charge-ordered state. Another is morerealistic, i.e., a view based on a separation between twophases. Quite recently, the phase separation between thecharge-ordered insulating state and the PM metallic state hasbeen claimed by photoemission spectroscopy experiments. 31 According to their measurements, holes are localized regu-larly in the former state, while they are itinerant and distrib-uted uniformly in the latter state. 31Since EFG depends sen- sitively on the charge distribution around the nucleus, anychange in the EFG value is related to either the structuralphase change or the change in electronic state. When acharge-ordered state is realized in the MLCO’s, /H9263Qchanges with varying T. However, the almost constant behavior of /H9263Qc for S1 and S2 evidences the absence of any charge ordering at least down to Tm2. In our NMR experiments, however, the coexistence of the SDW and FM order would be proposed,the detail of which is discussed as follows. The intensityratio of the S1 and S2 signals depended strongly on the oxy-gen content in MLCO’s with a triple subsystem. The NMRspectrum for S1 whose Kshowed almost T-independent be- havior is predominant in the sample with the large /H9254. The NMR spectrum S1 for Ca 3Co3.92O9below 17 K had a char- acteristic triangular shape, which is similar to that expectedfor a typical SDW ordered state. Therefore, we verify thepossibility of the SDW state. The NMR shape function Fin the SDW ordered state is expressed as F/H11008ln/H208491+ /H208811−x2/H20850//H20841x/H20841, /H208495/H20850 where x=/H20849H−/H9275//H9253/H20850//H20849Hint/H20850maxand /H20849Hint/H20850maxis the respective maximum amplitude of the internal ﬁeld.32The NMR spec- trum at 4.2 K for Ca 3Co3.92O9could be roughly ﬁtted by this equation as seen in Fig. 16/H20849a/H20850. Because the NMR spectrum for S1 above 10 K overlapped that for S2, we ﬁtted the NMRspectrum using a combination of Eq. /H208495/H20850and a Gaussian function. As can be seen from Fig. 16/H20849b/H20850, the NMR spectrum at 10 K with two components seems to be explained by thesetwo functions. The values of /H20849H int/H20850maxwere estimated to be 0.27, 0.20, and 0.10 T at 4.2, 10, and 17 K, respectively.Considering both the /H9262+SR and the present NMR data, the SDW ordered state is likely realized in the CoO 2layer, par- ticularly for MLCO’s with a large /H9254. Interestingly, the coexistence of coherent electrons and incoherent ones for the MLCO’s has been argued by photo-emission spectroscopy experiments. 33The enhancement of /H9267 with decreasing Tin the IC-SDW ordered state is more dis- tinct with increasing /H9254,24which implies that the electrons for S1 have an incoherent nature and those for S2 have a coher-ent nature. On the other hand, the partial electronic statescorresponding to the rock-salt layer may be formed by inco-herent electrons because the electrical conductivity in thislayer is insulating.65Cu63Cu PMS1 TFR Tm2 Tm1S2 FR 60 70 80 Frequency (MHz)59Co Spin-Echo Amplitude (arb. units) 300 K280 KCa3Co3.92O9240 K200 K150 K100 K20 K17 K10 K4.2 K 25 K 30 K 40 K 50 K 60 K 70 K 80 K 90 K 110 K 120 K 130 K 140 K H=6 . 1 0 6 5T FIG. 14. Frequency-swept59Co NMR spectra for Ca 3Co3.92O9 measured under 6.1065 T at various T’s. The inverted triangles in the ﬁgure represent the peak positions of the spectra. PM, FR, Tm1, Tm2, and TFRhave the same meaning as those in Fig. 4/H20849see Fig. 15 and text /H20850. The peaks observed at 68.91 and 73.82 MHz are the63Cu and65Cu NMR signals in an NMR coil, respectively. Tm1Tm2 S1 Tm2 Tm1S1Ca3Co3.92O9 Ca3Co3.92O900.51.01.5 (b)Whwhm(MHz) 0 100 200 300(a) T(K)I(arb. units) FIG. 15. /H20849Color online /H20850Tdependences of /H20849a/H20850the half-width at half maximum and /H20849b/H20850the integrated intensity of the59Co NMR spectrum corresponding to the S1 signal for Ca 3Co3.92O9.Tm1and Tm2have the same meaning as those in Fig. 4. The solid curves are guides to the eyes.PHASE SEPARATION IN THE CoO 2LAYER … PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-9For the same sample used in the /H9262+SR experiments, i.e., thec-axis-aligned Ca 3Co4O9.1sample, the Tdependences of Hint,K,Whwhm, and Ifor S2, whose signal shows the positive Weiss Tand FM Hint’s, are rather likely to correlate with the phase diagram determined by the /H9262+SR measurements. Therefore, the magnetic nature detected by means of thistechnique may be mostly due to the Tvariation in the mag- netism with the FM interaction. However, the existence of SDW order in the MLCO’s with a triple subsystem is notnecessarily denied because we observed an NMR spectrumin which both the existence of the SDW and FM orders couldpossibly be inferred; the degree of their competition wouldbe controlled by the oxygen content in Ca 3Co3.92O9.34−/H9254. The appearance and stability of the SDW phase have been theoretically discussed by the Hubbard model within a mean-ﬁeld approximation using parameters such as the electronﬁlling, the Hubbard on-site repulsion, and the nearest-neighbor hopping amplitude. 34,35Based on the phase dia- gram proposed by the extended Hubbard model on a trian-gular lattice, an increase in the on-site repulsion leads to acompetition between the SDW and FM order. 36The elec- tronic speciﬁc-heat coefﬁcient /H9253of Ca 3Co4O9has been re- ported to be as large as /H1101590 mJ /mol K2, which is about two times larger than /H9253of NaCo 2O4,37,38indicating Ca 3Co4O9is a strongly correlated electron material. Therefore, the com-petition can be interpreted by the strong correlation between3delectrons. In this model, the FM order is suppressed with increasing electron ﬁlling and the boundary between theSDW and FM order is almost electron-ﬁlling independent. 36 The trend that an increase in the electron ﬁlling leads SDWorder can be accounted for in the model calculation providedthere is a decrease in ﬁrst-neighbor repulsion with increasing /H9254. Also, the development of SDW order with decreasing n coincides with the phase diagram for Na xCoO 2, in which the onset Tof the SDW order observed for x=0.75 increases with x.3The ground state for the CoO 2layer in Ca3Co3.92O9.34−/H9254may be summarized with the phase diagram of Fig. 17.Next, we discuss brieﬂy the magnetic nature at Tm1,Tm2, andTFR. Below Tm1, the KS2versus Tcurve deviated from the Curie-Weiss law and bent downward. And also, theasymmetry of a weak transverse ﬁeld /H9262+SR spectrum that is proportional to the volume fraction of a PM phase decreasedbelow T m1. These results suggest that a magnetic order de- velops below Tm1. Because the values of KS2below Tm1was smaller than those expected from the Curie-Weiss law, ashort-range AF order coming from an interplane interactionis thought to develop below T m1. On the other hand, the origin of the change in KatTm2is still unclear for the samples of Ca 3Co3.92O9.34and Ca 3Co3.92O9.1. However, we can exclude the possibility that Tm2is a competition Tas reported for the rare-earth iron garnets. This is because aclear hysteretic loop is observed only below T FR. In addition, an anomalous enhancement in the Co-Co correlation in theCoO 2layer has been reported to occur at Tm2.39For these samples, the short-range FM order in the CoO 2layer may develop below Tm2, which may be caused by the frustration due to a 2D triangular lattice and the disorder. Because theintegrated intensity for S2 exhibited a minimum at aroundT m2, the great majority of magnetic moments would be al- ready aligned ferromagnetically at this temperature. In con-trast to these interpretations, the characteristic temperaturesT m1andTm2observed in Ca 3Co3.92O9may correspond to the onset Tof the short-range IC-SDW order and the long-range one, respectively, since an analysis of the NMR shape for S1at low Tbelow which the S1 and S2 signals are distinguish- able indicates the presence of a SDW order. B. Magnetism and transport properties An increase in /H9254in Ca 3Co3.92O9.34−/H9254may increase both S and/H9267due to a decrease in the carrier concentration /H20849n/H20850. This behavior can be understood by a simple model assuming aparabolic band, in which both of them vary monotonically asa function of n, 40i.e., they increase with decreasing n.I nt h e framework of the band picture, electrons inside the energyrange of a few k BTin width centered at the chemical poten- tial are attributable to the transport properties, where kBis the Boltzmann constant. Quite recently, it has been revealedthat the density of states /H20849DOS /H20850that arises from the coherent electrons located at the lower binding-energy region, while60 62 64 66 68 70T=4 . 2K T=1 0K Frequenc y(MHz)59Co Spin-Echo Amplitude (arb. units ) Ca3Co3.92O9 (a) (b)S1 S2S1 FIG. 16. /H20849Color online /H2085059Co NMR spectra for Ca 3Co3.92O9 measured under 6.1065 T at /H20849a/H208504.2 K and /H20849b/H2085010 K. The solid and dashed curves are the result of ﬁtting Eq. /H208495/H20850and a Gaussian func- tion to the data, respectively. The former curves represent F/H20849x/H20850 smeared over a range of 1/5 and 1/65 of /H20849Hint/H20850maxat 4.2 and 10 K, respectively. SDW FM oxygen contentoxygen vacancy δ 9.0 9.1 9.340.34 0.24 0 FIG. 17. /H20849Color online /H20850Schematic phase diagram of the CoO 2 layer in Ca 3Co3.92O9.34−/H9254proposed by the present NMR measurements.TAKAMI et al. PHYSICAL REVIEW B 81, 014401 /H208492010 /H20850 014401-10the DOS that arises from the incoherent electrons is at the higher binding-energy region.33Therefore, the contribution of the DOS with a coherent nature near the Fermi level EF dominates the transport properties at lower T’s. In particular, the narrow band with a sharp slope in the vicinity of EF, which is caused by the strong electron correlation, gives a steep increase in Sat lower T’s. With increasing T, the inco- herent electrons, in addition to the coherent ones, are alsoattributable to S. As already mentioned in the introduction, theoretical work has proposed the importance of the degen-eracy of spins and orbitals of the 3 delectrons of the Co ions on the enhancement of S. 22In the MLCO’s investigated in this work, the magnetic order is completed at low T, which means that the freedom of spins of the 3 delectrons is not frozen at higher T’s. Furthermore, Ca 3Co3.92O9.34exhibits a spin-state transition at around 380 K.19However, the spin state of Co3+still remains in the low-spin /H20849LS/H20850state and that of Co4+is changed from the LS to intermediate-spin /H20849IS/H20850 state with increasing T. Because the IS state of Co4+has higher degeneracy than the LS state of Co4+, this spin-state transition enhances the entropy of spins and orbitals of the3delectrons, resulting in a large Sat high T. In Ca 3Co3.92O9.34−/H9254, the number of electrons with coherent/incoherent nature in the CoO 2layer would be changed by controlling the oxygen content, which highlightsthe role of each electron to the transport properties. Sand /H9267 increase with /H9254in the whole Trange below 300 K.23This is probably because the slope of the DOS near EFdoes not signiﬁcantly depend on /H9254, in addition to a decrease in a ﬁnite DOS with increasing /H9254. Although Sof Ca 3Co3.92O9.34−/H9254at high Twould also increase with /H9254because of the contribu- tion of the incoherent electrons, the coherent electrons areresponsible for the metallic conductivity. Provided that a ﬁ-nite DOS in the vicinity of E Fbecomes steeper with decreas- ing/H9254, the/H9254dependence of the S/H20849T/H20850curve is exciting; that is, the enhancement of Ssurpasses the increase in /H9267with in- creasing /H9254up to the energy range where the coherent elec- trons are attributable to the transport properties. Furthermore,a large Swill still remain at high Tby the contribution of the incoherent electrons. If the increase in Sis larger than that in /H9267at high T, the good thermoelectric performance will also berealized. Therefore, in either side, in order to realize excel- lent thermoelectric performance, both a narrow band with astrongly energy-dependent DOS being formed by the elec-trons with coherent nature in the vicinity of E Fand a large entropy of spins and orbitals of the incoherent electrons areconcluded to be needed. V. CONCLUSION 59Co NMR measurements were conducted to study the local magnetic properties of misﬁt layered cobalt dioxideswith randomly oriented polycrystalline Ca 3Co3.92O9.34and Ca3Co3.92O9samples, together with a c-axis-aligned sample of Ca 3Co3.92O9.1of high quality. We successfully observed the59Co NMR spectra corresponding to signals from the Co both in the CoO 2layer and the rock-salt layer and clariﬁed the magnetic interactions that give rise to various magneticorders. Speciﬁcally, the separation between two phases wasfound in the CoO 2layer consisting of a crystallographically unique Co site and the degree of competition between themdepended on the oxygen contents in misﬁt layered cobaltdioxides with a triple subsystem. The coexistence of bothcoherent and incoherent electrons in the conducting layer isconsidered to be one of the origins of the excellent thermo-electric performance for misﬁt layered cobalt dioxides. ACKNOWLEDGMENTS This study was supported by the Grant-in-Aid for Scien- tiﬁc Research /H20849Grant No. 19340097 /H20850from the Japan Society for the Promotion of Science and by the Grant-in-Aid forScientiﬁc Research /H20849Grant No. 19014007 /H20850from the Ministry of Education, Culture, Sports, Science, and Technology ofJapan. 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PhysRevB.98.214201.pdf	PHYSICAL REVIEW B 98, 214201 (2018) Transverse conﬁnement of ultrasound through the Anderson transition in three-dimensional mesoglasses L. A. Cobus,1,W. K. Hildebrand,1S. E. Skipetrov,2B. A. van Tiggelen,2and J. H. Page1,† 1Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada 2Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France (Received 16 October 2018; published 6 December 2018) We report an in-depth investigation of the Anderson localization transition for classical waves in three dimensions (3D). Experimentally, we observe clear signatures of Anderson localization by measuring thetransverse conﬁnement of transmitted ultrasound through slab-shaped mesoglass samples. We compare ourexperimental data with predictions of the self-consistent theory of Anderson localization for an open mediumwith the same geometry as our samples. This model describes the transverse conﬁnement of classical wavesas a function of the localization (correlation) length, ξ(ζ), and is ﬁtted to our experimental data to quantify the transverse spreading/conﬁnement of ultrasound all of the way through the transition between diffusion andlocalization. Hence we are able to precisely identify the location of the mobility edges at which the Andersontransitions occur. DOI: 10.1103/PhysRevB.98.214201 Anderson localization can be described as the inhibition of wave propagation due to strong disorder, resulting in thespatial localization of wave functions [ 1–3]. In the localization regime, waves remain localized inside the medium on a typ-ical length scale given by the localization length ξ. Between diffusive and localized regimes there is a true transition, whichoccurs at the so-called mobility edge and exists only in threedimensions (3D) [ 4] for systems that respect time reversal and spin rotation symmetry (the so-called orthogonal symmetryclass) [ 5]. For conventional quantum systems, such as the electronic systems considered in Ref. [ 4], this transition to localization occurs when particle energy becomes less thanthe critical energy. In contrast, the localization of classicalwaves in 3D is only expected to occur in some intermediaterange of frequencies called a mobility gap : a localization regime bounded by two mobility edges (MEs) (one on eitherside) [ 2]. This is because localization in 3D requires very strong scattering ( k/lscript∼1, where kand/lscriptare wave vector and scattering mean free path, respectively), and strong scatteringis only likely to occur at intermediate frequencies where thewavelength is comparable to the size of the scatterers. Weakscattering, where localization is unlikely, occurs both at lowfrequencies where the wavelength is large compared to thescatterer size (Rayleigh scattering regime) and at high fre-quencies where wavelength is small compared with scatterersize or separation (the ray optics or acoustics regime). In theintermediate frequency regime, the scattering strength mayvary strongly with frequency due to resonances, and the pos-sibility of localization may be enhanced at frequencies where Current address: Institut Langevin, ESPCI Paris, CNRS UMR 7587, PSL University, 1 rue Jussieu, 75005 Paris, France;laura.cobus@espci.fr. †john.page@umanitoba.cathe density of states is reduced [ 6]. As a result, classical waves may even offer the opportunity to observe many mobilityedges (one or more ME pairs) in the same sample. While searches for Anderson localization in 3D have been carried out for both optical and acoustic waves, acoustic wavesoffer several important advantages for the experimental ob-servation of localization. Chief among these is the possibilityof creating samples which scatter sound strongly enough toenable a localization regime to occur [ 7–12]. Media which scatter light strongly enough to result in localization havenot yet been demonstrated, possibly due to the difﬁculty ofachieving a high enough optical contrast between scatterersand propagation medium [ 13,14]. In addition, effects which can hinder or mask signatures of localization can be bypassedor avoided in acoustic experiments. One of the most signiﬁ-cant of these effects is absorption, which has hindered initialattempts to measure localization using light waves [ 15]. With ultrasound, it is possible to make measurements which aretime, frequency, and position resolved, which enable the ob-servation of quantities which are absorption independent [ 16]. Inelastic scattering (e.g., ﬂuorescence), which has plaguedsome optical experiments [ 17], is also not expected to occur for acoustic waves. We have reported previously on several aspects of An- derson localization of ultrasound in 3D samples [ 7–10,18]. In general, we are able to make direct observations of lo-calization by examining how the wave energy spreads withtime in transmission through or reﬂection from a stronglyscattering medium [ 7,10,19,20]. In this work, we present a detailed experimental investigation of the Anderson transi-tion in 3D, using measurements of transmitted ultrasound.The media studied are 3D ‘mesoglasses’ consisting of smallaluminum balls brazed together to form a disordered solid.Results for two representative samples are presented: onethinner and monodisperse, and one thicker and polydisperse. 2469-9950/2018/98(21)/214201(14) 214201-1 ©2018 American Physical SocietyL. A. COBUS et al. PHYSICAL REVIEW B 98, 214201 (2018) Since we cannot perform measurements inside the mesoglass samples, our measurements are made very near the surface.Our experiments measure the transmitted dynamic transverse intensity proﬁle , which can be used to observe the transverse conﬁnement of ultrasonic waves in our mesoglass samples andto furthermore prove the existence of Anderson localizationin 3D [ 7]. By acquiring data as a function of both time and space, this technique enables the observation of (ratio)quantities in which the explicit dependence of absorptionon the measurements cancels out, so that absorption cannotobscure localization effects. Our experimental data are compared with predictions from the self-consistent (SC) theory of localization for open media[20,21]. This theory is described in Sec. I, where its develop- ment in the context of interpreting experiments such as theones described in this paper is emphasized. In Sec. IIwe explain the details of our experimental methods for observingthe transverse conﬁnement of ultrasound. Section IIIpresents experimental results for two mesoglass samples and theirquantitative interpretation based on numerical calculations ofthe solutions of SC theory for our experimental geometry.A major focus of this work is to show that this comparisonbetween experiment and theory enables signatures of local-ization to be unambiguously identiﬁed and mobility edges tobe precisely located. We aim to provide a sufﬁciently detailedaccount of our overall approach that future observations of 3DAnderson localization will be facilitated. I. THEORY To describe transmitted ultrasound in the localization regime, we use a theoretical model derived from the self-consistent (SC) theory of Anderson localization with aposition- and frequency-dependent diffusion coefﬁcient. SCtheory was developed by V ollhardt and Wölﬂe in the begin-ning of the 1980s [ 22–24] as a very useful and quantitative way of reformulating the scaling theory of localization [ 4,25]. Despite its many successes, the original variant of SC theoryhad a very approximate way of treating the ﬁnite size of asample Land the boundary conditions at its boundaries. In the return probability, which is the essential ingredient in theSC theory that suppresses diffusion, an upper cutoff Lwas introduced in the summation over all possible paths in themedium. This produced the correct scaling of localizationwith sample size but is clearly insufﬁcient if one aims atquantitatively accurate results. To circumvent this problem,van Tiggelen et al. demonstrated that constructive interference is suppressed by leakage through the boundaries of an openmedium, causing the return probability to become positiondependent near the boundaries and implying the existence ofa position-dependent diffusion coefﬁcient [ 26]. The position dependence of Dalso emerged later from perturbative dia- grammatic techniques [ 27] and the nonlinear sigma model [28]. Subsequent studies focused on the analysis of quanti- tative accuracy of SC theory in disordered waveguides [ 29], the experimental veriﬁcation of the position dependence of D [30], and different ways to improve the accuracy of SC theory deep in the localized regime [ 31,32]. It should be noted that most of the tests of SC theory with a position-dependent D have been, up to date, performed in 1D or quasi-1D disorderedsystems, leaving the question about its accuracy in higher- dimensional (e.g., 3D) media largely unexplored. Here we use self-consistent equations for the intensity Green’s function C(r,r /prime,/Omega1)=(4π/vE) /angbracketleftG(r,r/prime,ω0+/Omega1/2)G∗(r,r/prime,ω0−/Omega1/2)/angbracketrightand the position- dependent diffusion coefﬁcient D(r,/Omega1) derived in Ref. [ 27]: [−i/Omega1−∇ r·D(r,/Omega1)∇r]C(r,r/prime,/Omega1)=δ(r−r/prime),(1) 1 D(r,/Omega1)=1 DB+12π k2/lscript∗ BC(r,r,/Omega1), (2) where G(r,r/prime,ω) is the Green’s function of a disordered Helmholtz equation, C(r,r,/Omega1) is the return probability, vE is the energy transport velocity (assumed unaffected by local- ization effects), kis the wave number, the angular brackets /angbracketleft ···/angbracketright denote ensemble averaging, and DBand/lscript∗ Bare the diffusion coefﬁcient and transport mean free path that wouldbe observed in the system in the absence of localizationeffects: D B=vE/lscript∗ B/3. As compared to Ref. [ 27], Eqs. ( 1) and ( 2) are now generalized to allow for anisotropic scattering (/lscript∗ B/negationslash=/lscript) which can be done by repeating the derivation of Ref. [ 27] with /lscript∗ B/negationslash=/lscriptfrom the very beginning. The result is that/lscriptis replaced by /lscript∗ Bin Eq. ( 2) as follows from the same substitution taking place in the Hikami box calculation in asystem with anisotropic scattering [ 33]. Physically, the Fourier transform C(r,r /prime,t)=1 2π/integraldisplay∞ −∞d/Omega1C(r,r/prime,/Omega1)e−i/Omega1t(3) ofC(r,r/prime,/Omega1) gives the probability to ﬁnd a wave packet at a point ra time tafter emission of a short pulse at r/prime. The pulse should be, on one hand, short enough to be well approximated by the Dirac delta function δ(t) (so that adequate temporal resolution is not sacriﬁced), but, on theother hand, long enough to ensure the frequency independenceof transport properties [such as, e.g., the mean free path /lscript(ω) within its bandwidth]. These two quite restrictive conditionscan typically be best fulﬁlled at long times, when the energydensity C(r,r /prime,t) becomes insensitive to the duration of the initial pulse. A. Inﬁnite disordered medium To set the stage, let us ﬁrst analyze Eqs. ( 1) and ( 2)i n an unbounded 3D medium where Dbecomes position inde- pendent: D(r,/Omega1)=D(/Omega1). The analysis is most conveniently performed in the Fourier space: C(r,r/prime,/Omega1)=1 (2π)3/integraldisplay d3qC(q,/Omega1)e−iq(r−r/prime). (4) Equation ( 1) yields C(q,/Omega1)=[−i/Omega1+q2D(/Omega1)]−1, (5) whereas the return probability C(r,r,/Omega1)i nE q .( 2)i s expressed as C(r,r,/Omega1)=1 (2π)3/integraldisplay d3qC(q,/Omega1) =1 2π2/integraldisplayqmax 0dqq2C(q,/Omega1), (6) 214201-2TRANSVERSE CONFINEMENT OF ULTRASOUND THROUGH … PHYSICAL REVIEW B 98, 214201 (2018) where an upper cutoff qmaxis needed to cope with the un- physical divergence due to the breakdown of Eq. ( 1)a ts m a l l length scales. The need for the cutoff can be avoided if Eq. ( 1) is replaced by a more accurate calculation, which indicatesthat the cutoff is related to the inverse mean free path, a resultthat is physically intuitive. Equation ( 1) is unsatisfactory for length scales /lessorsimilar/lscript ∗ B, leading to the cutoff qmax=μ//lscript∗ B, with μ∼1. The precise value of μcannot be determined from the present theory, but it ﬁxes the exact location of the mobilityedge because one easily ﬁnds by combining Eqs. ( 2), (5), and (6) that D(0)=D B/bracketleftbigg 1−6μ π1 (k/lscript∗ B)2/bracketrightbigg . (7) Hence, a mobility edge (ME) at k/lscript=1 (Ioffe-Regel criterion) would correspond to μ=(π/6)(/lscript∗ B//lscript)2. In order to introduce deﬁnitions compatible with the exper- imental geometry of a disordered slab conﬁned between theplanes z=0 and z=Lof a Cartesian reference frame (see the next subsection for details), the integral in Eq. ( 6) can be performed by using a cutoff q max ⊥=μ//lscript∗ Bin the integration over only the transverse component q⊥of the 3D momentum q={q⊥,qz}: C(r,r,/Omega1)=1 (2π)2/integraldisplay∞ −∞dqz/integraldisplayqmax ⊥ 0dq⊥q⊥C(q⊥,qz,/Omega1).(8) This leads to an equation similar to Eq. ( 7): D(0)=DB/bracketleftbigg 1−3μ (k/lscript∗ B)2/bracketrightbigg . (9) Now a ME at k/lscript=(k/lscript)c=1 would correspond to μ= 1 3(/lscript∗ B//lscript)2. When ﬁtting the data, we use the link between μ and ME ( k/lscript)cthe other way around. Namely, ( k/lscript)cwill be a free ﬁt parameter to be adjusted to obtain the best ﬁt to thedata with μ= 1 3(k/lscript)2 c(/lscript∗ B//lscript)2. In the localized regime k/lscript < (k/lscript)c, an analytic solution of Eqs. ( 1) and ( 2) can be obtained for a point source emitting a short pulse at r/prime=0 andt/prime=0. To study the long-time limit, we set D(/Omega1)=−i/Omega1ξ2and obtain C(r,r/prime,t)=1 4πξ2\|r−r/prime\|exp(−\|r−r/prime\|/ξ), (10) where the localization length is ξ=6/lscript (k/lscript)2c/parenleftbigg/lscript /lscript∗ B/parenrightbiggχ2 1−χ4,χ < 1, (11) andχ=k/lscript/(k/lscript)c. When k,/lscript, and /lscript∗ Bare measured inde- pendently or ﬁxed based on some additional considerations,Eq. ( 11) provides a one-to-one correspondence between the value of ( k/lscript) cthat we obtain from ﬁts to data and ξ.I ti s then convenient to use ξas the main parameter obtained from a ﬁt to data. In Eq. ( 11), the right-hand side changes sign when the localization transition is crossed and takes negativevalues in the diffuse regime k/lscript > (k/lscript) c. Then Eq. ( 11) can berewritten as ζ=6/lscript (k/lscript)2c/parenleftbigg/lscript /lscript∗ B/parenrightbiggχ2 χ4−1,χ > 1, (12) where ζplays the role of a correlation length of ﬂuctuations that develop in the wave intensity when the localization tran-sition is approached. Equation ( 11) exhibits one of the problems of SC theory: In the vicinity of the localization transition it predicts ξ∝ (1−χ) −1and hence the predicted critical exponent is ν=1. This value is different from ν/similarequal1.57 established numerically for 3D disordered systems belonging to the orthogonal univer-sality class (see, e.g., Ref. [ 34] for a recent review). Recently, the same value of νhas been found for elastic waves in models that account for their vector character [ 35]. To our knowledge, no analytic theory exists that predicts a criticalexponent different from ν=1[5]. B. Disordered slab To compare theory to experimental data, we need to solve Eqs. ( 1) and ( 2) in a bounded disordered medium having the shape of a slab of thickness L, conﬁned between the planes z=0 and z=L. First of all, Eqs. ( 1) and ( 2)h a v et ob e supplemented by a boundary condition corresponding to noincident diffuse ﬂux (since the incident energy is provided bya point source at depth /lscript ∗ B): C(r,r/prime,/Omega1)−z0D(r,/Omega1) DB(n·∇)C(r,r/prime,/Omega1)=0,(13) where nis a unit inward normal to the surface of the slab at a point ron one of its surfaces; nis parallel (antiparallel) to thezaxis for the surface at z=0(z=L). This condition is a generalization of the one derived in Ref. [ 27] to a medium with an arbitrary internal reﬂection coefﬁcient Rintthat can be obtained using the approach of Ref. [ 36]. The so-called extrapolation length z0is given by z0=2 3/lscript∗ B1+Rint 1−Rint. (14) Next, the translational invariance in the ( x,y) plane im- poses D(r,/Omega1)=D(z,/Omega1). We obtain the solution of the sys- tem of Eqs. ( 1), (2), and ( 13) in a slab following a sequence of steps described below: (i) Equation ( 1) is Fourier transformed in the ( x,y) plane: C(r,r/prime,/Omega1)=/integraldisplay∞ 0d2q⊥ (2π)2C(q⊥,z,z/prime,/Omega1) ×e−iq⊥(ρ−ρ/prime), (15) where ρ={x,y}. (ii) The resulting equation for C(q⊥,z,z/prime,/Omega1), Eq. ( 2), and the boundary conditions ( 13) are rewritten in dimen- sionless variables ˜z=z/L,u=(q⊥L)2,˜/Omega1=/Omega1L2/DB,˜C= −i/Omega1L×C, andd=(D/D B)/(−i˜/Omega1): [1+ud(˜z,˜/Omega1)]˜C(u,˜z,˜z/prime,˜/Omega1) −∂ ∂˜z/bracketleftbigg d(˜z,˜/Omega1)∂ ∂˜z˜C(u,˜z,˜z/prime,˜/Omega1)/bracketrightbigg =δ(˜z−˜z/prime), (16) 214201-3L. A. COBUS et al. PHYSICAL REVIEW B 98, 214201 (2018) 1 d(˜z,˜/Omega1)=−i/Omega1+3 (k/lscript∗ B)2/lscript∗B L/integraldisplayumax 0˜C(u,˜z,˜z/prime,˜/Omega1)du, (17) ˜C(u,˜z,˜z/prime,˜/Omega1)±i˜/Omega1d(˜z,˜/Omega1)˜z0∂ ∂˜z˜C(u,˜z,˜z/prime,˜/Omega1)=0, (18)where umax=(μL//lscript∗ B)2and the signs ‘ +’ and ‘ −’i nE q .( 18) correspond to ˜z=0 and ˜z=1, respectively. (iii) Equations ( 16)–(18) are discretized on grids in zandu (we omit tildes above dimensionless variables from here on tolighten the notation): z n=(n−1)/Delta1z, with/Delta1z=1/(N−1) andn=1,...,N ;uν=(ν−1)/Delta1u, with/Delta1u=umax/(M− 1) and ν=1,...,M : (/Delta1z)2[1+uνdn(/Omega1)]Cnm(uν,/Omega1)−dn(/Omega1)[C(n+1)m(uν,/Omega1)−2Cnm(uν,/Omega1)+C(n−1)m(uν,/Omega1)]−/Delta1z 2d/prime n(/Omega1) ×[C(n+1)m(uν,/Omega1)−C(n−1)m(uν,/Omega1)]=/Delta1zδnm, (19) 1 dm(/Omega1)=−i/Omega1+3 (k/lscript∗ B)2/lscript∗B L/Delta1u/braceleftBiggM/summationdisplay ν=1Cmm(uν,/Omega1)−1 2[Cmm(u1,/Omega1)+Cmm(uM,/Omega1)]/bracerightBigg (20) /Delta1zC1m(uν,/Omega1)+i/Omega1d1(/Omega1)z0[C2m(uν,/Omega1)−C1m(uν,/Omega1)]=0, (21) /Delta1zCNm(uν,/Omega1)−i/Omega1dN(/Omega1)z0[CNm(uν,/Omega1)−C(N−1)m(uν,/Omega1)]=0. (22) Here d/prime n(/Omega1)=[dn+1(/Omega1)−dn−1(/Omega1)]/(2/Delta1z)f o r n= 2,...,N −1 whereas d/prime 1(/Omega1) and d/prime N(/Omega1) are assumed to be equal to d/prime 2(/Omega1) andd/prime N−1(/Omega1), respectively. (iv) We start with an initial guess for dn(/Omega1):dn(/Omega1)= 1/(−i/Omega1), corresponding to D=DB. Linear algebraic equa- tions ( 19), (21), and ( 22) are solved for Cnm(uν,/Omega1)a tﬁ x e d /Omega1for all m=2,...,N −1 and ν=1,...,M . An efﬁcient solution is made possible by the fact that the matrix of coefﬁ-cients of the system of linear equations ( 19), (21), and ( 22)i s tridiagonal; we obtain the solution with the help of a standardroutine zgtsl from LAPACK library [ 37]. Then, new values ford m(/Omega1) are calculated using Eq. ( 20)f o rm=2,...,N − 1.d1(/Omega1) anddN(/Omega1) are found by a linear extrapolation from d2(/Omega1),d3(/Omega1) and dN−2(/Omega1),dN−1(/Omega1), respectively. In prac- tice, this procedure is performed for m=2,... (N+1)/2 only since dm(/Omega1) is symmetric with respect to the middle of the slab. To increase the accuracy of representation of theintegral over uin Eq. ( 17) by a discrete sum in Eq. ( 20), we use ag r i dw i t hav a r i a b l es t e p /Delta1u:As m a l l /Delta1u 1=u1/(M1−1) is used for u/lessorequalslantu1and a larger /Delta1u2=(umax−u1)/(M2−1) foru1<u/lessorequalslantumax. The typical values of u1,M1, andM2used in our calculations are u1=umax/100,M1=M2=400. The number of sites in the spatial grid is typically N=2001. We checked that doubling M1,M2, andNdoes not modify the results by more than a few percent. (v) The solution described in the previous step is repeated iteratively, each new iteration using the values of dn(/Omega1) ob- tained from the previous one, until either a maximum numberof iterations is reached (1500 in our calculations) or a certaincriterion of convergence is obeyed [typically, we require thatnod n(/Omega1) changes by more than (10−5)% from one iteration to another]. (vi) With dn(/Omega1) obtained in the previous step, we solve Eqs. ( 19), (21), and ( 22) for the last time for all ν=1,...,M andm=m/primecorresponding to the position z/prime=/lscript∗ Bof the physical source describing the incident wave. The correspond-ing solution C nm/prime(uν,/Omega1) allows us to compute the Fourier transforms of position- and time-dependent transmission andreﬂection coefﬁcients T(q⊥,/Omega1) and R(q⊥,/Omega1), respectively (we temporarily reintroduce tildes above dimensionless vari-ables for clarity): T(q ⊥,/Omega1)=−D(z,/Omega1)∂ ∂zC(q⊥,z,z/prime=/lscript∗ B,/Omega1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=L =−˜CNm/prime(u,˜/Omega1) i˜/Omega1˜z0, (23) R(q⊥,/Omega1)=D(z,/Omega1)∂ ∂zC(q⊥,z,z/prime=/lscript∗ B,/Omega1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=0 =−˜C1m/prime(u,˜/Omega1) i˜/Omega1˜z0, (24) where we made use of boundary conditions ( 13) to express the derivative of Cat a boundary via its value. The above algorithm allows us to compute T(q⊥,/Omega1) and R(q⊥,/Omega1) for each /Omega1. The Fourier transforms of T(0,/Omega1) and R(0,/Omega1), for example, yield the total time-dependent trans- mission and reﬂection coefﬁcients studied in Ref. [ 21]. The position- and time-dependent intensity in transmission studiedin Ref. [ 7] and used to ﬁt experimental results (Sec. III B 4 of this paper) is given by a double Fourier transform T(ρ,t)=/integraldisplay ∞ −∞d/Omega1 2πe−i/Omega1t/integraldisplayd2q⊥ (2π)2e−iq⊥ρT(q⊥,/Omega1).(25) The dynamic coherent backscattering (CBS) peak R(θ,t) studied in Ref. [ 10] is obtained more simply as R(θ,t)=R(q⊥=k0sinθ,t) =/integraldisplay∞ −∞d/Omega1 2πe−i/Omega1tR(q⊥=k0sinθ,/Omega1).(26) As a ﬁnal, quite technical, but important remark, we de- scribe our way of performing integrations over /Omega1in Eqs. ( 25) and ( 26). These integrations can, of course, be performed by directly approximating integrals by sums and computingT(q ⊥,/Omega1) and R(q⊥,/Omega1) on a sufﬁciently ﬁne and extended 214201-4TRANSVERSE CONFINEMENT OF ULTRASOUND THROUGH … PHYSICAL REVIEW B 98, 214201 (2018) grid of /Omega1. However, this task turns out to be quite tedious because TandRare oscillating functions of /Omega1that decay very slowly as \|/Omega1\|increases. An accurate numerical integration then requires both using a small step in /Omega1and exploring a wide range of /Omega1, which is resource consuming. In order to circumvent this difﬁculty, we close the path of integration over/Omega1in the lower half of the complex plane and apply Cauchy’s theorem by noticing that T(q ⊥,/Omega1) andR(q⊥,/Omega1) have special points (poles or branch cuts) only on the imaginary axis.This allows us to deform the integration path to follow astraight line that is inﬁnitely close to the imaginary axis onthe right side of it from Im /Omega1=0t oI m /Omega1=− ∞ and then a symmetric line on the opposite side of the imaginary axisfrom Im /Omega1=− ∞ to Im/Omega1=0. For CBS intensity we obtain, for example, R(θ,t)=R(q ⊥=k0sinθ,t)=−i 2πlim /epsilon1→0+/integraldisplay∞ 0dαe−αt ×[R(q⊥=k0sinθ,/Omega1=−iα+/epsilon1) −R(q⊥=k0sinθ,/Omega1=−iα−/epsilon1)], (27) where we denoted Re /Omega1=±/epsilon1and Im /Omega1=−α. A similar expression is obtained for the transmitted intensity T(ρ,t) with an additional Fourier transform with respect to q⊥: T(ρ,t)=−i 2πlim /epsilon1→0+/integraldisplay∞ 0dαe−αt/integraldisplayd2q⊥ (2π)2e−iq⊥ρ ×[T(q⊥,/Omega1=−iα+/epsilon1) −T(q⊥,/Omega1=−iα−/epsilon1)]. (28) In the diffuse regime [ k/lscript/greatermuch(k/lscript)c],R(q⊥=k0sinθ,/Omega1= −iα±/epsilon1) is equal to a sum of Dirac delta functions repre- senting the so-called diffusion poles, and Eq. ( 27) is nothing else than the calculation of the integral in Eq. ( 26) via the theorem of residues. When Anderson localization effects startto come into play for k/lscriptapproaching ( k/lscript) cfrom above, the diffusion poles widen and develop into branch cuts. Finally,in the localized regime [ k/lscript < (k/lscript) c] the different branch cuts that were associated with different diffusion poles merge intoa single branch cut covering the whole imaginary axis. The advantage of Eqs. ( 27) and ( 28) with respect to Eq. ( 25) and ( 26) is obvious: The presence of the exponential function exp( −αt) under the integral limits the effective range of integration to small αfor the most interesting regime of long times t. This allows for an efﬁcient calculation of the long-time dynamics with a reasonable computational effort. Finally, for convenience with comparing theory to exper- imental data, the output of the SC calculations is scaled intime in units of the diffusion time, i.e., as t/τ D.F o ras l a b geometry, the diffusion time is related to the leakage rate ofenergy from the sample, and thus internal reﬂections play animportant role. The diffusion time is deﬁned as τ D≡L2 eff π2D, (29) where Leff≡L+2z0is the effective sample thickness. FIG. 1. (a) View of a monodisperse mesoglass sample (similar to sample H5). The sample surface has been lightly polished. (b) A polydisperse sample (sample L1). The difference in brazing between the two samples can be seen—the contacts between beads in sampleL1 are generally thicker than in sample H5. II. EXPERIMENTAL A. Mesoglass samples The ‘mesoglass’ samples examined here are solid disor- dered networks of spherical aluminum beads, similar to thosepreviously studied [ 7–12,38]. These samples are excellent media in which to observe Anderson localization, since theabsorption of ultrasound in aluminum is very weak, and thedisordered porous structure gives rise to very strong scatter-ing. The samples are slab shaped, with width much largerthan thickness. This geometry is ideal for our measurements;the relatively small thickness enables transmission measure-ments, while the large width avoids complication due toreﬂections from the side walls and facilitates the observationof how the wave energy spreads in the transverse direction, i.e., parallel to the ﬂat, wide faces of the sample. The samplesare created using a brazing process which has been describedin detail previously [ 12,38], resulting in a solid 3D sample in which the individual aluminum beads are joined togetherby small metal bonds (Fig. 1). Depending on several factors during the brazing process, the ‘strength’ of the brazing mayvary, resulting in thinner/thicker bond joints. This providesa mechanism for controlling the scattering strength in thesamples. The entire process is designed to ensure that thespatial distribution of the beads is as disordered as possible[12]. In this work, we study two types of brazed aluminum mesoglasses, which differ from each other in terms of beadsize distribution and brazing strength, as shown in Fig. 1.W e present experiments and analysis for an illustrative sampleof each type: Sample H5 is made from monodisperse alu-minum beads (bead diameter is 4 .11±0.03 mm) and has a circular slab shape with diameter 120 mm and thicknessL=14.5 mm. Sample L1 is made from polydisperse beads (mean bead radius is 3.93 mm with a 20% polydispersity), ismore strongly brazed than sample H5, and has a rectangularslab shape with cross section 230 ×250 mm 2and thickness L=25±2 mm. Ultrasound propagates through the samples via both lon- gitudinal and shear components, which become mixed due tothe scattering. Our experiments are carried out in large watertanks, with source, sample, and detector immersed in water,and thus only longitudinal waves can travel outside the sampleand be detected. As a result, there is signiﬁcant internalreﬂection. However, because the waves traveling inside thesample are incident on the boundary over a wide range of 214201-5L. A. COBUS et al. PHYSICAL REVIEW B 98, 214201 (2018) FIG. 2. Transverse conﬁnement experimental conﬁguration for a 3D slab mesoglass (the cross section of which is shown here). A beam is focused through a small aperture onto the surface of the sample (blue dashed lines, left). The spreading of the outputwave energy in the transverse direction (blue dashed line, right) is measured by translating a hydrophone parallel to the sample surface and acquiring the transmitted ﬁeld near the surface at manytransverse positions ρ. angles, the longitudinal ultrasonic waves outside the sample nonetheless include contributions from all polarizations insidethe sample. For each experiment, the mesoglass sample iswaterproofed, and the air in the pores between beads isevacuated. The sample remains at a low pressure (less than10% atmospheric pressure) for the entire duration of theexperiment, thus ensuring that the ultrasound propagates onlythrough the elastic network of beads. To assess the scattering strength in these samples, mea- surements of the average wave ﬁeld were performed, allowingresults for the phase velocity v p, the group velocity vg, and the scattering mean free path /lscriptof longitudinal ultrasonic waves to be obtained [ 39,40]. At intermediate frequencies, our data for vpand/lscriptlead to values of k/lscript∼1.7 and 2.7 for samples H5 and L1, respectively. These values of k/lscriptindicate very strong scattering and are close enough to the Ioffe-Regel criterionk/lscript∼1 to indicate that Anderson localization may be possible in these samples. B. Time- and position-resolved average intensity measurements To investigate the diffusion and localization of ultrasound in our samples, we measure the transmitted dynamic trans- verse intensity proﬁle . This quantity is a direct measure of how fast the wave energy from a point source spreads through thesample [ 7]. Our experiments measure the transmission of an ultrasonic pulse through the sample as a function of both timeand position. The experimental setup is shown in Fig. 2.O n the input side of the sample, a focusing ultrasonic transducerand cone-shaped aperture are used to produce a small point-like source on the sample surface. Transmission is measuredon the opposite side of the sample using a subwavelengthdiameter hydrophone. We denote the transverse position of thehydrophone at the sample surface, relative to the input point,as transverse distance ρ. Transmitted ﬁeld is measured at the on-axis point directly opposite the source ( ρ=0), as well as at several off-axis points ( ρ> 0). From the measured wave ﬁeld, the time and position dependent intensity, T(ρ,t), is determined (within an unimportant proportionality constant)by taking the square of the envelope of the ﬁeld. Becausetime-dependent intensities are measured at all points, they should be affected equally by absorption when compared atthe same propagation time. Thus, in the ratio of off-axis toon-axis intensity, absorption cancels [ 41]. We write this ratio, the normalized transverse intensity proﬁle, as T(ρ,t) T(0,t)=exp/parenleftBigg −ρ2 w2ρ(t)/parenrightBigg , (30) where the absorption-independent transverse width ,wρ(t), is deﬁned as w2 ρ(t) L2=−ρ2/L2 ln/bracketleftbigT(ρ,t) T(0,t)/bracketrightbig. (31) In the diffuse regime, the transverse intensity proﬁle is Gaus- sian [see Eq. ( 30)]; the transverse width is independent of transverse distance ρand increases linearly with time as w2(t)=4DBt[41]. Near the localization regime, however, w2 ρ(t) exhibits a slowing down with time due to the renormal- ization of diffusion, eventually saturating at long times in thelocalization regime [ 7,20]. Close to the localization transition, w 2 ρ(t) depends on ρ(although this dependence is weaker for largeL), meaning that the transverse intensity proﬁle deviates from a Gaussian shape. It is important to note that this ρ dependence means that the saturation of w2 ρ(t) in time cannot be simply explained by a time-dependent diffusivity D(t) (which would imply a Gaussian-shaped transverse intensityproﬁle with a ρ-independent width), but is a consequence of the position dependence of the diffusion coefﬁcient that is akey feature of Anderson localization in open systems [ 7]. Because the scattering in our mesoglass samples is so strong, the transmitted signals can be very weak, especially atlong times. This means that even very small spurious signalsor reﬂections can inﬂuence the data at long times, and it is thusimportant to ensure that only the signals that were transmittedthrough the sample are detected by the hydrophone. For eachexperiment, great care is taken to block any possible straysignals. A cone-shaped aperture (shown in Fig. 2) is placed at the focal point to block any side lobes from the sourcespot generated by the focusing transducer. A large bafﬂe,with an opening in its center for the sample, was placed inthe water tank between the source and detection side of thesample to block any signals from traveling around the sidesof the sample and eventually reaching the detector. Beforeeach experiment, the hole in the bafﬂe was blocked and thehydrophone scanned around the detection side of the tank, todetect any spurious signals from the source; if any were found,their travel path from source to detector was tracked down andblocked. These methods have been described in more detail inRefs. [ 11,12]. To improve statistics, for each input point, the transmitted ﬁeld was measured for four different ρvalues at thirteen different ( x,y) positions, (x,y)={(0,0),(±15,0),(±20,0),(±25,0), (0,±15),(0,±20),(0,±25)}mm, (32) 214201-6TRANSVERSE CONFINEMENT OF ULTRASOUND THROUGH … PHYSICAL REVIEW B 98, 214201 (2018) where xandydenote transverse positions of the detector in a plane parallel and close to the sample surface (typically a wavelength away), with ρ=/radicalbig x2+y2. Our experimental method is designed to facilitate ensemble averaging, which is especially important in the strong scat-tering or critical regimes where ﬂuctuations play an increas-ingly important role [ 42,43]. Conﬁgurational averaging was performed on the data obtained by translating the sample anddetermining the intensity at all sets of detector positions foreach source position. Typically 3025 source positions wererecorded for each experiment (a grid of 55 ×55 positions over the sample surface). The source positions were separated byabout one wavelength to maximize the number of statisticallyindependent intensity measurements that could be performedon a given sample and ensure that the averaging was notspoiled by spatial correlations [ 8]. To reduce the effect of electronic noise, each measurement of the acquired wave ﬁeldwas repeated many times and averaged together; typically,each signal was averaged 4000–5000 times. As we would liketo consider only the multiply scattered signals, any contri-butions from coherent pulse transport were removed by sub-tracting the average ﬁeld from each individual ﬁeld, i.e., wedetermine ψ MS(t,ρ in,ρout)=ψ(t,ρ in,ρout)−/angbracketleftψ(t,ρ in,ρout)/angbracketrightρin(33) and use ψMS(t,ρ in,ρout) to obtain the multiply scattered intensities. III. RESULTS, ANALYSIS, AND DISCUSSION A. Amplitude transmission coefﬁcient To quantify the frequency dependence of transmitted ultra- sound through our mesoglasses, we calculate the amplitudetransmission coefﬁcient T amp(f) from the time-dependent transmitted ﬁeld ψ(t,ρ=0). A Fourier transform converts ψ(t,ρ=0) into the frequency domain, resulting in /Psi1(f). The amplitude of /Psi1(f) is found, and then conﬁgurational averaging is performed on \|/Psi1(f)\|as described in Sec. II B. The same process, without the conﬁgurational average, isperformed on the reference ﬁeld—the input pulse travelingthrough water to the detector. The normalized amplitudetransmission coefﬁcient is then calculated as: T amp(f)=/angbracketleft\|/Psi1(f)transmitted \|/angbracketright \|/Psi1(f)reference \|. (34) Figure 3(a) shows Tamp(f) measured this way for sample L1 using a focused transducer source. It is important to empha-size the difference between this conﬁgurational average of theabsolute value of the ﬁeld (in which phase is ignored), and theaverage ﬁeld (in which phase coherence plays a signiﬁcantrole, and which gives the effective medium properties). The amplitude transmission coefﬁcient can also be mea- sured using a plane wave source, approximated by placingthe sample in the far ﬁeld of a ﬂat disk-shaped emittingtransducer. In this case, the transmitted ﬁeld ψ(t,x,y ) is mea- sured with the hydrophone over a large number of positions (x,y) in the speckle pattern [ ∼11 500 positions for the results shown in Fig. 3(b)],\|/Psi1(f,x,y )\|is averaged over all positions (x,y), and the normalized amplitude transmission coefﬁcient T amp(f) is calculated using Eq. ( 34) [Fig. 3(b)]. Note thatFIG. 3. Amplitude transmission coefﬁcient Tamp(f) as a function of frequency. Data shown are (a) for sample L1, taken using a point source, and (b) for sample H5, taken using a plane-wave source. Red arrows indicate the resonance frequencies of single, unbrazed4.11 mm aluminum beads. Vertical gray hatched bars indicate the frequency ranges of interest for transverse conﬁnement analysis. although the overall amplitude of Tamp(f) changes depending on whether the input is a plane-wave or a point source (dueto the normalization of T amp(f) which does not account for the ﬁnite lateral width of the input beams), the frequencydependence of T amp(f), which is the desired quantity for guiding the interpretation of the experimental results, does notdepend on the source used. In Fig. 3, the resonance frequencies of single, unbrazed 4.11 mm aluminum beads are shown with red arrows. Atlong wavelengths ( λ/greatermuchd, where dis the bead diameter), the beads move as a whole, and one might expect the vi-brational characteristics to be described by a Debye modelwith effective medium parameters [ 44]. By analogy with a mass-spring system, the beads act as the masses, and small‘necks’ connecting them act as the springs. The ﬁrst dip intransmission around 500 kHz corresponds to the upper cutofffrequency for these vibrational modes, which consist only oftranslations and rotations of the beads. Above the upper cutofffor this long-wavelength regime, when the wavelength be-comes comparable with the bead diameter, internal resonancesof the beads can be excited, and these bead resonances coupletogether to form pass bands near and above the individualbead resonant frequencies (Fig. 3). These pass bands are thus elastic-wave analogues of the “tight-binding” regimefor electrons; in the electronic case, tight-binding models ofAnderson localization have been extensively used, startingwith Anderson’s initial paper [ 1]. The width of each pass band is ﬁnite since the pass bands do not overlap when thecoupling between the beads (determined by the strength ofthe ‘necks’ between them) is weak. These coupled resonancesare the only mechanism through which ultrasound can prop-agate through the mesoglass in this part of the intermediatefrequency regime. Correspondingly, the substantial dips intransmission seen in Fig. 3are due to the absence of such coupled resonances and are not related to Bragg effects whichwould only be expected in media with long-range order, whichis not present here. For sample L1, the presence of smaller 214201-7L. A. COBUS et al. PHYSICAL REVIEW B 98, 214201 (2018) bead sizes has shifted the transmission dips in Tamp(f)t o higher frequencies and has lessened their depth compared tothe monodisperse sample H5 [ 7,8,10,45]. These ‘pseudogaps’ for L1 are probably also shallower due to slightly strongerbrazing between individual beads (Fig. 1)[46]. In this work, we focus our investigation of Anderson localization on thebehavior at frequencies near the transmission dips seen in thetwo samples around 1.2 MHz, as indicated by the gray hatchedbars in Fig. 3. B. Time-, position-, and frequency-resolved average intensity 1. Frequency ﬁltering To differentiate precisely between the diffuse, criti- cal, and localized regimes, it is desirable to examine thebehavior of the dynamic transverse proﬁle as the fre-quency is changed in very small increments. Frequency-dependent results were obtained by ﬁrst digitally ﬁltering themeasured wave ﬁelds over a narrow frequency band, by takingthe fast Fourier transform of ψ MS(t,ρ in,ρout), multiplying the resulting (frequency-domain) signal by a Gaussian of theform exp/bracketleftbig −(f−f 0)2/w2 f/bracketrightbig , (35) where f0is the central frequency of the ﬁlter and wfis the width, and calculating the inverse Fourier transform of the re-sulting product. By varying the central frequency of theGaussian window, intensity proﬁles can then be determinedfor each frequency. The width w fwas chosen with the goal of performing sufﬁciently narrow frequency ﬁltering to resolvethe change in behavior with frequency, without broadeningthe time-dependent features too much. For the calculation ofT(ρ,t), a typical width of w f∼15 kHz was used. Because the average transmitted intensity varies greatly with frequency, the impact of this dependence on the fre-quency ﬁltering procedure needs to be assessed. This effectis illustrated in Fig. 4, which shows that, after having been ﬁltered in frequency, the data may not be centered on f 0,t h e nominal central frequency of the ﬁlter. In other words, this“frequency-pulling” effect means that when the ﬁlter functionof Eq. ( 35) is applied to a region where intensity changes rapidly with frequency, the resulting quantity, T(x,y,t ), is heavily weighted by data to one side of the central frequency.To account for this shift, the frequency-dependent transmittedintensity is multiplied by the ﬁlter function, and the meanfrequency of the ﬁltered data f mis calculated from the ﬁrst moment of this product. The mean frequency fmis used to label each set of frequency-ﬁltered data instead of f0, which may not accurately represent the frequency content of thedata. After frequency ﬁltering, the procedure to determine the time-dependent intensity T(x,y,t ) is the same as indicated above, namely T(x,y,t ) is found by taking the square of the envelope of the time-dependent wave ﬁelds. Then, ensembleaveraging is performed by averaging the ﬁltered intensity overallN=3025 source positions. The standard deviation in this average is also calculated and divided by√ Nto give an estimate of the experimental uncertainty in the mean intensity[47]. The transmitted intensity proﬁles measured at the same transverse distance from the source position ρ=/radicalbig x2+y2FIG. 4. The frequency-pulling effect on a bandwidth-limited sig- nal caused by the frequency dependence of the average transmitted intensity. The average transmitted intensity (black line) is calculatedsimilarly to the amplitude transmission coefﬁcient of Fig. 3but from the average intensity instead of amplitude. The mean frequency of the ﬁltered data f mis shifted from the central (nominal) frequency of the ﬁlter function f0. [Eq. ( 32)] are averaged together, resulting in average inten- sity proﬁles T(ρ,t). Finally, the noise contribution to each averaged T(ρ,t) is estimated from the intensity level of the pretrigger part of the signal (the signal recorded before theinput pulse arrives at the sample input surface, i.e., for t<0.). This noise level is subtracted from the average time-dependentintensity, and w 2 ρ(t) is then calculated using Eq. ( 31). 2. Transverse conﬁnement data The spreading of wave energy in the sample is charac- terized by the time- and position-dependent transverse widthw 2 ρ(t) [see Eqs. ( 30) and ( 31)]. In the diffuse regime, our experimentally measured w2 ρ(t) and transmitted intensity pro- ﬁlesT(ρ,t) are well described by predictions from the dif- fusion approximation and may be ﬁt with diffusion theory toascertain parameters such as D B[12,16,41]. An example of such ﬁtting is shown in Fig. 5, where data at the low frequency of 250 kHz are reported for sample L1. The linear timedependence of the width squared, and the observation that thewidth squared is independent of transverse distance ρ, both clearly indicate that the transport behavior at low frequenciesin this sample is diffusive. The slight deviation from linearityinw 2 ρ(t) at early times is due to the ﬁnite bandwidth of these frequency-ﬁltered data (35 kHz), as well as to the ﬁnite areaof the source and detection spots. These ﬁnite spot sizes alsohave the effect of adding a small constant offset to w 2 ρ(t). As emphasized in Ref. [ 41], such a measurement of the transverse width provides a direct measurement of the Boltzmann diffu-sion coefﬁcient D Bwithout complications due to absorption and boundary reﬂections. The excellent ﬁt of diffusion theoryto the experimental time-of-ﬂight intensity proﬁle T(ρ,t) yields additional information about the transport mean freepath and the absorption time [ 41]. At higher frequencies, however, the data deviate from the behavior predicted by the diffusion approximation: Notably, 214201-8TRANSVERSE CONFINEMENT OF ULTRASOUND THROUGH … PHYSICAL REVIEW B 98, 214201 (2018) 1×10−3 1×10−4 1×10−5 FIG. 5. Experimental data (symbols) and ﬁts with diffusion the- ory (solid lines), for sample L1 at fm=250 kHz. The time de- pendence of the transmitted intensity is shown in (a). The time dependence of w2 ρ(t) is shown in (b). The ﬁtting of w2 ρ(t) with dif- fusion theory gives a measure of the Boltzmann diffusion coefﬁcient DB=1.45±0.02 mm2/μs. This value of DBalso allows good ﬁts to the time-of-ﬂight proﬁles T(ρ,t) to be obtained, as shown in (a). These ﬁts to T(ρ,t) give estimates of the transport mean free path /lscript∗ B≈8m ma n da b s o r p t i o nt i m e τA≈560μs. For clarity, error bars are only shown for every third data point. w2 ρ(t) no longer increases linearly with time but increases more slowly as time progresses, and neither the width squarednor the associated T(ρ,t) curves can be ﬁt with diffusion theory (c.f. Ref. [ 7]). In the following, we show the evolution of this behavior as a function of frequency, which is a controlparameter for selecting the disorder strength in a single sam-ple. Typical experimental results are shown for both samplesin Fig. 6(symbols). At these frequencies there are clear devi- ations from conventional diffusion, as the spreading of waveenergy is slower than would be expected if the behavior werediffusive, and the intensity may become conﬁned spatiallyas time increases. For sample H5, w 2 ρ(t) even saturates at long times for some frequencies, implying that the transversespreading of the intensity has halted altogether and suggestingthat Anderson localization may have occurred. To determinewhether or not this is the case, and to be able to discriminatebetween subdiffuse and localized regimes, we ﬁt our data withthe self-consistent theory of localization. 3. Self-consistent theory calculations As described in Sec. IB, our SC theory gives as output the temporally and spatially dependent transmitted intensityT(ρ,t), from which the associated transverse width w 2 ρ(t) canTABLE I. Acoustic parameters for mesoglass samples in the frequency ranges delineated by the gray hatched bars in Fig. 3. Sample L1 Sample H5 L(mm) 25 14.5 vp(mm/μs) 2.8 2.8 vg(mm/μs) 2.7 2.9 /lscript(mm) 1.1 0.76 k/lscript 2.7 1.7 Rint 0.67 0.67 /lscript∗ B(mm) 4 6 τA(μs) 170–900 100–300 be directly calculated. These SC theory calculations require a number of input parameters, many of which are ﬁxed, asthey have been determined from measurements of the averagewave ﬁeld. These ﬁxed input parameters are /lscript,k/lscript, andR int. For simplicity, we use a representative value for each of theseparameters in all SC theory calculations for each sample, asdetermined by an average value appropriate for the frequencyranges of interest (see the gray hatched bars in Fig. 3). Table Ishows values for these average scattering and transport parameters. Our SC calculations do not depend strongly on thevalues of /lscriptork/lscriptover the range of experimental values used to determine the averages reported in Table I. The internal reﬂection coefﬁcient R intwas estimated using a method based on the work of Refs. [ 36,41,48–50], and its impact on the data analysis is discussed in Appendix. In addition to these parameters determined from the av- erage ﬁeld, Table Ialso includes values for the parameters L,/lscript∗ B, and τA: the sample thickness Lwas measured with calipers and averaged over several sections of the sample, theBoltzmann transport mean free path l ∗ Bwas estimated from SC theory ﬁtting as described in Refs. [ 11,12], and the values of the absorption time τAresult directly from ﬁts of SC theory to the time-of-ﬂight proﬁles T(ρ,t) at the different frequencies of interest (see Appendix). The ﬁnal and most important parameter that must be speciﬁed to calculate T(ρ,t) andw2 ρ(t) using the SC theory isL/ξ (orL/ζ). As indicated in Secs. IAand III B 4 this parameter determines how close the predicted behavior is tothe localization transition, where L/ξ=L/ζ=0. The ﬁtting procedure to determine this parameter for a given sample at agiven frequency is described in the next section. 4. Comparison of data with self-consistent theory The goal in comparing our experimental data with theory is the determination of the localization (correlation) length ξ(ζ) as a function of frequency. This is achieved by ﬁtting eachset of frequency-ﬁltered data with many sets of SC theorypredictions, each calculated for a different ξorζvalue. The best ﬁt is found by minimizing the reduced chi-squared χ 2 red. In this way, each set of data, denoted by its unique centralfrequency f m, is associated with the theory set that ﬁts it best, denoted by its unique value of ξ(ζ). This process is described in detail in Appendix. Figure 6shows representative ﬁtting results for a few frequencies. The predictions of the theory set that best ﬁts the 214201-9L. A. COBUS et al. PHYSICAL REVIEW B 98, 214201 (2018) 1×10−4 1×10−5 1×10−6 1×10−7 1×10−8 1×10−7 1×10−8 1×10−9 1×10−10 1×10−11 FIG. 6. Experimental data (symbols) and ﬁts with SC theory (solid lines), for sample H5 [(a),(b)] and sample L1 [(c),(d)]. For some data points, the error bars are smaller than the symbols. Sample H5: The time dependence of the transmitted intensity at fm=1.0938 MHz is shown in (a). The time dependence of w2 ρ(t)a tfm=1.0938 MHz (upper three curves) and at fm=1.1094 MHz (lower three curves) is shown in (b). Forfm=1.0938 MHz, the best ﬁt result was for correlation length ζ=3.63 cm (diffuse regime). For fm=1.1094 MHz, the best ﬁt result was for localization length ξ=5.27 cm (localization regime). In the localization regime, w2 ρ(t) increases more slowly and saturates at long times, compared to the diffuse regime. The open symbols show data points that were not included in the ﬁts, due to a measurement artifact that is visible in the data at fm=1.1094 MHz and is discussed in Appendix. For clarity, only every 80th data point is shown for data in (a) and (b). Sample L1: The time dependence of the transmitted intensity at fm=1.1780 MHz is shown in (c). The time dependence of w2 ρ(t)a t fm=1.1780 MHz (upper two curves) and at fm=1.2175 MHz (lower two curves) is shown in (d). Data for only two (of three) ρvalues are shown, as the w2 ρ(t) curves for different ρvalues essentially overlap in this ﬁgure [the large sample thickness of L1 substantially weakens the ρdependence of w2 ρ(t); see Eq. ( 31) and following discussion]. For fm=1.1780 MHz, the best ﬁt result for correlation length is ζ=3.851 cm (diffuse regime). For fm=1.2175 MHz, the best ﬁt result for localization length is ξ=7.866 cm (localization regime) (d). For clarity, only every 100th data point is shown for data in (c) and (d). data are shown by the solid lines. For both samples, H5 (top plots) and L1 (bottom plots), the data are well ﬁt by the theoryat all times. (Note that the w 2 ρ(t) curves do not reach zero at t=0 due to the effect of the narrow frequency ﬁlter width.) Figure 7shows best theory ﬁts for a single value of ρat several different frequencies. This ﬁgure shows the evolutionofw 2 ρ(t) as the frequency is increased, starting from simple diffuse behavior, where w2(t) increases linearly with time, passing through a subdiffusive regime where w2 ρ(t) increases more slowly, reaching the critical frequency at the mobilityedge, where w 2 ρ(t) saturates in the limit as t→∞ , and ﬁnally crossing into the localized regime, where w2 ρ(t) saturatesat a constant value in the observation time window. Thus, this ﬁgure illustrates how w2 ρ(t) reveals the differences in wave transport that are encountered as an Anderson transitionfor classical waves is approached and crossed in a stronglydisordered medium, providing clear signatures of whether ornot, and when, Anderson localization occurs. Furthermore, bydetermining the best-ﬁt value of ξorζfor each frequency, an estimate of ξ(f) andζ(f) can be obtained, and thus the frequency/ies at which the mobility edge occurs ( L/ξ=0) can be identiﬁed. Results for ξ(f) andζ(f) for sample H5 are shown in Fig. 8, where a mobility edge can be identiﬁed atf m=1.101 MHz. For frequencies above fm=1.115 MHz 214201-10TRANSVERSE CONFINEMENT OF ULTRASOUND THROUGH … PHYSICAL REVIEW B 98, 214201 (2018) FIG. 7. The time evolution of w2 ρ(t) for sample H5, for one transverse distance ρ=20 mm. Fits of self-consistent theory (solid lines) are shown with the data (symbols). Results for ﬁve repre-sentative frequencies are shown. At 1.024 MHz (brown diamonds), transport is almost entirely diffusive; the slope of the linear ﬁt gives D=0.64 mm 2/μs. As frequency is increased, subdiffuse behavior is observed (magenta downward pointing triangles, green upward triangles), one then arrives at the mobility edge (blue squares), andﬁnally the localization regime is reached (red circles). Best-ﬁt results from ﬁtting the data with SC theory give the values of the correlation (localization) lengths (legend). (deep inside the transmission dip), the level of transmit- ted signal was not sufﬁciently above the noise for reliablemeasurements, and thus only one mobility edge could beidentiﬁed for sample H5. For sample L1, two mobility edgesare identiﬁable at the critical points where ξdiverges, and the FIG. 8. Results from the comparison of self-consistent theory to data, for sample H5 near the mobility edge. The position of themobility edge is identiﬁed as the frequency f mfor which the data is best ﬁt by the SC theory for L/ξ=0 (dotted lines). Top plot: the ratio of sample thickness to localization (correlation) length, withL/ξ(f m) represented by red solid symbols and L/ζ(fm) by blue open symbols. Bottom plot: the localization (correlation) length ξ(fm) (red solid symbols) and ζ(fm) (blue open symbols).FIG. 9. Localization (correlation) length ξ(ζ), as a function of frequency for sample L1. Two mobility edges are identiﬁed ( fm= 1.199 MHz and fm=1.243 MHz), where ζandξdiverge. Between the mobility edges there exists a localization regime. frequency range between them is identiﬁed as the localization regime (mobility gap with L/ξ > 0) (Fig. 9). Whereas only one mobility edge could be identiﬁed for sample H5, forsample L1 a measurement of ξall the way through the mobility gap was obtained. This was possible because sampleL1 is polydisperse, with stronger bonds between beads, andthus more signal is transmitted through the sample in thetransmission dips (see Fig. 3) than through sample H5. 5. Discussion Having identiﬁed the localization regime and mobility edge(s) for each sample, we can revisit Figs. 6and 7.F o r frequencies just below the mobility edge but not yet in thelocalization regime, the clear deviations from conventionaldiffusive behavior are seen, indicating subdiffusion when the renormalization of the diffusion coefﬁcient due to disorderhampers the transverse spread of waves but does not blockit entirely. As frequency is increased into the localizationregime, the increase of w 2 ρ(t) with time is initially slower and eventually saturates at long times. Figure 7shows w2 ρ(t)f o r ﬁve frequencies near the low-frequency edge of the dip intransmission just below 1.2 MHz. At frequencies where thetransmission dip becomes deeper, w 2 ρ(t) approaches satura- tion at earlier times, and the data are better ﬁt with theoreticalpredictions for larger L/ξ values. For sample L1, which is thicker, the range of times experimentally available is notlong enough to show a clear saturation of w 2 ρ(t)( a ss h o w ni n Fig. 6). However, since for each frequency, the best-ﬁt value of the theory to the data gives a measure of the localizationlength ξ(orζif outside the localization regime), we are still able to determine whether the localization scenario isconsistent with our data. In general, it is important to note that the existence of a transmission dip (Fig. 3), which is linked to a reduction in the number of coupled resonant modes when the couplingbetween bead resonances is weak, does not necessarilyimply the existence of a mobility gap, which is caused bythe interplay between interference and disorder. While it istrue that the density of states becomes smaller as the upperedge of a pass band is approached [ 46], and that all mobility edges shown in this work do coincide with the edges of atransmission dip, such a reduction in the density of states maymake localization “easier” to realize but should not be used onits own as an indication of localization. It is also worth noting 214201-11L. A. COBUS et al. PHYSICAL REVIEW B 98, 214201 (2018) FIG. 10. The frequency dependencies of the localization and correlation lengths ξandζnear the mobility edge, from ﬁts of data from sample H5 to the self-consistent theory. Critical frequency fc was found from the ﬁts to be 1.1011 MHz. The power law of ν=1 produced by the self-consistent theory is shown (black dotted line). Apower law with ν/similarequal0.95 provides a better ﬁt to the data (not shown). that the original evidence of Anderson localization of elastic waves in mesoglasses was found at frequencies outside thetransmission dips for these samples [ 7]. We also note that over the entire frequency range studied in this work, our estimatesof scattering strength k/lscriptare consistent with the Ioffe-Regel criterion for localization, which is often interpreted as k/lscript∼1. However, the localization regime only exists in a small sectionof this spectral region. Thus, a careful and thorough com-parison of theory and experiment is essential for determiningwhether signatures of localization are indeed present. Finally, Figs. 8and9imply that the critical exponent of the localization transition νmay be estimated from our results, since near a mobility edge f c, the localization (correlation) length is expected to evolve with fasξ(f)∝\|f−fc\|−ν. Our measured ξis shown as a function of \|f−fc\|in Fig. 10 near the mobility edge at fc=1.1011 MHz for sample H5. The increase of ξnearfcappears roughly linear, correspond- ing to a value of ν≈1 (shown for comparison in Fig. 10 as a dashed line). However, as discussed in Sec. IB,S C theory itself predicts that ν=1. One might thus argue that this mean-ﬁeld value is ‘built-in’ and that therefore our resultsfor the frequency dependence of ξdo not give an indepen- dent measurement of the critical exponent. Nonetheless, thisoutcome (Fig. 10)does give additional evidence that our data are consistent with SC theory predictions and lends additionalsupport to our determination of the locations of mobilityedges, which are independent of the exact value of the criticalexponent. IV . CONCLUSIONS The measurement of the transverse spreading of ultrasound in 3D slab mesoglasses is an excellent method to observethe dynamics of Anderson localization. In particular, in thiswork we have shown that the width of the transmitted dy-namic transverse intensity proﬁle, w ρ(t), is a sensitive and absorption-independent quantity with which to investigate lo-calization. The transverse width was measured as a function oftime and frequency for two different samples. At frequenciesapproaching the edges of the dips in transmission, we haveobserved that w 2 ρ(t) increases less rapidly than linearly with time, tending towards a saturation at long times at frequenciesdeeper into the transmission dips. This observation agreeswith the intuitive expectation that the spreading of waveenergy will slow down and eventually halt in the localizationregime. We were able to model the slowing of the spread ofacoustic energy using the self-consistent theory of localiza-tion. Our results show that our experimental measurementsagree with the theoretically predicted behavior for Andersonlocalization. The self-consistent theory of localization can provide a detailed quantitative model for our observations. This enabledus to extract several transport parameters of our mesoglasssamples. Numerical solutions of the SC theory were obtainedand compared to our measurements of the transverse inten-sity proﬁles. The comparison of theory and experiment wasperformed in a careful and systematic way, which enabledus to identify the critical frequency at which the mobilityedge occurs, f c. We were able to precisely identify fcfor both samples: For our thinner monodisperse sample, fc= 1.1011 MHz while for our thicker, polydisperse sample an entire mobility gap was observed, consisting of a localizationregime bounded by two mobility edges at f c=1.199 MHz andfc=1.243 MHz. The comparison of our data with predictions from SC theory is an important strength of thiswork, as it enabled not only the conﬁrmation of the existenceof localization regimes in both samples but also a completemeasurement of the correlation and localization lengths as afunction of frequency as the mobility edges were crossed intothe localization regimes. ACKNOWLEDGMENTS This work was supported by the Natural Sciences and En- gineering Research Council of Canada (NSERC) [DiscoveryGrants No. RGPIN/9037-2011 and No. RGPIN/6042-2016],the Canada Foundation for Innovation and the Manitoba Re-search and Innovation Fund (CFI/MRIF, LOF Project 23523),the Agence Nationale de la Recherche under Grant No.ANR-14-CE26-0032 LOVE, and the Centre National de laRecherche Scientiﬁque (CNRS) France-Canada PICS projectUltra-ALT. APPENDIX: DETAILS OF THE COMPARISON OF SELF-CONSISTENT THEORY WITH EXPERIMENTAL DATA In this Appendix, the procedures that were followed to ﬁt predictions of the self consistent theory to the measured trans-verse widths and time-of-ﬂight proﬁles are fully described.To ﬁt one data set with one set of theoretical predictions,we perform a least-squares comparison between experimentalw 2 ρ(t) curves and SC theory predictions. Fits are weighted by the experimental uncertainties. The diffusion time τD[see Eq. ( 29)] is a free ﬁt parameter and is sensitive to the reﬂection coefﬁcient; however, we have checked that the uncertainty 214201-12TRANSVERSE CONFINEMENT OF ULTRASOUND THROUGH … PHYSICAL REVIEW B 98, 214201 (2018) in our estimate of Rintdoes not pose a problem for the measurement of ξorζ. To check the reliability in the ﬁtting process, we also ﬁt the intensity proﬁles T(ρ,t) with theoretical predictions. Thus, each ﬁt is a global ﬁt of both w2 ρ(t) and its associated T(ρ,t). Since there are four different ρvalues, this yields three w2 ρ(t) and four T(ρ,t) curves which are ﬁt simultaneously with the same ﬁt parameters, weighted by experimental uncertainties.Two additional ﬁt parameters are needed only for T(ρ,t); a multiplicative amplitude scaling factor with no physicalsigniﬁcance, and the absorption time τ Awhich is included by multiplying the theoretical predictions of T(ρ,t)b ya n additional factor of exp( −t/τA) and which, as discussed, cancels out in the ratio used to calculate w2 ρ(t) and thus does not affect the w2 ρ(t) data. It is also worth noting several technical but important considerations for the comparison of theory with data. Atearly times tthe self-consistent theory calculations contain known inaccuracies which become worse for larger ρvalues. These early times are not included in the ﬁtting procedure,and thus the range of times used for ﬁtting is slightly differentfor different ρvalues. These ranges can be clearly seen in Figs. 6(a) and 6(b) where the theory curves begin at the earliest times used in the ﬁtting. Late times for which the noiseand ﬂuctuations in the data are large are also not included (thelatest time in the ﬁts was 275 μs for sample H5 and 400 μs for sample L1). Data for sample H5 suffer from an artifact in the acquired signals at some frequencies; just after 200 μs, the acoustic signal from the generating transducer has reﬂected from thefront surface of the sample and traveled back to the generatingtransducer. This signal induced a small voltage in the piezo-electric generator, which was picked up electromagneticallyby the sensitive detection electronics. The narrow-bandwidthfrequency ﬁltering applied to the data broadens this (originallybrief) signal in time, so a large range of times is affected bythis signal. While the artifact is only visible when the signalsare small (near the transmission dip), data for this range oftimes were not included for the ﬁtting at any frequencies forconsistency. Data from sample L1 did not suffer from thisartifact. There is a non-negligible effect on the experimental data caused by frequency ﬁltering (Sec. II B) that must be compen- sated for in the theory calculations of T(ρ,t). The ﬁltering op- eration is equivalent to the convolution of the time-dependentintensity with a function of the form exp[ −2(πw ft)2], which has the effect of ‘smearing out’ the time-domain signals (see,e.g., early times of Fig. 6). To properly account for this effect, our calculations for T(ρ,t) are convolved with this function before they are used to ﬁt our data. Estimation of ξand ζfrom SC theory ﬁtting As outlined in Sec. III B 4 ,ξ(f) andζ(f) are determined by comparing all sets of frequency-ﬁltered data (each with aunique central frequency f m) with all sets of calculated SC predictions (each with a unique value of ξorζ). For each ﬁt, the reduced chi-squared is recorded; the best ﬁt is the one withthe smallest χ 2 red. By ﬁltering the data in frequency with a very ﬁne resolution (for many, closely spaced, central frequenciesFIG. 11. The reduced chi-squared from ﬁtting SC theory to data for three representative frequencies, χ2 red, is shown as a function of the ratio of sample thickness to localization (correlation) length, L/ξ (L/ζ). For each frequency, the most probable value of ξor ζand its associated uncertainty are found via a parabolic ﬁt near the minimum point [Eqs. ( A1)a n d( A2)]. The three representative frequencies are fm=1.1780 MHz (diffuse regime, blue circles), fm=1.1968 MHz (very close to the mobility edge, green squares), andfm=1.2175 MHz (localized regime, red triangles). fm) and calculating many sets of theory over a wide range of closely spaced ξandζvalues, it is possible to estimate ξ(fm) andζ(fm) with precision. However, this method requires a great deal of time-intensive data processing and ﬁtting. Amore efﬁcient approach is to estimate the most probable value ofξorζfor each f m. To do this, we consider the reduced chi-squared results from the least-squares comparison of datawith theory. Figure 11shows χ 2 redfor three different sets of data (each at a different frequency fm)f o rs a m p l eL 1 . For example, in the localization regime, the best ﬁt, i.e., themost probable value ξ best, corresponds to the minimum of the function χ2 red∝(ξ−ξbest)2/σ2, (A1) and, for a sufﬁciently large data set, the uncertainty in the most probable value is given by the curvature of this function nearits minimum [ 51] σ 2=2/parenleftbigg∂2χ2 red ∂ξ2/parenrightbigg−1 . (A2) Similar expressions in terms of ζapply in the diffuse regime. This formalism can be applied to our results to estimate themost probable value of ξorζfor each frequency, based on the available data and theory, by ﬁtting a parabola to a few pointsaround the minimum value of χ 2 red[51] (Fig. 11). This method does not require the data to be ﬁltered with closely spacedvalues of f m, reducing the required calculation time (for the frequency ﬁltering of the data and ﬁtting theory to experiment)and amount of ﬁltered data. In addition, the parabola-ﬁttingtechnique gives an estimate of the uncertainty for the resultingestimates of ξ(ζ). 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PhysRevB.95.125115.pdf	PHYSICAL REVIEW B 95, 125115 (2017) Semilocal exchange hole with an application to range-separated density functionals Jianmin Tao,1,Ireneusz W. Bulik,2and Gustavo E. Scuseria2 1Department of Physics, Temple University, Philadelphia, Pennsylvania 19122-1801, USA 2Department of Chemistry and Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA (Received 14 November 2016; revised manuscript received 20 February 2017; published 13 March 2017) The exchange-correlation hole is a central concept in density functional theory. It not only provides justiﬁcation for an exchange-correlation energy functional but also serves as a local ingredient for nonlocal range-separateddensity functionals. However, due to the nonlocal nature, modeling the conventional exact exchange hole presentsa great challenge to density functional theory. In this work, we propose a semilocal exchange hole underlyingthe Tao-Perdew-Staroverov-Scuseria (TPSS) meta-generalized gradient approximation functional. Our model isdistinct from previous ones not only at small separation between an electron and the hole around the electronbut also in the way it interpolates between rapidly varying and slowly varying densities. Here the interpolation isdetermined by the wave-vector analysis on the inﬁnite-barrier model for a jellium surface. Numerical tests showthat our exchange-hole model mimics the conventional exact one quite well for atoms. As a simple application,we apply the hole model to construct a TPSS-based range-separated functional. We ﬁnd that this range-separatedfunctional can substantially improve the band gaps and barrier heights of TPSS, without losing much accuracyfor atomization energies. DOI: 10.1103/PhysRevB.95.125115 I. INTRODUCTION Kohn-Sham density functional theory (DFT) [ 1–3]i sa mainstream electronic structure theory due to its usefulaccuracy and high computational efﬁciency. Formally, it is anexact theory, but in practice the exchange-correlation energycomponent, which accounts for all many-body effects, hasto be approximated as a functional of the electron density.Development of exchange-correlation energy functionals fora wide class of problems with high accuracy has been thecentral task of DFT. Many density functionals have beenproposed [ 4–24], and some of them have achieved remarkable accuracy in condensed-matter physics or quantum chemistryor both. According to their local ingredients, density functionals can be classiﬁed into two broad categories: semilocal andnonlocal. Semilocal functionals make use of the local electrondensity, density derivatives, and/or the orbital kinetic energydensity as inputs, such as the local spin-density approxi-mation (LSDA) [ 25,26], generalized gradient approximation (GGA) [ 10,27,28], and meta-GGA [ 11,16,17,20,24]. Due to the simplicity in theoretical construction and numericalimplementation, as well as relatively low computational cost,semilocal functionals have been widely used in electronicstructure calculations [ 29–32]. Indeed, semilocal DFT can give a quick and often accurate prediction of many properties suchas enthalpies of formation or atomization energies [ 23,33–38], bond lengths [ 39,40], lattice constants [ 40–44], cohesive energies [ 45], etc. Semilocal DFT has achieved a high level of sophistication and practical success for many problems in chemistry, physics,and materials science, but it encounters difﬁculty in theprediction of reaction barrier heights, band gaps, chargetransfer, and excitation energies. Accurate description of theseproperties requires electronic nonlocality [ 46], which is absent Corresponding author: jianmin.tao@temple.edu; http://www.sas. upenn.edu/ ∼jianmint/in semilocal functionals. Nonlocality can be accounted for via mixing some amount of exact exchange into a semilocalDFT. This leads to the development of hybrid [ 8,13,33,47] and range-separated functionals [ 14,48]. The former involve the exact exchange energy or energy density, while the latterinvolve the exact and approximate semilocal exchange holes. There are three ways to approximate an exchange hole. It can be constructed from paradigm densities in which theexact exchange hole is known, such as the slowly varying density [ 4,49,50] (the paradigm of condensed-matter physics) and the one-electron density [ 7] (the paradigm of quantum chemistry). It can also be constructed from a density functionalwith the reverse-engineering approach [ 51–53]. A physically more appealing approach to approximate an exchange holeis from the density-matrix expansion [ 24]. Among the three general methods, the reverse-engineering approach is most frequently used. However, a semilocal exchange hole based on the reverse-engineering approach may not be in the gauge ofthe conventional exchange hole because a semilocal exchangeenergy density is usually not in the conventional gauge [ 54]. In the construction of a semilocal exchange hole, one mustimpose certain exact constraints on a hole to recover the under-lying exchange energy density, which is usually not in the same gauge of the conventional exchange energy density, due to the integration by parts performed in the construction of semilocalDFT. Examples include the Perdew-Burke-Ernzerhof (PBE)GGA [ 49,51] and Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA [ 52,53] exchange holes. Many range-separated functionals have been proposed [ 14,55–58], and some of them have obtained great popularity in electronic structure calculations. The exchange hole in the conventional gauge is of special interest. For example, the subsystem functional scheme pro-posed by Mattsson and coworkers [ 15,59–61] was developed from the conventional exchange hole of the edge electrongas [ 62]. In the present work, we aim to develop an exchange hole in the conventional gauge. The hole will reproducethe TPSS exchange energy functional by construction. To 2469-9950/2017/95(12)/125115(12) 125115-1 ©2017 American Physical SocietyTAO, BULIK, AND SCUSERIA PHYSICAL REVIEW B 95, 125115 (2017) ensure that our model hole is in the conventional gauge, we not only impose the exact conventional constraints in theconventional gauge (e.g., recovery of the correct short-rangebehavior without integration by parts) on the hole model butalso modify the TPSS exchange energy density by adding agauge function. The present gauge function is similar to the oneproposed by Tao et al. [54], but with a modiﬁcation so that the gauge-corrected exchange energy density or underlying holeis ensured to be negative even in the far density tail. Addinga proper gauge function to the exchange energy density willnot alter the integrated exchange energy, but it will improvethe agreement of the model hole with the exact conventionalone. Furthermore, the hole model can generate the exactsystem-averaged exchange hole accurately by replacing theTPSS exchange energy density with the gauge-corrected exactconventional exchange energy density (i.e., in TPSS gauge).As a simple application, we apply our semilocal exchangehole to construct a range-separated exchange functional. Ournumerical tests show that this range-separated functional,when combined with the TPSS correlation functional, canyield band gaps and barrier heights in much better agreementwith experimental values than the original TPSS functional,without losing much accuracy of atomization energies. II. EXACT CONVENTIONAL EXCHANGE HOLE For simplicity, let us ﬁrst consider a spin-unpolarized density ( n↑=n↓). For such a density, the exchange energy can be written as Ex[n]=/integraldisplay d3rn(r)/epsilon1x(r) =/integraldisplay d3rn(r)1 2/integraldisplay d3uρx(r,r+u) u, (1) where n(r)=n↑+n↓is the total electron density, /epsilon1x(r)i s the conventional exchange energy per electron, or, looselyspeaking, the exchange energy density, and ρ x(r,r+u)i s the exchange hole at r+uaround an electron at r.I ti s conventionally deﬁned by ρx(r,r+u)=− \|γ1(r,r+u)\|2/2n(r). (2) Hereγ1(r,r+u) is the Kohn-Sham single-particle density matrix given by γ1(r,r+u)=2N/2/summationdisplay iφi(r)∗φi(r+u), (3) withNbeing the number of electrons and φi(r) being the occupied Kohn-Sham orbitals. According to expression ( 1), one can regard the exchange energy as the electrostaticinteraction between a reference electron at rand the exchange hole at r+u. Therefore, strictly speaking, an exchange energy functional cannot be fully justiﬁed unless the underlyingexchange hole has been found. But this issue can be addressedwith the reverse-engineering approach [ 52]. The exchange hole for a spin-unpolarized density can be generalized to any spin polarization with the spin-scalingrelation [ 63] ρ x[n↑,n↓]=n↑ nρx[2n↑]+n↓ nρx[2n↓]. (4) Therefore, in the development of the exchange hole, we need to consider only a spin-compensated density. Performing thespherical average of the exchange hole over the direction ofseparation vector u, the exchange energy of Eq. ( 1) may be rewritten as E x[n]=/integraldisplay∞ 0du4πu2/integraldisplay d3rn(r)/angbracketleftρx(r,u)/angbracketrightsph 2u, (5) where /angbracketleftρx(r,u)/angbracketrightsphis the spherical average of the exchange hole deﬁned by /angbracketleftρx(r,u)/angbracketrightsph=/integraldisplayd/Omega1 u 4πρx(r,r+u). (6) This suggests that the exchange energy does not depend on the detail of the associated hole. Rearranging Eq. ( 5) leads to a simple expression Ex[n]=N/integraldisplay du4πu2/angbracketleftρx(u)/angbracketright 2u, (7) where /angbracketleftρxc(u)/angbracketrightis the system average of the exchange hole deﬁned by /angbracketleftρx(u)/angbracketright=1 N/integraldisplay d3rn(r)/angbracketleftρx(r,u)/angbracketrightsph. (8) Although the conventional exact exchange hole of Eq. ( 2) satisﬁes the sum rule/integraldisplay d3uρx(r,u)=− 1( 9 ) (the most important property of the exchange hole), the exact exchange hole transformed to a new coordinate system [ 64,65] does not. Nevertheless, the system-averaged hole alwayssatisﬁes the sum rule/integraldisplay d 3u/angbracketleftρx(u)/angbracketright=− 1. (10) This is the constraint that has been imposed in the development of a semilocal exchange hole. While the exchange energy isuniquely deﬁned, the exchange energy density /epsilon1 x(r)a sw e l la s the exchange hole ρx(r,r+u) are not. For example, both quan- tities can be altered by a general coordinate transformation orby adding an arbitrary amount of the Laplacian of the electrondensity, without changing the total exchange energy [ 54,66]. III. CONSTRAINTS ON THE EXCHANGE HOLE The conventional exchange hole is related to the pair distribution function gx(r,r/prime)b y n(r)ρx(r,r/prime)=n(r)n(r/prime)gx(r,r/prime). (11) In general, a semilocal exchange hole can be written as n(r)ρx(r,r+u)=n2(r)Jx(s,z,u f), (12) where J(s,z,u f) is the shape function that needs to be constructed, with s=\|∇n\|/(2kfn) being the dimensionless reduced density gradient, kf=(3π2n)1/3being the Fermi wave vector, z=τW/τ, anduf=kfu.H e r e τW=\|∇n\|2/8n 125115-2SEMILOCAL EXCHANGE HOLE WITH AN APPLICATION . . . PHYSICAL REVIEW B 95, 125115 (2017) is the von Weizs ¨acker kinetic energy density, and τis the Kohn-Sham orbital kinetic energy density deﬁned by τ(r)=/summationtextN/2 i\|∇φi(r)\|2. A. Constraints on the shape function We will seek a shape function that satisﬁes the following constraints: (i) On-top value J(s,z,0)=− 1/2. (13) (ii) Uniform-gas limit Junif(uf)=−9 2/bracketleftBigg sin(uf)−cos(uf) u3 f/bracketrightBigg . (14) The uniform-gas limit that will be imposed here is the nonoscillatory model [ 67][ E q .( 28)f o rs=0 andz=0]. (iii) Normalization 4 3π/integraldisplay∞ 0dufu2 fJ(s,z,u f)=− 1. (15) (iv) Negativity J(s,z,u f)/lessorequalslant0. (16) (v) Energy constraint 8 9/integraldisplay∞ 0dufufJ(s,z,u f)=−FTPSS x (s,z). (17) (vi) Small- ubehavior lim uf→0∂2J(s,z,u f) ∂u2 f=L(s,z). (18) L(s,z) is the curvature of the shape function that will be discussed below. (vii) Large-gradient limit lim s→∞J(s,z,u f)=JPBE(s,uf). (19) In the large-gradient limit, the TPSS enhancement factor approaches the PBE enhancement factor. Therefore, the TPSSshape function should also approach the PBE shape functionin this limit. Among these constraints, (vi) is for the conventional ex- change hole, while (vii) is a constraint used in the developmentof the TPSS functional. These two constraints will be discussedin detail below. In previous works [ 52,53], constraint (vi) was used with integration by parts and thus is not a constraint forthe conventional exchange hole, and constraint (vii) was notconsidered. B. Small- ubehavior and large-gradient limit Expanding the spherically averaged exchange hole up to second order in uyields /angbracketleftρx(r,u)/angbracketrightsph=−1 2n+1 12/bracketleftbigg 4/parenleftbigg τ−\|∇n\|2 8n/parenrightbigg −∇2n/bracketrightbigg u2+··· . (20) Since the Laplacian of the density tends to negative inﬁnity at a nucleus, the negativity of the exchange hole for small uwill beviolated. Therefore, we must eliminate it. In previous works, the Laplacian of the density is eliminated by integration byparts [ 52]. In order to model the conventional exchange hole, here we eliminate it instead with the second-order gradientexpansion of the kinetic energy density in the slowly varyinglimit, τ≈τ unif+\|∇n\|2/(72n)+∇2n/6. (21) This technique has been used in the development of the TPSS [ 17] and other functionals [ 18,68]a sw e l la si nt h e construction of electron localization indicator [ 69]. Substituting Eqs. ( 20) into Eq. ( 12) and eliminating the Laplacian ∇2nvia ( 21) yields the small- uexpansion of the shape function J(s,z,u f)=−1 2+1 6/parenleftBig −3 10τ τuni+9 10−5 6s2/parenrightBig u2 f+··· , (22) leading to L(s,z)=−1 3/parenleftBig3 10τ τuni−9 10+5 6s2/parenrightBig . (23) For one- or two-electron densities, L(s,z) reduces to L(s,z=1)=3 2/parenleftBig1 5−8 27s2/parenrightBig , (24) while for the uniform gas, L(s=0,z=0)=1 5. Note that lims→0L(s,z=1)=3/10, while lim s→0L(s,z=0)=1/5 (order-of-limit problem). In the large-gradient limit, the TPSS shape function should recover the PBE shape function [Eq. ( 19)]. This requires thatL(s,z) must be merged smoothly with the PBE small- u behavior, LPBE(s)=/parenleftBig1 5−2 27s2/parenrightBig . (25) We can achieve this with LTPSS=1 2erfc/parenleftbiggs2−s2 0 s0/parenrightbigg L(s,z) +/bracketleftbigg 1−1 2erfc/parenleftbiggs2−s2 0 s0/parenrightbigg/bracketrightbigg LPBE(s), (26) where erfc( x) is the complementary error function deﬁned by erfc(x)=1−erf(x)=2 π/integraldisplay∞ xdte−t2. (27) Heres0=6 is a switching parameter that deﬁnes the point at which the small- ubehavior smoothly changes from the TPSS to PBE. This choice of s0ensures that the small- ubehavior of our shape function is essentially determined by Eq. ( 22), while it merges into the PBE shape function in the large-gradientlimit [Eqs. ( 35) and ( 36)o fR e f .[ 67]]. 125115-3TAO, BULIK, AND SCUSERIA PHYSICAL REVIEW B 95, 125115 (2017) IV . SHAPE FUNCTION FOR THE TPSS EXCHANGE HOLE A. TPSS shape function The shape function for the TPSS exchange hole is assumed to take the following form: JTPSS(ufs,z)=/bracketleftbigg −9 4u4 f/parenleftbigg 1−e−Au2 f/parenrightbigg +/parenleftbigg9A 4u2 f+B+C(s,z)u2 f+G(s,z)u4 f +K(s,z)u6 f/parenrightbigg e−Du2 f/bracketrightbigg e−H(s,z)u2 f, (28) where A=0.757211, B=− 0.106364, and D=0.609650 are determined by the recovery of the nonoscillatorymodel [ 67] of the uniform electron gas, while the functions C(s,z),G(s,z), andK(s,z) are determined by constraints (iii), (v), and (vi). They can be analytically expressed in terms ofH(s,z)a s C=1 8/parenleftBig 4L+3A3+9A2H−9AD2−18ADH +8Bλ/parenrightBig , (29) G=−63 8λ3/bracketleftbigg FTPSS x+Aln/parenleftBigβ λ/parenrightBig +Hln/parenleftBigβ H/parenrightBig/bracketrightbigg −24 5λ7 2/parenleftbigg3A√ H+√β−√π/parenrightbigg +603 40Aλ3 −19 10Bλ2−11 10Cλ, (30) K=8 35λ9 2/parenleftbigg3A√ H+√β−√π/parenrightbigg −12 35Aλ4 −8 105Bλ3−4 35Cλ2−2 7Gλ, (31) where λ=D+H(s,z) andβ=A+H(s,z). Following the procedure of Constantin, Perdew, and Tao [ 52] in the construc- tion of the original TPSS shape function, here we determinethesdependence of H(s,z) by ﬁtting to the two-electron exponential density, because for two-electron densities, zis identically a constant everywhere in space. It depends onlyon the density gradient s. We determine the zdependence of H(s,z) with the wave-vector analysis of the surface energy in the inﬁnite barrier model, because in this model, the electrondensity, the kinetic energy density, and the exchange holeare analytically known and the surface energy is also knownaccurately.B.sdependence of H(s,z) In iso-orbital regions where z≈1 (e.g., core and density tail regions), we assume that the function H(s,z=1) takes the form Hiso−orb(s,z=1)=h0+h1s2+h2s4+h3s6 d0+d1s2+d2s4+d3s6. (32) Note that Hiso−orb(s,z) has only an even-order gradient dependence. This is because in the slowly varying limit, thespherical average of the exchange hole [Eq. ( 20)] depends only upon the even-order gradient terms [ 70]. In the large-gradient regime, H(s,z=1) of TPSS should recover H(s)[67] of PBE, H PBE(s)=p1s2+p2s4+p3s6 1+p4s2+p5s4+p6s6. (33) For any density between the two regimes, we take the interpolation formula, H(s,z=1)=1 2erfc/parenleftbiggs2−s2 0 s0/parenrightbigg Hiso−orb(s,z=1) +/bracketleftbigg 1−1 2erfc/parenleftbiggs2−s2 0 s0/parenrightbigg/bracketrightbigg HPBE(s).(34) Finally, we insert Eq. ( 34) into Eqs. ( 29)–(31) and perform the ﬁtting procedure by minimizing the following quantity: /summationdisplay iui/parenleftbigg/angbracketleftbig ρTPSS x(ui)/angbracketrightbig sph−/angbracketleftbig ρexact x(ui)/angbracketrightbig sph/parenrightbigg2 , (35) where /angbracketleftρx(u)/angbracketrightsphis the spherical system average of the exchange hole deﬁned by Eq. ( 8). We can express /angbracketleftρx(u)/angbracketrightsphin terms of the shape function as /angbracketleftρx(u)/angbracketrightsph= (1/N)/integraltext d3rn(r)2J(s,z,u f). For numerical convenience, we replace the integral with discretized summation. All theparameters for H(s,z=1) and H(s) are listed in Table I. Figure 1shows the system-averaged exchange hole for the two-electron exponential density evaluated with different holemodels compared to the exact one. We can observe from Fig. 1 that the present TPSS hole is slightly closer to the conventionalexact hole than the original TPSS hole, but it is much closerthan the PBE GGA and LSDA holes. C. Inﬁnite barrier model and wave-vector analysis for surface energy As discussed above, in iso-orbital regions, the sdepen- dence of H(s,z) is determined by ﬁtting the model hole to the conventional exact exchange hole for the two-electronexponential density. In the uniform-gas limit, our exchangehole should correctly reduce to the nonoscillatory model [ 67] of the LSDA. This requires H(s,z) to vanish in this limit. To TABLE I. Parameters of the TPSS shape function H(s,z=1) of Eq. ( 34) and the PBE shape function H(s)o fE q .( 33) determined by a ﬁt to the two-electron exponential density. H(s,z=1) of Eq. ( 34) H(s)o fE q .( 33) h0 h1 h2 h3 d0 d1 d2 d3 p1 p2 p3 p4 p5 p6 0.0060 2.8916 0.7768 2.0876 13.695 −0.2219 4.9917 0.7972 0.0302 −0.1035 0.1272 0.1203 0.4859 0.1008 125115-4SEMILOCAL EXCHANGE HOLE WITH AN APPLICATION . . . PHYSICAL REVIEW B 95, 125115 (2017) -0.3-0.25-0.2-0.15-0.1-0.050 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52πNunx(u)[a.u.] u(bohr)Exact TPSS-present TPSS-original PBE LSDA FIG. 1. System-averaged exchange hole for the LSDA, PBE GGA, and TPSS meta-GGA for the two-electron exponential den-sity. “TPSS-original” represents the original TPSS hole model of Constantin, Perdew, and Tao [ 52], while “TPSS-present” represents the present TPSS hole model. The area under the curve is theexchange energy (in hartrees): E LSDA x=− 0.5361,EPBE x=− 0.6117, ETPSS x=− 0.6250, and Eex x=− 0.6250. Both the original and present TPSS holes yield the same exchange energy due to the same energyconstraint. fulﬁll these considerations, we assume that H(s,z)=1 2erfc/parenleftbiggs2−s2 0 s0/parenrightbigg Hiso−orb(s,z=1)zm +/bracketleftbigg 1−1 2erfc/parenleftbiggs2−s2 0 s0/parenrightbigg/bracketrightbigg HPBE(s), (36) where mis an integer. In order to determine m, we follow the procedure of Ref. [ 52] to study the wave-vector analysis (WV A) of the surface energy. But instead of using the jelliumsurface model with a linearly increasing barrier, here weemploy the exactly solvable inﬁnite barrier model (IBM).Since the single-particle density matrix and hence the electrondensity of IBM is analytically known, this allows us to obtaininsight into the zdependence of H(s,z) from this model more easily. -0.3-0.2-0.100.10.20.30.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Γx(kr)[a.u.] kr FIG. 2. /Gamma1(k)o fE q .( 43) and smooth ﬁt.Let us consider a uniform gas of noninteracting electrons subject to an inﬁnite potential barrier perpendicular to the x axis (V→∞ forx< 0). The one-particle density matrix is given by [ 71,72] γ1(r,r/prime)=¯n/bracketleftbigg J(uf)−J/parenleftbig/radicalBig u2 f+4xfx/prime f/parenrightbigg/bracketrightbig /Theta1(x)/Theta1(x/prime), (37) where /Theta1(x) is a step function, with /Theta1(x)=1f o r x> 0 and/Theta1(x)=0f o rx/lessorequalslant0. Here ¯nis the average bulk valence electron density, xf=xkf,x/prime f=x/primekf,uf=\|r−r/prime\|kf, and J(ξ)=3j1(ξ)/ξ, (38) withj1(ξ)=sin(ξ)/ξ2−cos(ξ)/ξbeing the ﬁrst-order spher- ical Bessel function. The electron density can be obtained fromthe single-particle density matrix by taking u=\|r /prime−r\|=0 in Eq. ( 37). This yields n(x)=¯n[1−J(2xf)]/Theta1(x). (39) The WV A for the surface exchange energy density is given by [52] γx(k)=/integraldisplay∞ 0du8kfu2bx(u)sin(ku) ku, (40) where bx(u)=/integraldisplay∞ −∞dxn(x)/bracketleftbig ρx(x,u)−ρunif x(u)/bracketrightbig . (41) The exchange hole ρx(x,u) of IBM can be obtained from the one-particle density matrix of Eq. ( 37). With some algebra, we can express the WV A surface exchange energy as [ 73] σx=1 2/integraldisplay∞ 0dkrγx(kr), (42) where kr=k/kF,γx(kr) is given by γx(kr)=8 k2 f/integraldisplay∞ 0dufb(uf)u2 fsinc(kruf) =1 (πrs)3/Gamma1(kr), (43) and bx(uf)=−¯n2 2kf/integraldisplay∞ 0dxfix(xf,uf). (44) Here sinc( x)=sin(x)/x, and ix(xf,uf)=/summationtext6 l=1χl(xf,uf), withχl(xf,uf) being deﬁned by Eq. (3.18) of Ref. [ 73]. Figure 2shows the exact variation of /Gamma1(kr) withkr. Figure 3 shows the comparison of approximate /Gamma1(kr) with the exact curve (red) for different mvalues. The area under the curve is proportional to the surface exchange energy. From the electrondensity and density matrix of IBM given by Eqs. ( 37) and ( 39), the exact surface exchange energy can be calculated with theWV A of Eq. ( 42). Langreth and Perdew [ 73] reported that the value of σ x103r3 sis 4.0 a.u., where rsis the Seitz radius. This value is slightly smaller than the value obtained earlierby Harris and Jones [ 74] and Ma and Sahni [ 75] (4.1 a.u.). Our present work gives 3.99 a.u., which is closer to that ofLangreth and Perdew. 125115-5TAO, BULIK, AND SCUSERIA PHYSICAL REVIEW B 95, 125115 (2017) -0.3-0.2-0.100.10.20.30.4 0.2 1 1.8 2.6 3.4 4.2 5Γx(kr)[a.u.] krz2 z3z4 z5Exact FIG. 3. Analysis of zdependence of the WV A for the present TPSS hole of Eq. ( 28). The z3curve provides the best ﬁt to the peak region of the exact /Gamma1(kr)o fE q .( 43). D.zdependence of H(s,z) Thezdependence of H(s,z)[ E q .( 36)] can be determined by ﬁtting the TPSS hole to the wave-vector analysis. We startwith the speciﬁc expressions for the local ingredients of thehole model in IBM. From the electron density of Eq. ( 39), the reduced density gradient can be explicitly expressed as s(x f)=3 2xf\|sinc(2xf)−J(2xf)\| [1−J(2xf)]4/3. (45) The kinetic energy density can be obtained from the single- particle density matrix of Eq. ( 37). This yields τ(xf)=k2 f¯n/braceleftbigg3 10+1 2J(2xf)+9 4x2 f[sinc(2 xf)−J(2xf)]/bracerightbigg . (46) Finally, the von Weizs ¨acker kinetic energy density can be expressed as τW=9k2 f¯n 8x2 f/braceleftbigg[sinc(2 xf)−J(2xf)]2 1−J(2xf)/bracerightbigg . (47) Next, we calculate γxfrom the TPSS hole. Inserting the TPSS model hole into Eq. ( 41) yields bx(u)=/integraldisplay∞ 0dxn (x)/bracketleftbig ρTPSS x(x,u)−ρunif x(u)/bracketrightbig =˜n2 kf/integraldisplay∞ 0dxf[1−J(2xf)]{[1−J(2xf)] ×JTPSS(uf3/radicalbig 1−J(2xf),s(xf),z(xf)) −Junif(uf)}. (48) [Note that /Theta1(x) is implicit on the electron density.] Substitut- ing Eq. ( 48) into Eq. ( 40), we obtain γx(k)=8¯n 3π2/integraldisplay∞ 0dxf/integraldisplay∞ 0dufjx(uf,xf,kr), (49)-0.3-0.2-0.100.10.20.30.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Γx(kr)[a.u.] krTPSS-present TPSS-original PBE Exact FIG. 4. Comparison of the WV A for the present and original TPSS hole models as well as the PBE hole with the exact one.“TPSS-present” represents the present TPSS hole model, while “TPSS-original” represents the original TPSS model. where jx(uf,xf,kr)=[/rho12(xf)JTPSS(uf3/radicalbig /rho1(xf),s(xf),z(xf)) −/rho1(xf)Junif(uf)]sinc( kruf)u2 f (50) and/rho1(xf)=1−J(2xf). Rearrangement of Eq. ( 49) leads to the ﬁnal expression γx(k)=1 (πrs)3/Gamma1TPSS(kr), (51) where /Gamma1TPSS(kr)=2/integraldisplay∞ 0/integraldisplay∞ 0dxfdufjx(uf,xf,kr). (52) Figure 3shows the comparison of H(s,z) with different choices of mto the exact one. From Fig. 3, we see that the best ﬁt to the exact /Gamma1(kr) in the peak region is m=3. Figure 4 shows that, compared to the WV A of the LSDA, PBE, andoriginal TPSS holes, the WV A of the present model is closestto the exact one in the peak region. To further understand theoriginal and present TPSS models, we plot the TPSS shapefunction of the present and the original models in IBM atz=0.55, as shown by Figs. 5and6, respectively. From Figs. 5 and6, we observe that while the present model hole is always negative, the original TPSS hole can be positive in some rangeofu fands. To check our wave-vector analysis for the surface exchange energy, we have computed σxfrom σx=/integraldisplay∞ −∞dxn (x)/bracketleftbig /epsilon1x(n)−/epsilon1unif x(¯n)/bracketrightbig . (53) The results are shown in Table II. From Table II, we can see that the surface energy from the WV A of the TPSShole (both original and the present version) agrees very wellwith the surface energy calculated directly from the TPSSexchange functional [Eq. ( 53)]. Furthermore, the TPSS surface energy is closer to the exact value than those of the LSDAand PBE. The LSDA signiﬁcantly overestimates the surfaceexchange energy, while the PBE gives underestimation. These 125115-6SEMILOCAL EXCHANGE HOLE WITH AN APPLICATION . . . PHYSICAL REVIEW B 95, 125115 (2017) -2.5-2-1.5-1-0.50 01234J(uf,s,z=0.55) ufPresents=0 s=1 s=2 s=3 s=5 s=∞ FIG. 5. Present TPSS shape function of Eq. ( 28)f o rz=0.55. observations are consistent with those evaluated from the jellium surface linear potential model [ 45]. It is interesting to note that even though the original TPSS shape functionin a certain range is positive, the surface energy from theoriginal TPSS hole is the same as that from the present model.This result is simply due to the cancellation of the originalhole model between positive values and too negative values atcertain u fandsvalues, as seen from the comparison of Fig. 6 to Fig. 5. The IBM surface energy presents a great challenge to semilocal DFT. It is more difﬁcult to get it right than thesurface energy of the jellium model with ﬁnite linear potentialbecause the electron density at the surface of IBM is highlyinhomogeneous due to the sharp cutoff at surface and is toofar from the slowly varying regime where semilocal DFT canbe exact (e.g., TPSS functional). Figure 7shows a comparison of the differences of the system-averaged hole between the approximations and theexact curve for the LSDA, PBE, and the original and presentTPSS exchange hole models of the Ne atom, in which zis, in general, different from 0 (slowly varying density) and 1(iso-orbital density). The PBE and LSDA curves are plottedwith the hole models of Ref. [ 67]. From Fig. 7we can see that, except for the small region near the core, the present -3-2-1012 01234J(uf,s,z=0.55) ufOriginal s=0 s=1 s=2 s=3 s=5 s=∞ FIG. 6. Original TPSS shape function for z=0.55.TABLE II. Comparison of the surface exchange energies (in a.u.) of the IBM surface (expressed as σxrs3103) calculated directly with exchange energy functionals and with the WV A formula. The exact value (obtained in this work) is 3.99 a.u. Eq. ( 53) WV A integration LSDA 6.318 PBE 2.576TPSS 2.945 2.95 (original hole) 2.95 (present hole) TPSS hole model is closer to the exact one than the original TPSS hole model, but both TPSS models obviously improvethe system-averaged holes of the LSDA and PBE. V . TPSS HOLE IN THE GAUGE OF THE CONVENTIONAL EXACT EXCHANGE The shape function explicitly depends on the enhancement factor via the energy constraint of Eq. ( 17). The latter may be altered by adding an arbitrary amount of the Laplacian ofthe density without changing the total exchange energy. Thisambiguity of the exchange energy density [ 66] leads to the ambiguity of the semilocal exchange hole. Our primary goal ofthis work is to develop a semilocal exchange hole in the gaugeof the conventional exact exchange. This is partly motivatedby the fact that, in the development of range-separated densityfunctionals, the exact exchange part is usually provided in theconventional gauge. The exact exchange energy density in the conventional gauge can be conveniently evaluated with the Della Sala–G¨orling (DSG) [ 76] identity resolution e x conv(r)=1 2/summationdisplay μνQσ μνχμ(r)χ∗ ν(r), (54) where Qσis the spin block of the DSG matrix [ 54]. However, many semilocal exchange energy densities or enhancement -3-2-101234567 0 0.4 0.8 1.2 1. 62Δ2πNunx(u)[a.u.] u(bohr)TPSS-present TPSS-original PBE LSDA FIG. 7. Comparison of the difference of the system-averaged hole between the approximations and the exact curve for the Ne atom. “TPSS-present” represents the present TPSS hole model, while“TPSS-original” represents the original TPSS model. 125115-7TAO, BULIK, AND SCUSERIA PHYSICAL REVIEW B 95, 125115 (2017) factors of Eq. ( 17) are not in the gauge of the conventional exact exchange due to the constraints such as the Lieb-Oxford boundand the slowly varying gradient expansion (with integration byparts) imposed on the enhancement factor. For example, for thetwo-electron exponential density, the conventionally deﬁnedexact enhancement factor is less than 1 near the nucleus, whilethe TPSS enhancement factor is F TPSS x/greaterorequalslant1 by design. In the density tail region, the conventional exact enhancement factortends to inﬁnity, but the maximum value of F TPSS x is 1.804. To construct the TPSS exchange hole in the conventionalgauge, we can replace the original energy density constraint[Eq. ( 17)], which was used in the construction of the original TPSS exchange hole [ 52], with the TPSS exchange energy density or enhancement factor in the conventional gauge. Inthis gauge, the TPSS exchange energy density can be writtenas [54] e TPSS x(r)=eTPSS,conv x (r)+G(r), (55) where eTPSS x(r) is the standard TPSS exchange energy den- sity [ 17] (i.e.,λ=0.92) and eTPSS,conv x (r) is the TPSS exchange energy density in the exact conventional gauge (i.e., λ=1), withλbeing the general coordinate transformation parame- ter [54,64,65]. Here ex(r)=n(r)/epsilon1x(r). Equivalently, we can also write eex,tpssg x (r)=eex,conv x (r)+G(r), (56) where eex,tpssg x (r) is the exact exchange energy density in TPSS gauge and eex,conv x (r) is the exact conventional exchange energy density evaluated from the single-particle densitymatrix [Eqs. ( 2)–(6)]. Based on the uniform and nonuniform coordinate scaling properties of the exact exchange energydensity, Tao, Staroverov, Scuseria, and Perdew (TSSP) [ 54] proposed a gauge function G(r)=a∇·[f(r)∇˜/epsilon1], (57) f=n/˜/epsilon1 2 1+c(n/˜/epsilon13)2/parenleftbiggτW τ/parenrightbiggb . (58) Herea=0.015 and c=0.04 are determined by a ﬁt to the conventional exact exchange energy density of the H atom, andbis an integer which is chosen to be 4 due to the consideration of sodium jellium sphere clusters. ˜ /epsilon1=−/epsilon1 ex,conv x is the exact exchange energy density in the conventional gauge. This gaugefunction is integrated to zero, i.e.,/integraltext d 3rG(r)=0, as required. It satisﬁes the correct uniform coordinate scaling relation,G λ(r)=λ4G(λr), and nonuniform coordinate scaling relation Gx λ(x,y,z )=λG(λx,y,z ). However, in the far density tail ( r→∞ )o fa na t o m , the exact exchange energy density in the conventional gaugedecays as e ex,conv x ∼−n/2r, but the original TSSP gauge function decays as G(r)∼n. As a result, the exchange energy density in this gauge becomes positive in the density tail region.In order to ﬁx this deﬁciency, we impose a constraint on thedensity tail, lim r→∞G econvx=0. (59)-0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.050 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52πNunx(u)[a.u.] u(bohr)TPSS gTPSSExact TPSS TPSS(Exact,gTPSS) TPSS(Exact) FIG. 8. Comparison of the system-averaged holes for the two- electron exponential density. “Exact” represents the conventionalexact system-averaged hole ρ exact x(r,u) (red) from Eqs. ( 4)–(6), “TPSS” represents the present TPSS system-averaged hole (blue) f r o mE q s .( 28)–(31)a n d( 36) with Table Iandm=3, “TPSS(Exact)” represents the system-averaged hole (green) generated from the TPSS hole but with FTPSS x(r)o fE q .( 17) replaced by Fexact x(r), and “TPSS(Exact,gTPSS)” represents the system-averaged holegenerated from e ex,tpssg x (r)o fE q .( 56) (purple). This can be achieved by requiring that in the r→∞ limit,G decays as npwithp> 1. Here we choose p=3 2and take the same form of the TSSP gauge function, but with fgiven by f=/parenleftbig n/˜/epsilon17 3/parenrightbig3/2 1+c(n/˜/epsilon13)5/2/parenleftbiggτW τ/parenrightbiggb . (60) Herea=0.01799 and c=0.00494 are determined by ﬁtting the TPSS system-averaged hole in the conventional gauge tothe exact system-averaged hole of the two-electron exponentialdensity. The ﬁtting procedure is the same as that in thedetermination of the H(s,z=1) function. The parameter b= 4 remains the same as that in the original version [Eq. ( 58)]. our present gauge function retains all the correct properties thatthe original gauge function satisﬁes, including the nonuniformcoordinate scaling property. Figure 8shows the comparison of the present TPSS system- averaged exchange hole and the exact conventional system-averaged exchange hole calculated from the present TPSS holemodel but with F TPSS x (r)o fE q .( 17) replaced by Fexact x(r) with and without the gauge correction of Eq. ( 60) to the exact conventional one [Eqs. ( 4)–(6)]. From Fig. 8we can observe that the exact system-averaged exchange hole generated fromthe present TPSS hole model without the gauge correctionsigniﬁcantly deviates from the exact system-averaged hole.However, the agreement has been signiﬁcantly improved withour present gauge correction [Eq. ( 60)]. Figure 9shows the comparison of the TPSS exchange energy density evaluated with the TPSS functional without andwith the gauge correction to the exact conventional exchangeenergy density for the two-electron exponential density. FromFig. 9, we can observe that the effect of the present gauge correction deﬁned by Eq. ( 60) is small for the present TPSS hole. However, as observed in Fig. 8, it is important for the 125115-8SEMILOCAL EXCHANGE HOLE WITH AN APPLICATION . . . PHYSICAL REVIEW B 95, 125115 (2017) -0.5-0.4-0.3-0.2-0.10 0 0.5 1 1.5 2 2.5 3ex[a.u.] r(bohr)Exact TPSS Exact-gTPSS FIG. 9. Comparison of the exchange energy densities for the two-electron exponential density calculated with different approx-imations to the exact one. “TPSS” represents the TPSS exchange energy density calculated directly from the TPSS exchange energy functional; “gTPSS” represents the gauge-corrected TPSS exchangeenergy density. conventional exact exchange hole evaluated with the present TPSS hole. VI. APPLICATION TO RANGE-SEPARATED EXCHANGE FUNCTIONAL As a simple application, we apply the present TPSS hole model to construct a range-separated functional. In general,there are two ways to construct a range-separated functional,simply depending on the need. For example, we may employ asemilocal DFT as the long-range part, while the exact exchangeis used for the short-range part, as pioneered by Heyd,Scuseria, and Ernzerhof [ 14]. This kind of range-separated functional is developed largely for solids and is particularlyuseful for metallic solids because usual hybrids require muchlarger momentum cutoff for metallic systems with electronsnonlocalized. Nevertheless, this range-separated functional isalso accurate for molecules. We may also employ a semilocalDFT for the short-range part, while the exact exchange is usedfor the long-range part, as developed by Henderson et al. [67] on the basis of the PBE hole. These kinds of range-separatedfunctionals are usually developed for molecular calculationsbecause the improved long-range part of the exchange holewill improve the description of molecular properties. Manyrange-separated functionals have been proposed [ 48,77–81]. In the following, we will explore the TPSS hole-based range-separated functional with the TPSS exchange functional beingthe long-range (LR) part and the Hartree-Fock exchange beingthe short-range (SR) part, aiming to improve the too small bandgaps and reaction barrier heights of the TPSS functional. The idea of the construction of our TPSS-based range- separated functional is rooted in the construction of the usualone-parameter hybrid functionals, which, in general, can bewritten as E hybrid xc=aEHF x+(1−a)Esl x+Esl c, (61)TABLE III. Band gaps (in eV) calculated with the LSDA, PBE, HSE, TPSS, and TPSS-based range-separated functional with a= 0.25 and ω=0.10 (PW =present work) compared to experiments. ME stands for mean error and MAE stands for mean absolute error. LSDA PBE TPSS HSE PW Expt. C 4.17 4.2 4.24 5.43 5.48 5.48CdSe 0.31 0.63 0.85 1.48 1.82 1.90GaAs 0.04 0.36 0.6 1.11 1.44 1.52 GaN 2.15 2.22 2.18 3.48 3.5 3.50 GaP 1.56 1.74 1.83 2.39 2.53 2.35 Ge 0.13 0.32 0.8 0.99 0.74 InAs 0 0.08 0.57 0.85 0.41InN 0 0 0 0.72 0.75 0.69 InSb 0 0.47 0.73 0.23 Si 0.53 0.62 0.71 1.2 1.31 1.17ZnS 2.02 2.3 2.53 3.44 3.78 3.66 ME −0.89 −0.86 −0.76 −0.12 0.14 MAE 0.89 0.86 0.76 0.15 0.17 where ais the mixing parameter that controls the amount of exact exchange mixed into a semilocal (sl) functional. Following the prescription of Heyd, Scuseria, and Ernz- erhof (HSE) [ 14], we write the TPSS-based range-separated functional as Exc=aEHF,SR x+(1−a)Esl,SR x+Esl,LR x+Esl c,(62) where EHF,SR x is the Hartree-Fock (HF) exchange serving as part of the short-range contribution, while Esl,SR x is the TPSS exchange that provides the rest of the short-range contribution.E sl cis the TPSS correlation. The long-range contribution is provided fully by the TPSS exchange Esl,LR x. They are given, respectively, by /epsilon1HF,SR x=1 2/integraldisplay∞ 0du4πu2ρHF x(r,u)erfc(ωu) u, (63) /epsilon1sl,SR x=1 2/integraldisplay∞ 0du4πu2ρTPSS x(r,u)erfc(ωu) u, (64) /epsilon1sl,LR x=1 2/integraldisplay∞ 0du4πu2ρTPSS x(r,u)erf(ωu) u, (65) where ωis a range-separation parameter and erf( x)i st h ee r r o r function deﬁned by Eq. ( 27). From Eqs. ( 62)–(65), we can see that the amount of exact exchange mixing is controlled bytwo parameters, aandω. Determination of them is discussed below. To test this functional, we have implemented it into thedevelopmental version of GAUSSIAN 09 [82]. In the TPSS-based hybrid functional (TPSSh) [ 33],a=0.1 was ﬁtted to 223 G3 /99 atomization energies. In other words, the optimal value of ais 0.1 for TPSSh. If we consider only atomization energy, then the best value of ωin the TPSS-based range-separated functional should be zero ifa=0.1 is chosen. Since, in the range-separated functional, some amount of the exact exchange (here the long-rangepart) in the TPSSh is replaced by the TPSS functional, tocompensate for this, we need a value of alarger than 0.1. Then we can ﬁnd the best range-separated parameter ωby ﬁtting to some electronic properties. This situation is different fromPBE-based range-separated functionals, in which the mixing 125115-9TAO, BULIK, AND SCUSERIA PHYSICAL REVIEW B 95, 125115 (2017) TABLE IV . AE6 atomization energies (in kcal /mol) calculated with the LSDA, PBE, TPSS, TPSSh, HSE, and TPSS-based range- separated functional with a=0.25 and ω=0.10 (PW =present work) compared to experimental values [ 84]. ME stands for mean error and MAE stands for mean absolute error. LSDA PBE TPSS TPSSh HSE PW Expt. SiH 4 347.4 313.2 333.7 333.6 314.5 333.6 322.4 SiO 223.9 195.7 186.7 182.0 182.1 175.4 192.1 S2 135.1 114.8 108.7 105.9 106.3 101.9 101.7 C3H4 802.1 721.2 707.5 704.4 705.9 699.9 704.8 C2H2O2754.9 665.1 636.0 628.0 635.3 616.4 633.4 C4H8 1304 1168 1156 1154 1152 1152 1149 ME 77.4 12.4 4.1 0.75 −1.2−4.0 MAE 77.4 15.5 5.9 6.1 4.8 8.8 parameter a=1/4i nP B E 0[ 13] can be retained. To avoid possible overﬁtting, here we choose a=1/4, a value that was recommended by Perdew, Ernzerhof, and Burke [ 83] and adopted with the PBE0 functional [ 13]. The parameter ωis determined by a ﬁt to the band gap of diamond (C). Thisyields ω=0.1. Then we apply this range-separated functional to calculate the band gaps of 10 semiconductors. The results arelisted in Table III. From Table III, we see that the band gaps of this range-separated functional are remarkably accurate, witha mean absolute deviation from experiments of only 0.17 eV ,about the same accuracy as the HSE functional. We can also seefrom Table IIIthat the TPSS-based range-separated functional will be expected to yield a more accurate description forlarge band-gap materials and therefore provides an alternativechoice for band-gap and other solid-state calculations. Next, we apply our range-separated functional to calculate atomization energies of six molecules (AE6). The results arelisted in Table IV. From Table IV, we can see that our range- separated functional worsens the atomization energies of theTPSS functional for this special set only by about 3 kcal/mol.This error is still smaller than many other DFT methods suchas the LSDA and PBE. Reaction barrier heights are a decisive quantity in the study of chemical kinetics. However, semilocal functionalstend to underestimate this quantity. As another application,we apply our range-separated functional to calculate sixrepresentative reaction barrier heights (BH6), which consistof three forward (f) and three reverse (r) barrier heights. The results are listed in Table V. For comparison, we also calculated these barrier heights using the PBE, TPSS, TPSSh,and HSE. From Table V, we observe that our range-separated functional provides a substantially improved description forbarrier heights compared to the TPSS and TPSSh functionals. VII. CONCLUSION In conclusion, we have developed a conventional semilocal exchange hole underlying the TPSS exchange functional. Thehole is exact in the uniform-gas limit and accurate for compactiso-orbital densities. It satisﬁes the constraints that the TPSSexchange functional satisﬁes. It also satisﬁes the constraints onthe conventional exchange hole. The hole can be regarded as aninterpolation between the two-electron exponential density andthe IBM jellium surface. Numerical tests on H and Ne atomsshow that the hole mimics the conventional exact exchangehole quite accurately. In particular, with our present gaugefunction correction, the hole model can generate the exactsystem-averaged hole accurately. As an immediate application, we have employed the exchange hole model to construct a range-separated functional.Our tests show that this functional can yield accurate bandgaps, in particular for insulators, and reaction barrier heightswithout losing much accuracy for atomization energies. SinceTPSS is more accurate than PBE for many properties and sincethe PBE hole has been thoroughly explored in recent years,development of TPSS hole-based range-separated functionalsis of general interest. Recently, Arbuznikov and Kaupp [ 21] found that the gauge function has some effect on local hybridfunctionals. It is expected that our present gauge function canbe useful in the development of nonlocal functionals. ACKNOWLEDGMENTS We thank T. M. Henderson for providing the code for the exchange hole of the Ne atom. J.T. acknowledges supportfrom the NSF under Grant No. CHE 1640584. J.T. also ac-knowledges support from Temple start-up via John P. Perdew.I.W.B. and G.E.S. were supported by the U.S. Department ofEnergy, Ofﬁce of Basic Energy Sciences, Computational andTheoretical Chemistry Program under Award No. DE-FG02-09ER16053. G.E.S. is a Welch Foundation Chair (C-0036). TABLE V . BH6 reaction barrier heights (in kcal /mol) calculated with the PBE, TPSS, TPSSh, HSE, and TPSS-based range-separated functional with a=0.25 and ω=0.10 (PW =present work) in comparison with reference values [ 85,86]. Here f (r) =forward (reverse) barrier height. ME stands for mean error and MAE stands for mean absolute error. PBE TPSS TPSSh HSE PW Reference OH + CH 4→CH 3+H 2O −5.29 −0.97 1.50 1.96 4.86 6.54(f) 8.95 9.90 11.79 13.9 14.3 19.6(r) H+O H →O+H 2 3.69 −1.56 −0.15 7.06 1.75 10.5(f) −1.47 4.73 6.90 5.93 9.89 12.9(r) H+H 2S→H2+H S −1.20 −4.55 −3.72 1.03 −2.64 3.55(f) 9.40 12.72 13.4 12.4 14.4 17.3(r) ME −9.37 −8.34 −6.76 −4.66 −4.63 MAE 9.37 8.34 6.76 4.66 4.63 125115-10SEMILOCAL EXCHANGE HOLE WITH AN APPLICATION . . . PHYSICAL REVIEW B 95, 125115 (2017) [1] W. Kohn and L. J. Sham, Phys. Rev. 140,A1133 (1965 ). [2] R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989). [3] R. M. Dreizler and E. K. U. Gross, Density Functional Theory (Springer, Berlin, 1990). [4] J. P. Perdew and Y . Wang, P h y s .R e v .B 33,8800 (1986 ). [5] A. D. Becke, Phys. Rev. A 38,3098 (1988 ). [6] C. Lee, W. Yang, and R. G. Parr, P h y s .R e v .B 37,785(1988 ). [7] A. D. Becke and M. R. Roussel, Phys. Rev. A 39,3761 (1989 ). [8] A. D. Becke, J. Chem. 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PhysRevB.86.235115.pdf	PHYSICAL REVIEW B 86, 235115 (2012) Quasinormal modes of quantum criticality William Witczak-Krempa Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada and Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada Subir Sachdev Department of Physics, Harvard University, Cambridge, Massachusetts, 02138, USA (Received 23 October 2012; revised manuscript received 28 November 2012; published 12 December 2012) We study charge transport of quantum critical points described by conformal ﬁeld theories in 2 +1 space-time dimensions. The transport is described by an effective ﬁeld theory on an asymptotically anti-de Sitter space-time,expanded to fourth order in spatial and temporal gradients. The presence of a horizon at nonzero temperaturesimplies that this theory has quasinormal modes with complex frequencies. The quasinormal modes determinethe poles and zeros of the conductivity in the complex frequency plane, and so fully determine its behavior onthe real frequency axis, at frequencies both smaller and larger than the absolute temperature. We describe the roleof particle-vortex or S duality on the conductivity, speciﬁcally how it maps poles to zeros and vice versa. Theseanalyses motivate two sum rules obeyed by the quantum critical conductivity: the holographic computations arethe ﬁrst to satisfy both sum rules, while earlier Boltzmann-theory computations satisfy only one of them. Finally, we compare our results with the analytic structure of the O(N) model in the large- Nlimit, and other CFTs. DOI: 10.1103/PhysRevB.86.235115 PACS number(s): 74 .40.Kb, 11 .25.Hf I. INTRODUCTION The dynamics of quantum criticality1has long been a central subject in the study of correlated quantum materials.Two prominent examples of recent experiments are (i) theobservation of criticality in the penetration depth of a high-temperature superconductor at the quantum critical point ofthe onset of spin density wave order 2and (ii) the criticality of longitudinal “Higgs” excitations near the superﬂuid-insulatortransition of ultracold bosons in a two-dimensional lattice. 3 A complete and intuitive description of the low-temperature dynamics of noncritical systems is usually provided by their quasiparticle excitations. The quasiparticles are long-lived ex- citations that describe all low-lying states, and their collectivedynamics is efﬁciently captured by a quantum Boltzmannequation (or its generalizations). The Boltzmann equation thencan be used to describe a variety of equilibrium properties, such as the electrical conductivity, thermal transport, and thermoelectric effects. Moreover, such a method can alsoaddress nonequilibrium dynamics, including the approach tothermal equilibrium of an out-of-equilibrium initial state. A key property of strongly interacting quantum critical systems is the absence of well-deﬁned quasiparticle excita-tions. The long lifetimes of quasiparticles is ultimately thejustiﬁcation of the Boltzmann equation, so ap r i o r i it appears that we cannot apply this long-established method to suchquantum critical points. However, there is a regime where, in asense, the breakdown of quasiparticle excitations is weak: thisis the limit where the anomalous exponent, usually called η, of a particle-creation operator φis small (strictly speaking, φcreates particles away from the quantum critical point). The spectral weight of the φGreen’s function is a power-law continuum, but in the limit η→0, it reduces to a quasiparticle δfunction. By expanding away from the η→0 limit, one can extend to the Boltzmann method to quantum critical points,and such a method has been the focus of numerous studies. 4–14A typical example of such Boltzmann studies is the theory of transport at the quantum critical point of the N-component φ4ﬁeld theory with O(N)s y m m e t r yi n2 +1 dimensions; theN=2 case describes the superﬂuid-insulator transition of Ref. 3. Conformal symmetry emerges at the quantum critical point and the corresponding conformal ﬁeld theory (CFT)admits a ﬁnite dc charge conductivity even in the absenceof translation-symmetry breaking perturbations 4(such as disorder or umklapp scattering). This property follows fromthe presence of independent positive and negative chargeexcitations related by charge conjugation (particle-hole) sym-metry, which does not require conformal invariance. We shall,however, restrict oursevles to CFTs in the current work. TheBoltzmann analysis of transport was applied in the large- N limit of the O(N) model, 1,5,14and the structure of the frequency dependence of the conductivity σ(ω) is illustrated in Fig. 1(b). The low-frequency behavior is as expected for weakly interacting quasiparticles: there is a Drude peak whose heightdiverges as ∼N, and whose width vanishes as 1 /N, while preserving the total weight as N→∞ . It is not at all clear whether such a description of the low-frequency transport isappropriate for the N=2 of experimental interest: while it is true that the anomalous exponent ηremains small even at N= 2, it is deﬁnitely not the case that the thermal excitations of thequantum critical point interact weakly with each other. At highfrequencies, ω/greatermuchT(Tis the temperature), the predictions of the large Nexpansion for σ(ω) seem more reliable: the result asymptotes to a nonzero universal constant σ ∞whose value can be systematically computed order-by-order in the 1 /N expansion without using the Boltzmann equation. In this paper, we argue for a different physical paradigm as a description of low frequency transport near quantumcritical points, replacing the quasiparticle-based intuition ofthe Boltzmann equation. We use the description of quantum-critical transport based on the AdS/CFT correspondence 15to emphasize the physical importance of “quasinormal modes” in 235115-1 1098-0121/2012/86(23)/235115(21) ©2012 American Physical SocietyWILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) Generic correlated CFTIdeal gas of free particles 0quantum Boltzmann equationHolography on AdS4“Nearly perfect” quantum Liquid 0 1 1N 1ΩT1NΣ 1ΩT11.5Σ(a) (b)( c) FIG. 1. (Color online) (a) Perspective on approaches to the charge transport properties of strongly interacting CFTs in 2 +1 dimension. The quantum Boltzmann approach applies to the 1 /Nexpansion of the O(N) model: its starting point assumes the existence of weakly interacting quasiparticles, whose collisions control the transport properties. In the present paper, we start from the “nearly perfect” quantum liquid obtaine d in theNc→∞ limit of a SU(Nc) super Yang-Mills theory, which has no quasiparticle description. Holographic methods then allow expansion away from this liquid ( λis the ’t Hooft coupling of the gauge theory). (b) Structure of the charge conductivity in the quantum Boltzmann approach. The dashed line is the N=∞ result: it has a δfunction at zero frequency and a gap below a threshold frequency. The full line shows the changes from 1 /Ncorrections. (c) Structure of the charge conductivity in the holographic approach. The Nc=∞ result is the dashed line, and this is frequency independent . The full line is the conductivity obtained by including four-derivative terms in the effective holographic theory for γ> 0. the charge response function. Formally, the quasinormal mode frequencies are the locations of poles in the conductivity in thelower-half complex frequency plane, i.e., the poles obtainedby analytically continuing the retarded response function fromthe upper-half plane (UHP) to the second Riemann sheet in thelower-half plane (LHP). By considering a particle-vortex dual(or “S-dual”) theory whose conductivity is the inverse of theconductivity of the direct theory, we also associate quasinormalmodes with the poles of the dual theory, which are the zeros ofthe direct theory. Both the pole and zero quasinormal modesare directly accessible in AdS/CFT methods, 16–18and are related to the normal modes of excitations in the holographicspace: the normal modes have complex frequencies becauseof the presence of the “leaky” horizon of a black brane; seeFig. 2.We will show that knowledge of these modes allows a complete reconstruction of the frequency dependence ofthe conductivity, σ(ω), extending from the hydrodynamic regime with ω/lessmuchT, to the quantum critical regime with ω/greatermuchT. Moreover, these quasinormal mode frequencies are also expected to characterize other dynamic properties of thequantum critical system: the recent work of Bhaseen et al. 19 showed that the important qualitative features of the approach to thermal equilibrium from an out-of-equilibrium thermalstate could be well understood by knowledge of the structureof the quasinormal mode frequencies.Apart from the quasinormal modes, the long-time dynamics also exhibits the well-known 20classical hydrodynamic feature of “long-time tails” (LTT). The LTT follow from the principlesof classical hydrodynamics: arbitrary long-wavelength ur0r 10CFTBH JΜ AΜ FIG. 2. (Color online) AdS space-time with a planar black brane. The current ( Jμ) correlators of the CFT are related to those of the U(1) gauge ﬁeld ( Aμ) in the AdS (bulk) space-time. The temperature of the horizon of the black brane is equal to the temperature of the CFT. The horizon acts as a “leaky” boundary to the bulk Aμ normal modes, which consequently become quasinormal modes with complex frequencies. These quasinormal modes specify the ﬁnite temperature dynamic properties of the CFT. 235115-2QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) hydrodynamic ﬂuctuations lead to the algebraic temporal decay of conserved currents. The LTT depend only uponvarious transport coefﬁcients, thermodynamic parameters,and a high-frequency cutoff above which hydrodynamics doesnot apply. In the quantum-critical systems of interest here,this high-frequency cutoff is provided by the quasinormalmodes. Thus the LTT describe the dynamics for frequenciesω/lessmuchT, while the quasinormal modes appear at ω∼Tand higher. We emphasize that the value of the dc conductivity,σ(ω/T=0), is determined by the full CFT. The nonanalytic small-frequency dependence associated with the LTT canbe obtained from the effective classical hydrodynamicdescription which takes the transport coefﬁcients of the CFTtreatment as an input. The focus of the present paper will beon the quasinormal modes, and we will not have any newresults on the LTT; the description of the LTT by holographicmethods requires loop corrections to the gravity theory, 21 which we will not consider here. From our quasinormal mode perspective, we will ﬁnd two exact sum rules that are obeyed by the universal quantumcritical conductivity, σ(ω), of all CFTs in 2 +1 dimensions with a conserved U(1) charge. These are /integraldisplay ∞ 0dω[/Rfracturσ(ω)−σ∞]=0, (1) /integraldisplay∞ 0dω/bracketleftbigg /Rfractur1 σ(ω)−1 σ∞/bracketrightbigg =0. (2) Here,σ∞is the limiting value of the conductivity for ω/greatermuchT(in applications to the lattice models to condensed matter physics,we assume that ωalways remains smaller than ultraviolet energy scales set by the lattice). The ﬁrst of these sum ruleswas noted in Ref. 22. From the point of view of the boundary CFT, Eq. (1)is quite natural in a Boltzmann approach; it is similar to the standard f-sum rule, which we extend to CFTs in Appendix A. There we connect it to an equal-time current correlator, which we argue does not depend on IRperturbations such as the temperature or chemical potential.The second sum rule follows from the existence of a S-dual (or “particle-vortex” dual) theory 15,18,23–25whose conductivity is the inverse of the conductivity of the direct theory. Although itcan be justiﬁed using the direct sum rule, Eq. (1), applied to the S-dual CFT, whose holographic description in general differsfrom the original theory, we emphasize that it imposes a furtherconstraint on the original conductivity. To our knowledge, thesecond sum rule has not been discussed previously. All ourholographic results here satisfy these two sum rules. We showin Appendix Bthat the N=∞ result of the O(N) model in Ref. 4obeys the sum rule in Eq. (1), a feature that was not noticed previously. However, such quasiparticle-Boltzmanncomputations do not obey the sum rule in Eq. (2).T h e holographic computations of the conductivity are the ﬁrst results which obey not only the sum rule in Eq. (1), but also the dual sum rule in Eq. (2). In principle, the quasinormal mode frequencies can also be determined by the traditional methods of condensed matterphysics. However, they are difﬁcult to access by perturbativemethods, or by numerical methods such as dynamical mean-ﬁeld theory. 26One quasinormal mode is, however, very familiar; the Drude peak of quasiparticle Boltzmann transport,appearing from the behavior σ(ω)∼σ0/(1−iωτ), corre- sponds to a quasinormal mode at ω=−i/τ. In a strongly- interacting quantum critical system, we can expect from thearguments of Ref. 4that this peak would translate to a quasinormal mode at ω∼−iT. As we will see in detail below, this single Drude-like quasinormal mode does not, by itself,provide a satisfactory description of transport, and we need tounderstand the structure of the complete spectrum of quasinor-mal modes. And the most convenient method for determiningthis complete spectrum is the AdS/CFT correspondence. As we indicate schematically in Fig. 1(a), the AdS/CFT description becomes exact for certain supersymmetric gaugetheories in the limit of a large number of colors N cin the gauge group.27–29This theory has no quasiparticles, and in the strictNc=∞ limit the conductivity is frequency independent even at T> 0, as indicated in Fig. 1(c). Our quasinormal mode theory expands away from this frequency-independent limit, incontrast to the free particle limit of the Boltzmann theory [in thelatter limit, the Drude contribution becomes σ(ω)∼Tδ(ω)]. We describe the basic features of σ(ω) obtained in this manner in the following subsection. Because strong interactions arecrucial to the structure of σ(ω) at all stages, and there is no assumption about the existence of quasiparticles, we expectour results to be general description of a wide class of stronglyinteracting quantum critical points. A. Generic features of the ﬁnite- Tconductivity of a CFT The frequency dependent conductivity of a CFT in 2 +1 dimensions at ﬁnite temperature will naturally be a functionof the ratio of the frequency to the temperature, ω/T , which we will denote as w, with a factor of 4 πconvenient in the holographic discussion, w≡ω 4πT. (3) In general, we do not expect the conductivity of a generic CFT to be a meromorphic function of the complex frequency w, i.e., analytic except possibly at a discrete set of points whereit has ﬁnite-order poles, all in the LHP. (The latter conditionfollows from the causal nature of the retarded current-currentcorrelation function.) The absence of meromorphicity for theconductivity of an interacting CFT, or the presence of branchcuts, can be attributed to the LTT. 20,30In the present paper, we will not discuss LTT and focus on the meromorphic structureof the conductivity. On the one hand, such a descriptionshould be valid for CFTs that have a holographic classicalgravity description. 22For example, there is strong evidence that certain super Yang-Mills large- Ncgauge theories are holographically dual to classical (super)gravity and do not haveLTT, which are suppressed by 1 /N 2 ccompared to the leading meromorphic dependence.30On the other hand, we believe that understanding the meromorphic structure is a ﬁrst step tounderstanding the full analytic structure of generic CFTs, anddo not expect branch cuts from the LTT to signiﬁcantly modifythe poles and zeros of the quasinormal modes at frequenciesof order Tor larger. The meromorphic condition is tantamount to assuming that in response to a small perturbation, the system will relaxexponentially fast to equilibrium at ﬁnite temperature. Inaddition to LTT, we expect deviations from such behavior 235115-3WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) 2 1 1 2Ω 4ΠT 321Ω4ΠT (a)0.5 1.0 1.5Ω4ΠT0.20.40.60.81.01.2 Σ Σ (b) 2 2 ΩΩ (c)1 2 3 4 5 6ΩT 0.020.020.040.060.080.100.12 Σ Σ2T (d) FIG. 3. (Color online) (a) Poles (crosses) and zeros (circles) of the holographic conductivity at γ=1/12. (b) Real and imaginary parts of the holographic conductivity on the real frequency axis. (c) Poles and zeros of the O(N) model at N=∞ ; the zeros coincide with branch points, and the associated branch cuts have been chosen suggestively, indicating that the branch cuts transform into lines of poles and zeros after collisions have been included. (d) Conductivity of the O(N) model at N=∞ ; note the δfunction in the real part at ω=0, and the co-incident zero in both the real and imaginary parts at ω=2/Delta1. In these ﬁgures /Delta1/T=2l n [ (√ 5+1)/2], and the O(N) computation is reviewed in Appendix B. to occur at a thermal phase transition for instance, where power law relaxation will occur. In that case σis not expected to be meromorphic and branch cuts can appear. Anotherexception is free CFTs, such as the O(N) model in the limit where N→∞ , where we ﬁnd poles and zeros directly on the real frequency axis, as well as branch cuts, as shownin Fig. 3(c). We restrict ourselves to the ﬁnite-temperature regime of an interacting conformal quantum critical point witha classical gravity description and do not foresee deviationsfrom meromorphicity. 22 Moreover, we expect the universal conductivity to go to a constant as w→∞ :4,31 σ(w→∞ )=σ∞<∞,w∈R. (4) Such a well-deﬁned limit will generally not exist as one approaches complex inﬁnity along certain directions in theLHP. This is tied to the fact that σwill not necessarily satisfy the stronger condition of being additionally meromorphicat inﬁnity. In other words, s(z):=σ(1/z) is not necessarily meromorphic in the vicinity of the origin, z=0. If it were, σ(w) would be a rational function, the ratio of two ﬁnite-order polynomials, and would have a ﬁnite number of poles (and ze-ros). In our analysis, we shall encounter a class of CFTs whoseconductivity has an inﬁnite set of simple poles, and is thus notmeromorphic on the Riemann sphere C∪{ ∞ } . A familiar example of such a function is the Bose-Einstein distribution,n B(w)=1/(ew−1), which is meromorphic, but not at inﬁnity because it has a countably inﬁnite set of poles on the imaginaryaxis. In fact, n B(1/z) has an essential singularity at z=0.A further generic property that σsatisﬁes in time-reversal invariant systems is reﬂection symmetry about the imaginary-frequency axis: σ(−w ∗)=σ(w)∗, which reduces to evenness or oddness for the real and imaginary parts of the conductivityat real frequencies, respectively. In particular, this means thatall the poles and zeros of σeither come in pairs or else lie on the imaginary axis. Following this discussion, we can expressthe conductivity as σ(w)=/producttextzeros /producttextpoles=/producttext l/parenleftbig w−ζ0 l/parenrightbig /producttext p/parenleftbig w−π0p/parenrightbig/producttext n(w−ζn)(w+ζ∗ n)/producttext m(w−πm)(w+π∗m), (5) where ζdenotes zeros and πpoles; {ζ0 l,π0 p}and{ζm,πm}lie on and off the imaginary axis, respectively. In this sense, thepoles and zeros contain the essential data of the conductivity.Actually, since σ(w→∞ )/σ ∞=1 on the real axis, which also holds for all directions in the UHP, they entirely determineσ/σ ∞. In the current holographic analysis, all the poles and zeros are simple, excluding double and higher order poles. Wesuspect this is a general feature of correlated CFTs. If oneis interested in the behavior on the real frequency axis only,the expression for the conductivity arising from the AdS/CFTcorrespondence can be truncated to a ﬁnite number of polesand zeros: we will show in Sec. II Ethat this leads to reasonable approximations to the conductivity on the real frequency axis.Such a truncated form can be compared with experimentallyor numerically measured conductivities for systems describedby a conformal quantum critical point. 235115-4QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) As we will show in this paper, the holographic methods allow easy determination of the poles in the conductivity,which are identiﬁed as the frequencies of the quasinormalmodes of the theory on AdS 4in the presence of a horizon at a temperature T. Moreover, the zeros in the conductivity emerge as the frequencies of the quasinormal modes of a S-dual (or “particle-vortex” dual) theory.15,18,23–25We summarize our holographic results for a particular parameter value in Fig. 3, along with the corresponding results for the O(N) model at N=∞ . TheO(N) model has a pole at ω=0, corresponding to the absence of collisions in this model at N=∞ . This turns into a Drude-like pole on the imaginary axis, closest to thereal axis in the holographic result. We show in Appendix B that the O(N) model also has a pair of zeros on the real axis, and this is seen to correspond to zeros just below the realaxis in the holographic result. Finally, the O(N) model has a pair of branch points on the real axis; the location of thebranch cuts emerging from these branch points depends onthe path of analytic continuation from the upper half plane.We have chosen these branch cuts in a suggestive manner inFig. 3(c), so that they correspond to the lines of poles and zeros in the lower-half plane of the holographic result. So we see anatural and satisfactory evolution from the analytic structureof the collisionless quasiparticles of the O(N) model, to the quasinormal modes of the strongly interacting holographicmodel. The outline of our paper is as follows. The holographic theory on AdS 4will be presented in Sec. II. We will use the effective ﬁeld theory for charge transport introduced inRef. 24, expanded to include terms with up to four space-time derivatives. The quasinormal modes will be computed usingmethods in the literature. 17,18,33,34Section IIIwill turn to the traditional quantum Boltzmann methods where new resultsregarding the analytic structure are given; in particular, weﬁnd that the low-frequency Boltzmann conductivity can beaccurately represented by a single Drude pole. II. HOLOGRAPHIC ANALYSIS The AdS/CFT holographic correspondence we use arose from the study of nonabelian supersymmetric gauge theoriesin the limit of a large number of colors, for example withgauge group SU( N c),Nc→∞ . By taking an appropriate limit for the gauge coupling, such theories are strongly interactingyet they can be described by weakly coupled gravity in anAnti-de-Sitter (AdS) space-time with one extended additionalspatial dimension, and six or seven compactiﬁed ones. Theﬁxed-point CFT describing the strongly correlated gaugetheory can be seen as existing on the boundary of AdS.Different correlation functions on the boundary quantum CFT,such as the charge-current ones of interest to this work, canbe computed by using the bulk (semi-)classical gravitationaltheory. For instance, the current operator corresponding to aglobal U(1) charge in the CFT can be identiﬁed with a U(1)gauge ﬁeld in the higher dimensional gravitational bulk (seeFig. 2). We refer the reader to a number of reviews 1,35,36 with condensed matter applications in mind and proceed to the holographic description of transport in 2 +1 dimensional CFTs.These CFTs are effectively described by a gravitational bulk theory in 3 +1 dimensions. In the case of the supersymmetric ABJM model37in a certain limit with an inﬁnite number of colors, the holographic dual is simply Einstein’s generalrelativity in the presence of a negative cosmological constantresulting in an AdS 4space-time. Charge-transport correlations functions in the CFT can be obtained from those a U(1) probegauge ﬁeld with Maxwellian action in the AdS background.It was shown 15that the conductivity of the large- NcABJM model is frequency independent due to an emergent S duality.Reference 24discovered that deviations from self-duality are obtained by considering four-derivative corrections to theEinstein-Maxwell theory, which can potentially arise at order1/λin the inverse ’t Hooft coupling. The effective action for the bulk gravitational theory discussed in Ref. 24reads S bulk=/integraldisplay d4x√−g/bracketleftbigg1 2κ2/parenleftbigg R+6 L2/parenrightbigg −1 4g2 4FabFab+γL2 g2 4CabcdFabFcd/bracketrightbigg , (6) where gis the determinant of the metric gabwith Ricci scalar R;Fabis the ﬁeld strength tensor of the probe U(1) gauge ﬁeld Aaholographically dual to the current operator of a global charge of the CFT. (We use roman indices forthe 3+1 space-time, and greek ones for the boundary 2 +1 space-time.) Such an action was also considered in Ref. 38. The four-derivative contribution to charge-transport can beencoded in the last term, proportional to γ.C abcd is the (conformal) Weyl curvature tensor; it is the traceless part ofthe full Riemann curvature tensor, R abcd:Cabcd=Rabcd− (ga[cRd]b−gb[cRd]a)+1 3Rga[cgd]b. We observe that the γ term directly couples the probe U(1) gauge ﬁeld to themetric. Lis the radius of curvature of the AdS 4space while the gravitational constant κ2is related to the coefﬁcient of the two-point correlator of the stress-energy tensor Tμνof the boundary CFT (for a review, see Ref. 39), an analog of the central charge of CFTs in 1 +1D. The gauge coupling constant g2 4=1/σ∞dictates the inﬁnite- wconductivity, which we shall set to 1 throughout, effectively dealing with σ/σ∞. The crucial coupling in this theory is the dimensionless parameter control-ling the four-derivative term, γ; it determines the structure of a three-point correlator between the stress-energy tensor and theconserved current. Stability constraints in the theory imply 24 that\|γ\|/lessorequalslant1/12, and we explore the full range of allowed γvalues here. Positive values of γyield a low-frequency peak in the conductivity as shown in Fig. 1(c) or7(a),w h i l e negative values of γgive rise to a low-frequency dip illustrated in Fig. 7(b), as may be expected from a theory of weakly interacting vortices. Explicit computations of γdirectly from the CFT yield values39in line with these expectations. In the spirit of the effective ﬁeld theory approach of Ref. 24, we should also consider adding other terms to Eq. (6) involving ﬁelds other than Faband the metric tensor.40The most important of these are possible “mass” terms, whichtune the CFT away from the critical point at T=0. Such terms are not present in the CFT at T=0, but their values at nonzero Tare precisely such that the expectation value of the mass operator does not change, e.g., in the quantumcritical O(N) model of Appendix A,/angbracketleftˆφ 2 α/angbracketrightisTindependent.41 235115-5WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) The mass terms can be included in the holographic theory by allowing for a scalar dilaton ﬁeld /Phi1and this can modify charge transport via a term ∼/Phi1FμνFμν. In the holographic theory, in the absence of external sources, such a dilatondoes not acquire an expectation value at T> 0 when it is not present at T=0. And external sources coupling to the gauge ﬁeld only modify /Phi1at quadratic order, and so /Phi1can be neglected in the tree-level linear response. Thus even afterallowing for additional ﬁelds, γremains the only important coupling determining the structure of the charge transport atnonzero temperatures. In the absence of the gauge ﬁeld, which is here only a probe ﬁeld used to calculate the linear response, the metric that solvesthe equation of motion associated with S bulkis ds2=r2 L2[−f(r)dt2+dx2+dy2]+L2dr2 r2f(r), (7) where f(r)=1−r3 0/r3andris the coordinate associated with the extra dimension. The CFT exists on the boundaryof AdS, r→∞ , on the Minkowski space-time parameterized by (t,x,y ). We emphasize here that the holographic theory is naturally written in real time allowing direct extraction of theretarded current-current correlation function characterizing theconductivity. Equation (7)corresponds to a 3 +1D space-time with a planar black hole (BH) whose event horizon is located atr=r 0, and that asymptotically tends to AdS 4asr→∞ .W e thus refer to it as Schwarzchild-AdS, or S-AdS. The positionof the event horizon is directly proportional to the temperatureof the boundary CFT, T=3r 0 4πL2. (8) AsT→0, the black hole disappears and we are left with a pure AdS space-time, which is holographically dual to thevacuum of the CFT. The statement that the thermal states ofthe CFT can be accessed by considering a BH in AdS canbe heuristically understood from the fact that the BH willHawking radiate energy that will propagate to the boundaryand heat it up. It will be more convenient to use the dimensionless coordinate u=r 0/r, such that Eq. (7)becomes ds2=r2 0 L2u2[−f(u)dt2+dx2+dy2]+L2du2 u2f(u), (9) f(u)=1−u3. The boundary, r=∞ ,i sn o wa t u=0, while the BH horizon is atu=1. The equation of motion (EoM) for the probe gauge ﬁeld is the modiﬁed Maxwell equation ∇a(Fab−4γL2CabcdFcd)=0, (10) where ∇adenotes a covariant derivative with respect to the background metric, gab. As we are interested in the current correlator in frequency-momentum space, we Fouriertransform the gauge ﬁeld: A a(t,x,y,u )=/integraldisplayd3k (2π)3e−iωt+ik·xAa(ω,kx,ky,u),(11) where the coordinate uwas left untransformed since there is no translational invariance in that direction. We shall actuallysolve for the full udependence of Aa. We work in the radial gauge Au=0. Without loss of generality, we also set the spatial momentum to be along the xdirection, ( kx,ky)=(k,0). In the limit where k→0, appropriate to a uniform “electric” ﬁeld coupling to the global charge, the equation of motion forthe transverse component A yreads A/prime/prime y+h/prime hA/prime y+9w2 f2Ay=0, (12) where we have deﬁned the dimensionless frequency win Eq. (3), and primes denote derivatives with respect to u. The function h(u)i ss i m p l y fg, where g=1+4γu3takes the same form as f=1−u3.A sg(u) fully encodes the γ dependence, we wish to make its role more transparent byrewriting the above equation: A /prime/prime y+/parenleftbiggf/prime f+g/prime g/parenrightbigg A/prime y+9w2 f2Ay=0. (13) The term g/prime/g=12γu2/(1+4γu3) is seen to be proportional toγ, and as such, goes to zero as u→0 consistent with the fact that the Weyl tensor vanishes in the pure AdS space-time,which is said to be conformally ﬂat. The AdS/CFT correspondence provides an expression for the conductivity of the CFT in terms of the transverse gaugeﬁeld autocorrelator evaluated at the boundary, u=0, σ(ω)=iG yy ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle u=0, (14) where σ(ω) is the complex valued conductivity, and Gyy(ω,u) is the retarded Ayautocorrelation function. More speciﬁcally, one gets15,24 σ(w)=−i 3w∂uAy Ay/vextendsingle/vextendsingle/vextendsingle/vextendsingle u=0, (15) where Aysolves the equation of motion Eq. (13) with suitable boundary conditions, as discussed below. The above equation,central to our analysis, has the following heuristic explanation:A y(0) acts as a source for the current, while ∂uAy( 0 )i st h e corresponding response. We will see in Sec. II C that the quasinormal modes, i.e. the poles of conductivity in the LHP,correspond to driving frequencies at which a “response” existsin the limit of vanishing source strength. A. Direct solution of conductivity The real part of the conductivity on the real frequency axis (retarded correlator) was numerically obtained in Ref. 24. We extend their analysis from real to complex frequencies,w∈C. The boundary conditions necessary to solve Eq. (13) are imposed at the BH event horizon 24atu=1. To obtain them we examine the EoM near the horizon, which admits thefollowing two solutions: A y∼(1−u)±iw. These correspond to outgoing and ingoing waves from the point of view ofthe BH, respectively. The retarded correlator is obtained bychoosing the ingoing condition. To implement this in thenumerical solution, we factor out the singular behavior: A y= (1−u)−iwF(u), where F(u) is the sought-after function; it is regular at the horizon. From Eq. (15), we see that we are free to ﬁx one of the two boundary conditions, either for Ay(1) orA/prime y(1), to an arbitrary ﬁnite constant without altering the 235115-6QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) (a) σ(w;γ=1/12)} (b) ˆσ(w;γ=1/12)} (c) σ(w;γ=−1/12)} (d) ˆσ(w;γ=−1/12)}FIG. 4. (Color online) Con- ductivity σand its S-dual ˆ σ= 1/σin the LHP, w/prime/prime=/Ifracturw/lessorequalslant0, for \|γ\|=1/12. The zeros of σ(w;γ) are the poles of ˆ σ(w;γ). We fur- ther note the qualitative correspon- dence between the poles of σ(w;γ) and the zeros of ˆ σ(w;−γ). conductivity. We impose Ay(1)=F(1)=1. The appropriate boundary condition for F/primecan be obtained by examining the differential equation near u=1a si sd i s c u s s e di nR e f . 24and in Appendix C. All the poles of the conductivity are in the LHP, as it is obtained from the retarded current-current correlationfunction. The numerical result is shown in Figs. 4(a) and 4(c) for the two values of γsaturating the stability bound, γ=± 1/12, respectively. Figure 4(a) shows the conductivity forγ=1/12, which corresponds to particle-like transport with a Drude peak at small real frequencies as can be seenon the real w-axis, or more clearly in Fig. 3(b) or7(a). Such low-frequency behavior is dictated by a Drude pole, locatedclosest to the origin. The numerical solution also shows thepresence of satellite poles, the two dominant ones being shown.These are symmetrically distributed about the /Ifracturwaxis as required by time-reversal, and are essential to capture thebehavior of σbeyond the small frequency limit. In contrast, the conductivity at γ=− 1/12 in Fig. 4(c) shows a minimum atw=0 on the real axis, see also Fig. 7(b) for a plot restricted to real frequencies. The corresponding pole structure shows nopoles on the imaginary axis, in particular no Drude pole. Theconductivity at γ=− 1/12 is said to be vortex like because it can be put in correspondence with the conductivity of the CFTS dual to the one with γ=1/12, as we now explain. B. S duality and conductivity zeros Great insight into the behavior of the conductivity can be gained by means of S duality, a generalization of the familiarparticle-vortex duality of the O(2) model. S duality on the boundary CFT is mirrored by electric-magnetic (EM) dualityfor the bulk U(1) gauge ﬁeld, which we now brieﬂy review.Given the Abelian gauge theory for the U(1) bulk ﬁeld A a, we can always perform a change of functional variables inthe partition function to a new gauge ﬁeld ˆA aby adding the following term to Sbulk,E q . (6): S/prime=/integraldisplay d4x√−g1 2εabcd ˆAa∂bFcd, (16) with the corresponding functional integral for ˆAa. Performing the integral over ˆAawould simply enforce the Bianchi identity, εabcd∂bFcd=0, implying Fab=∂aAb−∂bAa, where εabcd is the fully-antisymmetric tensor in 3 +1D with εtxyu=√−g. If instead one integrates out Aaﬁrst, a new action in terms ofˆAaresults: ˆSbulk=−/integraldisplay d4x√−g1 8ˆg2 4ˆFabˆXabcd ˆFcd, (17) where we have deﬁned the ﬁeld strength of the dual gauge ﬁeld, ˆFab=∂aˆAb−∂bˆAa, and dual coupling ˆg4=1/g4.A n exactly analogous action holds for Aawithout the hats. The rank-4 tensors X,ˆXare shorthands to simplify the actions: Xabcd=Iabcd−8γL2Cabcd, (18) ˆXabcd=1 4εabef(X−1)efghεghcd, (19) with the rank-4 tensor Iabcd≡δacδbd−δadδbc, the identity on the space of two forms, e.g., Fab=1 2IabcdFcd.T h ei n v e r s e 235115-7WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) tensor of Xis then deﬁned via1 2(X−1)abcdXcdef=Iabef.I n t e r m so ft h e Xtensors, the EoM for Aaand ˆAasimply read ∇b(XabcdFcd)=0, (20) ∇b(ˆXabcd ˆFcd)=0. (21) It can be shown24that for small γ, the dual Xtensor has the following Taylor expansion: ˆXabcd=Iabcd+8γL2Cabcd+O(γ2), (22) =Xabcd/vextendsingle/vextendsingle γ→−γ+O(γ2). (23) We thus see that if γ=0,X=ˆXand the actions, and associated EoM, for Aand ˆAhave the same form. In that case, the two theories are related by an exchange between electricand magnetic ﬁelds: the standard EM (hodge) self-duality of electromagnetism. In contrast, in the presence of thefour-derivative term parameterized by γ, the EM self-duality is lost. However, at small γthe EM duality is particularly simple and will serve as a guide for any ﬁnite γ: the holographic theory forγmaps to the one for −γ, neglecting O(γ 2) contributions. Let us now examine the impact of this bulk EM duality, A→ˆA, on the boundary CFT. The holographic correspon- dence relates the bulk gauge ﬁeld Ato the current of a global U(1) charge of the CFT, J.I nt h es a m ew a y ,t h e dual gauge ﬁeld ˆAwill couple to the current ˆJof the S-dual CFT, which generically differs from the original CFT. Just as the conductivity of the original CFT, σ, is related to the J autocorrelator, the conductivity of the S-dual CFT, ˆ σ, will be obtained from the ˆJautocorrelator. The conductivities of the S-dual CFT pair are in fact the inverse of each other: ˆσ(w;γ)=1 σ(w;γ), (24) where we emphasize that this relation holds for the complex conductivities, σ=/Rfracturσ+i/Ifracturσ. We present the short proof here using results of Ref. 24. (We note that such a result was de- rived for a speciﬁc class of CFTs in Ref. 15.) We begin with the general form of the retarded current-current correlation func- tion:Gμν(ω,q)=/radicalbig qλqλ[PT μνKT(ω,q)+PT μνKL(ω,q)], with the orthogonal transverse and longitudinal projectors PT,L: PT tt=PT ti=PT it=0,PT ij=δij−qiqj/q2, and by orthog- onality: PL μν=[ημν−qμqν/(qλqλ)]−PT μν. The Minkowski metric was introduced, ημν=diag(−1,1,1), such that qλqλ= ηλλ/primeqλqλ/prime=−ω2+q2. Of interest to us is the holographic re- lation between the transverse correlator giving the conductivityand the bulk gauge ﬁeld correlator, G μν: /radicalbig q2−ω2KT(ω,q)=Gyy(ω,q)\|u=0=ωσ(ω,q)/i, (25) where σ(ω,q) is the frequency and momentum dependent conductivity. The same expression (with hats) holds in theS-dual theory. Using the action of EM duality on the bulk,Ref. 24showed the relation: K T(ω,q)ˆKL(ω,q)=1, (26) that relates the transverse current-current correlator of the original CFT to the longitudinal one of the dual CFT. Whencombined with the fact that in the limit of vanishing spatial mo-mentum, q→0, rotational invariance enforces K T(ω,q)=KL(ω,q), which is also naturally true with hats, we obtain ˆKT(ω,q=0)=1 KT(ω,q=0). (27) By virtue of Eq. (25) and its dual version, this concludes the proof of Eq. (24). The poles of the dual conductivity, ˆ σ=1/σ, then must correspond to the zeros of the conductivity, σ, and vice versa. As a consequence, we see that S duality interchanges the locations of the conductivity zeros and poles .T h i si si s consistent with the direct solution shown in Fig. 4. Take for example the theory at γ=1/12, Fig. 4(a): it will have a Drude pole on the imaginary axis, which gives rise to a Drude peak atsmall frequencies. Under S duality this pole becomes a Drude zero of ˆσ,F i g . 4(b), and the conductivity of the new theory will have a minimum at small frequencies. As we saw above, changing the sign of γcorresponds to an approximate S duality valid for \|γ\|/lessmuch 1. More generally, in terms of the “pole/zero-topology” or ordering, both operationsare equivalent. Indeed, if we consider the pole/zero structureof the positive frequency branch of the conductivity /Rfracturw/greaterorequalslant 0 (which is sufﬁcient by time reversal) and order the polesand zeros according to their norm, we get the following twoequivalence classes: pole−zero−pole−zero−··· → particle-like (e.g.,γ > 0), (28) zero−pole−zero−pole−··· → vortex-like (e.g.,γ < 0), (29) where the ﬁrst label (in bold) designates the Drude pole or zero. Both S duality and γ→−γinterchange these two analytic structures. This underlies the qualitative correspondence be-tween the pole structure of σ(w;γ) and that of ˆ σ( w;−γ); for example, compare Figs. 4(a) and4(d),o rF i g s . 4(c) and4(b). The correspondence quantitatively improves in the limit ofsmallγ. Explicitly, σ(w;γ)≈1 σ(w;−γ),\|γ\|/lessmuch 1, (30) holds because performing σ→1/σtogether with γ→−γ is approximately tantamount to two S-duality transformationsand is equivalent to the identity, modulo O(γ 2)t e r m s . Finally, we mention that for a given γit is not possible to ﬁnd a γ/primesuch that ˆ σ(w;γ)=σ(w;γ/prime). In other words, the dual of the boundary CFT with parameter γcannot correspond to the original CFT with a different parameterγ /prime. This can be seen as follows. We ﬁrst require that the relation hold true at zero frequency: ˆ σ(0;γ)=σ(0;γ/prime), which implies 1 /(1+4γ)=1+4γ/primeorγ/prime=(1 1+4γ−1)/4, where we have used σ(0;γ)=1+4γ(see Refs. 24and 38). Although for this value of γ/prime,ˆσ(w;γ) and σ(w;γ/prime) agree for both w,1/w=0, we have numerically veriﬁed that they always disagree at intermediate frequencies, the disagreementdecreasing as γ→0, in which limit γ /prime≈−γ. The absence of aγ/primesatisfying ˆ σ(w;γ)=σ(w;γ/prime) is in accordance with the fact that holgraphic action of the S-dual CFT contains termsbeyond C abcdFabFcd. The latter is only the ﬁrst term in the Taylor expansion in γ. 235115-8QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) We now turn to a better method of determining the poles and zeros, as the direct solution of Eq. (12) can only reliably capture the poles nearest to the origin. The main problem withthe direct solution of the differential equation for A y,E q . (12), is that the Fourier modes Ay(u;w) at the UV boundary, u=0, generically grow exponentially as the imaginary part of thefrequency /Ifracturwbecomes more and more negative making the numerical results unstable. Although an exception occurs atthe poles, where A y(u=0;ωpole) vanishes (see below), it is hard to untangle the true analytical structure from the numer-ical noise, hence the need for a more sophisticated approach. C. Quasinormal modes and poles We present an alternative and more powerful method of capturing the poles by considering the so-called quasinormalmodes (QNMs) of the gauge ﬁeld in the curved S-AdS 4 space-time. These modes are eigenfunctions of the EoM forA y,E q . (12): A/prime/prime n+h/prime hA/prime n+9w2 n f2An=0, (31) where Anis a QNM with frequency wn. The QNM have the special property that they vanish at the boundary: An→0a s u→0. From the expression for the conductivity, Eq. (15),w e can see that this will lead to wnbeing a singular point of the conductivity: σ(wn)∼∂uAn An/vextendsingle/vextendsingle/vextendsingle/vextendsingle u=0∼∂uAn(0) 0→± ∞ , (32) where ∂uAn(0) is generically ﬁnite at the QNM frequencies whereAn(0)=0. [In contrast, the conductivity zeros or QNM of the EM-dual Maxwell equation correspond to frequencies atwhich ∂ uA(0)=0b u tA(0) is ﬁnite.] The name quasinormal instead of normal is used because the eigenfunctions An diverge approaching the BH horizon, u=1. This follows from the above-mentioned asymptotic form near the horizon,A n∼(1−u)−iwn=(1−u)w/prime/prime n−iw/prime n, implying a divergence for frequencies in the LHP. As predicted by the AdS/CFTcorrespondence and veriﬁed by our numerical analysis, shownin Fig. 5, the QNMs indeed agree with the poles of the conductivity shown in Fig. 4and more precisely in Fig. 11. The QNMs are found by using a Frobenius expansion A y=uf(u)−iwM/summationdisplay m=0am(u−¯u)m, (33) where we have factored out the behavior near the event horizon, f(u)−iw∼(1−u)−iw, and near the boundary, u.W e have chosen to Taylor expand around ¯u=1/2;M+1i st h e number of terms in the truncated series. Substituting Eq. (33) in Eq. (12) yields a matrix equation for the coefﬁcients, am: M/summationdisplay m=0Blmam=0, (34) where the left-hand side is the coefﬁcient of ( u−¯u)l,0/lessorequalslantl/lessorequalslant M. Note that Blm=Blm(w) andam=am(w) both depend on the frequency, and although not explicitly shown, on γas well. For ﬁxed γ, this homogeneous system of linear equations has a solution at a set of frequencies {wn}at which det B(wn)=0.Or equivalently, when the smallest-normed eigenvalue of B, λmin, vanishes, which we ﬁnd more convenient to implement numerically. Plots of 1 /\|λmin\|(multiplied by an exponential function to improve the visibility) as a function of ware given in Fig. 5for\|γ\|=1/12. The QNMs are the bright spots. In obtaining the QNMs of the dual conductivity, ˆ σ=1/σ,w e have used the EoM for the dual gauge ﬁeld ˆA,E q . (21): ˆA/prime/prime y+/parenleftbiggf/prime f−g/prime g/parenrightbigg ˆA/prime y+9w2 f2ˆAy=0. (35) It differs from the one for Ay,E q . (13), by the negative sign.24 Note that this shows that γ→−γdoes not exactly correspond to S duality, because the former would give −g/prime/(1−4γu3)/negationslash= −g/prime/g, where g=1+4γu3. Whereas the direct solution only gives reliable answers up to /Ifracturw∼− 1, the QNM approach has a wider range of applicability and is numerically more stable giving us moreinsight into the analytic structure. We have performed a WKBanalysis in Appendix Dto determine the asymptotic QNMs for\|w\|/greatermuch 1. We next examine the transition that occurs when going from positive to negative values of γ. D. Pole motion and S duality The motion of the poles and zeros as γchanges sign is illustrated in Fig. 6forγ> 0. For γ< 0, one simply interchanges the zeros and poles, i.e., the crosses and circles.The pole/zero motion can be loosely compared with a “zippermechanism.” The arrows in Fig. 6show the nontrivial motion of a pair of poles or zeros as they become “zipped” to theimaginary axis. (A caveat regarding the arrows: by time-reversal symmetry, w→−w ∗, so we cannot say which pole goes to which once they become pinned to the imaginary axis.The arrows are just a guide.) For sufﬁciently small γ, each point on the imaginary axis located at wzip n=−in/2, where n is a positive integer, will have a pole and zero arbitrarily closeto it. When γ=0, they will “annihilate” as it should because the complex conductivity for γ=0 has no poles or zeros as it takes the constant self-dual value for all complex frequencies.It should be noted that since w=ω/4πT, the annihilation frequencies are ω zip n=−i2πnT, n =1,2,3,..., (36) i.e., the bosonic Matsubara frequencies in the LHP. Although this results seems natural, we do not have a clear explanationfor it and leave the question for future investigation. Finally,from the direct numerical solution of the EoM, we have lookedat the residue of the pole near w=−i/2 (closest to the origin), and have found that it decreases linearly with γ, consistent with theγ=0 limit. The motion of a pair of poles becoming attached to the imaginary axis bears some similarity to that found in arecent paper, 19where as the (dynamic) spontaneous symmetry breaking happens, a pair of QNM poles becomes glued tothe imaginary axis. In their case, one of the poles stays at theorigin, signaling a gapless Goldstone boson. We will see belowone peculiar limit where a conductivity pole hits the origin. 235115-9WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) (a)σ(γ=1/12) (b) ˆσ(γ=1/12) (c)σ(γ=−1/12) (d) ˆσ(γ=−1/12)FIG. 5. (Color online) Quasinormal modes (bright spots) of the transverse gauge mode for γ=\|1/12\|in the com- plex frequency plane, w=w/prime+iw/prime/prime.T h e QNMs correspond to the poles of the conductivity (a) and (c). EM duality yields the QNMs of the dual gauge mode, andthese correspond to the poles of the dual conductivity, ˆ σ(w)=1/σ(w), i.e., the zeros of σ(w), see (b) and (d). E. Truncations If one is interested in the behavior on the real frequency axis only, the expression for the conductivity arising from theAdS/CFT correspondence can be truncated to a ﬁnite numberof poles and zeros. For instance, in a parameter regime believedto be of interest to the a wide class of CFTs, the conductivity has a single purely imaginary pole, accompanied by satellite poles off the imaginary axis. By truncating the number of poleswe obtain an excellent approximation to the exact dependenceas we show in Fig. 7(a):n pcounts the number of poles/zeros, not counting the time-reversal partners. The truncated conductivity reads σnp(w)=(w−ζ0) (w−π0)np−1/productdisplay n=1(w−ζn)(w+ζ∗ n) (w−πn)(w+π∗n), (37) where 2 np−1 is the odd number of poles or zeros (the −1 follows because the Drude pole/zero is its own time-reversalpartner). The value of the zero ζ 0is obtained by ﬁxing σ(0)= σ0. Just like π0, it lies on the imaginary axis: ζ0 π0=σ0np−1/productdisplay n=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleπ n ζn/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (38) It is included so that the truncated conductivity goes to a ﬁnite constant as lim w→∞σ=σ∞>0. Figure 7(b) showsthe corresponding dual conductivity, ˆ σ(w)=1/σ(w), whose poles/zeros correspond to the zeros/poles of σ. Note that the real part of the dual Drude conductivity, ˆ σ=1/σ= (1−w/π0)/σ0, is trivially constant (for real frequencies). III. EMERGENCE OF DRUDE FORM IN LARGE- N CFT’S AND BEYOND In this section, we examine the conductivity of CFTs such as the critical point of the O(N) model in a perturbative 1 /N expansion away from the free theory obtained for N=∞ , with a focus on the emergent pole structure. We are thusapproaching a general correlated CFT from the free quantumgas limit, as illustrated in the left-hand side of Fig. 1,i n contrast to the holographic approach. Our main example,though not the only one, is the O(N)N LσM. We show that the small-frequency quantum critical conductivity in the large- N limit accurately satisﬁes the Drude form: σ(ω)=σ 0 1−iωτ. (39) The quantum Boltzmann equation (QBE) approach in the hydrodynamic regime thus captures the leading QNM atsmall frequencies, but is limited in that it misses the otherpoles and all the zeros. Although it would be desirable to 235115-10QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) 0.5 0.0 0.51.61.41.21.00.80.60.4 ww (a)γ=1 0−2→10−30.5 0.0 0.51.61.41.21.00.80.60.4 ww (b)γ=1 0−3→10−4 0.5 0.0 0.51.61.41.21.00.80.60.4 ww (c)γ∼0+ FIG. 6. (Color online) Illustration of the motion of the poles and zeros as γgoes to zero in three steps: γ=10−2→10−3→10−4→0+. In each panel, the motion is from bold to thin as γdecreases; with crosses representing poles while circles, zeros. (a) Blue thick markers are forγ=10−2, while the red thin ones for γ=10−3. (b) The red thick markers are for γ=10−3, while the green thin ones for γ=10−4. (c) “Zipped” pole-zero structure for γ∼0+, where only poles and zeros far from the origin will lie off the imaginary axis. have a method that captures the full analytic structure of the conductivity of CFTs such as the O(N) model, the Drude pole nonetheless contains essential information regarding thedc limit. In addition, we can use the Drude form to verifysmall-frequency conductivity sum rules. The fact that a single pole can capture the small-frequency complex conductivity at large but ﬁnite Ncan seem a priori surprising given that the QBE that is solved to obtain σis fairly complicated, including both elastic and inelastic scattering ofthe critical quasiparticles. Below, we shed light on previousanalyzes 1,4,14by providing a transparent form for the solution to the QBE, which leads to the emergent Drude behavior ofthe low-frequency conductivity. Although we focus mainly ontheO(N) model, we provide similar results for a particular gauged O(N) model as well as for a fermionic CFT. Let us ﬁrst consider the case of the pure O(N) model. We focus on the small frequency limit, ω/lessmuchT, where the conductivity σadopts the universal scaling form 1,4 σ=e2 ¯h×N/Sigma1I/parenleftbiggNω T/parenrightbigg , (40) where eis the quantum of charge, and the subscript Iin the scaling function /Sigma1reminds us that it is valid only at small frequencies, ω/lessmuchT. The factors of Nare such that the small-frequency conductivity becomes a delta functionatN=∞ , the free limit. For ω/lessmuchT, the conductivitycontains important contributions from the incoherent inelastic scattering processes between the bosons. When Nis large these scattering processes can be treated perturbatively in1/N. 1,5We now present the essence of the QBE approach and the results; further details can be found in Refs. 1,5, and14. Under an applied oscillatory electric ﬁeld that couples to the charge, the distribution functions of the bosonic posi-tive/negative ( +/−) charge excitations are modiﬁed to linear order according to f ±(k,ω)=nB(/epsilon1k)2πδ(ω)+sE·kϕ(k,ω). [Note that the O(N) model has many conserved charges; we pick one and couple the “electric ﬁeld” to it.] It can be shownthat the linearized QBE for the deviation ϕtakes the form: 1,14 −i˜ωϕ+g(p)=−F(p)ϕ+/integraldisplay dp/primeK(p,p/prime)ϕ(p/prime),(41) where we have rescaled the frequency, ˜ ω=Nω/T , deﬁned the dimensionless momentum p=k/T , and absorbed factors ofTandNinto the unkown function ϕ. The right-hand side is the linearized collision term arising from the interactionsbetween the quantum critical modes appearing at order 1 /N. In the NL σM formulation, the system consists of a vector ﬁeld coupled to a single Lagrange multiplier ﬁeld that enforcesthe unimodular constraint for the former. The collision termarises from interactions between the vector ﬁeld and theLagrange multiplier, the latter aquiring dynamics at order1/N. It contains two terms: the ﬁrst, depending on a function 235115-11WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) 0.5 1.0 1.5 2.0w0.60.81.01.21.4Σ Exact np7 np4 Drude (a)0.5 1.0 1.5 2.0w0.60.81.01.21.4Σ1Σ Exact np7 np4 Drude (b) FIG. 7. (Color online) Conductivity (a) and its dual (b), ˆ σ=1/σ, arising from a holographic treatment with a truncated number of poles, 2np−1. One pole lies on the imaginary axis, the Drude pole, while np−1 pairs have a ﬁnite real part. The Drude form is characterized by a single pole: σ=σ0/(1−iωτ). F[see Fig. 8(b)], encodes elastic scattering processes; Fis essentially a momentum dependent scattering rate. The secondterm involves an integral over a kernel Kand it encodes the inelastic scattering processes with the Lagrange multiplierﬁeld. On the left-hand side the function g(k/T )=T∂ /epsilon1knB(/epsilon1k) acts as “source” for the QBE, where /epsilon12 k=/Delta1(T)2+k2and /Delta1∝T. More details regarding this temperature dependent mass (inverse correlation length) can be found in Appendices A andB. Solving the equation numerically, we ﬁnd that to great precision the solution satisﬁes the simple form ϕ(p,˜ω)=g(p) i˜ω−F(p), (42) whereF(p) is a monotonous function whose behavior closely resembles that of F(p), Eq. (41), as can be seen in Fig. 8(b). The case F=Fwould be the exact solution in the absence of the kernel Kin the right-hand side of Eq. (41). (The latter complicates the analysis and prevents analytical solubility.)We see that the effect the kernel Kis to renormalize Fto F, which encodes all the information about the nontrivial inelastic scattering processes. The corresponding solution forthe conductivity is shown in Fig. 8(a); it can be obtained 1,14by integrating ϕ: σ(ω/lessmuchT)=e2 ¯hN×1 2π/integraldisplay/Lambda1/T 0dpp3ϕ(p,˜ω) /epsilon1p/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright /Sigma1I(˜ω),(43) where /Lambda1is a momentum cutoff that is used in the numerical solution. We note that as ϕdecays exponentially at large momenta, a cutoff can be safely used. Interestingly, theresulting conductivity is found to obey a Drude form to greataccuracy: /Sigma1 I(˜ω)=/Sigma1(0) 1−i¯τ˜ω, (44) where /Sigma1(0)=0.085 and ¯ τ=τ/T=0.775 are two universal numbers that characterize the entire low-frequency charge response. The former yields the dc conductivity while the latteris a dimensionless scattering rate: σ 0=e2 ¯h×N/Sigma1(0), (45) τ=N¯τ T. (46) 2 4 6 8 10 12 14Ω0.020.040.060.08I,I (a)0 2 4 6 8 10p0.51.01.52.02.5 F (b) FIG. 8. (Color online) (a) Universal scaling function for the small-frequency conductivity /Sigma1I(˜ω) of the quantum critical O(N) model. The solid lines correspond to the numerical solution of the nontrivial QBE, while the dashed ones to the Drude form ﬁt. (b) The momentum-dependent F(p) function entering the kernel of the QBE, Eq. (41), and the renormalized Ffunction determining the solution of the QBE, Eq. (42). 235115-12QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) The plot for the Drude form is shown with dashed lines in Fig. 8(a). The numerical solution and the Drude forms are nearly indistinguishable over the entire range 0 /lessorequalslant˜ω< 14.5. The emergent scattering rate 1 /τgives the location of the only pole of the conductivity in this limit: ωDrude=−iT N¯τ. (47) AsNgrows, the pole approaches the origin along the imaginary axis in the LHP; once it reaches it, the low-frequencyconductivity becomes a delta function, as shown by the arrowin Fig. 1(b).T h eN=∞ conductivity is singular and cannot be described by a meromorphic function. This is to be expectedsince it describes the transport of a free gas of bosons asopposed to a generic correlated CFT. Although a Drude-like low-frequency conductivity can be expected from the broadening of the zero-frequencydelta function by interactions, 4we do not have a complete understanding regarding the excellent quantitative agreementmentioned above. We observe that many different deviationfunctions ϕcan give rise to a conductivity that is very well characterized by the Drude form. For example, one could useϕ(p)=1+1/(1+p)i nE q . (43) and obtain a very accurate Drude form. At the same time, numerous choices wouldyield clear deviations. One ingredient that seems to contributeto the Drude form is the presence of a nonparametricallysmall temperature dependent mass for the excitations, /Delta1∼T. In contrast, in the Wilson-Fisher ﬁxed point accessed bydimensional expansion in ε=3−d, where dis the spatial dimension of the O(N) model, the mass in the QBE can be neglected at leading order in ε. The resulting conductivity does not agree as well with the single-pole form. A furtherexample can be found below where we consider a CFT ofDirac fermions. The QBE for the conductivity can again besolved by ignoring the temperature-dependent mass to leadingorder, 5and we ﬁnd that although the Drude form ﬁts well, it is not as a successful when compared with the large- N O(N) model. A full treatment of these questions is beyond the scope of the present paper and we leave it for futurework. At this point, we can compare these numerical results with those from the holographic analysis. In the latter, we takeγ=1/12, which saturates the stability bound on the particle- like side and should be the most appropriate to compare withthe almost free large- NO (N) quantum critical point. Indeed, the further γis from the bound, the closer the effective theory is to the strongly interacting “ideal quantum ﬂuid”limit found at γ=0. At γ=1/12, we ﬁnd that the Drude pole is located at w hol Drude≈− 0.26i[see Fig. 5(a) or11(a) ], which translates to ωhol Drude=−i4πwhol DrudeT≈−i3.27T.O n the other hand, the Drude pole of the O(2) model obtained by extending the result from the large- Nlimit, Eq. (47),i s located at ωDrude≈−i0.65T. The Drude pole from the QBE approach is thus located closer to the origin compared to theone arising from the holographic analysis. We thus predictthat higher 1 /Ncorrections to the QBE will push the pole further down in the LHP. This is not surprising because theextension of the large- Nresult to N=2 yields a ratio of the dc to high-frequency conductivities, σ 0/σ∞, that is larger thanwithin the holographic analysis: σ0 σ∞=N/Sigma1(0) /Sigma1(∞)N=2−−→2.13,large-NO (N) model, (48) σ0 σ∞=1+4γ=1.33,holography, (49) where we have used /Sigma1(∞)=(1−8η/3)/16N=2−−→0.039 98 as the large-frequency scaling function for the conductivityof the O(N) model at order 1 /N, with η∝1/Nbeing the anomalous dimension of the boson ﬁeld. 32It is expected that higher order 1 /Ncorrections will decrease this ratio and will thus push the Drude pole further away from the origin. A. Interactions spread the weight Using the above quasiexact Drude dependence, we can examine the sum rule for the low-frequency part of theconductivity. This is a limited version of the sum rules forthe full universal conductivity, Eqs. (1)and(2). The sum rule reads/integraldisplay ∞ 0d˜ω/Rfractur/Sigma1I(˜ω)=πD/ 4=0.172 350 6 ..., (50) where we have deﬁned the constant πD=/integraldisplay∞ /Theta1dx/parenleftbigg 1+/Theta12 x2/parenrightbigg1 ex−1=0.689 403 ..., (51) where /Theta1=2l n [ ( 1 +√ 5)/2] is twice the natural logarithm of the golden ratio. The integral involving the Bose-Einsteinfunction follows simply from the expression of the conduc-tivity in the free theory at N=∞ , see Appendix B.I n that limit, the low-frequency part of the conductivity reads/Rfracturσ I(ω)=(TπD/ 2)δ(ω). On the other hand, the Drude form, Eq.(44), satisﬁes the following relation: /integraldisplay∞ 0d˜ω/Rfractur/Sigma1I(˜ω)=/integraldisplay∞ 0d˜ω/Rfractur/braceleftbigg/Sigma1(0) 1−i¯τ˜ω/bracerightbigg =π 2/Sigma1(0) ¯τ=0.172 21 ..., (52) where in the last equality we have used the result given above for /Sigma1(0) and ¯ τ. We ﬁnd that the emergent Drude form satisﬁes the sum rule Eq. (50) within a margin of 10−4, leaving plenty of room for numerical uncertainty. We thus see that theinteractions generated at order 1 /Nspread the weight of the δfunction over a ﬁnite Drude peak, whose area corresponds exactly to that of the δfunction of the free theory at N=∞ . Not only is this an excellent check on the calculation, it alsoprovides a constraint between the location of the Drude poleand the value of the dc conductivity. We are effectively left witha single universal number characterizing the small-frequencybehavior of the complex conductivity at low frequencies. B. Flattening the conductivity with gauge bosons We now consider an interesting application of the above sum rule to a gauged O(N) model, where the gauge ﬁeld is Landau damped by a Fermi surface of spinons,14,42which breaks conformal invariance of the critical point. This ﬁeldtheory was shown to be relevant to the quantum critical Motttransition from a metal to quantum spin liquid, 42as well as for the quantum critical transition between a N ´eel-ordered 235115-13WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) 5 10 15 20 25 30Ω0.0020.0040.0060.0080.010I,I FIG. 9. (Color online) Universal scaling function for the conduc- tivity/Sigma1(˜ω) of the gauged O(N) model, with damped gauge ﬁeld. The solid lines correspond to the numerical solution of the nontrivialQBE, while the dashed ones to the Drude form. Fermi-pocket metal and a non–Fermi-liquid algebraic charge liquid, called a “doublon metal.”43It was shown14that the same scaling form, Eq. (40), holds as for the pure rotor model, Eq.(40), since only the static gauge ﬂuctuations contribute, the dynamical ones being strongly quenched by the Landaudamping. This phenomenon was referred to as a “fermionicHiggs mechanism.” 43The numerical solution to the QBE including the static gauge ﬂuctuations is shown in Fig. 9(for details, see Ref. 14). As in the case of the pure O(N)C F T ,i t obeys a Drude form, Eq. (39) with Drude parameters Eqs. (45) and(46), this time with numerical values: /Sigma1(0)=0.010,¯τ=0.092. (53) The dc conductivity /Sigma1(0) is smaller than in the ungauged O(N) model due to the additional scattering channel: the gauge bosons. The static gauge ﬂuctuations are actually quitestrong and thus appreciably decrease the scattering time. Thenumerical solution and the Drude form agree very well again.Note the large range of scaled frequencies over which theagreement occurs. The deviations between the Drude andnumerical solution seem slightly larger than in the purerotor theory probably due to numerical uncertainties. Thelow-frequency sum rule for the conductivity, Eq. (50), yields π 2/Sigma1(0) ¯τ=0.1720..., (54)differing from πD/ 4b yo n l y3 .5×10−4. We see that as we add Landau damped gauge bosons to the pure O(N) model, we ﬂatten the conductivity while keeping the emergent Drudeform. The interactions, again, preserve the weight of the Drudepeak. C. Fermionic CFT We now examine the conductivity in an interacting CFT of Dirac fermions that arises in a model for transitions betweenfractional quantum Hall and normal states. 5The ﬁeld theory consists of two Dirac fermions with masses M1andM2 coupled to a Chern-Simons gauge ﬁeld. The latter attaches ﬂux tubes to each Dirac fermion converting it to a Diracanyon with statistical parameter (1 −α), where α=g 2/(2π), gbeing the gauge coupling. The coupling αcharacterizes the strength of the long range interaction between the Diracquasiparticles mediated by the Chern-Simons ﬁeld. WhenM 1,M 2>0, the system is in a fractional quantum Hall state with Hall conductivity σxy=e2q2/[h(1−α)], where qeis the electric charge of each Dirac quasiparticle. The transition toan insulating state is obtained at the point where M 1changes sign while M2is taken to be large and constant. At the quantum critical point, the M1Dirac quasiparticles coupled to the Chern-Simons gauge ﬁeld yield a ﬁnite and universallongitudinal conductivity, whose small-frequency functionalform is analogous to Eq. (40): ˜σ qp xx(ω)=q2e2 α2h˜/Sigma1qp xx/parenleftbiggω α2T/parenrightbigg , (55) where 1 /α2plays the same role as Ndid in the O(N) model and is taken be large. To be more accurate, ˜ σis the response to the total electric ﬁeld, including a contribution from theemergent Chern-Simons ﬁeld. It can be simply related tothe physical conductivity. 5The superscript “qp” reminds us that this is the low-frequency contribution arising from thescattering of thermally excited quasiparticles with each other;it is simply a different notation for /Sigma1 I. A QBE was numerically solved5to leading order in α2, and the result is reproduced in Fig. 10(a) , while the corresponding Drude form ﬁt is shown in Fig. 10(b) . Again, both plots agree very well. The two universal Drude parameters extracted from 00.20.4 0 5 10 15~ xxqp ~ω (a)0 5 10 15Ω0.20.4xxqp (b)FIG. 10. (Color online) Universal scaling functions for the conductivity of interacting Dirac fermions (a) as com-puted by solving a QBE, 5(b) from the Drude form ﬁtted to (a). 235115-14QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) the ﬁt are /Sigma1qp xx(0)≈0.437,¯τ≈0.664. (56) The sum rule for the model is given in Ref. 5: /integraldisplay∞ 0d˜ω π/Rfractur/bracketleftbig˜/Sigma1qp xx(˜ω)/bracketrightbig =ln 2 2=0.3466..., (57) where ˜ ω=ω/(α2T). By using the Drude form /Sigma1qp xx(˜ω)= /Sigma1qp xx(0)/(1−i¯τ˜ω), we ﬁnd /integraldisplay∞ 0d˜ω π/Rfractur/bracketleftbig˜/Sigma1qp xx(˜ω)/bracketrightbig ≈0.33. (58) The agreement is again quite good. In summary, we have shown that the Drude form with its single pole captures well the low-frequency hydrodynamicconductivity of different CFTs, a fact that was not appreciatedbefore. We have also seen that such a description holds for adeformation of the O(N) model to include nearly static gauge modes. Low-frequency sum rules where veriﬁed in all themodels and serve as a useful guide in the study of interactionson the charge response. IV . CONCLUSIONS The main thesis of this paper is that charge transport of CFTs in 2 +1 dimensions is most efﬁciently described by knowledge of the poles and zeros of the conductivity inthe lower half of the complex frequency plane. Truncationto a small number of poles and zeros gives an accuratedescription of the crossover from the hydrodynamic physicsat small frequencies to the quantum-critical physics at highfrequencies, as was shown in Sec. II E. Such truncated forms can be used as a comparison ground with experimentally ornumerically measured charge response at conformal quantumcritical points. We also showed that the conductivity of CFTswith a global U(1) symmetry exactly obeys two sum rules,Eqs. (1)and(2), for the conductivity and its (S-dual) inverse. The holographic computations presented here are the ﬁrstto satisfy both sum rules, while earlier quantum Boltzmann-theory computations satisfy only one of them. In the holographic approach, the poles and zeros of the conductivity are identiﬁed with quasinormal modes of gaugeﬁeld ﬂuctuations in the presence of a horizon. These quasinor-mal modes are the proper degrees of freedom for describingquantum critical transport, replacing the role played by thequasiparticles in Boltzmann transport theory. We presentedresults for the quasinormal mode frequencies in an effectiveholographic theory for CFTs which kept up to four derivativeterms in a gradient expansion. We expect that the quasinormal modes will help describe a wide variety of dynamical phenomena in strongly-interactingquantum systems, including those associated with deviationsfrom equilibrium. 19The quasinormal mode poles and zeros should also help in the analytic continuation of imaginarytime data obtained from quantum Monte Carlo simulations. ACKNOWLEDGMENTS We are grateful for many enlightening discussions with B. Burrington, S.-S. Lee, A. Singh, and X.-G. Wen. Wealso wish to thank P. Ghaemi, A. G. Green, M. Killi, Y .-B. Kim, J. Maldacena, J. Rau, T. Senthil, and R. Sorkin foruseful conversations. This research was supported by theNational Science Foundation under grant DMR-1103860 andby the Army Research Ofﬁce Award W911NF-12-1-0227(S.S.) as well as by a Walter Sumner Fellowship (W.W.-K.).S.S. acknowledges the hospitality of the Perimeter Institute,where signiﬁcant portions of this work were done. Research atPerimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontariothrough the Ministry of Research & Innovation. APPENDIX A: CONDUCTIVITY SUM RULES Conductivity sum rules are familiar in condensed matter physics in systems with a ﬁnite lattice cutoff. The standardderivation starting from the Kubo formula for a generalHamiltonian, H, yields 44 I≡/integraldisplay∞ 0dω/Rfracturσ(ω)=−π 2lim q→01 q2V/angbracketleft[[H,ρ(q)],ρ(−q)]/angbracketright, (A1) where ρ(q) is the density operator at wave vector q, andV is the system’s volume. It is now our task to understand thestructure of the commutators on the right-hand side in thescaling limit appropriate for a CFT in 2 +1 dimensions. In quantum ﬁeld theory, the right-hand side of Eq. (A1) has the structure of an ultraviolet divergent Schwinger contactterm. 45The divergence is acceptable to us, because the sum rule in Eq. (1)is convergent only after the subtraction of the constant σ∞term. The important issue for us is whether the right-hand side of Eq. (A1) has any ﬁnite corrections that depend upon infrared energy scales such as the temperatureor chemical potential ( μ). If such ﬁnite corrections are absent, then the sum rules in Eqs. (1)and (2)follow immediately, because σ ∞is the value of the σ(ω)a tT=0 andμ=0, and the integral is independent of Tandμ. It is useful to analyze this issue ﬁrst for a simple CFT of free Dirac fermions. Here we can regularize the Dirac fermions ona honeycomb lattice (as in graphene). Fortunately, such a sumrule analysis for the honeycomb lattice has already been carriedout in Ref. 46. On a lattice with spacing a, Fermi velocity v F, temperature T, and chemical potential μ, they ﬁnd when T andμare smaller than the bandwidth that I=c1vF a+a2T3 v2 Ff(μ/T ), (A2) for some constant c1and function f. Observe that this is divergent in the continuum limit ( a→0a tﬁ x e d vF, T,μ), but the leading portion dependent upon Tandμ vanishes. So there is no dependence of Iof the CFT upon μ andT. Let us now carry out the corresponding analysis for the large-Nlimit of the O(N) rotor model. This is an interacting theory at ﬁnite N, and we will see that the scaling limit has to be taken carefully so that we remain properly in the vicinity of theconformal ﬁxed point in the presence of infrared perturbationslikeTor deviations from the critical point. We regularize the rotor model on a square lattice of sites i,j, spacing a, with the 235115-15WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) Hamiltonian H=ga2 2N/summationdisplay iˆπ2 iα+c2N 2g/summationdisplay /angbracketleftij/angbracketright(ˆφiα−ˆφjα)2, (A3) where ˆφiα, with α=1...N are the rotor co-ordinates, which obey the constraint /summationdisplay αˆφ2 iα=1( A 4 )at all sites i.T h e ˆπiαare their conjugate momenta with [ˆφiα,ˆπjβ]=iδαβδij a2. (A5) The coupling constant gis used to ﬁx the model in the vicinity of the critical point at g=gc, and we will take the continuum limita→0 at ﬁxed velocity candT.I nt h el a r g e Nlimit, the critical point is at 1 gc=/integraldisplay k∈BZ/integraldisplaydω 2π1 {ω2+2(c/a)2[2−cos(kxa)−cos(kya)]}. (A6) This determines gc≈3.11ac. If we move away from the critical point, or to nonzero temperatures, then the Lagrange multiplier enforcing the constraint Eq. (A4) induces an energy gap /Delta1(T) determined by 1 g=/integraldisplay k∈BZT/summationdisplay ωn1/braceleftbig ω2n+2(c/a)2[2−cos(kxa)−cos(kya)]+/Delta12(T)/bracerightbig, (A7) where ωnare the bosonic Matsubara frequencies. We will take the limit a→0a tﬁ x e d /Delta1(T) andT. In this limit, we have 1 g=1 gc−/Delta1(0) 4π. (A8) The density operator is ρ(q)=a2/summationdisplay ie−iq·rilαβˆφiαˆπiβ, (A9) where lαβis one of the antisymmetric generators of O(N) normalized so that Tr( l2)=− 1. Evaluating the commutator in Eq. (A1) , we ﬁnd [[H,ρ(q)],ρ(−q)]=−2c2 g/summationdisplay /angbracketleftij/angbracketrightˆφiαˆφjα\|eiq·ri−eiq·rj\|2. (A10) So taking the limit, the long-wavelength limit yields lim q→01 q2[[H,ρ(q)],ρ(−q)]=−c2a2 g/summationdisplay /angbracketleftij/angbracketrightˆφiαˆφjα. (A11) Using Eq. (A4) , we can now write the conductivity sum rule as I=πc2 2g−πc2a2 4gV/summationdisplay /angbracketleftij/angbracketright/angbracketleft(ˆφiα−ˆφjα)2/angbracketright=πc2 2g−πc2 2/integraldisplay k∈BZT/summationdisplay ωn[2−cos(kxa)−cos(kya)]/braceleftbig ω2n+2(c/a)2[2−cos(kxa)−cos(kya)]+/Delta12/bracerightbig.(A12) Evaluating the frequency summation, and then taking the limit a→0, we obtain the expansion I=πc2 2g−α1c a+α2/Delta12 ca−a2πc2 4/integraldisplay∞ 0d2k 4π2k2 √ c2k2+/Delta12(e√ c2k2+/Delta12/T−1)+··· , (A13) where α1≈0.75 and α2≈0.13. The crucial feature of this result is that there is no term ∼/Delta1, and all terms containing /Delta1vanish as a→0. A term ∼/Delta1does appear if we choose a general /Delta1, which does not obey Eq. (A7) and then evaluate Eq.(A11) . Thus the imposition of the constraint Eq. (A4) at all Twas important for the absence of such a term. The general features of Eq. (A13) are similar to Eq. (A2) , and so the same conclusions apply.APPENDIX B: ANALYTIC STRUCTURE IN THE N→∞ LIMIT OF THE O(N)M O D E L This Appendix notes a few features of the conductivity of theO(N) rotor model in the complex frequency plane, in the N→∞ limit. For the model in Eq. (A3) , the conductivity as a function of the complex frequency zfollows from Ref. 4: σ(z)=iTD z+iz 4π/integraldisplay∞ /Delta1d/Omega1(/Omega12−/Delta12) /Omega12(z2−4/Omega12)coth/parenleftbigg/Omega1 2T/parenrightbigg , (B1) 235115-16QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) where the contour of /Omega1integration determines the speciﬁc choice of the current correlator, and the Drude weight scaleslinearly with the temperature. We have deﬁned the numericalconstant D=1 8π/integraldisplay∞ /Delta1d/Omega1(/Omega12−/Delta12)/T2 /Omega1sinh2[/Omega1/(2T)](B2) whose value is given in Eq. (51). The retarded response function σR(z) is obtained by choosing zin the UHP, and the contour of integration along the real frequency axis. This function σR(z) is analytic in the UHP and has a pole at z=0 and branch points at z=± 2/Delta1. We can perform the analytic continuation of σR(z) into the lower-half plane by deforming the contour of /Omega1integration into the lower-half plane, so that it is always below the points ±z/2. Because of the presence of these branch points, the analyticcontinuation of σ R(z) into the lower-half plane is not unique, and depends upon the path of zaround the branch points. This is a key difference from the holographic results of the presentpaper, which had no branch points and a unique analyticcontinuation into the LHP. We expect that fully incorporating1/Ncorrections will make the O(N) model result similar to the holographic computation. We have already demonstratedthis for the case of the pole at z=0, which becomes a LHP Drude pole. However, a careful analysis of 1 /Ncorrections determining the fate of the branch points at z=± 2/Delta1has not yet been carried out. In any case, the physical value on the real axis σ R(ω+i0+) is unique, and was shown in Fig. 3(d). At the critical point, this is to be evaluated at /Delta1=/Theta1T, where /Theta1=2l n [ (√ 5+1)/2)]. Curiously, for this value of /Delta1, we ﬁnd zeros of the conductivity on the real axis branch points, with σR(±2/Theta1T+i0+)=0. So the structure of poles and zeros of the N=∞ conductivity has a remarkable similarity to the γ> 0 holographic results, aswas reviewed in Fig. 3. The pole at z=0o ft h e N=∞ theory corresponds to the closest pole on the negative imaginary axisof the holographic result, as we have already noted. And thezeros at z=± 2/Theta1T of theory correspond to the two zeros closest to the real axis in Fig. 4(b). Finally, we can verify that the sum rule in Eq. (1)is satisﬁed by Eq. (B1) : /integraldisplay ∞ 0dω/bracketleftbigg /RfracturσR(ω+i0+)−1 16/bracketrightbigg =0, (B3) w h e r ew eh a v eu s e d σ∞=1/16. Note that this result is obeyed only for/Delta1=/Theta1T, and not for other values of /Delta1, as is expected from the considerations in Appendix A. Also, as noted in the introduction, the inverse sum rule in Eq. (2)is not satisﬁed by Eq.(B1) . Although σ(ω) has a zero at ω=2/Delta1, the location of the branch point, this nevertheless leads to an integrabledivergence in /Rfractur[1/σ(ω)] at that point. We have indeed veriﬁed that the integral of /Rfractur[1/σ(ω)]−σ −1 ∞is ﬁnite (actually, it is greater than unity), proving that the conductivity of the criticalO(N→∞ ) model does not respect the S-dual sum rule. Let us also mention that the analytic structure of response functions of the O(N) model was also examined recently in Ref. 47away from the CFT critical point, but at T=0. In the ordered phase with broken O(N) symmetry, poles were found in the lower-half plane corresponding to the Higgs excitationsdamped by multiple spin-wave emission. APPENDIX C: DIFFERENTIAL EQUATION FOR THE NUMERICAL SOLUTION OF THE CONDUCTIVITY We ﬁrst factor out the singular part of Aynear the horizon: Ay=(1−u)−iwF(u). Making this substitution in the EoM forAy,E q . (12), we obtain the following differential equation forF: 0=F/prime/prime−/braceleftbigg3u2[1−4(1−2u3)γ] (1−u3)(1+4u3γ)−2iw 1−u/bracerightbigg F/prime +iw{(1+u+u2)[1+2u+4u2(3+4u+5u2)γ]−i(2+u)(4+u+u2)(1+4u3γ)w} (1−u)(1+u+u2)2(1+4u3γ)F. (C1) This is to be compared with the simpler form of the equation for the full Ay,E q . (12). The two boundary conditions at the horizon read F(1)=1, (C2) F/prime(1)=iw[i+2w+8γ(2i+w)] (1+4γ)(i+2w). (C3) The second condition follows from the solution of the differential equation near u=1:F(u)≈1−(1−u)Ϝ, with Ϝbeing the right-hand side of Eq. (C3) . The numerical solution is shown in Figs. 4and 11, where the poles and zeros in the LHP can be seen more precisely.APPENDIX D: WKB ANALYSIS FOR ASYMPTOTIC QUASINORMAL MODES The goal of the WKB analysis is to identify the QNMs of the gauge ﬁeld at large frequencies, \|w\|/greatermuch 1. According to the AdS/CFT correspondence, these frequencies can then beput in correspondence with the poles of the gauge correlationfunction G yyproportional to the conductivity, Eq. (14).T h e standard analysis examines the solutions to Eq. (D15) near (1) the black-hole singularity, (2) the event horizon, and (3)the asymptotic boundary. Matching of the solutions usuallygives an expression for a set of discrete QNM frequencies.Generically one obtains two solution for A y, with one vanishing as the boundary is approached. Discarding thenonvanishing one leads to a “quantization” condition on theQNMs. 235115-17WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) (a) σ(γ=1/12)} (b) ˆσ(γ=1/12)} (c) σ(γ=−1/12)} (d) ˆσ(γ=−1/12)}FIG. 11. (Color online) Conductivity σand its dual ˆ σ=1/σin the LHP, w/prime/prime=/Ifracturw/lessorequalslant0, for \|γ\|=1/12. There is a qualitative correspondence of the pole structure between σ(w;γ)a n d ˆσ(w;−γ). Note that the poles of ˆ σ(w;γ) are the zeros ofσ(w;γ). As mentioned in the main text, the EoM for the ycomponent of the gauge ﬁeld reads 0=A/prime/prime y+h/prime hA/prime y+9w2 f2Ay, (D1) h/prime h=f/prime f+g/prime g. (D2) The second equality follows from h=fg. We can change coordinates to bring this equation into a Schr ¨odinger form, which will be more convenient for the analysis of the QNMs.To do so, we want to transform away the linear-derivativeterm. One way involves changing variables to dx=du/ f ,a s we illustrate below. Before going into the WKB analysis, let us ﬁrst review the simplest scenario, γ=0, i.e., in the absence of the function g arising from the Weyl curvature coupling. The exact solutionis obtained by using the new (complex) coordinate z: dz du:=3 f=3 1−u3. (D3) This puts Eq. (D1) in the form ∂2 zAy+w2Ay=0( D 4 ) with solutions e±iwz. To apply the boundary condition, we need to examine the explicit form of z(u). Integrating Eq. (D3) ,w eobtain z(u)=3/summationdisplay p=13 f/prime(up)ln(1−u/up), (D5) where upare the 3 zeros of f. They are simply the cubic roots of unity: u3 p=1, i.e., u1=1, (D6) u2=− (1+i√ 3)/2,u 3=u∗ 2, (D7) which is trivially found by noting that 1 −u3=(1−u)(u2+ u+1). We give a few properties of the generating polynomial fand its roots that will be useful for future analysis. First, the derivative of fpermutes u2andu3, while leaving u1 invariant (up to signs): f/prime(u1)/3=−u1andf/prime(u2)/3=−u3. As a result, we get the following identities: 3/summationdisplay p=1up=0, (D8) 3/summationdisplay p=1un p f/prime(up)=/braceleftbigg −1i f nmod 3 =2, 0 otherwise.(D9) Recall that we need to apply an infalling boundary condi- tion,Ay≈(1−u)−iw, near the event horizon, u=1. Using Eq.(D5) , we ﬁnd that as u→1, e±iwz→C±×(1−u)∓iw(D10) 235115-18QUASINORMAL MODES OF QUANTUM CRITICALITY PHYSICAL REVIEW B 86, 235115 (2012) where C±=e±iw(ln 3+π/√ 3)/2. Hence, the boundary condition selects Ay=eiwz. This in turn yields σ=−i∂uAy 3wAy\|u→0= −i3iw 3w(1−u3)\|u→0=1. As expected the conductivity of the CFT holographically dual to the Einstein-Maxwell theory on S AdS 4is constant for all complex frequencies, hence self-dual. We now include a ﬁnite γ, which prevents analytical solubility, just like the 1 /Ncollision term did for the O(N) model. We wish to transform Eq. (D1) into a Schr ¨odinger form. To facilitate comparison with the literature, notably with Ref. 48 which serves as a guide for our analysis, we shall perform theWKB analysis starting with the coordinate r=1/uinstead ofu. This is the radial holographic coordinate introduced in the main body, with the difference that it is rescaled by r 0.W e deﬁne f=r2f=r2−r−1and the corresponding new tortoise coordinate (the analog of zintroduced above): dx dr=1 f. (D11) In terms of x, the EoM for Aybecomes d2Ay dx2+1 gdg dxdA dx+ν2Ay=0,ν=3w. (D12) We have deﬁned the rescaled frequency νto simplify the comparison with previous works. We note that in the limitwhere γ=0, the linear derivative term vanishes and we are left with a trivial harmonic equation as above. For ﬁnite γ, we can remove such a term by introducing two functions toparametrize A y: Ay=G(x)ψ(x), (D13) where in order for ψto satisfy an equation of the Schr ¨odinger form,Gneeds to satisfy the ﬁrst order differential equation: dG dx+1 2gdg dxG=0. (D14) This can be solved in general by G=1/√g=1//radicalbig 1+4γu3. The resulting “Schr ¨odinger” equation for ψis −d2ψ dx2+W(x)ψ=ν2ψ, (D15) where W=6γ(r3−1) r4(r3+4γ)2[2r6+(2γ−5)r3−14γ].(D16) The potential Wprevents the exact solubility of the equation, and as expected vanishes as γ→0. In that limit, G→1 andW→0, and the equation reduces to the harmonic one Eq. (D4) . Note that the potential vanishes at the boundary, r=∞ , just as the Weyl curvature does. The underlying idea of the WKB method is to examine the behavior of Ayorψon the Stokes line in the complex rplane deﬁned via: /Ifractur(νx)=0. (D17) The ﬁrst step is thus to identify this Stokes line by studying the behavior of the tortoise in terms of r. As above, the deﬁning FIG. 12. (Color online) The Stokes line, /Ifractur(νx)=0, in black in the complex rplane; r=0 corresponds to the intersection point of the two branches of the Stokes line. The color shading represents the value of /Ifractur(νx). The three branch cuts coming from the logarithms are clearly visible. relation for the tortoise can be integrated to give x(r)=1 33/summationdisplay p=11 f/prime(rp)ln(1−r/rp) (D18) =1 3[ln(1−r)+α∗ln(1−α∗r)+αln(1−αr)], (D19) where r1=1,r2=α,r 3=α∗=α2are the three cubic roots of unity, with α=(−1+i√ 3)/2; precisely the upintroduced above. Near r=0,∞, the tortoise scales like x≈−r2 2,r→0, (D20) x≈x0−1 r,r→∞, (D21) respectively, where we have introduced x0≡x(r→∞ )=2π√ 3 9e−iπ/3, (D22) which will play a central role in the WKB analysis.48Its value is well deﬁned due to the absence of monodromy at inﬁnity,even in the presence of the three branch cuts coming fromthe logarithms, see Fig. 12.T h ev a l u eo f x 0dictates that of ν viaνx0∈R:ν=ζeiπ/3, where ζ∈R. In particular, from this and Eq. (D21) , we see that the branch of the Stokes line that extends to inﬁnity follows the line r=ρeiπ/3, where ρis real. Near the origin, we have /Ifractur(eiπ/3x)≈− /Ifractur (eiπ/3r2)/2, which implies r=ρe−iπ/6,ρ∈R, in addition to r=ρeiπ/3. These two branches of the Stokes line cross at the origin as we showin Fig. 12. We now proceed to the WKB analysis by examining the solution to Eq. (D15) in the vicinity of r=∞,0,1. Nearr=∞ , the potential W(r) is irrelevant since W∼ 1/r. This is not surprising since we expect γto be irrelevant near the UV boundary and W∝γ. The equation becomes harmonic. We write the solution in terms of the shifted variable,x−x 0, and use Bessel functions although simple sines and cosines would sufﬁce; this allows us to compare with other 235115-19WILLIAM WITCZAK-KREMPA AND SUBIR SACHDEV PHYSICAL REVIEW B 86, 235115 (2012) QNM analyses.48We have ψ(x)=B+/radicalbig 2πν(x−x0)Jj∞/2[ν(x−x0)] +B−/radicalbig 2πν(x−x0)J−j∞/2[ν(x−x0)],(D23) where j∞=1 and J1/2(z)=√2/πsin(z)/√z,J−1/2(z)=√2/πcos(z)/√z. As we have discussed in the main text, we need to impose the vanishing of Ay=ψG at the boundary, which leads to ψ(x0)=0 since G(x0)=1. We thus have our ﬁrst constraint, B−=0. Nearr=0. Near the black-hole singularity, the potential diverges W(r)=21 4r4=21/4 4x2=j2 0−1 4x2, (D24) withj0=5/2. In the second inequality we have used x≈ −r2/2 near the singularity. We thus have the Bessel solution ψ(x)=A+√ 2πνxJ j0/2(νx)+A−√ 2πνxJ −j0/2(νx).(D25) We can match the solutions near r=∞ andr= 0 using the asymptotic expansion for z/greatermuch1:Ja(z)≈√2/(πz) cos[z−(1+2a)π/4]. Expanding near the origin, r=0, we obtain ψ(x)≈2A+cos(νx−α+)+2A−cos(νx−α−) (D26) =(A+e−iα++A−e−iα−)eiνx+(A+eiα++A−eiα−)e−iνx, (D27) where we have deﬁned α±=(1±j0)π/4. On the other hand, extending from r=∞ , we get ψ≈2B+cos[ν(x−x0)−β+] (D28) =B+e−iβ+eiν(x−x0)+B+eiβ+e−iν(x−x0), (D29) where β+=π/2. Matching both solutions by equating the ratios of the coefﬁcients of e±iνxyields another constraint: A+sin(νx0+β+−α+)+A−sin(νx0+β+−α−)=0. (D30) We turn to the behavior near r=1. We want to match the behavior on the Stokes branch r=ρeiπ/3with that near the black hole event horizon r=1. First, we have the small- z expansion Ja(z)≈zaw(z), where w(z) is an even and holo- morphic function, w(z)=0F1(a+1;−z2/4)/[2a/Gamma1(a+1)], where 0F1is an instance of the hypergeometric function. We will rotate from the branch r=ρeiπ/3,ρ∈R−tor=ρe−iπ/6, ρ∈R+.U s i n g x∼r2nearr=0, theπ/2r-rotation becomes aπx rotation: √ 2πe−iπνxJ±j0/2(e−iπνx)=e−i(1±j0)π/2√ 2πνxJ ±j0/2(νx) (D31) →2e−i2α±cos(νx−α±).(D32)Using this we have the following behavior on the r=ρe−iπ/6, ρ∈R+branch: ψ(x)∼2A+e−i2α+cos(−νx−α+) +2A−e−i2α−cos(−νx−α−) (D33) =(A+e−iα++A−e−iα−)eiνx +(A+e−i3α++A−e−i3α−)e−iνx. (D34) We know that at the horizon, ψ(x)∼eiνxin order to satisfy the infalling condition, consequently, A+e−i3α++A−e−i3α−=0. (D35) Combining Eqs. (D30) and(D35) , we ﬁnd get a condition that the homogeneous system of equations needs to satisfy inorder to have a solution: det/parenleftbigg e −i3α+ e−i3α− sin(νx0+β+−α+)s i n ( νx0+β+−α−)/parenrightbigg =0. (D36) This equation leads to the general solution for the asymptotic QNMs: 3wx 0=ξ−2πn, n ∈N&n/greatermuch1, (D37) where we have switched back to w=ν/3. We ﬁnd two solutions for the offset parameter ξ: ξ1=2itanh−1/bracketleftBigg 4√ 2+(1+i) 4√ 2+(−1−i)/bracketrightBigg ≈− 2.356−i0.173, (D38) ξ2=2t a n−1/bracketleftBigg i4√ 2+(1−i) 4√ 2+(1+i)/bracketrightBigg ≈0.785−i0.173.(D39) The offset and gap, deﬁned via w=[gap]−n[offset] for large n,a r eg i v e nb y offset =ξ 3x0, (D40) gap=2π 3x0=√ 3eiπ/3, (D41) where the offset obtained using ξ1,2is−0.283−i0.586 or 0.150+i0.164, respectively. Interestingly, we note that these results for the asymptotic QNMs are independent ofthe value of γ, as long as it is ﬁnite. 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PhysRevB.85.155401.pdf	PHYSICAL REVIEW B 85, 155401 (2012) Electronic structure of the indium-adsorbed Au/Si(111)-√ 3×√ 3 surface: A ﬁrst-principles study Chia-Hsiu Hsu,1Wen-Huan Lin,1Vidvuds Ozolins,2and Feng-Chuan Chuang1,2,* 1Department of Physics, National Sun Yat-sen University, Kaohsiung 804, Taiwan 2Department of Materials Science and Engineering, University of California, Los Angeles, California 90095-1595, USA (Received 27 October 2011; revised manuscript received 12 March 2012; published 2 April 2012) Electronic structures of the indium-adsorbed Au /Si(111)-√ 3×√ 3 surface were examined using ﬁrst- principles calculations at In coverages of 0, 1 /6, 1/3, 2/3, and 1 ML. The band structures of the numerous models were analyzed in detail. We found that the surface bands around the Mpoint exhibit notable Rashba-type spin-orbit splittings. In addition, our results show that the calculated bands of the lowest-energy model at 1 /3M L are in fair agreement with the identiﬁed bands in the angle-resolved photoemission study [J. K. Kim et al. ,Phys. Rev. B 80, 075312 (2009) ]. DOI: 10.1103/PhysRevB.85.155401 PACS number(s): 68 .35.B−,6 8.43.Bc, 73 .20.At I. INTRODUCTION Metal overlayers on a semiconductor surface have gener- ated huge research interest in recent years due to their low-dimensional electronic properties and potential applications in the microelectronics industry. One of the prototypical systems under intensive study is the Au overlayers on theSi(111) surface. 1–40Depending on the Au coverages and the annealing conditions, the Au/Si(111) system exhibits varioussurface reconstructions, such as 5 ×1, 5×2,√ 3×√ 3, 6×6, etc.1–40 Depending on the orientation of the reconstruction, a surface exhibits either two-dimensional (2D)20,40–43or one- dimensional (1D) metallic characteristics.34Recent studies have shown that the complex surface band structure of thePb/Si(111)-√ 7×√ 3 phase is governed by a simple 2D free- electron character,20,40–43while the Au/Si(111)-5 ×2 phase exhibits a 1D feature.18,30 The√ 3×√ 3(√ 3 hereafter) phase of Au/Si(111) has been studied extensively,1–21and the well-known conjugate honeycomb-chained-trimer (CHCT) model10,12,21,44,45for√ 3 is regarded as the lowest-energy model at Au coverage of 1 ML.The previous calculated band structure of this model is in fairagreement with the angle-resolved photoelectron spectroscopy(ARPES). 45However, there is a small discrepancy between the experimental data reported by Zhang et al.19and Altmann et al.18Both studies showed that the two bands S2andS3are degenerate at the /Gamma1point. While the results of Zhang et al.19 seem to indicate that S2andS3bands do not merge and leave a band opening of around 0.4 eV at the Mpoint, Altmann et al.18found that these bands do in fact merge at the Mpoint, at least within an uncertainty of about 0.1 eV imposed by thelifetime broadening. Recently, there has been a slew of very interesting reports concerning domain walls of the√ 3-Au surface.46–48The scanning-tunneling-microscopy (STM) study46found that submonolayer In adsorbates (0.15–0.4 ML) on the α-√ 3-Au surface eliminate the whole domain wall to yield a verywell ordered and homogeneous√ 3×√ 3( h -√ 3 hereafter) phase. More recently, Kim et al.48measured the surface band dispersions and Fermi surfaces before and after the Inadsorption on the Au/Si(111)-√ 3 using ARPES. They found that In adsorbates do not signiﬁcantly alter the surface bandstructure but shift the bands by about 200–500 meV . Moreover, result from core-level photoelectron spectroscopy by Kimet al. 48suggested that In adsorbates interact directly with the surface Si atoms rather than Au atoms. Thus, it is highly likelythat the In atoms adsorb in the middle of the Si trimers, assuggested by the STM study. 46 Moreover, strong Rashba-type spin-orbit splittings in the surface alloy on Si(111) and Ge(111) have attracted someresearch interest. 49–52In view of these experimental data for the In-adsorbed Au/Si(111)-√ 3 phase, a further theoretical study is required in order to clarify the adsorption structureof In atoms and to further check the effect of In adsorbateson the surface band dispersion, as well as to examine whetherthis surface alloy will produce strong Rashba-type spin-orbitsplittings. In this paper, we examined the atomic and electronic structures of the indium-adsorbed Au/Si(111)-√ 3 surface using ﬁrst-principles calculations. For some adsorption sitesand structural motifs, the surface band structures do not changedramatically. Instead, the whole band structures were shiftedby−329 to 850 meV . We found that the surface bands around the Mpoint exhibit notable Rashba-type spin-orbit splittings. The calculated bands for the lowest-energy modelat In coverage of 1 /3 ML are in fair agreement with the identiﬁed bands in the angle-resolved photoemission study. 48 The surface band dispersion of the lowest-energy structures atindium coverage of 2 /3 ML is quite interesting and may have further implications. The rest of this paper is arranged as follows: In Sec. II,t h e computational methods are discussed. Results and discussionof atomic and calculated band structures are presented inSec. III. Finally, our major ﬁndings in this work are sum- marized with a brief conclusion in Sec. IV. II. COMPUTATIONAL METHODS AND STRUCTURAL MODELS The calculations were carried out within the generalized gradient approximation53to density functional theory54using projector-augmented-wave potentials,55as implemented in Vienna Ab-Initio Simulation Package.56The kinetic energy cutoff was set to 500 eV (36.75 Ry), and the gamma-centered10×10×1 Monkhorst-Pack grid was used to sample the 155401-1 1098-0121/2012/85(15)/155401(7) ©2012 American Physical SocietyHSU, LIN, OZOLINS, AND CHUANG PHYSICAL REVIEW B 85, 155401 (2012) TABLE I. The relative surface energies /Delta1Es(meV per√ 3 cell) with respect to the CHCT model of proposed models. Eshiftis the energy shift (meV) of ARPES data to match our calculated band structures. δEo(meV) is the band opening at the Mpoint with SOC. The values in the parentheses are without SOC. /Delta1Es Eshift δEo Label Figure θIn θAu θSi (meV per√ 3) (meV) (meV) CHCT 1(a) 0 1 1 0 +250 258(311) H C T 1 ( b )011 8 3 CHCT-T4 1 /61 1 −149 CHCT-AT 1 /61 1 −12 CHCT-AS 1 /61 1 2 2 Substitutea1/65 /6 1 434 Distorted substituteb1/615 /6 754 CHCT-T4 1(c) 1 /31 1 −183 −100 261 (354) CHCT-AT 1(d) 1 /3 1 1 231 −329 170 (212) CHCT-AS 1 /3 1 1 243 Distorted substituteb2(a) 1 /312 /3 352 Substitutea2(b) 1 /312 /3 433 +850 133(79) Distorted substituteb2(c) 1 /32 /3 1 599 CHCT-2T4 3(a) 2 /31 1 7 CHCT-1T4-1AT 3(b) 2 /3 1 1 215 CHCT-2AS 3(c) 2 /3 1 1 444 CHCT-1AT-1AS 3(d) 2 /3 1 1 579 −300 118 (159) Distorted substituteb2/32 /3 1 555 Distorted substituteb2/312 /3 215 CHCT-2T4-1AT 1 1 1 172 CHCT-3AS 1 1 1 303 aThe CHCT motif is retained after In substitution. bThe CHCT motif is not retained after In substitution. surface Brillouin zones (SBZ) for the√ 3 phases. Moreover, for all our surface calculations, the theoretical Si bulk latticeconstant of 5.468 ˚A was adopted. We employed a periodically repeating slab consisting of three Si bilayers, a reconstructedlayer, and a vacuum space of ∼12˚A. Hydrogen atoms were used to passivate the Si dangling bonds at the bottom of theslab, and the positions of H atoms were kept ﬁxed. Similarly,the silicon atoms of the bottom bilayer were kept ﬁxed atthe bulk crystalline positions. The remaining In, Si, and Auatoms were relaxed until the residual force was smaller than0.01 eV /˚A. After calculating the total energies of the models, the relative surface energy /Delta1E swith respect to the lowest-energy model, CHCT, of the√ 3 phase at Au coverage of 1.0 ML is calculated next according to the relation /Delta1Es=Emodel−ECHCT−/Delta1θ InμIn−/Delta1θ SiμSi −/Delta1θ AuμAu. (1) In the above, ECHCT andEmodel are the total energies of the CHCT-√ 3 and the proposed models, respectively. μIn,μAu, andμSidenote the chemical potentials of the bulk phases, and /Delta1θ In,/Delta1θ Au, and/Delta1θ Sirepresent the differences in coverages in the surface layer for the proposed models with respect tothe CHCT model. The relative surface energies /Delta1E sof the models listed in Table Iare calculated by setting the bulk energies of Au, Si, and In to the values of their respectivechemical potentials. Both calculated band structures with andwithout the spin-orbit coupling (SOC) for the representative models are shown in the ﬁgures. Finally, we have manually created the atomic structures for various In coverages. In addition, we also randomly placed In,Au, and Si atoms on the substrate and then relaxed them totheir local minima. In total, we examined roughly 30 structuresfor the In coverages of 1 /3 and 2 /3 ML. Of these, selected low-energy models are shown in Table Iand Figs. 1–3. III. RESULTS AND DISCUSSION The well-known conjugate honeycomb-chained-trimer model10,12,19,21,44,45for the√ 3 phase is illustrated in Fig. 1(a), where Au atoms form the trimer. The corresponding bandstructure of the CHCT model is shown in Fig. 1(e).O u r calculated band structure along /Gamma1-M-/Gamma1is similar to that reported by Lee and Kang. 45The red dotted lines represent the ARPES data reproduced from Ref. 48. The experimental result is shifted by +250 meV in order to match the band merging feature at the Mpoint. However, the experimental bands S2and S3merging at the Mpoint is not replicated in the theoretical calculation. Rather, the band opening of 0.311 eV , δEo,a t theMpoint is observed. We further reexamined the band structures of the 1 ×1 model shown in Fig. 1(a) of Ref. 48and ﬁnd that the degeneracy of S2andS3at theMpoint is broken by the trimerization of the Au atoms and result in a band gap of0.311 eV at the Mpoint. However, further spin-orbit coupling calculations result in splitting and broadening of S 2andS3such 155401-2ELECTRONIC STRUCTURE OF THE INDIUM-ADSORBED ... PHYSICAL REVIEW B 85, 155401 (2012) FIG. 1. (Color online) (a) and (b) show the optimized atomic structures for the√ 3 phase, and (e) and (f) are their corresponding band structure along the /Gamma1-M-/Gamma1. (c) and (d) show models for a single In atom adsorbed on√ 3 corresponding to In coverage of 1 /3M L , while (g) and (h) are their corresponding band structures along the/Gamma1-M-/Gamma1.T h e√ 3 supercell is outlined with the red dashed lines. The values above the models are the relative surface energies (meV per√ 3 cell) with respect to CHCT model. Large red (medium gray) and blue (dark gray) and small golden (light gray) ﬁlled circles indicate In, Au, and Si atoms of the surface layer, respectively, and white spheres represent Si atoms below the surface layers. For the band structures, the solid lines indicate the results without SOC. The red circles and blue crosses in the band structures indicate opposite spinorientations, and the their sizes are proportional to contributions of the Au, In, and Si atoms at the surface layer. The dashed lines are the band structures including SOC. The red dotted lines are the ARPESdata reproduced from Ref. 48.that these S2andS3bands seem to be closer around the Mpoint (a gap of 0.258 eV). The calculated highest surface band, S1, differs from the experimental value by around 0.4 eV at the/Gamma1point. Next, the honeycomb-chained-trimer (HCT) model shown in Fig. 1(b) is found to be higher in energy by 83 meV per√ 3 where its band structure is shown in Fig. 1(f). Based on Fig. 1(f), it would seem that the band structure of the HCT model does not match the experimental result. Apparently, theband structure is sensitive to surface atomic reconstruction. After reexamining the√ 3 phase, we began to simulate the experimental studies46–48where the indium atoms were adsorbed on the√ 3 surface. We started with a single indium atom per√ 3 cell, which corresponds to a coverage of 1 /3 ML. Numerous structures were examined, and ﬁrst two lowest-energy are shown in Figs. 1(c) and 1(d). The model in Fig. 1(c) shows that the indium atom resides at a position higher thanthe Au atoms and is found among the Au trimers. Moreover,it bonds with the Au atoms of the three neighboring trimers.The position of this In atom is right on top of the T4 site withrespect to the underlying Si(111) substrate. Thus, we label itas the CHCT-T4 model. The In position in the model shown inFig. 1(c) is, in fact, the same as the site proposed by previous studies. 46,48Furthermore, the model in Fig. 1(c) has a lower relative energy than the CHCT model. Nonetheless, the bandstructure in Fig. 1(g) is in fair agreement with the experimental observations. S 3is not fully replicated in the calculations. The second model shown in Fig. 1(d) shows the indium atom residing on top of the Au trimer. It was therefore appropriatelylabeled as the CHCT-AT model. The band structure of thisCHCT-AT model at In coverage of 1 /3 ML agrees well with the experimental band when shifted by −329 meV , as shown in Fig. 1(h). The band dispersions of S 1,S2, andS3match the experimental bands. In addition, our calculations with andwithout the SOC exhibit band openings of 212 and 170 meV attheMpoint, respectively. It appears that the In atoms behave as the electron donors when they reside on top of Au trimers.Finally, the possible adsorption site with the next higher energyis found to be on top of the Si atom and then is labeledCHCT-AS. The relative energy of this model is included inTable I. However, since its band structure does not match the experiment, we will not present it in this study. We further explored other possibilities. In one possible scenario, the CHCT model is no longer retained after Inadsorptions. Numerous models were then examined, and weillustrate two models wherein one Si atom is replaced by oneIn atom at In coverage of 1 /3M L ,a ss h o w ni nF i g s . 2(a) and 2(b). For the ﬁrst In substitution model, the CHCT isbroken; thus the band structure in Fig. 2(d) does not match the experimental result. For the second In substitution model,the CHCT is retained after the Au atom was substitutedby an In atom. The band structure of the second modelshown in Fig. 2(e) reproduces the experimental S 2andS3 bands well, provided the experimental result is shifted by +850 meV . The S2andS3bands differ by 0.079 and 0.133 eV around the Mpoint, which is close to the experimental resolution limit of 0.1 eV .18Furthermore, we also explored the coverage where the Au atom is substituted by an In atom.One such model is shown in Fig. 2(c), where it appears that its band structures does not match the experimentalobservation. 155401-3HSU, LIN, OZOLINS, AND CHUANG PHYSICAL REVIEW B 85, 155401 (2012) FIG. 2. (Color online) (a) and (b) show models at In coverage of 1/3 ML where one In atom substitutes the Si atom for each√ 3 cell, and (d) and (e) are their corresponding band structures along the/Gamma1-M-/Gamma1. (c) depicts the model at In coverage of 1 /3M Lw h e r e one In atom substitutes one Au atom at the surface layer, and itscorresponding band structure is shown in (f). In the experiment at around In coverage of 0.15 ML, a sharp√ 3×√ 3 low-energy electron diffraction (LEED) pattern without any other diffraction features developed.46,48 Therefore, after determining the possible adsorption sites at 1/3 ML, we intuitively augmented our supercell to a 2√ 3×√ 3 unit such that one In atom in the supercell corresponds to 1 /6 (0.167) ML, approximately close to the experimental coverageof 0.15 ML. Moreover, the other possible models are thatones in which an In atom substitute either one Au atom ora Si atom on the surface. We have examined three sites andnumerous substitution models, and those with low energies arelisted in Table I. Our result is in agreement with the previous calculation by Gruznev et al. 46in which the CHCT-T4 model is the lowest-energy adsorption site. Since additional discussionsof the CHCT-T4 models at 1 /6 and 1 /3 ML can be found in the aforementioned study, 46we will not elaborate further here.Furthermore, we also noted that the In atom substitution of a Si atom and a Au atom in the√ 3 cell are energetically unfavorable, which also mirrors the experimental ﬁnding46 that the Si coverage and the Au coverage were found to be 1ML. The energies of the models at In coverage of 1 /6M La r e higher than that of the lowest-energy model shown in Fig. 1(a) at In coverage of 1 /3M L . The In coverage was increased to 2 /3 ML so that two indium atoms are in a√ 3 unit. Since we know the possible adsorption sites for the In atoms from the models with Incoverage of 1 /3 ML, these possible sites are enumerated to generate new structural models. In addition, an In atomalso substitutes position of Au or Si atoms. Furthermore, we performed random arrangement of atoms on the surface. Up to 30 structures were examined, and four low-energy structuralmodels are illustrated in Fig. 3. The model with the lowest energy in Fig. 3(a) has two indium atoms residing among the Au trimers in a way similar to the CHCT-T4 model in Fig. 1(a). This model is found to be identical to that illustrated by Kimet al. 48We note that the models at 1 /6 and 1 /3M Lh a v el o w e r energies than the model at In coverage of 2 /3 ML. The band structure shown in Fig. 3(e) of the lowest-energy model at In coverage of 2 /3 ML does not match the experimental result. However, the surface band dispersion of the lowest-energystructures at indium coverage of 2 /3 ML is quite interesting and may have further implications. The second-lowest-energymodel shown in Fig. 3(b) has one indium atom on top of an Au trimer with other indium atom among the Au trimers. In Fig. 3(f), the band crossing of S 2andS3at the Mpoint is reproduced, but the band S1dispersing upward at the /Gamma1point does not as shown in our calculation. The third model shown inFig. 3(c) contains two indium atoms are on top of the Si atoms, where we note that its corresponding band structure in Fig. 3(g) does not match the experimental result either. Furthermore, the fourth model in Fig. 3(d) has one indium atom residing on top of the Au trimer while the other indium atoms sit on top ofan Si atom. Its band structure, plotted in Fig. 3(h),a l s os h o w s the band crossing of S 2andS3at the Mpoint. However, an additional band dispersing at the /Gamma1point which emerges from our calculation is not seen in the experimental result. The In coverage was further increased to 1 ML. Nu- merous models were examined, and two low-energy modelsare listed in the Table I. The ﬁrst model CHCT-2T4-1AT has one additional In atom that is adsorbed on top of anAu trimer. The second model CHCT-3AS has all three Inatoms sit on top of the Si atoms. We found that the bandstructures of these two models are not in agreement withexperiments. 48 The inclusions of SOC in the band calculations showed that the SOC mainly causes the splitting of surface bands thatthe Au atoms contribute to. We further notice that for themodels to match the experimental dispersions of bands S 1, S2, andS3where their Eshiftare provided in Table I, the band gaps of S2andS3at theMpoint have to be less than 0.354 eV . Considerations of spin-orbit coupling in these systems showed that the splittings of the S2andS3bands leave gaps near theMpoint that are smaller and close to the experimental resolution limit of 0.1 eV . Finally, the splittings of the S1and S3bands around the Mpoint are found to be Rashba spin-orbit 155401-4ELECTRONIC STRUCTURE OF THE INDIUM-ADSORBED ... PHYSICAL REVIEW B 85, 155401 (2012) FIG. 3. (Color online) (a), (b), (c), and (d) show models for two In atoms adsorbed on√ 3a tI nc o v e r a g eo f2 /3 ML, and (e), (f), (g), and (h) are are their corresponding band structures along /Gamma1-M-/Gamma1. splitting57since the In-Au-Si surface layer formed a potential gradient at the surface. After investigating the models at different In coverages, we further discuss the stability as a function of In coverage.The relative surface energies of the models versus In coverageare plotted in Fig. 4(a). The plot shows that the system at In coverage of 1 /3 ML is most stable, while the experimental FIG. 4. (Color online) (a) The relative surface energies of models vs In coverage. (b) The relative surface energies (meV per√ 3 cell) of lowest-energy models at different In coverages vs the chemicalpotential of In. observations suggested rather that the In coverage is 1 /6 ML.46,48The lines connecting the lowest-energy models form a convex hull, implying the surface is less stable at In coverageof 2/3 ML. Moreover, a consistent trend was found. The lowest-energy models at coverages ranging from 1 /6t o1M L are those of In atoms sitting on the T4 sites. Next, we discuss the stability as a function of the chemical potential. The relative surface energies of the lowest-energymodels versus the chemical potential of In are plotted in Fig. 4. A quick inspection of Fig. 4reveals that for ( μ In−μbulk In) >0.177 eV the the most stable structure is the model at In coverage of 1 ML. Gradually, when −0.066 eV <(μIn−μbulk In) <0.177 eV the model at In coverage of 1 /3 ML exhibits the most stability. The bulk energy of In is within this range.However, when −0.299 eV <(μ In−μbulk In)<−0.066 eV , the most stable structure is the model at In coverage of 1 /6M L . We note that when the chemical potential differs from the bulkv a l u eb yo n l y −0.066 eV the model at In coverage of 1 /6M L has a lower energy than the model at In coverage of 1 /3ML. When ( μ In−μbulk In)<−0.299 eV , the surface exhibits the most stability without any In adsorption. The lowest-energymodel at 2 /3 ML seems to be less stable with respect to the chemical potential. 155401-5HSU, LIN, OZOLINS, AND CHUANG PHYSICAL REVIEW B 85, 155401 (2012) FIG. 5. (Color online) The empty-state (top) and ﬁlled-state (bottom) images of the (a) CHCT-T4 [Fig. 1(c)]a n d( b )C H C T - A T [Fig. 1(d)] models at 1 /3 ML. (c) and (d) are those of CHCT-2T4 [Fig. 3(a)] and CHCT-T4-AS [Fig. 3(b)]. The sample biases are +1.0 Va n d −1.0 V for empty (top) and ﬁlled (bottom) states, respectively. Finally, we calculated STM images of our models and compared them with the experimental observations.46In Figs. 5(a) and 5(b), our simulated STM images of 1 /3-ML models show one bright spot per√ 3 cell and thus do not exhibit the hexagonal pattern. Furthermore, the simulated STMimages of the lowest-energy model of 2 /3M La ss h o w ni n Fig. 5(c) exhibit the hexagonal pattern, which matches the experiment STM observations. However, the STM experimentwas performed at a low coverage of 0.15 ML (around 1 /6M L )and at room temperature (300 K). A plausible explanation for this discrepancy was proposed by Gruznev et al. 46in which the In atoms migrate actively and hop among neighboringT4 sites at 300 K and the STM observations in fact weretaken as the time-averaging images. Their further measurementat a lower temperature (125 K) veriﬁed one protrusion per√ 3 cell, meaning that one In atom sits on one√ 3 cell. In addition, the ARPES study by Kim et al.48was performed at a temperature ranging from 300 K down to 40 K, and the surfaceband dispersions have no signiﬁcant change. Moreover, ourcalculations showed a huge change in the band dispersionsat indium coverage of 2 /3 ML. Based on these facts, we can conclude that at 1 /6 ML, even though In atoms are active at the surface at 300 K, only one indium is within one√ 3 cell at any time; thus the band dispersion measurement48should be a mixture of dispersions from the CHCT and CHCT-T4models, while at a lower temperature, the indium atoms will befrozen, 46and thus the same mixture of dispersions is expected. Further experimental study at a higher coverage is needed dueto the interesting surface band dispersions at 2 /3M L .U s i n g the√ 3 as a templet, exotic band dispersions may be tailored by adsorbing different metals. IV . CONCLUSIONS In conclusion, atomic and electronic structures of the In-adsorbed Au/Si(111)-√ 3×√ 3 surface reconstruction were examined using ﬁrst-principles calculations at In coveragesranging from 1 /6 to 1 ML. The analysis of stability due to the chemical potential indicates that the model at In coverageof 2/3 ML is less stable. The T4 site was found to be the preferred adsorption site for indium atoms. The band structuresof the numerous models were analyzed in detail. Our resultsshow that the calculated bands for lowest-energy model at Incoverage of 1 /3 ML are in fair agreement with the identiﬁed bands in the angle-resolved photoemission study. Finally, thesurface bands around the Mpoint exhibit Rashba spin-orbit splitting since the In-Au-Si layer formed a potential gradientat the surface. ACKNOWLEDGMENTS F.C.C. was supported by the National Center of Theoretical Sciences (NCTS) and the National Science Council of Taiwanunder Grant No. NSC98-2112-M110-002-MY3. 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PhysRevB.103.075147.pdf	PHYSICAL REVIEW B 103, 075147 (2021) Fermion-enhanced ﬁrst-order phase transition and chiral Gross-Neveu tricritical point Yuzhi Liu ,1,2Zi Yang Meng ,3,1and Shuai Yin4 1Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 3Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 4School of Physics, Sun Yat-sen University, Guangzhou 510275, China (Received 3 December 2020; revised 9 February 2021; accepted 17 February 2021; published 26 February 2021) The ﬂuctuations of massless Dirac fermion can not only turn a ﬁrst-order bosonic phase transition (in the Landau sense) to a quantum critical point, but also work reversely to enhance the ﬁrst-order transition itself,depending on the implementation of ﬁnite-size effects in the coupling corrections. Here, we report a case study ofthe latter by employing quantum Monte Carlo simulation upon a lattice model in which the bosonic part featuringthe Landau-Devonshire ﬁrst-order phase transition and Yukawa coupled to the Dirac fermions. We ﬁnd that theparameter range for the ﬁrst-order phase transition becomes larger as the Yukawa coupling increases, and themicroscopic mechanism of this phenomena is revealed, at a quantitative level, as the interplay between the criticalﬂuctuations and the ﬁnite-size effects. Moreover, the scaling behavior at the separation point between the ﬁrst-order and the continuous phase transitions is found to belong to the chiral tricritical Gross-Neveu universality.Our results demonstrate that the interplay of massless Dirac fermions, critical ﬂuctuations, and the ﬁnite-sizeeffects could trigger a plethora of interesting phenomena, and therefore great care is called for when makinggeneralizations. DOI: 10.1103/PhysRevB.103.075147 I. INTRODUCTION Fluctuations play vital roles in both ﬁrst-order and continu- ous phase transitions [ 1–4]. It was realized that the self-similar ﬂuctuating modes are responsible for the scaling behaviors in the second-order phase transition by Wilson’s renormalization group theory [ 5], which consequently brought in the notion of the universality class—one of the organization principlesin statistical and condensed-matter physics. Moreover, ﬂuc-tuations can even change the order of the phase transitions.For example, the Coleman-Weinberg mechanism showedthat the ﬂuctuation of the gauge ﬁeld can turn a con- tinuous phase transition into a ﬁrst-order one [ 6,7]. On the contrary, the theory of the deconﬁned quantum crit-ical point proposed that the ﬂuctuations from the frac-tionized spinons and emergent gauge ﬁeld can rounda ﬁrst-order transition (in the Landau sense) betweentwo ordered phases into a continuous phase [ 8–20]. The Landau-Ginzburg (LG) model is the typical model to achieve a continuous phase transition. The Landau–de Gennes [ 21] model introduced the cubic term into LG, and the Landau-Devonshire model [ 21–24] increased the φ 6term based on LG w i t ham i n u s φ4term. Both of them are an effective model for the ﬁrst-order transition. In a similar spirit with the deconﬁnedquantum critical point, ﬂuctuations of Dirac fermions can alsosoften the Landau–de Gennes and the Landau-Devonshire ﬁrst-order transition in the bosonic sector into continuous ones, which are dubbed as the type-I [ 25–32] and type-II fermion-induced quantum critical points (FIQCP) [ 33].The aforementioned model studies [ 8–20,25–32] are usu- ally carried out numerically on ﬁnite lattice sizes. It iswell-known that the ﬁnite size (using Lto denote the lin- ear lattice size) provides a natural infrared truncation in thelong wavelength ﬂuctuation, and one shall perform ﬁnite-sizescaling (FSS) [ 34] to extract the critical properties of the uni- versality, i.e., treating Las a tunable relevant scaling variable to estimate the critical point and exponents. In particular, itwas shown that the scaling form of the FSS should be drasti-cally amended near the deconﬁned quantum critical point as aresult of the appearance of the dangerously irrelevant scalingvariable [ 15]. On the other hand, a controlled FSS analysis in the Dirac-fermion-induced quantum critical point is stillrare [ 35], and it is our ﬁrst motivation in this work to address this issue. The critical properties of the interacting Dirac fermion systems have attracted attention from the condensed-matterto the high-energy physics communities, not only in thediscussion of quantum electrodynamics with fermionic mat-ter [ 36–41], but also in that the Dirac fermion drives the Wilson-Fisher ﬁxed point into the chiral Gross-Neveu ﬁxedpoint [ 25,42–45]. Although enormous investigations have been devoted to this issue [ 35,42–44,46–62], the numerical veriﬁcation for type-II FIQCP is still lacking. Remarkably,recent studies based on the ﬁeld-theoretical effective modelpropose that type-II FIQCP can also feature new tricriticalbehaviors, controlled by the chiral tricritical point (CTP) [ 63]. This CTP separates the conventional Landau-Devonshire ﬁrst-order transition from the type-II FIQCP [ 33], and the universal 2469-9950/2021/103(7)/075147(9) 075147-1 ©2021 American Physical SocietyYUZHI LIU, ZI Y ANG MENG, AND SHUAI YIN PHYSICAL REVIEW B 103, 075147 (2021) scaling behavior near this CTP is quite different from its pure bosonic counterpart. A numerical veriﬁcation on such CTPconstitutes our second motivation. In this paper we hit the two birds with one stone by nu- merically investigating a lattice model, which consists of aspin (boson) part hosting the Landau-Devonshire ﬁrst-ordertransition, a massless Dirac fermion part, and the couplingbetween them. The numerical approach is based on the de-terminant quantum Monte Carlo (DQMC) method [ 35,46, 64–66]. Although the theory of the type-II FIQCP pre- dicts that the fermion ﬂuctuation can soften the bosonﬁrst-order transition into a continuous one [ 33], here we ﬁnd that apparently things can also go in the oppositedirection in that the range of the ﬁrst-order transition isextended due to the coupling with Dirac fermions. Toexplain this observation, we develop a modiﬁed mean-ﬁeld theory to study the effective coupling in the freeenergy and reveal that this anomalous phenomenon is in-duced by the interplay between critical ﬂuctuations and theﬁnite-size effects, in a quantitative manner. Moreover, wepinpoint the tricritical point separating the ﬁrst-order andthe continuous phase transitions, and numerically verify thatthis CTP acquires the critical exponents of chiral Gross-Neveu universality, conﬁrming the renormalization grouppredictions [ 63]. The rest of the paper is organized as follows. Section II introduces the lattice model and DQMC methodology. Thenumerical results are shown in Sec. III A , where we demon- strate the Dirac fermion-enhanced ﬁrst-order transition. Toexplain it, in Sec. III B a modiﬁed mean-ﬁeld analysis is presented. In Sec. IVthe position and critical exponents at the Gross-Neveu CTP are revealed with FSS upon numericaldata. Finally, a summary is given in Sec. V. II. LATTICE MODEL AND NUMERICAL METHOD The lattice model is comprised of Dirac fermions, Ising spins (bosons), and their coupling, on the square lattice. Asshown in Fig. 1(a), the bosonic part reads [ 67] H Boson=Ja/summationdisplay /angbracketleftp,q/angbracketrightσz pσz q−Jb/summationdisplay /angbracketleft/angbracketleftp,q/angbracketright/angbracketrightσz pσz q −/Gamma1z/summationdisplay pσz p−/Gamma1x/summationdisplay pσx p, (1) in which the Pauli matrices σz/x prepresent a local spin at the bosonic site p,Jarepresents the nearest antiferromagnetic (AFM) interaction, Jbrepresents the next-nearest ferromag- netic (FM) interaction, and /Gamma1xis the transverse ﬁeld, /Gamma1zis the longitudinal ﬁeld. The fermion part reads [ 35,54,68] HFermion =/summationdisplay /angbracketlefti,j/angbracketright,σf−tijeiσfθijc† i,σfcj,σf+μ/summationdisplay ini+H.c.,(2) in which ci,σf(c† i,σf) is the fermionic annihilation (creation) operator at the fermionic site iwith spin σf=± 1/2, and the phase θijis set to be θij=π/4, which allows a πmag- netic ﬂux on each fermionic plaquette and supports two Diracpoints in its energy bands [ 69].n iis density of fermion and FIG. 1. The lattice model and the ground-state phase diagram. (a) The gray lattice sites and red lattice sites respectively present thefermion and boson sites. One unit cell therefore contains two fermion sites and two boson sites. The solid line with the arrow indicates the fermion hopping. The dashed straight line means fermion-bosoncoupling as in Eq. ( 2). The dashed curve between the bosonic sites is bosonic interaction, and the dashed circle with arrow on each fermionic plaquette means the πﬂux. (b) The schematic ground-state phase diagram, AFM and FP phases of the Ising spins are separated by a continuous (blue line) or ﬁrst-order (red line) phase transition where they meet at the tricritical point (CTP) of the model. In theAFM and FP phases, fermion is inside the quantum spin Hall (QSH) and Dirac semimetal (DSM) states due to the coupling with the bosons. μis chemical potential. We set μ=0 for half ﬁlling of fermions. The coupling between Eqs. ( 1) and ( 2)i s HCoupling =/summationdisplay /angbracketleft/angbracketlefti,j/angbracketright/angbracketright,σfλijσz pc† i,σfcj,σf+H.c., (3) in which λijrepresents the coupling strength. Thus the total Hamiltonian is H=HBoson+HFermion +HCoupling . (4) Throughout the paper, we set tij=t=1 as the energy unit andλij=λthe same on every bond. The schematic phase diagram of the model, spanned by the axes of /Gamma1xand/Gamma1z, is shown in Fig. 1(b). In the absence of 075147-2FERMION ENHANCED FIRST-ORDER PHASE TRANSITION … PHYSICAL REVIEW B 103, 075147 (2021) the coupling to the Dirac fermions, i.e., λ=0, the pure spin model with ﬁxed JaandJbhas two phases [ 67].F o rs m a l l /Gamma1x and/Gamma1z, the system is in an AFM phase, while for large /Gamma1xor /Gamma1z, the system is in a fully polarized (FP) phase. By tuning /Gamma1x and/Gamma1z, there is a phase transition between these two phases. For small /Gamma1z, the phase transition is continuous and belongs to the (2 +1)D Ising universality class as denoted by the blue line. For large /Gamma1z, the phase transition is ﬁrst order, as denoted by the red line. It was shown that when Ja=Jb, this ﬁrst-order transition can be casted into the Landau-Devonshire effectivemodel with a negative quartic coupling [ 22,67]. In addition, there is a quantum tricritical point (CTP) separating the ﬁrstorder and the continuous phase transition. In the followingwe also perform the simulation at J a=Jb, since in this case the uniform part of /angbracketleftσz/angbracketrightcan be treated as a background ﬁeld rather than a dynamical ﬁeld, and HBoson has been solved with quantum Monte Carlo simulation in Ref. [ 67]. HFermion is theπ-ﬂux model which gives rise to two Dirac cones at ( π,0) and (0 ,π) in the Brillouin zone (BZ). As shown in Fig. 1(a), we couple a pair of next-nearest-neighbor fermion sites in HCoupling in which the sign relies on the spin at the bosonic site. In the AFM phase of Ising spins, a mass termcan be generated for fermions which gap out the Dirac points,transforming the Dirac semimetal (DSM) into a dynamicallygenerated quantum spin Hall insulator (QSH) [ 54]. In the FP phase, the bosonic ﬁeld still keeps the Dirac cone at ( π,0) and (0 ,π) intact but renormalized the high-energy parts of the bands in the BZ. Meanwhile, the original three-dimensional(3D) Ising universality class between the AFM phase and FPphase is replaced by the chiral Ising Gross-Neveu universalityclass. This result has been numerical revealed by some of thepresent authors in Refs. [ 35,54]. In the presence of the coupling to the Dirac fermions, there are two main theoretical predictions: one is that thefermion ﬂuctuation can drive the ﬁrst-order transition into acontinuous one [ 33]. This may indicate that the region of the ﬁrst-order transition should shrink as long as the coupling tothe Dirac fermion is introduced. Surprisingly, in the followingwe will show a contrary phenomenon has occurred in theactual DQMC simulation and provide an explanation. Theother is that the universality class of the tricritical point isdrastically changed by the gapless Dirac fermions [ 63]. The ﬁrst order and the continuous transition belonging to the chiralIsing Gross-Neveu universality class is separated by this CTPwhich is in the chiral tricritical Ising universality class. The computation of Hcan be carried out without a sign problem in the DQMC method, and we present the detailedimplementation in Appendix A. III. FERMION-ENHANCED FIRST-ORDER TRANSITION In Sec. III A we ﬁrst present numerical results showing the range of the ﬁrst-order phase transition in our modelis actually extended rather than shrunk, i.e., the ﬁrst-ordertransition line [the red line in our Fig. 1(b)]i nt h e /Gamma1 x−/Gamma1z phase diagram extends a bit towards larger values of /Gamma1xand /Gamma1zcompared with that of the bare spin model [ 67], seemingly contrary to the expectation. Then in Sec. III B we will give an self-consistent explanation based on a modiﬁed mean-ﬁeldanalysis to reconcile the puzzle.A. Numerical results In the DQMC simulation, we choose the parameter Ja= Jb=1 and explore the phase transition properties by scanning /Gamma1zfor different /Gamma1x. For ﬁnite-size systems, phase transition properties can be reﬂected by the behavior of the Binder ratio near thephase transition point. In the present case the Binder ratio isgiven as B 2=3 2/parenleftbigg 1−1 3/angbracketleftm4/angbracketright /angbracketleftm2/angbracketright2/parenrightbigg , (5) in which m2=1 L2/summationdisplay k,l(−1)α1−α2σz k,α1σz l,α2, (6) with Lbeing the lattice size, k,lbeing the unit cell in which α1,α2is the site of the unit cell, such that the staggered mag- netization of the AFM phase is measured. In the ordered phaseB 2→1, while in the disordered phase B2→0. In continuous phase transitions, if the scaling correction can be neglected,curves of the Binder ratio versus the tuning parameter fordifferent size cross at the critical point. In the ﬁrst-orderphase transition, negative values for the Binder ratio will bedeveloped [ 70]. We show the data of Binder ratio in Fig. 2. Without the coupling to the Dirac fermions, i.e., λ=0, Fig. 2(a) shows the Binder ratio for /Gamma1 x=4.4. The curves of the Binder ratio belonging to various system sizes cross at a point.This demonstrates that a continuous phase transition [of a (2+1) Ising universality] occurs for this set of parameters. Figure 2(b) is also without coupling to the Dirac fermions, it shows the curves of Binder ratio for /Gamma1 x=4.0, where different system sizes develop small negative values, signifying that theparameter set is close to the CTP of the pure boson model.This is consistent with the previous literature [ 67]. After in- troducing the coupling to the Dirac fermions with λ=0.3b u t still keeping /Gamma1 x=4.0, Fig. 2(c) shows that obvious negative values appear in the Binder ratio, indicating the appearance ofthe ﬁrst-order phase transition. Moreover, as Lincreases, the values of the Binder ratio tend to diverge, which is a typicalsignature of the ﬁrst-order transition. Such results reveal thatthe boson continuous phase transition is changed into a ﬁrst-order phase transition by coupling to the Dirac fermions. To further illustrate this result, we scan the /Gamma1 x–/Gamma1zphase diagram with different values of the fermion-boson couplingλ, and the phase diagram of pure boson model ( λ=0) is repeated and consistent with the result in Ref. [ 67]. The phase boundaries are obtained by inspecting the behavior of Binderratio as shown in Fig. 2.I nF i g . 3the blue point is the phase boundary for the pure spin model, and its tricritical point isdenoted by a blue triangle, while the black and red pointsare the phase boundaries in the presence of the coupling tothe Dirac fermions, with the coupling strength being λ=0.3 andλ=0.5, respectively. One ﬁnds that for ﬁxed /Gamma1 x,t h e value of /Gamma1zat the phase boundary increases as the coupling strength increases. Moreover, one ﬁnds that the regions forthe ﬁrst-order phase transition are extended as λincreases. Thus it seems that the ﬁrst-order phase transition is apparentlyenhanced by the ﬂuctuation from the Dirac fermions. 075147-3YUZHI LIU, ZI Y ANG MENG, AND SHUAI YIN PHYSICAL REVIEW B 103, 075147 (2021) FIG. 2. Binder ratio B2in different couplings λ. (a) Binder ratio of continuous phase transition for the case of λ=0.0,/Gamma1x=4.4. (b) Binder ratio close to the tricritical point for the case of λ=0.0, /Gamma1x=4.0. (c) Binder ratio of ﬁrst-order transition for the case of λ=0.3,/Gamma1x=4.0.FIG. 3. Phase diagram obtained by DQMC near tricritical points in the presence of different coupling λ=0.0 (blue), 0 .3 (black) ,and 0.5( r e d ) . /squarerepresents a ﬁrst-order transition,/bigtriangleupwith a green border represents the CTPs, and ◦represents a continue phase transition. B. Modiﬁed mean-ﬁeld theory for ﬁnite-size systems To understand the fermion-enhanced ﬁrst-order phase tran- sition, we here develop a modiﬁed mean-ﬁeld theory. In thistheory, we focus on the effective potential of the boson ﬁeldafter integrating out the fermion ﬂuctuations. The fermionﬂuctuations can be truncated from lower bound, since themomentum cannot be smaller than 1 /Lin the lattice model. We begin with the pure spin (boson) model, whose Hamiltonian is shown in Eq. ( 1) and mean-ﬁeld analyses is reported in Ref. [ 67]. The expectation value of σ zcan be decomposed as /angbracketleftσz i/angbracketright=/braceleftbiggs+φb(i∈A) s−φb(i∈B), (7) in which φbis the boson dynamical ﬁeld, sis the background ﬁeld for Ja=Jb, which is just the condition employed in the present work, and A,Brepresents the indices for the sublattice. Then by doing the following replacement, σz iσz j→σz i/angbracketleftbig σz j/angbracketrightbig +/angbracketleftbig σz i/angbracketrightbig σz j−/angbracketleftbig σz i/angbracketrightbig/angbracketleftbig σz j/angbracketrightbig , (8) one obtains the mean-ﬁeld bosonic Hamiltonian as [ 67] HMF Boson N=(J−s−J+φb−/Gamma1z)σz A−/Gamma1xσx A +(J−s+J+φb−/Gamma1z)σx B−/Gamma1xσx B −J−s2+J+φ2 b, (9) where J±=4(Ja±Jb). With J−=0o r Ja=Jb, this Hamil- tonian gives the bosonic free energy per unit cell according tofb≡−1 βNlogTr( e−βHMF Boson). At T=0, near the phase tran- sition, the free energy can be expanded as a function of φbas follows: fb=f0+r 2φ2 b+u 4φ4 b+v 6φ6 b+··· , (10) 075147-4FERMION ENHANCED FIRST-ORDER PHASE TRANSITION … PHYSICAL REVIEW B 103, 075147 (2021) in which the coefﬁcients f0,r,u, and vread f0=−/Delta1, /Delta1=/radicalBig /Gamma12x+/Gamma12z, r=1 2J+/parenleftbigg 1−/Gamma12 xJ+ /Delta13/parenrightbigg , u=(/Gamma12 x−4/Gamma12 z)/Gamma12 xJ4 + 8/Delta17, v=(12/Gamma12 x/Gamma12 z−8/Gamma14 z−/Gamma14 x)/Gamma12 xJ6 + 16/Delta111,(11) and the ellipsis represents the higher-order terms which are irrelevant and can be neglected. For /Gamma1x>2/Gamma1z,u>0 and the system hosts a continuous phase transition, while for/Gamma1 x<2/Gamma1z,u<0 and the system hosts the Landau- Devonshire ﬁrst-order phase transition [ 22]. In the continuous case with u>0, the order parameter φbdevelops as φb=√−r/ucontinuously when r decreases from its critical point r=0. In contrast, when u<0,φbjumps from zero to φb=±√−3u/4v at the transition point rt=3u2/16v. In addition, when r<0, the ordered phase is the only stable phase; when r>u2/8v, the disordered phase is the only stable phase. In betweenwhen 0 <r<u 2/8v, both phases can coexist. When rt<r<u2/8v, the disordered phase is more stable, and when 0 <r<rt, the ordered phase is more stable. To explore the effects induced by the coupling to the Dirac fermion, in principle one should consider the bosonand fermion ﬂuctuations simultaneously and investigate therenormalization ﬂow on all relevant and marginal operators.However, for the ﬁnite-size system, such a procedure is quitecomplex to implement. We have to take a step back andstudy the inﬂuence of the Yukawa coupling on the bosonfree energy. The mean-ﬁeld free energy density reads f≡ − 1 βNlog Tr( e−β(HMF Boson+HFermion+HMF Coupling)), in which HMF Coupling is the mean-ﬁeld version of the coupling Hamiltonian ( 4), with σzbeing approximated by its mean-ﬁeld expectation value Eq. ( 7). Note that in f,HFermion keeps intact as in Eq. ( 2), and contributions from both valleys are included. At zerotemperature, we have f=f b+(−)1 π/integraldisplay d2k/radicalBig λ2φ2 b+2t2k2, (12) in which the last term comes from the coupling with the Dirac fermions. In the thermodynamic limit, the range of integral/integraltext is from zero to /Lambda1(/Lambda1is the ultraviolet cutoff). By explicitly integrating out Eq. ( 12), one ﬁnds that the Yukawa coupling between the Dirac fermion and the boson ﬂuctuations can notonly change the coefﬁcients in Eq. ( 11), but also generate an additional nonanalytic term λ3\|φb\|3 6t2. This cubic term is traced back to the gapless Dirac points in the thermodynamic limit.Actually, at these singular points the fermion functional inte-gral is ill-deﬁned. However, in ﬁnite-size systems, ﬂuctuations are truncated from the IR limit by the system size L. In this case a fermion gap appears proportional to 1 /L. Subsequently, after inte- grating out the fermion ﬂuctuating modes with length scalefrom 1 /Lto/Lambda1, the nonanalytic term vanishes and the ef-FIG. 4. Phase diagram obtained from modiﬁed mean-ﬁeld the- ory for ﬁnite-size systems according to Eq. ( 13) with ﬁxing the coefﬁcient L=10,/Lambda1=1. Black (red) lines represent a ﬁrst-order (continue) phase transition, and dotted (solid) lines are phase bound- aries with the coupling of λ=0.0 (0.5), respectively. fective quadratic and quartic coupling in the boson part free energy reads r/prime=r−λ2(L/Lambda1−1) 4√ 2πLt, u/prime=u+λ4(L/Lambda1−1) 64√ 2t3/Lambda1π,(13) respectively, with the fermion hopping t=1. At ﬁrst we study the change of the phase boundary after turning on the coupling between the Dirac fermions and thedynamical boson ﬁeld. The phase transition occurs at r /prime=0. By substituting this condition and Eq. ( 11) into Eq. ( 13), one ﬁnds that for ﬁxed /Gamma1xandλ, at the phase transition, the value of/Gamma1zchanges as δ/Gamma1z=λ2(L/Lambda1−1)/Delta15 4√ 2πLtJ2 +/Gamma12x/Gamma1z, (14) in which L/Lambda1> 1 since the lattice constant is chosen to be 1 and other parameters are all positive. Thus, one ﬁnds that thevalue of /Gamma1 zincreases with λgrowing. The result is shown in Fig. 4. It is interesting to see that this result is qualitatively consistent with the DQMC numerical results shown in Fig. 3. Then we explore the phase transition properties via this modiﬁed mean-ﬁeld approach. By substituting Eq. ( 11)i n t o Eq. ( 13) and setting r/prime=0, one gets u/prime=u+/parenleftbigg /Lambda1−1 L/parenrightbigg/bracketleftbiggLλ4 64√ 2t3/Lambda1π−5J2 +/parenleftbig 3/Gamma12 x−4/Gamma12 z/parenrightbig λ2 48√ 2πt/Delta14/bracketrightbigg . (15) According to Eq. ( 15), Fig. 5explicitly shows the dependence ofu/primeonLandλ.F r o mF i g . 5(a) one ﬁnds that for small system size, L∼10,udecreases as Lincreases. This explains the en- hancement of the ﬁrst-order phase transition with the increaseof the system size, as shown in Fig. 2(c). In addition, Fig. 5(b) shows that for small system sizes, udecreases as λincreases. 075147-5YUZHI LIU, ZI Y ANG MENG, AND SHUAI YIN PHYSICAL REVIEW B 103, 075147 (2021) (a) (b) FIG. 5. Quartic term u/primedepending on λand Laccording to Eq. ( 15): (a) u/primevsLfor three types of bosonic quartic contribution u by ﬁxing λ=0.5, and (b) u/primevsλby ﬁxing L=10. This is consistent with the numerical result that the ﬁrst-order phase transition is enhanced for larger Yukawa coupling, asshown in Fig. 3. Therefore, as for the model investigated in the paper, the fermion-enhanced ﬁrst-order phase transition isrevealed numerically and understood analytically. We shallalso stress that although this mean-ﬁeld approach explainsthe enhancement of the ﬁrst-order phase transition for smallsystem sizes and small Yukawa coupling, there are also limita-tions in such analysis and open questions remain to be solved.For instance, for any ultraviolet value of u,u /primewill change back to positive values. This is contrary to the theoretical predictionthat there exists a tricritical point for ﬁnite Yukawa coupling.Such limitation can be traced back to the procedure that wedo not treat the boson and fermion ﬂuctuations on the samefooting, so the ultimate fate of u /primein the renormalization ﬂow is still largely unknown. Moreover, since the computationalcomplexity of DQMC scales with a high power with respecttoL, numerical calculations for even larger system sizes are increasingly difﬁcult to carry out. For larger λ, numerical results show apparent unstable results. The reason may be theeffects induced by the higher-order terms. Therefore, despite(a) (b) (c) FIG. 6. Data collapse of /angbracketleftm2/angbracketrightfor different critical exponents close to the CTP at ( /Gamma1∗ x=4.2,/Gamma1∗ z=3.6) with the fermion-boson coupling strength λ=0.3. (a) Chiral Ising Gross-Neveu CTP uni- versality class, (b) mean-ﬁeld Ising tricritical universality class, and (c) chiral Ising Gross-Neveu universality class. The critical expo- nents used are shown in Table I. 075147-6FERMION ENHANCED FIRST-ORDER PHASE TRANSITION … PHYSICAL REVIEW B 103, 075147 (2021) TABLE I. Different critical exponents for the tricritical point. Universality class νη φ ηψ ω Chiral Ising Gross-Neveu CTP in this work 0.49 0.75 Ising tricritical point (mean-ﬁeld) [ 67]1 /20 Chiral Ising Gross-Neveu [ 35] 1.0 0.59 0.05 0.8 Chiral Ising Gross-Neveu CTP from functional renormalization group [ 63] 0.435 0.736 0.036 that the boson quartic coupling changes to positive for large L andλ, according to the modiﬁed mean-ﬁeld method presented here, it is still a open question as to whether to explore theentire parameter region of the fermion-enhanced ﬁrst-orderphase transition. IV . CHIRAL GROSS-NEVEU TRICRITICAL POINT The fermion ﬂuctuations not only change the type of phase transition but also inﬂuence the critical properties. A prevalentexample is the chiral universality class, in which the Diracfermions drive the Wilson-Fisher ﬁxed point for the pureboson model into the Gross-Neveu ﬁxed point [ 25,42–44]. Persistent efforts, including both theoretical and numericalworks, have been devoted to unveil the critical properties inthese systems [ 35,42–44,48–62]. As a generalization of the chiral critical point, the CTP manifests itself when the usualbosonic tricritical point is coupled to the Dirac fermions.Similar to the critical point, it was shown that the fermionﬂuctuation can drive the bosonic tricritical behavior into a newuniversality class [ 63]. However, to the best of our knowledge, studies on the chiral tricritical point were hitherto limitedin the theoretical approach. Here we employ the DQMC toexplore the critical properties near the chiral tricritical point. We locate the position of the CTP at /Gamma1 ∗ zby the crossing point of the Binder ratio, which opportunely appears at a smallnegative value at /Gamma1 ∗ x, and the obtained CTPs are shown in Fig. 3as the triangles with a green border. There are two relevant directions near the CTP, one associated with the massterm r /primeand the other the quartic term u/prime, both of which are shown in Eq. ( 13). At the tricritical point, the former dominates. We compute the order parameter m2close to the CTP forλ=0.3 for various lattice sizes. As shown in Fig. 6(a), by rescaling the curves of m2versus /Gamma1z−/Gamma1∗ zaccording to the ﬁnite-size scaling form /angbracketleftm2/angbracketrightLz+η=f[L1/ν(/Gamma1z−/Gamma1∗ z)//Gamma1∗ z], with (/Gamma1∗ x=4.2,/Gamma1∗ z=3.6) being the value at the CTP, we ﬁnd the curves of m2collapse onto each other when the critical exponents ν=0.49 and ηφ=0.75, as shown in the ﬁrst row in Table I. As a contrast, we also plot the rescaled m2curves with the mean-ﬁeld tricritical exponents ν=1/2 and η=0 for the pure boson model (second row in Table I), which is at the upper critical dimension [ 67], and the chiral Ising critical exponents at the continuous transition (third row in Table I), determined from previous DQMC simulation [ 35], with the corresponding results show in Figs. 6(b) and 6(c), respec- tively. We ﬁnd that the collapse is obviously better in Fig. 6(a), and the exponents are close to the predicted chiral Ising Gross-Neveu CTP from functional renormalization group analysis[33], as shown in the fourth row in Table I, which provides strong evidence for the existence of CTP. The discrepancy ofthe critical exponent between the ﬁrst and fourth row in Table I may come from the truncation approximation in the functionalrenormalization group calculation in the previous literature[63] or the ﬁnite-size scaling in the DQMC simulation in this work. Comparing the exponents of the CTP with those of the Ising tricritical point as shown in Table I, one ﬁnds that a nonzero anomalous dimension of the boson ﬁeld for the CTPis developed, while it is zero for the Ising tricritical point.The reason is that (2 +1)D is the upper critical dimension for the Ising tricritical point, while for the CTP the upper criticaldimension cannot be determined from naive power countingof the dimension analysis, since near the Gaussian ﬁxed point,the dimensions of the quartic boson coupling and the Yukawacoupling are different. This behavior may also prohibit thestudy of the properties of the CTP from the usual perturba-tive renormalization group from the dimension regularization.From this point of view, our present work provides a solidveriﬁcation of the existence of the CTP and its related criticalproperties. V . SUMMARY In summary, we have numerically studied the phase transi- tions in the Landau-Devonshire model coupled to the Diracfermions. We ﬁnd that the interplay of critical ﬂuctuationsand ﬁnite-size effect can give rise to a fermion-enhancedﬁrst-order phase transition. This seems to be contrary to thetheory of the type-II FIQCP. By developing a modiﬁed mean-ﬁeld theory, we show that the reason for this anomalousphenomenon is the interplay between the fermion ﬂuctua-tions and the ﬁnite-size effects, and the fate of the type-IIFIQCP for larger system sizes remains to be addressed. More-over, we have numerically revealed the critical behavior nearthe chiral Ising Gross-Neveu tricritical point and obtained thecritical exponents therein. Our result demonstrates that theinterplay of massless Dirac fermions, critical ﬂuctuations, andthe ﬁnite-size effects could trigger a plethora of interestingphenomena and therefore great care is called for when makinggeneralizations. In the future it will be instructive to exploresimilar behaviors in other systems with ﬁnite Fermi surfacesother than Dirac cones and also interesting to study the fullscaling form in these fermion-boson coupled systems, includ-ing the other relevant directions. ACKNOWLEDGMENTS Y .Z.L. and Z.Y .M. acknowledge support from the RGC of Hong Kong SAR of China (Grants No. 17303019 and No.17301420), MOST through the National Key Research andDevelopment Program (Grant No. 2016YFA0300502), 075147-7YUZHI LIU, ZI Y ANG MENG, AND SHUAI YIN PHYSICAL REVIEW B 103, 075147 (2021) and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33000000).S.Y . is supported by a startup grant (Grant No.74130-18841229) from Sun Yat-sen University. We arethankful for the Computational Initiative at the Faculty ofScience and the Information Technology Services at the University of Hong Kong, and the Tianhe-1A and Tianhe-3prototype platforms at the National Supercomputer Centers inTianjin for their technical support and generous allocation ofCPU time. APPENDIX: DETERMINANT MONTE CARLO METHOD We use the determinant quantum Monte Carlo (DQMC) method to simulate the model, which is illustrated by Eq. ( 4). We start with the partition function Z=Tr{e−βH}=/summationdisplay [σz]ωB[σz]ωF[σz], (A1) where the conﬁguration space of [ σz] is comprised of an Ising ﬁeld. The bosonic part of the partition function is ωB=exp/bracketleftBigg −/parenleftBigg /Delta1τJa/summationdisplay l/summationdisplay /angbracketleftpq/angbracketrightσz p,lσz q,l−/Delta1τJb/summationdisplay l/summationdisplay /angbracketleft/angbracketleftpq/angbracketright/angbracketrightσz p,lσz q,l−γ/summationdisplay p/summationdisplay lσz p,l+1σz p,l−/Delta1τ/Gamma1 z/summationdisplay p/summationdisplay lσz p,l/parenrightBigg/bracketrightBigg (A2) from HBoson in Eq. ( 1), when the two-dimensional transverse-ﬁeld Ising model is mapped to a 3D classical model with γ= −1 2ln[tanh( /Delta1τ/Gamma1 x)]. Meanwhile, the fermion part of the partition function is ωF=/productdisplay σ=↑,↓det/bracketleftbig 1+Bσ MBσ M−1···Bσ 2Bσ 1/bracketrightbig . (A3) Due to the spin-staggered phase eiσφinHFermion term in Eq. ( 2), the spin-up determinant det[1 +B↑ MB↑ M−1···B↑ 2B↑ 1]i s complex conjugate to the spin-down one det[1 +B↓ MB↓ M−1···B↓ 2B↓ 1],which indicates a no-sign problem in the system. The Bσ lcan be decomposed into two parts: Bσ l=exp(−/Delta1τHFermion )e x p (−/Delta1τHCoupling ). (A4) For the HCoupling term, as for the fermion-boson coupling term in Eq. ( 3), it is separated into four parts such that in each part all the hopping terms commute with each other. For sampling the conﬁguration, we update one site Ising ﬁeld in which theacceptance ratio is expressed as r=ω B[σ/prime z] ωB[σz]ωF[σ/prime z] ωF[σz]. 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PhysRevB.102.054405.pdf	PHYSICAL REVIEW B 102, 054405 (2020) Suppression of magnetic ordering in Fe-deﬁcient Fe 3−xGeTe 2from application of pressure Dante J. O’Hara ,1,2,,†Zachary E. Brubaker ,2,3,5Ryan L. Stillwell,2Earl F. O’Bannon ,2Alexander A. Baker ,2 Daniel Weber ,4Leonardus Bimo Bayu Aji,2Joshua E. Goldberger ,4Roland K. Kawakami,1,4Rena J. Zieve,5 Jason R. Jeffries,2and Scott K. McCall2 1Materials Science and Engineering, University of California, Riverside, Riverside, California 92521, USA 2Lawrence Livermore National Laboratory, Livermore, California 94550, USA 3Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 4The Ohio State University, Columbus, Ohio 43210, USA 5University of California, Davis, California 95616, USA (Received 29 April 2020; revised 12 July 2020; accepted 14 July 2020; published 3 August 2020) Two-dimensional van der Waals magnets with multiple functionalities are becoming increasingly important for emerging technologies in spintronics and valleytronics. Application of external pressure is one method to cleanlyexplore the underlying physical mechanisms of the intrinsic magnetism. In this paper, the magnetic, electronic,and structural properties of van der Waals-layered, Fe-deﬁcient Fe 3−xGeTe 2are investigated. Magnetotransport measurements show a monotonic decrease in the Curie temperature ( TC) and the magnetic moment with increasing pressure up to 13.9 GPa. The electrical resistance of Fe 3−xGeTe 2shows a change from metallic to a seemingly nonmetallic behavior with increasing pressure. High-pressure angle dispersive powder x-raydiffraction shows a monotonic compression of the unit cell and a reduction of the volume by ∼25% with no evidence of structural phase changes up to 29.4(4) GPa. We suggest that the decrease in the T Cdue to pressure results from increased intralayer coupling and delocalization that leads to a change in the exchange interaction. DOI: 10.1103/PhysRevB.102.054405 I. INTRODUCTION The discovery of intrinsic ferromagnetism in the mono- layer limit of van der Waals (vdW) materials has resultedin many opportunities to study quasi-two-dimensional (2D)magnetism [ 1,2]. Properties, such as gate-tunable magnetism and giant tunneling magnetoresistance have been observedin mechanically exfoliated CrI 3[2–5], and room-temperature ferromagnetic ordering in large-area ﬁlms of monolayerMnSe 2and VSe 2have been reported, showing potential for spin-based technological applications [ 6,7]. Among the 2D magnets, Fe 3−xGeTe 2is of interest because of its high Curie temperature, TC, strong perpendicular magnetic anisotropy, and competing magnetic phases, all of which are tunable bycontrolling the concentration of Fe and the number of layers[8–15]. Measurements of the bulk parent compound may provide a better understanding of the single atomic sheets ofthese chalcogen-based vdW materials as well as key insightsneeded to develop more structurally and magnetically stable2D materials. “Chemical pressure,” generated by substituting Ni or Co into the Fe sites, has been shown to suppress ferromagnetismin Fe 3−xGeTe 2crystals [ 16,17]. Pressure offers a clean ap- proach to modifying the relative strengths of the exchange in-teractions by altering the interatomic separations of the atomic Corresponding author: dante.ohara.ctr@nrl.navy.mil †D. J. O’Hara is currently with the U.S. Naval Research Laboratory, Materials Science and Technology Division, U.S. Naval ResearchLaboratory, 4555 Overlook Ave. SW, Washington, D.C. 20375.planes without changing the chemical composition [ 18,19]. For example, hydrostatic pressure drives a spin-reorientationtransition in vdW Cr 2Ge2Te6by reducing the Cr-Te bond distance within individual unit layers, therefore, changing thespin-orbit interaction [ 20]. Here, we investigate the crystal structure, electronic, and magnetic properties of Fe-deﬁcient Fe 3−xGeTe 2as a function of temperature and pressure and observe a reduction of TC with increasing pressure up to 13.9 GPa. Independent determi- nations of TCbased on temperature-dependent measurements of magnetization, resistance ( Rxx), and anomalous Hall effect (AHE) provide consistent values of TCand conﬁrm the pres- sure dependence of TC. Pressure-dependent x-ray diffraction (XRD) provides a correlation of TCwith the lattice param- eters a(in-plane) and c(out-of-plane), yielding trends for Fe-deﬁcient Fe 2.75GeTe 2that are similar to previous results on stoichiometric Fe 3GeTe 2[21]. This paper indicates that the structure can be controlled with pressure, systematicallysuppressing the magnetic ordering of Fe 3−xGeTe 2until no longer detectable near 16 GPa. II. METHODS Crystals of Fe-deﬁcient Fe 2.75GeTe 2were grown from a Te ﬂux using a technique adapted from previous reports[15]. Initial ingredients of 80.4-mg Fe granules (2 equiva- lent (eq.), 99.98% purity, Alfa Aesar), 52.3-mg Ge powder(1 eq., 99.999% purity, Alfa Aesar), and 367.4-mg Te lumps(4 eq., 99.999% purity, Alfa Aesar) were heated in an aluminacrucible in an evacuated quartz ampoule to 950 °C, soaked for12 h, cooled to 875 °C at a rate of 60 °C/h, and to 675 °C 2469-9950/2020/102(5)/054405(10) 054405-1 ©2020 American Physical SocietyDANTE J. O’HARA et al. PHYSICAL REVIEW B 102, 054405 (2020) at a rate of 3 °C/h. The ampoule was quenched to air, and the hot ﬂux removed by centrifugation, yielding metallicmillimeter-sized crystals. Sample composition of a Fe:Ge:Teratio of 2.75:1:2 was conﬁrmed by Rutherford backscatteringmeasurements (Appendix Fig. 7). The ambient lattice parameters of the samples were mea- sured using a Bruker D8 Discover x-ray diffractometer withaC u Kαsource ( λ=1.5406 Å). Pressure-dependent angle dispersive x-ray diffraction scans were performed at sector16-BMD of the Advanced Photon Source at Argonne NationalLaboratory using synchrotron radiation monochromated bySi(111) to a wavelength of 0.4133 Å (30 keV). Detector orien-tation, distance, and x-ray wavelength were calibrated using aNational Institute of Standards and Technology CeO 2powder sample. Samples were powdered and loaded into a LawrenceLivermore National Laboratory (LLNL) membrane diamond-anvil cell (DAC) [ 22], a rhenium gasket was used to contain the sample, and neon was used as the pressure-transmittingmedium. The pressure was estimated by measuring the latticeparameter of Au powder, which was mixed with the sample,using the Au equation of state published by Fei et al. [23]. At select pressures, ruby ﬂuorescence spectra were collected,and the pressure was estimated by using the ruby calibrationof Dewaele et al. [24]. The XRD patterns were collected by an area detector and radially integrated into powder patternsusing DIOPTAS [25]. The CeO 2diffraction pattern at ambient pressure was used to determine the instrument parametersfor reﬁnements via GSAS-II [ 26,27]. All measurements were performed at room temperature. Bulk magnetization measurements were made in a su- perconducting quantum interference device (SQUID) magne-tometer (magnetic property measurement system, QuantumDesign) between 5 and 350 K with the caxis of the sample oriented parallel or perpendicular to the magnetic ﬁeld. Mag-netic measurements under pressure used a Cu-Be piston cell(Almax easyLab Mcell 10) up to 0.7 GPa. The sample wasimmersed in a pressure-transmitting medium of Fluorinert,and the pressure was determined using the superconductingtransition of a Sn manometer inside the pressure cell [ 28]. Resistivity and magnetoresistance (MR) measurements were performed in a 160 kOe superconducting magnet system(physical property measurement system, Quantum Design)using the four-probe AC transport option ( f=17 Hz) with the external magnetic ﬁeld applied parallel to the caxis of the sample. High-pressure measurements were performed on amonolithic polycrystalline sample (approximately 50 ×50× 10μm 3) using an eight-probe designer DAC [ 29–32] with steatite as the pressure-transmitting medium and ruby as thepressure calibrant (see the Appendix for further details). Themaximum pressure employed in this paper was 16.2 GPa. III. RESULTS AND DISCUSSION Iron-deﬁcient Fe 2.75GeTe 2is a weak itinerant ferromagnet that crystallizes into a hexagonal structure with the spacegroup of P6 3/mmc . The structure consists of two distinct Fe sites that are tetrahedrally coordinated to Ge and Te atomsand form sheets that are vdW bonded between each unit layer(Fig. 1, inset). The following lattice parameters were obtained from the ambient-pressure room-temperature XRD measure- FIG. 1. Ambient pressure temperature-dependent magnetization measurement showing TC=155 K and easy axis along c. Closed symbols are with the magnetic ﬁeld along the caxis, and open symbols are with the ﬁeld on the abplane. The inset: Ball-and-stick model showing Fe 3−xGeTe 2crystal structure from the side view where the green, orange, and purple balls represent Te, Fe, and Geatoms, respectively. ments a=3.9555(3) and c=16.3887(1) Å, consistent with the lattice parameters reported by May et al. [a=3.9421(9) andc=16.378(5) Å] [ 15]. The magnetic properties at am- bient pressure (Fig. 1) show a preferred out-of-plane magne- tization along the caxis of the crystal and a TCof∼155 K, which is close to reports for Fe 2.8GeTe 2(TC=154 K) [ 15]. The transition temperature is determined via differentiation ofthe temperature-dependent magnetization curves dM/dT. Chemical doping studies of Fe 3−xGeTe 2crystals demon- strated that TCis correlated with the quantity of Fe vacancies and the degree to which they distort the crystal structure [ 15]. With increasing Fe vacancies, the structure contracts alongtheaaxis whereas it expands along the caxis, leading to a decrease in T Cto 140 K, whereas fewer Fe vacancies leads to an increase in the TC[15]. Furthermore, substitution of Fe with either Co or Ni, leads to a gradual suppression in theferromagnetic ordering due to a transition to a glassy magneticphase [ 16,17]. The effect of substitution of Co or Ni on T C can be viewed as increasing the concentration of Fe vacancies, and, therefore, the evolution of TCwith Co or Ni substitution closely mimics the evolution of TCwith Fe vacancies [ 15–17]. Although prior work is used as a guide, the application of hydrostatic pressure compresses both crystal axes, potentiallyleading to different effects on the ordering temperature. TheXRD patterns from 0.7(2) to 29.4(4) GPa show a gradual shiftin peak position to higher 2 θangles indicative of a smaller unit cell [Appendix Fig. 9(a)]. All the diffraction peaks can be identiﬁed, indexed and reﬁned with peaks from the sampleFe 2.75GeTe 2, an impurity FeTe 2, the pressure marker Au, and the gasket (Re). Figures 2(a)–2(d) show monotonic decreases in the lattice parameters, c/aratio, and volume as functions of pressure with no evidence of a phase transition. The absolutecompression of the caxis is more than the aaxis across the pressure range of these measurements, which is likely a 054405-2SUPPRESSION OF MAGNETIC ORDERING IN … PHYSICAL REVIEW B 102, 054405 (2020) FIG. 2. Room-temperature XRD measurements showing the compression of the unit-cell parameters and volume. There is no indication of a phase transition, and the caxis is compressed about 4% more than the aaxis by 30 GPa. consequence of the weak interlayer vdW interaction (van der Waals gap at ambient pressure 2.95 Å). The pressure versus volume curve in Fig. 2(d) shows a reduction of approximately 25% near 30 GPa. This pressure evolution can be well ﬁt by aRose-Vinet equation of state with a bulk modulus ( K) of 52(8) GPa and pressure derivative ( K’)o f5 . 8 ( 1 )[ 33], comparable to other vdW crystals under high pressure, such as WSe 2[K= 72(5) and K/prime=4.6(5)] [ 34]. The detailed Rietveld reﬁnement results is presented in the Appendix [Fig. 9(b)]. A series of isobaric electrical resistance ( RxxandRxy) mea- surements are performed as a function of applied magneticﬁeld to develop an understanding on how compression ofthe structure affects the material’s electronic and magneticproperties. Figure 3(a) is a schematic of a designer DAC where the electrical leads are embedded in the diamondanvils [ 29–32]. The in-plane resistance R xx(T) for a series of pressures is shown in Fig. 3(b) normalized to R(T=300 K). The electronic properties change as a function of appliedpressure, evolving from a metallic state ( dR/dT>0) at ambient pressure to a seemingly nonmetallic state ( dR/dT< 0) at the highest pressures. Measurements on stoichiometricFe 3GeTe 2(bulk TC=220 K) denote a “kink” in the ρxx(T) curve that represents the transition from a ferromagnetic toparamagnetic phase [ 21]. This kink is most easily quantiﬁed with the temperature derivative of resistivity dρ xx/dT,b u t this feature becomes smeared out at pressures above 13.4 GPa (TC≈120 K) [ 21] likely due to deviatoric stress in the sample compartment at these pressures. A similar broadening of thedR xx/dTcurve for pressures above 4.1 GPa makes it chal- lenging to determine the TCusing this method. Although there is a clear trend indicating the Curie temperature decreaseswith increasing pressure, the broadening of the transitionwith increasing pressure limits a precise determination of T C, particularly at higher pressures. Figure 4presents both the symmetric ( Rxx) and antisym- metric components ( Rxy) of the MR curves as functions of pressure and temperature (see the Appendix for a detailed dis-cussion). The isothermal R xx(H) data have been normalized using the following expression: MR(%)=/Delta1R R=R−R0 R0×100%, (1) where Ris the resistance at a given magnetic ﬁeld and R0 is the resistance at zero ﬁeld. The symmetric MR curves show negative MR at low pressures—which is common inferromagnetic compounds because of the suppression of spinscattering via a magnetic ﬁeld [ 12,35,36] and an increase in magnitude of the MR up to 11 GPa followed by a gradualdecrease in magnitude at higher pressures [Fig. 4(a)]. The MR transitions from sublinear with Hto linear as temperatures increase and no saturation behavior is observed [Fig. 4(b)]. FIG. 3. Resistance of Fe 3−xGeTe 2measured at a series of increasing pressures. (a) Schematic of the designer DAC used for electrical measurements. (b) Normalized resistance measurements for cross comparison (offset for clarity). The change in slope is an indication of the Curie temperature in each trace. With increasing pressure, the paramagnetic region changes from metallic to nonmetallic. 054405-3DANTE J. O’HARA et al. PHYSICAL REVIEW B 102, 054405 (2020) FIG. 4. MR measurements. (a) Negative MR at 5 K for several pressures showing the greatest relative change at 11 GPa. (b) Negative MR at 7.8 GPa showing changes in shape (linear versus sublinear) as a function of temperature due to magnon scattering. Rxy(H) scans showing changes in magnitude as a function of temperature and pressure where (c) the AHE component shows suppression of magnitude fromcompression of the Fe 2.75GeTe 2crystal and (d) shows the raw data as a function of temperature at 4.1 GPa where the signal is dominated by the linear ordinary Hall component above TCand a low-ﬁeld saturation below TC. The transition to the weaker nonsaturating linear region is consistent with the presence of magnon scattering at elevatedtemperatures near T C[36–39], and, with increasing pressure, this transition temperature decreases (not shown). Additional insight is gleaned from the Rxy(H) data. For ferromagnetic conductors, the Rxy(H) has two components contributing to the signal as shown in the expression, Rxy=RHH+RAHE=RHH+RSM, (2) where RHHrepresents the ordinary Hall effect and RAHE,a n additional nonlinear ferromagnetic contribution known as theanomalous Hall effect (AHE), which is directly proportionaltoR S, a scattering coefﬁcient, and M(H), the magnetization [12,40]. At 5 K, there is a distinct linear region at high ﬁelds due to saturation of the magnetization. To isolate the AHEcontribution, we ﬁt the linear region where the R xy(H)s h o w s only linear behavior and above the ambient pressure magneticsaturation [ ∼5 kOe, Appendix Fig. 8(b)], spanning from 5 to 100 kOe [ 31]. This linear component is then subtracted from the measured signal to yield the AHE component. This isshown in Fig. 4(c) as a function of pressure and temperature, respectively. With increasing pressure, the overall saturationvalue of the R AHE signal decreases as pressure increases, consistent with the observations of Wang et al. [21] due to thedecrease in TCand the gradual suppression of the Fe magnetic moment. For the temperature dependence, we plot the totalR xy(H) including the AHE and ordinary Hall contributions [Fig. 4(d)], which suggests a change from a ferromagnetic to a paramagnetic state. At temperatures of 135 K and above (at4.1 GPa), the change in slope from a dominating nonlinearAHE contribution to a dominating linear contribution of R H is a signature of TC.TCis determined quantitatively by ﬁrst deﬁning Sas the slope of Rxy(H) from 0 to 1 kOe and plotting Sas a function of temperature for various pressures [Fig. 5(a)]. The raw data are shown in the Appendix (Fig. 11). Since the AHE component of Rxyis proportional to magnetization M, the slope Sis proportional to the susceptibility, χ=dM/dH, plus an offset from the ordinary Hall effect. Accordingly, theSversus Tcurves [Fig. 5(a)] have similar shapes as the M versus Tcurves from magnetic measurements (Fig. 1) where the ferromagnetic-to-paramagnetic transition with increasingtemperature is identiﬁed by a strong reduction in signal ( Sor M)a tT C. The derivative of S(i.e., dS/dT) is taken to determine TC and is shown in Fig. 5(b), designated by arrows and shows a decreasing trend with higher pressure. Furthermore, thereis no signature of a transition temperature in the dS/dTdata at 16.2 GPa, which suggests that the ferromagnetism may be 054405-4SUPPRESSION OF MAGNETIC ORDERING IN … PHYSICAL REVIEW B 102, 054405 (2020) FIG. 5. (a)-(b) TCdetermined from AHE data. Sis deﬁned as the initial slope of the Rxy(H<1 kOe) curve. The derivative shows a dip near/at TCwhere arrows indicate the local minimum of a Lorentzian ﬁt (this ﬁtting is also used to determine error bar for TC). The plotted data are offset for clarity. There is no measurable transition temperature above 13.9 GPa, therefore, an arrow denoting TCat 16.2 GPa is not shown in (b). (c) P-Tphase diagram showing TCdetermined by these approaches. The dashed line is a parabolic ﬁt of the data to 13.9 GPa for a guide to the eye. The TCshows a monotonic decrease at a decay rate of 7.4 K/GPa. The contour regions show MR∗=d2MR/dH2atH=60 kOe (in units of 10−12/Omega1/Oe2) where the change from zero is consistent with indications of ferromagnetic ordering. Pressure error bars are determined via a difference of pressure before and after temperature cycles, and details are discussed in the Appendix. suppressed or that pressure smears the transition until it is undistinguishable. A combination of the dM/dT,dRxx/dT, anddS/dTcurves are used to plot the TCfor pressures above ambient conditions (see Table Ifor quantiﬁed TCvalues from different methods) and are plotted on a temperature-pressurephase diagram. This pressure-dependent reduction of the T C is shown in Fig. 5(c) with different methods of determination and a parabolic extrapolation to higher pressures showinga monotonic decrease at a decay rate of ∼7.4 K/GPa. A contour map designating the ferromagnetic and paramagneticregions of the phase diagram using the second derivative ofthe local negative MR curvature at magnetic saturation [asseen in Fig. 4(b)] is denoted as MR ∗. The represented data are ﬁxed at a ﬁeld of 60 kOe and, for the same trend, is presentfor any large ﬁeld above saturation. Note that the MR ∗map of the phase diagram shows possible magnetic ordering atP>14 GPa, closely following the parabolic ﬁtting but is not direct evidence of ferromagnetic ordering [ 41]. For pressures below 2 GPa, bulk magnetization measurements conﬁrm thedownward trend in ordering temperature (Appendix Fig. 12). The absence of evidence of a T Cbeyond 13.9 GPa is con- sistent with Ref. [ 21] where a transition temperature could not be distinguished for pressures higher than 13.4 GPa. Itshould be noted that the application of pressure to this Fe-deﬁcient sample lowers the absolute temperature where theferromagnetic transition remains detectable as compared tothe stoichiometric sample in Ref. [ 21]. Figures 6(a)and6(b) show the T Cas a function of the unit- cell parameters compared to the results of pressurized stochio-metric Fe 3GeTe 2[21] and chemical doped Fe 3−xGeTe 2[15] which the TCdrops from ∼200 K close to 50 K as a function of compressed lattice parameters. In Ref. [ 15], the reduced TCcorrelates with the reduction of a, expansion of c, an increased Fe(I)-Fe(I) bond distance, and a decreased Fe(I)-Fe(II) bonddistance. In the present paper, pressure causes both aandcto decrease with corresponding reductions in both the Fe(I)-Fe(I)and the Fe(I)-Fe(II) atomic distances. This is accompanied bya reduced T Cand could be evidence of spin-lattice coupling from pressure-induced compression of the crystal. Figure 6 FIG. 6. (a)-(b) TCas a function of unit-cell parameters. The red and blue data are the results of pressure measurements whereas thegreen circles show a variation in xfrom 0 to 0.3 for Fe 3−xGeTe 2.T h e variation in the a lattice parameter is consistent for all three sets of measurements whereas the caxis parameter is not. 054405-5DANTE J. O’HARA et al. PHYSICAL REVIEW B 102, 054405 (2020) shows the aaxis is very closely correlated with the decrease inTCwith the link being less clear for the caxis. The TC decreases with decreasing aaxis in all three cases, and a linear extrapolation shows a reduction close to 5 K at 3.7 Å whereasfor the caxis, it deviates with opposite slope while applying chemical pressure. This provides evidence that intralayer ex-change coupling plays a larger role in the T Creduction than the interlayer exchange coupling. Another possibility is that because Fe 2.75GeTe 2is a weak itinerant ferromagnet, the 3 d-electron bandwidth of the spin density of states (spin-DOS) near the Fermi level shouldbroaden as volume decreases which will lead to a correspond-ing decrease in the Stoner factor and thereby reduce the Curietemperature based on the Stoner criterion for magnetic order-ing [ 42,43]. This should lead to a reduction in the splitting of the spin-DOS bands, thus, decreasing the magnetic momenteventually resulting in a nonmagnetic state. To conﬁrm this,we combine our bulk magnetization and AHE data at 5 K un-der pressure and use this to calculate a Rhodes-Wohlfarth ratio(RWR) [ 44,45]. RWR is deﬁned as p c/pswith pcobtained from the effective moment calculated from the Curie-Weisssusceptibility, p c(pc+2)=p2 eff, (3) andpsis the saturation moment obtained at low temperatures. RWR is 1 for localized systems and is larger in an itinerant(delocalized) system. Here, we take p sas the magnetization saturation obtained at 5 K and 50 kOe, so the calculatedRWR values are ∼4.0. The RWR shows that the system becomes more delocalized with pressure, which is consistentwith chemical doping studies on Fe 3−xGeTe 2[15,46] and supports the explanation of reduced TCin terms of the Stoner criterion. Thus, the reduction of the magnetic moment (and,subsequently, T C) is caused by the diminished 3 d-electron correlations likely due to the shortening of the Fe(I)-Fe(II) dis-tance from increasing pressure. A combination of a spin-waveor x-ray scattering experiment under pressure and theoreticalsupport will further address the itinerant nature of this systemand is needed to conﬁrm this observation. IV . CONCLUSIONS This paper is a systematic study of the effect of pressure on the resistance and MR of Fe-deﬁcient Fe 2.75GeTe 2.I nt h e absence of applied pressure, Fe-deﬁcient Fe 2.75GeTe 2shows negative MR and metallic behavior, and the magnetic easyaxis is along the cdirection. T Cdecreases linearly from 155 to∼50 K with increasing the pressure to 13.9 GPa. Although there is evidence that ferromagnetic ordering may exist abovethis pressure, conﬁrmation requires a low-temperature mea-surement of the crystal structure and a direct measurementof the magnetization under pressure. The electronic transportimplies a transition from a metallic to a nonmetallic state withincreasing pressure up to 16.2 GPa. R xy(H) measurements are used to quantify the evolution of TCup to nearly 14 GPa and are used in combination with other methods to generatea magnetic P-Tphase diagram. The effect of pressure on Fe 2.75GeTe 2illustrates the value of pressure as a tool to better understand the underlying mechanisms for magnetic orderingin vdW systems.ACKNOWLEDGMENTS This work was performed under the auspices of the U.S. Department of Energy (DOE) by LLNL under Contract No.DE-AC52-07NA27344. Part of the funding was providedthrough the LLNL Lawrence Graduate Scholar Program.D.J.O. acknowledges support from the GEM National Consor-tium Ph.D. Fellowship. Portions of this work were performedat HPCAT (Sector 16), Advanced Photon Source, ArgonneNational Laboratory (ANL). HPCAT operations are supportedby DOE-NNSA’s Ofﬁce of Experimental Sciences. The APSis a U.S. DOE of Science User Facility operated for theDOE Ofﬁce of Science by ANL under Contract No. DE-AC02-06CH11357. D.W., J.E.G., and R.K.K. acknowledgesupport from the Center for Emergent Materials, an NSFMRSEC under Grant No. DMR-1420451. Z.E.B. and R.J.Z.acknowledge support from NSF Grant No. DMR-1609855.We thank J. Beckham, J. R. I. Lee (LLNL), and C. Park(HPCAT) for technical assistance. D.J.O. was supported byNRC/NRL while ﬁnalizing the paper. APPENDIX The elemental composition of the sample was character- ized by RBS with a 2-MeV4He beam. Rutherford backscat- tering spectrometry (RBS) is a nondestructive method basedon high-energy ion scattering, providing depth-resolved infor-mation about the elemental composition of near-surface layers[47] (see Fig. 7). For RBS, the He ion beam was incident normal to the sample surface and backscattered into a detectorlocated at 165° from the incident beam. The analysis of RBSspectra was performed with the RUMP code [ 48]. Phase identiﬁcation and magnetic properties in ambient conditions were determined using XRD (Cu Kα, Bruker) and magnetometry. Figure 8(a)shows the ambient XRD pattern of the Fe 2.75GeTe 2polycrystalline sample with an observation FIG. 7. Rutherford backscattering spectra from Fe 2.75GeTe 2 sample. Symbols are experimental points, whereas solid lines are results of RUMP -code simulations. For clarity, only every tenth ex- perimental point is depicted. Surface edges of Fe, Ge, and Te are marked by arrows. 054405-6SUPPRESSION OF MAGNETIC ORDERING IN … PHYSICAL REVIEW B 102, 054405 (2020) FIG. 8. Ambient pressure measurements of Fe 2.75GeTe 2with (a) showing XRD with a preferred texture along the caxis. The inset: Photograph of Fe 2.75GeTe 2sample used in measurements (left) and 2D diffraction image showing polycrystallinity in the sample (right). Scale bar is 5 mm. Black lines are from the image plate. (b) M(H) measurements at 5 K showing magnetic anisotropy along the caxis of the crystal. of strong diffraction peaks along the (0 0 2l), indicating ah i g h c-axis orientation of the crystal. The indexing and reﬁnement of the peaks aligns with previous reports [ 15]. The insets show a laboratory photograph (scale bar is 5 mm)and a 2D diffraction image showing the crystallinity of thesample. Figure 8(b) depicts the M(H) loops at 5 K where the magnetization prefers to lie along the caxis and saturates at approximately 5 kOe. Angle-dispersive x-ray diffraction measurements under pressure were performed at room temperature using beam-line 16 BM-D (HPCAT) of the Advanced Photon Source atArgonne National Laboratory. A gas membrane-driven DACcomposed of two 500- µm diamond anvils was used to generate pressures up to 29.4(4) GPa [ 22]. A rhenium gasket was prein- dented to a thickness of 60 µm, and a 180- µm hole was drilled using a wire electric discharge machine in the center of thegasket to serve as a sample chamber. The sample was ground into a powder with a mortar and pestle under inert gloveboxconditions. The powders were then loaded into the DACsample chamber and mixed with Au powder, which servedas an x-ray pressure calibrant, and a ruby sphere was used forinitial pressure calibration. The Au bulk modulus ( K) and Au pressure derivative ( K’) are 167 GPa and 6, respectively [ 23]. Neon gas was used as the pressure-transmitting medium. Ne isessentially hydrostatic up to ∼15 GPa and at pressures above that the uniaxial stress remains low [ 49]. Incident x-rays with a monochromated energy of 30 keV ( λ=0.413 28 Å) were microfocused to a 12 ×5-μm 2spot. X-ray diffraction mea- surements were performed in a transmission geometry, anda MAR345 image plate was used as the detector with 120-sexposures at each pressure. The detector was calibrated usingCeO 2. The resulting 2D diffraction patterns from the detector FIG. 9. (a) XRD plotted as a function of pressure showing compression of the unit cell with higher 2 θangles. (b) Rietvield reﬁnement of powdered Fe 2.75GeTe 2at 0.7 GPa. †, *, and are FeTe 2, Au pressure marker, and Re gasket, respectively. All other unlabeled indexed peaks are the sample. The ^ symbol is indicating solidiﬁcation of the neon gas at higher pressure. 054405-7DANTE J. O’HARA et al. PHYSICAL REVIEW B 102, 054405 (2020) FIG. 10. Magnetoresistance measurements at a select pressure of 4.1 GPa and select temperature of 5 K. (a) Raw electrical resistancedata as a function of applied magnetic ﬁeld showing nonsymmetric MR about the zero ﬁeld. (b) Symmetrized and (c) antisymmetrized data showing both R xxand Rxycontributions in the raw R(H) measurement. were integrated to obtain conventional one-dimensional pow- der patterns using the program DIOPTAS [25]. Reﬁnements of the lattice parameters were performed using GSAS-II [ 26,27] and are shown in Fig. 9at 0.7(2) GPa. These samples contain a Fe-deﬁcient phase of nonmagnetic FeTe 2(orthorhombic, Pnnm ), which is considered in the XRD reﬁnement analysis but excluded from the main text of the paper. Ne is observedin Fig. 9(a) at pressures above ∼4.6 GPa, which is consistent with Ref. [ 49].Electrical transport studies under pressure were performed on a small polycrystal of Fe 2.75GeTe 2using an eight-probe designer DAC [ 29,30] with steatite as a pressure-transmitting medium and ruby as the pressure calibrant. The gasket wasmade of the nonmagnetic alloy MP35N, preindented to athickness of 40 µm, and a 100- µm diameter hole was drilled in the center of the gasket using a wire electric discharge ma-chine. The crystal was cleaved to an ∼10-µm thickness with a cross-sectional area of ∼50×50μm 2. Electrical contact to the sample was made via the exposed tips of the tungstenmicroprobes at the culet of the diamond anvil. To ensure goodcontact between the sample and the microprobes, steatite wasinitially precompressed into the gasket hole, and then thesample was placed on top of the steatite so that when theDAC was closed, the steatite pressed the sample against theleads. The pressure was determined by a single ruby, so nomeasurements of gradients were possible. However, based onprevious measurements in this designer DAC, gradients on the order of 5% are expected [ 32], which is consistent with values reported by Klotz et al. [49]. Pressure was determined at room temperature by averaging the shift of the R 1ruby ﬂuorescence peak before and after temperature cycles. Theuncertainty was determined by the difference of P maxand Pmin, where PmaxandPminare the pressures before and after temperature cycles. If the difference after temperature cyclesis larger than 5%, then this spread is chosen as the errorbar. Electrical transport measurements under pressure wereperformed as a function of temperature and magnetic ﬁeldusing the AC transport option in a Quantum Design physicalproperty measurement system. MR measurements were takenat each pressure using an excitation current of 0.316 mA at afrequency of 17 Hz. FIG. 11. (a)–(f) Antisymmetrized Rxy(H) measurements as a function of temperature at a given pressure. Rxycurve transitions from a nonlinear saturating curve indicative of ferromagnetism below TCto a linear nonsaturating curve above TC. 054405-8SUPPRESSION OF MAGNETIC ORDERING IN … PHYSICAL REVIEW B 102, 054405 (2020) TABLE I. Determination of magnetic ordering temperature at select pressures (n/a represents not available). Pressure (GPa) TC,1(K) TC,2(K) 2.5 143 n/a 4.1 130 128.2 7.8 102.5 97.811 77.4 79.7 13.9 n/a 50.6 16.2 n/a n/a The magnetoresistance Rxx(H) where the current is passed along the abplane of the sample and the magnetic ﬁeld is along the caxis of the sample is symmetrized by sweeping the magnetic ﬁeld over both the negative and the positive ﬁeldranges. The sum was then calculated between the resistancesover the positive and negative ﬁeld regions and divided by twoto extract the resistance solely due to the MR. In the samemanner, the Hall resistance R xy(H) where the current is passed perpendicular to the measured voltage and magnetic ﬁeld isantisymmetrized by taking the difference of the resistancesover the negative and positive magnetic ﬁeld regions anddividing by two. This procedure is displayed in the graphs inFig.10where the raw data show both R xxandRxysignals due to the irregular shape of the sample. For clarity in the maintext, the positive magnetic-ﬁeld values are only displayed.Temperature-dependent R xy(H) curves are shown in Fig. 11 denoting a sublinear saturating (below TC) curve transition to a linear nonsaturating curve (above TC). At ambient pressure, the TCis determined via the dif- ferentiation of the M(T) curve in Fig. 1of the main text. For pressures from 2.5 to 11 GPa, the TCof pressurized Fe2.75GeTe 2is quantiﬁed by taking dRxx/dT(labeled TC,1). For pressures from 4.1 to 13.9 GPa, the TCis determined via the dS/dT curve shown in Fig. 5(b) of the main text (labeled TC,2below). The quantiﬁed TCis shown in Table Ibelow. To perform hydrostatic pressures up to 0.7 GPa in a SQUID magnetometer, piston pressure cells were used. The QuantumDesign SQUID magnetometer is equipped with a rod attach-ment for these types of pressure cells. A Fe 2.75GeTe 2crystal FIG. 12. Pressure measurements using the magnetometer show- ing a drop in TCwith pressures up to 0.7 GPa. was placed inside a small cylindrical polytetraﬂuoroethylene cap along with a Sn manometer (as a pressure calibrant) anda pressure-transmitting medium of Fluorinert. The Fluorinertremains hydrostatic up to the maximum pressure in theseexperiments, and the deviatoric stress is known to be very lowfor pressures up to and above 0.7 GPa, at least, until ∼7G P a [49]. This was then placed in the middle of the Cu-Be pressure cell and held in place by extrusion disks and ceramic pistons.Samples were externally pressurized via a piston hydraulicpress (Mpress Mk2). After each pressurization, samples wereattached to the sample rod via threading and loaded into thechamber. The pressure cell was then cooled down slowlybelow the transition temperature of the superconducting Snmanometer and positioned accordingly. 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PhysRevB.93.195168.pdf	PHYSICAL REVIEW B 93, 195168 (2016) Electrical conduction mediated by ﬂuorine atoms in the pyrochlore ﬂuorides RbV 2F6and CsV 2F6with mixed-valent V atoms Hiroaki Ueda,1Kihiro Yamada,1,2Hirotaka Yamauchi,3Yutaka Ueda,3,4and Kazuyoshi Yoshimura1 1Department of Chemistry, Graduate School of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan 2Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, 611-0011, Japan 3The Institute for Solid State Physics, The University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan 4Toyota Physical and Chemical Research Institute, 41-1 Yokomichi, Nagakute, Aichi 480-1192, Japan (Received 31 March 2016; revised manuscript received 17 May 2016; published 31 May 2016) We have investigated structural, electrical, and magnetic properties of single crystals of modiﬁed pyrochlore ﬂuorides RbV 2F6and CsV 2F6, which have mixed-valent V atoms. At room temperature, they have orthorhombic structures. With increasing temperature, each of them exhibits two structural transitions, and electrical resistivityrapidly decreases accompanied with one of these structural changes. The changes of unit cell volume and electricalresistivity at these transition temperatures indicate that the structural instability and the charge ordering causestructural transitions of RbV 2F6and CsV 2F6, respectively. At low temperatures, CsV 2F6shows antiferromagnetic ordering at 5 K, and RbV 2F6shows two-step magnetic transitions. DOI: 10.1103/PhysRevB.93.195168 I. INTRODUCTION Most of ﬂuorides are electrically insulating because of their strong ionic characteristics. The electron afﬁnity of ﬂuorine isthe largest among those of all elements, and the chemicalbonds between ﬂuorine atoms and other kinds of atoms arehighly ionic. Hence all electrons in a ﬂuoride are localizedaround a ﬂuorine atom or an atom bonded with ﬂuorine atoms,and electrical conduction is suppressed. However, we havesome exceptions that have low electrical resistivity: Hg 3NbF 6, Hg3Ta F 6[1], and Ag2F[2]. In their crystal structures, Hg or Ag atoms form layers with metal-metal bonds, whichcause electrical conduction. Electron conduction mediated byﬂuorine atoms was not reported even in these electricallyconducting ﬂuorides. On the other hand, many transition-metal oxides exhibit electrical conduction. Their electrical conduction is due tostrong hybridization of oxygen porbitals and transition-metal dorbitals. Particularly, most of mixed-valent oxides are electrically conducting, since the Fermi level is located int h em i d d l eo f dbands. One of the most popular examples of mixed-valent oxides is magnetite Fe 3O4. In the structure, Fe2+and Fe3+are distributed on a pyrochlore lattice, and charge frustration was discussed by Anderson [ 3]. Oxides with a mixed-valent state on a pyrochlore lattice include somespinels (LiV 2O4[4,5], AlV 2O4[6], CuIr 2O4[7]), and modiﬁed pyrochlores (CsW 2O6[8,9],AOs2O6[10,11]), and most of them are electrically conducting. Although there are some ﬂuorides with mixed-valent ions, there are no detailed reports. One of such materials is themodiﬁed pyrochlore ﬂuoride system. Modiﬁed pyrochloreﬂuorides have a chemical formula of AMM /primeF6, where Ais an alkaline metal, Mis a divalent atom, and M/primeis a trivalent atom. [ 12]. Most of them contain two kinds of transition metals, MandM/prime[13]. Some of the modiﬁed pyrochlore ﬂuorides have the same transition metal M=M/primesuch as AFe2F6[14,15],ACr2F6[15–17], and AV2F6[16](A=Rb and Cs), in which the formal valence of Mis+2.5. In these compounds, divalent and trivalent cations are likely to order ina pyrochlore lattice, and hence they are electrically insulating.The former two systems AFe 2F6andACr2F6are undoubtedly insulating, because their colors are not black. However, AV2F6 was reported to be black, which indicates possible electrical conduction. In this paper, we report the structural, electrical, and magnetic properties of modiﬁed pyrochlore ﬂuorides RbV 2F6 and CsV 2F6. We found that they exhibit some structural tran- sitions at high temperatures. Furthermore, they are electricallyconducting, and show a jump in electrical resistivity withvarying temperature. We discuss the origin of these transitionsconcerning the formation of charge orderings. In addition, bothof them exhibit magnetic orderings at low temperatures, oneof which has two-step feature. II. EXPERIMENTAL DETAILS Single crystals of AV2F6(A=Rb and Cs) were grown usingACl-ﬂux methods [ 18]. As starting materials, we used V grains and halides VF 3,AF, and ACl. We did not use V powder but V grains, since V powder gives low-qualitysamples owing to the presence of a certain amount of oxideformed on the surface of V powder. These halides had beendried or puriﬁed before use. These starting materials weremixed with an appropriate ratio, and were heated and slowlycooled in a Ni crucible. All above procedures were conductedin a glove box ﬁlled with Ar gas. The ﬂux was removed usingwater. X-ray diffraction measurements at high temperatures were conducted in a small furnace ﬁlled with He gas usinga diffractometer with a Cu K αsource. The signals from CuKα2were numerically subtracted from raw data. Electrical resistivities of single crystals at high temperatures weremeasured using a conventional four-probe method under anAr atmosphere. Direct-current magnetization measurementswere performed using commercial superconducting quantuminterference device magnetometers (Quantum Design) in theResearch Center for Low Temperature and Materials Sciences,Kyoto University. Speciﬁc heat measurements were carried outu s i n gat w o - τrelaxation method (Quantum Design). In order 2469-9950/2016/93(19)/195168(7) 195168-1 ©2016 American Physical SocietyUEDA, Y AMADA, Y AMAUCHI, UEDA, AND YOSHIMURA PHYSICAL REVIEW B 93, 195168 (2016) FIG. 1. Crystal structures of AV2F6with space groups of Fd¯3m,Imma ,a n dPnma (spheres: A, octahedra: VF 6). The unit cells of orthorhombic Imma andPnma structures, which are indicated by lines, are 1 /√ 2×1/√ 2×1 of the cubic Fd¯3munit cell. While the cubic Fd¯3mstructure has single V site, orthorhombic structures have two V sites, which is indicated using the colors of octahedra. The difference between the Imma structure and the Pnma structure is the rotation of VF 6. to evaluate the lattice contributions, we measured the speciﬁc heats of nonmagnetic compound AZnGaF 6with a cubic modiﬁed pyrochlore structure. The lattice speciﬁc heat at eachtemperature was compensated for by the Debye temperatureθ D, assuming that θDinversely proportional to the square root of the formula weight. Magnetic entropies were obtained fromspeciﬁc heats divided by temperature after the subtraction ofthe lattice contribution. III. RESULTS AND DISCUSSIONS The obtained single crystals of RbV 2F6and CsV 2F6are black and have octahedral shapes. A typical size of thecrystal is approximately 2 mm in edge. The octahedral shapessuggest that RbV 2F6and CsV 2F6seem to be cubic when the crystal forms at high temperatures in the crystal growth. Asmentioned below, owing to the transitions to the orthorhombicstructure, six domains emerge in a single crystal, when thecrystal is cooled down to room temperature. Our singlecrystals of RbV 2F6and CsV 2F6consist of crystallite domains, which prevent us from conducting structural analysis usingsingle crystals. Hence powder x-ray diffraction measurementsare important to elucidate the structural transitions of thesecompounds. Powder x-ray diffraction patterns indicate that both RbV 2F6 and CsV 2F6are orthorhombic at room temperature. However, the details of extinction rules of these two are different. Thespace groups of RbV 2F6and CsV 2F6arePnma andImma , respectively. These space groups are consistent with a previousreport [ 16]. It is noteworthy that Pnma is a subgroup of Imma . Our x-ray investigations revealed that both RbV 2F6 and CsV 2F6become cubic Fd¯3mat high temperatures as described below. In Fig. 1, the three crystal structures of modiﬁed pyrochlore with three space groups are displayed. Figure 2demonstrates temperature Tdependence of the x-ray diffraction pattern of RbV 2F6. At high temperatures, the pattern is consistent with the cubic modiﬁed pyrochlorestructure, which has a space group of Fd¯3m. Below 600 K, most of the diffraction signals split, which indicates a structuralchange. The splitting of the 220 signal indicates that the crystalsystem is not cubic, and the splitting of the 202 signal indicatesthat it is not tetragonal. The diffraction pattern just below 600 Ksuggests that the structure has an orthorhombic space group ofImma with a 1 /√ 2×1/√ 2×1 unit cell compared with that of cubic phase. Below 540 K, the peak splittings becomemore distinct, and in addition, some diffraction signals suchas 201 and 210 appear in orthorhombic indices. The presenceof these new diffraction signals violate the extinction rule ofImma , and suggest that the space group becomes Pnma . Although CsV 2F6also becomes cubic at high temperatures, the details of structural changes are different from those of 2θ (degree)23 24 25 26 27 28 29 30 31 Intensity (arb. unit)×2 ×6 ×1Rb V 2F6 Cu K α 400 K420 K440 K460 K480 K500 K520 K540 K560 K580 K600 K620 K640 K660 K680 K220311 222 020 112+200 201 210 121211+ 013+ 103022 202Fd3m Im m a Pn m a FIG. 2. Temperature dependence of the powder x-ray diffraction pattern of RbV 2F6. To show the changes clearly, the intensities of 23◦∼25◦and 25◦∼27.5◦are enlarged by a factor 2 and 6, respectively. The broken lines indicate the boundary of the phases. 195168-2ELECTRICAL CONDUCTION MEDIATED BY FLUORINE . . . PHYSICAL REVIEW B 93, 195168 (2016) 2θ (degree)400 K420 K440 K460 K480 K500 K520 K540 K560 K580 K600 K620 K640 K660 K680 K700 K 27 28 29 30×1311 222 211 103 202 211+ 121013+ 103022+ 202 33 34 35×5400 220 004 220 004 37 38×5331 031+ 301123+ 213Intensity (arb. unit)Cs V 2F6Cu K α Fd3m I41/amd Imma FIG. 3. Temperature dependence of the powder x-ray diffraction pattern of CsV 2F6. To show the changes clearly, the intensities of 33◦∼35◦and 36 .5◦∼38◦are enlarged by a factor 5. The broken lines indicate the boundary of the phases. RbV 2F6reﬂecting the difference in the space group at room temperature. Figure 3demonstrates Tdependence of the x-ray diffraction pattern of CsV 2F6. As is the same as RbV 2F6,t h e high-temperature pattern is consistent with the cubic modiﬁedpyrochlore structure. However, low-temperature structures aredifferent from those of RbV 2F6. Below 640 K, some diffraction signals split, which indicates a structural change to a tetragonalstructure with a space group of I4 1/amd . No broadening of the 222 signal suggests that the system has a=b. Below 460 K, some diffraction signals, such as those at approximately 29 .5◦ and 37◦, become broader. These broadenings indicate that the value of ais not equal to that of band that the space group becomes Imma . The peak splitting of x-ray diffraction pattern is less distinct than that of RbV 2F6. No superlattice reﬂection was observed. These structural changes are summarized as temperature dependencies of the lattice parameters as shown in Fig. 4, which clearly demonstrate the differences in structural transi-tions between RbV 2F6and CsV 2F6.F o rR b V 2F6,a,b, andc have the same value at high temperatures reﬂecting the cubicstructure. With decreasing temperature, the lattice constantsjump approximately at 600 K, indicating this structuraltransition is of ﬁrst order. Please note that the unit cell oftetragonal phase is 1 /√ 2×1/√ 2×1 of that of cubic one. Slightly below the transition temperature,√ 2aandcof the orthorhombic unit cell have almost the same value and thevalue of√ 2bis substantially larger than those of them. With+++++++++++++++++++ 300 400 500 600 70010.510.6 10.410.7 T (K)Lattice constants (A°)Cs V 2F6 c2b 2a 32V2a ca Fd3m I41/amd Imma++++++++++++++++++++ 10.310.410.510.6 10.2 Lattice constants (A°) Rb V 2F6 c32V 2a2b a Fd3m Imma Pn m a FIG. 4. The lattice constants and the unit cell volumes of RbV 2F6 and CsV 2F6as a function of temperature. We note that ain the cubic unit cell corresponds to√ 2a,√ 2b,a n d3√ 2Vin the tetragonal or orthorhombic unit cell. The broken lines indicate the boundary of the phases. further decreasing temperature,√ 2aandcgradually separate with each other, and the temperature dependence of latticeparameters has an anomaly at 560 K, indicating a second-ordertransition. At 300 K, the difference of the lattice parametersis more than 0 .3˚A. In contrast, CsV 2F6has small difference of the lattice parameters at 300 K, which is approximately0.15˚A. In addition, temperature dependence of the lattice constants in CsV 2F6is qualitatively different from that in RbV 2F6. At high temperatures, a,b, and chave the same value. Approximately, at 650 K, cabruptly reduces, while the other two remain almost constant, which indicates the structurebecomes tetragonal with a space group of I4 1/amd through a ﬁrst-order transition. Again, please note that the unit cell of tetragonal phase is 1 /√ 2×1/√ 2×1 of that of cubic one. Below 480 K,√ 2aand√ 2bgradually separate from each other and the system becomes an orthorhombic structure witha space group of Imma . In addition to the difference in the space group at room temperature, one of the most signiﬁcant differences betweenthe structural changes of RbV 2F6and CsV 2F6is the directions of the lattice distortions. With decreasing temperature, thecubic Fd¯3mphase of RbV 2F6directly transforms into the Imma phase with√ 2b>√ 2a∼c. The cubic Fd¯3mphase of CsV 2F6goes into the Imma phase with√ 2b∼√ 2a> cthrough the tetragonal I41/amd phase. Although both 195168-3UEDA, Y AMADA, Y AMAUCHI, UEDA, AND YOSHIMURA PHYSICAL REVIEW B 93, 195168 (2016) compounds have the Imma phase, they have different ten- dencies:√ 2a∼cfor RbV 2F6and√ 2a∼√ 2bfor CsV 2F6. These lattice constants indicate that the cubic crystal ofRbV 2F6elongates along the [110] direction and that of CsV 2F6 shrinks along the [001] directions in their Imma phases. The different distorting directions of the Imma phase indicate that the origins of the structural phase transitions of RbV 2F6and CsV 2F6are different. Modiﬁed pyrochlore compounds have a large A-site ion, which means that the corner-sharing network of VF 6octahedra have large holes at the Asite. Structural instability owing to this large hole would be one possible origin of the structuraltransitions of AV 2F6compounds. Particularly, RbV 2F6has a smaller A-site ion and larger space around the ion than that of CsV 2F6. The upper panel of Fig. 4displays shrinkage along the [1¯10] and [001] direction, expansion along the [110] direction in the cubic indices, and volume reduction, accompanied withthe structural transition from Fd¯3mtoImma . These changes indicate that the large space around Aion is closely related to this structural transitions of RbV 2F6. The slight volume increase with structural transition from Imma toPnma of RbV 2F6is possibly due to the rotation of VF 6octahedra in the Pnma structure. In contrast, the lower panel of Fig. 4displays that CsV 2F6exhibits a small volume change accompanied with structural transitions, suggesting that the space around Aion has little effect on the structural transitions of CsV 2F6. In addition to the large hole mentioned above, charge or- dering would be another likely origin of structural transitions,particularly for CsV 2F6.R b V 2F6, and CsV 2F6have V atoms with a formal valence of +2.5. It is likely that charge orderings of V2+and V3+take place with the change of structure. If the charge orderings take place, the number of V sites wouldbecome two or more. In the cubic Fd¯3mstructure, corner- sharing VF 6chains along /angbracketleft110/angbracketrightare equivalent. Although the structure shrinks along the cdirection in the tetragonal I41/amd structure of CsV 2F6, the chains along [110] and [1¯10] in the cubic indices are equivalent as shown in Fig. 1. Hence the cubic Fd¯3mand tetragonal I41/amd structures have a single V site, indicating no charge ordering. In contrast,these two kinds of chains expand or shrink in the Imma structure, and VF 6octahedra rotate in the Pnma structure of RbV 2F6, and therefore orthorhombic Imma andPnma structures have two V sites as shown in Fig. 1. In the charge ordering states of Imma andPnma structures, V3+ions withS=1 form one-dimensional chains along the [1 ¯10] direction, and V2+ions with S=3/2 form those along the [110] direction, where cubic indices are used. Please note thatall V 4tetrahedra consist of two V2+and V3+, which satisﬁes the Anderson’s criteria of charge ordering in a pyrochlorelattice [ 3]. In our structural study of RbV 2F6mentioned above, the number of V sites changes from one to two owing to thestructural transition from Fd¯3mtoImma approximately at 600 K. For CsV 2F6, it changes owing to the transition from I41/amd toImma approximately at 480 K. Charge ordering is closely related to the electron transfer from one V ion to another one. This means that the formationof charge ordering has large effects on electrical resistivity.Although most of ﬂuorides are electrically insulating, blackcolors of RbV 2F6and CsV 2F6indicate possible electron conduction.400 500 600 700 800101102103104105 T (K)ρ (Ω cm) Rb V 2F6Cs V 2F6 1.0 1.5 2.0 2.5102103104105 1000/ T (K−1)ρ (Ω cm) Rb V 2F6Cs V 2F6 FIG. 5. Temperature dependence of electrical resistivities of RbV 2F6and CsV 2F6. The main panel shows ρ-Tcurves. At high temperatures, the values of ρof RbV 2F6and CsV 2F6are of the order of 102/Omega1cm. In the entire temperature range, ρexhibits semiconducting temperature dependence. Figure 5shows Tdependence of electrical resistivities ρof RbV 2F6and CsV 2F6measured at high temperatures. Surprisingly, electrical conduction is observed in these ﬂuo-rides, although the values of ρare high. For both compounds, ρgradually decreases with increasing T. Approximately at 650 K, ρof CsV 2F6suddenly drops by one order of magnitude, indicating an electronic phase transition. In contrast, RbV 2F6 exhibits small change of ρapproximately at 600 K. With further increasing T,ρgradually decreases again for both compounds. The jump of ρapproximately at 650 K observed for CsV 2F6 suggests that this transition is caused by charge ordering of V2+ and V3+. However, above-mentioned structural study suggests that this transition corresponds to the structural transition fromFd¯3mtoI4 1/amd with a single V site, and the number of V sites does not seem to change through this structural transition.The number of V sites is two in the Imma phase observed below 480 K, and there is no anomaly in ρat this temperature. It is natural to think that the electronic state of the I4 1/amd phase is similar to that of the Imma phase. Hence we think that theI41/amd phase has a very small orthorhombic distortion and that it has two V sites, which correspond to V2+and V3+. For RbV 2F6, the jump of ρapproximately at 600 K is very small compared with that of CsV 2F6. This temperature corresponds to the structural transition from Fd¯3mwith a single V site to Imma with two V sites, which is consistent with the charge ordering scenario. However, the change ofρat the transition temperature is much smaller than that of CsV 2F6. In the structural view point, the most signiﬁcant difference between RbV 2F6and CsV 2F6is the changes of lattice constants at their transition temperatures, whichare shown in Fig. 4.F o rC s V 2F6, the change of lattice constants is very small, indicating that V–V distances belowand above the transition temperature are almost the same.Owing to charge ordering, electrons on V ions are localizedandρdrastically increases. In contrast, for RbV 2F6,V – V 195168-4ELECTRICAL CONDUCTION MEDIATED BY FLUORINE . . . PHYSICAL REVIEW B 93, 195168 (2016) distances along the baxis are large and others are small in the orthorhombic structure below the transition temperature.Particularly, substantial contraction is observed in V chainsalong [112] in the orthorhombic index consisting of V 2+and V3+. The reduction of the V–V distance enhances electron transfer between V2+and V3+, which is likely to reduce ρin the charge ordering state. Hence the jump of ρis small for RbV 2F6. To elucidate the origin of electron transfer from one V atom to a neighboring one through F atoms bridging between them,we plot log 10ρas a function of 1 /Tin the inset of Fig. 5. In the high-temperature cubic phase, the linear behaviorssuggest that ρis described as the activation type resistivity ρ=ρ ∞exp(Ea/kBT), where Eais the activation energy, and kBis the Boltzmann constant. The activation energies Eaof CsV 2F6and RbV 2F6are approximately 0 .54 and 0 .43 eV, respectively. The lower activation energy of RbV 2F6is due to the shorter V–V distances in the cubic phase. These gapenergies are consistent with the fact that both compounds areblack in color. The values of ρ ∞are 0.0146 and 0 .015/Omega1cm for CsV 2F6and RbV 2F6, respectively. In the low-temperature phase, however, the data become noisy and have sampledependence (not shown here), which are possibly owing tothe domain formation. Below the transition temperatures,log 10ρ-1/T plots are not linear, indicating the change of mechanism of electron transport owing to the formation ofcharge ordering. In charge ordering systems, variable rangehopping is sometimes discussed. However, variable rangehopping type plots are not linear for low temperature phasesofAV 2F6. To compare the transport behaviors above and below transition temperature, we will discuss log10ρ−1/T plots also in low-temperature phases. In the low-temperaturephase, ρ ∞is larger and Eais smaller than those of the high-temperature phase for each compound. The obtainedparameters at low temperatures for CsV 2F6areρ∞=30/Omega1cm andEa=0.27 eV, and those for RbV 2F6areρ∞=10/Omega1cm andEa=0.18 eV. With the formation of charge orderings, ρ∞ becomes large. The V–V distance of RbV 2F6is much smaller than that of CsV 2F6at low temperatures, which is likely to cause the difference of ρ∞at low temperatures. It seems strange that the values of Eaat low temperatures are smaller than those at high temperatures. The resistivity is mainlygoverned by the carrier density and the mobility of the carriers.The temperature dependence of lattice parameters shown inFig. 4suggests that the V–V distance and the mobility of the carriers strongly depend on the temperature especially belowthe transition temperatures. This Tdependence of mobility is likely to violate the activation law, and the values of E aseem to reduce. The fact that ρstrongly depends on –V distances, indicates that the electron conduction between V atoms isenhanced by the orbital overlapping of V atoms and F atomsbridging between them. In addition to ρ, the magnetic susceptibility χgives information about the electronic state of V atoms, sincethe change of the electronic state affects the spin state. ForRbV 2F6and CsV 2F6, temperature dependencies of χare shown in Fig. 6. As shown in the right inset of Fig. 6,χdata measured under 1 T in both compounds are well-ﬁtted usingthe Curie-Weiss law except for low-temperature regions. Thevalues of effective magnetic moments p effare 3.21±0.060 200 400 600 8000.000.050.100.150.20 T (K)χ (emu/mol) Rb V 2F6Cs V 2F6 H=1 Tsingle crystals0 400 8000100200300 T (K)1/χ (mol/emu)Rb V 2F6 Cs V 2F6 01 0 2 00.00.10.2 T (K)χ (emu/mol ) Rb V 2F6Cs V 2F6 FIG. 6. Temperature dependence of magnetic susceptibility χof RbV 2F6and CsV 2F6. Inverse magnetic susceptibility 1 /χdata are plotted in the inset. These measurements are conducted using singlecrystals with random orientation. and 3.35±0.03, and the Curie-Weiss temperatures /Theta1are −19.5±0.4 K and 0 .7±0.3Kf o rR b V 2F6and CsV 2F6, respectively. The experimentally obtained values of peffare quite consistent with the ideal value of 3.39, which is theroot-mean-square of p effof V3+withS=1 and V2+withS= 3/2 assuming g=2. This fact suggests that these magnetic systems are well-described as 1:1 mixtures of V3+and V2+. It is noteworthy that the slope of the 1 /χ−Tplot does not change at the temperature where ρjumps for each compound, suggesting that the spin states of V below and above thetransition temperature are quite similar. Below the transitiontemperature, electrons of V 2+and V3+are localized and V2+ and V3+are ordered. Even above the transition temperature, thepeffis larger than the ideal value of pefffor the valence of +2.5, which is 2.96. At the transition temperature, the slope of 1/χ−Tdoes not change, and the peffis larger than the ideal value of pefffor the valence of +2.5, which is 2 .96. Although electrons transfer among V atoms at high temperatures, peff suggests that the valence of V is not +2.5 but a mixture of +2 and+3. It is likely that V2+and V3+are randomly distributed and their electrons transfer among them. At room temperature, V2+and V3+are ordered for both compounds. At low temperatures, their spins exhibit magneticorderings. As shown in the left inset of Fig. 6,χ−Tcurves of both compounds exhibit anomalies at low temperatures.However, these anomalies have remarkable difference betweentwo compounds. For CsV 2F6, the anomaly of χatTN≈8Ki s similar to those of conventional antiferromagnetically orderedsystems. In contrast, χof RbV 2F6increases stepwise at TN1≈ 13 K and TN2≈6 K with decreasing temperature. To elucidate the origin of the two-step magnetic transition of RbV 2F6, we have measured the magnetic ﬁeld, H, dependence of the magnetization Mand we plot M/H -Tas shown in the left inset of Fig. 7. Two magnetically ordered phases of RbV 2F6have ferromagnetic characteristics. At 7 T, two anomalies at TN1andTN2are not distinct in M/H . With decreasing magnetic ﬁeld, M/H below TN2≈6 K gradu- ally increases, which suggests the existence of spontaneous 195168-5UEDA, Y AMADA, Y AMAUCHI, UEDA, AND YOSHIMURA PHYSICAL REVIEW B 93, 195168 (2016) 012345670.00.20.40.60.81.0 H (T)M (μB/f.u.) Rb V 2F6 H//[111]2 8 16K 0 5 10 150.050.100.150.20 T (K)M/H (emu/mol )0.1 0.5 1 3 7Tsingle crystals 0.050.060.070.080.090.100 5 10 15T (K)χ (emu/mol )H=1 T[110] [100] [111] FIG. 7. Magnetization of RbV 2F6per formula unit as a function of magnetic ﬁeld at various temperatures. After cooling under zeroﬁeld condition, these measurements were conducted with increasing magnetic ﬁeld. In the left inset, M/H is plotted as a function of temperature under various magnetic ﬁelds, which were measured inﬁeld cooled conditions. In the right inset, χdata measured under magnetic ﬁelds with various directions are shown. The directions of the ﬁeld were indexed based on the cubic system. Owing to the orthorhombic domain formation in a single crystal, cubic [100] corresponds to orthorhombic [001], [110], and [1 ¯10] directions. Similarly, cubic [110] corresponds to orthorhombic [100], [010], and [112] directions. Cubic [111] corresponds to orthorhombic [101] and [011] directions. magnetization. The ﬁeld dependence of M/H between TN1 andTN2is small, but the step at TN1≈13 K is pronounced below 0 .1T . To determine the direction of spins in the magnetically ordered phases, the direction dependence of χ-Tis measured as shown in the right inset of Fig. 7. In general, the phase with spontaneous magnetization has large direction dependence ofχ=M/H . However, three χ-Tcurves of H/bardbl[100], [110], and [111] are different but similar to that of randomly orientedsingle crystals in Fig. 6. This similarity is likely due to the domain formation in the single crystal as mentioned before.A single direction in the cubic phase corresponds to severalorthorhombic directions. Even if the magnetic measurementsare conducted using a single crystal in cubic condition, thedata come from six orthorhombic domains. Hence Mof a single crystal with domain formation is similar to that of thepolycrystalline samples. In Fig. 7,M-Hcurves of H/bardbl[111] at various temperatures are exhibited. In the paramagnetic phase at 16 K ( >T N1), the M-Hcurve is linear. At 8 K ( <T N1),Mlinearly increases up to 1 .5 T, and rapidly increases around 2 T, and again increases linearly. At 2 K ( <T N2),Mjumps to 0 .05μBin the low-ﬁeld region, indicating the ferromagnetic nature, andtheM-Hcurve slightly bends approximately at 2 T. From these measurements, the difference between two magneticallyordered phases is clariﬁed. Below T N2,R b V 2F6is in a canted antiferromagnetic phase with small spontaneous magnetiza-0.00.51.01.52.02.53.03.5 0510152025 0510152025 0 1 02 03 04 05 06 0 T (K)C/T (J/mol K2) SM (J/mol K ) Rb V 2F6Cs V 2F6 lattice (Rb)(Cs)Rln 12 FIG. 8. Speciﬁc heat divided by temperature C/T (solid lines) and magnetic entropy SM(dashed lines) of RbV 2F6and CsV 2F6as a function of temperature. The lattice contribution estimated from thespeciﬁc heat of RbZnGaF 6or CsZnGaF 6(dotted lines) is used to calculate SM. tion. Between TN2andTN1, it is in an antiferromagnetic phase with a spin-ﬂop transition approximately at 2 T. For CsV 2F6,TN=8 K is approximately half of \|/Theta1\|=20 K. However, for RbV 2F6,TN1=13 K is much higher than \|/Theta1\|= 1K .T h ev a l u eo f /Theta1scales the average of various magnetic interactions Jbetween V spins. In the orthorhombic structure, there are several neighboring V–V bonds. If the magneticinteractions of them have different signs, T Npossibly becomes much higher than \|/Theta1\|. Compared with CsV 2F6,R b V 2F6has a crystal structure with lower symmetry, which means that RbV 2F6has many kinds of neighboring V–V interactions. The Imma structure of CsV 2F6has three kinds of neighboring V–V bonds, V2+–V2+,V3+–V3+, and V2+–V3+. While the Pnma structure of RbV 2F6has four –V bonds, V2+–V2+,V3+–V3+, and two V2+–V3+. Many kinds of neighboring Jof RbV 2F6seemingly make nearly degenerated magnetic structures and a two-stepmagnetic transition. Accompanied by the formation of magnetic ordering, the change in the magnetic entropy S Moccurs. To evaluate SM, we measured the speciﬁc heats Cof RbV 2F6and CsV 2F6. In Fig. 8, the temperature dependence of C/T andSMis displayed. For CsV 2F6,C/T exhibits a lambda-type anomaly at TN, and then remains almost constant above TN. The sharp lambdalike feature in C/T suggests that long-range ordering is established at this temperature. For RbV 2F6, two lambda-type anomalies are observed at TN1andTN2. Magnetic entropies SMof CsV 2F6and RbV 2F6rapidly increase at TN,TN1, andTN2reﬂecting the variation of C/T , and continue to increase above them. The total magneticentropy, which is the sum of the Rln(2S+1) values of V 2+ (S=3/2) and V3+(S=1), equals Rln 12≈20 J/mol K. Up to 60 K, SMof both compounds reach approximately the total magnetic entropy. These observations suggest that only themagnetic degree of freedom remains below 60 K for thesecompounds, and that charge and orbital degrees of freedom 195168-6ELECTRICAL CONDUCTION MEDIATED BY FLUORINE . . . PHYSICAL REVIEW B 93, 195168 (2016) are already relieved. This is consistent with our conclusion that the charge ordering takes place at high temperatures. IV . SUMMARY Our experiments revealed that two mixed-valent modiﬁed pyrochlore ﬂuorides RbV 2F6and CsV 2F6are electrically conducting, although the temperature dependence of resistiv-ities of these compounds are semiconducting. Accompaniedby a structural transition indicating the formation of chargeorderings of V 2+and V3+, the resistivity of each compoundabruptly increases. In this charge ordering state, all V 4 tetrahedra contain two V2+and V3+. These vanadium ions are antiferromagnetically coupled, which is indicated by thenegative values of /Theta1. At low temperatures, both systems exhibit magnetic orderings. Particularly, RbV 2F6exhibits two-step magnetic transitions. ACKNOWLEDGMENTS This work was supported by a Grant-in-Aid for Scientiﬁc Research (C) (Grant No. 24540345) from the Japan Societyfor the Promotion of Science. [1] W. R. Datars, K. R. Morgan, and R. J. Gillespie, Phys. Rev. B 28,5049 (1983 ). [2] H. Kawamura, I. Shirotani, and H. Inokuchi, Chem. Phys. Lett. 24,549 (1974 ). [3] P. W. Anderson, Phys. Rev. 102,1008 (1956 ). [4] S. Kondo, D. C. Johnston, C. A. Swenson, F. Borsa, A. V . Mahajan, L. L. Miller, T. Gu, A. I. Goldman, M. B.Maple, D. A. Gajewski, E. J. Freeman, N. R. Dilley, R. P.Dickey, J. Merrin, K. Kojima, G. M. Luke, Y . J. Uemura, O.Chmaissem, and J. D. Jorgensen, P h y s .R e v .L e t t . 78,3729 (1997 ). [5] C. Urano, M. Nohara, S. Kondo, F. Sakai, H. Takagi, T. Shiraki, and T. Okubo, Phys. Rev. Lett. 85,1052 (2000 ). [6] K. Matsuno, T. Katsufuji, S. Mori, Y . Moritomo, A. Machida, E. Nishibori, M. Takata, M. Sakata, N. Yamamoto, and H. Takagi,J. Phys. Soc. Jpn. 70,1456 (2001 ). [7] T. Hagino, T. Tojo, T. Atake, and S. Nagata, Phil. Mag. B 71, 881 (1995 ). [8] R. J. Cava, R. S. Roth, T. Siegrist, B. Hessen, J. J. Krajewski, and W. F. Peck, Jr., J. Solid State Chem. 103,359 (1993 ).[9] D. Hirai, M. Bremholm, J. M. Allred, J. Krizan, L. M. Schoop, Q. Huang, J. Tao, and R. J. Cava, Phys. Rev. Lett. 110,166402 (2013 ). [10] S. Yonezawa, Y . Muraoka, Y . Matsushita, and Z. Hiroi, J. Phys. Soc. Jpn. 73,819 (2004 ). [11] M. Br ¨uhwiler, S. M. Kazakov, N. D. Zhigadlo, J. Karpinski, and B. Batlogg, Phys. Rev. B 70,020503(R) (2004 ). [12] E. Banks, J. A. Deluca, and O. Berkooz, J. Solid State Chem. 6, 569 (1973 ). [13] M. J. Harris, M. P. Zinkin, Z. Tun, B. M. Wanklyn, and I. P. Swainson, P h y s .R e v .L e t t . 73,189 (1994 ). [14] S. W. Kim, S.-H. Kim, P. S. Halasyamani, M. A. Green, K. P. Bhatti, C. Leighton, H. Das, and C. J. Fennie, Chem. Sci. 3,741 (2012 ). [15] W. O. J. Boo, R. F. Williamson, K. N. Baker, and Y . S. Hongba, Mol. Cryst. Liq. Cryst. 107,195 (1984 ). [16] Y .-K. Yeh, Y .-S. Hong, W. O. J. Boo, and D. L. Mattern, J. Solid State Chem. 178,2191 (2005 ). [17] H. Ueda, A. Matsuo, K. Kindo, and K. Yoshimura, J. Phys. Soc. Jpn. 83,014701 (2014 ). [18] B. J. Garrard and B. M. Wanklyn, J. Cryst. Growth 47,159 (1979 ). 195168-7
PhysRevB.75.224429.pdf	Dominant role of thermal magnon excitation in temperature dependence of interlayer exchange coupling: Experimental veriﬁcation S. S. Kalarickal, X. Y . Xu,†K. Lenz, W. Kuch, and K. Baberschke‡ Institut für Experimentalphysik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany /H20849Received 20 March 2007; revised manuscript received 30 April 2007; published 27 June 2007 /H20850 Ultrathin Ni/Cu/Co trilayers were deposited in ultrahigh vacuum and the ferromagnetic resonance measured in situ as a function of both, temperature and out-of-plane angle of the external ﬁeld. The interlayer exchange coupling Jinterwas then unambiguously extracted at various temperatures, entirely from the angular dependence of the resonance ﬁeld positions. The temperature dependence of Jinter/H20849T/H20850follows an effective power law ATn,n/H110151.5. Analysis of the scaling parameter Ashows an oscillatory behavior with spacer thickness, as does the strength of the coupling at T=0. The results clearly indicate that the dominant contribution to Jinter/H20849T/H20850is due to the excitation of thermal spin waves and follows recently developed theory closely. DOI: 10.1103/PhysRevB.75.224429 PACS number /H20849s/H20850: 75.70.Cn, 76.50. /H11001g, 75.30.Ds, 75.30.Et I. INTRODUCTION The ferromagnetic/normal metal/ferromagnetic ultrathin ﬁlm trilayer is the fundamental component in multilayeredgiant magnetoresistive /H20849GMR /H20850materials. The parameter which governs the ferromagnetic /H20849FM/H20850and the antiferromag- netic /H20849AFM /H20850coupling in these trilayers, and hence the utility of the GMR material is the interlayer exchange coupling/H20849IEC /H20850parameter J inter. Considerable work has been done at theT=0 level see, e.g., Refs. 1and2. Though this parameter has been well studied, the dependence of Jinteron tempera- ture T, an extremely important aspect, has been much de- bated upon and not yet clearly understood.3–14 To elucidate the basic trilayer structure used for the inves- tigation in this work, a schematic diagram is shown in Fig. 1. Trilayers studied to date have comprised of ferromagneticlayers with different anisotropies. Among others, such sys-tems as Fe/Pd/Fe, 4Ni/Cu/Ni with differing Ni thicknesses to ensure in-plane and normal-to-plane anisotropies,15,16or Co/Cu/Ni with both in-plane anisotropies have beenstudied. 11,12 For uncoupled trilayers, one expects two ferromagnetic resonance /H20849FMR /H20850lines, corresponding to the two layers. When the interlayer exchange coupling is engaged, these twolines correspond to the so-called optical and acoustic reso-nance modes. Previous methods of determining the Tdepen- dence of J inter, given by Lindner and Baberschke,16correlate the change in Jinter, with the shift of the FMR position Hres between that of the ﬁrst layer and the Hresfor the optical mode. A complete angular dependence of the FMR spectrumfor each temperature was not taken. This method has thedrawback that the explicit and complicated temperature de-pendence of the parameters that affect H res, like the magne- tization and the anisotropy of the two ﬁlms, cannot be easilytaken into account. The work presented in this paper is a study on the Co/Cu/Ni system, with 1.8 monolayers /H20849ML/H20850C o ,6M Lo f Cu spacer, and 7 ML Ni on Cu /H20849001/H20850substrate. The present work provides an investigation of the temperature depen-dence of J interentirely determined from the angular depen- dence of the ferromagnetic resonance positions for weaklyFM coupled trilayers near the ordering temperature. TheT n,n/H110151.5 dependence of Jinteris very clear. This work also gives a different analysis of the data from Refs. 16and12to provide a complete picture for small spacer thicknesses inthe range of 4 to 9 ML. The main motivation for this work lies in the ongoing debate regarding the different contributions to the tempera- ture dependence of J inter. Three different sources have been attributed to this dependence. Early discussions regardingIEC and its temperature dependence focused solely on theelectronic band structure. 2,13The softening of the Fermi edge at higher temperatures makes the coupling less effective. Thesecond effect is the interface contribution, which uses thespin asymmetry of the electron reﬂection coefﬁcient withincreasing temperatures. In either of these contributions, thestrong coupling between the spins, which is the signature ofFM materials, have not been taken into account. 7The tem- perature dependence due to this third contribution, i.e., thecoupling within the individual layers, is manifest in the ex-citation of thermal spin waves. The decrease in the interlayerexchange coupling due to spin-wave excitation was calcu-lated recently in Ref. 12. To provide a background for the work presented in this paper, a brief overview of the theoryis given below. Schwieger and Nolting have used a microscopic Heisen- berg model to calculate the temperature dependence of low-energy spin-wave excitations. The difference in the free en-ergy for parallel and antiparallel orientation of magnetizationin the two ordered layers contributes to the temperature de-pendence of J inter. This basically depends on two parameters: /H20849i/H20850the direct exchange coupling Jintrawithin the ferromag- netic layer, yielding also its Curie temperature, and /H20849ii/H20850the interlayer exchange coupling Jinterbetween the two layers FM1 and FM2. The direct exchange between the spins ineach FM layer J intrais much stronger, being in the meV re- gime while the IEC coupling Jinterfor weak coupling in trilayers with spacer thickness d/H110153–9 ML, is in the /H9262eV range. To extract the effect of the magnetic contributionsalone for different spacer thicknesses, J interhas been normal- ized to the parameter J0/H11013Jinter/H20849T=0/H20850. The authors of Ref. 14 have discussed Jinter/J0as a function of an effective T1.5 power law. Figure 2shows the temperature dependences af- fected by these two parameters, as described by SchwiegerPHYSICAL REVIEW B 75, 224429 /H208492007 /H20850 1098-0121/2007/75 /H2084922/H20850/224429 /H208497/H20850 ©2007 The American Physical Society 224429-1and Nolting.14Figure 2/H20849a/H20850shows the temperature depen- dence of Jinternormalized to J0, for different Jintravalues. This graph shows the inﬂuence of different ferromagneticmaterials on the Tdependence of J inter. It is interesting to note that stronger direct exchange coupling in the FM layerresults in a weaker temperature dependence for J inter. The effects due to different magnetic materials are not taken into account in this work where the investigated sys-tem comprises Co and Ni. However, it is important to notethat J inter/H20849T/H20850also depends on the type of magnetic material used, which points at the competition between the thermal energy and the strength of the coupling. Figure 2/H20849b/H20850shows theTdependence of the normalized Jinterfor different J0 values. This also takes into account the properties of the spacer and interface at T=0. Overall, it can be seen that, an effective power law is followed. However, it is also clear thatthe results do not follow a straight line in T 1.5, i.e., the power is not exactly 3/2. The curvature and slope both depend onthe parameters J intraandJ0. It can be seen that the larger the J0value, the weaker the decrease of Jinterwith T. This trend has been veriﬁed in the work presented in this paper. All contributions due to the spacer, interface, and mag- netic layers, nevertheless give an effective power-law depen-dence on the temperature, J/H20849T/H20850/H110151−ATn,n/H110151.5. /H208491/H20850 As mentioned earlier, the differences between the above- mentioned mechanisms lie in their dependence on the spacerthickness. The spacer contribution, i.e., the electronic band-structure effect exhibits a linear dependence of Awith d. The interface contribution is independent of dwhile the contribu- tion due to spin-wave excitation gives a very weak depen-dence and oscillates with d. In connection with Fig. 2,i tc a n be summarized that J 0and the scaling parameter Ashould follow opposite trends as functions of spacer thickness.14 The interesting problem hence, lies in separating the T dependence of the above-mentioned mechanisms in ultrathinﬁlms. This question was partially addressed by Schwieger et al.for two AFM and one FM coupled trilayers. 11,12For AFM coupled samples, it was found that the temperature depen-dence increases with coupling strength. However, no ﬁnal conclusion could be made for the FM coupled samples andhence an overall picture was difﬁcult to extract. Another interesting aspect of nanomaterials is that the value of the Curie temperature T cfor these materials is very much below the bulk value and close to room temperaturedue to ﬁnite size effects. 17,18Hence the study of the inter- layer exchange coupling close to the ordering temperaturegains importance. Ferromagnetic resonance with its well es-tablished theory gives a unique possibility to study the tem-perature dependence of J interin detail. The paper is organized as follows. Section II presents the experimental and sample details as applicable to this paperand also gives some typical FMR data. Section III gives ashort summary of the data analysis. Section IV presents theresults and discussion and Sec. V gives the conclusions. II. EXPERIMENTAL DETAILS AND FMR DATA The in situ ultrahigh-vacuum /H20849UHV /H20850FMR spectrometer, and its capabilities have been described in detailelsewhere. 16,19In brief, this setup allows one to deposit ul- trathin, multilayered ﬁlms and measure its FMR spectrumwithout any contact of the layers to air. The ﬁlms that wereinvestigated comprised of a few atomic monolayers. At thesethicknesses, this technique becomes extremely crucial anduseful since contact with air would change the magneticproperties of the ﬁlm entirely. Also it is well known that theelectronic band structure and the magnetic moment per layerfor ultrathin ﬁlms is different from bulk or even nanometerthick ﬁlms. 20 Trilayers were prepared on single crystalline Cu /H20849001/H20850. The substrate was ﬁrst Ar+ion sputtered at 3 kV, followed by a longer duration of sputtering at 1 kV. Subsequent an-nealing at 820 K for 10 min gave a better surface quality.First 1.8 ML of Co were deposited on Cu /H20849001/H20850. Then 6 ML Cu spacer were deposited and the sample was annealed againat 420 K for 10 min. Recently, intermixing at the interface asa function of temperature has been discussed. 21In the present experiment the sample undergoes a double cycle of anneal-ing. This ensures that there is no further interdiffusion duringthe temperature-dependent measurements. Thereafter, FMRFIG. 1. /H20849Color online /H20850Geometry of the sample showing the relevant angles.FIG. 2. Normalized Jinteras a function of /H20849T/300 K /H208501.5for dif- ferent parameters. /H20849a/H20850Jinter/J0forJintra=50 and 90 meV and J0 =40/H9262eV, /H20849b/H20850Jinter/J0for J0=−22.5 and 40 /H9262eV and Jintra =90 meV. The data are taken from Ref. 14.KALARICKAL et al. PHYSICAL REVIEW B 75, 224429 /H208492007 /H20850 224429-2spectra were recorded at various temperatures between 250 and 420 K at a microwave frequency of 9 GHz. The out-of-plane angular dependence of the FMR parameters was mea-sured at room temperature. Then 7 ML of Ni were depositedon the spacer. The FMR measurements at various angles andtemperatures were then repeated. The pressure during depo-sition and measurement was always in the low 10 −10mbar range. All depositions were done at room temperature. Thethickness of the ﬁlms was monitored using medium energyelectron diffraction /H20849MEED /H20850. Experiments were done on trilayers with Ni as the topmost layer and also for samplescapped with 5 ML Cu. Samples were carefully annealed be-tween the FMR scans to ensure that there are no adsorptioneffects which could bring about a change in the anisotropy.For this work, the thicknesses were chosen so that both theFM layers have an easy axis in the ﬁlm plane. As mentioned previously, a typical spectrum for a trilayer sample would comprise of two modes. The relative positions of the optical and the acoustic modes with respect to themodes for the uncoupled ﬁlms, determine the type, FM orAFM, of the coupling. 15,16At 320 K, the in-plane magne- tized 1.8 ML Co layer with the 6 ML Cu cap, had the FMRposition at H res=198 Oe and a narrow FMR linewidth /H9004Hof 129 Oe. On the deposition of 7 ML Ni wit ha5M LC u cap, the optical mode was found at 151 Oe while the acousticmode was found at 1.9 kOe. The shift of H resof the optical mode to a lower value with respect to the Co line, showed aweak FM coupling for this spacer thickness of 6 ML. Thecorresponding /H9004Hvalues were 250 Oe and 370 Oe. A single Co ﬁlm has a much lower /H9004Hthan a Ni ﬁlm. Note that in standard literature, it is shown that for largercoupling the intensity /H20849oscillator strength /H20850of the acoustic mode increases and the optical mode weakens. 16,22Also, the optical mode has a larger relaxation rate than the acousticmode. 16,22Here, experimental evidence of the opposite limit, i.e., extremely weak coupling, is given. From the resonancepositions one can clearly identify that the low ﬁeld and nar-row line is the optical mode while the higher ﬁeld line and abroader line is the acoustic mode. The narrower line for theoptical mode can easily be explained qualitatively. For thedecoupled system, one has a narrow Co and a broad Ni reso-nance line. Now, when a weak coupling is switched on, thelines ﬁrst shift to lower ﬁeld positions. If the coupling wasincreased, the optical mode would have a larger /H9004Hthan the acoustic mode. However, the nature of the coupling being soweak, the linewidths retained their comparative values. Figure 3shows the FMR proﬁles for two different out-of- plane external ﬁeld angles /H9258H. These proﬁles were taken on a trilayer wit ha5M LC u cap, at room temperature. The ﬁgure shows the shift in the mode positions to higher ﬁeld valuesfor smaller /H9258Hvalues, as predicted by theory. The solid curves are ﬁts to the data to two Lorentzian functions, whichgive the H resand the /H9004Hmeasured as the width between the optima of the signal of the modes. For /H9258H=90°, the reso- nance positions were 0.138 and 1.88 kOe for the optical andthe acoustic modes, respectively. The /H9004Hwere 0.34 and 0.4 kOe, respectively. From a complete angular dependenceof the resonance positions one can determine J inter. Further details of extraction of Jinterfrom Hresversus /H9258Hwill be evident from the next section and Fig. 6below.These angular dependences of FMR spectra were then taken at different temperatures in a range between 250 K and420 K. As the temperature is reduced the spectra move tolower ﬁeld values. This is because of the temperature-dependent changes in magnetization, anisotropy, and J inter values. Below 250 K these changes pushed the spectra to such low ﬁelds that the optical mode was not visible. Hencethe spectra for these temperatures could not be consideredfor analysis. Figure 4shows FMR proﬁles for different temperatures for /H9258H=90°. Figure 4/H20849a/H20850shows the proﬁles for an uncapped trilayer while Fig. 4/H20849b/H20850shows the proﬁles for a capped one. As before, the blue dotted and red solid curves are ﬁts to thedata to two Lorentzian functions. In both cases, the changesin the anisotropy energy and magnetization push the spectratoward higher ﬁelds as Tis increased. Also, the linewidths ofFIG. 3. /H20849Color online /H20850FMR proﬁles for two different angles of /H9258H=90 and 40°, taken at T=307 K for the capped sample. The blue dotted and red solid curves are ﬁts to two coupled Lorentzians. FIG. 4. /H20849Color online /H20850FMR proﬁles for different temperatures for/H9258H=90°. /H20849a/H20850Proﬁles at 320 K and 330 K for an uncapped sample. /H20849b/H20850Proﬁles at 307 K and 365 K for the capped sample. The blue dotted and red solid curves are ﬁts to two coupled Lorentzians.DOMINANT ROLE OF THERMAL MAGNON EXCITATION IN … PHYSICAL REVIEW B 75, 224429 /H208492007 /H20850 224429-3both the modes are seen to increase with temperature. It has been well documented that as one approaches Tc, the line- width of a thin ﬁlm is seen to increase.23This is also the case here since the Tc’s of these layers are close to 400 K.20 The Hresand the linewidth at a particular /H9258HandTwere determined from the Lorentzian ﬁts. Thereafter, at each tem-perature the complete out-of-plane angle dependence of H res was ﬁt to theory to give the Jinterparameter at that particular temperature. III. FMR CONDITION FOR COUPLED TRILAYERS The extraction of Jinterfrom the angular dependence of FMR spectra has been described in detail elsewhere.16,19For the sake of completeness, however, it is outlined in briefbelow. The resonance condition for a coupled trilayer systemmay be determined using the Smit and Beljers method, 24 /H20873/H9275 /H20841/H9253/H20841/H208742 =F/H9258/H9258F/H9272/H9272−F/H9258/H92722 M2sin2/H9258. /H208492/H20850 Here/H9275is the resonance frequency, /H9253=g/H9262B//H6036is the gyro- magnetic ratio, and /H9258and/H9272are the azimuthal and the polar angles of the magnetization. Fis the free energy density and the subscripts stand for second partial derivatives with re-spect to the angles. Fincludes the contributions due to the anisotropies and the interlayer exchange and is given by F=F inter+/H20858 i=12 Fi, /H208493/H20850 where Finter=−JinterM1·M2 M1M2, /H208494/H20850 Fi=di/H20873−Mi·H−/H208492/H9266Mi2−K2/H11036isin2/H9258i/H20850 −K4/H20648i 8/H208493 + cos 4 /H9272i/H20850sin4/H9258i/H20874. /H208495/H20850 Here, the subscript istands for the two layers FM1 and FM2. K2/H11036is the intrinsic out-of-plane anisotropy constant due to surface effects and tetragonal distortion of the ﬁlm.The ﬁrst nonvanishing contribution to the in-plane aniso-tropy is K 4/H20648, which is the fourfold in-plane anisotropy con- stant. The thicknesses of the individual layers are given byd i. It is easy to see that the resultant expression for /H9275, albeit complicated, depends on the magnetizations and theanisotropies of the individual FM layers. At a glance, it mayseem as if there are several ﬁtting parameters. However, thepower of the in situ UHV FMR spectrometer can be utilized to reduce these parameters drastically, since the layers can bedeposited and measured step by step. The magnetization,anisotropies, and their temperature dependences for the ﬁrstFM layer are estimated from the FMR measurements beforethe deposition of the second FM layer. Parameters for Cowere obtained from the measurements taken of the Co layerprior to the deposition of Ni. For additional veriﬁcation, 7ML Ni were deposited separately on Cu /H20849001/H20850and the angular as well as the temperature dependences of FMR parameterswere measured. This gives one a handle on all the ﬁttingparameters required for the evaluation of J inter. The gvalue for 7 ML Ni is very close to the bulk value while for 1.8 MLof Co, it is known that there is an enhancement in the orbitalmomentum and hence gwas taken to be 2.21. 25The 4/H9266Meff=4/H9266M−2K2/H11036/Mvalues were taken to be 0.8 and 32.7 kG for Ni and Co, respectively. The resonance positions as given from the above equa- tions are obtained by numerical simulation. Figure 5shows results of the simulations to illustrate the inﬂuence of each of the parameters for K2/H11036Ni,K2/H11036Co,K4/H20648Ni, and JinteronHres/H20849/H9258H/H20850. The curves being symmetric with respect to /H9258H=0, the results are shown only for positive values of /H9258H. The black solid /H20849red dashed /H20850curves were obtained by varying one of the param- eters by +10% /H20849−10% /H20850. The lower resonance ﬁeld curves correspond to the optical mode. Several points are of note. First, a change in the K2/H11036parameter for either Ni or Co brings about a change in both, the acoustic and the opticalmodes. Figures 5/H20849a/H20850and5/H20849b/H20850show that the angular depen- dence of the acoustic mode is more sensitive to a change inK 2/H11036for either Ni or Co than the optical mode. Figure 5/H20849c/H20850 shows the insensitivity of the angular dependence in either mode to a relative change in K4/H20648Ni. For all the ﬁts to follow in this work, K4/H20648Cowas kept constant at zero. Figure 5/H20849d/H20850shows that the main effect of a relative variation of Jinteris seen on the optical mode. The effects taken together gave an esti-mated error of 10% in J inter. As was seen in Fig. 3, one can see that as the out-of-plane angle /H9258Happroaches zero, i.e., perpendicular to the ﬁlm plane, Hresincreases. This is more clearly understood from Fig. 6, which shows Hresvs/H9258HatT=355 K. The solid red circles are the positions for the acoustic mode and the openFIG. 5. /H20849Color online /H20850Calculated resonance positions as a func- tion of /H9258H. The simulations were done assuming K2/H11036Ni =6.9/H9262eV/atom, K2/H11036Co=−69 /H9262eV/atom, K4/H20648Ni=0.62 /H9262eV/atom, and Jinter=2.8/H9262eV/atom. The black solid /H20849red dashed /H20850curves were ob- tained by varying one of the parameters by +10% /H20849−10% /H20850. The varied parameter is indicated in each panel. The other parameterswere kept ﬁxed. From the simulation in panels /H20849a/H20850and /H20849b/H20850follows that the error bar for a ﬁt of K 2/H11036is 1%. Having this value ﬁxed the uncertainty for Jinterin panel /H20849d/H20850then is approximately 5%.KALARICKAL et al. PHYSICAL REVIEW B 75, 224429 /H208492007 /H20850 224429-4black circles for the optical mode. The solid curves give a ﬁt to the data according to the process described in the preced-ing section. The variation of H reswith/H9258His dependent on three pa- rameters, namely the magnetization, the anisotropy, and theinterlayer exchange coupling. For this work, the values of themagnetization and the anisotropy were determined from themeasurements taken on the single layers, as mentioned ear-lier. Taken together, the ﬁt gives J interfor the capped trilayer at 355 K to be 1.4 /H9262eV/atom. This small value of Jinterfor d/H110156 ML has in fact been predicted by Bruno.13 IV . RESULTS AND DISCUSSION The values of Jinteras obtained from the FMR data were analyzed as a function of temperature, and also compared topreviously obtained results. With the results obtained here, aclear picture emerges regarding the temperature dependenceofJ interfor both, FM and AFM coupling. Figure 7shows the values of Jintervs spacer thickness at three different temperatures of 270 K, 300 K, and 365 K asindicated. The solid curve is a calculation according toBruno, which takes into account the effects due to Cu spacer 13scaled on the yaxis to match the data as is done in Ref. 16. The inset shows the data for the spacer thickness of 6 ML on an enlarged scale. The oscillations of Jinterhave been previously discussed in Refs. 16and26for these sys- tems. These oscillations show the effect of the spacer thick-ness on J inter. Here, the focus is on temperature dependence. The values for the calculations and the data for d/HS110056M L have been taken from Lindner and Baberschke16and Schwieger et al.12The recent data are in concurrence with the previous results. The values of Jintercan be seen to decrease with increasing temperatures. Figure 8shows Jinterin absolute units of energy per atom, as a function of T3/2for three trilayer sets, the capped and two uncapped samples. The solid lines are linear ﬁts to thedata. 7The values of Tcorresponding to the T3/2values are given on the top axis of the graph. The data sets can be seento be linear in T 3/2. The uncapped trilayer samples have slightly higher values than the capped one. However, to setthese on a similar scale, J interneeds to be normalized to the zero intercept value of the linear extrapolation J0, and stud- ied as a function of temperature. These data can then becompared to previous observations in order to obtain a com-plete picture of the temperature dependence of J inter. Figure 9gives the normalized Jinter/J0vsT3/2for different spacer thicknesses. The solid symbols are the data for d=6 ML, with the solid circles being the data for the capped andthe squares and triangles being the data for the uncappedsamples. These are compared with the data given in Refs. 12 and16. The open triangles, open circles, and open squares are the data for spacer thicknesses of 4, 5, and 9 ML, respec-tively. These data were obtained by determining only twoJ intervalues from angular dependence and interpolating the others from the shift of the modes. Note that for the recentmeasurements the scatter and error bar of the data is largerbecause each value of J interwas obtained independently, however it conﬁrms the previous analysis in Refs. 12and16. With the results of these experiments, a complete pictureemerges wherein one can compare the temperature depen-dences of FM as well as AFM coupling. The change for theFIG. 6. /H20849Color online /H20850FMR resonance position vs the out-of- plane angle at T=355 K for the capped sample. The solid, red /H20849open, black /H20850circles are the data for the acoustic /H20849optical /H20850mode. The solid curves are ﬁts to the data as described in the text. FIG. 7. Jintervs spacer thickness at temperatures of 270 K /H20849open circles /H20850, 300 K /H20849solid triangles /H20850, and 365 K /H20849solid squares /H20850. The solid curve is a calculation according to Bruno /H20851Ref. 13/H20852. The inset shows data for the capped sample with d=6 ML. The error is on the order of 10%.FIG. 8. /H20849Color online /H20850Jinterin absolute units vs T3/2for three 7 ML Ni /6 ML Cu/1.8 ML Co samples. The solid circles are the datafor the capped sample while the solid triangles and squares are thedata for the uncapped samples.DOMINANT ROLE OF THERMAL MAGNON EXCITATION IN … PHYSICAL REVIEW B 75, 224429 /H208492007 /H20850 224429-5FM coupled layers is stronger than that for the AFM coupled layers. Moreover, as expected, there is a stronger decreasewith T 3/2for larger spacer thicknesses. Figure 10shows the connection between the scaling pa- rameter Aand the J0with relation to the spacer thickness. Figure 10/H20849a/H20850shows the values of Afrom Eq. /H208491/H20850vs spacer thickness. Figure 10/H20849b/H20850shows J0vs spacer thickness. The data for the 4, 5, and 9 ML have been obtained from ananalysis of the data given in Refs. 12and16. The dashed lines are guides to the eye meant to show the trend in thedata. Several points can be noted from this ﬁgure. Largevalues of Aimply severe suppression of coupling due to temperature. The trends in Fig. 10are an experimental evi- dence of the trends predicted by Schwieger and Nolting 14 reproduced in Fig. 2/H20849b/H20850. The larger the J0value, the weaker is the temperature dependence. A qualitative interpretation ofthe theoretical prediction can be easily given. The IECstrength between the two ferromagnetic layers is in compe-tition with the thermal energy kT. For strong coupling be- tween FM1 and FM2, the thermal energy can be neglectedand elevated temperatures would have little effect on J inter. On the other hand, for vanishing IEC between the two ferro-magnetic ﬁlms, the thermal energy and resulting spin-waveexcitations become very important. Hence, it is straightfor-ward that the parameter Ain Eq. /H208491/H20850increases for small IEC and decreases for stronger IEC. The data shown in Fig. 10 conﬁrms this theoretical prediction where the trend shown bythe ﬁt parameter Ais opposite to that shown by J 0. Another feature is that there is a deﬁnite oscillation in A, which indi- cates that there is a larger role of the spin-wave excitationthan of the spacer in the temperature dependence. 14A lineardependence would, on the other hand, have been a signature of the spacer effects. V . CONCLUSIONS An investigation into the temperature dependence of Jinter was undertaken through the study of ferromagnetic reso- nance positions in Ni/Cu/Co trilayer systems. The spacerthickness was chosen so that it was in the ultrathin limit andalso gave a weak exchange coupling between the two ﬁlms,a regime important for the fundamental understanding ofJ inter/H20849T/H20850. The interlayer exchange parameter Jinterwas evalu- ated entirely from the angular dependence of the FMR posi- tions of the optical and acoustic modes. The ﬁt parametersfor the temperature dependence were compared with the ex-trapolated J 0values and spacer thicknesses. The scaling pa- rameter Awas found to be neither independent nor a linear function of d, as would have been expected from a dominant interface effect or spacer electronic band structure contribu-tion. Instead, the oscillations in Agive an experimental veri- ﬁcation of the theory forwarded by Schwieger and Nolting. 14 It is a clear conclusion that the excitation of spin waves or inother words, the creation of thermal magnons is the domi-nant cause of the temperature dependence of J interin FM and AFM coupled trilayers. ACKNOWLEDGMENTS Discussions with S. Schwieger are acknowledged. Two of the authors /H20849S.S.K. and X.Y .X. /H20850thank the Institut für Experi- mentalphysik at Freie Universität Berlin for their hospitalityduring their stay at the department. This work was supportedin part by BMBF /H20849Contract No. 05KS4 KEB/5 /H20850and DFG Sfb 658 /H20849TP B3 /H20850. Current address: Magnetics and Magnetic Materials Laboratory, Colorado State University, Fort Collins, Colorado 80523. †Permanent address: Surface Physics Laboratory, Fudan University, Shanghai 200433, People’s Republic of China.‡Corresponding author. FAX: /H1100149 30 838-55048; bab@physik.fu- berlin.de; URL: http://www.physik.fu-berlin.de/ /H11011bab/ 1M. D. Stiles, in Ultrathin Magnetic Structures III , edited by B. Heinrich and J. A. C. Bland /H20849Springer-Verlag, Heidelberg,FIG. 10. /H20849a/H20850Fit parameter Avs spacer thickness. /H20849b/H20850J0vs spacer thickness. The dashed curves are mere guides to the eye.FIG. 9. /H20849Color online /H20850Normalized JintervsT1.5for different spacer thickness. The data points for the thicknesses of 4, 5, and 9ML were taken from Schwieger et al. /H20849Ref. 12/H20850, and were obtained from the extrapolation of the shift between the H resof Co and the optical mode.KALARICKAL et al. PHYSICAL REVIEW B 75, 224429 /H208492007 /H20850 224429-62005 /H20850,p .9 9 . 2K. B. Hathaway, in Ultrathin Magnetic Structures II , edited by B. Heinrich and J. A. C. Bland /H20849Springer-Verlag, Heidelberg, 1994 /H20850. 3J. Lindner, C. Rüdt, E. Kosubek, P. Poulopoulos, K. Baberschke, P. Blomquist, R. Wäppling, and D. L. Mills, Phys. Rev. Lett. 88, 167206 /H208492002 /H20850. 4Z. Celinski, B. Heinrich, J. F. Cochran, W. B. Muir, A. S. Arrott, and J. Kirschner, Phys. Rev. Lett. 65, 1156 /H208491990 /H20850. 5D. M. Edwards, J. Mathon, R. B. Muniz, and M. S. Phan, Phys. Rev. Lett. 67, 493 /H208491991 /H20850. 6Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev. Lett. 73, 336 /H208491994 /H20850. 7N. S. Almeida, D. L. Mills, and M. Teitelman, Phys. Rev. Lett. 75, 733 /H208491995 /H20850. 8J. Lindner, E. Kosubek, P. Poulopoulos, K. Baberschke, and B. Heinrich, J. Magn. Magn. Mater. 240, 220 /H208492002 /H20850. 9B. Heinrich, Z. Celinski, L. X. Liao, M. From, and J. F. Cochran, J. Appl. Phys. 75, 6187 /H208491994 /H20850. 10A. Layadi and J. O. Artman, J. Magn. Magn. Mater. 92, 143 /H208491990 /H20850. 11S. Schwieger, J. Kienert, K. Lenz, J. Lindner, K. Baberschke, and W. Nolting, J. Magn. Magn. Mater. 310, 2301 /H208492007 /H20850. 12S. Schwieger, J. Kienert, K. Lenz, J. Lindner, K. Baberschke, and W. Nolting, Phys. Rev. Lett. 98, 057205 /H208492007 /H20850. 13P. Bruno, Phys. Rev. B 52,4 1 1 /H208491995 /H20850. 14S. Schwieger and W. Nolting, Phys. Rev. B 69, 224413 /H208492004 /H20850.15B. Heinrich, in Ultrathin Magnetic Structures II , edited by B. Heinrich and J. A. C. Bland /H20849Springer-Verlag, Heidelberg, 1994 /H20850. 16J. Lindner and K. Baberschke, J. Phys.: Condens. Matter 15, R193 /H208492003 /H20850. 17P. Gambardella, S. Rusponi, M. Veronese, S. S. Dhesi, C. Grazioli, A. Dallmeyer, I. Cabria, R. Zeller, P. H. Dederichs, K.Kern et al. , Science 300, 1130 /H208492003 /H20850. 18K. Baberschke, Appl. Phys. A: Mater. Sci. Process. 62, 417 /H208491996 /H20850. 19K. Lenz, E. Kosubek, T. Tolinski, J. Lindner, and K. Baberschke, J. Phys.: Condens. Matter 15, 7175 /H208492003 /H20850. 20P. Srivastava, F. Wilhelm, A. Ney, M. Farle, H. Wende, N. Haack, G. Ceballos, and K. Baberschke, Phys. Rev. B 58, 5701 /H208491998 /H20850. 21E. Holmström et al. , Phys. Rev. Lett. 97, 266106 /H208492006 /H20850. 22B. Heinrich, in Ultrathin Magnetic Structures III , edited by B. Heinrich and J. A. C. Bland /H20849Springer-Verlag, Heidelberg, 2003 /H20850. 23Y . Li and K. Baberschke, Phys. Rev. Lett. 68, 1208 /H208491992 /H20850. 24J. Smit and H. G. Beljers, Philips Res. Rep. 10,1 1 3 /H208491955 /H20850. 25M. Tischer, O. Hjortstam, D. Arvanitis, J. Hunter-Dunn, F. May, K. Baberschke, J. Trygg, J. M. Wills, B. Johansson, and O.Eriksson, Phys. Rev. Lett. 75, 1602 /H208491995 /H20850. 26R. Hammerling, J. Zabloudil, P. Weinberger, J. Lindner, E. Ko- subek, R. Nünthel, and K. Baberschke, Phys. Rev. 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PhysRevB.84.165306.pdf	PHYSICAL REVIEW B 84, 165306 (2011) Voltage-controlled spin precession A. N. M. Zainuddin,S. Hong, L. Siddiqui, S. Srinivasan, and S. Datta† School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Received 20 July 2010; revised manuscript received 20 February 2011; published 4 October 2011) Spin-transport properties of a lateral spin-valve structure originating from spin precession in its two- dimensional semiconducting channel under the inﬂuence of Rashba spin-orbit (RSO) coupling are explored.The effect of the ﬁnite extent of the injecting and detecting contact pads, along the length of the channel,on the spin signals is studied in these structures using (1) a simple theoretical treatment leading to analyticalexpressions for spin-dependent voltages derived using the stationary phase approximation, and (2) a morerigorous theoretical treatment based on nonequilibrium Green’s function formalism to calculate these voltages,in a nonlocal spin-valve setup. Using both these approaches, it is found that the oscillation in spin voltages, whichis observed by varying RSO when the magnetization directions of the injector and detector are parallel to thecurrent ﬂow, reduces in amplitude and shifts in phase for contact pads having ﬁnite length when compared to thecorresponding results for a zero length (point-contact) limit. The amplitude and phase of the oscillation can berecovered to its point-contact limit if the RSO underneath the contacts is assumed to be zero. These models werecompared against a recent experiment, and it is found that certain aspects of the experiment can be described wellwhile some other aspects deserve further investigation. Factors that could have inﬂuenced the experiment andthereby could explain the discrepancy with the theory were analyzed. Conditions for observing Hanle oscillationin such a structure is discussed. Finally, the possibility of controlling the magnetization reversal via the gate isdiscussed, which could extend and quantify the ‘Datta-Das’ effect for voltage controlled spin-precession. DOI: 10.1103/PhysRevB.84.165306 PACS number(s): 85 .75.−d I. INTRODUCTION V oltage-controlled spin precession, proposed in 1990,1 posed two difﬁcult challenges: (1) spin-polarized injection into a semiconducting channel and (2) gate control of theRashba spin-orbit (RSO) interaction in the channel. 2The latter was demonstrated by Nitta et al. in 1997 using an inverted InGaAs/InAlAs quantum well with a top gate.3But spin-polarized injection into a semiconductor proved to be a more difﬁcult challenge4which has only recently been overcome through the combined efforts of many groupsaround the world. 5–8Very recently, Koo et al.9combined both ingredients, spin-polarized injection and gate-controlled RSO,into a single experimental structure using a high-mobilityInAs heterostructure with a top gate interposed betweenthe current contacts and the voltage contacts. The nonlocal voltage signal 10shows an oscillatory behavior when the contacts are magnetized along the direction of current ﬂow,but shows nonoscillatory behavior when they are magnetizedperpendicular to the current ﬂow, as expected from the theorypresented in Ref. 1. Furthermore, it was shown 9that the oscillation is described well by a single cosine functionwith an additional phase shift. The oscillation period was 2m ∗α(VG)L/¯h2, where m∗is the effective mass and α(VG) is the RSO measured independently from the Shubnikov–deHaas (SDH) beating pattern. For carriers ﬂowing in quasi-two-dimensional channels such periodic oscillation is believedto be washed out with increasing number of channels dueto the nontrivial intersubband coupling effect. 11–16However, Pala et al.17and recently Agnihotri et al.18showed that for two-dimensional channels of semi-inﬁnite width where periodic boundary conditions (PBCs) can be imposed insteadof hard wall boundary conditions (HBCs) along the widthdirection, such periodic oscillation can still persist although itdecays due to the averaging effect over an angular spectrum with increasing strength of the RSO interaction. Based onthis observation it would seem that the single cosine-like oscillation observed in Ref. 9is plausible, but the amplitude and phase require a more detailed consideration especiallysince the simple models view the contacts as point sources. The objective of this paper is to ﬁrst explore the inﬂuence of extended injecting and detecting contacts on RSO-modulatedspin signals. The model is then compared against the re-cent experiment 9and possible sources of discrepancies are discussed. We also discuss the possibility of controlling themagnetization switching via modulating spin-current. Wehope that our analysis will establish this gate-controlledspin-precession effect on a ﬁrm footing, so that it can be usedboth for fundamental studies as well as for various proposedapplications such as spin ﬁltering, magnetic recording andsensing, or quantum computing. 19 The organization of this paper is as follows. In Sec. II we provide an overview of our model for calculating spin-dependent voltages in a two-dimensional channel with a RSO.Here we will ﬁrst provide a simple analytical model which isan extension of the approach taken in Ref. 1to include the sum over the angular spectrum of electrons. The simple model isfollowed by a more rigorous nonequilibrium Green’s function(NEGF)-based model for electronic transport, with which wesimulate an actual nonlocal spin-valve structure. In Sec. III, we discuss spin voltages in the limit of injecting and detectingpoint contacts. Then in Sec. IVwe discuss how the spin voltage reduces in amplitude and changes in shape with the inﬂuenceof extended contacts. In Sec. Vwe compare our model with the experiment in Ref. 9, and we discuss possible reasons for discrepancies in Sec. VI. We brieﬂy discuss the magnetic- ﬁeld-controlled oscillation, Hanle effect, in such RSO-coupledchannels in Sec. VII. In Sec. VIII we discuss a scheme to 165306-1 1098-0121/2011/84(16)/165306(13) ©2011 American Physical SocietyZAINUDDIN, HONG, SIDDIQUI, SRINIV ASAN, AND DATTA PHYSICAL REVIEW B 84, 165306 (2011) manipulate the magnetization direction by modulating the gate voltage. Finally, we summarize our conclusions in Sec. IX. II. MODEL OVERVIEW We start from an effective mass Hamiltonian for a two- dimensional conductor having a RSO interaction and anegligible Dresselhaus spin-orbit (DSO) interaction of theform ( /vectorσ: Pauli spin matrices): H=−¯h 2 2m∗/parenleftbigg∂2 ∂x2+∂2 ∂y2/parenrightbigg +α(σXkY−σYkX). (1) A. Simple analytical model Equation ( 1) leads to the dispersion relation E=¯h2k2 2m∗±αk, k =+/radicalBig k2 X+k2 Y, (2) with the upper and lower signs corresponding to eigenspinors of the form {ψ±}={ 1±exp(iφ)}T, where tan φ≡−kX/kY. Here,XandYare the longitudinal (or transport) and transverse direction, respectively, following the coordinate system usedin Ref. 9, which is different from that used in Ref. 1. Assuming periodic boundary conditions in the transverse direction leadstok Ybeing conserved in the absence of any scattering mechanism and also to two values of kX(kX+andkX−) corresponding to the upper and the lower signs in Eq. ( 2), which are given by E=¯h2 2m∗/parenleftbig k2 X++k2 Y/parenrightbig +α/radicalBig k2 X++k2 Y =¯h2 2m∗/parenleftbig k2 X−+k2 Y/parenrightbig −α/radicalBig k2 X−+k2 Y, (3) and for small αwe can write kX−−kX+≈2m∗α ¯h2k0/radicalBig k2 0−k2 Y, (4) withk0≡√ 2m∗E/¯h. Equation ( 4) determines the frequency at which the spins would rotate while traveling at a certaink Ymode. It also suggests that the frequency of rotation would be higher for higher kYmodes. A similar expression for a one-dimensional channel was derived in Ref. 1[see Eq. ( 6)], and one can get the same by simply putting kY=0 in Eq. ( 4). To get the magnitude and phase of oscillation, we calculate the transmission tfor an electron injected from a point-contact injector and detected at a point-contact detector separated by achannel length L. Nonlocal voltages V X(Y)[see Fig. 1(a)], for the magnetizations of X(Y)-directed injecting and detecting ferromagnetic contacts being parallel and antiparallel, areproportional to \|t xx(yy)\|2and\|txx(yy)\|2, respectively, and are, henceforth, denoted by VX(Y),PandVX(Y),AP, respectively. In this paper, we present the results in terms of a quantitynamed “spin voltage,” which is denoted by /Delta1V X(Y)and is deﬁned as /Delta1VX(Y)=[VX(Y),P−VX(Y),AP]. These notations are similar to the ones used by Takahashi et al.20Throughout this paper the analytical expressions for spin voltages will bevalidated by comparing them with the results from a more (b)(a) FIG. 1. (Color online) Schematics of (a) a lateral structure under nonlocal setup where VX(VY) corresponds to the spin voltages when the injecting and detecting ferromagnetic (FM) contacts aremagnetized in the X(Y) direction, (b) NEGF-based model for the structure in (a) with /Sigma1 2and/Sigma13representing injecting and detecting FM contacts, /Sigma11and/Sigma14representing nonmagnetic (NM) contacts, and/Sigma1Land/Sigma1Rrepresenting the semi-inﬁnite regions outside the central region. rigorous model based on NEGF formalism for electronic transport. B. NEGF-based model A detailed description of the NEGF-based model can be found in Ref. 21. The inputs to this model are the Hamiltonian [ H] and the self-energy matrices [ /Sigma1] [Fig. 1(b)]. ForHwe use a discretized version of the one used in the simple model section [Eq. ( 1)], described in Ref. 21 assuming PBCs along Yas discussed above. We neglect all scattering processes, assuming both the mean free path andthe spin coherence length are longer than the longitudinaldimensions at low temperatures. To understand any signaldecay at higher temperatures will require a consideration ofboth momentum and spin relaxation processes, but we leavethis for future work. The self-energies for the ferromagnetic(FM) contacts ( /Sigma1 2,/Sigma13) have the form −(i/2)γ[I+PC/vectorσ·ˆn] where the polarization PC=(GM−Gm)/(GM+Gm) and ˆnis the unit vector in the direction of the magnet. Here GMandGmare the majority and minority spin-dependent conductances of the tunneling contacts. We note that thesespin-dependent interfacial conductances determine the spinaccumulation at the ferromagnetic-nonmagnetic interface bothin the diffusive and in the ballistic regimes. 22,23The constant γ=π(GM+Gm)¯h3/e2m∗is chosen to give a tunneling conductance equal to the experimental value. The nonmagnetic(NM) contacts ( /Sigma1 1,/Sigma14) are represented similarly with PC=0. SoGMandGmare the only two ﬁtting parameters used in this model. Finally, the long extended regions outside the channelat two ends [see Fig. 1(a)] are represented by two semi-inﬁnite contacts whose coupling is given by /Sigma1 L(R)=τL(R)gSτ† L(R), 165306-2VOLTAGE-CONTROLLED SPIN PRECESSION PHYSICAL REVIEW B 84, 165306 (2011) where τis the spin-dependent coupling matrix between the contact and the channel and gSis the surface Green’s function. The transmission functions are calculated from the NEGFmodel and contacts 3, 4, L, and R are treated as voltageprobes with zero current (following the approach introducedby Buttiker, see Sec. 9.4, in Ref. 24). We note that although we are not including any scattering processes explicitly, thevoltage probes introduce an effective spin scattering thatreduces the signal. This is due to the fact that for the chargecurrent to be zero in a voltage probe, two spin components,majority and minority spins, of the current become equal inmagnitude. Thus majority spins convert to minority spins andthereby spin relaxation takes place. To explain further aboutour method of calculating nonlocal spin voltages, we compareour NEGF-based calculation with an equivalent circuit modelin Appendix Afor a given structure. The results are consistent with those of the ballistic model of spin signal described inRef. 23. In the following sections we discuss the magnitude and phase of spin voltage for the structure shown in Fig. 1(a) featuring the effects of contacts based on both our simple andNEGF-based models. III. DEVICE WITH POINT CONTACTS We start our discussion by considering a point-contact injector and a point-contact detector. It is shown in AppendixBthat starting from the eigenspinors in Eq. ( 2) and assuming ballistic transport in the channel, the contributions to thevoltage signals for X- and Y-directed magnets coming from a particular Eandk Ycan be written as ( C0: constant) /Delta1VX0(E,kY)=C0/braceleftbigg s2+(1−s2) cos/parenleftbiggθL√ 1−s2/parenrightbigg/bracerightbigg ,(5a) /Delta1VY0(E,kY)=C0/braceleftbigg (1−s2)+s2cos/parenleftbiggθL√ 1−s2/parenrightbigg/bracerightbigg ,(5b) where s≡kY/k0=¯hkY/√ 2m∗EandθL=2m∗αL/¯h2. These contributions from different E,kYall act “in parallel,” giving a voltage equal to the average. At low temperatures wecan average the contributions from all transverse wave vectorsk Yover the Fermi circle ( E=EF) to write /Delta1VX(Y)=/integraldisplay+k0 −k0dky 2πk0/Delta1VX0(Y0)(EF,kY). (6) We note that Eq. ( 6) is equivalent to the conductance modula- tion expressions derived in Refs. 17and18for a two-terminal spin ﬁeld-effect transistor. Interestingly, the results obtainedfrom the integration in Eq. ( 6) look almost like a single cosine. This can be understood by noting that the argument θ L/√ 1−s2has a stationary point at s=0,25and we can use the method of stationary phase to write approximately /Delta1VX/similarequalC0 3π+C0√2πθLcos/parenleftBig θL+π 4/parenrightBig , (7a) /Delta1VY/similarequal2C0 3π. (7b) (a) (b) FIG. 2. (Color online) Spin voltages as a function of the RSO for both X-a n d Y-directed point injecting and detecting ferromagnetic contacts from (a) analytic expression in Eq. ( 7)f o r /Delta1VYand /Delta1VX, (b) NEGF-based model. Parameters: PC∼1a n d nS=2.7× 1012cm−2, and the spacing between two point contacts is 1.65 μm. As shown in Appendix Cthese approximations describe the results from the exact integration quite well for θL/greaterorsimilar2π which falls within the current experimental status.9 Although the simple model here makes no prediction about the amplitude C0, it does suggest that the peak-to-peak amplitude of the oscillation in /Delta1VXshould be 3 π/√2πθL times the spin-valve signal /Delta1VY. This is shown in Fig. 2(a) by plotting the analytical expression in Eq. ( 7) and is also evident from our numerical NEGF-based model as shownin Fig. 2(b). IV . DEVICE WITH EXTENDED CONTACTS In this section we consider injection and detection from contacts that are extended over the channel along x.I nt h e point-contact case, all the injected electrons travel across thesame length Lbefore reaching the detector. But with extended contacts, electrons will travel across a length depending on thepoint of injection and the point of detection under the contacts.This will give rise to a spread in the values of θ Lin Eqs. ( 5a) and ( 5b). We can write /tildewidest/Delta1VX=CiCdC0cos(θ0+θi+θd+π/4)√2π(θ0+θi+θd), (8) where CiandCdare numbers less than unity representing the averaging effects of the injecting and detecting contacts,respectively, and θ i,θdare the additional phase shifts intro- duced by the injecting and detecting contacts, respectively,in addition to θ 0, which is the phase shift corresponding to the channel length between the contacts. θi,θdorCi,Cdwill 165306-3ZAINUDDIN, HONG, SIDDIQUI, SRINIV ASAN, AND DATTA PHYSICAL REVIEW B 84, 165306 (2011) (a) (b) FIG. 3. (Color online) Spin voltages as a function of the RSO for both X-a n d Y-directed extended contacts with uniform injection and detection. (a) Analytic expression in Eq. ( 8)f o r/Delta1VXand in Eq. ( 7b)f o r/Delta1VY, (b) NEGF-model-based calculation for the same signals. Parameters: same as in Fig. 2for point contacts (solid), and LCi,d=0.2μm (dashed) and 0 .4μm (dotted) for extended contacts. depend on (A) the length of the injecting and detecting contacts and (B) how the RSO α(VG) varies under the contacts. We discuss these two points in the following sections. However,we note that extended contacts do not affect /Delta1V Y, because it is nonoscillatory. As a result, /Delta1VYcan be described by the same Eq. ( 7b) even with extended contacts. A. Length of the contacts It is shown in Figs. 3(a)and3(b)that, considering uniform injection and detection along the channel, the oscillatory signal(/Delta1V X) reduces in amplitude and shifts in phase with increasing contact lengths. Besides, the fact that the nonoscillatory signalstays almost the same with contact lengths is also veriﬁedfrom the NEGF calculation [see /Delta1V Yin Figs. 3(a)and3(b)]. Here the signal /Delta1VXis averaged over both the injecting and detecting contacts for which the amplitude degrades. As aresult the ratio /Delta1V Y//Delta1V X(peak-to-peak or “p-p”) is further increased from the point-contact limit. Our analytical result[Eq. ( 8)] also matches that from the NEGF model if we useθ i,d=m∗αLCi,d/¯h2andCi,d=sin(θi,d)/θi,dwhich can be justiﬁed if the electronic wave function is assumed to remainconstant under each contact. B. Variation of RSO under the contacts Since the contacts are metallic, and in addition to being ferromagnetic, it is possible for the gate electric ﬁeld to bescreened out under the contacts. In such a case, the RSOunderneath the contacts α 0might not follow the variation (a) (b) FIG. 4. (Color online) Spin voltages as a function of the RSO for both X-a n d Y-directed extended contacts where the RSO in the channel under the contacts does not vary in accordance with the channel outside the contacts. (a) Analytic expression in Eq. ( 8)f o r /Delta1VXand in Eq. ( 7b)f o r/Delta1VY, (b) NEGF-model-based calculation, for cases (1) when RSOs under the the contacts vary accordingly with the rest of the channel outside the contacts α0=α(solid), (2) RSOs under the contacts are ﬁxed at α0=4×10−12eV m (dashed), and (3) RSO is absent under the contacts α0=0 (dotted). Other parameters are the same as in Fig. 3. that the gate electric ﬁeld brings about in the “bare” channel region not placed underneath the contacts. Moreover it is alsopossible that, underneath the contacts, the local magnetic ﬁeldreduces the RSO. However, a detailed treatment of this issueis beyond the scope of this paper. Here we only show howvarious choices of RSOs under the contacts can change theshape and amplitude of the oscillatory /Delta1V X.I nF i g . 4(solid line) we ﬁnd that if RSO varies under the contacts, oscillationis washed out at higher α. However, for a ﬁxed RSO under the contact the situation [see Fig. 4(dashed)] improves, because nowθ i,d=2m∗α0LCi,d/¯h2under the contacts do not vary with the increase in RSO in the channel outside the contacts andhence C i,d=sin(θi,d)/θi,dhas a constant value which was otherwise decreasing with the increase in α0. In this case the ratio/Delta1VY//Delta1V X(p-p) reduces with decreasing α0and it reaches again the point-contact limit when α0is assumed to be zero under the contacts [see Fig. 4(dotted)]. V . COMPARISON WITH EXPERIMENT Next we compare our models against the experiment in Ref.9. To obtain a current level equal to the experimental value in the NEGF model we adjust the applied potential difference(μ 1−μ2) for contacts 1 and 2 [see Fig. 1(b)]. We use a contact conductance of GC=GM+Gm=4×1010/Omega1−1m−2based on the experimental parameters in Ref. 9andPC=(GM− Gm)/(GM+Gm)∼0.05 to match the spin-valve signal /Delta1VY. 165306-4VOLTAGE-CONTROLLED SPIN PRECESSION PHYSICAL REVIEW B 84, 165306 (2011) (a) (b) (c) FIG. 5. (Color online) Comparison with experiment in Ref. 9. (a) Experimental observation for nonlocal voltage in X-a n d Y- directed injector and detector. Reprinted with permission from science publishing group. (b) Simple qualitative model and (c) NEGFmodel. In all cases RSO under the contacts α 0is varied among three choices: (1) α0varies according to the channel α(VG) (solid), (2)α0is kept ﬁxed at α(VG=0) (dashed), and (3) α0is assumed zero (dotted). Parameters: PC∼0.05,nS=2.7×1012/cm2,GC= GM+Gm=4×1010/Omega1−1m−2,LCi=0.2μm, and LCd=0.25μm with 1 .65μm spacing in between. To be consistent with the notations used in Ref. 9we relate the measured oscillatory signal VX,P(p-p) [Fig. 5(a)left panel] to our calculated oscillatory spin voltage /Delta1VX,Pa:VX,P(p- p)=/Delta1VX/2 while the measured nonoscillatory spin-valve signal [Fig. 5(a) right panel] can be directly compared to our calculated spin voltage /Delta1VY. We note that to obtain the right shape of the oscillatory signal VX,P, we need to consider the contact length to be half of its actual lengthwith a spacing of 1 .65μm [Figs. 5(b) and5(c)] between them rather than the full contact length. But most importantly,although we ﬁnd the peak-to-peak amplitude of the oscillatorysignal V X,P(FM-FM, X) is to be equal to the spin-valve signal /Delta1VY(FM-FM, Y) in the experiment [see Fig. 5(a)]w e observe a much smaller signal for VX,Pcompared to /Delta1VYin our calculations [Figs. 5(b)–5(e)]. We obtain the closest agreement with the experimental results when we neglect thecontact averaging effect by assuming RSO to be zero underthe contacts [Figs. 5(b)–5(e)dotted lines], which leads to the point-contact limit as discussed in Sec. IVB. Such a condition gives the minimum calculated /Delta1V Y/VX,P(p-p) which, using m∗=0.05m0,α/similarequal10−11eV m, and L=1.65μm, is equal to 2.4, and is apparently larger than the experimentally observed value. In summary, our models (1) explain the observed period of the nonlocal voltage oscillation, (2) point out the fact that thephase requires a better understanding of αunder the contacts and show that a certain (nonunique) choice ﬁts the data, and(3) show that the amplitude is larger than expected. VI. DISCUSSION In this section we discuss a few possible sources of discrepancy that could have reduced the ratio /Delta1VY/VX,P(p-p) even below the point-contact limit. A. Dresselhaus spin-orbit coupling Although we have neglected DSO (see Sec. II) so far in our calculation, in this section we would like to investigate whethera signiﬁcant DSO along with RSO could have explained thediscrepancy with the experiment. In the experiment, DSOwas assumed to be negligible compared to RSO since thematerial has a narrow band gap. 26,27However, it was shown later28,29that DSO can become comparable to RSO in similar structures. As a result, further investigation of the inﬂuenceof DSO on RSO-modulated signals revealed that the choiceof crystallographic orientation of the channel material playsan important role in the Datta-Das effect. 15,30So in this work, we also incorporated the effect of DSO by includinga linear DSO term with the Hamiltonian Hin Eq. ( 1) to write H dso=H+β(σXkX−σYkY), where βis the linear DSO coefﬁcient. Here we are neglecting the cubic DSO term sinceit only modiﬁes the linear DSO term. 28In Fig. 6, we show our NEGF simulation that depicts the inﬂuence of DSO on the spinvoltages. The results indicate that the ratio /Delta1V Y/VX,P(p-p) would have increased more, if the DSO were comparable toRSO in the experiment. B. Boundary scattering Next, we discuss the role of boundary scattering on the RSO modulation. In our models we have assumed PBC intheYdirection making k Ya “good quantum number” like E. But when a real conﬁning potential is used for HBC, simple decoupling of different transverse wave vectors ( kY) is not allowed due to nontrivial “boundary scattering.” In thissection our numerical calculation shows that, although forsmaller number of transverse modes (in a narrow channel)the results are very different, for larger number of modes (in awider two-dimensional channel), use of HBC does not changethe conclusions described above with PBC in a signiﬁcantway. We show a comparison of HBC and PBC to calculate/Delta1T=\|t xx(yy)\|2−\|txx(yy)\|2for the structure shown in Fig. 1 in the point-contact limit. To include HBC, the Hamiltonianis written as H hbc=H+VC(y), where His the Hamiltonian 165306-5ZAINUDDIN, HONG, SIDDIQUI, SRINIV ASAN, AND DATTA PHYSICAL REVIEW B 84, 165306 (2011) FIG. 6. (Color online) Spin voltages in the presence of DSO cou- pling, β, in addition to RSO in the point-contact regime. Calculation is done with the NEGF-ased model. Solid lines correspond to spin voltages without DSO ( β=0), dashed line corresponds to the case whenβ=0.2α(VG=0), dotted line corresponds to the case when β=0.5α(VG=0). Parameters are same as in Fig. 5. given by Eq. ( 1) andVC(y) is a conﬁning potential of the form VC(y)=0f o r0 <y<W andVC(y)=∞ otherwise. Wis the channel width which is varied to include a different numberof modes in the channel. Figure 7shows our two-dimensional real-space NEGF simulation 24results. We see that for lower number of modes the results are quite different dependingon the choice of boundary conditions [see Figs. 7(a) and 7(b)] and indeed for channels with smaller widths, where HBC is more appropriate, RSO-induced oscillation looks nonsinusoidal. 11–16However, with increasing number of modes, HBC and PBC do not show much difference in results [seeFigs. 7(c)and7(d)] suggesting that our PBC-based conclusions (which are in agreement with Refs. 17,18) should hold quite well for HBC as well. C. Spin relaxation In this type of spin orbit material, the dominant spin- relaxation mechanism is believed to be that of the D’yakonov-Perel (DP) type. 31,32The effect of such a relaxation mechanism has been extensively studied in disordered two-dimensionalelectron gas (2DEG) under a quantum transport approach(see, for example, Refs. 11,33–35). But generally spin-orbit interaction effects in spin transport are taken into account ina semiclassical approach 26,36through their role in relaxing the nonequilibrium spin polarization. The spin-relaxationlength of the channel is then obtained from this approach.In the present experiment, the spin-relaxation length λ sfand the mean free path λmwere found to be ∼2μm37and ∼1.61±0.23μm(T=1.8K),9respectively. But the channel length (length between the injector and detector) of 1 .65μm was found to be shorter than both λsfandλm.9As a result, we believe, the DP spin-relaxation mechanism should not changeour conclusions in any signiﬁcant way as well. However, it isquite possible that high k Ycomponents are suppressed because they actually travel a longer length compared to the lowerones and hence have shorter effective spin coherence lengthswhich is not considered in a purely ballistic theory. As a result, (a) (b) (c) (d) FIG. 7. (Color online) /Delta1T=\|txx(yy)\|2−\|txx(yy)\|2as a function of RSO for different choices of boundary conditions with variousnumbers of conducting channels. Solid and dashed lines correspond to the results for periodic and hard-wall boundary conditions imposed along the width ( Y) direction, respectively. Red and black lines correspond to the results for the magnets directed along Yand Xdirections, respectively. The number of conducting channels are increased from panels (a)–(d) by varying the width of the channel.Parameters: n S=1×1012cm−2,Lch=0.5μm,PC∼1. we have included the effect of the spin-relaxation process phenomenologically through an exponential decay functionwith respect to the value of λ sf. We found that the difference in magnitude of VX,P(p-p) and /Delta1VYreduces with shorter λsf. This is because high kYmodes relax faster than the low kY modes which would reduce the angular averaging effect and the signal would become more and more one dimensional.However, even there we found, assuming point contact, thata reasonable agreement with the experiment requires a spincoherence length much smaller than the value mentioned inthe experiment. The details are explained in Appendix D. Another possibility for the discrepancy is that the P Cwe use was calibrated for the spin-valve signals obtained withY-directed magnets. The same magnets when forced into the X direction for the oscillatory signals may have a higher effectiveP C. However, to account fully for the discrepancy we needed to increase the PCvalue to ∼10% for X-directed magnets while keeping ∼5% for the Y-directed magnets. VII. SPIN PRECESSION IN MAGNETIC FIELD An important question to address would be whether it is possible to control the precession of spins, in an RSO-coupledballistic channel like the one in Ref. 9, with a magnetic ﬁeld of magnitude similar to the values used in observing Hanle 165306-6VOLTAGE-CONTROLLED SPIN PRECESSION PHYSICAL REVIEW B 84, 165306 (2011) signals in a diffusive channel having no RSO interaction. Typically Hanle signals are measured by applying a magneticﬁeld of a few hundred Gauss perpendicular to the directionof the injected spin direction. For example, when the injectedspins are either XorYpolarized in the x-ytransport plane, a magnetic ﬁeld B z, applied in the zdirection, will create a spin precession and generate Hanle voltage at the detector. Anexpression for the Hanle voltage due to varying B z, similar to the one due to varying RSO, for the structure shown inFig.1(a)can be obtained by including the 1 2gμBBzterm in the Hamiltonian in Eq. ( 1) and following a derivation procedure similar to that in Appendixes BandC, which ﬁnally gives /Delta1VX/similarequalaC0 3π+C0/radicalBig 2πθBz Lcos/parenleftBig θBz L+π 4/parenrightBig , (9a) /Delta1VY/similarequal2aC0 3π+bC0/radicalBig 2πθBz Lcos/parenleftBig θBz L+π 4/parenrightBig , (9b) where θBz L≈2m∗ ¯h2k0/radicalBigg (αk0)2+/parenleftbigg1 2gμBBz/parenrightbigg2 L, a=4c2 Bz/parenleftbig 1+c2 Bz/parenrightbig2,b=/parenleftbig 1−c2 Bz/parenrightbig2 (1+c2 Bz)2, cBz=αk0 1 2gμBBz+/radicalBig/parenleftbig1 2gμBBz/parenrightbig2+(αk0)2. Here the effect of the vector magnetic potential is neglected, which limits our analysis to small magnetic ﬁelds far from thequantum Hall effect regime. In a ballistic channel, similar to that in a diffusive channel, the oscillatory Hanle signal decays with an increasing B zdue to a spread in transit times of electrons, although the originof such spread in the former is not the same as it is in thelatter. In a diffusive channel the mentioned spread originatesfrom the differences in transit times corresponding to differentrandom-walk trajectories taken by the electrons while goingfrom the injector to the detector. In a ballistic channel it wouldbe the differences in transit times corresponding to differentelectronic transverse modes that would give rise to such spread.The consequent decay in the Hanle signal for an increasingmagnetic ﬁeld in a ballistic channel appears quantitatively through the dependence of the quantity θ Bz L, in the modulating prefactor of the oscillating terms in Eqs. ( 9a) and ( 9b), on Bz. At the same time its dependence on RSO strength α,i n addition to its dependence on Bz, suggests that Bzneeds to be larger in a material having strong α[such as an InAs quantum well (QW), where α∼8×10−12eV m] than that in a material having weak αto create any signiﬁcant change in θBz Lleading to a signiﬁcant decay in the Hanle signal. Such a scenariocan be interpreted in terms of an internal magnetic ﬁeldB RSO=2αk0/gμBdue to RSO, which acts in addition to Bz. This observation suggests that for Bzto have any effect on the Hanle signal its magnitude needs to be comparabletoB RSO. In the case of an InAs QW, for k0∼4×108m−1 (corresponding to a carrier density ns=2.7×1012cm−2) and \|g\|=15,9BRSO∼8 T, which necessitates the exertion of avery large Bzthat might even take the material into a quantum Hall regime. However, by tuning α38,39andns, the magnitude ofBRSOand, hence, the required Bzcan be made smaller. Indeed, it would be interesting to look for a Hanle signal aswell as a RSO-modulated signal in the same structure whereαcan be tuned through ∼0 to a higher value. In that case, one has to be careful about choosing the parameters to observeHanle oscillation near α∼0. For example, to rotate the spins by 2πwithin a ballistic channel length of L∼4μm, which can be obtained in InAs 2DEG samples, 40and a carrier density of∼1011cm−2,gμBBzhas to be varied from 0 to 0 .2m e V . On the other hand, to get a similar 2 πrotation by varying RSO with the same parameters, αneeds to be varied from 0 to 1×10−12eV m. VIII. VOLTAGE-CONTROLLED MAGNETIZATION REVERSAL Finally, we discuss the possibility of controlling mag- netization reversal by modulating spin-current41(Is), which could be an alternative way to demonstrate voltage controlledspin-precession effect. Recently, in lateral structures withmetallic channels, spin-torque 42,43induced magnetization re- versal has been demonstrated by pure spin-current.44,45Similar switching mechanism is yet to be seen in semiconductorlateral structures, although spin-torque switching is alreadyseen in semiconductor vertical structures (magnetic tunneljunctions). 46,47Moreover, spin-orbit coupling effect in spin- torque is a relatively new area where conventional spin-torquetheories are extended to include spin-orbit coupling inside theferromagnet 48–50and a few experiments51,52seem to show this. Here we discuss how one might design experiments involvingchannels with strong spin orbit coupling. Gate control oversuch channels would allow modulation of the RSO couplingcoefﬁcient which in turn would modulate the magnitude anddirection of I sin the channel. Reversing the sign of Isin the channel could in principle allow for reversible switchingof magnetization of a magnet on which this I sis exerting a torque. In Fig. 8we are showing different components of Isin x,y, and zdirections for each of the three different magnet conﬁgurations, namely, FM-FM, x,F M - F M , y, and FM-FM, z. They are calculated within our NEGF based model using thegeneralized current operator described in Ref. 53. We provide the equation in Appendix E. The variation of spin-current components with αimplies that the spin-torque exerted on the detecting magnet can be controlled, and thereby a switchingevent, with a gate. Moreover, since any component of I swhich is perpendicular to the direction of magnetization is going toexert a torque on the magnet, it might also be possible to switchthe magnet in a desired direction with a careful tuning of α. The magnitude of I scan be estimated from the equivalent circuit model shown in Appendix A, which is Is=/Delta1V PC2GMGm GM+Gm(per unit area), for the magnets in collinear conﬁguration. Here /Delta1Vis the spin-valve voltage ( VP−VAP).PCandGM(m)are related to the interface of the magnet to be switched. We alsoprovide a comparison of I scalculated from this expression against the same with that of the NEGF based model for variousP CandGC=GM+Gmvalues in Appendix E, for further clariﬁcation. 165306-7ZAINUDDIN, HONG, SIDDIQUI, SRINIV ASAN, AND DATTA PHYSICAL REVIEW B 84, 165306 (2011) (a)(b) (c) (d) FIG. 8. (Color online) (top) Spin-current components in x(solid black), y(dotted red), and z(dashed blue) directions with RSO calculated from the NEGF based model for magnets along x(FM-FM, x)( a )y(FM-FM, y)( b )a n d z(FM-FM, z) (c) directions. Parameters: Lch=1.65μm, LCi=0.2μm,LCd=0.25μm,W=8μm,PC∼5% and GM+Gm=4×1010/Omega1−1m−2. Charge current is maintained at 1 mA. (bottom) Different switching mechanisms are shown schematically in (d). Considering the spin-valve structure in Ref. 9,t h e Is at the detecting magnet would be ∼2.4×106Am−2,f o r /Delta1V∼6μV,PC∼5% and ( GM+Gm)∼4×1010/Omega1−1m−2. This value of Is, at present, would be few orders of magnitude smaller than those of metal based structures, for example thestructure in Ref. 44, mainly due to the smaller number of conducting modes and spin-polarization in semiconductorsthan in metals. As a result, if I sis insufﬁcient to switch a regular magnet, one could consider magnetic semiconductorssince a lower switching current was reported in Refs. 46,47 for the latter compared to a regular magnet. But, in general,ifI sis lower than the critical limit for easy-axis switching [Fig. 8(d), left], which is usually believed to be given by Eq. 18 in Ref. 41for monodomain magnets, one could also use hard-axis switching [Fig. 8(d), right]. A possible scheme could be to follow a two step process similar to the one introduced byBenett. 54In the ﬁrst step, the magnet is taken into its hard axis through an external means (e.g., B-ﬁeld), where it is unstable,and in the next step a small tilt due to the I sinduced spin-torque will tip the magnet to one of its easy axes once the externalﬁeld is removed. This idea of two step switching process isalso being used in various contexts. 51,55,56But here also the Isinduced torque has to overcome the thermal noise which depends on the temperature of operation57along with other few nonideal factors (see the Supplementary Information inRef. 56for a detailed analysis of hard axis switching). IX. CONCLUSION In summary, we have studied spin transport through a channel with RSO coupling. We provide both a simpleanalytical model as well as an NEGF-based model to calculatethe spin voltages in a nonlocal spin-valve structure. We discussthe effect of having extended contacts in addition to the effect of angular spectrum averaging of electrons ﬂowingin a two-dimensional channel. The extended nature of thecontacts is found to be detrimental to the oscillatory behaviorof spin signals. The model is used to analyze a recentexperiment 9and, the results are summarized in Sec. Vin addition, the Hanle oscillation in the presence of RSO is alsodiscussed. Finally, the possibility for gate controlled switchingof magnetization through spin-current modulation is discussedwhich could extend and quantify the ‘Datta-Das’ effect forvoltage controlled spin-precession. ACKNOWLEDGMENTS This work is supported by the Ofﬁce of Naval Research under Grant No. N0014-06-1-2005 and the Network forComputational Nanotechnology (NCN). Also ANMZ wouldlike to thank Angik Sarkar and Behtash-behinein for helpfuldiscussions. APPENDIX A: NONLOCAL VOLTAGE In this section we explain the nonlocal voltage calculated using an NEGF-based approach with a simple circuit model.As mentioned earlier that the contacts are adjusted to ﬁt theexperimental contact conductances, an equivalent conductancenetwork can be drawn for the structure shown in Fig. 1(b)[see Fig.9(a)]. Here the spin-dependent contact conductances are connected to their respective spin-dependent channels for twospins in the semiconducting 2DEG. The semi-inﬁnite leads/Sigma1 L(R)at two ends connect the two spin channels and thereby act as a spin-ﬂip conductance of ( q2/h)Meach. In Fig. 9(b)we show the nonlocal voltage /Delta1V=[μ3P−μ3AP]/qfrom the 165306-8VOLTAGE-CONTROLLED SPIN PRECESSION PHYSICAL REVIEW B 84, 165306 (2011) FIG. 9. (Color online) (a) Simple circuit model to illustrate the method of calculating nonlocal voltage in the spin-valve setup inNEGF, (b) nonlocal voltage /Delta1V calculated from the simple circuit model (solid lines) in (a) compared against the same from the NEGF-based model (circles) as a function of contact conductanceG C=(GM+Gm)/Omega1−1m−2andPC. Parameters: LCi=0.2μm, LCi=0.25μm, separation between the contacts 1 .65μm, carrier density nS=2.7×1012cm−2.NEGF model compared against the simple circuit model as a function of contact conductance GC=(GM+Gm)/Omega1−1m−2 for different contact polarization PC. In all cases we main- tained a current of 1 mA between contacts 1 and 2. We seethat the simple circuit agrees well with the NEGF-based model.One thing to note is that we are capturing the effect of large un-etched regions at two ends with /Sigma1 Land/Sigma1R. Since these are act- ing as spin-ﬂip conductances we believe that etching out theseregions would have signiﬁcantly improved the spin signals. 58 APPENDIX B: DERIVATION OF EQS. ( 5a) AND ( 5b) We start by writing the incident state {ψi}with a linear combination of {ψ+}and{ψ−} {ψi}=A{ψ+}+B{ψ−}=[/Psi1]/braceleftbigg A B/bracerightbigg , (B1) where [ /Psi1]≡[{ψ+}{ψ−}]. After propagating from x=0t o x=L, the ﬁnal state is written as ( θ+(−)=kX+(−)L) /angbracketleftψf/angbracketright=Aexp(iθ+){ψ+}+Bexp(iθ−){ψ−} =[/Psi1]/bracketleftbigg exp(iθ+)0 0e x p ( iθ−)/bracketrightbigg/braceleftbigg A B/bracerightbigg . (B2) Hence we can write, {ψf}=[t]{ψi}, with [t]=[/Psi1]/bracketleftbigg exp(iθ+)0 0e x p ( iθ−)/bracketrightbigg [/Psi1]−1, (B3) where [/Psi1]=1√ 2/bracketleftbigg 11 exp(iφ+)−exp(iφ−)/bracketrightbigg . (B4) Multiplying out the matrices leads to [t]≡/bracketleftbigg exp(iφ++iθ−)+exp(iφ−+iθ+)e x p ( iθ+)−exp(iθ−) {exp(iθ+)−exp(iθ−)}exp(iφ++iφ−)e x p ( iφ++iθ+)+exp(iφ−+iθ−)/bracketrightbigg exp(iφ+)+exp(iφ−). (B5) Setting φ+≈φ−≡φ(this amounts to ignoring the nonorthogonality of the +and−states), the expression simpliﬁes to [t]≡/bracketleftbigg exp(iθ+)+exp(iθ−) {exp(iθ+)−exp(iθ−)}exp(−iφ) {exp(iθ+)−exp(iθ−)}exp(iφ)e x p ( iθ+)+exp(iθ−)/bracketrightbigg 2. (B6) Note that [ t] can also be written as [t]=exp[i(θ++θ−)/2] exp( i[/vectorσ·ˆn]/Delta1θ/2), (B7) where /Delta1θ≡θ+−θ−=θL/√ 1−s2and ˆnis a unit vec- tor in the direction of the effective magnetic ﬁeld: ˆn= cosφˆx+sinφˆy. This form is intuitively appealing, showing the transmission [ t] as a product of a simple phase-shift exp{i(θ++θ−)/2}and a rotation around ˆnby/Delta1θ.A l s of o r the magnetic ﬁeld applied along the zdirection, which is in this case perpendicular to the x-ytransport plane, givingrise to the Hanle effect (discussed in the paper earlier), the transmission function remains the same except that ˆnnow becomes ˆn=cosφˆx+sinφˆy+gμBBz 2αk0ˆz. Forz-polarized contacts in the parallel conﬁguration, tzz=/braceleftbig10/bracerightbig [t]/braceleftbigg 1 0/bracerightbigg =t11, Tzz=\|t11\|2≈1+cos(θ+−θ−) 2, (B8) 165306-9ZAINUDDIN, HONG, SIDDIQUI, SRINIV ASAN, AND DATTA PHYSICAL REVIEW B 84, 165306 (2011) and in the antiparallel conﬁguration, t¯zz=/braceleftbig01/bracerightbig [t]/braceleftbigg 1 0/bracerightbigg =t21, T¯zz=\|t21\|2≈1−cos(θ+−θ−) 2. (B9) Forx-polarized contacts in the parallel conﬁguration, txx=1 2/braceleftbig11/bracerightbig [t]/braceleftbigg 1 1/bracerightbigg =t11+t22+t12+t21 2, Txx∼/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1+cosφ)e x p (iθ +)+(1−cosφ)e x p (θ−) 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ∼(1+cos2φ)+sin2φcos(θ+−θ−) 2, (B10) and in the antiparallel conﬁguration, t¯xx=1 2/braceleftbig1−1/bracerightbig [t]/braceleftbigg 1 1/bracerightbigg =t11−t22+t12−t21 2, T¯xx∼(1−cos2φ)−sin2φcos(θ+−θ−) 2. (B11) Fory-polarized contacts in the parallel conﬁguration, tyy=1 2/braceleftbig1−i/bracerightbig [t]/braceleftbigg 1 +i/bracerightbigg =t11+t22+i(t12−t21) 2, Tyy∼/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1+sinφ)e x p (iθ +)+(1−sinφ)e x p (iθ−) 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ∼(1+sin2φ)+cos2φcos(θ+−θ−) 2, (B12) and in the antiparallel conﬁguration, t¯yy=1 2/braceleftbig1+i/bracerightbig [t]/braceleftbigg 1 +i/bracerightbigg =t11−t22+i(t12+t21) 2, T¯yy∼(1−sin2φ)−cos2φcos(θ+−θ−) 2. (B13) Noting that tan φ≈−kX/kYandk2 0≈k2 X+k2 Ywe can write /Delta1VZ∼Tzz−T¯zz=C0cos⎛ ⎝2m∗αL ¯h2k0/radicalBig k2 0−k2 Y⎞ ⎠, /Delta1VX∼Txx−T¯xx =C0⎧ ⎨ ⎩k2 Y k2 0+/parenleftbigg 1−k2 Y k2 0/parenrightbigg cos⎛ ⎝2m∗αL ¯h2k0/radicalBig k2 0−k2 Y⎞ ⎠⎫ ⎬ ⎭, /Delta1VY∼Tyy−T¯yy =C0⎧ ⎨ ⎩/parenleftbigg 1−k2 Y k2 0/parenrightbigg +k2 Y k2 0cos⎛ ⎝2m∗αL ¯h2k0/radicalBig k2 0−k2 Y⎞ ⎠⎫ ⎬ ⎭. (B14)APPENDIX C: DERIVATION OF EQS. ( 7a) AND ( 7b) From Eqs. ( 6) and ( 5a), /Delta1VX=1 π/integraldisplay1 0dsV X0(s)=B π/integraldisplay1 0dss2 +B πRe/braceleftbigg/integraldisplay1 0ds(1−s2)e x p/parenleftbiggiθL√ 1−s2/parenrightbigg/bracerightbigg . Noting that the phase has a stationary point at s=0,25we expand it in Taylor’s series around s=0 to obtain /Delta1VX/similarequalC0 3π +C0 πRe/bracketleftbigg/integraldisplay0+/epsilon1 0ds(1−s2)e x p/braceleftbigg iθL/parenleftbigg 1+s2 2/parenrightbigg/bracerightbigg/bracketrightbigg /similarequalC0 3π+C0 πRe/braceleftbigg exp(iθL)/integraldisplay∞ 0dsexp/parenleftbigg iθLs2 2/parenrightbigg/bracerightbigg =C0 3π+C0 πRe/braceleftBigg exp(iθL)exp/parenleftbig iπ 4/parenrightbig √2θL/Gamma1/parenleftbigg1 2/parenrightbigg/bracerightBigg =C0 3π+C0√2πθLcos/parenleftBig θL+π 4/parenrightBig , as stated in Eq. ( 7a). (a) (b) FIG. 10. (Color online) (a) Numerical calculation (squares) of Eqs. ( 6)a n d( 5a) vs analytical expression (solid) in Eq. ( 7a)a sa function of α. (b) Numerical calculation (squares) of Eqs. ( 6)a n d (5b) vs analytical expression (solid) in Eq. ( 7b), as a function of α. 165306-10VOLTAGE-CONTROLLED SPIN PRECESSION PHYSICAL REVIEW B 84, 165306 (2011) Similarly from Eqs. ( 6) and ( 5b), /Delta1VY=C0 π/integraldisplay1 0ds(1−s2) +C0 πRe/braceleftbigg/integraldisplay1 0dss2exp/parenleftbiggiθL√ 1−s2/parenrightbigg/bracerightbigg /similarequal2C0 3π, as stated in Eq. ( 7b). In Fig. 10we compare stationary phase approximation with direct numerical integration. APPENDIX D: SPIN COHERENCE OF HIGHER MODES To include the effect of a ﬁnite spin-coherence length, we ﬁrst express our mode-space expressions for spin voltages[Eqs. ( 6)] in real space. The mode-space variables can be mapped onto the real space [see Fig. 11(a) ] in the following way: s=k Y k0=sinθ=y R, /Delta1VX=/integraldisplay+∞ −∞dy/braceleftbiggy2 R2+L2 R2cos/parenleftbigg2m∗αR ¯h2/parenrightbigg/bracerightbiggL2 R3, /Delta1VY=/integraldisplay+∞ −∞dy/braceleftbiggL2 R2+y2 R2cos/parenleftbigg2m∗αR ¯h2/parenrightbigg/bracerightbiggL2 R3. (D1) In Fig. 11(b) we see that the real-space expressions in Eqs. ( D1) are in exact agreement with the mode-space expression inEqs. ( 6). From Fig. 11(a) we also realize that higher k Ywill travel a larger length [ R(kY)>R(kY=0)=L] in the channel to reach the detecting contact. So a ﬁnite spin-coherence lengthλ sfshould gradually suppress the contribution from higher kY (a) (b) FIG. 11. (Color online) (a) Spin transport in real space where electrons of certain spin at higher kYmode travels a distance Rat an angle θwhich is greater than the distance Lthey travel at mode kY=0. (b) Spin voltages /Delta1VXand/Delta1VYfrom Eqs. ( D1)( s o l i da n d dashed) and ( 6) (circles). Parameters are same as in Fig. 2.in/Delta1VX,Y. Including an exponential decay term representing the suppression of higher kYwithλsf,E q s .( D1) can be rewritten as /Delta1VX=/integraldisplay+∞ −∞dy/braceleftbiggy2 R2+L2 R2cos/parenleftbigg2m∗αR ¯h2/parenrightbigg/bracerightbigg ×L2 R3exp/parenleftbigg−L λsf/parenrightbigg , /Delta1VY=/integraldisplay+∞ −∞dy/braceleftbiggL2 R2+y2 R2cos/parenleftbigg2m∗αR ¯h2/parenrightbigg/bracerightbigg ×L2 R3exp/parenleftbigg−L λsf/parenrightbigg . (D2) In Fig. 12we see that spin voltages are reduced in amplitude asλsfreduces from a value of λsf=2μm reported in the experiment9to a value of λsf=0.5μm. But in addition we note that the ratio /Delta1VY//Delta1V X(p-p) is reduced from its point-contact limit in the shorter λsfcase. To clarify the latter we show /Delta1VY(λsf=0.5μm) scaled up in amplitude to the value at /Delta1VY(λsf=2μm) within the experimental limit∼(8–13) ×10−12eV m by multiplying both /Delta1VY andVX,P=/Delta1VX/2f o r λsf=0.5μm with the factor f= /Delta1VY(λsf=2μm) /Delta1VY(λsf=0.5μm)[see Fig. 12(b) ]. However, from Fig. 12(b) we also realize that to make /Delta1VY≈VX,P(p-p) we need λsfto be much smaller compared to the value mentioned in the experiment.9 (a) (b) FIG. 12. (Color online) (a) Spin voltages /Delta1VYandVX,P= /Delta1VX/2 at different spin-coherence length λsf, (b) same voltages in (a) plotted within the experimental range of αin Ref. 9, and for the purpose of comparison, in all cases /Delta1VYat different λsfis scaled up in amplitude to the value at λsf=2μm and accordingly /Delta1VXare multiplied with the same scaling factors, respectively. 165306-11ZAINUDDIN, HONG, SIDDIQUI, SRINIV ASAN, AND DATTA PHYSICAL REVIEW B 84, 165306 (2011) APPENDIX E: SPIN CURRENT To obtain the spin current at a given terminal, we used the current operator described in Ref. 53(see Eq. 8.6.5, p. 317). The expression for spin-current density at any grid point canbe written as /vectorI s(ky)=Re(Tr(i/vectorσ[G(ky)/Sigma1in(ky)−/Sigma1in(ky)G†(ky) −/Sigma1i(ky)Gn(ky)+Gn(ky)/Sigma1† i(ky)])). (E1) Here, /vectorσis the Pauli spin matrix, /Sigma1inis the in-scattering func- tion,Gis Green’s function, Gn(≡−iG<) is the correlation function whose diagonal elements are electron density, and/Sigma1 iis the contact self-energy ( i=1, 2, 3, 4). Equation ( E1) is integrated over all the transverse modes ( ky) to obtain the total spin current at any energy. In Fig. 13we compare the spin-current ( Is,0) from Eqn. ( E1) with the expression Is,0=/Delta1V 0 PC2GMGm GM+Gmatα=0 and a good agreement is found. Here, Isincreases with the interfacial conductance of the magnet for a given spin-polarization and charge current. FIG. 13. (Color online) Comparison of spin-current ﬂowing into the detecting magnet in Fig. 1 calculated from NEGF equation(circles) against the same obtained from the equivalent circuit model (solid) in Appendix A, as a function of detecting contact conductance G Cd=(GM+Gm)a n dPC. Parameters: LCi=0.2μm, LCd=0.25μm. 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PhysRevB.103.205411.pdf	PHYSICAL REVIEW B 103, 205411 (2021) Editors’ Suggestion Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k·ptwo-band model B. Andrei Bernevig,1,Zhi-Da Song,1Nicolas Regnault,1,2and Biao Lian1,† 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 2Laboratoire de Physique de l’Ecole normale superieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France (Received 28 October 2020; revised 15 April 2021; accepted 16 April 2021; published 11 May 2021) We investigate the twisted bilayer graphene (TBG) model of Bistritzer and MacDonald (BM) [Bistritzer and MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011) ] to obtain an analytic understanding of its energetics and wave functions needed for many-body calculations. We provide an approximation scheme for the wave functionsof the BM model, which ﬁrst elucidates why the BM K M-point centered original calculation containing only four plane waves provides a good analytical value for the ﬁrst magic angle ( θM≈1◦). The approximation scheme also elucidates why most of the many-body matrix elements in the Coulomb Hamiltonian projected to the activebands can be neglected. By applying our approximation scheme at the ﬁrst magic angle to a /Gamma1 M-point centered model of six plane waves, we analytically understand the reason for the small /Gamma1M-point gap between the active and passive bands in the isotropic limit w0=w1. Furthermore, we analytically calculate the group velocities of the passive bands in the isotropic limit, and show that they are almost doubly degenerate, even away from the/Gamma1Mpoint, where no symmetry forces them to be. Furthermore, moving away from the /Gamma1MandKMpoints, we provide an explicit analytical perturbative understanding as to why the TBG bands are ﬂat at the ﬁrst magicangle, despite the ﬁrst magic angle is deﬁned by only requiring a vanishing K M-point Dirac velocity. We derive analytically a connected “magic manifold” w1=2/radicalbig 1+w2 0−/radicalbig 2+3w2 0, on which the bands remain extremely ﬂat as w0is tuned between the isotropic ( w0=w1) and chiral ( w0=0) limits. We analytically show why going away from the isotropic limit by making w0less (but not larger) than w1increases the /Gamma1M-point gap between the active and the passive bands. Finally, by perturbation theory, we provide an analytic /Gamma1Mpoint k·ptwo-band model that reproduces the TBG band structure and eigenstates within a certain w0,w1parameter range. Further reﬁnement of this model are discussed, which suggest a possible faithful representation of the TBG bands by atwo-band /Gamma1 Mpoint k·pmodel in the full w0,w1parameter range. DOI: 10.1103/PhysRevB.103.205411 I. INTRODUCTION The interacting phases in twisted bilayer graphene (TBG) are one of the most important new discoveries of the last fewyears in condensed matter physics [ 1–111]. The theoretical prediction that interacting phases would appear in this sys-tem was made based on the appearance of ﬂat bands in thenoninteracting Bistritzer-MacDonald (BM) Hamiltonian [ 1]. This Hamiltonian is at the starting point of the understandingof every aspect of strongly correlated TBG (and other moirésystems) physics [ 2–27]. Remarkably, it even predicts quite accurately the so-called “magic angles” at which the bandsbecome ﬂat, and is versatile enough to accommodate thepresence of different hoppings in between the AAand the AB stacking regions of the moiré lattice. The BM Hamiltonianis in fact a large class of k·pmodels, which we will call BM-like models, where translational symmetry emerges at asmall twist angle even though the actual sample does not havean exact lattice commensuration. bernevig@princeton.edu †biao@princeton.eduThis paper is the ﬁrst of a series of six papers on TBG [107–111], for which we present a short summary here. In this paper we investigate the spectra and matrix elements ofthe single-particle BM model by studying the k·pexpan- sion of the BM model at /Gamma1 Mpoint of the moiré Brillouin zone. In TBG II [ 107] we prove that the BM model with the particle-hole (PH) symmetry deﬁned in Ref. [ 43] is always stable topological , rather than fragile topological as revealed without PH symmetry [ 43–45,76]. We further study TBG with Coulomb interactions in Refs. [ 108–111]. In TBG III [ 108] we show that the TBG interaction Hamiltonian projectedinto any number of bands is always a Kang-Vafek type [ 71] positive semi-deﬁnite Hamiltonian (PSDH), and genericallyexhibit an enlarged U(4) symmetry in the ﬂat band limitdue to the PH symmetry. This U(4) symmetry for the lowesteight bands (two per spin valley) was previously shown inRef. [ 72]. We further reveal two chiral-ﬂat limits, in both of which the symmetry is further enhanced into U(4) ×U(4) for any number of ﬂat bands. The U(4) ×U(4) symmetry for the lowest eight ﬂat bands in the ﬁrst chiral limit was ﬁrstdiscovered in Ref. [ 72]. With kinetic energy, the symmetry in the chiral limits will be lowered into U(4). TBG in the secondchiral limit is also proved in TBG II [ 107] to be a perfect 2469-9950/2021/103(20)/205411(42) 205411-1 ©2021 American Physical SocietyBERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) metal without single-particle gaps [ 112]. In TBG IV [ 109], under a condition called ﬂat metric condition (FMC) whichis deﬁned in this paper [Eq. ( 20)], we derive a series of exact insulator ground /low-energy states of the TBG PSDH within the lowest eight bands at integer ﬁllings in the ﬁrst chiral-ﬂatlimit and even ﬁllings in the nonchiral-ﬂat limit, which canbe understood as U(4) ×U(4) or U(4) ferromagnets. We also examine their perturbations away from these limits. In the ﬁrstchiral-ﬂat limit, we ﬁnd exactly degenerate ground states ofChern numbers ν C=4−\|ν\|,2−\|ν\|,...,\|ν\|−4 at integer ﬁlling νrelative to the charge neutrality. Away from the chiral limit, we ﬁnd the Chern number 0 ( ±1) state is favored at even (odd) ﬁllings. With kinetic energy further turned on,up to second order perturbations, these states are intervalleycoherent if their Chern number \|ν C\|<4−\|ν\|, and are valley polarized if \|νC\|=4−\|ν\|. At even ﬁllings, this agrees with the K-IVC state proposed in Ref. [ 72]. At ﬁllings ν=±1,±2, we also predict a ﬁrst order phase transition from the lowestto the highest Chern number states in magnetic ﬁeld, whichis supported by evidences in recent experiments [ 14–16,24– 27]. In TBG V [ 110] we further derive a series of exact charge 0,±1,±2 excited states in the (ﬁrst) chiral-ﬂat and nonchiral- ﬂat limits. In particular, the exact charge neutral excitationsinclude the Goldstone modes (which are quadratic). This al-lows us to predict the charge gaps and Goldstone stiffness.In the last paper of our series TBG VI [ 111] we present a full Hilbert space exact diagonalization (ED) study at ﬁllingsν=−3,−2,−1 of the projected TBG Hamiltonian in the lowest eight bands. In the (ﬁrst) chiral-ﬂat and nonchiral-ﬂatlimits, our ED calculation with FMC veriﬁed that the exactground states we derived in TBG IV [ 109] are the only ground states at nonzero integer ﬁllings. We further show that in the(ﬁrst) chiral-ﬂat limit, the exact charge ±1 excitations we found in TBG V [ 110] are the lowest excitations for almost all nonzero integer ﬁllings. In the nonchiral case with kineticenergy, we ﬁnd the ν=−3 ground state to be Chern number ±1 insulators at small w 0/w1[ratio of AAandABinterlayer hoppings, see Eq. ( 4)], while undergoing a phase transition to other phases at large w0/w1, in agreement with the recent density matrix renormalization group studies [ 80,81]. Forν= −2, while we are restricted within the fully valley polarized sectors, we ﬁnd the ground state prefers ferromagnetic (spinsinglet) in the nonchiral-ﬂat (chiral-nonﬂat) limit, in agree-ment with the perturbation analysis in Refs. [ 72,109]. To date, most of our understanding of the BM-like models comes from numerical calculations of the ﬂat bands, whichcan be performed in a momentum lattice of many moiré Bril-louin zones, with a cutoff on their number. The ﬁner details ofthe band structure so far seem to be peculiarities that vary withdifferent twisting angles. However, with the advent of interact-ing calculations, where the Coulomb interaction is projectedinto the active, ﬂat bands of TBG, a deeper, analytic under-standing of the ﬂat bands in TBG is needed. In particular,there is a clear need for an understanding of what quantitativeand qualitative properties are not band-structure details. Sofar the analytic methods have produced the following results:by solving a model with only four plane waves (momentumspace lattice sites, on which the BM is deﬁned), Bistritzer andMacDonald [ 1] found a value for the twist angle for which the Dirac velocity at the K Mmoiré point vanishes. This is FIG. 1. Several quantitative characteristics of the Bistritzer and MacDonald model that require explanation. In particular, an analytic understanding of the active band ﬂatness is available only in thechiral limit w 0=0. However, the band is very ﬂat far away from the chiral limit. Several other features of the bands are pointed out. called the magic angle. In fact, the full band away from the KMpoint is ﬂat, a fact which is not analytically understood. A further analytic result is the discovery that, in a limit ofvanishing AAhopping, there are angles for which the band isexactly ﬂat. This limit, called the chiral limit [37], has an extra chiral symmetry. However, it is not analytically knownwhy the bands remain ﬂat in the whole range of AAcoupling between the isotropic limit ( AA=ABcoupling) and the chiral limit. We note that the realistic magic angle TBG is in be-tween these two limits due to lattice relaxations [ 113–116]. A last analytical result is the proof that, when particle-holesymmetry is maintained in the BM model [ 43], the graphene active bands are topological [ 42–47,76,117,118]. This leaves a large series of unanswered questions. Rather than listing them in writing, we ﬁnd it more intuitive to vi-sualize the questions in a plot of the band structure of TBGin the isotropic limit at the magic angle and away from it,towards the chiral limit. In Fig. 1we plot the TBG low-energy band structure in the moiré Brillouin zone, and the questionsthat will be answered in the current paper. To distinguish themwith the high symmetry points ( /Gamma1,M,K,K /prime) of the monolayer graphene Brillouin zone (BZ), we use a subindex Mto denote the high symmetry points ( /Gamma1M,MM,KM,K/prime M)o ft h em o i r éB Z (MBZ). Some salient features of this band structure are: (1)In the isotropic limit, around the ﬁrst magic angle, it is hardto obtain two separate ﬂat bands; it is hard to stabilize thegap to passive bands over a wide range of angles smallerthan the ﬁrst magic angle. In fact, Ref. [ 43] computes the active bands separated regions as a function of twist angle, andﬁnds a large region of gapless phases around the ﬁrst magicangle. (2) The passive bands in the isotropic limit are almost doubly degenerate, even away from the /Gamma1 Mpoint, where no symmetry forces them to be. Moreover, their group velocitiesseem very high, i.e., they are very dispersive. (3) While theanalytic calculation of the magic angle [ 1] shows that the Dirac velocity vanishes in the isotropic limit at AA-coupling w 0=1/√ 3 (in the appropriate units, see below), it does not explain why the band is so ﬂat even away from the Dirac point,for example on the K M-/Gamma1M-MM-KMline. (4) Away from the isotropic limit, while keeping w1=1/√ 3, the gap between the active and passive bands increases immediately, while thebandwidth of the active bands does not increase. (5) The ﬂatbands remain ﬂat, over the wide range of w 0∈[0,1/√ 3], 205411-2TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) FIG. 2. Matrix elements needed for the interacting problem. Speciﬁcally, the form factors M(η) m,n(k,q+G)=/summationtext α/summationtext Q∈Q±u∗ Q−G,α;mη(k+q)uQ,α;nη(k) of the Coulomb interaction are needed. They correspond to the overlap of the Bloch state at momentum k, on the momentum lattice Q,uQ,α;nη(k) with the Bloch state at momentum q+kon the momentum lattice Q+G, u∗ Q−G,α;mη(k+q). Here m,nare band indices, α=A,Bis the graphene sublattice index, ηis the valley index, Gis a reciprocal momentum, and Qis the honeycomb momentum lattice generated by the moiré reciprocal vectors shown in this ﬁgure. from chiral to the isotropic limit. Also, our observation (6) in Fig.1shows that since the gap between the active and passive bands is large in the chiral limit compared to the bandwidth ofactive bands, a possible k·pHamiltonian for the active bands might be possible. A further motivation for the analytic investigation of the TBG Bistritzer-MacDonald model is to understandthe behavior of the matrix elements M (η) m,n(k,q+G)=/summationtext α/summationtext Q∈Q±u∗ Q−G,α;mη(k+q)uQ,α;nη(k) as a function of G, which we call the form factor (oroverlap matrix ). These are the overlaps of different Bloch states in the TBG momentumspace lattice (see Fig. 2) and their behavior is important for the form factors of the interacting problem [ 108,109]. These will be of crucial importance for the many-body matrix elements[107,111] as well as for justifying the approximations made in obtaining exact analytic expressions for the many-bodyground states [ 109] and their excitations [ 110]. We provide an analytic answer to all the above questions and observations. We will focus on the vicinity of the ﬁrstmagic angle. We ﬁrst provide an analytic perturbative frame-work in which to understand the BM model, and show thatfor the two ﬂat bands around the ﬁrst magic angle, only avery small number of momentum shells is needed. We justifyour framework analytically, and check it numerically. Thisperturbative framework also shows that M (η) m,n(k,q+G)i s negligible for Gmore than two times the moiré BZ (MBZ) momentum—at the ﬁrst magic angle, irrespective of k,q. We then provide two approximate models involving a verysmall number of momentum lattice sites, the tripod model ( K M centered, also discussed in Ref. [ 1]), and a new, /Gamma1Mcenteredmodel. The tripod model captures the physics around the KM point (but not around the /Gamma1Mpoint), and we show that the Dirac velocity vanishes when w1=1/√ 3 irrespective of w0. The/Gamma1Mcentered model captures the physics around the /Gamma1M point extremely well, as well as the physics around the KM point. Moreover, an approximation of the /Gamma1Mcentered model with only six plane waves, which we call the hexagon model,has an analytic sixfold exact degeneracy at the /Gamma1 Mpoint in the isotropic limit w1=w0=1/√ 3, which is the reason for feature (1) in Fig. 1. By performing a further perturbation theory in these six degenerate bands away from the /Gamma1Mpoint, we obtain a model with an exact ﬂat band at zero energy onthe/Gamma1 M-KMline, and almost ﬂat bands on the /Gamma1M-MMline, answering (3) in Fig. 1. In the same perturbative model, the velocity of the dispersive bands—which can be shown to bedegenerate—can be computed and found to be the same withthe bare Dirac velocity (with some directional dependence),answering (2) in Fig. 1. Away from the isotropic limit, our per- turbative model, which we still show to be valid for w 0/lessorequalslantw1 (but not for w0/greatermuchw1), allows for ﬁnding the analytic energy expressions at the /Gamma1Mpoint, and seeing a strong dependence onw0answering (3) in Fig. 1. At the same time, one can obtain allthe eigenstates of the hexagon model at the /Gamma1M point after tedious algebra, which can serve as the starting point of a perturbative k·pexpansion of the two-active band Hamiltonians. With this, we provide an approximate two-bandcontinuum model of the active bands, and ﬁnd the mani- foldw 1(w0)=2/radicalBig 1+w2 0−/radicalBig 2+3w2 0withw0∈[0,1/√ 3], where the bandwidth of the active bands is the smallest, in this approximation. The radius of convergence for the k·p expansion is great around the /Gamma1Mpoint but is not particularly good around the KMpoint for all w0,w1parameters, but can be improved by adding more shells perturbatively, which weleave for further work. A series of useful matrix elementconventions are also provided. II. NEW PERTURBATION THEORY FRAMEWORK FOR LOW-ENERGY STATES IN k·pCONTINUUM MODELS In this section we provide a general perturbation theory for thek·pBM-type Hamiltonians that exist in moiré lattices. We exemplify it in the TBG BM model, but the generalcharacteristics of this model allow this perturbation theoryto be generalizable to other moiré system. The TBG BMHamiltonian is deﬁned on a momentum lattice of plane waves.Its symmetries and expressions have been extensively exposedin the literature (including in our paper [ 107]), and we only brieﬂy mention them here for consistency. We ﬁrst deﬁnek θ=2\|K\|sin(θ/2) as the momentum difference between K point of the lower layer and Kpoint of the upper layer of TBG, and denote the Dirac Fermi velocity of monolayergraphene as v F. To make the TBG BM model dimensionless, we measure all the energies in units of vFkθ, and measure all the momentum in units of kθ. Namely, any quantity E (k) with the dimension of energy (momentum) is redeﬁned as dimensionless parameters E→E/(vFkθ),k→k/kθ. (1) 205411-3BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) FIG. 3. (a) The Brillouin zones of two graphene layers. The gray solid line and red dots represent the BZ and Dirac cones of the top layer, and the gray dashed line and blue dots represent theBZ and Dirac cones of the bottom layer. (b) The lattice formed by adding q 1,2,3iteratively. Red and blue circles represent Q+and Q−, respectively. (c) Relation of graphene BZ and moiré BZ in the commensurate case. Here we take the graphene BZ reciprocal vectors b1=3bM1+2bM2,b2=−2bM1+5bM2. We will then work with the dimensionless single particle Hamiltonian for the valley η=+, which in the second quan- tized form reads [ 1,43,107] ˆH(+) 0=/summationdisplay k∈MBZ/summationdisplay sαβ/summationdisplay QQ/prime∈Q±HQα,Q/primeβ(k)c† k,Q,+,αsck,Q/prime,+,βs,(2) where MBZ stands for moiré BZ, the momentum kis mea- sured from the center ( /Gamma1Mas shown in Fig. 3) point of the MBZ, s=↑,↓is spin, and α,β denotes the two indices ofA,Bsublattices. Here the dimensionless ﬁrst quantized Hamiltonian HQα,Q/primeβ(k)i sg i v e nb y HQα,Q/primeβ(k)=δQ,Q/prime[(k−Q)·σ]αβ +3/summationdisplay j=1/parenleftbig δQ−Q/prime,qj+δQ/prime−Q,qj/parenrightbig (Tj)αβ,(3) where Tj=w0σ0+w1/bracketleftbigg cos2π 3(j−1)σx+sin2π 3(j−1)σy/bracketrightbigg , (4) withw0being the interlayer AAhopping and w1being the interlayer ABhopping, σ=(σx,σy), and σ0,x,y,zstand for the identity and Pauli matrices in the two-dimensional sub-lattice space. ktakes value in MBZ, and k=0corresponds to the/Gamma1 Mpoint in the moiré BZ. We deﬁne q1as the difference between the Kmomentum of the lower layer of graphene and the rotated Kof the upper layer, and q2andq3as the C3z andC−1 3zrotations of q1(see Fig. 3). The moiré reciprocal latticeQ0is then generated by the moiré reciprocal vectors bM1=q3−q1andbM2=q3−q2, which contains the origin. We also deﬁne Q+=q1+Q0andQ−=−q1+Q0as the moiré reciprocal lattices shifted by q1and−q1, respectively. Q∈Q±is then in the combined momentum lattice Q+⊕Q−, which is a honeycomb lattice. For valley η=+, the fermion degrees of freedom c† k,Q,+,αswith Q∈Q+andQ∈Q−are from layers 1 and 2, respectively. Since energy and momen-tum are measured in units of vFkθand kθ, we have that \|qi\|=1, and both w0andw1are dimensionless energies. It should be noticed that, for inﬁnite cutoff in the lattice Q, we have c† k+bMi,Q,ηαs=c† k,Q−bMi,ηαs/negationslash=c† k,Q,ηαs, as proved in Refs. [ 43,107]. In practice, we always choose a ﬁnite cutoff /Lambda1QforQ(/Lambda1Qdenotes the set of Qsites kept). We note that in the Hamiltonian ( 3) we have adopted the zero angle approximation [ 1,107], namely, we have approxi- mated the Dirac kinetic energy k·σ±θ/2(±for layers 1 and 2, respectively) as k·σ, where σ±θ/2are the Pauli matrices σrotated as a vector by angle ±θ/2 about the zaxis. With the zero angle approximation, the Hamiltonian ( 3) acquires a unitary particle-hole symmetry [ 43], which is studied in detail in another paper of ours [ 107]. In the absence of the zero angle approximation, the particle-hole symmetry is only broken upto 1% [ 107] near the ﬁrst magic angle, and is exact in the (ﬁrst) chiral limit w 0=0[106]. We also note that different variants of the TBG BM model exist in the literature, which furtherinclude nonlocal tunnelings, interlayer strains, or kdependent tunnelings [ 119–122]. However, we shall only focus on the BM model in Eq. ( 3) in this paper. It is the cutoff /Lambda1 Qthat we are after : we need to quantize what is the proper cutoff Q∈/Lambda1Qin order to obtain a fast convergence of the Hamiltonian. We devise a perturbationtheory which gives us the error of taking a given cutoff inthe diagonalization of the Hamiltonian in Eq. ( 3). For the ﬁrst magic angle we will see that this cutoff is particularly small,allowing for analytic results. A. Setting up the shell numbering of the momentum lattice and Hamiltonian We now consider the question of what momentum shell cutoff /Lambda1Qshould we keep in performing a perturbation theory of the BM model. In effect, considering an inﬁnite cutoff fortheQlattice, we can build the BM model centered around any point k 0in the MBZ, by sending k→k−k0,Q→Q−k0 (5) in Eq. ( 3); however, it makes sense to pick k0as a high- symmetry point in the MBZ, and try to impose a ﬁnite cutoff/Lambda1 Qin the shifted lattice Q. Two important shifted lattices k0 can be envisioned, see Fig. 4. These lattices will be developed and analyzed in Sec. III; here we only focus on the perturba- tive framework of Eq. ( 3), which is the same for either of these two lattices (and in fact, on a lattice with any k0center). We introduce a numbering of the “shells” in momentum space Qon this lattice. In the KM-centered lattice [Fig. 4(b)] which is a set of hexagonal lattices but centered at one of the“sites” (the K Mpoint, corresponding to the choice k0=−q1), the sites of shells nare denoted Ani, with n−1 being the minimal graph distance (minimal number of bonds traveledon the honeycomb lattice from one site to another) from thecenter A1 1, while igoes to the number of Qsites with the same graph distance n−1. The truncation in Qcorresponds to a truncation in the graph distance n−1. In particular, with lattice Qcentered at the KMpoint, the momentum hopping Tiin the BM Hamiltonian Eq. ( 3) then only happens between sites in two different shells n↔n+1 but not between sites in the same shell. The simplest version of this model, with a 205411-4TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) FIG. 4. Lattices centered around momentum k0on which one can calculate the TBG Hamiltonian. (a) The hexagon centered model(/Gamma1 M-centered model, in which we build “shells” by graph distance from the hexagon centered at the /Gamma1Mpoint. The circles denote the different shells, although going to a larger graph distance will makethe circles into hexagons. There are two different types of subshells in each shell, the Aand the Bsubshells in this model. The Ashells connect to the Bshells, but the Asites within a shell also contain hoppings within themselves. The Bsites hop only to Asites. (b) The triangle centered at the K M-point model in which we build shells by graph distance from the KMpoint centered at the origin. The circles denote the different shells, although going to a larger graph distance will make the circles into triangles. There are only one type of shells, theAshells in this model. The Asites within a shell do not hop to other sites within each shell. truncation at n=2, with sites A11andA21,A22,A23was used by Bistritzer and MacDonald to show the presence of a “magicangle”—deﬁned as the angle for which the Dirac velocityvanishes. We call this the tripod model. This truncated model(the tripod model) does not respect the exact C 2xsymmetry, although it becomes asymptotically good as more shells areadded. The magic angle also does not explain analytically theﬂatness of bands, since it only considers the velocity vanishingat one point K M. However, the value obtained by BM [ 1]f o r the ﬁrst magic angle is impressive: despite considering onlytwo shells (four sites), and despite obtaining this angle fromthe vanishing velocity of bands at only one point ( K Min the BZ), the bands do not change much after adding more shells.Moreover, they are ﬂat throughout the whole BZ, not onlyaround the K Mpoint. The Dirac velocity also does not change considerably upon introducing more shells. We now introduced a yet unsolved lattice, the /Gamma1M-centered model in Fig. 4(a), which corresponds to the choice k0=0in Eq. ( 5). This model, which we call /Gamma1Mcentered was not solved by BM, perhaps because of the larger Hilbert space dimensionthan the K M-centered one. It however respects all the symme- tries of the TBG (except Bloch periodicity, which is only fullyrecovered in the large cutoff /Lambda1 Qlimit) at any ﬁnite number of shells and not only in the large shell number limit. While notrelevant for the perturbation theory described here, we ﬁnd ituseful to partition one shell nin the /Gamma1 M-centered lattice into two subshells AnandBn, each of which has 6 nsites. The ﬁrst shell is A1 given by the six corners of the ﬁrst MBZ; then we deﬁne Anas the shell with a minimal graph distance 2( n−1) to shell A1, and Bnas the shell with a minimal graph distance2n−1t os h e l l A1.AniandBniwhere i=1,..., 6nis the index of sites in the subshell AnorBn. The partitioning in subshells is useful when we realize that the hopping Tiin the BM Hamiltonian Eq. ( 3) can only happen between AnandBn shells, between BnandAn+1 shells, and within anAnshell, butnotwithin the same Bnshell. In Appendix Awe provide an explicit efﬁcient way of implementing the scattering matrixelements of the BM Hamiltonian Eq. ( 3), and provide a block matrix form of the BM Hamiltonian in the shell basis deﬁnedhere. Written compactly, the expanded matrix elements inAppendix Aread (H An,An)Q1,Q2=/braceleftbigg TjifQ1−Q2=±qj, 0 otherwise(6) for the hopping terms, and similarly for HAn,Bnwhere Q1,Q2 are the initial and ﬁnal momenta in their respective shells. Fi- nally for k-dependent dispersion we take a linearized model: (Hk,An/Bn)Q1Q2=(k−Q1)·σδQ1Q2, (7) which is accurate in the small-angle low-energy approxima- tions we make. Recall that the momentum is measured in unitsofk θ=2\|K\|sin(θ/2) with θthe twist angle, while the energy (and Hamiltonian matrix elements) are in units of vFkθ.W e may now write the dimensionless BM Hamiltonian H(k)i n Eq. ( 3) in block form as H=⎛ ⎜⎜⎜⎝H kA1+HA1,A1HA1,B1 0 ··· H† A1,B1HkB1 HB1,A2 ··· 0 H† B1,A2HkA2+HA2,A2... ... 0......⎞ ⎟⎟⎟⎠ ≡⎛ ⎜⎜⎜⎜⎜⎜⎝M 1N1 00 ... 00 N† 1M2N2 0... 00 0 N† 2M3N3... 00 ... ... ... ... ... ... ... 0000 ... ML−1NL−1 0000 ... N† L−1ML⎞ ⎟⎟⎟⎟⎟⎟⎠,(8) where Lis the shell cutoff that we choose. In the above equation, the M,Nblock form of the matrix is a schematic, in the sense that both the /Gamma1 M-centered model Fig. 4(a)and the KM-centered model Fig. 4(b)can be written in this form, albeit with different Mn,Nn,n=1,...L. Also, each Mndepends onk, which for space purposes was not explicitly written in Eq. ( 8). B. General Hamiltonian perturbation for bands close to zero energy with ramp-up term In general, Eq. ( 8), with generic matrices Mi,Nirepresents anyHamiltonian with short range hopping (here on a momen- tum lattice), and not much progress can be made. However,for our BM Hamiltonians, we know several facts which render them special: (1) The Hamiltonian in Eq. ( 3) has very ﬂat bands, at close to zero energy \|E\|/lessorequalslant0.02v Fkθ. Numerically, the energy of the ﬂat bands /lessmuchw1andw0, since numerically we know that the ﬁrst magic angle happens at w1(orw0) around 1 /√ 3. (2) The block-diagonal terms Mncontain a ramping up di- agonal term Eq. ( 7), of eigenvalue \|k−Q\|.T h e kmomentum 205411-5BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) runs in the ﬁrst MBZ, which means that \|k\|/lessorequalslant1. Since Qfor thenth shell is proportional to n, higher order shells contribute larger terms to the diagonal of the BM Hamiltonian. We now show that, despite the higher shell diagonal terms being the largest in the BM Hamiltonian, they contribute ex-ponentially little to the physics of the low-energy (ﬂat) bands.This should be a generic property of the moiré systems. TheM n,NnBlock Hamiltonian Eq. ( 8) acts on the spinor wave function ( ψ1,ψ2,ψ3,...,ψ L−1,ψL) where the /Psi1n’s are the components of the wave function on the shells n= 1,2,3,..., L−1,L, and Lis the cutoff shell. Notice that they likely have different dimensions: in the /Gamma1M-centered model, ψ1is a 12-dimensional spinor (six vertices of the ﬁrst hexagon momentum Q—for subshell A1i,i=1,..., 6—times 2 for theαβindices), ψ2is also a 12-dimensional spinor (six legs coming out of the vertices of the ﬁrst hexagon momentumQ—for subshell B1 i,i=1,..., 6—times 2 for the αβin-dices), ψ3is a 24-dimensional spinor (12 vertices of the momentum Q—for subshell A2i,i=1,..., 12—times 2 for theαβindices), and ψ4is also a 24-dimensional spinor (12 legs coming out of the vertices of the previous momentumshell Q—for subshell B2 i,i=1,..., 12—times 2 for the αβ indices), etc. To diagonalize Hwe write down the action of H in Eq. ( 8) on the wave function ψ=(ψ1,ψ2,...,ψ L): M1ψ1+N1ψ2=Eψ1, ... N† n−1ψn−1+Mnψn+Nnψn+1=Eψn, ... N† L−1ψL−1+MLψL=EψL, (9) and solve iteratively for ψ1starting from the lastshell. We ﬁnd that ψL=(E−ML)−1N† L−1ψL−1, ψL−1=[E−ML−1−NL−1(E−ML)−1N† L−1]−1N† L−2ψL−2, ψL−2={E−ML−2−NL−2[E−ML−1−NL−1(E−ML)−1N† L−1]−1N† L−2}−1N† L−3ψL−3 ... (10) We notice three main properties: (1)Mn≈nfor large shells n/greatermuch1 is generically an invert- ible matrix with eigenvalues of the order ±nfor the nth shell. This is because Mnis just the ramp-up term, block diagonal with the diagonal being ( k−Q)·σforQin the nth subshell ofBtype; if the subshell is of Atype, then the matrix is still generically invertible, as it contains the diagonal term (k−Q)·σplus the small (since w0,w1≈1/√ 3) hopping Hamiltonian HAn,An(see Appendix A). Nonetheless, because the magnitude of the momentum term increases linearly with\|k−Q\|/greatermuch1 for momenta Qoutside the ﬁrst two shells n>2, while the hopping term has constant magnitude, H kAndomi- nates the BM Hamiltonian. (2) Since we are interested in the ﬂat bands E≈0(E≈ 0.02 in vFkθ), we can expand in E/Mnterms, especially after the ﬁrst n>2 shells, and keep only the zeroth and ﬁrst order terms. We use (E−M)−1≈−M−1−M−1EM−1(11) if the eigenvalues of Eare smaller than those of ME/lessmuchM. (3) For the ﬁrst magic angle, the off-diagonal terms are also smaller than the diagonal terms, for the ﬁrst magic angle,and for \|Q\|/greaterorequalslant2, we have that N n−1M−1 nN† n−1/lessmuch1f o r n/greaterorequalslant2 and for w0,w1≈1/√ 3 (more details on this will be given later). With these approximations, we obtain that the general so- lution is ψn=(EPn−Mn+Rn)−1N† n−1ψn−1m, (12) where Pnis deﬁned recursively as PL−n=NL−nM−1 L−n+1PL−n+1M−1 L−n+1N† L−n+1 (13)subject to PL=1 and Rnis RL−n=NL−nM−1 L−n+1RL−n+1M−1 L−n+1N† L−n +NL−nM−1 L−n+1N† L−n, (14) with RL=0,RL−1=NL−1M−1 LN† L−1,PL=1. This continues until the ﬁrst shell, where we have ψ2=[EP2−M2+R2]−1N† 1ψ1. (15) C. Form factors and overlaps from the general perturbation framework Notice that the wave function for the E≈0 bands decays exponentially (ψn≈1 nψn−1) over the momentum space Qas we go to larger and larger shells. This is due to the inversesin the linear ramp-up term M n∝nof Eq. ( 12) [a consequence of the Qterm in Eq. ( 7)]. This has immediate implications for the form factors. For example, in Refs. [ 108–110]w eh a v et o compute M(η) m,n(k,q+G)=/summationdisplay α/summationdisplay Q∈Q±u∗ Q−G,α;mη(k+q)uQ,α;nη(k) (16) form,nthe indices of the active bands, and for different G∈Q0. Notice that almost all \|G\|/lessorequalslant\|Q\|change the shells (with the exception of \|G\|=1): if Qis in the subshell An/Bn, while Gis of order \|G\|/greaterorequalslant2\|/tildewideb1\|with/tildewideb1the moire reciprocal vector, then Q−Gisnotin the subshell An/Bn. Hence, considering \|Q−G\|>\|Q\|without loss of generality, we have, for 2 \|/tildewideb1\|/lessorequalslant\|G\|/lessorequalslant\|Q\|: u∗ Q−G,α;mη(k+q)/lessorequalslant\|Q\|! \|(Q−G)\|!u∗ Q,α;nη(k+q) (17) 205411-6TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) for any m,n. Since the wave functions of the active ﬂat bands at (or close to) zero energy exponentially decay with the shelldistance from the center we can approximate M (η) m,n(k,q+G)≈/summationdisplay α/summationdisplay QorQ−G∈An,Bn,n/lessorequalslantn0 ×u∗ Q−G,α;mη(k+q)uQ,α;nη(k),(18) with n0a cutoff. For any k,q, the (maximum of any com- ponents of the) wave functions on the subshells A2,B2a r e of order 1 /3!,2!/4! times the components of the wave func- tions on the subshells A1,B1. Hence we can restrict to small shell cutoff in the calculation of form factor matrices n0=1 (meaning only the subshells A1,B1 are taken into account), while paying at most a 15% error. Conservatively, we can keepn 0=2 and pay a much smaller error <3%. Next, we ask for which Gmomenta are the function M(η) m,n(k,q+G) considerably small. Employing Eq. ( 17), we see that M(η) m,n(k,q+G) falls off exponentially with increas- ingG, and certainly for \|G\|>2\|/tildewideb1\|they are negligible. The largest contributions are for G=0 and for \|G\|=\|/tildewideb1\|, i.e., for Gbeing one of the fundamental reciprocal lattice vectors. We hence make the approximation: M(η) m,n(k,q+G)≈/summationdisplay α/summationdisplay QorQ−G∈A1,B1u∗ Q−G,α;mη(k+q) ×uQ,α;nη(k)/parenleftbig δG,0+δ\|G\|,\|/tildewideb1\|/parenrightbig . (19) This is one of the most important results of our perturbative scheme. In Refs. [ 108–111] we employ heavily an approx- imation called the “ﬂat metric condition” (see [ 110]f o rt h e link between this condition and the quantum metric tensor)to show that some exact eigenstates of the interacting Hamil-tonian are in fact, ground states. The ﬂat metric conditionrequires that Flat metric condition: M (η) m,n(k,G)=ξ(G)δm,n.(20) In light of our ﬁndings on the matrix elements Eq. ( 19), we see that the ﬂat metric condition is satisﬁed for \|G\|/greaterorequalslant2\|/tildewideb1\|, as the matrix element vanishes M(η) m,n(k,G)≈0→ξ(G)≈0 for\|G\|/greaterorequalslant2\|/tildewideb1\|.F o r G=0, the condition Eq. ( 20)i sa l - ways satisﬁed, even without anyapproximation Eq. ( 19), as it represents the block wave function orthonormality. Hence,the ﬂat metric condition Eq. ( 20) is almost always satis- ﬁed, with one exception: the only requirement in the ﬂatmetric condition is M (η) m,n(k,G)=ξ(G)δm,nfor\|G\|=\|/tildewideb1\|. There are six Gvectors that satisfy this condition, namely G=±/tildewideb1,±/tildewideb2,±(/tildewideb2−/tildewideb1). The overlaps are all related by symmetry. In Fig. 5(a) we plot the eigenvalues at q=0o ft h e M†M matrix. We see clearly that these eigenvalues are virtuallynegligible for \|G\|/greaterorequalslant2˜b i, and that for \|G\|=\| ˜bi\|they are at most 1 /3o ft h ev a l u ef o r \|G\|=0. D. Further application of general perturbation framework to TBG While Eqs. ( 12)t o( 14) represent the general perturbation theory of Hamiltonians with a linear (growing) ramping termfor almost zero energy bands, we need further simpliﬁcationsFIG. 5. The magnitude of the form factor (overlap ma- trix) M(η=+)(k,q+G), calculated for w0=0.4745 and w1= 0.5931. (a) The colored dots are the Gvectors we consider in M(η=+)(k,q+G). Different colors represent different length of G. (b) The eigenvalues of M(η=+)†(k,q+G)M(η=+)(k,q+G) as func- tions of k. In the left and right panels we choose q=0andq=1 2kM, respectively, where kMis the MMmomentum in the moiré BZ. to practically apply them to the TBG problem. However, the form of the ( k−Q)·σ+HAn,An, which is not nicely invert- ible (although it can be inverted), and the form of HBn−1,An (see Appendix Afor the notation of these matrix elements), which is not diagonal, makes the matrix manipulations dif-ﬁcult, and unfeasible analytically for more than two shells.Hence further approximations are necessary in order to makeanalytic progress. First, we want to estimate the order of magnitudes of P L−n andRL−nterms in Eqs. ( 13) and ( 14). Recall that our energy is measured in units of vFkθ, which for angle of 1◦is around 180 meV . We note the following facts: (1) The diagonal terms HkAnare of order \|n−\|k\|\|,w h i l e theHkBnare of order \|n+1−\|k\|\|with kin the ﬁrst Bril- louin zone ( \|k\|<1). Therefore, HkB1/greaterorequalslant1,HkA2>1, and all the other HkAn,HkBnare considerably larger. This shows that Mn+1in Eq. ( 8) is of order n, due to the dominance of the momentum term in relation to the hopping terms. (2)HAnBn andHBn−1Anare proportional to Tj,s ot h e ya r e of order α=w1/(vFkθ). Near the ﬁrst magic angle ( θ≈1◦, orw1≈1/√ 3 in units of vFkθ),α≈0.6/θwith the angle in degrees (hence smaller angles have larger α). By Eq. ( 8), this means the matrices Nn∼HBnAn+1are of order α. These facts allow us to estimate Pnin Eq. ( 13): Pn∝\|Nn\|2\|Mn+1\|−2\|Pn+1\|+1 ∝(vFkθ)2α2(vFkθn)−2\|Pn+1\|+1 =α2n−2\|Pn+1\|+1. (21) Forn/greaterorequalslant2 therefore Pn=1 up to a correction term no more thanα2n−2<0.1. Therefore we are justiﬁed (up to a 10% error) of neglecting all Pn,n/greaterorequalslant2 terms. Similarly, using these estimates and substituting into Rnin Eq. ( 14), we see that \|Rn\|/lessorequalslantα2 (n+1)2\|Rn+1\|+(vFkθ)α2 (n+1) /lessorequalslant0.04\|Rn+1\|+0.09(vFkθ) (22) when n/greaterorequalslant2 at the ﬁrst magic angle α≈0.6. Again this will allow us to neglect the Rnterm for n/greaterorequalslant2. This means that shells after the ﬁrst one can be neglected at the ﬁrst magic angle. More generally, only the ﬁrst Nshells will be needed for understanding the Nth magic angle. In order to see the validity of the above approximations more concretely, it is instructive to write down the two-shell 205411-7BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) (A1,B1,A2,B2) Hamiltonian explicitly, and estimate the con- tribution of the second shell. A1 and B1 are 12-dimensional Hilbert spaces while A2 and B2 are 24-dimensional Hilbertspaces, see Appendix A. Further shells are only a gen- eralization of the ones below. We write the eigenvalueequation: (HkA1+HA1,A1)ψA1+HA1,B1ψB1=EψA1, H† A1,B1ψA1+HkB1ψB1+HB1,A2ψA2=EψB1, H† B1,A2ψB1+(HkA2+HA2,A2)ψA2+HA2,B2ψB2=EψA2, HA2,B2ψA2+HkB2ψB2=EψB2. (23) We integrate out from the outer shell to the ﬁrst to obtain the equations (HkA1+HA1,A1)ψA1+HA1,B1ψB1=EψA1, H† A1,B1ψA1+{HkB1+HB1,A2[E−(HkA2+HA2,A2)−HA2,B2(E−HkB2)−1H† A2,B2]−1H† B1,A2}ψB1=EψB1, (24) and to ﬁnally obtain EψA1=(HkA1+HA1,A1+HA1,B1{E−HkB1−HB1,A2[E−(HkA2+HA2,A2) −HA2,B2(E−HkB2)−1H† A2,B2]−1H† B1,A2}−1H† A1,B1)ψA1. (25) Solving the above equation would give us the eigenstate en- ergies, as well as the reduced eigenstate wave functions ψA1. However, even for two shells above, this is not analyticallysolvable, hence further approximations are necessary. We im-plement our approximations here. (1) First, focusing on the ﬁrst magic angle of 1 ◦,f r o m numerical calculations we know that the energy of the activebands \|E\|<60 meV ≈0.3v Fkθ. Hence EH−1 kB1<0.3 and fur- thermore EH−1 kBn,EH−1 kAn<0.3n−1forn/greaterorequalslant2. This justiﬁes the approximation around the ﬁrst magic angle: (E−HkB1)−1=−H−1 kB1−EH−2 kB1(26) and (E−Hk(A,B)n)−1=−H−1 k(A,B)n−EH−2 k(A,B)n(27) forn/greaterorequalslant2.Region of validity of this approximation: this approximation is independent on w0,w1, the interlayer tun- neling. It, however, depends on θas well as on the energy range of the bands we are trying to approximate. For ex-ample, for θ=0.3 ◦, an energy range \|E\|/lessorequalslant60 meV would mean that \|E/vFkθ\|/lessorequalslant1. This gives \|EH−1 kBn\|,\|EH−1 kAn\|<n−1 and hence we would only be able to neglect shells larger than n=3. In particular, in order to obtain convergence for bands of energy Eat angle θ, we can neglect the shells at distance n=2+[E/vFkθ] (where xmeans the integer part ofx). Hence, as the twist angle is decreased, and if we are interested in obtaining convergent results for bands at a ﬁxedenergy, we will need to increase our shell cutoff to obtain afaithful representation of the energy bands. If we keep thenumber of shells ﬁxed, we will obtain faithful (meaning ingood agreement with the inﬁnite cutoff limit) energies onlyfor bands in a smaller energy window as we decrease thetwist angle. Notice that this approximation does not depend onw 0,w1and hence it is notan approximation in the interlayer coupling. (2) The second approximation is regarding w0,w1: be- causeα=w1/vFkθ≈0.6 at the ﬁrst magic angle, we can doa perturbation expansion in the powers of α. We remark that Hk,Bn,Hk,An∼n/greatermuchαforn/greaterorequalslant2 and θ=1◦. We also remark thatH−1 k,B1α/lessorequalslant0.6 for all kin the ﬁrst BZ (the largest value, H−1 KM,B1α=0.6 is reached for kat the KMcorner of the moiré BZ). As such, we ﬁnd terms of the following form scale as HAnBnH−1 kA,BnH† AnBn∼α2n−1(n/greaterorequalslant2), HBn−1AnH−1 kA,BnH† Bn−1An∼α2n−1(n/greaterorequalslant2), HB1A1H−1 kB1H† B1A1∼α2. (28) With Eqs. ( 26)–(28) one can see that in Eq. ( 25) the leading order contributions of the terms involving the second shell(A2,B2) are roughly ∼\|H A1,B1\|2\|HkB1\|−2\|HB1,A2\|2\|HkA2\|−1∼ α4/2∼0.05. It is hence a relatively good approximation to neglect shells higher than n=1 for angle θ=1◦. For exam- ple, at the KMpoint, neglecting the n=2 shell will induce a less than 10% percent error. Region of validity of this ap- proximation: Notice that as the twist angle is decreased, α increases. In general, the relative error of the nth shell is roughly HBn−1AnH−2 kAnH† Bn−1An∼α2/n2, so we can neglect the shells for which n/greatermuchαwhere /greatermuchshould be considered twice the value of α. Hence, for an angle of 0 .5◦(α=1.2) we can neglect all shells greater than 3, etc. For angle 1 /nof the ﬁrst magic angle we can neglect all shells above n+1. All the above remarks, which were made for the /Gamma1M- centered model, can also be extended to the KM-centered model in Fig. 4(b). In particular, the tripod model in Fig. 8(b), containing only the A1,A2 shells, is a good approximation to the inﬁnite model around the Dirac point, giving the correctﬁrst magic angle. E. Further approximation of the one-shell ( A1,B1) Hamiltonian in TBG In the previous section we claimed that, remarkably, a relatively good approximation of the low-energy BM modelcan be obtained by taking a cutoff of one shell, where we 205411-8TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) only consider the ﬁrst A subshell and the ﬁrst Bsubshell. The eigenvalue equations are (HkA1+HA1,A1)ψA1+HA1,B1ψB1=EψA1, H† A1,B1ψA1+HkB1ψB1=EψB1, (29) which can be solved for ψB1to obtain ψB1=(E−HkB1)−1H† A1,B1ψA1. (30) Eliminating ψB1we ﬁnd the eigenvalue equation for the ﬁrst Ashell (which includes the coupling to the ﬁrst Bshell): [HkA1+HA1,A1+HA1,B1(E−HkB1)−1H† A1,B1]ψA1=EψA1. (31) This is a 12 ×12 nonlinear eigenvalue equation in E.A tt h i s point we will make a few assumptions in order to simplify theeigenvalue equation. In particular, we would like to make thisa linear matrix eigenvalue equation. Since we are interestedclose to E=0 we may assume that E/lessmuchH kB1. This allows us to treat the Bshell perturbatively, obtaining /parenleftbig HkA1+HA1,A1−HA1,B1H−1 kB1H† A1,B1/parenrightbig ψA1=EψA1.(32) Our approximation Hamiltonian is HApprox1 (k)=HkA1+HA1,A1−HA1,B1H−1 kB1H† A1,B1.(33) We note that HApprox1 (k) is a further perturbative Hamiltonian for the n=1 shell ( A1,B1). For ksmall, around the /Gamma1M point, we expect this to be an excellent approximation of then=1 shell Hamiltonian [and since the n=1 shell is a good approximation of the inﬁnite shell, then HApprox1 (k)i s expected to be an excellent approximation of the full BMHamiltonian close to the /Gamma1 Mpoint]. The good approximation is expected to deteriorate as kgets closer to the boundary of the MBZ, since HA1,B1H−1 kB1H† A1,B1increases as kapproaches the MBZ boundary. This is because H−1 kB1has larger terms as kapproaches the MBZ boundary. However, we expect still moderate qualitative agreement with the BM Hamiltonian.We also predict that taking two shells ( A1,B1,A2,B2) would give an extremely good approximation to the inﬁnite shell BMmodel. F. Numerical conﬁrmation of our perturbation scheme The series of approximations performed in Secs. II Dand II Eare thoroughly numerically veriﬁed at length in Appendix B. We here present only a small part of the highlights. In Fig. 6we present the n=1,2,3 shell (one shell is made out ofA,Bsubshells) results of the BM Hamiltonian in Eq. ( 3), for two values of w0,w1. We virtually see no change between two and three shells (see also Appendix B), we verify this for higher shells and for many more values of w0,w1, around— and away from, within some manifolds ( w0,w1) explained in Sec. III—the magic angle. Hence our perturbation framework works well, and conﬁrms the irrelevance of the n>2 shells. The n=1 shell band structure in Fig. 6, while in excellent agreement to the n=2 shells around the /Gamma1Mpoint, contains some quantitative differences from the n=2 shell (equal to the inﬁnite cutoff) away from the /Gamma1Mpoint. However, the generic aspects of the band structure, low bandwidth, almostexact degeneracy (at n=1, becoming exact with machine precision in the n>2) at the K Mpoint are still present even FIG. 6. Comparison of the different cutoff shells of the BM model in Eq. ( 3), for two values of w0,w1. (more data available in Appendix B). We clearly see that n=2 has reached the inﬁnite cutoff limit (the band structure does not change from n=2a n d n=3, while n=1 (only one shell, A1,B1 subshells) shows excellent agree- ment around the /Gamma1Mpoint, and good agreement even away from the /Gamma1Mpoint (for example see the second row). in the n=1 case, as our perturbative framework predicts in Secs. II DandII E. Our approximations of the n=1 shell Hamiltonian in Sec. II Ehave brought us to the perturbative HApprox1 (k)i n Eq. ( 33). Around the ﬁrst magic angle we claim that this Hamiltonian is a good approximation to the band structure ofthen=1 shell, especially away from MBZ boundary. The n=1 shell is only a 15% difference on the n=2 shell and that the n=1 shell is within 5% of the thermodynamic limit, we then make the approximation that H Approx1 explains the band structure of TBG within about 20%. The approximationsare visually presented in Fig. 8(a), and the band structure of the approximation H Approx1 to the one-shell Hamiltonian is presented in Fig. 7. We see that around the /Gamma1Mpoint, the Hamiltonian HApprox1 (k)i nE q .( 33) has a very good match to the BM Hamiltonian Eq. ( 3), while away from the /Gamma1M point the qualitative agreement, small bandwidth, crossing at (close to) KM(the crossing is at KMfor the inﬁnite shell FIG. 7. Band structure of the approximation HApprox1 (k)t ot h e one-shell Hamiltonian, versus the inﬁnite limit approximation, forthew 0=w1=1/√ 3 magic point. The n=1 shell Hamiltonian band structure is undistinguishable from HApprox1 (k), and is plotted in Appendix B. 205411-9BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) FIG. 8. The two types of approximate models used for analytics. (a) The one-shell ( A1,B1) model which we have theoretically argued and numerically substantiated to represent a good approximation for values w0,w1/lessorequalslant1/√ 3. Analytically we will ﬁrst solve it by perturbation theory around the hexagon model, which involves onlytheA1 sites. The shell B1 will be added perturbatively to obtain H Approx1 (k)i nE q .( 33). (A second way to solve for this Hamiltonian will be presented later.) (b) The tripod model, which involves the two shells A1 (also known as the KMpoint) and A2. Due to the same considerations as for the /Gamma1M-centered model, this should be a good approximation for the inﬁnite shell model for w0,w1/lessorequalslant1/√ 3. This is the same model as solved by Bistritzer and MacDonald [ 1]. We ﬁnd that the magic angle at which the Dirac velocity vanishes at theK Mpoint is given by w1=1/√ 3,∀w0. cutoff by symmetry, but can deviate slightly from KMfor ﬁnite cutoff). In Appendix Bwe present many different tests which conﬁrm all aspects of our perturbative framework, differ-ent twist angles and AA,ABcoupling. We test the n= 1,2,3,4,... shells, and also further test the validity of the approximation H Approx1 (k)t ot h e n=1 shell Hamiltonian in Sec. II E. III. ANALYTIC CALCULATIONS ON THE BM MODEL: STORY OF TWO LATTICES We will now analytically study the approximate Hamil- tonian in Eq. ( 33). While in Secs. II D and II E we have focused on the /Gamma1M-centered lattice, the same approximations can be made in the KM-centered lattice, where the HApprox1 (k) changes to HApprox1 (k)=HkA1+HA1,A2H−1 kA2H† A1,A2.T h et w o types of approximations are schematically shown in Fig. 8 in the /Gamma1M- and KM-centered lattice. First, we start with the tripod model [Fig. 8(b)] to extend the Bistritzer-MacDonald calculation of the magic angle in the isotropic limit and ﬁnda “ﬁrst magic manifold,” where the Dirac velocity vanishes inthe tripod model (and is very close to vanishing in the inﬁniteshell BM model). We then solve the 1-shell /Gamma1 M-centered model [Fig. 8(a)], deﬁned by Eq. ( 33), which is supposed to faithfully describe TBG at and above the magic angle, asproved in Sec. II. This is a 12 ×12 Hamiltonian, with no known analytic solutions, formed by shell 1: A1,B1, where theBpart of the ﬁrst shell B1 is taken into account perturba- tively, as H A1,B1H−1 kB1H† A1,B1.A. The KM-centered “tripod model” and the ﬁrst magic manifold For completeness we solve for the magic angle in the model in the KM-centered model of Fig. 4by taking only four sites, one in shell A1 and three in shell A2. We call this approximation, depicted in Fig. 8(b), the tripod model. This model is identical to the one solved by Bistritzer andMacDonald in the isotropic limit. However, we will solvefor the Dirac velocity away from the isotropic limit, to ﬁnda manifold w 1(w0) where the Dirac velocity vanishes. The tripod Hamiltonian HTri(k,w0,w1), with kmeasured from the KMpoint, reads HTri(k,w0,w1) =⎛ ⎜⎝k·σ T1(w0,w1) T2(w0,w1) T3(w0,w1) T1(w0,w1)( k−q1)·σ 00 T2(w0,w1)0( k−q2)·σ 0 T3(w0,w1)0 0( k−q3)·σ⎞ ⎟⎠. (34) The Schrödinger equation in the basis ( ψA11,ψA21,ψA22,ψA23) reads k·σψ A11+/summationtext i=1,2,3Ti(w0,w1)ψA2i=EψA11,(35) TiψA11+(k−qi)·σψ A2i=EψA2i,i=1,2,3. (36) From the second equation we ﬁnd ψA2i=[E−(k−qi)· σi]−1TiψA11and plug it into the ﬁrst equation to obtain EψA11=k·σψ A11+3/summationdisplay i=1TiE+(k−qi)·σ E2−(k−qi)2TiψA2i ≈k·σψ A11−3/summationdisplay i=1Ti[(E+(k−qi)·σ] ×(1+2k·qi)TiψA2i, (37) where we neglect E2as small and expand the denominator to ﬁrst order in kto focus on momenta near the KMDirac point. Keeping only ﬁrst order terms in E,k(not their product as they are both similarly small), and using that \|qi\|=1,∀i= 1,2,3, we ﬁnd /parenleftbig 1−3w2 1/parenrightbig k·σψ A11=/bracketleftbig 1+3/parenleftbig w2 0+w2 1/parenrightbig/bracketrightbig EψA11 (38) and hence we ﬁnd that the Dirac velocity vanishes on a mani- fold of w0,w1given by w1=1√ 3and∀w0, which we call the ﬁrst magic manifold. The angle for which the Dirac velocity vanishes at the KMpoint is hence not a magic angle but a magic manifold. However, a further restriction needs to beimposed: w 0cannot be too large, since from our approxima- tion scheme in Secs. II DandII E,i fw0/greatermuch1/√ 3, the tripod model would not be a good approximation for the BM modelwith a large number of shells; hence we restrict ourselves tow 0/lessorequalslant1/√ 3, and deﬁne First magic manifold: w0/lessorequalslantw1=1√ 3. (39) The tripod model, Fig. 4(b), in which we found the ﬁrst magic manifold, does not respect the exact C 2xsymmetry of the lattice, although it becomes asymptotically accurate as thenumber of shells increases. The magic angle also does not 205411-10TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) explain analytically the ﬂatness of bands, since it only con- siders the velocity vanishing at one point. However, the valueobtained by BM for the magic angle is impressive; despiteconsidering only four sites and the K Mpoint, the bands do not change much after adding more shells, and they are ﬂatthroughout the whole Brillouin zone, not only around the K M point. Why is the entire band so ﬂat at this value? We answer this question by examining the /Gamma1M-centered model below. B. The /Gamma1M-centered hexagon model and the second magic manifold In Sec. II E we introduced a yet unsolved approximate model HApprox1 (k)i nE q .( 33), the /Gamma1M-centered model inFig. 4(a). This model respects all the symmetries of TBG, and we have showed in Appendix Bthat it represents a good approximation to the inﬁnite cutoff limit. As we can see inFig.15, the band dispersions of the n=1 shell model is very similar to that of n=2. After n=2 shells the difference to the inﬁnite cutoff band structure is not visible by eye. An analytic solution for the 12 ×12 Hamiltonian H Approx1 (k)i nE q .( 33)i snot possible at every k. We hence separate the Hamiltonian into HHex(k,w0,w1)=HkA1+ HA1,A1, then treat the smaller part HA1,B1H−1 kB1H† A1,B1perturba- tively, for w0,w1/lessorequalslant√ 3. We will try to solve the ﬁrst (largest) part of HApprox1 (k): the A1 shell model HHex(k,w0,w1)= HkA1+HA1,A1which we call the hexagon model: HHex(k,w0,w1)=⎛ ⎜⎜⎜⎜⎜⎝(k−q 1)·σ T2(w0,w1)0 0 0 T3(w0,w1) T2(w0,w1)( k+q3)·σ T1(w0,w1)0 0 0 0 T1(w0,w1)( k−q2)·σ T3(w0,w1)0 0 00 T3(w0,w1)( k+q1)·σ T2(w0,w1)0 000 T2(w0,w1)( k−q3)·σ T1(w0,w1) T3(w0,w1)0 0 0 T1(w0,w1)( k+q2)·σ⎞ ⎟⎟⎟⎟⎟⎠. (40) This is still a 12 ×12 Hamiltonian and its eigenstates cannot be analytically obtained at general k. In particular, it is also not illuminating to focus on a 12 ×12 Hamiltonian when we want to focus on the physics of the two active bands and thelow-energy physics of the dispersive passive bands. As suchwe make a series of approximations, which also elucidatesome of the questions posed in Fig. 1. We ﬁrst analytically ﬁnd a set of bands which can act as a perturbation theory treatment. 1. Energies of the hexagon model at k=0for arbitrary w0,w1 The only momentum where the hexagon model HHex(k,w0,w1) can be solved is the /Gamma1Mpoint. This is fortunate, as this point preserves all the symmetries of TBG,and is a good starting point for a perturbative theory. We ﬁndthe 12 eigenenergies of H Hex(k=0,w0,w1) given in Table I. By analyzing these energies as a function of w0,w1,w e can answer the question (1) in Fig. 1and give arguments for question (3) in Fig. 1. Numerically, at (and around) the ﬁrst magic angle—which as per the tripod model is deﬁned as TABLE I. Eigenvalues of the hexagon model in Eq. ( 40)a t/Gamma1M point ( k=0). The values for general w0,w1and for w0=w1=1√ 3are given, and Dege. is short for degeneracy. Band Energy at k=0for any w0,w1w0=w1=1√ 3Dege. E1 2w1−/radicalbig 1+w2 0 01 E2 −2w1+/radicalbig 1+w2 0 01 E3,4−1 2(/radicalbig 4+w2 0−/radicalbig 9w2 0+4w2 1)0 2 E5,61 2(/radicalbig 4+w2 0−/radicalbig 9w2 0+4w2 1)0 2 E7,8−1 2(/radicalbig 4+w2 0+/radicalbig 9w2 0+4w2 1) −√13/32 E9,101 2(/radicalbig 4+w2 0+/radicalbig 9w2 0+4w2 1)√13/32 E11 −2w1−/radicalbig 1+w2 0 −4/√ 31 E12 2w1+/radicalbig 1+w2 0 4/√ 31w1=1/√ 3—and in the isotropic limit w0=w1, the system exhibits two very ﬂat active bands, not only around the KM point but everywhere in the MBZ. It also exhibits a very small gap (sometimes nonexistent) between the active bands andthe passive bands, around the values w 0=w1=1/√ 3. The hexagon model HHex(k,w0,w1) explains both these obser- vations. We ﬁnd that the eigenenergies of HHex(k=0,w0= 1/√ 3,w1=1/√ 3), in the isotropic limit, are given in the third column of Table I. Remarkably, in the isotropic limit w0=w1, and at the ﬁrst magic angle w1=1/√ 3, the bands at the /Gamma1Mpoint are sixfold degenerate at energy 0. The two active bands are degenerate with the two passive bands abovethem and the two passive bands below them. This degen-eracy is ﬁne tuned, but the degeneracy breaking terms inthe next shells (subshells B1,A2,B2,etc.) are perturbative. Hence the gap between the active and the passive bands willremain small in the isotropic limit, answering question (1)in Fig. 1. From the tripod model, the two active bands have energy zero at the K Mpoint, and vanishing velocity at w1=1√ 3. Moreover, they also have energy zero at the /Gamma1Mpoint in the hexagon model (a good approximation for the inﬁnite caseat the /Gamma1 Mpoint). This now gives us twopoints ( /Gamma1M,KM)i n the MBZ where the bands have zero energy; at one of thosepoints, the band velocity vanishes. This gives us more analyticarguments that the band structure remains ﬂat than just theK Mpoint velocity, i.e., point (3) in Fig. 1. We further try to establish band properties away from the /Gamma1M,KMpoints by per- forming a further perturbative treatment of HHex(k,w0,w1) using the eigenstates at /Gamma1M. 2.k/negationslash=0six-band approximation of the hexagon model in the isotropic limit In the isotropic limit at w0=w1=1/√ 3, the sixfold de- generacy point of the hexagon model HHex(k,w0,w1)a t/Gamma1M 205411-11BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) prevents the development of a Hamiltonian for the two active bands. However, since the gap ( =√13/3) between the six zero modes E1,...,6(k=0,w0=1√ 3,w1=1√ 3) in Table Iand the rest of the bands E7,...,12(k=0,w0=1√ 3,w1=1√ 3)i s large at /Gamma1M, we can build a six-band k·pHamiltonian away from the /Gamma1Mpoint: H6-band ij (k)=/angbracketleftψEi\|HHex/parenleftbigg k,w0=w1=1√ 3/parenrightbigg −HHex/parenleftbigg k=0,w0=w1=1√ 3/parenrightbigg/vextendsingle/vextendsingleψEj/angbracketrightbig =/angbracketleftbig ψEi/vextendsingle/vextendsingleI6×6⊗k·/vectorσ/vextendsingle/vextendsingleψEj/angbracketrightbig , (41) where \|ψEj/angbracketrightwith j=1,..., 6 are the zero energy eigenstates ofHHex(k=0,w0=w1=1√ 3). We ﬁnd these eigenstates in Appendix C, where we place them in C3,C2xeigenvalue multiplets. The 6 ×6 Hamiltonian is the smallest effective Hamiltonian at the isotropic point, due to the sixfold degener-acy of bands at /Gamma1 M. The explicit form of the Hamiltonian H6-band(k)i sg i v e ni n Appendix C,E q .( C7). Due to the large gap between the six bands (degenerate at /Gamma1M) and the rest of the bands, it should present a good approximation of the hexagon model at ﬁnitekforw 0=w1=√ 3. The approximate H6-band(k) is still not generically diagonalizable (solvable) analytically. However,we can obtain several important properties analytically. First,the characteristic polynomial Det[E−H 6-band(k)]=0 ⇒/bracketleftbig 13E2−12/parenleftbig k2 x+k2 y/parenrightbig E+kx/parenleftbig k2 x−3k2 y/parenrightbig/bracketrightbig2=0.(42) Or, parametrizing ( kx,ky)=k(cosθ,sinθ), where \|k\|=k, we have [13E3−12k2E+k3cos(3θ)]2=0. (43) The characteristic polynomial reveals several properties of the six-band approximation to the hexagon model. (1) The exponent of 2 in the characteristic polynomial reveals that all bands of this approximation to the hexagonmodel are exactly doubly degenerate. This explains the almostdegeneracy of the ﬂat bands [point (3) in Fig. 1], but further- more it explains why the passive bands, even though highlydispersive, are almost degenerate for a large momentum rangearound the /Gamma1 Mpoint in the full model (see Fig. 14): they are exactly degenerate in the six-band approximation to thehexagon model; corrections to this approximation come fromthe remaining six bands of the hexagon model, which resideextremely far (energy√ 13/3), or from the B1shell, which we established is at most 20% in the MBZ—and smalleraround the /Gamma1 Mpoint. Thus, the almost double degeneracy of the passive bands pointed out in (2) of Fig. 1is explained. (2) Along the /Gamma1M-KMline we have kx=0,ky=kand hence the characteristic polynomial becomes /Gamma1M−KM:/parenleftbig 13E3−12k2 yE/parenrightbig2=0. (44) This implies two further properties: (1) The “active” bands of the approximation of the hexagon mode are exactly ﬂat atE=0 for the whole /Gamma1 M-KMline, thereby explaining their ﬂat- ness for a range of momenta; notice that our prior derivations FIG. 9. Band structure of the six-band approximation H6-bandto the hexagon model for the w0=w1=1/√ 3 magic point. (a) The six zero energy eigenstates at /Gamma1Mmarked by the red circle are used to obtain a perturbative Hamiltonian for the six lowest bands acrossall the BZ. As the six bands are very well separated from the other six, we expect a good approximation over a large part of the BZ. The active and passive bands in the dashed square are almost doublydegenerate. In the right panel, the six lowest bands of the hexagon model, for a smaller energy range, are shown. Notice the passive bands are undistinguishably twofold degenerate by eye (not an exactdegeneracy, they split close to K M, see left plot) Note the Dirac fea- ture of the passive bands. The active bands split at KMin the hexagon model, but the B1 shell addition makes them degenerate. (b) Theﬁrst order approximation to the hexagon model using the six zero energy bands at the /Gamma1 Mpoint gives exactly doubly degenerate bands over the whole BZ. It gives the correct velocity of the Dirac nodes,zero dispersion of active bands on /Gamma1 M-KM, and a small dispersion of active bands on /Gamma1M-MM, with known velocities. Along these lines, all eigenstates are kindependent. found that the active bands have zero energy at KM,/Gamma1Mand vanishing Dirac velocity at KMforw0=w1=√ 3; our cur- rent derivation shows that the approximately ﬂat bands alongthe whole /Gamma1 M-KMline originate from the doubly degenerate zero energy bands of the hexagon model. (2) The dispersive(doubly degenerate) passive bands, for w 0=w1=√ 3, have a linear dispersion E=±/radicalbig 12/13k (45) along/Gamma1M-KM, with velocity 2√3/13=0.960769, close to the Dirac velocity. This explains property (2) in Fig. 1. Note that the velocity is equal to 2 /[E9,10(k=0,w0=1/√ 3,w1= 1/√ 3)] or two over the gap to the ﬁrst excited state. This approximation is visually shown in Fig. 9. (3) Remarkably, the eigenstates along along the /Gamma1M-KM line can also be obtained (see Appendix D). Along this line, the eigenstates of all bands of the H6-bandHamiltonian ap- proximation to the hexagon model are kyindependent (see Appendix D)! (4) Along the /Gamma1M-MMline ( kx=k,ky=0) the character- istic polynomial becomes /Gamma1M−MM:(k+E)2(k2−13kE+13E2)2=0.(46) Hence the energies are E=−k, a highly dispersive (dou- bly degenerate) hole branch passive band of velocity −1; E=1 2(1+3√ 13)k(≈0.916025 k), another highly dispersive doubly degenerate electron branch passive band. This ex- plains property (2) in Fig. 1. Notice that this velocity is1 2(1+1 E9,10(k=0,w0=1/√ 3,w1=1/√ 3)). The third dispersion is 205411-12TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) TABLE II. Eigenvalues of the hexagon model in Eq. ( 40)a t /Gamma1Mpoint ( k=0) at the second magic manifold w1=√ 1+w2 0 2.T h e notation Dege. is short for degeneracy. Band Energy at k=0atw1=√ 1+w2 0 2Dege. E1,2 02 E3,4√ 10w2 0+1−√ w2 0+4 22 E5,6 −√ 10w2 0+1−√ w2 0+4 22 E7,8 −√ 10w2 0+1+√ w2 0+4 22 E9,10√ 10w2 0+1+√ w2 0+4 22 E11 −2/radicalbig 1+w2 0 1 E12 2/radicalbig 1+w2 0 1 E=1 2(1−3√ 13)k(≈0.0839749 k), a weakly dispersive dou- bly degenerate active band. This explains the very weak, but nonzero dispersion of the bands on /Gamma1M-MM. The eigenstates along this line can also be obtained (see Appendix D). The approximation is visually shown in Fig. 9. (5) Along the /Gamma1M-MM, the eigenstates of all bands of the H6-bandHamiltonian approximation to the hexagon model are kxindependent (see Appendix D)! (6) In the six-band model, eigenstates are independent of kon the manifold kx=ky. 3. Energies of the hexagon model at k=0away from the isotropic limit and the second magic manifold In the isotropic limit (which coincides with the magic angle of the tripod model), w0=w1=1/√ 3, due to the sixfold degeneracy of the /Gamma1Mpoint, it is impossible to obtain an approximate Hamiltonian that is less than a 6 ×6 matrix. Moving away from the isotropic limit, and staying in the rangeof approximations w 0,w1/lessorequalslant1√ 3, the hexagon model is a good starting point for a perturbative expansion. We now ask what values of w1,w0might have a “simple” expression for their energies. We see that if w1=√ 1+w2 0 2, the sixfold degeneracy at the /Gamma1Mpoint at zero energy for w1=1/√ 3 splits into a 2(enforced) +2(accidental) +2(enforced)-fold degen- eracy. There is an accidental twofold degeneracy of the active bands at zero energy, and a gap to the passive bands whichhave an symmetry enforced degeneracy. The twofold acci- dental degeneracy at zero energy along w 1=√ 1+w2 0 2is the important property of this manifold in parameter space. Theeigenvalues of the hexagon model in this case are given inTable II. Although the perturbative addition of the B1 shell will split the /Gamma1 Mpoint E1,2(k=0,w0,w1=√ 1+w2 0 2)=0 degen- eracy, we ﬁnd that this zero energy doublet of the hexagonmodel is particularly useful to calculate a k·pperturbation theory of the active bands, as many perturbative terms can-cel. In particular, we see that the gap between the activeband zero energy doublet and the passive bands [ E 3,4(k=0,w0,w1=√ 1+w2 0 2)] of the hexagon model becomes large in the chiral limit [ E3,4(k=0,w0=0,w1=√ 1+w2 0 2=1/2)= −1/2]. We note that this explains property (4) of Fig. 1: from the hexagon model, the gap between the active and thepassive bands is, in effect, proportional to w 1−w0. Since the bandwidth of the TBG model is known to be smallerthan this gap, we will use the /Gamma1 Mpoint doublet of states E1,2(k=0,w0,w1=√ 1+w2 0 2)=0 to perform a perturbative expansion. We deﬁne this paramter manifold as the “secondmagic manifold”: Second magic manifold: w 1=√ 1+w2 0 2,w0/lessorequalslant1/√ 3. IV . TWO-BAND APPROXIMATIONS ON THE MAGIC MANIFOLDS A. Differences between the ﬁrst and second magic manifolds We have deﬁned two manifolds in parameter space where the two active bands of the hexagon model are separated fromthe passive bands. Hence, we can do a perturbative expansionin the inverse of the gap from the passive to the active bands.We ﬁrst brieﬂy review the differences between the two magicmanifolds First magic manifold: w 0/lessorequalslantw1=1/√ 3. (1) For these values of w0,w1, the Dirac velocity at KM vanishes in the tripod model, which is a good approximation to the inﬁnite cutoff model. Hence the velocity at the KMpoint in the inﬁnite model should be small. The Dirac node is atE=0. (2) One end of the ﬁrst magic manifold, the isotropic point w 0=w1=1/√ 3 is also the endpoint of the second magic manifold, and exhibits the sixfold degeneracy at E=0a tt h e /Gamma1Mpoint in the hexagon model. (3) Away from the isotropic point, on the ﬁrst magic man- ifold, a gap opens everywhere between the six states of thehexagon model. At the /Gamma1 Mpoint, the sixfold degenerate bands at the isotropic limit split when going away from this limit,i n t oa2( s y m m e t r ye n f o r ced) -1-1-2 (symmetry enforced) degeneracy conﬁguration. Hence the two active bands, inthe hexagon model, split from each other in the ﬁrst magicmanifold. (4) The splitting of the active bands in the hexagon model in the ﬁrst magic manifold is corrected by the addition of theB1 shell as the term H A1,B1H−1 kB1H† A1,B1in Eq. ( 33). (5) The active bands, when computed with the full Hamil- tonian without approximation, are very ﬂat on the ﬁrst magicmanifold (much ﬂatter than on the second magic manifold),and there is a full, large gap to the passive bands (see Fig. 10). Second magic manifold: w 1=√ 1+w2 0 2,w0/lessorequalslant1/√ 3. (1) The hexagon model exhibits a doublet of zero energy active bands at /Gamma1Malong the entire second magic manifold. (2) One end of the second magic manifold, the isotropic point w1=w0=1/√ 3 is also the endpoint of the ﬁrst magic manifold, and exhibits a sixfold degeneracy at E=0a tt h e /Gamma1Mpoint in the hexagon model and a vanishing Dirac velocity in the tripod model. 205411-13BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) FIG. 10. Plots of the active bands band structure on the ﬁrst magic manifold, w1=1/√ 3,w0/lessorequalslant√ 3, for a large number of shells. In the second row, the gap to the passive bands is large and outside the range. The Dirac velocity is small for all values of w0/w1(it vanishes in the tripod model, but has a ﬁnite value once further shells are included), and the bands are extremely ﬂat. The ratio of active bands bandwidth tothe active-passive band gap decreases upon decreasing w 0/w1. (3) Away from the isotropic point, on this manifold, the bands do not have a vanishing velocity at the Dirac point. (4) The eigenstates of the active bands are simple (simpler than on the ﬁrst magic manifold) on this manifold, with simplematrix elements (as proved below). A perturbation theory canbe performed away from the /Gamma1 Mpoint and away from this manifold to obtain a general Hamiltonian for k,w0,w1.T h e B1 shell can then also be included perturbatively as the term HA1,B1H−1 kB1H† A1,B1in Eq. ( 33). (5) The active bands are not the ﬂattest on this manifold. They are much less ﬂat than on the ﬁrst magic manifold, dueto the fact that the Dirac velocity does not vanish (is not small)at the K Mpoint on the second magic manifold. B. Two-band approximation for the active bands of the hexagon model on the second magic manifold We now try to obtain a two-band model on the mani- foldw1=/radicalBig 1+w2 0/2,∀w0/lessorequalslant1/√ 3, for which we use the /Gamma1M-point HHex(k=0,w0,w1=√ 1+w2 0 2) as a zeroth order Hamiltonian and perform a k·pexpansion away from the /Gamma1M point. Figure 10shows that away from the isotropic limit, the gap that opens at the /Gamma1Mpoint between the formerly sixfold degenerate bands can be much larger than the bandwidth ofthe active bands even for modest deviations from the isotropiclimit. We have explained this from the behavior of the six-band approximation to the hexagon model, and from knowingthe analytic form of the /Gamma1 M-point energy levels in the hexagon model. We have also obtained the eigenstates of all the /Gamma1M- energy levels in Appendix E2. It is then sufﬁciently accurate to treat the manifold of the two/Gamma1M-point zero energy states at w1=√ 1+w02 2,∀w0/lessorequalslant1/√ 3 as the bases of the perturbation theory. To perform a two-band model approximation to the hexagon model, we take the unperturbed Hamiltonian to be HHex(k=0,w0,w1=/radicalBig 1+w2 0/2) (the hexagon model onthe second magic manifold) in Eq. ( 40). For this Hamilto- nian we are able to obtain all the eigenstates analytically in Appendix E2. The perturbation Hamiltonian, on the second magic manifold, is Hperturb (k,w0)=HHex⎛ ⎝k,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ −HHex⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =I6×6⊗k·/vectorσ. (47) The manifold of states which are kept as “important” are the two zero energy eigenstates of HHex(k=0,w0,w1=/radicalBig 1+w2 0/2), given in Eq. ( E7). This manifold will be de- noted as ψwith a band index m∈{1,2}. The manifold of “excited” states, which will be integrated out, is made up ofthe eigenstates Eqs. ( E8), (E9), (E10), and ( E11), each doubly degenerate, and Eqs. ( E12) and ( E13), each nondegenerate. This manifold will be denoted as ψwith a band index l∈ {3,4,..., 12}. We now give the expressions for the pertur- bation theory up to ﬁfth order. We here give only the ﬁnalresults, for the expression of the matrix elements computed inperturbation theory, see Appendix F2. We ﬁrst note that the ﬁrst order (linear in k) perturbation term is H (1) mm/prime(k,w0)=/angbracketleftψm\|Hperturb (k,w0)\|ψm/prime/angbracketright=0. This is a particular feature of the second magic manifold and rendersthe perturbation theory simple. Furthermore, it implies that,on the second magic manifold, the active bands of the hexagonmodel have a quadratic touching at the /Gamma1 Mpoint, as conﬁrmed numerically. Due to the vanishing of these matrix elements,one can perform quite a large order perturbative expansion. It can be shown that the nth order perturbation is proportional to 1 /(3w 2 0−1)n−1, with symmetry-preserving functions of k(see Appendix F2). Up to the ﬁfth order, the full two-band approximation to the hexagon Hamiltonian can 205411-14TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) be expressed as HHex 2-band⎛ ⎝k,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =d0(k,w0)σ0+d1(k,w0)(σy+√ 3σx), where d0(k,w0)=4w0 9/radicalBig w2 0+1/parenleftbig 1−3w2 0/parenrightbig2/bracketleftbigg/parenleftbig w2 0−3/parenrightbig −4/parenleftbig 29w6 0−223w4 0−357w2 0−9/parenrightbig 9/parenleftbig 1−3w2 0/parenrightbig2/parenleftbig w2 0+1/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig/bracketrightbigg ×kx/parenleftbig k2 x−3k2 y/parenrightbig (48) and d1(k,w0)=4w2 0 3/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig ×/bracketleftbigg −1+2/parenleftbig 35w4 0+68w2 0+9/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig 9/parenleftbig w2 0+1/parenrightbig/parenleftbig 3w2 0−1/parenrightbig2/bracketrightbigg ×/parenleftbig k2 x+k2 y/parenrightbig , (49) while the Pauli matrices σjhere are in the basis deﬁned in Appendix E2a (rather than the basis of graphene sublattice). In particular, we note that the eigenstates of the k·pmodel HHex 2-band (k,w0,w1=√ 1+w2 0 2) are independent of kup to the ﬁfth order perturbation within the hexagon model. C. Away from the second magic manifold: Two-band active bands approximation of the hexagon model We now want to perform calculations away from the sec- ond magic manifold, and possibly connect the perturbationtheory with the ﬁrst magic manifold. There are two ways ofdoing this, while still using the /Gamma1 M-point wave functions as a basis (we cannot solve the hexagon model exactly at any otherkpoint). One way is to solve for the wave functions at the /Gamma1 M point for all w0,w1, and use these states to build a perturbation theory that way. However, away from the special ﬁrst andsecond magic manifolds, the expression of the ground states iscomplicated. The second way is to use the eigenstates already obtained for the second magic manifold w 1=√ 1+w2 0 2and obtain a perturbation away from the second magic manifold.In this section we choose the latter. We take the unperturbed Hamiltonian to be H Hex(k= 0,w0,w1=/radicalBig 1+w2 0/2) (the hexagon model on the second magic manifold) in Eq. ( 40). For this Hamiltonian we are able to obtain all the eigenstates analytically in Appendix E2. The perturbation Hamiltonian, away the second magicmanifold, is Hperturb (k,w0,w1) =HHex(k,w0,w1)−HHex/parenleftbigg k=0,w0,w1=/radicalBig 1+w2 0 2/parenrightbigg =I6×6⊗k·/vectorσ+HHex/parenleftbigg k=0,0,w1−/radicalBig 1+w2 0 2/parenrightbigg .(50) We now give the expressions for the perturbation theory up to fourth order. We here give only the ﬁnal results, for theexpression of the matrix elements computed in perturbationtheory, see Appendices F2andF3. We ﬁrst note that the ﬁrst order Hamiltonian is H (1) mm/prime(k,w0,w1)=/parenleftbigg/radicalBig w2 0+1 2−w1/parenrightbigg (σy+√ 3σx).(51) Hence we ﬁnd there is now a linear order term in the Hamiltonian—as it should since the two states degenerate at/Gamma1 Mon the second magic manifold are no longer degenerate away from it. Because of this, many other terms in the furtherdegree perturbation theory become nonzero, and the pertur-bation theory has a more complicated form. We present alldetails in Appendix F3and here show only the ﬁnal result, up to fourth order. We can label the two-band Hamiltonian as H Hex 2-band (k,w0,w1)=d0(k,w0,w1)σ0 +d1(k,w0,w1)(σy+√ 3σx),(52) where the expressions of d0(k,w0,w1) and d1(k,w0,w1)a r e given in Eqs. ( F35) and ( F36) in Appendix F3. The pertur- bation is made on the zero energy eigenstates of HHex(k= 0,w0,w1=√ 1+w2 0 2). Ifw1=√ 1+w2 0 2, then the expressions reduce to our previous Hamiltonian Eq. ( F20). Notice that so far, remarkably the eigenstates are not kdependent, they are just the eigenstates of ( σy+√ 3σx). D. Two active bands approximation of the n=1 shell model HApprox1 (k) on the second magic manifold In Sec. IV B we have obtained an effective model for the two active bands of the hexagon model on the second magic manifold w1=√ 1+w02 2,∀w0/lessorequalslant1/√ 3u s i n gt h e /Gamma1M-point HHex(k=0,w0,w1=√ 1+w2 0 2) as zeroth order Hamiltonian. We expect this to be valid around the /Gamma1Mpoint. We know that a good approximation of the TBG involves at least n=1 shells: the A1 subshell, which is the hexagon model, and the B1 subshell, which is taken into account perturbatively in HApprox1 (k)o fE q .( 33). After detailed calculations given in Appendix F4, we ﬁnd the ﬁrst order perturbation Hamiltonian given by H(B1)(k,w0,w1)=1/producttext i=1,2,3\|k−2qi\|2\|k+2qi\|2 ×/summationdisplay μ=0,x,y,z/tildewidedμ(k,w0,w1)σμ, (53) 205411-15BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) FIG. 11. Plots of the ratio of the bandwidth of the active bands for the large number of shells to the analytic bandwidth /Delta1in Eq. ( 56), for different values of w0,w1, including the two magic manifolds. In the regime of validity of our approximations, we can see that this ratio is substantially above 90%. where/tildewidedμ(k,w0,w1) are given in Eqs. ( F39)–(F42)o fA p - pendix F4. This represents the ﬁrst order HApprox1 (k) projected into the zero energy bands of the hexagon modelon the second magic manifold. We note that the B1 shell perturbation expressions can only be obtained to ﬁrst order.Second and higher orders are particularly tedious and notilluminating. Note that, to ﬁrst order in perturbation theory onthe second magic manifold, only the term H A1,B1H−1 kB1H† A1,B1 contributes to the approximate two-band Hamiltonian. Also, we obtained the perturbation of HA1,B1H−1 kB1H† A1,B1forgeneric w0,w1projected into the second magic manifold /Gamma1Mpoint bands of the hexagon model. E. Two-band approximation for the active bands of the n=1 shell model HApprox1 (k) in Eq. ( 33) for any w0,w 1/lessorequalslant1√ 3 We are now in a position to describe the two active bands of the approximate Hamiltonain of the one-shell modelin Eq. ( 33),H Approx1 =HkA1+HA1,A1−HA1,B1H−1 kB1H† A1,B1by adding H(B1)(k,w0,w1)o fE q .( 54)t oHHex 2band(k,w0,w1)o f Eq. ( 53). We note that this is still perturbation theory per- formed by using the /Gamma1M-point HHex(k=0,w0,w1=√ 1+w2 0 2) as a zeroth order Hamiltonian: H2-band (k,w0,w1)=HHex 2-band (k,w0,w1)+H(B1)(k,w0,w1). (54) We now ﬁnd some of the predictions of this Hamiltonian. The energies of the two bands of Eq. ( 55)a t/Gamma1Mpoint are E±(w0,w1)=±/parenleftbigg−4/radicalBig w2 0+1w1+w2 0+w2 1+2 2/radicalBig w2 0+1/parenrightbigg (55) over the full range of w0,w1/lessorequalslant1/√ 3. Remarkably we ﬁnd an amazing agreement between the energy of the bands at /Gamma1M point and the numerics. We ﬁnd that the bandwidth of the ﬂatband at /Gamma1Mpoint is /Delta1(w0,w1)=2\|E±(w0,w1)\|. (56) This matches incredibly well with the actual values. In Fig. 11 we plot the ratio of actual active bandwidth at /Gamma1Mpoint from the large number of shell model to /Delta1in Eq. ( 56), for values w0<1/√ 3,w0<w1<1/√ 3. Note that even though we are sometimes going far from the second magic manifold values w0,w1=/radicalBig 1+w2 0/2 where the perturbation theory is valid, the ratio holds up well, and is actually never smaller than 0.8 or larger than 1. We are using w0<w1because the pertur- bation theory is around the manifold w0,w1=/radicalBig 1+w2 0/2/lessorequalslant 1√ 3for which w0<w1.F o rw1<w0the approximation be- comes worse, but is outside of the validity regime. For the two magic manifolds, also shown in Figs. 11and 12, the agreement is very good. We point out several consis- tency checks. First, remarkably, the set of approximations thatled us to ﬁnding a two-band Hamiltonian becomes exact at some points. (1) The /Gamma1 Mpoint bandwidth at w0=w1=1/√ 3 van- ishes/Delta1(1√ 3,1√ 3)=0. This degeneracy reproduces the exact result, in the one-shell model (see n=1i nF i g . 13, the sixfold degeneracy at the /Gamma1Mpoint). The approximate model of the one-shell HApprox1 of Eq. ( 33) also has an exact sixfold degen- eracy at the /Gamma1Mpoint at w0=w1=1/√ 3 (the two bands here being part of the sixfold manifold). It is remarkable that ourtwo-band projection perturbation approximation reproducesthis degeneracy exactly, especially since it is supposed notto work close to w 0=w1=1/√ 3—where the gap to the active bands is 0 and the /Gamma1Mpoint becomes sixfold degenerate. (2) At w0=w1=0, the bandwidth at /Gamma1Mis/Delta1(0,0)=2. This is again an exact result for the inﬁnite shell model . Indeed, at the /Gamma1Mpoint, the BM Hamiltonian with zero interlayer coupling has a gap =2\|q1\|=2. 205411-16TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) FIG. 12. /Gamma1Mpoint bandwidth of the active bands (large num- ber of shells) on the manifold /Delta1(w0,w1)=0(w1=2/radicalbig 1+w2 0−/radicalbig 2+3w2 0) of zero analytic bandwidth [Eq. ( 56)] divided by the bandwidth of the active bands in the chiral limit [( w0,w1)= (0,1√ 3)]. Note that this number is extremely small away from w0= w1=1√ 3, showing that our analytic manifold of smallest bandwidth [/Delta1(w0,w1)=0] also exhibits small bandwidth in the large cell number. Inset: The curve w1=2/radicalbig 1+w2 0−/radicalbig 2+3w2 0for which /Delta1(w0,w1)=0f o r0 /lessorequalslantw0/lessorequalslant1√ 3. Note that w1changes extremely little 1% (stays within 1% of1√ 3) during the entire sweeping of w0. (3) We now ask: what is the w0,w1manifold, under this approximation, for which the /Gamma1Mpoint bandwidth is zero? This is easily solved to give: Two-band model with zero bandwidth at /Gamma1M: w1=2/radicalBig w2 0+1−/radicalBig 3w2 0+2,w0∈/bracketleftbigg 0,1√ 3/bracketrightbigg . (57) Figure 12plots the ratio of the bandwidth of the full BM model on this manifold to the bandwidth at at the chiral limitw 0=0,w1=1√ 3(which is already really small!). We can see that, for most of the w0∈(0,1/√ 3), this ratio is below 0.1, showing us that we have identiﬁed an extremely smallbandwidth manifold. (4) What are the values of w 1on this manifold? Re- markably, as can be seen in Fig. 12,w1=2/radicalBig w2 0+1−/radicalBig 3w2 0+2 is an almost fully constant over the interval w0∈ (0,1/√ 3): it changes by around 1% only. Moreover, its values (0.578–0.586) are very close to 1 /√ 3≈0.57735. Hence our approximation explains the ﬂatness of the bands over the ﬁrst magic manifold ,0/lessorequalslantw0/lessorequalslant1√ 3,w1=1√ 3: This manifold is al- most the same as the one for which our analytical approximate calculation gives zero gap. Hence property (6) of Fig. 1is answered. (5) At w0=0, one has w1=2/radicalBig w2 0+1−/radicalBig 3w2 0+2= 2−√ 2i nE q .( 57), for which the bandwidth is 0 in our perturbative model. As we show in Appendix F5, this value ofw1coincides with the exact value for which the /Gamma1Mband- width is zero in the approximation Hamiltonian HApprox1 of Eq. ( 33). Furthermore, at w0=0, the value w1=2−√ 2a l s o coincides with the exact value of zero /Gamma1Mbandwidth in theno-approximation Hamiltonian of the n=1 shell Hamiltonian (ofA1,B1 subshells) (see Appendix F5). (6) At w0=0, the value w1=2/radicalBig w2 0+1−/radicalBig 3w2 0+2= 2−√ 2 for which the bandwidth of our approximate two-band model is projected to be zero is numerically very close to the value of 0.586 quoted for the ﬁrst magic angle in the chirallimit [ 37]. In fact, at w 0=0,w1=2−√ 2 the bandwidth of the active bands is half of that at w1=0.586. F. Region of validity of the two-band model and further ﬁne tuning The two-band approximation to the n=1 shell model has a radius of convergence in kspace in the ﬁrst MBZ. This radius of convergence is easily estimated from the followingargument. In Table II, the (maximum) gap, at the /Gamma1 Mpoint, between the active and the passive bands in the hexagon model(and in the region w 0/lessorequalslant1/√ 3) is at w0=0 and equals 1 /2. The distance, in the MBZ between /Gamma1MandKMpoints, equals 1. Hence we expect that our two-band model will work for\|k\|/lessmuch1/2, as our numerical results conﬁrm. The form factor matrices can be computed for this range of kanalytically, by using the full hexagon Hamiltonian in Eq. ( 52)p l u st h e B1 shell perturbation in Eq. ( 53). They will be presented in a future publication. The k=K Mpoint is outside the range of validity of the two-band model, and hence this does not capture the gaplessDirac point for all values of w 0,w1. However, with some physical intuition, we can obtain a two-band model that has agap closing at the K Mpoint. In Fig. 9we see that the hexagon model does not have a gap closing between the active bandsat the K Mpoint. However, in Figs. 18,19, and 20we see that HApprox1 (k)i nE q .( 33) has a gap closing close to, or almost at theKMpoint. This means that one of the main roles of the B1 shell is to close the KMgap, leading to the Dirac point. Hence we can use the two-band model of the ﬁrst order approximation to the hexagon model, Eq. ( 51), H(1) mm/prime(k,w0,w1)=(√ w2 0+1 2−w1)(σy+√ 3σx) along with the two-band model ﬁrst order approximation for the B1-shell H(B1)(k,w0,w1) to obtain a ﬁrst order two-band approxima- tion Hamiltonian: H(1)(k,w0,w1)+H(B1)(k,w0,w1). Note thatH(1)(k,w0,w1), the two-band ﬁrst order approximation to the hexagon model, has two ﬂat kindependent bands. We now impose the condition: H(1)(k=KM,w0,w1)+ H(B1)(k=KM,w0,w1)=0 to ﬁnd the manifold ( w1,w0)o n which this condition happens. Notice that, ap r i o r i , there is no guarantee that the result of this condition will give a manifoldthat is anywhere near the values of w 1,w0considered in this paper, for which our set of approximations is valid (i.e.,w 0,w1not much larger than 1 /√ 3). We ﬁnd H(1)(k=KM,w0,w1)+H(B1)(k=KM,w0,w1)=0 (58) ⇒ Two-band model degenerate at KM: w1=1 32/parenleftbig 63/radicalBig w2 0+1−/radicalBig 2977w2 0+1953/parenrightbig . (59) 205411-17BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) FIG. 13. Plots of the band structure for different parameters around the ﬁrst magic angle, and for different ranges of the yaxis. Notice no change from n=2t on=4, in agreement with the theoretical discussions. Remarkably, we note that as w0is tuned from 1 /√ 3t o0 , w1 only changes from (1 /√ 3)=0.57735 and3 32(21−√ 217)= 0.587726! Hence the isotropic point is included in this man- ifold, and w1changes by only about 2% as w0is tuned from the isotropic point to the chiral limit. We hence propose thismodel as a ﬁrst, heuristic k·pmodel for the active bands on thew 1(w0) manifold in Eq. ( 58). Importantly, this model will have (A) ﬂat bands with small bandwidth; (B) identical gapbetween the active bands at the /Gamma1 Mpoint with the TBG BM model; and (C) gap closing at the KMpoint (Fig. 14). V . CONCLUSIONS In this paper we presented a series of analytically justiﬁed approximations to the physics of the BM model [ 1]. These FIG. 14. Comparison between (a) the active bands of the BM model at the w0=0,w1≈0.588 point and (b) the bands of the two-band ﬁrst order approximation to HApprox1 (k)i nE q .( 33). Notice that the bandwidth at the /Gamma1Mpoint is virtually identical, that the bands are ﬂat, and that they close gap at the KMpoint.approximations allow for an analytic explanation of several properties of the BM model such as (1) the difﬁculty tostabilize the gap, in the isotropic limit from active to pas-sive bands over a wide range of angles smaller than the ﬁrstmagic angle. (2) The almost double degeneracy of the passive bands in the isotropic limit, even away from the /Gamma1 Mpoint, where no symmetry forces them to be. (3) The determina-tion of the high group velocities of the passive bands. (4)The ﬂatness of the active bands even away from the Diracpoint, around the magic angle which has w 1=1/√ 3. (5) The large gap, away from the isotropic limit (with w1=1/√ 3), between the active and passive bands, which increases imme-diately with decreasing w 0, while the bandwidth of the active bands does not increase. (6) The ﬂatness of bands over thewide range of w 0∈[0,1/√ 3], from chiral to the isotropic limit. Also, we provided a 2 ×2k·pHamiltonian for the active bands, which allowed for an analytic manifold on which the bandwidth is extremely small: w1=2/radicalBig w2 0+1−/radicalBig 3w2 0+2,w0∈[0,1√ 3]. However, the most important feature uncovered in this paper is the development of an analytic perturbation theorywhich justiﬁes neglecting most of the matrix elements [formfactors /overlap matrices, see Eq. ( 19)], which will appear in the Coulomb interaction [ 108]. The exponential decay of these matrix elements with momentum will justify the useof the “ﬂat metric condition” in Eq. ( 20) and allow for the determination of exact Coulomb interaction ground states andexcitations [ 108–111]. Future research in the BM model is likely to uncover many surprises. Despite the apparent complexity of the model andthe need for numerical diagonalization, one cannot help butthink that there is a 2 ×2k·pmodel valid over the whole area of the MBZ, for all w 0,w1around the ﬁrst magic angle. Our two-band model is valid around the /Gamma1Mpoint—for a large 205411-18TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) interval but not for the entire MBZ, although we can ﬁne tune to render the qualitative aspects valid at the KMpoint also. A future goal is to ﬁnd an approximate summation, based onour perturbative expansion, where outer shells can be takeninto account more carefully and possibly summed together ina closed-form series, thereby leading to a much more accuratek·pmodel. We leave this for future research. ACKNOWLEDGMENTS We thank Aditya Cowsik and Fang Xie for valuable discus- sions. B.A.B. thanks Michael Zaletel, Christophe Mora, andOskar Vafek for fruitful discussions. This work was supported by the DOE Grant No. DE-SC0016239, the Schmidt Fund forInnovative Research, Simons Investigator Grant No. 404513,and the Packard Foundation. Further support was providedby the NSF-EAGER No. DMR 1643312, NSF-MRSEC No.DMR-1420541 and No. DMR-2011750, ONR No. N00014-20-1-2303, Gordon and Betty Moore Foundation throughGrant GBMF8685 towards the Princeton theory program, BSFIsrael US foundation No. 2018226, and the Princeton GlobalNetwork Funds. B.L. acknowledge the support of PrincetonCenter for Theoretical Science at Princeton University in theearly stage of this work. APPENDIX A: MATRIX ELEMENTS OF THE /Gamma1M-CENTERED MODEL We introduce the shells in the /Gamma1M-centered model. The Anjsites of the nthAshell [see Fig. 4(a)] are situated at QAnj=(n−1)(q1−q2)+(j−1)(q2−q3)+q1,j=1,..., n, QAnn+j=C6QAnj=(n−1)(q1−q3)+(j−1)(q2−q1)−q3,j=1,..., n, QAn2n+j=C2 6QAnj=(n−1)(q2−q3)+(j−1)(q3−q1)+q2,j=1,..., n, QAn3n+j=C3 6QAnj=(n−1)(q2−q1)+(j−1)(q3−q2)−q1,j=1,..., n, QAn4n+j=C4 6QAnj=(n−1)(q3−q1)+(j−1)(q1−q2)+q3,j=1,..., n, QAn5n+j=C5 6QAnj=(n−1)(q3−q2)+(j−1)(q1−q3)−q2,j=1,..., n. (A1) There are 6 nAsites in the nth shell. The Bnjsites of the nthBshell [see Fig. 4(a)] are situated at QBnj=QAnj+q1,QBnn+j=QAnn+j−q3,QBn2n+j=QAn2n+j+q2, QBn3n+j=QAn3n+j−q1,QBn4n+j=QAn4n+j+q2,QBn5n+j=QAn5n+j−q2,j=1,..., n. (A2) There are 6 nBsites in the nth shell. The basis we take for the BM Hamiltonian in Eq. ( 3)i st h e n (A1,B1,A2,B2,..., AN,BN)=(A11,A12,A13,A14,A15,A16,B11,B12,B13,B14,B15,B16,A21,A22,...), (A3) where Nis the cutoff in the number of shells that we take. Each shell nhas 6 nAsites and 6 nBsites. The separation of shell n=1,...,∞intoAandBis necessary in the /Gamma1M-centered model due to the structure of the matrix elements. Unlike in the KM-centered model, where different shells hop from one to another but notwithin a given shell, in the /Gamma1M-centered model, the Ashells hop between themselves too. Explicitly, the nonzero matrix elements within the nthAshell are called HAn,An: HAn,An=Ann↔Ann+1:T2;An2n↔An2n+1:T1;An3n↔An3n+1:T3; An4n↔An4n+1:T2;An5n↔An5n+1:T1;An6n↔An6n+1:T3. (A4) In the Bshell there are no matrix elements between different Bsites, but there are matrix elements between the AandBsites in the same shell n. They are called HAn,Bnand the nonzero elements are HAn,Bn=Anj↔Bnj:T1;Ann+j↔Bnn+j:T3;An2n+j↔Bn2n+j:T2; An3n+j↔Bn3n+j:T1;An4n+j↔Bn4n+j:T3;An5n+j↔Bn5n+j:T2; j=1,..., n,n=1,...,∞. (A5) Last set of couplings are between the n−1thBshell Bn−1 and the nth shell AnareHBn−1,Anwith nonzero matrix elements given by HBn−1,An=Bn−1j↔Anj:T2,j=1,..., n−1;Bn−1j−1↔Anj:T3,j=2,..., n; Bn−1n+j↔Ann+j:T1,j=1,..., n−1;Bn−1n+j−1↔Ann+j:T2,j=2,..., n; Bn−12n+j↔An2n+j:T3,j=1,..., n−1;Bn−12n+j−1↔An2n+j:T1,j=2,..., n; Bn−13n+j↔An3n+j:T2,j=1,..., n−1;Bn−13n+j−1↔An3n+j:T3,j=2,..., n; Bn−14n+j↔An4n+j:T1,j=1,..., n−1;Bn−14n+j−1↔An4n+j:T2,j=2,..., n; Bn−15n+j↔An5n+j:T3,j=1,..., n−1;Bn−15n+j−1↔An5n+j:T1,j=2,..., n. (A6) 205411-19BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) FIG. 15. Plots of the band structure for different parameters around the ﬁrst magic angle, and for different ranges of the yaxis. Notice no change from n=2t on=4, in agreement with the theoretical discussions. The diagonal matrix elements are ( k−Q)σδQ,Q/primewhere the Q/prime,Q’s are given by the shell distance: We call these HkAnorHkBn depending on whether the Qis on the AorBshell. Note that the Hamiltonian within the Bshell is HkBnwhile the Hamiltonian within the Ashell is HkAn+HAn,An. We now have deﬁned all the nonzero matrix elements of the Hamiltonian. In block-matrix form, it takes the expression H=⎛ ⎜⎜⎜⎜⎜⎜⎜⎝H kA1+HA1,A1HA1,B1 000 ··· H† A1,B1HkB1 HB1,A2 00 ··· 0 H† B1,A2HkA2+HA2,A2HA2,B2 0 ··· 00 H† A2,B2HkB2 HB2,A3 ··· 000 H† B2,A3HkA3+HA3,A3··· ............⎞ ⎟⎟⎟⎟⎟⎟⎟⎠. APPENDIX B: NUMERICAL CONFIRMATION OF THE PERTURBATIVE FRAMEWORK What our discussion in Secs. II D andII Eshows is that: (1) For the ﬁrst magic angle, we can neglect all shells greaterthan 2, while having a good approximation numerically. (2) For the next, smaller, magic angle, we need to keep more shells in order to obtain a good approximation. We have testedthatmachine precision convergence can be obtained for the active bands by choosing a cutoff of 5–6 shells. We test thisnext, along with other conclusions of Secs. II D andII E.I n particular: (1) We ﬁrst conﬁrm our analytic conclusion that shells above n>2 do not change the spectrum for the ﬁrst magic angle (and for larger angles than the ﬁrst magic angle). Fig-ures 14,15, and 16show the spectrum for several values of w 0,w1around (or larger than) the ﬁrst magic angle character- ized by w0=1/√ 3f o rt h e KM-centered model and by w0= w1=1/√ 3f o rt h e /Gamma1M-centered model model in Sec. III.F o r theKM-centered model, the magic angle does not depend on w1but for the /Gamma1M-centered model it does, see Sec. III.F o r either w0orw1/lessorequalslant1/√ 3, we see that the spectrum looks com- pletely unchanged from n=2t o n=4 shells. From n=2 ton=4 shells, the largest change is smaller than 1%, and invisible to the naked eye. Above n=4 shells, the spectrumis numerically the same within machine precision. We con- ﬁrm our ﬁrst conclusion: To obtain a faithful model for TBG around the ﬁrst magic angle, we can safely neglect all shellsabove n =2.Keeping the n=2 shells gives us a Hamiltonian which contains the A1,B1,A2,B2 shells in Fig. 4(a),g i v i n g a Hamiltonian that is a 72 ×72 matrix, too large for analytic tackling. Hence further approximations are necessary, as perSecs. II DandII E, which we further numerically conﬁrm. (2) We conﬁrmed our perturbation theory predictions of Secs. II D and II E for angles smaller than the ﬁrst magic angle. In Fig. 17we conﬁrm the analytic prediction that at angle 1 /ntimes the ﬁrst magic angle, we can neglect all the shells above n+1. (3) We conﬁrmed our perturbation theory predictions Secs. II DandII Ethat—for the ﬁrst magic angle and below (w 0,w1/lessorequalslant1/√ 3)—keeping only the ﬁrst shell induces only a 20% error in the band structure. We have already established that keeping up to n=2 shells at the ﬁrst magic angle gives the correct band structure within less than 5%. Figures 14, 15, and 16also contain the n=1 shells band structure for a range of angles around and above the ﬁrst magic anglew 0,w1/greaterorequalslant1/√ 3. We see that the band structures differ little to very little, while keeping the main characteristics, from n=1 ton=2. In particular, in the chiral limit of w0=0 and for w1=1/2 (along what we call the second magic manifold ,s e e 205411-20TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) FIG. 16. Plots of the band structure for different parameters around the ﬁrst magic angle, and for different ranges of the yaxis. Notice no change from n=2t on=4, in agreement with the theoretical discussions. Sec. IV) the band structures do not visibly differ at all (see Fig.15,l o w e s tr o w )f r o m n=1t on=2.Hence for the ﬁrst magic angle, to make analytic progress, we will consider onlythe n=1shell, to a good approximation. This gives a 24 ×24 Hamiltonian, which is still analytically unsolvable. Hencefurther approximations are necessary, such as H Approx1 (k)i n Eq. ( 33). (4) We test the prediction that HApprox1 (k)i nE q .( 33) approximates well the band structure of TBG around (andfor angles larger than) the magic angle for a series ofvalues of w 0,w1/lessorequalslant1/√ 3, Figs. 18,19, and 20.W es e e remarkable agreement between HApprox1 (k) and the n=1 Hamiltonian. We also see good agreement with the large shelllimit. For values of the parameters w 0=0,w1=1 2in the second magic manifold (see Sec. IV), the HApprox1 (k) and the n=1,2,3,... shells give rise to bands undistinguishable by eye (see Fig. 19,l a s tr o w ) . We will hence use H Approx 1(k) as our TBG Hamiltonian. This is a 12 ×12 Hamiltonian that cannot be solved analytically. Hence further analytic approximations are necessary. APPENDIX C: EIGENSTATES OF THE HEXAGON MODEL AT THE /Gamma1MPOINT We provide the explicit expressions for the six-band model approximation for the hexagon model at w0=w1=1/√ 3. The basis we choose is made of simultaneous eigenstates of C3zandHfor the states \|ψj(k=0,w0=w1=1√ 3)/angbracketright=ψEjj=1,..., 6 in Eq. ( 41): ψE1=⎛ ⎜⎜⎜⎜⎜⎝ζ 1 e−i(2π/3)σzη1 ei(2π/3)σzζ1 η1 e−i(2π/3)σzζ1 ei(2π/3)σzη1⎞ ⎟⎟⎟⎟⎟⎠,ζ 1=1 2√ 2/parenleftbigg 1 1/parenrightbigg ,η 1=1√ 3(−2iσz−σy)ζ1=1 2√ 6/parenleftbigg −i i/parenrightbigg , (C1) 205411-21BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) FIG. 17. Plots of the band structure for different parameters far away from the ﬁrst magic angle: at half, a third, and a fourth of the ﬁrst magic angle. Notice that for an angle 1 /ntimes the magic angle we can neglect all shells above n+1, which conﬁrms our perturbation theory result. For the ﬁrst magic angle, above n=2 shells, the band structure goes not change. For half the magic angle, the band structure above n=3 shells does not change (but the band structure at n=2 shells is changed compared to the n=3 band structure). For a third of the magic angle, the band structure above n=4 shells does not change (but the band structure at n=2,3 shells is changed compared to the n=4b a n d structure. For a quarter of the magic angle, the band structure above n=5 shells does not change (but the band structure at n=2,3,4 shells is changed—dramatically—compared to the n=6 band structure. ψE2=⎛ ⎜⎜⎜⎜⎜⎝ζ 2 e−i(2π/3)σzη2 ei(2π/3)σzζ2 η2 e−i(2π/3)σzζ2 ei(2π/3)σzη2⎞ ⎟⎟⎟⎟⎟⎠,ζ 2=1 2√ 6/parenleftbigg 1 −1/parenrightbigg ,η 2=1√ 3(−2iσz−σy)ζ2=1 2√ 2/parenleftbigg −i −i/parenrightbigg , (C2) ψE3=⎛ ⎜⎜⎜⎜⎜⎝ζ 3 e−i(2π/3)(σz−σ0)η3 ei(2π/3)(σz−σ0)ζ3 η3 e−i(2π/3)(σz−σ0)ζ3 ei(2π/3)(σz−σ0)η3⎞ ⎟⎟⎟⎟⎟⎠,ζ 3=1/radicalBig 26(5−√ 13)/parenleftbigg2 3−√ 13/parenrightbigg , η3=1√ 3/parenleftbiggσy 2+3i 2σx+iσz/parenrightbigg ζ3=i/radicalBig 78(5−√ 13)/parenleftbigg 5−√ 13 1+√ 13/parenrightbigg , (C3) ψE4=⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ζ 4 e−i(2π/3)(σz−σ0)η4 ei(2π/3)(σz−σ0)ζ4 η4 e−i(2π/3)(σz−σ0)ζ4 ei(2π/3)(σz−σ0)η4⎞ ⎟⎟⎟⎟⎟⎟⎟⎠,ζ 4=1/radicalBig 26(5+√ 13)/parenleftbigg2 3+√ 13/parenrightbigg , 205411-22TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) FIG. 18. Plots of the band structure of HApprox1 for different parameters around the ﬁrst magic angle, and for different ranges of the yaxis. For convenience we also replot the n=1,2,3 shells band structure. Notice the good agreement of HApprox1 with the n=1 shell Hamiltonian, and, further on, the good approximation of the n=2,3 band structures by this Hamiltonian. For the chiral limit w0=9/10√ 3,w1=/radicalbig 1+w2 0/2, the approximate HApprox1 is a remarkably good approximation of the n=1 shell and a good approximation to the thermodynamic limit, albeit with the Dirac point slightly shifted. η4=1√ 3/parenleftbiggσy 2+3i 2σx+iσz/parenrightbigg ζ4=i/radicalBig 78(5+√ 13)/parenleftbigg 5+√ 13 1−√ 13/parenrightbigg , (C4) ψE5=⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ζ 5 e−i(2π/3)(σz+σ0)η5 ei(2π/3)(σz+σ0)ζ5 η5 e−i(2π/3)(σz+σ0)ζ5 ei(2π/3)(σz+σ0)η5⎞ ⎟⎟⎟⎟⎟⎟⎟⎠,ζ 5=1/radicalBig 26(5−√ 13)/parenleftbigg3−√ 13 2/parenrightbigg , η5=1√ 3/parenleftbiggσy 2−3i 2σx+iσz/parenrightbigg ζ5=−i/radicalBig 78(5−√ 13)/parenleftbigg 1+√ 13 5−√ 13/parenrightbigg , (C5) ψE6=⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ζ 6 e−i(2π/3)(σz+σ0)η6 ei(2π/3)(σz+σ0)ζ6 η6 e−i(2π/3)(σz+σ0)ζ6 ei(2π/3)(σz+σ0)η6⎞ ⎟⎟⎟⎟⎟⎟⎟⎠,ζ 6=1/radicalBig 26(5+√ 13)/parenleftbigg 3+√ 13 2/parenrightbigg , η6=1√ 3/parenleftbiggσy 2+3i 2σx+iσz/parenrightbigg ζ6=−i/radicalBig 78(5+√ 13)/parenleftbigg 1−√ 13 5+√ 13/parenrightbigg . (C6) 205411-23BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) FIG. 19. Plots of the band structure of HApprox1 for different parameters around the ﬁrst magic angle, and for different ranges of the yaxis, which helps us focus on different bands. For convenience we also replot the n=1,2,3 shells band structure. Notice the remarkable (almost undistinguishable by eye) agreement of HApprox1 with the n=1 shell Hamiltonian, and the, further on, good approximation of the n=2,3 band structures by this Hamiltonian. For the chiral limit w0=0,w1=1/2, the approximate HApprox1 is a remarkably good approximation of the thermodynamic limit—undistinguishable by eye—while for all other values it is a very good approximation. The Dirac point in the chiral limitw0=0,w1=/radicalbig 1+w2 0/2i sa t KMeven for the HApprox1 . The basis ψE1,ψE2hasC3z=1, the basis ψE3,ψE4hasC3z=ei2π/3, and the basis ψE5,ψE6hasC3z=e−i2π/3.T h e6b y6 Hamiltonian in Eq. ( 41) under these 6 basis takes the form H6-band ij/parenleftbigg k,w0=w1=1√ 3/parenrightbigg =⎛ ⎝02 A1k−A† 2k+ A† 1k+ 02 A3k− A2k−A† 3k+ 02⎞ ⎠, (C7) where k±=kx±iky,02is the 2 by 2 zero matrix, and A1=⎛ ⎜⎝2√ 13−13 13√ 5−√ 13√ 6√ 13+22−1√ 13(√ 13+5) 1 52(√ 13−13)/radicalbig√ 13+5/radicalBig 1 26(√ 13+4)−/radicalBig 3 13(√ 13+5)⎞ ⎟⎠, A2=⎛ ⎜⎝2√ 13−13 13√ 5−√ 13−1 52(√ 13−13)/radicalbig√ 13+5 √ 6√ 13+22−1√ 13(√ 13+5)−/radicalBig 1 26(√ 13+4)+/radicalBig 3 13(√ 13+5)⎞ ⎟⎠, A3=⎛ ⎜⎝1√ 132√ 13−5√ 6√ 13+22+√ 78√ 13+286+2 52√ 3 2√ 13−5√ 6√ 13+22+√ 78√ 13+286+2 52√ 3−2(√ 13+8)−√ 6√ 13+22+√ 78√ 13+286 26(√ 13+2)⎞ ⎟⎠.(C8) We note that ψE1,ψE2also serves as the Gamma point basis of the two-band approximation at w1=/radicalBig 1+w2 0/2 in Sec. IV. 205411-24TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) FIG. 20. Plots of the band structure of HApprox1 for different parameters around the ﬁrst magic angle, and for different ranges of the yaxis, which helps us focus on different bands. For convenience we also replot the n=1,2,3 shells band structure. Notice the remarkable (almost undistinguishable by eye) agreement of HApprox1 with the n=1 shell Hamiltonian, and the, further on, good approximation of the n=2,3 band structures by this Hamiltonian. For the chiral limit w0=0,w1=1/√ 3, the approximate HApprox1 is a remarkably good approximation of then=1 Hamiltonian, and a good approximation to the thermodynamic limit. The Dirac point is slightly moved away from the KMpoint. APPENDIX D: EIGENSTATES OF ALONG THE /Gamma1M-KMLINE kx=0 AND ON THE /Gamma1M-MMLINE ky=0 1. Eigenstates of H6-band ij [k=(0,ky),w 0=w1=1√ 3] On the /Gamma1M-KMline, the energies (already mentioned in the main text) are E6-band/bracketleftbigg k=(0,ky),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg −2/radicalbigg 3 13ky,−2/radicalbigg 3 13ky,2/radicalbigg 3 13ky,2/radicalbigg 3 13ky,0,0/parenrightbigg . (D1) The energies have eigenstates (not orthonormalized yet) ψ1;6-band/bracketleftbigg k=(0,ky),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg −1 200/radicalbigg 1 221(5570051 i√ 3−153112√ 13+1077176 i√ 39+17078669) , 191760161 i√ 3+166713618√ 13−59265370 i√ 39−527508405 200√ 2074(13477√ 13−45994), −2437915 i√ 3+698430√ 13+569554 i√ 39−3303424 100√ 22570(49√ 13−156),23i(26i−1222√ 3+86i√ 13+221√ 39) 1300√ 370,0,1/parenrightbigg , ψ2;6-band/bracketleftbigg k=(0,ky),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg1 200(−23)/radicalbigg 1 221(37641 i√ 3+808√ 13−2136 i√ 39−91159) , 205411-25BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) 23 100/radicaltp/radicalvertex/radicalvertex/radicalbt705768√ 13−8i/radicalBig 39(886369537 −160909896√ 13)+4606081 26962, 23[−881719 i√ 3+56(−687+3704 i√ 3)√ 13+52881] 600√ 22570(49√ 13−156),104(775 −596i√ 3)+529i(25√ 3+23i)√ 13 2600√ 370,1,0/parenrightbigg , ψ3;6-band/bracketleftbigg k=(0,ky),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg1 200/radicalbigg 1 221(5570051 i√ 3+8(19139 −134647 i√ 3)√ 13+17078669) , −191760161 i√ 3+166713618√ 13−59265370 i√ 39+527508405 200√ 2074(13477√ 13+45994), 2437915 i√ 3+698430√ 13+569554 i√ 39+3303424 100√ 22570(49√ 13+156),23(−1222 i√ 3+86√ 13−221i√ 39−26) 1300√ 370,0,1/parenrightbigg , ψ4;6-band/bracketleftbigg k=(0,ky),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg23 200/radicalbigg 1 221i(91159 i+37641√ 3+808i√ 13+2136√ 39), 23 100/radicaltp/radicalvertex/radicalvertex/radicalbt−705768√ 13+8i/radicalBig 39(160909896√ 13+886369537) +4606081 26962, 23i[52881 i+881719√ 3+56√ 13(3704√ 3+687i)] 600√ 22570(49√ 13+156),104(775 −596i√ 3)+529(23 −25i√ 3)√ 13 2600√ 370,1,0/parenrightbigg , ψ5;6-band/bracketleftbigg k=(0,ky),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg1 529/radicalbigg 2 51(710−19i√ 3),2 529/radicalbigg 2 1037(−2732+659i√ 3),−1 529/radicalbigg 185 61(2483 +5763 i√ 3),0,1 46(47−19i√ 3),1/parenrightbigg , ψ6;6-band/bracketleftbigg k=(0,ky),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg1 46/radicalbigg 185 17(5√ 3+11i),1 46/radicalbigg 185 1037(−57−71i√ 3),3(31−46i√ 3) 23√ 61,1,0,0/parenrightbigg . (D2) Fundamentally, what we notice is that the bands are kyindependent! 2. Eigenstates of H6-band ij [k=(kx,0),w 0=w1=1√ 3] On the /Gamma1M-MMline, the energies (already mentioned in the main text) are E6-band/bracketleftbigg k=(kx,0),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg −kx,−kx,1 26(3√ 13+13)kx,1 26(3√ 13+13)kx,−1 26(3√ 13−13)kx,−1 26(3√ 13−13)kx/parenrightbigg . (D3) The energies have eigenstates (not orthonormalized yet) ψ1;6-band/bracketleftbigg k=(kx,0),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg −219√ 3+115i 52√ 34,1609−63i√ 3 52√ 2074,3(1253 +41i√ 3) 52√ 22570,69(−5−3i√ 3) 52√ 370,0,1/parenrightbigg , 205411-26TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) ψ2;6-band/bracketleftbigg k=(kx,0),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg69/radicalBig 3 34 26,69(9−i√ 3) 52√ 2074,−23i(√ 3−151i) 52√ 22570,277−112i√ 3 26√ 370,1,0/parenrightbigg , ψ3;6-band/bracketleftbigg k=(kx,0),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg7(−10569 i√ 3+17434√ 13−2949 i√ 39+62876)√ 34(3√ 3−i)(323√ 13−65),481425 i√ 3+307265√ 13+145119 i√ 39+1454167 4√ 2074(323√ 13−65), ×9i(10385 i+10526√ 3+4333 i√ 13+736√ 39) 2√ 22570(61√ 13−247),69(169 i√ 3+8√ 13−45i√ 39+26) 52√ 370(8√ 13−29),0,1/parenrightbigg , ψ4;6-band/bracketleftbigg k=(kx,0),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg69(−1679 i√ 3+5303√ 13−457i√ 39+19129) 2√ 34(3√ 3−i)(323√ 13−65),69(6479 i√ 3+3374√ 13+1939 i√ 39+12004) 2√ 2074(323√ 13−65), 23i(16877 i+3295√ 3+4843 i√ 13+2705√ 39) 4√ 22570(61√ 13−247),−36205 i√ 3−14941√ 13+10699 i√ 39+64675 104√ 370(8√ 13−29),1,0/parenrightbigg , ψ5;6-band/bracketleftbigg k=(kx,0),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg69(−1679 i√ 3+5303√ 13−457i√ 39+19129) 2√ 34(3√ 3−i)(323√ 13−65),69(6479 i√ 3+3374√ 13+1939 i√ 39+12004) 2√ 2074(323√ 13−65), 23i(16877 i+3295√ 3+4843 i√ 13+2705√ 39) 4√ 22570(61√ 13−247),−36205 i√ 3−14941√ 13+10699 i√ 39+64675 104√ 370(8√ 13−29),1,0/parenrightbigg , ψ6;6-band/bracketleftbigg k=(kx,0),w0=w1=1√ 3/bracketrightbigg =/parenleftbigg69(1679 i√ 3+5303√ 13−457i√ 39−19129) 2√ 34(3√ 3−i)(323√ 13+65),69(−6479 i√ 3+3374√ 13+1939 i√ 39−12004) 2√ 2074(323√ 13+65), 23(−3295 i√ 3−4843√ 13+2705 i√ 39+16877) 4√ 22570(61√ 13+247),i(64675 i+36205√ 3+14941 i√ 13+10699√ 39) 104√ 370(8√ 13+29),1,0/parenrightbigg .(D4) Fundamentally, what we notice is that the bands are kxindependent! APPENDIX E: SOLUTIONS OF EIGENSTATES FOR THE HEXAGON MODEL We now solve the eigenvalue equation HHex(k,w0,w1)ψ=Eψ (E1) for the hexagon model in Eq. ( 40) in the basis ψ(k,w0,w1)=(ψA11,ψA12,ψA13,ψA14,ψA15,ψA16)(k,w0,w1) where each ψA1i(k,w0,w1) is a two-component spinor of Fig. 8, for different values of k,w0,w1. 1. Eigenstate solution at k =0 for arbitrary w0,w 1 The eigenvalue equation cannot be solved for general k,w0,w1and we hence concentrate on several cases. First, we only can solve only the k=0 point. Using \|/vectorqi·/vectorσ\|=1, we ﬁnd ψ6=E+q2·σ E2−1(T1ψ5+T3ψ1),ψ 4=E+q1·σ E2−1(T3ψ3+T2ψ5),ψ 2=E+q3·σ E2−1(T2ψ1+T1ψ3), /bracketleftbig (E+q3·σ)(E2−1)−E/parenleftbig T2 2+T2 1/parenrightbig −T2q1·σT2−T1q2·σT1/bracketrightbig ψ5=T2(E+q1·σ)T3ψ3+T1(E+q2·σ)T3ψ1, /bracketleftbig (E+q2·σ)(E2−1)−E/parenleftbig T2 1+T2 3/parenrightbig −T1q3·σT1−T3q1·σT3/bracketrightbig ψ3=T1(E+q3·σ)T2ψ1+T3(E+q1·σ)T2ψ5, /bracketleftbig (E+q1·σ)(E2−1)−E/parenleftbig T2 2+T2 3/parenrightbig −T2q3·σT2−T3q2·σT3/bracketrightbig ψ1=T2(E+q3·σ)T1ψ3+T3(E+q2·σ)T1ψ5,(E2) 205411-27BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) where shorthand notation Ti=Ti(w0,w1),ψi=ψA1i(k=0,w0,w1). Using the expressions of Tifrom Eq. ( 4), we rewrite the last three equations above as /bracketleftbig E(E2−1)σ0+q3·σ/parenleftbig E2−1+w2 0+2w2 1/parenrightbig −E/parenleftbig 2/parenleftbig w2 0+w2 1/parenrightbig σ0+w0w1(σx+√ 3σy)/parenrightbig/bracketrightbig ψ5 =/braceleftbigg E/bracketleftbigg/parenleftbigg w2 0−w2 1 2/parenrightbigg σ0−w0w1σx+i√ 3 2w2 1σz/bracketrightbigg +/parenleftbig w2 0−w2 1/parenrightbig q1·σ/bracerightbigg ψ3 +/braceleftbigg E/bracketleftbigg/parenleftbigg w2 0−w2 1 2/parenrightbigg σ0+w0w11 2(σx−√ 3σy)−i√ 3 2w2 1σz/bracketrightbigg +/parenleftbig w2 0−w2 1/parenrightbig q2·σ/bracerightbigg ψ1, /bracketleftbig E(E2−1)σ0+q2·σ/parenleftbig E2−1+w2 0+2w2 1/parenrightbig −E/parenleftbig 2/parenleftbig w2 0+w2 1/parenrightbig σ0+w0w1(σx−√ 3σy)/parenrightbig/bracketrightbig ψ3 =/braceleftbigg E/bracketleftbigg/parenleftbigg w2 0−w2 1 2/parenrightbigg σ0+w0w11 2(σx+√ 3σy)+i√ 3 2w2 1σz/bracketrightbigg +/parenleftbig w2 0−w2 1/parenrightbig q3·σ/bracerightbigg ψ1 +/braceleftbigg E/bracketleftbigg/parenleftbigg w2 0−w2 1 2/parenrightbigg σ0−w0w1σx−i√ 3 2w2 1σz/bracketrightbigg +/parenleftbig w2 0−w2 1/parenrightbig q1·σ/bracerightbigg ψ5, /bracketleftbig E(E2−1)σ0+q1·σ/parenleftbig E2−1+w2 0+2w2 1/parenrightbig −E/parenleftbig 2/parenleftbig w2 0+w2 1/parenrightbig σ0−2w0w1σx/parenrightbig/bracketrightbig ψ1 =/braceleftbigg E/bracketleftbigg/parenleftbigg w2 0−w2 1 2/parenrightbigg σ0+w0w11 2(σx+√ 3σy)−i√ 3 2w2 1σz/bracketrightbigg +/parenleftbig w2 0−w2 1/parenrightbig q3·σ/bracerightbigg ψ3 +/braceleftbigg E/bracketleftbigg/parenleftbigg w2 0−w2 1 2/parenrightbigg σ0+w0w11 2(σx−√ 3σy)+i√ 3 2w2 1σz/bracketrightbigg +/parenleftbig w2 0−w2 1/parenrightbig q2·σ/bracerightbigg ψ5. (E3) Plugging in the expressions for the energy E, we can obtain the relations between ψi. However, these are messy, and we choose to ﬁnd the eigenstates on several, simpler, manifolds in the w0,w1parameter space. 2. Eigenstate solution at k =0 for on the second magic manifold w1=/radicalbig 1+w2 0/2 We ﬁrst solve for the two zero eigenstates E1,2(k=0,w0,w1=√ 1+w2 0 2)=0 of Table I. Equation ( E2) becomes /parenleftbig 3w2 0−1/parenrightbig q3·σψ 5=/parenleftbig 3w2 0−1/parenrightbig 2(q1·σψ 3+q2·σψ 1), /parenleftbig 3w2 0−1/parenrightbig q2·σψ 3=/parenleftbig 3w2 0−1/parenrightbig 2(q3·σψ 1+q1·σψ 5), /parenleftbig 3w2 0−1/parenrightbig q1·σψ 1=/parenleftbig 3w2 0−1/parenrightbig 2(q3·σψ 3+q2·σψ 5). (E4) We now have two cases. a. Zero energy eigenstate solution at k=0for on the second magic manifold w1=/radicalbig 1+w2 0/2,w0/negationslash=1/√ 3 In this case 3 w2 0−1/negationslash=0 and Eq. ( E4) becomes q3·σψ 5=1 2(q1·σψ 3+q2·σψ 1);q2·σψ 3=1 2(q3·σψ 1+q1·σψ 5);q1·σψ 1=1 2(q3·σψ 3+q2·σψ 5), (E5) with solutions (for the two zero energy eigenstates) ψ1=(q3·σ)(q2·σ)ψ3; ψ5=(q2·σ)(q3·σ)ψ3; ψ4=−q1·σ[T3+T2(q2·σ)(q3·σ)]ψ3; ψ2=−q3·σ[T1+T2(q3·σ)(q2·σ)]ψ3; ψ6=−q2·σ[T3(q3·σ)(q2·σ)+T1(q2·σ)(q3·σ)]ψ3. (E6) The two independent zero energy eigenstates on the second magic manifold can be obtained by taking ψ3=(1,0) and ψ3= (0,1), respectively. However, they are not orthonormal and a further Gram-Schmidt must be performed to orthogonalize them. 205411-28TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) We obtain ψE1=0⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =/parenleftBigg −i(√ 3−i) 2√ 6/radicalBig w2 0+1,0,−6√−1√ 6,iw0 √ 6/radicalBig w2 0+1,1 √ 6/radicalBig w2 0+1,0,−(−1)5/6 √ 6, −(−1)5/6w0 √ 6/radicalBig w2 0+1,i(√ 3+i) 2√ 6/radicalBig w2 0+1,0,i√ 6,−6√−1w0 √ 6/radicalBig w2 0+1/parenrightBigg , ψE2=0⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =/parenleftBigg i(√ 3+i)w0 2√ 6/radicalBig w2 0+1,i(√ 3+i) 2√ 6,0,(−1)5/6 √ 6/radicalBig w2 0+1, −3√−1w0 √ 6/radicalBig w2 0+1,1√ 6,0,6√−1 √ 6/radicalBig w2 0+1,w0 √ 6/radicalBig w2 0+1,−i(√ 3−i) 2√ 6,0,−i √ 6/radicalBig w2 0+1/parenrightBigg . (E7) b. Nonzero energy eigenstate solutions at k=0for on the second magic manifold w1=/radicalbig 1+w2 0/2,w0/negationslash=1/√ 3 We can adopt the same strategy to build the other, nonzero energy orthonormal eigenstates. It is tedious (analytic diagonal- ization programs such as Mathematica fail to provide a result, hence the algebra must be performed by hand) to write the details,but the ﬁnal answer is, for the eigenstates of energies on the ﬁrst magic manifold given in Table II: ψ E3⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =1 4√ 6/radicalBig/parenleftbig w2 0+4)/parenleftbig 10w2 0+1/parenrightbig/bracketleftbig (√ 3+3i)/parenleftbig√ 3w2 0+i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,(√ 3+3i)/parenleftbig −2/radicalBig 10w2 0+1+iw0/radicalBig w2 0+1/parenrightbig , −(√ 3−3i)/parenleftbig 2/radicalBig w2 0+1+√ 3w0/radicalBig w2 0+4−iw0/radicalBig 10w2 0+1/parenrightbig ,−2i/parenleftbig√ 3/radicalBig w2 0+1/radicalBig w2 0+4+6w0/parenrightbig ,12w2 0,0, −(√ 3+3i)/parenleftbig −2/radicalBig w2 0+1+√ 3w0/radicalBig w2 0+4+iw0/radicalBig 10w2 0+1/parenrightbig ,(√ 3−i)/parenleftbig√ 3/radicalBig w2 0+1/radicalBig w2 0+4−6w0/parenrightbig , (√ 3−3i)/parenleftbig√ 3w2 0−i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,−2√ 3/parenleftbig 2/radicalBig 10w2 0+1−iw0/radicalBig w2 0+1/parenrightbig ,−12w0/radicalBig w2 0+4,−12(√ 3+i)w0/bracketrightbig , ψE4⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =1 4√ 6/radicalBig/parenleftbig w2 0+4/parenrightbig/parenleftbig 10w2 0+1/parenrightbig/bracketleftbig −(√ 3+i)/parenleftbig 2/radicalBig 10w2 0+1+iw0/radicalBig w2 0+1/parenrightbig ,−(√ 3+i)/parenleftbig 3√ 3w2 0+i/radicalBig 10w4 0+41w2 0+4/parenrightbig , 2/parenleftbig/radicalBig w4 0+5w2 0+4−6√ 3w0/parenrightbig ,(√ 3−i)/parenleftbig −2/radicalBig w2 0+1+3√ 3w0/radicalBig w2 0+4−iw0/radicalBig 10w2 0+1/parenrightbig , 4(−1)5/6/parenleftbig 2/radicalBig 10w2 0+1+iw0/radicalBig w2 0+1/parenrightbig ,4/radicalBig 10w4 0+41w2 0+4,i(√ 3+i)/radicalBig w4 0+5w2 0+4−6(√ 3−3i)w0, (√ 3+i)/parenleftbig 2/radicalBig w2 0+1+3√ 3w0/radicalBig w2 0+4+iw0/radicalBig 10w2 0+1/parenrightbig ,−2w0/radicalBig w2 0+1+4i/radicalBig 10w2 0+1, −(√ 3−i)/parenleftbig 3√ 3w2 0−i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,2(1+i√ 3)/radicalBig w4 0+5w2 0+4,−4w0/radicalBig 10w2 0+1+8i/radicalBig w2 0+1/bracketrightbig , (E8) 205411-29BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) ψE5⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =1 4√ 6/radicalBig/parenleftbig w2 0+4/parenrightbig/parenleftbig 10w2 0+1/parenrightbig/bracketleftbig (√ 3+3i)/parenleftbig√ 3w2 0+i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,(√ 3+3i)/parenleftbig 2/radicalBig 10w2 0+1+iw0/radicalBig w2 0+1/parenrightbig , (√ 3−3i)/parenleftbig −2/radicalBig w2 0+1+√ 3w0/radicalBig w2 0+4−iw0/radicalBig 10w2 0+1/parenrightbig ,2i/parenleftbig√ 3/radicalBig w2 0+1/radicalBig w2 0+4−6w0/parenrightbig ,12w2 0,0, (√ 3+3i)/parenleftbig 2/radicalBig w2 0+1+√ 3w0/radicalBig w2 0+4+iw0/radicalBig 10w2 0+1/parenrightbig ,i(√ 3+3i)/radicalBig w2 0+1/radicalBig w2 0+4−6(√ 3−i)w0, (√ 3−3i)/parenleftbig√ 3w2 0−i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,2√ 3/parenleftbig 2/radicalBig 10w2 0+1+iw0/radicalBig w2 0+1/parenrightbig ,12w0/radicalBig w2 0+4,−12(√ 3+i)w0/bracketrightbig , ψE6⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =1 4√ 6/radicalBig/parenleftbig w2 0+4/parenrightbig/parenleftbig 10w2 0+1/parenrightbig/bracketleftbig 26√ −1/parenleftbig 2/radicalBig 10w2 0+1−iw0/radicalBig w2 0+1/parenrightbig ,−(√ 3+i)/parenleftbig 3√ 3w2 0+i/radicalBig 10w4 0+41w2 0+4/parenrightbig , −2/parenleftbig/radicalBig w4 0+5w2 0+4+6√ 3w0/parenrightbig ,−(√ 3−i)/parenleftbig 2/radicalBig w2 0+1+3√ 3w0/radicalBig w2 0+4−iw0/radicalBig 10w2 0+1/parenrightbig , −4(−1)5/6/parenleftbig 2/radicalBig 10w2 0+1−iw0/radicalBig w2 0+1/parenrightbig ,4/radicalBig 10w4 0+41w2 0+4,(1−i√ 3)/radicalBig w4 0+5w2 0+4−6(√ 3−3i)w0, −(√ 3+i)/parenleftbig −2/radicalBig w2 0+1+3√ 3w0/radicalBig w2 0+4+iw0/radicalBig 10w2 0+1/parenrightbig ,−2w0/radicalBig w2 0+1−4i/radicalBig 10w2 0+1, −(√ 3−i)/parenleftbig 3√ 3w2 0−i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,−2i(√ 3−i)/radicalBig w4 0+5w2 0+4,4w0/radicalBig 10w2 0+1+8i/radicalBig w2 0+1/bracketrightbig , (E9) ψE7⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =1 4√ 6/radicalBig/parenleftbig w2 0+4/parenrightbig/parenleftbig 10w2 0+1/parenrightbig/bracketleftbig (√ 3+3i)/parenleftbig√ 3w2 0−i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,(√ 3+3i)/parenleftbig 2/radicalBig 10w2 0+1+iw0/radicalBig w2 0+1/parenrightbig , −/parenleftbig√ 3−3i/parenrightbig/parenleftbig 2/radicalBig w2 0+1+√ 3w0/radicalBig w2 0+4+iw0/radicalBig 10w2 0+1/parenrightbig ,−2i/parenleftbig√ 3/radicalBig w2 0+1/radicalBig w2 0+4+6w0/parenrightbig ,12w2 0,0, −(√ 3+3i)/parenleftbig −2/radicalBig w2 0+1+√ 3w0/radicalBig w2 0+4−iw0/radicalBig 10w2 0+1/parenrightbig ,(√ 3−i)/parenleftbig√ 3/radicalBig w2 0+1/radicalBig w2 0+4−6w0/parenrightbig , (√ 3−3i)/parenleftbig√ 3w2 0+i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,2√ 3/parenleftbig 2/radicalBig 10w2 0+1+iw0/radicalBig w2 0+1/parenrightbig ,−12w0/radicalBig w2 0+4,−12(√ 3+i)w0/bracketrightbig , ψE8⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =1 4√ 6/radicalBig/parenleftbig w2 0+4/parenrightbig/parenleftbig 10w2 0+1/parenrightbig/bracketleftbig −(√ 3+i)/parenleftbig 2/radicalBig 10w2 0+1−iw0/radicalBig w2 0+1/parenrightbig ,(√ 3+i)/parenleftbig 3√ 3w2 0−i/radicalBig 10w4 0+41w2 0+4/parenrightbig , −2/parenleftbig/radicalBig w4 0+5w2 0+4−6√ 3w0/parenrightbig ,−(√ 3−i)/parenleftbig −2/radicalBig w2 0+1+3√ 3w0/radicalBig w2 0+4+iw0/radicalBig 10w2 0+1/parenrightbig , −2(√ 3−i)/parenleftbig 2/radicalBig 10w2 0+1−iw0/radicalBig w2 0+1/parenrightbig ,4/radicalBig 10w4 0+41w2 0+4,/parenleftbig 1−i√ 3/parenrightbig/radicalBig w4 0+5w2 0+4+6/parenleftbig√ 3−3i/parenrightbig w0, −(√ 3+i)/parenleftbig 2/radicalBig w2 0+1+3√ 3w0/radicalBig w2 0+4−iw0/radicalBig 10w2 0+1/parenrightbig ,2w0/radicalBig w2 0+1+4i/radicalBig 10w2 0+1, (√ 3−i)/parenleftbig 3√ 3w2 0+i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,−2i(√ 3−i)/radicalBig w4 0+5w2 0+4,−4w0/radicalBig 10w2 0+1−8i/radicalBig w2 0+1/bracketrightbig , (E10) 205411-30TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) ψE9⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =1 4√ 6/radicalBig/parenleftbig w2 0+4/parenrightbig/parenleftbig 10w2 0+1/parenrightbig/bracketleftbig (√ 3+3i)/parenleftbig√ 3w2 0−i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,−(√ 3+3i)/parenleftbig 2/radicalBig 10w2 0+1−iw0/radicalBig w2 0+1/parenrightbig , (√ 3−3i)/parenleftbig −2/radicalBig w2 0+1+√ 3w0/radicalBig w2 0+4+iw0/radicalBig 10w2 0+1/parenrightbig ,2i/parenleftbig√ 3/radicalBig w2 0+1/radicalBig w2 0+4−6w0/parenrightbig ,12w2 0,0, (√ 3+3i)/parenleftbig 2/radicalBig w2 0+1+√ 3w0/radicalBig w2 0+4−iw0/radicalBig 10w2 0+1/parenrightbig ,−(√ 3−i)/parenleftbig√ 3/radicalBig w2 0+1/radicalBig w2 0+4+6w0/parenrightbig , (√ 3−3i)/parenleftbig√ 3w2 0+i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,−2√ 3/parenleftbig 2/radicalBig 10w2 0+1−iw0/radicalBig w2 0+1/parenrightbig ,12w0/radicalBig w2 0+4,−12(√ 3+i)w0/bracketrightbig , ψE10⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =1 4√ 6/radicalBig/parenleftbig w2 0+4/parenrightbig/parenleftbig 10w2 0+1/parenrightbig/bracketleftbig (√ 3+i)/parenleftbig 2/radicalBig 10w2 0+1+iw0/radicalBig w2 0+1/parenrightbig ,(√ 3+i)/parenleftbig 3√ 3w2 0−i/radicalBig 10w4 0+41w2 0+4/parenrightbig , 2/parenleftbig/radicalBig w4 0+5w2 0+4+6√ 3w0/parenrightbig ,(√ 3−i)/parenleftbig 2/radicalBig w2 0+1+3√ 3w0/radicalBig w2 0+4+iw0/radicalBig 10w2 0+1/parenrightbig , 43√ −1/parenleftbig w0/radicalBig w2 0+1−2i/radicalBig 10w2 0+1/parenrightbig ,4/radicalBig 10w4 0+41w2 0+4,i(√ 3+i)/radicalBig w4 0+5w2 0+4+6(√ 3−3i)w0, (√ 3+i)/parenleftbig −2/radicalBig w2 0+1+3√ 3w0/radicalBig w2 0+4−iw0/radicalBig 10w2 0+1/parenrightbig ,2w0/radicalBig w2 0+1−4i/radicalBig 10w2 0+1, (√ 3−i)/parenleftbig 3√ 3w2 0+i/radicalBig 10w4 0+41w2 0+4/parenrightbig ,2(1+i√ 3)/radicalBig w4 0+5w2 0+4,4w0/radicalBig 10w2 0+1−8i/radicalBig w2 0+1/bracketrightbig , (E11) ψE11⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =/bracketleftbigg(√ 3−3i)(w0+i) 12/radicalBig w2 0+1,1 12(−√ 3+3i),−1 2√ 3,−3√−1(w0+i) 2√ 3/radicalBig w2 0+1,(√ 3+3i)(w0+i) 12/radicalBig w2 0+1,1 2√ 3, 1 12(√ 3−3i),−(√ 3−3i)(w0+i) 12/radicalBig w2 0+1,−w0+i 2√ 3/radicalBig w2 0+1,1 12(−√ 3−3i),1 12(√ 3+3i),w0+i 2√ 3/radicalBig w2 0+1/bracketrightbigg , (E12) ψE12⎛ ⎝k=0,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠ =/bracketleftBigg (√ 3−3i)(w0−i) 12/radicalBig w2 0+1,1 12(−√ 3+3i),1 2√ 3,(√ 3+3i)(w0−i) 12/radicalBig w2 0+1,(√ 3+3i)(w0−i) 12/radicalBig w2 0+1,1 2√ 3, 1 12(−√ 3+3i),(√ 3−3i)(w0−i) 12/radicalBig w2 0+1,−w0−i 2√ 3/radicalBig w2 0+1,1 12(−√ 3−3i),1 12(−√ 3−3i),−w0−i 2√ 3/radicalBig w2 0+1/bracketrightBigg . (E13) c. Zero energy eigenstate solution at k=0for on the second magic manifold w1=/radicalbig 1+w2 0/2=w0=1/√ 3 There are six zero energies in Table Iat this point w1=/radicalBig 1+w2 0/2=w0=1/√ 3. They have already been given in Appendix C. 205411-31BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) APPENDIX F: PERTURBATION THEORY FOR H(1) mm/prime(k,w 0)=0,Em=0 MANIFOLD 1. Review of perturbation theory We review the perturbation theory being performed in the main text. This formalism was ﬁrst presented in Ref. [ 123], but we go to higher order in current perturbation theory. We have a Hamiltonian H0whose eigenstates we know, and is hence purely diagonal in its eigenstate basis. We also have a perturbation Hamiltonian H/prime, with both diagonal and off-diagonal elements. Among the eigenstates of H0we have a set of eigenstates separated by a large gap from the others, which cannot be closed by the addition of H/prime, and they represent the manifold we want to project in. These states are indexed by m,m/prime,m/prime/prime,m/prime/prime/prime,... while the rest of the eigenstates are indexed by l,l/prime,l/prime/prime,l/prime/prime/prime,.... These two form separate subspaces. We now want to ﬁnd a Hamiltonian Hmm/primewhich incorporates the effects of H/primeup to any desired order. We separate H/primeinto a diagonal part H1plus an off-diagonal partH2between these manifolds: H/prime=H1+H2, (H1)mm/prime=/angbracketleftψm\|H/prime\|ψm/prime/angbracketright;(H1)ll/prime=/angbracketleftψl\|H/prime\|ψl/prime/angbracketright;(H2)ml=/angbracketleftψm\|H/prime\|ψl/angbracketright;(H2)mm/prime=(H2)ll/prime=(H1)ml=0. (F1) We also have H\|ψm/angbracketright=Em\|ψm/angbracketright,H\|ψl/angbracketright=El\|ψl/angbracketright. (F2) We look for a unitary transformation: ˜H=e−S(H0+H/prime)eS, (F3) where S(=−S†) has only matrix elements that are off-diagonal between the subspaces, i.e., Sml=0. The unitary transformation is chosen such that the off-diagonal part of ˜His zero to the desired order ( Hml=0). Since we know S,H2are off-diagonal and H1is diagonal, we ﬁnd that Scan be obtained from the condition ˜Hoff-diagonal =∞/summationdisplay j=01 (2j+1)![H0+H1,S]2j+1+∞/summationdisplay j=01 (2j)![H2,S]2j=0 (F4) (the off-diagonal Hamiltonian is zero). Once Sis found, the diagonal Hamiltonian is ˜Hdiagonal =∞/summationdisplay j=01 (2j)![H0+H1,S]2j+∞/summationdisplay j=01 (2j+1)![H2,S]2j+1, (F5) where [ A,B]j=[[[[[A,B],B],B],...],B] where the number of B’s is equal to j. We then parametrize S=S1+S2+S3+··· , where Snis order nin perturbation theory, i.e., in H/prime(or equivalently, in H1orH2). The terms up to order 4 are derived in Winkler’s book [ 123], and for our simpliﬁed problem, they are presented in the main text. We have numerically checked their correctness. We here also present the ﬁfth order term: this term is tedious, but we usea particularly nice property of our eigenstate space that ( H 1)mm/prime=/angbracketleftψm\|H/prime\|ψm/prime/angbracketright=0,Em=0f o r m=1,2 property is true only forH/prime=I6×6⊗k·σand for the zero energy eigenstates ψm,m=1,2o f H0=HHex(k=0,w0,w1=/radicalBig 1+w2 0/2). To the desired order, we ﬁnd (S1)ml=H/prime ml El,(S1)lm=−H/prime lm El, (S2)ml=−/summationdisplay l/primeH/prime ml/primeH/prime l/primel ElEl/prime,(S2)lm=/summationdisplay l/primeH/prime ll/primeH/prime l/primem ElEl/prime, (S3)ml=/summationdisplay l/prime,l/prime/primeH/prime ml/primeHl/primel/prime/primeHl/primel ElEl/primeEl/prime/prime−1 3/summationdisplay l/primem/primeH/prime ml/primeHl/primem/primeHm/primel/parenleftbigg3 E2 lEl/prime+1 E2 l/primeEl/parenrightbigg , (S3)lm=−/summationdisplay l/prime,l/prime/primeH/prime ll/primeHl/primel/prime/primeHl/prime/primem ElEl/primeEl/prime/prime+1 3/summationdisplay l/primem/primeH/prime lm/primeHm/primel/primeHl/primem/parenleftbigg3 E2 lEl/prime+1 E2 l/primeEl/parenrightbigg . (F6) Due to our property ( H1)mm/prime=/angbracketleftψm\|H/prime\|ψm/prime/angbracketright=0,Em=0 on the second magic manifold, we ﬁnd that the fourth order S4is not needed in order to obtain the ﬁfth order diagonal Hamiltonian, as terms in the expression of the Hamiltonian that contain itcancel. We ﬁnd that the ﬁfth order Hamiltonian is ˜H (5) diagonal=−S2H0S3−S3H0S2−S1H1S3−S3H1S1−S2H2S2 −1 6/parenleftbig S1H0S1S2S1+S1H0S2S2 1+S1H0S2 1S2+S2H0S3 1+S1H1S3 1 +S1S2S1H0S1+S2S2 1H0S1+S2 1S2H0S1+S3 1H0S2+S3 1H1S1/parenrightbig 205411-32TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) +1 6/bracketleftbig H2S2S2 1+H2S2 1S2+H2S1S2S1+3/parenleftbig S1S2H2S1+S2S1H2S1+S2 1H2S2/parenrightbig −/parenleftbig S2S2 1H2+S2 1S2H2+S1S2S1H2/parenrightbig −3/parenleftbig S1H2S1S2+S1H2S2S1+S2H2S2 1/parenrightbig/bracketrightbig . (F7) The matrix elements of these terms give 1 6/bracketleftbig H2S2S2 1+H2S2 1S2+H2S1S2S1−/parenleftbig S2S2 1H2+S2 1S2H2+S1S2S1H2/parenrightbig/bracketrightbig mm/prime =−1 6/summationdisplay l,l/prime,l/prime/prime/summationdisplay m/prime/primeH/prime mlH/prime ll/primeHl/primem/prime/primeHm/prime/primel/prime/primeHl/prime/primem/prime+Hm/primel/prime/primeHl/prime/primem/prime/primeHm/prime/primel/primeHl/primelHlm ElEl/primeEl/prime/prime/parenleftbigg1 El+1 El/prime+1 El/prime/prime/parenrightbigg , (F8) 1 6/bracketleftbig 3/parenleftbig S1S2H2S1+S2S1H2S1+S2 1H2S2/parenrightbig −3/parenleftbig S1H2S1S2+S1H2S2S1+S2H2S2 1/parenrightbig/bracketrightbig mm/prime =−1 2/summationdisplay l,l/prime,l/prime/prime/summationdisplay m/prime/primeH/prime mlH/prime ll/primeHl/primem/prime/primeHm/prime/primel/prime/primeHl/prime/primem/prime+Hm/primel/prime/primeHl/prime/primem/prime/primeHm/prime/primel/primeHl/primelHlm ElEl/primeEl/prime/prime/parenleftbigg1 El+1 El/prime+1 El/prime/prime/parenrightbigg , (F9) −1 6/parenleftbig S1H0S1S2S1+S1H0S2S2 1+S1H0S2 1S2+S2H0S3 1+S1H1S3 1 +S1S2S1H0S1+S2S2 1H0S1+S2 1S2H0S1+S3 1H0S2+S3 1H1S1/parenrightbig =1 6/summationdisplay l,l/prime,l/prime/prime/summationdisplay m/prime/primeH/prime mlH/prime ll/primeH/prime l/primem/prime/primeH/prime m/prime/primel/prime/primeH/prime l/prime/primem/prime+H/prime m/primel/prime/primeH/prime l/prime/primem/prime/primeH/prime m/prime/primel/primeH/prime l/primelH/prime lm ElEl/primeEl/prime/prime/parenleftbigg1 El+1 El/prime+1 El/prime/prime/parenrightbigg , (−S2H0S3−S3H0S2−S1H1S3−S3H1S1−S2H2S2)mm/prime=/summationdisplay l,l/prime,l/prime/prime,l/prime/prime/primeH/prime mlH/prime ll/primeH/prime l/primel/prime/primeHl/prime/primel/prime/prime/primeHl/prime/prime/primem/prime ElEl/primeEl/prime/primeEl/prime/prime/prime. (F10) Hence ˜H(5) diagonal=/summationdisplay l,l/prime,l/prime/prime,l/prime/prime/primeH/prime mlH/prime ll/primeH/prime l/primel/prime/primeHl/prime/primel/prime/prime/primeHl/prime/prime/primem/prime ElEl/primeEl/prime/primeEl/prime/prime/prime−1 2/summationdisplay l,l/prime,l/prime/prime/summationdisplay m/prime/primeH/prime mlH/prime ll/primeHl/primem/prime/primeHm/prime/primel/prime/primeHl/prime/primem/prime+Hm/primel/prime/primeHl/prime/primem/prime/primeHm/prime/primel/primeHl/primelHlm ElEl/primeEl/prime/prime/parenleftbigg1 El+1 El/prime+1 El/prime/prime/parenrightbigg . (F11) 2. Calculations of the Hamiltonian matrix elements when ﬁrst order vanishes Here we calculate explicitly the perturbations of Hperturb (k,w0)=I6×6⊗k·/vectorσin Eq. ( 47)u pt oﬁ f t ho r d e r . a. First order The ﬁrst order perturbation can be easily seen to be zero: H(1) mm/prime(k,w0)=/angbracketleftψm\|Hperturb (k,w0)\|ψm/prime/angbracketright=0. (F12) b. Second order H(2) mm/prime(k,w0)=−/summationdisplay l=3...121 El/angbracketleftψm\|Hperturb (k,w0)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0)\|ψm/prime/angbracketright= −4w2 0/parenleftbig k2 x+k2 y/parenrightbig 3/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig(σy+√ 3σx).(F13) c. Third order H(3) mm/prime(k,w0)=/summationdisplay l,l/prime=3,...,121 ElEl/prime/angbracketleftψm\|Hperturb (k,w0)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0)\|ψl/prime/angbracketright/angbracketleftψl/prime\|Hperturb (k,w0)\|ψm/prime/angbracketright =4kxw0/parenleftbig w2 0−3/parenrightbig/parenleftbig k2 x−3k2 y/parenrightbig 9/parenleftbig 1−3w2 0/parenrightbig2/radicalBig w2 0+1σ0. (F14) 205411-33BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) d. Fourth order For the fourth order, there are two terms: First, H(41) mm/prime(k,w0)=−/summationdisplay l,l/prime,l/prime/prime=3,...,121 ElEl/primeEl/prime/prime/angbracketleftψm\|Hperturb (k,w0)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0)\|ψl/prime/angbracketright/angbracketleftψl/prime\|Hperturb (k,w0)\|ψl/prime/prime/angbracketright ×/angbracketleftψl/prime/prime\|Hperturb (k,w0)\|ψm/prime/angbracketright =8w2 0/parenleftbig w4 0+16w2 0−9/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig2 27/parenleftbig w2 0+1/parenrightbig3/2/parenleftbig 3w2 0−1/parenrightbig3(σy+√ 3σx). (F15) Second, H(42) mm/prime(k,w0)=/summationdisplay l,l/prime=3,...,12/summationdisplay m/prime/prime=1,21 ElEl/prime/parenleftbigg1 El+1 El/prime/parenrightbigg ×/angbracketleftψm\|Hperturb (k,w0)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0)\|ψm/prime/prime/angbracketright/angbracketleftψm/prime/prime\|Hperturb (k,w0)\|ψl/prime/angbracketright/angbracketleftψl/prime\|Hperturb (k,w0)\|ψm/prime/angbracketright =16w2 0/parenleftbig 17w2 0+9/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig2 27/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig3(σy+√ 3σx). (F16) Notice that so far, the eigenstates are not kdependent, they are just the eigenstates of ( σy+√ 3σx). e. Fifth order The ﬁfth order perturbation theory is not available in any book. Hence we derived it in Appendix F, for the special case for which the manifold mof states we project in has the ﬁrst order Hamiltonian H(1) mm/prime(k,w0)=0 and for which its energies are Em=0. The ﬁfth order also has two terms, just like the fourth order (see Appendix F). We ﬁnd /summationdisplay l,l/prime,l/prime/prime,l/prime/prime/primeH/prime mlH/prime ll/primeH/prime l/primel/prime/primeHl/prime/primel/prime/prime/primeHl/prime/prime/primem/prime ElEl/primeEl/prime/primeEl/prime/prime/prime=32kx/parenleftbig w2 0−3/parenrightbig2(2w2 0−1)w0/parenleftbig k2 x−3k2 y/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig 81/parenleftbig w2 0+1/parenrightbig3/2/parenleftbig 3w2 0−1/parenrightbig4σ0 (F17) and −1 2/summationdisplay l,l/prime,l/prime/prime/summationdisplay m/prime/prime/parenleftbig H/prime mlH/prime ll/primeHl/primem/prime/primeHm/prime/primel/prime/primeHl/prime/primem/prime+Hm/primel/prime/primeHl/prime/primem/prime/primeHm/prime/primel/primeHl/primelHlm ElEl/primeEl/prime/prime/parenleftbigg1 El+1 El/prime+1 El/prime/prime/parenrightbigg =−16kx/parenleftbig 11w4 0−94w2 0−9/parenrightbig w0/parenleftbig k2 x−3k2 y/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig 27/parenleftbig/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig4/parenrightbigσ0. (F18) We can clearly see the structure of the order nHamiltonian, as a perturbation in 1 /(3w2 0−1)n−1, with symmetry-preserving functions of k. The full two-band approximation to the hexagon Hamiltonian is, up to ﬁfth order, is HHex 2band⎛ ⎝k,w0,w1=/radicalBig 1+w2 0 2⎞ ⎠=4w2 0 3/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig/bracketleftBigg −1+2/parenleftbig 35w4 0+68w2 0+9/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig 9/parenleftbig w2 0+1/parenrightbig/parenleftbig 3w2 0−1/parenrightbig2/bracketrightBigg /parenleftbig k2 x+k2 y/parenrightbig (σy+√ 3σx) +4w0 9/radicalBig w2 0+1/parenleftbig 1−3w2 0/parenrightbig2/bracketleftbigg/parenleftbig w2 0−3/parenrightbig −4/parenleftbig 29w6 0−223w4 0−357w2 0−9/parenrightbig 9/parenleftbig 1−3w2 0/parenrightbig2/parenleftbig w2 0+1/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig/bracketrightbigg kx/parenleftbig k2 x−3k2 y/parenrightbig σ0 (F19) better expressed as HHex 2band/parenleftbigg k,w0,w1=/radicalBig 1+w2 0 2/parenrightbigg =d0(k,w0)σ0+d1(k,w0)(σy+√ 3σx), (F20) 205411-34TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) where d0(k,w0)=4w0 9/radicalBig w2 0+1/parenleftbig 1−3w2 0/parenrightbig2/bracketleftbigg/parenleftbig w2 0−3/parenrightbig −4(29w6 0−223w4 0−357w2 0−9) 9/parenleftbig 1−3w2 0/parenrightbig2/parenleftbig w2 0+1/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig/bracketrightbigg kx/parenleftbig k2 x−3k2 y/parenrightbig (F21) and d1⎛ ⎝(k,w0)=/radicalBig 1+w2 0 2⎞ ⎠=4w2 0 3/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig/bracketleftBigg −1+2(35w4 0+68w2 0+9)/parenleftbig k2 x+k2 y/parenrightbig 9/parenleftbig w2 0+1/parenrightbig/parenleftbig 3w2 0−1/parenrightbig2/bracketrightBigg /parenleftbig k2 x+k2 y/parenrightbig . (F22) 3. Calculations of the Hamiltonian matrix elements when ﬁrst order does not vanish We take the unperturbed Hamiltonian to be HHex(k=0,w0,w1=/radicalBig 1+w2 0/2) (the hexagon model on the second magic manifold) in Eq. ( 40). For this Hamiltonian we are able to obtain all the eigenstates analytically in Appendix E 2. The perturbation Hamiltonian, away from the second magic manifold, is Hperturb (k,w0,w1)=HHex(k,w0,w1)−HHex/parenleftbigg k=0,w0,w1=/radicalBig 1+w2 0 2/parenrightbigg =I6×6⊗k·/vectorσ+HHex/parenleftbigg k=0,0,w1−/radicalBig 1+w2 0 2/parenrightbigg . (F23) a. First order H(1) mm/prime(k,w0,w1)=/angbracketleftψm\|Hperturb (k,w0,w1)\|ψm/prime/angbracketright=⎛ ⎝/radicalBig w2 0+1 2−w1⎞ ⎠(σy+√ 3σx). (F24) Hence there is now a linear term in the Hamiltonian. Because of this, many other terms in the further degree perturbation theory become nonzero. b. Second order H(2) mm/prime(k,w0,w1)=−/summationdisplay l=3,...,121 El/angbracketleftψm\|Hperturb (k,w0)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0)\|ψm/prime/angbracketright =−4w2 0/parenleftbig k2 x+k2 y/parenrightbig 3/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig(σy+√ 3σx). (F25) The second order perturbation theory is unchanged! c. Third order There are now two third order terms, as the ﬁrst order perturbation terms do not vanish. First, H(31) mm/prime(k,w0,w1)=/summationdisplay l,l/prime=3...121 ElEl/prime/angbracketleftψm\|Hperturb (k,w0)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0)\|ψl/prime/angbracketright/angbracketleftψl/prime\|Hperturb (k,w0)\|ψm/prime/angbracketright =4kxw0/parenleftbig w2 0−3/parenrightbig/parenleftbig k2 x−3k2 y/parenrightbig 9/parenleftbig 1−3w2 0/parenrightbig2/radicalBig w2 0+1σ0−8w2 0/parenleftbig k2 x+k2 y/parenrightbig/parenleftbig/radicalBig w2 0+1−2w1/parenrightbig 9/parenleftbig 1−3w2 0/parenrightbig2(σy+√ 3σx). (F26) 205411-35BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) Second, H(32) mm/prime(k,w0,w1)=−1 2/summationdisplay l=3...12/summationdisplay m/prime/prime=1,2/angbracketleftψm\|Hperturb (k,w0,w1)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0,w1)\|ψm/prime/prime/angbracketleftψm/prime/prime\|Hperturb (k,w0,w1)\|ψm/prime/angbracketright+H.c. E2 l =−2/parenleftbig 17w2 0+9/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig/parenleftbig/radicalBig w2 0+1−2w1/parenrightbig 9/parenleftbig 1−3w2 0/parenrightbig2(σy+√ 3σx) (F27) (where H.c. is the Hermitian conjugate). The total third order Hamiltonian then reads 4kxw0/parenleftbig w2 0−3/parenrightbig/parenleftbig k2 x−3k2 y/parenrightbig 9/parenleftbig 1−3w2 0/parenrightbig2/radicalBig w2 0+1σ0−2/parenleftbig 7w2 0+3/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig/parenleftBig/radicalBig w2 0+1−2w1/parenrightBig 3/parenleftbig 1−3w2 0/parenrightbig2(σy+√ 3σx). (F28) d. Fourth order For the fourth order, there are now four terms: First, H(41) mm/prime(k,w0,w1)=−/summationdisplay l,l/prime,l/prime/prime=3,...,121 ElEl/primeEl/prime/prime/angbracketleftψm\|Hperturb (k,w0,w1)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0)\|ψl/prime/angbracketright ×/angbracketleftψl/prime\|Hperturb (k,w0,w1)\|ψl/prime/prime/angbracketright/angbracketleftψl/prime/prime\|Hperturb (k,w0,w1)\|ψm/prime/angbracketright =8w0/parenleftbig 7w2 0+3/parenrightbig kx/parenleftbig k2 x−3k2 y/parenrightbig/parenleftbig 2w1−/radicalBig w2 0+1/parenrightbig 27/parenleftbig 3w2 0−1/parenrightbig3σ0 +4w2 0/parenleftbig k2 x+k2 y/parenrightbig/bracketleftbig 2/parenleftbig w4 0+16w2 0−9/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig +/parenleftbig w2 0+1/parenrightbig/parenleftbig 5w2 0−7/parenrightbig/parenleftbig 2w1−/radicalBig w2 0+1/parenrightbig2/bracketrightbig 27/parenleftbig w2 0+1/parenrightbig3/2/parenleftbig 3w2 0−1/parenrightbig3(σy+√ 3σx). (F29) Second, H(42) mm/prime(k,w0)=/summationdisplay l,l/prime=3,...,12/summationdisplay m/prime/prime=1,21 ElEl/prime/parenleftbigg1 El+1 El/prime/parenrightbigg /angbracketleftψm\|Hperturb (k,w0,w1)\|ψl/angbracketright/angbracketleftψl\|Hperturb (k,w0)\|ψm/prime/prime/angbracketright ×/angbracketleftψm/prime/prime\|Hperturb (k,w0,w1)\|ψl/prime/angbracketright/angbracketleftψl/prime\|Hperturb (k,w0,w1)\|ψm/prime/angbracketright =16w2 0/parenleftbig 17w2 0+9/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig2 27/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig3(σy+√ 3σx). (F30) Third, we have, adopting the notation /angbracketleftψm\|Hperturb (k,w0)\|ψl/angbracketright=H/prime ml, etc., H(43) mm/prime(k,w0,w1)=−1 2/summationdisplay l,m/prime/prime,m/prime/prime/prime1 E3 l(H/prime mm/prime/primeH/prime m/prime/primem/prime/prime/primeH/prime m/prime/prime/primelHlm/prime+H/prime mlH/prime lm/prime/primeH/prime m/prime/primem/prime/prime/primeH/prime m/prime/prime/primem/prime) =−8w2 0/parenleftbig 35w2 0+23/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig/parenleftbig/radicalBig w2 0+1−2w1/parenrightbig2 27/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig3(σy+√ 3σx), (F31) H(44) mm/prime(k,w0,w1)=1 2/summationdisplay l,l/prime,m/prime/prime1 ElEl/prime/parenleftbigg1 El+1 El/prime/parenrightbigg (H/prime mlH/prime ll/primeH/prime l/primem/prime/primeHm/prime/primem/prime+H/prime mm/prime/primeH/prime m/prime/primelH/prime ll/primeH/prime l/primem/prime) =32kxw0/parenleftbig w2 0−15/parenrightbig/parenleftbig k2 x−3k2 y/parenrightbig/parenleftbig/radicalBig w2 0+1−2w1/parenrightbig 27/parenleftbig 3w2 0−1/parenrightbig3σ0 +4/parenleftbig 25w4 0+28w2 0+27/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig/parenleftbig/radicalBig w2 0+1−2w1/parenrightbig2 27/parenleftbig 1−3w2 0/parenrightbig3/radicalBig w2 0+1(σy+√ 3σx). (F32) 205411-36TWISTED BILAYER GRAPHENE. I. MATRIX ELEMENTS, … PHYSICAL REVIEW B 103, 205411 (2021) The full fourth order Hamiltonian reads −8kxw0/parenleftbig w2 0+21/parenrightbig/parenleftbig k2 x−3k2 y/parenrightbig/parenleftbig/radicalBig w2 0+1−2w1/parenrightbig 9/parenleftbig 3w2 0−1/parenrightbig3σ0 +4/parenleftbig k2 x+k2 y/parenrightbig/bracketleftbig 2w2 0(35w4 0+68w2 0+9)/parenleftbig k2 x+k2 y/parenrightbig −9/parenleftbig w2 0+1/parenrightbig/parenleftbig 10w4 0+9w2 0+3/parenrightbig/parenleftbig 2w1−/radicalBig w2 0+1/parenrightbig2/bracketrightbig 27/parenleftbig w2 0+1/parenrightbig3/2/parenleftbig 3w2 0−1/parenrightbig3(σy+√ 3σx).(F33) Ifw1=√ 1+w2 0 2, then the expressions reduce to our previous Hamiltonian. We can label the two-band Hamiltonian as HHex 2band(k,w0,w1)=d0(k,w0,w1)σ0+d1(k,w0,w1)(σy+√ 3σx), (F34) where d0(k,w0,w1)=4kxw0/parenleftbig w2 0−3/parenrightbig/parenleftbig k2 x−3k2 y/parenrightbig 9/parenleftbig 1−3w2 0/parenrightbig2/radicalBig w2 0+1−8kxw0/parenleftbig w2 0+21/parenrightbig/parenleftbig k2 x−3k2 y/parenrightbig/parenleftbig/radicalBig w2 0+1−2w1/parenrightbig 9/parenleftbig 3w2 0−1/parenrightbig3σ0 (F35) and d1(k,w0,w1)=/parenleftbigg/radicalBig w2 0+1 2−w1/parenrightbigg −4w2 0/parenleftbig k2 x+k2 y/parenrightbig 3/radicalBig w2 0+1/parenleftbig 3w2 0−1/parenrightbig−2/parenleftbig 7w2 0+3/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig/parenleftbig/radicalBig w2 0+1−2w1/parenrightbig 3/parenleftbig 1−3w2 0/parenrightbig2 +4/parenleftbig k2 x+k2 y/parenrightbig/bracketleftbig 2w2 0/parenleftbig 35w4 0+68w2 0+9/parenrightbig/parenleftbig k2 x+k2 y/parenrightbig −9/parenleftbig w2 0+1/parenrightbig/parenleftbig 10w4 0+9w2 0+3/parenrightbig/parenleftbig 2w1−/radicalBig w2 0+1/parenrightbig2/bracketrightbig 27/parenleftbig w2 0+1/parenrightbig3/2/parenleftbig 3w2 0−1/parenrightbig3, (F36) where the perturbation is made on the zero energy eigenstates of HHex(k=0,w0,w1=√ 1+w2 0 2). Notice that so far, remarkably the eigenstates are not kdependent, they are just the eigenstates of ( σy+√ 3σx). We did not obtain the ﬁfth order for this Hamiltonian: due to the fact that the ﬁrst order Hamiltonian does not cancel, this is not easy to do. 4. Calculations of the B1 shell ﬁrst order perturbation We now compute the shell B1 perturbation Hamiltonian: −HA1,B1H−1 kB1H† A1,B1(k,w0,w1) =−⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝T1(k−2q1)·σT1 \|k−2q1\|2 00000 0T3(k+2q3)·σT3 \|k+2q3\|2 0000 00T2(k−2q2)·σT2 \|k−2q2\|2 000 000T1(k+2q1)·σT1 \|k+2q1\|2 00 0000T3(k−2q3)·σT3 \|k−2q3\|2 0 00000T2(k+2q2)·σT2 \|k+2q2\|2⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠. (F37) We now compute the perturbation Hamiltonian: H (B1)(k,w0,w1)=/angbracketleftψm\|−HA1,B1H−1 kB1H† A1,B1(k,w0,w1)\|ψm/prime/angbracketright =1/producttext i=1,2,3\|k−2qi\|2\|k+2qi\|2[/tildewided0(k,w0,w1)σ0+/tildewidedx(k,w0,w1)σx+/tildewidedy(k,w0,w1)σy+/tildewidedz(k,w0,w1)σz], (F38) 205411-37BERNEVIG, SONG, REGNAULT, AND LIAN PHYSICAL REVIEW B 103, 205411 (2021) where /tildewided0(k,w0,w1)=4kx/parenleftbig k2 x−3k2 y/parenrightbig/parenleftbig k2 x+k2 y+4/parenrightbig/bracketleftbig/parenleftbig k2 x+k2 y/parenrightbig2−4/parenleftbig k2 x+k2 y/parenrightbig +16/bracketrightbig w0/parenleftbig/radicalBig w2 0+1+w1+1/parenrightbig/parenleftbig/radicalBig w2 0+1+w1−1/parenrightbig /radicalBig w2 0+1, (F39) /tildewidedz(k,w0,w1)=64kxky/parenleftbig k2 x−3k2 y/parenrightbig/parenleftbig 3k2 x−k2 y/parenrightbig w0/bracketleftbig/parenleftbig/radicalBig w2 0+1w1+w2 0/parenrightbig2+w2 0/bracketrightbig /parenleftbig w2 0+1/parenrightbig3/2, (F40) /tildewidedx(k,w0,w1)=−16/parenleftBig√ 3/radicalBig w2 0+1/braceleftBig −/bracketleftBig ky/parenleftBig 3k2 x−k2 y/parenrightBig/bracketrightBig2 +/bracketleftBig kx/parenleftBig k2 x−3k2 y/parenrightBig/bracketrightBig2 +64/bracerightBig/parenleftBig w2 0−w2 1/parenrightBig −2kxky/parenleftBig k2 x−3k2 y/parenrightBig/parenleftBig 3k2 x−k2 y/parenrightBig/parenleftBig/radicalBig w2 0+1w2 1+2w2 0w1+/radicalBig w2 0+1w2 0/parenrightBig/parenrightBig w2 0+1, (F41) /tildewidedy(k,w0,w1)=−16/braceleftBig/radicalBig w2 0+1/bracketleftBig −k2 y/parenleftBig 3k2 x−k2 y/parenrightBig2 +k2 x/parenleftBig k2 x−3k2 y/parenrightBig2 +64/bracketrightBig/parenleftBig w2 0−w2 1/parenrightBig +2√ 3kxky/parenleftBig 3k2 x−k2 y/parenrightBig/parenleftBig k2 x−3k2 y/parenrightBig/parenleftBig/radicalBig w2 0+1w2 1+2w2 0w1+/radicalBig w2 0+1w2 0/parenrightBig/bracerightBig w2 0+1.(F42) This gives the ﬁrst order term of HApprox1 (k) projected into the zero energy bands in the hexagon model on the second magic manifold. 5. Exact eigenvalues of the one-shell model at /Gamma1Mpoint Atw0=0 we ﬁnd the /Gamma1Mpoint eigenenergies of the Hamiltonian HApprox1 =HkA1+HA1,A1−HA1,B1H−1 kB1H† A1,B1in Eq. ( 33) to be the following: /parenleftbig −w2 1+4w1−2/parenrightbig 2,/parenleftbig w2 1−4w1+2/parenrightbig 2,/parenleftbig −w2 1+2w1−2/parenrightbig 2,/parenleftbig −w2 1+2w1−2/parenrightbig 2,/parenleftbig w2 1−2w1+2/parenrightbig 2,/parenleftbig w2 1−2w1+2/parenrightbig 2, /parenleftbig −w2 1−2w1−2/parenrightbig 2,/parenleftbig −w2 1−2w1−2/parenrightbig 2,/parenleftbig w2 1+2w1+2/parenrightbig 2,/parenleftbig w2 1+2w1+2/parenrightbig 2,/parenleftbig −w2 1−4w1−2/parenrightbig 2,/parenleftbig w2 1+4w1+2/parenrightbig 2.(F43) One sees the /Gamma1Mpoint has zero bandwidth at w1=2−√ 2, the same as that of the zero-bandwidth manifold w1=2√ w2 0+1−√ 3w2 0+2=2−√ 2i nE q .( 58) for the two-band model at w0=0. Furthermore, in the chiral limit w0=0, the value w1=2√ w2 0+1−√ 3w2 0+2=2−√ 2 for which the bandwidth is 0 in our two-band model is in fact exact for the no-approximation Hamiltonian of the n=1 shell Hamiltonian (of A1,B1 subshells). We ﬁnd its eigenvalues at /Gamma1Mto be /parenleftbig −/radicalBig 5w2 1−6w1+9−w1−1/parenrightbig 2,/parenleftbig −/radicalBig 5w2 1−6w1+9−w1−1/parenrightbig 2,/parenleftbig −/radicalBig 5w2 1−6w1+9+w1+1/parenrightbig 2, /parenleftbig −/radicalBig 5w2 1−6w1+9+w1+1/parenrightbig 2,/parenleftbig/radicalBig 5w2 1−6w1+9−w1−1/parenrightbig 2,/parenleftbig/radicalBig 5w2 1−6w1+9−w1−1/parenrightbig 2, /parenleftbig/radicalBig 5w2 1−6w1+9+w1+1/parenrightbig 2,/parenleftbig/radicalBig 5w2 1−6w1+9+w1+1/parenrightbig 2,/parenleftbig −/radicalBig 5w2 1+6w1+9−w1+1/parenrightbig 2, /parenleftbig −/radicalBig 5w2 1+6w1+9−w1+1/parenrightbig 2,/parenleftbig −/radicalBig 5w2 1+6w1+9+w1−1/parenrightbig 2,/parenleftbig −/radicalBig 5w2 1+6w1+9+w1−1/parenrightbig 2, /parenleftbig/radicalBig 5w2 1+6w1+9−w1+1/parenrightbig 2,/parenleftbig/radicalBig 5w2 1+6w1+9−w1+1/parenrightbig 2,/parenleftbig/radicalBig 5w2 1+6w1+9+w1−1/parenrightbig 2, /parenleftbig/radicalBig 5w2 1+6w1+9+w1−1/parenrightbig 2,/parenleftbig −/radicalBig 8w2 1−12w1+9−2w1−1/parenrightbig 2,/parenleftbig −/radicalBig 8w2 1−12w1+9+2w1+1/parenrightbig 2, /parenleftbig/radicalBig 8w2 1−12w1+9−2w1−1/parenrightbig 2,/parenleftbig/radicalBig 8w2 1−12w1+9+2w1+1/parenrightbig 2,/parenleftbig −/radicalBig 8w2 1+12w1+9−2w1+1/parenrightbig 2, /parenleftbig −/radicalBig 8w2 1+12w1+9+2w1−1/parenrightbig 2,/parenleftbig/radicalBig 8w2 1+12w1+9−2w1+1/parenrightbig 2,/parenleftbig/radicalBig 8w2 1+12w1+9+2w1−1/parenrightbig 2. 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PhysRevB.99.024307.pdf	PHYSICAL REVIEW B 99, 024307 (2019) Wavelet imaging of transient energy localization in nonlinear systems at thermal equilibrium: The case study of NaI crystals at high temperature Annise Rivière,1Stefano Lepri,2Daniele Colognesi,2and Francesco Piazza1,* 1Université d’Orléans, Centre de Biophysique Moléculaire (CBM), CNRS UPR4301, Rue C. Sadron, 45071 Orléans, France 2Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy (Received 27 August 2018; revised manuscript received 6 November 2018; published 14 January 2019) In this paper we introduce a method to resolve transient excitations in time-frequency space from molecular dynamics simulations. Our technique is based on continuous wavelet transform of velocity time series coupled toa threshold-dependent ﬁltering procedure to isolate excitation events from background noise in a given spectralregion. By following in time the center of mass of the reference frequency interval, the data can be easilyexploited to investigate the statistics of the burst excitation dynamics, by computing, for instance, the distributionof the burst lifetimes, excitation times, amplitudes and energies. As an illustration of our method, we investigatetransient excitations in the gap of NaI crystals at thermal equilibrium at different temperatures. Our resultsreveal complex ensembles of transient nonlinear bursts in the gap, whose lifetime and excitation rate increasewith temperature. The method described in this paper is a powerful tool to investigate transient excitations inmany-body systems at thermal equilibrium. Our procedure gives access to both the equilibrium and the kineticsof transient excitation processes, allowing one in principle to reconstruct the full picture of the dynamical processunder examination. DOI: 10.1103/PhysRevB.99.024307 I. INTRODUCTION Hamiltonian many-body systems with nonlinear interac- tions admit quite generally a special class of periodic orbits,whose amplitude-dependent frequency does not resonate byconstruction with any of the linear (normal) modes (NM) andwhose oscillation pattern is typically exponentially localizedin space. These modes, termed discrete breathers (DB) [ 1–3] or intrinsic localized modes (ILM) [ 4], have been shown theoretically to exist at zero temperature in a wide range ofsystems, including model lattices of beads and springs, suchas the celebrated Fermi-Pasta-Ulam (FPU) chain [ 5], real 2D and 3D crystals [ 6], both in the gap [ 7] and above the phonon spectrum [ 8], including cuprate high- T csuperconductors [ 9], boron nitride [ 10], graphene [ 11–13] and diamond [ 14], disordered media [ 15–17], and biomolecules [ 18] including proteins [ 19,20]. Nonlinear modes of this kind are surmised to play a subtle role in many condensed-matter systems. Forexample, DBs have been found to be connected to negative-temperature states (i.e., states for which the derivative ofentropy versus energy is negative) in the discrete nonlinearSchrödinger equation [ 21], which is relevant to the physics of Bose-Einstein condensates in optical lattices and arraysof optical waveguides. ILMs have also been surmised toaccelerate the kinetics of defect annealing in solids [ 22] and more generally to speed up heterogeneous catalysis processes[23,24]. If zero-temperature nonlinear excitations are well- established and fairly understood physical objects, whenit comes to systems at thermal equilibrium the scenario *Francesco.Piazza@cnrs-orleans.frproves far more complex and thorny [ 25]. Numerical techniques based on spectral analyses coupled to surfacecooling techniques have been proposed as means to detectspontaneous DB excitation in model nonlinear lattices[26]. More recently, other studies have also addressed this problem via equilibrium MD simulations, both in modelnonlinear chains [ 27] and in crystals with realistic potentials ranging from graphane [ 28,29] to crystals with the NaCl structure [ 30]. Experimental evidence for nonlinear localized excitations is no less a spinous matter. Nonlinear localized modes havebeen found experimentally at ﬁnite temperature in Josephsonladders [ 31] and arrays [ 32]. However, the oldest experi- mental evidence explained in terms of excitation of ILMs atﬁnite temperature in a crystal are the elusive tracks arisingfrom nuclear scattering events in muscovite mica [ 33]. Such dark lines, known since a long time [ 34] ,h a v el e dt ot h e suggestion that ILMs might act as energy carriers in crystalsalong speciﬁc directions with minimal lateral spreading andover long distances [ 35]. Recently, experimental evidence has been collected in support of this inference, as inﬁnite chargemobility has been measured at room temperature in muscovitemica crystals irradiated with high-energy alpha particles [ 36]. Indirect evidence for the nonequilibrium excitation of ILMs at ﬁnite temperature has been also gathered throughinelastic x-ray and neutron scattering measurements on α- uranium single crystals [ 37,38]. In particular, the authors of these studies speculate that the excitation of mobile modes,whose properties are consistent with those of ILMs, couldexplain the measured anisotropy of thermal expansion and thedeviation of heat capacity from the theoretical prediction athigh temperatures [ 39]. More recently, the same authors have published experimental evidence of the excitation of intrinsic 2469-9950/2019/99(2)/024307(13) 024307-1 ©2019 American Physical SocietyRIVIÈRE, LEPRI, COLOGNESI, AND PIAZZA PHYSICAL REVIEW B 99, 024307 (2019) localized modes in the high-temperature vibrational spectrum of NaI crystals [ 40], where ILMs have been predicted to exist atT=0 and characterized by many authors [ 7,30,41–43]. In 2011, the same authors published time-of-ﬂight inelasticneutron scattering measurements performed on NaI singlecrystals [ 44]. Their results seemed to point at the spontaneous thermal excitation of ILMs, moving back and forth betweenthe [111] and [011] orientations at intermediate temperaturesand eventually locking in along the [011] orientation aboveT=636 K. Further inelastic neutron scattering measure- ments on NaI crystals published in 2014 found no evidencefor thermally activated localized modes [ 45]. Even though these measurements conﬁrmed a very small peak within thegap, its intensity is so small—the authors argue—that it isnearly impossible to discern whether it is part of the inelasticbackground or whether it is indeed a true signature of a co-herent scattering event. However, in a subsequent paper [ 46], Manley and coworkers made it clear that the interpretationof the coherent scattering from NaI requires a correction ofthe incoherent background from the incoherent cross sectionof Na, which was not included in Ref. [ 45]. As the partial phonon DOS of Na displays a stretch of reduced intensity athigh temperatures in the spectral region corresponding to theT=0 gap, when this correction is made (as in Ref. [ 46]), the ILM feature becomes a little more pronounced. Combiningneutron scattering, laser ﬂash calorimetry and accurate x-raydiffraction data, the authors then argued that ILM localizationin NaI occurs in randomly stacked planes perpendicular tothe (110) direction(s) with a complex temperature dependence[46]. As a result, they suggested that spontaneous localiza- tion of ILMs should be regarded as some sort of collectivephenomenon rather than the random excitation of pointlikemodes. To this complex scenario, one should add that the expected relative fraction of light ions harboring a thermally excitedILM in NaI is relatively low. As an example, the predictionmade in Ref. [ 41] for ILMs polarized along the [111] orien- tation at T=636 K is about 8 .3×10 −4, which would make their direct observation a very hard matter. Taken together, the facts exposed above reveal a lively albeit rather intricate debate concerning the very existence ofthermal ILMs in crystals and the means to possibly spotlighttheir presence and characterize them. In order to addressthese questions, in this paper we develop a robust numericaltechnique based on continuous wavelet analysis, designedas a tool to pinpoint and characterize transient vibrationalexcitations, in general, in many-body system, and illustrate itin the case of NaI crystals. The paper is organized as follows.In Sec. II, we describe the MD simulation protocol and present our wavelet-based technique designed to pinpoint and charac-terize transient energy bursts in the time-frequency plane. InSec. III, we apply our technique to characterize transient exci- tation of energy in the gap of NaI crystals. In Sec. IV, based on the assumption that the population of transient energy burstsdetected in the gap may contain spontaneous excitation eventsof ILMs, we address the problem of how to sieve them outof the burst population. In Sec. V, we summarize our main ﬁndings and discuss possible improvements and extensions ofour method to detect and characterize spontaneous excitationof ILMs at thermal equilibrium.II. SIMULATIONS AND WA VELET ANALYSIS In order to illustrate our approach, we have used the molecular simulation (MD) engine LAMMPS [47] to simulate the equilibrium dynamics of a NaI crystal as a function oftemperature. The simulation box comprises N 3 ccubic unit cells with periodic boundary conditions (PBC) along the threeCartesian directions, each cell containing 4 Na +and 4 I−ions. For all simulations reported here, we have taken Nc=10, so that the total number of ions is 8000.1The choice of interatomic potentials is crucial. In order to determine thebest available choice, we have scrutinized a large body ofspecialized literature [ 48–57], which led us to reconstruct a total potential energy of the form U({r,R})=/summationdisplay i>jV++(\|Ri−Rj\|)+/summationdisplay i>jV−−(\|ri−rj\|) +/summationdisplay i,jV+−(\|Ri−rj\|), (1) where Riandridenote the position vectors of Na+and I− ions, respectively. Each pairwise contribution comprises three terms, V±±(r)=Q±Q± 4π/epsilon10r+WLR ±±(r)+PSR ±±(r). (2) The Coulomb energy has been computed via the Ewald method [ 58]. Instead of specifying a cutoff wave vector for the Ewald sums, we have set the relative error in the calculationof electrostatic forces to be less than 10 −5at any given time. We have veriﬁed that our results did not change by requiring amore accurate estimation. The potential energy W LRaccounts for a long-range potential of the (6,8) kind, namely, WLR ±±(r)=−C± r6−D± r8(3) corresponding to induced dipole-induced dipole interactions (C±) and induced dipole-induced quadrupole interactions (D±) computed via the Kirkwood-Muller methods, i.e., using experimental measurements of the ionic polarizability andmolar susceptibility [ 59,60]. The short-range term is well described by a Buckingham-type potential [ 61] of the form P SR ±±(r)=A±±exp(−r/ρ±±)( 4 ) restricted to the nearest-neighbor shell (5 Å cutoff). The values of the parameters in Eqs. ( 3) and ( 4) are listed in Table I. Since the lattice constant of NaI crystals is known exper- imentally and has been used, alongside other experimentallydetermined constants, to parametrize the potential energy ( 1) [48–57], we have used these measurements to set the dimen- sion of the unit cell at different temperatures and performedﬁxed-volume simulations. A typical simulation consisted ofa ﬁrst thermalization NVT stage of duration /Delta1t th, where 1We observe that PBCs with Nc=10 appears a safe choice to inspect energy localization on length scales of the order of half/oneunit cell. 024307-2WA VELET IMAGING OF TRANSIENT ENERGY … PHYSICAL REVIEW B 99, 024307 (2019) TABLE I. Parameters of the pair-wise short-range and long- range potential energies used in this study to simulate the dynamics a NaI crystal. For more information, see Refs. [ 48–57]. Short range Long range Pair kind A±,±(eV) ρ±±(Å) C±,±(eV Å6)D±,±(eV Å8) ++ 8500.74 0.29333 4.93337 3.55827 −− 384.924 0.50867 810.714 805.769 +− 736.498 0.40100 54.9164 47.0954 the system was brought to thermal equilibrium through a Nosé-Hoover thermostat [ 62,63] starting from zero initial atomic displacements and random velocities drawn from aMaxwell distribution. We have veriﬁed that /Delta1t th=5p sw a s sufﬁcient to correctly thermalize our system for tempera-tures larger than 400 K. Once the system is thermalized, werun constant energy trajectories (NVE) of duration /Delta1t pfor data production. It is interesting to remark that distortionsdriven by the localization of nonlinear vibrational modes areexpected to conserve volume, as it was found for the internaldistortions associated with ILM localization in the faultlikeplanar structures reported in Ref. [ 46]. 2The results pre- sented in the following refer to /Delta1tp=100 ps, which afforded a reasonable compromise between computational costs andsolid statistics. The time step used in the MD simulations was0.001 ps. Figure 1illustrates the comparison of the low-temperature phonon density of states computed by Fourier transformingthe velocity-velocity autocorrelation functions computed fromour LAMMPS NVT trajectories with the results from lattice dynamics calculations performed with the GULP package [ 64]. The excellent agreement validates our MD simulation pro-tocol and in particular the values of the phonon frequenciesthat deﬁne the gap at zero temperature, i.e., ω 1=16.104 ps−1 (upper edge of the acoustic band) and ω2=20.343 ps−1 (lower edge of the optical band). A. Wavelet imaging of transient energy bursts in the gap Wavelet analysis is the ideal tool to analyze nonstationary signals in the time-frequency domain in order to characterizetransient frequency components appearing at speciﬁc timesand perduring for ﬁnite lapses of time. As a matter of fact,Forinash and co-workers have shown 20 years ago that this 2The use of an NVT dynamics for production runs does not appear to make sense in this study. In fact, thermostats are, in principle, noth-ing but smart sampling techniques, designed to produce time series sampled from the canonical measure. However, there is absolutely no guarantee that the actual trajectories (i.e., the actual dynamics ) make any physical sense. In particular, all vibrational coherences are either (artiﬁcially) damped or completely destroyed, depending on the value of the relaxation time scale chosen for the speciﬁcthermostat. In practice, it is preferable to switch off the thermostat once the system has reached thermal equilibrium, so that no artiﬁcial noise is left to ﬁddle with the vibrational coherences that mightemerge in speciﬁc frequency regions.024680.00.20.40.60.8ZNa(E) (THz-1) E (THz)024680.00.20.40.6ZI(E) (THz-1) LAMMPS GULPNaI FIG. 1. Phonon DOS of NaI computed from NVE MD simula- tions ( T=38 K, LAMMPS , red staircases) and from lattice dynamics calculations ( T=77 K, GULP , blue lines). The energy E=hνis measured in units of the frequency ν. kind of tools can provide precious information on the dy- namics of discrete breathers at zero temperature in nonlinearchains [ 65]. Thus, it appears natural to extend this line of reasoning to explore transient nonlinear localization in realcrystals at thermal equilibrium. In this work, we have com-puted the Gabor transform [ 66] of the time series of atomic velocities, namely, G iα(ω,t)=/integraldisplay+∞ −∞e−(t−τ−/Delta1tp/2)2/ae−iωτviα(τ)dτ, (5) where viαis the velocity of the ith ion along the Cartesian direction α. We have set the resolution parameter a=20 ps2, optimized so as to maximize the resolution in both the timeand frequency domains. As an illustration of our analysis, Fig. 2shows typical density maps of \|G iα(ω,t)\|2computed from the velocity time series of two random Na ions at T=600 and 900 K. It can be appreciated that, as the temperature increases, transientenergy bursts pop up increasingly deep in the gap and persistwith lifetimes of the order of up to 10 ps, during which theirfrequency appears to drift to a various degree. In order toseparate energy bursts from the background and perform a fulltemperature-dependent statistical analysis of the excitationdynamics, it appears natural to impose a threshold P Gon the Gabor power so as to eliminate transient background noise. Tothis end, we deﬁne the ﬁltered normalized two-dimensional 024307-3RIVIÈRE, LEPRI, COLOGNESI, AND PIAZZA PHYSICAL REVIEW B 99, 024307 (2019) FIG. 2. Time-frequency density maps of the function \|Giα(ω,t)\|2in the gap region for two different Na ions at T=600 and 900 K along the three Cartesian directions ( x,y,z from top to bottom). Spectral power is color-coded from blue (low energy) to red (high energy). The two horizontal white lines mark the edges ω1,ω2of the gap region. excitation density ρiα(ω,t)a s ρiα(ω,t)=\|/tildewideGiα(ω,t)\|2 /integraltextω1 ω2\|/tildewideGiα(ω/prime,t)\|2dω/prime, (6) where /tildewideGiα(ω,t)=/braceleftBigg Giα(ω,t)f o r \|Giα(ω,t)\|2/greaterorequalslantPG 0 otherwise.(7) This deﬁnition allows us to compute the time-dependent mo- ments of ρiα(ω,t), which provide important information on the dynamics of transient energy excitation in the gap. In thepresent work, we concentrate on the ﬁrst moment, namely, /angbracketleftω iα(t)/angbracketright=/integraldisplayω2 ω1ωρiα(ω,t)dω. (8) As it can be seen from the top panel in Fig. 3, the choice of the threshold PGsets the resolution limit of individual burst events. After careful examination of many such events, we have ﬁxed PG=128 Å2, which ensures that consecutive bursts should be optimally resolved. Although the resultsreported in the following refer to this (rather conservative)choice, we have repeated our analyses with the two lower val-ues of P Gshown Fig. 3. While the actual ﬁgures may change slightly, we have veriﬁed that the relevant statistical and physi-cal properties of the burst excitation dynamics are unchanged. After the ﬁltering and integration procedure for a given ion i, the time series /angbracketleftω iα(t)/angbracketrightare piecewise composed of stretches of consecutive zeros (absence of a burst) and consecutivenonzero values, each representing a burst and extending overits corresponding lifetime. Such values describe the drift ofthe center-of-mass frequency of the burst since the moment ofits excitation until it collapses. From the support of these timeseries, it is then straightforward to obtain other restricted timeseries per burst , most importantly the sequences of kineticenergies and vibration amplitudes for each burst during its lifetime. III. RESULTS I: TRANSIENT ENERGY BURSTS IN THE GAP WITH INCREASING LIFETIMES Nonlinear localized vibrations in the gap of diatomic lat- tices detach from the bottom of the optical band [ 67], which means that their energy is almost entirely conﬁned to lightions. For a given Na ion, two key kinetics parameters describethe burst excitation dynamics, notably the lifetimes t nand the excitation times τn+1,n=0,1,2,... These two measures are illustrated in the middle panel in Fig. 3for a random typical excitation sequence. The excitation times are deﬁned as theintervals between consecutive excitation events. Together withthe lifetimes, they provide a rich wealth of information on thekinetics of burst excitation at a given temperature. However,irrespective of the kinetics, the temperature dependence of thesite-occupancy probability (SOP) P(T) describes the equilib- rium properties of this process. This can be simply computedas the fraction of Na ions harboring at least one burst in thegap along one of the Cartesian directions. 3The data, reported in Fig. 4(top left), can be ﬁtted by a simple equilibrium model of the kind P(T)=1 1+eβ/Delta1f, (9) where β=1/kBTand/Delta1f=/Delta1/epsilon1−T/Delta1sis the free energy of burst excitation per ion. The excellent ﬁt of the MD simulationdata gives /Delta1/epsilon1=0.54±0.01 eV and /Delta1s=9±0.2k B.T h e data reported in Fig. 4are obtained by averaging the site- 3In this work, we implicitly refer to the gap spectral region when we mention the excitation of a burst. 024307-4WA VELET IMAGING OF TRANSIENT ENERGY … PHYSICAL REVIEW B 99, 024307 (2019) τ1 τ2 τ3 t0t1 t2 t3 Δ1Δ2Δ− 2 F F∗BΔΔ1Δ−1 F BΔ− 1 Δ FIG. 3. (Top) Illustration of the ﬁltering procedure to isolate energy bursts with three different thresholds (units of Å2). (Middle) Scheme of the algorithm to identify lifetimes tnand excitation times τnduring the production run /Delta1tpfor a given ion from the time series of /angbracketleftωiα(t)/angbracketright deﬁned in Eq. ( 8). (Bottom left) Kinetic model based on a two-well landscape fails to reproduce the kinetics and equilibrium properties of burst excitation. (Bottom right) At least one intermediate state is required to rationalize the kinetics and equilibrium of the thermally activated process of burst generation. This proﬁle reproduces to scale a possible three-well landscape that is in agreement with our simulation data. The energy scale that controls the burst lifetimes in this picture is δ/epsilon1:=(/Delta1/epsilon11−/Delta1/epsilon1− 1)−/Delta1/epsilon1− 2(see extended discussion in the text). occupancy probabilities referring to bursts along individual Cartesian directions. However, we observe that the threeindividual SOPs are indistinguishable from one another (datanot shown), which appears natural in view of the symmetry ofthe crystal. It is interesting to note that the simple law ( 9) was found to describe the excitation of ILMs along [111] in Ref. [ 41], with /Delta1/epsilon1=0.608 eV and /Delta1s=4k B, corresponding to the four symmetry-equivalent Lpoints at the boundary of the Brillouin zone (BZ) from which an ILM can in principle detach with a[111] polarization. In our case, we only expect a small fractionof the bursts to possibly be transient excitations of ILMs. It is nonetheless interesting to observe that the excitation energythat we ﬁnd is close to a very good guess for an ILM in 3DNaI. Furthermore, the value /Delta1s=9k Bis close to the overall symmetry degeneracy of the L,K, andXpoints in the BZ taken together, i.e., 10, corresponding to the extra degeneracyassociated with the theoretical conversion points to ILMsalong [110] ( K) and along [100] ( X). Of course, if this interpretation has some truth to it, it seems that the three kindsof ILMs might be excited at the same time and possibly moveas units back-and-forth among them, as already speculated 024307-5RIVIÈRE, LEPRI, COLOGNESI, AND PIAZZA PHYSICAL REVIEW B 99, 024307 (2019) 0 0.2 0.4 0.6 0.8 1 500 600 700 800 900Site−occupancy prob. 15 35 55 75 500 600 700 800 900Average exc. time [ps]Δε1 = 0.12 eV Δε1 = 0.04 eV 3 4 5 400 500 600 700 800 900Average lifetime [ps] Temperature [K]Δε2− = 0.05 eV, μ = 0.001 Δε2− = 0.12 eV, μ = 0.002 16 17 18 19 20 21 500 600 700 800 900Frequency [ps−1] 0.15 0.2 0.25 0.3 0.35 500 600 700 800 900MSD [Å2] System average 0.07 0.09 0.11 0.13 500 600 700 800 900Average kin. energy [eV] Temperature [K]Equipartition FIG. 4. Analysis of the burst excitation equilibrium, kinetics, and dynamics. (Top left) Equilibrium burst site-occupancy probability at Na ions vs temperature from the simulations (ﬁlled circles) and ﬁt with the chemical equilibrium model ( 9). Best-ﬁt parameters are /Delta1/epsilon1= 0.54±0.01 eV ,/Delta1s=9±0.2kB. (Middle left) Average excitation times (see again Fig. 3) identiﬁed from the support of the ﬁltered integrated time series ( 8). Open squares represent the average values computed over all the pairs of consecutive excitation events, further averaged over x, y,a n dz. The crosses represent the values computed by ﬁtting the exponential tails of the distributions and rescaled so as to match the high-temperature averages. This set of data is likely to better approximate the true values at low temperatures. The two lines are plot best-ﬁt Arrhenius laws of the kind ( 12). Best ﬁt parameters are /Delta1/epsilon11=0.12±0.1e V ,k∞ 1=0.18±0.03 ps−1(solid line) and /Delta1/epsilon11=0.04±0.02 eV, k∞ 1=0.06±0.005 ps−1(dashed line). The true value of /Delta1/epsilon11(i.e., the average computed over a simulation long enough to sample very long excitation times) is expected to be in the interval [0 .04,0.12] eV. (Bottom left) Average lifetimes (see again Fig. 3) identiﬁed from the support of the ﬁltered integrated time series ( 8) (symbols) and ﬁts with the three-states model expression ( 18). The solid line is a three-parameter ﬁt, where the ﬂoating parameters are t∞,δ/epsilon1:=(/Delta1/epsilon11−/Delta1/epsilon1− 1)−/Delta1/epsilon1− 2,μ=k∞ −1/k∞ 1and/Delta1/epsilon1− 2is kept ﬁxed at 0.04 eV . The dashed line is a two-parameter ﬁt, where /Delta1/epsilon1− 2is kept ﬁxed at 0.1 eV , while this time the energy scale that physically controls the increasing trend, δ/epsilon1,i sk e p t ﬁxed at the previous best-ﬁt value, i.e., δ/epsilon1=0.07 eV (see text for the full discussion). (Top right) Average burst frequencies vs temperature. (Middle right) Average burst amplitude vs temperature (ﬁlled diamonds) and average amplitude of the ﬂuctuations of all Na ions in the system(dashed straight line). The solid line is a ﬁt with a function of the kind /angbracketleftA 2(T)/angbracketright=αT+βT4, intended as a guide to the eye. (Bottom right) Average burst kinetic energy vs temperature (ﬁlled pentagons), i.e., ensemble average of the individual burst energies. The dashed line marks the equilibrium value /angbracketleft/epsilon1kin/angbracketright=3kBT/2. At each temperature, the reported average frequencies, amplitudes, and kinetic energies represent the ensemble averages of the individual average values per burst . The latter are computed by averaging over the individual drift of each single burst, as identiﬁed from the support of the corresponding ﬁltered time series ( 8). We remind the reader that each burst is associated with a single Na ion and Cartesian direction. by Manley and co-workers for the interplay of [110] and [111] ILMs below 636 K [ 44]. We observe, however, that this kind of complex dynamics would appear exceedingly difﬁcultto disentangle, even in the framework of a computationalstudy like this, as conﬁrmed by the indistinguishability of theSOPs describing burst excitation along individual Cartesiandirections.From the point of view of chemical kinetics, the expression (9) describes the equilibrium between two species/states with an arbitrary number of intermediates. It is tempting to followthis lead to get some insight into the burst excitation pro-cess. In the simplest possible scenario, we would be dealingwith two states, FandB, describing random energy ﬂuctu- ations ( F) and energy ﬂuctuations within a burst ( B). In the 024307-6WA VELET IMAGING OF TRANSIENT ENERGY … PHYSICAL REVIEW B 99, 024307 (2019) framework of this simple mean-ﬁeld description, the time evolution of the site-occupancy probability would be given toa ﬁrst approximation by ∂P(T,t) ∂t=k1[1−P(T,t)]−k−1P(T,t), (10) where k1andk−1stand for the burst birth and death rates, respectively. In this picture, one immediately sees that theequilibrium site-occupancy probability is simply given by P(T)=1 1+k−1/k1, (11) where k−1/k1is the effective dissociation constant of the F−Bequilibrium. In a simple picture described by an energy landscape with two minima (Fig. 3, bottom left), the excitation energy /Delta1/epsilon1would just be the difference between the two excitation barriers /Delta1/epsilon11(F→B) and/Delta1/epsilon1− 1(B→F), deﬁned by Arrhenius-like laws of the kind k1=k∞ 1e−β/Delta1/epsilon11, (12) k−1=k∞ −1e−β/Delta1/epsilon1− 1. (13) In this model, /Delta1/epsilon1=/Delta1/epsilon11−/Delta1/epsilon1− 1</Delta1/epsilon11and/Delta1s= ln(k∞ −1/k∞ 1). However, a quantitative analysis of our data reveals that the best estimate of the excitation energy is/Delta1/epsilon1 1=0.12±0.1e V ,w h i c hi s lower than/Delta1/epsilon1(middle left panel in Fig. 4). It should be stressed that the numerical determination of average excitation times is a delicate matter,for long excitation times are clearly under-represented inthe population of recorded events (i.e., pairs of consecutiveexcitations). In fact, the population observed in a simulationis obviously cut off at τ=/Delta1t p. This means that the observed averages /angbracketleftτ(T)/angbracketrightare underestimated at the lower temperatures, where excitation times are longer. In order to gauge this effect,it is expedient to ﬁt the exponential tail of the numericaldistributions before the cutoff. The temperature trend ofsuch decay times, lower in value than the correspondingaverages, should nonetheless be a good representation of thetrue trend (i.e that of averages computed from inﬁnitely long simulations). The middle panel in Fig. 4shows that this seems, indeed, to be the case, placing the value of the excitationenergy/Delta1/epsilon1 1somewhere in the interval [0 .04,0.12] eV. The fact that /Delta1/epsilon11</Delta1/epsilon1rules out a simple two-minima picture. To complicate the picture further, it can be seen fromFig. 4(bottom left panel) that the average burst lifetimes are found to increase with temperature, in agreement with previous results of MD simulations in crystals with the NaClstructure at thermal equilibrium [ 68]. As a matter of fact, we found that the distribution of burst lifetimes extends to longerand longer times (up to lifetimes of the order of 20–30 ps) asthe temperature increases (see Fig. 5).These somewhat coun- terintuitive results are also incompatible with a two-well freeenergy landscape, which would predict /angbracketleftt(T)/angbracketright∝1/k −1and therefore lifetimes decreasing with temperature, as escapingfrom the Bstate becomes more and more favored at higher temperatures as prescribed by Eq. ( 13). Of course, one might invoke general nonlinear effects to explain the observed increase in self-stabilization of burstsat increasing temperatures. However, it is not clear how this10-510-310-2100 0 10 20Normalized histogram Lifetime [ps]T = 900 K T = 800 K T = 700 K T = 600 K T = 500 K FIG. 5. Distributions of burst lifetimes at ﬁve representative tem- peratures (symbols). The solid lines are plots of exponential ﬁts to the distribution tails. can be quantiﬁed in simple terms. In this paper, we explore another route that provides an effective description of theburst excitation dynamics and has the advantage of sketchinga general interpretative paradigm to combine equilibrium andkinetics observables. ILM/DB excitation is expected to be a thermally activated phenomenon, in view of the general existence of excitationthresholds in nonlinear lattices [ 69,70]. This has been con- ﬁrmed explicitly for spontaneous excitation of DBs in theframework of surface-cooling numerical experiments in 2DFPU lattices [ 71]. If one sticks to the physics of a thermally activated process occurring along some reaction coordinate,in order to rationalize the observed burst excitation process,it is necessary to introduce at least an intermediate state, F ∗, according to the kinetic model Fk1−−/arrowrighttophalf/arrowleftbothalf− k−1F∗k2−−/arrowrighttophalf/arrowleftbothalf− k−2B. (14) The state F∗could be interpreted as a precursor ﬂuctuation that can be either stabilized—this is where nonlinear effectscome into play in this picture—to yield a persistent burst, orit can decay back into the background. As we shall see in thefollowing, the obvious coming into play of nonlinear effectsas temperature increases is conﬁrmed by the observed trend ofthe burst average amplitudes. The scheme ( 14) corresponds to a three-minima landscape as illustrated in Fig. 3(bottom right panel). The relative equilibrium population of the Bstate, i.e., the burst site-occupancy probability Pin our analogy, can be simply computed by imposing the detailed-balance conditionsk 1Fe=k−1F∗ eandk2F∗ e=k−2Be. This yields immediately P≡Be Fe+F∗e+Be=1 1+k−2(k1+k−1) k1k2. (15) In this model, the burst lifetime is set by the rate k−2. With reference to the landscape depicted in the bottomright panel in Fig. 3, let us take /angbracketleftt(T)/angbracketright∝1/k −2and let us assume that k2andk−2are described by Arrhenius ex- pressions such as ( 12) and ( 13) [i.e., k2=k∞ 2exp(−β/Delta1/epsilon12), 024307-7RIVIÈRE, LEPRI, COLOGNESI, AND PIAZZA PHYSICAL REVIEW B 99, 024307 (2019) k−2=k∞ −2exp(−β/Delta1/epsilon1− 2)]. Then, comparing Eqs. ( 15) and ( 9), we are led immediately to the following expression: /angbracketleftt(T)/angbracketright=t∞/parenleftbigg1+μeβ/Delta1/Delta1/epsilon11 1+μ/parenrightbigg e−β(/Delta1/Delta1/epsilon11−/Delta1/epsilon1− 2), (16) where/Delta1/Delta1/epsilon11:=/Delta1/epsilon11−/Delta1/epsilon1− 1,μ=k∞ −1/k∞ 1, and t∞is the asymptotic, inﬁnite-temperature lifetime ( ∝1/k∞ −2) deter- mined uniquely by the kinetic (entropic) constants (see againthe three-well landscape pictured in Fig. 3). The function ( 16) is a monotonically decreasing function of temperature or features a minimum at low temperatures andan increasing trend for higher temperatures depending on therelative value of the relevant kinetic and energy scales. Moreprecisely, an increasing portion at high temperature will beobserved provided ( /Delta1/Delta1 1−/Delta1/epsilon1− 2)//Delta1/epsilon1− 2>μ, that is, /Delta1/epsilon11−/Delta1/epsilon1− 1 /Delta1/epsilon1− 2>1+k∞ −1 k∞ 1. (17) It should be observed that no bursts in the gap are observed in our simulations below 500 K (see again the top left panel inFig. 4). This is consistent with a barrier /Delta1/epsilon1 1in the 0.1 eV ballpark (at 500 K the average kinetic energy per particle would yield a rate k1≈0.1k∞ 1). Thus the three-wells free energy landscape sketched in Fig. 3should be considered as describing the stabilization of ﬂuctuations for temperatures/greaterorsimilar500 K. The two barriers should be imagined as being vanish- ingly small at lower temperatures, where, at most, ﬂuctuationsmight be described by a simple two-state F−F ∗equilibrium. This is the regime where bursts become short-lived and makeonly rare appearances in the gap, most likely, close to thebottom of the optical band (see again the left panel in Fig. 2). We see from the condition ( 17) that, physically, increasing burst lifetimes at high temperatures arise as a combinationof (i) slow decay kinetics of the intermediate state F ∗, (ii) large values of the energy describing the F−F∗equi- librium, /Delta1/epsilon11−/Delta1/epsilon1− 1, and small values of the energy barrier for the decay of the Bstate,/Delta1/epsilon1− 2. In particular, if the velocity constant of the F∗→Fde-excitation is much slower than the velocity of the ﬁrst excitation, F→F∗(i.e., a large positive entropy difference in favor of the F∗state), then the term proportional to μcan be neglected and the burst lifetime will be an increasing function of temperature over the wholephysically meaningful temperature range, as controlled solelyby the positive energy difference ( /Delta1/epsilon1 1−/Delta1/epsilon1− 1)−/Delta1/epsilon1− 2. From a practical standpoint, due to the short temperature stretch available to ﬁt the numerical data and the functionalform ( 16), it is not possible to ﬁt meaningfully all the unknown parameters in Eq. ( 16). However, the energy scale controlling the increasing trend is δ/epsilon1:=(/Delta1/epsilon1 1−/Delta1/epsilon1− 1)−/Delta1/epsilon1− 2. Hence the agreement of this simple kinetic mean-ﬁeld theory with thesimulations can be assessed by ﬁxing the unknown barrier/Delta1/epsilon1 − 2and ﬁtting a functional form of the kind /angbracketleftt(T)/angbracketright=t∞/parenleftBigg e−βδ/epsilon1+μeβ/Delta1/epsilon1− 2 1+μ/parenrightBigg (18) witht∞,μ, andδ/epsilon1free to ﬂoat. For example, with /Delta1/epsilon1− 2= 0.04 eV, we get μ=0.06±0.03,δ/epsilon1=0.07±0.03 eV, and t∞=10±2 ps. To obtain a more meaningful assessment,we repeated the ﬁt by ﬁxing the barrier to a different value, /Delta1/epsilon1− 2=0.1 eV, and kept δ/epsilon1=0.07 eV from the ﬁrst ﬁt. It is clear from Fig. 4that the theory still describes the simulation data in the observed temperature range. In this case, we getconsistent values of the two ﬂoating parameters left, namelyμ=0.013±0.03 and t ∞=11.5±0.2p s . The top right panel in Fig. 4shows the average frequency of bursts as a function of temperature. Of course, the loweredge of the phonon optical band is expected to soften, hence itis difﬁcult to disentangle nonlinear phonon frequencies frompossible ILM events from these average data as the gap getsprogressively colonized by soft nonlinear phonons. In thefollowing, we will discuss this point further and point to apossible strategy to get more insight as to ILM signatures. At variance with the average frequencies, an analysis of the average vibrational amplitudes of bursts in the gap reveala telltale sign of nonlinear effects. In the middle right panelin Fig. 4, we compare the mean square displacement (MSD) computed over all Na ions in the crystal with the averageMSD of Na ions hosting a burst (i.e., the mean over theburst population of the average MSD of each burst, the latterbeing computed over its corresponding lifetime). It is clearthat, starting from temperatures of the order 500 K, burstsclearly vibrate with increasing amplitudes, detaching fromthe harmonic ∝Tlaw. This seems to indicate that bursts of energy in the gap are intrinsically nonlinear excitations. Another rather puzzling piece of information comes from the analysis of the average burst kinetic energies (lower rightpanel in Fig. 4). These turn out to follow a linear trend, as the equipartition theorem would prescribe for each and every Naion in the system, however, the average energies seem to beproportional to an effective temperature that is about 100 K higher than the true one (see the dashed line in the lowerright panel of Fig. 4). In other words, during the lifetime of a burst, the corresponding Na ion has on average systematicallya higher energy than the average Na ion in the system. This isin agreement with the behavior of the MSD. If one surmisesthat the fraction of bursts that display characteristics typical ofILMs is non-negligible, a possible explanation of these effectsmight reside in the known tell-tale ability of ILMs to harvestenergy from the background by absorbing lower-energy radi-ation [ 2,71]. Pushing this line of reasoning further, the origin of the observed higher-than-average energies of bursts in thegap might reveal a sheer nonlinear self-stabilization processakin to the well-known ILM behavior during surface cooling[71] or akin to the properties of the so-called chaotic breathers [72,73]. IV . RESULTS II: SIEVING THROUGH THE POPULATION OF BURSTS FOR ILMS The wavelet-based procedure described in this work allows one to build and characterize ensembles of nonlinear exci-tations that increasingly populate the gap as the temperatureis raised. Even though these soft excitations display distinctILM-like features, such as the apparent ability to gathersome energy from the background and self-stabilize duringtheir lifetime beyond the equipartition law, it is hard to statewhether such bursts are indeed instances of ILM excitation. Infact, according to the general arguments developed by Sievers 024307-8WA VELET IMAGING OF TRANSIENT ENERGY … PHYSICAL REVIEW B 99, 024307 (2019) FIG. 6. Illustration of the procedure employed for sifting possi- ble ILM-like excitations through the whole ensembles of bursts in the gap. At each temperature, [100], [110], and [111] subensembles arecreated (transparent circles) by keeping only the bursts closer than 1% to the corresponding theoretical ILMs [ 43] (solid lines) in the frequency-amplitude plane. and co-workers in Ref. [ 41], the site-occupancy probability of athermal ILM is expected to be very low—about 0.02 for a [111] excitation in 3D NaI at T=900 K. While numerical analogues of exquisitely nonlinear experimental techniquessuch as discussed in Ref. [ 74] would be powerful tools to address this question, it also makes sense to turn to theoret-ical predictions for T=0 excitations as possible templates , against which the raw ensembles of gap bursts can be sifted . The theoretical ILM frequency-amplitude relations re- ported in Ref. [ 43] are shown as solid lines in Fig. 6for the three ILM polarizations, [100], [110] ,and [111]. At each temperature, we sifted through the whole collection of burstsand assembled three subpopulations by keeping only thoseexcitations whose distance from the theoretical curves wasless than 1%. Practically, for each burst, we recovered thethree theoretical frequencies corresponding to its measuredaverage amplitude. The burst was then kept under the ap-propriate polarization label if the relative difference betweenits average frequency and the theoretical frequency was lessthan 1%. We observe that this is a rather crude scheme, aseach burst is associated with a single Cartesian direction.Therefore, while this procedure makes perfect sense for the[100] polarization, it might be objected that by doing this weare not enforcing the additional correlations among differentCartesian directions required by the assumed polarizations.Of course, a burst found along xthat would correspond to a genuine ILM polarized along the [110] direction would mostlikely match to some extent a burst on the same ion along theydirection. However, this is a tricky matter, as the phase relation between the two directions might be such that the twobursts would not necessarily appear correlated, depending onTABLE II. Best-ﬁt values of the energy and entropy differences describing the equilibrium between energy ﬂuctuations and stabi- lized bursts according to the law ( 9), with /Delta1f=/Delta1/epsilon1−T/Delta1s.T h e excitations labeled according to different polarizations correspond tothe subpopulations sieved out at each temperature from the whole ensemble of bursts by keeping only the excitations that match the corresponding theoretical frequency-amplitude relations taken fromRef. [ 43] (see again Fig. 6). Excitation kind /Delta1/epsilon1(eV) /Delta1s(kB) All 0.54 ±0.01 9 ±0.2 [100] 1.16 ±0.06 11.4 ±0.8 [110] 0.32 ±0.01 2.5 ±0.2 [111] 1.06 ±0.04 11.9 ±0.5 the spectral resolution and on the burst lifetime itself. While conceiving the appropriate tool to enforce such constraintsas rigorously as possible, we are nonetheless reporting heresome interesting results obtained with the simplest sievingprocedure outlined above. Direct inspection of Fig. 6shows that the number of pu- tative ILM excitations increases with temperature. Moreover,it seems that the excitations that fall on the [110] theoreticalcurves are much more abundant than the [100] and [111]excitations, despite that the theory developed in Ref. [ 43]p r e - dicted the [111] modes to be the most stable ones. However,it should be remarked that the lifetime ≈3×10 −9s, predicted in Ref. [ 43] for the [111] modes based on the interaction with a (Bose-Einstein) thermal distribution of phonons, exceeds bytwo orders of magnitude the longest lifetimes assigned to aburst in the gap in this study (about 30 ps). The top left panel in Fig. 7compares the site-occupancy probabilities relative to the ILM subpopulations to the globalsite-occupancy probability of the whole burst database. Thedata are well ﬁtted by general chemical equilibria between twofree-energy minima (possibly separated by a number of in-termediates), embodied by expression ( 9). The corresponding free-energy differences are reported in Table II. It can be ap- preciated that putative ILM excitations along [100] and [111]appear to be rather in the minority with respect to generic burst excitations. Putative [110] modes seem to be morenumerous at low and intermediate temperature. Nonetheless,the population of these kind of excitations seem to increasewith temperature as that of the generic bursts, while [100] and[111] modes appear to be about three orders of magnitudeless than generic bursts at intermediate temperatures, whilesurging in number with temperature much more rapidly than[110] modes. This is reﬂected by the best-ﬁt value of theenthalpy and entropy differences (see Table II). Putative [100] and [111] ILM-like bursts seem far easier to excite from thepoint of view of entropy than [110] excitations, explaining themarked temperature dependence of their SOP. It is interestingto observe that the predictions made in Ref. [ 41] for [111] modes seem to underestimate the excitation entropy differ-ence (4 k Bversus 12 kB), which results in a reduced tem- perature dependence of their excitation equilibrium (dashedline in the top left panel of Fig. 7). This might indicate that in general at thermal equilibrium there might be more excitationchannels than merely speciﬁed by the symmetry-equivalent 024307-9RIVIÈRE, LEPRI, COLOGNESI, AND PIAZZA PHYSICAL REVIEW B 99, 024307 (2019) 0.06 0.08 0.1 0.12 0.14 0.16 400 500 600 700 800 900Average kinetic energy [eV] Temperature [K]All bursts ILM 100 ILM 110 ILM 11110−410−310−210−1100 400 500 600 700 800 900Site−occupancy probability Temperature [K]All bursts ILM 100 ILM 110 ILM 111 0 2 4 6 400 500 600 700 800 900Average lfetime [ps] Temperature [K]All bursts ILM 100 ILM 110 ILM 111 0 0.1 0.2 0.3 0.4 500 600 700 800 900Mean squared displacement [Å2] Temperature [K]All bursts ILM 100 ILM 110 ILM 111 FIG. 7. Burst analysis for the putative ILM subpopulations compared to the data for the whole burst ensemble. (Top left) Site-occupancy probabilities and ﬁts with the expression ( 9). The corresponding best-ﬁt parameters are reported in Table II. The green dashed line is the SOP computed in Ref. [ 41] for [111] ILM excitations. (Top right) Average lifetime. (Bottom left) Mean-square displacement. The dashed line represents the average computed over the whole set of Na ions in the crystal. (Bottom right) Average kinetic energy. The dashed line marks the equipartition result. As expected, this describes the average kinetic energy of Na ions when computed over the whole set of Na ions in the crystal. points at the boundary of the Brillouin zone ( Lpoints in the case of [111] modes). These might reﬂect interconversionevents or mixed-character modes, as already suggested inRef. [ 44]. An analysis of the lifetimes measured for putative ILM- like excitations also conﬁrms some of the predictions made in Ref. [ 43] (top right panel in Fig. 7). Excitations along [100] and [110] display lower-than-average lifetimes, whilethe lifetimes of [111] excitations increase rapidly with tem- perature, to last beyond average bursts at high temperatures. Interestingly, the lifetimes of [100] and [111] bursts seem todisplay a marked dependence on temperature, matched bytheir rapidly increasing SOP, while [110] excitations shownearly temperature-independent lifetimes, rhyming with a much more slowly increasing SOP (top left panel). This seems to point to a less marked nonlinear character for bursts sievedout along [110].Amplitudes and energies of bursts seem to trace a consis- tent picture (bottom panels in Fig. 7). While along [110], and to a lesser extent along [100], the data relative to the putativetheoretical subpopulations display trends that are consistentwith the average behavior of the whole burst database, the[111] subensemble demonstrates a substantially contrastingtrend. More speciﬁcally, excitations selected to lie along thetheoretical [111] dispersion law display systematically higher-than-average energies and larger-than-average amplitudes.This is consistent with a more marked nonlinear characterof these excitations, which in turn upholds the predictionsreported in Ref. [ 43] concerning the markedly higher lifetime of [111] ILMs. V . CONCLUSIONS AND DISCUSSION In this paper, we have introduced a method to resolve transient localization of energy in time-frequency space. Our 024307-10WA VELET IMAGING OF TRANSIENT ENERGY … PHYSICAL REVIEW B 99, 024307 (2019) technique is based on continuous wavelet transform of ve- locity time series coupled to a threshold-dependent ﬁlteringprocedure to isolate excitation events from background noisein a speciﬁc spectral region. A frequency integration in thereference spectral region allows us to track the time evolutionof the center-of-mass frequency of that region. These reduceddata, in turn, can be easily exploited to investigate the statisticsof the burst excitation dynamics. For example, this procedurecan be employed to characterize the distribution of the burstlifetimes and investigate the roots of the excitation process bylooking at the distribution of excitation times (time intervalsseparating consecutive excitation events). As an illustration of our method, we have employed the wavelet-based energy burst imaging technique to investigatespontaneous localization of nonlinear modes in the gap ofNaI crystals at high temperature. Our method allows oneto build a database of excitation events, and to measuretheir site-occupancy probability, average lifetime, energy, fre-quency, amplitude, and excitation times. It is highly likelythat such database contains subpopulations corresponding tospontaneous excitation of ILMs, provided a sufﬁcient numberof events is recorded, i.e., provided large enough systemsare considered and long-enough trajectories are simulated.Overall, the burst database shows rather clearly that the eventsrecorded are thermally excited. One way to rationalize theoverall excitation equilibrium and kinetics is in terms of areaction kinetic scheme involving chemical species equiva-lents, representing ﬂuctuations (F),bursts (B) along with a variable number of intermediates. The numerically measuredlifetimes and excitation times suggest that such kind of reac-tion scheme is associated with an energy landscape with asmany minima as different virtual species. It is possible thatthis analogy could be pushed even farther than this, throughthe identiﬁcation of the appropriate collective coordinates (thesupport of the energy landscape), which could allow oneto reconstruct the landscape from the simulations throughstandard free-energy calculation algorithms. The problem than one faces in the second logical stage of our method is how to single out events corresponding togenuine ILM excitation, as opposed to generic soft nonlinear phonon excitations. We observe that this is a rather formidabletask, as the fraction of such events is expected to be low,while their polarization and localization length can only beguessed from zero-temperature calculations. In this paper, wehave followed a very simple and minimalistic strategy, basedexplicitly on the zero-temperature predictions, to sift throughthe whole burst database at each temperature in the questfor ILM events. This procedure seems to succeed, at leastpartially, in the task of isolating events that display a markednonlinear character. In particular, events selected from theburst database by matching the theoretical T=0 frequency- amplitude relation for the [111] polarization seem to detachthe most from the average behavior of the entire databases,suggesting that at least some of these events might be genuineILMs along [111]. The corresponding site-occupancy prob-ability for these events is described by the same theoreticalexpression as suggested in Ref. [ 41], although we ﬁnd that there might be more excitation pathways for these modes thanmerely speciﬁed by the symmetry-equivalent points at theboundary of the Brillouin zone ( Lpoints). This might reﬂect interconversion events or mixed-character modes, as hinted atin Ref. [ 44]. From a general point of view, it is hard to state whether thermal populations of ILMs in crystals allow them to bedetected and characterized directly from equilibrium MD sim-ulations. It is possible that this would require, in general, somesort of an intrinsically nonlinear pump-probe technique toenhance selectively thermal populations of nonlinear excita-tions. A clever example of ampliﬁcation and counting of ILMexcitations is reported in Ref. [ 75] for quasi-one-dimensional antiferromagnetic lattices, where an original pump-probetechnique based on a four-wave mixing ampliﬁcation of theweak signal from the few large-amplitude ILMs is used tocount ILM emission events. In principle, an ILM generationand steady-state locking techniques such as further discussedin Ref. [ 74] could be implemented numerically to produce energy localization in a controlled fashion in atomic latticesat high temperature. In general, ILM localization is expected to be accompanied by a strain ﬁeld (sometimes referred to as the dc component)as a result of odd-order anharmonic terms. Moreover, as sug-gested in Ref. [ 46], the strain ﬁeld associated with thermal ex- citation of ILMs is expected to take the form of planar faultlikestructures with an occurrence frequency fof approximately one in every ten cells ( f=1/10). However, our method is based on the analysis of velocity time series. Therefore it isinsensitive in principle to static distortions associated withthe ILM displacement ﬁelds. Nonetheless, we observe that aspatial version of our method could be designed in principle to detect the features of the strain ﬁelds associated with ILMs,by Gabor transforming spatial-Fourier transformed time seriescorresponding to speciﬁc wave vectors. To make contact withthe results reported in Ref. [ 46], one should also consider larger systems including at least twice as many cells in eachdirections than the present study. Although we demonstrated here the power of wavelet- based imaging to investigate the dynamics of nonlinear ex-citations in the gap of NaI crystals, methods of the like canbe useful in many contexts where one wishes to characterizetransient energy excitation or energy transfer processes. Thelatter kind of phenomena, which is not investigated here, ap-pears to be a promising domain of application of our method,both at the classical and quantum level. For example, it wouldbe interesting to adopt a tool inspired to our method to char-acterize the dynamics of energy transfer and exciton-phononinteractions in light-harvesting complexes [ 76–78]. Wavelet- based methods could be used to characterize the dynamics ofvibrational energy transfer [ 79,80] in many complex system, including biomolecules. 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PhysRevB.74.024204.pdf	Thermodynamic properties of binary hcp solution phases from special quasirandom structures Dongwon Shin, Raymundo Arróyave, and Zi-Kui Liu Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Axel Van de Walle Engineering and Applied Science Division, California Institute of Technology, Pasadena, California 91125, USA /H20849Received 13 September 2005; revised manuscript received 25 April 2006; published 14 July 2006 /H20850 Three different special quasirandom structures /H20849SQS’s /H20850of the substitutional hcp A1−xBxbinary random solutions /H20849x=0.25, 0.5, and 0.75 /H20850are presented. These structures are able to mimic the most important pair and multi-site correlation functions corresponding to perfectly random hcp solutions at those compositions. Due tothe relatively small size of the generated structures, they can be used to calculate the properties of random hcpalloys via ﬁrst-principles methods. The structures are relaxed in order to ﬁnd their lowest energy conﬁgurationsat each composition. In some cases, it was found that full relaxation resulted in complete loss of their parentalsymmetry as hcp so geometry optimizations in which no local relaxations are allowed were also performed. Ingeneral, the ﬁrst-principles results for the seven binary systems /H20849Cd-Mg, Mg-Zr, Al-Mg, Mo-Ru, Hf-Ti, Hf-Zr, and Ti-Zr /H20850show good agreement with both formation enthalpy and lattice parameters measurements from experiments. It is concluded that the SQS’s presented in this work can be widely used to study the behavior ofrandom hcp solutions. DOI: 10.1103/PhysRevB.74.024204 PACS number /H20849s/H20850: 61.66.Dk I. INTRODUCTION Thermodynamic modeling using the calculation of phase diagrams /H20849CALPHAD /H20850method1,2attempts to describe the Gibbs energy of a system through empirical models whoseparameters are ﬁtted using experimental information. Thesedescriptions allow the extrapolation of a system’s thermody-namic properties to regions in the composition-temperature space that have not/cannot be accessed through experiments.These empirical models, however, are as good as the dataused to ﬁt them and are therefore limited by the availabilityof accurate experimental data. This limitation can be over-come by using theoretical calculations based on ﬁrst-principles methods, which are capable of predicting thephysical properties of phases with no experimental input. 3 Unfortunately, despite their predictive nature, these methodsare not yet able to calculate the thermochemistry ofmaterials—especially multicomponent, multiphasesystems—with the precision required in industry. A natural way of improving the predictive capabilities of empirical models while maintaining their applicability topractical problems is by combining ﬁrst-principles andCALPHAD techniques. Thanks to efﬁcient schemes forimplementing density functional theory /H20849DFT /H20850, 4the almost- routine use of ﬁrst-principles results within the CALPHADmethodology has become a reality. In this hybrid approach,the energetics obtained through electronic structure calcula-tions are used as input data within the CALPHAD formalismto obtain the parameters that describe the Gibbs energy of thesystem. 5 The ﬁrst-principles electronic structure calculations of perfectly ordered periodic structures are relatively straight-forward since they usually rely on the use of periodic bound-ary conditions. Problems arise, however, when attempting touse these methods to study the thermochemical properties ofrandom solid solutions since an approximation must be madein order to simulate a random atomic conﬁguration through aperiodic structure. The usual approaches that have been used in the past can be summarized as follows. /H20849i/H20850The most direct approach is the supercell method. In this case, the sites of the supercell can be randomly occupiedby either AorBatoms to yield the desired A 1−xBxcomposi- tion. In order to reproduce the statistics corresponding to arandom alloy, such supercells must necessarily be very large.This approach is, therefore, computationally prohibitivewhen the size of the supercell is on the order of hundreds ofatoms. /H20849ii/H20850Another technique, the coherent potential approximation 6/H20849CPA /H20850method, is a single-site approximation that models the random alloy as an ordered lattice of effec-tive atoms. These are constructed from the criterion that theaverage scattering of electrons off the alloy componentsshould vanish. 7In this method, local relaxations are not con- sidered explicitly and the effects of alloying on the distribu-tion of local environments cannot be taken into account. Lo-cal relaxations have been shown to signiﬁcantly affect theproperties of random solutions, 8especially when the con- stituent atoms vary greatly in size and, therefore, their omis-sion constitutes a major drawback. Although the local relax-ation energy can be taken into account, 7these corrections rely on cluster expansions of the relaxation energy of orderedstructures and the distribution of local environments is notexplicitly considered. Additionally, such corrections are sys-tem speciﬁc. /H20849iii/H20850A third option is to apply the cluster expansion approach. 9In this case, a generalized Ising model is used and the spin variables can be related to the occupation of eitheratom AorBin the parent lattice. In order to obtain an ex- pression for the conﬁgurational energy of the solid phase, theenergies of multiple conﬁgurations /H20849typically in the order of a few dozens /H20850based on the parent lattice must be calculated to obtain the parameters that describe the energy of anygiven A 1−xBxcomposition. This approach typically relies onPHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 1098-0121/2006/74 /H208492/H20850/024204 /H2084913/H20850 ©2006 The American Physical Society 024204-1the calculation of the energies of a few dozen ordered structures. In the techniques outlined above, there are serious limita- tions in terms of either the computing power required /H20849super- cells, cluster expansion /H20850or the ability to accurately represent the local environments of random solutions /H20849CPA /H20850. Ideally, one would like to be able to accurately calculate the thermo-dynamic and physical properties of a random solution withas small a supercell as possible so that accurate ﬁrst-principles methods can be applied. This has become possiblethanks to the development of special quasirandom structures/H20849SQS’s /H20850. The concept of SQS was ﬁrst developed by Zunger et al. 10 to mimic random solutions without generating a large super- cell or using many conﬁgurations. The basic idea consists ofcreating a small—4–48 atoms—periodic structure with thetarget composition that best satisﬁes the pair and multisitecorrelation functions corresponding to a random alloy, up toa certain coordination shell. Upon relaxation, the atoms inthe structure are displaced away from their equilibrium po- sitions, creating a distribution of local environments that canbe considered to be representative of a random solution, atleast up to the ﬁrst few coordination shells. Provided the interatomic electronic interactions in a given system are relatively short range, the ﬁrst-principles calcula-tions of the properties of these designed supercells can beexpected to yield sensible results, especially when calculat-ing properties that are mostly dependent on the local atomicarrangements, such as enthalpy of mixing, charge transfer,local relaxations, and so forth. It is important to stress thatthe approach fails whenever a property depends on long-range interactions. The SQS’s for fcc-based alloys and bcc alloys have been generated by Wei et al. 11and Jiang et al. ,12respectively. However, to the best knowledge of these authors, there hasbeen no investigation on the application of the SQS approachto the study of hcp substitutional random solutions. In thepresent work, we propose two SQS’s capable of mimickinghcp random alloys at 25, 50, and 75 at. %. The paper is or-ganized as follows. The proposed SQS’s are characterized in terms of their ability to reproduce the pair and multisite correlation func-tions of a truly random hcp solution. Subsequently, the struc-tures are tested in terms of their ability to reproduce, viaﬁrst-principles calculations, the properties of certain selectedstable or metastable binary hcp solutions, namely, Cd-Mg,Mg-Zr, Al-Mg, Mo-Ru, Hf-Ti, Hf-Zr, and Ti-Zr. To furtheranalyze the relaxation behavior of the structures, the distri-bution of ﬁrst nearest bond lengths as well as the radial dis-tribution for the ﬁrst few coordination shells is presented.Finally, for each of the selected binaries, the calculated andavailable experimental lattice parameters and enthalpy ofmixing are compared. Results from other techniques are alsopresented where available in order to further corroborate thepresent calculations. II. GENERATION OF SPECIAL QUASIRANDOM STRUCTURES In order to characterize the statistics of a given atomic arrangement, one can use its correlation function.13Withinthe context of lattice algebra, we can assign a “spin value,” /H9268= ±1, to each of the sites of the conﬁguration, depending on whether the site is occupied by A-o rB-type atoms. Fur- thermore, all the sites can be grouped in ﬁgures, f/H20849k,m/H20850,o fk vertices, where k=1,2,3,..., responds to a shape, point, pair, and triplet, ¼, respectively, spanning a maximum distance of m, where m=1,2,3,..., is the ﬁrst, second, and third-nearest neighbors, and so forth. The correlation functions /H9016¯k,mare the averages of the products of site occupations /H20849±1 for bi- nary alloys and ±1, 0 for ternary alloys /H20850of ﬁgure kat a distance mand are useful in describing the atomic distribu- tion. The optimum SQS for a given composition is the onethat best satisﬁes the condition /H20849/H9016¯k,m/H20850SQS/H11061/H20855/H9016¯k,m/H20856R, /H208491/H20850 where /H20855/H9016¯k,m/H20856Ris the correlation function of a random alloy, which is simply by /H208492x−1/H20850kin the A1−xBxsubstitutional bi- nary alloy, where xis the composition. We considered SQS’s of two different compositions, i.e., x=0.5 and 0.75. Unlike cubic structures, the order of a given conﬁguration in the hcp lattices relative to a given lattice site may bealtered with the variation of c/aratio. However, these new arrangements will not cause any change in the correlationfunctions, since one can thus use any c /aratio to generate the hcp SQS’s. As a matter of simplicity, the ideal c/aratio was considered in order to generate SQS’s. In the present work, we used the alloy theoretic automa- tion toolkit /H20849ATAT /H20850 3to generate special quasirandom struc- tures for the hcp structure of 8 and 16 sites. The schematicdiagrams of the created special quasirandom structure with16 atoms are shown in Fig. 1and the corresponding lattice vectors and atomic positions are listed in Table I. The correlation functions of the generated 8- and 16-atom SQS’s were investigated to verify that they satisﬁed at leastthe short-range statistics of an hcp random solution. As isshown in Table II, the 16-atom structures satisfy the pair correlation functions of random alloys up to the ﬁfth andthird nearest neighbor for the 50 at. % and the 75 at. % com-positions, respectively. On the other hand, Table IIshows that the SQS-8 for 75 at. % could not satisfy the randomcorrelation function even for the ﬁrst-nearest-neighbor pair.Thus, SQS’s with 16 atoms are capable of mimicking a ran-dom hcp conﬁguration beyond the ﬁrst coordination shell. It is important to note that in Table II, and contrary to what is observed in the SQS for cubic structures, some ﬁg-ures have more than one crystallographically inequivalentﬁgure at the same distance. For example, in the case of hcplattices with the ideal c/aratio, two pairs may have the same interatomic distance and yet be crystallographically in-equivalent. In this case, despite the fact that the two pairs /H208490,0,0 /H20850and /H20849a,0,0 /H20850;/H208490,0,0 /H20850and /H208491 3,2 3,1 2/H20850, have the same in- teratomic distance a, they do not share the same symmetry operations. This degeneracy is broken when the c/aratio deviates from its ideal value. For the sake of efﬁciency, the initial lattice parameters of the SQS’s were determined from Vegard’s law. By doing so,thec/aratio was no longer ideal. Afterwards, we checked the correlation functions of the new structures and found thatSHIN et al. PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-2they remained the same as long as the corresponding ﬁgures were indentical. The maximum range over which the correlation function of an SQS mimics that of a random alloy can be increased byincreasing the supercell size. As the size of the SQS in-creases, the probability of ﬁnding conﬁgurations that mimicrandom alloys over a wider coordination range increases ac-cordingly. The search algorithm used in this work consists ofenumerating every possible supercell of a given volume andfor each supercell, enumerating every possible atomic con-ﬁguration. For each conﬁguration, the correlation functionsof different ﬁgures, i.e., points, pairs, and triplets, are calcu-lated. To save time, the calculation of the correlations isstopped as soon as one of them does not match the randomstate value. This algorithm becomes prohibitively expensivevery rapidly. The generation of a larger SQS could be accom-plished by using a Monte Carlo–like scheme /H20849e.g., Abrikosov et al. 14/H20850, but this is beyond the scope of present work. In fact, the authors could generate a 32-atom SQS’s, and the averagetotal energy difference between 16-atom SQS’s and 32-atomSQS’s in the Cd-Mg system was around 2 meV per atom.The authors maintain a focus on 16-atom SQS, because thissize represents a good compromise between accuracy and thecomputational requirements associated with the necessaryﬁrst-principles calculations. It is also important to note that ﬁnding a good hcp SQS is more difﬁcult than ﬁnding an SQS of cubic structures withthe same range of matching correlations due to the fact that,for a given range of correlations, there are more symmetri- cally distinct correlations to match. Additionally, the lowersymmetry of the hcp structure implies that there are alsomany more candidate conﬁgurations to search through in or- der to ﬁnd a satisfactory SQS. Thus, the number of distinctsupercells is larger and the number of symmetrically distinctatomic conﬁgurations is larger, in comparison to fcc or bcclattices.TABLE I. Structural descriptions of the SQS- Nstructures for the binary hcp solid solution. Lattice vectors and atomic positionsare given in fractional coordinates of hcp lattice. Atomic positionsare given for the ideal, unrelaxed hcp sites. x=0.5 x=0.75 Lattice vectors Lattice vector /H208980− 1 − 1 −2 −2 0 −2 1 −1 /H20899/H2089811 1 −1 0 1 0− 4 0/H20899 Atomic positions Atomic positions −21 3−12 3−11 2A −1 3−22 311 2A −1 −1 −1 A −1 3−12 311 2A −2 0 −1 A 0− 32 A −11 32 3−11 2A 0− 31 A −3 −2 −1 A 0− 22 B SQS-16 −21 32 3−11 2A 0− 12 B −4 −2 −2 A 002 B −31 3−12 3−11 2A −1 3−2 311 2B −2 −2 −1 B −1 31 311 2B −11 3−12 31 2B −1 3−32 31 2B −3 −1 −1 B 0− 21 B −2 −1 −1 B −1 3−22 31 2B −11 32 31 2B 0− 11 B 1 32 31 2B −1 3−12 31 2B −21 3−12 31 2B 001 B −3 −1 −2 B −1 3−2 31 2B Lattice vectors Lattice vectors /H20898−1 1 1 1− 1 111 0/H20899/H2089811 − 10− 1 − 1 −2 2 0 /H20899 Atomic positions Atomic position SQS-81 32 31 2A −1 1 −1 A 1 32 311 2A −2 32 3−1 2A 101 A −12 312 3−1 2B 112 A −1 1 −2 B 011 B −2 32 3−11 2B 111 B 00 − 1 B 11 32 31 2B 00 − 2 B 11 32 311 2B1 3−1 3−11 2B FIG. 1. Crystal structures of the A1−xBxbinary hcp SQS-16 structures in their ideal, unrelaxed forms. All the atoms are at theideal hcp sites, even though both structures have the space group P1./H20849a/H20850SQS-16 for x=0.5. /H20849b/H20850SQS-16 for x=0.75.THERMODYNAMIC PROPERTIES OF BINARY hcp ¼ PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-3In order to verify the proposed 16-atom SQS’s are ad- equate for the simulation of hcp random solutions, the au-thors calculated other SQS’s at 75 at. % which have random-like pair correlations up to the third nearest neighbor but thathave slightly different correlations for the fourth nearestneighbor. The pair correlation function at 75 at. % of a trulyrandom solution would be /H208492/H110030.75−1 /H20850 2=0.25 and therefore the four SQS’s in Table IIIare worse than the one used in the present work. These structures were applied to the Cd25 at. % –Mg 75 at. % system and, as can be seen in TableIII, the associated energy differences are negligible. This is due to the fact that the energetics of this system are domi-nated by short-range interactions. Thus, as long as the mostimportant pair correlations /H20849up to the third nearest neighbors in hcp structure with ideal c/aratio /H20850are satisﬁed, the SQS’s can successfully be applied to acquire properties of randomsolutions in which short-range interactions dominate. III. FIRST-PRINCIPLES METHODOLOGY The selected hcp SQS-16 structures were used as geo- metrical input for the ﬁrst-principles calculations. The Vi-enna Ab initio Simulation Package 15/H20849V ASP /H20850was used to per- form the density functional theory electronic structurecalculations. The projector augmented wave method16was chosen and the general gradient approximation17was used to take into account exchange and correlation contributions toTABLE II. Pair and multisite correlation functions of SQS- Nstructures when the c/aratio is ideal. The number in the square bracket next to /H9016¯k,mis the number of equivalent ﬁgures at the same distance in the structure, the so-called degeneracy factor. Randomx=0.5 SQS-16 SQS-8 Randomx=0.75 SQS-16 SQS-8 /H9016¯2,1/H208516/H20852 0 0 0 0.25 0.25 0.16667 /H9016¯2,1/H208516/H20852 0 0 0 0.25 0.25 0.33333 /H9016¯2,2/H208516/H20852 0 0 0 0.25 0.25 0.33333 /H9016¯2,3/H208512/H20852 0 0 0 0.25 0.25 0 /H9016¯2,4/H2085112/H20852 0 0 0 0.25 0.25 0.16667 /H9016¯2,4/H208516/H20852 0 0 −0.33333 0.25 0.45833 0 /H9016¯2,5/H2085112/H20852 0 0 −0.33333 0.25 0.33333 0.33333 /H9016¯2,6/H208516/H20852 0 −0.33333 0.33333 0.25 0.16667 0.33333 /H9016¯2,7/H2085112/H20852 0 0 0 0.25 0.25 0.5 /H9016¯2,8/H2085112/H20852 0 0 0 0.25 0.1667 0.33333 /H9016¯3,1/H2085112/H20852 0 0 0.33333 0.125 −0.08333 0.16667 /H9016¯3,1/H208512/H20852 0 0 0 0.125 0.25 0.5 /H9016¯3,1/H208512/H20852 0 0 0 0.125 0.25 0.5 /H9016¯3,2/H2085124/H20852 0 0 0 0.125 −0.04167 0 /H9016¯3,3/H208516/H20852 0 0 0 0.125 −0.08333 0.16667 /H9016¯3,3/H208516/H20852 0 0 0 0.125 −0.08333 −0.16667 /H9016¯4,1/H208514/H20852 0 0 0 0.0625 0 0.5 /H9016¯4,2/H2085112/H20852 0 0 −0.33333 0.0625 −0.16667 −0.16667 /H9016¯4,2/H2085112/H20852 0 0 0 0.0625 0 0 /H9016¯4,3/H208516/H20852 0 0.33333 0.33333 0.0625 −0.16667 0 TABLE III. Pair correlation functions up to the ﬁfth and the calculated total energies of other 16 atoms SQS’s for Cd 0.25Mg 0.75 are enumerated to be compared with the one used in this work/H20849SQS-16 /H20850. The total energies are given in unit’s of eV/atom. abcd SQS-16 /H9016¯2,1/H208516/H20852 0.25 0.25 0.25 0.25 0.25 /H9016¯2,1/H208516/H20852 0.25 0.25 0.25 0.25 0.25 /H9016¯2,2/H208516/H20852 0.25 0.25 0.25 0.25 0.25 /H9016¯2,3/H208512/H20852 0.25 0.25 0.25 0.25 0.25 /H9016¯2,4/H2085112/H20852 0.20833 0.16667 0.16667 0.08333 0.25 /H9016¯2,4/H208516/H20852 0.5 0.5 0.5 0.16667 0.45833 /H9016¯2,5/H2085112/H20852 0.5 0.16667 0.33333 0.33333 0.33333 Symmetry preserved−1.3864 −1.3882 −1.3886 −1.3886 −1.3869 Fully relaxed−1.3874 −1.3887 −1.3889 −1.3893 −1.3883SHIN et al. PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-4the Hamiltonian of the ion-electron system. A constant en- ergy cutoff of 350 eV was used for all the structures, with5000 kpoints per reciprocal atom based on the Monkhorst- Pack scheme for the Brillouin-zone integrations. The k-point meshes were centered at the /H9003point. The convergence crite- rion for the calculations was 10 meV with respect to the 16atoms. Spin-polarization was not taken into account. Thegenerated SQS’s were either fully relaxed, or relaxed withoutallowing local ion relaxations, i.e., only volume and c/aratio were optimized. As will be seen below, the full relaxationcaused some of the SQS’s to lose the original hcp symmetry. IV. RESULTS AND DISCUSSIONS A. Analysis of relaxed structures The symmetry of the resulting SQS was checked using the PLATON code18before and after the relaxations. Both SQS’s have the lowest symmetry of P1, although all the at-oms are sitting on the lattice sites of hcp. The procedure wasveriﬁed by checking the symmetries of the generated unre- laxed SQS. Once all the sites in the SQS were substituted with one single atomic species, PLATON identiﬁed SQS’s as perfect hcp structures. All the atoms of the initial structuresare on their exact hcp lattice sites. However, upon relaxation the atoms may be displaced from these ideal positions. Ac-cording to the deﬁnition of an hcp random solution, all theatoms, in this case two different type of atoms, should be atthe hcp lattice points—within a certain tolerance—even afterthe structure has been fully relaxed. The default tolerance ofdetecting the symmetry of the relaxed structures allowed theatoms to deviate from their original lattice sites by up to20%. In principle, relaxations should be performed with respect to the degrees of freedom consistent with the initial symme-try of any given conﬁguration. In the particular case of thehcp SQS’s, local relaxations may in some cases be so largethat the character of the underlying parent lattice is lost.However, within the CALPHAD methodology, one has todeﬁne the Gibbs energy of a phase throughout the entirecomposition range, regardless of whether the structure isstable or not. In these cases, it is necessary to constrain therelaxations so that they are consistent with the lattice vectorsand atom positions of an hcp lattice. Obviously, the energeticcontributions due to local relaxations are not considered inthis case. The results of these constrained relaxations cantherefore be directly compared to those calculations usingthe CPA. In most cases, local relaxations were not signiﬁ-cant. However, in a few instances, it was found that thestructure was too distorted to be considered as hcp after thefull relaxation. However, this symmetry check was not suf-ﬁcient to characterize the relaxation behavior of the relaxedSQS. Furthermore, in some of the cases it may be possiblefor the structure to fail the symmetry test and still retain anhcp-like environment within the ﬁrst couple of coordinationshells, implying that the energetics and other properties cal-culated from these structures could be characterized as rea-sonable, although not optimal, approximations of randomconﬁgurations.1. Radial distribution analysis In order to investigate the local relaxation of the fully relaxed SQS, their radial distribution /H20849RD/H20850was analyzed. Through this analysis, the bond distribution and coordinationshells were studied to determine whether the relaxed struc- tures maintained the local hcp-like environment they weresupposed to mimic in the ﬁrst place. Additionally, this analy-sis permitted us to quantify the degree of local relaxations upto the ﬁfth coordination shells. The RD of each of the fully relaxed structures was ob- tained by counting the number of atoms within bins of10 −3Å, up to the ﬁfth coordination shell. In order to elimi- nate high frequency noise, the raw data was scaled andsmoothed through Gaussian smearing with a characteristicdistance of 0.01 Å. Pseudo-V oigt functions were then used toﬁt each of the smoothed peaks and the goodness of ﬁt was inpart determined through the summation of the total areas ofthe peaks and comparing them to the total number of atomsthat were expected within the analyzed coordination shells.The relaxation of the atoms at each coordination shell isquantiﬁed by the width of the corresponding peak in theﬁtted RD. The RD results of selected SQS’s are given in Fig. 2. The unrelaxed, fully relaxed, and nonlocally relaxed structuresare compared in each case as well as the smoothed bonddistributions and their ﬁtted curves. These results are repre-sentative of the RD’s obtained for the seven binary systemsat the three compositions studied. Figure 2/H20849a/H20850shows the RDs for the Hf-Zr SQS at the 50 at. % composition. As can be seen in the ﬁgure, the RDsfor the unrelaxed and nonlocally relaxed SQS are almostidentical, implying that in this system Vegard’s Law isclosely followed. Furthermore, the RD for the fully relaxedSQS in Fig. 2/H20849b/H20850shows a rather narrow distribution around each of the the bondlengths corresponding to the ideal orunrelaxed structure. The system therefore needs to undergovery negligible local relaxations in order to minimize its en-ergy. In the case of the Cd-Mg solution at 50 at. % /H20851Fig. 2/H20849c/H20850/H20852, the RDs of the unrelaxed and nonlocally relaxed SQS aremore dissimilar. Even in the nonlocally relaxed calculation,the original ﬁrst coordination shell /H20849corresponding to the six ﬁrst-nearest neighbors /H20850has split into two different shells /H20849of 4 and 2 atoms /H20850and the position of the peak is noticeably shifted. The ﬁrst two well deﬁned coordination shells of theunrelaxed structure have merged into a single, broad peak at3.14 Å upon full relaxation, as shown in Fig. 2/H20849d/H20850. This peak now encloses 12 ﬁrst nearest neighbors. As shown in Table II,/H9016 ¯2,1and/H9016¯2,4have two differnet types of pairs. However, since they have the same correlation functions, they cannotbe distinguished. In Fig. 2/H20849d/H20850it is also shown how the fourth and ﬁfth coordination shells merge at 5.40 Å, enclosing 18atoms. It can be expected that if the c/aratio of a relaxaed structure is close to ideal and the broadening of nearby shellsare wide enough that they merge, then the structure has al-most the same radial distribution of an ideal hcp structure,albeit with a large peak width. Figure 2/H20849e/H20850shows the RD for the Mg 50Zr50composition. Among the three RD’s presented in Fig. 2, this one is clearlyTHERMODYNAMIC PROPERTIES OF BINARY hcp ¼ PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-5the one that undergoes the greatest distortion upon full relax- ation. Even in the nonlocally relaxed structures there is abroad bondlength distribution around the peaks of the unre-laxed SQS. With respect to the fully relaxed SQS, it can beseen how the peaks for the ﬁfth and sixth coordination shells have practically merged. In this case, the local environmentof each atom within the SQS stops being hcp-like within theﬁrst couple of coordination shells. Although the two end FIG. 2. Radial distribution analysis of selected SQS’s. The dotted lines under the smoothed and ﬁtted curves are the error between the two curves. /H20849a/H20850RD of Hf 50Zr50/H20849/H9004Hmix/H110110/H20850./H20849b/H20850Smoothed and ﬁtted RD’s of fully relaxed Hf 50Zr50./H20849c/H20850RD of Cd 50Mg 50/H20849/H9004Hmix/H110210/H20850./H20849d/H20850 Smoothed and ﬁtted RD’s of fully relaxed Cd 50Mg 50./H20849e/H20850RD of Mg 50Zr50/H20849/H9004Hmix/H110220/H20850./H20849f/H20850Smoothed and ﬁtted RD’s of fully relaxed Mg 50Zr50.SHIN et al. PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-6members of this binary alloy have an hcp as the stable struc- ture, it is evident from this ﬁgure that the SQS arrangementis unstable and there is a tendency for the structure to distort.In this system, there is a miscibility gap in the hcp phase upto/H11011900 K and the RD reﬂects the tendency for the system to phase separate. The results from the peak ﬁtting for all the fully relaxed SQS’s are summarized in Table IV. It should be noted that regardless of the system and compositions, the sum of theareas under each peak should converge to a single value,proportional to 50 atoms. For each peak, the error was quan-tiﬁed as the absolute and normalized difference between theexpected and actual areas. The error reported in the table isthe averaged value for all the peaks in the RD. The broadnessof the peaks in the RD is quantiﬁed through the full width athalf maximum /H20849FWHM /H20850. In the table, the reported FWHM corresponds to the average FWHM observed for the coordi-nation shells enclosing a total of 50 atoms. Note that thealloys with the smallest FWHM are Hf-Zr and Cd-Mg. Aswill be seen later, Hf-Zr behaves almost ideally and Cd-Mgis a system with rather strong attractive interactions between unlike atoms that forms ordered hexagonal structures at the25 and 75 at. % compositions. 2. Bond length analysis In addition to the RD analysis, we performed the bondlength analysis /H20849A-A,B-B, and A-B/H20850for all the relaxed SQS’s. In Table Vthe bond lengths corresponding to the ﬁrst nearest neighbors for all the 21 SQS’s are presented. As ex-pected, in the majority of the cases the sequence d ii/H11021dij /H11021djjis observed throughout the composition range, where dijcorresponds to the bond distance between two different atom types. The two notable exceptions to this trend corre-spond to the Cd-Mg and Mg-Zr alloys. As will be mentionedbelow, the Cd-Mg system tends to form rather stable inter-metallic compounds at the 25, 50, and 75 at. % composi-tions, including two hexagonal intermetallic compounds. Thecalculated enthalpy of mixing in this case—shown in Fig.3/H20849a/H20850—is the most negative among seven binaries studied and the fact that the Cd-Mg bonds are shorter than Cd-Cd andTABLE IV . Results of radial distribution analysis for the seven binaries studied in this work. FWHM shows the averaged full width at half maximum and is given in Å. Errors indicate the difference in the number of atoms calculated through the sum of peak areas and thoseexpected in each coordination shell. Compositions Cd-Mg Mg-Zr Al-Mg Mo-Ru Hf-Ti Hf-Zr Ti-Zr FAHM 0.06±0.01 0.09±0.03 0.08±0.02 N/A a0.11±0.03 0.02±0.00 0.16±0.05 A75B25 Error, % 0.72 0.39 0.47 N/A 1.07 1.84 1.27 Symmetry PASS PASS PASS FAIL PASS PASS FAIL FWHM 0.07±0.02 0.15±0.02 0.15±0.07 0.13±0.01 0.16±0.02 0.03±0.01 0.09±0.06 A50B50 Error, % 0.30 1.42 1.28 1.90 0.35 1.84 2.39 Symmetry PASS FAIL FAIL PASS PASS PASS PASS FWHM 0.04±0.01 0.09±0.03 0.10±0.02 0.07±0.02 0.11±0.06 0.03±0.00 0.13±0.07 A25B75 Error, % 2.05 1.22 0.26 1.93 0.26 1.01 0.96 Symmetry PASS PASS PASS PASS PASS PASS PASS aThe radial distribution analysis of Mo 75 at. % –Ru 25 at. % was not possible since it completely lost its symmetry as hcp. TABLE V . First nearest-neighbor average bondlengths for the fully relaxed hcp SQS of the seven binaries studied in this work. Uncertainty corresponds to the standard deviation of the bondlength distributions. Compositions Bonds Cd-Mg Mg-Zr Al-Mg Mo-Ru Hf-Ti Hf-Zr Ti-Zr A100B0 A–A 3.07 3.18 2.87 2.75 3.13 3.13 2.87 A–A3.17±0.10 3.18±0.03 2.92±0.03 3.14±0.05 3.18±0.03 2.96±0.07 A75B25 A–B3.16±0.11 3.18±0.05 2.95±0.03 N/A 3.10±0.05 3.18±0.03 3.02±0.07 B–B3.18±0.10 3.12±0.10 2.96±0.03 3.09±0.06 3.18±0.04 3.04±0.06 A–A3.16±0.04 3.16±0.04 2.98±0.06 2.81±0.08 3.09±0.06 3.18±0.03 3.00±0.09 A50B50 A–B3.12±0.04 3.20±0.06 3.02±0.06 2.75±0.04 3.05±0.07 3.19±0.03 3.06±0.08 B–B3.15±0.03 3.14±0.08 3.07±0.08 2.75±0.04 3.00±0.06 3.20±0.03 3.12±0.08 A–A3.16±0.01 3.15±0.04 3.06±0.04 2.73±0.04 3.02±0.05 3.19±0.03 3.09±0.08 A25B75 A–B3.14±0.02 3.19±0.04 3.08±0.04 2.73±0.04 3.00±0.06 3.19±0.03 3.11±0.06 B–B3.15±0.01 3.18±0.04 3.11±0.03 2.71±0.04 2.95±0.05 3.20±0.04 3.17±0.06 A0B100 B–B 3.18 3.19 3.18 2.68 2.87 3.19 3.19THERMODYNAMIC PROPERTIES OF BINARY hcp ¼ PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-7Mg-Mg seems to reﬂect the tendency of this system to order. In the case of the Mg-Zr alloys, the Mg-Zr bonds are longerthan Mg-Mg and Zr-Zr, suggesting that this system has agreat tendency to phase separate, as indicated by the pres-ence of a large hcp miscibility gap in the Mg-Zr phase diagram. 19B. Enthalpy of mixing It is obvious that if an hcp SQS alloy is not stable with respect to local relaxations, its properties are not accessiblethrough experimental measurements. However, approximateeffective properties could still be estimated through CALPHAD modeling. In order to compare the energeticsand properties of the calculated SQS’s with the availableexperiments or previous thermodynamic models, only thenonlocally relaxed structures were considered whenever theSQS was identiﬁed as unstable. This effectively assumes thatthe structures in question are constrained to maintain theirsymmetry. The total energies of the structures undersymmetry-preserving relaxations are obviously higher sincethe relaxation energy is not considered. However, we canconsider these calculated thermochemical properties as anupper bound which can still be of great use when attemptingto generate thermodynamically consistent models based onthe combined ﬁrst-principles/CALPHAD approach. As mentioned earlier, obtaining thermodynamic properties of random alloys using cluster expansion or the CPA methodhas some drawbacks. These methods, however, have the ad-vantage of calculating the properties of random alloys at ar-bitrary and closely spaced concentrations. SQS’s in this caseare at a disadvantage since the size of the SQS itself limitsthe concentrations with randomlike correlations. Neverthe-less, if we can acquire the properties at these three composi-tions, we can sufﬁciently describe the tendency of the sys-tem. Furthermore, these SQS’s can be applied directly toother binary systems without any modiﬁcations. The enthalpies of mixing for these alloys were calculated at the 25, 50, and 75 at. % concentrations through the ex-pression /H9004H/H20849A 1−xBx/H20850=E/H20849A1−xBx/H20850−/H208491−x/H20850E/H20849A/H20850−xE/H20849B/H20850, /H208492/H20850 where E/H20849A/H20850andE/H20849B/H20850are the reference energies of the pure components in their hcp ground state. In the following sections, the generated SQS’s are tested by calculating the crystallographic, thermodynamic, andelectronic properties of hcp random solutions in seven binarysystems Cd-Mg, Mg-Zr, Al-Mg, Mo-Ru, Hf-Ti, Hf-Zr, andTi-Zr. The results of the calculations are then compared withexisting experimental information as well as previous calcu-lations. C. Cd-Mg In the Cd-Mg system, both elements have the same va- lence and almost the same atomic volumes. Consequently,there is a wide hcp solid solution range as well as order/disorder transitions in the central, low temperature region ofthe phase diagram. In fact, at the 25 and 75 at. % composi-tions there are ordered intermetallic phases with hexagonalsymmetries. Figure 3/H20849a/H20850compares the enthalpy of mixing calculated from the fully relaxed and symmetry preserved SQS with theresults from cluster expansion. 20The results by Asta et al.20 at 900 K are presented for comparison since it is to be ex- pected that these values would be rather close to the calcu-lated enthalpy of completely disordered structures. The pre- FIG. 3. Calculated and experimental results of mixing enthalpy and lattice parameters for the Cd-Mg system. /H20849a/H20850Calculated en- thalpy of mixing for the disordered hcp phase in the Cd-Mg systemwith SQS at T=0 K, cluster variation method /H20849CVM /H20850/H20849Ref. 20/H20850at T=900 K, and experiment /H20849Ref. 21/H20850atT=543 K. /H20849b/H20850Calculated lattice parameters of the Cd-Mg system compared with experimen-tal data /H20849Refs. 22–24/H20850.SHIN et al. PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-8vious and current calculations are also compared with the experimental measurements as reported in Hultgren21at 543 K. The ﬁrst thing to note from Fig. 3/H20849a/H20850is that the fully relaxed and symmetry preserved calculations are very closein energy, implying negligible local relaxation. Additionally,the present calculations are remarkably close /H20849/H110111 kJ/mol /H20850to the experimental measurements. By comparing the SQS en- thalpy of mixing with the results from the cluster expansioncalculations, 20it is obvious that the former is, at least in this case, more capable of reproducing the experimental measure-ments. Formation enthalpies of the three ordered phases in the Cd-Mg system, Cd 3Mg, CdMg, and CdMg 3are also pre- sented. The measurements from Hultgren21deviate from the calculated results from Asta et al.20and this work. Cd and Mg are known as very active elements and it is likely thatreaction with oxygen present during the measurements mayhave introduced some systematic errors. Furthermore, themeasurements were conducted at relatively low tempera-tures, making it difﬁcult for the systems to equilibrate. Nev-ertheless, experiments and calculations agree that these threecompounds constitute the ground state of the Cd-Mg system. Figure 3/H20849b/H20850also shows that the present calculations are able to reproduce the available measurements on the varia-tion of the lattice parameters of hcp Cd-Mg alloys with com-position, as well as the deviation of these parameters fromVegard’s Law. This deviation is mainly related to the ratherlarge difference in c/aratio between Cd and Mg. The c/a ratio of Cd is one of the largest ones of all the stable hcpstructures in the periodic table. D. Mg-Zr The Mg-Zr system is important due to the grain reﬁning effects of Zr in magnesium alloys. According to the assess-ment of the available experimental data by Nayeb-Hashemiand Clark, 19the Mg-Zr system shows very little solubility in the three solution phases, bcc, hcp, and liquid. In fact, thelow temperature hcp phase exhibits a broad miscibility gapup to 923 K, corresponding to the peritectic reaction hcp+liquid→hcp. 19 Our calculations yielded a positive enthalpy of mixing, conﬁrming the trends derived from the thermodynamicmodel developed by Hämäläinen et al. 25In the case of the full relaxation, however, it was observed that the Mg 50Zr50 SQS was unstable with respect to local relaxations. The in-stability at this composition and the large, positive enthalpyof mixing indicate that the system has a strong tendency tophase separate. By comparing the fully relaxed and the non-locally relaxed structures, we estimate that the local relax-ation energy lowers the mixing enthalpy of the random hcpSQS by about 2 kJ/mol in this system. Figure 4shows the calculated mixing enthalpy for the Mg-Zr hcp SQS with no local relaxations, as well as themixing enthalpy calculated from the thermodynamic modelby Hämäläinen et al. , 25which was ﬁtted only through phase diagram data. It is therefore remarkable that the maximumdifference between the CALPHAD model and the presenthcp SQS calculations is /H110113 kJ/mol. The CALPHAD model,however, does not correctly describe the asymmetry of the mixing enthalpy indicated by the ﬁrst-principles calculations.The results of the hcp SQS calculations for the Mg-Zr sys-tem have recently been used to obtain a better thermody-namic description of the Mg-Zr system 26and, as can be seen in the ﬁgure, this description is better at describing the trendsin the calculated enthalpy of mixing. E. Al-Mg As one of the most important industrial alloys, the Al-Mg system has been studied extensively recently.27–29This sys- tem has two eutectic reactions and shows solubility withinboth the fcc and hcp phases. However, the solubility rangesare not wide enough so there is only limited experimentalinformation for the properties of the hcp phase. The maxi-mum equilibrium solubility of Al in the Mg-rich hcp phase isaround 12 at. %. In Fig. 5/H20849a/H20850the calculated enthalpy of mixing is slightly positive. The fully relaxed calculations show that the SQSwith the 50 at. % composition was unstable with respect tolocal relaxations. This can be explained by the strong inter-action between Al and Mg, as evident from the tendency ofthis system to form intermetallic compounds at the middle ofthe phase diagram, such as /H9252-Al 140Mg 89,/H9253-Al 12Mg 17, and /H9255-Al 30Mg 23. At the 25 and 75 at. % compositions the SQS’s were stable with respect to local relaxations because bothelements have a close-packed structure. Furthermore, atthese compositions either the fcc or hcp phase take part inequilibria with some other /H20849intermetallic /H20850phase. Figure 5/H20849a/H20850 shows that the present fully relaxed calculations are in excel-lent agreement with the most recent CALPHADassessments. 27,28Note also that in this case, and contrary to what is observed in the Cd-Mg binary, the energy change FIG. 4. Calculated enthalpy of mixing in the Mg-Zr system compared with a previous thermodynamic assessment /H20849Ref. 25/H20850. Both reference states are the hcp structure.THERMODYNAMIC PROPERTIES OF BINARY hcp ¼ PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-9associated with local relaxation is not negligible, although it is still within /H110111 kJ/mol. Additionally, the calculated lattice parameters agree very well with the experimental measurements of Mg-rich hcpalloys, as can be seen in Fig. 5/H20849b/H20850. It is important to note that the lattice parameter measurements of metastable hcp alloysfrom Luo et al. 30/H2084977.4 and 87.8 Mg at. % /H20850are lying on the extrapolated line between the 75 at. % SQS and the pure Mg calculations. This is another example of how SQS’s can besuccessfully used in calculating the properties of an hcp solid solution system with narrow solubility range and mixed withnon-hcp elements, even in the metastable regions of thephase diagram. F. Mo-Ru The Mo-Ru system shows a wide solubility range within both the bcc and hcp sides of the phase diagram. In theRu-rich side, the maximum solubility of Mo in the hcp-Rumatrix is up to 50 at. %. The calculations at Mo 25Ru75and Mo 50Ru50retained the original hcp symmetry but Mo 75Ru25 did not. The instability of the Mo-rich SQS is not surprisingsince the Mo-rich bcc region is stable over a wide region ofthe phase diagram. As shown in Wang et al. , 34elements whose ground state is bcc are not stable in an hcp lattice andvice versa /H20849bcc Ti, Zr, and Hf are only stabilized at high temperature due to anharmonic effects /H20850. Thus hcp composi- tions close to the bcc-side would be dynamically unstableand would have a very large driving force to decrease theirenergy by transforming to bcc. Recently, Kissavos et al. 7calculated the enthalpy of mix- ing for disordered hcp Mo-Ru alloys through the CPA inwhich relaxation energies were estimated by locally relaxingselected multisite atomic arrangements. Enthalpy of forma-tion for hcp solutions were calculated from Eq. /H208493/H20850shown below. The enthalpy of mixing of the disordered hcp phasecan be evaluated accordingly based on the so-called latticestability 2Ebcc/H20849Mo/H20850−Ehcp/H20849Mo/H20850: /H9004Hf/H20849Mo 1−xRux/H20850 =Ehcp/H20849Mo 1−xRux/H20850−/H208491−x/H20850Ebcc/H20849Mo/H20850−xEhcp/H20849Ru/H20850 =Ehcp/H20849Mo 1−xRux/H20850−/H208491−x/H20850Ehcp/H20849Mo/H20850−xEhcp/H20849Ru/H20850 −/H208491−x/H20850Ebcc/H20849Mo/H20850+/H208491−x/H20850Ehcp/H20849Mo/H20850 =Hmixhcp/H20849Mo 1−xRux/H20850−/H208491−x/H20850/H20851Ebcc/H20849Mo/H20850−Ehcp/H20849Mo/H20850/H20852. /H208493/H20850 Usually, structural energy differences /H20849or lattice stability /H20850 between ﬁrst-principles calculations and CALPHAD showquite good agreement. However, for some transition ele-ments, the disagreement between the two approaches is quitesigniﬁcant. 35Mo is one such case, with the structural energy difference between bcc and hcp from ﬁrst-principles calcula-tions and the CALPHAD approach differing by over30 kJ/mol. After a rather extensive analysis, Kissavos et al. 7 arrived at the conclusion that in order to reproduce enthalpy values close enough to the available experimental data36the CALPHAD lattice stability /H2084911.55 kJ/mol /H20850needed to be used for the value of the bcc →hcp promotion energy. The SQS and CPA calculations are compared with the experimental measurements in Fig. 6. On the assumption that the experimental measurements by36are correct, the derived enthalpy of formation of the hcp Mo-Ru system from theﬁrst-principles calculated lattice stability with the SQS andCPA approach in Fig. 6/H20849a/H20850cannot reproduce the experimental observation at all since the ﬁrst-principles bcc →hcp lattice stability for Mo is 42 kJ/mol. Given this lattice stability, theonly way in which the ﬁrst-principles calculations within FIG. 5. Calculated and experimental results of mixing enthalpy and lattice parameters for the Al-Mg system. /H20849a/H20850Calculated en- thalpy of mixing for the hcp phase in the Al-Mg system comparedwith assessed data /H20849Ref. 27–29/H20850. Reference states are hcp for both elements. /H20849b/H20850Calculated lattice parameters of the hcp phase in the Al-Mg system compared with experimental data /H20849Refs. 23,24, and 30–33/H20850.SHIN et al. PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-10both the SQS and CPA approaches would match the experi- mental results would be for the calculated enthalpy of mixingto be very negative, which is not the case. In fact, as can beseen in Fig. 6/H20849a/H20850, the SQS and CPA calculations are very close to each other.On the other hand, the enthalpy of formation derived from the CALPHAD lattice stability in Fig. 6/H20849b/H20850shows a better agreement than that from the ﬁrst-principles lattice stability.It is important to note that the CALPHAD lattice stabilitywas obtained through the extrapolation of phase boundariesin phase diagrams with Mo and stable hcp elements and, therefore, are empirical. The reason why such an empiricalapproach would yield a much better agreement with experi-mental data is still the source of intense debate within theCALPHAD community and has not been resolved as of now.The main conclusion of this section, however, is that theSQS’s were able to reproduce the thermodynamic propertiesof hcp alloys as good as or better than the CPA method whileat the same time allowing for the ion positions to locallyrelax around their equilibrium positions. G. IVA transition metal alloys The group IV A transition metals Ti, Zr, and Hf have hcp structure at low temperatures and transform to bcc at highertemperatures due to the effects of anharmonic vibrations.When they form a binary system with each other, they showcomplete solubility for both the hcp and bcc solutions with-out forming any intermetallic compound phases in themiddle. The Hf-Ti binary is reported to have a low temperature miscibility gap and was modeled with a positive enthalpy ofmixing by Bittermann and Rogl. 37Figure 7/H20849a/H20850shows remark- able agreement between the fully relaxed ﬁrst-principles cal-culations and the thermodynamic model, which was obtainedby ﬁtting the experimental phase boundary data. Despite thefact that the local relaxation energies are rather large/H20849/H110114 kJ/mol /H20850, the lattice parameters in both cases agree be- tween each other and with the experimental results. 39–41 In the case of the Ti-Zr binary, although no low- temperature miscibility gap has been reported, Kumar et al.38 found that the enthalpy of mixing for the hcp solutions in this binary was positive through ﬁtting of phase diagram data.Our results conﬁrm this ﬁnding, although with even morepositive enthalpy. They are in fact similar in value to thosecalculated in the Hf-Ti alloys, suggesting that a low tempera-ture miscibility gap may also be present in this binary. In the Hf-Zr system no miscibility gap has been reported. The hcp phase was modeled as an ideal solution /H20849/H9004H mix =0/H20850in the CALPHAD assessment.42The present calcula- tions suggest that the enthalpy of mixing of this system is positive, although rather small. In this case, it is expectedthat any miscibility gap would only occur at very low tem-peratures. The three systems described in this section are chemically very similar, having the same number of electrons in the d bands. Electronic effects due to changes in the widths andshapes of the DOS of the dbands are not expected to be signiﬁcant in determining the alloying energetics. Chargetransfer effects are also expected to be negligible. The en-thalpy observed can then be explained by just consideringthe atomic size mismatch between the different elements. Aswas shown in Table V, the Hf-Zr hcp alloys are the ones with FIG. 6. Enthalpy of formation of the Mo-Ru system with both ﬁrst principles and CALPHAD lattice stabilities. Reference statesare bcc for Mo and hcp for Ru. /H20849a/H20850Enthalpy of formation of hcp phase in the Mo-Ru system from SQS’s /H20849this work /H20850and CPA /H20849Ref. 7/H20850. Total energy of hcp Mo is obtained from ﬁrst-principles calcu- lations in both cases. /H20849b/H20850Enthalpy of formation of hcp phase in the Mo-Ru system from SQS’s and CPA. Total energy of hcp Mo isderived from the SGTE /H20849Scientiﬁc Group Thermodata Europe /H20850lat- tice stability.THERMODYNAMIC PROPERTIES OF BINARY hcp ¼ PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-11the smallest difference in their lattice parameter, thus ex- plaining their very small positive enthalpy of mixing. As a ﬁnal analysis of the ability of the generated SQS to reproduce the properties of random hcp alloys, Fig. 8shows the alloying effects on the electronic DOS in Ti-Zr hcp al-loys. The ﬁgure also presents the results obtained through theCPA approach by Kudrnovsky et al. 43As can be seen in the ﬁgure, both calculations predict that the DOS correspondingto the occupied dstates are virtually insensitive to alloying. The overall shape of the d-DOS remains relatively invariant. Since Ti and Zr have the same number of valence electrons,the fermi level remains essentially unchanged as the concen-tration varies from pure Zr to pure Ti. On the other hand,alloying effects are more pronounced in the d-DOS corre- sponding to the unoccupied states. Figure 8shows how the broad peak at /H110114.5 eV of the d-DOS for Zr is gradually transformed into a narrow peak at /H110113.0 eV as the Ti content in the alloy is increased. The results from the CPA and theﬁrst-principles SQS calculations thus agree with each other,conﬁrming the present results. V. SUMMARY We have created periodic special quasirandom structures with 16 atoms for binary hcp substitutional alloys at threedifferent compositions 25, 50, and 75 at. %, to mimic thepair and multisite correlations of random solutions. The gen-erated SQS’s were tested in seven different binaries andshowed fairly good agreement with existing experimental ei-ther enthalpy of mixing and/or CALPHAD assessments andlattice parameters. Analysis of the radial distribution andbond lengths in the 21 calculated SQS’s, yielded a detailedaccount of the local relaxations in the hcp solutions and hasbeen proven a useful way of characterizing the degree relax-ation over several coordination shells. It should also be noted that when using enthalpy of mix- ing to derive formation enthalpy to compare with experimen-tal measurements, there can be a severe discrepancy betweentheoretical calculations and experimental data when the lat-tice stability, or structural energy difference, from ﬁrst-principles calculation is problematic such as the Mo-Ru FIG. 7. Enthalpy of mixing for the Hf-Ti, Hf-Zr, and Ti-Zr bi- nary hcp solutions calculated from ﬁrst-principles calculations andCALPHAD thermodynamic models. All the reference states are hcpstructures. /H20849a/H20850Calculated enthalpy of mixing for the hcp phase in the Hf-Ti system compared with a previous assessment /H20849Ref. 37/H20850. /H20849b/H20850Calculated enthalpy of mixing for the hcp phase in the Ti-Zr system compared with a previous assessment /H20849Ref. 38/H20850./H20849c/H20850Calcu- lated enthalpy of mixing for the hcp phase in the Hf-Zr system./H9004H mix/H112290. FIG. 8. Calculated DOS of Ti 1−xZrxhcp solid solutions from /H20849a/H20850 SQS and /H20849b/H20850CPA /H20849Ref. 43/H20850.SHIN et al. PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-12system in this work. This problem remains as an unsolved issue. These supercells can be applied directly to any substitu- tional binary alloys to investigate the mixing behavior ofrandom hcp solutions via ﬁrst-principles calculations withoutcreating new potentials, as in the coherent potential approxi-mation /H20849CPA /H20850or calculating other structures in the cluster expansion. Although the size of the current SQS’s is notlarge enough to generate a supercell which can satisfy itscorrelation function at more than just three compositions /H20849x =0.25, 0.5, and 0.75 in A 1−xBxbinary /H20850, calculations for these compositions can yield valuable information about the over-all behavior of the alloys.ACKNOWLEDGMENTS This work is funded by the National Science Foundation /H20849NSF /H20850through Grant No. DMR-0205232. First-principles calculations were carried out on the LION clusters at thePennsylvania State University supported in part by the NSFgrants /H20849DMR-9983532, DMR-0122638, and DMR-0205232 /H20850 and in part by the Materials Simulation Center and theGraduate Education and Research Services at the Pennsylva-nia State University. We would also like to thank ChristopherWolverton at Ford for critical proofreading of the manu-script. Earle Ryba is acknowledged for his valuable advicefor radial distribution analysis. Electronic address: dus136@psu.edu 1L. Kaufman and H. Bernstein, Computer Calculation of Phase Diagram /H20849Academic Press, New York, 1970 /H20850. 2N. Saunders and A. P. Miodownik, CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide /H20849Pergamon, Oxford, 1998 /H20850. 3A. van de Walle, M. Asta, and G. Ceder, CALPHAD: Comput. Coupling Phase Diagrams Thermochem. 26, 539 /H208492002 /H20850. 4W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 /H208491965 /H20850. 5C. 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B 43, 4622 /H208491991 /H20850.THERMODYNAMIC PROPERTIES OF BINARY hcp ¼ PHYSICAL REVIEW B 74, 024204 /H208492006 /H20850 024204-13
PhysRevB.82.045403.pdf	Electronic implementations of interaction-free measurements L. Chirolli,1,E. Strambini,2V. Giovannetti,2F. Taddei,2V. Piazza,2R. Fazio,2F. Beltram,2and G. Burkard1 1Department of Physics, University of Konstanz, D-78457 Konstanz, Germany 2NEST, Scuola Normale Superiore and Istituto Nanoscienze–CNR, I-56126 Pisa, Italy /H20849Received 9 April 2010; revised manuscript received 21 May 2010; published 7 July 2010 /H20850 Three different implementations of interaction-free measurements /H20849IFMs /H20850in solid-state nanodevices are discussed. The ﬁrst one is based on a series of concatenated Mach-Zehnder interferometers, in analogy tooptical-IFM setups. The second one consists of a single interferometer and concatenation is achieved in thetime domain making use of a quantized electron emitter. The third implementation consists of an asymmetricAharonov-Bohm ring. For all three cases we show that the presence of a dephasing source acting on one armof the interferometer can be detected without degrading the coherence of the measured current. Electronicimplementations of IFMs in nanoelectronics may play a fundamental role as very accurate and noninvasivemeasuring schemes for quantum devices. DOI: 10.1103/PhysRevB.82.045403 PACS number /H20849s/H20850: 03.65.Ta, 03.67.Lx, 42.50.Dv, 42.50.Pq I. INTRODUCTION Interaction-free measurements /H20849IFMs /H20850were ﬁrst intro- duced by Elitzur and Vaidman,1who showed that the laws of quantum mechanics allow to reveal the presence of an objectwithout disturbing it. The original proposal exploited the co-herent splitting and the subsequent recombination of thewave function of a photon entering a Mach-Zehnder /H20849MZ /H20850 interferometer. The disturbance induced by the object placedin one of the two arms of the interferometer /H20849an absorber in the original proposal /H20850manifests itself in the properties of the outgoing photon ﬂux. Upon suitable setting of the interfer-ometer parameters it was shown that even without absorption taking place its mere possibility does modify the state of theparticle emerging from the interferometer. As a result an ex-ternal observer will be able to gather information about thepresence or absence of the absorber, without the photon be-ing actually absorbed. The maximal success probability wasbound to be 50% in the original proposal. A way to improvethe efﬁciency of the scheme was put forward by Kwiat et al., 2who suggested to use coherently repeated interrogations. In their scheme a photon was repeatedly sent into a MZinterferometer, with an absorber placed in one of the twoarms. By properly tuning the MZ phase it was shown that itis possible to enhance the efﬁciency of the setup arbitrarilyclose to 1. Such a scheme can be thought as an application ofa discrete form of the quantum Zeno effect 3since every step can be considered as a measurement accompanied by statereduction. IFMs were experimentally realized using single-photon sources 2,4–6and in neutron interferometry.7The enhanced ef- ﬁciency of concatenated MZ interferometers schemes wastested in Ref. 8with a demonstrated improvement up to 73%. Its application was extended to the case of semitrans-parent objects with classical light. 9–12An important conse- quence of these works is that IFM can be interpreted in termsof deterioration of a resonance condition 9which does not necessarily need a quantum description /H20849“classical” optical coherence is sufﬁcient /H20850, at least for these optical realizations. The implementation of IFM in electronic devices deserves in our opinion a careful scrutiny since it constitutes an idealtest bed for the study of quantum-control and quantum-mechanics phenomena in mesoscopic systems. It is worth noting that, differently from the optical case, for electronicsystems there is no corresponding classical model to realizean IFM. In recent years advances in device fabricationopened the way to the observation of interference phenom-ena in electronic-transport experiments, suggesting importantopportunities for a variety of applications. The achievementsobtained in the context of two-dimensional electron gases inthe integer quantum-Hall-effect regime 13are of particular in- terest for what follows. Here, various experimental realiza-tions of the MZ /H20849Refs. 14–18/H20850and Hanbury-Brown-Twiss interferometers 19,20were successfully implemented. In addi- tion, quantized electron emitters were recently realized.21–24 The possibility to extend IFM to electronic systems seems therefore now at reach, paving the way to the development ofnovel noninvasive measurement schemes in mesoscopic sys-tems, with possible important implications for quantum in-formation processing. A ﬁrst application of IFM strategies to electronic systems was proposed in Ref. 25to detect the presence of a current pulse in a circuit by monitoring the state of a superconduct-ing qubit coupled to the circuit, without any energy exchangebetween the two. Subsequently, in the very same spirit of the original works, 1,2it was shown how to employ IFM to detect with unitary efﬁciency a source of noise acting on one arm ofan Aharonov-Bohm /H20849AB/H20850chiral ring without affecting the transmitted and reﬂected currents. 26In view of its /H20849unavoid- able /H20850presence in nanoelectronics, the proposal focused on the detection of external random ﬂuctuating electric or mag-netic ﬁelds, which represents the most common source ofnoise in nanoscale quantum devices. 27–29Therefore, in Ref. 26a classical ﬂuctuating electrical ﬁeld that randomizes the phase of the electron traveling through it played the role ofthe absorber in optical schemes. 1,2,4–12The resulting appara- tus operates as a sort of quantum fuse which opens or closes a contact depending on the presence or on the absence of thedephasing source. The results presented in Ref. 26show that the mechanism underlying the IFM does not depend, to alarge extent, on the type of disturbance which is induced inthe interferometer. In the present paper we extend our previous work 26on the electronic version of the IFM in several ways. First of all wePHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 1098-0121/2010/82 /H208494/H20850/045403 /H2084911/H20850 ©2010 The American Physical Society 045403-1introduce two alternative IFM implementations based on the integer quantum-Hall effect. The ﬁrst scheme closely re-sembles the optical setup of Ref. 8and uses a recent proposal 30for realizing concatenated MZ interferometers. The second scheme instead is based on the standardquantum-Hall interferometric architecture 14–20and assumes the presence of a quantized electron emitter.21–24As in Ref. 26, both setups are shown to be capable of detecting the presence of a localized dephasing source without affectingthe coherence of the probing signals. Finally we review theAB-ring implementation of Ref. 26and provide a detailed characterization of the scheme. The paper is organized as follows. In Sec. IIwe present a noise-sensitive coherent electron detector, based on the con-catenation of several MZ interferometers. We show that wecan detect the presence of a dephasing source affectingpropagation in one of the interfering electronic paths by mea-suring the output currents. We then study the coherence ofthe outgoing signal by computing the fraction of coherentsignal and show that an IFM measurement of the dephasingsource is achievable. In Sec. II C we embed the device de- scribed in Sec. IIin a larger Mach-Zehnder interferometer and study the visibility of the output currents, showing howthe coherence of the outgoing signal can be experimentallyaddressed. In Sec. IIIwe propose an implementation of IFM based on a single Mach-Zehnder interferometer that makesuse of a quantized electron source and concatenation in thetime domain. In Sec. IVwe present a double-ring structure in which a small chiral AB ring is embedded in one arm of alarger AB ring. We show that the current which ﬂowsthrough the whole device is a measure of the coherent char-acter of the detection. II. COHERENT DETECTION OF NOISE WITH IFMS A straightforward implementation of IFM along the lines developed originally in optics can be realized exploiting theedge-channel interferometric architecture of Ref. 30based on the integer quantum-Hall effect at ﬁlling factor /H9263=2. The feature of this architecture which is particularly relevant forour purposes is that it allows for successive concatenationsof different interferometers. In this scheme, beam splitters/H20849BSs /H20850are realized by introducing a sharp potential barrier which mixes the two edges. Populating initially only onechannel, at the output of a BS we ﬁnd electrons in a super-position state. Additional phase shifters /H20849PSs /H20850can be easily realized by spatially separating the two channels with the useof a top gate that can locally change the ﬁlling factor to /H9263=1: only one channel can traverse the region at /H9263=1 and the other is guided along its edges. This is schematicallyshown in Fig. 1, where a phase difference /H9278is introduced between the channels by changing the path of the incomingchannel. Based on this approach we can build an apparatus which implements an IFM scheme along the lines of the opticalsetup of Ref. 8. The proposed device, illustrated in Fig. 1, consists in a sequence of Ninterferometric elements in which output edges emerging from the nth interferometer are di- rectly fed into the input of the /H20849n+1/H20850th one. As we shall see,the apparatus allows one to detect the presence of a ﬂuctu- ating electromagnetic ﬁeld affecting the upper region of theHall bar /H20849depicted as a shaded area in Fig. 1/H20850, without any coherence loss of the transmitted currents. This is obtainedthanks to the action of the top gates of the setup which divertthe path of the i/H20849inner /H20850channel inside the Hall bar /H20849where the ﬂuctuating ﬁeld is supposed to be absent /H20850and thanks to the coherent mixing between the ichannel and the o/H20849outer /H20850 channel induced by the BSs. If no dephasing is present in theupper region of the bar, then the electron coherently propa-gates toward the next step, that is, nominally equal to theprevious one. By properly tuning the degree of admixture ofthe channel populations, it is possible to gradually transferthe electron from the ichannel to the ochannel at the end of a chain of Ninterferometers. The situation changes com- pletely when the dephasing ﬁeld is present in the shadedregion of Fig. 1. Indeed, as a result of a random-phase shift, the part of the wave function that propagates in the ochannel does not coherently add to the one propagating in the ichan- nel. Consequently the gradual transfer of electronic ampli-tude from itoodoes not take place. At the same time the electron that propagates into the channel not exposed to theﬂuctuating ﬁeld preserves its coherence. The presence or ab-sence of noise is revealed by the electron emerging from lead3 or 4, respectively, and, as we shall clarify in the following,the setup does preserve the coherence of the emerging elec-tronic signal. A. Detection of a dephasing noise source Electron propagation is described in the Landauer- Büttiker formalism of quantum transport.31–33The scattering matrix that describes transport in each block can be writtenasa b ν=1 ν=2BS BS ΦΦBS a b ν =1 ioε 12 4 3γBlock FIG. 1. /H20849Color online /H20850Schematic illustration of a noise-sensitive coherent electron channel consisting of N=2 representative blocks, implemented in a quantum-Hall bar at integer ﬁlling /H9263=2. Incoming electrons in contacts 1 and 2 are represented by their annihilationoperators aand outgoing electrons in contacts 3 and 4 by their annihilation operators b. Each block is constituted by a beam split- ter/H20849BS/H20850and a phase shifter /H20849PS/H20850. Each BS is characterized by a degree of admixture /H9253and mixes the incoming electron in the iand oedge states. The PS is constituted by an applied top gate /H20849yellow solid rounded rectangle with ﬁlling factor /H9263=1/H20850that spatially sepa- rates the edge channels and introduces a phase difference /H9278.A n external ﬂuctuating ﬁeld of strength /H9280/H20849shaded area /H20850introduces dephasing by randomly shifting the phase of the electron travelingin the oedge state.CHIROLLI et al. PHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 045403-2S/H20849/H9254/H20850=/H20873ei/H9278cos/H20849/H9253/2/H20850iei/H9278sin/H20849/H9253/2/H20850 iei/H9254sin/H20849/H9253/2/H20850ei/H9254cos/H20849/H9253/2/H20850/H20874, /H208491/H20850 where 0 /H11021/H9253/H110212/H9266parametrizes the degree of edge-channel mixing introduced by BS and /H9278is the phase shift between the two edge channels. The presence of a dephasing source isdescribed by a random-phase shift exp /H20849i /H9254/H20850. By using this scattering matrix it is possible to relate electrons exiting thechain of Nblocks to the incoming ones at the beginning of the chain, b=/H20863 i=1N S/H20849/H9254i/H20850a, /H208492/H20850 with a=/H20849ai,ao/H20850Tbeing the Fermionic annihilation operator describing incoming electrons in leads 1 and 2 /H20849connected to channels iando, respectively /H20850andb=/H20849bi,bo/H20850Tthe Fermionic annihilation operator describing outgoing electrons /H20849leads 3 and 4 /H20850. Contact 1 is biased at a chemical potential eV, reser- voirs 2, 3, and 4 are kept at reference potential. Setting thetemperature to zero, the current in contact 3 is I 3,N=e2V h/H20841/H20851SN/H2085211/H208412/H208493/H20850 while the current in contact 4 is I4,N=e2V h/H20841/H20851SN/H2085221/H208412/H208494/H20850 with SN=/H20863i=1NS/H20849/H9254i/H20850. Here we do not take into consideration the electron-spin degree of freedom. The effect of the ﬂuctuating ﬁeld can be taken into ac- count by averaging the phases /H9254iover a generic distribution of width 2 /H9266/H9280and zero mean. For simplicity we assume a uniform distribution. The outgoing currents depend now en-tirely on the degree of mixing /H9253of edge states in the BS and on the phase shift /H9278. The average current in contact 3 /H208494/H20850is given by /H20855I3/H208494/H20850/H20856/H9254/H110131 /H208492/H9266/H9280/H20850N/H20885 −/H9266/H9280/H9266/H9280 d/H9254I3/H208494/H20850,N /H208495/H20850 with d/H9254=d/H92541,..., d/H9254N. We deﬁne the two-component vectors e+=/H208491,0/H20850Tande−=/H208490,1/H20850Tthat allow us to express /H20841/H20851SN/H2085211/H208412=e+TSN†e+e+TSNe+, /H208496/H20850 /H20841/H20851SN/H2085221/H208412=e+TSN†e−e−TSNe+. /H208497/H20850 Introducing a representation of 2 /H110032 matrices in terms of Pauli operators, concisely written through the Pauli vector /H9268=/H20849/H92680,/H92681,/H92682,/H92683/H20850T, with /H92680=1, we can write e/H11006e/H11006T =/H208491/H11006/H9268Z/H20850/2/H11013p/H11006·/H9268, with /H20849p/H11006/H20850i=Tr /H20849e/H11006e/H11006T/H9268i/H20850/2. This allows us to calculate the average over phases /H9254ias a matrix prod- uct. By deﬁning matrix Qij=1 2/H20885 −/H9266/H9280/H9266/H9280d/H9254 2/H9266/H9280Tr/H20851S†/H20849/H9254/H20850/H9268iS/H20849/H9254/H20850/H9268j/H20852, /H208498/H20850 we can write the zero-temperature average current in output 3/H208494/H20850after Nblocks as/H20855I3/H208494/H20850,N/H20856/H9254=e2V hp/H11006·QN·/H20849e+T/H9268e+/H20850. /H208499/H20850 We point out that, due to the unitarity of S/H20849/H9254/H20850,Qijdeﬁned in Eq. /H208498/H20850preserves the trace. One can reduce the dimensional- ity of the problem and work with the Bloch representation of2/H110032 density matrices. The behavior of the output currents in the limit of large N is obtained by studying the eigenvalues of the 4 /H110034 matrix Q. Choosing the working point /H9278=0,Qassumes a diagonal block form that allows a direct solution: Q =U−1diag /H208511,sin /H20849/H9266/H9280/H20850//H9266/H9280,/H9261−,/H9261+/H20852U, with Uand/H9261/H11006given by Eqs. /H20849B2/H20850and /H20849B3/H20850in Appendix B. The currents in terminal 3/H208494/H20850can be then written as /H20855I3/H208494/H20850,N/H20856/H9254=e2V h1 2/H208731/H11006/H9261+Nu+−/H9261−Nu− u+−u−/H20874 /H2084910/H20850 with u/H11006given in Eq. /H20849B1/H20850in Appendix B. Figure 2/H20849left panel /H20850shows the current in terminal 3 ver- sus the phase shift /H9278for the case of no dephasing /H20849/H9280=0/H20850.W e can see that for large N,/H20855I3,N/H20856/H9254is approximately e2V/hfor almost all values of /H9278, and that only at /H9278=0 it drops very rapidly to zero. For such value the outgoing currents areindeed /H20855I 3,N/H20856/H9254=0 and /H20855I4,N/H20856/H9254=e2V/hindependently from N /H20849increasing Nfurther shrinks the dip at /H9278=0/H20850. This corre- sponds to having a very narrow resonance at the workingpoint /H9278=0 where interference gives rise to a gradual transfer of the electron wave function to the ochannel and all the current emerges from contact 4. Such a resonance is verysensitive to small deviations of the phase /H9278from the working point/H9278=0 and imply a large variation in the current re- sponse. In the case of strong dephasing /H20849/H9280=1/H20850the current is in- stead given by00 . 5 100.20.40.60.81 εφ=0 00.20.40.60.81 π −π 0ε=0 φCurrent 3 (e V/h)2 N=1 0 N=2 0N=5 0N=5 0 N=1 0 0N=1 5 0 FIG. 2. /H20849Color online /H20850Current in contact 3 for different number Nof blocks. /H20849Left panel /H20850/H20855I3,N/H20856/H9254of Eq. /H208499/H20850versus the phase shift /H9278 in the coherent case /H9280=0. By increasing Na narrow dip arises in the coherent case for /H9278=0 and all the current goes out in contact 4. /H20849Right panel /H20850/H20855I3,N/H20856/H9254versus the strength /H9280of the dephasing ﬁeld, at the working point /H9278=0. As /H9280increases, the current tends to go out all from contact 3, thus witnessing the presence of the dephasingﬁeld.ELECTRONIC IMPLEMENTATIONS OF INTERACTION-FREE … PHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 045403-3/H20855I3/H208494/H20850,N/H20856/H9254=e2V h1 2/H208511/H11006cosN/H20849/H9253/H20850/H20852. /H2084911/H20850 If the asymmetry of the BSs is properly tuned at the value /H9253=/H9266/N, the output currents are /H20855I3/H208494/H20850,N/H20856/H9254=e2V h1 2/H208511/H11006cosN/H20849/H9266 N/H20850/H20852 so that, in the limit of large N, one ﬁnds that /H20855I3,N/H20856/H9254=e2V/h and /H20855I4,N/H20856/H9254=0. The behavior of the current in contact 3 versus the dephasing strength /H9280is shown in Fig. 2, right panel. It is evident that the presence of a strong dephasing sourcechanges the interference response so that for N/H112711 all elec- trons exit the device from terminal 3, whereas in the coherentcase they would exit from terminal 4. Thus, in this respectthe system behaves like a “which-path” electronicinterferometer. 34Interestingly we note that Eq. /H2084911/H20850predicts that, for even N, the same behavior can be observed also in the highly asymmetric case when the electronic amplitude isdiverted to the noisy channel o, i.e., /H9253=/H9266/N+/H9266. In the next section we shall see however that, differently to the case /H9253 =/H9266/N, this last regime does not correspond to a true IFM effect since the coherence of the transmitted signals is totallywashed out. B. Coherence of the outgoing signal A key feature of the IFM detection of noise is that coher- ence of the output be preserved and this can open the way tonovel applications in quantum-coherent electronics. Depend-ing on whether the electron is mostly injected into the secureichannel by setting /H9253=/H9266/Nor into the ochannel exposed to dephasing, by setting /H9253=/H9266/N+/H9266, the coherence of the out- going signal can be asymptotically preserved or totally lost. An effective way to quantify the coherence of the outgo- ing signal can be obtained by deﬁning the fraction of coher-ent signal as F/H11013/H20841 /H20855t/H20856 /H9254/H208412+/H20841/H20855r/H20856/H9254/H208412, /H2084912/H20850 where we have set t=/H20851SN/H2085211andr=/H20851SN/H2085221so that /H20855t/H20856/H9254/H20849/H20855r/H20856/H9254/H20850is the averaged transmission amplitude to contact 3 /H208494/H20850.Ftakes values between 0 /H20849complete loss of coherence /H20850and 1 /H20849coher- ence fully preserved since in this case /H20841/H20851SN/H2085211/H208412+/H20841/H20851SN/H2085221/H208412=1/H20850. The two quantities /H20855t/H20856/H9254and /H20855r/H20856/H9254measure the coherence of the transmitted electrons into contacts 3 and 4, respectively,since they are proportional to the interference terms of suchelectrons with a reference, coherent, signal /H20849a thorough dis- cussion is given in Sec. II C /H20850. In Fig. 3we plot Ffor differ- ent choices of Nand /H9253. For/H9253=/H9266/N, the fraction of coherent signal initially decreases as a result of the disturbance in-duced by the ﬂuctuating ﬁeld /H20849degradation of coherence /H20850. For large values of /H9280, however, the dephasing of the tiny portion of the wave-function pertinent to the ochannel prevents the occurrence of destructive interference. As a result full, coher-ent transmission through the lower arm of the setup is estab-lished, yielding F/H112291 and thus indicating that an IFM is taking place in the setup. This can be understood as due tothe quantum Zeno effect 3associated with repetitive measure- ments that try to determine whether or not the electron is“passing” through the upper arm of the interferometer. 26For /H9253=/H9266/N, the outcome of such a measurement will be nega- tive with a very high probability /H20849i.e., the electron is found inthe lower arm /H20850preserving coherence. An interplay between these two regimes occurs for intermediate values of /H9280giving rise to a minimum in Fwhich sharpens for higher N/H20849see Fig. 3/H20850. This scenario changes completely for /H9253=/H9266/50+/H9266. Here electrons are mostly injected into the ochannel. For small values of /H9280the situation is analogous to the case /H9253 =/H9266/N, the behavior of Fbeing actually the same: the noise source induces a partial suppression of the destructive inter-ference yielding a consequent degradation of coherence. Asevident from Fig. 3however, in this case large values of /H9280 yields a drop of Fto zero indicating that no IFM is taking place here. This originates from the fact that the completesuppression of the destructive interference is accompaniedby a likewise complete loss of coherence due to the strongdephasing experienced by the electron. So far we have considered an ideal situation in which dephasing takes place only in the ochannel. Figure 4shows the behavior of Fversus the strength /H92801of the dephasing ﬁeld acting on channel o, when a ﬂuctuating ﬁeld of strength /H92802affects propagation in the ichannel. We see that a strong response corresponds to a slight increase in /H92802, with the co- herence of the outgoing signal being signiﬁcantly degraded. C. Detection of the coherent signal In this section we show that the fraction of coherent signal Fdeﬁned in Eq. /H2084912/H20850can actually be measured by embed- ding the Nconcatenated blocks in a Mach-Zehnder interfer- ometer, as schematically illustrated in Fig. 5. A voltage Vis applied to contact 1 while all other contacts are at referencepotential. A beam splitter /H20849BST in Fig. 5/H20850splits the current injected by contact 1 so that the transmitted portion enterstheN-block system from channel iwhile the reﬂected one follows a path whose length /H20849and phase /H9272/H20850can be arbitrarily adjusted. The current exiting the N-block system via channel iis then mixed with the signal of known phase at beam0 0.2 0.4 0.6 0.8 100.20.40.60.81 Ν=50 Ν=100Ν=50 Ν=150γ=π/Ν+πF ε FIG. 3. /H20849Color online /H20850Fraction of coherent signal Fof Eq. /H2084912/H20850 versus the strength /H9280of the ﬂuctuating ﬁeld. Choosing the degree of admixture of the BSs to be /H9253=/H9266/N, with most of the electron amplitude injected in the coherent ichannel, the outgoing signal initially partially dephases for small /H9280, reaches a minimum, and then recovers its coherence as /H9280approaches one /H20849IFM regime es- tablished /H20850. On the contrary, injecting most of the electron amplitude in the channel affected by random-phase shift by setting /H9253=/H9266/N +/H9266, the coherence of the outgoing signal is totally lost /H20849no IFM regime /H20850.CHIROLLI et al. PHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 045403-4splitter BSB. The two outgoing currents are collected by con- tacts 3 and 3 /H11032. Electrons exiting the N-block system from channel oare drained separately by contact 4. Assuming that both BST and BSB are 50/50 beam split- ters, the transmission probability for electrons to exit viacontact 3 is given by T 3/H20849/H9272/H20850=1 4/H20855/H20841t+ei/H9272/H208412/H20856/H9254=1 4/H20849/H20855T/H20856/H9254+1/H20850+1 2/H20841/H20855t/H20856/H9254/H20841cos/H20851arg/H20849/H20855t/H20856/H9254/H20850−/H9272/H20852, /H2084913/H20850 where we recall that tis the amplitude for electrons to exit from the Nconcatenated interferometers in channel iandT =/H20841t/H208412. The visibility of T3/H20849/H9272/H20850is deﬁned as the maximal nor- malized amplitude of the /H9272oscillation, namely, V3=2/H20841/H20855t/H20856/H9254/H20841 /H20855T/H20856/H9254+1. /H2084914/H20850 Figure 6shows function V3versus /H9280with/H9253=/H9266/N, for dif- ferent numbers of interferometers /H20849N/H20850.A t/H9280=0 the destruc- tive interference for /H9278=/H9266produces a zero amplitude signal t, leading to zero visibility. In the presence of the dephasingﬁeld the visibility rapidly increases and saturates to one,thereby revealing the coherence of the amplitude twith re- spect to the phase /H9272. Analogously, transmission probability T4is related to the amplitude rof electrons exiting from the Nconcatenated interferometers from channel o.T4can be measured by tun- ing the beam splitter CS in Fig. 5in order to swap inner and outer channels. One ﬁnds that T4/H20849/H9272/H20850=1 2/H20849/H20855R/H20856/H9254+1/H20850+/H20841/H20855r/H20856/H9254/H20841cos/H20851arg/H20849/H20855r/H20856/H9254/H20850−/H9272/H20852/H20849 15/H20850 with R=/H20841r/H208412and visibility V4is deﬁned analogously to Eq. /H2084913/H20850. If we label T¯3/H20849T¯4/H20850the mean value with respect to the phase of the transmission probability in 3 /H208494/H20850, we can write F=V32T¯ 32+V42T¯ 42. /H2084916/H20850 In order to allow only a small fraction of the electron wave function to propagate in the dephasing ochannel and realize the conditions that allow IFMs, it is necessary to set thedegree of admixture in the BS to the precise value /H9253=/H9266/N. This may represent a technical obstacle to an experimentalrealization since BSs are difﬁcult to be tuned all to the sameprecise degree of admixture and a high-efﬁciency IFM isobtained in the limit of large N. In the following we shall present a more robust architecture that allows one to over-0 0.2 0.4 0.6 0.8 100.20.40.60.81Fε =0.12 ε1ε =0.22ε=02 ε =0.32 FIG. 4. Fraction of coherent signal Fwhen the dephasing ﬁeld affects both channels, respectively, owith strength /H92801and iwith strength /H92802. The degree of admixture is set to /H9253=/H9266/N, with N=50, and most of the electron amplitude is injected in channel i.B y increasing /H92802the coherence is rapidly lost. BST BSBν=11 2 3 43’eiϕ N blocks ν=2N blocks1 BST BSB3’ 33’ 42 eiϕINSET CS FIG. 5. /H20849Color online /H20850Schematic representation of the proposal for an experimental realization of an N-block noise-sensitive elec- tron channel embedded in a Mach-Zehnder interferometer. Elec-trons entering the Hall bar from contact 1 split at the beam splitterBST. The electrons transmitted will traverse the N-block system and eventually go out from contact 4 or impinge onto BSB. The lattermix with those initially reﬂected at BST and interfere. The result ofthe interference can be collected in contact 3 or 3 /H11032. In the yellow solid rounded rectangles the ﬁlling factor is /H9263=1 and in the rest of the Hall bar the ﬁlling factor is /H9263=2. The coherence of the outgoing signal can be directly addressed by measurement of the visibility ofcurrent in contact 3 versus the tunable phase /H9272acquired during the propagation by the electron reﬂected at BST. Inset: schematics ofthe main picture.0 0.2 0.4 0.6 0.8 100.20.40.60.81 N=2 0 N=5 0 N=1 0 0 N=1 5 0 εV3 FIG. 6. /H20849Color online /H20850Visibility /H20851Eq. /H2084913/H20850/H20852of the current in contact 3 versus the strength of the dephasing ﬁeld for several num-bers of blocks N. In the coherent case /H9280=0 the current in contact 3 is zero and so is V3. Increasing /H9280the visibility approaches one. We set/H9253=/H9266/N.ELECTRONIC IMPLEMENTATIONS OF INTERACTION-FREE … PHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 045403-5come this difﬁculty by translating the spatial concatenation to the time-domain regime. III. MULTIPLE INTERFERENCE IN THE TIME DOMAIN In this section we show that it is possible to implement an IFM scheme based on the integer quantum-Hall MZ interfer-ometer of the type experimentally realized in Refs. 14–20by exploiting a quantizing electron emitter. 21–24Figure 7shows a schematic view of the MZ interferometer, which comprisestwo beam splitters, two electrodes coupled through quantumpoint contacts /H20849QPC1 and QPC2 /H20850, and a dephasing source affecting the propagation of electrons in the edge channel e tr. A small weakly coupled circular cavity is placed betweencontact 1 and QPC1. This produces a train of time-resolvedelectron and hole wave packets /H20849details of such single elec- tron source can be found in Appendix A /H20850. Every period com- prises a pair of electron and hole pulses, as shown in Fig. 10. QPC1 and QPC2 are controlled by the time-dependent exter-nal potentials U 1/H20849t/H20850andU2/H20849t/H20850. The system is operated as follows. In the ﬁrst period, QPC1 is opened during the ﬁrst half cycle letting the electronpulse to be injected into the MZ. It is closed during thesecond half so that holes will be reﬂected back into lead 1.The injected electron propagates with velocity vFalong the edge eblof the MZ until it meets the ﬁrst beam splitter BSL where it is split into two packets that follow two differentedge channels /H20849e trandebr/H20850of equal length Land ﬁnally reach the second beam splitter BSR after a time L/vF. Here the two packets interfere and then propagate along edges eblandetl of length L. Keeping QPC1 and QPC2 closed, the sequence repeats itself with the electronic wave packet being split andreunited many times at beam splitters BSL and BSR. Thispropagation is fully equivalent to a spatial concatenation ofdistinct MZ interferometers. At a chosen time, the electronpulse can be collected from leads 1 and 2 by opening QPC1and QPC2, respectively.Let us assume that an electron at time t +and a hole at time t−arrive at QPC1, with 0 /H11349t+/H11349T/2 and T/2/H11349t−/H11349T,Tbe- ing the period of the cycle. The electron injected throughQPC1 at time t +will appear at one of the two QPCs after a time t++N/H9004t, with /H9004t/H110132L/vF, after performing Nrounds. The two QPCs are then opened simultaneously. In the casewhere no dephasing ﬁeld is present, /H9280=0, it is possible to tune the MZ such that after Nrounds the electron pulse is at QPC2 and can be collected in contact 2. In the case of maxi-mal dephasing, /H9280=1, the electron pulse is at QPC1. Energy-level spacing inside the MZ can be estimated as /H9004E/H11011h//H9004t.Lcan be chosen to be large enough for a con- tinuum approximation of the level spacing to be valid. Thispicture allows us to describe the physics in the Landauer-Büttiker formulation, with no needs of the Floquet treatmentof this time-dependent problem. We introduce the electron annihilation operators /H20853eˆ tr,eˆbr,eˆbl,eˆtl/H20854that annihilate an elec- tron on the edge states /H20853etr,ebr,ebl,etl/H20854. In order to obtain the transport regime described in the previous section we musttune beam splitters BSL and BSR so that S BSL=SBSR=/H20873cos/H20849/H9253/2/H20850isin/H20849/H9253/2/H20850 isin/H20849/H9253/2/H20850cos/H20849/H9253/2/H20850/H20874 /H2084917/H20850 with /H20849eˆtr,eˆbr/H20850T=SBSL/H20849eˆbl,eˆtl/H20850Tand /H20849eˆbl,eˆtl/H20850T=SBSR/H20849eˆtr,eˆbr/H20850T, with the particular choice /H9253=/H9266/N. Concerning the dynami- cal phase acquired by propagating along the edge channels,arms of equal length Ldo not give rise to a relative phase shift, and the condition for the working point /H9278=0 depends only on the applied magnetic-ﬁeld intensity. IV . IFM WITH AN AHARONOV-BOHM RING In this section we review the implementation of the IFM scheme using an asymmetric AB ring proposed in Ref. 26 and discuss a scheme allowing the direct test of output-signalcoherence. This latter task can be performed by embeddingthe asymmetric AB ring in a larger, symmetric AB ring. Weshall examine the case in which the smaller ring is placed inthe upper arm of the larger one, as shown in Fig. 8. We shall use again the Landauer-Büttiker formalism of quantum transport and assume that the small asymmetric ABring supports a single channel. Following Ref. 26, we param- etrize the scattering matrix connecting the incoming to theoutgoing modes in node Aas S A=/H20873rAt¯A tAr¯A/H20874=/H20898a bcos/H20873/H9266 2/H9253/H20874bsin/H20873/H9266 2/H9253/H20874 bsin/H20873/H9266 2/H9253/H20874 a bcos/H20873/H9266 2/H9253/H20874 bcos/H20873/H9266 2/H9253/H20874bsin/H20873/H9266 2/H9253/H20874 a/H20899 /H2084918/H20850 with rA=a,tAthe 2/H110031 bottom left block, t¯Athe 1/H110032 top right block, and r¯Athe remaining 2 /H110032 bottom right block, with a=−sin /H20849/H9266/H9253/H20850//H208512+sin /H20849/H9266/H9253/H20850/H20852andb=/H208811−a2. Similarly, for node BBSL BSR V(t) 2U (t) 1U (t)ε 2 1QPC2 QPC1etr etl eblebr FIG. 7. /H20849Color online /H20850Mapping of concatenation in space to the time domain in a Mach-Zehnder interferometer. A time-dependentvoltage V/H20849t/H20850generates a current of well separated electrons and holes and the QPC1 lets only the electrons enter the Mach-Zehnderinterferometer. A dephasing ﬁeld of strength /H9280/H20849depicted by a shaded area /H20850may affect the dynamics of electrons in channel etr. Depend- ing on the presence /H20849/H9280/HS110050/H20850or absence /H20849/H9280=0/H20850of the dephasing ﬁeld, after performing Nrounds in the interferometer, the electrons are collected into contact 1 or contact 2, respectively.CHIROLLI et al. PHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 045403-6SB=/H20873r¯BtB t¯BrB/H20874. /H2084919/H20850 We further assume injection invariance under node ex- change. This conﬁguration was theoretically studied and ex-perimentally realized at low magnetic ﬁelds 35–37and can be understood as the result of Lorentz force. We label annihila-tion operators for incoming /H20849L/H20850and outgoing /H20849u,d/H20850modes in node Aasa L/H11013/H20849aL,au,ad/H20850Tand bL/H11013/H20849bL,bu,bd/H20850T, respec- tively, so that bL=SAaL. Analogously we label incoming and outgoing modes in node BasaR/H11013/H20849aR,au/H11032,ad/H11032/H20850Tand bR /H11013/H20849bR,bu/H11032,bd/H11032/H20850T, respectively, with bR=SBaR. Symmetry under cyclic exchange of nodes AandBimplies that /H20898bR bd/H11032 bu/H11032/H20899=SA/H20898aR ad/H11032 au/H11032/H20899. /H2084920/H20850 By rearranging the order of the vector components we obtain SB=SAT./H9253controls the asymmetry of nodes AandB, so that for/H9253=0 /H20849/H9253=1/H20850complete asymmetry is achieved, with the electron entering from the left lead being injected totally inthe lower /H20849upper /H20850arm, whereas for /H9253=1 /2 the injection is symmetric. An external magnetic ﬁeld is applied perpendicu-larly to the plane and is responsible for the magneticAharonov-Bohm phase acquired in the ring. At the sametime it yields the Lorentz force which leads to the ring asym-metry. Electron propagation in the two arms is described bymatrices S p/H20849/H9254/H20850=eikF/H5129diag /H20849ei/H9278/2+i/H9254,e−i/H9278/2/H20850, for transmission from left to right, and S¯p/H20849/H9254/H20850=eikF/H5129diag /H20849e−i/H9278/2+i/H9254,ei/H9278/2/H20850, for transmission from right to left. Here /H9278is the ratio of the magnetic-ﬁeld ﬂux through the asymmetric ring to the ﬂuxquantum, k Fis the Fermi wave number, /H5129is the length of the arms, and /H9254is an additional random phase. In the following we shall set kF/H5129=/H9266/2 and anticipate that a different choice does not change qualitatively our ﬁndings.As mentioned earlier, the asymmetric AB ring is embed- ded in a larger symmetric AB ring so that the phase that anelectron accumulates while traveling in the lower arm of thelarge ring represents a reference for the electron thattraverses the asymmetric ring. By tuning the magnetic ﬁeldthat pierces the larger ring, we can determine the visibility ofthe current which reﬂects the loss of coherence occurring inthe small asymmetric ring. We describe scattering at nodes LandRof the large ring by a scattering matrix 31 SL=/H20873rLt¯L tLr¯L/H20874=/H20898c/H20881g/H20881g /H20881gde /H20881ged/H20899/H2084921/H20850 with rL=c,tLthe 2/H110031 bottom left block, t¯Lthe 1/H110032 top right block, and r¯Lthe remaining 2 /H110032 bottom right block. The scattering matrix depends only on parameter g, which controls the lead-to-ring coupling strength via c=/H208811−2g,d =−/H208491+c/H20850/2, and e=/H208491−c/H20850/2, with /H9003/H20849j/H20850/H11013/H9003/H20849/H9254j,/H9254j/H11032/H20850 =Sp/H11032/H20849/H9254j/H20850/H9267¯AS¯ p/H11032/H20849/H9254j/H11032/H20850/H9267B. On the right node we have SR=SL†. Free propagation along the large-ring arms /H20849assumed to be of equal length L/H20850is accounted for by splitting the ring into two halves, each of which is described by the 2 /H110032 diagonal matrix P=eikFL/2diag /H20849ei/H9272/4,e−i/H9272/4/H20850, for propagation from left to right, and P¯=eikFL/2diag /H20849e−i/H9272/4,ei/H9272/4/H20850, for propagation from right to left. Here /H9272is the ratio of the magnetic-ﬁeld ﬂux through the larger symmetric ring /H20849/H9023/H20850to the ﬂux quan- tum. The overall amplitude for transmission /H9270from the left to the right lead is calculated through a multiple-scattering for-mula which takes into account all interference processes be-tween possible paths that electrons can take to go from theleft to the right. In the absence of decoherence one ﬁnds /H20849see Appendix C /H20850, /H9270=/H9270B/H208491−/H9003/H20850−1Sp/H11032/H9270A. /H2084922/H20850 A. Transmission in the presence of a dephasing ﬁeld We now assume a ﬂuctuating external ﬁeld /H20849dephasing source /H20850is placed in the upper arm of the small asymmetric ring. This can be described by deﬁning the partial transmis-sion amplitude of order Nwith t N=/H9270B/H20858 n=0N /H20863 j=0n /H9003/H20849n−j/H20850Sp,0/H11032/H9270A, /H2084923/H20850 where /H9003/H20849j/H20850/H11013/H9003/H20849/H9254j,/H9254j/H11032/H20850=Sp/H11032/H20849/H9254j/H20850/H9267¯AS¯ p/H11032/H20849/H9254j/H11032/H20850/H9267Bdepends on two random phases /H9254jand/H9254j/H11032, and Sp,0/H11032/H11013Sp/H11032/H20849/H92540/H20850. As in Sec. IIwe then choose the random phases from a uniform distributionof zero mean and width 2 /H9266/H9280and compute the averaged par- tial transmission probability as /H20855tN/H11569tN/H20856/H9254. It can be shown that the following recursive relation holds /H20855tN/H11569tN/H20856/H9254=/H20855tN−1/H11569tN−1/H20856/H9254+/H9014N. /H2084924/H20850 By iterating the procedure, the averaged transmission prob- ability /H20855T/H20856/H9254=lim N→/H11009/H20855tN/H11569tN/H20856/H9254can be written as /H20855T/H20856/H9254=/H20858N=0/H11009/H9014N. To compute such limit we introduce the Gell-Mann matrixA B γγ Φ ε L RΨ FIG. 8. /H20849Color online /H20850Schematic representation of a double-ring setup that allows to quantify via a current measurement the degreeof coherence of the signal going out from the small ring. A dephas-ing ﬁeld of strength /H9280/H20849depicted by a shaded area /H20850may affect the dynamics of electrons traveling in the upper arm of the small ringby randomly shifting their phase. The larger ring is pierced by amagnetic ﬂux /H9023and the small ring by a ﬂux /H9021. The nodes LandR of the large ring split the electron amplitude impinging on them ina symmetric way, whereas the nodes AandBof small ring split the electron amplitude in a non symmetric way according to the param-eter /H9253.ELECTRONIC IMPLEMENTATIONS OF INTERACTION-FREE … PHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 045403-7vector /H9018=/H20849/H90180,/H90181,...,/H90188/H20850T, with /H90180=/H208812/3/H110031, write /H9270B†/H9270B =pB·/H9018, with /H20849pB/H20850i=1 2Tr/H20849/H9270B†/H9270B/H9018i/H20850, and deﬁne the following decoherence matrix: Qij=1 2/H20855Tr/H20851/H9003†/H20849/H9254/H20850/H9018i/H9003/H20849/H9254/H20850/H9018j/H20852/H20856/H9254, /H2084925/H20850 which allows us to perform the average over the random phase as a matrix product. Similarly we deﬁne /H9003av=/H20855/H9003/H20849/H9254/H20850/H20856/H9254 and the decoherence map Pwith entries Pij=1 2/H20855Tr/H20851Sp†/H20849/H9254/H20850/H9018iSp/H20849/H9254/H20850/H9018j/H20852/H20856/H9254 /H2084926/H20850 that describes the average over the random phase in Sp,0/H11032./H9014N can be concisely written as /H9014N=/H20873pB·QN+/H20858 k=1N pk·QN−k/H20874·P·/H9270A†/H9018/H9270A /H2084927/H20850 with the vector /H20849pk/H20850i=1 2/H20851Tr/H20849/H9270B†/H9270B/H9003avk/H9018i/H20850+c.c. /H20852. By writing pk =Re /H20851/H92611k/H90111+/H92612k/H90112+/H92613k/H90113/H20852·pB, with /H9261ithe eigenvalues of /H9003av, Uthe matrix of the eigenvectors of /H9003av, and /H20849/H9011i/H20850jk =/H20849U/H9018j/H9018kU−1/H20850iithat satisfy /H20849/H90111+/H90112+/H90113/H20850/2=1, we can per- form the sum on Nobtaining /H20855T/H20856/H9254=pB·/H20849T−1/H20850·/H208491−Q/H20850−1·P·/H9270A†/H9018/H9270A /H2084928/H20850 withTbeing a 9 /H110039 matrix deﬁned by T=/H20858i=13Re/H20851/H208491 −/H9261i/H20850−1/H9011iT/H20852. The averaged transmission probability /H20855T/H20856/H9254is now function of the AB phase /H9272. B. Current as a measure of coherence The coherence of the signal transmitted through the small, asymmetric AB ring can be established by studying the trans-port properties of the entire device. We focus on the case ofstrong coupling /H20849g/H113511/2/H20850for which an electron approaching the large ring from node Lis mostly transmitted into the two arms of the large ring /H20849g=0.49 in the following. /H20850For clarity, we also set the magnetic ﬁeld and the arm length so that /H9278 =/H9266,/H9272=0,kF/H5129=/H9266, and kFL=/H9266. Actually, in a realistic experi- mental implementation it would be difﬁcult to realize suchconditions. We note however that the degree of coherencecould be studied by changing one of the parameters of thelarge ring /H20849e.g., k FL/H20850and measuring the visibility of the os- cillations of the output signal. For an applied bias voltage V, the zero-temperature cur- rent through the device of Fig. 8is given by I=e2V h/H20855T/H20856/H9254 /H2084929/H20850 with /H20855T/H20856/H9254as in Eq. /H2084926/H20850. Figure 9shows Ias a function of the noise parameter /H9280for various values of the small-ring asym- metry parameter /H9253. As the dephasing strength /H9280is increased, however, Iincreases with a behavior that strongly depends on the degree of asymmetry of the small ring. In the case ofmaximum decoherence /H20849 /H9280=1/H20850two different cases can be dis- tinguished. For /H9253=0.02 most of the electron amplitude that enters the small ring from the left will propagate into thelower arm of the small ring and coherently transmit intonode R. There it interferes constructively with the reference path, saturating the current to the maximum e 2V/h.O nt h e other hand for /H9253=0.98 most of the electron amplitude that enters from the left into the small ring will propagate into theupper arm of the small ring. There a dephasing ﬁeld ispresent and the signal that propagates through the small ringwill combine at node Rwith the reference path. The current exiting the device reaches a maximal value between zero ande 2V/h. We interpret this behavior as an IFM of the dephasing ﬁeld. The current exiting the device is proportional to thevisibility of the output signal of the small asymmetric ring. V . CONCLUSION Based on the idea ﬁrst suggested in Ref. 26and directly inspired to the original proposal of Elitzur and Vaidman,1in this paper we focused on studying and detecting the presenceof a classical external random ﬂuctuating electric or mag-netic ﬁeld, which represents a common dephasing source inquantum devices. The noise source randomizes the phase ofa propagating electron and plays the role of absorption inoptical schemes while the loss of coherence of the outgoingelectrons mimics photon absorption. The fraction of coherentoutput signal or alternatively the visibility of the outgoingsignal represents the ﬁgures of merit that qualify an IFM.The study of these quantities allowed us to point out thedifference between a which-path detection and an IFM: theformer allows only the detection of the presence of a dephas-ing source at the expense of the degradation of the visibilityof the outgoing signal, whereas the latter allows a coherentdetection of the dephasing source. Three distinct IFM schemes were investigated. The ﬁrst system is a concatenation of interferometers based on theinteger quantum-Hall interferometric architecture proposedin Ref. 30. The dynamics of electrons traveling along edgeγ=0.02 γ=0.98γ=0.5Current (e V/h)2 ε0 0.2 0.4 0.6 0.8 100.20.40.60.81 γ=0.2 γ=0.8 FIG. 9. /H20849Color online /H20850Plot of the current /H20851Eq. /H2084927/H20850/H20852in units of e2V/h, ﬂowing from the left lead to the right lead of the double-ring structure represented in Fig. 8, versus the strength /H9280of the dephas- ing ﬁeld, at several degree of asymmetry /H9253. For/H9253→1 we divert the electrons mostly toward the dephasing source and consequently wehave a reduction in the current ﬂowing in the device. For /H9253→0w e divert the electron mostly toward the dephasing-free region and thecoherent propagation gives rise to a maximal current ﬂowing in thedevice. Plot realized with g=0.49, /H9278=/H9266,/H9272=0, kF/H5129=/H9266, and kFL =/H9266.CHIROLLI et al. PHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 045403-8channels is exposed to the action of an external ﬂuctuating ﬁeld. We suggest to steer the propagation of one channeltoward the inner part of the Hall bar, where dephasing isminor or absent, and by separating and recombining manytimes the two channels we reproduce an electronic analogueof the high-efﬁciency scheme proposed in optics by Kwiat et al.in Ref. 2. We showed that, for a strong dephasing source, only an asymptotically negligible amount of coherent signalis lost by proper tuning the degree of admixture of the chan-nels at the beam splitters. Moreover, the effect is very robustagainst small ﬂuctuation about the exact value of the admix-ture required. Indeed, although the fraction of coherent sig-nal is reduced in magnitude by the averaging process, itsqualitative behavior is not affected by it. The second system we considered is based on a standard quantum-Hall electronic Mach-Zehnder interferometer andassumes the presence of a quantized electron emitter. A veryprecisely time-resolved electronic wave packet is sent into aMach-Zehnder interferometer in which an arm is affected byexternal classical noise. The packet travels at a precise speedand tests the region affected by noise many times, being splitand recombined until it is allowed to escape the interferom-eter to be collected. The entire sequence can be mapped tothe concatenation in the space domain that characterizes thenoise-sensitive coherent electron channel previously de-scribed: the same results and conclusions apply also to thissystem. The latter has the advantage that it is experimentallymuch easier to realize since it is based on a system alreadyavailable. The last system we considered is a double-ring structure based on the proposal suggested in Ref. 26. There, authors considered an Aharonov-Bohm chiral ring in which a local-ized source of noise affects one arm of the ring and studiedthe fraction of coherent signal that exits the device. However,such a quantity is not measurable in that setup. We suggest toembed the chiral AB ring in one arm of a larger AB ring andmeasure the total current ﬂowing through the device as aﬁgure of merit of the coherence of the output signal from thesmall chiral AB ring. Such a setup has the advantage to over-come the difﬁculties arising from concatenating many inter-rogation steps, necessary in order to achieve high efﬁciencyIFM in the noise-sensitive coherent electron channel. It alsoeliminates the need for very precise time-resolved electron-ics, on which the second proposal was based. We point out here that IFM can be designed also for the case of an electron absorber and the same results obtainedwith the dephasing source are found. The different imple-mentations described here can ﬁnd useful applications inquantum-coherent electronics and quantum computations,where the coherence of the signals is always threatened bythe presence of ﬂuctuating external ﬁelds. ACKNOWLEDGMENTS This work was supported by funding from the German DFG within Grant No. SPP 1285 “Spintronics,” from theSwiss SNF via Grant No. PP02-106310, and by the ItalianMIUR under the FIRB IDEAS project ESQUI. V.P. acknowl-edges CNR-INFM for funding through the SEED Program.APPENDIX A: ELECTRON-HOLE SWITCH Let us consider the mechanism suggested in Sec. IIIfor injecting and collecting electrons in the MZ interferometer.The system is depicted in Fig. 10/H20849a/H20850and is composed by a cavity formed by a circular edge state that is coupled to an edge channel by a QPC Vof transmission amplitude t˜and reﬂection amplitude r˜. It was experimentally demonstrated21,22that such a device, if periodically driven by a time-dependent potential V/H20849t/H20850, produces a periodic current composed by an electron in one half period and a hole in theother half period, see Fig. 10/H20849b/H20850. We wish to separate the electron and the hole by transmitting the electron through abarrier toward contact 3 and reﬂecting the hole into contact4. A time-dependent QPC Udriven by an external potential U/H20849t/H20850behaves like a beam splitter that mixes the incoming channels, from the contacts 1 and 2, into the outgoing chan-nels 3 and 4. If properly driven, it works as a switch thatseparates electrons and holes generated by the cavity intodifferent edge channels. Following Refs. 23and24we de- scribe the effect of the time-dependent potential QPC Uby a scattering matrix SU/H20849t/H20850=/H20873S31/H20849t/H20850S32/H20849t/H20850 S41/H20849t/H20850S42/H20849t/H20850/H20874. /H20849A1/H20850 In the symmetric case one has S31/H20849t/H20850=S42/H20849t/H20850and S32/H20849t/H20850 =S41/H20849t/H20850. From the unitarity of SU/H20849t/H20850follows that 1=/H20858 j/H20841Sjk/H20849t/H20850/H208412, /H20849A2/H20850 0=S32/H11569/H20849t/H20850S31/H20849t/H20850+S42/H11569/H20849t/H20850S41/H20849t/H20850. /H20849A3/H20850 The dynamics of the cavity can be described by a time- dependent scattering amplitude Sc/H20849t,E/H20850, which satisﬁes /H20841Sc/H20849t,E/H20850/H208412=1. In the adiabatic regime, keeping all the reser-V(t) U(t)3 2QPCVQPCU1 4a) b) el100 50 0 -50 -100 0 0.2 0.4 0.6 0.8 1I( eV / h )c2 ho t(ns) FIG. 10. /H20849Color online /H20850/H20849a/H20850Schematic representation of a time- dependent electron-hole switch. The cavity driven by the potentialV/H20849t/H20850is connected via QPC Vto a linear edge and produces a well separated pair of electron and hole per cycle. The potential U/H20849t/H20850 drives the QPC Uthat connects contacts 1 and 2 to contacts 3 and Fig. 4and periodically transmits the electron to contact 3 and re- ﬂects the hole to contact 4. /H20849b/H20850Time-resolved electron-hole current produced by the driven cavity in front of QPC V, as given by Eq. /H20849A5/H20850.ELECTRONIC IMPLEMENTATIONS OF INTERACTION-FREE … PHYSICAL REVIEW B 82, 045403 /H208492010 /H20850 045403-9voirs at the same chemical potential /H9262, the zero-temperature current in contacts 3 and 4 can be written as Ij/H20849t/H20850=/H20841Sj1/H20849t/H20850/H208412Ic/H20849t/H20850+e 2/H9266i/H20858 k=1,2Sjk/H20849t/H20850/H11509 /H11509tSjk/H11569/H20849t/H20850/H20849 A4/H20850 with j=3,4. Here Ic/H20849t/H20850is the current produced by the cavity, that can be written as23,24 Ic/H20849t/H20850=e 2/H9266iSc/H20849t,/H9262/H20850/H11509 /H11509tSc/H11569/H20849t,/H9262/H20850. /H20849A5/H20850 Ic/H20849t/H20850is plotted in Fig. 10/H20849b/H20850for a harmonic driving V/H20849t/H20850 =V0cos/H20849/H9024t/H20850, for the choice /H9024/2/H9266=1 GHz and /H20841t˜/H208412=0.1. By deﬁning S31/H20849t/H20850=/H20881T/H20849t/H20850and S41/H20849t/H20850=i/H208811−T/H20849t/H20850, it follows that I3/H20849t/H20850=T/H20849t/H20850Ic/H20849t/H20850and I4/H20849t/H20850=/H208511−T/H20849t/H20850/H20852Ic/H20849t/H20850, with T/H20849t/H20850related to the applied external potential U/H20849t/H20850. By choosing a proper modulation of T/H20849t/H20850, it is possible to separate the electrons from the holes. APPENDIX B: EIGENV ALUE PROBLEM Deﬁning u/H11006=1 2 tan /H20849/H9253/H20850/H208771 − sinc /H20849/H9280/H20850/H11006/H20881/H208511 + sinc /H20849/H9280/H20850/H208522−4sinc /H20849/H9280/H20850 cos2/H20849/H9253/H20850/H20878. /H20849B1/H20850 The matrix Uassumes the simple form U=/H2089810 0 0 01 0 0 00 u+u− 00 1 1/H20899/H20849B2/H20850 with sinc /H20849/H9280/H20850=sin /H20849/H9266/H9280/H20850//H9266/H9280that allows for a simple solution of the eigenvalue problem in terms of a Jordan decomposition,Q=U −1diag /H208511,sin /H20849/H9266/H9280/H20850//H9266/H9280,/H9261−,/H9261+/H20852U, with /H9261/H11006=1 2cos/H20849/H9278/H20850/H208511 + sinc /H20849/H9280/H20850/H20852 /H110061 2/H20881cos2/H20849/H9278/H20850/H208511 + sinc /H20849/H9280/H20850/H208522− sinc2/H20849/H9280/H20850. /H20849B3/H20850APPENDIX C: DOUBLE RING TRANSMISSION AND REFLECTION AMPLITUDES In the absence of decoherence, the transmission amplitude for electrons going from the left lead Lto the right lead R can be calculated through the following multiple-scatteringformula /H9270=/H9270B/H208491−/H9003/H20850−1Sp/H11032/H9270A /H20849C1/H20850 with/H9003=Sp/H11032/H9267¯AS¯ p/H11032/H9267Band Sp/H11032=/H20849Sp0 01/H20850. We deﬁne the following transmission matrices in nodes AandBthat take into account the lower arm of the larger ring, tA/H11032=/H20873tA0 01/H20874,tB/H11032=/H20873tB0 01/H20874, /H20849C2/H20850 t¯A/H11032=/H20873t¯A0 01/H20874,t¯B/H11032=/H20873t¯B0 01/H20874 /H20849C3/H20850 with tA/H11032andt¯B/H11032of dimension 3 /H110032, and t¯A/H11032andtB/H11032of dimen- sion 2 /H110033. Analogously we deﬁne the reﬂection matrices rA/H11032=/H20873rA0 00/H20874,r¯B/H11032=/H20873r¯B0 00/H20874, /H20849C4/H20850 r¯A/H11032=/H20873r¯A0 00/H20874,rB/H11032=/H20873rB0 00/H20874 /H20849C5/H20850 with rA/H11032andr¯B/H11032of dimension 2 /H110032, and r¯A/H11032andrB/H11032of dimen- sion 3 /H110033. The effective transmission amplitudes /H9270Aand/H9270B are given by the matrices /H9270A=tA/H11032/H208491−Pr¯LP¯rA/H11032/H20850−1PtL, /H20849C6/H20850 /H9270B=tL/H208491−Pr¯B/H11032P¯rR/H20850−1PtB/H11032 /H20849C7/H20850 with dimension, respectively, 3 /H110031 and 1 /H110033. 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PhysRevB.101.224515.pdf	PHYSICAL REVIEW B 101, 224515 (2020) Muon spin rotation and infrared spectroscopy study of Ba 1−xNaxFe2As2 E. Sheveleva ,1,B. Xu,1P. Marsik,1F. Lyzwa,1B. P. P. Mallett ,2K. Willa,3C. Meingast,3Th. Wolf,3 T. Shevtsova,4Y u .G .P a s h k e v i c h ,4and C. Bernhard1,† 1University of Fribourg, Department of Physics and Fribourg Center for Nanomaterials, Chemin du Musée 3, CH-1700 Fribourg, Switzerland 2The MacDiarmid Institute for Advanced Materials and Nanotechnology and The Dodd-Walls Centre for Photonic and Quantum Technologies, The University of Auckland, NZ-1010 Auckland, New Zealand 3Institute for Quantum Materials and Technologies - IQMT, Postfach 3640, DE-76021 Karlsruhe, Germany 4O. O. Galkin Donetsk Institute for Physics and Engineering NAS of Ukraine, UA-03680 Kyiv, Ukraine (Received 4 April 2020; revised manuscript received 29 May 2020; accepted 4 June 2020; published 30 June 2020) The magnetic and superconducting properties of a series of underdoped Ba 1−xNaxFe2As2(BNFA) single crystals with 0 .19/lessorequalslantx/lessorequalslant0.34 have been investigated with the complementary muon-spin-rotation ( μSR) and infrared spectroscopy techniques. The focus has been on the different antiferromagnetic states in the underdopedregime and their competition with superconductivity, especially for the ones with a tetragonal crystal structureand a so-called double- Qmagnetic order. Besides the collinear state with a spatially inhomogeneous spin- charge-density wave (i-SCDW) order at x=0.24 and 0.26, that was previously identiﬁed in BNFA, we obtained evidence for an orthomagnetic state with a “hedgehog”-type spin vortex crystal (SVC) structure at x=0.32 and 0.34. Whereas in the former i-SCDW state the infrared spectra show no sign of a superconducting response downto the lowest measured temperature of about 10 K, in the SVC state there is a strong superconducting responsesimilar to the one at optimum doping. The magnetic order is strongly suppressed here in the superconductingstate and at x=0.34 there is even a partial reentrance into a paramagnetic state at T/lessmuchT c. DOI: 10.1103/PhysRevB.101.224515 I. INTRODUCTION The phase diagram of the iron arsenide superconductors is characterized by a close proximity of the antiferromag-netic (AF) and superconducting (SC) orders [ 1,2]. This is exempliﬁed by the prototypical system BaFe 2As2(Ba-122) for which large, high-quality single crystals are readily avail-able. The undoped parent compound is an itinerant antiferro-magnet with a Neel temperature of T N≈135 K [ 1]. Upon electron or hole doping in Ba(Fe 1−xCox)2As2(BFCA) [ 3], Ba1−xKxFe2As2(BKFA) [ 4], or Ba 1−xNaxFe2As2(BNFA) [5,6], the AF order gets gradually suppressed and supercon- ductivity emerges well before the magnetic order vanishes.In this so-called underdoped regime, the AF and SC orderscoexist and compete for the same low-energy electronic states[7–10]. Upon doping, the superconducting critical tempera- tureT cand other SC parameters, like the condensate density nsor the condensation energy γsare enhanced whereas the AF order parameter (the staggered magnetization) is reduced.The full suppression of the static AF order is observed aroundoptimum doping at which T c,ns, andγsreach their maximal values [ 11,12]. A further increase of the doping leads to a decrease of Tcin the so-called overdoped regime for which the AF spin ﬂuctuations also diminish. This characteristicdoping phase diagram is one of the reasons, besides theunconventional s ±symmetry of the SC order parameter, why evgeniia.sheveleva@unifr.ch †christian.bernhard@unifr.chAF ﬂuctuations are believed to be responsible for the SC pairing [ 1]. Nevertheless, there exist other candidates for the SC pairing mechanism such as the nematic/orbital ﬂuctuations[13,14]. Even a phonon mediated pairing or a coupled spin- phonon mechanism is not excluded yet [ 15,16]. There is also a strong coupling between the spin, orbital, and lattice degrees of freedom that is exempliﬁed by thecoupled AF and structural phase transition from a tetragonalparamagnetic state with C 4symmetry at high temperature to an orthorhombic antiferromagnetic AF (o-AF) state with C2 symmetry [ 17–19]. For this o-AF state, which occupies major parts of the magnetic phase diagram, the spins are antiparallelalong (0; π) and parallel along (0; π), giving rise to a so- called single- Qor stripelike AF order [ 20]. Deviations from this o-AF order occur closer to optimum doping. For example,in BFCA the o-AF order and the associated lattice distortionsare reported to become incommensurate and to be stronglysuppressed by SC and eventually vanish below T c[21]. A different type of AF order, for which the lattice structure remains tetragonal ( C4symmetry), albeit with a fourfold enlarged unit cell, was recently observed in the hole-dopedBKFA and BNFA systems [ 22–28]. This tetragonal antifer- romagnetic (t-AF) order can be described in terms of a so-called double- Qorder due to a superposition of the single- Q states along (0; π) and ( π; 0). It can be realized either with a noncollinear magnetization of the single- Qcomponents, corresponding to a so-called orthomagnetic or “spin-vortex-crystal” (SVC) order, or with a collinear magnetization thatgives rise to an inhomogeneous state for which the Fe mag-netic moment either vanishes or is doubled [ 29–31]. The 2469-9950/2020/101(22)/224515(22) 224515-1 ©2020 American Physical SocietyE. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) latter state is accompanied by a subordinate charge density wave, forming a so-called spin-charge-density wave (SCDW)[26,32]. Experiments on BNFA [ 27] and BKFA [ 25]h a v e identiﬁed the SCDW order with the spins oriented along thecaxis direction [ 23], suggesting that spin-orbit interaction plays an important role [ 33]. It is still unknown which factors are most relevant for stabilizing these single- Qand double- QAF orders, and even an important role of disorder has been proposed [ 34]. In this context, it is interesting that a “hedgehog”-type orthomagnetic state has recently been iden-tiﬁed in underdoped CaK(Fe 1−xNix)4As4for which the K+ and Ca2+ions reside in separate layers that alternate along the caxis. It has been speculated that the SVC order is stabilized here by the broken glide symmetry across the FeAs planes orby a reduced cation disorder [ 35,36]. Of equal interest is the recent observation of yet another magnetic phase in BNFAthat occurs at 0 .3<x<0.37, i.e., between the i-SCDW phase and optimum doping [ 5]. The latter is accompanied by a tiny orthorhombic distortion and therefore has been discussed interms of an o-AF order with a very small magnetic moment[5]. Alternatively, it could be explained in terms of one of the SVC phases with tetragonal ( C 4) symmetry that is somewhat distorted or coexists with a small fraction of the o-AF phase. The above described questions have motivated us to further explore the complex magnetic phase diagram of the ironarsenides and its relationship with SC. Here, we present anexperimental approach using the complementary techniquesof muon spin rotation ( μSR) and infrared spectroscopy to study a series of BNFA single crystals that span the under-doped regime with its various magnetic phases. In particular,we provide evidence that the recently discovered AF phasethat occurs shortly before optimum doping likely correspondsto an orthomagnetic “hedgehog”-type SVC order. This paper is organized as follows. The experimental meth- ods are presented in Sec. II. Subsequently, we discuss in Sec. IIItheμSR data and in Sec. IV, the infrared spectroscopy data. We conclude with a discussion and summary in Sec. V. II. EXPERIMENTAL METHODS Ba1−xNaxFe2As2(BNFA) single crystals were grown in alumina crucibles with an FeAs ﬂux as described in Ref. [ 5]. They were millimeter-sized and cleavable yielding ﬂat andshiny surface suitable for optical measurements. Selectedcrystals were characterized by x-ray diffraction reﬁnement.For each crystal presented here, the Na content, x, was deter- mined with electron dispersion spectroscopy with an accuracyof about ±0.02 (estimated from the variation over the crystal surface). Figure 1shows the location of these crystals in the temperature versus doping phase diagram (marked with stars)that has been adopted from Ref. [ 5]. It also shows sketches of the various o-AF, i-SCDW, and SVC magnetic orders. Themagnetic and superconducting transition temperatures of thecrystals, or of corresponding crystals from the same growthbatch, have been derived from transport and from thermalexpansion and thermodynamic experiments as described, e.g.,in Ref. [ 5]. Except for the SC transition of the crystals in the SCDW state at x=0.24 and 0.26, the various magnetic and superconducting transitions have been conﬁrmed with theμSR and infrared spectroscopy measurements as described FIG. 1. Schematic phase diagram of Ba 1−xNaxFe2As2(BNFA) showing the different antiferromagnetic and superconducting phases and the location of the studied samples. below. The bulk SC transition of the crystal at x=0.24 is evident from additional speciﬁc heat data that are also shownbelow. TheμSR experiments were performed at the general purposes spectrometer (GPS) at the πM3 beamline of the Paul Scherrer Institute (PSI) in Villigen, Switzerland whichprovides a beam of 100% spin-polarized, positive muons.This muon beam was implanted in the crystals along thecaxis with an energy of about 4.2 MeV . These muons ther- malize very rapidly without a signiﬁcant loss of their initialspin polarization and stop at interstitial lattice sites with adepth distribution of about 100–200 μm. The magnetic and superconducting properties probed by the muons are thusrepresentative of the bulk. The muons sites are assumed tobe the same as in BKFA with a majority and a minoritysite that account for about 80% and 20% of the muons,respectively. As discussed in Appendix Aand also shown in Fig. 3(a) of Ref. [ 27], the majority site has a rather high local symmetry and is located on the line that connects theBa and As ions along the caxis(at the (0,0,0.191) coordinate of the I4/mmm setting [ 33]). The minority site is located at (0.4,0.5,0) and has a similar high local symmetry with thesame direction and qualitative changes of the local magneticﬁeld. The spin of the muons precesses in its local magneticﬁeld B μ, with a frequency νμ=(γμ/2π)Bμ, where γμ= 2π135.5 MHz/T is the gyromagnetic ratio of the muon. In aμSR experiment, one measures the time evolution of the spin polarization of an ensemble of (typically several million)muons, P(t). This is done via the detection of the asymmetry of the positrons that originate from the radioactive decay ofthe muons with a mean life time of τ μ≈2.2μs and which are preferentially emitted in the direction of P(t) at the instant of decay. This asymmetry is recorded within a time windowof about 10 −6–10−9s which allows one to detect magnetic ﬁelds ranging from about 0.1 Gauss to several Tesla. Mostof the zero-ﬁeld (ZF) and transverse ﬁeld (TF) experimentsreported here were performed in the TF geometry using theso-called upward (u) and downward (d) counters which have 224515-2MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) FIG. 2. Sketch of the geometry of the μSR setup at the GPS beamline of PSI showing the three pairs of position counters. Thesample, shown in purple, has its caxis aligned with the zaxis and the incoming muon beam. In the so-called transverse-ﬁeld (TF) geometry, the muon spin is rotated by 54 ◦with respect to the zaxis. The external ﬁeld for the transverse ﬁeld experiments Bextis applied parallel to the zaxis. higher and more balanced count rates than the forward (f) and backward (b) counters. The signal of the pair of fb-counters was only used in combination with the one fromthe ud-counters for the determination of the direction of B μ (as speciﬁed in the relevant ﬁgures). The initial asymmetry of the ud-counter in the so-called transverse-ﬁeld (TF) geometry,for which the muon spin is rotated by about 54 ◦(toward the upward counter) in the direction perpendicular to themomentum of the muon beam as shown in Fig. 2, is about 20%–22%. This variation of the initial asymmetry typicallyarises from a difference in the size and the exact positioning ofthe samples with respect to the positron counters as well as theso-called veto counter that is used for small samples to reducethe background signal due to muons that missed the sample.Further details about the μSR technique can be found, e.g., in Refs. [ 37–40]. The optical response was measured in terms of an in-plane reﬂectivity function R(ω) at a near-normal angle of incidence with a Fourier-transform infrared (FTIR) spectrometer BrukerVertex 70V in the frequency range from 40–8000 cm −1 with an in situ gold evaporation technique [ 41]. Data were collected at different temperatures between 10 and 300 Kusing a ARS Helitran cryostat. Room temperature spectraof the complex dielectric function in the near-infrared toultraviolet (NIR-UV) range of 4000–52 000 cm −1were ob- tained with a commercial Woollam V ASE ellipsometer. Thecombined ellipsometry and reﬂectivity spectra were used toperform a Kramers-Kronig analysis to derive the complexoptical response functions [ 42] which in the following are ex- pressed in terms of the complex optical conductivity σ(ω)= σ 1(ω)+iσ2(ω), or, likewise, the complex dielectric function, /epsilon1(ω)=/epsilon11(ω)+i/epsilon12(ω), that are related according to σ(ω)= i2π Z0ω/epsilon1(ω). Below 40 cm−1, we extrapolated the reﬂectivity data with a Hagen-Rubens model R(ω)=1−A√ωin the normal state and a superconducting model R(ω)=1−Aω4 below Tc. On the high-frequency side above 52 000 cm−1,w e assumed a constant reﬂectivity up to 225 000 cm−1that is followed by a free-electron ( ω−4) response.FIG. 3. Zero-ﬁeld (ZF) μSR data of the BNFA crystal with x≈ 0.24 and Tc≈12 K showing two magnetic transitions to an o-AF state below TN,1≈85 K and the i-SCDW state below TN,1≈38 K. [(a)–(c)] ZF- μSR curves taken at 50 K in the o-AF state and at 36 and 5 K in the i-SCDW state, respectively. [(d) and (e)] Temperature dependence of the precession frequencies and relaxation rates of the oscillatory signals from two different muon sites, respectively.(f) Temperature dependence of the normalized amplitudes of the oscillatory signals and the slowly relaxing, nonoscillatory signal. Be- lowT N, the latter arises mainly due to the nonorthogonal orientation ofBμandP, apart from a small background due to muons that missed the sample. III. MUON SPIN ROTATION - μSR We start with the discussion of the zero-ﬁeld (ZF)- μSR data for which only the internal magnetic moments contributeto the magnetic ﬁeld at the muon site B μ. Figure 3summarizes the (ZF)- μSR study of the BNFA crystal with x≈0.24 that exhibits a transition from a high- temperature paramagnetic state to an o-AF state at TN,1≈ 85 K and a subsequent transition to a t-AF and i-SCDW stateatT N,2≈38 K that is followed by a SC transition at Tc≈ 12 K. Figures 3(a)–3(c) display characteristic, time-resolved spectra of the evolution of the muon spin polarization, P(t), in the o-AF state at 50 K and in the i-SCDW state at 36 and5 K. They exhibit clear oscillatory signals that are indicativeof a bulk magnetic order. The solid lines show ﬁts with the 224515-3E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) FIG. 4. ZF- μSR spectra of the x≈0.24 crystal showing the spin reorientation at the o-AF to i-SCDW transition. (a) ZF- μSR spectra of the pairs of forward-backward (fb) and up-down (ud) positron counters (see the sketch in Fig. 2) at 50 K in the o-AF state. The absence of an oscillatory signal of the fb-counters conﬁrms that Bμis parallel to the caxis. (b) ZF- μSR spectra at 5 K in the i-SCDW state for which the fb-counters show a large oscillatory signal suggesting an in-plane orientation of Bμ. function: P(t)=P(0)2/summationdisplay i=1Aosc icos(γμBμt+/Phi1i)e−λit+Anon 3e−λ3t,(1) where Ai,Bμ,i,/Phi1i, andλiaccount for the relative amplitudes of the signal, the local magnetic ﬁeld at the muon sites,the initial phase of the muon spin, and the relaxation rates,respectively. The two oscillating signals with amplitudes A osc 1 andAosc 2arise from two muon sites with different local ﬁelds, as discussed in Ref. [ 27]. The nonoscillating signal Anon 3 results from the nonorthogonal orientation of PandBμ.I n addition, it contains a small contribution due to a nonmagneticbackground from muons that stopped outside the sample. Thelatter is typically less than 5% of the total signal. The temperature dependence of the obtained ﬁt parameters is displayed in Fig. 3(d) for the two precession frequencies, in Fig. 3(e) for the corresponding relaxation rates, and in Fig. 3(f)for the normalized amplitudes. All three parameters exhibit pronounced changes at T N,2≈38 K. As outlined in Ref. [ 27], the decrease of the precession frequency, the relaxation rate and the amplitude of the oscillatory signalbelow T N,2≈38 K are indicative of a transition from the o-AF to a t-AF and i-SCDW order. The only difference withrespect to BKFA in Ref. [ 27] is that the present BNFA crystal does not show any sign of a reentrance towards an o-AF statebelow T c, i.e., it remains in the i-SCDW state down to 5 K without any noticeable anomaly at Tc=12.3K . Figures 4(a) and4(b) reveal that the ZF- μSR data show clear signatures of a change of the Fe-spin direction from anin-plane orientation in the o-AF phase to a caxis orientation in the i-SCDW state. This is evident from the comparison ofthe amplitudes of the oscillatory signals of two different pairsof positron counters, i.e., of the upward (u) and downward(d) counters and the forward (f) and backward (b) counters (asketch of the counter geometry is shown in Fig. 2). According to the calculations in Appendix B, the in-plane oriented Fe spins in the o-AF phase give rise to a local magnetic ﬁeld atthe muon site, B μ, that is pointing along the caxis, Bμ//c.T h i s is because of the high symmetry of the majority muon siteFIG. 5. Speciﬁc heat curve of a BNFA crystal at x≈0.24 with the phonon contribution subtracted as described in Ref. [ 5]. In addition to two sharp peaks due to magnetic transitions to the o-AF state below TN,1and the i-SCDW state below TN,2, it exhibits a pronounced signature of bulk superconductivity. (Inset) Magniﬁca- tion of the low-temperature data from which the superconducting transition temperature of 12.3 K has been deduced using an entropy conserving construction (black line). which is on a straight line between the As and Ba (or Na) ions. To the contrary, the caxis oriented spins (on every second Fe site) in the i-SCDW phase cause Bμto be parallel to the ab plane, Bμ//ab[27]. From the sketch in Fig. 2it is seen that the former case with Bμ//c(o-AF phase) gives rise to a vanishing oscillatory signal for the fb-counters and a large oscillatorysignal for the ud-counters. Such a behavior is evident for theZF-μSR spectra in the o-AF phase at 50 K in Fig. 4(a).I n contrast, for the ZF spectra at 5 K in the t-AF and i-SCDWstate in Fig. 4(b), the fb-counters exhibit a large oscillatory signal that is characteristic of an in-plane orientation of B μ due to a caxis orientation of the spins. Note that this change of the direction of Bμis also evident from the TF- μSR spectra (not shown) for which in the o-AF phase the applied ﬁeld Bextand the ﬁeld from the magnetic moments Bmagare along the caxis, yielding Bμ=Bmag± Bext, whereas in the i-SCDW phase Bmagis along the abplane such that Bμ=√ B2 mag+B2 ext[27]. OurμSR data thus provide clear evidence that the BNFA crystal with x≈0.24 undergoes a transition from a bulk o-AF state below TN,1≈85 K with in-plane oriented spins to a bulk i-SCDW phase below TN,2≈38 K that persists to the lowest measured temperature of 5 K, even well below theSC transition at T c≈12 K. The bulk nature of the super- conducting state with Tc≈12 K is evident from the speciﬁc heat data shown in Fig. 5for which the phonon contribution has been subtracted as described in Ref. [ 5]. As detailed in the inset, the value of Tc=12.3 K has been determined by the midpoint of the speciﬁc heat jump using an entropyconserving construction (black line). The bulk nature of SCis evident from the more or less complete suppression ofthe electronic speciﬁc heat at very low temperature. Thespeciﬁc heat curves also exhibit two more strong peaks at 224515-4MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) higher temperature that are due to the magnetic transitions into the o-AF state and the i-SCDW state at TN,1≈75 K and TN,2≈45 K, respectively. Note that these magnetic transition temperatures are somewhat lower than the ones obtained fromtheμSR data in Fig. 4. Since the speciﬁc heat measurements have been performed on a smaller piece that was cleaved fromthe thick crystal measured with μSR (from the same side from which the infrared data have been obtained), the differenceof the T N,1andTN,2values is most likely due to a variation of the Na content that is within the limits of /Delta1x=±0.02 as determined with electron dispersion spectroscopy. The sameapplies for the rather large peak width at T N,1≈75 K that is also indicative of a signiﬁcant spread in the Na contentthat is, however, within /Delta1x=±0.02. Finally, note that the μSR data of the BNFA crystal with x≈0.26 (not shown) reveal a corresponding behavior as described above with twomagnetic transitions from an o-AF phase below T N,1≈80 K to a i-SCDW phase below TN,2≈42 K. Next, we discuss the ZF- μSR data of the BNFA crystal with x≈0.32 and a Neel temperature of TN≈45 K and Tc≈ 22 K. The nature of this AF order, which appears just shortlybefore static magnetism vanishes around optimal doping, re-mains to be identiﬁed. The thermal expansion measurementsof Ref. [ 5] have shown that this AF order is accompanied by a very weak orthorhombic lattice distortion. In addition, mag-netization measurements on Sr 1−xNaxFe2As2crystals with a corresponding magnetic order revealed an in-plane orientationof the magnetic moments [ 43]. Accordingly, these data have been interpreted in terms of an o-AF order with a very smallmagnetic moment [ 5] or a small magnetic volume fraction. To the contrary, our ZF- μSR data in Fig. 6establish that this magnetic order is very strong and not only due to a smallminority phase (at least at T>T c). Figure 6(a)conﬁrms that the ZF spectra below TN≈45 K exhibit a large oscillatory signal with a rather high precessionfrequency. The temperature dependence of the precessionfrequencies ν 1andν2(due to the two different muon sites) and of the corresponding amplitudes, as obtained from ﬁttingwith the function in Eq. ( 1), are displayed in Figs. 6(d) and 6(f), respectively. The value of ν 1increases steeply below TN≈45 K and reaches a maximum of νμ≈22 MHz at 25 K. The latter is only about 10% lower than the one obtainedfor the x≈0.24 sample in the o-AF state where it reaches a maximum of ν μ≈24.5 MHz at 40 K [see Fig. 3(d)]. Note that the precession frequency is proportional to Bμand thus to the magnitude of the magnetic moment, given that the muonsite remains the same (which is most likely the case). The μSR data are therefore incompatible with an o-AF state that has avery small magnetic moment. Instead, the observed magnetic state at x≈0.32 seems to be compatible with an orthomagnetic, so-called “hedgehog”-type SVC order for which the spins are oriented within the ab plane as sketched in Fig. 1and shown in the Appendix Bin Fig. 19. This “hedgehog”-type SVC is expected to give rise to a local magnetic ﬁeld that for half of the majority muonsites (the 4e-sites) is rather large and pointing along the caxis direction and vanishes for the other half (the 4f-sites). Notethat the loop-type SVC can be excluded since the magneticﬁeld is predicted to vanish for both the 4e- and 4f-type muonsites (see Fig. 18in Appendix B).FIG. 6. ZF- μSR data at x≈0.32,Tc≈22 K. (a) ZF- μSR spec- tra showing the formation of a bulk magnetic state below TN≈ 45 K. [(b) and (c)] Comparison of the ZF spectra of the fb- and ud-counters that indicate a caxis orientation of Bμand thus in- plane orientated spins. [(d) to (f)] Temperature dependence of the precession frequencies, ν1andν2, the relaxation rates, λ1andλ2,a n d the normalized amplitudes of the oscillatory and the nonoscillatorysignals, respectively. The comparison of the ZF- μSR spectra of the forward- backward (fb) and up-down (ud) pairs of positron countersin Figs. 6(b) and6(c) conﬁrms indeed a predominant caxis orientation of the local magnetic ﬁeld at the 4e muon site sincethe (fb)-signal has no detectable oscillatory component (ex-cept for a fast relaxing component of about 15%). Moreover,the amplitude of the high frequency oscillatory signal in the(ud) conﬁguration amounts to only about 30% as comparedto almost 55% in the o-AF state at x≈0.24 [see Fig. 3(f)]. This is roughly consistent with a large caxis oriented local magnetic ﬁeld at the 4e sites and a vanishing one at the 4fsites, especially since the potential depth of these 4e and 4fmay be slightly different and their population probability mayvary accordingly. The small fast relaxing signal (about 15%) in the (fb) conﬁguration in Figs. 6(b) and6(c), seems to be an indication that the magnetic moments are slightly canted along the c axis direction. Such a spin canting could be connected tothe small orthorhombic lattice distortion that was reported in 224515-5E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) Ref. [ 5] for corresponding BNFA crystals with x=0.3–0.36. The orthorhombic distortion removes the tetragonal symmetrywhereas it preserves the C2 rotation axis. Such a symmetryrestriction can induce a Dzyaloshinsky-Moriya interactions(DMI) that will lead to a canting of the SVC hedgehogstructure along the zaxis. The magnitude of this spin canting and the related in-plane component of the local ﬁelds will beproportional to the strength of the orthorhombic distortion.In Appendix C, we outline that among the different possi- ble orthorhombic lattice structures which can arise from theP4/mbm space group the only space groups with C2h rotating symmetry and with the C2 axis along the xyor xydirec- tions create the canted SVC “hedgehog” structures. Thesesymmetry constraints and the DMI allow for a superpositionof both the hedgehog-type SVC and the i-SCDW orders inthe lattice with orthorhombic distortions. An example of thesymmetry allowed and canted SVC hedgehog structures (e.g.,the orthorhombic distorted double- Qstructures) is shown in Appendix Cin Fig. 24. Note that such a magnetic degeneracy due to the effect of spin orbit interaction on quantum ﬂuctua-tions has been predicted in Ref. [ 44]. The assumption of a predominant, hedgehog-type SVC order at x≈0.32 and 0.34 also yields a reasonable estimate of the magnitude of the ordered magnetic moment. The cal-culations in Appendix Bpredict that ν 1is about 40% higher than for the single- QAF order (for the same magnitude of the magnetic moment). When comparing the values of theprecession frequencies with the one of the parent compoundatx=0, with ν μ≈29 MHz and a magnetic moment of about 1μB/Fe ion as reported in Refs. [ 45,46], we thus obtain an estimate of the magnetic moment of the hedgehog SVCphase at x≈0.32 of about 0 .5μ B/Fe ion. Likewise, the magnetic moment in the o-AF phase at x≈0.24 with νμ≈ 24.5 MHz at 40 K amounts to 0 .85μB/Fe ion. Figure 7shows the resulting doping dependence of the estimated magneticmoment and the corresponding Neel temperature which bothevolve continuously and tend to vanish around x=0.36–0.37. Notably, Fig. 6(d) reveals that the onset of SC at x≈0.32 is accompanied by a pronounced reduction of the precessionfrequency, from ν μ≈21 MHz at T/greaterorequalslantTc≈22 K to νμ≈ 15.5M H za t T<<Tc, and thus of the magnetic moment of the suspected SVC order. Figure 6(e) shows that there is also a clear increase of the relaxation rate below Tc≈22 K, which suggests that the magnetic order parameter becomesless homogeneous in the SC state. Nevertheless, the amplitudeof the magnetic signal does not show any sign of a suppressionbelow T c, suggesting that the magnetic order remains a bulk phenomenon even at T/lessmuchTc. Figure 8reveals that the suppression of the magnetic order due to the competition with SC becomes even more severe forthe BNFA crystal with x≈0.34. The magnetic signal in the ZF spectra develops here below T N≈38 K and the frequency and amplitude of the precession signal are rising rapidly tovalues of ν μ≈19.5 MHz and about 65%, respectively, at 30 K. These are characteristic signatures of a bulk AF orderthat seems to be of the SVC type for the same reasons asdiscussed above for the x≈0.32 crystal. Notably, the onset of SC below T c≈30 K at x≈0.34 gives rise to a much stronger suppression of the magnetic order than at x≈0.32. Not only the frequency is rapidly suppressed here but, as shown inFIG. 7. Doping dependence of the normalized values (to the ones atx=0) of the Neel-temperature, TN, the magnetic Fe moment estimated from μSR (derived as described in the text), and the spectral weight of the SDW peak from IR spectroscopy for the o-AF phase at x<0.3 and the suspected orthomagnetic SVC phase at x>0.3. Fig. 8(d), even the amplitude of the magnetic signal gets strongly reduced to about 25% below 20 K. This highlightsthat the magnetic order becomes spatially inhomogeneous FIG. 8. ZF- μSR data at x≈0.34. [(a) and (b)] ZF- μSR spectra in the magnetic state below TN≈38 K above and below Tc≈ 25 K, respectively. (c) Temperature dependence of the precessionfrequency and (d) of the normalized amplitudes of the oscillatory signal and the nonoscillatory but fast relaxing signal that both arise from regions with large magnetic moments. 224515-6MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) FIG. 9. TF- μSR spectra at 100G for x≈0.34. [(a)–(c)] TF- μSR spectra at T=45 K>TN≈38 K, TN>T=30 K/greaterorequalslantTc≈30 K, andT=5K/lessmuchTc, respectively. (d) Temperature dependence of the normalized amplitudes ( AfandAoff) of the magnetic signals and the nonmagnetic signal ( As) as described in the text. [(e) and (f)] Temperature dependence of the Gaussian relaxation rate σand the precession frequency νμof the nonmagnetic signal ( As) showing an enhanced relaxation and diamagnetic shift due to the superconduct-ing vortex lattice below T c≈30 K. with a large fraction of the sample reentering a paramagnetic state. A similar reentrance behavior of the AF order waspreviously only observed for BFCA crystals in the region veryclose to optimum doping [ 7]. This reentrance of large parts of the sample volume from a magnetic state at T∼T c≈30 K to a nonmagnetic state at T/lessmuchTc≈30 K is also evident from the 100G TF- μSR data in Fig. 9. The solid lines in Figs. 9(a)–9(c) show ﬁts with the function: P(t)=P(0)/bracketleftbig Afcos(γμBμ,ft+/Phi1f)e−λft +Aoff+Ascos(γμBμ,st+/Phi1s)e−1 2σ2t2/bracketrightbig . (2) The ﬁrst two terms describe the magnetic signal. The fast relaxing one with the normalized amplitude Afaccounts for the strongly damped or even overdamped oscillatory part andthe constant term with amplitude A offfor the nonoscillatory part of the magnetic signal that arises below TN. The third term represents the nonmagnetic signal with a Gaussian re-laxation rate, σ.T h ev a l u eo f σis much smaller than theone of λ fand is governed above Tcby the nuclear spins and below Tcby the SC vortex lattice. The frequency of this nonmagnetic signal is determined by the external magneticﬁeld, except for the diamagnetic shift in the SC state. Incontrast, the signal from the magnetic regions is governed bythe internal magnetic moments (the contribution of the 100GTF is considerably smaller) which yield a higher frequencyand a much faster relaxation such that this signal vanishing ona time scale of less than 0 .5μs. Figure 9(d) shows the temperature dependence of the nor- malized amplitudes of the magnetic signals A fandAoff, and of the nonmagnetic signal, As. It reveals that the magnetic volume fraction increases rapidly to about 80% at 30 Kand then decreases again below T cto about 35% at low temperature. Correspondingly, the nonmagnetic fraction isreduced to about 20% at 30 K and increases again to about65% well below T c. Figures 9(e) and 9(f) show that this nonmagnetic part exhibits clear signs of a SC response interms of a diamagnetic shift and an enhanced relaxation fromthe SC vortex lattice, respectively, that both develop belowT c≈30 K. From this Gaussian relaxation rate, the value of the in-plane magnetic penetration depth, λab, can be derived according to:σ 1.23[μs]=7.086×10−4 λ2 ab(nm−2), as outlined, e.g., in Refs. [ 47,48]. This yields a low temperature value of the mag- netic penetration depth of λab(T→0)≈350 nm and for the related SC condensate densityns m∗ ab=1 μ0e2λ2abofns m∗ ab=2.7× 1020m∗ ab me(cm−3), where μ0,e,m∗ ab, and meare the magnetic vacuum permeability, the elementary charge of the electron,and the effective band mass and bare mass of the electron,respectively. This value has to be viewed as an upper limit tothe penetration depth (lower limit to the condensate density),since the Gaussian function is symmetric in frequency spaceand thus does not capture the asymmetric “line shape” of thefrequency distribution due to a vortex lattice which has a tailtoward higher frequency. Also note that from these μSR data we cannot draw ﬁrm conclusions about the SC properties inthe magnetic regions for which the relaxation due to the SCvortex lattice is much weaker than the magnetic one. IV . INFRARED SPECTROSCOPY TheμSR study of the magnetic and superconducting prop- erties of the BNFA crystals presented in Sec. IIIhas been complemented with infrared spectroscopy measurements asshown in the following. Figure 10gives an overview of the temperature dependent spectra of the measured reﬂectivity R(ω) (upper panels) and of the obtained real part of the optical conductivity σ 1(ω)( l o w e r panels) for the crystals with x=0.22,0.24,0.26,0.32,and 0.34 that cover the different AF orders of the BNFA phasediagram in the normal and in the superconducting states (seeFig.1). The spectra are characteristic of a coherent electronic response, except for the ones at high temperature (300 K)some of which reveal a downturn of the conductivity towardzero frequency. The latter behavior is typical for so-called badmetals with strong electronic correlations [ 49]. Figure 11displays representative spectra of the infrared conductivity in the paramagnetic state at 120 K and inthe various AF states and shows their ﬁtting with a model 224515-7E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) FIG. 10. Temperature-dependent infrared optical response of BNFA crystals with 0 .22/lessorequalslantx/lessorequalslant0.34. The upper panels show the reﬂectivity spectra at different temperatures in the paramagnetic state and in the various AF phases. The lower panels display the corresponding spectra of the real part of the optical conductivity obtained from a Kramers-Kronig analysis, as described in Sec. II. function that consists of a sum of Drude, Lorentz, and Gaus- sian oscillators: σ1(ω)=2π Z0⎡ ⎣/summationdisplay jω2 pDjγDj ω2+γ2 Dj+/summationdisplay kγkω2S2 k/parenleftbig ω2 0k−ω2/parenrightbig2+γ2 kω2⎤ ⎦ +3/summationdisplay i=1SGie−(ω−ω0Gi)2 2γ2 Gi. (3) The ﬁrst term contains two Drude peaks that account for the response of the itinerant carriers, each described by a plasmafrequency ω pDjand a broadening γDjthat is proportional to the scattering rate 1 /τDj. The Lorentz oscillators in the second term with an oscillator strength Sk, resonance frequency ω0k, and linewidth γk, describe the low-energy interband transi- tions in the midinfrared region that are typically weakly tem-perature dependent [ 50]. The Gaussian oscillators in the third term with the oscillator strength S Gi, eigenfrequency ω0Gi, and linewidth γGi, represent the so-called pair-breaking peak that develops in the itinerant AF state due to the excitation of theelectronic quasiparticles across the gap of the spin densitywave (SDW) [ 42]. The sharp and much weaker feature around 260 cm −1corresponds to an infrared-active phonon mode, the so-called Fe-As stretching mode [ 51], that has not been included in the modeling. The upper row of panels in Fig. 11shows the spectra in the paramagnetic state at T>TNthat are described by a sum of two Drude bands, a broad and a narrow one, plus oneLorentzian oscillator. A similar model was previously usedto describe the spectra in the paramagnetic normal state ofcorresponding BKFA and BFCA crystals [ 5,52–58]. Based on the comparison with the electronic scattering rate of Ramanexperiments in the so-called A 1gandB2gscattering geometries of the incident and reﬂected laser beam [ 59], which allow one to distinguish between a less coherent response of the holelikebands and a more coherent one of the electronlike bands, weassign the broad Drude-peak in the infrared response to theholelike bands near the center of the Brillouin zone ( /Gamma1point) and the narrow Drude peak to the electronlike bands near theboundary of the Brillouin zone ( Xpoint), respectively. The lower panels of Fig. 11show corresponding spectra and their ﬁtting for the different AF phases in the normalstate above T c. The spectra are described by two additional Gaussian functions to account for the so-called SDW peak thatarises from the quasiparticle excitations across the SDW gap.The spectral weight of this SDW peak, shown by the greenshaded area, is a measure of the fraction of itinerant chargecarriers that contribute to the staggered magnetic moment ofthe SDW and thus is representative of the magnitude of the AForder parameter, see, e.g., Figs. 3 and 8 of Ref. [ 58]. Note that the total spectral weight deﬁned as SW ∞=/integraltext∞ 0σ1(ω)dω= π·n·e2 meis a conserved quantity where n,me, and eare the overall density, the mass and the amount of charge of theelectrons. For the present case, this so-called optical sumrule is also fulﬁlled, since the gain of partial spectral weightdue to the formation of the SDW peak is compensated bya corresponding loss of partial spectral weight of the Drudepeaks. Figure 12gives a full account of the obtained temperature dependence of the spectral weight (SW) of the SDW peakfor the series of BNFA crystals. Also shown, for comparison,are the corresponding data for the undoped parent compoundatx=0 that are adopted from Ref. [ 58]. The various AF and SC transition temperatures are marked with arrows in thecolor code of the experimental data. Figure 12reveals that the spectral weight of the SDW peak is continuously suppressedas a function of hole doping, x. Concerning the temperature dependence, for the sample with x=0.22, which exhibits an o-AF order and an orthorhombic ( C 2) structure below TN≈ 110 K, the SW of the SDW grows continuously below TN, without any noticeable anomaly due to the competition withsuperconductivity below T c≈15 K. For the samples with x= 0.24 and 0.26, the SW of the SDW peak exhibits a sudden, ad- ditional increase at the transition from the intermediate o-AF 224515-8MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) FIG. 11. Selected spectra of the optical conductivity in the paramagnetic and the different AF states and their ﬁtting using the model described in Eq. ( 3). The upper panels show the spectra in the paramagnetic state as described by the sum of a narrow and a broad Drude peak (dark and light blue lines) and a Lorentz oscillator that account for the free carriers and the low-energy interband transitions, respectively. The lower panels display the spectra in the various AF states, i.e., at x=0.22 in the o-AF state (left), at x=0.26 in the intermediate o-AF, and the i-SCDW states at low temperature, and at x=0.32 in the suspected SVC phase. The green shaded area shows the pair-breaking peak that arises from the excitations across the SDW gap and has been accounted for with a sum of two Gaussian functions. state with C2symmetry below TN,1≈85 K to the i-SCDW order with C4symmetry below TN,2≈40 K. A similar SW increase of the SDW peak in the i-SCDW state was previouslyreported in Ref. [ 26] for a corresponding BKFA crystal. For the BNFA samples at x=0.24 and 0.26, the infrared spectra show no sign of a bulklike superconducting response down tothe lowest measured temperature of 10 K. Note, however, thata bulk SC transition with T c=12.3Ka t x=0.24 is evident from the speciﬁc heat data in Fig. 5. Finally, at x=0.32 and 0.34 the SDW peak acquires only a rather small amount ofSW in the spin vortex crystal (SVC) state below T N≈45 and 40 K, respectively, that is assigned based on the μSR data as discussed in Sec. III. Nevertheless, as shown in Figs. 10and 12, a SDW peak can still be identiﬁed in the infrared spectra. Moreover, pronounced anomalies occur in the SC state belowT c≈20 and 25 K, respectively, where the SW of the SDW peak is reduced. This SC-induced suppression of the SDWpeak corroborates the μSR data which reveal a corresponding suppression of the ordered magnetic moment at x=0.32 andof the magnetic volume fraction at x=0.34 (see Figs. 6and 8, respectively). Figure 13shows for the example of the x=0.26 and 0.32 samples how the spectral weight and the scattering rate of thenarrow ( D1) and broad ( D2) Drude peaks are affected by the formation of the SDW. The scattering rate of the broad Drudepeak remains almost constant and therefore has been ﬁxedto reduce the number of ﬁt parameters. Figure 13(c) shows that the scattering rate of the narrow Drude peak is stronglytemperature dependent and exhibits a pronounced decreasetoward low temperature that is quite similar for both samples.The most signiﬁcant difference between the x=0.26 and x= 0.32 samples concerns the spectral weight loss of the Drude peaks in the AF state that occurs due to the SDW formation,see Figs. 13(a) and13(b) .A tx=0.26, the broad Drude peak shows a pronounced spectral weight loss in the AF statewhereas the spectral weight of the narrow Drude-peak remainsalmost constant or even increases slightly below T N. A similar behavior was reported for the undoped parent compound [ 60] 224515-9E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) FIG. 12. Temperature evolution of spectral weight of the SDW peak. Arrows mark the transitions into the o-AF state below TNat x=0 and 0.22, into the successive o-AF and i-SCDW states below TN,1andTN,2, respectively at x=0.24 and 0.26, and into the SVC phase below TNatx=0.32 and 0.34. and, recently for an underdoped Sr 1−xNaxFe2As2crystal that undergoes a corresponding transition from o-AF to i-SCDWorder [ 61]. To the contrary, for the x=0.32 sample in the assigned hedgehod SVC state, the major spectral weight lossinvolves the narrow Drude peak ( D 1), whereas the SW of the broad Drude peak ( D2) remains almost constant. Since the narrow Drude peak is believed to arise from the electron-like bands near X, and the broad Drude peak from the holelike bands near /Gamma1, the results in Figs. 13(a) and13(b) suggest that the o-AF and i-SCDW orders are giving rise to gaps primarilyon the holelike bands near /Gamma1, whilst the SVC order mostly causes a SDW gap on the electronlike bands near X. A different trend in the assigned SVC state, as compared to the one in the o-AF and the i-SCDW states, is also evidentfrom the doping dependence of the frequency of the SDWpeak. Figure 14(a) shows a comparison of the optical con- ductivity spectra in the AF state at 25 K and Fig. 14(b) the evolution of the Gaussian ﬁts of the SDW peak as detailed inFig.11. Whereas the SW of the SDW peak decreases contin- uously with hole doping (as was already discussed above andshown in Fig. 11), the peak frequency also decreases at ﬁrst in the o-AF and i-SCDW states, from about 700 cm −1atx= 0.22 to about 400 cm−1atx=0.26, but then increases again to about 600 cm−1in the SVC state at x=0.32 and 0.34. Since the SDW peak energy is expected to be proportional tothe magnitude of the SDW gap, this anomaly suggests that theaverage magnitude of the SDW gap in the SVC state exceedsthe one in the i-SCDW state. The combined evidence fromour infrared data thus suggests that the SVC order at x=0.32 and 0.34 involves an electronlike band around the Xpoint that has quite a large SDW gap but only a weak contribution tothe optical spectral weight. The latter point can be explainedeither in terms of a very low concentration or a large effectivemass of the charge carriers of this band. Additional information about the structural changes in the different AF phases has been obtained from the tempera-ture and doping dependence of the infrared-active phononFIG. 13. Temperature dependence of the Drude parameters at x=0.26 and 0.32. [(a) and (b)] Normalized spectral weight (with respect to the one at 150 K) of the broad Drude peak ( D2) and the narrow Drude peak ( D1), respectively. (c) Temperature dependence of the scattering rate of the narrow Drude peak, /Gamma1D1. mode around 260 cm−1that is summarized in Fig. 15.I t was previously reported for BKFA that this in-plane Fe-Asstretching mode develops a side band at a slightly higherenergy in the i-SCDW state [ 26,58]. This new feature was explained in terms of an enlarged unit-cell and a subsequentBrillouin-zone folding due to the presence of two inequivalentFe sites (with and without a static magnetic moment) in thei-SCDW state [ 26]. Figure 15conﬁrms that a corresponding phonon side band occurs in BNFA at x=0.24 and 0.26 in 224515-10MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) FIG. 14. (a) Doping dependence of the optical conductivity in the AF state at 25 K. A vertical offset has been added for clarity.(b) Evolution of the SDW peak as ﬁtted with the Gaussian functions that are described in Eq. ( 3)a n ds h o w ni nF i g . 11. The peak intensity decreases continuously as a function of hole doping, whereas thepeak frequency exhibits a partial recovery in the suspected SVC state atx=0.32 and 0.34. terms of an additional peak around 275 cm−1that develops right below TN,2≈40 K. Notably, such a satellite peak is not observed in the assigned SVC phase at x=0.32 and 0.34. This ﬁnding conﬁrms that the enlargement of the unit cell andthe subsequent BZ folding is unique to the i-SCDW order andemphasizes the distinct nature of the SVC order at x=0.32 and 0.34. Finally, we discuss how the onset of superconductivity affects the spectra of the infrared conductivity in the presenceof the different AF orders. Figure 16shows the corresponding changes to the optical conductivity due to the formation ofthe SC gap(s) for the samples in the o-AF state at x=0.22 and in the assigned SVC state at x=0.32 and 0.34. For the samples in the i-SCDW state at x=0.24 and 0.26 no sign of the formation of a SC gap and a related delta-function at zerofrequency due to a SC condensate has been observed down tothe lowest measured temperature of 10 K [see Figs. 10(c) – 10(f) ]. As shown in Fig. 5, this is despite of a bulk SC transition at T c≈12 K at x=0.24 measured with the speciﬁc heat. Our infrared data thus reﬂect a strong suppression ofthe superconducting response due to the competition with thei-SCDW order that is more severe than in the o-AF and theassigned SVC states. Finally, note that clear signatures of aSC energy gap have very recently been reported for a similarBNFA sample for which T cwas somewhat higher and the measurements were performed to a lower temperature of 5 K[61]. The upper panels of Fig. 16show the spectra in the normal state slightly above T cfor which the ﬁtting has already been shown in Fig. 11. The lower panels display the corresponding spectra and their ﬁtting in the SC state. Here the opticalconductivity at low frequency (below 50 cm −1atx=0.22, 120 cm−1atx=0.32 and 100 cm−1atx=0.34) is strongly suppressed due to the opening of the superconducting energygap(s). This SC gap formation has been accounted for usinga Mattis-Bardeen-type model that allows for isotropic gaps ofdifferent magnitude on the narrow and the broad Drude-bands. For the x=0.22 crystal, for which the SC state coexists with a strong o-AF order, there are clear signs of the SC gapformation below T c≈15 K, i.e., Fig. 16(b) reveals a strong suppression of the optical conductivity toward low frequency.The SC gap edge is also evident from the bare reﬂectivityspectrum in the inset of Fig. 10(a) . The obtained gap energies amount to 2 /Delta1 SC≈4.4 and 5.2 meV for the broad and narrow Drude bands, respectively, and ratios of 2 /Delta1SC/kBTc≈2.87 and 3.35 that compare rather well with the prediction of theweak coupling BCS theory of 2 /Delta1 SC/kBTc=3.54. A strong increase of the SC gap energy is observed for the samples in the SVC state at x=0.32 (Fig. 16,m i d - dle panel) and x=0.34 (Fig. 16, right panel) for which the overall shape of the SC spectra is quite similar to theone of optimally doped BKFA [ 52,58,62–64]. Here, 2 /Delta1 SC for the narrow and broad Drude bands amounts to 19.8 and 13 meV at x=0.32 and 30 and 12 meV at x=0.34, respec- tively (see also Table I). Similar to BaFe 2−xCoxAs2(BCFA) and BKFA [ 54,55,57,61] and also CaKFe 4As4(BCKFA) [ 65], the larger SC gap is assigned to the narrow Drude peak, whichsupposedly originates from the electronlike bands near the X point of the Brillouin zone. Finally, we derived the SC plasma frequency, /Omega1 2 pS, and the related ratio of the condensate density to the effective bandmass, ns m∗ ab, from the analysis of the missing spectral weight using the Ferrell-Glover-Tinkham (FGT) sum rule: /Omega12 pS=Z0 π2/integraldisplayωc 0+[σ1(ω,T∼Tc)−σ1(ω,T/lessmuchTc)]dω, (4) where the upper cutoff frequency ωchas been chosen such that the optical conductivity in the normal and SC states is almostidentical above ω c. Alternatively, the superﬂuid density has been determined from the analysis of the inductive response in the imaginarypart of the optical conductivity, σ 2, according to ns m∗ ab=/Omega12 pS=Z0 2πωσ2S(ω). (5) Here the contribution of the regular response to σ2, due to the excitation of unpaired carriers, has been subtracted asdescribed in Refs. [ 66,67]. Both methods yield consistent values of the SC condensate density, ns m∗ ab, and the related magnetic penetration depth, λ, that are listed in Table I. 224515-11E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) FIG. 15. Temperature and doping dependence of the in-plane Fe-As stretching phonon mode near 260 cm−1. A satellite peak is visible here in the i-SCDW state below TN,2≈40 K at x=0.24 and below TN,2≈42 K at x=0.26. No sign of such a satellite peak is observed in the other AF states, i.e., in the o-AF state at x=0.22 and the suspected SVC state at x=0.32 and 0.34. V . DISCUSSION AND SUMMARY We have performed a combined μSR- and infrared spec- troscopy study of the magnetic part of the phase diagram ofthe hole doped BNFA system. We have conﬁrmed that the so-called double- QAF state with an inhomogeneous spin-charge density wave (i-SCDW) order exists in a sizable doping rangewhere it persists to the lowest measured temperature, i.e., evenbelow T c. This is different from BKFA where the i-SCDW or- der exists only in a rather narrow doping regime and exhibits areentrance to an o-AF state at low temperature [ 24,26,27,58]. Otherwise, we observed the same signatures of the i-SCDWstate as in BKFA. This concerns the reorientation of the spinsfrom an in-plane direction in the o-AF state to an out-of-planeone in the i-SCDW state. We also observed a satellite peakof the infrared-active Fe-As stretching phonon mode, whichsignals a folding of the Brillouin zone due to an enlargedunit cell in the i-SCDW state. In the infrared spectra, at thelowest measured temperature of 10 K, no sign of a bulklikeSC response has been seen with infrared spectroscopy in the i-SCDW state of the BNFA crystals with x=0.24 and 0.26, for which a bulk SC transition is evident from speciﬁc heat.This suggests that the superconducting response is stronglysuppressed by the competition with i-SCDW order. We also obtained evidence for a new type of t-AF state that is likely a hedgehog-type spin-vortex-crystal (SVC) order.This new AF phase shows up at a higher hole doping level thanthe i-SCDW phase and persists until the static magnetism isfully suppressed at optimum doping. This additional magneticphase in the BNFA phase diagram was ﬁrst discovered withthermal expansion measurements where it shows up in termsof a very small orthorhombic distortion [ 5]. Accordingly, it has been interpreted in terms of an o-AF order that is eithervery weak, strongly incommensurate, or inhomogeneous. Tothe contrary, our μSR data establish that this AF state is bulklike, more or less commensurate and has a surprisinglylarge magnetic moment (at least at T>T c). Due to its almost TABLE I. Values of the SC gaps of the narrow and broad Drude bands and of the SC plasma frequency and magnetic penetration depth as obtained from the optical data in the SC state for the samples with x=0.22, 0.32, and 0.34. x (Na) 2 /Delta1SC narrow (meV) 2 /Delta1SC narrow/kBT 2/Delta1SC broad(meV) 2 /Delta1SC broad/kBT /Omega12 pS(cm−2) λ(nm) 0.22 5.21 3.35 4.46 2.87 4 .9×107227 0.32 19.84 10.46 13.14 6.93 6 .2×107202 0.34 30.75 14.27 12.65 5.87 5 .4×107216 224515-12MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) FIG. 16. Selected spectra of the optical conductivity and their ﬁtting slightly above and well below Tcin the o-AF state at x=0.22 (left) a n di nt h eS V Cs t a t ea t x=0.32 (middle) and x=0.34 (right). Note that no sign of a SC gap formation has been observed in the corresponding spectra of the samples with x=0.24 and 0.26. In the normal state at T≈Tc, the experimental spectra (black line) have been ﬁtted with two Drude-terms (blue solid lines), a Lorentzian (orange line) and two Gaussian peaks for the SDW pair-breaking peak as discussed in the text. In the SC state at T/lessmuchTc, a Mattis-Bardeen-type isotropic gap function has been added to each Drude-band. The values of the obtained SC gap energies are listed in Table I. tetragonal structure and since the μSR data reveal magnetic moments that are rather large and oriented along the FeAsplanes, we have assigned this new AF order to an orthomag-netic double- Qstate, in particular, to the hedgehog-type spin vortex crystal (SVC) structure. This SVC state was previouslyonly observed in the K,Ca-1144 structure where it is believedto be stabilized by the reduced disorder and/or the breakingof the glide-plane symmetry of the FeAs layers due to thealternating layers of Ca and K ions [ 35,36]. It is therefore in- teresting that this kind of SVC order also occurs in the presentBNFA system for which the Na and Ba ions are randomlydistributed. Another remarkable feature of this SVC orderis its very strong competition with superconductivity whichleads to a large reduction of the magnetic moment (at x= 0.32) and even of the magnetic volume fraction (at x=0.34). A similarly large suppression of magnetic order due to theonset of SC was so far only observed in BFCA crystals closeto optimum doping for which a very weak incommensurateo-AF order exhibits a reentrance into a nonmagnetic statebelow T c[21]. Another interesting aspect of our present work emerges from the comparison of the doping evolution of the magnitudeof the ordered magnetic moment as deduced from the localmagnetic ﬁeld in the μSR experiment and the SW of the SDW peak in the infrared spectroscopy data. The trends can be seenin Fig. 7which compares the doping evolution of the AF mo- ment, normalized to the one of the undoped parent compound,as seen with μSR (which probes the total ordered magneticmoment) and infrared spectroscopy (which is only sensitive to the itinerant moment). The solid blue symbols show thevalue of the AF order parameter as obtained from the localmagnetic ﬁeld at the muon site. The solid orange symbolsshow the corresponding values of the SW of the SDW peak. Inboth cases, the amplitude of the magnetic moment decreasescontinuously with doping, but the decrease is considerablystronger for the itinerant moments deduced from the infrareddata than for the total magnetic moment seen with μSR. This might indicate that the ordered magnetic order has a mixedcharacter with contribution from itinerant and from localizedmoments. The different trends of the optics and μSR data thus could be explained if the magnetic moments are more stronglylocalized as the hole doping increases. An alternative, and toour opinion more likely explanation is in terms of a changeof the effective mass of the itinerant charge carriers that aregapped by the SDW. The very small SW of the SDW peak inthe SVC state, as compared to the large local magnetic ﬁeld intheμSR experiment, thus implies that the SDW gap develops on a ﬂat band with a rather large effective mass. Note that sucha scenario, that the SDW develops on different parts of theFermi-surface in the SVC state, as compared to the o-AF andi-SCDW states, is consistent with the data in Fig. 13which show that the SDW peak obtains a major part of its SW fromthe narrow Drude peak, rather than from the broad one, as inthe o-AF and i-SCDW states. This scenario could be probed,e.g., by future ARPES studies on such BNFA crystals in thei-SCDW and SVC states. 224515-13E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) ACKNOWLEDGMENTS Work at the University of Fribourg was supported by the Schweizerische Nationalfonds (SNF) by Grant No. 200020-172611. K.W. acknowledges funding from the Alexander vonHumboldt Foundation. K.W. acknowledges valuable discus-sions with Frédéric Hardy. We thank Christof Neuruhrer andBernard Grobety for their technical assistance in performingthe EDX measurements. APPENDIX A: MUON SITE CALCULATION The space group symmetry of Ba 1−xKxFe2As2(BKFA) and Ba 1−xNaxFe2As2(BNFA) in the paramagnetic phase is I4/mmm with one formula unit ( Z=1) in the primitive cell. The Ba ions reside in the 1a – position (0,0,0), As in the 2e –position (0 ,0,zAs) and Fe in the 2d – position (0 ,1/2,1/4). Note that the crystallographic unit cell differs from theprimitive cell which is built by primitive translations: a 1= (−a/2,b/2,c/2)=(−τ,τ,τ c), a2=(a/2,−b/2,c/2)= (τ,−τ,τ c),a3=(a/2,b/2,−c/2)=(τ,τ,−τc). In the following, we analyze the position of the muon stoppingsites for a K content of x=0.2465 with the structural room temperature data: a=b=3.9343 Å, c=13.2061 Å, and z As=0.35408 Å. We assume here that these muon sites do not strongly change when the K content is varied or when Kis replaced by Na. We used a modiﬁed Thomas Fermi approach [ 68] that al- lows a direct determination of the self-consistent distributionof the valence electron density from which the electrostaticpotential can be restored. The local, interstitial minima of thiselectrostatic potential are identiﬁed as muon stopping sites. For the same purposes, we performed more elaborated ab initio calculations within the framework of density func- tional theory (DFT). We applied the all-electron full-potentiallinearized augmented plane wave method ( ELK code) [ 69] with the local spin density approximation [ 70] for the ex- change correlation potential and with the revised generalizedgradient approximation of Perdew-Burke-Ernzerhof [ 71]. The calculations were performed on a 9 ×9×6 grid which corre- sponds to 60 points in the irreducible Brillouin zone. In bothapproaches we used a supercell 2 a×2b×cand supposed x=0.25 (e.g., Ba 0.75K0.25Fe2As2). This allows one to explic- itly incorporate K ions which were positioned in the supercellat coordinates K(1) - ( a,b,0) and K(2) - (3 /2a,3/2b,1/2c). The DFT and modiﬁed Thomas Fermi approaches give almost the same answers. We observed three possible typesof muon sites. Two of them are located on the line along thecdirection connecting the nearest Ba or K and As ions at the coordinates (0 ,0,z μ) with zμ=0.191 for Ba and zμ= 0.170 for K. In the I4/mmm setting these muon sites have a 2e – local point symmetry (4 mm), i.e., the same as the As ions. We have veriﬁed that the dipolar ﬁelds from a givenmagnetic structure of the Fe moments have nearly the samemagnitudes at these two positions. Accordingly, in the dipolarﬁeld calculations we discuss only one type of muon stoppingsite. The third muon site is located in the Ba abplane close to the line connecting the As-As ions along the cdirection. In the I4/mmm setting it has a rather high 4j – local point symmetry(m2m) at the coordinates (0.4,0.5,0). Its electrostatic potential is roughly 20% less than the potential of the previous twosites. Accordingly, this site should be less populated in theμSR experiment. The probability of the occupation of this secondary site, as compared to the one of the majority site,we calculate to be 0.24 which agrees rather well with theexperimental amplitude ratio of A os 2/Aos 1≈0.2( s e eF i g .2o f the Ref. [ 27]). The qualitative changes of the local dipole ﬁelds on this minority site at the o-AF to t-AF transitionare very similar to the ones on the majority muon sites.Accordingly, in the following and in the paper we do notfurther discuss this minority muon site and focus instead onthe changes of the local dipole ﬁeld on the majority muon site. APPENDIX B: CALCULATION OF THE DIPOLAR FIELD AT THE MUON SITE To unify the description of the possible magnetic struc- tures in the tetragonal phase of Ba 1−xKxFe2As2(BKFA) and Ba1−xNaxFe2As2(BNFA) we used the space group P4/mbm N127 that is the subgroup of index 4 of the parent groupI4/mmm N139. This choice is dictated by the expected four- fold increase of the magnetic unit cell as compared to theparent I4/mmm primitive cell that is caused by the lowering of the translation symmetry. The P4/mbm subgroup has the same origin as the parent group I4/mmm and the basis ( a,b,c) that is rotated by 45 ◦in the abplane as compared to the I4/mmm basis ( a/prime,b/prime,c/prime) with a=b=2a/primeandc=c/prime.I nt h e P4/mbm setting the eight Fe atoms in the unit cell are in t h e–8 k( x,x+1/2,z) position. The 2e position of the As atoms and of the muon site in the I4/mmm notation are divided in the P4/mbm setting into the 4e – (0 ,0,z1(As/μ)) and 4f - (0 ,1/2,z2(As/μ)) positions. Respectively, the 4j position of the third muon site in the I4/mmm notation are divided in the P4/mbm setting into the 8j – ( x,y,1/2) and two 4g - ( x,x+1/2,0) positions. The primitive cell of BKFA in theP4/mbm setting is shown in Fig. 17. The symmetry consideration of the possible 2k- and 1k- (or double- Qand single- Q) magnetic structures is based on the so called representation analysis of the magnetic degreesof freedom that are real and located on the magnetic ionsand that are virtually assigned on the muon stopping sites[72–74]. The magnetic degrees of freedom, for a set of atoms at a given Wyckoff position, form a magnetic representationwhich is reducible and can be decomposed into irreduciblerepresentations (IR). The possible magnetic structures canbe presented in terms of a linear combination of magneticmoments L, which transform under the symmetry operations as basic functions of a given IR. This is in accordance withthe Landau concept that only one IR is realized at a phasetransition for which Lis a nonzero order parameter in the low symmetry phase. Purely based on symmetry arguments one can make the following strict predictions for the local magnetic ﬁeld that isseen in a zero-ﬁeld μSR experiment. The complex magnetic structure does not give rise to a ﬁnite magnetic ﬁeld at themuon site if the IR of its order parameter does not enterinto the decomposition of the magnetic representation for themuon site. 224515-14MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) FIG. 17. Sketch of the unit cell of Ba 1−xKxFe2As2in the tetrag- onal subgroup P4/mbm of the space group I4 /mmm. Atoms and muon stopping sites are in the positions; Ba 1- 2a (0,0,0), Ba 2–2 c (0,1/2,1/2), As 1/μ1–4 e( 0 ,0,z1(As/μ) with z1(As)=0.35408 andz1(μ)=0.188, As 2/μ2–4 f( 0 ,1/2,z2(As/μ)) with z2(As)= 0.14592 and z2(μ)=0.312, Fe – 8k ( x,x+1/2,z) with x=1/4, z=1/4,μ3- 4g (0.4,0.9,0) and 4g – (0.1,0.6,0) and 8j – (0.1,0.1,0.5). The enumeration of the Fe and μ1andμ2sites is indicated. This circumstance is illustrated below for the possible magnetic structures in the tetragonal phase of BKFA. For thefollowing analysis it is important to note that the lowering ofthe translation symmetry in the 2k structures is already ac-counted for by using a four times enlarged primitive unit cell.In the P4/mbm setting, thus we can perform the symmetry treatment for the Fe- and muon-site magnetic representationsfor the propagation vector K 0=(0,0,0). To represent the order parameters of the respective 2k- magnetic structures, which can arise in the I4/mmm setting with the propagation vectors k1=(1/2,1/2,0) and k2= (−1/2,1/2,0), we introduce the following linear combina- tions Lof the magnetic iron moments in the P4/mbm setting with K0=(0,0,0): /vectorF(±)=1/8[(/vectorm1+/vectorm2+/vectorm3+/vectorm4) ±(/vectorm5+/vectorm6+/vectorm7+/vectorm8)]; /vectorL(±) 1=1/8[(/vectorm1+/vectorm2−/vectorm3−/vectorm4) ±(/vectorm5+/vectorm6−/vectorm7−/vectorm8)]; /vectorL(±) 2=1/8[(/vectorm1−/vectorm2+/vectorm3−/vectorm4) ±(/vectorm5−/vectorm6+/vectorm7−/vectorm8)];/vectorL(±) 3=1/8[(/vectorm1−/vectorm2−/vectorm3+/vectorm4) ±(/vectorm5−/vectorm6−/vectorm7+/vectorm8)]. (B1) The magnetic order parameters Lconsist of the Fourier com- ponents of the magnetic propagation vector K0, in terms of the sublattice magnetic moments mαwithα=1–8. Similarly one can introduce linear combinations of the K0- Fourier components of the magnetic ﬁelds BI,II α(α=1–4) at the muon positions with 4e and 4f site symmetry that are enumerated byI and II, respectively. The respective staggered magnetic ﬁeldsat these muon sites have the form: /vectorF (I,II)=1 4/parenleftbig/vectorB(I,II) 1+/vectorB(I,II) 2+/vectorB(I,II) 3+/vectorB(I,II) 4/parenrightbig ; /vectorL(I,II) 1=1 4/parenleftbig/vectorB(I,II) 1+/vectorB(I,II) 2−/vectorB(I,II) 3−/vectorB(I,II) 4/parenrightbig ; /vectorL(I,II) 2=1 4/parenleftbig/vectorB(I,II) 1−/vectorB(I,II) 2+/vectorB(I,II) 3−/vectorB(I,II) 4/parenrightbig ; /vectorL(I,II) 3=1 4/parenleftbig/vectorB(I,II) 1−/vectorB(I,II) 2−/vectorB(I,II) 3+/vectorB(I,II) 4/parenrightbig .(B2) The quantities deﬁned in Eqs. ( B1) and ( B2) can serve as the basic functions of the irreducible representations of theP4/mbm group with propagation vector K 0=(0,0,0). The attribution of these basic functions to the IR of the tetragonalgroup P4/mbm is as shown in Table II. The possible eight noncollinear spin vortex crystal structures which are allowedby the double- Qmagnetic order are described by the τ 1−τ8 irreducible representations. The following examples illustrate how to read the data of Table II. The magnetic structures which can be realized with the iron order parameters of a given IR give rise to staggeredﬁelds at the muon sites that transform by the same IR. Forexample, the magnetic structure which transforms accordingto the IR τ 5−B1gconsists of the two order parameters L(−) 3x− L(−) 1yandL(+) 2z. According to Table II, both order parameters L(−) 3x−L(−) 1yandL(+) 2zdo not create ﬁnite dipolar ﬁelds at the 4e muon stopping sites. At the same time, at the 4f muonstopping sites they both create dipolar ﬁelds that are directedalong the caxis and have the same staggered structure L II 2z. This is a strict result if we take the iron coordinates in theform Fe – 8k ( x,x+1/2,z). However, there is the starting symmetry I4/mmm which we can reproduce by taking the iron coordinates as x=1/4,z=1/4 so that we get 8k (1/4,3/4,1/4). This additional, internal symmetry leads to the disappearance of the magnetic ﬁelds at some of the muonstopping sites. The magnetic structures (order parameters), which do not give rise to a ﬁnite magnetic ﬁeld at the muon site for x=1/4, z=1/4, are marked in “yellow.” The “pink” color denotes the magnetic structures (order parameters) that cannot be detectedbyμSR for the given 4e and 4f muon stopping sites, even for an arbitrary choice of the xandzcoordinates in Fe – 8k ( x,x+ 1/2,z). All of these structures are illustrated in Fig. 18. Note that structures marked in “yellow” can give rise to small, ﬁnite ﬁelds at the muon sites in the case of small, staticdeviations of the iron coordinates from the values x=1/4 and z=1/4. In this case, the local ﬁelds and the resulting μSR precession frequencies will be more or less proportional tothe amplitude of the deviations. Below we summarize the outcome of the dipole ﬁeld calculations for the magnetic order parameters with AF order 224515-15E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) TABLE II. Symmetry of the order parameters of the possible Fe-based magnetic phases and the symmetry and magnitude of the respective staggered magnetic ﬁelds from Eq. ( B2) at the muon sites in the tetragonal phase of Ba 1−xKxFe2As2inP4/mbm setting for the magnetic propagation vector K0=(0,0,0). for the case Fe – 8k (1 /4,3/4,1/4). The magnetic ﬁelds are given in units of MHz, corresponding to the μSR precession frequency, νμ=γμ 2πBμ, and the magnetic order parameters [linear combinations from Eq. ( B1)] in units of μB. The ﬁelds at the 4e muon sites with coordinates (0.0,0.0,0.1880) are ⎛ ⎝Bx By Bz⎞ ⎠=⎛ ⎝28.52 0 0 02 8 .52 0 00 −57.04⎞ ⎠⎛ ⎜⎝F(−) x F(−) y F(−) z⎞ ⎟⎠ +⎛ ⎝00 0 003 7 .36 03 7 .36 0⎞ ⎠⎛ ⎜⎜⎝L(−) 1x L(−) 1y L(−) 1z⎞ ⎟⎟⎠ +⎛ ⎝05 9 .96 0 59.96 0 0 00 0⎞ ⎠ ×⎛ ⎜⎝L(−) 2x L(−) 2y L(−) 2z⎞ ⎟⎠+⎛ ⎝00 3 7 .36 000 37.36 0 0⎞ ⎠⎛ ⎜⎜⎝L(−) 3x L(−) 3y L(−) 3z⎞ ⎟⎟⎠.(B3)The ﬁelds at the 4f muon sites with coordinates (0.5,0.0,0.312) are ⎛ ⎝Bx By Bz⎞ ⎠=⎛ ⎝−28.52 0 0 0 −28.52 0 00 5 7 .04⎞ ⎠⎛ ⎜⎝F(−) x F(−) y F(−) z⎞ ⎟⎠ +⎛ ⎝00 0 00 −37.36 0−37.36 0⎞ ⎠⎛ ⎜⎝L(−) 1x L(−) 1y L(−) 1z⎞ ⎟⎠ +⎛ ⎜⎝05 9 .96 0 59.96 0 0 00 0⎞ ⎟⎠⎛ ⎜⎝L(−) 2x L(−) 2y L(−) 2z⎞ ⎟⎠ +⎛ ⎝00 3 7 .36 000 37.36 0 0⎞ ⎠⎛ ⎜⎝L(−) 3x L(−) 3y L(−) 3z⎞ ⎟⎠. (B4) In the following Fig. 18, we show the magnetic structures which do not give rise to a magnetic ﬁeld at the muon site andthus to a ﬁnite μSR precession frequency. These structures are therefore not compatible with our experimental data in thet-AF state. Interestingly, all of them belong to so called loop-type SVC structures. In Fig. 19, we show the noncollinear double- Qstructures with in-plane oriented magnetic moments which create a ﬁnitedipolar magnetic ﬁeld at the muons sites of tetragonal BKFAor BNFA. All of them belong to the hedgehog-type SVC 224515-16MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) FIG. 18. SVC loop double- Qmagnetic structures in P4/mbm setting which preserve the C4 symmetry and do not create a magnetic dipole ﬁeld at the muon sites. Only the iron atoms are shown. The structures in (a)–(c) exhibit a FM order along the caxis, the ones in (d)–(f) a corresponding AFM order. structures. The indicated μSR precession frequencies have been obtained using Eqs. ( B2) and ( B3) under the assumption that each Fe ion has a magnetic moment of 1 μB[45,46]. These local ﬁelds are larger than the ones calculated for thesingle- Qmagnetic order in the o-AF state (see below and Fig. 21) as well as for the double- Qmagnetic order of the tetragonal i-SCDW state (see Fig. 20). For the double- Qmagnetic structures shown above, each Fe ion has the same magnetic moment which is assumed to FIG. 19. SVC hedgehog double- Qstructures with in-plane ori- ented moments in the tetragonal phase which create a ﬁnite magneticdipole ﬁeld along the caxis at the muons sites. Shown are only the iron atoms. (a) and (b) show the structures with AFM order along thecaxis, (c) and (d) the corresponding structures with FM or- der. The indicated μSR precession frequencies are calculated using Eqs. ( B3)a n d( B4). They are very similar and thus are likely within the error bar of a typical μSR experiment. FIG. 20. Inhomogeneous, double- Qmagnetic structure i-SCDW, L(−) 1z+L(−) 3z, with alternating zero and nonzero magnetic moment at the iron sites with P C42/ncm magnetic group symmetry according to Ref. [ 30]. The indicated μSR precession frequency has been calculated using Eqs. ( B3)a n d( B4). Shown are only the iron atoms. This is the double- Qmagnetic structure that is compatible with our μSR data. FIG. 21. Magnetic structures and their order parameters in the orthorhombic state. All phases preserve the same pattern (type) ofthe exchange interactions. (a) and (b) show the two domain state of the stripelike AF order that is realized in the orthorhombic phase; (c) a spin rotated phase with an arbitrary rotation angle α; (d) the pure out-of-plane magnetic order that has been suggested in Ref. [ 23] as the magnetic structure in the tetragonal AF phase. This structure breaks the C4 symmetry of the tetragonal crystal structure. Note that theμSR precession frequencies remain the same in accordance with Eqs. ( B3)a n d( B4), under the continuous rotation from the pure in- plane to the pure out-of-plane structure. 224515-17E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) amount to 1 μB. However, there exists also the possibility of a so-called inhomogeneous double- Qmagnetic structure for which the magnetic moment becomes zero for half of theFe sites. It is described by the P C42/ncm magnetic group symmetry and preserves the C4 symmetry. In our P4/mbm setting, this structure corresponds to the linear combinationL (−) 1z+L(−) 3z.I nt h e I4/mmm setting, it is described by the linear combination of the order parameter ηz(k1)+ηz(k2) which belong to different arms of the K13-star. This structureis shown below in Fig. 20. The calculations show that it yields a moderate μSR precession frequency that is lower than the one in the orthorhombic phase (see below and Fig. 21)i n agreement with the experimental data. In contrast to othermagnetic phases, the coexistence of the nonmagnetic ( S=0) and magnetic ( S/negationslash=0) sites may indicate an alteration of the iron spin states of the neighboring ions. A large variationof the iron spin state is indeed not uncommon to the parentcompounds of the iron superconductors for which the mag-netic moment varies from the high spin state with S=2 and a moment of 3 .5μ B/Fein Rb 2Fe4Se5to the low spin state with S=0 in FeSe. Finally we discuss the so-called single- Qmagnetic struc- tures which require an orthorhombic structure since theybreak the C4 symmetry. From the magnetic symmetry point of view, the symmetry reduction that takes place at the transitionfrom the paramagnetic tetragonal I4/mmm 1 /primephase to the magnetic orthorhombic CAmca (orFCmm/primem/prime) phase can be described as a condensation of the magnetic order parameterη xy(k1)o rη¯xy(k2)i nt h e I4/mmm setting. Here two order parameters with different translation symmetry form two dif-ferent orthorhombic domains. In our P4/mbm setting for the paramagnetic phase, these two domains of the orthorhombicmagnetic phase can be described as a condensation of the L (−) 3x andL(−) 1yorder parameters, respectively. The structure with the out-of-plane direction of the magnetic moments in the tetrag-onal AF phase can be obtained by a continuous rotation of themagnetic moments in the acplane. The respective magnetic structures are shown in Fig. 21. Note that in accordance with Eqs. ( B2) and ( B3) all of them give rise to the same μSR precession frequency which for a Fe moment of 1 μ Bamounts to 32.3 MHz. At last we mention the μSR precession frequency at the third muon site for the relevant magnetic structures under theassumption of a magnetic moment of 1 μ B/Fe. In the o-AF state [for the structure shown in Fig. 21(b) ] it amounts to about 8.9 MHz; whereas in the t-AF state (for the structureshown in Fig. 20) it is reduced to about 6.5 MHz. Moreover, the direction of the ﬁeld at this third muon site is parallel tothecaxis in the o-AF state and parallel to the abplane in the t-AF phase, similar to local magnetic ﬁeld at the main muonsite. APPENDIX C: IMPACT OF WEAK ORTHORHOMBIC DISTORTIONS ON THE SVC STRUCTURE AND ITS M A N I F E S T A T I O NI NT H E μSR EXPERIMENT Here we address the question whether our observation of a small in-plane component of the local magnetic ﬁeldat the muon sites ( ∼15%) can be connected with the small orthorhombic lattice distortions that are detected in the phase FIG. 22. (Left) Sketch of the orthorhombic distortion of the SVC hedgehog structure described by a superposition of τ1andτ5SVC order parameters [cos δ(L(−) 3x+L(−) 1y)+sinδ(L(−) 3x−L(−) 1y)] cosθand a small canting L(+) 2zalong the c direction. Here, δis the deviation angle of the Fe 1magnetic moment from the (110) direction and θis the out-of-plane deviation angle, both angles should be proportional to the value of lattice distortion. (Right) The lattice and SVC-state distortions are shown for the abplane cross-section (with \|a\|/negationslash=\|b\|). diagram of the Ref. [ 5] for Na contents in the range of x=0.32 and 0.34. The orthorhombic distortions remove the tetragonal symmetry but preserve the C2 axis. The symmetryrestrictions can induce a Dzyaloshinsky-Moriya interaction(DMI) according to which the SVC hedgehog structure maybe distorted and particularly may get canted along the caxis. Respectively an in-plane component of the local magneticﬁeld might arise at the muon stopping sites. In the following, we apply pure symmetry arguments to account for the impact of the weak orthorhombic distortionson the SVC magnetic structure and its manifestation in theμSR data. Orthorhombic distortions lower the initial D4hro- tation symmetry into subgroups with D2hrotation symmetry that are the highest subgroups without a C4 axis. In both subgroups, the C 2zaxis coincides with the previous C4 axis whereas the others C2 axes are directed: in the D2h(1) case along the previous 2 xand 2 yaxes; or in the D2h(2) case along the previous 2 xyand 2¯ xyaxes. Under the symmetry restriction of the D2h(1) case, the symmetry operations permute all eight Fe ions which im-poses equal magnitudes of the iron magnetic moments. In the following, we consider only the SVC hedgehog structures keeping in mind that the orthorhombic distortions are weak.We do not further discuss the SVC loop structures for whichweak distortions do not create a sizable magnetic ﬁeld at themajority muon stopping site. In the same IR of the D 2h(1) group, we get a superposition of two SVC hedgehog orderparameters with small canting along the caxis: for the IR(A g) the SVC( τ1) mixes with SVC( τ5) states with additional small L(+) 2zcanting; for IR(B1u) the SVC( τ4) mixes with SVC( τ8) with additional small L(−) 2zcanting. Note that the L(±) 2zcanting along the caxis does not induce a magnetic ﬁeld at the muon sites [see Eqs. ( B3) and ( B4)] for small values of the lattice distortions [ 5] and therefore very small shifts of the iron coordinates from (1 /4,3/4,z). Figure 22shows an example 224515-18MUON SPIN ROTATION AND INFRARED SPECTROSCOPY … PHYSICAL REVIEW B 101, 224515 (2020) FIG. 23. Sketch of the orthorhombic distortion of the SVC hedgehog (a) and (b) and the SVC loop (c) structures which is described by a superposition of SVC ( τ5) hedgehog and SVC ( τ3) loop order parameters α[(L(−) 3x−L(−) 1y)+(L(−) 1x−L(−) 3y)]± β[(L(−) 3x−L(−) 1y)−(L(−) 1x−L(−) 3y)]. The ( +)/(−) signs refer to hedgehog /loop distorted SVC structures. Here, α/negationslash=βdue to dif- ferent magnitudes of the magnetic moments pointing along theorthogonal directions. In the absence of orthorhombic distortions, we have α=β=1/2. Note that the canting along the caxis leads to a weak ferromagnetic moment α[F (+) z+L(+) 2z]±β[F(+) z−L(+) 2z] which is allowed by the given symmetry of distorted SVC structures and spin-orbital coupling. In the cases (b) and (c), we have \|a\|=\|b\|. of such an orthorhombic distortion of the double- QSVC hedgehog structure. TheμSR response transforms under the D2h(1) orthorhom- bic distortions in accordance with the mixed order parametersand Eqs. ( B3) and ( B4). For instance, for the order parameter [cosδ(L (−) 3x+L(−) 1y)+sinδ(L(−) 3x−L(−) 1y)] cosθ, we get mag- netic ﬁelds along the caxis for both muon sites: at the 4e muon sites B z(4e)=52.84 cos δcosθMHz and at the 4f muon sites Bz(4f)=52.84 cos δcosθMHz for 1 μB/Fe. The distorted SVC hedgehog structure thus induces a B zﬁeld at the 4f muon site that is strictly forbidden for the undistorted SVC hedge-hog state. The magnitude of this ﬁeld is proportional to theone of the lattice distortion. The stripe spin density wave state(SSDW) arises if δ=π/4 [SSDW of L (−) 3xtype Fig. 21(b) ] orδ=−π/4 [SSDW of L(−) 1ytype Fig. 21(a) ]. Both types of SSDW states create the same magnetic ﬁelds on the 4e and4f muon sites. However, a smooth transition from the SVCstate to the SSDW state is unlikely since the required largeorthorhombicity would renormalize the magnetic interactionssuch that a variety of complex intermediate phase transitionswould occur, as is also discussed in Ref. [ 75]. The D 2h(1) type orthorhombic distortions of the SVC hedgehog states are notsupported by our μSR data which show no clear sign of weak and large local magnetic ﬁelds that are directed along the c axis. Next, we consider the impact of the orthorhombic lattice distortions which arise under a reduction of the D4hsymmetry into the D 2h(2) case with twofold axes along z,xyand ¯xy directions. Here the eight Fe ions divide into two sets withfour ions in each and the D 2h(2) symmetry operations do not permute ions from different sets. This implies that themagnitudes of the iron magnetic moments are different forthe two sets of ions. In the same IR of the D 2h(2) group we have a superposition of SVC hedgehog and SVC looporder parameters: IR(Ag) the SVC( τ1) hedgehog mixes with SVC(τ7) loop states; IR(B1u) the SVC( τ4) hedgehog mixes with SVC( τ6) loop states; IR(B1g) the SVC( τ5) hedgehog mixes with SVC( τ3) loop states; IR(Au) the SVC( τ8) hedge- hog mixes with SVC( τ2) loop states. A small, additional canting along the caxis is allowed for the SVC structures from IR(B1g) and IR(Au). Under such a mixing the SVC distorted structures remain orthogonal but with nonequal magneticmoments that are pointing along two different orthogonaldirections. The mixing of different SVC states as a result ofspeciﬁc relations between magnetic interaction constants wasalso considered in Ref. [ 75]. An example of the distorted SVC hedgehog and SVC loop states of IR(B 1g) symmetry is shown in Fig. 23. The distorted SVC structures, both hedgehog and loop types, without canting along the caxis create a zcomponent of the dipole magnetic ﬁelds only at one of the two types of muonstopping sites (e.g., the ﬁeld at the other type of muon sitevanishes). The weak magnetic ﬁeld from the distorted SVCloop structures arises due to the inequality of the magneticmoments. The SVC hedgehog distorted structures with smallccanting give rise to a strong B zﬁeld at one of the muon stopping sites and a weak Bzﬁeld at the other one. The weak ﬁelds arise from the F(±) zorder parameters which equally contribute to both types of the majority muon stopping sites.Our experimental data do not support the observation of thedistorted SVC structures with ccanting as we do not observe FIG. 24. The distorted SVC hedgehog structure which is obtained by the superposition of the SVC( τ1) hedgehog, SVC(τ7) loop and a small amount of the i-SCDW L(−) 1z+L(−) 3zorder parameters in the form α{[(L(−) 3x+L(−) 1y)+ (L(−) 1x+L(−) 3y)] cosθ+(L(−) 1z+L(−) 3z)s i nθ}+β[(L(−) 3x+L(−) 1y)− (L(−) 1x+L(−) 3y)] (left). The rotation part of the lattice space group consists of the elements e,2¯xy,I,m¯xy. The angle θis the out-of-plane canting angle; here α/negationslash=βpresents the difference in the value of the magnetic moments that are pointing along the orthogonal directions. The dashed lines are drawn to stress the absence or presence of thesmall ccanting of the iron magnetic moments. The canted SVC structure remains orthogonal. The right panel shows the abplane cross-section. 224515-19E. SHEVELEV A et al. PHYSICAL REVIEW B 101, 224515 (2020) additional weak local magnetic ﬁelds along the cdirection. Whereas in the case of the D2h(2) type orthorhombic distor- tion we can not distinguish in μSR experiment the orthomag- netic SVC structures with inequality of the magnetic momentsand without ccanting from the undistorted tetragonal SVC hedgehog states. None of the above orthorhombically distorted SVC mag- netic structures creates an in-plane magnetic ﬁeld as it is seenin theμSR experiment. The latter can only arise under further symmetry lowering. We ﬁnd that among the subgroups of indexes 4 of the P4/mbm space group only the space groups with C2hrotation symmetry with C2 axes along either the xyor ¯xydirections can create the SVC hedgehog canted structures, which induce bothout-of-plane and in-plane ﬁelds at the muon stopping sites.The symmetry constrains and the DMI allow superpositionof both order parameters SVC hedgehog state and i-SCDWstate in the lattice with monoclinic distortions. An exampleof a possible SVC hedgehog structure with speciﬁc cantingalong the ca x i s ,w h i c hi sa l l o w e db yt h e C2hrotating sym- metry (e.g., the distorted double- Qstructure), is shown in Fig.24.The distorted SVC hedgehog structure shown in Fig. 24 creates a B zﬁeld that is quite strong at the 4e muon sites and vanishes at the 4f muon sites. Simultaneously, an in-plane component with equal magnitude of the magnetic ﬁeldis present at both the 4e and 4f muon stopping sites. TheμSR response from such an orthorhombically distorted SVC hedgehog structure thus agrees well with our experimentaldata. Interestingly, such a distorted SVC hedgehog structureuniﬁes the order parameters of the two neighboring phasesin the phase diagram of Fig. 1. Actually, the phase with an exotic superposition of the SVC and i-SCDW order param-eters requires only the tetragonal symmetry breaking, as itcan be realized through a complex interplay of the magneticanisotropy constants, see also Ref. [ 75]. 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PhysRevB.80.155103.pdf	Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states Tetsuyuki Ochiai1and Masaru Onoda2 1Quantum Dot Research Center, National Institute for Materials Science (NIMS), Tsukuba 305-0044, Japan 2Department of Electrical and Electronic Engineering, Faculty of Engineering and Resource Science, Akita University, Akita 010-8502, Japan /H20849Received 25 May 2009; revised manuscript received 13 July 2009; published 2 October 2009 /H20850 This paper investigates the topological phase transition in honeycomb lattice photonic crystals with and without time-reversal and space-inversion symmetries through extensive analysis on bulk and edge states. Inthe system with both the symmetries, there appear multiple Dirac cones in the photonic band structure, and themass gaps are controllable via symmetry breaking. The zigzag and armchair edges of the photonic crystals cansupport novel edge states that reﬂect the symmetries of the photonic crystals. The dispersion relation and theﬁeld conﬁguration of the edge states are analyzed in detail in comparison to electronic edge states. Leakage ofthe edge states to free space, which is inherent in photonic systems, is fully taken into account in the analysis.A topological relation between bulk and edge states, which has been discussed in the context of electronicquantum Hall effect, is also examined in the photonic system with leaky edge states. DOI: 10.1103/PhysRevB.80.155103 PACS number /H20849s/H20850: 42.70.Qs, 73.20. /H11002r, 61.48.De, 03.65.Vf I. INTRODUCTION A monolayer of graphite sheet, called graphene, has at- tracted growing interests recently.1,2Graphene exhibits a Dirac cone /H20849for each spin degree of freedom /H20850with a linear dispersion at each corner of the ﬁrst Brillouin zone, resultingin a variety of novel transport phenomena of electrons. Theystimulate theoretical and experimental studies taking accountof analogy to physics of relativistic electron, such as Kleintunneling 3and Zitterbewegung.4Moreover, semi-inﬁnite graphene and ﬁnite stripe of graphene /H20849called graphene rib- bon /H20850with zigzag edges support peculiar edge states with nearly ﬂat dispersion.5,6On the contrary, armchair edge does not support such an edge state. The ﬂat dispersion impliesthat the density of state /H20849DOS /H20850diverges at the ﬂat band en- ergy, in a striking contrast to the DOS in bulk. The diver-gence affects many physical properties of graphene ribbon. The above interesting phenomena of graphene arise mainly from the gapless Dirac cone in the dispersion relationof electron. The Dirac cone can be also regarded as the signalof a critical state in the context of the topological phasetransition, which has been originally discussed in the quan-tum Hall system. 7Phase transitions between topologically distinctive phases are accompanied by gap closings, whichoften emerge as gapless Dirac cones. Such kind of phasetransitions is sometimes governed by symmetry breaking.However, in graphene, the symmetry is “built in,” and thefreedom in designing and tuning of the system is limited.From a theoretical point of view, it is possible to implementsymmetry breaking via the energy difference between A- and B-site atomic orbitals, 8periodic magnetic ﬂux of zero average,9and Rashba spin-orbit interaction10in a model of graphene. However, their realistic implementations are not soeasy. It is worth noting that the Dirac cone is not limited in graphene, but can emerge in completely different physicalsystems. For example, also in a certain class of photoniccrystals /H20849PhCs /H20850, the photonic band structure exhibits Dirac cones at nonzero frequency values. We can expect many in-teresting phenomena in optics relevant to photonic Dirac cone, e.g., the pseudodiffusive scaling, 11Zitterbewegung,12 and extinction of coherent backscattering.13PhC has a great advantage in designing and tuning of the structure. That is,we can freely select optical substances and their shapes inPhC. In addition, static and dynamical tunings via externalﬁeld are available. We can comparatively easily control thesymmetry and its breaking of the system, and can investigatetheir effect on geometrical and topological properties of pho-tonic bands throughout in a wide range of parameter space.Consequently, PhC provides a unique platform to investigatesymmetry-breaking physics relevant to Dirac cone. Such aninvestigation may realize novel optical components and willbe valuable for feedback between optics of PhC and nano-electronics of graphene. When we investigate physics of Dirac cone in PhC based on analogy of graphene, we must be careful about differencebetween electron and photon. The difference stands out inﬁnite systems with boundary. In electronic systems the elec-trons near Fermi level are prohibited to escape to the outerregion via the work function, i.e., a conﬁning potential, andthe wave functions of the electrons are evanescent in theouter region. Therefore, to sustain an edge state, formation of the band gap in bulk is the minimum requirement. On theother hand, in the former system conﬁning potentials forphoton are absent at the boundary. Energy of photon is al-ways positive as in free space, and no energy barrier existsbetween the PhC and free space. The simplest way to conﬁnephotonic edge states in the PhC is to utilize the light cone.This restriction of the conﬁnement makes photonic systemsquit nontrivial in various aspects. In this paper, we study a photonic analog of graphene model, 14namely, two-dimensional PhC composed of the honeycomb lattice of dielectric cylinders embedded in abackground substance. The honeycomb lattice consists oftwo interpenetrating triangular lattices /H20849called AandBsub- lattices /H20850with the same lattice constant. This PhC exhibits multiple Dirac cones at the corners of the ﬁrst Brillouin zoneowing to its spatial symmetry. 15Here, we introduce two kinds of symmetry breaking, breaking of the space-inversionPHYSICAL REVIEW B 80, 155103 /H208492009 /H20850 1098-0121/2009/80 /H2084915/H20850/155103 /H208499/H20850 ©2009 The American Physical Society 155103-1symmetry /H20849SIS /H20850and the time-reversal symmetry /H20849TRS /H20850. The SIS is broken by using different optical substances betweenA- and B-site rods. The degree of the symmetry breaking is controllable via /H9255 A−/H9255B, the difference in dielectric constant between A- and B-site rods. The SIS breaking opens up a band gap and causes geometrically nontrivial Bloch statesaround the corners of the Brillouin zone. 16,17The TRS is efﬁciently broken by applying a magnetic ﬁeld parallel to thecylindrical axis. Nonzero static magnetic ﬁeld induces imagi-nary off-diagonal elements in the permittivity or permeabil-ity tensors, through the magneto-optical effect. The TRS breaking is crucial for the emergence of topo- logically nontrivial phases, each of which is characterized bya topological index called the Chern number. We clarify howsuch phases appear by tracing the change in bulk and edgestates through a phase diagram in the space of the twosymmetry-breaking parameters. We also present the photonicedge states visually by employing a ﬁrst-principles calcula-tion of the Maxwell equation. 18The energy leakage of photon at open boundary is also taken into account in theﬁrst-principles calculation to clarify its inﬂuence on the to-pological relation between bulk and edge states. This rela-tion, so called bulk-edge correspondence, 19has been studied in electronic quantum Hall systems, and nowadays becomesof high interest also in photonic systems without TRS. Inquantum Hall system, nontrivial topology of bulk states leadsto the emergence of chiral edge states, 20,21which are robust against localization effect. Recently, Haldane and Raghu22 proposed one-way light waveguide realized in PhCs withoutTRS. Explicit construction of such waveguides is demon-strated by several authors. 23–26Strictly speaking, these works investigate interface states localized by an interface sand-wiched by screened media. Such systems are similar to elec-tronic ones with conﬁning potentials. In contrast, this paperis focusing on systems with open boundary, and the leakageof photon to outer region is fully taken into account. Espe-cially one of important ﬁndings in this paper is that the prop-erty of leakage strongly depends on the type of edge, i.e.,whether zigzag edge or armchair edge. Furthermore, as forboth structure and controllable parameters, our honeycombPhC with open boundary is rather simpler than those dem-onstrated so far. Thus, we can systematically clarify how thebulk-edge correspondence is modiﬁed for leaky edge states. This property speciﬁc to optical systems, i.e., the coexistenceof leaky and nonleaky edge states, enables one-way lighttransport without preparing a particular kind of interface. Wealso visually demonstrate a clockwise one-way light trans-port for a rectangular-shaped honeycomb PhC, which hasboth zigzag and armchair edges. This paper is organized as follows. Section IIis devoted to present bulk properties of the PhC with and without TRSand SIS. A numerical method to deal with edge states isgiven in Sec. III. Properties of zigzag and armchair edge states are investigated in detail in Secs. IVandV, respec- tively. A one-way light transport along the edge of arectangular-shaped PhC is demonstrated in Sec. VI. Finally, summary and discussions are given in Sec. VII. II. DIRAC CONE AND BAND GAP Let us consider two-dimensional PhCs composed of the honeycomb array of circular cylinders embedded in air. Thephotonic band structure of the PhCs with and without TRS is shown in Fig. 1for the transverse magnetic /H20849TM /H20850polariza- tion. For comparison, the photonic band structure of thetransverse electric /H20849TE/H20850polarization is also shown for the PhC with TRS. The SIS holds in all the cases. Here, the dielectric constants /H9255 A/H20849B/H20850and radius rA/H20849B/H20850of the A/H20849B/H20850cylinders are taken to be 12 and 0.2 a, respectively. The magnetic permeability of the cylinders is taken to be 1 forthe PhC with TRS, and has the tensor form given by /H9262ˆ=/H20898/H9262 i/H92600 −i/H9260/H92620 00 /H9262/H20899,/H9262=1 , /H9260= 0.2, /H208491/H20850 for the PhC without TRS. The ﬁrst, second, and third rows /H20849columns /H20850stand for x,y, and zCartesian components, re- spectively. The cylindrical axis is taken to be parallel to the z axis. The imaginary off-diagonal components of /H9262ˆare re- sponsible for the magneto-optical effect and break the TRS.Thus, parameter /H9260represents the degree of the TRS break- ing. As mentioned in introduction, for the PhC with TRS the Dirac cone is found at the K point. In particular, the ﬁrst/H20849lowest /H20850and second TM bands are in contact with each other at the K point. They are also in contact with the K /H11032point because of the spatial symmetry. This property is quite simi-lar to the tight-binding electron in graphene. As for the Diracpoint at /H9275a/2/H9266c/H112290.55 of the TM polarization, the fourth band is in contact with the ﬁfth band at K /H20849and K /H11032/H20850, whereas the former and the latter are also in contact with the third andsixth bands, respectively, at the /H9003point. Concerning the TE polarization, the Dirac cones are not clearly visible, but areindeed formed between the second and third and between thefourth and ﬁfth. On the other hand, in the PhC without TRS, all the de- generate modes at /H9003and K are lifted. The point group of this PhC becomes C 6and the point group of kat the K point isMKM 00.10.20.30.40.50.60.70.8 0.250.270.29 K K' FIG. 1. /H20849Color online /H20850The photonic band structure of the hon- eycomb lattice PhCs of dielectric cylinders embedded in air. Solid/H20849dashed /H20850line stands for the TM band structure of the PhC with /H20849without /H20850TRS. The SIS holds in both the cases. The dielectric con- stant and radius of the cylinders are taken to be 12 and 0.2 a, re- spectively, where ais the lattice constant. The magnetic permeabil- ity of the cylinders is taken to be 1 for the PhC with TRS and isgiven by Eq. /H208491/H20850for the PhC without TRS. For comparison, the TE band structure of the PhC with TRS is also shown by doted line.TETSUYUKI OCHIAI AND MASARU ONODA PHYSICAL REVIEW B 80, 155103 /H208492009 /H20850 155103-2C3. They are Abelian groups, allowing solely one- dimensional representations. Therefore, the degeneracy isforbidden. The energy gap between the lifted modes is pro-portional to /H9260if it is small enough. As for the SIS breaking, it is given by the difference in the dielectric function, /H9255/H20849−r/H20850−/H9255/H20849r/H20850. In the honeycomb PhC con- cerned, the difference is equal to /H11006/H9004/H9255 /H20849/H9004/H9255/H11013/H9255A−/H9255B/H20850inside the cylinders and zero otherwise. Therefore, the degree of theSIS breaking is represented by /H9004/H9255. The SIS breaking lifts the double degeneracy at K, but not at /H9003when the TRS is pre- served. The energy gap between the lifted modes is propor-tional to /H9004/H9255. 17 Let us focus on the gap between the ﬁrst and second TM bands of the PhC as a function of the SIS and TRS breakingparameters. The phase diagram of the PhC concerning thegap is shown in Fig. 2. Here, the average dielectric constant and the radius of the cylinders are kept ﬁxed to /H20849/H9255 A+/H9255B/H20850/2=12 and rA=rB=0.2 a, respectively. At generic values of the parameters the gapopens. However, if we change the parameters along certaincurves in the parameter space, the gap remains to close. Thisproperty implies that at ﬁnite /H9260the gap closes only at certain values of /H9004/H9255. In Fig. 2there are four regions that are sepa- rated by the curves. The four regions are characterized by theChern numbers of the ﬁrst and second photonic bands. TheChern number is a topological integer deﬁned by C n=1 2/H9266/H20885 BZd2k/H20849/H11612k/H11003/H9011 nk/H20850z, /H208492/H20850 /H9011nk=−i/H20855unk/H20841/H11612k/H20841unk/H20856, /H208493/H20850 /H20855umk/H20841unk/H20856=1 A/H20885 UCd2rumk/H20849r/H20850/H9255/H20849r/H20850unk/H20849r/H20850=/H9254m,n /H208494/H20850 for each nondegenerate band. Here, BZ, UC, and Astand for Brillouin zone, unit cell, and the area of unit cell, respec-tively. The envelop function u nk/H20849r/H20850of the nth Bloch state at k is of Ez/H20849i.e., the zcomponent of the electric ﬁeld /H20850. For in- stance, in the upper region of Fig. 2,C1=−1 and C2=1,whereas in the right region C1=C2=0. At the gap closing point, the Chern number transfers between the upper andlower bands under the topological number conservation law. 7 We will see that the phase diagram correlates with a propertyof edge states in corresponding PhC stripes. This correlationis a guiding principle to design a one-way light transportnear PhC edges. 22 Finally, we should note that the Chern numbers given in Fig. 2are consistent under the time-reversal and the space- inversion transformations. We also note this phase diagram issimilar to that obtained in a triangular lattice PhC with an-isotropic rods. 25 III. CHARACTERIZATION OF EDGE STATES So far, we have concentrated on properties of the PhCs of inﬁnite extent in plane. If the system has edges, there canemerge edge states which are localized near the edges andare evanescent both inside and outside the PhC. In this sec-tion we introduce a PhC stripe with two parallel edges. Theedges are supposed to have inﬁnite extent, so that the trans-lational invariance along the edges still holds. The edgestates are characterized by Bloch wave vector parallel to theedges. Optical properties of the PhC stripe are described by the S matrix. It relates the incident plane wave of parallel momen-tum k /H20648+G/H11032to the outgoing plane wave of parallel momen- tum k/H20648+G, where GandG/H11032are the reciprocal lattice vectors relevant to the periodicity parallel to the stripe edges.27Both the waves can be evanescent. To be precise, the Smatrix is deﬁned by /H20873/H20849a+out/H20850G /H20849a−out/H20850G/H20874=/H20858 G/H11032/H20873/H20849S++/H20850GG/H11032/H20849S+−/H20850GG/H11032 /H20849S−+/H20850GG/H11032/H20849S−−/H20850GG/H11032/H20874/H20873/H20849a+in/H20850G/H11032 /H20849a−in/H20850G/H11032/H20874, /H208495/H20850 where /H20849a/H11006in/H20849out/H20850/H20850Gis the plane-wave-expansion components of upward /H20849+/H20850and downward /H20849−/H20850incoming /H20849outgoing /H20850waves of parallel momentum k/H20648+G, respectively. In our PhCs under consideration the Smatrix can be calculated via the photonic layer Korringa-Kohn-Rostoker method28as a function of par- allel momentum k/H20648and frequency /H9275.I ft h e Smatrix is nu- merically available, the dispersion relation of the edge statesis obtained according to the following secular equation: 0 = det /H20851S −1/H20852. /H208496/H20850 Strictly speaking, this equation also includes solutions of bulk states below the light line. If we search for the solutionsinside pseudogaps /H20849i.e.,k /H20648-dependent gaps /H20850, solely the disper- sion relations of the edge states are obtained. In actual cal-culation, however, the magnitude of det /H20851S/H20852becomes ex- tremely small with increasing size of the matrix. The matrixsize is given by the number of reciprocal lattice vectors takeninto account in numerical calculation. In order to obtain nu-merical accuracy, we have to deal with larger matrix. There-fore, this procedure to determine the edge states is generallyunstable. Instead, we employ the following scheme. Supposethat the Smatrix is divided into two parts S uand Slthat correspond to the division of the PhC stripe into the upperand lower parts. This division is arbitrary, unless the upper or-2 -1 0 1 2 ∆ε(=εA-εB)-0.4-0.200.20.4κC1=-1, C2=1 C1=C2=0 P1P2P3P4P5 C1=C2=0 C1=1, C2=-1 FIG. 2. Phase diagram of the honeycomb lattice PhC for the TM polarization. Phase space is spanned by two parameters, /H9004/H9255and/H9260, which represent the SIS and TRS breaking, respectively. The aver-age dielectric constant and the radius of the cylinders are kept ﬁxedto/H20849/H9255 A+/H9255B/H20850/2=12 and rA=rB=0.2 a, respectively. The Chern num- berCnof the ﬁrst and second bands is indicated.PHOTONIC ANALOG OF GRAPHENE MODEL AND ITS … PHYSICAL REVIEW B 80, 155103 /H208492009 /H20850 155103-3lower part is not empty. The following secular equation also determines the dispersion relation of the edge states: 0 = det /H208511−S−+lS+−u/H20852. /H208497/H20850 This scheme is much stable for larger matrix. As far as true edge states are concerned, the secular equa- tion has the zeros in real axis of frequency for a given real k/H20648. Here we should also mention leaky edge states /H20849i.e., reso- nances near the edges /H20850, which are not evanescent outside the PhC but are evanescent inside the PhC. Such an edge state isstill meaningful, because the DOS exhibits a peak there. Thepeak frequency as a function of parallel momentum k /H20648fol- lows a certain curve that is connected to the dispersion curveof the true edge states. To evaluate the leaky edge states, themethod developed by Ohtaka et al. 29is employed. In this method, the DOS at ﬁxed k/H20648and/H9275is calculated with the truncated Smatrix of open diffraction channels. The unitarity of the truncated Smatrix enables us to determine the DOS via eigenphase shifts of the Smatrix. A peak of the DOS inside the pseudogap corresponds to a leaky edge state. IV. ZIGZAG EDGE First, let us consider the zigzag edge. Figure 3shows four sets of the projected band diagram of the honeycomb PhCand the dispersion relation of the edge states localized nearthe zigzag edges. In Fig. 3the shaded regions represent bulk states, whereas the blank regions correspond to thepseudogap. Inside the pseudogap edge states can emerge. Inthe evaluation of the edge states, we assumed the PhC stripeofN=16, being Nthe number of the layers along the direc- tion perpendicular to the zigzag edges.Here, we close up the ﬁrst and second bands. Higher bands are well separated from the lowest two bands. Each setrefers to either of four points indicated in the phase diagramof Fig. 2. In accordance with the Dirac cone in Fig. 1, the projected band structure of point P 1also exhibits a point contact at k/H20648a/2/H9266=1 /3 and 2/3. The ﬁrst and second bands are separated for P2and P3, but are nearly in contact at k/H20648a/2/H9266=1 /3 for P4. This is because P4is close to the phase boundary. Except for the lower right panel, in which the TRSand the SIS are broken, the projected band diagrams and theedge-state dispersion curves are symmetric with respect tok /H20648a/2/H9266=0.5. This symmetry is preserved if either the TRS or the SIS holds. Let us consider symmetry properties of the bulk and edge states in detail. The time-reversal transformation implies /H9275n/H20849−k/H20648,−k/H11036;/H9004/H9255,−/H9260/H20850=/H9275n/H20849k/H20648,k/H11036;/H9004/H9255,/H9260/H20850, /H208498/H20850 where /H9275n/H20849k/H20648,k/H11036;/H9004/H9255,/H9260/H20850is the eigenfrequency of the nth Bloch state at given parameters of /H9004/H9255and/H9260, and k/H11036is the momentum perpendicular to the edge. In the case of /H9260=0, after the projection concerning k/H11036, the symmetry with re- spect to k/H20648=0 is obtained. This symmetry combined with the translational invariance under k/H20648→k/H20648+Gresults in the sym- metry with respect to k/H20648a/2/H9266=0.5. Similarly, the space- inversion results in /H9275n/H20849−k/H20648,−k/H11036;−/H9004/H9255,/H9260/H20850=/H9275n/H20849k/H20648,k/H11036;/H9004/H9255,/H9260/H20850. /H208499/H20850 The symmetry with respect to k/H20648a/2/H9266=0 and 0.5 is obtained at/H9004/H9255=0. When edge states are well deﬁned in PhCs with enough number of layers, their dispersion relation satisﬁes /H9275e1/H20849e2/H20850/H20849−k/H20648;/H9004/H9255,−/H9260/H20850=/H9275e1/H20849e2/H20850/H20849k/H20648;/H9004/H9255,/H9260/H20850, /H2084910/H20850 /H9275e1/H20849−k/H20648;−/H9004/H9255,/H9260/H20850=/H9275e2/H20849k/H20648;/H9004/H9255,/H9260/H20850, /H2084911/H20850 owing to the time-reversal and space-inversion transforma- tions, respectively. Here, /H9275e1and/H9275e2denote the dispersion relation of opposite edges of the PhC stripe. At /H9260=0, both /H9275e1and/H9275e2are symmetric under the inversion of k/H20648. In con- trast, at /H9004/H9255=0 they are interchanged. The resulting band dia- gram is symmetric with respect to k/H20648a/2/H9266=0 and 0.5 as in Fig.3. The upper left panel of Fig. 3shows two almost- degenerate curves that are lifted a bit near the Dirac point.This lifting comes from the hybridization between edgestates of the opposite boundary, owing to ﬁnite width of thestripe. The lifting becomes smaller with increasing N, and eventually two curves merge with each other. Since P 1cor- responds to /H9004/H9255=/H9260=0, we obtain /H9275e1=/H9275e2owing to Eqs. /H2084910/H20850 and /H2084911/H20850, irrespective of k/H20648. As is the same with in graphene, our edge states appear only in the region 1 /3/H11349k/H20648a/2/H9266 /H113492/3. However, the edge-state curves are not ﬂat, in a strik- ing contrast to the zigzag edge state in the nearest-neighbortight-binding model of graphene. In the upper right panel two edge-state curves are sepa- rated in frequency and each curve terminates in the samebulk band. On the contrary, in the lower two panels the dis-persion curves of the two edge states intersect one another ata particular point and each curve terminates at different 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0 0.2 0.4 0.6 0.8 k\|\|a/2π0.230.240.250.260.270.28 ωa/2πc P3Q1Q2 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0 0.2 0.4 0.6 0.8 1 k\|\|a/2πP4 FIG. 3. /H20849Color /H20850The projected band diagrams at point Pnin the phase diagram /H20849Fig.2/H20850and the dispersion curves of the edge states. The zigzag edge is assumed. The shaded regions represent bulkstates. The edge states are of the PhC stripes with 16 layers. Solidline stands for the light line. The surface Brillouin zone is taken tobe 0/H11349k /H20648a/2/H9266/H113491 in order to see the connectivity of the edge-state dispersion curves. The edge state localized near the upper /H20849lower /H20850 zigzag edge is represented by red /H20849blue /H20850dot. Violet dots stand for the edge states that are not simply categorized into the upper orlower edge owing to bonding or antibonding via /H20849approximate /H20850 degeneracy.TETSUYUKI OCHIAI AND MASARU ONODA PHYSICAL REVIEW B 80, 155103 /H208492009 /H20850 155103-4bands. For instance, in the lower left panel, the curve includ- ingQ1terminates at the upper band near k/H20648a/2/H9266=1 /3 and at the lower band near k/H20648a/2/H9266=2 /3. At other points in the pa- rameter space, we found that the two edge-state curves areseparated if the system is in the phase of zero Chern number.Otherwise, if the system is in the phase of nonzero Chernnumber, the two curves intersect one another. The wave function of the edge state at marked points Q 1 andQ2is plotted in Fig. 4. We can easily see that the edge states at Q1andQ2are localized near different edges. This property is consistent with the fact that at /H9004/H9255=0,/H9275e1and/H9275e2are interchanged under the inversion of k/H20648. The ﬁeld conﬁgu- ration at Q1is identical to that at Q2after the space-inversion transformation /H20849/H9266rotation /H20850. Since the SIS is preserved in this case, they are the SIS partners. It is also remarkable that theelectric ﬁeld intensity is conﬁned almost in the rods formingone particular sublattice. This ﬁeld pattern is reminiscent ofthe nonbonding orbital of the zigzag edge state in graphene.The edge state at Q 1/H20849Q2/H20850has the negative /H20849positive /H20850group velocity. Moreover, no other bulk and edge states exist at thefrequency. Therefore, solely the propagation from left toright is allowed near the upper edge, while the propagationfrom right to left is allowed in the lower edge. In this way aone-way light transport is realized near a given edge. Theone-way transport is robust against quenched disorder withlong correlation length, because the edge states are out of thelight line and the bulk states at the same frequency is com-pletely absent. 30This is also the case in the lower right panel of Fig. 3, although the frequency range of the one-way trans- port is very narrow. It should be noted that the noncorrelateddisorder would cause the scattering into the states above thelight line, where the energy leakage takes place. Detailedinvestigation of disorder effects is beyond the scope of thepresent paper. The results obtained in this section strongly support the bulk-edge correspondence, which was originally proven inthe context of quantum Hall systems 19and was discussed in the context of photonic systems recently.22Namely, the num- ber of one-way edge states per edge in a given two-dimensional omnidirectional gap /H20849i.e.,k /H20648-independent gap /H20850is equal to the sum of the Chern numbers of the bulk bandsbelow the gap. A negative sign of the sum corresponds to theinverted direction of the edge propagation. For instance, ifthe sum is equal to −2, the number of the one-way edgestates is 2, but they ﬂow in the opposite direction to the casethat the sum is equal to 2. In our case, at P 3, for instance, the Chern number of the lower /H20849upper /H20850band is equal to −1 /H208491/H20850. Accordingly, there is only one /H20849one-way /H20850state per edge in the gap between the ﬁrst and second bands. Moreover, noedge state is found between the second and third bands. Thisbehavior is consistent with the Chern numbers of the ﬁrstand second bands, according to the bulk-edge correspon-dence. This is also the case at P 4and at other generic points in the phase space. Finally, let us brieﬂy comment on the edge states in P1 andP2.I nP1the two edge states are completely degenerate atN=/H11009. For the system with narrow width, there appear the bonding and antibonding orbitals, each of which has an equalweight of the ﬁeld intensity in both the zigzag edges. As forthe edge states in P 2, the upper /H20849lower /H20850edge states are local- ized near the upper /H20849lower /H20850zigzag edge. V. ARMCHAIR EDGE Next, let us consider the armchair edge. The projected band diagram and the dispersion curves of the edge states areshown in Fig. 5. We assumed the PhC stripe with N=64. It should be noted that they are symmetric with respect to k /H20648=0 regardless of SIS and TRS. This property is understood by the combination of a parity transformation and Eq. /H208498/H20850. Under the parity transformation with respect to the mirrorplane parallel to the armchair edges, /H9275n/H20849k/H20648,−k/H11036;/H9004/H9255,−/H9260/H20850=/H9275n/H20849k/H20648,k/H11036;/H9004/H9255,/H9260/H20850. /H2084912/H20850 By combining Eqs. /H208498/H20850and /H2084912/H20850, we obtain the symmetric projected band diagram with respect to k/H20648=0. Concerning the edge states, the parity transformation results in /H9275e1/H20849k/H20648;/H9004/H9255,−/H9260/H20850=/H9275e2/H20849k/H20648;/H9004/H9255,/H9260/H20850. /H2084913/H20850 Therefore, by combining Eqs. /H2084910/H20850and /H2084913/H20850, we can derive that/H9275e1and/H9275e2are interchanged by the inversion of k/H20648, regardless of SIS and TRS /H9275e1/H20849−k/H20648;/H9004/H9255,/H9260/H20850=/H9275e2/H20849k/H20648;/H9004/H9255,/H9260/H20850. /H2084914/H20850 Equation /H2084914/H20850results in the degeneracy between /H9275e1and/H9275e2at the boundary of the surface Brillouin zone. Moreover, it is FIG. 4. /H20849Color online /H20850The electric ﬁeld intensity /H20841Ez/H208412of the true edge state at Q1/H20849left panel /H20850and Q2/H20849right panel /H20850in Fig. 3. The intensity maxima is normalized as 1. In the enlarged panels thePoynting vector ﬂow is also shown.PHOTONIC ANALOG OF GRAPHENE MODEL AND ITS … PHYSICAL REVIEW B 80, 155103 /H208492009 /H20850 155103-5obvious from Eq. /H2084913/H20850that the two edge states are com- pletely degenerate at /H9260=0 in the entire surface Brillouin zone. In the armchair projection the K and K /H11032points in the ﬁrst Brillouin zone are mapped on the same point k/H20648=0 in the surface Brillouin zone, being above the light line. Therefore,possible edge states relevant to the Dirac cone are leaky,unless the region outside the PhC is screened. Accordingly,the DOS of an armchair edge state at ﬁxed k /H20648shows up as a Lorentzian peak, in a striking contrast to that of a zigzagedge state being a delta-function peak. The dispersion rela-tion of the leaky edge states depends strongly on the numberof layers N. However, if Nis large enough, the N-dependence disappears. We found that at large enough N, the leaky edge states correlate with the Chern number fairlywell. In the case as P 3where the Chern numbers of the upper and lower bands are nonzero, we found a segment of thedispersion curve of the leaky edge state whose bottom is atthe lower band edge, as shown in the lower left panel of Fig.5. There also appears another segment of the dispersion curve which crosses the light line. Across the phase bound-ary, the upper band touches to and separates from the lowerband. After the separation as the case P 5, the bottom of the former segment moves from the lower band edge to the up-per band edge as shown in the lower right panel of Fig. 5.B y increasing /H9004/H9255, this segment hides among the upper bulk band /H20849not shown /H20850. We should note that the dispersion curve of the leaky edge states is obtained by tracing the peak fre-quencies of the DOS as a function of k /H20648. If a peak becomes a shoulder, we stopped tracing the curve and indicated shoul-der frequencies by dotted curve. This is the case for P 3and P5. For P3, the DOS changes its shape from peak to shoulder atk/H20648a/2/H9266/H11229/H110060.058. This is why the segment including Q3 andQ4seems to terminate around there. However, we candistinguish this shoulder in the region 0.058 /H11021/H20841k/H20648/H20841a/2/H9266 /H110210.1, accompanying an additional peak above it. The peak bringing the shoulder with it becomes an asymmetric peakfor /H20841k /H20648/H20841a/2/H9266/H110220.1 and crosses the light line. In the DOS spectrum of P5, we can ﬁnd two shoulders just below the peaks of bulk states in the region 0.04 /H11021/H20841k/H20648/H20841a/2/H9266/H110210.1. Again, they merge each other and become an asymmetricpeak for /H20841k /H20648/H20841a/2/H9266/H110220.1. Such an asymmetric peak consists of two peaks with different heights and widths, which comefrom the lifting of the degenerate edge states in the limit of /H9260=0. Actually, for P1and P2in which the edge states are doubly degenerate, we can see a nearly symmetric singlepeak for the leaky edge states in each case. As in the case of zigzag edge, the leaky edge states in the two-dimensional omnidirectional gap exhibit a one-way lighttransport if the relevant Chern number is nonzero. Here, weconsider the structure with two horizontal armchair edges.The incident wave with positive k /H20648coming from the bottom cannot excite the leaky edge state just above the lower bandedge, e.g., state Q 4in Fig. 5. However, the incident wave with negative k/H20648coming from the bottom can excite the leaky edge state, e.g., at Q3. This is because the leaky edge states with positive k/H20648are localized near the upper edge, while those with negative k/H20648are localized near the lower edge. In the latter case, the leaky edge states have negative groupvelocities, traveling from right to left. This relation becomesinverted for the plane wave coming from the top. The inci-dent plane wave with positive /H20849negative /H20850k /H20648can /H20849cannot /H20850ex- cite the leaky edge state localized near the upper armchairedge. This edge state has positive group velocity, travelingfrom left to right. In this way, one-way light transport isrealized as in the zigzag edge case. Under quenched disorderthe one-way transport is protected against the mixing withbulk states, because no bulk state exists in the omnidirec-tional gap. However, in contrast to the zigzag edge case,even the disorder with long correlation length could enhancethe energy leakage to the outer region. Figure 6shows the electric ﬁeld intensity /H20841E z/H208412induced by the incident plane wave whose /H9275andk/H20648are at the marked points /H20849Q3andQ4/H20850in Fig. 5. The intensity of the incident plane wave is taken to be 1 and the ﬁeld conﬁguration above y/a=8 is omitted. Although, the dispersion curve is symmetric with respect tok/H20648=0, the ﬁeld conﬁguration is quite asymmetric. Of par- ticular importance is the near-ﬁeld pattern around the loweredge. In the left panel the strongest ﬁeld intensity of order 40is found in the boundary armchair layer, whereas in the rightpanel it is found outside the PhC with much smaller inten-sity. In both the cases, the transmittances in the ydirection are the same and nearly equal to zero. Accordingly, no ﬁeldenhancement is observed near the upper edge /H20849not shown /H20850. The remarkable contrast of the ﬁeld proﬁles indicates that theleaky edge state with horizontal energy ﬂow is excited in theleft panel, but is not in the right panel. If the plane wave isincident from the top, the ﬁeld pattern exhibits an oppositebehavior. That is, the plane wave with /H9275andk/H20648atQ4from the top excites the leaky edge state localized near the upperedge, but at Q 3it cannot excite the leaky edge state. The resulting ﬁeld proﬁle at Q4is the same as the left panel of Fig. 6after the space-inversion transformation. This is 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0.230.240.250.260.270.280.29 ωa/2πc P1 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-0.2 -0.1 0 0.1 0.2 k\|\|a/2π0.230.240.250.260.270.28 ωa/2πc P3Q3Q4 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-0.2 -0.1 0 0.1 0.2 k\|\|a/2πP5 FIG. 5. /H20849Color /H20850The projected band diagrams at point Pnin the phase diagram /H20849Fig.2/H20850and the dispersion curves of the edge states. The armchair edge is assumed. The shaded regions correspond tobulk states. The edge states are of the PhC stripes with 64 layers.Solid line stands for the light line. The edge state localized near theupper /H20849lower /H20850zigzag edge is represented by red /H20849blue /H20850dot. Violet dots stand for the edge states that are not simply categorized intothe upper or lower edge owing to bonding or antibonding via /H20849ap- proximate /H20850degeneracy.TETSUYUKI OCHIAI AND MASARU ONODA PHYSICAL REVIEW B 80, 155103 /H208492009 /H20850 155103-6cause the states at Q3andQ4are the space-inversion part- ners. The property of each edge state is also understood as fol- lows. When we scan k/H20648from negative to positive along the dispersion curve of the leaky edge state, the localized centerof the edge state transfers from one edge to the other. Thecritical point is at the bottom of the dispersion curve, wherethe edge state merges to the bulk state of the lower band. It isextended inside the PhC, making a bridge from one edge tothe other. The entire picture is consistent with Eq. /H2084914/H20850. Finally, let us comment on the ﬁeld conﬁguration of other edge states. For P 1and P2, the edge states are degenerate between the upper and lower edges. Accordingly, the inci-dent plane wave coming from the bottom /H20849top/H20850of the struc- ture excites the leaky edge states localized around the bottom/H20849top/H20850edge. It is regardless of the sign of k /H20648. For P3andP5, the edge-state curve that crosses the light line corresponds toan asymmetric peak in the DOS, which is actually the sum oftwo peaks. It is difﬁcult to separate the two peaks, becausethey are overlapped in frequency. Thus, the edge states canbe excited by the incident wave coming from both top andbottom of the PhC. Concerning the edge states around k /H20648 =0 of P5, a similar contrast in the ﬁeld conﬁguration be- tween positive and negative k/H20648is obtained as in Q3andQ4.However, under quenched disorder this edge state readily mixes with bulk states that exist at the same frequency. VI. DEMONSTRATION OF ONE-WAY LIGHT TRANSPORT The direction of the one-way transport in the zigzag edge is consistent with that in the armchair edge. Let us consider arectangular-shaped PhC whose four edges are zigzag, arm-chair, zigzag, and armchair in a clockwise order. The one-way transport found at P 3in Fig. 2has to be clockwise in this geometry. To verify it certainly happens, we performed a numerical simulation of the light transport in the rectangular-shapedPhC. The multiple-scattering method is employed along witha Gaussian beam incidence. 31We assume N=32 for the zig- zag edges and N=64 for the armchair edges. The incident Gaussian beam is focused at the midpoint of the front arm-chair edge. The electric ﬁeld intensity /H20841E z/H208412at the focused point is normalized as 1 and the beam waist is 20 a. The frequency and the incident angle of the beam are taken to be /H9275a/2/H9266c=0.273 and /H92580=7.263°, which corresponds to the leaky edge state very close to the Q3point. The beam waist size is chosen to avoid possible diffraction at the corner ofthe PhC and not to excite the states near the Q 4point at the same time. Resulting electric ﬁeld intensity /H20841Ez/H208412is plotted in Fig. 7. The incident beam is almost reﬂected at the left /H20849armchair /H20850 edge, forming the interference pattern in the left side of thePhC. However, as in the left panel of Fig. 6, the leaky edge state is certainly excited there. This edge state propagatesupward, and is diffracted at the upper left corner. A certainportion of the energy turns into the zigzag edge state local-ized near the upper edge. This edge state propagates fromleft to right. The energy leakage at the upper edge is verysmall compared to that in the left and right edges. This zig-zag edge state is more or less diffracted at the upper rightcorner. However, the down-going armchair edge state is cer-tainly excited in the right edge. Obviously, the ﬁeld intensity FIG. 6. /H20849Color online /H20850The electric ﬁeld intensity /H20841Ez/H208412induced by the incident plane wave having /H20849k/H20648,/H9275/H20850atQ3/H20849left panel /H20850andQ4 /H20849right panel /H20850in Fig. 5. The incident plane wave of unit intensity comes from the bottom of the structure. In the enlarged panels thePoynting vector ﬂow is also shown. FIG. 7. /H20849Color online /H20850The electric ﬁeld intensity /H20841Ez/H208412induced by the time-harmonic Gaussian beam coming from the left of therectangular-shaped PhC. The four edges are either zigzag /H20849top and bottom /H20850or armchair /H20849left and right /H20850. The beam is focused at the mid point of the left armchair edge with the unit electric ﬁeld intensityand beam waist of 20 a.PHOTONIC ANALOG OF GRAPHENE MODEL AND ITS … PHYSICAL REVIEW B 80, 155103 /H208492009 /H20850 155103-7of the right edge reduces with reducing ycoordinate. This behavior is consistent with the energy leakage of the arm-chair edge state. Finally, the ﬁeld intensity almost vanishedat the lower right corner. In this way, the clockwise one-waylight transport is realized in the rectangular-shaped PhC. We also conﬁrmed that the incident beam with the same parameters but inverted incident angle /H20849− /H92580/H20850does not excite the counterclockwise one-way transport along the edges. Theincident beam is just reﬂected without exciting the relevantleaky edge state in accordance with the right panel of Fig. 6. VII. SUMMARY AND DISCUSSIONS In summary, we have presented a numerical analysis on the bulk and edge states in honeycomb lattice PhCs as aphotonic analog of graphene model and its extension. In theTM polarization the Dirac cone emerges between the ﬁrstand second bands. The mass gap in the Dirac cone is con-trollable by the parameters of the SIS or TRS breaking. Oncertain curves in the parameter space, the band touchingtakes place. These curves divide the parameter space intofour topologically distinct regions. Two regions are charac-terized by zero Chern number of the upper and lower bands,and the others are characterized by Chern number of /H110061. Of particular importance is the correlation between the Chernnumber in bulk and light transport near edge. Nonzero Chernnumber in bulk photonic bands results in one-way lighttransport near the edge. It is quite similar to the bulk-edgecorrespondence found in quantum Hall systems. In this paper we focus on the TM polarization in rod-in- air type PhCs. This is mainly because the band touchingtakes place between the lowest two bands and they are wellseparated from higher bands by the wide band gap, providedthat the refractive index of the rods are high enough. In rod-in-air type PhCs the TE polarization results in the bandtouching between the second and third bands. However, theDirac cone is not clearly visible, although it is certainlyformed. As for hole-in-dielectric type PhCs, an opposite ten-dency is found. Namely, the band touching between the low-est two bands takes place only in the TE polarization. In thiscase the distance between the boundary column of air holesand the PhC edge affect edge states. Therefore, we must takeaccount of this parameter to determine the dispersion curvesof the edge states.Concerning the TRS breaking, we have introduced imagi- nary off-diagonal components in the permeability tensor.This is the most efﬁcient way to break the TRS withoutdissipation for the TM polarization. Such a permeability ten-sor is normally not available in visible frequency range. 32 However, in GHz range it is possible to obtain /H9260of order 10. Such a large /H9260is necessary to obtain a robust one-way trans- port against thermal ﬂuctuations, etc. In the numerical setupwe assumed an intermediate frequency range with smaller /H9260. On the other hand, in the TE polarization, the TRS can beefﬁciently broken by imaginary off-diagonal components inthe permittivity tensor. In this case the PhC without the TRScan operate in visible frequency range. However, strongmagnetic ﬁeld is necessary in order to induce large imagi-nary off-diagonal components of the permittivity tensor.Thus, it is strongly desired to explorer low-loss optical mediawith large magneto-optical effect, in order to have robustone-way transport. Recently, another photonic analog of graphene, namely, honeycomb array of metallic nanoparticles, was proposedand analyzed theoretically. 33Particle plasmon resonances in the nanoparticles act as if localized orbitals in carbon atom.The tight-binding picture is thus reasonably adapted to thissystem, and nearly ﬂat bands are found in the zigzag edge.Vectorial nature of photon plays a crucial role there, givingrise to a remarkable feature in the dispersion curves of theedge states in the quasistatic approximation. In contrast, vec-torial nature of photon is minimally introduced in our model,but a full analysis including possible retardation effects andsymmetry-breaking effects has been made. Effects of theTE-TM mixing in off-axis propagation are an important issuein our system. In particular, it is interesting to study to whatextent the bulk-edge correspondence is modiﬁed. We hopethis paper stimulates further investigation based on the anal-ogy between electronic and photonic systems on honeycomblattices. ACKNOWLEDGMENTS The works of T.O. and M.O. were partially supported by Grant-in-Aid under Grants No. 20560042 and No. 21340075,respectively, for Scientiﬁc Research from the Ministry ofEducation, Culture, Sports, Science and Technology. 1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov,Nature /H20849London /H20850438, 197 /H208492005 /H20850. 2A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 /H208492007 /H20850. 3O. Klein, Z. Phys. 53, 157 /H208491929 /H20850. 4E. Schrödinger, Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl.24, 418 /H208491930 /H20850. 5M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Jpn. 65, 1920 /H208491996 /H20850. 6K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 /H208491996 /H20850.7J. E. Avron, R. Seiler, and B. Simon, Phys. Rev. Lett. 51,5 1 /H208491983 /H20850. 8G. W. Semenoff, Phys. Rev. Lett. 53, 2449 /H208491984 /H20850; W. Yao, S. A. Yang, and Q. Niu, ibid. 102, 096801 /H208492009 /H20850. 9F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 /H208491988 /H20850. 10C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 /H208492005 /H20850. 11R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 /H208492007 /H20850. 12X. Zhang, Phys. Rev. Lett. 100, 113903 /H208492008 /H20850. 13R. A. Sepkhanov, A. Ossipov, and C. W. J. Beenakker, EPL 85, 14005 /H208492009 /H20850.TETSUYUKI OCHIAI AND MASARU ONODA PHYSICAL REVIEW B 80, 155103 /H208492009 /H20850 155103-814D. Cassagne, C. Jouanin, and D. Bertho, Phys. Rev. B 52, R2217 /H208491995 /H20850. 15Y. D. Chong, X. G. Wen, and M. Solja čić, Phys. Rev. B 77, 235125 /H208492008 /H20850. 16M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. E 74, 066610 /H208492006 /H20850. 17M. Onoda and T. Ochiai, Phys. Rev. Lett. 103, 033903 /H208492009 /H20850. 18Although the tight-binding approximation is often employed in modeling of graphene, this approximation cannot visualize wavefunctions directly. 19Y. Hatsugai, Phys. Rev. Lett. 71, 3697 /H208491993 /H20850. 20B. I. Halperin, Phys. Rev. B 25, 2185 /H208491982 /H20850. 21X. G. Wen, Phys. Rev. B 43, 11025 /H208491991 /H20850. 22F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 /H208492008 /H20850. 23Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Solja čić, Phys. Rev. Lett. 100, 013905 /H208492008 /H20850. 24Z. Yu, G. Veronis, Z. Wang, and S. Fan, Phys. Rev. Lett. 100, 023902 /H208492008 /H20850.25S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 /H208492008 /H20850. 26H. Takeda and S. John, Phys. Rev. A 78, 023804 /H208492008 /H20850. 27J. B. 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PhysRevB.99.115432.pdf	PHYSICAL REVIEW B 99, 115432 (2019) Effect of a Chern-Simons term on dynamical gap generation in graphene M. E. Carrington* Department of Physics, Brandon University, Brandon, Manitoba, Canada R7A 6A9 and Winnipeg Institute for Theoretical Physics, Winnipeg, Manitoba, Canada (Received 20 December 2018; revised manuscript received 21 February 2019; published 25 March 2019) We study the effect of a Chern-Simons term on dynamical gap generation in a low-energy effective theory that describes some features of monolayer suspended graphene. We use a nonperturbative Schwinger-Dysonapproach. We solve a set of coupled integral equations for eight independent dressing functions that describefermion and photon degrees of freedom. We ﬁnd a strong suppression of the gap and corresponding increase inthe critical coupling as a function of increasing Chern-Simons coefﬁcient. DOI: 10.1103/PhysRevB.99.115432 I. INTRODUCTION Quantum electrodynamics in 2 +1 dimensions (QED2+1) has been studied for many years as a toy model for quantumchromodynamics (QCD). The main point is that QED 2+1is strongly coupled, and therefore, in spite of being Abelian,it can be used to study many interesting features of QCD[1–5]. In this paper, we are interested in reduced QED 3+1 (RQED) in which the fermions are restricted to remain in a two-dimensional plane but the photons which are responsiblefor the interactions between fermions are not. In the reducedtheory the Coulomb interaction between the electrons has thesame 1 /rform as in the (3 +1)-dimensional theory, instead of the logarithmic form obtained from QED 2+1. The theory is physically relevant for the description of what are calledDirac planar materials, which refer to condensed-matter sys-tems for which the underlying lattice structure produces afermionic dispersion relation that has the form of a Diracequation in some regimes. We are particularly interested ingraphene, where the fermions have an effective speed v F which is on the order of 300 times smaller than the speed of light. The unique band structure of graphene gives it highmobility, large thermal and electrical conductivity, and opticaltransparence, which are characteristics that are valuable intechnological applications. We study speciﬁcally suspendedsingle-layer graphene, where we deal with a single atomiclayer in the absence of scattering from a substrate, so that theintrinsic electronic properties of the system are accessed. Forsimplicity we will also work at half ﬁlling (which means zerochemical potential). In both QED 2+1and RQED the fermions couple to a three- dimensional Abelian gauge ﬁeld, and therefore, the Chern-Simons (CS) term can be added to the action. This termbreaks the time-reversal symmetry and gives a mass to thephoton. It is important in condensed-matter physics in thecontext of chiral symmetry breaking [ 6–9], high-temperature superconductivity [ 10], and the Hall effect [ 11]. CS terms can dynamically generate magnetic ﬁelds in QED 2+1[12], and *carrington@brandonu.camagnetic ﬁelds are thought to inﬂuence dynamical symmetry breaking in a universal and model-independent way throughwhat is known as magnetic catalysis (for a review see [ 13]). In this work we use RQED and study the inﬂuence of a CS term on phase transitions in graphene. We will showin Sec. II B that the CS term has the same properties under discrete symmetry transformations as a chirally symmetricmass term which was discussed in the context of graphene byHaldane [ 14]. One therefore expects that including a CS term in the photon part of the action could dynamically generatea Haldane-type mass for the fermions. In Appendix Bwe show that such a mass term in the Lagrangian of the effec-tive theory would correspond physically to including in theHamiltonian of the discrete theory a contribution that wouldgive counterclockwise hopping around the triangles that areformed by each sublattice of the graphene sheet. In [ 14]i t was originally proposed that such hopping could take placein response to an externally applied magnetic ﬁeld, and asmentioned above, the inﬂuence of magnetic ﬁelds on the phasetransition in graphene is a subject of much interest. Within aneffective theory description, a natural way to investigate thisis through the introduction of a CS term at the level of theLagrangian. The coupling constant and CS parameter are dimensionful scales in QED 2+1, but they are dimensionless parameters in RQED. In natural units the effective coupling can be writ-tenα=e 2/(4π/epsilon1vF), where vF∼c/300 is the velocity of a massless electron in graphene. The parameter /epsilon1/greaterorequalslant1 is related to the screening properties of the graphene sheet, and we takethe vacuum value /epsilon1=1. The Chern-Simons parameter will be denoted θ, and we consider θ∈(0,1). II. THE LOW-ENERGY EFFECTIVE THEORY A. Noninteracting Hamiltonian The carbon atoms in graphene are arranged in a two- dimensional hexagonal lattice. The hexagonal structure can beviewed as two sets of interwoven triangular sublattices (calledAandB). The geometry dictates each primitive cell has one atom from the Asublattice and one from the Bsublattice and that each lattice site has three nearest neighbors on the 2469-9950/2019/99(11)/115432(11) 115432-1 ©2019 American Physical SocietyM. E. CARRINGTON PHYSICAL REVIEW B 99, 115432 (2019) opposite sublattice. For each atom, three of the four outer electrons form hybridized σbonds with the three nearest neighbors. The fourth sits in the pzorbital, perpendicular to the hybrid orbitals, and forms a πbond. The simplest description of graphene is a tight-binding Hamiltonian for theπorbitals, H 0=−t/summationdisplay /angbracketleft/vectorn/vectorn/prime/angbracketrightσ[a† /vectornσb/vectorn/primeσ+H.c.], (1) where tis the nearest-neighbor hopping parameter and the operators a† /vectornσandb† /vectorn/primeσare creation operators for πelectrons with spin σon the AandBsublattices, respectively. We can rewrite the Hamiltonian as a momentum integral by Fourier transforming. Our deﬁnitions for the lattice vectorsand discrete Fourier transforms are given in Appendix A. From the dispersion relation for the noninteracting theorywe obtain six Kpoints, and our choice of two inequivalent ones, which we denote K ±,i sg i v e ni nE q .( A5). Using Eqs. ( A6) and (( A7)), we rewrite the Hamiltonian in ( 1)a s a momentum integral, and we expand around K±. We deﬁne a four-component spinor: /Psi1σ(/vectorp)=(aσ(/vectorK++/vectorp),bσ(/vectorK++/vectorp),bσ(/vectorK−+/vectorp),aσ(/vectorK−+/vectorp))T, (2) where the superscript Tindicates that the spinor should be written as a column vector. Using this notation, the tight-binding Hamiltonian becomes H 0=¯hvF/summationdisplay σ/integraldisplayd2p (2π)2¯/Psi1σ(/vectorp)(γ1p1+γ2p2)/Psi1σ(/vectorp),(3) where we have deﬁned ¯ hvF=3at/2 and our representation for the gamma matrices is given in Eq. ( A7). The Lagrangian of the effective theory (including minimal coupling to thegauge ﬁeld) then takes the form [ 15] L=/summationdisplay σ¯/Psi1σ(t,/vectorx)[iγ0Dt+i¯hvF/vectorγ·/vectorD]/Psi1σ(t,/vectorx), (4) where we deﬁne Dμ=∂μ−ieAμ(taking e>0). In the next sections we will discuss how to include interac- tions. At this point however, we note that while our effectivetheory can accurately describe the low-energy dynamics ofthe system and allow us to correctly include both frequencyand nonperturbative effects, it does not allow for the inclusionof screening from the σ-band electrons and localized higher- energy states. B. Symmetries We consider the discrete symmetries of the tight- binding Hamiltonian. The parity, time-reversal, and charge-conjugation transformations on the spinor in ( 2)a r e P/Psi1(/vectorp)P −1=γ0/Psi1(−/vectorp), (5) T/Psi1(/vectorp)T−1=iσ2γ1γ5/Psi1(−/vectorp), (6) C/Psi1(/vectorp)C−1=γ1¯/Psi1(/vectorp)T. (7) It is easy to check that the noninteracting theory is invari- ant under these symmetries. To see the physical content ofEqs. ( 5)–(7) we show the action of each on the spinor deﬁned in Eq. ( 2). The parity transformation takes the form /Psi1=⎛ ⎜⎝aσ(K++/vectorp) bσ(K++/vectorp) bσ(K−+/vectorp) aσ(K−+/vectorp)⎞ ⎟⎠P−→⎛ ⎜⎝bσ(K−−/vectorp) aσ(K−−/vectorp) aσ(K+−/vectorp) bσ(K+−/vectorp)⎞ ⎟⎠, (8) which tells us that the parity operator reverses the sign of the momentum and exchanges the sublattices. We note thatthis deﬁnition is different from the one commonly usedin QED 2+1, where the transformation P:(x,y)→(−x,−y) would correspond to spatial rotation. Because of the hexag-onal lattice structure of graphene, spatial rotation is not asymmetry of the system unless the sublattice indices areinterchanged. The time-reversal operator changes the sign of momentum and spin, and its action on a spinor is /Psi1=⎛ ⎜⎝a σ(K++/vectorp) bσ(K++/vectorp) bσ(K−+/vectorp) aσ(K−+/vectorp)⎞ ⎟⎠T−→⎛ ⎜⎝aσ(K−−/vectorp) bσ(K−−/vectorp) bσ(K+−/vectorp) aσ(K+−/vectorp)⎞ ⎟⎠ (9) (where we have not explicitly written the action of the factor iσ2, which ﬂips spin), and therefore, the time-reversal operator inverts the Kpoints (and spin) but does not act on the sublattice degrees of freedom. The action of the charge-conjugation operator is /Psi1=⎛ ⎜⎝aσ(K++/vectorp) bσ(K++/vectorp) bσ(K−+/vectorp) aσ(K−+/vectorp)⎞ ⎟⎠C−→⎛ ⎜⎜⎝−b† σ(K++/vectorp) −a† σ(K++/vectorp) a† σ(K−+/vectorp) b† σ(K−+/vectorp)⎞ ⎟⎟⎠. (10) We can also consider continuous symmetries of the low- energy effective theory. The action is invariant under theenlarged group of global symmetries generated by both γ 5 and the third spatial gamma matrix γ3, which is not part of the Lagrangian ( 4). The matrices T1=i 2γ3,T2=1 2γ5,T3=i 2γ3γ5(11) commute with the Hamiltonian. They also satisfy the com- mutation relations [ Ti,Tj]=i/epsilon1ijkTkand therefore form a four-dimensional representation of SU(2). Including T4= I/2 gives a representation of U(2). Physically this is a sym- metry in the space of valley and sublattice indices, where“valley” refers to the K ±points. The noninteracting theory has a global U(4) symmetry that operates in the space of [valley ⊗sublattice ⊗spin]. We call this a chiral symmetry, and using our representation of the gamma matrices (see Appendix A), the chirality quantum number corresponds to the valley index. C. Fermion bilinears One reason that fermion bilinears are interesting is that, close to the critical point, possible interactions of the low-energy theory are constrained to have the form of localfour-fermion interactions. For example, in the Gross-Neveumodel the basic interaction is a four-fermion contact betweenscalar or pseudoscalar densities, and in the Thirring model 115432-2EFFECT OF A CHERN-SIMONS TERM ON DYNAMICAL … PHYSICAL REVIEW B 99, 115432 (2019) TABLE I. Transformation properties of the bilinears deﬁned in Eq. ( 13) under P,C,T. PCT I +++ γμ ˜+− ˜+ γ3−++ iγ5−−+ iγμγν ˜+− ˜− iγμγ3 ˜−+ ˜− γμγ5 ˜−− ˜− γ3γ5++− the interaction is a contact between two conserved currents. We note that while short-range interactions are not relevantfor dynamics in a perturbative theory, they can be importantin a strongly coupled system. Mass scales are especially in-teresting because they are directly related to chiral symmetrybreaking and a possible semimetal /insulator transition. We use /Gamma1 (n)to indicate one element of the list, /Gamma1={I,γμ,γ3,iγ5,iγμγν,iγμγ3,γμγ5,γ3γ5},(12) withμ∈(0,1,2), which gives a complete basis in Dirac space. We deﬁne a set of fermion bilinears as G(n)=m(n)/integraldisplay d2x¯/Psi1(/vectorx)/Gamma1(n)/Psi1(/vectorx). (13) The terms constructed with scalar /pseudoscalar elements /Gamma1(n)∈{1,γ3,iγ5,γ3γ5}correspond to mass terms and will be denoted M,M3,M5, andM35. We look at the transformation properties of fermion bilin- ears under parity, time reversal, and charge conjugation. Weintroduce the notation γ ˜μ=(γ0,−γi), with i∈(1,2). T wo examples where this notation can be used are P¯/Psi1γμ/Psi1P−1= ¯/Psi1γ˜μ/Psi1andP¯/Psi1γμγ5/Psi1P−1=− ¯/Psi1γ˜μγ5/Psi1. Our results areshown in Table I, and transformations of the type discussed above are listed with a tilde over the sign. The ﬁrst of the twoexamples given above is written ˜+in the second row of the second column of Table I, and the second is the symbol ˜− in the seventh row of the second column. The mass termsM 3andM5can be accessed from the standard Dirac mass Mby a change of integration variables in the path integral. The mass M35is completely independent of the other three and is related to a model introduced by Haldane [ 14]. We remark that although actions constructed from an effectiveLagrangian with mass term M,M 3,o rM5will describe identical physics, the symmetries of a continuous theory arenot necessarily evident in the original discrete theory, whichmeans that equivalent continuous theories may correspond todifferent discrete theories. To see directly how mass terms are related to physical quantities in the discrete theory, we look at a speciﬁc example.We consider a term in the Hamiltonian of the form H 1=/summationdisplay /vectornσ[maa† /vectornσa/vectornσ+mbb† /vectornσb/vectornσ], (14) which would correspond to different densities of particles on theAandBsublattices and could be realized physically by placing the graphene sheet on a substrate. Fourier transform-ing to momentum space and expanding around the Kpoints, Eq. ( 14) becomes H 1=/summationdisplay σ/integraldisplayd2p (2π)2[m+¯/Psi1σ(/vectorp)γ0/Psi1σ(/vectorp)+m−¯/Psi1σ(/vectorp)γ3/Psi1σ(/vectorp)], (15) where we have deﬁned m±=1 2(ma±mb). The term in ( 15) with the factor m−is proportional to the M3mass term, which breaks parity. Writing it explicitly in terms of creation andannihilation operators, we obtain M3=/integraldisplay d2x¯/Psi1γ3/Psi1=/summationdisplay σ/integraldisplayd2p (2π)2{[a† σ(/vectorK++/vectorp)aσ(/vectorK++/vectorp)+a† σ(/vectorK−+/vectorp)aσ(/vectorK−+/vectorp)] −[(b† σ(/vectorK++/vectorp)bσ(/vectorK++/vectorp)+b† σ(/vectorK−+/vectorp)bσ(/vectorK−+/vectorp)]}, (16) which makes clear that the order parameter M3is proportional to the difference in electron densities for the AandBsublattices. A nonzero value of this order parameter corresponds physically to a charge density wave and lifts the sublattice degeneracy. Theterm in ( 15) that is proportional to m +is less interesting since it can be absorbed into a redeﬁnition of the chemical potential. The independent mass term M35=/integraldisplay d2x¯/Psi1γ3γ5/Psi1=/summationdisplay σ/integraldisplayd2p (2π)2{[a† σ(/vectorK++/vectorp)aσ(/vectorK++/vectorp)−a† σ(/vectorK−+/vectorp)aσ(/vectorK−+/vectorp)] −[(b† σ(/vectorK++/vectorp)bσ(/vectorK++/vectorp)−b† σ(/vectorK−+/vectorp)bσ(/vectorK−+/vectorp)]} (17) corresponds to a gap with opposite sign at the K−point, relative to M3. Mathematically, a triangular next-neighbor hopping term in the Hamiltonian of the discrete theory gives amass term proportional to M 35in the effective theory. This is shown in Appendix B. Physically it corresponds to a topolog- ically nontrivial phase generated by currents propagating onthe two different sublattices. Both the CS term and the M35 mass term violate time-reversal invariance (see Table I), and one therefore expects that one-loop radiative corrections tothe photon polarization tensor obtained from internal fermionswith a Haldane-type mass would generate an odd- Tpiece in the polarization tensor or that including a CS term in 115432-3M. E. CARRINGTON PHYSICAL REVIEW B 99, 115432 (2019) the photon part of the action would dynamically generate a Haldane-type mass for the fermions. In this paper we willintroduce a CS term into the action and study the effect of thisterm through dynamical mass generation on phase transitionsin graphene. D. The brane action Dynamical photons are included in RQED by constructing the brane action [ 16–18]. We start with the four-dimensional Euclidean action S=/integraldisplay d4x/bracketleftbigg1 4FμνFμν−1 2ξ(∂μAμ)2+ie¯/Psi1/A/Psi1/bracketrightbigg (18) and integrate out the four-dimensional gauge ﬁeld to obtain S→1 2/integraldisplay d4x/integraldisplay d4yJμ(x)Dμν(x−y)Jν(y), Dμν(x−y)=/integraldisplayd3K (2π)3/integraldisplaydk3 2πeik(x−y) ×/bracketleftbigg δμν−(1−ξ)kμkν K2+k2 3/bracketrightbigg1 K2+k2 3,(19) where we write k=(K,k3). We use capital letters for three vectors which include a timelike component, for example, K=(k0,/vectork)=(k0,k1,k2) and X=(x0,/vectorx)=(x0,x1,x2). To describe graphene we take J3=0,Jμ(x0,x1,x2,x3)=jμ(x0,x1,x2)δ(x3), μ∈(0,1,2), (20) which allows us to do the k3integral in ( 19) analytically and obtain Dμν(X−Y)=/integraldisplayd3K (2π)3eiK(X−Y) ×/bracketleftbiggδμν 2√ K2−(1−ξ)KμKν 4√ K2K2/bracketrightbigg .(21) Note that in this equation the indices μandνare∈(0,1,2), and therefore they should properly be written differently (as ¯μand ¯ν, for example), but to simplify the notation we use the same letters for these indices. We can rescale the gauge pa-rameter (1 −ξ)→2(1−¯ξ) and suppress the bar to remove the factor of 1 /4 in the last term in ( 21). We introduce a three-dimensional vector ﬁeld (which we again call A) and write the effective action S=/integraldisplay d 3X/bracketleftbigg1 2Fμν1√ −∂2Fμν+AμJμ+1 ξ∂·A1√ −∂2∂·A/bracketrightbigg , (22) which corresponds to ( 21) in the sense that if we integrate out the gauge ﬁeld, we reproduce the dimensionally reduced prop-agator. We redeﬁne the gauge-ﬁxing term to be ( ∂·A) 2/ξ, add the kinetic term for the fermions [see Eq. ( 4)], and add a CS term to obtain S=/integraldisplay d3X/bracketleftbigg ¯/Psi1i/D/Psi1+1 2Fμν1√ −∂2Fμν +1 2ξ(∂·A)2+iθ/epsilon1μνλAμ∂νAλ/bracketrightbigg . (23)We want to use this relativistic theory to describe graphene near the Dirac points. To do this, we replace the Euclideanmetric in the ﬁrst term of Eq. ( 23) with the noncovariant form g μν→Mμνwith M=⎡ ⎣10 0 0vF 0 00 vF⎤ ⎦, (24) so that ¯/Psi1i/D/Psi1becomes ¯/Psi1iγμMμνDν/Psi1. We obtain the Feyn- man rules (in Landau gauge) from the resulting action: S(0)(p0,/vectorp)=− (iγμMμνPν)−1, (25) G(0) μν(p0,/vectorp)=/parenleftbigg δμν−PμPν P2/parenrightbigg1 2√ P2, (26) /Gamma1(0) μ=Mμνγν. (27) From this point on we will not refer again to the original four-dimensional theory. We deﬁne new notation so that low-ercase letters denote the spatial components of three vectors[for example, P=(p 0,/vectorp)]. We also introduce some additional notational simpliﬁcations that will be used in the rest of thispaper: we will sometimes write all momentum arguments offunctions with a single letter [for example, S(P):=S(p 0,/vectorp)], we deﬁne dK:=dk0d2k/(2π)3, and we write Q=K−P. III. NONPERTURBATIVE THEORY We will include nonperturbative effects by introducing fermion and photon dressing functions and solving a set ofcoupled Schwinger-Dyson (SD) equations. A. Propagators and vertices In the nonperturbative theory the bare propagator S(0)(P)i nE q .( 25) is written with six dressing functions (Z+ P,A+ P,B+ P,Z− P,A− P,B− P), where we have used subscripts instead of brackets to indicate the momentum dependence[for example, Z + P:=Z+(p0,/vectorp)]. We deﬁne two projection operators χ±=1 2(1±γ3γ5). Using this notation, the fermion propagator has the form S−1(P)=[−i(Z+ Pp0γ0χ++vFA+ P/vectorp·/vectorγ)+B+ P]χ+ +[−i(Z− Pp0γ0χ−+vFA− P/vectorp·/vectorγ)+B− P]χ−, S(P)=1 Den+ P[i(Z+ Pp0γ0+vFA+ P/vectorp·/vectorγ)+B+ P]χ+ +1 Den− P[i(Z− Pp0γ0+vFA− P/vectorp·/vectorγ)+B− P]χ−, Den± P=p2 0Z±2 P+p2v2 FA±2 P+B±2 P. (28) We deﬁne the even and odd functions: X±=Xeven±Xodd→Xeven/odd=1 2(X+±X−),(29) where X∈(Z,A,B). In the notation of Sec. II C,Beven(0,0) is a standard Dirac-type mass (denoted M), which breaks chiral symmetry but not time-reversal symmetry, and Bodd(0,0) is a Haldane-type mass ( M35), which preserves chiral symmetry but violates time-reversal invariance. In the bare theory Z±= A±=1, and B±=0, and therefore the odd functions are zero. It is easy to see that ( 28) reduces to ( 25) in this limit. 115432-4EFFECT OF A CHERN-SIMONS TERM ON DYNAMICAL … PHYSICAL REVIEW B 99, 115432 (2019) The Feynman rule for the dressed vertex is /Gamma1ν(P,K)=1 4[H+ νσ(P)+H+ νσ(K)]γσ(1+γ5)+1 4[H− νσ(P)+H− νσ(K)]γσ(1−γ5), (30) where Pis the outgoing fermion momentum, Kis the incoming fermion momentum, and H±indicates the diagonal 3 ×3m a t r i x H±(P)=⎡ ⎣Z±(P)0 0 0 vFA±(P)0 00 vFA±(P)⎤ ⎦. (31) It is clear that ( 30) and ( 31) reduce to ( 27) in the limit Z±=A±=1. Equations ( 30) and ( 31) are the ﬁrst term in the full Ball-Chiu vertex [ 19]. We include only the ﬁrst term because calculations are much easier using this simpler ansatz and because in our previous calculation we found that the contribution of the additional terms is very small [ 20]. To deﬁne the dressed photon propagator we start with a complete set of 11 independent projection operators. Deﬁning nμ= δμ0−q0Qμ/Q2, we write P1 μν=δμν−QμQν Q2,P2 μν=QμQν Q2,P3 μν=nμnν n2,P4 μν=Qμnν,P5 μν=nμQν, P6 μν=/epsilon1μναQα,P7 μν=/epsilon1μναnαQ2 q2,P8 μν=/epsilon1μαβQαnβQν,P9 μν=/epsilon1ναβQαnβQμ, P10 μν=−/epsilon1μαβQαnβnνQ2 q2,P11 μν=−/epsilon1ναβQαnβnμQ2 q2. (32) Using this notation, the inverse dressed photon propagator can be written G−1 μν=2/radicalbig Q2/bracketleftbigg P1 μν+1 ξP2 μν/bracketrightbigg +2θP6 μν+/Pi1μν, (33) where the polarization tensor is written in a completely general way as the sum /Pi1μν=11/summationdisplay i=1aiPi μν. (34) We invert the inverse propagator and then impose the constraints that the polarization tensor be transverse and satisfy the symmetry condition /Pi1μν(Q)=/Pi1νμ(−Q). The surviving components of the polarization tensor give /Pi1μν(Q)=α(Q)P1 μν+γ(Q)P3 μν+/Theta1(Q)P6 μν+ρ(Q)/bracketleftbig P10 μν+P11 μν/bracketrightbig , (35) and the propagator is Gμν=GLP3 μν+GT/bracketleftbig P1 μν−P3 μν/bracketrightbig +GDP6 μν+GE/bracketleftbig P10 μν−P11 μν/bracketrightbig ,GL=2/radicalbig Q2+α (2/radicalbig Q2+α)(2/radicalbig Q2+α+γ)+Q2(2θ+ρ+/Theta1)2, GT=2/radicalbig Q2+α+γ (2/radicalbig Q2+α)(2/radicalbig Q2+α+γ)+Q2(2θ+ρ+/Theta1)2, GD=−(2θ+/Theta1)(2/radicalbig Q2+α+γ) (2/radicalbig Q2+α)[(2/radicalbig Q2+α)(2/radicalbig Q2+α+γ)+Q2(2θ+ρ+/Theta1)2], GE=(2θ+/Theta1)(2/radicalbig Q2+α+γ)−(2/radicalbig Q2+α)(2θ+ρ+/Theta1) (2/radicalbig Q2+α)[(2/radicalbig Q2+α)(2/radicalbig Q2+α+γ)+Q2(2θ+ρ+/Theta1)2]. (36) B. Fermion Schwinger-Dyson equations The inverse fermion propagator is written generically as S−1(P)=(S(0))−1(P)+/Sigma1(P), (37) where the fermion self-energy is obtained from the SD equation as /Sigma1(P)=e2/integraldisplay dK G μν(Q)MμτγτS(K)/Gamma1ν. (38) Comparing ( 37) and ( 38) with ( 28), we ﬁnd the operators that project out each of the fermion dressing functions. For example, PB+=1 4(1+γ5)→B+ P=Tr[PB+/Sigma1(P)]. (39) 115432-5M. E. CARRINGTON PHYSICAL REVIEW B 99, 115432 (2019) Performing the traces, we obtain the set of self-consistent integrals that give the six fermion dressing functions: Z± P=1−4απvF 2p0/integraldisplaydK Den K±q2GL Q2k0ZK±(ZK±+ZP±), A± P=1+4απvF 2p2/integraldisplaydK Den K±/braceleftBig k0GDE(/vectorq×/vectorp)ZK±(AK±+AP±−ZK±−ZP±) +GL Q2[q2(/vectork·/vectorp)AK±(ZK±+ZP±)+k0q0(/vectorp·/vectorq)ZK±(AK±+AP±+ZK±+ZP±) −q0(/vectorq×/vectorp)B+ K(AK±+AP±−ZK±−ZP±)]/bracerightBig , B± P=4απvF 2/integraldisplaydK Den K±q2GL Q2BK±(ZK±+ZP±). (40) We have used the notation /vectorq×/vectorp=q1p2−q2p1,GDE=GD+GEand dropped terms proportional to v2 F(relative to 1), which is the reason there are no terms containing factors GTin (40). From Eq. ( 40) it is easy to see that if we ﬁnd a solution for the plus dressing functions Z+,A+, and B+, then we automatically have a solution for the minus dressing functions of the form Z−=Z+,A−=A+, and B−=−B+. We expect therefore that we will always be able to ﬁnd a chirally symmetric and time-reversal-violating solution ( Beven=0 and Bodd/negationslash=0) if we initialize with Zodd=Aodd=Beven=0. We call this solution 1 and write the solutions for the nonzero dressing functions Z(1) even,A(1) even, and B(1) odd. We can also see immediately that a solution with Zodd=Aodd=Bodd=0 should not exist since setting all odd dressing functions to zero on the right sides of Eqs. ( 40)g i v e s Zodd(P)=Bodd(P)=0b u t Aodd(P)=4απvF 2p2/integraldisplaydK Q2BKeven Den Keven[q0GL(/vectorq×/vectorp)(ZKeven+ZPeven−AKeven−APeven) +GDEQ2(/vectorp·/vectorq)(AKeven+APeven+ZKeven+ZPeven)]. (41) In the vicinity of the critical point, however, where BKevenis small, we would have Aodd(P)≈0. We therefore expect to get rapid convergence if we start in the vicinity of the critical point and initialize with Zodd=Aodd=Bodd=0. We will call this solution 2. We can also show that the two solutions discussed above are approximately the same, except for the reversal of the even and odd parts of the Bdressing function. To see this we substitute on the right side of ( 40) Z(2) odd=A(2) odd=B(2) odd=0,Z(2) even=Z(1) even,A(2) even=A(1) even,B(2) even=B(1) odd, (42) which gives Z(2) even=Z(1) even,B(2) even=B(1) odd,A(2) even=A(1) even+4απvF 2p2/integraldisplayB(1) Kodd Den KQ2 ×/bracketleftbig q0GL(/vectorq×/vectorp)/parenleftbig A(1) Keven+A(1) Peven−Z(1) Keven−Z(1) Peven/parenrightbig −GDEQ2(/vectorp×/vectorq)/parenleftbig A(1) Keven+A(1) Peven+Z(1) Keven+Z(1) Peven/parenrightbig/bracketrightbig .(43) The ﬁrst two lines in ( 43) are consistent with ( 42), and the last line is approximately consistent when we are close to the critical point. This analysis agrees with our numerical results, which are presented in detail in Sec. V. In summary, for all values of ( α,θ) we have considered, we have found only two solutions, which have the form Solution 1: Z(1) even/negationslash=0,A(1) even/negationslash=0,B(1) odd/negationslash=0;Z(1) odd=A(1) odd=B(1) even=0; (44) Solution 2: Z(2) even≈Z(1) even,A(2) even≈A(1) even,B(2) even≈B(1) odd;Z(2) odd≈A(2) odd≈B(2) odd≈0. (45) The approximately equal to symbols in the second line indicate deviations from zero of less than 0.01 percent. Solution 1 preserves chiral symmetry but violates time-reversal invariance, and to the degree of accuracy noted above, solution 2 breakschiral symmetry but satisﬁes time-reversal invariance. C. Photon Schwinger-Dyson equations The two components of the polarization tensor denoted ρand/Theta1can be set to zero in the approximation v2 F/lessmuch1, which is consistent with what was done with the fermion dressing functions in Sec. III B . In this case Eqs. ( 36) become GL=2Q+α (2/radicalbig Q2+α)(2/radicalbig Q2+α+γ)+4Q2θ2,GD+GE≡GDE=−2θ (2/radicalbig Q2+α)(2/radicalbig Q2+α+γ)+4Q2θ2.(46) 115432-6EFFECT OF A CHERN-SIMONS TERM ON DYNAMICAL … PHYSICAL REVIEW B 99, 115432 (2019) TABLE II. Comparison of B+(0,0) from different approxima- tions for two different values of ( α,θ). Approximation ( α,θ)=(4.0,0.2) ( α,θ)=(3.4,0.6) (3,0,0) 0.00256085 0.00036157 (3,0,1) 0.00255782 0.00036102(2,0,0) 0.00256082 0.00036167 (2,0,1) 0.00255762 0.00036080 (1,0,0) 0.00256083 0.00036168(1,0,1) 0.00255762 0.00036080 These expressions involve only two components of the polar- ization tensor: α(p0,p) andγ(p0,p). We work with the more convenient expressions /Pi100=q2 Q2(α+γ), (47) Tr/Pi1=/Pi1μμ=α+Q2 q2/Pi100. (48) From the Schwinger-Dyson equation for the polarization ten- sor we obtain /Pi100=− 4απvF/integraldisplaydK Den K+Den Q+(ZK++ZQ+) ×/bracketleftbig v2 F(/vectork·/vectorq)AK+AQ++BK+BQ+−k0q0ZK+ZQ+/bracketrightbig +(+→− ), (49) where the notation ( +→− ) indicates a second integral with the same form as the ﬁrst but with all plus superscriptschanged to minus. Similarly, we obtain for the trace /Pi1 μμ=− 4απvF/integraldisplaydK Den K+Den Q+/braceleftbig 2v2 F(AK++AQ+) ×(BK+BQ++k0q0ZK+ZQ+)+(ZK++ZQ+) ×/bracketleftbig v2 F(/vectork·/vectorq)AK+AQ++BK+BQ+−k0q0ZK+ZQ+/bracketrightbig/bracerightbig +(+→− ). (50) Equations ( 40), (46), (49), and ( 50) form a complete set of self-consistent equations that involve only the approximationv 2 F/lessmuch1. Now we discuss some additional approximations for the photon propagator and dressing functions. From Eqs. ( 49)and ( 50) it is straightforward to show that /Pi1μμ=/Pi100+O/parenleftbig v2 F/parenrightbig , (51) and therefore to O(v2 F) we can set /Pi1μμ=/Pi100, which gives α(q0,q)=−q2 0 q2/Pi100. (52) From equation ( 52) we see that by making a Coulomb-like approximation we can set α(q0,q)=0. The full Coulomb ap- proximation involves setting q0=0 everywhere in the photon propagator. We summarize as follows: For approximation 1 we use ( Z,A,B,/Pi1)\| v2 F/lessmuch1and GL=q2Q2/Pi100−q4(/Pi1μμ+2Q) Q/bracketleftbig Q3/Pi12 00−/Pi1μμq2(Q/Pi100+2q2)−4(1+θ2)q4Q/bracketrightbig, GDE=2θq4 Q/bracketleftbig Q3/Pi12 00−/Pi1μμq2(Q/Pi100+2q2)−4(1+θ2)q4Q/bracketrightbig. For approximation 2 we use ( Z,A,B,/Pi1)\| v2 F/lessmuch1and/Pi1μμ= /Pi100and GL=q2q2 0/Pi100−2q4Q Q/bracketleftbig q2 0Q/Pi12 00−2q4/Pi100−4(1+θ2)q4Q/bracketrightbig, GDE=2θq4 Q/bracketleftbig q2 0Q/Pi12 00−2q4/Pi100−4(1+θ2)q4Q/bracketrightbig. For approximation 3 we use ( Z,A,B)\| v2 F/lessmuch1and/Pi1\| (v2 F,q0/q)/lessmuch1 and GL=q2 Q[Q/Pi100+2(1+θ2)q2], GDE=−θq2 Q2[Q/Pi100+2(1+θ2)q2]. For approximation 4, ( Z,A,B,/Pi1)\| v2 F/lessmuch1(GL,GDE)\| q0=0, GL=1 /Pi100+2(1+θ2)q, GDE=−θ q[/Pi100+2(1+θ2)q]. FIG. 1. The ratio of the odd mass divided by the even one for solution 2 [see Eq. ( 45)]. The left panel shows the ratio as a function of the coupling with θ=0.6, and the right panel shows the dependence on θwithα=4.0. 115432-7M. E. CARRINGTON PHYSICAL REVIEW B 99, 115432 (2019) When θ=0 approximations 1 and 3 reduce to the full back- coupled calculation of Ref. [ 21], and approximation 4 reduces to the Coulomb version of that calculation. We also consider using analytic results for the polarization components /Pi100and/Pi1μμobtained from the one-loop expres- sions using bare-fermion propagators. This is a commonlyused approximation and is based on the vanishing fermiondensity of states at the Dirac points. Finally, we note that although Eq. ( 46) indicates that G DE is of the same order as GLfor values of θof order 1, we expect that the contribution of this term to the fermiondressing functions will be small. To understand this point,recall that the propagator component G Tdoes not contribute in Eq. ( 40) because it drops out in the limit v2 F/lessmuch1. Likewise, in the second line of ( 40) the term proportional to GDEis proportional to a difference of the form Z−A, and the ﬁrst two lines of this equation show that this difference is oforder v F. In summary, the full set of possible approximations we have discussed above can be written using the notation (n,m,l), where (1) n∈(1,2,3,4) for approximation 1, 2, 3, or 4 [as deﬁned following Eq. ( 52)], (2) m∈(0,1), where m=0 means the polarization components /Pi100and/Pi1μμare obtained from their self-consistent expressions (back coupled)andm=1 means we use their analytic one-loop approxima- tions, and (3) l∈(0,1), where l=0 means G DEis set to zero andl=1 means GDEis included. There are, in principle, 16 possible calculations, corre- sponding to approximations n=(1,2,3,4)×m=(0,1)× l=(0,1). Approximations n∈(1,2,3) and l∈(0,1) agree to very high accuracy. We show some results for the values ofB +(0,0) which verify this in Table II. From this point on we will consider only approximations (3,0,0), (4,0,0), and (4,1,0). IV . NUMERICAL METHOD We need to solve numerically the set of eight coupled equations ( 40), (49), and ( 50) for the dressing functions Z±,A±,B±,/Pi1 00, and/Pi1μμ. The functions /Pi100and/Pi1μμare renormalized by subtracting the zero-momentum value /Pi1renorm 00 (P)=/Pi100(P)−/Pi100(0), /Pi1renorm μμ (P)=/Pi1μμ(P)−/Pi1μμ(0). FIG. 2. Beven(0,0) and Bodd(0,0) as functions of the parameter θ with coupling α=4.0.FIG. 3. Beven(0,p) as a function of momentum at ﬁxed α=4.0. We work in spherical coordinates and deﬁne cos( θ)=/vectorp· /vectork/(pk), so that the integrals have the form /integraldisplay dK=1 (2π)3/integraldisplay∞ −∞dk0/integraldisplay∞ 0dk k/integraldisplay2π 0dθf(k0,k,θ) =1 (2π)3/integraldisplay∞ 0dk0/integraldisplay∞ 0dk k/integraldisplay2π 0 ×dθ[f(k0,k,θ)+f(−k0,k,θ)]. (53) We use an ultraviolet cutoff /Lambda1on all momentum integrals and deﬁne dimensionless variables ˆ p0=p0//Lambda1, ˆp=p//Lambda1, ˆk0= k0//Lambda1, and ˆk=k//Lambda1. We also use generically ˆB=B//Lambda1for all components and representations of the masslike fermiondressing function. The hatted momentum and frequency vari-ables range from 10 −6to 1, and to simplify the notation we suppress all hats. We use a logarithmic grid in the k0 andkdimensions to increase sensitivity to the infrared. We use Gauss-Legendre integration. Dressing functions are in-terpolated using double linear interpolation, using grids of220×200×16 points in the k 0,k, andθdimensions. In the calculation of /Pi1μνwe use an adaptive grid for the k0integral to more efﬁciently include the region of the integral wherek 0∼p0, /integraldisplay1 10−6dk0=/integraldisplayp0 10−6dk0+/integraldisplay1 p0dk0. (54) The integrands for the fermion dressing functions are smoother, and the adaptive grid is not needed. FIG. 4. Beven(0,0) as a function of coupling for different approx- imations and different values of the parameter θ. 115432-8EFFECT OF A CHERN-SIMONS TERM ON DYNAMICAL … PHYSICAL REVIEW B 99, 115432 (2019) FIG. 5. The dressing functions Z+andZ−as functions of p0, with pheld ﬁxed to its maximum and minimum values, for two values of αandθ=0.6. V. R E S U LT S Unless stated otherwise, all results in this section are obtained with the approximation (3,0,0). In Refs. [ 20,21] we learned that using the Lindhard screen- ing function, instead of calculating the photon polarizationtensor using a self-consistently back-coupled formulation,produces an artiﬁcially large damping effect which increasesthe critical coupling. This result can be understood as arisingfrom the fact that large fermion dressing functions ZandA are neglected in the denominator of the integral that gives theLindhard expression for the polarization tensor. In this workwe ﬁnd that higher values of θincrease the critical coupling, and this result can be understood in the same way as resultingfrom increased screening. We have found (for all values of θandαconsidered) only two types of solutions [see Eqs. ( 44) and ( 45) ] .U pt ov e r y small corrections, there is an odd mass solution (solution 1)and an even mass solution (solution 2), but no solutions forwhich both the even and odd mass parameters are nonzero. InFig. 1we show the absolute value of B odd(0,0)/Beven(0,0) for solution 2. As claimed following Eq. ( 45), this ratio is always less than 0.01%. From this point on we show only results from solution 2. In Fig. 2we show the condensates Beven(0,0) and Bodd(0,0) as a function of θat ﬁxed coupling, and in Fig. 3we show the dressing function Beven(p0,p) as a function of momentum at ﬁxed p0=0, using different values of θ. Figures 2and3show FIG. 6. The dressing functions Z+andA+as functions of p, with p0held ﬁxed to its maximum and minimum values, for two values of αandθ=0.6.FIG. 7. The dressing functions Z+andA+as functions of p0for α=3.4a n dθ∈(0,0.6). clearly that the condensate decreases as a function of θ, which implies that the critical coupling will increase as θincreases. The dependence of the critical coupling on the parameter θis seen explicitly in Fig. 4, which shows the condensate as a function of αfor different values of θ, using different approximations. In order to understand what drives this behavior, we look at the momentum dependence of the dressing functions Zand A. Figures 5and 6show the dressing functions Z+andA+ as functions of p0andpwith the other variable held ﬁxed to its maximum or minimum value. The two values of α that are shown are α=2.85, which is close to the critical coupling for the value of θ=0.6 that is chosen, and α=3.4, which is relatively far from the critical coupling. One seesthat the Zdressing function does not change much, but the A function does change and is responsible for the experimentallyobserved increase in the Fermi velocity at small frequencies asone approaches the critical coupling. To see explicitly how this effect is inﬂuenced by the parameter θ,w es h o wi nF i g . 7the fermion dressing functions Z +andA+as functions of p0for two different values of θ. Figure 7shows that once again it is A+(p0,0) which changes the most and that the largest effect is obtained with the highervalue of θ. In Fig. 8we show /Pi1 00as a function of momentum for α=3.4 and two different values of θ. For comparison the Lindhard expression is also shown. Maximal screening isobtained with the Lindhard approximation, and the smallest FIG. 8. The component /Pi100as a function of p0andpforα=3.4 andθ∈(0,0.6). 115432-9M. E. CARRINGTON PHYSICAL REVIEW B 99, 115432 (2019) TABLE III. Extrapolated values of the critical coupling for dif- ferent approximations and different values of the Chern-Simonsparameter. Approximation θ (3,0,0) (4,0,0) (4,1,0) 0.0 2.07 1.99 3.19 0.6 2.84 2.80 4.20 screening effect occurs when we set θto zero. This is con- sistent with our results in Figs. 2and4, which show that the critical coupling increases with θ. We ﬁt the data shown in Fig. 4using Mathematica , and the resulting function is extrapolated to obtain the value ofthe critical coupling for which B even(0,0) goes to zero. Our results are collected in Table III. The result for approximation (4,1,0) with θ=0 is taken from [ 20], and the result for approximation (4,0,0) with θ=0 is taken from [ 21]. VI. CONCLUSIONS Chern-Simons terms have been widely studied in condensed-matter physics in the context of chiral symmetrybreaking, the Hall effect, and high-temperature superconduc-tivity. In this work we showed that they are also relevant tothe study of phase transitions in graphene. We worked witha low-energy effective theory that describes some features ofmonolayer suspended graphene. We used reduced QED 3+1, which describes planar electrons interacting with photons thatcan propagate in three spatial dimensions. We studied theeffect of a Chern-Simons term in this theory. We found twoclasses of solutions: in the odd sector the theory dynamicallygenerates a time-reversal-violating Haldane-type mass, and inthe even sector a mass term of the standard Dirac type is gen-erated. We studied the dependence of the Dirac mass on theChern-Simons parameter θand showed that it is suppressed asθincreases, which means that the critical coupling at which a nonzero Dirac condensate is generated increases with θ.W e showed that this effect can be understood physically as arisingfrom an increase in screening. ACKNOWLEDGMENTS This work was supported by the Natural Sciences and Engineering Research Council of Canada and the HelmholtzInternational Center for FAIR. The author thanks C. S. Fis-cher, L. von Smekal, and M. H. Thoma for hospitality atthe Institut für Theoretische Physik, Justus-Liebig-UniversitätGiessen, and for discussions. APPENDIX A: NOTATION Our deﬁnitions of the lattice vectors are a1=a{−√ 3,0,0}, a2=a 2{−√ 3,−3,0}, a3=a 2{3√ 3,3,0}. (A1)Using these deﬁnitions, the volume of the lattice cell is S= 3√ 3a2 2. The vectors that generate the positions of the nearest- neighbor lattice points are δ1=a 2{−√ 3,1,0}, δ2={0,−a,0}, δ3={√ 3,1,0}, (A2) and the reciprocal lattice vectors are b1=2π a/braceleftbigg −1√ 3,1 3,0/bracerightbigg , b2=2π a/braceleftbigg 0,−2 3,0/bracerightbigg , b3=2π a/braceleftbigg −1√ 3,−1 3,0/bracerightbigg . (A3) The six Kpoints are Ki=3a 2π⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1√ 31 4√ 30 1√ 3−1 −1√ 3−1 −4√ 30 −1√ 31⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (A4) and we choose our two inequivalent Kpoints as K +=−K−=/braceleftbigg −8π 3√ 3a,0/bracerightbigg . (A5) We deﬁne the Fourier transform a/vectornσ=√ S/integraldisplay BZd2k (2π)2ei/vectork·/vectornaσ(/vectork), /summationdisplay /vectornei(/vectork−/vectork/prime)·/vectorn=(2π)2 Sδ2(/vectork−/vectork/prime). (A6) Our representation of the γmatrices is γ0=⎛ ⎜⎝0010 0001 1000 0100⎞ ⎟⎠,γ1=⎛ ⎜⎝00 0 −1 00 −10 01 0 0 1 000⎞ ⎟⎠, γ2=⎛ ⎜⎝00 0 i 00 −i0 0−i00 i 00 0⎞ ⎟⎠,γ3=⎛ ⎜⎝00 −10 000 1 100 0 0−10 0⎞ ⎟⎠, γ5=⎛ ⎜⎝10 0 0 01 0 000 −10 00 0 −1⎞ ⎟⎠. (A7) APPENDIX B: HALDANE-TYPE MASS We consider a term in the Hamiltonian which would give counterclockwise hopping around the triangles that are 115432-10EFFECT OF A CHERN-SIMONS TERM ON DYNAMICAL … PHYSICAL REVIEW B 99, 115432 (2019) formed by each sublattice. We write H2=t2/summationdisplay/bracketleftbig i/parenleftbig a† x1ax2+a† x2ax3+a† x3ax1/parenrightbig/bracketrightbig +t2/summationdisplay/bracketleftbig i/parenleftbig b† y1by2+b† y2by3+b† y3by1/parenrightbig/bracketrightbig +H.c.,(B1) where {/vectorx1,/vectorx2,/vectorx3}and{/vectory1,/vectory2,/vectory3}indicate the AandBsites on one hexagonal cell and the sums are over all AandBtriangular sublattices. We will take the origin of the coordinate system tobe at/vectorx 1=(0,0). Using our deﬁnitions of the lattice vectors, the corners of the triangular AandBsublattices which form the hexagon with /vectorx1in the lower left corner are /vectorx1=(0,0),/vectorx2=−/vectora1√ 3=a(1,0), /vectorx3=−/vectora2√ 3=a 2(1,√ 3), /vectory1=−2 3/vectora1+1 3/vectora2=a 2√ 3(√ 3,−1),/vectory2=−2 3/vectora1−2 3/vectora2=a√ 3(√ 3,1), /vectory3=−1 3/vectora1−2 3/vectora2=a√ 3(0,1). (B2) Fourier transforming to momentum space and expanding around the Dirac points, we obtain H2=t2C/integraldisplayd2p (2π)2{[a† +(p)a+(p)−a† −(p)a−(p)] −[b† +(p)b+(p)−b† −(p)b−(p)]} =t2C/integraldisplayd2p (2π)2[¯/Psi1(p)γ3γ5/Psi1(p)], (B3) where we have deﬁned the constant C=2[sin(2 φ)− 2s i n (φ)], with φ=8π/(3√ 3). Equation ( B3) shows that the Hamiltonian ( B1) corresponds to a mass of the form M35in the effective theory. [1] R. D. Pisarski, Phys. Rev D 29,2423 (1984 ). [2] T. W. Appelquist, M. Bowick, D. Karabali, and L. C. R. Wijewardhana, Phys. Rev. D 33,3704 (1986 ). [3] A. Lopez and E. Fradkin, P h y s .R e v .B 44,5246 (1991 ). [4] V . P. Gusynin and P. K. Pyatkovskiy, Phys. Rev. D 94,125009 (2016 ). [5] A. V . Kotikov and S. Teber, Phys. Rev. D 94,114011 (2016 ). [6] T. Appelquist, M. J. Bowick, D. Karabali, and L. C. R. Wijewardhana, Phys. Rev. D 33,3774 (1986 ). [7] T. Matsuyama and H. Nagahiro, Mod. Phys. Lett. A 15,2373 (2000 ); Gravitation Cosmol. 6, 145 (2000). [8] A. Bashir, A. Raya, and S. Sánchez-Madrigal, J. Phys. A 41, 505401 (2008 ). [9] K. I. Kondo and P. Maris, Phys. Rev. Lett. 74,18(1995 );Phys. Rev. D 52,1212 (1995 ). [10] A. P. Balachandran, E. Ercolessi, G. Morandi, and A. M. Srivastava, Int. J. Mod. Phys. B 04,2057 (1990 ). [11] E. Witten, La Rivista del Nuovo Cimento 39,313(2016 ).[12] Y . Hosotani, Phys. Lett. B 319,332(1993 ). [13] I. A. Shovkovy, in Strongly Interacting Matter in Magnetic Fields , Lecture Notes in Physics V ol. 871 (Springer, Berlin, 2013), p. 13. [14] F. D. M. Haldane, P h y s .R e v .L e t t . 61,2015 (1988 ). [15] V . P. Gusynin, S. G. Sharapov, and J. P. Carabotte, Int. J. Mod. Phys. B 21,4611 (2007 ). [16] E. C. Marino, Nucl. Phys. B 408,551(1993 ). [17] E. V . Gorbar, V . P. Gusynin, and V . A. Miransky, Phys. Rev. D 64,105028 (2001 ). [18] E. V . Gorbar, V . P. Gusynin, V . A. Miransky, and I. A. Shovkovy, Phys. Rev. B 66,045108 (2002 ). [19] J. S. Ball and T. W. Chiu, P h y s .R e v .D 22,2542 (1980 );22, 2550 (1980 ). [20] M. E. Carrington, C. S. Fischer, L. von Smekal, and M. H. Thoma, P h y s .R e v .B 94,125102 (2016 ). [21] M. E. Carrington, C. S. Fischer, L. von Smekal, and M. H. Thoma, P h y s .R e v .B 97,115411 (2018 ). 115432-11
PhysRevB.88.024115.pdf	PHYSICAL REVIEW B 88, 024115 (2013) First-principles study of helium, carbon, and nitrogen in austenite, dilute austenitic iron alloys, and nickel D. J. Hepburn,D. Ferguson, S. Gardner, and G. J. Ackland† Institute for Condensed Matter and Complex Systems, School of Physics and SUPA, The University of Edinburgh, Mayﬁeld Road, Edinburgh EH9 3JZ, United Kingdom (Received 22 January 2013; published 22 July 2013) An extensive set of ﬁrst-principles density functional theory calculations have been performed to study the behavior of He, C, and N solutes in austenite, dilute Fe-Cr-Ni austenitic alloys, and Ni in order to investigatetheir inﬂuence on the microstructural evolution of austenitic steel alloys under irradiation. The results showthat austenite behaves much like other face-centered cubic metals and like Ni in particular. Strong similaritieswere also observed between austenite and ferrite. We ﬁnd that interstitial He is most stable in the tetrahedralsite and migrates with a low barrier energy of between 0.1 and 0.2 eV . It binds strongly into clusters as well asovercoordinated lattice defects and forms highly stable He-vacancy (V mHen) clusters. Interstitial He clusters of sufﬁcient size were shown to be unstable to self-interstitial emission and VHe ncluster formation. The binding of additional He and V to existing V mHenclusters increases with cluster size, leading to unbounded growth and He bubble formation. Clusters with n/m around 1.3 were found to be most stable with a dissociation energy of 2.8 eV for He and V release. Substitutional He migrates via the dissociative mechanism in a thermal vacancy populationbut can migrate via the vacancy mechanism in irradiated environments as a stable V 2He complex. Both C and N are most stable octahedrally and exhibit migration energies in the range from 1.3 to 1.6 eV . Interactions betweenpairs of these solutes are either repulsive or negligible. A vacancy can stably bind up to two C or N atoms withbinding energies per solute atom up to 0.4 eV for C and up to 0.6 eV for N. Calculations in Ni, however, showthat this may not result in vacancy trapping as VC and VN complexes can migrate cooperatively with barrierenergies comparable to the isolated vacancy. This should also lead to enhanced C and N mobility in irradiatedmaterials and may result in solute segregation to defect sinks. Binding to larger vacancy clusters is most stablenear their surface and increases with cluster size. A binding energy of 0.1 eV was observed for both C and N toa [001] self-interstitial dumbbell and is likely to increase with cluster size. On this basis, we would expect that,once mobile, Cottrell atmospheres of C and N will develop around dislocations and grain boundaries in austeniticsteel alloys. DOI: 10.1103/PhysRevB.88.024115 PACS number(s): 61 .72.−y, 61.82.Bg, 71 .15.Mb, 75 .50.Bb I. INTRODUCTION Steel, in its many forms, is the primary structural material in current ﬁssion and fusion systems and will be so for theforeseeable future. Carbon (C) and nitrogen (N) are bothcommonly found in steel, either as important minor alloying elements or as low-concentration impurities. In body-centered cubic (bcc) α-iron ( α-Fe), it has been shown experimentally that C interacts strongly with vacancy point defects and moreweakly with self-interstitial defects 1,2and can form so-called Cottrell atmospheres around dislocations,3inﬂuencing yield properties and leading to strain aging of the material. First-principles ( ab initio ) calculations, as summarized in a recent review by Becquart and Domain, 4support these ﬁndings and demonstrate that N exhibits similarly strong interactions. As such, both of these elements have a signiﬁcant inﬂuenceon microstructural evolution in bcc Fe, even down to verylow concentrations, and a detailed understanding of theirinteractions and dynamics in steels is worthy of development,more generally. Helium (He) is produced in signiﬁcant quantities in the high neutron-irradiation ﬂuxes typically experienced by theinternal components of ﬁssion reactors and in the structuralmaterials for fusion systems by ( n,α) transmutation reactions. In combination with the primary point defect damage typical ofirradiated environments, the presence of He plays a critical rolein the microstructural evolution of these materials. As a resultof its low solubility in metals, He becomes trapped in regions of excess volume, such as dislocations, grain boundaries, and,most strongly, vacancies and vacancy clusters. 5–12As such, it aids the nucleation, stabilization, and growth of voids (Hebubbles), resulting in swelling of the material. 10,13–16The formation of He bubbles has also been implicated in high-temperature embrittlement of materials. 10,17,18It is therefore of critical importance to gain a deep understanding of thebehavior of He in these materials and the part it plays in theunderlying mechanisms of microstructural evolution. First-principles electronic structure calculations offer the most accurate means to develop an atomic level understandingof the dynamics and interactions of solutes and point defects in solids. As such, they play a central role in the development of a theoretical understanding of the microstructural evolution ofirradiated materials, as part of a multiscale modeling approach,such as that used in the FP6 project, PERFECT, 19and the FP7 project, PERFORM60.20 The behavior and interactions of He in a number of bcc and face-centered cubic (fcc) metals have been studiedusing density functional theory (DFT) techniques. 4,19,21–30 This database of He kinetics and interactions is essential for the interpretation of complex experimental results, such asthose present in thermal He desorption spectra. A case inpoint is the work of Ortiz et al. , 31who have developed a rate theory model based on DFT calculations of the kinetics and 024115-1 1098-0121/2013/88(2)/024115(26) ©2013 American Physical SocietyHEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) interactions of point defects, He and C in bcc Fe (Refs. 25,31, and33) The model successfully reproduces and interprets the existing experimental desorption results.8It is interesting to note that agreement with experiment was only possible oncethe effects of C were included, even though only 150 at. ppmof C was necessary; this again indicates the sensitivity of themicrostructural evolution to C concentration. To date, however,there have been no ab initio studies of He in austenite, that is fccγ-Fe, or austenitic FeCrNi alloys. This, primarily, is a result of the difﬁculty in describing the paramagnetic state ofthese materials. Ab initio calculations have also been used to extensively study C (Refs. 4,31,32, and 34–38) and N (Refs. 4and 35) in bcc Fe. These calculations show excellent agreement withexperimentally veriﬁable parameters, such as the migrationenergy barrier for C diffusion, where ab initio values of 0.86 eV (Refs. 34and 37), 0.87 eV (Ref. 38), and 0.90 eV (Ref. 35) are in good agreement with the experimental value of 0.87 eV (Refs. 2and39). For N, an equally good agreement is seen for the migration barrier, where a value of 0.76 eVwas found by ab initio calculations 35and a value of 0.78 eV was found experimentally.40Calculations in austenite are, however, limited primarily to solute dissolution, diffusion, andtheir inﬂuence on the electronic structure, local environment,and stacking fault energies, 34,41–46although calculations of vacancy-C binding have been performed.47 In this work we present a detailed study of the energetics, kinetics, and interactions of He, C, and N solutes in modelaustenite and austenitic systems using DFT. A full treatmentof paramagnetic austenite and FeCrNi austenitic alloys wouldnaturally take into account the magnetic and compositiondependence of the variables under study, and while ab initio techniques are now becoming available to model the para-magnetic state 48–51and calculations in concentrated alloys are certainly achievable,52their complexity precludes a broad study of all the necessary variables relevant for radiationdamage modeling. Previous studies have, instead, either takenferromagnetic (fm) fcc nickel (Ni) as a model austeniticsystem 19,22,53or modeled austenite using a small set of stable, magnetically ordered states, as in our previous work.54The advantage is that a more detailed study is possible, but the levelof approximation involved is certainly not ideal and careful useshould be made of the results obtained. Here, we follow thesame approach used in our previous work, 54performing our calculations in the two most stable ordered magnetic statesof fcc Fe. In addition, we present and compare the results ofcorresponding calculations in fm Ni in order to make moregeneral conclusions in Fe-Ni-based austenitic alloys. In Sec. IIwe present the details of our calculations. We then proceed to present and discuss our results for He, C,and N solutes in defect-free austenite and dilute Fe-Cr-Niaustenitic alloys in Sec. IIIand their interactions with point defects and small vacancy clusters in Sec. IVbefore making our conclusions. II. COMPUTATIONAL DETAILS The calculations presented in this paper have been per- formed using the plane-wave DFT code, V ASP ,55,56in the gen- eralized gradient approximation with exchange and correlationdescribed by the parametrization of Perdew and Wang57and spin interpolation of the correlation potential provided by theimproved V osko-Wilk-Nusair scheme. 58Standard projector augmented wave potentials59,60supplied with V ASP were used for Fe, He, C, N, Ni, and Cr with 8, 2, 4, 5, 10, and 6 valenceelectrons, respectively. First-order ( N=1) Methfessel and Paxton smearing 61of the Fermi surface was used throughout with the smearing width, σ, set to 0.2 eV to ensure that the error in the extrapolated energy of the system was less than1 meV per atom. A 2 3k-point Monkhorst-Pack grid was used to sample the Brillouin zone and a plane-wave cutoff of 450 eV .These pseudopotentials or exchange-correlation schemes areidentical to a wide body of previous work, where they werechosen to ensure reasonable magnetic moments and atomicvolumes. All calculations used supercells of 256 ( ±1,±2,...)a t o m s , with supercell dimensions held ﬁxed at their equilibriumvalues and ionic positions free to relax. For the relaxationof single conﬁgurations, structures were deemed relaxed oncethe forces on all atoms had fallen below 0.01 eV /˚A. For the nudged elastic band 62(NEB) calculations used to determine migration barriers an energy tolerance of 1 meV or betterwas used to control convergence. Spin-polarized calculationshave been performed throughout this work with local magneticmoments on atoms initialized to impose the magnetic stateordering but free to relax during the calculation. The relaxedlocal magnetic moments were determined by integrating thespin density within spheres centered on the atoms. Sphere radiiof 1.302, 0.635, 0.863, 0.741, 1.286, and 1.323 ˚Aw e r eu s e d for Fe, He, C, N, Ni, and Cr, respectively. We have performed our calculations in both the face- centered tetragonal (fct) antiferromagnetic single layer (afmI)and double layer (afmD) collinear magnetic reference statesfor austenitic Fe (at T=0 K), which we refer to as afmD Fe and afmI Fe, respectively, in what follows, using the same methodology as our previous work. 54Both of these structures consist of (ferro-)magnetic (001) fcc planes, whichwe refer to as magnetic planes in what follows, but withopposite magnetic moments on adjacent planes in the afmIstate and an up,up,down,down ordering of moments in adjacentmagnetic planes in the afmD state. The fcc fm and fct fmstates were found to be structurally unstable and spontaneouslytransformed upon addition of a whole range of defects andsolutes. 54The fcc ferromagnetic high-spin (fm-HS) state was, however, found to be stable to isotropic effects andwe have performed a select few calculations in this statefor comparison with other work in the literature. 34We have previously attempted to use randomly disordered moments torepresent paramagnetism 49,51,54and found that but for migra- tion processes (and some relaxations) the spins spontaneouslyreorient, making it impossible to deﬁne a reference state. Thisis probably due to the low-symmetry conﬁgurations requiredand the low paramagnetic transition temperature in Fe. Wehave also performed a number of calculations in fcc fm Ni,which we refer to, simply, as Ni in what follows, where theseresults were not available in the literature. We take the latticeparameters for afmI Fe as a=3.423 ˚A and c=3.658 ˚A, those for afmD Fe as a=3.447 ˚A and c=3.750 ˚A and take a= 3.631 ˚A for fm-HS Fe. Calculations in Ni have been performed with an equilibrium lattice parameter of a=3.522 ˚A. The 024115-2FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) corresponding magnitudes for the local magnetic moments in bulk, equilibrium afmI, afmD, and fm-HS Fe were determinedas 1.50, 1.99, and 2.57 μ B, respectively,54and a local moment of 0.59 μBwas found in bulk, equilibrium Ni. Convergence tests indicated that local moments were determined to a fewhundredths of μ B. We use elastic constants for our reference states, as deter- mined previously,54or determined here using the same tech- niques. For fm-HS Fe, we ﬁnd C11=40 GPa, C12=240 GPa, andC44=− 10 GPa, which clearly shows instability to shear strains and tetragonal deformations, as C/prime=C11−C12= −200 GPa. It is, however, stable to isotropic deformations as the bulk modulus, B=187 GPa, is positive. For Ni, we ﬁndC11=272 GPa, C12=158 GPa, and C44=124 GPa, which gives C/prime=114 GPa and B=196 GPa, and shows that this material is stable to any strain deformations. We have determined the solution enthalpy for carbon in Fe and Ni using diamond as a reference state. The diamondstructure was determined using the same settings as ourother calculations but with sufﬁcient k-point sampling to ensure absolute convergence of the energy. We found alattice parameter of a=3.573 ˚A, in good agreement with the standard experimental value. We deﬁne the formation energy, E f, of a conﬁguration containing nXatoms of each element, X, relative to a set of reference states for each element using Ef=E−/summationdisplay XnXEref X, (1) where Eis the calculated energy of the conﬁguration and Eref Xis the reference state energy for element, X. We take the reference energies for Fe, Ni, and Cr to be the energies peratom in the bulk metal, that is, Fe in either the afmI, afmD, orfm-HS states, as appropriate, Ni in its fcc fm ground state, andCr in its bcc antiferromagnetic (afm) ground state. Details ofthe Fe and Ni reference states are given above, whereas for Cran equilibrium lattice parameter of 2.848 ˚A was found with a corresponding local moment of magnitude 0.87 μ B.F o rH e , C, and N the reference states were taken to be the nonmagneticfree atom, as calculated in V ASP . In a similar manner, we deﬁne the formation volume at zero pressure, Vf, of a conﬁguration relative to the bulk metal by Vf=V(0)−nbulkVbulk, (2) where V(0) is the volume of the conﬁguration at zero pressure, nbulkis the number of bulk (solvent) metal atoms in the conﬁguration and Vbulkis the volume per atom in the defect- free bulk metal, which we found to be 11.138, 10.712, 11.970, and 10.918 ˚A3in afmD Fe, afmI Fe, fm-HS Fe, and Ni, respectively. For our calculations, V(0) was determined by extrapolation from our calculations at the ﬁxed equilibriumvolume using the residual pressure on the supercell and thebulk modulus for the defect-free metal. We deﬁne the binding energy between a set of nspecies, {A i}, where a species can be a defect, solute, clusters of defects and solutes, etc., as Eb(A1,..., A n)=n/summationdisplay i=1Ef(Ai)−Ef(A1,..., A n),(3)where Ef(Ai) is the formation energy of a conﬁguration containing the single species, Ai, and Ef(A1,..., A n)i s the formation energy of a conﬁguration containing all ofthe species. With this deﬁnition an attractive interactionwill correspond to a positive binding energy. One intuitiveconsequence of this deﬁnition is that the binding energy ofa species, B, to an already existing cluster (or complex) of species, {A 1,..., A n}, which we collectively call C,i sg i v e n by the simple formula Eb(B,C )=Eb(B,A 1,..., A n)−Eb(A1,..., A n). (4) This result will be particularly useful when we consider the additional binding of a vacancy or solute to an already existingvacancy-solute complex. We have quantiﬁed a number of uncertainties in the forma- tion and binding energies presented in this work. Test calcula-tions were performed to determine the combined convergenceerror from our choice of k-point sampling and plane-wave cutoff energy. For interstitial C and N solutes in a defect-freelattice, formation energies were converged to less than 0.05 eVand formation energy differences, such as migration energies,to less than 0.03 eV . For interstitial He the convergence errorswere half of those for C and N. For conﬁgurations containingvacancies or self-interstitial defects, formation energies wereconverged to 0.03 or 0.07 eV , respectively, while bindingenergies were converged to 0.01 eV , except for the bindingof He to a vacancy, where the error was 0.03 eV . The zero-point energy (ZPE) contributions to the formation energy, which can be signiﬁcant for light solute atoms, havenot been calculated in this work. We performed calculationsof the ZPE for He, C, and N solutes in a number of test sites inafmD and afmI Fe, keeping the much heavier Fe atoms ﬁxed,which is equivalent to assuming they have inﬁnite mass. 34The results showed that the ZPE contributions were consistentlyaround 0.10 eV in all cases, which we take as an estimateof the ZPE error on the formation energies of conﬁgurationscontaining C, N, and He. The variation with site was, however,surprisingly low at 0.01 eV , which we take as an estimate of theZPE error in formation energy differences, binding energies,and the solution enthalpy for C, given that ZPE contributionin graphite is very similar to in Fe (Ref. 34). Performing calculations in a ﬁxed supercell of volume, V, results in a residual pressure, P, for which an Eshelby- type elastic correction to the total and, therefore, formationenergy 63,64ofEcorr.=−P2V/2B, can be applied. As such, Ecorr.also serves to indicate the likely ﬁnite-volume error. For many of the conﬁgurations considered here these correctionsare negligible compared to other sources of error. Where theyare signiﬁcant, however, their relevance is discussed at theappropriate points in the text. III. SOLUTES IN THE DEFECT-FREE LATTICE A. Single solutes The formation energies for substitutionally and interstitially sited He, C, and N solutes in the sites shown in Fig. 1are given in Table Ifor Fe and Table IIfor Ni. We found that the Eshelby corrections were negligible for substitutional He but could beas high as 0.02 eV in magnitude for interstitial He and 0.04 eV 024115-3HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) 124 6 35 0 x,[100]z,[001] y,[010] FIG. 1. Substitutional (0) and interstitial octahedral (1), tetrahe- dral (2-3), and crowdion (4-6) positions (in black) in afmD Fe. The Fe atoms are shown in white with arrows to indicate the local moments. Magnetic planes are included to aid visualization. The afmD Fe state,which has the lowest symmetry, is shown to uniquely identify all distinct positions. In afmI Fe and Ni, positions 2 and 3 are equivalent by symmetry, as are 5 and 6. In Ni, position 4 is also equivalentto 5 and 6. for interstitial C and N. The corresponding uncertainties in for- mation energy differences were around half of these values. Wediscuss the results for He ﬁrst, followed by those for C and N. 1. He solute We found that He exhibits a large, positive formation energy in all sites but is most stable substitutionally, whichTABLE II. Formation energies, Ef, in eV , for substitutionally and interstitially sited He, C and N atoms in Ni. The layout and data content is as in Table I. He C N Conﬁg. Ef(/Delta1E f) Ef(/Delta1E f) Ef(/Delta1E f) sub (0) 3.185 −5.386 −4.562 (—) (—) (—) octa (1) 4.589 −8.422 −7.520 (0.129) (0.000) (0.000) tetra (2-3) 4.460 −6.764 −6.497 (0.000) (1.659) (1.023) /angbracketleft110/angbracketrightcrow. (4-6) 4.651 −6.795 −5.970 (0.191) (1.628) (1.550) is consistent with existing DFT studies of He in other bcc and fcc metals.4,19,21–30The standard explanation is that, as a closed-shell noble-gas element, bonding interactions should beprimarily repulsive, leading to insolubility and a preference forsites with the largest free volume. 21,26This result distinguishes He from other small solutes, such as C and N, which are morestable interstitially but also distinguishes it from substitutionalalloying elements, such as Ni and Cr with formation energydifferences between substitutional and interstitial sites in Feof 3.0 eV and above. 54,65 In Fe, the inﬂuence of substitutional He on the local magnetic moments of atoms in its ﬁrst-nearest-neighbor (1nn)shell was found to be similar to those for a vacancy, beinggenerally enhanced relative to the bulk moment and by upto 0.38 μ Bhere. This is similar to He in bcc Fe (Refs. 25 and26). Indeed, we found that if the He atom was removed from the relaxed substitutional conﬁguration with no furtherrelaxation, the local 1nn Fe moments changed by less than TABLE I. Formation energies, Ef, in eV , for substitutionally and interstitially sited He, C and N atoms in austenite, as shown in Fig. 1.T h e formation energies in bold are for the most stable states. For He, which is most stable substitutionally, the most stable interstitial site is also highlighted. The formation energy differences, /Delta1E f(in brackets), to the most stable interstitial conﬁgurations are also given, in eV . Where the conﬁguration was found to be unstable the conﬁguration to which it relaxed is given. The substitutional N conﬁguration in the fct afmD state relaxed to one with an octa N at 1 nn to a vacancy. He C N afmD Fe afmI Fe afmD Fe afmI Fe afmD Fe afmI Fe Conﬁg. Ef(/Delta1E f) Ef(/Delta1E f) Ef(/Delta1E f) Ef(/Delta1E f) Ef(/Delta1E f) Ef(/Delta1E f) sub (0) 4.024 4.185 −6.981 −6.244 rlx (other) −5.153 (—) (—) (—) (—) (—) octa (1) 4.669 5.026 −8.797 −8.856 −8.602 −8.621 (0.206) (0.059) (0.000) (0.000) (0.000) (0.000) tetra uu (2) 4.529 as −6.535 as −6.917 as (0.066) tetra ud (2.261) tetra ud (1.685) tetra ud tetra ud (3) 4.464 4.967 −6.644 −6.272 −7.044 −6.737 (0.000) (0.000) (2.153) (2.585) (1.558) (1.884) [110] crow. (4) rlx (3) 5.271 −6.764 −6.412 rlx (3) −6.006 (0.303) (2.033) (2.445) (2.614) [011] crow. uu (5) 4.827 as −7.354 as −7.000 as (0.364) [011] crow. ud (1.443) [011] crow. ud (1.602) [011] crow. ud [011] crow. ud (6) 4.802 5.188 −7.487 −6.744 −7.218 −6.328 (0.338) (0.221) (1.310) (2.113) (1.384) (2.293) 024115-4FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) 0.03μB. In contrast to the vacancy, however, where 1nn Fe were displaced inwards by 0.09 and 0.02 ˚Ai na f m Da n d afmI Fe, respectively,54the respective displacements around a substitutional He were, on average, outwards by 0.02 and0.04 ˚A. This contrast can also be seen in the formation volumes, which were found to be 0.74 V bulkand 0.96 Vbulkfor a vacancy, compared to 1.17 Vbulkand 1.38 Vbulkfor substitutional He in afmD and afmI Fe, respectively. Results in Ni were very similarto Fe, with enhanced moments in the 1nn shell around both avacancy and substitutional He, a contraction of 0.04 ˚Ai nt h e 1nn shell around a vacancy, and an expansion of 0.02 ˚A around substitutional He. The formation volume for substitutional He,at 1.02 V bulk, was again found to be greater than that for the vacancy, at 0.66 Vbulk. The large formation energy difference, of around 2 eV , between substitutional He and the underlying vacancy inFe and Ni (see Sec. IV), which must be due to chemical interactions, may seem at odds with the relatively inertbehavior of He mentioned above. However, similar results inbcc Fe have been reproduced using simple pair potentials, 66,67 which demonstrates that such a large energy difference, once distributed over 1nn and 2nn bonds, is commensurate with therelatively small forces observed on the neighboring Fe atomsaround substitutional He. In Fe, interstitial He was found to be most stable in the tetrahedral (tetra) site, the octahedral (octa) site being the nextmost stable and lying 0.206 and 0.059 eV higher in energy inthe afmD and afmI states, respectively. There is no consistentordering of the octa and tetra sites in ab initio studies of other fcc metals, with the octa site being most stable in Ag (Ref. 21), Al (Ref. 23), and Pd (Ref. 24and24) and the tetra site being most stable in Cu (Ref. 21) and Ni (Refs. 19,21, and 22), as our results for Ni conﬁrm. In both Fe and Ni, however, He favorsthe tetra site, which gives a strong indication that the tetrasite will also be the most stable interstitial site in concentratedFe-Ni-based austenitic alloys. The other interstitial sites considered here lie no more than 0.364 eV above the tetra site, suggesting many low-energymigration paths for interstitial He, that is, in the absence ofany lattice defects that can act as strong traps. The bilayerstructure in afmD Fe breaks the symmetry of the octa site anda He atom placed there was found to spontaneously relax in the[00¯1] direction (as deﬁned in Fig. 1), to between layers of the same spin by 0.55 ˚A. It is, perhaps, surprising that in both afmI Fe and Ni, an octa-sited He was also found to be unstable tosmall displacements in many directions. We present the resultsof these calculations in Table III. It is particularly clear in the afmI Fe data that lower energy conﬁgurations were found along all of our test directions, withHe relaxing to between 0.23 and 0.62 ˚A from the symmetrical position. The picture is less clear in afmD Fe, where He wasgenerally found to relax to the lowest local energy minimumbut other metastable positions were found. In Ni, the drop inenergy is far less pronounced than in Fe but is still present,with He relaxing to stable positions 0.29 ˚A from the center along /angbracketleft100/angbracketrightdirections and 0.54 ˚A along /angbracketleft110/angbracketrightdirections. These conﬁgurations are important, certainly as intermediatestates for the migration of interstitial He, but also as potentialtransition states and already suggest a low migration-energybarrier. We study these possibilities in detail in Sec. III B .TABLE III. Formation energies, Ef, and formation energy dif- ferences, /Delta1E f, in eV , to the most stable tetra site (in brackets) for octa-sited He atoms in Fe. He is either sited symmetrically (sym.) or has been displaced off center, in which case the direction of thedisplacement is used to label the conﬁguration and the displacement length after relaxation, /Delta1r,i sg i v e n ,i n ˚A. The symmetrical position is as shown in Fig. 1and directions determined from that point with the coordinate system shown. When no stable local energy minimum was found the state to which the conﬁguration relaxed is given. afmD Fe afmI Fe Ni Ef Ef Ef Conﬁg. ( /Delta1E f)/Delta1r (/Delta1E f)/Delta1r (/Delta1E f)/Delta1r rlx 5.208 4.617octa sym. 0.00 0.00octa [00 ¯1] (0.241) (0.157) rlx 5.105 4.607octa [100] 0.39 0.29octa [00 ¯1] (0.138) (0.147) 4.812 5.026 asocta [001] 0.30 0.50(0.348) (0.059) octa [100] 4.669 as asocta [00 ¯1] 0.58(0.206) octa [001] octa [100] rlx 5.079 4.589octa [110] 0.54 0.54octa [00 ¯1] (0.112) (0.129) 4.799 5.035 asocta [011] 0.58 0.23(0.335) (0.068) octa [110] rlx as asocta [01 ¯1]octa [00 ¯1] octa [011] octa [110] rlx 5.029 rlxocta [111] 0.62tetra ud (0.062) tetra rlx as asocta [11 ¯1]octa [00 ¯1] octa [111] octa [111] For completeness, we also tested for the presence of stable off-center positions for tetra-sited He but relaxation alwaysreturned He to the symmetrical position. The displacements of 1 nn Fe atoms around interstitially sited He were, unsurprisingly, found to be greater than forthe substitutional site. A tetra-sited He in afmI Fe displacedits neighbors by 0.23 ˚A. In afmD Fe, displacements of 0.22 and 0.32 ˚A were found for tetra uu- and tetra ud-sited He, respectively. The magnetic moments on the 1 nn Fe atomswere quenched relative to the bulk moments by 0.24 μ Bin afmI Fe and by 0.16 μBfor the tetra uu site in afmD Fe but enhanced by 0.15 μBfor the tetra ud site. We attribute this difference to the greater free volume into which 1 nn Fe atomsaround a tetra ud site may be displaced. We found formationv o l u m e so f0 . 8 2 V bulkand 0.99 Vbulkfor tetra-sited He in afmI Fe and tetra-ud-sited He in afmD Fe, respectively. Once again,results in Ni were similar to Fe, with a 0.24 ˚A displacement and moment quench of 0.09 μ Bin 1 nn Ni atoms around a tetra-sited He and a formation volume of 0.78 Vbulk. In the most stable octa conﬁguration in Fe, the local geometry is complicated by the displacement of He from the symmetrical position. For that reason, we deﬁne a localunit cell surrounding the octa site using the positions of itssix 1 nn metal atoms, which lie at the centers of the cellfaces, and report on the lattice parameters of that cell. In both 024115-5HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) afmI and afmD Fe, the local lattice parameter along [100] and [010] directions, a1nn, is increased by 0.31 ˚A relative to the bulk equilibrium lattice, with the local lattice parameter alongthe [001] direction, c 1nn, exhibiting an increase of 0.26 ˚A in the afmD state and 0.29 ˚A in the afmI state. The local moment on the 1nn Fe atom that He is displaced towards issigniﬁcantly quenched by 1.04 and 0.41 μ Bin the afmD and afmI states, respectively. In contrast, the other 1 nn moments are moderately enhanced by between 0.03 and 0.17 μB.I n Ni, the most stable off-center octa position is along /angbracketleft110/angbracketright directions from the symmetrical position. The resulting localunit cell, which exhibits a very slight shear, has c 1nn/negationslash=a1nn, witha1nnincreased by 0.31 ˚A relative to bulk and c1nnby 0.28 ˚A. Local 1 nn Ni moments were found to be quenched by between 0.02 and 0.08 μB. These ﬁndings suggest that the relative stability of tetra over octa He, which is opposite to the order suggested by free volume arguments,21,26may be best ascribed to the relative ease with which a tetra He may lower its purely repulsiveinteractions with neighboring atoms by local dilatation. Tofurther investigate this hypothesis in Fe we split the formationenergy for unrelaxed and relaxed substitutional octa and tetraHe conﬁgurations into three terms, in a similar manner to that in the work of Fu et al. 38The ﬁrst is the formation energy, Edef. f, of any underlying, atomically relaxed, defects, e.g., a single vacancy for substitutional He. The second is themechanical energy, E mech. f , required to deform the Fe matrix containing those relaxed defects to the exact positions found inthe conﬁguration under study. The third is the energy changefrom chemical interactions, E chem. f , upon insertion of the solute into its ﬁnal position with no further relaxation. We also deﬁne the insertion energy, Eins. f,a st h es u mo f Emech. f and Echem. f , i.e., the formation energy for insertion of a solute into any position in a relaxed Fe matrix containing any relevantdefects. We take the insertion energy as a more appropriatemeasure of site preference than the (total) formation energy,E f. The results are given in Table IV. TABLE IV . Mechanical deformation energy, Emech. f, and chemical bonding energy, Echem. f, contributions to the total formation energy, Ef, and the insertion energy, Eins. f, for unrelaxed and relaxed substitutional, tetra and octa He solute conﬁgurations in afmD and afmI Fe, in eV . The most stable octa conﬁguration was used in bothstates and the tetra ud conﬁguration was used for the afmD state. Conﬁg. Emech. f Echem. f Eins. f afmD Fe +He sub, unrelaxed 0.136 2.150 2.286 sub, relaxed 0.167 2.045 2.212 tetra, unrelaxed 0.000 6.778 6.778 tetra, relaxed 1.330 3.134 4.464octa, unrelaxed 0.000 5.804 5.804 octa, relaxed 0.755 3.914 4.669 afmI Fe +He sub, unrelaxed 0.023 2.662 2.685 sub, relaxed 0.155 2.073 2.228 tetra, unrelaxed 0.000 6.774 6.774tetra, relaxed 0.999 3.968 4.967 octa, unrelaxed 0.000 6.081 6.081 octa, relaxed 0.855 4.171 5.026The substitutional site is clearly the most favored, even in the unrelaxed state and by at least 2.25 eV once relaxed. Inthe unrelaxed lattice, an octa He is signiﬁcantly more stablethan a tetra He, as expected from purely repulsive interactionsgiven the relative proximity of 1 nn Fe in the two sites.Under relaxation the chemical bonding energy is signiﬁcantlyreduced and to a far greater degree in the tetra site. The positivemechanical deformation energy is also greater for tetra He butthe net result is still to stabilize tetra over octa He. These resultsclearly show that the relative stability of He in tetra and octasites can be understood as resulting from a balance betweenthe energy required for local dilatation of the Fe matrixcoupled with a purely repulsive Fe-He interaction, which wesuggest could be easily modeled using a simple pair potential. In bcc Fe, the relative stability of tetra over octa He has been explained as resulting from strong hybridization of He pstates with Fe dstates. 21,26However, we do not ﬁnd the evidence for such strong hybridization to be convincing. We suggest that arepulsive nonbonding mechanism also applies to bcc Fe andexplains the difference in a much simpler manner. The mag-netic and polarization effects discussed by Seletskaia et al. 26 and Zu et al.21are a simple consequence of these nonbonding interactions and not He p-state, Fe d-state hybridization. Formation energy calculations21,26show that octa-sited He is higher in energy both before and after relaxation, despitethe relaxation energy for octa He being greater than for tetraHe. This results, primarily, from the very short 1 nn Fe-Heseparations in the octa site when compared to those for thetetra site and the relative strengths of the resulting repulsiveinteractions. The fact that purely repulsive pair potentials forFe-He interactions in bcc Fe are capable of reproducing therelative stability 66,67gives further support to our claim. 2. C and N solutes The results for C and N solutes (in Tables IandII)s h o w that both elements clearly favor the octa interstitial site inboth Fe and Ni. Experimental observations show this to bethe preferred site for C in an Fe-13wt%Ni-1wt%C austeniticalloy. 68One exception worth comment is that of substitutional C in afmD Fe, for which the insertion energy, Eins. f, which as discussed for He provides a more appropriate measure ofsite preference, is comparable to that for octa C. On furtherinspection we found that, due to the asymmetries in the afmDstate, the initially on-lattice C atom relaxed to 0.77 ˚Af r o mt h e lattice site. While this displacement is certainly signiﬁcant, theC atom remains closer to the substitutional site than to an octaposition at 1 nn to the (vacated) lattice site and has been namedto reﬂect this difference. Relaxation of the substitutional Nconﬁguration also resulted in displacement away from thelattice site but convergence was to a conﬁguration with theN atom in an octa site at 1 nn to a vacancy. We performedcalculations to test for the presence of any stable off-center octaconﬁgurations for C and N but none were found, in contrast tothe results for He. We discuss the inﬂuence of octa C and N solutes on the local lattice geometry in an identical manner to octa-sitedHe, that is, using a 1nnandc1nn. The results are presented in Table Vfor both Fe and Ni, including results in fm-HS Fe, which was shown to be mechanically unstable in our previous 024115-6FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) TABLE V . Lattice parameter differences ( /Delta1a 1nnand/Delta1c 1nn,i n˚A) between those for the unit cell surrounding octa C and N solutes ( a1nnand c1nn) and the bulk equilibrium lattice parameters and the local c1nn/a1nnratio. Linear expansion coefﬁcients ( /Delta1a/ (axf X)a n d/Delta1c/(cxf X)) for the dependence of the lattice parameters on the fractional atomic solute composition, xf X, for solute X. For afmD and afmI Fe, the linear expansion coefﬁcient for an effective lattice parameter, deﬁned by aeff.=(a2c)1/3is also given. Fractional formation volumes, Vf/Vbulkfor octa-sited C and N solutes are given. The solution energy, Esol. f,G, taken to dissolve graphite in each of the reference states is given, in eV . For comparison, our calculations in bcc fm Fe give Esol. f,G=0.700 eV . afmD Fe afmI Fe fm-HS Fe Ni C N CN C NCN /Delta1a 1nn 0.321 0.303 0.305 0.276 0.174 0.145 0.183 0.170 /Delta1c 1nn 0.080 0.048 0.154 0.127 c1nn/a1nn 1.016 1.013 1.023 1.023 /Delta1a/ (axf X) 0.266 0.265 0.341 0.327 0.072 0.034 0.243 0.263 /Delta1c/(cxf X) −0.057 −0.044 0.026 0.042 /Delta1a eff./(aeff.xf X) 0.158 0.162 0.236 0.232 Vf/Vbulk 0.53 0.54 0.78 0.76 0.214 0.102 0.73 0.79 Esol. f,G 0.323 0.263 −0.164 0.697 work,54but not to the isotropic strain exerted locally by an octa-sited solute. We include this extra state here to comparewith the work of Jiang and Carter. 34It is immediately clear that the geometrical inﬂuence of octa C is rather similar toocta N, although with slightly smaller dilatations for N. Localexpansion is observed in all our reference states, although theexpansion of cin afmD and afmI Fe is much less than for a.A s a result, the local c/aratio is signiﬁcantly reduced relative to the bulk material, to 1.02 around a C solute in both afmD andafmI Fe, which is in good agreement with the 3% tetragonaldistortion found by Boukhvalov et al. , 42and to 1.01 and 1.02 around an N solute in afmD and afmI Fe, respectively. The magnetic inﬂuence of octa C and N solutes is, again, very similar with signiﬁcant quenching of the local momentson 1 nn solvent atoms seen in all reference states, as expectedfor magnetic atoms under compression. In both afmD andafmI Fe the effect is most pronounced in those neighbors lyingwithin the same magnetic plane as the solute, which also showthe most signiﬁcant displacement, resulting in a quench of0.72(0.66) μ Bfor C(N) in afmD Fe and of 1.25(1.37) μB in afmI Fe. In fm-HS Fe, 1 nn moments are quenched by 0.48(0.57) μBaround C(N) and in Ni a quench of 0.42 μBwas observed for both C and N. In addition to this local inﬂuence, we have investigated the dependence of the lattice parameters of our reference stateson the fractional atomic compositions, x f Candxf Nfor C and N, respectively. For low concentrations, as studied here, thelattice parameters change linearly as a function of the fractionalcomposition. 69In this case, quantities such as /Delta1a/ (axf C), where /Delta1ais the difference between the lattice parameter with and without solute atoms present, are dimensionless constantsthat completely specify the linear expansion. Our calculationshave been performed in supercells at the equilibrium latticeparameters, so we determine the linear expansion coefﬁcientsby extrapolating to zero stress using the residual stress thatbuilds up on the supercell upon addition of a solute and aknowledge of the elastic constants (see Sec. II). In afmD and afmI Fe we have also calculated the linear expansioncoefﬁcients for an effective lattice parameter, a eff.=(a2c)1/3, as a means to compare more directly with experiment.The results (in Table V) show that local expansion around the solutes leads to a net expansion of the cell overall. TheafmD state of Fe does, however, exhibit a small contractionincand the afmI state shows very little expansion in c, when compared to that for a. Once again, this shows that the addition of C and N acts to reduce the c/a ratio, bringing the lattice back toward perfect fcc. In austenite, experimental results byCheng et al. 69and presented by Gavriljuk et al.41show that /Delta1a/ (axf C) lies between 0.199 and 0.210, with /Delta1a/ (axf N) being slightly greater at between 0.218 and 0.224. Our results inafmD and afmI Fe are in broad agreement with these valuesbut do not differentiate between C and N. Results in fm-HSFe are signiﬁcantly different from experiment, which againshows the unsuitability of this state for modeling austenite.It is interesting to note that our results for Ni are consistentwith those for austenite and do exhibit more expansion dueto N than for C. Experimental results in austenitic FeCrNialloys 41are comparable to those for pure austenite and the general agreement with our results strengthens the case forusing afmD and afmI Fe or using Ni as model systems foraustenite and austenitic alloys. We have determined the solution energy at ﬁxed equilibrium volume, E sol. f,G, taken to dissolve graphite into our four bulk states (Table V). We have done this by calculating the solution energy relative to diamond and then applying the commonlyaccepted experimental energy difference of 20 meV /atom between the cohesive energies of diamond and graphite atT=0K( R e f . 70). For comparison, we have calculated E sol. f,G for C in bcc fm Fe and ﬁnd a value of 0.70 eV , which is in good agreement with the experimental value of between 0.60and 0.78 eV found by Shumilov et al. 34,71 The solution energies in all three Fe states are signiﬁcantly lower than for bcc fm Fe. This is consistent with the relativelyhigher solubility of C in austenite than in ferrite and withthe well-known experimental result that C stabilizes austeniteover ferrite, as seen in the phase diagram. The effect is mostpronounced in the fm-HS state, where the reaction is exother-mic, in good agreement with previous DFT calculations. 34 In combination with the results discussed above, this impliesthat at sufﬁcient concentrations, C will act to stabilize the 024115-7HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) fm-HS state over the others, just as was found for Ni in fcc Fe-Ni alloys.72The same conclusions follow for N by a direct comparison of the formation energies for octa sited N (seeTable I), for which we found values of −8.252 and −9.018 eV in bcc fm and fcc fm-HS Fe, respectively. Experimental results for the solution enthalpy of C in austenite 73yield a value of Esol. f,G=0.37 eV at the concen- tration studied here, which agrees to within 0.1 eV withour calculations in afmD and afmI Fe but not with thosefor the fm-HS state and again supports their suitability asreference states for paramagnetic austenite. Our calculationsin the ferromagnetic state for Ni are in good agreementwith previous DFT calculations of Siegel and Hamilton. 74 However, as they report, this value is higher than those foundexperimentally in high-temperature, paramagnetic Ni, whichlie between 0.42 and 0.49 eV . It is worth noting that theircalculations in nonmagnetic (nm) Ni, which they use to modelthe paramagnetic state, underestimate the experimental rangeat between 0.2 and 0.35 eV . We conclude that the calculatedsolution enthalpy for C in Ni is particularly sensitive to theunderlying magnetic state. B. Solute migration As a ﬁrst step in the calculation of migration energies for He, C, and N solutes we investigated whether a 32-atom cellwould be sufﬁcient for this purpose. To do this we recalculatedthe formation energies for substitutional and interstitial He andC in afmD Fe using a 32-atom cell. We compare these withour 256-atom cell calculations (Table I) in Table VI. There is a signiﬁcant size effect on the formation energies in the 32-atom cell, except for substitutional He and octa C,where the formation energies are within errors of those inthe 256-atom cell. The formation energies are greater in the32-atom cell, as expected from volume-elastic effects, bybetween 0.06 and 0.12 eV for interstitial He and by between0.00 and 0.37 eV for interstitial C. Formation energy differ-ences to the most stable interstitial conﬁguration also exhibita signiﬁcant size effect, with the smaller cell underestimatingthem by between 0.04 and 0.06 eV for He conﬁgurations andoverestimating them by between 0.10 and 0.37 eV for C. Itis reasonable to assume that the migration energy, which isitself a formation energy difference, will suffer from similarsize effects. For C, the choice of cell size actually changes the relative stability of the [110] crowdion and tetra ud conﬁgurations. Thisis important as these two are transition states on two distinctmigration paths for C (as will be shown in what follows). Thesmall cell would, therefore, give the wrong minimum energypath (MEP), as found previously for C in fm-HS Fe (Ref. 34). Closer inspection of the [110] crowdion conﬁguration showedthat the periodic boundary conditions in the smaller cellapplied unphysical constraints on the displacements of Featoms at 1 nn to C and along the crowdion axis generally,which resulted in a signiﬁcant buckling, moving the C atomtowards the tetra uu site, that is along [001], by 0.71 ˚A. In the larger cell these constraints are not present, resulting in asigniﬁcantly lower formation energy and while there is still asmall displacement towards the tetra uu site of 0.18 ˚At h i si s to be expected given the asymmetry present in the afmD state.TABLE VI. Comparison between calculations in 32-atom and 256-atom cells in afmD Fe of the formation energies, Ef,i ne V ,f o r substitutional and interstitial He and C solutes and formation energy differences, /Delta1E f, in eV , to the most stable interstitial conﬁgurations, highlighted in bold. The layout and data content of each column is as in Table I. The column headed “32 atom” contains the results for the 32-atom cell and the column headed “Error” contains the differencebetween the 32-atom and 256-atom results, which we take as an estimate of the ﬁnite volume error in the 32-atom cell. He C 32 atom Error 32 atom Error Ef Ef Ef Ef Conﬁg. ( /Delta1E f)( /Delta1E f)( /Delta1E f)( /Delta1E f) 4.039 0.015 −6.911 0.070Sub (0)(—) (—) (—) (—) 4.730 0.061 −8.798 −0.001octa (1)(0.151) ( −0.055) (0.000) (0.000) 4.607 0.078 −6.395 0.140tetra uu (2)(0.028) ( −0.038) (2.403) (0.142) 4.579 0.115 −6.544 0.100tetra ud (3)(0.000) (0.000) (2.255) (0.102) −6.396 0.368[110] crow. (4) rlx (3) rlx (3)(2.402) (0.369) 4.897 0.070 −7.209 0.145[011] crow. uu (5)(0.318) ( −0.046) (1.589) (0.146) 4.866 0.064 −7.346 0.141[011] crow. ud (6)(0.287) ( −0.051) (1.452) (0.142) As a ﬁnal test, we investigated the case of C in fm-HS Fe, where Jiang and Carter have determined a migration barrierin a 32-atom cell. 34They found that the /angbracketleft110/angbracketrightcrowdion site is an intermediate site for C migration, lying only 0.01 eVbelow the transition state energy and 0.98 eV above the stableocta site. Our calculations in a 32-atom cell agree well with thisﬁnding, with an energy difference of 1.01 eV between the /angbracketleft110/angbracketright crowdion and octa sites for C. However, when we repeated thecalculations in a 256-atom cell, we found that a conﬁgurationwith C in the /angbracketleft110/angbracketrightcrowdion was structurally unstable and spontaneously transformed as a result of the nonisotropic stresson the Fe lattice. By contrast, the isotropic stress from anocta-sited C only led to local relaxation of the Fe matrix andmaintenance of the crystal structure. We conclude that the 32-atom cell effectively imposed artiﬁcial constraints that alloweda seemingly sensible migration barrier to be determined. Overall, we ﬁnd that the ﬁnite size effects in the 32-atom cell are too signiﬁcant and while some intermediate cellsize between the two investigated here may be sufﬁcient,we have performed our migration energy calculations in the256-atom cell. 1. Interstitial He migration The migration of interstitial He is relevant in the initial stages after He production by transmutation and α-particle irradiation and at sufﬁciently high temperatures for He toescape from defect traps. The migration of He betweenadjacent tetra sites (that is, between sites at 1 nn on the 024115-8FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) ud uu1 2 3 FIG. 2. Possible migration paths for interstitial He in the afmD Fe lattice. Paths are shown for 1 nn jumps from initial to ﬁnal tetrapositions (black circles) via off-center octa intermediate transition state positions (gray circles). The Fe atoms (white circles) are shown with arrows to indicate the local moments. The symmetry of the afmDstate leads to two distinct tetra sites (uu and ud) and three distinct 1 nn jumps, as shown. In the afmI state paths 1 and 3 are equivalent but still distinct from path 2. Coordinate axes are as in Fig. 1. cubic sublattice of tetra sites) can proceed along many distinct paths, with their corresponding transition states deﬁning theenergy barrier for the transition. A direct path would lead toan intermediate state with He in the crowdion position butthe energy differences to the tetra conﬁgurations (in Tables I andIII) suggest that the direct path is not the MEP and that the transition state has He in an off-center position. We showrepresentative paths for the three distinct 1 nn jumps in afmDFe in Fig. 2. We have performed NEB calculations for He migration in Fe along these paths and show the formation energy differenceto the most stable interstitial conﬁguration against a suitablychosen reaction coordinate in Fig. 3, with the corresponding migration barrier heights given in Table VII. It is immediately Tetra udI1Tetra udI2Tetra uu/udI3Tetra uu Reaction Coordinate00.10.20.30.4ΔEf (eV)afmD Fe afmI Fe Path 1 Path 2 Path 3 FIG. 3. Formation energy difference, /Delta1E f, to the lowest energy tetra conﬁguration along the distinct migration paths for interstitialHe in Fe, as shown in Fig. 2. Positions of the tetra conﬁgurations and the intermediate conﬁgurations, I i, along path iare labeled. In the afmI state, the data for path 3 has been omitted as it is equivalent topath 1. The arrows indicate the expected lowering of the migration barrier heights if a reorganization of magnetic moments is allowed along the migration path.TABLE VII. Migration energy barrier height, E m,i ne V ,f o rt h e migration of interstitial He along the distinct paths identiﬁed in Fig. 2. In afmD Fe, path 2 is asymmetrical and the direction of migration has been identiﬁed by the initial and ﬁnal tetra sites. In Ni, all paths areequivalent and the migration energy is given by the formation energy difference between the octa [110] and tetra He conﬁgurations from Tables IIandIII. Path,i afmD Fe afmI Fe Ni 1 0.335 0.070 0.129 2 (ud to uu) 0.349 0.119 – 2 (uu to ud) 0.283 – – 3 0.160 – – obvious that the results for the two Fe reference states differ signiﬁcantly. This is not surprising, however, given that typicalmagnetic effects can be of the same order of magnitude asthe migration barrier height (see Table I). The high barriers along paths 1 and 2 in afmD Fe are because the lowestenergy tetra site is between layers of unlike moment andso not adjacent to the lowest energy octa intermediate site,which lies off-center between like-spin layers (Table III). A wholesale reorganization of spins would lower the barriersalong these paths and would be preferred in the paramagneticstate. This problem is not present for path 3, resulting in asigniﬁcantly lower barrier, which is more consistent with thosefound in the afmI state, where a more uniform distributionof energies around the octa site exists (see Table III). Path 1 in the afmI state and path 3 in the afmD state both show adouble-peaked structure with weakly stable octa [001] and octa[00¯1] intermediate states, respectively. These intermediates are equidistant from four tetra sites, resulting in the same energybarrier for 1 nn and 2nn jumps on the tetra sublattice. The samecannot be said for migration along path 2, which proceeds via a(near-)octa [110] transition state in both reference states. In theafmI state, there appears to be a very shallow minimum at I 2, that is the off-center octa [110] conﬁguration, but with a depthof 0.007 eV this may well be just an artifact of the convergencecriteria as it is less than the expected error for formation energydifferences. The data also exhibits a shoulder between the tetraud and I 2sites, which we suggest results from close proximity to the octa [111] conﬁguration. It seems reasonable to suggestthat the barriers for 2nn jumps that cross a magnetic plane willbe close in energy to those for path 2, given the additional datain Table III. Overall, our ﬁndings suggest an energy barrier for interstitial He migration in austenite that is below 0.35 eV andmore likely in the region between 0.1 and 0.2 eV . Such low migration barriers are typical of all metals for which data are available. Ab initio calculations ﬁnd a value of 0.10 eV for fcc Al (Ref. 23), 0.07 eV for fcc Pd (Ref. 24), and 0.06 eV for bcc Fe (Ref. 25), W (Ref. 27), and V (Ref. 30). Experimental validation of these results is not forthcoming, primarily due to the low temperatures involvedand the complications of He interactions within the material. Inbcc W, Wagner and Seidman 75estimate the migration enthalpy to be between 0.24 and 0.32 eV , with He being immobile upto temperatures of at least 90 K, which is consistent with thevalue of 0.28 eV found for 3He migration by Amano and Seidman.76The discrepancy between ab initio and experiment 024115-9HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) was explained by Becquart and Domain as being due to the presence of strong He-He binding, as found in theirab initio calculations, resulting in the formation of less mobile interstitial He clusters for all but the lowest concentrations. 27 This is consistent with the work of Soltan et al. ,77who found He to be mobile in W and Au at temperatures below 5 K butwith increasing suppression of mobility as a function of Heconcentration. To this data we add the results of our own investigation into He migration in Ni. Following on from the results inafmD and afmI Fe, we make the reasonable assumption thatthe most stable off-center octa He conﬁguration is a goodcandidate for the transition state for interstitial He migration.The additional uncertainty on the inferred migration barrierheight from this assumption should be less than 0.01 eV .From the results presented in Table IIIthis is the off-center octa/angbracketleft110/angbracketrightconﬁguration, with a corresponding migration barrier height of 0.13 eV , which compares well with theexperimental value of 0 .14±0.03 eV measured by Philipps and Sonnenberg, 78corresponding to a migration activation temperature of 55 ±10 K. This barrier height also compares well with our best estimate for austenitic Fe. We thereforetentatively suggest that the barrier height for interstitial Hemigration in austenitic Fe-Ni-based alloys will be in the range0.1 to 0.2 eV , resulting in free, three-dimensional diffusionwell below room temperature. We accept that there is a veryreal possibility of signiﬁcant local composition dependence inthese concentrated alloys but we speculate that the effectivebarrier height will still be in the given range. 2. Substitutional He migration The diffusion of substitutional He generally proceeds via the dissociative and vacancy mechanisms.7,13,79Direct ex- change with a neighboring solute atom provides an alternativemechanism 79but is highly unlikely to contribute signiﬁcantly to diffusion due to the large activation energy for the process.For example, our best estimate of the barrier height in Niis 3.50 eV , which compares well with that found using anembedded atom model (EAM) potential of 3.1 eV by Adamsand Wolfer 79and means that substitutional He is, essentially, immobile. For many applications, substitutional He is best considered as an interstitial He atom strongly bound to a vacancy pointdefect, with a binding energy, E b(HeI,V). The dissociative mechanism for substitutional He migration proceeds by thedissociation of He from a vacancy followed by interstitialmigration until it becomes trapped in another vacancy. Assuch, the diffusion coefﬁcient by this mechanism is inverselyproportional to the vacancy concentration. 7,8,79If thermal vacancies dominate, the activation energy is given by7,8,13,79 Ediss. A=Em(HeI)+Eb(HeI,V)−Ef(V), (5) where Em(HeI) is the migration energy for interstitial He. However, if there is a supersaturation of vacancies, forexample, under irradiation, then the diffusion is dominatedby the dissociation step and E diss. A=Em(HeI)+Eb(HeI,V), (6)which is, essentially, the dissociation energy for substitutional He from its vacancy, and the diffusion coefﬁcient will remaininversely proportional to the vacancy concentration. 8,79 The diffusion of a substitutional solute by the vacancy mechanism in an fcc lattice is usually well described bythe ﬁve-frequency model of Lidiard and LeClaire. 80,81Ak e y assumption of this model is that when a vacancy binds at1 nn to a substitutional solute, the solute remains on-lattice.However, this is not the case for He, which we ﬁnd relaxes to aposition midway between the two lattice sites to form a V 2He complex. The possibility of solute-vacancy exchange at 2nn isalso not included in this model, a point to which we return inthe following discussion. Given the strong binding between a vacancy and substitu- tional He at 1 nn, which we discuss in Sec. IV A , we assume that the migration of the V 2He complex, as a single entity, dominates the diffusion by the vacancy mechanism,82with a migration energy, Em(V2He). The diffusion coefﬁcient will be proportional to the V 2He concentration, which is, in turn, proportional to the vacancy concentration and depends on thebinding energy between a substitutional He and a vacancy,E b(HeS,V). The resultant activation energy for substitutional He migration by the vacancy mechanism is given by7 Evac. A=Em(V2He)−Eb(HeS,V)+Ef(V) (7) when thermal vacancies dominate and by Evac. A=Em(V2He)−Eb(HeS,V)( 8 ) when there is a supersaturation of vacancies.25 We have determined the migration energies for the V 2He complex using a combination of NEB and single conﬁgurationcalculations. In afmD and afmI Fe, where more than onedistinct V 2He complex exists, we have calculated the migration energy along all of the distinct paths where the migrating Featom retains the sign of its magnetic moment. In previouswork, 54we found unrealistically high migration barriers along paths where the moment changed. We label the migration pathsfor V 2He migration by the initial and ﬁnal conﬁgurations, which are deﬁned in Fig. 4, and present the corresponding migration energies in Table VIII. The migration energies lie approximately 0.2 eV higher than those for the corresponding single vacancy migrationin afmD and afmI Fe (Ref. 54) and in Ni, where we found a vacancy migration energy of 1.06 eV , in good agreementwith other DFT calculations 22,53and with the experimental average value83of 1.04±0.04. We suggest that this results from the additional energy required to move the He atomfrom its central position in the V 2He complex back towards the lattice site during migration, as observed in all cases. Wealso contrast these results with those for divacancy migration.In afmD Fe, afmI Fe, and Ni we ﬁnd migration energies forthe divacancy along the 1b →1b path of 0.370, 0.221, and 0.473 eV , respectively, which are signiﬁcantly lower than thosefor the V 2He complex. In this case the difference arises not only from the energy required to move He to an on-lattice siteduring migration but also from its hindrance of the migratingFe atom. Vacancy-He exchange at 2nn provides an alternative migration path for substitutional He to that of V 2He migration. We found energy barriers for 2nn exchange as low as 0.47 and 024115-10FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) 1c1b 1a2b 2a 2c FIG. 4. Conﬁgurations for A-B pairs of interacting substitution- ally sited solutes and defects in afmD Fe. Species A is shown in black and species B in gray along with the conﬁguration label. Fe atoms are shown in white with arrows to indicate the local moments. Coordinateaxes are as in Fig. 1. 0.55 eV in afmD and afmI Fe, respectively, and a value of 0.94 eV in Ni. While these results are lower than the migrationenergies for V 2He, the repulsive interactions between a vacancy and substitutional He at 2nn (see Sec. IV A ) mean that the equilibrium concentrations of such conﬁgurations willbe signiﬁcantly lower than the V 2He concentration, resulting, we believe, in a much lower contribution to total diffusion.While this does strengthen our position that V 2He migration dominates substitutional He diffusion by the vacancy mech-anism, a model including all the relevant migration paths isnecessary to answer this question conclusively. Using the results presented here and in Sec. IV, we evaluate the expressions in Eqs. (5)–(8)and present the results in Table IX. The results clearly differentiate between the two mecha- nisms and show a strong correlation to corresponding results inbcc Fe (Ref. 25). When thermal vacancies dominate we predict that diffusion will proceed predominantly by the dissociativemechanism. If a supersaturation of vacancies exists then thevacancy mechanism clearly has the lowest activation energy.However, the vacancy concentration also plays a critical role indetermining which mechanism dominates through the distinctway it enters the expressions for the diffusion coefﬁcients. Forsufﬁciently low but still supersaturated vacancy concentrationsthe dissociative mechanism may become dominant. This is,however, most likely to be the case at low temperatures TABLE VIII. Migration energies, Em(V2He), in eV , for the V 2He complex. The migration paths are labeled by the initial and ﬁnal conﬁgurations, as deﬁned in Fig. 4. Path afmD Fe afmI Fe Ni 1b→1b 1.033 as 1c →1c 1.197 1c→1c 0.910 0.898 – 1a→1b 1.216 – – 1b→1a 1.211 – –TABLE IX. Activation energies for substitutional He migration, in eV , by the dissociative, Ediss. A, and vacancy, Evac. A, mechanisms for thermal [Eqs. (5)and(7)] and supersaturated [Eqs. (6)and(8)] vacancy concentrations. For afmD and afmI Fe we give the range ofpossible values corresponding to the distinct migration paths in these states. afmD Fe afmI Fe Ni Ediss. A,E q . (5) 0.599–0.788 0.853–0.902 1.405 Ediss. A,E q . (6) 2.411–2.600 2.810–2.859 2.756 Evac. A,E q . (7) 2.066–2.413 2.232–2.251 2.192 Evac. A,E q . (8) 0.254–0.601 0.275–0.294 0.841 where diffusion by either mechanism is likely to be negligible. As such, we suggest that vacancy-mediated diffusion isthe most important mechanism in conditions of vacancysupersaturation. For the case of Ni, Philipps, and Sonnenberg 6ﬁnd an activation energy for He diffusion of 0 .81±0.04 eV from isothermal, He-desorption spectrometry experiments. Theyattribute this result to the diffusion of substitutional He by thedissociative mechanism, hindered by thermal vacancies, fromwhich they infer an energy for dissociation of He of 2.4 eV . Ourresults agree that substitutional He migration will proceed bythe dissociative mechanism in a thermal vacancy population.There is, however, a 0.6-eV difference between our calculatedactivation energy (Table IX) and experiment. We also ﬁnd a dissociation energy for He from the substitutional site of2.756 eV , which is in excess of the inferred experimental value.This large discrepancy suggests that the inferred experimentalmechanism may not be correct. Ab initio calculations show that interstitial He atoms bind strongly to one another in Ni(Ref. 22). As discussed earlier, just such a mechanism was responsible for the suppression of interstitial He migrationin W and may also explain the experimental result in Ni.Alternatively, the He bombardment used in the experimentalsetup may have resulted in a supersaturation of vacancies, inwhich case our calculated activation energy, at 0.84 eV , wouldbe in good agreement with experiment. 3. Interstitial C and N migration The migration of interstitial C and N in both Fe and Ni goes from octa site to adjacent octa site. In afmD Fe, there arethree distinct migration paths, depending on where the initialand ﬁnal octa sites lie. We label these as “in-plane,” when theocta sites lie in the same magnetic plane, “uu,” when the octasites lie in adjacent magnetic planes with the same sign ofmagnetic moment and “ud,” when the octa sites lie in adjacentmagnetic planes with the opposite sign of magnetic moment.In afmI Fe, only the in plane and ud paths are distinct and inNi, all paths are equivalent. Each of these distinct migrationpaths will be symmetrical about an intermediate state lyingin the plane that bisects the direct path between the two octasites. In what follows, we consider the tetra and /angbracketleft110/angbracketrightcrowdion sites as candidate intermediate states. Possible migration pathsfor in-plane migration in afmD Fe are shown in Fig. 5, as an example. For C, the results in Tables IandIIshow that the crowdion conﬁgurations are the lowest lying of the possible intermediate 024115-11HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) ud uu FIG. 5. Possible migration paths for interstitial C and N in afmD Fe. Paths are shows for migration from initial to ﬁnal octa sites (blackcircles) lying in the same magnetic plane via tetra ud, [110] crowdion, and tetra uu intermediate sites (gray circles). Fe atoms (white circles) are shown with arrows to indicate the local moments. Migrationbetween octa sites in adjacent magnetic planes have not been show for clarity. Coordinate axes are as in Fig. 1. states. We have performed NEB calculations in afmD Fe for C migrating from the octa site to all of the distinct crowdionsites in order to determine the energy proﬁles along thesepaths. We ﬁnd a single maximum in the energy at the crowdionconﬁgurations. We ﬁnd this is also the case for N migrationvia the crowdion conﬁguration in Ni, as discussed below. Onthis basis and given the signiﬁcant local dilatation necessary toform a crowdion, we make the assumption that there will be asingle energy maximum at all /angbracketleft110/angbracketrightcrowdion sites so that the MEPs and barrier heights for C migration in afmD and afmIFe and in Ni can be determined from the data in Tables Iand II. The same can also be said for N migration in afmD Fe along uu and ud paths. For all other cases, however, the tetra sites arelower in energy and we have performed NEB calculations withclimbing image 84to investigate the migration energy proﬁles along these paths. In afmD Fe, our calculations conﬁrm that the tetra ud site is the energy barrier for N migration. In afmI Fe, however, thereis evidence of a shallow minimum, 0.015 eV deep, around thetetra conﬁguration. Results in Ni, by contrast, show a cleardouble-peaked structure in the energy proﬁle. We present theresults in Fig. 6and include the results for migration via the crowdion site for comparison. The results show that the tetraN conﬁguration is a stable local minimum, with a depth of0.273 eV relative to the transition state, and not a saddle point,like the crowdion conﬁguration. Despite this, the MEP for Nmigration is still via the tetra site. We summarize our results for the energy barriers and MEPs for interstitial C and N migration in Table X. In the Fe reference states there is a signiﬁcant spread in the migration barrierheights for C migration, both along distinct migration paths andbetween the two states. In-plane migration clearly exhibits ahigher energy barrier in both states, which results directly fromthe tetragonal distortion of the lattice and the subsequentlyhigher energy necessary to form the [110] crowdion transitionstate. The data also suggests a signiﬁcant dependence onthe local magnetic order, just as was seen for interstitialHe migration. The large spread in barrier heights means wecannot make any deﬁnitive predictions, except that diffusion isthree-dimensional. However, in any thermodynamic average,Octa I Octa Reaction Coordinate00.511.52ΔEf (eV)<110> Crowdion Tetra FIG. 6. Formation energy difference, /Delta1E f, to the lowest energy octa conﬁguration for N migration in Ni via tetra and /angbracketleft110/angbracketrightcrowdion intermediate states. the lower-energy paths will dominate, which suggests an effective barrier height around 1.4 eV in afmD Fe and 2.1 eV inafmI Fe. The afmD Fe value is reasonably consistent with theexperimentally determined activation energies for C migrationin austenite of 1.626 eV (Ref. 85) and 1.531 eV (Ref. 86). In Ni, we ﬁnd that C migrates via the crowdion site with an energy barrier height of 1.63 eV . This contrasts with the32-atom cell, where the tetra pathway is preferred. 74Once again, this demonstrates the inadequacy of using a 32-atomcell for solute migration in fcc Fe and Ni. Experimental results,using a variety of techniques applied both above and belowthe Curie temperature, T C=627 K, for Ni, yield activation energies in the range 1.43 to 1.75 eV (Ref. 87), consistent with our results. The experimental results also suggest that theinﬂuence of magnetism on the migration barrier (and enthalpyof solution) for C is no more than 0.2 eV and suggests this isthe likely error in using fm Ni results to estimate those in theparamagnetic state. Experimental results for C in Fe-Ni austenitic alloys, as discussed by Thibaux et al. , 88show only slight changes in C mobility as a function of Ni composition in the range from20 to 100 wt% Ni. They also report an activation energyof 1.30 eV in an Fe-31 wt% Ni austenitic alloy. Overall,our results, in conjunction with the experimental results wehave discussed, suggest that the migration energy barrier for Cmigration will lie in the range 1 .5±0.2 eV across the whole composition range for Fe-Ni austenitic alloys. For N, the migration barrier lies between 1.38 and 1.60 eV in afmD Fe, with a value of 1.90 eV in afmI Fe. As with C, theafmD Fe results are, on average, lower than those for the afmIstate. The result of an Arrhenius ﬁt to combined experimentaldiffusion data for N migration in austenite gave a similar valueof 1.75 eV (Ref. 87). In Ni, we ﬁnd a barrier height of 1.30 eV , which is in excess of the experimental activation energyreported by Lappalainnen and Anttila 89of 0.99±0.12 eV . In light of the signiﬁcant variation in experimental results for Cmigration in Ni, these two results are in reasonable agreementand certainly to within the 0.2 eV we have suggested earlieras a likely error when using ferromagnetic Ni to model the 024115-12FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) TABLE X. Migration energy barrier heights, Em, in eV for interstitial C and N migration in afmD and afmI Fe and in Ni. Migration is between adjacent octa sites via a transition state/intermediate (TS/I), which is speciﬁed in the table, along all of the distinct paths for each particular reference state. Where the transition state/intermediate is only an intermediate state on the migration path, its name has been marked with an asterisk. C migration N migration afmD Fe afmI Fe Ni afmD Fe afmI Fe Ni Path Em TS/I Em TS/I Em TS/I Em TS/I Em TS/I Em TS/I in plane 2.033 [110] crow. 2.445 [110] crow. 1.628 /angbracketleft110/angbracketrightcrow. 1.558 tetra ud 1.899 tetra∗1.296 tetra∗ uu 1.443 [011] crow. uu as ud as in plane 1.602 [011] crow. uu as ud as in plane ud 1.310 [011] crow. ud 2.113 [011] crow. ud as in plane 1.384 [011] crow. ud 1.899 tetra∗as in plane paramagnetic state. Overall, these results show that N migrates with a signiﬁcantly lower barrier in Ni than in austenitic Fe andwe would expect to ﬁnd an intermediate value in Fe-Ni-basedalloys, more generally. C. Solute-solute interactions We have performed calculations to investigate the interac- tions between pairs of He atoms in substitutional and tetra sitesin afmD and afmI Fe. Conﬁgurations with single substitutionaland tetra-sited He atoms at up to 2 nn separation were foundto consistently relax to a vacancy containing two He atoms.While this does not yield any useful binding energy data itdoes indicate that there is little or no barrier for this process andplaces a lower limit on the capture radius of a substitutional Heof around 3 ˚A. Results for pairs of interacting substitutional and tetra-sited He atoms, as identiﬁed in Figs. 4and7, respectively, are given in Table XI. Substitutional He pairs show consistent results across the Fe reference states with a strong positive binding at 1 nnand slightly repulsive interactions at 2 nn (Table XI). In our calculations, He atoms at 1 nn relax directly towards oneanother by between 0.38 and 0.44 ˚A, resulting in a consistent He-He separation of between 1.67 and 1.69 ˚A. While still close to the lattice sites, these displacements are in stark contrast tothe insigniﬁcant displacements observed at 2 nn. SubstitutionalHe pairs in Ni are similar: At 1 nn the He atoms are displacedtowards one another by 0.37 ˚A to a He-He separation of 1.75 ˚A. 1526 4 3 FIG. 7. Conﬁgurations for interacting tetra-sited solutes in afmD Fe. Conﬁgurations are labeled by the indices attached to theappropriate solute atom positions, shown in black. Fe atoms are shown in white with arrows to indicate the local moments. Coordinate axes are as in Fig. 1.The resultant binding energy, at 0.657 eV , is less than in Fe but is in similar proportion to the substitutional He to vacancybinding energy (see Sec. IV A ). At 2 nn He remains on-lattice with a repulsive binding energy of −0.16 eV . Pairs of tetra-sited He atoms exhibit signiﬁcant binding energies of up to 0.7 eV in afmD Fe and 0.6 eV in afmIFe. Such strong interactions are consistently observed in bccand fcc metals. Previous ab initio calculations found binding energies of 0.47 eV in Ni (Ref. 22) ,0 . 7e Vi nP d( R e f . 24) and Al (Ref. 23), 0.4 eV in bcc Fe (Ref. 25), and 1.0 eV in W( R e f . 27). At 1 nn separation, the He atoms in afmD and afmI Fe are displaced from the tetra sites only slightly underrelaxation. The resulting He-He “bonds” all lie along one of theaxes of the unit cell with lengths in a small range from 1.62 to1.68 ˚A, which is consistent with those found for substitutional He pairs at 1 nn. At 2 nn and 3 nn the He atoms displacesigniﬁcantly towards one another under relaxation from the TABLE XI. Formation and binding energies in eV for interacting pairs of He atoms in substitutional (S) and tetra (T) sites in afmD andafmI Fe. The conﬁgurations are labeled as in Figs. 4and7for S-S and T-T pairs, respectively. In the afmD reference state the binding energies between tetra-sited pairs of He atoms have been calculatedrelative to two isolated tetra ud He. For interacting pairs of tetra-sited He atoms at 2 nn and 3 nn separation the conﬁgurations are labeled by the initial He positions, which due to the signiﬁcant displacementsunder relaxation should not be taken as the ﬁnal positions. Eshelby corrections for S-S pairs were found to be negligible but were −0.09 eV for T-T pairs with a resulting increase in the T-T binding energies of up to 0.05 eV . afmD Fe afmI Fe A-B/Conﬁg. Ef Eb Ef Eb S-S/1a 7.115 0.934 7.423 0.946 S-S/1b 7.112 0.937 as S-S/1c S-S/1c 6.976 1.073 7.419 0.950 S-S/2a 8.197 −0.149 8.570 −0.201 S-S/2b 8.109 −0.060 8.493 −0.123 T-T/1-2 8.614 0.313 9.692 0.242 T-T/1-3 8.831 0.096 as T-T/2-4 T-T/2-4 8.215 0.712 9.463 0.472 T-T/1-4 rlx T-T/2-5 9.403 0.531T-T/1-5 8.700 0.227 as T-T/2-6 T-T/2-6 8.643 0.284 9.428 0.506 T-T/2-5 8.487 0.440 9.340 0.594 024115-13HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) 1b 1c2a2b 1a 2c FIG. 8. Conﬁgurations for A-B pairs of interacting octa-sited interstitials in afmD Fe. Species A is shown in black and speciesB in gray along with the conﬁguration labels. Fe atoms are shown in white with arrows to indicate the local moments. The lowest symmetry afmD state is shown to uniquely identify all of the distinctconﬁgurations. Some of these conﬁgurations will be symmetry equivalent in the afmI state. Coordinate axes are as in Fig. 1. initial tetra sites, resulting in He-He separations from 1.51 to 1.65 ˚A. These displacements are sufﬁciently large to take the He atoms either to the edge of their initial tetrahedral regionsor into the adjacent octahedral region via one of the faces ofthe tetrahedron. This is most pronounced for the 3 nn T-T/2-5conﬁguration, which in afmI Fe relaxed to a conﬁguration withthe He atoms within the octahedral region and symmetricallyopposite the central position along the [111] axis. The situationis similar for the afmD state but one He atom is signiﬁcantlycloser to the central position. It is worth noting that this isthe most stable conﬁguration in afmI Fe and the second moststable in afmD Fe. The large binding energies result, simply,from the cooperative dilatation of the lattice and the reductionof repulsive He-Fe interactions, which are naturally greatestwhen the two He atoms are in close proximity. The results at2 nn and 3 nn separations show that the local dilatation of thelattice around a single interstitial He results in an attractiveforce to other interstitial He atoms up to at least 3 ˚A away and encourages the formation of clusters. To investigate interstitial cluster formation further we have determined the most stable conﬁgurations with three and fourHe atoms in afmD and afmI Fe. For a ﬁxed number of He atomswe found many distinct conﬁgurations with similar energiesbut the most stable clusters were reasonably predictable froma simple pair interaction model, given the data in Table XI.I n afmD Fe, the most stable He 3conﬁguration found had two He atoms in a 2-4 formation (see Fig. 7) with the third occupying the nearest octa site. In afmI Fe, the most stable was an L-shaped 1-2-3 cluster. In the most stable He 4clusters, all He atoms occupied tetra sites in a rectangular-planar formationwith 1 nn edges, such as a 1-2-3-4 cluster. This is, in fact, themost stable arrangement found in afmI Fe, whereas in afmDFe a square-planar conﬁguration with all He atoms in tetra udsites was the most stable. The total binding energies for theTABLE XII. Total binding energies, in eV , for the most stable interstitial He clusters containing up to 4 He atoms. Results in Ni are from the work of Domain and Becquart.22Eshelby corrections were found to be −0.19 and −0.34 eV for He 3and He 4clusters, respec- tively, with corresponding increases in the binding energies of 0.14 and 0.27 eV . Cluster afmD Fe afmI Fe Ni He2 0.712 0.594 0.47 He3 1.537 1.374 1.25 He4 2.637 2.561 most stable clusters are given in Table XIIalong with results in Ni (Ref. 22). The strong clustering tendency of interstitial He is clearly demonstrated by the data. Application of the Eshelby correc-tions only enhances this effect. The binding energy for anadditional He, that is, E b(Hen)−Eb(Hen−1), increases with nfor the small clusters studied here. We would expect this, however, to plateau to an additional binding energy of around1 eV per He atom in afmD and afmI Fe and in Ni, given thatthe cooperative dilatation of the lattice that gives rise to thebinding happens locally. Such strong clustering can not onlyresult in an effective reduction in interstitial He mobility asHe concentration increases but is also a critical ﬁrst step in thespontaneous formation of Frenkel-pair defects, as observed ingold. 90Indeed, the most stable He 4conﬁgurations found here show a signiﬁcant displacement of the nearest Fe atom to thecluster off lattice by 0.94 ˚A in afmD Fe and 1.36 ˚A in afmI Fe. We consider this possibility further in Sec. IV B in the context of V mHenclustering. Interactions between pairs of octa-sited C and N atoms at up to 2 nn separation in afmD and afmI Fe are given inTable XIII. The interactions are, generally, repulsive at both 1 nn and 2 nn, with a reasonable consistency between the tworeference states, although the repulsion is slightly stronger inthe afmI state. For C, the pair binding energy is between −0.1 and−0.15 eV at 1 nn and more repulsive at 2 nn at up to around −0.2 eV . By contrast, N pairs exhibit stronger repulsion than C TABLE XIII. Formation and binding energies, in eV , for interact- ing pairs of octa-sited C and N interstitials. The conﬁgurations areas labeled in Fig. 8. Eshelby corrections were −0.03 and −0.06 eV in afmD and afmI Fe, respectively, with resultant increases in the binding energy of 0.02 and 0.03 eV . afmD Fe afmI Fe A-B/Conﬁg. Ef Eb Ef Eb C-C/1a −17.490 −0.104 −17.572 −0.141 C-C/1b −17.487 −0.106 N/A C-C/1c −17.559 −0.034 −17.561 −0.151 C-C/2a −17.420 −0.174 −17.487 −0.226 C-C/2b −17.617 0.023 −17.655 −0.058 N-N/1a −17.010 −0.195 −17.037 −0.204 N-N/1b −17.035 −0.170 N/A N-N/1c −17.131 −0.074 −17.020 −0.221 N-N/2a −17.054 −0.150 −17.075 −0.167 N-N/2b −17.212 0.008 −17.172 −0.070 024115-14FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) at 1 nn, at around −0.2 eV and weaker repulsion at 2 nn, where the binding energy is at most around −0.15 eV . Calculations for C-C pairs at up to 4 nn separation in afmI Fe found amaximal binding energy 0.02 eV . We conclude that there willbe no appreciable positive binding of C-C and N-N pairs inour reference states for austenite. Experimental determinations of C-C and N-N interaction energies in austenite are discussed in a review by Bhadeshia. 91 If quasichemical theory, which only includes 1 nn interactions,is used to interpret the existing thermodynamic data, then a pairbinding energy of −0.09 eV is found for C and −0.04 eV for N. Our results for C in afmD and afmI Fe are in good agreement with this value and while we do ﬁnd a repulsive interactionbetween N-N pairs, we ﬁnd a stronger repulsion than for C,which is contrary to the results of this analysis. A more detailed analysis can be performed using M ¨ossbauer spectroscopy data to study the distribution of C atoms in the Fe matrix, whichcan be compared with the results of Monte Carlo simulationsto determine the solute interaction energies at 1 nn and 2 nn. 92 Such an analysis yields C-C binding energies of −0.04 eV at 1 nn and less than −0.08 eV at 2 nn and N-N binding energies of−0.08 and −0.01 eV at 1 nn and 2 nn, respectively.92Our results follow the same pattern for the relative strengths ofrepulsion but are signiﬁcantly in excess of the results of thisanalysis. The agreement is still impressive, however, given the level of extrapolation between our two ordered magnetic states at 0 K and temperatures where paramagnetic austenite is stable. For C in fm Ni, Siegel, and Hamilton 74found C-C binding energies at 1 nn and 2 nn of 0.01 and −0.01 eV , respectively, using comparative DFT calculations to those performed inthis work. They, furthermore, show that this negligible levelof binding is consistent with the experimental estimates of theC-C pair concentration as a function of total C concentration. 93 From the data presented above we would suggest that C- C and N-N interactions in Fe-based austenitic alloys will berepulsive at 1 nn and 2 nn, with binding energies in the rangefrom−0.1 to −0.2 eV . We would, furthermore, expect the level of repulsion to be reduced as a function of increasing Niconcentration. D. Interactions with substitutional Ni and Cr solutes in Fe As an initial step in the investigation of the interactions of He, C, and N with substitutional Ni and Cr solutes inaustenite we have calculated the formation energies for singlesubstitutional Ni and Cr and present the results in Table XIV. 94 On this basis, the results of our calculations of the interactions TABLE XIV . Formation energies, Ef, in eV and magnetic moments, μ,i nμBfor substitutional Ni and Cr solutes in austenitic Fe. The sign of the moments indicates whether there is alignment (positive) or anti-alignment (negative) with the moments of the atoms in the same magnetic plane. fct afmD fct afmI Conﬁg. Ef μE f μ Sub. Ni 0.083 0.039 0.145 −0.301 Sub. Cr 0.263 0.843 0.061 1.120TABLE XV . Formation and binding energies, in eV , for substi- tutional Ni/Cr (species A) to substitutional He, tetra He and octa C/N (species B) with conﬁgurations labeled as in Figs. 4,9and10, respectively. Eshelby corrections to Efwere found to be −0.02 eV when interstitial solutes were present but negligible for all other quantities. afmD Fe afmI Fe A-B/cfg Ef Eb Ef Eb sub Ni-sub He/1a 4.032 0.076 4.212 0.117 sub Ni-sub He/1b 4.035 0.073 as 1c sub Ni-sub He/1c 4.018 0.090 4.233 0.097 sub Ni-tetra He/1b 4.496 0.051 4.979 0.133 sub Ni-tetra He/2a 4.500 0.047 5.078 0.034 sub Ni-tetra He/2d 4.480 0.067 5.062 0.050 sub Ni-octa C/1a −8.692 −0.021 −8.717 0.006 sub Ni-octa C/1b −8.673 −0.040 as 1c sub Ni-octa C/1c −8.729 0.016 −8.643 −0.069 sub Ni-octa N/1a −8.439 −0.080 −8.417 −0.058 sub Ni-octa N/1b −8.432 −0.087 as 1c sub Ni-octa N/1c −8.445 −0.074 −8.349 −0.126 sub Cr-sub He/1a 4.353 −0.065 4.284 −0.038 sub Cr-sub He/1b 4.358 −0.070 as 1c sub Cr-sub He/1c 4.433 −0.145 4.341 −0.095 sub Cr-tetra He/1b 4.609 0.118 4.883 0.145 sub Cr-tetra He/2a 4.746 −0.019 5.005 0.023 sub Cr-tetra He/2d 4.781 −0.054 5.024 0.004 sub Cr-octa C/1a −8.647 0.114 −8.845 0.050 sub Cr-octa C/1b −8.628 0.094 as 1c sub Cr-octa C/1c −8.730 0.197 −8.826 0.030 sub Cr-octa N/1a −8.597 0.258 −8.729 0.169 sub Cr-octa N/1b −8.566 0.227 as 1c sub Cr-octa N/1c −8.574 0.235 −8.704 0.145 between He, C, and N solutes and substitutional Ni and Cr solutes in afmD and afmI Fe are presented in Table XV. In both Fe reference states, substitutional He binds weakly to Ni, by around 0.1 eV , and has a repulsive interaction withCr of−0.1 eV . The similarity to vacancy-substitutional Ni/Cr binding is striking. 54The similarity also extends to the local moments on 1 nn atoms surrounding the substitutional Heand vacancy, as was discussed in Sec. III. These results are also consistent with Ni and Cr acting as slightly oversizedand undersized solutes, respectively, when interacting withpoint defects in afmD and afmI Fe, as discussed previously. 54 We would expect the interactions of other transition metalsolutes with substitutional He to be readily inferred from theirinteractions with vacancies. Interstitial He binds weakly to Ni by, on average, 0.09 eV at 1 nn and 0.05 eV at 2 nn in the Fe reference states. We alsoobserve weak positive binding with Cr, but only at 1 nn, wherethe binding energy is, on average, 0.13 eV . Closer observationsof the conﬁgurations revealed that He relaxed slightly awayfrom Ni, but toward Cr at 1 nn. Ni also remained closer to thelattice site than Cr. These geometrical results are consistentwith Ni and Cr behaving as oversized and undersized solutes,respectively, despite both exhibiting binding to interstitial He,although the binding to Cr is marginally greater. The level 024115-15HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) 1b1a2a 2b2c 2d FIG. 9. Conﬁgurations for interactions between a substitutionally sited species (A) and a tetrahedrally sited species (B) in afmDFe. Species A is shown in black, and species B is shown in gray. Conﬁgurations are labeled by the position of the tetrahedrally sited species, as shown. Fe atoms are shown in white with arrows to indicatethe local moments. Coordinate axes are as in Fig. 1. of binding suggests that Ni and Cr may act as weak traps for migrating interstitial He at low concentrations. However,with increasing concentration and, therefore, likelihood thatHe remains in similar local environments as it migrates, a directstudy of the local composition dependence of the migrationenergy becomes necessary. From the binding energy data wecan speculate, however, that such a dependence will also beweak and maintain our earlier suggestion that the activationenergy for interstitial He migration will lie in the 0.1- to 0.2-eVrange in concentrated Fe-Cr-Ni alloys. The interactions of octa C and N with substitutional Ni and Cr are reasonably consistent in both afmD and afmI Fe.For C, interactions with Ni are minimal, although slightlyrepulsive, at 1 nn, whereas positive binding is observed withCr on the order of 0.1 eV . The interactions of N are similar tothose of C but signiﬁcantly stronger and exhibit a repulsion ofaround −0.1 eV to Ni and attraction to Cr of around 0.2 eV . The repulsive interactions with Ni are consistent with thelower solubility of C and, particularly, N in fcc Ni, comparedto afmD and afmI Fe (see Tables Iand II), and suggests that the interactions are cumulative. In the case of Cr, suchcumulative interactions would encourage the formation ofCr-C/N complexes and the precipitation of Cr-carbonitrides, asobserved experimentally in nonstabilized austenitic stainlesssteels, 95under conditions where these elements are mobile, that is, at high temperatures or in irradiated environments. IV . SOLUTE INTERACTIONS WITH POINT DEFECTS In this section we consider the interactions of He, C, and N with a single vacancy (V), in small vacancy-solute clusters,V mXn, and with the [001] self-interstitial (SI) dumbbell in afmD and afmI Fe and in Ni. We present the formation (andbinding) energies of the underlying and most stable defectsand defect clusters in Table XVI, as found previously. 54Pairs1b 1c1a2a 2b4a4b 4c FIG. 10. Conﬁgurations for interactions between a substitution- ally sited species A and octa-sited species B in the fct afmD reference state. Species A is shown in black and species B in gray along withthe conﬁguration labels. Fe atoms are shown in white with arrows to indicate the local moments. The lowest symmetry afmD state is shown to uniquely identify all of the distinct conﬁgurations. Some ofthese conﬁgurations will be symmetry equivalent in the afmI state. Coordinate axes are as in Fig. 1. TABLE XVI. Formation energies, E f, in eV , for the vacancy, the most stable di-, tri-, tetra-, and hexa-vacancy clusters, as found by Klaver et al.54and the [001] SI dumbbell in afmD and afmI Fe and in Ni. Total binding energies, Eb, in eV , are given for the vacancy clusters in brackets below the formation energies. The results in Fe are consistent with those found previously.54Results in Ni compare well to other DFT calculations.21,22,53,96,97Eshelby corrections were found to be negligible except for the tetra-vacancy in Ni at −0.03 eV , the hexavacancy at −0.06−0.03 and −0.05 eV in afmD Fe, afmI Fe and Ni, respectively, and the dumbbell at −0.05,−0.08 and −0.10 eV in afmD Fe, afmI Fe and Ni, respectively. The only non-negligible effect on binding energies was for the hexavacancy, where increases of 0.05,0.03, and 0.03 eV apply in afmD Fe, afmI Fe and Ni, respectively. Defect afmD Fe afmI Fe Ni vacancy 1.812 1.957 1.352 3.443 3.840 2.688di-vacancy(0.181) (0.075) (0.016) 4.790 5.285tri-vacancy(0.646) (0.587) 6.479 7.097 4.956tetra-vacancy(0.768) (0.733) (0.451) 8.378 9.210 6.865hexa-vacancy(2.493) (2.534) (1.245) [001] SI dumbbell 3.196 3.647 4.135 024115-16FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) TABLE XVII. Formation and binding energies in eV for vacancy (species A) to substitutional He, tetra He and octa C/N (species B) with conﬁgurations labeled as in Figs. 4,9and 10, respectively. Conﬁgurations with a single solute atom in the substitutional position(Sub.), where stable, are also considered as an interstitial solute interacting with a vacancy. In afmD Fe, the vacancy-tetra He binding energies were calculated relative to tetra ud He. Binding energiesbetween octa C and N solutes and a vacancy at 3 nn and 4 nn separations were investigated but did not exceed 0.03 eV . The only non-negligible Eshelby corrections found were for the bindingenergies between a vacancy and interstitial solutes at no more than 0.02 eV in magnitude. afmD Fe afmI Fe A-B/Conﬁg. Ef Eb Ef Eb V-sub He/1a 5.216 0.620 5.519 0.623 V-sub He/1b 5.221 0.615 as V-sub He/1c V-sub He/1c 5.180 0.656 5.538 0.604V-sub He/2a 5.940 −0.103 6.293 −0.151 V-sub He/2b 5.854 −0.018 as V-sub He/2c V-sub He/2c 5.919 −0.083 6.253 −0.111 V-tetra He/Sub. 4.024 2.251 4.185 2.740 V-tetra He/2a 6.328 −0.053 as V-tetra He/2b V-tetra He/2b 6.408 −0.133 6.932 −0.008 V-tetra He/2c 6.377 −0.101 as V-tetra He/2d V-tetra He/2d 6.233 0.042 6.883 0.041 V-C/Sub. −6.981 −0.004 −6.244 −0.655 V-C/1a −7.165 0.180 −7.276 0.377 V-C/1b −7.040 0.056 as V-C/1c V-C/1c −7.268 0.283 −7.186 0.287 V-C/2a −7.031 0.046 as V-C/2b V-C/2b −7.018 0.033 −6.948 0.049 V-N/Sub. unstable −5.153 −1.510 V-N/1a −7.230 0.440 −7.275 0.612 V-N/1b −7.065 0.275 as V-N/1c V-N/1c −7.217 0.427 −7.161 0.498 V-N/2a −6.887 0.097 as V-N/2b V-N/2b −6.883 0.092 −6.769 0.106 of vacancies were consistently most stable at 1 nn separation. The most stable tetravacancy cluster consists of a tetrahedralarrangement of vacancies at 1 nn to each other. The most stabletrivacancy cluster is formed from this by placing an atom nearthe tetravacancy center. Finally, the most stable hexavacancyis an octahedral arrangement of vacancies with 1 nn edges. A. Vacancy-solute interactions We present the formation and binding energies for conﬁg- urations containing a single vacancy and solute atom, at up to2 nn separation, in Table XVII . 1. V-He binding We observe strong binding of between 0.60 and 0.66 eV for V-Sub He pairs at 1 nn in both Fe reference states. This issigniﬁcantly greater than the binding between vacancy pairs 54 and represents the simplest case of enhanced vacancy bindingby He, as we discuss in what follows. We ﬁnd that He doesnot remain on-lattice at 1 nn to a vacancy but relaxes to aposition best described as at the center of a divacancy. With this perspective, the V-Sub He binding represents the signiﬁcantenergetic preference of He for the greater free volume availableat the center of a divacancy over a single vacancy. At 2 nn,the interactions are repulsive at around −0.1 eV , which is slightly greater than that observed between vacancy pairs. 54 He remains on-lattice in these conﬁgurations, which explainsthe lack of enhanced binding at 2 nn separation. The situationin Ni is very similar, where we ﬁnd binding energies of 0.356and−0.127 eV at 1 nn and 2 nn, respectively. Interstitial He binds strongly to a vacancy to form a substitutional He conﬁguration. The same is also true inNi, where we ﬁnd a binding energy of 2.627 eV , in goodagreement with previous work. 22Conﬁgurations with tetra He at 1 nn to a vacancy are unstable. At 2 nn, however, weﬁnd stable conﬁgurations with weak repulsive or attractivebinding, depending on the conﬁguration. The fact that nostable conﬁgurations were found with tetra He at up to 2 nnfrom a substitutional He atom demonstrates that the additionof a single He to a vacancy signiﬁcantly increases the captureradius for interstitial He. We expect this effect to increase withthe subsequent addition of He, given the additional pressureand dilatation that would be exerted on the surrounding lattice. 2. V-C and V-N binding C binds to a vacancy by up to 0.38 eV at 1 nn in the Fe reference states and weakly at 2 nn. This level of bindingagrees well with previous experimental and theoretical workin austenite and austenitic alloys. 47We ﬁnd that V-N binding is signiﬁcantly stronger than for C with binding energies in therange from 0.3 to 0.6 eV at 1 nn and around 0.1 eV at 2 nn.For both C and N, the substitutional conﬁguration is stronglydisfavored. As discussed in Sec. III, the substitutional C and N conﬁgurations in afmD Fe were found to be unstable andthe conﬁguration labeled V-C/Sub in Table XVII has C in a stable position off lattice by 0.77 ˚A. Overall, these results bear a strong similarity to those found in bcc Fe (Ref. 35), where binding energies of 0.47 and 0.71 eV were found for C and Nat 1 nn to a vacancy, respectively. Results in Ni are broadly similar to those in Fe. We ﬁnd a V- C binding energy of 0.062 eV at 1 nn and 0.121 eV at 2 nn. V-Nbinding is, again, stronger, than C, with energies of 0.362 and0.165 eV at 1 nn and 2 nn, respectively. We also ﬁnd a strongrepulsion from the substitutional site. We note that the V-Cbinding at 1 nn seems anomalously low, given the other resultsbut no problems were found with this calculation and othertest calculations found the same stable structure and energy. The signiﬁcant V-C and V-N binding energies suggest that the relatively less mobile solutes could act as vacancy traps,much as was found in bcc Fe (Refs. 1,2,38, and 98). This would certainly be the case if dissociation of the complexwas required before the vacancy could freely migrate but thepossibility of cooperative migration also exists. In the fcclattice there are many possible migration pathways that wouldavoid the dissociation of this complex, including some thatwould maintain a 1 nn separation. A complete study of thesepossibilities is beyond the scope of this work but preliminarycalculations in Ni show that the energy barriers for C and Njumps that would maintain a 1 nn separation to the vacancy 024115-17HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) TABLE XVIII. Total binding energies, Eb, in eV for the most stable V mXnclusters found in afmD and afmI Fe, where Xis He, C, or N. Results in Ni are also given for C and N and can be found in the literature for He.19,22Eshelby corrections to Ebfor V mHenclusters were found to be below 0.05 eV in magnitude except for VHe 5and V mHenwithmandnequal to 3 or 4, which were below 0.1 eV and VHe 6,w h i c hw a s 0.2 eV . For C and N clusters, the corrections were below 0.02 eV in magnitude except for those with six vacancies or with four or more Natoms, where the corrections were up to 0.1 eV for most but were 0.2 eV for VN 6in afmI Fe and VN 5in Ni and 0.3 eV for VN 6in Ni. Cluster afmD Fe afmI Fe Cluster afmD Fe afmI Fe Ni Eb Eb Eb Eb Eb VHe 2.251 2.740 VC 0.283 0.377 0.121VHe 2 3.845 4.627 VC 2 0.484 0.795 0.422 VHe 3 5.674 6.588 VC 3 0.423 0.484 −0.206 VHe 4 7.452 8.609 V 2C 0.499 0.550 0.211 VHe 5 9.239 10.305 V 4C 1.107 1.307 0.718 VHe 6 10.845 12.015 V 6C 3.546 3.253 1.531 V2He 2.907 3.363 VN 0.440 0.612 0.362 V2He2 5.575 6.430 VN 2 0.981 1.295 0.872 V2He3 7.791 8.990 VN 3 1.264 1.341 0.877 V2He4 10.197 11.682 VN 4 1.371 1.514 0.933 V3He 3.711 4.237 VN 5 1.439 1.516 0.651 V3He2 6.458 7.461 VN 6 1.474 1.482 0.246 V3He3 9.323 10.857 V 2N 0.743 0.933 0.558 V3He4 11.750 13.685 V 4N 1.364 1.573 1.047 V4He 4.475 4.993 V 6N 3.224 3.466 2.033 V4He2 7.504 8.542 V4He3 10.565 12.120 V4He4 13.606 15.711 V6He 6.566 7.191 are around half the value for the isolated solutes at around 0.75 eV . In contrast, vacancy jumps that maintain a 1 nnseparation were found to be signiﬁcantly higher than those forthe isolated vacancy but jumps from 1 nn to 2 nn separationand back exhibited lower or comparable energy barriers. Whilethese calculations are preliminary, they do indicate the distinctpossibility of cooperative vacancy-solute motion that wouldavoid dissociation of the complex. The implications for anabsence of vacancy pinning and for the enhanced diffusionof C and N solutes in the presence of vacancies in austeniticalloys makes this an interesting subject for further study. B. Vacancy-solute clustering Small-vacancy-He (V mHen) clusters have been found to be highly stable both experimentally8,9,11,12and using DFT techniques19,22–25,28in a number of metals and are, therefore, critically important as nuclei for void formation. Experimentalevidence in bcc Fe 1,2has also shown that C can act as a vacancy trap through the formation of small, stable V mCnclusters, which has been conﬁrmed in a number of DFT studies.31,35–38 Small V mNnclusters have also been shown to exhibit similar stability.35 In this section we present the results of a large number of DFT calculations to ﬁnd the most stable V mXnclusters, where Xis He, C, or N, in afmD and afmI Fe. A comprehensive search for the most stable conﬁguration was only practicable for thesmaller clusters. For larger clusters, a number of distinct initialconﬁgurations, based around the most stable smaller clusters,were investigated to improve the likelihood that the most stablearrangement was found. The total binding energies for the most stable conﬁgurations can be found in Table XVIII . 1. V mHe nclusters The geometries of the relaxed V mHenclusters were constrained by the tendency to maximize He-He and He-Fe separations within the available volume and, therefore,minimize the repulsive interactions. In a single vacancy wefound that this led to the following structures: two He formeddumbbells centered on the vacancy with He-He bond lengthsaround 1.5 ˚A; three He formed a near-equilateral triangle with bond lengths of between 1.6 and 1.7 ˚A; four He formed a near-regular tetrahedron with bond lengths between 1.6 and1.7˚A; ﬁve He formed a near-regular triangular bipyramid with bond lengths between 1.6 and 1.8 ˚A; and six He formed a near- regular octahedron with bond lengths between 1.6 and 1.8 ˚A. In clusters with more than one vacancy, a single He atom relaxedto a central position. Additional He tended to form clusterssimilar to those seen in a single vacancy but now around thecenter of the vacancy cluster. The trivacancy case is interestingbecause previous DFT calculations in austenite 54found that a conﬁguration consisting of a tetrahedral arrangement ofvacancies with one Fe atom near the center of the void,which can be considered as the smallest possible stacking faulttetrahedron (SFT), was more stable than the planar defect ofthree vacancies with mutual 1 nn separations. The addition ofa single He atom was enough to reverse the order of stabilitywith a difference in the total binding energy of 0.8 eV inafmI Fe, in favor of the planar defect. We suggest that thisresult should readily generalize, with planar defects being 024115-18FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) 1 234 5 6 No. He atoms, n012345Eb(He,VmHen-1) (eV)V6Hen V4Hen V3Hen V2Hen VHen Hen (a) afmD Fe, He binding1 234 5 6 No. He atoms, n012345Eb(He,VmHen-1) (eV)V6Hen V4Hen V3Hen V2Hen VHen Hen (b) afmI Fe, He binding 1 234 5 6 No. Vacancies, m012345Eb(V,Vm-1Hen) (eV)VmHe4 VmHe3 VmHe2 VmHe Vm (c) afmD Fe, V binding1 234 5 6 No. Vacancies, m0123456Eb(V,Vm-1Hen) (eV)VmHe4 VmHe3 VmHe2 VmHe Vm (d) afmI Fe, V binding 0 1 234 5 No. Vacancies, m012345Eb(SI,Vm+1Hen) (eV) Vm VmHe VmHe2 VmHe3 VmHe4 (e) afmD Fe, SI binding0 1 234 5 No. Vacancies, m-10123456Eb(SI,Vm+1Hen) (eV) Vm VmHe VmHe2 VmHe3 VmHe4 (f) afmI Fe, SI binding FIG. 11. (Color online) Binding energies, in eV , for a He atom, V , or SI to an existing cluster to form one with the V mHenstoichiometry in afmD Fe [panels (a), (c), and (e)] and afmI Fe [panels (b), (d), and (f)] Fe. Interstitial He cluster data have been included in panels (a) and (b)for completeness. more stable than SFTs with sufﬁcient addition of He. That said, however, planar defects have been found54to be less stable than three-dimensional protovoids and this situation isunlikely to change with the addition of He due to the greaterfree volume of the latter clusters. In Sec. III A1 the addition of a single He to a vacancy was found to have very little effect on the local magnetism. Theaddition of He to vacancy clusters was generally found to havevery little effect on the total magnetic moment of the supercells containing the cluster. The only exception was for the singlevacancy in afmD Fe, although it took the addition of six Heatoms to signiﬁcantly change the magnetic moment. Even inthe absence of vacancies, a cluster of at least three He atomswas necessary to inﬂuence the total magnetic moment. In Fig. 11we present results for the binding energy of either a He atom, vacancy (V), or [001] self-interstitial dumbbell 024115-19HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) (SI) to an already existing cluster to form one with the V mHen stoichiometry. These results show that He consistently binds strongly to an existing cluster and that the strength of the bind-ing only increases with m. For a ﬁxed value of n, this binding energy will converge to the formation energy for interstitialHe (see Table I)a smincreases and is well on the way to doing that for n=1. For ﬁxed mthe additional He binding energy appears to plateau as nincreases, although it should diminish eventually as the pressure within the cluster builds. The binding energy for an additional vacancy is also consis- tently positive. The presence of He signiﬁcantly increases thisadditional binding for all values of m, which is consistent with the observation that it aids the nucleation, stabilization, andgrowth of voids in irradiated environments. 10,13–16For ﬁxed n, the data shows that the vacancy binding energy is tending to aplateau as mincreases and is consistent with the fact that all of these curves should converge to the vacancy formation energy. The SI binding energy can be related to the vacancy binding energy as E b(SI,Vm+1Hen)=Ef(SI)+Ef(V)−Eb(V,VmHen),(9) which implies that the spontaneous emission of an SI from an existing cluster will be energetically favorable if and onlyif the binding of the newly created vacancy is greater thanthe Frenkel pair formation energy. The data show that theSI binding energy clearly decreases as He concentration isincreased at ﬁxed mand for sufﬁciently high concentration will become negative. Indeed, it is energetically favorable foran interstitial He cluster with four He atoms in afmI Fe, andmost likely for ﬁve He atoms in afmD Fe, to spontaneouslyemit an SI defect. This mechanism was proposed to explain theobservation of He bubbles in Au samples after subthresholdHe implantation 90and could also explain observations of He trapping in Ni (Ref. 99), where the He was introduced by natural tritium decay to avoid implantation-produced defects.Our results show that this would, most likely, occur in austeniteand austenitic alloys and could lead to bubble formation, withthe potential for blistering in the presence of, even low-energy,bombardment by He ions, as seen in W (Refs. 100and101).As a whole, the binding energy data is qualitatively similar to DFT results in Al (Ref. 23) and Pd (Ref. 24) and is quantitatively similar to results in bcc Fe (Ref. 25) and Ni (Refs. 19and22). This observation gives us conﬁdence that our results are not only applicable to austenite but to austeniticalloys more generally. The binding energy data above has also been used to determine the dissociation energy, that is the energy ofemission, of He, V , or SI from a V mHencluster using the simple ansatz that the dissociation energy, Ediss.(X), for species, X,i s given by Ediss.(X)=Eb(X)+Em(X), (10) where Em(X) is the migration energy for isolated species, X. We present results for the dissociation energies in Fig. 12, using the migration energies in Table XIX. There is a strong and distinct dependence on the He to vacancy ratio, n/m , for the dissociation energies of the three species. Both graphs exhibit a clear crossover between the Heand V curves at around n/m=1.3 and another between the He and SI curves at about n/m=6. An identical He-V crossover ratio was found in bcc Fe (Ref. 25) and fcc Al (Ref. 23). For n/m below 1.3 the clusters are most prone to emission of a vacancy, between 1.3 and 6 He has the lowest dissociationenergy, and above 6 SI emission is the preferred dissociationproduct. The slope of the curves ensures that emission ofthe species with the lowest dissociation energy will make theresulting cluster more stable. At sufﬁciently high temperaturesthat these processes are not limited by kinetics this should leadto the formation of the most stable clusters, which have ann/m value at the He-V crossover, where our results predict a minimum dissociation energy of around 2.8 eV in both afmDand afmI Fe. 2. V mCnand V mNnclusters In fct afmD and afmI Fe and in Ni we considered V Xn clusters with octa-sited C and N at 1 nn to the vacancy and conﬁgurations where C and N are close enough to form C-C and N-N bonds within the vacancy. Our results for VC 2 0 1 234 5 6 n/m012345Ediss(X) (eV)X = He X = V X = SI (a)afmD Fe0 1 234 5 6 n/m01234567Ediss(X) (eV)X = He X = V X = SI (b)afmI Fe FIG. 12. (Color online) Dissociation energies, Ediss.(X), in eV , for species Xfrom a V mHencluster, where Xis a He, V , or SI. Results are presented for (a) afmD Fe and (b) afmI Fe versus the He to vacancy ratio, n/m . The solid curves are simple polynomial ﬁts to the data and are present to aid visualization. 024115-20FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) TABLE XIX. Migration energies, Em(X), in eV , for species, X, where Xis He, V , or SI. For He, the lowest values from Table VII were used. For V , the lowest vacancy migration energies from Klaver et al.54were used. The SI migration energies were calculated here as that for a102dumbbell SI migrating between two lattice sites at 1 nn separation within a magnetic plane using identical settings to Klaver et al.54 Species,X afmD Fe afmI Fe He 0.160 0.070 V 0.743 0.622 SI 0.196 0.254 and VN 2clusters are given in Table XX. We found that V X2 clusters with octa-sited C and N are most stable when the C/N atoms are as far apart as possible, that is, opposite oneanother across the vacancy. For these conﬁgurations the totalbinding energy is more than the sum of the binding energiesfor each single solute to the vacancy, indicating either somechemical or cooperative strain interaction. We found that C-C dumbbells centered on the vacancy are stable, with bondlengths between 1.38 and 1.48 ˚A, that is, much shorter than the separations between octahedral sites. The most stable liealong /angbracketleft100/angbracketrightdirections and binding over and above that for octa-sited C was found in afmD Fe and Ni. The enhancementin binding upon forming a C-C dumbbell is not, however, aspronounced as was seen in bcc Fe (Refs. 25and35–37). We also found stable conﬁgurations with N-N dumbbells in Fe andNi with bond lengths between 1.34 and 1.49 ˚A, although they exhibit a much lower, and generally negative, total bindingenergy compared to conﬁgurations with octa-sited N atoms.For VX 3clusters, we investigated all possible conﬁgura- tions with three octa-sited C or N solutes in addition to thosewith a C-C dumbbell and an octa-sited C solute in one ofthe four octa sites perpendicular to the dumbbell axis anda conﬁguration with three C atoms close enough for C-Cbonding. Although we do ﬁnd stable conﬁgurations with C-Cbonding in either a dumbbell or triangular arrangement in bothFe and Ni, these arrangements are the least stable and exhibitsigniﬁcant, negative total binding energies. The most stablearrangements consist of three octa-sited C atoms placed as farapart as possible, for example in three 1a sites relative to thevacancy as in Fig. 10. However, the total binding energies for the most stable VC 3clusters (see Table XVIII ) are less than for VC 2, which implies that a vacancy can only bind up to two C atoms within a vacancy. A vacancy may still, however,bind more than at 2 nn octa sites but we did not investigatethis possibility due to the strongest binding being at 1 nn to thevacancy and due to the large number of possible conﬁgurations. The most stable VN 3clusters have the same geometry as found with C but, in contrast, are more stable than VN 2 clusters. Beyond this point, we found that the total bindingenergy only increases for up to four N atoms in afmI Feand Ni but increases all the way up to six N atoms in afmDFe. That said, however, the binding energy per N atom onlyincreases up to a VN 2cluster in all reference states. The equilibrium concentrations of clusters with more than two Natoms, which can be calculated using the law of mass action, 38 would very likely be negligible, even at room temperature.Despite their magnitude, the Eshelby corrections do not changethese conclusions but would result in the total binding energyincreasing all the way up to six N atoms in afmI Fe, as wasfound for afmD Fe. TABLE XX. Formation and total binding energies, in eV , for the interactions of a vacancy with two octa sited C or N solutes. Conﬁgurations with C or N in octa sites at 1 nn to the vacancy are labeled by the positions of the two solutes as in Fig. 10. When both octa solutes are in the same plane as the vacancy the conﬁgurations are additionally labeled by their relative orientation i.e. opposite (opp.) or adjacent (adj.) toone another. Doubly mixed dumbbells centered on the vacancy site were also considered as conﬁgurations of an interacting vacancy with two octa solutes and the total binding energies were calculated accordingly. Eshelby corrections to both E fandEbwere found to be no more than 0.03 eV in magnitude. afmD Fe afmI Fe Ni Conﬁg. Ef Eb Ef Eb Ef Eb VC 2clusters 1a-1a (opp.) −16.226 0.444 −16.551 0.795 −15.692 0.199 1b-1c −16.166 0.385 −16.466 0.711 as 1a-1a (opp.) 1a-1a (adj.) −15.908 0.127 −16.163 0.408 −15.396 −0.097 1a-1b −15.832 0.050 −16.116 0.361 as 1a-1a (adj.) 1a-1c −16.104 0.323 as 1a-1b as 1a-1a (adj.) [100] dumb. −16.087 0.305 −16.224 0.469 −15.915 0.422 [001] dumb. −16.265 0.484 −16.262 0.507 as [100] dumb. [110] dumb. −15.407 −0.375 −15.238 −0.517 −15.380 −0.113 [111] dumb. rlx [001] dumb. −15.565 0.072 VN 2clusters 1a-1a (opp.) −16.373 0.981 −16.579 1.295 −14.559 0.872 1b-1c −16.124 0.732 −15.932 0.648 as 1a-1a (opp.) 1a-1a (adj.) −16.113 0.720 −16.237 0.953 −14.279 0.591 1a-1b −16.051 0.658 −16.037 0.753 as 1a-1a (adj.) 1a-1c −16.228 0.835 as 1a-1b as 1a-1a (adj.) [100] dumb. −14.451 −0.942 −14.487 −0.796 −13.616 −0.071 024115-21HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) We investigated site preference and binding for single C and N solutes to the most stable di-, tetra-, and hexavacancyclusters in afmD and afmI Fe and in Ni. For the V 2C cluster in afmD Fe, we considered all 1 nn octa sites to the three distincttypes of 1 nn divacancy as well as conﬁgurations with C at thecenter of all 1 nn and 2 nn divacancy clusters. For the octa sites,C was found to bind to existing divacancy clusters with similarbinding energies to a single vacancy, that is with E b(C,V2)i n the range from 0.03 to 0.32 eV . The most stable of these, whichwas also found to be the most stable V 2C cluster, contained the most stable divacancy and bound C more stably than to asingle vacancy. We found that C was repelled from the centerof a 1 nn divacancy lying within a magnetic plane but bound tothe other two 1 nn divacancies with energies similar to thosefound in octa sites. As in bcc Fe (Refs. 31,36, and 37), the most preferred site for C was at the center of a 2 nn divacancy,withE b(C,V2)=0.35 eV . However, this was not sufﬁcient to overcome the difference in stability between 1 nn and 2 nndivacancies in afmD Fe (Ref. 54) and did not, therefore, form the most stable V 2C cluster, in contrast to in bcc Fe. The analysis above motivated the use of only the most stable 1 nn divacancy in the remaining calculations alongwith conﬁgurations containing solutes at the center of 2 nndivacancies. For N in afmD Fe, the order of site preferencemirrors that for C. An N solute is capable of stabilizing a2 nn divacancy conﬁguration but the most stable V 2Nc l u s t e r shared the same geometry as for C with a binding energy tothe underlying divacancy of 0.56 eV , which is, again, in excessof the binding to a single vacancy. The situation in afmI Fe and Ni was found to be rather similar to that of afmD Fe. For both C and N, the site atthe center of a 1 nn (in-plane) divacancy was disfavored. Themost stable conﬁguration generally contained an octa-sitedsolute bound t oa1n nd i v acancy. The only exception was in afmI Fe, where a conﬁguration with C at the center of a 2 nndivacancy lying within a magnetic plane had a greater totalbinding energy but only by 0.03 eV . This most likely resultedfrom the much smaller energy difference between 1 nn and2 nn divacancies in afmI Fe of compared to afmD Fe (Ref. 54) and to Ni, where we ﬁnd an energy difference of 0.1 eV infavor of the 1 nn divacancy. In the most stable clusters, thebinding of the solutes to the underlying divacancy was, onceagain, in excess of the binding to a single vacancy. For the binding of C and N to the most stable tetravacancy, we found that the central position was extremely disfavored.We investigated all conﬁgurations with solute atoms in anocta site at 1 nn to at least a single vacancy. We also performedcalculations with solute atoms placed initially at random withinthe protovoid but found that these relaxed to octa sites alreadyconsidered. Conﬁgurations with only a single vacancy at 1 nnto the solute were found to be the most stable. The total bindingenergies for these conﬁgurations are given in Table XVIII . Using these results we found that the binding of C and N tothe tetravacancy was in excess of that to a divacancy and asingle vacancy, in all cases except for N in afmI Fe, wherethe binding to the tetravacancy and divacancy reversed order,although they differ by only 0.02 eV . For the hexavacancy, the central octa site was unstable for both C and N in afmD Fe. In afmI Fe and Ni it wasstabilized by symmetry but still strongly disfavored. Thisrepulsion is, however, signiﬁcantly less than was observed for the tetravacancy. Closer observation showed that while thenearest neighboring Fe and Ni atoms to the solutes movedvery little under relaxation in the tetravacancy, the contractionin bond length was between 25% and 30% in the hexavacancyfrom an initial separation of around 3 ˚A. This demonstrates how important the formation of strong chemical bonds withcharacteristic bond lengths is to the stability of conﬁgurationscontaining C and N in Fe and Ni. We investigated the stability of conﬁgurations with C and N in all octa sites at 1 nn to at least one vacancy in thehexavacancy cluster. We found that there were additional stablesites, lying along /angbracketleft100/angbracketrightaxes projected out from the center of the hexavacancy. For C, these sites were found to lie betweenthe ﬁrst vacancy reached along these axes and the next octasite out. They are close to but distinct from the octa sites andwe, therefore, refer to them as octa-b sites. For N, stable siteswere found between the center of the hexavacancy and the ﬁrstvacancy reached along the /angbracketleft100/angbracketrightaxes and we refer to these as off-center sites. We found that C was, consistently, moststable in an octa-b site, whereas N preferred octa sites withtwo vacancies at 1 nn, although an off-center site along [00 ¯1] was the most stable in afmD Fe. Once again, the binding energy between the solutes and hexavacancy was greater than for all smaller vacancy clusters.We summarize these results for E b(X,Vm)i nF i g . 13, which clearly shows the increase in binding energy as the vacancycluster becomes larger. It also clearly shows that in the samereference state, the binding energy for N is consistently greaterthan for C and that the binding energies in afmI Fe lie abovethose in afmD Fe. The one anomalous point is the bindingenergy for C to a hexavacancy in afmD Fe, which is much largerthan the trends would suggest. Other conﬁgurations with C inan octa-b site in afmD Fe exhibited similar levels of bindingand no problems with any of these calculations or instabilitiesin the relaxed structures could be found. 1 234 5 6 No. Vacancies, m00.20.40.60.81Eb(X,Vm) (eV) FIG. 13. (Color online) Binding energy, Eb(X,Vm), in eV , where Xis C (solid symbols and solid lines) or N (open symbols and dashed lines) in afmD Fe (red circles), afmI Fe (green squares) and Ni (blue diamonds). The binding energies were calculated for the most stable clusters. 024115-22FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) TABLE XXI. Formation and binding energies in eV for [001] self-interstitial dumbbell (species A) - solute (species B) interactions. For octa-sited C and N solutes the conﬁgurations are labeled as in Fig. 10. He interactions were investigated with He sited substitutionally and tetrahedrally with conﬁgurations labeled as in Figs. 4and9, respectively. Conﬁgurations with substitutional He in 1b and 1c positions relative to a [001] dumbbell SI were unstable to defect recombination and interstitial He kickout. In Ni, the [001]-tetra He binding energy was observedto be 0.20 eV . 22Eshelby corrections were found down to −0.1 eV for conﬁgurations containing substitutional He but the related increases in Ebwere no more than 0.02 eV . For conﬁgurations containing interstitial solutes Ecorr.could be as low as −0.2 eV with corresponding increases inEbup to 0.08 eV . afmD Fe afmI Fe Ni A-B/Conﬁg. Ef Eb Ef Eb Ef Eb [001]-tetra He/1a unstable unstable [001]-tetra He/1b 7.734 −0.075 as 1a [001]-tetra He/2a 7.494 0.166 8.517 0.098 [001]-tetra He/2b 7.761 −0.036 as 2a [001]-tetra He/2c 7.609 0.117 8.435 0.180[001]-tetra He/2d 7.514 0.146 as 2c [001]-sub He/1a 7.045 0.176 7.743 0.089 7.085 0.235 [001]-sub He/2a 7.164 0.057 7.815 0.017 7.316 0.003[001]-sub He/2b 7.050 0.171 7.653 0.179 unstable [001]-sub He/2c 7.095 0.126 as 2b as 2b [001]-C/1a −5.563 −0.037 −5.007 −0.202 −4.300 0.012 [001]-C/1b −4.604 −0.997 −3.975 −1.234 [001]-C/1c −4.459 −1.141 as 1b [001]-C/2a −5.585 −0.015 −5.064 −0.145 −4.322 0.034 [001]-C/2b −5.527 −0.074 as 2a as 2a [001]-C/4a −5.626 0.025 −5.266 0.057 −4.303 0.015 [001]-C/4b −5.652 0.051 −5.287 0.078 −4.362 0.075 [001]-C/4c −5.642 0.041 as 4b as 4b [001]-N/1a −5.106 −0.300 −4.444 −0.529 −3.198 −0.188 [001]-N/1b −4.461 −0.945 −3.721 −1.252 [001]-N/1c −4.197 −1.209 as 1b [001]-N/2a −5.290 − 0.116 −4.762 −0.211 −3.400 0.015 [001]-N/2b −5.251 −0.155 as 2a as 2a [001]-N/4a −5.425 0.019 −5.027 0.054 3.402 0.017 [001]-N/4b −5.430 0.024 −5.043 0.069 −3.481 0.096 [001]-N/4c −5.458 0.052 as 4b as 4b C. [001] dumbbell SI-solute interactions We investigated the binding of He, C, and N solutes to a [001] dumbbell in afmD and afmI Fe and in Ni and present theresults in Table XXI. We found that interstitial He, placed initially 1 nn to a [001] SI dumbbell, either spontaneously displaced underrelaxation to a 2 nn site or exhibited a repulsive binding energy in Fe. At 2 nn, however, a positive binding energy was observed, up to almost 0.2 eV , as was found in Ni (Refs. 19 and22). Eshelby corrections do not qualitatively change these results and would only act to enhance the binding at 2 nn.This positive binding energy is comparable to that in bcc Fe(Ref. 25) but while signiﬁcant, it is only likely to result in mutual trapping at low temperature, given the high mobilityof the two species. Taken as a model for the binding ofinterstitial He to other overcoordinated defect sites, such asnear dislocations and grain boundaries, however, this resultdoes show that He would be likely to be trapped at such sites,leading to interstitial He cluster formation and spontaneousbubble nucleation and growth, as discussed earlier. It isworth mentioning that bubble nucleation by this mechanism would happen much more readily at grain boundaries where,due to their disorder, vacancies can be formed without theadditional SI. A substitutional He atom in the 1b and 1c sites (see Fig. 4) to a [001] SI dumbbell resulted in the spontaneous recombination of the vacancy and SI and the kickout of aninterstitial He atom. At all other 1 nn and 2 nn sites except2b in Ni, however, stable complexes with binding energies of up to around 0.2 eV were formed. Barriers to recombination for these complexes, while positive, were not calculated inthis work. These results do, however, show that substitutionalHe and most likely other V mHenclusters can act as trapping sites for SI dumbbells in austenite and austenitic alloys witha capture radius extending out to at least 2 nn. We can alsospeculate that, once trapped, recombination will be likely to occur. Both C and N are either repelled from 1 nn and 2 nn sites to a [001] SI dumbbell or show very little positive binding, muchas was observed in bcc Fe (Ref. 35). Eshelby corrections do 024115-23HEPBURN, FERGUSON, GARDNER, AND ACKLAND PHYSICAL REVIEW B 88, 024115 (2013) not change this conclusion in Fe but would result in binding of around 0.1 eV at 2 nn in Ni. Motivated by the result thatC does exhibit positive binding to the most stable SI andsmall SI clusters in bcc Fe (Ref. 19) at further separation, we investigated this possibility here and found sites with bindingenergies from 0.05 to 0.1 eV at 4 nn to the dumbbell, whichwould only be enhanced by Eshelby corrections. These sitescan be related to the corresponding ones in bcc Fe by a Baintransformation 102and the binding almost certainly results from strain ﬁeld effects in both cases. The fact that such bindingwas found to increase with interstitial cluster size 19means that Cottrell atmospheres3of C and N are very likely to form around other overcoordinated defects, such as dislocations andgrain boundaries, in both ferritic and austenitic alloys underconditions where these species are mobile. V . CONCLUSIONS An extensive set of ﬁrst-principles DFT calculations have been performed to investigate the behavior and interactions ofHe, C, an N solutes in austenite, dilute Fe-Cr-Ni alloys, andNi as model systems for austenitic steel alloys. In particular,we have investigated the site stability and migration of singleHe, C, and N solutes, their self-interactions, interactionswith substitutional Ni and Cr solutes, and their interactionswith point defects typical of irradiated environments, payingparticular attention to the formation of small V mXnclusters. Direct comparison with experiment veriﬁes that the two- state approach used to model austenite in this work is reason-ably predictive. Overall, our results demonstrate that austenitebehaves much like other fcc metals and is qualitatively similarto Ni in many respects. We also observe a strong similaritybetween the results presented here for austenite and thosefound previously for bcc Fe. We ﬁnd that interstitial He is most stable in the tetrahedral site and migrates via off-center octahedral transition states witha migration energy from 0.1 to 0.2 eV in austenite and 0.13 eVin Ni. The similarity of these results and the weak interactionswith Ni and Cr solutes in austenite suggests a migration energyin Fe-Cr-Ni austenitic alloys in the 0.1- to 0.2-eV range. Inter-stitial He will, therefore, migrate rapidly from well below roomtemperature until traps are encountered. Its strong clusteringtendency, with an additional binding energy approaching 1 eVper He atom in austenite and 0.7 eV in Ni, will lead toa reduction in mobility as interstitial He concentrationincreases. Interactions with overcoordinated defects, whichare on the order of a few tenths of 1 eV , will result in thebuildup and clustering of interstitial He at dislocations andgrain boundaries. The most stable traps, however, are vacancyclusters and voids, with binding energies of a few eV . Thestrength of this binding means that growing interstitial Heclusters eventually become unstable to spontaneous Frenkelpair formation, resulting in the emission of a self-interstitialand nucleation of a VHe ncluster. The binding of additional He and vacancies to existing V mHenclusters increases signiﬁcantly with cluster size, leading to unbounded growthand He bubble formation in the presence of He and vacancyﬂuxes. The most stable clusters have a helium-to-vacancyratio, n/m , of around 1.3, with a dissociation energy for the emission of He and V of 2.8 eV in austenite and Ni.Generally, we assume that V mHenclusters are immobile. For the simplest case of substitutional He, however, migrationis still possible. In a thermal vacancy population, diffusionby the dissociative mechanism dominates, with an activationenergy of between 0.6 and 0.9 eV in Fe and 1.4 eV in Ni.In irradiated environments, however, the vacancy mechanismdominates and diffusion can proceed via the formation andmigration of the stable V 2He complex, with an activation energy of between 0.3 and 0.6 eV in Fe and 0.8 eV in Ni. We ﬁnd that C and N solutes behave similarly, both in austenite and Ni, although the interactions of N are stronger.The octahedral lattice site is preferred by both solutes, leadingto a net expansion of the lattice and a reduction of the c/a ratio in the afmD and afmI Fe reference states. Both solutesalso stabilize austenite over ferrite and favor ferromagneticover antiferromagnetic states in austenite. Carbon migratesvia a/angbracketleft110/angbracketrighttransition state with a migration energy of at least 1.3 eV in austenite and of 1.6 eV in Ni. For N, migrationproceeds via the crowdion or tetrahedral sites, depending onpath, with a migration energy of at least 1.4 eV in austeniteand 1.3 eV in Ni. Pairs of solute atoms are repelled at 1 nnand 2 nn in austenite and do not interact in Ni. Both C andN interact very little with Ni solutes in austenite but bind toCr, which may act as a weak trap and encourage the formationof Cr-carbonitrides under conditions where the solutes aremobile. Carbon binds to a vacancy by up to 0.4 eV in austeniteand 0.1 eV in Ni, with N binding more strongly at up to0.6 eV in austenite and 0.4 eV in Ni. While this may suggestthat C and N act as vacancy traps, as in bcc Fe, preliminarycalculations in Ni show that VC and VN clusters may diffusecooperatively with an effective migration energy similar tothat for the isolated vacancy. This also raises the possibilityof enhanced C and N mobility in irradiated alloys and theirsegregation to defect sinks. A vacancy can bind up to twoC atoms and up to six N atoms in austenite (or four in Ni),although the additional binding energy reduces signiﬁcantlyabove two. Covalent bonding was observed between solutesin a vacancy but did not lead to any enhanced stability, as seenin bcc Fe. Both C and N show a strong preference for sitesnear the surface of vacancy clusters and the binding increaseswith cluster size, suggesting that they will decorate the surfaceof voids and gas bubbles, when mobile. A binding energy of0.1 eV was observed to a [001] SI dumbbell in austenite andNi, which we would expect to increase with interstitial clustersize, as in bcc Fe, resulting in Cottrell atmospheres of C and Naround dislocations and grain boundaries in austenitic alloys. Along with previous work, these results provide a complete database that would allow realistic Fe-Cr-Ni austenitic alloysystems to be modeled using higher-level techniques, suchas molecular dynamics using empirical potentials and kineticMonte Carlo simulations. As such, they play a critical role ina multiscale modeling approach to study the microstructuralevolution of these materials under irradiation in typical nuclearenvironments. ACKNOWLEDGMENT This work was part sponsored through the EU-FP7 PERFORM-60 project, the G8 funded NuFUSE project, andEPSRC through the UKCP collaboration. 024115-24FIRST-PRINCIPLES STUDY OF HELIUM, CARBON, AND ... PHYSICAL REVIEW B 88, 024115 (2013) dhepburn@ph.ed.ac.uk †gjackland@ed.ac.uk 1A. 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PhysRevB.103.224415.pdf	PHYSICAL REVIEW B 103, 224415 (2021) Quasi-one-dimensional uniform spin-1 2Heisenberg antiferromagnet KNaCuP 2O7 probed by31Pand23Na NMR S. Guchhait,1Qing-Ping Ding ,2M. Sahoo,3A. Giri,4S. Maji,4Y . Furukawa ,2and R. Nath1,* 1School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, India 2Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 3Department of Physics, University of Kerala, Kariavattom, Thiruvananthapuram 695581, India 4School of Physical Sciences, Indian Association for the Cultivation of Science, Kolkata 700032, India (Received 17 March 2021; revised 21 May 2021; accepted 2 June 2021; published 14 June 2021) We present the structural and magnetic properties of KNaCuP 2O7investigated via x-ray diffraction, magneti- zation, speciﬁc heat, and31Pand23Na NMR measurements and complementary electronic structure calculations. The temperature-dependent magnetic susceptibility and31PNMR shift could be modeled very well by the uniform spin-1 2Heisenberg antiferromagnetic chain model with a nearest-neighbor interaction J/kB/similarequal58.7K . The corresponding mapping using ﬁrst-principles electronic structure calculations leads to JDFT/kB/similarequal59 K with negligibly small interchain couplings, further conﬁrming that the system is indeed a one-dimensionaluniform spin- 1 2Heisenberg antiferromagnet. The diverging trend of NMR spin-lattice relaxation rates (311/T1 and231/T1) implies the onset of a magnetic long-range ordering at around TN/similarequal1K .F r o mt h ev a l u eo f TN, the average interchain coupling is estimated to be J/prime/kB/similarequal0.28 K. Moreover, the NMR spin-lattice relaxation rates show the dominant contributions from uniform ( q=0) and staggered ( q=±π/a) spin ﬂuctuations in the high- and low-temperature regimes, respectively, mimicking one-dimensionality of the spin lattice. We have alsodemonstrated that 311/T1in high temperatures varies linearly with 1 /√ H, reﬂecting the effect of spin diffusion on the dynamic susceptibility. The temperature-dependent unit cell volume could be described well using theDebye approximation with a Debye temperature of /Theta1 D/similarequal294 K, consistent with the heat capacity data. DOI: 10.1103/PhysRevB.103.224415 I. INTRODUCTION Quantum ﬂuctuations play a pivotal role in deciding the ground state properties in low-dimensional spin systems [ 1,2]. In particular, in uniform one-dimensional (1D) spin-1 2Heisen- berg antiferromagnetic (HAF) chains, quantum ﬂuctuationsare enhanced due to a low spin value and reduced dimen-sionality which preclude magnetic long-range order (LRO)[3]. Often, the interchain and /or intrachain frustration am- pliﬁes the effect of quantum ﬂuctuations, leading to variousintriguing low-temperature features. Further, spin chains arethe simplest systems which can be easily tractable from bothexperimental and computational point of views as they havea relatively simple and well-deﬁned Heisenberg HamiltonianH=J/summationtext iSiSi+1, where SiandSi+1are the nearest-neighbor (NN) spins and Jis the exchange coupling between them. Transition metal oxides offer ample opportunities for ﬁndingspin chains with different exchange geometries. Copper (Cu 2+)-based oxides are proven to be excellent model compounds and are extensively studied because oftheir interesting crystal lattice and low spin (3 d 9,S=1/2) value. The Cu2+chains formed by the direct linkage of CuO 4units can be categorized into two groups. One is the chains formed by the edge sharing of CuO 4units and another formed by the corner sharing of CuO 4units. The chains of *rnath@iisertvm.ac.inedge-sharing CuO 4units have a Cu-O-Cu angle nearly 90◦ and are having competing NN ( J1) and next-nearest-neighbor (NNN) ( J2) interactions [ 4]. For AF J2, these chains are frustrated, irrespective of the sign of J1, and host a wide variety of ground states, controlled by the J2/J1ratio [ 5]. The prominent manifestation of frustration in 1D spin-1 2chains encompasses a spin-Peierls transition in CuGeO 3[6], a chi- ral state in NaCu 2O2[7], LiCu 2O2[8], LiCuVO 4[9], and Li2ZrCuO 4[10], and the realization of a Majumdar-Ghosh point in Cu 3(MoO 4)(OH) 4[11]. In these compounds, J1and J2are comparable in strength, which generates a strong frus- tration within the chain. On the contrary, in Sr 2CuO 3, chains are formed by the corner sharing of CuO 4units and is an ideal realization of spin-1 2uniform HAF chains [ 12–16]. Because of the nearly 180◦Cu-O-Cu angle, the AF J1prevails over J2, largely reducing the in-chain frustration and making the chains uniform. Another family of 1D compounds is the copper phosphates (Sr,Ba) 2Cu(PO 4)2,( B a,Sr,Pb)CuP 2O7, and (Li ,Na,K)2 CuP 2O7which contain isolated CuO 4units [ 17–22]. Though there is no direct linking of CuO 4units, the interaction among Cu2+ions takes place via an extended path involving the corner /edge sharing of CuO 4and PO 4tetrahedra. The mag- netic properties of all these compounds are described well bythe 1D uniform spin- 1 2HAF model with intrachain coupling J/kB(=J1/kB) in the range ∼30–160 K. (Sr ,Ba) 2Cu(PO 4)2 has emerged to be the best realization of uniform spin-1 2HAF chains showing one-dimensionality over a large temperature 2469-9950/2021/103(22)/224415(14) 224415-1 ©2021 American Physical SocietyS. GUCHHAIT et al. PHYSICAL REVIEW B 103, 224415 (2021) Na(c) Chain cNa c a Na K(a) b(b) b aPO 4 bca J' J OuC 4J'' FIG. 1. (a) A three-dimensional view of the crystal structure of KNaCuP 2O7that shows well-separated spin chains. (b) Two uniform spin chains of Cu2+running along the adirection featuring the intrachain coupling ( J) and the frustrated interchain network of J/prime[dCu-Cu/similarequal 5.772(2) Å] and J/prime/prime[dCu-Cu/similarequal5.676(2) Å]. (c) A section of the crystal structure showing the coupling of Na atoms with Cu2+ions. range ( kBT/J/greaterorequalslant6×10−4), similar to Sr 2CuO 3(kBT/J/greaterorequalslant 2×10−3)[13,17]. Spin chains based on organometallic com- plexes are another class of compounds portraying interesting1D physics [ 23]. When the spin chains are embedded in a real material, a weak residual coupling between the chains comesinto play at sufﬁciently low temperatures and the ground stateis decided based on the hierarchy of coupling strengths. Theseinterchain couplings often form a frustrated network betweenthe chains and either forbid the system to cross over to a LROstate or stabilize in a exotic ground state [ 24]. Thus, the quest for novel states in spin chains necessitates the search for newmodel compounds with nontrivial interchain geometries. Herein, we investigate the magnetic behavior of potassium sodium copper (II) diphosphate (V) (KNaCuP 2O7), which has a monoclinic crystal structure with space group P21/n.T h e lattice parameters and unit cell volume ( Vcell) at room tem- perature are reported to be a=5.176(3) Å, b=13.972(5) Å, c=9.067(3) Å, β=91.34(2)◦, and Vcell=655.6(5) Å3[25]. The crystal structure of KNaCuP 2O7is presented in Fig. 1. Distorted CuO 4plaquettes are corner shared with four PO 4 tetrahedra forming isolated magnetic chains stretched alongtheadirection. In each CuO 4plaquette, Cu-O bond lengths are within the range 1.93–1.98 Å, while in each PO 4tetrahe- dra, the P-O bond length varies within the range 1.48–1.63 Å. These chains are well separated from each other and the Naand K atoms are located in the interstitial positions betweenthe chains. Thus, P is located almost symmetrically betweentwo Cu 2+ions within a chain and is strongly coupled with the magnetic Cu2+ions. The Na and K atoms are also positioned symmetrically between the chains, providing a weak inter-chain coupling and making a complex three-dimensional (3D)structure. Further, the chains are arranged in such a way thateach CuO 4plaquette in one chain has two identical neighbors in each adjacent chain. With AF J,J/prime, and J/prime/primethis leads to a frustrated interchain geometry. Figure 1(b) presents a sketch of the spin lattice illustrating the leading intrachain ( J) and thefrustrated interchain couplings ( J/prime,J/prime/prime) between two neighbor- ing chains. Moreover, only one Cu site in the crystal structureand the presence of inversion centers in the middle of eachCu-Cu bond imply that the anisotropic Dzyaloshinskii-Moriya(DM) interaction vanishes by symmetry. Figure 1(c) shows a section of the crystal structure demonstrating the couplingof the Na atom with three neighboring chains. The magneticproperties of this compound are not available to date. Our experimental results reveal the uniform spin- 1 2chain character of the spin lattice with an intrachain couplingJ/k B/similarequal58.7 K. The magnetic LRO is suppressed to TN/similarequal1K due to weak and frustrated interchain couplings. The exper-imental assessment of the spin lattice is further supportedby the complementary electronic structure calculations. Thedynamical properties of the spin system are also extensivelyinvestigated via 31Pand23Na NMR spin-lattice relaxation measurements. II. METHODS A blue-colored polycrystalline sample of KNaCuP 2O7was synthesized by the traditional solid state synthesis proce-dure. A stoichiometric amount of CuO (Aldrich, 99 .999%), NaH 4PO 5(Aldrich, 98%), and KHPO 4were ground thor- oughly and heated at 450◦C for 24 h in air. Subsequently, the sample was ﬁred at 570◦C for 24 h and at 600◦C for 48 h, followed by intermediate grindings and palletizations. Finally,the main phase was found to be formed at 600 ◦C. At each step, the phase purity of the sample was checked by doinga powder x-ray diffraction (XRD) experiment at room tem-perature using a PANalytical powder diffractometer equippedwith Cu Kαradiation ( λ avg/similarequal1.541 82 Å). The temperature (T)-dependent powder XRD was performed on the phase pure sample in the temperature range 15 K /lessorequalslantT/lessorequalslant300 K, using a low-temperature attachment (Oxford PheniX) to the x-raydiffractometer. A Rietveld analysis of the XRD patterns was 224415-2QUASI-ONE-DIMENSIONAL UNIFORM … PHYSICAL REVIEW B 103, 224415 (2021) performed using the FULLPROF software package [ 26], taking the initial structural parameters from Ref. [ 25]. Magnetization ( M) was measured as a function of tem- perature (2 K /lessorequalslantT/lessorequalslant350 K), in the presence of an applied magnetic ﬁeld H=1 T. Magnetization isotherms ( MvsH) were also measured at two different temperatures ( T=2 and 300 K) by varying Hfrom 0 to 9 T. All these measurements were carried out using a vibrating sample magnetometer(VSM) attachment to the physical property measurementsystem (PPMS, Quantum Design). Speciﬁc heat ( C p)w a s measured as a function of temperature (2–100 K), by usingthe thermal relaxation method in PPMS, on a sintered pellet inzero magnetic ﬁeld. Magnetic spin susceptibility of a uniformAF chain lattice of Heisenberg spins was obtained from thequantum Monte Carlo (QMC) simulations performed with theLOOP algorithm [ 27]o ft h e ALPS simulation package [ 28]. Simulations were performed on a ﬁnite lattice ( L=200) size. The pulsed NMR experiments were performed on the 31Pnucleus with nuclear spin I=1 2and gyromagnetic ratio γ 2π=17.237 MHz /T and the23Na nucleus with I=3/2 and γ 2π=11.26 MHz /T.31PNMR measurements were done in different radio frequencies of 121, 85, 39, 21, and 11.6 MHzwhile 23Na NMR measurements were done in 79 MHz. The NMR spectrum at different temperatures was obtained bychanging the magnetic ﬁeld in a ﬁxed frequency. A largetemperature range of 1 .6K/lessorequalslantT/lessorequalslant300 K was covered in our experiments. A temperature-dependent NMR shift K(T)= [H ref/H(T)−1] was calculated from the resonance ﬁeld of the sample Hwith respect to the resonance ﬁeld of a nonmag- netic reference sample ( Href). The spin-lattice relaxation rate 1/T1was measured by the conventional single saturation pulse method. The ﬁrst-principles electronic structure calculations have been performed within the framework of density functionaltheory (DFT) using the plane-wave basis with a projectoraugmented-wave (PAW) potential [ 29,30] as implemented in the Vienna ab initio simulation package ( V ASP )[31,32]. The generalized gradient approximation (GGA) implementedwithin the Perdew-Burke-Ernzerhof (PBE) prescription [ 33] has been chosen for the exchange-correlation functional. Aplane-wave cutoff of 500 eV was set to obtain good con-vergence of the total energy and a kmesh of 5 ×2×3 was used for the Brillouin zone (BZ) integration. Maximallylocalized Wannier functions (MLWFs) for the low-energyCud x2−y2model Hamiltonian have been constructed using V ASP 2WANNIER and WANNIER 90 codes [ 34], providing the hopping parameters required to identify the various exchangepaths. The missing correlation in GGA calculations are in-cluded within the GGA +Umethod for all the spin-polarized calculations, where standard values of Uand Hund’s coupling J H[35] were chosen for Cu with Ueff(=U−JH)=6.5e Vi n the Dudarev’s scheme [ 36]. III. RESULTS A. X-ray diffraction The powder XRD patterns of KNaCuP 2O7along with the Rietveld reﬁnement are shown in Fig. 2for two different temperatures ( T=300 and 15 K). All the XRD patterns downFIG. 2. Powder XRD patterns (open circles) at room temperature (300 K) and 15 K for KNaCuP 2O7. The solid line is the Rietveld ﬁt, the vertical bars mark the expected Bragg peak positions, and the lower solid line corresponds to the difference between the observedand calculated intensities. to 15 K could be reﬁned using the same crystal structure (monoclinic, space group P21/n), which indicates that there is neither any structural transition nor lattice distortion. Theappearance of sharp and high-intensity peaks with no extrareﬂections further reﬂects the high-quality and phase puresample. From the reﬁnement, the goodness of ﬁt is achievedto be χ 2∼7.4 and ∼8.2f o r T=300 and 15 K, respec- tively. The reﬁned lattice parameters and unit cell volume are[a=5.1846(1) Å, b=13.9904(2) Å, c=9.0777(2) Å, β= 91.286(2) ◦, and Vcell/similarequal658.281 Å3] and [ a=5.1731(1) Å, b=13.9110(2) Å, c=9.0515(1) Å, β=91.484(2)◦, and Vcell/similarequal651.20 Å3]f o r T=300 and 15 K, respectively. The reﬁned structural parameters at room temperature are in closeagreement with the values reported earlier [ 25]. Moreover, V cell/similarequal658.281 Å3at room temperature is found to have an intermediate value between K 2CuP 2O7(∼721.88 Å3), Li2CuP 2O7(∼585.24 Å3), and Na 2CuP 2O7(∼612.88 Å3), as expected based on the ionic radii of K1+,L i1+, and Na1+ [37]. Hence, one may also expect the magnetic parame- ters of KNaCuP 2O7to have values between K 2CuP 2O7and (Li,Na) 2CuP 2O7, as a change in volume brings in a change in the interatomic distances. The obtained temperature-dependent lattice parameters ( a,b,c, and β) and unit cell volume ( V cell) are plotted in Fig. 3. The lattice constants a, b, and care found to be decreasing in a systematic way, while the monoclinic angle βis increasing with decreasing 224415-3S. GUCHHAIT et al. PHYSICAL REVIEW B 103, 224415 (2021)(deg) FIG. 3. The lattice constants ( a,b,a n d c), monoclinic angle ( β), and unit cell volume ( Vcell) are plotted as a function of temperature from 15 to 300 K. The solid line in the bottom panel represents the ﬁt using Eq. ( 1). temperature. These lead to a overall decrease of Vcellwith temperature. The variation of unit cell volume with temperature can be expressed in terms of the Grüneisen ( γ) ratio as γ= Vcell(∂P ∂U)Vcell=αVcellK0 Cv, where αis the thermal expansion co- efﬁcient, Cvis the heat capacity at constant volume, K0is the bulk modulus, and U(T) is the internal energy of the system [38]. Assuming both γandK0are independent of temperature, Vcell(T) can be written as [ 39] Vcell(T)=γU(T) K0+V0, (1) where V0is the unit cell volume at T=0 K. According to the Debye model, U(T) can be written as U(T)=9NkBT/parenleftbiggT θD/parenrightbigg3/integraldisplayθD T 0x3 (ex−1)dx, (2) where Nis the number of atoms per unit cell, kBis the Boltz- mann constant, and θDis the average Debye temperature [ 40]. The variable xinside the integration stands for the quantity ¯hω kBTwith phonon frequency ωand Planck constant ¯ h.T h e ﬁt of the experimental Vcell(T) data by Eq. ( 1) is shown as a solid line in the lower panel of Fig. 3. The obtained best ﬁt parameters are θD/similarequal294 K, V0/similarequal651.19 Å3, andγ K0/similarequal 1.14×10−4Pa−1. FIG. 4. Upper panel: χvsTof KNaCuP 2O7in an applied ﬁeld of 1 T and the red solid line is the best ﬁt using Eq. ( 4). The dashed line represents the impurity contribution, χimp(T)=χ0+Cimp T+θimp, ob- tained from the ﬁt. The spin susceptibility χspin(T) is obtained by subtracting χimp(T) from χ(T). The dashed-dotted line illustrates the QMC data with J/kB=55.7Ka n d g=2.1. Lower panel: Inverse magnetic susceptibility (1 /χ) as a function of Tand the solid line is the Curie-Weiss ﬁt. B. Magnetization The magnetic susceptibility [ χ(T)≡M/H]o f KNaCuP 2O7measured in an applied ﬁeld H=1T i s shown in the upper panel of Fig. 4. At high temperatures, χ(T) follows the standard paramagnetic behavior and then passes through a broad maximum at around Tmax χ/similarequal35 K. This broad maximum is a clear signature of the short-rangeordering. At low temperatures, it shows a upturn which couldbe due to extrinsic paramagnetic impurities, defects, and /or uncorrelated spins at the open end of the ﬁnite chains in thepowder sample [ 41,42]. No indication of any magnetic LRO was found down to 2 K. The inverse susceptibility 1 /χ(T) is shown in the bottom panel of Fig. 4. The data in the paramagnetic regime are ﬁtted by the Curie-Weiss (CW) law χ(T)=χ 0+C T+θCW. (3) Here,χ0is the temperature-independent susceptibility, which includes Van Vleck paramagnetic susceptibility (due to openelectron shells of Cu 2+ions) and core diamagnetic suscep- tibility (due to the core electron shells), Cis the Curie constant, and θCWis the CW temperature. The ﬁt in the temperature range T/greaterorequalslant100 K yields the parameters χ0/similarequal 2.01×10−4cm3/mol Cu2+,C/similarequal0.425 cm3K/mol Cu2+, andθCW/similarequal+ 33 K. Using the value of C, the effective 224415-4QUASI-ONE-DIMENSIONAL UNIFORM … PHYSICAL REVIEW B 103, 224415 (2021) moment can be estimated as μeff=(3kBC/NAμ2 B)1 2, where NAis the Avogadro’s number and μBis the Bohr mag- neton. Our experimental value of Ccorresponds to μeff/similarequal 1.84μB/Cu2+. This value of μeffis slightly greater than the ideal value 1 .73μBfor spin-1 2and is typical for Cu2+-based compounds [ 43,44]. The positive value of θCWindicates the AF exchange coupling between the Cu2+ions. The core diamagnetic susceptibility ( χcore) of the compound was cal- culated to be −1.15×10−4cm3/mol by adding the core diamagnetic susceptibility of Na+,K+,C u2+,P5+, and O2− ions [ 45]. The Van Vleck paramagnetic susceptibility ( χvv) was estimated to be ∼3.16×10−4cm3/mol by subtracting χcorefromχ0, which is very close to the value reported for other Cu2+-based compounds [ 13,17,46]. In order to understand the spin lattice, χ(T) was ﬁtted by the uniform spin-1 2Heisenberg chain model, taking into account the temperature-independent ( χ0) and extrinsic para- magnetic contributions. For the purpose of ﬁtting, one canwriteχ(T) as the sum of three parts, χ(T)=χ 0+Cimp T+θimp+χspin(T). (4) Here, the second term accounts for the paramagnetic impu- rity contributions, with θimpbeing the interaction strength between the impurity spins and χspin(T) represents the spin susceptibility of a spin-1 2uniform Heisenberg AF chain. We have used the expression of χspin(T) given by Johnston et al. [47], which predicts the spin susceptibil- ity accurately over a wide temperature range 5 ×10−25/lessorequalslant kBT/J/lessorequalslant5. Our experimental data in the whole measured temperature range were ﬁtted well by Eq. ( 4), reﬂecting the purely 1D character of the compound. As shown inFig. 4(upper panel), the best ﬁt yields the intrachain cou- pling J/k B/similarequal55.5K ,χ0/similarequal2×10−4cm3/mol Cu2+,Cimp/similarequal 0.0089 cm3K/mol Cu2+,θimp/similarequal1.74 K, and Landé g-factor g/similarequal2.1. The value of Cimpcorresponds to an impurity con- centration of nearly ∼2.1%, assuming impurity spins S=1 2. A slightly larger value of g(>2) is typically observed from electron-spin-resonance (ESR) experiments on Cu2+-based compounds [ 21]. The intrinsic χspin(T)o fK N a C u P 2O7obtained af- ter subtracting the temperature-independent and paramag-netic impurity contributions from χ(T) is also shown in Fig. 4(upper panel). We also simulated χ spin(T)u s i n ga QMC simulation considering a uniform chain model withJ/k B=55.7 K and g=2.1 [see Fig. 4(upper panel)]. The simulated data without any additional term repro-duceχ spin(T) perfectly in the whole temperature range. Indeed, our estimated quantities χmax spinJ/NAg2μ2 B/similarequal0.1464 andχmax spinTmax χ/g2/similarequal0.035 12 cm3K/mol (where χmax spin= 0.004 38 cm3/mol is the maximum in χspin atTmax χ in Fig. 4) are quite consistent with the theoretically predicted values χmax spinJ/NAg2μ2 B=0.146 926 279 and χmax spinTmax χ/g2= 0.035 322 9 cm3K/mol [ 47,48], endorsing the 1D spin-1 2uni- form HAF nature of the spin lattice in KNaCuP 2O7. The magnetization isotherms ( MvsH) measured at two end temperatures ( T=2 and 300 K) are shown in Fig. 5.F o r T=300 K, Mincreases linearly with H, as expected for typi-FIG. 5. Magnetization ( M) of KNaCuP 2O7as a function of mag- netic ﬁeld ( H) at two different temperatures. The solid line is the ﬁt to the magnetic isotherm at T=2 K, as described in the text. cal AFs at high temperatures. On the other hand, for T=2K , the behavior is found to be nonlinear and Mreaches a value ∼0.064μB/Cu2+at 9 T which is far below the saturation value 1 μB. This is because our maximum measured ﬁeld of 9 T is far below the expected saturation ﬁeld Hs=2J/gμB/similarequal 78.5 T, taking J/kB/similarequal55.5K[ 21]. Further, the magnetization data at T=2 K were ﬁtted well using the phenomenolog- ical expression for a spin chain, Mchain=αH+β√ H.T h e obtained parameters α/similarequal5.46×10−7andβ/similarequal5.02×10−5 are comparable with the values reported for the spin-1 2chain compound Bi 6V3O16[49]. C. Speciﬁc heat The temperature-dependent speciﬁc heat Cp(T) measured in zero applied ﬁeld is shown in Fig. 6. No anomaly associated FIG. 6. Cpof KNaCuP 2O7as a function of temperature in the absence of magnetic ﬁeld. Inset: Cp/TvsT2at low temperatures. 224415-5S. GUCHHAIT et al. PHYSICAL REVIEW B 103, 224415 (2021) with the magnetic LRO was noticed down to 2 K, consistent with the χ(T) data. In a magnetic insulator, there are two major contributions to the speciﬁc heat: phonon excitationsand a magnetic contribution. In the high-temperature region(T>J/k B),Cpis mainly dominated by phonon excitations, whereas the magnetic part contributes only in the low-temperature region. In the low-temperature regime, C p(T) can be ﬁtted by Cp= γT+βT3, where the cubic term accounts for the phononic contribution to the speciﬁc heat ( Cph) and the linear term rep- resents the magnetic contribution to the speciﬁc heat ( Cmag). In the inset of Fig. 6,Cp/Tis plotted against T2which follows a linear behavior in the low-temperature regime. Fora gapless spin- 1 21D HAF chain, Cmag(T) at low tempera- tures is expected to be linear with temperature and the linearcoefﬁcient ( γ) provides a measure of J/k B. From the the- oretical calculations, Johnston and Klümper have predictedthe relation γ theory=2R 3(J/kB)for low temperatures T<0.2J/kB [47,50]. Using the value of J/kB/similarequal55.5 K, it is calculated to beγtheory/similarequal0.1J/mol K2for KNaCuP 2O7.T h e Cp/Tvs T2data in the temperature range T/lessorequalslant10 K were ﬁtted by the above equation and the extracted parameters are γexpt/similarequal0.107 J/mol K2andβ/similarequal0.0018 J /mol K4.T h ev a l u eo f γexptis indeed very close to γtheory . Following the Debye model, one can write β=12π4mR/5θ3 D, where mis the total number of atoms in the formula unit and Ris the universal gas constant [40]. From the value of βthe corresponding Debye tempera- ture is estimated to be θD/similarequal235 K, which is close to the value obtained from the VcellvsTanalysis [ 51]. D. NMR NMR is an extremely powerful local tool used to inves- tigate the static and dynamic properties of a spin system. InKNaCuP 2O7, P is coupled strongly while Na, which is located in between the chains, is coupled weakly to the Cu2+ions (see Fig. 1). Therefore, one can extract information about Cu2+ spins by probing at the31Pand23Na nuclear sites. 1.31PNMR spectra As presented in Fig. 7, we obtained a narrow and single spectral line at high temperatures, as expected for an I= 1/2 nucleus. The line shape is asymmetric and the central line position shifts with temperature. The asymmetric lineshape reﬂects either asymmetry in the hyperﬁne coupling oranisotropic spin susceptibility. As the temperature is lowered,the linewidth also increases. Further, there are two inequiv-alent P sites in the crystal structure and both of them arecoupled to the Cu 2+ions. Thus, our experimentally observed single spectral line in the whole measured temperature rangeimplies that the local environment of both the P sites is nearlythe same. Indeed, a careful analysis of the crystal structurereveals that the atomic positions of both the P sites are veryclose to each other. Further, no signiﬁcant line broadening orchange in line shape was observed down to 1.6 K, ruling outthe low-temperature magnetic LRO.FIG. 7. Field sweep31PNMR spectra of KNaCuP 2O7at different temperatures measured in 121 MHz. The dashed line indicates the reference ﬁeld position. 2.31PNMR shift The temperature-dependent NMR shift [31K(T)] extracted from the central peak position is shown in Fig. 8. Similar toχ(T),31K(T) also passes through a broad maxima at around 40 K, a footprint of the 1D short-range correlations.The noteworthy characteristic of 31K(T)i st h a t31K(T) has a great advantage over the bulk χ(T). At low temperature χ(T) shows a Curie tail which originates mostly from either extrinsic paramagnetic impurities or defects in the powdersample. In contrast, the NMR shift is completely insensitive tothese contributions and probes only the intrinsic spin suscep-tibility, as the 31Pnucleus is coupled only to the Cu2+spins in the chain. Thus, the31K(T) data allow us to do a more accurate analysis of χspinthanχ(T). Moreover, the effect of impurity and defect contributions appears in the form ofNMR line broadening. Therefore, the linewidth as a functionof temperature should follow the bulk χ(T). One can express 31K(T) in terms of χspin(T)a s 31K(T)=K0+/parenleftbigg31Ahf NAμB/parenrightbigg χspin(T), (5) where K0is the temperature-independent chemical shift and 31Ahfis the average hyperﬁne coupling between the31Pnu- cleus and Cu2+ions. The plot of31Kvsχspinwith Tas an indirect variable is shown in the lower panel of Fig. 8. Here, 224415-6QUASI-ONE-DIMENSIONAL UNIFORM … PHYSICAL REVIEW B 103, 224415 (2021) FIG. 8. Upper panel:31PNMR shift (31K) vs temperature in 121 MHz. The solid line is the ﬁt using Eq. ( 5). Inset: Full width at half maximum (31FWHM) vs T. Lower panel:31Kvsχspinmeasured atH=1 T in the Trange 2–300 K. The solid line is a linear ﬁt. Inset:31FWHM vs χand the solid line is a linear ﬁt. χspinatH=1 T is taken from Fig. 4. The plot exhibits a nice straight line over the whole temperature range. From theslope of the linear ﬁt, the total hyperﬁne coupling constant iscalculated to be 31Ahf/similarequal2151.2O e/μB. In order to establish the spin lattice and to extract the ex- change coupling,31K(T) data were ﬁtted using Eq. ( 5), taking the expression of χspin(T) for a spin-1 2uniform Heisenberg AF chain model [ 47]. It is apparent from Fig. 8that Eq. ( 5) provides an excellent ﬁt to the data in the entire temperaturerange 1 .6K/lessorequalslantT/lessorequalslant300 K, unambiguously corroborating the 1D character of the spin lattice. While ﬁtting, the value ofhyperﬁne coupling was kept ﬁxed to A hf/similarequal2151 Oe /μB, obtained from the31K-χanalysis. The obtained best ﬁt pa- rameters are K0/similarequal52.74 ppm, J/kB/similarequal58.7 K, and g/similarequal2.17. Theoretically, χspin(T)o r K(T) for a spin-1 2uniform HAF chain is predicted to show a logarithmic decrease(lnT −1) at low temperature ( T<0.1J/kB) and reaches a ﬁnite value at T=0K [ 52]. The exact value of spin sus- ceptibility at zero temperature can be estimated as χspin(T= 0)=NAg2μ2 B Jπ2[47,53]. Experimentally, χ(T) and17OK(T) data of Sr 2CuO 3and31PK(T) data of (Sr ,Ba) 2Cu(PO 4)2and K2CuP 2O7, at very low temperatures, are reported to show such a logarithmic decrease [ 13,17,18]. For Sr 2CuO 3with J/kB/similarequal2200 K, the decrease was observed at T/similarequal0.01J/kBinχ(T)[13] and at kBT/J/similarequal0.015 in K(T)[16]. Simi- larly, for (Sr ,Ba) 2Cu(PO 4)2(J/kB/similarequal160 K) and K 2CuP 2O7 (J/kB/similarequal141 K) the decrease in K(T) was observed be- low T/similarequal0.003J/kBand 0 .028J/kB, respectively [ 17,18]. However, in KNaCuP 2O7,31K(T) attains a ﬁnite value ∼1334 ppm at 1.6 K, without any logarithmic decrease. More-over, this value is found to be larger than the theoretically expected value K theo(T=0K )=K0+Ahfg2μB Jπ2/similarequal1234 ppm, taking J/kB/similarequal58.7K ,31Ahf/similarequal2151 Oe /μB, and g=2.17. In our case, the lowest measured temperature of 1.6 K cor-responds to ∼0.03J/k Bonly. This implies that one may need to go further below 1.6 K in order to see the low-temperaturedecrease in 31K(T). The full width at half maximum (31F W H M )o ft h e31P NMR spectra as a function of temperature is shown in the inset of the upper panel of Fig. 8. It displays a broad maximum at around 35 K and a Curie tail below 10 K, suggesting that 31FWHM traces the bulk χ(T), as expected. The31FWHM vs χplot (see, lower inset of Fig. 8) is quite linear above 6 K. 3.31Pspin-lattice relaxation rate311/T1 The31Pspin-lattice relaxation rate311/T1was measured at the ﬁeld corresponding to the central peak position ateach temperature. The longitudinal magnetization recoveriesat three selected temperatures are shown in the upper panel ofFig. 9.A s 31Pis an I=1/2 nucleus, one can ﬁt the recoveries by a single exponential function 1−M(t) M(∞)=Ae−t/T1, (6) where M(t) is the nuclear magnetization at a time tafter the saturation pulse and M(∞) is the equilibrium ( t→∞ ) mag- netization. Indeed, all the recovery curves could be ﬁtted wellby Eq. ( 6) (see the upper panel of Fig. 9) and the curves show a linearity over more than two decades when the yaxis is plotted in log scale. The extracted 311/T1as a function of temperature measured at different frequencies are shown in the lower panelof Fig. 9. For the data at 121 MHz, 311/T1is almost constant forT>90 K which is typical due to the random movement of the paramagnetic moments [ 54]. As the temperature is lowered further,311/T1decreases in a linear manner down to 20 K and then exhibits a temperature-independent behaviorbetween 20 and 4 K. At very low temperatures ( T<4K ) , 311/T1increases rapidly, which indicates the slowing down of the ﬂuctuating moments as the system approaches the mag-netic LRO at T N. From the low-temperature trend of311/T1, the magnetic LRO is expected to set in at around TN∼1K . 4.23NaNMR spectra Since23Na is a quadrupolar nucleus with I=3/2, the NMR line should have three lines: the central line correspond-i n gt ot h e I z=+ 1/2←→ − 1/2 transition and two equally spaced satellite lines corresponding to Iz=± 3/2←→ ± 1/2 transitions on either side of the central line. The23Na spectra as a function of temperature are presented in Fig. 10.A th i g h temperatures, the line is very narrow and slightly asymmetric.As the temperature is lowered, the linewidth increases andtwo broad humps or satellites on both sides of the centralline become prominent [ 55]. However, the overall line shape 224415-7S. GUCHHAIT et al. PHYSICAL REVIEW B 103, 224415 (2021) FIG. 9. Upper panel: Longitudinal magnetization recovery curves at three selective temperatures measured on the31Pnuclei and the solid lines are ﬁts using Eq. ( 6). Lower panel:31PNMR spin-lattice relaxation rate (311/T1) as a function of temperature measured in different frequencies. The xaxis is shown in log scale in order to highlight the features in different temperature regimes. Inset:1/( 31K31T1T)v sTfor 121 MHz. remains invariant down to 1.6 K. Further, the position of the central line does not shift at all with temperature, whichconﬁrms a weak hyperﬁne coupling of 23Na with the Cu2+ ions due to a negligible overlap of orbitals. This also justi- ﬁes why the interchain interaction via Na is so weak. Thespectrum at T=15 K could be ﬁtted well with K iso/similarequal− 60 ppm (isotropic shift), Kaxial/similarequal20 ppm (axial shift), Kaniso/similarequal 50 ppm (anisotropic shift), η=0 (asymmetry parameter), andνQ/similarequal0.57 MHz [nuclear quadrupole resonance (NQR) frequency]. The quadrupole frequency is almost temperatureindependent in the whole temperature range, which essentiallyexcludes the possibility of any structural distortion in thestudied compound. The 23FWHM with temperature, obtained from the ﬁt, is shown in the left inset of Fig. 10. It passes through a broad maximum and then shows a low-temperatureCurie tail, identical to the bulk χ(T). The 23FWHM vs χplot (see, right inset of Fig. 10) is linear above 10 K.FIG. 10. Field sweep23Na NMR spectra of KNaCuP 2O7at dif- ferent temperatures. The solid line is the ﬁt of the spectrum at T= 15 K and the satellites are marked by arrows. Left inset:23FWHM vs T. Right inset:23FWHM vs χand the solid line is a linear ﬁt. 5.23Naspin-lattice relaxation rate231/T1 231/T1was measured by irradiating the central line of the23Na spectra, choosing an appropriate pulse width. The recovery of the longitudinal magnetization was ﬁtted well bythe following double stretch exponential function [ 56,57] 1−M(t) M(∞)=A[0.1e x p (−t/T1)β+0.9e x p (−6t/T1)β], (7) relevant for the23Na (I=3/2) nuclei. Here, βis the stretch exponent. The upper panel of Fig. 11depicts recovery curves at three different temperatures. The obtained231/T1vsTis shown in the lower panel of Fig. 11. The overall temperature dependence behavior of231/T1is nearly identical to that observed for311/T1(T). For T>150 K,231/T1is almost temperature independent. It decreases linearly below 150 Kdown to 30 K and remains constant between 30 and 4 K. Be-low 4 K, 231/T1shoots up and from the low- Tdiverging trend one expects a peak at around TN/similarequal1 K, similar to311/T1.T h e exponent βas a function of Tis presented in the inset of the upper panel of Fig. 11. The absolute value of βvaries between 224415-8QUASI-ONE-DIMENSIONAL UNIFORM … PHYSICAL REVIEW B 103, 224415 (2021) FIG. 11. Longitudinal magnetization recovery curves at three se- lective temperatures measured on the23Na nuclei and the solid lines are ﬁts using Eq. ( 7). Inset: The exponent βas a function of T. Lower panel:231/T1as a function of T. Inset: The ratio of relaxation rates 231/T1and311/T1vsTmeasured at H/similarequal7T . 0.63 and 0.84. Such a reduced value of βillustrates that there could be a Na deﬁciency, as Na is the lightest element in thecompound. E. Electronic structure calculations First-principles electronic structure calculations in the framework of DFT have been carried out to identify thedominant exchange paths, the various exchange couplings,and the resulting spin model. In order to get insights on theelectronic structure of KNaCuP 2O7, we have started with the non-spin-polarized calculations [see Fig. 12(a) ]. Our calcu- lations revealed O pstates are completely occupied while K, Na, and P states are empty, consistent with the nominalionic formula K 1+Na1+Cu2+P25+O72−, indicating Cu is in the 3 d9conﬁguration. As a consequence, the Fermi level is dominated by four Cu dbands arising from the four Cu atoms in the four formula unit cells of KNaCuP 2O7[see Fig. 12(a) ]. In the local frame of reference, i.e., assuming that the Cu atomis residing at the origin and choosing the zaxis along the long Cu-apical O bond, the xandyaxes along the Cu-O bonds in the basal plane, we ﬁnd that these bands at the Fermi levelare predominantly of Cu d x2−y2character. The band structure FIG. 12. (a) Non-spin-polarized band dispersion along various high-symmetry directions. The inset shows the crystal ﬁeld splitting.(b) Wannier function of the effective Cu d x2−y2orbital. shows a strong dispersion parallel to the chain direction Z- BandD-Ybut is nearly dispersionless perpendicular to the direction of the chains, indicating a strong 1D character ofthis system. In order to evaluate the Cu intersite exchange strengths, we have calculated exchange interactions using the “four-state”method [ 58] based on the total energy of the system with few collinear spin alignments. If the magnetism in the sys-tem is fully described by the Heisenberg Hamiltonian ( H=/summationtext ijJijSi·Sj), the energy for such a spin pair can be written as follows, E=J12S1·S2+S1·h1+S2·h2+Eall+E0, (8) 224415-9S. GUCHHAIT et al. PHYSICAL REVIEW B 103, 224415 (2021) TABLE I. Exchange parameters of KNaCuP 2O7obtained from DFT calculations: Cu-Cu distances d(in Å), electron hoppings ti (in meV), AFM contributions to the exchange JAFM i=4t2 i/Ueff(in K), and total exchange couplings Ji(in K) from the generalized gradient approximation plus interaction term U(GGA +U) mapping procedure with Ueff=6.5e V . dCu-Cu ti JAFM i Ji J 5.17 98 69 59 J/prime5.67 2.17 ∼0.1 ∼0.1 J/prime/prime5.77 0.14 ∼0.1 ∼0.1 where we consider the exchange interaction J12between spins at sites 1 and 2. h1=/summationtext i/negationslash=1,2J1iSi,h2=/summationtext i/negationslash=1,2J2iSi, Eall=/summationtext i/negationslash=1,2JijSi·Sj, and E0contains all other nonmag- netic energy contributions. The second (third) term in Eq. ( 8) corresponds to the coupling of the spin 1 (2) with all otherspins in the unit cell excluding spin 2 (1). E alltakes into ac- count the exchange couplings between all spins in the unit cellexcept from spins 1 and 2. The exchange interaction strengthbetween sites 1 and 2 is obtained by considering four collinearspin states (i)1 ↑,2↑, (ii) 1 ↑,2↓, (iii) 1 ↓,2↑, and (iv) 1 ↓,2↓as J12=E↑↑+E↓↓−E↑↓−E↓↑ 4S2. (9) The ﬁrst (second) sufﬁx of energy ( E) represents the spin state of site 1 (2). The estimated exchange interactions along withthe corresponding Cu-Cu distances [as depicted in Fig. 1(b)] are tabulated in Table I. The NN exchange interaction is found to be the strongest one and AFM ( J/k B=59 K) which is in excellent agreement with the experiment. The other exchangeinteractions J /primeandJ/prime/primeare abysmally small (0.1 K) and are AFM, adding interchain frustration to the system. Further, thecalculated mean-ﬁeld Curie-Weiss temperature θ CW=29 K compares well with the experiment [ 35]. Finally, the Cu dx2−y2Wannier function has been plotted for KNaCuP 2O7in Fig. 12(b) . The tails of the Cu dx2−y2orbital are shaped according to the O px/pyorbitals such that Cu dx2−y2forms strong pdσantibonds with the O px/pytails in the basal plane. We see that the Cu-Cu hopping primarilyproceeds via the oxygen. The dominant intrachain AFM ex-change interaction Jis mediated via the Cu-O-P-O-Cu path, while the other interchain exchange interactions are mediatedvia the long Cu-O bond along the apical oxygen (2.32 Å),thereby rendering them to be weak. IV . DISCUSSION We have demonstrated that KNaCuP 2O7is a good ex- ample of a 1D spin-1 2uniform HAF. KNaCuP 2O7formally belongs to the family of A2CuP 2O7(A=Na, Li, and K) compounds, although they have different crystal structures.KNaCuP 2O7has a monoclinic structure with space group P21/nin contrast to a monoclinic unit cell with space group C2/cfor (Na ,Li) 2CuP 2O7and an orthorhombic unit cell with space group Pbnm for K 2CuP 2O7[37]. For (Na ,Li) 2CuP 2O7, slightly distorted CuO 4plaquettes are corner shared with PO 4tetrahedra, making spin chains with an intrachain ex-change coupling J/kB/similarequal28 K and magnetic LRO at TN/similarequal 5K[ 19,21]. Here, the neighboring plaquettes are tilted toward each other by an angle of about 70◦and 90◦for Na and Li compounds, respectively, resulting in a buckling of the spinchains. This modulation in spin chains is responsible for aweaker intrachain coupling and magnetic LRO at a relativelyhigh temperature. On the other hand, for K 2CuP 2O7, the ar- rangement of CuO 4plaquettes is more planar and the chains are strictly straight, which gives rise to a pronounced 1Dmagnetism with a larger intrachain coupling J/k B/similarequal141 K and without any magnetic LRO down to 2 K [ 18]. For KNaCuP 2O7, though the CuO 4plaquettes are arranged in the same plane, similar to K 2CuP 2O7, they are more distorted with four different Cu-O bond distances ( ∼1.932–1 .987 Å). Further, the Cu-Cu interchain distances are slightly reducedfor KNaCuP 2O7(∼5.6767–7 .01 Å) compared to K 2CuP 2O7 (∼5.879–7 .388 Å). Because of the difference in the struc- tural arrangements, the intrachain (NN) exchange couplingof KNaCuP 2O7(J/kB/similarequal58.7 K) has an intermediate value between K 2CuP 2O7and (Na ,Li) 2CuP 2O7. Further, the interchain couplings, which are unavoidable in experimental compounds, drive the system into a LRO state ata ﬁnite temperature. However, when the interchain couplingsform a frustrated network, the ground state is modiﬁed signif-icantly and in many cases forbids the compound from goingto a LRO state. The magnetic LRO at a very low temperature(T N/similarequal1 K) in KNaCuP 2O7evidences extremely weak as well as frustrated interchain exchange couplings. With this valueofT N, the compound exhibits one-dimensionality over a large temperature range kBTN/J/similarequal1.7×10−2. One can tentatively estimate the average interchain coupling ( J/prime) of a quasi-1D HAF chain by putting the appropriate values of JandTNin the simple expression obtained from the mean-ﬁeld approxi-mation [ 59,60] J /prime/kB=3.046TN zkAF/radicalBig ln/parenleftbig5.8J kBTN/parenrightbig +0.5l nl n/parenleftbig5.8J kBTN/parenrightbig. (10) Here, kAFrepresents the AF wave vector and z=6i st h e number of nearest-neighbor spin chains. Numerical calcula-tions for a 3D model yield k AF/similarequal0.70. For KNaCuP 2O7, using J/kB/similarequal58.7 K and TN/similarequal1 K, the average interchain coupling is estimated to be J/prime/kB(=J/prime/prime/kB)/similarequal0.28 K. This value is of the same order of magnitude as that obtained fromthe electronic structure calculations. The spin-lattice relaxation rate 1 /T 1provides useful infor- mation on the spin dynamics or dynamic susceptibility of aspin system. It helps to access the low-energy spin excitationsby probing the nearly zero-energy limit (in the momentumspace) of the local spin-spin correlation function [ 61]. Quite generally, 1 T1Tis written in terms of the dynamic susceptibility χM(/vectorq,ω0)a s[ 54] 1 T1T=2γ2 NkB N2 A/summationdisplay /vectorq\|A(/vectorq)\|2χ/prime/prime M(/vectorq,ω0) ω0, (11) where the sum is over the wave vector /vectorqwithin the ﬁrst Brillouin zone, A(/vectorq) is the form factor of the hyperﬁne in- teraction, and χ/prime/prime M(/vectorq,ω0) is the imaginary part of the dynamic 224415-10QUASI-ONE-DIMENSIONAL UNIFORM … PHYSICAL REVIEW B 103, 224415 (2021) susceptibility at the nuclear Larmor frequency ω0. Thus, 1 /T1 has contributions from both uniform ( q=0) and staggered (q=±π/a) spin ﬂuctuations. For 1D spin-1 2chains, theory predicts that the uniform component leads to 1 /T1∝Twhile the staggered component gives 1 /T1=const [ 62,63]. Typi- cally, q=±π/aandq=0 components dominate the 1 /T1 data in the low-temperature ( T/lessmuchJ/kB) and high-temperature (T∼J/kB) regimes, respectively [ 17]. Thus, the experimen- tally observed linear decrease and temperature-independentbehavior of 1 /T 1in the intermediate-temperature ranges re- ﬂect the dominance of q=0 and q=±π/acontributions, respectively. As discussed earlier,31Pis located symmetrically between two adjacent Cu2+ions along the chain. Similarly,23Na is coupled, though weakly, to four Cu2+ions from three neigh- boring chains. Therefore, the staggered components of thehyperﬁne ﬁelds from the neighboring Cu 2+ions are expected to be canceled out at both the31Pand23Na sites. Accordingly, one should be able to probe the low-energy spin excitationscorresponding to the q=0 mode separately from the stag- gered q=±π/amode. However, in our case, there is still a signiﬁcant contribution from q=±π/awhich dominates the low-temperature 1 /T 1data. One possible source of the remnant staggered ﬂuctuations could be the unequal hyperﬁnecouplings arising due to the low symmetry of the crystalstructure. Further, the linear and constant temperature regimesare found to be different for 311/T1and231/T1, which is likely due to a subtle difference in the hyperﬁne form factorsassociated with the 31Pand23Na nuclei. In Eq. ( 11)f o r q=0 andω0=0, the real component of χ/prime M(q,ω0) represents the static susceptibility χ(orK). Therefore, 1 /(χT1T) should be temperature independent. As shown in the inset of thelower panel of Fig. 9,1/( 31K31T1T) indeed demonstrates the dominant contribution of χto 1/(31T1T). However, a slight increase in 1 /(31K31T1T) below ∼5 K indicates the growth of AF correlations with decreasing T. Moreover, when the ratio of231/T1at 79 MHz ( H/similarequal7.0147 T) and311/T1at 121 MHz (H/similarequal7.0203 T) is plotted against temperature (see the inset of the lower panel of Fig. 11), it results in an almost constant value above ∼40 K and then increases rapidly towards low temperatures. In order to detect the effect of an external magnetic ﬁeld on the spin dynamics, we have measured311/T1vsTat different frequencies /ﬁelds. As seen in the lower panel of Fig. 9,311/T1shows a strong frequency dependency in the high-temperature regime and the absolute value of311/T1 decreases with an increase in frequency. This difference is narrowed down as the temperature is lowered, and belowabout 20 K, the data sets in different frequencies overlap witheach other. It is established that the long-wavelength ( q∼0) spin ﬂuctuations in a Heisenberg magnet often show diffusivedynamics. In 1D spin chains, such a spin diffusion leads toa1/√ Hﬁeld dependence of311/T1[64,65]. Thus, the strong ﬁeld dependency of311/T1at high temperatures appears to be due to the effect of spin diffusion where long-wavelength q= 0 ﬂuctuations dominate. Moreover, the weak ﬁeld dependencyof 311/T1at low temperatures also reﬂects that the relaxation is dominated by the staggered ( q=±π/a) ﬂuctuations below 20 K.FIG. 13. Upper panel:31PNMR spin-lattice relaxation rate (311/T1) as a function of applied magnetic ﬁeld ( μ0H)a tT=80, 125, and 200 K. The solid lines are the ﬁts using 1 /T1=a+ b/√μ0H.I n s e t :311/T1vs 1/√μ0H. Lower panel: Temperature dependence of Dsdeduced from311/T1. The solid line is the ﬁt using Ds∼1/T2. The classically expected value at high temperatures is also shown as a dashed line. The contribution of spin diffusion to 1 /T1can be written as [15,16,66] 1 Tsd 1T=A2 hf(q=0)γ2 nkBχ(T,q=0) μ2 B√2gμBDsH/¯h, (12) where Dsis the spin-diffusion constant. Thus, the slope of the linear311/T1vs 1/√ Hplot at a ﬁxed temperature should yield Ds. In the upper panel of Fig. 13, we have plotted 311/T1vsHfor three different temperatures ( T=80, 125, and 200 K) which are ﬁtted by 1 /T1=a+b/√μ0H, where aandbare the constants. To magnify the linear behavior, 311/T1is plotted against 1 /√μ0Hin the inset of the upper panel of Fig. 13. Using the value of χ(T) obtained from the NMR shift measurement and the slope ( b)i nE q .( 12), the diffusion constant at each temperature is determined. The tem-perature dependence of D sdeduced from311/T1is presented in the lower panel of Fig. 13. It increases moderately with de- creasing temperature, as expected in the region dominated bytheq=0 ﬂuctuations. The value of D sin high temperatures (T>100 K) is of the same order as the classically ex- pected value, Ds=(J/¯h)√2πS(S+1)/3=9.64×1012s−1 [66]. This is indeed consistent with the previous reports on other Heisenberg spin-chain compounds [ 15,65,67,68]. Fur- ther, the temperature-dependent Dscould be ﬁtted by Ds∼ 224415-11S. GUCHHAIT et al. PHYSICAL REVIEW B 103, 224415 (2021) 1/T2, similar to17ONMR in Sr 2CuO 3[16]. However, it is not clear whether such a behavior of Ds(T) can be accounted for by the 1D spin-1 2chain model. V . CONCLUSION Our results demonstrate that KNaCuP 2O7is an excel- lent 1D spin-1 2HAF model system with a nearest-neighbor only exchange. The magnetic susceptibility, magnetizationisotherm, and 31PNMR shift data show good agreement with the theoretical predictions for a 1D spin-1 2HAF chain with intrachain coupling J/kB/similarequal58.7 K. The value of intra- chain coupling is further conﬁrmed from the complementaryelectronic structure calculations and the subsequent QMCsimulations. From the 31Kvsχspinplot, the hyperﬁne cou- pling of31Pwith the Cu2+ion is estimated to be31Ahf/similarequal 2151.2O e/μB. The presence of magnetic LRO at a very low temperature provides evidence of extremely weak andfrustrated interchain couplings and one-dimensionality over alarge temperature range k BTN/J/similarequal1.7×10−2. The moderate value of the exchange coupling allowed us to access the spinexcitations of the spin- 1 2Heisenberg chain at both low- andhigh-temperature limits. The change of slope in311/T1(T) and231/T1(T) at around T∼20–30 K explains the crossover regime of the dominant contributions from the uniform ( q= 0) and staggered ( q=±π/a) spin ﬂuctuations. Our results also established that the dynamic spin susceptibility has astrong diffusive contribution at high temperatures. However,the nature of the temperature-dependent diffusion constant D s is not yet understood. ACKNOWLEDGMENTS The authors acknowledge I. Dasgupta for discussions re- garding the theoretical work. S.G. and R.N. would like toacknowledge BRNS, India for ﬁnancial support bearing Sanc-tion No. 37(3) /14/26/2017-BRNS. 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PhysRevB.99.134102.pdf	PHYSICAL REVIEW B 99, 134102 (2019) Valley Hall phases in kagome lattices Natalia Lera,1Daniel Torrent,2P. San-Jose,3J. Christensen,4and J. V . Alvarez1 1Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, Madrid 28049, Spain and Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera (INC) 2GROC, Institut de Noves Tecnologies de la Imatge (INIT), Universitat Jaume I, Castellon 12071, Spain 3Materials Science Factory, Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), Sor Juana Ines de la Cruz 3, 28049 Madrid, Spain 4Department of Physics, Universidad Carlos III de Madrid, Leganes 28916, Madrid, Spain (Received 19 December 2018; revised manuscript received 11 February 2019; published 8 April 2019) We report the ﬁnding of the analogous valley Hall effect in phononic systems arising from mirror symmetry breaking, in addition to spatial inversion symmetry breaking. We study topological phases of plates andspring-mass models in kagome and modiﬁed kagome arrangements. By breaking the inversion symmetry itis well known that a deﬁned valley Chern number arises. We also show that effectively, breaking the mirrorsymmetry leads to the same topological invariant. Based on the bulk-edge correspondence principle, protectededge states appear at interfaces between two lattices with different valley Chern numbers. By means of a planewave expansion method and the multiple scattering theory for periodic and ﬁnite systems, respectively, wecomputed the Berry curvature, the band inversion, mode shapes, and edge modes in plate systems. We alsoﬁnd that appropriate multipoint excitations in ﬁnite system gives rise to propagating waves along a one-sidedpath only. DOI: 10.1103/PhysRevB.99.134102 I. INTRODUCTION The unusual properties of fabricated metamaterials origi- nate from their designed patterns and geometry as opposed to their chemical composition. Speciﬁcally, when created withperiodic structures, the study of wave propagation can betreated similar to electrons in periodic potentials [ 1–6]. In this way, topological properties studied in electronic band struc-tures [ 7] can be transferred to classical metamaterials. Inspired by topological electronic systems, the search for protected modes in classical wave phenomena has been active in recent years in areas such as photonics [ 8,9], acoustics [ 10,11], and elastic media [ 12,13]. The bulk-boundary correspondence principle has been proved to hold also in these areas byshowing how topological protected waves arise at the edge ofsystems containing topologically inequivalent phases. In par-ticular, mechanical metamaterials present several advantages: (1) the ﬂexibility to create patterns and to modify band struc- tures in metamaterials is much richer than in real solids [ 14]. (2) In electronic systems topological features are easier todetect when they occur close to the Fermi energy, whichis hard to shift and control. On the other hand, mechanicalsystems can be excited in a wide range of frequencies, andthe excitation can be easily tuned to the frequency of the topological mode. We consider mechanical metamaterials with time-reversal symmetry, establishing analogy with the quantum valley Halleffect (QVHE) [ 15–18]. The QVHE may arise in systems where intervalley scattering is suppressed and the valley de-gree of freedom is well-deﬁned [ 15,16]. A prototypical exam- ple is the hexagonal Brillouin zone, where opposite corners ofthe hexagon are not related by reciprocal vectors and the lifetime of electrons in each valley is long [ 19]. This fact givesrise to nonequivalent points at opposite momenta. The two valleys act as independent degrees of freedom and thereforeas a pseudospin. In QVHE each valley degree of freedomeffectively behaves as a Chern insulator. Mixing the valleysdegrees of freedom will destroy the effect. This approachhas been successfully achieved in spring-mass models andplate topology [ 20–24], as well as in photonics [ 25–28]o r acoustics [ 11,29–31], by breaking the spatial inversion sym- metry. The existence of topological modes have been shownexperimentally [ 14,32,33], along with unusual properties in the absence of backscattering [ 22,34]. In continuous systems like plates, wave guiding through edge modes could have ap-plications for mechanically isolating structures or transferringenergy and information through elastic waves. In this paper, we focus on the kagome lattice, which has a graphenelike structure with degenerate Dirac conesat inequivalent points of the Brillouin zone. Recent interestin metamaterials based on kagome arrangement suggest fu-ture applications [ 11,35–37]. The wide range of crystalline symmetries and the underlying C 3symmetry of this system provides a playground to test mechanical topology as well asdistinguishing basic features that are relevant to topologicalmechanics. We study discrete spring-mass models in the linear regime in addition to continuum systems such as plates. The formersystems allow analytic computations which capture the essen-tials of topology in easy models with couplings between fewneighbors. The solutions for plate systems are long rangedwaves that propagate through the inﬁnite medium coupling alldegrees of freedom in the system. The understanding of topo-logical modes could lead to relevant engineering applications,in particular, efﬁcient and controlled wave guiding. Plates willbe described in the linear regime by Kirchhoff-Love theory. 2469-9950/2019/99(13)/134102(16) 134102-1 ©2019 American Physical SocietyNATALIA LERA et al. PHYSICAL REVIEW B 99, 134102 (2019) To endow the plate with a crystalline structure, we attach a lattice of resonators on top. Modiﬁcations of the unit cellproperties might open gaps in the phononic band structurewith nontrivial topology. Remarkably, the methodology usedin this paper to describe ﬂexural waves in plates is not basedon commercial software but on the multiple scattering theory(MST) developed in Ref. [ 38]. The structure of the paper is as follows. In Sec. II,w e describe brieﬂy the methodology for studying ﬂexural wavesin plates. In Sec. III, we describe the distorted kagome lattice, its symmetries, and the parameter space used in this paper. InSec. IV, topology arising from spatial inversion symmetry is deduced from the spring-mass model and explained via platephysics, we employ ribbons to create topological protectededge states and design ﬁnite systems with interesting prop-erties, like one-sided wave propagation. In Sec. V, we study the effects of mirror symmetry breaking in a kagome lattice.In Sec. VI, we conclude this paper. II. PLATE PHYSICS AND METHODOLOGY In this section, to present the system and derive the nota- tion, we brieﬂy introduce the classical theory of ﬂexural wavesfor thin plates and describe the methodology, following theapproach taken by Torrent et al. [38] and Chaunsali et al. [24]. We consider a thin plate coupled to a lattice of resonators.The equation of motion for the deformation ﬁeld, wis a fourth-order derivative in real space and we look for solutionsharmonic in time: w(/vectorr,t)=w(/vectorr)e iωt. (D∇4−ω2ρh)w(/vectorr)=−/summationdisplay /vectorRακ/vectorRα(w(/vectorRα)−z(/vectorRκα))δ(/vectorr−/vectorRα), (1) where D=Eh3 12(1−ν2)is the plate stiffness, ρis the volume mass density of the plate, his thickness, and the sum runs over all resonator sites /vectorRαwithin the unit cell. Resonator masses and spring constants are, respectively, mαandκαand their displacements are z(/vectorRα), in the direction perpendicular to the plate. The equation for each resonator is, ω2mαz(/vectorRα)=−κ/vectorRα(w(/vectorRα)−z(/vectorRα)) (2) A. Plane wave expansion In the plane wave expansion method (PWE), the lattice is inﬁnite in two dimensions and the displacement ﬁeld can bewritten in terms of Bloch waves, w(/vectorr)=/summationdisplay /vectorGW(/vectorG)e−i(/vectorG+/vectork)·/vectorr, (3) where /vectorG=n1/vectorg1+n2/vectorg2are the reciprocal lattice vectors, n1,2 are integers and /vectorgjare the basis of vectors fulﬁlling /vectorai·/vectorgj= 2πδij, with /vectoraibeing the lattice vectors. The result is either a search for zeros of a complex function as described inRef. [ 38] or a generalized eigenvalue problem as described in Ref. [ 24]. For completeness, we highlight some steps of the derivation.Method 1. Substituting the resonator equation, Eq. ( 2)i n t o the plate equation, Eq. ( 1), we get /parenleftbigg ∇ 4−ω2ρh D/parenrightbigg w(/vectorr)=−/summationdisplay /vectorRαmα Dω2 αω2 ω2α−ω2w(/vectorRα)δ(/vectorr−/vectorRα), (4) where ω2 α(ω)=κα/mαandtα=mα Dω2 αω2 ω2α−ω2. Due to the sys- tem’s periodicity, we omit the /vectorRdependence on masses and spring constants. Substituting the Bloch Ansatz ( 3) into the previous equation, deriving each independent term in theFourier summation and integrating over the unit cell we obtain /parenleftbigg \|/vectork+/vectorG\| 4−ω2ρh D/parenrightbigg W/vectorG=/summationdisplay /vectorG/prime,αtα Acei(/vectorG/prime−/vectorG)·/vectorRαW/vectorG/prime,(5) where ais the lattice parameter and Acis the area of the unit cell. We have used the following identities: /integraldisplay UCe−i(/vectorG/prime−/vectorG)·/vectorrd/vectorr=Acδ(/vectorG/prime−/vectorG); /integraldisplay UCf(/vectorr)δ(/vectorr−/vectorRα)d/vectorr=f(/vectorRα)( 6 ) Now, we write the expected solution expanded on a the Fourier basis, Wβ=/summationdisplay /vectorG/primeW/vectorG/primeei/vectorG/prime·/vectorRβ, (7) and substitute W/vectorGfrom Eq. ( 5), Wβ=/summationdisplay /vectorG1 \|/vectork+/vectorG\|4−ω2ρh D1 Ac/summationdisplay αei/vectorG·(/vectorRα−/vectorRβ)tαWα. (8) Therefore a set of Nequations with Nunknowns can be written, where Nis the number of resonators per unit cell. We ﬁnd solutions of this system as the zeros of the determinant ofthe following matrix: A αβ(/vectork)=δα,β−γβ/Omega12a2 1−/Omega12//Omega12α/summationdisplay /vectorGe−i/vectorG·(/vectorRα−/vectorRβ) \|/vectork+/vectorG\|4a4−/Omega12a2,(9) where we have introduced the dimensionless variables /Omega12= ω2ρa2h/Dandγα=mα ρa2h. We evaluate for each /vectorkand deduce its/Omega1(/vectork) solutions. The null space of the Amatrix correspond to mode shapes at the resonator points. Method 2. We substitute Bloch waves from Eq. ( 3)i nt h e plate Eq. ( 1). Equating for each mode and integrating over the unit cell, we get Ac(D\|/vectork+/vectorG\|4−ω2ρh)W/vectorG =/summationdisplay ακα⎛ ⎝z(/vectorRα)−/summationdisplay /vectorG/primeW/vectorG/primee−i(/vectorG/prime+/vectork)·/vectorRα⎞ ⎠ei(/vectorG+/vectork)·/vectorRα.(10) Using Bloch’s theorem for the resonators, we can refer all resonator displacements to the ones of the one unit cell, 134102-2V ALLEY HALL PHASES IN KAGOME LATTICES PHYSICAL REVIEW B 99, 134102 (2019) z(/vectorRα)=z(/vectorR0α)e−i/vectork·/vectorRα. We substitute in the previous equation, (\|/vectork+/vectorG\|4a4−/Omega12)W/vectorG =/summationdisplay αγα/Omega12 αei/vectorG·/vectorRα⎛ ⎝z(/vectorR0α)−/summationdisplay /vectorG/primeW/vectorG/primee−i/vectorG/prime·/vectorRα⎞ ⎠, (11) and in the resonator equation ( 2), −/Omega12z(/vectorR0α)=/Omega12 α⎛ ⎝/summationdisplay /vectorGW/vectorGe−i/vectorG·/vectorRα−z(/vectorR0α)⎞ ⎠. (12) Where we have used the same dimensionless variables /Omega1and γthan in method 1. Now Eqs. ( 11) and ( 12) are rewritten in matrix form of dimension NG+N, where NGis the number of reciprocal vectors taken for the computation (calculationsin this paper are made with N G=49) and Nis the number of resonators per unit cell: /parenleftbigg P11 P12 P21 P22/parenrightbigg/parenleftbiggW/vectorG z(/vectorR0,α)/parenrightbigg =/Omega12/parenleftbigg Q11 0 0 Q22/parenrightbigg/parenleftbiggW/vectorG z(/vectorR0,α)/parenrightbigg , (13) where P11,ij=a4\|/vectork+/vectorGi\|4δi,j+/summationdisplay αγα/Omega1αei(/vectorGj−/vectorGi)·/vectorRα, P12,iα=−γα/Omega12 αei/vectorGi·/vectorRα=P∗ 21,αi,P22,αβ=γα/Omega12 αδα,β,(14) Q11,ij=δi,j,Q22,αβ=γαδα,β. In Eq. ( 14), we use i,jindices for the NGreciprocal vectors andα, β for the Nresonators of the unit cell. The generalized eigenvalue problem gives us the band structure, /Omega1(/vectork), and the mode shape by substituting W/vectorGinto Eq. ( 3). B. Edge states in ribbons We consider ribbons of resonators arranged periodically in the/vectorr1direction. However, the plate is still inﬁnite, so the unit cell in direction /vectorr2is inﬁnite, where /vectorriform a basis in 2D. The unit cell is inﬁnite in size but with ﬁnite number of resonatorspresent in the supercell, see Fig. 1. Unlike electronic systems where wave functions decay exponentially in space, ﬂexuralwaves decay slowly in the plate and an inﬁnite large unit cell FIG. 1. Schematic representation of a ribbon in an inﬁnite plate. r1andr2are a basis of lattice. The two red parallel lines delimit one supercell, the supercell is inﬁnite in size. The unit cell is presented with two different topological phases as we will see later in the text.will account for long range waves along the /vectorr2direction. The discrete summation over n2/vectorg2in Eq. ( 3) transforms into an integral 1 Ac/summationdisplay G2→1 2πa/integraldisplay∞ −∞dg2. (15) Applying this transformation to Eq. ( 9),Aα,βmatrix simpliﬁes to depend only on k1. The governing equations are described in Ref. [ 38]. Our main interest creating ribbons consist of studying boundary states between two phases. The interface iscontained in the supercell. Bands are computed from the zeros of the A(/vectork) matrix determinant and its null space contains the eigenmodes, i.e., the w(/vectorR α) weight over the supercell resonators. C. Multiple scattering method For ﬁnite clusters in an inﬁnite plate, we use multiple scattering theory (MST). The governing equations are Eqs. ( 1) and ( 2) where the number of /vectorRαis ﬁnite. The Green’s function of the plate equation without resonators, G0(/vectorr), is used as a basis to expand the solution of the resulting wave. A system ofself-consistent equations lead to the solution of the ﬁeld w(/vectorr) under some harmonic incident ﬁeld ψ 0(/vectorr,t)=ψ0(/vectorr)eiωt+ϕ, w(/vectorr)=ψ0(/vectorr)+/summationdisplay αTαψe(/vectorRα)G0(/vectorr−/vectorRα), (16) where ψeis the incident ﬁeld at scatterer α, which allows to deduce the value of Tα=tα 1−itα/(8k2).ψe(/vectorRα) can be solved from the system of equations, ψe(/vectorRα)=ψ0(/vectorRα)+/summationdisplay β(1−δα,β)TβG0(/vectorRα−/vectorRβ)ψe(/vectorRβ). (17) We compute the resulting ﬁeld w(/vectorr) by substituting the so- lution of ψeback into Eq. ( 16). The incident ﬁeld is the external excitation of the system and is taken as a point sourceψ 0(/vectorRα)=G0(/vectorRα−/vectorx0), we also consider multipoint dephased excitations ψ0(/vectorRα)=/summationtext jG0(/vectorRα−/vectorxj)eiϕjand solutions with- out input ﬁeld ψ0(/vectorRα)=0 that we call natural excitations of the system. III. KAGOME LATTICE, DISTORTIONS, AND SYMMETRIES The standard kagome lattice consists of three sets of straight parallel lines intersecting at lattice sites as shown inFig. 2. This ﬁgure also shows the unit cell chosen in this paper as a parallelogram with lattice vectors /vectora 1=a(1,0)/vectora2=a/parenleftbigg cos/parenleftbiggπ 3/parenrightbigg ,sin/parenleftbiggπ 3/parenrightbigg/parenrightbigg . (18) The normalized masses and resonator frequencies are γα=10 and/Omega1α=4π, respectively, for the three resonators of the unit cell. The lattice sites in the unit cell form an equilateraltriangle of side a/2. In this paper, we consider distortions of the standard kagome lattice with two parameters: fthat 134102-3NATALIA LERA et al. PHYSICAL REVIEW B 99, 134102 (2019) FIG. 2. (a) Undistorted kagome lattice. The unit cell is indicated in a green box of side a. The unit cell contains three resonators forming an equilateral triangles. (b) Parameters used in the paper for deformations of kagome lattice and its effect in the unit cell. They are characterized by an angle αand a uniform expansion factor f. (c) Brillouin zone. controls the size of the triangle with respect to the lattice parameter which will remain unchanged, and αthe rotation angle of the equilateral triangle with respect to its center, seeFig. 2. The positions of the three sites in the unit cell are /vectorR n=f·b(cos(/Theta1n+α),sin(/Theta1n+α)), (19) where b=a 2√ 3,/Theta1n=nπ 3−7π 6, and nlabels the lattice sites n={1,2,3}. The undistorted kagome lattice is deﬁned for f=1 andα=0. Kagome lattice in our parameter space have several sym- metries. For a constant f, there are three equivalent lattices for every αcorresponding to {α,α+2π 3,α−2π 3}, meaning all systems in this parameter space have C3symmetry. For a given angle and f<1, the lattice with f/prime=2−fis equivalent as well. However, lattices with f<1 and 2 >f>1 are distin- guished by triangles pointing in opposite directions as shownin Figs. 3(a) and 3(b). Playing with parameters it is possible to create subtle differences in lattice structure, as shown inFigs. 3(a) and3(c). The arrangement of resonators is the same but the unit cell where each resonator belongs are differentin each case. Such conﬁgurations are therefore physicallyindistinguishable. The undistorted kagome lattice f=1 and α=0h a v e C 6symmetry, inversion symmetry both with cen- ters in the middle of hexagons, C3symmetry with center in the middle of triangles and three mirror symmetries given by the FIG. 3. Real-space arrangement of resonators for several de- formation parameters. The unit cell is highlighted. Notice (a) and(c) look similar but the chosen cell is different. Notice the breaking of spatial inversion symmetry is the three cases.following normal to the mirror line vectors: /vectorm1=(1,0), /vectorm2=/parenleftbigg−1 2,√ 3 2/parenrightbigg , /vectorm3=/parenleftbigg1 2,√ 3 2/parenrightbigg , (20) see Fig. 2. The elastic systems have time-reversal symmetry as well. The interrelation of all these symmetries give manyinteresting features and we will explore some of them. Due to the symmetries of the lattice, some qualitative band features are independent of the system (springs or plates). Forinstance, the gap closings at Kpoint of the Brillouin zone will be relevant through the article and they are representedin Fig. 4in parameter space. Each red and dashed line cor- responds to a gap closing in conelike shape. At momentumK, there are Dirac points, and opening the gap gives rise to FIG. 4. Gap closings in parameter space at Kpoints. Solid red lines are the closing of the ﬁrst gap. Dashed black lines are theclosings of the second gap. The second gap is a partial gap in k space. Topological transitions studied in this paper are marked with a ﬁve- and a four-pointed stars. The driving parameters are fand α, respectively, and the symmetry breaking is spatial inversion and mirror symmetry, respectively. 134102-4V ALLEY HALL PHASES IN KAGOME LATTICES PHYSICAL REVIEW B 99, 134102 (2019) interesting phenomena. Spring-mass model approach is being used in kagome lattice to explain band inversion topology andthey constitute a ﬁrst step towards topology in plates. IV . INVERSION SYMMETRY BREAKING AND TOPOLOGY A. Spring-mass model We design a spring-mass model where masses are located at sites of the kagome lattice, i.e., circles in Fig. 2and each blue line connecting neighboring masses are springs. Themasses have only one degree of freedom, they move in thedirection perpendicular to the plane. The three springs insidethe unit cell have spring constant κ 1and the springs connect- ing neighboring unit cells have constant κ2. The equations of motion read m¨u1=−κ1(u1−u2)−κ1(u1−u3) −κ2(u1−u2e−i/vectork·/vectora1)−κ2(u1−u3e−i/vectork/vectora2), m¨u2=−κ1(u2−u1)−κ1(u2−u3) −κ2(u2−u1ei/vectork/vectora1)−κ2(u2−u3e−i/vectork/vectora3), m¨u3=−κ1(u3−u2)−κ1(u3−u1) −κ2(u3−u2ei/vectork/vectora3)−κ2(u3−u1ei/vectork/vectora2). (21) where /vectora3=/vectora2−/vectora1. Solving the temporal part as a harmonic function u1(t)=u1eiωtand introducing the dimensionless variable β=κ1−κ2 κ1+κ2, which plays a role analogous to the dis- tortion fof the preceding section, and /Omega12=2mω2/(κ1+κ2), the equation of motion reads −/Omega12⎛ ⎝u1 u2 u3⎞ ⎠ =⎡ ⎢⎣⎛ ⎜⎝−41 +e−i/vectork/vectora11+e−i/vectork/vectora2 1+ei/vectork/vectora1 −41 +e−i/vectork/vectora3 1+ei/vectork/vectora2 1+ei/vectork/vectora3 −4⎞ ⎟⎠ +β⎛ ⎜⎝01 −e−i/vectork/vectora11−e−i/vectork/vectora2 1−ei/vectork/vectora1 01 −e−i/vectork/vectora3 1−ei/vectork/vectora2 1−ei/vectork/vectora3 0⎞ ⎟⎠⎤ ⎥⎦⎛ ⎝u1 u2 u3⎞ ⎠. (22) Forβ=0, we recover the dispersion relation of the undistorted kagome lattice (analogous to f=1) with Dirac cones at KandK/primepoints of the Brillouin zone. Two bands cross linearly at Dirac frequency and the third band haslarger frequency. For β/negationslash=0, the gap opens up at Kand K /primepoints, gapping the system. Because C3is a symmetry of the lattice, its eigenvectors are eigenvectors of the sys-tem. C 3rotation center located in the middle of the trian- gle of the unit cell gives the following matrix form for C3 symmetry, ˆC3=⎛ ⎝010 001 100⎞ ⎠. (23)Thus the eigenvalues are {1,ei2π 3,e−i2π 3}and its corresponding eigenvectors, u0 C3=1√ 3⎛ ⎝1 11⎞ ⎠,u + C3=1√ 3⎛ ⎝e−iπ 3 eiπ 3 −1⎞ ⎠,u− C3=1√ 3⎛ ⎝eiπ 3 e−iπ 3 −1⎞ ⎠. (24) These eigenvectors diagonalize the dynamical matrix (which plays the same role than a Hamiltonian) for β/negationslash=0 atKandK/primepoints. For β=0, the two states degenerate at K(K/prime) point are u0 C3andu+ C3(u− C3). For β> 0, i.e., κ1>κ 2, u+ K=u+ C3and u− K=u0 C3, and reversed for β< 0:u+ K=u0 C3 andu− K=u+ C3. Due to the three mirror symmetries, each K point is related to K/prime=−K, and its eigenvectors are the mirror symmetric of uK. Notice that a mirror in real space given by /vectorm1in Eq. ( 20) transforms in momentum space as (kx,ky)→(−kx,ky) and thus Ktransforms into K/primeby mirror symmetry. The same is true for the other two mirror planes.This is straightforward for the kagome Brillouin zone, whichis a regular hexagon and the three mirror planes are the threeperpendicular to each pair of parallel sides, see Fig. 2. Notice that the superindex indicates different things de- pending on the subindex. When the subindex makes referencetoKpoint, the plus and minus signs correspond to the bands above and below the Dirac frequency. The subindex C 3refers to the symmetry and the plus minus or zero superindexcorrespond to its eigenvalues. We see there is a crossing of eigenvectors at β=0( t h eg a p must close at the transition), see Fig. 5(a). To capture the basic topology of this system, let us derive an effective model neareach valley. The effective model at Dirac frequency can bewritten in the basis of the two crossing eigenvectors. The thirdeigenvalue at Kis away in frequency from Dirac frequency and does not have dependence at ﬁrst order with β, see Fig. 5. The effective model is then D K,ij=/angbracketleftbig ui K/vextendsingle/vextendsingleH/vextendsingle/vextendsingleuj K/angbracketrightbig , (25) where i,j={ +,−}, we expand the dynamical matrix near each valley KandK/primeand the result is Dη=/parenleftbigg 1.5(1−β) vD(−ηkx+iky) vD(−ηkx−iky)1 .5(1+β)/parenrightbigg , (26) FIG. 5. (a) Frequency level crossing at Kpoint for model in Eq. ( 22) as a function of β. (b) Representation of Moperator in Eq. ( 33). The arrows indicate how each component transforms, the color indicates different signs. 134102-5NATALIA LERA et al. PHYSICAL REVIEW B 99, 134102 (2019) FIG. 6. Band structure of ﬂexural waves plotted along a the hexagonal Brillouin zone for several fvalues and α=0. where vD=√ 3 4aandη=± 1f o r ±Kand/vectorkis measured from each valley /vectork=(±4π 3a+kx,ky). This is a well known model for graphene with a staggered potential [ 19,39,40]a l s o studied in topological mechancis [ 20–23]. The whole system has time-reversal symmetry, one valley is transformed intothe other with a time-reversal transformation, remember thattime-reversal symmetry transforms /vectork→−/vectorkand thus, K→ −K=K /primeorη→−η. However, each valley is independent from the other, since there are no direct scattering termscoupling them (see Appendix B). Separately each valley dy- namical matrix effectively behaves as a Chern insulator withbroken time-reversal symmetry [ 19] where βis a symmetry breaking term responsible for the topological gap, and analo-gous to the magnetic ﬁeld in quantum Hall phases. OppositeChern numbers are computed near each valley when β/negationslash= 0. Since the valleys are disconnected, a well-deﬁned valleyChern number arises. The total system is still time-reversalsymmetric and therefore total Chern number is zero. Notice that the eigenvalues of the dynamical matrix in Eqs. ( 25) and ( 26) are the square of the actual normalized frequencies /Omega1as in the Hamiltoninan in Eq. ( 22). In any case, the eigenvectors (or normal modes) and conclusions abouttopology hold. We can compute the subspace generated by the two Dirac crossing vectors reads M=/vextendsingle/vextendsingleu 0 C3/angbracketrightbig/angbracketleftbig u0 C3/vextendsingle/vextendsingle−/vextendsingle/vextendsingleu+ C3/angbracketrightbig/angbracketleftbig u+ C3/vextendsingle/vextendsingle=1 2⎛ ⎝011 101110⎞ ⎠ −i 2√ 3⎛ ⎝01 −1 −101 1−10⎞ ⎠. (27) This matrix corresponds to the gap-opening operator in the low energy model and is proportional to the linear term inthe perturbation evaluated at Kpoint and its imaginary part is schematically represented in Fig. 5(b). It gives the spatial inversion symmetry breaking term in the full spring-massmodel. As we will see, this result is relevant for plates with attached resonators. If we model the strength of springs bythe distance between resonators, we obtain an analogous gapclosing with the same inversion of eigenvectors. This is areasonable assumption since the closer two resonators are inspace, the stronger their motion is mutually affected. In ourcase,β> 0 means that κ 1is stronger and in a plate system is analogous to a contraction of the sites’ distance in the unitcell, i.e, f<1. In the same way, β< 0 is analogous to f>1. B. Plate model and valley Chern number To reproduce previous results from spring-mass systems, we study plates with a kagome arrangement of resonatorsand model spring strength with distance between resonators.Fixing α=0 and varying faround 1 allow us to model the variation of spring constants within the unit cell ( κ 1) with respect to the springs connecting different cells ( κ2). The corresponding band structures are shown in Fig. 6.A t both sides of the transition, the band structure is the same[see their similar spatial distribution in Figs. 3(a) and 3(c)]. However, the topological invariant (the valley Chern number)is different as we will see. At the transition point f=1, the two bands form Dirac cones at ﬁrst order in momentumaround KandK /prime, i.e., the frequency bands are linear in kxand kywhen expanded at KandK/primepoints /vectork=(±4π 3a+kx,ky). The Dirac frequency is /Omega1Da=2.5, the frequency where the two bands touch. For f/negationslash=1, spatial inversion symmetry is broken, while the remaining symmetries are still present (see Fig. 3). The broken inversion symmetry allows us to deﬁne a valley Chernnumber, as previously stated in the spring-mass model. InFig. 7, the computed Berry curvature of ﬁrst band is plotted. The Berry curvature in 2D kspace is B=−i/angbracketleft∂ xuk\|∂yuk/angbracketright+i/angbracketleft∂yuk\|∂xuk/angbracketright, (28) FIG. 7. In color scale, the Berry curvature of the lower band over the ﬁrst Brillouin zone. The Berry curvature is localized at KandK/prime points with different signs for different phases. Blue is negative and yellow is positive. 134102-6V ALLEY HALL PHASES IN KAGOME LATTICES PHYSICAL REVIEW B 99, 134102 (2019) FIG. 8. Mode shapes. Real part of w(/vectorr) for different bands and phases. Notice the analogy of the ﬁrst row with the eigenvalue of thespring model u 0 C3=1√ 3(1,1,1)tor in the second row with the real part Re {u+ C3}=1√ 3(0.5,0.5,−1)t. Notice the band inversion. Mode shapes are not periodic due to the phase e−i/vectorK·/vectorrin Eq. ( 3). where ukis the eigenvector of one band at momentum k. The eigenvector is computed from the PWE method as thenull space of Amatrix in Eq. ( 9). We can see that the Berry curvature is localized near KandK /primewith opposite sign and it changes at the transition.For further analogy with the spring system, we compute the mode shapes in real space at the Kpoint for the two lower bands in Fig. 8, which closely resemble the eigenvectors involved in the transition u± K. Moreover, band inversion is clearly visible. The mode shapes switch energies at both sidesof the transition in the same way than eigenvectors in thespring-mass model [Fig. 5(a)]. C. Edge states in ribbons In this section, we study the interface states appearing between two lattices with distinct valley Chern numbers,which are topologically protected [ 21,39,40], i.e., with zigzag interfaces. In analogy to graphenelike lattices, the edges go-ing along the directions /vectora 1,/vectora2, and/vectora2−/vectora1we call zigzag edges. The edge mixing valleys are then called an armchairinterface in kagome lattice. We will compute vertical edgesalong direction /vectora 2−1 2/vectora1in our deﬁnition of the unit cell. We create ribbons in a supercell along /vectora2direction and periodic in/vectora1direction. Even ribbons with valley topological phases in electronic system do not have gapless edges states, becausevalleys are not well deﬁned in vacuum, unless the boundaryis with another topological phase with opposite valley Chernnumber [ 39]. The same reasoning is true for plates. Therefore boundary states appear at the interface between two phaseswith opposite signed topological invariants. Such interfaceis contained in the supercell of the ribbons as shown inFig. 1. Two types of interfaces can be made, which are depicted in Figs. 9and 10. Schematic real-space supercell is highlighted, a solid black line separates two topologicalphases distinguished by opposite valley Chern numbers. Thebands are limited by the free-wave dispersion relation, outside FIG. 9. Ribbon of resonators over an inﬁnite plate. The system contains a domain wall between two phases with opposite valley Chern numbers. At the top left, real-space ribbon representation. The horizontal line separates the two phases and black arrows indicate that theribbon is inﬁnite in horizontal direction. In red, resonators in one supercell. At the top right there is the band structure of the ﬁnite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve /Omega1a=( kx π)2. Two mid gap bands appear. At the bottom, mode shapes or, in other words, real-space displacement ﬁeld along the supercell sites w(/vectorRα) for different frequencies and momenta as indicated with colored dots on the band structure. 134102-7NATALIA LERA et al. PHYSICAL REVIEW B 99, 134102 (2019) FIG. 10. Ribbon of resonators over an inﬁnite plate. The system contains a domain wall between two phases with opposite valley Chern numbers (see Fig. 7). At the top left, real-space ribbon representation. The horizontal line separates the two phases and black arrows indicate that the ribbon is inﬁnite in horizontal direction. In red, resonators in one supercell. At the top right there is the band structure of the ﬁnite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve /Omega1a=(kx π)2. One mid gap band appears. At the bottom, mode shapes or, in other words, real-space displacement ﬁeld along the supercell sites w(/vectorRα) for different frequencies and momenta as indicated with colored dots on the band structure. that region there are no bulk solutions of the plate equation. The two types of interfaces exhibit a band of boundary stateslocalized at the domain wall. In Fig. 9, a second band appears containing edge states at the top of the ribbon which arenontopological. An analogous band is present in Fig. 10with edge states at the bottom of the ribbon as can be seen in themode shapes. The topological edge modes are robust against certain types of perturbations that do not mix valleys, like zigzagedges (see Appendix B). We have conﬁrmed this fact by corroborating that these states are not removed by the additionof general perturbations to the boundary. However, thereare perturbations mixing valley degrees of freedom such anarmchair boundary [ 9] that will destroy the protection as can be seen in Fig. 11, see Appendix Bfor the derivation. Notice the change in the unit cell parameter, now in the direction ofperiodicity it is a /prime=√ 3a. D. Finite systems Now, we study a ﬁnite cluster of resonators on top of an inﬁnite plane where multiple scattering theory described inSec. IIand developed in Ref. [ 38] applies. The cluster of resonators contains two phases separated by a zigzag interfacewith Z shape, Fig. 12. Topological protected state appears at mid gap frequency. Notice that the horizontal interface isequivalent to the domain wall in Fig. 9, thus the frequency is tuned to ﬁnd topological edge modes, in this case, /Omega1a= 2.51. Figure 13shows an edge state without backscattering, this mode is being computed without external input ﬁeld,i.e.,ψ 0=0. The vector of coefﬁcients ψe(/vectorRβ)i nE q .( 17) is the right-singular vector whose single value is zero. Thismethod computes natural excitations of the system at a givenfrequency. Moreover, in the same cluster, we ﬁnd appropriate mul- tipoint excitation with dephasing in time. A two-point exci-tation ψ 0(/vectorRα)=G0(/vectorRα−/vectorx1)+G0(/vectorRα−/vectorx1)eiϕwhere point sources are located at the horizontal domain wall, /vectorx1= (−1,0)aand/vectorx2=(1,0)a. The dephasing ϕis varied until FIG. 11. Ribbon of resonators over an inﬁnite plate. The system contains a domain wall between two phases with opposite valleyChern numbers (see Fig. 7). On the left, real-space ribbon represen- tation. The vertical line separates the two phases and black arrows indicate that the ribbon is inﬁnite in vertical direction. The interfaceis armchairlike. The band structure does not show localized modes within gap frequencies, bands that appear isolated at gap frequencies are bulk modes. 134102-8V ALLEY HALL PHASES IN KAGOME LATTICES PHYSICAL REVIEW B 99, 134102 (2019) FIG. 12. Schematic representation of a cluster of resonators on top of an inﬁnite plate. The cluster is designed with a Z-shaped interface. propagating waves in one direction only are tuned. The results are shown in Fig. 14and are similar to those presented in Ref. [ 24]. V . MIRROR SYMMETRY BREAKING AND TOPOLOGY A. Spring-mass model Now we consider a model with mirror symmetry at α= π/6 and consider two continuous deformations that break FIG. 13. MST simulations of an arrangement of resonators with two phases separated by a zigzag domain wall. The frequency is tuned so that the mode is in a gap and corresponds to topologicaledge states. mirror symmetry. Changing αtowards one side or the other will give two phases differentiated by different eigenvectorsofC 3symmetry. The spring-mass model is constructed by changing the relative spring constant between green and bluesprings as indicated in Fig. 15. The equations of motion read m¨u1=−γ(u1−u2)−γ(u1−u3)−κ1(u1−u2e−i/vectork/vectora2)−κ1(u1−u3e−i/vectork/vectora3)−κ2(u1−u2e−i/vectork/vectora1)−κ2(u1−u3e−i/vectork/vectora2), m¨u2=−γ(u2−u1)−γ(u2−u3)−κ1(u2−u1ei/vectork/vectora2)−κ1(u2−u3ei/vectork/vectora1)−κ2(u2−u1ei/vectork/vectora1)−κ2(u2−u3e−i/vectork/vectora3), m¨u3=−γ(u3−u2)−γ(u3−u1)−κ1(u3−u2e−i/vectork/vectora1)−κ1(u3−u1ei/vectork/vectora3)−κ2(u3−u2ei/vectork/vectora3)−κ2(u3−u1ei/vectork/vectora2), (29) introducing the relative difference β=κ1−κ2 κ1+κ2, we rewrite the system of equations in matrix form −/Omega12⎛ ⎝u1 u2 u3⎞ ⎠=γ/prime⎛ ⎝−211 1−21 11 −2⎞ ⎠⎛ ⎝u1 u2 u3⎞ ⎠+⎛ ⎜⎝−4 e−i/vectork/vectora2+e−i/vectork/vectora1e−i/vectork/vectora3+e−i/vectork/vectora2 ei/vectork/vectora2+ei/vectork/vectora1 −4 ei/vectork/vectora1+e−i/vectork/vectora3 ei/vectork/vectora3+ei/vectork/vectora2e−i/vectork/vectora1+ei/vectork/vectora3 −4⎞ ⎟⎠⎛ ⎝u1 u2 u3⎞ ⎠ +β⎛ ⎜⎝0 e−i/vectork/vectora2−e−i/vectork/vectora1e−i/vectork/vectora3−e−i/vectork/vectora2 ei/vectork/vectora2−ei/vectork/vectora1 0 ei/vectork/vectora1−e−i/vectork/vectora3 ei/vectork/vectora3−ei/vectork/vectora2e−i/vectork/vectora1−ei/vectork/vectora3 0⎞ ⎟⎠⎛ ⎝u1 u2 u3⎞ ⎠, (30) where /Omega12=2mω2/(κ1+κ2) and γ/prime=2γ/(κ1+κ2)T h e eigenvectors at Kpoint are the same eigenvectors of C3 symmetry. Now, they eigenvectors crossing at Dirac frequency areu± K=u± C3.T h eg a pc l o s e sa t Kpoint forming a Dirac cone atα=π/6. Notice the closing occurs on ﬁrst or second gap depending on f(see Fig. 4). In any case, the Dirac cones are made of states with complex conjugate eigenvalues of C3 symmetry. Moreover, they are interchanged at the transition: u± K=u± C3forβ> 0 and u± K=u∓ C3forβ< 0 see Fig. 16; and interchanged again at the other valley K/prime. We compute the effective model for this band crossing system as in Eq. ( 25). The result is Dη=/parenleftbigg3γ/prime+1.5(1−β)ηvDei2π/3(kx+iky) vDηe−i2π/3(kx−iky)3γ/prime+1.5(1+β)/parenrightbigg ,(31)where vD=√ 3 2a. By rotating /vectork=(kx,ky) reference system byπ/3, the dynamical matrix can be written with the same structure than Eq. ( 26), Dη=/parenleftbigg3γ/prime+1.5(1−β)vD(ηk/prime x+ik/prime y) vD(ηk/prime x−ik/prime y)3 γ/prime+1.5(1+β)/parenrightbigg . (32) This result illustrates that the mirror symmetry break- ing in the original model is analogous to an inversionsymmetry in graphenelike systems where βis the pseu- domagnetic ﬁeld in quantum valley Hall effect. Instead ofinducing nonequivalent sublattice potential, here the potentialis between eigenstates of the system and C 3, i.e., the βhas opposite sign for the two different eigenstates crossing atDirac frequency in the same way that a sublattice potential 134102-9NATALIA LERA et al. PHYSICAL REVIEW B 99, 134102 (2019) FIG. 14. In color scale, the absolute value of the place displacement. MST simulations of an arrangement of resonators with two phases separated by a zigzag domain wall (no mixing valleys) in Z shape. The frequency is tuned so the modes are topological edge states. The reddots correspond to the two excitation points /vectorx 1=(−1,0)aand/vectorx2=(1,0)a. The temporal dephasing is ϕ=0 (the two points are excited simultaneously) and ϕ=π(antiphase excitation), respectively. distinguishes A and B lattices by a diagonal σzterm in graphenelike Hamiltonians, where σzis a Pauli matrix. The subspace generated by the two Dirac eigenstates cross- ing at Kand deﬁned as follows: M=/vextendsingle/vextendsingleu+ C3/angbracketrightbig/angbracketleftbig u+ C3/vextendsingle/vextendsingle−/vextendsingle/vextendsingleu− C3/angbracketrightbig/angbracketleftbig u− C3/vextendsingle/vextendsingle=i√ 3⎛ ⎝01 −1 −101 1−10⎞ ⎠(33) is equal to M=i(ˆC3−ˆCt 3), where ˆC3is the symmetry rotation operator. The Moperator differentiates between states rotating in different directions (notice the opposite sign of ˆC3). This matrix is proportional to the linear term in βatKpoint and it is schematically represented in Fig. 5(b). This gives us the mirror symmetry breaking effect in real-space lattice vectors. B. Plate model and valley Chern number In this section, we plot several band structures around α= π/6. Notice that in Fig. 4the gap closes for all fatα=π/6a t Kpoint. For f<2√ 3, the second and third bands are degener- ate at Kpoint. For f>2√ 3, the ﬁrst and second bands form the Dirac cone. The two transitions have equivalent topology. In the spring-mass model, this corresponds to varying the valueofγthat tunes the frequency of u 0 C3but does not affect theother two crossing states. However, the gap opening at Kwhen α/negationslash=π/6 is not complete for small f.F o rl a r g e f,Kpoint is not the minimum of the second band, although topologicalstates come from what happens at Kpoint, the gap is complete and we show the results of for f=1.5. The band structures of plates with different arrangements of resonators are plottedin Fig. 17. At equidistant points in parameter space from the transition points the band structures are the same, howevertheir topology is not. At the transition point, a Dirac coneatKpoint is formed which induces the breaking of mirror symmetry. The Dirac frequency is /Omega1a=2.7. Forα/negationslash=π/6, mirror symmetry is broken and as we show in the effective model we can deﬁne a Berry curvature as inEq. ( 28). The result is shown in Fig. 18. The eigenvectors in the Brillouin zone are computed from PWE method as the nullspace of Amatrix in Eq. ( 9) at the appropriate frequency as de- scribed in Ref. [ 38]. We can see that the Berry curvature is lo- calized near KandK /primewith opposite sign and it changes at the transition, consistently with the effective spring-mass model. C. Edge states in ribbons We compute the edge states of a ribbon with an interface and ﬁnd two crossing bands in the middle of the gap. The FIG. 15. Distorted kagome lattice for two fvalues and α=π/6. Notice that spatial inversion is not a symmetry of the system. Increasing slightly αshortens green links and enlarges blue links. The spring system models changes in distance with appropriate changes in β. 134102-10V ALLEY HALL PHASES IN KAGOME LATTICES PHYSICAL REVIEW B 99, 134102 (2019) FIG. 16. Frequency levels at Kpoint for the model in Eq. ( 30)a s a function of βand for γ/prime=1. crossing indicates that the two bands have different symmetry. In Fig. 19, edge states appear in the boundary of the two phases, due to the different valley Chern numbers. In this tran-sition, there are two crossing bands with different symmetriesthat are topologically protected. The different symmetries canbe observed in the modes in Fig. 19. They are symmetric or antisymmetric respect to the domain wall. Notice site twomaps onto itself under inversion at the domain wall andsite three and one maps onto one another. This symmetryin the eigenvectors reﬂect the inversion symmetry presentin real space in the ribbon due to the fact that phases areequidistant in real space from the transition point in parameterspace. In other words, the two phases are characterized byα=π/6±φ, where α=π/6 is the transition point and φ=0.1. This ribbon symmetry is also present in ribbons with two phases breaking inversion symmetry in honeycomblattice like in Ref. [ 21]. Since the two phases are equidistant from the transition point, there is an inversion that givessymmetric and antisymmetric edge modes with respect to thedomain wall (see Appendix A). Unlike honeycomb lattice in kagome arrangement each number site has its inversionpoint. In graphene, the spatial inversion is clearly seen inthe eigenvectors u A=(1,0)tanduB=(0,1)tthan transform into one another by appropriate inversion in real space. Inour case, Eq. ( 32), the eigenvectors at a given frequency and at each side of the domain wall are related by spatial FIG. 18. Berry curvature of the lower band over the ﬁrst Bril- louin zone. Berry curvature is localized at Kand K/primepoints with different signs for different phases. Blue is negative and yellow is positive. inversion too uβ>0 K=1√ 3⎛ ⎝e−i2π 3 1 ei2π 3⎞ ⎠=eiπ 3u+ C3uβ<0 K=1√ 3⎛ ⎝ei2π 3 1 e−i2π 3⎞ ⎠ =eiπ 3u− C3. (34) Site 2 maps into itself, while sites 1 and 3 interchange and appropriate combinations. The result shows symmetric andantisymmetric modes, as observed in the ribbon eigenvectors(Fig. 19). Notice inversion symmetry is not present in domain walls in ribbons with phases of kagome lattice with brokeninversion symmetry shown in Figs. 9and 10. Modes are not symmetric or antisymmetric and neither do the eigenvectorsatKinvolved in the transition ( u 0 C3andu+ C3) exhibit inversion symmetry, as expected. Valley topology is not protected against perturbations mix- ing the valleys. For instance, a vertical interface (armchairtype) mixes the valleys, and the edge states disappear, asshown in Fig. 20. The band displayed at frequencies that correspond to the bulk gap in the periodic system are notlocalized at the edge, as shown in Fig. 20. These are not topological states. D. Finite systems We design a ﬁnite structure of resonators over an inﬁnite plate and compute the real part of w(/vectorr). A similar result occurs for natural modes of the system, as in Fig. 13. We also ﬁnd two-point time-dephased excitation at mid gap frequency, so FIG. 17. Band structure of deformed kagome lattice for several αvalues and f=1.5. 134102-11NATALIA LERA et al. PHYSICAL REVIEW B 99, 134102 (2019) FIG. 19. Ribbon of resonators over an inﬁnite plate. The system contains a domain wall between two phases with opposite valleyChern numbers (see Fig. 7). At the top left, real-space ribbon representation. The horizontal line separates the two phases and the UC is highlighted in red. Black arrows indicate that the ribbon isinﬁnite in horizontal direction. In red, resonators in one supercell. At the top right there is the band structure of the ﬁnite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve /Omega1a=( kx π)2. Two crossing bands appear in the gap, they have different symmetry under domain wall spatial inversion as seen at the bottom. At the bottom, mode shapes, i.e., real-space displacementﬁeld along the supercell sites w(/vectorR α) for different frequencies and momenta as indicated with colored dots on the band structure.one-sided propagation is achieved, see Fig. 21, the red dots are the points where the external excitation force is applied, /vectorx1=(−1,0) and /vectorx2=(1,0). Different dephasing ϕexcites different directional waves. VI. CONCLUSION We have studied two types of topological transitions in me- chanical metamaterials based on the distorted kagome lattice,namely inversion symmetry or mirror symmetry breaking.In spring-mass systems, we derived a dynamical matrix foreach valley that effectively behaves as a Chern insulator. Wehave identiﬁed, in the microscopic model, the operator actingas a pseudomagnetic ﬁeld which is controlled by relativevalues of springs’ strengths. We also exploit this ﬁnding forﬂexural waves in plates coupled to resonators. In this context,the “magnetic ﬁeld” is controlled by the distance betweenresonators. The main manifestation of the valley Hall effectin our system is the presence of protected boundary stateslocated at interfaces between domains with opposite signedvalley Chern numbers. These interfaces must have appropriateedges as shown in simulations of ribbons and ﬁnite clus-ters of resonators with zigzag domains. We also illustratedhow mixing valleys with armachair-type interfaces producesback-scattering and destroys the topological modes. However,we also claim that a lattice lacking inversion symmetry atthe transition despite intact mirror symmetry exhibits thesame type of valley topology of broken mirror symmetry.We compute a similar effective model for springs and ﬁndprotected edge states with different symmetry. We ﬁnd simpletwo-point excitation generating one-sided ﬂexural waves inﬁnite systems that can propagate through desired bends in2D space. It is well known that the dynamics of spring-masssystems is dissimilar in several ways to the one of interactingresonators coupled to plates. For instance, interaction betweenthe resonators is long-ranged and the dynamical matrix isfrequency dependent on the latter. However, throughout thiswork we have established a common origin to their topologi-cal properties. We hope all these ﬁndings help enlightening the FIG. 20. Ribbon of resonators over an inﬁnite plate. The system contains a domain wall between two phases with opposite valley Chern numbers (see Fig. 7). On the left, real-space ribbon representation. The vertical line separates the two phases whose interface is armchair type and the UC is highlighted in red. Black arrows indicate that the ribbon is inﬁnite in the vertical direction. The band structure does not show localized modes within gap frequencies, bands that appear isolated at gap frequencies are bulk modes. 134102-12V ALLEY HALL PHASES IN KAGOME LATTICES PHYSICAL REVIEW B 99, 134102 (2019) FIG. 21. MST simulations of an arrangement of resonators with two phases separated by a zigzag domain wall in Z shape. The frequency is tuned so the mode is in a gap and correspond to topological edge states. The red dots correspond to the two excitation points. The dephasingisϕ=πon the left and ϕ=− 0.36πon the right. path towards future applications in wave guiding and related ﬁelds. ACKNOWLEDGMENTS N.L and J.V .A. acknowledges ﬁnancial support from MINECO grant FIS2015-64886-C5-5-P. N.L. acknowledgesﬁnancial support from the Spanish Ministry of Economy andCompetitiveness, through The María de Maeztu Programmefor Units of Excellence in R&D (MDM-2014-0377), andalso hospitality from the Universitat Jaume I in Castellonwhere part of this work was done. D.T. acknowledges ﬁ-nancial support through the “Ramón y Cajal” fellowshipunder grant number RYC-2016-21188. P.S.-J. acknowledgesﬁnancial support from the Spanish Ministry of Economyand Competitiveness through Grant No. FIS2015-65706-P(MINECO /FEDER). J.C. acknowledges the support from the European Research Council (ERC) through the Starting GrantNo. 714577 PHONOMETA and from the MINECO through aRamón y Cajal grant (Grant No. RYC-2015-17156). APPENDIX A: HONEYCOMB RIBBONS WITH BROKEN INVERSION SYMMETRY As computed in Ref. [ 21], the analogous to quantum val- ley Hall effect guarantees boundary modes localized at theinterface between two phases. Inversion symmetry is brokenby different masses of resonators in the two dimensional unitcell and two types of interface can be created (with zigzagboundary). In this Appendix, we examine the symmetry ofthe boundary modes. As explained in the main text, the ribbonstructure has inversion symmetry at the domain wall providedthe two phases are equally large and masses are the same, seeFig. 22, full circles correspond to γ=11 and empty circles toγ=9 (the same at each side of the domain wall), all resonators have the same spring constant and their frequencyis/Omega1 R=4π. Dirac frequency for γ=10 is/Omega1D=2.9. In Sec. V, ribbons such as the one shown in Fig. 19contain different inversion symmetries at the domain wall, site 2 mapsinto itself from an inversion center different from where site3 maps into site 1, and at the same time different from theinversion center where site 1 maps into site 3.The symmetry of the boundary eigenvectors in presented in Figs. 23and24for light and heavy boundaries, respectively. In the soft boundary, Fig. 23, the mid gap band correspond to antisymmetric modes. Symmetric boundary modes are lostin the bulk band structure. However, we can compute andplot the extended and symmetric boundary mode. In the hardboundary, Fig. 24, the mid gap band merges with bulk bands near k x=π. At each side, the symmetry is different, for kx<π modes are antisymmetric under inversion symmetry and for kx>π modes are symmetric. APPENDIX B: INTERVALLEY SCATTERING AND EDGES In this Appendix, we explain weather a given boundary preserves the valley degree of freedom or not. The quantumvalley Hall effect relies on conservation of valley index for theappearance of topological edge states. Kagome lattices have ahexagonal Brillouin zone with inequivalent KandK /primepoints. These two momenta are not related by reciprocal latticevectors, /vectorK= 4π 3aand/vectorK/prime=−/vectorK, thus /vectorK−/vectorK=2 3(/vectorG1+/vectorG2) andKandK/primepoints are not connected by a linear combination of reciprocal vectors with integer coefﬁcients. Where /vectorG1= 2π a(1,1√ 3) and /vectorG2=π√ 3a(0,1) are the reciprocal vectors and athe lattice parameter of the kagome lattices. However, a perturbation, like a boundary, might mix the two valleys. In general, the overlapped ﬁeld of the two valleys reads /angbracketleftψK/prime\|ψK/angbracketright=ψ∗ K/primeψK=u∗ K/primeuKei(K−K/prime), (B1) FIG. 22. Schematic of hexagonal arrangement of resonators hav- ing two different masses (ﬁlled or empty circles represent heavy and light masses) with soft (light-light) and hard (heavy-heavy)interfaces. The solid black line is the domain wall. In green star marker, the inversion symmetry centers of the ribbon. The arrows represent the inﬁnity of the ribbon in horizontal direction. 134102-13NATALIA LERA et al. PHYSICAL REVIEW B 99, 134102 (2019) FIG. 23. Ribbon of resonators over an inﬁnite plate. The unit cell highlighted in red contains a domain wall between two phases with different valley Chern numbers. (Top) Band structure of the ﬁnite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve /Omega1a=(kx π)2. (Bottom) Real-space displacement ﬁeld along the supercell sites for different frequencies and momenta (eigenvectors) as indicated with colored dots on the band structure. The two lines represent sites A and B. where the function u/vectorkis a periodic function in the unit cell, and also its product, u∗ K/primeuK=/summationdisplay n1,n2aK,K/prime,n1,n2ei(n1/vectorG1+n2/vectorG2)·/vectorr. (B2) Thus /angbracketleftψK/prime\|ψK/angbracketright=e2 3(/vectorG1+/vectorG2)/summationdisplay n1,n2hK,K/prime,n1,n2ei(n1/vectorG1+n2/vectorG2)·/vectorr =h/prime K,K/prime,n/prime 1,n/prime 2ei(n/prime 1/vectorG/prime 1+n/prime 2/vectorG/prime 2)·/vectorr, (B3) where /vectorG/prime j=1 3/vectorGjandj={1,2}. The overlapped ﬁeld has the original hexagonal symmetry ( /vectorG/prime jhave the same ratio with the reciprocal lattice vectors) but the period is three times thelattice parameter a /prime=3a. Now, we evaluate the overlapped ﬁeld integral along a period a/prime. Imagine a domain wall along an arbitrary direction φ, we deﬁne a orthonormal basis of the two-dimensional plane: ˆ e=(cos(φ),sin(φ)) and ˆ e⊥=(−sin(φ),cos(φ)). The integral along the perpendicular direction should be ﬁnitesince the two phases are gapped and the overlapped ﬁeldmust in few lattice parameters inside each phase. Now, weare left with the integral in the direction of the domainwall, /integraldisplay 3a 0drˆeh/prime K,K/prime,n/prime 1,n/prime 2ei(n/prime 1/vectorG/prime 1+n/prime 2/vectorG/prime 2)·/vectorr. (B4) Making use of the Fourier series expansion in the direction of ˆe, i.e., in one dimension, eikr=eim2π ar, (B5)where kandrare in the direction of ˆ eandmis an integer. Thus the overlap integral reads /summationdisplay mAmh/prime K,K/prime,n/prime 1,n/prime 2/integraldisplay3a 0drei(n/prime 1/vectorG/prime 1+n/prime 2/vectorG/prime 2)/vectorreim2π ar. (B6) The integral is /integraldisplay3a 0drei2π 3a(n/prime 1cos(φ)+n/prime 1+n/prime 2√ 3sin(φ)+3m)r=i3a 2π1−ei2πI I,(B7) where Iis I=n/prime 1cos(φ)+n/prime 1+n/prime 2√ 3sin(φ)+3m. (B8) The overlap vanishes when Iis an integer, i.e., sin( φ)m u s t cancel the factor1√ 3. The solution is thus sin( φ)=√ 3 2or sin(φ)=0. The domain walls that preserve the valley degree of freedom are those at direction ˆ e=(cos(φ),sin(φ)) such thatφ=msπ 3, where msis an integer. Domain walls in the directions with suppressed over- lap of intervalley modes are called zigzag. Any otherdirection mixes valleys, the overlap is nonzero and wecalled them armchair. The reason for this names arethe appearance of honeycomb lattices, for which thisderivation is valid. We conserve the names for kagomelattices. 134102-14V ALLEY HALL PHASES IN KAGOME LATTICES PHYSICAL REVIEW B 99, 134102 (2019) FIG. 24. Ribbon of resonators over an inﬁnite plate. The unit cell highlighted in red contains a domain wall between two phases with different valley Chern numbers. (Top) Band structure of the ﬁnite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve /Omega1a=(kx π)2. 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PhysRevB.87.014431.pdf	PHYSICAL REVIEW B 87, 014431 (2013) Change in interface magnetism of an exchange-coupled system due to the presence of nonmagnetic spacers Amitesh Paul,1,N. Paul,2Jaru Jutimoosik,3Rattikorn Yimnirun,3Saroj Rujirawat,3Britta H ¨opfner,2Iver Lauermann,2 M. Lux-Steiner,2Stefan Mattauch,4and Peter B ¨oni1 1Technische Universit ¨at M ¨unchen, Physik Department E21, Lehrstuhl f ¨ur Neutronenstreuung, James-Franck-Strasse 1, D-85748 Garching b. M ¨unchen, Germany 2Helmholtz-Zentrum Berlin f ¨ur Materialien und Energie GmbH, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany 3School of Physics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima, and Thailand Center of Excellence in Physics (ThEP Center), Commission on Higher Education, Bangkok, Thailand 4J¨ulich Centre for Neutron Science Forschungszentrum J ¨ulich GmbH, Außenstelle am FRM-II c/o TU M ¨unchen, Lichtenbergstraße 1, D-85747 Garching b. M ¨unchen, Germany (Received 27 April 2012; revised manuscript received 3 September 2012; published 28 January 2013) We report on the effect of nonmagnetic spacer layers on the interface magnetism and the exchange bias in the archetypical [Co /CoO] 16system. The separation of the magnetic bilayers by Au layers with various thicknesses dAu/greaterorequalslant25 nm leads to a threefold increase of the exchange bias ﬁeld ( Heb). Reﬂectometry with polarized neutrons does not reveal any appreciable change in the domain population. This result is in agreement with the observationthat the granular microstructure within the [Co /CoO] bilayers is independent of d Au. The signiﬁcant reduction of the magnetic moments in the Co layers can be attributed to interfacial disorder at the Co-Au interfaces.Element-speciﬁc x-ray absorption spectroscopy attributes part of the enhancement of H ebto the formation of Co3O4in the [Co /CoO] bilayers within the multilayers. A considerable proportion of the increase of Hebcan be attributed to the loss of magnetization at each of the Co-Au interfaces with increasing dAu. We propose that the interfacial magnetism of ferro- and antiferromagnetic layers can be signiﬁcantly altered by means ofmetallic spacer layers thus affecting the exchange bias signiﬁcantly. This study shows that the magnetism inmagnetic multilayers can be engineered by nonmagnetic spacer layers without involving the microstructure ofthe individual layers. DOI: 10.1103/PhysRevB.87.014431 PACS number(s): 75 .70.Cn, 75 .60.Jk I. INTRODUCTION As new magnetic hard-disk-drive products are designed for higher storage densities in magnetic recording materi-als, the “superparamagnetic effect” has become increasinglyimportant. 1As the grains become smaller (50–100 nm), due to thermally activated ﬂuctuations, the magnetization of the grains may become unstable. One approach to delaysuperparamagnetism is to increase the magnetic anisotropy orthe unidirectional anisotropy. The exchange bias phenomenon can be described as a form of a unidirectional magnetic anisotropy that arises dueto the interfacial exchange coupling between a ferromagnet(FM) and an antiferromagnet (AF) and can effectively delaythe superparamagnetic limit. 2In most usual cases, the AF ordering temperature is lower than that of the FM, belowwhich one observes a horizontal shift of the hysteresisloop. However, temperature-dependent competition betweeninterfacial exchange and AF anisotropy energies can resultin bias ﬁelds even for materials with higher AF orderingtemperature. 3Conventionally, a cooling ﬁeld ( HFC) provides the unidirectional anisotropy while the shift is observedopposite to the applied ﬁeld ( H a) direction. Over the last decade many salient features of the exchange bias effect havebeen clariﬁed. It turns out that only a very small percentageof moments at the AF interface are pinned while the restof the moments rotate rigidly with the FM. It also turnsout that it is energetically favorable to form domains in theantiferromagnet. They account for the lowering of the energycost associated with the reversal of the FM that determines the strength of the bias ﬁeld ( H eb).4–6Exchange bias is also associated with many salient features such as coercivityenhancement, 7,8asymmetric hysteresis loops,9,10and training effects.11 One of the interesting problems in multilayer physics is the inﬂuence of the interface between the magnetic ﬁlm and the nonmagnetic spacer on kinetic, magnetic, and magnetooptical properties of thin-ﬁlm systems. Informationconcerning effects of (a) an underlayer grain morphologyand a grain crystallographic orientation (texture of thegrains) on magnetic properties, 12(b) induced magnetic mo- ments via s-dhybridization,13(c) interface alloying,14and (d) canted magnetic structure are intrinsic to interfacesbetween magnetic-nonmagnetic magnetic layers. 15These are highly relevant to systems that are used as magnetic ﬁeldsensors, read heads, or memory devices. It may be noted that exchange bias systems are often coated with a Au ﬁlm, in order to protect them against furtheroxidation. 16Moreover, Au is often used as metallic leads for spin-valve structures. Thus the Au /FM (or AF) interfaces and their effect on exchange bias cannot be ignored. In general, the introduction of a nonmagnetic (NM) metallic spacer such as Cu, Ag, or Au between the FM and FM /AF layers modiﬁes the interface coupling between them. Therefore it is of greatinterest to obtain information about the spin directions in thevicinity of the interfaces. This aspect, however, remains largelyunexplored. In fact, there are no studies on the impact of the 014431-1 1098-0121/2013/87(1)/014431(18) ©2013 American Physical SocietyAMITESH PAUL et al. PHYSICAL REVIEW B 87, 014431 (2013) AF/Au or FM /Au interface magnetism including the effects of roughness and interdiffusion on the exchange bias phenomena.The aim of this study is, therefore, to systematically investigatethe magnetisation of exchange coupled bilayers of Co /CoO that are separated by nonmagnetic Au spacer layers. Contrary to the expectations, we show here that the ex- change bias ﬁeld increases gradually with increasing thicknessof the Au spacer layer. As expected, the magnetization reversalmechanism remains asymmetric for the two branches of thehysteresis loops, however, it shows signiﬁcantly increasedcoerciveness along both branches with increasing thicknessof the Au layers. These effects occur despite the fact that thediameter of the magnetic grains attains a similar size as the Auspacer thickness and that the FM domains show no signiﬁcantvariation in their size. It appears that the impact of the metallicAu spacer adjacent to an AF or FM is very signiﬁcant forexchange bias systems in general as it can alter the interfacialmagnetism. II. SAMPLES AND MEASUREMENTS A. Sample preparation Over the years, Co /CoO has served as a prototypical exchange bias system, even though it is not actually techno-logically practical. In fact, very recent extensive investigationsa r eo nt h es a m eA F /FM combination. 16–19It is ideal for investigation due to its large biasing ﬁeld,6very distinct asymmetry of magnetization reversal,5large enough training effects,11and most interestingly, the AF moment conﬁguration can be frozen-in in a variety of ways during the processof ﬁeld cooling 20without affecting the overall structure as the AF ordering temperature is far below room temperature(negligible interdiffusion at the interfaces). We have investigated multilayers of the composition SiO 2/ [Co(11 .0n m )/CoO(5 .0n m )/Au(25 ,30,50 nm)] N=16and com- pare them with SiO 2/[Co(11 .0n m )/CoO(7 .0n m ) ] N=20/ Au(50 nm). A schematic of the layer structure is shown inFig. 1. During deposition, the Ar pressure in the magnetron sputtering chamber was 3 ×10 −3mbar. The process was started at a base pressure of 1 ×10−7mbar. We employ an Co_1 Co_25,_30,_50 FIG. 1. (Color online) Schematic of the layer structures, namely, Co_1 having no spacer layer and Co_25, Co_30, and Co_50 having the bilayers separated by Au spacer layers.ultraviolet light assisted oxidation at an O 2pressure of 200 mbar at 50◦C for 1 hour.21 B. Measurement techniques 1. Magnetometery Conventional in-plane magnetization loops are mea- sured using a superconducting quantum interference device(SQUID) MPMS and a physical property measurement system(PPMS) from Quantum design. We use a cooling ﬁeld H FC= +4.0 kOe within the sample plane for all specimens inducing an exchange bias as the system is cooled down to 10 K. 2. X-ray scattering and microscopy X-ray diffraction patterns from the samples conﬁrm the [111] fccstructure for the Au and Co layers. The microstruc- tural characterization was performed using cross-sectionaltransmission electron microscopy (XTEM). Studies withtransmission electron microscopy have been carried out oncross-sectional samples prepared by standard mechanical(diamond) polishing followed by Ar +ion milling at 4 kV for about 1 hour. A conventional bright-ﬁeld imaging modewas used. 3. Polarized neutron scattering Polarized neutrons are an excellent probe for investigating the in and out of plane correlations of the ferromagneticdomains. Depth-sensitive polarized neutron scattering mea-surements are performed at the neutron reﬂectometer TREFFat FRM II using polarization analysis. The specular as well asthe off-specular data were measured. The neutron wavelengthwas ﬁxed at λ=4.73˚A. Details on the technique and a corresponding review can be found elsewhere. 23In the experiment, four different cross sections are measured, namely,non-spin-ﬂip (NSF) ( R ++andR−−) and spin-ﬂip (SF) ( R+− andR−+) channels . Here, the subscripts +and−designate polarizations of the neutron beam parallel or antiparallel tothe guide ﬁeld, respectively. The specimens are ﬁeld cooled inH=4.0 kOe to 10 K inside a cryostat at the instrument. The NSF intensities provide the amplitude of the projection of themagnetization along the polarization direction of the neutrons(M /bardbl), while the SF intensities provide information about the magnetization components perpendicular to the polarizationdirection (M ⊥). The latter contributions are exclusively of magnetic origin. 4. X-ray absorption spectroscopy An increase in the bias ﬁeld Hebcan originate from the formation of defects within the antiferromagnetic Co xOylayer or from deviations in the stoichiometry during the course ofthe oxidation of Co to CoO 21leading to a stronger pinning of the domain walls at the defect sites thus resulting in anincrease of H eb.24To verify the formation of such defect sites that can be inadvertently related to the degree of oxidation ofthe Co layer (few nanometers), it is necessary to investigate theproportion and stoichiometry of the CoO layers in the system.Such a detailed examination of the chemical species can beeffectively done by x-ray absorption spectroscopy (XAS). 014431-2CHANGE IN INTERFACE MAGNETISM OF AN EXCHANGE- ... PHYSICAL REVIEW B 87, 014431 (2013) TABLE I. Samples and their saturation magnetization and exchange energy. The bilayers Co /CoO of sample Co_1 are not separated by Au spacer layers. MFM Magnetic moment E Composition Label (emu cm−3) μB/Co(FM) (erg cm−2) [Co(11 .0 nm)/CoO(7 .0 nm)] 20/Au(50 nm) Co_1 1694 ±100 2.01 ±0.2 0.75 ±0.05 [Co(11 .0 nm)/CoO(5 .5 nm)/Au(25 nm)] 16 Co_25 1132 1.34 0.72 [Co(11 .0 nm)/CoO(5 .0 nm)/Au(30 nm)] 16 Co_30 992 1.18 0.73 [Co(11 .0 nm)/CoO(5 .0 nm)/Au(50 nm)] 16 Co_50 726 0.86 0.79 XAS is generally used to obtain information about the local arrangement of atoms around the absorbing atoms. In particu-lar, the x-ray absorption near-edge structure (XANES) regioncorresponds to the excitation of core electrons to unoccupiedbound states or to low lying continuum states. It thus turns outthat the angular momentum and site projected partial densityof empty states, with some broadening, resemble the XANESabsorption spectra. The Co K-edge XANES measurements were performed in the ﬂuorescent mode with a 13-component Ge detectorat the x-ray absorption spectroscopy beamline (BL-8) of theSiam Photon Source (electron energy of 1.2 GeV , beam current120–80 mA), Synchrotron Light Research Institute, Thailand.A double crystal monochromator Ge (220) was used to scanthe energy of the synchrotron x-ray beam with energy steps of0.30 eV . Further, we performed Co L 2,3edge XAS measurements on the specimens. The XAS spectra result from Co 2 p−→ 3ddipole transitions (2 p3/2and 2p1/2core-shell electrons to unoccupied 3 dorbitals).25Comparing with the ab initio calculations of the L-edge and K-edge structure of Co, CoO, and Co 3O4, it is possible to identify the individual constituents of magnetic species in the system. The absorption cross section is measured by collecting the energy selective ﬂuorescence yield using a commercialXES300 spectrometer with an energy resolution of 0.89 eV atthe CISSY end station of the high-ﬂux beamline U49 /2-PGM1 installed at the Berliner Elektronenspeicherring Gesellschaftf¨ur Synchrotronstrahlung GmbH (BESSY). The photon energy is swept through the L3 (778 eV) and L2 (798 eV) edges of Co. The detector consists of a multichannel plate inconjunction with a resistive anode assembly. We integratethe x-ray emission spectroscopic signal to get the ﬂorescencesignal. In principle, x-ray magnetic dichroism (XMCD) can selectively probe the induced magnetic moment of Au in Co/Au multilayers and separate it into spin and orbital terms. 26However, XMCD (sensitive to p,d, andf-electron polarization) is a surface sensitive technique as the probingdepth in the soft x-ray regime is ∼5.0 nm in the electron yield (EY) mode and ∼100 nm in the ﬂuorescence yield (FY) mode. FY dichroism measurements are extremely sensitive to saturation and self-absorption effects, complicating theevaluation. Thus it is almost impossible to investigate theinterface of an ML with a thicker spacer at deeply buriedinterfaces (as in the present case). Alternatively, by using the low-temperature nuclear orienta- tion (LTNO) technique, one can detect the average magnitudeand alignment of the nuclear spins which can be due to theinduced nuclear polarization in the nonmagnetic Au spacer ( s- moment polarization). 15Canting of the induced Au magnetic moments was found to originate at the AF(FM) /Au/AF(FM) interface as well as canting of the Co moments (reducingthe net moment of the uncompensated spins) was observedearlier in AF /Au/FM interfaces. However, a detailed in- spection of the interface magnetization (depending uponthe structure of the interface) was limited by the level ofresolution available with the technique, and it also requiresmilli-Kelvin sample environment, which is not commonlyavailable. III. RESULTS AND DISCUSSION A. Magnetization The labeling of the samples along with the saturation magnetization per unit volume ( MFM) and the magnetic moment per Co (FM) atom is given in Table I. The exchange coupling energy27per unit surface area is usually given by E=−JEMAFMFMtFMcosδ =−HebMFMtFMcosδ. The unidirectional anisotropy energy is characterized by the exchange coupling constant JE. The unidirectional anisotropy Kudis included in JEMAFMFMin terms of the exchange ﬁeld Heb=JEMAF. Here, tFMis the thickness of the FM layer and δis the angle between MFMand the easy axis of the FM. MFMandMAFare the respective magnetizations. We deﬁne the exchange bias shift Heb= (HC2+HC1)/2 and the coercive ﬁeld HC=(HC2−HC1)/2, where HC1andHC2are the coercive ﬁelds on the decreasing and increasing branches of the hysteresis loop, respectively.Also given in Table Iare the exchange coupling energy Eas obtained from the respective FM layer thickness, the exchangebias ﬁeld values and the saturation magnetizations for the MLsfrom the magnetization measurements. 1. Hysteresis loops Figure 2(a) shows the hysteresis loops as measured with a SQUID for an in-plane cooling ﬁeld and longitudinalmagnetization measurements at 10 K for the sample Co_1.The results are reproduced from Ref. 32. For comparison, hysteresis loops for the samples Co_25, Co_30, and Co_50 areshown in Figs. 2(b)–2(d). Clearly seen is the usual asymmetry in the magnetization reversal and the disappearance of theasymmetry after the ﬁrst ﬁeld cycle. The room-temperature(RT) data [triangles in Figs. 2(b)–2(d)] show that the saturation ﬁeld is around 100 Oe [for clarity, see the inset of Fig. 2(b)]. 014431-3AMITESH PAUL et al. PHYSICAL REVIEW B 87, 014431 (2013) FIG. 2. (Color online) SQUID magnetization hysteresis loops for the (a) [Co /CoO] 20ML (from Ref. 32)a n df o r[ C o /CoO/Au] 16ML for Au layer thicknesses of (b) 25 nm: Co_25 (c) 30 nm: Co_30 and (d) 50 nm: Co_50. The measurements are done at room temperature (triangles) and at 10 K (after cooling down in HFC=+4 kOe). The inset in (b) shows the RT data in lower ﬁeld values. The blue dotted lines indicate the switching ﬁeld HC-Oduring the ﬁrst ﬁeld cycle. The thin solid lines are guides to the eye. For Co_25 and Co_30, the coercive ﬁelds at RT and the exchange bias ﬁelds at 10 K are approximately 20 Oe and≈−580 Oe /≈−670 Oe, respectively. The corresponding RT data for Co_50 shows that the coercive ﬁeld has increased to40 Oe. Such a broadening of the hysteresis loop at RT can begenerally attributed to defects within the magnetic layers. Wepoint out that the exchange bias ﬁeld along the cooling ﬁeldaxis is estimated to be around −1000 Oe for the 50 nm spacer ML, as compared to ≈−400 Oe for the ML specimen with no spacer.The hysteresis loops in Fig. 2(a) show at least three kinks near−780,−1400,and around −1676 Oe along the decreasing branch. These kinks are an indication for CoO layers havingdifferent oxidation levels. Similar kinks can also be seen duringthe ﬁrst cycle in Figs. 2(b)–2(d). The last switching ﬁelds show an increasing magnitude with increasing thickness ofthe spacer layer. In a previous work, Paul et al. have found very similar characteristics while varying the oxidation conditions for thebottom and top Co layer in a Co /CoO/Co based spin-valve system. 21Note that similar subloops in oxidized Co dots were initially attributed to the effect of the aspect ratio forpatterned samples, 28even though they have been commonly observed in nonpatterned specimens as well. Intuitively, avarying stoichiometry of the Co xOylayers, that may also depend on the number of bilayers, affects the strength of theexchange coupling between the AF and FM layers. Thereforean optimized stoichiometry can lead to an enhancement ofthe switching ﬁelds. Of course, the grain size may affect theswitching ﬁelds as well. The net magnetization in the Co_1 ML (for example) shows a decrease of 5% after the ﬁrst switching ﬁeld along thedecreasing branch of the hysteresis loop. This correspondsto 1 FM layer out of the 20 FM layers composing the ML,indicating that one of the 20 layers has already switchedwhile the other 19 layers are on the verge of ﬂipping.A similar argument can explain the magnetization data ofthe other samples, i.e., by a layer-by-layer ﬂipping of theheterostructure. 2. Magnetization versus temperature Figures 3(a) and3(b) show the temperature dependence of the magnetization M of the samples Co_25 and Co_50 as Co_25 Co_50(a) (b) FIG. 3. (Color online) ZFC and HFC magnetizations as a function of increasing temperature ( T) in a small external ﬁeld of H= 100 Oe for (a) Co_25 ML and (b) Co_50 ML. 014431-4CHANGE IN INTERFACE MAGNETISM OF AN EXCHANGE- ... PHYSICAL REVIEW B 87, 014431 (2013) (a) (b) FIG. 4. (Color online) (a) SQUID magnetization hysteresis loops for the Co_1 and Co_25 MLs showing the sub-loops shifts. The measurements are done at 220 K (after cooling down in HFC= +4 kOe). (b) The temperature variation of the coercive ﬁelds and the exchange bias ﬁelds for the two MLs. measured at 100 Oe using a PPMS. The merging point of the zero ﬁeld cooled (ZFC) and ﬁeld cooled (HFC, H=4k O e ) data provide the blocking temperature TBof the system. TB characterizes the onset of instabilities of the AF as thermal excitations creep in. The similar TBof both polycrystalline specimens indicates that their grain sizes are very similar.29 However, we ﬁnd three distinct steps in Co_25 before theloops merge at T B=240 K. In Co_50, the steps are smeared out. The samples show also a signiﬁcant difference in the macroscopic magnetization during ZFC and HFC. The ZFCvalues at low Tare smaller for Co_25 than for Co_50. The HFC values are larger for Co_25 than for Co_50. These resultsindicate that the anisotropy in the Co_50 sample is larger. 3. Initial domain conﬁgurations due to Au spacer Apart from the local inhomogeneities (roughness, defects) due to variations in the AF crystallite /grain sizes and con- comitant domain size distribution, a distribution of local TB is typically observed. It is well known that a thicker AF layer leads to an increased stability of the AF domains.29 Above a critical thickness (as in the present case), this maylead to splitting of the hysteresis loop into two subloopsshifted in opposite directions when measured just around theblocking temperature. This subloop shifts and the temperaturevariation of the coercive ﬁelds can be seen in Figs. 4(a) and 4(b), respectively, for the Co_1 and Co_25 MLs. A marked difference is seen as we compare the MLs with and withoutthe spacers. It is clear that the Co_1 ML does not show thesubloop shifts. This clearly indicates that these shifts in theCo_25 ML are due to the presence of Co-Au and/or CoO-Auinterfaces as they together are responsible for a FM imprintonto an AF. Thus there is a particular difference in the initialAF-FM domain conﬁgurations in such systems, which canFIG. 5. (Color online) (a) Representative ac susceptibility mea- surements at different frequencies are shown for the Co_25 ML.(b) The ﬁeld derivative of the magnetization as a function of ﬁeld measured at various temperatures without applying an ac ﬁeld is plotted for the Co_25 ML. They show the evolution of multiple switching with temperature along both branches of the hysteresis loop. be a topic of future investigation. Usually, an imprint of the FM domain structure onto the AF during zero-ﬁeld coolingprocedure divides the AF into two types of regions locallyoriented in opposite directions. 6Note that in the present case, the cooling ﬁeld is above the saturation ﬁeld of the FM andthe FM orders before the AF. Here, a proportion of the AFspins/domains (affected by the thermal activation) is aligned bythe cooling ﬁeld, while another proportion remains unaligned.After ﬁeld sweeping, this proportion gets realigned along thedirection opposite to that initially set during the ﬁrst ﬁeldcooling. 4. Susceptibility Susceptibility data of Co_25 are shown in Fig. 5(a).T h e in-phase susceptibility (Re χac=dM/dH a) data measured at 10 K and at a driving ﬁeld of 10 Oe (rms) after HFCfrom RT indicates also the occurrence of three reversal steps 014431-5AMITESH PAUL et al. PHYSICAL REVIEW B 87, 014431 (2013) FIG. 6. (Color online) Dependence of HebandHCon the thick- ness of the spacer layer as obtained from the SQUID magnetization hysteresis loops for [Co /CoO/Au] MLs. The coercive ﬁelds HC1 and the ﬁrst switching ﬁelds HC-Oshow an increase with increasing thickness. The dotted lines are guides to the eye. (indicated by arrows) along the decreasing branch and two reversal steps along the increasing branch. The response fromthe samples hardly shows any frequency (10 Hz–10 kHz)dependence. A much lower signal along the decreasing branchindicates that the domain dynamics along this branch is a slowprocess, at least slower than the response time correspondingto the 10 Hz of ac ﬁeld. The reversal steps are more evidentfollowing the ﬁeld derivative of the magnetization dM/dH aas a function of ﬁeld in Fig. 5(b) following the data measured at various temperatures without an ac ﬁeld. The evolution of theswitching ﬁelds with temperature is consistent with the datain Fig. 3(a). The behavior for all other samples is very similar and is therefore not shown. 5. H eband H Cwith Au spacer thickness The plot of HebandHCversus the spacer layer thickness in Fig.6shows an increasing magnitude with increasing spacer layer thickness. Also plotted is HC1and the ﬁrst switching ﬁeldHC-O. While an increase in HCcan be associated with an increased number of nonpinned hysteretic AF grains,an increase in H ebindicates an increase in the number of pinned domains or a stronger pinning by each domain in thepolycrystalline specimens. Microstructural investigation couldhelp in understanding such behavior further. B. Microstructure From the perspective of magneto-electronics, device char- acteristics are controlled by the magnetic evolution due to grainstructure modulation. Each bit usually contains hundreds ofgrains. Magnetic recording relies on the statistical averagingover these grains to obtain a satisfactory signal to noiseratio. As the bit size continues to decrease, the grain sizeneeds to be reduced too. The reduction can be achievedby controlling the surface properties of the coated and/orthe noncoated substrate. However, eventually, the grains willbecome superparamagnetic. Thus a control over grain sizeis essential. Sputtered species have a high kinetic energyand surface mobility allowing rearrangements in the structureduring ﬁlm growth. It was reported earlier that the exchange bias ﬁeld can be increased with the number of bilayers with successive FM-AFinterfaces. This is due to decreasing grain–size–mediated FM-AF exchange coupled domains stacked in successivelayers with gradually smaller sizes. 12For polycrystalline specimens, within the random anisotropy model, the exchangeinteraction averages over the anisotropy of the individualgrains. This would, in general, increase the effective exchangelength. However, with an increasing number of smaller grains(with an increase in the number of bilayers), as the exchangelength is reduced to the order of individual grain sizes ( ≈50 nm in the present case), the random anisotropy model will breakdown. This will lead to the formation of individual exchangecoupled grains—exchange coupled to the uncompensated AFmoments preferably located at the grain boundaries. The spinalignment in individual FM domains is determined, domain bydomain, by the spin directions in the AF grains. This is unlikethe case of nanocrystallites where the grain sizes ( /lessorequalslantexchange length) concomitantly reduces the average anisotropy ofthe system and make them soft (lowered coercivity). 30In exchange-coupled systems, the rotatable anisotropy ﬁeld valueis proportional to the magnetization of the small AF grains, itincreases with the exchange coupling strength, which in turnincreases the coercivity. 31An increase in the coercivity with smaller AF grain sizes is basically due to an increase in thenumber of rotatable grains (proportional to the sum of theprojections of these magnetizations along the bias direction). Paul et al. 32have reported earlier on the magnetization reversal for (i) a continuous sequence of successive FM-AFlayers (no spacer layers) and that for (ii) a sequence of FM-AFbilayers that are separated by a nonmagnetic spacer layer(Au). The main difference in their magnetization reversalmechanisms is the following: the separated multilayers (ML)showed a usual asymmetric reversal—a nonuniform (domainwall motion and domain nucleation) reversal for the decreasingbranch ( H FCanti–/bardblHa) of the hysteresis loop and a uni- form (coherent rotation) reversal for the increasing branch(H FC/bardblHa). In contrast, the continuous multilayer showed symmetric and sequential reversal (nonuniform) for bothbranches of the hysteresis loop. In this regard, it is interesting to note that in contrast to the unlike case of a continuous ML (case (i) above), in a sequence of bilayers Co-CoO that are interrupted by the presence ofthick Au layers, the evolution of the grains may be interrupteddepending on the thickness of the Au spacer layer, as thegrain size is limited by the layer thickness. 33For a thick enough Au layer, the grain structure of the underlayer is notpropagated to the next Co layer. This is similar to a decouplingof the intergranular interactions. 34It is therefore unlikely that the thickness of the nonmagnetic spacer will inﬂuence themagnetic grains as they are all nucleated on a similar spacerlayer. Therefore one may speculate that the magnetic behaviordoes not change with an increasing thickness of the spacerlayer. The aspect of grain structure evolution can be veriﬁedby cross-sectional TEM. Figure 7shows XTEM micrographs depicting repetitions of three layered structures with sharp interfaces for the MLswith (a) 25- and (b) 50-nm of spacer thickness. The thicknessof the individual layers is in agreement with the nominalthickness. Magniﬁcations of a trilayer interface show theexistence of columnar grains with a width of ≈25 nm and ≈50 nm for the 25-nm and 50-nm sample, respectively. 014431-6CHANGE IN INTERFACE MAGNETISM OF AN EXCHANGE- ... PHYSICAL REVIEW B 87, 014431 (2013) (a) 50 nm25 nm (b)25 nm11 nm5 nm 5 nm 11 nm 50 nm25 nm 50 nmAuCoCoOAuCoCoO FIG. 7. (Color online) XTEM micrographs of [Co /CoO/Au] 16ML for Au layer thicknesses of (a) 25 and (b) 50 nm. Vertically correlated Au grains are visible for both MLs. There are no visible differences for the Co-CoO grains, which are basically unaffected by the size of the underlying Au grains. A schematic of the granular layer structure is shown alongside. Note that the almost square-shaped Au grains are vertically correlated. The results conﬁrm the common observation insputtered and evaporated thin ﬁlms that the grain size is ofthe same order as the ﬁlm thickness. The grains of the Colayer, however, are approximately 11 ×20 nm and are very similar for Co_25 and Co_50. A similar size of grains d∼ 11.5 nm is also estimated from the width of the Co peak fromx-ray diffraction measurements. Therefore there is no visiblemicrostructural difference in the Co layers. An increased coercivity in exchange coupled systems is a clear indicator for a dominance of domain wall pinning, as theAF domain walls act as pinning sites for the neighboring FMdomains. 22Thus if we presume the grains to evolve (decrease) with increasing number of layers in a ML stack then anincrease in the number of AF domain walls or increased grainboundaries is expected for those domain walls to form. In thecase, that the evolution is interrupted (as in the present case),the number of AF domains will remain similar. In any case,this would concomitantly inﬂuence the FM domains. When comparing the ML microstructures, particularly for Co_25 and Co_50, the enhanced coercivities of the FM layersdo not appear to correlate in a systematic way with the AFgrains. Due to a distribution of grain size, one can expect exchange decoupled AF grains (associated with individual grain spins) at the interface and exchange coupled FM grains.The FM and AF layer coupling can be via exchange anddipole-dipole interactions. However, the additional anisotropygiving rise to the enhanced coercivity can also have itsorigin within the bulk of the AF layer due to the grainstructure that affects the AF magnetocrystalline anisotropy. 29 Hence the enhanced coercivity might be a combination of theeffects in both the bulk and interfacial grain spins of the AFlayers. Since the XTEM pictures do not show a signiﬁcant variation of the grain structures with an increase in the spacerlayer thickness, the coupling of the interfacial grains can beconsidered to be responsible for the increase of the coercivity.It may be possible that due to the different oxidation states of the AF layer, i.e., CoO, Co 3O4, and Co 2O3, the individual grains are coupled differently to the FM grains. Such differentoxidation states may originate from changes in the depositionconditions within the chamber while depositing a thickerspacer layer. CoO in a stoichiometric relationship Co : O =1: 1 is not the only binary oxide phase that forms under readilyattainable oxygen partial pressures. The thermodynamicallyfavored form of the cobalt oxide is often Co 3O4. In contrast to the two cobalt oxides mentioned above, the metastable formCo 2O3may be difﬁcult to form. C. Specular and off-specular neutron scattering 1. Scattering geometry The neutron scattering geometry is shown in Fig. 8.W e deﬁne the ML surface in the x-yplane and the zaxis along the surface normal. In the specular scattering geometry sample plane 22 sample plane k i k f z y x i 22f f f f φ Α lx ly M M FIG. 8. (Color online) Schematic of the neutron scattering geom- etry. In reﬂection geometry, the beam is collimated in the reﬂection plane and relaxed along the yaxis, whereas in the GISANS geometry scattering along the yaxis is resolved. Here, /vectorkiis the incident wave vector at an angle αi. The scattered wave vector /vectorkfmakes an angle αfand 2θfalong two different scattering planes. The grey shaded region represents the coherence ellipse covering several (or single) domains (shaded in green) and the mean magnetization making an angleφAwith the polarization axis, which is along the yaxis. 014431-7AMITESH PAUL et al. PHYSICAL REVIEW B 87, 014431 (2013) FIG. 9. (Color online) NSF intensity maps ( R−−) from Co /CoO/Au MLs measured on HADAS /TREFF at saturating ﬁeld along the decreasing branch of the hysteresis loops for Co_1 and Co_25 and Co_50 ML samples after ﬁeld cooling at 4.5 kOe and measured at 10 K. The color bar encodes the scattered intensity on a logarithmic scale. (i.e., angle of incidence αiequal to the exit angle αf), the reﬂectivities follow from energy and in-plane momentum conservation laws as normal wave-vector transfers /vectorQ⊥are probed. However, when the in-plane translational symmetryis broken by interface waviness (roughness) or by magneticdomains on a length scale shorter than the in-plane projection of the neutron coherence length l /bardblalong /vectorQ/bardbl(=/vectorQx,/vectorQy) then the off-specular scattering contributions along the in-plane momentum transfer vector ( /vectorQ/bardbl) arise. At grazing incidence, there can be three scattering geome- tries: specular reﬂection, scattering in the plane of incidence(off-specular scattering), and scattering perpendicular to theplane of incidence (Grazing Incidence SANS). We can esti-mate the extent of correlation lengths from the three equationsof momentum transfers along the three different axis owing tothe scattering geometry for small angles: /vectorQ z=/vectorQ⊥=2π λ[sin(αi)+sin(αf)]/similarequal2π λ(αi+αf), (1) /vectorQx=/vectorQ/bardbl=2π λ[cos(αf) cos(2 θf)−cos(αi)] /similarequal2π λ/parenleftbiggα2 i 2−α2 f 2−2θ2 f/parenrightbigg , (2) /vectorQy=/vectorQ/prime /bardbl=2π λcos(αf)s i n ( 2 θf)/similarequal4π λ(θf). (3) Here, the incident wave-vector deﬁned by /vectorki, makes an angle αiin thex-zplane with respect to the xaxis, while the scattered wave vector /vectorkfmakes angle αfin thex-zplane and also 2 θfin thex-yplane (relevant for diffuse scattering). Different length scales ξ=2π /vectorQranging from nanometers to micrometers can be accessed by using different scattering geometries in most practical cases. Specular scattering provides the scatteringpotential of the ML perpendicular to the ﬁlm plane. The typicalprobed length scales are in the range 3 nm <ζ< 1μm. Off-specular scattering scans provide the lateral correlations along /vectorQ x(500 nm <ξ< 50μm), whereas grazing incidence SANS scans probe the surface (3 nm <ξ< 100 nm) along /vectorQy. From the above equations, one may also note that for agiven geometry when αi∼αf∼θf/lessmuch1, the projection /vectorQy∼ /vectorQz/greatermuch/vectorQx. 2. NSF scattering The scattering-length densities (SLD) of a magnetic spec- imen are given by either the sum or difference of the nuclear(ρ n) and magnetic ( ρm) components. The ±signs refer to the spin-up and spin-down states of the incident neutronbeam with respect to the magnetization of the sample. Thenon-spin-ﬂip (NSF) scattering amplitude provides informationaboutρ n±ρmcosφA, and the spin-ﬂip (SF) channels measure ρ2 msin2φA, if the domain size is larger than the projection of the neutron coherence length along the sample plane ( l/bardbl). Here, φA is the angle between the magnetization Mand the applied ﬁeld Ha, which corresponds usually to the neutron quantization axis. a. Intensity maps. Next, we show the specular and off- specular NSF intensity maps in Fig. 9for the Co_1, Co_25, and Co_50 samples corresponding to the channel R−−.T h e intensity along the diagonal αi=αfis the specular reﬂection along the scattering vector Q⊥. In the experimental geometry, only/vectorQxis resolved whereas the signal along /vectorQyis integrated because the collimation along the yaxis is relaxed. The NSF intensities are shown at a saturating ﬁeld along the decreasingbranch of the respective hysteresis loops where the MLs arein the single domain state. The observed superlattice peaksfrom the specimens (see Fig. 9) conﬁrm the periodicity of the multilayer structure. The off-specular scattering along theBragg sheets occurs due to pronounced structural verticalcorrelation of each of the MLs. b. Specular scattering. The neutron reﬂectivity does not only carry information on the mean magnetization directionbut also on the layer-by-layer vectorial magnetization. Incorroboration to the drop of the net magnetization at the ﬁrstswitching ﬁeld ( H a=0.75 kOe) along the hysteresis loop of the Co_1 ML, the ﬁts to the neutron reﬂectivity data, indeed,show the switching of one out of the twenty FM layers at anapplied ﬁeld H/similarequal1.0 kOe. Similar to the Co_1 ML, 32we ﬁnd layer-by-layer ﬂipping for the Co_25 and C0_50 MLs as well, 014431-8CHANGE IN INTERFACE MAGNETISM OF AN EXCHANGE- ... PHYSICAL REVIEW B 87, 014431 (2013) FIG. 10. (Color online) Specular reﬂectivity patterns (solid symbols) along with their best ﬁts (open symbols) for the NSF [ R++(red) and R−−(black)] and SF [ R−+(green) and R+−(blue)] channels measured at a saturation ﬁeld, for the MLs with different spacer layer thicknesses. /vectorQz=2π λ[sin(αi)+sin(αf)], where αiandαfare the incident and exit angles, respectively. The ﬁts shown here are done by considering model A for the MLs Co_1 and Co_25 and model C for the ML Co_50. The corresponding nuclear (black) and magnetic (red) SLDs are shownalongside. which is indicated by their multiple switching ﬁelds along the respective hysteresis loops. Figure 10shows the specular reﬂectivity data (NSF and SF) corresponding to the three MLs at a saturation ﬁeld(H a=−4.0 kOe) on a logarithmic scale. The relative variation of the multilayer Bragg peak intensities due to differentperiodicities of the MLs is quite evident here. Earlier, the layermagnetizations for Co_25, measured at their ﬁrst switchingﬁeld by Paul et al. , 20revealed that at least four layers from the stack have ﬂipped and the remaining twelve layers are atthe onset of ﬂipping. Here, we ﬁnd the layer magnetizationsfor the Co_50 ML, also remain collinear at its ﬁrst switchingﬁeld, whereby nine of the sixteen layers have ﬂipped withthe ﬁeld. The value of the mean magnetization angle φ Afor the individual layers in the stack (0◦or 180◦with respect to theﬁeld) are taken from the ﬁtted values of the specular patterns (NSF and SF). We do not ﬁnd any signiﬁcant increase in theSF specular signals conﬁrming their nonuniform reversal thatis expected for these MLs, as we measure along the decreasingbranch of the ﬁrst ﬁeld cycle. The best ﬁts to the reﬂectivitydata revealed a good agreement with the nominal thicknessesand the ρ mandρnvalues as listed in Table II. The other parameters such as interface roughness are kept similar for allsamples. The respective nuclear and magnetic SLD values areplotted alongside. Note that the bulk value of the Co moment is ∼1.73μ B/ atom, here μBdesignates the Bohr magneton.35The estimated magnetic moment from the corresponding values of ρmas obtained from the least square ﬁt to the Co_1 ML neutronreﬂectivity proﬁle measured at saturation is ∼1.66μ B/atom 014431-9AMITESH PAUL et al. PHYSICAL REVIEW B 87, 014431 (2013) TABLE II. Fit parameters extracted from the PNR results. ρnandρmdesignate the nuclear and magnetic scattering length densities, respectively. In sample Co_1, there are no spacer layers between the Co /CoO bilayers. Also given are the respective magnetic moments as calculated from the magnetic scattering length densities and the exchange energy E. The magnetic moments are calculated following model A: considering no dead layer, model B: considering 1.0 nm of dead layer, and model C: considering reduced moment for the entire magneticlayer. The Au layer in Co_1 protects the sample against oxidation. Multilayer Au CoO Co Co-Au (dead layer) error Co_1 thickness (nm) 52.6 7.1 11.0 – ±0.2 (A) ρn(×10−6˚A−2) 4.5 4.5 2.3 – ±0.2 ρm(×10−6˚A−2) 0.0 0.0 4.1 – ±0.1 Co_1 thickness (nm) 52.6 7.1 10.0 1.0 ±0.2 (B) ρn(×10−6˚A−2) 4.5 4.5 2.3 2.3 ±0.2 ρm(×10−6˚A−2) 0.0 0.0 4.1 0.0 ±0.1 magnetic moment ( μB/atom) 1.66 ±0.1 E( e r gc m−2) 0.62 ±0.1 Co_25 thickness (nm) 22.5 5.5 11.0 ±0.2 (A) ρn(×10−6˚A−2) 4.5 4.5 2.3 ±0.2 ρm(×10−6˚A−2) 0.0 0.0 4.1 ±0.1 magnetic moment ( μB/atom) 1.66 ±0.1 E( e r gc m−2) 0.92 ±0.1 Co_50 thickness (nm) 48.0 5.0 11.0 ±0.2 (A) ρn(×10−6˚A−2) 4.5 4.5 2.3 ±0.2 ρm(×10−6˚A−2) 0.0 0.0 4.1 ±0.1 Co_50 thickness (nm) 48.0 5.0 10.0 1.0 ±0.2 (B) ρn(×10−6˚A−2) 4.5 4.5 2.3 2.3 ±0.2 ρm(×10−6˚A−2) 0.0 0.0 4.1 0.0 ±0.1 Co_50 thickness (nm) 48.0 5.0 11.0 ±0.2 (C) ρn(×10−6˚A−2) 4.5 4.5 2.3 ±0.2 ρm(×10−6˚A−2) 0.0 0.0 3.5 →reduced ±0.1 magnetic moment ( μB/atom) 1.45 ±0.1 E( e r gc m−2) 1.35 ±0.1 ±0.05. Note that this is 17.4% less when compared with the moment obtained from the magnetometric measurementsusing the SQUID /PPMS (see Table I). One may recall thatρ m=MFM2.853×10−9˚A−2cm3emu−1. The magnetic moment for the Co_50 ML as obtained from the PNR dataﬁts, is ∼1.45μ B/atom±0.05 (13% reduction from the Co_1 value). The reduced magnetic moment of the Co layers asobtained from the magnetometry measurements lead us toinfer that there can be plausible magnetic dead layers at theCo-Au interfaces as we increase the Au spacer thickness. Suchformation of dead layers on magnetron sputtered samples arecommonly attributed to the interdiffusion that occurs duringthe deposition process. 36 c. Models for ﬁtting. In order to verify the formation of weakly coupled noncollinear domains at the interface, wecompare the PNR proﬁles for the Co_1 and Co_50 specimens.These systems were chosen for comparison because thechanges of the magnetic moment are maximum for thesetwo MLs. First, we compare the NSF simulated data (on a linear scale) over a certain range of /vectorQ zwhere the changes are explicit, considering different probable models. The simulations inFig.11are shown for both the MLs as we consider three models with (A) no magnetic dead layer (closed symbols), (B) 1.0 nmof magnetic dead layer (open symbols) at the Co-Au interfaces(see Table II), and a third model (C) with reduced moment throughout the entire Co layer thickness for the Co_50 ML(lines). The Co_1 ML obviously does not have Au spacersafter each Co-CoO bilayer rendering model (C) irrelevant for it. One can clearly distinguish the impact of the models onthe proﬁles and therefore our inferences from the ﬁts can beconsidered unambiguous. d. Spin asymmetry. Furthermore, the measured spin- asymmetry (SA) proﬁle is plotted versus /vectorQ zin Fig. 12.T h e spin asymmetry is expressed as the ratio of the difference andsum of R ++andR−−reﬂectivities measured at a saturation ﬁeld of −4.5 kOe. This normalized difference is sensitive to the magnetization proﬁle across the ﬁlm and is less sensitiveto interface roughness. We follow the ﬁt qualities in Figs. 11and 12for the Co_1 ML and Co_50 ML proﬁles using the model A, B, and C. Note the different ranges of the /vectorQ zin Fig. 11chosen for the two samples in order to compare the differences ofmodel ﬁts. One can see from both ﬁgures that the ﬁt qualitydeteriorates for the case with model B (dead layer) in caseof Co_1 ML. This conﬁrms that there are no dead layersin this specimen. All Co layers (in each bilayer repetition)have an uniform magnetization throughout the entire thickness of the layer. A very similar situation is encountered for the Co_25 ML as well. However, from the Co_50 ML proﬁle,one can see that a slight improvement in the ﬁt qualityhas been achieved by using model C, i.e., by consideringa 13% reduction in the moment for the entire Co layer(11.0 nm). No signiﬁcant improvement in the ﬁt quality canbe achieved by using model B (dead layer at the Co-Auinterface). 014431-10CHANGE IN INTERFACE MAGNETISM OF AN EXCHANGE- ... PHYSICAL REVIEW B 87, 014431 (2013) FIG. 11. (Color online) NSF simulations [ R++(red) and R−− (black)] with /vectorQzfor the (a) Co_1 and (b) Co_50 MLs. The simulations are shown to compare for the models considering (A) no magnetic dead layer (closed symbols), (B) 1.0 nm of magnetic dead layer (open symbols) at the Co-Au interface, and a third model (C) with reducedmagnetic moment throughout the entire Co layers for the Co_50 ML (lines). Note the different ranges of /vectorQ zchosen for the two samples in order to show the differences in model ﬁts distinctly. FIG. 12. (Color online) Spin asymmetry (SA) (black square) with /vectorQzin order to compare the magnetization in the (a) Co_1 and (b) Co_50 MLs. The simulations are shown for the models consider- ing (A) no magnetic dead layer (black line), (B) 1.0 nm of magnetic dead layer (red line) at the Co-Au interface, and a third model (C)with reduced magnetic moment throughout the entire Co layers for the Co_50 ML (blue line).FIG. 13. (Color online) Simulated SA is plotted with /vectorQzfor the Co_50 ML considering different degrees of reduction in magnetiza- tion of the Co layer as obtained from the PNR data and also from thePPMS data. Figure 13shows the simulated SA versus /vectorQzfor various reductions of the magnetic moment of the Co layer in Co_50ML. One can see that when the moment (or ρ m) is reduced by 57% (which is estimated from the PPMS measurements)a strong deviation is encountered as compared to the bestﬁt which is simulated considering only a 13% reduction inthe magnetic moment. Note that these values are comparedfor the apparent saturation ﬁeld measurements, thus one canrule out the possibility of canting in the ﬁlm plane (howeverout-of-plane canting may be possible). e. Discrepancies in magnetic moment. In the present case, from the changes in M FMas obtained from PPMS (see Table I) and as obtained from PNR (see Table II), the exchange coupling Ecan be calculated. It turns out that E∼0.75±0.05 erg cm−2is almost independent of the spacer layer thickness of the MLs as obtained from the PPMSmeasurements. The Evalues, as obtained from the PNR measurements however, show a two times increase for theCo_50 ( ∼1.35±0.1e r gc m −2) ML as compared to the Co_1 ML. This of course follows from the respective difference inreduced magnetizations (particularly for the Co_50 ML) asobtained from the two techniques used. Discrepancies in the estimates of the magnetic moment are commonly reported for SQUID based magnetometers andPNR measurements. 37This becomes more visible, probably for oxidized layers, due to plausible inhomogeneities. Mea-surements at TREFF were done with a 2.0 mm beam divergingby∼0.1 ◦at a distance of 1500 mm from the 15-mm sample along /vectorQx. The neutron coherence lengths lx(along /vectorQx) and ly(along /vectorQy)38thereby turn out to be few micrometers and few angstrom, respectively, which can be estimated using the uncertainties in /vectorQxand/vectorQyas lx∼1 /Delta1Qx∼1 π λ/radicalbig (αi/Delta1αi)2+(αf/Delta1αf)2, (4) ly∼1 /Delta1Qy∼1 2π λ/Delta1θf. (5) Here, lxbeing /lessmuchthan the illuminated sample area ( ∼2.0– 0.65 mm), the intensities on the detector are an incoherentsum of the coherently scattered intensities from the coherentellipse. This can make a signiﬁcant difference for sampleswith laterally and vertically inhomogeneous magnetic entities 014431-11AMITESH PAUL et al. PHYSICAL REVIEW B 87, 014431 (2013) FIG. 14. (Color online) SF intensity maps ( R+−) along with their simulations within DWBA from Co /CoO/Au MLs measured on HADAS /TREFF at the ﬁrst switching ﬁelds along the decreasing branch of the hysteresis loops for Co_1 and Co_25 and Co_50 ML samples after ﬁeld cooling at 4.0 kOe and measured at 10 K. The color bar encodes the scattered intensity on a logarithmic scale. varying from one coherence volume to the other. The PPMS measurements, on the other hand, are from a signal averagedover 5 t FM-mm3sample volume. 3. SF scattering a. Intensity maps. Figure 14shows intensity maps for the Co_1, Co_25, and Co_50 samples corresponding to thechannel R +−. The SF intensities are shown at a ﬁeld that is close to the ﬁrst switching ﬁelds along the decreasing branchof the respective hysteresis loops. These intensities eventuallydisappear at saturation, demonstrating their magnetic origin.A small contribution from the NSF intensities appears in theSF channels due to a reduction of the efﬁciency of ≈5% of the polarizer and the analyzer components. Note that no Braggsheets are visible in the SF channels unlike that in the NSFchannels. Here, we consider three possible scenarios for the Co_1 ML for ﬁelds close to the coercive ﬁeld: (i) Paul et al. 32have shown earlier that the reﬂectivity proﬁle near coercivity isbest simulated considering an almost equal number of layersoriented along the applied ﬁeld direction and opposite to it.(ii) The magnetization is close to zero due to the formation of amultidomain state with random orientation of the domains, and(iii) the magnetization is oriented along an axis perpendicularto the polarization axis, corresponding to a coherent rotationof magnetization. In all these three cases, the projection of thelongitudinal magnetization onto the neutron polarization axis(yaxis) is proportional to /angbracketleftcosφ A/angbracketright(=0), while the projectionof the transverse component with respect to the polarization axis onto the xaxis is proportional to /angbracketleftsin2φA/angbracketright. However, in the case of a random distribution of domain magnetizationdirections, the dispersion is /angbracketleftcos 2φA/angbracketright–/angbracketleftcosφA/angbracketright2/negationslash=0. For a coherent rotation this dispersion is essentially zero. Thus onecan distinguish between a situation of random distributionof domains and that between a coherent rotation. In case ofdomains that are smaller than the neutron coherence lengthalong the xaxis, off-specular scattering is expected as well. The situation becomes more involved when an equal number oflayer magnetizations is oriented along and opposite—but arestrictly collinear—to the polarization axis. It is then difﬁcult(or even impossible) to infer the domain size as there is noSF off-specular scattering in absence of ﬂuctuations aroundthe mean magnetization Mdirection even if the domains are smaller than the neutron coherence length. The absence of well deﬁned Bragg sheets in Fig. 14,i nt h e off-specular scattering from Co_25 and Co_50 MLs indicatesa lack of vertical correlations. In contrast to the Co_1 ML,both Co_25 and Co_50 MLs show a signiﬁcant increase in the off-specular intensities. They occur due to ﬂuctuations of the magnetization of the domains around the mean magnetizationangle indicating an instability that is induced in the system atthe onset of ﬂipping of the magnetization of the layers. Flippingis likely when the size of the domains becomes comparable tothe width of the domain walls. b. DWBA simulations. The specular and the off-specular intensity is simulated within the distorted wave Born 014431-12CHANGE IN INTERFACE MAGNETISM OF AN EXCHANGE- ... PHYSICAL REVIEW B 87, 014431 (2013) approximation (DWBA).38The simulations are conducted by taking into account spin-dependent reﬂection and refraction.Finally, the cross section is convoluted with the instrumentalresolution function (see Fig. 14). Inhomogeneities of the ML like magnetic roughness at the interfaces are taken intoaccount to ﬁrst order starting from an ideal multilayer withﬂat interfaces. We assume for all measurements that the meanmagnetization is collinear with the neutron polarization axis,which is along the yaxis. Note that the coherence area is substantially extended along the xaxis (see Fig. 8). This area is restricted via the uncertainty in the momentum transfers(/Delta1Q x,y∼2π lx,y) along the xandydirections. The uncertainties are a consequence of the angular divergences due to the beam collimation opted in the measurements. The off-specular scattering gradually disappears when the ﬁeld becomes larger than the ﬁrst switching ﬁeld (see Fig. 14). Within our model, we allow Mto ﬂuctuate from domain to domain around the mean angle by /Delta1φA=30◦averaged over the coherence volume for Co_25 and Co_50. Theseﬂuctuations can be longitudinal /angbracketleftcos(δφ A)/angbracketright(/Delta1M/bardblM)a s well as transverse /angbracketleftsin2(/Delta1φA)/angbracketright(/Delta1M⊥M). The structural parameters are obtained from the ﬁts to the specular patterns. Transmission and reﬂection amplitudes show singularities at the points of total reﬂection, i.e., at the critical edges.Figure 14clearly shows these singularities, i.e., the Yoneda wings, which in turn are accompanied by an enhancementof the diffuse scattering. Such enhancements can be seen inthe SF maps in cases when the domains are smaller than theneutron coherence length along the xaxis, i.e., as and when the coherence ellipse covers several domains. One usuallyencounters an asymmetry of the scattered neutrons in theSF channels due to the inverse population of the incomingand the outgoing neutrons selected by the polarizer (differentcritical edges for up and down neutrons) and ﬂipped by the spinﬂippers. 38One can also see, particularly for the Co_50 data and its simulation, that the Yoneda wings are associated withstreaks running parallel to the α iandαfaxes. These streaks are commonly observed when the SLD values of one of theconstituents of the ML form a shallow potential well (Co) withrespect to a wider and higher SLD value (Au). The effect isrelated to the difference in the phases of the transmitted andreﬂected waves. It is well known that a decreasing domain size leads to a concomitant increase in the number of grain boundaries (asdomains can be associated with the grain size) and therebyan increase in the number of uncompensated spins in theAF as in the case of Co_1 type (nonseparated) MLs. 12,32 However, for the Co_1 ML, the magnetic correlation length cannot be properly estimated. This is because, at the reversalpoint, 50% of the layers are directed along the applied ﬁeldand the remaining 50% are directed opposite to the appliedﬁeld direction. Thus the net magnetization is close to zero.Furthermore, there are no indications of small scale variationsaround the mean magnetization angle close to the critical angleof total reﬂection (even at its reversal point) and also that thesedomains are either vertically uncorrelated (no Bragg sheets areobserved in the SF channels) and/or larger than the neutron coherence length projected along /vectorQ x. Whereas, in the cases Co_25 and Co_50 ML specimens, the typical FM verticallyuncorrelated domain sizes are of ≈1–2μm (estimated from the observed enhanced SF scattering intensities around thetotal reﬂection edges in each of the specimens), which areconsistent with previous measurements on similar samples. 39 Note that we could not observe any appreciable change in thedomain size with the spacer layer thickness, at least not for theseparated MLs. Generally, as the grain sizes become small enough that they are comparable to the domain wall width, where domainwalls can form within one grain, the magnetization directioncorresponds to the anisotropy direction varying from grain tograin. For grain sizes below the critical size, one can opt forthe random-anisotropy model, which takes into account themagnetic alignment between the grains that competes withthe anisotropies of the individual grains. The spontaneousspatial magnetic correlations, extended over many individualgrains, thus depend strongly on grain size. 40,41Interestingly, nonevolving domain sizes in our separated MLs are incorroboration with the underlying grains (which are only offew nm in size) as they are also of very similar dimensionsirrespective of the separation between the magnetic layers.This actually, in a way, conﬁrms that the grain structurevariation that was evident for the continuous multilayer wasrestricted in case of the separated multilayers. This informationis signiﬁcant enough as a variation in the domain sizes wouldhave had an effect on the exchange bias as well. D. X-ray absorption spectroscopy 1. K-edge spectroscopy Figure 15(a) shows a comparison of the measured Co K- edge XANES spectra from the MLs (solid symbols) and thereference spectra from each of the possible constituents thatcan produce the absorption edge for example, CoO, Co 3O4, and Co metal. By considering CoO, Co 3O4, and Co metal as the parent components, the XANES spectra of the three CoMLs are ﬁtted (lines) with a superposition of XANES proﬁlesof the parent components using the linear combination analysis(LCA) method. The ﬁtting was performed using the package ATHENA42with the LCA tool. The ﬁts are shown in Fig. 15(a) together with the measured XANES spectra. In this way, weestimate the weighted proportions of Co xOyand Co layers. Further, we calculate the Co metal, Co 3O4and CoO spectra (open symbols) using the FEFF 8.2 code, which is based on ab initio multiple scattering calculations.43The calculated spectra are shown along with the measured spectra for the MLs. For Co metal (hexagonal), a=2.5074 ˚A and c= 4.0699 ˚A are used as the lattice parameters. Whereas a= 4.2667 ˚A is used for the lattice constant of CoO (rocksalt) structures. For Co(CoO) metal, a cluster of 40(57) atoms[radius of 4.5(5.0) ˚A] is used to calculate the self-consistent ﬁeld mufﬁn-tin atomic potentials within the Hedin-Lundqvistexchange potential and a 80-atom cluster with a radius of 6.0 ˚A is used for full multiple scattering calculations. They includeall possible paths within a larger cluster radius of 7.0 ˚A (147 atoms). Next, we vary the proportions of each of the constituents in the calculated ( ab initio ) spectra according to the Rratio obtained from the proportional ﬁts and compare them (opensymbols) with measured XANES spectra. One can easily see 014431-13AMITESH PAUL et al. PHYSICAL REVIEW B 87, 014431 (2013) (a) (b)30 / / FIG. 15. (Color online) (a) Comparison of the measured Co K- edge normalized XANES spectra of the Co /CoO MLs (solid symbols) and their ﬁts (lines) using the LCA method using the package ATHENA . Also included are the reference spectra for CoO, Co 3O4, and Co metal. Theab initio calculated XANES spectra for the reference materials using the FEFF 8.2 code are also included. A weighted proportion of the species, with various proportions of Co metal, Co 3O4,a n dC o O as obtained from the ﬁts and are used to calculate the ML spectra, arealso plotted (open circles). (b) The ratios Rfrom the K-edge spectra are plotted for the total thickness of the MLs using two possible scenarios discussed in the text. that the calculated XANES spectra are in very good agreement with the corresponding features in the measured spectra of theMLs in both energy positions and shapes. This conﬁrms thepresence of multiple constituents in the MLs from ab initio calculations. The ratio of the signal, Ris determined by evaluating the ratio between the Co-signal and the CoO- or the CoO +Co 3O4signal as obtained from the ﬁts. Here, we have considered two possible scenarios for the AF layer in comparing the ratios(i) with CoO +Co 3O4content and (ii) only with CoO content. We have plotted these ratios in Fig. 15(b) as a function of the total ML thickness. In case of Co_1, the layer thicknessesbeing little different from the other MLs, the ratio cannotbe strictly compared for the same thickness ratio. A betteragreement with the data is obtained while considering scenario(i). The goodness-of-ﬁt parameter ( Rfactor) decreases by 5–30%. This indicates that the Co MLs are composed ofphase-separated regions that differ in the proportion of theirrespective constituents (Co metal, Co 3O4, and CoO). From the ratio Rin Fig. 15(b) , it is interesting to note that the XANES spectra show an increasing proportion of oxide(AF) material, which is largely compensated by a decreasingproportion of Co in the Co_25, Co_30, and Co_50 MLs. Aplausible change in the deposition pressure and temperature,with increasing deposition time (while growing thicker Aulayers), might have caused an ≈4% increase in the Co 3O4 content. Co 3O4has an ordering temperature ( TN=40 K) lower than CoO, which can vary depending upon the FM layerin its proximity. 44Coupling of the uncompensated AF spins within the Co 3O4proportion may be quite different from that within the CoO proportion, as they have different crystallinestructures which can even lead to different anisotropy axes.Therefore the presence of multiple constituents with differentmagnetic ordering temperatures in a way corroborates with themagnetization loops and the multiple switching ﬁelds that hasbeen discussed in the magnetization section. Apart from the effect of the Co 3O4content, in general, an increase in the exchange bias ﬁeld as has been observed here,may be associated with (a) an increase in the AF proportion(the AF thickness of our MLs is below a typical critical AFthickness of ≈10 nm), 45(b) a decrease in the FM proportion (increasing the surface to volume effect), and (c) plausibleformation of smaller AF domains 46(domains are preferably stabilized at the grain boundaries) with an increase in the totalﬁlm thickness. 12A≈30% change in the AF-FM thickness ratio corroborates well with the 35% change in the exchangebias ﬁeld for the corresponding MLs. The XAS data deﬁnitelyprovide important clues to the fact that there are indeedchanges in the magnetic layer thickness or proportions thathave occurred due to the spacer layer. This information is alsosigniﬁcant enough to proceed further with the investigation. 2. L-edge spectroscopy L-edge spectra from the Co /CoO multilayers as measured at RT in the remanent state are shown in Fig. 16(a) .A s common for transition metals and transmission metal oxides,the spectra are dominated by two peaks separated by a fewmilli-electron-volts. The two main peaks L 2,3arise from the spin-orbit interaction of the 2 pcore shell. The total intensity of the peaks is proportional to the number of empty 3 dvalence states above the Fermi level. While spectra from a metal showtypically two broad peaks reﬂecting the width of the emptydbands, oxides exhibit a multiplet structure arising from the spin and orbital momentum of the 3 dvalence holes in the electronic ground state and from the coupled states formed 014431-14CHANGE IN INTERFACE MAGNETISM OF AN EXCHANGE- ... PHYSICAL REVIEW B 87, 014431 (2013) (a) 30 (b) B I(L3) I(L2)+ I(L3)B=-------------- FIG. 16. (Color online) (a) Plot of the L-edge XAS for the multilayers. (b) Ratios of the area under the absorption peaks L3 andL2. The lines are guide to the eye. after x-ray absorption between the 3 dvalence holes and the 2pcore holes.47 In our MLs we observe two broad peaks with broadened bases, a typical signature of the localized character of the3dstates. We do not observe any ﬁne structure (negating hybridization of the dorbitals with the sorbital of the Au spacer). We neither observe a shift in the absorption energiesnor a change in separation of the peaks that amounts to 15.3 eVfor the MLs investigated. Therefore the amount of core-holescreening by delocalized valence electrons is negligible. 48 The branching ratios B=I(L3)/[I(L2)+I(L3)] (see Ref. 49) are calculated from the area under the L2andL3 peaks as shown in Fig. 16(b) using the IFEFFIT package.42The advantage of using Bis the minimization of the effects of line broadening by the ﬁnite lifetime of the transitions andexperimental broadening contributions. The changes in Bbeing a measure for the amplitude of the angular part of the spin-orbit operator showing a 12% decreasewith increasing spacer layer thickness which further partiallycorroborates with the increasing Co valency or changes ofthe local magnetic moment. Theoretical and experimentalstudies have shown that the ratio of a 3 dtransition metal atom generally increases with its magnetic moment. However,a clear relation, or a sum rule, relating these two quantities has not been established, and the absolute value of the magneticmoment cannot be obtained directly from this ratio. 50Recently, XMCD spectra from Co-Au multilayers have demonstratedthat changes in the neighborhood of the Co atoms can suppressits magnetism due to impurities and interdiffusion. 51The increased thickness of the Au layers might have lead to anenhanced concentration of Au impurities around the Co atoms.The reduced magnetization of Co with the increasing thicknessof the spacer layer is in agreement with the PPMS [seeFigs. 3(a) and3(b)] and PNR measurements [see Figs. 12(a) , 12(b) , and 13]. E. Interface magnetism Furthermore, we discuss the various possibilities that can be responsible for the observed magnetic behavior with spacerthickness such as (a) exchange coupling across the spacer,(b) interfacial dilution, and (c) perpendicular anisotropy. a. Exchange coupling. In the present scenario, the unusual thickness dependence of the spacer layer on exchange biastherefore raises the question of whether there is RKKY typecoupling or magnetostatic coupling or any other mechanismthat might determine the enhanced exchange coupling, besidesthe variations in relative proportion of AF-FM. In magneticmultilayers, magnetic moments can be looked upon as im-mersed in a sea of the conduction electrons of the spacerlayer which gives rise to damped long-range oscillation ofthe interlayer exchange coupling as a function of the spacer thickness. 52,53The magnetostatic interaction between two FM ﬁlms, separated by a nonmagnetic spacer, is caused by the strayﬁelds ( magnetostatic coupling ) with antiparallel magnetiza- tions. However, following N ´eel’s theory (in presence of a cor- related roughness), an interlayer coupling can be induced thatis ferromagnetic in nature and decreases exponentially. 54In the possible coupling mechanisms discussed above, the couplingstrength deﬁnitively dampens down at around 2.0–5.0 nm ofspacer thickness, again ruling out such a possibility in our case. Possible long-range interaction across a spacer layer is common in magnetic multilayers. For example, Gierlingset al. 13investigated the effect of a Au spacer across a similar Co-CoO system where they could ﬁnd induced magneticmoments in Au by local s-dhybridization with the dband of the nearest Co atoms. A canted magnetic structure in theﬁlm plane, thus realized at the interface across a Au spacerof/lessorequalslant1.0 nm, reduces the exchange coupling. Very recently, Meng et al. 17and Valev et al.55have reported an interlayer coupling between CoO and Fe separated by at least 4.0 nm(10 monolayers) of Ag and 3.5 nm of Cu spacer layer,respectively. The pinning centers deep inside the AF layer,contributing to the exchange bias ﬁeld are also indicativeof the long range aspect of it. 56Note that in our case, the thickness of the Au spacers are at least an order of magnitudelarger. Moreover, PNR shows that there is no in-plane cantedspin structure. This may rule out spin-canting due to possiblepin-hole formation between the two consecutive Co-CoOlayers on either side of the Au layer. b. Interfacial dilution. In this regard, it is natural to think of interfacial dilution for a magnetic/nonmagnetic interface.In an earlier case, a decrease of the thermal stability of the 014431-15AMITESH PAUL et al. PHYSICAL REVIEW B 87, 014431 (2013) AF was conjectured.14It was shown that the bias ﬁeld can be slightly increased (only by around 100 Oe) by Cu dilution inIrMn based exchange biased system. Moreover, such a dilutioneffect affects the blocking temperature of the system as well. Inthe present case, we can rule out any dilution effect as we do notobserve any signiﬁcant change in the blocking temperaturesfor our multilayers. We may also rule out diffusion of Auimpurities into the Co and CoO layers as Co and Au areimmiscible (positive heat of mixing) at or below RT and wesee no magnetic dead layers by PNR. c. Perpendicular magnetic anisotropy. In ultrathin ﬁlms, perpendicular magnetic anisotropy (PMA) effects may com-monly result from interface and/or magnetoelastic effects apartfrom more intrinsic magnetocrystalline anisotropy. Magnetoe-lasticity is dominant with decreasing ﬁlm thickness which canmake PMA restricted to low thicknesses (typically 1 nm).St¨ohr has shown that the orbital moment on a Co atom becomes anisotropic (below 10 monolayers or ∼2.5 nm) through quenching effects by the anisotropic ligand ﬁeldsof the neighboring Au atoms (which can be as thick as28.0 nm). 47Recently, Paul et al. have also observed strong PMA in [Co(2.0 nm) /Au(2.0 nm)] 32multilayers. Note that the Co layer thicknesses were restricted to 8 monolayers insteadof usual range of 1–2 monolayers. 57 The thickness of the Co layers in the present case are 11.0 nm ( ∼44 ML) thus one should not expect any PMA in this range. However, for the Co_50 ML, as comparedto the Co_25 ML, we ﬁnd ≈13% decrease in the magnetic moment (when measured along the sample plane, either byPPMS, PNR, or L-edge spectroscopy). The reduced magnetic moment indicates that there can be a relative increase in PMA.Schematics of possible scenarios for the magnetic structures ofthe layers are shown in Fig. 17. As an example, we have shown the cases for two the MLs namely, Co_25 and Co_50 at theFM- interfaces. From the K-edge spectroscopy, we observe the following. (i) For FM(Co)-Au interfaces: ≈30% decrease in the proportion of Co thickness (increase in the Au layerthickness can affect the interface with the Co layer). This willreduce the effective magnetic Co thickness from 11.0 nm toaround 8.0 nm. Such a decrease can be due to canting ofthe Co moments at the interfaces. This would then obviouslyincrease the probability of PMA. (ii) For AF(CoO +Co 3O4)- FIG. 17. (Color online) The schematics of the layer structures at the Co-Au and Au-CoO interfaces of the two MLs Co_25 and Co_50. The arrows represent the out-of-plane FM spins (red) and the in-plane FM spins (green).Au interfaces: ≈4% increase of the Co 3O4proportion within the AF layers. This can, on the other hand, increase the numberof uncompensated spins within the AF. On the other hand, theycan have increased un-oxidized proportions of Co (Co xOy). Since the absolute thickness of such an unoxidized layer is verysmall (below 10 monolayers), with increased possibility of Auat its neighborhood, the unoxidized Co magnetic momentscan turn out of plane. Thus one can argue that the observedincrease in the bias ﬁeld can be attributed to the canting ofthe Co moments at the Co-Au interface (effective reductionin the FM layer thickness) and/or increased proportion ofuncompensated moments within the CoO layers. Presumably, the uncompensated AF moments within the CoO are located at approximately 1–2 nm from the interfaceand a canting of those spins would have reduced the bias ﬁeld asa result of net reduction in the number of uncompensated spins.Similarly, with an induced magnetism in Au at the Co-Au(FM-Au) interface, there would have been a decrease in thebias ﬁeld (effective increase of the FM layer). On the otherhand, an induced magnetism within the AF, adjacent to a FM(AF-FM interface), can only reduce the bias ﬁeld rather thanincreasing it. 58However, the effect of an induced magnetism at the CoO-Au (AF-Au interface) interface would have beeninteresting to investigate. Thus we can rule out any inducedmoment either at the FM-Au or AF-FM interfaces or cantingof the AF moments. Paul et al. have recently reported that with the application of a perpendicular cooling ﬁeld (perpendicular to the ﬁlmplane) one can induce an exchange bias in Co /CoO/Au MLs which is directed out of plane. 20This unconventional exchange biasing was possible mainly due to the difference in uniaxialanisotropy energies of the Co ( ∼5×10 5erg cm−3)59and the CoO ( ∼25×107erg cm−3)60layer apart from the possible intrinsic tendency of PMA at the Co-Au interfaces. In thepresent context, we performed similar measurements on ourMLs. An increasing tendency of induced bias in the out-of-plane direction would essentially conﬁrm the increasing out-of-plane canted proportion of the Co moments, with increasingspacer thickness. In other words, the larger the number of out-of-plane uncompensated moments is the larger the reductionof the moments in the ﬁlm-plane will be. Figure 18shows the longitudinal magnetization measured at 10 K for an out-of-plane cooling ﬁeld ( H FC=+4.0k O e ) FIG. 18. (Color online) Magnetization loops for the substrate, Co_1, Co_25, and Co_50 MLs for out-of-plane cooling ﬁeld. 014431-16CHANGE IN INTERFACE MAGNETISM OF AN EXCHANGE- ... PHYSICAL REVIEW B 87, 014431 (2013) for the Co_1, Co_25, and C_50 MLs. The signal can be compared with the background signal from a Si substratemeasured with the same conditions in the PPMS, showinga typical linear paramagnetic slope. One may note that ahysteresis (opening up of the loop) is seen only for the Co_25and Co_50 MLs and not for the Co_1 ML. This is expectedsince the Co_1 ML does not contain any Au spacer layer.This obviously indicates the increased tendency of PMA withincreased spacer layer thickness. Out-of-plane canting of theCo moments have resulted in the net reduction in the momentsin the ﬁlm-plane. Additionally, we ﬁnd a distinct but smallvertical shift of the hysteresis loops for all of our MLs (and notfor the substrate). Vertical shifts are related to uncompensatedmoments at the FM-AF interfaces or noncollinear magneticstructure at interfaces. 61Depending upon their origin (which, however, remains unclear) that can be in the AF and/orin the FM, they can be, in principle, correlated or uncor-related to the H ebvalues. Thus nonmagnetic spacers are shown to affect the interface magnetism without changing themicrostructure. Canting of the Co moments or induced magnetism of the Au layer can be looked upon as due to s-dhybridization. However, whether the hybridization is at the Co-Au interfaceor at the CoO-Au interface is beyond the scope of the availabletechniques. We suggest that a deeper insight into the impactof the AF /Au or FM /Au interface magnetism including the effects of roughness and interdiffusion on the exchange biasphenomena has to be undertaken for a better understanding. IV . CONCLUSION We observe a systematic increase in the exchange bias ﬁelds and the coercive ﬁelds with increasing thicknesses of the Aulayer that are immersed between the Co /CoO bilayers which may be an important route to improve future devices using theexchange bias. The structural evolution of the ferromagneticgrains as seen by XTEM measurements is interrupted bygrowing Au layers of appropriate thickness. The grains in the Au layers are of the order of the Au layer thickness. TheAu layer decouples the structural and magnetic properties ofthe magnetic bilayers thus inhibiting the evolution of domainsacross the heterostructure. Evidence of this is provided byoff-specular polarized neutron scattering. Interestingly, themagnetic moment per atom in the FM layers is seen to decreasewith increasing thickness of the Au spacer layer. This isconﬁrmed by PPMS and PNR measurements. Subloop shiftsof the hysteresis around the blocking temperature indicates adifferent initial AF-FM domain conﬁguration for samples withAu spacers (as compared to that without spacers). The increase in the bias ﬁeld, to some extent, accounts for the relative proportions of the FM and AF species as inferredfrom the XANES and the XAS measurements. However, alarger extent of the increment is owed to reduced magneticmoment of the Co layer as inferred from the magnetometryand PNR measurements. Such a reduction is plausibly owedto the out-of-plane orientation tendencies of the Co momentsat the Co-Au interfaces. By performing perpendicular ﬁeldcooling, we could demonstrate an increasing tendency of theCo moments to orient out-of-plane which effectively explainsthe in-plane decrease of the magnetic moment with increasedAu spacer thickness. Perpendicular ﬁeld cooling is thus seenas a novel way to characterize the uncompensated spins at theinterface of such exchange coupled systems. 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PhysRevB.89.235311.pdf	PHYSICAL REVIEW B 89, 235311 (2014) Atomistic modeling of coupled electron-phonon transport in nanowire transistors Reto Rhynerand Mathieu Luisier† Integrated Systems Laboratory, ETH Z ¨urich, Gloriastr. 35, 8092 Z ¨urich, Switzerland (Received 18 March 2014; revised manuscript received 6 May 2014; published 13 June 2014) Self-heating effects are investigated in ultrascaled gate-all-around silicon nanowire ﬁeld-effect transistors (NWFETs) using a full-band and atomistic quantum transport simulator where electron and phonon transport arefully coupled. The nonequilibrium Green’s function formalism is used for that purpose, within a nearest-neighborsp 3d5s∗tight-binding basis for electrons and a modiﬁed valence-force-ﬁeld model for phonons. Electron-phonon and phonon-electron interactions are taken into account through speciﬁc scattering self-energies treated in theself-consistent Born approximation. The electron and phonon systems are driven out of equilibrium; energy isexchanged between them while the total energy current remains conserved. This gives rise to local variationsof the lattice temperature and the formation of hot spots. The resulting self-heating effects strongly increasethe electron-phonon scattering strength and lead to a signiﬁcant reduction of the ON-current in the consideredultrascaled Si NWFET with a diameter of 3 nm and a length of 45 nm. At the same time, the lattice temperatureexhibits a maximum close to the drain contact of the transistor. DOI: 10.1103/PhysRevB.89.235311 PACS number(s): 73 .63.−b,72.10.−d,63.22.−m,63.20.kd I. INTRODUCTION The continued miniaturization of the transistor dimensions according to Moore’s scaling law [ 1]h a sl e dt oa ni m p r e s s i v e evolution of the electronic device functionalities. By reducingthe size of the transistors a signiﬁcant improvement of theirelectrical performance is obtained. On the negative side, sincetheir supply voltage has stopped scaling as fast as theirdimensions, heat dissipation has kept increasing from onegeneration to the other [ 2]. Consequently, the power density of integrated circuits (ICs) is dangerously approaching the150 W /cm 2limit up to which air can be used to cool the device temperature [ 3]. The recent replacement of the two- dimensional planar Si metal-oxide-semiconductor ﬁeld-effecttransistors (MOSFETs) by three-dimensional FinFETs [ 4] has momentarily stabilized the increase in heat dissipation and ICpower consumption. FinFETs indeed show a decrease of theirpassive power component as compared to two-dimensional(2D) MOSFETs due to the better electrostatic control providedby their triple gate conﬁguration. In the future FinFETs might evolve towards ultra- scaled gate-all-around nanowire ﬁeld-effect transistors (GAANWFETs) [ 5–10]. Because of the superior electrostatic control of a surrounding gate the electrical performance of GAANWFETs outperforms that of FinFETs. While the static anddynamic aspects of NWFETs have received a lot of attention,their electrothermal properties have not been thoroughlyinvestigated so far, although they might be the limiting factorin such devices [ 11]! Nanowires exhibit a reduced thermal conductivity as compared to bulk structures [ 12–14], which represents a fertile ground for the formation of localized hotspots and self-heating effects. The narrow dimensions of ultrascaled NWFETs make it difﬁcult to measure an internal temperature distribution ora power dissipation proﬁle [ 15]. Hence, it is challenging to experimentally investigate the inﬂuence of self-heating and rhyner@iis.ee.ethz.ch †mluisier@iis.ee.ethz.chhot spots on the characteristics of future nanotransistors. Asa technology enabler physics-based device simulations canbe used to support the experimental work and compute theelectrothermal properties of a given structure. However, theselected simulation approach must go beyond the compu-tationally efﬁcient classical drift-diffusion (DD) [ 16] model or the semiclassical Boltzmann transport equation (BTE)[11,17]. It must correctly cover all the quantum mechanical phenomena present at the nanoscale, especially tunneling,energy quantization, and geometrical conﬁnement. To accountfor these effects and treat thermal transport at the phononlevel a full-band and atomistic device simulator capable ofhandling both electrons and phonons is needed. There havebeen some attempts to combine electron and phonon transportin an atomistic basis, but they have been restricted to molecularjunctions with a small number of atoms [ 18,19]. A fully coupled electron and phonon transport approach based on the nonequilibrium Green’s function formalism(NEGF) is therefore proposed here. It can deal with three-dimensional nanowire transistors composed of several thou-sand atoms [ 20]. The electron properties are expressed in asp 3d5s∗tight-binding basis while the phonon ones are described in a modiﬁed valence-force-ﬁeld model. The NEGFformalism provides a natural treatment of the electron-phononand phonon-electron interactions through scattering self-energies solved in the self-consistent Born approximation.These scattering self-energies drive both the electron andphonon populations out of equilibrium and allow for theconsideration of coupled electrothermal transport phenomenasuch as self-heating or localized hot spots. The resultingimprovement in the simulation accuracy can be compared tothat brought by the extension of the drift-diffusion approachwith an energy-balance and electrothermal model [ 21]. As an application, self-heating effects are investigated in a Si GAA NWFET with a diameter of 3 nm, a totallength of 45 nm, and composed of more than 15 000 atoms.These results are compared to the case where the electronsare coupled to equilibrium phonons at room temperature.It is shown that for reasonably high electron currents thepower dissipated by phonon emission leads to a signiﬁcant 1098-0121/2014/89(23)/235311(12) 235311-1 ©2014 American Physical SocietyRETO RHYNER AND MATHIEU LUISIER PHYSICAL REVIEW B 89, 235311 (2014) increase of the phonon population through the entire device. Furthermore the nonequilibrium phonon distribution causes astrong enhancement of the electron-phonon coupling strengthand therefore a noticeable reduction of the electron current. Tobetter quantify the self-heating, an effective lattice temperatureis introduced and calculated in the selected NW structure. Itsspatial distribution demonstrates the formation of hot spotsthat are clearly related to the shape of the phonon population. The paper is organized as follows: In Sec. II, the simulation approach is introduced, starting from the electron and phononNEGF equations, their interactions, and the calculation ofenergy currents. Details about the numerical implementationare given in the Appendix. In Sec. IIIthe fully coupled electron-phonon transport model is applied to a Si GAANWFET where self-heating effects are investigated and aneffective lattice temperature extracted. The paper is concludedin Sec. IVand an outlook on possible future works is proposed. II. THEORY Electron and phonon transport are treated in the framework of the NEGF formalism under steady-state conditions, i.e., allthe Green’s functions are solved in the energy (frequency)domain and not as a function of the time. The targetedstructures are Si circular nanowires surrounded by an oxidelayer that does not take part in the transport calculations. Theelectrons and phonons can only enter or escape the simulationdomain at both ends of the nanowire and not at its surface. Inparticular, thermal losses through the oxide are not included. A. Electron model The NEGF equations for electrons are expressed in a nearest-neighbor tight-binding basis where the lesser ( G<), greater ( G>), and retarded ( GR) Green’s functions have the following form in a nanowire structure [ 22]: /summationdisplay l((E−V(Rm))δlm−Hml−/Sigma1RB ml(E)−/Sigma1RS ml(E))GR ln(E)=δmn, (1) G≷ nm(E)=/summationdisplay l1l2GR nl1(E)/parenleftbig /Sigma1≷B l1l2(E)+/Sigma1≷S l1l2(E)/parenrightbig GR† ml2(E),(2) /Sigma1R nm(E)=1 2(/Sigma1> nm(E)−/Sigma1< nm(E)) +iP/integraldisplaydE/prime 2π/Sigma1> nm(E/prime)−/Sigma1< nm(E/prime) E−E/prime. (3) In Eq. ( 3),Pdenotes the Cauchy principal integral value. The indices n,m,l,l 1, and l2run over all atomic positions. The matrices E(diagonal, injection energy), V(Rn) (diagonal, self-consistent electrostatic potential at position Rn),Hmn (tight-binding matrix elements, on-site energy if m=n, nearest-neighbor coupling between atom mandnotherwise), /Sigma1B mn(E) (electron boundary self-energy, different from 0 only if atoms mandnare directly connected to the semi-inﬁnite leads, computed as in Ref. [ 23]),/Sigma1S mn(E) (electron-phonon scattering self-energy between atoms mandnmodeling the coupling to the phonon system), and Gnl(E) (electron Green’s functions between atoms nandl) are of size Norb×Norb, whereNorbis the number of orbitals of the tight-binding model. In this work a sp3d5s∗basis without spin-orbit coupling is used to describe the Si properties [ 24], i.e., Norb=10. The deﬁnition and the interpretation of the tight-binding Hamiltonian blocksH mncan be found in Ref. [ 25]. In this approach, each atom is treated individually so that the size of the linear systemof equations in Eqs. ( 1) and ( 2) is equal to N A×Norb,NA being the total number of atoms in the Si channel. Hard wall boundary conditions are applied at the nanowire surface [ 26]. B. Phonon (thermal) model For the phonons the NEGF equations look as follows [ 27]: /summationdisplay l/parenleftbig Mmω2δlm−/Phi1ml−/Pi1RB ml(ω)−/Pi1RS ml(ω)/parenrightbig DR ln(ω)=δmn, (4) D≷ nm(ω)=/summationdisplay l1l2DR nl1(ω)/parenleftbig /Pi1≷B l1l2(ω)+/Pi1≷S l1l2(ω)/parenrightbig DR† ml2(ω), (5) /Pi1R nm(ω)=1 2(/Pi1> nm(ω)−/Pi1< nm(ω)) +iP/integraldisplaydω/prime 2π/Pi1> nm(ω/prime)−/Pi1< nm(ω/prime) ω−ω/prime. (6) Similar to the electron case the indices n,m,l,l 1, andl2run over all the atomic positions. The matrices ω2(diagonal, ωis the phonon frequency), /Phi1mn(dynamical matrix block correspond- ing to the second derivative of the harmonic potential withrespect to mandn),/Pi1 B mn(ω) (phonon boundary self-energy between atom mandn, only different from 0 when mandnare directly connected to the semi-inﬁnite leads, computed withthe same “shift-and-invert” scheme as the electron boundaryself-energy [ 23], except that the structure of the involved matrices changes due to the presence of beyond nearest-neighbor connections), /Pi1 S mn(ω) (phonon-electron scattering self-energy between atoms mandndescribing the coupling to the electron system), and Dnl(ω) (phonon Green’s functions between atoms nandl) are of size 3 ×3 where 3 is the number of degrees of freedom per atom, i.e., the number of directionsalong which atoms can oscillate ( x,y, andz). The entries of the dynamical matrix /Phi1 mnare approximated as /Phi1ij mn=d2Vharm dRimdRj n, (7) the second derivative of the valence-force-ﬁeld (VFF) har- monic potential energy Vharmwith respect to the ithandjth components ( x,y, andz) of the atom positions RmandRn. For an accurate reproduction of the phonon band structure ofgroup IV semiconductors, the VFF potential energy V harmmust include at least four bond interactions. More information aboutthe construction of the dynamical matrix and the harmonicforce constants of Si can be found in Refs. [ 28,29]. Here again, the N Aatoms composing the simulated structures have an individual treatment. The size of the system to be solvedin Eqs. ( 4) and ( 5) is therefore 3 ×N A. The Si atoms at the nanowire surface can freely oscillate. 235311-2ATOMISTIC MODELING OF COUPLED ELECTRON-PHONON . . . PHYSICAL REVIEW B 89, 235311 (2014) C. Electrothermal coupling To derive the coupling between the electron and phonon population it is convenient to start from the total Hamiltonianoperator in the second quantization, ˆH(t)=/summationdisplay nm/summationdisplay σ1σ2Hσ1σ2 mnˆc† mσ 1(t)ˆcnσ2(t) +1 2/summationdisplay n/summationdisplay iMnˆ˙ui n(t)ˆ˙ui n(t) +1 2/summationdisplay nm/summationdisplay ij/Phi1ij mnˆui m(t)ˆuj n(t) +/summationdisplay nm/summationdisplay σ1σ2/summationdisplay i∇iHσ1σ2 mnˆc† mσ 1(t)ˆcnσ2(t)/parenleftbigˆui n(t)−ˆui m(t)/parenrightbig . (8) In Eq. ( 8) the indices i,j, and σrefer to the real space directions ( x,y, and z) and the atomic orbitals ( s,p,d, ands∗), respectively. The operator ˆc† mσ 1(t)(ˆcmσ 1(t)) creates (annihilates) an electron with orbital σ1at position Rmand at time t, while ˆui m(t) is the phonon quantized displacement operator along the direction iat time tand at Rmwith respect to the equilibrium atom position. The ﬁrst term onthe right-hand side of Eq. ( 8) is directly included in the tight-binding block H nmin Eq. ( 1). The second (phonon- kinetic) and third (phonon-harmonic) terms appear in thedynamical matrix block /Phi1 nmin Eq. ( 4). The last term in Eq. ( 8) connects the electron and phonon populations and is treated as a perturbation that is cast into the electron-phonon(/Sigma1) and phonon-electron ( /Pi1) scattering self-energies. The presence of lattice vibrations where atoms oscillate aroundtheir equilibrium position R 0 m→Rm(t)=R0 m+um(t) with the displacement vector um(t) induces the electron-phonon interactions [ 30,31]. To account for the atom oscillations the tight-binding Hamiltonian matrix Hnmis expanded in a Taylor series around the equilibrium bond vector ( R0 n−R0 m)t ot h e lowest order in the oscillations un(t)−um(t): Hmn≈H0 mn+/summationdisplay iδHmn δ(R0 n,i−R0 m,i)/parenleftbig ui n(t)−ui m(t)/parenrightbig ≈H0 mn+/summationdisplay i∇iHmn/parenleftbig ui n(t)−ui m(t)/parenrightbig . (9) The transformation of the second term on the right-hand-side in Eq. ( 9) into the second quantization leads to the last operator in Eq. ( 8), representing the electron-phonon coupling. It still remains to determine an expression for the electron-phononand phonon-electron scattering self-energies in Eqs. ( 1), (2), (4), and ( 5), respectively. To do that an equation of motion is derived for the time-dependent electron Green’s functionG σ1σ2nm(t,t/prime), which is proportional to the expectation value /angbracketleftˆcnσ1(t)ˆc† mσ 2(t/prime)/angbracketright, and for the time-dependent phonon Green’s function Dij nm(t,t/prime), which is proportional to /angbracketleftˆui n(t)ˆuj m(t/prime)/angbracketright. The Hamiltonian operator ˆH(t)i nE q .( 8) is used for that purpose. As a next step the Wick’s decomposition technique[32] is applied to truncate the arising inﬁnite hierarchy of the equations of motion, the expectation value of two operatorsdepending on three operators whose expectation value dependson four operators, and so forth. Langreth’s theorem [ 33] is recalled to replace the general Green’s functions witharguments on a complex time contour by real-time retarded,lesser, and greater Green’s functions. Finally, after Fouriertransforming the time difference t−t /prime, the steady-state form of the electron-phonon and phonon-electron scattering self-energy is obtained. For a detailed description of the derivation,see Appendix A. The greater or lesser components are deﬁned as /Sigma1 ≷σ1σ2 nm (E)=i/summationdisplay l1l2/summationdisplay ij/summationdisplay σ3σ4/integraldisplay∞ −∞d(/planckover2pi1ω) 2π∇iHσ1σ3 nl1 ×G≷σ3σ4 l1l2(E−/planckover2pi1ω)∇jHσ4σ2 l2m/parenleftbig D≷ij l1m(ω) −D≷ij l1l2(ω)−D≷ij nm(ω)+D≷ij nl2(ω)/parenrightbig , (10) /Pi1≷ij nm(ω)=2spin·i/summationdisplay l3l4/summationdisplay σ1σ2σ3σ4/integraldisplay∞ −∞dE 2π/parenleftbig ∇iHσ3σ1 l3n ×G≷σ1σ4 nl4(/planckover2pi1ω+E)∇jHσ4σ2 l4mG≶σ2σ3 ml3(E) −∇iHσ3σ1 l3nG≷σ1σ2 nm (/planckover2pi1ω+E)∇jHσ2σ4 ml4G≶σ4σ3 l4l3(E) −∇iHσ1σ3 nl3G≷σ3σ4 l3l4(/planckover2pi1ω+E)∇jHσ4σ2 l4mG≶σ2σ1 mn (E) +∇iHσ1σ3 nl3G≷σ3σ2 l3m(/planckover2pi1ω+E)∇jHσ2σ4 ml4G≶σ4σ1 l4n(E)/parenrightbig . (11) Because spin-orbit coupling is not considered in the present calculations spin degeneracy is modeled via a factor twolabeled 2 spin. The lesser self-energies /Sigma1<and/Pi1<are related to in-scattering processes, the greater ones /Sigma1>and/Pi1>to out-scattering [ 34]. More precisely, the lesser electron-phonon self-energy /Sigma1<(E) describes for positive phonon energies (/planckover2pi1ω> 0) the in-scattering of an electron from an occupied state G<(E−/planckover2pi1ω) at energy E−/planckover2pi1ωinto an empty state at E. This happens through the absorption of an available phononwith energy /planckover2pi1ωwhose occupancy is given by D <(ω). In the case /planckover2pi1ω< 0 since D<ij nm(−ω)=D>ji mn(ω) it follows that an electron in the occupied state G<(E+\|/planckover2pi1ω\|)a tE+\|/planckover2pi1ω\|is transferred to Eby a phonon emission. The probability of such transition depends on the availability of an empty phononstate at frequency ω, which is given by D >(ω). For the greater electron-phonon self-energy /Sigma1>(E) the situation is reversed, a positive (negative) phonon frequency ωcorresponding to the out-scattering of an electron with energy Einto a state with energy E−/planckover2pi1ω(E+\|/planckover2pi1ω\|) through phonon emission (absorption). The phonon in- and out-scattering processes described by /Pi1<(ω) and/Pi1>(ω) behave slightly differently. An electron transition from an occupied state at energy E,G<(E), to an empty state at E+/planckover2pi1ω,G>(E+/planckover2pi1ω), requires the absorption of a phonon with energy /planckover2pi1ωand contributes to a decrease of the phonon population at this frequency (out-scattering).In-scattering involves an electron transition from E+/planckover2pi1ωto Ethrough phonon emission, locally increasing the phonon count. The scattering self-energies /Sigma1(E) and/Pi1(ω) couple the electron and phonon baths because /Sigma1(E) depends on D(ω) and/Pi1(ω)o n G(E). It clearly appears that the absorption 235311-3RETO RHYNER AND MATHIEU LUISIER PHYSICAL REVIEW B 89, 235311 (2014) or emission of a phonon does not only affect the electron population, but also the phonon one, which is not the case ifthe/Pi1self-energies are ignored, as in most electron-phonon scattering treatments, e.g., Refs. [ 35–38]. It is also important to realize that the energy that is lost by the electrons doesnot vanish, but is captured by the phonons so that energyconservation is ensured. A careful veriﬁcation of this propertyis critical for the accuracy of the results. Equations ( 1)–(6), (10), and ( 11) must be solved iteratively until convergence between the Green’s functions and thescattering self-energies is reached. This process is calledself-consistent Born approximation. There is a second self-consistent loop between the Schr ¨odinger and Poisson equa- tions. Once convergence is achieved, the charge and currentdensities as well as the distribution of the phonon populationare calculated as in Refs. [ 22,27]. Furthermore, the electron and phonon energy currents ﬂowing between the s thandsth+1 slab (unit cell) of the simulated structures can be computed as Iel,s→s+1=2spin /planckover2pi1/summationdisplay n∈s/summationdisplay m∈s+1/summationdisplay σ1σ2/integraldisplay∞ −∞dE 2πE/parenleftbig Hσ1σ2 nmG<σ 2σ1 mn (E) −G<σ 1σ2 nm (E)Hσ2σ1 mn/parenrightbig , (12) and Iph,s→s+1=/planckover2pi1/summationdisplay n∈s/summationdisplay m∈s+1/summationdisplay ij/integraldisplay∞ 0dω 2πω/parenleftbig /Phi1ij nmD<ji mn(ω) −D<ij nm(ω)/Phi1ji mn/parenrightbig . (13) In Eqs. ( 12) and ( 13), the atom position Rnis located in thesthslab and Rmin the sth+1 one. A slab contains an ensemble of Nconsecutive atomic layers along the direction of the current ﬂow. For example, N=4 for transport along the /angbracketleft100/angbracketrightcrystal axis or N=6f o r/angbracketleft111/angbracketright. The total energy current must be conserved and constant through the entire device sothatI el,s→s+1+Iph,s→s+1remains the same for all possible s. D. Simpliﬁcations and implementation As already mentioned in Refs. [ 22,27] the electron-phonon (/Sigma1) and the phonon-electron ( /Pi1) self-energies in Eqs. ( 10) and ( 11) are exact, but difﬁcult to implement from a numerical point of view. To investigate fully coupled electron-phonontransport in realistic nanowire structures some simpliﬁcationsmust be applied to the calculation of /Sigma1and/Pi1. According to the arguments in Ref. [ 22] the electron- phonon scattering self-energies /Sigma1 nm(E) are limited to on-site interactions only, i.e., n=m, but they remain blocks of size Norb×Norb, /Sigma1≷σ1σ2 nn (E)=i/summationdisplay l∈NN(n)/summationdisplay ij/summationdisplay σ3σ4/integraldisplay∞ −∞d(/planckover2pi1ω) 2π∇iHσ1σ3 nl ×G≷σ3σ4 ll (E−/planckover2pi1ω)∇jHσ4σ2 ln/parenleftbig D≷ij ln(ω) −D≷ij ll(ω)−D≷ij nn(ω)+D≷ij nl(ω)/parenrightbig . (14) Reducing Eq. ( 14) to its simplest expression means omit- ting the nondiagonal phonon Green’s function Dnl(ω) and Dln(ω). However, ignoring Dnl(ω) and Dln(ω) leads to anunderestimation of the electron-phonon coupling strength that should be avoided. Standard recursive Green’s function (RGF) algorithms [ 39] are fully capable of producing Dnl(ω) and Dln(ω) where Rl andRnare nearest-neighbor positions, but the inclusion of these terms complicates the situation. The additional difﬁcultycomes from the fact that to ensure energy conservation, besidethe diagonal phonon-electron self-energies, /Pi1 ≷ij nn(ω)=−i/summationdisplay l/summationdisplay σ1σ2σ3σ4/integraldisplay∞ −∞dE 2π/parenleftbig ∇iHσ3σ1 ln ×G≷σ1σ2 nn (/planckover2pi1ω+E)∇jHσ2σ4 nlG≶σ4σ3 ll (E) +∇iHσ1σ3 nlG≷σ3σ4 ll (/planckover2pi1ω+E)∇jHσ4σ2 lnG≶σ2σ1 nn (E)/parenrightbig , (15) also the nondiagonal phonon-electron self-energies /Pi1nl(ω) must be taken into account, /Pi1≷ij nl(ω)=i/summationdisplay σ1σ2σ3σ4/integraldisplay∞ −∞dE 2π/parenleftbig ∇iHσ3σ1 ln ×G≷σ1σ2 nn (/planckover2pi1ω+E)∇jHσ2σ4 nlG≶σ4σ3 ll (E) +∇iHσ1σ3 nlG≷σ3σ4 ll (/planckover2pi1ω+E)∇jHσ4σ2 lnG≶σ2σ1 nn (E)/parenrightbig . (16) In Eq. ( 16), it is sufﬁcient to consider the case where lis a nearest neighbor of n. To calculate Dnl(ω) as needed in Eq. ( 14), the RGF algorithm used to solve Eqs. ( 4) and ( 5)m u s t be extended to produce not only diagonal, but also nondiagonalentries, as described in Ref. [ 40]. A closer look at the parallel implementation of the NEGF equations is given in AppendixB. Note ﬁnally that in Eqs. ( 3) and ( 6), the principal integral term has been neglected in all the calculations reported in thispaper. It contributes only to an energy renormalization, but notto relaxation or phase breaking and previous studies have alsoshown that this simpliﬁcation does not signiﬁcantly affect thedevice current [ 41,42]. III. RESULTS A. Structure deﬁnition As a simulation example, the Si GAA NWFET schematized in Fig. 1is considered. The diameter of the NW measures 3 nm and it is surrounded by an oxide layer with a thicknesst ox=3n mo fH f O 2characterized by a relative dielectric constant /epsilon1r=20 for an equivalent oxide thickness EOT = 0.58 nm. The gate length Lgis set to 15 nm while the n-doped (donor concentration, ND=1×1020cm−3) source and drain extensions measure 15 nm. The drain current ﬂows alongthexdirection of the NWFETs, which is aligned with the /angbracketleft100/angbracketrightcrystal axis; yandzare directions of conﬁnement. All the simulations are performed at room temperature ( T 0) with a supply voltage VDD=0.6 V . Room temperature means that the electrons (phonons) ﬂowing into the NWFETs fromthe contacts obey an equilibrium Fermi-Dirac (Bose-Einstein)distribution function characterized by a temperature T 0= 300 K. In contrast the outﬂowing electrons and phonons arerearranged due to scattering and the electrostatic potential 235311-4ATOMISTIC MODELING OF COUPLED ELECTRON-PHONON . . . PHYSICAL REVIEW B 89, 235311 (2014) yxz SourceDrainLg diametertoxGate FIG. 1. (Color online) Schematic view of the n-type Si GAA NWFET simulated in this work. The gate length Lgmeasures 15 nm while the source and drain extensions have a length of 15 nm anda donor doping concentration ﬁxed to N D=1×1020cm−3.T h eS i channel has a diameter of 3 nm and is surrounded by HfO 2dielectric layers ( /epsilon1R=20) of thickness tox=3 nm. The transport direction xis aligned with the /angbracketleft100/angbracketrightcrystal axis; yandzare directions of conﬁnement. The total number of Si atoms in this structure is 16 019. and therefore have a different distribution function and temperature. The lowest conduction sub-bands (CB) and the ﬁrst phonon branches of the free standing silicon nanowire are presentedin Figs. 2(a) and 2(b), respectively. Due to geometrical conﬁnement along the yandzdirections the sixfold degenerate CB minimum of bulk Si is splitted into a group of foursub-bands at /Gamma1(/Delta1 4) and two single bands at kx=± 2.08 nm−1 (/Delta12). The transport effective mass is equal to m∗=0.29m0for the/Delta14group and m∗=0.92m0for the /Delta12bands. Quantum conﬁnement does not only increase the band gap value from1.12 to 1 .62 eV , but also the transport effective mass from 0 .2t o 0.29m 0. For the phonons in Fig. 2(b) the group velocity of the purely longitudinal (LA) and transverse (TA) acoustic modes 1.522.5Electron Energy [eV] -3 -2 -1 0 1 2 302468 Reduced Wavevector [kxLx]Phonon Energy [meV](a) (b)Δ4Δ2 LA TA FIG. 2. (Color online) (a) Electron band structure for the same Si nanowire as in Fig. 1. The local minima are indicated with /Delta14and /Delta12where the subscripts deﬁne the degeneracy of the corresponding energy point. (b) Phonon band structure for the same Si nanowire as in (a). The purely longitudinal (LA) and transverse acoustic (TA)branches are indicated in the plot.0.811.21.4 x [nm]Energy [eV] 0 10 20 30 400.020.040.06(a) (b)DrainCB edge phonon emission Gate Sourcepower dissipation generation ratehighest phonon FIG. 3. (Color online) (a) Energy- and position-resolved electron current in the Si GAA NWFET of Fig. 1atVgs=0.6Va n d Vds= 0.6 V . Red indicates high current concentrations and green no current. The dashed blue line refers to the position of the conduction bandedge. (b) Energy- and position-resolved phonon energy current at the same bias conditions as in (a). Red indicates positive currents and blue negative ones. The black dashed line refers to the location with the highest phonon generation rate. is reduced to 4600 m /s and 6300 m /s as compared to the bulk values of 5421 m /s and 8905 m /s, respectively. The inﬂuence of these modiﬁed electrothermal properties is investigated inthe next subsections. In particular, the lower group velocitiesof the acoustic phonon branches make it difﬁcult to evacuatethe dissipated heat from nanowires and cause a strong increasein the lattice temperature. B. Electrothermal Effects To illustrate the electrothermal effects occurring in an ultrascaled Si nanowire transistor, a speciﬁc bias point hasbeen selected with a gate-to-source voltage V gs=0.6 V and a drain-to-source voltage Vds=0.6 V . The standard scatter- ing approach where the electrons interact with equilibriumphonons characterized by a Bose-Einstein distribution and aconstant temperature T 0=300 K, as in Ref. [ 22] is compared to the fully coupled electron and phonon transport modelintroduced in Sec. II. The electrical currents are labeled I d,scatt in the standard case and Id,selfin the fully coupled one. At Vgs=0.6 V and Vds=0.6V ,Id,scatt=9.32μA and Id,self=6.06μA. As explained later, the current reduction comes from self-heating effects. The energy- and position-resolved electron and phonon currents are reported in Fig. 3for the considered bias point with self-heating. In subplot Fig. 3(a), red indicates high current concentrations, green no current. It can be observed thatelectrons lose energy while ﬂowing from the source (left) to thedrain (right) contact. This happens through phonon emission.As a consequence, phonons are created, as shown in subplot(b) where red indicates a positive phonon energy current andblue a negative one. The current magnitude is higher aroundthe bulk optical phonon frequency and around the frequencythat corresponds to the transverse acoustic plateau in bulk Si. 235311-5RETO RHYNER AND MATHIEU LUISIER PHYSICAL REVIEW B 89, 235311 (2014) 8101214 -0.500.5 0 10 20 30 408101214 x [nm]Energy Current [ μW] standard scatt. self-heating (c)(b)(a) Electrons + Phonons (Total)PhononsElectronsenergy loss FIG. 4. (Color online) (a) Electron component of the energy current ﬂowing through the same Si GAA NWFET as in Fig. 3. The standard scattering (equilibrium phonons, solid blue line) and the self-heating (nonequilibrium phonons, green dashed line) casesare shown. (b) Same as in (a) but for the phonon component of the energy current. (c) Same as in (a), but for the total energy current (electron +phonon). In nanowires, the emitted phonon has the same probability to propagate towards the source or drain extension. Hence, thecurrent ﬂow vanishes at the location with the highest phonongeneration rate. There, the formation of a hot spot is expected. By looking at the electron and phonon energy currents, as in Fig. 4, it is conﬁrmed that (i) electrons lose energy between source and drain and (ii) close to the end of thenanowire, there is a position with no phonon current. Thefundamental difference between the standard scattering theoryof Ref. [ 22] and the coupled electron-phonon model presented here becomes also visible in Fig. 4. The power dissipated by electrons can only be captured by the phonons if thelatter are driven out of equilibrium. In this case, the total(electron +phonon) energy current is conserved all along the transport axis of the nanowire, as demonstrated in Fig. 4(c). With equilibrium phonons, the energy lost by the electronssimply vanishes and energy conservation is broken. The totalenergy current is larger on the source than on the drain side. Itis worthwhile noting that the phonon energy current is positiveclose to the end of the device, but negative in the rest of thesimulation domain. Another important difference between equilibrium ( ph eq) and nonequilibrium ( phneq) phonons is shown in Fig. 5where the spatially resolved low frequency ( /planckover2pi1ω< 30 meV , labeled “acoustic”) and high frequency ( /planckover2pi1ω> 30 meV , “optical”) phonon populations are reported as well as the ratio betweenph neqandpheq. It can be seen that the acoustic phonon generation remains almost constant throughout the entirenanowire structure while the emission of optical phonons islarger close to the drain side. At the location of the highestgeneration rate, the optical phonon population increases bya factor of 10 as compared to the standard electron-phononscattering theory. Close to the source, there is a growth bya factor 5 of the number of optical phonons. Since electrons0 20 401234 x [nm]Acoustic Pop. [arb. units]0 20 400123 x [nm]Optical Pop. [arb. units] 0 10 20 30 4004812 x [nm]Pop. Growth Factorstandard scatt. self-heatingstandard scatt. self-heating Popacoustic self-heating/Popacousticst. scatt. Popoptical self-heating/Popopticalst. scatt.(c)(a) (b) FIG. 5. (Color online) (a) Low frequency (or acoustic) and (b) high frequency (or optical) phonon population in the same Si GAA NWFET as before. The blue solid lines refer to the standard scattering case, the green dashed lines to the self-heating case.(c) Growth factor for the optical (green dashed line) and acoustic (blue solid line) phonon populations between self-heating and the standard scattering theory. interact more strongly with such phonons, as explained in Ref. [ 22], a higher optical phonon population causes more scattering events and therefore a reduction of the drain currentfromI d,scatt=9.32μAd o w nt o6 .06μA. The energy- and position-resolved effective electron generation rate Gel−eff(E,Rn), as depicted in Fig. 6, gives a different perspective on the involved physics. It isdeﬁned as G el−eff(E,Rn)=1 /planckover2pi1Tr[G> nn(E)·/Sigma1< nn(E)−/Sigma1> nn(E)· 0 10 20 30 4011.21.41.6 x [nm]Energy [eV] 1.2 1.4 1.6-10-505 Energy [eV]Gel-effsource [arb. units] 1 1.2 1.4 1.6-2-1012 Energy [eV]Gel-effdrain [arb. units]CB edge(a) (c) (b)highest el-current injection source region drain regionphonon absorption phonon emission electron creation electron annihilationelectron creation electron annihilationphonon absorption phonon emission~60meV phonon emission FIG. 6. (Color online) (a) Conduction band edge (solid blue line) of the Si GAA NWFET at Vgs=0.6Va n d Vds=0.6V .T h e source and drain regions as well as the energy location with the highest spectral electron current [see Fig. 3(a), dashed black line] are indicated. (b) Energy-resolved effective electron scattering rate inthe source region ( G source el−eff(E)∼/summationtext n∈source1 /planckover2pi1Tr[G> nn(E)·/Sigma1< nn(E)− /Sigma1> nn(E)·G< nn(E)]). The dashed black line corresponds to the highest spectral electron current as in (a). (c) Same as in (b), but in the drainregion. 235311-6ATOMISTIC MODELING OF COUPLED ELECTRON-PHONON . . . PHYSICAL REVIEW B 89, 235311 (2014) G< nn(E)]. A positive (negative) value indicates that in- (out-) scattering occurs at energy Eand atom position Rn.I n other words, with Gel−eff(E,Rn)<0, electrons with energy Eare annihilated at position Rn, with Gel−eff(E,Rn)>0 electrons with energy Eare created at Rn. In subplot Fig. 6(b),Gel−eff(E,Rn) is shown in the source region (0 nm /lessorequalslantx/lessorequalslant15 nm). At energies 1 .354 eV /lessorequalslantEout/lessorequalslant1.424 eV corresponding to the maximum of the electron ﬂow, as shown in Fig. 3, electrons are annihilated through phonon emission and optical phonon absorption.Hence, in-scattering happens for 1 .27 eV/lessorequalslantE in,1/lessorequalslant1.353 eV (phonon emission) and for 1 .437 eV /lessorequalslantEin,2/lessorequalslant1.595 eV (optical phonon absorption). The momentum of the scattered electrons might change its direction so that the resulting back-scattering effect even-tually reduces the current magnitude [ 43]. As indicated in Fig. 6(a) back-scattering has a higher probability to occur in combination with phonon absorption (50%). In the caseof phonon emission the potential distribution prevents theback-scattered electrons from reaching the source contact andreducing the current magnitude. Unless they absorb a phonon,their only way out of the device is towards the drain side.As mentioned earlier, in the nonequilibrium case, the opticalphonon population grows, thus increasing the in-scatteringprobability in the energy range 1 .437 eV /lessorequalslantE in,2/lessorequalslant1.595 eV . This causes the current reduction between Id,scattandId,self.I n Fig. 6(c), the out-scattering of high energy electrons through phonon emission in the drain region (30 nm /lessorequalslantx/lessorequalslant45 nm) can be clearly identiﬁed. However, because the electrons havepassed the critical length of the transistor [ 43], no further current reduction is induced by these scattering events. C. Effective lattice temperature To further quantify self-heating an effective lattice temperature ( Teff) is introduced. Because the considered NWFET structure is ultrascaled and in a nonequilibriumstate the concept of temperature is questionable especiallyits direct relation to the thermodynamical quantity. Based onexisting calculations of temperatures in molecular junctions[44,45] two approaches are proposed here to evaluate T eff. They are compared to each other to validate the effectivetemperature concept. Both methods are intuitive measures ofan atomistic temperature and coincide with the temperaturein the thermodynamical limit. (1) Population approach ( T pop eff). In the ﬁrst approach the temperature of a Bose-Einstein distribution function isadjusted to reproduce the same total phonon population[N tot ph(Rn)] as obtained with the NEGF calculations, Ntot ph(Rn)=/integraldisplay∞ 0d(/planckover2pi1ω) 2πNBose(/planckover2pi1ω,T eff)LDOS( ω,Rn)2/planckover2pi1ω /planckover2pi12 =/integraldisplay∞ 0d(/planckover2pi1ω) 2πiTr[D< nn(ω)]2/planckover2pi1ω /planckover2pi12, (17) with the Bose-Einstein distribution function NBose(/planckover2pi1ω,T eff)=1 e/planckover2pi1ω/kBTeff−1and the local density of states LDOS( ω,Rn)=Tr[Ann(ω)] where Ann(ω)= i[DR nn(ω)−DA nn(ω)]=i[D> nn(ω)−D< nn(ω)] is the spectral function. The variable Rndeﬁnes the lattice site at which the effective temperature Teffis extracted.(2) Probe approach ( Tprobe eff ). The second method is inspired by the fact that, experimentally, a temperature probe contactsthe structure until thermal equilibrium is reached, i.e., untilno net energy exchange occurs between the probe and thestructure. The temperature probe is modeled by artiﬁcialphonon scattering self-energies /Gamma1 <> nn(ω) chosen in such a way that no net energy current ﬂows at the lattice site Rn, i.e., in- and out-scattering compensate each other, /integraldisplay∞ 0d(/planckover2pi1ω) 2π/planckover2pi1ωTr[/Gamma1> nn(ω)·D< nn(ω)] =/integraldisplay∞ 0d(/planckover2pi1ω) 2π/planckover2pi1ωTr[D> nn(ω)·/Gamma1< nn(ω)]. (18) These calculations are done in a postprocessing step. First the phonon Green’s functions are computed without the /Gamma1 self-energies, as highlighted in the previous section. They arethen used to solve Eq. ( 18). For that purpose, the /Gamma1 <> nnare assumed to have the following form (similar to Ref. [ 44] and B¨uttiker probes [ 46]): /Gamma1> nn(ω)=−i(NBose(/planckover2pi1ω,T eff)+1)Ann(ω)vcoup,(19) /Gamma1< nn(/planckover2pi1ω)=−iNBose(/planckover2pi1ω,T eff)Ann(ω)vcoup. (20) The strength of the vcoupcoupling between the probe and the atomic system is not relevant since it cancels out in Eq. ( 18). Again, the temperature of the Bose-Einstein distribution inEqs. ( 19) and ( 20) is adjusted to fulﬁll Eq. ( 18). The value of the effective temperature Tpop effandTprobe eff averaged over a nanowire slab is reported in Fig. 7.F o r the coupled electrothermal transport model the structure isdivided into 83 slabs and each slab contains 193 atoms.Beside V gs=0.6 V the cases Vgs=0.4 V and Vgs= 0.0 V are also presented. The good agreement between the two computational approaches supports the deﬁnition of theeffective lattice temperature. Two important facts should be 0 10 20 30 40300350400450500 x [nm]Effective Lattice Temperature [K]Vgs= 0.0 V Vgs= 0.4 V Vgs= 0.6 V Teff, max FIG. 7. (Color online) Effective lattice temperature averaged over a nanowire slab in the structure described in Fig. 1. It is calculated according to the population (solid blue lines with symbols, Tpop eff)a n d the probe approach (green dashed lines, Tprobe eff). Three gate biases Vgs=0.0V ,Vgs=0.4V ,a n d Vgs=0.6 V are considered. 235311-7RETO RHYNER AND MATHIEU LUISIER PHYSICAL REVIEW B 89, 235311 (2014) 0.0 0.1 0.2 0.3 0.4 0.5 0.610-510-410-310-210-1100101 Vgs [V]Id [μA] 024681012ballistic standard scatt. self-heating self-heating Id [μA] FIG. 8. (Color online) Transfer characteristics Id-VgsatVds= 0.6 V of the Si GAA NWFET in Fig. 1. The ballistic (blue solid line), standard scattering ( Id,scatt, red dashed line), and self-heating (Id,self, green dashed-dotted line) currents are plotted. The inﬂuence of self-heating is indicated by the double arrow. Note that ballistic simulations do not converge at high gate voltages. emphasized. At low Vgs, when the electrical current is too small to generate phonons at a high rate, the temperature remainsconstant and equal to 300 K in the entire nanowire structure. AtV gs=0.4 V and Vgs=0.6 V , the effective lattice temperature considerably increases and exhibits a peak close to the drainside, in accordance with the results from Figs. 3(a), 3(b), and 5. The peak location corresponds to the point where the phononenergy current changes its sign and where the optical andacoustic phonon populations reach a maximum. The valuesofT effatVgs=0.4 V and Vgs=0.6 V indicate self-heating effects. In the standard electron-phonon scattering theory, Teff would not increase with Vgs, but always stay equal to 300 K. D. Device characteristics Finally, the intrinsic transfer characteristics of the investi- gated Si GAA nanowire transistor are plotted in Fig. 8. Three different currents can be identiﬁed: (i) in the ballistic limit oftransport ( I d,bal), (ii) computed with the standard scattering method ( Id,scatt), and (iii) with self-heating ( Id,self). Despite the short gate length of 15 nm Fig. 8shows that the transistor does not operate close to its ballistic limit, neither with anequilibrium nor with a nonequilibrium phonon distribution.Turning on electron-phonon scattering reduces the currentmagnitude by about 45% at V gs=0.4 V as compared to the ballistic case. Driving the phonons out-of-equilibrium furtherdecreases the current by another 30% at V gs=0.6V ,a s indicated by the double arrow in Fig. 8. Hence, the total current reduction is roughly 50% in the presence of self-heating. Two other physical quantities can be extracted from the coupled electrothermal transport simulations: the electricalpower dissipated as heat and the maximum effective latticetemperature in the nanowire. The ﬁrst one is deﬁned as thedifference between the electrical energy current at source anddrain. The second corresponds to the lattice temperature atthe location with the highest phonon generation rate. Both00.51Power [μW] 0 0.1 0.2 0.3 0.4 0.5 0.6300400500 VgsTemperature [K]Dissipated Power Teff, max (b)(a) FIG. 9. (Color online) Evolution of the dissipated power (a) and the maximum effective temperature (b) as a function of Vgsfor the Si GAA NWFET of Fig. 1. quantities are shown in Fig. 9as a function of Vgs.T h e threshold voltage at which the dissipated power and themaximum temperature start to rapidly increase, V gs=0.3V , is directly related to the point in Fig. 8where self-heating starts to affect the current magnitude. After this turn-on, thedissipated power almost linearly increases up to V gs=0.6V where it reaches a value larger than 1 μW. This, combined with an effective lattice temperature close to 500 K, suggests thatthermal management will be a critical issue in future integratedcircuits made of GAA NWFETs. IV . CONCLUSION AND OUTLOOK Fully coupled electron-phonon transport has been treated in a full-band and atomistic device simulator based on thenonequilibrium Green’s function formalism formulated in anearest-neighbor tight-binding basis for electrons and in amodiﬁed valence-force-ﬁeld basis for phonons. In thisapproach it has been possible to drive not only the electrons butalso the phonons out-of-equilibrium to investigate self-heatingeffects in a Si gate all-around nanowire transistor with adiameter of 3 nm, a total length of 45 nm, and composedof more than 15 000 atoms. The simulation results have beencompared to the case where electrons interact with equilibriumphonons characterized by a constant temperature of 300 K. Ithas been found that self-heating signiﬁcantly increases thelattice temperature that can be mapped to the nonequilibriumphonon population. In addition, the higher phonon populationhas caused a strong enhancement of the electron-phonon cou-pling strength and a strong reduction of the electron current. Itis therefore essential to take thermal management into accountto design future electronic circuits relying on GAA NWFETs. As future works, the inﬂuence of anharmonic phonon- phonon scattering on self-heating effects should be inves-tigated. The optical phonon population might artiﬁciallyaccumulate in nanowires due to the missing decay of highfrequency particles into low frequency ones. The redistributionof the phonon population towards more acoustic componentsis expected to decrease the electron-phonon coupling strength 235311-8ATOMISTIC MODELING OF COUPLED ELECTRON-PHONON . . . PHYSICAL REVIEW B 89, 235311 (2014) close to the source contact and lead to a slight increase of the current. Currently phonons can only escape at both endsof the nanowire and not at its surface, which could inducean overestimation of the lattice temperature values. The effectof the poor thermal conductivity of the surrounding oxide ispartially compensated by the strongly reduced oxide thicknessin these ultrascaled nanostructures. Hence, thermal losses at the gate contacts probably affect the temperature distribution and will be accounted for in a future study.ACKNOWLEDGMENTS This work was supported by SNF Grant No. PP00P2_133591, by a grant from the Swiss National Su-percomputing Centre (CSCS) under Project No. s363, byNSF Grant No. EEC-0228390 that funds the Network forComputational Nanotechnology, by NSF PetaApps Grant No.0749140, and by NSF through XSEDE resources provided bythe National Institute for Computational Sciences (NICS). APPENDIX A: SCATTERING SELF-ENERGIES To calculate the scattering self-energies the starting point is the contour-ordered Green’s function in the interaction picture because a systematic perturbation theory can be applied to it [ 33,39]: Gσ1σ2 nm(τ,τ/prime)/bracketleftbig Dij nm(τ,τ/prime)/bracketrightbig =−i /planckover2pi1/angbracketleftbig TCe−i /planckover2pi1/integraltext Cdτ/prime/primeˆHint(τ/prime/prime)ˆcnσ1(τ)/bracketleftbigˆui n(τ)/bracketrightbigˆc† mσ 2(τ/prime)/bracketleftbigˆuj m(τ/prime)/bracketrightbig/angbracketrightbig , (A1) where G[D]is the electron [phonon] Green’s function, TCthe contour ordering operator, Cdescribes the Keldysh contour, and the brackets /angbracketleft ···/angbracketright indicate the nonequilibrium ensemble average [ 47]. The Hint(τ) term is the not-exactly solvable perturbation Hamiltonian according to the last term in Eq. ( 8): ˆHint(τ/prime/prime)=/summationdisplay nm/summationdisplay σ1σ2/summationdisplay i∇iHσ1σ2 mnˆc† mσ 1(τ/prime/prime)ˆcnσ2(τ/prime/prime)/parenleftbigˆui n(τ/prime/prime)−ˆui m(τ/prime/prime)/parenrightbig . (A2) The second quantized electron creation ˆc† nσ(τ/prime/prime) and annihilation ˆcnσ(τ/prime/prime) operators as well as the quantized lattice displacement ˆui n(τ/prime/prime) evolve according to the corresponding unperturbed Hamiltonian terms also described in Eq. ( 8). The noninteracting electron [phonon] Green’s function can therefore be deﬁned as G0,σ1σ2 nm (τ,τ/prime)/bracketleftbig D0,ij nm(τ,τ/prime)/bracketrightbig =−i /planckover2pi1/angbracketleftbig TCˆcnσ1(τ)/bracketleftbigˆui n(τ)/bracketrightbigˆc† mσ 2(τ/prime)/bracketleftbigˆuj m(τ/prime)/bracketrightbig/angbracketrightbig . (A3) The scattering self-energies result from the expansion of the exponential in Eq. ( A1) to the second order. The ﬁrst- order term vanishes since the expectation value of an odd number of quantized lattice displacements is zero, /angbracketleftui n(τ)/angbracketright= /angbracketleftui1n1(τ1)ui2n2(τ2)ui3n3(τ3)/angbracketright=0. The irreducible scattering self-energy functional can be identiﬁed by writing the Dyson equation for the electron, Gnm(τ,τ/prime)=G0 nm(τ,τ/prime)+/integraldisplay Cdτ1/integraldisplay Cdτ2/summationdisplay n1m1G0 nn1(τ,τ 1)/Sigma1n1m1(τ1,τ2)Gm1m(τ2,τ/prime), (A4) and phonon Green’s function, Dnm(τ,τ/prime)=D0 nm(τ,τ/prime)+/integraldisplay Cdτ1/integraldisplay Cdτ2/summationdisplay n1m1D0 nn1(τ,τ 1)/Pi1n1m1(τ1,τ2)Dm1m(τ2,τ/prime). (A5) In the self-consistent Born approximation the noninteracting Green’s functions occurring in the expressions for the scattering self-energies are replaced by the full Green’s functions as will be shown in the next section. 1. Electron-Phonon Scattering Self-Energy ( /Sigma1) To evaluate the electron-phonon scattering self-energy /Sigma1in Eq. ( A4) the exponential in Eq. ( A1) is expanded up to the second order, Gσ1σ2 nm(τ,τ/prime)=−i /planckover2pi1/angbracketleftbigˆcnσ1(τ)ˆc† mσ 2(τ/prime)/angbracketrightbig +1 2/parenleftbigg−i /planckover2pi1/parenrightbigg3/integraldisplay Cdτ1/integraldisplay Cdτ2/angbracketleftˆHint(τ1)ˆHint(τ2)ˆcnσ1(τ)ˆc† mσ 2(τ/prime)/angbracketright. (A6) Note that for brevity the contour-ordering operator TCis omitted. By comparing Eqs. ( A4) and ( A6) it appears that the ﬁrst term is equal to G0,σ1σ2nm (τ,τ/prime), while the second one contains information about the scattering self-energy. By replacing ˆHint(τ/prime/prime) with 235311-9RETO RHYNER AND MATHIEU LUISIER PHYSICAL REVIEW B 89, 235311 (2014) its value in Eq. ( A2), the following expression is obtained: 1 2/parenleftbigg−i /planckover2pi1/parenrightbigg3/integraldisplay Cdτ1/integraldisplay Cdτ2/summationdisplay n1m1n2m2/summationdisplay σ3σ4σ5σ6/summationdisplay ij/angbracketleftbig ∇iHσ3σ4 m1n1∇jHσ5σ6 m2n2ˆc† m1σ3(τ1)ˆcn1σ4(τ1)ˆc† m2σ5(τ2)ˆcn2σ6(τ2)ˆcnσ1(τ)ˆc† mσ 2(τ/prime)/angbracketrightbig ×/angbracketleftbig/parenleftbigˆui n1(τ1)ˆuj n2(τ2)−ˆui n1(τ1)ˆuj m2(τ2)−ˆui m1(τ1)ˆuj n2(τ2)+ˆui m1(τ1)ˆuj m2(τ2)/parenrightbig/angbracketrightbig . (A7) Since the electron and phonon operators commute with each other, it is not important how they are arranged with respect to each other. To evaluate the expectation values /angbracketleft ···/angbracketright Wick’s decomposition technique [ 32] is used and only the relevant connected terms are kept 1 2/parenleftbigg−i /planckover2pi1/parenrightbigg3/integraldisplay Cdτ1/integraldisplay Cdτ2/summationdisplay n1m1n2m2/summationdisplay σ3σ4σ5σ6/summationdisplay ij/parenleftbig ∇iHσ3σ4 m1n1∇jHσ5σ6 m2n2/angbracketleftbigˆcnσ1(τ)ˆc† m1σ3(τ1)/angbracketrightbig/angbracketleftbigˆcn1σ4(τ1)ˆc† m2σ5(τ2)/angbracketrightbig/angbracketleftbigˆcn2σ6(τ2)ˆc† mσ 2(τ/prime)/angbracketrightbig +∇iHσ3σ4 m1n1∇jHσ5σ6 m2n2/angbracketleftbigˆcnσ1(τ)ˆc† m2σ5(τ2)/angbracketrightbig/angbracketleftbigˆcn2σ6(τ2)ˆc† m1σ3(τ1)/angbracketrightbig/angbracketleftbigˆcn1σ4(τ1)ˆc† mσ 2(τ/prime)/angbracketrightbig/parenrightbig ×/angbracketleftbig/parenleftbigˆui n1(τ1)ˆuj n2(τ2)/angbracketrightbig −/angbracketleftbigˆui n1(τ1)ˆuj m2(τ2)/angbracketrightbig −/angbracketleftbigˆui m1(τ1)ˆuj n2(τ2)/angbracketrightbig +/angbracketleftbigˆui m1(τ1)ˆuj m2(τ2)/parenrightbig/angbracketrightbig . (A8) The contraction of the quantized lattice displacements is straight forward, whereas for the electron operators only two connected pairings remain. They can be merged together by interchanging the indices and introducing a factor two. Recalling the deﬁnitionof the unperturbed Green’s function in Eq. ( A3) yields i/planckover2pi1/integraldisplay Cdτ1/integraldisplay Cdτ2/summationdisplay n1m1n2m2/summationdisplay σ3σ4σ5σ6/summationdisplay ijG0,σ1σ3 nm 1(τ,τ 1)∇iHσ3σ4 m1n1G0,σ4σ5 n1m2(τ1,τ2)∇jHσ5σ6 m2n2G0,σ6σ2 n2m(τ2,τ/prime) ×/parenleftbig D0,ij n1n2(τ1,τ2)−D0,ij n1m2(τ1,τ2)−D0,ij m1n2(τ1,τ2)+D0,ij m1m2(τ1,τ2)/parenrightbig . (A9) By comparing Eqs. ( A4) and ( A9) the electron-phonon scattering self-energy can be identiﬁed as /Sigma1σ1σ2 nm(τ1,τ2)=i/planckover2pi1/summationdisplay n1m1/summationdisplay σ3σ4/summationdisplay ij∇iHσ1σ3 nn1Gσ3σ4 n1m1(τ1,τ2)∇jHσ4σ2 m1m ×/parenleftbig Dij n1m(τ1,τ2)−Dij n1m1(τ1,τ2)−Dij nm(τ1,τ2)+Dij nm 1(τ1,τ2)/parenrightbig . (A10) The noninteracting Green’s functions can be replaced by the full Green’s functions due to the implicit inclusion of higher order perturbation terms in Eq. ( A4). To replace the complex-time contour arguments by real-time arguments Langreth’s theorem [ 33] C(τ1,τ2)=A(τ1,τ2)B(τ1,τ2)→C≷(t1,t2)=A≷(t1,t2)B≷(t1,t2) is used. The consideration of steady-state situations allows for the Fourier transformation of the time difference t1−t2. The electron-phonon scattering self-energy ﬁnally takes the following form: /Sigma1≷σ1σ2 nm (E)=i/summationdisplay n1m1/summationdisplay ij/summationdisplay σ3σ4/integraldisplayd(/planckover2pi1ω) 2π∇iHσ1σ3 nn1G≷σ3σ4 n1m1(E−/planckover2pi1ω)∇jHσ4σ2 m1m/parenleftbig D≷ij n1m(ω)−D≷ij n1m1(ω)−D≷ij nm(ω)+D≷ij nm 1(ω)/parenrightbig .(A11) 2. Phonon-Electron Scattering Self-Energy ( /Pi1) For the calculation of the phonon-electron self-energy /Pi1in Eq. ( A5) the same approach as in the last section can be followed. However, a different solution based on the energy conservation condition is proposed here. The energy lost (gained) by theelectrons [ +(−)Q e] must be absorbed (emitted) by the phonons [ −(+)Qph]o ri no t h e rw o r d s QeandQphmust compensate each otherQe+Qph=0 with Qe=1 /planckover2pi1/summationdisplay nm/integraldisplaydE 2πETr(/Sigma1> nm(E)·G< mn(E)−G> nm(E)·/Sigma1< mn(E)), (A12) 235311-10ATOMISTIC MODELING OF COUPLED ELECTRON-PHONON . . . PHYSICAL REVIEW B 89, 235311 (2014) and Qph=1 /planckover2pi1/summationdisplay nm/integraldisplayd(/planckover2pi1ω) 2π/planckover2pi1ωTr(/Pi1> nm(ω)·D< mn(ω)−D> nm(ω)·/Pi1< mn(ω)). (A13) Each element composing the out-scattering rate ETr(/Sigma1> nm(E)·G< mn(E)) in Eq. ( A12) has a corresponding element in the in-scattering rate /planckover2pi1ωTr(D> nm(ω)·/Pi1< mn(ω)) in Eq. ( A13) so that they cancel each other: 1 /planckover2pi1/summationdisplay nm/summationdisplay σ1σ2/integraldisplaydE 2πE⎛ ⎝i/summationdisplay n1m1/summationdisplay ij/summationdisplay σ3σ4/integraldisplayd(/planckover2pi1ω) 2π∇iHσ1σ3 nn1G>σ 3σ4 n1m1(E−/planckover2pi1ω)∇jHσ4σ2 m1m/parenleftbig D>ij n1m(ω) −D>ij n1m1(ω)−D>ij nm(ω)+D>ij nm 1(ω)/parenrightbig⎞ ⎠G<σ 2σ1 mn (E) =1 /planckover2pi1/summationdisplay n2m2/summationdisplay i1j1/integraldisplayd(/planckover2pi1ω/prime) 2π/planckover2pi1ω/prime/parenleftbig D>i1j1 n2m2(ω/prime)/Pi1<j1i1 m2n2(ω/prime)/parenrightbig . (A14) The same relationship can be established between the in-scattering rate in Eq. ( A12) and the out-scattering rate in Eq. ( A13), leading to the following expression for the phonon-electron scattering self-energies: /Pi1≷ij nm(ω)=2spin·i/summationdisplay n1m1/summationdisplay σ1σ2σ3σ4/integraldisplaydE 2π/parenleftbig ∇iHσ3σ1 n1nG≷σ1σ4 nm 1(/planckover2pi1ω+E)∇jHσ4σ2 m1mG≶σ2σ3 mn 1(E) −∇iHσ3σ1 n1nG≷σ1σ2 nm (/planckover2pi1ω+E)∇jHσ2σ4 mm 1G≶σ4σ3 m1n1(E)−∇iHσ1σ3 nn1G≷σ3σ4 n1m1(/planckover2pi1ω+E)∇jHσ4σ2 m1mG≶σ2σ1 mn (E) +∇iHσ1σ3 nn1G≷σ3σ2 n1m(/planckover2pi1ω+E)∇jHσ2σ4 mm 1G≶σ4σ1 m1n(E)/parenrightbig . (A15) APPENDIX B: NUMERICAL IMPLEMENTATION The computational burden is too large to simulate the device described in Sec. IIIon a single processor or on small clusters. The results presented in this work are obtained by using NCPU=4500 cores. The NCPU are distributed according to the number of electron energy ( Nel E∼1000) and phonon frequency ( Nph ω∼120) points that are retained in Eqs. ( 1)–(6), respectively. This means that around 90% of the cores solve the electron system and 10% the phonon one. First the ballistic solution is calculated by setting the scattering self-energies to zero. Then, at the beginningof each self-consistent Born iteration, the CPUs dealing with phonon Green’s functions send their D <>(ω)t ot h e CPUs dedicated to the electrons. The latter ones solveEqs. ( 14)–(16) to evaluate /Sigma1 <>(E) and/Pi1<>(ω) and then send the phonon-electron self-energies /Pi1<>(ω) back to the phonon CPUs. The scaling performance of the fully coupled approach (self-heating) described in this work and of the standardscattering approach of Ref. [ 22] is reported in Fig. 10for a reduced nanowire system with d=3 nm, 7141 atoms, N el E= 895, and Nph ω=31. It is shown that the simulation time for one Born iteration in the fully coupled case is about two timeslonger than in the standard scattering case where no /Pi1 <>(ω) are calculated. Note that in the self-heating simulations morecores need to be allocated ( ∼120) than in the standard scattering case to be able to simultaneously solve the electronand phonon system. As a consequence, the scaling behaviorof the fully coupled simulation approach is not as good as in the standard case due to the increase of interprocessorcommunication. 500 1000 2000 400040080016003200 Number of CoresWalltime [s]self-heating standard scatt. ideal scaling ~ 2x FIG. 10. (Color online) Parallel execution time on a CRAY XE6 for the calculation of one self-consistent Born iteration in the standard scattering (blue solid line with circles), as in Ref. [ 22], and the self- heating case (dashed green line with squares), i.e., the solution ofEqs. ( 1)–(6)a n d( 14)–(16) for all electron energies ( N el E) and phonon (Nph ω) frequencies. 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PhysRevB.78.125306.pdf	Magnetoconductance of rectangular arrays of quantum rings Orsolya Kálmán,1,2Péter Földi,2Mihály G. Benedict,2,and F. M. Peeters3 1Department of Quantum Optics and Quantum Information, Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, Hungary 2Department of Theoretical Physics, University of Szeged, Tisza Lajos körút 84, H-6720 Szeged, Hungary 3Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium /H20849Received 22 April 2008; revised manuscript received 30 June 2008; published 4 September 2008 /H20850 Electron transport through multiterminal rectangular arrays of quantum rings is studied in the presence of Rashba-type spin-orbit interaction /H20849SOI /H20850and of a perpendicular magnetic ﬁeld. Using the analytic expressions for the transmission and reﬂection coefﬁcients for single rings we obtain the conductance through such arraysas a function of the SOI strength, of the magnetic ﬂux, and of the wave vector kof the incident electron. Due to destructive or constructive spin interferences caused by the SOI, the array can be totally opaque for certainranges of k, while there are parameter values where it is completely transparent. Spin resolved transmission probabilities show nontrivial spin transformations at the outputs of the arrays. When pointlike random scatter-ing centers are placed between the rings, the Aharonov-Bohm peaks split, and an oscillatory behavior of theconductance emerges as a function of the SOI strength. DOI: 10.1103/PhysRevB.78.125306 PACS number /H20849s/H20850: 73.23.Ad, 03.65. /H11002w, 85.35.Ds, 71.70.Ej I. INTRODUCTION Magnetoconductance oscillations of quantum rings made of semiconducting materials1exhibiting Rashba-type spin- orbit interaction2–4/H20849SOI /H20850have been intensely studied in the past few years. These effects are manifestations of ﬂux- andspin-dependent quantum interference phenomena. In view ofthe possible spintronic applications and the conceptual im-portance of these interference effects in multiply-connecteddomains, closed single-quantum rings /H20849without attached leads /H20850 5–8as well as two- or three-terminal ones were investigated9–22extensively. Additionally, the conductance properties of a linear chain of rings have also beendetermined. 23 In this paper we present a method that enables one to calculate the conductance and the spin transport properties oftwo-dimensional rectangular arrays consisting of quantumrings with Rashba-type SOI /H20849Ref. 24/H20850and with a perpendicu- lar magnetic ﬁeld. Such arrays, fabricated from, e.g., anInAlAs/InGaAs based two-dimensional electron gas/H208492DEG /H20850, 25have been studied in a recent experiment26and in a subsequent theoretical work27to demonstrate the time- reversal Aharonov-Casher effect.28Here we present a more general survey of the magnetoconductance properties of suchdevices, including the perturbative treatment of the magneticﬁeld which still allows us to analytically solve the scatteringproblem in case of two-, three-, and four-terminal rings,which are then used as building blocks of larger arrays. Wealso present results related to the spin-resolved transmissionproperties of the network, which is an issue that has not beenaddressed so far. Our method is based on analytic results andcan be used for an arbitrary conﬁguration. For the sake ofdeﬁniteness, we consider 3 /H110033, 4/H110034, and 5 /H110035 rectangular arrays, 26,27which are closed in the vertical and open in the horizontal direction. Additionally, we study the magnetocon-ductance properties and spin-resolved transmission prob-abilities of the same array geometry with only one inputchannel. We also investigate to what extent the conductanceproperties are modiﬁed by the presence of pointlike random scattering centers between the rings. In our calculations we assume that the rings are narrow enough to be consideredone dimensional and the transport of the electrons throughthe arrays is ballistic. We determine the magnetoconductancein the framework of the Landauer-Büttiker formalism. 29 Rectangular arrays26,27—depending on the number of in- put leads—consist of two-, three-, and four-terminal rings/H20849see Fig. 1/H20850, where the two- and three-terminal ones are situ- ated on the boundary of the arrays as shown in Fig. 2with or without the input leads displayed by dashed lines. The trans-mission and reﬂection properties of two- and three-terminalrings have been determined in previous works 10–12,14,30–35but the effect of the magnetic ﬁeld on the spin degree of freedomhas not been taken into account for an arbitrary geometry.Additionally, the most general boundary condition that is re-quired by this two-dimensional problem has not been inves-tigated so far. Therefore in Sec. II we ﬁrst consider a perpen-dicular magnetic ﬁeld as a weak perturbation, then, in orderto account for all possible reﬂections and transmissions whenbuilding up the array from single rings, we generalize ourprevious results to the case when electrons can enter/exit onany of the terminals of a three-terminal ring /H20849results for two- and four-terminal rings are presented in Appendix /H20850. Next, in Sec. III A the individual rings are used as building blocks ofthe arrays by ﬁtting the wave functions and their derivativesin the points where neighboring rings touch each other. Mag-netoconductance properties are presented here as a functionof the wave number kof the incoming electron, the magnetic ﬂux, and the SOI strength. Spin resolved transmission prob-abilities on the output side of the arrays are also derived. InSec. III B we investigate the effect of random Dirac-deltascattering potentials in between the rings. II. BUILDING BLOCKS OF TWO-DIMENSIONAL ARRAYS: SINGLE QUANTUM RINGS In this section we consider a single narrow quantum ring31 of radius alocated in the xyplane in the presence of RashbaPHYSICAL REVIEW B 78, 125306 /H208492008 /H20850 1098-0121/2008/78 /H2084912/H20850/125306 /H2084910/H20850 ©2008 The American Physical Society 125306-1SOI /H20849Ref. 24/H20850and a perpendicular magnetic ﬁeld B.I fBis relatively weak, then the interaction between the electronspin and the ﬁeld, i.e., the Zeeman term, can be treated as aperturbation and the relevant dimensionless Hamiltonianreads 11,36 H=/H20875/H20873−i/H11509 /H11509/H9272−/H9021 /H90210+/H9275SO 2/H9024/H9268r/H208742 −/H9275SO2 4/H90242/H20876+Hp, /H208491/H20850 where /H9272is the azimuthal angle of a point on the ring, /H9021 denotes the magnetic ﬂux encircled by the ring, /H90210=h/eis the unit ﬂux, and /H9275SO=/H9251//H6036ais the frequency associated with the spin-orbit interaction. /H6036/H9024=/H60362/2m/H11569a2characterizes the kinetic energy with m/H11569being the effective mass of the elec- tron, and the radial spin operator is given by /H9268r=/H9268xcos/H9272+/H9268ysin/H9272. The perturbative term Hpis given by11 Hp=/H9275L /H9024/H9268z, where /H9275L=g/H11569eB /4m, with g/H11569andmbeing the effective gy- romagnetic ratio and the free-electron mass, respectively. The energy eigenvalues of the unperturbed Hamiltonian are E0/H20849/H9262/H20850/H20849/H9260/H20850=/H20873/H9260−/H9021 /H90210/H208742 +/H20849−1 /H20850/H9262/H20873/H9260−/H9021 /H90210/H20874w+1 4/H20849/H9262=1 , 2 /H20850, /H208492a/H20850 and the corresponding eigenvectors in the /H20841↑z/H20856,/H20841↓z/H20856eigenba- sis of /H9268zread /H9274/H20849/H9262/H20850/H20849/H9260,/H9272/H20850=ei/H9260/H9272/H20873e−i/H9272/2u/H20849/H9262/H20850 ei/H9272/2v/H20849/H9262/H20850/H20874, /H208492b/H20850 where u/H208491/H20850=−v/H208492/H20850=cos /H20849/H9258/2/H20850,u/H208492/H20850=v/H208491/H20850=sin /H20849/H9258/2/H20850, and tan /H20849/H9258/2/H20850=/H9024 /H9275SO/H208491−w/H20850, /H208493/H20850 with w=/H208811+/H9275SO2//H90242. The matrix elements of Hpin the basis of these eigenstates are obtained as /H20855/H9274/H20849/H9262/H20850/H20841Hp/H20841/H9274/H20849/H9262/H20850/H20856=/H20849−1 /H20850/H9262+1/H9275L /H9024cos/H9258=/H20849−1 /H20850/H9262+1/H9275L /H90241 w, /H20855/H9274/H208491/H20850/H20841Hp/H20841/H9274/H208492/H20850/H20856=/H9275L /H9024sin/H9258. In the ﬁrst-order approximation one neglects the off-diagonal elements; this is reasonable if they are small, i.e., if /H9275L//H9024 /H11270k2a2, where kdenotes the wave number of the incident electron, which is described as a plane wave. Within thisapproximation, the eigenspinors are not perturbed and theirdirection is still speciﬁed by the angle /H9258given by Eq. /H208493/H20850./c103/c49/c102/c73 /c114/c73 /c102/c73/c73/c114/c73/c73 /c103/c49/c103/c50/c102/c73 /c114/c73 /c102/c73/c73/c102/c73/c73/c73 /c114/c73/c73/c114/c73/c73/c73 /c103/c49/c103/c50/c103/c51 /c102/c73 /c114/c73 /c102/c73/c73/c114/c73/c73/c114/c73/c73/c73 /c102/c73/c73/c73/c102/c73/c86 /c114/c73/c86(b)(a) (c) FIG. 1. The notations used for the spinor part of the wave func- tions in the case of /H20849a/H20850two-, /H20849b/H20850three-, and /H20849c/H20850four-terminal rings.11/c102(11) /c73 /c114(11) /c73/c73 /c73/c73/c73/c73/c7312/c73 /c73/c73/c73/c73/c73 21/c102(21) /c73 /c114(21) /c73/c73 /c73/c73/c73/c73/c73/c73/c86 22/c73 /c73/c73/c73/c73/c73/c73/c8613 23/c73 /c73/c73/c73/c73/c73 /c73 /c73/c73/c73/c73/c73 31/c102(31) /c73 /c114(31) /c73/c73/c73/c73/c73 /c73/c7332/c73/c73/c73/c73 /c73/c73/c114(13) /c73/c73/c73 /c114(23) /c73/c73/c73 /c114(33) /c73/c7333/c73/c73/c73/c73 /c73/c73/c73/c86 FIG. 2. The geometry of the device in the simplest case of a 3 /H110033 array with three or one /H20849without leads displayed with dashed lines /H20850input terminals. The notations can easily be generalized to larger arrays.KÁLMÁN et al. PHYSICAL REVIEW B 78, 125306 /H208492008 /H20850 125306-2The energy eigenvalues including the ﬁrst-order corrections are given by E/H20849/H9262/H20850/H20849/H9260/H20850=E0/H20849/H9262/H20850/H20849/H9260/H20850+/H20849−1 /H20850/H9262+1/H9275L /H90241 w. Imposing the condition of energy conservation k2a2 =E/H20849/H9262/H20850/H20849/H9260/H20850determines the possible values of /H9260, /H9260j/H20849/H9262/H20850=/H20849−1 /H20850/H9262+1/H20875w 2+/H20849−1 /H20850jq/H20849/H9262/H20850/H20876+/H9021 /H90210, where /H9262,j=1,2 and q/H20849/H9262/H20850=/H20881q2+/H20849−1 /H20850/H9262/H9275L /H90241 w, /H208494/H20850 with q=/H20881/H20849/H9275SO /2/H9024/H208502+E//H6036/H9024, where E=/H60362k2/2m/H11569denotes the energy of the incoming electron. The corresponding foureigenspinors read /H9274j/H208491/H20850/H20849/H9260j/H208491/H20850,/H9272/H20850=ei/H9260j/H208491/H20850/H9272/H20873e−i/H9272/2cos /H20849/H9258/2/H20850 ei/H9272/2sin /H20849/H9258/2/H20850/H20874, /H208495/H20850 /H9274j/H208492/H20850/H20849/H9260j/H208492/H20850,/H9272/H20850=ei/H9260j/H208492/H20850/H9272/H20873e−i/H9272/2sin /H20849/H9258/2/H20850 −ei/H9272/2cos /H20849/H9258/2/H20850/H20874. /H208496/H20850 The wave functions belonging to the same energy in the different sections of the ring are linear combinations of theseeigenspinors. The building blocks of the rectangular arrays we investi- gate are two-, three-, and four-terminal quantum rings /H20849see Fig. 1/H20850, where, in general, the boundary conditions allow both incoming and outgoing spinor valued wave functions ateach terminal: /H9023 i=fieikxi+rie−ikxi/H20849i=I,II,III,IV /H20850, where xi denotes the local coordinate in terminal i. Note that the am- plitudes fI,rI,fII,¯refer to two-component spinors, e.g., fI=/H20873/H20849fI/H20850↑ /H20849fI/H20850↓/H20874. For the sake of deﬁniteness, we focus on a general three- terminal ring, shown in Fig. 1/H20849b/H20850. The scattering problem in the case of a ring with four terminals /H20851Fig. 1/H20849c/H20850/H20852can also be solved analytically, as presented in Appendix, where we alsogive the results for a general two-terminal ring /H20851Fig. 1/H20849a/H20850/H20852. The outgoing spinors /H20849r i,i=I,II,II /H20850are connected to the in- coming ones /H20849fi/H20850by 2/H110032 matrices, which can be determined by requiring the continuity of the wave functions and van-ishing net spin current densities /H20849Grifﬁth conditions /H20850 11,13,32,37 at the junctions. For the same boundary conditions as in Ref. 35, i.e., for fII,fIII=0 in Fig. 1/H20849b/H20850, the reﬂection matrix which connects rIto the incoming spinor fIis given by R↑↑fI=/rho1/H208491/H20850cos2/H20849/H9258/2/H20850+/rho1/H208492/H20850sin2/H20849/H9258/2/H20850−1 , R↑↓fI=/H20849/rho1/H208491/H20850−/rho1/H208492/H20850/H20850sin /H20849/H9258/2/H20850cos /H20849/H9258/2/H20850, R↓↑fI=R↑↓fI,R↓↓fI=/rho1/H208491/H20850sin2/H20849/H9258/2/H20850+/rho1/H208492/H20850cos2/H20849/H9258/2/H20850−1 , /H208497/H20850 where /rho1/H20849/H9262/H20850=8ka /y/H20849/H9262/H20850/H20853−i/H20849q/H20849/H9262/H20850/H208502sin /H208492q/H20849/H9262/H20850/H9266/H20850 −kaq/H20849/H9262/H20850/H20851sin /H20849q/H20849/H9262/H20850/H92531/H20850sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92531/H20850/H20850 + sin /H20849q/H20849/H9262/H20850/H92532/H20850sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92532/H20850/H20850/H20852 +ik2a2sin /H20849q/H20849/H9262/H20850/H92531/H20850sin /H20849q/H20849/H9262/H20850/H20849/H92532−/H92531/H20850/H20850 /H11003sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92532/H20850/H20850/H20854, and y/H20849/H9262/H20850=8 /H20849q/H20849/H9262/H20850/H208503/H20853cos /H20851/H20851/H20849−1 /H20850/H9262+1w+2/H9278/H20852/H9266/H20852+ cos /H208492q/H20849/H9262/H20850/H9266/H20850/H20854 −1 2ika /H20849q/H20849/H9262/H20850/H208502sin /H208492q/H20849/H9262/H20850/H9266/H20850+4k2a2q/H20849/H9262/H20850cos /H208512q/H20849/H9262/H20850/H9266/H20852 −2k2a2q/H20849/H9262/H20850/H20853cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532+/H92531/H20850/H20852− cos /H208492q/H20849/H9262/H20850/H9266/H20850 + cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532/H20850/H20852+ cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20852/H20854 +ik3a3/H20853sin /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532+/H92531/H20850/H20852− sin /H208512q/H20849/H9262/H20850/H9266/H20852 + sin /H208512q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20852− sin /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532/H20850/H20852/H20854, with/H9278=/H9021//H90210. The matrices describing the connection be- tween the outgoing spinors rII,rIIIand the input fI—the so- called transmission matrices—are given by /H20849TnfI/H20850↑↑=e−i/H9253n/2/H20849/H9270n/H208491/H20850cos2/H20849/H9258/2/H20850+/H9270n/H208492/H20850sin2/H20849/H9258/2/H20850/H20850, /H20849TnfI/H20850↑↓=e−i/H9253n/2/H20849/H9270n/H208491/H20850−/H9270n/H208492/H20850/H20850sin /H20849/H9258/2/H20850cos /H20849/H9258/2/H20850, /H20849TnfI/H20850↓↑=ei/H9253n/2/H20849/H9270n/H208491/H20850−/H9270n/H208492/H20850/H20850sin /H20849/H9258/2/H20850cos /H20849/H9258/2/H20850, /H20849TnfI/H20850↓↓=ei/H9253n/2/H20849/H9270n/H208491/H20850sin2/H20849/H9258/2/H20850+/H9270n/H208492/H20850cos2/H20849/H9258/2/H20850/H20850, /H208498/H20850 where n=1,2, indicating the two possible output channels, and /H92701/H20849/H9262/H20850=8kaq/H20849/H9262/H20850 y/H20849/H9262/H20850ei/H92531/2/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850 /H11003/H20853−kasin /H20849q/H20849/H9262/H20850/H20849/H92532−/H92531/H20850/H20850sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92532/H20850/H20850 +iq/H20849/H9262/H20850/H20851e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850sin /H20849q/H20849/H9262/H20850/H92531/H20850 − sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92531/H20850/H20850/H20852/H20854, /H92702/H20849/H9262/H20850=8kaq/H20849/H9262/H20850 y/H20849/H9262/H20850ei/H92532/2/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850 /H11003/H20853kae−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850sin /H20849q/H20849/H9262/H20850/H92531/H20850sin /H20849q/H20849/H9262/H20850/H20849/H92532−/H92531/H20850/H20850 +iq/H20849/H9262/H20850/H20851e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850sin /H20849q/H20849/H9262/H20850/H92532/H20850 − sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92532/H20850/H20850/H20852/H20854. Note that the boundary conditions applied to obtain the RfI andTnfImatrices above are similar to that of Ref. 35. How- ever the magnetic ﬁeld induced shift of the spin Zeemanlevels leads to a doubling of the parameters according to Eq./H208494/H20850. This modiﬁes signiﬁcantly the physical transport proper- ties of the device. Let us point out that having obtained the matrix elements above is enough to handle the problem with both incomingMAGNETOCONDUCTANCE OF RECTANGULAR ARRAYS … PHYSICAL REVIEW B 78, 125306 /H208492008 /H20850 125306-3and outgoing waves on all terminals of the ring as shown in Fig. 1/H20849b/H20850. Namely, we can consider the three inputs fi/H20849i =I,II,III /H20850separately and determine the corresponding reﬂec- tion and transmission matrices. The outputs in the super-posed problem will consist of contributions from all inputs:the reﬂected part of the spinor which enters on the same leadand the transmitted parts of the other two inputs into therespective lead, r I=RfIfI+T2fIIfII+T1fIIIfIII, rII=T1fIfI+RfIIfII+T2fIIIfIII, rIII=T2fIfI+T1fIIfII+RfIIIfIII. /H208499/H20850 Considering fII/H20849fIII/H20850as the only input, the reﬂection and transmission matrices are the same as those for the input fI, except for the appropriate changes in the angles, since in thereference frame of f II/H20849fIII/H20850, the angles of the output leads are measured from the lead through which fII/H20849fIII/H20850enters the ring. In order to get the contributions to the output spinorsfor the input f II/H20849fIII/H20850in the reference frame of fI, the matrices need to be rotated /H20851see Fig. 1/H20849b/H20850/H20852by the angle of /H92531/H20849/H92532/H20850, MfII=U/H92531M/H92531↔/H92532−/H92531 /H92532↔2/H9266−/H92531fIU/H92531−1, /H2084910/H20850 MfIII=U/H92532M/H92531↔2/H9266−/H92532 /H92532↔2/H9266−/H92532+/H92531fIU/H92532−1, /H2084911/H20850 where M=R,T1,T2and U/H9253n=/H20873e−i/H9253n/20 0 ei/H9253n/2/H20874,n=1 , 2 . The above approach is also valid in the case of the two- and four-terminal rings. Using the reﬂection and transmis-sion matrices as presented in Appendix, the more generalproblem of having both incoming and outgoing waves on allterminals can easily be treated. All possible reﬂections andtransmissions can thus be taken into account when formingtwo-dimensional arrays of such rings. III. RECTANGULAR ARRAYS OF QUANTUM RINGS A. Magnetoconductance properties Based on the analytic results presented in Sec. II and in Appendix we may build M /H11003M two-dimensional rectangular arrays of quantum rings, where both perpendicular electricand magnetic ﬁelds are present, so that the former one can beused to change the strength of the SOI. 3Here we focus on of 3/H110033, 4/H110034, and 5 /H110035 arrays and assume that neighboring rings touch each other. In addition, we limit ourselves toarrays that are closed in the vertical and open in the horizon-tal direction, as shown in Fig. 2. Two types of such arrays will be investigated: /H20849i/H20850the electron can enter/exit the array through any of the rings in the horizontal direction and /H20849ii/H20850 the electron can enter the array through one ring only /H20849no leads are attached to the other rings on the entrance side /H20850butcan exit through any of the rings on the opposite side /H20849Fig.2 without the dashed curves /H20850. In both cases the conductance is derived from the linear set of equations resulting from the ﬁt of the wave functions /H9023 i/H20849kl/H20850/H20849i=I,II,III,IV and k,l =1,..., N, where Nis the number of rings along one direc- tion in the array /H20850and their derivatives /H11509xi/H20849kl/H20850/H9023i/H20849kl/H20850in the points, where the rings touch each other, for example, /H9023III/H2084911/H20850/H20841xIII/H2084911/H20850=0=/H9023I/H2084912/H20850/H20841xI/H2084912/H20850=0, /H11509xIII/H2084911/H20850/H9023III/H2084911/H20850/H20841xIII/H2084911/H20850=0=−/H11509xI/H2084912/H20850/H9023I/H2084912/H20850/H20841xI/H2084912/H20850=0. /H2084912/H20850 Here we used the notations of Fig. 2./H20849Note that the negative sign in Eq. /H2084912/H20850is a consequence of the opposite direction of the local coordinates in leads III of ring /H2085311/H20854and I of ring /H2085312/H20854./H20850Equation /H2084912/H20850leads to fIII/H2084911/H20850+rIII/H2084911/H20850=fI/H2084912/H20850+rI/H2084912/H20850, fIII/H2084911/H20850−rIII/H2084911/H20850=−fI/H2084912/H20850+rI/H2084912/H20850, from which follows that fIII/H2084911/H20850=rI/H2084912/H20850, rIII/H2084911/H20850=fI/H2084912/H20850, i.e., the spinor entering /H20849exiting /H20850ring /H2085311/H20854on terminal III is equal to the spinor exiting /H20849entering /H20850ring /H2085312/H20854on terminal I. The spinors rIII/H2084911/H20850andrI/H2084912/H20850can be given with the help of the reﬂection and transmission matrices of a three-terminal ringaccording to Eq. /H208499/H20850. For a small number of rings the resulting set of equations can be solved analytically; however already for an array of3/H110033 rings shown in Fig. 2, it consists of 60 equations, which is preferably solved by numerical means, althoughanalytic solutions exist in principle. /H20849For larger arrays the number of equations scales practically with the number ofrings. /H20850After having determined the output spinor valued wave functions r III/H208491N/H20850,rIII/H208492N/H20850,..., rII/H20849NN /H20850, where Nis the number of rings in the horizontal direction, the Landauer-Büttiker29 formula G=G↑+G↓, where G↑=e2 h/H20849/H20841/H20849rIII/H208491N/H20850/H20850↑/H208412+/H20841/H20849rIII/H208492N/H20850/H20850↑/H208412+¯+/H20841/H20849rII/H20849NN /H20850/H20850↑/H208412/H20850, G↓=e2 h/H20849/H20841/H20849rIII/H208491N/H20850/H20850↓/H208412+/H20841/H20849rIII/H208492N/H20850/H20850↓/H208412+¯+/H20841/H20849rII/H20849NN /H20850/H20850↓/H208412/H20850 are used to calculate the conductance of the arrays, averaged over the two /H9268zeigenspinor inputs. We note that our method of using single rings as building blocks can easily be used todetermine the conductance of arrays of arbitrary—not neces-sarily rectangular—conﬁguration as well. Figure 3shows a contour plot of the conductance /H20849ine 2/h units /H20850of rectangular arrays of 3 /H110033, 4/H110034, and 5 /H110035 quan- tum rings for zero magnetic ﬂux as a function of the SOIKÁLMÁN et al. PHYSICAL REVIEW B 78, 125306 /H208492008 /H20850 125306-4strength /H9275SO //H9024andka. The values of kaare varied around kFa=20.4, corresponding to a Fermi energy of 11.13 meV in case of an effective mass m/H11569=0.023 mof InAs and rings of radius a=0.25 /H9262m. In two-dimensional electron systems within an InAs quantum well, the value of /H9251can be varied2,3 up to 40 peV m. The different arrays show similar behavior for larger values of the SOI strength; there are slightly down-ward bending stripes /H20849initially around even values of ka/H20850 where the devices are completely opaque for the electronsand also conducting regions which are initially around oddvalues of kaand have complex internal structure. Comparing our results to the case of a single ring with diametricallycoupled leads, 11it can be seen that the overall periodicity as a function of kais determined by single-ring interferences. The increasing number of the rings causes modulations su-perimposing on the single-ring behavior. This point is prob-ably the most apparent if we recall 11that zero conductance areas are simply lines on the ka−/H9275SO //H9024plane for a single two-terminal ring, while in our case there are stripes, thewidth of which is slightly increasing with the size of thearray. This effect is related to the increasing number of con-secutive partially destructive interferences that ﬁnally lead toessentially zero currents at the outputs. Additionally, if weconsidered an inﬁnite network, the periodic boundary condi-tions would allow only discrete values of kafor a given SOI strength with nonzero conductance. These conducting linesin the inﬁnite case are situated on the ka− /H9275SO/H9024plane around the middle of the conducting stripes shown in Fig. 3. Thus the results presented in this ﬁgure demonstrate a tran-sition between the conductance properties of a single ringand that of an inﬁnite network. In Sec. III B we will analyzethe effect of pointlike scatterers on the nonconducting stripesshown in Fig. 3. Focusing on small values of /H9275SO //H9024, Fig. 3shows a nar- rowing of the nonconducting regions until they eventuallydisappear when no SOI is present. Here the conductance stilldepends on ka, but its minimal values are not zeros and a periodic behavior can be seen; for a network of N/H11003Nrings, there are Nminima as the value of kais increased by 1. This size-dependent modulation is related to the horizontal extentof the device; if we compare the conductance of the networksto that of rings of the same size and number without verticalconnections, the same periodic behavior can be seen aroundzero SOI. Figure 4shows the normalized magnetoconductance of networks of 3 /H110033, 4/H110034, and 5 /H110035 quantum rings for ka =19.6 as a function of the SOI strength and the magnetic ﬂux/H9021 /H20849measured in units of /H9021 0/H20850. When /H9275SO //H9024is zero, Aharonov-Bohm /H20849AB /H20850oscillations appear. For larger values00.20.40.60.81 19202122232425ka(a) 00.20.40.60.81 19202122232425ka(b) 00.20.40.60.81 0123456789 1 0ωSO/Ω19202122232425ka(c) FIG. 3. /H20849Color online /H20850The conductance G/G0/H20849G0=e2/h/H20850of/H20849a/H20850 3/H110033, /H20849b/H208504/H110034, and /H20849c/H208505/H110035 rectangular arrays with 3, 4, and 5 input terminals, respectively, for zero magnetic ﬂux as a function ofthe SOI strength and ka.00.20.40.60.8 -6-4-20246Φ/Φ0(a) 00.20.40.60.8 -6-4-20246Φ/Φ0(b) 1 0123456789 1 0 ωSO/Ω-6-4-20246Φ/Φ0 00.20.40.60.8(c) FIG. 4. /H20849Color online /H20850The conductance G/G0/H20849G0=e2/h/H20850of/H20849a/H20850 3/H110033, /H20849b/H208504/H110034, and /H20849c/H208505/H110035 rectangular arrays with 3, 4, and 5 input terminals, respectively, for ka=19.6 as a function of the SOI strength and the magnetic ﬂux /H9021 /H20849in units of /H90210=h/e/H20850.MAGNETOCONDUCTANCE OF RECTANGULAR ARRAYS … PHYSICAL REVIEW B 78, 125306 /H208492008 /H20850 125306-5of/H9275SO //H9024both AB and Aharonov-Casher28oscillations can be seen in the magnetoconductance. As Fig. 4was plotted for a certain value of ka, the effect of the bending nonconducting stripes seen in Fig. 3can also be seen as the suppression of the conductance oscillations when such a stripe is reacheddue to the change in the SOI strength and their appearanceagain when the stripe is left. We note that for larger values ofkathis bending effect is less pronounced. Figures 5and6show the conductance of a 5 /H110035 network with a single input lead in the middle /H20849i.e., attached to ring /H2085331/H20854using the notations of Fig. 2/H20850as a function of kaand /H9275SO //H9024 /H20849Fig. 5/H20850and the magnetic ﬁeld and /H9275SO //H9024 /H20849Fig. 6/H20850. The overall structure of these plots remains the same as inthe case when the current can enter through all the rings onthe left-hand side, but the different boundary conditionsmodify the ﬁne structure of the plots. Our method allows the calculation of the spin directions for the different output terminals, and we found that spin-dependent interference in the array results in nontrivial spintransformations. Figure 7shows the spin-resolved transmis- sion probabilities for a 5 /H110035 ring array with a single input lead. The incoming spin state is chosen to be /H20841↑ z/H20856, i.e., the spin-up eigenstate of /H9268z, and the contour plots show the probabilities of the /H20841↑x/H20856,/H20841↑y/H20856, and /H20841↑z/H20856outputs at ring /H2085355/H20854on the right-hand side. The fact that the /H20841↑z/H20856input spinor changes its direction /H20849as it is seen in Fig. 7, it can be trans- formed into /H20841↑x/H20856or /H20841↑y/H20856/H20850is due to the SOI induced spin rota- tions. The actual values of the spin-resolved transmissionprobabilities are determined by the spin-dependent interfer- ence phenomena. Figure 8shows the zcomponent of the normalized output spinors and visualizes that spin-resolvedresults depend on the input side geometry as well. As we cansee, the spin components change in the whole availablerange between −1 and 1, and their behavior is rather differentfor the cases when the electron can enter the array throughany of the ﬁve terminals or only through the one attached toring /H2085331/H20854. This phenomenon together with other spin- dependent interference effects 38–44can lead to spin sensitive quantum networks. B. Effect of pointlike scatterers Now we will investigate to what extent the conductance properties are modiﬁed by the presence of random scatterers.Although high mobility samples have already become avail-able /H20849such that at cryogenic temperatures transport is found to be ballistic over tens of microns /H20850, considering also the effects caused by scattering events provides a more realisticdescription for most cases. To this end we introduce pointlikescattering centers between the rings. Note that attachingleads to rings and different rings to each other may lead toscattering, which is why the scattering centers are chosen to00.20.40.60.81 0123456789 1 0ωSO/Ω19202122232425ka FIG. 5. /H20849Color online /H20850The conductance G/G0/H20849G0=e2/h/H20850of a 5/H110035 rectangular array with a single input lead attached to ring /H2085331/H20854 for zero magnetic ﬂux as a function of the SOI strength and ka. 00.20.40.60.81 0123456789 1 0 ωSO/Ω-6-4-20246Φ/Φ0 FIG. 6. /H20849Color online /H20850The conductance G/G0/H20849G0=e2/h/H20850of a 5/H110035 rectangular array with a single input lead attached to ring /H2085331/H20854 forka=19.57 as a function of the SOI strength and the magnetic ﬂux/H9021 /H20849in units of /H90210=h/e/H20850.00.20.40.6 -6-4-20246Φ/Φ0(a) 00.20.40.6 -6-4-20246Φ/Φ0(b) 00.20.40.6 0123456789 1 0ωSO/Ω-6-4-20246Φ/Φ0(c) FIG. 7. /H20849Color online /H20850The probabilities of the /H20849a/H20850/H20841↑x/H20856,/H20849b/H20850/H20841↑y/H20856, and /H20849c/H20850/H20841↑z/H20856outputs at ring /H2085355/H20854o fa5/H110035 rectangular array with one input lead /H20849attached to ring /H2085331/H20854/H20850forka=19.6 as a function of the SOI strength and the magnetic ﬂux /H9021 /H20849in units of /H90210=h/e/H20850. The incoming spin state is chosen to be /H20841↑z/H20856.KÁLMÁN et al. PHYSICAL REVIEW B 78, 125306 /H208492008 /H20850 125306-6be placed in the junctions. At the end of this section we shall return to the question to what extent the transmission prop-erties depend on the positions of the scattering centers. At each point jwhere two rings touch each other, we consider an additional Dirac-delta potential of the form /H9257j/H9254/H20849j/H20850. Here /H9257jrepresent independent normally distributed random variables with zero mean and root-mean-square de-viation D. By tuning Dwe can model weak disturbances /H20849small D/H20850as well as the case when frequent scattering events completely change the character of the transport process/H20849corresponding to large values of D/H20850. As shown in Fig. 9, the most general consequence of these random scattering events is the overall decrease in theconductance. However, for strong enough disturbance, moreinteresting effects can be seen, namely, the splitting of theAB peaks. Note that the scattering has the most dramaticeffect for the AB resonances, i.e., /H9021=n/H9021 0, and the least for the antiresonance condition, i.e., /H9021=/H20849n+1 /2/H20850/H90210. We want to stress that the model we considered /H20849random elastic- scattering processes in single-electron approximation /H20850is similar to the case when the Al’tshuler-Aronov-Spivak/H20849AAS /H20850effect 45is expected to survive in a single ring. Our results for a more complex geometry indicate similar physi-cal consequences of the scattering events: introduction ofnew peaks in the AB oscillations. In fact, the Fourier spec-trum of the conductance shown in Fig. 10clearly indicates that for strong enough disturbance, the peaks correspondingto oscillations with a period of 2 /H9021//H9021 0are stronger than the AB peaks. Let us note that phenomena related to the AASeffect have recently been predicted for a single ring 46and were detected in the case of ring arrays.26 Finally we return to the stripes shown in Fig. 3, where the conductance is negligible. According to Sec. III A, destruc-tive interference is responsible for the appearance of thesestripes. Therefore we expect that when scattering events de-stroy phase coherence, conductance should increase. This ef-fect can be seen in Fig. 11, where the conductance is plotted as a function of the SOI strength for different root-mean-square deviations Dof the random variables. As it is shown by this ﬁgure, for most values of /H9275SO //H9024, the conductance is signiﬁcantly increased in this region, although it is negligiblein the exact ballistic case /H20849D=0 /H20850. On the other hand, how- ever, Gis practically zero around /H9275SO //H9024=7.9, independently from the strength of the disturbance. This effect is related tosingle-ring interferences: having investigated the currentsand spinor valued wave functions in the network, we foundthat for this parameter set /H20849ka, /H9275SO, and/H9021/H20850, the input rings /H20849/H2085311/H20854–/H2085351/H20854/H20850are essentially totally opaque for the electrons, i.e., they basically do not enter the second column of thenetwork. Clearly, in this case scattering centers in the junc-tions cannot modify the transmission properties. However,this kind of effects appears only for certain special parametersets. We found that the positions of the scattering centers fora single ring are important, but in a system of two rings thiseffect is already remarkably weaker. The transmission prop-erties of larger arrays are usually determined by global /H20849i.e.,FIG. 8. The spin transformation properties of a 5 /H110035 array with input leads attached to all rings and only to ring /H2085331/H20854/H20849black and gray curves, respectively /H20850. The zcomponent of the normalized spin states transmitted via the output terminals attached to ring /H2085325/H20854 /H20849solid line /H20850and ring /H2085345/H20854/H20849dashed line /H20850. The incoming spin state is chosen to be /H20841↑z/H20856. FIG. 9. /H20849Color online /H20850The conductance G /H20849in units of G0 =e2/h/H20850o fa5 /H110035 rectangular array with and without pointlike ran- dom scatterers between the rings as a function of the magnetic ﬂux/H9021 /H20849in units of /H9021 0=h/e/H20850forka=20.2 and /H9275SO //H9024=13.0.FIG. 10. Fourier spectra of the data shown in Fig. 9. Notice that the relative weight of peaks corresponding to 2 /H9021//H90210oscillations increases when scattering effects are introduced.MAGNETOCONDUCTANCE OF RECTANGULAR ARRAYS … PHYSICAL REVIEW B 78, 125306 /H208492008 /H20850 125306-7involving all the rings /H20850interferences when for strong enough disturbance the positions of the scattering centers play usu-ally no signiﬁcant role. IV. SUMMARY In this paper we calculated the spin-dependent transport properties of two-dimensional ring arrays. We applied gen-eral boundary conditions for the case of single-quantumrings, which allowed the construction of arrays of suchrings as building blocks. The magnetoconductance of two-dimensional arrays of 3 /H110033, 4/H110034, and 5 /H110035 quantum rings exhibited Aharonov-Bohm and Aharonov-Casheroscillations. 28We also determined the spin-resolved trans- mission probabilities of the arrays and found signiﬁcant spinrotations depending on the SOI strength. We introducedpointlike random scattering centers between the rings, which,for strong enough disturbance, resulted in the splitting of theAB peaks. We note that an array of quantum rings with local /H20849ring by ring /H20850modulation of the SOI can lead to novel effects in spin state transformation of electrons. 47 ACKNOWLEDGMENTS This work was supported by the Flemish-Hungarian Bi- lateral Programme, the Flemish Science Foundation /H20849FWO- Vl/H20850, the Belgian Science Policy, and the Hungarian Scientiﬁc Research Fund /H20849OTKA /H20850under Contracts No. T48888, No. M36803, and No. M045596. P.F. was supported by a J.Bolyai grant of the Hungarian Academy of Sciences. Wethank J. Sólyom for enlightening discussions. APPENDIX Here we present the detailed analytic expressions of the scattering problem for general two- and four-terminal rings,in which SOI and a perpendicular magnetic ﬁeld are pre-sent, the latter of which is considered as a perturbation.As we have shown in Sec. II, it is sufﬁcient to consideronly one input terminal and determine the connection be-tween the input and output states, i.e., the reﬂection andtransmission matrices, since the more general boundary con-dition of having inputs on all terminals is just a superpositionof such cases with an appropriate rotation of the matrices/H20851see Eqs. /H2084910/H20850and /H2084911/H20850/H20852. Considering f Ias the only input /H20851i.e., fi/HS11005I=0, in Figs. 1/H20849a/H20850and1/H20849c/H20850/H20852, requiring the continuity of the wave functions, and applying Grifﬁth boundaryconditions 32,37at the junctions in both cases, we can obtain the reﬂection matrices RˆfIand R˜fIof the two-terminal ring and of the four-terminal ring, respectively. Both can be writ-ten in a form analogous to that of R fIof the three-terminal case given by Eq. /H208497/H20850with /rho1ˆ/H20849/H9262/H20850=4k2a2 yˆ/H20849/H9262/H20850/H20853sin /H20849q/H20849/H9262/H20850/H92531/H20850sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92531/H20850/H20850 +iq/H20849/H9262/H20850sin /H208492q/H20849/H9262/H20850/H9266/H20850/H20854 and /rho1˜/H20849/H9262/H20850=2ka y˜/H20849/H9262/H20850/H20853k3a3/H20851cos /H208492q/H20849/H9262/H20850/H9266/H20850+ cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92532−/H92531/H20850/H20850− cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92532/H20850/H20850+ cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92531/H20850/H20850 − cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92532+/H92531/H20850/H20850− cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92533/H20850/H20850+ cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92532/H20850/H20850− cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20850/H20852 +2ik2a2q/H20849/H9262/H20850/H20851sin /H208492q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92532/H20850/H20850− 3 sin /H208492q/H20849/H9262/H20850/H9266/H20850+ sin /H208492q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92531/H20850/H20850+ sin /H208492q/H20849/H9262/H20850/H20849/H9266−/H92532+/H92531/H20850/H20850/H20852 +4ik2a2q/H20849/H9262/H20850/H20851sin /H208492q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20850− sin /H208492q/H20849/H9262/H20850/H20849/H9266−/H92533/H20850/H20850/H20852−4ka/H20849q/H20849/H9262/H20850/H208502/H20851cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92533/H20850/H20850+ cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92532/H20850/H20850 + cos /H208492q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20850− 3 cos /H208492q/H20849/H9262/H20850/H9266/H20850/H20852−8i/H20849q/H20849/H9262/H20850/H208503sin /H208492q/H20849/H9262/H20850/H9266/H20850/H20854, respectively. Here yˆ/H20849/H9262/H20850=k2a2/H20853cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20852− cos /H208512q/H20849/H9262/H20850/H9266/H20852/H20854+4ikaq/H20849/H9262/H20850sin /H208512q/H20849/H9262/H20850/H9266/H20852−4 /H20851q/H20849/H9262/H20850/H208522/H20853cos /H20851/H20851/H20849−1 /H20850/H9262+1w+2/H9278/H20852/H9266/H20852+ cos /H208512q/H20849/H9262/H20850/H9266/H20852/H20854, y˜/H20849/H9262/H20850=1 6 /H20849q/H20849/H9262/H20850/H208504/H20853cos /H20851/H20851/H20849−1 /H20850/H9262+1w+2/H9278/H20852/H9266/H20852+ cos /H208512q/H20849/H9262/H20850/H9266/H20852/H20854−3 2ika /H20851q/H20849/H9262/H20850/H208523sin /H208512q/H20849/H9262/H20850/H9266/H20852+2 4k2a2/H20851q/H20849/H9262/H20850/H208522cos /H208512q/H20849/H9262/H20850/H9266/H20852 −4k2a2/H20851q/H20849/H9262/H20850/H208522/H20853cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533/H20850/H20852+ cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532/H20850/H20852+ cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20852+ cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92531/H20850/H20852 + cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92532/H20850/H20852+ cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532+/H92531/H20850/H20852/H20854−8ik3a3q/H20849/H9262/H20850sin /H208512q/H20849/H9262/H20850/H9266/H20852+4ik3a3q/H20849/H9262/H20850/H20851sin /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92532/H20850/H20852FIG. 11. The conductance G/H20849in units of G0=e2/h/H20850o fa5 /H110035 rectangular array with pointlike random scatterers between the ringsfor different root-mean-square deviations Das a function of the SOI strength for ka=19.6 and /H9021=0.3/H9021 0.KÁLMÁN et al. PHYSICAL REVIEW B 78, 125306 /H208492008 /H20850 125306-8− sin /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533/H20850/H20852+ sin /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532+/H92531/H20850/H20852+ sin /H208512q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20852/H20852+k4a4/H20853cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92532−/H92531/H20850/H20852+ cos /H208512q/H20849/H9262/H20850/H9266/H20852 + cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92531/H20850/H20852− cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533+/H92532/H20850/H20852− cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532+/H92531/H20850/H20852− cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92533/H20850/H20852 + cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92532/H20850/H20852− cos /H208512q/H20849/H9262/H20850/H20849/H9266−/H92531/H20850/H20852/H20854, where the angles /H9253iare deﬁned in Figs. 1/H20849a/H20850and1/H20849c/H20850. The transmission matrices TˆfIof the two-terminal ring and T˜ nfI/H20849n =1,2,3 /H20850of the four-terminal ring can be given in an analogous form to that of the transmission matrices TnfIof the three- terminal one given by Eq. /H208498/H20850with /H9270ˆ/H20849/H9262/H20850=4ikaq/H20849/H9262/H20850 yˆ/H20849/H9262/H20850ei/H92531/H20849/H20849−1 /H20850/H9262+1w/2+/H9278/H20850/H20851sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92531/H20850/H20850−e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850sin /H20849q/H20849/H9262/H20850/H92531/H20850/H20852, and /H9270˜1/H20849/H9262/H20850=4kaq/H20849/H9262/H20850 y˜/H20849/H9262/H20850ei/H92531/2/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850/H11003/H20853ik2a2/H20851sin /H20849q/H20849/H9262/H20850/H208492/H9266−2/H92533+2/H92532−/H92531/H20850/H20850− sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92531/H20850/H20850+ sin /H20849q/H20849/H9262/H20850/H208492/H9266−2/H92532+/H92531/H20850/H20850 − sin /H20849q/H20849/H9262/H20850/H208492/H9266−2/H92533+/H92531/H20850/H20850/H20852−2kaq/H20849/H9262/H20850/H20851cos /H20849q/H20849/H9262/H20850/H208492/H9266−2/H92532+/H92531/H20850/H20850− 2 cos /H20849q/H20849/H9262/H20850/H208492/H9266−/H92531/H20850/H20850+ cos /H20849q/H20849/H9262/H20850/H208492/H9266−2/H92533+/H92531/H20850/H20850/H20852 +4i/H20849q/H20849/H9262/H20850/H208502/H20851e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850sin /H20849q/H20849/H9262/H20850/H92531/H20850− sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92531/H20850/H20850/H20852/H20854, /H9270˜2/H20849/H9262/H20850=4kaq/H20849/H9262/H20850 y˜/H20849/H9262/H20850ei/H92532/2/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850/H20853−2kaq/H20849/H9262/H20850/H20851e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850cos /H20849q/H20849/H9262/H20850/H92532/H20850−e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850cos /H20849q/H20849/H9262/H20850/H208492/H92531−/H92532/H20850/H20850 + cos /H20849q/H20849/H9262/H20850/H208492/H9266−2/H92533+/H92532/H20850/H20850− cos /H20849q/H20849/H9262/H20850/H208492/H9266−/H92532/H20850/H20850/H20852+4i/H20849q/H20849/H9262/H20850/H208502/H20851e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850sin /H20849q/H20849/H9262/H20850/H92532/H20850− sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92532/H20850/H20850/H20852/H20854, /H9270˜3/H20849/H9262/H20850=4kaq/H20849/H9262/H20850 y˜/H20849/H9262/H20850ei/H92533/2/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850/H20853ik2a2e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850/H20851sin /H20849q/H20849/H9262/H20850/H92533/H20850+ sin /H20849q/H20849/H9262/H20850/H208492/H92531−/H92533/H20850/H20850− sin /H20849q/H20849/H9262/H20850/H208492/H92532−/H92533/H20850/H20850 + sin /H20849q/H20849/H9262/H20850/H208492/H92532−2/H92531−/H92533/H20850/H20850/H20852−2kaq/H20849/H9262/H20850e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850/H208512 cos /H20849q/H20849/H9262/H20850/H92533/H20850− cos /H20849q/H20849/H9262/H20850/H208492/H92531−/H92533/H20850/H20850− cos /H20849q/H20849/H9262/H20850/H208492/H92532−/H92533/H20850/H20850/H20852 +4i/H20849q/H20849/H9262/H20850/H208502/H20851e−i/H9266/H20849/H20849−1 /H20850/H9262+1w+2/H9278/H20850sin /H20849q/H20849/H9262/H20850/H92533/H20850− sin /H20849q/H20849/H9262/H20850/H208492/H9266−/H92533/H20850/H20850/H20852/H20854, respectively. benedict@physx.u-szeged.hu 1M. 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PhysRevB.100.104418.pdf	PHYSICAL REVIEW B 100, 104418 (2019) Numerical observation of a glassy phase in the three-dimensional Coulomb glass Amin Barzegar,1Juan Carlos Andresen,2Moshe Schechter,2and Helmut G. Katzgraber3,1,4 1Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA 2Department of Physics, Ben Gurion University of the Negev, Beer Sheva 84105, Israel 3Microsoft Quantum, Microsoft, Redmond, Washington 98052, USA 4Santa Fe Institute, Santa Fe, New Mexico 87501, USA (Received 1 December 2018; revised manuscript received 26 August 2019; published 13 September 2019) The existence of an equilibrium glassy phase for charges in a disordered potential with long-range electrostatic interactions has remained controversial for many years. Here we conduct an extensive numerical study of thedisorder-temperature phase diagram of the three-dimensional Coulomb glass model using population annealingMonte Carlo to thermalize the system down to extremely low temperatures. Our results strongly suggest that,in addition to a charge order phase, a transition to a glassy phase can be observed, consistent with previousanalytical and experimental studies. DOI: 10.1103/PhysRevB.100.104418 I. INTRODUCTION The existence of disorder in strongly interacting elec- tron systems—which can be realized by introducing ran-dom impurities within the material, e.g., a strongly dopedsemiconductor—plays a signiﬁcant role in understandingtransport phenomena in imperfect materials and bad metals,as well as in condensed matter in general. When the density of impurities is sufﬁciently large, electrons become localized via the Anderson localization mechanism [ 1] and the long-range Coulomb interactions are no longer screened. This, in turn,leads to the depletion of the single-particle density of states(DOS) near the Fermi level, as ﬁrst proposed by Pollak [ 2] and Srinivasan [ 3], thus forming a pseudogap. Later, Efros and Shklovskii [ 4] (ES) solidiﬁed this observation by describing the mechanisms involved in the formation of this pseudogap.The ES theory explains how the hopping (DC conductivity)within a disordered insulating material is modiﬁed in thepresence of a pseudogap, also referred to as the “Coulombgap.” Numerous analytic studies have predicted, [ 5–14], as well as experimental studies observed [ 15–29], the emergence of glassy properties in such disordered insulators, leading to the so-called “Coulomb glass” (CG) phase. Experimentally,to date, none of the aforementioned studies have observed atrue thermodynamic transition into a glass phase but ratherhave found evidence of nonequilibrium glassy dynamics, i.e.,dynamic phenomena that are suggestive of a glass phase, suchas slow relaxation, aging, memory effects, and alterations inthe noise characteristics. Theoretically, more recent seminalmean-ﬁeld studies by Pankov and Dobrosavljevi ´c[12], as well as Müller and Pankov [ 30], have shown that there exists a marginally stable glass phase within the CG model whosetransition temperature T cdecreases as Tc∼W−1/2for large enough disorder strength W, and is closely related to the for- mation of the Coulomb gap. Whether the results of the mean- ﬁeld approach can be readily generalized to lower space dimensions is still uncertain. However, as we show in thiswork, the mean-ﬁeld results of Ref. [ 12] quantitatively agreewith our numerical simulations in the charge-ordered regime (see Fig. 1) with similar values for the critical disorder W c where the charge-ordered phase is suppressed. The critical temperatures Tcfor the glassy phase, on the other hand, are substantially smaller than in the mean-ﬁeld predictions. This,in turn, suggests that the mean-ﬁeld approach of Ref. [ 12]i n - cludes the ﬂuctuations of the uniform charge order collectivemodes, but not of the glassy collective modes. There have been multiple numerical studies that attempt to both understand the DOS, as well as the nature of thetransitions of the CG model. In fact, there has even beensome slight disagreement as to what the theoretical modelto simulate should be with some arguing for lattice disorderto introduce randomness into the model [ 31,32] and others suggesting that the disorder should be introduced via random biases. Numerically, a Coulomb gap in agreement with the EStheory has been observed in multiple studies. However, thereis no consensus in the vast numerical work [ 31,33–52]o nt h e existence of a thermodynamic transition into a glassy phase.Nonequilibrium approaches suggest the existence of glassybehavior; however, thermodynamic simulations have failed to detect a clear transition. In this paper we investigate the phase diagram of the CG model using Monte Carlo simulations in three spatial dimen-sions. For the ﬁnite-temperature simulations we make useof the population annealing Monte Carlo (PAMC) algorithm[53–57] which enables us to thermalize for a broad range of disorder values down to unprecedented low temperatures pre- viously inaccessible. In addition, we argue that the detectionof a glass phase requires a four-replica correlation length, ascommonly used in spin-glass simulations in a ﬁeld [ 58,59]. Our main result is shown in Fig. 1. Consistently with previous numerical and analytical studies [ 12,47,60] we ﬁnd a charge ordered (CO) phase for disorders lower than W c=0.131(2) where electrons and holes form a checkerboard-like crystal.This is in close analogy with the classical Wigner crystal [ 61] which happens at low electron densities where the potentialenergy dominates the kinetic energy resulting in an ordered 2469-9950/2019/100(10)/104418(12) 104418-1 ©2019 American Physical SocietyAMIN BARZEGAR et al. PHYSICAL REVIEW B 100, 104418 (2019) CO CG Plasma 00.020.040.060.080.10.120.14 00 .20 .40 .60 .811 .21 .4T W FIG. 1. Phase diagram of the three-dimensional Coulomb glass model. There is a charge order (CO) phase for W/lessorsimilar0.131 where electrons and holes form a checkerboard-like crystal. For W/greaterorsimilar0.131 the system undergoes a glassy transition into the Coulomb glass(CG) phase, albeit at considerably lower temperatures than in the CO phase. The dashed lines indicate extrapolations where numerical simulations are not available. arrangement of the charges. It should however be noted that atW=0 the lattice model, unlike in the continuum case, is not a standard Wigner crystal [ 62] because the system exhibits a pseudogap in the excitation spectrum (unrelated tothe Coulomb gap) prior to entering the charge-ordered phase.For disorders larger than W cwe ﬁnd strong evidence of a thermodynamic glassy phase restricted to temperatures whichare approximately one order of magnitude smaller comparedto the CO temperature scales. This, in turn, suggests that,indeed, a thermodynamic glassy phase can exist in experimen-tal systems where typically off-equilibrium measurementsare performed. It also resolves the long-standing controversywhere numerical simulations were unable to conclusivelydetect a thermodynamic glassy phase while mean-ﬁeld theorypredicted such a phase. We note that for the disorder strengthvalues studied, we are unable to discern a monotonic decreasein the critical temperature, as suggested by mean-ﬁeld theory. The paper is structured as follows. In Sec. IIwe introduce the CG model, followed by the details of the simulation inSec. III. Section IVis dedicated to the results of the study. Concluding remarks are presented in Sec. V. II. MODEL The CG model in three spatial dimensions is described by the Hamiltonian H=e2 2κ/summationdisplay i/negationslash=j(ni−ν)1 \|rij\|(nj−ν)+/summationdisplay iniφi, (1) where κ=4π/epsilon10,ni∈{0,1}, andνis the ﬁlling factor. The disorder φiis an on-site Gaussian random potential, i.e., P(φi)=(2πW2)−1/2exp (−φ2 i/2W2). At half ﬁlling ( ν= 1/2) the CG model can conveniently be mapped to a long- range spin model via si=(2ni−1). The Hamiltonian can be made dimensionless by choosing the units such that e2/κ=1anda=1i nw h i c h ais the lattice spacing. We thus simulate H=1 8/summationdisplay i/negationslash=jsisj \|rij\|+1 2/summationdisplay isiφi, (2) where si∈{ ± 1}represent Ising spins. III. SIMULATION DETAILS In order to reduce the ﬁnite-size effects we use periodic boundary conditions. Special care has to be taken to deal withthe long-range interactions. We make inﬁnitely many periodiccopies of each spin in all spatial directions, such that each spininteracts with all other spins inﬁnitely many times. We usethe Ewald summation technique [ 63,64], such that the double summation in Eq. ( 2) can be written in the following way: 1 2N/summationdisplay i=1N/summationdisplay j=1sisj/bracketleftbig f(1) ij+f(2) ij+f(3) ij+f(4) ij/bracketrightbig , (3) where the terms fijare deﬁned as f(1) ij=1 4/prime/summationdisplay nerfc(α\|rij+nL\|) \|rij+nL\|, (4) f(2) ij=π N/summationdisplay k/negationslash=0e−k2/4α2 k2cos(krij), (5) f(3) ij=π 3Nri·rj, (6) f(4) ij=−α 2√πδij. (7) Here, erfc is the complimentary error function [ 65],αis a regularization parameter, and k=2πn/Lis the reciprocal lattice momentum. The vector index nin Eq. ( 4) runs over the lattice copies in all spatial directions and the prime indicatesthatn=0 is not taken into account in the sum when i=j. For numerical purposes, the real and reciprocal space sum-mations, i.e., Eqs. ( 4) and ( 5), respectively, are bounded by \|r ij+nL\|<rcandk<2πnc/L. The parameters α,rc, and ncare tuned to ensure a stable convergence of the sum. We ﬁnd that 2 <α< 4,nc/greaterorsimilar4L, and rc=L/2 are sufﬁcient for the above purpose. We use population annealing Monte Carlo (PAMC) [53–57] to thermalize the system down to extremely low temperatures. In PAMC, similarly to simulated annealing(SA) [ 66], the system is equilibrated toward a target tempera- ture starting from a high temperature following an annealingschedule. PAMC, however, outperforms SA by introducingmany replicas of the same system and thermalizing them inparallel. Each replica is subjected to a series of Monte Carlomoves and the entire pool of replicas is resampled accordingto an appropriate Boltzmann weight. This ensures that the sys-tem is equilibrated according to the Gibbs distribution at eachtemperature. For the simulations we use particle-conservingdynamics to ensure that the lattice half ﬁlling is kept constant,together with a hybrid temperature schedule linear in βand linear in T[57]. We use the family entropy of population annealing [ 55] as an equilibration criterion. Hard samples are resimulated with a larger population size and number ofsweeps until the equilibration criterion is met. Note that we 104418-2NUMERICAL OBSERV ATION OF A GLASSY PHASE IN … PHYSICAL REVIEW B 100, 104418 (2019) TABLE I. PAMC simulation parameters used for the ﬁnite- temperature simulations in the CO phase ( W/lessorequalslant0.131). Lis the linear system size, R0is the initial population size, Mis the number of Metropolis sweeps, T0is the lowest temperature simulated, NTis the number of temperatures, and Nsais the number of disorder re- alizations. Note that the values in the table vary slightly for different values of the disorder W. LR 0 MT 0 NT Nsa 42 ×10410 0.05 401 5000 65 ×10410 0.05 601 5000 81 ×10520 0.05 801 2000 10 2 ×10520 0.05 1001 1000 12 5 ×10530 0.05 1201 500 have independently examined the accuracy of the results, as well as the quality of thermalization for system sizes up toL=8 using parallel tempering Monte Carlo [ 67]. Both data from PAMC and parallel tempering Monte Carlo agree withinerror bars. We investigate the phase diagram of the CG modelusing ﬁxed values of the disorder width, i.e., vertical cuts ontheW-Tplane. Further details of the simulation parameters can be found in Tables IandIIfor the CO and CG phases, respectively. IV . RESULTS A. Charge-ordered phase To characterize the CO phase, we measure the speciﬁc heat capacity cv=Cv/N(only used to extract critical exponents; see Appendix Bfor details), staggered magnetization ms=1 NN/summationdisplay i=1σi, (8) where σi=(−1)xi+yi+zisiandN=L3the number of spins, as well as the disconnected and connected susceptibility ¯χ=N/bracketleftbig/angbracketleftbig m2 s/angbracketrightbig/bracketrightbig , (9) χ=N/bracketleftbig/angbracketleftbig m2 s/angbracketrightbig −/angbracketleft\|ms\|/angbracketright2/bracketrightbig . (10) In addition, we measure the Binder ratio g[68], g=1 2/parenleftBigg 3−/bracketleftbig/angbracketleftbig m4 s/angbracketrightbig/bracketrightbig /bracketleftbig/angbracketleftbig m2s/angbracketrightbig/bracketrightbig2/parenrightBigg , (11) TABLE II. PAMC simulation parameters used for the ﬁnite- temperature simulations in the CG phase ( W>0.131). For details see the caption of Table I. Note that the values in the table vary slightly for different values of the disorder W. LR 0 MT 0 NT Nsa 42 ×10420 0.004 401 100000 65 ×10430 0.004 601 50000 81 ×10540 0.004 801 30000 10 2 ×10560 0.004 1001 200000.5124 0.11 0 .115 0 .12 0 .125 0 .13ξ/L TL=4 L=6 L=8 L=1 0 L=1 2 W=0.05(b) 0.250.512 0.122 0 .126 0 .13 0 .134ξ/L TL=4 L=6 L=8 L=1 0 L=1 2 W=0.0(a) 1234 −0.06−0.03 0 0 .03 0 .06Tc=0.1187(3) ν=0.87(14)ξ/L L1/ν(T−Tc)L=6 L=8 L=1 0 L=1 2 P3(x) (d) 0.40.81.21.6 −0.05 0 0 .05 0 .1Tc=0.1284(1)ν=0.76(4)ξ/L L1/ν(T−Tc)L=6 L=8 L=1 0 L=1 2 P3(x)(c) FIG. 2. Finite-size correlation length per system size ξ/Lver- sus temperature Tfor various disorder strengths. (a) No disorder, (b) small disorder ( W=0.05). In both cases we observe a crossing of the data for different system sizes, suggesting a phase transition between a disordered electron plasma and a CO phase. (c), (d) Finite- size scaling analysis used to determine the best estimates for thecritical temperature T c, as well as the critical exponent νat the aforementioned disorder values. Note that the smallest system size is left out of the analysis for better accuracy. The transition temperatureT cof the CO phase decreases as the disorder grows. and the ﬁnite-size correlation length ξ/L[69–71], deﬁned via ξ=1 2s i n (\|kmin\|/2)/parenleftbiggχ(0) χ(kmin)−1/parenrightbigg1/2 , (12) where kmin=(2π/L,0,0) is the smallest nonzero wave vector and χ(k)=1 N/summationdisplay ij[/angbracketleftσiσj/angbracketright]e x p ( ik·rij) (13) is the Fourier transform of the susceptibility. Furthermore, /angbracketleft ···/angbracketright represents a thermal average and [ ···]i sa na v e r a g e over disorder. According to the scaling ansatz, in the vicinityof a second-order phase transition temperature T c,a n yd i - mensionless thermodynamic quantity such as the Binder ratioand the ﬁnite-size correlation length divided by linear systemsize will be a universal function of x=L 1/ν(T−Tc), i.e., g=˜Fg(x) and ξ/L=˜Fξ(x), where νis a critical exponent. Therefore, an effective way of probing a phase transition isto search for a point where gorξ/Ldata intersect. Given the universality of the scaling functions ˜F gand ˜Fξ, if one plots gorξ/Lversus x=L1/ν(T−Tc), the data for all system sizes must collapse onto a common curve. Because we aredealing with temperatures close to T c, we may approximate this universal curve by an appropriate mathematical functionsuch as a third-order polynomial f(x)=P 3(x) in the case of ξ/Lor a complimentary error function f(x)=1 2erfc( x) when studying the Binder cumulant. Hence, by ﬁtting f(x)t ot h e data with Tcandνas part of the ﬁt parameters, we are able to determine their best estimates. The statistical error bars ofthe ﬁt parameters are calculated by bootstrapping over thedisorder realizations. In Fig. 2we show the simulation data 104418-3AMIN BARZEGAR et al. PHYSICAL REVIEW B 100, 104418 (2019) TABLE III. Critical parameters of the plasma-CO phase transition at different disorder values. The exponents, except for ν, change with disorder. Note that at T=0, the exponents αandγhave been calculated in a different way (see text in Appendix B). Model WT c να / ν β / ν ¯γ/ν γ/ν CG 0.000 0.1284(1) 0.76(4) 0.550(2) 0.42(1) 2.41(1) 2.05(2) CG 0.050 0.1187(3) 0.87(14) 0.418(25) 0.305(19) 2.67(2) 1.79(3) CG 0.131(2) 0.000 0.71(5) 0.006(31) 0.154(5) 2.88(1) 1.55(4) as well as the ﬁnite-size scaling (FSS) plots for ξ/Lat two different disorder values. Crossings can clearly be observedwhich signals a phase transition into the CO phase. Simulatingmultiple values of W, we observe a phase transition between a disordered electron plasma and a CO phase for W<0.131(2), consistent with previous studies [ 12,47,60]. The CO phase is a checkerboard-like crystal [ 61], where electrons and holes form a regular lattice as the potential energy dominates thekinetic energy at low temperatures. We have also conducted zero-temperature simulations us- ing simulated annealing to determine the zero-temperaturecritical disorder W cthat separates the CO from the CG phase. We average over Nsa=2048 different disorder realizations for disorders W>0.10 and Nsa=512 for W/lessorequalslant0.10. Each disorder realization is restarted at least at 20 different initialrandom spin conﬁgurations and at each temperature stepequilibrated N eqMonte Carlo steps. If at least 15% of the runs reach the same minimal energy conﬁguration, we assumethat the chosen N eqwas large enough and that the reached conﬁguration is likely the ground state. If less than 15% of theconﬁgurations reach the minimal state, we increase N eqand rerun the simulation until the 15% threshold is achieved. Forthe largest simulated system size ( L=8) and large disorders, typical equilibration times are N eq=227Monte Carlo sweeps. To estimate Wc, we use the Binder ratio deﬁned in Eq. ( 11) which by deﬁnition quickly approaches 1 when T→0 within the CO phase. Therefore, in order to retain a good resolutionof a putative transition, we use an alternative quantity /Gamma1which is deﬁned in the following way [ 49]: /Gamma1=− ln(1−g). (14) Close to W c, we may assume the following ﬁnite-size scaling behavior for /Gamma1: /Gamma1=˜F/Gamma1[L1/ν(W−Wc)]. (15) Asgis restricted to 0 /lessorequalslantg/lessorequalslant1 with a step-function-like shape, we may use a complementary error function1 2erfc(x−μ σ)t o represent the universal scaling function ˜F/Gamma1in which x= L1/ν(W−Wc) and Wc,ν,μ,σare the ﬁt parameters. The ﬁt is shown in Fig. 3where we obtain Wc=0.131±0.002 and ν=0.71±0.05. In Table III(Appendix B) we list the values of the critical exponents for the plasma-CO phase transition for variousdisorder values Wafter a comprehensive FSS analysis of different observables. Note that we have used the methods de-veloped in Ref. [ 72] to compute the exponents αandγatT= 0. An important observation one can promptly make is thatthe exponents—except for νwhich is universal—vary with disorder. This can be attributed to the fact that the perturba-tions at large length scales are contested between random-ﬁeldﬂuctuations which have static nature and dynamic thermalﬂuctuations [ 73–75]. At W=0, the perturbations are purely thermal, while at T=0, the random ﬁeld completely domi- nates. At such large length scales, the interactions within thecharge-ordered phase resemble the random-ﬁeld Ising model(RFIM) [ 76–79] with short-range bonds; namely, screening takes place. This can be understood by remembering that thedynamics of the system is constrained by charge conservation.In the spin language, excitations are no longer spin ﬂipsbut spin-pair ﬂip-ﬂops owing to the conservation of totalmagnetization. For instance, one can create a local excitationwhile preserving charge neutrality by moving a number ofelectrons out of a subdomain in the CO phase. The excessenergy of such a domain scales like its surface, similarly to theshort-range ferromagnetic Ising model. It is worth mentioningthat the Imry-Ma [ 80] picture gives a lower critical dimension of 2 for discrete spins with short-range interactions. Hencethree-dimensional Ising spins, such as in the RFIM, are stableto small random ﬁelds as we also ﬁnd here. Returning to the discussion of the critical exponents, we note that scaling relations such as γ=β(δ−1)=(2−η)ν, (16) as well as the modiﬁed hyperscaling relation (d−θ)ν=2−α=2β+γ, (17) 0123456 −0.50 0 .511 .522 .53Wc=0.131(2) ν=0.71(5)Γ x=L1/ν(WWc)L=4 L=6L=8 −log[1−1 2erfc(x−μ σ)] T=0.0 FIG. 3. Zero-temperature simulation results for the plasma-CO phase transition. The quantity /Gamma1deﬁned in Eq. ( 14)i su s e dt o perform a ﬁnite-size scaling analysis. We conclude that the CO phase terminates at Wc=0.131(2). The statistical error bars are estimates by bootstrapping over disorder instances. erfc( x)i st h e complimentary error function which is used to ﬁt the Binder ratio data (see text). 104418-4NUMERICAL OBSERV ATION OF A GLASSY PHASE IN … PHYSICAL REVIEW B 100, 104418 (2019) can be utilized to obtain estimates for the critical exponents η, θ, andδ. For instance, using the values in Table III, we see that η(W=0.0)=−0.05(2) and η(W=0.05)=0.22(1). Near criticality, the correlation functions decay as a power of dis-tance, i.e., G(x)∼1/\|x\| d−2+η. The fact that the exponent ηis slightly negative for W=0.0 shows that correlation between the spins remains in effect over a much longer distance in theabsence of disorder. Physically this is plausible, as disordertends to decorrelate the spins. B. Coulomb glass phase To examine the existence of a glassy phase in the CG model, we measure the spin-glass correlation length deﬁnedin Eq. ( 12), however, for a spin-glass order parameter, namely ξ SG=1 2s i n (\|kmin\|/2)/parenleftbiggχSG(0) χSG(kmin)−1/parenrightbigg1/2 . (18) Here, the spin-glass susceptibility χSGhas the following deﬁ- nition [ 71]: χSG(k)=1 NN/summationdisplay i=1N/summationdisplay j=1[(/angbracketleftsisj/angbracketright−/angbracketleft si/angbracketright/angbracketleftsi/angbracketright)2]eik·(ri−rj).(19) It is important to note that /angbracketleftsi/angbracketright/negationslash=0 because the Hamiltonian [Eq. ( 2)] is not symmetric under global spin ﬂips. Therefore, at least four replicas are needed to compute the connectedcorrelation function in Eq. ( 19). We start with the partition function of the system, using Eq. ( 2): Z=/summationdisplay {si}exp⎡ ⎣−β⎛ ⎝1 8/summationdisplay i/negationslash=jsisj \|rij\|+1 2/summationdisplay isiφi⎞ ⎠⎤ ⎦. (20) We may now expresses any combination of the spin moments in terms of the replicated spin variables sα iin the following way: /angbracketleftbig s11...s1k1/angbracketrightbigl1.../angbracketleftbig sm1...smkm/angbracketrightbiglm =1 Zn/summationdisplay {sα i}e−βn/summationtext α=1H[{sα i}] s1 11...s1 1k1···sn m1...sn mkm =1 n!n/summationdisplay α1...α n/angbracketleftbig sα1 11...sα1 1k1···sαn m1...sαn mkm/angbracketrightbig , (21) where n=l1+···+ lmis the total number of replicas and replica indices α1,...,α nare all distinct. As a special case, one can show (/angbracketleftsisj/angbracketright−/angbracketleft si/angbracketright/angbracketleftsj/angbracketright)2=2 4!4/summationdisplay α,β/angbracketleftbig sα isα jsβ isβ j/angbracketrightbig −2 4!4/summationdisplay α,β,γ/angbracketleftbig sα isα jsβ isγ j/angbracketrightbig +1 4!4/summationdisplay α,β,γ,λ/angbracketleftbig sα isβ isγ jsλ j/angbracketrightbig . (22) Using the above expression, the spin-glass susceptibility [Eq. ( 19)] can be written in terms of the replica overlaps as0510152025 0.02 0 .04 0 .06 0 .08 0 .1ξSG/L[Two Replicas] TW=0.80.190.20.210.220.230.24 0.004 0 .005 0 .006 0 .007 0 .008 0 .009ξSG/L[Four Replicas] TL=4 L=6 L=8 L=1 0 FIG. 4. Spin-glass correlation length divided by system size ξSG/Lcalculated using two replicas at W=0.8 versus temperature T.N oc r o s s i n gi so b s e r v e dd o w nt ov e r yl o wt e m p e r a t u r e s .T h e inset shows the same quantity using four replicas where a transition is clearly visible. Here, data points for different system sizes cross approximately at the temperature indicated by the dashed line. Thissuggests that in the presence of external ﬁelds four-replica quantities need to be used to characterize phase transitions in glassy systems. follows: χSG(k)=N 64/summationdisplay α<β[/angbracketleftqαβ(k)q∗ αβ(k)/angbracketright] −N 64/summationdisplay α4/summationdisplay β<γ[/angbracketleftqαβ(k)q∗ αγ(k)/angbracketright] +N 34/summationdisplay α<β4/summationdisplay γ<λ[/angbracketleftqαβ(k)q∗ γλ(k)/angbracketright]. (23) Once again, the indices α,β,γ, andλmust be distinct. Here, q∗ αβ(k) represents the complex conjugate of qαβ(k), and qαβ(k) is the Fourier-transformed spin overlap, i.e., qαβ(k)=1 NN/summationdisplay i=1sα isβ ieik·ri. (24) To underline the signiﬁcance of this matter, we have shown in Fig. 4the spin-glass correlation length calculated using two replicas, as has been done in some previous numericalstudies of the CG [ 31,81]. The inset shows the same quantity computed using four replicas. While the two-replica versionof the ﬁnite-size correlation length shows no sign of a CGtransition, the four-replica expression captures the existenceof a phase transition into a glassy phase. We have performed equilibrium simulations for W∈ {0.15,0.30,0.50,0.80,1.2}.I nF i g . 5we plot the four-replica spin-glass correlation length as a function of temperature atselected disorder values. Our results strongly suggest thatthere is a transition to a glassy phase which persists forrelatively large values of the disorder. This is signiﬁcant inthe sense that it conﬁrms the phase transition via replicasymmetry breaking as predicted by mean-ﬁeld theory. The 104418-5AMIN BARZEGAR et al. PHYSICAL REVIEW B 100, 104418 (2019) 0.180.190.20.21 0.004 0 .005 0 .006 0 .007 0 .008ξSG/L TL=4 L=6 L=8 L=1 0 W=1.2(d) 0.230.240.250.260.27 0.004 0 .005 0 .006 0 .007 0 .008ξSG/L TL=4 L=6 L=8 L=1 0 W=0.5(c)0.240.260.280.3 0.004 0 .006 0 .008 0 .01ξSG/L TL=4 L=6 L=8 L=1 0 W=0.3(b) 0.30.40.5 0.005 0 .006 0 .007 0 .008 0 .009ξSG/L TL=4 L=6 L=8 L=1 0 W=0.15(a) FIG. 5. Spin-glass ﬁnite-size correlation length ξSG/Las a func- tion of temperature Tat various disorder strengths W.( a )W=0.15, (b)W=0.30, (c) W=0.50, and (d) W=1.20. For W/greaterorsimilar0.15 the data for different system sizes cross, indicating a plasma-CG phase transition. Corrections to scaling must be considered to reliably estimate the value of the critical temperature Tc(see text for details). nontriviality of our ﬁndings can be better understood if one juxtaposes the CG case with that of ﬁnite-dimensional spinglasses lacking time-reversal symmetry due to an arbitrarilysmall external ﬁeld where the existence of de Almeida–Thouless [ 82] transition, except for a few rare cases [ 83,84], has been ruled out by numerous studies [ 58,85–89]. For the random-ﬁeld Ising model the droplet picture of Fisher andHuse [ 85,86] can be invoked to show the instability of the glass phase to inﬁnitesimal random ﬁelds. Yet, the CG modelis different in two signiﬁcant ways: typical compact domainsare not charge neutral, and therefore cannot be ﬂipped, andthe long range of the interactions, while it does not affectthe domain wall formation energy in the ordered phase, maybe signiﬁcant in the more complex domain formation ofthe glass phase. It is worth emphasizing here that properequilibration is key in observing a glassy phase in the CGsimulations. For instance, in Fig. 8of Appendix Awe show an example of a simulation where the crossing in the spin-glasscorrelation length is completely masked due to insufﬁcientthermalization. Some corrections to scaling must be considered in the anal- ysis in order to estimate the position of the critical temperatureand the values of the critical exponents. In the vicinity of thecritical temperature T cand to leading order in corrections to scaling, we may consider the following FSS expressions forthe spin-glass susceptibility χ SGand the ﬁnite-size two-point correlation length divided by the linear size of the system,ξ SG/L: χSG∼CχL2−η[1+AχL−ω+BχL1/ν(T−Tc)], (25) ξSG/L∼Cξ+AξL−ω+BξL1/ν(T−Tc), (26) where Aχ,Bχ,Cχ,Aξ,Bξ, and Cξare constants. In order to ﬁnd the critical temperature Tcas well as the critical exponents ν, η,ω, we perform the following procedure.(i) Estimation of Tc: Given any pair of system sizes (L1,L2)w eh a v e L1=¯L−/Delta1L/2,L2=¯L+/Delta1L/2, (27) in which /Delta1L=L2−L1and¯L=(L1+L2)/2. Using Eq. ( 26), to the leading order in /Delta1L/¯Lwe ﬁnd ξSG(Li,T) Li∼ξSG(¯L,T) ¯L −(−1)i/Delta1L 2¯L/bracketleftbigg ωAξ¯L−ω−Bξ ν¯L1/ν(T−Tc)/bracketrightbigg , (28) where the index ican take values i=1,2. One can now use Eq. ( 28) to determine the temperature T∗(L1,L2) at which the curves of ξSG/Lcross; in other words, ξSG(L1,T∗)/L1= ξSG(L2,T∗)/L2and T∗(L1,L2)∼Tc+/Theta1ξ¯L−ω−1/ν=Tc+/Theta1ξ¯L−φ. (29) Here Tcis the true critical temperature in the limit L→∞ and /Theta1ξis a constant. In Fig. 6(a), we show the Tcestimate for the case W=0.50. The best-ﬁt curve is obtained by minimizing the sum of the square of the residuals, χ2=N/summationdisplay i=1(T∗ i−Tc−/Theta1ξ¯L−φ i)2, (30) where iruns over all pairs of linear system sizes. Now we vary Tc, minimizing χ2along the way with respect to the remaining parameters. Since /Theta1ξappears linearly in the model, it can be eliminated [ 90] to reduce the optimization task to one free parameter, i.e., φ: /parenleftbigg∂χ2 ∂/Theta1ξ/parenrightbigg Tc=0⇒˜/Theta1ξ(Tc,φ)=/summationtextN i=1(T∗ i−Tc)¯L−φ i/summationtextN i=1¯L−2φ i.(31) Because there are ﬁve data points with three parameters in the original model, we have two degrees of freedom. There-fore, the probability density function (PDF) is proportional to e −χ2/2. To determine the conﬁdence intervals, we calculate the cumulative distribution function (CDF) [ 91]: Q(Tc)=/integraldisplayTc e−1 2χ2(T/prime c)dT/prime c. (32) As an example, in Fig. 6(b) we have shown the 68% con- ﬁdence interval as well as the best estimate for the criticaltemperature. (ii) Estimation of ω:F r o mE q .( 26) we observe that ξ SG(Tc)/L∼Cξ+AξL−ω. (33) Thus, using the best estimate of Tcfrom the previous step, we expect the data points of ξSG(Tc)/Las a function of L−ω to follow a straight line when ωis chosen correctly. We can therefore vary ωand measure the curvature until it vanishes at the optimal value. We have demonstrated this in Figs. 6(c) and6(d). Note that the error bar for ωis calculated using the bootstrap method. 104418-6NUMERICAL OBSERV ATION OF A GLASSY PHASE IN … PHYSICAL REVIEW B 100, 104418 (2019) 00.20.40.60.81 0.0045 0 .0048 0 .0051 0 .0054 0 .0057 0 .006Tc=0.00534+0.00018 −0.00029(b)CDF( Tc) Tc00.20.40.60.81 0.0045 0 .0048 0 .0051 0 .0054 0 .0057 0 .0061.21.622.42.8 1.21 .41 .61 .822 .22 .4dξSG(Tc)/LdT ∼L1/ν ν=0.74(5) (e)log[dξSG(Tc)/LdT] log(L)1.21.622.42.8 1.21 .41 .61 .822 .22 .4 −3.2−2.8−2.4−2−1.6 1.21 .41 .61 .822 .22 .4χSG(Tc)∼L2−η η=0.81(2) (f)log[χSG(Tc)] log(L)−3.2−2.8−2.4−2−1.6 1.21 .41 .61 .822 .22 .40.00520.00560.0060.00640.0068 56789(a)T∗(L1,L2)=Tc+Θ ξ¯L−φ Tc=0.00534(29)T∗(L1,L2) ¯L=(L1+L2)/20.00520.00560.0060.00640.0068 56789 0.220.240.260.280.3 00 .05 0 .10 .15 0 .2ω=1.24(28)Cξ=0.2534(2) (d)ξSG/L L−ωT=0.00400 T=0.00480 T=0.00534T=0.00600 T=0.00700 Cξ+AξL−ω 0.220.240.260.280.3 00 .05 0 .10 .15 0 .2−0.15−0.1−0.0500.050.10.15 00 .40 .81 .21 .62ω=1.24(28) ξSG(Tc)/L∼Cξ+AξL−ω(c)Curvature ω−0.15−0.1−0.0500.050.10.15 00 .40 .81 .21 .62 FIG. 6. Process of estimating the critical exponents, as well as the critical temperature Tcof the plasma-CG phase transition for W=0.5. Other values of Ware analyzed using the same procedure. (a) The temperatures where ξSG/Lcurves of different systems sizes cross are used to determine the critical temperature Tc. The crossing temperatures decay toward the thermodynamic limit Tc. (b) The cumulative distribution function (CDF) is constructed by minimizing χ2with respect to /Theta1ξandφwhile holding Tcconstant. The shaded region shows the 68% conﬁdence interval and the green vertical line indicates the best estimate of Tc. (c) The value of Tcobtained in the previous step is used to determine ω.A tT=Tcand optimal ω,ξSG/Lis linear as a function of L−ω; i.e., it has zero curvature as demonstrated in panel (d). (e) The critical exponent νis estimated using the derivative of ξSG/Lwith respect to temperature which scales as L1/νwhen evaluated at Tc. Some deviations are evident for the smallest system size. (f) The spin-glass susceptibility χSGatT=Tcwhich scales as L2−ηis used to determine the best estimate of the exponent η. (iii) Estimation of νandη: It is straightforward to show from Eqs. ( 25) and ( 26) that to the leading order in corrections, χSG(Tc)=CχL2−η(1+AχL−ω), (34) d dT(ξSG/L)(Tc)=BξL1/ν(1+DξL−ω), (35) in which the best estimates obtained for Tcandωare used. We see that the above quantities simply scale as χSG(Tc)∼L2−η andd dT(ξSG/L)(Tc)∼L1/νfor large enough L. Therefore, a linear ﬁt in logarithmic scale will yield the exponents νand ω. This is shown in Figs. 6(e)and6(f), respectively. The above procedure has been repeated for all other values of the disorder W. The results are summarized in Table IV of Appendix B. We observe that within the error bars, the TABLE IV . Critical parameters of the plasma-CG phase transi- tion for various values of the disorder W. The exponent νandωare independent of Wwithin error bars highlighting their universality whereas the exponent ηvaries as the disorder strength increases. WT c νωη 0.300 0.00446(25) 0.62(5) 1.26(7) 0.56(1) 0.500 0.00534(29) 0.74(5) 1.24(28) 0.82(5) 0.800 0.00590(56) 0.64(2) 1.28(20) 0 .97(5) 1.200 0.00600(16) 0.65(3) 1.33(21) 1 .09(1)critical exponents νandωare robust to disorder which under- lines the universality of these exponents. Nevertheless, largersystem sizes—currently not accessible via simulation—wouldbe needed to conclusively determine the universality class ofthe model. The fact that we observe stronger corrections toscaling for smaller disorder shows that the energy landscape isrougher due to competing interactions where ﬁnite-size effectsare accentuated. For larger values of W, on the other hand, the system becomes easier to thermalize as the disorder dominatesthe electrostatic interactions. V . CONCLUSION We have shown that, using the four-replica expressions for the commonly used observables, the CG model displaysa transition into a glassy phase for the studied system sizes,provided that large enough disorder and sufﬁciently lowtemperatures are used in the simulations (see Fig. 1for the complete phase diagram of the model). Previous numericalstudies—including a work [ 48] by a subset of us—have failed to observe the glassy phase. In this study, we areable to present strong numerical evidence for the validityof the mean-ﬁeld results in three space dimensions, whichpredicts transition to a glassy phase at large disorder viareplica symmetry breaking. Moreover, we corroborate theresults of previous studies for the low-disorder regime wherea CO phase, similar to the ferromagnetic phase in the RFIM,is observed. Interestingly, for large disorder values, the CG 104418-7AMIN BARZEGAR et al. PHYSICAL REVIEW B 100, 104418 (2019) and the RFIM are different, as the RFIM does not exhibit a transition into a glassy phase (see, for example, Ref. [ 79] and references therein). A possible reason is the combination ofthe constrained dynamics (magnetization-conserving dynam-ics) and the long-range Coulomb interactions not present inthe RFIM. These two factors can increase frustration suchthat a glassy phase can emerge. Our ﬁndings open the pos-sibility of describing electron glasses through an effective CGmodel both theoretically and numerically. Because most of theelectron glass experiments are performed in two-dimensionalmaterials, it would be desirable to investigate these results intwo-dimensional models. Our preliminary results in two spacedimensions show no sign of a glass phase. ACKNOWLEDGMENTS The authors thank V . Dobrosavljevi ´c, A. Möbius, W. Wang, and A. P. Young for insights and useful discussions. We alsothank Darryl C. Jacob for assistance with the simulations.We thank the National Science Foundation (Grant No. DMR-1151387) for ﬁnancial support, Texas A&M University foraccess to HPC resources (Ada and Terra clusters), Ben GurionUniversity of the Negev for access to their HPC resources,and Michael Lublinski for sharing with us his CPU time. Thiswork is supported in part by the Ofﬁce of the Director of Na-tional Intelligence (ODNI), Intelligence Advanced ResearchProjects Activity (IARPA), via MIT Lincoln Laboratory AirForce Contract No. FA8721-05-C-0002. The views and con-clusions contained herein are those of the authors and shouldnot be interpreted as necessarily representing the ofﬁcial poli-cies or endorsements, either expressed or implied, of ODNI,IARPA, or the US Government. The US Government is au-thorized to reproduce and distribute reprints for Governmentalpurpose notwithstanding any copyright annotation thereon. APPENDIX A: EQUILIBRATION In this Appendix, we outline the steps taken to guarantee thermalization. The data for this work are predominantlygenerated using population annealing Monte Carlo (PAMC).In order to ensure that the states sampled by a Monte Carlosimulation are in fact in thermodynamic equilibrium, i.e.,weighted according to the Boltzmann distribution, one needsto strive against bias by controlling the systematic errorsintrinsic to the algorithm due to the ﬁnite population size. Fortunately, PAMC offers a convenient way to study and tune the systematic errors to a desired accuracy. It can beshown [ 55] that the systematic errors in a PAMC simulation are directly proportional to the equilibration population size ρ f which has the following deﬁnition: ρf=lim R→∞Rvar(βF). (A1) Here, Ris the population size and Fis the free energy. ρf is an extensive quantity deﬁned at the thermodynamic limit although in reality it converges at a large but ﬁnite R. Because ρfis computationally expensive to measure as it requires multiple independent runs, one may alternatively study theentropic family size ρ sdeﬁned as ρs=lim R→∞Re−Sf, (A2)2345 2345(b)log10(ρs) log10(R)M=1 0 M=2 0 M=4 0 M=6 0 NT=2 0 1 22.533.54 2345(c)log10(ρs) log10(R)M=1 0 M=2 0 M=4 0 M=6 0 NT=4 0 1 1.51.82.12.42.73 2345(e)log10(ρs) log10(R)M=1 0 M=2 0 M=4 0 M=6 0 NT=8 0 11.51.82.12.42.7 2345(f)log10(ρs) log10(R)M=1 0 M=2 0 M=4 0 M=6 0 NT= 1001012345 12345log10(ρf) log10(ρs)L=6 L=8 L=1 0 W=0.5(a) 1.622.42.83.23.6 2345(d)log10(ρs) log10(R)M=1 0 M=2 0 M=4 0 M=6 0 NT=6 0 1 FIG. 7. Equilibration of a PAMC simulation. (a) Equilibration population size ρfversus entropic family size ρsfor a CG simulations atW=0.5. 100 instances for each system size have been studied. Evidently, ρsis greatly correlated to ρfwhich controls the systematic errors in thermodynamic quantities. Because ρfis computationally expensive to measure, one may instead use ρsas the measure of thermalization. (b)–(f) ρsversus the population size Rfor system size L=8 at various number of temperatures NTand Metropolis sweeps M.W h e n ρsconverges, the system is guaranteed to be in thermal equilibrium. As seen from the plots, convergence is achieved faster as the number of temperatures and sweeps is increased. However,for extremely large values of N TandM, marginal improvement in equilibration is gained at the cost of extended run time of the simulation. where Sfis the family entropy of PAMC. As shown in Fig. 7(a),ρsis well correlated with ρfwhich is why we can reliably use ρsas the measure of equilibration. ρssimilarly toρfconverges at a ﬁnite R. The population size at which the convergence is achieved is a function of the number oftemperatures N Tas well as the number of Metropolis sweeps M. Optimization of PAMC is studied in great detail in the context of spin glasses [ 56,57] much of which can be carried over to the CG simulations. As an example we show inFigs. 7(b)–7(f) how we choose the optimal values of the PAMC parameters. We observe that the convergence of ρ sis attained faster as the number of temperatures and sweeps isincreased. However, beyond a certain point, any further increase solely prolongs the simulation time while contributingnegligibly to lowering the convergent value of ρ s. A good rule of thumb for checking thermalization, as seen in Fig. 7,i s thatρsand as a result ρfconverges when ρs/R=exp(−Sf)< 0.01. We ensure that the above criterion is met for every 104418-8NUMERICAL OBSERV ATION OF A GLASSY PHASE IN … PHYSICAL REVIEW B 100, 104418 (2019) 0.220.240.260.280.3 0.004 0 .006 0 .008 0 .01ξSG/L TL=6 L=8 L=1 0 W=0.5(a) 0.230.240.250.260.27 0.004 0 .005 0 .006 0 .007 0 .008ξSG/L TL=6 L=8 L=1 0 W=0.5(b) FIG. 8. Importance of proper thermalization in observing a CG phase transition. Panel (a) shows a simulation where some instances have not reached thermal equilibrium whereas panel (b) illustrates the same simulation in which all of the instances have been thor-oughly thermalized. instance that we have studied. This matter has been inves- tigated thoroughly in Ref. [ 55]. It is worth mentioning here that proper equilibration is crucial in observing phase tran-sitions, especially in subtle cases like the CG model. Wehave illustrated this matter in Fig. 8. Figure 8(a) shows a simulation where the system has been poorly thermalized inwhich ρ s/R∼0.1 on average across the studied instances. By contrast in Fig. 8(b) the same simulation is done with careful equilibration; that is to say, the criterion ρs/R< 0.01 is strictly enforced for every instance. It is clear that the observation of a crossing is contingent upon ensuringthat every instance has reached thermal equilibrium. This, inturn, could explain why simulations using parallel temperingMonte Carlo, e.g., Ref. [ 47], see no sign of a transition. APPENDIX B: FINITE-SIZE SCALING RESULTS In this Appendix we list the estimates for the critical parameters of the plasma-CO as well as the plasma-CGphase transitions. Because the CO phase is essentially anantiferromagnetic phase in the spin language, multiple criticalexponents such as ν,α,β, andγcan be measured numerically. We have estimated these quantities using FSS techniques,speciﬁcally by a FSS collapse of the data for different systemsizes onto a low-order polynomial, as explained in the maintext. To estimate the exponent νwe have used the ﬁnite- size correlation length per linear system size ξ/L[Eq. ( 12)]. Because this is a dimensionless quantity, in the vicinity of thecritical point it scales as ξ/L=F ξ[L1/ν(T−Tc)]. (B1) Other critical exponents such as α,γ, and βcan be estimated by performing a FSS analysis using the peak val-ues of the speciﬁc heat c v=Cv/N, connected susceptibility02468 1.822 .22 .42 .6α/ν=0.418(28) β/ν=0.305(19)γ/ν=1.79(3)¯γ/ν=2.67(2)log[F(T∗ c,L)]∼x νlog(L) log(L)cv msχ ¯χ FIG. 9. Finite-size scaling analysis for the plasma-CO phase transition at W=0.05. The peak values of the speciﬁc heat capacity cv, connected and disconnected susceptibilities χand ¯χ, as well as the inﬂection point value of the staggered magnetization are used to estimate the critical exponents α,β,γ,a n d ¯γ, respectively. According to Eqs. ( B2)a n d( B3), the above quantities scale as a power law in the linear system size Las clearly seen from the ﬁgure. χ, and the disconnected susceptibility ¯ χas well as the inﬂec- tion point value of the staggered magnetization mswhich scale as following: cmax v∼Lα/ν,minﬂect s∼L−β/ν. (B2) χmax∼Lγ/ν,¯χmax∼L¯γ/ν. (B3) As we can see in Fig. 9the above scaling behaviors are very well satisﬁed. The best estimates of the critical parameters forvarious values of the disorder are listed in Table III. Note that with the exception of the universal exponent ν, other critical exponents vary with disorder which can be due to the trade-off between large-scale thermal and random-ﬁeld ﬂuctuations.Because at T=0 the system has settled in the ground state, one cannot use thermal sampling to measure the variance ofenergy and staggered magnetization which are proportional tothe heat capacity and susceptibility, respectively. Instead, wehave used the techniques developed by Hartmann and Youngin Ref. [ 72]. For the plasma-CG transition we have calculated the crit- ical exponents νandη, as well as the correction to scaling exponent ω, using the procedure explained in Sec. IV B . Table IVlists the estimates of the critical parameters. 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PhysRevB.86.134502.pdf	PHYSICAL REVIEW B 86, 134502 (2012) Effect of iron content and potassium substitution in A0.8Fe1.6Se2(A=K, Rb, Tl) superconductors: A Raman scattering investigation A. M. Zhang, K. Liu, J. B. He, D. M. Wang, G. F. Chen, B. Normand, and Q. M. Zhang* Department of Physics, Renmin University of China, Beijing 100872, People Republic of China (Received 24 January 2012; revised manuscript received 11 August 2012; published 1 October 2012) We have performed Raman-scattering measurements on high-quality single crystals of the superconductors K0.8Fe1.6Se2(Tc=32 K), Tl 0.5K0.3Fe1.6Se2(Tc=29 K), and Tl 0.5Rb0.3Fe1.6Se2(Tc=31 K) as well as of the insulating compound KFe 1.5Se2. To interpret our results, we have made ﬁrst-principles calculations for the phonon modes in the ordered iron-vacancy structure of K 0.8Fe1.6Se2. The modes we observe can be assigned very well from our symmetry analysis and calculations, allowing us to compare Raman-active phonons in theAFeSe compounds. We ﬁnd a clear frequency difference in most phonon modes between the superconducting and nonsuperconducting potassium crystals, indicating the fundamental inﬂuence of iron content. By contrast,substitution of K by Tl or Rb in A 0.8Fe1.6Se2causes no substantial frequency shift for any modes above 60 cm−1, demonstrating that the alkali-type metal has little effect on the microstructure of the FeSe layer. Several additionalmodes appear below 60 cm −1in Tl- and Rb-substituted samples, which are vibrations of heavier Tl and Rb ions. Finally, our calculations reveal the presence of “chiral” phonon modes, whose origin lies in the chiral nature ofthe K 0.8Fe1.6Se2structure. DOI: 10.1103/PhysRevB.86.134502 PACS number(s): 74 .70.−b, 74.25.Kc, 63 .20.kd, 78 .30.−j I. INTRODUCTION Iron pnictide superconductors display the highest super- conducting transition temperatures yet known outside cupratesystems. Unsurprisingly, their discovery almost four years agoignited an enduring drive both to search for new supercon-ducting materials and to explore the fundamental physicalproperties of these systems, especially, the pairing mechanism.Until recently, ﬁve such systems had been synthesized andstudied, namely, LnFeAsOF (known as “1111,” with Ln ≡La, C e ,P r ,N d ,S m , ...), 1AEFe2As2andAFe2As2(“122,” with AEan alkaline earth and Aan alkali metal),2AFeAs (“111”),3 Fe(Se,Te) (“11”),4and Sr 2VO 3FeAs (“21311”).5 FeSe is of particular interest among these systems for a number of reasons. The most important is that it does notcontain the poisonous element As. In addition, its transitiontemperature, T c, displays a very strong pressure dependence. At ambient pressure, Tc≈8K ,6much lower than in the 1111 and 122 systems, but a maximum Tcof 37 K can be reached under a pressure of approximately 6 GPa.7It has been shown6 that the microscopic effect of the applied pressure is to alterthe separation of the Se atoms from the Fe planes, and this verystrong dependence opens the possibility of raising T cby the introduction of internal chemical pressure. The ﬁrst successfulexecution of this program was reported in Ref. 8, where a potassium-intercalated FeSe superconductor was synthesized and found to have T c≈31 K, a value comparable to that in the 122 materials. In parallel with intensive efforts to synthesize further examples of AxFe2−ySe2systems, the electronic and magnetic properties of these compounds have been studied extensively.Infrared optical conductivity measurements indicated that thenonsuperconducting system is a small-gap semiconductorrather than a Mott insulator. 9In superconducting samples, early nuclear magnetic resonance (NMR) measurements foundvery narrow linewidths, singlet superconductivity with nocoherence peak, and only weak spin ﬂuctuations, but no signof magnetism. 10Angle-resolved photoemission spectroscopy(ARPES) measurements were initially inconclusive, but now11 indicate three electron-like Fermi surfaces (two around the /Gamma1 point and one around the Mpoint) with full gaps on at least two, but again no evidence for magnetic order. Both Raman12and infrared9spectroscopy ﬁnd large numbers of phonon modes beyond those expected in a 122 structure, and transmissionelectron microscopy (TEM) 13reveals a well-deﬁned surface vacancy ordering. The ﬁrst piece of the puzzle concerning the true nature of theAxFe2−ySe2materials was revealed by neutron diffraction experiments.14First, these determine that the predominant structure is dictated by a real Fe content of 1.6 in thesuperconducting crystals. This gives a regular, 1 /5-depleted Fe vacancy ordering pattern with a√ 5×√ 5 unit cell. Second, the magnetic properties of this phase are perhaps the most unusualof any known superconductor, featuring an ordered spinstructure of antiferromagnetically coupled four-spin blocks,av e r yh i g hN ´eel temperature of 520 K, and an extraordinarily large local moment of 3.31 μ Bper Fe site.14The magnetic transition has now been conﬁrmed by bulk measurements,15 while M ¨ossbauer spectroscopy has been used to verify the large local moment, giving results of 2.9 and 2.2 μBin two separate studies.16These observations demonstrate directly that the AFeSe superconductors are completely different fromFeAs-based and cuprate superconductors in at least tworespects. One is that a bulk ordering of the Fe vacanciesplays a key role in determining the electronic and magneticproperties of the system. The other is the apparent (micro-or mesoscopic) coexistence of long-range antiferromagneticorder with superconductivity. The question of coexistence is the other piece of the puzzle. It involves reconciling the NMR and ARPES results, whichappear to originate from a homogeneous, nonmagnetic bulksuperconductor, with the data from all of the other techniquescited above, which are the signatures of a magnetic insulatorwith a complex structure. The nature of this coexistenceor cohabitation has been the focus of almost all recent 134502-1 1098-0121/2012/86(13)/134502(11) ©2012 American Physical SocietyZHANG, LIU, HE, WANG, CHEN, NORMAND, AND ZHANG PHYSICAL REVIEW B 86, 134502 (2012) experimental investigations of the AxFe2−ySe2materials. A clear consensus has emerged in support of a phase separationbetween antiferromagnetic and superconducting regions, butoccurring on microscopic length scales. Phase separation hasbeen reported in optical 17and ARPES experiments,18the former authors attributing a much larger direct band gap(0.45 eV) to the majority insulating phase than that deducedin Ref. 9. Very recent NMR measurements have detected clear signals from a majority magnetic phase as well as aminority superconducting one. 19The appearance of phase separation occurring at nanometer scales has been detectedby M ¨ossbauer, 20x-ray,21and in-plane optical spectroscopy measurements.22Scanning tunneling microscopy (STM) has been used to image this nanoscopic phase separation directly inepitaxially grown ﬁlms. 23Estimates of the volume fraction of the magnetic and insulating phase by these techniques remainclose to the value of 90% reported by muon spin resonance(μSR) measurements. 24The minority (10%) superconducting phase must clearly be percolating to give the appearanceof bulk superconductivity. Several studies 18,19,23,25indicate that the ordered vacancy conﬁguration is present only inthe AF phase, consistent with the unusual magnetism beingan essential component in stabilizing this structure, whilethe superconducting phase is structurally homogeneous withstoichiometric FeSe planes (some authors 18,23,25suggesting AFe2Se2). Finally, there are only two experimental reports concerning the question of whether this coexistence is collab-orative or competitive; our own data from two-magnon Ramanscattering 26and additional results from neutron diffraction27 suggest a strong competition, in that 5–10% of the magneticvolume is suppressed by an apparent proximity effect at theonset of superconductivity. Returning to the question of sample synthesis, rapid progress followed the ﬁrst report of K xFe2−ySe2, with several groups achieving superconductivity by substitution of alkali-type metals including Rb, Cs, and Tl. 28The purpose of substitution by ions of equal valence but different radii is toalter the chemical pressure to control the electronic properties.An example is the maximum T cof approximately 56 K achieved by the substitution of rare-earth ions in the 1111system. 29It is thus somewhat surprising that substitution of K by Rb, Cs, or Tl in the new superconductors leavesT cessentially unaltered at around 30 K.28This result poses another fundamental question, concerning why superconduc-tivity should be so robust in the AFeSe system, and its answer requires a careful investigation into the effect of Tl, Rb, and Cssubstitution on the microstructure of the FeSe layers in thesematerials. In this paper, we address these questions through a Raman-scattering study. We have measured the spectrain three high-quality superconducting crystals of Tl- andRb-substituted K 0.8Fe1.6Se2, and in one nonsuperconducting crystal with an altered Fe content. For each crystal, we observedouble-digit numbers of phonon modes, dramatically differentfrom a normal 122 structure but consistent with an orderedvacancy structure. We perform ﬁrst-principles calculations forthe zone-center phonons in K 0.8Fe1.6Se2(Tc=32 K) in order to assign the observed modes by the symmetries and frequencieswe measure. The resulting assignment is very satisfactory,demonstrating that this vacancy-ordered structure is indeedthe majority phase of our samples. From this understanding, we ﬁnd the effect of a varying Fe content to be detectableas frequency shifts of the Raman modes above 60 cm −1,a s these are vibrations involving Fe and Se atoms. By contrast,the effects of Tl and Rb substitution are not discernibleabove 60 cm −1, indicating that K-layer substitution causes no substantial distortion of the FeSe layer. Below 60 cm−1, additional modes associated with vibrations of the heavier Tland Rb ions can be observed in the substituted samples. Inour calculations, we also ﬁnd some unconventional “chiral”phonon modes, which arise due to the chiral nature of the√ 5×√ 5 Fe-vacancy structure, and we consider their implications for coupling to possible chiral electronic and magneticmodes. The structure of the manuscript is as follows. In Sec. II,w e discuss our sample preparation and measurement techniques.In Sec. III, we present the full theoretical analysis for comput- ing the phonon spectrum from the known lattice structure ofthe insulating and magnetic majority phase, and we discuss thenature of the predicted modes. With this frame of reference,we may then understand our Raman-scattering results, whichare presented in detail in Sec. IV. Section Vcontains a short summary and conclusion. II. MATERIALS AND METHODS The FeSe-based crystals used in our measurements were grown by the Bridgman method. The detailed growth proce-dure may be found elsewhere. 30The accurate determination of crystal stoichiometry has been found to be a delicate issue,which is crucial in establishing the proper starting point forunderstanding both the magnetism and the superconductivity.We have obtained highly accurate results for our crystal com-positions by using inductively coupled plasma atomic emissionspectroscopy (ICP-AES), and have obtained results com-pletely consistent with the neutron diffraction reﬁnement. 14 The crystals we used in this study were K 0.8Fe1.6Se2(Tc≈ 32 K), Tl 0.5K0.3Fe1.6Se2(Tc≈29 K), and Tl 0.5Rb0.3Fe1.6Se2 (Tc≈31 K), all of which were superconducting with similar transition temperatures, and also the nonsuperconductingcompound KFe 1.5Se2. The precise chemical formula for this series of compounds is thought to be AxFe2−x/2Se2(A=K, Rb, Cs, Tl),14,31and our results agree with this deduction. The above discussion of phase separation notwithstanding,x-ray diffraction patterns obtained for our crystals show nodiscernible secondary phases, indicating that the volume frac-tion of the superconducting minority phase is low in all cases.The resistivities of the samples were measured with a QuantumDesign physical properties measurement system (PPMS), andthe magnetization by using the PPMS vibrating sample magne-tometer (VSM). The sharp superconducting and diamagnetictransitions, which were found for all three superconductingcrystals, are presented in Sec. IVto accompany a more detailed discussion of phase separation. These results indicate that allof the crystals used in our Raman measurements are of veryhigh quality, which in a phase-separation context means thatthe nanoscale percolation of the minority phase is good andhomogeneous. All measurements were made by ﬁrst cleaving the crystals in a glove box, to obtain ﬂat, shiny ( ab)-plane surfaces. 134502-2EFFECT OF IRON CONTENT AND POTASSIUM ... PHYSICAL REVIEW B 86, 134502 (2012) FIG. 1. (Color online) Atomic displacement patterns for selected Raman-active Agmodes of K 0.8Fe1.6Se2, with frequencies of (a) 75.1, (b) 130.5, (c) 159.2, (d) 212.6, (e) 268.5, and (f) 286.1 cm−1. Fe atoms connected by red lines have right-handed chirality in thisrepresentation. The freshly cleaved crystals were sealed under an argon atmosphere and transferred into the cryostat within 30 secondsfor immediate evacuation to a work vacuum of approximately10 −8mbar. Raman-scattering measurements were performed with a triple-grating monochromator (Jobin Yvon T64000) in apseudobackscattering conﬁguration. The beam of the 532-nmsolid-state laser (Torus 532, Laser Quantum) was focused intoa spot on the sample surface with a diameter of approximately20μm. The beam power was reduced to avoid heating, and was kept below 0.6 mW during our measurements at the lowest temperatures; the real temperature in the spot was deduced from the intensity relation between the Stokes and anti-Stokesspectra. The polarization determination for the phonons whosesymmetries we assign as A gandBgin the spectra shown in Sec. IVwas performed by adjusting the polarization of the incident and scattered light, rather than by a formal symmetryanalysis, as discussed in detail in Ref. 12. FIG. 2. (Color online) Atomic displacement patterns for selected Raman-active Bgmodes of K 0.8Fe1.6Se2with frequencies of (a) 66.7, (b) 106.2, (c) 149.0, (d) 238.3, and (e) 279.0 cm−1. Fe atoms connected by red lines have right-handed chirality. III. FIRST-PRINCIPLES DYNAMICAL ANALYSIS A full understanding of our Raman-scattering results, and in particular of the effects caused by iron content andpotassium substitution, requires a complete phonon modeassignment. The ordered pattern of Fe vacancies 14explains quite naturally the large number of optical phonons observedin light-scattering experiments. However, the large unit cellmeans that a detailed vibration analysis is somewhat involved.We begin with the results from neutron diffraction, 14which gives the structural space group of K 0.8Fe1.6Se2asI4/mand the Wyckoff positions of the atoms as 8 hfor potassium, 16 i for iron, 4 dfor the iron vacancies, and 16 ifor selenium. The corresponding symmetry analysis allows a total of 17 Agor Bgmodes.12 We have calculated the nonmagnetic electronic structure and the zone-center phonons of K 0.8Fe1.6Se2from ﬁrst princi- ples by performing density functional calculations. We use the VIENNA ab initio simulation package,32,33which makes use of the projector augmented wave (PAW) method33combined with 134502-3ZHANG, LIU, HE, WANG, CHEN, NORMAND, AND ZHANG PHYSICAL REVIEW B 86, 134502 (2012) FIG. 3. (Color online) Atomic displacement patterns for selected infrared-active phonon modes of K 0.8Fe1.6Se2with frequencies of (a) 119.1, (b) 212.3, (c) 253.4, and (d) 308.5 cm−1. Fe atoms connected by red lines have right-handed chirality. a general gradient approximation (GGA), implemented with the Perdew-Burke-Ernzerhof formula,34for the exchange- correlation potentials. The nonmagnetic K 0.8Fe1.6Se2system was modeled by adopting a parallelepiped supercell containingeight Fe atoms plus two Fe vacancies, ten Se atoms, andfour K atoms plus one K vacancy. The Brillouin zone of thesupercell was sampled with an 8 ×8×8k-space mesh and the broadening was taken to be Gaussian. The energy cutofffor the plane waves was 400 eV . Both the shape and volume ofthe cell and the internal coordinates of all the ions were fullyoptimized until the forces on all relaxed atoms were below0.01 eV /˚A. The frequencies and displacement patterns of the phonon modes were calculated using the dynamical matrix method 35 in which the derivatives were taken from the ﬁnite differencesin atomic forces at a ﬁxed atomic displacement of 0.01 ˚A. All 22 atoms in the supercell were allowed to move from theirequilibrium positions in all directions ( x,y,z), leading to a6 6×66 matrix. The phonon frequencies and displacement patterns are given by diagonalizing this matrix. Convergencetests carried out by comparing the different kpoints assured that the ﬁnal results were well converged both in their overallenergetics and in the phonon spectrum (yielding accuracies oforder 2 cm −1). The 22-atom supercell has 63 optical modes. However, to illustrate the displacement patterns of the phononmodes deduced from real-space translational invariance, inFigs. 1–3we show our results in the 44-atom I4/mcell. Calculated phonon frequencies for prominent modes of all symmetries are listed in Table I. The experimental frequencies are discussed in Sec. IV. As expected, the majority of the FIG. 4. (Color online) Atomic displacement patterns for chiral phonon modes [(a) 67.0, (b) 86.2, and (c) 301.3 cm−1]a n df o rt h e breathing mode [(d) 269.6 cm−1]o fK 0.8Fe1.6Se2. Fe atoms connected by red lines have right-handed chirality. modes are vibrations related to the Fe and Se atoms in the primary structural unit, and this includes all but one of theexperimentally relevant modes (see Table I). Vibrations of the K atoms appear only at low energies, reﬂecting the weakrestoring forces they encounter far from the FeSe planes. Inthe real material, these atoms are thought to be rather mobile. The atomic displacement patterns of the assigned A gand Bgmodes are shown, respectively, in Figs. 1and 2.T h e displacement arrows are to scale between panels, and it isclear that the largest atomic motions are in the cdirection, while in-plane motion is more restricted. The right-handcolumns of Table Idetail the character of the calculated eigenmodes, showing whether they correspond to atomicmotions primarily in the FeSe plane, perpendicular to it, orin a genuine combination of both. In addition to the dominantA gandBgmodes, we also compute a number of Egphonons over the same frequency range and list some selected modesin Table I; these twofold degenerate modes correspond to in-plane atomic motions, although they may obtain weaknormal components due to the presence of the vacancies. Forreasons of light-scattering selection rules, these modes arenot usually observed in Raman experiments for approximatelytetragonal materials. The optical phonons we have calculated include not only the Raman-active modes but also similar numbers of infrared-active ones. Four selected examples are also listed in Table I and compared to recent experimental measurements, whiletheir displacement patterns are illustrated in Fig. 3.A s f o r the Raman-active modes, most of the infrared modes arevibrations of the Fe and Se atoms. In the tetragonal 122iron arsenide superconductors, there exist just two ideal, one-dimensional Raman-active phonon modes of the FeAs plane,whose symmetries are A 1gand B 1g,37and, similarly, just two infrared-active modes. To the extent that K 0.8Fe1.6Se2can be considered as an ordered, 1 /5-depleted version of this system, it is clear that the symmetry reduction and expansion of the 134502-4EFFECT OF IRON CONTENT AND POTASSIUM ... PHYSICAL REVIEW B 86, 134502 (2012) TABLE I. Symmetry analysis for space group I4/mand assignment of selected optical modes in K 0.8Fe1.6Se2.T h e“ =,” “⊥,”a n d“ /negationslash” symbols denote respectively eigenmode directions parallel, perpendicular, and at an angle to the FeSe plane of the crystal. WyckoffOptical modes Atom position Raman active Infrared active K8 h 2Ag+2Bg+2Eg Au+4Eu Fe 16 i 3Ag+3Bg+6Eg 3Au+6Eu Se 4 eA g+2Eg Au+2Eu Se 16 i 3Ag+3Bg+6Eg 3Au+6Eu Cal. freq. Expt. freq. Symmetry Index Atoms Direction of eigenmode (cm−1)( c m−1)K ( 8 h)F e ( 1 6 i)S e ( 4 e)S e ( 1 6 i) 66.7 61.4 Bg1Bg Se /negationslash 75.1 66.3 Ag1Ag Se ⊥ 106.2 100.6 Bg2Bg K ⊥ 130.5 123.8 Ag2Ag Se /negationslash 159.2 134.6 Ag3Ag Se ⊥ /negationslash 149.0 141.7 Bg3Bg Se /negationslash 212.6 202.9 Ag4Ag Se ⊥ /negationslash 238.3 214.3 Bg4Bg Fe ⊥ 268.5 239.4 Ag5Ag Fe /negationslash 286.1 264.6 Ag6Ag Fe,Se /negationslash ⊥ 279.0 274.9 Bg5Bg Fe /negationslash 83.3 Eg K,Se ⊥ /negationslash 102.4 Eg K,Se ⊥= = 143.4 Eg Se == 208.7 Eg Se = /negationslash 242.5 Eg Fe,Se /negationslash = 284.9 Eg Fe,Se /negationslash = 119.1 102.2aAu K ⊥ 212.3 208.3aAu Se ⊥ /negationslash 253.4 236.3aAu Fe /negationslash 308.5 Au Fe,Se /negationslash ⊥ 67.0 Chiral Se /negationslash 86.2 Chiral K = 301.3 Chiral Fe = 269.6 Breathing Fe /negationslash aReference 36. unit cell allow many more Raman- and infrared-active optical modes to exist in the alkali-intercalated FeSe superconductors. Table IIcontains the full details of the eigenvectors for the atomic displacements corresponding to all of our selectedmodes. While much more specialized than the polarizationinformation, we provide this data for completeness andspeciﬁcity concerning the representations in Figs. 1−4. A full understanding of experimental data concerning resistivity,pairing, and anomalies in ARPES, inelastic neutron scattering,and magnetic Raman signals depends on an accurate knowl-edge of the phonon spectrum, and the quantitative intensity(see Sec. IV) and polarization information we provide can be used to calculate the interactions between speciﬁc phononmodes and the itinerant electrons or spin ﬂuctuations of thecharge and spin sectors. As an immediate example of this, in our calculations, we also ﬁnd some novel “chiral” phonon modes, whose atomicdisplacement patterns are shown in Fig. 4. These are nonde- generate and primarily in-plane modes in which all the Fe or Seatoms in a single plane of the structural unit have a net rotationabout the center of the cell, an apparent angular momentumcanceled by the atomic displacements in neighboring unit cells of the same FeSe layer. The presence of these chiral modesis a direct consequence of chiral symmetry-breaking in theAFeSe system when the 1 /5-depleted vacancy structure is adopted; the√ 5×√ 5 unit cell14itself has an explicit left- or right-handed structural chirality.38The chiral phonon modes are not active in the Raman channel, and therefore are notobserved in the Raman measurements performed here. Wesuggest, however, that these chiral modes may be observedby different spectroscopic techniques in circular polarizationconﬁgurations. The presence of chiral modes is of particular interest as a possible probe of chiral electronic or magnetic excitations,which break time-reversal symmetry. Such excitations havebeen discussed in the parent compounds of cuprate supercon-ductors, where an A 2component attributed to chiral spin exci- tations was detected by Raman scattering.39At the theoretical level, a chiral d-density-wave state has also been proposed in the underdoped state of cuprate superconductors.40Such modes, involving chiral electronic motion, may couple pref-erentially to the chiral phonon vibration in the 1 /5-depleted 134502-5ZHANG, LIU, HE, WANG, CHEN, NORMAND, AND ZHANG PHYSICAL REVIEW B 86, 134502 (2012) TABLE II. Eigenvectors of the most important atoms involved in selected vibrational modes of K 0.8Fe1.6Se2(see Table I). Directions x,y, andzare those shown in Figs. 1–4. Cal. freq. Eigenvector ( xyz ) (cm−1) Symmetry K(8 h)F e ( 1 6 i)S e ( 4 e)S e ( 1 6 i) 66.7 Bg (−0.07 0.24 0.20) 75.1 Ag (0.00 0.00 0.34) 106.2 Bg (0.00 0.00 0.39) 130.5 Ag (0.20 0.12 0.20) 159.2 Ag (0.00 0.00 0.29) ( −0.07 0.22 0.11) 149.0 Bg (0.11 0.13 0.29) 212.6 Ag (0.00 0.00 0.40) ( −0.04 0.11 0.23) 238.3 Bg (0.00 0.04 0.31) 268.5 Ag (0.10 0.27 0.12) 286.1 Ag (0.15 0.08 0.23) (0.00 0.00 0.24) 279.0 Bg (−0.15 0.12 0.19) 83.3 Eg (0.00 0.00 0.17) (0.12 0.14 0.05) 102.4 Eg (0.00 0.00 0.28) ( −0.12 0.22 0.00) ( −0.03 0.24 0.01) 143.4 Eg (−0.25 0.18 0.00) ( −0.11 0.29 0.04) 208.7 Eg (0.34−0.04 0.00) (0.24 −0.04 0.15) 242.5 Eg (0.19−0.03 0.22) ( −0.06 0.26 0.00) 284.9 Eg (0.24−0.02 0.07) ( −0.19 0.18 0.00) 119.1 Au (0.00 0.00 −0.46) 212.3 Au (0.00 0.00 −0.45) (0.13 0.22 0.08) 253.4 Au (0.03 0.27 −0.16) 308.5 Au (0.01 0.17 0.22) (0.00 0.00 −0.23) 67.0 Chiral (0.20 0.23 0.06) 86.2 Chiral (0.02 0.42 0.00) 301.3 Chiral (0.01 0.29 0.00) 269.6 Breathing ( −0.08 0.27 0.09) AFeSe system, in the same way as phonons of B1gsymmetry couple to the d-wave superconducting order parameter in cuprates, and thus Raman phonon spectroscopy may be usedto detect their presence. Without such a coupling to phononmodes, any chiral electronic or magnetic excitations in theAFeSe system may also be subject to direct detection by the polarized spectroscopies, such as ARPES and neutronscattering, also applied in the study of cuprates. IV . RAMAN-SCATTERING MEASUREMENTS We begin the presentation of our Raman-scattering results by recalling the basic features the low-temperature spectra,w h i c ha r es h o w nf o rK 0.8Fe1.6Se2in Fig. 5. The measurements were performed at 9 K in polarization conﬁgurations whichseparate the A gandBgchannels. At least thirteen Raman- active modes are observed, all located below 300 cm−1;12at least ten infrared-active modes have also been measured9in the same frequency range. This abundance of optical modes arisesdue to the symmetry reduction caused by Fe vacancy ordering,which we have identiﬁed as being from D 4hto C 4h.12The space group of the undepleted, 122-type structure, I4/mmm , is reduced to I4/m, a process in which all in-plane, two- fold rotation axes and all mirror planes perpendicular to the(ab) plane are lost. Both the phonon mode energies and the polarizations observed in Fig. 5are in excellent agreement with the calculations of Sec. IIIfor the spectrum of optical modes (see Table I), as also are the measured infrared modes. 36A. Fe content In Fig. 6, we compare the Raman modes in superconducting K0.8Fe1.6Se2with those in nonsuperconducting KFe 1.5Se2.T h e modes in the two samples show a general similarity in intensityand location, which implies a similarity in the microstructuresand symmetries of their FeSe layers. However, it is also evidentthat changing the Fe content does cause a signiﬁcant shiftin frequency for most of the modes. Because these modesare vibrations involving the Fe and Se ions (see Tables I andII), this reﬂects some signiﬁcant differences between the FeSe layers in the two samples. As noted above, neutrondiffraction measurements conﬁrm that the majority phase ofthe system at an Fe stoichiometry of 1.6 (20% Fe vacancies) forms the ideal, four-fold-symmetric, 1 /5-depleted,√ 5×√ 5 vacancy-ordering pattern. These measurements also indicate14 that the same ordering pattern is maintained for the samplewith an Fe content of 1.5, despite the increase to 25% Fevacancies; in this case, the 16i Fe positions are only partiallyoccupied. This occupation means a random distortion of theFe-Se bonds, which is responsible for the shifts in modefrequencies. Our results are thus in agreement with thosefrom neutron diffraction, conﬁrming that the electronic andmagnetic properties of the AFeSe system are rather sensitiveto the vacancy content of the FeSe layers, even if the overalllayer structure is not. These changes should thus be consideredas disorder effects rather than microstructural effects. Alteringthe Fe content from 1.6 to 1.5 causes our sample to become aninsulator with a small gap, which is estimated by infrared 134502-6EFFECT OF IRON CONTENT AND POTASSIUM ... PHYSICAL REVIEW B 86, 134502 (2012) FIG. 5. (Color online) Raman spectra for K 0.8Fe1.6Se2, measured in the AgandBgchannels at 9 K. The mode assignment is made on the basis of the symmetry analysis and the ﬁrst-principles calculations of Sec. III. The corresponding atomic displacement patterns can be found in Figs. 1and2. Blue arrows indicate unassigned modes. experiments to be 30 meV (see Ref. 9) and by transport measurements to be approximately 80 meV .31 B. K substitution Raman spectra for the three superconducting crystals K0.8Fe1.6Se2,T l 0.5K0.3Fe1.6Se2, and Tl 0.5Rb0.3Fe1.6Se2are shown in Fig. 7. In contrast to the case of changing Fe content discussed above, the modes above 60 cm−1exhibit no substan- tial shift in frequency (although there are clear differences inrelative intensities). This suggests that substitution within thepotassium layers (at ﬁxed Fe content) has little effect on theFeSe layer, and essentially none on the ordering pattern of theFe vacancies. This substitution does, however, cause certainother changes to occur. The most notable is the presence ofsome additional phonon modes, which appear below 60 cm −1. These modes can be attributed unambiguously to vibrationsof the heavier Tl and Rb ions, which are absent in the spectraof K 0.8Fe1.6Se2(see Figs. 5and6) and KFe 1.5Se2(see Figs. 6 and8). The other important alteration is the dramatic intensity enhancement of the mode at 180 cm−1, which has Agcharacter but cannot (see Fig. 5) be assigned well from the calculations of Sec. III; we discuss this feature in detail below. The additional low-frequency modes induced by K sub- stitution are shown in Fig. 8. These become weaker but not narrower with decreasing temperature, eventually disappear-ing at 9 K. This behavior is similar to that of the 66 cm −1 SeAgmode, which largely follows the Bose-Einstein thermal factor.12By comparison with K 0.8Fe1.6Se2, these modes may readily be identiﬁed as vibrations of heavier Tl and Rb ions.It should also be noted here that there exist two possibleWyckoff positions for the Aions, namely 2 aand 8h, and that no Raman-active modes are allowed for atoms in the2apositions. No structural transition is found below the N´eel temperature (520 K) in neutron-diffraction studies of K 0.8Fe1.6Se2,14and the temperature-dependent Raman spectra in Fig. 8show that it is reasonable to assume the same behaviorFIG. 6. (Color online) Comparison between Raman spectra of K0.8Fe1.6Se2and KFe 1.5Se2. Labels eiandesdenote, respectively, the polarizations of the incident and scattered light. in the Tl- and Rb-substituted crystals. We therefore deduce that the additional modes are allowed due to changes of thelocal symmetry in the (Tl,K/Rb) layer, for which a randomoccupation of 2 aand 8hsites by Aions is the most likely possibility. C. Discussion We begin our discussion with the anomalous 180 cm−1 mode. This shows not only a curious temperature dependence of its intensity between samples, but also of its frequency atT c. The fact that this mode cannot be assigned properly by our symmetry analysis and ﬁrst-principles calculations suggeststhat it may be a local mode. One of the most likely candidatesfor this would be a nanoscopic region where the Fe vacancyis ﬁlled, creating a locally regular square lattice. Indeed, theAsA 1gmode in the 122 compounds occurs at a frequency of 182 cm−1in SrFe 2As2.37 A locally regular square lattice is also one of the leading candidates suggested in the phase-separation description ofthe FeSe superconductors. As noted in Sec. I, several authors have proposed that the superconducting minority phase isAFe 2Se2,18,23,25while others19agree with the stoichiometric FeSe planes but not with the Acontent. This scenario, that the 180 cm−1mode we observe is not merely a local ﬁlled vacancy, but the leading ﬁngerprint of a 122-like (A xFe2Se2) minority phase, would also be consistent with the jump we observein the frequency of this mode at T c, which suggests a strong coupling of this speciﬁc mode to the superconducting orderparameter. Further evidence in favor of this interpretation couldbe found in the B 1gmode of the 122-type structure, which appears at 204 cm−1in SrFe 2As2.37Our results do contain a Bgcomponent very close to this frequency, but we caution that it is accompanied by a very strong Agsignal, and may only be a shadow of this mode arising due to a disorder-inducedmixing of local symmetries. 41 In the general context of phase separation, it is clear that our samples have both a robust structural and magneticorder (from the neutron diffraction studies performed on the 134502-7ZHANG, LIU, HE, WANG, CHEN, NORMAND, AND ZHANG PHYSICAL REVIEW B 86, 134502 (2012) FIG. 7. (Color online) Raman spectra of the three superconduct- ing crystals at room temperature. Additional modes, indicated by blue arrows, appear below 60 cm−1in the Tl- and Rb-substituted samples. Dotted lines are guides indicating the peak positions. same crystals) and a clear superconducting component. We have been unable to ﬁnd any evidence for the presence ofsecondary phases in x-ray and neutron-scattering studies, 14,42 and we show in Fig. 9that the resistive and diamagnetic transitions at the onset of superconductivity are sharp andcontinuous in all three samples. However, as pointed out bymany authors, none of these results is sufﬁcient to excludeminority phases with a low volume fraction, and the data forthe superconducting transitions show only that the percolationof the superconducting fraction is complete and homogeneous(which would be consistent with a microscale phenomenon). In the most extreme version of a phase-separation scenario, only the 122-like ( A xFe2Se2) and 245 ( A0.8Fe1.6Se2) phases would exist, and altering Fe content would affect only theirratio. While this situation would account for a loss ofsuperconductivity on reducing the Fe content, it does seemto require at least one further low-Fe phase in the dopingrange of our samples. Our results do not support this scenario.It would predict that changes in the Fe stoichiometry in Fig. 6 should appear only as alterations in phonon intensity, ratherthan to the phonon frequencies as we observe. Our resultsdeﬁnitely indicate continuous alterations to a single majorityphase, and show further that some vacancy-disorder effectsare clearly (if not strongly) detectable. Thus we conclude thatour Raman phonon spectra contain no unambiguous evidencefor a robust, 122-like minority phase, and we suggest rather aphase-separation scenario in which the minority phase is oneof homogeneous vacancy disorder. Returning now to the anomalous phonon modes, both scenarios (a secondary 122-like phase and locally ﬁlled Fevacancy sites) can account qualitatively for the frequenciesof additional phonon modes beyond our dynamical analysis.While both also explain the anomalous behavior of the180 cm −1mode at Tc, neither accounts directly for the anomalous intensity of this mode. To explain this, we notethat the assignment of the mode as a (local or bulk) version ofthe 122 A 1gmode means that it involves a c-axis displacement of the Se atoms. These are the FeSe modes most stronglyFIG. 8. (Color online) Raman spectra of superconducting and nonsuperconducting AFeSe systems at selected temperatures. The dotted line indicates the location of additional modes induced by K substitution and the red arrows indicate modes showing large changes of intensity between different crystals. affected (see Figs. 7and 8)b yA-induced changes in the local microstructure, and we suggest that these alter the modeintensity in the same way as for the 66 cm −1mode.12 Away from the nature of the phase separation, we comment also on the distinctive low-energy background observed forthe four crystals we have measured (see Fig. 8). The spec- trum of semiconducting KFe 1.5Se2at room temperature rises strongly at low frequencies, whereas the low-energy part forK 0.8Fe1.6Se2is rather ﬂat. All of the background contributions fall with decreasing temperature. We suggest that the low-frequency enhancement may be due to electronic Ramanscattering. Because KFe 1.5Se2is a small-gap semiconductor, the smaller Coulomb screening effect relative to a normal metalwould allow stronger charge-density ﬂuctuations and hence alarger electronic Raman scattering contribution. Finally, one of the most surprising features of the K 0.8Fe1.6Se2material that makes up the majority of our samples is its apparently high degree of structural order. Thisoccurs despite its depleted nature, which one would expectto be prone to atomic disorder. Evidence for disorder canin fact be found in the polarized Raman spectra at roomtemperature (see Fig. 10). The Tl- and Rb-substituted samples show larger phonon widths compared to K 0.8Fe1.6Se2, which 134502-8EFFECT OF IRON CONTENT AND POTASSIUM ... PHYSICAL REVIEW B 86, 134502 (2012) FIG. 9. (Color online) Superconducting and diamagnetic tran- sitions for the three superconducting crystals used in the Raman- scattering investigations. implies that more disorder is induced by the substitution. Given that no substantial shifts occur in the mode frequenciesfor the three samples, this disorder can be attributed tothe random occupation and motion of the K, Tl, and Rbions. The breaking of local symmetry and periodicity intheAlayer acts to shorten the phonon lifetimes also in the FeSe layer. Overall, it appears that the high stability of the√ 5×√ 5 vacancy-ordered structure, which assures the constant frequencies of the phonon modes we observe in allour superconducting samples, may be a consequence of thevery speciﬁc magnetically ordered state it allows. V . SUMMARY To conclude, we have measured Raman spectra in single- crystalline samples of the superconductors K 0.8Fe1.6Se2, Tl0.5K0.3Fe1.6Se2, and Tl 0.5Rb0.3Fe1.6Se2as well as in their in- sulating derivative compound KFe 1.5Se2. A symmetry analysis and ﬁrst-principles calculations of the zone-center phonons, both based on the√ 5×√ 5 vacancy-ordering pattern of the K 0.8Fe1.6Se2unit cell, allow an excellent assignment of the observed phonon modes. We illustrate the correspondingatomic displacement patterns and demonstrate the presence ofchiral phonon modes. We observe a clear frequency shift in all phonons be- tween superconducting K 0.8Fe1.6Se2and nonsuperconducting KFe 1.5Se2, showing the effect of further Fe vacancies within the√ 5×√ 5 structure on the microscopic properties of theFIG. 10. (Color online) Polarized Raman spectra of the three superconducting crystals, showing that phonon widths are generally lower in K 0.8Fe1.6Se2. FeSe layers. By contrast, the frequencies of modes involving Fe and Se ions are little affected on substituting K by Tl orRb. However, this substitution does induce additional Tl andRb modes below 60 cm −1. Our measurements also contain a number of anomalies, which may be purely effects ofthe intrinsic vacancy disorder or may be explained in partby the presence of the weak minority phase responsible forsuperconductivity. Our results reveal the complex effects ofFe vacancies in the FeSe plane, laying the foundation fora full understanding of the distinctive structural, electronic,magnetic, and superconducting properties of the A xFe2−ySe2 series of materials. ACKNOWLEDGMENTS We thank W. Bao and Z. Y . Lu for helpful discussions. This work was supported by the 973 program of the MoST of Chinaunder Grant Nos. 2011CBA00112 and 2012CB921701, by the NSF of China under Grant Nos. 11034012, 11174367, and 11004243, by the Fundamental Research Funds for CentralUniversities, and by the Research Funds of Renmin Universityof China (RUC). Computational facilities were provided bythe HPC Laboratory in the Department of Physics at RUC.The atomic structures and displacement patterns were plottedusing the program XCRYSDEN .43 *qmzhang@ruc.edu.cn 1Y . Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya, and H. Hosono, J. Am. Chem. Soc.128, 10012 (2006); X. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and D. F. Fang, Nature (London) 453, 761 (2008); G. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong, 134502-9ZHANG, LIU, HE, WANG, CHEN, NORMAND, AND ZHANG PHYSICAL REVIEW B 86, 134502 (2012) P. Zheng, J. L. Luo, and N. L. Wang, Phys. Rev. Lett. 100, 247002 (2008). 2M. Rotter, M. Tegel, and D. 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PhysRevB.84.054533.pdf	PHYSICAL REVIEW B 84, 054533 (2011) Inﬂuence of spin-dependent quasiparticle distribution on the Josephson current through a ferromagnetic weak link A. M. Bobkov and I. V . Bobkova Institute of Solid State Physics, Chernogolovka, Moscow reg., 142432 Russia (Received 27 January 2011; revised manuscript received 9 May 2011; published 12 August 2011) The Josephson current ﬂowing through weak links containing ferromagnetic elements is studied theoretically under the condition that the quasiparticle distribution over energy states in the interlayer is spin dependent. It isshown that the interplay between the spin-dependent quasiparticle distribution and the triplet superconductingcorrelations induced by the proximity effect between the superconducting leads and ferromagnetic elements ofthe interlayer leads to the appearence of an additional contribution to the Josephson current. This additionalcontribution j tcan be extracted from the full Josephson current in experiment. The features of the additional supercurrent jt, which are of main physical interest are the following: i) We propose the experimental setup, where the contributions given by the short-range triplet component (SRTC) and long-range triplet component (LRTC)of superconducting correlations in the interlayer can be measured separately. It can be realized on the basis ofa S/N/F/N/S junction, where the interlayer is composed of two normal metal regions with a spiral ferromagnetlayer sandwiched between them. For the case of tunnel junctions, the measurement of j tin such a system can provide direct information about the energy-resolved anomalous Green’s function components describing SRTCand LRTC. ii) In some cases the exchange ﬁeld-suppressed supercurrent can be not only recovered but alsoenhanced with respect to its value for a nonmagnetic junction with the same interface resistances by the presenceof a spin-dependent quasiparticle distribution. This effect is demonstrated for the S/N/S junction with magneticS/N interfaces. In addition, it is also found that under the considered conditions the dependence of the Josephsoncurrent on temperature can be nontrivial: At ﬁrst, the current rises upon the temperature increasing and only afterthat starts to decline. DOI: 10.1103/PhysRevB.84.054533 PACS number(s): 74 .45.+c, 74.50.+r I. INTRODUCTION The interplay between superconductivity and ferromag- netism in layered mesoscopic structures offers an arena ofinteresting physics to explore. By now it is already wellknown that so-called odd-frequency triplet-pairing correla-tions are generated in hybrid superconductor/ferromagnet(S/F) structures. 1,2The essence of this pairing state is the following. The wave function of a Cooper pair /angbracketleftψσ1(r1,t1)ψσ2(r2,t2)/angbracketrightmust be an odd function with respect to permutations of the two electrons. Consequently, in the mo-mentum representation the wave function of a triplet Cooper pair has to be an odd function of the orbital momentum forequal times t 1=t2, that is, the orbital angular momentum Lis an odd number. Thus, the triplet superconducting condensate is sensitive to the presence of impurities, because only the s-wave (L=0) singlet condensate is not sensitive to the scattering by nonmagnetic impurities (Anderson theorem). S/F hybridstructures are usually composed of rather impure materials.Therefore, according to the Pauli principle, equal-time tripletcorrelations should be suppressed there. However, anotherpossibility for the triplet-pairing exists. In the Matsubara representation the wave function of a triplet pair can be an odd function of the Matsubara frequency and an even function ofthe momentum. Then the sum over all frequencies is zero andtherefore the Pauli principle for the equal-time wave functionis not violated. These are the odd-frequency triplet-pairingcorrelations, which are realized in S/F structures. If there is no a source of spin-ﬂip processes in the considered structure (that is, the magnetizations of all themagnetic elements, which are present in the system, are alignedwith the only one axis) then the Cooper pairs penetratinginto the nonsuperconducting part of the structure consist of electrons with opposite spins. Their wave function is the sumof a singlet component and a triplet component with zerototal spin projection S z=0 on the quantization axis. The resulting state has a common origin with the famous Larkin-Ovchinnikov-Fulde-Ferrel (LOFF) state 3,4and can be referred to as its mesoscopic analog. This mesoscopic LOFF statewas predicted theoretically 5,6and observed experimentally.7–11 In this state the Cooper pair acquires the total momentum 2Qor−2Qinside the ferromagnet as a response to the energy difference between the two spin directions. HereQ∝h/v F, where his an exchange energy and vFis the Fermi velocity. A combination of the two possibilities resultsin the spatial oscillations of the condensate wave function/Psi1(x) in the ferromagnet along the direction normal to the SF interface. 12/Psi1s(x)∝cos(2Qx) for the singlet Cooper pair and /Psi1t(x)∝sin(2Qx) for the triplet Cooper pair. The same picture is also valid in the diffusive limit. However, there is an extradecay of the condensate wave function due to scattering in thiscase. In the regime h/greatermuch\|/Delta1\|, where /Delta1is a superconducting order parameter in the leads, the decay length is equal to themagnetic coherence length ξ F=√D/h , while the oscillation period is given by 2 πξF.H e r e Dis the diffusion constant in the ferromagnet, ¯ h=1 throughout the paper. Due to the fact that the decay length ξFis rather short (much less than the superconducting coherence length ξS=√D//Delta1 ), the sum of /Psi1s(x) and/Psi1t(x) (corresponding to Sz=0) can be considered as a short-range component (SRC) of the pairing correlationsinduced by the proximity effect in the ferromagnet. The situation changes if the magnetization orientation is not ﬁxed. The examples are domain walls, spiral ferromagnets,spin-active interfaces, etc. In such a system not only the 054533-1 1098-0121/2011/84(5)/054533(18) ©2011 American Physical SocietyA. M. BOBKOV AND I. V . BOBKOV A PHYSICAL REVIEW B 84, 054533 (2011) singlet and triplet Sz=0 components exist, but also the odd-frequency triplet component with Sz=± 1a r i s e si nt h e nonsuperconducting region. The latter component penetratesthe ferromagnet over a large distance, which can be of the orderofξ N=√D/T in some cases. So, this triplet component can be considered as the long-range triplet component (LRTC).Various superconducting hybrid structures, where the LRTCcan arise, were considered in the literature (see Refs. 2,13,14, and references therein). In addition, the creation of the LRTCwas theoretically predicted in structures containing domainwalls, 15spin-active interfaces,17,18spiral ferromagnets,19–21 and multilayered SFS systems.22,23There are several experimental works, where the long-range Josephsoneffect 24–26and the conductance of a spiral ferromagnet attached to two superconductors27were measured. These results give quite convincing evidence of LRTC existence. Practically all the papers discussed above are devoted to the investigation of an odd-frequency triplet component underthe condition that the energy distribution of quasiparticles isequilibrium and spin independent. However, as it was shownrecently, 28the creation of spin-dependent quasiparticle distri- bution in the interlayer of SFS junction leads to the appearenceof the additional contribution to the Josephson current throughthe junction. This additional supercurrent ﬂows via vector part N j,tof the supercurrent-carrying density of states, which does not contribute to the Josephson current in a junction withs-wave superconductor leads if the quasiparticle distribution in the interlayer is spin independent. Below we brieﬂy describehow this effect arises. The energy spectrum of the superconducting correlations is expressed in a so-called supercurrent-carrying density of states(SCDOS). 29–32This quantity represents the density of states weighted by a factor proportional to the current that each statecarries in a certain direction. Under equilibrium conditions thesupercurrent can be expressed via the SCDOS as 33 j∝/integraldisplay dεN j(ε) tanhε/2T, (1) where εstands for the quasiparticle energy, tanh ε/2T=ϕ(ε) is the equilibrium distribution function, and Nj(ε)i st h e SCDOS. In the presence of spin effects the SCDOS becomesam a t r i x2 ×2 in spin space and can be represented as ˆN j=Nj,s+Nj,tσ, where σiare the Pauli matrices in spin space. The scalar in the spin-space part of the SCDOS, Nj,s,i s referred to as the singlet part of the SCDOS in this paper andvector part N j,tis referred to as the triplet part. Nj,tis directly proportional to the triplet part of the condensate wave function.It is well known that the spin supercurrent cannot ﬂow throughthe singlet superconducting leads. Therefore, N j,tdoes not contribute to the supercurrent in equilibrium. Having in mindthat the triplet part of the SCDOS is an even function ofquasiparticle energy, one can directly see that this is indeedthe case. Otherwise, if the distribution function becomes spindependent, that is, ˆ ϕ(ε)=ϕ 0(ε)+ϕ(ε)σ, the supercurrent car- ried by the SCDOS triplet component Nj,tin the ferromagnet is nonzero because the scalar product Nj,t(ε)ϕ(ε) contributes to the spinless supercurrent in this case.28 As is obvious from what was discussed above, the spin- independent nonequilibrium quasiparticle distribution doesnot result in an additional contribution to the supercurrentﬂowing via N j,t. However, it is worth noting here that the effect of the spin-independent nonequilibrium distribution functionhas been considered as well. 33,34It was shown that in the limit of small exchange ﬁelds h/lessmuch\|/Delta1\|the combined effect of the exhange ﬁeld and the nonequilibrium distribution functionis also nontrivial. For instance, part of the ﬁeld-suppressedsupercurrent can be recovered by adjusting a voltage betweenadditional electrodes, which controls the distribution function. In the present paper we continue investigation of the interplay between the triplet correlations and spin-dependentquasiparticle distribution. As was explained above, the simul-taneous presence of the triplet correlations and spin-dependentquasiparticle distribution in the interlayer results in theappearence of the additional contribution to the supercurrentﬂowing via N j,t. In the present paper we concentrate on two features of this additional supercurrent, which are of mainphysical interest and propose appropriate mesoscopic systems,where they can be observed: (i) The additional supercurrent allows for direct measure- ment of the energy-resolved odd-frequency triplet anomalousGreen’s function in the interlayer. The point is that, for junc-tions with low-transparency interfaces between the supercon-ductor and the interlayer region, N j,tis directly proportional to the triplet part of the anomalous Green’s function in theinterlayer. By measuring the “nonlocal” conductance (that is,the derivative of the critical current with respect to voltage V, which is applied to the additional electrodes attached to theinterlayer region and controls the value of spin injection intothe interlayer), one can experimentally obtain the value of thetriplet part of the anomalous Green’s function in the interlayeras function of energy. As was discussed in the introduction,the triplet correlations induced by the proximity effect inS/F structures are odd in Matsubara frequency, that is, thecorresponding two-particle condensate wave function taken atcoinciding times is zero. Therefore, the direct measurementof the energy-resolved anomalous Green’s function is of greatinterest. Here we propose an experimental setup, which allows for extracting from the current short-range triplet component(SRTC) and LRTC contributions and their separate obser-vation. By measuring the “nonlocal” conductance one canseparately obtain the values of LRTC and SRTC of theanomalous Green’s function in the interlayer as functionsof energy. It is based on a multilayered S/N/F/N/S junction,where a layer made of a weak ferromagnetic alloy havingan exchange ﬁeld /Delta1/lessmuchh/lessmuchε Fis sandwiched between two normal metal layers. The direction of the F layer magnetizationis assumed to be nonuniform in order to have a possibilityof LRTC investigation. The leads are made of dirty s-wave superconductors. While all the experiments described in the introduction give unambigous signatures of the fact that the odd-frequencytriplet correlations do exist in hybrid SF systems, they do notallow for direct investigation of how the triplet anomalousGreen’s function depends on energy. For example, the Joseph-son current in equilibrium is only carried by the scalar part ofthe SCDOS N j,s. Surely, it is modiﬁed by the presence of the triplet component (and, in particular, manifests weakly decay-ing behavior if LRTC is present in the system). However, N j,s is not directly proportional to the triplet anomalous Green’s 054533-2INFLUENCE OF SPIN-DEPENDENT QUASIPARTICLE ... PHYSICAL REVIEW B 84, 054533 (2011) function, but can contain it only in a nonlinear way. The other measurable quantity in equilibrium is the local density of states(LDOS), where the odd-frequency triplet component manifestsitself as a zero-energy peak. This effect has been studied forSF bilayers and for SN bilayers with magnetic interfaces. 35–40 However, the LDOS is also not directly proportional to the triplet anomalous Green’s function. The oscillating behaviorof the critical temperature as a function of an SF bilayer width(see, for example, Ref. 11and references therein) is also an ex- cellent ﬁngerprint of the triplet correlations (one-dimensionalLOFF state) presence. However, the order parameter in thesinglet superconductor S is related only to the singlet partof the anomalous Green’s function, which is modiﬁed by thepresence of triplet correlations, but does not allow for theirdirect observation. On the other hand, if the quasiparticledistribution is spin dependent, quantities, which are directlyproportional to the triplet anomalous Green’s function, start tocontribute to experimentally observable things.The Josephsoncurrent under the condition of spin-dependent quasiparticledistribution in the interlayer is one of them. (ii) It is well known that ferromagnetism and singlet superconductivity are antagonistic to each other. In an over-whelming majority of situations it results in the suppression ofthe Josephson current through the system with ferromagneticelements with respect to the system with the same interfaceresistances but without ferromagnetic elements. This is alsovalid even if the LRTC is formed in the system. In thepresent paper we show that in some cases the exchangeﬁeld-suppressed supercurrent can be not only recovered butalso enhanced with respect to its value for a nonmagneticjunction with the same interface resistances by the presenceof a spin-dependent quasiparticle distribution. That is, roughlyspeaking, in some cases the spin-dependent quasiparticle dis-tribution can overcompensate for the suppression of proximity-induced superconducting correlations by ferromagnetism. Wedemonstrate that such an effect can be observed in an S/N/Sjunction with magnetic interfaces. The paper is organized as follows. In Sec. IIthe considered model systems are described and the theoretical frameworkto be used for obtaining our results is established. In Sec. III we present the results of the Josephson current calculationfor a multilayered S/NFN/S system under a spin-dependentquasiparticle distribution and demonstrate how to obtain fromthese data information about the structure of the odd-frequencytriplet correlations. Section IVis devoted to consideration of an SNS junction with magnetic SN interfaces under similarconditions for the quasiparticle distribution in the interlayer.We summarize our ﬁndings in Sec. V. In Appendix Awe present the results for the anomalous Green’s function inthe interlayer and all the parts of the Josephson currentfor S/NFN/S junction, calculated in the framework of aparticular microscopic model of the N/F/N layer. Appendix B is devoted to a particular microscopic model of the magneticS/N interface, which we assume to be more appropriate for theinvestigation of current enhancement in the SNS junction. II. MODEL AND GENERAL SCHEME OF CALCULATIONS The ﬁrst system we consider is a multilayer S/NFN/S Josephson junction shown schematically in Fig. 1. It consistsof two s-wave superconductors (S) and an interlayer composed of two normal layers NlandNrwith a ferromagnetic layer F, sandwiched between them. The xaxis is directed along the normal to the junction and the yandzaxes are in the junction plane. The coordinates of FN interfaces are x=∓dF/2, while SN interfaces are located at x=∓ (dF+dN)/2. That is, the full length of the F layer is dF, while the length of each N layer isdN/2. The middle F layer is supposed to have the exchange ﬁeldhsatisfying the condition /Delta1/lessmuchh/lessmuchεF. The exchange ﬁeld of the F layer is assumed to be nonhomogeneous,which allows for the existence of triplet pairs with oppositespins (SRTC) and triplet pairs with parallel spins (LRTC)in the interlayer. h=h(0,sin/Theta1(x),cos/Theta1(x)), that is, the magnetization vector rotates in the F layer (within the junctionplane). For simplicity we suppose that the rotation angle has asimple xdependence: /Theta1(x)=/Theta1 /primex,−dF/2<x<d F/2, (2) where /Theta1/primedoes not depend on coordinates. The additional electrodes are supposed to be attached to the N layers in orderto make it possible to create a spin-dependent quasiparticledistribution in the interlayer. We use the formalism of quasiclassical Green–Keldysh functions 41and assume that the superconductors and all the internal layers are in the diffusive regime. The fundamentalquantity for diffusive transport is the momentum average of thequasi-classical Green’s function ˇg(r,ε,t)=/angbracketleftˇg(p f,r,ε,t)/angbracketrightpf. It is an 8 ×8 matrix form in the product space of Keldysh, particle-hole, and spin variables. In the absence of an explicitdependence on a time variable the Green’s function ˇg(r,ε)i n the interlayer obeys the Usadel equation D π∇(ˇg∇ˇg)+[ετ3σ0ρ0−ˇh,ˇg]=0, (3) where τi,σi, andρiare Pauli matrices in particle-hole, spin, and Keldysh spaces, respectively, and τ0,σ0, and ρ0stand for the corresponding identity matrices. For simplicity, thediffusion constant Dis supposed to be identical in all three internal layers. The matrix structure of the exchange ﬁeld is asfollows ˇh=hσρ 0(1+τ3)/2+hσ∗ρ0(1−τ3)/2. (4) The exchange ﬁeld hrotates in the F layer according to the model described above. In the N layers h=0. Usadel equation ( 3) should be supplied with the normaliza- tion condition ˇg2=−π2τ0σ0ρ0and is subject to Kupriyanov– Lukichev boundary conditions42at the S/N and N/F interfaces. The barrier conductances of the left and right S/N interfacesare assumed to be identical for simplicity and are denoted asG T. Then the boundary conditions at S/N interfaces take the form ˇgN∂xˇgN=−αGT 2σN[ˇgN,ˇgS]. (5) Here ˇgNis the solution of Usadel equation ( 3)a tt h el e f t( x= −(dN+dF)/2) or right ( x=(dN+dF)/2) S/N interface. α=+ 1(−1) at the left (right) interface. σNis the conductivity of the N layers and σFis the conductivity of the F layer (deﬁned for later use). ˇgSstands for the Green’s functions at the superconducting leads. Due to the fact that we are mostly 054533-3A. M. BOBKOV AND I. V . BOBKOV A PHYSICAL REVIEW B 84, 054533 (2011) interested in the case of low-transparent S/N interfaces below, we can safely neglect the suppression of the superconductingorder parameter in the S leads near the interface and takethe Green’s functions at the superconducting side of theboundaries to be equilibrium and equal to their bulk values. Inthis case ˇg K S=/parenleftbigˇgR S−ˇgA S/parenrightbig tanhε 2T, (6) ˇgR,A S=−iπκ cosh/Theta1R,A Sτ3σ0+iπκ sinh/Theta1R,A Siσ2 ×/bracketleftbigg e−iαχ 2τ1+iτ2 2+eiαχ 2τ1−iτ2 2/bracketrightbigg , (7) cosh/Theta1R,A S=−κiε/radicalbig \|/Delta1\|2−(ε+κiδ)2, (8) sinh/Theta1R,A S=−κi\|/Delta1\|/radicalbig \|/Delta1\|2−(ε+κiδ)2, where κ=+ 1(−1) for the retarded (advanced) Green’s func- tion,χstands for the order parameter phase difference between the superconducting leads, and δis a positive inﬁnitesimal. The second model system, which we consider in order to study the enhancement of a ﬁeld-suppressed supercurrentunder the spin-dependent distribution, is an S/N/S junctionwith magnetic S/N interfaces. The full length of the normalregion is d N,t h exaxis is normal to the junction plane, and the interfaces are located at x=∓dN/2. As in the previous case, additional electrodes are attached to the interlayer regionfor the creation of a spin-dependent quasiparticle distributionin the interlayer. The Green’s function in the N layer obeysEq. ( 3) provided that ˇh=0. However, the boundary conditions contain additional terms with respect to Eq. ( 5) because the transmission properties of spin-up and spin-down electronsinto a ferromagnetic metal or a ferromagnetic insulator aredifferent, which gives rise to spin dependent conductivities(spin ﬁltering) and spin-dependent phase shifts (spin mixing)at the interface. The generalized boundary conditions for thediffusive limit can be written in the form 43,44 ˇgN∂xˇgN=−αGT 2σN[ˇgN,ˇgS]−αGMR 2σN[ˇgN,{ˇmα,ˇgS}] +αGφπ 2σN[ˇmα,ˇgN], (9) where ˇgNis the Green’s function value at the normal side of the appropriate S/N interface (at x=∓dN/2). As above, ˇgS stands for the Green’s function in the superconducting lead and is expressed by Eqs. ( 6)–(8).ˇmα=mασρ0(1+τ3)/2+ mασ∗ρ0(1−τ3)/2, where mαis the unit vector aligned with the direction of the left ( α=+ 1) or right ( α=− 1) SN interface magnetization. {...}means an anticommutator. The second term accounts for the different conductancesof different spin directions and G MR∼GT,↑−GT,↓.T h e third term, ∼Gφ, gives rise to spin-dependent phase shifts of quasiparticles being reﬂected at the interface. It is worth notinghere that boundary conditions ( 9) are valid only for small (with respect to unity) values of transparency and a spin-dependentphase shift in one transmission channel. 44However, for the case of plane diffusive junctions we can safely consider ˜Gφ=GφξS/σN>1 due to a large number of channels. Gφhasbeen calculated for some particular microscopic models of the interface40,43and can be large enough even if the conductance GT→0. In Appendix Bwe calculate Gφ,GT, andGMRfor S/N interface composed of an insulating barrier and a thin layerof a weak ferromagnetic alloy. We suppose this microscopicmodel to be the most appropriate to the considered problem. In what follows we assume that the S/N interfaces are low transparent for both the considered systems, that is, ˜G T≡ GTξS/σN/lessmuch1 and ˜GMR≡GMRξS/σN/lessmuch1. To calculate the Josephson current through the junction in the leading orderof the interface transparency ˜G Tit is enough to obtain the retarded and advanced Green’s functions in the leading order ofthe transparency. If one makes use of the following deﬁnitionsfor the Green’s function elements in the particle-hole space (allthe matrices denoted by ˆ ...are 2×2 matrices in spin space throughout the paper), ˇg R,A=/parenleftbiggˆgR,A ˆfR,A ˆ˜fR,A ˆ˜gR,A/parenrightbigg , (10) then one can obtain from Usadel equation ( 3) and the appropriate boundary conditions, Eq. ( 5)o r( 9), that the diagonal in particle-hole space elements of ˇgR,Aare zero order in˜GTquantities and take the following forms in the interlayer: ˆgR,A=−iκπ (11)ˆ˜gR,A=iκπ. The off-diagonal in particle-hole space elements of the Green’s function are of the ﬁrst order in ˜GTand should be obtained from the linearized Usadel equations, which areto be derived from Eq. ( 3). It is convinient to represent the off-diagonal elements in the following forms: ˆf R,A=fR,A siσ2+fR,A tσiσ2, (12)ˆ˜fR,A=−iσ2˜fR,A s−iσ2˜fR,A tσ, where fR,A s andfR,A tdenote the singlet and triplet parts of the anomalous Green’s function, respectively. For the case weconsider (the magnetization vectors of all the ferromagneticlayers and spin-active interfaces, which are present in thesystem, are in the junction plane) the out-of-plane xcomponent of the triplet part is absent and the linearized Usadel equations for the anomalous Green’s function {f R,A s,fR,A t}can be written as follows: 2εfR,A s−2hfR,A t−iκD∂2 xfR,A s=0, (13) 2εfR,A t−2hfR,A s−iκD∂2 xfR,At=0. According to the general symmetry relation45ˆ˜fR,A(ε)= ˆfR,A∗(−ε) the singlet and triplet parts ofˆ˜fR,Acan be expressed via the corresponding parts of ˆfR,Aas follows ˜fs(ε)=−f∗ s(−ε), (14)˜ft(ε)=f∗ t(−ε). Linearized Usadel equations ( 13) should be supplemented by the appropriate boundary conditions, which are to be 054533-4INFLUENCE OF SPIN-DEPENDENT QUASIPARTICLE ... PHYSICAL REVIEW B 84, 054533 (2011) obtained by linearization of Eq. ( 5)o r( 9) and at the S/N interfaces take the form ∂xfR,A N,s=−αGT σNiκπ sinh/Theta1R,A Se−iαχ/ 2+αGφ σNiκmαfR,A N,t; (15) ∂xfR,A N,t=αGφ σNiκmαfR,A N,s, where fR,A N,s andfR,A N,tare the singlet and triplet part values of the anomalous Green’s function at the normal side of the S/Ninterface. G φ/negationslash=0 only if the S/N interface is spin active. It is worth noting here that in the linear order in ˜GTand ˜GMR the term proportional to GMRdoes not enter the boundary conditions. Equations ( 13) and ( 15) allow for the calculation of the retarded and advanced Green’s functions to the leading orderin transparency. However, it is not enough for obtainingthe electric current through the junction, which should becalculated via the Keldysh part of the quasi-classical Green’sfunction. For the plane diffusive junction the corresponding expression for the current density reads as follows j=−σ N e/integraldisplay∞ −∞dε 8π2Tr4/bracketleftbiggτ0+τ3 2(ˇg(x,ε)∂xˇg(x,ε))K/bracketrightbigg ,(16) where eis the electron charge. The expression is written for the normal layer, but it is also valid for the ferromagneticregion with the substitution σ FforσN.(ˇg(x,ε)∂xˇg(x,ε))Kis the 4×4 Keldysh part of the corresponding combination of the full Green’s function. It is convenient to calculate the currentat the S/N interfaces. Then the required combination of theGreen’s functions can be easily found from the Keldysh partof boundary conditions ( 5)o r( 9). In addition, we express the Keldysh part of the full Green’s function via the retarded and advanced components and the distribution function: ˇg K= ˇgRˇϕ−ˇϕˇgA. Here argument ( x,ε) of all the functions is omitted for brevity. The distribution function is diagonal in particle-hole space: ˇ ϕ=ˆϕ(τ 0+τ3)/2+σ2ˆ˜ϕσ2(τ0−τ3)/2. Then to the leading (second) order in transparency, Tr4/bracketleftbiggτ0+τ3 2(ˇg(x,ε)∂xˇg(x,ε))K/bracketrightbigg =αGTiπ σN/bracketleftBig/parenleftbig sinh/Theta1R S+sinh/Theta1A S/parenrightbig tanhε 2T/parenleftbig fR N,seiαχ/ 2+˜fA N,se−iαχ/ 2/parenrightbig −/parenleftbig fR N,s˜ϕ(0) 0−ϕ(0) 0fA N,s/parenrightbig sinh/Theta1A Seiαχ/ 2+/parenleftbig˜fR N,sϕ(0) 0−˜ϕ(0) 0˜fA N,s/parenrightbig sinh/Theta1R Se−iαχ/ 2 −/parenleftbig fR N,t˜ϕ(0)−ϕ(0)fA N,t/parenrightbig sinh/Theta1A Seiαχ/ 2+/parenleftbig˜fR N,tϕ(0)−˜ϕ(0)˜fA N,t/parenrightbig sinh/Theta1R Se−iαχ/ 2 −2iπ/parenleftbig cosh/Theta1R S+cosh/Theta1A S/parenrightbig/parenleftBig tanhε 2T−ϕ(0)+(1) 0/parenrightBig/bracketrightBig +2αGMRiπ σN/bracketleftbig 2iπ/parenleftbig cosh/Theta1R S+cosh/Theta1A S/parenrightbig mαϕ(0)+(1)/bracketrightbig , (17) where fR,A N,s andfR,A N,t are taken at the normal side of the appropriate S/N boundary. ϕ0andϕrepresent the scalar and vector parts of the distribution function ˆ ϕ=ϕ0+ϕσ, which is also taken at the normal side of the appropriate S/Nboundary. The superscripts ... (0)and...(0)+(1)of the distribution functions mean that the corresponding quantity is calculatedup to the zero and the ﬁrst orders of magnitude in the interfaceconductance ˜G T, respectively. To calculate the current through the junction one should substitute Eq. ( 17) into Eq. ( 16). The resulting expression for the current can be further simpliﬁed by taking intoaccount the general symmetry relations between the Green’sfunction elements 45expressed by Eq. ( 14) and the ones given below: fA s(ε)=fR s(−ε), fA t(ε)=−fR t(−ε), (18) ˜ϕ0(ε)=−ϕ0(−ε), ˜ϕ(ε)=ϕ(−ε).Then the expression for the current density takes the form j=/integraldisplay∞ −∞dε 2πe/braceleftBig αGT/parenleftBig Im/bracketleftbig fR N,seiαχ/ 2/bracketrightbig tanhε 2TRe/bracketleftbig sinh/Theta1R S/bracketrightbig +Re/bracketleftbig fR N,seiαχ/ 2/bracketrightbig ˜ϕ(0) 0Im/bracketleftbig sinh/Theta1R S/bracketrightbig +Re/bracketleftbig fR N,teiαχ/ 2/bracketrightbig˜ϕ(0)Im/bracketleftbig sinh/Theta1R S/bracketrightbig −πcosh/Theta1R S/bracketleftbig ϕ(0)+(1) 0 (ε)+ϕ(0)+(1) 0 (−ε)/bracketrightbig /2/parenrightBig +αGMRπcosh/Theta1R Smα/bracketleftbig ϕ(0)+(1)(ε)+ϕ(0)+(1)(−ε)/bracketrightbig/bracerightbig . (19) The additional contribution to the current, which is absent for a spin-independent distribution function, is given by the third term. As was mentioned in the introduction, this term (connected to the triplet part of the SCDOS) is directlyproportional to the triplet anomalous Green’s function at theinterface. The ﬁfth term also results from the vector part ofthe distribution function, but under the considered conditionsit does not contribute to the current, as it is shown below. Itis worth noting here that, as seen from Eq. ( 19), the singlet part of the SCDOS is expressed only via the singlet part 054533-5A. M. BOBKOV AND I. V . BOBKOV A PHYSICAL REVIEW B 84, 054533 (2011) of the anomalous Green’s function. However, it does not mean that the triplet correlations do not contribute to thecurrent for the case of a spin-independent distribution function.They do contribute, as was demonstrated by a number ofexperiments discussed in the introduction. The point is thatfor the considered case of the tunnel junction, f sin general contains long-range contributions resulting from the LRTC (ifthey are present in the system). It is worth emphasizing thatall the aforesaid concerns only the calculation of the current atthe interface. If one would calculate the current at an arbitrarypoint of the interlayer, the corresponding expression wouldcontain f tquadratically. Surely, the current by itself does not depend on the xcoordinate, as it is required by the current conservation. The distribution function ˆ ϕ(0)+(1)entering the current given by Eq. ( 19) should be calculated by making use of the kinetic equation, which is obtained from the Keldysh part of Usadelequation ( 3). As we need the distribution function only up to the ﬁrst order in the interface conductance, all the termsaccounting for the proximity effect (which are of the secondorder in ˜G T) drop out and the kinetic equation takes an especially simple form (we do not take into account inelasticrelaxation in the interlayer): ∇ 2ˆϕ−i D[h(x)σ,ˆϕ]=0, (20) where the exchange ﬁeld h(x) is determined above for the ferromagnetic layer and vanishes for all the normal regions. The kinetic equation should be supplemented by the boundary conditions at the S/N interfaces and the interfaceswith additional electrodes, attached to the normal regions ofthe interlayer in order to create a spin-dependent quasiparticledistribution. While the boundary conditions at the interfaceswith additional electrodes are discussed below for a particularconsidered system, the boundary conditions at the S/N inter-faces are obtained from the Keldysh part of Eqs. ( 9)o r( 5) and to the ﬁrst order in the interface conductance take the form ∂ xˆϕ(0)=αiG φ 2σN[mασ,ˆϕ(0)], ∂xˆϕ(1)=−αGT 2σN/parenleftbig cosh/Theta1R S+cosh/Theta1A S/parenrightbig/parenleftBig tanhε 2T−ˆϕ(0)/parenrightBig −αGMR σN/bracketleftBig/parenleftbig cosh/Theta1R S+cosh/Theta1A S/parenrightbig tanhε 2Tmασ −cosh/Theta1R Smασˆϕ(0)−cosh/Theta1A Sˆϕ(0)mασ/bracketrightbig +αiG φ 2σN[mασ,ˆϕ(1)]. (21) For the case of a multilayered N/F/N interlayer, Eq. ( 20) should be also supplemented by boundary conditions at theN/F interfaces, which are to be obtained from Keldysh part ofKupriyanov–Lukichev boundary conditions ( 5) and are given in Appendix Afor a particular microscopic model. III. S/NFN/S JUNCTION Now we consider the particular systems. This section is devoted to the S/NFN/S Josephson junction. The modelassumed for the exchange ﬁeld of the F layer is alreadydescribed above. The anomalous Green’s function is foundFIG. 1. S/N/F/N/S junction under consideration with the addi- tional electrodes, which are proposed to be used for the creation of aspin-dependent quasiparticle distribution in the interlayer. up to the ﬁrst order in S/N conductance ˜GTaccording to Eqs. (13), (15), and boundary conditions at the F/N interface, which should be easily obtained from Eq. ( 5) for a given conductance of this interface. We assume that the magnetization of theF layer rotates slowly, that is, /Theta1 /primeξF/lessmuch1, while /Theta1/primeξS∼1 or even larger than unity. This assumption seems to bequite reasonable. 2Therefore, upon calculating the anomalous Green’s functions, we disregard all the terms proportional to/Theta1 /prime√D/h and higher powers of this parameter, while keeping the terms, where /Theta1/primeenters in the dimensionless combination /Theta1/prime√D/\|ε\|. To this accuracy the triplet part of the anomalous Green’s function can be represented as fR t=(0,fy,fz), fy=sin/Theta1(x)fSR(x)−cos/Theta1(x)fLR(x), (22) fz=cos/Theta1(x)fSR(x)+sin/Theta1(x)fLR(x), where the zaxis is aligned with the direction of the exchange ﬁeld in the middle of the F layer (at x=0).fSR(fLR)i s formed by the Cooper pairs composed of the electrons withopposite (parallel) spins. We are interested in the values ofthe triplet component at the S/N interfaces, where sin /Theta1(x)≡ −αsin/Theta1≡−αsin[/Theta1 /primedF/2]. The particular expressions for fSRandfLRand singlet component fsdepend strongly on the conductance of the F/N interface and in general arequite cumbersome. To give an idea of their characteristicbehavior we have calculated them for the most simple modelof absolutely transparent F/N interfaces. The correspondingexpressions are given in Appendix A. f SRis rapidly decaying in the interlayer, while fLRis slowly decaying. Let us consider fSRat the left S/N interface (the left interface is chosen just for deﬁniteness). It can be rewritten inthe form f SR=fl SRe−iχ/2+fr SReiχ/2, (23) where fl SRis generated by the proximity effect at the left S/N interface itself and fr SRcomes from the right S/N interface. It can be shown that for a thick-enough ferromagnetic layer 054533-6INFLUENCE OF SPIN-DEPENDENT QUASIPARTICLE ... PHYSICAL REVIEW B 84, 054533 (2011) dF/ξF/greatermuch1fr SR/fl SRis proportional to the small factor e−dF/ξF. On the other hand, if fLRis represented as fLR=fl LRe−iχ/2+fr LReiχ/2, (24) fr LRdoes not contain the small factor e−dF/ξFin the leading approximation and, therefore, fLRdescribes the LRTC. As is explicitly demonstrated in Appendix A, the characteristic decay length of fLRin the F layer is \|λt\|−1, where λt=/radicalbig /Theta1/prime2−2i(ε+iδ)/D. It is much larger than ξFfor the considered case ξF/lessmuch/Theta1/prime−1. To the considered accuracy the singlet component of the anomalous Green’s function also decays at the distance ∼ξFin the F layer, just as SRTC fSRdoes, because it is also composed of the electron pairs with antiparallel spin directions. Indeed,iff R sat the left boundary is also represented as fR s=fl se−iχ/2+fr seiχ/2, (25) thenfr s∝e−dF/ξFin the limit dF/ξF/greatermuch1. Therefore, the main contribution to the Josephson current of Eq. ( 19)i s given by the LRTC component fLRof the anomalous Green’s function. However, this contribution is nonzero only for thecase of the spin-dependent quasiparticle distribution. In thestandard case of a thermal spin independent quasiparticledistribution the current is determined by the singlet componentf s. Consequently, it contains only the term proportional to the small factor e−dF/ξF. At ﬁrst glance, it contradicts to the well known fact that the equilibrium Josephson current containsthe contribution generated by the LRTC, if it is present inthe system. 1,2In fact, if one calculates the current at the S/N boundary, then fsshould be modiﬁed by presence of LRTC and should contain a slowly decaying term, whichprovides the appropriate contribution. It is indeed the case forthe system we consider. However, the corresponding term isproportional to ( /Theta1 /primeξF)2and is disregarded in our calculation. It should deﬁnitely be taken into account upon calculatingthe Josephson current for the case of a spin independentquasiparticle distribution, because in spite of the small factor(/Theta1 /primeξF)2it can result in a large-enough current contribution due to the absence of the suppression factor e−dF/ξF.A t the same time we can safely disregard this term, becausefor the considered case of a spin-dependent quasiparticledistribution the main contribution to the Josephson currentis given by the f LRterm, which contains neither ( /Theta1/primeξF)2nor the ferromagnetic suppression factor e−dF/ξF. To generate a spin-dependent quasiparticle distribution in the interlayer, additional electrodes are attached to the normalregions of the interlayer. While in this paper we propose someparticular way of such a distribution creation, it is not importanthow it is obtained. The main point is to have a vector partϕ(ε) of the distribution function in the interlayer generated anyway. For example, it can be created by a spin injection intothe interlayer. If this is the case, the results discussed belowqualitatively survive. In this paper we assume that each of the normal regions of the interlayer is attached to two additional normal electrodes N l(r) bandNl(r) t(see Fig. 1). In their turn, the electrodes Nl bandNr bhave insertions PlandPrmade of a strongly ferromagnetic material. Let the voltage Vl(r) b−Vl(r) t=Vl(r)be applied between the electrodes Nl(r) bandNl(r) t.H e r e Vl(r) b andVl(r) tare the electric potentials of the outer regions of theNl(r) bandNl(r) telectrodes with respect to the potential of the superconducting leads. It is worth noting here thatthe superconductor is assumed to be closed to a loop andthe voltage between the superconducting leads is absent. The conductances of the N l(r)/Nl(r) bandNl(r)/Nl(r) tinterfaces are denoted by gl(r) bandgl(r) t, respectively. Further, for deﬁniteness we consider the left normal region of the interlayer with the corresponding additional electrodes.We choose the quantization axis z lalong the magnetization of the left ferromagnetic insertion Pl, and the deﬁnitions RPl↑, RPl↓stand for the Plregion resistivities for spin-up and spin- down electrons. Then under the conditions that (i) the Nllayer resistance RNand the resistance of Nl bpart, which is enclosed between NlandPlcan be disregarded compared with 1 /gt andRPl↓, and (ii) 1 /RPl↓/lessmuchgl t/lessmuch1/RPl↑, one can believe that the voltage drops mainly at the Plregion for spin-down electrons and at the Nl/Nl tinterface for spin-up electrons. Also, the dissipative current ﬂowing through the Nl b/Nl/Nl t system is small and can be disregarded. Consequently, it is obtained that the electric potentials for spin-up and spin-downelectrons in the N l bregion enclosed between PlandNlare different and practically constant over this region. While thespin-up electrons are at the electric potential V l bin this region, the potential for spin-down electrons is approximately Vl t. To simplify the calculations we assume that Vl=Vr= Vb−Vt≡2V. The left and the right additional electrodes differ only by the direction of the magnetization of the Pl andPrinsertions. For later use we deﬁne the unit vectors aligned with the PlandPrmagnetizations as MlandMr, respectively. To satisfy the electroneutrality condition theelectric potential of the superconducting leads should be equalto (V t+Vb)/2. Then the electric potentials for spin-up and spin-down electrons in the Nl bregion enclosed between Pland Nlcounted from the level of the superconducting leads are V↑=(Vb−Vt)/2=VandV↓=(Vt−Vb)/2=−V.D u et o the fact that one can disregard the voltage drop inside thisregion, the distribution functions for spin-up and spin-downelectrons in this region are close to the equilibrium form (withdifferent electrochemical potentials). For the general case (ifthe quantization axis does not align with the P lmagnetization) the distribution function becomes a matrix in spin space andtakes the form ˆϕ l=ϕ0σ0+ϕtMlσ, ϕ0=1 2/bracketleftbigg tanhε−eV 2T+tanhε+eV 2T/bracketrightbigg , (26) ϕt=1 2/bracketleftbigg tanhε−eV 2T−tanhε+eV 2T/bracketrightbigg . The same form of the distribution function is valid for the Nr b part enclosed between PrandNrwith the substitution Mrfor Ml. Now we can obtain the distribution function in the Nl andNrregions of the interlayer, which enters the current of Eq. ( 19). For the considered case gt/lessmuch1 the dissipative 054533-7A. M. BOBKOV AND I. V . BOBKOV A PHYSICAL REVIEW B 84, 054533 (2011) current ﬂowing through Nl b/Nl/Nl tjunction is negligible and, therefore, the y-dependence of the distribution function in the Nl(r)region can be disregarded. Then under the condition σF/lessmuchσNthe distribution functions ˆ ϕ(0)in the NlandNr regions calculated up to zero order in the S/N conductance ˜GTare spatially constant and equal to ˆ ϕland ˆϕr, respectively. Indeed, Eq. ( 20) for the distribution function at h=0 and Eq. ( 21) for the boundary conditions at S/N interfaces (corre- sponding to Gφ=0) are satisﬁed by this solution. Boundary conditions ( A3) at F/N interfaces are satisﬁed approximately due to the smallness of the distribution function gradient underthe condition σ F/lessmuchσN. If one goes beyond the approximation σF/σN/lessmuch1, the distribution function in the NlandNrregions acquires gradient terms proportional to the parameter σF/σN. If the F/N interface is less transparent than what is consideredin Appendix A, the distribution function gradient in the N layer is even smaller and the condition σ F/lessmuchσNis not so necessary. Although the distribution function in the middle F layer does not enter current expression ( 19), it is interesting to discuss here how it behaves. For simplicity we consider the limiting case /Theta1/prime→0, when the exchange ﬁeld in the ferromagnet is practically constant and the qualitative physicalpicture is more clear. According to Eq. ( 20) and boundary conditions ( A3) the scalar part of the distribution function ϕ 0 is constant over the F layer and coincide with its value in the Nl andNrregions. The vector component parallel to the exchange ﬁeld of the ferromagnet is a linear function of the xcoordinate, which matches the constant values ϕtMl(r)hl(r)/hat the F/N interfaces. Here hl,r≡h(x=∓dF/2) are the exchange ﬁeld values at the left and right N/F interfaces. The vector compo- nent perpendicular to the exchange ﬁeld of the ferromagnet decays from the F/N interfaces into the ferromagnetic regionat the characteristic length ξ Foscillating simultaneously with a period 2 πξF, as can be obtained from Eq. ( 20) and boundary conditions ( A3). Strictly speaking, the distribution function in the N layers only takes the form of Eqs. ( 26) if one assumes no spin relaxation there. Spin-relaxation processes reduce the vectorpartϕ tof the distribution function of Eqs. ( 26). The reduction can be roughly estimated as ϕsr t=ϕt/(1+τesc/τsr). Here ϕsr tis the vector part of the distribution function in the presence of spin-relaxation processes, while ϕtis deﬁned by the second of Eqs. ( 26).τesc=σNdy/D(gb+gt)i sa n effective time, which an electron spends in the N layerbefore escaping. τ sris the characteristic spin-relaxation time. So spin-relaxation processes do not qualitatevely inﬂuencethe distribution fuction if τ esc/τsr/lessmuch1. This condition seems to be not restrictive in real materials. For example, let usassume that the N layers are made of Al in the normal state,where λ sr=√Dτ sr=450μm46has been reported. Then the condition τesc/τsr=σNdy/(gb+gt)λ2 sr/lessmuch1 can be valid in a wide range of the values of dimensionless parameter ( gb+ gt)ξS/σNcharacterizing the joint conductance of Nl(r) b/Nl(r) andNl(r) t/Nl(r)interfaces. Now we turn to the discussion of the Josephson current through the junction. It is expressed by Eq. ( 19). As for the considered case of nonmagnetic S/N interfaces GMR=0, the last term in this formula is absent. Due to the fact that the scalarpartϕ(0) 0of the distribution function in the interlayer [Eq. ( 26)]is an odd function of quasiparticle energy, the part of the current generated by the term ∝cosh/Theta1R S[ϕ(0) 0(ε)+ϕ(0) 0(−ε)]/2a l s o vanishes. Further, to avoid the ﬂowing of a quasiparticlecurrent through the junction we assume that \|eV\|</Delta1 and the temperature is low ( T/lessmuch/Delta1). Under these conditions the linear in the xcoordinate part of ϕ (1)(it is this term that provides the ﬂowing of the quasiparticle current through the junction)is zero in each of the N regions of the interlayer, as dictatedby boundary conditions ( 21). Therefore, ϕ (1)is approximately constant in each of the N layers. We comment on the values ofthese constants below. The ﬁrst two terms in Eq. ( 19) represent the contribution of the SCDOS singlet part, which takes place as for the caseof spin-independent quasiparticle distribution, just when thisdistribution is spin dependent. We refer to this contributionasj s. The particular expressions for jscan be easily found in the framework of a given microscopic model of the NFNinterlayer after substitution of the particular expressions forthe singlet part of the anomalous Green’s function and thescalar part of the distribution function [Eqs. ( 26) and ( 18)] into Eq. ( 19). The third term in Eq. ( 19) contains the current ﬂowing through the SCDOS triplet part and is nonzero onlyfor the case of spin-dependent quasiparticle distribution. Thiscontribution is the main result of the present section. If onesubstitutes the particular expressions for the triplet part of theanomalous Green’s function [Eqs. ( 22)] and the vector part of the distribution function [Eqs. ( 26) and ( 18)] this contribution takes the form j l,r t=−jSRhl,rMl,r h+αjLR(Ml,r×hl,r)ex h, (27) where exis the unit vector along the xdirection. The currents jSRandjLRare generated by the SRTC and the LRTC of the anomalous Green’s function, respectively. Consequently,if the F layer is thick, that is ξ F/lessorsimilardF, the current jSR(as well as js) is small due to the factor e−dF/ξF, while jLRis not suppressed by this factor. The particular expressions for js, jSR, andjLRcalculated in the framework of the microscopic model considered here are given in Appendix A. It is seen from Eq. ( 27) that the values of the current contribution jtat the left and right S/N interfaces can be different, that is, in general, jl,r t=jt±ja. However, under the condition that the superconducting leads are closed into a loop,the currents at the left and right S/N interfaces must be equalto each other. It appears that the distribution function in the N layers acquires additional terms ϕ (1) l,r, which are proportional to GT. Under the condition Vl=Vrwe obtain that ϕ(1) l=ϕ(1) r. Then, according to Eq. ( 19) this term results in the current contribution, which exactly compensates for ja. Therefore, the Josephson current jtﬂowing through the junction can be simply calculated as jt=(jl t+jr t)/2. It is obvious from Eqs. ( 26) that jsis an even function of voltage Vapplied to the additional electrodes and jt is an odd function of this voltage. Therefore, it is easy to extract in experiment contributions jsandjtfrom the full Josephson current: js(V)=(j(V)+j(−V))/2, while jt(V)=(j(V)−j(−V))/2. Further, it is seen from Eq. ( 27) that by choosing the appropriate orientation of PlandPr magnetizations, one can, in principle, measure either jSR 054533-8INFLUENCE OF SPIN-DEPENDENT QUASIPARTICLE ... PHYSICAL REVIEW B 84, 054533 (2011) orjLRcurrent contributions. For this reason it makes sense to discuss all the current contributions, js,jSR, and jLR, separately. In the tunnel limit all of them manifest sinusoidaldependence on the superconducting phase difference χ, that is,j s,SR,LR =jc s,SR,LR sinχ. Therefore, we discuss only the corresponding critical currents, jc s,jc SR,jc LR, below. Figure 2represents these contributions, calculated in the framework of the microscopic model discussed in Appendix A, as a function of voltage V. First, it is worth noting that current components jSRandjLR, carried by the triplet part of SCDOS, are nonzero only for V/negationslash=0. That is, indeed, the triplet part of SCDOS contributes to the current only if a spin-dependentquasiparticle distribution is created in the interlayer. Theexchange ﬁeld his chosen to be not very strong, h=10/Delta1. Such a choice is in general agreement with the characteristicvalues of the exchange ﬁeld in weak ferromagnetic alloys.However, the results discussed below qualitatively survive forthe case of more strong exchange ﬁelds. Roughly speaking,increasing of the exchange ﬁeld inﬂuences the results in thesame manner as increasing the F layer length d F. Panels (a), (b), and (c) of Fig. 2correspond to different lengths of the N and F regions forming the interlayer. Belowall the lengths are expressed in units of the superconductingcoherence length ξ S. The magnetic coherence length ξF= ξS√/Delta1/h is approximately three times shorter than ξS.F o r panels (a) and (b) the ferromagnetic layer is not thick ( dF= 1). They differ by the length of the normal layer: Panel(a) corresponds to d N=2, and for panel (b) dN=1. As expected, upon increasing dNthe magnitude of all the current components decreases not very sharply. The correspondingdecay length is considerably larger than ξ F. On the other hand an increase of dFsuppresses current components jsandjSR exponentially with the characteristic decay length ξF.I ti s natural because they ﬂow via the singlet component and SRTCof the anomalous Green’s function. These components arecomposed of the Cooper pairs with opposite spin directionsand, correspondingly, decay rapidly into the depth of theferromagnetic region. It is seen from the ﬁgure that for panels(a) and (b) j SRandjLRare of the same order, while jsis even larger. This is not the case for panel (c), where dF=2. For this parameter range jsandjSRare already suppressed. However, for a certain voltage range (small-enough voltages)j LRis not suppressed and the dependence of its magnitude ondFis the same as on dN. For larger voltages the value ofjLRis also suppressed. It is interesting to note that this suppression takes place for all the panels of Fig. 2irrespective of the F layer length. It is obvious that the insensitivity of jLR to the length of the ferromagnetic region is a result of the fact that it is carried by Cooper pairs composed of the electronswith parallel spins. However, the characteristic behavior ofthis component upon varying V(sharp maximum at small voltages and subsequent suppression) requires an additionalexplanation. Such an explanation is closely connected to theparticular shape of the anomalous Green’s function LRTC andis given below upon discussing the LRTC. Further, the dependence of all three current components on the length of the ferromagnetic layer is studied in moredetail. Panels (a), (b) and (c) of Fig. 3demonstrate this dependence for three different voltages V. For panel (a) the particular value of this voltage is chosen to be V=0.05/Delta1.FIG. 2. Current components jc s(solid line), jc SR(dashed line), andjc LR(dotted line) as functions of voltage V, applied between the additional eletrodes. The currents are measured in arbitrary units. (a)d F=1,dN=2, (b)dF=1,dN=1( c )dF=2,dN=1. All lengths are measured in units of ξS. The other parameters are the following: h=10/Delta1,/Theta1/primeξS=0.2,T=0. This value approximately corresponds to the maximum of jLR in Fig. 2. For panel (b) V=0.1/Delta1. Current jLRgradually declines at this voltage region. Finally, the plots shown inpanel (c) correspond to V=0.5/Delta1, where j LRis already greatly suppressed. First, it is worth noting that the decaylength of j sandjSRisξFto a good accuracy for any voltage region. Also, it is seen from Fig. 3thatjsandjSRoscillate upon increasing dFwith the period 2 πξF(irrespective of the particular voltage). For js, which is nonzero even for a spin-independent quasiparticle distribution, these oscillationsare well studied. They are a hallmark of the mesoscopic LOFFstate, as was mentioned in the introduction. j SRis absent for a spin-independent quasiparticle distribution, but is carriedby the same pairs of electrons with opposite spin directions,just as j s, and, consequently, also manifests the LOFF state oscillations. While the oscillation period is the same for jsand 054533-9A. M. BOBKOV AND I. V . BOBKOV A PHYSICAL REVIEW B 84, 054533 (2011) FIG. 3. Current components jc s(solid line), jc SR(dashed line) andjc LR(dotted line) as functions of dF/ξS(logarithmic scale). (a)V=0.05/Delta1,( b )V=0.1/Delta1,( c )V=0.5/Delta1. The other parameters are the following: h=10/Delta1,/Theta1/primeξS=0.2,dN=1,T=0. jSR, there is a phase shift between their oscillations, which depends on the particular value of the voltage V. Unlike jsandjSR,jLRdoes not manifest oscillating behavior. Its decay length is not connected to ξFand crucially depends on V. This decay length lLRis maximal for the voltage region, where jLRhas maximal value [( lLR≈2ξS≈6ξFfor panel (a)] and declines upon increasing V. As was mentioned in the introduction, the dependence of the anomalous Green’s function in the interlayer onthe quasiparticle energy can be partially extracted from theJosephson current measurements. It can be done due to thefact that voltage Venters current expression ( 19) only via distribution functions ( 26). Then, according to Eqs. ( 19), (27), and ( 22), by taking the derivatives of the Josephson currentsj s,jSR, andjLRwith respect to the voltage applied between the additional electrodes, at T→0 one obtaines that djs/dV∼Im/bracketleftBigg fr s(V)/radicalbig /Delta12−(eV+iδ)2/bracketrightBigg , djSR/dV∼Im/bracketleftBigg fr SR(V)/radicalbig /Delta12−(eV+iδ)2/bracketrightBigg , (28) djLR/dV∼Im/bracketleftBigg fr LR(V)/radicalbig /Delta12−(eV+iδ)2/bracketrightBigg . Herefr SR,fr LR, and fr sare determined by Eqs. ( 23), (24), and ( 25), respectively. That is, indeed, imaginary parts of the anomalous Green’s function components comingfrom the opposite interface corresponding to all threetypes of superconducting correlations can be extractedfrom the Josephson current measurements. However,under the condition \|eV\|</Delta1 , it can be done only for subgap energies \|ε\|</Delta1 . It is worth noting here that the derivatives of current components j s,jSR, and jLRwith respect to Vgive us the corresponding anomalous Green’s function components only in the tunnel limit ˜GT/lessmuch1. In the general case these derivatives are proportional tothe appropriate components of the SCDOS, which areexpressed via the anomalous Green’s function in a morecomplicated way. Panels (a), (b), and (c) of Fig. 4represent combinations F s≡Im[fr s(ε)//radicalbig /Delta12−(ε+iδ)2],FSR≡ Im[fr SR(ε)//radicalbig /Delta12−(ε+iδ)2], and FLR≡Im[fr LR(ε)//radicalbig /Delta12−(ε+iδ)2], calculated in the framework of our microscopic model, as functions of the quasiparticle energyεmeasured in units of /Delta1. In each panel different curves correspond to different lengths d NanddF(see caption to Fig. 4). It is seen that the value of the normal region length does not inﬂuence qualitatively all three components of theanomalous Green’s function. As expected, F sandFSRare strongly suppressed upon increasing of dF. On the other hand, FLRis only very weakly sensitive to the changing of dF.I ti s dominated by the sharp dip at low energies, which is followedby wider peaks, where F LRchanges sign. The width δεof the dip is∼√ /Delta1D/Theta1/prime. It is the characteristic shape of FLRthat is responsible for jLRbehavior upon varying V, shown in Fig. 2. The point is that at low-enough temperatures only the part of FLR belonging to energy interval [ −\|eV\|,\|eV\|] contributes to jLR. Consequently, upon increasing of V,jLRgrows sharply up toV∼(1/2)√ /Delta1D/Theta1/prime, and after that starts to decline due to the opposite sign contribution of the peaks. It appears that thecontributions of the dip and the peaks mainly compensate eachother, which leads to strong supression of j LRfor large-enough V. The dependence, discussed above, of jLRdecay length on Vis also closely connected to the fact that only the part of FLR, belonging to energy interval [ −\|eV\|,\|eV\|], “works” upon creating jLR. Indeed, the characteristic decay length of FLR(ε) dcr F(ε)∼\|λt(ε)\|−1. Therefore, jLRdecay length ∼1//Theta1/primefor small voltages and gets shorter for larger voltages due to theincreased contribution of higher energies. 054533-10INFLUENCE OF SPIN-DEPENDENT QUASIPARTICLE ... PHYSICAL REVIEW B 84, 054533 (2011) FIG. 4. Combinations (a) Fs,( b )FSR,a n d( c ) FLRas functions of quasiparticle energy ε//Delta1 . In each panel ( dF=1,dN=2) for solid curves, ( dF=1,dN=1) for dashed curves, and ( dF=2,dN=1) for dotted curves. The other parameters are the same as in Fig. 2. The possibility of extracting singlet and triplet components of the proximity-induced anomalous Green’s function in theinterlayer is not the only motivation to study the Josephsoncurrent under a spin-dependent quasiparticle distribution.Being an easily controllable parameter, voltage Vgives a possibility of obtaining highly nonlinear characteristics j(V) with a number of 0- πtransitions, which can be essential for superconducting electronics. As was already mentionedabove, by choosing the appropriate orientation of P landPr magnetizations, one can, in principle, “turn off” either the jSRorjLRcurrent contribution. Then the full current through the junction is given by the joint contribution of jLRandjs orjSRandjs, respectively. The corresponding full currents are demonstrated in Fig. 5. Panel (a) reperesents the case of a short-enough ferromagnetic layer dF=1, while panel (b) corresponds to dF=2. It is seen from panel (b) that in this case the main contribution to the current is given by jLR, at least for small enough voltages. It is worth noting thatit may be not easy to adjust P landPrmagnetizations in such a way that only one of the components, jSRorjLR,FIG. 5. Joint currents jc s+jc SR(solid curves) and jc s+jc LR (dashed curves) as functions of eV //Delta1.( a ) dF=1,dN=2, (b)dF=2,dN=1. The other parameters are the same as in Fig. 2. ﬂows. In fact, it is not necessary in order to obtain highly nonlinear j(V) characteristics. For this purpose it is enough to create any spin-dependent quasiparticle distribution in theinterlayer region. The internal structure of the anomalousGreen’s function F LRcan also be studied separately for long-enough ferromagnetic interlayers. IV . S/N/S JUNCTION WITH MAGNETIC INTERFACES In this section the Josephson current is studied for a S/N/S junction with magnetic S/N interfaces under the condition ofa spin-dependent quasiparticle distribution in the interlayer.The model is already described in Sec. II. The anomalous Green’s function in the interlayer is found up to ﬁrst orderin S/N conductance ˜G Taccording to Eqs. ( 13) and ( 15) (assuming h=0). In general, the condensate penetrating into the interlayer region comprises two types of electron pairs:with opposite electron spins and with parallel electron spins.However, due to the absence of ferromagnetic elements inthe interlayer region they have the same characteristic decaylength. Both types of pairs occur in the system if there is areason for spin-ﬂip there. For example, it is the case if themagnetization vectors of the both interfaces are not parallel, m l∦mr.I fml\|\|mr, then only the pairs with opposite electron spins, generated by the singlet superconductor, occur in theinterlayer region. To make the formulas less cumbersome wegive ﬁnal expressions only for the case m l\|\|mr≡m. Then, at the left ( α=+ 1) and right ( α=− 1) S/N interfaces, the singlet 054533-11A. M. BOBKOV AND I. V . BOBKOV A PHYSICAL REVIEW B 84, 054533 (2011) part of the anomalous Green’s function takes the following form fR s=fs1e−iαχ/ 2+fs2eiαχ/ 2, fs1=2iπG Tsinh/Theta1R S σNZ/bracketleftBigg λN+1 λN/parenleftbiggGφ σN/parenrightbigg2/bracketrightBigg sinh[2 λNdN], fs2=4iπG Tsinh/Theta1R S σNZ/bracketleftBigg λN−1 λN/parenleftbiggGφ σN/parenrightbigg2/bracketrightBigg sinh[λNdN], Z=4λ2 Nsinh2[λNdN]+8/parenleftbiggGφ σN/parenrightbigg2 (cosh2[λNdN]+1) +4/parenleftbiggGφ σN/parenrightbigg4sinh2[λNdN] λ2 N, (29) where λNis determined below in Eq. ( A4). The triplet component of the anomalous Green’s function has only a zcomponent and takes the form fR t=(0,0,fz), fz=fz1e−iαχ/ 2+fz2eiαχ/ 2, fz1=4πGφGTsinh/Theta1R S σ2 NZ ×/parenleftBigg/bracketleftBigg 1+1 λ2 N/parenleftbiggGφ σN/parenrightbigg2/bracketrightBigg sinh2[λNdN]+2/parenrightBigg , fz2=8πGφGTsinh/Theta1R S σ2 NZcosh[λNdN], (30) where Zis determined in Eqs. ( 29). Physically, fs1andfz1 are generated by the proximity effect at the same S/N interface andfs2,fz2are extended from the opposite S/N interface. Just as in the previous section, to generate a spin-dependent quasiparticle distribution in the interlayer, additional elec-trodes are attached to it. The principal scheme is the sameas before except for the fact that there is only one normalregion in the considered system. Therefore, we assume thatthe interlayer is attached to two additional normal electrodes,N bandNt, and electrode Nbhas an insertion Pmade of a strongly ferromagnetic material. The unit vector alignedwith the magnetization of Pis denoted by M.A g a i n ,i f voltage 2 Vis applied between the electrodes N bandNt, then the electric potentials for spin-up and spin-down electronsin the N bregion, enclosed between Pand the normal interlayer, counted from the level of the superconducting leadsareV ↑=(Vb−Vt)/2=VandV↓=(Vt−Vb)/2=−V.T h e distribution functions for spin-up and spin-down electrons inthis region are close to the equilibrium form (with differentelectrochemical potentials). In matrix form the distributionfunction is expressed by Eqs. ( 26) with the substitution Mfor M l. Now we can obtain the distribution function in the inter- layer, which enters the current [Eq. ( 19)]. Again, for simplicity we assume that gt/lessmuch1. Consequently, the dissipative current ﬂowing through the Nb/N/N tjunction is negligible and, therefore, the ydependence of the distribution function inthe interlayer region can be disregarded. For simplicity we assume below that M\|\|m. Under this condition the distribution function ˆ ϕ(0)in the interlayer calculated according to Eq. (20)a th=0 supplemented by boundary conditions at S/N interfaces, Eqs. ( 21), is spatially constant and equal to its value coming from the Nbregion. If M∦mthen the spatially constant distribution function does not satisfy boundaryconditions ( 21) anymore. In this case the problem becomes two dimensional and much more complicated. Now we are able to calculate the Josephson current through the junction according to Eq. ( 19). After substitution of the expression for the singlet part of the anomalous Green’sfunction [Eqs. ( 29)] and the scalar part of the distribution function [Eqs. ( 26) and ( 18)] into ﬁrst two terms of Eq. ( 19), the contribution of the SCDOS singlet part takes the form j s=2iG2 Tsinχ eσN/integraldisplay∞ −∞/Delta12dε˜ϕ0(ε) ×λNsinh[λNdN]/parenleftbigg 1−/bracketleftBig Gφ σNλN/bracketrightBig2/parenrightbigg [(ε+iδ)2−/Delta12]Z(ε), (31) where Z(ε) is as determined in Eqs. ( 29). The current ﬂowing through the SCDOS triplet part and expressed by the third term in Eq. ( 19) takes the form [in order to obtain this expression one should substitute Eqs. ( 30), (26), and ( 18) into Eq. ( 19)] jt=4G2 TGφsinχ eσ2 N/integraldisplay∞ −∞/Delta12dε˜ϕt(ε) ×cosh[λNdN] [(ε+iδ)2−/Delta12]Z(ε). (32) As opposed to the problem of the S/NFN/S junction considered in the previous section, it is seen from Eq. ( 32) that jtvalues at the left and right S/N interfaces are equal to each other. As for the case of the S/NFN/S junction, the part of the current of Eq. ( 19) generated by the term ∝cosh/Theta1R S[ϕ(0) 0(ε)+ ϕ(0) 0(−ε)]/2 vanishes due to the fact that the scalar part ϕ(0) 0 of the distribution function in the interlayer [Eqs. ( 26)] is an odd function of quasiparticle energy. Further, under theconditions \|eV\|</Delta1 andT/lessmuch/Delta1, the last term, generated by∝αG MRcosh/Theta1R Sm[ϕ(0)(ε)+ϕ(0)(−ε)]/2, also vanishes because this expression is an odd function of quasiparticleenergy at \|ε\|</Delta1 and is absent elsewhere. Taking into account thatm l\|\|mr\|\|Mone can obtain from Eqs. ( 21) that∂xˆϕ(1)=0 at the S/N interfaces. Therefore, ˆ ϕ(1)is approximately constant in the interlayer. Moreover, this constant is to be equal to zeroin order to satisfy the condition j l=jr. Therefore, the full Josephson current ﬂowing through the junction is given by thesum of singlet [Eq. ( 31)] and triplet [Eq. ( 32)] the SCDOS contributions. As for the previous case of the S/NFN/S junction, j c s is an even function of voltage Vapplied to the addi- tional electrodes and jc tis an odd function of this voltage. Therefore, contributions jc sandjc tcan be extracted from an experimentally measurable Josephson current and, so,it makes sense to discuss them separately. Panel (b) ofFig. 6demonstrates the full critical Josephson current and its contributions j c sandjc tas functions of Vfor a typical 054533-12INFLUENCE OF SPIN-DEPENDENT QUASIPARTICLE ... PHYSICAL REVIEW B 84, 054533 (2011) set of parameters (see caption to Fig. 6for speciﬁc values). Functions Fs(ε)≡Im[fs2(ε)//radicalbig /Delta12−(ε+iδ)2] andFt(ε)≡ Im[fz2(ε)//radicalbig /Delta12−(ε+iδ)2] are represented in panel (a) of Fig. 6for the same set of parameters. As seen from the deﬁnitions given in Eqs. ( 29) and ( 30), these functions are proportional to the singlet and triplet components of theanomalous Green’s function, coming from the opposite S/Ninterface, and can be experimentally found by differentiatingthe currents j c sandjc twith respect to voltage V,a si tw a s explained in the previous section. The characteristic shape of FsandFtdictates how jsandjt behave upon varying V. Upon discussing the characteristic features of jsandjtwe consider only V> 0 and, corre- spondingly, ε> 0f o r FsandFt. The main characteristic features of FsandFt, which are responsible for the current behavior, are proximity- induced dips at εφ∼Gφξ2 S/Delta1/σ NdN (for the parameter region εφ</Delta1 ). These dips are followed by an abrupt changing of sign of the corresponding quantity.Figure 7shows F sandFtevolution with Gφ(left column) and withdN(right column). It is seen that upon Gφincreasing the proximity-induced dip shifts to higher energies. If the junctionbecomes shorter the dip also shifts to the right and its integralheight increases due to the fact that the proximity effect ismore pronounced for short junctions. According to Eqs. ( 19) and ( 26), at low-enough tempera- tures only the part of F sbelonging to energy intervals [ −∞,− \|eV\|] and [ \|eV\|,+∞ ] contributes to jc s. Consequently, upon increasing V, the absolute value of jc sgrows up to V∼εφ and after that starts to decline due to the sign changing of Fs(ε)a tε=εφ. Analogously, only the part of Ft, belonging to energy interval [ −\|eV\|,\|eV\|], contributes to jc t. Therefore, theFIG. 6. (a) Functions Fs(dotted line) and Ft(solid line) as functions of ε//Delta1 for S/N/S junction with magnetic interfaces. (b) Full critical current (solid line) and its contributions jc s(dotted line) and jc t(dashed line) as functions of eV //Delta1. For the both panels, dN=0.5ξS,GφξS/σN=0.35, and T=0.1/Delta1. FIG. 7. FsandFtas functions of ε//Delta1 for S/N/S junction with magnetic interfaces. The upper row represents Fs, while the lower row demonstrates Ft. For panels (a) and (c) dN=ξSand different curves correspond to different values of GφξS/σN=0.1 (black solid curve), 0.3 (dotted curve), 0.7 (dashed curve), and 1.1 (gray solid curve). In panels (b) and (d) GφξS/σN=0.5 and different curves correspond to different dN/ξS=2 (black solid curve), 1 (dotted curve), 0.6 (dashed curve), and 0.4 (gray solid curve). T=0.1/Delta1. 054533-13A. M. BOBKOV AND I. V . BOBKOV A PHYSICAL REVIEW B 84, 054533 (2011) FIG. 8. Full critical Josephson current for the S/N/S junction with magnetic interfaces as a function of eV //Delta1. Panel (a) demonstrates the case of a low-temperature junction, where the proximity effect is well pronounced: T=0.01/Delta1,dN=0.3ξS.GφξS/σN=0.15 (dashed curve), 0 .3 (solid curve), and 0 .45 (dotted curve). Panel (b) corresponds to a longer junction at the same temperature: T=0.01/Delta1,dN=3ξS. GφξS/σN=1 (dashed curve), 2 (dotted curve), 3 (solid curve), and 4 (dashed-dotted curve). Panels (c) and (d) represent the same results as panels (a) and (b), respectively, but at a higher temperature, T=0.1/Delta1. For all the panels the gray solid line represents jnm(V) for the corresponding set of the parameters. absolute value of jc talso grows up to V∼εφand declines after that. The described behavior is characteristic for theabsolute value of j c sandjc tjust for eV >0, just as for eV<0. However, due to the fact that jc sis a symmetric and jc tis an antisymmetric function of V, the total Josephson current is highly nonsymmetric with respect to V, as seen in Figs. 6and 8. While for eV <0 the contributions of jc sand jc tpartially compensate each other, leading to suppression of the full current and 0 −π-transition at some ﬁnite V,t h e y are added for eV >0, resulting in the considerable current enhancement. The value of eV, where the peak in the criticalcurrent is located, can be used for experimental estimateof spin-mixing parameter G φ, characterizing the magnetic interface, because eV p∼εφ. For short-enough junctions with dN<ξSthe current value can even exceed the critical current value for the S/N/S junction with nonmagnetic S/N interfaces(G φ=0) and the same S/N interface conductance GTfor some voltage range. It is worth noting here that such an enhancementis possible only for ﬁnite V, when the triplet part of the SCDOS F tcontributes to the current. At V=0 the critical Josephson current through the S/N/S junction with magneticinterfaces G φ/negationslash=0 is always lower than the corresponding current for S/N/S junction with nonmagnetic interfaces butthe same interface conductance G T(this statement is valid for the entire range of parameters we consider). Let us denote the value of the critical current for the S/N/S junction with nonmagnetic interfaces and interfaceconductance G Tbyjnm(V). In the framework of the micro- scopic model of S/N interface considered in Appendix B,a comparison between j(V) andjnm(V) physically correspondsto a comparison between the Josephson currents in the system with a thin magnetic layer between N and I and with nosuch a layer. The value j nm(V=0) is shown in Fig. 6(b) by the horizontal line. It is seen that a small excess of thetotal current j coverjnm(V=0) takes place for some voltage range. However, the excess can be much greater; the currentat ﬁnite Vcan exceed the equilibrium current j nm(V=0) for a nonmagnetic S/N/S junction more than twice. Such a caseis illustrated in Fig. 8(a). Maximal excess can be expected for short junctions with ε φ≈/Delta1, where the proximity effect in Ftis most pronounced and the proximity- induced dip at εφ merges with the coherence peak at /Delta1, thus greatly enhancing theFtvalue in the subgap region. In addition, the temperature should be low enough to avoid temperature smearing of theeffect. For longer junctions with d N/greaterorsimilarξSthe current does not exceed the equilibrium value jnm(V=0) because of a weaker proximity effect in the interlayer region, as illustrated inFig. 8(b). For all the panels of Fig. 8the gray solid line represents j nm(V) for the corresponding set of the parameters. It is worth noting here that dependencies jnm(V)o nVare qualitatively very similar to the current discussed in Ref. 30 for a nonmagnetic S/N/S junction under nonequilibriumquasiparticle distribution in the normal interlayer. Indeed,atG φ=0 the triplet part of the SCDOS Ftis absent and, consequently, the vector part of the distribution function ofEqs. ( 26) does not contribute to the current. The singlet part of this distribution function is formally equivalent to thenonequilibrium distribution 30for a narrow normal interlayer (a wire or a constriction). Full quantitative agreement between 054533-14INFLUENCE OF SPIN-DEPENDENT QUASIPARTICLE ... PHYSICAL REVIEW B 84, 054533 (2011) our results for jnm(V) and the results of Ref. 30cannot be reached because they are obtained for somewhat differentparameter ranges. Panels (c) and (d) of Fig. 8show the results for the current at a higher temperature, T=0.1/Delta1. Panel (c) corresponds to a shorter junction with d N=0.3ξS, while panel (d) demonstrates the case of longer junction with dN=3ξS. It is seen that, for a short junction, where the effect ofcurrent enhancement is well pronounced at low temperatures,raising of the temperature suppresses the effect. The reasonis that the distribution function of Eqs. ( 26) smears upon raising of temperature. Consequently, not only the part of F t corresponding to \|ε\|<εφ, but also some regions of higher energies, where Fthas the opposite sign, are involved in the current jtnow. This leads to partial compensation of Ftparts with different signs. Although the maximal value of the critical Josephson current, which can be reached at a ﬁnite V, is suppressed by temperature, the current dependence on Tat a particular voltage Vcan be quite interesting. Figure 9demonstrates how the current depends on temperature at several speciﬁed valuesof voltage Vfor the case of a short junction with d N=0.3ξS. The gray solid line represents the dependence of jnm(V=0) on temperature and is given for comparison of our resultswith the equilibrium nonmagnetic case. It is well known thatthe Josephson current for the equilibrium nonmagnetic S/N/Sjunction declines upon raising of temperature rather sharply,as demonstrated by the gray solid curve. On the other hand,the current at ﬁnite Vfor a S/N/S junction with magnetic interfaces can even grow up to some temperature and only afterthat start to decline. The qualitative explanation of this fact isthe following. The main contribution to the Josephson currentin a nonmagnetic equilibrium S/N/S junction is given by a highpeak of the SCDOS located at low energies. Consequently, thetemperature smearing of the equilibrium distribution functiontanhε/2Tcrucially reduces the current. At the same time, the main contribution to j tis given by the energies up to ε∼ \|eV\|and under the condition that \|eV\|<εφthe temperature smearing of the distribution function involves higher energies,where the absolute value of F tis even larger, in the current transfer. In addition, at some voltage ranges the junction canmanifest 0 −πtransition upon varying temperature. V . SUMMARY In conclusion, we have theoretically investigated the Josephson current in weak links, containing ferromagneticelements, under the condition that the quasiparticle distributionin the weak link region is spin dependent. Two types of weaklink are considered. The ﬁrst system is a S/N/F/N/S junctionwith a complex interlayer composed of two normal metalregions and a middle layer made of a spiral ferromagnet,sandwiched between them. The second considered system isa S/N/S junction with magnetic S/N interfaces. In both casesa spin-dependent quasiparticle distribution in the interlayerregion is proposed to be created by attaching additionalelectrodes with ferromagnetic elements to the interlayer regionand applying a voltage Vbetween them. The interplay of the triplet superconducting correlations, induced in the interlayerby the proximity with the superconducting leads, and a spin-FIG. 9. Full critical Josephson current for the S/N/S junction with magnetic interfaces as a function of temperature for severaldifferent voltages V.d N=0.3ξS, black solid curve: GφξS/σN= 0.3, eV//Delta1=0.85; dashed curve: GφξS/σN=0.3, eV//Delta1=0.7; dotted curve: GφξS/σN=0.3, eV//Delta1=0.5; dashed-dotted curve: GφξS/σN=0.15, eV //Delta1=0.83. Gray solid curve represents the temperature dependence of jnm(V=0) for the corresponding set of parameters. dependent quasiparticle distribution results in the appearence of the additional contribution to the Josephson current jt, carried by the triplet part of the SCDOS. It is shown that jtis an odd function of V, while the standard contribution js, carried by the singlet part of the SCDOS, is an even function. So jtcan be extracted from the full Josephson current measured as a function of V. Further, it is demonstrated that derivative djt/dV can provide direct information about the anomalous Green’s function describing the superconductingtriplet correlations induced in the interlayer. We show that inthe S/N/F/N/S junction the contributions given by the SRTCand LRTC of superconducting correlations in the interlayercan be measured separately. For a S/N/S junction with magnetic interfaces it is also obtained that the critical Josephson current at some ﬁnite Vcan considerably exceed the current ﬂowing through the equilib-rium nonmagnetic S/N/S junction with the same S/N interfacetransparency. This enhancement is due to the fact that the tripletcomponent of the SCDOS “works” under a spin-dependentquasiparticle distribution, giving the additional contribution tothe current, while it does not take part in the current transferfor spin-independent quasiparticle distribution. In addition,we have studied the temperature dependence of the criticalcurrent in the S/N/S junction with magnetic S/N interfaces.As opposed to the case of an equilibrium nonmagnetic S/N/Sjunction, where the current is monotonouosly suppressed bytemperature, in the considered case at a ﬁnite voltage Vit can at ﬁrst rise with temperature and only then start to decline. The dependence of the full critical current on Vis typically highly nonlinear and strongly nonsymmetric with respect toV=0 due to the interplay of j sandjt. This also leads to appearence of a number of 0- πtransitions in the system upon varying controlling voltage V. ACKNOWLEDGMENT The authors acknowledge the support by RFBR Grant No. 09-02-00779. 054533-15A. M. BOBKOV AND I. V . BOBKOV A PHYSICAL REVIEW B 84, 054533 (2011) APPENDIX A: MICROSCOPIC CALCULATION OF THE ANOMALOUS GREEN’S FUNCTION AND A JOSEPHSON CURRENT IN THE S/NFN/S JUNCTION In this Appendix we calculate the anomalous Green’s functions fs,fSR, and fLRand the corresponding current contributions js,jSR, and jLRin the framework of the most simple microscopic model for the N/F/N interlayer.We assume the N/F interfaces to be absolutely transparent.This approximation simpliﬁes the calculations signiﬁcantly,but does not inﬂuence qualitatively our main conclusions. Forthis case the boundary conditions at x=∓d F/2 take the form ˇgN=ˇgF, (A1) σN∂xˇgN=σF∂xˇgF. As far as we need only the anomalous Green’s func- tions to ﬁrst order in the S/N interface transparency, theabove boundary conditions should be linearized. Then for retarded and advanced Green’s functions they read asfollows: ˆf R,A N=ˆfR,A F, (A2) σN∂xˆfR,A N=σF∂xˆfR,A F. The boundary conditions for the distribution function at the N/F interface to the considered accuracy take theform ˆϕ F=ˆϕN, (A3) σF∂xˆϕF=σN∂xˆϕN. The singlet part of the anomalous Green’s function, cal- culated according to Eqs. ( 13), (15), and ( A2), at the left (α=+ 1) and the right ( α=− 1) S/N interfaces, takes the following form: fR s=iπG T σNλNtanhφNsinh/Theta1R Se−iαχ/ 2+iπG Tsinh/Theta1R S 2σFcosh2φN/bracketleftbiggcos(χ/2) cosh φ+ λ+sinhφ++ρcoshφ+−iαsin(χ/2) sinh φ+ λ+coshφ++ρsinhφ+ +cos(χ/2) cosh φ− λ−sinhφ−+ρcoshφ−−iαsin(χ/2) sinh φ− λ−coshφ−+ρsinhφ−/bracketrightbigg , (A4) where λ±=√h/D (1∓i),λN=√−2i(ε+iδ)/D,φ±=λ±dF/2,φN=λNdN/2, and, ρ=(σN/σF)λNtanhφN. The results for triplet components fSRandfLRare the following fSR=−iπG Tsinh/Theta1R S 2σFcosh2φN/bracketleftbiggcos(χ/2) cosh φ+ λ+sinhφ++ρcoshφ+−iαsin(χ/2) sinh φ+ λ+coshφ++ρsinhφ+ −cos(χ/2) cosh φ− λ−sinhφ−+ρcoshφ−+iαsin(χ/2) sinh φ− λ−coshφ−+ρsinhφ−/bracketrightbigg , fLR=−iπG Tsinh/Theta1R S 2σFcosh2φN/braceleftbigg/Theta1/primeisin(χ/2) cosh φt ρcoshφt+λtsinhφt/bracketleftbiggsinhφ+ λ+coshφ++ρsinhφ+−sinhφ− λ−coshφ−+ρsinhφ−/bracketrightbigg −α/Theta1/primecos(χ/2) sinh φt ρsinhφt+λtcoshφt/bracketleftbiggcoshφ+ λ+sinhφ++ρcoshφ+−coshφ− λ−sinhφ−+ρcoshφ−/bracketrightbigg/bracerightbigg , (A5) where λt=/radicalbig /Theta1/prime2−2i(ε+iδ)/Dandφt=λtdF/2. The fact that fSRrapidly decays in the ferromagnetic region and, consequently, represents the SRTC can be easilyseen from Eqs. ( A5) in the limit of a thick-enough F layer: d F/ξF/greatermuch1. To leading order in the parameter e−dF/ξFfor quantities fl SRandfr SR, deﬁned by Eq. ( 23), one obtains from Eqs. ( A5) fl SR=−iπG Tsinh/Theta1R S 2σFcosh2φN/bracketleftbigg1 λ++ρ−1 λ−+ρ/bracketrightbigg , (A6) fr SR=−iπG Tsinh/Theta1R S σFcosh2φN/bracketleftbiggλ+e−λ+dF (λ++ρ)2−λ−e−λ−dF (λ−+ρ)2/bracketrightbigg .At the same regime the corresponding components of fLRtake the following form fl LR=iπG Tsinh/Theta1R S 2σFcosh2φN/bracketleftbigg1 λ++ρ−1 λ−+ρ/bracketrightbigg ×/Theta1/prime(λtcosh[2 φt]+ρsinh[2 φt]) (ρcoshφt+λtsinhφt)(ρsinhφt+λtcoshφt), fr LR=−iπG Tsinh/Theta1R S 2σFcosh2φN/bracketleftbigg1 λ++ρ−1 λ−+ρ/bracketrightbigg ×/Theta1/primeλt (ρcoshφt+λtsinhφt)(ρsinhφt+λtcoshφt). (A7) 054533-16INFLUENCE OF SPIN-DEPENDENT QUASIPARTICLE ... PHYSICAL REVIEW B 84, 054533 (2011) As seen, fr LRdoes not contain the small factor e−dF/ξFin the leading approximation and, therefore, fLRdescribes the LRTC. The characteristic decay length of fLRin the F layer is \|λt\|−1. To the considered accuracy the singlet component of the anomalous Green’s function also decays at the distance∼ξ Fin the F layer, just as the SRTC fSRdoes. In the regime dF/ξF/greatermuch1 it can be obtained from Eq. ( A4) that fl s=iπG T σNλNtanhφNsinh/Theta1R S +iπG Tsinh/Theta1R S 2σFcosh2φN/bracketleftbigg1 λ++ρ+1 λ−+ρ/bracketrightbigg , fr s=iπG Tsinh/Theta1R S σFcosh2φN/bracketleftbiggλ+e−λ+dF (λ++ρ)2+λ−e−λ−dF (λ−+ρ)2/bracketrightbigg .(A8) Substituting Eqs. ( A4) and ( A5) together with the expressions for the vector part of the distribution function [Eqs. ( 26) and (18)] into Eq. ( 19) one can ﬁnd js=G2 Tsinχ 8eσF/integraldisplay∞ −∞i/Delta12dε˜ϕ0(ε) [(ε+iδ)2−/Delta12] cosh2φN ×/bracketleftbigg1 λ+tanhφ++ρN−tanhφ+ λ++ρNtanhφ+ +1 λ−tanhφ−+ρN−tanhφ− λ−+ρNtanhφ−/bracketrightbigg , (A9) jSR=−G2 Tsinχ 8eσF/integraldisplay∞ −∞i/Delta12dε˜ϕt(ε) [/Delta12−(ε+iδ)2] cosh2φN ×/bracketleftbigg1 λ+tanhφ++ρN−tanhφ+ λ++ρNtanhφ+ −1 λ−tanhφ−+ρN+tanhφ− λ−+ρNtanhφ−/bracketrightbigg ,(A10) jLR=−G2 Tsinχ 8eσF/integraldisplay∞ −∞i/Delta12dε˜ϕt(ε)/bracketleftbig /Delta12−(ε+iδ)2/bracketrightbig cosh2φN ×/braceleftbigg/Theta1/prime ρ+λttanhφt/bracketleftbiggtanhφ+ λ++ρtanhφ+ −tanhφ− λ−+ρtanhφ−/bracketrightbigg −/Theta1/primetanhφt ρtanhφt+λt ×/bracketleftbigg1 λ+tanhφ++ρ−1 λ−tanhφ−+ρ/bracketrightbigg/bracerightbigg .(A11) APPENDIX B: MICROSCOPIC MODEL OF MAGNETIC S/N INTERFACE Let us introduce the electronic scattering matrix Seassoci- ated with electrons with spin σof the nth transmission channel. We assume that the interface does not rotate an electron spin,that is, the scattering matrix is diagonal in spin space: S e nσ=/parenleftbigg rl nσtr nσ tl nσrr nσ,/parenrightbigg , (B1)where rl(r) nσdenotes the reﬂection amplitude at the left (right) side of the interface and tl(r) nσis the transmission amplitude from the left (right) side to the right (left) side of the interface.Taking into account the constraints on S eresulting from the unitarity condition SeSe†=1 and time-reversal symmetry, one can show that without any loss of generality Seis entirely determined by the following parameters: the transmissionprobability T n, the degree of spin polarization Pn, and the spin-mixing angle dϕl(r) n. These parameters are deﬁned as Tnσ=\|tnσ\|2=Tn(1+ σPn) and arg[ rl(r) nσ]=ϕl(r) n+σ(dϕl(r) n/2). These parameters can be straightforwardly calculated in the framework of amicroscopic model describing the interface. Here we modelthe S/N interface by an insulating barrier I (with a transparencyT/lessmuch1) and a thin layer of a ferromagnetic metal, which is located between I and the normal interlayer. This layeris supposed to provide a required value for the spin-mixingangle. Given that the exchange ﬁeld in the ferromagneticlayer is small with respect to the Fermi energy h/lessmuchε F, in the framework of this microscopic model Tn=T,Pn≈ 0, and dϕn≈2wFh/vF, where wFis the length of the ferromagnetic layer and vFis the corresponding Fermi velocity. The main parameters entering magnetic boundary condi- tions ( 9) are connected to the microscopic parameters Tn,Pn, anddϕl(r) nin the following way:44 GTS=2Gq/summationdisplay nTn, (B2) GMRS=Gq/summationdisplay nTnPn, (B3) GφS=2Gq/summationdisplay n(Tn−1)dϕn, (B4) where Sis the junction area and Gq=e2/his the quantum conductance. It is worth noting here that boundary conditions(9) are the expansion in small T n,Pn, anddϕnof the more general boundary conditions44and, consequently, Eq. ( 9)i s valid only if all these parameters are considerably less thanunity. However, because of summation over a large numberof transmisson channels, it does not mean that the parametersG T,GMR, andGφmust be small. Let us estimate the value of dimensionless ˜Gφ=GφξS/σN, which can be obtained in the framework of our microscopic model. For T/lessmuch1, ˜Gφ≈−NξSGq SσNdϕ∼−ξS ldϕ, (B5) where Nis the number of transmission channels and lis the mean free path. dϕmeans the average value of the spin-mixing angle dϕn. For rough estimates it is possible to take dϕ≈ 2wFh/vF. Our main results for a S/N/S junction with magnetic interfaces are calculated for ˜Gφ∼1. From Eq. 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PhysRevB.82.085419.pdf	Full counting statistics in disordered graphene at the Dirac point: From ballistics to diffusion A. Schuessler,1P. M. Ostrovsky,1,2I. V . Gornyi,1,3,4and A. D. Mirlin1,4,5,6 1Institut für Nanotechnologie, Karlsruher Institut für Technologie, 76131 Karlsruhe, Germany 2L. D. Landau Institute for Theoretical Physics, RAS, 119334 Moscow, Russia 3A. F . Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia 4DFG Center for Functional Nanostructures, Karlsruher Institut für Technologie, 76131 Karlsruhe, Germany 5Inst. für Theorie der Kondensierten Materie, Karlsruher Institut für Technologie, 76131 Karlsruhe, Germany 6Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia /H20849Received 8 June 2010; published 12 August 2010 /H20850 The full counting statistics of the charge transport through an undoped graphene sheet in the presence of smooth disorder is studied. At the Dirac point both in clean and diffusive limits, transport properties of agraphene sample are described by the universal Dorokhov distribution of transmission probabilities. In thecrossover regime, deviations from universality occur which can be studied analytically both on ballistic anddiffusive sides. In the ballistic regime, we use a diagrammatic technique with matrix Green’s functions. For adiffusive system, the sigma model is applied. Our results are in good agreement with recent numerical simu-lations of electron transport in disordered graphene. DOI: 10.1103/PhysRevB.82.085419 PACS number /H20849s/H20850: 73.63. /H11002b, 73.23. /H11002b, 73.22.Pr I. INTRODUCTION Electron transport in graphene remains a ﬁeld of intense experimental and theoretical activity.1,2The hallmark of graphene is the massless Dirac character of low-energy elec-tron excitations. This gives rise to remarkable physical prop-erties of this system distinguishing it from conventional two-dimensional metals. The most remarkable effects arise whenthe chemical potential is located in a close vicinity of theneutrality /H20849Dirac /H20850point. In particular, a short-and-wide sample /H20849with width Wmuch exceeding the length L/H20850of clean graphene exhibits at the Dirac point pseudodiffusive chargetransport 3with “conductivity” 4 e2//H9266hand with counting sta- tistics /H20849characterizing ﬂuctuations of current /H20850equivalent to that of a diffusive wire.4–8In particular, the Fano factor /H20849the shot noise power divided by the current /H20850takes the universal value5–7F=1 /3 that coincides with the well-known result for a diffusive metallic wire.4,8This is at odds with usual clean metallic systems, where the conductance /H20849rather than con- ductivity /H20850is independent of Land the shot noise is absent /H20849F=0/H20850. The reason behind these remarkable peculiarities of transport in clean graphene at the Dirac point is linearly van-ishing density of states. This implies that the current is me-diated by evanescent rather than propagating modes. Theabove theoretical predictions have been conﬁrmed in mea-surements of conductance and noise in ballistic grapheneﬂakes. 9–11Recent advances in preparation and transport stud- ies of suspended graphene samples also indicate that the sys-tem may be in the ballistic regime. 12,13 Effects of impurities on transport properties of graphene are highly unusual as well. In contrast to conventional met-als, ballistic graphene near the Dirac point conducts betterwhen potential impurities are added. 14–16Quantum interfer- ence in disordered graphene is also highly peculiar due to theDirac nature of the carriers. In particular, in the absence ofintervalley scattering, the minimal conductivity 17/H11011e2/his “topologically protected” from quantum localization.18The exact value of the conductivity of such a system at the Diracpoint depends on the type of intravalley scattering /H20849random scalar or vector potential, or random mass, or their combina-tion /H20850. For the case of random potential only /H20849which is experi- mentally realized by charged scatterers /H20850the conductivity, in fact, increases logarithmically with the length L, in view of antilocalization. 14,19–22 In our previous work,16we have studied the evolution of conductance of a short-and-wide graphene sample from theballistic to the diffusive regime. We have also shown that theleading disorder-induced correction to the noise and fullcounting statistics in the ballistic regime is completely gov-erned by the renormalization of the conductance. This im- plies, in particular, that the Fano factor 1/3 remains unaf-fected to this order. Indeed, the experiments 10,23give Fano factor values in the vicinity of 1/3 at the Dirac point fordifferent system lengths L. One could thus ask whether de- viations from this value should be expected at all. In this work we present a detailed analysis of the shot noise and the full counting statistics in samples with long-range /H20849no valley mixing /H20850disorder. We show that to second order in the disorder strength a correction to the universalcounting statistics of the ballistic graphene does arise. Wecalculate this correction and demonstrate that it suppressesthe Fano factor below the value 1/3. For the case of randomscalar potential, we also analyze the opposite limit of large L when the system is deep in the diffusive regime. Generaliz-ing the analysis of weak-localization effects on the countingstatistics by Nazarov, 24we ﬁnd that the Fano factor returns to the value of 1/3 from below with increasing L. The approach to 1/3 is, however, logarithmically slow. These results com-pare well with recent numerical works 20,21and particularly with the most detailed study by Tworzydlo et al.22 The structure of the paper is as follows. In the Sec. II,w e describe the general matrix Green’s function formalism andits application to the problem of full counting statistics. Themodel for graphene setup and disorder is introduced in Sec.III. We proceed with applying matrix Green’s function method to the calculation of the distribution of transmissionPHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 1098-0121/2010/82 /H208498/H20850/085419 /H2084914/H20850 ©2010 The American Physical Society 085419-1probabilities of the clean graphene sample in Sec. IV. In Sec. Vwe evaluate perturbative disorder corrections to the full counting statistics in ballistic regime. Diffusive transportthrough disordered graphene is considered in Sec. VIwithin the sigma-model approach. The paper is concluded by Sec.VIIsummarizing the main results. Technical details of the calculation are presented in three appendices. II. MATRIX GREEN’S FUNCTION FORMALISM We begin with the general presentation of the matrix Green’s function approach to the full counting statistics of aquasi-one-dimensional system. This formalism was devel-oped by Nazarov in Ref. 25. Consider a quasi-one-dimensional sample attached to two perfect metallic leads. Transport characteristics of the systemare encoded in the matrix of transmission amplitudes t mn, where the indices enumerate conducting channels /H20849quantized transverse modes /H20850in the leads. Eigenvalues of the matrix tˆ†tˆ determine transmission probabilities of the system /H20849we use the “hat” notation for matrices in the space of channels /H20850. Our main goal is to calculate the distribution of these transmis-sion probabilities. The full counting statistics of the chargetransport is given by the moments of this distribution or,equivalently, by the distribution itself. The ﬁrst two momentsof the transferred charge provide the conductance /H20849by Land- auer formula /H20850and the Fano factor G=e 2 hTrtˆ†tˆ,F=1−Tr/H20849tˆ†tˆ/H208502 Trtˆ†tˆ. /H208491/H20850 The starting point of our consideration is the relation be- tween the matrix of transmission probabilities and theGreen’s function of the system t mn=i/H20881vmvnGmnA/H20849x,x/H11032/H20850, /H208492/H20850 Here vm,nare velocities in the mth and nth channels. The Green’s function is taken in the mixed representation withreal-space coordinates in the longitudinal direction and chan-nel indices in transverse direction. The positions xand x /H11032 should be taken in the left and right leads, respectively, in order to obtain the transmission matrix of the full system. The conjugate matrix tˆ†is related to the retarded Green’s function by a similar identity. The Green’s functions are deﬁned in the standard way as /H20849/H9280−Hˆ/H11006i0/H20850GˆR,A/H20849x,x/H11032/H20850=/H9254/H20849x−x/H11032/H208501ˆ/H208493/H20850 with energy /H9280and Hamiltonian Hˆ, the latter being an opera- tor acting both on xand in the channel space. With the help of Eq. /H208492/H20850we can express all the moments of transmission distribution in terms of the Green’s functions Tr/H20849tˆ†tˆ/H20850n=T r /H20851vˆGˆA/H20849x,x/H11032/H20850vˆGˆR/H20849x/H11032,x/H20850/H20852n, /H208494/H20850 where vˆis the velocity operator and xandx/H11032lie in the left and right leads, respectively. For the ﬁrst moment, n=1, the above identity establishes an equivalence of the Landauerand Kubo representations for conductance. The complete statistics of the transmission eigenvalues can be represented by the generating functionF/H20849z/H20850=/H20858 n=1/H11009 zn−1Tr/H20849tˆ†tˆ/H20850n=T r /H20851tˆ−1tˆ†−1−z/H20852−1. /H208495/H20850 Once this function is known, all the moments of transmission distribution are easy to obtain by expanding the generatingfunction in series at z=0. An efﬁcient method yielding the whole generating function was proposed in Ref. 25.I t amounts to calculating the matrix Green’s function deﬁnedby the following equation: /H20873/H9280−Hˆ+i0 −/H20881zvˆ/H9254/H20849x−xL/H20850 −/H20881zvˆ/H9254/H20849x−xR/H20850/H9280−Hˆ−i0/H20874Gˇ/H20849x,x/H11032/H20850=/H9254/H20849x−x/H11032/H208501ˇ. /H208496/H20850 The parameter zhere corresponds to the source ﬁeld mixing retarded and advanced components of the matrix Green’sfunction. The positions x LandxR, where the source ﬁeld is applied, lie within the left and right leads, respectively. Wewill refer to this speciﬁc matrix structure as the retarded-advanced /H20849RA /H20850space and denote such a matrices with the “check” notation. The main advantage of the matrix Green’s function de- ﬁned by Eq. /H208496/H20850is the following concise expression for the generating function of transmission probabilities: F/H20849z/H20850=1 /H20881zTr/H20875/H2087300 vˆ0/H20874Gˇ/H20849xR,xR/H20850/H20876=1 /H20881zTr/H20875/H208730vˆ 00/H20874Gˇ/H20849xL,xL/H20850/H20876. /H208497/H20850 The validity of this equation can be directly checked by ex- panding the Green’s function in powers of zwith the help of perturbation theory and comparing this expansion termwisewith Eq. /H208495/H20850. The equivalence of these two expansions is provided by the identity Eq. /H208494/H20850. Another and, probably, most intuitive representation of the full counting statistics is given by the distribution func-tion of transmission probabilities P/H20849T/H20850. This function takes its simplest form when expressed in terms of the parameter /H9261 related to the transmission probability by T=1 /cosh 2/H9261.I n terms of /H9261the probability density is deﬁned by the identity P/H20849T/H20850dT=P/H20849/H9261/H20850d/H9261. The deﬁnition of the generating function, Eq. /H208495/H20850, implies a trace involving all transmission probabili- ties. With the distribution function of these probabilities wecan express F/H20849z/H20850by the integral F/H20849z/H20850=/H20885 0/H11009P/H20849/H9261/H20850d/H9261 cosh2/H9261−z. /H208498/H20850 The function F/H20849z/H20850has a branch cut discontinuity in the com- plex zplane running from 1 to + /H11009. The jump of the function across the branch cut determines the distribution function/H20849see Ref. 16for derivation /H20850 P/H20849/H9261/H20850=sinh 2/H9261 2/H9266i/H20851F/H20849cosh/H9261+i0/H20850−F/H20849cosh/H9261−i0/H20850/H20852./H208499/H20850 In other words, Eq. /H208498/H20850solves the Riemann-Hilbert problem deﬁned by Eq. /H208499/H20850. The generating function F/H20849z/H20850can be related to the “free energy” of the system in the “external” source ﬁeld. The freeSCHUESSLER et al. PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-2energy is deﬁned in terms of the functional determinant /H9024=T rl n Gˇ,F=/H11509/H9024 /H11509z. /H2084910/H20850 The free energy can be calculated by standard diagrammatic methods and hence provides a very convenient representa-tion of the full counting statistics. It is also convenient toparametrize the argument of the free energy by the angle /H9278 according to z=sin2/H20849/H9278/2/H20850. Thus we have three equivalent representations of the full counting statistics by the functions F/H20849z/H20850,P/H20849/H9261/H20850, and/H9024/H20849/H9278/H20850.I n this paper we calculate the transport characteristics of a dis-ordered graphene sample in terms of its free energy /H9024/H20849 /H9278/H20850. The two other functions can be found with the help of iden-tities F/H20849z/H20850=/H208792 sin/H9278/H11509/H9024 /H11509/H9278/H20879 /H9278=2 arcsin /H20881z, /H2084911/H20850 P/H20849/H9261/H20850=/H208792 /H9266Re/H11509/H9024 /H11509/H9278/H20879 /H9278=/H9266+2i/H9261. /H2084912/H20850 The ﬁrst of these relations directly follows from Eq. /H2084910/H20850 while the second one is the result of the substitution of Eq./H2084911/H20850into Eq. /H208499/H20850. The two most experimentally relevant quantities con- tained in the full counting statistics, namely, conductance andFano factor, Eq. /H208491/H20850, can be expressed in terms of any of the functions introduced above. Then the following expressionsfor the conductance and the Fano factor hold: G= /H208792e2 h/H115092/H9024 /H11509/H92782/H20879 /H9278=0, /H2084913/H20850 F=1 3−/H208792 3/H115094/H9024//H11509/H92784 /H115092/H9024//H11509/H92782/H20879 /H9278=0. /H2084914/H20850 We will apply the matrix Green’s function formalism out- lined in this section to the problem of full counting statisticsof a disordered graphene sample. Our strategy is as follows.First, we calculate the matrix Green’s function of a cleanrectangular graphene sample and obtain the full counting sta-tistics with the help of Eq. /H208497/H20850. Then we introduce disorder in the model perturbatively. Evaluation of the free energy bydiagrammatic methods yields disorder corrections to the fullcounting statistics of a clean sample. III. MODEL We will adopt the single-valley model of graphene. More speciﬁcally, we will consider scattering of electrons onlywithin a single valley and neglect intervalley scatteringevents. Indeed, a number of experimental results show that inmany graphene samples the dominant disorder scatters elec-trons within the same valley. First, this disorder model issupported by the odd-integer quantization 1,17,26of the Hall conductivity, /H9268xy=/H208492n+1/H208502e2/h, representing a direct evidence27in favor of smooth disorder which does not mix the valleys. The analysis of weak localization also corrobo-rates the dominance of intravalley scattering.28Furthermore, the observation of the linear density dependence1of graphene conductivity away from the Dirac point can be ex-plained if one assumes that the relevant disorder is due tocharged impurities and/or ripples. 19,29–32Due to the long- range character of these types of disorder, the intervalleyscattering amplitudes are strongly suppressed and will beneglected in our treatment. Finally, apparent absence of lo-calization at the Dirac point down to very lowtemperatures 17,26,33points to some special symmetry of dis- order. One realistic candidate model is the long-range ran-domness which does not scatter between valleys. 18,34 The single-valley massless Dirac Hamiltonian of electrons in graphene has the form /H20849see, e.g., Ref. 2/H20850 H=v0/H9268p+V/H20849x,y/H20850,V/H20849x,y/H20850=/H9268/H9262V/H9262/H20849x,y/H20850. /H2084915/H20850 Here/H9268/H9262/H20849with/H9262=0,x,y,z/H20850are Pauli matrices acting on the electron pseudospin degree of freedom corresponding to thesublattice structure of the honeycomb lattice, /H9268/H11013/H20853/H9268x,/H9268y/H20854 and the Fermi velocity is v0/H11015108cm /s. The random part V/H20849x,y/H20850is, in general, a 2 /H110032 matrix in the sublattice space. Below we set /H6036=1 and v0=1 for convenience. We will calculate transport properties of a rectangular graphene sample with the dimensions L/H11003W. The contacts are attached to the two sides of the width Wseparated by the distance L.W eﬁ xt h e xaxis in the direction of current, Fig. 1, with the contacts placed at x=0 and x=L. We assume W /H11271L, which allows us to neglect the boundary effects related to the edges of the sample that are parallel to the xaxis /H20849at y=/H11006W/2/H20850. Following Ref. 5, metallic contacts are modeled as highly doped graphene regions described by the same Hamiltonian/H2084915/H20850. In other words, we assume that the chemical potential E Fin the contacts is shifted far from the Dirac point. In particular, EF/H11271/H9280, where /H9280is the chemical potential inside the graphene sample counted from the Dirac point. /H20849All our results are independent of the sign of energy, thus we assume /H9280/H110220 throughout the paper. /H20850We also assume zero tempera- ture, that is justiﬁed provided TL/H112701. With the boundary conditions speciﬁed above, we are able to calculate explicitly the matrix Green’s function Eq. /H208496/H20850for a clean graphene sample /H20851V/H20849x,y/H20850=0/H20852at zero energy. This calculation is outlined in Appendix A /H20851see Eq. /H20849A7/H20850/H20852. Using this Green’s function, we will study disorder effects in theframework of the diagrammatic technique for the free en-ergy.y W/2 0 −W/2 x 0 L FIG. 1. /H20849Color online /H20850Schematic setup for two-terminal trans- port measurements. Graphene sample of dimensions L/H11003Wis placed between two parallel contacts. We assume W/H11271Lthroughout the paper.FULL COUNTING STATISTICS IN DISORDERED … PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-3IV. ELECTRON TRANSPORT IN CLEAN GRAPHENE In this section we apply the matrix Green’s function for- malism developed in Sec. IIto the case of clean graphene. These results will play the role of the zeroth approximationfor our perturbation theory. The matrix Green’s function is derived in Appendix A. The generating function for the full counting statistics isgiven by Eq. /H208497/H20850. With the Green’s function Eq. /H20849A7/H20850,w e obtain F 0/H20873sin2/H9278 2/H20874=W sin/H9278 2Tr/H20875/H208730/H9268x 00/H20874Gˇ/H208490,0;0 /H20850/H20876=W /H9266L/H9278 sin/H9278. /H2084916/H20850 The corresponding dependence of the free energy on the source ﬁeld /H9278follows from integration of Eq. /H2084911/H20850. This yields a simple quadratic function /H90240/H20849/H9278/H20850=W/H92782 4/H9266L. /H2084917/H20850 This remarkably simple result reveals the convenience of the source ﬁeld parametrization z=sin2/H20849/H9278/2/H20850. The clean sample responds linearly to the external ﬁeld /H9278. The distribution of transmission probabilities given by Eq. /H2084912/H20850is just a con- stant, P0/H20849/H9261/H20850=W//H9266L, in terms of /H9261. This means the distribu- tion acquires the Dorokhov form4characteristic for disor- dered metallic wires P0/H20849T/H20850=W 2/H9266L1 T/H208811−T. /H2084918/H20850 Hence electron transport in clean graphene at the Dirac point is often called pseudodiffusive. Let us now calculate an energy correction to the pseudod- iffusive transport regime. In the vicinity of the Dirac point/H20849we assume /H9280L/H112701/H20850, we can account for ﬁnite energy /H9280by means of perturbation theory. The linear term is absent dueto particle-hole symmetry of the Dirac point. The lowestnonvanishing correction appears in the /H92802order and is given by the single diagram in Fig. 2,/H9024/H9280=W/H92802 2/H20885 0L dxdx /H11032/H20885 −/H11009/H11009 dyTrG/H20849x,x/H11032;y/H20850G/H20849x/H11032,x;−y/H20850. /H2084919/H20850 This integral of the product of two Green’s functions is cal- culated in Appendix B. The result takes the form /H9024/H9280=W /H9266L/H20849/H9280L/H208502 sin/H9278 2/H11509 /H11509/H9278/H20877cos/H9278 2/H20875/H9274/H20873/H9266+/H9278 2/H9266/H20874+/H9274/H20873/H9266−/H9278 2/H9266/H20874/H20876/H20878, /H2084920/H20850 where /H9274is the digamma function. As explained above, the free energy /H9024/H9280contains informa- tion about the full counting statistics, i.e., all moments of thetransfered charge. In particular, from Eqs. /H2084913/H20850and /H2084914/H20850we obtain the following results for the conductance and Fanofactor: G=4e 2 /H9266hW L/H208511+c1/H20849/H9280L/H208502/H20852,F=1 3/H208511+c2/H20849/H9280L/H208502/H20852, /H2084921/H20850 c1=35/H9256/H208493/H20850 3/H92662−124/H9256/H208495/H20850 /H92664/H110150.101, /H2084922/H20850 c2=−28/H9256/H208493/H20850 15/H92662−434/H9256/H208495/H20850 /H92664+4572/H9256/H208497/H20850 /H92666/H11015− 0.052. /H2084923/H20850 These expressions coincide with the results of Ref. 16ob- tained within an alternative /H20849transfer-matrix /H20850approach. V. DISORDERED GRAPHENE: BALLISTIC LIMIT Let us now include the random part V/H20849x,y/H20850of the Hamil- tonian /H2084915/H20850into consideration. There are in total four differ- ent types of disorder within the single-valley Dirac model:V 0is the random potential /H20849charged impurities in the sub- strate /H20850,VxandVycorrespond to the random vector potential /H20849e.g., long-range corrugations /H20850, and Vzis the random mass. We will assume the standard Gaussian type of disorder char-acterized by the correlation function /H20855V /H9262/H20849r/H20850V/H9263/H20849r/H11032/H20850/H20856=2/H9266/H9254/H9262/H9263w/H9262/H20849/H20841r−r/H11032/H20841/H20850. /H2084924/H20850 The functions w/H9262/H20849r/H20850depend only on the relative distance r and are strongly peaked near r=0. Thus we deal with isotro- pic and nearly white-noise disorder. However, in order toaccurately treat ultraviolet divergencies arising in our calcu-lation, we keep a small but ﬁnite disorder correlation length.The results will be expressed in terms of four integral con-stants characterizing the disorder strength /H9251/H9262=/H20885drw/H9262/H20849/H20841r/H20841/H20850. /H2084925/H20850 Within the speciﬁed Gaussian disorder model, perturba- tive corrections to the free energy are given by the loopdiagrams. The ﬁrst- and second-order corrections are shownin Fig. 3. Dashed lines in these diagrams denote disorder correlation functions Eq. /H2084924/H20850.1 2 /epsilon1/epsilon1 FIG. 2. Lowest energy correction to the free energy of the system.SCHUESSLER et al. PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-4A. First-order correction The ﬁrst-order correction to the free energy /H9024/H20849/H9278/H20850is given by the loop diagram containing two Green’s functions andone impurity line, Fig. 3/H20849a/H20850. The Green’s function at coinci- dent points diverges. That is why we keep a ﬁnite correlationlength calculating the ﬁrst-order diagram. Assuming theseparation between two vertices of the diagram is given bythe vector /H9254, we obtain the expression /H9024a=/H20885d/H9254/H20858 /H9262w/H9262/H20849/H20841/H9254/H20841/H20850/H9024a/H20849/H9262/H20850, /H2084926/H20850 /H9254/H9024a/H20849/H9262/H20850=/H9266/H20885drTr/H20851/H9268/H9262Gˇ/H20849r,r+/H9254/H20850/H9268/H9262Gˇ/H20849r+/H9254,r/H20850/H20852./H2084927/H20850 Now we substitute the Green’s function from Eq. /H20849A7/H20850and expand the correction to the free energy in powers of /H9254. For the four possible disorder types, this yields /H9024a/H208490/H20850,/H20849z/H20850=/H9266W 2L2/H20885 0L dx/H20900−1 2 sin2/H9266x L/H11006/H20873/H9254x2−/H9254y2 6/H92542+/H9254y2/H92782 /H92662/H92542/H20874/H20901, /H2084928/H20850 /H9024a/H20849x/H20850,/H20849y/H20850=/H9266W L2/H20885 0L dx/H20900−1 4 sin2/H9266x L/H11006/H208731 12+/H9254x2−/H9254y2 /H92662/H92544/H20874/H20901. /H2084929/H20850 In these expressions we encounter two types of divergent terms: one with negative power of /H9254and one with an integral of sin−2/H20849/H9266x/L/H20850, which diverges at x=0 and x=L. These terms, however, are free of the source parameter /H9278and hence do not change any observables. The /H9278-dependent terms are ﬁnite and, after integrating over /H9254in Eq. /H2084926/H20850, yield the simple result35 /H9024a= const + /H20849/H92510−/H9251z/H20850W/H92782 4/H9266L. /H2084930/H20850 It provides a linear /H20849in/H9251/H9262/H20850correction to the free energy of the clean sample, Eq. /H2084917/H20850, merely changing the overall pref- actor /H20849conductance /H20850but preserving the quadratic dependence on/H9278and hence the form of the Dorokhov distribution. Thus the linear disorder correction does not destroy the pseudod-iffusive character of transport in graphene at the Dirac point.B. Second-order corrections Since the lowest disorder correction Eq. /H2084930/H20850preserves the form of the Dorokhov distribution, we proceed withhigher order corrections. Our aim is to ﬁnd a deviation fromthe pseudodiffusive transport. The second-order correction tothe free energy is due to the diagrams in Figs. 3/H20849b/H20850and3/H20849c/H20850. The diagram with parallel impurity lines /H20851Fig. 3/H20849b/H20850/H20852yields /H9024 b=/H20885d/H9254d/H9254/H11032/H20858 /H9262,/H9263w/H9262/H20849/H20841/H9254/H20841/H20850w/H9262/H20849/H20841/H9254/H11032/H20841/H20850/H9024b/H20849/H9262/H9263/H20850, /H2084931/H20850 /H9024b/H20849/H9262/H9263/H20850=2/H92662/H20885drdr/H11032Tr/H20851/H9268/H9262G/H20849r,r+/H9254/H20850/H9268/H9262G/H20849r+/H9254,r/H11032/H20850 /H11003/H9268/H9263G/H20849r/H11032,r/H11032+/H9254/H11032/H20850/H9268/H9263G/H20849r/H11032+/H9254/H11032,r/H20850/H20852. /H2084932/H20850 Using the Green’s function from Eq. /H20849A7/H20850, we expand the correction to the free energy in powers of /H9254and/H9254/H11032. Then we drop all /H9278-independent terms and average with respect to the directions of /H9254and/H9254/H11032. The following four contributions to the free energy are nonzero: /H9024b/H2084900/H20850=/H9024b/H20849zz/H20850=−/H9024b/H208490z/H20850=−/H9024b/H20849z0/H20850=W/H92782 64L4/H20885 0L dxdx /H11032/H20885 −/H11009/H11009 dy /H11003/H209001 sin2/H9266/H20849x+x/H11032+iy/H20850 2L+1 sin2/H9266/H20849x−x/H11032+iy/H20850 2L+ c.c./H20901. /H2084933/H20850 After integrating with respect to xandx/H11032the above expres- sion vanishes. Thus we conclude that the diagram in Fig.3/H20849b/H20850gives no contribution to the free energy, /H9024 b=0 . /H2084934/H20850 Let us now consider the diagram in Fig. 3/H20849c/H20850with crossed impurity lines. This diagram contains no Green’s functions atcoincident points and hence does not require regularization.We can replace disorder correlation functions w /H9262by equiva- lent delta functions and obtain /H9024c=/H20858 /H9262/H9263/H9251/H9262/H9251/H9263/H9024c/H20849/H9262/H9263/H20850, /H2084935/H20850 /H9024c/H20849/H9262/H9263/H20850=/H92662/H20885drdr/H11032Tr/H20851/H9268/H9262Gˇ/H20849r,r/H11032/H20850/H9268/H9263Gˇ/H20849r/H11032,r/H20850/H208522. /H2084936/H20850 With the Green’s function Eq. /H20849A7/H20850we ﬁnd the following contribution to the sum in Eq. /H2084935/H20850: /H9024c/H2084900/H20850=/H9024c/H208490z/H20850=/H9024c/H20849z0/H20850=/H92662W 64L4/H20885 0L dxdx /H11032/H20885 −/H11009/H11009 dycosh2/H9278y L /H11003/H208981/H20879sin/H9266/H20849x+x/H11032+iy/H20850 2L/H208792−1/H20879sin/H9266/H20849x−x/H11032+iy/H20850 2L/H208792/H208992 , /H2084937/H208501 21 21 4 (a)( b)( c) FIG. 3. Loop diagrams for the disorder corrections to the ground-state energy /H9024,/H20849a/H20850ﬁrst-order, /H20849b/H20850and /H20849c/H20850second order.FULL COUNTING STATISTICS IN DISORDERED … PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-5/H9024c/H20849zz/H20850=/H9024c/H2084900/H20850+/H92662W 8L4/H20885 0L dxdx /H11032/H20885 −/H11009/H11009 dycosh2/H9278y L /H11003/H20879sin/H9266/H20849x+x/H11032+iy/H20850 2Lsin/H9266/H20849x−x/H11032+iy/H20850 2L/H20879−2 ./H2084938/H20850 Two-dimensional integrals with respect to xand x/H11032are straightforward due to periodicity of the integrand. As a re-sult, the free energy is expressed as a single yintegral /H9024 c=/H92662W 8L2/H20885 −/H11009/H11009 dycosh /H208492/H9278y/L/H20850 sinh2/H20849/H9266y/L/H20850/H20875/H20849/H92510+/H9251z/H208502coth/H20879/H9266y L/H20879 −/H20849/H92510+3/H9251z/H20850/H20849/H92510−/H9251z/H20850/H20876. /H2084939/H20850 This integral diverges at y=0. Expanding near this point, we ﬁnd that the integrand behaves as /H20849L//H20841y/H20841/H208503+2/H92782L//H20841y/H20841. The most singular part is /H9278independent and hence unobservable. Integral of the second term diverges logarithmically and mul-tiplies /H92782. This gives a logarithmic correction to the conduc- tance of the system preserving the pseudodiffusive form ofthe transmission distribution. Let us cutoff the logarithmicintegral at some ultraviolet scale y=athat is the smallest scale where the massless Dirac model with Gaussian white-noise disorder applies, e.g., the scale of the disorder correla-tion length or lattice spacing in graphene. The upper cutoff isalready embedded in the integrand of Eq. /H2084939/H20850: the small y expansion is valid for y/H11351L. Thus we can isolate the diver- gent part of the integral Eq. /H2084939/H20850and the remaining /H9024 ccor- rection, which has a nontrivial dependence on /H9278. /H9024c=W/H92782 4/H9266L/H20853/H20849/H92510+/H9251z/H208502/H208512l n /H20849L/a/H20850+/H92751/H20849/H9278/H20850/H20852 +/H20849/H92510+3/H9251z/H20850/H20849/H92510−/H9251z/H20850/H92752/H20849/H9278/H20850/H20854. /H2084940/H20850 Since the logarithmic term in the free energy contains an ultraviolet parameter adeﬁned up to a model-dependent con- stant, the functions /H92751,2/H20849/H9278/H20850are ﬁxed up to an arbitrary con- stant. With this accuracy, we ﬁnd /H92751/H20849/H9278/H20850=/H92663 2L/H92782/H20885 −/H11009/H11009 dycosh/H9266y L sinh3/H20879/H9266y L/H20879/H20873cosh2/H9278y L−1−2/H92782y2 L2/H20874 = const − /H9274/H20849/H9278//H9266/H20850−/H9274/H20849−/H9278//H9266/H20850, /H2084941/H20850 /H92752/H20849/H9278/H20850=−/H92663 2L/H92782/H20885 −/H11009/H11009 dycosh /H208492/H9278y/L/H20850−1 sinh2/H20849/H9266y/L/H20850 = const + /H92662/H9278cot/H9278−1 /H92782. /H2084942/H20850 The logarithmic correction in Eq. /H2084940/H20850can be included into an effective Ldependence of the disorder strength pa- rameters /H9251/H9262by renormalization group /H20849RG /H20850methods. The model of two-dimensional massless Dirac fermions subjectto Gaussian disorder and its logarithmic renormalization ap-peared in various contexts. In particular, disorder renormal-ization in graphene was considered in Refs. 16,32, and 36. One-loop RG equations for effective disorder couplings asfunctions of a running scale /H9011are /H11509/H92510 /H11509ln/H9011=2/H20849/H92510+/H9251z/H20850/H20849/H92510+/H9251x+/H9251y/H20850, /H2084943a /H20850 /H11509/H9251x /H11509ln/H9011=/H11509/H9251y /H11509ln/H9011=2/H92510/H9251z, /H2084943b /H20850 /H11509/H9251z /H11509ln/H9011=2/H20849/H92510+/H9251z/H20850/H20849−/H9251z+/H9251x+/H9251y/H20850. /H2084943c /H20850 Parameters deﬁned in Eq. /H2084925/H20850serve as initial conditions for the RG equations at an ultraviolet scale a. Integrating Eq. /H2084943/H20850up to the largest scale that is the system size L,w e obtain effective disorder couplings /H9251˜/H9262=/H9251/H9262/H20849L/H20850and automati- cally take into account all leading logarithmic contributionslike the one in Eq. /H2084940/H20850. This allows us to replace the disor- der parameters in the free energy by their renormalized val-ues and drop the logarithm from Eq. /H2084940/H20850. Collecting in this way the contributions in Eqs. /H2084917/H20850,/H2084930/H20850, and /H2084940/H20850, we obtain the ﬁnal expression for the free energy up to the second orderin renormalized disorder parameters, /H9024=W /H92782 4/H9266L/H208511+/H9251˜0−/H9251˜z+/H20849/H9251˜0+/H9251˜z/H208502/H92751/H20849/H9278/H20850 +/H20849/H9251˜0+3/H9251˜z/H20850/H20849/H9251˜0−/H9251˜z/H20850/H92752/H20849/H9278/H20850/H20852. /H2084944/H20850 Thus we have established a deviation from pseudodiffu- sive transport regime /H20849/H9024/H11011/H92782/H20850in the second order in disor- der strength. C. Corrections to the distribution function Let us now derive a correction to the Dorokhov distribu- tion function of transmission probabilities. In the /H9261represen- tation, the distribution function is given by Eq. /H2084912/H20850. Using the result in Eq. /H2084944/H20850, we obtain P/H20849/H9261/H20850=W /H9266L/H208511+/H9251˜0−/H9251˜z+/H20849/H9251˜0+/H9251˜z/H208502p1/H20849/H9261/H20850 +/H20849/H9251˜0+3/H9251˜z/H20850/H20849/H9251˜0−/H9251˜z/H20850p2/H20849/H9261/H20850/H20852. /H2084945/H20850 Similarly to /H92751,2, the functions p1,2/H20849/H9261/H20850are deﬁned up to a model-dependent constant. From Eqs. /H2084941/H20850and /H2084942/H20850we ob- tain p1/H20849/H9261/H20850= const − 2 Re/H11509 /H11509/H9261/H20875/H9261/H9274/H208732i/H9261 /H9266/H20874/H20876, /H2084946a /H20850 p2/H20849/H9261/H20850= const +/H92662 2 sinh2/H208492/H9261/H20850. /H2084946b /H20850 The functions p1andp2are shown in Fig. 4./H20849When the only disorder is /H92510, correction to the distribution function is given by the sum p1+p2also shown in the ﬁgure. /H20850The func- tions p1andp2cannot be used for direct calculation of trans- mission moments due to their divergence at /H9261=0. This diver-SCHUESSLER et al. PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-6gence signiﬁes the breakdown of perturbative expansion in small values of disorder couplings close to /H9261=0 /H20849that is T =1/H20850. Comparing disorder correction with the distribution in the clean sample, we conclude that the result in Eq. /H2084945/H20850is valid provided /H9261/H11271/H9251˜. The deviation from pseudodiffusive transport regime can be experimentally demonstrated as a correction to the Fanofactor F=1 /3 characteristic to the diffusive systems. Diver- gence of the functions p 1,2at/H9261=0 prevents us from calcu- lating transmission moments from the distribution functionEq. /H2084945/H20850. However, we can obtain transport characteristics from the free energy Eq. /H2084944/H20850instead. With the help of Eq. /H2084914/H20850, we ﬁnd the Fano factor up to quadratic terms in the renormalized disorder strength, F=1 3−16/H9256/H208493/H20850 /H92662/H20849/H9251˜0+/H9251˜z/H208502+8/H92662 45/H20849/H9251˜0+3/H9251˜z/H20850/H20849/H9251˜0−/H9251˜z/H20850 /H110151 3− 0.194 /H9251˜02− 0.388 /H9251˜0/H9251˜z− 7.212 /H9251˜z2. /H2084947/H20850 Remarkably, any weak disorder, irrespective of its matrix structure, suppresses the Fano factor. /H20851Note that the energy correction Eq. /H2084921/H20850is also negative. /H20852The correction to the Fano factor increases with increasing sample length Ldue to renormalization Eq. /H2084943/H20850. At some length l, referred to as the mean free path, one of the renormalized disorder couplingsreach a value of order unity and the perturbative RG treat-ment breaks down. This signiﬁes the crossover from ballisticto diffusive transport regime. Disorder correction to the Fanofactor becomes strong in this crossover region. To go beyondthe mean-free-path scale we resort to other methods designedfor diffusive systems. VI. DISORDERED GRAPHENE: DIFFUSIVE LIMIT When the system size exceeds the mean free path, the sample exhibits diffusive electron transport. On a semiclas-sical level, the system can be characterized by its conductiv-ity per square in this limit. At the ballistics-diffusion cross-over the conductivity of graphene is close to the quantumvalue e 2/h. This signiﬁes strong interference corrections to transport characteristics making semiclassical picture inad-equate. These quantum effects lead to one of the four pos-sible scenarios depending on the symmetry of disorder. /H20849i/H20850If the only disorder is random potential /H20849 /H92510/H20850, the sys- tem possesses time inversion symmetry H=/H92682HT/H92682and falls into symplectic symmetry class AII.37,38Quantum correc- tions to the conductivity are positive, leading to good metal-lic properties /H20849large dimensionless conductivity /H20850at large scales. /H20849ii/H20850In the case of random vector potential /H9251x,y, the only symmetry of the problem is chirality, H=−/H92683H/H92683, signifying the chiral unitary symmetry class AIII. Such disorder pro-duces no corrections to the conductivity to all orders and canbe effectively gauged out at zero energy. 16From the point of view of its transport properties, the system remains effec-tively clean and ballistic at all scales. /H20849iii/H20850If the only disorder is random mass /H20849 /H9251z/H20850, the Hamil- tonian has a Bogolyubov-de Gennes symmetry H=/H92681HT/H92681 characteristic for the symmetry class D. Upon renormaliza- tion Eq. /H2084943/H20850the disorder coupling gets smaller and the sys- tem becomes effectively clean. This means the absence ofthe mean-free-path scale and hence of the diffusive transportregime. /H20849iv/H20850In the generic case, when more than one disorder type is present and all symmetries are broken, the symmetry classis unitary /H20849A/H20850and transport properties are the same as at the critical point of the quantum Hall transition. We will concentrate on the ﬁrst case /H20849random potential /H20850 when the system eventually acquires a large parameter—dimensionless conductivity—and can be quantitatively de-scribed by the proper effective ﬁeld theory—sigma model ofthe symplectic symmetry class. Our consideration in this partof the paper is closely related to that of Ref. 24. Derivation of the sigma model with the source ﬁelds z from Eq. /H208496/H20850is sketched in Appendix C. The symplectic sigma model operates with the matrix ﬁeld Qof the size 4N/H110034N, where Nis the number of replicas. Apart from replica space, matrix Qhas retarded-advanced /H20849RA /H20850and particle-hole /H20849PH/H20850structures. The former is similar to the matrix Green’s function while the latter is introduced in or-der to account for time-reversal symmetry of the problem.We will denote Pauli matrices in RA space by /H9011 x,y,z.T w o constraints are imposed on Q, namely, Q2=1 and Q=QT. This yields the target space Q/H33528O/H208494N/H20850/O/H208492N/H20850/H11003O/H208492N/H20850 characteristic for symplectic class systems. The sigma-modelaction is 39 S/H20851Q/H20852=/H9268 16/H20885drTr/H20849/H11612Q/H208502. /H2084948/H20850 Here/H9268is the dimensionless /H20849in units e2/h/H20850conductivity of the two-dimensional disordered system. The source ﬁeld isincorporated into boundary conditions, Q/H20841 x=0=/H9011z,Q/H20841x=L=/H9011zcos/H9278+/H9011xsin/H9278. /H2084949/H20850 The free energy of the system in the source ﬁeld /H9278is ex- pressed through the N→0 limit of the sigma-model partition function as /H9024= lim N→01 N/H208731−/H20885DQe−S/H20851Q/H20852/H20874. /H2084950/H20850p2 p1p1/Plusp2 0 1 2 3 4 5/Minus15/Minus10/Minus5051015 Λp/LParen1Λ/RParen1 FIG. 4. Functions p1andp2entering the disorder correction to the distribution of transmission probabilities Eq. /H2084945/H20850. In the case of random scalar potential /H20849/H92510/H20850, the distribution is determined by the sum p1+p2only.FULL COUNTING STATISTICS IN DISORDERED … PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-7In a good metallic sample with /H9268/H112711, the Qintegral in Eq. /H2084950/H20850can be evaluated within the saddle-point approxima- tion. The action Eq. /H2084948/H20850is minimized by the following con- ﬁguration of the ﬁeld Q: Q0=U−1/H9011zU,U= exp/H20873i/H9011y/H9278x 2L/H20874. /H2084951/H20850 Replacing the integral in Eq. /H2084950/H20850with the value of the inte- grand at the saddle point, we obtain the semiclassical resultfor the full counting statistics, /H9024 0= lim N→0S/H20851Q0/H20852 N=W/H9268/H92782 4L. /H2084952/H20850 This yields the Dorokhov distribution of transmission prob- abilities in diffusive two-dimensional system.4In order to ﬁnd corrections to this result, we take into account ﬂuctua-tions of the ﬁeld Qnear its saddle-point value Q 0. This is equivalent to the calculation of a Cooperon loop, Fig. 5, carried out in Ref. 24. Small ﬂuctuations of Qnear the saddle point Q0are pa- rametrized by the matrix Bas/H20849we write expressions involv- ingBup to the second order /H20850 Q=U−1/H9011z/H208731+B+B2 2/H20874U,B=/H208730 b −bT0/H20874. /H2084953/H20850 This parametrization of Qautomatically fulﬁls the conditions Q2=1 and Q=QT. The sigma-model action expanded up to the second order in Btakes the form S/H20851Q/H20852=S/H20851Q0/H20852−/H9268 16/H20885drTr/H20875/H20849/H11612B/H208502−/H92782 4L2/H20853/H9011x,B/H208542/H20876. /H2084954/H20850 Curly braces denote anticommutator. Let us separate Binto the parts commuting and anticommuting with /H9011x. These two parts do not couple to each other in the quadratic action Eq./H2084954/H20850and only the former one couples to the source parameter /H9278. Thus we can constraint the matrix Bby requiring its com-mutativity with /H9011x. In terms of bthis yields b=−bTand the action becomes S/H20851Q/H20852=S/H20851Q0/H20852+/H9268 8/H20885drTr/H20875/H11612b/H11612bT−/H92782 L2bbT/H20876. /H2084955/H20850 This quadratic form is diagonalized in momentum represen- tation. Component of momentum perpendicular to the leadstakes quantized values /H9266n/Lwith positive integer ndue to geometrical restrictions /H20851boundary conditions Eq. /H2084949/H20850ﬁxb =0 at the interfaces with metallic leads /H20852. Momentum parallel to the leads is continuous and unrestricted. For each value ofthe momentum there are N/H208492N−1/H20850independent matrix ele- ments in b. Calculating the Gaussian integral in Eq. /H2084950/H20850we obtain the free energy /H9024=/H9024 0−W 2/H20858 n=1/H11009/H20885dqy 2/H9266ln/H20849/H92662n2+q2L2−/H92782/H20850 =W 2L/H20875/H9268/H92782 2−/H20858 n=1/H11009 /H20881/H92662n2−/H92782/H20876. /H2084956/H20850 In the result /H20851Eq. /H2084956/H20850/H20852, the sum diverges at large n. The situation is similar to what we have encountered in the bal-listic regime. Expanding the sum in powers of /H9278, we see that the most divergent term is /H9278independent while the next term multiplies /H92782and diverges logarithmically. This is nothing but the weak antilocalization correction. It renormalizes theconductivity but does not deform the full counting statistics.Logarithmically divergent sum is cut at n/H11011L/l, where lis the mean free path. At larger values of nthe diffusive ap- proximation /H20849gradient expansion in the sigma model /H20850breaks down. In terms of renormalized conductivity, the free energyreads /H9024=W 2L/H20875/H9268˜/H92782 2−/H20858 n=1/H11009/H20873/H20881/H92662n2−/H92782−/H9266n+/H92782 2/H9266n/H20874/H20876,/H2084957/H20850 /H9268˜=/H9268+1 /H9266lnL l/H110151 /H9266lnL l. /H2084958/H20850 The bare value of conductivity, /H9268, is of order one and hence negligible in comparison with the large renormalizing loga-rithm. The sum over nin Eq. /H2084957/H20850is convergent and provides the deviation from semiclassical Dorokhov statistics of trans-mission probabilities. In fact, a more rigorous procedure is to perform ﬁrst a renormalization of the sigma model from the mean-free-pathscale lto the scale /H11011L. Then the free energy can be calcu- lated perturbatively. It turns out, however, that this yields aresult identical to the one obtained above within the pertur-bative analysis at the scale l. Indeed, the RG equation d /H9268/dln/H9011=1 //H9266will lead exactly to the renormalization of conductivity /H9268/H21739/H9268˜, see Eq. /H2084958/H20850. The consequent evaluation of the perturbative contribution to /H9024yields Eq. /H2084956/H20850with/H9268 replaced by /H9268˜and the sum restricted to a ﬁnite /H20849independent ofL/H20850number of terms. In other words, the renormalization shifts the logarithmical contribution to /H9268from the second to the ﬁrst term in square brackets in Eq. /H2084956/H20850.FIG. 5. Cooperon correction to the free energy in the diffusive limit.SCHUESSLER et al. PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-8Let us derive the distribution function P/H20849/H9261/H20850from the free energy Eq. /H2084957/H20850. Applying Eq. /H2084912/H20850, we obtain the result in the form P/H20849/H9261/H20850=W L/H20851/H9268˜+p/H20849/H9261/H20850/H20852, /H2084959/H20850 p/H20849/H9261/H20850=1 /H9266/H20858 n=1/H11009/H20875Re/H9266+2i/H9261 /H20881/H92662n2−/H20849/H9266+2i/H9261/H208502−1 n/H20876. /H2084960/H20850 At small values of /H9261, the sum in Eq. /H2084960/H20850is determined by the term with n=1. In the opposite limit, the sum can be estimated by the corresponding integral with the help ofEuler-Maclaurin formula. Thus we obtain the asymptotic ex-pressions p/H20849/H9261/H20850=/H20902/H208811 8/H9266/H9261/H9261/H112701 −1 /H9266ln/H9261/H9261/H112711./H20903/H2084961/H20850 The function p/H20849/H9261/H20850is shown in Fig. 6. It is qualitatively simi- lar to the numerical result of Ref. 20. Deviation from the semiclassical transport can be demon- strated by the correction to the Fano factor. With the help ofEq. /H2084914/H20850, we obtain F=1 3−2/H9256/H208493/H20850 /H92663/H9268˜=1 3−0.244 ln/H20849L/l/H20850. /H2084962/H20850 A similar correction to the Fano factor was found numeri- cally in Ref. 22. We compare the numerical results with Eq. /H2084962/H20850below. In the case of weak scalar disorder /H20849described by the cou- pling/H92510/H20850, the system undergoes a continuous crossover from ballistic to diffusive transport regime as the size Lgrows. In both limiting cases, we encounter nearly Dorokhov distribu-tion of transmission probabilities with small corrections, Eqs./H2084947/H20850and /H2084962/H20850, on both sides of the crossover. In the ballistic limit, we can formally introduce a dimensionless conductiv-ity as /H9268=/H20849L/W/H20850G//H20849e2/h/H20850. Then the corrections to the Fano factor are expressed in terms of the conductivityF=1 3−/H20902/H2087316/H9256/H208493/H20850 /H92662−8/H92662 45/H20874/H20849/H9266/H9268−1/H208502/H9266/H9268−1/H112701 2/H9256/H208493/H20850 /H92663/H9268/H9268/H112711./H20903 /H2084963/H20850 This Fano factor as a function of conductivity is shown in Fig. 7together with numerical results from Ref. 22.I nt h e numerical simulations, a single valley of graphene was mod-eled using a ﬁnite-difference approach. By construction, dis-order in Ref. 22has the symmetry of scalar potential which does not mix the valleys. It is this symmetry /H20849class AII /H20850 which is considered in the present section. Our results per-fectly agree with the numerics in the diffusive limit /H20849see Fig. 7/H20850in the range /H9266/H9268/H114073. On the ballistic side, the deviation is due to the nonuniversality of the ballistic transport. Speciﬁ-cally, the function F/H20849 /H9268/H20850depends crucially on the microscopic details of disorder. In the numerical analysis of Ref. 22, the model with strong scatterers was used while in the presentpaper we adopt the model of weak Gaussian white-noise dis-order. For theoretical predictions on electron transport in thepresence of strong scatterers see Ref. 40. An earlier numerical study of Ref. 20, based on the transfer-matrix description of the Dirac problem, reported thevalue of the Fano factor in the range 0.29–0.30 /H20849for different samples /H20850with the conductivity, /H9266/H9268, of the largest systems varying from 6 to 10. This is consistent with our predictionsfor the diffusive transport regime /H20849see Fig. 7/H20850. The behavior ofFin the ballistic regime is different due to the reasons described above /H20849strong vs weak disorder /H20850. A nonmonoto- nous dependence F/H20849 /H9268/H20850at the Dirac point was also observed in Ref. 21. The Fano factor is 1/3 both in the clean and strongly disordered limits. In the crossover from ballistics to diffu-sion, the Fano factor strongly deviates from this universalvalue signifying the breakdown of the /H20849pseudo /H20850diffusive de- scription characterized by Dorokhov distribution of transmis-sion probabilities.0 1 2 3 4 5/Minus0.6/Minus0.4/Minus0.20.00.2 Λp/LParen1Λ/RParen1 FIG. 6. Correction to the distribution of transmission eigenval- ues in the diffusive limit.1 2 3 4 5 6 7 80.150.200.250.300.35 ΠΣF FIG. 7. Fano factor as a function of conductivity. Solid lines show ballistic and diffusive results in Eq. /H2084963/H20850. Dashed line corre- sponds to the asymptotic value F=1 /3. Solid symbols are numeri- cal results from Ref. 22, the size of rectangles corresponds to the error estimate.FULL COUNTING STATISTICS IN DISORDERED … PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-9VII. SUMMARY We have studied the full counting statistics of the charge transport through an undoped graphene sheet in the presenceof weak and smooth /H20849not mixing valleys /H20850disorder. We have identiﬁed deviations from the Dorokhov distribution of trans-mission probabilities both in ballistic /H20851Eqs. /H2084945/H20850and /H2084946/H20850/H20852 and diffusive /H20851Eqs. /H2084959/H20850and /H2084960/H20850/H20852regimes. In the former case, corrections are model dependent while in the latter caseonly the symmetry of disorder matters. We have consideredGaussian white-noise disorder in the ballistic regime and po-tential disorder /H20849symplectic symmetry class /H20850in diffusive limit. Deviation from /H20849pseudo /H20850diffusive transport always re- sults in a negative correction to the Fano factor, F/H110211/3. Our results are in good agreement with recent numerical simula-tions of electron transport in disordered graphene, see Fig. 7. ACKNOWLEDGMENTS We are grateful to R. Danneau, P. San-Jose, and M. Titov for stimulating discussions and to C. Groth for providing uswith the numerical data of Ref. 22. The work was supported by Rosnauka under Grant No. 02.740.11.5072 and by theEUROHORCS/ESF EURYI Award scheme /H20849I.V .G. /H20850. APPENDIX A: MATRIX GREEN’S FUNCTION The full counting statistics of the electron transport is conveniently expressed in terms of the matrix Green’sfunction 25in the external counting ﬁeld z=sin2/H20849/H9278/2/H20850, Eq. /H208496/H20850. For the clean graphene sample attached to perfect metal- lic leads, Fig. 1, this Green’s function satisﬁes the following equation: /H20873/H9262/H20849x/H20850−/H9268p+i0 −/H9268x/H20881z/H9254/H20849x/H20850 −/H9268x/H20881z/H9254/H20849x−L/H20850/H9262/H20849x/H20850−/H9268p−i0/H20874Gˇ0/H20849r,r/H11032/H20850 =/H9254/H20849r−r/H11032/H20850,/H9262/H20849x/H20850=/H208770, 0/H11021x/H11021L, +/H11009,x/H110210o r x/H11022L./H20878 /H20849A1/H20850 Since the operator in the left-hand side of the above equationcommutes with the ycomponent of the momentum, we will ﬁrst calculate the Green’s function in the mixed coordinate- momentum representation, Gˇp/H20849x,x/H11032/H20850. Inside the sample this function satisﬁes /H20875i/H9268x/H11509 /H11509x−/H9268yp/H20876Gˇp/H20849x,x/H11032/H20850=/H9254/H20849x−x/H11032/H20850. /H20849A2/H20850 We will look for a general solution of this equation in the form Gˇp/H20849x,x/H11032/H20850=e/H9268zp/H20849x−L/2/H20850Me/H9268zp/H20849x/H11032−L/2/H20850,M=/H20877M/H11021x/H11021x/H11032 M/H11022x/H11022x/H11032./H20878 /H20849A3/H20850 The chemical-potential proﬁle together with the inﬁnitesimal terms /H11006i0 in Eq. /H20849A1/H20850deﬁnes the boundary conditions for the Green’s function. The counting ﬁeld zcan also be incor- porated into the boundary conditions. In terms of M/H11124we thus obtain /H2087311 i/H20881zi/H20881z 001− 1 /H20874e−/H9268zpL /2M/H11021=0 , /H208731− 1 0 0 −i/H20881z−i/H20881z11/H20874e/H9268zpL /2M/H11022=0 . /H20849A4/H20850 Delta function in the right-hand side of Eq. /H20849A2/H20850yields a jump of the Green’s function at x=x/H11032which provides the relation M/H11022−M/H11021=−i/H9268x. /H20849A5/H20850 The matrices M/H11124, and hence the Green’s function, are com- pletely determined by Eqs. /H20849A4/H20850and /H20849A5/H20850, M/H11124=−i 2/H20849cosh2pL−z/H20850/H20898cosh pL z−sinh 2 pL 2i/H20881ze−pLi/H20881z z+sinh 2 pL 2cosh pL i/H20881zi /H20881zepL i/H20881zepLi/H20881z − cosh pL −z−sinh 2 pL 2 i/H20881zi /H20881ze−pL−z+sinh 2 pL 2− cosh pL/H20899/H11006i/H9268x 2. /H20849A6/H20850SCHUESSLER et al. PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-10Fourier transform in pyields the Green’s function in the full coordinate representation. To facilitate further calculations,we decompose this Green’s function into the following prod-uct of matrices: Gˇ 0/H20849x,x/H11032;y/H20850 =1 4LUˇ/H20849x/H20850/H9011ˇ/H20898icosh/H9278y 2Lsinh/H9278y 2L sinh/H9278y 2L−icosh/H9278y 2L/H20899 RA /H11003/H208981 sin/H9266 2L/H20849x+x/H11032+iy/H208501 sin/H9266 2L/H20849x−x/H11032+iy/H20850 1 sin/H9266 2L/H20849x−x/H11032−iy/H208501 sin/H9266 2L/H20849x+x/H11032−iy/H20850/H20899 /H9268 /H11003/H9011ˇUˇ−1/H20849x/H11032/H20850, /H20849A7/H20850 /H9011ˇ=/H20873/H9268z0 01/H20874 RA,Uˇ/H20849x/H20850=/H20898sin/H9278/H20849L−x/H20850 2Lcos/H9278/H20849L−x/H20850 2L icos/H9278x 2Lisin/H9278x 2L/H20899 RA. /H20849A8/H20850 Here we have used the source angle /H9278deﬁned by z =sin2/H20849/H9278/2/H20850. The matrices Uˇ/H20849x/H20850and Uˇ−1/H20849x/H11032/H20850operate in the retarded-advanced space only and hence commute with anydisorder operators placed between the Green’s functions. As a result, factors UˇandUˇ −1drop from expressions for any closed diagrams. The matrices /H9011ˇin the above equation allow us to decompose the Green’s function into a direct product ofthe two operators acting in the RA space and in the sublatticespace. APPENDIX B: ENERGY CORRECTION TO THE FULL COUNTING STATISTICS In this appendix we evaluate the diagram in Fig. 2for the lowest energy correction to /H9024/H20849/H9278/H20850. Substituting Green’s func- tion Eq. /H20849A7/H20850into Eq. /H2084919/H20850and performing rescaling of inte- gration variables we obtain /H9024/H9280=WL/H92802 4/H20885 01 dxdx /H11032/H20885 −/H11009/H11009 dycosh /H20849/H9278y/H20850 /H11003/H208751 cosh /H20849/H9266y/H20850− cos/H9266/H20849x+x/H11032/H20850 −1 cosh /H20849/H9266y/H20850− cos/H9266/H20849x−x/H11032/H20850/H20876. /H20849B1/H20850 The ﬁrst /H20849second /H20850term in square brackets depends only on sum /H20849difference /H20850ofxandx/H11032. This allows us to integrate over the difference /H20849sum /H20850of these variables. After some shifts ofvariables the remaining integral takes the form /H9024/H9280=−WL/H92802 2/H20885 01 duusin/H9266u 2/H20885 −/H11009/H11009dycosh /H20849/H9278y/H20850 cosh2/H20849/H9266y/H20850− sin2/H20849/H9266u/2/H20850 =−WL/H92802 2 sin /H20849/H9278/2/H20850/H20885 01 duusin/H20849/H9278u/2/H20850 cos/H20849/H9266u/2/H20850. /H20849B2/H20850 The last expression is the result of yintegration. It can be performed, e.g., by closing the integration contour and sum-ming up residues in the upper half plane of imaginary y.I n order to make this sum convergent, one has to add a weak-damping factor by an inﬁnitesimal imaginary shift of /H9278. We proceed with the last integral in Eq. /H20849B2/H20850by repre- senting 1 /cos/H20849/H9266u/2/H20850as a Fourier series /H9024/H9280=2WL/H92802 sin/H20849/H9278/2/H20850/H11509 /H11509/H9278/H20885 01 ducos/H9278u 2/H20858 n=0/H11009 /H20849−1/H20850ncos/H20851/H9266u/H20849n+1 /2/H20850/H20852. /H20849B3/H20850 Convergence of this Fourier series should also be justiﬁed by a proper damping factor. This does not change the ﬁnal resultof the calculation hence we omit such extra factors for sim-plicity. Performing the integration over uwe obtain /H9024 /H9280=2WL/H92802 sin/H20849/H9278/2/H20850/H11509 /H11509/H9278cos/H9278 2/H20858 n=0/H11009/H208751 /H9266/H208492n+1/H20850+/H9278 +1 /H9266/H208492n+1/H20850−/H9278/H20876. /H20849B4/H20850 The sum over ndiverges logarithmically. However, this di- vergence is independent of /H9278and hence does not inﬂuence any observable quantities, which are expressed as derivativesof the free energy. We can easily get rid of the divergent partby subtracting a similar sum over nwith /H9278=0. This yields the ﬁnal result /H9024/H9280=2WL/H92802 sin/H20849/H9278/2/H20850/H11509 /H11509/H9278cos/H9278 2/H20858 n=0/H11009 /H11003/H208751 /H9266/H208492n+1/H20850+/H9278+1 /H9266/H208492n+1/H20850−/H9278−2 /H9266/H208492n+1/H20850/H20876 =−W /H9266L/H20849/H9280L/H208502 sin/H9278 2/H11509 /H11509/H9278 /H11003/H20877cos/H9278 2/H20875/H9274/H20873/H9266+/H9278 2/H9266/H20874+/H9274/H20873/H9266−/H9278 2/H9266/H20874+4l n2+2 /H9253/H20876/H20878. /H20849B5/H20850 Here /H9274is the digamma function and /H9253is the Euler- Mascheroni constant. The last expression yields Eq. /H2084920/H20850of the main text /H20849where we drop the unobservable constant /H20850. APPENDIX C: DERIVATION OF THE SIGMA MODEL In order to carry out a parametrically controlled derivation of the sigma model, it is convenient to consider a modiﬁedFULL COUNTING STATISTICS IN DISORDERED … PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-11problem with n/H112711 ﬂavors of Dirac fermions. To perform the disorder average of the free energy, we also introduce N replicas. /H20849Alternatively, one can use supersymmetry. As we will treat the sigma model perturbatively, the two approachesare fully equivalent. /H20850 The derivation of the sigma model starts with the fermi- onic action generating the matrix Green’s function Eq. /H208496/H20850. S/H20851 /H9278,/H9278/H11569/H20852=/H20885dr/H20858 a,b,/H9251/H9278a/H9251†/H20849/H20853i0/H9011z−/H9268p−/H20881z/H9268x /H11003/H20851/H9011+/H9254/H20849x/H20850+/H9011−/H9254/H20849x−L/H20850/H20852/H20854/H9254ab−Vab/H20849r/H20850/H20850/H9278b/H9251. /H20849C1/H20850 Here/H9011/H11006=/H20849/H9011x/H11006i/H9011y/H20850/2 are matrices operating in RA space. This action is the functional of two independent Grassmanntwo-component /H20849in /H9268space /H20850vector ﬁelds /H9278and/H9278/H11569. Lower indices, aand b, refer to ﬂavors while the upper index /H9251 enumerates replicas. Overall, there are 4 nNindependent Grassmann variables in the Lagrangian. The random matrixV abis symmetric, that insures the time-reversal symmetry of the model. We assume Gaussian white-noise statistics for thematrix Vdeﬁned by the correlator /H20855V ab/H20849r/H20850Vcd/H20849r/H11032/H20850/H20856=2/H9266/H92510 n/H20851/H9254ac/H9254bd+/H9254ad/H9254bc/H20852/H9254/H20849r−r/H11032/H20850. /H20849C2/H20850 Using the time-reversal symmetry, we rewrite the action in terms of the single four-component ﬁeld /H9274/H20849and its charge- conjugate version /H9274¯, that is linearly related to /H9274/H20850 /H9274=1 /H208812/H20873/H9278 i/H9268y/H9278/H11569/H20874,/H9274¯=i/H9274T/H9268y/H9270x=1 /H208812/H20849/H9278†,i/H9278T/H9268y/H20850./H20849C3/H20850 This introduces an additional PH structure of the ﬁelds. Pauli matrices operating in PH space are denoted by /H9270x,y,z. Bar denotes the charge conjugation operation which has two im- portant properties: /H9274¯1/H92742=/H9274¯2/H92741and /H20849/H92741/H9274¯2/H20850T=/H9270x/H9268y/H92742/H9274¯1/H9268y/H9270x. The action takes the following form in terms of /H9274: S/H20851/H9274/H20852=/H20885dr/H20858 a,b,/H9251/H9274¯ a/H9251/H20849/H20853i0/H9011z−/H9268p−/H20881z/H9268x /H11003/H20851/H9267+/H9254/H20849x/H20850+/H9267−/H9254/H20849x−L/H20850/H20852/H20854/H9254ab−Vab/H20849r/H20850/H20850/H9274b/H9251./H20849C4/H20850 In this expression we have introduced the notation /H9267/H11006 =/H20849/H9011x/H9270z/H11006i/H9011y/H20850/2. Now we are ready to average e−Sover the Gaussian dis- order distribution with the correlator Eq. /H20849C2/H20850. This yields an effective action with the quartic term. Using the above-mentioned properties of charge conjugation, we recast theaction in the form S/H20851 /H9274/H20852=/H20885dr/H20875/H20858 a,/H9251/H9274¯ a/H9251/H20853i0/H9011z−/H9268p−/H20881z/H9268x /H11003/H20851/H9267+/H9254/H20849x/H20850+/H9267−/H9254/H20849x−L/H20850/H20852/H20854/H9274a/H9251 +2/H9266/H92510 n/H20858 a,b,/H9251,/H9252Tr/H9274a/H9251/H9274¯ a/H9252/H9274b/H9252/H9274¯ b/H9251/H20876. /H20849C5/H20850Next, we decouple the quartic term introducing an auxiliary 8N/H110038Nmatrix Rby the Hubbard-Stratonovich transforma- tion. This yields the action S/H20851R,/H9274/H20852=/H20885dr/H20875n/H92532 8/H9266/H92510TrR2+/H20858 a,/H9251,/H9252/H9274¯ a/H9251/H20849i/H9253R/H9251/H9252 −/H20853/H9268p+/H20881z/H9268x/H20851/H9267+/H9254/H20849x/H20850+/H9267−/H9254/H20849x−L/H20850/H20852/H20854/H9254/H9251/H9252/H20850/H9274a/H9252/H20876. /H20849C6/H20850 Parameter /H9253is an arbitrary number at this stage, its value will be ﬁxed later. Matrix R/H9251/H9252couples to the product /H20858a/H9274a/H9251/H9274¯ a/H9252. This allows us to impose the corresponding symme- try constraint on the matrix R:R=/H9268y/H9270xRT/H9268y/H9270x. Finally, we integrate out the fermionic ﬁelds and obtain the action oper-ating with the matrix Ronly, S/H20851R/H20852=n 2Tr/H20873/H92532R2 4/H9266/H92510−l n /H20853i/H9253R−/H9268p−/H20881z/H9268x /H11003/H20851/H9267+/H9254/H20849x/H20850+/H9267−/H9254/H20849x−L/H20850/H20852/H20854/H20874. /H20849C7/H20850 The bold “ Tr” symbol implies the full operator trace includ- ing integration over space coordinates. Derivation of the sigma model proceeds with the saddle- point analysis of the action Eq. /H20849C7/H20850in the absence of the source ﬁeld z. We ﬁrst look for a diagonal and spatially con- stant matrix Rminimizing the action. The saddle-point equa- tion is identical to the self-consistent Born approximation/H20849SCBA /H20850equation for the self-energy − i /H9253R, −i/H9253R=2/H9266/H92510/H20885dp /H208492/H9266/H208502/H20849i/H9253R−/H9268p/H20850−1. /H20849C8/H20850 We ﬁx /H9253to be the imaginary part of the SCBA self-energy, /H9253=/H9004e−1 //H92510with/H9004being ultraviolet energy cutoff /H20849band- width /H20850. Then the saddle-point conﬁguration for the matrix R is simply R=/H9011z. This ﬁxes the boundary conditions for the matrix Rat the contacts. Since the leads are very good metals and ﬂuctuations of Rare strongly suppressed there, R=/H9011zfor x/H110210 and x/H11022L. The matrix R=/H9011zis not the only saddle point of the action Eq. /H20849C7/H20850. Other conﬁgurations minimizing the action can be obtained by rotations R=T−1/H9011zTwith any matrix Twhich commutes with /H9268pand preserves the constraint R =/H9268y/H9270xRT/H9268y/H9270x. Matrix T, and hence R, is trivial in /H9268space. This allows us to reduce the dimension of Rto 4N/H110034N operating in /H9011,/H9270, and replicas only. The saddle manifold generated by matrices TisO/H208494N/H20850/O/H208492N/H20850/H11003O/H208492N/H20850. Let us now restore the source term in the action and es- tablish boundary conditions for R. The matrix Rhas a jump at the interfaces with the leads due to the delta functions inthe action Eq. /H20849C7/H20850. However, we can eliminate these jumps by a proper gauge transformation. Let us perform a rotation R=AR˜A−1with an x-dependent matrix A. The action acquires the following form in terms of R˜:SCHUESSLER et al. PHYSICAL REVIEW B 82, 085419 /H208492010 /H20850 085419-12S/H20851R˜/H20852=n 2Tr/H20875/H92532R˜2 2/H9266/H92510−l n/H20873i/H9253R˜−/H9268p+i/H9268xA−1 /H11003/H20877/H11509A /H11509x+i/H20881z/H20851/H9267+/H9254/H20849x/H20850+/H9267−/H9254/H20849x−L/H20850/H20852A/H20878/H20874/H20876./H20849C9/H20850 The source ﬁeld drops from this action if we choose Asuch that the expression in curly braces vanishes. This yields A=/H209021 x/H110210 1−i/H20881z/H9267+ 0/H11021x/H11021L /H208491−i/H20881z/H9267−/H20850/H208491−i/H20881z/H9267+/H20850x/H11022L./H20903/H20849C10 /H20850 Note that the matrix R˜, deﬁned with the help of the above matrix A, fulﬁls the condition R˜=/H9270xR˜T/H9270x. Since delta func- tions disappear from the action, we can infer that R˜is con- tinuous at the interfaces with the leads. In the left lead we have R=R˜=/H9011z. This is the left boundary condition for the matrix R˜. The right boundary condition is ﬁxed by the iden- tities R˜=A−1RAandR=/H9011zforx/H11022L. This yields R˜/H20849L/H20850=/H208491−2 z/H20850/H9011z+iz3/2/H9011x+/H20881z/H208492−z/H20850/H9011y/H9270z. /H20849C11 /H20850 We can further simplify this bulky expression by performing a constant rotation R˜=B−1QBwith the matrixB=/H9270z−/H9270y 2/H208812/H20851/H208491−z/H20850−1 /4/H208491+/H9011z/H9270z/H20850−i/H208491−z/H208501/4/H208491−/H9011z/H9270z/H20850/H20852. /H20849C12 /H20850 After such a rotation the action and boundary conditions be- come S/H20851Q/H20852=n 2Tr/H20875/H92532Q2 2/H9266/H92510−l n /H20849i/H9253Q−/H9268p/H20850/H20876, /H20849C13 /H20850 Q/H208490/H20850=/H9011z,Q/H20849L/H20850=/H9011zcos/H9278+/H9011xsin/H9278. /H20849C14 /H20850 Thus we have reduced the boundary conditions to the form Eq. /H2084949/H20850. The matrix Bis chosen such that BTB=/H9270x. Hence the matrix Qobeys the symmetry constraint Q=QT. The last step of the sigma-model derivation is the gradient expansion in Eq. /H20849C13 /H20850. 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The reason for this discrepancy is that in the latter case ultraviolet diver-gency was regulated by introducing a ﬁnite correlation length intheydirection only. This resulted in the angle average /H20855 /H9254y2//H92542/H20856 =1 being twice larger than for the isotropic model adopted in thecurrent paper, /H20855 /H9254y2//H92542/H20856=1 /2. 36I. L. Aleiner and K. B. Efetov, Phys. Rev. Lett. 97, 236801 /H208492006 /H20850. 37M. R. Zirnbauer, J. Math. Phys. 37, 4986 /H208491996 /H20850; A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 /H208491997 /H20850. 38F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 /H208492008 /H20850. 39The sigma-model action contains also an imaginary Z2topologi- cal term /H20849Ref. 18/H20850omitted in Eq. /H2084948/H20850. This term is crucial for ensuring topological protection from Anderson localization atsmall /H9268. 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PhysRevB.103.075136.pdf	PHYSICAL REVIEW B 103, 075136 (2021) Magnetic response trends in cuprates and the t-t/primeHubbard model Julian Mußhoff,1,2Amin Kiani,1and Eva Pavarini1,3 1Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany 2Department of Physics, RWTH Aachen, Germany 3JARA High-Performance Computing, RWTH Aachen University, 52062 Aachen, Germany (Received 28 October 2020; revised 9 December 2020; accepted 27 January 2021; published 22 February 2021) We perform a systematic study of static and dynamical magnetic properties of the t-t/primeHubbard model in a parameter regime relevant for high-temperature superconducting cuprates. We adopt as solution method thedynamical mean-ﬁeld theory approximation and its real-space cluster extension. Our results show that large t /prime/tsuppresses incommensurate features and eventually leads to ferromagnetic instabilities for sufﬁciently large hole doping x. We identify isosbestic points which separate parts of the Brillouin zone with different scaling behaviors. Calculations are compared to available nuclear magnetic resonance, nuclear quadrupole resonance,inelastic neutron scattering, and resonant inelastic x-ray scattering experiments. We show that while many trendsare correctly described, e.g., the evolution with x, some aspects of the spin-lattice relaxation rates can apparently only be explained invoking accidental cancellations. In order to capture the material dependence of magneticproperties in full, it may be necessary to add further degrees of freedom. DOI: 10.1103/PhysRevB.103.075136 I. INTRODUCTION High-temperature superconducting cuprates (HTSCs), a representative system of which is shown in Fig. 1, remain puz- zling decades since their discovery [ 1–3]. Spin ﬂuctuations have been early on suggested as possible keys to unravel thenature of superconductivity. Magnetic properties have beenthus investigated via a number of different techniques, rangingfrom elastic and inelastic neutron scattering (INS), inelastic resonant x-ray scattering (RIXS), to magnetic susceptibil- ity measurements, nuclear magnetic resonance (NMR), andnuclear quadrupole resonance (NQR) experiments [ 2–49]. Theoretical investigations have followed. They are based ona bonanza of strategies, from phenomenological approachestoab initio methods based on density-functional theory to techniques for solving representative many-body models [50–64]. In the last few years, important steps forward have been made by reanalyzing the problem with state-of-the-artmethods [ 65–81]. One of the paradigmatic—and most studied—models used for HTSCs is the single-band Hubbard Hamiltonian, assumedto describe the low-energy electronic states stemming fromthe CuO 2planes, shown in Fig. 2. From the electronic structure point of view, the justiﬁcation of such a modelrelies on the fact that the Cu 3 dx 2−y2–like band cross- ing the Fermi level is a generic feature of cuprates [ 62,63]. In addition, for magnetism, the one-band model descrip-tion is grounded on the single spin-ﬂuid scenario, whichemerges from Knight shift and susceptibility measurementsin YBa 2Cu3O7−δ[33,37,51]. Within the single-band Hubbard model, band-structure calculations have shown [ 62] that key aspects of the material dependence are captured by changes inthe hopping-integral range, r∼t /prime/t. In this picture, the actual value of the ratio t/prime/tis controlled by ˜ εs, the energy of theaxial orbital [ 62,63]. Remarkably, many electronic properties in the doped single-band Hubbard model turned out to be verysensitive to the value of t /prime/t, for example the strength of an- tiferromagnetic correlations [ 61]. Recent ground-state studies of the Hubbard model based on the density-matrix renormal-ization group approach indicate that a ﬁnite t /primemight be crucial for ground-state properties, superconductivity [ 72,74,75], as well as for stripe order [ 75,76]. Furthermore, investigations of the t-t/prime-Jmodel, the large- Ulimit of the doped Hubbard model, have identiﬁed spectroscopic signatures of t/primein charge and spin dynamics of one-dimensional antiferromagnets [ 73]. In parallel to these successes, however, some problems came to light. The single spin-ﬂuid picture has been chal-lenged in La 2−xSrxCuO 4and HgBa2CuO 4+δ[43,44], based on recent reanalyses of NMR and NQR experiments. This,in turn, raises questions on the description of magnetic prop-erties based on the single-band Hubbard model. The validityof the single-ﬂuid scenario relies not only on its power ofdescribing the qualitative picture but also on the extent towhich it captures essential differences in the magnetic prop-erties of the various families of cuprates. Despite past andpresent successes, as well as impressive theoretical advances[50–81], a systematic investigation of two-particle magnetic properties in this direction, to the best of our knowledge,is still missing. It is thus time to reanalyze the problem.The purpose of the present work is to ﬁll holes in thiscontest. To this end, we calculate the evolution of static and dynam- ical magnetic response with the number xof holes in the CuO 2 plane, from the underdoped all the way to the less explored highly overdoped regime, progressively increasing t/prime/tand the strength of the Coulomb interaction U.W ee m p l o ya sa method the single-site and the cluster dynamical mean-ﬁeldtheory (DMFT) approach, adopting quantum Monte Carlo 2469-9950/2021/103(7)/075136(18) 075136-1 ©2021 American Physical SocietyMUßHOFF, KIANI, AND PA V ARINI PHYSICAL REVIEW B 103, 075136 (2021) FIG. 1. The crystal structure of the single-layered high- temperature superconducting cuprate HgBa2CuO 4.T h r e eC u O 2 planes (described in more detail in Fig. 2)a r es h o w n . (QMC) impurity solvers. The results obtained show that pro- gressively increasing t/primeandxsuppresses antiferromagnetism, favoring ﬁrst incommensurate instabilities around the Mpoint and eventually ferromagnetic correlations. We ﬁnd that thenature of magnetic correlations changes very strongly enteringthe overdoped regime. We identify isosbestic points whichseparate regions of the Brillouin zone with different scalingbehaviors. We show that the magnetic trends do not changequalitatively with increasing U, provided that one stays away from the U/lessmuchU cregime, where Ucis the critical value for the Mott transition at half ﬁlling; a large Umakes how- ever ferromagnetic instabilities more likely in the overdopedregime. We show that while many aspects of the experimentaldoping dependence, for example uniform susceptibility andKnight shift measurements or the resonance mode in theunderdoped regime, are well captured, others are not—in par-ticular concerning experimental NMR and NQR spin-latticerelaxation rates. For the realistic description of such propertiesit might be necessary to go beyond the simple t-t /primeHubbard model. The paper is organized as follows. In Sec. II, we present the method employed. In Sec. IIIwe present the results, ﬁrst for the static and then the dynamical magnetic properties. Fi-nally we give our conclusions in Sec. IV . Additional technicaldetails can be found in Appendices AandB. FIG. 2. The CuO 2plane in the middle of Fig. 1.C u :S m a l l spheres; O: large spheres. The hopping integrals t,t/prime,t/prime/primeof the single- band Hubbard model are also shown. II. MODEL AND METHOD We describe the low-lying states via the single-band Hub- bard model H=−/summationdisplay ii/primeσti,i/primec† iσci/primeσ+U/summationdisplay ini↑ni↓. (1) Here c† iσ(ciσ) creates (annihilates) an electron at site iwith spinσand ni=c† iσciσ, and Uis the screened Coulomb interaction. The parameter ti,i/primeis the hopping integral be- tween sites iand i/prime. For high-temperature superconducting cuprates (see Fig. 2) the key terms are the nearest-neighbor and next-nearest-neighbor hopping integrals, tand−t/prime.T h i s leads to the band dispersion ε(k)=−2t(coskx+cosky)+ 4t/primecoskxcosky.It has been previously established [ 62] that realistic values are t∼0.4 eV , with t/prime/tranging from t/prime/t∼ 0.17 for La 2−xSrxCuO 4tot/prime/t∼0.33 for YBa 2Cu3O7−δor HgBa2CuO 4+δ. Here we thus study the magnetic properties fort/prime/tin the range 0 /lessorequalslantt/prime/t<0.4, for hole-doping corre- sponding to 0 <x<0.4. This covers the full range from underdoped to the heavily overdoped regime and well be-yond; optimal doping is around x∼0.16 in many cuprate families [ 3]. More controversial is the estimate of the screened Coulomb repulsion. Spin-wave measurements could be takenas evidence of a relatively weak direct (screened) Coulomb in-teraction, U∼3 eV; this is due to the fact that the behavior of the experimental spin-wave dispersion appears not compatiblewith the antiferromagnetic J 1-J2Heisenberg model derived from the Hubbard model in second-order perturbation theory.Its description requires [ 5] either a ferromagnetic (negative) value of J 2or higher-order interactions, for example a ring- type four-spin superexchange term [ 82,83], negligible in the very large Ulimit. A relatively small Uis also supported by constrained random-phase approximation (cRPA) calcu-lations [ 84,85]. On the other hand, a small ferromagnetic J 2 can arise from the standard ferromagnetic intersite Coulomb exchange coupling and /or multiorbital superexchange effects. Indeed, ferromagnetic couplings J2∼−10 meV , sufﬁciently large, have been obtained theoretically using a ﬁrst-principleslinear-response approach [ 54]. Furthermore, cRPA calcula- tions often overestimate screening effects. Much larger values 075136-2MAGNETIC RESPONSE TRENDS IN CUPRATES AND THE … PHYSICAL REVIEW B 103, 075136 (2021) ofU, up to 10 eV , have been estimated via the constrained local-density approximation (cLDA) approach [ 86–89]. This technique, on the other hand, tends to overestimate theCoulomb repulsion, in part due to the fact the more localizedfunctions are typically used as basis, in part because fewerscreening channels are considered [ 89]. Taking all this into consideration, in this paper we present results for several val-ues of Uin the interval between the cRPA and cLDA estimates and discuss the most important effects of increasing Ufor magnetism. We solve the Hamiltonian ( 1) via the dynamical mean- ﬁeld theory (DMFT) and its real-space cluster extension(cDMFT) [ 64,90–92]. In this context, some additional re- marks on the choice of the screened Coulomb parameter arein place. Within paramagnetic dynamical mean-ﬁeld theory,as m a l l Uis hard to conciliate with a relatively large ex- perimental [ 93–96] gap of ∼2 eV . More speciﬁcally, for the hopping parameters used in this work, the critical Ufor the metal-insulator transition is about U c∼4.5 eV . Since a con- sistent picture of the whole phase diagram cannot be fullyrecovered if U<U c, we ﬁrst systematically explore the case U∼7 eV. This value yields at half ﬁlling a gap ∼2e V in paramagnetic DMFT calculations, i.e., a gap in line withphotoemission spectroscopy [ 93], photoinduced absorption spectroscopy [ 94], and optical conductivity measurements [95], as well as with the reported observation of upper Hub- bard bands [ 3,96]. Next we study the effects of varying Uin the range from 3 to 11 eV , all values adopted in the literature.DMFT is exact in the inﬁnite-coordination limit, in which theself-energy is momentum independent. In the case of the t-t /prime Hubbard model it is therefore an approximation. For magnetic properties, nonlocal effects become important in particularapproaching a phase transition [ 97]. Thus, in the most rele- vant cases we compare DMFT results with those of 2- and4-site cellular DMFT (cDMFT) calculations, which have beenshown to capture key effects of spatial ﬂuctuations [ 78,98]. For the quantum impurity solvers we chose two similar but complementary approaches. The ﬁrst is the Hirsch-Fye(HF) quantum Monte Carlo (QMC) method [ 99], in the implementation presented in Ref. [ 100]. The second is the hybridization-expansion continuous-time QMC method (CT-HYB) [ 101], in the implementation of Refs. [ 102,103]. The bottleneck, in both approaches, is the calculation of the localsusceptibility tensor [ 92], which is performed at the end of the self-consistency DMFT loop. This is deﬁned as χ α(τ)=/angbracketleftbig Tcα1(τ1)c† α2(τ2)cα3(τ3)c† α4(τ4)/angbracketrightbig −/angbracketleftbig Tcα1(τ1)c† α2(τ2)/angbracketrightbig/angbracketleftbig Tcα3(τ3)c† α4(τ4)/angbracketrightbig . (2) Here Tis the time order operator, τ=(τ1,τ2,τ3,τ4)a r e the imaginary times; α=(α1,α2,α3,α4) and αj=mjσjij are collective orbital ( mj), spin ( σj), and site ( ij). The cal- culation is performed in different ways, depending on thesolver. In Hirsch-Fye QMC simulations we compute it di-rectly in Matsubara frequency space. This yields χ α(ν), where ν=(νn,−νn−ωm,νn/prime+ωm,−νn/prime),νnandνn/primeare fermionic andωmbosonic Matsubara frequencies, the Fourier transform ofχα(τ). To reduce the computational time we obtain the Fourier transform of the Green’s function matrix Gα,α/prime(τ,τ/prime) by shifting the discontinuity at τ=τ/primeto the border, andapply the semianalytical Filon-trapezoidal approach [ 100]. In the CT-HYB QMC solver we perform the calculations incompact polynomial representations (Legendre and numeri-cal polynomial basis); when necessary we transform to theMatsubara frequency representation. More details on the ap-proach adopted in our general implementation can be found inRefs. [ 102,103]. Next we use the (c)DMFT lattice Green’s function G αiαj(k;iνn), obtained from the noninteracting Hamiltonian and the (c)DMFT self-energy, and compute the bubble contri-bution to the lattice and local susceptibility. They are deﬁnedrespectively via the tensors χ α 0(q;iωm)=−βδnn/primeδσ2σ3δσ1σ41 Nk ×/summationdisplay kGα3α2(k+q;iνn+iωm)Gα1α4(k;iνn), (3) where β=1/Tis the inverse temperature and χα 0(iωm)=1 Nq/summationdisplay qχα 0(q;iωm). (4) The associated bubble longitudinal lattice magnetic suscepti- bility is given by χ0(q;iωm)=(gμB)2 4/summationdisplay α(−1)σ1+σ3δσ1σ2χα 0(q;iωm)δσ3σ4, (5) where σj=±1f o rs p i n s ↑and↓, respectively. From the tensors given in Eq. ( 3) we build square matrices, e.g., χα(iωm)=[χ(iωm)]NN/primewith elements N=α1n,α2n,N/prime= α3n/prime,α4n/prime, so that for the magnetic susceptibility only the terms σ1=σ2=σandσ3=σ4=σ/primeare taken into account [100]. In this case, the (bare) local susceptibility is zero everywhere except for the impurity block, i.e., ij=i1for DMFT and ij={ic}for cluster DMFT calculations. In the last step, we obtain the lattice susceptibility χ(q;iωm) solving the Bethe-Salpeter equation in the local-vertex approximation [90,104] [χ(q;iωm)]−1≈[χ0(q;iωm)]−1−/Gamma1(iωm). (6) The local vertex itself is given by /Gamma1(iωm)=[χ0(iωm)]−1−[χ(iωm)]−1, (7) where χ(iωm) is the local susceptibility tensor obtained from QMC simulations. Finally, the full longitudinal lattice mag-netic susceptibility is obtained as χ(q;iω m)=(gμB)2 4/summationdisplay α(−1)σ1+σ3δσ1σ2χα(q;iωm)δσ3σ4.(8) In the hybridization-expansion continuous-time QMC ap- proach the Bethe-Salpeter equation is solved in the compactpolynomial representation ( l,l /prime) instead of in the Matsub- ara fermionic frequencies ( n,n/prime) representation [ 103]. The Hirsch-Fye approach is better suited in the weak-interactionand large-cluster cDMFT regime, while the continuous-timesolver yields the /Delta1τ=β/L→0 limit and it is best suited 075136-3MUßHOFF, KIANI, AND PA V ARINI PHYSICAL REVIEW B 103, 075136 (2021) 0 0.5 1 1.5 2 t′=0.2 t4m1/ χ⊥t′=0.4 t4m 0 0.5 1 1.5 2 0 500 1000 1500Γ LXM1/ χ\|\| T (K)0 500 1000 1500 T (K) FIG. 3. Static inverse transverse and longitudinal susceptibility χ(q; 0) as a function of temperature, U=7 eV . Left panels: t/prime= 0.2t. Right panels: t/prime=0.4t. Triangles: /Gamma1andMpoints. Gray pen- tagons: Xpoint. Black circles: Local (indicated with L in the plot) susceptibility. Above the critical temperature it shows the Curie- Weiss behavior. Dotted lines: Curie-Weiss ﬁt at high temperature, and associated low-temperature extrapolation [ 97]. Black circles: 4m,w h e r e mis the magnetization per site. Special points: /Gamma1= (0,0,0),X=(π,0,0),M=(π,π, 0). for dynamical response calculations. By combining the two approaches we can study in detail different aspects of theproblem. Finally, data on the real axis are obtained via analyticcontinuation using the maximum-entropy approach. III. RESULTS A. Static susceptibility for x=0 We start by analyzing the lattice spin susceptibility at half ﬁlling ( x=0), in both the paramagnetic and antiferromag- netic phases. This sets the stage for analyzing in the nextsections the ﬁnite- xcase. The principal results are collected in Figs. 3,4and5. In the paramagnetic phase ( T>T N), the DMFT static susceptibility has a Curie-Weiss-like behavior[64] in all considered cases, reﬂecting the mean-ﬁeld approx- imation. This is shown in Fig. 3for representative qvalues. The ﬁgure also shows the transition to the antiferromagneticphase at the critical temperature T N. The calculations yield the (expected) mean-ﬁeld behavior of the transverse and longitu-dinal susceptibility, with χ /bardbl(q; 0) going to zero in the T→0 limit and χ⊥(q; 0) remaining constant below TN; here /bardbland ⊥indicate the direction of the applied magnetic ﬁeld with respect to the ordered magnetic moments. The temperaturedependence is mostly determined by the local vertex /Gamma1(iω n). As we have previously shown, e.g., in Ref. [ 100] for layered vandadates, in the paramagnetic insulating phase, the “bub-ble” term of the static lattice susceptibility at half ﬁlling isapproximately χ 0(q;0 )≈(gμBμeff)2 Ur0/braceleftbigg 1−1 2U/bracketleftbigg Jr0(0)+1 2Jr0(q)/bracketrightbigg +.../bracerightbigg . (9) 0 1 2 3 4χ/χA t′ = 0.2 tT = 2320 K T = 1934 K T = 1450 K T = 1160 K T = 774 K T = 580 K T = 462 K qISqIX 0 1 2 3 4χ/χA t′ = 0.4 tT = 2320 K T = 1934 K T = 1450 K T = 1160 K T = 774 K T = 580 K T = 462 K 0 1 2 3 4 Γ X M S ΓZχ/χA T=774 Kt′ = 0.10 t t′ = 0.15 t t′ = 0.20 t t′ = 0.25 t t′ = 0.30 t t′ = 0.35 t t′ = 0.40 t FIG. 4. Static lattice magnetic susceptibility χ(q; 0) along high- symmetry lines of the Brillouin zone, normalized to the atomic susceptibility χA∼1/4kBT. Special points: M=(π,π, 0),S= (π/2,π/2,0),X=(π,0,0),Z=(0,0,π). Top: t/prime∼0.2t.C e n t e r : t/prime∼0.4t. Bottom: Results at ﬁxed temperature, but for different values of t/prime/t. -0.4 0 0.4 Γ X M S ΓZT=774 Kt′ = 0.10 t t′ = 0.15 t t′ = 0.20 t t′ = 0.25 t t′ = 0.30 t t′ = 0.35 t t′ = 0.40 tJSE (q) (meV) FIG. 5. Effective superexchange couplings for the susceptibil- ities shown in the bottom panel of Fig. 4. Special points: M= (π,π, 0),S=(π/2,π/2,0),X=(π,0,0),Z=(0,0,π). 075136-4MAGNETIC RESPONSE TRENDS IN CUPRATES AND THE … PHYSICAL REVIEW B 103, 075136 (2021) In this equation the effective magnetic moment is deﬁned asμeff=√S(S+1)/3 and the value S∼1/2 is obtained independently via the equal-time correlation function (seeAppendix A). The effective superexchange (SE) couplings can be obtained from the inverse of the susceptibility as fol-lows: J r0(q)=([χ(q;0 ) ]−1−[χ(0)]−1)(gμB)2=JSE(q)/2r2 0, (10) where r0is a renormalization parameter and where the Fourier decomposition reads JSE(q)≈2J1(cosqx+cosqy)+4J2cosqxcosqy+··· . (11) I nt h ev e r ys m a l l t/Ulimit, the superexchange parame- ters take the second-order expression J1∼J(2) 1=4t2/Uand J2∼J(2) 2=4t/prime2/U. Increasing the ratio t/U, higher-order terms, e.g., those arising from the ring exchange coupling,J r, can contribute [ 82,83]. For clarity, let us discuss explic- itly the numbers in some cases. For U=7e Vw eh a v e J(2) 1∼4t2/U∼91 meV . In this situation the 4th order term J1r=24t4/U3∼1.8 meV is negligible in comparison; J2r= 4t4/U3is also small with respect to J(2) 2∼4m e V( t/prime=0.2t). The 4th order terms start to become relevant for U∼Ucand smaller, i.e., in the same regime in which charge ﬂuctuationsand double occupancies start to increase in addition. For U∼ U c∼4.5e V ,w eh a v e J(2) 1∼142 meV and J1r∼6.7m e V , while J(2) 2∼5.7 meV and J2r∼1.1 meV . Experimental es- timates of the J1andJ2parameters have been obtained by ﬁtting inelastic neutron scattering results [ 5,11] and magnetic susceptibility [ 105] or Raman scattering data [ 106–109]. The second-order perturbation theory value J(2) 1∼91 meV ( U= 7 eV) is slightly smaller than typical experimental estimates,while J (2) 1∼142 ( U=4.5 eV) is slightly larger then the value for La 2CuO 4[5]. Finally, we ﬁnd that the scaling factor has values from r0∼0.9t o r0∼1.0 in the complete range of parameters considered here. Including the local DMFT vertex we obtain (see Appendix Afor a simple derivation) the static mean-ﬁeld expression [ 64,100] χ(q;0 )≈(gμB)2μ2 eff kBT+μ2 effJSE(q). (12) This approximate formula well describes our numerical data, shown in Figs. 3and 4. On lowering the temperature, we ﬁnd a divergency at the Mpoint, the signature of an insta- bility toward antiferromagnetism, as expected in this regime[4,7,16]. This can be seen in both Figs. 3and4. The ﬁgures show that we are well inside the Heisenberg-model limit ofthe Hubbard Hamiltonian; in this situation increasing t /prime/t enhances frustration, hence reduces the dynamical mean-ﬁeldcritical temperature T N. The effective degree of frustration f=J2/J1can be extracted from the susceptibility via the expression [ 100] f≈1 2×χ(0,π;0 )−1−χ(π/2,π/2; 0)−1 χ(π/2,π/2; 0)−1−χ(π/2,0; 0)−1. (13) ForU=7 eV we ﬁnd that f∼0.036 for t/prime=0.2tand f∼ 0.157 for t/prime=0.4t; hence in all cases the system remainsin the weak frustration regime, with fclose to the value obtained in second-order perturbation theory, indicating thatcharge ﬂuctuations and higher-order processes such as thering-exchange are not yet playing a crucial role. In additionwe ﬁnd that fis weakly temperature dependent. Remarkably, we ﬁnd that the qualitative behavior of the static susceptibilityand the effective frustration degree change little if we reduce Ufrom 7 eV to 4.7 eV , i.e., approaching the metal-insulator transition. Going back to Fig. 4, at the nodal point, S=(π/2,π/2), located in the middle of the /Gamma1Mline, the susceptibility is close to the atomic value, χ(S;0 )∼(gμ B)2μ2 eff/T∼χA, since the effective superexchange coupling JSE(q=S) is ba- sically zero. Instead, at the antinodal points, X=(π,0) and Y=(0,π), the susceptibility is close to but slightly differs fromχA, since the term proportional to J1in the Fourier series JSE(q) does not contribute; thus, the susceptibility depends in ﬁrst approximation only on t/prime(and not on t) at these qvectors. Such a t/primedependence is shown in detail in the bottom panel of Fig. 4for a representative temperature. In addition, since at the Xpoint the J2term is not frustrated, the susceptibility increases with lowering the temperature. As a consequence,the ratio χ/χ Aexhibits temperature-independent isosbestic points [ 110], e.g., one at qIS=Sand one at a vector qIX close to Xalong the /Gamma1-Xdirection (and symmetry-equivalent qvectors). This can be seen in the upper panels of Fig. 4.A t the isosbestic points the susceptibility is close to the atomiclimitχ A. The exact position of qIXdepends on t/prime, so that the distance between SandqIXincreases with increasing t/prime/t;f o r t/prime=0,qIX=X. This may be seen comparing the top and middle panels of the ﬁgure. Finally, at ( π/2,qx) and ( qy,π/2) the magnetic susceptibility is not inﬂuenced by t/prime, since JSE(q) in ﬁrst approximation depends only on the term proportionaltoJ 1at such a qvector. This in turn gives rise to isosbestic points as a function of t/primeatqx=π/2 and qy=π/2. In the bottom panel of Fig. 4they are hard to see, but they can be seen clearly in Fig. 5, which shows the associated effective superexchange coupling, extracted via Eq. ( 12). B. Dynamical susceptibility for x=0 Let us now switch to the antiferromagnetic phase [ 97], i.e., T<TN. Below the transition the static susceptibility splits into transverse and longitudinal components, as shownin Fig. 3. While the static transverse susceptibility is tem- perature independent, the longitudinal goes to zero in theT→0 limit. In Fig. 6we show the spin-wave dispersion, obtained from the static susceptibility, well below the mag-netic transition temperature, i.e., in the regime in which theorder parameter is close to the saturation value m∼1/2. The ﬁgure shows that dynamical mean-ﬁeld theory calculationsbasically yield the Holstein-Primakoff spin-wave dispersionfor the J 2-J1Heisenberg model in the small-frustration limit. This can be understood as follows. In the insulatingantiferromagnetic phase the DMFT local self-energy is in ﬁrstapproximation close to the Hartree-Fock shift; i.e., it takes theform/Sigma1 σ(ωn)≈−μ+piUm, where mis the magnetization; the shift changes sign ( pi=±) for neighboring sites i.I nt h i s approximation, at sufﬁciently low temperature and at linear 075136-5MUßHOFF, KIANI, AND PA V ARINI PHYSICAL REVIEW B 103, 075136 (2021) t′=0 U=7.0 eV 0 0.1 0.2 0.3ω (eV)t′=0 U=7.0 eV t′=0.2 t U=7.0 eV 0 0.1 0.2 0.3ω (eV)t′=0.2 t U=7.0 eV t′=0.2 t U=4.7 eV 0 0.1 0.2 0.3ω (eV)t′=0.2 t U=4.7 eV t′=0.4 t U=7.0 eV Γ XM Γ 0 0.1 0.2 0.3ω (eV)t′=0.4 t U=7.0 eV FIG. 6. Spin-wave spectra (in eV) for ﬁxed tand for represen- tative values of t/prime. The spectra are obtained from the transverse dynamical susceptibility calculated with the dynamical mean-ﬁeld theory approach (intensity maps) and standard Holstein-Primakoff spin-wave theory calculated using the superexchange parametersfrom second-order perturbation theory (white lines). The high- symmetry points are /Gamma1=(0,0),X=(π,0), and M=(π,π ). order in J1, one can show (see Appendix B) that /bracketleftbigg1 χ0(q;iωm)−1 χ0(iωm)/bracketrightbiggii/prime σ−σ−σσ≈2J1fq(1−δii/prime),(14) where fq=(cosqx+cosqy)/2. Solving the associated Bethe-Salpeter equation we have χ⊥(q;iωm)∼(gμB)2 J1(1−fq) ω2m+4J2 1/parenleftbig 1−f2q/parenrightbig, (15) which yields the conventional spin-wave dispersion. The magnon bandwidth for U∼4.7 eV and t/prime=0.2tis in reason- ably good agreement with the experimental results of Ref. [ 5] for La 2CuO 4. The smaller experimental magnon bandwidth [12] reported in YBa 2Cu3O6.15is in line with the calculation for larger t/prime, taking into account that the interlayer coupling is neglected here. Summarizing, at half ﬁlling, in all ranges of parameters considered, the DMFT static susceptibility is close to the onethat can be obtained from the associated Heisenberg modelin the small- t/Ulimit. In addition, the DMFT spin-wave spectrum is very close to the corresponding expression forconventional spin-wave theory in the weak-frustration regime. Remarkably, this remains true also for Uvalues very close to the metal-insulator transition, as can be seen in Fig. 6, although deviations start to appear. Neutron scattering data athalf ﬁlling are sufﬁciently well described for U∼4t o5e V ; increasing Uup to 7 eV does not alter the qualitative behavior, but merely reduces the spin-wave dispersion in an almostuniform way, only slightly modifying the effective frustrationparameter f. In addition, the effect of high-order couplings and charge ﬂuctuations remains small even for U∼4.7e V . The main effect of reducing Uis that the spin-wave band- width is larger due to the smaller excitation energy for chargeﬂuctuations. The spin-wave energy at Xis as high as at S= (π/2,π/2), as Fig. 6shows, indicating that high-order terms such as the ring-exchange correction are not sufﬁciently largefor explaining experimental ﬁndings alone; a ferromagneticterm, e.g., from Coulomb exchange, would still be requiredfor a realistic description. Instead, a larger t /primeis compatible with a smaller spin-wave dispersion going from La 2CuO 4to YBa 2Cu3O6.15. So far, although not all details are captured, the trends are correctly described. C. Uniform and local susceptibility for x>0 Let us now analyze the results in the doped Mott insulat- ing phase. For x∼0, when the metallic contribution is still negligible (two-pole approximation for the self-energy), theDMFT static lattice magnetic susceptibility is approximately(Appendix A) given by the Curie-Weiss-like form χ(q;0 )≈(gμ Bμeff)2(1−x) T+μ2 eff(1−x)JSE(q). (16) In this regime the dominant spin-spin correlations remain antiferromagnetic, albeit with square local moments reducedto∼μ 2 eff(1−x); this is due to the fact that double occupancies remain much smaller than in the uncorrelated limit, /angbracketleftni↑ni↓/angbracketright= 0.25(1−x)2, which would yield /angbracketleftSi zSi z/angbracketright∼μ2 eff(1−x2)/2 instead. For small but ﬁnite x, the behavior of the uniform suscepti- bility deviates very quickly from Eq. ( 16), however. Still, the temperature dependence remains similar, χ(q;0 )∝1/[T+ Jeff(q)]α, with α∼1f o r xnot too large. We ﬁnd that, while the effective local magnetic moment decreases linearly evenforxas large as 0.4, the bubble term χ 0(0; 0) increases with xdue to the growing relevance of the metallic contribution. The result of the competition between opposite effects is thenonmonotonic behavior of χ(0; 0) shown in Fig. 7.T h el e f t panels of the ﬁgure show that at a given (sufﬁciently low)temperature, χ(0; 0) ﬁrst increases, a maximum is reached atx c∼0.25 for t/prime∼0.2t, and then χ(0; 0) decreases. In the right panel of Fig. 7,w es h o wh o w xcincreases with t/prime/t, going from xc=0.15 for t/prime=0.1ttoxc=0.4f o r t/prime=0.35t; for larger t/prime=0.4tthe magnetic susceptibility diverges at x=0.30. For La 2−xSrxCuO 4, characterized by t/prime∼0.2t, this be- havior is in very good agreement with reported magneticsusceptibility [ 47,48] measurements—including the value of the turning point x c. NMR Knight shift measurements [27–29,34] also show an increase with increasing x; unfor- tunately, the x>xcregime was not systematically explored, 075136-6MAGNETIC RESPONSE TRENDS IN CUPRATES AND THE … PHYSICAL REVIEW B 103, 075136 (2021) 0 1 2 3 4 0 1000 2000χ(0;0) T (K)x=0.00 x=0.10 x=0.15 x=0.20 x=0.25 0 1000 2000 T (K)x=0.25 x=0.30 x=0.40 0 5 10 15 0.1 0.2 0.3 0.4χ(0;0) x 0 5 10 15 0.1 0.2 0.3 0.4χ(0;0) xt′=0.10t t′=0.15t t′=0.20t t′=0.25t t′=0.30t t′=0.35t FIG. 7. Left: Static uniform magnetic susceptibility χ(0;0 ) f o r t/prime=0.2tand several values of x, as a function of the temperature, for temperatures above the pseudogap regime. The susceptibilityincreases with xup to x c∼0.25 (ﬁrst panel); for x>xc,χ(0;0 ) i t decreases (second panel). Right: χ(0; 0) as a function of xfor several values of t/prime, at ﬁxed temperature, T∼387 K. The maximum is at xc (diamonds), whose value increases with t/prime. however. For YBa 2Cu3O6+yan increase of Knight shifts with hole doping up to y∼1 (slightly overdoped regime) has also been reported [ 35–39]. Similar trends appear in HgBa2CuO 4−δ[111]. In Tl 2Ba2CuO 6+y[34,40,41], which is considered to be heavily overdoped, the opposite behavior isobserved, as one would indeed expect in the present descrip-tion decreasing xwhile starting from x>x c. While further systematic experiments would help in clarifying this point, thedescription based on the t-t /primeHubbard model appears therefore to capture the trends in the observations so far. One important conclusion is that the nonmonotonic xde- pendence is speciﬁc of the /Gamma1point and the qvectors around it. The local susceptibility, the average over the qvectors, merely decreases with xgoing from x=0t ox=0.4. This is shown in Fig. 8. The ﬁgure compares, in addition, single-site calcu- lations ( χ1SC) with 2- and 4-site cluster results ( χ2SC,χ4SC) and shows that differences are minor. At a ﬁxed temperature,for a given t /prime, we ﬁnd χ1SC>χ 2SC>χ 4SCifxis sufﬁciently small, while the opposite is true ( χ1SC<χ 2SC<χ 4SC)f o r large x. The same reversal is found for xﬁxed and t/primein- creasing. The effect remains however very small, as the ﬁgureshows; picture and trend remain unchanged. The static localsusceptibility, in the temperature regime analyzed, scales to avery good approximation as χ(0)∼(gμ B)2μ2 eff(1−x) T+T0(x), (17) where T0(x) increases with increasing xand decreases with increasing t/prime. More speciﬁcally, T0(x)∼16 K for x=0 and t/prime=0.2t; keeping t/prime/tﬁxed and increasing x,T0(x)∼200 K forx=0.1 and T0(x)∼920 K for x=0.4. For t/prime=0.4t the corresponding values are T0(x)∼6Kf o r x=0,T0(x)∼ 190 K for x=0.1, and T0(x)∼630 K for x=0.4. We em- phasize once more that the scaling with xis very different for local and uniform susceptibility. Extracting the analogueofT 0(x)a t q=0 with a similar procedure would yield a characteristic scale ﬁrst decreasing ( x<xc) and then ( x>xc) increasing with x.F o r x<0.2 such a scaling has been indeed identiﬁed early on from analysis of uniform susceptibilitymeasurements [ 49]. 0 2 4 6 0 1000 2000t'=0.2 tχ(0) T (K) 0 2 4 61SC 2SC 4SC t'=0.4 tχ(0)x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 FIG. 8. Static local magnetic susceptibility for t/prime=0.2t(top) andt/prime=0.4t(bottom) as a function of temperature and for several x values from single-site DMFT, two- and four-site CDMFT calcula- tions (labeled as 1SC, 2SC, and 4SC in the caption). D. Static x>0 susceptibility: q dependence Let us now analyze in detail the entire qdependence. Figure 9collects the most important results. The top panels show the case t/prime=0.2t. In the ﬁgure the value of xincreases from x=0.1t ox=0.4 going from left to right. The ﬁrst two panels on the left show underdoped and slightly underdopedregimes, x∼0.10 and x∼0.15. For x=0.10 the expected dominant instability is antiferromagnetic, as in the half-ﬁllingorx=0 limit, and the susceptibility is still not far from the Curie-Weiss-like form, although with reduced local mo-ments. Around x=0.15 the picture changes, however. For T<460 K peaks at incommensurate vectors appear. This can be seen in Fig. 9, second top panels from the left. There are two types of potential instabilities, the one at q XM, a vector close to Malong the XMdirection, and the one at q/Gamma1M, a vector close to Malong the /Gamma1Mhigh-symmetry line. The associ- ated critical temperatures TC(q/Gamma1M) and TC(qXM), obtained via linear extrapolation from the inverse susceptibility, are bothof the order of ∼T N/10, where TN=TC(M)f o r x=0. It is important to point out that the mean-ﬁeld critical temperaturesjust discussed are excellent estimates of the actual strength ofthe effective magnetic coupling J eff(q), as Eq. ( 16), Fig. 5, and the surrounding discussion illustrate. The trends thus suggest that in the ground state static incommensurate structures could be realized in this regime.Further increasing xprogressively suppresses the magnetic response around the Mpoint, giving rise to a depression in M. This can be seen moving from left to right in Fig. 9,t o p panels. It can be noticed that the reduction of the magneticresponse is not uniform in qandx.A tt h e /Gamma1point, as we have already discussed, for t /prime=0.2tthe susceptibility at ﬁrst 075136-7MUßHOFF, KIANI, AND PA V ARINI PHYSICAL REVIEW B 103, 075136 (2021) Γ 2X2M 0 6 12 χ(q;0)x=0.10χ(q;0)x=0.15 Γ 2X2Mx=0.20 Γ 2X2Mx=0.25 Γ 2X2Mx=0.30 Γ 2X2Mx=0.40 Γ 2X2MΓ X2 XY2Y Γ X2 XY2Y Γ X2 XY2Y Γ X2 XY2Y Γ X2 XY2Y Γ X2 XY2Y 0 6 12 Γ X M ΓZχ(q;0)T =2320 K T =1450 K T =1160 K T =774 K T =580 K T =462 K T =387 K T =290 K T =232 K Γ X M ΓZΓ X M ΓZqXM qΓM Γ X M ΓZΓ X M ΓZΓ X M ΓZ Γ 2X2M 0 15 30χ(q;0)x=0.10χ(q;0)x=0.15 Γ 2X2Mx=0.20 Γ 2X2Mx=0.25 Γ 2X2Mx=0.30 Γ 2X2Mx=0.40 Γ 2X2MΓ X2 XY2Y Γ X2 XY2Y Γ X2 XY2Y Γ X2 XY2Y Γ X2 XY2Y Γ X2 XY2Y 0 6 12 Γ X M ΓZχ(q;0) Γ X M ΓZqXMqΓM Γ X M ΓZΓ X M ΓZΓ X M ΓZΓ X M ΓZ FIG. 9. Static lattice magnetic susceptibility for t/prime=0.2t(top panels) and t/prime=0.4t(bottom panels) for representative temperatures and along high-symmetry lines. From left to right xincreases from 0.1 to 0.4. The three-dimensional plots and the contour plots on top of each ﬁgure show χ(q;0 )f o r T∼290 K. For t/prime=0.4tandx=0.3o r x=0.4 the temperature chosen is right above the ferromagnetic transition. The special points are /Gamma1=(0,0,0),M=(π,π, 0), 2 M=(2π,2π,0),X=(π,0,0), 2 X=(2π,0,0),Y=(0,π,0), 2 Y=(0,2π,0), Z=(0,0,π). increases and then drops again (Fig. 7). For what concerns the incommensurate features, the extrapolated critical tempera-tures T C(q/Gamma1M) and TC(qXM) strongly decrease; at x=0.2 their value is already very small, making it less likely that staticincommensurate spin structures can be realized for x/greaterorequalslant0.2. Two observations are in place. First, even in the U=0 limit the susceptibility develops peaks at incommensuratevectors around M, as was often pointed out; for complete- ness, this is shown in Appendix A. Such peaks qualitatively evolve with xin a way similar to that in the ﬁnite- Ucase, although they do differ in many aspects, as may be seencomparing Fig. 9to Fig. 18in Appendix A. Second, the phe- nomenological nearly antiferromagnetic Fermi-liquid theorysusceptibility [ 52], with a maximum at the antiferromagnetic vector ( π,π ), approximates the results in Fig. 9o n l yu pt o max x∼0.1. Approaching optimal doping ( x∼0.15) and going well beyond, the qdependence qualitatively changes. Still, the change is only abrupt entering the overdoped regime(x∼0.2 and larger in the ﬁgure). Let us now analyze the effect of increasing t /primefrom t/prime=0.2t tot/prime=0.4t. The main results as a function of xare shown in Fig. 9, bottom panels, and show that changes are large. Forx∼0.10 the static lattice susceptibility has a maximum at theMpoint, as for t/prime=0.2t. The response at Mis weaker, however, in line with the fact that we are still in the Curie-Weiss-like limit and frustration is increasing. For x∼0.15 again we ﬁnd a change in behavior. At low temperature even-tually incommensurate peaks develop around M; this time, however, the area around Mis strongly asymmetric for x= 0.15, as can be seen in the two-dimensional contours in the inset. Furthermore, the q /Gamma1Mfeature clearly dominates in the- low temperature regime, but the associated mean-ﬁeld criticaltemperature, again obtained as linear extrapolation, is as lowas∼T N/30; again, TN=TC(M)f o r x=0. Increasing x,t h e asymmetry around the Mpoint increases, qXMmoves toward Xandq/Gamma1Mtoward /Gamma1, while at the same time the peak at q/Gamma1M grows taller; this eventually leads to dominant ferromagnetic instabilities for sufﬁciently large x. The increase in relevance of ferromagnetic correlations with xis also present for U=0 (Appendix A), but the effect is less strong. The trends obtained so far are qualitatively in line with the picture emerging from experimental facts. Summariz-ing, antiferromagnetic ﬂuctuations at ( π,π ) dominate up to x∼0.1. They are then suppressed increasing x, eventually 075136-8MAGNETIC RESPONSE TRENDS IN CUPRATES AND THE … PHYSICAL REVIEW B 103, 075136 (2021) 0 1 0 1000 20002m/n T (K)1/χ(0;0) 0 1 -2 0 2A(ω) ω (eV) FIG. 10. Left: Normalized magnetization (symbols), deﬁned as 2m/n=(n↑−n↓)/(n↑+n↓), and inverse static uniform susceptibil- ity (symbols and lines) for t/prime=0.4tforx∼0.30 (circles) and x∼ 0.40 (squares). The fully polarized state with m=0.35 is reached in the T→0 limit. The static susceptibility diverges T∼430 K for x∼0.30 and T∼610 K for x∼0.40. Right: Spin-resolved spectral function for x=0.3 at 230 K. Black line: Noninteracting density of states. Full lighter line: Majority spin. Dashed line: Minority spin. becoming unimportant in the overdoped regime. For x∼0.15 incommensurate features the become dominant, to be quicklywashed out further increasing xor suppressed by increasing t /prime. The static mean-ﬁeld critical temperature for incommensurateinstability at q /Gamma1Mis about TN/10 for t/prime=0.2tandTN/30 for t/prime=0.4t. Incommensurate stripes and spin waves are best known in the underdoped regime for the La 2−xBxCuO 4family, but have also been found in other cuprate families. Fort/prime∼0.4tandx/greaterorsimilar0.30 we ﬁnd a ferromagnetic phase (Fig. 10). A ferromagnetic phase was predicted by Kopp et al. [69] in overdoped cuprates via quantum critical scaling theory. Experimentally, in La 2−xSrxCuO 4a potential low-temperature ferromagnetic phase [ 18](T<2 K) was reported at x=0.33. Ferromagnetic ﬂuctuations in the overdoped regime were re-cently found in (Bi,Pi) 2Sr2CuO 6+δ[79]. For the t-t/primeHubbard model, U→∞ Nagaoka ferromagnetism was obtained in Ref. [ 77]f o r t/prime/t=0.1, and it was shown to be suppressed for negative t/prime/t=−0.1. In this picture, the majority spin shows small mass renormalization, while the minority spin ishighly correlated. Our results are Nagaoka-like, as the spec-tral functions in Fig. 10show. Furthermore we ﬁnd that the ferromagnetic state is favored by large xandt /prime/t, everything else staying the same. If t/prime/tis too small, the extrapolated Curie temperature becomes negative; i.e., even in mean-ﬁeldtheory no actual transition is expected. Ferromagnetism forﬁnite Uwas also obtained very recently in Ref. [ 80] via a dy- namical cluster approximation study; its origin was explainedby mapping, via bonding and antibonding orbitals for 4-siteplaquettes, the one-band Hubbard model into an equivalenttwo-orbital Hubbard model [ 81] with effective Coulomb pa- rameters ˜U=˜U /prime=˜J=U/2. Figure 8shows that for the static local susceptibility there are no qualitative changes in cDMFT calculations up to four-site clusters. Analyzing cluster effects in detail as a function ofqwe ﬁnd that nonlocal effects are more sizable approaching a phase transition and around the associated critical qvector, reducing the transition temperatures [ 97]. Thus for t /prime=0.2t nonlocal correlations appear most important for x/lessorequalslant0.1 and close to the Mpoint, where they decrease the value of the susceptibility; for t/prime=0.4tthey are instead stronger for large xaround the /Gamma1point. At incommensurate vectors the effects Γ X2 XY2Yt′ = 0.40 t Γ X2 X t′ = 0.35 t Γ X2 X t′ = 0.30 t Γ X2 X t′ = 0.25 t Γ X2 X t′ = 0.20 t 0 10 20 Γ X M ΓZχ(q;0) qIS qIXqΓMqXM Γ X 2XY2Yt′ = 0.40 t Γ X2 X t′ = 0.35 t Γ X2 X t′ = 0.30 t Γ X 2X t′ = 0.25 t Γ X2 X t′ = 0.20 t Γ X 2XY2Yt′ = 0.15 t Γ X2 X t′ = 0.10 t 0 10 20 Γ X M ΓZχ(q;0) t′ = 0.10 tt′ = 0.15 tt′ = 0.20 tt′ = 0.25 tt′ = 0.30 tt′ = 0.35 tt′ = 0.40 t FIG. 11. Static lattice magnetic susceptibility for several t/prime/t. Calculations are performed for U=7e Va n d T∼230 K. Top: x∼0.15. Bottom: x∼0.20. appear instead weaker. Overall, they do not affect in a quali- tative way the trends so far. E. Dependence of the static x>0 susceptibility at ﬁnite q on t/prime/tandU In Fig. 11we analyze the effects of systematically increas- ingt/primefor representative xvalues, below and above optimal doping. A similar behavior is found for all xvalues. The ﬁgure shows that the isosbestic point at qIS, which we already discussed for x=0, moves toward Mwith increasing x.T h e two isosbestic points which, for x=0, were on the left and right of X(see Fig. 5) now collapse toward X, where a valley is formed (see label qIXin the ﬁgure). The susceptibility changes strongly with increasing t/prime, but in an opposite way for qvec- tors between qIXandqIS(it decreases) and for vectors outside this region (it increases). This is because of the sum rule [ 110] yielding the local susceptibility χ(0).In addition, while the incommensurate feature at qXMonly slightly moves to the left when t/primeincreases, progressively losing in strength, the one at q/Gamma1Mmoves rapidly away from M. Eventually it crosses qIS 075136-9MUßHOFF, KIANI, AND PA V ARINI PHYSICAL REVIEW B 103, 075136 (2021) Γ X2 XY2YU = 3 eV Γ X 2XU = 5 eV Γ X 2XU = 6 eV Γ X2 XU = 7 eV Γ X 2XU = 8 eV Γ X2 XY2YU = 10 eV 0 2 4 6 Γ X M ΓZχ(q;0)U=3 eV U=5 eVU=6 eVU=7 eV U=8 eV U=10 eV Γ X2 XY2YU = 3 eV Γ X2 XU = 5 eV Γ X2 XU = 7 eV Γ X2 XU = 9 eV Γ X2 XU = 11 eV 0 4 8 12 Γ X M ΓZχ(q;0)U= 3 eV U= 5 eV U= 7 eVU= 9 eV U=11 eV FIG. 12. DMFT static lattice magnetic susceptibility χ(q;0 )f o r x∼0.25,T∼460 K, and different values of the Coulomb interac- tion. Top: t/prime=0.2t. Bottom: t/prime=0.4t. and approaches /Gamma1, this time gaining height, and dominating for large xvalues. This trend is perhaps more clear if we observe the evolution of the two-dimensional maps on the topof the panels. With increasing t /prime/tthe four incommensurate maxima in χ(q; 0) around Mturn into a ring; eventually the ring changes into incommensurate maxima around the cornersof the Brillouin zone. The switch occurs for larger xift /primeis smaller, or, seen the other way around, for smaller t/primeifxis larger. Last, we analyze the effects of varying the value of the screened Coulomb parameter Uf r o m3t o1 1e V .T h em a i n conclusions are collected in Fig. 12and in Fig. 13.I nF i g . 12 we display results for x∼0.25, at which value two types of incommensurate features are present, and a possible instabil-ity toward ferromagnetism appears. Around Mthe response is slightly suppressed with increasing U, as one would ex- pect when superexchange interactions between local momentsdominate, but otherwise the behavior does not change quali-tatively. Indeed, the larger differences are observed for U= 3 eV , which yields a metallic solution at half ﬁlling. Further0 6 12 χ(q;0)t′=0.2 tU=7 eV U=4.7 eV x=0.10 x=0.15 x=0.20 x=0.25 x=0.40 0 6 12 Γ X M Γχ(q;0)t′=0.4 t Γ X M Γx=0.10 x=0.15 x=0.20 x=0.25 x=0.40 FIG. 13. DMFT static lattice magnetic susceptibility χ(q;0 ) a t T∼387 K for several xand two representative Uvalues. Top: t/prime= 0.2t. Bottom: t/prime=0.4t. reducing UtoU=1 eV yields a result which is closer to an enhanced noninteracting response, shown in Appendix A.T h e exact position of qXMis also moving with U, but the shift is small. The most remarkable effect of increasing Uis that the response between qISand/Gamma1increases very fast—much faster than expected from the reduction of the antiferromagneticsuperexchange coupling, which can instead be seen for x=0, Fig. 4. Furthermore q /Gamma1Mrotates by 45 degrees and progres- sively moves toward /Gamma1. This can be seen most clearly from the two-dimensional contour plots in the upper panels of theﬁgure. Within the present description, the fact that ferromag-netism was found in La 2−xSrxCuO 4forx∼0.33, although at very low temperatures [ 18], would suggest that the effective Ucannot be too small. Increasing the value of t/primetot/prime=0.4t the dominant features are always along /Gamma1Mand quickly move to/Gamma1with increasing U, favoring a ferromagnetic instability at sufﬁciently low temperature. Another important point is that even as a function of U we observe isosbestic points along the X-M-/Gamma1direction. This may be seen clearly in the top panel of Fig. 12. They reﬂect the fact that the local susceptibility depends weakly on Utill local moments persist—and this still happens well below Uc;s m a l l deviations start to appear at U∼3 eV . The ﬁgure shows that, as a consequence, the effect of Uchanges across the isosbestic points. The qXMpeak is more prominent the smaller Uis, while the opposite happens around /Gamma1. The ﬁgure thus conﬁrms that the actual nature of the magnetic response is strongly q dependent. While around Mit is dominated by antiferromag- netic superexchange between local moments even for largexand relatively small U, around /Gamma1it is metallic-like. The evolution with xis emphasized in Fig. 13, where we compare results for U=7 eV and U∼U c. Here one may notice in addition that, as a function of x, the isosbestic points are only approximate and tend to disappear for large x. F. NMR relaxation rate 1 /T1 Next we calculate the NMR /NQR spin-lattice relaxation rate. It is deﬁned via the relation [ 112] 1 T1T=γ2 21 Nq/summationdisplay qF⊥(q)F⊥(−q) lim ω→0/parenleftbiggχ/prime/prime ⊥(q;ω) ω/parenrightbigg ,(18) 075136-10MAGNETIC RESPONSE TRENDS IN CUPRATES AND THE … PHYSICAL REVIEW B 103, 075136 (2021) 0246T [χ″(ω) / ω]ω → 0x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 0246T [χ″(ω) / ω]ω → 0x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 0246T [χ″(ω) / ω]ω → 0x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 0246 0 1000 2000T [χ″(ω) / ω]ω → 0 T (K)x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 0246 0 1000 2000T [χ″(ω) / ω]ω → 0 T (K)x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 0246 0 1000 2000T [χ″(ω) / ω]ω → 0 T (K)x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 FIG. 14. Local ( η=0) contribution to the relaxation rate, calculated via DMFT (ﬁlled circles), 2S-cDMFT (rhombs), and 4S-cDMFT (triangles) for t /prime=0.2t(top), t/prime=0.4t(bottom), and several values of x. where γis the nuclear gyromagnetic ratio and Fα(q) the form factor for a magnetic ﬁeld in direction α.A ta63Cu site the form factor is given by Fα(q)=Aα+4Bfq; here fq= (cosqx+cosqy)/2,Bis the (transferred) contact hyperﬁne ﬁeld, and Aαthe sum of the direct hyperﬁne interaction terms [51]. Experimentally, the relaxation rate anisotropy R= T1c/T1abwas recently [ 44] found to be temperature indepen- dent, ranging from 1 to 3.4. This suggests that the temperaturedependence of the relaxation rate should be captured alreadywell by the local contribution. In DMFT and cDMFT calcula-tions, this term can be obtained directly via the self-consistentquantum impurity problem, i.e., without solving the Bethe-Salpeter equation in addition. The result is displayed in Fig. 14 for two representative t /primevalues and several xvalues. The ﬁgure shows that the local relaxation rate ﬁrst increases withthe temperature, reaches a maximum, and slowly decreases;forx=0.4f o r t /prime=0.2tthe curve looks basically ﬂat at high temperature; for t/prime=0 qualitatively similar results were obtained in Ref. [ 113]. The maximum of the relaxation rate corresponds to T∼T0(x) and it is thus more pronounced for small xand larger t/prime(lower panel, x=0.4), i.e., when T0(x) is smaller, as we have previously discussed. The ﬁg- ure also shows, however, that for small x, the value of themaximum and the temperature at which it is reached are not well captured by single-site DMFT. The maximum decreaseswith increasing cluster size; this happens because for smallxthe integrand is large at the Mpoint, i.e., where nonlocal correlations are most important. Figure 14reasonably well reproduces some of the trends seen in experiments; for example it captures the decrease inrelaxation rate in the normal state with increasing xobserved in La 2−xSrxCuO 4and YBa 2Cu3O6+y[27,28,33,34,36]. There are remarkable differences, however. In La 2−xSrxCuO 4, NQR experiments found a basically x-independent plateau at about 700 K [ 42]. In YBa 2Cu4O8a ﬂattening of the relaxation rate occurs at about 400 K [ 45]. In Fig. 14, while all curves become close at very high temperatures, no such collapse toone universal value around 700 K for t /prime=0.2tis observed, or at lower temperature for larger t/prime/t, and a real ﬂattening is only seen for x=0.4 and t/prime=0.2. This can be understood as follows. Our results show that the local relaxation rate andsusceptibility satisfy approximately a local Korringa law K=T (0) 1T[χ(0)]2≈(0.43)2, (19) where 1 T(0) 1T=lim ω→0/parenleftbiggχ/prime/prime ⊥(ω) ω/parenrightbigg . (20) Indeed, from Eq. ( 17), one may see that (1−x) χ(0)T0(x)≈1+T/T0(x) (gμBμeff)2. (21) This linear behavior is shown in the bottom panel of Fig. 15. We ﬁnd that the relaxation rate, instead, scales approximatelyas follows: (1−x)/radicalBig T(0) 1T T0(x)≈√ K1+T/T0(x) (gμBμeff)2. (22) This is shown in the top panel of Fig. 15. Hence, the ratio of Eq. ( 22) and Eq. ( 21) yields, squared, the local Korringa ratio. ForT/greatermuchT0(x) the tails of 1 /T1depend on xvia the effective moment, which decreases with xincreasing. The ﬂattening in Fig.14fort/prime=0.4tandx=0.2 is thus an effect of T0(x) be- ing large. Nonlocal effects, on the other hand, increase T0(x), as may be seen in Fig. 14. This makes the curves look more ﬂat for a given x; it does not, however, cancel out the xdepen- dence of the tails. Furthermore, experimentally [ 34,36,44], the 63Cu relaxation rates are visibly larger in La 2−xSrxCuO 4than in Tl 2Ba2CuO 6or YBa 2Cu3O6+y. A trend in this direction does not emerge in Fig. 14simply increasing t/prime/tfor a given x, however. In the picture so far, it can only be ascribed to the differences in hole doping, with Tl 2Ba2CuO 6being in the overdoped regime. In Fig. 16we summarize the effects of the form factor F⊥(q). To this end we ﬁrst split the63Cu relaxation rate into three components, which we label as 1 /T(η) 1, withη=0,1,2. They are obtained as 1 Tη 1T=1 Nq/summationdisplay qwη(q) lim ω→0/parenleftbiggχ/prime/prime ⊥(q;ω) ω/parenrightbigg , (23) withwη(q)=(−2fq)η.T h eη=0 component gives the local contribution to the relaxation rate shown in Fig. 14.T h el e f t 075136-11MUßHOFF, KIANI, AND PA V ARINI PHYSICAL REVIEW B 103, 075136 (2021) 01224 0 5(1-x)/ ξT0(x) T/ T0(x)x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 01224 0 5(1-x)/ ξT0(x) T/ T0(x)x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 01224 0 5(1-x)/ ξT0(x) T/ T0(x) 03060 0 5(1-x)/ χT0(x) T/T0(x)x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 03060 0 5(1-x)/ χT0(x) T/T0(x)03060 0 5(1-x)/ χT0(x) T/T0(x)x=0.10 03060 0 5(1-x)/ χT0(x) T/T0(x)x=0.10 x=0.15 x=0.20 x=0.25 x=0.30 x=0.40 FIG. 15. Top: Normalized inverse square root of the spin-lattice relaxation rate. Here ξ=[1/T(0) 1T]1/2. Bottom: Normalized inverse Knight shift. The notation is the same as in Fig. 14. panels of Fig. 16show that the η=0 and η=2 components of the relaxation rate yield a similar contribution, since the re-sponse function is weak at the Xpoint; the η=1 term remains small in comparison, and tends to become negative increasing Uand t /prime, decreasing the anisotropy and 1 /Tab 1. Including theη=1,2 terms has stronger effects, however, through the hyperﬁne ﬁelds. While AcandAabare typically considered weakly material dependent, the transferred ﬁeld B(extracted by ﬁtting the experimental Knight shifts) was found to bestrongly affected by the environment and doping [ 31,34,56]. Theoretically, this is supported by electronic-structurecalculations showing that also Bdepends on the energy ˜ ε sof the axial orbital [ 53]; for single-layered materials, in ﬁrst ap- proximation, Bthus increases for the same reasons for which t/primeincreases [ 62]. In phenomenological theories, to explain the fact that in YBa 2Cu3O7and La 2−xSrxCuO 4theKc63Cu Knight shift is temperature independent below Tc, an acciden- tal cancellation 4 B+Ac∼0 is typically assumed. Based on these premises, values of Btwo or even three times larger were estimated for Tl 2Ba2CuO 6+y[31,34], with the maximum value for the sample with no superconducting phase. Figure 16 shows (top right panel) that for ﬁxed B, the in-plane relax- ation rate is larger for smaller t/prime/t, while the opposite can happen if the ﬁeld is along c(bottom right panel). Increas- ingBof a factor two, everything else staying the same, can-2 0 2 4 U=4.7 eV t′ = 0.2 t t′ = 0.4 t1/T1(η) -2 0 2 4 U=4.7 eV t′ = 0.2 t t′ = 0.4 t1/T1(η) 0 2 4 1/T1ab 1/T1c 0 2 4 1/T1ab 1/T1c 0 2 4 1/T1ab 1/T1c 0 2 4 1/T1ab 1/T1c -2 0 2 4 0 1 2 U=7 eV1/T1(η) η-2 0 2 4 0 1 2 U=7 eV1/T1(η) η0 2 4 0 0.5 1 4B=-Ac 2B/\|A\|0 2 4 0 0.5 1 4B=-Ac 2B/\|A\|0 2 4 0 0.5 1 4B=-Ac 2B/\|A\|0 2 4 0 0.5 1 4B=-Ac 2B/\|A\| FIG. 16. Left: Contributions 1 /T(η) 1to the relaxation rate for x=0.15 (close to optimal doping), t/prime=0.2tandt/prime=0.4tat 580 K, in the temperature regime T∼aT0(x), with a∈(1,2). Right: 1 /Tc 1 and 1/Ta 1as a function of y=2B/\|Ac\|. They are deﬁned in units ofAcas 1/Tc 1=r2 A/T(0) 1+y2/T(2) 1−2rAy/T(1) 1and 1/Tab 1=1 2(1+ r2 A)/T(0) 1+y2/T(2) 1+y(1−rA)/T(1) 1,w h e r e rA=Aab/\|Ac\|andAc∼ −5Aab. The vertical line corresponds to 4 B=−Ac. increase sizably the relaxation rate; this is because the only term that can reduce it, the linear η=1 contribution, is small in comparison to the quadratic η=2 term. Furthermore, for sufﬁciently large BandUone could, in principle, even reverse the sign of the anisotropy. On the other hand, we ﬁnd thatincreasing x, everything else staying the same, reduces 1 /T (0) 1 and 1/T(2) 1, reducing the average relaxation rate, and makes 1/T(1) 1more negative, reducing the anisotropy. In conclusion, in the picture emerging from these results, if we assume thatthe experimental Bvalues are approximately correct, a smaller relaxation rate in Tl 2Ba2CuO 6+yshould be mostly ascribed to the fact that this system is well inside the overdoped regime. More complicated is to conciliate the theoretical results with the x-independent plateau at 700 K in La 2−xSrxCuO 4. In this system, Bis often considered almost doping inde- pendent, in order to explain the fact that the perpendicularKnight shift does not drop below T c, and does not change much in absolute value. A certain amount of xdependence is still compatible with NMR experiments, however [ 56]. An increase of Bcould in principle compensate the decrease associated with the reduction in effective local moment. Ithas to be noticed, however, that a universal plateau wouldrequire a (second) accidental cancellation and a sufﬁcientlylarge T 0(x), a delicate equilibrium of factors. If this is the case, it should be possible to observe that the universality is brokenby measuring spin-lattice relaxation rates with magnetic ﬁeldin different directions. G. Bosonic spin excitations Bosonic spin excitations in cuprates have been intensively studied, and have evidenced features common to severalcuprates [ 2,3]. Among those are resonance peaks [ 7,14,15] around Min the range 50–70 meV as well as incommensu- rate low-energy excitations [ 6,8,10]. With time, evidence of a seemingly “universal” X-shaped behavior of spin excitations 075136-12MAGNETIC RESPONSE TRENDS IN CUPRATES AND THE … PHYSICAL REVIEW B 103, 075136 (2021) x=0.10 0 0.5 1 1.5 Γ XM Γ0 0.5 1 t′=0.2 tx=0.15 t′=0.4 t XM Γx=0.20 XM Γx=0.25 XM Γ FIG. 17. Dynamical susceptibility (intensity maps) for t/prime=0.2t(top) and t/prime=0.4t(bottom) and increasing xin the paramagnetic phase, U=7 eV . The special points are /Gamma1=(0,0),X=(π,0), and M=(π,π ). The spectra do not change much, further increasing xto 0.4. in underdoped cuprates accumulated, with perhaps the excep- tion of HgBa2CuO 4+δ[46]. Theoretically, the xdependence of spin excitations has been studied with various techniques and models[22,65,114,115]. Recently, it has been shown via the dual bo- son approach [ 68] that in the underdoped region the dominant spin excitations remain close to the Mpoint. Our results are in line with this conclusion, as one can see in Fig. 17, left panels. For small xthe low-energy spectra have a form similar to the one we obtained for x=0( s e eF i g . 6) with a maximum at Mwhich persists till optimal doping. As in the x=0 case, we ﬁnd that the spectra are very similar decreasing UtoU c, leaving a slightly larger dispersion aside. Figure 17also shows that, at sufﬁciently low frequency, the calculated modes reﬂectthe behavior of the static susceptibility and the q-resolved re- laxation rate. Finally, the spectrum is qualitatively very similarfort /prime=0.2tandt/prime=0.4t, although the intensity at the M point decreases in absolute value increasing t/prime. The energy of the maximum at Mis compatible with the resonance modes. Increasing xbeyond the underdoped regime the situation changes. Although a shadow of the original mode stays, al-ready at optimal doping the maximum weight starts to moveaway from the Mpoint. One can then identify incommensurate features at q XMandq/Gamma1M, as for the static susceptibility. For x=0.25 the weight is already mostly at /Gamma1. Qualitatively the trend remains the same for t/prime=0.2tandt/prime=0.4t,b u t when t/primeis larger, the ﬁgure shows that the intensity moves faster toward the /Gamma1point. This indicates that the bosonic spin excitations, within the present modeling, are not, at the core,really universal, although the shade of the small xspectra does persist even for large x; below x=0.15 the spectra look very similar, however. IV . CONCLUSIONS We have studied the static and dynamical magnetic properties of the t-t/primeHubbard model in a parameter regime rel- evant for high-temperature superconducting cuprates. Whenpossible, we complement numerical results with approxi-mate analytic expressions. Our calculations conﬁrm previousconclusions [ 61,65–68,70–77] showing that the electronic properties are very sensitive to the value of t /prime/t. In addition, we ﬁnd a sharp change in behavior entering the overdopedregime. At half ﬁlling ( x=0), the calculated spin-wave spectra are close to those obtained from standard spin-wave theory, bothin the paramagnetic and magnetic phase. This remains trueeven for Uapproaching the insulator-to-metal transition; in this regime, the spin-wave spectrum is enhanced, however,due to the smaller charge ﬂuctuation energy. The trends with t /prime/tare approximately in line with experimental observations so far. Forx/negationslash=0, the nonmonotonic evolution of the uniform susceptibility, reported for thermodynamics experiments inLa 2−xSrxCuO 4, is fully captured by the model. The turning point tends to move to larger xby increasing t/prime. The case of overdoped Tl 2Ba2CuO 6+yappears a further conﬁrmation of the trend. Also captured is the tendency toward the forma-tion of incommensurate structures for small xand in systems characterized by a relatively small t /prime.F o rv e r yl a r g e xandt/prime ferromagnetic instabilities are favored instead. The results obtained show that the nature of the magnetic response is strongly qdependent. Isosbestic points mark re- gions of the Brillouin zone exhibiting different scaling withthe parameters U,t /prime/t,T. Thus, scaling laws obtained, e.g., from the uniform susceptibility and Knight shifts, should notbe automatically extended to experiments probing other partsof the Brillouin zone, or to local responses. Ferromagneticinstabilities are suppressed for sufﬁciently small U. The material dependence of the experimental relaxation rates appears more problematic to describe. While qualita-tively the temperature and xdependence are in line with experiments, some remarkable observations are not quantita-tively reproduced. In particular, the universal ( x-independent) high-temperature plateau in La 2−xSrxCuO 4would require ac- cidental cancellations. Instead, the fact that the relaxation rateis smaller in Tl 2Ba2CuO 6+ythan La 2−xSrxCuO 4could be ascribed to overdoping. The difﬁculties in describing trendsin spin-lattice relaxation rates can be due to the intrinsic com-plexity of NMR /NQR experiments, e.g., the fact that some of 075136-13MUßHOFF, KIANI, AND PA V ARINI PHYSICAL REVIEW B 103, 075136 (2021) the current assumptions on hyperﬁne ﬁelds are incorrect, or that further channels have to be explicitly taken into account[43,44]. Finally, bosonic excitations appear robust under changes in t /primeup to close to optimal doping. For larger xthe intensity shifts toward /Gamma1going through incommensurate features, although the shadow of the antiferromagnetic mode remains for muchlarger x. These results complement those obtained recently with different techniques; e.g., for t /prime=0.3tan intensity trans- fer toward /Gamma1was found in Ref. [ 22] using the determinant quantum Monte Carlo approach; for small xandt/primea stable structure of paramagnons at Mwas obtained in Ref. [ 68]v i a the dual-boson method. In conclusion, together with the successes, we discussed some limitations of the single-band picture, which indicatethat experimental observations, in particular the descriptionof NMR /NQR relaxation rates, require a more realistic mod- eling [ 62,63] to fully account for the differences between families of superconducting cuprates. ACKNOWLEDGMENTS The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V . [ 116] for funding this project by pro- viding computing time on the GCS Supercomputer JUWELSat Jülich Supercomputing Centre (JSC). We acknowledge ﬁ-nancial support from the Deutsche Forschungsgemeinschaftthrough RTG 1995 and the former research unit FOR 1346. APPENDIX A: LATTICE MAGNETIC SUSCEPTIBILITY CLOSE TO HALF FILLING, T>TN In the small- t/Uand small- xlimit, neglecting the metallic contribution (two-pole approximation), an approximate formof the local self-energy is [ 117,118] /Sigma1 σ(iνn)∼Un−σ+n−σ(1−n−σ)U2r2 0 iνn+μ−Bσ−U(1−n−σ),(A1) where nσ=n/2=(1−x)/2 is the number of parti- cles with spin σ,μthe chemical potential, and Bσ=/summationtext ijtij/angbracketleftc† iσcjσ(2ni,−σ−1)/angbracketrightis a shift which increases with x; in the paramagnetic phase all quantities are spin independent.The factor r 0is obtained ﬁtting the numerical self-energy; for r0=1 and n=1, Eq. ( A1) equals the atomic self-energy at half ﬁlling. Within this approximation, setting μ/prime=μ−B− nU/2 and n=1−x, the Green’s function takes the two-pole form Gσ(k;iνn)=E+ k−Ux+μ/prime iνn−E+ k−E− k−Ux+μ/prime iνn−E− k E+ k−E− k, (A2) where E± k=Ux−μ/prime+1 2(εk−xU−B±/Delta1Ek), (A3) /Delta1Ek=/radicalBig (εk−xU−B)2+(1−x2)U2r2 0, (A4)andεkis the band dispersion. After performing the Matsubara sum, in the limit of large βUwe obtain χ0(q;0 )≈(gμBμeff)2 r0U/braceleftbigg c1(x)−c2(x) 2U/bracketleftbigg Jr0(0)+1 2Jr0(q)/bracketrightbigg/bracerightbigg , where μeff∼√S(S+1)/3 and c1(x)=1−x2 d3/2, (A5) c2(x)=1−x2 d5/2/parenleftbigg 1−5b2 d/parenrightbigg , (A6) with d=1−x2+b2andb=x/r0+B/Ur0. The effective exchange coupling is deﬁned as Jr0(q)=JSE(q)/2r2 0.I nt h e small- xlimit the coefﬁcients become c1(x)=1+o1(x2) and c2(x)=1+o2(x2). At ﬁrst order in x, the associated local susceptibility is thus given by χ0(0)≈(gμBμeff)2 r0U/bracketleftbigg 1−1 2UJr0(0)/bracketrightbigg . (A7) Next we approximate the total local susceptibility with the atomic susceptibility in the large βU/greatermuch1 limit, assuming negligible double occupations. Thus χ(0)∼(gμBμeff)2 kBT(1−x). (A8) Consequently, the vertex function is given by /Gamma1(0)=[χ0(0)]−1−[χ(0)]−1 ≈1 (gμBμeff)2/braceleftBigg r0U/parenleftbigg 1+Jr0(0) 2U/parenrightbigg −kBT 1−x/bracerightBigg .(A9) The lattice magnetic susceptibility takes then the form χ(q;0 )≈(gμB)2μ2 eff(1−x) kBT+μ2 eff(1−x)r0Jr0(q). (A10) This formula is a generalization of the one derived in Ref. [ 100] for the case of half ﬁlling. For comparison, the noninteracting susceptibility is shown in Fig. 18for increasing x. APPENDIX B: STATIC AND DYNAMICAL LATTICE SUSCEPTIBILITY AT HALF FILLING FOR T/lessmuchTN In the magnetic phase the local self-energy matrix can be approximated by its Hartree-Fock contribution. Thus/Sigma1 i σ(iνn)≈−μ+siσmU, where m∼1/2 is the magnetiza- tion, iis the site, and si=±1, alternating for neighboring Cu sites; the number of sites in the unit cell is ni=2. As a conse- quence, for a given spin quantum number σ, we can write the associated ni×niGreen’s function matrix as follows: Gσ(k;iνn)=1 Dk(iνn)/parenleftbigg iνn−γk−σmU αke−ikxa αkeikxaiνn−γk+σmU/parenrightbigg , (B1) where Dk(iνn)=(iνn−γk)2−(α2 k+(mU)2). (B2) 075136-14MAGNETIC RESPONSE TRENDS IN CUPRATES AND THE … PHYSICAL REVIEW B 103, 075136 (2021) x=0 Γ X 2X M 2M 0 0.3 0.6 0.9x=0.1 Γ X 2X M 2Mx=0.15 Γ X 2X M 2Mx=0.20 Γ X 2X M 2Mx=0.25 Γ X 2X M 2Mx=0.30 Γ X 2X M 2Mx=0.40 Γ X 2X M 2M x=0 Γ X 2X M 2M 0 0.3 0.6 0.9x=0.1 Γ X 2X M 2Mx=0.15 Γ X 2X M 2Mx=0.20 Γ X 2X M 2Mx=0.25 Γ X 2X M 2Mx=0.30 Γ X 2X M 2Mx=0.40 Γ X 2X M 2M FIG. 18. The noninteracting magnetic response function for t/prime=0.2t(top panels) and t/prime=0.4t(bottom panels) in the T→0 limit. Hereαk=−2t(coskx+cosky) andγk=4t/primecoskxcosky,s o thatεk=αk+γk. Let us introduce the energies E± k=γk±/radicalBig α2 k+(mU)2=γk±/Delta1α k. (B3) The elements of the Green’s function matrix can then be expressed as Gii/prime σ(k;iνn)=/summationdisplay p=±wii/prime σkp iνn−Ep k. (B4) The weights are given by w11 σkp=1 2⎛ ⎝1−pσmU/radicalBig α2 k+(mU)2⎞ ⎠=w22 −σkp (B5) and w12 σkp=p 2αk/radicalBig α2 k+(mU)2e−ikxa=/bracketleftbig w21 σkp/bracketrightbig∗. (B6) We can now calculate the elements of the lattice susceptibility tensor χ0;ii/prime σσ/primeσ/primeσ(q;iωm)=−1 βNk/summationdisplay knGσ ii/prime(k;iνn)Gσ/prime i/primei(k+q;iνn+iωm). (B7) Summing over the fermionic Matsubara frequency this ex- pression simpliﬁes to the sum given below: χ0;ii/prime σσ/primeσ/primeσ(q;iωm)≈−1 Nk/summationdisplay k/summationdisplay pp/primewii/prime σkpwi/primei σ/primek+qp/primeIpp/prime k,q(iωm), (B8) where Ipp/prime k,q(iωm)=βnF/parenleftbig Ep k/parenrightbig/bracketleftbig nF/parenleftbig Ep k/parenrightbig −1/bracketrightbig δωm,0δ/parenleftbig Ep k,Ep/prime k+q/parenrightbig (B9) +nF/parenleftbig Ep k/parenrightbig −nF/parenleftbig Ep/prime k+q/parenrightbig iωm+Ep k−Ep/prime k+q/bracketleftbig 1−δωm,0δ/parenleftbig Ep k,Ep/prime k+q/parenrightbig/bracketrightbig . (B10) Here nF(ε) is the Fermi distribution function. This formula shows that the elements of the static susceptibility tensor goto zero in the zero-temperature limit. Assuming that the quan-tization axis ˆ zis also the magnetization axis, the longitudinal and transfer susceptibilities are deﬁned as follows: χ/bardbl 0(q;iωm)=(gμB)2 4/summationdisplay σ1 2/summationdisplay ii/primeχ0;ii/prime σσσσ (q;iωm)eiφii/prime q,(B11) χ⊥ 0(q;iωm)=(gμB)2 4/summationdisplay σ1 2/summationdisplay ii/primeχ0;ii/prime −σσσ−σ(q;iωm)eiφii/prime q, (B12) where φii/prime q=(1−δii/prime)(−1)iqxa. Summing over the sites and spin quantum numbers we thus obtain χα 0(q;iωm)∼−(gμB)2 41 Nk/summationdisplay k/summationdisplay pp/prime=±vα,pp/prime k,qIpp/prime k,q(iωm)eiφii/prime q, (B13) where α=/bardbl,⊥. The weights are deﬁnes as v/bardbl,pp/prime k,q=1 2/parenleftbigg 1+pp/primeαkαk+q+(mU)2 /Delta1α k/Delta1α k+q/parenrightbigg , (B14) v⊥,pp/prime k,q=1 2/parenleftbigg 1+pp/primeαkαk+q−(mU)2 /Delta1α k/Delta1α k+q/parenrightbigg . (B15) In the low-temperature limit only the Ipp/prime k,qterms with p/prime=− p contribute. This has consequences for the behavior of thedynamical susceptibility. Let us consider ﬁrst the case of the longitudinal response function. The weight v/bardbl,pp k,qis ﬁnite for every qvector; it takes its maximum value at the /Gamma1point (v/bardbl,pp k,0=1). The weight v/bardbl,p−p k,q, however, is of order 4 t2/U2 and goes to zero at the /Gamma1point. The situation is opposite for the transverse susceptibility. The weight v⊥,p−p k,qis maximum (v⊥,p−p k,M=1) at the Mpoint and, furthermore, it remains close to one for all values of q. In the low-temperature limit (in which m∼1/2), setting t/prime=0 for simplicity, at ﬁnite frequency we obtain in the small- t/Ulimit the approximate expression χ0;ii/prime σ−σ−σσ(q;iωm)≈/bracketleftbigg −aii/prime σ(q) iωn−U+aii/prime −σ(q) iωn+U/bracketrightbigg e−iφii/prime q,(B16) where a11 σ(q)=a22 −σ(q)≈1 4/bracketleftbigg 1−σ/parenleftbigg 1−2J1 U/parenrightbigg/bracketrightbigg2 , (B17) a12 σ(q)=a21 −σ(q)≈−J1 Ufq, (B18) 075136-15MUßHOFF, KIANI, AND PA V ARINI PHYSICAL REVIEW B 103, 075136 (2021) and fq=(cosqx+cosqy)/2. By inverting the susceptibility matrix with the elements deﬁned above we thus obtain atlinear order in J 1the matrix with elements /bracketleftbigg1 χ0(q;iωm)−1 χ0(iωm)/bracketrightbiggii/prime σ−σ−σσ≈2J1fq(1−δii/prime)e−iφii/prime q. 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PhysRevB.98.155318.pdf	PHYSICAL REVIEW B 98, 155318 (2018) Magnetic ﬁeld dependence of the electron spin revival amplitude in periodically pulsed quantum dots Iris Kleinjohann,1Eiko Evers,2Philipp Schering,3Alex Greilich,2Götz S. Uhrig,3Manfred Bayer,2and Frithjof B. Anders1 1Lehrstuhl für Theoretische Physik II, Technische Universität Dortmund, Otto-Hahn-Straße 4, 44227 Dortmund, Germany 2Experimentelle Physik IIa, Technische Universität Dortmund, Otto-Hahn-Straße 4a, 44227 Dortmund, Germany 3Theoretische Physik I, Technische Universität Dortmund, Otto-Hahn-Straße 4, 44227 Dortmund, Germany (Received 8 June 2018; revised manuscript received 24 August 2018; published 24 October 2018) Periodic laser pulsing of singly charged semiconductor quantum dots in an external magnetic ﬁeld leads to a synchronization of the spin dynamics with the optical excitation. The pumped electron spins partially rephaseprior to each laser pulse, causing a revival of electron spin polarization with its maximum at the incidence timeof a laser pulse. The amplitude of this revival is ampliﬁed by the frequency focusing of the surrounding nuclearspins. Two complementary theoretical approaches for simulating up to 20 million laser pulses are developed andemployed that are able to bridge between 11 orders of magnitude in time: a fully quantum mechanical descriptionlimited to small nuclear bath sizes and a technique based on the classical equations of motion applicable for alarge number of nuclear spins. We present experimental data of the nonmonotonic revival amplitude as functionof the magnetic ﬁeld applied perpendicular to the optical axis. The dependence of the revival amplitude onthe external ﬁeld with a profound minimum at 4 T is reproduced by both of our theoretical approaches and isascribed to the nuclear Zeeman effect. Since the nuclear Larmor precession determines the electronic resonancecondition, it also deﬁnes the number of electron spin revolutions between pump pulses, the orientation of theelectron spin at the incidence time of a pump pulse, and the resulting revival amplitude. The magnetic ﬁeld of4 T, for example, corresponds to half a revolution of nuclear spins between two laser pulses. DOI: 10.1103/PhysRevB.98.155318 I. INTRODUCTION Manipulation of the resident electron spins in singly charged semiconductor quantum dots (QDs) using laser pulsesis considered a promising route for optically controlled quan-tum functionality [ 1]. The well-localized electron spins ex- hibit an increased coherence time, which is primarily lim-ited by the hyperﬁne interaction between the electron spinand the surrounding nuclear spins at cryogenic temperatures[2–7]. Periodic optical pumping in an external magnetic ﬁeld leads to a synchronization of the electron spin precessionfrequencies to the pumping periodicity by nuclear frequencyfocusing. Floquet’s theorem predicts resonance or mode-locking conditions [ 1] that have been investigated using a classical representation of the spin dynamics [ 8,9]a sw e l l as a perturbative quantum mechanical treatment of the spinsystem [ 10]. At resonance, the electron spins partially rephase prior to each laser pulse, causing a constructive interference.Since each electron is well localized within its own bathof nuclear spins, the electron spin and the nuclear spinsevolve as a coupled system, reaching a stroboscopic stationarystate after long pumping [ 1]. This quasistationary state of a periodically pumped ensemble of QDs strongly differs fromthe equilibrium starting point and is characterized by thesynchronization of the evolution of electronic and nuclearspins, implying a ﬁnite revival amplitude of the electron spinpolarization. Although the electronic resonance condition in steady state is well established [ 1,8–10], the dependency of the revival am- plitude on the applied magnetic ﬁeld has not been thoroughlyinvestigated yet. In this paper, we approach the subject in a threefold way. After brieﬂy presenting recent experimentalmeasurements of the revival amplitude, we devise a fullquantum mechanical approach to the numerical calculationof a periodically pulsed QD. The results of the quantum me-chanical exploration are supplemented by a classical approach[11]. The theoretical approaches have to face the challenge of a wide variation of timescales in the pulsed QD system. Shortlaser pulses with a duration of 2 to 10 ps have to be combined with free dynamics of 13 .2 ns between the laser pulses to a repetitive propagation in time. Our approaches achieve thesimulation of up to 20 million laser pulses, hence coveringa total simulation time up to 0 .2 s and bridging 11 orders in magnitude. This huge computational effort is necessary toreach a converged steady-state of the spin dynamics, which iscrucial to analyze the revival amplitude and its dependence on the external magnetic ﬁeld. Both theoretical treatments are based on the central spin model (CSM) [ 12] containing the hyperﬁne interaction be- tween the resident electron spin and the surrounding nuclearspins as well as the Zeeman effect. The quantum mechanicalapproach includes the full quantum mechanical time evolutionof the density operator and hence focuses on a rather smallnuclear bath of N=6 nuclear spins. However, it has been established [ 10,13–15] that even for a low number of nuclei the generic spin dynamics [ 2] can be reproduced. The time evolution between laser pulses is captured by the exact solu-tion of a Lindblad equation, accounting for the decay of theoptically excited trion and the dynamics of the CSM. 2469-9950/2018/98(15)/155318(19) 155318-1 ©2018 American Physical SocietyIRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) The laser pulses are quantum mechanically described by unitary transformations. For this purpose, we ﬁrst treat thelaser pulse in the limit of vanishing duration considering thepulses as instantaneous. However, a main advantage of ourquantum mechanical approach is the possibility to lift thisapproximation and turn toward pulses with arbitrary durationand shape. In the later part of the paper, Gaussian pump pulseswith a width of several picoseconds, which are based on theexperiment, serve as a step toward modeling the inﬂuenceof more general pulse shapes onto the spin dynamics. Wedemonstrate that taking into account the ﬁnite width has aprofound inﬂuence on the magnetic-ﬁeld-dependent revivalamplitude at large external magnetic ﬁeld. The electron spinprecession period of the order of 10 ps in a magnetic ﬁeld ofabout 10 T becomes as short as the laser pulse duration. In theclassical treatment, in turn, a classical approximation of thelaser pulses is employed that neglects the trion excitation but,however, respects the quantum uncertainty of the electronicspin components. The classical approach allows us to treatspin baths of up to 700 effective nuclear spins, calculatingpulse sequences up to a million laser pulses in the limit ofinstantaneous pulses, and hence corroborates the quantummechanical results with larger nuclear spin baths. We present results on the ﬁeld dependency of the re- vival amplitude in pump-probe experiments with an expandedrange up to 10 T for the magnetic ﬁeld applied perpendicularto the optical axis, whereas former experiments had been lim-ited to 6 T only [ 9]. The data for two different (In,Ga)As/GaAs QD samples show a characteristic minimum of the revivalamplitude at roughly 4 T. Our theoretical approaches disclosethat the nuclear Larmor frequency [ 10] plays a crucial role in understanding these experimental data, e.g., 4 T roughlycorresponds to the external magnetic ﬁeld where the nuclearspins perform half a revolution between two succeeding pumppulses. The nuclear Larmor precession determines the elec-tronic resonance condition and thus the number of electronspin revolutions between two pump pulses. Since the numberof electron spin revolutions in between two pump pulses alsodetermines the alignment of the electron spin immediatelybefore a pump pulse, we connect the nuclear resonance con-dition directly to the revival amplitude. In the experiments, the properties of the QDs, such as the electron gfactor and the trion excitation energy, vary in the ensemble. Detuned QDs are not efﬁciently pumpedand practically do not contribute to the spin polarization.The mode-locking condition [ 1] in such ensembles, however, causes a synchronization of the electron spin dynamics inperiodically pumped QDs with slightly different gfactors [1,9]. In this paper, the theoretical approaches focus on an ensemble with ﬁxed gfactor and trion excitation energy, but the quantum mechanical treatment includes variationsof the hyperﬁne coupling, accounting for slightly differentcharacteristic dephasing timescales T ∗in each QD. For completeness, we note that there have been extensive studies of the electron-nuclear interaction on the single-QDlevel; see Refs. [ 7,16,17]. The spin coherence of electrons and also holes and in particular its limitation due to couplingto the nuclear bath were studied by echo-type experiments[18,19]. Requirements for the nuclear spin system to reduce the detrimental effect on the electron spin coherence wereformulated [ 20]. Sophisticated strategies were implemented to suppress the carrier spin dephasing, both in gate-deﬁned QDs[21] and in self-assembled QDs using pulse sequences for dynamic decoupling [ 22,23] or coherent population trapping that is sensitive to the nuclear Overhauser ﬁeld [ 24,25]. Using the latter technique, it has recently become possible to monitorthe evolution of the nuclear spin bath and to demonstrate anextension of the electron spin dephasing time by an order ofmagnitude in self-assembled QDs [ 26]. Vice versa, also the impact of the electron spin on the nuclear spin coherence hasbeen studied [ 27]. Here, we focus on a different problem, namely the contribution of the nuclear spin bath to the syn-chronization of the electron spin precession about an externalmagnetic ﬁeld with the periodically pulsed excitation laserthat orients the spins. We monitor the electron spin coherencein a QD ensemble over times covering 11 orders of magnitudeas a function of magnetic ﬁeld strength. The paper is organized as follows. We start by presenting measurements of the revival amplitude obtained in pump-probe experiments in Sec. II. Then, we turn toward the theoretical calculations. The CSM underlying both the quan-tum mechanical and the classical approach is introduced inSec. III. In Sec. IV, we devise the full quantum mechanical approach to periodically pulsed QDs. The results obtained bythe quantum mechanical approach with instantaneous pumppulses are illustrated in Sec. V. These results are compared to the classical approach in Sec. VI. In Sec. VII, we extend the quantum mechanical description to pump pulses with Gaus-sian envelope. The last section summarizes our theoretical andexperimental results and draws conclusions. II. EXPERIMENTAL RESULTS First, we present the experimental results of the magnetic ﬁeld dependency of the revival amplitude. We study twodifferent samples of singly charged (In,Ga)As/GaAs QD en-sembles using a pump-probe Faraday rotation setup similarto the one in Ref. [ 1]. A Ti:sapphire laser emits pulses of 2.5 ps duration with a repetition period of 13 .2 ns. To polarize the electron spins via trion excitation [ 28], the pump pulses are circularly ( σ +/−) polarized. Switching the polarization between σ+andσ−with a frequency of 84 kHz enables us to perform synchronous detection using a lock-in ampliﬁer. Thesamples are cooled to 4 .7 K in a cryostat, which is equipped with a superconducting split-coil solenoid and allows us toapply magnetic ﬁelds of up to 10 T. We align the magneticﬁeld perpendicular to the light propagation vector, which isparallel to the growth axis of the sample (V oigt geometry).Directing the probe beam through the sample, the Faradayellipticity change is detected by an optical differential bridge. The two samples were grown by molecular-beam epitaxy on a (001)-oriented GaAs substrate. Each sample features20 layers of self-assembled InGaAs QDs with a dot densityof approximately 10 10cm−2. Each QD layer is followed by 16 nm of GaAs. Then, a Si-donor δlayer is deposited with a density similar to the QD density, providing therefore onaverage one electron per dot, so that the QDs are singlycharged. This sheet of donors is followed by a GaAs barrierof 44 nm before the next layer of QDs is grown, leading to atotal separation of 60 nm between two adjacent dot layers. 155318-2MAGNETIC FIELD DEPENDENCE OF THE ELECTRON … PHYSICAL REVIEW B 98, 155318 (2018) FIG. 1. Photoluminescence (PL) spectra of the two studied sam- ples. The spectra are taken at a temperature of 4 .7 K with a photon excitation energy of 1 .631 eV. The laser pulses used in the pump- probe experiment are shown in red. (a) Sample 1 and (b) sample 2. After the epitaxial growth, the samples were thermally annealed to homogenize the QD ensembles. In addition tohomogenizing the dot sizes, the annealing also led to a furtherexchange of Ga and In atoms between the InGaAs QDs andthe surrounding GaAs barriers [ 29] so that the In content in the QDs is reduced. Besides, the thermal annealing shiftsthe emission energy of the sample to higher values. For bothsamples, the rapid thermal annealing time was chosen to be30 s. Sample 1 was annealed at 945 ◦C and sample 2 at 880◦C. The photoluminescence spectra of both samples as well asthe spectra of the exciting pulsed laser in the pump-probeexperiments are shown in Fig. 1. Sample 1 is the sample used in Refs. [ 1,30,31], which we resonantly excite in the low-energy ﬂank of the ground-state transition at a photonenergy of 1 .386 eV. The recombination of electron-hole pairs with the electron in excited QD conﬁned states above theground state shows up as additional peaks toward higherenergies. Sample 2 has a lower central emission energy andis resonantly excited at a photon energy of 1 .376 eV. In the Faraday rotation measurements, the pump and the probe pulse trains were split from the same laser source. Wetake pump-probe traces for both samples by incrementingthe transit time of the pump pulses through the sample viaa mechanical delay line. These traces are the experimen-tal manifestation of the steady-state spectra of the coupledelectron-nuclear system after several million pump pulses inFig. 5(a). The signal for each delay step is integrated over 100 ms. Starting from 1 T, we record the dynamics of theelectron spin projection onto the optical axis for magneticﬁelds up to 10 T in steps of 0 .5 T. Figure 2shows a selection of these spectra. At delay t=0, the pump pulse aligns the electron spins. Toward negative and positive delays, the totalspin polarization decreases due to a dephasing of the spinensemble. The decay of the total spin polarization is super-imposed by an oscillating function which reﬂects the Larmorprecession. We ﬁt the negative side of the spectrum with an inhomoge- neously (Gaussian) decaying cosine function [in accordanceFIG. 2. Dynamics of the electron spin projection onto the optical axis in different magnetic ﬁelds. The Faraday ellipticity measured for sample 2 is plotted vs the pump-probe delay. The dynamics areobtained at a temperature of 4 .7 K. The curves are shifted vertically for clarity. with Eq. ( 10)]: Sz(t)=Acos(ωt)e x p/parenleftbigg −t2 6T∗2/parenrightbigg . (1) From these ﬁts, we can extract the revival amplitude A,t h e Larmor frequency ω, and the dephasing time T∗. The Larmor frequencies show a linear dependence on the magnetic ﬁeldwith an electron gfactor of g e=0.57. The dephasing time T∗ obviously decreases with increasing magnetic ﬁeld. Because of the ﬁnite spectral width of the pulses, a distribution ofelectron gfactors is excited that is translated into a spread of precession frequencies. This spread increases linearly withincreasing magnetic ﬁeld, leading to an enhanced dephasing.The measured T ∗dependence follows to a good approxima- tion the expected 1 /Bextbehavior. We note that this dephasing does not impact the discussion of the revival amplitude, as wedetermine the amplitude right before a pump pulse. In Fig. 3, we plot the revival amplitude in arbitrary units as function of the magnetic ﬁeld. Note that the measuredrevival amplitude depends strongly on the experimental setupand thus cannot be compared quantitatively to the theoreticalresults in the later sections. Samples 1 and 2 both show a nontrivial magnetic ﬁeld dependency but with similar characteristics. The amplitudesdecrease toward higher magnetic ﬁelds, which we attributeto the varying In and Ga contents in each dot, leading toa nuclear g-factor spread and a dephasing of the nuclear spins (see Sec. VDfor further explanation). Additionally, for both samples we see an oscillatory behavior of the revivalamplitude with a main minimum between two maxima, ontowhich smaller ﬂuctuations are superimposed. The centralminimum for sample 1 is positioned at about 4 .2 T, whereas the two maxima occur around 2.5 and 5 .7 T, respectively. For sample 2, the maxima and the minimum occur at slightlyhigher magnetic ﬁelds as if the oscillatory period is increased. 155318-3IRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) FIG. 3. Dependence of the revival amplitude on the external magnetic ﬁeld. The revival amplitudes (the symbols) were extracted from the pump-probe spectra for the two different samples at tem-perature 4 .7 K. The lines are guides to the eye. III. CENTRAL SPIN MODEL (CSM) In order to describe the experimental ﬁndings, the theo- retical approaches in this paper target the spin dynamics ofa QD ensemble subject to periodic laser pump pulses. Thedynamics are separated into three parts: First, the electron spininteracting with its nuclear spin environment is accounted forby a CSM (also called Gaudin model [ 12]). Hereby, we restrict our description to the hyperﬁne inter- action and neglect other effects such as the nuclear dipole-dipole interaction or the nuclear-electric quadrupolar inter-action [ 32–36]. The two latter typically are some orders of magnitude smaller than the hyperﬁne interaction [ 6] and only are relevant on timescales much larger than the pulse repe-tition time T R=13.2 ns. It has been shown [ 33–35] that the nuclear-electric quadrupolar interactions induce an additionalelectronic dephasing time of the order of 300 ns in the absenceof an external magnetic ﬁeld. The effect of this interaction issuppressed in a ﬁnite magnetic ﬁeld due to its competitionwith the nuclear Zeeman energy: the spin-noise spectrum[37] can be ﬁtted by a frozen Overhauser ﬁeld approxima- tion for external ﬁelds exceeding 40 mT. The characteristicdephasing time associated with these competing interactionsincreases with the external ﬁeld and arrives at values of2–4μs[15,18,19]f o rB ext>3 T. Since the timescale induced by the nuclear-electric quadrupolar interactions is about 300times larger than T Rin a large external magnetic ﬁeld, the nuclear-electric quadrupolar interactions only provide a smallperturbative correction and can be omitted relative to theleading-order effect presented here. The second ingredient for the spin dynamics in the QD is the light-matter interaction of the classical laser ﬁeld. Thethird part comprises the radiative decay of the laser-inducedtrion state. At the end, we average over different realizationsof QDs to obtain the spin dynamics in a QD ensemble. The CSM [ 2–4,12,13,38,39] comprises a bath of Nnuclear spins coupled to the electron spin via hyperﬁne interaction.Its Hamiltonian H CSM consists of three terms, the hyperﬁne interaction HHF, the electron Zeeman effect HEZ, and the nuclear Zeeman effect HNZ: HCSM=HHF+HEZ+HNZ. (2)These three parts can be written in terms of the electron spin operator /vectorSand the nuclear spin operators /vectorIk HHF=N/summationdisplay k=1¯h−2Ak/parenleftbig SxIx k+SyIy k+SzIz k/parenrightbig =N/summationdisplay k=1¯h−2Ak/bracketleftbigg SxIx k+1 2/parenleftbig S+I− k+S−I+ k/parenrightbig/bracketrightbigg ,(3) HEZ=¯h−1geμBBextSx, (4) HNZ=¯h−1gNμNBextN/summationdisplay k=1Ix k. (5) While the negatively charged QDs studied in this paper are described by isotropic coupling constants Ak, positively charged QDs require the extension to an anisotropic couplingbetween electron and nuclear spins [ 5,14,40]. Using the spin ladder operators S ±=Sy±iSzandI± k=Iy k±iIz k,t h eh y - perﬁne interaction can be rewritten as an Ising term parallelto the external magnetic ﬁeld (in the xdirection) and two spin-ﬂip terms. In the Zeeman terms, g eandgNdenote the electron and the nuclear gfactor, respectively. The constants μBandμN are the Bohr magneton and the nuclear magneton. Note that we choose one effective value gNμNfor all nuclei. Different types of nuclei have been treated, for instance, in Ref. [ 10]b u t are beyond the scope of the present work. In our calculations,we use an electron gfactor g e=0.555, which is typical in experimental studies of InGaAs QDs [ 1,31]. This leads to an angular electron Larmor frequency ωe=μBgeBext/¯h of roughly 97 .6×109rad/sa tBext=2 T. For the nuclear spins, we choose a precession 800 times slower with theratioz=g NμN/(geμB)=1/800. This value is based on the weighted average of the nuclear magnetic moments of Gaand As [ 41,42] and has been calculated in Ref. [ 10]. Thus, the nuclear angular Larmor frequency ω N=μNgNBext/¯his roughly 122 ×106rad/sa tBext=2 T and gN≈1.27. The Hamiltonian HCSMis diagonal in the spin xbasis ex- cept for the spin-ﬂip terms in HHF. In the following, we denote the electron spin xbasis, i.e., the eigenbasis of HEZ,b y\|↑/angbracketrightand \|↓/angbracketright. Therefore, it is Sx\|↑/angbracketright = ¯h/2\|↑/angbracketrightandSx\|↓/angbracketright = − ¯h/2\|↓/angbracketright. For the sake of simplicity, we also treat the nuclear spins asspins 1 /2 even though in real QDs the nuclei have spin 3 /2 (Ga and As) and spin 9 /2( I n )[ 41,42]. The assumption of nuclear spins 1 /2 restricts the dimension of the density matrix in our quantum mechanical approach to 2 ×2 Nwith two spin states for the electron and each nucleus, respectively. The hyperﬁne coupling constants Akarise from the Fermi contact interaction. Therefore, their values are determinedby the electron wave function \|ψ(/vectorR k)\|2at the position of a nucleus [ 2]. The hyperﬁne interaction HHFcan be interpreted in terms of an additional magnetic ﬁeld that acts on the elec-tron spin and is caused by the nuclear spins. This additionalmagnetic ﬁeld is called the Overhauser ﬁeld: /vectorB N=(geμB¯h)−1N/summationdisplay k=1Ak/vectorIk. (6) 155318-4MAGNETIC FIELD DEPENDENCE OF THE ELECTRON … PHYSICAL REVIEW B 98, 155318 (2018) The additional magnetic ﬁeld that is caused by the electron spin and acts on nuclear spin k, in turn, is termed Knight ﬁeld: /vectorBk,Kn=(gNμN¯h)−1Ak/vectorS. (7) The ﬂuctuation of the Overhauser ﬁeld /vectorBNleads to a dephas- ing of the electron spin with a characteristic time T∗[2] (T∗)−2=¯h−4N/summationdisplay k=1A2 k/angbracketleftbig I2 k/angbracketrightbig . (8) In the experiment, the dephasing time typically is of the order o f1t o3n s[ 1,30] if ﬁtted proportional to exp ( −t2/(T∗)2) as in Ref. [ 2]. The deﬁnition of T∗in Eq. ( 8), however, leads to a dephasing with envelope ( 10) such that T∗takes values in the range of 0.4 to 1.2 ns. These experimental valuesinclude additional dephasing mechanisms, e.g., the electrongfactor spread as discussed above. Since the two theoretical approaches in this paper only comprise the electron dephasingdue to the hyperﬁne interaction, we adjust the characteristicdephasing time T ∗to the experimental values of the overall dephasing time, mimicking other effects as well. IV . QUANTUM MECHANICAL APPROACH TO PERIODICALLY PULSED QDS The scope of this work is to calculate the approach of the spin dynamics to steady state in a periodically driven QDensemble. In order to access this limit numerically with afull quantum mechanical simulation, several million pumppulses are required. Since the computational time grows ex-ponentially with the Hilbert space dimension, we restrict ourcalculation to a rather small bath size of N=6 nuclear spins. A. Hyperﬁne coupling constants Ak A real QD typically contains of the order 105nuclear spins with couplings Akthat are given by a distribution function p(Ak). It has already been shown that a representation of the system with a reduced number of nuclear spins is ableto reproduce the generic spin dynamics of a larger system[14]. To compensate for the small number of nuclear spins and to simultaneously include ﬂuctuations induced by theslightly different QDs in the ensemble, we consider N C=100 realizations of the CSM. These realizations differ in theirset of hyperﬁne coupling constants {A k}. During the whole pulse sequence, the conﬁgurations are treated as independentrepresentations of a single QD and the results are merged onlyat the end. As a side product, the computation time scalesonly linearly with N C. To distinguish the conﬁgurations, we introduce an index j∈{1,...,N C}; e.g.,Ak,jis the coupling constant for nuclear spin kin conﬁguration j. For brevity, the index jwill be omitted, when we consider a single conﬁguration only. Since the details of the distribution function have a weak inﬂuence on the steady-state dynamics [ 9], we choose the coupling constants uniformly distributed in the range [0 .2; 1]. In this way, we exclude very small couplings to nuclear spins,which have minor impact on the electron spin. The randomlydistributed coupling constants A k,jlead to an ensemble aver- FIG. 4. Inﬂuence of the hyperﬁne coupling constants Ak.( a )T h e numerically calculated time evolution of the electron spin component /angbracketleftSz(t)/angbracketright.A tt=0, the spin is directed in the negative zdirection. The Gaussian envelope function according to Eq. ( 10) is indicated by black dashed lines. Small deviations are due the ﬁnite number of nuclei N=6(NC=100). (b) The distribution p(T∗ j) of the de- phasing time T∗ jwithin a single conﬁguration for different numbers Nof nuclei. Here, we scale the coupling constants to T∗=1n s i n 1000 sets with NC=100 conﬁgurations each, in order to obtain an approximately continuous distribution p(T∗ j). aged dephasing time (T∗)−2=1 NC¯h4NC/summationdisplay j=1N/summationdisplay k=1A2 k,j/angbracketleftbig I2 k,j/angbracketrightbig , (9) where /angbracketleftI2 k,j/angbracketright=/angbracketleftI2/angbracketright=3 4¯h2for nuclear spins 1 /2. In our calcu- lations, we set T∗=1 ns based on the experiments and scale the coupling constants accordingly. Figure 4(a) shows the time evolution of the electron spin component /angbracketleftSz(t)/angbracketrightcalculated by averaging all NCconﬁgu- rations [see Eq. ( 27)]. Small deviations from the Gaussian envelope function [ 2] /angbracketleftSz(t)/angbracketrightenv=S0exp/parenleftbigg −t2 6T∗2/parenrightbigg (10) are caused by the ﬁnite number of N=6 nuclear spins. Since we ﬁx the ensemble-averaged dephasing time in Eq. ( 9),T∗varies in the different conﬁgurations j, mimicking an ensemble of quantum dots. This variation is depicted inFig.4(b) for different numbers of nuclei N. For that purpose, 155318-5IRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) we deﬁne the ratio T∗ j/T∗via /parenleftBigg T∗ T∗ j/parenrightBigg2 =1 ¯h4N/summationdisplay k=1(T∗Ak,j)2/angbracketleftbig I2 k,j/angbracketrightbig (11) for each of the NC=100 conﬁgurations entering the deﬁni- tion of T∗in Eq. ( 9). The distribution p(T∗ j) is obtained from 1000 such sets containing NC=100 conﬁgurations each. The distribution p(T∗ j) clearly reveals a self-averaging effect for increasing Nif normalized ak,j=T∗Ak,jare used. For simulations with a large number of nuclei N, one has to replace our approach by ak,j=T∗ jAk,j, where T∗ jmust be randomly generated from a distribution p(T∗ j) with a ﬁxed width corresponding to the experimental variations of theQDs. B. Instantaneous laser pump pulses In order to describe the time evolution during a laser pump sequence, the CSM has to be extended by the trion state\|T/angbracketright = \|↑↓⇑/angbracketright x, which is excited by the circularly polarized pump pulses [ 43]. Since we consider σ+-polarized light only, we omit the trion state \|↑↓⇓/angbracketrightz. Hence, the electronic subspace comprises three possible states, i. e., {↑z,↓z,T}, and the full density matrix has dimension (3 ×2N)×(3×2N). Here, we choose the spin basis along the optical axis in the zdirection. The states \|↑/angbracketrightzand\|↓/angbracketrightzcan be transformed into the magnetic ﬁeld eigenbasis, \|↑/angbracketrightand\|↓/angbracketright,v i a\|↑/angbracketrightz=(\|↑/angbracketright + \|↓/angbracketright )/√ 2 and \|↓/angbracketrightz=(\|↑/angbracketright − \|↓/angbracketright )/√ 2, respectively. At ﬁrst, we treat the laser pulses in the limit of vanishing duration, hence considering them as instantaneous. Later, inSec. VII, we will lift this approximation. The impact of an instantaneous πpulse, which resonantly excites the trion state, is given by ρ→U PρU† Pwith the unitary pulse operator UP=\|T/angbracketright/angbracketleft↑\|z−\|↑/angbracketrightz/angbracketleftT\|+\|↓/angbracketrightz/angbracketleft↓\|z. (12) This unitary transformation of the density operator ρcorre- sponds to a complete exchange of the \|↑/angbracketrightzpopulation and the\|T/angbracketrightpopulation. Meanwhile, the \|↓/angbracketrightzpopulation remains unaffected by the pulse. Note that the pulse operator UPdoes not have any effect on the nuclear spin conﬁgurations at all. C. Lindblad approach Because of the trion decay after each pump pulse, a unitary time evolution between pulses would have to include theparticipating photons. Since we are not interested in the reso-nance ﬂuorescence, we treat the trion decay in the frameworkof an open quantum system, i. e., by a master equation inLindblad form [ 44] for the time evolution of the density operator ρbetween two succeeding pump pulses dρ dt=−i ¯h[H,ρ ]+γ(s†sρ+ρs†s−2sρs†)=Lρ(13) and treat the photon emission by a spontaneous Markov process with rate γ. The term including the commutator of the von Neumann equation contains the unitary part of the timeevolution, namely the spin dynamics captured by the CSM.Here, the Hamiltonian Hincludes H CSM and the trion statewith excitation energy ε H=HCSM+ε\|T/angbracketright/angbracketleftT\|. (14) The term proportional to γin the Lindblad equation accounts for the trion decay. The decay rate γis set to 10 ns−1based on experimental data for a trion lifetime of about 400 ps [ 1]. The operators s=\| ↑ /angbracketrightz/angbracketleftT\|ands†=\|T/angbracketright/angbracketleft↑\|zmap the trion state to the spin-up state along the optical axis and vice versa.The whole time evolution of ρcan be written in terms of a super operator, the so-called Liouville operator L.F o ra time-independent L, the solution to Eq. ( 13) is given by an exponential function ρ(t)=e −Ltρ(0), (15) which is valid for the times between two pulses, where ρ(0) is the density operator right after the pulse. However, theactual calculation of this solution would involve diagonal-ization of the matrix representation of L, which has dimen- sion (3 ×2 N)2×(3×2N)2. In order to circumvent this time- consuming task, we develop an alternate approach that isdescribed below. Within this method, we only have to treatmatrices of much smaller dimension (2 ×2 N)×(2×2N). D. Partitioning of the density operator To solve the Lindblad equation, we ﬁrst transform into the frame rotating with the Larmor frequency ωNof the nuclear spins. Hereby, we eliminate the nuclear Zeeman term in theHamiltonian. The transformed Lindblad equation reads ˙˜ρ=i ¯h[˜H,˜ρ]−γ(˜s†˜s˜ρ+˜ρ˜s†˜s−2˜s˜ρ˜s†), (16) where the transformed operators ˜O=URFOU† RFare denoted by a tilde ( ˜ ) and URF=exp/braceleftBigg −iωN ¯h/parenleftBigg Sx+/summationdisplay kIx k/parenrightBigg t/bracerightBigg . (17) The new Hamiltonian in the rotating frame is given by ˜H=(ωe−ωN)Sx+HHF+ε\|T/angbracketright/angbracketleftT\|. (18) In this frame, the electron precesses with the reduced frequency ωe−ωN, while the hyperﬁne interaction remains unaffected by the transformation. The operator ˜sin the rotat- ing frame, deﬁned in the basis along the external magneticﬁeld, is ˜s=1 √ 2(e−iωNt/2\|↑/angbracketright+eiωNt/2\|↓/angbracketright)/angbracketleftT\|. (19) Inserting ˜sand its conjugate ˜s†, the Lindblad equation yields ˙˜ρ=−i ¯h[˜H,˜ρ]−γ(\|T/angbracketright/angbracketleftT\|˜ρ+˜ρ\|T/angbracketright/angbracketleftT\|) +γ/angbracketleftT\|˜ρ\|T/angbracketright(\|↑/angbracketright/angbracketleft↑\|+\|↓/angbracketright/angbracketleft↓\| +e−iωNt\|↑/angbracketright/angbracketleft↓\|+eiωNt\|↓/angbracketright/angbracketleft↑\|). (20) This Lindblad equation in the rotating frame allows us to separate the trion decay from the remaining dynamics. In theelectron-nuclear tensor space spanned by the basis \|e,K/angbracketright, 155318-6MAGNETIC FIELD DEPENDENCE OF THE ELECTRON … PHYSICAL REVIEW B 98, 155318 (2018) one can deﬁne a reduced density operator ˜ ρTT=/angbracketleftT\|˜ρ\|T/angbracketright acting only on the nuclear spin conﬁgurations \|K/angbracketright, while the electronic state e∈{ ↑,↓T}has been ﬁxed to the trion state T. Apparently, the dynamics of this operator obeys ˙˜ρTT=−2γ˜ρTT (21) and its matrix representation has the dimension 2N×2N determined from the nuclear Hilbert space only. The analytic solution to Eq. ( 21) is an exponential decay of the trion population for arbitrary nuclear spin conﬁgurations ˜ρTT(t)=˜ρTT(0)e−2γt(22) that decouples from the electronic subsystem. Therefore, there is no nuclear dynamics in this sector of the density matrix. We partition the density operator into the remaining eight reduced density operators ˜ ρe,e/prime=/angbracketlefte\|˜ρ\|e/prime/angbracketrightand ﬁrst focus on the four contributions involving the trion, namely the trion co-herence suboperators /angbracketleftT\|˜ρ\|↑/angbracketright,/angbracketleftT\|˜ρ\|↓/angbracketright,/angbracketleft↑\|˜ρ\|T/angbracketright, and/angbracketleft↓\|˜ρ\|T/angbracketright. Their differential equations, which we obtain from Eq. ( 20), decouple from those of the suboperators without trion. Asa result, the elements of trion coherence suboperators decayexponentially with γand we do not have to further investigate them since γT R/greatermuch1. We now concentrate on the time evolution for the sub- operator ˜ ρScomprising the four reduced-density operators involving no trion. The matrix representation of ˜ ρShas the dimension (2 ×2N)×(2×2N) and only contains the spin-up and spin-down state for the electron. We insert the analyticsolution for the operator ˜ ρ TT(t), i.e., Eq. ( 22), into the Lind- blad equation ( 20) to determine the time evolution of ˜ ρS: ˙˜ρS+i ¯h[˜HS,˜ρS]=γ˜ρTT(0)e−2γt(\|↑/angbracketright/angbracketleft↑\|+\|↓/angbracketright/angbracketleft↓\| +e−iωNt\|↑/angbracketright/angbracketleft↓\|+eiωNt\|↓/angbracketright/angbracketleft↑\|),(23) where ˜HS=(ωe−ωN)Sx+HHFis the projection of ˜Honto the spin-spin subspace. The differential equation for ˜ ρSwas divided into the homogeneous part on the left-hand side anda source term stemming from the trion decay on the right-hand side of the equation. It can be solved by combiningthe solution for the homogeneous part of the equation and aparticular solution for the full inhomogeneous equation. Sincethe homogeneous part equals a von Neumann equation, it issolved by unitary time evolution ˜ρ S,h(t)=e−i˜HSt/¯h˜ρS,h(0)ei˜HSt/¯h. (24) A particular solution to the full equation can be obtained by the ansatz ˜ρS,nh(t)=˜χ0e−2γt+˜χ+e(iωN−2γ)t+˜χ−e(−iωN−2γ)t.(25) For further details on the numerical calculation of ˜ χ0,˜χ+, and ˜χ−, see Appendix A. Finally, the solution to the Lindblad Eq. ( 23) is given by ˜ρS(t)=˜ρS,h(t)+˜ρS,nh(t), (26)where the initial condition directly after a pump pulse at t=0 yields ˜ ρS,h(0)=˜ρS(0)−(˜χ0+˜χ++˜χ−). To evaluate the solution just before the next laser pulse at t=TR, the contribution of ˜ ρS,nh(TR) can be neglected due to the exponential decay with decay rate γ, since γTR/greatermuch1. Therefore, the effect of the trion decay is a correction of thedensity operator ˜ ρ S(0) in the electronic sector right after the pulse into ˜ ρS,h(0), which allows the calculation of the time evolution until the next pulse by a single unitary transforma-tion substituting t→T Rin Eq. ( 24). By iterating the elementary building block that combines the effect of a single instantaneous pump pulse and the timeevolution for T R, we calculate the effect of pulse sequences with up to 20 million laser pulses. Note that we presentan exact approach to the spin dynamics of a QD subject tosequential pulses for a ﬁnite nuclear spin bath. In contrast, theperturbative approach presented in Ref. [ 10] only includes up to one spin ﬂip of the nuclear spin system between subsequentpulses. Our approach is limited to a smaller nuclear spinbath sizes but allows for the simulation of N P>107pump pulses, while the approach in Ref. [ 10] was restricted to approximately 104pulses due to CPU run time limitations. V . RESULTS OF THE QUANTUM MECHANICAL APPROACH In this section, we present results for the time evolution of the electron spin polarization along the optical axis and inparticular the electron spin revival amplitude obtained by thequantum mechanical approach. We explicitly make contact tothe nonmonotonic magnetic ﬁeld dependency of electron spinrevival amplitude found in the experiment and presented inFig. 3. Additionally, we can directly access the nuclear spins in the numerical calculations, which, in contrast, is impossiblein the pump-probe experiments. As a signature of the nuclearstate, we present a detailed analysis of the distribution ofOverhauser ﬁelds. A. Evaluation of numerical results We calculated the quantum mechanical time evolution for each single conﬁguration of a QD represented by a ﬁxed butrandom selection of {A k}. After iterating pump pulse and time evolution of duration TRup to the desired number of laser pulses, the average over many calculations of this sortdescribes the ensemble of QDs. In this way, the electronicexpectation value of the spin polarization is given by /angbracketleftS i(t)/angbracketright=N−1 CNC/summationdisplay j=1/angbracketleftSi(t)/angbracketrightj, (27) where /angbracketleftSi(t)/angbracketrightj=Tr[Siρj(t)] with i∈{x,y,z }denotes the quantum mechanical expectation value in conﬁguration jat timet. In addition to the electron spin, we are also interested in the effect of the pump sequence on the alignment of the nuclearspins. The distribution of B x Nalong the external magnetic ﬁeld axis [ 8–10] as deﬁned in Eq. ( 6) can be obtained from the 155318-7IRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) conﬁguration average Bx N=N−1 CNC/summationdisplay j=1/angbracketleftbig Bx N,j/angbracketrightbig =N−1 CNC/summationdisplay j=1Tr/bracketleftbig Bx N,jρj/bracketrightbig =N−1 CNC/summationdisplay j=1/summationdisplay e,K/angbracketlefte,K\|ρj\|e,K/angbracketrightBx K,j, (28) where e∈{ ↑,↓,T}labels the electron degree of freedom andKdenotes the conﬁguration of the Nnuclear spins with quantization axis in the xdirection. Equation ( 28) can be interpreted such that the value Bx K,j=/angbracketleftK\|Bx N,j\|K/angbracketright (29) occurs with probability pK,j=N−1 C/angbracketlefte,K\|ρj\|e,K/angbracketright. (30) Accumulating all probabilities for a ﬁxed value Bx N, p/parenleftbig Bx N/parenrightbig =1 NCNC/summationdisplay j=1/summationdisplay e,K/angbracketlefte,K\|ρj\|e,K/angbracketrightδ/parenleftbig Bx N−Bx K,j/parenrightbig (31) deﬁnes the continuous distribution p(Bx N)f o rBx N[10], whose integral is normalized to unity by construction [ 8]. B. Electron spin revival amplitude First, we investigate the effect of a sequence of pump pulses with separation TR=13.2 ns on the electron spin polarization along the optical axis ( zdirection). In Fig. 5(a), the evolution of /angbracketleftSz/angbracketrightin the time interval between two pulses is depicted for different numbers NPof applied pump pulses. The dephasing after the ﬁrst pump pulse (red curve) is ap-proximately a Gaussian and determined by the dephasing timeT ∗deﬁned in Eq. ( 9). The time evolution of /angbracketleftSz/angbracketrightafter a large number of pump pulses corresponds to the experimentalmeasurements in Fig. 2. Note that the fast Larmor oscillation in the external magnetic ﬁeld B ext=1.95 T is not resolved on the timescale in Fig. 5(a) leading to the colored areas. In Fig. 5(b), the electron Larmor oscillation of both transversal spin components directly before the arrival of the next pumppulse is presented: The ycomponent almost vanishes for pumping with instantaneous ideal πpulses. After roughly 10 pump pulses, a revival of spin polariza- tion has established with a maximum just before the nextpump pulse. The amplitude of this initial revival (approxi-mately 0.077) is independent of the external magnetic ﬁeld,since it originates from a purely electronic steady state (seeAppendix B)[10]. We deﬁne the revival amplitude as the spin polarization S ⊥(NPTR)=/radicalBig /angbracketleftSz(NPTR)/angbracketright2+/angbracketleftSy(NPTR)/angbracketright2(32) afterNPpulses right before the next pulse. This deﬁnition is motivated by the experimental procedure, where an envelopefunction is ﬁtted to the measured oscillating signal in orderto obtain the revival amplitude (see Sec. II). For the numer- ical calculations, this procedure is not necessary, as we candirectly read off the amplitude via Eq. ( 32). FIG. 5. (a) Time evolution of the electron spin component /angbracketleftSz/angbracketright. Various colors show the time evolution after different numbers NP of pump pulses. The time axis starts with the arrival of the NPth pump pulse and ends before the arrival of the next pump pulse afterthe repetition time T R=13.2 ns. (b) Time evolution directly before the next pump pulse. In addition to the electron spin component /angbracketleftSz/angbracketright (solid lines), the electron spin component /angbracketleftSy/angbracketrightis depicted (dashed lines) to show the spin precession. Starting from the initial value originating from the purely electronic steady state, the revival amplitude evolves furtherupon increasing the number of pump pulses. This evolution,however, is dependent on the external magnetic ﬁeld. Thegrowth of revival amplitude for B ext=1.95 T is shown in Fig. 6(red curve). In addition, the evolution of the revival amplitude for other external magnetic ﬁelds is pictured. Fordistinct magnetic ﬁelds, an increase or a decrease of amplitudewith the number of pump pulses N Pcan be observed, but the rate of change with NPbecomes much slower compared to the initial revival obtained after 10 pulses, especially forstronger external magnetic ﬁelds. The magnetic ﬁeld depen-dency of the revival amplitude results from a synchronizationof the dynamics of all spins including the nuclei with thepump pulses, i.e., from the nuclear focusing. The periodicallypulsed electron spin transmits the effect of the pump pulsesto the nuclear spins via the hyperﬁne coupling. Therefore,the nuclear spins gradually align along the external magneticﬁeld, which in turn focuses the electron Larmor frequency andthereby leads to either an ampliﬁcation or a reduction of theinitial revival. To analyze the magnetic ﬁeld dependency of the revival amplitude S ⊥(NPTR) in more detail, we plot the converged 155318-8MAGNETIC FIELD DEPENDENCE OF THE ELECTRON … PHYSICAL REVIEW B 98, 155318 (2018) FIG. 6. Evolution of the electron spin revival amplitude S⊥(NPTR) with the pulse number NP. Various colors show the development for different external magnetic ﬁelds Bext. revival amplitude after up to 20 million pump pulses as a function of the external magnetic ﬁeld. The result in Fig. 7 (blue curve) shows a nonmonotonic behavior with maxima atapproximately 2 and 6 T and minima at 4 and 8 T, respectively.This behavior can also be observed in the spin component/angbracketleftS z(NPTR)/angbracketright(red crosses), which matches S⊥(NPTR)f o re x - ternal magnetic ﬁelds above 2 T. Therefore, we observe thatthe contribution of the spin component /angbracketleftS y(NPTR)/angbracketrightnearly vanishes at the incidence time of a pump pulse, which wasalready indicated in Fig. 5(b). The nonmonotonic behavior ofS ⊥(NPTR) and/angbracketleftSz(NPTR)/angbracketrightis caused by the resonance of the nuclear spins, which depends on the external magneticﬁeld and is investigated in the next section by means of theOverhauser ﬁeld distribution. FIG. 7. Magnetic ﬁeld dependency of the electron spin re- vival amplitude. The converged revival amplitude S⊥(NPTR)a f t e r 1.5×106/lessorequalslantNP/lessorequalslant20×106is depicted as the blue curve. The exact number of pump pulses depends on the magnetic ﬁeld Bext.T h e dominating electron spin component /angbracketleftSz(NPTR)/angbracketrightin the full expres- sion for S⊥(NPTR) is indicated by red crosses. Furthermore, the revival amplitudes calculated from the Overhauser ﬁeld distributions according to Eq. ( 41) are added as green diamonds.Compared to the experimental results in Fig. 3, the revival amplitude shows a more pronounced oscillatory behaviordemonstrating two equally pronounced maxima in the mag-netic ﬁeld range up to 10 T. The results have a minimumin common at roughly 4 T enclosed by the maxima at lowerand higher external magnetic ﬁelds. In the experiments, theamplitude of the maxima decreases with stronger externalmagnetic ﬁelds. This effect is not visible in Fig. 7, indicat- ing that certain aspects are not yet captured. Among theseare (i) some sample dependencies as depicted in Fig. 3 (see Sec. VD) as well as (ii) the approximation of an instanta- neous pump pulse which is inappropriate for larger magneticﬁelds (see Sec. VII). C. Overhauser ﬁeld distribution During the pulse sequence, the nuclei align along the ex- ternal magnetic ﬁeld axis in such a way that the electron spinperforms a certain number of revolutions during the Larmorprecession between two successive pump pulses. The numbermof electron spin revolutions during T Ris determined by the combination of external magnetic ﬁeld and Overhauser ﬁeld m=geμBTR 2π¯h/parenleftbig Bx N+Bext/parenrightbig . (33) We adjust Bextsuch that for a zero Overhauser ﬁeld the electron spin performs an integer number of revolutions Bext=m/prime2π¯h geμBTR(34) withm/prime∈Z. Since we start in the high-temperature limit with an initial density operator ρ∝ˆ1, the initial Overhauser ﬁeld distribution p0(Bx N) is approximately a Gaussian due to the central limit theorem. For constant Ak, the distribution would be binomial. The numerical result for the initial distribution isshown in Fig. 8(a)forN=6 andN C=100 (blue curve). The ﬁnite-size noise arises from the mismatch between the discretebut random eigenvalue spectrum of the operator B x N:A sNC is larger, the eigenvalue spectrum becomes more continuous and the distribution will be smoother, even for small N.T h e distribution p0(Bx N)f o rN→∞ approaches a Gaussian [ 2] p/parenleftbig Bx N/parenrightbig =/radicalbigg 3 2πexp/bracketleftBigg −3 2/parenleftbigggeμBT∗ ¯hBx N/parenrightbigg2/bracketrightBigg , (35) in accordance with the central limit theorem and is added to Fig.8for comparison (the red dashed curve). During the pump sequence, p(Bx N) evolves into a peaked structure. The Overhauser ﬁeld distribution calculated afterN P=1.5×106pulses is shown in Fig. 8(b) for a ﬁxed ex- ternal magnetic ﬁeld, Bext=1.95 T. The maxima coincide with an integer number of electron spin revolutions during TR (marked by gray dashed lines). In order to reduce the ﬁnite-size noise, we deﬁne a relative Overhauser ﬁeld distribution prel/parenleftbig Bx N/parenrightbig =p/parenleftbig Bx N/parenrightbig −p0/parenleftbig Bx N/parenrightbig p0/parenleftbig Bx N/parenrightbig , (36) accounting for the normalized difference of p(Bx N)t ot h e initial distribution p0(Bx N)[10]. 155318-9IRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) FIG. 8. Overhauser ﬁeld distribution p(Bx N)f o rBext=1.95 T [i.e.,n=1i nE q .( 37)]. The Gaussian distribution given by Eq. ( 35) is indicated by the red dashed curve. The results for a quantum mechanical system with N=6a n d NC=100 are drawn as solid lines (blue). (a) The initial distribution p0(Bx N) before the ﬁrst pulse. We added p0(Bx N) obtained for NC=105in green for comparison as well. (b) p(Bx N) after 1.5 million pump pulses. Overhauser ﬁelds that correspond to an integer number of electron Larmor revolutionswithin T Rare indicated by the gray dashed vertical lines. The relative distributions of the Overhauser ﬁeld in Figs. 9(a)–9(f)reveal a dependency of the peak position on the external magnetic ﬁeld. In Ref. [ 10], a nuclear resonance condition Bext=nπ¯h 2TRgNμN≈n×1.95 T (37) attributed to the nuclear Zeeman term ( 5) was proposed, where ncounts the number of quarter turns of the nuclear spins within TR.A ne v e n nfavors a half-integer number mof elec- tron spin revolutions (half-integer resonance) while an odd n favors an integer number mwithin TR(integer resonance). The relative Overhauser ﬁeld distribution for integer ndeﬁned in Eq. ( 37) displays peaks [ 10] at either the gray dashed lines (half-integer mfor even n) or the green dotted lines (integer mfor odd n). For noninteger n, the relative Overhauser ﬁeld distribution has peaks at both positions, the gray dashed andthe green dotted ones [see Fig. 9(b)]. For higher magnetic ﬁeld, such as 7 .80 T ( n=4), that have not been treated in Ref. [ 10], we observe effects additional to Eq. ( 37). Here, we would have expected half-integer peaks, but peaks betweenthe integer and the half-integer positions occur.D. Analysis of the steady-state revival amplitude So far, we presented the results of a very expensive nu- merical calculation to iteratively solve the combination of ashort laser pulse that has been treated as instantaneous and thepropagation of an open quantum system between two pulsesrepetitively up to 20 million times. Now we present a simpli-ﬁed analysis that reveals the essential connection between therevival amplitude and the Overhauser ﬁeld distribution. We make use of the fact that at larger magnetic ﬁelds, (i) the effect of the Knight ﬁeld deﬁned in Eq. ( 7)i sw e a k compared to the nuclear Zeeman term and (ii) the collectiverotation of all nuclear spins around the external ﬁeld directiononly very weakly changes the transversal component of thetotal effective magnetic ﬁeld that the central spin is observing.The major additional contributions to the external magneticﬁeld arise from the xcomponent of the Overhauser ﬁeld that is quasistatic on the timescale of T R. While the quantum mechanical calculation presented above accounts for the fulldynamics of the problem, we explore a frozen Overhauserﬁeld approximation in this section, assuming a quasistaticOverhauser ﬁeld distribution. This is justiﬁed analytically by inspecting the magnitude of the individual A kentering the Hamiltonian or by the explicated demonstration of a very slowchange of the revival amplitude with the number of pulses aspresented in Fig. 6. To derive the relation between Overhauser ﬁeld distribution and revival amplitude, we start by treating a single conﬁgura-tionKof nuclear spins (in conﬁguration j). For this purpose, we consider the spin component /angbracketleftS z(NPTR)/angbracketright, which matches the revival amplitude S⊥(NPTR)i nF i g . 7forBext/greaterorequalslant2T almost perfectly. Since a pump pulse does not act on the thenuclear spins, we can relate the expectation value of the spincomponent /angbracketleftS z/angbracketrightb K,jbefore and /angbracketleftSz/angbracketrighta K,jafter the NPth pump pulse by [ 9,10] /angbracketleftSz(NPTR)/angbracketrighta K,j=1 2/parenleftbigg /angbracketleftSz(NPTR)/angbracketrightb K,j−¯h 2/parenrightbigg . (38) For the time evolution between pump pulses, we neglect the effect of the trion decay under the assumption γ/lessmuchωeand consider the nuclear spins as frozen. Note that the nuclearspins still rotate around the external magnetic ﬁeld, but theseadditional components are small and oscillating compared tothe total effective ﬁeld in xdirection and only will generate a very small perturbative effect in an external ﬁeld that is twoorders of magnitude larger than the Overhauser ﬁeld. Thisleads to the simpliﬁed relation /angbracketleftS z((NP+1)TR)/angbracketrightb K,j =/angbracketleftSz(NPTR)/angbracketrighta K,jcos((ωe+ωK,j)TR), (39) where we introduced the electron Larmor frequency ωK,j= geμBBK,j/¯hin the Overhauser ﬁeld BK,j. Iterating Eqs. ( 38) and ( 39) and assuming a steady state with constant revival amplitude /angbracketleftSz(NPTR)/angbracketrightb K,j=/angbracketleftSz((NP+1)TR)/angbracketrightb K,j, we obtain /angbracketleftSz(NPTR)/angbracketrightb K,j=−cos((ωe+ωK,j)TR)¯h 4−2 cos ((ωe+ωK,j)TR). (40) 155318-10MAGNETIC FIELD DEPENDENCE OF THE ELECTRON … PHYSICAL REVIEW B 98, 155318 (2018) FIG. 9. Relative Overhauser ﬁeld distribution prel(Bx N) for distinct external magnetic ﬁelds Bext. Overhauser ﬁelds that correspond to an integer (half-integer) number of electron Larmor revolutions are indicated by gray dashed (green dotted) vertical lines, respectively. The number NPof pump pulses is in the range 1 .5×106/lessorequalslantNP/lessorequalslant20×106. The total revival amplitude /angbracketleftSz(NPTR)/angbracketright=/summationdisplay K,jpK,j/angbracketleftSz(NPTR)/angbracketrightb K,j (41) in this approximation results from the sum over all nuclear conﬁgurations Kand coupling sets jweighted by their prob- ability pK,jintroduced in Eq. ( 30). We make use of the fact that the electron spin dynamics is very fast in comparison toa very slow change of nuclear spin distribution encoded inprobabilities p K,j. Equation ( 41) is the central result of this section: It re- lates the steady-state revival amplitude obtained in a frozenOverhauser ﬁeld approximation and the probability p K,jfor a speciﬁc Overhauser ﬁeld conﬁguration K,j to the total revival amplitude. The quality of this approximation relies onthe separation of timescales: While the electronic steady-stateis reached rather fast after only a few pulses as demonstratedin Fig. 5(a), the Overhauser ﬁeld distribution and therefore the probability p K,jevolves very slowly on the scale of thousands of pulses—see also Ref. [ 10]. For the calculation of /angbracketleftSz(NPTR)/angbracketrightaccording to Eq. ( 41), we use the weights pK,jobtained from the full numerical simulation. We added the results as function of the externalmagnetic ﬁeld into Fig. 7as green diamonds. They match the amplitude of the full quantum mechanical calculation verywell except for magnetic ﬁelds below 1 T, where the triondecay must be properly taken into account and the frozenOverhauser approximation becomes less justiﬁed. This agree-ment clearly demonstrates that the revival amplitude is fullydetermined by the Overhauser ﬁeld distribution p rel(Bx N). Thus, the maxima (minima) of the revival amplitude coincidewith odd (even) nin the resonance condition ( 37), respec- tively. For the continuous Gaussian distribution in Eq. ( 35), the initial revival /angbracketleftS z(NPTR)/angbracketright/¯h=−0.077 of the electronicsteady state [ 10], that has been deduced in Appendix B, results directly from Eq. ( 41). At the end of a very long pulse sequence, the Overhauser ﬁeld distribution has a peaked structure. We divide thesepeaks into two subgroups: one corresponding to the integerresonance and one for the half-integer resonance. Assum-ingδpeaks for each subgroup distribution, we obtain the value/angbracketleftS z(NPTR)/angbracketright=− 1/2 for the integer case and the value /angbracketleftSz(NPTR)/angbracketright=1/6 for the half-integer case from Eq. ( 40), respectively (cf. Ref. [ 9]). These values are independent of the resonance Larmor frequency ωK,j. Thus, the weights pK,jdo not enter the full revival amplitude in Eq. ( 41). Since the steady-state amplitudes have opposite signs forthe different resonance conditions a destructive interferencebetween these two subsets is found [ 9] and the ﬁnal value depends on the ratio between the fractional weights of theseparts. Compared to the initial value /angbracketleftS z(NPTR)/angbracketright/¯h=−0.077 (NP≈10) in the electronic steady state, the electron spin component \|/angbracketleftSz(NPTR)/angbracketright\|either increases (integer case) or decreases (half-integer case). As the peaks in our numericalcalculation of the Overhauser ﬁeld retain a ﬁnite width, therevival amplitude results from a superposition of the con-tributions from both resonances explaining the evolution ofS ⊥(NPTR) as depicted in Fig. 6. By means of the resonance condition for the Overhauser ﬁeld distribution, we are now able to understand why thebehavior of the revival amplitude is more complex in theexperiment (cf. Fig. 3) than presented in Fig. 7.W eh a v e simpliﬁed our theoretical model to a single type of nucleiwith a single average gfactor, whereas in real samples the gfactor differs between the elements In, Ga, and As as well as between the respective isotopes of an element. We ob-served the magnetic ﬁeld dependency of the revival amplitudestemming from the resonance condition ( 37) for different values of the nuclear gfactor g Ndetermining the number 155318-11IRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) of nuclear spin revolutions in the time TR(not shown here). For increasing gN, the minimum of the revival amplitude shifts to lower magnetic ﬁelds. Results presented in Ref. [ 10] indicate that each type of nucleus leads to a separate resonancecondition in the form of Eq. ( 37). The different kinds of peaks in the Overhauser ﬁeld distribution are more or lesspronounced depending on the external magnetic ﬁeld andwhich resonances of the various nuclear species are closest.As a result, the behavior of the revival amplitude is expectedto become more complex when involving several types ofnuclei. Since the individual gfactors of most nuclei induce a minimum of revival amplitude between 3.7 and 5 .2 T accord- ing to Eq. ( 37)(n=2), the combined behavior results in a minimum at around 4 T for both samples in Fig. 3. Additional nonmonotonic behavior distinguishing the samples can beattributed to the different concentration of the nuclear speciesin the QDs, e.g., due to the different thermal annealing of thesamples. At higher external magnetic ﬁelds, the resonancecondition for the different nuclear species disperses morestrongly leading to a decrease of the total revival amplitudefor both samples in Fig. 3. VI. CLASSICAL APPROACH TO THE QUANTUM DYNAMICS OF PERIODICALLY DRIVEN QDS A nonmonotonic dependence of the revival amplitude on the external magnetic ﬁeld is also obtained in an advancedclassical approach simulating the quantum dynamics. In thisapproach, the central electron spin and the nuclear spin bathare treated as classical vectors, but the average is takenover Gaussian distributed initial conditions which mimicsthe quantum mechanical dynamics [ 13,45–48]. The details of the approach are developed and analyzed in detail inRefs. [ 11,49]. We calculate the full time evolution of the classical equa- tions of motion of the CSM ( 2) for generically distributed dimensionless hyperﬁne couplings {A k} Ak=Ce−kζ, (42) where ζreplaces the parameter γin Ref. [ 11]. In the numer- ics,Cis chosen such that AQ:=/radicalBig/summationtext kA2 kis set to unity, i.e., all energies are measured in units of AQ. In order to enable a quantitative comparison to the experiment, we set ¯ h/A Q= 0.79 ns, which implies that for bath spins I=3/2 the charac- teristic time reads T∗=¯h/[AQ√I(I+1)]=0.41 ns accord- ing to Eq. ( 8), in good agreement with the experiment [ 1,30] (cf. Sec. III). The average over 104–105random initial conﬁgurations is used to approximate the quantum mechanical behaviorof a single quantum dot [ 46,48]. The initial values of each conﬁguration are drawn from a Gaussian distribution withvanishing average value and a variance reﬂecting the spinlength, i.e., 1 /4 for each component of the central electron spin and 5 /4 for each component of a nuclear spin with spin I=3/2. The full time evolution of the Overhauser ﬁeld is simu- lated efﬁciently by the spectral density approach developed inRef. [ 49]. It allows us to consider an inﬁnite spin bath, while the number of effectively coupled spins is ﬁnite and given byN eff≈2/ζ[11,49,50]. In our calculations, we use between 44 and 74 auxiliary vectors, where the exact number dependsonN P(cf. Refs. [ 11,49]), to represent bath sizes of up to Neff=667. The Zeeman effect of the magnetic ﬁeld applied to the central spin is taken into account by adding a term hSx withh=geμBBext/¯h, while the Zeeman effect of the nuclear spins is reduced by the factor z/lessmuch1 according to h→zh.T h e valuez=1/800 represents a good estimate [ 10] as discussed in Sec. III. Note that we are considering a single quantum dot here, not an ensemble, but the extension to an ensemble ofQDs is straightforward. The quantum mechanical description of the pump pulses is involved as is evident from the above discussion. In the ap-proximating classical simulation, we pursue two aims. On theone hand, we aim at a transparent description in the classicalapproach. On the other hand, it should mimic the quantummechanical properties best. In previous work [ 11], we found that the following assumption leads to convincing results. Inparticular, it leads to nonmonotonic revival amplitudes. In our pulse description, the pulse affects the vector of the central spin instantaneously. Independent of the directionprior to the pulse, right after the pulse the vector of the centralspin becomes /vectorS→⎛ ⎝X Y 1/2⎞ ⎠. (43) This means that we assume the pulse to be perfect in the sense that it produces maximum alignment along the zaxis. The values of XandYare chosen randomly for each pulse from a Gaussian distribution with vanishing mean value 0and variance 1 /4. This randomness is introduced to respect Heisenberg’s uncertainty relation for the electron spin, whichforbids a perfect alignment. Additionally, it ensures that theexpectation value for the spin length /angbracketleft/vectorS 2/angbracketrighttakes the correct value of 3 /4. To consider this sort of classical pulse mimick- ing quantum mechanics is motivated by viewing the pulse as aquantum mechanical measurement with a deﬁnite outcome forthezcomponent. In Ref. [ 11], this type of pulse was denoted as pulse model II. It represents an extension of the pulsesstudied in previous works [ 8,11]. The used values of the parameters, e.g., T ∗, differ slightly from those used in Sec. IVbut still correspond to the values typical for (In,Ga)As/GaAs quantum dots as measured inSec. II. Hence, the results can be compared at least qualita- tively. Simulating up to 10 6pulses, we are able to reliably ex- trapolate a value for the saturated revival amplitude Slim.T h e explicit value is calculated by ﬁtting the function S⊥(NPTR)=Slim,0(1−e−NPTR/τ)+S0 (44) to the data. Eventually, the revival amplitude is given by Slim=Slim,0+S0. This analysis is carried out for various external magnetic ﬁelds up to 10 T for two different effectivebath sizes N eff≈200 (ζ=0.01) and 667 ( ζ=0.003). An illustration of the ﬁt procedure for various external magneticﬁelds is depicted in Fig. 10. The time required to approach the saturation value scales linearly in the inverse size of the bath ∝ζ∝1/N effand quadratically in the magnetic ﬁeld ∝B2 extas analyzed in 155318-12MAGNETIC FIELD DEPENDENCE OF THE ELECTRON … PHYSICAL REVIEW B 98, 155318 (2018) FIG. 10. Evolution of the revival amplitude S⊥(NPTR) as func- tion of the pulse number NPfor various external magnetic ﬁelds Bext, averaged over 25 200 random initial conﬁgurations with Neff=200. The solid black lines show the ﬁts ( 44), yielding the saturated value Slimof the revival amplitude. Ref. [ 11]. Hence, the simulations become very tedious for large magnetic ﬁelds and large bath sizes. Thus, we have torestrict ourselves to moderate bath sizes in this study, but theystill exceed the bath sizes which can be addressed quantummechanically by two orders of magnitude so that they yieldcomplementary information. The results are compiled in Fig. 11. The nonmonotonic dependence of S limon the external magnetic ﬁeld Bextshows a pronounced minimum at around 4 T similar to what is foundin Fig. 7for the quantum mechanical approach although the details are different. A less pronounced minimum occurs ataround 8 T, which is much narrower than what is found quan-tum mechanically. Additionally, there is a maximum slightlybelow 1 T. Such a maximum is also found in the quantummechanical approach but shifted to larger magnetic ﬁelds,which may result from the difference in bath sizes and fromthe difference between the full quantum mechanical dynamicsand the classical simulation. The comparison to the experimental data in Fig. 3also reveals strong similarities such as the pronounced minimum atabout 4 T and weaker structures at around 8 T. The minimum FIG. 11. Dependence of the saturated revival amplitude Slimon the external magnetic ﬁeld. The saturated revival amplitudes are de- termined by ﬁtting Eq. ( 44) to the data from the classical simulations. The lines are guides to the eye.FIG. 12. Distribution of the Overhauser ﬁeld xcomponent ob- tained from the classical simulations. The calculations were per-formed for an ensemble of 25 200 random initial conﬁgurations with N eff=200. The vertical dashed lines indicate the integer resonance condition. at 8 T is very narrow and requires many data points to be resolved correctly. Note also the similarity to the experimentaldata for the revival amplitude published in Fig. 20 in Ref. [ 9]. The position of the maximum to the left of the minimumdiffers because there are additional experimental features. Wepresume that they result from the different species of nucleipresent in the samples as discussed in Sec. VD. Concomi- tantly, there are ﬁve different g Nfactors which one should consider, while our theoretical treatments deal with one aver-ageg Nfactor only. In addition, the classical simulation does not treat an ensemble of QDs; i.e., the effects of a spread inT ∗and in the electronic gfactor are not yet included. We emphasize that in the classical simulation, the buildup of the revival amplitude is solely due to the frequency focusingof the nuclei, i.e., of the formation of a comblike structurein the distribution of the Overhauser ﬁeld. The contributionfrom the electronic steady-state condition is not included.Figure 12shows the almost stationary distribution of the x component of the Overhauser ﬁeld for two external magneticﬁeldsB ext=0.93 T and 3 .71 T. As an aside, we note that it is not the xcomponent of the total magnetic ﬁeld /vectorB, external and Overhauser, which matters [ 11], but its length \|/vectorB\|. The ﬁrst magnetic ﬁeld corresponds to the blue curve for the revivalamplitude in Fig. 10, the second ﬁeld to the red curve. Both distributions show a comblike structure of nuclear focusingwith peaks corresponding to the integer resonance condition, i.e., for an integer number of electron spin revolutions withinthe interval T Rbetween two pulses. But the width and con- comitantly the height of the peaks differ substantially. Thisexplains the much smaller revival amplitude for the magneticﬁeld close to 4 and 8 T. In general, we ﬁnd that a largervalue of the revival amplitude corresponds to sharper peaks.We do not observe additional peaks at the Overhauser ﬁeldscorresponding to half-integer resonances. The buildup of nuclear frequency focusing in the classical simulations has been studied in detail in Ref. [ 11]. For the pulse model ( 43), it was found that the buildup rate scales approximately with 1 /N eff. No perfect scaling was found so that there remains a dependence on the bath size; this is also 155318-13IRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) manifest in Fig. 11where the long-time minima and maxima are more pronounced for larger Neff. For the dependence on the external magnetic ﬁeld, a nonmonotonic behavior wasfound [ 11]. However, the overall time required to reach a stationary Overhauser ﬁeld distribution, and hence a saturatedrevival amplitude, scales approximately with B 2 ext(cf. Fig. 10). What is the reason in the classical simulations for the nonmonotonic dependence on the external magnetic ﬁeldshown in Fig. 11? It does capture the interplay of electronic and nuclear precessions. An important additional clue is ob-tained from setting X=Y=0 in each classical pulse ( 43), i.e., from neglecting the uncertainty in the spin orientation.Then, the revival amplitude saturates at S lim=1/2, totally independently of the value of the applied external magneticﬁeld (cf. Ref. [ 11]). The distribution of the xcomponent of the Overhauser ﬁeld displays very sharp peaks at positionscorresponding to the integer resonance condition, leading to aperfect refocusing of the electron spin precession before eachnext pulse, i.e., to a maximum revival amplitude. Hence, it isindeed the quantum uncertainty, mimicked by the randomnessofXandYin the classical simulations, which is decisive for the ﬁnite peak widths shown in Fig. 12which imply the reduced revival amplitude and eventually the nonmonotonicbehavior depicted in Fig. 11. An additional piece of information, in which way the randomness in XandYacts against perfect nuclear focusing, results from the following observation for a magnetic ﬁeldaround 4 T. Including only the ﬂuctuations in the ycomponent results in half-integer resonances while including only theﬂuctuations in the xcomponent results in integer resonances. Hence, they act against each other and the reduced nuclearfocusing is an effect of destructive interference. Clearly, itwill be attractive to clarify this issue further by analyticalconsiderations. VII. NONINSTANTANEOUS PUMP PULSES IN THE QUANTUM MECHANICAL APPROACH So far, we only took into account instantaneous πpulses that resonantly excite the trion in the quantum mechanical ap-proach and affect the electron spin in the classical simulation.Experimental pump pulses, however, have a ﬁnite durationof a few picoseconds [ 1,30,31]. A deviation from a perfect resonance condition due to the electronic Zeeman energy aswell as the spin precession during the pulses might affectthe steady-state revival amplitude at large external magneticﬁelds. Furthermore, an extension to arbitrary pulse shapes willopen a new door for more complex pulse sequences in thefuture. As a ﬁrst step for more realistic pulses, we consider Gaussian pump pulses in the quantum mechanical approach.Thus, we need to replace the unitary pulse operator U Pby a new operator U/prime P. This operator U/prime Pis obtained by integrating the equation of motion for the unitary time evolution duringthe pulse duration. For this purpose, we use the light-matterHamiltonian in the rotating-wave approximation [ 44] H L(t)=f(t)e−iωLt/¯h\|T/angbracketright/angbracketleft↑\|z+H.c., (45) where ωLdenotes the laser frequency. The Gaussian pulse shape is included in the (complex) envelope function f(t).During the pump pulse, the total Hamiltonian is given by H(t)=HL(t)+HCSM+/epsilon1\|T/angbracketright/angbracketleftT\|. The trion decay is ne- glected, as the decay rate γ=10 ns−1is slow compared to the duration TPof the pulse. First, we transform into the frame rotating with the laser frequency ωLand eliminate the fast oscillation with ωLin the Hamiltonian. Introducing the detuning δ=ωL−/epsilon1, we obtain the transformed Hamiltonian H/prime(t)=eiωL\|T/angbracketright/angbracketleftT\|t/¯h(H(t)−ωL\|T/angbracketright/angbracketleftT\|)e−iωL\|T/angbracketright/angbracketleftT\|t/¯h =f(t)\|T/angbracketright/angbracketleft↑\|z+f∗(t)\|↑/angbracketrightz/angbracketleftT\| +HCSM−δ\|T/angbracketright/angbracketleftT\|. (46) Second, we discretize the Hamiltonian H/prime(t) in small time steps, deﬁning intervals for which f(tn) can be considered approximately as constant. In our numerics, we typicallydivide a single laser pulse in 1000 time steps so that /Delta1t≈ 22 fs for a total pulse duration T P≈22 ps. The unitary time evolution is approximated by operators U(tn)=e−iH/prime(tn)/Delta1t/¯h(47) and their Hermitian conjugates U†(tn), where /Delta1t=tn+1−tn is the step width in time. Neglecting the Trotter error, which vanishes for /Delta1t→0, the unitary transformation is given by the product of all individual transformations U/prime P=e−iωL\|T/angbracketright/angbracketleftT\|TP/¯h/productdisplay nU(tn) =e−iωL\|T/angbracketright/angbracketleftT\|TP/¯hU(TP)...U (t2)U(t1). (48) Note that the additional exponential factor accounts for the back transformation from the rotating frame. The transforma-tion of the density matrix into the rotating frame is omitted. The pulse action is described by ρ(T P)=U/prime Pρ(0)U/prime† P, (49) where ρ(0) and ρ(TP) are the density operator before and after the pulse respectively. Since the unitary transformationis obtained initially and stored as one unitary complex matrix,modiﬁed pulses just come at the expense of two additionalcomplex matrix multiplications in the numerical implementa-tion. After each pulse, the time evolution is again calculated using the Lindblad equation discussed in Sec. IV C with ρ(T P) obtained via Eq. ( 49) as input. Since the pump pulse now has a ﬁnite duration TP, we evaluate the time evolution via Lindblad equation for a reduced duration TR−TP. For the calculations presented below, we choose a Gaussian pulse shape with f(t)=f∗(t), whose iterated area corre- sponds to a πpulse. The full width at half maximum (FWHM) is adjusted to 6 ps. This width is slightly larger than in theexperiments but renders possible effects on the spin dynamicsmore visible. The duration of the pulse is set to T P≈22 ps, within which we consider the part of the pulse up to which theenvelope f(t) has decayed to a hundredth of its maximum. For the laser frequency, we restrict ourselves to ω L=/epsilon1, such that the trion is resonantly excited without detuning ( δ=0), and leave the investigation of the inﬂuence of the detuning ina quantum dot ensemble to future studies. 155318-14MAGNETIC FIELD DEPENDENCE OF THE ELECTRON … PHYSICAL REVIEW B 98, 155318 (2018) FIG. 13. Evolution of the electron spin revival amplitude with the pulse number NPfor Gaussian pump pulses in the quantum mechanical approach. Various colors show the development for different external magnetic ﬁelds Bext. We use the same parameters as before but replace the instantaneous πpulses by Gaussian shaped pulses with a ﬁnite width. The pulse-number-dependent revival amplitude forsuch Gaussian pulses is shown in Fig. 13for the same external magnetic ﬁeld values as in Fig. 6. In comparison with the result for instantaneous pump pulses in Fig. 6, a slower rate of change is observed. Therefore, a larger number of pumppulses is required to reach a converged steady-state revivalamplitude. Especially for higher magnetic ﬁelds, the pumppulses become less efﬁcient, as the electron spin precessesduring the pulse duration. To investigate the inﬂuence of the Gaussian pump pulses on the magnetic ﬁeld dependency, the converged revival am-plitude S ⊥(NPTR) is again plotted as function of Bext.T h e result in Fig. 14(red curve) shows some difference to the data for instantaneous pump pulses taken from Fig. 7which we added for comparison (blue curve), even though the overallqualitative behavior remains the same. There are still twomaxima and two minima respectively in the magnetic ﬁeldrange up to 10 T, but the maximum amplitude has decreased.Besides, the amplitude of the second maximum is smallerthan the amplitude of the ﬁrst maximum. Note that the revivalamplitude for the data point at B ext=9.75 T is not com- pletely converged (see Fig. 13, green curve) and therefore the minimum at about 8 T is not very pronounced. However, weagain observe that the revival is weaker for higher externalmagnetic ﬁelds. This behavior matches the overall decreaseof the revival amplitude with the external magnetic ﬁeld inthe experimental data in Fig. 3. Thus, the ﬁnite pulse duration is another aspect which has to be included for a realisticdescription of the experiments. We augment the analysis by adding the spin component \|/angbracketleftS z(NPTR)/angbracketright\|as green crosses to Fig. 14. While the spin polarization in the zdirection agrees well with the revival amplitude S⊥(NPTR) in the interval 2 T /lessorequalslantBext/lessorequalslant6T , s i g - niﬁcant deviations are found for Bext<2 T as well as for Bext>6 T. In these regions, the spin component /angbracketleftSy(NPTR)/angbracketright does not vanish. FIG. 14. Magnetic ﬁeld dependency of the electron spin revival amplitude calculated by the quantum mechanical approach with Gaussian pump pulses. The revival amplitude S⊥ G(NPTR) and the spin component \|/angbracketleftSz G(NPTR)/angbracketright\|are taken after a number of pump pulses 2 .5×106/lessorequalslantNP/lessorequalslant20×106large enough such that they have converged. The exact value of NPdepends on the magnetic ﬁeld. For comparison, we added the revival amplitude S⊥ I(NPTR) with instantaneous pump pulses taken from Fig. 7. For further investigation of the Gaussian pump pulses, we inspect the relative Overhauser ﬁeld distribution in Fig. 15 (red solid lines). Here, we present the distributions for thesame external magnetic ﬁelds as in Fig. 9for the instantaneous laser pulses. The previous results for instantaneous pumppulses are added to Fig. 15for comparison as blue dashed lines. For external magnetic ﬁelds up to 2 .93 T, we do not observe signiﬁcant differences in the distributions for the twotypes of pump pulses. However, for the ﬁrst external magneticﬁeld with even nin Eq. ( 37)(B ext=3.90 T), for which we found peaks at the green dotted lines for the instantaneouspump pulses, we also ﬁnd tiny peaks at the gray dashed posi-tions for the Gaussian pump pulses. For even higher externalmagnetic ﬁelds, the differences become more signiﬁcant. For n=4(B ext=7.80 T), one kind of peaks is more pronounced than the other. The peaks for external magnetic ﬁeld with oddn(B ext=5.85 T and Bext=9.75 T) are slightly shifted to the right from their original position at the gray dashed lines.Therefore, the shape of the pump pulses seems to inﬂuencethe resonance condition for the Overhauser ﬁeld distributionand thus the electron spin revival amplitude. VIII. SUMMARY AND CONCLUSION We investigated the magnetic ﬁeld dependency of the revival amplitude of the electron spin polarization alongthe optical axis in a periodically pulsed QD ensemble. Thesteady-state resonance condition leads to a signiﬁcant revivaldirectly before each pump pulse. This has been qualitativelyexplained by the mode locking of the electron spin dynamics,comprising a synchronization of the electron spin precessionimposed by the periodic pumping and an enhancement by the 155318-15IRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) FIG. 15. Relative Overhauser ﬁeld distribution prel(Bx N) for various external magnetic ﬁelds Bext. The considered pulse sequence consists of Gaussian pump pulses (red solid lines). Overhauser ﬁelds that correspond to an integer (half-integer) number of electron spin revolutions during TRare indicated by gray dashed (green dotted) vertical lines respectively. The number NPof pump pulses is in the range 2 .5×106/lessorequalslant NP/lessorequalslant20×106. For comparison, we added the results for instantaneous pump pulses taken from Fig. 9as blue dotted lines. nuclear frequency focusing that develops on a much longer time scale [ 1,30,31]. The nonmonotonic magnetic ﬁeld dependency of the re- vival amplitude, however, had not been theoretically under-stood. In this paper, our simulations of the CSM subject to upto 20 million laser pulses are able to link this nonlinear ﬁelddependency to the nuclear Zeeman effect. The quantum mechanical calculations are based on an extension of the CSM including the trion excitation due tothe pump pulses. The time evolution between two successivepump pulses including the trion decay is described by aLindblad equation for open quantum systems that is exactlysolved for each pulse interval. Although our approach cantreat arbitrary pulse shapes and durations, we focus on π pulses in this paper. In order to achieve pulse sequences with up to 20 million pump pulses in our quantum mechanical approach, we restrictourselves to a small bath of N=6 nuclear spins due to CPU time limitations. Even though in real QDs an electron spincouples to the order of 10 5nuclear spins, it is already estab- lished that the generic spin dynamics of the CSM can alreadybe accessed by a relatively small number of nuclei [ 14,15]. We simulated a distribution of different characteristic timescalesT ∗ jin a QD ensemble by the treatment of NC=100 conﬁg- urations with distinct hyperﬁne coupling constants Ak,j.T h e number of pump pulses required to reach a converged revivalamplitude grows with increasing external magnetic ﬁeld. To support the demanding quantum mechanical compu- tations, we also perform a classical simulation of the CSMwhich simulates a bath of up to 670 effectively coupled spins[11]. This simulation is set up such that it approximates the quantum mechanical dynamics as closely as possible, but theintermediate trion excitation and its subsequent fast decay arenot built into the classical treatment.Both approaches cover up to 11 orders of magnitude in times: from a single laser pulse with the duration of 2–10 ps,the laser repetition time of 13.2 ns, to 20 million pulsesreaching a total simulation time of approximately 0 .2s . O u r key ﬁnding is that the stationary revival amplitudes exhibita nonmonotonic behavior as function of the applied externalmagnetic ﬁeld. There are minima of the revival amplitude at4 and 8 T, which roughly match the experimental data. In the quantum mechanical approach, the steady-state res- onance conditions favor an integer or a half-integer numberof electron spin revolutions between two pump pulses andeventually lead to a rearrangement of the Overhauser ﬁelddistribution function similar to the one found in Refs. [ 9–11]. The minimum of the revival amplitude is reached in thecase of the half-integer resonance, whereas the maximumcorresponds to the integer resonance. In the simulations, we only included a single average nuclear gfactor but were able to link the revival minima to the nuclear gfactor by variation of its value. However, it has been indicated [ 10] that in real QDs the different nuclear species yield separate resonance conditions. Since the nuclear gfactor is isotope dependent, the experimental response is not uniquebut sample dependent. The mechanism generating the magnetic ﬁeld dependency in the classical simulations works similarly, but with oneimportant difference. No peaks at the Overhauser ﬁelds ofthe half-integer resonances occur. Instead, the peaks corre-sponding to the integer resonances become broad and lesspronounced for an even number of nuclear quarter turns.Hence, the nuclear frequency focusing is not very efﬁcientand the revival amplitudes are small again due to a partialdestructive interference. We have also extended the quantum mechanical theory from instantaneous laser pulses to pulses with a ﬁnite width 155318-16MAGNETIC FIELD DEPENDENCE OF THE ELECTRON … PHYSICAL REVIEW B 98, 155318 (2018) of 6 ps. The pulses have a Gaussian shape with an area corresponding to the instantaneous πpulses. In this way, we take into account the possible detuning of the resonancefrequency in a strong magnetic ﬁeld by the Zeeman effect aswell as the electron spin rotation during the pulse duration.Deviations from the instantaneous pulses occur at higherexternal magnetic ﬁeld, when the electron spin rotation isnon-negligible during the pulse duration. Here, the pulse isless efﬁcient and the formation of a revival is less pronounced.Besides, the resonance condition for the Overhauser ﬁeld isslightly shifted for higher external magnetic ﬁelds. Even though we restricted ourselves to resonant Gaussian πpulses, the effects of arbitrary pulse shapes as well as the de- tuning of laser frequency become accessible by our approachand present an interesting ﬁeld for future research. Finite pulselengths, detuned laser frequencies, and pulse shapes whichdo not correspond to πpulses will be addressed with our approach to design specially tailored and optimized pulsetrains for quantum coherent control. Furthermore, we stressthat the theoretical approaches developed and used in thiswork can be applied to a considerable variety of experimentson QDs subject to optical pulses. The pulse trains need not beperiodic but could be varied to a large extent. ACKNOWLEDGMENTS We are grateful for fruitful discussions on the project with A. Fischer and N. Jäschke. We acknowledge the supply ofthe quantum dot samples by D. Reuter and A. D. Wieck(Bochum). We also acknowledge ﬁnancial support by theDeutsche Forschungsgemeinschaft and the Russian Founda-tion of Basic Research through the transregio TRR 160 withinthe Projects No. A1, No. A4, No. A5, and No. A7 as well asﬁnancial support by the Ministry of Education and Scienceof the Russian Federation (Contract No. 14.Z50.31.0021,leading researcher M. Bayer). M.B. and A.G. acknowledgethe support by the BMBF in the frame of the Project Q.com-H(Contract No. 16KIS0104K). The authors gratefully acknowl-edge the computing time granted by the John von NeumannInstitute for Computing (NIC) under Project HDO09 andprovided on the supercomputer JUQUEEN at the Jülich Su-percomputing Centre. APPENDIX A: PARTICULAR SOLUTION FOR THE LINDBLAD EQUATION To obtain a particular solution to the Lindblad Eq. ( 23), we need to calculate the operators ˜ χ0,˜χ+, and ˜ χ−in the ansatz ( 25). For this purpose, we insert Eq. ( 25) into Eq. ( 23). Separating the terms according to the three different expo-nents in the exponential functions yields the conditions ( α∈ {0,+,−}) [iω N(δα,+−δα,−)−2γ]˜χα=−i ¯h[˜HS,˜χα]+γrα˜ρTT(0). (A1) Here,δα,+andδα,−denote the Kronecker symbol. The opera- torsrαare deﬁned as r0= \|↑/angbracketright/angbracketleft↑\| + \|↓/angbracketright/angbracketleft↓\| ,r+= \|↓/angbracketright/angbracketleft↑\| , and r−= \|↑/angbracketright/angbracketleft↓\| .Equation ( A1) can be solved by transforming into the eigenbasis of ˜HS=SDS†, where Dis diagonal. We introduce ˜χ/prime α=S†˜χαSand consider the transformed Eq. ( A1) element- wise. Rearranging for the elements of ˜ χ/prime α, we obtain ˜χ/prime α=Gα◦(S†(rα˜ρTT(0))S)( A 2 ) with a Schur product denoted by ◦. The elements of operator Gαare given by (Gα)a,b=γ{−2γ+iωN(δα,+−δα,−) +i((D)a,a−(D)b,b)}−1. (A3) Finally, the operators ˜ χαresult from transforming from the eigenbasis of ˜HSback into the original basis. Altogether, this approach allows us to diagonalize ˜HSand prepare the three operators Gαbefore the simulation of a pulse sequence. During the pulse sequence, the operator ˜ ρTT(0) after each pump pulse has to be inserted in Eq. ( A2). The results for ˜ χ/prime αare transformed via ˜ χα=S˜χ/prime αS†and then enter the time evolution of ˜ ρSin Eq. ( 26). APPENDIX B: REVIV AL AMPLITUDE OF THE ELECTRONIC STEADY STATE Even before the nuclear spins are affected by the pump pulses, a revival amplitude appears due to a purely electronicsteady state [ 10]. The evolution of this electronic revival can be understood by iteration of the pump pulse [cf. Eq. ( 38)] and the evolution for the time T R[cf. Eq. ( 39)]. Similar to the calculation of the revival amplitude in Eq. ( 41), we ﬁrst consider a single nuclear conﬁguration Kfor a set jof couplings. The iteration of Eqs. ( 38) and ( 39) yields /angbracketleftSz(NPTR)/angbracketrightb K,j=−NP/summationdisplay i=1¯h 2i+1{cos((ωe+ωK,j)TR)}i(B1) afterNPpump pulses. If we assume our external magnetic ﬁeld to ensure an integer number of electron spin revolutionsbetween two pump pulses, ω eTRis an integer multiple of 2 π and can be omitted in the cosine. The full revival amplituderesults from integrating over ω K,jweighted by its Gaussian FIG. 16. Evolution of the electron spin revival amplitude for small numbers NPof pump pulses, i.e., without nuclear frequency focusing. Various colors show the amplitudes for different external magnetic ﬁelds Bext. The black curve is calculated analytically from Eq. ( B2). The analytic revival amplitude \|/angbracketleftSz(NPTR)/angbracketright∞\|= \|1/2−1/√ 3\|≈0.077 in the limit NP→∞ is indicated by a gray dashed horizontal line. 155318-17IRIS KLEINJOHANN et al. PHYSICAL REVIEW B 98, 155318 (2018) distribution as we do not consider nuclear focusing. Since the width of the Gaussian distribution of ωK,jis proportional to the inverse T∗, it is large compared to the periodicity of the cosine in Eq. ( B1) that is determined by the inverse ofTR. Hence, we substitute the integration of Eq. ( B1) overωK,jby an integration over one period of the cosine (ωK,j∈[0; 2π/TR]): /angbracketleftSz(NPTR)/angbracketright=−NP/summationdisplay i=1¯h 2i+2/integraldisplay2π/TR 0dωK,j[cos(ωK,jTR)]i =−⌊NP/2⌋/summationdisplay i=1¯h 24i+1(2i)! (i!)2. (B2) Since the integral over ωK,jyields zero for odd i, we trans- form the index of summation i→i/2 in the second line of Eq. ( B2). From a physical point of view, the contributions to the revival amplitude from different Overhauser ﬁelds canceleach other for every second pulse. Thus, the revival amplitude increases with NPin steps of two. Note that we obtain /angbracketleftSy(NPTR)/angbracketrightb K,j=−NP/summationdisplay i=1¯h 2i+1[cos((ωe+ωK,j)TR)]i−1 ×sin((ωe+ωK,j)TR) (B3) for the spin component in the ydirection. Thus, the integration in analogy to Eq. ( B2) yields /angbracketleftSy(NPTR)/angbracketright=0 and we can stateS⊥(NPTR)=\| /angbracketleftSz(NPTR)/angbracketright\|in the analytic calculation. The limit NP→∞ yields the ﬁnal revival amplitude of the electronic steady state /angbracketleftSz(NPTR)/angbracketright∞=1/2−1/√ 3≈ −0.077. In Fig. 16, the growth of revival amplitude up to the 10th pump pulse is illustrated. The deviations of the numericalcalculations (colored symbols) from Eq. ( B2) (black curve) are only minor and due to the ﬁnite number of nuclear spins(N=6). [1] A. Greilich, R. Oulton, E. A. Zhukov, I. A. Yugova, D. R. Yakovlev, M. Bayer, A. Shabaev, A. L. Efros, I. A. Merkulov,V . Stavarache, D. Reuter, and A. Wieck, Phys. Rev. Lett. 96, 227401 (2006 ); A. Greilich, A. Shabaev, D. R. 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1.1703153.pdf	Vacancy and Interstitial Cluster Production in NeutronIrradiated α Iron J. R. Beeler Jr. Citation: Journal of Applied Physics 37, 3000 (1966); doi: 10.1063/1.1703153 View online: http://dx.doi.org/10.1063/1.1703153 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/37/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multivacancy clusters in neutronirradiated silicon J. Appl. Phys. 78, 6458 (1995); 10.1063/1.360530 Further Comments on ``Vacancy and Interstitial Clusters in NeutronIrradiated Nickel'' J. Appl. Phys. 38, 3033 (1967); 10.1063/1.1710052 Comments on the Identification of Vacancy and Interstitial Clusters in NeutronIrradiated Nickel J. Appl. Phys. 38, 1973 (1967); 10.1063/1.1709792 Vacancy and Interstitial Clusters in NeutronIrradiated Nickel J. Appl. Phys. 37, 3317 (1966); 10.1063/1.1703202 Primary Damage State in NeutronIrradiated Iron J. Appl. Phys. 35, 2226 (1964); 10.1063/1.1702823 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:21JOURNAL OF APPLIED PHYSICS VOLUME 37, NUMBER 8 JULY 1966 Vacancy and Interstitial Cluster Production in Neutron-Irradiated n: Iron* J. R. BEELER, JR. General Electric Company-NMPO, Cincinnati, Ohio (Received 15 November 1965) Vacancy cluster and interstitial c~u.ster production.in a iron was computed by simulating atomic collision cascad.es on a computer. Each collisIOn was determmed by the Erginsoy-Vineyard interatomic potential for a . Iron. The computed densities. of . displa~ement spikes produced by primary knock-on atoms with energl~s abo:,~ 3 keV corr~lated quahtahvely wIth the degrees of irradiation hardening observed by Harries et al. m fe.r~tlc ~teel specrmens i?r :five different neutron energy spectra. The computed total displaced ~tom. densIt~es dl,d n.ot correlate w:th the degrees of irradiation hardening observed by Harries et al. Anneal mg SImulatIOns mdicated that ?Ispla~ement spikes produced by primary knock-on atoms with energies below 2.5 keV should not contnbute Importantly to irradiation hardening in specimens irradiated at the temperatur~ ~",60°C) adopted ~y H~rries et al. The volume of collided atoms involved in a collision cascade us~ally exhibIted a m~rked ?ncutatI?n along. (110) directions as did the associated displacement spike. SpIkes ~re ther~fore onented m the pnmary slip planes of a iron and each could serve in toto as a barrier to dIslocatIOn motton. 1. INTRODUCTION THE n:echanical proper~ies of a~ irradiated metallic speCImen are determmed by mteractions among dislocations, impurity atoms, vacancies, and inter stitials.1 These interactions depend importantly upon the structure and deployment of defect clusters and defect-impurity-atom complexes at the atomic dimen sion scale. of observation. This suggests that a damage computatIOn method should give the spatial distribution of vacancies and interstitials produced by irradiation, at the level of atomic dimension, if it is to be of signifi cant service in the interpretation of mechanical property changes caused by irradiation. This paper describes the application of such a method in computing the number of atomic displacements produced per unit exposure in neutron-irradiated a-iron specimens. The development and use of this approach was motivated by an interest in the interpretation of irradiation effects data on the mechanical properties of metals. The calculations were performed by simulating atomic elastic collision cas cades in the bcc atomic array of a iron, at OOK, on an IBM 7094 computer. Following Brinkman, the collec tion of vacancies and interstitials produced by such a cascade in iron is called a displacement spike.2 The Erginsoy-Vineyard interatomic potential3 was used to characterize atomic collisions in the cascade simulations. Each cascade was initiated by a primary knock-on atom (PKA) directly dislodged from a normal atom position by a neutron collision. The energy spec- * This paper originated from work sponsored by the Fuels and Materials Development Branch, Atomic Euergy Commission under Contract AT(40-1)-2847. A preliminary version appears in Trans. Am. Nucl. Soc. 8, 5 (1965). 1 D. S. Billington and J. H. Crawford, Jr., Radiation Damage in Solids (Princeton University Press, Princeton, New Jersey 1961)' G. J. Dienes and G. H. Vineyard, Radiation Effects i~ SoUck (Interscience Publishers, Inc., New York, 1957); D. K. Holmes in T/ze Interaction of Radiation with Solids, R. Strumane J. NihouI' R. Gevers, and S. Amelinckx, Eds. (North-Holland 'Publishing Co., Amsterdam, The Netherlands, 1964). 2 J. A. Brinkman, Am. J. Phys. 24, 246 (1956). 8 C. Erginsoy, G. H. Vineyard, and A. Englert, Phys. Rev. 133, A595 (1964). trum and spatial distribution of these cascade-initiating PKA were obtained by simulating neutron collision chains in an a-iron rod 6.3 cm long and 0.28 cm in diameter. As shown elsewhere,4 the PKA energy-spec trum results for this particular rod also apply to rods with diameters in the range 0.05 to 0.5 cm and with l:ngt?s greater tha? 5 cm. The defect spatial distribu tIon m such rods IS nearly uniform. All experimental results we will quote were obtained from rods with di mensions in the size range defined above. The compu tational procedure used accounts for the effects of the neutron energy spectrum, neutron angular distribution specimen size, crystal structure, and current damag; state upon the production and distribution of new damage.4 An estimate of the damage state produced at a finite irradiation temperature T was made by simulat ing the annealing of displacement spikes initially produced at OOK, on a computer at temperature T. It is thought that defects produced by irradiation cause hardening in the same way as do solute atom aggregates,5 namely, by impeding the movement of dislocations. As pointed out by Fleischer,6 any deviation from perfect crystal structure acts as a barrier to dis location motion with an asymmetric distortion tending to be a more effective barrier than a symmetric one. Figure 1 is a schema of a dislocation line held in a mini mum potential energy configuration by barriers. Our results, in conjunction with experimental data of FIG. 1. SchCUla of a dislocation line held in a minimum energy con:figIlra tion by barriers. • J. R. Beeler, Jr., J. Appl. Phys. 35, 2226 (1964). 6 R. L. Fleischer, in The Strength of Metals (Reinhold Pub lishing Corporation, New York, 1962). 6 R. L. Fleischer, Acta. Met. 11, 203 (1963); J. App!. Phr.s. 33, 3504 (1962); Acta Met. 10, 835 (1962); Acta. Met. 8, 598 (1960). 3000 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:21SIMULATION OF NEUTRON DAMAGE IN Fe 3001 Harries et at.,7 indicate that the principal source of ir radiation hardening in neutron-irradiated ferritic steels is closely associated with the density of displacement spikes produced by PKA with energies above 3 keY. This model for the source of irradiation hardening is similar in concept to the PKA threshold energy model proposed by Harries et at.,7 and is closely related to the cluster population model of Williamson and Edmondson.8 These models have the common feature that they select certain defect aggregates in the primary damage state as being "effective" with respect to irradiation harden ing. As will be shown, it appears that this approach is superior to that of assuming the total number of dis placed atoms governs the degree of irradiation hardening. The shape of a displacement ,spike produced by a collision cascade, initiated by lS-keV PKA, in ex iron is depicted in Fig. 2. Each block section in this figure approximates th~ volume of the damaged region pro duced in four successive {OO2} planes. The thickness of each section, therefore, corresponds to two lattice con stants ("'-'5.7 A). A physical interpretation of this three-dimensional, stacked-block representation of a FIG. 2. Block section representation of a 1S-keV displacement spike in a iron. displacement spike will be given later. The geometrical cross sections of the distorted crystal region barriers, associated with displacement spikes in an irradiated specimen, might manifest themselves in a plane cross section through a specimen as shown in Fig. 3. This figure was prepared by selecting sections from the spike in Fig. 2 at random and then randomly positioning them on a plane. 2. COMPUTATIONAL MODEL AND PROCEDURE Detailed descriptions of the computational model and procedure have been published previously.4.9 However, for the convenience of the reader, it is appropriate to remark that: (1) It was assumed that a collision cascade can be represented as a branching sequence of binary 7 D. R. Harries, P. J. Barton, and S. B. Wright, J. Brit. Nucl. Energy Soc. 2, 398 (1963) j S. B. Wright and D. R. Harries (private communication). 8 G. K. Williamson and B. Edmondson, quoted by D. Harries, K. Bagley, r. Bell, W. Gibson, J. Gillis, P. Pfeil, and S. Wright in 1964 Geneva Conference Paper. 8 J. R. Beeler, Jr., and D. G. Besco, J. App!. Phys. 34, 2873 (1963). FIG. 3. Displacement spike geometrical cross sec tions in a iron. Plane of the figure is parallel to a (001) plane. ~ ~ ~~ ~ ~ ~ ~ atomic collision events. (2) All collision events were de termined by the Erginsoy-Vineyard3 potential. (3) Only displacement events associated with vacancy-inter stitial pair configurations which are stable at OaK, according to the Frenkel pair-stability criteria of Erginsoy et at., were counted in the displaced atom tally. It should be emphasized that collision cascade data which appeared in an earlier paper4 on the primary damage state in neutron-irradiated ex iron were based on collision cascades detelmined by a modified Bohr potential and a hard-sphere scattering approximation. Qualitatively, the collision cascade characteristics given by these two treatments are the same, but important quantitative differences do exist. Ninety-six independent collision cascades were simu lated for each of seven PKA energies in the range 0.5 to 20 keY. Each cascade-initiating PKA was started from a normal lattice site with an initial direction selected at random from an isotropic distribution. The symmetry section chosen for initial direction sampling was such that the average solid-angle resolution per direction ray was 0.0082 sr. This resolution is equivalent to selecting 1536 initial directions at random from the whole space of 471" sr. 3. DISPLACEMENT SPIKE CHARACTERISTICS A fundamental entity in the discussion of a displace ment spike is the distribution of collided atoms10 in the associated collision cascade. Because it is the part of a crystal directly affected by a collision cascade, the volume of collided atoms is the appropriate volume for use in computing the defect density in the associated displacement spike. Some collided atoms are hit with sufficient force to be knocked out of their normal posi tions in the metal lattice, while others are just shaken up a bit, so to speak, but not hit sufficiently hard to be forced out of their normal-position potential energy well. A collided-atom map for part Df a 5-keV cascade appears in Fig. 4. This map is a [OOlJ projection of all collided-atom positions in four successive (002) planes. Filled circles represent the initial positions of atoms which were hit hard enough to be forced out of their 10 In a strict sense a "collided atom" must be defined here as an atom which suffered a collision in the particular cascade simulation given by our binary collision approximation. A less exact but more informative description, appropriate to more than 99% of the instances concerned, would designate any atom receiving more than O.OOS eV in a collision as a collided Ilotom. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:213002 J. R. BEELER, JR. 130 100 :0 0000. 0 0 0·0 • .':;~ o~o o:.~.~ ~oX· ••• 0 000 ·o~o·o· o.o.o.~~ ~~ ••• o • °o:.~. • o o~. "0 0 o. 0 o. ~g?~ 00 ~oo 0 o· .0. ••• ~.to o· • •• 0 .0. 0 000 o. 120 t 110 'is' B ~ 90 8 0 90 100 110 120 130 ta/2J[100]- FIG. 4. Collided-atom distribution map for the center section of a 5-keV displacement spike in a iron. normal positions. Filled circles bounded by a square outline denote the positions of vacancies, i.e., the initial position of atoms which were permanently displaced. All other filled circles represent atom positions which were vacated and subsequently reoccupied, although not necessarily by the atom initially located at the posi tion concerned. Any atom receiving more than twice the sublimation energy (2Es= 8 eV in a ironll) was con sidered to be forced out of its normal position, at least temporarily. An average of about 3 eV was deposited per atomic site in a collided-atom volume in a iron and 13or----,----'1---------, O Vacancies :. I I " . • Interstitials }// "" -J- -~. t I // 31 /1 :/ -0 / 120 / 0 ~ Ii t 110' ---/~- --i = ~ L_ ~ / I ' I B / ~ • Ii _~ ., CL I' I , .U2 I, 100 -"----- 1)--'\ ID~4. I: \. ,~·ll ! ,,'-"--b~3~\_ .............. ' ----...;.. ~.---90 - 809=0--~1~00~--~--~~-~130 FIG. 5. Vacancy and interstitial-atom deployment in the section shown in Fig. 4. Vacancies are denoted by D and interstitials by •. 11 D. E. Harrison, Jr., and D. P. Magnuson, Phys. Rev. 122, 1421 (1961). the effective permanent displacement energy threshold was found to be ",,45 eV.12 Figure 5 is a map of the vacancy (open square) and interstitial-atom (filled circle) deployment in the same region concerned in Fig. 4. The dashed line drawn through the peripheral defects in Fig. 5 defines the way the shapes of the block sections in Fig. 2 were deter mined. The displacement-spike volume thus defined approximates the region of an elastic strain set up by the vacancies and interstitials within a displacement spike. It represents the shape of a type of barrier which should interact strongly with the core of a dislocation. The general shape and structure of a collided-atom volume is perhaps best illustrated by a three-dimen sional scale model built in accordance with the com puter printout for a 5-keV collision cascade history. Photographs of this model as seen when viewed along the [OOIJ and [liiJ directions appear in Fig. 6. The exterior surface of the region is very irregular and the volume is, in the topological sense, multiply connected, i.e., it exhibits a structure like that of Swiss cheese. It is within a region such as this that the previously men tioned average energy deposition of 3 eV per atomic volume pertains. As discussed elsewhere,13 this particu lar structure plays an important role in determining the multiplicity of displacement spike overlap required to produce a saturated displaced atom density at a given temperature. The average displacement density pro duced by a collision cascade at OaK was 3.2 at. %, in dependent of the initiating PKA energy. Another important general feature of the collided atom volume in a iron was the tendency for it to de velop primarily along (110) directions. This character istic is clearly shown in Fig. 6. The shape of the dis placement spike associated with a collided-atom volume was usually similar to the collided-atom-volume shape and hence exhibited some type of (110) orientation. In this regard, the "tail" on the displacement spike of Fig. 2 points in the [ilOJ direction. At larger energies the (110) orientation of a collided-atom volume was even more pronounced than that shown in Fig. 6. This is illustrated by the top view of a (00l) section at the (a) (b) FIG. 6. Collided-atom volume for a 5-keV spike. _(~) Top view, i.e., along [OOiJ direction. (b) View along the [111J ~irection. The typical Swiss cheese structure and (110) orientatIOn of a collided-atom volume are clearly illustrated in this figure. 12 J. R. Beeler, Jr., Bull. Am. Phys. Soc. 10,361 (1965). 13 J. R. Beeler, Jr., in Lattice Defects and Their Interaction, R. R. Hasiguti, Ed. (to be published). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:21SIMULATION OF NEUTRON DAMAGE IN Fe 3003 center of a to-keY-cascade collided-atom volume given in Fig. 7. Vacancies and interstitials tended to be segregated in a displacement spike. As a consequence of this segre gation, a marked imbalance existed between the va cancy and interstitial populations within individual subregions of a spike, even though the total vacancy popUlation and the total interstitial population were equal, for the spike as a whole. Roughly speaking, defect segregation occurred in the form of an interstitial-rich blanket at the spike surface which enclosed a vacancy rich interior. Qualitatively, this deployment is the same as that predicted by Brinkman.2 Three-dimensional spike models and damage maps, such as those in Figs. 2 and 5, respectively, can be used to frame a more accu rate description of vacancy-interstitial segregation. As mentioned before, the spike volume was approximated as a stack of block sections with a common thickness (two lattice constants), as illustrated in Fig. 2. When each section of a spike was examined individually, it was found that interstitials tended to be deployed preferentially at the periphery of the section containing the centroid of the spike, whereas the converse was true for vacancies. This type of defect segregation is illus trated in Fig. 5 which describes the damage distribution in a section containing the centroid of a 5-keV displace ment spike. The local imbalance between the vacancy and the interstitial populations, mentioned previously, is also illustrated by Fig. 5-note that the damage map contains 18 vacancies and 10 interstitials. In a section containing the centroid of a spike, the vacancy popula tion was usually from 1.5 to 2.5 times the interstitial population. In sections positioned progressively farther from the centroid, the relative number of interstitials increased and sections at the ends of a spike were interstitial-rich. One would expect this type of initial defect deployment to favor interstitial-cluster and FIG. 7. Center section of a lO-keV-cascade collided-atom volume directed along [110]. vacancy-cluster formation as a result of any subsequent point-defect migration. This indeed occurred in simula tions of point-defect annealing in a displacement spike at a finite temperature. It is possible that atomic rearrangement processes and solid-state chemical reaction rates could be en hanced within the collided-atom volume by virtue of the intense local excitation of the crystal lattice in this region (3 eV per site). In this regard, Vineyard14 has computed isotherms in collided-atom volumes of low energy cascades (100-500 eV) as a function of time. It is believed that precipitate nucleation, oxidation, and corrosion, for example, are directly enhanced by this excitation, at least in the case of large cascades.IS The importance of this aspect of a collision cascade in the case of precipitate nucleation, for example, could be en hanced by the preferred (110) orientation of cascade regions since {110} planes are primary slip planes in a iron. 4. DISPLACEMENT PRODUCTION The average total number of displacements v(E) in a displacement spike produced at OOK by a PKA with energy E was obtained by taking a simple average over the 96-spikes run for that energy. This procedure gave· v (E) = K(E)E, (1) where K(E) = 12.33[1-0.0411 (log.E)], (2) with E being expressed in keY. K(E) is the average number of displacements per unit PKA energy and is called the displacement efficiency. Figure 8 gives v(E)=K(E)E from the present study and that pre- 104 .. ---....-----1'---, THOMPSON & WRIGHT """\ -- 10 100 1000 PRIMARY KNOCK-ON ENERGY, KEY FIG. 8. Average number of displacements v(E)=K(E)E in a displacement spike produced by a primary knock-on atom with energy E. 14 G. H. Vineyard, Discussions Faraday Soc. 31, 1 (1961). 16 J. Moteff, in Symposium on Radiation Effects, AIME, 8-10 Sept. 1965, Asheville, North Carolina (Gordon and Breach Publishers, Inc., New York, to be published). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:213004 J. R. BEELER, JR. dicted by the Kinchin-Pease model,16 The effect of crystal structure upon displacement production is ex plicitly considered in the present study but not in the Kinchin-Pease model. Hence, the difference between the two curves in Fig. 8 represents the general effect of crystal structure on depressing displacement produc tion relative to that predicted for astructure less solid with the same average atomic density. The dashed curve above 20 keV is an estimate of veE) for the cas cades of mixed elastic and inelastic atomic collisions. It is based on the method of Thompson and Wright,l7 The displacement function veE) combined with the number of PKA produced per unit volume per unit neutron exposure gives the total displacement density per unit neutron exposure. In particular, the total dis placement density d(EN) for a given neutron energy EN is where y(EN) is the number of PKA (displacement spikes) produced per unit volume per unit exposure of neutrons with energy EN, and feE; EN)dE is the frac tion of these PKA produced with an energy in the interval dE at E. Both y(EN) and feE; EN) were com puted by simulating neutron collision histories through the specimen concerned in a Monte Carlo calculation.4 The quantities y and d are plotted in Fig. 9 as func tions of neutron energy for a square-base iron column 6.35 cm long and having a base dimension of 0.28 cm. The solid curves are for elastic neutron scattering only. Above 1 MeV, both densities are significantly di minished by inelastic neutron scattering. The dis- NIUIrUn nero. Mev FIG. 9. Displacement density d and displacement spike density y per unit neutron exposure as functions of neutron energy. Speci men was a square-base column of a iron with a base dimension of 0.28 em and a length of 6.35 em. 16 G. H. Kinchin and R. S. Pease, Rept. Progr. Phys., 18, 1 (1955). 11 M. W. Thompson and -So B. Wright, J. Nuc1. Materials 16, 146 (1965). TABLE I. Vacancy cluster size distribution. Fraction of vacancies contained in n-vacancy clusters. Cluster PKA energy (keV) size (n) 0.5 1 2.5 5 10 15 20 1 0.352 0.326 0.328 0.336 0.350 0.356 0.351 2 0.208 0.194 0.211 0.191 0.202 0.193 0.203 3 0.129 0.125 0.125 0.145 0.135 0.129 0.131 4-6 0.254 0.243 0.196 0.244 0.202 0.196 0.203 7-9 0.058 0.111 0.091 0.057 0.072 0.079 0.070 2:10 0 0 <0.04- 0.055 0.038 0.046 0.042 • The fraction lay in the range 0.02-0.04. placed atom and displacement spike densities obtained when the effects of mixed inelastic and elastic neutron scattering are accounted for are given by the dashed curves. These curves can also be used to predict the density of vacancy clusters and interstitial clusters pro duced directly by a collision cascade at OaK in the manner explained below. The number of vacancies and the number of inter stitials produced in a displacement spike are equal, their common value being called the number of atomic displacements. The vacancy popUlation in a displace ment spike was distributed among the different n vacancy clusters as described by Table 1. This table gives the fractions of all vacancies which appear in n vacancy clusters as a function of the energy of the initiating PKA. Combination of the information given in Table I with the PKA energy spectra as a function of neutron energy gave the useful result that the density dn (v) of n-vacancy clusters per unit neutron exposure can be closely approximated by (4) for neutron energies above 0.1 MeV, where d is the total displacement density per unit neutron exposure. The values of an(v) are listed in Table II. The interstitial cluster densities are given by (5) and the values of an(i) by Table III. These data, to gether with the displacement spike density y, provide most of the information needed to discuss neutron ir radiation hardening in pure a-iron specimens irradiated at OaK. 5. COMPARISON WITH EXPERIMENT Harries et al.7•8 have made an extensive study of the neutron-irradiation hardening of ferritic steels. The TABLE II. Values of an(V) for the relation dn(v) =an(v)d. n 1 2 3 0.343 0.100 0.0438 n 4-6 7-9 2:10 0.0422 0.00958 0.00379 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:21SIMULATION OF NEUTRON DAMAGE IN Fe 3005 TABLE III. Values of an (i) for the relation dn(i)=an(i)d. n 1 2 3 0.942 0.0275 0.001 ordering of the relative increases in the lower yield point they observed for five different neutron spectra, along with the associated average neutron energies for these spectra, are given in Table IV. In each instance, the integrated exposure was that required to produce a common density of 58Ni (n,p) 58CO threshold detector reactions. As indicated by footnotes (a) and (b) in Table IV, their irradiations were performed in both graphite-moderated and heavy-water-moderated re actors. The average neutron energies given in Table IV are for neutrons with energies above 10 keV. The results of our absolute method damage calcula tions were used to compare several suggested irradiation hardening models from the standpoint of obtaining the closest match to the experimental results of Harries et al. In this regard, absolute damage calculations were done for the particular specimen size used by Harries et al. on the basis of neutron spectra computed by Wright18 for the irradiation facilities listed in Table IV. Wright's Monte Carlo calculations accounted for the detailed geometry and material makeup of the speci men's environment in the reactor. As in the experi mental case, the integrated exposure for each spectrum calculation was selected such that the same number of computed 58Ni (n,p) 58CO threshold detector reactions resulted in each instance. The following possible meas ures of irradiation hardening were considered: 1. The total number of displaced atoms. 2. The number of displaced atoms produced by neutrons with energies greater than 1 MeV. 3. The total number of displacement spikes produced. 4. The number of displacement spikes produced by PKA with energies above a certain critical value Ec. TABLE IV. Relative irradiation hardening (increase in lower yield point) observed by Harries et al. for ferritic steel. Average Relative Reactor neutron irradiation spectrum energy (MeV) hardening BEPO emptY" 0.332 1.00 PLUTO Mk-IIIb 0.591 0.756 PLUTO empty 0.397 0.741 BEPO hollow 0.868 0.515 Herald rig 0.969 0.449 • BEPO is graphite moderated. b PLUTO is heavy-water moderated. 18 S. B. Wright, in Radiation Damage in Solids (International Atomic Energy Agency, Vienna, 1962), Vol. II. TABLE V. Relative hardening predicted by two displacement density (d) models. Elastic neutron scattering. Neutron Experi- spectrum d(all) d(~1 MeV) mental BEPO empty 1.00 1.00 1.00 PLUTO Mk-III 0.522 0.637 0.756 PLUTO empty 0.474 0.573 0.741 BEPO hollow 0.558 0.714 0.515 Herald rig 0.495 0.649 0.449 If one assumes that irradiation hardening is measured by the total displacement density, then the ordering of the relative magnitudes of irradiation hardening given by the column labeled d(all) in Table V is predicted. This ordering is not in accord with that given by experi ment (Tablt; IV). On the basis of the average neutron energy, the first three neutron spectra (Table IV) fall into a low-energy category relative to the last two. In each category, the predicted ordering given by the total displacement density model is the same as that given by experiment, but the model fails to serve as a general measure of irradiation hardening. If it is assumed that only those displaced atoms pro duced by neutrons with energies above 1 MeV measure irradiation hardening, then the ordering given by the column labeled d(~ 1 MeV) is predicted. This prediction also is not in accord with experiment. Hence, neither of the two commonly suggested irradiation-hardening measures, based on the number of displaced atoms, gives results which are even in qualitative agreement with the experiments of Harries et al. The two measures of irradiation hardening based on displacement spike densities are in good agreement with experiment. As indicated by the column labeled y(all) in Table VI, the ordering of the total displacement spike densities among the five neutron spectra is almost the same as that for the observed increases in the lower yield stress. The single exception is associated with the PLUTO empty spectrum. If it is assumed that the dominant irradiation-hardening contribution is made by spikes associated with PKA energies greater than ",,3 keV, the observed ordering is predicted. The frac- TABLE VI. Relative hardening predicted by two displacement spike density (y) models. Neutron spectrum y(all) y(E>2.5)B y(E>2.7)B y(E>3.6)B y(E>4.9)B BEPO 1.00 1.00 1.00 1.00 1.00 empty PLUTO 0.418 0.654 0.517 0.529 0.550 Mk-III PLUTO 0.565 1.003 0.506 0.502 0.501 empty BEPO 0.288 0.404 0.460 OA98 0.532 hollow Herald 0.244 0.360 0.412 0.454 0.489 rig -Correct order • PKA energy (E) in keV. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:213006 J. R. BEELER, JR. TABLE VII. Fraction (F) of PKA produced with energies above 3 keV in an iron specimen 0.28 cm in diameter irradiated by neutrons with energy EN. EN (MeV) F(E>3 keV) 0.01 0.02 0.05 0.10 0.32 0.50 1.00 1.50 2.00 4.00 6.00 8.00 9.00 o o 0.12 0.52 0.73 0.84 0.91 0.92 0.93 0.94 0.94 0.96 0.96 tion of the PKA produced with energies above 3 keY is given in Table VII as a function of neutron energy. These data pertain to the specimen size concerned in Fig. 9. The columns in Table VI labeled y(>2.5), y(>2.7), y(>3.6), and y(>4.9) show the orderings predicted when it is assumed that irradiation hardening depends only on the density of displacement spikes associated with PKA having energies above those stated within the parentheses (in keY). Correct ordering is achieved over a PK,A energy interval, .about 1 keY wide, centered about 3 keY. The possible significance of this result is directly related to the characteristics of displacement spike annealing which are discussed in the next section. In any event, of the two types of irradiation-hardening measures which have been suggested, a measure based 1.0 0.8 .Q 0.6 :Ii! ~ 0.4 0.2 1.0 0.8 t 0.6 ~ All displacements Displacements Ike. to neutrons aIIoYB 1 Mev 3 Experiment All spikes 82 03 02 Experiment Spikes ooe tD PKA's abo" 2.7 keY 0.8 1.0 0 0.2 0.4 0.8 0.8 1.0 Experiment Experinent KEY: 1 -BEPO Empty, 2 -PLUTO Mk-III, 3 -PLUTO Empty, 4 • BEPO Hollow, 5 -Herald Ail. FIG. 10. Comparison of relative irradiation hardening com puted according to four different models with the expe~entaI observations of Harries· et at. The coordinates of each pomt are experimental relative hardening (abscissa) 'and calculated relative hardening (ordinate). Numbers give ordering (rank) along the "experimental" axis. Ordering is the same along both axes only for the model concerned in the lower right plot. on a density of displacement spikes appears to be in far better accord with experiment than is one based on a density of displaced atoms. This is clearly illustrated by Fig. 10 which gives a graphical summary and com parison of the predictions given by the four different measures considered. 6. ANNEALING SIMULATIONS Eyre's experiments indicate that vacancies are im mobile in a iron up to about 250°C and, hence, that any defect annealing below this temperature is the result of interstitial migration.19 Lucasson and Walker20 have shown that interstitials are mobile in a iron at and above -153°C. In view of these results, the migration of interstitials in displacement spikes produced at OOK was simulated in an attempt to estimate the damage state produced at a finite irradiation temperature in the range -153° to 250°C. This damage state is that appro priate to the irradiation temperature (60°C) used by Harries et al. Interactions between interstitials and between an interstitial and a vacancy were accounted for within localized interaction regions, defined below, in the annealing simulations. Otherwise, it was assumed that each interstitial migrated independently, via a sym metric random walk on the bec lattice of a iron, in a field of immobile vacancies. These simultaneous inter stitial random walks were generated by a Monte Carlo program. The initial defect distribution used in each displacemen t spike annealing calculation was a primary damage state produced in one of the collision cascade simulations. An attractive interstitial-interstitial inter action was assumed for separation distances less than or equal to one lattice constant, and zero interaction was assumed for separation distances greater than one lattice constant. It was assumed that the interstitial vacancy interaction could be described in terms of the Erginsoy-Vineyard recombination region.3 When ever an interstitial entered the recombination region centered about a stationary vacancy, the two defects were im mediately annihilated; otherwise their interaction was assumed to be zero. The recombination region asso ciated with a vacancy cluster was taken to be the super position of the Erginsoy-Vineyard recombination regions for the individual members of the cluster. Interstitial clusters containing three or more members were assumed to be immobile. The annealing process was simulated for a time inter val !::.t= 100 T, where T is the average time between the jumps of a migrating interstitial. Nearly all of the features of the change in the defect distribution observed at this time had in fact developed during the first 30-50 T sec of the annealing history. It thus appears that the history length was sufficient to describe the role of 19 B. L. Eyre and A. F. Bartlett, Phil. Mag. 12, 261 (1965). 20 P. C. Lucasson and R. M. Walker, Phys. Rev. 127, 485 (1962); Phys. Rev. 127, 1130 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:21SIMULATION OF NEUTRON DAMAGE IN Fe 3007 13 0 12 0 0 0 0 0 14 ~o 0 Y'O 100 0 0 100 110 120 130 Uoq]- FIG. 11. [oolJ projection of vacancy (0) and interstitial (0) positions in four successive (002) planes of a displacement spike at t=O (primary damage state). Small circles denote positions of vacancies in the 14-vacancy cluster which lie in places above the four concerned in this figure. The defect distribution evolving from this state as a result of annealing appears in Fig. 12. short-range interstitial migration and that the primary damage state exerts a very strong influence on the char acter of the annealed state. A pictorial representation of aimealing results in one section of a displacement spike is provided by Figs. 11 and 12. The primary damage state for the section is given in Fig. 11, and that which evolved from this initial state during the time interval 6.t=37 T appears in Fig. 12. These figures illustrate the occurrence of rapid interstitial cluster formation when a large vacancy cluster (one containing 10 or more vacancies) is present in the primary damage state. ' The general results of the annealing simulations perti nent to this discussion were: (1) Displacement spikes which contained large vacancy clusters were relatively immune to annihilation via interstitial-vacancy re combination. (2) Sizeable interstitial clusters, contain ing from five to seven members, were observed to form during the annealing of spikes containing large vacancy clusters but were rarely observed when the spike did not contain a large vacancy cluster. Large vacancy clusters were only slightly reduced in size by interstitial-vacancy recombination events, whereas smaller aggregates tended to suffer either annihilation or severe attrition. By virtue of the vacan cies in large clusters having escaped annihilation during thermal annealing, an equal number of interstitials also survived the annealing process. These interstitials were usually collected into immobile clusters. Within a time interval of less than 50 T, clusters containing as many as seven interstitials would be formed. This is such a rapid rate at 60°C, for example, that one could almost consider such clusters to be directly formed by the associated collision cascade. This result is significant because it shows that the existence of interstitial dusters and a rapid interstitial clustering rate can be explained solely on the basis of the initial interstitial distribution in a displacement spike, even though clusters of more than two interstitials were rarely pro duced directly by a collision cascade. The clusters so rapidly formed during short-range migration of inter-130 120 I ~11 0 13 ~~ 0 "'b 10 P 9 0 90 100 110 120 130 Qo~- FIG. 12. [001 J projection of defect positions in the same region concerned in Fig. 11 after an annealing time of t=37r. Note for mation of the 6·interstitial cluster. r is the average jump time for an interstitial. The only alteration from this distribution pro duced by the remainder of the annealing process was the removal of the interstitial in the lower right portion of the figure. stitials could serve as nuclei for aggregates visible in transmission electron microscopy. 7. DISCUSSION The purpose of this section is to suggest a possible explanation for the qualitative correlation between the irradiation-hardening measurements of Harries et al. for ferritic steel and our computed densities for high energy (>3 keY) displacement spikes in pure a iron. For convenience; we refer always to the 60°C irra diation temperature used by Harries et al. However, the discussion is based on the characteristics of short-range migration of interstitials produced in the primary damage state and should be applicable to any irradia tion temperature in the range -153° to 250°C. A. Pure a Iron As shown by Table I, large vacancy clusters rarely occurred in spikes produced by PKA with energies less than 2.5 keY. Approximately the same fraction of the vacancies in a displacement spike are contained in large vacancy clusters for energies above 2.5 keY. Because of this circumstance, the density of spikes con taining large vacancy clusters can be considered to be proportional to the density of spikes produced by PKA with energies above 2.5 keY. This is the same class of displacement spikes whose density correlates with the degree of hardening observed by Harries et al. It is im portant to emphasize that we do not intend to suggest that large vacancy clusters directly provide the domi nant irradiation-hardening contribution. Rather, the occurrence of large vacancy clusters signifies that a particular type of defect deployment exists in a dis placement spike. When this deployment exists, anneal ing via interstitial migration tends to produce markedly more sizeable interstitial clusters and, consequently, far fewer recombination events than is the case when large vacancy clusters are absent. It appears that only when large vacancy clusters are directly produced by a col lision cascade will the associated displacement spike [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:213008 J. R. BEELER, JR. contain a sufficient number of appropriately deployed interstitials and vacancies, after annealing via inter stitial migration, to contribute significantly to irradia tion hardening. The damage state predicted by the cascade and annealing simulations for pure a iron irradiated at 60°C is one primarily composed of interstitial clusters and vacancy clusters. Attardo and Galligan21 have observed such a damage state in neutron-irradiated, pure plati num using field ion microscopy. Because spikes pro duced by PKA with energies less than 2.5 keV rarely contained large vacancy clusters, we conclude that such spikes succumb to thermal annealing during irradiation at 60°C on the basis of the annealing simulation results. In this case, only spikes produce by PKA with energies above 2.5 keV could contribute to irradiation hardening in pure a iron. B. Ferritic Steel The nature of the immobile vacancy population in the annealing simulations for pure a iron was not com pletely consistent with that thought to occur in a ferritic steel. In particular, Eyre concluded that the vacancies in his a-iron specimens were immobile below 250°C because of impurity trapping. In his view, inter stitial-vacancy recombination did not occur between an interstitial and a trapped vacancy. According to Johnson's calculations,22 the migration energy for an interstitial in a iron is 0.33 eV and that for a vacancy is 0.66 eV. At 60°C, the interstitial jump rate would be about 105 times that for a vacancy if Johnson's results are realistic. Hence, for the short annealing time we consider (that for '" 100 jumps per interstitial) it is permissible to ignore vacancy motion. The inconsistency mentioned above is connected with the circumstance that the annealing simulations did not consider the negation of the recombination interaction for an impurity-trapped vacancy. In most instances, the course of displacement spike annealing via short-range interstitial migration in a ferritic steel should be nearly identical with that for pure a iron due to the way the pertinent impurities are distributed in the steel. Most of the carbon in a ferritic steel is contained in precipitates. This is also true for any other interstitial impurity in an a-iron matrix, the type of impurity which dominates vacancy trapping. Because of this circumstance, only a few percent of the spikes produced would envelop an appreciable number of vacancy-trapping impurities. This being the case, nearly all of the spikes in a ferritic steel should behave essentially as those in pure a iron at the particular time and during the particular time interval pertinent to our annealing calculations. In these instances only high- 21 M. J. Attardo and J. M. Galligan, Phys. Rev. Letters 14, 641 (1965). 22 R. A. Johnson, Phys. Rev. 134, A1329 (1964). energy spikes (>3 keV) produced with large vacancy clusters should survive thermal annealing during irra diation at 60°C. There is an additional characteristic of short-range interstitial migration which should preferentially en hance the importance of those high-energy spikes con taining large clusters (LC) to irradiation hardening after prolonged annealing. The maximum size of inter stitial clusters formed during the annealing of LC high energy spikes was larger than that formed during the annealing of other spikes. The magnitude of this dis parity was at least a factor of two. Hence, cluster nuclei formed by short-range interstitial migration at the loca tions of their native LC high-energy spikes are con siderably larger than are the nuclei formed by inter stitials at the locations of other spikes. A small fraction of the interstitials produced by a cascade are mobile at the end of the short-range migration stage and have also moved away from their native displacement spike. By virtue of their size, the larger nuclei at the LC high energy spike locations will collect these migrating foreign interstitials at a faster rate than their subordi nates at other locations. As is well known, large assem blies grow at the expense of smaller ones in a nucleation and growth process. Hence, the locations of interstitial clusters grown by the collection of foreign interstitials on homogeneously formed nuclei should be, pre dominantly, the original locations of LC high-energy spikes. This suggests that the density of interstitial clusters grown during prolonged annealing should be proportional to the density of LC high-energy spikes, provided the initial displacement density was unsa tura ted. 8. SUMMARY 1. Collision cascade simulations indicate that dis placement spikes in ex iron should exhibit a preferred (110) orientation. This tendency was especially promi nent in spikes produced by primary knock-on atoms (PKA) with energies equal to or greater than 5 keV. In these instances a major cross section of each spike lies in a primary slip plane; hence, the spike could act in toto as a barrier to dislocation motion. 2. Annealing simulations indicate that nearly all of the displacement spikes which survive thermal anneal ing at irradiation temperatures between -1530 and 250°C should be those produced by PKA with energies above 3 keV. This should be the case both for pure a iron and for ferritic steel specimens. 3. The displacements pike density appears to be a far better index of irradiation hardening in a iron and ferritic steels than is the total displaced atom density. If it is assumed that only spikes produced by PKA with energies above 3 keV can contribute to irradiation hardening as a consequence of (2), then the predicted irradiation-hardening magnitudes correlate qualita tively with the measured magnitudes of Harries et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:21SIMULATION OF NEuTRON DAM.\GE IN Fe 3009 The computed total displaced atom densities did not correlate with the irradiation-hardening data of Harries et at. 4. Sizeable interstitial clusters were not produced directly by a simulated collision cascade. However, the existence of interstitial clusters in specimens irradiated at temperatures in the range -153° to 250°C can be explained in terms of the initial deployment of inter stitials produced in a spike. This deployment favors rapid homogeneous nucleation of sizeable interstitial clusters within the original collision cascade volume. JOURNAL OF APPLIED PHYSICS 5. Specimens irradiated at temperatures between -153° and 250°C should contain a damage state made up predominantly of interstitial clusters and vacancy clusters. ACKNOWLEDGMENTS The computer programs used in this study were written by N. R. Baumgardt and D. G. Besco, and the data processing was done by C. M. Schnur. The author wishes to thank Dr. J. Moteff and Dr. W. F. Schilling for helpful suggestions and good advice. VOLUME 37. NUMBER 8 JULY 1966 Variation of the Gain Factor of GaAs Lasers with Photon and Current Densities YASUO NANNICHI Central Research Laboratories, Nippon Electric Co., Limited, Kawasaki, Japan (Received 19 July 1965; in final form 4 January 1966) The variation of the gain factor with the threshold current was studied in two cases, viz., (1) a reflective film was applied on one end of a GaAs laser, and (2) antireflective films were applied on both ends of the laser. In (1) the threshold current is reduced to one third as compared with the case in which no film is applied. The gain factor increases 30%. In (2) the threshold current becomes eleven times greater and the gain factor is reduced to one fourth. These phenomena were analyzed in the light of spontaneous and stimulated lifetime of electrons in the p region. A formula was obtained giving the gain factor as a function of the density of photons and of cur rent. When the current is constant, the gain factor is inversely proportional to (P+ 1), where P is the density of photons. At the threshold current the gain factor is inversely proportional to the sum of quasi-Fermi levels, F nand F p. The saturation effect of a light amplifier at a fixed current observed by Crowe and Craig and also the variation of the gain factor with the threshold current can be calculated by this formula. I. INTRODUCTION RECENTLY, the variation of the threshold current of GaAs diode lasers was observed when an Ag film was deposited on one end.1 The Ag film was insu lated from the junction with a thin SiO film. The gain coefficient g at the threshold current fe was found to be proportional to fe• The result was consistent with those obtained by other investigators2•3 who observed the variation of the threshold current with the lengths of the diode. consistent interpretation to the phenomena mentioned above. I realized that the above results were not quite correct with respect to shorter and less lossy diode lasers in which larger variation of the threshold was observed. The gain factor at the threshold current is not constant under these conditions. The results obtained by Crowe and Craig, viz., the decrement effect of the gain factor with the increase of light intensity at a fixed current seem to be in line with the above res).llts. A theory has been worked out herewith which gives 1 Y. Nannichi, Japan. J. App!. Phys. 4, 53 (1965). 2 M. Pilkuhn and H. Rupprecht, Proc. IEEE 51, 1243 (1963). S M. Pilkuhn, H. Rupprecht, and S. Blum, Solid-State Electron. 7,905 (1964). II. EXPERIMENTAL A reflective film of Ag was deposited on one end and antireflective films of SiO were deposited on both ends of the GaAs lasers.1 The observed variation of threshold currents of lasers at the temperature of liquid nitrogen with the application of films is shown in Table I. The simultaneous equations, gi =(3(f ci)m=a-ln(RliR2i)!j L, TABLE I. Variation of threshold currents and gain factors. Sample JcX10-s No. p(cm- S) (A/cm2) 7372 1.4,X10'8 2.05/ 0.85- 7494 1.7 X 10'8 2.75 / 1.26- 8032 4.0X10'8 0.96b/10.6c R,=R 2=0.25, unless indicated. • R. =0.25. R, = 1.0. b R. =R, =0.32. • R. =R, =0.02. L Cl I3X1Q2 (em) (cm-l) (em/A) 0.044 - 9.5 3.6/4.9- 0.033 -6.7 3.5/4.2& 0.033 -23 4.4b/l.2° [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 19:56:21
1.1727740.pdf	On the Existence of Conformers of Cyclobutyl Monohalides. II. Temperature Dependence of the Infrared Spectra of Bromocyclobutane and Chlorocyclobutane Walter G. Rothschild Citation: The Journal of Chemical Physics 45, 1214 (1966); doi: 10.1063/1.1727740 View online: http://dx.doi.org/10.1063/1.1727740 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/45/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Photoelectron Spectroscopy of HighTemperature Vapors. II. Chemical Bonding in the Group III Monohalides J. Chem. Phys. 57, 3194 (1972); 10.1063/1.1678738 FarInfrared Spectra of Ring Compounds. VII. The RingPuckering Vibration in Chlorocyclobutane, Bromocyclobutane, and Cyanocyclobutane J. Chem. Phys. 56, 1706 (1972); 10.1063/1.1677428 Vibrational Spectra and Structure of FourMembered Ring Molecules. II. Chlorocyclobutane, Chlorocyclobutaned 1, and Chlorocyclobutaned 4 J. Chem. Phys. 46, 4854 (1967); 10.1063/1.1840647 On the Existence of Conformers of Cyclobutyl Monohalides. III. Assignments of the Fundamentals of Bromocyclobutane and Chlorocyclobutane J. Chem. Phys. 45, 3599 (1966); 10.1063/1.1727378 On the Existence of Conformers of Cyclobutyl Monohalides J. Chem. Phys. 44, 2213 (1966); 10.1063/1.1727013 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:551214 W. F. EDGELL AND R. E. MOYNIHAN this can be shown to have a second-order effect on the P-R maxima separation and the average of (a (x) ) at the P and R maxima. Vibrational anharmonicity and second-order Coriolis effects when sufficiently pro nounced will also similarly modify the envelopes. THE JOURNAL OF CHEMICAL PHYSICS ACKNOWLEDGMENT This work was supported by the U.S. Atomic Energy Commission under Contract AT(11-1)-164 with Purdue Research Foundation. VOLUME 45, NUMBER 4 15 AUGUST 1966 On the Existence of Conformers of Cyclobutyl Monohalides. II. Temperature Dependence of the Infrared Spectra of Bromocyc1obutane and Chlorocyc1obutane WALTER G. ROTHSCHILD Scientific Laboratory, Department of Chemistry, Ford Motor Company, Dearborn, Michigan (Received 7 March 1966) The infrared spectra of bromocyclobutane and chlorocyclobutane vapor between 250 and 3100 cm-! are reported as a function of temperature between 30° and 172°C. The spectra of the low-temperature ( -185°C) solids are also given. The data are described in terms of two conformers which are present in each halide. The energy difference between the two conformers of bromocyclobutane was measured to be about 1 kcal/mole. The conformers differ by their average dihedral angle: The more stable conformer is in a bent ring conforma tion ("equatorial"), the less stable one is in an essentially planar ring conformation. The sets of energy levels of the ring-puckering motion of the two conformers are contiguous, there is no tunneling between the con formers. The conformations are sufficiently different as to lead to two widely separated carbon-halogen stretching fundamentals for each cyclobutyl halide. One stretching fundamental is based on the equatorial, the other (towards higher wave numbers) is based on the planar ring conformation. The data were evaluated with the help of (1) calculations of the dipole moment as a function of the ring conformation, (2) computer calculations of vapor band envelopes including rotation-vibration interactions, (3) quantum-mechanical computer calculations of the energy levels, probability distribution, transition moments, infrared intensities, and average dihedral angles of the ring-puckering mode, and (4) some simple, qualitative considerations of the contributions of exchange interactions to the measured and calculated energy differences between the two conformers. I. INTRODUCTION IN a previous publication! some spectroscopic data of bromocyclobutane were discussed which, in combi nation with estimates of van der Waals and London forces, seem to indicate that bromocyclobutane is per manently bent in the equational position (e) or, at most, planar in high excited states of the out-of-plane bending (ring-puckering) motion.! The available evi dence did not admit the existence of an "axial" con former3 (a) which is conceivably attained by bending the carbon ring from positive to negative dihedral angles (see Fig. 1). In order to test further the above predictions, a study of the temperature dependence of the infrared spectrum was undertaken. The following alternatives may be expected to be answered by such a study. (1) If a compound exists only as one conformer, the appearance of its infrared spectrum should be inde pendent of the temperature. (2) In the case that two 1 W. G. Rothschild, J. Chern. Phys. 44, 2213 (1966) (I). 2 The carbon atoms move in a perpendicular direction to the carbon ring, the motion of each carbon atom being 180° out of phase with respect to that of its adjacent carbon atoms. 3 W. G. Rothschild and B. P. Dailey, J. Chern. Phys. 36, 2931 (1962) . conformers of different molecular structure are present, the total infrared spectrum would be a superposition of the individual spectra of the conformers. If the mo lecular structures of the conformers differ appreciably, e a FIG. 1. Equatorial (6) and axial (a) conformations of cy c1obutyl-X. The angle 'Y is the dihedral angle. many vibrational transitions may be "split" into two components, each component belonging to one of the conformers only.' If the molecular structures are less diverse, only a few vibrational transitions can be ex pected to be resolved into such components; in general, among them are the carbon-halogen stretching fre quencies.5•6& If the spectrum of the compound is then scanned at different temperatures, the ratio of the 4 K. Kozima and K. Sakashita, Bull. Chern. Soc. Japan 31, 796 (1958). 6 F. F. Bentley, N. T. McDevitt, and A. L. Rozeck, Spectro chim. Acta 20,105 (1964). 6 (a) J. K. Brown and N. Sheppard, Trans. Faraday Soc. 50, 535 (1954); (b) J. D. Roberts and R. H. Mazur, J. Am. Chern. Soc. 73, 25G9 (1951); (c) Org. Reactions 9,358 (1957). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:55CONFORMERS OF CYCLOBUTYL MONOHALIDES. II 1215 , , , , -r n Jll ..... r ht-f 1'\ If 10 !'IV" bl -, -I I I .. rr .. I .. I " " L_ _K' LMI!A ~ 3000 2500 2000 1800 1600 1400 em-I 1200 1000 800 600 400 200 (a) ,00 , , , ,,.. '1111-1 i< !f r- \.. { IJ,... I 10 " .. . 1\ / 11 T I I .. .. " I r .. .. -, LM ... • 3000 2500 2000 1800 1600 1400 em-I 1200 1000 800 600 400 200 (b) FIG. 2. Infrared spectrum of bromoeyclobutane vapor at (a) room temperature and (b) heated at 120°C between 3100 and 265 em-l. intensities of the two components of a vibrational mode should change in accordance with the Boltzmann distri bution of the conformers. This report describes experiments on the temperature dependence of the infrared spectra of bromocyclobutane and chlorocyclobutane vapor and on the spectra of the solidified compounds at low temperatures. The con clusions are supported by computations of band en velopes, of the dipole moment as a function of confor mation, of the energy eigenvalues and amplitudes of the ring-puckering mode, and finally on some general con siderations of nonbonded repulsion energies. n. EXPERIMENTAL The spectra of the vapors were scanned with an infrared spectrometer, Perkin-Elmer Model 521, be tween 250 and 3100 cm-1 in a ll-cm-path-Iength cell equipped with CsI windows. The cell could be warmed by heating tape. The temperature of the vapor in the absorption path was assumed to equal that of the body of the cell (measured by a thermocouple). A sidearm containing a few cubic centimeters of liquid halide was attached to the cell and kept at a constant temperature (room temperature or below). A cell of identical di mensions and temperature, but evacuated, was inserted into the reference beam of the spectrometer. The spectra of solid bromocyclobutane and chloro cyclobutane were scanned between 250 and 3100 cm-1 with a Beckman infrared spectrometer, Model IR-12. The compound was condensed slowly from its vapor phase onto a CsI plate which was affixed to a cold finger inside a liquid-nitrogen-cooled Dewar equipped with CsI windows. The temperature of the sample was about -185°C. The preparation of the halides has been described.6b•e Their purity was checked frequently by their reported infrared survey spectra.6b III. EXPERIMENTAL RESULTS Photographs of the recording of the infrared spectrum of bromocyclobutane vapor taken at 31 ° and 120°C are showllin Figs. 2(a) and 2 (b), respectively. This spectral This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:551216 WALTER G. ROTHSCHILD 100.-------------------, ~ 80 z ;! .... i 60 '" z .. 0:: ... ... 40 z \oJ o 0:: ~ 20 -. em FIG. 3. Spectral region of 580 to 420 cm-1 of bromocyclobutane vapor at three different temperatures. (The curves do not cross.) region comprises all fundamentals save the two lowest ones, namely the carbon-bromine deformation (a") at 248 cm-I and the ring-puckering mode (a') at 144 cm-I• Both fundamentals are too weak to be useful in these experiments. Comparison of Figs. 2(a) and (b) shows that the intensities of all transitions have decreased in the high-temperature scan. Part of this decrease is due to the lower density of the vapor at the higher temper ature (about 30%) and a small amount of emission from the hot sample,u The noteworthy aspect ex hibited by the spectra is, however, the relative change of the ratio of the intensities of the bands at 487.5 and 551 cm-I: If the temperature is raised, the intensity of the band at 487.5 cm-I decreases, whereas that of the band at 551 cm-I (effectively) increases. Three repre sentative scans of this pair (under scale expansion), out of a total of seven scans at different temperatures between 30° and 172°C, are shown in Fig. 3. Assuming that the band at 551 em-I is due to a less stable conformer,s the energy difference t:.E between its zero energy level and that of the equatorial confor mation can be estimated by measuring the ratio of the intensities of the two bands (the areas under the peaks) and plotting the logarithm of this ratio versus the reciprocal absolute temperature.4.9 The plot is shown in Fig. 4. The slope yields t:.E~1 kcal/mole. The spectrum of solidified bromocyclobutane is shown in Fig. S. For the purpose of this study, the most notable difference between this spectrum and the vapor spectra [see Figs. 2 (a) and 2 (b) ] is the disappearance of the band at 551 cm-I in the spectrum of the low-temper ature solid. The other transitions in the spectra of the vapor are also present in the spectrum of the solid. Some bands, for instance those at 90S, 940, and 965 cm-I, appear to be more intense in the spectrum of the solid, however, this effect is not caused by a variation 7 S. F. Kapff, J. Chern. Phys. 16,446 (1948). 8 The wavenumber of 551 coincides with that of a possible difference band arising from the 144-cm-1 level. However, the fraction of molecules in the 144-cm-1level decreases with increasing temperatures. 9 G. J. Szasz, N. Sheppard, and D. H. Rank, J. Chern. Phys. 16, 704 (1948). of the Boltzmann distribution with temperature since these bands appear with comparable intensities in the room-temperature spectrum of the liquid.6b The appearances of the spectra of chlorocyclobutane vapor as a function of temperature and that of solidified chlorocyclobutane were completely analogous to those of the bromo compound. In chlorocyclobutane, the weak band which became more intense when the tem perature of the vapor was raised and which vanished in the spectrum of the low-temperature solid was de tected at 631 cm-I• The adjacent, stronger band which became much less intense at elevated temperatures but which persisted in the spectrum of the solid was found at 532.5 cm-I• No attempt was made to determine t:.E since the 631-cm-1 band overlaps slightly with another transition towards higher wavenumbers. IV. DISCUSSION A. Assignments of the 551-and 631-cm-1 Bands of Bromocyc1obutane and Chlorocyc1obutane The disappearance of the SS1-cm-1 band in the spec trum of the low-temperature solidified bromocyclo butanelO and of the corresponding band at 631 cm-I in solid chlorocyclobutane, as well as the inverse intensity variations of the components at (551; 487.5) and (631; 532.5) cm-I in the vapor spectra as functions of the temperature are strong indications that two conformers exist in both halides.4.9 At a first glance, it is surprising that there are no other pairs of adjacent transitions which show inverse intensity variations with temperature. This seems par ticularly disturbing in view of the relatively large fre quency interval of 63 cm-I between the two components in bromocyclobutane and the even larger value (98 cm-I) for those in the chloro compound. However, the 0.35 0.30 ,-0.25 ... "... ~ 0.20 ; 0.15 0.10 • • • • 2.2 24 2.6 2.8 3.0 3.2 3.4 3.6 (11Th 10' FIG. 4. Plot of the logarithm of the ratio of the intensities of the 487.5-cm-1 band (1) and the 551·cm-1 band (I') of bromo cyclobutane vapor. The slope yields AE= 1.02 kcal/mole. 10 To which state the compounds had solidified, whether glassy or crystalline, was of lesser importance in these experiments and herefore was not determined. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:55CONFORMERS OF CYCLOBUTYL MONOHALIDES. II 1217 200 FIG. 5. Spectrum of solidified bromocyclobutane, at -185°C, between 3100 and 250 em-I. The sudden rise in the background near 650 cm-1 is spurious. 487.5-cm-1 component is undoubtedly the carbon bromine stretching fundamental of the equatorial con formation of bromocyclobutanel (and the 532.5-cm-1 band is the corresponding carbon-chlorine stretch in chlorocyclobutane) ,11 and it has been a general experi ence that the carbon-halogen stretches are quite sensi tive to the surrounding molecular geometry.5.6a The band at 55l cm-I in bromocyclobutane is therefore assigned to the carbon-bromine stretch of the less stable conformer. The band at 631 cm-I in chlorocyclo butane is assigned to the carbon-chlorine stretch of the less stable conformer of the chloro derivative. In order to invalidate these assignments, it would be necessary to assign these observed bands to another fundamental of the respective cyclobutyl halides since the corresponding Raman shifts, at 536 em-I (liquid bromocyclobutane) and at 617 cm-I (liquid chi oro cyclobutane), are relatively intense,u The only other fundamentals of bromo-or chlorocyclobutane which might possibly be assigned to this low spectral range are motions involving the carbon-carbon bond and the CH2 groups. For instance, in cyclobutanone,12a a ring deformation has been assigned to a frequency of 670 cm-I, and in cyclobutaneI2b a CH, rocking vibration to 627 cm-I• An assignment of the 631-cm-1 band in chlorocyclobutane to one of these modes encounters, however, two very severe objections. (1) In C,H7X, there are six fundamentals involving motions of the (ring) carbon atoms and three funda mentals involving CH, rocking vibrations, but only one 11 W. G. Rothschild (unpublished data, 1964-1965). 12 (a) K. Frei and Hs. H. Giinthard, J. Mol. Spectry. S, 218 (1960); (b) R. C. Lord and 1. Nakagawa, J. Chern. Phys. 39, 2951 (1963). of these nine fundamentals would show two widely separated components. (2) Substitution of chlorine by bromine would de crease the wavenumber of a motion involving mainly the carbon ring or the CH2 groups from 631 to 551, whereas the same substitution decreases the carbon halogen stretching frequencies only from 532.5 to 487.5 cm-I• It is therefore believed that the assignments of the 631- and 55l-cm-1 bands to the carbon-halogen stretches of the respective less stable conformers are correct. Recently, attention has been drawn to the depend ency of infrared solvent shifts of carbon-halogen stretching frequencies on rotational isomerism.13 In this respect, it is noteworthy that the solvent shifts vapor-t liquid of the 487.5-and 55l-cm-1 bands of bromocyclo butane are widely different, namely 2.5 and 13 em-I, respectively,l4 Comparable shifts were observed for the chloro compound. Solvent shifts of such different mag nitude may ensue from conformers which differ greatly with respect to their dipole momentI5 or steric environ ment.13 If such were the case, one might also anticipate that appreciable "splittings" of vibrational modes into two components should occur in, at least, a fe:w of the fundamental modes. This, as described above, was not observed. Furthermore, computations of the dipole moment of bromocyclobutane from its charge distri bution show that the dipole moment varies by no more than 2% among the three extreme ring conformations, 13 L. H. Hillen and R. L. Werner, Spectrochim. Acta 21, 1055 (1965). 14 Unpublished data. See also W. G. Rothschild, Spectrochim. Acta 21,852 (1965). Figures 1 and 2 of this reference show the solvent shifts in polyethylene. 16 H. E. Hallam and T. C. Ray, J. Chern. Soc. 1964, 318. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:551218 WALTER G. ROTHSCHILD t '" u z e i en z ~ 710 700 690 710 700 690 680 em-I FIG. 6. Band contours exhibiting multiple central Q branches for a fundamental transition in bromocyclobutane and in a deuterobromocycIobutane. namely, "equatorial," "planar," and "axial" (see Ap pendix II). However, it is conceivable that the carbon bromine stretch at 487.5 em-I (or the carbon-chlorine stretch at 532.5 em-I) couples to a different degree with other vibrational modes than the SSl-cm-1 (or the 631-cm-l) component. This would account for the widely different solvent shifts of the component bands.13 B. Assignments of Some Other Bands There is a band near 700 em-I in bromocyclobutane and in a-deuterobromocyclobutane which exhibits a complicated band envelope with two closely spaced central Q branches. This is shown in Fig. 6. It is con ceivable that the two Q branches belong to the two conformers. A band of very similar contour was found in chlorocyclobutane at 715.7 cm-I• Of the other phe nomena which might cause multiple Q branches, (1) the bromine isotope effect can certainly be excluded,16 (2) combination or hot bands are expected to have broader Q branches since the rotational constants of the lower and upper vibrational levels are usually quite different,17 (3) the presence of Fermi resonance cannot be established by the data but is not considered to be the cause,18 and (4) the degree of vibration-rotation interaction exhibited by the 700-cm-1 bands does not account for the observed separation between the Q branches (see Appendix I). Obviously, the ambiguity of Points (2) and (3) does not permit to draw a final conclusion, but a tentative assignment of the multiple Q branches in the 700-cm-I bands to two conformers is in agreement with the available data. 16 C. H. Townes and A. L. Scltawlow, Microwave Spectroscopy (McGraw-Hill Book Co., Inc., New York, 1955), p. 645. 17 G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules (D. Van Nostrand Co., Inc., Princeton, N.J., 1951), p. 484. Band envelope computations on bromocyclobutane show that the central Q branch is shifted as well as broadened if the sets of the rotational constants of the upper and lower vibrational levels differ by more than 1 %-2% (see Appendix I). 18 G. J. Szasz and N. Sheppard, J. Chern. Phys. 17,93 (1949). C. Structure of the Less Stable Conformer If, on the basis of the prevalent evidence, the exist ence of two conformers is accepted, one may attempt to predict the molecular structure of the more energetic conformer from the data. As discussed previously, an axial conformer of excess zero-point energy of .lE~300 cm-l or less with respect to the (most stable) equatorial conformer does not exist.I,3 The value of 1 kcal/mole (~3S0 cm-l), which was obtained for bromocyclo butane from the temperature variations of the (551; 487.S)-cm-l bands, represents therefore the lower limit of .lE. On the other hand, unless one wishes to make the assumption that potential functions describing non bonded interactionsI9,2°fail for cyclobutane monohalides, the value of .lE~1 kcal/mole is already more than a reasonable upper limit of .lE. In fact, 1 kcal/mole is of the order of magnitude computed for the energy differ ence between the equatorial and the planar ring confor mations.1 In the light of the collective evidence, it seems thus reasonable to assign an essentially planar ring structure to the more energetic conformer. As a consequence, the potential function of the ring-puckering motion may not only cause a gradual flattening of the equatorial conformation with increasing vibrational quantum number (see Ref. 1, Fig. 1) but also a more or less sudden switch from the equatorial to a chiefly planar conformation near a certain vibrational level. Such a potential has been indicated21: It possesses a flat maxi mum below which all levels belong to the equatorial conformer, and above which all levels belong to an essentially planar ring conformation. In this connection, there is a piece of evidence in the microwave spectrum of bromocyclobutane which seems to indicate that indeed there may be a relatively large increase in the planarity of the ring with the third excited vibrational level of the ring-puckering motion. The pertinent part of the rotational spectrum3 is shown in Fig. 7. One notices that the nearly equal frequency interval of about 40 Mc/sec between the rotational transition SI.~615 of successively higher excited vi brationallevels of the ring-puckering mode breaks after \ 19108 SWEEP 18968 M. ,,,. I FIG. 7. Pure rotational transition 514->616 of CaHeCD79Br for the vibrational levels 11=0, 1, 2 (and 3?) of the ring-puckering mode. 19 R. A. Scott and H. A. Scheraga, J. Chern. Phys. 42, 2209 (1965) . 20 H. E. Simmons and J. K. Williams, J. Am. Chern. Soc. 86, 3222 (1964). 21 See Ref. 1, Footnote 16. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:55CONFORMERS OF CYCLOBUTYL MONOHALIDES. II 1219 vibrational level v=2, but that there follows another, still weaker transition (v=3?) after a relatively larger interval of 63 Mc/sec. The same phenomenon was noticed for the transition 615-716, where the interval jumped from about SO to 84 Mc/sec after level v=2. Since a shift of the rotational transitions towards higher frequencies is mainly related to a decrease of the di hedral angle,!·22 it follows that if the transition at 19108 Mc/sec belongs to vibrational level v=3, the molecular structure of this level is relatively much more planar than those of the vibrational levels below. To give a more detailed picture, computations of the energy levels, transition frequencies, and infrared in tensities of the ring-puckering mode of bromocyclo butane were performed (see Appendix III). The po tential function was adjusted to express the experi mental data and theoretical considerations, that is, (1) it contains a flat maximum at about 1 kcal/mole above the lowest vibrational level, (2) vibrational levels of the ring-puckering mode below the maximum belong to equatorial ring conformations (vibrational quantum numbers v = 0, 1, and 2), whereas all levels above the flat maximum belong to essentially planar ring confor mations. According to this potential function, the existence of two conformers must be understood in the following terms. At room temperature, the first five to seven levels of the ring-puckering motion-that is, all three levels below the potential maximum and a few levels above it-are populated to an appreciable degree. The respective values of the vibrational quantum number and Boltzmann factor are: 0, 1; 1, 0.502; 2, 0.273; 3, 0.177; 4, 0.141; 5, 0.097; 6, 0.063, etc. Therefore, on an average, a fraction of approximately (1 +0.502+ 0.273)/2.253=0.8 of the number of molecules is in equatorial conformations, whereas a fraction of 0.2 is in planar conformations. For the solidified compound (-185°C), the fraction of molecules in equatorial conformations rises to nearly unity. At an elevated temperature of 120°C, the fraction of molecules in equatorial conformations drops to 0.7 and that of molecules in near-planar conformations rises to 0.3. During the time of one period of the ring-puckering motion, the higher-lying fundamentals complete many vibrations. Their normal frequencies must therefore be influenced by the particular motion of the ring puckering mode.23 The two observed frequencies of bromocyclobutane at 487.5 and 551 cm-1 may then be assigned to the carbon-bromine stretching fundamental averaged over an essentially equatorial (the 487.S-cm-1 band) and near-planar ring conformation (the SSI-cm-1 band), respectively. The corresponding assignments for the carbon-chlorine stretch in chlorocyclobutane are 532.5 cm-1 (equatorial) and 631 cm-1 (planar). 22 See Ref. 3, p. 2939. 23 K. Monter, E. Schafer, and E. Wolff-Mitscherlich, Z. Elektro chern. 65, 1 (1961). The set of energy levels of the ring-puckering motion belonging to the equatorial conformation does not over lap with the set of energy levels belonging to the planar conformation. There is no tunneling between the con formers. It is interesting to note that the flattening of the carbon ring is more gradual in bromocyclobutane than in chloro-or fluorocyclobutane in the first few levels (v:S;2) of the ring-puckering mode.1.24 It seems appropriate to add a few remarks concerning the soundness of the value of the energy difference between the equatorial and planar conformations. In the potential functions19.2o used to compute the inter action energies,! the contributions of exchange interac tions between the electrons from bonds adjacent to the bonds about which torsional motion occurs were not considered because of the difficulty of the problem.25-27 Instead, the interaction energies were computed only on the bases of van der Waals and London forces between the nonbonded atoms, the so-called nonbonded interactions.19 It has been reported that exchange inter actions may be expected to increase the repUlsion energy above that computed from van der Waals forces be tween atoms by at least 50%.19.28 On the other hand, if such exchange interactions were largely predominent (and if they lead to repulsion terms), then one might predict that an axial conformer should be either of essentially the same energy content as the equatorial conformer or be even less stable than the planar con formation. This may be shown by some qualitative arguments on the expected net degree of overlap29 brought about by the carbon-bromine bond as function of the ring conformation. The flexing of the carbon ring to form, sequentially, the equatorial~planar-axial conformations causes the following changes in the di rection cosines between certain bonds (see Fig. 8): (1) a monotonic decrease between the carbon-bromine and the cis carbon-hydrogen bond29a which lies across the ring diagonal, (2) a monotonic decrease between the carbon-bromine bond and the two nonadjacent carbon carbon bonds, (3) a monotonic increase between the trans carbon-hydrogen bond29a at the carbon carrying the halogen atom and the nonadjacent carbon-carbon bonds, (4) a monotonic increase between the two trans carbon-hydrogen bonds29& in the plane of symmetry of the molecule. All the other bond direction cosines be tween nonadjacent bonds average out to be at a mini- 24H. Kim and W. D. Gwinn, J. Chern. Phys. 44, 865 (1966). 25 E. B. Wilson, Jr., Advan. Chern. Phys. 2, 391 (1959). 26L. Pauling, Proc. Natl. Acad. Sci. U.S. 44, 211 (1958). 27 L. Pauling, The Nature of the Chemical Bond (Cornell Univer sity Press, Ithaca, N.Y., 1960), 3rd ed., p. 130. 28 M. Cignitti and T. L. Allen, J. Phys. Chern. 68,1292 (1964). 29 "Overlap" means here proximity of bonds among which ex change interaction may take place. Although this is a very quali tative definition, it seems reasonable to assume that if exchange interaction is significant, it should increase when the respective centers approach each other. 29& N ole added in proof: "cis" and "trans" with respect to the carbon-halogen bond. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:551220 WALTER G. ROTHSCHILD equatorial planar axial FIG. 8. Relative positions of some nonadjacent bond orbitals in three extreme ring conformations of cyclo butyl-X. mum in the planar conformation and therefore need not be considered. If the influence of the carbon bromine bond on the exchange interactions differs little from that of a carbon-hydrogen bond,30 the sum of the effects of the interactions via (1), (2), (3), and (4) are nearly identical in the equatorial and axial conformer. Therefore, the energy difference between the equatorial and axial conformers would essentially vanish. If the presence of the carbon-bromine bond, however, engenders a preponderance of the repulsion terms via (1) and (2) over those via (3) and (4), the resultant repulsion energy would be maximized in an axial conformation rather than in a planar confor mation. [If the interactions via (1) and (2) were preponderantly attractive, the axial conformer should be the most stable conformer, in contradiction to the data.] V. CONCLUSIONS The temperature variations of the infrared spectra of bromocyclobutane and chlorocyclobutane can be ex plained reasonably on the premise that two conformers, separated by an order of magnitude of 1 kcal/mole, are present in each halide. The energy levels of the ring puckering vibrations of the two conformational struc tures do not overlap: the switch from one conformer to the other takes place at a higher excited vibrational level of the ring-pUckering motion. The concept of conformer here is to be understood in terms of molecular populations which are distributed, by their respective Boltzmann factors, between two sets of carbon ring structures which differ sufficiently enough to possess two widely spaced carbon-halogen stretching fundamentals. The two ring conformations differ by their average dihedral angle: one structure is essentially planar, the other is nonplanar with the halogen atom bent away from the cis hydrogen atom29a situated on the carbon atom across the carbon ring diagonal (equatorial conformer, the most stable one). 30 Substitution of a hydrogen atom by a halogen atom has rel atively little effect on the barrier to internal rotation. See, for instance, E. B. Wilson, Jr., Tetrahedron 17, 191 (1962), Table 2. ACKNOWLEDGMENTS It is my pleasure to acknowledge the continuing interest of Dr. R. C. Taylor, University of Michigan, and the generous help of Mr. C. F. Farran, University of Michigan, who scanned the spectra of the solids. I am also grateful for the fruitful discussions with Dr. L. L. Lohr, Jr., of our laboratory. APPENDIX I: CALCULATIONS OF BAND ENVE LOPES FROM THE ROTATIONAL CONSTANTS. INCLUSION OF ROTATION-VIBRATION INTERACTION A computer program which calculates the band en velopes of asymmetric rotors has been described by Haller.31 The rotational constants of the lower and upper vibrational levels are assumed to be equal. In this approximation, the band contour of a symmetric rotor exhibits a strong linelike Q branch (II band) at 11=110, where 110 is the frequency of the vibrational tran sition, or a series of equidistant lines (..L band) with maximum absorption very near 110.32 For bromocyclo butane, a nearly accidentally symmetric rotor, the band contours of A and C bands show indeed a very strong Q line at 110 if the band envelopes are calculated. Ro tation-vibration interaction and centrifugal distortion (nonrigid rotor) tend to broaden the central Q branches and to shift them off the 11=110 position towards shorter or longer wavenumbers.32 The effect of rotation-vibration interaction was in corporated into the program by using different sets of rotational constants for the lower and upper vibrational levels. Since the total storage requirements of the en larged program exceeded the available space in 32K memories, the program was divided into three core loads and executed with chaining as it appeared desir able to retain the large range of rotational quantum numbers.31 The inclusion of centrifugal distortion is nearly im possible because of the complexity of the problem. Fortunately, an order-of-magnitude estimate shows that the frequencies of the rotational transitions are far more influenced by using slightly different sets of ro tational constants for lower and upper vibrational levels than by the effect of centrifugal distortion.33 Since the rotational constants for the upper vibra tionallevel generally are not known, they have to be approximated by trial and error. This procedure, how- 31 I. Haller, "Computation of Contours of Vibration-Rotation Bands of Asymmetric Rotor Molecules," Proc. Intern. Symp. Mol. Struct. Spectry., Ohio State Univ., Columbus, Ohio, 1964 (to be published). 32 See Ref. 17, Chap. IV. 2. 33 Distortion coefficients are of the order of 4B3j,,> (em-I), see Ref. 17, p. 14. Assuming the rotational transition.:lJ = 1 from level J = 39, and the values B = 0.330 and 0.333 cm-I for upper and lower vibrational levels, respectively, and vo= 700 em-I, the contribution of centrifugal distortion to the rotational frequency amounts to ----0.1 em-I, whereas that of rotation-vibration inter action amounts to ",4 em-I. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:55CONFORMERS OF CYCLOBUTYL MONOHALIDES. II 1221 FIG. 9. Computed A-and C-type band contours and two hybrid band contours of bromocyclobutane. Rotational constants (in em-I) are, for the ground vibrational level, 0.33368, 0.054354, and 0.049654; for the upper level, 0.33595, 0.054241, and 0.049534. Slitwidth 0.5 em-I, temperature 303°K. The abscissa is in units of em-I counted from the center of the band. The ordinate is in units of relative transmit tance. ever, converges fairly rapidly to envelopes which most closely resemble the observed contours under the experi mental conditions (slitwidth and temperature). Changes of the rotational constants of the upper vi brational level which go beyond certain limits cause increasing distortions of the computed envelope, such as an excessive broadening and shift of the central Q branch. The computations as described here were then applied to the band envelope of the 700-cm-1 band shown in Fig. 6. Since the band has a polarized Raman shift,!1 only the A-and C-type band envelopes need be com puted. They are shown in Fig. 9. The rotational con stants of the lower vibrational level are (in cm-I) 0.33368, 0.054354, and 0.049654.3 Those of the upper level were chosen to be 0.33595,0.054241, and 0.049534. To reproduce the observed band contour, the computed bands were superimposed to hybrid bands, CIA +c2C, with various ratios of the coefficients CI and C2. Although it was possible to attain a computed band envelope which, in essence, resembled the observed contours, only one of the two observed central Q branches could be reproduced by the computations for all values of CI and C2. Two representative hybrid band envelopes, 0.5(A+C) and 0.75A+0.25C, are shown in Fig. 9. APPENDIX II: CALCULATION OF THE DIPOLE MOMENT OF BROMOCYCLOBUTANE AS A FUNC- TION OF THE DIHEDRAL ANGLE A calculation of the charge distribution based on the inductive effect34,35 puts the following charges ei (in units of 10-10 esu) on the atoms of bromocyclobutane (see Fig. 10). The dipole moment was calculated from "'(I') = LeiX, (I') , where Xi is the coordinate vector of Atom i 34 R. P. Smith, T. Ree, J. L. Magee, and H. Eyring, J. Am. Chern. Soc. 73, 2263 (1951). 36 R. P. Smith and E. M. Mortensen, J. Am. Chern. Soc. 78, 3932 (1956). c I I I 30 20 10 o -10 -20 -30 A at dihedral angle ')'. The values of Xi were calculated using the known bond parameters.3 The resulting dipole moments I", I are 2.20,2.18, and 2.23 D for an equatorial (,),=30°), planar (,),=0°), and axial (1'=-30°) con formation of the carbon ring, respectively. The measured value of the dipole moment of liquid bromocyclobutane is 2.09 D.36 APPENDIX m: COMPUTATION OF THE SPEC TRUM OF THE RING-PUCKERING MODE The frequencies, probability distribution, transition moments, and infrared intensities, as well as the average dihedral angle of the ring-puckering mode were com puted using the nondimensional Hamiltonian37 (1) where V=!fi.BLci~i=M.BV'. The dimensionless coord i- -0.955 0.015 0.015 0.015 FIG. 10. Computed charge distribution of bromocyclobutane (in units of 10-10 esu). 36 J. D. Roberts and V. C. Chambers, J. Am. Chern. Soc. 73, 5030 (1951). 37 E. Heilbronner, Hs. H. Giinthard, and R. Gerdil, Helv. Chim. Acta 39, 1171 (1956). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:551222 WALTER G. ROTHSCHILD TABLE 1. Energy levels, transition frequencies, transition moments, overlap integrals, and infrared intensities of the potential function VW =1fip( -2r+6e+2~·). Vibrational quantum Transition number and energy frequencya levels (in units of 1fiP) m~ (em-I) 0 -23.1767 0 144 -13.6533 2 128 2 -5.2045 2 3 90 3 0.7260 3 4 48 4 3.9562 4 5 76 5 8.9924 5 6 89 6 14.9302 6 7 101 7 21.6046 7 8 159 8 32.0966 8 9 217 9 46.4471 9 10 483 10 78.3580 0 2 272 3 217 2 4 139 3 5 125 4 6 166 6 8 260 2 5 215 a ifiP=15.121 em-I. nate ~ is a measure of the nonplanarity of the carbon ring: the ring is planar at ~=O. The solutions of Eq. (1), E., are in units of ili,B, where ,B is an arbitrary multiplier. The calculations were performed on a Philco-212 computer.3S The state vector of the motion (the ampli tude) was taken to be a linear combination of basis vectors (2) The coefficients a"" are the matrix elements of the orthogonal matrix A, where A-1XA = o;jE;. As set of basis vectors Uk, the set of harmonic oscillator wave functions was chosen. The nonzero coefficients Ci of Va) are C2= -2, c3=6, and C4 = 2. The matrix elements of the Schrodinger equation have been tabulated.37•39 The size of the secular determinant was 20X20.39 The probability distribution if;n2(~) and the energy levels En (n=vibrational quantum number) are shown 38 The program was written in FORTRAN IV. Requests for cards should be addressed to the author or to the Quantum Chemistry Exchange Program, Program No. 74, University of Indiana, Bloomington, Ind. 47405. 89 R. L. Somorjai and D. F. Hornig, J. Chern. Phys. 36, 1980 (1962). Relative (Vtm I ~ I Vtn) (Vtm IVtn) ir intensity m~ -0.1816 -0.00004 1 0.2668 0.00021 0.97 -0.2968 -0.00027 0.46 -0.4356 -0.00027 0.34 0.4608 0.00037 0.48 0.4876 0.00057 0.44 -0.4860 -0.00093 0.32 0.6032 0.00120 0.48 0.8903 0.00809 0.59 -0.7322 -0.01810 0.35 0.0325 0.00001 0.06 0.0674 0.00008 0.10 -0.1767 -0.00025 0.25 0.1306 0.00027 0.08 -0.0934 -0.00031 0.04 0.1751 0.00120 0.11 0.1088 0.00015 0.15 in Fig. 11. Table I lists the values of En, the transition moments where the overlap integrals between States m and n as check on the orthogonality of the state vectors, and finally the . TABLE II. Comparison between observed and computed average dihedral angles of bromocyclobutane as a function of vibrational excitation of the ring-puckering motion. Vibrational quantum (Vtn I ~ I Vtn)/ (Vtn IVtn) number n 0 -2.366 1 -2.147 2 -1. 780 3 -0.753 4 -0.451 5 -0.456 B See Ref. 1. Computed Observed a (0.511 =29.3°) 29.3° 0.464 =26.6° 27.0° 0.384 =22.0° 24.6° 0.163 =9.3° 0.0974=5.6° 0.0985=5.6° This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:55CONFORMERS OF CYCLOBUTYL MONOHALIDES. II 1223 infrared intensities Imn = J.Lmn2J1mn[exp( -Em/kT)], where Jlmn=En-Em (in cm-I).40 The spectrum plotted from the calculated values is shown in Fig. 12. The weaker branch of the spectrum 5 4 3 2 o -100 -80 -60 -40 -20 -0 --20 -4 -3 -2 -I 0 2 3 { FIG. 11. Energy levels and probability distribution of the ring puckering mode of bromocyclobutane. The potential is V W = !ht/(-2r+6~3+2~), in terms of the dimensionless coordinate~. The planar ring conformation is at ~=O. Negative values of ~ correspond to positive dihedral angles (equatorial conformation). The energy levels are in units of !ht/. The heavy horizontal lines denote the classical region. 40 See Ref. 17, p. 261. 1.0 -0-1 1-2 >-l-0.8 - e;; Z '" 8-9 I-0.6 -2-3 ~ / 4-5 7-8 '" > f= 0.4 -6-7 cr 3-4 2-4 ..J '" a: 0.2 - 2-5 1-3 6-8 1 I I 0-2 11"""-- 1 40 80 120 160 200 240 280 -I em FIG. 12. Computed spectrum of the ring-puckering motion of bromocyclobutane. The numbers are the vibrational quantum numbers for the lower and upper energy levels of the transition. centered near 80 cm-l arises from the crowding of the energy levels at the plateau41 (see Fig. 11). The main absorption, between 120 and 160 cm-\ agrees with the observed spectrum.l,ll It is noteworthy that the n--tn+2 overtones arising from the lowest levels are very weak and therefore should not be detectable under ordinary conditions. This also agrees with the observations. Although the assumed potential can give no more than a very approximate representation of the actual problem, the computations show the complexity of the spectra of low-frequency vibrations that possess ex tremely anharmonic potential functions. Finally, Table II shows the computed values of the average dihedral angle, (I/In I l' I I/In), for the first six vibrational levels of the ring-puckering mode. The quantity (I/In I l' I I/In) is defined here to be a quantity which is proportional to the computed expecta tion value of the coordinate ~ for energy level n, (I/In I ~ I I/In)/ (I/In I I/In). The proportionality factor is ad justed to yield (1/10 I l' 11/10)=0.511, the observed value of the dihedral angle in the ground state,! and to give (I/In I 0 Il/In)=O. 41 The exact depth of the shallow right-hand-side potential well is immaterial for the arguments presented here. If the well should, at all, be sufficiently deep to contain one energy level, this level will be so close to the barrier top as to belong to an es sentially planar conformation (see Ref. 39). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sat, 20 Dec 2014 16:51:55
1.1727841.pdf	Pseudonatural Orbitals as a Basis for the Superposition of Configurations. I. He2 + C. Edmiston and M. Krauss Citation: The Journal of Chemical Physics 45, 1833 (1966); doi: 10.1063/1.1727841 View online: http://dx.doi.org/10.1063/1.1727841 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/45/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Perturbation expansion theory corrected from basis set superposition error. I. Locally projected excited orbitals and single excitations J. Chem. Phys. 120, 3555 (2004); 10.1063/1.1630952 Basis set superposition problem in interaction energy calculations with explicitly correlated bases: Saturated second and thirdorder energies for He2 J. Chem. Phys. 104, 3306 (1996); 10.1063/1.471093 On the performance of atomic natural orbital basis sets: A full configuration interaction study J. Chem. Phys. 93, 4982 (1990); 10.1063/1.458635 Pseudonatural Orbitals as a Basis for the Superposition of Configurations. II. Energy Surface for Linear H3 J. Chem. Phys. 49, 192 (1968); 10.1063/1.1669809 Superposition of Configurations and Natural Spin Orbitals. Applications to the He Problem J. Chem. Phys. 30, 617 (1959); 10.1063/1.1730019 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Sun, 30 Nov 2014 10:18:19VALENCE LEVELS OF BERYLLIUM OXIDE 1833 Triplet Sigma 3 5 HSCFCU) = HuN+ LC2Jiu-Kiu)+ LCJju-K ju) +4J1 .. -2Kh·, i-I f-4 3 5 HSCFC7r) =H .. .N+ LC2Jiu-Kiu) + LCJju-!Kju)+4h,O-K lr°-K1 .. 2, i=l j=4 3 5 3 5 ECb3"2;+) = LCEiuN +Eiu) +! LCEjuN + Eju) +2CElrN +Elr) +! LLKjuiu. ;=1 1=4 i=1 j=4 THE JOURNAL OF CHEMICAL PHYSICS VOLUME 45, NUMBER 5 1 SEPTEMBER 1966 Pseudonatural Orbitals as a Basis for the Superposition of Configurations. I. He 2 + C. EDMISToN* University of Wyoming, Laramie, Wyoming AND M. KRAUSS National Bureau of Standards, Washington, D. C. (Received 1 December 1965) The use of pseudonatural orbitals (PNO) is proposed to improve the rate of convergence in the super position of configurations (SOC). Natural orbitals are determined for selected electron pairs in the Hartree Fock field of the n-2 electron core and are then used as the basis for the total SOC calculation. Since these natural orbitals are not natural for the n-electron system they are considered false or pseudonatural orbitals when used in the n-electron problem. The PNO basis has been applied to He2 + and Hs to test the convergence. Complete results are reported here only for He2+. The PNO's are quite successful in speeding up the convergence of the SOC and rendering the calculation of correlation energy quite practical in genera!. Gaussian-type orbitals (GTO) are used throughout and were not a serious impediment to obtaining quantitative accuracy. In fact the large number of unoccupied Hartree-Fock orbitals consequent upon the use of a GTO basis permit a straightforward determination of the PNO orbitals. I. INTRODUCTION THE calculation of accurate electronic energies of atomic and molecular systems can be based upon an initial determination of the Hartree-Fock CHF) energy and a subsequent calculation of the remainder or correlation energy.1 Recently the HF energy has been approached within chemical accuracy (,.....,0.1 e V) for many atoms and diatomic molecules.2 It is evident that the list of accurate calculations will soon extend to polyatomic molecules as well. The principal practical deterrent has been the difficulty in evaluating poly atomic energy integrals over Slater-type orbitals (STO). Although recent progress3 in this regard may obviate the need for any alternative, the use of Gaussian-type orbitals (GTO) has proved valuable for polyatomic * Supported by the National Science Foundation Grant 6P-3896. 1 A standard review of the correlation problem is P.-O. L6wdin, Advan. Chern. Phys. 2,207 (1959). 2 Reference is to the many papers of Roothaan and his colleagues, Clementi, Nesbet, and many others. These papers are too numer ous to cite. See the recent review of B. M. Gimarc and R. G. Parr, Ann. Rev. Phys. Chern. 16,451 (1965). 3 M. Karplus and I. Shavitt, J. Chern. Phys. 38, 1256 (1963); F. E. Harris and H. H. Michels, ibid. 42, 3325 (1965). calculations.4 Accurate electronic energies are possible without the use of prohibitively large GTO basis sets. Since the use of the GTO basis is essentially a solved clerical problem, there is little to prevent the accumula tion of polyatomic HF results but the economics of large computers. Accurate calculation of the correlation energy for complex systems, however, is still an unsolved problem. Of the possible approaches to a solution, only one is discussed here, the superposition of configurations (SOC). The trial function is a SOC where each configu ration is a symmetry-adapted linear combination of Slater determinants. This technique is one of the oldest5 and simplest means of determining the correlation energy. However, the basis of one-electron orbitals from which the configurations are constructed must be carefully chosen if slow convergence is to be avoided.6 4 M. Krauss, J. Res. Nat!. Bur. Std. A68, 635 (1964); J. W. Moskowitz and M. C. Harrison, J. Chern. Phys. 42,1726 (1965); J. W. Moskowitz, ibid. 43,60 (1965). 5 E. A. Hylleraas, Z. Physik 48, 469 (1928). 6 A. C. Hurley, J. E. Lennard-Jones, and J. A. Pople, Proc. Roy. Soc. (London) A220, 446 (1953) ; P.-O. L6wdin, Phys. Rev. 97, 1474 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Sun, 30 Nov 2014 10:18:191834 C. EDMISTON AND M. KRAUSS Much recent work has been devoted to natural orbitals which provide a basis for most rapid convergence.7 No simple procedure has yet been developed for the determination of such orbitals for the n-electron prob lem, but approximate natural orbitals have been deter mined directly for two-electron systems in several ways.s Analysis of a SOC two-electron function is simplest and is used here. Electron correlation is essentially a two-electron inter action in the averaged n-2 electron field and the two electron or geminal functions provide a basis for an n-electron calculation. The natural orbitals for the two electron problem are the natural orbitals of the n electron problem only in the geminal or separated-pair approximation.9 Insofar as a product of geminal func tions is a good approximation to the wavefunction, then the one-electron functions deduced by analysis of the pair correlations will provide a basis for constructing configurations which are more rapidly convergent than those obtained from the unoccupied HF set. Such orbitals are not necessarily good approximations to the true natural orbitals and are denoted as pseudonatural orbitals (PNO). This procedure is essentially an adaptation of the early work of Fock, Wesselow, and Petrashen,lO and the suggestion of Hurley, Lennard-Jones, and Pople6 that the geminals are best represented by a linear combination of doubly occupied orbitals. The use of SOC trial functions in this context has been related by N esbetll to the Bethe-Goldstone equation and an im portant criteria for the accuracy of the function is shown to be the vanishing of single excitations. By forcing this condition on a limited trial function which spans a sufficient number of pair functions to cover the physical space of the system, a set of orbitals, the PNO's, are deduced whose nodal properties are more relevant to the correlation of the system than those of the excited HF molecular orbitals. In order to test this procedure on a nontrivial and quantitative level but avoid all possible complications, application has been made to the three-electron prob-lems for He2+ and H3. The calculation of the He2+ energy primarily provides a check on the quantitative accuracy of the results and the H3 energy surface has long been one of the goals of molecular quantum mechanics. In this paper only a few of the H3 results are presented. The results for the linear H3 surface are given in a subsequent paper. The main purpose at present is to outline the determination of the PNO's and their subsequent use. In particular, the relatively rapid convergence of the SOC is emphasized even when an atomic GTO basis is chosen. II. PSEUDONATURAL ORBITALS: DEFINITIONS Consider a wavefunction of the form 'lI(1, "', n) =Acp(1, 2) rr,p,(i) , i>3 where,pi are one-electron spin orbitals, cp( 1, 2) is a two electron pair function including spin, and A is the anti symmetrizer. The pair function is held strongly orthogo nal to all the orbitals which are chosen to be accurate approximations to solutions of the Hartree-Fock equa tions. Note that the latter choice is not necessary but was adopted because of the simplification in the com putations. In addition, it is assumed that the n-2 electron core is transformed to a proper symmetry state so that'lI actually is represented by a sum of determi nants. The final-state symmetry is determined by the symmetry of cp and its coupling to the core. Although there are instances where triplet-coupled electron pairs might provide a suitable base, in most cases of concern the correlation is primarily between singlet-coupled electrons and <p is always considered as a singlet. The function cp is determined from the usual secular equation which results from variation of a SOC trial function. Each configuration is constructed from an augmented basis of HF molecular orbitals. Without disturbing the core all possible single and pair excita tions are made in the original ,pI and ,p2 orbitals. For He2+, 'lI(1, 2,3) would consist of the following terms: (1/V2)[I iCJ"yO!jai31CJ"uO! 1-I iCJ"y(3jCJ"yO!lCJ"uO! IJ; (1/V2) [I iCJ"uO!jCJ"u(31CJ"uO! 1-1 iCJ"u(3jCJ"u0!1CJ"uO! IJ; i,j= 1-n, i,j=2-n, i,j= 1-m, i,j=1-m. 7 A reasonably complete survey of the necessary references is given by W. Kutzelnigg, J. Chern. Phys. 40, 3640 (1964). 8 A. P. Yutsis, Va. I. Vizbaraite, T. D. Strotskite, and A. A. Bandzaitis, Opt. Spectry. 12, 83 (1962); W. Kutzelnigg, Theoret. Chim. Acta 1, 343 (1963); C. E. Reid and Y. Ohm, Rev. Mod. Phys. 35, 445 (1963); D. D. Ebbing and R. C. Henderson, J. Chern. Phys. 42, 2225 (1965); G. Das and A. C. Wahl, Bull. Am. Phys. Soc. 10, 102 (1965). 9 C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 35, 457 (1963); also see Ref. 7. 10 V. Fock, M. Wesselow, and M. Petrashen, Zh. Eksperim. i Teor. Fiz. 10, 723 (1940). 11 R. K. Nesbet, Phys. Rev. 109, 1632 (1958). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Sun, 30 Nov 2014 10:18:19PSEUDONATURAL ORBITALS. I. He2+ The natural-orbital solutions for the cfJ singlet state are obtained by following the procedure of Hurley, Lennard-Jones, and Pople6 and L6wdin and Shulp2 for the diagonalization of the first-order density matrix. The space function is expanded in terms of the molecular orbitals Xk cfJ(1,2) = LCkkXk(1)Xk(2) + LCkl[Xk(1)XI(2) +xl(1)Xk(2) J. k l~k This defines a matrix C, where L! Ckk !2+2L ! Cki!2= 1. k l~k The transformation which diagonalizes C is equivalent to the one which diagonalizes the first-order density matrix. When a doubly occupied HF orbital is corre lated, the elements of C can be symmetry adapted and the matrix partitioned according to the molecular orbital symmetry. For interorbital calculations the diagonal elements are zero and the solutions occur in degenerate TABLE I. Parameters for basis orbitals, Nj(x, y, z) exp ( -ar2). Symmetry type Orbital 1 2 3 4 5 6 7 8 9 1 2 3 j(x, y, z) z z 1 1 1 1 1 1 1 (x,y) (x,y) (x,y) a 0.518272 2.228416 0.160274 0.447530 1. 297177 4.038781 14.22123 62.24915 414.4665 0.6180 1.9950 8.430 pairs from which symmetry-adapted orbitals can be selected. Only the doubly occupied singlet-coupled orbitals are considered here. Interorbital or other intraorbital correlations are described using PNO's determined from a HF pair in the same principal shell. The PNO's for the principal shell span the correlation region and it is presumed that only a few of the PNO's are required to approximate an accurate natural orbital. PNO's of the same sym metry from separated shells must also be orthogonalized but they are only slightly perturbed if they are well localized. This argues for the use of optimally localized9 HF orbitals as the basis for the n-2 electron field, when two or more doubly occupied orbitals are under consideration. Evidently this procedure represents a compromise in finding the most rapidly convergent set of configurations. Only a few pairs are well represented but the correlation function for these pairs will span the important regions of space such that the correlation of all pairs is ade- 12H. Shull and P.-O. L6wdin, Phys. Rev. 101, 1730 (1956). ..... -II) -..... 0: --.t< I -I N ..... .; ~ --I -t-oo b i::: --I 1835 -N ...: N n :i ,; This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Sun, 30 Nov 2014 10:18:191836 C. ED MIST 0 NAN D M. K R A U S S TABLE III. Comparison of symmetry-ordered energy contributions to closed orbital pair correlation in He2+, Hz, H2, and H,+.· He2+(R=2.0626) Symmetry type 0'. 0.21 Uu 0.07& ru 0.12 r. 0.068 Sum PNO energies 0.47 Total energy -4.98415 a.u. Correlation energy 1.71 • All correlation energies are in electron volts. quately described by far fewer configurations than are required in an uncritical use of unoccupied HF orbitals. III. GAUSSIAN-TYPE ORBITAL BASIS The GTO basis must be so large that it is impractical to consider variation of all the effective nuclear charges. Fortunately, an accurate variation has already been completed for the first-row atoms by Huzinaga.13 For H and He atomic bases it was then decided that the atom s-function exponent values would be retained and the exponents of the p functions varied to minimize the molecular HF energy. The exponent values are listed in Table I. The linear-expansion coefficients for the large number of s functions provide the necessary flexibility for the determination of the molecular orbitals. The HF results are given in Table II for He2+' Approximate HF results were reported earlier for H3.4 The calculation of the PNO orbitals of 0' symmetry uses only the HF atomic basis. For computational simplicity no attempt was made to augment this basis with additional 0' orbitals or to vary the orbital expo nents. The 11" basis was chosen by varying only a single scale factor for the 2p bases reported by Huzinaga; the effective charge was found to be 2.5. Hz(R=1.8) H2(R=1.4) Hs+(R=1.8) 0.34 0.26 0.34 0.079 0.54 1.00 0.23 0.33 0.25 0.03, 0.04. 0.030 0.68 1.17 1.62 -1. 64934 a.u. -1.17069 a.u. -1.26195 a.u. 1.56 1.03 1.36 The GTO's are particularly advantageous for the calculation of the PNO orbitals since they provide a large number of unoccupied HF orbitals. It did not prove necessary to augment the HF atomic basis set for the determination of the u-type correlation energy. Although the Slater-type orbitals require only about one-half the basis orbitals that are needed if the GTO's are used, the difference in the size of the basis using STO or GTO bases may be much smaller for the SOC. The present evidence is that the GTO basis adequate for the HF calculation is adequate for the determination of the PNO basis. Whether a small STO basis can provide the flexibility for the determination of the PNO is still an open question. IV. SOC CALCULATION The PNO calculations yield the pair correlation energies for substitution from a 1ul pair which is in the field of a 10'u molecular orbital. Comparison of these values, broken down into their symmetry components, with the results for H3+ and H2 dramatically exhibits in the O'u results the exclusion effectl4 and the additional correlation energy that results from the poor asymptotic behavior of HF energies. The results in Table III were TABLE IV. PNO expansion coefficients for He2+ at R=2.0626 a.u. PNO-He2+(R=2.0626) Symmetry orbital 10'. 20'. 30'. 40'. 10' .. 2uu 3uu 0' 1 -0.02611 -0.30717 -0.17803 0.18782 0.00168 0.09661 0.35284 2 -0.00948 -0.24045 -0.27893 -0.33538 0.00587 0.21069 0.60684 3 0.04119 0.00382 -0.16908 0.35996 0.07634 0.14864 -0.13303 4 0.27861 0.29662 -0.50730 0.22692 0.36233 0.76119 -0.61615 5 0.25179 -0.25677 0.42485 -1.19441 0.31319 -0.35061 -0.07709 6 0.11237 -0.30695 0.22816 0.74670 0.13775 -0.37918 0.34453 7 0.03434 -0.04248 0.05364 0.16884 0.03986 -0.06199 0.06077 8 0.00692 -0.00980 0.00746 0.01355 0.00807 -0.01142 0.00739 9 0.00092 -0.00099 0.00109 0.00207 0.00106 -0.00131 0.00099 'II' lru 2'11'u In'. 211'. 1 0.45006 0.63843 0.31933 0.90813 2 0.24753 -0.72021 0.46735 -0.52497 3 0.02650 -0.20340 0.05646 -0.30231 13 S. Huzinaga, J. Chern. Phys. 42,1293 (1965). U V. McKoy and O. Sinanoglu, J. Chern. Phys. 41, 2689 (1964). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Sun, 30 Nov 2014 10:18:19PSEUDONATURAL ORBITALS. 1. He2+ 1837 obtained by exciting the orbital pair into excited orbitals of only one symmetry type at a time but the total energies result from complete mixing of all configura tions, including in the case of Hs and He2+ the intershell ones. The sum of PNO energies exceed the actual corre lation energy for the two-electron cases because all pair excitations were not mixed simultaneously and because many of the doubly occupied PNO configurations were not included in the final SOC. Only four Ug, three Uu, two 1I"u, and one 1I"g PNO's were used in both H2 and Hs+. The tabulation of all the linear-expansion coefficients for the He2+ PNO would be burdensome but the more important are found in Table IV ordered according to occupation number for the equilibrium separation. Note that the first Uy PNO is very close to the HF molec ular orbital and in this application, the first UU PNO is identical to the HF molecular orbital. For the three-electron problem, construction of sym metry-adapted configurations for the final SOC can be found by inspection or by the use of projection operators. Energy matrix elements can readily be con structed by the computer program from a knowleqge of these configurations. The basic types of configurations were illustrated in a prior communication.16 Additional configurations of these types were included to exhaust the possibilities of the Gaussian basis. One new type permitted single excitation from the HF configuration. This type provides a means for relaxation of the open shell electron by an SOc. It is equivalent to self-con sistent optimization of the luu orbital within the given Gaussian basis with correlation of the remaining elec trons. The final set for which energies are reported included 45 configurations. They are symbolized by their orbital occupation in Table V. One group is represented by double-orbital excitations from the pair luy2• The energy contributions of these configurations would equal those obtained from the original PNO calculations if all the PNO's were used. Little is lost by truncating the set of PNO's. Only configurations that contribute at least 0.01 eV are included. The contribution of a configura tion is assessed by determining the energy improvement between the result for the nine most important configu rations and that obtained by adding the given con figuration to the base nine. The resultant energies and expansion coefficients are given in Table VI for a range of internuclear separations of He2+. In Table VII the energy con tributions of the configurations for R = 2.0626 a.u. are listed. It should be noted that for large inter nuclear distances additional configurations should be included or substitutions made. However, the present 45 provide the necessary flexibility for the Gaussian basis if the energy improvement criteria is valid. V. DISCUSSION It is unfortunate that the accuracy of this procedure cannot be tested by comparison with an experimental 16 C. Edmiston and M. Krauss, J. Chern. Phys. 42,1119 (1965). TABLE V. SOC configurations: S refers to configuration where two orbitals are singlet coupled and T to the same orbitals triplet coupled with over-all doublet symmetry. 1 2 3 4 5 6 7 8 9 10 11 12 13 17 S 14 T 18 S 15 T 16 T 19 S 25 T 20 S 26 T 21 S 27 T 22 S 28 T 23 S 29 T 24 S 30 T 33 S 31 T 34 S 32 T Configuration 10-0 10-" 20-" b·" b·o 211"" 211"0 35 36 37 38 39 40S 43 T 41 S 44T 42 S 45 T value for the dissociation energy of He2+. The interrela tion between this dissociation energy and the dissocia tion energies in neutral excited He2 is of little value since the neutral states possess large potential maxima.ls Identification of the Rydberg state of He which can 16 M. L. Ginter, J. Chern. Phys. 42, 561 (1965); J. C. Browne, ibid. 42, 2826 (1965). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Sun, 30 Nov 2014 10:18:191838 C. EDMISTON AND M. KRAUSS TABLE VI. Results of SOC calculation for He.+. Internuclear separation 1.75 1. 9376 Energy -4.97131 -4.98306 configuration 1 0.9931 0.9923 2 -0.0240 -0.0245 3 -0.0227 -0.0247 4 -0.0186 -0.0182 5 -0.0032 -0.0031 6 -0.0059 -0.0068 7 -0.0021 -0.0027 8 0.0031 0.0038 9 0.0010 0.0005 10 -0.0002 -0.0007 11 0.0023 0.0032 12 -0.0344 -0.0353 13 0.0073 0.0086 14 -0.0038 0.0430 15 -0.0808 -0.0791 16 0.0157 0.0149 17 -0.0021 -0.0017 18 -0.0007 -0.0015 19 -0.0249 -0.0102 20 -0.0066 -0.0138 21 -0.0188 -0.0293 22 0.0120 0.0070 23 0.0025 0.0021 24 0.0010 0.0011 25 -0.0113 -0.0093 26 -0.0010 -0.0029 27 0.0031 -0.0033 28 0.0032 0.0022 29 -0.0013 -0.0014 30 0.0004 0.0003 31 -0.0002 -0.0002 32 -0.0009 -0.0008 33 0.0003 0.0023 34 -0.0001 -0.0006 35 -0.0351 -0.0341 36 -0.0164 -0.0175 37 -0.0049 -0.0050 38 -0.0029 -0.0029 39 0.0013 0.0013 40 0.0338 0.0342 41 0.0058 0.0060 42 0.0044 0.0039 43 -0.0101 -0.0091 44 0.0017 0.0018 45 -0.0028 -0.0023 yield the molecular ion upon collision with ground state He yields a high upper bound to the He2+ energy,I1 An accurate bound could only be obtained if the ejected electron's energy would be analyzed. At present no accurate experimental information on the dissociation energy of He2+ exists and comparison is limited to other calculations. At equilibrium internuclear separation the present results are in excellent agreement with the calculations of Reagan et al.ls The dissociation energy obtained in this work is D.= 2.24 e V. At smaller distances the present results are yet more accurate, but the results 17 F. J. Comes, Z. Naturforsch. 17a, 1032 (1962). 18 P. N. Reagan, J. C. Browne, and F. A. Matsen, Phys. Rev. 132,304 (1963). 2.0626 2.1876 2.375 4.98415 -4.98207 4.97551 0.9919 0.9916 0.9911 -0.0280 -0.0300 -0.0325 -0.0229 -0.0225 -0.0221 -0.0180 -0.0179 -0.0178 -0.0032 -0.0033 -0.0034 -0.0074 -0.0081 -0.0091 -0.0033 -0.0036 -0.0040 0.0044 0.0048 0.0054 0.0019 0.0022 0.0025 -0.0015 -0.0017 -0.0020 0.0022 0.0020 0.0019 -0.0362 -0.0370 -0.0382 0.0096 0.0105 0.0118 0.0827 0.0886 0.0945 -0.0417 -0.0351 -0.0314 0.0131 0.0111 0.0089 -0.0005 -0.0005 -0.0009 -0.0021 -0.0023 -0.0027 0.0100 0.0128 0.0146 -0.0165 -0.0173 -0.0184 -0.0292 -0.0280 -0.0270 -0.0030 -0.0048 -0.0062 0.0017 0.0016 0.0016 -0.0005 -0.0014 -0.0022 -0.0049 -0.0036 -0.0023 -0.0038 -0.0039 -0.0040 -0.0078 -0.0075 -0.0067 -0.0001 -0.0006 -0.0010 -0.0014 -0.0014 -0.0013 0.0004 0.0005 0.0007 -0.0002 -0.0002 -0.0002 -0.0007 -0.0007 -0.0007 0.0019 0.0017 0.0016 -0.0010 -0.0010 -0.0009 -0.0333 -0.0325 -0.0313 -0.0182 -0.0189 -0.0198 -0.0050 -0.0050 -0.0050 -0.0030 -0.0031 -0.0032 0.0013 0.0013 0.0013 0.0345 0.0346 0.0348 0.0060 0.0059 0.0055 0.0036 0.0033 0.0028 -0.0084 -0.0078 -0.0068 0.0019 0.0019 0.0019 -0.0021 -0.0019 -0.0015 are worse at larger distances. No thorough study of this effect has been made, but apparently it is due to lack of sufficient terms to determine accurate asymptotic values with regard to both HF and correlation terms. Since this study was primarily a test of the efficacy of the GTO-PNO procedure the accurate results at the internuclear separation were accepted as a proof of the usefulness of the procedure and an additional effort for the asymptotic distances was not considered worth while. The present results provide the data for a detailed investigation of the origin of correlation in the He2+ molecule in the neighborhood of the equilibrium sepa ration. The configurations which form the basis for the PNO calculation actually contribute only 0.45 eV This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Sun, 30 Nov 2014 10:18:19PSEUDONATURAL ORBITALS. I. He2+ 1839 out of a total of l.71-eV correlation energy at R= 2.0626 a.u. Nonetheless, the set of PNO provides a rapidly convergent set of configurations. Consider the interesting configurations 14, 15, and 16 which represent a double excitation in spin orbitals but only a single space orbital excitation. These three configurations exhaust the contribution for such excitations with this basis. The triply coupled pair contributes 0.66 eV and would have a considerable effect on the natural orbitals for the three-electron system. These configurations can be ascribed to intershell correlation which, in effect, exchanges the orbital spin while introducing additional nodes in the gerade pair. The calculation of properties dependent upon the spin density such as the Fermi contact term are very dependent on these terms. The single and triple spin-orbital excitation terms do not contribute significantly. However, all possible configurations of this type were not considered. With the PNO's one can never avoid the suspicion that the convergence in a set of configurations is not mono tonic; configurations including, say, the ninth {}'g PNO may be significant but they are never considered. The main group of intrashell configurations are those used to determine the PNO's. These configurations can be categorized as either in-out (Configurations 2, 3, and 5), left-right (4 and 6), and angular (35-38) correlationI9 which graphically describes the correlation with respect to the molecular axis. The rapidity of convergence of the PNO configurations is illustrated in Table VII for the above-mentioned configurations. Additional intrashell terms that were considered fall into two categories. Configurations 12 and 13 are significant and compensate for the reduction in the left-right correlation due to the exclusion effect. The second group are single excitations, in Configurations 17 and 18, and double excitations, in Configurations 31-34, which test the efficacy of the diagonalization of the first-order density matrix. The latter configurations contribute essentially nothing. The intershell configurations cannot easily be categor ized. However, all angular-correlation terms can be summed to give 0.43 eVfor R=2.0626. The PNO terms by themselves give a misleading picture of the correla tion. The intershell configurations must be added to the total and they contribute 0.41 eV to the total energy. The intershell contributions are practically independent of internuclear separation for the small range considered. If the configurations 14 and 15 are 19 A. D. McLean, A. Weiss, and M. Yoshimine, Rev. Mod. Phys. 32, 211 (1960). TABLE VII. Approximate contributions, tiE, of the con figurations to the correlation energy (in electron volts) for He2+, R=2.0626 a.u. Configuration 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 tiE 0.120 0.076 0.060 0.004 0.016 0.000 0.003 0.001 0.000 0.001 0.130 0.015 0.555 0.078 0.027 0.000 0.001 0.014 0.056 0.112 0.001 0.001 Configuration 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 tiE 0.000 0.003 0.003 0.008 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.131 0.063 0.011 0.005 0.000 0.184 0.023 0.003 0.007 0.000 0.000 added to the intershell total, it is evident the PNO configurations were not necessarily the best group to determine the approximate natural orbitals. Nonethe less, the present PNO choice allows calculation of the intershell correlation with a practicable number of configurations. Some 24 of the 45 configurations can be neglected with only a loss of about 0.02 eV in the total energy. The ordering of the significant configurations by the PNO occupation number does not hold as rigorously as for the intrashell terms. But the triplet-coupled terms and those involving the fourth PNO in a group could be neglected with the loss of only 0.02 eV. Again, it must be cautioned that the fifth and higher PNO's were not considered and could hold some surprises. The results for the simplest three-electron molecule shows that for open-shell systems the correlation energy is fragmented among many configurations and none dominate. The PNO basis supports a SOC which is rapidly convergent even using GTO's. In fact, without the PNO or some analogous technique it would be hopeless to proceed further than the HF limit with the GTO's. ACKNOWLEDGMENT We thank Joice Doolittle for assistance in the calcu lations. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Sun, 30 Nov 2014 10:18:19
1.1725730.pdf	Desorption from Metal Surfaces by LowEnergy Electrons Dietrich Menzel and Robert Gomer Citation: J. Chem. Phys. 41, 3311 (1964); doi: 10.1063/1.1725730 View online: http://dx.doi.org/10.1063/1.1725730 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v41/i11 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 41, NUMBER-II 1 DECEMBER 1964 Desorption from Metal Surfaces by Low-Energy Electrons DIETRICH MENZEL * AND ROBERT GOMER Institute for the Study of Metals, and Department of Chemistry, University of Chicago, Chicago, Illinois (Received 22 June 1964) The effect of low-energy (lS-200-eV) electrons on hydrogen, oxygen, carbon monoxide, and barium adsorbed on tungsten has been investigated by a field-emission technique. Desorption cross sections u were determined from work function and Fowler-Nordheim pre-exponential changes and are significantly smaller than would be expected for comparable molecular processes. Marked variations in cross section with binding mode within a given system were found. Thus uH=3.S 10-,20 cm2 and SX10-,21 cm2 for processes tentatively interpreted as the splitting of molecularly adsorbed H2 and desorption of H, respectively; uo=4.S X 10-19 cm2 for a loosely bound state and uQ~2XlO-,21 cm! for all other states; uB.<2X10- cm2 under all conditions. In the case of CO (reported in detail elsewhere), three binding modes observed pre viously could be confirmed and differentiated by their different cross sections: UVir"in=3X10-19 cm2; u~= S.8X 10-,21 cm2, Ua =3X 10-18 cm2; conversion by electrons of virgin to!3 uv,Y-1O-19 cm2• These results are interpreted in terms of transitions from the adsorbed ground state to repulsive portions of excited states, followed by de-exciting transitions which prevent desorption. Arguments are made to show that the excita tion cross sections should be essentially "normal," i.e., ",10-16 to 10-17 cm2, and that the much smaller over-all cross sections observed are due to high transition probabilities to the ground state, estimated as 1014 to 1015 secl. A detailed calculation for the case of exponentially varying transition probabilities and repulsive upper states is presented and discussed, and the variations in cross section with binding mode made plausible. It is shown that low-energy electron impact constitutes a sensitive tool for studying chemisorption. ALTHOUGH the statics and kinetics of chemisorp ..tl. tion have been investigated extensively, most studies have been confined to processes caused by mo mentum transfer to the adsorbate nuclei by phonons (e.g., thermal activation) or massive particles (e.g., ion sputtering). With the exception of field desorption,l which occurs by transition to a field-deformed ionic state, very little attention seems to have been paid to processes involving electronic transitions. The most obvious means of causing these are photons or slow electrons. In the former case energies of 5 eV or above are not readily available at high intensity, and ob servable photon-induced processes may therefore be confined to dissociation into the vibrational continuum of the electronic ground state2 by adsorbate-surface dipole transitions. The threshold energies for this type of transition equal the heats of adsorption, 2-5 eV, but the transition probabilities are undoubtedly low and these processes may therefore be very difficult to observe in any case. For electrons. on the other hand, achievable intensity increases with energy (in the 1O-200-eV range) and primary excitation cross sections should be comparable to those observed in molecules,3 i.e., ",1O-1L10-17 cm2• Thus low-energy electron im pact becomes of theoretical interest for adsorption * Present address: Lehrstuhl flir Physikalische Chemie, Tech nische Hochschule, Darmstadt, Germany. I R. Gomer and L. W. Swanson, J. Chern. Phys. 38, 1613 (1963) . 2 W. J. Lange and H. Riemersma, Trans. Nat!. Vacuum Symp. 9, 197 (1962). 3 J. D. Craggs and H. S. W. Massey, Handbuch der Physik, edited by S. Fliigge (Springer-Verlag, Berlin, 19S9), Vo!' 37/1, p.314. studies. In addition, desorption by slow electrons can have practical importance for high-vacuum devices like ion gauges and mass spectrometers. Despite these facts relatively little work has been reported. Plumlee and Smith4 found evolution of 0+ from Mo surfaces with an efficiency of 10-6 ion/electron at 11000K and 10-8 ion/electron at 3000K for 300-eV electrons. Ion energies up to 10 eV were observed, but no depletion of the surface layer was noted. Young5 found that 0+ evolved with an efficiency of 10-5 ions/ electron from oxidized Cu, Ni, Mo, Ta, and Ti surfaces. Moore6 studied the effect of 20-300-e V electrons on CO adsorbed on Mo ribbons and observed only 0+, with an efficiency up to 10-4 ion/electron. In order to obtain measurable signals he had to use electron currents so large that the Mo substrate was heated to 900°K. Moore's thresholds for 0+ production, 17.0- 18.3 eV, are approximately 7.0 eV lower than that of the corresponding gas-phase reaction. These investiga tions were all carried out mass-spectrometrically and yielded only information on ionic desorption products. The surfaces were rather poorly characterized in all cases and the results are therefore chiefly of qualitative interest. Indirect evidence for an effect also comes from field ion microscopy. Mulson and Mtiller7 and Ehrlich and Hudda8 attribute depletions and changes in adsorbed 4 R. H. Plumlee and L. P. Smith, J. App!. Phys. 21,811 (1950). 5 J. R. Young, J. App!. Phys. 31, 921 (1960). 6 G. E. Moore, J. App!. Phys. 32, 1241 (1961). 7 J. F. Mulson and E. W; Miiller, J. Chern. Phys. 38, 261S (1963) . 8 G. Ehrlich and F. G. Hudda, Phil. Mag. 8,1587 (1963). 3311 Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3312 D. MENZEL AND R. GOMER layers to e!ectrons released by gas atoms ionized in the vicinity of the field emitter. These effects occur in the presence of an applied field of ",4 V / A. In addition, a number of studies motivated mainly by practical considerations have been reported. Marmet and Morrison9 and RobinslO investigated the effects of adsorbed gases in ion sources for mass spectrometers, and 'HartmannlJ and Redheadl2 studied the effects of electron-impact desorption in Bayard-Alpert ioniza tion gauges. Degras, Peterman, and Schramml3 believe that they can use electron-impact desorption to dis tinguish between adsorbed and absorbed gases in stainless steel. Very recently Redheadl4 and the present authors/6 working independently and by quite different tech niques, have investigated electron desorption more quantitatively and have attempted to provide theo retical models to account for their results. Redhead bombarded a Mo ribbon with electrons, and energy analyzed the resultant ion current. Although his method permits only the direct determination of ions, he was able to infer the rate of neutral desorption from the rate of ionic desorption by regarding the ion current as a measure of the total adsorbate coverage. His results indicate the existence of at least two distinct binding states of 0 on Mo. For the more labile state his efficiencies are 6.SX 10-4 atom/electron and 1.3X 10-5 ion/electron, and for the more tightly bound state 3.9XlO-7 atom/electron and 7.0XlO-9 ion/electron. He explains these results, and his ion energy distribu tions, by a mechanism which consists of primary ex citation to the ionic state, followed either by ionic desorption or, in the majority of cases, Auger transition to an excited neutral state. This mechanism is very similar to that proposed by us. Redhead also investi gated the desorption of CO but found it to be too rapid for quantitative measurements. A preliminary account of some of the present work has appeared in this journal.l5 Our specific interest in electron desorption originated in field desorption: It was found in all cases investigated that the pre-expo nential term of the desorption rate constant decreased dramatically with increasing field, or perhaps better with the decreasing temperature of the measurements necessitated by increasing field.16•l7 A theoretical calcu lation of the desorption rate constant,l based on a rather crude one-dimensional model, indicates that 9 P. Marmet and J. D. Morrison, J. Chern. Phys. 36, 1238 (1962) . 10 J. L. Robins, Can. J. Phys. 41, 1385 (1963). 11 T. E. Hartman, Rev. Sci. Instr. 34, 1190 (1963). 12 P. A. Redhead, Vacuum 12, 267 (1962). 13 D. A. Degras, L. A. Petermann, and A. Schramm, Ref. 3, p.497. 14 P. A. Redhead, Vacuum 13, 253 (1963); Can. J. Phys. 42, 886 (1964). 15 D. Menzel and R. Gomer, J. Chern. Phys. 40, 1164 (1964). 16 H. Utsugi and R. Gomer, J. Chern. Phys. 37, 1706 (1962). 17 L. W. Swanson and R. Gomer, J. Chern. Phys. 39, 2813 (1963). transition from the neutral to the ionic state at the cross-over point is rapid, i.e., that electron tunneling from the adsorbate into the metal is fast. If this is correct the inverse process, electron tunneling from the metal into the adsorbate, should also be rapid. We felt that this conclusion would be confirmed, at least qualitatively, by small over-all cross sections for elec tron desorption: If the cross section for ejection of an electron from the adbond is comparable to excitation cross sections in molecules, a much smaller over-all cross section for desorption could result only from rapid "healing" of the broken or excited adbond by electrons tunneling into it from the metal. In other words, the relevant electronic transitions would have to be rapid relative to the time required by the adsorbate to move through the critical recapture zone. If this general mechanism is correct, one should expect the electronic transition probability, the dwell time of the adsorbate, and hence the over-all desorp tion cross section to vary with the mode of binding in a given adsorbate-substrate system. Consequently elec tron desorption might constitute a sensitive probe for distinguishing between different adsorption states, and we were also anxious to explore this possibility. METHOD In studying desorption one has the choice of looking at the products or the residual surface layer. Generally speaking the first method requires macroscopic sur faces and thus introduces some uncertainties since sub strate characterization is usually difficult, except on a microscale. However, it has great sensitivity for ionic products, which can be mass-and energy-analyzed quite directly. On the other hand small over-all cross sections make it very difficult to determine neutral desorption products directly since geometry, intensity, and background limitations virtually interdict the use of mass spectrometry for their subsequent ionization and analysis, except in very favorable cases, or by very elaborate special designs. The second method suffers from the disadvantage of lower sensitivity, and the fact that it cannot sort out desorption products or yield information on their energy. However, it has the very great advantages of being applicable to micro specimens and of being capable of yielding detailed information on the surface before and during reaction. The present experiments are based on this alternative, and utilize a tungsten field emitter as substrate. The qualitative effects of electron impact were determined visually by examining the field-emission patterns. More quantitative information was obtained from emission data and the Fowler-Nordheim equa tion.lsa For present purposes this can be written in 18 (a) R. Gomer, Field Emission and Field Ionization (Harvard University Press, Cambridge, Massachusetts, 1961), p. 19 (Eq. 51). (b) Ibid., p. 50. (c) Ibid., p. 114. (d) Ibid., p. 175. Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsDESORPTION FROM METAL SURFACES 3313 logarithmic form as In(i/V2) = InA-kcf>i/V, ( 1) where i is the total emitted current, V the applied voltage, A is a constant,1sa cf> the work function of the emitting area, and k a constant which includes the field-voltage proportionality. Work functions of a gas covered emitter can thus be obtained in the usual way by comparing the slopes of In(ijV2) vs 1jV plots for clean and adsorbate covered emitters if the work function of the former is known. It is obvious that the use of Eq. (1) is limited, strictly speaking, to regions of constant cf> and k, but in practice only minor errors are introduced by averaging over an emitter if emission for the clean and covered states comes from approxi mately the same regions of the tip, so that the initial cf> and k assignments apply in both cases. With the exception of electropositive adsorbates this is usually the case in chemisorption. Cross sections can be determined from emission changes as follows: For a given adsorption state j (2) where no' is the electron flux in electrons per square centimeter· second, 0" j the desorption cross section in square centimeters, and N j the coverage of state j in adparticles per square centimeter. If the relation be tween coverage N j and work function cf> is linear over the desorption range-almost certainly an adequate approximation for small coverage changes-we may write (3) where Cj is a constant and cf>"" the work function when N;=O. Combination of Eqs. (2) and (3) then yields for the cross section where i is the current density in amperes/square centi meter and cf>o and cf>l are, respectively, the work func tions at times 0 and t. Thus the correctness of the postu lated first-order reaction and the value of 0" j can be obtained from plots of log(cf>t-cf>,,') vs t. It has long been known that the Fowler-Nordheim pre-exponential A is coverage-sensitive, over and above its relatively unimportant direct cf> dependence. If the relation between A and N is known explicitly, regardless of cause, it can obviously be exploited for coverage and also cross-section determinations. There is good theoretical reason to expect an expo nential relation A=Aoexp(-gN) (5) in the case of electronegative adsorbates of small polarizability,1sb and recent work in this laboratory19 19 A. Bell and R. Gomer (to be published). has in fact confirmed this for CO adsorbed on W, with remarkably good agreement between experimental and calculated values of the constant g. Since the changes in A turn out to be of interest for the present work, it is worthwhile to indicate the origin of Eq. (5). In the presence of an applied negative field F the total work function is cf>(F) =cf>+47rNexF/c, (6) where ex is the polarizability of the adsorbate and F j c the effective polarizing field. While c, effectively the dielectric constant of the adlayer, depends on N, its variation from N=O to N=Nmax is of the order of 20% for CO and less for H. If the second term in Eq. (6) is small compared to cf>, the Fowler-Nordheim exponent may be expanded as follows: exp( -6.SX lO7) (cf>+47rNexF /c)! F ""'[ exp( -6;X 107)cf>JI exp( -6.S~107) 67rexcf>Wl (7) Since the second exponential in this expression does not contain F, it appears as part of A in accordance with Eq. (5), with g given by g= 1.2SXlO-15excf>ljc cm2jmolecule (S) for ex in cubic angstroms and cf> in electron volts. It should be noted that the dependences of cf>! and C on N are slight in the first place and more or less cancel each other out. Thus in the case of CO on W the term cf>!jc varies from '"'-'2.0 at zero coverage to ,...,1.96 at maximum coverage.17 If ex is large c and its variation with coverage cannot be neglected, of course. For those cases where the present approximations are adequate, the time dependence of InA can be used to obtain desorption cross sections. It is easy to show that 3.6SX lO-19 I 10g(Ai/ A"J ) O"j= it oglOlog(At/Aro) , (9 where Ai, At, and Aoo are, respectively, the Fowler Nordheim pre-exponentials corresponding to times 0, t, and infinity (meaning desorption of the reactive species only). The results of Eq. (9) can also be com pared with those of Eq. (4) where both cf> and A changes occur to provide a check on self-consistency. Since the reference value of A invariably refers to the adsorbate free substrate, it is convenient to define a quantity B as B= In(Ao/ A), (10) with Ao the value of A for a clean substrate. Eq. (9) then takes the form 3.6SX 10-19 Bi-Boo O"j= it loglo Bt-B oo (11) Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3:314 D. MENZEL AND R. GOMER FIG. 1. Schematic diagram of field-emission tube for electron desorption. A field-emission tube, looking from rear toward screen; B getter bulb; I gun assembly; 2 chemical gas source with conducting shield; 3 tip loop with potential leads; 4 collector electrode; 5 anode connection; 6 screen; 7 Ta getter wire. with the subscripts having the same significance as for Eq. (9). EXPERIMENTAL The apparatus used is shown schematically in Fig. 1. It consists essentially of a low-temperature field emis sion tube18C equipped with a simple electron gun in addition to the usual chemical and/or sublimation gas sources. In order to insure high vacuum the sealed-off tube was immersed in liquid helium or hydrogen during experiments. The electron gun is shown in Fig. 2. It consisted of a W cathode, beam-forming electrode (usually close to cathode potential), and anode. The cathode assembly consisted of a 3 mm long hairpin of 0.002-in.-diam W wire, spotwelded to 5 mm long O.Ol-in.-diam. Pt wires, spotwelded in turn to ,,-,5-10 mm long 0.015 in. diam Nichrome supports. The use of Nichrome thermal barriers was necessary to prevent sudden catastrophic onset of resistive heating of the tungsten loop. The insertion of platinum wires prevented slow and con tinuing gas liberation by excessive heating of the ni- FIG. 2. Schematic cross section of electron gun drawn to scale indicated. I anode, O.OOI-in. Pt foil; 2 focusing cylinder, O.OOI-in. Pt foil; 3 O.OO2-in. diam W filament; 4 O.OIO-in. diam Pt wire; 5 O.OIS-in. diam Nichrome wire. Leads to electrodes for electrical heating are indicated by the letter L. chrome barriers; the outgassing of Pt proceeded rapidly and completely. All electrodes were constructed from O.OOl-in. Pt foil and could be outgassed by resistive heating. The gun exit was generally placed ,,-,5 mm from the tip, with which it was lined up visually. In most tubes a Pt receiving anode which could be out gassed resistively was used to prevent the electron beam from hitting the glass walls of the tube, in order to minimize desorption of physically adsorbed gas. The receiver was run 10-15 V positive with respect to the tip to prevent the bulk of secondaries from reaching the latter. In the few cases where the receiver was omitted the inner walls of the tube (conductivized by the tin oxide method) were at tip potential or + 10 V. An electron gun rather than a source in the form of a spiral filament surrounding the field-emitter assembly was chosen because the former, if constructed of pure W, heated the field emitter appreciably at the tempera tures required for adequate thermionic emission. On the other hand the problems of contamination and out- Imm W FIG. 3. Microprobe for current density determination. W: O.OO5-in. W wire etched to resemble emitter, beaded with Nonex shield coated with Aquadag. A small Pt sleeve is fitted over the Nonex coating to prevent charge buildup on the end of the Nonex which cannot be covered with Aquadag .. gassing militated against the use of oxide cathodes, either as a simple spiral or in the gun. Tubes were outgassed, loaded, and sealed off on greaseless high-vacuum lines, with the usual precau tions. After immersion in liquid helium or hydrogen, the emitter tip and gun were cleaned by resistive heating, allowed to cool, and the tip dosed with gas from the source. After low-temperature spreading the gun electrodes were outgassed once more before actual experiments. Blank runs were carried out by observing the effect of gun operation on clean and dosed tips with and without electrons reaching the latter. In this way it was established that the effects attributed to electron impact were not caused either by heat from the gun filament or by contamination. Since con tamination of the tip limits the ultimate sensitivity of the method special precautions and construction features were required for different systems and these are discussed individually later. Guns were calibrated after the completion of experi ments by replacing the field emitter assembly by a microprobe (Fig. 3) consisting of a 0.OO5-in.-diam wire Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsDESORPTION FROM METAL SURFACES 3315 TABLE I. Performance data for a typical electron gun. a Common for run and calibration Vth, VA V.oll VIDe iheat (V) (V) (V) (V) (A) 5 20 17 -0.5 0.97 10 20 15 -0.5 0.98 12 20 14 -0.5 0.97 15 20 18 -0.5 0.97 20 20 25 -0.5 0.97 30 20 35 -0.5 0.97 40 20 50 -0.5 0.96 50 20 60 -0.5 60 20 70 -0.5 0.95 80 20 90 -0.5 0.95 100 20 110 -0.5 0.95 130 20 145 -0.5 0.95 160 20 180 -0.5 0.95 200 20 210 -0.5 g.95 240 33 260 -0.5 .95 300 45 320 -0.5 0.95 Run only iA iool1 ilOOf) (AX 1()6) (AX 1()6) (AX 1()6) 26.5 1.0 2.5 20.5 11.8 1.3 11.3 2.8 5.0 11.8 2.9 4.3 9.5 2.5 4.2 9.5 3.5 4.2 6.2 3.0 3.2 6.5 3.0 3.6 5.5 2.5 3.1 5.0 2.0 3.0 5.0 2.2 3.2 4.8 2.0 3.1 5.0 2.0 3.4 4.1 2.0 3.4 5.0 3.5 5.4 6.5 2.5 5.3 Calibration only iprobe (AX 1()9) 0.03 0.4 3.2 3.6 7.0 11.2 9.6 12.8 11. 7 11.9 17.5 17.5 33.5 43.3 51.5 53.5 j (AI cm2X 1()4) 0.01 0.14 1.1 1.3 2.5 4.0 3.4 4.6 4.2 4.3 6.3 6.3 12.0 15.5 18.4 19.2 a Abbreviations used in the Table: V 'iv, field emitter voltage during electron impact; VA, gun anode voltage; V.oll, voltage on collector electrode behind tip (see Fig. 1); V foe, voltage on focusing electrode; iheat, cathode heating current; iA, anode current; icon, colJector current; iloOIh current to field emitter assembly; iprobe, probe current; j, current density at probe. bent to resemble the emitter assembly loop, covered with a thin coating of Nonex glass, and then etched to a point resembling an actual emitter both in size and shape. The Nonex coating of the probe was made conducting by painting it with Aquadag, and a fine Pt sleeve was then slipped over its end. The size of the probe was determined with an optical microscope. When sealed into the tube the probe coincided within 0.5-1 mm with the original tip location. Experiments in which the probe was moved, and the reproducibility of results from emitter to emitter indicated that beam profiles were uniform to ",SO% over 3-4 mm, so that this procedure could not have introduced large errors into absolute intensity determinations. After position ing the probe the tubes were sealed off under vacuum and cooled with liquid H2 to establish ultrahigh vacuum and prevent any ion formation. Probe currents were then measured under all the conditions used in actual desorption experiments, with the Nonex and Pt sleeves o 10 20 30 40 Probe Current x 109 Amperes FIG. 4. Total extracted current versus probe current for a typical electron gun at 80 eV. kept at probe potential. Representative data are shown in Table I. It is seen that beam intensities in the tip region were generally 10-3 to 10-4 A/cm2 over an energy range from 10-300 V. Figure 4 shows the total extracted beam current vs probe current at 80 eV. It is seen that the curve is linear up to probe-current densities of 1.3X 10-3 A/cm2, indicating that space charge spreading is unimportant below this value. It may be worthwhile to mention that resistive heating of the tip by the electron beam can readily be calculated to be less than 1°-lOOK at the highest beam intensities used for 100-eV electrons, depending on the value of the thermal conductivity used (2-30 W cm-1·deg-1 at 21°K). The absence of diffusion or major changes even with physisorbed layers, or of nonlinear intensity effects indicates that this estimate is substantially correct. Work functions and B values were determined from current-voltage measurements and the Fowler-Nord heim relation, Eq. (1). Since desorption rates were small and in some cases near the limit of detectability, great care had to be exercised. In particular, the de sorption of physisorbed gas from the screen by field emitted electrons had to be reduced below the (in most other work negligible) amount normally en countered by minimizing the amount of gas on the screen and by using very low total emission currents (10-10 to 10-8 A). These were measured with a vibrating electrometer and a 100-MQ load. Voltages were pro vided by a regulated power supply and measured with a digital voltmeter operating on the S-kQ tap of a SO-MQ precision potentiometer across the high-voltage source. Since a large number of work function measurements were required, their calculation was performed on an IBM 1620 computer. A program constructed for this purpose by L. Schmidt of this laboratory, computed Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3316 D. MENZEL .\ND R. GOMER a b c d e f g h FIG. 5. Field-emission patterns for electron desorption of O2 from W: (a) Pattern of clean W emitter. Compression of pattern is caused by the proximity of the gun and source shield which are at emitter potential during field emission. Gun is at lower left, source at lower right. V =S.8 kV, </>=4.50 eV. (b) Fully covered with oxygen at 20oK, then heated to 8soK for 3 min. V = 11.72 kV, </>=6.39 eV, B=4.0S. (c) After impact of 3.7X 1018 electrons/cm2 at SO eV. V =9.73 kV, </>=6.16 eV, B=3.3S. (d) Tip of (Sc) redosed at 20oK. V=11.71 kV, </>=6.40 eV, B=4.10. (e) Tip dosed as in (5b) followed by impact of 4.7XI0'8 electrons/em2 at 100 eV. V=9.64 kV, </>=5.79 eV, B=2.33. (f) Tip of (Se) after 3 min at 400°K. Spots disappear. V= 10.14 kV, </>=S.95 eV, B 2.46. (g) Tip of (Sf) after 3 min. at 540oK. Incipient diffusion of 0 from shank. V =9.89 k V, </> = 5.87 e V, B = 2.40. (h) Tip of (Sg) after S min at 670oK. Further diITusion. V =9.71 kV, </>=S.79 eV, B= 2.35. (i) Tip of (5h) after redosing at 20oK. V = 10.60 kV, </>= 5.91 eV, B =3.10. slopes and intercepts of Fowler-Nordheim In(i/V2) vs 1/V plots as well as 4> and B relative to that of the clean emitter (4)=4.50 eV, B=O). The program tirst found averages based on all data points and then re computed all quantities with rejection of points de viating from the first line by more than very narrow preset limits. If more than one out of the 8 to 10 cur rent-·voltage points normally obtained for a given work function determination was rejected by the computer the work function was regarded as unreliable. The precision of 4> wa.s estimated to be 0.1 %-0.2%, I.e., ",,0.01 eV. Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsDESORPTION FROM METAL SURFACES 3317 Tip temperatures were controlled with a servo mechanism, differing somewhat from that previously described,18d in that the sensing signal was generated in a Kelvin bridge. Emitter loop resistances were read out continuously by a digital ohmmeter consisting of two voltage-to-frequency converters (one measuring current, the other the voltage generated between the potential leads) and a ratio counter which divided the voltage by the current. RESULTS In this section results and specific experimental features for the systems studied are discussed individually. Oxygen on Tungsten The gas source was identical to that used by Gomer and Hulm20 and consisted of an electrically heatable Pt crucible filled with copper oxide prepared in situ. Cooling with liquid hydrogen was found to be adequate for maintenance of high vacuum. The initial state for desorption experiments was obtained by depositing O2 and spreading it at "-'20oK. The tip was then heated to 70°-SO OK to remove excess physisorbed O2• If this was omitted some spurious effects were observed, ap parently because of slight radiative heating from the gun filament. Emission patterns showing the effect of electron impact on a virgin layer are shown in Fig. 5. It is seen that the area hit by electrons shows en hanced emission [Figs. 5 (c), 5 (e)] and decreased work function. At the same time a considerable in crease in granularity is apparent, suggestive of pro truding oxygen molecules or tungsten-oxygen com plexes. This phenomenon will be discussed later. Figure 5(d) shows that redosing raises the work func tion to its original value and restores the pattern to its prebombardment appearance. This indicates that 6.5 3.5 6.0 3.0 8 2.5 2.0 20 40 t (min) FIG. 6. Work function <f> and pre-exponential B vs impact time for an oxygen covered tip bombarded by lOO-eV electrons. The total number of electrons impinged is also shown. 20 R. Gomer and J. K. Hulm, J. Chern. Phys. 27, 1363 (1957). FIG. 7. The data of .10 Fig. 6 plotted as log(<f>-S.72) and log (B-2.20) versus im pact time. • o log (8-2.20) .01 0!--..,2:!::O,.--4:'::0,.--60:';:--::!-:ao;:----J o. I f (min) 8 the effect of electron impact is largely desorption since redosing of thermally oxidized tips does not restore work functions or patterns.20 Heating after electron impact leads to roughly the same conclusion. Figures 5(f) and 5(h) indicate that partial equilibration of the pattern occurs on heating to 670°K. The incomplete equilibration of 100 and its vicinals is probably due to incipient oxidation at 6700K on the high coverage regions not hit by electrons which may prevent diffusion. Redosing with oxygen after heating restores the symmetry, but does not raise the work function to its original high value, as shown by Fig. 5 (i). It should also be noted that the granularity resulting from electron impact [Fig. 5(c)] disappears gradually on heating to 120° to 3000K and is essentially gone at 400°K. While the presence of granularity tends to introduce some scatter into the work function measurements, Fig. 6 clearly indicates the decrease in f/J and increase in B resulting from electron impact at 100 eV. Figure 7 shows correspond ing plots of log(f/J-5.72) and log(B-2.2) vs time. The plots are linear and have essentially identical slopes, giving cross sections of 4.5 10-19 cm2 and 4.1 10---19 cm2, respectively. The limiting values of 4>00= 5.72 eV and Bro=2.2 indicate the presence of adsorbed states with much lower cross sections. This conclusion is supported by the fact that no desorption could be detected if the initial value of f/J was below 5.7 eV, either because of small initial oxygen doses, or because of thermal desorption. An upper limit for desorption from these states was estimated to be (J :S 2X 10-21 cm2 for 20° < T:S 600 oK from the uncertainties in 4> and B measurements. Desorption fell below the detectable limits at elec tron energies lower than 25 e V and no attempt to determine thresholds or detailed energy dependence was made. Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3318 D. MENZEL AND R. GOMER a b c d e FIG. 8. Field emission patterns for electron impact on a hydrogen-covered W tip. (a) Clean W, pattern compressed by gun at lower left. (b) Full hydrogen layer prepared by dosing at 20oK. V=6.460 kV, <1>=4.83 eV, B=2.1O. (c) Same tip after 116 min electron impact at 100 eV (4X 1019 electrons/em2). V =6.211 kV, <1>=4.85 eV, B= 1.13. (d) Same tip after an additional 86 min of impact (7X 1019 eleetrons/em2 in toto). IT=6.115 kV, <1>=4.77 eV, B=1.09. (e) Same tip after addilional100 min of impact (1O.6X1019 electrons/em' in toto). V=6.084 kV, <1>=4.72 eV, B= 1.04. It is seen that the present results agree qualitatively with those of Redhead on MO.14 His cross section for the labile state (1.3 X 10-18 cm2) is approximately three times higher than ours, while his value for the tightly bound state, 6X 10-22 cm, is quite consistent with our estimated limit. In view of the fact that differ ent though similar substrates are involved and in view of the uncertainties in our absolute electron density measurements the agreement between these very different methods is quite good. Hydrogen on Tungsten The gas source was identical to that of Gomer, Wortman, and Lundy21 and consisted of a Pt crucible loaded with zirconium hydride, prepared in situ. In addition the tube was permanently connected by means of a 9-mm-o.d. Pyrex sidearm to a 100-cc getter bulb containing a Ta filament. After seal-off a Ta film was deposited on the getter walls which were kept at 77°K during deposition. Experiments were at first carried out in liquid helium. However it was found that onlv crude upper limits of cross sections could be estimated 21 R. Gomer, R. Wortman, and R. Lundy, J. Chern. Phys. 26, 1147 (1957). in this way because of a slow hydrogen contamination of the tip.15 Apparently this was due to two effects. First, the vapor pressure of H2 at 4.2°K is of the order of 10-7 mm Hg, so that high vacuum can be main tained only as long as all the hydrogen in the tube is present as a physisorbed layer, for which the vapor pressure is much lower than that of bulk H2. Second, desorption by heat from the gun and by field-emitted electrons seems particularly easy with H2. On the other hand, the low vapor pressure of H2 prevents it from leaving the emission tube and being permanently adsorbed on the getter in finite time at 4.2°K. For these reasons later experiments were conducted in liquid H2 as follows. The tip was cleaned and the hydrogen source activated. After times ranging from 5-30 min all the excess hydrogen had been permanently pumped by the getter, and adsorption from all causes reduced by at least an order of magnitude over rates achieveable at 4.2°K. With a properly prepared tube and getter it was even possible to warm to 77°K after activating the source without contaminating the tip with any gas other than hydrogen. The use of liquid H2 as coolant also permitted substantially longer running times since its heat of vaporization is much greater than that of He. Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsDESORPTION FRO~VI METAL SURFACES 3319 FIG. 9. Electron desorption from hy drogen layer at 0<1. (a) Hydrogen covered tip after heating to 385°K for 60 sec. V=6.300 kV, </>=4.84 eV, B= 1.06. (b) Same tip after 83 min of im pact of 100-eV electrons. (2.4X 1019 electrons/cm2). V=6.136 kV, </>=4.78 eV, B=0.76. a The results of electron impact on hydrogen-covered emitters are shown in Figs. 8 and 9. The bombarded region extends over most of the tip, so that less con trast between the enfiladed and defiladed zones is noticeable than in the case of oxygen. It is seen that the patterns after electron impact closely resemble those obtained after thermal desorption. Figure 8 (c) lies somewhere between Figs. 17 and 18 of Ref. 21 while Fig. 8( d), representing more desorption, re sembles Fig. 19 of Ref. 21. The resemblance can also be seen by comparing Fig. 8(c) to Fig. 9(a), which corresponds to pure thermal desorption after heating to 385cK. Figure 9(b) shows the results of electron impact on a partially desorbed layer, i.e., that of Fig. 9(a). The work function and pre-exponential changes for a fully covered tip at 200K are shown in Fig. 10. It is seen that there is an initial period during which ¢ increases slightly while B decreases rapidly, followed by a period in which ¢ decreases while B stays essen tially constant. It is very interesting to note that this behavior closely resembles the changes observed on pure heating. Figure 11 shows ¢ vs B for electron im pact (squares) and for thermal desorption (circles). The latter are replotted from Fig. 24 of Ref. 21. Figure 50 '0 11.1019 2.5 4.90 470 4.60 ,.0 o 50 'DO 150 200 I(mln) FIG. 10. Work function</> and pre-exponential B vs time for 100 eV electron impact on a fully hydrogen-covered tip. Arrows indicate", and B changes on redosing at 20oK. b 12 shows semilogarithmic ¢ and B plots. In the second regime ¢ approaches the clean W value, 4.5 eV, and thus we have plotted log(¢-4.5) vs t. It is seen that the curve is quite linear and leads to a cross section of 5X 10-21 cm2• If log(¢-4.50) for this regime is back extrapolated to 1=0 a value of ¢(O) extr=4.99 eV is obtained. It is possible to plot the small ¢ change for the first regime as log(¢extr-¢) vs I. This plot is also shown in Fig. 12 and leads to a cross section of 3.5X 10-20 cm2. Since B approaches a value of 1.1 at the end of the first regime and stays sensibly constant thereafter a plot of 10g(B-1.1) vs t was made. It closely parallels the log (¢extr-¢) vs t plot and gives a cross section of 3.3 X 10-20 cm2 if the first point is dis regarded. The reason for doing this appears presently. These results indicate that there are at least two adsorbed species present on the surface with consider ably different cross sections. The ¢ and B behavior strongly suggests that the first regime corresponds to desorption or dissociation of a molecularly adsorbed species. This is supported by the fact that such a species would have a positive contact potential since it would probably be adsorbed with some electron transfer to the metal, and would have a much larger effect on B than H because of its much greater polar- 490f 165 itO 50 20 480 1·0 400 370 1> 'V 470 460 550 492 4 50----.J0-=-5 -~ -:,70----L, =-5 ----c2:'cO---:275 ----="3 0 FIG. 11. Comparison of </> vs B curves for hydrogen on W emitters for thermal and electron impact desorption. Solid circles heating curve redrawn from Fig. 24, Ref. 21 (numbers given ar~ 60-sec heating temperatures). Squares, electron impact at 200K (Fig. 10). Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3320 D. MENZEL AND R. GOMER ne (electrons/em!) 6 8 10 12 ;I; lOIS L I~ ~ ~Og(.",,-.) . log (8 -I I) OIO'~~~'5~O~~~~'OO~-I-(m-.,,-)7015~O~~~2~onO~~~25~O~-" FIG. 12. Semilogarithmic cf> and B vs time plots for 100-eV electron desorption from a fully hydrogen-covered W tip at 20oK. Solid circles, log (cf>-4.5) . Dashed line extrapolates the linear part of this curve into Regime 1. Open circles, log[cf>(t) extr-cf». Squares,log(B-l.1). izability. The second regime undoubtedly corresponds to desorption of atomic Hi as indicated by the small effect on B (because of low polarizabili ty) and the negative contact potential associated with this ad species. It is quite likely that the molecular species corresponds, at least in part, to adsorption on single sites at high coverage, and that electron impact re moves the protruding H atom, leaving the site filled with the remaining one. This view is supported by the fact that CPextr is higher than that obtainable by ad sorption, and by the fact that redosing experiments after electron desorption never raise B to the initial value while they increase cP above that obtainable by virgin adsorption (Fig. 10). The very beginning stages of desorption may corre spond to removal of physically adsorbed H2. As pointed out later, it is possible that direct momentum transfer 6 4 2 ne (electron5/cm2) I 2 x 10'9 FIG. 13. Semilogarithmic cf> and B plots for 100-eV electron desorption from a partiaJly hydrogen covered tip, prepared by heating a fully covered tip to 385°K. Open circles, log(cf>-4.5) for impact at 20°K. Open squares, log (cf>-4.5) for impact at 150oK. Solid cir cles: 10gB for impact at 20oK. Solid squares: 10gB for impact at 150°K. is effective here. In any case the behavior seems to differ from the main portion of Regime 1, as shown by the curves of Figs. 10 and 12. Figure 13 shows plots of log(cp-4.S) and 10gB vs t for electron desorption from a H2-covered tip heated to 38S cK before electron impact. In this case desorp tion was carried out both at 20° and at IS0°K. It is seen that the loge cP-4.S) plots coincide, indicating that there is very little temperature effect. The cross section obtained from these curves is 7.3 10-21 cm2, which is roughly equal to the value obtained for the atomic desorption from a virgin tip. The 10gB plots yield fairly good straight lines and lead to values of 1.3 X 10-20 at 200K and 1.9X 10-20 cm2 at IS00K. These values lie very much closer to the cross sections at tributed to molecular desorption and suggest that there may be some quasimolecular hydrogen remaining after heating to 38SoK which does not show up in the cP variation but does show up in the B variation. This conclusion is purely speculative and considerably more work would be required to establish this point. TABLE II. Summary of CO results (80-eV electron impact).' Process Virgin desorption (cm') 2-5XlO-19 Virgin-/3 conversion (cm') ;:::10-19 • For explanation of symbols see Ref. 17. /3 desorption (cm') 5-8XlO-'1 a desorption (cm') 3XlO-18 In view of the small cross sections only very qualita tive information on their energy dependence could be obtained. The cross sections decreased with energy below 100 eV, and barely detectable desorption still occurred at IS eV. CO on Tungsten This work will be reported in detail separately. We give here only the main results for comparison with the other data. The general scheme of adsorption states postulated previously17 could be confirmed by means of the widely differing cross sections for electron de sorption, and fairly detailed information on the inter conversion among states could be obtained. The cross sections at 80 eV are summarized in Table II. Baon W The Ba source was identical to that used by Utsugi and Gomer16 and consisted of a short section of Fe-clad Ba wire, notched to permit escape of Ba. A rough work function vs relative coverage curve, constructed by spreading successive doses of Ba over the tip and Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsDESORPTION FROM METAL SURFACES 3321 TABLE III. Results of 100-eV electron impact for Ba on W. Starting conditions Impact conditions Time of Current ct> dct>/dO Ttip Run (eV) (J (eV) OK 1 2.22 0.85 -2.2 77 2 2.33 0.80 -2.2 385 3 2.40 2 0.2 77 measurmg the resultant work functions is shown in Fig. 14. No particular pains were taken to ensure con stant size of the doses or complete equilibration over the emitter shank since this curve was intended only for a rough comparison with the data of Moore and Allison22 or Becker.23 It is interesting to note, however, that the agreement with both is reasonable. Desorption was attempted at three coverages and two temperatures as shown in Table III. With current densities of 10-3 A/cm2 and bombardment times up to 7 h (total number of electrons """'102°/cm2) no effects could be found under any conditions. In these experi ments the major portion of the emitting area of the tip was hit by electrons so that emission changes would have been noticeable electrically as well as visually despite the fact that emission from the unbombarded portion of the tip would have been greater than from the bombarded region if desorption had occurred (except for Run 3, where desorption would have in creased emission). Assuming that work function changes of 0.05 eV on the emitting portions of the tip could have been detected visually or electrically (a 4> 3.0 @ t / 2.0 2 6 Number of Sa Doses FIG. 14. Work function vs amount of barium on W. Solid curve, Becker, Ref. 23. Squares, Moore and Allison, Ref. 22. Solid circles, this work. Numbers and arrows indicate coverage regions where electron impact was carried out. 22 G. E. Moore and H. W. Allison, J. Chern. Phys. 23, 1609 (1955). 23 J. A. Becker, Trans. Faraday Soc. 28, 151 (1932). impact density Total Act> u (min) (A/cm2) electrons/ cm2 (eV) (cm2) 540 6.2XlO-4 1.3X 1020 <0.04 :::; 1. 7X1Q-22 320 6.4XI0-4 7.7X1019 <0.04 :::;2.9XIQ-22 315 4.2XlO-4 4.9XI019 <0.05 :::; 1. 3XIQ-21 conservative estimate), cross sections for desorption must be (T::; 10-22 cm2 under all conditions. SUMMARY OF RESULTS Table IV summarizes the results for hydrogen, oxygen, carbon monoxide, and Ba adsorbed on tungsten. It is seen that the cross sections are several orders of magnitude smaller than those for electron-impact ex citation or ionization in the relevant molecules.3 (For Ba on W (T is, in fact, zero for all practical purposes.) Although our results on energy dependence are still very meager, they indicate a similar dependence to that found in molecules: a steep increase in cross section from threshold values up to about 40 eV, fol lowed by a shallow maximum near 80-100 e V and a slow decrease thereafter. Within the limits of our ex periments cross sections were essentially temperature independent. DISCUSSION Before discussing the possible mechanisms of de sorption consistent with these results, it may be worth while to indicate the possible role of direct energy transfer to the adsorbate nuclei by the impinging electrons. The maximum possible transfer for a free particle of mass m is given by tJ.E = 2Ei(me/m) = 2E;/1838M, (12) where Ei is the incident electron energy and M the molecular weight of the particle. The effective mass of the latter goes up rapidly with increasing binding energy, so that Eg. (12) gives an upper limit, valid only for very low binding energies. It is seen that the maximum energy transfer to His ,....,,0.1 eV for 100-eV electrons, and correspondingly less for more massive absorbates. Thus direct energy transfer is quite in sufficient to affect chemisorption, but can and appar ently does result in removal of physically adsorbed gas. In connection with the hydrogen results one more comment is necessary. The heat of chemisorption of H2 at very high coverage apparently goes down to very low values, ,....,,0.2 eV, relative to H2 (not H) m Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3322 D. MENZEL AND R. GOMER TABLE IV. Summary of electron impact desorption results (for 80-1OO-eV electrons). System Conditions ~/W fully covered, 200K 02/W </><5.7,20°-600°K H2/W fully covered, 200K (Regime 1) (Regime 2) H2/W heated to 385° after dosing (20°-150°) Ba/W 8=0.8-2 CO/W virgin CO CO/W {j-CO CO/W a-CO CO/W conversion v-+{j • as far as can be ascertained. the gas phase. If the adsorbed state consists exclusively of H atoms this corresponds to an atomic heat of adsorption of the order of 2.5 eV, and in that case direct energy transfer is obviously unimportant. If, as some of our results suggest, there is weakly bound molecular H2 present on the surface, it is conceivable that a fraction of it is in fact removed by direct energy transfer. If this were the case however, the cross sec tions should be of the order 10-16 to 10-17 cm2• Thus direct energy transfer is apparently not involved ex cept possibly in the removal of physisorbed H2 on W. This also suggests that the main effect of electrons on chemisorbed H2 is rupture of the H-H bond. The most striking feature of all the experiments is the remarkably small over-all cross sections for de sorption. This could result from one of two causes: (1) The cross sections for the initial excitation are very small. (2) The rate of de-excitation is very large and prevents desorption of all but a small fraction of the excited adparticles. As will be seen there is good theo retical reason for the second of these possibilities, but since the work was undertaken in part to verify this point, it is worthwhile to examine the first in more detail. Before doing so it is useful to indicate some of the relevant electronic states and transitions on a potential energy diagram for the metal-adsorbate system. We limit the discussion for the moment to the case where I -c/J is large. Figure 15 shows the lowest bonding (henceforth ground) state, M+A, derived from neutral metal (M) and adsorbate (A), an antibonding state (M+A): derived from the same separated states; and an ionic state derived from positively charged adsorbate and negatively charged metal, M-+A+. At large dis tances the latter lies I -c/J volts above the ground state and, in its attractive region, is probably describable by a classical image potential, Vi= -3.6/x in angstrom-Cross section for desorrtion (cm2 Approached state 4. 5X 10-19 </>=5.72 <2X10-21 3.5X10-2O </>=4.99 5X10-21 clean surface (</>=4.50) 7.3X10-21 clean surface (</>=4.50) <2X1Q-22 2-5 X 10-19 </>=5.14 5-8X1Q-21 clean surface (?). 3X10-18 depends on coverage 2::10-19 depends on coverage electron-volt units. Figure 15 also shows some of the possible vertical excitations. If the excitations could be regarded as one-electron jumps (of localized electrons) it is obvious that their probability should be comparable to that of similar processes in atoms or molecules, so that the cross sec tions should be of the order of 1'V1O-16 cm2• The follow ing simple valence bond argument shows that the more complicated nature of the excitations in the present case cannot affect the cross sections very much as long as the electrons involved are reasonably localized. Let .. ___ -- M .A I-~ M+A G H --____________ 1 X FIG. 15. Potential-energy diagram for adsorption. Bonding state M+A, antibonding state (M+A):, ionic state M-+A+. Two excited copies of the bonding state M+A, and M+A, are shown intersecting the antibonding and ionic curves, respec tively. Vertical arrows indicate some of the possible transitions, and dashed arrows the subsequent possibilities for desorption in the excited states or transitions to the bonding state. Ez is the excitation for the transition to the antibonding state indicated by the middle vertical arrow, and E' the electronic excitation of the system after transition to the bonding curve M+A. I, ioni zation potential of A, </>, work function of M. Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsDESORPTION FROM METAL SURFACES 3323 the ground state consist of the basis states (1) =M+A, (2)=M++A-, (3)=M+++A--, and so on, so that its wavefunction can be written (13) where the g's are the coefficients of the "pure" state wavefunctions in the ground state. Let the excited state in question, say the "ionic" state, consist of the "pure" states (1), and (4) =M-+A+, and so on, so that its wavefunction can be written (14) Then the transition probability from the ground to the excited state will contain, inter alia, terms like gl2el Pl4, which correspond to one-electron transitions and differ from Pl4, the transition probability between the pure states (1) and (4), only by the intensities gl2e42. Since some of the basis states suitable for one electron transitions are bound to be present in sub stantial amounts, the factors gi2e/ cannot reduce the PijS by much more than an order of magnitude; if there are several terms of this kind the total probability may approach that of a single jump even more closely. While the argument has been made specifically for transitions from the bonding to the ionic state, it must apply equally well to other transitions. Thus there is good reason to assume that primary excitations occur with high probability and that the observed small over all desorption cross sections result from rapid secondary transitions leading to readsorption. As pointed out previouslyl the curves shown in Fig. 15 are merely the lowest members of families of curves which correspond to electronic excitations of the metal but not of the adbond. Thus the lowest-lying ionic and antibonding curves are intersected by higher members of the bond ing curve family (here called M+A), and these intersections provide the means for making adiabatic transitions in which there is no interchange between nuclear and electronic energy. It is now easy to see what processes can occur in electron impact. We consider first a primary excitation to the ionic state. If the excitation takes the system above the dotted line in Fig. 15 desorption of A + can occur; the excitation threshold for A+ is obviously I-cfJ+H a• However, the possibility of crossing to a bonding curve M+A exists: If the zero of this curve lies above the threshold, desorption will have been prevented; if its zero lies below the threshold, the adsorbate will have sufficient kinetic energy to desorb as a neutral. The particular sequence of events just described is identical to the mechanism proposed in dependently by Redhead.12•l4 An entirely similar se quence of events can follow excitation to an anti bonding curve. If the location of the various curves on the potential energy diagram is such as to place the antibonding curve above the lowest ionic curve in the \ , , , c A FIG. 16. Hypothetical energy levels for two electrons involved in the bonding scheme of Fig. 15. Curve (1) is the term value for the transition bonding-->antibonding on the assumption that this requires the promotion of an electron from the bond to the Fermi level 1'. Curve (2) is the term value for promotion of a second electron from A to 1', when the system is in the anti bonding state before ionization. Franck-Condon region (the opposite is shown in Fig. 15), a transition to the ionic from the antibonding curve can also occur. A somewhat simplified but physically appealing picture of these transitions can be obtained by repre senting the states in terms of electron energy levels relative to the Fermi energy }J.. If the ground state is considered, for simplicity, to consist of the basis states (1) and (2) already mentioned, and the antibonding state to consist of (1), so that the transition from bond ing to antibonding states can be regarded as the promotion of an electron from the bond to the Fermi level, the energy of this electron level relative to }J. can be obtained by subtracting the energy difference llE between the bonding and antibonding curves (shown as dashed lines in Fig. 16) from }J. (Curve 1, Fig. 16). The energy level of the adsorbate electron which is promoted from A to the metal at }J. to give the ionic state M-+A+ can be represented analogously, by plotting the algebraic difference between the anti bonding and ionic curves on a base line 1-cfJ below }J. (Curve 2, Fig. 16). The difference between the bonding and ionic states could also be represented directly in similar fashion. It is now easy to see the meaning of the excitations and the subsequent de-exciting transitions. Figure 17 (a) shows an excitation to the antibonding curve, with an energy change Ex, shown also in Fig. 15. Figure 17(b) shows the reversion to an excited ground state curve in terms of a horizontal transition, indicated as an electron tunneling from the metal into the bond. The net electronic excitation E' of the system after this. transition arises from the presence of a hole in the Fermi sea, as indicated, and an extra electron at }J.. The remainder of the original excitation energy Ex- E' = EA appears as kinetic energy of the adsorbate. Similar representations can be made for all the other Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3324 D. MENZEL AND R. GOMER (a) (b) / / / / / I I I I Eo. I I J' I , , , , ! M A M A Xo xl FIG. 17. Schematic diagram showing (a) excitation to the antibonding curve of Fig. 15 and (b) transition to':curve M+A. The adsorbate has moved from Xo in (a) to Xl in (b), so that the net electronic excitation after the transition in (b) is E', not Eu. exciting and de-exciting transitions. If it happens that the bonding level which must be filled lies between bands of the metal, the appropriate transitions would be of the Auger type. The model also indicates why retransitions to antibonding curves are unlikely even if energetically permitted: Unless such a retransition occurs immediately after the transition to the bonding curve, it can no longer occur in first order, since the motion of the adsorbate will have moved the electron level relative to the hole in the Fermi sea created by the tunneling into the bond. The filling of this hole by any number of mechanisms not involving the bond is therefore far more likely than a retransition to a lower excited state. l While these transitions can be thought of as one electron jumps only in crude approximation, so that the "tunneling" is to be taken as having pictorial rather than completely valid physical significance, all the arguments become perfectly precise if we replace the words "electron level" by "quantum state of the system." In particular all the energy arguments carry over unchanged. Although a detailed quantum-mechanical calculation of transition and desorption probabilities is beyond present means, it may be worthwhile to give the re sults of a semiempirical calculation which serves to indicate trends and point out what information would be required for a detailed theory. We consider only the simplest case, illustrated once more in Fig. 18. Excitation to a repulsive (or weakly bonding) curve at Xo is followed either by desorption along this curve or by transition to the ground state. If the transition occurs at a distance from the surface x<xc recapture results; if the transition occurs at x2::xc the adsorbate will have sufficient kinetic energy to be desorbed along the ground-state curve. The posi tion of Xc depends on the relative location of the ground and excited-state curves, as seen from Fig. 18. A very similar treatment, considering only the case of desorp tion without de-excitation, has been given by Redhead.14 The total probability of desorption, regardless of mode, PT, is given by PT= exp(_jXC dX), (15) xo VT and the probability of desorption without de-excitation, PE, by (16) where v is the velocity along the excited-state curve and T the mean life with respect to transitions to the ground state. It should be noted that the second integral is larger, hence PE smaller than PT. Equa tions (15) and (16) can be integrated only if the shape of the excited state curve and T are known. Before attempting a detailed solution we shall give a feeling for the quantities involved by a rough order of magnitude estimate. We assume that desorption preventing transitions occur with uniform probability T-1 in the zone Llx= Xc-Xo and with zero probability outside it. If the excited state curve is considered linear over the range LlX, its slope SE can be approxi mated by SE ~Ha/ LlX, as can be seen from Fig. 18, since the change in potential energy of the excited state in LlX must be less or equal to the heat of adsorp tion Ha. Solution of Eq. (15) then yields -lnPT2:: (2m/H a)!LlX/T. (17) Since the observed total desorption cross section UT is given by (18) where Uex is the excitation cross section, T can be esti mated from UT if Uex is known or guessed at. Taking Ha,-...,2 eV, Llx,-...,l A, and uex",10-17 cm2, one obtains Rec",ture - - - - - -::= Critical Curve / '" ./ /' _ - - -Desorption ofter Transition // / / Ho Be-bit (I~e-u) / /L __ -= ,±~L I I I 1 Xo Xc FIG. 18. Schematic potential energy diagram showing energy and distance relations for transitions leading to recapture and desorption. Xo distance at which vertical (Franck-Condon) excitation occurs. Xc critical distance for transition. Dotted curves drawn for X<XC, leading to recapture, and for x~xc leading to desorption along the ground-state curve. Ha heat of adsorption. The antibonding curve is assumed to have the form V = Be-bx• u is defined in the text. Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsDESORPTION FROM METAL SURFACES 3325 very roughly TH;(; 10-16, TO~ 10-15, and TCO~ 10-15 sec for the states with highest desorption probability, i.e., largest T. While these values may well be in error by an order of magnitude, they indicate that the transitions leading to de-excitation must be very fast if the excitation cross sections are reasonably large. It is perhaps even more illuminating to consider the effect of T, or better TI t:.x, on PT. Figure 19 shows -IOgloPT as a function of T I t:.x for hydrogen and oxygen with Ha=3 eV in both cases. It is seen that PT is es sentially unity when T I t:.x> 10-13 secl A but decreases rapidly when TI t:.x< 10-14• This means that the ob served desorption cross sections will be appreciably affected by T I t:.x only if this quantity is less than 10-13, or alternately that (J" ex must demonstrably exceed (J"T by a factor of 10 or more before any conclusions about T I t:.x can be drawn from the observed cross sections. The fact that the variations of (J"T within a given system often exceed 10-100 suggests that this is probably the case and that the conclusion that T< 10-13 sec in all the cases investigated is probably correct. Although the preceding arguments are based on a rather simple model the fact that the critical range of the ordinate of the repulsive curve is limited to Ha or less and is therefore relatively small, means that the critical zone t:.x is also small, so that the assumptions of linearity and constant T cannot be in very serious error. It is nevertheless interesting to carry out a somewhat more detailed analysis. If it is assumed that repulsive terms predominate for the excited state in the region of interest it can be represented by V.=Be-bx, (19) where Band b are constants of dimension energy and reciprocal distance. An evaluation of T would require a 20.-------,-------nr------, 18 16 14 12 6 4 2 011::3 =;;;..--==--------L--------,!16 FIG. 19. Logarithmic plot of total desorption probability PT as a function of r/Ax for Hand 0 with H,,=3 eV. Values shown are based on Eq. (17). 4.0 3.0 .5 1.0 2.0 5.0 9.9 100 1.0 2.0 3.0 FIG. 20. F(u, p) as function of u for various values of p as indicated on the curves. For definitions see text. detailed knowledge of the initial and final state wave functions. It can probably be approximated by the form (20) where TO and a are constants. Equation (20) can be roughly justified by regarding the transition as a tunneling process through a potential barrier, whose effective height varies only slowly, but whose width increases linearly with distance from the surface. On this basis a would be given, very crudely, by a= 1.4El (1)-1 where E is the height of the tunneling barrier, say 5-10 eV, and TO ____ 10-16 secl• With these assumptions one obtains from Eqs. (15) and (16) (mI2B)1 --__ ,e-(a-b/2)xoF(p, 00) Tob (21) and (mI2B)1 b e-(a-b/2)xoF(p, u), TO (22) where m is the mass of the adsorbate particle, p= alb, u=b(x.-xo), and F(p, u) = jUe-PU(l-e---tl)--ldu, o (23) and pep, 00) the integral of Eq. (23) from 0 to 00: pep, 00) =lI"lr(p)/r(p+!). (24) The function F(p, u) is equivalent to the incomplete {:1 function and was evaluated by us on an IBM 7094 computer. It is shown for some representative values of p and u in Figs. 20 and 21. Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3326 D. MENZEL AND R. GOMER a. .536 ~ 1.0 . 0914 .010 °0~----J.10~--~2~.o~--~3~.o----~4.~0----~50 FIG. 21. F(u, p) as function of p for various values of u, as indicated on the curves. For definitions see text. A number of interesting conclusions can be drawn from Eqs. (21) and (22). First it is seen that PE and PT increase with increasing T, and B, and decrease as m increases; this behavior is obviously required of a physically meaningful theory. We note next that PE is independent of u, while PT depends sensitively on it. The reason is that desorption without de-excitation requires the adsorbate to run the entire gamut of possible transitions without making any, and is there fore insensitive to u, while PT depends on the width of the recapture zone. Since F (u. p) increases as u increases (Fig. 20) PT decreases with increasing u as physically required. The parameter p= alb which is a measure of the relative x dependence of T and V E appears both in F (u, p) and in the term exp[ -(a-b/2)xoJ, which can be rewritten as exp[ -bxo (p -!)]. Since F(u, p) decreases with increasing p, it is seen that both PE and PT increase, as p increases other factors, notably b, being constant. If p>!, PE and PT increase with increasing Xo, other factors being constant. Since a change in x~ generally implies a change in u, the effect on PT (but not on PE, which is independent of u) must be analyzed in more detail. Before doing so it is worthwhile to examine the ratio of PEl PT, or more conveniently 10gPE/logP T. This is 10gPE/logP T= [1T1r(p) /r(p+!) J[F(p, u) J-t, (25) so that At the same time u is subject to the inequality Be-bxo(1-e-u) ~Ha, (27) as can be seen from Fig. 18. Relations (26) and (27) place some bounds on the possible values of p and u, if PE and PT are known separately. If one takes for instance Redhead's values for oxygen on molybdenum14 one obtains with p= 1 (his choice) a value of u=0.53 from the graphical solution of Eq. (26). Substitution in the inequality (25) with Ha"'-'3 eV and B= 104 eV yields bxo~9.5. If Redhead's choice of b=4.15 (1)-1 is made this yields xo~2.3 A. If a larger value of p, say p= 1.6 is taken, one obtains u=0.3 and bxo~6.8. While these considerations show the general reason ableness of the mechanism they cannot be used for much more, since PE and PT can be obtained from the experimental desorption cross section only if the ex citation cross section is known separately . The effect of Xo on desorption probabilities, and its relation to u are of interest in estimating variations in cross section with binding modes differing in energy and configuration within the same adsorbate-substrate system. We note first that a lateral displacement of a family of bonding curves affects Xo and Xc equally and therefore leaves u unchanged. We see next that if two bonding curves differing in Ha are drawn from the same zero of energy at x= 00, the curve corresponding to the higher Ha value lies below the more weakly binding curve at all values of x, (unless the shape of the curves is very peculiar) particularly near the equilibrium distance X=So. Therefore, for a common Xo, Xc and hence u will be greatest for the most tightly binding curve. Since an increase in Ha may also imply a decrease in Xo it is seen that increased binding energy will lead, on both the counts of Xo and u to a decrease in desorption cross sections. Thus it is easy to show that a decrease in Xo of 0.1 1 can change PE from. say, 10-4 to 10-0 if a-b/2=2 to 3(1)-1 if the excited state is unchanged. As noted, the effect on PT can be even greater, since u may increase simultaneously. As Fig. 20 indicates F(u, p) varies most rapidly with u when both u and p are small. From these considerations it is possible to draw the following conclusions. If a common excited state exists for two binding modes, desorption will be more probable from the looser, weaker mode. If both binding modes are possible in the same spatial region of the surface, transitions from one binding mode to the other via excitation to the common excited state followed by de-excitation into the other mode will be more probable for loosc-+tight than for tight-+loose. In this connec tion it is interesting to note that the granularity ob served on electron impact in many cases. for instance with oxygen, may correspond to a transition from tight binding to loose adsorption. If each bright spot is con sidered to correspond to a single 0 atom (or at any rate to a single event) it can be seen from Fig. 5 (c) that 3.7 X 1018 electrons/ cm2 cause approximately 10-50 events of this kind on an area of ",-,10-10 cm2 (the visible tip surface). If the 0 coverage is estimated to be "'-'1016 atoms/cm2 the cross section for this process is approxi mately 10-22 cm2, i.e., very small indeed. On the other hand the conversion of virgin CO (looser) to {3 CO (tighter) on tungsten seems to occur with a relatively Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsDESORPTION FROM METAL SURFACES 3327 high cross section ((j;::: 10-19 cm2) , at least equal to that of virgin desorption. If the difference in binding modes is determined by substrate geometry the excited state is probably also affected. If the substrate configuration permits the adsorbate to "burrow" into the surface, the excited state will also tend to be shifted inward, but probably not as much. Consequently, even assuming that B and b stay constant from one situation to the other, there may still be a slight increase in u. The chief effect however is probably a decrease in T, since the probability of transitions must increase with the num ber of substrate atoms in contact with the adsorbate. The gist of these conclusions is that configurational changes can very easily account for the observed variations in cross section with binding mode, even if the original excitation cross section were totally unaffected. It may also be worthwhile to point out the effect of a change in Xo on u if ground-and upper-state curves are fixed, for instance if Xo is varied through vibrational excitation. It is not difficult to show that, for small AXIJ, where SE and SG are the slopes of the repulsive and ground-state curves, respectively. If SG«SE Eq. (28) reduces to Aur-...lbAxo(eu-1) . (29) The effect of a change Axo on PT is then given approxi matelyby 10gPT(Xo+AXo) r-...I [b(p 1) A ] exp --2 '-1Xo 10gPT(xo) where the derivative, evaluated at Uo=U(XIJ) can be found from Fig. 20. Since aF jau and hence a InF jau are positive, the effect of a change in Xo on PT will be counteracted by the resultant change in u. Conse quently a change in Xo will generally have a larger effect on PE than on PT, if the potential curves are fixed. In view of the great sensitivity of PE to Xo it may be worthwhile to indicate the effect of spreading Xo over the vibrational ground state of the bonding curve. Averaging over the oscillator wavefunction yields where So is the distance coresponding to the minimum in the ground state curve and ~2=hj27rmv is the square of the classical turning point. In spite of the sensitivity of desorption to Xo there is very little if any temperature effect. This can be readily understood when it is considered that an appreciable shift in effective Xo due to vibrational excitation re quires a large activation energy. Even at the highest temperatures used the population in such a state will be very small relative to the low vibrational states, and any gain in cross section from a change in Xo will be swamped by the decrease resulting from this Boltz mann factor. The results obtained in this work and by Redhead indicate the general validity of the mechanism out lined here and show that the de-exciting transitions are in general quite fast. It is difficult to go beyond this generality. It is impossible to determine theoretically which of the possible excitation and de-excitation channels are most important in a given case, and the experimental results are subject to alternative inter pretations. Thus ionic desorption products can come from direct excitation to an ionic curve or they can arise by excitation to an antibonding curve followed by transition to the ionic state. Skewed energy dis tributions result in either case. A great deal of informa tion would come from accurate threshold energies for neutral desorption, since this would show whether or not excitation to an antibonding state is importantly involved. Unfortunately it is very difficult to obtain this information accurately by any of the presently available methods. In the absence of a direct deter mination of neutral products its measurement depends on determining the small difference between two large quantities, namely, the adsorbate coverage before and after desorption, and is thus subject to errors like readsorption, which limit the sensitivity of the deter mination, both in our method and in Redhead's. (The situation for ion yields by the latter method is quite different, since it involves the direct measurement of a small quantity, not the difference between large ones). All that can be said is that PT decreases with decreasing electron energy, and thus does not seem to depend on exchange collisions, for which cross sections increase with decreasing energy. The fact that Redhead's 0 yields agree reasonably well with ours, and exceed his 0+ yields by a factor of ",10 indicates either that (a) transitions from the ionic to the bonding curve are very probable or that (b) excitation to the ionic state (by whatever mechanism) is less probable than excitation to an antibonding state, with the transitions from either to the bonding state about equally probable. Thus many details remain unresolved at this time. It is also worth pointing out again that desorption is not the only possible electron-impact-induced process in adsorption systems. Thus, the change from one bind ing state to another via an excited state is only a special case of possible reactions after excitation to an upper potential surface, from which crossing into various lower states may occur. It is quite possible that metastable Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions3328 D. MENZEL AND R. GOMER states or those requiring an activation energy can be populated in this way in certain systems. We turn next to the case where I -cf> is small, i.e., electropositive adsorption. It has been postulated that binding involves almost wholly delocalized electrons, in such systems, i.e., that the barrier for tunneling to and from the adsorbate is so transparent that transi tion times are extremely short. If this is so, one would expect electron impact to be ineffective in causing desorption or even rearrangement, since (a) the factors g?ej2 will be very small so that excitation cross sections will be small, and (b) tunneling into the bonding states should be extremely rapid. By the same token one should expect no effect in metallic adsorption, which is a special case of electropositive adsorption. The fact that slow electrons do not affect a metal surface (except indirectly by heating it) seems fairly well established. Our results with Ba on W seem to confirm this view since no desorption could be detected. In fairness it must be pointed out that the mass of Ba alone decreases PT by a factor of loa relative to oxygen, if all other factors were constant (although its large size and number of electrons probably leads to a large excitation cross section of the free Ba atom). A more stringent test of this hypothesis would perhaps be supplied by very small cross sections in the case of N a or Li adsorption. Finally a word should be said about the results of Mulson and Mtiller7 and Ehrlich and Hudda8 at high fields. Although it is not certain that electron desorp tion is involved at all and cross sections have not been determined, it appears that electron desorption may be somewhat more efficient in the presence of high positive fields than at zero field. The preceding dis cussion shows that this is not surprising for a number of reasons. First, the pure ionic curve is now without question the lowest repulsive curve, and is steeper because of the external field so that the available time for de-exciting transitions is reduced. Second, if ex citation to the ionic state is followed by a transition to an M+A curve the resultant vibrational excitation of the latter may be sufficient to permit "ordinary" field desorption, since the applied fields are such that only a relatively small vibrational excitation is required for this. CONCLUSION Electron desorption seems to show that the reforma tion of bonding states on a metal surface is very rapid. Although it is not possible at this time to distinguish between possible mechanisms in detail, this information is already useful in that it substantiates earlier conclu sions about the mechanism of field desorption, and helps to explain the rapidity of chemisorption on metals. Despite the small cross sections for desorption the differences encountered for different bonding states make electron desorption a very useful probe for study ing the nature of adsorption and for elucidating the existence and behavior of different adsorption states. ACKNOWLEDGMENTS We wish to thank the National Science Foundation for partial support of this work under grant NSF G-196l8. The authors also wish to acknowledge general support of the Institute for the Study of Metals by the Advanced Research Projects Agency and the U. S. Atomic Energy Commission. One of us (D. M.) wishes to acknowledge gratefully a Fulbright travel grant. Downloaded 14 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
1.1726891.pdf	Raman Spectral Studies of the Effects of Temperature on Water and Electrolyte Solutions G. E. Walrafen Citation: The Journal of Chemical Physics 44, 1546 (1966); doi: 10.1063/1.1726891 View online: http://dx.doi.org/10.1063/1.1726891 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/44/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Raman Spectral Studies of the Effects of Solutes and Pressure on Water Structure J. Chem. Phys. 55, 768 (1971); 10.1063/1.1676144 Raman Spectral Studies of the Effects of Temperature on Water Structure J. Chem. Phys. 47, 114 (1967); 10.1063/1.1711834 Raman Spectral Studies of Water Structure J. Chem. Phys. 40, 3249 (1964); 10.1063/1.1724992 Raman Spectral Studies of the Effects of Electrolytes on Water J. Chem. Phys. 36, 1035 (1962); 10.1063/1.1732628 Raman Effect of Concentrated Electrolytic Solutions J. Chem. Phys. 7, 380 (1939); 10.1063/1.1750451 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:52THE JOURNAL OF CHEMICAL PHYSICS VOLUME 44, NUMBER 4 15 FEBRUARY 1966 Raman Spectral Studies of the Effects of Temperature on Water and Electrolyte Solutions G. E. WALRAFEN Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey (Received 16 September 1965) All known intermolecular Raman bands of water, viz., the hydrogen-bond bending and stretching bands, and the librational bands, decrease rapidly in intensity with temperature rise. In contrast, the librational intensities of water in electrolyte solutions exhibit very small variations with temperature. The intensity decreases observed for pure water indicate that hydrogen bonds are broken by increase of temperature, but the near constancy observed for solutions indicates that primary hydration is not greatly affected, even at temperatures near the normal boiling points of some of the solutions studied. Integrated Raman intensities of the hydrogen-bond-stretching vibrations of pure water at 152-175 cm-1 were redetermined in the temperature range of -6.0° to 94.7°C. The new intensity data, which are more accurate than the old [ef., J. Chem. Phys. 40, 3249 (1964) J, yield the values ClJfO=5. 6 kcal/mole and ASo""19 cal/deg·mole for the process B->U, where B refers to water molecules which contribute intensity to the 152-175-cm-1 Raman band, and U refers to molecules which make very little or no contribution. Interpreted in terms of non hydrogen-bonded monomeric defects in a tetrahedral liquid lattice, the above flHo yields a value of 2.8 kcal/mole H bond in reasonable agreement with Scatchard's value of 3.41 kcal/mole H bond. The value of Clso for the process B--->U also leads to interesting comparisons with known entropies, but the calculated heat capacity of water is only in fair agreement with accepted values. The observed insensitivity of the solution librational intensities to changes of temperature indicates that primary hydration is involved almost exclusively. This conclusion complements the previous ob servations involving linearity of librational intensity with electrolyte concentration [cf., J. Chern. Phys. 36, 1035 (1962)], since both observations can be explained by primary hydration. In addition, molar librational intensities (obtained from the temperature studies) confirm the large anionic effects reported previously, with Br->CI-, but they also indicate that the effects produced by NH4+ are much smaller than those arising from Li+, Na+, and K+. It is apparent, therefore, that at least some cationic effects can be observed in the Raman spectra. INTRODUCTION On a time scale of ",,10-13 sec, liquid water appears to possess an intermolecular structure which involves TWO photoelectric Raman spectral investigations tetrahedral hydrogen bonding. This structure is readily concerned with the structures of water and electro- disrupted by increase of temperature. The disruption, lyte solutions have been conducted in this laboratory.1.2 which involves nearest-neighbor structure, is thought The work now reported represents a continuation of to produce a new species which engages in few or no those investigations. The results of other Raman studies hydrogen bonds of the type which most effectively involving water and solution structure have also contribute to the intermolecular Raman intensities. appeared recently, vid., Refs. 3-10, but they refer The non hydrogen-bonded species, however, is con largely to intramolecular Raman bands. Intermolecular sidered to be bound by other forces, but not by the Raman intensities have been of prime concern here. directional covalent interactions which lead to tetra- Much progress has been made by the workers cited hedral structure. The tetrahedral species, on the other above, but differences in interpretation have developed. hand, is thought to resemble ice, at least on a local Nevertheless, conclusions resulting largely from experi- scale. ence gained in this laboratory are introduced. If certain electrolytes are added to water, the tetra- 1 G. E. Walraien, J. Chem. Phys. 36,1035 (1962). hedral structure is also disrupted, but in varying 2 G. E. Walrafen, J. Chem. Phys. 40, 3249 (1964). degrees depending upon the electrolyte, and upon the 3 W. R. Busing and D. F. Hornig, J. Phys. Chern. 65, 284 concentration. The disruption arises, in part, from the (1?~~)W. Schultz and D. F. Hornig, J. Phys. Chern. 65, 2131 formation of strongly hydrated units. The primary (1961). hydration numbers, however, have not yet been deter- 6 R. E. Weston, Spectrochim. Acta 18, 1257 (1962). mined from Raman data. Nevertheless, they appear 6 H. A. Lauwers and G. P. Van der Kelen, Bull. Soc. Chirn. Belges 72,477 (1963). to be virtually independent of temperature, and of 7 J. Clifford, B. A. Pethica, and W. A. Senior, Conference on electrolyte concentration. (Solutions containing small Forms of Water in Biologic Systems (New York Academy of amounts of water at high temperatures and pressures Sciences, New York, 1964). 8 Z. Kecki, Roczniki Chern. 38, 329 (1964). are excluded.) ~ T. T. Wall, doctoral dissertation, Princeton University, In terms of Raman observations, breakdown of the 1963; T. T. Wall and D. F. Hornig, J. Chern. Phys. 43, 2079 tetrahedral structure with increase of temperature is (1965). 10 W. A. Senior and W. K. Thompson, Nature 205, 170 (1965). apparent from the rapid decreases observed in all 1546 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:52RAMAN SPECTRUM OF WATER 1547 intermolecular intensities. Breakdown of the water structure upon electrolyte addition also produces inten sity decreases of the intermolecular water bands. How ever, the weak librational bands of pure water are replaced by the intense librational bands of hydrate water. The total intensity within the librational-fre quency region of pure water is thus observed to increase rapidly with electrolyte concentration, although the higher-frequency components arising from the tetra hedral structure are observed to disappear. Similarly, electrolyte addition can produce large increases in intramolecular Raman intensities, and small increases in intramolecular intensities are also produced by temperature decrease. (See Refs. 1-5 for exceptions which involve F-, Fermi resonance, frequency com parisons, etc.) Some effects observed in the spectra appear to be highly specific to the Raman method. For example, effects produced by Cl-and Br are large and readily evident from qualitative examinations of the spectra, but effects produced by cations are, in general, small. However, it would be unreasonable to explain the latter observations in terms of unhydrated cations, (although NH4+, for example, may be essentially un hydrated). Apparently, bands which involve large vibrational polarizability changes are very effective in contributing to the Raman intensity, and in regard to this, evidence is now available which indicates that cations, as well as anions, produce effects in the spectra, but in widely varying degrees. Finally, it should be emphasized that a short time scale, comparatively high energies, and selection rules which depend upon vibrational polarizability changes are important characteristics of the Raman method. Thus, in regard to time scale and energy, the Raman method is most nearly compatible with infrared and inelastic-neutron-scattering methods, but not necessar ily with other methods, e.g., nuclear magnetic reso nance, which have been employed in the study of water and electrolyte solutions. EXPERIMENTAL All Raman data were obtained with a Cary Model 81 spectrophotometer, but modifications of that instru mentll were extended to allow for the present studies of water and electrolyte solutions. The vertical lamp housing now employed, is shown in Fig. 1 with the thermostatted water jacket, and a removable Raman tube, in position. A resistance-heated Raman tube, and a second thermostatted Raman tube, vid., Fig. 2, are also shown in operating position. The arrangement depicted in Fig. 1 was employed to study the librational intensities in the temperature range of 25° to "-'90°C. Intensities at ,...,.,980 cm-1 from a 1.79M. standard solution of Na2S04 at 25.0o±0.2°C were determined before and after the completion of a 11 G. E. Walrafen, J. Chern. Phys. 43, 479 (1965). FIG. 1. Thermostated jacket and removable Raman tube in vertical lamp housing. The innermost tube is the Raman tube; it is held by the large standard-taper joint, and positioned ac curately by use of marks (not shown). The space between the Raman tube and the inner wall of the jacket was filled by puri fied water, to transfer heat and to reduce reflections from surfaces. The thermostated jacket also contained purified water which was pumped from the constant· temperature bath, vid., upper arrows. Temperatures in the range of 25°-90°C were controlled to about ±O.2°C. The vertical lamp housing was described pre viously jl1 the 45° prism has now been replaced by a front-surfaced mirror. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:521548 G. E. WALRAFEN FIG. 2. Thermostated Raman tube (left) and resistance heated Raman tube (right). Brine and purified water were pumped from the constant-temperature bath, vid., upper arrows, mto the thermostat ted tube. Dry nitrogen was admitted as shown by the lower arrows to prevent frosting of the Ramanltube and entrance optics. The large outermost tube, and the Raman tube were centered by the transitelinsert. (See Ref. 11). The resistance-heated Raman tube was made of quartz, and was wrapped with Nichrome wire. The optic flat (dotted) was raised to provide for uniform heating of the sample. Temperatures were measured with the mercurial thermometer held in a poly ethylene insert (not shown), which prevented loss of water. Temperatures were maintained to about ±0.5°C from """90°- 140°C. given set of intensity determinations. Ratios of inte grated librational intensities, to the average integrated intensity of the standard were thus obtained. The ability to interchange Raman tubes in the same jacket, of course, facilitated the measurements. The resistance-heated Raman tube shown in Fig. 2 was employed in the study of librational intensities in the temperature range of "'-'700-"'-'140°C. No external standard was employed here, but the intensity varia tions of the Raman lamps were found (from the low temperature librational studies) to show no significant time dependence due to lamp darkening within the time required for the study. The integrated librational intensities were simply scaled to match the relative integrated librational intensities in the neighborhood of 70°C. The second thermostated Raman tube shown in Fig. 2 was employed exclusively for studies of pure water. Again no standard was employed because significant lamp darkening was not expected to occur during the measurements, and because other determinations indicated negligible short-term intensity variations. Apparently, increased stability was obtained by use of the 100-V, 100-A dc power supply described previously,!l and by the addition of a large Sola constant-voltage transformer used exclusively with the Cary Model 81 electronic components. The water studied in this work was carefully purified. Distillation, removal of organic material, and de-ion ization were followed by filtrations, first through 0.22-J.! and then through 0.01-J.! Millipore filters. Brine was employed in the water jacket to produce temperatures from -6.0° to 25°C, d., Fig. 2, but above 25°C, the brine was replaced by purified water. (The brine was ultrafiltered, and made by solution of Fisher certified reagent-grade NaCl in water treated as above, except that only the first filtration was employed.) Supercooling below -6.0°C, however, was not possible in this work, but the inability to attain lower tempera tures was almost certainly unrelated to the water purity. Freezing of the pure water probably resulted from the use of Pyrex glassware which nearly always possesses some surface imperfections. Dry nitrogen was passed between the optic :fIat of the Raman tube and the entrance optic and also around the sides of the Raman tube, vid., Fig. 2. This procedure prevented frosting at low temperatures, which can produce sig nificant decreases in intensities. The thorough purification of water mentioned above proved to be of great importance, and it was partly responsible for the improved accuracy in the 152-175- cm-I band intensities. In the earlier work a good grade of distilled water was forced through glass filters of uItrafine porosity, viz., "-'1 J.!. The background slopes from that work, however, have been found to be much greater than those obtained recently, and the previous integrated intensities are now thought to be too small, particularly at the higher temperatures. The present intensities, of course, are more accurate because the reduced background slope revealed area which was previously lost. Indeed, it is now possible to begin a spectral scan at 15 cm-1 with a 10-cm-l slitwidth. However, an iterative method for obtaining the inte grated intensities also uncovered unsuspected intensity. In the iterative method, background slopes were first estimated by extrapolating slopes on either side of the Raman band. The resulting area was then transferred with dividers to a horizontal base line and examined for asymmetry. The high-frequency portions of all con tours studied appeared to be of normal shape, but abnormal low-frequency contours were often found in the first trial. The background slope was then readjusted by trial and error until a symmetric contour resulted for the Raman band. Of course, a reasonable background near the 4358-A exciting line, i.e., one involving a smoothly decreasing slope was demanded. Fortunately, the iterative method readily yielded half-widths which were virtually independent of temperature (and in close agreement with infrared half-widths shown later), although constancy of half-width had not been made a This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:52RAMAN SPECTRUM OF WATER 1549 condition of the method. In the early work, decreases of half-width with temperature increase were observed, but the large slopes precluded more accurate back ground estimates, despite the fact that the decreases were intuitively unexpected. In regard to the iterative method, the weak 60-cm-1 Raman band should be mentioned, although that band does not greatly affect it. Because more accuracy was attainable from the high-frequency half of the 152-175- cm-I contour, the 60-cm-1 band was in effect resolved, since the low-frequency half of the 152-175-cm-1 band was essentially determined by band symmetry. Further, the temperature dependence of the 60-cm-1 band intensity is known to be the same, at least qualitatively, as that of the 152-175-cm-1 band. Backgrounds of the librational bands from the solu tions were easily determined because the slopes were comparatively small. (The bands are displaced from the exciting line by at least ,..,.,450 cm-I.) Slopes on either side of a given librational band were simply extrapolated to yield a smooth base line. In the case of NOa-and SO,-2, the deformation bands of the anions were so much more narrow than the librational bands that they could accurately be removed. The spectra gave the appearance of sharp spikes on the broad librational bands. The concentrated aqueous solutions studied were also purified by filtration. Unlike water, however, the high viscosities precluded the use of Millipore filters. Thus, it was necessary to use glass filters of ultrafine or fine porosity in conjunction with a heating mantle to reduce viscosity. Nevertheless, the filtered solutions were free of turbidity, when viewed through columns 35 cm in length, and they were also free of color under those conditions. Analyses of chlorides and bromides were accom plished by the Volhard method, and the LiNOa solu tions were also analyzed volumetrically, but by ion exchange. Solutions of LbS04 and Ca(NOa)2 were analyzed gravimetrically. Densities were determined as described previously at 25°C,12 and the data were then used in the determinations of densities at temperatures above 25°C. Measurements of density at elevated tem peratures simply involved observations of the expan sion, by means of a cathetometer, of a solution con tained in a uniform small bore tube. RESULTS AND DISCUSSION A. Water 60-cm-1 Band of Water A weak broad band of water (and of heavy water) centered at ,..,.,60 cm-l has been reported previously, vid., Ref. 2, and that band is thought to arise from the hydrogen-bond-bending vibrations of water. In terms 11 G. E. Walrafen, J. Chern. Phys. 40, 2326~(1964). i >I- iii ill I- ~ ~ -~-5'C ~ ------ ~67'C ~ ~ ) ../ // ...----/ ) ~ 'I 0' ~ --" 400 350 300 250 200 150 100 50 0 _CM-l FIG. 3. Raman spectra of water in the low-frequency region at two temperatures. Note that the band ~ontour at ""60. cm-1 is (slightly) concave downward at ",,-5 C. The bas.e lines 3;re estimated (not iterated). (These spectra were obtamed earlier with the horizontal Cary housing, and with water forced~through O.2-p MiIIipore filters.) of tetrahedral water structure, it has been assigned to the Vaal and V4aI deformations of the C2v model. The 6O-cm-1 band, and the neighboring 152-175-cm-1 band, at two temperatures, are shown in Fig. 3. (The base line was estimated; the iterative method was not employed here.) The rapid intensity decrease of the 60-cm-1 band with temperature rise parallels the de crease in the intensity of the 152-175-cm-1 band. The 60-cm-1 band has not yet been reported in the infrared spectrum of water, but the far-infrared region poses many difficulties. A band near 60 cm-I, however, has been reported in inelastic-neutron-scattering spectra.Ia 152-175-cm-1 Band of Water Intensity variations of the 152-175-cm-1 Raman band of water with temperature rise, and with electro lyte addition have been reported previously,2 and that band is almost certainly produced by the hydrogen bond-stretching motions. (A similar band occurs in the heavy-water spectrum.) In terms of the C2~ model of 18 D. J. Hughes, H. Palevsky, W. Kley, and E. Tunkelo, Phys. Rev. 119, 872 (1960). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:521550 G. E. WALRAFEN >l ii; Z "' I- ~ 400 300 200 10.aOc ----_ ...... ,.,. ,/ 400 300 200 100 I I I / 100 . 400 300 200 100 400 300 200 '100 FIG. 4. Raman spectra of water in the low-frequency region at a series of temperatures. (Purified water, 0.01-1' Millipore filter, and the vertical Raman tube were employed.) Note the large intensity differences evident in the two upper spectra cor responding to -1.8° and 94.7°C. Also note that the approach to the exciting line is about the same as that of Fig. 3 despite the fact that the amplification was greater. tetrahedral water structure, components designated VIal, V2al, v7bl, and v9b2 all contribute to the 152-175-cm-l band. Raman spectra of the 152-175-cm-l band are shown in Fig. 4 at a series of temperatures. (The base lines shown in that figure were obtained by the iterative method.) Infrared spectra of water and heavy water in the low-frequency region are shown in Fig. 5. (The spectra were kindly supplied by the Beckman Instrument Company.) More work in the low-frequency infrared region, however, is in progress here with a Beckman IR-ll spectrophotometer. Hence, the spectra shown now should be considered preliminary, although broad absorptions near ,.....,,167 and ,.....,,170 cm-l have recently been observed several times in this laboratory, at ambient temperatures, and also at temperatures approaching O°c. (See also Refs. 14 and 15.) In addi tion, inelastic-neutron-scattering spectra confirm the reality of a band in this spectral region.l3 450-780-cm-l Bands of Water Bands observed in the Raman and infrared spectra of water in the 450-780-cm-l region, viz., at 450 and ,.....,,780 cm-l (Raman), and at 705 cm-l (infrared), have been considered previously.2 These bands decrease in frequency by a factor of,....."V'1, in heavy water, and hence they arise from librational motions. The three librational components have been designated V5a2, vfibl, and vSb2, according to C2v symmetry, and these designations complete the vibrational assignments of the tetrahedral water structure. The librational bands have also been observed to decrease in intensity with temperature rise. Further, similar bands have been observed in inelastic-neutron-sca ttering spectra.l3 Two-Species Model of Water Structure The intermolecular bands of water discussed thus far involve all normal vibrations of the tetrahedral structure (approximated by C2• symmetry), but break down of tetrahedral structure with increase of tem perature gives rise to a second species, thought to be a non hydrogen-bonded monomer. A thermodynamic two species treatment of water, therefore, involves an equilibrium between water molecules, designated as B, which are bound by tetrahedrally directed covalent forces, and between water molecules, designated as U, bound by other forces which do not make appreciable contributions to the intermolecular Raman intensities. The intermolecular Raman intensities, which are thought to involve nearest-neighbor structure almost exclusively, are considered proportional to the B con centration, i.e., the Raman intensity changes essentially reflect changes in the nearest-neighbor potential field surrounding a given water molecule. The definition of the hydrogen bond employed in the thermodynamic method, therefore, is an operational one; it involves only those interactions which contribute significantly to the intermolecular Raman intensities. Some approximations employed in the thermody namic method have been mentioned previously,2 but in addition it should be noted that !:J.Ho is also assumed to be independent of temperature, i.e., !:J.cop=O. Of course, it is known that LlCop for the sublimation of ice at O°C is about -0.5 to -1.0 calf deg· mole; LlHo is thus only approximately independent of temperature. However, it is doubtful that refinements in the method involving LlCop are justified by the present intensity data, even when the data are corrected as described later. 14 A. E. Stanevich and N. G. Yaroslavsky, Dok!. Akad. Nauk SSSR 137, 60 (1961) [English trans!': Soviet Phys.-Doklady 6, 224 (1961)]. 15 C. H. Cartwright, Phys. Rev. 49,470 (1936). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:52RAMAN SPECTRUM OF WATER 1551 z o ~ FdlhG. 5. Far-infra~ed spectra Ii ofbwather &: an eavy water, kmdly supp ed y teo Beckman Instrument Company. :2 ..: + E 350 D20 310 The uncorrected intensity data are shown in Fig. 6. When the data are treated by the thermodynamic method (see Ref. 2 for details) they yield the equation shown in the figure, and hence the uncorrected I1Ho and I1So values of -5.1 kcaljmole and -17 caljdeg· mole, respectively, for the process U--+B. [The 152-175- cm-I appelation for the hydrogen-bond-streatching band is obvious from the l)(t) values of the figure.] In Fig. 7 corrected intensities are shown in terms of iE, the fraction of bound molecules contributing inten sity. The unbound fraction is, of course, given by 1-iE. The corrections applied involve changes in density and in Raman intensity with temperature. The latter in clude all terms shown in Fig. 7 except p(t) and A, d., Ref. 16. The constant A normalizes the corrected in tensities to unity at very low temperatures. The density term p(t) corrects for loss of intensity due to expansion. It should be noted here that the values of l)i(t) from Fig. 6 are not involved in the corrections, but they indicate that the values of I(t) are reasonably accurate, since no significant temperature dependence is evident. (See also the widths evident in Fig. 5.) The interpolated l)(t) data from Fig. 6, however, were employed in the corrections. The corrected /1Ho and I1So values are larger than the uncorrected values, viz., -5.6 kcaljmole and "'-'-19 caljdeg·mole, respectively. Thermodynamic Tests of the Two-Species Model If the tetrahedral water lattice is completely dis rupted, two hydrogen bonds on the average, are broken for each H20 molecule liberated. In this case, I1Ho for the process B--+U is divided by 2 to give the hydrogen- 16 L. A. Woodward and D. A. Long, Trans. Faraday Soc. 45, 1131 (1949). 270 230 190 150 110 70 __ CM-l bond enthalpy. Scatchard et al. have reported a value of 3.41 kcal/mole H bondI7 which is highly regarded, and Grunberg and Nissan have reported a range of 3.23 to 3.71 kcaljmole H bond.Is It is obvious, there fore, that division of I1Ho by two yields a value of 2.8 kcal/mole H bond in reasonable agreement with Scatchard's value, and also with the range of Grunberg and Nissan. These agreements lend strong support to the present method. 130 ~~ l~i°:--:o~~ --;;--"'-<>--;;~--2---n __ """":~ 0.20 0.16 e H 0.12 0.08 0.04 ~·~0~~~~~~L-~-L~~~~~ (a) (b) (c) FIG. 6. Integrated intensities 1(1), Raman frequencies ,,(I), and half-widths "i (t) t obtained for water in the temperature range of -6.0° to 94.7°C. 1 (I) is the uncorrected intensity. (a) Least squares; (b) SUbjective interpolation; (c) least squares, )ogIG{1 (1)/[0.320-1 (t)]1 = (1116.2/T) -3.710 7• 17 G. Satchard, G. M. Kavanagh, and L. B. Ticknor, J. Am. Chem. Soc. 74, 3715 (1952); 74, 3724 (1952). 18 L. Grunberg and A. H. Nissan, Trans. Faraday Soc. 45, 125 (1949). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:521552 G. E. WALRAFEN 1.o..---..;:7__.- .......... -.-----.-----.-----.----...,.------, 0.9 2 0.8 ...... ';3 r:::! 0.1 ~ .., "",- "\ \ ~ ........ 2\ ~ 0.6 fa 1212.2 /--LOGIO -f-= -T -4.2192 FIG. 7. Values of the fraction of asso ciated water fB as a function of tem perature. (/B is proportional to the cor rected integrated intensity.) The dashed lines refer to extrapolations according to the least-squares equation shown in the figure. Note that fB=0.6 2 at O°C. Values of t.Ho and t.So refer to the process U->B. (t.Ho= -5.e kcal/mole, t.So ",,-19 cal/deg.mole.) / 1-B ~~ ~ 0.5 I ... CI>-l. ~ 0.4 '--« II 0.3 ..... 1Jl 0.2 0.1 --------.------jo~0~--~-----7----~~----~~---~~----~200 The !:J.So value of ",19 caljdeg·mole is suggestive of a process resembling vaporization. Further, it is smaller than !:J.So for the sublimation of ice at ooe, viz., 34.6 cal/deg·mole, whereas the value obtainable from previ ous work2 is larger. The present value of !:J.So, therefore, is probably of sufficient accuracy to allow for the fol lowing comparison. If liquid water is considered to possess entropy con tributions only from the Band U species, (and the small ideal entropy of mixing is omitted because of uncertainty in !:J.SO) the standard entropy is given by (1) At 2Soe, S°(1)= 16.716 cal/deg·mole, and /B=0.4126 (from the equation of Fig. 7). Hence, when !:J.S°-;::::;;-19 cal/deg·mole, SO(u)-;::::;;2S and SO(B)-;::::;;6 caljdeg·mole. But 5°(g) = 45.106 and 5°(8)-;::::;;10 caljdeg·mole at 25°C. Thus the calculated 5°(u) and 50(B) values, although not oj quantitative significance, are most nearly related to the entropies of water vapor and of ice, respectively. Similarly, if water possesses enthalpy contributions only from the Band U species, the standard enthalpy IS (2) and differentiation with respect to temperature yields COp(l)= COP(U)+/B!:J.cop+!:J.HO(d/B/dT). (3) Substitution into Eq. (3) of djB/dT=UB(1-jB)!:J.HO]/RP (4) leads to COp (1) = COP(U)+/B!:J.co p+UB(1-/B)/ R](!:J.Ho/T)2. (5) However, if !:J.Cop is taken to be zero At 25°e, for /B=0.4126 and !:J.Ho= -5600 caljmole, COp(1)=43+Co p(u)' but this result is much too high since CO P(l) = 17.996 cal/ deg· mole. Nevertheless, the present result represents a significant advance over the standard heat capacity obtainable from the previous !:J.HO,2 and it should be emphasized that the heat capacity represents a stringent test of the data. Comparisons With Other Methods An important investigation involving light scattering in water has recently been reported by Mysels.19 In that work a network containing predominantly filled cavities was found to be most compatible with the light-scattering data. In terms of the present work it would appear that the network refers to the tetrahedral liquid lattice, and the entitites filling the cavities, to non hydrogen-bonded monomers. However, Mysel's work has recently been criticized.20 Danford and Levy21 have interpreted their extensive and accurate x-ray data in terms of "interstitial" molecules filling an icelike tetrahedral framework, but they assumed that /B=0.8 at their ambient temperature, and it can be shown that their model refers to /B values which are not so temperature dependent as those obtained here. In regard to the degree of association /B at various temperatures it should be mentioned that Wada22 has been able to explain various properties of water by assuming that /B=O.2 at 1000e. (An icelike state and close-packed non hydrogen-bonded monomers were assumed to exist.) From Fig. 7 it is apparent that the corresponding value of /B is "-'0.1. Luck23 has reviewed 19 K. J. Mysels, J. Am. Chern. Soc. 86, 3503 (1964). 20 J. P. Kratohvil, M. Kerker, and L. E. Oppenheimer, J. Chern. Phys. 43, 914 (1965). 21 M. D. Danford and H. A. Levy, J. Am. Chern. Soc. 84, 3965 (1962). 22 G. Wada, Bull. Chern. Soc. Japan 34,955 (1961). 23 W. Luck, Fortschr. Chern. Forsch. 4, 653 (1964). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:52RAMAN SPECTRUM OF WATER 1553 various reported values of is at oDe, and he generally favors the high values. It would appear, however, that the sound-absorption value of Hall,24 who also employed the two-species treatment, agrees well with the value of 0.62 now obtained. Further, the asymptotic approach of is to unity at very low temperatures is in agreement with the rapid rise in the viscosity of supercooled water at temperatures approaching (and probably below) -24°e,26 since the water lattice should be more rigid when the defect concentration is small, i.e., when the non hydrogen-bonded fraction is small. Another recent two-state theory of water structure by Davis and Litovitz26 is also of considerable importance. Those workers described an interesting mechanism by which non hydrogen-bonded species may be formed. Although their model does not involve the breakage of all hydrogen bonds at low temperatures, it is suggested that further disruption occurs at high temperatures. Other note worthy papers concerned with two-state models of water structure are those of Frank et al.,?:i-3I of Marchi and Eyring,32 of Nemethy and Scheraga,33.34 of Pauling,36.36 and of Forslind.37 Other Models of Water Structure The models of water structure proposed by various workers are so numerous that it is not possible to consider all of them here, but many of them can be encompassed by (1) models in which molecules of H20 are considered to be engaged in interactions of suffi ciently different character as to allow for measurable differences, e.g., the two-species model described previ ously, and (2) models in which no distinguishable features are considered to exist between H20 molecules. Few, if any, attempts have been made to reconcile (1) and (2) above. However, a recent Raman investi gation of HDO in H20 and in D20 reported by Wall and Hornig9 is employed here to illustrate at least a plausible means of producing agreements. In the above investigation, the frequency of the symmetric valence vibration of steam was used as a 24L. Hall, Phys. Rev. 73,775 (1948). 2& J. Hallett, Proc. Phys. Soc. (London) A82, 1046 (1963). 26 C. M. Davis, Jr., and T. A. Litovitz, J. Chern. Phys. 42, 2563 (1965). 27 H. S. Frank and M. W. Evans, J. Chern. Phys. 13, 507 (1945). 18 H. S. Frank and W. Y. Wen, Discussions Faraday Soc. 24, 133 (1957). 28 H. S. Frank, Proc. Roy. Soc. (London) A247, 481 (1958). ao H. S. Frank and A. S. Quist, J. Chern. Phys. 34, 604 (1961). 81 H. S. Frank, Desalination Research Conference Proceedings, Washington, 1963, NAS-NRC Publication 942,141. 32 R. P. Marchi and H. Eyring, J. Phys. Chern. 68, 221 (1964). II G. Nernethy and H. A. Scheraga, J. Chern. Phys. 36, 3382 (1962) • 34 G. Nernethy and H. A. Scheraga, J. Chern. Phys. 41, 680 (1964). 36 L. Pauling and R. E. Marsh, Proc. Nat!. Acad. Sci. (US) 38, 112 (1952). ae L. Pauling, Hydrogen Bonding, edited by D. Hadzi (Per garnon Press, Ltd., London, 1959). 37 E. Forslind, Acta Poly tech. Scand. 115, 9 (1952). criterion to determine the fraction of water not engaged in hydrogen bonds. With that criterion the non hydro gen-bonded fraction was determined to be less than 5% at 25°C. However, from the present work it appears that the standard entropy of the U form is much less than that of steam. Further, the U form is considered to be more dense than the B form.22 Therefore, the criterion involving low-pressure steam is unrealistic. But steam at the critical point approaches the normal density of liquid water more closely than does steam at 1 atm. Accordingly, the symmetric stretching frequency of high-pressure steam should provide a better experi mental criterion. Saumagne38 has recently investigated the infrared spectrum of water in the critical region, and he observed two frequencies at 3545 cm-I (vI,aI) and at 3650 cm-I (V3bI)' Further, the low-frequency tail of the 3545-cm-I band was observed to extend at least to 3300 cm-I• Now, a criterion of 3500 cm-I, arrived at from an observation to be described later (instead of the 3600- cm-I criterion used by Wall and Hornig9), yields a value for the fraction of non hydrogen-bonded H20 molecules of roughly 30% at 27°C. In contrast, the value for water at 25°C obtained by the present thermo dynamic method is 59%. There is, of course, consider able uncertainty in Wall and Hornig's method which involves the transferring of a crystal correlation to the liquid case, and in regard to this it is evident that the fraction of non hydrogen-bonded molecules obtained by that method is extremely sensitive to the frequency criterion for distances above 2.80 A. Further, the re ported agreement9 between the nearest 0-0 distance from Raman and x-ray distributions is to be expected, and it does not imply the existence of only one species. X-ray radial distributions have repeatedly been inter preted in terms of two species.39--42 Another recent (infrared) spectral investigation of HDO in H20 and D20 reported by Falk and Ford43 should be mentioned. Here an even more extreme con clusion, viz., that there is no spectroscopic evidence for appreciable concentrations of non hydrogen-bonded monomers, was reached. This investigation is in dis agreement with the conclusion of Wall and Hornig9 who studied the same types of solutions. Further, from the considerable spectroscopic evidence for non hydro gen-bonded monomers presented here, the conclusion of Falk and Ford43 can only be regarded as erroneous. It may be possible to reconcile the data with the two species model, however. The near coincidence of the critical-point frequencies iI8 P. Saurnagne, doctoral dissertation, University of Bordeaux, 1961. au J. D. Bernal and R. H. Fowler, J. Chern. Phys. 2, 559 (1934). 40 J. Morgan and B. E. Warren, J. Chern. Phys. 6,666 (1938). 410. Ya. Sarnoilov, Zh. Fiz. Khim. 20, 1411 (1946). 420. Ya. Sarnoilov, Discussions Faraday Soc. 24,141 (1957). 48 M. Falk and T. A. Ford, Syrnp. Mol. Structure Spectry. Columbus, June 14-18, 1965. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:521554 G. E. WALRAFEN with those of liquid water suggests that uncoupled fundamental and overtone bands of HDO may contain unresolved components from both the U and B species, e.g., Wall and Hornig9 observed some band asymmetry. Thus, the Raman9 and infrared4s methods involving HDO are probably of insufficient sensitivity to resolve contributions from two species. Small frequency shifts, asymmetries, and intensity variations with temperature are all that can be expected. Finally, the model of Pople44 which suggests that hydrogen bonds can be bent in the liquid should be mentioned, because the non hydrogen-bonded molecules described in the present work may relate to large devia tions from hydrogen-bond linearity. Thus, the covalent character of tetrahedral units contributes significantly to the intermolecular Raman intensities and it leads to high directionality, but the dipole-dipole interactions, for example, remaining after large deviations from linearity have occurred probably would not contribute significantly. The non hydrogen-bonded molecules, then, would refer to molecules restrained by predominantly noncovalent interactions, and the lower entropy of the non hydrogen-bonded molecules, relative to the entropy of low-pressure steam, apparently reflects those restraints. B. Electrolyte Solutions Librational Bands of Water in Electrolyte Solutions Decreases in the intensities of the librational Raman bands from pure water with increase of temperature have been reported previously.! In addition, the libra tional Raman intensities of electrolyte solutions have been investigated as functions of electrolyte concentra tion.2 The temperature dependences of the librational intensities from electrolyte solutions were known previ ously (from unpublished work) to be small, but it was considered desirable to determine them quantitatively. The temperature-dependence data are shown in Fig. 8 for various solutions in the temperature range of 25°- 140°C, and those data were corrected for density and for temperature effects in a manner similar to that employed previously. However, because no appreciable changes of frequency with temperature were observed, several terms were omitted, and the average of the librational band "centers" (the bands are asymmetric!) from all the solutions, viz., 578 cm-I, was employed. It should also be noted that molar librational intensities are plotted in the figure. Many important conclusions may be drawn from the data shown in Fig. 8. The most obvious conclusion is, of course, that all of the temperature dependences are small. It is also evident that the data are not particularly sensitive to the stoichiometric water-to-electrolyte con centration. In addition the larger effects of Br-com- 44 J. A. Pople, Proc. Roy. Soc. (London) A205, 163 (1951). pared to CI-, which were observed previously,! are readily evident. Smaller effects involving the cations are found upon close inspection of the data for Li+, Na+, and K+, but the effects of NH4+ are particularly interesting. In that case the molar intensities are correspondingly smaller than those of the other chlorides or bromides. The molar intensities of LiNOs, Ca(NOsh, and Li2S04 are also small. The molar intensities from the NH4Cl and NH4Br solutions suggest that NH4+ is essentially un hydrated, and this is in agreement with the Raman work of Vollmar.45 For the chlorides and bromides the molar librational intensities increase in the order N&+, Li, Na+ or K+. A similar trend Li+, Na+, K+ was observed previously for values of S",y.l The new data are compared with the old in Table I. (The molar librational intensities defined by this work are essentially proportional to the molar intensity enhancements Szy defined previously, because the librational intensities of pure water are almost negligible, and thus decrease the slopes only slightly.) Agreements within the expected accuracy of the method are apparent. [In Table I, values of Sx/ are compared with 1(25). Szy' = 85Sxy, and the proportionality factor was determined from an average taken from chlorides and bromides of Li+, Na+, and K+. 1(25) is the un corrected molar librational intensity at 25°C.] In previous work,! librational intensities of water in electrolyte solutions were found to be linear in the electrolyte concentration even to concentrations near saturation. But, because Raman intensities have re peatedly been found to be linear in species concentra tion,46 it is apparent that the librational intensities must be linear in electrolyte concentration, and also in the concentration of units involving primary hydra tion, since water intensities were measured. Further, TABLE I. Comparisons between values of S' z. and 1(25); and 1(25) values for other solutions. xy LiBr S'x. 8.4X10-2 1(25) 8.2X10-2 xy LiCI S'x. 3.8X10-2 1(25) 3.6X10-2 xy NH.Br 1(25) 4.6XI0-2 XnYm LiNO. 1(25) 2.2XlO-2 NaBr 1. IX 10-1 1.1X 10-1 NaCI 5.0X10-2 7.0X10-2 NH.CI 2.6XI0-2 Li2SO. 5.5XIQ-2 KBr 1.3X10-1 1.1X 10-1 KCI 6.3XIQ-2 5.9XlO-2 Ca(NO.h 3.8XlO-2 45 P. M. Vollmar, J. Chern. Phys. 39,2236 (1963). 46 T. F. Young, L. F.Maranville,and H. M. Smith, The Structure of Electrolytic Solutions, edited by W. J. Hamer (John Wiley & Sons, Inc., New York, 1959). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:52RAMAN SPECTRUM OF WATER FIG. S. Corrected relative integrated molar librational intensities as functions of tempera ture. (Molar refers to the stoichiometric elec trolyte concentration, but the relative inte grated intensities refer to librational bands of water.) p (I) / p (25) is, of course, unity when t= 25°. This ratio corrects for density changes, but it allows all intensities to be compared on the same basis since the individual densities vary widely, cf., Fig. 7 where only p(t) was used. 0, observed; --, least squares. (a) 1-2.33M Li.S0 4, [H.OJ/[Li,S04]= 22.2; 2-5.53M Ca(NOa)., [H20]/[Ca (NOa).] = 6.7; 3--6.S3M LiNOa, [H20 ]/[LiNO a] = 6.7. (b) 1-7.54M NH.Br, [H20 ]/[NH,Br] = 4.2; 2-5.6SM NH,Cl, [H20 ]/[NH4Cl] = 7.6. (c) 1-4.63M KBr, [H20]/[KBr]=9.S; 2-4.0SM KCl, [H20]/[KCl]= 11.9. (d) 1-7.12M NaBr, [H20]/[NaBr] = 6.2; 2-5.33M NaCl, [H20]/[NaCl]=9.2; (e) 1-1O.55M LiBr, [H20 ]/[LiBr] = 3.S; 2-13.67M LiCi, [H20]/[LiCl] = 2.9. 2 ..... 811-.<:"'" I Cl> I --.:::m L-...J 1.2 ~~-I-~ I-~[~I-'---I--'I--'-- [-r-I-rl-'---I-'I_ 1.0 f-'""o---oo--., ---;::--_--2 , ',-1 O.Sf- 0.6 r,o:r----"---;:,--O"o----oo '--2 0.4 - 0.21--- - - - - OL-~I_L_I~I_~I_L_[~[ __ L_[~[ __ ~I __ L_[~[~ 1.2 ~--r-_r_-r_-.,.-...,--.___,_-...,--,__--,--...,_--, 1.0 ."Qtr---"'-_~,--o."---_~ '---1 O.S o 0.6 o ",2 o 0.4 0.2 OL-~ _ _L_L-~_~_~~_~_~~_~~ 1.0,----,--...,_-,---,-...,_-,----,--...,--,---;-...,--, O.S 0.6 0.4 0.2 ~---~~-\\-~~r--~O~--4C--~v_£ ',_ 1 o \ &-----~S~ __ ~ __ ~~_ '--2 OL-~ _ _L_L-~_~_L-~_~_~~_~~. 20 30 40 50 60 70 SO 90 100 110 120 130 140 t (Oc) 1555 (c) (d) (e) the primary hydration numbers must be essentially independent of concentration. The librational Raman intensities are apparently only very sensitive to inter actions involving the first hydration sphere. The small temperature dependences observed here for the librational intensities are also consistent with the conclusion that interactions beyond the first hydra tion sphere are not involved to any great extent. Secondary hydration would be expected to be sensitive to temperature variations within the temperature range investigated here, whereas the small decreases observed with increasing temperature are nearly within experi mental accuracy. In regard to this, it should be noted that the intensities from Liel and LiBr solutions were obtained at temperatures only a few degrees below the normal boiling points. Of course, it seems reasonable to expect that changes in the primary hydration should occur in solutions, saturated at elevated temperatures, and under pressure, but work involving high pressure has begun only recently in this laboratory. There can now be little doubt, however, that strongly hydrated units exist in electrolyte solutions, and that the concept of ionic hydration is a useful one, at least as far as Raman methods are concerned. (The conclusion of Wa1l9 to the contrary, is not consistent with the present data.) Models of Hydrate Structure If the over-all point group of an ion and its local environment were to be considered, information involv- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:521556 G. E. WALRAFEN TABLE II. Properties expected of the H'OH molecule of C, symmetry. The protons are considered to be nonequivalent because of interactions. ========_----------- ------c-_--_-_---= Description ,,(H'-OH) ,,(H'O-H) o (H'-O-H) H'OH C. Symmetry Species Polarization Activity a' p Raman, ir a' p Raman, ir a' p Raman, ir ='-======-:-==---=--- ----- ing primary (and secondary) hydration numbers, inter actions with ions of opposite charge, and structure, would be required, but such information is, in general, unavailable. However, if attention is fixed on a single molecule of H20 engaged in hydration, or on the com bined symmetry of the molecule and its nearest neigh bors, the symmetries of both can be approximated. In the first of two treatments to be employed here, it is convenient to consider the intramolecular vibra tions of a molecule of H20 engaged in hydration in terms of C. symmetry, and the vibrations of that molecule against its nearest neighbors by the same symmetry. This treatment is employed because it emphasizes anion-water interactions, which in many cases studied thus far are predominant. Further, the intramolecular vibrations can be separated from the intermolecular vibrations; the intramolecular vibrations are excluded in the latter case. A molecule of water engaged in hydration is here designated H'OH. That H20 molecule and its neighbors, are then designated. X-HOH and X-HOH, I y where X refers to the anion and Y to the cation. The characteristics of the intramolecular H'OH vibrations treated according to C. symmetry are given in Table II. Tables III and IV indicate characteristics TABLE III. Properties expected of the intermolecular anion water hydration model of C. symmetry. X refers to the anion. H'OH vibrations excluded. v,-libration. X , , II H "'./ 0 C, Symmetry Description Species Polarization Activity v (X-HOH) a' p Raman, ir v,(HOH) a' P Raman, ir JI,(HOH) a" dp Raman, ir expected of the intermolecular vibrations, with the H'OH vibrations excluded. The X-HOH vibrations of Table III refer to the very large effects produced by certain anions such as CI-and Br-where covalent interactions are important, and they are pertinent to the data involving NH4Cl and NH4Br, where the effects of N&+ are obviously very small. If NH4+ is unhydrated as suggested before, the anionic effects completely predominate as required by the model. The vibrations described in Table IV are important in concentrated solutions, and they are relevant to the effects produced by the chlorides and bromides of Li+, Na+, and K+ where the anionic effects as well as the cationic effects are significant. They also refer to the effects produced by LiNOa, Li2S04, and Ca(NOah, TABLE IV. Properties expected of the intermolecular anion water-cation hydration model of C. symmetry. Y refers to the cation. H'OH vibrations excluded. v,-Iibration. Description v (X-H) v(Y-D) Il(X-HO-Y) v,(HOH) v,(HOH) v,(HOH) X H H "'./ o , Y C. Symmetry Species Polarization a' p a' P a' p a' P a" dp a" dp Activity Raman, ir Raman, ir Raman, ir Raman, ir Raman, ir Raman, ir where cationic and anionic effects are small, and thus more nearly equa1.47 It is also apparent that large anions may involve at least two types of hydrated water, i.e., water bound to the cations, and water not bound to them. A large singly charged anion, for example, could be involved in one X-HOH I Y interaction, and perhaps in several X-HOH interac tions. (A parallel statement applies to cations which are considered next.) In the second treatment, the H20 molecules are considered to retain their usual symmetry, and the interaction of a given H20 molecule with a cation is then approximated by C2• symmetry. Here, the cation 47 Ca2+-H.O-NO.- interactions are thought to be predominantly ionic. D. E. Irish and G. E. Walrafen J. (to be published). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:52RAMAN SPECTRUM OF WATER 1557 is thought generally to approach the ~20 I?olecule in the vicinity of the oxygen atom, and In this cas~ the intermolecular vibrations, again with the H20 Vibra tions excluded are described in Table V. Interactions , . of this type, however, are relatively weak for the catlOns studied in this work, although they can assume great importance in other cases where polarized intermolec ular bands have been observed.48 In regard to experimental verifications of the models, the librational polarizations indicated by Tables III-V are observed to agree with the high depolarizations of the librational bands of water in solution.! The inten sification and sharpening observed for the intramolec ular valence band at '"'-'3450 cm-1 upon addition of Cl and Br also strongly suggest polarization, d., Table II and Raman measurements indicating polarization fo; concentrated solutions have been reported.3 Several bands predicted by the intermolecular models, however, have not yet been observed, e.g., X-HOH stretching, and one valence vibration of H'OH. In regard to valence vibrations, the recent artic.le by Senior and ThompsonlO would appear to contaIn some attractive features, i.e., Vial' and V2al' modes of the H'OH molecule of C. symmetry could be unresolved but their hypothesis concerning the 3630-cm-1 band cannot be taken seriously in view of the high depolar ization of the 3630-cm- 1 band and the rapid decrease in the 152-175-cm- 1 band intensity with temperature. Further, Saumagne37 has definitely observed a shou~der in the infrared spectrum near 3600 cm-t, thus the assign ments based on the absence of that band are not valid. In addition, Fermi resonance between a combination of an inter-and intramolecular vibration, with the same intramolecular vibration, is extremely improbable when the inter-and intramolecular frequencies differ by nearly 3300 cm-l• Also, the 3225-cm-1 ban? is no,,: generally ascribed to the overtone of V2al In Fermi resonance with Vial, but the intensity of 2v2al decreases TABLE V. Properties expected of the inte~mole.cular cation water hydration model of C2• symmetry. H20 VIbratIOns excluded. v,-libration. Description v (Y-OH 2) v,(HOH) v,(HOH) H H "'-./ o , , y Co. Symmetry Species Polarization a, P b, dp b2 dp Activity Raman, ir Raman, ir Raman, ir '8 Here the cation-oxygen hyd:ation interactions. are ,:,ery large and they involve covalent catIOn-oxygen honds With catIOns such'as Zn2+, vid., D. E. Irish, B. McCarroll, and T. F. Young, J. Chern. Phys. 39, 3436 (1963). TABLE VI. Comparisons between librational frequencies of this work and frequencies reported for solid hydrates.·.b Raman aqueous solutions Infrared solid hydrates 13.67M LiCI 420 660 5.33M NaCl 420 630 BaCI2·2H2O 5200 6900 4.0SM KCl 450 650 5.6SMNH,Cl ",400 ",650 10.55M LiBr 460 670 7.12M NaBr 460 650 NaBr·2H 2O 470d 625d 4.63M KBr 440 660 7.54M NH,Br 450 650 6.S3M KiNOa ",420 ",600 5.53M Ca(NOa)2 ",460 ",650 2 .33M Li2(SO,) ",450 "'620 • See Ref. 49. b Alllibrational frequencies of tbis work are uncertain by :1=50 em-I. unless marked by"'. in which case they are uncertain by :1=75 em-I. eo· .. HOH .. ·Cllibration. d O ... HOH ... Br libration. very rapidly with temperature rise, whereas the corre sponding decrease in V2al is extremely small. Thus, eve.n in this case it is not certain that Fermi resonance IS involved exclusively. The contour in the valence region is a very complex one and recent close examinations suggest that at lea;t four bands may be involved, viz., 3225, 3450, '"'-'3500, and 3630 cm-l• The '"'-'3500-cm- 1 band appears to exhibit an increase of intensity with temperature rise whereas the 3450-cm- 1 band intensity decreases slightly, again suggesting the possibility of two species, d. the '"'-'3500-cm-1 band in liquid water with the '"'-'3545-cm- 1 band of steam at the critical point. The evidence for the existence of the '"'-'3500-cm- 1 band is not obvious, however, but careful examinations reveal that the contour is concave upward in this region at temperatures near O°C, and concave downward near 100°C, and juxtaposition of bands does not appear to explain the curvature changes. (Also the '"'-'3500-cm- 1 band is too broad to arise from mercury.) Comparisons With Librational Frequencies From Hydrated Solids Van der Elsken and Robinson49 have contributed important information related to the librational fre quencies of hydrated solids. Their data are valuable because they were able to relate certain frequencies to O· .• HOH· .. Cl and O· .. HOH· .. Br librations. Table VI contains comparison between some of their data, and frequencies obtained from presents analyses of librational contours according to two components. The agreements (even when Ba2+ is involved in one case) are reasonable, and they strengthen the C. model of this work. Further, the librational frequencies of the solids and of the solutions differ from the corresponding 49 J. van der Elsken and D. W. Robinson, Spectrochim. Acta, 17, 1249 (1961). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:521558 G. E. WALRAFEN frequencies of pure water, in that the ",780-cm-1 component of pure water is absent. FUTURE WORK Work is now in progress in this laboratory with high pressure Raman cells, and a test cell has recently with stood hydrostatic pressures to 1700 atm. The cell windows were of sapphire, 1 in. in diameter and 1 in. thick, but the available diameter for passage of radia tion was only 0.5 in. Nevertheless, a large high-pressure Raman assembly for use with the four vertical lamps is now under construction. It will employ 17 windows, 16 of which will be placed in four vertical groups, with each group facing one of the four Raman lamps. The remaining window will admit Raman radiation to the entrance optic, d., Fig. 1. Work involving an argon-ion laser to be used with the Cary Model 81 spectrophotometer is also in progress. The laser is expected to produce radiation continuously at 4880 A, and at power levels approaching 1 W. The spectral sharpness of the laser radiation, of course, should allow for a close approach to the exciting line, and the physical narrowness should also allow for the use of small high-pressure Raman cells. The high-pressure work will probably involve the 152-175-cm-1 region of pure water, as well as the librational region of the electrolyte solutions. High pressure Raman work on water and electrolyte solutions should provide much valuable new structural informa tion. ACKNOWLEDGMENT The author is grateful to R. Popiel for assistance in this work. THE JOURNAL OF CHEMICAL PHYSICS VOLUME 44, NUMBER 4 15 FEBRUARY 1966 N uc1ear Magnetic Relaxation of Polymer Solutions. Side-Chain Motion ROBERT ULLMAN Scientific Laboratory, Ford Motor Company, Dearborn, Michigan (Received 10 August 1965) In this paper, the magnetic relaxation of nuclei of spin! which are attached to polymer molecules is calculated. The computation is based on the idea that the relaxation process is a consequence of magnetic dipole interaction between spins attached to the same segment. The particular point of this paper is to treat the relaxation as a consequence of the relative motion of the spins with respect to the polymer segment to which they are attached. The polymer segment, itself, undergoes a complicated Brownian motion which has been analyzed in a previous paper in terms of the Zimm-Rouse model of chain macromolecules. INTRODUCTION IN a recent paper,! henceforth to be referred to as I, a theory of nuclear magnetic relaxation of polymer solutions was presented which was based on the Zimm Rouse model2,3 of a chain polymer molecule. This model, which has had considerable success in account ing for the steady-state viscosity, dynamic viscoelas ticity, and dielectric relaxation of dilute polymer solutions was adopted for the magnetic relaxation problem. The near-neighbor interactions are dominating in nuclear magnetic relaxation while long-range inter action between chemical groups playa more important 1 R. Ullman, J. Chern. Phys. 43, 3161 (1965). 2 P. E. Rouse, Jr., J. Chern. Phys. 21, 1272 (1953). 3 B. H. Zimm, J. Chem. Phys. 24, 269 (1956). role in determining dynamic mechanical and dielectric behavior. The suitability of the Zimm-Rouse model is less certain in the nuclear magnetic relaxation case, since in this model the detailed local geometry of a polymer chain is replaced by a connected set of elastic springs, which do not represent a real molecular structure very well on a local (1-10 A) scale. Never theless, subject to the appropriate choice of parameters, it seems likely that the model would provide a useful basis for comparison with experiment and would make it possible to provide a more complete interpretation of magnetic relaxation data on polymer solutions. It was pointed out in I that nuclear magnetic relaxation of polymer molecules in solution containing nuclei of spin! only (hydrogen, fluorine) takes place because of the relative motion of the spins and that This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.135.12.127 On: Fri, 21 Nov 2014 18:28:52
1.1708942.pdf	Theory of Thermionic Converter ExtinguishedMode Operation with Applications to Converter Diagnostics Daniel R. Wilkins and Elias P. Gyftopoulos Citation: Journal of Applied Physics 38, 12 (1967); doi: 10.1063/1.1708942 View online: http://dx.doi.org/10.1063/1.1708942 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/38/1?ver=pdfcov Published by the AIP Publishing [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 93.180.53.211 On: Tue, 18 Feb 2014 05:42:5012 DALE L. HAMILTON 1.75 2,; <1 0:: f- ([) f-.1.25 z <1 Z ~ W CL .75 o w N --' <1 ~ .25 o z ,~'/' ,/~ : 2 4 6 8 10 12 14 ELECTRIC FIELD (KV I eM ) FIG. 4. Normal ized strain as a func tion of applied elec tric field which clearly shows the ex istence of a thresh old field. does not cause a remnant effect which clearly indicates that a threshold field required for polarization reversal exists. This threshold effect is more clearly illustrated in Fig. 4. This figure is a plot of normalized (computed from Bragg's law) strain versus electric field and shows a threshold field required for polarization reversal of approximately 3 kVjcm. This agrees well with thresh old switching field values obtained by Pulvari4 from electric measurements. CONCLUSIONS The existence of a distinct threshold field required for polarization reversal is one of the most important properties of ferrielectric materials and was never ob served in ordinary ferroelectrics. Before this investiga tion the only means of detecting a threshold field was through direct electrical methods. The ability to detect a threshold with x-ray techniques provides an addi tional method. More important, the change in Bragg diffraction conditions shows that a small, but distinct, physical crystallographic structure rearrangement oc curs as a ferrielectric material is switched from one polarization state to another. Although it was previ ously known that small relative shifts of atomic posi tions do occur, it was not known that remnant structure rearrangements of the unit cell occur. Even though the detailed process of the switching mechanism which produces this effect is not presently completely under stood, it appears that the origin of the threshold field required for polarization reversal can be traced to minute remnant rearrangements of the structure. ACKNOWLEDGMENT I wish to express my sincere gratitude to Professor Charles F. Pulvari for suggesting this topic and for his guidance and discussions during the work. JOURNAL OF APPLIED PHYSICS VOLUME 38, NUMBER 1 JANUARY 1967 Theory of Thermionic Converter Extinguished-Mode Operation with Applications to Converter Diagnostics* DANIEL R. WILKINst AND ELIAS P. GYFTOPOULOS Department of Nuclear Engineering and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts (Received 11 April 1966; in final form 24 June 1966) An analysis of thermoionic converters operating in the extinguished mode is presented. Expressions for the forward and reverse saturation output current densities, and for the open circuit voltage are derived for the first time from a single set of transport equations and boundary conditions. Agreement between theo retical and experimental results is established. It is shown that the output current density cannot exceeed a certain upper limit which depends only upon the emitter temperature and the interelectrode spacing, and is independent of the emitter work function and the cesium pressure. It is shown that, under certain operat ing conditions, measurements of the forward and reverse saturation output-current densities and of the open circuit voltage can be used to infer values of the emitter temperature, emitter work function, collector work function, and electron and ion mobilities. 1. INTRODUCTION THE purposes of this paper are to present a unified analysis of the extinguished mode of cesium thermionic-converter operation, and to demonstrate the utility of extinguished mode measurements in con verter diagnostics. * This work was supported in part by the Joint Services Elec tronic Program (Contract DA36-039-AMC-03200(E» and the National Science Foundation (Grant GK-1165). t Present address: General Electric Company, Special Purpose Nuclear Systems Operation, Pleasanton, Calif. The output-current characteristics of a thermionic converter frequently exhibit two distinct branches as shown schematically in Fig. 1. The upper branch is referred to herein as the "ignited mode" of operation; the lower branch as the "extinguished mode." For a wide variety of operating conditions, the lower branch exhibits forward and reverse saturation current densi ties and an open circuit voltage as indicated in Fig. 1. It is toward the analysis of this extinguished mode output-current characteristic that the present study is directed. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 93.180.53.211 On: Tue, 18 Feb 2014 05:42:50THEORY OF THERMIONIC CONVERTERS 13 J / Ignited Mode Jfor v FIG. 1. Schematic of output-current characteristic of a thermionic converter operating in the collisional regime. Several features of the extinguished mode have been analyzed in previous studies.1-7 Shavit and Hatso POUlOS,l Warner and Hansen,2-4 and Warner5 derived expressions for the "forward saturation current density Jfor" (see Fig. 1). Wilkins6 derived an expression for the open circuit voltage Voc. Houston7 presented an ex pression for the "reverse saturation current density J rev" (see Fig. 1) which is applicable in the limi t ~f negligible collector back emission. In this paper, the forward and reverse saturation current densities, and the open-circuit voltage are deriv~d for the first time from a single set of transport equatlOns and boundary conditions. In addition the implications of these results in thermionic-con;erter diagnostics are emphasized. The transport equations are the same as those derived in Ref. 8 and used to analyze thermionic converters operating in the ignited mode.9 Thus, a single unified description of the entire collisional regime of thermionic-converter operation is achieved. The paper is divided into four parts. First, the plasma transport differential equation8 are presented in a form suitable for extinghished mode analyses. The boundary conditions which must be satisfied by the solutions ~f these equations are also given. Second, expressions for the forward saturation current density are derived and interpreted for use in converter diagnostics. Third. ex pressio.ns f?r the reverse saturation current density' and open-ClrcUlt voltage are derived and their use in con verter diagnostics is discussed. Fourth, the results of 1 A. S?avit an~ G:. N. Hatsopoulos, Proceedings of the Thermionic Converston Spec2al1st Conference Cleveland Ohio October 1964 pp. 206--213. ., " , 2 L. K. Hansen and C. Warner, Ref. 1, pp. 310-315. 3 C. Warner and L. K. Hansen, 23rd Annual Phys. Elec. Con ference, M. I. T., Cambridge, Mass., March 1963, pp. 400-405. 4 L. ~. Hans~n ~nd C. Warner, Proceedings of the Thermionic Converswn Spcczahst Conference Gatlinburg Tenn. October 1963 pp. 44-50. "" 5 C. Warner, Ref. 4, pp. 51-56. 6 D. R. Wilkins, Ref. 1, pp. 275-283. 7 J. M. Houston, Proc. 24th Annual Conference on Phys. Elec., M. I. T., Cambridge, Mass. (March 1964) pp. 211-223. 8 D. R. Wilkins and E. P. Gyftopoulos J. Apr..\. Phys. 37 3533 (1966). ,t·, 9 D. R. Wilkins and E. P. Gyftopoulos J. App\. Phys. 37 2892 (1966). " the above investigations are compared with experi mental data and agreement between theory and experi ment is established. The methods and equations of this paper can also be used to derive complete output-current characteristics of thermionic converters operating in the extinguished mode. Such characteristics, however, are not included herein. 2. TRANSPORT EQUATIONS AND BOUNDARY CONDITIONS 2.1. Transport Equations . Cesium plas~as in thermionic converters operating m the extmgUlshed mode have several characteristic properties which, when reflected in the plasma trans port differential equations of Ref. 8, lead to mathe matical simplifications. First, the electron and ion densities, ne and ni, respectively, are sufficiently low that charged-particle interactions may be neglected. Second, inelastic collisions are negligible because of the low electron densities and temperatures involved. Third, since the net electron current is small compared to the random electron current throughout most of the plasma, the plasma electron temperature may be assumed constant and equal to the emitter temperature. Thus, if the heavy-particle temperature gradients are neglected and the interelectrode plasma is assumed neutral, the transport equations reduce to a set of two equations of the form: (1) Je=Ji+J= -MeO[kTe(dn/dx)+enE], for Te= TE, (2) where Ja and Ta(a=e, i) are the uniform current density and temperature of species a, respectively; n is the charged particle density; MaO is the mobility of species a in the absence of charged-particle collisions·8 J is the output current density; and E is the electric field of the plasma. 2.2. Boundary Conditions The solutions of Eqs. (1)-(2) involve integration constants which may be evaluated through the use of boundary conditions. These boundary conditions are obtained by writing electron and ion current balances across the Debye sheaths at the plasma-electrode inter faces. The exact fonn of a particular balance depends upon the polarity of the sheath. For convenience, a shea th polari ty is called accelerating or retarding if the ~heath a~cele:a tes or retards an electron traveling m the dIrectlOn from the emitter to the collector respectively. ' ~oundary conditions for accelerating and retarding emItter and collector sheaths are given in Table 1. In this table, V E S and V c s are the emi tter and collector sheath voltage drops, respectively; T E and Teare the emi tter and collector temperatures, respectively; J r [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 93.180.53.211 On: Tue, 18 Feb 2014 05:42:5014 D. R. WILKINS AND E. P. GYFTOPOULOS TABLE I. Boundary conditions for the plasma transport equations. Accelerating emitter sheath Retarding emitter sheath J, =JE -J,(O) exp( -eVEs/kTE) J, =JE exp( -eVEs/kTE) -J,(O) J, =iE exp( -eVEs/kTE) -/,(0) J, =1 E -1,(0) exp( -eVEs/kTE) Accelerating collector sheath Retarding collector sheath J, =J,(d) -Je exp( -eVes/kTe) J, =J,(d) exp( -eVcs/kTE) -Je J, =l,(d) exp( -eVcs/kT,) Ji =l,(d) and I r are the plasma electron and ion random current densities, respectively; IE and IE are the electron and ion emission current densities from the emitter, re spectively; I c is the collector back emission; the nota tions H(O) and H(d) are used to denote any x-dependent quantity H(x) evaluated at the emitter edge (x=O) and at the collector edge (x=d) of the plasma, respec tively; and surface ionization at the collector is neg lected. The boundary conditions are not exact since they do not account for the non-Maxwellian, aniso tropic nature of the charged-particle distribution func tions in the interelectrode space. Although first-order corrections which account for these effects can be in cluded, the resulting relations are not sufficiently different to justify the added complexity. It should also be noted that no electron kinetic energy flux balances are included in Table 1. This is consistent with the con stancy of the electron temperature. The electron emission current densities from the emitter and collector are given by the relations: J E=ATE2 exp( -eCPE/kTE) and Jc=ATc2 exp(-ecpc/kTc), (3) where A = 120(A/cm 2. °K2), and CPE and CPc are the emitter and collector work functions, respectively. The ion-emission current density from the emitter is given by the approximate Saha-Langmuir equation: where pCs, mi, and Vi are the pressure, mass, and ionization potential of cesium, respectively. The ap proximation is valid for e(V i-CPE)>>kT E, which is generally true for thermionic converters. 2.3. Extinguished Mode Analyses Equations (1) and (2) and the boundary conditions of Table I provide the basis for the analysis of the extinguished mode. Such analyses proceed as follows: (a) sheath polarities are specified; (b) the charged particle density and electric field profiles, and the sheath voltage drops are determined from Eqs. (1) and (2) and the corresponding boundary conditions; (c) the output current vs output voltage relation is com pu ted; and (d) the operating conditions for which the specified sheath polarities prevail is established, i.e., the region of validity of step (a) is defined. For operating conditions outside this region of validity alternate sheath polarities are considered. In Sees. 3 and 4 the above procedure is used to derive expressions for several characteristic quantities asso ciated with extinguished mode output-current charac teris tics, namely I for, I rev, and V oc. 3. FORWARD SATURATION CURRENT DENSITY At low output voltages, an extinguished mode out put-current characteristic saturates at a forward satura tion current density Ifor as shown in Fig. 1. An expres sion for this limiting output-current density is derived below. ~hen the output voltage V«O, the collector sheath becomes accelerating, Ii ---t 0, and Ie = I = I for. Thus, Eqs. (1) and (2) and the boundary conditions (Table I) for an accelerating collector sheath yield I,(x) =h,,[i+Re'(1-x/d)J; (5) where v", is the average thermal speed of particles of species Q!. Note that Re' is closely related to the inter electrode spacing measured in electron-neutral mean free paths. For hard-sphere collisions, Re' =i[TE/(T E+ Ti)JnOlTeod, where no is the neutral cesium density and IT eO is the electron-neutral cross section. Equation (5) must be combined with the boundary conditions at the emitter edge of the plasma to yield an expression for I for' Two possibilities exist since the emitter sheath may be either accelerating or retarding. 3.1. hor for Accelerating Emitter Sheaths For accelerating emitter sheaths, Eq. (5) and the boundary conditions of Table I yield an expression for lIor which may be written in two convenient forms, namely: Ilor/ h= 20[ (1 +(2)LOJ, 0=/1!/2(1 + R.'); (6a) lIor= [Ir /(1 +Re')J[(1 +(2)t-OJ; (6b) where /1 is the ion-richness ratio given by the relation /1= (mi/me)!I E/ IE, (7) Jr=env e/4, and n is the charged-particle density in a neutral plasma in thermodynamic equilibrium with the emitter and is given by the reI a tion n= (pcs/kTE)!(27rmekTE/h2)l exp( -eVi/2kTE). (8) Equations (6) are valid, i.e., the emitter sheath is accelerating, provided the ion-richness /1 is greater than a critical value {JeT given by the relation {3er== (1+R.')/(2+Re'). (9) The meaning of Eq. (6a) is that the ratio Ilor/ I E [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 93.180.53.211 On: Tue, 18 Feb 2014 05:42:50THEORY OF THERMIONIC CONVERTERS 15 depends only upon the ion-richness ratio fJ and the number of electron-neutral mean-free paths R.' across the plasma. This conclusion has been reached inde pendently by Shavit and Hatsopoulos1 and Hansen and Warner.2 The form of Eq. (6a), however, is different from that derived by the previous authors due to different approximations regarding the plasma electron distribu tion function. For 0»1, Eq. (6a) takes the simpler form (6c) In other words, under this condition the forward satura tion current density is electron-emission-limited and depends strongly upon the emitter work function. On the other hand, for 0«1, Eq. (6b) becomes ffor"",f, 1(1 +R.'), for 0«1. (6d) This implies that hor is determined by the plasma properties and is independent of the emitter work function. Equations (6c) and (6d) , for {3?;{3cr, are useful in converter diagnostics. For example, when Eq. (6c) is applicable and TE is known, a measurement of hor yields f E and hence the emitter work function. When Eq. (6d) is applicable and R.' is known, then a measure ment of ffor yields fT . Since fT is extremely sensitive to the emitter temperature, this measurement provides a means of accura tely determining T E in devices in which the emitter is not accessible for temperature measurements. Also, when Eq. (6d) is applicable, a plot of experimental data on 11hur vs d should yield a straight line. If pCs is known, the intercept of this line at d=O yields 1/f., while the slope yields the electron mobility. 3.2. 110r for Retarding Emitter Sheaths For retarding emitter sheaths the combination of Eq. (5) with the corresponding boundary conditions yield ffor for fJ~fJcr' The results may again be written in two convenient forms, namely: fror/f E= [{3!/(1 + Re')][(l +Re')/(2+Re')]!, (lOa) Jfor= [Jr /(1 +Re')][(l + R/)/(2+R.')Jt =~crU,/(1+Re'). (lOb) The meaning of these equations is that the forward saturation current density for ~~~cr is determined by ion emission and plasma effects rather than by electron emission and plasma effects. The relative importance of these effects is brought forth by Eq. (lOa). For R.'»l, the factor (3"1 is the probability that an electron sur mounts the emitter sheath barrier which arises from insufficient ion emission, and the factor 1/(1+ Re') is the probability that an electron diffuses through the plasma to the collector. Equation (lOa) has been re ported previously by Warner and Hansen.3 Note also that Eq. (lOb) for fJ~fJcr may be used in converter diagnostics in the same manner as discussed previously in connection with Eq. (6d) for f3?;f3cr. 3.3. Implication of the ltor Results The forward saturation current density is the largest current density which can be achieved under conditions of extinguished mode operation. The upper limit of this density is given by Eq. (lOb). This upper limit depends only upon pCs, T E, and d and is independent of the emitter work function. Furthermore, for given practical emitter temperature and interelectrode spac ing, there is an optimum cesium pressure at which the upper limit of fror is a maximum. This maximum cannot be exceeded regardless of the choice of emitter work function or cesium·pressure or both. Consequently, the surface ionization scheme for electron space-charge neutralization is limited. 4. REVERSE SATURATION CURRENT-OPEN CIRCUIT VOLTAGE 4.1. Output Current Characteristics The reverse saturation current f •• v and the open circuit voltage Voc can he found from the output current characteristics for output voltages in the vicinity of Vo. and higher. For such output voltages, the currents through the converter satisfy the inequality f.jJi«Ji.eO!Ji.P, (11) and the collector sheath is, in general, retarding. Under these conditions, integration of Eqs. (1) and (2) yields I r(X) =f;[1 + R/(l-xld)], R/=eiJ id!4IJ.Pk(T E+Ti) ; Vp= (kTs/e) In(1+R/)i (12) (13) v cs= (kTE/e) In[(milme)!J;/ (Je+lc)]; (14) where R/ is a quantity analogous to R/ [Eq. (5)J and is closely related to the interelectrode spacing measured in terms of ion-neutral mean-free paths, and V p is the plasma voltage drop. Note that in the absence of collector back emission (J c=O) the collector sheath is retarding provided felfi«milm.)~. This condition is always satisfied in the range of output voltages under consideration. The output-current characteristics are derived by combining Eqs. (12)-(14) with the emitter sheath boundary conditions. Provided f e«f E, these character istics, for either accelerating or retarding emitter sheaths, are given by the relation: f =fr exp[ -e(V+cpc-cpp)lkTE]- (Ii+J C), (15) where V is the output voltage, and CPP is the chemical potential, measured relative to the Fermi level of the emitter,~of a neutral plasma in thermodynamic equilib- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 93.180.53.211 On: Tue, 18 Feb 2014 05:42:5016 D. R. WILKINS AND E. P. GYFTOPOULOS rium with the emitter. This potential is given by the relation CPP = Vi/2+ (kTE/2e) In[4(27rme/h2)!(kTE)!pcsJ. (16) Although Eq. (16) is valid regardless of the emitter sheath polarity, the ion current Ji depends on that polarity. Ji for Accelerating Emitter Sheaths For accelerating emitter sheaths, Eq. (12) and the boundary conditions (Table I) yield that the ion current is given by the relation J;= [Ir! (1 + R/)J[(1 + R/)/ (2+ R/)Ji, (17) where I/=enfJ;/4. Equation (17) is valid, namely the emitter sheath is accelerating, provided the ion rich ness ratio {J is greater than a critical value {Jor' given by the relation {Jcr' = (2+ R/)/ (1 + Ri'). (18) Equation (17) has been derived previously by Houston.7 J; for Retarding Emitter Sheaths For retarding emitter sheaths, namely {J'5:{Jcr, Ji is given by the relation J;= [I// (1 + R/)J[(l +712)t-71J; 71=!(1+R;'){J!. (19) Two limiting forms of Eq. (19) are of particular interest, namely; Ji"'-'Ie, for 71»1; (19a) Ji"'-'Ir/(1+R;'), for 71«1. (19b) 4.2. Reverse Saturation Current Density The reverse saturation current density Jrev is derived from Eq. (15) for V»O. Thus, (20) where J i is given either by Eq. (17) for {J"2{Jcr' or by Eq. (19) for {J'5:{Jcr'. For J C«J;, the reverse saturation current density is given directly by Eq. (17) or (19). The formal simi larity of Eqs. (17) and (19) to Eqs. (lOb) and (6b), re spectively, reflects the fact that, for J c«Ji, the for ward and reverse saturation current densities merely correspond to different particle species reaching the collector. Because of this similarity, Egs. (17) and (19) are useful in converter diagnostics in the same manner as Eqs. (lOb) and (6b), respectively. In particular, if JC«Ji, measurements of Jrev can be used to infer values of TE, CPE, and )1,io. In general, if Ji is known, either from measurements at very low To or from theory, Eq. (20) permits a determination of J c from a measurement of J rev' Thus, the collector work function CPc follows if T c is known. 4.3. Open Circuit Voltage The open-circuit voltage V oc follows directly from Eq. (15) for J=O. Thus, Voc=CPp+ (kTE/e) InU/jJrcvJ-cpc. (21) This result is only slightly different from that reported in Ref. 6. The open-circuit voltage is particularly useful in con verter diagnostics when coupled with measurements of Jrev' Specifically if TE and pCs (and hence CPP and J/) are known, measurements of V oc and J rev yield the collector work function. 5. COMPARISON OF THEORY AND EXPERIMENT 5.1. Comparison of Theoretical and Experimental lfor Results Warner and Hansen3 have reported experimental da ta on hor for the case of {J < {J cr= 1. By plotting their results, for fixed T E, on a 1/ J for VS d plot, and utilizing a theoretical expression similar to Eg. (lOa), they were able to infer a value for the electron-neutral cross sec tion of (J e~200 A2. Because of uncertainties in their estimate of the emitter work function, the value in ferred for (J eo was considered approximate. Although the procedure employed by Warner and Hansen is correct, it does not recognize an important feature of the theoretical expression for Jfor for {J<{Jcr; namely that Jffr is independent of the emitter work function. This independence is brought forth by Eg. (lOb), and permits a determination of (Jeo which is not subject to errors in the estimated emitter work function. 12 + PCs ; 0.9Torr ~ 10 + 0 8 0 '" ~ ~6 -'" x "') TE(OK) 4 /8 0 1293 0 1422 2 ;{~ + 1550 x 1682 0 20 40 60 80 100 d (mils) FlG. 2. Comparison of theoretical J,/fror vs d [solid line-Eq. (lOb)] with experimental data from Ref. 3. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 93.180.53.211 On: Tue, 18 Feb 2014 05:42:50THEORY OF THERMIONIC CONVERTERS 17 1.0 0.8 0 0 0 0 0 0 0 0 0 '" 0.6 8% 0 OJ '::-- .!? TE (OK) Pes (Torr) /3 OJ 0.4 [J 1700 0.68 1.72 x 1730 0.46 3.50 + 1900 0.96 21.0 0.5 1.0 1.5 .2.0 2.5 e = (3 VZ/2( I + R~ ) FIG. 3. Plot of theoretical hoJfE vsO [solid line-Eq. (6a)]. Superimposed also are experimental data reported in Ref. 1. The theory is not strictly applicable to these data. Figure 2 shows the Warner-Hansen data3 on a Jr/hor plot as suggested by Eq. (lOb). The data for the several emitter temperatures indeed fall reasonably close to a single straight line when plotted in this manner. From the slope of this line a value of U eo= 180± 100 A2 is inferred if the average background gas temperature is assumed to be 1200oK. This value confirms the Warner Hansen estimate.3 The present value (u eo= 180± 100 N) should also be compared with: (a) the values ueo~40- 1000 A2 obtained from various theoretical and experi mental studies and tabulated by EoustonlO; (b) the value U eO = 400 A2 suggested by H oustonlO as an appro priate average of existing data; and (c) the values U eo~260-1S00 A2 inferred from ignited-mode measuremen ts. 9 The theoretical Jfor/J E VS d rdation for (3'2,(3or [Eq. (6a)] is shown in Fig. 3. No appropriate experi mental data for truly collision dominated operation is available for comparison with this result. Shavit and Hatsopoulos,1 however, have reported experimental data on approximate values of Jfor for (3'2,(3or and Re''.5. 5. For lack of more appropriate measurements these data are plotted in Fig. 3 assuming that ueo=400 N. The agreement between theory and experiment for (3= 1. 72, 3.5, and 21 is surprisingly good considering the small number of mean-free paths across the plasma, and the fact that the data do not represent truly Jfor. The agreement for (3= 108 is less favorable. 5.2. Comparison of Theoretical and Experimental Jrev Results Houston7 has reported experimental data for Jrev ob tained under operating conditions for which the emission is ion-rich ((3>(3cr'~1) and back emission from the 10 J. M. Houston, Ref. 1, pp. 300-309, > .. ~ Pes (Torr) Te (OK) I 0 0.08 522 n + 0.48 572 m 0 1.80 623 d = 1.04 mm 10.5 L.._--L __ L...._-L __ -'--_......l.._..........J 0.45 0.55 0.65 0.75 FIG. 4. Comparison of theoretical Jrov vs liTE [solid lines Eq. (17)J with experimental data from Ref. 5. collector is negligible. He compared his results with Eq. (17) and found that the best agreement between theory and experiment was obtained using the ion mobility llio=O.32X 1019/no cm2/V ·sec. Based on this value of the ion mobility, Houston's comparison of theory and experiment is shown in Fig. 4. In this figure, the solid curves are the theoretical predictions of Eq. (17) and the datum points are experimental. The agreement is indeed excellent over a 8000K range in emitter temperature and for an order of magnitude variation in cesium pressure. Houston7 has compared the inferred value for IJ.io with independent measure ments and found satisfactory agreement. 3.0r------~-----~----~ 2.5 2.0 <3 :: 1.5 >~ 1.0 0.5 m --;:----.,~--&--u--~~ n~ o 0 o -o 0 I 0 TE" 1655 oK. 575 OK :s Jc !S 625 oK. d " I mm noTE" 1863 oK. T, '" 5750 K. d., mtn m A T( = 1912 OK, Tc = 6750 K. d. I mill OL---___ ~ _____ ~ _____ ~ 10'2 10" 1.0 10 pc. (Tor,.) FIG. 5. Comparison of theoretical Voc vs Pc, [solid lines-Eqs. (21) and (17)J with experimental data from Ref. 9. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 93.180.53.211 On: Tue, 18 Feb 2014 05:42:5018 D. R. WILKINS AND E. P. GYFTOPOULOS 5.3. Comparison of Theoretical and Experimental Voe Results Figure 5 shows plots of open-circuit voltage vs cesium pressure for several emitter temperatures. The data were obtained by Reicheltll and correspond to operating conditions for which the emission is ion-rich and the coll~ctor back emission is negligible. The solid curves are the theoretical predictions of Eqs. (21) and (17) for the same ion mobility as above. Collector work func tions for the cesium-covered nickel collector are deter mined by scaling the work-function data of Rump et al.12 into the cesium pressure region of interest, as described in Ref. 6. The theoretical curves are bounded on the left at the cesiLlm pressure for which R/ = 1.0. The agreement between theoretical and experimental results ~ "0 ~ 1.5 1.0 0.5 o pc. = 0.48 mm Hg., Tc = 573 oK, d = 1.04 mm c pc. = 0.45mm Hg., Tc = 575 oK, d = 1.0 mm 1600 1800 2000 TE (OK) FIG. 6. Comparison of the theoretical Voe vs TE [solid line-Eqs. (21) and (17)J with experimental data from Refs. 9 and 5. 11 W. Reichelt, Los Alamos Scientific Laboratory (private com munication August 1964). 12 B. S. Rump, J. F. Bryant, and B. L. Gehman, Ref. 3, pp. 232-238. is good. Maximum discrepancies are approximately equal to five percent. Figure 6 shows a plot of open-circuit voltage vs emitter temperature for PCB"-'O.45 Torr. The low temperature data were obtained by Reicheltll and the high-temperature data by Houston.7 In each case the emission was ion-rich, the collector temperature was sufficiently low that back emission was negligible, and cf>:;~1.81 eV. The solid line in Fig. 6 is the theoretical prediction of Eqs. (21) and (17). The agreement of the theoretical curve with both sets of data is excellent. Errors are less than several percent over a 7000K range in emitter temperature. 6. CONCLUSIONS Theoretical expressions for the forward and reverse saturation current densities and open-circuit voltage of cesium thermionic converters operating in the collisional extinguished mode are derived for the first time from a single set of transport equations and boundary condi tions. The theoretical results are in good agreement with experimental measurements. The forward saturation current density may be elec tron emission limited, ion emission limited, or plasma limited, depending upon the operating conditions. Furthermore, this quantity cannot exceed an absolute upper limit which depends only upon the emitter tem perature and the interelectrode spacing, and is inde pendent of the emitter work function and cesium pressure. The forward and reverse saturation current densities and open-circuit voltage are useful in diagnostics. Under appropriately selected operating conditions, measure ments of these quantities may be used to infer values of the emitter temperature, emitter work function, col lector work function, electron mobility, and ion mobility. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 93.180.53.211 On: Tue, 18 Feb 2014 05:42:50
1.1705099.pdf	Calculation of Exchange Second Virial Coefficient of a Hard‐Sphere Gas by Path Integrals Elliott H. Lieb Citation: Journal of Mathematical Physics 8, 43 (1967); doi: 10.1063/1.1705099 View online: http://dx.doi.org/10.1063/1.1705099 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/8/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantum‐Mechanical Second Virial Coefficient of a Hard‐Sphere Gas at High Temperatures J. Chem. Phys. 51, 4675 (1969); 10.1063/1.1671845 Erratum: ``Exchange and Direct Second Virial Coefficients for Hard Spheres'' J. Chem. Phys. 46, 1224 (1967); 10.1063/1.1840810 Exchange and Direct Second Virial Coefficients for Hard Spheres J. Chem. Phys. 45, 499 (1966); 10.1063/1.1727597 On the Percus—Yevick Virial Coefficients for a Hard‐Sphere Gas J. Chem. Phys. 38, 1262 (1963); 10.1063/1.1733842 Consistency and the Fifth Virial Coefficient for a Hard‐Sphere Gas J. Chem. Phys. 36, 1680 (1962); 10.1063/1.1732797 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:43JOURNAL OF MATHEMATICAL PHYSICS VOLUME 8. NUMBER 1 JANUARY 1967 Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals* ELLIOTT H. LIEB Department of Physics, Northeastern University, Boston, Massachusetts (Received 21 February 1966) By direct examination of the path (Wiener)-integral representation of the diffusion Green's function in th~ presence of an opaque sphere, we are able to obtain upper and lower bounds for that Green's f?-nctlOn. These bounds are asymptotically correct for short-time, even in the shadow region. Essen tIally, we have succeeded in showing that diffusion probabilities for short-time intervals are concen trated ma~nly on the optical path. By integrating the Green's function, we obtain upper-and lower bound estlInates for the exchange part of the second virial coefficient of a hard-sphere gas. We can show that, for high temperature, it is asymptotically very small compared to the corresponding quan tity for an ideal gas, viz., B ... h/Boexch = exp {-h3(a/ A)2 + O[(a/ A)W]}, where A is the thermal wavelength and a is the hard-sphere radius. While it was known before that B •• ch/Boe.ch is exponentially small for high temperatures, this is the first time that a precise asymptotic formula is both proposed and proved to be correct. I. INTRODUCTION FOR a gas of particles that interact via a two body potential, the calculation of the second virial coefficient! involves an analysis of only a two body problem. This simplification holds for quantum as well as for classical mechanics, but there the similarity between the two kinds of mechanics ends. Classically, the second virial coefficient depends neither on particle mass, m, nor on statistics and, for a one-component gas, is given by the simple configuration integral: Bcl(T) =!N J drl1 -exp [-,8v(r)]), (1.1) where vCr) is the pair potential, N is Avogadro's number, and,8 = (kT)-l. Quantum-mechanically, no such simple formula as (1.1) exists, for the calculation of B(T) requires either a detailed knowledge of the solutions of the two-particle SchrOdinger equation at all energies, or, alternatively, a solution of the corresponding diffusion problem. Thus, while the problem of cal culating the second virial coefficient may not be as profound as the original many-body problem from which it arose, it does require the answer to interesting questions about the classical analysis * This paper was supported by the U. S. Air Force Office of Scientific Research under Grant No. 508-66 at Yeshiva University, New York. 1 The nth-virial coefficient is the temperature-dependent coefficient of 11-.. +1 in the series Pv = RT[1 + B(T)v-1 + C(T)v-2 + ... ]. Here, P is the pressure, T is the temperature, R is the gas constant, and v is the volume per mole of gas. In terms of N (Avogadro's number), II = Np-l and R = Nk, where p is the particle number density and k is Boltzmann's constant. 43 of the three-dimensional diffusion equation. To be come familiar with the problem is to realize how difficult it is to calculate quantum corrections to (1.1).2 The true physicist will doubtless inquire whether quantum corrections to (1.1) are in fact significant, and the answer is that for helium they are quite important. Even for temperatures as high as 60oK, the quantum corrections in helium are about a third of the total.3 For a hard-sphere gas, the quantum corrections do not drop to a tenth of the total until a temperature of about 12000K is reached.4 Since experimental values of the second virial coeffi cient are used in attempting to determine the effec tive inter-atomic helium potential, these quantum corrections are certainly worthy of consideration. There is alsoa a pronounced difference between the second virial coefficient of Re3 and Re4, es pecially below 60°K. Assuming (as is always done) that the interaction potential is the same for the two isotopes, the difference could conceivably come from three sources: (a) the atomic mass difference; (b) the difference in nuclear spin which affects the statistical weights; and (c) the difference between Fermi-Dirac and Bose-Einstein statistics. For an ideal (noninteracting) quantum gas (b) and (c) are everything [see Eq. (1.11) below], and one might be tempted to conclude; that, for helium too, the isotopic mass difference was relatively unimportant. Numerical calculations have, however, indicated the 2 Hug~ E. J?eWitt, J. Math. Phys. 3, 1003 (1962). 8 J. Kilpatnck, W. Keller, E. Hammel, and N. Metropolis, Phys. Rev. 94, 1103 (1954); J. Kilpatrick, W. Keller, and E. Hammel, ~"bid. 97, 9 (1955). 4 F. Mohling, Phys. Fluids 6, 1097 (1963). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:4344 ELLIOTT H. LIEB reverse. Above about 4 oK, almost all of the difference in the two second virial coefficients is a mass effect.3 This difference is about 10% at 600K and drops only to the order of 5% at room temperature. In other words, on the one hand the mass effect is unusually large for helium, while on the other hand the effects due to statistics and spin decrease very rapidly with increasing temperature. For an ideal gas, these latter effects decrease as T-!, but for helium the decrease is far more rapid. Under the assumption that the repulsive part of the helium interaction potential can be effectively replaced by a hard core, it has been proved5 that the statistical and spin effects decrease at least exponentially fast with increasing temperature (for high temperatures). The suppression of exchange effects is so rapid that 200K may be considered to be a high temperature for which asymptotic formulas are reasonably valid. I t is the purpose of this paper to prove that the exponential law for the hard-sphere gas mentioned above is more than just an upper bound, that it is in fact correct. The true coefficient appearing in the law [cf., (1.13) below] is, however, different from that of the bound given in Ref. 5, although the correct value was stated there, without proof, in a footnote. To the casual reader, the problem must seem almost trivial. In the first place, we have eschewed calculating the true equation of state, and have, instead, contented ourselves with examining only the second virial coefficient-a simple matter of a two-body problem. Secondly, we are examining only the effects of spin and statistics. Thirdly, we are confining ourselves to high temperatures. That there is no simple perturbation theory for this problem must appear strange. But it is a fact that, in many respects, the problem is similar to the classical problem of diffraction of waves (of short wavelength) around a sphere into the dark zone, a problem which has exercised mathematicians for years. The mathematical statement of the problem is as follows: The quantum-mechanical second virial coef ficient may be written as the sum of a direct and 5 S. Larsen, J. Kilpatrick, E. Lieb, and H. Jordan, Phys. Rev. 140, A129 (1965). While it was realized in Ref. 4 that exchange effects are small at high temperatures, no proof of this assertion nor statement of its exponential character were offered. For further results on the hard sphere problem, see the following papers: M. Boyd, S. Larsen, and J. Kilpatrick, J. Chem. Phys. 45, 499 (1966); S. Larsen, K. Witte, and J. Kil patrick, J. Chem. Phys. (to be published). Recently, J. B. Keller and R. A. Handelsman, Phys. Rev. 148,94 (1966), have calculated the first few terms in a high-temperature power series for the direct second virial coefficient of a hard-sphere gas. an exchange part, (1.2) where Bdiroct = tN J dr [1 -2iA3G(r, r; .8)], (1.3) Bexch = Tv'2 A3N(2S + 1)-1 J drG(r, -r;.8), (1.4) and (1.5) In (1.4) the -sign is for bosons and the + sign is for fermions. S is the total spin of the atom (the nuclear spin alone in the case of helium), it is to be noted that the spin enters only into Boxch' Thus, (b) and (c) mentioned above go together. The function G(r, r'; t) is the diffusion Green's function (also known as the Bloch function), and it satisfies [-DV'~ +v(r) + ajat]G(r,r', t) = 0, (for t > 0) (1.6) with the initial condition lim GCr, r'; t) = oCr -r'). (1.7) In addition, G satisfies appropriate boundary condi tions in r, such as vanishing on the walls of a box. In our case, we are interested in the limit of an infinite volume which means that G satisfies (1.6) for all r but vanishes when r ~ 00. It is to be noted that boundary conditions need only be defined with respect to r. Despite this fact, and despite the fact that (1.6) refers really only to r, G automatically turns out to be a symmetric function of rand r' for all t. Equation (1.6) describes diffusion in a potential vCr), with r' the source point, t the elapsed time, and D the diffusion constant. For quantum-mech anical purposes, t is interpreted as .8, v is the inter particle potential, and D is related to the mass of a single atom by (1.8) Thus, (I.9) In the case of no interaction (v = 0), G is given by Go(r, r', t) = (7ratri exp [-(r -r,)2 j at], (1.10) and when this is inserted into (1.3) and (1.4), we obtain the result: (1. 11 a) (l.11b) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:43VIRIAL COEFFICIENT OF HARD-SPHERE GAS 45 For a hard-sphere potential, vCr) = co, for r:::; a, (1.12) = 0, for r> a, Eq. (1.Ub) is a very misleading approximation to Bexob for high temperatures. Weare to prove that, for small t or A, (1.13) The proof consists in obtaining upper and lower bounds for G(r, r'j t) by means of Wiener, or path integrals. These bounds are valid for all tempera tures, and we could, in fact, give a more detailed estimate than is indicated in (1.13). The bounds are, however, complicated functions of t, and it seems neither necessary nor desirable to go beyond the asymptotic formula in (1.13). Before giving the proof, it is worthwhile mention ing an alternative formulation of the problem which, at first sight, seems to offer an immediate solution. For a particle in a box, we can write '" G(r, r' j t) = L exp ( -ten) Y;n(r) y;~(r'), (1.14) n-l where en is the nth energy level and y;" is the cor responding normalized eigenfunction. When (1.14) is inserted into (1.3) and (1.4), it is seen that knowledge of the energy levels alone is required. When the box is very large compared to the range of the potential, the virial coefficient can be ex pressed in terms of the bound-state energy levels (if any) and the scattering phase shifts of the potential, viz. Bdi rect = -v'2 N A 3 L (2l + l)B z, (1.15a) all I Bexcb = B~xcb =r= (2S + 1)-1 v'2 N A3 X {L -L}(2l + l)Bz, (1. 15b) Z even lodd where + (A2/rr2) 1'" e-lI.'k'/2>-TJI(k)k dk. (1.16) In (1.16) the sum is over negative energy levels (if any), while the integral contains the phase shift TJ,-all for the appropriate angular momentum, l. For the case of no bound state, the above formula for the second virial coefficient in terms of the phase shifts was apparently first stated by Gropper and by Beth and Uhlenbeck,6 and a derivation of it can be found in Ref. 3. For the hard-sphere potential, there are no bound states, and it would appear that (1.16) and (1.15) should give the answer simply, especially as the phase shifts are given by the elementary formula (1.17) For small A, however, we see that large values of k are important in (1.16). For very large k, the sum on l in (1.15b) may be performed with the aid of Watson's transformation, and it is similar to the problem of diffraction around a sphere at short wavelength.7 Apart from certain technical conver gence difficulties connected with the fact that we are really interested in the diffracted field on a diameter (that is to say a caustic), there is another more important problem.s This problem is that there may also be contributions to (1.16) from small k, a region where Watson's transformation is not of great use. Finite k contributions would, from (1.16), be expected to give a power series in A for small A. But it is a fact that there is a remarkable cancellation between even and odd l in (1.15b) so that every term in this power series vanishes. The final result, as shown in (1.13), is a function that vanishes faster than any power as A ~ O. If the potential were finite, instead of a hard core, this power series would not vanish. Thus, in summary, (1.16) and (1.15b) is a difficult starting point for hard spheres, despite the simplicity of the phase shifts and the existence of Watson's transformation. Our approach is to go back to (1.4) and, as we mentioned before, to estimate GCr, -rj t) directly through its expression in terms of a Wiener integral. Such integrals play an important theoretical role in analysis but, unless the integrand is Gaussian, it is difficult to obtain numerical answers from them. There have, of course, been rare exceptions such as Feynman's treatment of the Polaron problem.9 Nevertheless, the analysis presented here is one of the very few cases, if not the only one, in which both an upper and a lower bound to a function is obtained with path integrals. The path integral approach also has the great virtue of transparency because it brings out the close connection between the diffusion equation, (1.6), and a random walk 6 L. Gropper, Phys. Rev. 51, 1108 (1937); E. Beth and G. Uhlenbeck, Physica 4, 915 (1937); see also G. Uhlenbeck and E. Beth, ibid. 3, 729 (1936). 7 B. Levy and J. Keller, Commun. Pure App!. Math .. 12, 159 (1959), where the relevent asymptotic formulas are gIven on p. 201. See also J. Keller, J. Opt. Soc .. ~. 52,.116 (1962). 8 I am indebted to Dr. S. Larsen for pomtmg thiS out to me. 9 R. Feynrnan, Phys. Rev. 97, 660 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:4346 ELLIOTT H. LIEB problem. For these reasons, we believe the sequel might also possess an intrinsic mathematical value. II. LOWER BOUND BY PATH INTEGRALS The solution to (1.6) and (1.7) is easily shown to be unique and to satisfy the relation G(r, r'; t) = J dz G(r, z; tl)G(Z, r'; t2) (2.1) for any positive tl and t2 such that tl + t2 = t. If the time interval t is divided into n + 1 intervals of duration ll, so that t = (n + 1)1l, then, from (2.1), G(r, r'; t) = ~~ J dZ G(r, ZI; Il)G(zl' Z2; ll) X G(Zn_l, zn; Il)G(zn' r', ll) = lim J dZ Go(r, ZI; ll)e-~'(z,) n"'''' conditional Wiener measures on 0 as defined by Ginibre.ll The crucial point to note is that Pr,r':' is concentrated on the paths that are bounded and continuous on [0, t]. Denote integration of P r, r' : , integrablefunctionals, F, on 0, by f F(w)Pr,r':' (dw), where w denotes a generic path in O. Let O,(w) = {1 if Iw(r) I > a foraH 0 < r ~ t, o otherwise. Then 0, is Pr,r':' integrable and This result is essentially given, with different nota tion, by Ray.12 Since Pr,r':' is concentrated on the bounded, continuous paths on [0, t], it follows that P r, r' : ,-almost everywhere on o. The function e is defined in Eq. (2.3) below. Hence, applying the (2.2) dominated convergence theorem we have where dZ = dZI dZ2 ••• dzn• The heuristic justification for (2.2) is that, if a = 4D were zero, then G(r, r'; ll) = li(r -r') exp [-llv(r)], whereas if v = 0 then G = Go, which is very nearly c5(r -r') for small ll. The combination Go(r, r'; ll) exp [-llv(r)] is, hopefully, a good approximation to G [at least as far as the integral in (2.2) is concerned] for very small ll. Formally, this combination satisfies (1.6) to leading order in II for those values of rand r' such that G(r, r'; ll) significantly contributes to (2.2). The fact that (2.2) is correct for a large class of bounded potentials has been known for some time. Weare interested, however, in the hard-core po tential [see Eq. (2.3) below] for which a special proof is apparently required. We remark that Ginibre has previously used (2.2) for the hard-core case , but without giving an explicit proof.lO I am indebted to Professor D. Babbitt for the proof in the hard-core case, which is outlined as the following. Take D = 1-for convenience, and let ~ be the se~ of functions (paths) from [0, co) into (Ra, where (Ra is the one-point compactification of (R3, the three-dimensional Euclidean space. Let IPr,r':'; r, r' E (R3, t > O} denote the family of 10 J. Ginibre, J. Math. Phys. 6, 1432 (1965). See especially Eqs. (A1.6)-(Al.l0). !~r;! J {n e[ w(n ~ 1) ]}Pr,r,jdw) = J O,(w)Pr,r,:,(dw). By definition of P r,r':" the left side of this equation is identical to the right side of (2.2) for the hard core case [ef. Eq. (2.3) below]. Having established (2.2), we use it as the rigorous starting point for our analysis. The limit n -+ co in (2.2) defines a conditional Wiener integral or path integral (conditional because both ends, rand r', are fixed). The n-fold integral in (2.2) bears to the path integral essentially the same relationship as a finite sum bears to the ordinary Riemann integral. Brushla has remarked that "it is usually impossible to do this" (evaluate the path integral) "by the direct method of finding an ex plicit formula for the finite dimensional integral and then passing to the limit of a continuous integral". Contrary to this dictum, we find, in fact, upper and lower bounds to the finite integral in (2.2) and then pass to the limit n -+ co. In this way, we obtain upper and lower bounds to G(r, r'; t). Weare interested in the case that v is a hard core, (1.12), and hence the factor exp [(-ll)v(z)] in (2.2) is equal to the simpler expression 11 r Ginibre, J. Math. Phys. 6, 238 (1965); see the Ap pendlX. 12 D. Ray, Trans. Am. Math. Soc. 77, 299 (1954). 13 S. Brush, Rev. Mod. Phys. 33, 79 (1961). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:43VI RIAL COEFFICIENT OF HARD-SPHERE GAS 47 FIG. 1. Important quanti ties for calculating the path integral [cf. Eq. (2.9) et seq.]. The opaque sphere having radius a is shown centered at the origin, O. A slightly larger, concentric sphere of radius b is also shown. The vectors rand r' r are the observation and source points, respectively, and the curve from r to r' via rl and r .. is the shortest path from r to r' lying entirely outside the larger sphere. The straight line rl - r is divided into (/ + 1) equal parts by the vectors Pl , ... , PI; the arc (J from rl to r .. is divided into (m -1) equal arcs by the vectors r2,... rm_l; and the straight line r' -rm is divided to (n + 1) equal parts by the vectors Pl' , ... , p,.'. O(Z) = 1, for z > a = 0, for z ~ a (2.3) for all A. Hence, the integrand in (2.2) is over a simple product of Go functions, but the integration range for each Zi is restricted to z > a. Such an integral is impossible to calculate. Since the integrand is positive, however, it is easy to obtain a lower bound to (2.2) by restricting the integration range still further, in such a manner that the restricted integral can be calculated exactly. To do this, we must define certain geometric quantities as shown in Fig. 1. The plane of Fig. 1 is the r, r' plane, and 0 is the center of the sphere of radius a. A larger, con centric sphere of radius b > a is shown, and it is assumed that b < minimum (r, r'). (2.4) The two straight lines, (r, rl) and (rm, r'), together with the circular arc (rl' roo) delineate the path which would be followed by a piece of string drawn taut between rand r'. Thus, rl• (r -rl) = 0 and r .. · (r' -rm) = O. The angles rp, 0, and q/ are the angles between rand rl, rl and r .. , and r .. and r', respectively, whence (2.5) is the angle between rand r'. Note that the angle o may be zero and that the shortest path from r to r' may consist of only one straight line that does not touch the sphere of radius b. In that case rl and rn are not defined, but the subsequent analysis remains valid with trivial modifications. In any event, r' Sb = r sin rp + r' sin rp' + bO (2.6) is the distance from r to r' along the shortest path lying outside of a sphere of radius b. An intuitive discussion of (2.2) is useful at this point in order to motivate the subsequent analysis. This and the following paragraph are entirely heu ristic and are not part of our proof. It will be recalled that we are interested in G(r, r'j t) for small t. In this regime, the Go factors in (2.2) give a large weight to that " path" (or sequences of points Zl, '" , zn) from r to r' which is of shortest length. That path is, moreover, traversed with constant speed (Le., IZi+1 -zil/ A = const) and is, in fact, the path of classical geometrical optics. Alterna tively, we may say that a Brownian particle, which is observed to go from r to r' in a short time, most likely went by way of the Newtonian, non-Brownian, trajectory. As the time increases, the optimum path ceases to have such a preponderant weight and other paths contribute more and more to (2.2). For the case of no interaction, however, we see from (1.10) that G is always proportional to the maximum of the integrand, namely exp [-S2/ at], where S is the distance from r to r'. When v ¢ 0, this simple relationship will not hold for all time, but for short time it is clear that the "optical" path is strongly preferred if v is finite. Thus, for finite v, the factors exp [-AV] in (2.2) contribute approxi mately the average potential along the optical path and G(r, r' j t) """" (1I'at)-1 exp [-S2/at] X exp [ -t f vCr + p.(r' -r) dp.} (2.7) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:4348 ELLIOTT H. LIEB For the hard-core case, (2.7) is patently nonsense. Instead, the fictitious Brownian particle traverses the shortest allowed path from r to r' with constant speed and we are thus led to the conjecture G(r, r'i t) 1"0.1 (1rat)-J exp [-S!/atJ (2.8) for small t and for rand r' > a. The reason for previously introducing the slightly larger fictitious sphere of radius b is that a single path, even the optimum one, cannot by itself contribute to the integral in (2.2). The path must also be associated with a nonvanishing measure. In other words, the path must be at the center of a tube which in turn lies wholly in the allowed region. The path which just skims the surface of the sphere of radius a does not have this property, but a path of slightly greater length, lying along the larger sphere, does. We return now to our proof. To find a lower bound we now, divide the line (r, rl) into l + 1 equal parts, designated by the vectors PI, ... , PI' Likewise, divide (rm, r') into n + 1 equal parts, designated by pL ... , p~. The arc (rl, rm) is to be divided into m -1 equal arcs, of angle 0 = 8/(m -1), and designated by r2, ••• , rm-l-We define where CI = (,ra LlI)-w+Il(1ra Llm)-!(m-l)(1ra Ll,.)-l<n+!), C 2 = exp { -~; [r sin cp + r' sin cp' + 2b(m -1) 1 -: cos oJ} sm 0 ' (2.13) {II F,(X, Y, X') = exp -(aLlI)-l ~ Ix; -x;_d2 with + Ixd2 + Iy, -xzl2 ] m -(aLlm)-1 2: Iy; -y;_,12 i=2 -(aLln)-{ ~ Ix~ -x~_,12 + Ix~12 + Iy,. -XwJ} , F2(y) = exp {-~ f y; .u;} , aLlm ;_1 (2.14) (2.15) S'; = r sin cp + r' sin cp' + b(m -1) sin 0, so that (2.9) u; = 2r; -r;_1 -rHI, for j = 2, ... , m -1, Sb = lim S';. Associated with these three divisions, we define the time intervals Lll = tr sin cp/(l + I)S;;', Ll,. = tr' sin cp' /(n + I)S;;', Llm = tb sin 0/ S;;" (2.10) whence (l + I)LlI + (m -I)Llm + (n + I)Ll" = t. Furthermore, in (2.2) let there be l + m + n variables of integration and we take the limit l, m, n --t 00. We make the following changes from the z. variables to Xi, Yi, and x~: Zi = Pi + Xi (i = 1, ... , l), (i = 1, ... ,m), (2.11) Zi+l+m = p~ + x~ (i = 1, ... ,n). We also use the symbol Glm .. to designate the integral in (2.2) before taking the limit on l, m, andn. Ul = rl -r2 + (rl -r)b sin o/r sin cp, (2.16) Um = rm - rm-l + (rm -r')b sin o/r' sin cp'. We come now to the important point for which Eqs. (2.9)-(2.16) were preparations. From (2.11), it is clear that, by restricting the integration vari ables Xi, Yi, and x~ to the regions Ixd < c, ly,l < c, and Ix~1 < c, (2.17) where c = b -a, we can, on the one hand, satisfy the hard-sphere condition (2.3) and, on the other hand, obtain a lower bound for Glm". We also note that lu.1 = 2b(1 -cos 0), for i = 2, ... , m -1 = b(1 -cos 0), for i = 1 or m. (2.18) Thus, in the region, (2.17), we can replace the factor F 2 (Y) by the bound F2(y) ;::: exp {-2(aLlm)-1 t: c IU;I} = exp {-4c(m -1)(1 -cos o)S~/at sin o} (2.19) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:43VIRIAL COEFFICIENT OF HARD-SPHERE GAS 49 We also note that lim C2 = exp {-S~/at}. (2.20) We must now calculate the quantity (which is in dependent of rand r') and we note that, in the limit l, m, n -? co, this is the Wiener integral for a well-known Green's func tion. Namely, consider the solution to (1.6) and (1. 7) with zero potential but with rand r' in the interior of a sphere of radius c and with G = 0 boundary conditions on the surface of the sphere. If we denote this Green's function by Gc(r, r'; t) then, in the limit l, m, n -? co, C4 = Gc(O, 0; t). (2.21) To compute Gc, it is convenient to use the ex pansion (1.14). Each tfn(r) is a spherical harmonic times a spherical Bessel function but, since we are interested only in the point r = r' = 0, only S wave (spherically symmetric) solutions will be rel evant. For S waves, the normalized radial func tions are simply (27rc)-! sin kr/r, the energies are e(k) = iae, and k = n7r/c with n = 1, 2, 3, Thus, (2.22) (2.23) Our lower bound for G(r, r'; t) is the product of C2, C3, and C4, each of which depends on rand r' and/or the radius b (or c = b -a): G(r, r'; t) 7r {S~ + 2Sbc8 (a!7r)2t} >-exp- --. 2c3 at 2c (2.24) The inequality (2.24) is generally valid, even if the geodesic from r to r' around the sphere of radius b is a straight line. In that case the term 2Sbc8/at is to be omitted. The next step is to determine c so that the right hand side of (2.24) is maximized. This is a tedious problem since the dependence of Sb on c is com plicated. Furthermore, b must always be less than rand r'. To calculate B.x•h, however, we are in terested in having r = -r' and, from (1.4), it is clear that r /"V a is the important region to consider in the integral. For our purpose-the proof of (1.13)-it is sufficient, as well as legitimate, to take c = r -a. The distance Sb is then simply 7rr, while 8 is simply 7r for all r > a. Thus, ~~XCh = 8 J G(r, -r; 2A2/7ra) dr excb (2.25) where (2.26) The second inequality in (2.25) is obtained by noting that l ~ a2 , and by changing variables to p = (27r2) 1/3 A -4/3a1/3(r -a). The inequality (2.25) is plainly of the form stated in (1.13). To make it more definite, however, we can obtain a lower bound to the integral in (2.25) in the following way: Replace the integration region by (0, 1) instead of (0, co); in this region, the terms p2 and p in the exponent may be replaced by unity. Weare thus left with an integral of the form n dpp -3. exp (-t7r2np·2) = (7r2n)-1 exp (-h2n). Collecting the various factors, we obtain Bexch { 7r3 (a)2 37r2 ( a)! -0 -> exp --- - - 2V;- Bexch 2 A 2 A (2.27) as our final lower bound for Bexcb' m. UPPER BOUND BY PATH INTEGRALS Weare interested in computing the path integral, (2.2), when the factors exp [-Av(z)] are omitted, but when the integration ranges are restricted to Iz;1 > a for all i. The lower bound to (2.2) was obtained in Sec. II by restricting the integration range still further, namely, to a tube lying just outside the sphere. At first sight it would seem that the opposite procedure-integrating over too great This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:4350 ELLIOTT H. LIEB a region-should yield a suitable upper bound. Indeed, when rand r' are in each other's line of sight (i.e., when the straight line between the two points does not interest the sphere), then the simple expedient of integrating over all space yields an upper bound which is at once useful and accurate for small time (high temperature), viz: G(r, r'; t) < Go(r, r'; t). (3.1) While (3.1) is true for all rand r', it is quite mis leading when the two points are in each other's shadow. A more sensitive extension of the integra tion range is required; but, unfortunately, allowing the paths to penetrate the sphere only slightly does not render the integral any more tractable than the original. In order to make the integration fea sible, it appears to be necessary to extend the integra tions to all space; but then the upper bound so obtained, (3.1), is virtually useless. Our resolution of the dilemma is to integrate over all space, but at the same time to include an additional weight factor in the integrand of (2.2) so that paths which penetrate the sphere are ef fectively suppressed. As in Sec. II, we consider the "taut string" shown in Fig. 1, except that this time we take c = 0 (i.e., radius b = radius a). Otherwise, every thing is the same as given in Eqs. (2.10)-(2.16). The first step in obtaining an upper bound is to integrate over the variables X and X' (alternatively, z. for i=l, ... , land i=l+m+l, ... , l+m+n) over all space. We then pass to the limit land n -(Xl and obtain G(r, r'; t) < lim Gm(r, r'i t), (3.2) m-H. where with D - ( t )-!( t )-!( A )-1("'-1) 1 -1I"a 1 1I"a 2 1I"aL.l", , FlY) = exp {-(atl)-I IYI12 -(at2)-1 IYml2 -(aAmfl fly; -Y;_d2 } , ;-2 and tl = (l + 1) AI = tr sin <pIS':, t2 = (n + 1) A .. = tr' sin <p' IS':. (3.3) (3.4) (3.5) (3.6) The quantities C2 and F2 are as given in (2.13) and (2.15), respectively (with b = a, of course). The integration range in (3.3) is R: Iy. + ril > a, for i = 1, ... , m. (3.7) Since the r, are different, one from another, the integration range for each i is different. To over come this complication, we integrate (3.3) over all space after first replacing the function F2(Y) by another positive function, F2(Y), which has the property that F2(Y) 2:: F2(Y) for Y in the allowed region, R, while F2(Y) is generally less than F2(Y) for paths which penetrate the sphere. First note that the vectors Uj, given in (2.16), are parallel to rj: u. = 2(1 -cos o)r;, for i = 2, '" , m -1 = (1 -cos o)r;, for i = 1 or m. (3.8) In the allowed region, R, we have a2 :::; Iy. + r.12 = ly.12 + 2Yi·r; + a2 • Thus, in R, y.·u, 2:: -ly;12 (1 -cos 0), for i = 2, '" , m -1 2:: -! ly.12 (1 -cos 0), ~ i=1 m m. ~~ Hence, in, R {I -cos 0 F2(y) :::; F2(y) = exp a Am (3.10) Now, the integral over all space of the product F2(Y)F3(Y) is a simple m-dimensional Gaussian integral, which can be evaluated by using the well known formula 1"" dXI •. '1'" dXN exp {-.f x.AiiX;} -co _00 1.,,-1 (3.11) for any symmetric, positive definite N-square matrix A. Applying this formula to Gm (with F2 replaced by F2), we obtain Gm(r,r'i t) < C2[1I"~~t2IB"'ITI, (3.12) where IB"'I is the determinant of the tri-diagonal m-square matrix This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:43VI RIAL COEFFICIENT OF HARD-SPHERE GAS 51 .1. t~ + cos 0 -1 -1 2 cos 0 -1 o -1 2coso-l -1 B'" = o The exponent! in (3.12) instead of ! as in (3.11) comes about because each of the m variables of (3.13) =-1 ·2 cos 0 -1 2 cos 0 -1 -1 .1. t: + cos a column as well as in the mth row and column and obtain integration is three dimensional. In order for (3.12) to be valid, it is necessary IBml = (cos a + ~l"')( cos a + ~2"')U m-2 that B ... be positive definite. If a = 0, that criterion is surely satisfied and (by continuity) B", is positive ( ) definite for 0 < 0 < ~, where ~ is the smallest value - 2 cos 0 + ~~ + ~: U ",-3 + U ",-4, of a for which IBm I = O. (3.14) To evaluate IB"'I, we expand in the first row and where U", is the m-square determinant 2 cos a -1 -1 2 cos a -1 0 -1 2 cos 0 -1 U", = Det 0 Since U", obviously satisfies the recursion relation ship U", = 2 cos aU ... -l -U",-2, (3.16) it follows that U '" (cos a) is the Chebyshev poly nominal of the second kindl4 (in the variable cos a), whence U", = sin (m + 1) a/sin a. (3.17) Combining (3.17) with (3.14) and, recalling that (J = (m -l)a, we obtain 14 A. Erdelyi, Ed., Higher Transcendental Functions (Mc Graw-Hill Book Co., Inc., New York, 1953), Vol. II, Chap. 10, p. 183. -1 (3.15) . 2 cos a -1 -1 2cosa IB'" I .1.! sin (J (1 1) " = t lt 2sina -.1.", t: + ~ cos (J -sm (Jsm a. (3.18) Now, recalling the definitions (2.10), (3.6) and the fact that r cos I" = a = r' cos 10', (3.18) is equivalent to t1t2 IBml = tasin (I" + 10' + (J). (3.19) .1.... S: cos I" cos 101 But I" + 10' + (J = '" = angle between rand r' [cf. (2.5)]. Thus, combining (3.19) with (2.13), (3.3), and (3.12) and passing to the limit m -+ 00, we have our upper bound G(r, r" t) < [Sa cos 10. cos 10' ]1 e {_ S!}. (3.20\ I rata sm '" xP at ) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.31.134 On: Wed, 03 Jun 2015 11:09:4352 ELLIOTT H. LIEB Formula (3.20) has the essential feature that we have sought, namely, the factor exp {-(shortest distance from r to r' around the sphere) 2/ at I. It also has the factor (n-at) -I, characteristic of Go. The factor (Sa cOS!p cos !p'/a sin if;), while it is usually of the order of unity, can be embarrassingly large when if; '" 7/". Unfortunately, it is precisely the case of diametric juxtaposition of rand r' that is of interest in calculating Bexoh' Plainly, some slight improvement is required before inserting (3.20) into (1.4). It is interesting to note, however, that the diver gence in our upper bound at if; = 7/" is not entirely unexpected. This is because many paths of the same length come together at that angle. In other words, if; = 7/" can be regarded as a caustic. Our upper bound concentrated essentially on only one path around the sphere and, since that one path is not sufficient at if; = 7/", difficulties were encountered there. It is noteworthy that precisely the same divergence is encountered in the classical asymptotic expansion for diffraction around a sphere.7 A simple artifice to overcome the annoying (sin if;)-I factor is the following: Let OQ be a vector of length q < a perpendicular to r and let s' be the sphere of radius b = a -q centered at the point Q. This sphere is clearly tangent to the original sphere, s, (of radius a) at the single point (a/q)OQ and otherwise lies entirely inside the larger sphere, s. Also, let G.,Cr, r'j t) be the Green's function for the exterior of s', just as G(r, r'j t) is the Green's func tion for the exterior of s. From (2.2), we see at once, G(r,r'j t) < G.,(r,r'j t) (3.21) for all points rand r'. We can, in turn, say that G., is less than the right-hand side of (3.20), where the quantities !p, !p', if;, and S are now measured relative to the sphere s' centered at Q. For our purposes, we want r' = -r with r > a. Relative to the sphere s', we have the following simple geometric inequalities for all r > a: 7/"(a -3q) < S < 7/"T, sin if; = 2 2rq 2 > fJ.. r + q r In addition, cos !p cos!p' < 1, whence G(r, -rj t) < r3(ataq)-i X exp {-7/"2(a -3q)2jatl, for any 0 < q < a/3 and for all r > a. (3.22) (3.23) We can now evaluate B,xoh as given by (1.4). To do so, we divide the integration range f: dr into two parts: f!a dr and f2: dr. In the former range, we use the bound (3.23), while in the later range, we use the very simple bound Go as in (3.1). Thus, BB~xOh = 8 J G(r, -r; 2A2/7/"a) dr exoh < 8 1.20 dr 47/"T2r3[2A2qa/7/"f! X exp {-7/"3(a -3q//2A2} + 8100 dr 4'1f'T2(2A2)-i exp {-27/"T2 j A21. (3.24) 20 In the first integral, take q = A2/(27/"3a), assuming that (A/a)2 < 27/"3/3. The second integral is clearly Order {exp [-87/"(a/ A)2]1 and is therefore expo nentially small compared to exp [-!7/"3(a/ A)2]. While an upper bound to this second terms can be easily found, there is little point in doing so. Evaluating the first integral in (3.24) and com bining it with the second, we obtain our final upper bound: Bexoh { 7/"3 (a)2 + I [1 10 7(a)6] 3 -0 -< exp --- n - 2 7/" -+ -Bexoh 2 A 3 A 2 -(2!)3 (~r + o[ exp (h3 -87/")(~rJ}· (3.25) IV. CONCLUSIONS By means of the discrete version of the Wiener integral, (2.2), we have obtained upper and lower bounds to the diffusion Green's function in the presence of an opaque sphere [Eqs. (2.24) and (3.20), respectively]. These bounds are useful for short time (high temperature), especially when the source point and the observation point are in each other's shadow. The bounds enable us to calculate lower and upper bounds to the exchange part of the second virial coefficient of a hard-sphere gas. These bounds, respectively, given in (2.27) and (3.25), permit us to assert that the correct B,xoh diminishes with temperature much more rapidly than the non interacting B~xoh' in a manner given by the equation ~~::: = exp {-~3 (~r + o[ (~y]}. ACKNOWLEDGMENTS The author thanks Dr. S. Larsen, Dr. J. Kil patrick, and H. Jordan, who first stimulated my interest in the problem. Thanks are also due to Dr. E. Hammel for the hospitality of his depart ment at Los Alamos, where these conversations occurred. I am also indebted to Dr. S. Larsen for many valuable comments during the course of this work. This article is copyrighted as indicated in the article. 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1.1710113.pdf	ZeroField Solutions and Their Stability in the OneDimensional LowPressure Cesium Diode Peter Burger Citation: Journal of Applied Physics 38, 3360 (1967); doi: 10.1063/1.1710113 View online: http://dx.doi.org/10.1063/1.1710113 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/38/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Low field mobility and thermopower in onedimensional electron gas J. Appl. Phys. 68, 1070 (1990); 10.1063/1.346746 Exchange formalism applied to onedimensional triplet excitons: Observation of restricted and temperaturedependent k to k′ scattering in the zerofield optically detected magnetic resonance spectrum of 1,2,4,5tetrachlorobenzene J. Chem. Phys. 72, 6485 (1980); 10.1063/1.439150 Exact Relativistic Solution for the OneDimensional Diode J. Appl. Phys. 40, 3924 (1969); 10.1063/1.1657117 Theory of LargeAmplitude Oscillations in the OneDimensional LowPressure Cesium Thermionic Converter J. Appl. Phys. 36, 1938 (1965); 10.1063/1.1714378 Theoretical CurrentVoltage Curve in LowPressure Cesium Diode for ElectronRich Emission J. Appl. Phys. 35, 728 (1964); 10.1063/1.1713449 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:433360 DORE ET AL. (2) Independence of emzsswn work function on the work function of the substrate metals. Further data ob tained here using Ba as a substrate material have emphasized that both the photoelectric work function and the thermionic work function have little dependence on the work function of the substrate. The present theory predicts this near independence for both cases. (3 ) Nonuniform deposits. By predicting a thickness dependent work function, the theory makes under standable the inadequacy of simple Fowler emission theory for very thin films. A correct representation of such cathodes is of a surface with a distribution of work functions. (4) Electron-bombardment and thermionic effects on photoemission. The large enhancement of photoemission that has been measured when the cathode is bombarded by electrons or heated can be understood in terms of JOURNAL OF APPLIED PHYSICS the model presented. The enhancement results from decreased barriers to electron flow through the BaO owing to charge buildups in the BaO layer. From these points of agreement, it appears that the basic theory of the BaO-coated emitter is sufficiently proven to act as a guide to attempts at improving the performance of such structures. In particular, the de velopment of a technique for enhancing shallow donor densities should help to extend the photothreshold for these emitters in the infrared region. ACKNOWLEDGMENTS The authors gratefully acknowledge the assistance of John P. Papacosta who constructed the tubes and to Robert A. Mueller and David J. Dionne for assistance in making measurements and preparing data. VOLUME 38, NUMBER 8 JULY 1967 Zero-Field Solutions and Their Stability in the One-Dimensional Low-Pressure Cesium Diode PETER BURGER Institute for Plasma Research, Stanford University, Stanford, California (Received 31 May 1966; in final form 9 March 1967) The dc potential solutions which have zero slope at the emitter are calculated and examined for the low pressure cesium diode. It is assumed that ions and electrons are emitted thermionically at the emitter with a given ratio of ion-to-electron saturation currents and a positive dc potential is applied to the collector. Furthermore, the potential is nowhere negative in the diode and electron saturation current flows through it. This assumption limits the range of a's to 0<", < 1, where", is the ratio of ion-to-electron saturation currents times the square root of the ratio of their masses. It was found by Auer and Hurwitz that monotonically increasing potential solutions can exist only for the range of ",'5,0<",<0.405. In the range 0.35<",<0.405 we found large amplitude oscillations in the diode by computer simulation methods. The static solutions for a<0.35 were found stable. For 1>",>0.405 the static potential solutions have a long zero slope region within the diode or become periodic in space. All these solutions were found unstable and nonexistent in the diode that oper;t.tes under time-varying conditions just as predicted by Auer and Hurwitz. INTRODUCTION The model of the onc-dimensional low-pressure cesium diode consists of a planar, thermionic emitter of ions and electrons which is opposed by a planar nonemitting collector. A dc potential difference is present between the two plates that is set up by the combined effects of the applied potential difference between Fermi levels and the work functions of the surfaces. The diode plates are assumed to be non reflecting to incoming particles. Both the electron and ion saturation currents are constant in time. Randomiz ing collisions are absent, i.e., the trajectories of electrons and ions are determined only by the injection velocities at the emitter and by the electric fields that act upon them in the diode space. The electric field is determined by Poisson's equation from the charge distribution in the diode. The possible dc states of this device were investigated by Auer and Hurwitzl and by McIntyre2; the non existence of a dc state under some conditions was dem onstrated by Burger using computer simulation meth ods. The theory of large amplitude oscillations in this device was given by Norris4 and by Burger,o and a correspondence between theory and experiment was shown by Cutler and Burger.6 The present paper deals with static potential distributions for which the electric field is zero at the emitter and the potential is nonnega tive; it also investigates the stability of these zero-field solutions. The need for this work has arrived from two sources. First, in a report by Breitwieser and Notting- * Present address: Dept. of Electrical Eng. University College, London, England. 1 P. L. Auer and H. Hurwitz, J. App1. Phys. 30, 161 (1959). 2 R. G. McIntyre, J. Appl. Phys. 33,2485 (1962). 3 P. Burger, J. Appl. Phys. 35, 3048 (1964). 4 W. T. Norris, J. Appl. Phys. 35,3260 (1964}. 6 P. Burger, J. Appl. Phys. 36, 1938 (1965). 6 W. H. Cutler and P. Burger, J. Appl. Phys. 37, 2867 (1966). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:43ZERO-FIELD SOLUTIONS IN A CESIUM DIODE 3361 ham,7 single-species space-charge theory was used for the calculation of the zero-field solutions in the cesium diode because of the lack of data on the solutions for electrons and ions. Numerical data for the two-species case are given. Secondly, in the paper by Cutler and Burger6 only the presence of oscillations were demon strated but the region of stability of the diode was not determined. The region for stable diodes is determined in this paper. It was already supposed by Auer and Hurwitzl that solutions of the periodic type (0.405 < a < 1) are all unstable because of switching between possible dc states, but they had no means for demonstrating the instability. Even though the switching occurs because of the existence of a "temporary dc state"4,5 rather than the possibility of many self-consistent dc states, the predictions of Auer and Hurwitz were correct; the dc solutions for the range 0.405 <a< 1 are indeed unstable.8 Our stability check is based on the zero-field solutions (electric field is zero at the emitter). For the range of a's O<a< 1 these zero-field solutions are the most sensitive to large-signal, low-frequency fluctuations. The reason for this fact is twofold. First, if we lower the collector potential from its zero-field value, a potential minimum will appear at the front of the emitter and saturation electron current will no longer flow through the diode. In this case the instability stops (because of the lack of a temporary dc state) as shown by experi ments6 and also checked by computer simulation techniques. Secondly, if the collector potential is in creased from its zero-field value, an accelerating electric field will appear at the emitter for the electrons and this will reduce the time the electrons will spend in an average in the diode. This field also has a dampening effect on the oscillations. This effect was also found by experiment6 and is demonstrated in this paper. Conse quently, the stability of the zero-field solution implies stability of the diode for all collector potentials. The unstable behavior of the zero-field solutions on the other hand could be stopped both by decreasing or increasing the collector potential from its zero-field value. The stability of the zero-field solutions is examined by computer simulation techniques described in earlier papers.a,5,6 This technique gives the large-signal time dependent behavior of the diode; therefore, we are able not only to demonstrate the instability but also to show the effect of diode parameters on the oscillations. We examine the current-vs-time curves for different param eters of the cesium diode and compare them to the electron saturation current which should flow in a stable diode. 7 W. B. Nottingham and R. Breitwieser, NASA TN D-3324, March (1966). 8 Auer's a is defined as the ratio of densities of ions and electrons at the front of the emitter, and therefore it is not the same as our a. We defined a on the basis of the injected saturation currents; therefore, when the anode is highly positive and the ions are all returned to the emitter, Auer's a is twice the value of ours. This is why he has the value 0.81 instead of 0.405. DISTANCE FIG. 1. The possible types (A, B, and C) of static zero-field solutions in the low-pressure cesium diode which allow electron saturation current to flow through the diode. STATIC ZERO-FIELD SOLUTIONS As we can observe in the papers by Langmuir,9 Auer and Hurwitz,! and McIntyre,2 the static potential vs distance curves for a plasma diode can be calculated by quadratures once the form of the potential curve is established. We are going to deal with a particularly simple form of potential curves (see curve A in Fig. 1) that has zero slope at the emitter and is monotonically increasing throughout the diode. Other forms, such as curve C in Fig. 1, are also possible dc solutions which allow electron saturation current to flow in the diode. We were never able to find these types of solutions within the computer-simulated diode. There is a good physical reason for such periodic type of solutions to be unstable. If we calculate the spatial derivative of the space-charge functions for tY1>e-C solutions, we find that it is discontinuous. It can be proved in generapo that the derivative of the space-charge function of a periodic dc potential solution in a collisionless diode is always discontinuous. Even a very small amount of collisions would destroy such a discontinuity because of large diffusion currents generated at the point of dis continuous derivatives. If collisions are absent, we could expect that rf perturbations will have the same effect on these discontinuities when the diode operates under time-varying conditions. We infer the instability of the type-C solutions by examining the stability of the transition states between type-C and type-A solutions (see type-B curves in Fig. 1). These transition states (type-B solutions) have potential functions which have zero slopes (zero field) at points where the second derivative also vanishes (zero charge). From this point the potential could be continued with zero slope to infinity, or to a finite distance and then the solution continued to a given diode potential (Vd). Hence we could construct solu tions for the same diode potential V d and different lengths (see curves Bl, B2, Ba, etc" in Fig. 1). From the point of view of stability, type-B curves are more 9 I. Langmuir, Phys. Rev. 33, 976 (1929). 10 P. Burger,Ph. D, dissertation, Stanford University, Stanford, California (1964). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:433362 PETER BURGER 10 8 -' 9; ..... z 6 UJ ..... EJ (l.. 0 UJ t: -' '-l a:: :0: a:: EJ z 2 o o 2 '-l 6 8 N~AMALIZED DISTANCE 10 FIG. 2. Normalized diode potential e Vd/ k T vs normalized diode distance d/ADB of the zero-field solutions for the range O::5d/ADB::5 10. The parameter a= J.iMl/2/ J •• mI/2. For a::50.4, all solutions are of type A. promising than type-C curves, because the derivative of the space-charge function of type-B curve is con tinuous everywhere in the diode. We have calculated type-B solutions (Fig. 4) but in the next section we find that they are always unstable. In calculating the dc solutions, we use the normaliza tion procedure of Langmuir,9 i.e., the normalized poten tial 'I) is given by \ e \ V / k T where V is potential in volts, I e \ is the electronic charge, T is the emitter temperature and k is Boltzmann's constant. Distance is normalized to the electron Debye length ~= X/ADB' The Debye length is given by the electron saturation curren t 1 se: AnB= (eo/I e \ lse)I/2[(kT)3/211"m],1/4 (1) wl)ere m is the electron mass and eo is the dielectric constant in vacuum. Since the emission velocities of both electrons and ions have Maxwellian distributions and the potential is monotonic everywhere in the diode, the space-charge density as a function of potential can be determined easily.1.2,10 Poisson's equation p= -eo(d2V /dx2) in normalized form becomes d2'1)/dr=!F-('I) -(o:e-~d/2) F+-('I)a-'I)) (2) where 0: is defined as (3) 'l)d is the normalized diode potential, the ion mass is M, ion saturation current is lsi. The functions F-(TJ) and F+(TJ) are defined by the following equation (4) where erf ( ) is the error function so 21~1/2 erf (TJ1/2) = -e-t2dt. 11"; 0 Instead of converting Eq. (2) to a quadrature it was simpler in our case to integrate this equation in the form of a differential equation with bour;.dary condition dTJ/d~=O at ~=O. Integration starts at ~=O and pro ceeds in steps until TJ becomes 'l)d, the diode potential. At this point ~ is determined. After calculating the func tion H'I)d) for many different values of 'l)d (and one value of 0:), the function is inverted by interpolation and we arrive at the desired function TJd= f(~d, 0:), where TJdis the normalized diode potential that is necessary for a given normalized diode distance ~d and given 0: to produce the zero-field solution in the diode. The results are plotted in Figs. 2 and 3 for the range of ~d, O<~d< 10, and 0< ~d< 100, respectively. Numerical results are given in the Appendix. In Figs. 2 and 3 the range of 0: for which type-A solutions are possible is 0<0:<0.404. For 0:>0.404 the zero-field solutions in the diode have forms Band C. The limiting cases (type B) are shown in Fig. 4. These solutions have to obey the condition p(TJO) = E('I)O) =0 for a potential '1)0 with O<'I)O<TJd' After TJo has been found for a given 0:, TJd can be evaluated. As we have already remarked, all these solutions were found un stable, and hence nonexistent in a collisionless diode; therefore, they are shown here only for reference. These solutions might become important in ce!>ium diodes, in which a small amount of electron neutral collisions could stabilize the potential, but the effect of the collisions on the form of the potential function would not be very large. A subsequent paper analyzes this problem. Before turning our attention to the problem of stability of the static solutions, an equation for large ~d is given. The approximate solution for ~d as a function 300 250 ...J a: ;: 200 z w ..... E:) a.. Cl 150 w a :::! ..J a: ~100 z 50 o t-1lIIIi~;,e:~:t:==:::J=::±=±=:l==:::J=::±=::J-'O.405 o 10 20 30 ~o 50 60 70 60 90 100 NORMALIZED DISTANCE FIG. 3. Normalized diode potential vs normalized distance of the zero-field solutions for the range 0::5djADB::5100. For 0:::5 0.404, all solutions are of type A. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:43ZERO-FIELD SOLUTIONS IN A CESIUM DIODE 3363 of fJd and a can be derived from the quadrature form of Eq. (2) (see, e.g., McIntyre2). We can get the equation in the form fJ~(3413/47r113) (~d-C .. )4/3=0.739(~d-C",)4/3. (5) The values for the constants c'x are given in Fig. 5 as a function of the parameter a. These constants were determined from data calculated for ~d> 100 using graphical interpolation. Equation (5) can be used for ~d~ 100 with less than 3% error in fJd for the range 0<a<0.4. STABILITY OF THE LOW-PRESSURE CESIUM DIODE NEAR ZERO-FIELD CONDITIONS We have used the digital computer in the pastS,S to simulate the large-signal behavior of the low-pressure cesium diode with success. This method consists of calculating the trajectories of a large number of charged sheets in the one-dimensional space of the diode. The positively and negatively charged sheets are injected at the plane of the emitter with velocities that are r~nd?ml~ selected according to the Maxwellian velocity dIstrIbutIOn law of a thermionic emitter. The trajec tories of the sheets are calculated in small time steps. At every time step the electric field distribution is re calculated in the diode space from the known positions of the sheets using a difference equation form of Pois son's equation. The boundary condition that the integral of the field is equal to the negative of the diode potential is satisfied at every time step. The trajectories of the sheets are calculated always with the fields that act on them at the particular time step when the calculations are made. 5 q --J cr: t- Z 3 w t-El <L 0 W '" -' 2 cr: E a: '" z o a·OA .406 __ -+0.4' _.-+----!-0.42 ~ _ __t----t·0.43 f-..-----¥/ . __ -+------1-·0.44 :.t::::':==4'==~=f.O.45 -1-_---+----1-0.475 :::......-+-----+-----1-0.5 a I ~~:;:;::~~~~~i;~;;;;~.O'6 .... .:0.10.6 o 10 20 30 N~AMALIZEO DISTANCE FIG. 4. Normalized diode potential vs normalized diode distance of tyPe.B ze!o-~eld solutions. The curve with a=O.4.is type·A solution and IS gIven for comparison only. 25 20 c. 15 10 '. i / I l7 /' ....----------- 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 a FIG. 5. The values of C" as a function of a in the approximate expression given for large diode distances [Eq. (5)]. It was shown in earlier papers·'s that the computer simulated diode gives essentially the time-dependent behavior of the one-dimensional diode if the effects of numerical errors are minimized. The principal numerical errors are the fini teness of the time step and the small number of sheets that can be handled by the computer. In our computations we have used time steps of the order of hDB/V e, where hDB is the electron Debye length near the emitter, and iie is an average electron thermal velocity defined as iie= (2kT/m)1/2• Our diodes con tained typically 6000 sheets. Repeated tests proved that the errors caused by these two sources (size of time step, number of sheets) were negligible; the use of either a smaner time step or a larger number of sheets gave identical results to those presented here. A different type of ",numerical" error arises from the fact that if physical ion-to·electron mass ratios were used in computer calculations, then the calculations would take excessive computer time. We have used mass ratios from M/m=4 to values as large as 256 in the past. It became evident that mass ratios larger than 50 all gave very similar results if we scaled the low frequency oscillations to the average transit time of the ions Ti [see Eq. (6)]. We have used an ion-to·electron mass ratio M/m=64 throughout our calculations. We are certain that the results we present here are valid for larger mass ratios also, and they describe the behavior of an experimental diode well. Before discussing the results of our "computer ex periments," a few observations about the oscillating tendencies of the diode are in order. It was demonstrated in Ref. 6 that the large-amplitude low-frequency oscilla tions are inhibited when the potential becomes negative in the diode space so that saturation electron current cannot flow. The oscillations can be suppressed also by applying a large positive potential on the collector causing the field to become negative at the emitter. When the smallest positive potential was applied that could draw saturation current through the diode, the current through the diode became unstable. The small est positive potential that draws saturation electron current through the diode is the diode potential of the zero-field solution. These solutions were calculated in the preceding section, and the diode potentials were [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:433364 PETER BURGER TABLE I. Numerical results for the computer-simulated diode working near zero-field conditions. ~d '" ')'d J av/ J.e Jrd J.e Trf/Ti (peak-to-peak) Remarks 10 0.30 5.30 1.00 0.00 Stable 0.35 3.50 1.00 0.10 2.00 Marginal 0.40 2.10 0.90 0.18 1.20 0.42 1.80 0.40 ~.4O ",1.00 Unstable 30 0.30 42.60 1.00 0.00 Stable 0.35 37.20 0.95 0.13 0.68 Marginal 0.40 20.30 0.85 0.50 0.80 Very large amplitude irregular os- cillations 30.00 0.90 0.30 0.75 Stabilizing effect of raising collector 40.00 0.98 0.20 0.60 potential is shown 50.00 1.00 0.00 Stable 0.42 3.00 0.40 ~.60 ",1.00 Unstable 50 0.30 98.20 1.00 0.00 Stable 0.35 90.80 1.00 0.10 0.80 Marginal 0.40 82.50 0.95 0.20 0.60 0.42 3.00 0.40 ",0.60 ,..,.,1.00 Unstable 100 0.30 280.00 1.00 0.00 Stable 0.40 240.00 0.98 0.20 0.50 Marginal 280.00 1.00 0.07 0.20 Stabilizing effect of larger 71d 300.00 1.00 0.00 Stable 0.42 5.00 0.40 ",0.60 ,..,.,1.00 Unstable given as functions of diode distance and parameter a. We now examine the time-varying operation of the diode with the diode potentials set at the calculated values of the zero-field solutions. Even though we are interested in the stability of the zero-field solutions here, we can state in general that, if the diode is stable in this state, then it will be stable for all values of collec tor potentials. Both increasing and decreasing the collector potential will have a stabilizing effect on the diode because we draw less than electron saturation current in the first case, and we create a negative field at the emitter in the second case. 10, where the stable and unstable regions of the diode are indicated. We can choose our output quantities in the computer simulated diode at will. We have chosen the current through the diode as the function of time for our output because this quantity is easily measured in a real experi ment and is a good indicator of the diode's stability. We normalize current to the electron saturation current; hence for stable operation the normalized current value should be 1. When the average diode current is signifi cantly lower than the electron saturation current then the zero-field solution can not exist in the diode; hence it has to be unstable. Our results are summarized in Table I, where numerical results are shown, and in Fig. In Table I both the average current and the peak-to peak values of the oscillating current are shown with the average oscillation period normalized to the average ion transit time 'ri. The average ion transit time is given by , 'ri=d/ (2kT/M) 1/2, (6) where d is the diode distance and M is the mass of ions. Similarly, the average electron transit time 'r.= d/(2kT/m)l/2. The peak-to-peak value of the oscillating current gives an indication of the percentage variation in the diode current. When the collector potential is near to the value of the zero-field solution, the current wave forms tend to be irregular (see Fig. 7). When the potential is raised, the current takes up a triangular waveform (see Fig. 8) while the amplitude of the oscilla tion decreases. The same tendency from irregular waveforms to triangular waveforms was observed when the collector potential was raised in an earlier experi ment.6 The amplitudes also decreased when the collector potential was raised. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:43ZERO-FIELD SOLUTIONS IN A CESIUM DIODE 3365 1.0 o a 8 12 16 20 2'i 28 NORMRL! ZED Tl ME FIG. 6. Normalized current J / Joe vs normalized time t(kT/m)112/d in the computer-simulated diode for the zero-field parame~ers d/XDB=30, eVa/kT=42.6, «=0.3. The time step for calculations was 1/30 normalzed time unit and there are an aver age of 4000 sheets in the diode. Ion-to-electron mass ratio M/m=64. A few representative current-vs-time curves are shown in Figs. 6-9. We have constructed similar figures for all the cases run, but it would be superfluous to present all of them here. In these figures, time is normal ized to the average electron transit time. As mentioned earlier, the ion-to-electron mass ratio was 64; hence, the average ion transit time is 8 normalized time units. A variety of waveforms can be produced by changing 1Jd or a for a given diode distance ~d. (The same variety of waveforms was found in our experimental diode.6) But our concern here is to distinguish between stable and unstable cases. We have chosen the condition that the average normalized diode current should be 1 for stable cases, and should be significantly lower for unstable cases. The current-vs-time curve shown in Fig. 6 .for diode parameters ~d= 30, a= 0.3, 1Ja= 42.6, is eVIdently stable. We calculated the potential distribu tion in the computer-simulated diode for the stable cases and found that they were indeed the zero-field solutions. This gave further proof of the stability of these de states. The stable states were found for the range of a's 0~a~0.35. The exact range was affected slightly by the diode distance ~d as shown in Fig. 10. In general, we can state that for a~0.35, the diode is stable regardless of diode distance and diode potential and electron saturation current can be drawn in thi~ 1.0 0-§ 0118 0:: => <.:7 0.6 a lLl ::::! 0.4 -' a: >: 15 0.2 z a 0 ~ 8 12 16 20 2'! 28 NORMRL! ZED TJ ME FIG. 7. Normalized current vs normalized time in the computer simulated diode for the zero-field parameters d/XDB = 30, e Val kT= 2~.3, «,:,0.4. The average ion transit time is eight normalized time umts. 1.0 I- ~ 0.8 § <.:7 0.6 Cl lLl ::::! 0.'1 -' a: >: :is 0.2 z a o 8 12 16 20 21t 28 NORMALIZED TI ME F!G. 8. The effec~ ot increasing collector potential on the current vs-tJme. charactenstlcs of the simulated diode (cf. Fig. 7). Same diOde parameters as in Fig. 7 except eVa/kT=40. region of a's when the zero-field potential is applied across the diode. In the region 0.35~a~0.405, large-amplitude oscilla tions start in the diode. The transition between stable and unstable regions is very sudden for the medium sized diodes (20~~d~50) and at transition the ampli tude of the oscillations is large. The largest oscillations were found in the diode with ~a=30, a=0.4, and the current-vs-time curve of this diode with 1Jd= 20.3 is shown in Fig. 7. We tried to inhibit these oscillations by raising the collector potential. The results are shown in Table I, and in Fig. 8, where a normalized diode potential1Jd= 40 was applied across the diode. When the diode potential was raised to the value 1Jd= 50 the diode became stable, and saturation electron cu'rrent was flowing through it (the characteristics were similar to the one shown in Fig. 6). The normalized electric ~eld value w~s. 0.4 at. the emitter when the diode opera tIOn was stabIlIzed WIth 1Jd= 50. The normalized electric field is given by 8=eE"ADB/kT, where E is the field in volts/m, and "DB is the electron Debye length in m. For very short (~d ~ 20) or long separation lengths (~d= 5?) the oscillati~ns seem to start less violently a~d WIth smal~er amplItudes. In these cases the average dIOde current IS close to the electron saturation current even when the oscillations start. We have inhibited the 1.0 .... z ~ 0.8 => c.J 0.6 Cl w ::::! 0.'1 -' a: >: :B 0.2 z a a of 8 12 16 20 21t 28 NORMRLIZED TJ ME . FIG. 9. Normali~ed current vs normalized time characteristics III the computer-simulated diode with parameters d/ADB = 100 eVa/~T~5, «=~.42. ~n attempt to find the type-B zero-field s?lutiOn III the diOde wI.th eValkT=1.8 failed, and even with this hlgh~r collec~or potential, electron saturation current does not flow III the diOde. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:433366 PETER BURGER a FIG. 10. The regions of stable and unstable zero-field solutions in the low-pressure cesium diode. For a>0.405, type-A solutions are not possible, and the diode does not have a stable zero-field solution. In the range 0.35<a<0.4, large-amplitude oscillations start in the diode when the type-A zero-field solutions are set up. oscillations for a long diode ~d= 100, a=0.4 and found that a smaller percentage increase in the collector potential was sufficient to stop the oscillations. The diode reached a stable state with the potential 7Jd= 300 instead of the calculated zero field value of 7Jd= 238. When the diode was stabilized the normalized electric field value was 0.5 at the emitter. For a~0.4OS static theory predicts that saturation current should flow with the zero-field potential dis tributio.n of type B in the diode. These distributions predict a very small collector potential (see Fig. 5). We could not find these potential distributions in the computer-simulated diode. When we programmed the computer-simulated diode with the parameters ~d, a, Tid as shown for the type-B solutions in Fig. 4, we found that the average diode current was substantially lower than saturation current, and large-amplitude oscilla tions were present in the diode. A representative cur rent-vs-time curve is shown in Fig. 9 for diode param eters ~d= 100, a= 0.42, 7Jd= 5. The potential distribution in the diode also has large-amplitude oscillations; its form approaches the type-B solution only temporarily for the short time interval when the normalized current is near to 1 (see Fig. 9) . For the larger part of the cycle a potential minimum forms near the emitter that returns electrons to the emitter and does not allow saturation electron current to flow in the diode. Accord ing to our definition of stable operation (normalized current::::: 1) , the type-B solutions are all unstable, and we expect that they are never present in a low-pressure cesium diode. The results are summarized in Fig. 10, where the stability of the zero-field solutions is shown. The solu tions for 1>a>0.4OS (type B) are all unstable. The solutions for aSO.3S are stable, and in the range 0.35< a<0.4OS the average diode current is near to saturation current; therefore we called this region "marginal" in Fig. 10. In this marginal region we should expect to measure a larger collector potential for electron satura tion current than predicted by static theory because stability can be reached only with a larger collector potential than given by the zero-field solution. If the calculated zero-field potential is applied for this range of a's, low-frequency noise should be present in the diode and the average diode current should be less than the electron saturation current of the emitter. It is expected, however, that a small amount of electron neutral collisions will stabilize these cases, and hence, in an experimental device, one might not be able to detect oscillations in this region. A subsequent paper will deal with the effect of electron neutral collision on the un stable operation of the cesium diode. CONCLUSIONS We have calculated values for the normalized diode potential Tid as a function of normalized diode distance ~d and parameter a for the zero-field potential dis tributions in the low-pressure cesium diode. The results of computer calculations have shown that the given de curves are stable for a~0.3S. In the range 0.3SSaS 0.405 the diode exhibits low-frequency oscillations; however, the average diode current remains near the saturation electron current. For 1>a>0.4OS, type-B and type-C solutions are possible but they were found unstable. In this range of a's a zero-field de solution does not exist in the diode. Stable saturation current can be drawn across the unstable diodes only with a large negative field present at the emitter, which can be obtained by a higher than calculated collector poten tial. Consequently, the de curves given in Figs. 2 and 3 and in the Appendix are valid for aSO.3S, but only approximate for the range O.3SSaSO.4OS. The curves for 1 >a>0.40S (Fig. 4) can not be applied to collision less cesium diodes at all. ACKNOWLEDGMENTS The author wishes to thank Ronald A. Breitweiser for suggesting this problem and Dr. Donald A. Dunn for reading the manuscript and suggesting corrections. This work was supported by NASA Grant (Project 0254) NSG-299-63 at Stanford University. APPENDIX The numerical values for the normalized diode poten tial vs normalized distance of the zero-field solutions are given below for two ranges of normalized diode distance. For the range 0.lS~dSl0, ~d is increased by 0.1; and for the range lS~dS100, it is increased by 1. For ~d> 100, consult the text [Eq. (5)] for an ap proximate expression. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:43ZERO-FIELD SOLUTIONS IN A CESIUM DIODE 3367 0.0 0.1 0.0025 0.2 0,0096 NORMALt?.ED DIODE PCYtENTIAL VALUES FOR THE ZERO-FrELD SOLUTIONS 0.1 ~ ~d ~ 10 0.05 0.1 0,15 0.2 0.25 0,3 0.35 0.38 0.4 0.0023 0.0091 0,0022) 0.0020 0.0086 0.0080 0.0019 0.0018 0.0076 0,0070 0,0017 0,0016 0,0014 0,0014 0.0065 0,0060 0,0056 0,0054 0.3 0,0213 0.0201 0.0188 0.0176 0.0164 0,0152 0,0140 0.0128 0.0121 0.0177 0.4 0,0373 0.0350 0.0327 0.0305 0,0283 0,0261 0.0240 0,0218 0.0206 0.0198 0,5 0.0573 0.0537 0.0500 0.0465 0,0430 0.0395 0,0361 0.0328 0,0308 0.0295 0,6 0,0812 0.0758 0.0705 0,0653 0.0602 0,0551 0,0502 0,0453 0,0425 0.0406 0.7 0.1089 0.1014 0.0940 0.0868 0,0797 0.0727 0,0659 0,0593 0,0554 0,0529 0.8 0.1401 Q.l3ill 0.1203 O,llOR 0.1014 0,0922 0,0833 0.0746 0,0695 0,0662 0,9 0.1748 0.1619 0.1494 0.1371 0.1250 r 0.1133 0.1020 0.0909 0.0845 0.0803 1.0 0.2127 0.1961 0.1809 0.165G 0.1506 0.1360 0.J21S 0.1083 0.1003 0.0951 1.1 0.2539 0.2342' 0.2150 0.1962 0.1179 0.1601 0.1430 0.1264 0.1168 0.1106 1.2 0.2981 0.2744 0.2513 0.2288 0.2068 0.1856 0.1651 0.1454 0.1340 0.1266 1.3 0.3452 0'.3173 0.2899 0.2632 0.2373 0.2122 0.1881 0.1650 0.1516 0.1430 1.4 0.3953 0.3626 0.3307 0.2995 0.2693 0.2400 0.2120 0.1852 0,1697 0.1597 1.5 0.4481 0.4104 0.3735 0.3375 0.3026 0.2690 0.2366 0.2059 0.1882 0.1168 1.6 0.5036 0.4605 0.4184 0.3172 0.3373 0.2989 0.2621 0.2271 0.2070 0.1941 1.7 0.5617 0.5130 0.4652 0.4186 0.3734 0.3299 0.2882 0.2487 0.2261 0.2116 1.8 0.6224 0.5676 0.5139 0.4615 0.4106 0.3617 0.3149 0.2707 0.2455 0.2293 1.9 0.6855 0.6245 0.5645 0.5060 0.4492 0.3945 0.3423 0.2930 0.2650 0.2471 2.0 0.7510 0.6835 0.6170 0.5520 0.4889 0.4281 0.3702 0.3157 0.2g48 0.2650 2.1 0.8189 0.7446 0.6713 0.5995 0.5297 0.4226 0.3987 0.3381 0.~047 0.2830 2.2 0.8890 0.8077 0.7273 0.6485 0.5718 0.4980 0.4278 0.3619 0.3247 0,3010 2.3 0.9614 0.8728 0.7851 0.6989 0.6150 0.5342 0.4574 0.3854 0.3449 0.3191 2.4 1.036 0.9399 0.8446 0.7508 0.6593 0.5112 0.4875 0.4092 0.3652 0.3372 2.5 ·1.112 1.009 0.9058 0.8041 0.7048 0.6091 0.5181 0.4331 0.3855 0.3553 2.6 1.191 1.080 0.9687 0.8589 0.7514 0.6477 0.5492 0.4573 0.4060 0.3735 2.7 1.272 1.153 1.033 0.9151 0.7992 0.6872 0.5808 0.4818 0.4265 0,3916 2.8 1.355 1.227 1.100 0.9727 0.8481 0.7275 0.6129 0.5064 0.4471 0.4097 2.9 1.439 1.304 i.167 1.032 0.8981 0.7686 0.6455 0.5312 0.4618 0.4278 3.0 1.526 1.382 1.237 1.092 0.9493 0.8106 0.6786 0.5563 0.4885 0.4459 3.1 1.614 1.462 1.308 1.154 1.002 0.8534 0.7123 0.5815 0.5093 0.4640 3.2 1.104 1.544 1.381 1.217 1.055 0.8970 0.7464 0.6070 0.5301 0.4820 3.3 1.796 1.6Z7 1.455 1.282 l.ll0 0.9416 0,7811 0.6327 .0.5510 0.5000 3.4 1.890 l.n2 1.531 1.348 1.166 0,9870 0.8163 0.6585 0.5719 0.5180 3.5 1.985 1.799 1.609 1.416 1.223 1.033 0.8521 0.6846 0.5929 0.5359 3.6 2.082 1.888 1.688 1.485 1.281 1.081 0.8884 0.7110 0.6140 0.5538 3.7 2.181 1.978 1.769 1.556 1.341 1.129 0.9253 0.7375 0.6350 0.5711 3.8 2.282 2.070 1.851 1.628 1.402 1.178 0.96.28 0.7643 0.6562 0.5895 3.9 2.384 2.163 1.935 1.701 1,464 1.~28 1.001 0.7913 0.6774 0.607S 4.0 2.487 2.258 2.021 1.776 1.527 1.2"79 1.040 0.8186 0.6987 0.6251 4.1 2.592 2.355 2.108 1.853 1.592 1.331 1.079 0.8461 0.7201 0.6428 4.2 2.699 2.453 2.196 1.931 1.658 1.385 1.119 0.8739 0.7415 0,6605 4.3 2.807 :;!.552 2.287 2.010 1.726 1.439 1.160 0.9020 0.7630 0.6182 4.4 2.917 2.654 2.378 2.091 1.794 1.494 1.201 0.9303 0.7845 0,6958 4.5 3.028 2.756 2.471 2.173 1.864 l.551 1.244 0,9590 0.8062 0.7135 4.6 3.140 2.880 2.566 2.257 1.936 1.608 1.286 0.9879 0.8279 0.1311 4.7 3,254 2.966 2.662 2.342 2.009 1.667 1.330 1.017 0.8498 0.7487 4.8 3.369 3.073 2.759 2.429 2.083 1.727 1.375 1.047 0.8717 0.7662 4.9 3.486 3.181 2.858 2.517 2.159 1.788 1.420 1.077 0.8937 0.7838 5.0 3.604 3.291 2.959 2.607 2.236 1.851 1.466 1.107 0.9158 0.8012 5.1 3.723 3.402 3.060 2.698 2.314 1.914 1.513 1.134 0.9381 0.8188 5.2 3.844 3.$15 3.164 2.790 2.394 1.979 1.561 1.169 0.9804 0.8363 5.3 3.966 3.628 3.269 2.884 2.475· 2.:015 1.610 1.200 0.9829 0.8538 5.4 4.089 3.744 3.375 2.980 2.558 2.113 1.660 1.232 1.006 0.8713 5.5 4.214 3,860 3.482 3.077 2.642 2.181 1.711 1.265 1,028 0.8888 5.6 4.$39 3.978 3.591 3.175 2.128 2.252 1.763 1.297 1.0!51 0.9063 5.7 4.457 4.097 3.701 3.274 2.815 2.323 1.815 1.331 1.014 0.9238 5.8 4.595 4.217 3.812 3.375 2.903 2.396 l.g69 1.364 1.097 0.9412 5.9 4.124 4.339 3,925 3.478 2.993 2.470 1.924 1.398 1.121 0.9587 6.0 4.854 4.618 4.039 3.582 3.084 2.546 1.980 1.433 1.144 0.9762 ~d 0.0 6.1 4.986 6.2 5.119 NOWIALIZED DIODE POTEKl'IAL VALVES FOR THE ZERO·FIELD Sct.1JTIQfS 0.1. ~d. 10 (Continued; 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.38 0,4 4.586 4.155 3.687 3.177 2.623 2.038 1.469 1.168 0.9937 4.711 4.271 3.793 3.271 2.701 2.096 1.504 1.192 1.011 6.3 5.253 4.838 4.389 3.901 3.366 2.781 2.156 1.541 1.216 1.029 6.4 5.388 4.965 4.;i08 4.010 3.463 2.862 2.216 1.578 1.240 1.046 6.5 5.525 5.094 4.628 4.120 3.561 2.945 2.278 1.615 1.265 1.064 6.6 5.662 5.224 4.750 4.232 3.661 3.029 2.342 1.654 1.289 1,082 6.7 5.801 5.355 4.873 4.345 3.761 3.114 2.406 1.693 1.314 1.010 6.8 5,940 5.487 4.996 4.459 3.864 3.201 2.412 1.732 1,340 1.117 6.9 6.081 5.621 5.122 4.574 3.967 3.289 2.540 1.773 1.365 1.134 7.0 6.223 5.755 5.248 4.691 4.072 3.379 2.608 1.814 1.391 1.152 1.1 6.366 5.891 5.375 4.809 4.178 3.470 2.678 1.856 1.417 1.170 7.2 6.509 6.028 5.504 4.928 4.286 3.562 2.749 1.899 1.443 1.188 7.3 6.654 6.165 5.633 5.048 4.394 3.656 2.821 1.942 1.469 1.205 7.4 6.800 6.304 5.764 5.169 4.504 3.751 2.895 1.987 1.496 1.223 7.5 6.947 6.444 5.896 5.292 4.616 3.841 2.971 2.032 1.523 1.241 7.6 7.095 6.585 6.029 5.415 4.728 3.945 3.047 2.078 1.551 1.259 7.7 7.244 6.727 6.163 5.540 4.842 4.044 3.126 2.125 1.579 1.277 7.8 7.394 6.810 6.298 5.666 1.957 4.~45 3.205 2.173 1.607 1.295 7.9 7.544 7.013 6.434 5.793 5.073 4.247 3.286 2.222 1.635 1.313 8.0 1.697 7.159 6.571 5.921 5.190 4.350 3.368 2.273 1.664 1.331 8.1 7.849 7.304 6.709 6.051 5.308 4.454 3.452 2.324 1.693 1.350 8.2 8.003 7.451 6.848 6.181 5.428 4.560 3.538 2.376 1.723 1.368 8.3 8.158 7.599 6.989 6.312 5.549 4.661 3.624 2.429 1.753 1.386 8.4 8.313 7.748 7.130 6.445 5.671 4.775 3.712 2.484 1.783 1.405 8.5 8.170 7.898 7.272 6.578 5.79,1 4.884 3.802 2.540 1.814 1.423 8.6 8.627 8.019 7.415 6.713 5.918 4.995 3.893 2.596 1.846 1.441 8.7 8.786 8.200 7.560 6.848 6.043 5.107 3.985 2.655 1.877 1.460 8.8 8.945 8.353 7.705 6.985 6.169 5.220 4.078 2.714. 1.910 1.419 8.9 9.105 8.506 1.851 7.122 6.297 5.334 4.173 2.775 1.943 1.497 9.0 9.266 8.661 7.998 7.261 6.425 5.449 4.270 2.836 1.976 1.516 9.1 9.128 8.816 8.146 7.101 6.554 5.566 4.367 2.900 2.010 1.535 9.2 9.591 8.07;! 8.295 7.541 6.685 5.683 4.466 2.964 2.044 l.tl54 9.3 9.755 9.130 8.445 7.683 6.816 5.802 4.566 3,030 2.080 1.573 9.4 9.919 9.281:1 8.596 7.825 6.949 5.922 4.668 3,098 2.115 1.592 9.5 10.08 9.447 8.747 7.969 7.082 6.043 4.771 3.166 2.152 1.812 9.6 10,25 9.607 8.900 8.113 7.217 6.165 4,875 3.237 2.189 1.631 9.7 10.42 9._ 9._ 8._ 7.= 6._ 4._ 3._ 2._ 1.~ 9.8 10.59 9.9 10.75 10.09 9.363 8.552 7.627 6.538 5.195 3.456 2.304 1.689 10.0 10.92 10.25 9.520 8.700 7.765 6.664 5.304 3.532 2.344 1.109 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 10:40:433368 ~d 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15' 16 11 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3S 36 37 38 39 40 41 42 43 44 45 46 47 4_ 49 50 PETER BURGER NORMALIZED PlODE POTENTIAL VALUES FOR 'l1tE ZERO-FIELD SOLUTIONS 1 < Sd < loa C< 0.0 0.05 ! 0.1 0.15 0.2 0.25 , 0.3 0.35 0.38 I .2127 .:t966 ,1809 .1654 .1504 .1360 .1188 ,1054 ,0981 .7510 .6834 .6170 .5520 .4869 .4281 .3702 .3157 ,2847 1. 525" 1.382 1.237 1.092 .9493 ,glOG' .6786 .5539 .4'1376 2.487 2.2$8 2.021 1.776 1.527 1.279 1.040 .8186 ,6981 ! 3.604 3~291 2.959 2.607 2.236 1. 851 11. 466 I 1. 107 .9158 ~.854 4.462 4.039 3.582 3.084 2.546 1.98011.433 1.144 6.223 5.755 5.248 4.691 4.072 3.379 2.608 1.814 1.391 7.697 7.158 6.571 5.921 5.190 4.350 3,369 2.273 1.664 9.267 8.661 7.998 7.261 6.42& 5.449 ~.270 2.837 1.976 10.92 10.25 9.520 8.700 7.761;) 6.664 5.304 3.532 2.314 12e66 11.93 11.13 10.23 9.202 7.984 6.458 4.372 2.791 14.48 13.69 12.82 11.-84 10.73 9.400 )7, 720 5,353 3.348 16.37 15.52 14.58 13.54 12.34 10.90 9.080 6.458 4.041 18.32 17.42 16.42 15.30 14.02 12.49 10.53 7.676 4.895 20.34 19.38 18.32 17.14 15.78 14.15 12.06 8.994 5.895 22.42 'L,n 20.29 19.04 17~61 15.88 13.67 10.40 7.017 24.58 23.49 22.32 21.01 19.50 17.69 15.36 11.89 8.258 26.75 25.64 24.41 23.03 21.45 19.55 17.11 13.46 9.594 29.00 27.84 26.55 25.12 23.47 21.48 18.93 15.11 11.02 31.30 30.08 28.75 27.26 25.54 23.47 20.81 16.82 -1.2.52 33.65 32.39 31.00 29.45 27.66 25.52 22.75 18.60 14.10 36.04 34.73 33.30 3L69 29.84 27;62 24.75 20.44 15.75 38.48 37.13 35.64 33.98 32.07 29.77 26.81 22.35 17.47 40.97 39:51 38.04 36.32 34.34 31.98 28,92 24.31 19.26 43.50 42.06 40.47 38.70 36.67 34.23 31.08 26.33 21.11 46.07 44.58 42.95 41.13 39.04 36.53 33.29 28.41 23.02 48.68 47.15 45.48 43.60 41.45 38.87 35.56 30.53 24.99 51.34 49.76 48.04 46.12 43.91 41.27 37.86 32~70 27.01 54.03 52.41 50.65 48.67 46,41 43.10 40.21 34.93 29.08 56.76 55.10 53.29 51.27 48.95 46.17 42.60 37.2Q 31.21 59.52 57.83 55.97 53.90 51.53 48.69 45,04 39.51 33.38 62.32 60.59 58.69 56.57 54.1S 51.25 47.52 41.87 35.61 65.16 63.39 61.45 59.28 56.81 53.84 50.03 44.28 37,88 68.03 66.22 64.24 62.03 59,50 56.48 52.59 46.72 40.19 70.94 69.09 67.06 64.81 62.23 59.15 55.19 49.21 52~5S. 73.88 71.99 69.92 67.62 6-;\.99 61.86 57.82 51.73 44.95 76.85 74.92 72,82 70.47 67,79 64.60 60.49 54.30 47.39 79.85 77.89 75.74 73.36 70.63 67.38 63.20 56.90 49.88 82.88 80.88 78.70 76.27 73.50 70.19 65.94 59.54 52.40 85~95 83.n 81.69 79~22 76,39 73.03 68.72 62.21 54.96 89~O4 86.97 84.71 82.20 79.33 75.91 71~53 64.92 57,56 92.17' 90.06 87.76 85.21 82.29 78.82 74.37 67.67 60.20 95.32 93.18 90.84 88.~4 85.28 81.76 77.24 70.45 62.87 98.50 86.32 93.95 91.31 88 •. 31 84.73 80.15 73.26 65.57 101.7 99.5Q 97.09 94.41 91.36 87.73 83.09 76.U 68.32 104.9 102.7 100.3 97.54 94.44 90.77 86.06 7B~98 71.09 108,2 105.9 103.5' '100.7 97.56 93.83 89.06 81.89 73.90· 111.5 ~O9.2 106,7 103.9 100.7 96,92 92.09 84.83 76,74 114.8 112.5 109 •• 107.1 103,9 100.1 95.15 .87.80 79.61 118.2 115.8 113.2 110.3 107.1 103.2 98.23 90;80 82.52 ,0940 .2650 .4718 .6250 .8013 .9763 1.152 1.331 ,1.516 1.709 1.913 2.131 2.368 2.630 2.926. 3.269 3.681 4.192 4.843 5.669 6.632 7.812 9.096 10.47 11.92 13.45 15.06 16.73 18.47 20.28 22.14 24.07 26.04 28.07 SO.16 32.29 34.47 36.70 38.97 41.29 43.65 46.05 48.49 50.97 53'.50 56.05 58.65 61.28 63.95 66.65 d 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 63 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 8~ 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 NORMALIZED DIODE P(YI''ENTIAL VALUES FOR 'I1IE 1.ERO~F1ELD SOLUTIONS 1 <' ~d <: 100 (Continued) 01 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.38 121.5 119.1 116.5 113.61110.3 106.4 101.3 93.83 85.45 124.9 122.5 119.8 116.9 113.5 109.6 104.5 96.90 88.42 128.4 125.9 123.2 120.2- 116.8 112.8 107.7 99;98 91.42 131.8 129,3 126.6 123.5 120.1 116,0 110.9 103.1 94.44 13-5,3 132.7 130.0 126.9 123.4. 119.3 H4.l 106.2 97.50 138.8 136.2 133.4 130.3 126.8 122.6 117.31109.4 100.6 142.3 13~.7 136.S 133.7 130.2 126.0 120.6 112.6 103,7 145.8 143.2 140.3 137.2 133.6 129.3 123.9 115.8 106.8 149.4 146.7 143.8 140.6 137.0 132.7 127.2 U9.1 110.0 153.0 150.3 147.a 144.1 140.4 136.1 130.6 122.4 113.2 156.6 153.8 150.9 147.6 143.91139•5 134.Q 125.7 116.4 160.2 1S1.5 154.5 151.2 147.4 143,0 137.4 129.0 119.7 163.9 161.1 158.1 154.7 151.0 146.5 140.8 132.3 122.9 167.5 164.7 161.7 158,3 154.5 150.0 144.3 135.7 126.2 171.2 168.4 165.3 162.0 158.0 153.5 147.7 139.1 129.5 174.9 172.1 169.0 165,5 161.6 157.0 151.2 142.5 132.9 178'.7 l'75.8 172.6 169.2 165.2 160.6 154.7 146.0 136.3 182.4 179.5 176.3 172.8 168.9 164.2 158.3 149.5 139.7 IS6.2 183.3 j 180.1 176.5 172.5 167.8 161.8 153.0 H3.1 190.0 187.0 183.8 180.2 176.2 171.5 165.4 156.5 146.5 193.8 190.8 187.6 184.0 180.0 175.1 169.0 160.0 150.Q 197.7 194.7 191,4 187.7 183.5 178.8 172.7 163.6 153.5 201.5 198.5 195.2 191.5 187.4 182.5 176.3 167.2 157.0 205.4 202,3 199,0 195.3 191.1 18&.2 180.0 170.8 160.6 209.3 206.2 202.8 199.1 194.9 190.0 183.7 174.4 164.1 213.3 210.1' 206.7· 202,9 198.7 193.7 181.4 178.1 167.7 217.2 214.0 210.6 206.8 202.5 197.5 191.1 181.7 171.3 221.1 217.9 214.5 210.7 206.4 201.3 194.9 185.4 174.9 225.1 221.9 218.4 214.6 I 21.0.2 205.1 198.7 189.1 <78.6 229.1 225.9 222.3 218.5 i 214.1 209.0 202.$. 192.9 182.3 233.1 229.9 226.3 222.4 218.0 212.8 206,3 196.6 1.85.9 237.2 233.9 230.3 226.4 221.9 216.7 210.1 200.4 ~89.7 241.2 237.9 234.3 230.3 225.9 220.6 214.0 204.2 193 •. 4 245.3 242.0 238.3 234.3 229.8 224.5 217.9 208.0 197.2 249.4 246,0 242,3 238.3 233.8 228.5 221.8 211.9 200.9 253.5 250.1 246.4 242.3 237.8 232.4 225.7 215.7 204.7 257.6 254.2 250,5 246.4 241.8 236.4 229.6 219.6 208.5 26).7 258.3 254.5 250.1 245.8 240.4 233.6 223.5 212.4 265.9 262.4 258.7 254.5 249.9 244'.4 237.5 227,4 216.2 270.1 266.6 262.8 258.6 253.9 248.4 241.5 231.3 220.1 274.3 270.7 266,9 262,71258.0 252.5 245.5-235.3 224.0 278.5 274.9 271.1 266,9 262.1 256.5 249.5 239.3 227.9 282.7 279.1 275.3 271,.0 266.2 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1.1705035.pdf	Journal of Mathematical Physics 7, 1310 (1966); https://doi.org/10.1063/1.1705035 7, 1310 © 1966 The American Institute of Physics.Global Covariant Conservation Laws in Riemannian Spaces. II Cite as: Journal of Mathematical Physics 7, 1310 (1966); https://doi.org/10.1063/1.1705035 Submitted: 09 November 1965 . Published Online: 22 December 2004 Bohdan Shepelavey ARTICLES YOU MAY BE INTERESTED IN Conservation Laws for Free Fields Journal of Mathematical Physics 6, 1022 (1965); https://doi.org/10.1063/1.1704363 Global Covariant Conservation Laws in Riemannian Spaces. I Journal of Mathematical Physics 7, 1303 (1966); https://doi.org/10.1063/1.1705034JOURNAL OF MATHEMATICAL PHYSICS VOLUME 7, NUMBER 7 JULY 1966 Global Covariant Conservation Laws in Riemannian Spaces. II BOHDAN SHEPELA VEY General Electric Company, HMED, Syracuse, New York (Received 9 November 1965) Using the idea of tensor integration, the vector field developed in Part I of this report, and the full Bianchi identities, it is shown that in a general Riemannian space there are four global covar iantly-conserved tensors. The ranks of these tensors are three, four, five, and six. The traces of the first two of these tensors yield the generally covariant equivalent of the familiar linear and angular momentum. The remaining four traceless tensors describe, residually, the gravitational field. With each covariantly conserved tensor one can associate a number of independent invariants. Such in variants are conserved in the ordinary sense. Among these are two types of rest energies and two types of angular momentum magnitudes obtained from the trace and traceless tensors. Examples of global, conserved tensors are derived for a Schwarzschild metric with the electron mass and a metric of a point electron. It is shown that the rest energy of the Coulomb field diverges as "In(l/r) at the origin and the second rest energy, that is, the rest energy of the gravitational field diverges as In r as r approaches infinity. When cutoffs are introduced at the Schwarzschild radius ro, at the classical electron radius rl, and at the radius of the visible universe r2, the rest energy of the gravitational field contained in the shell of thickness rl -ro is approximately 100 times that of the electron rest energy. It is twice this value in the entire visible universe. Since the gravitational field is described by the traceless tensors and the former forms a heavy, compact cloud around the point particle, it is conjectured that the traceless tensors represent the internal degrees of freedom of the elementary particles. 1. INTRODUCTION THE objective of this report is to demonstrate, by an actual construction and a specific appli cation, the existence of a finite and unique set of global, covariantly conserved tensors in a general Riemannian space. The requisite mathematical tools for this task have been developed in Part I of this report and they are readily recognized to be a key factor in Part II, although no emphasis is being put on them here. The local covariant conservation laws, which go beyond the commonly accepted laws, are formulated in Sec. 2. They are derived from the full Bianchi identities and the vector field of Part I in a form of a fourth-rank tensor and its first three moments, all of them with an identically vanishing covariant divergence. The use of a fourth-rank tensor in a similar capacity is suggested by Trautman,l but not in a generally covariant context. In Sec. 3, tensor integration and the Gauss theorem for such integration is used to convert the locally conserved tensors into the corresponding global tensors whose ranks, due to integration, are lowered by one unit. These global tensors are also covariantly conserved provided that certain surface integrals vanish. It is shown that, in spaces where this provision is satisfied, the algebraic structure of the four global tensors admits of a decomposition 1 A. Trautman, in Gravitation, An Introduction to Current Research, L. Witten, Ed. (John Wiley & Sons, Inc., New York, 1962), Chap. 5, pp. 183-188. into two trace tensors of ranks one and two and four traceless tensors of ranks three, four, five, and six. All of these tensors are separately conserved. The trace tensors are the linear and angular momenta of the matter fields. The traceless tensors are the zeroth, first, second, and third moments of the fourth-rank energy-momentum tensor of the gravi tational field plus the matter field. In Sec. 5 the third-rank global tensor is obtained for the Riemannian spaces of the Schwarz schild metric and the metric of a point electron. This enables one, for the first time, to calculate in a manifestly covariant manner the total rest energy of the gravitational field. The final section concludes this report with an interpretation of the traceless tensors, besides the linear and angular momentum, as the additional degrees of freedom of a generally covariant dy namical system. 2. LOCAL CONSERVATION LAWS 2.1 Bianchi Identities The field equations of the theory of general relativity are R"' -!g~'R = (87rkN)T~', (1) where R~P is the Ricci tensor, R is the scalar curva ture, and T~' is the energy-momentum tensor of the matter. In the absence of matter, that is, when T~P is identically zero, Eq. (1) describes gravitational radiation. It is then typical of gravitational radiation 1310 CONSERVATION LAWS IN RIEMANNIAN SPACES. II 1311 that, if it carries any energy and momentum, a covariant quantity describing the energy and mo mentum of gravitational radiation must assume a form which is different from the terms of Eq. (1) (that is it cannot be a second-rank symmetric ten sor). This is significant in the fact that, in Lorentz covariant theories,2 the matter tensor T~' describes completely and adequately the energy and the mo mentum state of a dynamical system. In such theo ries the symmetric tensor T~' satisfies (2) From Eq. (2) and the symmetry of T~' it follows that Equations (2) and (3) define ten local conservation laws which, in Lorentz covariant theories, can be easily integrated. In the theory of general relativity one could proceed to obtain ten conservation laws in exactly the same way. It is known that Eq. (1) satisfies a generally covariant equivalent of Eq. (2) --.! (W' -,1 ~'R) = 87rk --.! T~' = o. ox" 2Y c4 ox~ (4) Similarly, it is shown later that, using the symmetry of T1 " and the vector field XI' introduced in Part I, a generally covariant equivalent of Eq. (3) also holds. From these generally covariant local con servation laws, ten global conservation laws can be obtained in complete analogy to the Lorentz case by means of the tensor integration developed In Part 1. Although the initial objective here, as well as in many other investigations,3-5 was to exhibit the ten conserved quantities of the Lorentz group in the theory of general relativity, it is obvious that having done this the case of the generally covariant conservation laws cannot be considered closed. There still remains the question of energy and momentum transfer by gravitational radiation. If it is assumed that the energy and momentum of a dynamical system are solely expressed by and derived from 2 L. Landau and E. Lifshitz, The Classical Theory of Fields, translated from the Russian by M. Hamermesh (Addison Wesley Publishing Corporation, Inc., Reading, Massachusetts, 1951), p. 80. a Ref. 2, pp. 316-323. 4 J. Rayski, "Conservation Laws in General Relativity," Bull. Polish Acad. Sci. 9, 33 (1961). fi C. Mj3ller, Tetrad, Fields and Conservation Laws in General Relativity, in Proc. Intern. School Phys. "Enrico Fermi,"June-July 1961. the tensor T"', one has to conclude that gravitational radiation cannot possess energy and momentum. At present the experimental evidence is inconclusive and can neither deny nor confirm this assumption. However, unless there is experimental evidence to the contrary, one would like to believe that gravi tational radiation, just like other types of radiation, is a carrier of energy and momentum. When this point of view is adopted, it is clear that there ought to be another conserved tensor besides T"·. The same question put in a more formal way is whether in the Riemannian geometry there are other tensors besides R'" -ty'" R with a vanishing covariant divergence. In order to look for such tensors, it is best to discard the tensor TI" which is not an object of the Riemannian geometry and to consider the tensor (c4/87rk)(R'" -tg"'R) instead. Its vanishing divergence expressed by Eq. (4) is a direct con sequence of the contracted Bianchi identies. At this point it is natural to go back to the general statement of the Bianchi identities and see if they yield a vanishing divergence of a tensor less re stricted than R'" -tt'R. If the Riemann curvature tensor and the Ricci tensor are defined as (5) then the Bianchi identities take the form It is seen by inspection that, when Eq. (6) is contracted on T and (T, one obtains a vanishing divergence of a linear combination of the Riemann curvature tensor, namely From the first term in (7) one concludes that this linear combination should possess all the sym metries of the Riemann curvature tensor. When the index (T is lowered, however, the Ricci tensor terms do not exhibit the required symmetry in the pair of (TP indices. This situation can be easily corrected if one considers two contracted versions of Eq. (7), Adding Eqs. (7) and (8) together one obtains (9) 1312 BOHDAN SHEPELAVEY where the tensor T p.p ~ is defined as has to state the acceptability criteria for such a con servation law. In generally covariant field theories, it seems reasonable to require that a conserved Tp./ == (-c4/87rk) IRp./ + gppR: -o~R.p + o;Rpp -g.pR: + !R(g.po: -gppo:»). (10) quantity be It can be verified directly that the sum of the Ricci tensor terms in Eq. (10) possess the same symmetries as the Riemann curvature tensor; consequently TP'P~ + T'PP~ = 0, (11) TP'P~ + TP'~P = 0, (12) TP'P~ -TP~P' = 0, (13) TjJ,/ + T,p/ + Tpp/ = O. (14) The gravitational constant in the definition of Tp,/ in Eq. (10) was introduced in order to make Tpvp~ assume the dimensions of energy. One observes that the contracted Tp.p~ coincides with the Einstein tensor R,p -!g.pR times the gravitational constant, and, modulo the field equations in Eq. (1), it is equal to the energy-momentum tensor of the matter. Tm" = T.p = (c4/87rk)(R,p -!g,pR). (15) Thus, the matter tensor is a trace of a higher-rank tensor Tp./. The former's vanishing divergence can be thought of as following from the local conser vation law satisfied by Tp./ as expressed by Eq. (9). The tensor T P'/ can be invariantly decomposed into its traceless and trace tensors just like the Riemann tensor, 6 Tp • ." = (-c4/87rk)lCp./ + p~[.gpJp + pp[pg'J~ + fiR(g,pgp~ -gppg.~»), (16) where C P'/ is the Weyl conform tensor, P p, is the traceless part of the Ricci tensor Rp. -tgp,R, and the square brackets around the indices indicate the antisymmetric part [J,Lv] = !(J,LV -VJ,L). Gravi tational radiation is characterized by a vanishing of all parts in Tp,/ except for the Weyl tensor Cp./, so that the latter may be interpreted as the energy momentum tensor of the gravitational field (radia tion). It is covariantly conserved only when nothing else but gravitational field is present. When matter is present then the sum of Cp,/ and the matter tensor in the form prescribed by Eq. (16) is co variantly conserved together. Having found a more general conservation law than that of Eq. (4), it is of interest to ascertain whether it is unique. Before this is undertaken one 8 J. Ehlers and W. Kundt, Ref. 1, Chap. 2. (1) a covariant quantity, (2) with a vanishing covariant divergence, (3) containing quadratic terms in the first deriv- atives of the field variables. To these requirements may be added the traditional one that the conserved quantity should not contain higher than the first derivatives of the field variables. For the gravitational field gp. it cannot be reconciled with the first two requirements which, here, are considered more important, therefore exemption from higher derivatives is dropped.7 Moreover, it should be noted that some conserved quantities, homogeneously linear in higher deriv atives of the field variables, have recently been dis covered by Lipkin,8 but since they do not seem to be of practical importance9 they are excluded here by the requirement (3). Within the Riemannian geometry there is only one tensor that satisfies requirements (1) and (3), namely, the Riemann curvature tensor. But the tensor Tp./, being a linear combination of the latter and possessing its symmetries, satisfies (1) and (3) as a matter of course. In addition it satisfies (2) in view of the Bianchi identities. Thus the uniqueness of Tp./ is established. 2.2 Angular and Higher Moments In Lorentz-covariant theories the symmetry of the matter tensor and its vanishing divergence led to a conservation law for the angular momentum in Eq. (3). This approach can be utilized in generally covariant theories. A new element that is needed for this purpose is the position vector. In Riemannian geometry no such vector exists; however, in Part I a vector field XI' was derived which can be used in the capacity of a position vector. It is defined by the differential equation oXP = dXJJ + rJJ dxa XfJ = dX"". (17) ou du afJ du du 7 If another affine connection, e.g., that of a flat space r;..,. is admitted into the Riemannian geometry, then the tensor </>;.., = r;., -r;., offers a possibility to eliminate higher than the first derivatives. However, attempts have so far failed to prove the existence or nonexistence of a quadratic expression in </> with a vanishing divergence. S D. M. Lipkin, J. Math. Phys. 5, 696 (1964). 9 T. W. B. Kibble, J. Math. Phys. 6, 1022 (1965). CONSERVATION LAWS IN RIEMANNIAN SPACES. II 1313 If the curve u is a geodesic x'" = x"'(u) determined by the equations (18) where E = -1,0, +1 if dx"'jdu is timelike, null, or spacelike, then the vector X", is dX""'" X"'=u- . du '" (19) The vector X'" depends on two points U1 and U2 and a geodesic u that passes through these two points. If U1 is made to coincide with the origin of the coordinate system and the point u~ is allowed to wander over the entire domain, then with each point of the domain one can associate a vector X"'. Next, it is necessary to determine the covariant derivative of X'" at any point of the domain if it is to be used in the role of the position vector. For this purpose, consider the solution of Eq. (18) which assumes the form x'" = x"'(x~, p"', u), (20) where x~ and p'" are constants of integration. More explicitly, x'" can be expanded as a power series10 in u, + 1 A'" 3" P "I + 3! "p-yup pp (21) Here, A'" are the r's and their derivatives evaluated at x'" = x~. They are obtained by repeated differ entiation of Eq. (18). The constants of integration are sufficient to pass the geodesic curve through any two desired points. Since one of the points is to be the origin, x~ must be set equal to zero (x~ = 0). The other constants p'" describe the direction of the geodesic at the origin, that is dx"" p"'---du .. -0' Any point may be specified by prescribing either its coordinates x'" or alternatively by stating the corresponding values of u and p"'. Consequently, x'" may be considered as functions of x~, p"', and u [as is shown in Eq. (20)], but in our case x~ are fixed so that x'" depend only on p'" and u. It follows from Eq. (21) by direct calculation [and therefore must also be true of Eq. (20)] that 10 J. L. Synge and A. Schild, Tensor Calculus (The Univer sity of Toronto Press, Toronto, 1952), p. 60. axP axP axP axP p"l -= U -or -= u -p (22a) ap"l au fJp -y au "I , , a2x'" 2 a2x'" p ap"l ap" = u au2 p..,. (22b) Here p.., = aujax"ll .. _o and, since the right-hand sides of Eq. (22) are obtained at pet = const, it follows that dx'" au I p"'p" = -- = 6~. du ax" .. -0 In view of Eq. (20) the covariant derivative of X" may be written as 6X'" au 6X'" ap"l 6X'" -=---+--. (23) 6x" ax" 6u ax" 6'p'Y Each term in Eq. (23) can be evaluated by sub stitution of u(ax"'jau) for X"', where dx"'jdu is now written as ax'" j au due to the fact that x'" is considered also a function of the integration constants p'" as is indicated in Eq. (20). The covariant derivatives in Eq. (23) are 6X'" _ ax'" u(a2xl> rl> ax" axP) 6u -au + au2 + "p au au ' 6XI> = ~ (u axl» + rl> u ax" axP • op'Y ap'Y au "p au fJp'Y With the repeated use of Eqs. (22), the expression may be converted to 6XI> = axl> u2 ap" (a2xl> r'" ax" axP). 6'p'Y ap'Y + p" ax'Y au2 + "P au au (24) (25) last (26) Due to Eq. (18) the last terms in Eqs. (24) and (26) vanish so that the final result is 6XI> au ax'" ap -y axl> dx'" -=--+--=-= 6'" (27) 6x" ax" au ax" ap'Y dx" '" This result suffices to make the intended use of the vector X"'. In analogy to Eq. (3), the angular momentum of the energy-momentum tensor T""P' can be formulated as It is satisfied, in view of Eqs. (9), (13), (27), and the distributive character of the covariant derivative indicated by the symbol D".. It should be noted that the trace of Eq. (28) reduces to the generally covariant equivalent of Eq. (3), (29) 1314 BOHDAN SHEPELAVEY The other three symmetries of T~vpv in Eqs. (11), (12), and (14) can be utilized to write down similar expressions to that in Eq. (28); however, they are nothing more than linear combinations of Eq. (28) so that they need not be considered. Since the divergence of TJl.vpv in Eq. (9) contains three free indices, it is possible to formulate higher moments of TJl.vpv with a vanishing divergence. One can easily verify that Dk L [(TkJl.PVXV -ThV~XP)XT] = 0, (30) JIoH tensor density. It was shown in Part I that the integral of the divergenceless tensor density ~~ (free indices suppressed on ~~) can be written as o (1 ~ l' r r ° • ; k) oxo 3! -& JJ ~ oX oX ox v. where ijk refer to the spatial components. The same result can be expressed more compactly, o ° 3 0'11'_ D p~ = El!..-+ ~ =-0 ~ oxo ~ ox' - , (33) where Lm stands for a sum of three terms in which JJ.JlT are cyclicly permuted. Since the diver gence of the parentheses vanishes by Eq. (28), it is sufficient to show that where in order to prove Eq. (30). But the last expression is zero in view of Eq. (14). Finally the divergence of the third moment of T~vpV also vanishes if it is defined by Dk L L [(Tk~pVXV -TkvV~XP)X'X'] = O. (31) Again one needs only prove that L L (T'P.pvxv -Tm~xp)XT = o. ptlf J,I."T Summing this expression first on PUE, we get L {(T'~PVXv -TfVV~XP)XT + (TP~v·X' -TP"~XV)XT + (Tv~"XP -TVVP~X')XTI = o. Thus it is zero because terms cancel in pairs due to Eq. (13). It is not possible to formulate higher moments of T~vpv than the third, because in Eq. (31) there are no more free indices left in the tensor TJl.vpv for mixing with the index of the vector XI'. This is also true of the tensor TJIoV in Eq. (29), consequently the higher moments conservation laws corresponding to Eqs. (30) and (31) do not exist in the Lorentz covariant theories. 3. CONSERVED GLOBAL TENSORS Anyone of the locally conserved quantities in Eqs. (9), (28), (30), or (31) can be converted into a global tensor if it is integrated over some volume of the Riemannian space. Such integration requires in the integrand a factor of the Jacobian of trans formation gl, where g is the absolute value of the metric tensor determinant. This factor, being covari antly constant (D~i = 0), can be pulled inside the divergence to make the tensor in question a pI' = i! ~ III ~J1. ox' ox; oxk • v. Equation (32) [or (33)] is the global conservation law, or rather the global equation of continuity, which states that the covariant rate at which the amount of the quantity ~ changes in the spatial volume V3 is equal to the flow of that quantity through the surface V2 bounding the volume V3• If the flow through some surface is zero for each component of the tensor, then the amount of the tensor within the corresponding volume remains covariantly constant in time. The above statement implies that associated with this volume there are n quantities which are con served in the ordinary sense. Here n is the number of independent invariants that can be formed from the tensor p. Clearly, from Eq. (34), it follows that ~ I.(p) = ali ~ = aI, = 0 i = 1 ... n (35) oxo. iJp oxo iJxo, '" I, (p) being the independent invariants of p. From now on only those spaces, in which the surface integrals at the spatial infinity vanish for the tensors of Eqs. (9), (28), (30), and (31), are considered. In the stipulated spaces there are four covariantly conserved tensors which exceed the corresponding tensors of the Lorentz group in rank and number. Since only the latter are well understood, many questions relating to the algebraic properties, phys ical meaning and importance of the four tensors remain to be answered. Although no thorough investigation of them has been undertaken so far, CONSERVATION LAWS IN RIEMANNIAN SPACES. II 1315 it is possible to comment on the more obvious algebraic structure as well as to make some inferences about the importance of these conservation laws. Thus, when matter is completely absent (T!" = 0), the Ricci tensor vanishes in view of the field equa tions (1), and the only surviving part in the energy tensor T!'P"u is the Weyl tensor C'''''''. In such a space filled with the gravitational radiation (field) only, none of the conserved tensors vanish identi cally. Consequently, the residual part of each conserved tensor pertains to the gravitational field. When matter is present, it is possible to split the conserved tensors invariantly into the matter part consisting of the same type of linear or angular momentum as in the Lorentz-covariant theory and the new, higher-rank traceless tensor which consists of the residual gravitational field plus those con tributions of the matter fields which interact with the gravitational radiation. This is well examplified by the global tensors derived from Eqs. (9) and (28). The first of these, pl"P, is a third-rank tensor antisymmetric in lip, There are six components in T!""u without a single zero index, so that p!"P consists of 14 components. It can be split into its traceless and trace parts as follows: pl"P = q!"P + Hrl'p" -gPPp'), (37) where p" = g!'.p!'v" is the linear energy-momentum vector of the matter fields. From the definition of ql"P in Eq. (37), it follows that ql"P is a sum of two terms (38) Here cm is the integral of the conformal tensor density gie°l"p. It represents the gravitational field contribution to the energy and momentum of the system. m'<PP is the integral of the tensor density entirely defined by the Ricci tensor so that it represents the contribution of the matter fields to the traceless energy-momentum tensor ql'Vp. The decomposition of pl"P into two mutually orthogonal tensors in Eq. (37) (that is, a 10 component tensor q'H" and a four-component vector pi» is invariant with regard to the general coordinate transfor mations. A number of significant consequences can be drawn from this invariance. First, the tensors q!'vp and pI' are conserved separately, (39) Secondly, the magnitude of pP, p = (Ipppp!)l, and the magnitude of qP'P, q = (\qI'VPql'vpD', being two independent invariants of pm, are both constants of the motion a axoP=O; (40) In analogy to p, which is the rest energy (or rest mass) of the dynamical system consisting of the matter fields, q can be interpreted as the rest energy of the gravitational field and those parts of the matter fields which interact with it. Thus, in gen erally covariant theories, each dynamical system, is characterized not by one but by two rest masses. Thirdly, there can be no exchange of the linear energy-momentum (pP) between the gravitational field and any of the matter fields due to the fact that the energy and momentum of the gravitational field is always expressed by the traceless tensor ql"p. Exchange of energy and momentum between the matter fields and the gravitational field is allowed by means of the tensors cl'VP and m'<P", for neither of them is individually conserved although their sum is. The latter part of the third conclusion has been known in various forms,l1 namely, that the lowest observable interaction mode between the gravi tational radiation and a test particle is through quadrupole oscillations. The global tensor derived from Eq. (28) expressing the conservation of angular momentum is a fourth rank tensor v. (41) It is antisymmetric with regard to the transposition of the first and the second pair of indices, (42) The maximum number of independent components in pH"" cannot exceed the product of the components of ra'p· and X' or 14 X 4 = 56. Among these components, however, nine are identically zero, plOlO = pl020 = pl030 = p2020 = p2030 = 0, p2121 = pa030 = paUl = p3232 = 0, and five differ only by a sign, p2120 = _p2021, p2312 = _p1232, pal30 = _pSOSI, pUS1 = _p3132, p3230 = _paon. ------: 11 B. DeWitt, in Ref. 1, p. 340. 1316 BOHDAN SHEPELAVEY Consequently, pTPPd consists of 42 independent com ponents. When it is contracted on lilT, a generally covariant equivalent of the angular momentum in the Lorentz theory is obtained. Thus the trace of the tensor pHPd is the angular momentum of the matter fields. The other trace of p.,Pd, g,ppTPPd, consists of p.d and another part dependent on the Weyl conform tensor C, (44) In analogy to the previously considered tensor, pTPPV can also be decomposed into its traceless and trace parts (45) where qTPPd is the traceless tensor with regard to the index pair lilT. Again, the tensor qTPPV is a sum of two different terms, the residual term that is the integral of the conform tensor density g!(COVPVXT - ComXp ) which may be interpreted as the angular momentum of the gravitational field, and the matter term-the integral, whose density consists only of the Ricci tensor, (46) From Eq. (45) it follows that the six-component tensor pTP and the 36-component tensor qTPPV are conserved separately (47) Their magnitudes, being two independent invariants of pTPPV, are constants of the motion (48) Conclusions drawn about the energy and momentum tensor ppvp are equally valid for the angular mo mentum tensor pHPV when appropriate terms re ferring to momentum are substituted with terms that refer to angular momentum. In a generally covariant dynamical system, the angular momentum of the matter fields is described by the familiar six-component, antisymmetric, covariantly conserved tensor pTP. The angular mo mentum of the gravitational field and of the matter fields which interact with the former is described by a new traceless, fourth-rank, covariantly con served tensor q'vpa. There can be no exchange of the angular mo mentum pTP between the gravitational field and the matter fields due to the traceless character of qTPPd. The exchange of the angular momentum between the gravitational radiation and the matter fields is allowed by means of the tensors cTPPd and mTPPd, since neither of them is conserved. Therefore, with a generally covariant dynamical system, one associates not only two rest energies or masses but also two types of angular momentum. Magnitudes of these momenta are two independent constants of the system. The remaining two con served tensors derived from Eqs. (30) and (31) obviously cannot be decomposed into the traceless tensors and the lower-rank trace tensors of the Lorentz group. In the Lorentz-covariant theories, there are no known conserved tensors expressing either the second or the third moments of the energy and momentum to fulfill the role of traces. It is concluded, then, that both these tensors are of the same character as the previously discussed q tensors, that is they govern the exchange of the second and the third energy-momentum moments between the gravitational field and the matter fields. This is also borne out by the fact that traces of rank one and two of these tensors do not reduce to any tensors of the Lorentz group but vanish identically. The second moment tensor of Eq. (30) is of the fifth rank. It will be written as -TOVVPXP)XT 5x' 5xi DXk, (49) where, it is recalled, LpVT stands for a sum of three terms in which MilT are cyclicly permuted. Using the symmetry arguments below one can show that (50) so that qTaPPV is indeed traceless. A number of symmetries of q'vpPv follow directly from its definition in Eq. (49). (51) They can be easily checked by writing out the sum LpVT explicitly. The number of independent components of qTaPP' cannot exceed the product of the components of TOpp• and X'XT or 14 X 10 = 140. However, in view of the above symmetries, many components may be linearly related or may vanish identically. CONSERVATION LAWS IN RIEMANNIAN SPACES. II 1317 The last conserved tensor describing the third moment of the energy and momentum is a six rank tensor qHI'PPE, where v. -TO"I'XP)XTX' ox; ox; oxk• (53) In view of the sums LI'PT and Lpu. in the definition of qHI'PPE, it is obvious that qHI'PPE does not change when either }J.VT or PUE are cyclicly permuted as in Eq. (51). When the sums are written out explicitly, one recognizes by inspection the following sym metries: qTUI'PPE is completely antisymmetric in the three indices }J.lIT and completely symmetric in the remaining three indices pUE. Obviously, all traces of the antisymmetric index pairs }J.1I, }J.P, VP vanish. The symmetric index triple PUE generates 20 distinct components and the antisymmetric one only four, consequently, the number of independent com ponents of qHI'PPE should not exceed the product of these, that is 80 components. 4. SUMMARY In generally covariant theories, a dynamical sys tem is characterized by four global tensors which express conservation of the system's energy and mo mentum, and their first three moments. The first two tensors are reducible into traceless and trace tensors. The latter constitute the generally covariant equiv alents of the conserved tensors of the Lorentz theoryj however, contrary to common expectations, they say nothing about the energy or momentum of the gravitational field but describe exclusively the matter fields. The gravitational field energy and momentum as well as its first three moments are contained in the four traceless tensors of the third, fourth, fifth, and sixth rank, respectively. The same tensors contain also the contribution from the matter fields via the Ricci tensor terms. Each traceless tensor is covariantly conserved, but the gravitational field and the matter field parts in it are not conserved separately. This fact allows for interaction of the gravitational field and matter fields with an exchange of energy and momentum or any of its first three moments between them. Conservation of the second and third moments of the energy and momentum arise in consequence of the high rank of the energy-momentum tensor. The latter is required by the nature of the gravi tational field in the general relativity theory. One can easily see by considering quadrupole or higher order multipole radiation that momentum trans-ferred to a test particle by such radiation is not confined to one direction, as is the case with the linear momentum, but is distributed in different directions simultaneously. Such distribution can be described only by a tensor of higher rank. More over, the net linear momentum in anyone direction imparted by the quadrupole or higher-order multi pole radiation to the test particle is zero. This explains the traceless character of the four q tensors. 5. EXAMPLES OF GLOBAL CONSERVED TENSORS Implications and usefulness of the global conserved tensors are easily shown by obtaining examples of some of these tensors in specific Riemannian spaces. This also serves other purposes. First, the method of tensor integration is demonstrated. Secondly, it is possible to show that some of the assumptions made, such as the one about vanishing of certain surface integrals at the spatial infinity, do not lead to a trivial class of conserved tensors. Finally, the numerical results have an important bearing on the physical interpretation of spaces under consideration. Also, they may shed some light not only on the relative magnitudes of the matter tensors and the q tensors, but, hopefully, indicate their relative importance in general. Two considered spaces are that of a neutral mass m or the Schwarzschild metric and that of a mass m with an electric charge e. The metric and the curvature tensor need be specified only for the latter since they become identical to those of the neutral space when the charge is set equal to zero (e = 0). This metric and the corresponding nonzero r's are12-14 + dT2 + 2(d2 + . 2 d2 \ 1 ( /) + ( / 2) T () sm () !Ph -To r roT1 T r~1 = !gll a1gu, r~2 = sin-2 (}r!a = rgooj r~2 = r;1 = r~a = r;1 = l/r, r;a = -sin () cos (}j r~a = r;2 = cot (). (54) Here ct, r, (), cp are the coordinates. The two constants 12 P. G. Bergmann, Introduction to the Theory of Relativity (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1953). 13 R. C. Tolman, Relativity, Thermodynamics and Cosmology (Clarendon Press, Oxford, England, 1934). 14 H. Weyl, Space-Time-Matter (Dover Publications, Inc, New York, 1950). 1318 BOHDAN SHEPELAVEY To and Tl are the Schwarzschild radius and the charge radius, respectively. (55) where k is the Newtonian gravitational constant which already appeared in Eq. (1). The six non vanishing components of the curvature tensor are R ° = (!:'!. _ 3Torl)(1 _ ?:2 + rorl)-1 . 011 r3 r4 r r2 , ordinary integration. The r integration does not, but it can be performed by taking the covariant derivative of Eq. (60) with regard to T and solving the resulting differential equation for pi ,0. This yields , - 4 ( )! I 2( )-lTO' d· p ,0 -11" -Yoo r -Yoo iO r, (61) _Y _ 1 _ ?:2 + rOri. 00 - 2 r r R 0 _ R 1 __ ! (?:2 _ 2rorl) 022 -122 -2 r r2' Substituting the expressions for TOi iO from Eq. (56) (57), the three components are R ° -! (?:2 _ 2rorI) . 2 f} -R 1. 033 = 2 r r2 sm -133, R 2 _ (?:2 _ rorI) . 2 f} 233 - 2 sm . r r The nonvanishing components of TI"pu containing at least one zero index are (57) All components of the form Ti/o i ~ 0 vanish identically so that the surface integrals of Eq. (32) are zero for all four tensor densities and for any arbitrary choice of the spatial surface V2• In view of Eq. (57) the energy-momentum tensor pl"P consists only of three components pi iO, i = 1,2, 3 which are given by piiO = ~ III r2 sin f}TO\o{Or(of} ocp + ocp 00) + 00( Or ocp + ocp or) + ocp( or 00 + 00 or)}. (58) If anyone of the six integrals in Eq. (58) is denoted by p!iO then it can be shown that (DiD; -D;Di)p~'o = (DiD; -D;Di)DkP~'o = 0, i, j, k, s = 1,2,3 i ~ j ~ k, n = 1, ... ,6, (59) so that all tensor integrations or, 00, ocp commute in this space. Consequently, pi iO are piiO = III r2sin OTOi;o or 00 ocp I 2 Oi = 411" r T ;0 or. (60) In the last integral the integrations over cp and 0 have been performed over the entire space. The result is 411", since they happen to coincide with the p\o = -mc2 ( -YoO)'CFl -F2), (62) p220 = p330 = -mc2(-Yoo)!(-tFI + F2). The functions F, are two integrals F == In [~(-Y )t + !. -!] + c 1 ro 00 ro 2 1, F2 == (~y {In [( -Yoo)! + (rO;I)! (63) where CI and C2 are the constants of integration which can be identified with the lower limit of the integral in Eq. (61) if it is definite. The trace of p' ,0 and its traceless tensor are 3 Po = L p\o = -mc2 ( -Yoo)'F2• ;=1 q\O = -mc2( -YOO)tCFI -tF2); (64) q220 = q330 = -mc2( -Yoo)t( -tFI + jF2). The magnitudes or rest energies of po and q' ,0 are p = mc2/F2/; q = mc2 /(WFI -(!)'F 2/. (65) In the Riemannian space of the Schwarzschild metric rl is zero so that F2 vanishes. But the rest energy of the gravitational field contained in the spherical shell of thickness r -r L does not vanish, and according to Eq. (65) is equal to _ 2(;).)t In r(1 -roM! + r -tro qSh-mc2 1 . o rL(1 -rO/rL) + rL -tro (66) Here C1 in Fl is chosen so as to coincide with the lower limit rL of the integral in Eq. (61). The lowest value that rL can assume is the Schwarzschild radius ro. The upper limit r may be made to approach infinity, in which case the rest energy of the gravi tational field diverges logarithmically. In order to obtain a finite result, it is necessary either to intro-CONSERVATION LAWS IN RIEMANNIAN SPACES. II 1319 duce a cutoff or to inquire about the energy contained in shells of finite thickness. First, the energy between ro and rl is calculated, where it is assumed that ro in Eq. (55) is produced by the smallest known mass m, that is, the mass of the electron and rl is of the order of nuclear size, or more accurately, the classical electron radius of Eq. (55) with e, m being the electron charge and mass. For this case, one obtains approximately (67) The energy of the gravitational field within the volume of the size of a nuclear particle is two orders of magnitude greater than the rest energy of the particle which produces the field. In the second calculation, let the upper limit r be extended up to the radius of the visible universe, that is, r = r2 = 1028 cm. In fact, it may be argued that this is the maximum that the upper limit should assume, since the regions beyond this point are not causally connected with the field-producing particle. The rest energy of the gravitational field within the volume of the size of the visible universe is only about twice of the value in Eq. (67), that is, (68) One concludes from this that the gravitational field energy is concentrated in the immediate neighbor hood of the Schwarzschild radius roo In the Riemannian space of the charged particle with the metric in Eq. (54) the field is characterized by two rest energies, that of the matter field which in this case is the Coulomb field and that of the gravitational field and matter field. The rest energy of the Coulomb field contained in a spherical shell of thickness r" -r is The upper limit r" can be extended to infinity with no ill consequences. However, the lower limit r, on approaching zero, yields a divergent result. This is not surprising since the Coulomb self-energy is known to diverge as 1/r. What is new here is that the general covariance removes one degree of divergence so that the Coulomb energy p in Eq. (69) diverges only logarithmically. To get a finite result for p a cutoff has to be introduced. Since the classical electron radius rl is indicative of the size of charge distribution, the lower limit may be chosen to be on the order of rl, say rl/n, where n is close to unity. With r" = CD one obtains ( )! 1 + (n -!)(:!:Q)! 2 r1 In r1 = nmc2• (70) p = me ;:;; 1 _ !. (:!:Q)t 2 rl In the calculation of the second rest energy q it is necessary to introduce also the upper cutoff r .. = r2. With these limits on the integral in Eq. (61) q becomes In the above calculation the cutoffs were imposed on both integrals FI and F2 of Eq. (64), however, each function F requires only one cutoff. If the function FI were to be calculated with the upper cutoff only, then q in Eq. (71) should be augmented by (72) The removal of the upper cutoff on F2 changes its value only infinitesimally since F 2 converges at Although the Riemannian space of the charged particle is considerably different from the space of a neutral particle, e.g., the charge removes the Schwarzschild interior region of r < ro, the quali tative features of both these spaces are the same. First, there is the field-producing particle repre sented by a singularity in the interior of the Schwarzschild radius or in the interior of the charge distribution. The latter is assumed to be a point charge due to the lack of a more satisfactory theo retical or experimental charge model. In view of its singular nature, this particle is not governed by the field equations. The singular particle is surrounded by two types of fields, the matter field provided the particle carries a matter "charge" and the gravitational field. The rest energy of the matter field is on the order of the rest energy of the singularity, whereas the rest energy of the gravitational field is two orders of magnitude greater than either of the other two. The "heavy" gravitational cloud surrounding the particle is mostly contained in a volume of the size of a nuclear particle. 1320 BOHDAN SHEPELAVEY 6. INTERPRETATION Appearance of the q tensors on the scene raises some questions in regard to their importance relative to the trace tensors, delineation of domains in which they are of primary significance, and their physical meaning. Answers to some of these questions are interrelated and have to be discussed jointly to a degree. It is shown in Sec. 3 that the class of four con served tensors divides into two subclasses, one containing tensors of the Lorentz theory, and the other consisting of high rank, traceless q tensors. Since tensors in each subclass are conserved sepa rately any measurement or knowledge of a tensor in one subclass does not extend to, or say anything about, a similar tensor in the other subclass. Thus the q tensors can be interpreted as those degrees of freedom which are necessary to specify a generally covariant dynamical system in addition to the familiar linear and angular momentum. Although it is not possible to ascribe different levels of importance to various degrees of freedom, the q tensors do seem to be more fundamental in the following sense. In a Riemannian space none of the q tensors need be zero when all matter fields vanish. At the same time none of the q tensors can be made to vanish in the presence of any nonzero matter field without violating the field equations. Usefulness of the q tensors in those theories, where the gravitational field has to be dealt with explicitly, is rather evident and need not be elabo-rated on here. What is most interesting, and at the same time least certain, is the speculation that the q tensors are the internal degrees of freedom of the elementary particles. This is strongly suggested by the picture of the particle which emerged from the specific examples of the conserved tensors con sidered in the previous section. Thus, can the second rest energy q account for heaviness of some ele mentary particles and is its conservation synony mous with the conservation of heavy particles? Is there any connection between the higher moments q tensors and the various particle spins? However, any such specific identifications are premature at this time. The conjecture that there is a connection between the q tensors and the internal degrees of freedom of the elementary particles can be proved or disproved only after these tensors are exhaustively studied and analyzed for their formal structure and symmetries, after they are applied to more realistic Riemannian spaces where cutoffs need not be introduced, and after one develops a set of observ abIes of these tensors. The correspondence, or a lack thereof, between the q tensors and the internal degrees of freedom of the particles will then be easily recognized. ACKNOWLEDGMENTS The author is indebted to Professor Rohrlich and others of Syracuse University for reading the first draft of the material appearing in Parts I and II and for their valuable criticisms and suggestions.
1.1725309.pdf	Integrated Intensity Measurements of the 1.9μ Bands of CO2 in the Temperature Range 1400° to 2500°K J. C. Breeze and C. C. Ferriso Citation: The Journal of Chemical Physics 40, 1276 (1964); doi: 10.1063/1.1725309 View online: http://dx.doi.org/10.1063/1.1725309 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/40/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermal conductivity of krypton in the temperature range 400–2500 K J. Chem. Phys. 63, 5052 (1975); 10.1063/1.431211 Integrated Intensity Measurements on the Fundamental and First Overtone Band Systems of CO Between 2500° and 5000°K J. Chem. Phys. 43, 3253 (1965); 10.1063/1.1697300 Integrated Intensity Measurements of the 5.3μ Fundamental and 2.7μ Overtone Bands of NO between 1400° and 2400°K J. Chem. Phys. 41, 3420 (1964); 10.1063/1.1725743 Spectral Emissivities and Integrated Intensities of the 1.87, 1.38, and 1.14μ H2O Bands between 1000° and 2200°K J. Chem. Phys. 41, 1668 (1964); 10.1063/1.1726142 ShockWave Integrated Intensity Measurements of the 2.7Micron CO2 Band between 1200° and 3000°K J. Chem. Phys. 39, 2619 (1963); 10.1063/1.1734073 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.189.203.83 On: Fri, 12 Dec 2014 06:57:27THE JOURNAL OF CHEMICAL PHYSICS VOLUME 40. NUMBER 5 1 MARCH 1964 Integrated Intensity Measurements of the 1.9-1-' Bands of CO2 in the Temperature Range 1400° to 25000K* J. C. BREEZE AND C. C. FERRISO Space Science Laboratory. General Dynamics Astronautics, San Diego, California 92112 (Received 28 October 1963) Measurements have been made of the total integrated band intensity of the 1.9-J' bands of CO2 in the temperature range 1400° to 2500oK. The gas was heated to the high temperature by a shock wave reflected from the rigid end plate of a shock tube. The experiment determines the total integrated band emission as a function of optical path length. The total emission is related to the integrated band intensity in a simple way. The intensity in the 1.9-J' region of the CO2 spectrum arises from three combination bands, namely the (v3+4v2), (va+2v2+vl), and (VJ+2Vl) bands. These band systems are in strong Fermi resonance. The bands have not been resolved; the total integrated intensity of the three bands was measured as a function of temperature. The temperature dependence of the absolute intensity is discussed in terms of a simple model using the harmonic oscillator approximation to the CO: molecule. The results indicate that the intensity in the 1.9-J' resonant triplet of CO2 originates in the (V3+2Vl) band. An extrapolation of the data using the derived temperature dependence gives an integrated band intensity of 2.07 (cm-2 atm-1) at STP for the total 1.9-J' C02 band. INTRODUCTION THE intensity in the 1.9-1-' region of the CO2 spec trum arises from three combination bands, namely, (V3+2vI), (V3+2v2+Vl), and (V3+4v2). Barker and Wul resolved the three bands at room temperature and assigned the band centers as 4860, 4982, and 5110 cm-l, respectively. Weber, Holm, and Penner,2 using the method of pressure broadening, measured the intensities of the three bands at room temperature. Their measured integrated intensities at 3000K are given as 0.426, 1.01, and 0.272 (cm-2'atm-l) for the (V3+2vI), (v3+2v2+Vl), and the (V3+4v2) bands, respec tively. The bands are located very near to each other and are expected to be in strong Fermi resonance. Unlike infrared fundamental bands, which are, in the harmonic oscillator approximation, independent of temperature, the integrated intensities of combination and overtone bands are highly temperature-dependent. Recently measurements of the integrated intensities of the fundamental 4.3-1-' band of CO2 and 2.7-1-' band of H20 have shown them to be temperature-independent3 while similar measurements on the 2.7-J.I. combination bands of CO2 have shown a marked temperature varia tion.' Furthermore, the particular variation of the integrated intensity with temperature of a complicated resonant multipet can be used to unscramble the in tensity contributions of the various components.' In the present study, a reflected shock wave tech nique' has been used in the determination of the inte- * This work was supported by the Advanced Research Projects Agency through the Office of Naval Research and by General Dynamics/ Astronau tics Research Funds. ' 1 E. F. Barker and T. Wu, Phys. Rev. 45,1 (1934). 2 D. Weber, R. J. Holm, and S. S. Penner, J. Chern. Phys. 20, 1820 (1952). 3 C. C. Ferriso, J. Chern. Phys. 37, 1955 (1962); C. C. Ferriso and C. B. Ludwig, J. Quant. Spectry. Rad. Transfer 4 (1963). • J. C. Breeze and C. C. Ferriso, J. Chern. Phys. 39 (1963); Report No. GDA63--{)240, General Dynamics/Astronautics San Diego, California, May 1963. ' grated intensity of the 1.9-1-' CO2 combination bands in the temperature range 1400° to 25000K. Since the intensities of the bands are almost completely mixed the bands have not been resolved, and the combined integrated intensity of the three bands is measured as a function of temperature. The observed temperature variation has been used to ascertain the important intensity component in the 1.9-1-' CO2 resonant triplet. In the present study a rapid response infrared de tector measures, as a function of time, the total band emission (in the 1.9-1-' region) from CO2 heated by a reflected shock wave. The detector views the test gas through a collimated optical system, in a direction parallel to the tube axis. Thus, the length of the test gas seen by the detector increases with time as the reflected shock recedes from the end plate. An absolute energy calibration of the detector and optical system is obtained using a standard blackbody source. The integrated band intensity is obtained from the limiting slope of a plot of the integrated emissivity versus optical path length. For a constant-velocity reflected shock, the optical path length is directly proportional to time measured from the instant of shock reflection. The spectral region of interest is isolated using a mono chromator, the spectral slitwidth of which is wide enough to include contributions from all parts of the band. As the detector measures the total integrated emission directly it is not necessary to pressure broaden the band. EXPERIMENTAL APPARATUS Shock Tube and Shock Measuring System The apparatus used in these experiments has been described previously' and only a brief description is given here. The shock tube is constructed of stainless steel and has a 2.33 in. internal diameter. The tube which is divided into an 11-ft driver section and a 29~ 1276 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.189.203.83 On: Fri, 12 Dec 2014 06:57:271.9-1" BANDS OF CO2 1277 ft. low-pressure section, is closed by an end plate in the center of which is a CaFz window one inch in diameter and a half inch thick. The tube can be evacuated to about 10-.1 cm Hg and has a leak rate of approximately 0.5 }l Hg per minute. The test gas used is pure CO2 (Matheson Company, Coleman grade). The minimum purity of this grade gas is 99.99% CO2 and it is used without further purification. The incident shock velocity is measured over several lengths of the low pressure section. The shock velocity at the end plate is obtained from an extrapolation of a plot of shock velocity versus distance from the diaphragm station. Infrared Detection System A schematic of the optical system is shown in Fig. 1. The system consists of external optics, a 10000K (IR Industries model 406) blackbody calibration source, and a Perkin-Elmer Model 98 monochromator fitted with a NaCI prism. The collecting optical system con sists of a 21 ° off-axis, 80-mm diameter paraboloid with a 203-mm focal length and a two-position, 70-mm square plane mirror which can be rotated to view either the shock tube or the blackbody source. The mono chromator exit image is brought to a focus on an Au doped germanium infrared cell (Westinghouse, Type 812) by a small Cassegrain system. The monochromator and optical system are sealed from the atmosphere and purged with dry nitrogen. An absolute energy calibra- 1 ion of the infrared detection system is obtained by rotating the two-position plane mirror and filling the identical monochromator aperture with radiation from the blackbody source. Experimental Procedure In operation the low-pressure section of the tube is evacuated to 10-5 cm Hg and after isolation from the pumping unit is filled to the desired test pressure with pure CO2• The initial channel pressure is adjusted so that the final reflected shock pressure P6, is of the order of 3 to 3.5 atm. A typical oscillogram of the detector output for the 1.9-}l CO2 bands is shown in Fig. 2. The lower trace in Fig. 2 is the output from the heat resistance gauge situated in the end plate and records the instant of arrival of the incident shock at the end plate. The BLACKBODY CALIBRATION o~, SHOCK TUBE r-"'::'-""...-:--1'"--'---.., Au-Ge DETECTOR CaFaWINDOW OPTICAL PATH FIG. 1. Schemate of the optical and detector systems. HEAT GAUGE 20 fLsec 1.9 fL C02 COMBINATION BANDS REFLECTED SHOCK TEMPERATURE 1,874°K PURE C02 FIG. 2. Typical radiation oscillogram. central trace represents the zero energy level and is obtained by photographing the oscilloscope sweep before recording the radiation trace. The upper trace of Fig. 2 represents the detector response to the energy emitted by the shock. Since the gas behind the in cident shock is at a relatively low temperature (",,8000K) the energy emitted in the 1.9-}l region by this gas (represented by the displacement of the initial level portion of Fig. 2) is extremely small. The rise in detector output represents the detector response to the energy emitted from the reflected shock region, the length of which increases with time. Determination of the Test Gas Parameters Computations of the test gas parameters for the incident and reflected shock waves were made using the Los Alamos Shock Parameter Code GNX-7 (courtesy of G. L. Schott, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico). A schematic of the wave diagram in the shock tube is shown in Fig. 3. The standard notation for state func tions is used: the subscript 1 refers to the initial down stream conditions; 2 to conditions behind the incident shock; 3 to conditions behind the contact surface; 4 to initial upstream conditions; and 5 to conditions behipd the reflected shock. The initial pressure PI, the initial temperature TI, and the incident shock velocity V., are sufficient to define the final reflected shock parameters. Spectroscopic Measurements In order to generate a constant spectral response over the width of the band, the monochromator is used with entrance and exit slits of 1250-and 650-}l mechanical slitwidths, respectively. The monochromator is centered at approximately 4980 cm-l. The shape of the monochromator slit function was measured by irradiating the slit with light from the standard black body source and recording the detector response. A bandpass filter placed in the optical path between the monochromator and the blackbody source isolates a spectral region which is narrow in comparison to the total spectral band pass of the monochromator. The This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.189.203.83 On: Fri, 12 Dec 2014 06:57:271278 J. C. BREEZE AND C. C. FERRISO 10 to TIME m ... 10 Ms =3.779 FOR PURE C02 10 DISTANCE (FEET) CD 20 ~IGH-PRESSURE lDIAPHRAGM OBSERVATION WINDCM'V LSHOCK TUBE LOW-PRESSURE SECTION SECTION FIG. 3. Schemate of wave diagram. particular filter used in the slit function determination is an interference filter having a maximum transmission at 1.981 /-I and a width at half-height of 188 cm-l• The monochromator is scanned as a function of wavelength and the detector response recorded. The experimental results are plotted in Fig. 4, which also includes the shape of the slit function corrected for the finite width of the filter bandpass. The width of the level portion of the slit function is 780 cm-l and is therefore approxi mately twice the width of the 1.9-/-1 CO2 bands which at room temperature have a total width of approxi mately 400 cm-I.I The base width of the slit function is 2150 cm-I and is therefore sufficiently narrow so that any appreciable contributions from the 2.7-/-1 CO2 combination bands are excluded. The spectral base width is also narrow enough to avoid overlapping the resonant peak region of the Au-doped germanium detector. For the particular detector used the resonant peak occurs in the region 6000 to 6700 cm-I. Outside of this peak in the region 4000 to 6000 cm-l (i.e., region covering the 1.9-/-1 CO2 bands) the detector sensitivity is sensibly constant. INTEGRATED INTENSITY MEASUREMENTS The experiment measures the integrated emission from the shock-heated CO2 as a function of time. The total emission can be related to the integrated band intensity in the following way. Since the gas behind the incident shock is at a rela tively low temperature the amount of energy emitted by this gas is negligibly small in comparison with the energy emitted by the high-temperature test region behind the reflected shock wave. Consider conditions at time t, where t is measured from the instant of shock reflection at the end plate. Thus the total length 1 of emitting gas is given by (1) where U, is the reflected shock velocity referred to the laboratory system of coordinates. Of the energy in the 1.9-/-1 region emitted by the column of test gas of length 1 and temperature To, an amount of energy Jt will, after passage through the monochromator, be incident on the detector; Jt= ( e(v, To)RO(v, To)g( I vLv I ,b)dv, (2) Jband where e(v, Ts) is the spectral emissivity of the test gas at temperature To, RO(v, To) is the spectral radiance of a blackbody at temperature To and g( I vO-v I , b) is the spectral transmission of the monochromator at wavelength v when it is centered at 11°. Here b represents all other instrumental constants. Since g( 111°_11 I , b) is constant over the width of 1.9-/-1 bands (d. Fig. 4) then Eq. (2) may be rewritten as Jt=l e(v, To)RO(II, Ts)dll=BV t, (3) band where Vt is the voltage response to energy Jt and B is the calibrated sensitivity of the detector in the mono chromator system. Rewriting (3) where RO(v, T5) is the average value of RO(v, T5) in the band wavenumber interval. Over most of the tem perature range used in the present experiments taking RO(II, T5) out of the integral and replacing it by its average value leads to an uncertainty of less than 3%. However, at temperatures below 16000K, the uncer tainty in this approximation is much larger and is of the order of 7%. The calibration energy is given by JB=l RO(II, TB)g( I pLv I, b)dp=BVB, (5) <1. where RO(v, TB) is the spectral radiancy of a blackbody at temperature TB corresponding to the temperature of the calibration source and ~v is the total spectral slitwidth. The integral h is evaluated graphically from a plot of the product of the blackbody radiance, RO(v, TB) and the experimental slit function as a func tion of v. An effective slitwidth is defined such that [RO( 4980, TB) ] (6) Over the range (1050° to HOOOK) of blackbody tem peratures used ~Peff remains constant. From (3), (4), (5), and (6) we have the integrated This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.189.203.83 On: Fri, 12 Dec 2014 06:57:271.9-,u BANDS OF CO2 1279 co FIG. 4. The monochromator slit func- tion. ~ 0.6 emissivity 6 z :::> u.. 0.4 I-:J en 0.2 =(Vt)[RO(4~80, TB)]..1Veff , VB RO(v, T5) (7) where X, the optical path length, is equal to P5U5t, and P5 is the pressure of the CO2 test gas. The integrated band intensity a is defined to be a1.9(T) = r k(v, T)dv, Jband (8) where k(v, T) is the absorption coefficient. Using E(V, T)=1- exp[ -k(v, T)X], which assumes that 6.0 5.0 4.0 T.Q f-n ><3.0 ... 2.0 1.0 J. 9 P. C02 COMBINATION BANDS T •• 2247 OK 2 3 4 5 6 Px Le02 (em aIm) FIG. 5. A vs X for the 1.9-,u CO2 bands. 7 r-0~-0-0-0'"-"\ I 0 \ I 0 \ I \ I 0\ " 0 \ o EXPERIMENTAL SLIT , FUNCTION -----CURVE CORRECTED FOR FINITE BAND PASS OF SOURCE (1.981 fL FI LTER ) " <;\ o / \ I \ " 0 , \ \ \ ~ k(v, T) is independent of X then A(T5,X)= r E(V, T6)dv Jband =1 [1-exp{-k(v, T5)X}]dv. (9) band Expanding the exponential term in (9) and taking the derivative with respect to X, then for small values of X we have dA(T5, X) dX ddX r E(V, T6)dv= r k(v, T5)dv. (to) Jband Jband Since X=P 6U6t, then from (7), (8), and (to) aband (T) = (dV t)[RO (4980, _ T B) ]..1veff • dt V BP5U6RO(v, T6) (11) Thus a the integrated intensity of the total 1.9-J.I CO2 band may be determined from the slope at small optical path lengths of A (T6, X) versus X. EXPERIMENTAL INTENSITY RESULTS Equation (11) is used in the determination of the value of the integrated intensity. The voltage-time relationship is obtained from the radiation traces and the value of a(T) is determined from the slope of a plot of A versus X at small values of X. A typical plot of A versus X is shown in Fig. 5, from which it is seen that A is essentially linear in X indicating the test gas is optically thin. The displacement of the curve from the origin is due to the difficulty in obtaining the true zero level on the radiation trace. A 2-J.lsec detector response time also contributes to this displacement. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.189.203.83 On: Fri, 12 Dec 2014 06:57:271280 J. C. BREEZE AND C. C. FERRISO 9.0 8.0 7.0 6.0 'E '0 5.0 '" 'E 1.9 f.L COMBINATION BANDS o o o u J!: 4.0 IzV, + v. I FIG. 6. Integrated intensity of the 1.9-/01 CO2 combination bands. d 3.0 2.0 A WEBER. HOLM AND PENNER o PRESENT EXTRAPOlATED VALUE FOR STP 1.0 0.5 100 500 1000 1500 T OK However, since it is the slope of the curve and not the absolute values which determine the integrated band intensity, the initial displacement of the curve does not affect the measured values of a( T). The measured integrated intensities for the 1.9-tL CO2 bands in the temperature range 1400° to 25000K are shown in Fig. 6. In order to refer all the values of a (T) to the same number density, the measured values have been nor malized to refer to a standard density. In general, the integrated intensity of a particular band, a/, can be written at any temperature, TOK, in the following way, using a/(T) =a;O(TO)cpi(T) (TOIT) or ai(T) =a/(T) (TlTO) =aN;(T), (12) where ai(T) is referred to a standard density, STP; TO is 273.2°K, aP is the value of the integrated inten sity at STP; and CPi( T) is a temperature variation depending on the particular band. The room tempera ture value of Weber, Holm, and Penner2 is also in cluded in Fig. 6. This value has been corrected to refer to density at STP. The estimated uncertainty in the results is less than ±20%. The vibrational relaxation time for CO2 at STP is of the order of a few microseconds.5 At the conditions used in the present experiments the vibrational relaxa tion time is less than a microsecond and does not therefore influence the results. At the higher tempera tures there is a possibility of dissociation of the carbon dioxide. However, the work of Brabbs, Belles, and Zlatarich6 on the rate of dissociation of CO2 at elevated • F. D. Shields, J. Acoust. Soc. Am. 29, 450 (1956). 6 T. A. Brabbs, F. E. Belles, and S. A. Zlatarich, J. Chern. Phys. 38, 1939 (1963). 2000 2500 temperatures indicates that no appreciable dissociation takes place during the first 100 tLsec. The maximum recording time used in the present studies is of the order of 100 tLsec, thus only the undissociated CO2 concen tration is used in the data reduction. Ideally the reflection of a shock at the closed end of a shock tube provides a quantity of stationary high temperature gas. In the real case the reflected shock interacts with the boundary layer formed behind the incident shock. The effects of the boundary layer become increasingly important as the reflected shock moves away from the region near the end plate. Measurements of the reflected shock pressure7 and densitr show that these parameters, which are initially in good agreement with the values computed from simple shock theory, increase with time behind the reflected shock. This indicates that the reflected shock accelerates as it leaves the end plate causing a rise in reflected shock temperature. The reflected shock tem peratures measured by Johnson and Britton,9 while being in good agreement, are slightly lower than the computed values. The experimental measurements have been made on reflected shocks in essentially monatomic gases. Although the effects of the boundary layer interaction are more marked in polyatomic gases, Strehlow and Cohen1o conclude that measure ments which are confined to the first centimeter or so 7 G. Rudinger, Phys. Fluids 4,1463 (1961). 8 w. C. Gardiner, Jr., and G. B. Kistiakowsky, J. Chern. Phys. 34,1080 (1961). DC. D. Johnson and D. Britton, J. Chern. Phys. 38, 1455 (1963) . 10 R. A. Strehlow and A. Cohen, "The Limitations of the Re flected Shock Technique for Studying Fast Chemical Reactions and its Application to the Observation of Relaxation in Nitrogen and Oxygen," Report No. 1059, Aberdeen Proving Ground, Maryland, December 1958. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.189.203.83 On: Fri, 12 Dec 2014 06:57:271.9-1' BANDS OF CO2 1281 of reflected shock travel are not adversely affected by the boundary layer. For this reason only the first 100 J.Lsec or so of reflected shock time are used in the inte grated band intensity measurements. The radiation traces show some curvature which could be caused bv small increases in the reflected shock parameters a:s the wave recedes from the end plate. The slope of the radiation traces determined for the first few micro seconds of reflected shock time is approximately 3% less than that averaged over the lOO-J.Lsec period. The effect of this uncertainty in the slope of the radiation trace is less than the uncertainty caused by the ± 1 % error in shock velocity measurement. The experimental results show a very strong tem perature dependence, cf>( T), increasing by a factor 4.5 in going from 3000 to 25000K. This effect is similar to that noted previously4 for the 2.7-J.L combination bands of CO2 which are shown to increase by a factor of ap proximately 2 over the same temperature range. A theoretical treatment of the temperature dependence of the 2.7-J.L combination bands of CO2 has been given by Malkmus.u This treatment is now extended to the 1.9-J.L CO2 combination bands. TEMPERATURE DEPENDENCE OF THE 1.9-1' CO2 BANDS When higher-order terms are considered in the expression for the electric dipole moment for the CO2 molecule then nonzero dipole matrix elements are found for transitions for which ~vl=2, ~va= 1, ~v2=~1=0, ~vl=l, ~V2= 2, ~va= 1, ~1=O, and ~v2=4, ~va= 1, ~v2=~I=O. These three transitions give rise to the (21'1+l'a), (Vl+21'2+1'3), and (41'2+l'a) band systems which to gether make up the 1.9-IL CO2 band. Apart from the above three bands there are other bands which con tribute to the CO2 intensity in the 1.9-IL region. How ever, these bands are expected to be much less intense than the above three bands. By analogy with the work of Malkmusll for the 2.7-J.L CO2 bands and Crawford and Dinsmorel2 for diatomic molecules, then, for the harmonic oscillator approxima tion neglecting rotational fine structure and the effects of Fermi resonance amongst the bands, the square of the matrix element for the (21'1+lIa) band is propor tional to (13) Similarly the square of the matrix element is propor- 11 W. Malkmus, "Infrared Emissivity of Carbon Dioxide (2.7-1' Band)" J. Opt. Soc. Am. (to be published). 12 B. L. Crawford and H. L. Dinsmore, J. Chern. Phys. 18, 983 (1950). tional to (Vl+ 1) [( v2+2)2- [2J (V3+ 1) (14) for the (1'1+2112+l'a) band and to (15) for the (4112+1'3) band. The average values of (14) and (15) taken over all possible values of I for a fixed value of V2 are propor tional to (16) and (v2+2) (V2+3) (V2+4) (v2+5) (va+l), (17) respectively. The integrated absorption intensity (in~ cm-2 atm-I) for a particular transition is given bi3 - I I I 87ra(32w' N 7' a(vIV21Va-t'l V21 Va) = gl 3hcQv [-w v (t'lv21Va)][ (-hCW')-'J X exp . 1-exp --kT kT ' (18) where NT is the total number of molecules per unit volume and pressure, Qv is the complete vibration partition function, Wi is the frequency (in cm-1) corre sponding to the changes in quantum numbers, (32 is a factor corresponding to the matrix element of the elec tric dipole moment associated with the given transition, and gl is the statistical weight of the state associated with the I quantum number. For the harmonic oscillator approximation to the CO2 molecule Qv= [1-exp( -,),Wl)J-I[l- exp( -,),W2).]-2 X[l-exp(-,),wa)]-1 (19) and Wv= hc(WIV1+W2V2+Wa Va), where Wl= 1351.2 cm-t, w2=672.2 cm-t, and Wa= 2396.4 cm-1 are the frequencies associated with the three fundamental modes of vibration and ')'= hc/kT. Thus from (18) and (19) I I I 87r3W' ~y T(32 a(vIV21va-VIV21va)= gl 3hc X exp[ -')'(WIV1+W2V2+w3Va) J[1-exp( -,),Wl) ] X [1-exp ( -,),W2) J2[1-exp ( -,),wa) ] X[l-exp( -,),w')]. The integrated band intensity is given by the sum of the integrated absorption intensities for all the vibra- 13 S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley Publishing Company, Reading, Massachusetts, 1959). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.189.203.83 On: Fri, 12 Dec 2014 06:57:271282 ]. C. BREEZE AND C. C. FERRI SO tional transitions composing the band, thus 00 00 00 "2 aCT) = L L L L a(vlviv3-Vl'V2I'V3'). (21) "1=0 V2=O "3=0 1=1 or ° Using Eqs. (13), (16), and (17) the sums for each of the bands are formed. Separating out the tempera ture-dependent terms, ¢( T), gives integrated band intensities proportional to [1-exp( -'YWI) J-2[1-exp( -'YW3) J-I X[1-exp{ -1' (2Wl+W3) I J (22) [1-exp( -'YW1)]-1[1- exp( -'YW2)]-Z X[1-exp( -'YW3)]-1[1- exp{ -1' (Wl+2wZ+W3) I J (23) and [1-exp( -'Ywz) J-4[1-exp( -'YW3)]-1 X[1-exp{ -1'( 4W2+W3) I J (24) for the (2Vl+V3), (Vl+2v2+V3), and (4V2+V3) bands, respectively. DISCUSSION The temperature variations, ¢(T) are plotted in Fig. 6. There are other temperature-dependent terms in :the integrated band intensities besides those given in (22), (23), and (24). For example, w', the centroid of the band, will vary with temperature; however, these terms are small and inclusion of them would not substantially alter the present discussion. From Fig. 6 it is seen that, to within the limits of the experimental accuracy, the results follow the calculated temperature dependence for the (2Vl+V3) band. The effect of Fermi resonance is to distribute the intensity among the resonating bands without causing any change in the combined total intensity of the three bands. The agree ment between the experimental values and the calcu lated curve for the (2Vl+V3) band indicates that the CO2 intensity in the 1.9-M resonance multiplet is in trinsic to the (2Vl+V3) band. Thus if Fermi resonance did not occur the intensities of the higher-order (Vl+ 2V2+V3) and (4V2+V3) combination bands would be negligible in comparison to the intensity of the lower order (2Vl+V3) band. This is in agreement with the result found4 for the 2.7-M CO2 combination bands where the intensity of the higher-order band (i.e., 2V2+V3) was found to be negligible in comparison to the lower-order (Vl+V3) band. The value of a1.9° was obtained by extrapolating each of the present experimental values to STP using the calculated temperature dependencies for the (2Vl+V3) band and averaging the results. The value of 2.07±20% (cm-2 atm-I) for a1.9° so obtained is in good agreement with the value of 1.89±20% (cm-2 atm-I) obtained by Weber, Holm, and Penner.2 The agreement between the present extrapolated value for a1.9° and the room temperature value of Weber, Holm, and Penner provides strong evidence of the correctness of the temperature dependence of the combination bands. The theoretical curves shown in Fig. 6 have been plotted using an a1.9° of 2.07 (cm-2 atm-I) and ¢ (T) from Eqs. (22), (23), and (24). To within the limits of uncertainty of the present experimental results, the variation with temperature of the 1.9-M CO2 integrated band intensity can be reasonably approximated with a very simple harmonic oscillator model based on only one combination band. ACKNOWLEDGMENTS The authors wish to acknowledge the great assistance of J. L. Anderson in obtaining the experimental results and Dr. J. A. L. Thomson for his helpful discussions. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.189.203.83 On: Fri, 12 Dec 2014 06:57:27
1.1713824.pdf	Pressure Theory of the Thermoelectric and Photovoltaic Effects Milton Green Citation: Journal of Applied Physics 35, 2689 (1964); doi: 10.1063/1.1713824 View online: http://dx.doi.org/10.1063/1.1713824 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Kinetic theory calculations of pressure effects of diffusion J. Chem. Phys. 97, 2671 (1992); 10.1063/1.463959 Theory of Pressure Effects on Alkali Doublet Lines J. Chem. Phys. 30, 1556 (1959); 10.1063/1.1730237 Electron Theory of Thermoelectric Effects J. Appl. Phys. 12, 519 (1941); 10.1063/1.1712935 The Photovoltaic Effect J. Chem. Phys. 9, 486 (1941); 10.1063/1.1750942 The Photovoltaic Effect J. Chem. Phys. 9, 377 (1941); 10.1063/1.1750913 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.209.144.159 On: Wed, 17 Dec 2014 17:39:38POTEKTIAL DISTRIBGTION IN THIl\ OXIDE FILMS 2689 voltage characteristics of metal-oxide-metal diodes is found quite generally for different oxides.1.6 For oxides with higher dielectric constant than SiO, V m generally occurs at lower voltages, and a correlation between V m and K! has been reported.l Band gaps for anodic oxide films are not well known; for AIz03, Eo> 8 V, for Ta205, a value of 4.6 e V has been found30 while a value of 3.0 has been reported for Ti(h31 In Table II, a correlation between V m2 and Ee is shown. For AIz03, Ee is derived from electron emission measurements.32 For Zr02 and SiO, Eo is not accurately known but a steep drop in electron emission from Zr-Zr02-Au diodes occurs at 4.3 30 L. Apker and E. A. Taft, Phys. Rev. 88, 58 (1952). 31 R. H. Bube, Photocond'uctivity of Solids (John Wiley & Sons, Inc., New York, 1960), p. 233. 32 T. W. Hickmott (to be published). V, just as it does for Ta205 diodes at 4.6 V.2 The higher the dielectric constant, the lower V m and Eo are, and the empirical relations V",2= 1O.3-0.18K(V)2 can be derived from Table II. If the model of Fig. 9 is correct, the impurity levels and hole levels are closely connected and their separation is determined by the dielectric constant of the insulator. ACKNOWLEDGMENT It is a pleasure to acknowledge many stimulating conversations with F. S. Ham. D. MacKellar kindly provided facilities for SiO evaporation. JOURNAL OF APPLIED PHYSICS VOLUME 35, NUMBER 9 SEPTEMBER 1964 Pressure Theory of the Thermoelectric and Photovoltaic Effects MILTON GREEN Burroughs Corporation, Defense and SPace Group, Paoli Research Laboratory, Paoli, Pennsylvania (Received 7 June 1963; in final form 16 April 1964) The theory is based upon the hypothesis that free charge carriers--electrons and holes-and phonolls exert pressures inside a solid. Gradients of such pressures exert motive forces on the carriers. On this basis, the hole cu rren t density / p, in the absence of a magnetic field, is assumed to be / p=upE-/lpgrad P p-/lp",gradP "', where Up, /lp, and P p are, respectively, the conductivity, mobility, and pressure of holes; /lop", is the inter action mobility between holes and phonons; P", is phonon pressure; and R is the electrostatic field. A similar expression is obtained for electrons by exchanging the subscript p for n. (The two mobilities associated with electrons, however, are negative.) The theory is applied to the nondegenerate semiconductor, with the assumption that the equation of the ideal gas law applies. (Thus, Pp= pkT, Pn=nkT, where k is the Boltzmann constant, T is temperature Kel vin, and p and n are concentrations of holes and electrons, respectively.) It is also assumed- for small cur rents-that deviation from the equilibrium pressures can be neglected. Assumptions concerning the phonon effect are quite general; the contribution from this source to the hole current density I"~ is given by /"", = -up(kT /e)op grad In T, where eis magnitude of electronic charge. The dimensionless quantity op, the phonon-dragging coefficient for holes (a temperature- and material-dependent parameter), is not amenable to calculation by the theory, in its present form, and must be determined experimentally. Again, a similar expression exists for electrons. I. INTRODUCTION IN this paper, thermoelectricity and voltaic photo electricity are treated mainly from a field theory approach. By this is meant that the problem is dealt with in terms of such electrical point-to-point parame ters of a circuit as electric fields, conductivity, charge carrier concentrations,"" mobilities, space charge, and however, is considered completely as a field theory. On the other hand, there is an abundance of literature on the statistical approach to thermoelectricity. Herring5 has collected a fairly large bibliography, as has Price.2 In behalf of the field theory treatment, it can be said that the fundamental physical processes involved are more easily understood,6 since the concepts are con crete, simpler, and also more familiar. current density. r,... Theoretical treatises involving, in part, such an ap proach as taken here have appeared.1-4 None of these, 1 F. W. G. Rose, E. Billig, and J. E. Parrott, J. Electron. Control 3,481 (1957). 2 P. J. Price, Phil. Mag. 46, 1252 (1955). 3 P. J. Price, Phys. Rev. 104, 1223, 1245 (1956). I J. Tauc, Phys. Rev. 95, 1394 (1954); Rev. Mod. Phys. 29, 30XJ19S7). The mathematical formulation of the flow equations, taken up in Sec. II, begins with the usual forces that act upon charge carriers-namely, electrostatic po- 5 C. Herring, Phys. Rev. 96, 1163 (1954). 6 Rose et at. 1 state, "The usual theoretical treatment of this effect (thermoelectricity) involves statistical techniques which do not readily lend themselves to a clear exposition of the subject." [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.209.144.159 On: Wed, 17 Dec 2014 17:39:382690 MILTON GREEN tential gradients, charge-carrier concentration gradi ents, and temperature gradients. A linear combination of these forces, when multiplied by the appropriate co efficients specific to each carrier, then gives the general equation for the current density of that carrier, appli cable to Ohmic flow. By substitution (for the general coefficients) of coefficients restricted to a nondegenerate semiconductor, followed by a rearrangement of terms, an equation is obtained and interpreted. This equation becomes the basis of the pressure theory. In Sec. III, the theory is applied to steady-state, open-circuit conditions, including isothermal equilib rium. The theory leads to the well-known mass-action law for isothermal equilibrium-namely, the product of the carrier concentrations (holes and electrons) must be a constant. A general treatment of steady-state, closed-circuit theory is given in Sec. IV, and the concept of pressure-electromotive force (or pressuremotive force) is introduced. The electrostatic "reaction" to the various pressuremotive forces is discussed in Sec. V, and the thermoelectric power (Seebeck coefficient) is touched upon briefly in Sec. VI. By their very nature, all electrical phenomena in solids must be associated with one or more equivalent electric circuits. In general, the equivalent electric circuits contribute to the understanding of the phe nomena. Therefore, Sec. VII is devoted to the various parameters describing the equivalent circuit of the phenomena. II. FLOW EQUATIONS-MATHEMATICAL FORMULATION The mathematical formulation of the pressure theory is based principally upon the interpretation of a com bined Fick and Soret diffusion; the interpretation is implied by experimental observation, and suggested in part by analogy with the kinetic theory of gases. The hole current density J p of a semiconductor, in which there exists a concentration gradient grad p, a thermal gradient grad T, and an electrostatic field E, is generally given by (1) where Dp and D7' are, respectively, the Fick and Soret diffusion constants, and (J p = e!J.pp is the hole conduc tivity. With the introduction of the Einstein relation and the Price equation7 for the Soret constant, Eq. (1) transforms into Ip=(JpE-!J.p gradkpT- ("(p-1)!J.pkp gradT =(Jp{E- (kT/e)[grad InkpT (2) + ("(p-l) grad InT]}, where "(p is the Soret parameter for holes. Similar ex pressions exist for electrons, but with negative values for e and !J. n. Equation (2) has been intentionally written in the 7 Stated in Eqs. (3) through (5), p. 1253, of Ref. 2. form shown to point out certain interpretive observa tions. First, by analogy with the kinetic theory of gases, the expression kTp can be interpreted as a pressure specifically P p, the pressure of a hole gas. Secondly, the dimensionless quantity 'Yp leaves wide room for in terpreta tion, 8 since it can be considered as a function of dimensionless ratios, such as p/po, T/To, '!'"p/'AP, etc. Obviously, when 'Yp has a value of approximately unity, the last member of Eq. (2) becomes negligible. If, however, it is assumed that "I p can be much greater than unity, then an interpretative means exists by which the phonon-drag effect can be introduced into the equations, and since "(p can also be considered to be a function of the physical dimensions9 of the semiconduc tor, it would be rather difficult to prove that 'Yp is in contradiction with experiment, or with the statistical theory" of the phonon drag. Therefore, in this paper, it is assumed that a very large "(p (and similarly, "In) cor responds to the phonon-drag effect, and conversely. The expression "1--1 will be designated by 0, and re ferred to as the phonon-drag coefficient. The definitions Pp=pkT and op="(p-1 transform Eq. (2) into Jp=(Jp{E-[(kT/e) (grad InPp+op grad InT)]} (3) = (Jp(E+Ep). The expression Ep= -(kT/e) (grad InPp+op grad InT) (4) will be referred to as the pressuremotive field of holes. Similar expressions, modified as stated above, exist for electrons. III. STEADY-STATE, OPEN-CIRCUIT CONDITIONS A. Isothermal Equilibrium For isothermal equilibrium, I p=J,,=O, and grad InT=O. (5) Hence, from Eq. (3) and its electron equivalent, (6) or E= -gradV = (kT/e) grad lnp= -(kT/e) grad Inn, (7) which integrates to V -V(O)= -(kT/e) In[p/p(O)]= (kT/e) In[n/n(O)], (8) 8 Rose et al.1 arrive at a value of ! for both IP and In, for non degenerate semiconductors. This same result is obtained by V.A. Johnson and K. Lark-Horovitz, Phys. Rev. 92, 226 (1953). 9 The phonon-drag effect becomes dependent upon the size of the specimen when the mean free path of the phonons and di mensions of the specimen become comparable. (See Sec. VI of Herring.') An analogous effect is also observed in gas transport phenomena when the mean free path and the dimensions of the container or of the transport tube become comparable. [See, for example, S. Dushman, Vacuum Technique (John Wiley & Sons, Inc., New York, 1949).J [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.209.144.159 On: Wed, 17 Dec 2014 17:39:38T IT E R 1\1 0 E L E (' T R T C }\ \f]) r II 0 T 0 VOL T 1\ ICE F FEe T S where V is electrostatic potential, p(O) and nCO) are the external emf's are applied to the circuit,ll so that concentrations at the point designated by (0), and V(O) is the potential at that point. Further, from Eq. (8), !E.dR=O, (16) it is found that pn = p(O)n(O) = const, (9) which is the mass action law, a condition for isothermal equilihrium. B. The Open-Circuit, Irreversible Steady State The conditions for the open-circuit, irreversible steady state arc [=Jl'+f,,=() or fl'=-f,,'l-O. (to) Hence, when Eqs. (3) and (4) and their electron equiv alents are substituted in Eq. (10), the resultant elec trostatic field E is given by (11) or R= p[fLp(gradP p+opP p grad InT) -,un(gradPn+onP n grad InT)], (12) where p is the local resistivity. The negative of E in Eq. (12) will be called the pressuremotive field,1O and will be designated by Ep= -E= (up/u)Ep+ (un/u)En = -p(kT /e)[up grad InP p-O"n grad InP n + (O"pop-unO n) gradlnTJ. (13) IV. STEADY-STATE CLOSED-CIRCUIT CONDITIONS: GENERAL TREATMENT The steady-state series closed circuit can now be analy?'ed. The total current density is, of course, given by 1= [p+ 1,,= up (J':+ Ep)+un(E+ En) = u(E+ Ep). (14) It is apparent from Eq. (14) that Eq. (10) is satisfied by the first equality in Eq. (13), which leads to E+Ep=O. For the steady-state closed circuit, I is everywhere solenoidal, and Eq. (14) is valid. Solving Eq. (14) for E+ Ep, and taking the line integral around a series path through the conductors, where R is the displacement vector. The case under consideration is that wherein no 10 The pressuremotive field E1' is defined by Ep=,,-I("pE p+unHn), and only for open-circuit conditions does Ep= -E, as stated in Eq. (13). For closed-circuit conditions in which I~O, Erpf·-E. because E is the gradient of a negative potential (-grad V), and must vanish when integrated around any closed path. Equation (15) then bccomes (17) The left member of Eq. (17) is evidently an emf, and will be designated by Op and referred to as the pressure motive jorce,12 or simply pmf (or emf). For isotropic material,13 l~, El', and I are parallel to one another, and normal to the equipotential surfaces of OP and of V, which are also parallel to one another, so that, by conventional electric circuit theory, Eq. (17) reouces to (18) where f1 is the total current through a cross section of any of the conductors, and R is the total series resistance. V. PRESSUREMOTIVE FORCES AND OPPOSING ELECTROSTATIC POTENTIALS For an open circuit, since E= -Ep , from Eq. (14), an electrostatic field must exist wherever a concentration gradient or a temperature gradient exists, except for the case14 where Ep=O and -Rp/En=un/u p. (19) Hence, by Eg. (15), for the case in which 1=0, the electrostatic potential (or "reaction") opposing the pmf is derived from J: E-dR= -J: VV·dR= -J: Ep'dR (20) or V(B)-V(.4)= op(A -7 B). (21) As stated above, the potential V(B)-V(A) exists across a concentration or temperature gradient, singly or combined, and, in general, increases or decreases monotonically unless the situation described by 11 When there are external emf's in the circuit, the left member of Eq. (16) does not vanish, but is equal to the algebraic sum of the emf's. 12 The integral fAB Ep·dR will be referred to as the pmf from A to B, or GP(A ----t B); that is, conventional circuit terminology will be used. 13 For anisotropic material, Eq. (14) becomes a tensor equation; U is then the conductivity tensor. In this case, none of the vector quantities E, Ep, or I need be parallel. 14 The most likely situations in which Eq. (19) is satisfied are: (1) Up""U n, corresponding to p""nb; or (2) ap/aT=ban/aT. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.209.144.159 On: Wed, 17 Dec 2014 17:39:38MILTON GREE~ TABLE I. Open-circuit, steady-state, electrostatic potential across thermal or concentration gradient. Type of gradient Electrostatic reaction potential Remarks Concentration, isothermal (kT/ e) In (nI/n2) = (kT / e) In (pd h) nl, PI = carrier concentration at point 1, and n2, p2=carrier concentration at point 2. n,h =n2P2=ni2(T) Concentration, isothermal (kT/e) In (n/ni) = (kT Ie) In(n,J p) =e-ll f\F-i';i 1 \Vith respect to intrinsic con cent ration ni J~F(n) = Fermi energy Fi = Fermi energy corresponding to intrinsic concentration Concentration, isothermal Thermal (only) (kT/e) In(n"pp/ni2)=e-J(RFn-FF,,) (k/e)(lHp)!!.T [p type] (k/e)(l+<,,,)t>T [n type] Across a p-n junction Homogeneous, extrinsic semiconductor; exhaustion temperature range (no concentration gradient). These expressions amount to 86.3 (1+0) }1Vj"K. Thermal, in temperature range in which donors and acceptors ionize -«~pie) In (T2/1\) [p type] (£,Je) In (T2!T,) [n type] Extrinsic semiconductor; c=activation energy. For Ge, <=0.013 eV; for (Plus the corresponding expressions for thermal gradient, above.) Si, <=0.045 eV. These values cor respond, respectively, to 0.65 and 2.25 m V r K at 20Q K for the expres sions at left. np product, isothermal (kT /e)[ (b-1)/ (H 1)] In (U2/Ul) (Other investigators have also derived this equation.) Homogeneous semiconductors, all types, photoelectron-hole pair in jection; b is the mobility ratio. Thermal, with n-p pair production (kT /e)[ (b-1) / (b+ 1) ] In (udul) + (k/e)[(bYn-Yp)/(H l)]!!.T Intrinsic semiconductor; T = appro priately weighed temperature between 1'2 and 1'1, with !!.T=T2-T1. Eq. (19) occurs, in which case the electrostatic potential will go through a minimum15 at the point where Eq. (19) is satisfied. The potentials that are derived from Eq. (21) and correspond to various physical situations are listed in Table I, and are illustrated graphically for silicon in Fig. 1. (The equation shown in Table I for "np product gradient, isothermal" has also been derived by van Roosbroeck.1fl) The curves plotted in Fig. 1 (a) are qualitative, and correspond to the potentials (V gn and V up) across a temperature gradient which has one end held at OaK and assumes no "spillover"-that is, the equilibrium concentrations prevail. The various curves correspond to different doping levels-the purer the sample, the greater the change in potential with a change in temperature. The curves in Fig. 1 (b) show, quantitatively, the potential across an isothermal concentration gradient namely, a p-n symmetric junction with doping as a parameter. The data for constructing Fig. 1 were taken from Morin and Maita17 and Putley and Mitchell.ls The dashed sections of the curves, Fig. l(b), are ex trapolations. The extrapolations to the origin at OaK are what one might intuitively expectl~ in order to satisfy the third law of thermodynamics. 15 For }1p>}1n, the potential passes through a maximum rather than a minimum; the conditions noted above14 still hold, however. 16 W. van Roosbroeck, Phys. Rev. 91, 285 (1953). 17 F. ]. Morin and ]. P. Maita, Phys. Rev. 96, 28 (1954). 18 E. H. Put ley and W. H. Mitchell, Proc. Royal Soc. (London) 72, 193 (1958). ,. For other theoretical interpretations, see W. Shockley, An example of the open-circuit potentials and the bucking pmf's existing in a p-n couple are illustrated in Fig. 2 (positive potential plotted downwards). Figure 2(b) illustrates the electrostatic potential for a p-n-p couple in isothermal equilibrium. The potential well for electrons in the n region decreases with tem perature (except in the vicinity of OOK); therefore, the well is shallower for temperature 1'2 than for tempera ture T1(T2> T1). (On the other hand, the concentration product np is greater at T2 than at Tl') The potential diagram corresponding to the temperature distribution of Fig. 2(c) is shown in Fig. 2(d). Similar changes in the potential profiles can be pro duced by optical generation of electron-hole pairs; however, in the nonequilibrium case, Fig. 2(d), the profile bends oppositely to the thermal generation (dashed curve in the gradient region between p' and p of Fig. 2). This assumes that the np product has a dis tribution similar to the temperature distribution of Fig. 2(c). VI. THERMOELECTRIC POWER (SEEBECK COEFFICIENT) The thermoelectric power of a couple is the rate of change, with respect to temperature, of the open circuit, thermally generated voltage. For the situation mectrons and Holes in Semiconductors (D. Van Nostrand Co., Inc., Princeton, New Jersey, 1951), p. 473, Fig. 16-7; see also A. K. Jonscher, Principles of Semiconductor Device Operation (John Wiley & Sons, Inc., New York, 1960), p. 9, Fig. 14. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.209.144.159 On: Wed, 17 Dec 2014 17:39:38THERMOELECTRIC A.'lD PHOTOVOLTAIC EFFECTS 2693 illustrated in Fig. 2, the thermoelectric power is simply where V J is the electrostatic potential across the junc tion. The thermoelectric power can, therefore, be de rived from the slopes of the appropriate V vs T curves, such as are illustrated in Figs. 1 (a) and 1 (b). Over certain ranges of temperature, the first term of Eq. (22) generally dominates. Therefore, this term provides an approximate method of measuring the temperature dependence of V J. VII. EQUIVALENT CIRCUITS: POTENTIAL DIAGRAMS Since the discussion is concerned mainly with elec trical concepts, it should be possible to represent the entire complex by an equivalent circuit or circuits. Once reduced to the equivalent circuit representation, the phenomena of thermoelectricity and photoelec tricity are easily understood. This section illustrates and discusses the "equivalent circuit at a point," the equivalent circuit of a semiconductor couple (p-n-p c -J" t (0) to Q. -J" -1 1.2 1.0 I O.B (b) ~ 0.6 "0 > 0.4 <= Q "£ 0.2 ~ Degrees Kelvin Sample Nu No -NA ZG -131° 2,1012 139b 1.3 x 10'7 126b 2.2 x la' 140b 2.7,10' O~--r--r--~-.~~--r--r--~ a 100 200. 300 400. 500 600 700 BOO remperalure (OK) FIG. 1. (a) Qualitative curves (for silicon) of electrostatic po tential vs temperature across a temperature gradient (reference temperature, OOK); Von and Vpp are for n-type and p-type mate rials, respectively, and parameter is N D-N A. (See text.) (b) Voltage of symmetric junction of silicon vs absolute tempera ture, with impurity content as parameter. Heading IV D-N A refers to difference between donor and acceptor concentrations of specimens. Dashed portions of curves are extrapolated. Experi mental data to construct curves is that of Morin and Maita,17 and Putley and Mitchell!8 (see text). FIG. 2. Qualitative, pressuremotive, and potential diagrams for open-circuit jrn couple configuration of (a), and for temperature or excess isothermal pair-concentration (np=const) corresponding to (c). Solid curve of (d) corresponds to thermal injection, dashed curve to isothermal photoinjection. structure), and the potential diagrams for closed circuits. A. Equivalent Circuit at a Point in a Semiconductor Two aspects of the "equivalent circuit at a point" are illustrated in Fig. 3. The two circuits correspond, respectively, to the first two associated equations beneath each. The fields Ep and En are defined by the other two equations; the sources for Ep and En are Eqs. (13) and (14), respectively. The interpretation is that each point of the material is represented electri cally by two parallel generators Ep and E", with series conductances Up and Un, respectively. The electrostatic field E is represented by a charged parallel capacitor, since E represents energy stored in the dielectric of the '--------1' • E ~---------1, + E I = Ip In 1=6 (~E + ~ E • E) () p () n = ~(Ep. E) ~ an(~+E) = 6 ( Ep • E ) ~ =-¥ [17-4 ~+~ 'V.k TJ En:!}. ['i7-LPn i-5n \?..t-.T] FIG. 3. Equivalent circuit representation at a point (distributed parameters). (Diagrams correspond to current density equations below each.) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.209.144.159 On: Wed, 17 Dec 2014 17:39:382694 MILTO~ GREE:--J I··IG. 4. Potential diagram (center) and equivalent. circuit (bottom) for p~n couple with closed-circuit load (top). Primed regions are at a higher temperature T'. semiconductor. All of these quantities are, of course, distributed parameters, such as those encountered in transmission lines. At points where the pressure gradients of holes and electrons are oppositely directed, such as at P-tt junc tions, the pmf fields Ep and En have the same sense, and, hence, In and I p add constructively. The converse is, of course, true when the pressure gradients are directed alike, such as occurs at a temperature gradient in homogeneous material. Since the electrostatic field E can be less than, equal to, or greater than, either Ep or En, the sense of I may be either the opposite or the same as that of E. B. Potential Diagrams and Equivalent Circuit of a p-n Couple The closed-circuit situation, with load RL and the temperature conditions of Fig. 2(c) imposed, is illus trated in Fig. 4. (The potential diagram, positive po tential up, is shown in the center, and the equivalent circuit at the bottom.) Before the circuit is closed, a situation such as that shown in Fig. 2 (d) exists in the semiconductor. Thus, since the total current fJ is zero, the voltages V p and V pi, V n and V n', and V L (as desig nated in the equivalent circuit diagram in Fig. 4) across the respective capacitances are each zero, because, in these regions, which are isothermal and homogeneous, no emf is generated. In the two junction regions J and J', and the gradient regions gn and gp, where emf's are generated, the total current is zero, because the emf's are just balanced by the opposing potentials arising from the static charges on the re spective capacitances of these regions. When the circuit is closed, there is a transient current flow in which a redistribution of the static charges across the various capacitances takes place, and, hence, there is a readjust ment of the static potentials. The various emf's-&J, &.1', &yn, and &up-will be affected more or less (de pending upon the temperatures of operation and "conditions" at the juncation) by a certain amount of temperature change which will occur, and by deviations from the equilibrium concentrations which will occur, even after the original temperature distribution is re established. Thus, even in steady-state operation, both the internal emf and the internal resistance of the couple may depend upon the current. For photovoltaic cells, this variation is known to be the case.20•21 In the steady state, the total current fJ flows through the series circuit such that, at the J junction, pressure (thermal) energy is transformed at the rate of &.1fJ J per sec into V.1fJ J of electrostatic energy per sec (in the process of moving the charge carriers against the potential gradient of V.1), and into fJ2R.1 J heat energy per sec (in overcoming the resistance to the flow of current through the J region). A similar situation occurs in the g nand g p regions. In the J' region, the electro static potential V.1' is greater than the junction pmf &.1', so that the situation arising is the reverse of that of the J region; namely, electrostatic energy is converted to pressure energy while also overcoming the resistance to current flow through the junction. The remainder of the generated electrostatic energy overcomes the re sistance to current flow in the homogeneous regions, and the balance is delivered to the load RL. VII. CONCLUDING REMARKS The hypothesis of the theory presented has been based mainly upon concepts derived from the theory of gases, particularly for ideal gases. This basis restricts the theory more or less to nondegenerate semiconduc tors. The development of the theory also assumes that, even in the nonequilibrium state, local deviations from equilibrium are negligible. Later discussion notes that this assumption is not entirely justifiable, and that, under certain conditions, large deviations from equi librium could and do occur. Nevertheless, the concepts developed here are applicable to many situations; and even where the quantitative applicability is not good, the qualitative applicability explains many phenomena which are felt to have been, heretofore, neither clearly explained nor correctly interpreted. 20 W. G. Pfann and W. van Roosbroeck, ]. Appl. Phys. 26, 534 (1955). 21 M. B. Prince, J. App1. Phys. 26, 534 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.209.144.159 On: Wed, 17 Dec 2014 17:39:38
1.1728275.pdf	Calculations of the Thermoelectric Parameters and the Maximum Figure of Merit for Acoustical Scattering Louis R. Testardi Citation: Journal of Applied Physics 32, 1978 (1961); doi: 10.1063/1.1728275 View online: http://dx.doi.org/10.1063/1.1728275 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermoelectric figure of merit calculations for semiconducting nanowires Appl. Phys. Lett. 98, 182104 (2011); 10.1063/1.3585659 Thermoelectric figure of merit of superlattices Appl. Phys. Lett. 65, 2690 (1994); 10.1063/1.112607 Figure of merit for thermoelectrics J. Appl. Phys. 65, 1578 (1989); 10.1063/1.342976 Erratum: Calculations of the Thermoelectric Parameters and the Maximum Figure of Merit for Acoustical Scattering J. Appl. Phys. 33, 1018 (1962); 10.1063/1.1777164 Effect of Impurity Scattering on the Figure of Merit of Thermoelectric Materials J. Appl. Phys. 30, 1922 (1959); 10.1063/1.1735090 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Tue, 23 Dec 2014 22:49:371978 K. WALTERS 3·0 2·0 !J (·0 0 40 n(cps) 80 (20 FIG. 11. Predicted (full line) and observed (181) relations between iI and n. (/=14.4 g cm2j L=4.08 cmj K=140 d em/rad.) results. Figures 5-12 illustrate the excellent agreement between the predicted curves and the observed results for both amplitude ratio and phase lag in four different experiments. The agreement between theory and experi ment is thought to be well within experimental accuracy, estimated from the variation in the original dynamic viscosity and rigidity results (Figs. 2 and 3). Although the idealized spectrum shows agreement with experiment in respect of both amplitude ratio and phase lag (d. Walters·), the interpretation is not free from ambiguity. No experimental results were available for frequencies less than 10 cps, and as a consequence, the long time end of the spectrum cannot be defined 3·0 ,,2·0 'V " .. '" ... (·0 0 40 80 (20 n(eps) FIG. 12. Predicted (full line) and observed (®) relations between c and n. (I = 14.4 g cm2j L=4.08 cmj K = 140 d cm/rad.) with any precision. The most we can say is that, at a point on the relaxation time scale near r= 1. 75 sec, there is a large concentration of viscosity, which, as far as the experiments of Markovitz et at. are concerned, is adequately described by a Maxwell element. This lack of resolution in the spectrum is due entirely to the limited frequency range which was observed. ACKNOWLEDGMENT The author wishes to express his gratitude to Dr. H. :l\Iarkovitz for allowing him access to his experimental results. JOURNAL OF ..... PPLIED PHYSICS VOLUME 32, NUMBER 10 OCTOBER, 1961 Calculations of the Thermoelectric Parameters and the Maximum Figure of Merit for Acoustical Scattering LOUIS R. TESTARDI The Franklin Institute Laboratories, PhilcuJelphia, Pennsylvania (Received May 3, 1961) The calculations of Chasmar and Stratton [R. P. Chasmar and R. Stratton, J. Electronic and Control 7, 52 (1959)] for the determination of the maximum thermoelectric figure of merit are extended for the case of acoustical scattering. Graphical data are presented for the determination of the material parameter, the optimum values of several quantities, and the degradation of the figure of merit for nonoptimum conditions. The variations of the Seebeck coefficient and the electrical conductivity computed from exact statistics are compared with experimental results for several alloys of thermoelectric interest. Good agreement is found except for high electrical conductivities. Other anomalies are noted. INTRODUCTION THE maximization of the performance of a thermo electric device from thermodynamic principles shows that, for a thermoelement at the operating temperature, the quantity Z=S2u/K should be as large as possible'! The Seebeck coefficient S, the electrical conductivity u, and the thermal conductivity K, are not, however, internally constrained in any way from thermodynamics alone. Semiconductors now represent the principal searching ground for materials of higher 1 A. F. Ioffe, Semiconductor Thermoelements and Thermoelectric Cooling (Infosearch, London, 1957). figure of merit z. In part, their merit lies in that the parameters S, u, and K can be varied by the addition of impurities (doping agents) thereby allowing an optimization of z for a given material. In practice, this additional latitude gives rise to the necessity of doping studies to establish the maximum figure of merit for the material; in principle, the maximum figure of merit is formally determined on writing the constraint relations among the thermoelectric parameters from transport theory. For an extrinsic semiconductor, numerical solutions of the transport equations can be obtained if specific assumptions are made for the form of the relation between energy and wave vector and [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Tue, 23 Dec 2014 22:49:37THERMOELECTRIC PARAMETERS FOR ACOUSTICAL SCATTERING 1979 between relaxation time and wave vector for the charge carriers. The maximum figure of merit on doping can then be related to parameters characterizing the nondegenerate properties of the material, and the sets of 5, (T, and K for the material at any extrinsic doping level can be generated from these parameters by varying the position of the Fermi level. Under the above assumptions, the material parameters can be determined from a single known set of 5, (T, and K for an extrinsic semiconductor in an unknown state of doping. It thus results that the knowledge of the thermoelectric parameters of one specimen determines its complete, extrinsic, doping behavior insofar as the method by which the Fermi level is varied in practice does not affect a breakdown of the conditions initially assumed to make the solution tractable. THE TRANSPORT EXPRESSIONS We consider the case of an extrinsic semiconductor with spherical constant energy surfaces whose charge carriers have a relaxation time T of the form T a: f-! (acoustical scattering), where f is the kinetic energy. For this case, it can be shown thatZ k[2Fl{-lJ) ] 5= (±)----71 , e F 0(71) (1) (T=neJ.l., (2) 7r!F 0(71) J.l.=---J.l.o, (3) 2Ft(f/) n= 47r-!(27rmkT / h2)~F t(f/), (4) K=Kg+A(TT, (5) .\.= 3Fo(f/)Fz(f/)-4F12(f/) (~y, Fo2(f/) e (6) where n is the carrier concentration, m the density of states effective mass, e the electronic charge, J.l. and J.l.o the conductivity mobilities in the actual and non degenerate states, respectively, Kg the lattice thermal conductivity, A the Lorenz number, and T the absolute temperature. Fr(f/) is the Fermi-Dirac integral of order rand 71 is the Fermi level in units of kT measured from the band edge and taken as negative in the energy gap and positive in the band. The assumptions of spherical energy surfaces and acoustical scattering have entered into Eqs. (1), (3), (4), and (6) while Eq. (5) assumes further that the thermal conductivity is composed of lattice and Lorenz electronic contributions only. The thermoelectric figure of merit Z may then be 2 A. H. Wilson, The Theory of Metals (Cambridge University Press, New York, 1953), 2nd ed., pp. 12, 196-204. f" I' 7 :IE u 10 :3 :IE S " 8 oi" ~ 'elE oj_ ~ I t;-~ "~_~~~_--~IO" "eIE ~ 'l ~ !!-. t' ~ .=, < 10-- 30 90 t50 2tO 270 330 390 450 SEEBECK COEFFICIENT I p. v/oc I FIG. 1. Curves for obtaining material parameter and other quantities from Seebeck coefficient. expressed in the form (0 T ) 52(7]) Z 300 = [2eM(27rm~k-30-0~/h-2)~!F-o-(f/-)J--~1+-3-00A' (7) The figure of merit is a function of the material parameter M=J.l.o(m/m)!(T/300)!/K g and the reduced Fermi level 11, while the maximum figure of merit Zmax, is determined by M alone. For various values of f/, Eqs. (1), (3), (4), (6), and (7) (here, also for various M) have been solved numerically with an electronic computer, and the results are given in Figs. (1) and (2). DISCUSSION OF CALCULATIONS Chasmar and Stratton,3 departing from a material parameter4 similar to M, have obtained the maximum figure of merit for a number of scattering laws. For the case of acoustical scattering, the value of the Seebeck coefficient uniquely determines the quantities A, and (T/J.l.o(Tm/300m)! according to Fig. 1. Thus, from the measurements of 5, (T, K and ~ in the extrinsic state, and Eq. (5), one can obtain the values for J.l.o(m/m)l (T/300)% and Kg from which Zmax can be determined using Fig. 2. The solution of Eq. (7) for Zmax also gives the values of 5 and 11 at Zmax and these are presented in Fig. 3. The optimum value of 11 also determines the ratio 11/ (Tm /300m)1 at Zmax and this quantity appears in Fig. 2. Since, in the as-grown state, the ratio 1l/(Tm/300m)1 is determined by the value of the Seebeck coefficient (see Fig. 1), the fractional change in the carrier concentration to reach optimum doping can be computed. In the preparation of thermoelectric materials by slow directional freezing, the thermoelectric parameters 3 R. P. Chasmar and R. Stratton, J. Electronics and Control 7, 52 (1959). 4 The material parameter of Chasmar and Stratton 13=0.8952 XlO-&Jf. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Tue, 23 Dec 2014 22:49:371980 LOUIS R. TESTARDI "'I,. _10-2 10" . 'I' ~ ! ~ '""~ I "I~ N 'EIE ~ FIG. 2. zmax(T/300) and optimum value of n/(Tm/300m)1 vs material parameter M =p.o(m* /m)l(T /3(0)I/K o. are often markedly dependent upon the position along the ingot and the exact optimum conditions may prevail only for a vanishingly short segment. Figure 4, obtained from the numerical solution of Eq. (7), indicates what fractional departure is S or n from their optimum values will reduce the Z to 0.9zmax. The degradation of Z on the degenerate and nondegenerate sides of optimum doping are approximately symmetrical for the case chosen. The very slow fall in z as S or n departs from the optimum value, of course, allows a greater portion of an inhomogeneous ingot to be utilized, but also indicates that the detrimental effects of minority carriers (through ambipolar diffusion and the reduction of the Seebeck coefficient) may be reduced by lowering the Fermi level down the de generate side of optimum doping. With all non degnerate properties given for a particular material, the magnitude of the minority carrier effects depends only on the position of the Fermi level. By operating at the z=0.9z max point, the increase in 1) (computed from Fig. 3) leads to an exponential reduction in the number of minority carriers and it may then be determined if the actual z has been improved. 3 For acoustical scattering, the maximum value of S2(J" occurs for 1)"" +0.65. At this point, 5",,167 p'vrC, n/(Tm/300m)J",,3.1XI019/cm3 and (J"",,4.3,uo(Tm /300m)!. For maximum heat pumping capacity the optimum doping level may more nearly coincide with that giving the maximum value in S2(J" than Zmax. Figure 4 shows the fractional reduction in z from the value at Zmax which occurs for the operation at (S2(J") max' The curve also indicates the importance of a sensitive as well as accurate method for the measure ment of the thermal conductivity. Note added in proof. For M ~ cr.>, Z is limited by the minority carrier reduction of Sand ambipolar diffu sion. Setting Eg/kT=E/ (reduced energy gap) and r= (p.om!)p/ <.uom!)n, an approximate (",5%) expres sion for Zma.~ in this case is ( T) [ 7.5(1-r)2J"'1 Zmax 300 = 1Q-3[O.115Ey'2.25+ 1J 1+ Eo' (+ and -for p-and n-type thermoelements, respec tively) in the range 5::;Eo'::;80 and l::;r::;lO. APPLICATION OF THE CALCULATIONS The procedure outlined above for the determination of the maximum figure of merit bears, of course, several shortcomings. For a new material one generally will not know the scattering law, the band structure, or whether the specimen under test is truly extrinsic. The inhomogeneity along the ingots of present day materials may often be used to determine the latter, and, in the case of extrinsic conduction, may also lead to a knowl edge of the scattering law when the above calculations are extended to other scattering laws. Band structures, other than simple spherical, need not alter the validity of the calculations based on a simple spherical band if the expressions for S and A can be reduced to the forms given in Eqs. (1) and (6) and the functional dependence on 1) of Eq. (2) is not changed, though in anisotropic materials the calculations will pertain to the crystalline direction in which M was obtained. The solution of Eq. (7) by numerical methods also leads to sets of Sand Z which, themselves, uniquely determine Zmax and M. Thus, it is possible from a measurement of Sand Z in an extrinsic semiconductor to deduce Zmax directly, without obtaining the material parameter. The factor ,uo(m/m)!, however, can generally be obtained with greater accuracy and ease than Kg. In addition, ambipolar diffusion may invalidate Eq. (5) before appreciable reductions in the Seebeck coefficient from mixed conduction are observed. The separate determinations of ,uo(m / m): and Kg should give a more lucid indication of the validity of the assump tions in this analysis and a better estimate of the maximum figure of merit. The calculations have been applied to p-type com positions in the Bi2Tea-Sb2Te3 pseudo-binary alloy system. The temperature dependences of Sand (J" observed in this laboratory have been found to be in satisfactory accord with the assumptions of acoustical scattering and a band structure of the type proposed by Drabble5 for Bi2Tea. The behavior of the thermo electric properties with doping at room temperature has been studied for several compositions. The results 460,-------------------------. 380 U ~340 :!, J 300 260 zzo -2 "7OPI -3 FIG. 3. Optimum values of Sand '1 vs ::;max(T/300). ------ 5 J. R. Drabble, Proc. Phys. Soc. (London) 72, 380 (1958). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Tue, 23 Dec 2014 22:49:37THERMOELECTRIC PARAMETERS FOR ACOUSTICAL SCATTERING 1981 for the alloy 70%SbzTe3+30%Bi2Te3 shown in Fig. 56 agree substantially with the variations predicted by Eqs. (1)-(4) for a material with the factor J.l.o(m/m)! = 325 cm2/v sec. (Within the experimental error, the observed thermal conductivities were also in agreement with predicted behavior). The discrepancy for 5<110 }.Iv;oC may arise as the effects of a band structure which is more complicated than the simple form assumed. With the large density of states effective mass usually required for a good thermoelectric material, the states of partial degeneracy are reached only with the inclusion in the sample of large numbers of impurities or defects. The possibility of consequent new scattering mechanisms or impurity band conduction increases and the analysis may not be adequate here. The substitution of the inhomogeneity along a lowered ingot for otherwise discriminately doped samples has been found successful, at least in the case when adjacent samples have Seebeck coefficients which differ by about 10%-15%. In this manner, it is possible to observe variations in the factor }.Io(m/m)~ of ingots grown in different manners. Smaller variations in this factor ("-' 5%-10%) are also found in ingots prepared in (ostensibly) identical manners. The calculated value of Zmax for the p-type alloy 70%Sb 2 Te3+30%BizTe3 is found to be 2.4X 10-3 (OK)-l to 2.6X 1Q-3(OK)-1 which is in good agreement with experimental observations. The measured values values of 5 and u for the n-type alloy 70%Bi2Te3+30%BizSe3 are also shown in Fig. 5. Although the iodine doped alloys of this composition seemed to exhibit acoustical scattering, the substitution of Cu2Br2 for the iodine led to several anomalies. Initially, the figures of merit obtained with the CuZBr2 doping agent were higher than those obtained with iodine, but the variations of the thermoelectric parameters were not consistent with acoustical scatter ing and the parameters were later found to be markedly 0.8 r------------------, 0.7 J 0.95 .: i 0.4 0.90 'M ... ~ ... 0.1 0.85 i c: "- 0.80 !i 0.75 a ~_....L-L_L___"_ _ ___'___'___L _ __"____l 0.70 024 6 8 W ~ M ~ • ~ Z .... 17/3(0) .10' FIG. 4. Curves of degradation in z for nonoptimum conditions. tJ.S/Sopt= !Smax-SO.9z!max/Szmax, tJ.n/nopt= !nzoptO-no.9zmaxil 6 Measurements made with the current flowing in the cleavage planes. 3~r----------------------' EXPERIMENTAL: 320 a ~ ~Nt~A~ U X IODINE DOPE ~280 ~ 80 , THEORETICAL: -~t (~fE~fNb) -N< , __ -ra:<-f 4~0' 10' ELECTRICAL CONDUCTIVITY (OHM-CMf' FIG. 5. Doping curves of Seebeck coefficient and electrical conductivity for two alloys. unstable on aging. Other anomalous behaviors have been reported for this composition.7.8 Figure 5 also shows the theoretical variations of 5 and u for the scattering laws To:. t-!, To:. e!, and To:. tH. At least on the degenerate side of optimum doping, the variations in 5 and u over the experimental range are reasonably distinct in the three cases. CONCLUSIONS The use of exact statistics to calculate a maximum figure of merit from a known set of thermoelectric parameters of an extrinsic semiconductor may be of limited utility without a knowledge of the band structure or scattering mechanism. Plausibility argu ments established from other material properties may be substituted in lieu of this ignorance, and it may also occur that the result is not too sensitive to some varia tion of the unknown parameters. In practice, several samples in different states of degeneracy will generally be prepared (or available from an inhomogeneous ingot) and it can be determined if the calculations are applicable for a given scattering law. The results from several alloys of thermoelectric interest have been found to confirm the assumptions on which our calcu lations have been based if the specimens are not too degenerate. In this case the calculations may be of further use in the comparison of thermoelectric materials prepared in different manners and different as-grown states of degeneracy, and in providing some information of the processes involve in aging, irradiation, or diffusion in these materials. ACKNOWLEDGMENTS The author is indebted to Dr. F. J. Donahoe for hi" advice and review of the manuscript, G. ':\lc Connell for assistance in the measurements, and the sponsors of The Thermoelectric Effects Program for their financial support. ., N. FuschiJIo, J. N. Bierly, and F. J. Donahoe, J. Phys. Chern. Solids 8,430 (1959). 'I. G. Austin and A. Sheard, J. Electronics 3, 236 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Tue, 23 Dec 2014 22:49:37
1.1725390.pdf	Consideration of ``Spherical Hot Spots'' Arising from Pion Capture in Explosives Using Thermal Initiation Theory Joseph Cerny and J. V. Richard Kaufman Citation: The Journal of Chemical Physics 40, 1736 (1964); doi: 10.1063/1.1725390 View online: http://dx.doi.org/10.1063/1.1725390 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/40/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Computational study of 3-D hot-spot initiation in shocked insensitive high-explosive AIP Conf. Proc. 1426, 255 (2012); 10.1063/1.3686267 Development of a Simple Model of “HotSpot” Initiation in Heterogeneous Solid Explosives AIP Conf. Proc. 620, 991 (2002); 10.1063/1.1483704 Modelling ‘hot-spot’ initiation in heterogeneous solid explosives AIP Conf. Proc. 505, 887 (2000); 10.1063/1.1303610 Shock and Hot Spot Initiation of Homogeneous Explosives Phys. Fluids 6, 375 (1963); 10.1063/1.1706742 Initiation of Explosions by Hot Spots J. Chem. Phys. 36, 1949 (1962); 10.1063/1.1701302 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:19THE JOURNAL OF CHEMICAL PHYSICS VOLUME 40, NUMBER 6 15 MARCH 1964 Consideration of "Spherical Hot Spots" Arising from Pion Capture in Explosives Using Thermal Initiation Theory JOSEPH CERNY* AND J. V. RICHARD KAUFMAN Explosives and Propellants Laboratory, Feltman Research Laboratories, Picatinny Arsenal, Dover, New Jersey (Received 24 June 1963) The process whereby relatively high energy density, "spherical" zones of radiation heating may arise from slow ,..--meson irradiations is described, and the effects of such microscale hot spots on six explosives (lead azide, lead styphnate, mercury fulminate, RDX, TNT, and PETN) was investigated. No explosions or signs of thermal decomposition were observed with any of the explosives. Analysis of these results by the hot-spot model of explosive initiation and thermal growth was attempted. The conclusions were (1) that this model can not explain the experimental results observed for RDX, in that it predicts initiation, and (2) that the previous experiments attempting explosive initiation by the micro scale thermal effects of ionizing radiation have not investigated those explosives most susceptible to initiation by this mechanism. INTRODUCTION ALTHOUGH attempts to initiate explosives with ~ ionizing radiation have been reported with null results,t-3 no detailed analysis of these experiments with the reasonably successful hot-spot model of explosive initiationHi has been presented. Additionally, some of the more reactive secondary high explosives have not been stringently investigated. Both spherical and cylin drical symmetries for the idealized radiation damage produced by particle irradiation are possible, but at high energy densities only the latter has so far been studied. The cylindrical temperature spikes arising from densely ionizing fission fragments have provided the best test to date of the resistance of explosives to initi ation by isolated microscale events. Our interest has been an investigation of the effects of relatively high energy density spherical zones of radiation damage in explosives, interpreted in the context of the hot-spot model. A usual, but low energy density, example of such an event is a thermal spike arising from a knock-on atom. High, local energy densities in roughly spherical shape can be obtained from slow pion bombardment of solids. The formation and destruction of the mesonic atoms formed by the atomic capture of 7r-mesons can result in the emission of "-'12-17 charged particles from a single lattice site. Within a reasonable radius about this site, the overlap of the differential energy loss of these * Now at the Department of Chemistry and Lawrence Radiation Laboratory, University of California, Berkeley, California. 1 F. P. Bowden and A. D. Yoffe, Fast Reactions in Solids (Academic Press Inc., New York, 1958), Ch. 7. 2 F. P. Bowden and K. Singh, Proc. Roy. Soc. (London) A227, 22 (1954). 3 The explosions observed during irradiation of nitrogen iodide are not considered relevant to a general discussion of explosive initiation by this mechanism due to the anomalous properties of this material.2 4 J. Zinn, J. Chern. Phys. 36, 1949 (1962). 6 T. Boddington, preprint "The Growth and Decay of Hot Spots, etc.," presented at the Ninth Symposium on Combustion, 1962. 6 F. P. Bowden and A. D. Yoffe, Endeavour 21, 125 (1962). particles, assumed degraded to heat, should create a quite intense, "spherical" hot spot. This process will be considered in detail. Three primary and three secondary explosives have been bombarded with slow pions. The subsequent be havior of the "initial" temperature profiles arising from the energy deposition has been followed on the hot-spot model and compared with experiment. IRRADIATION ARISING FROM ,..-CAPTURE AND ABSORPTION Mesonic atoms,1 formed by the atomic capture8 of low-energy, negatively-charged mesons, become a mo mentary "irradiation source" capable of producing rela tively high energy densities centered about the captur ing site. The meson capture into Bohr-like orbits is usually assumed to occur in the vicinity of the K-shell electrons, resulting in a mesonic orbit corresponding to ~14 for muons (mu_=207me) and n"'17 for pions (m,,_= 273me). After capture the mesons cascade to lower levels and interact with the nucleus (7r-, u-) or, if the nuclear interaction is weak enough, decay (u-). The free decay times of 7r-and u-mesons are 2.SX 10-8 sec and 2.2X 10-6 sec, respectively. However, a greater irradiation energy-density arises from the capture and absorption of 7r-mesons than from u-mesons since the former (1) have a greater mass and hence greater bind ing energy in the mesonic atom (see Part B of this section) and (2) more importantly, produce much more severe nuclear disruption following absorption (see Part C of this section). For this reason only pion processes are considered further; a detailed discussion of the rela tive atomic capture in chemical compounds, the cascade process, and the nuclear absorption follows. 7 M. B. Stearns, Progr. Nucl. Phys. 6, 108 (1957). 8 The terms capture and absorption follow Stearns' usage.7 Capture is the designation for a meson going into a mesonic orbit about the nucleus, while absorption denotes the disappearance of the meson through interaction with the nucleus. 1736 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:19PION CAPTURE IN EXPLOSIVES 1737 A. Capture in Chemical Compounds No clear systematics have yet arisen on the distribu tion of captured mesons among the constituents of chemical compounds. The sole theoretical prediction for this distribution,9 usually referred to as the "Fermi Teller Z law," indicates that the relative capture proba bilities should be roughly proportional to the atomic numbers. Recent experiments by BaijallO indicate a cap ture probability of Zr in various compounds where r falls in the range r= 1 to r= 1.4 (r= 1 corresponds to the Z law). However, other investigatorsll report that the relative atomic captures follow more closely the simple stoichiometry of the compound. In our calcula tions of the distribution of captured pions among the elements of the explosives, we considered two different capture probabilities-one independent of Z and the other proportional to ZI. Capture of mesons by hydrogen atoms forms a special case of the above. The small neutral system so created moves through the lattice9,12 and, in general, transfers the meson to a more highly charged nucleus before the hydrogen mesic K shell is reached. No more than 1 % capture of mesons by hydrogen atoms in hydrocarbons has been observed,I3 and for our purposes capture by hydrogen can be ignored. B. Cascade ProcessI4 The 11"-mesons captured in the n= 17 shell have two primary modes of losing energy in cascading to lower orbitsI5-I7j they can undergo Auger transitions (a radi ationless transition to a lower state ejecting an orbital electron) or radiative transitions. In competition with the cascade is the process of nuclear absorption which is very strong for pions in states of low angular momen tum. The chemical constituents of the investigated ex plosives divide naturally into quite low-Z elements (C,N,O) and high-Z elements (Hg, Pb). Auger proc esses (ejecting K electrons when energetically possible) dominate as the deexcitation mechanism through the n=4-->n=3 transitionI5,I8 in the low-Z elements, then giving way to radiative transitions for the last part of 9 E. Fermi and E. Teller, Phys. Rev. 72, 399 (1947). 10 J. S. Baijal "Atomic Capture of ~--Mesons in Chemical Compounds and'the 'Fermi-Teller Z-Law'," University of Cali fornia Radiation Laboratory Rept. UCRL-10429 (1962). 11 J. C. Sens, R. A. Swanson, V. L. Telegdi, and D. D. Yovano vitch, Nuovo Cimento (10) 7, 536 (1958). 12 W. K. H. Panofsky, R. L. Aamodt, and J. Hadley, Phys. Rev. 81, 565 (1951). 13 J. Tinlot and A. Roberts, Phys. Rev. 95, 137 (1954). 14 The same general behavior is expected for the cascade follow ing either 11"-or u-capture; results of investigations on both types of mesons are included in this section. In particular, discussion of n = 2->n = 1 transitions in the light elements is based on results using muons. IS C. R. Burbidge and A. H. de Borde, Phys. Rev. 89, 189 (1953) . 16 R. A. Ferrell, Phys. Rev. Letters 4,425 (1960). 17 M. A. Ruderman, Phys. Rev. 118, 1632 (1960). 18 Y. Eisenberg and D. Kessler, Phys. Rev. 123, 1472 (1961). the cascade. In fact, only one radiative transition is probable, since nuclear absorption in these elements takes place predominantly from the 2p level,7·19 Hence the deexcitation cascade in the light elements involves the ejection of "-'14 low-energy electrons ("-'0.05 to 10 keY) from the capturing site. As the Z of the capturing nucleus increases, however, radiative transitions be come more importantI5 and only 7 or 8 Auger electrons (of "-'1 to SO keY energy) can be expected from atomic capture at mercury and lead nuclei. Specific numerical calculations are presented in the following section. This nearly complete Auger deexcitation in the light elements has not been directly observed due to the very short ranges of most of the ejected electrons. Evidence supporting this process follows, since it is apparent that the electronic vacancies in these low-Z atoms must be quickly filled, compared to the time for radiative trans~ tions, to support this type of cascade. (In fact, addI tional electron "depletion" in these elements should occur, since the filling of electronic K-shell vacancies by L electrons results in further Auger ejection.20) Although the vast majority of the Auger electrons from carbon, nitrogen, and oxygen are experimentally undetectable, those few originating in the radiation dominated n= 3-+n= 2 and n= 2-+n= 1 transitionsl4 possess enough energy to be observed. Auger electrons from both these transitions in the light elements of emulsions have been found2I-23 in numbers consistent with theory.I5.I8 Additionally, a marked decrease in the L mesonic x-ray yield relative to the K yield is observed in these elements24 as expected by general theoretical prediction and implying increasing competition by Auger transitions.25 The presence of sufficient K elec trons to support these final Auger transitions implies adequate replenishment during the cascade.26 Also of importance to our subsequent calculations is the time required for the cascade process. Fermi and Teller9 estimated the time for a muon to go from +2- keY kinetic energy to the lowest mesonic orbit and found it to be of the order of 10-13 sec in both conductors and insulators. The same order of magnitude is obtained from detailed calculations for the cascade process alone 19 M. Camac, A. D. McGuire, J. B. Platt, and H. J. Schulte, Phys. Rev. 99, 897 (1955). 20 R. L. Platzman, Symposium on Radiobiology edited by J. J. Nickson (John Wiley & Sons, New York, 1952), pp. lO?ff. 21 A. Pevsner, R. Strand, L. Madansky, and T. ToohIg, Nuovo Cimento (10) 19,409 (1961). . . .. 22 A. O. Vaisenberg, E. A. Pesotskaya, and V,. A .. SmIrmts~ll, Soviet Phys.-JETP 14, 734 (1962) [Zh. Ekspenm. 1 Teor. F1Z. 41, 1031 (1961) J. 23 J. E. Cuevas and A. G. Barkow, Nuovo Cimento (10) 26, 855 (1962). . 24 J. L. Lathrop, R. A. Lundy, V. L. Telegdi, and R. Wmston, Phys. Rev. Letters 7, 147 (1961). 26 Absolute agreement with the theoretical mesonic K and L x-ray yield predictions has not as yet been observed, most proba bly due to lack of knowledge of the relative population of (n, t) states during capture.24 However, Ferrelll~ and Ruderm.anl: have indicated that the only competing mechamsm for deexcltatlOn of the low-lying levels is the Auger process. 26 A. H. de Borde, Proc. Phys. Soc. (London) A67, 57 (1954). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:191738 J. CERNY AND J. V. R. KAUFMAN TABLE I. Prong distribution of pion stars in carbon and nitrogen.· No. of prongs per star Light nucleus 0 2 3 4 5 Carbon (944 stars), 16.3 13.9 23.3 39.1 7.1 0.3 percent Nitrogen (430 stars) , 13.7 14.7 20.0 34.0 15.6 2.0 percent • Reference 30. in carbon, nitrogen, and oxygen.15 It has been suggested27 that trapping of mesons might occur in some insulators; however, Culligan et al.28 find no difference within their experimental accuracy in the decay probabilities of pions stopped in Teflon as compared to aluminum, where no such trapping should occur.9,17 C. Nuclear Absorption The nuclear absorption of pions, as observed in photo graphic emulsions, frequently results in the production of "stars." The difference in the number, energy, and nature of star products resulting from absorption in light as compared to heavy nuclei again makes such a separation convenient. Pion absorption events occurring in the light ele ments of nuclear emulsions (C, N, 0) have been inves tigated and we use the results of Menon et al.29 as repre sentative. Additionally, Ammiraju and Lederman30 have used a diffusion chamber to isolate and investigate pion absorption reactions in carbon and nitrogen. Both results can be summerized as follows: One to five prong stars are observed from absorption reactions on these elements with the distribution peak ing at three prongs. Table I gives the experimental prong distribution of Ref. 30. The dominant reaction in each of these elements is (1) (2) (3) (Though the reaction products are written as a particles and protons, unambiguous distinction between hydro gen isotopes and between helium isotopes was very diffi cult or impossible.) Other data on reactions (1) through (3) are given in Table II. It is apparent that low average proton and a-particle energies are observed even though the pion rest mass ("-'140 MeV) has been converted to energy. 27 R. Huby, Phil. Mag. 40,685 (1949). 28 G. Culligan, D. Harting, N. H. Lipman, L. Madansky, and G. Tibell, Nuovo Cimento (10) 20,351 (1961). 2U M. G. K. Menon, M. Muirhead, and O. Rochat, Phil. Mag. 41, 583 (1950). 30 P. Ammiraju and L. M. Lederman, Nuovo Cimento (10) 4, 283 (1956). Absorption reactions on heavy nuclei show different behavior. Reactions with silver and bromine nuclei have been observed which 29 (a) show one to five prong stars with the percentage distribution falling with increasing prong number (average number of prongs per star= 1.1), (b) possess an alp ratio of 0.3 with Ea"2.16 MeV (data from one-prong stars) and E1'"2.9.6 MeV (data from one-and two-prong stars), and (c) are, in general, describable by compound nucleus theory. Although pion absorption reactions with mercury and lead nuclei have not been investigated, the result (c), above, leads one to expect qualitatively similar behavior to that observed with silver and bromine nuclei. By contrast to these results, only 2.4% of muon cap ture events in photographic emulsions are accompanied by charged particle emission (decay events are included in this number) ,31 >65% of which corresponds to single a particle or proton emission. DISCUSSION OF A TYPICAL EVENT In this paper we are considering a spherical zone of radiation "heating" and now should establish a repre sentative event to use in further discussion. As noted in the preceding section, the capture and absorption of pions in light elements, as opposed to heavy elements, possesses: (1) a greater frequency and lower energy of Auger electrons arising from the cascade, and; (2) a higher average prong number, a greater fre quency of alp emission, and lower average particle energies following nuclear absorption. Hence, the cumu lative effect of the greater number of charged particles leaving a low-Z capturing site, aided by the almost dif fusive motion of the low-energy Auger electrons, sug gests that events occurring on light elements will tend to produce a region of radiation damage more closely approaching spherical symmetry. In addition, the en ergy density present in a spherical region about the capturing site will be greater in events arising from light rather than heavy elements. (This comparison ignores the effect of the recoil fragment following pion absorp tion in heavy nuclei.29 Since this recoil will produce a TABLE II. Frequencies and reaction-product average energies for dominant reactions in the light elements. Reaction Ea on (MeV)b (1) C (2) N (3) 0 -Reference 30. b Reference 29. 5.8 5.0 7.5 Ea (MeV)· "-'7.5 "-'4.7 Frequency of reaction Ep in particular (MeV)b nuc!eus(%) S.6 25.0- ~1S.S- ,,-,S (21±35)b 31 H. Morinaga and W. F. Fry, Nuovo Cimento (9) 10, 30S (1953) . This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:19PION CAPTURE IN EXPLOSIVES 1739 cylindrical zone of damage proceeding away from the original capturing site, its effects are not in accord with the model we wish to consider here.) Among the light elements, then, an event on nitrogen was chosen as "typical" since it is the only element common to all explosives. The representative cascade on nitrogen was assumed to involve 14 (n, l= n-1)~(n-1, l= n-2) transitions from n=17 to n=3. Using the Klein-Gordon equation and values of the K and L shell ionization potentials appropriate to carbon to allow for screening by the meson,15 Auger electron energies were obtained ranging from 69, 86, and 107 eV to 1.92, 3.76, and 8.46 keV for the first three and last three transitions, respectively. Transitions with An> 17 were ignored since these adjust ments would have negligible effect on our subsequent cal culations. Mesonic x-rays from the two possible radiative transitions (25 and 126 keV)7 should have no interac tions near the capturing site. Lastly, the pion is assumed to react in its dominant mode with the nitrogen nucleus, producing three a particles of energy 3, 5, and 7 MeV, respectively (so that E .. =5 MeV). The time required for the entire energy deposition process within a reasonable spherical volume enclosing the capturing site is of interest. As will be seen later, a radius of <400 A is appropriate. This limiting time is of the order of 10-13 sec-that of the Auger cascade since the times involved in (1) nuclear absorption and charged particle emission, and (2) stopping of the lower energy electrons and passage of the higher-energy elec trons and a particles outside this region can both be easily estimated as appreciably less than 10-13 sec. In summary, the typical irradiation process chosen is the capture and absorption of pions on nitrogen result ing in the ejection of 14 low-energy electrons and 3 low energy a particles from the capturing site with the charged particles stopping within or traversing a region of 400 A radius in 10-13 sec. EXPERIMENTAL PROCEDURE Materials and Mounting Techniques Three primary and three secondary explosives were investigated with safety requirements necessitating dif ferent mounting techniques for the two groups. Targets of du Pont colloidal lead azide, normal lead styphnate, and mercury fulminate (using a 1957 Pic atinny Arsenal laboratory sample) varying in thickness from 0.44 g/cm2 to 0.73 g/cm2 were prepared by incor porating these explosives in 2-cm Ld. plastic disks sealed on both sides by a thin Mylar film. These disks were then mounted individually in circular slots cut in rec tangular styrofoam blocks. The high explosives investigated were 2,4,6-trinitro toluene (TNT), Grade I; pentaerythritol tetranitrate (PETN), high purity (>99%); and hexahydro-1,3,5- trinitro-s-triazine (RDX), Type B (containing 7.7% HMX). These were pressed without binder at 20000 TABLE III. Irradiation summary. Stopped pions in the Number of different targets Explosive targets (in millions) Lead azide 3 1.7, 2.2, 2.3 Lead styphnate 2 2.9,4.4 Mercury fulminate 2 1.5,2.1 TNT 2.9 PETN 2 2.9,4.3 RDX 2 2.8,3.9 or 5000 psi into 1.27 em diameter by 0.81 em long cylin ders of average thickness 1.27, 1.23, and 1.16 g/cm2, respectively. The cylinders of explosives were then placed in styrofoam holders similar to those mentioned above. Irradiation Procedure The negative pion beam of the Columbia University Nevis cyclotron was used. For protection in the event of explosion, the targets in their styrofoam holders were mounted in the center of a metal-bound wooden box provided with viewing ports. The pions entered through a 0.15 mm thick Inconel foil, 5 em in diameter, which was sufficiently strong to contain any possible explosive fragments. An additional magnet had been placed in a standard beam path to produce a suitably converging beam, and the position of maximum pion flux through an area of one square inch was established. Measurements were also made to establish a curve of the amount of poly ethylene absorber versus number of stopped pions. Roughly 600 stopping pions/sec g (C7H7)n were avail able. Targets, either singly or doubly in tandem, were placed in the protective box and adjusted so that the first target was at the predetermined position of maxi mum flux. Maximum stopping rates were achieved by varying the polyethylene absorber to correct for the entrance foil to the box and changes in the air path. The beam was monitored during the 1-to 2-h runs by following coincidence counts from a counter telescope preceding the target system. At the end of the irradia tion, the flux/in~ at the target position was remeasured and agreed within 7% with the initial flux. RESULTS No explosions or visible signs of thermal decomposi tion occurred with any of the explosives. Table III lists the explosives and the number of pions stopped in each. The absolute values of the stopped pions are accurate to only ±25%. Two comparisons of irradiated to "standard" samples were made. X-ray diffractograms of irradiated lead azide, PETN, and RDX showed no detectable differ- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:191740 J. CERNY AND J. V. R. KAUFMAN 10' }' .. ~ " .. 10' u c .. , " 8. E 10' :? "0 ~ . Radius, A MII·3lG3' FIG. 1. The solid lines depict the initial temperature increase above the experimental equilibrium temperature arising from typical, pion-induced hot spots in lead azide and RDX. The dashed line shows the contribution of the three alpha particles alone in the initial temperature profile of RDX. ences from the unirradiated material. The thermal de composition of lead azide was investigated at 230°C using standard methods82; no appreciable differences in behavior of the irradiated vs the unirradiated explosive were observed outside the normal experimental fluctu ations. DISCUSSION Basically, we have also1,2 found that explosives are not initiated by isolated, high energy density irradiation processes. However, we wish to consider this situation further and determine whether, on kinetic and heat transfer grounds, such behavior should be expected from explosives. This investigation will be in the context of the hot-spot theory.4-6 Two conclusions appear: one, that most of the experimental effort so far has not been directed toward those explosives most likely to initiate on this model due to local, irradiation-induced heating; and, two, that the hot-spot model is apparently inade quate to explain the growth of reaction to explosion for this type of thermal event. The latter point arises, since we will show that predictions on this model disagree significantly with experiment. (Previous discussionl,2 of the expected thermal effects of fission-fragment bom bardment of explosives has relied heavily for its inter pretation on the 10-8-10- 5 cm diameters for hot spots observed at intermediate temperatures and, in addition, attempted no analysis in accord with heat-transfer theory.) A detailed calculation has been carried through on two of the six explosives: lead azide, because it has been frequently investigated, and RDX, which on close con sideration appeared to be the one most likely to initiate. Figure 1 shows the "initial temperature"-radius profiles (which in reality are energy-radius profiles) of the typi- 32 B. Reitzner, J. V. R. Kaufman, and E. F. Bartell, J. Phys. Chern, 66,421 (1962). cal hot spots arising from pion capture on nitrogen for both lead azide and RDX. The subsequent behavior of these profiles will be followed. Table IV indicates the number of typical events which occurred in these explo sives. These initial temperature profiles for the hot spot were obtained by determining the energy loss of the Auger electrons and alpha particles within successive spherical shells about the capturing site. The electron range-energy relation of GlockerB8 determined for 1 < E < 300 ke V electrons was used after modifica tion to a general form.84 The diffusive motion of these very low energy electrons (all < 10 keV) was compensated for by taking the average range as one-half the practical range84 as given by Glocker's expression. Alpha-particle differential energy loss values were taken from recent calculations of proton and alpha-particle ranges in ex plosives.85 Also shown on Fig. 1 is the contribution due to the alpha particles alone in establishing the initial temperature profile in RDX; by difference the effects of the Auger electrons can be seen. One theoretical paper [R. A. Mann and M. E. Rose, Phys. Rev. 121, 293 (1961) ] has suggested that the mesons (actually calculated for muons in carbon) are captured by the process of Auger capture when their kinetic energy is around 8 keV and that the resultant initial population peaks at nrv7 rather than n"'14. Should this be the case (also see Ref. 18), only "-'3-5% of the events in Table IV would be accompanied by a 14-electron cascade as discussed previously (plus the Auger capture electron); a somewhat greater per centage, by a 13-electron cascade; etc. A minimum of > 3000 events in accordance with our description of a typical event would still have been present in all explo sive samples. GENERAL HOT SPOT CALCULATIONS If the exothermic decomposition of these explosives follows a first-order reaction, then the future behavior TABLE IV. Number of 7I"-+14N-t3a+2n events in lead azide and RDX (in thousands). Total Explosive Run stops Lead azide 1700 Lead azide Lead azide RDX RDX 2 2200 3 2300 2800 2 3900 Total stops on Total number of nitrogen atoms typical events Stoic Z-Iaw Stoic Z-law 1460 575 274 108 1890 1970 1120 1560 745 780 1090 1520 355 370 210 293 140 147 205 286 33 L. Katz and A. S. Penfold, Rev. Mod. Phys. 24, 28 (1952). 34 F. Seitz, Phys. Fluids 1, 2 (1958). 3. J. Cerny, M. S. Kirshenbaum, and R. C. Nichols, Nature 198,371 (1963). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:19PION CAPTURE IN EXPLOSIVES 1741 of this initial hot spot is governed4,6 by (aT/at) -k'il2T= (Q/C)nZ exp( -E/RT), (4) where T is the absolute temperature, t is the time, k is the thermal diffusivity, Q is the heat of reaction, C is the specific heat, n is the mass fraction of unconsumed reactant, Z is the Arrhenius preexponential factor, E is the activation energy, and R is the gas constant. Men tioned later will be the density, p, and the thermal con ductivity, A. Equation (4) has not been solved analyti cally. Solutions to Eq. (4) in spherical geometry are desired which are valid for all r; solutions consonant with the initial temperature-radius profiles as shown in Fig. 1 and taking into account reactant consumption were obtained in the following manner. The problem is attacked by determining the temper ature profiles at a series of characteristic times, r. The heat-conduction equation for an inert material-the left side of Eq. (4)-is solved over each time interval, and the heat of reaction is assumed to be liberated batchwise and instantaneously at the end of each time interval. Close approximations to subsequent tempera ture profiles in inert materials from such complex initial conditions can be obtained by an extension of the Schmidt method of solution of unsteady heat-flow prob lems.36 Consider the sphere divided into a series of con centric, spherical shells each of thickness t:.r. Then, if a time period /10 (which will be the T above) equal to (t:.r)2j2k is chosen, the temperature in any shell at the end of this t:.e is determined by a geometric weighting of the temperatures of both shells bounding the shell under consideration at the beginning of the time interval. The average temperature of each shell over the particular r is used in calculating the temperature increase, if any, due to exothermic chemical reaction.31 Reactant con sumption is, of course, followed explicitly. PHYSICAL DATA Table V presents the thermochemical data and the parameters used in the calculations. Considerable effort has been spent in attempting to utilize reasonably relia ble data. Two sets of kinetic data, both reported as uni molecular decompositions,38,39 are given for RDX in the table; the one for the solid near the melting point, kinetics A, was used in the detailed calculations. Lead azide decomposes autocatalytically.40 The kinetics given 3& T. C. Patton, Ind. Eng. Chern. 36, 990 (1944). 37 Due to the absence of appropriate data, it is assumed that the kinetic expressions obtained for the decompositions of explo sives at """'130o-300°C can be extrapolated to these high tempera tures and very short times, Normal experimental errors in the original determinations of E and Z will not affect our conclusions. 38 D. Gross and A. B. Amster, Eighth Symposium (International) on Combustion (The Williams and Wilkins Company, Baltimore, 1962), p. 728. 39 A. J. B. Robertson, Trans. Faraday Soc. 45, 85 (1949). 40 J. Jach, "The Thermal Decomposition of a-Lead Azide," Brookhaven National Laboratory, Report BNL 6032. TABLE V. Thermochemical data and parameters used in the calculations. Property (1) k(cm2/sec) from p(g/cm3) C(cal/g degK) A (cal/sec cm degK) (2) Q(cal/g) (3) (A) Z (sec1) E (kcal/mole) Temperature range (B) Z(sec1) E (kcal/mole) Temperature range (4) Ar (1) (5) T (sec) Lead azide l.14X 10-3 4.10- 0.09- 4. 2X 10-4- 3970 1012/ 36.31 195°-253°C RDX l.12X 10-3 1.66b 0.264b 4.90XlO-4b 1280d,e 102\02 (from QZ) b 57.2b 170°-200°C, solid 1018.6K 47.5& 213°-299°C, liquid lO 10 0.438XIo-n 0.447X10-11 -A. F. Belajev and N. Matyushko, Compt. Rend. URSS 30, 629 (1941). b Reference 38. o P. Gray and T. C. Waddington, Proc. Roy. Soc. (London) A23S, 106 (1956). d Q. to gaseous H.O. eM. Tonegutti, Z. Ges. Schiess-u. Sprengstofiw. 32, 93 (1937), f Reference 40. K Reference 39. are those of the final decay, which agrees within experi mental error to those of the maximum rate. Further, it is assumed that k and C are temperature independent and that the average heat capacity of the products of reaction is the same as that of the original explosive. Rough comparisons of the latter with an average reaction product temperature of 9000K indicate that this approximation is very reasonable. Although the crystal density of lead azide and the pressed density of RDX differ by 15% from the values given in Table V, this difference has no effect on the calculations so long as a consistent density is used. The important physical parameters in the calculation are k and C, and the effects of the temperature dependence of C will be discussed later. It is apparent from the method of calculation that k determines r; if the value for k in Table V differs somewhat from the true average value, this will, to a first approximation, simply alter the time interval between the temperature profiles to be shown. Further Considerations Several additional features inherent in the calcula tions require discussion. As noted in Table V, T values of O.44X 10-11 sec were used, and a specific considera ation of the first time interval, Tl, is necessary. It is obvious that the complex molecular processes taking This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:191742 ]. CERNY AND ]. V. R. KAUFMAN Radius, A FIG. 2. Temperature profiles for lead azide at various times !lrising from a pion-induced hot spot. The characteristic time, T, 1S 0.438 X 10-11 sec; T23 corresponds to 10-10 sec. place during T1 can only be poorly approximated. The initial energy deposition occurred in 10-13 seconds, so that this order of time was the actual beginning of Tl; similar starting times are used in temperature-spike theory41 although in our case the majority of the excita tion is in electronic energy. In order to discuss this problem using thermal concepts, it was assumed that alll•42 the electronic ionization and excitation is very ra pidly converted to lattice vibration in these materials.41 With molecular vibration frequencies of ",1013/sec, it is then possible to have the original energy profile smoothed out by perhaps Tl/2 to lend some validity to the concept of temperature. As noted earlier, only the average temperatures over Tl were used in calculating the heat liberated by reaction. The applicability of macro scopic thermal parameters to these microscopic events was assumed here as elsewhere43 due to the lack of con tradictory information. Another point is that the modified Schmidt method does not permit simple estimation of the temperatures at the core of the sphere. These temperatures were ob tained insofar as possible by requiring their reasonable behavior with time in conjunction with an over-all heat balance for each time interval. Although truly satis factory results were not obtained at the later time inter vals, the minor adjustments at the core were not at all essential in establishing the general behavior of the tem perature profiles. Calculated Results for Lead Azide Figure 2 shows the thermal decay of the hot spot in lead azide. It is apparent from the figure that only minor chemical reaction has occurred. If the autocata lytic thermal decomposition behavior of lead azide per sists at high temperatures, then even the minor heat 41 F. Seitz and ]. S. Koehler, Solid State Phys. 2, 307 (1956). 42 W. Kauzmann, Quantum Chemistry (Academic Press Inc., New York, 1957), p. 696. 43V.1. Goldansky and Y. M. Kagan, Chemical Effects of Nu clear Transformations (International Atomic Energy Agency London, 1961), Part I, p. 47. ' liberation observed here could be an overestimation, since the important rate at these very short times would be the initial rate. Calculated Results for RDX Conversely to those for lead azide, the results of the calculations for RDX predict an initiation should result from a typical hot spot in this explosive. These results are shown in Fig. 3. Since the (calculated) step-function behavior of the complete decomposition in successive spherical shells noted in Fig. 3 cannot be simply handled by the Schmidt method, additional heat balances have been incorporated in the calculation to smooth out the temperature profiles. Although the results of the hot spot model require initiation, it is obvious that the method of calculation can not give any true account of the progress of the decomposition in distance and time. Initiation of RDX is predicted for this model using either set of kinetics; it should be noted that the first order decomposition assumed by the model is in accord with experiment.38.39 That this result would be invali dated by temperature-dependent changes of k or C is unlikely. By far the more important parameter is C; and calculations show that the true average C could be greater than that given in Table V by a factor of two, and initiation would still be predicted (kinetics A). The heat of fusion is not known but, as estimated from avail able data on secondary explosives, it would only de crease the calculated temperatures by < 100 deg C. Since the predicted results are not in agreement with experiment, the question of an overestimate of the initial temperature profile must be considered again. The escape of many of the slow (subexcitation) elec trons44 from the zone of interest, although less likely in molecular than in atomic materials,45 could markedly decrease the calculated energy deposition; unfortu nately, no estimate of the importance of this effect in . , ~ . ~ 10 ~ ... ROX 10·~-'---:'-o----'---L---'--.L~--,LL,,--..I.L--'--L-'-......J o 20 40 80 100 120 140 Radius, A FrG. 3. Indicative temperature profiles for RDX at various times arising from a pion-induced hot spot. The characteristic time, T, is 0.447 X 10-11 sec. 44 R. L. Platzman, Rad. Res. 2, 1 (1955). 46 M. Burton, W. H. Hamill, and J. L. Magee, Proc. Intern. Peaceful Uses At. Energy 29, 391 (1958). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:19PION CAPTURE IN EXPLOSIVES 1743 organic solids is known to us. However, it should be noted that the result of the calculation for RDX-pre dicted initiation-is not critically dependent on the present values of the initial temperatures. (This can be seen from the last part of this section.) Also, only the average behavior of the nuclear absorption was con sidered, which lead to a total Ea of 15 MeV. Much lower, total alpha-particle energies are observed29 (Ea totals of 7.3 and 9.0 MeV appear in two of eight cases studied in detail) whose greater energy deposition might compensate for such energy escape from this zone though of course in a proportionately reduced numbe; of cases. The failure of the hot-spot model under these experi mental conditions has been established. Apparently more must be involved in the growth of reaction to explosion for these microscale events than can be de scribed by simple thermal factors (related conclusions have been presented by Zinn4 and Mader46). Although the hot-spot theory has had some success in predicting explosion for larger hot spots at lower temperatures, there appears to be no fundamental reason why our experimental conditions should require a new mecha nism. It may well be that the previous success of the hot-spot model represents agreement with a large-scale limit of a more satisfactory theory. General trends of the predictions of this model useful for confirmatory experiments follow. GENERAL COMPARISON Additional insight into the relative sensitivity of these explosives to initiation by microscale radiation heating on this model can be obtained from Table VI. This table gives the temperature of a shell necessary to cause various temperature increases due to exothermic reac tion during the time interval T. A representative value of T= O.44X 10-11 sec was used. The initial temperature profiles of Fig. 1 in conjunction with Table VI enable one to immediately expect different behavior from lead azide and RDX. (Statements in this section will gen erally refer also to expected results from fission-frag ment irradiation.) The initial temperature-cylindrical radius hot spots arising from a fission-fragment track in lead azide and RDX-has been estimated by similar methods to those outlined above. Such tracks would possess greater temperatures than the spherical profiles of Fig. 1 by a maximum factor of ~2.5 over the impor tant radial distances. Of these two explosives, only lead azide has been investigated. One can readily see, in agreement with these experiments,l that on this model fission-fragment irradiations of lead azide at room tem perature or at 290°C will not cause initiation. (The minimum explosion temperature of lead azide is 315°C.2) It is obvious that close proximity to the macroscopic explosion temperature has no direct relevance if initi- 46 C. L. Mader, "The Hydrodynamic Hot Spot and the Shock Initiation of Homogeneous Explosives," Los Alamos Scientific Laboratory Report LA 2703. TABLE. VI. Temperature required to produce various tem perature mcreases due to decomposition in O.44X 10-11 sec for several explosives. Temperature (OC) of the explosive required to produce a temperature increase of Explosive- Kinetics' Q/C,oC 5°C Q/2C,oC (Q/C-5), °C Lead azide 4400 1970 10 200 No solution RDX-kinetics 4850 705 980 1120 A RDX-kinetics 4850 755 1150 1380 B TNT-1 4090(Q,b Co) 3400 No solution No solution TNT-2 4090 1920 6330 (~24 000) PETN-auto- 5330 (Q, C)b 500 690 775 catalytic PETN-"zero 5330 945 1800 2480 time" Type of Explosive- decomposition E Z, Kinetics observed kcal/~ole sec-1 Lead azide autocatalytic 36.3 10'2 RDX-Kinetics A first order 57.2 1()2L2 RDX-Kinetics B first order 47.5 10'8.6 TNT-l° first order 37.0 10'0.7 TNT-2d 41.1 10'2.6 PETN-autocatalytic· autocatalytic 52.3 1023.1 PETN-"zero time"· autocatalytic 38.6 10, •. 3 a Temperature dependence of the kinetics is given in the lower half of this table. b W. R. Tomlinson, Jr., "Properties of Explosives of Military Interest" Picatinny Arsenal Technical Report 1740, Rev. 1 (1958). ' o Reference 38. d Reference 47. • Reference 48. ation due to these microscale processes is under con sideration. Included in Table VI are the most recent kinetic data for TNT88,47 and PETN ;48 complete kinetic data are not available for lead styphnate and mercury fulminate. A comparison of the TNT and lead azide data in Table VI shows that no initiation would be predicted for TNT under our experimental conditions. The paramount importance of the kinetics in "pre dicting" if initiation might arise from thermal degrada tion of ionizing radiation can be seen from Table VI. Additionally, to be consistent with the assumption that the kinetic data can be extrapolated to these very short times, it is the initial rate of reaction which should be approximated. No difficulty arises from this in consider ing the extrapolated kinetics of the first-order decompo sitions of TNT88,47 and RDX89 (and HMX89). However, 47 J. Zinn and R. N. Rogers, J. Phys. Chern. 66, 2646 (1962). 48 M. A. Cook and M. T. Abegg, Ind. Eng. Chern. 48, 1090 (1956) . This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:191744 J. CERNY AND J. V. R. KAUFMAN care should be taken where possible to use the initial rates of autocatalytic reactions such as are observed for PETN and lead azide, especially if predicted initiation appears probable, until the decomposition can be con sidered to be proceeding at a faster rate. Cook and Abegg determined the temperature dependence of the PETN decomposition at "zero time" and when the reac tion was well under way; these two sets of kinetic data are denoted as "zero time" and autocatalytic, respec tively, in Table VI. The marked difference in these reaction rates changes the expected behavior of PETN from predicted initiation on the latter kinetics to an uncertain situation for which detailed calculations would be necessary on the former. The importance of the kinetics of the RDX decom position in testing the hot-spot model by microscale irradiation processes has been shown (and it appears THE JOURNAL OF CHEMICAL PHYSICS that HMX39 should behave in a similar manner). Unfor tunately, apart from the investigations reported here, no high energy density irradiations (pions or fission fragments) of explosives whose decomposition kinetics provide a test of the hot-spot model have been reported. Further experiments are planned to extend the present results; accordingly, a high-temperature irradiation of RDX and HMX with fission fragments is in progress. ACKNOWLEDGMENTS We wish to thank Dr. Warren F. Goodell and the Nevis Laboratory of Columbia University for their generous assistance with the 7I"--meson irradiations. It is a pleasure to thank Dr. Fred P. Stein, Dr. Paul Levy, and Dr. Peter J. Kemmey for many valuable discus sions. Lastly, we wish to thank Lt. James F. Mallay for his help with various phases of the calculations. VOLUME 40, NUMBER 6 15 MARCH 1964 Effects of Electron Transfer on High-Resolution NMR Spectra * CHARLES S. JOHNSON, JR., AND JOHN c. TULLyt Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut 06520 (Received 4 November 1963) The 15.1-Mc/sec NMR spectra of isopropylquinone and 2,5-diethoxyquinone in equilibrium with the corresponding semiquinone radicals in pyridine show line positions and intensities that depend on the radical concentrations. These spectra are compared with curves calculated by the density matrix technique with the assumption that modulated isotropic hyperfine interactions provide the only relaxation mechanism. Satisfactory agreement is found for the methyl protons in isopropylquinone from which we estimate that T,> 100T2 for the methine protons; however the line shapes for the methyl protons in 2, 5-diethoxyquinone suggest that other relaxation mechanisms may be present. I. INTRODUCTION NUCLEAR magnetic resonance linewidths can be used to determine the rate of electron transfer between molecules in solution, and under certain con ditions can provide information about the spin density distributions in the molecules which contain unpaired electrons.l---6 Unfortunately, the situation is not always clear cut since the magnetic nuclei in molecules usually interact by means of the well-known indirect electron * This research was supported in part by a grant (NSF-GP 1203) from the National Science Foundation. t National Science Foundation Undergraduate Summer Participant. 1 C. R. Bruce, R. E. Norberg, and S. 1. Weissman, J. Chern. Phys. 24, 473 (1956). 2 H. M. McConnell and H. E. Weaver, Jr., J. Chern. Phys. 25, 307 (1956). 3 R. G. Pearson, J. Palmer, M. M. Anderson, and A. L. Alfred, Z. Electrochem. 64, 110 (1960). 4 R. W. Kreilick and S. 1. Weissman, J. Am. Chern. Soc. 84, 306 (1962) . 6 M. W. Dietrich and A. C. Wahl, J. Chern. Phys. 38, 1591 (1963). 6 C. S. Johnson, Jr., J. Chern. Phys. 39,2111 (1963). coupling. Complex NMR spectra are then obtained in which the absorptions can not be assigned to specific nuclei.1 Another complication arises from the un certainty of the relative importance of the dipole-dipole part of the hyperfine interaction as a nuclear relaxa tion mechanism. In this paper we report an investigation of the effects of modulated hyperfine interactions on the NMR spectra of strongly coupled nuclear spin systems. NMR spectra have been obtained at 15.1 Mc/sec for iso propylquinone [R2=CH(CH 3h; R5=H] and 2,s-di ethoxyquinone (R2= R5= OC2H5) in equilibrium with the corresponding semiquinone radicals as shown III Reaction (1): o· 0 0 O. &R+ (yRJ! =*= (yRi-&R2 (1) R~ R~ R~ R~ 0- 0 0 5 0- 7 P. L. Corio, Chern. Rev. 60, 363 (1960). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 18:45:19
1.1703121.pdf	HighTemperature Deformation of Rutile N. E. Farb, O. W. Johnson, and P. Gibbs Citation: Journal of Applied Physics 36, 1746 (1965); doi: 10.1063/1.1703121 View online: http://dx.doi.org/10.1063/1.1703121 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/36/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Study on high-temperature deformation and practical application of ultra high strength steel BR1500HS in hot stamping AIP Conf. Proc. 1532, 819 (2013); 10.1063/1.4806916 High-load, high-temperature deformation apparatus for synthetic and natural silicate melts Rev. Sci. Instrum. 78, 075102 (2007); 10.1063/1.2751398 Deformation-induced nanoscale high-temperature phase separation in Co–Fe alloys at room temperature Appl. Phys. Lett. 90, 201908 (2007); 10.1063/1.2740476 Deformation apparatus for use in highresolution, hightemperature studies of mantle rheology Rev. Sci. Instrum. 64, 211 (1993); 10.1063/1.1144437 HighTemperature Superconductivity Phys. Today 44, 22 (1991); 10.1063/1.881302 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sun, 21 Dec 2014 01:19:131746 S. O'HARA AND G. M. McMANUS cient. It was found by x-ray anomalous transmission photographs that the strain in these iron bands was perpendicular to the trace of the original solid-liquid interface. This result is similar to that obtained by SchwuttkelO on impurities in silicon and germanium crystals. CONCLUSIONS It has been shown that the growth of a low-disloca tion-density oxide crystal, ZnW0 4, can)e accomplished by the Czochralski method using the same careful 10 G. H. Schwuttke, Ref. 5, p. 497. JOURNAL OF APPLIED PHYSICS growth conditions previously developed for the growth of semiconductor crystals. The effect of thermal stresses in the growing crystal on the dislocation density was found to be critical for this oxide. Chemical and x-ray methods showed that the predominate slip plane is (100). ACKNOWLEDGMENTS We would like to express our thanks to Dr. W. A. Tiller for suggesting this problem. The work was sup ported by the Advanced Research Projects Agency, Contract AF49-(638)-1029. VOLUME 36, NUMBER 5 MAY 1965 High-Temperature Deformation of Rutile* N. E. FARB,t o. W. JOHNSON, AND P. GIBBS University of Utah, Salt Lake City, Utah (Received 25 August 1964; in final form 16 November 1964) The creep of 44 rutile (TiOz) single crystals was studied as a function of stress (0'), temperature (T), ambient atmosphere, and impurity. For constant ambient and impurity in the range 1100° to 1230°C and O'=3.9-7.26kg/mm2, the steady-state deformation rate Ecould be fitted quite well to f=A exp (BO'-AH/kT) , under conditions of constant or decreasing temperature. However, an "hysteresis" was observed for speci mens tested with increasing temperature; E at a given temperature was irreversibly reduced after creep at a higher temperature. Nitrogen and argon atmospheres produced similar results. The "constant" A ranged between 1010 and 1019 for undoped specimens in N2, depending on purity. A decreased by a factor of 10' in O2, and fell to 100 and 1011 for doping with Fe and AI, in 02, respectively. B ranged: 1.0--2.2 in N2 and 0.7- 1.4 in O2• AH ranged: 5.7-7.7 eV in Nz, 4.0-7.5 eV in O2,2.3 and 5.8 eV for doping with Fe and AI, in O2, respectively. In all cases, less than 10% scatter was observed for tests with the same atmosphere and purity. Significant changes in activation energy as a function of stress were found only for the highest-purity speci mens (about 30-ppm cation impurities); in these crystals AH increased abruptly by ~2 eV at a stress be tween 3.9 and 4.5 kg/mm2, with accompanying changes in A and B. I. INTRODUCTION CREEP studies of Al20a and MgO prior to 1957 have been reviewed by Wach tman.1 Chang2 measured the steady-state creep in sapphire and ruby at high temperatures (above 1500°C) and attempted to explain his results on the basis of oxygen self-diffusion. Rogers, Baker, and Gibbs3 obtained somewhat lower activation energies for steady-state creep in sapphire at lower temperatures. Cumerow4 measured the steady-state creep of MgO from 1456° to 1700°C, and found a variation in activation energy between specimens from 3.5 to 7.0 eV. Cumerow suggested that this variation in activation energies could result from changes in the * Supported in part by the U. S. Air Force Materials Laboratory. t Present address: Autonetics, Anaheim, California. Submitted in partial fulfillment of the requirements for a Ph.D. in Physics at the University of Utah. 1 J. B. Wachtman, Jr., Creep and Recovery (American Society for Metals, Cleveland, 1957), p. 344. 2 Roger Chang, J. Appl. Phys. 31, 484 (1960). 3 W. G. Rogers, G. S. Baker, and P. Gibbs, Mechanical Properties of Engineering Ceramics (Interscience Publishers, Inc., New York, 1961), p. 303. 4 R. L. Cumerow, J. Appl. Phys. 34, 1724 (1963). apparent activation energy for diffusion of oxygen and magnesium defects due to a change in concentration of impurities. Hirthe and Brittain5 have measured the steady-state creep in Ti02 at temperatures below 1050°C in a con trolled atmosphere. Activation energies ranging from about 1.5 to 4.0 eV were measured, but the variation between specimens was too large to draw any conclu sions about effects of ambient. The defect concentration in rutile is strongly dependent upon the partial pres sure of oxygen in the ambient. It has been suggested&-9 that the predominant defect in rutile, at least in some temperature range, may be titanium interstitials or even a Ti complex involving two or more ions,lO rather than 6 W. M. Hirthe, Am. Ceram. Soc. Bull. 41, 311 (1962). 6 J. Yahia, Phys. Rev. 130, 1711 (1963). 7 J. H. Becker and W. R. Hosler, J. Phys. Soc. Japan Suppl. II 18, 152 (1962). 8 T. Hurlen, H. KjIlosdal, J. MarkaIi, and N. Norman, WADC TR 58-296, ASTIA No. AD 155638. 9 R. D. Carnahan and J. O. Brittain, J. Appl. Phys. 34, 3095 (1963). 10 J. B. Wachtman, and L. R. Doyle, Phys. Rev. 135, A276 (1964). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sun, 21 Dec 2014 01:19:13HIGH-TEMPERATURE DEFORMATION OF RUTILE 1747 oxygen vacancies, as had been widely assumed. Although the identity of the defect remains somewhat in doubt, it is well established that concentrations of the order of one percent are readily achieved in reducing atmos pheres. Thus, precise control of ambient composition is of extreme importance in the study of deformation and diffusion processes in this material. II. EXPERIMENTAL PROCEDURE The design of the creep-testing equipment is illus trated in Fig. 1. All high-temperature components were constructed of single-crystal sapphire or high-density alumina. The reactivity of Ab03 and Ti02 at the tem peratures used in these experiments is reported to be extremely small.ll The specimens were deformed in four point bending with the load applied by a rectangular sapphire bar having two knife edges, 0.396 em apart, with a centrally located groove on the upper side. Force was applied to this bar by a single-crystal alu minum oxide pull-rod, t in.X 10 in. long, in which a central knife edge has been cut in a slot 0.100 in. XO.650 in. The lower knife edges are fabricated from a sapphire disk t in.X 1 in. with a i in. central hole and knife edges spaced 1.650 cm apart. The specimen and the upper knife edges were accurately positioned in the slot and between the lower two knife edges by an aluminum jig. The load was applied to the lower end of the sapphire rod through a damping spring by weights. Deflection of the pull-rod was measured with SUPER } t== 1 LOAD WEIGHTS l..... . ATTACHED IfERE) -{CORE OF SCHAEVITZ \DIFFERENTIAl. TRANSRlRMER FIG. 1. Creep-testing apparatus. 11 S. M. Lang, C. L. Fillmore, and L. H. Maxwell, J. Res. Nat!. Bur. Std. 48, 298 (1952). a Shaevitz linear-variable differential transformer, operated at 500 cps and 3.2 V. The output signal, after demodulation, was recorded on a Brown strip-chart recorder, with a sensitivity of ±0.5 J.!, and long-term drift less than 2 J.!. This was equivalent to a sensitivity, in measurement of the outer fiber strain for the spec i ment used, of 5XlO-7 sec-I. The knife-edge disk was mounted on an alumina pedestal which was cemented to the center of a steel heat baffle which served as a heat shield for the strain transducer. A closed-one-end, high-density alumina tube served as the outer container for the ambient gas, which was introduced between the alumina tubes at the base of the pedestal. A stainless steel diaphragm fitted with an asbestos gasket and a molded ceramic ring served as a low-temperature seal. The whole furnace assembly could be raised or lowered over the center pedestal. The furnace, heated by two Super-Kanthal heating elements, was capable of a temperature of 1475°C at the sample, with an oxygen gas flow of 5 ft3/h. The temperature could be controlled to ±1.5°C using a Brown proportional controller, recalibrated for 4 Pt-Pt 13% Rh thermocouples in series. The temperature was controlled manually to ± 1°C using variable trans formers during the creep tests. The flow of ambient gas was controlled by two-stage pressure regulators coupled to Airco No. 805-1603 dual-range flow meters. The dew point was lowered to less than -75°C by use of concentrated H2S04 and activated alumina desiccators. Rectangular beam specimens (0.75X1.5X20 rom) were cut from single crystal Ti02 boules obtained from the Linde Company. Specimens were oriented so that the c axis was parallel to the length of the beam, and (except as noted below) the a axes were perpendicular to the faces of the speci mens. The boules were cut using 250 grit resinoid-bonded diamond cut-off wheels, 0.125 mm oversize, then hand lapped to final size (±0.OO5 rom) using 1200 mesh emery in water. All specimens were marked so that their position and orientation in the boule were known. All specimens were cleaned prior to testing by washing in hot benzene, concentrated H2S04, distilled water, and absolute ethyl alchol. Specimens were preheated for 9 h at 1133°C, in the atmosphere in which they were to be tested. One hour was allowed after the lowering of the furnace so that the specimen could reach temperature and compositional equilibrium with the atmosphere. This time was judged sufficient, since most defects influence resistance, and resistance measurements in the absence of stress indicated no change in conductivity more than 6 min after change in ambient atmosphere at the creep-testing temperature. No differences be tween the conductivity or the creep characteristics of rutile specimens tested in pure nitrogen or argon were found. For this reason, nitrogen was used throughout the measurements described below. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sun, 21 Dec 2014 01:19:131748 FARB, JOHNSON, AND GIBBS Specimens were stressed at the minimum level of 1.6 kg/mm2 maximum outer-fiber tensile stress, which resulted from the weight of the pull-rod during tem perature changes, thus minimizing the total strain during the sequence of measurements. This unloading and reloading procedure did not significantly affect the strain rates. All measurements were made after pre straining the specimens 1.2% outer fiber strain; the strain involved in each strain-rate determination was about 0.15%. III. RESULTS Preliminary tests with specimens from boule No.1, oriented with the c axis parallel to the length of the beam, and the a axes at 45° to the faces of the beam, showed slip on {012} planes as determined from slip traces which were visible on all faces of the specimens. Specimens from boules No.2, No.3, and No.4, which were oriented with the a axes perpendicular to the beam faces, showed slip traces only on the compres sion and tension surfaces. Etching of these specimens in hot H2S04 showed the plane of zero bending to be approximately in the center of the specimen, with etch-pit rows aligned in the (101) directions. There also appeared a tendency for the pits to line up in rows perpendicular to the (101) directions, which, presumably, indicates rearrangement of dislocations by climb, suggestive of early states of polygonization. These observations appear to establish that the slip system operating when specimens are stressed in the manner described for boules No.2, No.3, and No.4 is on {lOl} planes, in (101) directions. Typical creep curves for specimens below lOOOoe and above llOOoe are shown in Fig. 2. The usual three stages of creep are clearly evident for the specimen deformed at lower temperature. The first two stages were still present but somewhat supressed when the speci mens were deformed at higher temperatures. Both the strain rate and the total strain involved in stages 1 and 2 varied widely at temperatures below 1000oe, even I 2.4 ! I IB ~ I.' -------;-o.--'-~-- ., --:~! .. StageD 4. 60 7' 10. FIG. 2. Typical creep cruves in low-temperature and high-temperature ranges. 120 ! c: I ! to: I .. g ~ .. T ! 100 10 6.7 8oule43 0. AImosphere .,.0 6.61 kQ/rnm' (max. outor fiber .tr ••• ) 6.8 6.9 7.0 7·' 1fT (°1('.,0", 7.2 FIG. 3. Apparent activation-energy plot, showing effect of increasing and decreasing temperature. 7.3 for the same atmosphere and applied stress. In fact, the total time involved in stage 1 was found to vary as much as two orders of magnitude for specimens deformed under identical circumstances. Furthermore, a decrease in temperature of 300e after deforming a specimen at temperatures below l0000e generally resulted in a transient creep closely resembling the creep character istic of stage 1. Strain rates observed above HOOoe were much more consistent. Strain rates in stage 3, the so-called steady-state region, were constant to within a factor of two for different specimens at given tem perature, atmosphere, and applied stress. Most of this difference was attributable to variation in impurity concentration, as explained below. Fracture of speci mens frequently occurred in tests below 830°C. The lowest temperature at which measurable creep occurred was 572°e. In the temperature range HOOoe to 1230oe, the data could be well represented (except as noted below) by E=A exp(Bu-tJ.Hlkt), (1) where tJ.H is the apparent activation energy, and E and u are the maximum outer-fiber strain and stress, re spectively. A unique activation energy was obtained only for measurements made with decreasing tempera ture. Measurements (7 specimens, boule No.3) in an O2 atmosphere, using temperature increments, yielded a change in slope of a InE vs liT plot from an average of 3.9 eV for the lower temperatures to 2.2 eV for the upper temperatures. Further experiments indicated an "hysteresis" associated with creep rate, as indicated by Fig. 3. That is, the slope of the lnE vs liT plot con- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sun, 21 Dec 2014 01:19:13HIGH-TEMPERATURE DEFORMATION OF RUTILE 1749 I t .-0: I 8 oS ~ .. T¥ ! 100 BouI.-4 7 Spec I",.,. (1IY«Qge va .... , OaA~ ~::1~:;.01 mm'/Ieo tr (kgtm"" (1IIGIt 0UIIr flbeI' '_1 FIG. 4. Strain rate versus stress, showing constancy of creep param eter B over 2 orders of magnitude in strain rate. sistently decreased as the temperature was increased, but was essentially constant for decreasing tempera tures. Similar results were obtained whether tempera ture increments or decrements were used first. Eight specimens (boule No.3) were then tested using tem perature decrements only. The slope of lne vs l/T plots were very consistent, although a systematic difference in strain rate between specimens was observed. Statis tical analysis of the data yielded an activation energy of 4.1±0.4 eV (the indicated error represents 5% uncertainty limits). Comparison of these results with the data obtained in the previous temperature incre ment tests showed a close correspondence between the activation energy determined from the initial slope of the increment tests and the decrement slopes indicated above, which was within the uncertainty of the data. In view of the consistency of these results, all subse quent measurements were made utilizing temperature decrements only. The activation energy for the same stress and tem perature range, but with a N2 ambient atmosphere, was determined (8 specimens, boule No.3) to be 5.7±O.S eV. These results and others to be discussed are summarized in Table r. Although the "constants" A, B, and AH in Eq. (1) depended on specimen purity, ambient atmosphere, and one case, stress (see below), they were constant with in experimental error for a particular specimen over the temperature range 1100° to 1230°C when tested using temperature decrements, as is well demonstrated by Figs. 3, 4, and 5. Stress dependence of the creep rate for each specimen TABLE r. Creep parameters for various atmospheres, purity, and stress levels. Error limits, where indicated, are 95% confidence limits, calculated using standard statistical techniques. Stress Boule Atmos- level Imp. A B 6.H number phere kg/mm' added (sec)-' mm'/kg eV 3 0, 6.67 1.07 Xl0' 0.69±0.05 4.08±0.39 3 N, 6.67 1.01 Xl0'. 1.00±0.13 5.70±0.52 4 0, 5.1 } 1.12 Xl011 1.42 ±0.08 7.05±0.30 4.5 4 0, 3.9 2.2 Xl0' 1.42±0.08 5.00±0.30 4 N, 5.1 1.1 Xl019 2.22±0.30} 7.78±0.57 4 N, 4.5 1.1 Xl019 1.74±0.15 4 N, 3.9 4.9 Xl012 1.74±0.15 5.87±0.39 4 0, 7.26 Fe 2.52 Xl00 0.66±0.06 2.34±0.13 4 0, 5.5 Al 1.6 Xl011 1.46 5.78 was determined at the conclusion of the activation energy measurement by increasing the stress in incre ments of about O.S kg/mm2 outer fiber stress. Typical results are presented in Fig. 4. In all cases the behavior was accurately described by Eq. (1), although syste matic variation of the value of the constant B was observed with changes in ambient atmosphere and and impurity concentration. Also, apparently abrupt changes in the value of B were observed for specimens from boule No.4 in a N2 atmosphere when stressed at levels above 5.1 kg/mm2• At about this value of the stress, B increased from 1.74 mm2/kg to 2.22 mm2/kg. In all other cases, B was independent of stress. Specimens from boule No.4 were found to be con siderably "softer" than those from boule No.3; that T ! 10 6.7 6_1 6.9 7.0 o Boule 4, undoped, Ot almo •. A Boule 4, ""doped, Nt atmo •. o louie 4, Fe-doped, Ot .""0 •. o Boule 4, AI-doped, Oa .Imo •• 7.1 7.2 FIG. 5. Activation-energy plot for various atmospheres and impurities. Stress ranged from 4.5 kg/mm' for undoped specimens to 7.2 kg/=' for Fe-doped specimens. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sun, 21 Dec 2014 01:19:131750 FARB, JOHNSON, AND GIBBS is, a smaller stress was required to achieve comparable creep rates. The differences in orientation were in all cases less than 2°, and not sufficient to account for the observed difference in creep rates. These two boules were obtained from the same supplier; however, they represented different batches. Emission spectrographic analysis showed that the average cation impurity level was approximately 50% higher in boule No.3 than in boule No.4, in which the total cation impurity con centration was in the range of 30 ppm. Detailed im purity analyses of boules No.3 and No.4 are given in Table II with a typical analysis of Johnson-Mathey TABLE II. Average impurity content (in ppm) for several specimens taken from different locations in the houles, and for Johnson-Mathey polycrystalline Ti02• Johnson Mathey Boule No.3 Boule No.4 Sn Ca Mg AI Fe eu Si Ni Mn Cr V Ag 12 12 20 0.20 8 0.60 3.4 5.2 1 5 0.3 o 12 14 0.13 5.7 0.8 0.34 3.4 4.9 5.9 0.4 o 8.6 10 0.09 4.1 0.6 0.25 2.7 4.1 4.1 0.2 "Spec-pure" polycrystalline Ti02 for comparison. It was not possible to identify any particular metal impurity as being more significant than the others. Systematic variation in all three creep parameters A, B, and ~ were found between specimens from boules No.3 and No.4; these results again are summarized in Table I. Activation energies observed in boule No.4, the purer of the two specimens, were generally higher than those observed in boule No.3 by as much as 3 eV. Furthennore, variation of activation energy with stress, as noted above, was observed in specimens from boule No.4, and not in boule No.3. It should be noted that although the maximum stress for the boule No.4 speci mens was less than that for the boule No.3 specimens, the strain rate obtained in boule No.4 specimens was higher than observed in any of the other work. Further evidence for the importance of impurity con centration in determining creep behavior was obtained from comparison of creep rates, under identical experi mental conditions, for specimens from different loca tions within the same boule. Specimens cut from the center of the boule consistently showed a higher creep rate, by as much as a factor of two, than those cut from nearer the surface of the boule. Insufficient data were available, however, to pennit detection of statistically significant differences in activation energy and stress dependence. Spectrographic analysis indicated a some what lower impurity concentration for the "softer" specimens near the center of the boule. The importance of the role of impurity implied by the above results suggested that an attempt should be made to "dope" specimens with a high concentration of impurity, to determine, if possible, what the limiting behavior would be. Four specimens were prepared, two at a time, by heating for 10 h at lOOO°C in a sealed, evacuated Vycor tube, containing different amounts of FeCla• This treatment produced a reddish-brown dis coloration of the specimens which was considerably more pronounced for one batch than for the other (presumably indicating a higher Fe concentration). To produce reasonable creep rates, a stress level of 7.2 kg/mm2 was required for these specimens. The values of all three creep parameters were significantly lower than those observed for the "pure" specimens, as indi cated in Table I. The value of A indicated in Table I is for the two specimens with highest Fe concentration. The observed value of A for the other two specimens was somewhat larger. Both Band !J.H appeared to be independent of impurity concentration in this range, and the values indicated in Table I apply to all four specimens. In addition, one specimen was similarly doped using AlCla• The observed values of the creep parameters for this specimen (see Table I) were all intennediate between those observed for the "pure" and the Fe-doped specimens. IV. DISCUSSION OF RESULTS The data presented in the previous section and sum marized in Table I appear to suggest the following generalizations: 1. Heat treatment in a nonoxidizing atmosphere results in consistent and reproducable increases in the value of all three creep parameters. 2. An increase in the general level of impurity con centration results in a general decrease in the observed values of the three creep parameters. 3. An increase in applied stress above a certain level in "pure" specimens apparently results in a change in the rate-limiting step in the creep process, with an increase in activation energy of about 2 eV. This is accompanied by modest changes in the value of the creep parameter A and perhaps some change in the parameter B, although this was observed in only one case at very high stress levels. As can be seen from Table I, the increase in the ob served value of !J.H accompanying a change from oxidiz ing to nonoxidizing ambient, is quite uniform, ranging between 0.8 and 1.6 eV. The change in the observed value of B is even more consistent, being approximately 0.3 mm2/kg in all cases. The value of A changes by about 102• These changes undoubtedly result from an increase in the concentration of point defects, either oxygen vacancies or titanium interstitials. In a sense, the result of adding impurities to the crystals appears to be opposite to that obtained by increasing the point defect concentration, since the change in all three creep parameters is in the opposite direction in the two cases. This suggests that the action of the impurities might be that of producing impurity-defect complexes, thus reducing the concentration or mobility of vacancies and interstitials. The impurity analysis carried out as part of this work revealed only the concentration of cation impurities. The possibility of important effects [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sun, 21 Dec 2014 01:19:13HIGH-TEMPERATURE DEFORMATION OF RUTILE 1751 due to anion impurities should not, of course, be overlooked. The change in creep process, which apparently occurs at high stress levels, probably is not due to the stress directly, but rather to the high strain rate which results. The stress levels at which the transition apparently occurred were exceeded in a number of the less pure specimens. Such behavior might result from a change from extrinsic to intrinsic behavior for one of the point defects, when some critical strain rate was exceeded, this increasing the observed activation energy by the energy of formation of the point defect. The increasing activation energy observed for in creased defect concentrations is rather difficult to ex plain on the basis of the usual climb-limited processes. The activation energy for formation or motion of one of the point defects involved may change with a change in the Fermi level of the crystal, which is certainly changed with changing defect concentration. The "saturation" activation energy exhibited by the Fe doped specimens probably represents the activation JOURNAL OF APPLIED PHYSICS energy for interstitial Fe diffusion, but the exact nature of the creep-limiting process is still not clear. The linear dependence of Eon (l which is expected for the rate limi tation of creep by a Cottrell-type atmosphere was certainly not observed. Probably the most significant result of this work has been the demonstration of the extreme complexity of the creep behavior of rutile. Qualitative variation in creep behavior was shown to accompany changes in impurity concentration, atmosphere, stress (or strain rate), and temperature, which would seem to rule out the possibility of a single, simple rate-limiting process. Fundamental advance in the analysis of creep processes in such crystals apparently must await detailed infor mation on impurity and defect kinetics. ACKNOWLEDGMENT The authors wish to thank M. L. Gonshor of Kennecott Research Center for his valuable assistance in performing the spectrographic analyses. VOLUME 36. NUMBER 5 MAY 1965 Fatigue Hardening of Polycrystalline Copper, Nickel, and Aluminum R. J. HARTMANN M ax-Planck-Institut jiir MetaUjorschung, Stuttgart, Germany (Received 20 April 1964, in final form 19 November 1964) Fatigue hardening of polycrystalline copper, nickel, and aluminum has been determined by measuring the changes in the area of the hysteresis loop. It can be described as a superposition of two types of hardening, called V I and Vu. These two types can far better be resolved by the specific irreversible work of deformation than by the increment of subsequent stress amplitudes. Hardening V I occurs mainly at high strain amplitudes and is not peculiar to fatigue. Saturation (I) is reached between N = 10 and N = 1()4, depending on strain amplitude and stacking fault energy. Hardening Vu is effective only at low and medium strain amplitudes. It is characterized by a very low hardening coefficient. Saturation (II) is reached between N = 104 and N = 106• Intermediate overloading during saturation (II) results in a pronounced softening, indicating that harden ing V II is caused by the production of small-scale obstacles. I. INTRODUCTION Results of Previous Investigations CYCLING an annealed specimen between constant limits of plastic or total strain results in a rapid hardening for the first few percent of the life of the specimen. This initial hardening decreases quickly until a stage of low or zero hardening is reached, which is called saturation hardening.1 The beginning of this stage is connected with the occurrence of micro cracks, which later on propagate until the specimen fractures. This general picture of fatigue hardening holds at high as well as low strain amplitudes, although the hardening proc esses differ considerably. 1 W. A. Wood and R. L. Segall, Proc. Roy. Soc. (London) 242, 180 (1957). At high amplitudes, hardening is accompained by a pronounced increase of internal strains.2 In single crys tals, it is strongly orientation-dependent.3 The surface slip marks are very similar to those observed in unidirec tional straining,2 and the specimen fractures perpendicu lar to the direction of maximum normal stress.4 In this case, hardening is governed primarily by long-range in teractions of dislocations. 2 At low strain amplitudes, the internal strains increase only slightly.2 Hardening is associated with a pro nounced decrease of the average coherency length. 2 It shows little orientation dependance, and only small dif- 2 R. J. Hartmann and E. Macherauch, Z. MetallIc 54, 197 (1963). 3 M. S. Paterson, Acta Met. 3, 491 (1955). 4 C. Laird and G. C. Smith, Phil. Mag. 7, 847 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sun, 21 Dec 2014 01:19:13
1.1702680.pdf	Oxygen Outgassing Caused by Electron Bombardment of Glass Jack L. Lineweaver Citation: Journal of Applied Physics 34, 1786 (1963); doi: 10.1063/1.1702680 View online: http://dx.doi.org/10.1063/1.1702680 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stress relaxation in electron bombarded silicate glasses J. Appl. Phys. 55, 3315 (1984); 10.1063/1.333368 Outgassing Caused by Electron Bombardment of Glass J. Appl. Phys. 31, 51 (1960); 10.1063/1.1735417 ElectronBombardment Damage in OxygenFree Silicon J. Appl. Phys. 30, 1232 (1959); 10.1063/1.1735298 Outgassing of Glass J. Appl. Phys. 26, 1238 (1955); 10.1063/1.1721882 Negative Charging of Glass Fibers under Electron Bombardment J. Appl. Phys. 22, 1387 (1951); 10.1063/1.1699876 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Mon, 22 Dec 2014 01:25:39JOURNAL OF APPLIED PHYSICS VOLUME 34. KUMBER 6 JtTNE 1963 Oxygen Outgassing Caused by Electron Bombardment of Glass JACK L. LINEWEAVER Corning Glass Works, Corning, New York (Received 3 October 1962) A system employing a mass spectrometer as a continuous flow gauge has been used to study the oxygen evolved from aluminum-coated glass as a result of electron bombardment. The outgassing from most glasses is found to fit the empirical equation Q=Q.,(1-exp-t/K). In this equation Q is the sum of the oxygen released during the bombardment time t and that evolved during a subsequent thermal outgas and Q .. is the maximum amount of oxygen expected from a sample bombarded for long times. Experimental results from Code 8603 glass indicate that Q .. is a measure of the range of 10-to 27-keV electrons in glass, K varies inversely with electron current per unit mass of glass affected, and that electron current density may have a secondary effect on electron range in bulk glass. Oxygen outgassing data are presented from 12 commercial glasses subjected to 150 p.A of 20-ke V electrons bombarding a 3-X i-in. area. A mechanism of oxygen release is proposed which involves the dissipation of electron energy, the charge produced in the glass by the electrons, and the availability of nonbridging oxygen atoms in the glass structure. INTRODUCTION THE widespread use of glass in the construction of electron devices employing oxide cathodes has always posed numerous questions with regard to the effect of the glass on cathode life. It was reported in an earlier paper on this work! that studies of aluminum coated glass bombarded with 20-keV electrons showed oxygen to comprise at least 95%of the gas evolved. Being of particular interest in the use of cathode-ray tubes, this work has continued and includes more complete studies of glasses used in electron device manufacture. The present paper describes the latest techniques used for studying oxygen outgassing of glass under electron bombardment and relates the outgassing characteris tics of ten glasses to two parameters of an empirical formula. A mechanism of oxygen release is proposed which is based on changes in the glass that have been observed as a result of electron bombardment. Studies of the dependence of oxygen outgassing an electron energy and current density are included. APPARATUS AND METHOD The techniques reported earlier! have been modified slightly for the present work. In brief, glass samples are ground to 0.060 in. and polished on the bombardment side. A SOOO-A nitrocellulose film is applied followed by 1000 A of vacuum evaporated aluminum. The nitro cellulose film is volatilized during subsequent thermal cycles leaving a conductive aluminum coating loosely adhering to the glass. Temperature control of the sample is provided by 40-gauge Chromel-Alumel thermocouples located on the bare side of the samples at the center of each intended electron bombardment pattern (raster). Small spots of Saureisen cement and waterglass are used to attach the thermocouples to the samples. The thermocouple leads are imbedded in the solder glass frit panel to funnel seal of the sample tube. The cathode-ray sample-tube arrangement is shown schematically in Fig. 1. The sample and funnel coatings are operated at 1 B. J. Todd, J. L. Lineweaver, and J. T. Kerr, J. App!. Phys. 31, 51 (1960). ground potential with the cathode at negative high potential to eliminate voltage isolation problems be tween the sample and the thermocouples. The sample tube is connected by way of a relatively high conductance system, Fig. 2, directly to the source of a mass spectrometer. Kinetics and calibration of the complete system have been reported.! Gettering of oxygen within the sample tube is limited by the neck aperture and by the use of reasonably inert materials Electron Gun Graphite Coating , :O-cent8ring Magnet I Solder Glass 'Frlt Seal ,---,,-_Tin Oxide Coating Anode Button --rllt]!!!!:;==='=';"""'"",,,,~';"==i!;!!f1- t,~ Sold.r Glass , Frlt Seal _Sample Electrode Graded Seal Evaporated Aluminum FIG. 1. Cathode-ray sample tube., 1786 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Mon, 22 Dec 2014 01:25:39OXYGEN OUTGASSING BY ELECTRON BOMBARDMENT OF GLASS 1787 in the sample cavity so that at least 85% of the oxygen evolved is measured by the mass spectrometer. The one-inch magnetically operated mercury valve, Fig. 2, is used in place of the silver chloride valve of the earlier apparatus.! The purpose of this valve is to iso late the mass spectrometer from atmospheric pressure when sample tubes are changed or processed. Mercury in the reservoir seals the mating surfaces of a glass ball joint. During all outgassing measurements, the mercury diffusion pump is operating and the cold traps are cooled to -78°C. After a period of about one week, sufficient mercury is cryogenically pumped into the right trap to lower the mercury level below the lip of the dome. When this happens, the level is restored by opening the stopcock to the reservoir. Each time the pressure in the system on the right of the mercury valve is increased to an atmosphere, the mercury in the right trap is pushed into the trap drain where it is easily re moved from the system. The other reservoir and trap drain service the mercury pump in the same manner. Small amounts of mercury are maintained between the stopcocks and the vacuum in an attempt to hide the stopcock grease from the system. OUTGASSING MEASUREMENTS The bombarding electron current is measured by placing a meter in series with the sample electrode, Fig. 1, and ground. The significance of this measurement is considered in the Appendix. The bombarding electron energy is taken as the potential difference between the sample and the cathode of the electron gun. Samples of aluminum-coated glass are maintained at 200°C during bombardment by placing a small furnace around the panel end of the sample tube. The amount of oxygen evolved during the electron bombardment QE is determined by integration of the flow data recorded by a strip chart recorder on the output of the mass spec trometer. When the electron beam is turned off, oxygen continues to be evolved from the sample if the sample temperature is maintained at the temperature of bom bardment. To accelerate the depletion of this thermally 1m NERCURY _ SOFT IRON f ~ ALPERT VALVE ~TOPU"PS lli,ONIZATION GAUGE COLD TRAP \,'1G, 2. c::onnectin~ vacuum system, u 200 • II) ... 1i j teo I 120 FIG. 3. Outgassing of I Code 8603 glass. ~ 80 z ... ~ 0 o· 6 12 18 24 BOMBARDMENT TIME-HRS. evolved oxygen QT, the sample temperature is increased to 350°C. Figure 3 is a plot of the oxygen outgassing data from Code 8603 glass bombarded with 150 p.A of 20-keV elec trons using a 3-X i-in. raster. These data were obtained using a new area of glass for each of the QE points. Following each bombardment, the sample was baked out until the oxygen had been depelted. If one assumes a quantity of oxygen of uniform dis tribution within a volume of glass, and if a constant energy or force is applied to cause it to become available for removal, one would expect the Q data, where TABLE 1. Q", and K for glasses bombarded with 150 p.A of 20-keV electrons using a 3-X I-in. raster. Corning glass code Q.,[.u(Hg)liters at 25°C] K(h) 0081 247 23.4 8603 227 12.4 9019 178 23.7 9010 170 20.6 0041 83 5.9 0120 72 7.7 7740 60 5.5 0129 60 5.4 7070 51 12.5 7800 49 3.2 Q=QE+QT, to fit an equation of the formula Q=Q", X (1-exp-t/ K). In this equation, t is the actual time of bombardment, Q is the total amount of oxygen that can be produced and measured, and K is a measure of the time dependence of the phenomenon. The Q data from most glasses have been found to fit this equation rather well. In these cases the Q", and K parameters provide a concise means of describing the outgassing characteris tics under a given set of conditions. COMMERCIAL GLASSES Twelve commercial glasses were bombarded using 150 p.A of 20-keV electrons and a 3-X i-in. raster with bom bardment time t as the only variable. The data from ten of these were found to fit the empirical formula rather well. The Q", and K parameters are listed in Table 1. These values were determined with the aid of an elec tronic coml>uter as the 'V~lues th:;\t min~mi?e the avera~e [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Mon, 22 Dec 2014 01:25:391788 JACK L. LINEWEAVER ~ on 60 COl 11 .. '10 :z: ~ I 0 20 I ~ 6 o FIG. 4. Outgassing of Code 1723 and 8870 glasses. fractional deviations of the Q data. For values of t< 4K, the average deviation is less than 12% for any particu lar glass. For values of t> 4K, the observed Q data are somewhat larger than one would expect for several glasses. The oxygen data from Code 1723 and 8870 glasses ~o not fit the empirical equation. These data are plotted m Fig. 4. Glasses have been measured which evolve less than 1 JL(Hg)liter as a result of 24 h of bombardment. This is considered to be the background of the system. OXYGEN RELEASE MECHANISM An examination of glass samples after electron bom bardment has revealed the following, as depicted by the schematic cross section in Fig. 5: 1. The surface is displaced from the original surface plane of the glass in the direction of the bombarding electrons. As observed with a Zeiss interference micro scope, the displacement increases with increasing Q and is in the order of tenths of microns as Q approaches Qcc. 2. There is a very sharp density change with depth in the bombarded area of the glass as evidenced by inter ference colors. Judging from the brilliance of the trans mitted fringes, the transition from the less to more dense layers is less than 100 A thick. 3. The depth at which the transition in 2 occurs in creases progressively with bombardment time and ap proaches the estimated depth of electron penetration as Q approaches Q",. The depth of 20-keV electron pene tration in Code 7740 glass is estimated at 2.7 IJ. based on the Thomson-Widdington law and the work of Spear.2 4. The familiar electron browning begins at the bottom of the layer described in 2 and 3 and is continu ous to the depth of electron penetration with x-ray browning going much deeper. This was determined by etching away the bombarded surface in steps and meas uring the sample thickness loss and visible transmission increase between each step. Measurements of thickness changes were made with a Sheffield gauge. The sample was carefully indexed so that all measurements were taken at the same spot. S. It is not possible to produce reboil in a heavily bombarded area of the glass when the glass is reheated 2 W. E. Spear, Proc. Phys. Soc. (London) B68,991 (1955). in an open flame. (Reboil is the term used to describe the evolution of gas, in the form of bubbles, in molten glass.) 6. As much as 10% of the total oxygen, from the affected volume of some glasses, is removed as a result of electron bombardment. Obviously, the oxygen within the glass structure itself is removed. . In proposing any oxygen release mechamsm, the structure of a simple soda-silica glass is considered. Figure 6 is a two-dimensional picture of such a structure as deduced by Warren3 and is in agreement with the theoretical deductions of Zachariason.4 In three dimen sion each silicon atom is actually surrounded by four oxy~en atoms. It is noted that when Na2~ is added t~ Si02 the oxygen from the Na20 attaches Itself to a SI ato~ thus breaking one link of the normal Si-Q-Si bond. This leaves two oxygen atoms bonded to only one Si with the two Na atoms in the nearby interstices to provide charge neutrality. It is these nonbridging oxygen atoms, or oxygen half-ions, that are impor~ant in the release mechanism. Figure 7 (A) is a simplified section of Fig. 6. The high-energy electrons penetrate the aluminum coating, enter the glass, and dissipate their energy by ionization and excitation of the atoms of the glass struc ture. They come to rest at some depth within the glass producing a net negative charge. This charge and the grounded aluminum electrode set up a field in the glass layer between these two in a direction necessary to move positive sodium ions toward the negative charge region. (Reference is made to sodium ions for simplicity since these are generally considered to be the major charge carrier in glass.) A potential equilibrium is reached in which the arrival of primary electrons is balanced by the arrival of sodium ions together with the diffusion of electrons back to the surface electrode. As the sodium ions become separated from the oxygen half-ions their net positive charge can be satisfied by oxygen half-ions at a lower level [Fig. 7(B)] or electrons diffusing back toward the aluminum electrode. Metallic ions which have become neutralized by electrons exist within the glass in an elemental form and produce the color com monly referred to as electron browning. Bombarding electrons reionize these atoms, or separate sodium atoms from their new oxygen half-ion position, and they move I Raster : ----I Boundries r Z t :: '''K''ft "", .. Do, .. J FIG. 5. Schematic cross section of bombarded glass. 3 B. E. Warren, J. Appl. Phys. 8, 645 (1937). 'W. H. Zachariasen, J. Am. Chern. Soc. 54, 3841 (1932). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Mon, 22 Dec 2014 01:25:39o x Y G E N 0 U T GAS SIN G BYE LEe T RON B 0 MBA R D MEN T 0 F G LAS S 1789 TABLE II. Q~ and K for Code 8603 glass bombarded with vari- ations of sample current (J A) and electron energy (V p) using a 3-X i-in. area. Vp(keV) JA(p.A) Q~[p.(Hg)liters at 25°C] K(h) 10 150 78 5.5 75 48 5.4 25 56 20.9 20 150 227 11.7 75 243 26.9 25 298 118. 27 150 468 30.2 a step deeper into the glass, concentrating themselves at a depth approaching that of the effective primary elec tron penetration as Q approaches Q", [Fig. 7 (Cn Concurrent with the movement of the sodium, oxygen atoms are freed from the oxygen half-ion positions. They can move into vacated half-ion positions nearer the surface (provided sufficient sodium ions are on hand to provide charge neutrality) and stop migrating tempo rarily [Fig. 7 (B)], or they can loose their electrons to the aluminum electrode and become detected by the mass spectrometer [Fig. 7 (Cn Therefore, as the brown region moves deeper into the glass, more and more of the network's removable oxygen is left between this layer and the aluminum electrode in the form of ions. During the thermal outgassing cycle following electron bombardment, the excess primary electrons diffuse to the surface electrode along with all negative oxygen ions which cannot find half-ion positions to satisfy. This leaves a layer of glass far different in composition and properties from the original sample between the browned region and the aluminum. This layer is de ficient in sodium and oxygen half-ions [Fig. 7 (Cn The remaining loose structure would tend to pull itself to gether and reform the normal Si02 tetrahedron thus re ducing the density and lowering the surface plane. Several researchers in this laboratory have observed that reboil resistance is inversely proportional to the helium diffusion rate. Altemose5 has shown that the _5100 ONe FIG. 6. Schematic in two dimensions of a soda-silica glass after Warren.3 • V. O. Altemose, J. App!. Phys. 32, 1309 (1961). eoforo 8olllllard",em 00 eN. Durfll, BolIIII.rdlllant 4IttoIf Ion rnr.ill AI Aftar Bakoout FIG. 7. Schematic of oxygen release mechanism. addition of soda to glass reduces the diffusion rate. Conversely, the removal of soda from the structure should increase the diffusion rate and reduce reboil. Removal of the more weakly bound oxygen from the structure, which may cause reboil given sufficient ther· mal energy, may also contribute to the reduction of reboil. ELECTRON-ENERGY AND CURRENT DENSITY DEPENDENCE The dependence of electron energy and current den sity on the oxygen outgassing of Code 8603 glass has been studied. Electron energies V p of 10, 20, and 27 keY with total sample currents I A of 25, 75, and 150 p,A bombarding a 3-X i-in. area were used. The Qoo and K values determined are listed in Table II. From the work of Kanter,6 the average electron en· ergy loss in the 1000-A aluminum conductive coating has been determined to be 460, 270, and 220 eV for primary energies of 10, 20, and 27 keY, respectively (the 27-keV value was extrapolated from the 2-to 20-keV data). This leaves average electron energies of 9.5, 19.7, and 26.8 keY entering the glass. According to the Thomson-Widdington law and the work of Spear,2 the depth of electron penetration x, in cm is determined by the relation (1) where V p is in volts, d is the density in g cm-a, and {3 is a constant of 6.2X 1011 V2g-1cm2 for a borosilicate glass. The mass of glass is affected M is given by M=Axd, (2) where A is the bombarded area in cm2• Combining (1) and (2), M=AVl/{3· (3) In this work, A is held constant at 14.5 cm2. Assuming the value of {3 to be reasonably good for all glass, the 9.5-, 19.7-, and 26.8-keV electrons effect 2.1, 9.1, and 16.9 mgof glass, respectively, as determined by (3). Since Qoo is expected to be a measure of the removable oxygen in the glass affected, one would expect the Qoo 6 H. Kanter, Phys. Rev. 121,677 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Mon, 22 Dec 2014 01:25:391790 JACK L. LINEWEAVER values at the three energy levels to be in the ratio 2.1:9.1: 16.9 or 1:4.3:8.0. If one simply takes the aver· age of the Q", values at the three energies from Table II (60.5, 256, and 468 JL(Hg)liters at lO, 20, and 27 keY, respectively) the ratio of these values is 1:4.2: 7.8 and is in reasonably close agreement with the mass ratios. However, it is interesting to note that, with the ex ception of the 150-JLA lO-keV value, Q", tends to increase with decreasing electron current density at a given energy. Such a phenomenon could be explained by con sidering the repulsion of the primary electrons by the charge built up within the glass at different current den sities. Under equilibrium conditions, the primary elec trons that have entered the glass are diffusing back to the aluminum electrode at the same rate that new pri maries are arriving. The time required for this equilib rium is apparently a matter of seconds since the net sample current becomes reasonably steady almost im mediately after the electron beam is turned on. There fore, the total charge within a given glass increases as the bombarding electron current increases. This charge has a retarding effect on the normal penetration of the primary electrons and, an increase in the primary elec· tron current would reduce the effective depth of pene tration of primary electrons. This would result in a re duction in the depth of the most negative potential in the glass. From the proposed release mechanism, the depth of most negative potential would become the effective depth for the removal of oxygen since oxygen ions produced at a greater depth would be repelled by the more negative charge level. The time dependence factor K of the oxygen out gassing has been found to be a function of the beam cur rent per unit mass of glass effected. Since the effective depth of electron penetration of the primary electrons is expected to vary with current density, the mass at all energies and current densities is unknown. However, a reasonably good correlation seems to exist if one uses the values computed from (3). Figure 8 is a plot of K 100~----------------------------~ 50 I~I--~----~--~--~----~--~--~ FIG. 8. Outgassing time dependence factor K as a function of current.per unit mass of glass affected IA/M. T ABLE III. Ratio of electron current to mass of glass affected (lA/ M) for various bombardment conditions. Vp(keV) IA(p.A) IA/M(mA/g) K(h) 27 150 8.88 30.2 20 25 2.75 118. 75 8.25 29.6 150 16.5 11.7 10 25 11.9 20.9 75 35.8 5.4 150 71.5 5.5 as a function of I AI M. The values used are listed in Table III. ACKNOWLEDGMENTS The author particularly wishes to express gratitude to Dr. J. T. Kerr for being instrumental in proposing the oxygen release mechanism and to L. H. Pruden for his efforts in collecting the data. APPENDIX. SAMPLE CURRENT MEASUREMENT The sample current measured in this work is that of the primary electrons minus backscattered and second ary electrons which leave the aluminum sample coating and are collected bv the funnel coating. It is generally accepted that when" electrons leave a surface being bom barded with primary electrons, the secondary electrons are those with energies less than 50 eV, whereas the backscattered (rediffused and reflected primary elec trons) may be as energetic as the primary electrons. According to Young,1 an initial electron energy of about 2 keY is required to penetrate a lO00-A aluminum film. Therefore, all of the secondary electrons must originate in the aluminum and only backscattered electrons origi nating in the glass with initial energies greater than 2 keY are collected by the funnel coating. Although the ratio of secondary to primary electrons may vary with primary energy, the ratio would be the same for all samples at a given primary energy. The backscattered electrons having energies approaching the primary elec tron energy may originate in the glass. Archard8 re ports that the backscattered fraction depends upon the atomic number, and roughly upon the density, of the material bombarded. Therefore, the primary electron current necessary to maintain a given sample current would be expected to depend upon the glass sample. A more complete understanding of the controlling oxygen release factors is necessary in determining the importance of considering these currents. As a matter of interest however, the ratio of secondary (including true secondary plus backscattered electrons) to primary electron current IslIp has been measured for four 7 J. R. Young, Phys. Rev. 103, 292 (1956). 8 G. D. Archard, J. App!. Phys. 32, 1505 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Mon, 22 Dec 2014 01:25:39OX Y G EN 0 IT T GAS SIN G BYE LEe T RON B 0 MBA R n MEN T 0 F G LAS S 1791 T ABLE IV. Glass densities and electron current ratios. Corning glass code Density 7070 2.13 0.12 1.14 7740 2.23 7800 2.36 8603 2.36 0081 2.47 9019 2.59 1723 2.63 9010 2.64 0129 2.78 0041 2.89 0120 3.05 0.23 1.31 8870 4.28 0.34 1.52 8363 6.22 0.35 1.54 JOURNAL OF APPLIED PHYSICS samples with large differences in density. The measure ment was made in a cathode ray tube similar to that of Fig. 1. The aperture and neck coating between the aperture and the electron gun were electrically iso lated from the funnel coating. In addition to the I A meter which measures the net sample current, a meter was placed in series with the funnel coating and ground for measuring Is. The primary current was taken as IA+ls. The measurements were made using a 3-XI-in. raster, electron energies of 10, 20, and 28 keV and IA values of 25, SO, 75, and 150 p.A. No major differences were noted in the IslIp ratios for either I A or V p varia tions. Table IV lists the values determined, the I pi I A ratio, and the densities of all glasses mentioned in this paper. VOLUME 34, NUMBER 6 JUNE 1963 Threshold Currents for Line Narrowing in GaAs Junction Diodes SUMNER MAYBURG General Telephone &' Electronics Laboratories, Inc., Bayside 60, New York* (Received 18 January 1963) The threshold currents required to produce narrowing of the)mitted radiation from GaAs junctions can be estimated from the requirement that the quasi-Fermi levels lie near the band edges, as was first sug gested by Bernard and Duraffourg. The observed threshold currents are obtained and the temperature de pendence is deduced to be Tt. VALUES of the threshold current to produce signifi cant line narrowing of the emitted radiation from GaAs junctions have been reported by various workers to be in the range 1()3-1()4 AI cm2 at liquid nitrogen tem peratures.1 Threshold currents at helium temperatures have been reported to be ten to twenty times smaller. We wish to propose a model that yields the correct order of magnitude for the threshold current and pro vides a temperature dependence for the threshold cur rent of T!, A first prerequisite for line narrowing is that stimu lated emission occur more often than spontaneous emis sion. In conventional lasers which make use of transi tions between atomic levels, the number of different transitions allowable is restricted. The production of stimulated emission by resonance in a cavity simul taneously produces monochromatic radiation corre sponding to the energy of separation of the atomic levels involved. However, when, as in the case of GaAs diodes, the * The work reported herein was supported by the U. S. Army Engineer Research and Development Laboratories, Fort Belvoir, Virginia. 1 R. N. Hall, G. E. Fenner, J. D. Kinglsey, T. J. Soltys, and R. O. Carlson, Phys. Rev. Letters 9, 366 (1962). M. Nathan, W. P. Dumke, G. Burns, F. H. Dill, Jr., and G. Lasher, Appl. Phys. Letter 1, 62 (1962). T. Quist, R. H. Rediker, R. J. Keyes, W. E. Krag, B. Lax, A. L. McWhorter, and H. J. Zeigler, Appl. Phys. Letters 1, 91 (1962). transition involves one or both of the band edges, the production of stimulated emission in excess of spon taneous emission is no longer a sufficient condition to produce highly monochromatic radiation. For example, the fraction of filled levels in the conduction band is given by the Fermi function f= [eCE-EFl/kT +1]-1, where E is the electron energy and Ep' is the Fermi level. In the conduction band of a nondegenerate semicon ductor E»EF and the fraction f of states filled is small. Therefore, one could not be assured that a photon mov ing through the crystal would necessarily find an elec tron in the conduction band and an empty state in the valence band such that both levels have the same value of crystal momentum and the energy difference be tween these levels is exactly the energy of the stimu lating photon. Photons cannot easily make adjustment for differences in crystal momentum between the con duction and the valence band.2 In order to conserve mo mentum for transitions involving different states of crystal momentum, a phonon or an impurity atom or a crystal defect must become a party to the transition. Transitions involving these third parties necessarily 2 E. Spenke, Electronic Semiconductors (McGraw-Hill Book Company, Inc., New York, 1958), p. 242. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Mon, 22 Dec 2014 01:25:39
1.1696792.pdf	On the Theory of ElectronTransfer Reactions. VI. Unified Treatment for Homogeneous and Electrode Reactions R. A. Marcus Citation: J. Chem. Phys. 43, 679 (1965); doi: 10.1063/1.1696792 View online: http://dx.doi.org/10.1063/1.1696792 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v43/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsrHE JOURNAL OF CHEMICAL PHYSICS VOLUME 43, NUMBER 2 15 JULY 1965 On the Theory of Electron-Transfer Reactions. VI. Unified Treatment for Homogeneous and Electrode Reactions* R. A. MARcust Department of Chemistry, Brookhaven National Laboratory, Upton, New York, and Noyes Chemical Laboratory, University of Illinois, Urbana, Illinois (Received 8 March 1965) A unified theory of homogeneous and electrochemical electron-transfer rates is developed using statistical mechanics. The treatment is a generalization of earlier papers of this series and is concerned with seeking a fairly broad basis for the quantitative correlations among chemical and electrochemical rate constants predicted in these earlier papers. The atomic motions inside the inner coordination shell of each reactant are treated as vibrations. The motions outside are treated by the "particle description," which emphasizes the functional dependence of potential energy and free energy on molecular properties and which avoids, thereby, some unnecessary assumptions about the molecular interactions. 1. INTRODUCTION ATHEORETICAL calculation of the rates of homogeneous electron-transfer reactions was de scribed in Part I of this series1 and the method was subsequently extended to electrochemical electron transfer rates.2 The calculation was made for reactions involving no rupture or formation of chemical bonds in the elementary electron-transfer step. In this sense these electron transfers are quite different from other types of reactions in the literature. This property, together with the assumed weak electronic interaction of the reactants, introduced several unusual features: "nonequilibrium dielectric polarization" of the solvent medium,3 possible nonadiabaticity, unusual reaction coordinate, and an approximate calculation of the reaction rate without use of arbitrary adjustable parameters. Applications of the theoretical equations were made in several subsequent papers.2,4 The mechanism of electron transfer was later examined in more detail in Part IV using potential-energy surfaces and statisti cal mechanics,5 (In Part I the solvent medium outside the inner coordination shell of each reactant had been treated as a dielectric continuum. The free energy of reorganization of the medium, accompanying the for mation of an activated complex having nonequilibrium * This research was performed in part under the auspices of the U. S. Atomic Energy Commission while the author was a visiting Senior Scientist at Brookhaven National Laboratory. It was also supported by a fellowship from the Alfred P. Sloan Foundation and by a grant from the National Science Foundation. A portion of the work was performed while the author was a member of the faculty of the Polytechnic Institute of Brooklyn, and was pre sented in part at the 146th Meeting of the American Chemical Society held in Denver in January 1964. t Present address: Noyes Chemical Laboratory. 1 R. A. Marcus, J. Chern. Phys. 24,966 (1956). 2 R. A. Marcus, ONR Tech. Rept. No. 12, Project NR 051-331 (1957); cf Can. J. Chern. 37, 155 (1959) and Trans. Symp. Elec trode Processes, Phila., Pa., 1959, 239-245 (1961). 3 R. A. Marcus, J. Chern. Phys. 24, 979 (1956). 4 R. A. Marcus, J. Chern. Phys. 26, 867, 872 (1957); Trans. N. Y. Acad. Sci. 19, 423 (1957). & R. A. Marcus, Discussions Faraday Soc, 29, 21 (1960). dielectric polarization, was computed by a continuum method.) In Part IV, changes in bond lengths in the inner coordination shell of each reactant were also included, and the statistical-mechanical term for the free energy change in the medium outside was replaced only in the final step by its dielectric continuum equivalent. A number of predicted quantitative correlations among the data were made on the basis of Part IV. They have received some measure of experimental support, described in Part V and in a recent review article.6,7 A more general basis for these correlations is described in the present paper, which also presents a unified treatment of chemical and electrochemical transfers. The form of the final equations for the rate constants is comparatively simple, a circumstance which leads almost at once to the above correlations. (It permits extensive cancellation in computed ratios of rate constants.) This simplicity has resulted from several factors: (1) Some of the more complex aspects of the rate problem are rephrased so that they affect only a pre-exponential factor (p) appearing in the rate con stant, a factor that appears to be close to unity. (2) Little error is found to be introduced when the force constants of reactants and products are replaced by symmetrical reduced force constants. (3) An important term (X) in the free energy of activation is essentially an additive function of the properties of the two redox systems in the reaction. The electron transfer rate constants can vary by many orders of magnitude: For example, known homogeneous electron-exchange rate constants vary by factors of more than 1015 from system to system, and electrochemical rate constants derived from elec trochemical exchange currents vary by about 108 at any given temperature.6 (An electron-exchange reac tion is one between ions differing in their valence 6 R. A. Marcus, J. Phys. Chern. 67,853,2889 (1963). 7 R. A. Marcus, Ann. Rev. Phys. Chern. 15, 155 (1964). 679 Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions680 R. A. MARCUS state but otherwise similar.) Thus, small factors of 2 or 3 are of relatively minor importance in any theory which is intended to cover this wide range of values. Some approximations in this paper are made with this viewpoint in mind. In the present paper classical statistical mechanics is employed for those coordinates which vary appre ciably during the course of the reaction. This classical approximation is a reasonable one for orientational and translational coordinates at the usual reaction tempera tures and, in virtue of the above remark, for the usual low-frequency vibrations in inner coordination shells. Because of cancellations which occur in computations of ratios of rate constants this approximation could be weakened for deriving the predicted correlations, even when the quantum corrections would not be small. In calculations of absolute values of the electron transfer rate constants a classical approximation will introduce some error when the necessary changes in bond lengths to effect electron transfer are so small as to be comparable with zero-point fluctuations. How ever, in this latter case, the vibrational contribution to the free energy of activation is itself small and does not account for any large differences in reaction rates in redox reactions which have been investigated experi mentally. Hence, for our present purpose and, in the interests of simplicity, this particular possible quantum effect may be ignored. 2. ORGANIZATION OF THE PAPER The paper is organized in the following way: Individual and over-all rate constants are distin guished in Sec. 3, potential-energy surfaces for weak overlap electron transfers are discussed in Sec. 4, and formal expressions for the rate constants are given in Sec. 5. The latter expressions arise from a generalization of activated complex theory.s The approximate relation of certain surface integrals appearing in Sec. 5 to more readily evaluated volume integrals is described in Sec. 6, where certain complicating features are re phrased so as to cast some of the difficulties into an evaluation of one of the pre-exponential factors p. In Sec. 6 a linear dependence of an effective potential energy function (governing the configurational distri butions in the activated complex) on the potential energies of reactants and products is established [Eq. (13)]. The rate constants are expressed in Sec. 7 in terms of the contribution of the coordinates of the solvent molecules in the medium and of the vibrations in the inner coordination shell of each reactant to the free energy of formation of the activated complex. To deduce from Eq. (13) a simple dependence of the free energy of activation on differences in molecular 8 R. A. Marcus, J. Chern. Phys. 41, 2624 (1964). The U in the present Eqs. (1) to (3) was denoted there by ut. parameters, the contributions of the above two sets of coordinates are treated differently (Sec. 8), since one set already has a desired property while the other does not. Changes in bond force constants accompanying electron transfer are responsible for this difference in behavior. However, it is shown later in Appendix IV that the introduction of certain "reduced force con stants" circumvents the difficulty, with negligible error in typical cases. The contributions of the two sets of coordinates are computed in Secs. 9 and 10. The medium outside the inner coordination shell of each reactant is treated by a "particle description."9,lo The latter is a considerable generalization over the custom ary permanent-dipole-induced-dipole treatment of polar media and serves to emphasize the functional depend ence of the free energy of activation on various properties and to facilitate thereby the analysis leading to the predicted correlations. The standard free energy of reaction and the cell potentials are introduced in Secs. 11 and 12, and are used in Sec. 13 to evaluate a quantity (m) closely related to the electrochemical and chemical transfer coefficients. The final rate equations are summarized in Sec. 14. The additive property of A, mentioned in the previous section, is discussed in Sec. 15 and further established in Sec. 16. The significance of the characteristic scalar quantity (m) appearing in the potential-energy func tion of the activated complex is deduced in Sec. 17. Deductions from the final equations are made in Sec.18. In Sec. 19 the present paper is compared with earlier papers of this series, and the specific generalizations made here are described. Detailed proofs are given in various appendices. In Appendix VIII it is established that under certain conditions the correlations derived above should apply not only for rate constants of elementary steps but also for the over-all rate constant of a reaction occurring via number of complexes of the reactants with other ions in the electrolyte. 3. INDIVIDUAL AND OVER-ALL RATE CONSTANTS Many chemical and electrochemical redox reagents are ions which possess inner coordination shells and which may form complexes with ions of opposite sign. Any such complex is "inner" or "outer" according as the latter ions do or do not enter the inner coordination or shell of the reactant. To a greater or lesser extent, all such complexes normally contribute to the measured rate of the redox process. For this reason both a rate 9 R. A. Marcus, J. Chern. Phys. 38, 1335 (1963). 10 R. A. Marcus, J. Chern. Phys. 39, 1734 (1963). The notation differs somewhat from the rresent paper: 1', U, Uf, and p,o there become Vo', Uo, U" and Pa here. A typographical error occurs in Eq. (13): The fs should be deleted. No equations deduced from (13) need correction. Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 681 constant for the over-all reaction, involving all com plexes, and a rate constant for each individual step, involving a specific complex with a given inner coordi nation shell or involving a specific pair of complexes in a bimolecular step, have been defined in the literature. They equal the over-all reaction rate divided by the stoichiometric concentration (or product of such con centrations in the bimolecular case), in the case of an over-all rate constant, and the reaction rate divided by the concentration of the particular complex (or product of such concentrations in the bimolecular case), in the case of an individual rate constant. Often the individual rate constants are measured experimentally. Frequently, however, only the over-all rate constant is determined in the experiment. The derivation up to and including Sec. 6 applies to over-all as well as to individual rate constants. The Secs. 7 to 17 apply only to the individual rate constants. To calculate the over-all rate constant from the expres sion derived for the individual one in these latter sections, one must take cognizance of any reactions leading to the formation and destruction of the com plexes and must average over the behavior of all complexes, as in Appendix VIII. 4. POTENTIAL-ENERGY SURFACES The potential energy of the system is a function of the translational, rotational, and vibrational coordi nates of the reacting species and of the molecules in the surrounding medium. A profile of the potential energy surface is given in Fig. 1 in the case of homoge neous reactions. (The related electrochemical plot is considered later.) The abscissa, a line drawn in the above many-dimensional coordinate space, represents any concerted motion of the above types leading from any spatial configuration (of all atoms) that is suited to the electronic structure of the reactants to one suited to that of the products. Surface R denotes the potential-energy profile when the reacting species have the electronic structure of the reactants, and Surface P corresponds to their having the electronic structure of the products. If the distance between the reacting species is sufficiently small there is the usual splitting of the two surfaces in the vicinity of this intersection of Rand P. If the electronic interaction causing the splitting is sufficient, the system will always remain on the lowest surface as it moves from left to right in Fig. 1. Thus, the system has moved from surface R to surface P adiabatically, in the usual sense that the corresponding motion of the atoms in the system is treated by a quantum-mechanical adiabatic method. On the other hand, if the electronic interaction causing the splitting is very weak, a system initially on Curve R will tend to stay on R as it passes to the right across the intersection. The probability that as a result of R NUCLEAR CONFIGURATION FIG. 1. Profile of potential-energy surface of reactants (R) and that of products (P) plotted versus configuration of all the atoms in the system. The dotted lines refer to a system having zero electronic interaction of the reacting species. The adiabatic surface is indicated by a solid line. this nuclear motion the system ends up on Curve P is then calculated by treating this motion non adiabatically.ll It should be noted that the system can undergo this electron transfer either by surmounting the barrier if it has enough energy or by tunneling of the atoms of the system through it if it has not. We confine our attention to the case where the systems surmount the barrier. Some atom tunneling calculations have been made, however.I2 Since the abscissa in Fig. 1 is some combination of translational, rotational, and vibrational coordinates, this "reaction coordinate" is rather complex: The sur faces Rand P intersect, and the set of configurations describing this intersection form a hypersurface in configuration space. The exact motion normal to this hypersurface depends on the part being crossed. In some parts it involves changes in bond distances in the inner coordination shells of the reactants, in other parts it involves a change of separation distance of the 11 See, for example, L. Landau, Physik. Z. Sowjetunion 1, 88 (1932); 2, 46 (1932); C. Zener, Proc. Roy. Soc. (London) A137, 696 (1932); A140, 660 (1933); C. A. Coulson and K. Zalewski, ibid. A268, 437 (1962). The present situation has been summarized in Ref. 7, where the definition of nonadiabaticity was also dis cussed. Reference should also have been made there to the work of. E. C. G. Stueckelberg, Helv. Phys. Acta. 5, 369 (1932); d., H. S. W. Massey, in Encyclopedia of Physics, edited by S. Fliigge (Springer-Verlag, Berlin, 1956), Vo!' 36, p. 297. 12 N. Sutin and M. Wolfsberg, quoted by N. Sutin, Ann. Rev. Nuc!. Sci. 12, 285 (1962). These authors discussed the possibil ity of tunneling of the atoms in the inner coordination shell. Possible quantum effects which include atom tunneling in the medium outside this shell have been treated by V. G. Levich, and R. R. Dogonadze, Proc. Acad. Sci. USSR, Phys. Chern. Sec. [English transl. 133, 591 (1960)]; Collection Czechoslov. Chern. Comm. 26, 193 (1961) [trans!., O. Boshko. University of Ottawa, Ontario.] Any conclusions concerning the contribution of atom tunneling depend in a sensitive way on the assumed values for the bond force constants and lengths in the inner coor dination shell, properties on which data are now becoming avail able, and on the assumed value for a mean polarization frequency for the medium. [Atom tunneling is different from electron tunneling.>, the latter being a measure of the splitting in Fig. 1 (Ref. 7).J Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions682 R. A. MARCUS NUCLEAR CONFIGURATION FIG. 2. Same plot as Fig. 1 but for an electrode reaction. The finite spacing between the many-electron levels of a finite elec trode is enormously magnified, and only three of them are indi cated. The splitting differs from level to level. reactants, and in still others it involves reorientation of polar molecules in the medium. Analogous remarks apply to electrode reactions except that the intersection region is more complex because of the presence of many electronic energy levels in the metal. A blown-up portion of this region is indi cated in Fig. 2. The diagram consists of many potential energy surfaces, each for a many-electron state of the entire macrosystem. All the surfaces are parallel since they differ only in the distribution of electrons among "single-electron quantum states" in the metal. (Only one distribution of the electrons among these single electron quantum states correspond to each surface in Fig. 2 if the energy level of the entire macrosystem is nondegenerate. It corresponds to several distributions in the case of degeneracy.) There is a probability distribution of finding the macrosystem in any many electron energy level indicated in Fig. 2. As a conse quence of a Fermi-Dirac distribution of the electrons in the metal, most electrons which are transferred to or from the many-electron energy levels in the metal will behave as though they go into or from a level which is within k T of some mean energy level, and hence practically equal to it. Thus, except for the calculation of the transition probability associated with the transi tion from Surface R to Surface P in the intersection region, the situation is in effect very similar to that in Fig. 1. We return to this point in the following section. In the present paper we confine our attention in electrode reactions, as in homogeneous reactions, to reaction paths involving a surmounting of the barrier. 5. EXPRESSION FOR THE RATE CONSTANT We consider any particular pair of reactants (or a reactant, in the case of intramolecular electron trans fer). These "labeled" reactants may be any two given molecules in solution or one molecule and the electrode, and each may form complexes to various extents with other ions and molecules. In effect, we need to calculate the probability that the vibrational-rotational-transla-tional coordinates of the entire system are such that the system is in the vicinity of the many-dimensional intersection hypersurface in configuration space. It is assumed below that the distribution of systems in the vicinity of the intersection region of Figs. 1 or 2 is an equilibrium one. The usual equilibrium-type derivation of the rate of a homogeneous or heteroge neous reaction in the literature employs a special form for the kinetic energy, a form consistent with the set of configurations of the activated complex being de scribable by a hyperplane in configuration space. A more general curvilinear formulation has been given recently.8 Upon integrating over a number of coordi nates which leave the potential energy invariant one obtains (1), (2), and (3) for homogeneous bimolecular reactions, homogeneous unimolecular reactions, and heterogeneous reactions, respectively8,13: k (kT)'l exp( -U /kT) R2(mt)-!dS bi= 87l" ' , s Q k .=(kT)!l exp( -U /kT) (mt)-!dS lim 27l" S Q ' -(kT)!l exp( -U /kT) (mt)-!dS khet-27l" sQ' (1) (2) (3) In these equations mt is the effective mass for motion normal to the hypersurface S, R is the distance between the two reactants (normally between their centers of mass), Q is the configuration integral for the reactants, and dS is the area element in a many-dimensional internal coordinate space,13 Both mt and R may vary over S. In (1) to (3) integration has already been performed over several coordinates, as follows: (i) in Q, the center of mass of each reactant; (ii) in the numera tor of (1), the center of mass of one reactant and the orientation of the line of centers of the two reactants; (iii) in the numerator of (2), the center of mass of the reactant, and (iv) in the numerator of (3), the two coordinates of this center parallel to the solution-solid interface. Thus, these coordinates are to be held fixed in the internal coordinate space in (1) to (3). In adapting these equations to electron-transfer reactions one should consider the possibility of the reaction occurring nonadiabatically and, in the case of electrodes, should consider the existence of many levels which may accept or donate an electron to a reactant in solution. In the framework of a classical treatment of the motion of the nuclei in (1) to (3), a factor K 13 In these equations S is an abbreviation for Sint (made for brevity of notation), since several integrations over "external coordinates" have been performed and there remains only the integration over a hypersurface in internal coordinate space.8 Similarly, the symbols S', V, and V' discussed later should bear a subscript int, which is omitted here for brevity. Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 683 ca~ be shown to appear in the integrand (Appendix I) ; K is a momentum-weighted average of the transition probability from the R to the P surface per passage through the intersection region. (It is momentum weighted since the transition probability depends on the momentum.) K can vary over S. Normally, we take K as approximately equal to unity when the reactants are near each other, introducing thereby the assump tion that the reaction is adiabatic. In the case of (3) the situation is somewhat more complex because of the presence of the many electrode levels. At present there is, in the literature, no theoreti cal calculation of the transfer probability from a level R to a continuum (essentially) of levels P, per passage through the intersection range, for the entire range of transfer probabilities from 0 to 1. Such a calculation would take into account the fact that in an unsuccessful passage through the intersection region the system can also revert to other R levels different from the original one. At present only the limiting case of very small transfer probability has been considered in the litera ture.l4 In this case transfers to and from each of the levels have been treated independently using perturba tion theory; they do not interfere at this limit. When the transfer probability in electrode reactions is fairly large when ion and electrode are close a different approach must be employed. IS Here, we t~ke advantage of the fact that for a metal electrode most of the electron transfers occur to and from levels near the Fermi leveps: In the terminology of a one-electron model, most of the levels several kT below the Fermi level are fully occupied and cannot accept more elec trons. The Boltzmann factor discourages transfer to the rather unoccupied levels several k T above the Fermi level. Conversely, transfers from the occupied levels below the Fermi level are discouraged by a higher over-all energy barrier to reaction while transfer from a higher level is discouraged by the fact that most of the higher levels are unoccupied. To illustrate this point more precisely, let nee) be the density of the "one electron model levels" for the electrode and f( e) the Fermi-Dirac distribution, fee) = lexp[(e-.aB)/kT]+l}-l, (4) where e is the energy of one of these levels and where .aB is the electrochemical potential of electrons in the metal. Both e and jiB depend on the electrostatic poten tial of the metal cp: .a= p.-ecp, (5) 14 R. R. Dogonadze and Y. A. Chizmadzhev, Proc. Acad. Sci. USSR, Phys. Chern. Sec., English Trans!. 144, 463 (1962) 145, 563 (1962); V. G. Levich and R. R. Dogonadze, Intern. C~mm. Electrochem. Thermodyn. Kinet., 14th Meeting, Moscow (1963) preprints. This work is reviewed in Ref. 7. ' 16 This approximation was used but not discussed in Ref. 2. where e (0) is the value of e at cp = 0 and p. is the chemical potential.I6 The probability that electron transfer from the electrode to the ion or molecule in solution will occur from a "one-electron model" level of energy e would b.e expected to depend on e by a factor roughly propor tlOnal to n(e)f(e) exp(e/2kT), (6) the third factor arising in the region where the "electro chemical transfer coefficient" is 0.5, a common value.6 Since nee) is a weak function of e the last two factors in (6) largely determine the most probable value of e. The maximum of (6) is then easily shown to occur at e= .aB' Similarly, contribution to electron transfer from an ion in solution to a particular level e would be expected to vary with e as in n(e)[l-f(e)] exp( -e/2kT), (7) which also has a maximum at e= .aB, of course. Because of this circumstance (that most contribu tions arise from levels e near jiB), we approximate the situation in Fig. 2 by replacing the set of Surfaces R by one surface and P by another surface, corresponding to an electronic energy in the electrode given by jiB as above. IS If electron transfer accompanies each passage ~hrough the intersection region in Fig. 2 the reaction is referred. to as "adiabatic," purely by analogy with the term m the homogeneous reaction. The reaction rate is given by (3), where the equation of S depends on the electrostatic potential. On the other hand when the transfer probability per passage is very ~eak a term K. shoul? be introduced in the integral, K being a velOCity-weighted transition probability appropriately s~mmed over all energy levels in the electrode (Appen diX I). A value for K in this weak interaction limit has ~een discussed elsewhere.7 When a complete cal cula~lOn for the transfer probability from and to a contmuum of electrode levels becomes available it can be used to estimate K. Normally, however, we assume the electrode reaction to be "adiabatic" and so take K"'l on the average. 6. RELATION OF THE SURFACE INTEGRALS (1) TO (3) TO VOLUME INTEGRALS Although some deductions can be made from the surf~ce integ:als in (1) to (3) when the equation of the mtersectlOn surface S is simple, we find it con venient to express the surface integral in terms of volume one. The same aim was followed in Part IV but. in a less precise way. The principal equation denved in this section is (26), which is later used in conjunction with Eqs. (1) to (3) to obtain an expres sion for kr•te• 16 For example, C. Herring, and M. H. Nichols Rev Mod Phys.21, 185 (1949). ' . . Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions684 R. A. MARCUS Let Ur be the potential-energy function for the reactant and UP be that for the products. As mentioned earlier the intersection of the Rand P surfaces in Fig. 1 (and 2) forms a hypersurface in configuration space. This hypersurface is called the "reaction hypersurface." Its equation is given by (8). It is a hypersurface in the entire coordinate space and also in the internal coordinate space since (8) is independent of the external coordinates8 Ur-UP=O (for points on reaction hypersurface). (8) This surface is a member of a family of hypersurfaces in configuration space, represented by (9), where c is a constant: (9) The surface (8) can be obtained from the surface (9) by lowering the P surface in Figs. 1 or 2 by an amount c. We employ a coordinate system q1 to qn used in the derivation of (1) to (3) and recall that one coordinate, qN, in the internal coordinate space was chosen to be a coordinate constant on the hypersurface (8). Let qN be zero there. In fact, each member of the family of hypersurfaces (9) is made a coordinate hypersurface for qN. We consider any of the integrals in (1) to (3), include the factor K in the integrand, and write dS as dS'dRP The factor K depends primarily on R. In the following expression the same symbol K is used to denote this K, averaged over S'.18 Each of the integrals in (1) to (3) can be rewritten as L KR{l, exp( -k~ ) (mt)-tdS'}R, (10) where a is 2, 0, or ° according to whether (1), (2), or (3) is the equation involved. We wish to relate the above integral over S' to a volume integral (11) over the internal coordinate space at fixed R, as in (18) and finally as in (26)19: i, exp(-~;)dV" (11) where U* is a function to be determined; RotdV' is an element of volume of this internal coordinate space at fixed R.8 17 This factoring of dS (or as it was called there dS int) was described in Ref. 8. 18 The K appearing in (10) is now a symbol representing r(mt)-texp(~~}s' / f(mt)-lexp(~f)dS" where K is the original kappa. 19 These "internal coordinates" were defined8 as those coor dinates for which integration was not performed in obtaining (1) to (3). To establish (18), we first note from Appendix II that the distribution in volume which is centered on S' (but not confined to S', of course) isf, given by (12)20 1= exp( -~;) / f exp( -~;)dV" (12) where (13) and m is a parameter which varies with the coordinate R and which is determined in Sec. 13. On S', one sees from (8), U* equals Ur for any given R. We then recall from Ref. 8 that dV' and dS' are related by (14), and we introduce a quantity l(qN, R) defined by (15) : (14) where aNN is conjugate to an element aNN in the line element of the many-dimensional configuration space. On recalling from Ref. 8 that mt equals aNN/gNN, where gNN is conjugate to an element gNN in the line element of the corresponding mass-weighted configura tion space, the S' integral in (10) can then be rewritten as in (16), where «gNN) i) is a suitable average over S'21 f exp( -k~)(mt)-!dS'= «gNN)t) exp[ -1(0, R)J. (16) In deriving (16) we have also used the fact that U* equals Ur on S'. Finally, the integral in (11) can be rewritten as f exp[ -1(qN, R)]dqN, in virtue of (14) and (15). On the basis of a Gaussian expansion Eq. (17) can be derived (post). f exp[ -1(qN, R)]dqN =[211"1"(0, R)]! exp[ -1(0, R)], (17) where 1"(0, R) is d21(qN, R)/dqN2, evaluated on S' (and hence at qN=O). One then obtains, from (10), 20 If, for any R, a distribution functionj* is stated to be centered on S', we mean that it is centered on the set of configurations which lie at the intersection of the hypersurface S and of the hypersurface R=constant. Occasionally, in some part of the internal coordinate space the two hypersurfaces may be "cotan gential," but this circumstance does not alter the argument. At these parts of space the value of U' equals UP and (12) becomes "exact" for computing relative probabilities of various configura tions, rather than approximate. 21 {(gNN)i) in (16) is defined as f (gNN)l(aNN)-t exp( ~i)dS' / f (aNN)-l exp( ~i)dS" Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 685 (16), and (17), L KR{i, exp( -k~)(mt)-idS'}R =/. KRa (CgNN) !) exp[ -F(R)/kTJdR R [21TI"(0, R)J! ' (18) where F(R) is the configurational free energy of a system having the potential-energy function U* for this separation distance R ( F(R») f (U) exp -----:;;y:- = exp -kT dV'. To complete the proof of (18) we must verify (17). We recall from the definition of qN that Ur -UP depends only on qN on any hypersurface (9). To ensure centering of the system on 5', i.e., at qN = 0, meR) is to be selected so that (20) is satisfied: (20) where ( ) denotes average with respect to the distri bution function j. On using (12), (14), and (15), Eq. (20) becomes f exp[-I(qN, R)J(ur-Up)dqN=O. (21) Because of the centering of j, expansion of I(qN, R) about qN = ° is permissible, as is one of Ur -UP I(qN, R) =1(0, R)+qNI'(O, R) +[(qN)2/2 IJI"(O, R)+···, (22) Ur-Up=O+qN(Ur-Up)'+.··, (23) where ' indicates a derivative with respect to qN, evaluated at qN = 0. We retain only leading terms in each case. Insertion of (22) and (23) into (21) followed by integration reveals that 1'(0, R) vanishes. Intro duction of (22) into the left-hand side of (17) then establishes (17). Some of the terms in (18) can be expressed in terms of quantities of more immediate physical significance. It may be shown from (12), (14), (15), (22) and the vanishing of I' (0, R) that for small s's: «5s)2)= «aNN)-1(5qN)2)= «aNN)-1 )«5qN)2), (25) where «aNN)-I) is a suitable average of (aNN)-1.23 We make use of the fact that «gNN)!)«aNN)-I)! has units of (mass)-!, and denote it by (m)-!, and that the integrand in (18) has a maximum at some value of R, denoted in (26) by R. [When R becomes large K tends to zero and when R is small the van der Waals' repulsion makes F(R) large.J On treating the integrand as a Gaussian function of R, (18) becomes = KpRa(m)-! exp[ -F(R) /kTJ, (26) where K is evaluated at this value of R and where p is a ratio (27) whose value should be of the order of magnitude of unity: p= [(C5R)2)/ (COS)2)J!, (27) where « 5R) 2) is the mean square deviation in the value of R; p and K can be calculated from more specific models when the various integrals defining them can be evaluated. 7. RATE CONSTANT IN TERMS OF llF* Let Fr be the configurational free energy associated ~ith the Q of Eqs. (1) to (3) as in (28). Thereby, it IS the free-energy contribution for an equilibrium dis tribution of "V' coordinates" when the reactants are very far apart but fixed in position, Fr=kT lnQ. (28) Let peR) be the corresponding quantity when the reactants are a distance R apart. We then have wr= Fr(R) -Fr, (29) where wr can be called the reversible work to bring the reactants from fixed positions infinitely far apart to the cited separation distance. We also introduce llF(R) : llF(R)=F(R)-Fr(R). (30) Equations (1) to (3) for krate now yield (31) to (33), when (26) and (28) to (30) are used, kbi= KpZbi exp( -wr/kT) exp[ -llF(R) /kTJ, (31) (24) kuni= Kp(kT /21Tm)! exp( -llF/kT) , (32) where «qN)2) is the mean-square deviation of qN.22 The mean-square deviation of the perpendicular dis tance s from the reaction hypersurface is given by (25) 22 This average, «oqlV)2), is defined as f (oqlV) 2 fdV'. It is readily shown that (qN) vanishes. khet= KpZhet exp(-wr/kT) exp[-llF(R)/kTJ, (33) 23 This average is defined here as f (alVN)-1(oqN)2 exp( ~~)dV' / f (oqN)2 exp( ~~)dV" For the proof that ds2 equals (aNN)-1(dqN)2, see Ref. 32 Appen- dix III. ' Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions686 R. A. MARCUS where Zbi and Zhet are given by (34) [In Eq. (32) flF is simply F-Fr, there being only one reactant.] Zbi is in fact the collision number of two uncharged species in solution when they have unit concentration, when their reduced mass is m, and when their collision diameter is R. Zhet is the collision number of an uncharged species with unit area of an interface (here, the electrode), when it has unit concentration and when its mass is m. F and Fr in (31) to (33) involve an integration over the orientation of each reactant. The integrand in Fr is independent of these coordinates and, in the case of the "outer-sphere electron-transfer mechanism" discussed here, the integrand in F* is assumed to be independent of them also. (For purposes of deriving many of the correlations in Sec. 16, this assumption could be weakened because of cancellations.) Integra tion over these coordinates is regarded as having been performed in (31) to (34), since the orientational factors now cancel in flF(R). Thus, in the subsequent calculation of F and pr each reactant may be regarded as fixed not only in position, as before, but in orienta tion also. 8. DISTRIBUTION FUNCTION AND THE FREE ENERGY The main purpose of this section is the derivation of Eqs. (47) to (49). Equation (19) for F(R) can be rewritten as in (35), with the aid of (12), (13), and (20): F(R) = (Ur)+kT(lnf), (35) where (Ur)= f U1dV', (lnf)= f (lnf)fdV '. (36) Since -k (lnf) is the configurational entropy of a system having the distribution function f* and since (Ur) is the mean potential energy of a nonequilibrium system having a potential-energy function Ur but a distribution function of f* inappropriate to this Ur, we see that F(R) is also equal to the configurational free energy of this non equilibrium system. In obtaining an expression for F(R) it is convenient to divide, as one usually does in related problems, the internal coordinates at the given R into two groups: V'i coordinates describing the positions of the atoms in the inner coordination shells of the reactants, and V'o coordinates describing the positions of the atoms of the medium relative to each other and to those in the inner coordination shells. It is also convenient to write U as the sum of two terms, Ui and Uo, one describing the intramolecular interactions of the atoms in each coordination shell, the other describing the interactions of the atoms of the medium with each other and with those of the inner coordination shells. Thus, U i depends only on the V'i coordinates; Uo depends primarily on the V'o coordinates, but also depends on the V'i ones, (37) The quantities U;* and Uo* are defined in terms of U{, etc., to be given by (13), with i and 0 subscripts, respectively. Then, U* is the sum of Ui* and Uo . The volume element dV' is written as dVI=dVlidV'O, (38) where dV'i is defined as the product of the differentials (llidqi) of the V'i coordinates. Thereby, dV'o con tains the Jacobian appearing in dV'. It may vary, therefore, with the V'i coordinates. In calculating F and Fr we may evaluate the integrals appearing in them by first integrating over the V'o coordinates and then over the V'i ones. This procedure is convenient since the V'i ones perform small oscillations while the others can undergo con siderable fluctuations. With this procedure in mind, we define new quantities fi* and fo , the former de pending only on the V'i coordinates, the latter depend ing on the V'o coordinates and parametrically on the V'i ones: fo= exp[(xo- Uo)/kT], (39) fi= exp( -~;) / f exp( -~;)dVli' (40) where Oi= U;+xo, (41) exp( -~~)= f exp(-~~)dVlo. (42) One then obtains f=fofi. (43) Quantities fo', f/, 0/, and xo' can be defined, by replacing the by an r superscript in (39) to (42). However, xo' is simply Fo', the V'o contribution to the configurational free energy of the reactants for the given value of the V'i coordinates We also introduce Fo, defined by Fo= (Uo')'o+kT(lnfo).o, (45) where the average ( )'0 is computed with respect to fo . Fo * is the V'o contribution to the free energy of the nonequilibrium system having the potential energy function Ur and the distribution function fo . The first and second terms in (45) are the energy and entropy contributions, respectively. To obtain an expression for Oi, the function largely controlling the V'i coordinate distribution, we first Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 687 obtain (46) by introduction of (39) into (45) and by use of (13) with subscript o's added. Equations (41), (46), (37) and, with subscript i's added, (13) then yield (47), since Ul and UiP are independent of the V' 0 coordinates: On multiplying numerator and denominator of (40) by exp[Ui(Q.) /kTJ, introducing this expression for Ji into (48b) , then using (50), integrating,24 and finally introducing an expression for Oi(Q) [Eq. (47) evaluated at Q= QJ, Eq. (51) follows: Xo = Fo +m(Uo'- Uop).o, Oi= Ut+Fo+m(ur- Up).o. (46) F(R)=Ut(Q.)+Fo(Q.)-m(Ur(Q.)-Up(Q.) )'0 (47) -!kT In[(27rkT) ni/I hk* IJ, (51) Equations (35), (37), (43), and (45) yield (48a) , when one notes that Ut and Ji* are independent of the value of the V' 0 coordinates. Equation (48b) then follows from (20), (47), and (48a): F(R) = (Ut+Fo)'i+kT(lnJ')'i, (48a) (48b) where the average( )'i is computed with respect to the distribution function J.. The free energy pr(R), given by -kT In f exp(-k~)dVI evaluated at R, can also be shown to be given by expressions similar to (48) but with the asterisks replaced by r's (49) To evaluate krate, we compute I1F* from (30), (48), and (49), and use (47). The similarity of (48) and (49) and later of (51) and (52) is an example of the fact that properties of the [r J system can be obtained from those of the [J system by setting m= O. The origin of this behavior is seen in the original Eq. (13) defining the [J system. 9. VIBRATIONAL CONTRmUTION TO I1F(R) While it is not necessary to introduce the harmonic approximation, the expressions are appreciably simpli fied by it. There is evidence that the approximation is adequate for many reactions of interest. It is recalled that the generalized coordinates were denoted by qi. Let the first ni of these be vibrational coordinates of the reacting species, i.e., the V/ coordi nates, and let q.i denote the value of the jth vibra tional qi occurring at the minimum of Oi. We have 0, = Oi(Q.) +! fJik (qi_q .j) (qk_q .k) i,k=l where Q-Q. denotes a column vector whose elements are qi_q.i. F denotes a square matrix whose elements are Ji;' The superscript T denotes a transpose (a row vector here), and the dot indicates the scalar product of this row vector with the column vector F(Q-Q.). where 1 hk * 1 is the determinant of the hk 's. If qri denotes the value of a vibrational qi occurring at the minimum of U{, it can be shown that pr is given by (52) after a quadratic expansion of O/(Q) about Qr, pr(R) = Ut(Qr)+F/(Qr) -!kT In[(27rkT)ni/lh,: IJ, (52) where hkr= (a20t/aqiaqk) at Q=Qr' (53) Equation (54) is then obtained from (51) and (52) by noting that (Ur(Q.)-UP(Q.) )'0 vanishes (Appen dix V), that U/ equals U/-For at any Q, and that O/(Q.) can be expanded about the value of Ot at Qr: F(R) -pr(R) =!(Q.T_Qr T). Fr(Q.-Qr) +I1Fo(Q.) +!kT In(1 hk IIlhkr \), (54) where I1Fo(Q)=Fo(Q)-F/(Q) (atanygivenR). (55) It is shown later that at any given Rand Q I1F 0 * (Q) equals m2x,,(Q), where Ao(Q) is given by (69), and that 11 Fop(Q), which is Fop(Q)-For(Q) , equals (m+1)2x,,(Q). We then obtain (56) from (47)26 Oi= Ot+m(O/- OiP) -m(m+ 1)x,,(Q). (56) Since 0. is a minimum at Q= Q., the first variation in Ui* vanishes for any arbitrary infinitesimal oQ. In Appendix VI it is found that Ao may be neglected in obtaining Since the oqi are selected to be independent, the coefficient of OQT vanishes. Hence, Q.= [(m+ 1) Fr -mFpJ-l[ (m+ 1) FrQr-mFpQpJ, (58) and the first term in (54) becomes HQ.T_Ql) ·Fr(Q.-Qr) =!m2I1QT.FI1Q, (59) 24 We use Eq. (2) in R. Bellman, Introduction to Matrix An alysis (McGraw-Hill Book Company, Inc., New York, 1960), p. 96, to obtain the last term of (51). 25 On recalling the definition of Oir and rlap, and adding and subtracting mkT(lnfo).o it follows that Ui' in (47) can be written as (m+l)O;'-mO,p plus f1Fo+m(f1Fo-f1Fov). Equation (56) then follows. Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions688 R. A. MARCUS where (60) F= Fp[ (m+ 1) Fr-mFp]-lFr[ (m+ 1) Fr-mFp]-lFp, (61) and the equality of Fr, Fp, and [(m+l)Fr-mFpJ-l with their transposes have been used. On differentiating (56) twice and noting that an a posteriori calculation shows that the last term in (56) may be ignored in the differentiation we find (62), for use in the In term in (51) !ik=a2U;/aqiaqk= (m+l)!ikr-m!ikP, (62) Later it is shown that Eqs. (54) and (59) can be simplified considerably to a good approximation by introduction of symmetrical and antisymmetrical func tions of the force constants and then neglecting terms involving the antisymmetrical ones kik= 2!ik1ikP /(fikr+!ik P), (63) lik= (fikr-!ikP)/(fikr+!ik P). (64) The first of these quantities was chosen so as to have dimensions of a force constant and the second of these so as to be dimensionless. 10. ORIENTATION AND OTHER CONTRIBUTIONS TO tlF For purposes of generality we employ the particle description of the potential energy in a macrosystem. 9.10 It introduces fewer assumptions than those normally used in condensed polar media. Because of its compara tive generality it also permits a simultaneous formula tion of the theory of homogeneous intermolecular electron transfers, electron transfers at electrodes, and intramolecular electron transfers. In this description the system consists of particles each of which is a reacting molecule or any electrode present, the latter including as part of it any strongly bound layer of ions or solvent. The remainder of the system, the medium, can then be regarded as one giant particle. The potential energy is the sum of an intraparticle term (the energy when the particles are isolated, each having the given intraparticle coordinates) and an interparticle term (the energy change when the particles are brought together for the given values of the intra particle coordinates). The solvent particle possesses a "cavity" for each reactant particle, which the latter fills when they are brought together. The intraparticle terms below contain the electronic and potential energy of the reactants and of the medium. The interparticle term is, in the first approximation, the sum of interparticle polar terms and of interparticle electron correlation (i.e., exchange repulsion and London dispersion) energies.9 It can then be expanded in powers of the permanent charge density Pa 0 of the reactants. The usual approximations in the literature correspond to neglect of powers higher than the second, together with the assumption of specific forms for these terms.9 In terms of the symbols Ui and Uo introduced earlier, we have where Uo= U(0)+U(I)+U(2). (37) (65) In (37) Ui is the intraparticle term for the reactants and Uo is the sum of the intraparticle term for the medium and of the interparticle term. U (0), U (1) , and U(2) depend functionally on zeroth, first, and second powers of Pa 0 and, respectively, on second, first, and zeroth powers of PMo, the permanent charge density of the medium.9 U(O) also contains the intra particle term for the medium and the electron correla tion interparticle term. U i and pa ° depend only on the intraparticle coordinates, VIi, of the reactants, and PMo depends only on those of the medium, V'0.9 The distribution function /0 * defined in (39) can be shown to be similar to that which occurs when the permanent charge distribution on a reactant A is Pa 0, given by (66) for all A: 0_ 0+ (0 0) Pa -par m par -Pap , (66) where Par ° is the permanent charge distribution of Molecule A when it is actually a reactant and Pap ° is that when it is a product. The proof is given in Appendix III and utilizes the facts that U (1) is a linear functional of pa ° and that U (2) is insensitive to the usual transla tional-rotational fluctuations in condensed media, for reasons noted there, unlike the U(O) and U(1). Normally, as will be seen later, m will be close to -i. The V'o contribution to the free energy of formation of a system with a nonequilibrium V'o distribution, tlFo , at any given R and at any given Q, has been evaluated elsewhere on the basis of the particle descrip tion described above and of an assumption of (at most) partial electric saturation1o: tlF 0 = Fop m(r-p) -F m(r-p). (67) In (67) Fop and F denote the polar contributions to the free energies of two hypothetical equilibrium and di electrically unsaturated systems, each having a pa ° equal to m (Par ° -Pap 0) on each reactant. The first system is an "optical polarization" system,9 i.e., a system whose medium responds to these Pa a's only via an electronic polarization. The second system responds via all polarization terms. Both Fop and F are quadratic functions of the m (Par ° -Pap 0) 'so It can be shown26 that Fop-F depends on the square 26 According to Eqs. (10) and (11) of Ref. 10 Fop-F equals [(U(1)2)- (U(1) )2J/2kT. The latter depends only on the second power of the charge distribution, since U (1) is a linear functional of the first power. Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 689 of the permanent charge distribution on the reactants, temperature, and pressure. Hence, in this case m(Pa,o-Pap 0). We may then describe the dependence of AFo * by Fp-FT= AFo/. (72) (68) where and the averaging function is27: ex (-(Uor+UOP))avl flex (_ (u/+uoP))av l p 2k TOP 2k T o· (70) To use Eq. (68) and those derived earlier, an expres sion is needed for m. It is derived below after some preliminary analysis involving the standard free energy of reaction, the electrochemical cell potential, and the activation overpotential. 11. STANDARD FREE ENERGY OF REACTION The configurational free energy of the system when the reacting species are labeled reactant molecules, fixed in position but far apart, was denoted by FT. The corresponding quantity when the pair refers to labeled product molecules was denoted by Fp. The momentum and translational contributions of each member of the reacting pair to the free energy of the initial state cancels that in the final state in these reactions in volving no change in total number of moles of redox species. Thus, the difference Fp-Fr is equal to the free energy of reaction when a pair of labeled reactants form a pair of labeled products in the prevailing medium. This free energy of reaction in the prevailing medium can be expressed in terms of "standard" chemical potentials. The chemical potential J.ti can be written as J.tio/+kT In c;, where J.tiol is the "standard" chemical potential, defined here as the value of J.ti at C;= 1. Because of the labeling, Fp-pr does not contain a con tribution from entropy of mixing of the reactants. Since it is these mixing terms which contribute the k T In Ci to J.ti, we therefore have (71) P where Lp and Lr denote summation over products and reactants. There are one or two terms in each sum, according as the reaction is unimolecular or bimolecular. The right-hand side of (71) is AFo/, the "standard" free energy of reaction for the prevailing medium, 27 If the dielectric unsaturation approximation is used, one can show10 that (UJ+U op)/2 would be replaced by U(O) in Eq. (70). Within the range of validity of the partial dielectric saturation approximation, the average of the fluctuation term (69) would be the same if (UJ+U op)/2 were replaced by Uo, by UJ or by U.v. We have simply selected some mean value for the exponent, symmetrical in T and p. It equals -kT In K, where K is the equilibrium "constant" measured under these conditions. Both AFol and K can vary with electrolyte concentration, with temperature, and with pressure. 12. ACTIVATION OVERPOTENTIAL AND ELECTRODE-SOLUTION POTENTIAL DIFFERENCE For electrode systems, the counterpart of (72) is obtained by considering the free energy of Reaction (73) for a labeled molecule at any fixed position in the body of the solution, but far from the electrode, M Red+M= Ox+M(ne), (73) where Ox and Red denote the oxidized and reduced forms of the labeled molecule in the body of the solution. This free energy change, which accompanies the transfer of n electrons from the ion or molecule to the electrode at a mean energy level discussed in Sec. 5, has a number of contributions, such as one from the change in elec tronic energy, one from the change in ion-solvent interactions in the vicinity of the ion, and one from the change in vibrational energy. Let Fr now denote the configurational free energy of the system containing the electrode and a labeled reactant, the latter fixed in a position far outside the electrode double layer. Let Fp denote the corresponding quantity when labeled molecule is a product, the electrode having gained n electrodes as in (73). The term Fp-Fr is linear in the metal-solution potential difference, as may be seen from the discussion in Sec. 5, and thereby in the half-cell potential E. (E is defined to be the half-cell potential corrected for any ohmic drop and concentration polarization.) We have then (74) where A is independent of E, and where we have used a standard convention regarding the sign of E. [This convention is one which makes Reaction (73) increas ingly spontaneous with increasing positive Eo', a quantity defined later.] Because of the labeling the entropy-of-mixing term of the oxidized molecules and that of the reduced molecules are again absent in FT and Fp. When the system is at electrochemical equilibrium and when the probability of finding the labeled species as a reactant is the same as that for finding it as a product, Fp-Fr must vanish. Also, E then has its equilibrium value, which is Eo' for the case of equal concentrations of the labeled species. [Eo' is the "standard" oxidation poten tial or, as it is sometimes called, the formal oxidation potential of the half-cell; Eo' is defined in terms of the Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions690 R. A. MARCUS equilibrium half-cell potential Ee by (75) for any ratio of concentrations (Red)/(Ox)] Ee=Eo'+(kT/ne) In[(Red)/(Ox)]. (75) One then obtains, from (74), O=!::J.+neEo'. Hence, Fp-pr= ne(E-Eo'). (76) (77) We observe from (77) that E-Eo', rather than the activation overpotential E-Ee, plays the role of the "driving force" in these reactions. The same role is played by !::J.Fol in the homogeneous reaction. In terms of formal electrochemical potentials of the product and reactant ions and in terms of the electro chemical potential of the electrons in the electrode we have, incidentally, for Reaction (73), Fp-Fr= iip 0/_ iir 01 +niie. (78) 13. EQUATION FOR m We first note that !::J.Fol can be written as the alge braic sum of the following terms: The free energy change when the reactants are brought together to the separation distance R, wr; the free energy of reorganiza tion of the reacting system at this R, !::J.F; the free energy difference of reactants and products in this reorganized state, which equals «Up+kT ln1)-(Ur+kT lnf») because of cancellation of momentum and of transla tional contributions; minus the free energy of reorgani zation of the product system at this R to the above state, -!::J.Fp; and minus the free energy change when the products are brought together to the separation distance R, -wp• Thus, (79) is obtained when (20) is used, (Homogeneous) !::J.FO/=wr+!::J.F(R) -!::J.Fp(R) -wp. (79) The electrochemical equation corresponding to (79) is (80), as one may show from (77), (Electrochemical) ne(E-Eo') =wr+!::J.F(R) -wP-!::J.Fp(R). (80) Here, !::J.Fp is obtained from !::J.F, and wP from wr by interchanging rand p superscripts and, at the same time, interchanging -m and m+ 1. To establish this result it suffices to note from (13) that U and all its associated properties are unaffected by such a trans formation, but the properties of the reactants become those of the products. Upon introducing Eqs. (54) and (60) for !::J.F(R), using (68) for !::J.F 0 (Q .), and upon introducing the counterpart of this equation for !::J.Fp(R) , the equation for m is obtained. The final equations for the reaction rate become quite simple when one notes that to an excellent approximation terms involving the ljk'S de fined in (64) can be neglected. The proof is given in Appendix IV. 14. SUMMARY OF FINAL EQUATIONS On using the results of Appendix IV and referring to Eqs. (31) to (33), it is found that the rate constant for a bimolecular homogeneous reaction or a uni molecular electrochemical reaction is given by krate= KPZ exp( -!::J.F/kT) , (31), (33) where Z is given by (34), !::J.F* by (81) and (82), and P by (27). The rate constant of an intramolecular electron transfer reaction, on the other hand, is given by Eq. (32), with !::J.F* given by (81) but with the work terms wr and wP omitted: Homogeneous: wr+wp A !::J.Fol (!::J.FO/+WP-wr)2 !::J.F=-2-+4+-2-+ 4A ,(81) Electrochemical: wr+wp A ne(E-E ') !::J.F=--+-+ 0 2 4 2 (neE-neEo' + wP- wr) 2 + 4A (82) In (81) and (82) A is given by (83) where Ai is given by (84) and Ao is given by (69) at Q = Q . On introducing the symmetrical force constants one finds Q.=Qr+m(Qr-Qp). Since Ao depends but weakly on Q. and since m is usually close to -!, it suffices to evaluate Ao at Q=!(Qr+Qp) in the typical case. This result is used in deriving Eq. (88a), (84) The reduced force constants kik are defined in (63) and the !::J.q/s are differences in equilibrium values of bond coordinates (e.g., independent bond lengths and angles), q/-qp. It is expected that typically p should be about unity. As noted earlier, Z is essentially the collision number, being about 1011 liter mole-1·sec 1 and 104 cm sec1 for homogeneous and electrochemical reactions, respec tively. In Ref. 6 the above equations were written in an Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 691 equivalent form (S5) Homogeneous:- (2m+1).=tJ.Fo'+wp-w2, l Electrochemical: -(2m+1». =ne(E-Eo') +wP-wr. (S6) The value of m defined by (S6) can be shown to differ very slightly from that in the preceding sections, due to the approximation of neglect of the lik's, but the final equations obtained when (S6) is introduced into (S5) are identical with (Sl) and (S2). According to Eqs. (Sl) and (S2) tJ.F* depends on tJ.Fo, or on neE according to (S7a) and (S7b) when wr and wP are held constant. (iMF/atJ.FO')w=!(1/2)') (tJ.Fo'+wp-w r), (S7a) (atJ.F/aneE)w=!+ (1/2),) (neE-neEo'+wp-wr). (S7b) We refer to these slopes as "transfer coefficients at constant work terms." The second term in (S7a) and (S7b) can be calculated when A is known, and this in turn can be estimated from the experimental value of tJ.F* at tJ.Fo,=O, or at E=Eo' using (Sl) or (S2), when the work terms can be estimated or are negligible. Typically, this second term is found to be small, so that these "transfer coefficients" are then 0.5. Equations (S7a) and (S7b) are based on the neglect of the antisymmetrical functions lik defined in (64). When these functions are not neglected, the transfer coefficient is not exactly 0.5 for zero (tJ.Fo'+wp-w r) /A or zero (neE-neEo'+wP-w")/A, but is given instead by Eq. (A14) in Appendix IV. When these two sources of deviation from a 0.5 value are small, we may add them and so obtain (S7c) and (S7d) instead of (S7a) and (S7b) : (atJ.F/atJ.FO')w=!+ (1/2A) (tJ.Fo'+WP-W"+!Ai(I.» , (S7c) (atJ.F/aneE)w=!+ (1/2A) X (neE-neEo'+wp-wr+!Ai(I.». (S7d) As noted in Appendix IV the (I.) term could cause a deviation from the 0.5 value by 0.04 when the force constants in the products are all twice as large (or all twice as small) as the corresponding ones in the re actants and when Ai/A is about !. Smaller differences in force constants would lead to even smaller deviations than 0.04. This source of deviations would be difficult to detect experimentally, since there are other sources of deviation as well. In the case of homogeneous reactions, force constants on one reactant may stiffen and those in the other weaken, so that the average value of (l.) may be even less than that for the above case, and the deviation from the 0.5 value arising from this source correspondingly smaller. In summary, the transfer coefficient at constant w's is expected to be close to !, reflecting a type of sym metry of the Rand P surfaces in the vicinity of the reaction hypersurface (compare also Sec. 17). A source of deviation from this symmetry arises from a difference in corresponding force constants in reactants and products. It appears as an (ls) term in (S7) and has been shown to be small. A second source of deviation arises when the R or P surface is appreciably lower than the other, and is reflected in the presence of the tJ.Fo, and ne(E-Eo') terms in (S7). This source of deviation, too, is normally small. The leading term in (S7), !, arises from the quadratic nature of both the V' 0 and the V'i contributions to tJ.F. 15. PROPERTIES OF THE REORGANIZATION TERM A For use in subsequent correlations, we examine an additivity property of A and the relation between the values of A in related homogeneous and electrochemical systems. We consider first the (hypothetical) situation when R is very large, so large that the force field from one reactant does not influence the other. On noting that Ao is given by (69) and that the fluctuations around each reactant are now independent (large R), Ao can be written as the sum of two independent terms, one per reactant. It then follows that when R is large the value of Ao for a reaction between reactants from two different redox systems A and E, Aoab, is the arith metic mean of the values Aoaa and Aobb of the respective systems: (R large). (SSa) Furthermore, in the electrochemical case there is only a contribution from one ion (assuming that any dis tortion of atomic structure of the electrode yields only a relatively minor contribution to tJ.Fo ). Denoting the values of Xo for the electrochemical redox system A and for the homogeneous redox system A by Aoe! and XoeJr. respectively, we have (R large). (SSb) Relations similar to (SSa) and (SSb) also hold for Ai, independent of R, as may be seen from (S4): Part of the sum for Ai is over the bonds of the first reactant and the remainder is over those of the second one. While the kik'S of one reactant in the activated complex depends slightly on the fact that there is a neighboring reactant, this influence is taken to be weak. In the absence of specific interactions, Eqs. (SSa) and (SSb) would also hold for smaller R, since in the equation for Ao each ion would merely see another charge, -mtJ.e, and the surrounding medium, in both the homogeneous and electrode cases. In the homo- Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions692 R. A. MARCUS geneous case, the -m!:i.e is centered on the other ion. In the electrode case it is an image charge on the electrode.28 To obtain some estimate of deviations from (88a) due to differences in ion size (one type of "specific effects") we examine in the next section the evaluation of Ao in the dielectric continuum approximation. 16. DIELECTRIC CONTINUM ESTIMATE AT !:i.Fo* The present section on a continuum estimate of !:i.Fo* is included partly for what it can reveal approximately about certain aspects of the statistical mechanical value for !:i.Fo * and partly for making some approximate numerical calculations. It does not form a necessary part of the present electron-transfer theory itself, of course, for the latter rests on statistical mechanics. We note that !:i.Fo * can be regarded as the sum of two contributions, !:i.Fsol and !:i.Fatm· !:i.Fsol is defined as the contribution if the atmospheric ions have not adapted themselves to the change m (Pa/ -Pap 0), and !:i.F "tm is defined as the contribution due to their adaptation ("reorganization"). !:i.Fsol in an electrolyte medium will not have exactly the same value it has at infinite solution, since the local dielectric properties near the reactants will be altered somewhat by the presence of salt. These two contributions are estimated in Appendix VII by treating the medium as a dielectric continuum, the ion atmosphere as a continuum, and the reactants as spheres, and by neglecting dielectric image effects.29 We obtain (89) and (90) for the value of !:i.Fsol for a medium treated as dielectrically unsaturated continuum outside the inner coordination shell of each reactant. If partial saturation occurs, Eq. (67) still applies.9 If one then introduces "differential" rather than "integral" dielectric constants, as defined in the Appendix, and treats them approximately as constants Eqs. (89) and (90) again apply but now Dop and D. are mean values of these differential constants Homogeneous: ( 1 1 1)( 1 1 ) !:i.Fsol=m2(ne)2 -+------, 2al 2a2 R Dop D. (89) 28 Quantum-mechanical calculations in support of the classical image law are given by R. G. Sachs and D. L. Dexter, J. App!. Phys.21, 1304 (1950). At a distance of 5 A from the electrode the computed energy of an ion in vacuum may be estimated from their results to be about 8% higher than that estimated from the image law. Experimental evidence for the validity of the image law at distances of 5 A has been offered by L. W. Swanson and R. Gomer, J. Chern. Phys. 39,2813 (1963) (d. p. 2835). 29 The dielectric image contribution to !!J.F .01 is estimated to be negligible: It makes essentially no contribution to the value of Fm(r_p)OI> since this hypothetical system has a low diectric con stant equal to the optical dielectric constant throughout. Its contribution to Fm(r-p) is only about 8% of the value of the term containing liD. in (90). Since this term is only a negligible fraction of the 1/ DOI> term in (90), the dielectric image contri bution to !!J.F.ol* can be neglected. We note later that !!J.Fatm* is apparently much smaller than !!J.F.ol' Dielectric image effects may be estimated from electrostatic calculations to contribute about 8% to w' when two charges of equal magnitude meet. where ne is !:i.e, the charge transferred from one reactant to the other; al and a2 are the radii of the two reactants computed at intramolecular coordinates qi= q.i (the radii are of spheres each of which includes any inner coordination shell) ; R is taken to be the sum al+a2; Dop is the square of the refractive index of the medium; and D. is the static dielectric constant of the medium Electrochemical: !:i.Fsol= m2(ne)2(~_~)(_1 -~), (90) 2 al R Dop D. where R is twice the distance from the center of the ion to the electrode surface and al is the radius of the reactant (and hence of the product) computed at qi= qoi. The value obtained in Appendix VII for !:i.F* "tm in the electrically unsaturated region (i.e., in the Debye Huckel region for the atmosphere around the ion and, in electrode systems, for the diffuse part of the double layer) is given by Eq. (91) for homogeneous systems for the case of al = a2 ( = a), and by (92) for electrode systems. The value for !:i.Fatm for partially electrically saturated systems can also be obtained from (67). Once again, one introduces "differential" quantities. If the latter are replaced by "mean" values near the central species Eqs. (91) and (92) are again obtained, but with D. and Ie reinterpreted; Ie is given by (A23) in Appendix VII. A more reliable procedure, however, would be to use the position-dependent value of Ie in solving this particular linearized Poisson-Boltzmann equation, since the electric fields in electrolyte media die out fairly rapidly, namely as r-1 exp( -ter). Equa tions (91) and (92) are based on the solution of a linearized Poisson-Boltzmann equation with a local mean X Homogeneous: m2(ne)2 !:i.Fatm= D.R X[XR+ exp[ -le(R-a) ] (1+x2a2/2) 1+lea+ exp[ -le(R-a) Jx2a3/3R Electrode: !:i.Fatm=![rhs of (91)]. 1 J. (91) (92) Calculated as above, !:i.F "tm is much smaller than !:i.Fsol and is also expected to be less than the salt effects on wr and wp• Even at high Ie it is only m2(ne)2 (R-a)/D.aR. Since R"'"'2a, its value there is about m2(ne)2/D.R, which is only about 2% of !:i.Fsol' Parenthetically, we note that this term arising from (91) and (92) just cancels the D. term in (89) and (90), respectively. Added electrolyte can influence the rate constant, we conclude, principally by affecting wr, wP, and (by affect ing dielectric properties) !:i.Fsol. Comparison of (89) with (90) reveals that AD for an Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON.TRANSFER REACTIONS. VI 693 isotopic exchange reaction has twice the value of Xo for an electrode reaction involving this redox couple when the value of R is the same in each case. It is recalled that R is the value for which Kp exp( -I1FjkT) had a maximum. If one presumes this R to be the distance of closest approach of the "hard spheres" and assumes the reactant to just touch the electrode, then R is the same in each case. In Eq. (89) al = a2 for an isotopic exchange reaction since these are the radii evaluated for q= q., it is recalled, and since typically the transi tion state should be symmetrical in this respect. (From the equation cited the actual q.i's can be computed and the presumed symmetry verified for typical conditions.) It may be seen from (89) that Xo is essentially equal to the sum of two terms, one per reactant, and that for the same R the value of Xo for the homogeneous reaction in any redox system A equals twice its value for the electrochemical case. While the presence of the R term makes Xo not quite additive, the deviation from additivity can be shown to be small: On denoting the radii for ions of the two systems by a and b we obtain (93) . aLl. aa bb _ [1-(bja) J2 2(...!.. _~) Xo 2(XO +Xo )-4b[1+ (bja) J(ne) Dop D.' (93) Even if bja is !, a fairly extreme case, the ratio of the above difference to Xobb is (1-bja)2 2(1+bja) , i.e., -h. In virtue of this result, Xo has been treated as an additive function in applications6.7 of the equations of this paper. 17. SIGNIFICANCE OF m The parameter m was chosen in Sec. 13 so as to satisfy the centering condition (20), a condition which led to the vanishing of 1'(0). On differentiating I(qN, R) given in (15) and setting qN=O one finds: m=_<aur>/<~(ur_uP» (94) aqN aqN ' where the average ( ) is over the distribution function on the reaction hypersurface S' at the given R, From (94), -m is seen to be the mean slope at the reaction hypersurface S', (aUrjaqN), of the R surface, for the given separation distance R, divided by the sum of the mean slopes, (aUrjaqN) and (-aUpjaqN) of the Rand P surfaces at S'. If the intersection surface S' at this R passed through the stable configurations of the reactants, on the average, then (aUr jaqN) would be zero. If it passed through those of the products instead (aupjaqN) would be zero. In these two cases one sees from Eq. (94) that m would be 0 and -1, respectively. When in the vicinity of S' the Rand P surfaces are, on the average, mirror images of each other about S', (aUrjaqN) equals (-aUPjaqN) and one sees that m= -!. Values of m close to -! are typical6•7 and are to be expected, one sees from (86), when I1Fo, is near zero or when E is close to Eo' (typically of the order of or less than 10 kcal mole-lor 0.25 V, respectively). 18. DEDUCTIONS FROM THE FINAL EQUATIONS Equations (31) to (33), together with the additivity property of X (Sec. 15), and the relation between the electrochemical and chemical X's described earlier lead to the following deductions, if K and p are about unity, or if they satisfy milder conditions in some cases.30 (a) The rate constant of a homogeneous "cross reaction," k12, is related to that of the two electron exchange reactions, kn and k12, and to the equilibrium constant K12, in the prevailing medium by Eq. (96), when the work terms are small or cancel, kl2 OXl+ Red2~ Red1+ OX2, (95) k12= (knk22Kld)i, (96) where lnf= (lnK12) 2 4ln(knk22jZ2) (97) Frequently, f is close to unity. (b) The electrochemical transfer coefficient at metal electrodes is 0.5 for small activation overpotentials318 (Le., if 1 nFTJ 1 < II1Fo * I, where I1Fo * is the free energy of activation for the exchange current) ,31b when the work terms are negligible. (c) When a substituent in the coordination shell of a reactant is remote from the central metal atom and is varied in a series, a plot of the free energy of activation I1F* versus the "standard" free energy of reaction in the prevailing medium I1Fo, will have a slope of 0.5, if I1PO' is not too large (i.e., if II1Fo, 1 is less than the intercept in this plot at I1Fo, = 0). In this series, for a sufficiently remote substituent, X and the work terms are constant but I1Fo, varies, as in (87a). The slope of the I1F-versus-I1Fo, plot has been termed the chemical transfer coefficient,6 by analogy with the electrochemical terminology. (d) When a series of reactants is oxidized (reduced) by two different reagents the ratio of the two rate constants is the same for all members of the series in 30 For example, it suffices for some of the deductions that Kp be constant in a given series of reaction or that it have a geometric mean property. 31 (a) We have phrased this condition for the case that (Ox) = (Red). For any other case, 7J should be replaced by E-Eo'. (b) The exchange current cited refers to the value observed when (Ox) = (Red). Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions694 R. A. MARCUS the region of chemical transfer coefficients equal to 0.5 [i.e., in the region where I !1FOf I < (!1F) AFo °'=0 in each case]. (e) When the series of reactants in (d) is oxidized (reduced) electrochemically at a given metal-solution potential difference the ratio of the electrochemical rate constant to either of the chemical rate constants in (d) is the same for all members of the series, in the region where the chemical and (work-corrected) electro chemical transfer coefficient is 0.5. (f) The rate constant of a (chemical) electron exchange reaction, kex, is related to the electrochemical rate constant at zero activation overpotential,3la kel, for this redox system, according to Eq. (98) when the work terms are negligible: (98) where Zsoln and Zel are collision frequencies, namely about 1011 mole-I. secl and 104 cm secl. When the work terms are not negligible, or do not cancel in the comparison, the deductions which depend on this condition refer to rate constants, to KI2 and to an electrochemical transfer coefficient corrected for these terms. Again, a minor modification of the transfer coefficients from the value of t in (b) or (c) can also arise from the antisymmetrical force constant term ([8 > in Eqs. (87) and (AI4). It is shown in Appendix VII that under certain conditions these expected correlations apply to over-all rate constants as well as to those involving only one pair of reactants. 19. GENERALIZATION AND OTHER IMPROVEMENTS Some of the extensions or improvements in the present paper, compared with the earlier ones in this series, are the following: (1) Use is made of a more general expression for the reaction rate as the starting point. (2) A more detailed picture of the mechanism of electrode transfer is given for the electrochemical case. (3) The derivation is now given for both electrode and homogeneous reactions, and in a single treatment. (4) The statistical-mechanical treatment of polar interactions, based in Part IV on the interactions of permanent and induced dipolar molecules in the medium, was replaced by a more general particle description of polar interactions, through the use of the potential-energy function (37) and (65). (5) The equivalent equilibrium distribution made plausible in Part IV was proved more rigorously here. (6) The functional form (68) for !1Fo, obtained in Part IV only by subsequently treating the medium as a dielectric continuum, was derived here using a statistical-mechanical treatment of nonequilibrium polarization systems. (7) The basic equation for krau. has been converted to a simple form [e.g., (31) and (81)], a form used in Part V, by neglecting the anti symmetrical function of the force constants, a neglect which has only a minor effect numerically. (8) The symmetry arguments used in Part IV to convert the kT/h and a portion of a !1F term to the Z factor in (31) have now been given more rigorously. (9) The ion atmospheric reorganization term was but mentioned in Part IV. It is now incorporated into !1Fo . The nonpolar contribution to w" and wP is also formally included. (10) The contribution of a range of separation distances to the rate constant is included. The results in the present paper may be compared with earlier papers in this series. In Part I, !1Fo was computed for homogeneous reactions at zero ionic strength, and dielectric continuum theory was used. Equation (89) was obtained. The actual mechanism of electron transfer was discussed there, but without the detailed description which the use of many-dimen sional potential energy surfaces provides. The latter was used in later papers of this series, a use which added to the physical picture. The counterpart of Part I for electrode systems was also derived and applied to the data in a subsequent paper.2 In the earliest papers, the dielectric continuum equivalent of the equivalent equilibrium distribution was derived by a method apparently different from that used in the present paper. The distribution selected was the one which minimized the free energy subject to the constraint embodied in Eq. (20), or really embodied in the dielectric continuum counter part of (20). In Appendix IX this method is in fact shown to yield the same result for the equivalent equilibrium distribution as the functional analytic one used in Appendix II. It is entirely equivalent. APPENDIX I. NONADIABATIC ELECTRON TRANSFERS Several estimates are available for the probability of nonadiabatic reactions, per passage through the inter section region of two potential energy surfaces, and have been referred to and discussed in Ref. 7. In each case the motion along the reaction coordinate was assumed to be dynamically separable from the remain ing motions. (For conditions on separability see, for example, Ref. 32 and references cited therein.) The probability of electron transfer per passage through the intersection region in Fig. 1 will depend in the first approximation on the momentum PN conjugate to the reaction coordinate qN, as, for example, in the Landau Zenerl1 equation. While the value of K is not so simply represented in more rigorous treatments, we simply write it as K(PN). In the above treatments the reaction coordinate was assumed to be orthogonal to the others ;12 R. A. Marcus, J. Chern. Phys. 41,603 (1964). Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 695 in mass-weighted configuration space, so that gNi van ishes for i~N (and so, therefore, does gNi) in the kinetic energy. On recalling the derivation of Equations (1) and (2)8 and on introducing the above assumptions, the rate constant is given, one can show, by Eqs. (1) or (2), but with the integrand multiplied by K: K=------------- This K can depend on all the other coordinates, qi(i~N) at the given value of qN characterizing the intersection surface S. The denominator in the above equation is easily shown to equal kT, and so to be independent of the qi. In the discussions of K(PN) in the literature, the derivation of the Landau-Zener equation, for example, the reaction coordinate has been assumed to be recti linear; gNN is then a constant and the integral in the numerator then becomes independent of the qi and may be removed from the integral in Eqs. (1) and (2). There appears to be no treatment in the literature for nonadiabatic reactions involving many closely spaced energy surfaces as in Fig. 2, covering the range of K(PN) from 0 to 1. If K (pN) is sufficiently small, the transition to each P surface from the initial R surface may be assumed to be independent, as mentioned earlier, and the reverse transition to the initial R surface during this passage may also be neglected. In this case only does the method of Levich and co-workers in this connection become appropriate. (For references see Ref. 7.) In this case the above K appears in the inte grand of Eq. (3) and care is taken to sum over all levels in an appropriate fashion, as done by Levich et al. (see Ref. 7 for bibliography). One can then evaluate the K appearing in Eq. (33) and defined earlier. Usually, however, we assume that K(PN) is close to unity (within some small numerical factor, say) for the PN'S of interest. APPENDIX II. PROOF OF EQ. (13) FOR THE CENTERED DISTRIBUTION The centering is of both a horizontal type (horizontal in terms of Fig. 1 or 2) and of a vertical type, repre sented by Eqs. (A1) and (A2), respectively: !furdV'= !f Upd V', (A1) !fUdV'= !furdV'. (A2) Suppose, for possibly more general applications, that there are n linear equations of constraint of the type represented by (A3). Here, we are especially interested in the case n= 1, !fyjdV' = 0, j=1, ... , n. (A3) For any temperature and U, this integral is a linear functional of Yj. Although one can find functions, U, other than Yj (and other than linear combinations of Yj) for which f1udV' vanishes at some temperature T, the y/s are the only ones for which this integral is specified to vanish for all T. That is, there are only the n equations of constraint (A3) on the U* in 1. The space functions Y for which ff Y dV' is real and finite form a linear vector space over the field of real scalars. Moreover, the integral, denoted by J ( Y), is a linear functional on this space. For some subspace M of it, the integral vanishes. The functions Yj( j = 1 to n) form a basis for M. If there exists some function w for which J (w) does not vanish, then an elementary theorem33 of functional analysis shows that any func tion x can be written as x=w[J(x)/ J(w)]+y, (A4) where Y belongs to M. In the present instance w= 1 is such a function. On applying (A4) to the function x= U-Ur and using (A2) one sees that x=y, i.e., that x belongs to M and can so be written as a linear combination of the functions Yj. In the present case, M is one dimensional, the only Yj being Ur-UP, since (A1) is the only equation of constraint. Thus, x, i.e., U-Ur, equals Ur-UP multiplied by a real scalar, and Eq. (13) is established. APPENDIX III. DISTRIBUTION OF Yo' COORDI NATES IN THE ACTIVATED COMPLEX We first note that U(2) in Eq. (65) does not depend on PMo, the pO of the "medium," and so is insensitive to the usual rotational and translational fluctuations of the solvent molecules, unlike U(O) and U(1). Since Uo * is given by (13), with 0 subscripts added, one term in Uo * is ur(2) +m[ur(2) -Up(2)]. Since this can be extracted from the integral in the denominator of the above distribution function because of this insensitivity to the V'o coordinates, it cancels a corresponding term extracted from the numerator. The distribution func tion fo * then becomes (AS) : exp( -{U(O) +ur(1) +m[Ur(1) -UP(1) ]}/kT) ! exp(-{u(0)+ur(1)+m[ur(1)- UP(1)]l/kT)dV'0 (AS) Since U(1) is linearly dependent on the Pao of each reactant, UT(1) +m[UT(1) -Up(1)] equals the U(1) for a system in which each reactant has a Pa 0, Pa 0, 33 A. E. Taylor, Introduction to Functional Analysis (John Wiley & Sons, Inc. New York, 1958), p. 138. Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions696 R. A. MARCUS given by (66). Next, on multiplying the numerator and denominator of (A5) by the exponential of the -U(2)/kT corresponding to these Pao,S and placing it under the integral sign, we see that the distribution function fo * is the same as that corresponding to the Pa o's given by (66). APPENDIX IV. SIMPLIFICATION OF EQ. (54) AND THE EQUATION FOR krate We introduce the quantities kjk and Ijk defined in ( 63) and ( 64). The first was chosen so as to have dimensions of a force constant, and the second so as to be dimensionless. Principally, it is the diagonal stretching contributions which are usually important. Purely for simplicity of argument we confine our attention to the diagonal terms. We denote the new force constants by f:, f.p, and their symmetric and antisymmetric combinations cited above by k. and 1 •. In terms of k. and 1., f: equals k./ ( 1-1.) and f.p equals k./ (1 + 1.). To make use of the symmetry of the resulting equations we use the parameter e, equal to (m+t). We obtain (A6) from (54) and (60): .1F.= H e-t) 22:). (.1q. 0) 2(1-1.) (1 +2el.)-2 +tkTE In[(1+2el.)/(1+1.)J, (A6) • where .1F;* is defined as .1F(R) -.1Fo (R). Similarly, we find (A7) by noting that it is obtained from (A6) by replacing -m by m+l and interchanging rand p subscripts (see Sec. 13) .1F;P= He+t) 2 Ek.(.1q. 0)2(1 +1.) (1 +2el.)-2 • +tkTE In[(1+2el.)/(1-1.)J, (A7) where .1Fip is .1Fp(R) -.1Fo p(R). In terms of e, Eq. (79) can be written as (AS), upon introducing Eq. (6S) for .1Fo* and its counterpart for .1Fop[ = (m+l)2A"J where (A9) Most of the data are obtained in the vicinity of .1FRol=0.6,7 We consider this region first. Near the point (l.=0, .1FRol=O) one readily verifies from the equations below that e is close to zero and that it vanishes at that point. We let 1] denote e or 1., and 0 denote the "order of." (1] is a small quantity in the vicinity of this point.) One then finds from (A6) to (AS) -2eX-!Xi(I.)+0(1]3) = .1FRol+kTEl., CAlO) • where Xi= t Ek.(.1q. 0) 2, • Furthermore, according to (79) .1F equals .1FP+ .1FRo,. Hence, .1F= H.1F+.1F) = H.1F+.1Fp) +t.1FR 01. On introducing (A6) and (A7) one finds .1F= t.1FRo'+X(e2+t) +XiO(1]4) +tkTE( 41:1.+1.2). • (A12) On introducing (AlO) for e one finds that (A12) becomes .1F= t.1FRol+tX+ (1/4X) (.1FRo'+tX i(l. »2 +tkT[El.L (kT/X) (El.)2J+0(1]4). (A13) The same expression obtains for electrode processes, with the .1Fo, in .1FRol replaced by ne(E-Eo'). In an isotopic exchange reaction which involves mere interchange of charges in the electron transfer step, the term (1.) vanishes by symmetry. In other reactions there will be some tendency for it to vanish, for while I. increases on one reactant on going from State R to State P (due to an increase of charge), it will tend to decrease on the other. As a somewhat extreme case involving no compensation, consider two reactants one of which has vanishing I. and also vanishing con tribution to Xi. (Hence, we include the possibility that this "reactant" is an electrode.) For the other molecule let the force constants k: and k.p differ by as much as a factor of 2. Then one finds (t. )"'t. If X;/X"'t then Xl(l.)2/16X is only about 1% of Aj4, the main term at .1FRo,=O. When Aj4 has its usual value of 10 to 20 kcal mole-I, say 10, and when the reactant has a coordination number of six, then the kT term in (A13), is estimated to be about 4% of the X/4 term at room temperature. We consider next the effect of nonvanishing (1.) on the derivative (iJ.1F'/iJ.1Fo'ho'x, at .1FRo,=O, the region of greatest interest. This derivative equals (A14) In the case cited above the Xi(I.)/4X term is about +0.04. Thus, the derivative differs by only S% for this case. Hence, the (18) term may be neglected when e (and hence I.1FRo,/X I) is small. When I.1FRo,/X 1 is not small, one finds that (A13) should be replaced by (A14a), to terms correct to first order in the I. .1F= t.1FR o'+tX+tc.1FR °')2 Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 697 The term containing (lB) is still small: A fairly extreme case is one where the activated complex resembles the reactants (m=O) or the products (m=1). At each extreme I t!,.Fnol/A I is about unity, since the expression for E( = -t!,.Fn 01 /2A) is but slightly affected and since I E I equals! when m is 0 or 1. In the interval O~ I t!,.Fnol/A I ~1 the last term in (A14a) has a maximum at I t!,.Fnol/A I = (!)!. At this point it equals about 1 kcal mole-I for the values of (lB), AiA, and A/4 cited above. When one does not neglect second and higher-order terms in IB' and solves (A6) to (A8) numerically in this region one obtains the same result: The lB terms may be neglected. APPENDIX V. SMALLNESS OF (Ur(Q) -Up(Q) )0 If (ur-up).o at any Q is expanded about its value at Q. and if it can be shown that the linear term suffices, it follows that (Ur-up).o averaged over ji* equals the value at Q. plus the average of the linear term. In virtue of (SO) the averaged linear term vanishes and, in virtue of (20), the average of (Ur-Up).o over j;* vanishes. Hence, (ur (Q.) -UP (Q.) )'0 also vanishes. To show by a posteriori calculation that the linear term in the expansion suffices we make use of some notation introduced after (54). After use of (37), of the equality of (Uo"-Uop)'o with Fo-FoP, of (68), of the definition of Ot and O,P, and of their quadratic expansions about Qr and Qp, respectively, of the essen tial equality of the vibrational entropy of reactants and products, and of the justifiable neglect of the antisymmetrical functions (64) (Appendix IV) one finds (A15) for any given Q. (Ur-Up)'o= -(2m+1)Ao(Q) -t!,.F01 +! 2:)ii(qL qri) (qi_qri) i,i -!L:kij(qLqpi) (qi_q/). (A15) i,i The quadratic term, kiiqiqi, is seen to cancel. A linear expansion of Ao(Q) about Xo(Q) is sufficient, for even the linear term is small (compare Appendix VI) . Hence, he linear term in an expansion of (ur (Q) -UP (Q) )'0 suffices. The vanishing of (ur(Q.) -Up(Q.) )0 then follows. APPENDIX VI. JUSTIFICATION OF NEGLECT OF axo/aqi IN THE DERIVATION OF EQ. (58) It is shown here that the error in neglecting the dependence of Ao on Q in deducing (58) from (56) is minor. Since the arguments in Appendix IV reveal that the error in neglecting the antisymmetrical functions (64) is minor, we may simplify the present analysis by neglecting them. To this purpose all force constants may be replaced by the symmetrical ones, kik, defined by Eq. (63). Let Ao be a column matrix whose components are aXo/aqi: Xo(Q) =>'o(Q.) + L:(aAo/aqi) (qLq.i) + .. , i =Ao(Q.)+AoT. (Q-Q.)+" '. (A16) The first variation in an expansion of O;(Q) about Q. is found from (56) 00.= oQT{(m+1)K(Q.-Qr) -mK(Q-Qp) -m(m+1)Ao], (A17) where the elements of K are the kik'S. On setting 00. equal to zero, one obtains, instead of (58); Q=m(m+1)K-IA o+ (m+1) Qr-mQp. (A18) Equation (54) for t!,.F then becomes t!,.F= (m2/2) [t!,.QT+ (m+1) (K-lAo)T] ,K[t!,.QT + (m+ 1)K -lAo] +m2>'0(Q.)+!kTln I/;k III kik I. (A19) For present purposes it suffices to consider the case where t!,.Fo1 is small. An expression for t!,.Fp can be obtained from (A19) by replacing m by -(m+1) and t!,.Q by -t!,.Q. On letting t!,.F-t!,.Fp equal zero (since t!,.Fo1 is zero) the resulting equation is solved for m, which is thereby found to be -!. A simple numerical estimate then shows that the presence of the K-IAo terms have negligible effect: Other than the In term the rhs of (A19) is given by i>.(Q.)+!t!,.>'o+~AoT(KT)-I·Ao, (A20) where t!,.Ao is the total change in Ao when Qr is changed to Qp. Typically Ao/4 is of the order of 5 kcal mole-I and is inversely proportional to ion size. When the mean bond length changes by as much as 0.15 A (compare the probable Fe-O bond length difference in Fe2+ and FeH hydrates) and when the radius of the reactant including inner coordination shell is 3 A, t!,.>'0/4 is about !(0.15/3), i.e., about 0.25 kcal mole-I. The third term in (A20) is even less. For example, if one considers the stretching of bonds only, and if the stretching ki/s for metal-oxygen bonds in a hydrated cation are taken to be the same one finds i'XAoT(KT)-IAo= (t!,.Ao/Ai)2iAi. (A21) (Similar remarks apply to other coordination com plexes.) Since Ai/4 is of the order of 10 kcal mole-I for the cited case (A21) is about 0.006 kcal mole-I. APPENDIX VII. CALCULATION OF t!,.Fo IN CONTINUUM APPROXIMATION When dielectric unsaturation and electric unsatura tion prevail there is, respectively, a linear response of Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions698 R. A. MARCUS the solvent polarization and of the charge density of ions in an ion atmosphere to the charging of the central ion (or ions), and not merely to a small change in its charge. In real systems, some partial dielectric satura tion outside of the coordination shells may occur and, at appreciable concentration of added electrolyte, the response of the atmospheric ions is certainly nonlinear. (The region of linear response of an ion atmosphere to a charging of the "central ion" is confined to the Debye-Huckel region.) We introduce the partial saturation approximation, wherein only a linear response to a small change in charge of the central ion (ions) is assumed. The special case of unsaturation is automatically included, there fore. We are interested, typically, in changes of magni tude, mt.e, i.e., about t an electronic charge unit. Equation (67) was derived for both the partially saturated and for unsaturated systems, but in the former case the definition of FOPm(r_p) and Fm(r-p) has to be interpreted carefully. To calculate Fm(r-p) appearing in (67) and to take partial saturation into account, one considers two charge distributions: (i) The original charge distribu tion of the reactants and the medium for the cited R. (ii) A hypothetical charge distribution in which the reactants' charge distribution is altered from (i) by an amount m (Par 0 -pap 0), in a hypothetical system which has responded linearly to this change. To obtain the properties of the hypothetical system in Fm(r-p) one subs tracts the above two charge distributions on the reactants and also substracts the portions of the re maining charge distributions, induced or otherwise, which did not respond. One now has in this hypothetical [m(r-p)] system reactants which have permanent charges given by the distribution m (Par 0 -pap 0) and are imbedded in a medium of solvent and atmospheric ions which has linear "response functions" describing the above response. For example, if we use a continuum model, then the effective dielectric susceptibility of the solvent is the proportionality constant x( r) in34 BP(r) = -x(r) BE(r), (A22) where BP and BE are the change in polarization and in electric field at r. The effective dielectric constant describing the response to this BE is Ds (r) equal to 1+41lx(r). The quantities x(r) and Ds(r) can be tensors. Then, again, if p (r) is the charge distribution in the ion atmosphere and, if one wishes, in the electrical double layer at the electrode-solution interface, and if per) is approximated by a continuum expression per) = LC;'''Ci exp( -ciif;!kT) , i where Ci is the charge of Species i in this atmosphere, 34 R. A. Marcus, J. Chern. Phys. 38, 1858 (1963). C/-O is its concentration at infinity, and 1/; is the potential at r relative to the value at infinity, then Bp(r) = -(LciCOei2c-ei'i/kT/kT)01/I(r). i On recalling that the Debye kappa is defined as the square root of the proportionality constant of per) and -1/;(r) in linear systems, the quantity which plays the same role in this hypothetical system is x' (r) . x'(r) = [Lcicoei2 exp( -ei1/;/kT) ]!. i (A23) To calculate FOPm(r_p) we recall that this system responds only via the electronic polarization of the medium, and so K' vanishes for this system and x'(r) becomes X'e(r), the proportionality constant replacing x(r) in (A22). The medium in this hypothetical system behaves as though it had a dielectric constant D'op(r) equal to 1+41Txe. If we take D'op to be approximately a constant, for simplicity, then FOPm(r_p) is easily calculated. We neglect dielectric image effects.29 POPm(r-p) is the sum of the free energy of solvation of the central species when they are far apart, plus the free energy change when they are brought together in this "op" medium. The former is given by the Born formula (it is not the free energy of solvation of the bare ion, but of the coordi nated ion) and the latter by the Coulombic term. Hence, Fop m(r-p) = -[( mt.e) 2(1_ ~)+ (mt.c) 2(1_ ~)] 2al D op 2a2 D op (mt.c) 2 -D'opR. (A24) The Fm(r-p) term is the sum of its value when the ion atmosphere does not respond [ (mt.c )2( 1) mt.e( 1)] (mt.e) 2 -~ 1-D's + 2a2 1-D's -D'sR ' (AZS) and the contribution due to their response via K' (r) , t.Fatm. On taking K' to be approximately a constant near the central series the leading terms of the second contribution are36 _ (mAc)2[K'R+ exp[ -x'(R-a)](1+K'2 a2/Z) J D'.R l+K'a+ exp[ -x'(R-a)]x'2 a3/3R 1, (A26) when al=a2. The difference of (A24) and (A2S) is the value of Fop-F when the atmosphere does not respond, and 35 Since dielectric image effects are being neglected one may m~rely use the ~xpressions obtained by G. Scatchard and J. G. Kirkwood, PhYSik. Z. 33, 297 (1932), for the contribution to the free energy of interaction of a pair of ions with their atmosphere due to a response described by >C. We may merely replace >C by >c' and D. by Do' under the approximations stated. Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 699 was called !1FsoJ. In (89) to (92) we have omitted the prime superscripts for brevity. In the case of electrode systems, there is only one ion, but there is also the image charge of opposite sign in the electrode.28 Instead of (A24) to (A26) one finds, (electrode) Fop (_ ) = _ (m!1e) 2( 1-_1_) _ (m!1e) 2 m r P 2a D'op 2D'opR (A27) and that Fm(T-P) is the sum of (A28) and of one-half (A26) , _ (m!1e)2(1_~)_ (m!1e) 2 2a D'. 2D'.R· (A28) In this way (90) and (92) of the text were obtained. APPENDIX VIII. CORRELATIONS OF OVER-ALL RATE CONSTANTS Equations (31), (33), (81), and (82) describe the rate constant for any reactants with intact, specified inner coordination shells. !1Fo, there refers to the change for those species. Consider now the rate constants ex pressed in terms of the stoichiometric concentration of each redox reagent. The region of (81) linear in !1Fo, is the most important one in terms of the correlations made in Part V, and we restrict our attention here to such cases for each elementary redox step (A29) below. We consider only the case where the dissociation or formation of any important complex does not contribute appreciably to the reaction coordinate near the inter section surface: We make use of (81) and note that its derivation was based on intact coordination shells in a system near the intersection surface; the properties of the "reactants" or "products" appearing in Eq. (81) refer to those with such shells, even though they might be unstable. We consider the homogeneous case first. Let m denote the totality of any ligands Xl, X2, ••• in a reacting member of the A redox system having mi ligands of Type Xi, m= (ml' m2, ••• , mi, ••• ). Let n play the same role for the B system n= (nl, n2, ••• , ni, ••• ). Let the reactants and products be denoted by rand p superscripts, respectively. A typical contribution to the over-all redox reaction is (A29). Let it have a bimolecular rate constant kmn for the forward step kmnr AmT + BnT~AmP+ BnP. (A29) The over-all second-order rate constant kab then in volves a weighted sum over the rates of all bimolecular mn contributions, per unit stoichiometric concentra tions of A T and of BT: kab= L)mnr(A m') (BnT) /L)Amr) L(Bnr), m,n m n where ( ) denotes concentration. If 7rmr and 7rnr denote the probabilities that an A r species exists as Am' and that a Br one exists as Bnr, respectively, i.e., if m n then (A23) becomes kab= Lkmnr7rm'7rn'. (A30) m,n Let F mT + F n' denote the free energy of the system containing a labeled Amr and a labeled Bnr molecule far from each other, fixed in the medium, under the prevailing conditions, Let the corresponding property be FmP+FnP when the two labeled molecules are AmP and BnP. We subdivide Fmr+FnT such that Fmr depends on the properties of Amr and its environment alone. It is therefore independent of the nature of Bnr• We note that the 7r'S can be expressed in terms of these F's, if we assume, as we do, that the complexes AmT and Bnr have an equilibrium population, exp( -FmT/kT) 7rmr= L exp( _ Fmr/kT) ' etc. (A31) In virtue of their definition these F's depend on the concentration of X/so The free energy of any reaction (A32) in the prevailing medium is in fact Fm,r-Fmr: Am'+ L(m'i-mi)X,~Am,r, i (A32) Each kmn is given by a pair of equations of the type (31), (81), where for A we write Amn and recall the additivity of A (A33) On using (A32) the !1Fo, for Step (A29) is seen to be (A34) On neglecting !1Fmn o'2/4Amn in (81) as discussed earlier one obtains (A3S) , using (A30) to (A34) : kab=ZKab!L exp{ -[wmn'+WmnP+!(Am+An) J/2kTI m,n X (7rmP7rmT7rnP7rnr)i, (A3S) where Kab is given by (A36) and is, in fact, easily demonstrated to be the formal equilibrium constant of the reaction in the given medium, expressed in terms of the stoichiometric concentrations m n Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions700 R. A. MARCUS This equilibrium constant is, by definition, L(AmP) L(BnP)/L(A m') L(Bn'). m n m 110 From (A35) one can at once derive an expression for the isotopic exchange rate constant. On considering the A redox system a typical contribution to the exchange will be (A37) when m and m' describe any two complexes. The over-all rate constant, kaa, is then obtained by multiplying kmm,' by 7rm'7rm'P and summing over all m and m'. The result is given by (A38), and is then counterpart of (A35) : kmm,r Am'+ Am,p~AmP+ Am", kaa= L kmm,'7rm'7rm'p. m,ml (A37) (A38) kaa is obtained from (A35) by noting that Kaa is unity kaa=Z L exp{ -[Wmm"+Wmm,P+HAm+Am,)]/kT} m,mI X (7rmP7rm'7rm'P7rm") 1. (A39) When the work terms can be neglected one finds kab=ZKahlL exp( -Am/4kT) (7rmP7rm') 1 m n (A41) m linearly on E, as in (74). F MP and F M' are independent of the properties of A. They depend only on those of the electrode and the electrical double-layer region (A43) where Am is independent of E. When electrochemical equilibrium exists (E equals E. then), it does so for each m. Adding to the free energy difference (A43) the mixing term, kT In(AmP)/(A m'), the result must equal zero at equilibrium. We thereby obtain from (A43) the value of each Am, (A44) Equation (A45) is finally obtained for the free-energy difference FmP+FMP- Fm'-FMr=ne(E- E.) -kTln(AmP)/(A m'). (A45) Utilizing the fact that E. is related to Eo' according to (75), where (Ox) now equals Lm(AmP) and (Red) equals Lm(Am'), (A45) can be rewritten as FmP+FMP- Fm'-FM'=ne(E-Eo') -kT In7rmP/7rm'. (A46) From (A31) and (A46) one obtains: exp[ -ne(E-Eo')/kT) = exp[ -(FMP-FM')/kT] Lm exp( -FmP/kT) X Lexp( _ Fmr /kT) (A47) m From (A40) and (A41) one then obtains kab= (kaakbbKah) 1. For the over-all electrochemical rate constant of the (A42) forward reaction in (73), kel' we have On considering next the electrochemical case, let M denote the electrode, M' describing its state before electron transfer and Mp after. As in the text we assume that the acquisition or loss of an electron by the elec trode has essentially no effect on the force constants or equilibrium bond distances in any adsorbed layer of ions or molecules. (To be sure, one or more electrons on the electrode may be fairly localized when the reacting species is near it, and this number changes when the species gains or loses electrons.) We regard different compositions of the adsorbed layer as corre sponding to different domains of the coordinates in many-dimensional space. The free energy of a system having a labeled Am' molecule far from the electrode and fixed in position is written as Fm'+FM', the corresponding term when the molecule is AmP (and the electrode has lost n electrons thereby) is FmP+FMP. The free energy of Reaction (73) for the case where the reactant is Am' is then given by (A43) , since the translational contribution for Am cancels in computing FmP-Fm'. The change depends (A48) where kmr is the rate constant for (Am') going to (AmP) at the given E. For each m, the km' is given by an equation analogous to (82), with ne(E-Eo') replaced by ne(E-Eo') -kT In7rmP/7rmr [compare Eqs. (77) and (A46)]. One then obtains kel=Zel exp[ -ne(E-Eo')/2kT] XL exp[ -(Wm'+WmP+!Am)/2kT](7rmP7rm') 1. m (A49) The work terms naturally depend on E. When they can be neglected one has kel = Zel exp[ -ne(E- Eo') /2kT] XL exp( -Am/4kT) (7rmP7rmr) 1. (A50) m In the light of Eqs. (A40) to (A42) , (A49), and (A50) , we see that the correlations (a) to (f) in Sec. 18 Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsELECTRON-TRANSFER REACTIONS. VI 701 still hold, even when applied to over-all rate constants but, as one sees from (A42) , (a) is now restricted to the region of chemical transfer coefficient equal to ! [i.e., tof .. · .... 1 in (96)]. APPENDIX IX. ALTERNATIVE DERIVATION OF (13) As we have seen in the text, the configurational distribution of the V'i and V'o coordinates in the activated complex is not one which is appropriate to Surface R nor one appropriate to Surface P. That is, it is appropriate to neither electronic structure (the initial or final one) of a reacting species. Cognizance of this nonequilibrium distribution of solvent molecules was taken in Part I, using a dielectric continuum treatment of systems possessing nonequilibrium dielec tric polarization. An expression for the free energy of a system with arbitrary polarization was minimized, subject to an energy equation of constraint, the di electric continuum counterpart of (20). In this Appen dix we show that this method, formulated now in terms of statistical mechanics yields the same result as the method used in Appendix II. The configurational contribution to free energy of a nonequilibrium system described by a potential energy Ur and a distribution function 1, where 1 is to be THE JOURNAL OF CHEMICAL PHYSICS determined, is given by (AS1) to an additive constant Fnon= 11urdV'+kT 11 InjdV'. (AS1) Minimizing (AS1) subject to the energy equation of constraint (AS2) and to CAS3) , Icur-UP)1dV=O, CAS2) !1dV=1, (AS3) we obtain CAS4) , where a and m are Lagrangian multipliers: 1 (ur+m(ur- Up) +kT InJ+a) 5jdV'=0. (AS4) Setting the coefficient of 51 equal to zero, and evaluating a from (AS2) we find j= exp( -~;) / 1 exp( -~;)dV" where U equals ur+mCur-up). This equation was also obtained by the method in Appendix II. Once again, m is determined by the energy condition (AS2). VOLUME 43, NUMBER 2 15 JULY 1965 Current Oscillations in Solid Polystyrene and Polystyrene Solutions* A. WEINREB, N. ORANA, AND A. A. BRANER Department of Physics, The Hebrew University of Jerusalem, Israel (Received 19 March 1965) Application of a dc voltage across a plate of polystyrene gives rise to oscillatory currents which reach considerably high negative values. The dependence of current intensity (maximum, minimum, and plateau values) on various parameters (voltage, dimensions of samples and electrodes, nature of dissolved solute, etc., as well as repetitive use) is treated. The pattern of oscillation is found to depend on all these parameters, too. The length of the oscillation period decreases very quickly with increasing voltage. It depends also very strongly on the nature and pressure of the surrounding gas. 1. INTRODUCTION IN trying to measure the extremely low dark con ductivity of polystyrene we found a prohibitively strong influence of air on the measured intensities of the currents. The variation in current intensity which usually follows any mechanical handling of a plastic was also found to be strongly influenced by the presence of air. In order to avoid the effect of air the "chamber" * Performed under the auspices of the U. S. Atomic Energy Commission, Contract NYO-2949-6. which houses the investigated specimen was evacuated. Upon evacuation the following effect was observed: The current oscillates with a definite pattern reaching high negative values although a dc voltage is applied. I The period of oscillation as well as its pattern depends strongly on the voltage. It depends also strongly on the nature and pressure of the surrounding gas. These oscillations present a serious obstacle in measuring 1 A. Weinreb, N. Ohana, and A. A. Braner, Phys. Letters 10, 278 (1964). Downloaded 17 Jun 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
1.1714512.pdf	Preparation and some Properties of Mg2Ge Single Crystals and of Mg2Ge pn Junctions H. Kroemer, G. F. Day, R. D. Fairman, and J. Kinoshita Citation: Journal of Applied Physics 36, 2461 (1965); doi: 10.1063/1.1714512 View online: http://dx.doi.org/10.1063/1.1714512 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/36/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Silicon fiber with p-n junction Appl. Phys. Lett. 105, 122110 (2014); 10.1063/1.4895661 The physics and backward diode behavior of heavily doped single layer MoS2 based p-n junctions Appl. Phys. Lett. 102, 093104 (2013); 10.1063/1.4794802 The peculiar transport properties in p-n junctions of doped graphene nanoribbons J. Appl. Phys. 110, 013718 (2011); 10.1063/1.3605489 Optoelectronic properties of p-n and p-i-n heterojunction devices prepared by electrodeposition of n- ZnO on p-Si J. Appl. Phys. 108, 094502 (2010); 10.1063/1.3490622 Currentvoltage characteristics of amorphous silicon PN junctions J. Appl. Phys. 51, 4287 (1980); 10.1063/1.328246 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:14JOURNAL OF APPLIED PHYSICS VOLUME 36, NUMBER 8 AUGUST 1965 Preparation and some Properties of Mg2Ge Single Crystals and of Mg2Ge p-n Junctions* H. KROEMER, G. F. DAY, R. D. FAIRMAN, AND J. KINOSHITA Central Research Laboratory, Varian Associates, Palo Alto, California A technique was developed for the preparation of high-quality sing.le cryst.als of Mg:Ge wit? co~tr?lled dopings. It utilizes the reaction of stoichiometric amounts of the constituents m a graphite cru?Ible ms~de a hermetically sealed tantalum bomb followed by the solidification of the molten compound mto a smgle crystal by lowering the bomb thr~ugh a temperature gradient (Bridg~an technique). S.everal cryst~ls without intentionally added dopants were grown as well as crystals to whIch the donors antimony, arsemc, boron, the acceptor sodium, or the apparently insoluble element ~rani.um, had been added. . .. A technique was also developed for the preparation of p-n JunctIOns o~ n-t~pe Mg2Ge, U~IlIZI~g the alloying to and diffusing into the Mg2Ge of a thin evaporated gold film. p-n JunctIOns prepared m thIS way exhibit undesirably large reverse currents, but high breakdown field strength and very prono~nced su~ace passivation properties, which might make Mg2Ge a desirable material for MOS (metal-oxlde-semlcon ductor) devices. 1. INTRODUCTION AND SUMMARY IT is, unquestionably, desirable to increase the ~umber of useful semiconductors beyond those now muse. Among the different and essentially unexplored possi bilities, the Mg2X1V semiconductors and particularly Mg2Ge have one of the more promising combinations of known properties. The Mg2XIV compounds are simple binary com pounds, forming directly, rather than perit:ctically, by crystallization from a melt that. has a ~ery slmp~e ph~se diagram.l The lattice structure IS the SImple antlfiuonte structure containing only one formula unit (= 3 atoms) per primitive cell, a de~irable feat~re f?r.l~ttices w~th high degrees of perfectIOn, and hIgh ngId~ ty (= hl~h mobility). The lattice structure and the mteratomlC spacing suggest that the binding forces are largely covalent 2,3 a necessity for good stoichiometry, low effective'masses, and low amplitudes of the lattice vibrations. The lattice structure has a high degree of symmetry and a center of inversion. This should lead to vanishing piezoelectricity, easier technology, the possibility of degenerate bands with negative masses, and other interesting features. Being a compound, Mg2Ge was considered likely to have stronger optical phonon scattering and, there fore, a higher avalanche threshold, than elemental semiconductors. The energy gaps and the electron mobilities are about 0.7 eV and 400 cm2 V-I sec' for both Mg2Si4 and Mg2Ge.5 These values should rule out these ~ubstances for bipolar transistors, but hardly f.or any~hmg e.lse. Perhaps the most significant fact IS that It had m the past been possible to prepare single crystals with as few * This work was sponsored by the U. S. Air Force under Contract No. AF33 (657)-11015. . 1 M. Hansen, Constitution of Binary Alloys (McGraw-HIll Book Company, Inc., New York, 1958). 2 H Krebs Acta Cryst. 9,95 (1956). 3 H: Welk~r, Ergeb. Exakt. Naturw. 29, 280 ~1956). 4 R. G. Morris, R. D. Redin, and G. C. Damelson, Phys. Rev. 109, 1909 (1958). • See Ref. 4, p. 1916. as 3X 10'5 electrons per cc with little effort compared to that required to give similar purities in most III-V com pounds. S That such low carrier concentratio~s are obtained means, of course, that the compound IS not doped by its constituents, at least not to an extent larger than this (chemically speaking) very low number. It is nevertheless important to have a melt of good , , stoichiometry; otherwise small inclusions are formed that consist of the eutectic between the compound and whatever constituent is in excess.6 These earlier crystals had been obtained by a simple Bridgman technique. It is not possible to grow them i~ a conventional Czochralski furnace because of the hIgh vapor pressure of magnesium and the high reactivity of its vapor. The exact magnitude of this vapor pressure does not seem to be known for Mg2Ge, although it is known for Mg2SF and Mg2Sn.8 For Mg2Si at its mp (1102°C), the dissociation pressure is about 130 Tor~. The pressure of elemental Mg at this temperature IS about 700 Torr; therefore, the activity of Mg in Mg2Si must be about 0.2. Since the activity of Mg in Mg2Sn at its mp is even larger, about 0.6, the activi~y of M? in Mg2Ge is not likely to be smaller than that III Mg2Sl. If one assumes 0.2 also in Mg2Ge, the higher mp of the latter (1115°C) should lead to an even higher disso ciation pressure than that of Mg2Si. The purities thus achieved are probably limited by the purity of the best commercially available pure magnesium (99.99%, Dow Chemical Company) that has been used for these crystals. Zone refining of mag nesium should improve the over-all purity. Of those elements that had been studied by the earlier authors,6 the third-column elements, aluminum, gallium, indium, thallium, and scandium, had been found to act as donors rather than as acceptors. Apparently, they enter the magnesium rather than the germanium sublattice. The first-column elements, copper, silver,S and gold, act as acceptors. Arsenic, and probably the other fifth-column 6 G. C. Danielson (private communicatioI?-)' 7 K. Grjotheim, O. Herstad, S. P~truCCl, R .. Skarbo, and J. Toguri, Rev. Chim. Acad. Rep. Populaire Roumame 7, 217 (1962). 8 S. Ashtakala and L. M. Pidgeon, Can. J. Chern. 40,718 (1962). 2461 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:142462 KROEMER ET AL. elements as well, act as donors, as one might expect. Iron also seems to be a donor. Apart from these technological data, several physical properties are known5 which need not concern us at this time. However, no p-n junction studies appear to have been performed in Mg2Ge or any of the other Mg2X1V compounds. To perform such a study was one of the main objectives of the present work. In order to make such a study meaningful, single crystals of the highest quality had first to be prepared. As a result of this study, it can be stated that high quality Mg2Ge is easily grown and that it is quite capable of producing well-behaved p-n junctions. These junctions show larger reverse currents than those of germanium. This is undesirable and should rule out Mg2Ge as a competitor for germanium for all of those applications where a low reverse current is important. However, because of factors beyond our control, it was not possible to continue this study to a point where it could be determined whether these high currents are fundamental to Mg2Ge or whether they are due either to the diffusion technique employed in the production of the junctions or to uncontrolled impurities in the Mg2Ge. On the positive side of the ledger, the resulting diodes showed internal breakdown field strengths of the order 300 000 V / cm, which is as high as the Zener field strength in narrow germanium junctions and much higher than the avalanche field strength in wide ger manium junctions. Also, the diodes are readily passi vated by a natural surface oxide layer which appears to be so dense and stable as to make Mg2Ge a possible candidate for insulated-gate field-effect transistors. Finally, the diodes appear to be fast, although the evidence for this is somewhat incomplete. 2. MAGNESIUM GERMANIDE PREPARATION 2.1. Bomb Design Our initial attempts at producing Mg2Ge crystals were essentially by the same technique as that used by the group at Iowa State University.5.6 Stoichiometric amounts of the constituents were reacted above the mp of the compound in a graphite bomb with a conical bottom and a tight-fitting tapered graphite plug at the top. This bomb was then lowered through a tempera ture gradient to permit controlled solidification of the ingot. We encountered substantial magnesium losses in all cases, apparently by diffusion through the graphite. The magnesium loss could be minimized by exceeding the mp as little as possible, but it could never be avoided. By going to growing rates below 1 cm/h, single crystals could be achieved in spite of the mag nesium loss, which then showed up as a layer of Ge Mg2Ge eutectic on top of the crystal. The different thermal expansion coefficient of this layer often led to severe cracking of the ingot. This, in turn, could be avoided by using an initial magnesium excess but, basically, the use of graphite bombs remained un satisfactory. Therefore, they were finally abandoned in favor of a graphite crucible inside a welded metal container. Such a setup had been used by LaBotz and Mason,9 who used a stainless steel bomb. We used 99.9% pure tantalum instead, because it appeared to be a material less likely to contribute contaminants. It is relatively free of contaminants itself; it has an ex ceedingly low vapor pressure, and a thermodynamic analysis showed that the total vapor pressure of its inevitable surface oxides would also be, at most, 10-10 Torr, which is completely negligible. The bomb design that finally evolved, and that was completely satis factory, is shown in Fig. 1. This bomb may be evacuated and baked out prior to a dry-argon backfill. After that, it is closed by pinching off the !-in. section and by heliarc welding. The outer walls of the graphite liner are relieved in order that minor leakage of the melt through pores in the graphite cannot feed tantalum contamina tion back into the melt. A graphite lid on the liner is used to reduce the possibility of contamination of the components during welding or bakeout. Several runs were made with this bomb design, all of which led to single crystals. The loading procedure is as follows: 99.99% pure sublimed magnesium (Dow Chemical Company) is etched in IN HN0 3, rinsed in three portions of distilled water and dried with a methanol rinse, followed by a drying period in vacuum. Intrinsic polycrystalline germanium is etched in CP-4, rinsed three times in distilled water, and dried in the same manner as the magnesium. The carefully dried materials are weighed out to the exact stoichiometric weight ±O.2 mg. A calculated excess of magnesium is used to correct for the vapor in the dead space above the crystal. The bomb is then evacuated, baked, backfilled with argon, and sealed as mentioned above. For all but the first few runs, a conventional Bridg man-type furnace setup was used. The temperature set-point on the controller was determined by raising the temperature until the bottom of the bomb read 1140°C by optical pyrometry. The growing procedure which was finally developed is as follows. The bomb is placed in the uniform-tem perature zone of the furnace and the system is sealed and purged with dry argon. The bomb is then rapidly heated to about 650°C and thereafter slowly from 650°C to about 900°C, over a period of 3 h. The reaction of Mg with Ge is exothermic and can cause spattering of the charge, as well as reaction with the crucible, if not properly controlled. This slow heating is in contrast to the procedure of LaBotz and Mason,9 who heated the bomb as quickly as possible. After 900°C has been reached, the full power is applied, which heats the bomb to 1140°C over a period of 2 h. The bomb is then held at 1140°C for 1 h. At this stage, the bomb equilibrates at a 9 R. J. LaBotz and D. R. M'lson, J. Electrochem. Soc. 110, 121 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:14Mg2Ge SINGLE CRYSTALS AND p-n JUNCTIONS 2463 temperature above the mp of Mg2Ge to assure that the entire mass is fluid and homogeneous. With the furnace held at 1140°C, the bomb is then lowered through the thermal gradient near the lower end of the furnace. In the region of the Mg2Ge freezing isotherm, this gradient is gOC/cm. A dropping rate of about 0.75 cm/h is used to produce single-crystal boules. After the bomb has passed through the thermal gradient, the furnace is cooled slowly to room temperature. Initially, the cool down was achieved by manually turning down the furnace; later on, it was controlled by a program regula tor, to a cooldown period as long as 24 h, to reduce the thermal shock. 2.2. Results 2.2.1. Crystallographic Perfection A total of 25 crystals were grown. The first 13 of these were exploratory runs, to work out the details of the process. They ranged from being complete failures to being single crystalline. We discuss here only the remaining 12 crystals. The majority of these were single crystals. However, their crystallographic perfection was not the same as that of, say, a high-quality germanium crystal. In particular, all of the crystals contained a few very small inclusions and some mosaic structure. The inclusions consisted of small graphite particles, a few microns in diameter, which were apparently due to abrasion from the crucible during loading. In the later runs, their number and size were greatly reduced by careful crucible loading procedures, to the extent that they represented at most 1 ppm of the total crystal volume. They could probably be avoided completely if vitreous carbon crucibles were used; but in any event their presence appears to be of little consequence at the present stage of refinement of this work. The mosaic structure appears as individual blocks, of millimeter dimensions or larger, that show a small random deviation from the average orientation. The deviation is so small that it does not affect the cleaving of the crystal, thereby giving the appearance of a perfect single crystal. Only upon closer inspection does it show up as a slight waviness of the cleavage surface; over sufficiently long distances the cleavage plane appears perfectly even. An x-ray study showed that the average fluctuations of the orientations about their mean value are of the order of 1 0. One run, No. 24, showed some signs of segregation of added arsenic at some of the small-angle grain bound aries. Upon etching with bromine-methanol (see Sec. 2.2.4.), the mosaic structure was revealed as if some arsenic had precipitated along the low-angle grain boundaries. This etch has not revealed the mosaic structure on other crystals of Mg2Ge. This mosaic structure is probably due to polygoniza tion, i.e., to the lining-up of originally randomly dis tributed dislocations into dislocation walls, during the 0.034" Sheet Graphite LId 0.50" 00, 0.020" Wall Tantalum Tube Welds I. 00" 00, 0.020" Wall Tantalum Tube Tolerances: leave 0.010" total between Crucible and Tantalum Can. Make end caps to fit snugly into the 1" and 112" tubing FIG. 1. Tantalum bomb for Mg2Ge. slow annealing of the crystal.lO The Burgers vectors in such walls usually do not average out, leading to a small-angle grain boundary. This is a very common phenomenon in crystals grown by the Bridgman tech nique, as opposed to the Czochralski technique, because the inevitable wall strains of the former lead to large dislocation densities. Because of the fairly large block size, individual p-n junctions rarely lie across a small-angle grain boundary (which might not matter, anyway); and, as the cleavage experience shows, the mosaic structure certainly does not prevent the selection of a desired crystal orientation, unless the desired accuracy is better than about 1°. For these reasons, the mosaic structure appears to be of little consequence at the present stage of refinement of this work. If, ultimately, more perfect single crystals should be required, a suitably modified Czochralski technique might have to be used. Because of the mag nesium partial pressure, it would have to be a hot-wall technique, as for gallium arsenide; but because of the reactivity of magnesium vapors, the walls would have to be made of some material other than quartz or ceramic, such as graphite, tantalum, etc. 2.2.2. Crystal Stability Mg2Ge is attacked by water, including, under certain circumstances, the water vapor in the air. The litera ture abounds with remarks about the aerial instability of Mg2Ge. Some of these statements are quite exag gerated, such as: "Atmospheric oxidation is limited to surface layers on Mg2Si and Mg2Sn, whereas Mg2Ge breaks down to a powder." This is true only for crystals of very poor quality. They have to be kept in a desic- 10 See, e.g., R. A. Smith, Semiconductors (Cambridge University Press, New York, 1959), pp. 49-51, where further references are given. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:142464 KROEMER ET AL. cator, and even then some disintegration occurs. High quality crystals, even fairly heavily doped ones, were stable under the atmospheric conditions in our labora tory, which is not air-conditioned. Although the atmospheric stability of Mg2Ge was not studied specifically, certain incidental observations suggest that it is not only a question of purity but also, and perhaps more important, one of crystal perfection. A few of the earlier crystals disintegrated along well-defined, origin ally invisible, cracks, all apparently following the (111) cleavage planes. Since Mg2Ge cleaves extremely easily, one might expect such cracks to be initiated during cooling, if such cooling is too fast; and one might speculate whether a good deal of the instability reported in the literature is not due simply to thermal shock because of insufficiently slow cooling. Also, sawed surfaces tend to be much less stable than cleaved surfaces on the same crystal. Any attack that occurs on cleaved surfaces occurs at the cleavage steps, but does not propagate. 2.2.3. Crystal Handling Because of the attack by water on Mg2Ge crystals, they cannot be cut with a water-cooled diamond saw; therefore, a nonreacting cutting fluid has to be used, such as kerosene-based "Diala" (Shell Oil Company). It attacks or softens most mounting waxes, but in the case of "Green Pitch" (American Optical Company) the dissolution by the cutting fluid is so slow as to be negligible during the cutting. Of the epoxies, "Araldite 7072" (Ciba Corporation) can be employed as a thermo plastic material if used without hardener. For the actual cutting, the same saw, saw blades, and cutting speeds are used as for other semiconductors, such as germa nium, silicon, and GaAs. After cutting, the wafers are hand lapped on400-grit, and then on 600-grit paper, again using "Diala" as a lapping fluid. Approximately 50 J.I. are removed from each side to reduce saw damage. The wafers are then rinsed in alcohol and polished, first with a 9-J.l., then a 3-J.I. oil-based diamond paste. Thereafter, the wafers are rinsed again in alcohol and are ready for further handling. Wafers prepared in this way exhibit a high gloss and a high degree of atmospheric stability. 2.2.4. Etching Mg2Ge is reactive in aqueous acid and alkaline systems and in normal distilled water. Bromine methanol (4 drops bromine in 250 cc methanol) pro duces a bright chemical polish similar to CP-4 on germanium. The etching time varies with the purity of Mg2Ge. Very fine, damage-free surfaces that are easily wetted by molten metals can be achieved by polishing the Mg2Ge on a metallographic cloth wheel that is impregnated with a continual flow of bromine-methanol, the wheel running at fairly low speed. Apparently a very thin oxide surface is obtained by flooding the cloth with straight methyl alcohol before lifting the Mg2Ge wafer from the cloth, thus quenching the bromine activity. The composition of a post-alloy cleanup etchant for p-n junctions which has produced very stable surfaces is 200 H20 :2H2S04 :1H202, at room temperature. Etch ing time varies from 30 sec to several minutes. The etch is very reactive once it gets started; it is stopped by quenching in methyl alcohol. The etchant leaves a film of a presumably heavily oxidized surface which protects, but in no way appears to affect, the I-V characteristics of the diodes measured after such etching. No etchant was found for revealing p-n junctions and Mg2Ge regrowth layers. Sandblasting was used to delineate grain boundaries. 2.2.5. Basic Semiconductor Properties The basic semiconductor properties of the 12 runs that followed the exploratory runs are summarized in Table I, which serves as the basis for the following dis cussion of these runs. Only Nos. 16 and 18 were poly crystalline. In run No. 16 a sodium vapor bubble had produced a hollow chimney through the crystal, resulting in a coarse-grained polycrystal. Run No. 18 was no Mg2Ge run at all, but an attempt to synthesize the hypothetical cross-substituted compound Mg4InSb. No such compound seems to exist. The dopant densities given in column 2 of Table I are the amounts of impurities intentionally added, in number of atoms per cm3 of ingot. The data in columns 3-6 were obtained from Hall effect and resistivity measurements, with thermoelec tric probing used as a cross check on the sign of the Hall effect. All Hall effect measurements were taken by the Van der Pauw technique,!! the resistivity measure ments either by this technique or with a four-point probe. The carrier densities given are Hall carrier densities. The electrical data for all ingots refer to data on selected individual wafers cut from the ingots. As one might expect, there were various degrees of fluctuation within an ingot. For some of the runs we have indicated a range of resistivities over the entire ingot, excluding the extreme ends, as determined by four-point probe measurements. No comparable Hall effect range studies have been performed, but the range of Hall carrier densities might be expected to be of the same order. Several additional comments are in order concerning most of the individual runs. The first seven runs in Table I (Nos. 14-20) were still performed in graphite bombs. Since substantial magnesium losses are known to have occurred, the melt was far from stoichiometric during most of these runs. The resistivity of the four undoped runs, Nos. 14, 15, 17, and 19, was very non- 11 L. J. Van der Pauw, Philips Res. Rept. 13, 1 (1958). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:14Mg2Ge SINGLE CRYSTALS AND p-n JUNCTIONS 2465 TABLE I. Basic semiconductor properties of Mg2Ge ingots. (1) (2) (3) Run Doping added No. (cm-') Type 14 none n(p) 15 none n(p) 16 8.1 XI019Na p 17 none n(p) 18 See text 19 none n 20 3.5 XI019 Sb n 21 none n 22 3.5 XI017 Sb n 23 6XIOI'U n 24 4.8 XI018 As n 25 7.5 XlO18 B n (a) Very nonuniform. (b) Single sample only. (4) (5) (6) (7) p ,.. cm'/ n Re- !l-cm V-sec cm-3 marks <20 (a) 2.44 380 6.7 XI01' (b) <20 (a) ",I ",100 6 XIO" (c) ~20 (a) See text <20 (a) 3.4-3.7 X 10-' 175 101' (d) 0.49 320 4XI0 1• (b) (0.4-0.5) 0.046 390 3.5 XI017 (d) (0.044 -0.049) 0.73 355 2.4XIOI. (b) 0.01 300 2.2 XI018 (c) 0.018 385 9 X 1017 (c) (c) No quantitative data on uniformity. but probably uniform within ±20%. (d) See text for details on density values. uniform; at least the first three contained both n-type and p-type regions,12 separated by bands of resistivity up to 20 !J-cm. These bands are undoubtedly highly compensated and do not represent true high purities. It is not possible to single out any crystal, or portion thereof, for which it can be said that it might represent the kind of over-all impurity levels that one can achieve. Still, one of the highest Hall mobilities measured by us 380 cm2 IV-sec, was measured on a piece from run No. 14: This Hall mobility is within the limit of error of the highest val~e reported by Redin et at.5 for their sample llB-1, whIch had a Hall carrier density of about 4.2X lOt6 cm-3• It is substantially higher than the Hall mobility (""'.300) for their lowest electron density sample (7B-3), which must, therefore, be considered more heavily compensated. This indicates that this early crystal of ours compares with the best published results. No reliable Hall effect measurements exist for the heavily compensated high-resistivity bands. Run No. 16 indicates that sodium is an acceptor probably on magnesium sites, rather than an interstitiai donor. The same is probably true for the other alkali metals, with the possible expection of lithium. The low Hall density indicates that most of the sodium was lost by evaporation. The raw Hall effect data of run No. 20 indicate a Hall carrier density of only 1019 cm-3• However, at ~al~es so close to degeneracy, all donors might not be lOrn~ed, and the donor d~nsity should be even higher, possIbly equal to the denSity of added antimony atoms. Subsequent use of this very uniform ingot as a master doping alloy for run No. 22 indicates the actual • 1~ HaJI effect and thermoelectric probing often gave different mdlca~lOns as to conductivity type, indicating near-intrinsic behaVior. antimony concentration of 3.5X 1019 cm-3, as given in the table. The five remaining runs were all performed in tantalum bombs, with 99.99% pure sublimed magne sium (Dow Chemical Company) and controlled cool down. All were single crystals, essentially free of cracks and of a eutectic excess. The resistivity and mobility values of run No. 21 are disappointingly low and the carrier concentration disappointingly high compared to, say, the above-mentioned section of run No. 14. Apparently, some contamination was introduced in advertently. It is believed, though, that this can be avoided and that the tantalum bomb method is capable of producing entire ingots with the higher purity of this individual piece from run No. 14. Run No. 22 was another antimony-doped run with a lower doping. The doping was achieved by adding a piece from heavily antimony-doped run No. 20. The total antimony concentration added depends on what is assumed to be the concentration in run No. 20. On the basis of the Hall carrier concentration of run No. 20, the doping level of run No. 22 should be 1017 cm-3• on the basis of the total amount of antimony initi~lly added to run No. 20, it should be 3.5X1017 cm-3• The resistivity, determined by four-point probe measure ments, was extremely uniform over the entire 36-mm length and the entire diameter. Several Hall effect measurements were performed. Only the highest mobility set is given in Table I. These data are con sistent only with the assumption that all the antimony originally added to run No. 20 was actually incorporated into the crystal, but only part of it was ionized. The mo~ilities on other wafers were only slightly lower, typically between 344 and 365, except for the topmost wafer, with a value of only 325 cm2/V-sec. The higher mobilities than those in undoped run No. 21 are con firmation of the assumption that the latter run is not indicative of the quality that can be achieved with the tantalum bomb technique. Run No. 23 was performed in the hope that uranium might act. as a radiative recombination center, possibly even leadmg to laser action.13 However, uranium with its big atom is apparently nearly insoluble in Mg2Ge. Undissolved uranium was found at the bottom tip of the ingot. The crystal itself was very fragile, and only one wafer was measured in detail. This wafer has a better purity than the undoped run No. 21, and there is no reason to assume that even this electron density is due to dissolved uranium. Based on the limited evidence of run No. 24, arsenic seems to behave differently from antimony, at least at those high concentrations. The crystal did not cleave readily and, as stated in Sec. 2.2.1, some precipitation, apparently of arsenic, was found. These two observa tions could be explained by the assumption that the solubility of arsenic in Mg2Ge is retrograde, going below 13 R. L Bell, J. Appl. Phys_ 34, 1563 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:142466 KROEMER ET AL. the initially added concentration at lower temperatures at which noticeable diffusion can still take place. The last run, No. 25, was performed in the hope that boron might act as an acceptor, rather than as a donor, as aluminum, gallium, and indium do. The latter three elements apparently occupy magnesium sites rather than germanium sites in the Mg2Ge lattice. Both the electronegativity14 (2.0) and the tetrahedral covalent radius (0.88 A) of boron are not only closer to the values for germanium (1.8; 1.22 A) than to those for magnesium (1.2; 1.40 A), but they lie on that side of the germanium values which is opposite to the side on which the magnesium values lie. This situation is different from that for the other third-column elements; boron should, therefore, have a much stronger tendency to occupy germanium positions, as an acceptor, than those other elements. As it turned out, boron is also a donor. The ingot, again, was quite uniform. The electron density values are substantially below the amount of boron added, probably because a substantial fraction had reacted with the graphite crucible, but possibly because of incomplete ionization. One interesting fact about all the runs made in tantalum bombs is their high doping uniformity. This is rather different from the behavior of germanium. It could be explained in at least three ways: (a) The segregation coefficients of all these impurities may be close to unity. This is unlikely. (b) The solubility limit has been reached. However, at least for the most uni form run of all, run No. 22, this cannot be true. (c) Impurity equilibration by diffusion. This would require very large diffusion constants of the order of 10-4 cm2/sec. In germanium, the diffusion constants of most impurities are not nearly this large near the mp. As is shown in Sec. 3.1., the diffusion constant of gold is much larger in Mg2Ge than in Ge; possibly, the same is true for other impurities. 2.2.6. Miscellaneous Properties Photoconductivity measurements have been made on thin slices of material from run No. 15, using a short-duration flash lamp. The conductivity of the samples decayed exponentially to its dark value with a characteristic time of approximately 6 JJ.sec at 300oK. Lowering the sample temperature increased the photo conductive lifetime and greatly enhanced the photo conductance. This indicates that the measured lifetime is a trapping time and not a true minority-carrier lifetime. An attempt was made to observe the Gunn effect15 in a long (",7X lX 1 mm) sample cut from run No. 19, using pulse equipment which worked successfully with n-type GaAs. No current instabilities were observed up to the maximum applied field of 3000 V / cm. 14 L. Pauling, The N atuTe of the Chemical Bond (Cornell Univer sity Press, Ithaca, New York, 1960), 3rd ed., Tables 3-8 and 7-13. 16 J. B. Gunn, IBM J. Res. Develop. 8, 141 (1964). 3. p-n JUNCTIONS 3.1. Junction Preparation We have prepared numerous p--n-junction diodes in Mg2Ge, using several crystals and various techniques. The best diodes were obtained from run No. 22, a single crystal, n-type doped with antimony to a level of 3.5X1017 cm-B• The best technique was one of diffusion into the crystal of evaporated copper, silver, and gold. Most of the work was done with gold, which is particularly convenient, and we here discuss only gold-evaporated diodes, made on run No. 22. This technique, as it finally evolved, consists of the following steps. Wafers parallel to a (111) plane and of a convenient thickness, typically 0.5 mm, are cut or cleaved from an n-type Mg2Ge single crystal. The wafers are lapped and mechanically polished as described in Sec. 2.2.3., and chemically polished with quenched bromine-methanol as described in Sec. 2.2.4. After a final methanol rinse the wafers are immediately transferred into a vacuum evaporator, where gold is evaporated onto them. The evaporator contained an electron-beam heater, and electromagnetic shutter (meter movement), and a sample heating filament. After evacuating to 10-6 Torr, the sample heater is gradually turned on to bake out the sample for a few minutes prior to evaporation; the heater is left on during the evaporation itself. The sample heater power is adjusted empirically to a value below that which would cause evaporated gold to alloy to a clean wafer of Mg2Ge, i.e., to below about 335°C sample temperature. The electron-beam voltage is then gradually turned on to a value that causes a deposit of about 1 JJ. of gold in about 2 min. As soon as the gold source has reached its full temperature, the shutter is opened. Usually, the evaporation is terminated by the exhaustion of the gold, the amount of which is chosen to result in an evaporation thickness of about 1 JJ.. The gold is alloyed and diffused into the Mg2Ge in a tube furnace with a purified hydrogen atmosphere. It was found that alloying at 600°C for 15 min produced the best diodes, although neither of these numbers is critical. The furnace is preheated to this temperature, and the sample heating is timed by moving the sample into and out of the hot furnace, inside the hydrogen gas stream. After alloying, the wafers are grooved with a diamond saw into small squares, typically 20{}-250 JJ. across, to produce a "waftle-iron" array of many small individual mesa diodes on a common base. The wafers are then etched in the sulfuric-peroxide etch (Sec. 2.2.4.) for a few minutes. The minimum depth of grooving necessary to produce electrically separate diodes is 100 JJ., with subsequent etching times of 3--4 min. The etching re moves the saw damage and produces a final groove about 125 JJ. deep. The final mesa size is also reduced by the etching to about 10{}-150 JJ. across. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:14Mg2Ge SINGLE CRYSTALS AND p-n JUNCTIONS 2467 These p-n junctions are undoubtedly diffused junc tions, apparently about 100 fJ. deep. Combined with a 1S-min diffusion time, this indicates a very large diffu sion constant of about 10-8 cm2jsec at 600°C. Diodes have also been made with evaporated copper layers, rather than with gold. These diodes have prop erties quite similar to those of the gold diodes, with somewhat higher reverse currents. 3.2. Basic Electrical Properties 3.2.1. General Various electrical properties were measured for a large number of diodes. These were quite variable, particularly with respect to their reverse currents. Current-voltage oscillograms were regularly taken, and for the better diodes, i.e., the ones with lower reverse currents, they were often evaluated quantitatively. Some of the better diodes were subsequently encap sulated in a standard microwave diode package, and reverse bias capacitance measurements were performed on them. We present here a complete set of data, i.e., forward and reverse current, and reverse capacitance, for one of the diodes. 3.2.2. Forward Current The forward current-voltage characteristics of many of the diodes could be fitted to the diode equation I=I 8{exp[q(V-IR)jnkTJ-1}, (1) where R is the bulk series resistance of the diode (assumed constant), and n is some number between 10 1-------+'----- n -1.27 Is' 0.01 mA R·6.83n Diode No. 172a1 Area 3.8 x 104 cm2 V-IR- 0.01 '--_-'--_-'--_-'-_---L.._--,L-_---,'-,_--' a 0.1 0.2 0.3 0.4 0., 0.6 0.7 Voll FIG. 2. Room-temperature forward current of diode No. 172a-1. i!! i!l. Diode No. l72a-1 Area 3. 8 x 10-4 cm2 Ohmic 50 kn ! lOOI-----------+----~_r---1 :;; Volt 3 r·63\1Ax V V I Volt v- FIG. 3. Room-temperature reverse current of diode No. 172a-1. 1 and 2. During the early phases of this work those diodes that could be fitted at all to this equation had their best fit for n values very close to 2.0, and the fit obtainable with exactly 2.0 was always very close to the best obtainable fit. Since there is a simple theory16 for n= 2.0 based on space-charge layer recombination, this observation was considered proof for the applic ability of this theory. However, during the later phases of this work, an increasingly large number of diodes were encountered that required smaller n values. Figure 2 shows a diode that can be described best by setting n= 1.27. We are not aware of any technological differ ences between the diodes that might account for the difference,17 nor does there seem to be a good theoretical explanation for such an excellent fit to an n= 1.27 curve over such a wide range. This n value is the lowest that was observed. Other properties for this particular diode are reported below. 3.2.3. Reverse Current and Capacitance The reverse currents are one of the most variable properties of these diodes. They were not only large, but also depended on the reverse bias. Both facts suggest that the lifetimes must be very short and that the majority of the current must be due to carrier generation inside the space-charge regions, as is independently suggested by the forward current behavior. In such a case the reverse current of a presumably linearly graded diffused junction should be proportional to the cube root of the voltage, and the reverse current at 1 V should double at 8 V. This 1-V current and the current-doubling voltage were consistently measured. The majority of 16 C. T. Sah, R. N. Noyce, and W. Shockley, Proc. IRE 45, 1228 (1957). 17 This particular diode was alloyed for 30 min rather than 15 min, but lower n values were observed for IS-min diodes as well. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:142468 KROEMER ET AL. • FrtshlyAs PlCkIg.t • following Day ' . . . 3 IVo" -v-C -14.SSl#l ,oL------t-----l--~ ........ ~___1----__d DiodeNo.lna-i Area3.811O- 4tm2 Volt v- FIG. 4. Reverse bias capacitance of diode No. 172a-I. the doubling voltages seem to be on the low side, but there were several diodes around or above 8 V. These, then must be considered as the diodes most nearly , nondefective. Figures 3 and 4 show the reverse-current and junction-capacitance behavior of the selected diode, which is quite typical. It is seen that both the current and the capacitance follow a cube-root law, as expected. However, the capacitance (but not the current) should level out at the value corresponding to the built-in voltage. Instead, it actually increases above the theoretical straight line. This excess capacitance was found not be be stable with time. It is very unlikely that it is due to the diffusion capacitance of free minority carriers. There seems to be no way to fit the experimental curve to this assumption with physically reasonable parameters; thus, the ques tion of the nature of this excess capacitance must be left open. In view of this discrepancy the excellent fit to a cube-root law above a few tenths of 1 V is, perhaps, surprising, but this feature was shared by other diodes as well. Over the more restricted voltage range, 0.7-11 V an even better fit could be obtained with an exponent of -0.30 rather than -0.33, but it is doubtful whether this small change has any physical significance. In the most important voltage range above 1 V the data certainly cannot be fitted better by an abrupt-junction square-root law or a law between a square-root law and a cube-root law. Under these circumstances, we propose to accept the cube-root behavior above 1 V as phys ically significant and as evidence for a linearly graded junction, and to consider the time-dependent excess capacitance as a superimposed separate phenomenon of unknown origin. The positive deviation of the current at higher voltages is believed to be surface leakage. It might be avoidable by a better post alloy surface treatment. No such treatment was encountered during a limited search for it. That portion of the current that fits the cube-root portion of the characteristic is strongly tem-perature dependent. No quantitative determination of this dependence was made. The "excess current" essentially persists down to liquid-nitrogen tempera tures. All diodes exhibited this excess current, although in a few it did not set in until higher voltages had been reached than those in Fig. 2. 3.3. Interpretation There is little quantitative information that can be deduced from the forward characteristics. The n values larger than unity indicate space-charge layer recombina tion!6; the reason for their deviation from the value two is not known. The main source of a quantitative interpretation is the capacitance measurements. The capacitance per unit area of the diode of Fig. 2 can be expressed as C/ A =3.8X lO+4X (1 V /V)i(pF /cm2). Using a value of 20 for the static dielectric constant of Mg2Ge,18 the space-charge layer width becomes, from this, w=4.6X 10-5 X (V /1 V)t em, independent of any assumptions about impurity dis tributions. If the junction is a linearly graded junction, the maximum field strength occurs at the center, and is given by E=!(V/w)=3.3X1Q4X (V/1 V)f(V/cm). The breakdown of this diode occurred somewhere above 20 V; at 20 V, the field strength would be about 2.4X 105 V /cm. This value is of the same magnitude as the field strength at which Zener breakdown occurs in narrow germanium junctions (2.75 X 105 V / em) and substantially above the variable field strengths at which avalanche breakdown occurs in wide germanium junc tions,19 thus confirming the theoretical speculations to this effect made at the beginning. It is not known whether this breakdown is a Zener or an avalanche breakdown. Again assuming a linearly graded junction, the acceptor density gradient can be computed from the junction width dNA/dx= (12e/q)(V/w)!:~1.4X102! cm---4. This is a very steep gradient, considering that the junction must be located at least 100 fJ. deep. If one assumes that the gold diffusion follows a complementary error function, one is led to impossibly large values of the gold surface concentrations. This indicates that gold cannot obey a complementary error function distribu tion but must diffuse by some anomalous mechanism, such as has been encountered in GaAs and GaP.20 18 D. McWilliams and L. C. Davis (private communication). 1. D. R. Muss and R. F. Greene, J. App!. Phys. 29,1534 (19581' 20 For complete references, see the most recent paper on thIS topic: L. L. Chang and G. L. Pearson, J. App!. Phys. 35, 374 (1964). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:14Mg2Ge SINGLE CRYSTALS AND p-n JUNCTIONS 2469 The space-charge layer generation rate can be com puted from the capacitance and the reverse current data. The reverse current density for the diode of Fig. 2 can be written as j=1.7X1Q- 2X(V/l V)1(A/cm 2). The resulting generation rate is g= jjqw=2.3X1022 cm-3 sec1• This is a very large rate. If one assumes that it is produced by a single energy level in the forbidden gap and that the Hall-Shockley-Read theory21.22 is applic able to the process, one should obtain g=nl/(nTp+PTn)' Here, T p and Tn are the lifetimes for holes and electrons for sufficiently heavy doping, and n* and p* are related to the location of the recombination level. An upper limit for the lifetimes can be estimated by assuming a recombination level located at the intrinsic Fermi level. In this case, n=p=ni' And, if one also assumes Tp=Tn=T, one obtains g=n;j2T. Unfortunately, the intrinsic carrier density in Mg2Ge is not known with any accuracy. Redin et al.5 discuss the intrinsic behavior of several crystals above room temperature. A linear extrapolation to 3000K of their Fig. 7 leads to an intrinsic density of 7X 1014 cm-3• Inserted into the above equation, this leads to an upper limit for the minority carrier lifetimes of about 15 nsec. For a recombination level separated from the intrinsic Fermi level by an energy 5E, this value would have to be divided by cosh (oE/kT). 3.4. Diode Transient Recovery This short lifetime postulated is in agreement with the observation that the diodes are not noticeably photosensitive. It also suggests that the long photo conductive decay times reported in Sec. 2.2.6. are indeed trapping times, as was already suggested by the observation that the photoconductive decay times increase with decreasing temperature, while recombina tion times should decrease.21.22 Such short lifetimes cannot be measured by the common techniques for measuring bulk lifetimes. There fore, some diode recovery tests were performed on some different diode, No. 149a, using the Tektronix Type-S Diode Recovery Plug-In unit. For this diode, an analysis of capacitance and current vs voltage data along the same lines as shown above for diode 172a-1 indicated an upper limit for the lifetime of 9 nsec. The turn-on characteristic of this diode was different from that of conventional diodes that exhibit significant minority carrier storage. Within the limitations of the 21 R. N. Hall, Phys. Rev. 87,387 (1952). 22 W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952). equipment (12 nsec, 20 mA), there was no high impedance transient in this diode. In fact, the diode actually showed a low-impedance transient, the duration of which decreased with increasing forward current. At 1-mA forward current the full forward voltage was approached about exponentially, with a time constant of about 150 nsec. At 2 mA, the time constant halved; it decreased further at higher forward currents, although possibly somewhat more slowly than exactly inversely proportional to the current. This is the behavior one would expect if a fixed number of traps had to be filled, the number being 150X1Q-12 C/q=109, or about 4 X 1012/ cm2 of junction area. The turn-off measurements were not as well defined; in that case, the traps are emptying at their own rate, and the resulting current cannot be separated from the normal reverse current, which also changes with time because the voltage does and because the reverse characteristics of this diode are fairly soft. At any rate, the turn-off data are compatible with the interpretation of the turn-on data. If one assumes that all of the traps are located within the space-charge layer, their volume density must be equal to 4X 1012 cm-2, divided by the space-charge layer width around zero bias. If one assumes a built-in potential of about 0.5 V, one obtains a trap density of a little over 1017 cm-3• The nature of these traps, and whether or not they are related to the doping impurities is, of course, unknown. Since the trap density does not exceed the doping density it certainly cannot be con sidered unreasonably high. This trap density is also compatible with the low lifetime estimate. If one makes the ad hoc assumption of a capture cross section of the order 10-15 cm2, a density of 1017 cm-3 would lead to a lifetime around 1 nsec, entirely compatible with the rather indirect estimate (::::;9 nsec) from the diode characteristics. 3.5. Passivation It was pointed out in Sec. 2.2.3. that Mg2Ge is attacked by water, including under some circumstances, the water vapor in the air. Because of this, an important question from the beginning of this work was what influence this might have on the stability of the junction characteristics. The surprising answer was that they are very stable. Diodes that come out of the alloying furnace, and that have not yet been etched, often already have a rectification ratio of over 100:1. Both before and after etching, the characteristics of the com pletely exposed junctions do not fluctuate appreciably. Breathing down on the diodes has no noticeable effect. Exposure to room air for many hours, even days, has not changed the diode characteristics in any obvious way, even though the crystal itself may start to discolor in places. Perhaps the most amazing observation is that a junction which had been cross sectioned and biased for delineation by selective plating retained a good rectify- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:142470 KROEMER ET AL. ing characteristic, even under the plating bath. It should be noted that all of these are qualitative observations. Quantitative measurements would have very probably revealed some changes in all of these cases, but there is no question about the ruggedness of Mg2Ge. It is believed that a classical chemical passivation situation exists here, due to the magnesium in the Mg2Ge. It is well known that pure magnesium, like aluminum, can be passivated very readily and this property seems to extend to Mg2Ge. This is further supported by the observation that the sulfuric acid peroxide etch has a definite incubation time, after which vigorous action begins; but after removal from the etch, the wafer passivates again, which is a typical behavior for passive metals. A passivated surface is much more stable in air than a freshly cleaved one; the conflicting reports about the stability of Mg2Ge may simply reflect different degrees of surface passivation. The exact mechanism of the passivation was not studied; therefore, it remains unknown. The simplest possible assumption is that the passivating film is essentially magnesium oxide MgO with the germanium having gone into solution as Ge02. If one makes this assumption, it is interesting to speculate about how such MgO films might compare with the Si02 films that form on silicon. It is the ease of formation and the stability of the latter films that are responsible more than any other single fact for the advancement of the silicon device technology over the germanium device technology, including such developments as silicon integrated circuits and insulated-gate field-effect tran sistors. Only recently has it become apparent that these. Si02 films are far from being perfect. For example, they contain large densities of traps that slowly exchange electrons with the silicon, causing substantial drift and hysteresis, especially in MOS devices, such as insulated gate field-effect transistors.23 In view of the fact that the Si02 films are actually glasses with a wide-open pseudolattice, these defects are not surprising at all. In this respect MgO is very different from Si02. It crystallizes in a sodium chloride lattice with a density of 3.58, substantially more than the sum of the densities of magnesium (1.74) and of solid oxygen (1.43)24 and even more than twice the 23 For a complete survey and further references see the papers in the September, 1964 issue of IBM J. Res. Develop. 24 We owe this comparison to Dr. R. L. Longini. density of magnesium itself, which is already hcp. This has to be compared with the density for quartz (2.66), which is only slightly in excess of that of silicon alone (2.42), which has already a wide-open diamond lattice that is only half as densely packed as a close-packed lattice. Therefore, if the oxidation-passivation of Mg2Ge could be brought under control, it appears not unlikely that the resulting oxide layer could be much more nearly perfect and free of defects than Si02 on silicon. If this were so, Mg2Ge might be capable of a better performance than silicon in some MOS devices, such as insulated-gate field-effect transistors, at least at fre quencies sufficiently below the very high capability limit of silicon so that the lower mobilities of Mg2Ge do not rule out the latter, but can be offset by better performance otherwise. 3.6. Search for Recombination Radiation A careful attempt was made to observe infrared emission from forward-biased Mg2Ge junctions. A large-area diode was prepared by alloying an n-type Mg2Ge wafer which had an evaporated gold film on one side. A quartz lightpipe was used to electrically isolate the diode from a lead sulfide photodetector which was kept at room temperature. Infrared signals were ob served with the sample at room temperature and under liquid nitrogen. Emission, which appeared to be thermal in nature, was observed only for very large current pulses through the diode. Only a slight further increase in pulse size was required to destroy the diode and cause visible sparking. A slight lag of the emitted light behind the current pulse and unreproducibility of the light intensity further evidenced the thermal origin of the emission. ACKNOWLEDGMENTS The writers wish to acknowledge the constant en couragement of E. W. Herold, and numerous discussions with many of their colleagues, particularly with Professor G. Pearson of Stanford University. L. Garbini, A. Kaufman, and J. Mooney provided valuable analytical services. But most of all, they wish to thank C. Casau who provided the kind of assistance without which this work would not have been possible. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Mon, 17 Nov 2014 08:49:14
1.1708618.pdf	Properties of Clean Silicon Surfaces by Paramagnetic Resonance M. F. Chung and D. Haneman Citation: Journal of Applied Physics 37, 1879 (1966); doi: 10.1063/1.1708618 View online: http://dx.doi.org/10.1063/1.1708618 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/37/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Silicon surface electrical properties after lowtemperature in situ cleaning using an electron cyclotron resonance plasma J. Vac. Sci. Technol. B 12, 3010 (1994); 10.1116/1.587551 Effects of HF cleaning and subsequent heating on the electrical properties of silicon (100) surfaces Appl. Phys. Lett. 60, 1108 (1992); 10.1063/1.106459 Properties of silicon surface cleaned by hydrogen plasma Appl. Phys. Lett. 58, 1378 (1991); 10.1063/1.105211 Paramagnetic Resonance and Optical Properties of Amethyst J. Chem. Phys. 42, 2599 (1965); 10.1063/1.1696338 Paramagnetic Resonance in Electron Irradiated Silicon J. Appl. Phys. 30, 1195 (1959); 10.1063/1.1735292 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:16JOURNAL OF APPLIED PHYSICS VOLUME 37. NUMBER 4 15 MARCH 19(,6 Properties of Clean Silicon Surfaces by Paramagnetic Resonance M. F. CHUNG AND D. HANEMAN School of Physics, University of New South Wales, Australia (Received 16 July 1965; in final form 25 October 1965) Silicon crystals crushed in ultrahigh vacuum (~1O-jj Torr) display an electron spin resonance signal close to g= 2.0055 with a width of 7.0 De. The signal is strongly affected by exposure to 10--2 Torr of molecular hydrogen (increase ",,60%) and oxygen (increase ""80%), indicating that it is associated with surfaces. From surface area measurements, the ratio of dangling bonds to surface atoms was found to be approxi mately 1 to 5. High-vacuum heat treatment causes an irreversible decrease in the surface resonance at 380°C (1-h heating), but the signal is still increased by gas exposure. Above approximately 610°C the rela tively weak remaining signal is now decreased by oxygen exposure, indicating a second surface transforma tion . which correlates with that observed in this temperature region by low-energy electron diffraction. It is concluded that the surface structures for both cleaved and annealed clean silicon surfaces involve dangling bonds, the concentrations being of order 20% and 2%, respectively. Consequences with respect to surface atom arrangemen ts are discussed. I. INTRODUCTION IT has been known since 19541 that silicon samples subjected to damage, such as sandblasting or crushing, exhibit a paramagnetic resonance. For n-or p-type samples crushed in air, the resonance line appears at g= 2.0061±0.OO02, has a spin-lattice reJaxation time Tl of 10-5 sec, and a width of 6 Oe.2 It was reportedly not eliminated by treatment of the samples for a few minutes in concentrated HCI, HN03, or HF, which do not remove much silicon but affect the surface chemical layers. However removal of 10-4 cm of surface by an HF plus HN03 etch caused the resonance to disappear. Further studies of this signal by Walters and Estle3 indicated a g value of 2.0055±0.0002, the uncertainty representing a real variation from sample to sample, and values of T 1 in the range 3 X 10-2 to 3 X 10-4 sec. . Samples were also crushed in vacuum or specific am bients by sealing into Pyrex containers with polystyrene balls. The resonance from samples treated in this way was not altered by exposure to air. From results such as these it was concluded that the paramagnetic centers introduced into the silicon by the above mechanical damage were not at the silicon surface but distributed. in a layer = 10-4 cm thick. Recently, Muller et al.4 studied samples crushed in ultrahigh vacuum by a magnetically operated hammer. It was stated in the publication that for these samples no EPR signal broad or narrow was found within the sensitivity of 1012~H spins/gauss of the spectrometer. However, heating in an ambient containing oxygen caused the line at 2.0055, referred to as 2.006, and a narrow line at g= 2.0029 (width 1.8-3.2 Oe) to ap pear. Heating samples crushed in air to 400°-700°C in a moderate vacuum caused another resonance at g=2.0024 (width 0.8 Oe) to appear, as had been re ported previously by Kusumoto and Shoji.5 In this work we have found that in fact a signal does appear when silicon is crushed in ultrahigh vacuum, as had indeed been found by Muller.6 The signal has been studied in relation to surface area, heat treatment, and exposure to various gases. It is concluded that both cleaved and annealed silicon surfaces have "dangling bonds" but the transition between the surface struc tures of these types of clean surface as found by low energy electron diffraction,? is associated with changes in the relative density of dangling bonds. II. ION BOMBARDED AND ANNEALED SURFACES The first experiments were carried out on surfaces cleaned by ion bombardment and annealing. In order o ----', ( ... -.. _- <\-_c~_ ~------.---~::: b c--J .-----.. , a. FIG. 1. Apparatus for ion bombarding and annealing silicon strips and transferring to quartz tube for EPR studies. A-silicon crystals; B-quartz tube, 3 mm i.d.; C-crystal holder clip; 1 R. C. Fletcher, W. A. Yager, G. L. Pearson, A. N. Holden, D-electron guns. W. T. Read, and F. R. Merritt, Phys. Rev. 94, 1392 (1954). 0 H. Kusumoto and M. Shoji, J. Phys. Soc. Japan 17, 1678 2 G. Feher, Phys. Rev. 114, 1219 (1959). (1962). 3 G. K. Walters and T. L. Estle, J. App!. Phys. 32, 1854 (1961). 6 A. Steinemann, Battelle Institute (private communication). 4 K. A. Muller, P. Chan, R. Kleiner, D. W. Ovenall, and M. J. 7 J. J. Lander, G. W. Gobeli, and J. Morrison, J. App!. Phys. Sparnaay, J. App!. Phys. 35, 2254 (1964). 34, 2298 (1963). 1879 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:161880 M. F. CHUNG AND D. HANEMAN to obtain as much surface area as possible in the small sample chamber volume (3 mm i.d., by = 1 em long) in which the crossed microwave and magnetic fields were sufficiently homogeneous, single crystals of silicon (300 n'cm, p-type) were prepared in the form of long, thin (111) plates, 3XO.2XO.Ol em thick. They were held at one end in outgassed molybdenum clips in an outbaked ultrahigh vacuum Pyrex glass system, and subjected to usual cleaning-type argon ion bombard mentS (500 eV, 100 p.A/ cm2) and annealing (700o-8oo0e) cycles, using two electron guns with filaments shielded from the crystals as in Fig. 1. The holder was then retracted and rotated through 90° and the crystals fell via the funnel mouth through a Pyrex-quartz graded seal into the special quartz tube for insertion into the EPR cavity. The tube was sealed off at a well outgassed constriction and resonance scans made within half an hour. In no case could any signal be detected in these ex periments. The surface area was approximately 3 cm2. From the data for crushed powders to be described below we believe that a signal was present but the surface area was too small by a factor of about 50 for it to be detected with the instrument used, of sensitivity 2 X 1011 spins/ G (Varian V -4500 EPR spectrometer with a 100 kc/sec modulation at a frequency of ap proximately 9400 Me/sec). III. SAMPLES CRUSHED IN ULTRA HIGH VACUUM Single-crystal high-purity specimens from the same crystal as the samples used above were placed in Pyrex vacuum systems and sealed off after pressures of order 10-9 Torr were obtained. The pressures in the sealed off systems, Fig. 2, were of order 10-8 Torr, con sisting mainly of carbon monoxide according to mass spectrometer tests. In some cases sealed off tubes with appendage getter ion pumps which kept the pressure below 10-9 Torr before and after crushing were used, FIG. 2. Sealed-off vacuum tube. A-quartz tube containing powder specimens, B-glass' hammer, C-gas inlet break seal, D-air inlet break seal. 8 H. E. Farnsworth, R. E. Schlier, M. George, and R. M. Burger, J. Appl. Phys. 29, 1150 (1958). I -- crushed S, In vacuum , , , , , , -----__ .xpo.ur~ to H. " I I I I , ' .. ,' , FIG. 3. Trace of electron spin resonance signal from vacuum· crushedsilicon before (full line) and after (broken line) exposure to molecular hydrogen at 10-2 Torr. with no significant effect on the results. The samples were crushed by inverting the sealed-off tube and gently jerking the glass slug up and down onto them for several minutes. Later tests indicated an average particle size in the resulting powder of about 5 p.. The powder was shaken into the attached quartz tube which was lowered into the resonant cavity. In all, some hundred samples with no detectable signal were tested and in every case a clear resonance line was obtained after crushing, with a signal-to-noise ratio of order 100: 1. The powder area was about 70 cm2. Muller et al.' quoted an average sample area of about 1/7th of that used here, with an instrument sensitivity of about tth, so that in some cases the signal may have been below detection limit. The g value of our vacuum resonance was approximately 2.0055 with a line width at room temperature of about 7.0 Oe, (±l-G variation between samples) so that it is similar to that obtained by Fletcher et al.,! Feher,2 Walters and Estle,3 and our own results upon crushing in air. To help determine whether the signal was associated with defects below the surface, at the surface, or per haps a combination, the samples were exposed to various gases. IV. EXPOSURE TO HYDROGEN Molecular hydrogen was introduced by heating a palladium tube on the system with a coal gas flame. Unexpectedly the result was an increase in signal height of about 60% as in Fig. 3, but hardly any change in linewidth. Exhaustive tests for possible spurious causes of the increases were carried out. Spectroscopically pure hydrogen as sources gave the same results. Dummy runs were made with the crushing action followed by letting in hydrogen, in tubes with no samples or with uncrushed samples. No signals were observed, in the range of g=0.5 to 5. To explore the possibility that the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:16PROPERTIES OF CLEAN SILICON SURFACES 1881 7 6 5 ., u .:! ~ III Ne u ~ U • "0 ::I: -o . Z " t xtO ! 8 .--.<Q .. .... 0° .",. ." 0."';.0° 200 Temperature JC .e· • • I I X I I ~ I / / .e ~. .' .. e without annealll'lg ---,..--x-- after annealing at 593Gc. for. ~ hour .... "0'" .. 0 ...... after annealing at 80S' c . for ~ hour 'c 500 FIG. 4. Graph of desorbed hydrogen pressure versus temperature of heating crushed silicon containing adsorbed hydrogen. Note dif ferences in desorption curves between samples which had been exposed to hydrogen after heat treatment at (a) room temperature, (b) 593° C for t h, (c) 808°C for t h. powder had perhaps been heated nonuniforrnly during crushing, and that the hydrogen gas was equalizing temperature throughout the powder with a net average cooling and consequent signal increase, the specimens were cooled in liquid nitrogen for several hours before letting in room-temperature hydrogen at 10-2 Torr. In this way no cooling by the gas was possible. Still the signal increased. It was closely proportional to the inverse of the attenuation in the range from 5 dB to 40 dB, indicating no saturation. The measure ments were made with attenuation in the range 15-20 dB. Experiments were also performed by locking the klystron frequency onto the sample cavity resonant frequency instead of the reference cavity (as is the usual case). In this way possible tuning effects due to any slight frequency changes caused by introducing gas into the sample cavity were checked, and found to be negligible. Experiments on some dozen samples satis fied us that the effect was genuine. It is surprising both because the signal increases, and also because the in crease is so large whereas the adsorption of molecular hydrogen9 is reported to be less than 2%. To confirm the adsorption reports we carried out adsorption tests on the crushed powders. The areas were measured by the BET method, by adsorbing krypton at liquid-nitrogen temperature. Hydrogen at 10-2 Torr was then admitted to the sample and the amount adsorbed was measured by heating the powder and checking the desorption with a mass spectrometer calibrated for hydrogen pressure in the particular system. Dummy runs to check for effects from the chamber walls were also made. Typical results are 9 J. T. Law, J. Chern. Phys. 30, 1568 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:161882 M. F. CHUNG AND D. HANEMAN shown in Fig. 4. From them the hydrogen coverage on silicon crushed in high vacuum was found to be less than 3% of a monolayer to within the accuracy of the BET area determination which we regard to be reliable to within a factor of 2 in our case. The number of spins per surface atom was, from the area figures, 1 to S. The number of spins was estimated by comparison with known samples of O.l%-KCl pitch. If one estimates the heat of adsorption from Fig. 4, curve (a), and using the expressionlO p/(2mkT)I=nkT exp( -X/kT)hj., where p is the equilibrium pressure during desorption, n is the number on the surface, X is the heat of adsorp tion, and j. is the partition function for adatom vibra tion normal to the surface, and setting j.= 1, which may be in error, the result for the heat of adsorption is 38 kcal/mole. This is an intermediate figure for adsorp tion. However some evidence that the adsorption is molecular comes from the fact that exposure to atomic hydrogen results in monolayer coverages,9 whereas the coverage here is only of order 3%. The difference in desorption from surfaces annealed before exposure to H 2 is referred to in Sec. IX. The results for all samples are shown in Table I. The differences in the increase in signal height between the various samples from exposure to hydrogen at = 10-2 Torr is perhaps due to different packing of the powder, so that different proportions of the area were exposed. Removing the sample tube for further shaking was not satisfactory owing to the difficulty of replacing it with- TABLE J. Effect of exposure to H2 and aIr, % change in signal height. Sample Upon exposure to H2 Upon exposure to No. at 10~ Torr air after H2 Sample at liquid-N2 temperature Liquid-N 2 temperature +60% -50% 2 Sample at liquid-N2 temperature +11% 3 Sample at room temperature +53% 4 Powdered in H2 -24% 5 Powdered in H2 -17% 6 Powdered in H2 -35% 7 +65% -20% 8 +50% -35% 9 +63% -31% 10 +80% 11 +42% On pumping out H2 afterwards -8% -20% 10 G. Ehrlich, Brit. J. App!. Phys. 15, 349 (1964). ·7 10 ·s ., ·3 ·2 lOG,o 'P (torr) FIG. 5. Heights of EPR signal after exposure to hydrogen for 15 min at various pressures. ·1 out having to change the spectrometer tuning. In fact it was essential to prevent any vibration of the tube when any of the gas inlet processes were carried out. Table I also shows the effect of subsequently exposing the samples to. air, by gently crushing a thin glass dendrite tube attached to the svstem. In all cases the signal was reduced but not quit~ to the original value. The reduction applied also to samples which had been powdered in hydrogen, and also at room or liquid nitrogen temperatures. Subsequent evidence suggests that the decrease was due to water vapor. If the hydrogen was pumped out with an oil diffusion pump with liquid-nitrogen trap after exposure, there was only a small decrease (recovery) in the signal, as indicated in the last row of Table I. The above results all apply to equilibrium coverages. Exposure at pressures of less than 10-4 Torr of hydrogen had almost no effect on the signal as shown in Fig. S. V. EXPOSURE TO O2, N2, AIR, CO, CO2, Ar, Kr, H20 Other vacuum-crushed samples were exposed to oxygen of "mass-spectrometer grade" with impurity content less than 1 part in 105• Again the signal height increased, as shown in Fig. 6, with a slight decrease in width to 6 Oe. Results for 5 samples are shown in Table II. The increases were more than for hydrogen, with a spread of values for the percentage increase in height. Exposure of these samples to air caused a small decrease of signal, the decrease being significantly less than that caused by exposing hydrogen-"covered" samples to air. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:16PROPERTIES OF CLEAN SILICON SURFACES 1883 TABLE II. Effect of exposure to O2, % change in signal height. Sample No. Upon exposure to O2 at 10-2 Torr 12 Liquid-N 2 temperature 13 14 15 16 +47% Room temperature +75% +108% +75% +148% Upon exposure to air after O2 -13% -14% -13% Exposure of fresh sample surfaces to pressure of 10-1 Torr of the inert gases Ar and Kr, and to nitrogen and to mixtures of CO and CO2 had no detectable effect on the EPR signal. However, room air following the above gases caused signal increases of 20% to 60%. Samples cooled to liquid nitrogen and then exposed to air suffered a reduction in signal height of about 50% as shown in Table III. These results suggested that water vapor could be important. Therefore triply distilled de-ionized water was sealed into a side tube of the vacuum system after boiling to expel dissolved air. Three freshly crushed samples were then exposed to water vapor obtained by puncturing an internal glass seal between the water chamber and the tube. In each case the signal height decreased, by 10%, 16%, and 37%, respectively, as shown in Table IV. It is noted that the components of water vapor, namely hydrogen and oxygen, both separately cause an increase in signal so that if any dissociation of the water took place, the net effect on the signal would be the difference between the increase and decrease com- '. : . / r---- ! :r;---- ·V -- crushtd SI t I • I , I : I ------- txposU"t' to 0:0 I, : , , 'J FIG. 6. Trace of EPR signal from vacuum-crushed silicon before (full line) and after (broken line) exposure to oxygen at 10-2 Torr. TABLE III. Effect of heat treatment and subsequent exposure to air, % change in signal height. Annealing tem- Sample perature and No. time Upon exposure to air 17 Room temperature Liquid-N 2 Temp. -50% 18 Room temperature Liquid-N 2 Temp. -43% 19 Room temperature After argon +40% 20 Room temperature After mixture of CO and CO2, etc. +20% 21 Room temperature After Kr +50% 22 Room temperature After N2 +58% 23 Room temperature +30% 24 379°C for! h in O2 0% 25 607°C for! h in 0% vacuum 26 861°C for 1 h in -50% vacuum 27 400°C for 1 h in +126% ~ glass tube, under vacuum (Other samples in quartz tubes) ponents. The fact that a net decrease was observed suggests that if any dissociation of H20 takes place on crushed Si samples, it is certainly less than 50%. Subsequent exposure of the water-vapor covered surfaces to air caused increases of 0%, 41%, and 65% as in Table IV and Fig. 7. Apparently the oxygen in the TABLE IV. Effect of exposure to water vapor and air, % change in signal height. Sample Upon exposure to H2O Upon exposure No. at 1 Torr to air after H2O 28 -10% 0% 29 -16% +65% 30 -37% +41% air displaces the water vapor from its sites, if the latter had covered the surfaces completely. VI. EFFECTS AT LIQUID-NITROGEN TEMPERATURES Measurements at low temperatures were made by blowing cooled nitrogen gas past the sample tube. At liquid-nitrogen temperatures the active number of spins increased by approximately a factor of 4, as indicated by integration of the signal shown in Fig. 8(a) and 8(b), and by comparison with calibrated spin samples. However after adsorption of O2 and H2 onto the crushed surfaces, the number of spins only increased [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:161884 M. F. CHUNG AND D. HANEMAN -- crushtd SI '" .... ----...... C'lIpasurr to H20 •••••••• C'xpaslSC' to iillr FIG. 7. Trace of EPR signal from vacuum-crushed silicon before (full line) exposure to gases, after (broken line) exposure to water vapor and after a subsequent (dotted line) exposure to air. by factors of approximately 1.7 and 3.5, respectively, as in Fig. 8(c) and 8(d) and Fig. 8(e) and 8(f). Up to three hours was allowed for the samples to equilibrate, though usually much less time was sufficient. These changes indicate strong interactions with the gas as discussed below. VII. DISCUSSION OF CLEAN-SURFACE DATA Before describing the effects of annealing we briefly review some of the above data. A strong EPR signal appears on crushing silicon in a vacuum of 10-c 10-9 Torr. The resulting surface area, of order 100 cm2, contains about 1017 atom sites, the surfaces being mainly (111), the favored cleavage plane for silicon. The gas contained in as much as a liter at 10-8 Torr is onlv of order 1011 molecules so that it is insufficient by about six order of magnitude to cover the new surfaces with a monolayer, even if the components were active. Ex tensive experience with these kinds of vacuum systems and procedures confirms that the surfaces are essentially clean if no leaks are present. Many of the sealed-off tubes carried an ionization gauge which was run briefly before crushing and briefly just prior to a gas experiment to check that no leakage had occurred. Operating the gauge had no effect on the signal. The question arises whether the EPR signal is due to the surface or wholly or in part to the interior. Certainly the crushing process imposes severe mechani cal stresses and shocks to the material. No doubt many dislocations and cracks are formed and perhaps va cancies, interstitials, and other defects. The fact that the signal is so sensitive to gas suggests very strongly that at least a good part of it must be associated with surfaces. The centers could be on the surface itself, interacting directly with the gas. Another possibility is that they are some small distance below the surface, in which case they could be affected by gas if the latter altered the surface charge density, thereby altering the surface space-charge layer whose signifi cant extent in the semiconductor surface region is over 100 A. This would alter the occupation probability of all states in this surface region and thus account for a change in signal strength after gas adsorption. This second possibility is, we believe, not operative in view of the low-temperature behavior of the EPR signal. Before gas adsorption the signal intensity (obtained by integration) increases at liquid-nitrogen temperatures by approximately 4 times in accordance with the Boltzmann factor. This indicates that the positions of the band edges with respect to the Fermi level had not changed much at low temperature. After oxygen adsorption the signal, measured under identical conditions, increases at liquid-nitrogen temperature by a factor of only 1. 7, and after hydrogen adsorption by 3.5 as in Fig.~8."The distance of the band edges from the Room Iemeergtyr,.l..J...a.uJ.st N2 amJ~:gJJaI gain 200 { a} gain 200 (c) gain 200 ( e) gain 50 (b) {f} FIG. 8. Appearance of EPR signal at room temperature (Ieft hand side) and liquid-nitrogen temperature (right-hand side) (a) after vacuum crushing (c) after exposure to oxygen (e) after exposure to hydrogen. Three separate samples were used. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:16PROPERTIES OF CLEAN SILICON SURFACES 1885 Fermi level should not change differently with respect to temperature in the presence of gas than in the absence of gas. A qualification to this is a possible increase in coverage at low temperatures. -However this should cause the signal to increase even more, whereas in fact it increases at low temperatures to an extent signifi cantly less than in the clean state. This difference be tween the temperature behavior of clean and gas exposed surfaces indicates a strong interaction between the centers giving rise to the signal and the gas atoms and molecules that affect them. Such an interaction cannot be explained if the centers are even a few atom layers distant from the adsorbed gas, except by invoking rather far-fetched and artificial mechanisms. We there fore consider the evidence as pointing strongly to a direct interaction between the adsorbed gas and the sites giving rise to the EPR signal. The most natural and logical explanation for these surfaces sites are the dangling bonds expected on straightforward models of cleaved (111) surfaces. How, then, does one explain the data of Feher2 and Walters and Estle,3 where the signal from samples crushed in air or vacuum was insensitive to ambients and even acids, unless a layer of 10-4 em was dissolved in which case the signal disappeared? We believe that the above-reported signals could be due to surfaces covered with oxygen. The reason they are observed is that the surface area is increased by orders of magnitude by sandblasting, crushing, etc., bringing a formerly undetectable signal up to detection limit. Treatment with HCI, HF, or HNOa will still leave at least a monolayer thickness of oxide. A sandblasted or polished surface contains myriads of tiny cracks and fissuresll all contributing surface. Therefore, to decrease appreciably the actual surface area it is necessary to etch away the cracked layer, of order 10-4 cm deep. The effects of exposure to air would not be noted for samples crushed in vacuum unless it was a very good vacuum and clean, outgassed, crushing chamber, since we have found that air causes very little effect to a surface that has already seen as little as 10-3 Torr or less of oxygen (Table III). The positive sign of g-ge, where ge is the free electron value of 2.0023, implies that the surface centers are holes if they are nonlocalized, or that, if localized, they are due to electronic shells that are more than half completed.12 This is quite consistent with surface dangling bonds. Furthermore the relatively large ratio of spins to surface atoms, 1 to 5 (to within an accuracy of about 2, see Sec. IV) is in good accord with the reso nance coming from surface atoms. 11 E. N. Pugh and L. E. Samuels, J. Electrochem. Soc., 108 1043 (1961). 12 G. K. Walters, J. Phys. Chern. Solids 14, 43 (1960). VIII. EFFECTS OF HEAT TREATMENT Heat treatment of freshly crushed samples was carried out by immersing the quartz tube into a coil furnace for a given time, usually an hour, and replacing in the spectrometer cavity for room-temperature measure ments. The quartz tube had been thoroughly outgassed before system seal-off by more than 12-h heating at 800°-900°C, reaching a pressure of = 10-8 Torr during outgassing. The removal and replacement of the tube in the cavity meant that the distribution of powder in the cavity was not quite identical each time. Practice runs without heating indicated an error from this procedure of up to about 10%. The results of heat treatment were several. The signal discussed above, called a, was unaffected by heating for one hour each at temperatures up to 380°-400°C. After heating to higher temperatures there was a permanent reduction in signal, as indicated in Fig. 9. Furthermore, a very broad new resonance usually appeared, called {3, which was barely detectable after annealing at 380°C for one hour, becoming progressively stronger after higher-temperature treatment as the original signal became weaker, for the range up to about 700°C, as in Fig. 10. On recrushing the powder the original signal a reappeared, but the broad one {3 was still present at about the same strength. In a few cases, however, another phenomenon was observed. On heating at =600°C for 2 h another reso nance appeared, of width 1 Oe, as in Fig. 11. On exposing the powder to oxygen at 10-2 Torr the signal a in creased but the sharp signal, called /" decreased. 100 '0 20 200 '00 BOO Annealing Temper.tul'e' Ie FIG. 9. Room-temperature measurement of EPR signal a from vacuum-crushed silicon after heating for 1 h at various tempera tures. Critical temperatures are at ",,380°C (irreversible decrease in number of spins) and 610°C (reversal in effect of O2 on signal height). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:161886 M. F. CHUNG AND D. HANEMAN + ( g.lln 100 (~) tb) Olin 500 (el (dl 9,;lIn 500 9'lIn 500 leI I" FIG. 10. Usual appearance of EPR signal from outgassed vacuum-crushed silicon after heat treatment at (a) room tem perature (no change), (b) 380cC for 1 h (broad signal (3 barely detectable), (c) 570°C for 1 h, (d) 688°C for 1 h, (e) 810°C for lh. In (f) the signal is shown after recrushing the powder (e). Note reappearance of signal Ci, and continued presence of signal (3. Pumping out the oxygen did not affect the new value of ct but restored 'Y to its original height. This signal 'Y at g= 2.0024 appeared to be the one reported by Kusumoto and Shoji5 after heating in vacuum, samples that had been crushed in air. Their signal also dis appeared on exposure to air but was restored by re-evacuation. Further evidence regarding the line 'Y has been dis cussed by Kusumoto and Shoji, and MUller et al.4 Kusumoto and Shoji regarded it as due to broken surface bonds, MUller et al. as associated with internal oxygen coming to the surface. The latter was supported by a report by Fletcher and Feher13 of a similar narrow resonance line after heat treating silicon to 3S0o-S00°C. The fact that the signal was not always observed in our experiments leads us to believe that it could in fact be due to some variable impurity content, perhaps sites at the surface which were vacated by oxygen as it became more mobile during heating. This explains why 13 R. C. Fletcher and G. Feher, Bull. Am. Phys. Soc. 1, 125 (1956). the signal appears on vacuum heating samples crushed in air and therefore covered with oxide, and sometimes on samples crushed in vacuum, depending on sufficient cleavages occurring at planes containing oxygen. It also explains why exposure to sufficient oxygen destroys the signal. The fact that the signal is restored by pumping the air or oxygen away infers that the sites in question have now a very low affinity for oxygen. Further re search on the origin of this signal needs to be done. The clean-surface signal-heightct increased by a factor of approximately 4 on cooling to liquid-nitrogen temper atures, as shown in Fig. 12, and the width decreased to about S.S Oe. This indicates a signal approximately in versely proportional to temperature, as expected from Curie's law. However, after heating the crushed surfaces in vacuum to 400°-600°C, the signal was less than doubled on cooling from room temperature to liquid nitrogen temperature. This indicates departure from Curie's law due to interactions and indicates significant changes in the nature of the spin centers after gas adsorption or heat treatment above 400°C. Some re corder traces of signals are shown in Fig. 12, displaying q~ln 100 I.) rb) 500 qoun 500 Ie) Idl !iI.un SOD Ie) FIG. 11. Occasional appearance of EPR signal from outgassed vacuum-crushed silicon after heat treatment at (a) room tempera ture (no change), (b) 600°C for 2 h (note new sharp signal/" linewidth "" 1 Oe). The signal in (c) results from exposing (b) to oxygen at 10-' Torr. On pumping out the oxygen, signal/, is re stored, in (d). In (e) signal/, has disappeared after exposure to air. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:16PROPERTIES OF CLEAN SILICON SURFACES 1887 ROOM TEMPERATURE LIQUID N~ TEMPERATURE g.illn 25 Cb) gem 100 gain 100 Cel Cd) CjlOiln SOD 9-'" 500 Col Cfl FIG. 12. Appearance of EPR signal a at room temperature (left hand side) and at liquid-nitrogen temperature (right-hand side) (a) after vacuum crushing, (c) after subsequent annealing at 380°C for one hour, (e) after further annealing at 455°C for 1 h. The signals refer to one sample. Note values of gain. the increased noise both as a result of heating and of liquid -ni trogen measuring conditions. IX. EXPOSURE OF HEAT-TREATED SURFACES TO GASES Although the EPR signal was permanently reduced by heat treatment above about 480oe, it was still sufficiently strong, even after 700oe, to check the effects of gas adsorption. For samples that had been heated up to about 6100e for an hour, the effect of oxygen was to cause an increase in signal height, as for the room-temperature specimens. However samples that had been annealed above this temperature showed, on the contrary, a decrease in signal on exposure to oxygen at 10-2 Torr. The results are shown in Table V. As before, there appeared to be some variability in the magnitude of the effect from specimen to specimen. A decrease was also observed on exposing one sample that had been heated at a nominal S800e for 34 h. In this case it was suspected that the temperature may have TABLE V. Effect of annealing and subsequent exposure to gas. H, at 10-' Torr 0, at 10-' Torr Vacuum Vacuum Ann. Ann. Sample temp. After After H, Sample temp. After After 0, No. and time H, and air No. and time 0, and air 31 580°C +10% +215% 33 700°C -15% -7% 34 h IOh 32 608°C 0% +25% 34 700°C -15% !h 3h 35 580°C +63% -68% 15 h 36 580°C -34% 0 34 h 37 608°C +178% 0 ! h 38 631°C -27% i h 39 607°C +58% i h 40 609°C +93% -45% 1 h 41 620°C -21% -28% 1 h 42 635°C -17% -20% 1 h risen higher to the critical temperature during the very long anneal, accounting for its behavior. Results for two samples exposed to hydrogen are also shown in Table V. A feature of interest is the desorption curve for hydrogen from surfaces that had been heated, as shown in Fig. 4. Note the difference in the desorption between surfaces previously heated at 808°e and those heated at less than 610oe. The stronger affinity for the 808°e heated surface is consistent with the reduction in EPR signal upon adsorption onto this surface, indicating interaction different from that with surfaces where the signal increased. It is worth noting that the desorption for the unannealed surface is much faster than exponen tial, particularly in the region around soooe. This is believed to be correlated with a change in surface structure in this temperature region. X. DISCUSSION A. Clean Surfaces Heat treatment of silicon affects bulk as well as surface properties.2.13.14 The samples tested here had all, before crushing, been heated to 800°-900°C for at least 12 h in high vacuum. No EPR signal was observed after this treatment indicating that the concentrations of ionized impurities and possible oxygen aggregates in the bulk were below detection limit. The signal that appeared on crushing was therefore a surface signal as discussed above. Subsequent heat treatment was much less severe than that already given to the samples so that no new pure bulk contributions to the signal would 14 C. S. Fuller, J. A. Ditzenberger, N. B. Hannay, and E. Buehler, Phys. Rev. 96, 833A (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:161888 M. F. CHUNG AND D. HAN EM AN have appeared. The changes observed are therefore associated with the surface, as is confirmed by the sensitivity of the signals to interaction with gases. P s sem in Fig. 9, there are at least three different sets of properties for samples subjected to heat treat ment, corresponding to at least two different critical temperatures. Up to approximately 380ce the number of surface spins and the nature of their interaction with oxygen appear to be unaffected. ~amples heated beyond this temperature suffer a progressive decrease in the height of the main signal according to the temperature of heating (time, 1 h). (In the case of air-crushed samples, the signal decrease starts at a lower tempera ture and proceeds faster.) 1 he behavior of this signal on exposing the surfaces to oxygen appears to undergo a reversal at a critical heating temperature of approxi mately 6100e (time, order 1 h). On exposing the surfaces subsequently at room temperature to oxygen, the signal height decreases rather than increases, as in Fig. 11. This suggests a change in the nature of the surface. These results may be compared with the results of diffraction of low-energy electrons from cleaved surfaces of silicon, as reported by Lander, Gobeli, and Morrison.7 T he diffraction patterns indicated a rectangular surface mesh. On heating, the pattern changes. "The t orders of the pattern of the cleaved silicon surface disappeared when the temperature of the specimen reached about 5000e and the background intensity increased mark edly. At about 6000e new fractional orders began to appear and the background intensity decreased. After a long anneal (minutes) at about 7000e or a short anneal (seconds) at 8oooe, observation of the pattern at room temperature showed that the new pattern was very well resolved. It was in all cases the Si(111)-7 pattern characterized by 1/7 orders". The correlations between the transformations noted in the low-energy electron diffraction (LEED) results and those found in the EPR behavior seem significant. The onset of deterioration in the LEED cleaved surface pattern at "about soooe" would appear to be associated with the irreversible decrease in the number of surface spins. Our heating times of 1 h would cause this effect to appear at a lower temperature, in fact the onset is at 3800e and is well advanced at soooe. The appearance of the new diffraction pattern characterized by 1/7th orders in the region 600o-700oe, depending on annealing time, is well correlated with the change in behavior of the surface signal to the presence of gaseous oxygen. Lander et al. also report a change from the 1/7th order pattern to a ith order one "with prolonged an nealing (many minutes) below about 6oooe." Some of the powders may therefore have had surfaces with ith order structures as well. Badly fractured surfaces did not give good LEED patterns, probably due to high densities of steps and other variations in topography. This does not mean that within these regions the surfaces were necessarily dif-ferent from the larger flat areas. The qualities of the surfaces on the powders in terms of flatness were no doubt various, but it is assumed that the net surface was made up of small areas whose structure corre sponded with that of larger flat regions. B. Surface Models 1. Cleaved Surfaces The above results restrict the possibilities for surface atom arrangements capable of fitting LEED intensity data as understood at present.l5,16 The structure of silicon surfaces produced by crushing, which we take as equivalent as far as small areas are concerned, to those produced by cleavage, is such that the surface atoms are associated with unpaired electrons. These could well be the dangling bonds predicted on simple bond rupture processes for separating tetrahedrally bonded substances along (111) planes. The data do not preclude a density as high as nearly one dangling bond per surface atom. Although no models have been fitted in full detail to the cleaved surface diffraction data, the suggestion by Lander et al. of a surface struc ture with double bonds is not supported by the EPR data. Evidence concerning the degree of atom displace ment on cleaved surfaces has been obtained in our laboratory by mating such surfaces in high vacuumP These data, which show that the base region of a split in germanium can be made to heal, suggest that the atom displacements from their equilibrium positions are not gross as they are restored in the presence of a mating surface and pressure. If these results are appli cable to silicon, they combine with the dangling bond data to suggest a structure for cleaved surfaces based on relatively nondrastic displacements from the ideal arrangement. The arrangement of atoms proposed by Lander et aU would satisfy these requirements if the displacements were rather less than they proposed, and the double bonds between atoms replaced by single bonds, with appropriate addition of dangling bonds. 2. Annealed Surfaces After annealing up to temperatures of 800oe, the density of surface spins is greatly reduced as in Fig. 9, but is still detectable. There are several possibilities. One is that there are at least two types of centers con tributing to the resonance. One is destroyed by the annealing, leaving the other active, and it is such that its contribution is reduced by interaction with O2 and H2• However, this behavior is dominated at room tem- 15 J. J. Lander and J. Morrison, J. App!. Phys. 34, 3517 (1963). 16 N. R. Hansen and D. Haneman, Surface Sci. 2, 566 (1964) and related papers in this volume. 17 D. Haneman, W. D. Roots, and J. T. P. Gra,nt (to be published). . [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:16PROPERTIES OF CLEAN SILICON SURFACES 1889 perature by the first centers, unpaired surface atom bonds, whose signal is enhanced by the above gases. A seccnd possibility is that the surface structure exhibiting fth or 1/7th orders has in fact a small resonance of order 2 spins per 100 surface atoms. The relatively large unit cells necessary to give fth-or 1/7th-order diffraction patterns do indeed contain order 50 atoms so that the suggestion of a center associated with groups of this size is not implausible. A third possibility is that there is a degree of free spin associated with every surface atom, but due to resonance and overlap with subsurface bonds and/or with each other, the net effect is only a fraction of a "spin" per surface atom. This hypothesis is in better accord with the fact that the signal is relatively broad (llH =7 Oe) with a spin-spin relaxation time T2 of order 10-8 sec. Furthermore, no hyperfine structure from the 4.7% abundant 29Si isotope was detected. This suggests that the centers are not highly localized on surface atoms, as is indicated also by the apparent homogeneous broadening of the lines. The picture, then, is one of unpaired electrons or holes in the surface bonds, the net hole density being of order 2 per 100 surface atoms on the annealed surface and having limited mobility. Crudely one may think of surface atom bonds which are fully dangling for about 1/50 of the time, and are overlapping with other bonds or perhaps filled by conduction electrons, for the remainder of the time. Alternatively, due to resonance with lower bonds, the net effect is only equivalent to a small fraction of an unpaired electron per surface atom. These surface holes mav be associated with surface conduction. This picture is -consistent with a model proposed by one of us previously.16 These possible interpretations have assumed that the vacuum heat treatment did not result in appreciable surface contamination. The pressure indicated by an ionization gauge did rise to order 10-7 Torr during heating. However the sensitivity of the signal to subse quent gases, the fact that the signal was unaffected unless the temperatures exceeded 380°C, and the general reproducibility of the results for a number of tubes are taken as good evidence that the surfaces remained essentially clean. C. Adsorption of Gases We do not have as yet a fully satisfactory explanation of the apparently large (=60%) increase of the cleaved surface EPR signal after only 2%-3% coverage of molecular hydrogen. One suggestion is that the hydro gen is adsorbed on favored sites and these are the ones, rather than the whole surface, which are causing the resonance, i.e., the actually sensitive regions are fully covered. This seems difficult to support in view of the density of order 1 spin per 5 surface atoms. Furthermore oxygen, which does form monolayers, increases the resonance also, by even more (=80%). We think that in the case of hydrogen the adsorbed molecules are polarized and attract the surface dangling electrons, i.e., their partial overlap with subsurface bonds and inter action with conduction electrons is reduced, resulting in an enhanced signal. A quantitative treatment is not at tempted at this stage, nor of the oxygen adsorption, pending more detailed information from various sources regarding surface atom arrangements. ACKNOWLEDGMENT The authors thank Professor Alexander, J. Harle, and Professor R. Aitcheson of Sydney University for pro vision of EPR facilities and C. Dehlsen for technical assistance. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.111.210 On: Fri, 19 Dec 2014 04:02:16
1.1727621.pdf	Uranium Mononitride: Heat Capacity and Thermodynamic Properties from 5° to 350°K Edgar F. Westrum and Carolyn M. Barber Citation: J. Chem. Phys. 45, 635 (1966); doi: 10.1063/1.1727621 View online: http://dx.doi.org/10.1063/1.1727621 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v45/i2 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 25 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 45, NUMBER 2 15 JULY 1966 Uranium Mononitride: Heat Capacity and Thermodynamic Properties from 5° to 3500K* EDGAR F. WESTRUM, JR., AND CAROLYN M. BARBER Department of Chemistry, University of Michigan, Ann Arbor, Michigan (Received 25 February 1966) The low-temperature heat capacity of UN was determined by adiabatic calorimetry and found to have a normal sigmate temperature dependence, except for the presence of an anomaly near 52 oK associated with antiferromagnetic ordering of the electron spins. At 298.15°K the heat capacity (Cp), entropy (SO), enthalpy function [(HO-HOo)/T], and Gibbs energy function [-(Go-HOo)/T] are, respectively, 11.43, 14.97, 7.309, and 7.664 calj(gfm.oK). I. INTRODUCTION THE existence of three uranium nitrides, UN, U2N3, and UN2, has been well established, but few thermo dynamic and thermochemical properties have been reported. Recent redeterminations of the melting point of uranium mononitride have raised the previously reported melting temperature to 28s0°C at and above 2.5 atm pressure of nitrogen.1.2 Interest in UN as a potential reactor fuel has therefore increased. Its high melting point, high enthalpy of formation ,3 high density, high thermal conductivity [0.54 W / (cm. DC) at 298°K compared with 0.03 for U02 and 0.25 for UC], appreciable electrical conductivity, and good phase stability (even under neutron irradiation) pro vide a highly desirable combination of refractory characteristics. Its thermal, electronic, and bonding behaviors are of particular interest in comparison with those of other uranium chalcogenides and pnictides (US, USe, UC, and UP, for example) which also possess the sodium chloride structure. The evaluation of such data may provide explanation of the apparent bulk instability of the UO phase. II. EXPERIMENTAL A. Preparation and Characterization of the Sample Uranium mononitride is usually prepared by hydrid ing uranium, decomposing it to form powdered metal, and subsequently reacting this with nitrogen or ammonia. However, the uranium metal in this sample was not pulverized by hydriding but reacted directly • This research was supported in part by the U.S. Atomic Energy Commission. 1 R. W. Endebrock, E. L. Foster, and D. L. Keller, "Compounds of Interest in N ucIear Reactor Technology," in Nuclear M etaUurgy, J. T. Waber, P. Chiotti, and W. N. Miner, Eds. (American Institute of Mechanical Engineers, New York, 1964), Vol. 10, p.557. 2 W. M. Olson and R. N. R. Mulford, J. Phys. Chern. 67, 952 (1963). 3 P. Gross, C. Hayman, and H. Clayton, Thermodyn. NucI. Mater., Proc. Symp. Vienna, 1962, 653 (1962). with ammonia in a vertical Vycor flow furnace at 850°C, using a reaction time of about 24 h to obtain complete reaction of the metal. The uranium dinitride thus produced was converted to mononitride under vacuum (final value 0.23 torr) in a graphite crucible heated within a graphite resistance furnace at a temper ature of 1325°C for 2 h. Stanford Research Institute had prepared the sample (N-19) at the request of W. Hubbard of the Argonne National Laboratory. Through his interest and the generosity of the Labor atory, the material was made available for these measurements. The analytical data provided by Stan ford Research Institute and the Argonne National Laboratory are given in Table I. Calculation of the proximate constitution of the sample requires a knowledge of the form in which oxygen is present. The oxygen could well be totally present as UO, which is isostructural with UN, but x-ray-diffraction data taken at Stanford Research Institute utilizing synthesized calibration standards have been interpreted4 as indicating that oxygen is present partly (0.8 wt%) as a surface contaminant in the form of U02 and partly (1.4 wt%) as UO in solid solution with UN. Although we do not endeavor to judge the reliability of the x-ray result without more information on the basis of the calibration, we feel confident in ascribing the oxygen in the sample to uranium monoxide in solid solution in the nitride for several reasons. First, the presence of a separate, fairly pure UOz phase would be expected to show the co operative, antiferromagnetic-paramagnetic transition near 300K.6 The absence of any such anomaly in the region of 30° suggests essentially complete absence (i.e., less than 0.1 wt%) of the dioxide phase. Moreover, the amount of monoxide present is within the limits of its solubility in the mononitride.6 A further argument in favor of this interpretation is found in the 4 Stanford Research Institute, "High Purity Uranium Com pounds" (report submitted to Argonne National Laboratory, 1963) . & E. F. Westrum, Jr., and J. J. Huntzicker (unpublished work). 6 H. M. Feder (personal communication). 635 Downloaded 25 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions636 E. F. WESTRUM, JR., AND C. M. BARBER TABLE 1. Analysis and characterization of UN calorimetric sample. Average Substance Amount present (wt%) Source (wt%) U 94.49,94.56 SRI- 94.52 N 5.41,5.14 SRI 5.28 S.31,S.34b ANL 0 0.20,0.20 SRI 0.20 0.20,0.21 ANLe C 0.05 SRI 0.05 0.066,0.062,0.053,0.053 ANL H (0.04) SRI 0.0003 0.0002 to 0.0004 ANL Fe (0.01--{).1) (emission SRI 0.05 spectrograph) 0.05 (calorimetric) SRI Al 0.003-{).03 SRI 0.01 Mn 0.0003-{).003 SRI ---- Total 100.12 -SRI Stanford Research Institute.- b The~e recent nitrogen determinations by Holt of Argonne National Laboratory nsing an inert-gas fusion manometric method previously described [B. D. Holt and H. T. Goodspeed, Anal. Chern. 35, 1510 (1963)] are of higher precision and support the previously selected average value. e ANL, Argonne National Laboratory. proximate analysis given later which, on this ?as~s, shows the uranium nitride in this sample to be stOIchIO metric. Although a large stoichiometry range at suf ficiently high temperatures has been postulated for UN,7 Olson and Mulford2 noted no deviation from constancy in the lattice parameter. To establish the form of the iron in the mononitride, a mixture of UN and enough iron to form UsFe was heated at 1400°C for 2 h.4 The resultant x-ray patterns showed some iron, possibly some UFez, and an uni dentified compound, but no reduction in the intensity of the UN line. If UsFe or UFez had formed, a large fraction of the UN would have decomposed. The x-ray pattern of a sample prepared from uranium hydride plus iron powder at 600°C confirm.ed.l!6Fe to. be ~he predominant compound. Upon mtndmg thiS With ammonia at 850°C, the x-ray pattern showed only the lines for the UNz phase. Since Fe4N is not stable under the conditions used for forming the uranium dinitride phase, the conclusion follows that the iron present is largely elemental iron. . The proximate composition is, therefore, determmed as 95.5 mole % uranium mononitride (UNl.Oo), 3.2 mole % uranium monoxide, 1.1 mole % uranium monocarbide, and 0.2 mole % elemental iron. B. Cryostat and Calorimeter Determinations on uranium mononitride were made by the quasiadiabatic technique using the Mark III 7 R. Benz and M. G. Bowman, J. Am. Chern. Soc. 88, 264 (1966). cryostat, Calorimeter W -17 A (which has been pre viously described8) and thermometer (laboratory designation A-3) which is believed to r~pr?duce ~he thermodynamic temperature scale to wlthm 0.03 K above the oxygen point. All determinations of mass, temperature, resistance, voltage, and time are re ferred to calibrations or standardizations made by the National Bureau of Standards. The heat capacity of the empty calorimeter, thermometer, and heater assembly was determined in a separate series of measurements. Corrections to the data were made for the differing quantities of Apiezon-T grease (used to provide thermal contact between the heater-thermometer-calorimeter assembly), of Cerroseal (In-Sn) solder (used to seal the sample space), and of purified helium gas (used to facilitate thermal equilibration) present in the two series of determinations. For heat-capacity measure ments on the sample, 146 torr of helium gas was ad mitted to the sample space. The calorimetric sample massed 130.902 g (in vacuo) and represented more than 55% of the total measured heat capacity at all temper atures. A density of 14.32 g/cc9 for UN was used to obtain the buoyancy adjustment. III. RESULTS The heat capacity of the sample is presented in Table II in chronological sequence so that the temper ature increments used in the measurements may usually be deduced from the differences in the adjacent (mean) temperatures. These results are presented in terms of the defined thermochemical calorie of 4.1840 J, an ice-point temperature of 273.15°K, and a gram formula mass (gfm) of 252.037. These data have been adjusted for curvature and for the presence of 1.1 mole % of uranium monocarbide1o and 0.2 mole % of elemental ironll on the basis of values previously re ported. These adjustments total less than 0.2% of the heat capacity above 30°K. Because the 3.2 mole % of uranium monoxide believed to be present in the calorimetric sample is in solid solution, is isostructural, and is reported to have a lattice constant only 0.82% larger than that of uranium mononitride,9 it was con sidered to have a heat-capacity contribution equal to that of the mononitride. It is further presumed to have little influence on the temperature or the enthalpy of transition. The data in the region of the transition are presented in Fig. 1. The smoothed heat capacities and the thermo- 8 E. F. Westrum, Jr., and N. E. Levitin, J. Am. Chern. Soc. 81, 3544 (1959). 9 R. E. Rundle, N. C. Baenziger, A. S. Wilson, and R. A. McDonald J. Am. Chern. Soc. 70, 99 (1948). 10 E. F. Westrum, Jr., E. Suits, and H. K. Lonsdale, in Ad vances in Thermophysical Properties at Extreme Temperatures and Pressures, S. Gratch, Ed. (American Society of Mechanical Engineers, New York, 1965), p. 156. II A. Eucken and H. Werth, Z. Anorg. Chern. 188, 152 (1930). Downloaded 25 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsTHE R MOD Y N A M I CPR 0 PER TIE S 0 FUR A N I U M M 0 NON I T RID E 637 dynamic functions derived from these data are pre sented in Table III at selected temperatures. The smoothed heat capacities are obtained by a digital computer program and checked by comparison with large scale plots of the data. In spite of the relatively low purity of the sample, the heat-capacity values are believed to be characterized by a probable error decreasing from 0.3% above 600K to less than 0.2% above 2000K. The integrations also were performed by a digital computer. These functions are believed to have a probable error of less than 0.3% at temperatures above lOOoK. The enthalpy of Runs A, B, C, and D noted in Table II accorded with calculated enthalpy increments to within 0.07%. No adjustment has been made for isotope mixing or nuclear spin contributions to the entropy and Gibbs energy functions; hence, these values are practicable for use in chemical thermo dynamic calculations. TABLE II. Heat capacity of uranium mononitride.- T T Series I 17.13 0.373 18.87 0.462 118.03 7.019 20.78 0.579 124.99 7.330 22.85 0.728 133.41 7.686 25.06 0.914 142.59 8.054 27.64 1.167 151.92 8.399 30.48 1.501 161.23 8.716 33.66 1.788 170.55 9.017 37.36 2.211 179.92 9.285 38.62 2.358 189.16 9.535 42.07 2.759 198.10 9.756 46.07 3.222 206.88 9.962 !!.Ht Run A 215.56 10.156 !!.Ht Run B 224.51 10.335 61.97 4.051 233.79 10.511 68.62 4.482 243.07 10.695 75.38 4.879 252.21 10.821 82.54 5.299 261.31 10.958 89.99 5.666 270.37 11.083 98.98 6.115 279.40 11.199 106.90 6.505 288.55 11.319 116.26 6.938 297.75 11.430 306.84 11.532 Series III 316.34 11.642 325.97 11. 736 35.46 2.002 335.82 11.822 42.55 2.822 345.88 11.884 46.44 3.270 49.04 3.570 Series II 51.65 3.825 53.67 3.498 5.69 0.069 57.53 3.737 6.09 0.078 59.37 3.871 6.84 0.089 61.14 3.998 7.70 0.104 62.84 4.119 8.66 0.122 9.77 0.149 Series IV 11.04 0.172 12.41 0.202 33.52 1.780 13.89 0.248 37.57 2.243 15.46 0.302 40.84 2.623 a Units: calories, gram-formula mass, Kelvin degrees. T 43.67 46.19 48.48 50.60 52.61 54.62 56.59 58.47 60.27 2.952 3.240 3.503 3.756 3.666 3.537 3.667 3.804 3.937 Series V 25.64 0.976 28.86 1.348 !!.Ht Run C 64.81 4.251 70.30 4.585 Series VI 32.08 1.619 !!.HtRunD 61.80 4.045 Series VII 44.94 3.096 46.05 3.223 47.10 3.342 48.12 3.469 49.10 3.574 49.82 3.658 50.28 3.703 50.74 3.744 51.18 3.828 51.62 3.888 52.06 3.895 52.50 3.670 52.96 3.478 53.42 3.472 53.88 3.487 :.:: o E 4 .5 I /" I I ,/ I I ,'" I I I I I , , I , , ,'/ I :' / .' o~~~ ____ ~ ____ ~ __ ~ ____ ~ ____ ~ __ ~ o 40 r. OK FIG. 1. Heat capacity of UN in the region of the antiferromag netic-ferromagnetic transition. The points represent individual determinations, and the dashed curve is the estimate of the lattice contribution. IV. DISCUSSION The combination of refractory qualities with high electrical and thermal conductivities which characterize uranium nitride is partly a consequence of its electronic configuration. However, an unambiguous assignment of the electron configuration is certainly not possible on the basis of the limited, existing magnetic-sus ceptibility data.12-l4 Allbutt et al.13 found no field dependence in the susceptibility between 80° and 320°K. Their magnetic-moment values were sensibily constant at 3.11±0.OOS Bohr magnetons (J..!B) and corresponded to a Curie-Weiss 8 equal to -32soK. These values accord well with 8= -3100K and a moment of 3.0 J..!B reported by Trzebiatowski and co workers,12 and with that of Didchenko and Gortsema14 (3.04 J..!B). These are in poor agreement with the theoretical value (3.62 jJ.B) for Sj3 ions. Comparison with data on PuC suggests that despite the minute differences in interatomic distances a partial quenching of the orbital moment of UN occurs apparently as a consequence of the larger but less stable Sf shell of 12 W. Trzebiatowski, R. Troc, and J. Leciejewicz, Bull. Acad. Polon. Sci. Ser. Sci. Chim. 10,395 (1962). 13 M. Allbutt, A. R. Junkison, and R. M. Dell, Ref. 1, p. 65. I'R. Didchenko and F. P. Gortsema, Inorg. Chem. 2, 1079 (1963) . Downloaded 25 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions638 E. F. WESTRUM, JR., AND C. M. BARBER TABLE III. Thermodynamic functions of uranium mononitride.· T 5 10 15 20 25 30 35 40 45 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 350 273.15 298.15 0.060 0.148 0.286 0.527 0.913 1.406 1.946 2.517 3.099 3.684 3.914 4.572 5.157 5.691 6.188 6.661 7.113 7.543 7.949 8.327 8.676 8.996 9.290 9.558 9.806 10.034 10.244 10.439 10.619 10.785 10.937 11.079 11.211 11.336 11.455 11.951 11.12 11.43 0.056 0.123 0.207 0.319 0.476 0.685 0.942 1.238 1.568 1.925 2.599 3.253 3.902 4.541 5.166 5.779 6.378 6.964 7.538 8.100 8.648 9.184 9.707 10.216 10.713 11.197 11.669 12.128 12.577 13.014 13.440 13.855 14.260 14.656 15.042 16.848 13.98 14.97 0.14 0.65 1. 71 3.68 7.22 12.99 21.36 32.50 46.55 63.51 100.48 142.97 191.7 245.9 305.4 369.6 438.5 511.8 589.3 670.7 755.7 844.1 935.6 1029.8 1126.7 1225.9 1327.3 1430.7 1536.0 1643.1 1751.7 1861.8 1973.2 2086.0 2199.9 2785.8 1897 2179 • Units: calories, gram-formula mass, Kelvin degrees. 0.028 0.058 0.093 0.135 0.187 0.252 0.331 0.426 0.534 0.655 0.924 1.210 1.506 1.808 2.113 2.418 2.723 3.027 3.329 3.628 3.925 4.219 4.509 4.796 5.080 5.360 5.636 5.908 6.176 6.441 6.702 6.960 7.213 7.463 7.709 8.888 7.040 7.664 uranium. It is reasonable to predicate the existence of a transition in bond character from UC through UN and UP to US and USe. UC has been postulated as covalent and US as ionicY That UN does indeed have a transitional nature is in some respects demonstrated by the Curie-Weiss magnetic-susceptibility curve, which is intermediate between those of UC and US. Consequently, the bonding and electronic structure in UN may be expected to be significantly different from that in UC. No magnetic transformation has been found for UC either by means of heat-capacity meas urementsIO,15,16 or by resistivity measurements.17 As 16 R. J. L. Andon, J. F. Counsell, J. F. Martin, and H.}. Hedger, Trans. Faraday Soc. 60, 1030 (1964). 16 J. D. FaIT, W. G. Witteman, P. L. Stone, and E. F. Westrum, Jr., Ref. 10, p. 162. 17 P. Costa and R. Lallement, Phys. Letters 7, 21 (1963). Costa et al. have suggested,t8 the difference in behavior between UN and UC may be due to a larger band population in the nitride. This would have the effect of stabilizing the f states in this compound.19,20 Magnetic-susceptibility and neutron-diffraction data21 are said to have confirmed the existence of an antiferromagnetic transition in uranium mononitride near 4soK. Moreover, Costa et al.18 observed a change in slope in the thermoelectric power and a large decrease in the resistivity-temperature coefficient near SOcK. The existence of a discontinuity in the heat-capacity curve near 4soK has also been reported by Martin.22 Estimation of the entropy and enthalpy associated with the antiferromagnetic anomaly is reasonably difficult. However, utilizing a Debye e-versus-temper ature plot to assist in drawing a smooth curve beneath the transition for the lattice heat-capacity contribu tion yields a value for the enthalpy of transition of 7.2 caI/gfm and a corresponding entropy of transition of 0.17 cal/(gfm· OK). This probably minimal value may be compared with the entropy of the US transition [1.17 caI/ (gfm. OK) at 179°K]23 and that of the USe transi tion [1.05 cal/(gfm. OK) at 160.S0K]24 in spite of the signifi.cant differences in temperature. The UN entropy at 298°K has been estimated as 13 calj(gfm· OK) in the compilation of Rand and Kuba scheweski.25 The high electrical conductivity of UN accords with the appearance of a component of heat capacity linear in temperature below 23°K. Analysis of tht> data on a Cp/T-vs-T2 plot shows that the low-temperature heat capacity is well represented as Cv~Cp=0.0110T+ (3.86XlO-5) Ta. This equation was used for the ex trapolation of the thermal data to OaK. The coefficient (I' = 0.011) of the linear or electronic term is directly related to the density of states: 1'= (2IPk2/3q) (d'//dE').o, in which q is the number of electrons in the band per atom, eo is the Fermi level in electron volts, and (dv'/dE').o is the density of states per atom.26 By Stoner's method, the density of states for UN [ex pressed as the number of states per atom per electron volt, (dv' / de') ] is (dv'/de') =8.8788X10-2(I'X104) =9.73. -,---=-~.:" 18 P. Costa, R .. Lallement, F. Anselin, and D. Rossignol, Ref. 1, p.83. 19 H. Bilz, Z. Physik 153, 338 (1958). 20 P. Costa and R. Lallement, }. Phys. Chern. Solids 25, 559 (1964). 21 N. A. Curry and R. A. Anderson, Atomic Energy Research Establishment, Harwell, England (unpublished observations reported by Allbutt et al.13). 22 J. F. Martin, National Chemical Laboratory, Teddington, England (unpublished observation reported by Allbutt et al.11). 23 E. F. Westrum, Jr., and R. W. Walters (unpublished results). 24 Y. Takahashi and E. F. Westrum, Jr., J. Phys. Chern. 69, 3618 (1965). 26 M. H. Rand and O. Kubaschewski, The Thermochemical Properties of Uranium Compounds (Oliver and Boyd, London, 1963), p. 41. 26 E. C. Stoner, Acta Met. 2, 259 (1954). Downloaded 25 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsTHE R MOD Y N A M I CPR 0 PE R TIE S 0 FUR A N I U M M 0 NON I T RID E 639 Although relatively large, this value is compar able to those for other isostructural uranium compounds. 10,16,23,24 In addition to establishing an approximate value for the coefficient of the electronic heat-capacity con tributions and revealing the thermal and magnetic anomalies near 52°K, the present results provide definitive thermodynamic data at higher temperatures. Although the impurity content of the sample is rela tively high and the proximate composition is limited by the precision of the nitrogen analyses, the close simi larity of heat capacities of the impurities (UC, UO) with that of UN minimizes the uncertainty in the adjusted results as may be seen in the analogous case of heat-capacity measurements in two laboratorieslO.15.16 on impure but well-characterized samples of uranium carbides from three independent sources. Nevertheless, THE JOURNAL OF CHEMICAL PHYSICS further measurements on pure uranium mononitride are an obvious desideratum in the regions where the effect of impurities on the heat capacity cannot be accurately assessed, i.e., near the thermal anomaly and below 20°K. ACKNOWLEDGMENTS The partial financial support of the U. S. Atomic Energy Commission and the loan of the calorimetric sample by Dr. Ward Hubbard of the Chemical En gineering Division of Argonne National Laboratory are recognized with gratitude. The assistance of John T. S. Andrews, J. J. Huntzicker, and Dr. H. L. Clever and of the Analytical Chemistry Division of Argonne National Laboratory is greatly appreciated. One of us (C.M.B.) thanks the National Science Foundation for research participation awards. VOLUME 45, NUMBER 2 15 JULY 1966 Exchange Effects in the 3A2~IE Absorption Transition of the NiH Ion in Fluoride Compounds* W. W. HOLLOWAY, JR., AND M. KESTIGIAN SPerry Rand Research Center, Sudbury, Massachusetts (Received 11 March 1966) The effect of the exchange interaction between nickel ions on the structure and position of the 8A2-->lE Ni2+ ion absorption transition has been studied experimentally in fluoride compounds. The spectra observed in these materials are found to depend on the concentration of the nickel ion component and the crystal structure. INTRODUCTION STRUCTURE has been reported in the low-tempera ture absorption spectra of the aA2~E transition of the NiH ion in NiF2,1 KNiF3,2 and RbNiF3,a which has been attributed to the exchange interaction between nickel-ion pairs. The splitting of the major lines of this structure in NiF2 and KNiFa have been found to be proportional to the magnetic ordering temperature of the crystal.1.2 In this publication, we report effects on the spectra of the 3A~E NiH transition in several fluoride hosts due to variations in the composi tion and structure of the crystals. The ion-ion exchange interaction and the crystal structure are found to be * This work, supported by the U.S. Office of Naval Research, Contract No. Nonr-4127 (00), is part of Project Defender under the joint sponsorship of the Advance Research Projects Agency, the Office of Naval Research, and the Department of Defense. 1 M. Balkanski, P. Mach, and R. G. Shulman, J. Chern. Phys. 40,1897 (1964). I K. Knox, R. G. Shulman, and S. Sugano, Phys. Rev. 130, 512 (1963); S. Sugano and Y. Tanabe, Magnetism, Treatise Mod. Theory Mater. 1, 243 (1963). a W. W. Holloway, Jr., and M. Kestigian, Phys. Rev. Letters 15, 17 (1965). very important in determining the spectrum of this transition. EXPERIMENTAL Crystal specimens for the fluoride materials used in these experiments were prepared by the horizontal Bridgman technique in an HF or inert-gas atmos phere. High purity of the starting materials, particu larly the NiF2, was found to be essential to good crystal growth. Samples 0.5 cm on a side were typically obtained. X-ray photographs revealed that these mate rials contained less than 1 % of secondary phases. In the mixed crystals prepared, the concentrations re ported are those of the starting materials. The crystal samples were mounted on a copper cold finger which was attached to the coolant reservoir of an optical vacuum Dewar. Temperature measurements were made with a thermocouple fixed to the sample. The absorption spectra reported here were measured on a Perkin-Elmer 112 recording spectrometer equipped with a tungsten-filament lamp light source and an 8-20 response photomultiplier detector. The resolution of the present experimental arrangement was estimated Downloaded 25 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions
1.1697147.pdf	Electron Spin Resonance of Tetraphenylporphine Chelates Jacques M. Assour Citation: J. Chem. Phys. 43, 2477 (1965); doi: 10.1063/1.1697147 View online: http://dx.doi.org/10.1063/1.1697147 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v43/i7 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 43. NUMBER 7 1 OCTOBER 1965 Electron Spin Resonance of Tetraphenylporphine Chelates JACQUES M. AssouR RCA Laboratories, Princeton, New Jersey (Received 20 May 1965) ESR studies of three paramagnetic tetraphenylporphine chelates: vanadyl, cobalt, and copper reveal distorted crystal-field surroundings which are more pronounced in the cobalt derivative. The spin-Hamil tonian parameters and the 3d energy levels of the cobalt derivative are greatly influenced by axial distor tions. The bonding scheme in these complexes indicates strong in-plane u bonding characteristic of organo metallic square-bonded complexes, and little or no in-plane 1(' bonding. Out-of-plane 1(' bonding is more significant in the vanadyl and cobalt derivatives than in the copper complex. ESR of magnetically concen trated samples indicate a substantial reduction in the dipolar and exchange interactions between neighboring metal ions in comparison to those found in phthalocyanines. A tentative explanation for the reduction of the dipolar forces is that the crystalJographic packing of the phenyl rings above and below the molecule might effectively shield the paramagnetic ion from its nearest metal-ion neighbors. I. INTRODUCTION THIS report presents the results of an electron spin resonance (ESR) investigation of three paramag netic tetraphenylporhine metal chelates: vanadyl, cobalt, and copper. This study consisted of the meas urement of the spin-Hamiltonian parameters of solu tions and polycrystalline specimens of the above chelates, and a discussion on the energy levels of each metal ion including the type of metal-ligand bonding in these covalent complexes. Reports of recent studies!-4 on the x-ray structure of tetraphenylporphine (abbreviated TPP) molecular crystals have revealed in detail the nonplanarity of the TPP molecule, and have strikingly shown that the a-, {3-, "1-, and I)-phenyl rings in the molecule (Fig. 1) are tilted and twisted out of the porphine nucleus. In the copper TPP (CuTPP) crystal, for example, the phenyl groups have been found to be almost perpen dicular to the molecular plane and have been inter preted! as evidence that the phenyl groups are elec tronically isolated from the conjugated porphine system. On the other hand, according to Ingram and co-workers,6 ESR investigations of CuTPP and its para-chloro derivative (p-CICuTPP) implied that considerable overlap exists between the 3d unpaired electron of the CuH ion and the atomic orbitals of the peripheral chlorines which are separated by at least 9 A from the copper atom. This magnetic interaction was interpreted as evidence for a substantial intramolecular delocalization of the unpaired electron orbital via the 1I'-orbital system of the conjugated tetraphenylporphine molecule. The ESR results,6 contrary to x-ray anal yses,l·2 indicate that the phenyl groups are electronically coupled to the entire aromatic resonating system of the TPP molecule. 1 E. B. Fleischer, J. Am. Chern. Soc. 85, 1353 (1963). I J. L. Hoard, M. J. Hamor, and T. A. Hamor, J. Am. Chern. Soc. 85, 2334 (1963); 86, 1938 (1964). 3 S. Silvers and A. Tulinsky, J. Am. Chern. Soc. 86, 927 (1964). 4 E. B. Fleischer, C. K. Miller, and L. E. Webb, J. Am. Chern. Soc. 86, 2342 (1964). 6 D. J. E. Ingram, J. E. Bennett P. George, and J. M. Gold stein, J. Am. Chern. Soc. 78, 3545 (1956). The effect of substituents, such as p-chloro, etc., on the degree of covalent bonding between the square bonded metal ion and the nearest-neighbor ligands in the tetraphenylporphine chelates can be most effec tively studied by ESR techniques. Previous ESR H H H H H H M: METAL ATOM FIG. 1. Structure of the tetraphenylporphine molecule. studies on the analogous porphyrin6 and phthalo cyanine7 molecules have examined in detail the (J bonding and 1I'-bonding schemes in these molecules. The present study is aimed at understanding the bonding properties of the unsubstituted tetraphenyl porphine molecules, and at forming a basis for com parison with investigations to be reported on the substituted chelates. 6 (a) E. M. Robert, W. S. Koski, and W. S. Caughey, J. Chern. Phys. 34, 591 (1961); (b) D. Kivelson and S. K. Lee, ibid. 41, 1896 (1964). 7 (a) E. M. Roberts and W. S. Koski, J. Am. Chern. Soc. 83, 1865 (1961); 82, 3006 (1960); (b) D. Kivelson and R. Neiman, J. Chern. Phys. 35, 149 (1961); (c) S. E. Harrison and J. M. Assour, ibid. 40, 365 (1964); (d) J. M. Assour and W. K. Kahn, J. Am. Chern. Soc. 87,207 (1965). 2477 Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions2478 JACQUES M. ASSOtJR TABLE 1. Crystal structure and cell dimensions of tetraphenylporphines. Compound Phase a(A) (A) c(A) a(O) IW) I' (0) Za Ref. H2TPP Tetragonal 15.12 15.12 13.94 4 2 Orthorhombic 12.0 19.2 14.7 4 b Tric1inic 1 3 CuTPP Tetragonal 15.04 15.04 13.99 4 1 NiTPP Tetragonal 15.04 15.04 13.92 4 4 ZnTPP Triclinic 6.03 9.89 13.0 101 108 93 1 4, b Orthorhombic 14.8 17.2 14.6 4 4 • Molecules per unit cell. b J. M. Goldstein, Ph.D. dissertation, University of Pennsylvania, 1959. II. STRUCTURE OF TETRAPHENYLPORPHINES Available x-ray datal-4 on tetraphenylporphines show that these complexes can be grown in either tetragonal, triclinic, or orthorhombic structures. The occurrence of three crystalline modifications for such large molecules is apparently common since the analogous phthalo cyanine compounds are also known to exist in more than one polymorph. s The crystal structure and cell dimensions of several TPP molecules are given in Table I. Although the number of molecules per unit cell have been determined for each crystalline modifica tion, the relative orientations of the molecular and crystallographic axes are yet to be reported. The orientation of the molecules in these crystals presents a difficult problem because of the apparent flexibility of the porphine skeleton and its ease of deformation2,3 under different crystallographic environments. In pthalocyanines,7o.d for example, the physical stacking of the neighboring molecules along the b axis was found significant in causing additional electrostatic interactions between the central paramagnetic ion and distant out-of-plane ligands. In all metallotetraphenylporphines, the metal atom is at the center of the porphine nucleus and is sur rounded by four nitrogen at,oms. The metal-nitrogen distance is approximately 2 A. The four nitrogens are nonplanar. The deviation from planarity differs for each chelate and is dependent on the crystallographic packing of the molecules in each polymorph. The degree of nonplanarity of the organic skeleton and its influence on the strength of the metal-nitrogen bonding in each chelate are considered here. Tetraphenylporphines were prepared by a method similar to that reported by Rothemund and Menotti.9& and by Horeczy and co-workers.9b The synthesis of the vanadyl and copper derivatives was straightfor ward, as reported in the literature. On the other hand, several difficulties were encountered during the prepara tion of the cobalt derivative. These were due mainly to the ease of oxidation of the Co2+ ion to the diamagnetic 8 F. H. Moser and A. L. Thomas, Phthalocyanine Compounds (Reinhold Publishing Corporation, New York, 1963). 9 (a) P. Rothemund and A. R. Menotti, J. Am. Chern. Soc. 63, 267 (1941); 70, 1808 (1948); (b) J. T. Horeczy, B. N. Hill, A. E. Walters, H. G. Schultze, and W. H. Bonner, Anal. Chern. 27, 1899 (1955). C03+ ion which effectively reduced the purity of our samples. The numerous crystalline modifications of the tetra phenylporphines, which are dependent on the method of sample preparations from a wide variety of solutions,4 and the complexities experienced in growing well defined single crystals led us to confine the bulk of our investigations to polycrystalline samples and solutions. Magnetically diluted paramagnetic specimens were prepared by dissolving 1 mg of the metal chelate and 600 mg of the diamagnetic H2TPP derivative in chloro form (CHCla). The solution was then dried in vacuum. This method yielded a uniform dilution of the para magnetic salt in the host compound. ESR studies with solutions were performed with molar concentrations ranging from 5XlO-3M to 10-2M with solvents such as CHCla, CS2, pyridine, and trichloroethylene. The ESR spectra were measured with a Varian spectrometer Model 4500. The magnetic-field modula tion was at 100 kc/sec while the microwave frequency was about 9500 Mc/sec. The magnetic field was determined with a Harvey-Wells NMR gaussmeter used in conjunction with a Hewlett-Packard 524 D counter. In all measurements reported here the deriva tive of the absorption curve was recorded. Solutions examined at room temperature were placed in a Varian aqueous-solution sample cell. III. EXPERIMENTAL RESULTS A. Magnetically Concentrated Crystals ESR of phthalocyanines, and notably copper phthalo cyanine,ro have shown considerable dipolar and anisot ropic exchange interactions between nearest metal ions that are 4.79 A apart. In the metallotetraphenylpor phines, however, one would expect strong magnetic interactions between nearest neighbors, particularly since it is known that in CuTPP crystals the shortest Cu-Cu distance4 is 8.30 A. Surprisingly, the ESR spectra of VOTPP, CoTPP, and CuTPP displayed in each case the hyperfine structure of the paramagnetic ion. In order to verify that the decrease in dipolar interactions is characteristic of the physical stacking of the paramagnetic TPP molecules, and is not due to 10 J. M. Assour and S. E. Harrison, Phys. Rev. 136, 1368 (1964). Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsESR OF TETRAPHENYLPORPHINE CHELATES 2479 FIG. 2. ESR of O.01M vanadyl TPP solution in chloroform mea sured at room temperature. a magnetic dilution of the ions by diamagnetic mole cules incorporated as impurities, the spectrum of pure CuTPP single crystals grown by Adler and Longoll was measured. Indeed, the spectra of several single crystals revealed the copper hyperfine resonances as previously reported by Ingram and co-workers.5 It should be emphasized, however, that evidence of slight dipolar broadening was found and which caused the resonance lines to overlap such that it was difficult to analyze the data accurately. Only by magnetically diluting the neighboring ions were the resonance lines resolved completely. The interactions leading to the substantial reduction of the dipolar broadening between neighboring ions in tetraphenylporphines are presently little understood and in need of exploration. However, one can tenta tively speculate that the out-of-plane crystallographic phenyl rings4 might shield the paramagnetic ion from its nearest metal-ion neighbors, and effectively reduce the magnetic interactions in the crystal. Of course, this hypothesis must await a detailed ESR analysis of magnetically concentrated single crystals. B. Vanadyl Tetraphenylporphine The spectrum recorded at room temperature for a lO-2M solution of VOTPP in CHCla is shown in Fig. 2, and, as expected, the magnetic interaction between the unpaired electron and the vanadium nucleus (I = t) is evident. At room temperature, the resonance spec trum can be interpreted by an "isotropic" spin Hamil tonian given by X.= (go)iJH.S.+(a)S·I, ( 1) where (a)=!(A+2B), (2) 11 We gratefully acknowledge the assistance of A Adler and F. Longo for providing us with pure CuTPP single crystals. These co-workers have also provided the single crystals of H2TPP that were measured by x ray in Ref. 2. A first-part account of their work appeared in the J. Am. Chern. Soc. 86, 3145 (1964). 3400 3600 3800 H(GAUSS) and (3) A and B are the nuclear splitting constants of the vanadium nucleus. Since experimentally the hyperfine splittings were found unequal, the average Hamiltonian parameters were determined with the application of second-order perturbation theory and are expressed by Hm=Ho-am- (a2/2Ho) [(63/4) -m2], (4) where Hm is the resonance value of the applied magnetic field, Ho = hvo/ (go )iJ, and m is the nuclear spin quantum number along the z-axis. For VOTPP dissolved in CHCla, (go)= 1.9797 and (a )=89.4X10-4 cm-I• A spectrum of VOTPP dissolved in CS2 was measured and found quite similar to that shown in Fig. 2. The isotropic Hamiltonian parameters determined for the CS2 solution at room temperature are (go)= 1.981 and (a)=91XlO-4 cm-I. These two solvents, CHCla and CS2, were particularly chosen because ESR spectra6b of vanadyl porphyrin dissolved in CS2 and CHCla glasses displayed extra hyperfine structure arising from the magnetic interaction between the V4+ unpaired electron and the pyrrole nitrogens. Such an interaction, however, has not been observed in vanadyl phthalo cyanineI2 or in other substituted vanadyl porphyrins.6 Figure 3 shows the spectrum recorded at T= 77 oK for the same solution of VOTPP in CHCla. The inter pretation of the resonance spectrum follows closely the methods developed by Sands13 and Bleaney.14 The spectrum is composed of two sets of vanadium hyperfine lines that correspond to gil and gJ.. The two weak peaks at low-field and the three highest-field peaks correspond to gil while the central strong peaks are associated with gJ.. A similar low-temperature spectrum was observed for VOTPP dissolved in CS2• In both solvents we did not detect any extra hyperfine structure due to the 12 J. M. Assour, J. Goldmacher, and S. E. Harrison, J. Chern. Phys.43, 159 (1965). 13 R. H. Sands, Phys. Rev. 99, 1222 (1955). 14 G. Bleaney, Phil. Mag. 42, 441 (1951). Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions2480 JACQUES M. ASSOUR 2600 2800 3600 3800 H(GAUSS) FIG. 3. ESR of 0.01M vanadyl TPP solution in chloroform measured at T=77°K. nitrogens. The values of'gll and A were readily deter mined from the spectra of the frozen solutions, while gJ. and B were calculated from Eqs. (2) and (3) together with (go) and (a). The Hamiltonian param eters are listed in Table II. In order to duplicate the crystallographic environ ment in the solid more closely, VOTPP was mag netically diluted in H2TPP to a ratio of 1: 1000. The ESR spectrum shown in Fig. 4 was recorded at T= 300oK; an identical spectrum was also recorded at liquid-nitrogen temperature. The spectrum displays the general features outlined above for the frozen solutions. The spectrum was interpreted with the following spin Hamiltonian reflecting the axial sym metry X=.B[gIIH.S.+gJ.(H.S.+HyS y) ] +AlzS.+B(I",S,'+IyS y). (S) The experimental Hamiltonian parameters are sum marized in Table II. C. Copper Tetraphenylporphine The spectrum of 0.007 M solution of CuTPP in CHCla is shown in Fig. S. The spectrum recorded at room temperature is composed of four resolved copper resonance lines and extra superhypernne structure arising from the interaction of the 3d electron with the four pyrrole nitrogens (Fig. 1). A detailed resolution of the copper component, m= -!, revealed nine nitrogen lines with an equal separation of llH = 16 G. The isotropic Hamiltonian parameters determined from this spectrum are (go)= 2.1073, (acu)= 97.7XlO-4 cm-I, and (aN )= 1S.9X 10-4 cm. TABLE II. Summary of electron spin resonance data. Compound Diluent IA I IBI AN BN gO gJ. (10-4 em-I) (10-4 em-I) (10-4 em-I) (10-4 em-I) VOTPP CHCla 1 . 966±0 . 0003 1.985±0.0005 161±1 55±1 CS. 1. 965±0. 0003 1 . 990±0 . 0005 159±1 57±1 H2TPP 1. 966±0. 0003 1. 985±0. 0005 161±1 55±1 CuTPP CHCla gu(1)=2.187±0.003 gJ.(1)=2.067±0.0005 A(1)=-218±1 B(I)=-39±1 14.5 16.4 gl (2)=2. 181±0.003 gJ.(2)=U· A(2)=-218±1 B(2)=U H2TPP 2. 193±0.003 2.071±0.0003 A=-202±1 B=-29±1 14.5 16.1 CoTPP H2TPP 1. 798±0.001 3.322±0.001 197±1 315±1 2.034±0.003 2.505±0.003 115±1 92±1 CHCIa 1.848±0.003 gJ.(I) = 3. 330±0. 003 187±1 B(I)=380±2 gJ.(2)=3.198±0.003 B(2)=358±2 gJ.(3)=3.066±0.OO3 B(I)=298±2 • II, undetermined. Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsESR OF TETRAPHENYLPORPHINE CHELATES 2481 z o ~ 0: o C/) m ~ w ~ u. o 3000 3600 3800 4000 H(GAUSS) FIG. 4. ESR of vanadyl TPP magnetically diluted in metal-free TPP. Spectrum recorded at room temperature. A similar spectrum was also recorded for CuTPP dis solved in CS2• In either solution, the expected nuclear interaction between the unpaired electron and each copper isotope was not detected. One can reason that at room temperature the ESR signal of the less abun dant copper isotope is obscured by the Jandom motion of the paramagnetic centers. Upon freezing the solution at liquid-nitrogen tem perature, the recorded spectrum revealed a complicated pattern of both copper and nitrogen hyperfine lines as shown in Fig. 6. The low-field copper transitions with m=!, !, and -! are clearly resolved to allow the z o ~ 0: o C/)>-- ......... ~-- FIG. 5. ESR of O.OO7M copper TPP ~ solution in chloroform measured at room w temperature. ~ ~ ~ !;i > it: w o 2900 calculation of gil and Acu while the high-field compo nents strongly overlap and do not permit direct determination of gJ. and Beu. The low-field copper component with m=! has been amplified to show the distribution of the nitrogen lines. There are 18 visible narrow lines that are equally spaced with a separation ilH=7.2 G. Since the Cu2+ unpaired electron interacts with the four nearest nitrogen (N) atoms and since each of the four nitrogens has a nuclear spin 1=1, a pattern of (21+1)4=81 superhyperfine lines might be expected. The number of lines is considerably reduced if the array of N atoms obeys certain symmetry rules. 3100 3300 3500 H(GAUSSI" Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions2482 JACQUES M. ASSOUR z o ii: 0: o CJ) m <t III :I: l- LL o III > ~ rr III a mI"% Transition Amplified Five Times 3583 2701 H F or example, the puckered configuration of the porphine skeleton suggests that there are two pairs of equivalent nitrogen atoms. Therefore, for each N pair, five reso nance lines with intensities in the ratio 1: 2: 3: 2: 1 are expected, and a total pattern of 52= 25 lines would be observed. If all the N nuclei are magnetically equiva lent, a pattern of only nine lines in the ratio 1 :4: 10: 16: 19: 16: 10:4: 1 should be observed. In Fig. 6 the number of N lines and their intensities cannot be fitted to any of the expected patterns for different sets of N pairs. The observed superhyperfine lines, however, appear to alternate in their intensities which allows us to group them into two sets, each consisting of nine lines with equal spacing ~H= 14.4 G. Each nitrogen set can be interpreted as arising from a different Cu2+ center. Additional support for this grouping of the superhyperfine lines is the following. First, we find that the ratio of intensities determined for the intense set of nine lines is in fair agreement with that expected for four equivalent N nuclei. Second, the superhyperfine splitting ~H= 14.4 G is typical for N ligands in analo gous covalent copper complexes.7 Third, the spectra of solid CuTPP (see below) show only nine equally spaced superhyperfine lines with ~H = 14 G suggesting that the four N are equivalent. The occurrence of two alleged Cu2+ paramagnetic centers in the frozen solu tion of CuTPP in chloroform can be explained in two possible ways: (1) since there are two copper isotopes 63CU and 65CU each with a nuclear spin of t, and since FIG. 6. ESR of 0.007 M copper TPP solution in chloroform measured at T= 77°K. their magnetic moments differ by about 6%, the magnetic dipole transitions associated with each isotope will occur at a different value of the applied magnetic field. The observed separation in gauss between the m=! components of 63CU and 65CU is found consistent with that expected from a comparison of the magnetic moments of the two isotopes. The difference in intensity between the two groups of superhyperfine lines can be related to the fact that the natural abundance of 65CU is about half that of 63CU; (2) the flexibility of the porphine skeleton and its ease of adaptability to the crystalline environment might very well give rise to two Cu2+ centers in frozen solutions where the local crystal fields at randomly oriented molecules are markedly different (see spectrum of CoTPP in frozen solution). This latter interpretation appears to be more favorable because, as mentioned earlier, the ESR spectrum of CuTPP diluted in H2TPP shows only one Cu2+ center whereas no evidence of the nuclear inter action of each copper isotope was found. If either interpretation for the origin of the two sets of superhyperfine lines is correct, the equal splitting of the nitrogen lines implies that the 3d electron interacts equally with the four nitrogens. One can then reason ably state that the nonplanarity of the tetraphenyl porphine ring either does not influence significantly the bonding between the copper and the surrounding nitrogens or is too small to be detected by ESR experi ments. In reality, the degree of nonplanarity of the Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsESR OF TETRAPHENYLPORPHINE CHELATES 2483 ~ i= ... a:: ~ FIG. 7. ESR of copper TPP mag- c( netically diluted in metal-free TPP. ~ Spectrum recorded at T=77°K. ~ AMPLIFICATION INCREASED 10 TIMES o r----~_ III ~ Ii :> ii III o 2700 2900 3100 3300 3500 H (GAUSS) porphine nucleus is such that the distance between each nitrogen atom and the horizontal plane which includes the copper atom is 0.04 A. This deviation has apparently no influence on the symmetry of the crystal field (D4h) surrounding the Cu2+ ion. The spectrum of the frozen solution can be interpreted with the following spin Hamiltonian: JC= iJ[ (gil (1)+ gll(2» H.Sz + (gJ.(!)+gJ.(2» (H",S.+HySy) ] + (Acu(l)+ Acu(2»I.S. + (Bcu(!)+B cu(2» (I",SIt+I1IS1I) +ANI.S.+BN(I",S",+I1IS1I). (6) The best-fitted experimental data along the x, y, and z axes are given in Table II. CuTPP was also diluted in H2TPP, and the spectrum of a polycrystalline sample was recorded at T= 300° and T= 77°K. Figure 7 shows the spectrum at T= 77°K; an identical spectrum was recorded at room temperature. The spectrum is composed of five re solved copper components. The copper components m= -! and -t are each split into nine equally spaced nitrogen lines with a separation !J.H = 16.5 G. The low-field copper components centered around 2800 and 2998 G are also split into nine superhyperfine lines equally spaced with !J.H = 14 G. The ratio of their intensities is found in fair agreement with that expected from theory. Detailed resolution of the superhyperfine lines, particularly those of high-field copper com ponents, failed to show the expected nuclear interaction between the 3d electron and both 63CU and 65CU. Once more, the equally spaced nitrogen lines indicate that in the solid the nonplanarity of the porphine nucleus has little influence on the ESR data, and the unpaired electron is coupled equally to the four pyrrole nitrogens. The spectrum is interpreted by the following spin Hamiltonian reflecting the axial symmetry: JC=iJ[gIIH.S.+gJ.(H",S",+HyS y) + AcuI.S.+Bcu(I",S",+ IySy) ] +ANI.S.+BN(I",S",+IySII)· (7) The experimental data are listed in Table II. The parameters measured for the solid are slightly different from those reported by Ingram and co-workers.5 These workers determined their parameters from the spectra of concentrated crystals. A comparison between the Hamiltonian parameters determined from our spectra for concentrated and dilute CuTPP powder has indeed shown the expected difference. This differc_"e was found dependent on the broadness and overlap of the copper resonances in the concentrated crystal. D. Cobalt Tetraphenylporphine The oxidation state and the electronic environment of the Co2+ ion in CoTPP were found to be severely affected by the method of sample preparation. The first specimen examined was a fine polycrystalline powder of CoTPP diluted in H2TPP. The powder was thoroughly dried from CHCla in a vacuum atmosphere, and packed in a quartz tube which was evacuated and sealed. At room temperature no ESR signal was observed due to a short spin-lattice relaxation time as found in cobalt phthalocyanine.7d At T= 77°K the Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions2484 JACQUES M. ASSOUR resonances of the Co2+ ion were well resolved as shown in Fig. 8. The total spectrum consists of three separate sets of hyperfine lines. The portion marked A in the figure is due to a CuTPP paramagnetic impurity. Efforts to remove it by chromatography were unsuc cessful. The total width of the copper resonance is insignificant in comparison to that of the cobalt reso nance and has no influence on our present discussion. The low-field set of intense lines is composed of eight cobalt lines as expected from the magnetic interaction of the unpaired electron with the cobalt nucleus (l = i). The hyperfine lines are not equally spaced and their splitting increases from 194 to 311 G. This progressive hyperfine splitting is due to substantial second-order effects. From the spectrum gJ.=3.322 and B=395X 10-4 cm-1 are determined. The high-field set of lines that belongs to gil is com posed of only three weak visible peaks. These peaks and their corresponding nuclear transitions are ampli fied in the figure. The remaining g II peaks are masked by the strong gJ. resonances. The unequal spacings of the high-field peaks are of the order of 350 G. The spin Hamiltonian parameters determined from the spectrum are gil = 1.798 and A=197X1fr-4 cm-l• The deviation of gil from the free electron value g= 2.0023 implies that strong spin-orbit interactions are operative in the CoTPP molecule as found in cobalt phthalocyanine. When the same polycrystalline specimen was exposed to air, a different spectrum was recorded at T= 77°K as shown in Fig. 9. Once more, no resonance absorption z o ii: 0:: g III « 11.J ~ ... o 11.J ~ ~ ::> cr 11.J o 1100 1700 2300 81 8' 1 7.' 1 2900 H (GAUSS~ 3500 FIG. 8. ESR of cobalt TPP mag netically diluted in metal-free TPP. The powder was first evacuated and then sealed. Spectrum recorded at 77°K. was observed at room temperature. The new feature of this spectrum is the appearance of a second C02+ hyperfine structure. The new low-field lines consisting of eight peaks are numbered successively. These peaks are almost equally spaced with !lll",80 G. From the spectrum we find that gJ. has decreased by about 30% to 2.505, while B is reduced by a factor of 4 to 92 X 10--4 cm-l. Similar variations are found for the high-field parameters where gil has increased by about 13% to 2.034 and became greater than 2.0023, while A has decreased by 40% to 115 X 10-4 cm-l• These variations of the spin Hamiltonian parameters are not new and have been observed previously for the a and {3 cobalt phthalocynine polymorphs.7d Currently, the origin of the second Co2+ center is a matter of speculation and studies are under way to identify it. There are, however, two possible origins for this new center. The first can be explained in terms of the numerous crystalline modifications of the tetra phenylporphine chelates. In the polycrystalline powder used here the cobalt dilution is about 1: 1000 and consequently the crystal structure of the host com pound H2TPP would be expected to dominate that of the cobalt derivative. Since the complex H2TPP is known to exist in at least three polymorphic modi fications,' one can infer that more than one crystalline environment exists in our sample, a result which will distort the crystal-field symmetry of the Co2+ ion and lead to a different splitting of the 3d energy levels. It should be emphasized, however, that since the condi- ~I 4100 FIG. 9. ESR of cobalt TPP mag netically diluted in metal-free TPP. The powder was exposed to air. The prime numbers refer to the C02+ Center No. 2 while the SUbscripts identify the peaks associated with hand gg. Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsESR OF TETRAPHENYLPORPHINE CHELATES 2485 z o Ii: a: o !B FIG. 10. ESR of cobalt TPP < solution in chloroform measured w at T=77°K. Each set of numbers i= (1, I, I') represents a different Cot+ center. g; I wi T IL 900 __ ~~~ ____ ~~ ____ ~~~-L~~~ ____ ~~ ____ -=~~ tions governing the polymorphic transitions of H2TPP are not known it is rather difficult to explain the effect of vacuum or air on the change of crystallographic phases. The second explanation for the origin of the Co2+ center is dependent on the electronic configuration of the divalent cobalt ion. The crystal-field splitting parameters of square-bonded cobalt complexes have been found to be quite sensitive to axial distortions brought about by extra ligating molecules. Iii It is then conceivable that when the CoTPP was exposed to air, it absorbed gaseous molecules such as oxygen. These molecules can interact with the CoTPP along the axial positions and cause a considerable tetragonal distortion in the pseudo square-planar crystal-field surrounding the ion. This distortion effectively reduces the spin orbit admixtures between the ground and the excited states and thus increases the value of gil as observed here. Previous studies on the cobalt phthalocyanine poly morphs have shown that the a phase was completely converted to the fJ phase when heated at 300°C. Furthermore, when the sample was cooled to room temperature, there was no new crystalline transition, i.e., the sample retained its fJ crystalline structure. We have repeated a similar experiment with CoTPP. After exposure to air, the polycrystalline sample was heated at about 100°C for a period of 5 h. When the powder was examined by ESR, the signals of the second Co2+ center (Fig. 9) completely disappeared while the reso nances of the first Co2+ center (Fig. 8) became stronger. On the other hand, when the sample was once more exposed to air, the signal of the second Co2+ center reappeared. These experiments were repeated three times and in each case the same results were obtained. Furthermore, no apparent change in the spin Hamil tonian parameters before and after heat treatment was found. The sensitivity of the spin Hamiltonian parameters to environment can be most dramatically shown with the low-temperature spectrum of CoTPP dissolved in 16 J. M. Assour, J. Am. Chern. Soc. (to be published). chloroform as shown in Fig. 10. At low temperature the low-field hyperfine lines were identified as those arising from three Co2+ centers. The hyperfine lines of each center are numbered progressively as a function of the magnetic field. The set of numbers 1, I, and l' have been used to identify the three cobalt centers. The intense and relatively broad central line is char acteristic of a Cu2+-TPP impurity. The high-field cobalt components consist of only three hyperfine lines. The interpretation of the low-field lines was based on the characteristic nuclear hyperfine splittings observed earlier in the spectra of solids. An agreement with this assignment of the hyperfine lines was found in the spectra measured for solutions with various concentra tions. For example, when the molar concentration was decreased by about 50%, the number of low-field lines was reduced to only eight. These are identified in Fig. 10 by 1', 2', etc. As the molar concentration was increased, complicated hyperfine patterns were observed similar to those shown in Fig. 10. The spin Hamiltonian parameters derived from the spectrum of the frozen solution are summarized in Table II. IV. THEORY AND DISCUSSION Although the molecular framework of each TPP chelate has been shown by x-ray analysis to be markedly different, the magnetic data measured here for each cation reflects a fourfold axial symmetry. In the molec ular plane, which is assumed to be in the xy plane, the metal ion is surrounded by four pyrrole nitrogens arranged with N-N distance about 2.9 A, whereas along the out-of-plane axial positions the only known ligand is an 0 atom in VOTPP. In solutions, the flexibility of the TPP molecular framework might allow other ligands or molecules to occupy positions above and below the molecular plane as implied from the resonance data. The ligands symmetry. surrounding the metal ion can therefore be effectively described as that of a distorted octahedron, more specifically as D4h• Porphyrin systems are covalent complexes wherein a strong ligand field is acting on the cation. According Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions2486 J A C QUE S 11. .\. S SOU R to crystal-field theory, a strong octahedral field splits up the 3d orbitals of the free metal ion into two groups of orbitals t2g and ego The application of a tetragonal distortion along the z axis lowers the symmetry of the crystal field to D4h and splits up the eg and t2g orbitals into two orbitals each, in the following order of de creasing energy bl>al>b 2>eg• The relative energies of these 3d orbitals strongly depend on the type of axial distortion, i.e., either an axial compression or elonga tion. Furthermore, within the treatment of ligand-field theory, interactions arising from the contributions of the neighboring ligands greatly influence the ordering and the energy levels of the free metal ion. These inter actions are related to the fact that in covalent com plexes a portion of the charge density on the cation is shared by the ligands. The combined effect of both the crystal-field distortion and the delocalization of the metal orbitals in these organometallic molecules makes it rather difficult to calculate the relative energies of the 3d levels and one must rely entirely on empirical data. 3d9CuH Configuration The ESR data measured for the Cu2+ ion are con sistent with the assignment of the 3d unpaired electron to the bl ground state. Using appropriate7c antibonding molecular orbitals, the spin Hamiltonian parameters for the CuH ion in Eq. (7) are: 8'Aa2fJ2[ (a') 1 a' fJ' ] gll=2.0023--- 1--S---T(n) , Llil a 2 afJ gJ.=2.0023- 2'Aa2fJ2[1_(a')S_~ a'o'T(n)], LlJ. a Y1 ao A = PI -,*,aLK-2'Aa2[(4fJ2/ LlII) +t(02/ LlJ.)]1. B= P[i-a2-K -M('Aa202/ LlJ.)], (8) (9) (10) (11) 'A is the spin-orbit coupling constant of the free Cu2+ ion, P=2fJfJn'Ycu(d",'-IJ' I y-3 1 d",'--II'), 'YCu is the magnetic moment of copper, fJ is the Bohr magneton, and fJ .. is the nuclear magneton. K is the Fermi contact term, Llil = E(bl) -E(b2), and LlJ.= E(bl) -E(eg). a and a' are the bonding coe:fficient s7C of the bl molecular orbital, fJ and fJ' are those of the b2 orbital, 0 and 0' are those of the eg orbital. S is the overlap integral between the d:r;'_y' orbital and the normalized nitrogen IT orbitals: (12) and (13) where n= (2/3)1 for trigonal hybridization, R is the metal-nitrogen internuclear distance, Z. and Zp are the effective nuclear charges on the nitrogen atom. The nitrogen superhyperfine structure is expressed as AN= (ta')2(2fJfJn'YN) [ -~o(r)+h(r-3)p], (14) BN= (!a') 2 (2fJfJn'YN) [ -ho(r) -fi(y-3)p]; (15) where 'YN is the magnetic moment of nitrogen, oCr) is the 2s electron density at the nitrogen nucleus, and yp is the radius of the 2p nitrogen orbital. The evaluation of Sand T(n) depends on Rand the nuclear charges. The average metal-nitrogen distance measured4 for several metallotetraphenyl porphines is about 2.0 A. Using the following effective nuclear chargesl6: Z(Cu2+; 3d) = 8.2, Z(N; 2s) =4.5 and ZeN; 2p) =3.54; we findl7 S=0.092 and T(n) =0.33. From Eqs. (14) and (15) the nitrogen bonding coefficient a' can be calculated directly. Using 'YN = 0.4036 and oCr) = 33.4X1Q24 cm-3 after Maki <tnG McGarvey,18 a' = 0.6. From the normalization condition on the ground-state bIorbital a'~(1-a2)!+aS, we find a=0.83. The magnitude of a is similar to those determined for copper porphyrin and phthalocyaninei and is indicative of considerable in-plane IT bonding, i.e., the copper atom is strongly bonded to the pyrrole nitrogens. Further analysis of the data is severely hampered by the lack of accurate estimates of the excitation energies Llil and LlJ.. These energies, which are characteristic of the d-d transitions, are generally determined from the optical absorption spectra. The visible and ultraviolet spectra of several metallotetraphenylporphines were measured by Thomas and MartelF9 and by Dorough and co-workers.20 Of interest are the spectra of the copper and nickel derivatives dissolved in benzene which exhibited a weak absorption band near 5800 and 5600 A, respectively. These bands have been inter pretedl9 as forbidden d-orbital transitions. We have remeasured the absorption spectra of CuTPP in benzene and in several neutral and polar solvents such as chloro form, dioxane, pyridine, etc. The weak band of CuTPP dissolved in benzene was observed at 5800 A but in the other solvents the peak of this band was shifted by 80 A. An additional weak band was also observed at 5000 A and similarly was shifted by about 50 A. These spectral shifts are inconsistent with those ob served in copper acetylacetonate which afforded the identification21 of the 3d-orbital transitions of the CuH ion. It is significant to note that these weak bands were also observedl9.20 in the spectra of the zinc, platinum, and palladium derivatives. Our interpretation of the visible spectra failed to confirm the identification of the weak bands as d-d transitions. In the porphyrins and phthalocyanines the ligand 1r4r transitions (with ex tinction coefficient E'" 105) completely mask the d-d IB D. Hartree, The Calculation of Atomic Structure (John Wiley & Sons, Inc., New York, 1957). 17 H. H. Jaffe and G. O. Doak, J. Chern. Phys. 21, 196 (1953); 21, 258 (1953). 18 A. H. Maki and B. R. McGarvey, J. Chern. Phys. 29, 31, 35 (1958). 19 D. W. Thomas and A. E. Martell, Arch. Biochern. and Biophys. 76,286 (1958). 20 G. D. Dorough, J. R. Miller, and F. M. Huennekens, J. Am. Chern. Soc. 73,4315 (1951). 21 B. R. McGarvey, J. Phys. Chern. 60, 71 (1956). Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsESR OF TETRAPHENYLPORPHINE CHELATES 2487 transitions (E"-' 1(2). The lack of knowledge about the 3d excitation energy has led to conflicting results6•7 in the interpretation of the ESR spectra of these organo metallic complexes. To proceed with our analysis of the data we make use of the differences ~gll and ~gJ. to solve for the ratios {32 / ~ II and fN ~J.. In principle, these ratios can be also calculated from Eqs. (10) and (11). We feel, however, that calculations based on the experimental values of Acu and Bcu might lead to incorrect results for the following two reasons: (1) as was pointed out by Abragam and Pryce,22 and confirmed here, the constants Acu and Bcu are very sensitive to small variations in P and K; (2) a phenomenon yet to be explained for copper complexes is that the nuclear splitting constants, and particularly Bcu, are found7•23 to vary for the same complex when measured in the solid phase and solutions; a result leading to uncer tainties in the interpretation of the data. Needless to say, in copper complexes, quadrupole interactions might be significant (B/Q~4 where Q is the quadrupole constant), and these have been seldom considered7•23 when determining the constant Bcu. Choosing A = -828 cm-I for the free Cu2+ ion S = 0.092, T(n) =0.33, and assuming the contributi~ns of the terms associated with T(n) to be much less than unity in reasonable agreement with previous findings,? .18.23 the following ratios {32 / ~ II = 0.45 and fN ~J.= 0.6 are found. These ratios when substituted in Eqs. (10) and (11) allow the calculation of P and K. First, a comment concerning the sign of the constants Acu and Bcu is in order. A comparison between the average splitting (a)eu=t(A+2B) measured for solu tions and polycrystalline samples indicates that Acu and Bcu must have the same sign. If these constants differ in sign, we would expect an average spacing of about 50 G contrary to that determined experimentally. Moreover, since the sign of Acu is negative for all Cu2+ complexes, our results imply that the sign of Bcu is also negative. Substituting the experimental values of Acu and Beu together with the ratios {32/ ~II and 52/ ~J. in Eqs. (10) and (11), we obtain P=0.037 cm-I and K=0.32 in good agreement with those estimated by Abragam and Pryce. A plot of the bonding coefficients (3 and 5 as a function of the excitation energy is shown in Fig. 11. A maximum value of ~J. = E (bl) -E (eo) = 17 000 cm-I is obtained if it is assumed that 5= 1, i.e., zero out-of-plane 11" bonding. Similarly, assuming zero in-plane 11" bonding, ({3= 1), ~ II = E (bl) -E (b2) = 22 400 cm-I. This approximation shows that the excitation energy ~II>~.I.; the e level • • 0 IS raIsed above the b2 level. This energy scheme has been also found in copper etioporphyrin and phthalo cyanine.7s •• Since in copper acetylacetonatel8 the inter pretation of the ESR and optical absorption data led 22 A. Abragam and M. H. L. Pryce, Proc. Roy. Soc. (London) 206, 164 (1961). 23 H. R. Gersmann and J. D. Swalen J. Chern Phys 36 3221 (1962). ,. ., 1.00 0.95 0.90 flO :; 0.B5 <II. O.BO 0.75 0.70 0.65 O.B FIG. 11. The metal-orbital bonding coefficients {3 and 0 as a function of the excitation energy. to an energy scheme wherein the b2 level is raised above the eo level, we examine below the possibility of an energy cross over between the b2 and eo levels in CuTPP as a function of the bonding coefficients {3 and o. In order for the b2 level to be raised above the e level, it is required that {3 be at least 0.85 and 0=:1 (see Fig. 11). A comparison of the new magnitude of {3 and that of a implies that the amount of in-plane u bonding is equal to the in-plane 11" bonding in CuTPP. From symmetry considerations, however the new bonding scheme appears to be inconsisten~ since the d",_y' orbital points directly towards the nitrogens along the x and y axes and therefore its delocalization is expected to exceed that of the d"y' orbital which transects the dz'_y' orbital. In vanadyl porphyrin6 and phthalocyanine,12 the experimental evidence confirmed the total localization of the d:xy orbital on the metal ion leading to zero in-plane 11" bonding. It is believed her~ t.hat in pseudo square-planar complexes with strong lIgand field, the d:.y orbital is highly stabilized whereas the d,,'_y' orbital is highly delocalized and results in a strong in-plane u bond. On the other hand, the degree of out-of-plane u and 11" bonding associated with d3.'-r' and dz •• llz orbitals are a function of both the 1I"-electron conjugation of the whole organic framework and the basicity of extra out-of-plane ligands. The following bonding scheme is then suggested for the CuTPP molecule. Strong in-plane u bonding with a = 0.83 and a' = 0.6. Negligible in-plane 11" bonding with (3~1 and ~II = E(bl) -E(b2) "-'22000 cm-I• Little out of-plane 11" bonding with 5:::; 1 and ~J.= E(bl) -E(e g):::; 17 000 cm-I. The bonding property of the d3.'-r' orbital is undetermined from the ESR data because this orbital has no apparent influence on the magnetic property of the Cu2+ ion. 3dl V4+ Configuration ESR experiments of covalent (VOH) complexes have confirmed the placing of the vanadium unpaired elec tron in the b2 orbital. Assuming that in VOTPP the Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions2488 JACQUES M. ASSOUR ground state is b2; the relevant spin Hamiltonian parameters after Kivelson and Lee6b are (19) }. is the spin-orbit constant of the free V4+ ion, P= 2(3(3n'Yv(dxy 1 r-31 dxy), "Iv is the magnetic moment of vanadium, (3 is the Bohr magneton, and (3n is the nuclear magneton. K is the Fermi contact term t.1I=E(b2)-E(b l), and t.J.=E(b 2)-E(e g). 0" is th~ bonding coefficient of the oxygen atom. S, TI, and TIo are the overlap integrals defined as and S=2(d z'_II'I-UI), TI = 2 (dxy 1 Pyl), (20) Assuming the V-N internuclear distance R= 2 A the V-O distance6b Rl=1.6 A, Z(VOHj 3d) =7.1; ZeN; 2s) =4.5, ZeN; 2p) =3.54, and Z(O, 2p) =4.06; we find S=0.32, TI=0.13, and TIo=0.12. The evaluation of the bonding parameters is parallel to that discussed for CuTPP. The magnitude of the spin-orbit coupling constant of the vanadyl ion is chosen as that of the VH( 3d3) valence state,24 i.e., }.= 165 cm-l• The bonding coefficient a is taken to approxi mate that of CuTPP, although in VOTPP it might be larger since the charge density on the V4+ ion available for bonding with the nitrogens has been decreased by virtue of the stronger V-O bond. Increasing the magni tude of a leads only to a larger excitation energy t.1I between the bl and b2 levels. The absence of nitrogen hyperfine structure in solutions and in the solid allows us to set the in-plane 'If bonding coefficient (3 equal to unity. The existence of zero in-plane 'If bonding in vanadyl complexes has been also confirmed directly from the ESR spectra.6b Substituting the above con stants in Eq. (16), we find t.1I=E(b 2)-E(b l)Rd 20000 cm-1• In Eqs. (17) we have arbitrarily set (O')TI (0") 6" V2~ 5" TIo«1 to obtain an upper limit for the ratio IN t..L, or t..L= E(bl) -E(e g)::; 19 000 cm-l assuming 0= 1. In vanadyl complexes, ° is less than unity because the orbitals drc. and dllo are :apable of 'If bonding with the oxygen 2P,. and 2PII orbltals. The effect of the vanadium oxygen bonding is known to decrease the excitation energy t.J.. An estimate of the reduced t.J. is unfortunately a matter "T. M. Dunn, J. Chem. Soc. (London) 1959,623. of speculation since it depends in a critical manner on the unknown bonding coefficients 0, 0', and 0". The two other parameters to be correlated with theory are A and B. Once more, a comparison between the average spacing (a)=tcA+2B) measured for solutions and the splittings A and B measured in frozen solutions and solids reveals that the signs of A and B must be t~e same, although their absolute sign cannot be determmed from the present data. Using the ratios (32/t.1I and 02/t..L determined from Eqs. (16) and (17), we find K=0.75 and P= -0.011 cm-l assuming A B>O, whereas K=0.75 and P=+O.Ol1 cm-1 if A: B<O. These values compare favorably with those measured for vanadyl complexes.6•12 The excitation energy t.1I = E(b2) -E(bl) ",20 000 cm-1 is found in agreement with that postulated for the CuTPP complex. Furthermore, the rise of the eg level above the b2 level in VOTPP is consistent with the bonding scheme proposed earlier for the square-bonded tetraphenylporphines. It is shown below that in the CoTPP derivative the small energy separation between the ground state at and the first-excited state eg indi cates once more that the eg level is above the b2 level. 3d'"CoH Configuration The interpretation of the experimental data clearly shows that the square-bonded C02+ ion in CoTPP has highly distorted surroundings. The variations of the spin Hamiltonian parameters in solution and in the solid suggest that the unpaired electron is located in the al orbital. The spatial distribution of this al orbital has been shown to be most affected by out-of-plane tetragonal distortions in the cobalt phthalocyanine complex.7d Although the eg orbitals (d,.. and dll.) are equally ~ffected by axial ligands, an unpaired electron located m a doublet level is inconsistent with the spectra observed here. It is therefore assumed that the al level is the ground state. First-order calculations in terms of}. show that there is no spin-orbit admixtures of the excited states into the ground state and gil is equal to the free electron value 2.0023. Experimentally, however, the relatively low value of gil'" 1.8 for the CoH Center No.1 in the solid and solutions suggests that second-order calcu lations in terms of }.2 are necessary to interpret the data. These calculations are considerably simplified if the ligand orbitals are neglected. The 3d orbitals chosen for the cobalt ion are then those wavefunctions which are formed by a linear combination of the 3d or bitals transforming as the octahedral symmetry group. The first-order g factors are gll=2.0023, (21) gJ.= 2.0023+ (6}'/t.) , (22) and those of second-order calculations are gil = 2.0023-3 (A/t.) 2, (23) gJ.= 2.0023+6(V t.) -6(}'/ t.)2j (24) Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsESR OF TETRAPHENYLPORPHINE CHELATES 2489 where 11=E(al)-E(e q), and X is the spin-orbit coupling constant of the CoH ion. The ratio (X/11) determined from gil = 1. 798 for COH Center No. 1 is about 18% larger than that calculated from Eq. (24). This discrepancy is not unreasonable considering the rough approximations employed in the theory, and in particular the inadequate treatment of the anomalous magnetic behavior of CoH complexes. The spin-orbit constant for the free CoH ion is X=530 cm-I; however, in covalent complexes the magnitude of X is reduced by about 20% to 40%. Choosing X= 400 cm-I, we find 11= E(al) -E(eg) = 1250 cm-I• A comparison of the ratio (X/11) from gJ.= 2.505 of Center No.2 and from gJ.= 3.32 of Center No.1 reveals that 11 increased approximately fourfold in going from Center No.1 to Center No.2. The nuclear splitting constants derived from first order calculations are A=P[4-K-¥(X/11)], (25) B=P[ -t-K+¥(A/11)]i (26) where P=2{3{3n'Yco(da.'-r' 1 ,--31 da.'_r') and K is the Fermi contact term. Substituting the ratio (X/11) de termined for Center No.1, 1 A 1 =0.0197 cm-I, and 1 B 1 =0.0395 cm-I, we find that P=0.025 cm-I and K = 0.5 if we assume that A and B differ in sign. On the other hand, substituting (X/11) for Center No.2, 1 A 1 =0.0115 cm-I, and 1 B 1 =0.0092 cm-I, we find K=0.33 and P=0.38 cm-I• In the development of the theory of hydrated cobalt salts, Abragam and Pryce25 have estimated P=0.022 cm-I and K=0.325 for the COH ion. Our results, at least for the second COH center, are in disagreement with those of the hydrated cobalt salts. It is found here that the variations in the magnitude and ratio of A and B as one goes from Center No.1 to Center No.2 cannot be explained in a reasonable manner in terms of one set of constants P and K which are presumably characteristic of a given 3d ion. The change in K from Center No.1 to Center No.2 is about 34%. Similar variations of the unpaired s-electron effects were found in MnH bonded in ZnS phosphors where K was 30% smaller than that in hydrated salts. The tenfold increase in P in the second CoH center is yet to be explained. It is realized that a more refined theory is necessary to explain and inter pret the anomalous magnetic behavior of square-bonded COH complexes. These interpretations, however, depend 25 A. Abragam and M. H. L. Pryce, Proc. Roy. Soc. (London) 206, 73 (1951). intimately on future detailed x-ray analyses of the different possible CoTPP polymorphs. The small energy separation 11 between the al and eq levels in CoTPP strongly suggests that the eg level is raised above the b2 level. As suggested earlier for the square-bonded planar complexes, the dzy orbital is con centrated mainly in the molecular plane and its in-plane 11" bonding character is believed to be uninfluenced by axial distortions. V. SUMMARY ESR studies of three unsubstituted paramagnetic tetraphenylporphine chelates: vanadyl, copper, and cobalt reveal distorted crystal-field surroundings which are more pronounced in the cobalt derivative. The spin Hamiltonian parameters and the 3d energy levels are found to be greatly influenced by axial distortions in the cobalt chelate as found in cobalt phthalocyanine. The bonding scheme in these complexes indicates strong in-plane (]" bonding characteristic of analogous organo metallic square-bonded complexes, and little or no in plane 11" bonding. Out-of-plane 11" bonding is significant in the vanadyl and cobalt derivatives, and most likely so in the copper complex. The influence of the phenyl groups on the bonding properties of the porphine nucleus is not readily determined from the present data, however, experiments on the chloro-and methoxy substituted tetraphenylporphines are planned to eluci date the role of the substituents on the 1I"-electron system in these molecules. ESR spectra of magnetically concentrated samples indicate a substantial reduction in the dipolar and exchange interactions between neighboring paramag netic ions in comparison to those found in phthalo cyanines. A tentative explanation for the reduction of the dipolar forces is that the crystallographic packing of the phenyl groups above and below the molecule might effectively shield the metal ion from its nearest metal-atoms neighbors. Detailed analyses of the ESR spectra of concentrated crystals are underway to ex amine this effect more closely. Note added in proof: The author is grateful to Dr. E. B. Fleischer for pointing out a critical error for the Cu-Cu distance previously reported in Ref. 4. The correct distance is 8.20 A instead of 3.76 A. ACKNOWLEDGMENTS The author wishes to acknowledge the assistance of Dr. J. Goldmacher and Mr. L. Korsakoff in preparing the tetraphenylporphine complexes. Downloaded 20 Sep 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
1.1754243.pdf	ELECTRICAL TRANSPORT AND CONTACT PROPERTIES OF LOW RESISTIVITY n TYPE ZINC SULFIDE CRYSTALS Manuel Aven and C. A. Mead Citation: Applied Physics Letters 7, 8 (1965); doi: 10.1063/1.1754243 View online: http://dx.doi.org/10.1063/1.1754243 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/7/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electrical properties of metal contacts on laser-irradiated n-type GaN Appl. Phys. Lett. 82, 580 (2003); 10.1063/1.1537515 Low resistance graded contacts to ntype ZnSe Appl. Phys. Lett. 68, 370 (1996); 10.1063/1.116719 Extremely low resistivity erbium ohmic contacts to ntype silicon Appl. Phys. Lett. 55, 1415 (1989); 10.1063/1.101611 Interface studies and electrical properties of plasma sulfide layers on ntype InP J. Appl. Phys. 63, 150 (1988); 10.1063/1.340482 Electrical Effects of the Dissolution of nType Zinc Oxide J. Appl. Phys. 39, 4089 (1968); 10.1063/1.1656929 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.143.23.241 On: Thu, 29 May 2014 05:33:12Volume 7, Number 1 APPLIED PHYSICS LETTERS 1 July 1965 so that there is reasonable assurance that the basic excitation mechanism is understood. The theory should be applicable to most short-pulse gas lasers where heavy-particle diffusion effects are not im portant. I D. A. Leonard, Appl. Phys. Letters (previous Letter). 2 L. E. S. MathiasandJ. T. Parker, Appl. Phys. Letters 3,16 (1963). 3W. R. Bennett, Applied Optics, Supplement 2, Chemical Lasers, p. 13 (1965). 4D. T. Stewart and E. Gabathuler, Proc. Phys. Soc. 72, Pt. 2, 287 -289 (1958). 'Private communication with R. J. Spindler, Avco Research and Development Division. 6S. C. Brown, Basic Data of Plasma Physics, Technology Press and Wiley, 115-116 (1959). 7Ibid., p. 57. ELECTRICAL TRANSPORT AND CONTACT PROPERTIES OF LOW RESISTIVITY n-TYPE ZINC SULFIDE CRYSTALS1 (Hall effect; barrier height; work function; n-type conductivity; E) This Letter describes some electrical contact and transport properties of ZnS single crystals having room-temperature resistivities in the range of 1 to 10 ohm-cm. Previous electrical transport measure ments on ZnS have been done mainly at high tem peratures2 or under photoexcitation.3 Electrical contacts to ZnS which display ohmic characteristics at room temperature have been described by Alfrey and Cooke.4 A serious limitation to a more extensive investigation of the electrical properties of ZnS has been the difficulty in providing ZnS crystals with contacts which would stay ohmic at low tem peratures. It has also been difficult to dope ZnS n-type without simultaneously introducing large concentrations of native acceptor defects. The investigation of the nature of electrical con tacts to ZnS was carried out by cleaving the crystals in a vacuum and immediately evaporating a layer of the desired metal on the cleaved surface. Barrier heights were measured using the voltage variation of the capacitance, the volt-ampere characteristic, and the barrier photoresponse. The detailed pro cedure has been described previously.5 In all cases the measurements agreed within approximately 0.1 e V. The barrier heights for a number of metals are shown in Fig. 1 plotted against the electronega tivity of the metal. The previously reported results on CdS (ref. 5) are also shown in Fig. 1 for compari son. It can be seen that, with the exception of AI, 8 Manuel Aven General Electric Research Laboratory Schenectady, New York C. A. Mead California Institute of Technology Pasadena, California (Received 10 May 1965) the points all fall within approximately 0.1 e V of the straight line of unity slope and intercept of 0.3 eV. This result suggests that a meaningful formulation would be CPR = Xm -X where CPR is the barrier height, Xm is the electro negativity of the metal, and X is a type of "electron affinity" of the semiconductor, as given by the intercept in a plot such as Fig. 1. (The zero of this quantity is arbitrary and does not corre- 2.0 • ZnS x CdS > ~ m -e- f-" :I: C,!) w :I: 0::: W 0:: 0::: « !II 1.0 o '----' ........ _-'-----L~~----'-...J..:C: ...J..._L--*_L--....I....---J 2.0 Xm ELECTRONEGATIVITY (eV) Fig. 1. Barrier heights of various metals on ZnS and CdS as a function of the electronegativity of the metal. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.143.23.241 On: Thu, 29 May 2014 05:33:12Volume 7, Number 1 APPLIED PHYSICS LETTERS 1 July 1965 spond to the vacuum level.) As noted previously,5 the usual formulation using the metal work function results in a much larger scatter, and hence its use fulness is questionable in the present case. It is possible to argue that this will be the case for any semiconductor where the barrier heights are strong ly dependent on the metal. In these materials, the binding is largely of an ionic type, the wave func tions are localized,6 and the conduction and valence bands are best identified with the positive and nega tive ions.7 Under these conditions the electron trans fer across an interface is an atomic process and hence the interface energetics should be more de pendent upon atomic properties like the electro negativity than upon gross average properties like the work function. The present results demonstrate graphically why it is difficult to make ohmic metal contacts to ZnS. In contrast to many of the Group IV and III-V materials where the barriers are fixed by a large density of surface states, the problem in electroding ZnS stems mainly from its very low electron affinity which, according to Fig. I, is 1.14 eV below that of CdS. The best electrical contacts to n-type ZnS would, therefore, be provided by a metal with a low electron affinity. A satisfactory contact must also meet other specifications, e.g., it should be mechan ically compatible with the crystal and should not introduce any undesirable dopants. Contacts with the best overall performance were obtained by etching the ZnS crystals in hot (250°C) pyrophosphoric acid and immediately scribing on the contacts, with an In wire dipped in Hg. After adding more In to each contact,8 they were fired in 350°C in H2 atmosphere. Hall-effect measurements between about 100° and 4000K revealed the presence of two types of levels near the conduction band of ZnS, as shown in Fig. 2a. Hexagonal or cubic crystals doped with Al by firing at 1050°C in liquid ZnAI alloy showed a level 0.014 eV below the conduction band edge. I-doped cubic crystals exhibited the same level when fired at 9S0°C in liquid Zn. In view of the high donor concentration in these crystals (> 1 018 em-a) the 0.014-eV level probably represents a hydrogenic donor level whose ionization energy has been lowered by impurity banding. In I-doped cubic crystals and less strongly AI-doped crystals fired in liquid Zn above lOSO°C the Fermi level has varied between 0.10 and 0.29 eV. In I-doped crystals there appears to be a relatively stable level at 0.10 e V. Freeze-out on a O.IS-e V level in an AI-doped crystal is shown in Fig. 2a. Such behavior is quite similar to that seen in n-type CdS and ZnSe which have also been reported to display shallow9,1l as well as deeplO,ll levels near the conduction band edge, with some variability in the ionization energies of the deep levels. In samples where the ionized donor concentration was measured using the voltage varia tion of the capacitance of a surface barrier, the re sults agreed well with the Hall measurements. The temperature dependence of the Hall mobility of an n-type ZnS crystal is shown in Fig. 2b. The figure also shows the mobility calculated for the case of scattering by polar optical modes (this has been found to be the dominant scattering mechanism In all pure II-VI compounds in this temperature 0) 1O'2f-------- 400 ( cm2) p. Vsec 200 / / / / / /p.po / ED=0.014 eV b) IOO2~--~--+---~--~--~--~8 Fig. 2. (a) Temperature dependence of the electron concen tration in ZnS. Circles: Al-doped hexagonal ZnS fired at l050°C; squares: I-doped cubic ZnS fired at 950°C. (6) Temperature dependence of the Hall mobility of an Al-doped hexagonal ZnS crystal fired at I050°C. The dashed curve represents the calculated mobility of ZnS assuming scattering by the polar optical modes to be the predominant mobility limiting mech anism. The dotted curve is the mobility calculated for scattering by charged impurities. 9 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.143.23.241 On: Thu, 29 May 2014 05:33:12Volume 7, Number 1 APPLIED PHYSICS LETTERS 1 July 1965 range), and for the case of scattering by charged impurities. The concentrations of ionized donors and acceptors used in the calculations were obtained from the Hall data on the same crystal. It can be seen in Fig. 2b that the behavior of the experimental mobility can be understood in terms of these two scattering mechanisms. The presented data show that ZnS, like its close homologues CdS and ZnSe, can be made in low resistivity n-type form, and that it displays an energy level structure and mobility behavior similar to these compounds. The observed difference of 1.14 eV in the electron affinities of CdS and ZnS is close to the difference in the band-gap energies of these materials (2.4 eV and 3.6 eV, respectively, at room temperature, from electrical measurements12.13), indicating a vacuum level which is in both cases approximately the same energy above the valence band edge, a fact not surprising since the valence band can be identified with a sulfur ion in each case. The authors express their appreciation to W. G. Spitzer for many helpful suggestions, to H. M. Simpson for the preparation of surface barriers, to J. S. Prener for providing the cubic ZnS crystals, and to Miss E. L. Kreiger for evaluation of the transport data. ISponsored, in part, by the A. F. Cambridge Res. Lab. [Con tract No. AF-19 (628)-4976] and, in part, by the Office of Naval Research [Contract No. Nonr-220(42)]. 2M. Aven, Electrochem. Soc., Extended Abstracts 11, 46 (1962). 3F. A. Kroger, Physica 22, 637 (1956). 'G. F. Alfrey and]. Cooke, Proc. Phys. Soc. (London) B70, 1096 (1957). 5W. G. Spitzer and C. A. Mead,]. Appl. Phys. 34, 3061 (1963). 6C. A. Mead, Appl. Phys. Letters 6, 103 (1965). 7N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals, Dover, New York, p. 70 (1964). 8R. Smith, Phys. Rev. 97, 1525 (1955). 9M. Aven and B. Segall, Phys. Rev. 130,81 (1963). lOR. H. Bube and E. L. Lind, Phys. Rev. 110, 1040 (1958). "H. H. Woodbury and M. Aven, Radiation Damage of Semi conductors, Dunod, Paris, France, p. 179 (1964). 12M. Balkanski and R. D. Waldron, Phys. Rev. 112, 123 (1958). 13W. W. Piper, Phys. Rev. 92, 23 (1953). PREPARATION OF FERRITE FILMS BY EV APORATION (electron beam melting; E) Several methods of preparing thin ferrite films have been developed in recent years, e.g. pyrolytic spraying of organic complexes with proper metallic ion contents,l evaporation of the metal constituents and subsequent oxidation at elevated temperatures,2 spraying suspensions of hydroxides against hot substrates,3 and cathodic sputtering.4•5 However, in addition to being time-consuming some of the above methods are hazardous. For example: 1) metal organic complexes are very reactive especially in moist air and most of them are extremely toxic; 2) long annealing times are necessary to convert the evaporated metals into ferrites; 3) it is extremely difficult to prepare mixed ferrite films using the hydroxide spraying method; 4) very pure poly crystalline magnetite and hematite had to be cut and ground into discs to successfully sputter these materials; in addition, it took approximately 30 min to obtain a 1000-A film (33 A/min); 5) the same ob- 10 A. Baltz Franklin Institute Research Laboratories Philadelphia 3, Pennsylvania (Received 2 June 1965) jections as in number 2 exist, e.g. long annealing times are necessary to convert the metal films into ferrites. In this Letter results are presented on the direct evaporation of ferrite powders in an oxygen atmosphere. Pressed ferrite powders were placed into a water cooled copper trough. The vacuum system was pumped until it reached the 1O-6-torr range (ap proximately 20 min). Then by bleeding in oxygen, the pressure was adjusted to 5 JL. The evaporation was carried out using a Denton DEG-80 1 Electron Beam Gun.7 While in general electron guns have to operate at a maximum pressure of 10-4 torr, the unique design of this gun makes it possible to oper ate at pressures as high as 10 JL. The materials were deposited onto single-crystal rock-salt and glass substrates held at room temperature. The evapora tion rate was 1000 A/min. The films deposited onto rock-salt substrates This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.143.23.241 On: Thu, 29 May 2014 05:33:12
1.1702671.pdf	FieldEffect Light Modulation in Germanium B. O. Seraphin and D. A. Orton Citation: Journal of Applied Physics 34, 1743 (1963); doi: 10.1063/1.1702671 View online: http://dx.doi.org/10.1063/1.1702671 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simulation of tunneling field-effect transistors with extended source structures J. Appl. Phys. 111, 114514 (2012); 10.1063/1.4729068 Ballistic current in metal-oxide-semiconductor field-effect transistors: The role of device topology J. Appl. Phys. 106, 053702 (2009); 10.1063/1.3197635 Physics of strain effects in semiconductors and metal-oxide-semiconductor field-effect transistors J. Appl. Phys. 101, 104503 (2007); 10.1063/1.2730561 Performance and potential of germanium on insulator field-effect transistors J. Vac. Sci. Technol. A 24, 690 (2006); 10.1116/1.2167978 Growth and transport properties of complementary germanium nanowire field-effect transistors Appl. Phys. Lett. 84, 4176 (2004); 10.1063/1.1755846 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.176.129.147 On: Wed, 17 Dec 2014 11:13:52JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 6 JUNE 1963 Field-Effect Light Modulation in Germanium B. O. SERAPHIN AND D. A. ORTON Research Department, Michelson Laboratory, China Lake, California (Received 10 September 1962) Infrared light that is sent through a germanium prismatoid so that it is reflected several times internally, becomes modulated in intensity if the space-charge layers along the reflecting surfaces are changed by means of the field effect. The phase angle between the modulation signal and the field voltage changes by 11" when the surface is changed from n to p type, or vice versa, by change of the gaseous ambient. The modulation signal shows twice the frequency of the field voltage in the transition region between nand p type. This, together with other observations such as the waveform analysis of the modulation signal and the dependence of the modulation depth upon field voltage, indicates that the modulation is caused by free-carrier absorption in the space-charge layer, which follows roughly the master curve of the field-effect surface conductivity. The results are interpreted in terms of the field-effect mobility. INTRODUCTION THE electrical conditions in the surface of a semi conductor are known to be sensitive to the gaseous ambient; the electric charge of ions adsorbed onto the surface is compensated by an equal space charge layer of opposite sign which extends up to 10--4 cm into the material. Surface conductance measure ments show that this space-charge layer can be varied by a proper cycle of gaseous ambients from an accumu lation of majority carriers through a position of equal numbers of holes and electrons to an inversion layer characterized by an accumulation of minority carriers.1 Mobile carriers in the space-charge layer absorb infrared light.2 Internally reflected light passes twice through the space-charge layer.s The internally reflected light is modulated when the space-charge layer is changed by variations in an electric field perpendicular to the surface. This paper reports on measurements of the modulation as a function of surface conditions. By application of a proper cycle of gaseous ambients, the point of operation is moved through the different types of space-charge layers and the resulting modulation is measured with respect to its magnitude, its harmonic composition, and its phase relation to the applied elec tric field. The results fit the model of a semiconductor surface as deduced from field-effect surface conductance measurements and establish a correlation between the observed modulation and the field-effect mobility. The technique of using the modulation of infrared light by the absorption of free carriers, developed to a high degree by Harrick,4 was employed. His extensive investigations have probed out carrier distributions in semiconductors in the bulk as well as in the surface and have shown how most of the surface parameters can be obtained from an analysis of internally reflected infrared light. 1 R. H. Kingston, J. App!. Phys. 27,101 (1956). 2 K. Lehovec, Proc. Inst. Radio Engrs. 40, 1407 (1952); A. F. Gibson, Proc. Phys. Soc. (London) B66, 588 (1953); R. Newman, Phys. Rev. 91, 1311 (1953); and H. B. Briggs and R. C. Fletcher, Phys. Rev. 91, 1342 (1953). a,N. J. Harrick, Phys. Rev. 125, 1165 (1962). 4 N. J. Harrick, J. Phys. Chem. Solids 14, 60 (1960). Bibliog raphy of this review article lists N. J. Harrick's papers on this subject. FIELD-EFFECT MOBILITY AND MODULATION The field-effect mobility5 is defined as Ji.FE=d(fl.G)/dQ, (1) where fl.G is the conductance of the sample per square of surface and Q is the net charge per unit area of sur face, induced by the transverse electrical field. If the change dQ is caused by a change dV of the voltage V across the capacity C per unit area between field plate and sample, Eq. (1) can be written as Ji.FE= (ljC)[d(fl.G)/dV]. (2) This expression represents the slope of the curve which plots the surface conductance fl.G as a function of the total charge Q in the surface per unit area and which is referred to later on as the master curve of the field effect.6 It has the shape of a slightly unsymmetrical parabola. The field-effect mobility, as the slope of this curve, has negative values on the side of negative induced charge (n side) and positive values on the side of positive induced charge (p side). The surface conductance fl.G is produced by a surface excess fl.P and .fl.N per unit area of mobile holes and electrons, respectively, fl.G= e(!Lpfl.P+!LnAN). (3) We replace the hole and electron mobilities, Ji.P and !Ln, by an average mobility Ji., which makes the master curve symmetrical to the fl.G axis (4) With this approximation, the field-effect mobility is written (5) If, on the other hand, a light beam of intensity 10 and a wavelength for which the bulk of the material is transparent, travels from the interior towards the surface at an angle () greater than the critical angle for total internal reflection, it traverses the space-charge 5 J. R. Schrieffer, Phys. Rev. 97, 641 (1955). 6 H. C. Montgomery and W. L. Brown, Phys. Rev. 103, 865 (1956). 1743 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.176.129.147 On: Wed, 17 Dec 2014 11:13:521744 B. O. SERAPHIN AND D. A. ORTON L FIG. 1. Block diagram of the experimental setup for the field effect light modulation: (L) collimated light beam; (FE) field electrodes separated from the 30 n· cm n-type germanium prisma toid by a 0.25-mil Mylar strip; (AG) sine-wave generator; (PCI) , (PC2) matched PbS cells; (DA) differentia~ amplifier; (WA) wave analyzer; (DT-CO) double-trace oscIlloscope; (TMV) tunable microvoltmeter; and (I) integrator. layer twice upon reflection. This causes an absorption of the light by the free carriers of this region and makes the reflection less than totaF by an amount Ill: I R=lo-M=lo exp{ -(2/cos8)[kpIlP+knIlNJ}. (6) In the case of germanium in the wavelength region 2 to 5 p., the absorption cross sections kp and kn are both smaller than 10-15 cm2•8 Since the number of free carriers per unit area of the surface rarelv exceeds 1014 cm-2, (kp!:!.P+k nIlN)«1 is valid in m~st cases. Therefore, Eq. (6) reduces to M/lo= (2/cos8)[kpIlP+knIlNJ (7) We again equalize the contributions of holes and electrons by assuming kp=kn=k. Fortunately, this approximation works in the opposite direction to the previous one: In going from Eq. (3) to (4), we reduced the contribution of the electrons to the conductance and enhanced that of the holes. The absorption cross section kp, on the other hand, is larger than kn in this wavelength region,8 so that replacing kp and kn by an average value k means enhancing the contribution of the electrons to the free-carrier absorption and reducing that of the holes. The processes involved are too complex, of course, to give to this argument quantita tive significance. In particular, our assumption of equal absorption cross sections for holes and electrons is rather questionable. Due to hole absorption bands, there is a larger difference between kp and kn than there is between the mobilities. This can shift the infrared absorption 7 There is some doubt about the validity of the l/cos6 depend ence. Due to the fact that a standing-wave pattern is set up in the sample 'Yhe.re the electric field at the surface depends upon the angle C!f InCidence, the angular dependence of the absorption may not qUite follow l/cos6. (Private communication by N. J. Harrick.) 8 H. B. Briggs and R. C. Fletcher, Phys. Rev. 87, 1130 (1952); Phys. Rev. 91, 1342 (1953). minimum with respect to the surface conductance minimum. The general shape of the absorption curve, however, is similar to that of the master curve of the field effect. The qualitative conclusions, which we derive from this general shape is, therefore, permissible. With the above approximation, the intensity loss upon total internal reflection is written M/lo= (2k/cos8) (IlP+!:!.N). (8) If the space-charge layer is now changed by a change in the field voltage V, the reflected beam is modulated according to d(M/lo)= (2k/cos8)d(IlP+!:!.N), (9) or, by using the field-effect mobility from Eq. (5), d(M/lo)= (2kC/ep. cOS8)·j).FE·dV. (10) If the point of operation is now moved along the master curve by changing the gaseous ambient of the surface, one expects the following features of the modulation from the dependence of the field-effect mobility upon the surface condition: (a) The modulation depth is large in the outer branches of the master curve and small around the minimum. (b) The phase angle between the modulation signal and the field voltage changes by 7r when the surface is changed from n to p type, or vice versa, by change of the gaseous ambient because of the different sign of the field-effect mobility in the two branches of the master curve. (c) The modulation signal has twice the frequency of the field voltage around the minimum of the master curve, because of the change of sign of the field-effect mobility in the minimum. EXPERIMENTAL PROCEDURE The modulation of the infrared light was accom plished by internal total reflection in germanium prismatoids of the kind used by Harrick9 and others.10 The prismatoids were cut from 30 n-cm n-type single crystals, with dimensions that give four internal reflec tions. The samples were mechanically polished, etched for 15 sec in CP 4, and rinsed in deionized distilled water. This procedure is suitable for taking off the damaged surface layer without spoiling the optical finish too drasticallyY Figure 1 shows a schematic diagram of the experi mental setup. The two prismatoids face the collimated light of a battery-operated tungsten filament lamp and are adjusted with respect to the light beam so that an 9 N. J. Harrick, J. Phys. Chern. Solids 8, 106 (1959); Phys. Rev. Letters 4, 224 (1960); and J. Phys. Chern. 64, 1110 (1960). 10 L. H. Sharpe, Proc. Chern. Soc. (London) 1961, 461; and J. Fahrenfort, Spectrochim. Acta 17, 698 (1961). 11 T. M. Donovan and B. O. Seraphin, J. Electrochern. Soc. 109, 877 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.176.129.147 On: Wed, 17 Dec 2014 11:13:52FIELD EFFECT LIGHT MODULATION IN GERMANIUM 1745 equal amount of light enters each prismatoid. After four internal reflections, the two light beams fall onto matched PbS cells that are connected to a differential amplifier. The output of the differential amplifier can be connected to one trace of a double-trace oscilloscope, to a wave analyzer, or a tunable microvoltmeter with an integrator behind it. One prismatoid had two field electrodes attached to its long faces, separated from the sample by a i-mil Mylar strip. One terminal of a sine wave audio generator was connected to both field electrodes, with the other terminal connected to the prismatoid. The frequency was 85 cps, using values for the electric field up to 3 X 105 V / cm. The field-electrode terminal of the audio generator was at the same time connected to the other trace of the double-trace oscilloscope, so that the phase relation between field signal and modulation signal could be seen on the screen. The sample holder was enclosed in a glass housing with provisions made for feeding ozone, wet oxygen or nitrogen, borontrifluoride, or acetone into it. The fundamental absorption edge of the germanium on one side and the sensitivity of the PbS cell on the other limited the effective bandwidth to wavelengths between approximately 1.8 and 3.5 !J.. This is a spectral region in which the optical properties of germanium depend only very little upon the wavelength, so that the whole region contributes almost equally to the effect. EXPERIMENTAL RESULTS The magnitude of the modulation and its phase rela tion to the field signal as well as its composition with respect to the 8S-cps first harmonic and 170-cps second harmonic were measured as a function of the surface condition. Figure 2 shows the results of such a run with the surface starting on the p side and being driven through the flat-band position onto the n side by exposure to water vapor. The upper trace of the right-hand column shows the sinusoidal field voltage set at a value of 200-V peak-to-peak. The lower trace shows the differential output of the balanced photocells. The left-hand column represents the waveform analysis of the modulation signal. The highly p-type surface in a gives a linear modula tion which is out of phase with the field signal. The positive peak of the field voltage drives the surface towards the flat-band position, thereby decreasing the number of free carriers in the surface. After 3-min exposure to wet oxygen, the positive voltage peak drives the surface close enough towards the minimum of the master curve to flatten out the modulation signal and to show a second harmonic coming up at 170 cps in the waveform analysis. The trend is more pronounced after 5-min exposure to wet oxygen. Trace d shows the situation close to the minimum of the master curve. The magnitude of the modulation, as seen from the increased sensitivity of the lower trace 85 I 170 cps I (a) (b) (c) (d) (e) (0 Upper trace deilection lOOV/cm FIG. 2. Magnitude, phase relation to the applied field, and wave form analysis of the modulation signal during the transition from a p-type inversion layer to an n-type accumulation layer. (a) ozone 4 min (p-type surface), 20 m V / cm lower trace deflection; (b) 3-min wet oxygen, 10 mV/cm; (c) 5-min wet oxygen, 10 mV /cm; (d) 6-min wet oxygen,S mV /crn; (e) 10-min wet oxygen, 2 mV/cm; (f) 17-min wet oxygen, 10 mY/em. amplifier, is further decreased, but the modulation shows twice the frequency of the field signal. The surface is moved back and forth around the minimum, with the field-effect mobility changing sign twice within one cycle of the electric fieJd. The wave analysis shows a strong 170-cps component. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.176.129.147 On: Wed, 17 Dec 2014 11:13:521746 B. O. SERAPHIN AND D. A. ORTON 85 I 170 cps I (a) (b) Upper trace deflection 100 V/cm -120-V de bias Low'er trace deflection 1 mV/em +120-V de bias FIG. 3. The effect of a positive and negative de bias on the modulation signal, with the unbiased surface close to the flat-band position. The remaining traces show the situation beyond the minimum on the n side. The magnitude of the modula tion is increasing again, with the largest contribution from the in-phase half-wave now. The modulation is becoming linear again, but now in phase with the field signal, as expected in an accumulation layer, in which the positive peak of the field voltage now drives the surface to higher carrier densities in the space-charge region. Figure 3 shows the influence of a positive and negative de bias on the surface. The no-bias point is located slightly to the left of the minimum. Negative bias drives the modulation back into the out-of-phase condi tion of the p-type inversion layer, positive bias puts the surface slightly to the right of the minimum, and the slightly superior in-phase half-wave indicates an accumulation layer. Figure 4 shows the modulation depth in a run from n type to p type and back, applying a positive, a negative, and no dc bias in each of the three different surface conditions produced by three different gaseous ambients. Similar to the method used in gaining the master curve of surface conductivity, the photocell output as a function of field voltage can be plotted to illustrate the significance of the modulation signal as the gradient of an absorption master curve. Figure 5 finally shows the modulation depth as a function of the field voltage, with the surface in a strong n-type condition. Consequently, the modulation depth is a linear function of the voltage, with peak values of 4% modulation depth or 1% modulation per reflection. Close inspection of the traces in Fig. 2 shows that the peak of the modulation trace in the case of the inversion layer (photograph of top) is slightly delayed with respect to the peak of the field signal. This effect does not appear in the case of an accumulation layer where both peaks coincide. We have studied this under higher vertical deflection and horizontal sweep rates, as well as for frequencies up to a few kilocycles per second, and found that the time constant of this delay is in the order of 100 f,Lsec, the lifetime of the minority carriers in our particular material. DISCUSSION The experimental results suggest that the observed modulation is produced in the space-charge layer inside the surface. Magnitude, phase relation to the field signal, as well as the doubling of the frequency around the flat-band position of the surface are in good agree ment with what one would expect from the established model of a semiconductor surface which is moved from [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.176.129.147 On: Wed, 17 Dec 2014 11:13:52FIELD-EFFECT LIGHT MODULATION IN GERMANIUM 1747 n type to p type or vice versa. Measurements of the field effect which were made on the same material under identical conditions with identical surface treatments showed that the surface potential was changed by up to 10 kT / e. The related change in carrier densities in the space-charge region fits the modulation experiment assuming absorption cross sections of the order of magnitude observed by others.8 To explain the observed modulation, we have con sidered several possibilities, other than carrier modu lated absorption. They all fail to explain all of the observed features. An absorbing dielectric, for instance, not bonded to the surface and therefore moving in and out of the region where the penetrating radiation is found, gives rise to a modulation at twice the frequency of the modulating field if moving in an unpolarized condenser and at the field frequency for a polarized condenser. The phase of this signal, however, is different from the phase of the carrier modulated absorption. Modulation effects due to optical transitions involving surface states, as investigated by Harrick,a enhance the carrier modulated absorption on one side of the master curve and oppose it on the other side. Since this establishes a phase relationship contrary to the experi mental observations, the latter cannot be explained by surface state effects only. The close relation between modulation and field- I Division =IOOO/LV 1 Sa pie exposed to BF! Sample exposed to Acetone vopor '-// -' -150 0+150-1500+150 D.C. Bias / / / Sample etched inCP4 -150 0+150-'1 FIG. 4. The effect of a dc bias on the modulation signal for different surface conditions during a run from an n-type surface to a p-type surface and back, plotted to show the significance of the modulation signal as the gradient of an absorption master curve. %r-----------------------------------~ t ~ .... C J4 I .. J ~ 10. . ... ~ 2 i '3 ... o ,2 Sampl •• 'ch.d In CP4 In-t". surface) ISO 300 Modulalln, Yolta,. 01 es cpa FIG. 5. The modulation depth in an n-type accumulation layer as a function of the peak-to-peak modulating voltage between field electrode and sample. effect mobility suggests an expansion of the present work to higher frequencies. Montgomery's results on the frequency dependence of the field-effect mobility in an inversion layer12 show that the minority carriers are no longer effective in the space-charge region if one cycle of the electric field is shorter than the lifetime. The delay of the modulation with respect to the applied field, which we observed in the case of an inversion layer but not in the case of an accumulation layer, seems to confirm Montgomery's result. Building up the space charge layer and establishing equilibrium between valence and conduction band via recombination centers is governed by the time constant of the usual lifetime processes. In the case of the accumulation layer, the space charge consists predominantly of majority carriers and only a negligible change in electron density is required to establish equilibrium between the bands. This delay effect demonstrates the ability of the field effect light modulation to provide more information on nonequilibrium conditions within the surface than measurements of the field-effect mobility 'alone. Work is presently underway to extend the frequency range of this investigation by use of faster detectors, as well as microwave techniques. According to Montgomery's measurements, which extended upto SO Mc/sec, the field effect mobility in an inversion layer changes sign at the inverse-of-the-lifetime frequency and grows up to values greater than the bulk mobility beyond this frequency. He interprets his results with a theory which suggests that the field-effect mobility should have the value of the majority carrier bulk mobility up to frequencies in the order of 1012 CpS.12 If the close relation between modula tion and field-effect mobility holds at higher frequencies too, this mechanism, with appropriate modifications of the system,points towards interesting applications in the field of light modulation at very high frequencies. The small value of the modulation depth of about 1% per 12 H. C. Montgomery, Phys. Rev. 106, 441 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.176.129.147 On: Wed, 17 Dec 2014 11:13:521748 B. O. SERAPHIN AND D. A. ORTON reflection is no serious objection to the technical applica tion of the effect. It can be shown that by the use of interferometric techniques in properly matched multi layer systems, a minute change in an optical system can be amplified to a considerably larger modulation.13 13 B. O. Seraphin, J. Opt. Soc. Am. 52, 912 (1962). ACKNOWLEDGMENTS We are indebted to T. M. Donovan for help with the preparation of the samples, as well as to Dr. H. E. Bennett for advice on the optical part of the experiment. Dr. N. J. Harrick has made some valuable comments, which are included in the discussion part of this paper. JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 6 JUNE 1963 Solubility of Zinc in Gallium Arsenide J. O. McCALDIN North American Aviation Science Center, Canoga Park, California (Received 3 January 1963) The distribution of tracer zinc-65 between the vapor and solid GaAs was studied. For dilute concentrations [Zn~J of zinc in the s?lid, the dis!ribution coefficient K is a constant (Henry's law); at higher zinc concen trations, K falls off mversely WIth [Zn.J. These observations can be interpreted simply in terms of an ionization equilibrium Zn. ---> Zn.-+e+. Based~on this interpretation, the present measurements indicate an intrinsic carrier concentration n; of about 4X10l8 cm-a for GaAs at lO00°C. This value is roughly six times larger than n; estimated by extrapolation of Hall measurements; the latter, it is suggested, may reflect the presence of only the more mobile carriers. ~h~ solubility of ~c was also studied as the arsenic pressure in the system was changed from the dis SOCiatIOn pressure (estimated 10-3 atm) to one atmosphere. The zinc solubility was observed to increase three to fourfold with the increase in arsenic pressure. This result is in semiquantitive agreement with calculations for the mass action equilibrium of simple stoichiometric defects in GaAs. I. INTRODUCTION IMPURITY solubility studies in semiconductors are favored by a relatively simple model for interpreta tion. The work of Reiss, Fuller, and Morin! on lithium in germanium and silicon showed how the Fermi level in the semiconductor host is a governing factor for the lithium solubility. In compound semiconductors like GaAs, where an additional thermodynamic degree of freedom is present, the solubility of an impurity de pends not only on the Fermi level but also on the stoi chiometric balance of the compound, e.g., the Ga-to-As ratio in GaAs. The stoichiometry of many of these compounds may be easily controlled, however, by fixing the vapor pressure of a volatile component, e.g., the arsenic pressure over GaAs. In many compound semiconductors, furthermore, the effects of the Fermi level and of stoichiometry should be simply additive on a property like an impurity solu bility. Consider, for example, the location of the Fermi level. This depends on the various ionization processes in the crystal and is dominated by those processes which involve relatively large concentrations. In GaAs at elevated temperatures, the intrinsic carrier concen tration is of the order of 1018 cm-3 and fixes the location of the Fermi level, unless a chemical impurity is intro duced at high concentration. Stoichiometric defects like vacancies and interstitials, being present at con- 1 H. Reiss, C. S. Fuller, and F. J. Morin, Bell System Tech. J. 35, 535 (1956). siderably lower concentrations, e.g., 1015 cm-B, would, therefore, not influence the Fermi level; they do, how ever, still influence properties like solubility by par ticipating in the solubility reaction.2 Thus a model for interpretation of solubility data in GaAs obtained by simply superposing the ionization equilibrial and the stoichiometric equilibria2 seems reasonable. Such a model is discussed later in this paper. The early work of Whelan et al.,a on the behavior of Si in GaAs indicated several of the possiblities in solubility studies. In their interpretation they regarded the silicon as having a separate solubility on each sub lattice of the host compound. Thus the net doping depended on the difference in silicon solubilities on the two sublattices. Quantitative agreement between this interpretation and experiment was obtained by them, particularly in regard to the influence of the Fermi level in controlling the two silicon solubilities. Late!:. experiments on Ge in GaAs by the present author and by Harada4 showed that stoichiometry could also be important, controlling the semiconductor type. These experiments have since been put on a 2 D. G. Thomas, Semiconductors, edited by N. B. Hannay (Reinhold Publishing Corporation, New York, 1959), Chap. 7. 3 J. J¥. Whelan, J. D. ~truthers, and J. A. Ditzenberger, Proceedmgs of the lnternat~onal Conference on Semiconductor Physics, Prague 1960 (Czeckoslovak Academy of Sciences Prague 1961), pp. 943-945. ' , 4 J. O. McCaldin and Roy Harada, J. Appl. Phys. 31, 2065 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.176.129.147 On: Wed, 17 Dec 2014 11:13:52
1.1728272.pdf	Effect of Electron Bombardment on the NearInfrared Fluorescence of Single Crystal CdS B. A. Kulp Citation: Journal of Applied Physics 32, 1966 (1961); doi: 10.1063/1.1728272 View online: http://dx.doi.org/10.1063/1.1728272 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Transport properties of single-crystal CdS nanoribbons Appl. Phys. Lett. 89, 223117 (2006); 10.1063/1.2398891 PhotoHall effect of CdS:Li single crystals J. Appl. Phys. 47, 3360 (1976); 10.1063/1.323093 Thermoelectric and photothermoelectric effects in semiconductors: CdS single crystals J. Appl. Phys. 44, 138 (1973); 10.1063/1.1661848 Effects of Electron Bombardment on SingleCrystal CdSe at 77°K J. Appl. Phys. 37, 4936 (1966); 10.1063/1.1708168 THE GROWTH OF WURTZITE CdTe AND SPHALERITE TYPE CdS SINGLECRYSTAL FILMS Appl. Phys. Lett. 6, 73 (1965); 10.1063/1.1754172 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.102.42.98 On: Sat, 22 Nov 2014 16:02:461966 J. D. LIVINGSTON AND C. P. BEAN creasing the volume fraction of precipitate or by de creasing the average particle size. For positive K in the approximation of S= 00, we derive Cv=k(l+kT/KV) per particle for T«KV/k. Thus, superparamagnetic particles under optimum conditions might also make a measurable contribution to /" the linear temperature coefficient of the specific heat. It should be noted that the above treatment has neglected the effects of any interaction between particles. Note added in proof. An article on this subject has recently appeared7 in which anisotropy has been repre- 7 K. Schroder, J. App!. Phys. 32, 880 (1961). sented approximately by an effective field. For the case of large Sand kT«p.H, it is shown there that the expo nential drop of specific heat as T -t 0 takes the form of the Einstein specific heat function. ACKNOWLEDGMENT Prior to the appearance of the paper by Schroder and Cheng, this problem had been called to our attention by N. Kurti, who had also raised the question in discussion at the 1958 International Magnetism Conference.8 8 N. Kurti, J. phys. radium 20, 221 (1959). JOURNAL OF APPLIED PHYSICS VOLUME 32. NUMBER 10 OCTOBER. 1961 Effect of Electron Bombardment on the Near-Infrared Fluorescence of Single-Crystal CdS B.A. KULP Aeronautical Research Laboratory, Wright-Patterson Air Force Base, Ohio (Received April 24, 1961) Under electron bombardment at 40°C, two fluorescence bands in the near infrared are observed in many CdS crystals. The bands are at about 8500 A and 1.05 1-1. The 8500-A band is reduced in intensity by electron bombardment at 100 and 275 Kev and by exposure to x radiation. The 1.05-1-1 band is not greatly affected by these irradiations. Heat treatment for! hr at 200°C partially restores the 8500-A band. The effect is interpreted as a redistribution of electrons over the existing defects. The defect responsible for the 8500-A band is believed to be copper in a particular ionization state. The 1.05-1-1 band is observed to appear after heat treatment at 200°C if it is not present originally. This fact makes the origin of this latter band uncertain. INTRODUCTION THE use of an energetic electron beam to produce defects in CdS has produced some interesting results concerning edge emission and another fluores cence band in CdS. Collins! produced edge emission with a 2oo-kev beam of electrons, and on the basis of the effect of heat treating in a sulfur atmosphere, con cluded that edge emission was a result of sulfur vacan cies. Kulp and Kelley,2 on the other hand, measured the threshold for the production of edge emission by electron bombardment as being 115 kev, and proposed that edge emission was a result of sulfur interstitials. Further, the latter authors proposed that sulfur vacan cies are the center for a fluorescence band at 7200 A at nOK. Kulp and KelleyS have observed that the 1.4-1-' quenching band is reduced in intensity following elec tron bombardment at 100 kev of thin platelet-type crystals of CdS. On this basis and the effect of tempera ture on quenching and on edge emission, they propose that the center, or at least part of it, for the l.4-p. quenching band is the sulfur interstitial. While the 1 R. J. Collins, J. App!. Phys. 30, 1135 (1959). 2 B. A. Kulp and R. H. Kelley, J. App!. Phys. 31, 1057 (1960). 3 B. A. Kulp and R. H. Kelley, J. App!. Phys. (to be published). conclusions reached by analyzing the results of electron bombardment of CdS may be classified as somewhat speculative, nontheless, electron bombardment repre sents a new approach to the problem of identifying the centers responsible for the many and varied fluorescent and photoconductive properties of CdS. EXPERIMENTAL Platelet-type single crystals 25 to 250 I-' thick, grown by vapor-phase deposition by Greene according to the methods commonly used in this laboratory,4 were used throughout the experiments described here. The plate lets were from several crystal growing runs. While no intentional doping agents were added, spectrographic analysis showed copper, aluminum, magnesium, and calcium present in concentrations of 1 to 10 ppm and sili con and iron in smaller concentrations. The crystals were bombarded with 275-kev electrons from a Van de Graaff accelerator and with loo-kev electrons from a Cockroft-Walton type accelerator at a dc level of 0.25 to 10 l-'a/cm2• The fluorescent spectra were taken with a Perkin Elmer glass-prism spectrometer with a PbS detector. 4 D. C. Reynolds and S. J. Czyzak, Phys. Rev. 79, 1957 (1950). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.102.42.98 On: Sat, 22 Nov 2014 16:02:46E LEe T RON B 0 MBA R D MEN TAN D N EAR - I N F R ARE D FLU 0 RES C ENe E 1967 RESULTS A. Spectra Under Electron-Bombardment Excitation Figure 1 shows the fluorescence spectrum typical of a group of CdS crystals as a function of electron-beam current. The intensity of the 8500-A fluorescence in creases much more rapidly than the 1.05-J.t band as the current is increased. Figure 2 shows the behavior of the fluorescence spectrum of these crystals under electron bombardment, 275 kev, 2.5 f.La/cm2 at 40°C after 1 X 1017, 2.5XlO17, 7XlO17, and 5XlOI8 electrons/cm 2, respectively. The elimination of the band at 8500 A is clearly observed while the band at LOS f.L increases somewhat in intensity. Generally, the 8500-A band is reduced in intensity by a factor of 10 after bombard ment by about 1018 electrons/cm2• In another crystal the resistivity was measured using sputtered platinum electrodes and a voltage gradient of 100 v / cm. The dark resistivity increased from 3 X 104 to 5 X 109 ohm-cm after 1018 electrons/ cm2 struck the crystal. The 8500-A band was removed and the 1.05-f.L band decreased slightly during this bombardment. Similar increases in dark resistivity were observed in other crystals as the 8500-A band was removed. The 8500-A and the 1.05-f.L bands have been found in several batches of platelets and some bulk crystals grown with no intentional doping. The dark resistivity of these crystals is in the range of 104 to 107 ohm-cm. The 8500-A band was not found in a batch grown with 0.01% CuS added to the charge. The resistivity of these crystals was about lOll ohm-cm. The 1.05-f.L band was found in these crystals at room temperature. The 8500-A band was likewise not observed in silver-doped platelets, but was observed in a large crystal which showed both silver and copper impurities. Crystals grown from ultra-pure powder did not have either of the fluorescence bands mentioned here. The resistivity of the silver-doped platelets was 109 ohm-cm. That of the ultra-pure crystal was 107 ohm-cm. iu > ~ oJ ... a: ,: le;; Z ... I ~ .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 WAVELENGTH, MICRONS FIG. 1. Effect of electron beam current on near-infrared fluo rescence of CdS. Spectra taken with (1) O,Sl'a/cm'j (2) 2.Sl'a/ cm'j and (3) 5I'a/cm2, 275-kev electron excitation. ... > ~ oJ ... a: ~. l-e;; Z ... I ~ .6 .7 .8 .9 1.0 1.1 WAVELENGTH, MICRONS FIG. 2. Effect of electron bombardment on the near-infrared fluorescence of CdS. Spectra taken with 275-kev 2.5 ~/cm2 electron excitation. (1) Original; (2), after 1017 electrons/cm 2; (3) after 2.SX1017j (4) after 6XlO17j and (5) after 5XlO18 electrons/ cm'. The decrease in intensity of the 85OO-A fluorescent band under irradiation has been observed to take place with 100-kev electrons and with exposure to x rays from a tungsten target F A60 tube for 20 hr at 50 kv, 40 rna. The phenomenon was observed in all crystals having the broad peak at 8500 A regardless of the in tensity of the 1.05-f.L band. There was variation in the time required to produce the effect. One group of crystals required about 10 times as much radiation to produce the effect as those shown in Fig. 2. In rare cases, the 1.05-f.L band decreased as fast as the 8500-A band for the first 5XlO17 electrons; subsequent bom bardment, however, reduced the 8500-A band with no further decrease in the 1.05-f.L band. B. Spectra Under Band-Gap Light Excitation Figure 3 shows the fluorescence spectrum of a crystal taken with band-gap light excitation (a 100-w Sylvania Zr arc lamp through a CUS04 solution filter which cut off at 6200 A) before and after bombardment with 1018 electrons/ cm2 at lOO-kev energy. Figure 4 shows the spectra with band-gap light excitation and with electron excitation taken after the 85OO-A band had been re duced by a factor of 10 by bombardment by 1018 elec trons/cm2 at 275 kev. There is a definite difference in the spectra depending on the nature of the exciting radiation. Dependence of spectra on exciting radiation has been previously noted by Leverenz5 and others. In the case of band-gap light there are peaks at 7400 and 6900 A which become quite intense following electron bombardment. These peaks are not observed in these crystals before bombardment or while under electron bombardment in the current range 0.25 to 10 f.La/cm2. In other groups of crystals not showing the 8500-A band, however, the 7400-A and the 6900-A peaks have been observed under electron bombardment. The 6800- and 7400-A bands have been previously observed under electron bombardment at room temperature by Bleil and Snyder. 6 6 H. W. Leverenz, Luminescence of Solids (John Wiley & Sons, Inc., New York, 1950), e.g., p. 197. 6 C. E. Bleil and D. D. Snyder, J. App!. Phys. 30, 1799 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.102.42.98 On: Sat, 22 Nov 2014 16:02:461968 B. A. KULP ... > ;:: .. ..J ... 0:: ,.: .... in z ... .... ~ WAVELENGTH, MICRONS FIG. 3. Effect of. electron. bombardment on the fluor~~cence spectrum when excIted by lrght. (1) before; (2) after 10 elec trons/cm2 at 1OO-kev energy. DISCUSSION Two possible mechanisms for the disappearance of a fluorescence band can be readily postulated: (1) actual removal of the fluorescence center, and (2) insertion of a competing center which would poison the fluorescence, for example, nickel in CdS.6 The removal of the fluores cence could be accomplished in several ways: (1) by removing the atom or impurity from the lattice site it occupied, (2) by removing impurity atoms from the crystal altogether, or (3) by changing the character. of the impurity by adding or removing an electron. RadIa tion damage by high-energy electrons is capable of doing all three of the above, however, 100-kev electr?ns and especially x rays would be capable of performmg only the third type of damage. Further evidence that this is the mechanism comes from the fact that the cross section for the process is very large-about 10-18 cm2 compared to '" 10-22 to 10-23 cm2 for displacement of an atom from a lattice point with electrons.7 The actual mechanism for the irreversible (under isothermal conditions) redistribution of electrons over the existing impurity levels could be ~xplai~:d ~s follows: It is assumed that one or more Impunties In the crystals can exist in several ionization states. Such a model has been proposed by Woods and Wright8 for sulfur and cadmium vacancies, and for Cl and Cu occupying sulfur and cadmium lattice sites, respec tively. In addition, Woodbury and Tyler9 have estab lished that gold and copper impurities in germanium introduce a series of levels in which the impurity center has charges which may vary from +1 to -3 electronic charges. Schockley and LastlO have shown that the probability of such an impurity being in a particular charge state depends upon the position of the Fermi level. Thus, if the Fermi level is below one of the levels A characteristic of one charge state of a many 6 H. W. Leverenz, Luminescence of Solids (John Wiley & Sons, Inc., New York, 1950), p. 333. . 7 F. Seitz and J. S. Koehler, Solid State Physics (AcademIC Press Inc., New York, 1956), Vol. 2, p. 332. 8 J.' Woods and D. A. Wright, Solid State Physics, Brussels Conference (Academic Press, Inc., London, 1960), Vol. 2, Part 2, p. 880. 57) 9 H. H. Woodbury and W. W. Tyler, Phys. Rev. 205, 84 (19 . 10 W. Schockley and J. T. Last, Phys. Rev. 107, 392 (1957). charge-state impurity, then A will have essentially unit probability of occupancy while level B nearer the con duction band will have small probability of occupancy. As the Fermi level moves up, the probability of oc cupancy will shift continuously according to a Fermi function until level B has unit probability of occupancy when the Fermi level is several kT above level A. In this case, since it was observed that the d~rk resistivity of the crystal increased as the level was bemg emptied, it is necessary to consider that the impurity involved is a hole-trapping impurity. Using the model of Rosell in which he proposes a steady-state hole Fermi level and a hole-demarcation level under excita tion it can be seen that a redistribution of the holes can' take place under deeply penetrating radiation. There are, of course, many trapping centers in addition to the one responsible for the 8500-A fluorescence band which are important in establishing the properties of the crystals. It can take an appreciable length of time before steady-state conditions are reached. Such long time buildup or decay of photoconductivity and fluores cence is common in CdS. Upon removal of the ionizing radiation, the tendency of the crystal to return to the original state is prevented because the levels are too widely separated for thermal transitions to take place. Thus, the crystal is now in a state where the impurity has one less hole trapped at it than originally. The net result is that the 8500-A fluorescence band is no longer detectable and the crystal has a higher dark resistivity than originally. Whether there is a fluorescence band farther in the infrared which is due to the second level is not known since the wavelength limit of the equip ment presently available is 2.75 IJ.. The appearance of one or two shorter wavelength fluorescence bands after bombardment when the crys tals are excited with light indicates that there has been a redistribution of the electrons over several levels. It would seem that if these bands were associated with the 8500-A band, they should be observed under elec tron bombardment. One is tempted to suggest that the 6900-and 7400-A bands are the result of surface states or states near the surface which would be strongly excited by the light, but only weakly excited by pene trating radiation. However, in many crystals these ... ~ .... .. ,,,, ..J I \ WI\. I 0:: I \ ,: .... iii z ... .... ~ \ .6 .7 .8 .9 WAVELENGTH. MICRONS FIG. 4. Effect of exciting radiation on fluorescence of CdS crystal after electron bombardment. (1) light; (2) 275-kev 2.5 p.a/ cm2 electrons. 11 A. Rose, Phys. Rev. 97, 322 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.102.42.98 On: Sat, 22 Nov 2014 16:02:46E L E C T RON B 0 MBA R D MEN TAN D N EAR -I N F R ARE D FLU 0 RES C E N C E 1969 bands are observed under electron excitation and hence the difference in the spectra under light and electron excitation must be attributed to a difference in occupa tion of states brought about by the difference in ioniza tion densities of the two types of exciting radiation. Light causes very intense ionization in a thin layer, while electrons cause more uniform but not so intense ionization throughout the crystal. It would seem that the results shown in Fig. 1 are contrary to the model; however, since the temperature of the crystal increases as the electron beam increases in intensity, there is a tendency for the effect of the increased ionization on the steady-state Fermi level to be neutralized by the rise in temperature. HEAT TREATMENT If the above model is correct, one might expect that heating the crystal to a few hundred degrees Centigrade would return the crystal to its original state. Figure 5 shows the spectrum with electron excitation of a crystal before and after bombardment and after heat treatment for t hr at 200°C in air. The partial recovery of the 8500-A band is shown. The intensity of this band was subsequently reduced to its value before heat treatment by bombardment with 3X1017 electrons/cm2• Heat treatment of a similar crystal which had not been bom barded caused a slight decrease in the S500-A band and no change in the 1.05-,u band. While the heat treatment experiment is not conclusive, it is consistent with the model proposed. It should be noted here that heat treatment for t hr at 2000 and at 300°C has been observed to produce the 1.05-,u fluorescence band in many crystals which do not have it. In the case of the crystal in Fig. 5, the intensity of the 1.05-,u band is sufficiently high before heat treatment so that no change in its intensity is ob served. Figure 6, however, shows the effect of the same heat treatment on another crystal in which the 85OO-A band had been reduced in intensity by x radiation. The w > ~ ...J W II: >-" .... iii z w .... ~ . 6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 WAVELENGTH, MICRONS FIG. 5. Effect of heat treating on fluorescence of CdS after bombardment. (\) Original; (2) after 1018 electrons/cm2 at 275 key; (3) after! hr at 200°C. Spectra taken with 275-kev 2.5 "a/cm2 electron excitation at 40°C. w > >= « ...J w II: >.... iii z w .... ~ FIG. 6. Effect of heat treating. (1) Original; (2) after prolonged exposure to x rays; (3) after! hr at 200°C. Spectra taken with 275-kev 2.5 "a/cm2 electron excitation at 40°C. appearance of the 1.05-,u band after heat treatment is very evident as is the recovery of the S500·A band. When crystals with spectra similar to that shown in Fig. 6 are heat treated before bombardment, the 1.05-,u band appears strongly and the 85OO-A band decreases somewhat in intensity. The reason for the appearance of the 1.05-,u band is not known at the present. DEFECT RESPONSIBLE FOR THE 8500-A BAND Fluorescence bands at 8200 A and 1.02,u in copper activated CdS phosphors have been reported by Grillot and Guintinjl2,13 and confirmed by Garlick and Dum belton.l4 Grillot and Guintini found that, depending on the method of preparation, the S200-A and/or the 1.02-,u bands are found in CdS: Cu at room tempera ture. Avinor15 has found that with an excess of coacti vator (indium or gallium) over the copper activator, two bands appear at about 8300 A and 1.01,u. With small amounts of coactivator the 1.01-,u band appears alone. The bands observed here are quite broad and their shape and the position of the peaks depend on the relative intensities of the two bands. It seems reasonable to say that the band at S500 A is a copper band. The appearance of the 1.05-,u band upon heat treatment indicates that the origin of this band needs further investiga tion . ACKNOWLEDGMENTS The author wishes to thank L. C. Greene, of this Laboratory, for supplying the crystals used in this investigation, and to R. G. Schulze for making the re sistivity measurements and for discussions concerning the manuscript . 12 E. Grillot and P. Guintini, Compt. rend. 236, 802 (1953). 13 E. Grilliot and P. Guintini, Compt. rend. 239, 419 (1954). II G. F. J. Garlick and M. J. Dumhlcton, Proc. Phys. Soc. (London) B67, 442 (1952). 15 M. Avinor, thesis, University of Amsterdam, 1959. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.102.42.98 On: Sat, 22 Nov 2014 16:02:46
1.1728971.pdf	A Ductile, HighField, HighCurrent Ternary Superconducting Alloy R. M. Rose and J. Wulff Citation: Journal of Applied Physics 33, 2394 (1962); doi: 10.1063/1.1728971 View online: http://dx.doi.org/10.1063/1.1728971 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in HighField Superconductivity Phys. Today 39, 24 (1986); 10.1063/1.881053 Enhanced highfield current carrying capacities and pinning behavior of NbTibased superconducting alloys J. Appl. Phys. 57, 4415 (1985); 10.1063/1.334564 Highfield superconductivity in the NbTiZr ternary system J. Appl. Phys. 51, 3316 (1980); 10.1063/1.328039 HighField Superconductivity of Alloys in Porous Glass J. Appl. Phys. 42, 46 (1971); 10.1063/1.1659624 NiobiumThorium Eutectic Alloy as a HighField, HighCurrent Superconductor J. Appl. Phys. 34, 1771 (1963); 10.1063/1.1702677 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.222.12 On: Thu, 27 Nov 2014 00:42:172394 LETTERS TO THE EDITOR The author wishes to express his appreciation to Dr. B. Sujak for valuable discussions. The author thanks also the Polish State Commission for Peaceful Utilization of Nuclear Energy for financial support. 1 J. Kramer. Metalloberflache 9A. 1 (1955). 'T. C. Ku and W. T. Pimbley, J. Appl. Phys. 32, 124 (1961). I W. T. Pimbley and E. E. Francis, J. Appl. Phys. 32, 1729 (1961). • B. Sujak, Acta Phys. Polon. 20, 889 (1961). • J. C. Fisher and I. Giaever' have observed a photovoltaic effect when sandwiches AI-AhO. -AI were heating at 4OQ°C, and then cooled to room temperature. 'J. C. Fisher and I. Giaever. J. Appl. Phys. 32,172 (1961). 7 T. Lewowski, Acta Phys. Polon. 20. 161 (1961). 8 R. H. Kingston, J. Appl. Phys. 27, 101 (1956). 'W. H. Brattain and J. Bardeen, Bell System Tech. J. 32, 1 (1953). 10 J. T. Wallmark and R. R. Johnson, R. C. A. Rev. 18, 512 (1957). 11 Bohun" suggests also in his last report, that the process of adsorption and desorption of water vapor on the gold or aluminum surface may play an important role in some of the observed exoemission phenomena. This problem was also mentioned earlier by Sujak" in his report concernini the emission of exoelectrons from hydrates. 12 A. Bohun, Czechoslov. J. Phys. 11,819 (1961). 11 B. Sujak, Z. angew. Phys. 10, 531 (1958). A Ductile, High-Field, High-Current Ternary Superconducting Alloy R. M. ROSE AND J. WULFF Metals Processing Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts (Received February 26, 1962) OF the two types of high-field, high-current superconductors discovered recently, the solid-solution alloy type such as Nb-Zrl-a has proven to be of greater interest as a magnet material than the intermetallic compounds, such as NbaSn,4 or VaGa .• This is primarily due to the brittleness of these compounds, and the difficulty of making long lengths of wire, either filled or coated with the compound. Nb-Zr alloys can be fabricated directly into wire, although meticulous control of the metallurgical variables is necessary, if long lengths of fine (O.OlO-in. diameter) wire are to be produced. In addition, the current-carrying capacity of alloys such as 25% (atomic) Zr drops off sharply above 70 kG, making higher fields difficult to attain.a Efforts in this Laboratory to control properties of these alloys have also led to a study of ternary alloys containing Nb and Zr, whose rate of work-hardening is not as great as the binary alloy. In the course of this work, Nb-Zr-Ta alloys having a nominal electron-to-atom ratio of 4.756 not only proved to be more readily workable than Nb-Zr binary alloys, but also showed a higher current carrying capacity above 70 kG. The three alloys reported here (25% Zr-2% Ta-73% Nb, 25% Zr-5% Ta-70% Nb, 25% Zr-10% Ta-65% Nb, all atomic percent) were made by electron- Alcm'(OFF SCALE) 10' LEGEND' --,--nIOO C I- 10 ffi II: a: 13 ..J " (J ;:: ii: (J FIG. 1. Critical current densities of short samples of Ta-Nb-Zr wire. Specimen 1. 0 2% Ta, 25% Zr, 73% Nb. Specimen 2. A 5% Ta, 25% Zr, 70% Nb. Specimen 3. 0 10% Ta, 25% Zr, 65% Nb, annealed at 800°C for 15 min, after swaging. Specimen 4. X 10% Ta, 25% Zr, 65% Nb. beam melting. The i-in.-diam ingots were then homogenized by heat treatment in vtu;uo at 1500°C for 14 h. After grinding to ! in. in diameter, they were readily cold-swaged to 0.037 -in.-diam wire. After heat treatment7 of some of the specimens, all were cold drawn to 0.0098 in. in diameter. Current and voltage leads were attached by ultrasonic soldering with indium metal, and the samples were then mounted on Bakelite rod with "Teflon" tape. Tests were run, with the current transverse to the field of a 2-in. Bitter solenoid. Measurements were made at 4.2°K, using currents up to 100 A and fields to 90 kG. Voltage across the speci men was measured with a milli-microvoltmeter; less than 10-7 V were always measurable. In most cases the current was quenched sharply as the specimen went normal. The 2 and 10 atomic percent Ta material showed no real improvement over ordinary 25% Zr binary alloys (see Fig. 1.), in superconducting properties, although the 10% Ta alloy proved to be appreciably more ductile. Heat treatment of the 10% alloy wire, by an intermediate anneal at 800°C, appreciably raised the critical current curve at fields less than 70 kG. Of greater interest, however, are the results obtained with the 5% Ta alloy, which in the cold-worked state, without intermediate heat treatments, possesses considerably greater current carrying capacity between 70 and 90 kG than any solid-solution-type alloy heretofore reported. Cold-working to fine wire from larger diameter ingots and intermediate heat treatments, are both expected to further improve this property. Consequently, the new alloy offers the prospect of producing, in superconducting solenoids, magnetic fields of 90 kG or better. This research was supported by the Office of Naval Research under contract N-onr 1841 (78) with MIT authorized by ARPA Order No. 214-61. IT. G. Berlincourt, R. R. Hake, and O. H. Leslie, Phys. Rev. Letters 6, 671 (1961). • J. E. Kunzler, Bull. Am. Phys. Soc. 6, 298 (1961). • P. R. Aron and H. C. Hitchcock, Phys. Rev. Letters (to be published). 'J. E. Kunzler, E. Buehler, F. S. L. Hsu, and J. H. Wernick, Phys. Rev. Letters 6, 89 (1961). • J. H. Wernick, F. J. Morin, F. S. L. Hsu, O. Dorsi. J. P. Maita, and J. E. Kunzler, High Magnetic Fields (MIT Press, Cambridge, Massachu setts and John Wiley & Sons, Inc., New York, 1962), p. 609. • B. T. Matthias, Progress in Low Temperature Physics (North-Holland Publishing Company, Amsterdam, 1957). p. 138. 7 G. O. Kneip. Jr., J. 0. Betterton, Jr., D. S. Easton, and J. O. Scar brough, J. Appl. Phys. 33, 754 (1962). Hall Effect in Single-Crystal TiC* JOHN PIPER Union Carbide Research Institute, Tarrytown, New York (Received February 21, 1962) THE Hall coefficient R and the resistivity p have been meas ured for single crystals of titanium carbide in the tempera ture range 4.2° to 313°K. The crystals used were grown by the Verneuil method and were nonstoichiometric with an approxi mate composition of TiCo.94. The current and magnetic field were aligned in the < 100> directions of the NaCl type crystal. Figure 1 shows the temperature dependence of Rand p for one such sample. A second sample cut from a different boule showed approximately the same temperature dependence. Measurements were made at 4.2°K and continuously from 77° to 313°K, with an uncertainty in the temperature dependence as indicated by the widths of the curves in Fig. 1. Actual values of Rand p, which a;e more uncertain because of uncertainties in geometry, are gIVen in Table I for two samples. A magnetic field dependence of R was not observed, and hence an upper limit of less than 1 % from 1 to 12 kG at 4 OK is indicated for the effect. A very small transverse magnetoresistivity, of the order of 5 parts in 10' for a field of 17 kG at 77°K, was detected. The effect was no larger at 4.2°K. The resistivity of the single crystals is substantially higher than that of hot-pressed TiC for which the room-temperature value1.2 is usually below 1.0 ~rl-m. In fact, crushing and hot-pressing the single crystal material to 85% density was found to lower the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.222.12 On: Thu, 27 Nov 2014 00:42:17LETTERS TO THE EDITOR 2395 -25 X 10-10 ................ "." U ..... "'5 ... z \3.20 X io·IO ;;: ... ~ o .J ..J ~~15 XIO-IO o 50 Hall Coefficient ~ 100 150 200 250 TEMPERATURE IN OK ::u '" CJ) r:;; 1.8 :::! < =i 1.7 ~ 1: b 1.6 I 3 .. 1.5 ii 300 350 FIG.!' Temperature dependence of the resistivity and Hall coefficient of a single crystal of TiC. The dashed lines represent interpolations be tween 77' and 4.2'K. resistivity to 0.90 /Ln-m. The values of R for the single crystals are also somewhat larger in magnitude than those reported,,3 for polycrystalline samples (between -7 and -12XIQ-'O mS/C). It is noted that the temperature coefficient of R for the single crystals is opposite in sign to that reported by Tsuchida et aI.' for their hot-pressed samples. The reason for this discrepancy is not clear since this author's measurements on hot-pressed samples show a temperature dependence of R which is very similar to that of the single crystals. However, the presence of nitrogen impurity has been found to greatly reduce the temperature de pendence of R in the hot-pressed material. Thermoelectric measurements on similar TiC crystals have been reported by Hollander4 to indicate an n-type transport in accord with the negative Hall coefficients of Table I. TABLE J. Hall coefficients and resistivities of two TiC single crystals. Resistivity (1l1l-m) 300'K 77'K Hall coefficient (m'/C) 300'K 77'K Sample 1.78±O.08 1.54±O.07 -(14.1±O.4) XIO-lO -(24.8±O.8) XIO-lO II 1.70±O.08 1.43±O.07 -(13.7±OA) XIO-lO -(27.6±O.8) XlO-lO The transport properties of TiC are consistent with a simple two-band model. Transition metals, and probably their inter stitial compounds such as TiC, are characterized by two over lapping bands6; a relatively narrow a band with a large density of states, and an s-like conduction band with a much smaller density of states and a correspondingly smaller effective mass. Because of the large effective masses of the a electrons, the bulk of the charge transport may be expected to occur in the conduc tion band. This is in accord with the small magnetic field de pendence of Rand p, which is characteristic of a single s-like band. Assuming single-band transport, the conduction electron con centration is proportional to ](1 with values from 0.05 per Ti atom at 4°K to 0.08 at 273°K. The temperature dependence of the electron concentration is determined by the total density of states, which is dominated by the a band. The increase in the number of conduction electrons with temperature indicates an increasing Fermi energy and a large density of a states which is rapidly decreasing with increasing energy. Due to the complicated substructure of a bands6 this does not imply that the Fermi level is close to the top of the band. Further information, such as more of the density of states curve, is necessary before the position of the Fermi level with respect to the a band may be determined. The general characteristics of this band model are not critically dependent upon the assumption of negligible a-band transport. If this assumption is relaxed somewhat, the result is merely a less rapid decrease in the density of d states with increasing energy. The author is indebted to A. D. Kiffer of Linde Company, Division of Union Carbide Corporation, who grew the single crystals. * This work was accomplished under ARPA support under ARGMA contract DA-30-069-0RD-2787. 1 P. Schwartzkopf and R. Kieffer, &/ractory Hard Metals (The Macmillan Company, New York, 1953), p. 88. 2 T. Tsuchida, Y. Nakamura, M. Mekata, H. Sakurai, and H. Takaki, J. Phys, Soc. Japan 16, 2453 (1961) . • A. Munster and K. Sage!, Z. Physik. 144, 139 (1956). 'L. E. Hollander, J. Appl. Phys. 32, 996 (1961). • See, for example, A. H. Wilson, The Theory of Metals (University Press, Cambridge, England, 1954), p. 271. • J. Callaway, Phys. Rev. 121, 1351 (1961). A Method for Measuring the Thickness of Epitaxial Silicon Films* WILLIAM C. DASH General Electric Research Laboratory, Schenectady, New York (Received January 2. 1962) FOR proper control of the uniformity of epitaxial silicon films it is desirable to have a simple way to determine the thick ness. This can be done by the method described here which uses the properties of stacking faults in the epitaxial layers. The appearance of a lightly etched epitaxial silicon film on a (111) surface is shown in Fig. 1 (a). It can be seen that in addition to etch pits at dislocations, there are triangles, straight lines, and also more complicated aggregates of the two latter types of etch figures. It has been shown by Schwuttkel using x-ray. techniques and by Queisser and Washburn' using electron microscopy that these etch figures occur at {11I} stacking faults in the epitaxial film. When they emerge at the surface of the crystal the stacking faults nucleated at the substrate are very uniform in size. There fore, the length of either the straight line etch figures or one side of the triangular etch figures gives a direct measure of the thick ness of the epitaxial film if the orientations of the faults and the substrate are known. For a (111) substrate the thickness of the epitaxial film is 0.816 I, where I is the length of one side of an equilateral triangle which is the base of a regular tetrahedron with its apex at the substrate. Figure 1 (b) is a photograph of the same area as that in Fig. 1 (a) after removal of about 25/L of the epitaxial film by mechanical polishing, followed by etching to reveal the structure. It can be seen that all of the stacking faults in Fig. 1 (a) are recognizable in Fig. 1 (b), but some adjacent pairs have become disentangled into simpler structures. Continuing the polishing and etching down to the interface indicates that the faults are nucleated there. How ever, there is as yet no clear idea of their origin. There is excellent agreement between the thickness found by measuring the etch figures and that determined by breaking the epitaxial wafer and staining. The agreement is limited only by the accuracy of measuring the size of .the etch figures and the thickness of the fractured and stained edge of a specimen. It is important that the largest simple stacking fault structures be measured, since smaller faults are occasionally seen as a result of nucleation sometime after the growth has begun. Schwuttkel has suggested that some of the stacking fault structures observed on epitaxial films are nucleated by scratches on the substrate. Although no experiments have been carried out to nucleate these deliberately, a linear array of stacking faults which appears to be generated by a scratch can be seen in the micrographs in Fig. 1. If further work proves that scribing or some other treatment of the substrate surface can be used to produce the faults, the thickness of the epitaxial wafer subse· quently grown could be determined wherever appropriate, such [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.222.12 On: Thu, 27 Nov 2014 00:42:17
1.1702753.pdf	Crystallinity and Electronic Properties of Evaporated CdS Films J. Dresner and F. V. Shallcross Citation: Journal of Applied Physics 34, 2390 (1963); doi: 10.1063/1.1702753 View online: http://dx.doi.org/10.1063/1.1702753 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Optical and Structural Properties of Thermally Evaporated CdS Thin Films AIP Conf. Proc. 1372, 313 (2011); 10.1063/1.3644462 Optical properties of polycrystalline CdS films J. Appl. Phys. 51, 668 (1980); 10.1063/1.327323 Dynamic quenching of photocapacitance in CdS:Cu evaporated thin films J. Appl. Phys. 50, 483 (1979); 10.1063/1.325638 Effect of Neutron Irradiation on the Acoustic Performance of Evaporated CdS Films J. Appl. Phys. 39, 5987 (1968); 10.1063/1.1656102 Structure of CdS Evaporated Films in Relation to Their Use as Ultrasonic Transducers J. Appl. Phys. 38, 149 (1967); 10.1063/1.1708945 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.255.6.125 On: Thu, 11 Dec 2014 01:54:41JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 8 AUGUST 1963 Crystallinity and Electronic Properties of Evaporated CdS Films J. DRESNER AND F. V. SHALLCROSS RCA Laboratories, Princeton, New Jersey (Received 15 November 1962) The effect of several processing methods on the crystallinity and electronic properties of evaporated CdS films has been investigated. Diffusion of Cu or Ag at temperatures above 450°C has yielded films composed of crystals of controllable size ranging from 10-5 to 1 em in diameter. The electron mobility depends strongly on the reorientation of the crystallites but is only slightly affected by their size. The best films obtained have shown mobilities for photogenerated carriers of 300 cm2V-I secl, characteristic of the bulk crystal. Typical values of the total trap density are in the range of 1019 to 1021 cm-3, compared to 1014 to 1016 in single crystals. The resistivity of the high mobility films can be controlled by the addition of Cl or Ga at the proper stage of the processing. In the photoconductive films, the mobility may vary with the illumination level by an order of magnitude. In films processed at temperatures above 400°C, a model of conducting crystallites separated by thin in sulating barriers is insufficient account for the observed results. INTRODUCTION THIS paper presents a study of the crystalline structure and electronic properties of thin CdS films as a function of post-evaporation processing. The properties which are of technological interest are the resistivity p, the total density of traps Nt, and most importantly, the electron mobility p.. It had previously been shown1 that while p could be varied through nine orders of magnitude by controlling the evaporation parameters and by annealing the sulfur vacancies in the film, the Hall mobilities obtained were always of the order of 2 cm2V-1secl compared with the mobilities in good quality single crystals ranging from 100 to 300 cm2V-1sec-1• The processing methods studied in this paper involve the diffusion of acceptor impurities into the films after evaporation. Several processes of this type have long been known,2,3 but little is understood about their effect on the transport properties of carriers. It has also been shown by Gilles and Van Cakenberghe4 that thin films of CdS can be recrystallized by use of a flux or catalyst such as Ag, Cu, Pb, or In. In the following work, it is shown that diffusion of Ag or Cu even at relatively low temperatures induces drastic changes in the crystallinity of the films and that films treated in this manner can exhibit mobilities for photogene rated carriers equal to those found in single-crystal CdS. PREPARATION AND PHYSICAL PROPERTmS All films were deposited by vacuum evaporation on glass substrates at 170aC under a residual gas pressure of approximately 10-5 Torr and ranged in thickness from 3 to 5 p.. The evaporant consisted of pure crystal line <:d? (obtained from the Eagle-Picher Company) contammg a total of 1 ppm of spectrographically detect-able impurities. Before the beginning of the evaporation, the glass substrate was outgassed by baking under high vacuum at 400aC for one hour. Some films were recrystallized by the Cakenberghe process. After the CdS deposition, these were covered by an evaporated film of Ag or Cu about 100 A thick and then baked in argon between 4700 and 520aC. In some cases, a second layer of 100 A of In was applied before the bake in order to introduce donors and reduce the resistivity of the sample. Most other films were treated by packing in CdS powders and baking in air or argon at temperatures ranging from 250a to 500aC from 1 to 90 h. These powders were luminescent grade CdS doped with Cu or Ag as acceptors and with Ga or CI as donors with con centrations from 100 to 300 ppm. Spectrographic anal ysis showed that Cu diffused into the CdS films, but that bakes of several hours at 450aC were insufficient to bring the Cu concentration in equilibrium with the surrounding powder. From electrical measurements it was determined that diffusion of CI into the films ~as very slow below 450aC and increased rapidly for baking temperatures above 500aC. At that temperature, bakes of five hours yield films with resistivities below 103 Q cm. In some cases, the donor concentration was increased by diffusion of a thin evaporated Ga layer into the film. Some films received a second bake in an atmosphere of Cd vapor or NH4CI vapor but the effect was relatively small. The adherence of powder grains to the sample and the substrate can be minimized by use of a dry argon atmosphere during the bake. The condensation of im purity films on the substrate was eliminated by using a quartz or Pyrex substrate rather than soft glass. How ever, a small amount of oxygen in the ambient was found to accelerate the microrecrystallization which I J. Dresner and F. V. Shallcross Solid·State Electron 5 205 (1962). ' " occurs during the powder bake in the presence of Cu or 2 R. H. Bube, Photoconductivity of Solids (John Wiley & Sons Ag. Inc., New York, 1960), pp. 96, 171. 'Af' 3 F. Gans, U. S. Patent 2,651,700 (1953); P. Goercke, German ter processmg, the samples were more yellow in P~tent 919,.727 (1955). color than when deposited, indicating removal or com- J. M. Gilles and J. Van Cakenberghe, Nature 182, 862 (1958). pensation of the excess cadmium. Samples treated by 2390 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.255.6.125 On: Thu, 11 Dec 2014 01:54:41E LEe T RON I CPR 0 PER TIE S 0 F E V A PO RAT E D CdS F I L M S 2391 the Cakenberghe method were found to consist of crystals as large as 1 cm2 easily visible under polarized light, whose appearance has been described in detail previously. 1 Samples baked in Ag-or Cu-doped powders are composed of crystallites having a maximum di ameter of about 10 I-' visible under the polarizing micro scope. Crystallites smaller than 1 I-' were studied by analysis of x-ray diffractometer tracings. Samples baked in undoped CdS powders did not show crystallites large enough to be detected under the polarizing microscope (",0.51-')' Table I shows the effect on the crystallinity of a series of identical samples resulting from progressively more intensive bakes in CdS(Cu) powder and compares them with the results obtained on a film processed by the Cakenberghe method. The second column gives the ranges of crystal sizes for each film. The third column describes the distribution of orientations of the crystal lites as obtained from x-ray diffraction. The table gives the fraction of the crystallites having a particular set of planes parallel to the substrate, out of all crystallites which can yield reflections for sinO<0.79 for Cu Ka radiation. These values were calculated by comparing the intensity of the observed lines with those for a randomly oriented sample, using the x-ray intensity data of Swanson et al.5 and of Ulrich and Zachariasen6 and includes an empirical correction for absorption and temperature factor based on data for Oot planes. A given set of planes includes all equivalent planes and those giving reflections at the same angle (e. g., 103, 103, 113, etc ... ), so that the fraction of crystallites listed for a given orientation is a function of the multi plicity of the set of planes. In a randomly oriented sample, there are thus 1/6 as many crystallites with OOt-type planes parallel to the substrate as for each lOt type. A simpler description of the crystalline changes is given in the fourth column, which lists the most prob able inclination of the c axis of the crystallites from the perpendicular to the substrate. As deposited, the film shows a high degree of pre ferred orientation with the hexagonal c axis perpendicu lar to the substrate. As the crystals begin to grow under increasingly stronger processings, the intensity of the lOt and Ht reflections increases, especially for {~3, while the oot intensity becomes very small, indicating a tipping of the c axis away from the normal. The data of the third column of Table I shows that the preferred orientation in the powder-baked samples is similar to that in the film recrystallized by the Cakenberghe process, but with a broader distribution of crystal orientations. It is of interest that the relatively mild treatment of the second sample is sufficient to cause a strong change in the orientation of the crystallites while having only a slight effect on their size. The last column gives the highest electron mobility 6 H. E. Swanson, R. K. Fuyat, and G. M. Ugrinic, NBS Circular 539, V.4, (1955). 6 F. Ulrich and W. Zachariasen, Z. Krist. 62, 260 (1925). TABLE I. Crystallite size, orientation, and mobility in evaporated CdS films. Most Orientation probable JIo Crystallite (planes II to inclination cm2 V-I Process size substrate) of c axis sec-1 oot 0.6 O· 102 0.05 Evaporated on 170·C 0.1-0.3 I' substrate; no other treatment 103 0.02 105 0.3 170·C substrate; 0.2-0.5 I' oot 0.05 17· 81 102 0.009 103 0.02 baked in CdS (Cu) 90 h at 250·C 104 0.11 105 0.4 106 0.4 oot 0.04 17· 104 102 0.02 170·C substrate; 1-2 JIo baked in CdS (Cu) 1.5 h at 400·C 103 0.09 104 0.10 105 0.3 106 0.4 114 0.02 116 0.06 205 0.02 oot 0.01 28· 340 101 0.005 170·C substrate; 7-10 JIo baked in CdS (Cu) 4.5 h at 400·C 102 0.14 103 0.2 104 0.2 105 0.12 106 0.04 112 0.009 114 0.12 116 0.10 203 0.02 205 0.06 oot 0.0003 25· 105 103 0.02 170·C substrate; 0.1--0.5 em recrystallized by Cakenbergbe process 104 0.11 (Ag flux) at 520·C 105 0.2 106 0.3 116 0.4 (measured by the Hall effect) for photogenerated car riers under intense illumination for these particular samples. This method was chosen for comparing the samples because the mobility often decreases in darkness as is discussed below. It is of interest that the mobility increases by a factor of 16 during the first stage of proc essing and thereafter changes only by another factor of 4 under a more intense bake which considerably enhances the crystal growth. The mobility measured for the fourth sample is also higher than that of the Cakenberghe film, although the latter is composed of crystals more than two orders larger in size. The mobility thus appears to vary much more with the orientation of the crystallites than with their size. TRAP CONCENTRATIONS The effect of the various processing methods on the trap concentration was studied by the method of thermally stimulated currents7 using gap cells with Au or In electrodes. After cooling to 77 OK, the samples were subjected to strong illumination and then heated in darkness at a rate {3. By measuring it, the thermally stimulated current in excess of dark current, one obtains the number of traps emptied during a temperature rise t::..T t::..N = (it/GVe{3)t::..T, 7 R. H. Bube, Ref. 2, pp. 292-299. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.255.6.125 On: Thu, 11 Dec 2014 01:54:412392 J. D RES l\i ERA )J D F. \-. S HAL L C R 0 S S 0.06 0.05 0.04 ... s.. 0.Q3 0.02 0.01 CdS 252G(CdS:Cu) {3= 0.12°K/SEC. Nt = 1.4 xl020 cm-3 o 100 140 180 220 260 300 340 380 T(°K) FIG_ 1. Thermally stimulated current for a CdS film baked in CU-doped powder. The numbers along the curve indicate the position of the Fermi level in eV. where e is the electronic charge, V the volume of the sample, and G the photoconductive gain, i. e., the charge flowing through the circuit per released free carrier. The assumption made here is that the lifetime of thermally released carriers equals that for photogenerated carriers. The value of G may typically vary by two orders of magnitude over the temperature range used in these experiments because of changes in the recombination kinetics or in the probability of penetration of the various barriers which may be present in the sample, and was determined by measuring the photocurrent i~ as function of temperature under a known illumination. By combining these two measurements one obtains: 0.16 0155 0.14 0.12 .~O.IO '-W fit Nt=--dT V,B i~ , CdS 2558 (CdS: Ag.Go) {3= 0.12°K/SEC. Nt= 4.3xI020cm-3 -k !. ~0348 200 T(OK) 400 FIG. 2. Thermally stimulated current for a film baked ·in CdS:Ag, Ga powder. The numbers along the curve indicate the position of the Fermi level. where W is the flux of quanta incident on the gap. In these experiments the samples were irradiated with band gap light and W measured with a thermopile. The trap depths were determined by calculating the position of the Fermi level for various points along the thermally stimulated current curve. These values mav be too high (by an amount not exceeding 0.1 eV for th~ deepest traps) because of the possible effect of barriers on the measured bulk conductivity. The problem of intercrystalline barriers is discussed in greater detail below. Figure 1 shows the weighted thermally stimulated current for a CdS film baked in Cu-doped CdS powder. The numbers along the curve give the depth of the Fermi level in eV from the bottom of the conduction band. A broad distribution of traps is obtained on which is superimposed a dense level at 0.25 eV. In this case the distribution is cut off at 350oK, the temperature where the dark current becomes large compared to the thermally released current. Except in the most resistive samples, the method could be used only to study traps with energies smaller than 0.4 eV because of this limitation. Broad trap distributions were observed on nearly all samples and are dominant even when discrete peaks are observed as shown in Fig. 2, which gives the data obtained on a film doped with Ga and Ag. The area under the peak is approximately equal to that in the underlying broad distribution. Table II summarizes the results obtained in these experiments, including the samples processed in sulfur vapor which were dis cussed in Ref. 1. The size of crystals ranged from less than 1 J.I. in the first two samples to several millimeters in the samples recrystallized by the Cakenberghe proc ess. In all cases, the concentration of traps is much higher than that of 1014_1016 cm-3 observed in single crystal CdS. For the traps shallower than 0.35 eV, Nt varies only slightly with the type of processing and the crystallite size. However, for the two samples of suf ficient resistivity for traps as deep as 0.6 eV to be measured, the recrystallized film showed a considerably lower value of Nt. MOBILITY MEASUREMENTS This section describes in greater detail the effect of processing on the mobility, and the introduction of donor impurities to control the resistivity of the films. Table III gives the results obtained on groups of samples baked in doped CdS powders or recrystallized by the Cakenberghe process. The mobility was measured by the Hall effect, using samples of the type described in Ref. 1, with evaporated In contacts. Measurements were performed under intense illumination (approximately 1 W / cm2) and also after 30 min in darkness. With the exception of the first group of samples, all films were deposited on substrates at 170°C. The first line shows the low values of the mobility obtained for unprocessed films deposited on substrates from 100° to 200°C. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.255.6.125 On: Thu, 11 Dec 2014 01:54:41E L E C T RON I CPR 0 PER TIE S 0 F E V A PO RAT E D CdS F I L M S 2393 Values of the resistivity are not given since they depend upon the evaporation parameters.l The next line shows the results for samples having received a relatively mild bake in CdS: Cu powder. These films have the crystal linity of the second and third lines of Table I, and show few crystallites larger than 1 p.. The next group gives the effect of more intense bakes in CdS: Cu (fourth line of Table I). These films show mobilities under illumina tion as high as 340 cm2V-1secI, but lower by a factor of 10 in darkness. They are very photoconductive with ratios of resistivity from light to dark as high as 108• Such ratios are higher than those obtained for samples recrystallized by the Cakenberghe process with Ag or Cu flux, shown in the next group. The mobility for photogenerated carriers is high, but it could not be measured in darkness for any of the samples because of the high noise level present. This suggests that the crystals grown by this process are separated by barrier regions of relatively high resistivity. The next line shows the results obtained for two samples recrystallized with In in addition to Ag. Both samples were insensitive to light and exhibited low mobility, despite the formation of large crystal domains. Microscopic examination shows that in films prepared by the Cakenberghe proc ess, excess metallic impurities are deposited in clusters along the crystal boundaries.8 Excess In in those regions could then result in highly doped layers, constituting low resistance paths shorting out the Hall voltage. Such shorts might not occur in regions with excess Ag or Cu, which tend to increase the resistivity of CdS. In the next line, results are shown for a group of samples baked in powders containing Ga as well as Cu or Ag for as long as five hours at 500°C. No crystal growth took place in any sample and the mobility re mained small. The relatively high dark resistivity sug gests that diffusion of Ga into the sample is slow at that temperature. A more successful method of introducing TABLE II. Trap densities in CdS films. Energy range Sample N,(cm-') (eV) S vapor bake 102<'-1021 0.17--0.33 230°C S vapor bake 1()20 0.38-0.61 400°C Baked in CdS(Cu) 1.4XlOw 0.10--0.25 2 h at 400°C Baked in CdS(Ag, Ga) 4.3X 1020 0.11-0.35 2 h at 400°C Baked in CdS(Cu, Ga) 2X1021 0.10--0.31 2 h at 400°C Recrystallized 6X1019 <0.2 Ag flux: 1017-1018 0.4-D.6 Recrystallized 6X1019 0.06-0.15 Ag, In flux: donors consists in baking the samples at 500°C in CdS: Cu, Cl powders for five hours. This procedure yields films which combine a high value of }J-under illumination with relatively low dark resistivity. An other useful method consists in first baking the sample in CdS: Cu to induce microrecrystallization and then diffusing an evaporated film of Ga. The results obtained with a Ga film of 40% light transmission are shown in the last line. In such conducting films the high mobility is independent of illumination. In principle this method gives the greatest degree of control of the dark re sistivity, since the thickness of the evaporated Ga film can be monitored accurately. Table III shows the large variations in p. with illumination level which can occur for photoconductive samples. Abnormally low values of the mobility can be caused by barriers at contacts, intercrystalline barriers and scattering by inhomogeneities in the sample. Con- TABLE III. Hall mobility and resistivity of CdS films. Crystallite Processing Samples size Unprocessed 6 <lp Mild bake in CdS:Cu' 4 0.5-2p Strong bake in CdS:Cu· 3 7-lOp Recrystallized 4 1-5mm Ag or Cu flux: Recrystallized 2 1-5mm Ag+ln flux: 1-5 mm Baked in CdS:Cu, Ga 4 <1p or CdS:Ag, Ga Baked in CdS:Cu, CI 3 7-lOp Baked in CdS:Cu 7-10p +evaporated Ga • See text. 8 R. Addiss, U. S. Government Technical Rept. ASD-61-11 (1962). Mobility (cm2 V-I seCI) IIluminated Dark 2-10 1-5 35-100 3-25 160--340 15-30 70-230 0.25 0.25 4 4 7-18 4-11 240-300 1()""25 240 260 Resistivity (0 em) Illuminated Dark 2()""200 lOC105 3-7 lOL109 lOL105 lOL108 2X1Q4 2XIQ4 3 3 l()i1-104 lOC106 2.6-2.9 4O()""1300 0.12 0.12 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.255.6.125 On: Thu, 11 Dec 2014 01:54:412394 ]. DRESNER AND F. V. SHALLCROSS IOOf- 3 5 .-3018 x_ 307 7 9 - - II 13 FIG. 3. Mobility versus temperature. Upper. curv~s: ~trong illumi nation; Lower curves: very weak IllummatIOn. tact effects were eliminated here by the method of measurement. The effect of intercrystalline barriers which are wiped out by strong illumination has been reported in CdSe films9 and might be expected to come into play in our CdS films where the boundaries between crystallites are well defined. However, in the following discussion, limited to samples powder-baked at tempera tures in excess of 400°C, we show that intercrystalline barriers are insufficient to account for the large changes of fJ, with light. The measurement of mobility in samples consisting of conducting crystallites separated by thin insulating barriers has been studied by VolgerlO and Petritz.u Under these conditions, the magnitude of the Hall con stant R obtained by the measuremen.t is that of the crystallites. However, the calculated values of fJ, is low since the bulk conductivity is determined by the bar riers. If the insulating layers are treated as Schottky barriers under small applied potentials,!1 one obtains for the bulk mobility fJ,*= p.e-<I>lkT, where fJ, is the mobility in the crvstallites and q, is the barrier height. This model might b~ expected to apply to powder-baked layers where the crystal diameter is approximately 10 fJ, and the voltage per barrier smaller than 10 m V during the measurements. The temperature dependence of fJ, was studied for two samples baked in CdS: Cu, CI at 400°C, with the re sults shown in Fig. 3. Although both samples received 9 A. B. Fowler, J. Phys. Chern. Solids 22, 181 (1961). 10 J. Volger, Phys. Rev. 79, 1023 (1950). 11 R. L. Petritz, Phys. Rev. 104, 1508 (1956). I~ 10 10 FIG. 4. Free carrier density versus mo bility (varied by il lumination) at con stant temperature (20°C). similar processing and exhibited roughly the same photoconductive characteristics, they differ in the de tails of the mobility dependence with temperature. The upper curves, measured under intense illumination, show a high value of fJ" decreasing with temperature. The lower curves were measured under very weak illumination, adjusted to make the sample sufficiently conductive to permit measurements under conditions where the temperature varied slowly with time. The illumination was in both samples small enough to keep the value of fJ, close to the dark mobility. Sample 301B shows a monotonic decrease in fJ, with temperature, similar to that under strong illumination, in contradic tion to the barrier model. Sample 307 exhibits a more complex behavior. Although the decrease of fJ, at low temperature does not follow an exponential dependence, the presence of barriers cannot be excluded for this sample. The decrease in fJ, at high temperature, which occurs to a lesser degree in the other curves of Fig. 3, was found to coincide with a similar decrease in the photocurrent and can thus be attributed to the release of free holes from the sensitizing centers.12 The variation of fJ, with light intensity at constant temperature was studied for a third sample processed in the same manner. Figure.4 shows n, the free electron density in the crystallites, as a function of the bulk mobility. The greater part of the change in fJ, (by a factor of five) takes place under constant n, i. e., the electron Fermi level remains stationary. It is, therefore, difficult to attribute the change of fJ, in this part of the curve to an increase in the probability of barrier penetra tion. Thus, in some samples, at least, a model of con ducting crystallites surrounded by insulating barriers is insufficient to account for the observed data. It is probable that the mobility is controlled in part by scattering from ionized impurities12 or aggregates of impurities.13 In cases where two carrier effects are present, the dependence of fJ, on temperature and illumi nation can become exceedingly complex.14 That such effects must be present in these films can be seen from the data of Fig. 4, where the constancy of n implies that the mobility is controlled by the occupancy of the Cu acceptors in that part of the curve.12 12 R. H. Bube and H. E. MacDonald, Phys. Rev. 121, 473 (1961). 13 L. Weisberg, J. App!. Phys. 33, 1817 (1962). 14 R. H. Bube and H. E. MacDonald, Phys. Rev. 128, 2071 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.255.6.125 On: Thu, 11 Dec 2014 01:54:41E L E C T RON I CPR 0 PER TIE S 0 F E V A P 0 RAT E D CdS F I L M S 2395 CONCLUSION The principal result of this work is the demonstration that the diffusion of Cu in evaporated CdS films induces drastic changes in their crystallinity and can yield electronic mobilities equal to those in the bulk crystal for intrinsic as well as photogenerated carriers. The high mobility is dependent upon reorientation of the crystal lites and is nearly independent of their size. The con centration of the shallower traps (Et<0.35 eV) is relatively unaffected by the type of processing. The effects of both Cu and Ag diffusion on the properties of the films are not understood and require further study. Films processed in this manner may find applications in evaporated diodesl and triodes.ls In cases where a relatively high conductivity is permissible, full use can 15 P. K. Weimer, Proc. IRE 50, 1462 (1962). be made of the high value of J.I.. In applications where the traps cannot be filled either by the application of light or addition of donors, J.I. can still be made one order higher t?an in the unprocessed films. At ~he present time, the bakmg processes appear to be supenor to the macro scopic recrystallization for the production of devices, since the intercrystalline barriers in the latter might form a source of objectionable electrical noise. Further more, the relatively low temperature (250°C) at which micro recrystallization takes place may also constitute a practical advantage in thin film circuitry. ACKNOWLEDGMENTS The authors are indebted to Dr. P. K. Weimer and Dr. L. Wiesberg for helpful discussions and to V. L. Frantz for preparing the CdS films used in this study. JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 8 AUGUST 1963 Electron-Beam Probing of a Penning Discharge D. G. Dow (Received 15 November 1962) An electron beam probe has been used to study the distribution of electric fields within the cold-cathode Penn~ng discharge. Three basically different potential distributions have been observed, in addition to t~e ~gh pressure mod:, which could not be studied with this technique. At the lowest pressures, the poten hallS .roughly parabohc. As the pressure increases, a region develops in the center of the discharge, in which ther~ IS no electric field, that is, the potential profile is flat. Upon further increase in pressure, this central portIOn enlarges until it fills the majority of the discharge. At a pressure in the vicinity of 10-4 mm Hg the mode abruptly changes to one in which there is very little electric field throughout the discharge, although there are reasons to suspect a narrow region of high electric field near the anode. At still higher pressures (about 10-3 mm Hg) there is another abrupt change into the high pressure mode which acts essentially like a magnetically confined glow discharge. Due to instr~mentation difficulties, and end effects of considerable magnitude, it was not possible to measure numencal values for the electric field and potential, but their qualitative behavior has been plotted over a range of pressure, voltage, and magnetic field which is characteristic of the device. 1. INTRODUCTION AND HISTORY THE magnetically confined cold-cathode discharge, commonly called a Penning discharge, is con figurationally one of the simplest forms of gas discharge. The basic Penning cell consists of a cylindrical anode with two flat cathodes, one at each end, immersed in a magnetic field. Historically it has been recognized that there are two basic modes of this discharge. Under typical laboratory conditions, a high pressure mode exists at .pressures above approximately 10--a mm Hg. Below thIS pressure, the device operates in a variety of modes, which is the subject of this report. The high pressure mode has been fairly well studied by many investigatorsl-a and is essentially a magnetically con fined version of a common glow discharge, the basic ~John Backus, J. Appl. Phys. 30,1866 (1959). (19&Jfn Backus, and Norman E. Huston, J. Appl. Phys. 31, 400 3 Francis F. Chen, Phys. Rev. Letters, 8, 234 (1962). difference being only that the mobility transverse to the magnetic field is inhibited by the field. In this high pressure mode most of the discharge space is filled with a neutralized plasma having a sheath at the cathode typical of ~he glow disch~rg~. Below about 10-3 mm Hg the behavlOr of the deVIce IS markedly different. This report is concerned only with the properties of the dis charge in this lower pressure range. A number of investigators have studied the external characteristics of the Penning discharge at low pressure, and . have speculated. about the mode of operation.4-6 Dunng the course of mvestigations at this laboratory it became clear that these treatments were not consistent with one another, and that the device behavior must be appreciably more complicated than previously sus pected. For this reason it was decided that a detailed 4 J. C. Helmer and R. L. Jepsen, Proc. IRE, 49, 1920 (1961) • W. Knauer, J. Appl. Phys. 33, 2093 (1962). . 6 R. L. Jepsen, J. Appl. Phys. 32, 2619 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.255.6.125 On: Thu, 11 Dec 2014 01:54:41
1.1703182.pdf	Thermal Conductivity of SnTe between 100° and 500°K D. H. Damon Citation: Journal of Applied Physics 37, 3181 (1966); doi: 10.1063/1.1703182 View online: http://dx.doi.org/10.1063/1.1703182 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/37/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experiments on thermal contact conductance between metals below 100 K AIP Conf. Proc. 1573, 1070 (2014); 10.1063/1.4860824 Piezoresistance in SnTe J. Appl. Phys. 41, 811 (1970); 10.1063/1.1658757 Transport Properties of Thin Films of SnTe J. Vac. Sci. Technol. 6, 558 (1969); 10.1116/1.1315682 EVIDENCE THAT SnTe IS A SEMICONDUCTOR Appl. Phys. Lett. 4, 93 (1964); 10.1063/1.1753977 Evidence for the Existence of Overlapping Valence and Conduction Bands in SnTe J. Appl. Phys. 34, 3083 (1963); 10.1063/1.1729124 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:16JOURNAL OF APPLIED PHYSICS VOLUME 37, NUMBER 8 JULY 1966 Thermal Conductivity of SnTe between 100° and SOOoK* D. H. DAMON Westinghouse Research Laboratories, Pittsburgh, Pennsylvania (Received 15 February 1966; in final form 14 March 1966) The thermal conductivity, electrical conductivity, and Seebeck coe~c~ent. of several specimens of SnTe have been measured between 100° and 500oK. The thermal c?n.duc.hVlty IS we~kly dependent ~:m both t perature and hole concentration. The total thermal conductivity IS separated mto an electromc and a l=~ice thermal conductivity. Because of the large concentrations of Sn vacancies ~n the ~amples, t~e phon~ns are scattered both by three-phonon umklapp processes and by the Sn vacancies; this results m a lathce thermal conductivity that varies with temperature more like T-' rat~H:r t~an T-I. The Lore?z number relating the electrical conductivity and the electronic thermal conductlvlty IS a.n un~sual functIOn of h?le concentration. The Lorenz number is larger than the Sommerfeld ~a~ue Lo,. vanes ~Ith hole c?n~entrahon p, and has a maximum value of about 1.3 Lo a~ p = 2?< 1()20 cm-a• This IS .conSlstent Wlth the vanatlOn of the electrical conductivity and the Seebeck coefficient Wlth hole concentratIOn. INTRODUCTION THE theory of the heat transport by lattice waves in solids at high temperatures when the phonon mean free path is limited both by three phonon colli sions and by collisions of the phonon with point im perfections has been developed by Klemens,1 Callaway,2 Ambegaokar 3 and Parrot.' Klemens, Tainsh, and Whiteli found that the theory correctly predicted some of the characteristics of the thermal conductivity of Cu and Ag alloys. It has been used to interpret measure ments of the thermal conductivity of Ge--Si alloys6 and a number of other heavily doped semiconductors. 7 Vishnevskii and Sukharevskii8 studied the effect of foreign cations and cation vacancies on the thermal conductivity of MgO at high temperatures. Most of the previous investigations have been con cerned with the effect of substitutional foreign atoms. The aim of the present investigation was to study the applicability of the theory to the scattering ~f phonons by vacancies at high temperatures, For thIS purpose SnTe would seem to be an ideal material. Two inde pendent investigations9,I0 have established that Sn~e prepared at high temperatures and normal pressures IS nonstoichiometric, being deficient in Sn. The Sn va cancies act as doubly charged acceptors and the true hole concentration p is related to the value of the Hall constant at 77°K, R77, by p= 0.6/ R77e.n Therefore, the Sn vacancy concentration [V snJ can be simply deter- * Sponsored in part by the U. S. Air Force Office of Scientific Research. I P. G. Klemens, Phys. Rev. 119, 507 (1960). 2 J. Callaway, Phys. Rev. 113, 1046 (1959). a V. Ambegaokar, Phys. Rev. 114, 488 (1959). 4 J. E. Parrot, Proc. Phys. Soc. (London) 81, 726 (1~63). 6 P. G. Klemens, G. K. White, and R. J. Tainsh, Phil. Mag. 7, 1323 (1962). 6 B. Abeles, Phys. Rev. 131, 1906 (1963). 7 J. R. Drabble and H. J. Goldsmid, Thermal Conduction in Semiconductors (Pergamon Press, Inc., New Y!?!"k, 19?1). 81. I. Vishnevskii and B. Va. Sukharevskll, SOVIet Phys.- Solid State 6, 1708 (1965). 9 R. Mazelsky and M. Lubell, Advan. Chern. Ser. 39, 210 (1963). 10 R. F. Brebrick, J. Phys. Chern. Solids 24, 27 (1963). . 11 B. B. Houston, R. S. Allgaier, J. Babiskin, and P. G. Sieben mann, Bull. Am. Phys. Soc. 6, 60 (1964). mined by measuring the Hall constant. Moreover, [V SnJ can be varied over a fairly wide range of values (from ",5X 1019 to ",5X 1020 cm-3) by heat treating the samples in either Te-rich or Te-deficient atmo spheres.10,12 In this way one can study the effect of a known and variable vacancy concentration on the lattice thermal conductivity. In order to study the lattice thermal conductivity, one must subtract the electronic component from the measured thermal conductivity; the electronic com ponent is related to the electrical conductivity u by the relation K.=LuT, where L is the Lorenz number. Un fortunately, the Lorenz number depends on the elec tronic band structure, and may depart from the Sommerfeld value appropriate for the highly degenerate case of a metal. The principal features of the band structure of SnTe seem to be well establishedl3,14; it is a semiconductor with two overlapping valence bands, the band edges being separated by a few tenths of an electron volt. However, the electrical conductivity, the Seebeck coefficient and the Hall coefficient are not fully understood quantitatively. In particular, the See beck coefficient shows an unusual dependence on hole concentration.13 At room temperature the Seebeck coefficient first decreases as p increases reaching a minimum value for p~ 1.5 X 1020 cm-3 and then in creases to a maximum value for p~5X1020 cm-3• As p is further increased, the Seebeck coefficient again de creases. Brebrick and Straussl5 have published a de tailed analysis of the Seebeck coefficient using the two valence band model. Although their model reproduced the qualitative features of the observed variation of the Seebeck coefficient with hole concentration, it was quantitatively unsatisfactory. In particular, it could 12 A. Sagar and R. C. Miller, in Proceedings of the 1962 Inter national Conference on Physics of Semiconductors, Exeter, A. C. Stickland, Ed. (The Institute of Physics and The Physical Society, London, 1962). 13 J. A. Kafalas, R. F, Brebrick, and A. J. Strauss, Appl. Phys. Letters 4, 93 (1964). 14 J. R. Burke, Jr., R. S. Allgaier, B. B. Houston, J. Babiskin, and P. G. Siebenmann, Phys. Rev. Letters 14, 360 (1965). 16 R. F. Brebrick and A. J. Strauss, Phys. Rev. 131, 104 (1963). 3181 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:163182 D. H. DAMON High Vacuum r---~"1-~:-it-vr-- Shield Heater -t--+1H+-tt--+7t-- Sample Heater t--lI--4i'S1-tt--M-- Sample -v,~7<S44-+I-44-- Plug ~~4<7"~I--J.4I-- Si nk FIG. 1. Schematic drawing of the apparatus. The encircled numbers locate the positions of 4 copper-{;onstantan thermocouple beads. The temperature difference between 1 and 2 is reduced to zero by adjusting the shield heater. The temperature differences 2-3 and 3-4 are measured. The chamber is evacuated to a pressure of ",10-5 Torr and is immersed in a suitable temperature bath. A heater wrapped on the sink is used to reach intermediate temperatures. not explain the very small minimum value of S. This small minimum can be understood if one is willing to assume that another scattering mechanism is present. Even though one does not yet understand the scatter ing mechanism which causes the observed dependence of the Seebeck coefficient on hole concentration, one can relate the Lorenz number, the Seebeck coefficient, and the electrical conductivity in a phenomenological manner. It is shown that this variation implies a de parture of the Lorenz number from the Sommerfeld value, depending on p and reaching a maximum value for a value of p of about 2X 1020 em-a. The therrhal conductivity, electrical resistivity, and Seebeck coefficient of four samples of SnTe were meas ured between 100° and SOOoK. Since we do not know the Lorenz number, we cannot effect a unique sepa ration of the thermal conductivity into electronic and lattice components. However, we know from theory that departures from the Sommerfeld value are least at lowest temperature, and using this value at 1000K we obtain values of the lattice thermal conductivity which show a proper dependence on vacancy concentration. Extrapolating these values to higher temperatures we deduce the electronic thermal conductivity and the Lorenz number. We find that this procedure, while not absolutely certain, does lead to self-consistent results and that the measured thermal conductivity can be understood in terms of: (1) a lattice thermal conductivity determined by a combination of three-phonon umklapp processes and point-defect scattering which turns out to be strong for vacancies in SnTe; and (2) a Lorenz number whose dependence on hole concentration is consistent with the behavior of other transport properties of SnTe. EXPERIMENTAL PROCEDURE Figure 1 is a schematic drawing of the apparatus used to measure the electrical resistivity p, thermal conduc tivity K, and Seebeck coefficient S, of SnTe. The samples, rectangular parallelepipeds about 1XO.3XO.3 cm, were soldered between the plug and the sample heater. Temperatures were measured with four copper constantan thermocouples located as shown in the drawing. The thermal conductivity was measured by the standard stationary-heat-flow method.16 Bauerle17 has discussed the use of this apparatus in detail. The heat flux through the sample was calculated from the power generated in the heater corrected for any small drift of the average temperature and for the radiative transfer. The thermal resistance of the solder layers17 was subtracted from the total measured thermal resistance to obtain the thermal resistance of the sample. Measurements of pure germanium have been made with this apparatus in order to check the reli ability of the correction for the thermal resistance of the solder layer. These results showed that the error due to uncertainty in this correction could be kept below S% for samples with thermal conductivities as large as 2 W· cm-1 °K-I provided that the electrical resistance of the solder-sample contact could be kept very small. For this reason the electrical resistivity of the SnTe samples was first measured using a four-probe technique and then remeasured in the thermal conductivity appa ratus. The difference between the results of the four probe and two-probe measurements never exceeded 2%, most of which could be ascribed to uncertainties in the determination of sample geometry. The values of K given below are accurate to ±2% except possibly at the highest temperatures where the correction for the radiative transfer was about lS% of the total heat flow. The radiative heat transfer was studied as a function of the area of the surface of the sample, the emissive character of the surface, the temperature, and the temperature difference t:.T= T1-T4 (see Fig. 1). Meas urements were made on two specimens by the stationary heat flow method. One specimen was a stainless steel (N o. 304) spool. The disks forming the ends of the spool were soldered to the heater and plug. The rod forming the barrel of the spool was 0.S6 mm in diameter and 1.1 16 N. Pearlman, in Methods of Experimental Physics, K. Lark Horovitz and V. A,. Johnson, Eds. (Academic Press Inc., New York, 1959), Vol. 6A. 17 J. E. Bauerle, in Thermoelectricity, Science and Engineering, edited by R. R. Heikes and R. W. Ure, Jr. (Intersceince Pub lishers, Inc., New York, 1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:16THERMAL CONDUCTIVITY OF SnTe 3183 em long. The area of the rod from which heat could be radiated was therefore about 0.2 cm2, considerably smaller than the surface area of a typical SnTe speci men <,.....,1.2 cm2). At 4200K the total thermal conduc tancewas 1.78X 10-3 W OK-I; of thisO.34X 10-3W °K-I was due to conductance through the stainless steel rod. A second sample was fabricated from a hollow stain less steel cylinder 1.3 em long and 1.23 em in diameter with a wall thickness of 6X 10-3 cm. The cylinder was filled with powdered Zr02' A small hole was drilled through the wall of the cylinder so that the air would be removed. In a separate experiment it was found that the effective thermal conductivity of this powder, loosely packed in a vacuum, was 3X 10-5 W· cm-1 OK-I. The heat transported through the Zr02 powder was there fore negligihle compared to that conducted through the stainless steel. The surface area of this sample from which heat would be radiated was about S cm2, con siderably larger than the area of a typical SnTe speci men. This specimen was first measured with its outer surface gold-plated (thickness, ",,2X 10-5 em); it was then remeasured after being blackened with soot from an acetylene flame. Measurements on these samples were carried out between 3000 and SOOoK. Some difficulty was encoun tered since a typical time constant for the sample and heater is 2X 103 sec. Measurements were made at in tervals of about 20 min for a period of about 4 h after the apparatus had reached stationary conditions. For both specimens the total measured thermal con ductance, K, could be represented by K =Ko+eT3 in dependent of aT for l°K<aT<SoK. This has an obvious interpretation: K 0 represents the conductance through the stainless steel and eP is the conductance due to radiation. This relation cannot be exact since the thermal conductivity of stainless steel is feebly tempera ture-dependent. However, the experimental accuracy did not permit further analysis. The values of Ko yielded values of the thermal conductivity of No. 304 stainless steel in reasonably good agreement with pre viously published values considering, for example, the difficulty of making an accurate measurement of the wall thickness of the cylinder. The value of e was 1.3 times larger for the gold-plated hollow cylinder than for the thin rod. It was about 1.1 times larger for the blackened cylinder than for the gold-plated cylinder. The measurements made on the thin rod were assumed to give the heat radiated from the heater to the sink. The differences between these measurements and the measurements made on the thin-walled cylinder then gave the heat radiated from the surface of the cylinder. In this way one finds that the heat radiated from the blackened surface was only about 1.5 times greater than the heat radiated from the gold surface. This small difference is perhaps not surprising remembering that the gold layer was very thin and not polished. The average of the values found for the two surfaces was used for SnTe. This is, of course, uncertain. However, TABLE 1. Values of the Hall coefficient Rn, electrical con ductivity U77 at 77 OK, and the Sn vacancy concentration [V Sn], for each of the samples. R77 U77 [VsnJ X 10--19 Sample (cm3 C--1) (g--l em--I) (cm--3) a 2.34XHr-2 2.4X104 7.9 b 1.81 X 10--2 2.0X1()4 10.3 c 7.9 X1O--3 2.1Xl()4 24 d 3. 12 X 10--3 1.7 X 1()4 60 since we have shown that the fraction of the total radiative heat transfer due to radiation from the surface of the samples is small this uncertainty should not introduce appreciable error. The low-temperature radia tion corrections were found by extrapolation. Measurements were also made with the thermal shield unbalanced. At 485°K it was found that increasing T1-T2 (see Fig. 1) from OaK (actually < 0.01 OK) to 0.4°K increased the radiative heat transfer by 35%. This shows the effectiveness of the thermal shield. Measurements of the heat radiation were also made dynamically. For example, without any specimen in the chamber the heater was suspended from the shield on a thread. After stationary conditions were reached with t:.T= 6°K and T1-T2=0 the sample heater was turned off and t:.T was measured as a function of time. These results were unsatisfactory. It was found to be difficult to keep T1-T2 equal to zero and to prevent the sink temperature, Ta, from decreasing. These special test samples do not perfectly reproduce the environment of the SnTe samples, and some un certainties remain. As previously mentioned, the correc tion for the radiative transfer was only about 15% of the total heat flow at 500oK; therefore, it does not seem lik.ely that these uncertainties would introduce more than 1% or 2% error into the thermal conductivity of SnTe. Single-crystal SnTe is very brittle; if the samples are soldered to a copper heater and plug they break up upon cooling to nOK. It was found that over the tem perature range 100° to SOOoK the thermal expansion of aluminum was, quite fortuitously, a fair match to that of SnTe. It required some care to eliminate the electrical resistance at the solder-aluminum contacts; however, this was successfully accomplished using 75Pb-25Sn solder and Aluten SiB flux manufactured by Eutectic Welding Alloys Corporation, New York, New York. The samples were cut from the same single-crystal ingot in the form of rectangular parallelepipeds 1XO.5 X 0.5 cm. Each sample was heat treated in an appro priate atmosphere,1O,12 in some cases for as long as 1300 h, to produce a sample with a desired hole con centration. Homogeneity was checked by reducing the sample to lXO.3XO.3 em by a succession of lappings. Between lappings:the Seebeck coefficient was measured at room temperature and was found to be independent of the size of the sample. After all measurements were [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:163184 D. H. DAMON FIG. 2. Measured values of the Seebeck coefficient S plotted against temperature T for each of the samples. completed, each sample was sliced into three bars and Hall coefficient measurements were made. These meas urements also showed that the samples were homo geneous; for example, for sample b Hall constant values of 1.86, 1.76, and 1.85X 10-2 cma C-l were found. Table I lists the values of the Hall coefficients and electrical conductivities at 77°K and the Sn vacancy concentra tion for each sample. EXPERIMENTAL RESULTS In Fig. 2 the measured values of the Seebeck coeffi cients S are plotted against temperature. These re- 12~~~----~-----'-----r--~~ 10 K ..... I If ..... I E u 6 {r '" ~. N ~ x 4 :.: ?:: 2 00 500 FIG. 3. Measured values of the thermal conductivity K and the Wiedemann-Franz law electronic thermal conductivity, Ke' = (.fJ/3) (k/e)2aT, calculated from the measured electrical con ductivity a plotted against temperature. sults are in excellent agreement with previously pub lished values.12,13,15 The measured values of the thennal conductivity " are plotted against temperature in Fig. 3. Also shown in this figure are the values of "o'= (~/3)(k/e)2(fT, the Wiedemann-Franz law elec tronic thennal conductivity for a degenerate gas of holes, calculated from the measured values of the elec trical conductivity. Figure 4 shows the difference, "-,,e', plotted against absolute temperature. For samples a and b, "-,,e' decreases with increasing tem perature slightly faster and considerably slower, re spectively, than 1'-0.5. The difference between the values of "-Ko' for samples a and b should be particu larly noted. Sample b contains only about 30% more 7~--__ ~-----r------'-----~--' I :.< 0 '"i' E u 16 ~ N ~ ° A" -OJ " :.: '--v--J 500 FIG. 4. The difference between the measured thermal conductivity and Ke' = (.fJ/3)(k/e)2 a-T, K-Ke', plotted against temperature . holes and vacancies than sample a. Consistent with this difference in vacancy concentration, assuming that the vacancies scatter phonons, the value of K-"o' for sample a at 1000K is slightly larger than it is for sample b. However, at 5000K the value of "-Ko' for sample a is considerably smaller than it is for sample b. This result certainly suggests that the quantity K-Ke' is not the true lattice thennal conductivity for all the samples throughout the temperature range 100° to 500°K. In Fig. 5, K-Ke' at 1000K is plotted against [V SnJ; at this temperature K-"o' is proportional to [V Sn]-l. In order to be sure that the lengthy heat treatments of samples a, b, and d did not produce changes in the lattice (other than changing the vacancy concentration) which would affect the lattice thennal conductivity, an untreated single-crystal sample was annealed for 1300 h at 873°K (about 80% of the melting temperature). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:16THERMAL CONDUCTIVITY OF SnTe 3185 Table II shows that the electrical resistivity, the ther moelectric power, the Hall coefficient, and the thermal conductivity were all unaffected by this annealing. Therefore, the differences between the thermal conduc tivities of these samples can be attributed to the differ ences between their vacancy and hole concentrations. DISCUSSION We first show that for SnTe the Debye tempera ture (}D is approximately 1300K and that, therefore, high-temperature approximations can be used to discuss the results of this investigation. The Klemens theory of lattice thermal conductivity is then used to explain the temperature dependence of K-Ke' for sample a and the result K(l00)-Ke'(l00) ex: [V Sn]-l. Finally, the differ ence between the thermal conductivities of samples a and b is shown to be a result of the variation of the Lorenz number with hole concentration. A. Model for the SnTe Lattice In the absence of specific heat data one might hope to estimate a value of () from measurements of the elastic constants according to18 () B~0.7 (h/k) (B/ p) 'I (3/ 47r V m)l, (1) where B is the bulk modulus, p the density, and V m the volume occupied by one molecule. Houston and Straknal9 have shown that the elastic constants of SnTe vary with p. However, B is only weakly dependent on p and the use of B=4.2X 1011 dyn/em-2found by them for a sample with (R77e)-I= 1.24X 1020 should provide a rea sonably reliable estimate of (}D. Using V m= 6.31X 10-23 ..... ':.:: 8 o ..... , E u ~ ~ 4 N S FIG. 5. The difference between the measured thermal conduc tivity and /C.' = (",2/3) (k/e)2uT at lOooK, K(100) -K.' (100), plotted against Sn vacancy concentration [V Bn]. The dashed curve shows the best fit of Eq. (3) to the data if K .. =13:r-1 W· cm-1 °K-l is the lattice thermal conductivity of stoichiometric SnTe at T>6"" 130oK. 18 c. Zwikker, Physical Properties of Solid Materials (Inter science Publishers, Inc., New York, 1954), Chap. IX. 19 B. B. Houston and R. E. Strakna, Bull. Am. Phys. Soc. 9, 646 (1964). TA1~LE IT. An untreated single-crystal sample cut from an ingot pulled from a stoichiometric melt of SnTe was annealed at 873°K for 1300 h. Values of the Hall coefficient R, electrical resistivity p, thermoelectric power S, and thermal conductivity Ie, show that annealing SnTe without changing the vacancy concentration does not change the thermal conductivity. ",300 100 ",300 100 Before annealing p (Il·cm) 12.5X1Q-- S.1X1Q-- p (fl·cm) 12.4X1Q- S.2X1Q-5 Ie R S [cm:eJ [=~l (/LV/"K) C J 35.9 0.091 19.0 0.080 +7.8XlO-3 After annealing 36.5 19.4 Ie R [cm:eJ [C;3) 0.093 0.079 +7.5X1Q-' cm-3 computed by using 6.32 A for the lattice parameter of the fcc cell, Eq. (1) yields (}B~ 130oK. Alternatively, B might be eliminated in favor of the melting tempera ture T m by use of the relation18 (2) where "I is the Griineisen constant, No is Avogadro's number, and Cv is the heat capacity per mole of SnTe. Using "1=2.0 and Cv=6R, Eq. (2) yields Tm~1000oK which may be compared to the measured maximum melting temperature of SnTe,1O 1079°K. It is clear that the use of T m will yield nearly the same value of (}D. Bolef's measurements20 of the coefficient of linear thermal expansion a provide further information about On. He found a to be 2.0X 10-5 (OK)-l at room tempera ture and to decrease to half this value at about 2SoK . This suggests that (}D is not much greater than 100oK. This value of a together with B=4.2X1011 dyn/cm-2 yields "1= (3BaV mNo)/C v= 1.95. The available experi mental data, therefore, suggest that the SnTe lattice be described in tenns of the following parameters: (}n=130oK, "1=2.0, and Vm=6.3Xl0-23 cm-3 which defines a "lattice constant," a3= V m or a~4X 10-8 em, and a spherical Brillouin zone (47r/3)(QD/27r)3=V m-l with QD~ 108 em-I, the wavenumber at the Debye cutoff. B. Lattice Thermal Conductivity For T>() we assume that the phonon scattering by holes is negligible2l and the phonon mean free path is limited by three-phonon umklapp processes and by point-defect scattering. Combining the effects of both scattering processes, Klemensl obtained for the lattice 20 D. Bolef (private communication). 21 J. M. Ziman, Electrons and Phonons (Oxford University Press, London, 1960), p. 319 ff. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:163186 D. H. DAMON thermal conductivity KL, the formula (3) with (4) where WD is the Debye frequency, v the velocity of sound, n the fractional concentration of point defects (n= [V Sn]V m for SnTe), S2 is a parameter describing the strength of the point-defect scattering, and Ku is the lattice thermal conductivity of the perfect crystal. In the limit of strong point-defect scattering [(WO/WD)~O], Eq. (3) becomes (5) Substituting the values of v, a3, and qD given above, we have and Wo 6.22X1012 WD S[V so]!K,.! (6) (7) Since we expect Ku 0: T-l at these temperatures, Eq. (7) accounts for the temperature dependence of K-Ke' for sample a and for K(100)-K.'(100) ex: [V Snr! (Fig. 5). Using the data shown in Fig. 5, Eq. (7) yields Kul/S=S.4X102 erg! cm-i sect. We must attempt to estimate values of Ku and S to see if this value of Kut/S is reasonable and to find out if the condition wo/wD«l is satisfied. Using 8D"",130oK, Ku may be calculated from the Leibfried-Schloemann formula22 KuT=O.9S(k/h)3M a63=31 W· cm-I, where M is the mass of one molecule of SnTe. It is known23 that this expression yields values of KuT that are larger than experimental values by a factor be tween 2 and 3 for the elements and for binary com pounds whose constituents have nearly equal masses. Therefore, the value of KuT probably lies between 10 and 15 W·cm-1• Another estimate of KuT may be obtained from Keyes24 semiempirical formula relating KuT to T m312p9.13 A-716, where A is the mean atomic mass. For SnTe this yields KuT"'" 13 W· cm-I which, as Keyes points out, may be too small (large) by a factor four (two) if the bonding is purely covalent (ionic). These considerations suggest thatK"T= 13 W· cm-I is a reason able value for stoichiometric SnTe. Together with our previous result this requires S2=4.4 and wo/wD=O.29 for sample a at 100oK. The approximation wo/wD«1 is 22 G. Leibfried and E. Schloemann, Nockr. Akad. Weiss. Gottingen, Math.-Phys. Kl. 2a, No.4, 71 (1954). The expression given in the text was taken from Ref. 23. 23 P. G. Klemens, Solid State Phys. 7, 1 (1958). 24 R. W. Keyes, Phys. Rev. 115, 564 (1959). therefore not fully justified. Using Ku = 13 T-I W· cm-I deg-I, Eq. (3) was fit to the measured values of K(100) -K.'(100) with a constant value of S2=3.3. This fit is shown by the dashed curve in Fig. S. According to Klemens,25 point-defect scattering may be considered to be the combined effects of scattering due to the mass difference SI, the change in the force constants at the defect site S2, and the strain field caused by the dilation or contraction of the lattice about the defect Sa. The scattering parameter S2 repre sents the total point-defect scattering and is given by S2=SI2+ (S2+S3)2. The strength of the scattering due to the mass difference is measured by26 S 1 = f.M / M ""'!, where f.M is the change in the mass of a SnTe molecule upon introducing a Sn vacancy and M is the mass of the molecule.27 Therefore if S2"",3, then only a small frac tion of this can be accounted for by the mass difference. The effect of the change in the force constants upon introducing a vacancy is difficult to calculate but a crude estimate of the effect of the strain field may be obtained. S3 which represents the scattering due to the strain field surrounding the temperature is given by25 S32"",3X 102 (f.Rj R)2, where f.R is the displacement of each of the atoms nearest to the vacancy caused by the introduction of the vacancy and R is the nearest-neighbor distance. The lattice parameter of SnTe decreases linearly with Sn vacancy concentration9•10,12 from a= 6.323 A for [V sn]=7X 1019cm-Stoa=6.297 Afor[V Sn]=S.3X 1020 cm-3 j therefore there is some reason to expect that an appreciable strain field surrounds each vacancy. Vegard's law may be used to obtain a rough estimate of f.R/ R. We assume the following: Each complete molecule occupies a volume V m, each defective mole cule, i.e., each unpaired Te atom, occupies a volume V d, and the third power of the measured lattice param eter is a weighted average of these volumes. A simple calculation then yields (V m-V d)/V m. If f.R/ R is one third of this quantity, then one finds (f.R/R)2""'0.016 and S; "" 4.8. While this suggests that S2 may well have a value of 3 for vacancies in SnTe, it must be noted that the calculation probably overestimates f.R/ Rand moreover the term S2S3 that appears in S2 should be negative25 for a vacancy with f.R/R<O. Klemens' theory therefore provides a partial account of the experimental values of K-Ke'. The data establish 25 P. G. Klemens, Proc. Phys. Soc. (London) A68, 1113 (1955). 26 This expression is taken from Ref. 1 and differs from that in Ref. 25 by a factor 2VJ. 27 Throughout this discussion we have treated the SnTe lattice as having two atoms per unit cell. Considering the near equality of the masses of Sn and Te, one might also use a monatomic cell. This will make very little difference (factors of 21/8 or 21/6 in some of the expressions) since the theory is strictly applicable only to an elastic continuum, i.e., in the long-wavelength limit. The details of the construction of the cell do not matter much so long as one keeps the correct number of vibrational modes and maintains a consistent viewpoint. In using the monatomic cell, V m, a, qD, and n would have different values and one would find a different value of S2in fitting Eq. (3) to the data. One would then takeS1=1. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:16THERMAL CONDUCTIVITY OF SnTe 3187 a value of K,j S2 which is consistent with reasonable values of K," a~d S2. Obviously there is some freedom in choosing Ku and S2 (keeping Ku/S2 constant) but this freedom is not unlimited. In particular, one cannot choose considerably smaller values of both Ku and S2; otherwise the condition WO/WD< 1 would not be satisfied and the theory would not yield KL ex [V Sn]-!' It must be ~entioned that there are serious objections to the use of this theory. The theory assumes that the point defects are randomly distributed. The vacancy concentrations are large (approaching 4% for the sample with the largest hole concentration). It would not be unreasonable to expect at least partial ordering of the vacancies. The effect of N -processes has been ignored. Parrot28 has described the effect of N -processes at high temperatures; however, in view of the un certainties that are encountered in the next section, it did not seem worthwhile to pursue these questions. C. The Electronic Thermal Conductivity The fonnula29 for the electrical conductivity of a cubic crystal, where E is the electron energy and the integration dS is over a constant energy surface, may be written rr= -f dE rr(E) (afo/dE) (9) defining the function rr(E). One then attempts to specify only the behavior of rr(E) and does not deal separately with the density of states and the relaxation time. The function rr(E) is transfonned into another function rr(e) by the substitution E=kTe+t. Klemens30 has shown that if the transport properties in the presence of a temperature gradient are describable in tenns of the same function rr(e) (equivalent to the as sumption of the existence of a unique relaxation time) then the Seebeck coefficient S and the Lorenz number L are given by S= (k/e) (Kl/Ko) (10) L=~= k2{K2_(~1)2}, rrT e2 Ko \Ko (11) 28 Ref. 4; see also Ref. 6. 29 A. H. Wilson, Theory of Metals (Cambridge University Press, London, 1953), 2nd ed., p. 197. 30 P. G. Klemens, Proceedings of the 4th Conference on Thermal Conducti'IJtiy, San Francisco, 1964 (U. S. Naval Radiological Defense Lab., San Francisco, 1964), Paper lA. Equation (10) does not include a phonon-drag contribution. It does not seem likely that this is important for SnTe for T>l00oK because: (1) 11 "" 130 OK, and (2) the carrier concentrations are large. where f<Xl dfo u=Ko= -rr(e)-de, _<Xl de Therefore, if -rr(e)(d!o/de) is considered to be a dis tribution function, then the Seebeck coefficient is proportional to the first moment of this distribution function and the Lorenz number is proportional to the second moment about the mean. This fonnalism is, of course, simply another way of writing the familiar ex pression for rr, S, and L. It emphasizes the necessary correlation between these quantities. In dealing with materials for which one has little or no infonnation about the effective masses and/or the scattering proc esses it permits the extraction of some infonnation from experimental results without making specific assump tions about the effective masses and relaxation times. The rr(E) curve.for a p-type semiconductor with a single parabolic valence band and T a: (Eo-E)-i, where Eo is the energy of the band edge, is rr(E)=O E>Eo rr(E)=a(Eo-E) E<Eo, (a) (b) ~ :0 'c '2 :::J ::J (:' ~ ~ ~ :e :e 2 '" '" Q .... '" -; O~----~--~~- [1 EO [ (arbitrary units) [ (arbitrary units) 5 3 4 'c :::J (:' 3 ~ ~ 2 (c) w '" 0 FIG. 6. (T(E) function as defined by Eqs. (8) and (9). (a) A standard valence band with T=ToE-;; (b) two overlapping valence bands whose edges are separated by an energy E1 -Eo, T",F;-t for both bands; (c) the (T(E) function used in this discus sion of the transport properties of SnTe. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:163188 D. H. DAMON 40 Q §; 20 2· Vl o '40 Q 20 §; :1. Vl o -20 12 14 16 18 ~ (arbitrary units J FIG. 7. Calculated values of t~e Seebeck coefficient, S, using Eq. (10) and the u(E) function sh()wn in Fig. 6(c) plotted against (a) temperature T, (b) Fermi energy r, both T and r in the arbi trary units ()f energy shown in Fig. 6(c). where a may be temperature dependent. As usual in dealing with an extrinsic semiconductor one assumes that ajojaE is such that no other bands need be con sidered. This u(E) curve is shown in Fig. 6(a). Figure 6(b) shows the u(E) curve for a p-type semiconductor with two overlapping parabolic valence bands, the band edges being separated by an energy Eo-E1• This is the model treated by Brebrick and Strauss.l5 As previously mentioned, it does not give a good quantitative account of the dependence of the Seebeck coefficient on hole concentration. Therefore, some of the available experi mental information is used to modify this model. There are two experimental results that seem most important: (1) the minimum value of S as a function of hole concentration is almost zero (Smin"-'O.S p.V;oK at 2oo0K), and (2) the electrical conductivity at nOK as a function of hole concentration has a local minimum value12 at very nearly the same hole concentration for which S has its minimum value. Both results can be explained if the u(E) function has a local minimum value for SOme value of E below Eo. This local minimum must be fairly sharp so that there will be no minimum in a plot of u vs p at high temperatures where a 10/ aE must be wide enough to smear out the minimum in u(E). Therefore, although du(E)/dE should be nega tive for a valence band [Fig. 6(a)], one must ex pect du(E)/dE to be positive over a small range of values of E. Since ajo/ de is an even function of e only the odd part of u(e) will enter into K1, and to first order, Kl and thus S, will be proportional to some average value of du(E)/dE (for a highly degenerate metal the relation Sa:. [Cd/dB) Inu(E)]E=, is well known). When the Fermi level falls near the minimum in the u(E) curve, the positive and negative values of du (E) / dE will tend to cancel and the Seebeck coefficien t will be small. The simplest function having these properties is shown in Fig. 6(c). If the two-valence band model is correct, then the abrupt change in u (E) at E= El must be ascribed to the effect of a scattering mechanism. This function is, of course, overly simplified. How ever, it is very convenient since all the integrals entering into Eqs. (10) and (11) are of the form JEb ajo En-dE, '4 aE which is easily evaluated numerically. As we shall see, it is quite adequate for a semiquantitative discus sion of the Seebeck coefficient and Lorenz number. This u(E) function is specified by the following param eters: du/dE=a for E>E1, du/dE=b for E<E1, il=Eo-E1, and 5rr(E), which for convenience is also written as (l-c)ail, where c is a parameter [see Fig. 6(c)]. The values of Sand L depend on bfa, c, and il. Henceforth, all energies are measured in oK. Values of b/ a, c, il, and the energy-scale factor which will convert the arbitrary energy unit used in Fig. 6 into OK can be determined by fitting the experimental values of S to those calculated from Eq. (10) as a function of t and T under the following assumptions: (1) As in the two-valence-band model, p is a mono tonically increasing function of t. (2) Not only is the function form ofu(E) independent of T and [V snJ but bfa, c, and Il are constant. (3) The temperature dependence of t may be ignored. Thus we attempt to ascribe the major features of the temperature and hole concentration dependence of S to the shape of the u(E) curve and the variation of ajo/aE with temperature. Figures 7(a) and 7(b) show the calculated values of S plotted against t and T (both t and T in arbitrary units) for b/a=3 and c=0.4. Apart from the' negative values of S which are not ob served experimentally, comparison of Fig. 7 (a) with Fig. 2 shows that the calculation yields values of S in excellent qualitative agreement and rough quantitative agreement with the experimental results if the energy scale factor is chosen to be about 400, i.e., T= 1 corre sponds to 4000K and .:l = 4800°K. Figure 7 (b) should be compared to Fig. 1 of Ref. 13 and Fig. 4 of Ref. 12. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:16THERMAL CONDUCTIVITY OF SnTe 3189 This value of b. is almost exactly equal to the separation between the two valence-band edges suggested by Brebrick and Strauss. Figure 8 shows the quantity u/akT plotted against t. As anticipated, this calcula tion leads to a minimum in the electrical conductivity as a function of p. However, it suggests that this mini mum should become more pronounced at lower tem perature, whereas the experimental results show no minimum in the electrical conductivity at 4.2°K as a function of hole concentration. The simplest way to correct this deficiency is to allow c to increase towards 1 as the temperature decreases, i.e., flu(E) vanishes. This not only eliminates the minimum in u but helps eliminate the negative values of S at low temperatures when t=E1• Figure 9 shows values of L calculated from Eq. (11) again using b/ a= 3 and c= 0.4. Two qualitative features of these curves should be noted: (1) L is an increasing function of temperature, and (2) L has a well-defined maximum value for some value of t just smaller than E1• It is now possible to give a good qualitative account of the measured values of K shown in Fig. 3. For all samples, K is rather insensitive to temperature. This is a result of two factors. First, due to the strong point defect scattering, KL varies as approximately r---! rather than T-l. Second, the Lorenz number is an in. creasing function of T so that ice increases with tempera ture. At high temperatures the thermal conductivity of samples band c is appreciably larger than that of sample a even though all three samples have nearly the same electrical conductivity and the concentration of point defects is larger in samples band c. This is a result of unusually large values of L, especially for sample b. In Fig. 9, (K-KL)/uT is plotted against T, where K and u are the measured thermal and electrical conductivities and KL is computed from Eq. (3). It is T = 1/4 10 12 14 16 18 20 ~ (arbitrary units) FIG. 8. Calculated values of a/akT using the u(E) function shown in Fig. 6(c) plotted against r. Both a and r are in the arbitrary units shown in Fig. 6(c). 3.2r----.,---.--.----,.----,--, ';"-3.0 - "" o 2. ~ N > 00 2.6 ~ X ---J 2.4 2. 2'--__ -'-__ --'-__ ~ ___ '----' o .75 T (arbitrary units) 3.4~--._---r--~---r_-__, ';": 3.2 o N > 3.0 Xl ~ . 2.8 (/'0 2.6 b c 2. 4'--__ -'-~_-L __ ~ ___ .L.-__ ..... o 300 400 500 T (OK) FIG. 9. (a) Calculated values of L=K,/uT using Eq. (11) and the u(E) function shown in Fig. 6(c) plotted against T in the arbitrary units shown in Fig. 6(c). (b) Values of (K-KL)/uT plotted against T. K is the measured thermal conductivity, u is the measured electrical conductivity, and KL is the lattice thermal conductivity calculated from Eq. (3). seen that this CT(E) curve even yields a rough quantita tive agreement with the experimental results. Ideally one should like to use the measured values of Ke to deduce some information about the second moment of -CT(f) (iJjo/ih). However, this depends upon making an accurate separation of K into Ke and KL. While it has been shown that Klemens' theory probably gives reasonably good values for KL, one certainly cannot trust the theory to yield accurate values. No pretense is made that the u(E) curve used in these calculations is realistic; in particular, the dis continuity and assumptions 2 and' 3 are unrealistic. Nevertheless, we have been able to calculate values of S that are in fair agreement with measured values. Moreover, considering CT, S, and L as functions of p or t at fixed T, this model predicts that Pa-min <P v-mil< <PL-max. < Pa-max., where Pa-min is the value of p for which a has its minimum value, etc. This prediction is in good agreement with the experimental results. We claim that no matter what model of the band structure and the scattering mechanisms may be devised to explain the electrical conductivity and Seebeck coeffi cient, this model will necessarily yield values of L which will vary with T and t in a similar fashion to those shown in Fig. 9. The u (E) curve can easily be made more realistic, the discontinuity may be smoothed out, and by considering specific models of the band structure and scattering mechanisms, the values of a and b may be made to vary with T and [V Sn]. There would seem to be little value [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:163190 D. H. DAMON in doing this until the discontinuity in the u(E) curve can be explained. Using the two-valence-band model of Brebrick and Strauss in which the effective mass of the holes in the band with the higher hole energies was about ten times greater than the effective mass of the holes in the first valence band, one could suppose that interband scattering would be the necessary additional scattering mechanism as suggested by Houston and Allgaier.u It is easily seen that the discontinuity in the u(E) curve may describe the effect of interband scatter ing in a crude way, i.e., the mobility of the light-mass holes is substantially reduced when the light-mass holes can be scattered into heavy mass states. Moreover, recognizing the effectiveness of heavy-mass carriers in screening charge centers as pointed out by Robinson and Rodriguez,31 one could argue that the effect of the interband scattering should become of less importance at low temperatures where impurity scattering would dominate. The Shubnikov-De Haas oscillations in SnTe observed by Burke et at.14 show that there is indeed a second valence band; however, the effective mass of the carriers in this band is very small. If this should be the only other valence band, then interband scattering32 cannot be used to explain the experimental results. SUMMARY AND CONCLUSIONS The measured thermal conductivity of SnTe has been separated into a lattice and an electronic thermal conductivity. The lattice thermal conductivity can be described by a theory due to Klemens which treats the phonons as being scattered both by umklapp proc esses and point-defect scattering with the following conclusions: (1) The lattice thermal conductivity of pure stoi chiometric SnTe for T> 1000K should be about 13]'-1 W·cm-1°K-1. 31 J. E. Robinson and S. Rodriguez, Phys. Rev. 135, A779 (1964). 33 Recent measurements show that the Nernst-Ettingshausen coefficients for SnTe samples with p near 2XIQ20 cm-a are un usually large, thus indicating the presence of a scattering mecha nism which has a strong dependence on the energy of the carriers: B. A. Efimova, V. 1. Kaidanov, B. Va. Moizhes, and I. A. Chernik, Soviet Phys.-Solid State 7, 2032 (1966). (2) Phonon scattering by the Sn vacancies is strong. Most of this scattering is due to the strain field sur rounding the vacancy. The electronic thermal conductivity has been discussed in terms of a formalism that emphasizes the necessary correlations between the electrical conductivity, See beck coefficient and the electronic thermal conductivity. Assuming that the two-valence-band model provides a correct description of the basic features of the band structure and using the function u(E) defined by Eqs. (8) and (9) we conclude that: (1) The existence of a minimum in u as a function of p at nOK implies that u(E) must also have a local minimum value below the edge of the first valence band. (2) The existence of both a minimum and a maximum in S as a function of p is a consequence of the necessity of having both positive and negative values of du(E)/dE. (3) The Lorenz number for a material characterized by such a function u(E) must have a maximum value as a function of p. Qualitatively the measured values of the thermal conductivity show that this is the case for SnTe. Quantitatively fair agreement with the measured values of S can be obtained using an overly simplified function u(E). This function also yields values of L that appear to be in reasonable agreement with the experimental results. The mechanism responsible for this function remains unexplained. If the two-valence-band model is correct, then one must assume that some scattering mechanism reduces the mobility of the holes whose energies lie above an energy that is very close to the edge of the second valence band. The data also suggest that this scattering mechanism is more effective at 3000K than at 100°K. ACKNOWLEDGMENTS The author is indebted to Dr. P. G. Klemens, Dr. A. Sagar, and Dr. R. C. Miller for helpful discussions, to Miss B. Kagle for computer programming, and to P. Piotrowski for assistance with the measurements. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sun, 23 Nov 2014 15:02:16
1.1714608.pdf	Measurement of Nonlinear Polarization of KTaO3 using Schottky Diodes D. Kahng and S. H. Wemple Citation: Journal of Applied Physics 36, 2925 (1965); doi: 10.1063/1.1714608 View online: http://dx.doi.org/10.1063/1.1714608 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/36/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ferromagnetic-nonmagnetic and metal-insulator phase transitions at the interfaces of KTaO3 and PbTiO3 J. Appl. Phys. 116, 153709 (2014); 10.1063/1.4898738 Induced polarized state in intentionally grown oxygen deficient KTaO3 thin films J. Appl. Phys. 114, 034101 (2013); 10.1063/1.4813324 Schottky diodes using poly(3hexylthiophene) J. Appl. Phys. 74, 2957 (1993); 10.1063/1.355319 Surface Barrier Diodes on Semiconducting KTaO3 J. Appl. Phys. 38, 353 (1967); 10.1063/1.1708981 NONLINEAR DIELECTRIC PROPERTIES OF KTaO3 NEAR ITS CURIE POINT Appl. Phys. Lett. 2, 185 (1963); 10.1063/1.1753725 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 06:11:28JOURNAL OF APPLIED PHYSICS VOLUME 36. NUMBER 9 SEPTEMBER 1965 Measurement of Nonlinear Polarization of KTaOa using Schottky Diodes D. KAHNG AND S. H. WEMPLE Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey (Received 18 January 1965; in final form 24 May 1965) Capacitance vs bias voltage data are presented for Au-KTaO. surface barrier Schottky diodes. Sub stantial deviations from the normal Schottky capacitance relationship have been observed and attributed to a field-dependent dielectric constant in the depletion layer. From the capacitance data obtained at room temperature, E vs E and P vs E curves have been calculated for KTaO, and found to be consistent with previous measurements made using conventional techniques at 4.2°K. INTRODUCTION METAL-SEMICONDUCTOR surface barrier Schottky diodes1,2 fonned by vacuum deposition of gold on the cubic perovskite, potassium tantalate (KTaOa) have been investigated with particular at tention given to effects produced by polarization satu ration in the depletion layer. The semiconducting prop erties of KTaOa are described in detail elsewhere. s When slightly reduced, KTaOa is an n-type oxygen deficient 5d band semiconductor with a room-tempera ture electron mobility of 30 cm2/V-sec. Available samples have carrier concentrations in the range 3.5X 101LL2X 1019 cm-a. The "optical" bandgap of KTaOa is 3.5 eV, and the small-signal relative dielectric constant K is given by the following empirical expression in which T is the temperature in °Ka: K=48+5.7X 1041 (T-4). (1) More detailed discussions of dielectric and possible ferroelectric behavior of KTaOa are given by Hulm, Matthias, and Lont; Bell, di Benedetto, Nutter, and Waugh,· and one of the authors.a,6 Metal-semiconductor diodes are often well described by the Schottky theory1 which predicts the following dependence of diode capacitance per unit area C on applied bias voltage VB: (2) In Eq. (2) E is the assumedjield-independent permittivity of the semiconductor, qN is the assumed uniform charge density in the depletion layer, V D is the diffusion po tential, and q is the electronic charge. If the charge density (qN) is nonuniform, the slope (dl dV B) (1/C2) = 2/ qN E 1 W. Schottky, Z. Physik 118, 5 (1942). 2 H. K. Henisch, Rectifying Semi-Conductor Contacts (Oxford University Press, London, 1957). 3 S. H. Wemple, Phys. Rev. 137, A1575 (1965). 4 J. K. Hulm, B. T. Matthias, and E. A. Long, Phys. Rev. 79, 885 (1950). 6 R. O. Bell, B. di Benedetto, P. B. Nutter, and T. S. Waugh, "Nonlinear Microwave Dielectric Materials," Report No.8, Fourth Quarterly Progress Report (15 June 1962-15 October 1962). Contract No. DA36-039-SC-89126, Raytheon Company, Waltham, Massachusetts. 6 S. H. Wemple, thesis, MIT (1963). can be used to calculate qN at the space charge edge. This permits measurement, for example, of donor profiles in epitaxiallayers.7 In this paper we are concerned with effects on capaci tance measurements produced by a field dependent E, i.e., polarization saturation. The appropriate general ization of the Schottky relation, Eq. (2), is presented, and the result is used to analyze capacitance vs bias measurements on Au-KTaOa diodes. This analysis yields the electric field dependence of both the polari zation and the dielectric constant of KTaOa for electric fields in a range not readily attainable by other methods (1OL106 V /cm). EXPERIMENTAL DETAILS AND RESULTS All Au-KTaO a diodes were formed by thin film depo sition of 20-mil or 5-mil nominal diameter gold dots onto freshly cleaved [100J surfaces of KTaOa. Cleaving was performed in air just prior to placement in the vacuum chamber. Both an oil diffusion pump vacuum system and a Vac-Ion, Vac-Sorb system were found to give identical results. Film thicknesses were approx imately 5000 A for all diodes. The counterelectrode was an amalgam of indium-gallium rubbed onto a clean back surface. Total series resistance was lQ--100 n of which the KTaOa bulk resistance was a negligible part. Capacitance measurements were made at 100 kc/sec using a Boonton capacitance bridge and a biasing system described elsewhere.8 Most of the meas urements were made at room temperature. In Fig. 1 is shown a typical (1/C)2 vs VB result. The solid curve was obtained by a least-squares fitting of the data. Substantial deviations from the ideal Schottky theory are evident for large reverse bias voltages. For small voltages the curve appears to approach a linear asymptote. This is shown more clearly in Fig. 2, where the data of Fig. 1 are plotted on an expanded scale near zero bias. Substituting the asymptotic slope of Fig. 2 into Eq. (2) and using a room-temperature di electric constant of 242, we obtain the carrier concen tration N listed in Table I for diode A-I. Table I also shows typical results for several diodes having dif- 7 C. O. Thomas, D. Kahng, and R. C. Manz, J. Electrochem. Soc. 109, 1055 (1962). 8 D. Kahng, Solid-State Electron. 6, 281 (1963). 2925 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 06:11:282926 D. KAHNG AND S. H. WEMPLE XIOI~ 2.8 -- 2.4 ... I V , I ./ /1 /" ./ / /' V V 2.0 1.6 MI.2 0.8 0.4 V. 0_4 0 4 8 12 16 20 24 28 32 36 40 44 BACK VOLTAGE ON DIODE (VOLTS) FIG 1. Dependence of diode capacitance in F /cm2 on bias voltage. Diode area lS 3.6X10-4 cm2• ferent areas and carrier concentrations. A total of nearly 20 diodes have been investigated. The N" column in Table I gives carrier concentrations calculated from a four-terminal resistivity (p) measure ment and the room-temperature Hall mobility (P). Agreement with the diode results is good considering the fluctuations in carrier concentration known to occur. For example, diodes formed on the same cleaved surface may give concentrations differing by as much as a factor of two. In some samples nonuniformities in N can be observed visually since the blueness of the crystal increases with increasing N.3 Values of the diffusion potential V D listed in Table I show rather wide variations which are not presently understood. A more detailed study of the barrier characteristics is in progress in an effort to determine the importance of such factors as the image force cor rection, Fermi level shifts, interfacial surface layers, and surface-state density variations.2 In this paper we are mainly concerned with explaining the curvature in Fig. 1. After deriving the appropriate equations in the next section, we show that polarization saturation in the depletion layer gives a reasonable explanation for the observed curvature. DERIVATION Taking a one-dimensional planar geometry for a Schottky diode, we can use Gauss' law to write the Diode C-2 C-1 B-2 B-1 A-1 TABLE I. Typical results for several diodes having different areas and carrier concentrations. N VD Diode area (em-a) (V) N. (em2) 9.3XlOts 1.85 8XI018 2XlO-a 8.1 X lOIS 1.66 8X1()18 1.6X 10-4 7.0XlO17 1.25 5X1017 2Xlo--a 6.3 X 1017 1.38 5XlO17 1.6X 10--4 1.6Xl()18 2.00 3.6X10--4 following expression for a space charge of width X and unity cross section: Dx,'h= EuEx,A+PE= -iA qn(x')dx'. (3) In Eq. (3) x is the distance from the metal-semicon ductor interface, Ex, A is the electric field at that point, PE is the corresponding polarization which is assumed to be a function of E only, Dx,A is the displacement, qn(x') is the charge density at x=x', and EO is the free space permittivity. By definition we can write the dielectric permittivity of the semiconductor as E",A = dD",A/ dE", A, (4) where E is allowed to be a function of the electric field Ex ,A' From Eq. (3) we have iJD",x/iJA= -qn(A). Combining Eqs. (S) and (4) gives iJE",x/iJA= qn(A)/ E",x, At the metal-semiconductor interface x= 0 so that dDo,x/dEo,A= EO,X, iJDo,A/iJX= -qn(X), iJEo, A/iJX= qn(X)/ EO,)'. (S) (6) (4a) (Sa) (6a) Now the total voltage drop (V T= V D+ V B) across the depletion layeris (7) Differentiating, we obtain iJ V T t dE",), (fA = -Ex,x-J 0 --;:dx. (8) S~nce the electric field is zero at the space charge edge, E).,A=O. Substituting Eq. (6) into Eq. (8) gives (fVT fA dx -=-qn(A) -, (fA 0 E",>. (9) The reciprocal capacitance per unit area is defined by the following expression: 1 I a V T I I (f V T (fA I C = iJDo,>. = dX aDo,).. . (10) Using Eqs. (10), (9), and (Sa), we find (11) This should be compared with 1/C=A/E for the field- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 06:11:28XI019 2.0 1.8 -NONLINEAR POLARIZATION OF KTaOs 2927 1.6 ... ~ AU-KTQ03 DIODE ~ ~ f- ~ ,..... f- /' ~ V" -~ ~ 1.4 1.2 0.8 0.6 0.4 0.2 - 0.1 1 I I -I -0.5 o 1.5 I 1.5 2 2.5 3 3.5 4 BACK VOLTAGE ON DIODE (VOLTS) FIG. 2. Dependence of diode capacitance in farads on bias voltage for diode of Fig. 1 near zero bias. independent E case. If we take the following derivative or, using Eq. (15), (12) JEX;" 1>-dx dEx,>-= -qN -=E>-,>--E o,).. Eo X 0 Ex,). (19) and substitute Eqs. (11) and (9) into (12), we obtain Since E).,).=O, 10(1)1 2 0 tdx oVT C2 =qn(A) OA}O E",).· (13) 1>-dx Eo,).=-qN -. o Ex,). (20) Assuming that the charge density is uniform qn(x) = qN, Eq. (3) becomes Because Eo,o= 0, Eq. (17) can be written as follows: and oDx,)./ox= -(oDx,A/oA) = -qN. From Eq. (4a) dDo,). oDo,>-OA --=EO,).=----. dEo,). OA oEo,>- Combining Eqs. (15) and (16) gives fEO'X 1). dA' dEo,).=qN -=Eo.).-Eo,o. Eo,o 0 EO,).' Similarly, if A is held constant dDx,). oD",,>-oX --=E"',).=---- dE",,>- ox oE",,). (14) >-dA' Eo,)..=-qN r _,. } 0 EO,).. (21) (15) Combining Eqs. (21), (20), and (13) gives (16) (17) (18) 1 0 ~ T (~2) 1 = qN2 EO'),,' From Eqs. (20) and (11) we obtain Eo,)..= -(qN/C). (22) (23) It is interesting to note that Eq. (23) is also obtained when E is independent of E. Finally, from Eqs. (3) and (4), the polarization P becomes rEO,X (EO,)..' ) P= EO} 0 -:--1 dEo,)..'. (24) Equations (22)-(24) allow calculation of E, E, and P [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 06:11:282928 D. KAHNG AND S. H. WEMPLE 28 24 " ~ 20 .... oJ => 16 § ~ 12 i I 0. 8 4 o 280 240 ... 200 z '" Ii; 160 z o <> <> 120 0: ... ~ 80 oJ '" £5 40 o ---------/ V ./ 04 0.8 1.2 ELECTRIC FIELO (VOLTS/CM) XIO. -..... i'-.... -~ ~ -............. t--- 0.4 0.8 1.2 ELECTRIC FIELD (VOLTS/CM) x 10. FIG. 3. Calculated dependence of polarization and dielectric constant on electric field for the diode of Fig. 1. at the metal-semiconductor interface from experi mentally determined (ljC)2 vs VB data for the case N = constant. 2S0 260 r- 240 220 ~ ~I - 200 ISO • ~ • - • • ANALYSIS OF=RESULTS Using Eqs. (22)-(24) of the last section and (ljC)2 vs VB data, like that of Fig. 1, computer calculations of E vs E and P vs E have been performed. The results for the diode of Fig. 1 are shown in Fig. 3. Figure 4 shows a composite plot of all the E vs E results. The indicated points serve only to show the degree of scatter from diode to diode. All the curves are similarly shaped and in reasonable agreement considering the range of carrier concentrations (25: 1) and dot sizes (4: 1) included in the experiment. Some or all of the scatter may be due to concentration fluctuations within the very thin de pletion layer. Other possible sources of scatter include the surface roughness effects discussed by Goodman,9 and difficulties in measuring the dot areas accurately. The reasonableness of the polarization saturation model is perhaps best evaluated by comparing the room temperature saturation behavior of KTaOa obtained from diode measurements with the saturation behavior of bulk KTaOa at 4.2°K (Fig. 5) obtained from a high resistivity (p> 1010 O-cm) sample using conventional bridge methods.6 At this temperature K= 4430. Com parison of Figs. (4) and (5) can be made in terms of the coefficients in a Devonshire free energy expansion KTaO~ 295°K 160 ~ '140 120 f-~ FIG. 4. Composite plot of field depend ence of dielectric constant at room temperature for sev eral diodes. The in dicated points were taken rather arbi trarily and serve only to indicate the de gree of scatter. 100 .~ ~ f- 80 60 f- 40 20 f- o o 0.2 0.4 '0.6 O.S 1.0 E-(106VOLTS leM) 1.2 1.4 1.6 9 A. M. Goodman, J. Appl. Phys. 34, 329 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 06:11:28NO)lLINEAR POLARIZATION OF KTaOa 2929 (G) having the following form for a cubic lattice1o: Differentiating Eq. 25 we obtain aG E=-=xP+~pa+·· " ap 1 a2G --= EO-= Eo(x+3~P2)+ .. " K-l ap . (25) (26) (27) where K= e/ EO, and the other quantities are as previously defined. Using Eqs. (26) and (27) and the data of Figs. 4 and 5, the saturation parameter ~ can be readily evalu ated. The results are ~=9X109(V-m5jCS) at 4.2°K and ~= (4±1)XlO9 at 295°K. Data are insufficient to de termine higher-order saturation terms. The reasonable agreement between values of ~ based on Schottky diode saturation behavior and more direct E vs E measure ments suggests that the saturation model correctly explains the curvature in (1jC)2 vs VB data. The factor of two discrepancy may result from the quite different temperatures of observation. In the Devonshire thermo dynamic theory of ferroelectricity,ro the temperature dependence of the dielectric behavior is assumed to be contained in the parameter X, and the other coefficients such as ~ are assumed to be temperature independent. The measurements reported here for KTaOs suggest that this assumption is not altogether valid. Drougard, Landauer, and Youngll have reported an even stronger temperature dependence of ~ in BaTiOs. Because the phase transition is first order in this material, ~<O. Drougard et at. find that ~ decreases on cooling rather than increases as observed in KTaOs. An explanation of these results awaits a detailed theory of ferroelectricity. All of the Au-KTaOs diode results discussed thus far have been obtained at room temperature. Limited data taken at higher temperatures (<::< lOO°C) show a reduction of dielectric constant as expected from Eq. (1). 10 A. F. Devonshire, Phil. Mag. 40, 1040 (1949); 51, 1065 (1951). 11 M. E. Drougard, R. Landauer, and D. P. Young, Phys. Rev. 98, 1010 (1955). 5000 ~ KTa03 4.2"K 4000 3000 2000 o \. \ \ \. " " ~ ~ 2 4 6 8 10 12 14 ELECTRIC FIELD-k~lS FIG. 5. Dependence of dielectric constant on electric field at 4.2°K. Difficulties at low temperatures were encountered as a result of a very rapid decrease in reverse breakdown strength with decreasing temperature. The source of this deterioration is not understood at present. CONCLUSIONS The investigation of Au-KTaOs surface barrier diodes reported in this paper shows that (ljC)2 vs VB data can be used to obtain P vs E and E vs E for fields in the 10L 106 V j em range. The results obtained for KTaOs at 295°K are consistent with the previously measured saturation behavior of KTaOs at 4.2°K. ACKNOWLEDGMENTS The authors with to thank Professor P. J. Warter, Jr., of Princeton University for many discussions as well as for the computer program. Thanks are also extended to E. W. Chase for assistance in the experiments and for performing the gold evaporations, W. Bonner for growing some of the KTaOs crystals, and also to W. Belruss of MIT for growing some of the crystals used in this investigation. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 06:11:28
1.1707826.pdf	Comparison of a Numerical Method and the WKB Approximation in the Determination of Transmission Coefficients for Thin Insulating Films Beverly A. Politzer Citation: Journal of Applied Physics 37, 279 (1966); doi: 10.1063/1.1707826 View online: http://dx.doi.org/10.1063/1.1707826 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/37/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A method to extract absorption coefficient of thin films from transmission spectra of the films on thick substrates J. Appl. Phys. 111, 073109 (2012); 10.1063/1.3700178 Sound transmission through a thin baffled plate: Comparison of a light fluid approximation with the numerical solution of the exact equations and with experimental results J. Acoust. Soc. Am. 100, 2735 (1996); 10.1121/1.416830 New method for determining the nonlinear optical coefficients of thin films Appl. Phys. Lett. 61, 145 (1992); 10.1063/1.108199 Method for Doping Thin Insulating Films and the Comparison between the Electrical Characteristics of Undoped and Doped Films J. Appl. Phys. 42, 2081 (1971); 10.1063/1.1660491 A New Method for Doping of Thin Insulating Films and the Comparison between the Electrical Characteristics of Undoped and Doped Films J. Vac. Sci. Technol. 6, 605 (1969); 10.1116/1.1315700 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23ADSORPTION, DIFFUSION, AND NUCLEATION 279 and t=h+t2= (xNDOx)el;hlkT+(x'l,2/DoJeQ2fkT, (3) where at and a2 are jump distances, '}'1 and '}'2 are vibra tional frequencies, a1 and (.\:2 are factors that include entropies, and k is Boltzmann's constant. If X12/Do1 »xND~, the first term of the right side of Eq. (3) greatly exceeds the second term at high temperatures, whereas the second term dominates at lower tempera tures when the rapidly increasing magnitude of eQ21kT overpowers the (xNDo)eQllkT term, since Q~>Ql. From the data of Fig. 3, if D01=Dol1 Xl/X2=36. However, it is more likely that this ratio is smaller, since DOl -::'DIJa.8 In all of the experiments that resulted in a change in Q as a result of adsorption of nitrogen or oxygen, the value of Q increased and the value of Do also increased. If Do is written, as usual, Do= aryell.Slk, where AS is the activation entropy, then these results indicate an in crease in the activation entropy assuming a and')' re main somewhat constant. It is not clear at this time whether or not this assumption is justified. The large increase in activation energy for diffusion over the {OOl}-type region because of oxygen adsorp tion, compared with the small increase (if any) for dif fusion over the {011}-type region, must be due to the fact that oxygen is bound to the tungsten surface abundantly in the former region and sparsely (probably only at ledge step sites) in the former. . The effect of adsorbed gases on nucleation density can be understood to be a result of the increased surface diffusion activation energy, or decreased mobility of the adsorbate, similar to the effect of lowering the tempera ture at which the nucleation is carried out. The nuclea tion rate [la is proportional to e(mQ"d-Qd)/kT where Qad is the atom-surface binding energy, Qd is the activa tion energy for surface diffusion, and m is a positive integer. Although Qad was not measured in these experi ments, Qad= CQd with c usually equal to 4-12 19 so that an increase in Qd implies an increase in (mQad-Qd) and hence an increase in I. 18 D. Walton, J. Chern. Phys. 38, 2182 (1962). 19 G. Ehrlich, J. Chern. Phys. 31, 1111 (1959). JOURNAL OF APPLIED PHYSICS VOLUME 37, NUMBER I JANUARY 1966 Comparison of a Numerical Method and the WKB Approximation in the Determination of Transmission Coefficients for Thin Insulating Films* BEVERLY A. POLI'l'ZER Electronic Research Braf/eFt, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Oltio (Received 18 February 1965; in final form 16 August 1965) A direct numerical solution of the Schriidinger equation for the case of electron tunneling through thin film metaHnsulator-metal sandwiches is described. Curves of transmission coefficient vs"energy are ob tained for an image-force barrier model by this method and are then compared, for applied fields ranging from 1()3 to 1()3 V 1m, to analogous curves obtained by application of the WKB approximation. For a constant applied field, the WKB treatment predicts transmission coefficients which are smoothly varying functions of energy and monotonically decreasing functions of insulator thickness at all energies. On the other hand, the corresponding numerically computed quantities show definite periodic oscillations in energy and also thickness "resonances." The dependence of these oscillations on energy and thickness is shown to be the result of the partial reflection and interference of electron waves as the electron beam penetrates the barrier region representing the insulator film. The numerically computed transmission curves indicate that these reftection effects are significant at very low energies and at energies approaching the barrier maximum and that the resulting interference becomes significant at high energies and fields where the electron wavelength becomes comparable to the dimensions of the barrier. At the low energies, the major portion of the reflection is shown to originate from the regions adjacent to the metal-insulator interfaces j at the high energies, the reflection is attributed mostly to regions contiguous to the first metal-insulator interface and the top of the potential barrier. In all cases, however, the conditions for validity of the WKB treatment are seen to rule out these effects. Finally, the numerical results confirm the expected breakdown of the WKB connection formulas at those energies where a major portion of the barrier region is reflecting. INTRODUCTION IN the last few years, there has been a revived interest in the problem of electron transport * Information contained in this paper is the· result of research performed within the Electronic Research Branch, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio. U. S. Air Force Office of Aerospace Research Project 4150 is the programming authority for this work. through thin « 100 A) insulating films sandwiched between metal conducting layers. This problem was first examined in 1933 when Sommerfeld and Bethel became concerned with the electric tunnel effect in~ volved in the behavior of electrical contacts. Their 1 A. Sommerfeld and H. Bethe, Handbuch der Physik, edited by H. Geiger and K. Scheel (Julius Springer-Verlag, Berlin, 1933), Vol. 24, p. 450. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23280 BEVERLY A. POLITZER Vacuum Level I rr : . I I I I Melal r --1--- IllIUlalor --t- MetallI ---l o x----- FIG. 1. Potential energy distribution corresponding to Eqs. (2a) , (2b) , and (2c) in a biased, symmetrical metal-insulator metal sandwich. Broken line represents trapezoidal barrier with out double image-force correction. Distances shown between join points Xl and X2 and the metal-insulator interfaces at x=O and x=d are greatly exaggerated for the sake of illustration. results were later extended to an intermediate voltage range by HoIm,2 and more recently other investigators3•4 have generalized the theory further with the introduc tion of a solid dielectric layer and its associated physical properties (e.g., dielectric constant, effective electron mass, and energy band considerations). Finally, other recent treatments,5,6 in an attempt to reconcile experi mental measurements of tunnel currents with theo retically predicted results, have recognized the pres ence of electron traps in the insulating layer. In spite of the variety of approaches taken to this problem over the last 30 years, one aspect of the calculations which remains common to all is the WKB approximation to the solution of the Schrodinger wave equation. This approximation has been used as a start ing point for evaluating the transmission coefficients for the particular potential barriers assumed. Although the criteria for validity of the WKB ap proximation are extensively discussed in the literature, 7 the actual validity of the approximation in tunnel emission studies has, for the most part, been ignored. In general, the solutions to the Schrodinger equations associated with the assumed, rather complicated, po tentials cannot readily be written down in closed, 2 R. Holm, J. Appl. Phys. 22, 569 (1951). 3 J. G. Simmons, J. Appl. Phys. 34,1793 (1963). 4 R. Stratton, J. Phys. Chern. Solids 23, 1177 (1962). • J. C. Penley, Phys. Rev. 128, 596 (1962). • C. E. Drumheller, "Transverse Conductivity at High Fields in Thin Dielectric Films," National Aerospace Electronics Con f erence 1963 N ational Conference Proceedings (IRE Professional Group on Aerospace Electronics, Dayton Ohio), p. 485. 7 E. C. Kemble, The Fundamental Principle of Quantum Me fha'nics (Dover Publications, Inc., New York, 1958), p. 95. analytical form. Therefore, approximations, such as the WKB approximation, had to be used to obtain any quantitative information concerning barrier trans mission properties since tedious numerical methods were impractical before the advent of high-speed computers. The purpose of this paper is to describe a direct numerical solution to the SchrOdinger equation. This numerical solution is used to more accurately deter mine the transmission coefficients associated with an image-force barrier model of the metal-insulator-metal sandwich. The results of numerical calculations are compared· over a range of applied fields and electron energies to those results obtained by the WKB treatment. The numerical calculations presented here were done for a range of variables characteristic of the thin-film AI-AbOrAl sandwiches used for some of the tunnel investigations recently described in the literature. As a representation of the actual physical potentials ex isting in biased structures of this sort, the image-force barrier is, of course, somewhat naive; and, no doubt, distortion of the potential barrier due to space charge, traps, gradations in stoichiometry, etc., should be taken into account. Nevertheless, there is every indication that the addition of such minor details in structure to the assumed barrier will not significantly alter the general character of the numerical results obtained for this simplified potential. Furthermore, the compu tational methods described here may easily be applied not only to this simplified potential, but to any poten tial which can be expressed analytically. POTENTIAL ENERGY PROFILE Figure 1 is a plot of potential energy (electron potential) U(x) vs the distance x into a symmetrical metal-insulator-metal sandwich, the first metal elec trode being biased negatively at the voltage Vapp with respect to the second. The origin of the x axis is arbi trarily chosen at the metal I-insulator boundary. All energies are referred to the zero of energy in metal I. The potential energy is taken to be equal to zero in the first metal and -e Vapp in the second; the penetration of the applied field into the metal layerS and detailed potential structure near the metal-insulator surfaces9 are neglected. In the insulator the potential assumed is a simple, double image-force modification of a trape zoidal barrier, that is, only the image forces associated with the primary, positive, electron image charges in both the negative and positive electrodes are taken into account. In a recent studylO this has been shown to be a good approximation for the case of nonzero fields to the infinite-image potential treated by Som- 8 H. Y. Ku and F. G. Ullman, J. Appl. Phys. 35, 265 (1964). 9 J. Bardeen, Phys. Rev. 49, 653 (1936). 10 D. A. Naymik, "The Computation and an Application of the Psi Function" (to be published). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23COMPARISON OF NUMERICAL AND WKB TRANSMISSION COEFFICIENTS 281 merfeld and Bethe.ll Thus, the potential in the insulator in rationalized mks units is U(X)= 1]+ 'P-eFx-e2dj16n.EoX(d-x) , (1) whereU (X) = electron potential energy, 1]= Fermi energy in the metals, 'P=zero-field, rectangular barrier height, d=thickness of insulator film, F=applied electric field, Eo= relative dielectric constant of insulating film, and Eo=permittivity of free space (= 8.8SX 10-12 F jm). At distances close to the metal surfaces, the classical image potential is invalid and the value of U(x) [Eq. (l)J approaches -00. To maintain continuity throughout the sandwich, the potential in the insulator is joined to the potential in the metals at points Xl and X2, where U(x) is equal to zero and -eVapp, respec tively. In a typical physical situation (e.g., 'P= 2 eV, Eo=7.S, d=SO A, and 1]=S eV), points Xl and X2 are approximately 0.08 A away from the metal-insulator interfaces for fields up to 109 V jm. Thus, the potential energy throughout the sandwich can be subdivided into three areas of interest defined by the following equations: Region I U(x)=O, X::; Xl, (2a) total energy E, is traveling toward the +x direction in the one-dimensional potential illustrated in Fig. 1 and defined by Eqs. (2a), (2b), and (2c). Then in Regions I and III where the potential energy is con stant, the desired solutions to the Schrodinger equation are plane waves of the following form: 1/t!=Aeik1"+Be-ik1", X~Xl, (3a) hu=Ceiks", X~X2, (3b) where kl=propagation constant in Region I = [(2mjh2)EJ1, ka=propagation constant in Region III = [(2mjh2) (E+eVapp)Ji, and where A, B, and C are the complex amplitudes of the incident, reflected, and transmitted waves, respectively. In Region II, bounded by points Xl and X2, an analytical solution to the associated Schrodinger equation d2if;ujdX2= (2mjh2)[U(x)-EJPu, Xl~X~X2, (4) cannot be derived easily. In this closed interval then, let the wave function be written in a general form such Region II U(x)=1]+'P-eFx-e2dj16noEox(d-x), as (S) Xl::;X::;X2, (2b) Region III U(x)= -eVapp, X~X2' (2c) CALCULATION OF TRANSMISSION COEFFICIENT BY DIRECT NUMERICAL INTEGRATION Analysis and Approach A computer program has been written for the IBM 7094 Computer to generate tables of transmission and reflection coefficients as functions of energy and ap plied field for the potential defined by Eqs. (2a), (2b), and (2c). Although the program deals only with sym metrical sandwiches (such as illustrated in Fig. 1), the modifications necessary to handle the case of dissimilar metal electrodes can be easily incorporated. The varia ble parameters employed in the program are 1], 'P, E., and d. As the first step in evaluating the transmission coeffi cient, it is necessary to cycle the applied field through seven decades (lOa to 109 V jm) and to determine the join points Xl and X2 for each value of the field. Then the energy at the top of the potential barrier Umax(F) is computed, and Emax(F) , the maximum electron energy of interest, is thereby defined for the remainder of the calculation. Finally, the transmission and reflec tion coefficients are evaluated at closely-spaced energy steps in the energy range 0 to Emax. The analysis on which this evaluation was based follows. Suppose a beam of electrons, each having 11 A. Sommerfeld and H. Bethe, Handbuch der Physik, edited by H. Geiger and K. Scheel (Julius Springer-Verlag, Berlin, 1933), Vol. 24, p. 450. where if;ur is the real part of the complex wave function if;II, and if;IIi is the imaginary part; both !J;IIr and if;IIi are, of course, unknown functions' of X which inde pendently must satisfy the SchrOdinger equation through out the interval. If continuity of the wave functions and their first derivatives with respect to x is enforced at the points Xl and X2, the following two systems of linear equations are derived: [if;UrJ"="l =Ar'Pl+ Br'Pl-A i'P2+ Bi'P2, [if;m] "'-"'1 = Ar'PZ-Br'P2+ A i'Pl+ Bi'Pl, (6) [fUrJ"="l = -Arkl'PZ- Brkl'P2-A ikl'Pl+ Bikl'Pl, [fmJ"="l =Arkl'Pl- Brkl'Pl-A ikl'P2-Bikl'P2, and [if;IIrJ"="2 = C,ol-Ci82, [if;mJ"="2= C,oz+C i8l, [fIIrJ"="2= -Crka8z-C;kafh, [fIIiJ"="2=C rka8l-Cika8z. (7) Here Ar and Ai, Br and B;, Cr and C; are the real and imaginary parts of the complex coefficients A, B, and C, respectively; 'PI = cosk1Xl, 'P2= sink lXI, 81= COSkaX2, and 82=sinkax2 are known constants for a particular E and a given applied electric field. [if;IIr]", [!J;Ili]", [furJ", and [fmJ" represent the numerical values of the real and imaginary parts of the wave functions and of their first derivatives (with respect to x), all evaluated at a particular X in the interval Xl::;X~XZ. For a numerical integration of Eq. (4) over the region Xl::;X~XZ, one may hypothetically determine [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23282 BEVERLY A. POLITZER 180 160 14.0 , 12.0 10.0 . I ~ ao g ./§ :.~ § Q 103-106V/m ENERGY (IN) - 2.0 3.0 zero-field rectollQU lor barrier heiQhl 7.0 FIG. 2. Ratio of transmission coefficients computed by WKB ap proxim~tion to th~se computed by direct, n~m~rical int~gration of SchrOdinger equation are plotted for energIes m tunneling range. Values assumed for parameters: '1=5 eV, 1"=2 eV, e.=7.5, d=25 A. Large dots represent computed points and are shown where exact shape of curve is uncertain or where curve changes rapidly. Short lines through these dots represent final computed points at energy of barrier maxima. Curves are discontinued at energies where T(E)NUM is less than 10-37• Omitted curve for 107 V 1m closely follows low-field curve (loa to 106 Vim). Oscilla tions appearing here are the mantfestation of reflection and interference phenomena indicated by the numerical computations (see Figs. 4 and 5). the starting values of the wave function at the point Xl by guessing at the values of the complex coefficients A and B in Eq. (6). However, it is important to re member that the values of A and B must be chosen in such a way that the final solution in Region III represents only the physically desirable transmitted wave. That is, the number D, representing the ampli tude of a reflected wave in a very general solution to the SchrOdinger equation for X~X2 given by (8) must be identically zero. If it were possible to choose A and B in this way, Eq. (7) could then be used, along with the numerically integrated values of the wave function at point X2, to evaluate the complex number C. For practical purposes however, it was convenient in the computer calculations to turn this boundary value problem into an initial-value problem by working backwards. The amplitude of the reflected wave in Region III was assumed to be zero [i.e., D=O in Eq. (8)J, and the amplitude C of the transmitted wave was taken, for simplicity, as unity with Ci=O. The second-order wave equation, Eq. (4), was reduced to two sets of two simultaneous first-order differential equations, one set having 1fIlr and fIlr as the dependent variables, and the other having 1fIIi and fIIi. A back ward (negative AX) numerical integration of the trans formed system was then started at the point X2, the initial values of [1fIIrJz=:tt, [1fIIiJz=".. [fIIrJz="'2' and [fIIiJ",=Z2 needed for the integration being obtained from Eq. (7). At Xl the numerical integration was stopped. The known, integrated values of 1fIIr, 1fIIi, fIlr, and fIIi for this point were then substituted into the linear system given by Eq. (6) and the unknowns Ar, Ai, Br, and Bi were determined. Once the complex coefficients A, B, and C were determined, the transmission coefficient T(E) was readily computed according to the following relation: (9) where k3 and kl are the propagation constantsprevi ously defined for the particular energy. The trans mission coefficient as given by Eq. (9) is the ratio between the probability current densities associated with the plane waves representing the transmitted and incident beams in the two metal electrodes. At this energy E the reflection coefficient R(E) was also evalu ated using (10) Numerical Methods and Accuracy A brief description of the numerical methods used in the calculations and of the accuracy of the results follows. A fourth-order, Adams-Moulton predictor corrector methodl2 was used for the numerical inte gration of the reduced system of first-order differential equations referred to previously. The predictor-correc tor method employed a variable step size Ax, the magnitude of which was controlled by the local trunca tion error at every Xi in the closed interval XI~X~X2. A Runge-Kutta-Blum integration routine13 was used to generate the starting values for the Adams-Moulton formulas at the beginning of the integration and when ever the step size was changed. A minimum of 1500 to 2000 integration steps was needed to preserve a local truncation error of 10--8 throughout the range of energies and fields considered. With this number of steps, the round-off error produced was less than 10--6• This small round-off error was readily obtainable by (1) starting the variable-step integration using the very efficient Blum modification of the Runge-Kutta method and (2) carrying 16 sig nificant figures for the computed values of the depend ent variables (1fIlr, 1fIIi, fIlr, and fIIi) throughout the Adams-Moulton calculation. As mentioned previously, the computation begins by determining the join points for each value of the applied field. Two third-degree polynomials (one in Xl, the other in X2) were generated by equating the irnage force potential to those values achieved at these join points. The roots of these polynomials were then rapidly obtained in the computer calculation by using an iterative technique of Bairstow.14 In the process of evaluating Emax (F) , the same iteration method was 12 IBM Share Program #450. 13 E. K. Blum, "A Modification of the Runge-Kutta Fourth Order Method" in Proceedings of the Mathematics Committee of Univac Scientific Exchange Meeting, (Remington Rand Univac, St. Paul, Minnesota, 1957), Appendix H. 14 F. B. Hildebrand, Introduction to Numerical Analysis (Mc Graw-Hill Book Co., Inc., New York, 1956), p. 472. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23COM PAR ISO N 0 F N U MER I CAL AND W K B T RAN S MIS S ION CO E F F I C lEN T S 283 18.0 zero-field rectangular 16.0 barrier height 140 .:, . ii 12.0 103TO 106V1m~1 10.0 I f\ ~ll ! FIG. 3. Same as Fig. 2 with d=50 X. Since T(E)NUM for 50 X < T(E)NUM for 25 X, energies at which T(E)NUM falls below 10-37 are higher than those in Fig. 2. Note also that the frequency of oscillation in the case of 109 V 1m is greater than that in the corre sponding case in Fig. 2. 8.0 Fermi level / Ifl!\! 6.0 109 Vim 81 / .:t ~i! 4.0 10 Vim ;: .... : 'i ' 2.0 ': I, 'I O~----~~~~~~-A~~~~~--~~- -2.0 ~ -4.0 -6.0 L-__ L-__ ~-L ____ ~ ______ L-____ ~ ____ ~ __ ~ 1.4 2.0 3.0 4.0 5.0 6.0 7.0 used to solve the appropriate fourth-degree polynomial for Xm where Xm is the x at which U(x) achieved a maximum in the interval Xl~ x~ X2. In both applica tions of Bairstow's method, the relative errors were always less than 10-6• This relative error is based on a comparison of the sum and product of the iterated roots to the appropriate coefficients (from mathematical theory of polynomials) of the normalized polynomials. Several independent tests of the accuracy of the numerical integration were performed. First, the second order SchrOdinger equation [Eq. (4)J was integrated directly on an analog computer and the results checked against the digital solutions for the reduced system of first-order differential equations. Because of scaling problems and the limited accuracy inherent in analog computation, only a rough comparison of the shape of the Y;n function was obtainable by this method. Con sequently, the analog solutions were redone on the IBM 7094 computer using a program15 which simulates digitally the components and logic usually employed in analog computation. In this way, verification of the actual numerical values of the wave function was made to five significant digits, the maximum of digits written out by the analog-simulator program. As a significant check on the accuracy of the approach and numerical computations, the sum of T(E) and R(E) was monitored throughout the calculation. In all cases where this test was applicable, the sum turned out to be 1.0±10-8• A test of this nature was not appli cable, of course, in those cases where T(E) was less than 10-8 and R(E) was unity to eight places, eight being 15 F. J. Sansom, R. T. Harnett, and H. E. Petersen, "MIDAS Modified Integration Digital Analog Simulator," ASNCC In ternal Memo 63-24, Wright-Patterson Air Force Base, Ohio (June 1963). ENERGY (eV)- the maximum number of significant digits normally carried in computer arithmetic. CALCULATION OF TRANSMISSION COEFFICIENT BY THE WKB APPROXIMATION In order to determine quantitatively the extent to which the numerically computed transmission coeffici ents differed from those obtained by the application of the WKB approximation, a computer program was written to calculate T(E)wKB using the following well known relation16: The classical turning points Xtl and Xt2, defining the limits on the integral for a given E and F, were ob tained by setting E equal to U(x) in Eq. (2b) and solving the resulting polynomial with the Bairstow iteration technique. Again the relative errors were less than 10-6• Very accurate values of Xtl and Xt2 were needed in the computation since U(x), for most cases, rises very steeply in the vicinity of the turning points. A difference of only a fraction of an angstrom may represent a considerable difference in the area under the [U(x)-EJ curve; likewise, this resulting difference in the value of the integral is further magnified by the exponential nature of the T(E)wKB expression. Finally, the integration in Eq. (11) was performed numerically by applying Simpson's formula over the interval Xtl~X~Xt2. A convergence of 10-5 (corresponding to five significant figures in the computed values of the 16 N. F. Mott and 1. N. Sneddon, Wave Mechanics and Its Applications (University Press, Oxford, 1948), p. 23. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23284 BEVERLY A. POLITZER ENERGY (eV) FIG. 4. Computed values of transmission coefficients (WKB and numerical) for tunneling energies close to barrier maximum and for two values of insulator thickness differing by ",>./4 (for a 5-eV electron). Values of parameters assumed for calcula tion: F=1()9 Vim, '7=5 eV, \1'=2 eV, E.=7.5. Line through large dot represents final computed point at energy of barrier maxima. The thickness "resonances" seen here are an indication of inter ference phenomena. integral) was maintained for all values of applied field and electron energies. RESULTS AND CONCLUSIONS The results of some typical numerical and WKB calculations are graphically compared and summarized in Figs. 2 and 3. Values of the variable parameters assumed for these figures are '1)=5 eV, e.=7.5, and cp= 2 eV, with d= 25 A (Fig. 2) and 50 A (Fig. 3). The ratio ,of the transmission coefficients computed according to both methods is represented by the value of the ordinate, 10 log[T(E)wKBI T(E) NUM]. Ana lyzed in terms of a percent difference, such as 1001 (TNUM-TWKB)/TNUM I, the ordinate values cover a range of approximately 0% to 3000%. In those cases where T(E)NuM<lo-37, the magnitudes of lInr and lIni exceeded the numerical range allowed by the computer. When this occurred, for example, at low energies for low applied fields and at higher energies as the film thickness increased, the calculations were discontinued. The results for these cases are therefore not represented in the figures. Since the barrier shape is insensitive to changes in applied voltage throughout the low-field range of 1()3 to 106 V 1m, the computed values of the ordinate for these fields were identical to two digits. Consequently, the curves coincide in Figs. 2 and 3. In addition, the curves for F= 107 V 1m have been omitted because they follow the low-field curves quite closely. The oscillations appearing in these curves are caused solely by the oscillatory nature of T(E)NUM. To prove this point, in Figs. 4 and 5 the computed values of T(E)NUM and T(E)WKB have been plotted separately for an applied field of 109 V 1m and for several film thicknesses. Note the following: first, for all values of d, T(E)NUM oscillates around the corresponding WKB transmission curve which is itself a smoothly varying function of energy. Second, the energies at which the numerical and WKB curves coincide are, in fact, the same energies at which the value of the ordinate for these cases passes through zero in Figs. 2 and 3. The features cited here are generally characteristic of all the remaining cases represented in Figs. 2 and 3. That is, whereas the computed values of T(E)WKB were always smoothly varying functions of electron energy, oscillations appeared in the numerically computed transmission curves. Furthermore, these oscillations were found to increase in frequency as the electron energy approached Emax(F), as d increased, and as the fields increased to the range of values where tunnel emission becomes significant (note Figs. 2-5) For the most part, oscillatory transmission coeffici ents have been encountered previously only in situa tions where particles pass over a potential barrier [i.e., cases where E> U(x) for all x]. An example of this is thermionic emission at metal-vacuum interfaces. Here, the periodic deviations in the Schottky effect are the result of oscillatory reflection coefficients associated with electrons passing over an image-force type barrier similar to that in Fig. 1.17 As is shown later, how ever, for any generalized barrier and for any particle energy, there is a large amount of partial reflection of de Broglie waves whenever the potential gradient is large; and if in this situation the particle wavelength is comparable to the dimensions of the system, the resulting interference of the reflected waves will be manifested in an oscillatory transmission coefficient. In support of the statement above, the oscillatory reflection coefficient in the case of thermionic emission has been explained by Herring and Nichols18 in terms of the interference between electron waves reflected from the "potential hump" and from the metal-vacuum interface. Accordingly, they show the total reflection 57 59 al 63 65 6.7 6.9 ENERGY (611) FIG. 5. Computed values of. transmission cpefficients. (WKB and numerical) for insulator thicknesses of 25 A and 75 A. Other assumed parameters same as Fig. 4. Increased frequency of oscil lation in T(E)NUH for d=75A is an indication of higher-order interference effects. 17 E. Guth and C. J. Mullin, Phys. Rev. 59, 575 (1941). 18 C. Herring and M. H. Nichols, Rev. Mod. Phys. 21, 249 (1949). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23CO M PARISON OF NUM ERI CAL AN D WKB TRANSM ISSION COEFF I CI ENTS 285 coefficient for the system to have a definite dependence on the distance of the "potential hump" from the metal-vacuum interface. In Figs. 4 and 5, the oscilla tory transmission coefficient associated with electron tunneling may also be explained in terms of the partial reflection and interference of electron waves. It will be shown, however, that the reflecting regions in this case include not only the first metal-insulator interface and the top of the potential barrier but also the second metal-insulator interface where the potential plunges toward -00. In support of the interference interpretation here, Figs. 4 and 5 show the transmission at a given thick ness, or electron "path length," to be a periodic func tion of electron wavelength. The difference between the two film thicknesses (1.3 A) chosen for Fig. 4 is approximately a AI 4 difference in path length for a Fermi energy electron (S eV in this example). As ex pected, the numerical transmission curves for these two films oscillate around each other, the transmission "peaks" of one nearly coinciding with the "minima" of the other. This thickness "resonance" effect is even better illustrated in Fig. S where one sees the computed value of T(E)NUM for the 7s-A film exceeding that for the 2s-A film at several energies in the range plotted. Lending further support to this interference interpre tation, the frequency of oscillation of T(E)NUM in creases (or equivalently, the interference order in creases) as the electron wavelength approaches the dimensions of the system. Figures 2 and 3 show this occurring in the case of a constant d as the electron wavelength decreases with increasing energy or in creasing field. Figures 4 and S show the frequency of oscillation to increase, over a given energy range and a constant applied field, as d increases. We also see in Fig. S, however, that T(E)WKB is a monotonically decreasing function of film thickness for all possible tunneling energies. [Equation (11) demon strates quite clearly that the magnitude of the trans mission coefficient is merely inversely proportional to the size of the "classically forbidden" area between the turning points.] In effect, T(E)WKB, therefore, masks out the reflection and interference effects which appear when the more exact, numerical treatment is applied to the double image-force barrier. This fact, however, should not be surprising; in all situations, the conse quence of assuming wave functions of the WKB type is to exclude all possible reflection of electron waves and, in addition, all of the physical manifestation of these reflections, i.e., interference phenomena, periodic oscillations in the transmission coefficient, transmission resonances, etc.19 The proof of the last statement lies 19 This problem was first realized in 1939 when H. M. Mott Smith [Phys. Rev. 56, 668 (1939)J attempted to justify theo retically the observed periodic deviations from the Schottky line for thermionic tantalum and tungsten cathodes. In this case the WKB approximation alone, by the very nature of its assump tions, could not predict the fluctuating reflection coefficient caused by reflection of thermions from steeply rising regions of the surface potential. 9 8 I I 7 XI X2 "> 6 ..! 5 >-(!) 4 0: \oJ Z 3 \oJ 2 region of WKB validity I 0 0 8 16 24 32 40 48 0 X (Al FIG. 6. Representation of the regions of validity of the WKB wave function in the interval X1:SX:SX2 for electron energies be tween 0 and Ems.x. Values assumed for tl].e parameters: F= 109 Vim, 7]=5 eV, '1'=2 eV, E,=7.5, d=50A. The WKB validity criterion, inequality (12), was considered satisfied when the left side of (12) was :SO.I. Regions where the validity criterion is violated are designated as reflecting regions21 and are indicated by cross-hatching. Note that at the low energies, i.e., 0 to ",3.4 eV, the major portion of the reflection originates from regions adjacent to the metal-insulator interfaces. At high energies, in the range 5.7 eV to Ems.x, reflection can be attributed mainly to the first metal-insulator interface and the top of the potential barrier. in the nature of the so-called WKB validity criterion, mhidU(x)/dxi -----~«1 for E> U(x). (12) [2m[E- U(x)]]! [A similar condition is written for the case where U (x) > E.] Basically, (12) states that wave functions of the WKB type are valid mathematically in a region which is several wavelengths away from a turning point and where the potential varies slowly with X.2ll Reasoning by analogy, however, it can be shown21 that the condition expressed by (12) may be thought of as a quantum mechanical criterion for vanishing reflection of the de Broglie waves associated with the particles under consideration. On the basis, then, of the last statement, a con siderable amount of reflection is anticipated in a physical system which violates the WKB validity criterion. Thus, by evaluating Inequality (12) at every 20 L. 1. Schiff, Quantum Mechanics (McGraw-Hill Book Co., Inc., New York, 1955), p. 186. 21 Written in an equivalent form dealing with the electron wave length, (12) becomes Idh h I dX 2 ... «h. This inequality demands that the magnitude of the fractional change in the wavelength in a distance equal to A/2 ... be small compared to the wavelength itself. In the analogous optical situation (the transmission of photons through a medium of varying index of refraction), such an inequality characterizes the condition for reflection or nonreflection of electromagnetic waves. (Here, of course, the gradient in the index of refraction plays a similar role to the potential distribution.) By analogy then, (12), the WKB criterion for validity, expresses the quantum-mechanical condition for vanishing reflection of de Broglie waves in a system characterized by U(x). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23286 BEVERLY A. POLITZER x in the interval Xl:S;X:S;X2, one may determine, for each energy in the range 0 to Emax (F) , those portions of the barrier which contribute most to reflection effects. Figure 6 illustrates the results of such a nu merical investigation for an applied field of 109 Vim and for the same variables assumed in Fig. 4. For the purposes of these calculations, the WKB validity cri terion for each energy E was considered satisfied at those x's where the numerical value of the left side of (12) was :S;O.1. Those intervals of x :where the value of the left side of (12) exceeded 0.1 were considered to be reflecting intervals. It is easily seen from this representation that at low electron energies, in par ticular 0 to 3.4 eV, the major portion of the reflection comes from the vicinity of the two metal-insulator interfaces. At these energies, that part of the interval Xl:S; x:S; X2 which is responsible for reflection increases from approximately 4% at 0 eV to approximately 50% at 3.4 eV. At higher electron energies, e.g., 5.7 eV to Emsx, Fig. 6 shows the major reflecting region to be adjacent to the first metal-insulator interface and the top of the potential barrier. At approximately 5.7 eV, neady 46% of the interval Xl:S;X:S;X2 violates the WKB validity criterion; at Emax, this quantity is 30%. These results agree qualitatively with the informa tion contained in Figs. 2 and 3. Note first that with the exception of the individual oscillations and minor structure, all the curves appearing in Figs. 2 and 3 have roughly the same shape. [It is important to remember that the oscillations in T(E)NUM, and hence in the ratio T(E)WKBIT(E)NUM, are merely an indica tion of interference effects which enter the picture only at higher energies and fields.J This similarity in shape points out that regardless of the applied field or film thickness involved, the WKB treatment fails at ap proximately the same energies in the range 0 to Emax(F). It is easily seen also that these energies at which the WKB treatment fails coincide with those energies for which a large portion of the interval Xl:S; x:S; X2 is re flecting (see Fig. 6). In the cases where the value of the ordinate 10 10g[T(E)wKBIT(E)NuMJ has been com puted for most of the energy range, it is apparent that T(E)WKB and T(E)NUM diverge as E goes from 0 to ",3 eV, or as a larger portion of the interval Xl:S;X:S;X2 becomes reflecting. In the intermediate energy range, the interval responsible for reflection decreases and the value of the ordinate in Figs. 2 and 3 goes through zero. Finally, as E approaches Emax the magnitude of· the reflecting region again increases and T(E)WKBI T(E)NUM rapidly becomes very large, the two quanti ties beginning to differ by an order of magnitude. Superimposed upon the failure of the WKB wave function to satisfy the Schrodinger equation at low and high energies is the breakdown of the WKB con nection formulas at these energies. Again this is proba bly reflected in the similarity in shape of the curves appearing in Figs. 2 and 3. Briefly, these connection formulas are needed to handle a generalized barrier penetration problem and eventually arrive at Eq. (11); the formulas accomplish this by connecting the asymp totic WKB solutions on both sides of a turning point, across the regions adjacent to the turning point where the solutions become invalid. In general, the connection formulas break down in the following situations22: (1) when the particle energy E approaches U max the top of the potential barrier, and U(x), therefore, cannot appropriately be approximated by a linear potential as is assumed in the derivation of the connection formulas; and (2), where the slope of the potential around the turning points is large or infinite as it is, for example, when the turning points are close to discontinuous join points. In the application of the WKB approximation to the double image-force barrier, the second situation certainly accounts in part for the large errors observed at the lower energies. At these low energies the corre sponding turning points Xtl and xt21ie very close to the discontinuous join points Xl and X2. Likewise, the in crease in error at energies approaching the barrier maximum may be definitely explained by the first situation. In fact, as Fig. 6 clearly illustrates, at F= 109 V 1m it is absurd to attempt the application of the connection formulas [in the form of Eq. (l1)J to any electron energy between approximately 5.7 eV and Emax. At these energies, the WKB solutions do not successfully describe the electron wave function any where in the interval encompassing the entire classically forbidden region [i.e., where E< U(x)J and a con siderable amount of the classically allowed region ad jacent to both turning points. In the low-field cases at these very high energies, the increasingly important interference effects tend to obscure this evidence of connection formula failure. Here the rapid descent of these curves toward a zero value of the ordinate is an indication of the beginnings of oscillation in T NUM rather than of decreasing error at these energies. ACKNOWLEDGMENTS The author wishes to thank Dr. K. G. Guderley for a helpful and stimulating discussion at the beginning of this work. In addition, the author acknowledges S. M. Call and J. L. Politzer for their painstaking criticism of this manuscript. 22 E. C. Kemble, The Fundamental Principle of Quantum Me chanics (Dover Publications, Inc., New York, 1958), p. 100. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 178.118.80.235 On: Wed, 02 Apr 2014 13:43:23
1.1704366.pdf	Bosons and Fermions R. Penney Citation: Journal of Mathematical Physics 6, 1031 (1965); doi: 10.1063/1.1704366 View online: http://dx.doi.org/10.1063/1.1704366 View Table of Contents: http://aip.scitation.org/toc/jmp/6/7 Published by the American Institute of PhysicsGEOMETRIZATION OF A MASSIVE SCALAR FIELD 1031 our conditions reduce to Tp.. = T'M (17a) Tp..l. = 0, (17b) Too> 0, (17c) T < 0, (17d) Tp."T.<x = !T2g~, (27) (T~ -!Tg~){T<x1l1 "I -T""III1} = 0. (28) and Eq. (16). Aside from the boundary condition, these re strictions are the conditions previously3 found for the massless meson. The symmetry, and the van- JOURNAL OF MATHEMATICAL PHYSICS ishing divergence of T 1" are trivial conditions since the Einstein tensor obeys such identities. v. CONCLUSIONS We have found necessary and sufficient conditions which must be imposed upon a Riemannian geometry in order that we may consistently interpret the geometry in terms of a massive "meson" field. Analogously to the development of the Maxwell field in terms of geometry,2 the present analysis permits a geometrical interpretation of a classical field of physics. Further analysis of the geometrodynamical con sequences of our conditions may be expected to lead to deeper understanding of geometrodynamics2 itself. VOLUME 6, NUMBER 7 JULY 1965 Bosons and Fermions R. PENNEY Scientific Laboratory, Ford Motor Company, Dearborn, Michigan (Received 21 December 1964) It is proven that one cannot construct boson creation and annihilation operators from a finite number of fermion operators. The proof follows from the isomorphism of the fermion algebra and the algebra of Dirac matrices. I. INTRODUCTION IN the present analysis, we wish to address our selves to the problem of "Inaking bosons from fermions." Before proceeding further, we must clarify this concept. As usual, we consider a fermion field to be de scribed by a set of annihilation and creation opera tors in the Fock scheme. The anticommutation rules for these operators are the usual ones. We wish to investigate the possibility of combining such opera tors to produce a set of boson creation and annihila tion operators. The connection between boson and fermion opera tors has been studied by Case,l who showed that one could not, for example, produce a theory of gravitons using quadrilinear combinations of the operators for a two-component neutrino field. Our investigation will be more restrictive than Case's since we will consider only a finite number of fermion 1 K. M. Case, Phys. Rev. 106, 1316 (1957). operators, but more general in that we allow more general combinations of the fermion operators. We intend to prove that one cannot form a boson creation operator from a finite number of fermion operators. Our result may help to explain, for ex ample, why the creation operators for Cooper pairs2 in the BCS theory of superconductivity retain their Fermi-Dirac statistics. II. TWO-FERMION PROOF We consider the possibility of constructing a com bination of two fermion creation and annihilation operators to Inake a boson creation operator. Let us suppose, therefore, that we have two operators Al, A2 and their Hermitian conjugates obeying the rules (1) Z J. Bardeen, L. N. Cooper, and J. R. Schriefi"er, Phys. Rev. 108, 1180 (1957). 1032 R. PENNEY We then define the combinations BI == Al + A!, B2 == i(AI -A!), B3 = A2 + A~, B4 = i(A2 -A~), (2) (3) and we thereby summarize the properties of AI, A2 as B"B. + B.B" = 20"., B~ = B". (4) (5) The set of operators B" is thus seen to be isomor phic to the Dirac 'Y-matrices. As usual, therefore, we may form the Clifford algebra3 of the B" with the members (6) and generically denote the 16 members by r '" where r I is the unit operator. We may now use all of the well-known properties of the r" to solve our problem. We ask whether a function of the r '" say fer ,,) exists with the property: [f, fl-= 1, (7) which is the minimal property of a boson operator which we must demand. First we note that, most generally, (8) where the a" are c-numbers. The properties of the r" ensure that no powers of r" occur. We separate the unit element so that 16 f = al + L a"r '" (9) ,,-2 16 f = aT + La!r", (10) .. -2 and demand that 16 16 L L a"a~[r"r.]_ = l. (11) ,,-2 ,,-2 The terms in ai, a! obviously commute with all others, so our remaining sums do not include rl· Since r I does not occur in the series, we may use the well-known fact that each r" commutes with eight other r. and anti commutes with eight others. Thus every term in our series has the property [r"r.]_ = 0, or = 2r"r •. (12) (13) 3 P. Roman, Theory of Elementary Particles (North-Holland Publishing Company, Amsterdam, 1961), 2nd ed., p. 114. Our demand therefore reduces to (14) where the primes denote the omission of the van ishing terms. Now we recall that r"r. = 1 if and only if fJ. = v. But all terms for which fJ. = II are excluded in the primed sums because they gave zero for the com mutators. Next, we realize that r"r. is proportional to some rp ~ 1, so our sum reduces to the form 16 1 LQ"r" =-. ,,-2 2 (15) Renaming QI = -t here, we see that we are demanding (16) which, due to the linear independence of the r ", would demand that QI vanish. Thus our demand is absurd, and we have proved that we cannot con struct a boson from two fermions. ill. N-FERMION PROOF It is easy to see that method of proof can be ex tended to any finite number of fermions. We simply form Bj = Aj + A~, Bj+1 = i(Aj -A~) for each fermion operator, and see that B,Bj + BjB, = 20ij, B~ = Bj• (17) (18) (19) (20) We thereby have the properties of the fermion operators contained in a Clifford algebra of (2N)2 numbers. The basic properties we have used in the proof for N = 2 are the same for any N, and the proof follows trivially. Note that we are in general allowing for products of 2N fermion operators in our combinations. IV. APPARENT CONTRADICTION OF THE THEOREM It is important to realize that, in proving our theorem, we have assumed nothing about the states upon which our operators act. The fact that one is able to construct boson creation operators from fermion operators, as illustrated by Case, I is due to further assumptions concerning the states utilized in a particular theory. For example, if one supposes BOSONS AND FERMIONS 1033 that the boson operators act only upon states of the form of a "Dirac filled sea," in which all negative k states are filled for large k, and all positive k states are empty for large enough k, then one is able to construct boson operators. The well-known analysis4 of the neutrino theory of light uses the fact that neutrinos may be con sidered to occupy all negative energy states, and assumes that all positive energy states for high energy are empty. By this assumption, Born and Nath5 were first able to construct creation and an nihilation operators for photons from those for neu trinos. Thus, our theorem illustrates the important point that the commutation rules for operators may seem to differ depending upon the assumptions concern ing the states upon which the operators act. Lieb and Mattis,6 for example, have found that certain density operators for a one-dimensional electron gas "model" have bosonlike commutation rules, due to the existence of a filled Dirac sea, as first realized by Born and Nath.5 Actually, the commutation rules of operators should be independent of any assumptions concern ing the states upon which the operators act. Thus, we are faced with an apparent dilemma which must be resolved. To understand the problem, we may consider a trivial example. Let us suppose we have a single creation operator of the fermion type, with its annihilation operator. The only irreducible representation of the concomi tant algebra, as is well known, is of the form of 2 X 2 matrices, a = (8 A), a* = (~ 8), (21) which therefore operate upon states of the form .p = 'Pi &) + 'P2(~)' (22) Now it is possible for us to assume that our opera tors only act upon those states for which a.p = 0, (23) in which case, our operators, acting upon such states obey aa* -aa = 1, (24) as is easily checked. The point is that our choice of states upon which the operators act is a projection of the whole space, and the "altered" commutation 4 M. H. L. Pryce, Proc. Roy. Soc. (London) A165, 247 (1938). 5 M. Born and N. S. N. Nath, Proc. Indian Acad. Sci. A3, 318 (1936). 6 E. Lieb and D. Mattis, J. Math. Phys. 6, 304 (1965). rules are true only in the sense that the operators themselves are altered. In the present example, our restriction of states allowed only part of the opera tors to operate, and the apparent "boson" rules we obtained were meaningless. In a similar manner, the apparent change of com mutation rules for the operators used by Case,i or Lieb and Mattis,6 obtains because of assumptions concerning the Hilbert space wherein the operators perform. To Hlustrate this point quite clearly, we consider certain operators, first used by Born and and Nath.5 Let ak, at be a denumerable set of annihilation and creation operators for fermions obeying and define R f = L atak+i' k--R R f = L at+iak, k--R where R is a large number. (25) (26) (27) (28) (29) Using the commutation rules, we easily calculate R R [f, f]-= L atal: -L at+iak+l' (30) k--R k--R which reduces to Now, the right-hand side of Eq. (31) has the pos sible values 0, ±I, depending upon the assumptions concerning the underlying Hilbert space. As ex amples we may consider three possible subspaces. One subspace contains a finite number of occupied states, in which event we may always take R large enough to obtain O. Another (unrealistic) subspace has a "filled sea" of positive energy states, with negative energy states empty, and we obtain -1. The last subspace is the usual "filled sea" of negative energy states, which gives + 1 for our commutator. Thus we really have no contradiction of the the orem proved in the present analysis. Nonetheless, one can make boson operators from fermion opera tors, provided one operates only within a projected region of Hilbert space, and that no operations in volved remove one from that particular region of Hilbert space. 1034 R. PENNEY As long as one considers only a finite number of fermions, one may not construct bosons. If, however, one allows a "filled sea" of fermions, it is possible to obtain bosonlike operators. Our theorem is not true in the limit of N = <Xl. V. CONCLUSIONS We have shown that one cannot construct boson creation and annihilation operators from a finite number of fermion operators. Incidentally, we have seen that the commutation rules for fermion creation operators are summarized in a Clifford algebra, a result which has apparently not been noticed before Using the isomorphism of the fermion operators with the Clifford algebra, one can deduce the irreducible representations very quickly. For one fermion opera- JOURNAL OF MATHEMATICAL PHYSICS tor, the algebra is the Pauli algebra, of course, and that fact is commonly used. We have also seen that the existence of a filled Dirac sea with an infinite number of fermions allows one to construct boson operators. As long as one is careful to stay within the Hilbert subspace contain ing the filled sea, the commutation rules for the boson operators remain valid. Thus, the boson opera tors constructed by Born and Nath5 and recently rediscovered by Lieb and Mattis6 are not in con tradiction with our theorem. ACKNOWLEDGMENTS The author is indebted to A. W. Overhauser, A. D. Brailsford, and D. R. Hamann for several enlightening suggestions and conversations. VOLUME 6, NUMBER 7 JULY 1965 Exact Eigenstates of the Pairing-Force Hamiltonian. 11 R. W. RICHARDSON Courant Institute of Mathematical Sciences, New York University, New York, New York (Received 14 December 1964) The restrictions on a previously reported class of exact eigenstates of the pairing-force Hamiltonians are removed and it is indicated that all the eigenstates of this Hamiltonian can be included in this class. Explicit expressions are given for the expectation values of one-and two-body operators in the exact, seniority-zero eigenstates of this Hamiltonian. In particular, a simple expression for the occu pation probabilities of the levels of the single-particle potential is given. This expression may be easily evaluated for realistic nuclear systems. I. INTRODUCTION IN a previous paper,l the exact eigenstates of the pairing-force Hamiltonian for finite systems were studied. This study was motivated by the wide use of this Hamiltonian as a model Hamiltonian in nuclear physics.2 Some of the results of this study were subsequently applied to pairing models of some even isotopes of lead.3 This application indicated that there is a considerable improvement in the accuracY of the model's description of the excitation spectra of nuclei when exact eigenvalues of the * This work was supported by the AEC Computing and Applied Mathematics Center, Courant Institute of Mathe matical Sciences, New York University, under contract AT(30-1)-1480 with the U. S. AtOlniC Energy Commission. 1 R. W. Richardson and N. Sherman, Nucl. Phys. 52, 221 (1964) (to be referred to as I). 2 A. M. Lane, Nuclear Theory (W. A. Benjamin, Inc., New York, 1964), Part I, and the references cited therein. 3 R. W. Richardson and N. Sherman, Nucl. Phys. 52, 253 (1964). Hamiltonian are used instead of the currently fash ionable approximations to these eigenvalues.2 Sim ilar improvements in the description of other nuclear properties are to be expected from the use of the exact eigenstates of this Hamiltonian. The study of these eigenstates is continued in this paper. The principal result of I was the demonstration of the existence of a new "restricted class" of eigen states of the pairing-force Hamiltonian which can be written in a particularly simple form. That is, the wavefunction of an N-pair state in this class was shown to be that of a state of N independent pairs in which each pair interacts through an effec tive pairing interaction. This result is given below in Eqs. (1.1)-(1.12). The states of this class are restricted by the set of subsidiary requirements that the N single-pair functions which make up an N-pair wavefunction must be distinct. In this paper, we will discuss these restrictions and indicate how they
1.1753777.pdf	OBSERVATION OF THE BAND GAP IN THE ENERGY DISTRIBUTION OF ELECTRONS OBTAINED FROM SILICON BY FIELD EMISSION A. M. Russell and E. Litov Citation: Applied Physics Letters 2, 64 (1963); doi: 10.1063/1.1753777 View online: http://dx.doi.org/10.1063/1.1753777 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/2/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Negative differential conductance observed in electron field emission from band gap modulated amorphous-carbon nanolayers Appl. Phys. Lett. 89, 193103 (2006); 10.1063/1.2378492 Energy distributions of field emission electrons from silicon emitters J. Vac. Sci. Technol. B 23, 687 (2005); 10.1116/1.1885007 Observation of valence band electron emission from n-type silicon field emitter arrays Appl. Phys. Lett. 75, 823 (1999); 10.1063/1.124525 Field emission energy distribution analysis of wide-band-gap field emitters J. Vac. Sci. Technol. B 16, 689 (1998); 10.1116/1.589882 Bias voltage dependent field-emission energy distribution analysis of wide band-gap field emitters J. Appl. Phys. 82, 5763 (1997); 10.1063/1.366442 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 90.217.114.134 On: Wed, 30 Apr 2014 00:05:46Volume 2, Number 3 APPLIED PHYSICS LETTERS 1 February 1963 Each was found to produce a single sharp beat sig nal whose frequency agreed with the frequency pre dicted by Eq. (1) within the limit of accuracy of the electronics (± IS kc). Several of these beat signals were examined with a Collins 51 J-4 receiver for possible small devia tions from the predictions of Eq. (1). The photo multiplier signal and the transducer driving signal were applied simultaneously to the receiver input, and the receiver i. f. output was observed on an oscilloscope. Interference between the photomul tiplier rf signal and the appropriate harmonic of the driving signal should then produce an audio beat signal at a frequency equal to the deviation of the oIX ical beat frequency from the value predicted by Eq. (1). Thermal variations in the optical path lengths of the two arms of the interferometer result in random variations in the phase of the optical beat signal. This phase variation gives rise to an apparent random beat note of about 0.5 cps which sets a lower limit to the real frequency difference which can be detected. Since this random beat note was the only interference effect observed, we conclude that the Debye-Sears frequency shifts agree in absolute value with the predictions of Eq. (1) to within 0.5 cps. The authors thank the Polarad Electronics Cor poration for lending us the spectrum analyzer used in this experiment. lThis research was supported in part by the National Sc ience Foundation and the Army Research Office, and in part jointly by the U. S. Army Electronics Materiel Agency, the Office of Naval Research, and the Air Force Office of Scientific Research. 2 P. Debye and F. W. Sears, Proc. Nat. A cad. Sci. U. S. 18, 409 (1932). 3R• Lucas and P. Biquard, J. Phys. Radium 3, 464 (932). 4 C. V. Raman and N. S. Nagendra Nath, Proc. Indian Acad. Sci. Sect. A 2,406 (1935). 5 C. V. Raman and N. S. Nagendra Nath, Proc. Indian Acad. Sci. Sect. A 2,413 (1935). 6 C. V. Raman and N. S. Nagendra Nath, Proc. Indian Acad. Sci. Sect. A 3, 75 (1936). 7 c. V. Raman and N. S. Nagendra Nath, Proc. Indian Acad. Sci. Sect. A 3, 119 (936). 8L. Brillouin, Ann. phys. 17, 88 (1922). 9G• W. Willard, J. Acoust. Soc. Am. 21, 101 (1949). lOR. Bar, Helv. Phys. Acta 8, 591 (1935). IlL. Ali, Helv. Phys. Acta 9, 63 (1936). OBSERVATION OF THE BAND GAP IN THE ENERGY DISTRIBUTION OF ELECTRONS OBTAINED FROM SILICON BY FIELD EMISSION 1 A. M. Russell and E. Litov Department of Physics, University of Ca lifornia Riverside, California (Received 28 September 1962: in final form 23 November 1962) Electrons can be obtained by field emission from a metal or semiconductor at room temperature. Such electrons will have in the vacuum the same total energy they had within the solid. Their energy distribution can be measured by allowing them INDEXING CATEGORIES A. elemental semiconductor c. field emiss ion spec- A. Si trometer B. electron emiss ion E B. energy distribution (band gap) 64 to pass through a retarding potential equal to the accelerating potential and then varying the potential of the collector over a narrow range. 2.3 Harrison 4 has shown that the details of the density of states cannot be expected to appear in the energy dis tribution obtained from a metal because of the in verse relationship between the electron momentum and the density of electronic states. These con clusions do not, however, apply to the band gap in a semiconductor where the electron population is zero. In this case no field-emitted electrons are expected in the energy range corresponding to the location of the band gap at the surface. A field-emission spectrometer has been constructed which uses phase sensitive detection to measure the energy ill stribution of electrons obtained by field This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 90.217.114.134 On: Wed, 30 Apr 2014 00:05:46Volume 2, Number 3 APPLIED PHYSICS LETTERS I February 1963 emISSIOn from the surfac e of a semiconductor. 5 The distribution is obtained directly because the detector yields the derivative of the collector current as a function of retarding potential. 6 Initial measurements made on Si emitters yielded energy distributions which were similar to those of a metal except that a greater collector bias was required for the collection of the most energetic electrons.7 These measurements all indicated the existence of a p-type surface layer similar to that which has been observed by Allen and Law.8 Higher accelerating voltages eventually caused the tip to fracture while under ultra high vacuum, exposing material within the emitter. Electrons emitted from such a,tip, though they did not produce the symmetric patternS normally encountered in field emission, were found to have a markedly different distribution in energy. Figure I is a photograph of the energy distribution obtained from the fractured Si emitter as a function of retarding potential. Each large division on the horizontal axis corresponds to a change in retarding potential of .4 V. The gap separating the two peaks is 1.2 V wide, in good agreement with the known n =~-• ~, ! ., L~ j ,~ ~~ l., rJ ~' ! ~ I.: I'~ -.' ~ Fig. 1. Oscilloscope display of the energy dis- tribution of electrons obtained from the surface of Si by field emission. The retarding potential or electron energy is measured along the horizontal axis with each large division corresponding to .4 V. The emission on the left from the conduction band is separated from the va lence band emiss ian on the right by the band gap. Ef----------- :: . SEMICONDUCTOR BARRIER VACUUM Fig. 2. A field-emission energy diagram far Si showing a nearly degenerate n-type surface and emission from both the conduction and va lence bands. band gap for Si. The energy distribution thus appears to show emission from both the valence and con duction bands. The fact that the emission from the conduction band is about the same magnitude as that from the valence band can be accounted for by a Si surface which is almost but not quite degenerate as shown in Fig. 2. The bottom of the conduction band in this case is still somewhat above the Fermi energy. The emission from the conduction band is then limited by its population which is due to the high energy tail of the Fermi distribution at room temperature. The bottom of the conduction band and the top of the valence band are shown with finite slope at every point rather than the anticipated discontinuities because of the limited re solution of the spectrometer. This is due to lack of perfect geometry in the experime ntal tube, the time constant of the detector, and the finite amplitude of the modulation required by the phase sensitive detector system.9 A dis tribution obtained from W under similar conditions yielded a width at half-maximum of about .6 V as compared to the high resolution measurements of Young and Muller 3 which gave a half-width of .2 V at room temperature. The Si distribution may appear wide because no emission is obtained from the immediate region of the Fermi level where the maximum of the curve would normally be expected. There is thus no maximum from which to calculate a distribution width and it appears that all of the emission from Si which is detected arises from that part of the distribution well below the half-maximum 65 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 90.217.114.134 On: Wed, 30 Apr 2014 00:05:46Volume 2, Number 3 APPLIED PHYSICS LETTERS 1 February 19113 which would be obtained if the band gap were popu lated as in the case of a metal. This situation is shown graphically in Fig. 3. Why the forbidden band is so accurately depicted when the resolution is lower than Young and Muller's by a factor of three is not clear. Instability in the emission, unfortunately, pre cluded the possibility of making critical bias measurements 7 so that the actual location of the Fermi energy with respect to the measured dis tribution could not be determined. It is hoped that the development of techniques for obtaining surfaces which are both clean and stable will eliminate this difficulty. ISupported, in part, by the Office of Naval Research. 2E. W. Muller, Z. Physik 120,261 (1943). 3R. D. Young and E. W. Muller, Phys. Rev. 113, 115 (959). 4w. A. Harrison, Phys. Rev. 123,85 (1961). SA. M. Russell, Rev. Sci. Inslr. 33, to be published. 6L. B. Leder and J. A. Simpson, Rev. Sci. Inslr. 29, 571 (1958). 7 A. M. Russell, Phys. Rev. Letters 9, to be published. BF. G. Allen, T. M. Buck, and J. T. Law, J. Appl. Phr: 31,979 (1960). A. M. Russell and D. A. Torchia, Rev. Sci. Inslr. 33, 442 (1962). 66 dl dV ...J <:( :z (.!) u; 0: ~ '-' W f W o n " " " " " " " , ' I I I I , 1 I \ , , ---l 1_.6 eV , 1 I \ I \ !, \ , ----i 1.2 eV r COLLECTOR BIAS v Fig. 3. A sketch of the observed energy distribution obtained from 5i indicating, also, the distribution which might have been expected from a metal. The figure shows that the band gap may be observed even though it is substantia lIy greater than the expected half.width of the corresponding energy distribution obtained from a meta I. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 90.217.114.134 On: Wed, 30 Apr 2014 00:05:46
1.1713207.pdf	Nature of Spontaneous Oscillations in a Cesium Diode Energy Converter W. T. Norris Citation: Journal of Applied Physics 35, 3260 (1964); doi: 10.1063/1.1713207 View online: http://dx.doi.org/10.1063/1.1713207 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Oscillations in the Thermal Cesium Plasma Diode J. Appl. Phys. 37, 2867 (1966); 10.1063/1.1782142 Cesium-Diode Thermionic Converter for Laboratory Experiments Am. J. Phys. 34, 122 (1966); 10.1119/1.1972807 Optimization of Efficiency of a CesiumDiode Converter J. Appl. Phys. 33, 3491 (1962); 10.1063/1.1702434 LowFrequency Oscillations in Cesium Thermionic Converters Phys. Fluids 4, 1054 (1961); 10.1063/1.1706439 LowFrequency Oscillations in a Filamentary Cathode Cesium Diode Converter J. Appl. Phys. 32, 321 (1961); 10.1063/1.1735997 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:36JOURNAL OF APPLIED PHYSICS VOLUME 35, NUMBER 11 NOVEMBER 1964 Regular Articles Nature of Spontaneous Oscillations in a Cesium Diode Energy Converter* W. T. NORRlst Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts (Received 12 May 1964) In an attempt to understand the nature of the oscillations that are observed in plasma diodes the usual collision-free model is examined. The equations describing the model are simplified by neglecting the time dependent term in the Boltzmann equation for the electrons. Even this simplification does not readily yield quantitative predictions about the frequency or form of the oscillations, or even about the conditions under which they are likely to occur. However we introduce a criterion of stability which allows us to predict a possible instability in the steady state condition which arises because the electrons can redistribute their charge more rapidly than the ions. The simplified equations and the criterion together lead to a good qualitative and a partial quantitative understanding of a particular case. The particular case can be regarded as the epitome of the general case, and the arguments used in it seem capable of the necessary extension to cover all instances of this sort of sponta neous oscillation. INTRODUCTION THE phenomenon of spontaneous oscillations in a plasma diode is not yet well understood. Consider a diode, whose cathode is a hot refractory metal and whose anode is cold. The diode is filled with the vapor which has a low ionization potential: the usual vapor is that of cesium. The cathode emits, under such circumstances, both ions and electrons; there are normally many more electrons emitted than ions. If a suitable resistor is connected across the diode, a current flows in the circuit thus formed. Usually this is a steady direct current: a typical relationship between the cur rent through and the voltage across the diode is shown in Fig. 1. Sometimes there is an alternating current of high frequency (between 1 kc/sec and 1 Mc/sec) super posed on the direct current and of amplitude about one half of the direct current. When those oscillations occur the mean voltage corresponds to one of those on the upper flat part of the curve in Fig. 1. There have been many observations of such oscilla- CURRENT TO COLLECTOR NEGATIVE COLLECTOR 'POTENTIAL FIG. 1. Typical current voltage characteristic of a diode. * This work, which is partly based on an Sc.D. thesis, Depart ment of M.echanical Engineering, MIT (28 August 1962) was supported m part by the U. S. Army, Navy, and Air Force under Contract DA 36-039-AMC-03200(E); the National Science Foundation (Grant G-24073); and the U. S. Air Force (ASD Contract) (AF 33 1616-7624). t Present address: Central Electricity Research Laboratories Clecve Road, Letherhead, United Kingdom. ' tions, and these have been discussed by Houston.! Normally they occur when the cesium vapor pressure is very low, the collisions between particles are infre quent, and the mean free paths of the particles are much greater than the width of the diode. There are some exceptions.2 The period of these oscillations is nearly equal to the time it would take an ion to cross the diode if the ion had the mean velocity of atoms in a vapor at the same temperature as the cathode of the diode we are consider ing; the frequency is the reciprocal of the transit time of the ions. The origin and nature of these oscillations has been the subject of some considerable speculation. There are several theories3,4 about them. The theory presented in this paper proposes to find an unstable condition and to examine what happens when it is disturbed. The mathematics can become very arduous but there is a conveniently simple example which serves to illustrate the main idea without the encumbrance of complicated equations. ASSUMPTIONS AND EQUATIONS We contemplate, in common with other authors,4-7 a diode with plain parallel electrodes of infinite extent. One electrode, the emitter or cathode, is a source of charged particles; there are ions which are positively charged and there are electrons which have a negative charge of the same magnitude and which are very light 1 J. M. Houston, Proc. 22nd Phys. Electron. Conf., MIT, 1962. 2 N. D. Morgulis, C. M. Levitsky, and L. N. Groshev, Rad. Eng. Electron. Phys. 7, 330 (1962). 3 S. Birdsall and K. Bridges, J. App!. Phys. 32, 2611 (1961). • Paul Mazur, J. App!. Phys. 33, 2653 (1962); 33, 3387 (1962); K. G. Hernqvist and F. M. Johnson, Advan. Energy Conversion 2,601 (1962). 5 P. L. Auer, J. App!. Phys. 31,2096 (1960). 6 A. L. Eichenbaum and K. G. Hernqvist, J. App!. Phys. 32, 16 (1961). 7 R. G. McIntyre, J. App!. Phys. 33, 2485 (1962). 3260 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:36SPOi\[TANEOUS OSCILLATIONS Ii\[ A CESIUM DIODE 3261 in comparison with the ions. These particles are injected into the diode with specified velocities normal to the emitter. Mostly we consider the velocities of the parti cles to be distributed according to a Maxwell-Boltzmann law, but not always. We suppose that the particles do not collide with each other, since this is approximately true in most cases when the phenomenon is observed and the as sumption eases an already complicated analysis. This model takes no cognizance of neutral atoms, since in observed cases it seems they do not influence much the motions of the ions or electrons. The forces on the particles in the space between the electrodes are those due to the average electric field; we regard the large number of particles as a charge continuum for the pur pose of computing the electric field. Particles striking either electrode are immediately absorbed. The charges arriving at the second electrode, the collector, or anode constitute the current through the diode. The potential is uniform on any plane parallel to the electrodes. To find the field at such a plane it is sufficient to take account only of charges on one side of that plane, not forgetting charges actually on the surface of the electrodes. There are three equations which govern the behavior of the model. (The symbols are listed in below. Many of them are the same as those used in other papers on this SUbject.) t=time T= temperature characterizing emitted beams of charge ;r= distance from the emitting electrode n=density of particles V = potential at x 1 = velocity and space distribution function of one species of particle p= momentum of a particle in x direction m = mass of a particle k = Boltzmann's constant e= electronic charge EO= dielectric constant of space ~= x/ Xl, where X12= ~o(kT)!2t/ e(7rme)!ie If= -eV/kT i = current of one species of particle F=density of one species of particle as a fraction of density in emitted beam. Fe and Fi can be expressed either as functions of ~ or of If. /3= (i,o/ieO) (mjm e)!= ratio of charge densities in emitted beam 1) = a reduced potential used to modify if; R = nil' nOi for monoenergetic beam €= a small parameter Subscripts signifies reference to ions e signifies reference to electrons o signifies reference to the emitted beam of particles except in EO. Poisson's Equation: (1) Boltzmann's Equation which has to be written twice, once for each species of charge: aj/at= (p/11t)(aj/ap)+(e/~o)(ov/ax). (2) We also have to specify the spacing of the diode and the condition of the electrodes, which in turn are de termined by the characteristics of the external circuit connected to the diode. The equations are quite general; their solution is often difficult. We shall mention the steady-state solutions (when 01/01=0) and propose an approximate approach to the solution of the time dependent equations, which is what we need for the description of the oscillations. STEADY-STATE SOLUTIONS The steady-state equations have been widely studied and we quote from the solutions.5•7 In the particular case where particles are emitted with a Maxwell Boltzmann distribution, Eqs. (1) and (2) can be com bined and become (3) We have introduced the reduced parameters from the list of symbols. F. and Pi are similar functions of each species giving the density of each species of particle as a fraction of the density of that species in the emitted beam. Fe and Fi have values lying between 0 and 2. The value 2 occurs in the case when the collector potential is infinite and all of one species of charge is returned to the emitter: Then F= 2 at the emitter surface. When (3 is unity the density of ions and electrons in the emitted beams is the same. The mean energies of each species of particle are the same since each species has the same temperature but, since they are lighter, the electrons have a higher mean velocity. Thus the electron current is many times greater than the ion current. It is to be noticed that the density of both ions and electrons at any point depends only on the potential at that point and on the relative positions of maxima and minima of the potentiaL It does not depend on their masses nor explicitly on the value of x, the distance from the emitter. Consider the hypothetical, and in fact impossible, steady-state distribution of potential across the space which is illustrated in Fig. 2. The density of electrons is given by: F.(If) = e-4'[1+erf (lfm-4')!], in ranges d-a and b--m of Fig. 2; = cli'[1 +erf (4'm-If)l- 2erf (lfa-If)!], (4) in range a-f-b ; =e-"'[l-erf (lfm-If)l], in range m-e. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:363262 W. T. NORRIS FL EMITTER m d~----~------~--~--e FL COLLECTOR FIG. 2. Hypothetical potential distribution in a diode. The distance across the diode is plotted horizontally, the negative of the potential vertically. The point e could be anywhere below the point m, andf anywhere below a and b. This equation is sketched in Fig. 3. The expression for Pi is similar except (-if;) is substituted for (if;). The shapes of the solutions of (3) are sketched in Fig. 4. This figure shows families of curves for three values of {3. Each member of the family is for a different value of the potential difference between the electrodes. The figure is drawn wide enough to show any wavy solutions that may occur. In the group of curves for {3;t:O some of the detailed behavior of the solutions when the potentials of the emitter and collector are nearly the same has been omitted. The wavy solutions sometimes cross. This implies a current-voltage relation for the diode as illustrated in Fig. S. The negative slope part of the curve is unstable and in a real device analogous to the model a character istic current-voltage curve would show a hysteresis as illustrated in Fig. 6. There is no question here of oscillations occurring, unless a resonant circuit is used. There is always at least one stable point when any passive element is connected to the diode. Howevtr, this hysteresis may be associated with the phenomenon of ignition. t ELECTRON DENSITY -POTENTIAL. !It FIG. 3. Sketch of variation of electron density with potential. Points on this figure correspond with points on Fig. 2. Even when there are collisions such wavy solutions may exist, although the potential depressions will ac cumulate charges which lose energy in collisions and the trough and hump structure will be flattened, but, providing some vestige of the wavy potential distri bution remains when there are collisions between parti cles, then a hysteresis will occur. One of the two stable states on the hysteresis loop may well provide a potential well for the production of extra ions and alter the field at the emitter surface or otherwise cause a higher current flow in the diode. This hysteresis is related to that studied by Eichenbaum and Hernqvist.6 It has been proposed that certain of the wavy so lutions do not exist, or that for certain of the possible loads that may be put on the circuit, i.e., for certain ratios of current and voltage, no steady-state solution exists: There can only be time-dependent ones. The reader is referred to these suggestions themselves.4 Finally it should be noted that once the emission rates of ions and electrons are known and the spacing is fixed, then the value of the electric field at the emitter determined completely the form of the potential and charge density distributions across the diode. CRITERION OF STABILITY If, in a steady state, a small change of charge (ef fected by some external agent) on the emitter causes a change in the potential of the collector and in the current flowing through the diode such that the external circuit reacts in a manner which aggravates the alteration we made to the charge on the emitter, then the state is unstable. From the point of view of our model we can regard a change in charge on the emitter as a small change in the electric field at the emitter, since this field is directly proportional to the charge density on the emitter surface. We have made no statement about how long we wait for the effect of the change to take place. This is a key point in the argument. A rigorous analysis would follow out all the details of the effect of the change. We might even examine the response to a periodic variation of the emitter field. Such procedures would be long-winded, beyond the scope of the paper, and seem to be unneces sary. It seems as if two sorts of instability will arise and that a discussion of these will tell us all we wish to know. GENERAL INSTABILITY We imagine our change of charge to have been effected and that the potential and charge distributions across the diode have settled down with exactly the new quantity of charge on the emitter. Since the charge density on the emitter determines the field next to its surface and since we know the spacing of the diode, we can compute the potential of, and the current to, the collector. Then we can assess the response of the rest of the circuit, usually supposing it has no reactance and consists only of batteries and resistors. If the original [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:36SPONTANEOUS OSCILLATIONS IN A CESIUM DIODE 3263 (a) (b) (el FIG. 4. Sketches of solutions of Poisson-Vlasov equations: (a) f'l=O, no ions present at all; (b) f'l<1, some ions present; (c) f/ = 1, equal numbers of ions and electrons in emitted beam. For f/> 1 the curves are the inverse of f'l < 1. change of charge is increased by the circuit then the state is unstable, but not if it is decreased. The diode is, in effect, regarded as a circuit element with the voltage-current relationship of the steady state. As we indicated above, we then get only the sort of instability that changes to a stable state, but no osema tions. ELECTRON INSTABILITY Let us consider an oscillatory state. Suppose we have been able to find, at some particular time in the cycle, the form of the functions f and the' potential at all the points across the diode. Using a finite difference method we can program a computer to follow out the details of the subsequent changes as closely as we wish. Let us FIG. 5. Current voltage characteristic when solutions of Eq. (5) cross. I Ii choose as a time interval for this computation a period much less than a transit time for an ion, but greater than the transit time for an electron. We may do this since the electrons, being lighter, cross the diode more quickly than the ions. If we make this <:hoice, then at the end of our interval ions will not have moved much, but most of those electrons in the space will have started from the emitter during the interval. We are dealing with oscillations whose period is of the order of a transit time for an ion, and therefore during the interval we have specified, none of the distributions of charge or potential will have changed greatly. And, since most of the elec trons in the space at the end of the interval start from the emitter during the interval, the electric field has been nearly constant during their flight and we may FIG. 6. Hysteresis arising from the characteristic of Fig. S. Ii [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:363264 W. T. NORRIS suppose their distribution is the same as it would have been had there been a truly steady potential distri bution. The approximation, then, is to ignore the time dependent term in the equation for the electrons .and to suppose that their density at any point is determme.d by the potential there, as in the steady state and as IS given in Eq. (4). We can now discuss a new type of instability which we call an "electron instability." Consider some steady state condition and compute the various distributions of charge density and potential. Then with this par ticular ion density distribution fixed in the diode, we may be able to find other distributions of the electrons such that the boundary conditions, the steady-state Boltzmann equation of the electrons, and Poisson's equations are satisfied. Using our criterion of stability, or one of the exten sions we mention below, we can examine the stability of the various configurations of electron change density that we have found. The ion density is always kept fixed exactly as it was in the original steady state we considered. It may happen that under these conditions the original steady-state distribution of the electrons is unstable, but that there is another distribution which is stable. We suggest then, in view of the fact that the electrons arc so much lighter and more mobile than the ions and in the light of the remarks at the begin ning of this section, that in a device they will alter their distribution to the stable (or rather "electron stable") state virtually immediately and certainly before the ions have time to alter their distribution significantly. The electrons are supposed to have nearly zero mass. Their mass must be finite in order for them to be able to distribute themselves in the potential field. As long as they are very light, however, their effect is quite independent of their mass, rather as a small viscosity is necessary to cause the separation of the boundary layer from a body moving in a fluid, but the drag is insensitive to the actual magnitude of this viscosity. The new condition is by no means equilibrium. The ion density will begin to change under the influence of the altered electric field. In order to discover the nature of the oscillation we must follow the subsequent developments. We consider the ions to be always changing their distribution and never to be in equilibrium. The electron distribution, however, though it too is always changing, is that which satisfies the steady-state Boltzmann equation. It is to be expected that, if there is only one steady state equilibrium condition and if this is electron un stable, then the various unsteady potential and charge density distributions will be continually reappearing; that is we shall have oscillations. We then have two problems. One is to fmd out if CURRENT RETARDING VOLTAGE FIG. 7. Current-voltage relation with superposed load line. and when such instabilities occur. The second, if we do find such instabilities, is to describe the subsequent oscillations. EXTENDED CRITERIA OF STABILITY Our first approximation was to ignore the time it takes for electrons to alter their density as the electric field changes. We propose now to simplify the problem a little more and to introduce two rather different formu lations of the instability criterion. Consider firstly a load consisting only of batteries and resistors. For given emission rates for any diode we can draw the diode voltage-current relationship as we did in Fig. 5 and superpose on it a load line for the circuit. The crossing points of load line and character istic determine the equilibriums and the type of crossing the general stability at each point. In Fig. 7, A and C are stable, but the point B is unstable. For each state of general equilibrium we may compute the ion distribution. With this ion density fixed and varying only the electron density and the potential, a new current-voltage relationship for the diode can be drawn. Once more superposition of the load line will show the electron equilibrium states. One of these will correspond to the general equilibrium state with which we began. The nature of the crossing of the curve will show whether this is a stable state or not. For the second extension of the stability criterion, we consider an even more special case where the load con sists of batteries only. We consider electron instability. A graph may be drawn showing the potential of the collector versus the potential gradient of the emitter; the ion density is fixed, as is usual when we consider electron instabilities. Figure 8 is such a graph. The horizontal line on Fig. 8, which is at the level of the collector potential determined by the batteries in the circuit, fixes the (electron) equilibrium conditions. By following the original recipe for determining stability, it is easily seen that if the curve cross the line LM with a positive slope, then we have a stable condition, as far as the electrons are concerned, and if the slope is nega tive we have an unstable condition. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:36SPONTANEOUS OSCILLATIONS IN A CESIUM DIODE 3265 Consider the case of the crossing with a negative slope. If we put a small negative charge on the emitter, the potential gradient becomes more negative which in creases the potential at the collector. The battery tries to correct the altered potential difference between the electrodes, but does so by putting more negative charge on the emitter which increases the initial alteration and we have an instability. Similar arguments for the case of putting on positive charge and for the other crossover points demonstrate the matter completely. In Older to examine the electron stability of any steady state, we merely need to know the gradient of the curve (the emitter field vs collector potential) for fixed ion density at the general steady-state equilibrium condition. MATHEMATICAL DESCRIPTION In this section we are still concerned with a load consisting of batteries. Suppose we have some steady state where "'om, FeW, and Fi(~) are known. For the purposes of illustration (Fig. 9) we consider the case where", has a maximum (i.e., with the sign convention used here, there is a potential barrier for electrons to surmount). Equation (3) applies. With this special distribution of ions we wish to calcu late the shape of neighboring potential distributions where the electron density has changed from the original condition, but where Poisson's equation (including both species of charge) and Boltzmann's equation for the electrons are still satisfied. Let "'W+t1)W be such a neighboring potential distribution where t is smalL The ion distribution FiCt) is the same. Let Fe CO be the new electron distribution. F.W=F.("'+eq). Equation (3) becomes d2("'+fT/)/de={3F iW-F.("'+t1)). (5) If E is very small we may use Taylor's theorem to expand the expression F(",+t1), ignoring terms in higher powers of t than the first, subtract Eq. (3) and show that COLLECTOR POTENTIAL yJ COLLECTOR !:.. _...!!' _ ~OTEN~L . FIXED BY EXTERNAL BATTERIES POTENTIAL GRADIENT AT EMITTER FIG. 8. Variation of potential of collector with potential gradient at the emitter. FERMI LEVEL EMITTER SURFACE FIG. 9. Potential changes when the electron distribution is dis turbed slightly from the steady state; the upper curve is the dis turbed state. 1)max and "'max are the values of 1) and", where the original", W is a maximum. The steady state is unstable if any solution of (6), for which 1)=0 when t=O, has an odd number of zeros in the range O<t<~ diode. If the original "'(~) is such that there are no positive values of "', then 1)mnx=O. When we are near "'max each term of (5) becomes in finite, but they are of opposite sign and their difference remains finite. It is useful then to consider 1)' = 'r}max-'r}, ra ther than 1) i tseH. Without knowing the details of any steady state it is impossible to determine whether it exhibits electron instability or not. However, the sign of aF./ a", shows whether or not 1) has periodic solution. If there is only one maximum of FeC"'), aF./a", is negative; there are no zeros of 7]. In Fig. 10 example (a) is stable. But in cases (b) and (c) periodic solutions exist for Eq. (6) To find where the zeros occur would call for a calculation in any particular instance; an instability is likely. In case (d) of Fig. 10 although aF./ a", is positive, it decreases very rapidly and we do not expect oscillations whilst admitting the possibility. This instability is well exemplified by a special case which we consider next. EXAMPLE To illustrate more clearly this sort of instability there is an example which is sufficiently simple and free of complicated curves that would need a long calculation to explain completely. Consider the case where {3= 1; the density of charge in both emitted beams is the same. Suppose further that the diode is connected to a battery which keeps the emitter and collector surfaces at the same potential. There is only one solution to the steady-state equa tions. In this the ion and electron densities and the potential are uniform across the diode. This steady [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:363266 W. T. NORRIS (0) (b) eel Cd) EMITTER COLLECTOR FIG. 10. Steady-state distribution curves: (a) Shows no electron insta bility, (b) and (c) prob ably will show electron instability, and (d) is probably electron stable. state, however, displays electron instability. This can be shown by applying any of our criteria of stability. There is only one stable equilibrium distribution for the electrons (for fixed ion density). Figure 11 (a) shows the distribution of potential and charge density across the diode in the steady state. Figure 11 (b) shows the electron and potential distributions for the only electron stable condition with the given uniform ion distribution. In this condition the electrons are accelerated from the emitter and slowed down to their original speed at the collector: the electron density falls towards the middle of the diode so that, since we have a uniform ion density, there is a net positive charge all through diode space. This gives the correct curvature of the potential variation across the diode. According to our theory the steady state will collapse to this condition immediately, and then, since the ions are not in equi librium, all the various distributions will continue to change. Even in this simple case it is difficult to follow the details of the time-varying state. Nevertheless the two important factors remain. First, there is only one steady-state condition if the diode is set up in the way we describe. Second, this steady-state condition is in fact unstable. There is only one state of equilibrium and if ever the system reaches it, a slight disturbance will cause its disappearance. Since this equilibrium condition will always collapse whenever it is reached, we must conclude we have a time-varying condition. The simplest consequence is that we have an oscillatory condition. SUBSEQUENT OSCILLATIONS The calculations after the collapse of the general equilibrium state calls for the simultaneous solution of the steady-state Boltzmann equation for electrons, the time-dependent equation for the ions, and Poisson's equation for the total charge distribution. Finite difference methods can be applied. The pro cedure is to calculate the potential distribution when the electrons are in their equilibrium distribution, with the original distribution of the ions fixed; then to calcu late how each of the ions moves in this modified potential distribution. The change in the velocity and density distribution during a suitable short interval of time can be computed. For this new ion distribution there will be a slightly different electron and potential distri bution, which can be found. The ions are then allowed to move in this new distribution of potential for another short period. The process is repeated. A new instability may arise after a certain length of time. Eventually we expect the various distributions to reappear. The oscil lations would have settled down. A computer program was set up, but suffered from being too crude to give good results. It became apparent, GENERAL EQUIL I BRIOM POTENTIAL ION AFTER ELECTRONS HAVE REVERTED TO S TABLE STATE EMITTER COLLECTOR ~ OISTRIBUTION OENS ITY ELECTRON DENSITY FIG. 11. Potential and charge distributions: (a) Steady state, (b) after electron collapse. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:36SPONTANEOUS OSCILLATIO:-.rS IN A CESIUM DIODE 3267 however, that the limits, as it were, of the potential distribution were somewhat like those illustrated in Fig. 12. There are always a relatively large number of ions near the emitter. Condition (a) is (electron) stable when there are few ions in the space between the elec trodes. The low potential barrier of state (a) permits more ions to flow into the diode. Condition (b) then becomes more stable. But the higher potential barrier will reduce the flow of ions into the diode space and the return of state (a) will be encouraged. This can be understood better in terms of an even more simplified model. We suppose that the ions always have a uniform distribution of charge across the diode. This reservoir is being drained at the collector at a rate proportional to the density and being fIlled by ions flowing over a potential barrier near the emitter. We measure the ion density by R, the fraction of the density in the emitted beam. Using a simplified relationship between electron density and potential we can construct a graph of collector potential versus emitter field. This is shown in Fig. 13 for a number of different values of R. We also show how the potential extremes vary with emitter field. The width of the diode was 5.0 (non dimensional units). . Figure 13(a) is the curve for the steady-state con dition, R= 1.0. This is electron unstable and the dis tributions revert to point A. There is a potential barrier FIG. 12. Motive dia gram showing limiting potential distributions during oscillations. now against ion flow: R decreases. Figure 13(b) shows the case when R=O.90. Points A and B are stable: Cis not stable. There is no reason for a changeover from A to B. Figure IO(c) is when c=O.85. A is now only just stable and in 10(d) when R=0.8, point A has vanished and B is the stable point. There is now no potential barrier against the ions and they reaccumulate in the space. B remains stable until R= 1.0 (in this simple calculation) . A more sophisticated model would, we believe, pro duce only a slightly different picture of the oscillations. CONCLUSIONS We have identified the onset of oscillations with a special sort of instability of a steady-state condition, which we have called electron instability. It arises because the electrons are so much lighter and more mobile than the ions, and consequently redistribute their charge much more quickly. R =1.0 2 0/ R' 0.90 '/I AT COLLECTOR FIG. 13. Variation of po tential of collector with po tential gradient at the emitter for a selection of values of uniform ion charge density across the diode. !Jt AT EMITTER R = 0.85 :: AT EMITTER A IjtAT .,;-/ COLLECTOR .... ,... .... /' / / // IjtMINIMUM / ·2 2 '/I /- -I -2 d'/J Ijt AT COLLECTOR d( AT EMITTER -to A ./ ./ ./ ,. /"""' ./ /' '/I MINIMUM ./ / / R' 0.8 ~ AT EMITTER to -2 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:363268 W. T. NORRIS The range of conditions under which such instabilities exist is not properly circumscribed nor have we been able to calculate the frequency of oscillations that result from the collapse due to this instability. We were, how ever, able to cite a specific example when such an in stability does exist and to make what seems to be a plausible and moderately detailed description of the consequent oscillations. ACKNOWLEDGMENTS The M.LT. Computation Center helped considerably in the numerical work that was done in examining the nature of the oscillations. The author would like to thank Professor G. N. Hatsopoulos for his advice and encouragement. He would also like to thank the Commonwealth Fund for the support of a Harkness Fellowship while he was at M.LT. JOURNAL OF APPLIED PHYSICS VOLUME 35. NUMBER 11 NOVEMBER 1964 Transformation of Small-Signal Energy and Momentum of Waves* R. J. BRIGGst Department oj F:lectrical J·:ngineering and Research Laboratory of Electronics, Massachusetts Institute oj Technology, Cambridge, Massachusetts (Received 28 April 1964) The transformation of the small-signal energy and momentum of a wave between two inertial reference frames as first given by Sturrock are derived by using simple perturbation theory and the appropriate trans formations of length, time, current, and electromagnetic fields. The approach allows a straightforward generalization to the case of relativistic linear transformations and to nonrelativistic transformations be tween two reference frames that rotate with respect to each other. These rotating and linear transformations allow one to make very general statements about the frequency and wavenumbers for which negative-energy waves are obtained in a rotating and translating medium, as, for example, an electron beam in Brillouin flow. 1. INTRODUCTION THE transformation of small-signal energy and momentum of waves between two inertial refer ence frames has been considered in several recent papers.1-3 Such a transformation makes it possible to evaluate the energy, power, and momentum in the most convenient reference frame. Of even greater importance is the fact that the knowledge of the transformation allows one to make some very general statements about the phase velocities required for negative-energy waves when the medium is passive in one particular reference frame, as first shown by Sturrock. In this paper the transformation of the energy and momentum of a wave is derived by a method that is significantly different from the ones used by Sturrock, 1 Pierce,2 and Musha.3 In the present derivation the only assumption made is that a small-signal energy conserva tion principle and momentum conservation principle are obeyed in every reference frame. The only physical transformation laws used are those of length and time and of the electromagnetic fields, currents, and charges. An interesting feature of the derivation is the fact that the energy transformation can be derived without * T~is work was supported in part by the U. S. Army, Navy, and Air Force under Contract DA36-039-AMC-03200(E); and in part by the National Science Foundation (Grant G-24073). t Present address: Lawrence Radiation Laboratory Livermore California. ' , 1 P. A. Sturrock,'J. App\. Phys. 31, 2052 (1960). 2 J. R. Pierce, J. App!. Phys. 32, 2580 (1961). 3 T. Musha, J. App!. Phys. 35, 137 (1964). introducing the concept of momentum (and vice versa). The nonrelativistic results derived originally by Sturrock are extended to the relativistic case. Also, the method is adapted to include the case of a nonrelativistic transformation between two reference frames that rotate with respect to each other. These extensions allow application of the transformation of energy and momentum to relativistic electron beams and to elec tron beams in Brillouin flOW.4•5 2. TRANSFORMATION OF ENERGY AND POWER OF WAVES The general type of system that we consider consists of a lossless waveguide structure containing a lossless medium, both uniform in the longitudinal (z) direction. If the medium is nonlinear, it is assumed that perturba tions are small enough so that the equations of motion can be linearized and a single traveling wave of the form exp[j(wl-!3z)] can exist in the system. We assume that a small-signal energy-conservation principle is obeyed in any reference frame, so that as(z,t)/ az+aw(z,t)/ at= 0, (1) where s(z,t) is the small-signal power (or energy flow) in the +z direction and w(z,t) is the small-signal energy per unit length. The simpler case of a one-dimensional • A. Bers and R. S. Smith, Quarterly Progress Report No. 69, Research Laboratory of Electronics, MIT, 15 January, 1963, pp. 11-15. 5 G. C. Van Hoven and T. Wessel-Berg, J. App\. Phys. 34, 1834 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 15:09:36
1.1777023.pdf	Energy Band Structure of Gallium Antimonide W. M. Becker, A. K. Ramdas, and H. Y. Fan Citation: J. Appl. Phys. 32, 2094 (1961); doi: 10.1063/1.1777023 View online: http://dx.doi.org/10.1063/1.1777023 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v32/i10 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2094 WILLIAM PAUL Brooks, Dr. H. Ehrenreich, Dr. W. E. Howard, and Dr. G. Peterson. The measurements on GaP were carried out by Mr. R. Zallen and on the lead salts by Dr. L. Finegold and Mr. M. DeMeis. All of us are grateful to Mr. J. Inglis and Mr. A. Manning for necessary machine work and to Mr. D. Macleod for fashioning the samples used in the optical and electrical investigations. The samples of GaP measured in the new data reported were generously given us by the Monsanto Chemical Company and by Dr. W. G. Spitzer. For the PbS samples, we are indebted to Professor R. V. Jones of Aberdeen University and Dr. W. D. Lawson of the Radar Research Establishment; for the PbSe samples, to Dr. W. D. Lawson and Dr. A. C. Prior of R. R. E. and Dr. A. Strauss of Lincoln Laboratory, and for the p-type PbTe samples, to Dr. W. W. Scanlon of Naval Ordnance Laboratory. JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 32, NO. 10 OCTOBER. 1961 Energy Band Structure of Gallium Antimonide* W. M. BECKER, A. K. RAMDAS, AND H. Y. FAN Purdue University, Lafayette, Indiana Resistivity, Hall coefficient, and magnetoresistance were studied for n-and p-type GaSb. The infrared absorption edge was investigated using relatively pure p-type, degenerate n-type, and compensated samples. Infrared absorption of carriers and the effect of carriers on the reflectivity were studied. The magneto resistance as a function of Hall coefficient for n-type samples at 4.2°K gave clear evidence for a second energy minimum lying above the edge of the conduction band; the energy separation is equal to the Fermi energy for a Hall coefficient of 5 cm3/coulomb. The shift of absorption edge in n-type samples showed that the conduction band has a single valley at the edge, with a density of-state mass mdl =0.052 m. By combining the results on the edge shift, magnetoresistance, and Hall coefficient, it was possible to deduce: the density-of-states mass ratio mdjmdl = 17.3, the mobility ratio ~2/~1=0.06, and the energy separation 1l=0.08 ev between the two sets of valleys at 4.2°K. Anisotropy of magneto- I. INTRODUCTION INFORMA TION on the band structure of GaSb has been obtained from various investigations. Roberts and Quarrington1 found that the intrinsic infrared ab sorption edge extrapolated to 0.704 ev at 2900K and 0.798 ev at 4.2°K and had a temperature coefficient of -2.9X1O-4 ev;oC in the range 100o-290°K. The shape of the absorption edge led the authors to suggest that either the minimum of the conduction band or the maximum of the valence band is not at k=O. Ramdas and Fan2 attributed the absorption at high levels to direct transitions but found a temperature dependent absorption tail indicative of indirect transitions. They reported also effective mass values obtained from in frared reflectivity measurements: me= 0.04 m and mh=0.23 m. From studies of the resistivity and Hall coefficient in the intrinsic and extrinsic temperature * Work supported by Signal Corps contract. 1 V. Roberts and J. E. Quarrington, J. Electronics 1, 152 (1955-56). 2 A. K. Ramdas and H. Y. Fan, Bull. Am. Phys. Soc. 3, 121 (1958). The value of hole effective mass reported was in error and should have been mh=0.23 m. The experimental data used are shown in Fig. 8. resistance, observed at 300oK, showed that the higher valleys are situated along (111) directions. The infrared reflectivity of n-type samples can be used to deduce the anisotropy of the higher valleys; tentative estimates were obtained. Infrared reflectivity gave an estimate of 0.23 m for the effective mass of holes. The variation of Hall coefficient and transverse magnetoresistance with magnetic field and the infrared absorption spectrum of holes showed the presence of two types of holes. Appreciable anisotropy of magneto resistance was observed in a p-type sample, indicating that the heavy hole band is not isotropic; this was confirmed by the infrared absorption spectrum of holes. The results on the absorp tion edge in various samples seemed to indicate that the maximum of the valence band is not at k=O. However, it appears likely that transitions from impurity states near the valence band produced ahsorption beyond the threshold of direct transitions. ranges, Leifer and Dunlap3 deduced EG(T=0)=0.80 ev, me=0.20 m and mh=0.39 m. Zwerdling et aZ.4 ob served magneto-optical oscillations in the intrinsic infrared absorption which indicated that the absorption at high levels corresponded to direct transitions. By attributing the oscillations to Landau levels in the con duction band, an electron effective mass m.= 0.047 m was obtained. Sagar5 studied the temperature and pressure dependences of the Hall coefficient of n-type samples. The results were explained by postulating a second band with a minimum above the minimum of the conduction band. The second band was assumed to have minima along <111) directions by analogy with germanium, and piezoresistance effect was observed which supports the suggestion that the band has many valleys. Assuming the valleys to have the mass parame ters as in germanium, Sagar estimated a density-of states ratio of 40 and an energy separation of 0.074 ev at room temperature between the two conduction bands. The two-band model has since been used by other authors to interpret measurements on resistivity 3 H. N. Leifer and W. C. Dunlap, Jr., Phys. Rev. 95, 51 (1954). 4 S. Zwerdling, B. Lax, K. Button, and L. M. Roth, J. Phys. Chem. Solids 9, 320 (1959). 5 A. Sagar, Phys. Rev. 117, 93 (1960). Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsENE R G Y BAN D S T R U C T U REO F GAL L I U M A ~ TIM 0 N IDE 2095 and Hall coefficient,S infrared reflectivity,7 and pressure dependence of piezoresistance.8 Taylor9 observed that the infrared absorption edge shifted with pressure at a rate of 1.57XlO~5 ev/atm up to 200 atm. Recently, Edwards and Drickamer10 reported measurements ex tending to higher pressures. They found a rate of shift of 0.0120 ev/kilobar up to 18 kilobars which changed to 0.0073 ev/kilobar between 18 and 32 kilobars. Further more, the rate of shift leveled off and became negative in the range 32~50 kilobars. The results were explained by assuming that the conduction band has a similar structure as in germanium with a minimum at k=O, a set of (111) minima and a set of (100) minima lying at successively higher energies; with increasing pressure the (111) minima move away from the valence band at a slower rate than the k = 0 minimum and the (100) minima move toward the valence band. Finally, Cardonall reported recently the observation of optical reflectivity peaks in the visible and ultraviolet regions which seem to be analogous to the peaks observed in germanium that had been attributed to L3-Ll transitions. The brief summary shows that a large amount of information has been obtained about the band structure of GaSb, especially regarding the conduction band. However, qualitative confirmation of the evolving band model and quantitative determination of the important parameters are still needed. Furthermore, little informa tion is yet available about the valence band. The gal vanomagnetic and infrared studies reported below are presented and discussed with emphasis on the band structure. II. GALVANOMAGNETIC STUDIES A. n-type GaSb The results of magnetoresistance measurements on n-type samples give clear cut evidence for the existence of a second conduction band which is separated by a small energy from the lowest conduction band. 1\Ieas urements were made on samples which had Hall co efficients ranging from -3 to -110 cmB/coui. In these samples, the conduction electrons do not freeze out but become degenerate at sufficiently low temperatures. For the range of magnetic field used, the transverse magnetoresistance showed HZ dependence as indicated by Fig. 1. Figure 2 shows the results plotted against the Hall coefficient R for the various samples. The magneto resistance of the samples with IRI >5 cmB/coul de creased with decreasing temperature and became quite small at 4.2°K. This behavior is expected for carriers which become more degenerate with decreasing tem- 6 A. J. Strauss, Phys. Rev. 121, 1087 (1961). 7 M. Cardona, J. Phys. Chem. Solids 17, 336 (1961). 8 R. W. Keyes and M. Pollak, Phys. Rev. 118, 1001 (1960). 9 J. H. Taylor, Bull. Am. Phys. Soc. 3, 121 (1958). 10 A. L. Edwards and H. G. Drickamer, Phys. Rev. 122, 1149 (1961). 11 M. Cardona, Z. Physik. 161,99 (1961). 0.1 ~------+H----------1 I [110] H (ITI] 0.01~--~----,~+------- , 300'K 77'K • 4.2"K 0.001L. _____ L-____ --' 10' 10' 10" H in Oersleds FIG. 1. Transverse magneto resistance as a function of magnetic field in n-type samples. The 3000K and 77°K data were obtained on a sample having R(3()()OK)= -4 cm3/coul and the 4.2°K data were given by a sample having R(3000K)= -3.2 cm3/coul. perature, in an energy band with spherical surfaces of constant energy in k space.12 On the other hand, the samples with small Hall coefficients showed much larger magnetoresistance at 4.2°K than at room tem perature. The 4.2°K curve shows a sharp rise with de creasing Hall coefficient. This is a clear indication that a second type of carrier comes in at sufficient carrier concentrations.12 We estimate that the rise begins at R", -5 cm3/ coul, corresponding to an electron concen tration of n= 1.25X 1018 cm~3 and a Fermi level of t= (h2/2m) (31r2n)i= 3.63X lQ-15ni X (m/m) ev=4.21X1Q-3(m/m) ev. (1) Taking m/m=O.047,4 we get t=O.0895 ev as the energy at which the second band lies above the mini- mum energy of the conduction band. . '1'", .!! I!! ~ 0 '0 ~ :I: .e "-... q 12~------------------------------~ 10 8 6 4 2 \ \ \ ~--\ \ 1[110] Hl. [110] • 3000K x 77"K -4.2"K O2 1~------~~--~~==~~2~0--------f.0 - R (em'/coulomb) 01 4.2°K FIG. 2. Transverse magnetoresistance at three different tem peratures plotted against Hall coefficient at 4.2°K, for n-type samples. The dotted curve is calculated for 4.2°K using (24) ano the values of mddmd1, jJ.2/jJ.] , ti given by (25) and (26). 12 A. H. Wilson, The Theory oj M elals (University Press, Cambridge, England, 19.'53). Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2096 BECKER, RAMDAS, AND FAN 0.092 0.091 R (3000K) = -113 cm'/coulomb H~20.000 Oersteds 1[110] • Hl.[IIO] [Hl FIG. 3. Variation of transverse magnetoresistance with field orientation for n-type sample at 3()()OK. Except for samples with small electron concentrations and at very low temperatures, we have to consider conduction in two bands, of which the higher band may be one of many valleys. In the weak field approxi mation which is justified by the H2 dependence shown in Fig. 1, the magnetoresistance is characterized by the three parameters b, c, and d which can be determined from the measurements. These parameters are related to the components, UafJ"Y~, of the magnetoconductivity tensor. The factors which determine the anisotropy of the transverse magnetoresistance and the magnitude of longitudinal magnetoresistance are given by13 -O"o(b+c) = 0" XXyy+ 20" XllXY, -O"od=O"xxxx- (O"xxyy+20"xyxy), (2) where x, y, z are the cubic axes of the crystal and 0"0 is the conductivity for H = o. The magnetoconductivity tensor, therefore O"o(b+c) and O"od of the individual valleys of various bands are additive. A valley with spherical surfaces of constant energy gives (b+c)=d=O if the relaxation time is isotropic, which is usually a fair assumption. Thus the lower band should make little contribution to (b+c) and d. Indeed, at low tem peratures all the parameters of the lower band are negligible as shown by the smallness of the magneto resistance effect of the samples with IRI >5 cma/coul. We would then expect to find b+c= (b2+C2) (0"02/0"0), d=d2(0"02/0"0). (3) Subscripts 1 and 2 will be used to indicate the lower and the higher bands, respectively. The parameter b is important for the transverse magnetoresistance. The following relation holds: (4) where IJ.H is the Hall mobility. Combining two such /3 C. Herring and E. Vogt, Phys. Rev. 101,944 (1956). equations, one for each band alone, we get 0"01 0"02 0"010"02 1 b=-b 1+-b2+--(IJ.HI-IJ.H2)2_, 0"0 0"0 0"02 c2 (5) where c is the velocity of light. Thus, even with two spherical bands having b1""-"0 and b2~O at low tempera ture there can be still a large b and a corresponding '. . transverse magnetoreslstance due to the last term. ThIS is the cause for the sharp rise shown at 4.2°K in Fig. 2. Figure 3 shows the variation of transverse magneto resistance observed on an n-type sample at room tem perature. A small longitudinal magnetoresistance, (Ap/ pH2) nollO, was also observed. The results gave (I:::.p/ pH2)nollO= (b+c+td)""-"9.2X 10-12 oe-2; b= 2.21X 10-10 oe-2; d= 1O.SX 10-12 oe-2. (6) According to (3), these parameters are associated with the high band and they indicate that the high band consists of (111) valleys. The effects were small in mag nitude, the longitudinal magnetoresistance being about 1/25 of the transverse magnetoresistance. At 4.2°K, the ratio of the two is less than 1/30 and the longitudi nal effect could not be detected at H = 7000 oe. The fact that the ratio is very small does not necessarily mean that the higher band as well as the lower band has small anisotropy, since the parameter b can be much larger than b1 and b2 while (b+c) and d are de termined by (b2+C2) and d2. The longitudinal magneto resistance is given by where L2 depends on the anisotropy of the higher band. For (111) valleys, b" a: L2=Aj(2K+l)(K-l)2/K(K+2)2, (S) •• 8000 4000 •• .. --. • _-r • 3000K x 77°K • 4.2°K 20002~ ----,5~--,J,IO~---;;2!;:.0--~ -R(cm'/coulomb) 01 4.2°K FIG.4(a). Hall mo bility at three dif ferent temperatures plotted against Hall coefficient at 4.2°K, for n-type samples. Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsE :'J ERG Y B;\ N [) S T Rue T U REO F G i\ L L 1 {) M i\:'J T 1 1\1 0 N IDE 2097 where K is the anisotropy factor, K = mtTt/mtTI, and the factor A depends on the variation of relaxation time with energy and is of the order of unity. The Hall mobility of the crystal is given by (TOI (T02 JLIl= -JLIlI +-JLII2. (9) (To (To Thus (10) Equations (8) and (9) may be used to determine K from the data on the longitudinal magnetoresistance and Hall mobility, provided JLl/JL2 and nl/n2 are known. However, the right-hand side of (lO) is very sensitive to the mobility ratio which has not been determined for the sample at room temperature. The carrier and mobility ratios at 4.2°K have been evaluated and are given in (25) for the samples of low Hall coefficients. Anisotropy of magneto resistance was not detected and an upper limit ( ,1p )1l0 / (JLIl)2 - -<3X1o-3 pH2 110 C (11) was deduced from consideration of the experimental accuracy. Substituting the values in (lO), we get (12) At 4.2°K, the carriers are highly degenerate and A", l. The limit for L2 for very large K is t. Thus the anisot ropy of the higher band may be very large although magnetoresistance anisotropy was not detected. The variation of the Hall coefficient with temperature is shown in Fig. 4(a) and 4(b). The data for 3000K and T('\() 5000'r-~~rO~-----~7r7-------~~n I , ~4000-~~_ - ~3000 " -~ .. _ - "5 ---------x ~rooo- -a:. 100-- --~ -10 -- ~ ~ ! a! , 50- J- f--~-----; i - 5 .... A-____ .... R. ----- ..... ! . a: I 200 1~--~~--~~1--~----,b-1--~2 10 20 10"T (OKf' FIG. 4(b). Hall mobility and Hall coefficient as functions of tem perature for n-type samples. Data for different samples are indi cated by different symbols. R(30QOK); +55 em' I coulomb 10 10 H in Oe,.,.ds FIG. 5. Transverse and longitudinal magnetoresistances and Hall coefficient as functions of magnetic field for a p-type sample at 77°K. nOK are in general agreement with previous measure ments reported by Sagar5 and Strauss.6 As pointed out by these authors, the temperature dependence of R which is more pronounced in samples of smaller electron concentrations is qualitatively understandable on the basis of increasing share of carriers in the higher, low mobility band with increasing temperature. However, the increasing of R(To with decreasing R, shown by the low temperature curves in Fig. 3, cannot be explained by the two-band conduction, since the R(TO varied in the range of \R\ >5 cm3/coul where all the electrons are in the lower band at 4.2°K. We suggest that the behavior is caused by impurity scattering which should be the dominant scattering mechanism. In fact, the drop of R(TO with decreasing temperature shown in Fig. 4(a) and Fig. 4(b) can only be attributed to the effect of impurity scattering. The n-type samples used in these experiments were doped with Te, starting with purest obtainable material which was p-type with ",2XlO17 acceptors/cm3• Therefore we may expect in these degenerate samples a charge impurity concen tration that exceeds the carrier concentration by 2N A ",4 X lO17 cm-:J. The simple theory of impurity scattering predicts for degenerate carriers a mobility JL ex: nl/N, (13) where N is the concentration of charged centers. For uncompensated samples n= N the formula gives JL ex:. n-i. Actually, the observed mobility of various degenerate semiconductors varies more slowly with the carrier concentration. Assuming JL ex:. n-i for the un compensated case, we may expect for our samples (14) Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2098 BECKER, RAMDAS, A~D FAN According to this expression, the mobility will increase with increasing n up to n=4N A",8X1017 cm-3 which corresponds to R",-7.8 cm3/coul. The explanation appears therefore to be reasonable. It should be pointed out that the data on Se-doped samples reported by Strauss6 differed from the Te-doped samples in the variation of Hall mobility with R. The thorough under standing of scattering mechanisms in various kinds of samples requires further study. B. p-type GaSb Figure 5 shows the Hall coefficient and magneto resistance as functions of the magnetic field for a p-type sample. The Hall coefficient and the transverse mag netoresistance decreased with increasing field. The weak field Hall mobility of (he sample was 2700 cm2/v sec. The condition (WIJr)2= (eHr/mc)2", (JJ.1I/c)2~ (JJ.JIH/c)2«1 (15) is valid over the whole range of H. The decrease of the Hall coefficient and the transverse magnetoresistance is therefore a clear indication for the presence of two types of holes, similar to the case of germanium and silicon. The decrease of Rand iJ.p/ pH2 occurs when the condition no longer holds for the small number of light holes with large r / m. This effect was not observed at room temperature where the relaxation times of the carriers were much shorter, as can be judged from the measured Hall mobility of 690 cm2/v-sec against 2700 cm2/v-sec at 77°K. The fact that two types of holes are present at 77°K indicates that the two branches of the valence band must merge or come close together at the maximum. It is interesting to note that a decrease in R or iJ.p/ pH2 has not been observed for n-type samples. This does not mean that the light hole mass is necessarily much smaller than the light electron mass. The im purities in n-type samples remain charged with de creasing temperature and the scattering effect prevents the relaxation times of the electrons from reaching sufficiently large values with decreasing temperature; as pointed out above, we expect at least a charged impurity concentration of 2"V A",4X 1017 cm-3 even though some of the samples used had electron concen trations smaller by an order of magnitude. In the p-type sample, on the other hand, the holes freeze out with decreasing temperature and the charged impurity concentration at 77 oK is small judging by the value of R. The longitudinal magnetoresistance shown in Fig. 5 remained substantially constant, indicating that the effect was associated with the heavy holes, the effect being dependent on (b+c) and d which are additive for various types of carriers according to Eq. (2). Figure 6 shows the variation of the transverse magnetoresistance with the field orientation. From these data and (he longitudinal effect we get the following values: (b+c)=9.1XlO-l2, d=-8.9XlO-12, b=45.1XlO-12• 0.085.-------------------, R(300 °K)= + 55cml/coulomb H~13,400 Oersteds I [I 10J, H 1. [IIOJ 0.080 0.075 0.070lL-__ -'---____ '--~:__---',:__-----:-'--_::' iii 001 110 III III [HJ FIG. 6. Variation of transverse magnetoresistance with field orientation for a p-type sample at 77°K. Figures 5 and 6 give data for the same sample. The relation (b+c)~ -d is consistent with a band with valleys along (100) directions.14 On the other hand, it is also consistent with the behavior of the warped valence bands in germanium and silicon.15 III. INFRARED ABSORPTION A. Effect of Carriers 1. Carrier Absorption in n-Type Sample The long wavelength absorption in n-type gallium antimonide is shown in Fig. 7. Beyond ""S JJ. the ab sorption increases smoothly as a function of wave length. An absorption band can be seen in the range between 2 and 5 JJ. with a peak at 3.3 JJ.. This feature is very similar to the absorption band observed in n-type silicon16 and in n-type gallium arsenide,17 which has been attributed to transitions from the conduction band minimum to higher lying minima. According to this interpretation, the observed absorption band indicates the presence of energy band minima at ~0.25 ev above the minimum of the conduction band. This could be the (100) minima postulated by Edwards and DrickamerlO to explain the pressure effect on the absorption edge. However, they estimate the postulated minima to be 0.4 ev above the minimum of the conduction band against the value of 0.25 ev indicated by the absorption band. 2. Carrier E..ffect on Reflectivity Some time ago, W. G. Spitzer measured in this labo ratory the reflectivity of n-type GaSb samples for the purpose of determining the carrier effective mass ac cording to the method reported by Spitzer and Fan.18 The measured reflectivity for one of the samples is 14 M. Glicksman, Progress in Semiconductors (Heywood, London, 1958), Vol. 3, p. 3. l' j. G. Mavroides and B. Lax, Phys. Rev. 107, 1530 (1957). 16 W. Spitzer and H. Y. Fan, Phys. Rev. 108, 268 (1957). 17 W. G. Spitzer and J. M. Whelan, Phys. Rev. 114, 59 (1959). 18 W. G. Spitzer and H. Y. Fan, Phys. Rev. 106,882 (1957). Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsENE R G Y B A :\' D S T Rue T U REO F GAL L I U MAN TIM 0 N IDE 2099 :ihown in Fig. 8. It is convenient to express the result of this type of experiment in terms of the effective mass for electric susceptibility, ms which is defined by (16) where X is the electric susceptibility which can be ob tained from the optical measurements and n is the carrier concentration. Using the approximation n= 1/Rec, values of ms/m 0.043, 0.039, 0.041 were obtained for three samples with Hall coefficients of -4.6, -3.4, -2.5 cm3/ caul, respectively; the accuracy of the determination did not justify a definite conclusion about the variation of ms/m with Hall coefficient. The values of ms/m are appreciably smaller than the effective mass value, ml = 0.047 m, given by the mag neto-optical measurement and the density-of-states mass value, mdl = 0.052 m, estimated below for the lowest conduction band. The discrepancy is under standable in the light of the two-band model. Con sidering two conduction bands we have -X= nle2/mSlw2+n2e2/mS2w2. (17) The Hall coefficient is given by where 1+XYYH R=R 1-- (1+xy)2 1 fJ.1l1 1 +XYYH ---- nlee fJ.I (1 +xy)2' Combining (17) and (18), we get (18) _x=_1_ ~[fJ.HI 1+XYYIl(1+xmSI)] (19) Rec mSlw2 fJ.l (1 +xy)2 . mS2 . The term in the brackets should account for the dif ference between the value of ml and the values of ms cited above. The value of the term may be expected' to be larger than unity, making the values of ms appre ciably smaller than mI' FIG. 7. Transmis sion of n-type gal lium antimonide Trv80oK, thickness =0.12 em, R(77°K) = -129 cm3/coul. 50 10 T=80 OK 15.0 Wavelength (microns) 40 ....-N-type ><--x P -type / / , I .f ...... X / x __ ~ 1< -X-X-X~)(-- 10L-~5.~O----~IO~------1~5--~~~2~O~~ Wavelength (Microns) FIG. 8. Effect of free carriers on reflectivity of n-type and p-type gallium antimonide at 300oK: n-type sample, R(3000K)=-3.4 cm3/coul, p-type sample R(3000K)= +0.5 cm3/coul. The value of fJ.Hl/ fJ.l may be expected to be close to unity when there is sufficiently large carrier density for the Fermi level to be well inside the lower band. If the ratio of density-of-states masses, md2/mdl, and the energy separation .:l are known for the two bands and if Y~YH is known in addition, then x can be calculated from the Hall coefficient by using Eq. (18). With the additional knowledge of the ml value, the value of X obtained from optical measurements provides an estimate of mS2/mSI which gives in turn md2/mS2 = (md2/mdl)/(ms'1/'msl) in view of mdl=mSI. In case the higher band has many valleys of ellipsoids of revo lution, the value of mS2/md2 is a measure of the ellip ticity. Such an estimate depends on the reliable knowl edge of the various parameters involved. Recently, Cardona7 reported that his reflectivity curve measured at room temperature can be fitted by taking the values .:l = 0.08 ev; ratio of density of states equal to 40, y= i, as estimated by Sagar, and by assuming' for the second band (111) valleys with an ellipticity the same as in germanium. As shown below, we obtained from 4.2°K data on intrinsic absorption edge and galvano magnetic effects a much higher value for the ratio of density of states. Values for y and .:l were also obtained, the value of y being also much smaller than the value t. Optical determination of X at 4.2°K has yet to be made. The values of y and .:l at room temperature may be significantly different from the estimates obtained from 4.2°K data. Nevertheless, calculations were made using the values given by (23), (25), (26) in conjunction with the room temperature optical measurements. The calculated values of md2/mS2 are given in Table I for two values of mJ (0.047 m and 0.052 m). Assuming that the second band has four (111) valleys each character ized by a longitudinal effective mass ml and a trans verse effective mass mt we have Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2100 BECKER, RAMDAS, AND FAN 500 Wavelength (Microns) FIG. 9. Absorption spectrum of p-type gallium antimonide. R(3000K)=+5.1 cm3/coul, R(77°K)= 11.0 cm3/coul. where r=ml/mt. The calculated values of r are also given in Table I. It should be emphasized that the calculated results given in the table should be regarded as no more than order of magnitude estimates. 3. Carrier Absorption in p-Type Sample The long wavelength absorption in p-type gallium antimonide is shown in Fig. 9. The curve is similar to that observed in p-type samples of Ge,19 InAs20 and GaAs21 indicating that the absorption is produced by interband transitions within the valence band. This interpretation finds confirmation from the· galva no magnetic measurement of p-type samples which showed the existence of two types of holes. The effective mass ratio of the two types of holes may be estimated from the analysis of the interband transitions if the energy TABLE I. Calculation of mdz/ms2 for the higher conduction band. Values mdz/mdl=17.3, /Lz//LI=0.06, 1l=0.08ev, are used. The values of r=ml/m, are calculated assuming the conduction band has four (111) valleys of ellipsoids of revolution. Experimental data (3000K) R(cm'/coul) ms/m -4.6 -3.4 -2.5 0.043 0.039 0.041 For mJ/m=0.047 For mJ/m=0.052 md./ms2 r mdz/ms2 r 2.84 3.37 3.13 4.1 5.84 4.58 3.41 3.94 3.67 6.75 11.4 8.85 19 W. Kaiser, R. J. Collins, and H. Y. Fan, Phys. Rev. 91, 1380 (1953); H. B. Briggs and R. C. Fletcher, Phys. Rev. 91, 1342 (1953). 20 F. Matossi and F. Stern, Phys. Rev. 111, 472 (1958). 21 R. Braunstein, J. Phys. Chern. Solids 8, 280 (1959). hands have spherical surfaces of constant energy. The ahsorption coefficient is given then by the expression where mll and mL are the effective masses of the heavy holes and light holes, respectively. For hv»kT, the second term is negligible, and In(av-t) vs hv plot should be a straight line, the slope of which gives mdml/. If the expression is valid, the slope of such a plot should be proportional to (1/kT). Although the data give approximately straight-line plots, the slope does not change much with temperature indicating that at least one of the hole bands is not spherical. B. Intrinsic Absorption Edge The intrinsic absorption edge in pure p-type samples is shown in Fig. 10. The steep rising part of the edge corresponds to 0.725 ev at 300oK, O.SO ev at "'SOoK, 0.S1 ev at "'4.2°K. These values are taken to be the threshold energies, hVd, for direct transitions to the lowest conduction band. The value for liquid helium temperature agrees with that obtained by Zwerdling4 et al. Figure 10 shows also the absorption edges at liquid helium temperature for two n-type samples of different carrier concentra tions. The n-type samples show the shift of edge ex- 10000 5000 2000 I / 100 / /5 ) 5 I I I ·i 200 I I ~ I 100 / / § 50 I 0. I i / ... / 20 10 2 10.6 10 Pho!oo Energy I e v ) FIG. 10. Absorption edge in gallium antimonide. (1) p-typ~, T",300oK, R(3000K)=51 cm'/coul; (2) p-type, T~80oK, R(77°K)=380 cm'/coul; (3) p-type, T"-'4.2°K; (4) degenerate n-type, T~4.2°K, R(4.2°K)= -5.55 cm'/coul; (5) degener<lte n-type, T~4.2°K, R(4.2°K)=-3.19 cm3/coul; (6) p-type, com pensated, T"-'80oK; (7) p-type, compensated, T~80oK. Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsENE R G Y BAN D S T Rue T U REO F GAL L I U MAN TIM 0 N IDE 2101 pected from the filling of the conduction band. The positions of the shifted edge as calculated for a single ban9, model using the values 0.047 m and 0.052 m for the electron density-of-state mass mdl are shown by the vertical lines. The sample of lower electron concentra tion, R= -5.55 cms/coul, should have carriers only in the lower band according to Fig. 2. The data indicate that the edge position calculated for md= 0.047 m gives too large a shift whereas (22) gives reasonable agreement. From the edge shift, 0.075 ev, of this sample, we get for the height of Fermi level in a sample of R = -S cmS / coul 0.07S(S.S5/5)1= 0.08 ev= A, (23) .:1 being the energy difference between the minima of the second and the first bands. The edge shift in the sample of higher electron con centration, R= -3.19 cms/coul is considerably smaller than the estimates based on the single-band model, as is expected. The Fermi level given by the edge shift is 0.09 ev which provides an estimate of the electron concentration in the lower band: nl= 1.48X 1018 cm-a. This information can be combined with Hall mobility and magnetoresistance data to determine the ratios of mobilities and of density-of-states masses for the two conduction bands. The parameter b for magnetoresist ance is given by the expression (S) with bl",,-,O for liquid helium temperature. We have seen that (b+c) and d of the upper band are quite small compared to the third term of the right-hand side. Therefore, either the band is nearly isotropic or its parameters b, c, d must be small. In any case, we can neglect the term b2 compared with the last term. It can be shown then b (1-YH)2 --=xy , (RcrO)2 (1 + xy Hy)2 (24) where x and yare defined as in (18). We shall use the approximation YH=y which should not cause great error. Equations (18) and (24) can be used to calculate the values of x and y. Using the data nl = 1.48X 1018 cm-a, R= :-3.19 cms/coul and b/(Rcro)2=0.14 we get J.LdJ.Ll=0.06, n2/nl=2.67. (2S) Using the value of the Fermi level, r=0.09 ev, and the value .:1=0.08 ev, we get md2/mdl= (n2/nl)lr/Cr-.:1) = 17.3. (26) The shape of the absorption edge as shown in Fig. 10 suggests that the absorption begins with indirect transi tions. Mter a steep drop, the absorption tails off ex tending to much longer wavelengths than is expected and the effect is more pronounced at room temperature than at the liquid nitrogen temperature. The behavior is similar to the case of phonon-assisted indirect transi-tions in germanium and silicon with phonon-absorption transitions becoming reduced at lower temperatures. Furthermore, the absorption edge of the n-type samples is also difficult to reconcile with the assumption that the minimum of the conduction band and the maximum of the valence band coincide in k space. If the shift of the edge at high absorption level corresponds to the rise of the Fermi level in the conduction band, then the assumption predicts a very steep edge at 4.2°K with the absorption dropping as exp[h(v-vt)/kT] where hvt corresponds to transitions at the Fermi level. The measured curves are much too sloping in comparison with this prediction. On the other hand, a sloping curve may be produced with the help of indirect transitions if the conduction band minimum and the valence band maximum are at different points in k space. Measurements were made on p-type samples con taining large and nearly equal concentrations of acceptor and donor impurities, in order to observe the impurity enhancement of indirect transitions. The impurity compensation kept down the carrier concen tration and the background absorption due to carriers. The results are also shown in Fig. 10. It is seen that the absorptions in the two compensated samples appear to extend to about the same hv, ",,0.72 ev, for both samples. This seems to be an evidence against the possibility that the effect was the result of the lowering of the conduction band minimum by the presence of impurities. The data may be interpreted in the following way. The room temperature curve shows a clear change of slope at 0.627 ev. Subtracting the background carrier absorption ac from the observed absorption a and plotting (a-a c)! against hv, two straight line portions can be recognized which extrapolate to 0.627 ev and 0.690 ev, respectively. These two energy values appear to correspond to the onsets of indirect transitions with phonon absorption and phonon emission. Thus, we get hViCR.T.) =0.658 ev, hVp=0.031 ev, where hv i is the energy gap and hv p is the phonon energy. We note that infrared measurements give 0.029 ev for the long wavelength, optical transverse mode.n The data for the compensated samples indicate hVi(L.iY.)""O.72 ev. This gives (hVd-hvi)=O.08 ev which is reasonably close to the estimate 0.067 ev for the same quantity at room temperature. Also, the threshold for phonon emission transitions should be hVi(L.N.)+hvp=O. 75 ev; curve 2, Fig. 10, indeed rises sharply near this energy. The rounding-off in the curve as it merges with the background absorption could be caused by impurity induced indirect transition. Furthermore, the n-type samples, curves 4 and 5, Fig. 10, having their thresholds 22 G. S. Picus, E. Burstein, B. W. Henvis, and M. Hass, J. Phys. Chern. Solids 8, 282 (1959). Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2102 BECKER, RAMDAS, AND FAN of direct transitions at 0.885 and 0.90 ev are expected to have thresholds at 0.805 and 0.82 ev for impurity induced indirect transitions. Curves 4 and 5 in Fig. 10 show that the estimates are reasonably consistent with the experimental data. Thus, the suggested interpreta tion provides a satisfactory explanation for all the ab sorption edge observations. However, this interpreta tion requires that the valence band have off-center maxima, since we are certain that the lowest conduction band has its minimum at k= O. Furthermore, the presence of light holes at 78°K requires two degenerate bands at each maximum. It appears from the examina tion of the symmetry properties of the Brillouin zones23 that such a band structure is unlikely. An alternative interpretation may be suggested. The behaviors which cannot be understood on the basis of 23 G. Dresselhaus, Phys. Rev. 100, 580 (1955); R. H. Parmenter, Phys. Rev. 100, 573 (1955). direct transitions may be caused by excitations from impurity slates in the range of ",0.08 ev from the valence band. We would have to assume that there are sufficient such states even in the purest p-type samples used. Transitions from these states produce the tail absorption seen at room temperature. At low tempera tures, the states are depleted of electrons, resulting in a sharper absorption edge. In compensated samples, the states are occupied by electrons even at low tempera ture, giving a tail absorption. Finally, transitions from these states to the Fermi level in n-type degenerate samples begin at smaller photon energy than the direct transitions from the valence band, thus producing a sloping absorption edge. Measurements are being made with higher resolution. Preliminary results obtained indeed favor the second interpretation. Thus, we may accept tentatively that the maximum of the valence band is at k=O, having a warped heavy hole band which is degenerate with a light hole band. JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 32, NO. 10 OCTOBER. 1961 Lattice Absorption in Gallium Arsenide W. COCHRAN Cavendish Laboratory, Cambridge, f:ngland AND S. ]. FRAY, F. A. JOHNSON, J. E. QUARR!NGTON, AND N. WILLIAMS Royal Radar l~stablishment, Great Malvern, England A series of detailed measurements of the lattice absorption bands of gallium arsenide has been made over the wavelength range 10-40 p. and over the temperature range 20-292 oK. These results can be interpreted in terms of mUltiple phonon interactions involving five characteristic phonon energies. These results, along with the known elastic constants, have enabled us to supply all the relevant data for a computation of the complete phonon spectrum using an extension of the shell model. DURING the last nine months, measurements have been made of the lattice absorption spectrum of gallium arsenide! at the Royal Radar Establishment. The object of this work was to obtain as much informa tion as possible about the vibrational spectrum of this material and is part of a general study of the lattice spectrum of 3-5 semiconductors. This is a continuation of similar investigations on silicon,2 germanium,3 and indium antimonide.4 Lattice absorption bands arise from the direct inter action of infrared photons and phonons in the crystal lattice. In the 3-5 semiconductors the strongest of these interactions is between a photon and a single long wavelength optical phonon. This type of interaction is 1 S. J. Fray, F. A. Johnson, J. E. Quarrington, and N. Williams (to be published). 2 F. A. Johnson, Proc. Phys. Soc. (London) 73, 265 (1959). 3 S. J. Fray, F. A. Johnson, J. E. Quarrington, and N. Williams (to be published). • S. J. Fray, F. A. Johnson, and R. H. Jones, Proc. Phys. Soc. (London) 76, 939 (1960). responsible for the reststrahlen bands in these materials. However, the more important bands from our point of view are those that arise from the interaction of a photon with a pair of phonons. Two mechanisms are available for this type of coupling; one through anharmonic forces· and the other through second-order electric moments.6 The anharmonic mechanism depends on the production of a single long wavelength optical phonon as an intermediate state, followed by its splitting into a pair of phonons. The second-order electric moment mechanism depends on the fact that a charge is induced on a particular atom when a neighboring atom is dis placed from its equilibrium position and also that a dipole moment is produced when this atom is itself displaced. In either case, energy and wave vector must be con served between the initial photon and the two resulting phonons. The equations for the conservation of energy • D. A. Kleinman, Phys. Rev. 118, 118 (1960). 6 M. Lax and E. Burstein, Phys. Rev. 97, 39 (1955). Downloaded 23 Sep 2013 to 131.94.16.10. This article is copyrighted as indicated in the abstract. 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1.1729121.pdf	Barrier Height Studies on MetalSemiconductor Systems W. G. Spitzer and C. A. Mead Citation: Journal of Applied Physics 34, 3061 (1963); doi: 10.1063/1.1729121 View online: http://dx.doi.org/10.1063/1.1729121 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhancement of Schottky barrier height in heterodimensional metal-semiconductor contacts Appl. Phys. Lett. 70, 441 (1997); 10.1063/1.118175 New method of determination of a metalsemiconductor barrier height Appl. Phys. Lett. 25, 600 (1974); 10.1063/1.1655327 Surface States and Barrier Height of MetalSemiconductor Systems J. Appl. Phys. 36, 3212 (1965); 10.1063/1.1702952 MetalSemiconductor BarrierHeight Measurement by the Differential Capacitance Method—Degenerate OneCarrier System J. Appl. Phys. 35, 3351 (1964); 10.1063/1.1713221 Metal—Semiconductor Barrier Height Measurement by the Differential Capacitance Method—One Carrier System J. Appl. Phys. 34, 329 (1963); 10.1063/1.1702608 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:25JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 10 OCTOBER 1963 Barrier Height Studies on Metal-Semiconductor Systems W. G. SPITZER* Bell and Howell Research Center, Pasadena, California AND C. A. MEAD California Institute of Technology, Pasadena, California (Received 16 April 1963 ; in final form 16 May 1963) Photovoltaic and space-charge capacitance measurements have been used to study the height of the Schottky barrier at the metal-semiconductor interface of a series of metals evaporated onto "vacuum cleaved" samples of n-type CdS and n-and p-type GaAs. Although the barrier heights for metal-CdS samples increase with increasing metal work function as predicted by simple theory, significant deviations were noted. The barrier heights measured on metal-GaAs samples at different temperatures show very little dependence on the metal and appear to be fixed relative to the valence band edge by surface states. The results are compatible with the model in which the photoresponse, for photon energies less than the semi conductor energy gap, arises principally from photoemission of carriers from the metal into the semicon ductor; however, the results are sensitive to the method of surface preparation and comparisons with other work are difficult. I. INTRODUCTION THE ~tudy o~ the photovoltaic respons~ of surface- barner rectIfiers has produced conSIderable in formation on the transport of hot electrons (and holes) in metal films. In most of these studies, the system consists of a metal film deposited in some way on a semiconductor surface. In these systems, photocurrent is observed where the incident photon energy hll exceeds the energy gap Eg of the semiconductor. The source of this photocurrent is the band-to-band photoexcitation of carriers in the semiconductor under the metal film. It is anticipated, and has been observed experimentally, that this photocurrent is proportional to the intensity of the light transmitted by the metal film.! However, photocurrent is also observed in many cases for hll <Eg• The mechanisms responsible for this photocurrent could be excitation from defect levels in the semiconductor, localized states close to the metal-semiconductor inter face, or conduction electrons in the metal which have sufficient energy to surmount the potential barrier at the interface. Much of the recent work has been done with a view towards establishing photoemission from the metal film as an operating mechanism. In a few cases, studies of the spectral photoresponse with different metals for hll<E g and the dependence of this response upon the thickness of the metal film have given information on the attenuation lengths of hot electrons1.2 or holes3 of approximately 1 eV excess kinetic energy. In addition to the range of the hot carriers, a second parameter of interest in the photoemission process is the height of the potential or Schottky barrier and its . * Present address:. Ele~trical Engineering I?epartment, Univer Slty of Southern Cahforma, Los Angeles, Cahfornia. 1 C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. E. LaBate, Phys. Rev. 127, 2006 (1962). 2 w. G. Spitzer, C. R. Crowell, and M. M. Atalla, Phys. Rev. Letters 8, 57 (1962). 3 C. R. Crowell, W. G. Spitzer, and H. G. White, App!. Phys. Letters 1, 3 (1962). dependence on the work function of the metal film <pM, the electron affinity of the semiconductor x, and the concentration and distribution of surface states at the interface. There is some information available concern ing barrier heights for different metals and semicon ductors.I-8 In most cases, however, the papers are concerned with only one or two metals and one semi conductor. Any attempt to compare the work of differ ent investigators is difficult since different methods of both semiconductor surface preparation and metal film deposition have been employed. At the present time, the only detailed study of barrier heights known to the authors is the work of Archer and Atalla6 for a series of metals on silicon. The silicon surface was prepared in a vacuum chamber by cleavage and the metal film de posited by evaporation. In a number of cases, deliberate exposure. of the cleaved surface to oxygen prior to evaporatIOn of the metal substantially altered the resulting barrier height. The barrier heights were determined from the variation of the differential capacitance of the space charge region with applied bias. Crowell et at.! demonstrated that photoresponse meas urements of the same structures gave barriers which were compatible with those deduced from capacity measurements although the observed heights seemed to correlate with oxygen-contaminated cases of Archer and Atalla. . The pres~nt w?rk reports an experimental investiga tIOn of barner heIghts from vacuum deposited metals on "1 d' "I f c eave -Ill-vacuum samp es 0 n-type CdS, n-type GaA.s, and p-type GaAs. The height of the Schottky barner was measured by using: (1) the spectral re sponse of the photovoltage, (2) voltage dependence of 4 R. Williams and R. H. Bube, J. App!. Phys. 31, 968 (1960). : G. W. Mahlman, Phys. Rev. Letters 7, 408 (1961). R. J. Archer and M. M. Atalla, Ann. N. Y. Acad Arts Sci 101, 697 (1963). . . . ; R. Williams, Phys. Rev. Letters 8, 402 (1962). C. A. Mead and W. G. Spitzer, Appl Phys Letters 2 74 (1963). ", 3061 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:253062 W. G. SPITZER AND C. A. MEAD the differential capacitance of the space-charge region, and (3) forward biased I-V characteristic of the diode. Some difficulties associated with the interpretation are indicated in addition to those already reported. The results are compared with those of Archer and Atalla and others, and with the conventional model of a surface-barrier rectifier. Some results are interpreted in terms of Fermi level pinning by surface states. II. EXPERIMENTAL A. Material and Fabrication of Samples The CdS sulfide was single-crystal n-type material, not purposely doped, and with carrier concentration values ranging from 1015 to 1017 cm-3• The samples were cleaved parallel to the optic axis, which was determined visually with the aid of a polarizing microscope. The n-type GaAs samples were all cut from a pulled, Te doped single crystal grown in the (Ill) direction. The (110) plane, which is the cleavage plane, was deter mined in a conventional manner by using an optical goniometer after etching the sample surface with a preferential etch. The free-electron concentration, as determined from the Hall coefficient, was 3.8X 1017 cm-3 at both room temperature and 77°K. The p-type GaAs samples were obtained from a Zn-doped single crystal with a hole concentration of 4.8X 1016 cm-3 at room temperature and 5.0X 1015 cm-3 at 77°K. Devices were fabricated from small bars of single crystal material approximately 2X2 mm in cross sec tion. The samples were notched and then cleaved in the vacuum system with a small wedge which was struck with a magnetically operated hammer. The vacuum system consisted of an oil-diffusion pump, water-cooled chevron baffle, and an anti migration trap employing Linde 13x zeolites. Before evaporation the background pressure was typically 1 X 10-7 Torr and the pressure rose by a factor of between 2 and 10 during evaporation depending on the metal being evaporated. Evaporation of the metal was commenced before the crystal was cleaved in order to eliminate contamination of the crystal surface by residual gasses. Upon removal from the vacuum system, the cleaved surface was examined under a microscope, and usually consisted of several flat areas separated by multiple-cleavage steps and damaged areas. The flat areas were isolated electrically by flaking off a small amount of the crystal on all sides. Contact was made by a pointed O.13-mm-diam gold wire probe. All units were checked on a I-V curve tracer to display the rectification characteristic. Prior to cleaving, Ohmic contacts were made to the ends of the bars. The contacts were made on the CdS by cleaving a small section near the end of the bar in air and immediately soldering with indium. Contacts to the n-and p-type GaAs were made by soldering a freshly abraded surface with indium doped with Te or Zn, respectively. In the case of GaAs occasional high resistance contacts were encountered. Therefore, wires were soldered on both ends of the bar and the unit checked in the 1-V tracer. Only those showing very low impedance were processed further. B. Methods of Measurement and Interpretation The postulated energy-level diagram for a surface barrier rectifier has been given a number of times in the literature and is not reproduced here. In the usual model the height of the potential barrier CPR, measured with respect to the Fermi level is given by (1) where <PM is the work function of the metal film, X is the electron affinity of the semiconductor, and Ao is the potential drop across the metal-semiconductor spacing at the interface. It is almost certain that in the many experiments employing chemically prepared semicon ductor surfaces the contact between the semiconductor and metal is not an intimate one. Archer and Atalla have pointed out that even for an intimate contact, the work functions would not necessarily be the same as the vacuum values because of changes in the surface-dipole contributions. In addition, Rose9 has considered the variations introduced by the different positions that the first metal atoms can occupy with respect to the semi conductor surface. It is also known that if there exists a large concentration of surface states at the semicon ductor-metal interface, the interior of the semiconduc tor becomes screened from the metallO and the height of the potential becomes independent of <PM. This point is considered further in the next section. It is of interest to consider each of the techniques employed here to obtain quantitative information on the barrier height. 1. The Spectral Dependence of the Photoresponse Photomeasurements were made on a Gaertner model L234 quartz monochromator and focused-tungsten source. Calibration reference was a Reeder vacuum thermocouple. For photomeasurements the light was chopped at 50 cps at the entrance slit and the photo voltage was amplified by a narrow-band amplifier with 4-MQ input impedance and synchronously detected. The light from the exit slit was directly incident on the metalized side of the sample (front wall cell configur ation). All photomeasurements were made with the sample at room temperature. To eliminate all possibility of difficulty due to scat tered light, all data used to determine barrier heights were obtained with a 2-mm-thick GaAs filter in front of the entrance which effectively removed all radiation of wavelength shorter than ",0_95 J.I.-Comparison runs made on typical samples with and without the GaAs filter gave essentially identical barrier heights. 9 A. Rose, Concepts in Photocondttctivity and Allied Problems (John Wiley & Sons, Inc., New York, to be published). 10 J. Bardeen, Phys. Rev. 71, 717 (1949). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:25BAR R I E R H E I G H T SST U DIE SON MET A L -S E M I CON D U C TOR S Y S T EMS 3063 As discussed previously, when measuring the photo response for hv<E g, the response per absorbed photon in the metal film is the quantity of interest. However, it was demonstratedl that the fraction of the incident energy absorbed by most metals is approximately in dependent of wavelength for the spectral region of interest in the present work. The form of the photoresponse has been considered by Crowell et al.,l and it is concluded that if OIL> 1 and Olt> 1, where 01 is the absorption coefficient of the metal, L the electron attenuation length, and t the metal thickness, then the photoresponse has the approximate form j"'-'I'R (AE)e-t/L R = COl d (/!lE). o -1/L+0l (2) The spectral dependence of R depends upon the energy dependence of L. If L»t, then the familiar Fowler type of dependence is obtained (3) Quinnll has theoretically estimated the energy de pendence of the electron mean free path for electron electron scattering in a metal and concludes that 1+ (<PB+AE)/E o I=K----- (<pB+AE)2 ' (4) where Eo is the Fermi energy and AE is the excess energy of the electron over the top of the barrier. Recent Monte-Carlo calculationsl2 of I starting from published values of L indicate that for the metal-film thicknesses and photon energy range considered here ('" 1 eV), I and L can have quite different values. In the present work, it was occasionally necessary to attempt measurements of barriers where <PB~ 0.4 eV. In these cases, the photoresponse is weak and it is necessary to make the measurements at photon energies in the range hv=0.6 to 1.2 eV which is substantially larger than <PB. If t~ L, then attempts to extrapolate the data according to Rl rxhv-<PB can lead to a substantial underestimation of <PB because of the energy dependence of L. However, if this difficulty is present, then the data is concave towards the photon energy axis. An example of such a curve can be seen in Fig. 1 for the case of Au on p-type GaAs. The shape of the curve is in general agreement with the energy dependence given by Quinn and the previously reported values of the electron range in gold, but uncertainty as to the details of the transport process and in particular the role of phonon scattering makes exact correlation difficult. 11 John J. Quinn, Phys. Rev. 128, 1453 (1962). 12 R. N. Stuart, F. Wooten, and W. E. Spicer, Phys. Rev. Letters 10, 7 (1963); F. Wooten, R. N. Stuart, and W. E. Spicer, Bull. Am. Phys. Soc. 8, 254 (1963). 1.0 0.8 0.6 ~ 0.4 0.2 0.465 0 0.7 0.8 0.9 1.0 1.1 hv FIG. 1. Photoresponse per incident photon of Au on p-type GaAs. Vertical scale in arbitrary units. 2. Differential Capacitance Measurements In this measurement, the change in potential energy in crossing the space-charge region V 0 is obtained from the 1/0=0 intercept of a 1/C2 vs V plot, where C is the space-charge capacity and V is the applied dc reverse bias voltage. The dependence of C on V was determined on a modified Boonton model 74C-S8 capacitance bridge. The bridge operating frequency was 100 kc and the applied ac voltage was less than 2S m V. In those cases where measurements were made at 77°K, the sample was inserted directly into liquid nitrogen immediately after the room-temperature data had been taken, without breaking contact to the sample. In order to obtain the barrier height <PB, it is necessary to add (Ec-EF) or (EF-E.) to Vo depending upon whether the semiconductor bands bend up or down at the interface. Ee, E., and EF are the conduction band edge, valence band edge, and Fermi energies in the bulk semiconductor. The Ec-EF (or EF-E.) values are obtained from the carrier concentrationl3 and the relation RH=±1/ne, where RH is the Hall coefficient. Published values of the density of states effective masses,14 md, were used. Goodmanl5 has recently considered the assumptions which are made in relating the intercept of the capaci tance plot to the height of the Schottky barrier. The parameters one reads from the bridge circuit and their relation to the actual device-equivalent circuit, carrier trapping effects, variation of effective surface area with depletion layer width, and minority-carrier concentra tion within the space-charge region arising from inver sion layers were all considered in the light of Goodman's 13 See, e.g., W. Shockley Electrons and Holes in Semiconductors (D. Van Nostrand, Inc., Princeton, New Jersey, 1950), p. 242. 14 H. Ehrenreich, J. App!. Phys. Supp\. 32, 2155 (1961); E. D. Palik, S. Teitler, and R. F. Wallace, J. App!. Phys. Supp\. 32, 2133 (1961); C. Hilsum and A. C. Rose-Innes, Semiconducting III-V Compounds (Pergamon Press, Inc., New York, 1961), p. 62; J. J. Hopfield and D. G. Thomas Phys. Rev. 122, 35 (1961). 15 A. M. Goodman, J. App!. Phys. 34, 329 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:253064 W. G. SPITZER AND C. A. MEAD treatment, and with one exception some simple arith metic calculations indicated that these effects should cause little difficulty in the present measurements, i.e., the errors introduced being $0.01 eV or less. The exception noted above is the effect of trapping in the CdS measurements. However, if the diode is biased in the forward direction to flood the electron traps prior to making the capacitance measurements, and if the sample is protected from light then, as described by Goodman, the 1/C2 vs V plots are linear and quite reproducible at low-reverse bias ($1 V). Under these conditions, the drift in C was never more than 2% and in most cases was much less. It is of interest to note that the treatment of all of the above effects predict that the most reliable data are those obtained in the forward bias condition or at small reverse bias. An effect, not discussed by Goodman, occurs when the metal layer is very thin. Under relatively high reverse bias conditions, the leakage current can become appreciable, and this current flowing through the edge on spreading resistance of the metal layer causes portions of the metal area far from the contact probe to be less reverse biased than those near the contact. The net effect is a capacitance which changes less rapidly with voltage than expected. Since there is no voltage drop in the absence of applied bias, the zero-bias capacitance should be quite accurate. Hence the indi cated value of the barrier height is larger than the true barrier height. Under suitable conditions the 1/C2 vs V plot can still approximate a straight line, and it is difficult to determine how much the result has been TABLE 1. A summary of CdS photovoltaic and capacity data; all energies are in e V. t1E is the energy difference in the CdS crystal between the conduction band edge and the Fermi energy. Values of t1E followed by (p) or (H) were determined from resis tivity or Hall measurements. Other values of t1E were estimated from 1/0 vs V plots. Metal Au Cu Ni Mo Al . Ag Pt Photo barrier 0.75±0.01 0.75 0.80 0.78 0.77 0.78 0.79 0.78 0.75 0.36±0.02 0.36 ",0.4-0.5 0.54(5) 0.55 0.58 0.84 0.82 0.88 Vo==1/0 intercept M~ 1/0 (F:c-E/,,) barrier 0.66 0.79 0.75 0.65 0.60 0,32 0.20 0.30 0.38 0.30(77 OK) 0.50(77°K) 0.09 0.08 0.10 0.12(H) 0.16(p) 0.05 0.12(H) 0.16(p) Ohmic contact 0.45 0.16(p) 0,40 0.16(p) 0.70 0.71 0.68 0.16(p) 0.16(p) 0.16(p) 0.75 0.87 0.85 0.77 0.76 0.37 0.32 0.54 0.61 0.56 0.86 0.87 0.84 affected. For this reason samples which showed high forward resistances (few hundred ohms) and relatively high-leakage currents (;G0.1 rnA at 1 V) on the I-V curve tracer were not used for capacitance measure ments. The above consideration is particularly impor tant for a system in which CPB is small, $0.5 eV, and for metals where L is short as in the cases of Cu and Al. Because of the low photosensitivity and the desire for t<L in order to obtain the simple Fowler plot, it is reasonable to prepare samples with thin metal films, of the order of 100 A. Therefore, in such cases, it can be observed that photoresponse and capacity data are not necessarily taken on the same sample. Since, in the present work, the sample is cleaved in the stream of the evaporating metal in the vacuum system, contamination of the interface is effectivelv eliminated. Where a surface layer is present Goodma~ has shown that under suitable conditions Vo= CPB-(Ec-EF)+[nett2/2d]+[2etn Vo]!t/ tt, (5) where t is the semiconductor dielectric constant, t the effective thickness of the surface layer, and tt the di electric constant of the surface layer. In the measure ments which use n-type GaAs the correction terms (the last two terms in the above equation) may be appreci able depending upon the values of t and tt. For the p-type GaAs and the CdS the carrier concentrations are reduced by an order of magnitude or more and the correction terms are ",0.01 eV or less. 3. Diode Forward Characteristic Measurements The I-V characteristic in the forward direction where V> few tenths of a volt is of the form I=Ioexp (eV /akT), where a~ 1. The plot of log I vs V is extra polated to V =0 and the CPB deduced from [0 and the Richardson emission equation. There is considerable difficulty in obtaining any better than order of magni tude accuracy in 10 even at room temperature. At forward currents greater than 1-10 rnA the series resistance coming from the bulk semiconductor and, in some cases, the spreading resistance of the metal film start to limit the current, and for 1$10 J.lA the contri bution from leakage is often important. Therefore, the CPB from this measurement was only checked to see if reasonable (within "'0.1 eV) agreement was obtained with the CPB from the other methods. In almost all cases, such agreement was obtained at room temperature. At lower temperatures the log [ vs V curve for GaAs shifted to larger voltages but the slope did not indicate an appreciable change even at n°K. At the present time this behavior is not understood and casts doubt on the CPB obtained by this procedure. III. EXPERIMENTAL RESULTS AND DISCUSSION A. Cadmium Sulfide Table I summarizes the results of the present meas urements on n-type CdS. The measurements for Au and [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:25BAR R I E R H E I G H T SST U DIE SON MET A L - 5 E M I C 0 ~ D U C TOR 5 Y 5 T EMS 3065 TABLE II. A summary of CdS photovoltaic and capacity data for samples processed as indicated. Metal Process Photobarrier Au Cleave and etch 0.82±0.02 Cleaved 1.15±0.02 Cleaved Cleaved Cleave and etch Lapped and etched 0.88±0.33 Cleaved 0.82±0.05 eu Cleaved 0.60±0.01 Cu have been indicated in a previous paper.8 The !lE(H) was calculated from Hall measurements as described previously with md= 0.5mo. The !lE(p) was estimated from resistivity data assuming the electron mobility,16 }.Ie = 250 cm2/V sec. The other values of !lE were obtained from the slope of the 1/C2 vs V plot and the area of contact. As discussed in a previous paper, 8 the latter method can be inaccurate, however, in some cases it was the only practical measurement. Comparison of the photobarrier values with those obtained from capacity measurements clearly show the necessity of taking the Fermi energy into account. The agreement between the two types of 'PB measurements, except for a few isolated cases, is as good as the agree· ment among the various values obtained for a single metal from either type of measurement, i.e., a few hundredths of a volt. The barrier height shows a strong dependence upon the particular metal used ranging from 0.85±0.03 eV for Pt to an Ohmic contact ('PB<O.lO eV) for AI. In changing the metal work function by '" 1.1 v the barrier height changes by at least 0.75 V. In view of our previous remarks, there exists an almost surprisingly good relation between the two quantities. It should be noted, however, that there are other quantities which show a strong empirical relation to 'PB. For example, an even better correlation exists between 'PB and the elec tronegativity values given by Pauling17 and suggests a possible role played by the partially ionic nature of the semiconductor-metal bond in determining the value of 'PB. Several conclusions can be drawn from the CdS data. It has been proposed18,19 that the photoresponse for hll<E g is due to impurity excitation in the CdS, or the formation of a p-n junction with excitation from the impurity levels (i.e., the Cu 3d level) in the p region. In a recent letterS the present authors pointed out that on the basis of the Au and Cu results neither explanation would suffice to explain the values of 'PB for vacuum cleaved samples. The complete list of data given in Table I substantiates this latter viewpoint. If the photo- 16 W. W. Piper and D. T. F. Marple, J. Appl. Phys. 34, 2237 (1963). 17 L. Pauling, The Nature of the Chemical Bond (Cornell Uni versity Press, Ithaca, New York, 1960), Chap. 3. 18 E. D. Fabricius, J. Appl. Phys. 33, 1597 (1962). 19 H. G. Grimmeiss and R. Memming, J. Appl. Phys. 33, 2217 (1962). l/D intercept t1E 1/ D barrier 0.76±0.01 0.10 0.86 ""'4.5 1.05 0.17 1.22 Over 2.0 ~2.5 ~2.5 0.70±0.02 0.17 0.87 (0.5±0.3-1/c->' vs V not straight line) response were due to some impurity present in the CdS prior to evaporation of the metal film it would be difficult to explain the systematic variation in 'PB nor would there be any a priori reason for the agreement of the values of 'PB from. the two types of measurements. If the response were due to impurities in the metal evaporated, which is very unlikely with the purity material used, then in addition to the above objections there would be no reason for any correlation between 'PB and the metal work function. The above discussion does not, however, apply to the CdS-metal system when the cleaved surface has been exposed to the atmosphere prior to the evaporation of the metal film. The data for a number of samples in which the CdS surface was prepared as indicated are given in Table II. In the present case, elaborate pre cautions were not taken to insure reproducibility of atmospheric conditions, time of exposure, purity of etching solution, etc. It is apparent that the results are much less reproducible. In some cases, 'PB is similar to the vacuum-cleaved samples. In other cases, the two measurements of 'PB give different results, and often the 1/C2 vs V data predict very large barriers. In the light of these measurements difficulties in comparing data obtained by different investigators employing different techniques of sample preparation becomes apparent. Goodman15 has published Vo values for some Au-CdS samples. The CdS was etched (6M HCI) and the Au was electroplated. The values of 'PB deduced from capac ity measurements for three cases are 0.93, 1.08, and 0.93 eV. These values are all larger than the largest value obtained on the Au-CdS vacuum-cleaved samples but within the range of 'PB for the other samples which gave "reasonable" results, i.e., eliminating those which gave barriers of several volts and probably involve some type of interfacial dielectric layer. In more recent work Goodman20 has reported a 'PB=0.68 eV for Au evapor ated on an etched surface. The results obtained here may also be compared to the earlier work of Williams and Bube4 in which the Cu-CdS system gave 'PB= 1.1 eV from photoresponse measurements while some experiments on the quantum yield of photocurrent as a function of the CdS conduc tivity indicated a 'PB"'O.4 eV. The CPB can be estimated 20 A. M. Goodman, Bull. Am. Phys. Soc. 8, 210 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:253066 W. G. SPITZER AND C. A. MEAD 0.34 0-~O~.2~--~OL---~O~.2----~O~4~---O~.6~--~O.~B----~1.0· V FIG. 2. Capacitance of Au on p-type GaAs (same sample as Fig. 1). Vertical scale in arbitrary units but same for both curves. from the forward diode characteristic given in Fig. 3 of this same paper and is "'0.6-0.7 eV. Again, these films were electroplated so comparison to the present meas urements is difficult. It is of interest to note that the ",o.4 eV is close to the <(JB measured here, however, in view of the photoresponse and diode values, this agreement is probably accidental. B. Gallium Arsenide Tables III and IV summarize the results of the meas urements of the n-type and p-type GaAs units, respec tively. The Ec-EF and EF-Ev values were determined by room temperature and liquid nitrogen Hall coefficient measurements. Figures 4,5,6, and 7 show photoresponse and capacitance plots for Al samples. It is noted that the room temperature and 77°K plots of 1/C2 vs V have nearly the same slopes. This result was expected for the n-type sample since according to simple theory the slope is given by d(1/C2)/ dV = 2/ qN DeA 2, (6) where N D is the ionized donor concentration, A the area of contact, and € the semiconductor dielectric constant. Hall-coefficient measurements at the two temperatures 5.4 ~---IC PI 5.2 c-0= Si x "GAAs 50 I-b • Cd S 00 Pd - -f>.'3i 4.8 l:r--~ x-x Ni - /rO-t. Au 4.61- tr-i>. 0-0 x-x Cu 4.41- A--i> 0-0 )I-l( Ag 4.2k-- 0---0 lH( AI 4:0 1 1 1 1 0 0.2 0.4 0.6 0.8 1.0 "'8n FIG. 3. A comparison of barrier heights obtained for various metals on n-type CdS, Si, and GaAs. TABLE III. Photo-and capacity-barrier heights obtained on vacuum-cleaved n-type GaAs samples. For all samples Ec-EF=O at room temperature and = -0.03 eV at 77°K. Values for 77°K immediately follow room-temperature values for the same samples. Metal Au Pt Be Ag Cu Sn Ba Al Photo barrier 0.90 0.88 0.86 0.84 0.88 0.82 0.81 0.88 0.89 0.78 0.76 0.82 0.88 0.83 0.67 0.63 0.80 0.79 l/C" barrier 0.93 0.95 0.98 0.98 0.93 0.90 0.90 0.82 0.95 0.90 0.94 0.94 0.83 0.90 0.85 0.68 0.74 0.73 0.68 0.94 0.81 0.92(77°) 0.78 0.85(W) 0.80 0.78 0.85(7n showed no change in N D. It may also be remarked that the concentration of compensating acceptor levels N A is an order-of-magnitude less than N D for these samples. For the p-type sample the bulk ionized acceptor con centration (assumed equal to the hole concentration) decreases by approximately one order of magnitude between room temperature and 77°K. The slight change in slope of the 1/C2 curve indicates only a small change in ionized acceptor concentration in the space-charge region. 20r----r---,----~---r-_,----.--_, te; 10 - 1.2 h. FIG. 4. Photoresponse of typical AI on p-type GaAs sample. Vertical scale arbitrary. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:25BAR R I E R HE I G H T SST U DIE SON MET A L -S E M I CON D U C TOR S Y S T EMS 3067 TABLE IV. Photo-and capacity-barrier heights obtained on 8.--.----.----.-----.-----, vacuum cleaved, p-type GaAs samples. For all samples Ep-Ev =0.13 eVat room temperature =0.04 eVat 77°K. Values for 77 oK immediately follow room-temperature values for the same samples. Metal Au Pt Be Ag Cu Sn Al Photo- barrier 0.46 -0.38 -0.3 0.55 0.52 0.45 0.44 0.54 0.56 1/['2 1/0' intercept barrier 0.34 0.47 0.45 0.49 0.42 0.46(77°) 0.42 0.46(77°) 0.44 0.48(7n 0.44 0.48(7n 0.37 0.41(7n 0.40 0.44(77°) 0.48 0.52(77°) 0.46 0.50(W) 0.48 0.52(77°) 0.49 0.53(W) 0.58 0.71 0.69 0.73(77°) 0.58 0.71 0.52 0.65 0.58 0.62(W) 0.50 0.63 0.61 0.65(77°) 0.50 0.63 0.57 0.61(W) 0.44 0.57 0.52 0.56(77°) 0.47 0.60 0.53 0.57 (770 ) 0.52 0.65 0.58 0.62(W) 0.56 0.69 0.61 0.65(77°) The room-temperature value of ipB for all metals, with the exception of Sn, on n-type GaAs, is between ",0.80 and 0.98 eV. This is to be contrasted to the strong dependence of ipB on ipM for the same metals on CdS. OJ ~2 o~~~-~-~-~~-~--~ -0.6 -OA -0.2 0 0.2 V FIG. 5. Capacity data on sample of Fig. 4. Vertical scale arbitrary but same for both curves. 6 2 o~-a-----~-----~--~---~ 0.9 1.0 U 1.2 h~ FIG. 6. Photoresponse of typical AI on n-type GaAs sample. Vertical scale ar bi trary. The agreement between ipB for the two types of meas urement is not as good as previously noted for CdS. The room temperature and liquid-nitrogen carrier concen trations for the n-type GaAs are 4X1017 cm-3 and, as previously indicated, the correction terms in Eq. (5) may be as large as several hundredths of a volt, making Vo+ (Ec-E F) exceed ipB by this amount. It may be noted that ipB from photomeasurements does show a tendency to be somewhat less than ipB from capacity measurements. The lack of sensitivity of ipB on ipM for the n-type samples is also observed for the p-type samples as v FIG. 7. Capacity data on sample of Fig. 6. Vertical scale arbitrary but same for both curves. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:253068 W. G. SPITZER AND C. A. MEAD indicated in Table IV. Because of large leakage current for some materials it was difficult to obtain reliable photodata and good capacity measurements could only be made at low temperature. However, for the cases of AI, Au, and Sn, the leakage currents were substantially lower, capacity measurements were made both at room temperature and 77°K, and reliable photodata were obtained. Figures 1 and 2 show photoresponse and capacity data for an Au sample. The photoresponse curve is not a straight line but concave, as previously described. Except for this sample, a major discrepancy is noted in the barriers deduced from the two measure ments. The 'PB (photo) consistently agrees much more closely with Vo than with 'PB (capacity), and the room temperature value of Ep-E.=0.13 eV. The carrier concentration, p= 4.8X 1016 cm-3, was checked on Hall samples taken from the GaAs crystal immediately above and below the section used for the cleavage samples. Less than 10% difference was noted in the carrier concentration. At the present time, the authors do not have a satisfactory explanation for this discrepancy. Because of the curvature of the photoresponse curve for Al the correct V 0 may be uncertain, to at most, 0.05 e V. Photoresponse data -for thin Al samples (AI thickness '" 200 A), where the data lie on a straight line as simple Fowler theory predicts, also indicate the same discrepancy. Regardless of whether the previously described work function model of a surface barrier rectifier applies or if the Fermi energy is pinned at the interface by a large concentration of surface states, the barriers measured on n-type and p-type material, 'PBn and 'PBp, should give (7) where Eo is the semiconductor energy gap. This assumes that if surface states are important, they are the same on both the n-and p-type surfaces when the metal film has been deposited. Al is the only metal for which we have 'PBn and 'PBp measurements at both temperatures. ~Bn(3000K)=0.79 eV and ~Bp(3000K)=0.63 eV giving Eo~1.42 eV. This result is slightly higher than the values usually given,21 Eo= 1.35-1.40 eV, however, it has already been noted that 'PBn (capacity) may be a few hundredths of a volt too large. At liquid-nitrogen temperature ~Bn=0.87 eV and ~Bp=0.61 eV giving an Eo= 1.48 eV compared to Eo= 1.46-1.48 eV in the literature. Values of 'PBp(3000K)- 'PBp(77°K) measured for Sn-GaAs samples and given in Table IV are in reasonable agreement with the same quantity measured for the AI-GaAs samples. The agreement between the various Eo values is regarded as satisfactory, particu larly in view of the variability of the 'PB values between different samples. 21 T. S. Moss, Optical Properties of Semiconductors (Academic Press Inc., New York, 1959), p. 224; Semiconducting Ill-V Com pounds (Pergamon Press, Inc., New York, 1961). The similarity of the results obtained here and those previously reported for silicon are shown by Fig. 3 where 'PM is plotted against 'PBn for both silicon and GaAs. The silicon data are taken from Archer and Atalla. It is observed that the CPBn for the GaAs samples are con sistently larger by 0.15-0.30 eV than for the silicon which brackets the difference in the forbidden energy gaps for the two semiconductors. This would indicate that the Fermi level is pinned at the surface at an energy above E. which is nearly the same for each system. Moreover, the fact that CPBp shows very little change between 300° and 77°K indicates that the surface states responsible for fixing the Fermi level position remain fixed with respect to the valence band edge. According to the theory of Bardeen,lO at a surface state concentration »1013 cm-2 the barrier height becomes insensitive to the metal work function. The results of the GaAs and the previous work for Si indicate that this condition is close to being realized for a number of different metal contacts. Because of the techniques employed in making the diodes the surface states are assumed to be in intimate contact with the semicon ductor and hence are what is commonly called "fast states". The only previously available data for GaAs are those of Williams7 for Sn on p-type material. Again, in this case, the semiconductor surface was etched and the metal film electrodeposited. The CPBp was determined from the same measurements used here and values of 0.84±O.05 eV (capacity), 0.75 eV (photoresponse), and 0.79 eV (/-V characteristic) were obtained. The CPBp in the present study is significantly lower than the above values; however, it is of interest to note that as in the measurements reported here, the above values have CPBp (photoresponse) < CPBp (capacity) by "'0.1 eV. The GaAs measurements reported here have not demonstrated that cpBn is independent of the position of the bulk Fermi level. Increasing n = 4 X 1017 cm-3 by over an order of magnitude gives units in which the tunnel current can start to be appreciable, and de creasing n by the same amount gives samples in which compensa tion is important, that is, N Donor ",!Y Acceptor. Therefore, the total variation of Ec-EF"'0.15 eV which is not large compared to the spread in values of CPBn( "'0.05 eV) for units of a given metal-GaAs system. However, since (Ec-EF) at the surface is verv nearly the same for the n-and p-type samples, it ap pears that the assumption CPB is independent of EF is reasonable. In the case of the Au-silicon system, Archer and Atalla have shown this assumption to be a valid one. Early vacuum-photoelectric emission and work function data have been reported22 for n-type GaAs with ground and broken surfaces. The broken surface consisted primarily of (110) regions. The work function reported is 4.69 eV and the energy difference between 22 D. Haneman and E. W. J. Mitchell, J. Phys. Chern. Solids 15, 82 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:25BAR R I E R H E I G H T SST U DIE SON MET A L -S E M leo N Due TOR S Y S T EMS 3069 EF and Ev at the surface is ",0.3 eV. This value for (EF-Ev).urface is quite close to the values obtained here considering that the comparison is between a "free" surface and one with a metal film covering it. The more recent work of Gobeli and Allen23 on vacuum cleaved GaAs give a minimum-energy separation 23 G. W. Gobeli and F. G. Allen, Bull, Am. Phys. Soc. 8, 189 (1963). (EF-Ev).urface=O.72 eV, a value which clearly does not correspond to the metal-GaAs system. ACKNOWLEDGMENTS The authors wish to express their appreciation to D. Reynolds who supplied the CdS crystals, R. Willard son and W. Allred who furnished the GaAs, and H. M. Simpson who fabricated the samples. JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 10 OCTOBER 1963 Microplasma Breakdown in Germanium M. POLESHUK AND P. H. DOWLING Philips Laboratories, Irvington-on-Hudson, New York (Received 11 April 1963; in final form 27 May 1963) Interpretation of breakdown results in Ge diodes is frequently complicated by effects associated with surface excess current. When these effects are minimized, breakdown is observed within the junction at a "breakdown center," starting at a definite voltage VB, and is accompanied by the onset of microplasma pulses. In anyone diode, there may be a number of centers, each having its characteristic value of VB and producing characteristic microplasma pulses. The minimum value of VB determines the breakdown voltage of the diode and it is possible to increase the latter radically by etching away centers having lower values of VB. Observations were made at temperatures from -253° to 27°C on Ge alloy junctions (n-type base re sistivities from 0.54 to 5.4 n-cm). The properties of the pulses are discussed in some detail: the effect of raising the voltage above VB, the effect of light, and the temperature coefficient of VB. Values of the last are sufficiently high to suggest that suitable diodes can be used as cryogenic thermometers capable of reading smaller changes than 0.01°C at -253°C. Various aspects of the microplasma breakdown are discussed: the mechanism for triggering a pulse and that for "turning it off," the role of the spreading resistance, the possible role of a negative resistance at breakdown, and the effect of microplasma breakdown on the measurement of carrier multiplication at voltages in the vicinity of breakdown. I. INTRODUCTION THERE has been a considerable amount of work on the breakdown of semiconducting diodes, but most of this has been on silicon diodes at room temper ature. Less work has been done on germanium diodes,r-6 also mainly at room temperature, but this is frequently marred by a failure to ensure that actual breakdown was being observed rather than the effects of excess surface current ("soft knee," "soft Zener" volt-ampere characteristic, or "surface breakdown"). In a rather extensive series of observations on ger manium diodes, we find that when care is taken to minimize the excess surface current, true breakdown within the junction area is observable at room tempera ture and is indicated by the onset of characteristic pulses which are qualitatively the same as those ob- I K. B. McAfee, E. J. Ryder, W. Shockley, and M. Sparks, Phys. Rev. 83, 650 (1951). 2 K. B. McAfee and K. G. McKay, Phys. Rev. 92, 858 (1953). 3 S. L. Miller, Phys. Rev. 99, 1234 (1955). 4 R. D. Knott, I. D. Colson, and M. R. P. Young, Proc. Phys. Soc. (London) B68, 182 (1955). • D. R. Muss and R. F. Greene, J. App!. Phys. 29, 1534 (1958). 6 T. Tokuyama, Solid-State Electron. S, 161 (1962). served at the breakdown of silicon diodes. These are, however, of such short duration that they are extremely difficult to resolve. At low temperatures, the excess surface currents present no problem and the pulses at breakdown usually become of longer duration so that their properties can be studied readily. In Sec. II we describe the experimental diodes and measuring circuit used during this investigation. Char acteristic properties of soft-knee and actual junction breakdown are reported in Secs. III and IV, respec tively. In Sec. V we demonstrate the presence of micro plasma breakdown centers within germanium junctions and discuss the effect of their removal on the breakdown voltage of the diode. Section VI deals with the inter pretation of breakdown phenomena in terms of various breakdown criteria. In Sec. VII we discuss breakdown at low temperatures including the pulse properties, pulse-triggering mechanisms, diode volt-ampere char acteristics, the possible role of spreading resistance in limiting breakdown current, and the temperature dependence of microplasma breakdown voltage. Appli cation of some of these results to the measurement of cryogenic temperatures with a micropulsing diode is described in Sec. VIII. Section IX is devoted to a dis- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.250.144.144 On: Tue, 16 Sep 2014 02:15:25
1.1713223.pdf	Influence of Thermal Stresses on the Infrared Stimulability of ThalliumDoped Potassium Iodide Single Crystals Zoltan Kun Citation: Journal of Applied Physics 35, 3357 (1964); doi: 10.1063/1.1713223 View online: http://dx.doi.org/10.1063/1.1713223 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Infrared excitation spectrum of thalliumdoped silicon Appl. Phys. Lett. 33, 294 (1978); 10.1063/1.90343 Thalliumdoped silicon ionization and excitation levels by infrared absorption J. Appl. Phys. 46, 2130 (1975); 10.1063/1.321852 Optical Absorption of F Bands in ThalliumDoped Potassium Halides J. Chem. Phys. 32, 1885 (1960); 10.1063/1.1731051 Unusual Ring Structure of VacuumEvaporated, ThalliumDoped Selenium J. Appl. Phys. 28, 279 (1957); 10.1063/1.1722728 PhotonInduced Diffusion in Thin Films and Single Crystals of Potassium Iodide J. Chem. Phys. 22, 921 (1954); 10.1063/1.1740216 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 15 Dec 2014 15:37:38FLU 0 RES C ENe E 0 F G a A s U ~ D E R I N TEN SEE LEe T R 0 ~ E X CIT A T ION 3357 peak frequency or spectrum shape, the pseudo-Fermi level (E1') must have been at or below the conduction band edge (Ee). Therefore, an upper limit for the concentration of electrons in the conduction band (n) during these experiments can be obtained by calculating n for intrinsic GaAs with El'=Ec. This calculation,6 with the effective mass of the electron taken7 as 0.07, 6 A. J. Dekker, Solid State Physics (Prentiss-Hall, Inc., Engle wood Cliffs, New Jersey, 1957), p. 308. 7 C. HUsum and A. C. Rose-Innes, Semiconductor [ll-V Com pounds (Pergamon Press, Inc., New York, 1961), p. 60. gives n~SX1016. Since equilibrium requires that g1'=n, where l' is the recombination iifetime, the re combination lifetime must have been less than 5 X 1016/ 8X 1026"-'6 X 10-10 sec. ACKNOWLEDGMENTS I am grateful to P. Emtage, R. C Miller, and F. M. Ryan for useful discussion and advice, to S. Scuro and his staff for preparing the samples, and to R. E. Gmitter and R. Buige for aid in performing the measurements. JOURNAL OF APPLIED PHYSICS VOLUME 35. NUMBER 11 NOVEM BER 1964 Influence of Thermal Stresses on the Infrared Stimulability of Thallium-Doped Potassium Iodide Single Crystalst ZOLTAN KUN Zenith Radio Corporation, Chicago, lllitwis (Received 30 April 1964) It was observed that the room-temperature infrared stimulability (IRS) of one KI crystal containing 1.2XlO-a mole% TI was twice as high as that of another with 4.5X lO-' mole% TI concentration. An experi ment was designed to study this apparent discrepancy. X-ray rocking curves of the as-grown material showed that the crystal containing the lesser amount of Tl was plastically deformed, probably due to thermal stresses developed during crystal growth. Both crystals were subsequently annealed at 600°C. The change in the x-ray rocking curves for the originally deformed crystal suggested that there was an extensive rearrangement of the defect structure, mainly under the influence of stress fields around dislocations . .Etch pit studies sup ported this conclusion. In the heavily doped crystal the annealing changed the misorientations of the sub structure. This suggested subgrain boundary movement. Extrinsic ionic conductivity measurements showed similar behavior for both samples: In the as-grown state there were some free vacancies. Annealing almost eliminated them and increased the conductivity. After the annealing, the intensity of IRS of the lightly doped crystal dropped to about one-fifth of its original value; in the heavily doped crystal it did not change. Various models are discussed consistent with results for mechanically deformed samples. It is concluded that a certain kind of stress of a thermal nature is associated with the excessive IRS of the lightly doped sample. INTRODUCTION THE observation was made in our laboratory that the room-temperature infrared-stimulated lumi, nescence of one potassiumiodidesinglecrysta1containing 1.2X 1(}-3 mole% thallium dopant was twice as high as that of another crystal with 4.5 X 1(}--3 mole% thal lium dopant. This was somewhat unexpected since the intensity of phosphorescence of the potassium iodide thallium phosphors in this concentration range was reported to be proportional to their thallium content.1 It was therefore decided to investigate the nature of this apparent discrepancy. Semiconductor research reveals that lattice imperfections in single crystals in fluence their electronic properties llnd since the light absorption and emission in alkali halide phosphors are also electronic processes, it was suspected that the This paper was presented as a research abstract at the 93rd Annual. Meeting of the AIME, February, 1964, New York City. t ThIS work was supported by the U. S. Army Engineering and Research Development Laboratories. 1 F. Seitz, J. Chem. Phys. 6, 150 (1938). lattice defect structure would be an important param eter. Thus etch pit observations, x-ray rocking curve, and ionic conductivity measurements were carried out to determine the state of lattice defects in the as-grown crystals. A heat treatment was then applied with the anticipation that it would change the defect structure. Subsequent to the treatment, the above measurements were repeated. Optical absorption and infrared stim ulability (IRS) measurements were made before and after the thermal treatment. No attempt was made to identify the energy storing mechanism which caused the inconsistent IRS. However, on the basis of comparing our experimental observations with other works in this field, several possible models are discussed. THEORETICAL The potassium iodide single crystals studied in this work were grown from the melt by crystal pulling technique. In crystals produced this way, one important source of defect is thermal stresses due to uneven cool~ [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 15 Dec 2014 15:37:383358 ZOLTAN KUN SEED FIG. 1. Isotherms in a potassium iodide single crystal pulled from the melt. ing.2 The cooling takes place partly by axial and partly by radial heat losses. Owing to the poor thermal con ductivity of alkali halides, a substantial temperature gradient is quickly established between the inner "core" and the outer "skin" of the crystal. The calculated change of temperature in the neighborhood of the liquid solid interface is shown in Fig. 1, from an equation given by Billig2: T/T m=[I-(ah/2) (r/a) 2] exp[( -Z/a) (2ah)l], where T (OK) is the temperature of a chosen point of the crystal and T m(OK) is the melting temperature of the particular material. Z and r represent, respectively, the axial distance from the solid-liquid interface and the radial distance from the vertical axis of the crystal j h is the ratio of the emissivity (H) to the thermal con ductivity (K). (H is estimated to be 0.3 and K is given as 0.05 W / cm °C at O°C for potassium iodide.) This is a very rough approximation. It is derived from Carslaw's and Jaeger's expression3 of the tempera ture of an infinitely long cylinder which is heated to a constant temperature (T m) and then loses heat at a constant rate. The first part is the parabolic radial heat loss and the second is in the axial direction, which is exponential. The thermal expansion of potassium iodide is con siderable (a=45XlO-6°C-I). Thus, the differential thermal dilatation of the "core" and "skin" will cause stresses. For example, at 1 cm above the interface there is a 7° cm-I radial temperature gradient. The resultant strain, expressed as aAT, is 3.15X 10-4• Taking the average elastic modulus as E= 1011 dyn/cm2, the stress at this point is S=EaAT, or 315 g/mm2• This is high enough to cause plastic deformation.4 The local temperature gradients may vary due to accidental cooling. Consequently the stress and the strain distribution can be different from place to place within the single crystal. These stresses are relieved by a subsequent annealing heat treatment. The result will 2 E. Billig Proc. Roy. Soc. (London) A235, 37 (1956). 3 H. S. C~rslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, Oxford, England, 1959), 2nd ed., Chap. 7, p.188. 4 R. 1. Garber and L. M. Polyakov, Zh. Eksperim. i Teor. Fiz. 36, 1625 (1959) [English trans!.: Soviet Phys.-JETI' 9, 1158 (1959)]. be a redistribution of dislocations and probably of point defects.5 It is clear that after the heat treatment the crystal should be cooled at a rate consistent with its thermal conductivity in order to avoid the reintro duction of stresses.6 EXPERIMENTAL The purpose of this work is to reveal the changes, if any, of the IRS of the as-grown crystals when they were annealed and to show that the elimination of ther mal stresses is related to the cause of these changes. Thus the defect structure after treatment was compared to that of the as-grown samples. Absolute values were not calculated. The microscopic observations of etch pits and x-ray rocking curve measurements were applied to show over-all changes in the defect structure. The ionic conductivity was measured to indicate possible changes of point defect densities. The observation of thallium bands in the optical absorption curves served as a check of thallium concentration changes during annealing. 7 The single crystals were grown in our laboratory by the Czochralski method in argon atmosphere. They were " in. long and their cross-sectional area was ap proximately 4 sq cm. The thallium concentrations were determined by spectral analysis of samples taken from both the top and the lower ends of the crystals. A piece was cleaved from the center of each crystal for the various measurements; it was perpendicular to the direction of crystal pulling and had the same area as that of the crystal's cross section and a thickness of approximately 1.5 mm. One-half of the sample was divided into three parts for ionic conductivity speci mens. The other half underwent microscopic examina tion, x-ray rocking curve measurements, optical absorp tion, and IRS testing. The dislocation structure of the samples was ex amined by microscope, applying the etch pit technique. The etchant used was recommended by CookS (pro pionic acid plus 1.75 wt.% barium). The x-ray rocking curves were taken at room temperature on a modified Norelco quartz analysis unit using CuKa radiation. The Geiger tube of the unit was connected to a strip-chart recorder. The diffraction peaks were continuously re corded as the specimen was rocked through the dif fraction angle. For ionic conductivity measurements Acheson # 154 colloid graphite was painted on the two larger faces of the sample. Then it was placed between two copper electrodes in an evacuated Pyrex glass con tainer. The container was heated in a dc-powered elec trical resistance furnace, and the temperature was in dicated by a Chromel-Alumel thermocouple inside the 5 A. H. Cottrell, Dislocations and Plastic Flow in Crystals (Clarendon Press, Oxford, En!;land, 1953), Chap. 15...p. ~80. 6 V. D. Kuchin, Izv. VysshIkh. Uchebn. Zavedenu Flz. 1958, 117 (1958). 7 W. Koch, Z. Physik 57, 638 (1929). 8 J. S. Cook, J. App!. Phys. 32, 2492 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 15 Dec 2014 15:37:38I ~ F R ARE D S TIM U LAB I Ll T Y 0 F T 1-D 0 P ED K I CRY S TAL S 3359 container near the sample. The resistance was measured as the function of temperature with a model 10200 Beck man ultraohmmeter. Optical absorption measurements were made by a 14R model Cary double-pass recording spectrophotom eter in the 1200-to 200-m}.! wavelength interval. With highly absorbing samples, the differential signal may become too large for the normal range of the instru ment. In this case it can be effectively reduced by plac ing a neutral density filter into the reference compart ment. This step introduces significant error due to the fact that the filters in the reference beam cause the instrument to respond with a widening of the mono chromator exit slit. With the resultant wider bandpass, the resolution of sharp absorption peaks is lost. Thus their heights cannot be measured with great accuracy and repeatability. The IRS at room temperature was measured in a fluorimeter setup shown in Fig. 2. The stimulating in frared source was a 6-V incandescent bulb. Its output was passed through a collimating lens and then through a Corning # 7 -69 glass infrared filter (bandpass above 5% from 7200 to 11 000 A). The stimulated emission from the crystal was passed through a Corning # 4-72 glass filter (bandpass above 5% between 3500 to 5800 A) and then onto the end-on multiplier phototube (EM I/US-9536S). The output of the phototube was fed to a Keithley 610A model electrometer. A millivolt recorder was used in conjunction with the electrometer to record the phosphorescence and the IRS levels. The constancy of the stimulating infrared light was assured by means of a lead storage cell and a rheostat. The voltage on the bulb was measured by a vacuum tube voltmeter, and it was kept at such a constant value that it would cause no detectable signal above the photomultiplier dark current when there was no sample in the compartment. The dynode supply voltage was obtained by a series of DYNODE SUPPLY PHOTOTUBE OUTPUT FIG. 2. Fluorimeter setup for IRS measurements. FIG. 3. X-ray rocking-curves of the lightly doped sample be fore and after heat treatment. i -BEFORE H.lI' -----AFTER H.T , minl,ltes voltage regulating tubes with an adjustable output. It was maintained at 1190 V in this experiment. The sample and various optical components of the system were enclosed in light-tight compartments inter connected by shutters as indicated in the diagram. This made it possible to measure dark current, sample phos phorescence, and IRS without exposing the excited sample and phototube to outside light. Each sample was irradiated with the uv source for a period of 10 min. The cell shutters were then closed and the sample was rotated into the measuring position. The phosphores cnce level was recorded for approximately 30 sec be ginning 3 min after the uv excitation cutoff. At exactly 3.5 min after uv cutoff, the infrared shutter was opened and the stimulated emission was recorded for at least 10 min. After all these tests were completed on the as-grown samples, a heat treatment followed. Those halves of the samples which underwent optical and x-ray tests were annealed at 600°C for 4 h in purified nitrogen at mosphere. They were then cooled to room temperature at the rate of 1.3°C/min. (Kuchin6 calculated that 2.12°C/min cooling rate is the upper limit for stress free potassium iodide.) Subsequent to this the various examinations were repeated and the results were compared. For conductivity measurements, new samples were cleaved from these heat-treated halves adjacent to the previous conductivity samples. RESULTS AND DISCUSSION The x-ray rocking curves taken from the (100) (cleavage) plane before and after the heat treatment are shown in Figs. 3 and 4. Since rocking curves were re corded from both sides of the crystals, there are two curves to be considered for each sample in each state. It is clear from Fig. 3 that several changes took place in the lightly doped sample during heat treatment. On one side the broad single peak which had 12', 24" angu lar half-width in the as-grown state became 6', 12" after heat treatment. The intensity of reflection also in creased. On the other side the twin peaks were replaced [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 15 Dec 2014 15:37:383360 ZOLTAN KUN 1----- INTENSITY II"b,'r~Y ""'s) minulu FIG. 4. X-ray rocking curves of the heavily doped sample be fore and after heat treatment. by a narrow, single maximum. The rocking curves of the sample with more doping (Fig. 4) showed one noticeable change only, on one side of the sample. The double peaked maximum became a single peak after heat treatment. The existence of the two reflection maxima of both samples indicated that there was a substructure present in the as-grown state with two preferred orientations. When the two peaks approached each other or disap peared completely during annealing, this was because of the reduction of misorientation. This in turn seems to suggest that there were stresses across the sub grain boundaries which were relieved during heat treatment by moving the boundaries.5,9 The broad reflection maximum in Fig. 3 closely re sembles the shape of those rocking curves which can be obtained from lightly deformed crystals. The de crease of angular width during annealing could mean one or more of the following changes in the defect structurelO-I2: (a) reduction of lattice tilt, (b) relief of dislocation strain, (c) increase of the subgrain size, (d) decrease of the uniform lattice bending. Since in lightly deformed samples the same dislocations are re sponsible for tilt and strain,II the decrease of their den sity should mean considerable reduction of the reflec tion broadening. The tiny peak appearing on one side of the main maximum after annealing suggests that lattice bending was reduced by the formation of sub grain boundaries5 and a second orientation began to appear due to the increase of sub grain size. In these processes (except for the growth of subgrains) the driving force is stresses acting on dislocations, as dis tinct from those across the sub grain boundaries. It would require more elaborate x-ray equipmentl! to determine to what extent one or the others of these changes took place in the lightly doped sample. How ever, the microscopic examination and the change of 9 w. T. Read, Dislocations in Crystals (McGraw-Hill Book Company, Inc., New York, 1953), Chap. 14, p. 197. 1. A. D. Kurtz, S. A. Kulin, and B. L. Averbach, Phys. Rev. 101, 1285 (1956). 11 M. J. Hordon and B. L. Averbach, Acta Met. 9, 237 (1961). 12 E. F. Vasamillet and R. Smoluchowski, J. App!. Phys. 30, 418 (1959). the intensity maximum also suggest the extensive re arrangement of defect structure. The magnitude of the rocking curve maximum is de termined by the reflecting power of the crystal. The reflecting power, in turn, is a function of the primary and secondary extinctions. The defect structure is just one of the numerous factors which influence the extinc tion. However, it has been established that as the de formation of the crystal progresses the intensity of re flection will become less and less.I3 Therefore it is reason able to conclude from the intensity change that heat treatment reduced the disorder due to deformation in the lightly doped sample. Microscopic examination supported the assumption of dislocation movement during heat treatment. Areas in which there was a fairly uniform distribution of etch pits became free of them during thermal treatment and other areas formed with high etch pit densities. The ionic conductivity was measured as a function of temperature between room temperature and 300°C. In this temperature range the conductivity is structure sensitive. The number of charge carriers is determined by the positive bivalent impurity ion concentrationI4 and only the mobility of the carriers changes with the changing temperatures. However, it is in apparent con tradiction with the observationl5 that often the semi logarithmic plots of conductivity versus the reciprocal of absolute temperature are not represented by a straight line. All but the most carefully cooled (over several days) samples seem to have changing slopes in dicating varying activation energies. We shall come back to this phenomenon when discussing our results. The log IT vs T-I plots are shown in Fig. 5. The lightly doped sample had higher conductivity in the as-grown state than the heavily doped one. Heat treatment raised the conductivity of both samples proportionally. The TEr.f>ERATURE "T('K-') FIG. 5. Ionic conduc tivities vs liT of the lightly and heavily doped sam ples before and after heat treatment. 13 A. Guinier, X-Ray Crystallographic Technology (Hilger and Watts Ltd., London, 1952), Chap. 7, p. 192. 14 A. B. Lidiard, Encyclopedia of Physics, edited by S. Fliigge (Springer-Verlag, Berlin, 1957), Vo!' XX, p. 246. 15 D. B. Fischbach and A. S. Nowick, J. Phys. Chern. Solids 2, 226 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 15 Dec 2014 15:37:38I N F R ARE D S TIM U LAB I LIT Y 0 F T 1-D 0 P E D K I CRY S TAL S 3361 plots were not straight but they curved upwards with increasing temperature. This tendency was stronger in the as-grown state. The above described behavior could well be explained in terms of the following model. Our crystals were cooled fast after crystal growing. Thus, there was no time for the formation of an equilibrium number of neutral complexes at low temperatures consisting of a bivalent impurity cation and a cation vacancy. Conse quently there is a larger number of free vacancies in these crystals than in those cooled over several days.15 On the low-temperature end of the curves, the more gradual slope represents relatively low activation energy which consists of only the migration energy of the frozen-in vacancies. The steeper parts indicate higher activation energies. It is suggested15 that there are two . components which combine to give the higher value. They are the migration energy of a positive ion vacancy and the half of the binding energy between this positive ion vacancy and a bivalent impurity ion. It may be confusing that the number of free charge carriers was reduced by slow cooling following annealing whereas the conductivity increased. An explanation which seems plausible is the increase of mobility after annealing. In the crystal which is randomly stressed, the diffusion of the ions is hindered by the stress fields which act as scattering centers.16 When the annealing relieved these stresses the resulting unhindered mobil ity became the "true" mobility of the charge carrier. The optical absorption spectra are shown in Fig. 6. The absorption band due to thallium was observed in both samples at about 28S-m.u wavelength. Absorption at the thallium band peaks was greater than 99% (density> 2 on the absorption scale of the instrument). Since the absorption scale of the spectrophotometer does not cover such high densities, the earlier described neutral density filters were used. It is likely that the insignificant change of the thallium band is rather due to this instrumental error than the heat treatment. (The absorption curves after heat treatment are almost identical with those before. Thus only one pair of curves is shown in Fig. 6.) The absorption spectrum curves were not corrected ABSORPTION lOGf ~ I I I i I I I I I I ,t -- LIGHT DOPING ----- HEAVY DOPING zoo 400 600 000 WAVELENGTH mp FIG. 6. Optical absorption spectra of Tl-doped KI samples. 16 H. Kanzaki, K. Kido, and T. Ninomiya, J. Appl. Phys. Suppl. 33, 482 (1962). I R. STIMUlABlLITY 18 12 .\ I.R STIMIJI.AllON BEGINS UGtn<:~~:H.~T.== HEAVY<:~~~H~~' =~=-~== 06 ''\'~ l}\ ~ ":~~==:':~::'::=-===~:::=~:::=i=~=: 3 5 7 9 II minute, TIME AFTER U.y' CUTOFF FIG. 7. Infrared stimulability decays vs time of the lightly and the heavily doped samples in the as-grown and in the annealed state. for the difference in thicknesses since the slit settings of the instrument were slightly different for the two samples. Thus their absorption curves are not strictly comparable. However, it is clear that the thallium band of the heavily doped sample is considerably broader. (At these high absorption values, because of the work ing principles of the instrument, it is the width which changes with the activator concentration.) Infrared stimulability decays versus time in Fig. 7 were plotted from the average of three or more stimu lability measurements. In general, the deviation from the plotted value was within ± 10% at any point. These variations were due to slight changes in the infrared source intensity and differences in the positioning of the samples for excitation and measurement during the repeated tests. Since the measured intensities were small, the changes in the dark current of the photo multiplier also introduced some error. However, care was taken to check the dark current level regularly during the measurements. Slight differences in the timing of the various procedures also caused some varia tion since the crystals were at room temperature and the excited state decayed rapidly. It is dearly shown that the IRS of the lightly doped sample was a good deal higher before heat treatment than that of the heavily doped sample. After treatment the lightly doped sample dropped to about one-fifth of its original value and the other did not change much. It was noted, however, that there was a definite change in the shape of the IRS decay of both samples after annealing. The phosphorescence levels before infrared stimula tion are shown in the lower left-hand corner of Fig. 7. Since the magnitudes were very small and the changes were of the same order as the instrumental fluctuations, no conclusions were drawn about these values. The data for the heavily doped sample is corrected by a linear thickness factor from 1.196 to 1.475 mm, the latter value being the thickness of the lightly doped sample. Previous experiments indicated that the cor rection is valid in case of thin samples with small variations in thickness. Considering the IRS and optical absorption curves [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 15 Dec 2014 15:37:383362 ZOLTAN KUN together it seems clear that the heat treatment reduced the IRS of the lightly doped sample without reducing its thallium content in any corresponding proportion. Thus it is reasonable to suggest that annealing elimi nated some causes which resulted in IRS in addition to and other than that due to the thallium activator. We noticed from the change of the x-ray rocking curves that there was an extensive redistribution of the defect structure in the lightly doped sample during heat treat ment. This in turn suggests that previous to this treat ment the defect distribution was metastable, which in some way was connected with the cause of dispropor tionate IRS. It was indicated earlier that such a meta stable structure was formed most likely by thermal stresses due to nonuniform cooling. None of the measurements in this work gives direct information as to what kind of energy storing centers are created by internal stresses. However, there are several published papers on this matter and they will be briefly reviewed here. Ueta et alP found that the thermal glow luminescence of deformed KCI crystals was enhanced and the glow curve was quite simple regardless of whether the crystal was x rayed or uv irradiated. The authors tenta tively suggested that it was due to the recombination of electron and hole trapped by vacancy clusters. Hersh18 studied the thermoluminescence and optical absorption of x-rayed potassium chloride in the unde formed and deformed state. He found that deformation enhanced the intensity of thermoluminescence and there was a slight shift of the wavelength of absorption peaks. It was clear from the dichroic spectrum as well as from paramagnetic resonance measurements that deforma tion helped to create a large number of Cb-molecule ions. On the other hand, they were shown to be also present, in lesser quantities, in the impure undeformed potassium chloride. Hersh's conclusion was that the major center is self-trapped hole which is formed through the creation of Cb-molecule ions. But the presence of th~se molecule ions is not directly related to impurity, neIther is it shown to be an intrinsic property of pure, annealed crystals. Recently Hersh and Hadley19 reported that there is similarity between the IRS of impurity activated phos phors and undoped crystals which were plastically de formed at room temperature. It confirmed the presence of the previously mentioned molecule ions both in the doped and in the undoped but deformed material. On the basis of present work we could not say which of these models, if any, would explain the behavior of the lightly doped crystal. However, the reviewed papers offer fair support to our case that stresses do activate alkali halides. What appears to be new as compared to 17 M. Ueta et at., Proc. Intn!. Conf. Crystal Lattice Defects (J. Phys. Soc. Japan 18, Supp!. II, 286, 1963). 18 H. N. Hersh (unpublished paper) (Torino lecture). 19 H. N. Hersh and W. B. Hadley, Phy~. Rev. Letters 10 437 (1963). ' previous works is that this optical activity is additive to that due to doping. CONCLUSIONS Evidence in this study suggests that the excessive IRS of the lightly doped sample was associated in some way with the thermal stresses. On the other hand, it appears from the behavior of the heavily doped sample that not all kinds of thermal stresses are active in this way. X-ray data of the lightly doped sample seem to indicate that a metastable defect distribution was as sociated with stresses acting mostly on dislocations. Ionic conductivity measurements indicated higher density of single cation vacancies in the as-grown state as compared to that in the annealed samples. This observation could be remotely associated with higher IRS on the basis of Ueta's suggestion that the mecha nism is recombination of electron and hole trapped by vacancy clusters. Similarly to single crystals of semiconductors the alkali halide phosphor crystals are sensitive to lattice imperfections. Thus the conditions of crystal growing do influence their optical properties. Unlike semicon ductors, such "accidents" may improve the desirable properties. However, it is uncontrolled. It may vary randomly from place to place, even within a small section of a single crystal, thus causing confusing discrepancies. ACKNOWLEDGMENT Thanks are due to Dr. Robert Robinson and Dr. Charles T. Walker for helpful discussions and to Stanley Polick and Guy Falco for assisting with the measure ments. I am also indebted to Dr. Herbert N. Hersh for letting me use his unpublished manuscript and to L. W. Tresselt for growing the crystals. APPENDIX The point was raised by several reviewers that the additional IRS observed in the lightly doped sample might arise from other impurities in the crystals, which are not purposely added during growth but may exist in concentrations comparable to the added impurity. On the basis of this argument, the lowered IRS follow ing heat treatment and slow cooling would be assumed to result from a precipitation of such impurities. It appears therefore desirable to investigate if stresses are necessary for the "extra" IRS. At the end of the previous experiment the lightly doped sample was in the annealed state. Then its low IRS was presumably due to the thallium impurity activator only. It was believed that if the higher IRS were associated with stresses, a room-temperature de formation should "cause" IRS in addition to that present in the annealed sample. Such an observation, of course, would not exclude the possibility that impurities [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 15 Dec 2014 15:37:381 N F R ARE D S TIM U L A 13 I LiT Y 0 F T I - D 0 P E D K I CRY S TAL S 3363 play a role. It would only indicate that stresses are necessary conditions. The lightly doped sample was deformed with 230 g/sq-mm compressive stress in a suitable die. The com pression was obtained by the top ram moving while the bottom one was stationary. Appropriate marking was applied on the two faces of the sample for identifi cation during the various measurements. X-ray rocking curves, birefringence, and IRS measurements were made at room temperature. The techniques were similar to those included in the paper except for a few modifica tions. These will .be described later. Both the "top" and the "bottom" sides were x rayed (Fig. 8). The rocking curve obtained from the top side consisted of one broad, low intensity reflection peak. There was even less reflection from the lower side which resulted in a broad and low intensity curve, indicating a greater degree of deformation than of the top side. The birefr~gence patterns exhibited slip along the (110) and (110) planes. The fluorimeter setup was modified in order to im prove its sensitivity for phosphorescence measurements. This in turn meant that the intensity of the infrared light had to be reduced to bring the IRS onto the same scale as the phosphorescence. Thus the curves ob tained in this setup are not strictly comparable with the earlier ones. However, they yield valuable information within this complementary experiment. INTENSITY (orbilrarYllnitsl --TOP SIDE -----BOTTOM SIDE 24.B 37.2 49.6 minutes ROCKING ANGLE FIG. 8. X-ray rocking cl!rves obtai~ed from the lightly doped sample after plastic deformatIOn by compression. FIG. 9. Infrared stimulability decays vs time of the lightly doped sample before (in the annealed state) and after plas tic deformation. I R STiMULABILITY --I R STIMULATION BEGINS 05 0.' TOP/BEFORE P. -. "AFTER P. ------ /8£FORE po -- BOTTOM, AFTER p_ 6 8 10 IIIlnult~ TIME AFTER U.V. CUTOFF The IRS (Fig. 9) of the "bottom" side was obviouslv higher in the pressed state than in the annealed stat~. The IRS of the "top" side did not change. The rate of IRS decay of both sides was faster in the deformed state. It would be difficult to evaluate the phosphorescence levels in these measurements for two reasons. One was that the phosphorescence was measured at room tem perature and the means of excitation was ultraviolet (instead of the more penetrating x ray); thus the inten sity of emission was very low. The other reason was the small variation of the dark current from one measure ment to the other. This was in the same order of magni tude as the change of phosphorescence from the an nealed to the deformed state. (The phosphorescence curves are omitted from Fig. 9.) It appears from this experiment that plastic defor mation can increase the IRS of the thallium-activated potassium iodide. The fact that it was observed on the more deformed side only suggests that it is related to the amount of deformation. In the deformed state the decay of IRS was fast in the initial stage. This was similar to the IRS decay of both samples in the as grown state, shown in Fig. 7. In conclusion we can say that plastic deformation is a necessary condition for the phenomenon we observed. However, this work did not indicate if the impurities playa part in it and what this part may be. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 15 Dec 2014 15:37:38
1.1754066.pdf	STIMULATED BRILLOUIN SCATTERING IN LIQUIDS1 E. Garmire and C. H. Townes Citation: Applied Physics Letters 5, 84 (1964); doi: 10.1063/1.1754066 View online: http://dx.doi.org/10.1063/1.1754066 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/5/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Forward stimulated Brillouin scattering J. Appl. Phys. 63, 5220 (1988); 10.1063/1.340383 Stimulated Brillouin scattering in plastics J. Appl. Phys. 43, 3631 (1972); 10.1063/1.1661780 SteadyState Gain of Stimulated Brillouin Scattering in Liquids J. Appl. Phys. 40, 178 (1969); 10.1063/1.1657027 MULTIPLE STIMULATED BRILLOUIN SCATTERING FROM A LIQUID WITHIN A LASER CAVITY Appl. Phys. Lett. 11, 42 (1967); 10.1063/1.1755020 Stimulated Brillouin Scattering: Measurement of Hypersonic Velocity in Liquids J. Acoust. Soc. Am. 41, 1301 (1967); 10.1121/1.1910472 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 75.102.71.33 On: Mon, 24 Nov 2014 14:09:27Volume 5, Number 4 APPLIED PHYSICS LETTERS 15 August 1964 STIMULATED BRILLOUIN SCATTERING IN LIQUIDS1 (loser exc itot ion; interferometer detection; E) Stimulated Brillouin scattering of intense laser light, with a build-up of coherent hypersonic waves, has been observed in a number of liquids in an arrange ment which allows multiple Brillouin scattering processes and rather precise measurement of the velocity of hypersonic waves. Stimulated Brillouin scattering, which has already been reported in solids,2 can be considered parametric generation of an acoustic wave and a scattered light wave from an initial light wave.3 ,4 The nonlinear coupling of the three waves is typically electrostrictive. In solids, a single scattering of the incident light was observed, with a shift to lower frequencies equal to the frequency of the acoustic wave. In the present experiment with liquids, however, as many as eight orders of successively scattered light waves appear. Each order is generated backward from the incident wave and finds its way back to the laser cavity where it is amplified. This component again enters the liquid giving rise to its own Brillouin scattering, which appears as the next order. A giant-pulse ruby laser provided the incident light in an arrangement shown in Fig. 1. A glass flat with parallel sides was introduced into the beam as an additional optical resonator in order to separate longitudinal modes a nd produced a single mode with a frequency spread less than 0.04 cm -1 • Liquids were placed in a cell at or near the focal point of a lens, and the frequencies of light generated o near 6943 A were studied with a Fabry-Perot inter- ferometer placed at sites A, B. or C. figure 2 shows a typical interferometer pattern. Stimulated Brillouin scattering was also observed with the ruby laser under normal (not Q-switched) operation, but the giant pulse system was used for measurements of frequency shifts because of its high spectral purity. Photographs of the Fabry-Perot rings observed at sites A and C were generally similar. Both the stray laser light and the Brillouin components were too weak to be detected through the Fabry-Perot placed at B. This evidence, along with the multiple orders of Brillouin shifts, shows that the B!illouin components are amplified in the ruby and follow the path indicated by the dashed line in Fig. 1. The width of the dashes represents the relative intensity of the Brillouin 84 E. Garmire and C. H. Townes Massachusetts Institute of Technology Cambridge, Massachusetts (Received 6 July 1964) component as it is amplified. The intensity at B is too weak to be detected, while after amplification the signal at A may be large. Once the Brillouin component is amplified, it may be as strong as the original laser frequency and in fact acts somewhat like light from another mode of the laser, re-entering the liquid and causing another Brillouin scattering. This process can occur a number of times, since the frequency shift for Brillouin scattering in liquids in around 0.2 cm -1. Thus a number of orders are well within the ruby line width and may be amplified. This is not the case for the solids previously reported, 2 where the shifts were around 1 cm-1• The reiterative effect in this arrangement is different from that normally causing the higher orders of stimulated Raman scattering. As long as the liquid is outside a cavity (and the backward scatter~d light wave is weak) there can be no "anti-Stokes" wave, since the required backward-going acoustic wave is not present. With sufficiently intense effects, one might expect multiple-order stimulated Brillouin scattering by higher order processes within the liquid and with out further amplification of the first-order wave. This has not been observed. For, if the first order stimulated Brillouin sc attering occurs at 1800 to the incident light, the second order would occur at 00 , and one would expect alternating orders of Brillouin components at either A or C, which do not occur. If the Brillouin components amplified in the ruby are from spontaneous Brillouin back-scattering, rather than from stimulated Brillouin scattering, then Brillouin shifts to higher frequencies would occur with the same intensity as shifts to lower A C '-y----l '---r--! t t A:I:::::I:~:-~fr:-:~f-·-~-k. ~ -tJ A::u.Ss -tj- ~;;~ ROTATI NG RUBY MODE j. PLATE LENS L~1U~~ PLATE PRISM SELECTOR I 7 Fig. 1. Experimental arrangement, showing path of stimulated Brillouin scattering. A Fabry-Perot inter- ferometer was placed at sites A, B, or C. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 75.102.71.33 On: Mon, 24 Nov 2014 14:09:27Volume 5, Number 4 APPLIED PHYSICS LETTERS 15 August 1964 Fig. 2. Fabry-Perot interferograms from site A with water in the liquid cell. Left: Single frequency of laser with intens ity be low thresho Id for stimu lated Brillouin scattering. Right: Above threshold, with three Brillouin components from water in addition to the original laser frequency. Interorder spacing is 0.701 cm-l. frequencies; such is not observed. Furthermore, the large variation in relative intensity of the Brillo\Iin components and the original laser light with a rather small variation of input laser power shows conclusively that stimulated emission dominates. There is a threshold power density below which the Brillouin components were not observed at all. This was not necessarily identical with the threshold for initiation of amplification of the Brillouin com~ ponents, since the sensitivity of detection was not high. In carbon disulfide, stimulated Brillouin scattering was observed when the focal point of. a 17-cm focal length lens was 8 em beyond the cell. This gave a threshold of 30 MW/cm, the lowest observed in any of the liquids studied. The threshold in benzene was 1200 MW / cm2, and most of the liquids had comparable thresholds. Surprisingly, nitrobenzene had a much higher threshold than any other liquid studied. It is nearly as strong a spontaneous Brillouin scatterer as carbon disulfide and has almost 100 times less acoustic absorption in the measured ultrasonic region. The high threshold indicates that nitrobenzene may be unusually lossy to gigacycle acoustic waves. Many weak Brillouin scatterers had thresholds comparable with some of the strongest scatterers, presumably because of their lower acoustic losses. Water is a notable example. Stimulate(i Brillouin scattering should prove valuable in the study of such weak scatterers where it is difficult to obtain sufficient spontaneous signal. In carbon disulfide, nitrobenzene, toluene, benzene and acetone, stimulated Raman scattering occurred along with the stimulated Brillouin scattering. The Raman threshold in these cases is appreciably lower than that for Brillouin scattering. For water, carbon tetrachloride, and methanol, stimulated Raman scattering did not occur at the experimental power levels. Stimultaneous Raman scattering did not appear to markedly affect the Brillouin scattering. From the measured frequency shift L1v, the hyper sonic acoustic velocity v of the liquids can be calculated from the equation L1v = 2v o(v/c)n, where Vo is the frequency of the incident light, and n is the refractive index. Table I lists the frequency shifts measured about 22°C, the calculated sound velocities, and results of measurements of spon taneous Brillouin shifts. 5 The values agree within stated experimental errors. Stimulated Brillouin scattering is a useful method of measuring hypersonic velocities to a high degree of accuracy. The; directionality of the amplified Brillouin components, the many orders, the line sharpening of stimulated Brillouin scattering, and the sharp frequency of the single mode laser all contribute to an inherent accuracy which sli.ould allow measurement of velocities one or two orders of magnitude better than the rough measurements made here. The most accurate experimental means for determining the frequency shift would probably be observation of the microwave beats from a photo cathode mixing the laser and the Brillouin com ponents. This method would be especially convenient since all the components are of comparable intensity and in a single directional beam. As in stimulated Brillouin scattering in solids, intense hypersonic acoustic waves are generated in the liquids. Contrary to the case of crystals, neither the liquid nor the cell in which it is con tained are damaged by the stimulated Brillouin effect. This simplifies observations, and detailed studies can be made of the acoustic properties of the liquids. Probabl y Brillouin components due to solids with sufficiently small sound velocities can also be amplified by this method. It may thus allow convenient excitation of ultrasonics and stimulated Brillouin effects in crystals without their fracture. 85 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 75.102.71.33 On: Mon, 24 Nov 2014 14:09:27Volume 5, Number 4 APPLIED PHYSICS LETTERS 15 August 1964 Table I. Brillouin Shifts, Calculated Acoustic Velocities, and Previously Measured Velocities Liquid Brillouin 1 Calculated sound Previous Shift, cm- velocity, m/sec results,a m/sec CCl4 .141 1007 ± 7 1040 ± 27 Methanol .139 1100 ± 11 Acetone .153 1174 ± 7 1190 ± 40 CS2 .1925 1242 ±6 1265 ± 22 Hp .1885 147l ± 8 1509 ±25 Aniline .2575 1699 ± 8 aacoustic velocities given by K. F. Herzfeld and T. A. Litovitz in Absorption and Dispersion of Ultrasonic Waves (Academic Press, New York, 1959), p 362, which are calculated from measurements of spontaneous Brillouin scattering. We thank Boris Stoicheff and Raymond Chiao for very helpful discussions and the loan of equip ment. lWork supported in part by the National Aeronautics and Space Administration under Research Grant No. NsG-330, and in part by the Air Force Cambridge Research Laboratories, Office of Aerospace Research, under Contract AF 19(628)-4011. 2R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, Phys. Rev. Letters 12, 592 (1964). 3R. Y. Chiao, E. Garmire, and C. H. Townes, Proc. Enrico Fermi Intern. School of Physics, Course XXXI, 1963 (to be published). ~. M. Kroll, Bull. Am. Phys. Soc. 9, 222 (1964). 5K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press, New York, 1959), p 362. DIRECT OBSERVATION OF OPTICALLY INDUCED GENERATION AND AMPLICATION OF SOUND (EfT) We observed optically induced sound in two different experiments. In the first, a plane traveling wave of 45 Mc/ sec sound in water diffracts a light beam entering at the Bragg angle; Bragg reflections yields a second light beam whose frequency is shifted by 45 Mc/sec. The two beams interact to produce an observable change in the amplitude of the sound wave. In the second experiment, two light beams whose frequencies differ by 57 Mc/sec are made to intersect at the proper angle, again in water, but without initial sound present, and we observe the sound wave generated at the beat frequency. 86 A. Korpel, R. Adler and B. Alpiner Zenith Radio Corporation Chicago, Illinois (Received 6 July 1964) Garmire, Pandarese and Townes 1 and KroU2 have calculated this interaction process. Briefly, two intersecting light beams with frequencies CUI and cu2 and wave vectors k l' k2 produce a wave of electro strictive pressure of frequency I CU 1 -CU2 I and with a wave vector k I -k 2' If the angle between the two light beams is so chosen chat the phase velocity of the pressure wave (CUI -cu2)/1 ki -k21 equals the sound velocity in the medium, a sound wave arises. Amplification of thermal sound (stimulated Brillouin scattering) was observed by Chiao, Townes and Stoicheff.3 The process belongs to the class of This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 75.102.71.33 On: Mon, 24 Nov 2014 14:09:27
1.1702672.pdf	Solubility of Zinc in Gallium Arsenide J. O. McCaldin Citation: Journal of Applied Physics 34, 1748 (1963); doi: 10.1063/1.1702672 View online: http://dx.doi.org/10.1063/1.1702672 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Diffusion mechanism of zinc and beryllium in gallium arsenide J. Appl. Phys. 69, 3547 (1991); 10.1063/1.348497 Model for the diffusion of zinc in gallium arsenide Appl. Phys. Lett. 55, 2117 (1989); 10.1063/1.102080 Diffusion of zinc in gallium arsenide: A new model J. Appl. Phys. 52, 4617 (1981); 10.1063/1.329340 Diffusion and Solubility of Zinc in Gallium Phosphide Single Crystals J. Appl. Phys. 35, 374 (1964); 10.1063/1.1713321 Solubility and Diffusion of Zinc in Gallium Phosphide J. Appl. Phys. 34, 231 (1963); 10.1063/1.1729074 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sat, 20 Dec 2014 15:41:221748 B. O. SERAPHIN AND D. A. ORTON reflection is no serious objection to the technical applica tion of the effect. It can be shown that by the use of interferometric techniques in properly matched multi layer systems, a minute change in an optical system can be amplified to a considerably larger modulation.13 13 B. O. Seraphin, J. Opt. Soc. Am. 52, 912 (1962). ACKNOWLEDGMENTS We are indebted to T. M. Donovan for help with the preparation of the samples, as well as to Dr. H. E. Bennett for advice on the optical part of the experiment. Dr. N. J. Harrick has made some valuable comments, which are included in the discussion part of this paper. JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 6 JUNE 1963 Solubility of Zinc in Gallium Arsenide J. O. McCALDIN North American Aviation Science Center, Canoga Park, California (Received 3 January 1963) The distribution of tracer zinc-65 between the vapor and solid GaAs was studied. For dilute concentrations [Zn~J of zinc in the s?lid, the dis!ribution coefficient K is a constant (Henry's law); at higher zinc concen trations, K falls off mversely WIth [Zn.J. These observations can be interpreted simply in terms of an ionization equilibrium Zn. ---> Zn.-+e+. Based~on this interpretation, the present measurements indicate an intrinsic carrier concentration n; of about 4X10l8 cm-a for GaAs at lO00°C. This value is roughly six times larger than n; estimated by extrapolation of Hall measurements; the latter, it is suggested, may reflect the presence of only the more mobile carriers. ~h~ solubility of ~c was also studied as the arsenic pressure in the system was changed from the dis SOCiatIOn pressure (estimated 10-3 atm) to one atmosphere. The zinc solubility was observed to increase three to fourfold with the increase in arsenic pressure. This result is in semiquantitive agreement with calculations for the mass action equilibrium of simple stoichiometric defects in GaAs. I. INTRODUCTION IMPURITY solubility studies in semiconductors are favored by a relatively simple model for interpreta tion. The work of Reiss, Fuller, and Morin! on lithium in germanium and silicon showed how the Fermi level in the semiconductor host is a governing factor for the lithium solubility. In compound semiconductors like GaAs, where an additional thermodynamic degree of freedom is present, the solubility of an impurity de pends not only on the Fermi level but also on the stoi chiometric balance of the compound, e.g., the Ga-to-As ratio in GaAs. The stoichiometry of many of these compounds may be easily controlled, however, by fixing the vapor pressure of a volatile component, e.g., the arsenic pressure over GaAs. In many compound semiconductors, furthermore, the effects of the Fermi level and of stoichiometry should be simply additive on a property like an impurity solu bility. Consider, for example, the location of the Fermi level. This depends on the various ionization processes in the crystal and is dominated by those processes which involve relatively large concentrations. In GaAs at elevated temperatures, the intrinsic carrier concen tration is of the order of 1018 cm-3 and fixes the location of the Fermi level, unless a chemical impurity is intro duced at high concentration. Stoichiometric defects like vacancies and interstitials, being present at con- 1 H. Reiss, C. S. Fuller, and F. J. Morin, Bell System Tech. J. 35, 535 (1956). siderably lower concentrations, e.g., 1015 cm-B, would, therefore, not influence the Fermi level; they do, how ever, still influence properties like solubility by par ticipating in the solubility reaction.2 Thus a model for interpretation of solubility data in GaAs obtained by simply superposing the ionization equilibrial and the stoichiometric equilibria2 seems reasonable. Such a model is discussed later in this paper. The early work of Whelan et al.,a on the behavior of Si in GaAs indicated several of the possiblities in solubility studies. In their interpretation they regarded the silicon as having a separate solubility on each sub lattice of the host compound. Thus the net doping depended on the difference in silicon solubilities on the two sublattices. Quantitative agreement between this interpretation and experiment was obtained by them, particularly in regard to the influence of the Fermi level in controlling the two silicon solubilities. Late!:. experiments on Ge in GaAs by the present author and by Harada4 showed that stoichiometry could also be important, controlling the semiconductor type. These experiments have since been put on a 2 D. G. Thomas, Semiconductors, edited by N. B. Hannay (Reinhold Publishing Corporation, New York, 1959), Chap. 7. 3 J. J¥. Whelan, J. D. ~truthers, and J. A. Ditzenberger, Proceedmgs of the lnternat~onal Conference on Semiconductor Physics, Prague 1960 (Czeckoslovak Academy of Sciences Prague 1961), pp. 943-945. ' , 4 J. O. McCaldin and Roy Harada, J. Appl. Phys. 31, 2065 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sat, 20 Dec 2014 15:41:22SOLUBILITY OF ZINC IN GALLIUM ARSENIDE 1749 quantitative basis by Vieland and Seide!5 in agreement with the interpretation of Whelan et al.3 A somewhat simpler system for solubility studies was introduced by Merten and Hatcher6 in their study of Zn in InSb. In this case only a single solubility is in volved, as the zinc impurity normally occupies only the sites of one sublattice of the crystal. They observed a str?ng con~entration dependence of the solubility, WhICh they mterpreted as due to certain kinetics of the experiment. An alternative possibility that is discussed later in this paper is that their measurements represent true (equilibrium) solubilities, which can be interpreted in terms of the Fermi level. They sought, but did not find an effect of stoichiometry. Other studies by Fuller and Wolfstirn7 have dealt with Li in GaAs. Two species, a substitutional and an interstitial form of Li, seem to be important here. Fermi level effects, but not stoichiometric effects, were observed. The possible importance of stoichiometry in these systems is suggested by diffusion experiments. Cunnell and Gooch8 have shown that zinc diffusion in GaAs is anomalous; they suggest that stoichiometry may be an important factor. The present authorS has shown that p-:n junctions may be made to move through Ge-doped GaAs at different rates by varying the stoichiometry and Vieland10 has shown that the diffusivities of many impurities in GaAs are strongly influenced by stoichiometry. The present experiment was undertaken to measure concurrently both the Fermi level and the stoichiometric effects in a simple system. Provision was made to cover a wide range of these two variables in order to definitely establish their relative importance. A preliminary ac count of some early results of the present experiment has already been presented.H II. EXPERIMENTAL PROCEDURE To compare the concentration of zinc in solution in gallium arsenide with its concentration in the surround ing vapor a simple experimental technique was used. The tracer zinc-65 was equilibrated between the solid gallium arsenide phase and a gas phase in a quartz capsule, as shown in Fig. 1 (a). When the equilibration was completed, a cold finger was applied to one end of the quartz capsule, causing the gaseous zinc to condense there, as shown in Fig. l(b). The capsule was then broken in two and the zinc-65 counted in each half of the capSUle. 5 L. ~. Vieland and T. Seidel, J. App!. Phys. 33, 2414 (1962). 6 Ulnch Merten and A. P. Hatcher J. Phys. Chern Solids 23 533 (1962). ,., 7 C. S. Fuller and K. B. Wolfstirn J. App!. Phys. 33 2507 (1962). " 8 F. A. Cunnell and C. H. Gooch, J. Phys. Chern Solids 15 127 ~~. . , 9 J. O. McCaldin, Bull. Am. Phys. Soc. 6, 172 (1961). 10 L. J. Vieland, J. Phys. Chern. Solids 21, 318 (1961). 11 J. O. McCaldin, Bul!. Am. Phys. Soc. 7, 235 (1962). Material GaAs Zine Arsenic TABLE I. Purity of starting materials. Remarks Intrinsic, pulled crystal from Monsanto. Their Hall measurements give Jl. = 5800 cm2/Vsec and n=5X 10'5 em-3• Supplied by Asarco as high purity. Spectrographic analysis detected only the following: [Mg] [Si], and [Cu]<l ppm, and [Pb]<0.5 ppm: Ch~mical analysis in?icated. [Cd]=0.14 ppm. Supphed by Asarco as high E.unty. Spectrographic analysis detected only LMg]<0.5 ppm and [CuJ<O.1 ppm. A wafer of GaAs about 0.05 cm thick and weighing about 100 mg was used in each capsule. Extra arsenic could be added to the capSUle. Thus the arsenic pressure under which the zinc equilibration occurred could be independently specified. The completeness of the equilibration was checked in several of these experiments. In preliminary experi ments the equilibration had been accomplished by solid state diffusion of the zinc, but it was then observed that evaporation of the gallium arsenide across the diameter of the capsule afforded a more rapid equilibration. The evaporation method was used in all of the experimental runs reported in this paper. Various samples of the evaporated material taken from the same capsule showed variations in zinc concentration of about ten percent, which was considered adequate equilibration for the present purposes. An important consideration in this experiment was the freedom of the system from unwanted impurities particularly oxygen. The quartz capsules were baked overnight in flowing hydrogen at 1l00°C prior to use. Care was taken in the handling of the other materials to minimize their exposure to oxidizing environments. In particular, the arsenic was received in evacuated capsules and stored in desiccators except for the few minutes required to load it into the capsules. Data on the chemical purity of the starting materials are shown in Table I. The zinc-65 was prepared by neutron irradiation at Oak Ridge National Laboratory of the pure zinc de- EVAPORATED GaA. 1000·C-~ 1020·C f,=~=· ~~._~~ GaA. 1000·C GaA. + Zn (a) lb) EXTRA A. TRACER ZINC FIG. 1. E~perimental !lrrangement. Equilibration at lO00°C occurred durmg evaporatlOn of the GaAs across the diameter of the quartz capsule. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sat, 20 Dec 2014 15:41:221750 ]. O. McCALDIN t--..-.........:::-:;..:-::-. -------INTRINSIC ----o o o 10000 0.17' AlMOS ARSENIC . PRESSURE o TEMPERATURE·JOOOOC· o -4 ~ ~. ~ LOG CONCENTRATION ZINC IN SOLID SOLUTION eM-3 FIG. 2. The distribution of zinc between the solid GaAs phase and the vapor phase. The distribution coefficient is the ratio of zinc concentrations in the two phases. Corresponding values of the Fermi level based on the analysis in the teJ<t are also shown. scribed in Table I. A weighed amount of zinc was sealed in an evacuated quartz capsule for each irradiation. The activity produced was then measured in a conven tional scintillation well counter with spectrometer inter posed between counter and scaler. In cases where the activity was so great as to overload the well counter, the zinc-65 source was removed various distances from the scintillation crystal and the count rate recorded. A comparison of the various zinc-65 sources was then made on a 1/ R2 plot for consistency. Since zinc con centrations reported in this paper span almost five orders of magnitude, it was necessary to use several zinc-65 sources of different isotopic dilution. Another consideration in the experiment was the possibility of the formation of a second condensed phase. Two such phases are known12 in the Ga-As-Zn ternary system: ZnAs2 and Zn3As2. However, only the latter would be apt to form under the present conditions. An estimate of the pressures of zinc and arsenic needed to form condensed Zn3As2 at 1000°C can be had from the work of Silvey, Lyons, and Silvestri.13 They show a total vapor pressure of about 0.3 atm over stoichiometric CdaAs2 at its melting point (721°C), and it seems likely14 that the same pressure applies to ZnaAs2 at its melting point (1015°C). From this information one can calculate that a concentration of zinc vapor of 1.3XlO18 cm-a would have been required at an arsenic pressure of 0.17 atm to form condensed ZnaAS2. The highest zinc vapor concentration used was 5 X 1017 cm-s• Furthermore, no change in the behavior of the dis tribution coefficient of zinc is apparent, in the results to be discussed, at the highest zinc concentrations, where ZnaAs2 might form. Also the work of Silvey et al.,13 indicates that the compounds in the zinc-arsenic system are dissociated in the vapor phase. Thus the only complication intro- 12 Werner Koster and Werner Ulrich, Z. Metallk. 49, 361 (1958). 13 G. A. Silvey, V. J. Lyons, and V. J. Silverstri, J. Electrochem. Soc. 108, 653 (1961). 14 V. J. Lyons (private communication). duced by polyatomic vapor species in the present experi ment arises from the arsenic itself, as is discussed later. m. RESULTS A. Zinc Solubility at Arsenic Pressures Near One Atmosphere The first experimental runs were performed under arsenic pressures in the range ordinarily used in GaAs preparation, i.e., a few tenths of an atmosphere. The most extensive data were obtained for 0.17-atm total arsenic pressure j these data are presented in Fig. 2. The ordinate in the figure is the distribution coefficient K which is the ratio of the zinc concentration [Zn8] in the solid phase to the concentration [Zng] in the gas phase. Thus K is a measure of zinc solubility in GaAs. At low zinc concentrations K is constant (Henry's law), as might be expected for intrinsic GaAs. At higher values of [Zn.], however, K, and hence the zinc solubility, decrease. This situation is reminescent of the well-known behaviorl of lithium in silicon and ger manium, so that the present results might be interpreted simply in terms of ionization equilibria. Following the analogy we write (1a) for the combination of a zinc atom Zng in the gas phase with a vacant gallium site V Ga in the crystal to yield a zinc atom on a gallium site ZnGa. The corresponding mass action is (lb) where PZu is the external zinc pressure. The zinc atom in the crystal ordinarily ionizes (2a) (2b) where e+ is a hole and p the hole concentration. Com bining Eqs. (lb) and (2b), and approximating the total zinc concentration in the crystal as (3) since the ionization of the zinc is almost complete, we can represent the distribution coefficient In the present example, we are treating the total arsenic pressure, and hence [V GaJ, as constant. Relation (4) appears to fit, at least qualitatively, the experimental data of Fig. 2. In intrinsic GaAs, where the hole concentration p is constant, the distribution coeffi cient K is constant. At high doping levels, however, K varies inversely with p"'[Zn.], as expected. The relation between hole concentration and zinc concentration can be represented over the full range of [Zn.] by p=t{[Zn.]+([Zn.]+4n?)i}, (5) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sat, 20 Dec 2014 15:41:22SOLUBILITY OF ZINC IN GALLIUM ARSENIDE 1751 where ni is the hole concentration in intrinsic GaAs, as shown in standard texts. IS Thus the distribution coeffi cient becomes 10gK = 10gKo-log{ ([Zn,J/2ni) +[1+ ([Zn.J2/4nI2)Jl} , (6) where Ko is the distribution coefficient at "infinite dilution", i.e., in intrinsic GaAs. In fitting relation (6) to the present data, there are two adjustable parameters, Ko and ni. These parameters have been chosen to give a good fit in Fig. 2, yielding Ko= 17,800 and ni=4X 1018 cm-3• The result for ni is about 6 times larger than would be expected from Hall measurements, a point that is discussed in a later section of this paper. In general, however, the curve seems to fit within the random variations which are evident in the experimental points. Arsenic pressures other than 0.17 atm were also studied in these early experimental runs. Some measure ments made with 1.0-atm arsenic are shown in Fig. 3. The zinc solubility seems to be somewhat enhanced by the sixfold increase in arsenic pressure. However, the enhancement is comparable to fluctuations in the data points. Furthermore, the time required for equilibration is greater, the greater the arsenic pressure, and about one month was needed for equilibration at one atmos phere pressure. Thus the high arsenic pressure ex periments were discontinued in favor of low arsenic pressures. B. Zinc Solubility at the Dissociation Pressure Early data in this experiment had clearly indicated the influence of the ionization equilibrium (i.e., the Fermi level) on the zinc solubility. However, the in fluence of stoichiometry which one might expect through the factor [V GaJ in relation (1b) evidently :.: t-=' z ILl <:; ;:;: II. :g 1000 <.) • z o i= ~ III ir I CI) i5 I 001!;:;9:-----"-..l...-.L.l-L.l.U:2:!:0:---L--L--L-L...L.l....L.lJ 2 I LOG ZINC CONCENTRATION IN SOLID, CM-a FIG. 3. Effect on the distribution coefficient of increasing the arsenic pressure from 0.17 to 1.0 atms. Data at 1.0 atm were ob tained only in the extrinsic region. 10 W. Shockley, Electrons and Holes in Semiconductors (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1950). o o o 0 • INTRINSIC • • • o • ~'~17:--~~~~1~8--~~~~19~~~~~~2~O--~ LOG CONCENTRATION ZINC IN SOLID SOLUTION FIG. 4. Effect on the distribution coefficient of decreasing the arsenic pressure from 0.17 atm to the dissociation pressure. would not be clearly demonstrated until a large change in arsenic pressure was introduced in the system. This was accomplished by performing experimental runs at the dissociation pressure, omitting the extra arsenic shown in the capsule of Fig. 1 (a). Unfortunately the dissociation pressure is not known accurately. The available data have been summarized recently, however, by Silvestri and Lyons.16 Their com parative plot of the somewhat conflicting data of various experiments suggests an average value of 10-3 atm for the dissociation pressure at lOOO°C. This figure is used in the present paper as the best now available. The distribution coefficient observed at the dissocia tion pressure is highlighted in Fig. 4, which permits comparison with K for 0.17 atm pressure. Evidently K is reduced about threefold by the change in pressure. The reduction camiot be determined accurately because of scatter in the data, and because K at the dissociation pressure seems to fall off more rapidly at high zinc con centrations than the above theory, relation (6), would anticipate. This last point is discussed later in the paper. Nevertheless, the qualitative fact that zinc solubility decreases as the arsenic pressure is decreased is ex hibited clearly in Fig. 4. The simple mass-action treatment described in part A indicates that the zinc solubility should behave in this way. The intrinsic distribution coefficient Ko can be represented as [V GaJ divided by k1k2ni and, therefore, varies directly as the vacancy concentration. Thus the threefold shift in distribution coefficient in Fig. 4 suggests a similar change in vacancy concentration, [V Ga]. The vacancy concentration [V GaJ can be related to the arsenic pressure by consideration of the Schottky equilibrium in the crystal, a point that has been well discussedP The result is (7) 16 V. J. Silvestri and V. J. Lyons, J. Electrochem. Soc. 109,963 (1962). . 17 J. J. Lander, in Semiconductors, edited by N. B. Hannay (Reinhold Publishing Corporation, New York, 1959), Chap. 2, p.71. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sat, 20 Dec 2014 15:41:221752 J. O. McCALDIN TABLE II. Partial pressures (atm) of arsenic vapor species at 1000°C, as calculated from Stull and Sinke." Total pressure 1 atm 0.17 10-3 Tetramers 0.937 0.142 1.4X 10-' Dimers 0.072 0.028 8.6XlO-4 Monomersb 4.28XlO-6 2.67X 10-6 4.8XlO-7 a See reference 18. b The concentrations of monomers at the three pressures stand in the ratio 1.6:1:0.18. where P A. represents the concentration of monatomic arsenic vapor. The actual arsenic vapor present under the experimental conditions described above is quite complex, consisting of tetramers and dimers as well as monomers. The relative concentrations of these species can be calculated from available thermodynamic data,18 however, and the results of such a calculation are presented in Table II. Since the present study involves equilibrium only, the only important specie is the one appearing in equilibrium (7), i.e., the monomer. The variations in its relative concentration should also occur in K, according to the present analysis. Table II indicates that the change from 0.17 to 1.0 atm pressure should cause a 60% increase in K, which is roughly what is observed in Fig. 3. The table also indicates that the change from 0.17 atm to the dissociation pressure should cause a 5.6-fold decrease in K, which may be compared to the observed threefold decrease. In view of the various uncertainties in these calculations, the agreement between analysis and experiment appears to be semiquantitative as far as the effect of arsenic pressure is concerned. DISCUSSION The solubility of zinc in GaAs is evidently subject to two influences. First, the zinc concentration itself is a strong influence, capable of causing the solubility to decrease some 30-or 4O-fold, an effect we interpret here in terms of the Fermi level. Secondly, the stoichiometric balance in the host crystal is a somewhat weaker in fluence, capable of causing a solubility change of 3-or 4-fold in the conveniently accessible range of pressures in this system. Furthermore, the two influences seem to act independently of one another, judging from the parallelism of the solubility curves shown in Fig. 4. This fact supports the assumption of the model used here that the two influences are simply additive. First we consider the influence of the zinc concentra tion itself. The observed concentration effect can be explained with some elementary semiconductor theory,! as has been shown; the essential point of the explanation is that high concentrations of dopant cause the Fermi level to move, thereby shifting all ionization equilibria in the crystal. A consequence of this interpretation is 18 D. R. Stull and G. C. Sinke, Thermodynamic Properties of the Elements (American Chemical Society, Washington, D. C., 1956). that the intrinsic carrier concentration ni at lO00°C is about 4X1018 cm-3, a value something like six times larger than is suggested by Hall measurements. No Hall measurements appear to have been made on GaAs at lO00°C, probably due to its volatility; however, one can extrapolate from the data of Whelan and Wheatley19 and Folberth and Weiss20 to estimate nc::7X1017 cm-3 at 1000°C, certainly within a factor of two. Thus the present experiment indicates a value of ni definitely larger than the Hall concentration ni. A similar result was obtained in the experiments of Merten and Hatcher6 on the solubility of zinc in InSb near its melting point. In the case of lnSb, Hall measure ments of different observers21.22 agree that nC::1.6X1018 cm-3 near the melting point. Yet the measured distribu tion coefficients of Merten and Hatcher, when analyzed in the same way as the present data, yield a value of n,~3X1019 cm-3, i.e., about 20 times larger than the Hall concentration. Incidentally, our relation (6) can be fitted to their data about as well as to our own data; the principal difference is that their data fall more in the intrinsic region, ours mostly in the extrinsic region. Merten and Hatcher interpreted the discrepancy in the solubility and Hall values of ni in terms of kinetic processes in their experiment, i.e., due to nonattain ment of equilibrium in their "solubility" measurements. In the present solubility measurements precautions were taken to insure a close approach to equilibrium, as described above. Still, a concentration ni is measured which is several times the Hall ni. We do not believe the discrepancy is due to nonattainment of equilibrium in the present measurements. Also we consider the Merten and Hatcher results still open to the interpreta tion that they too have a near-equilibrium measurement. We offer an alternative explanation of the discrepancy by applying the argument about kinetics to the Hall measurement, rather than to the solubility measure ments. The Hall measurement reflects a transport property, not an equilibrium property, and, therefore, can be strongly influenced by various relaxation times, competing reaction paths, and the like. Indeed Auker man and Willardson23 have already discussed a case in GaAs where the Hall measurement reflects carrier transport in two conduction bands, with the higher mobility band being strongly weighted in the statistical averaging that the Hall measurement does. Ehrenreich24 in his review of the band structure of GaAs concludes that GaAs has, in addition to the [OOOJ minimum, a second band in the [l00J direction with minima "'0.36 eV above the [OOOJ minimum. The second con- 19 J. M. Whelan and G. H. Wheatley, J. Phys. Chern. Solids 6 169 (1958). ' 20 O. G. Folberth and H. Weiss, Z. Naturforsch. lOa, 615 (1955). 210. Madelung and H. Weiss, Z. Naturforsch. 9a, 527 (1954). 22 H. J. Hrostowski, F. J. Morin, T. H. Geballe, and G. H. Wheatley, Phys. Rev. 100, 1672 (1955). 23 L. W. Aukerman and R. K. Willardson, J. Appl. Phys. 31 939 (1960). ' 24 H. Ehrenreich, Phys. Rev. 120, 1951 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sat, 20 Dec 2014 15:41:22SOLUBILITY OF ZINC IN GALLIUM ARSENIDE 1753 duction band has a high density-of-states so that at high temperatures it may be more populated than the [OOOJ minimum. If one extrapolates the Aukerman and Willardson measurements to 1000°C, the second band is estimated to be populated with 2.5 times as many conduction electrons as the first band, which is the one emphasized by Hall measurements. This factor is not large enough to account for the above discrepancies in ni. However, the numbers used in calculating the factor of 2.5 are not known sufficiently well to make a jUdgment yet. Also there are other features of both the GaAs and lnSb band structures which are important for such a calculation and about which very little is known. Let us turn now to the influence of stoichiometry on zinc solubility. Merten and HatcherS also looked for this effect. The negative result of their search for the effect seems now to be clearly due to the small variations in antimony concentration employed. They changed the antimony concentration by twofold, compared to a change of arsenic concentration in the present experi ments by about a thousand-fold. The magnitude of the stoichiometry effect is severely reduced in the present example by the presence of complexes in the vapor phase; as Table II indicates, a total change of a thousandfold in the arsenic vapor con centration results in only a ninefold change in the vapor specie of interest. More favorable systems to obtain a quantitive measure of the stoichiometry effect on solu bility probably can be found in various oxides, where dimers are the only vapor complex. We have no plans for such studies, however. One other point in the present experiments needs comment. In Fig. 4 the solubility curve for low arsenic pressure seems to fall off more rapidly at high zinc concentration that either the theoretical curve or the comparison experimental data. This tendency is illu strated in the figure by a dashed line. A similar tendency is present in Merten and Hatcher's data. If this tendency should indeed be real, it could be supporting evidence for a defect proposed by Ruehrwein and Epstein,25 namely, a zinc atom on an arsenic site ZnAs. This defect, unlike other defects currently being considered, would be favored by both low arsenic pressures and high zinc concentrations. CONCLUSIONS The present investigation of zinc in GaAs reveals a strong concentration dependence of the solubility, similar to earlier results6 on zinc in InSb. The present measurements at least are reasonably close to equi librium, as judged by specimen homogeneity. The con centration dependence is interpreted as a Fermi level effect, with the consequence in the simple analysis given here that the intrinsic carrier concentration in GaAs at lOOO°C is about 4XlO18 cm-s. Extrapolated Hall measurements indicate a value roughly six times smaller, which may be due to the strong statistical weighting that a Hall measurement gives to high mobility carriers. The present study also shows unmistakably that the stoichiometric balance in the host crystal affects the impurity solubility. A solubility increase of three-or fourfold was observed in this system as the total arsenic pressure was increased roughly a thousandfold. These results are in semiquantitative agreement with a simple analysis of the stoichiometric defects thought to be important in this system. ACKNOWLEDGMENTS The author wishes to thank T. Crockett for advice concerning tracer technique and H. Reiss for a reading of the manuscript. Thanks are also due R. Rayburn for assistance with some of the experiments and E. Eisel for quartz work. Also the author wishes to thank U. Merten for a pre publication copy of reference 6. 25 R. A. Ruehrwein and A. S. Epstein, paper presented at the May 1962 meeting of the Electrochemical Society. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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1.1696845.pdf	Calculation and Interpretation of the 129I Isomer Shifts in the Alkali Iodide Lattices W. H. Flygare and D. W. Hafemeister Citation: The Journal of Chemical Physics 43, 789 (1965); doi: 10.1063/1.1696845 View online: http://dx.doi.org/10.1063/1.1696845 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/43/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the calculation of Mössbauer isomer shift J. Chem. Phys. 127, 084101 (2007); 10.1063/1.2761879 129I Mössbauer spectra of the tellurium iodides J. Chem. Phys. 68, 3067 (1978); 10.1063/1.436173 Interpretation of the 57Fe Isomer Shift by Means of Atomic Hartree–Fock Calculations on a Number of Ionic States J. Chem. Phys. 55, 141 (1971); 10.1063/1.1675500 Interpretation of Quadrupole Splittings and Isomer Shifts in Hemoglobin J. Chem. Phys. 47, 4166 (1967); 10.1063/1.1701594 Calculation and Interpretation of the NMR Chemical Shift and Its Pressure Dependence in the Alkali Halide Lattices J. Chem. Phys. 44, 3584 (1966); 10.1063/1.1727269 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 08 Dec 2014 14:29:44THE JOURNAL OF CHEMICAL PHYSICS VOLUME 43, NUMBER 3 1 AUGUST 1965 Calculation and Interpretation of the 1291 Isomer Shifts in the Alkali Iodide Lattices W. H. FLYGARE AND D. W. HAFEMEISTER* Departments of Chemistry and Physics, University of Illinois, Urbana, Illinois (Received 20 July 1964) (Revised Ms received 26 April 1965) . Starti~g with an ideal alkali halide lattice composed of closed-shell ions, the overlap and electrostatic mteractlOns ar~ computed .between the nearest and next-nearest neighbors in the alkali iodide lattices. The electr~stahc pertur~ahon on the s electr~ns is found to be small relative to the overlap deformation of the free-lOn wavefunctlOns: Both effects ~redlct the same correct dependence for the relative isomer shifts, howe.v~r .. The de!ormed free-lOn wavefunctlOns are used to compute the relative isomer shifts at 1291 in the alkali lOdlde lattices. The value of the nuclear parameter in 1291 is aR/R=0.5X10-4. INTRODUCTION RECENT experimental data on the isomer shift at the iodine nucleus in the alkali iodide lattices1 have focused attention on the electronic structures of these simple crystals. The isomer shift depends pri marily on the difference in the contact term or the s electron density at the nucleus between the absorber and emitter in a Mossbauer experiment. Thus, if the absorbing nucleus is placed in different chemical en vironments, the s-electron density will be different resulting in different isomer shifts relative to the emitter. The chemical shift as measured in nuclear magnetic resonance, however, depends primarily on differences in p-electron density. (Contributions from non-p electrons are negligible.) Thus, the isomer shift and chemical shift are excellent probes of different aspects of the electronic structure of the alkali halide lattices. This paper is an analysis of the isomer shift and the electronic structure of the alkali iodide lattices. The usual and successful interpretation of the elec tronic structure in the alkali halide lattices is to assume complete transfer of charge from the alkali atom to the halogen atom which completes a closed-shell configura tion on both the cation and anion. This starting point has both a classica12,3 and a quantum-mechanical4 justification. Any covalent bonding in these crystals would require sp3d2 hybrid atomic orbitals due to the octahedral site symmetry in the lattices. In the iodine atom, for instance, the formation of an spSd2 hybrid would require excitation of one 5s and three 5p elec trons into the Sd and higher atomic orbitals. As the promotional energies are so very large, there will be very little covalent character in the bonding of an alkali halide lattice. Thus, as a first approximation, the * Present address: Los Alamos Scientific Laboratory, Los Alamos, New Mexico. 1 H. deWaard, G. dePasquali, W. H. Flygare, and D. W. Hafemeister, Rev. Mod. Phys. 36, 358 (1964); D. W. Hafemeister, G. dePasquali, and H. deWaard, Phys. Rev. 135, B1089 (1964). 2 M. Born and J. E. Mayer, Z. Physik. 75, 1 (1932), 3 D. Cubicciotti, J. Chern. Phys. 31, 1646 (1959); Erratum: 33, 1579 (1960). 'P.-O. Liiwdin, "A Theoretical Investigation into Some Properties of Ionic Crystals," thesis, Uppsala, 1948; J. Chern. Phys. 18, 365 (1950). See also R. S. Knox, The Theory of Excitons configuration of the outer electrons in the positive and negative ions in all alkali halide lattices will be S2p6. As a constant configuration throughout the alkali halide series leads to constant values of the isomer shifts' some mechanism for distorting the closed-shell configu'ration must be found. Any systematic application of the electronegativity differences between the alkali metals and iodine leads to predicted increases in the isomer shift (decrease in s-electron density at the nucleus) from Li to Cs. As Fig. 1 shows that the isomer shift increases, reaches a maximum, then decreases in the Li to Cs series, the application of electronegativities to obtain the associated transfer of charge and resultant covalent character is not deemed a good approach. The method used here to compute the isomer shift is to start with the free-ion Hartree-Fock wavefunc Hons and deform the free-ion functions by the overlap and electrostatic effects. Both nearest- and next nearest-neighbor interactions are found to be important in the alkali iodide series. The calculated relative shifts agree well with the experimental values. ISOMER SHIFTS The measured isomer shifts 0 in 129I in the alkali iodide lattices where positive 0 indicates movement of ~he ~mitter and absorber towards each other, are given m Flg. 1 and Table I. The Mossbauer experimentsl are perf?rmed by plac.ing the ~mitting and absorbing nuclei m dIfferent chemIcal enVIronments and observing the electronic perturbations on the nuclear energy levels (isomer shifts) in the emitting nucleus relative to the absorbing nucleus. An expression for 0 in terms of the source and ab sorber s-electron densities at the nucleus, 1 1/;.(0) 12 and 1 l/;a(O) 12, respectively, can be obtained by con sidering the electrostatic interaction between the s electrons and a nucleus with a uniform charge density. A relativistic calculation6 yields (C) 27rao2-2P22Pe'l (1 + p) 0= E'Y ZI-2p[r(2p+ 1) J22p(2p+3) (2p+ 1) X(Rex2p-Rgnip)[\l/;a(0) 12-11/;.(0) 12J, (1) ----- ij G. Breit, Rev. Mod. Phys. 30, 507 (1958). (Academic Press Inc., New York, 1963). 789 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 08 Dec 2014 14:29:44790 w. H. FLYGARE AND D. W. HAFEMEISTER "J '" III '3" -.055f- >-8 I- ~ ! iii z '" .., !Y 0 1= -.050 z N 1 0 .. II: ., l-I- U II: '" ;: -.045 ! ..J .., .... I !t oil III J: ., ." Z r~l iii ! 0( W II: U = !::! -.035 " 0 3 II 19 37 5S LiI No.1 KI Rbi C51 .. INCREASINC ELECTRONECATIVITY 1.0 .9 .& .& .7 .95 .75 FIG. 1. The isomer shift at 1291 in the alkali halide lattices. Also plotted is the direction of increasing Ss-electron density in the iodine ion. where E'Y is the 'Y-ray energy, aD the Bohr radius, p= (1-a2Z2)!, a is the fine-structure constant, Z is the nuclear charge, and Rex and Rgnd the nuclear radii for the excited state and the ground state, respectively. For the case of 1291 (Z=53, A=129, R=1.2X10-13 Ai cm=6.06XlO-13 em) Eq. (1) becomes 0= 2.23X 10-23 ( .6R/R)[] lfa(O) ]2_ ] If.(O) ]2J em/sec, (2) where .6R/R= (Rex-Rgnd)/Rgnd. Hafemeister, de Pasquali, and DeWaardl have shown that a positive o corresponds to an absorber s-electron density at the nucleus greater than that of the ZnTe source. Calcula tions based on the Debye model show that the differ ences in the second-order Doppler shift between the various alkali iodides at 80cK is of the order of 0.001 em/sec. Therefore, this contribution to the isomer shift is neglected in our discussion. Relative values of the iodine ion s-electron density at the nucleus, ] If(O) 12, can be obtained by a direct comparison of the isomer-shift data as .6R/ R in Eq. (2) is a nuclear parameter and is constant for the same nucleus in different chemical environments. If the same emitter or source is used to study a series of absorbers, the value of ] If(O).]2 in Eq. (2) will be constant throughout the series and if the values of the root-me an-square radii of the nuclei are assumed to be independent of chemical environment, Eq. (2) may be written as 0= C1Ilfa(0) 12-C2, C1 = 2.23 X 10-23 .6R/ R, C2=2.23XlO-23( .6R/R) ] If.(O) 12• (3) The s electrons are the sole contributors to the electron TABLE 1. Experimental relative isomer shifts at the iodine nucleus. Isomer shift Referenced to LiI Lattice ~ .<l~ LiI -0.038±0.OO25 0 NaI -0. 046±0. 0025 o. 008±0. 005 KI -0. 051±0. 0025 O. 013±0. 005 RbI -0. 043±0. 0025 O. 005±0. 005 CsI -0. 037±0. 0025 O.OOO±O.005 density at the nucleus. Thus, the isomer shift in the 1291-case is primarily dependent on the difference in the s-electron density at the nucleus in the emitter and absorber. The difference in the isomer shifts in two different alkali halide crystals cancels the constant source term [C2 in Eq. (3) J giving .6oab= oa-Ob= C1[] lfa(O) ]2_ Ilfb(O) 12J, (4) where Ilfa(O) 12 and Ilfb(O) 12 are the 1291-s-electron density at the nucleus in lattices a and b, respectively. The values of .6oab referenced to the LiI lattice are given in Table 1. Thus, the experimental results show that the 5s-electron density at 1291-in KI is smaller than the other four lattices with the densities increasing going up or down in the series (see Fig. 1). In order to compute the relative values of .6oab in Eq. (4) we must compute the values for the s-electron density at the 1291 nucleus in these lattices. The wave functions in Eq. (4), lfa and lfb, are the true ground state wavefunctions for the lattices a and b, respectively. The ground-state wavefunction for the alkali halide crystals has been discussed by L6wdin.4 In general, the crystal wavefunction If for a non vibrating crystal can be written as an antisymmetrized product of doubly occupied crystal orbitals, ¢i. If may be written as a single determinant of the form If= (l/n!)l X ] ¢1(1)a(1)¢1(2).B(2)¢2(3)a(3)·· ·¢n/2(n)/3(n) /, (5) where ¢i are the orthonormal doubly occupied spacial orbitals in the crystal and n is the number of electrons in the crystal. The question now arises as to the appropriate choice of the crystal wavefunctions, ¢j. Ideally, ¢j would be obtained from the variational principle and the single determinant function in Eq. (5). The best single determinant wavefunctions are obtained by the self consistent-field procedure by solving the Hartree-Fock equations for the entire crystal, that is (6) where Ho(k) is the Hartree-Fock operator for the crystal, ¢j(k) the doubly occupied orbital in Eq. (5), and ~j is the one-electron orbital energy. Thus, if the This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 08 Dec 2014 14:29:44ISOMER SHIFTS IN ALKALI HALIDE LATTICES 791 Hartree-Fock equations were solved, 1/; would be the best single-determinant crystal wavefunction. The com plete solution to Eq. (6) is presently not feasible, however, and we are forced to consider a tractable alternative. As a first approximation,4 we neglect overlap and consider the cf>i to be the free-ion Hartree Fock atomic orbitals for the ions in the crystal which are known for all ions in the alkali halide lattices. The free-ion wavefunctions violate the orthonormal re quirement of the one-electron orbitals in Eq. (5), however, and must be orthonormalized. Lowdin4 has introduced the symmetric orthogonaliza tion technique which describes the atomic orbitals cf>j in terms of the free-ion atomic orbitals 'Po: as a ='Pj-!'~:::'PaSai+iL'PaSapSpj-+"" (7) ex ex,{3 where 'Pa satisfy the free-ion Hartree-Fock equations and S is the overlap matrix in the free-ion basis summed over all neighbors in the crystal, that is (8) where Oaj is the Kronecker delta. Thus, the cf>i in Eq. (7) are an orthonormal set of atomic orbitals which use the free-ion basis set and acknowledge the nonzero overlap between the neighbors in the crystal. The electron density is easily obtained using the symmetric orthogonalization technique and is per) = 2Lcf>1' (r)cf>I'(r) = 2L(1+S)atl-I'Pa (r)'Ptl(r) I' ail =2L I 'Pa(r) !2-2LSa{3'Pa(r)'Ptl(r) a,/'l =2L 1 'Pa(r) 12[1.0+ L(S".)2] ex • where and the sums are over all n/2 free-ion atomic orbitals in the crystal. As aU Sa. are zero if a and v are on the same center, it is clear that per) increases in the regions near the nucleus. The terms involving Sa. for near neighbors in the last term of Eq. (9) are multiplied by the free-ion product cf>,,(r)cfJ,(r) (a and v are on different centers) which is extremely small when ~o from one of the centers. Thus, the negative term in Eq. (9) is only important when r is large. Thus, the symmetric orthogonalization method leads to in creased electron density at the nucleus and decreased electron density in the regions between the ions in the lattice. As only s electrons have a nonzero density at the nucleus, a good approximation to the s-electron density at the iodine nucleus in the alkali iodide lattices is 11/;,(0) 12=2Lcf>I'(0)cfJl'(0) I' ns =2L 1 'PI'(O) 12[1.0+ 2:(SI") 2], (10) , where the sum over p. is over the s orbitals in 1-and the sum over v is over all neighboring ions. It is shown in the following papers that the overlap of an inner shell on one ion with the outer shell on another ion is at least on order of magnitude less than the outer-shell outer-shell overlaps. Thus, only the 5s orbital in Eq. (10) for 1-will participate appreciably in the overlap deformation. Using this approximation, Eq. (10) becomes 11/;1-(0) \2=2! 'P18(0) 12+2 ! 'P2.(0) [2+2\ 'P3.(0) 12 +21 'P4s(0) \2+21 'Po.(O) 12[1.0+ L(S.8.)2] (11) with only the last term varying from lattice to lattice. The inner-shell s-electron densities which are approxi mately constant from lattice to lattice cancel in the difference equation for the relative isomer shifts in Eq. (4). Thus, we are primarily interested in the over lap deformation of the 5s orbitals. Equation (4) can be rewritten to give Mab=oa-ob=Cd21 'P5.(0) a 12[1.0+ L(S5 .. )2] -21 'P5s(Oh 12[1.0+ 2:(S581')2]} , (12) I' where the first term is for the a lattice and the second term the b lattice. The values for the sums of overlaps over the neighboring atoms are: NaCI-type lattice nearest neighbors 2:( S .. )2= 6[ (S8" )2+ (S.pu)2], next-nearest neighbors CsCl-type lattice (13) nearest neighbors L(S .. )2=8[( S .. ,) 2+ (S.pu)2], next-nearest L (S •• )2= 6[ (S.s,)2+ (S.pu)2]. neighbors The appropriate values for the sums of overlap integrals are easily determined for nearest and next-nearest 6 D. W. Haferneister and W. H. Flygare, J. Chern. Phys. 43, 795 (1965). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 08 Dec 2014 14:29:44792 W. H. FLYGARE AND D. W. HAFEMEISTER TABLE II. The results of the overlap deformation on the I free-ion functions [see Eqs. (11) and (12)]. Sum of nearest and Nearest Next-nearest next-nearest Lattice neighbors neighbors neighbors LiI 0.000803 0.05743 0.05923 NaI 0.001311 0.03010 0.03141 KI 0.005601 0.01224 0.01784 RbI 0.006348 0.00812 0.01447 CsI 0.008948 0.0154 0.02434 neighbors from the tables in the following paper.6 The tabulated values of A.llM :and A.llll [Eqs. (14) and (16) and Tables II and IV in the following paper J are used here. The results are shown in Table II. The internuclear distances and other necessary data for the alkali iodide lattices is given in Table III. It is clear that the differences in the overlap terms in Table II follow qualitatively the relative isomer shifts as given in Eq. (12). Before proceeding, however, we must estimate the effects on the Ss-electron density at the 1-nucleus due to the attractive electrostatic interactions. The electro static perturbation can be described as follows c/>j= [1-2::CBP)2J!~l+ LBP~jP, (14) P P where c/>j is the jth atomic orbital in the deformed ion, ~jO is the free-ion orbital, ~jP is the pth excited-state atomic orbital, and (BP)2 is the probability of existence of the excited atomic orbital. The (BP)2 coefficients arise from the electrostatic perturbation. The overlap and electrostatic effects cannot be exactly treated in an independent manner as proposed here. The errors introduced, however, will be very small for small deformations. The more exact method would be to perform an iterative computation where first the free ions in the crystal would be corrected for orthonormality due to the overlap effect and then the electrostatic where the sums are over the excited states of the halide ions p and the excited states of the alkali ions k. EHO(EMO) and EHP(EMk) are the ground-and excited state energies of the halide (alkali) ions respectively. By introducing the classical polarizability and the average excitation energy in a straightforward manner Margenau7 has shown that the coefficient in Eq. (18) reduces to aH and aM are the halogen- and metal-ion dipole perturbations could be computed using these ortho normal nonoverlapping functions. The electrostatic deformation would require renormalization and thus the process would have to be carried to a limiting set of coefficients. This method converges very fast, how ever, due to the small deformations which justifies treating the deformations independently. The electrostatic or multipole-multipole interaction between two ions at Centers A and B can be expressed as follows for two nonoverlapping charge distributions: +L (_l)N+IMI (L+N)! (1 )N+L+l V= E~OMLL (L+ 1M j) !(N+ 1 M I)! fAB x TML(i)AUMN(jh, (15) where the TML(i) A and UMN(jh* represent the multi pole moments at Center A and Center B, respectively, and fAB is the internuclear distance: TML(i) A = Le,;riL PML(cos8 i)A exp(iMc/>i) , i U MN(jh*= Lei,f PMN(cos8j)B exp(iMc/>;). (16) j The permanent monopole-induced-dipole interactions go to zero due to the octahedral site symmetry of the alkali halide lattices. The first nonzero terms affecting the halogen ions in the lattice will be the alkali-in duced-dipole-halogen-induced-dipole. The next-near est-neighbor terms will be the halogen-induced-dipole halogen-induced-dipole. As an example, consider the nearest-neighbor interaction which is [see Eq. (15) and (16)J V(L= 1, N = 1) = [e2/(fHM)3J(XHXM+YHYM-2zHZM) , (17) where H is the halogen ion and M is the metal ion. The operator in Eq. (17) projects out excited ionic states in the interacting ions as shown by Margenau.7 As an example, consider the coefficient in Eq. (14), BHMP, for an atomic orbital in the halogen ion interacting with an atomic orbital on the metal ion. Margenau7 has shown the coefficient to be equal to polarizabilities respectively and EH and ~M are the average halogen- and metal-ion excitation energies. fHM is the internuclear distance. The coefficient giving the excitation of electrons from the metal ion due to the metal-induced-dipole-halogen-induced-dipole, BMH, is identical to BHM in Eq. (19). The square of these coefficients gives BHM2= BMH2= 3aHaMEHEM/[2(fHM) 6(EH+EM)2]. (20) 7 H. Margenau, Rev. Mod. Phys. 11, 1 (1939). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 08 Dec 2014 14:29:44ISOMER SHIFTS IN ALKALI HALIDE LATTICES 793 TABLE III. Data used to compute the overlap and electrostatic deformations on the free-ion wavefunctions." NN NNN a. (halogen) lXb (alkali) nearest next-nearest Lattice rAB rAA Xl()24 em' E.XI01• erg Xl()24 em' EbXI012 erg neighbors neighbors LiI 3.00 4.27 6.43 13.6 0.03 180.3 6 12 NaI 3.237 4.57 6.43 13.6 0.41 96.5 6 12 KI 3.533 4.99 6.43 13.6 1. 33 68.6 6 12 RbI 3.671 5.18 6.43 13.6 1. 98 52.8 6 12 CsI 3.956 4.56 6.43 13.6 3.34 48.1 8 6 • The equilibrium internuclear spacings are taken from D. Cubicciotti, J. Chern. Phys. 31, 1646 (1959). trons in the metal ion due to other metal-ion interac tions BMM and the excitation of electrons in the halogen ion due to other halogen-ion interactions BHR are easily obtained from an equation similar to Eq. (19). The results are BMM2= 3aM2/8(rMM) 6, BRR2=3aR2/8(TRH)6. (21) (22) TMM is the metal-ion-metal-ion next-nearest-neighbor distance and THR is the halogen-ion-halogen-ion next nearest-neighbor distance. The values of rHM, rMM= rHR are listed in Table III for several systems which are studied here. The remaining parameters in Eqs. (20), (21), and (22) must be chosen with great care. Tessman, Kahn, and Shockley8 have given experi mental values for the polarizabilities for a number of ions in the lattices. Their values for the polarizabilities are used here and are listed in Table III. Sternheimer9 has shown that the primary component of the experi mental polarizability is the dipole polarizability which is produced by an excitation of n~(n+1)s electrons (5~s in the I-case) and, therefore, the excitation of s electrons is negligible. Thus, ER in Eq. (34) ought to correspond closely to the 5~6s excitation in the iodine ion (or other halogen ions). As this transition is allowed optically, an approximate value can be ob tained from the lowest energy absorbtion in the crystal. The values of the first absorbtions in the alkali halide crystals are summarized by Mayer.IO These values will be used for the average excitation energy of the halogen ions in Eqs. (19)-(22) and are also listed in Table III. As ratios of the energies appear in the perturbation coefficients, the choice of excitation energies is not critical. The values of BRM2 and BRH2 from Eqs. (19) (22) for single p orbitals on the various atoms in the alkali-halide lattices are given in Table IV. The next important term in the multipole-multipole expansion in Eq. (15) is the point-charge-induced quadrupole term. A development very similar to that given above for the induced dipole-induced dipole co- 8 J. R. Tessman, A. H. Kahn, and W. Shockley, Phys. Rev. 92, 890 (1953). 9 R. M. Sternheimer, Phys. Rev. 96, 951 (1954); 107, 1565 (1957); T. P. Das and R. Bersohn, ibid. 102, 733 (1956). 10 J. E. Mayer, J. Chern. Phys. 1, 270 (1933). efficients yields CMH2= e4/ (rMH) 12[ (aq) M/EMJ2, CHM2= & / (rMH)l2[ (aq) H/ EH]2, CH~= e2/ (rHH) 12[ (aq) H/ EH]2, CMM2= e2/ (rml) 12[aq) M/EM]2. (23) (24) (25) (26) (aq) M and (aq) H are the quadrupole polarizabili ties of the metal and halogen ions respectively. CMH2 is the excitation of the metal ion due to the nearest-neighbor halogen point charge and CHM2 is the excitation of the halogen-ion electrons due to the metal-ion point charge. CHH2 and CMM2 are the next-nearest-neighbor terms. The quadrupole polarizabilities are related to the Sternheimer9 antishielding factors, 'Yro' and have been given.with considerable accuracy for some of the lighter atoms. Calculations of CHM2 and CMH2 for some of the lighter atoms indicated these terms were considerably smaller than the induced-dipole-induced-dipole effects. We therefore excluded these effects from our analysis. Interactions in the lattice involving the nuclear electric quadrupole moment (quadrupole relaxation phenomena in nuclear magnetic resonance) will be very sensitive to the quadrupole polarizability terms, however, due to the Sternheimer antishielding factor. The isomer shift which is interpreted here, does not involve the nuclear electric quadrupole moment and is, therefore, relatively insensitive to the point-charge-induced quadrupole ~effect. The resultant electronic excitations TABLE IV. The calculated values of (Bp), in Eq. (14) for the iodine ion in the alkali iodides.' H Lattice BHM'XI03 BRH"XI03 ~BHM2 ~ BHR" 3X~ NN NNN P LiI 0.03 2.60 0.00018 0.031 0.093 NaI 0.37 1.72 0.0022 0.020 0.066 KI 0.92 1.01 0.0055 0.012 0.054 RbI 1.22 0.82 0.0073 0.0098 0.051 CsI 1.43 1. 74 0.0114 0.010 0.066 • The (BRM)' are the nearest-neighbor interactions and (BRR)' are the next nearest-neighbor interactions. The primary excitation of electrons arises in the 5p sheil of 1-. The results are summed over all nearest neighbors (NN) and next-nearest neighbors (NNN). The total number of Sp electrons excited is given in tbe column headed by 3X~pR. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 08 Dec 2014 14:29:44794 W. H. FLYGARE AND D. W. HAFEMEISTER TABLE V. Calculated values of C1 in Eqs. (3) and (4). Calculated from Eq. (12) Experimental ( ~Oab ) ~O referenced C, 1 "'58 (0) I' Lattice to LiI C, LiI 0 0 NaI 0.008 0.054 0.14X1Q-26 KI 0.013 0.081 0.15X1Q-26 RbI 0.005 0.087 0.05XIQ-'6 CsI 0.000 0.068 due to the electrostatic effect of several initially closed shell ions in the alkali iodides are listed in Table IV. The result of the electrostatic interaction is to deplete primarily the Sp-shell electrons in the 1-ion. The number of electrons originally in the Sp closed shell which have been lost to excited states are given in Table IV. As the Sp excitation is primarily to the 6s shell, the direct effect on the s-electron density at the nucleus will be negligible as the 6s-electron density at the nucleus will be much smaller than the Ss density. As a result of the depletion of Sp electrons, however, the nucleus will be less shielded and the Ss electrons will have a greater probability of being at the nucleus. Thus, the electrostatic as well as the overlap effects both increase the s-electron density at the nucleus in the same relative order. In order to understand the order-of-magnitude im portance of the electrostatic effect in the determination of the Ss-electron density at the nucleus we use the Fermi-Sergre formula for each outer-shell Ss electron 1cf>68(0) 12=(ZzY/l"a02Pn3)[1-(d~n/dn)]. (27) Z is the nuclear charge, z is the effective nuclear charge seen by the electron contributing to 1 cf>S8(0) 12, ao is the Bohr radius, Pn is the effective quantum defect of the nth state, and ~n=n-Pn is the quantum defect of the nth state. We wish to determine whether 1 cf>68(0) 12 in Eq. (27) changes in the various alkali iodide lattices due to the small changes in p-electron density shown in Table IV. Shirleyll has discussed the usual interpretation of the effective nuclear charge for an outer electron which is Z= 1 +m, where m is the charge on the atom or ion. This interpretation leads to no change in 1 cf>58(0) 12 in the alkali iodide lattices as the electrostatic effects did not perturb the Ss electrons. A more liberal interpreta tion of z is to consider the partial shielding of each electron in the outer shells instead of using the total shielding (z = 1 + m) as suggested in the original work 11 D. A. Shirley, Rev. Mod. Phys. 36, 339 (1964). on Eq. (27). Thus, we will use Slater's method12 of ob taining the effective nuclear changes in Eq. (27). The necessary information regarding the number of p electrons in the outer shell of the iodine ion is obtained from the coefficients in Table IV where total p-electron excitation is given. The resultant changes in the values of 1 cf>S8(0) 12 in the alkali iodides using Eq. (27) and Slater's effective charges is negligible compared to the direct overlap effect. We have also calculated the effect of the shielding of the iodine Ss electrons by the Sp popUlation with Mayer's Hartree-Fock functions13 for 1-and 1°. This method agrees within 30% with the above method. Thus, we feel justified in using the overlap values in Table II to compute the relative isomer shifts. It is clear from Table IV and Eqs. (27) and (4) that the small electrostatic effect does predict the same trend as the overlap effect. That is, the larger the number of Sp electrons excited the larger the Ss electron density at the nucleus. As remarked above, the overlap effect in Table II does give the correct dependence on the relative isomer shifts. In order to compute C1 in Eq. (12) we must know the free-ion value of I (/'68(0) 12 for the iodine ion. 1 (/'68(0) 12=1.1X1026 cm-3 is used here.I3 Combining this information with Table I gives an average value of C1=0.12XlO-26 (see Table V). The resultant value of the nuclear parameter is ~R/R=0.SXlO-4. CONCLUSION The main conclusions of this work are: (1) The isomer shift at 1291-in the alkali halide lattices is primarily caused by the overlap deforma tions of the free-ion wavefunctions. (2) Electrostatic perturbations between neighboring atoms do not perturb directly the Ss electrons in 1291-. Thus, the electrostatic effect has a relatively small effect on the isomer shift. (3) Next-nearest neighbors as well as nearest neigh bors must be included in describing the isomer shift in 1291-. (4) The value of the change in the nuclear radius is ~R/ R = 0.5 X 10-4, for the 26.8-ke V state of 1291 and is not what is expected on the basis of nuclear shell theory.! ACKNOWLEDGMENTS We are grateful for the partial support in this re search by the National Science Foundation, Office of Naval Research, and the Materials Research Labora tory at the University of Illinois. 12 See, for instance H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry (John Wiley & Sons, Inc., New York, 1944). '3 D. F. Mayers (private communication). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 08 Dec 2014 14:29:44
1.1702773.pdf	On Tunneling Equations of Holm and Stratton C. K. Chow Citation: Journal of Applied Physics 34, 2490 (1963); doi: 10.1063/1.1702773 View online: http://dx.doi.org/10.1063/1.1702773 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On a dissipative form of Camassa–Holm equation J. Math. Phys. 51, 092704 (2010); 10.1063/1.3451108 Finite propagation speed for the Camassa–Holm equation J. Math. Phys. 46, 023506 (2005); 10.1063/1.1845603 On the integrable perturbations of the Camassa–Holm equation J. Math. Phys. 41, 3160 (2000); 10.1063/1.533298 Stratton Award Phys. Today 18, 79 (1965); 10.1063/1.3047217 Stratton Awards Phys. Today 15, 60 (1962); 10.1063/1.3057916 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.120.209 On: Wed, 03 Dec 2014 03:59:232490 COMMUNICATIONS FIG. 2. Ma$s spectrum showing the sputtered copper ion peaks and part of th~ir high energy tails. Argon ion energy 1000 eV. Mag netic scanning used with Nier source accelerating potential of 1200 V. Figure 1 is a mass spectrum of ionized sputtered neutrals, ob tained with the synchronous source-detector, for a copper target bombarded with 2-keV argon ions. The prominent peaks in the spectrum are the copper isotopes. The remainder of the peaks are primarily the isotopes of iron, nickel, and chromium sputtered from the Inconel target holder. Some small background peaks arising from adsorbed gases sputtered from the target surface are also present and tend to make the isotopic ratios for the target and target holder materials incorrect. On the basis of peak heights, the relative concentration of nickel and chromium in the Inconel is about 1/3, whereas the actual value is about 10/1. In this case, the discrepancy cannot be explained by the presence of small back ground peaks. Nor can it be explained by the relative sputtering yields measured for the pure metals, since at the bombarding energy used the sputtering yield is greater for nickel than for chromium. There is a possibility that the relative sputtering yield for these elements is dependent upon the matrix in which they are located. The width of the ionized sputtered neutral peaks in }'ig. 1 indicates that the ionized sputtered neutrals arriving at the collector had a small energy spread, probably less than 1 eV. This does not imply that the energy spread of the neutrals is this small,S for the probability of ionizing them is proportional to (E)-I, where E is their initial energy, Thus the more energetic neutrals are strongly discriminated against. For sputtered ions no appreciable energy discrimination is introduced by the instrument, and an estimate of their energy spread may be made by placing a retarding potential on the target or by measuring the width of recorded peaks. Using both of these methods it has been determined that for 2-keV argon ions bombarding a Cu target over 1 % of the sputtered copper ions have energies in excess of 350 eV. Similar values for energy spread were obtained for copper and beryllium ions sputtered from a copper-beryllium target. These results for the energy spread I I I I I I h II I J I ~~l~J~ 63-BOc:l<grounJ! IL6s-cu ~I..I\..JL 53-Cu L.-.... 55'Background FIG. 3, Mass spec trum showing sputtered copper ion peaks along with background peaks originating from re sidual gases ionized in the Nier source. Experi mental conditions same as for Fig. 2 except with electron beam on in Nier source. agree only with those of one other author,S Others have observed spreads of much lower value.1.2,4 A spectrum of sputtered ions is shown in Fig. 2. Figure 3 shows the modified spectrum obtained with the electron beam in the Nier source switched on. Superimposed on the sputtered copper ion peaks in Fig. 3 are a number of narrow background peaks which originate from residual gases ionized in t.he Kier source, A measurement of the separation between the background peaks and the copper peaks at M /e values 63 and 65 enables one to de termine the most probable initial energy of the sputtered ions. In this case, the most probable energy was found to be 9 eV, The authors acknowledge helpful discussions with L. F. Herzog of Nuclide Corporation and A. L. Southern of the Solid State Division, Oak Ridge National Laboratory. The authors are also indebted to T. J. Eskew of Nuclide Corporation for the design of many of the electronic circuits. ' This work was carried out on an instrument developed by Nuclide Analysis Associates for the Oak Ridge ]l;ational Laboratory, operated by Union Carbide Nuclear Company for the G. S. Atomic Energy Commission. l R. E. Honig, J. Appl. Pbys. 29, 549 (1958). 2 R. C. Bradley, J. Appl. Phys. 3(), 1 (1959). • Henry K Stanton, J. Appl. Phy •. 31, 678 (1960). 4 F. A. White, J. C. Sheffield, and F. M. Rourke, J. Appl. Phys. 33, 2915 (1962). , A. J. Smith, L. A. Cam bey, D. J. Marshall, and E. J. Michael (to be published). • O. Almen and K. O. Nielsen, Nucl. Instr. 1, 302 (1957). 7 A. O. C. Nier, Rev. Sci. Instr. 18,398 (1947). • R. V. Stuart and G. K. Wehner, 11M2 Vacuum Symposium TransactiO/IS (The Macmillan Company, New York, 19(2), p. 160; K. KOllitski and H-, 1';, Stier, Z. Naturfsch. 17a, 346 (1962). On Tunneling Equations of Holm and Stratton C. K. CHOW Burroughs Corporation, Burroughs Labora/ories, Paoli, Penttsylvania (Received !O December 1962; in final form 15 April 19(3) By extending previous theory, Stratton,' giv~s a new equation for tunnel current through a thin insulating film. Concerning Holm's equation,' Stratton states that "there are further approxi mations [in Holm's calculation] whose accuracy is difficult to assess." It is generally difficult to evaluate the accuracy of ap proximations. This note attempts to compare the equations of Holm and Stratton by examining the approximations underlying their derivations, and by pointing out some similarities and dis crepancies in their results, An insight into the accuracies of their approximations may thus be gained. Of course, the comparison can only be made in the domain where both equations are appli cable. Holm's basic equation is intended primarily for the case of trapezoidal barrier shape and zero temperature; Stratton's equa tion, on the other hand, encloses the broader class of arbitrary barrier shapes and temperatures. Both Stratton and Holm define the basic integral for current density J as (1) where In is mass of electron, • is charge of electron, It is Planck's constant, E. is the energy associated with the x-directed mo mentum (x being in the direction of tunneling), p(E.) is the supply function derived from the Fermi-Dirac distribution, and D(E.) is the tunneling probability. Both also use the WKB approxima tion for D(E.), D(Ex)=CX P{ -[4rr(2m)']/h f' (<p(x)+>J-E.)idx} (2) where ",(x) is the barrier potential, measured from the Fermi level >J of the positively biased electrode, and x, and Xz are the c1assicaJ turning points. To perform the integrations, Holm and Stratton use different approximations. Holm replaces I"(x) in (2) by a constant ip, which [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.120.209 On: Wed, 03 Dec 2014 03:59:23COMMUNICATIONS 2491 he chooses to he the average value of ",(x); namely, (3) As a consequence, the integrals of both (1) and (2) can be exactly evaluated as finite sums of elementary functions. This results in the basic equation of Holm, although Holm retains only the dominating term J =3e(2m)'/Ash'{(",o-!eV) exp[ -A (",o-!eV)]! -(",o+!eV) exp[ -A (",o+!eV)]!}, (4) where A = 411"(2m)!/It, "'0 is the harrier height of the rectangular barrier at zero applied voltage V, and S=X,-XI. Note that an additional term originally included by Holm, is omitted here. This term is negligible for low applied voltages and is cancelled out if the algebraic manipulation is carried a little further.' In other words, Holm, in his derivation, replaces the true barrier shape for a given applied voltage by a rectangular shape of the same average height. Although Holm considers only the trapezoidal barrier, his method of rectangular shape approxima tion could be used for an aribtrary shape. The extension has been carried out by Simmons} This approximation would be reasonable if ",(x) is nearly constant; or, more precisely, the approximation is good if is small with respect to unity. This is certainly valid for cases of trapezoidal potential barrier at low voltage. On the other hand, Stratton expands 10gD(E x) in a Taylor series, and then retains only the linear term in Ex, such that where 10gD(D x) == -[bl+CI(1/-E x)], bl=[h(2m)I]/h (X21 [",ex)J!dx, 1:1'11 CI = [h(2m)!]/h i:'1 dx/[",(x)]!. (5) (6) (7) The points Xl1 and X21 are values of x for which ",(x) =0, with respect to the Fermi level 1/. Equations (5), (6), and (7) are, re spectively, (8), (9), and (10) of Stratton. l Stratton then replaces the limits of integration in (1), above, by ± 00, and follows Murphy and Good4 in carrying out the integration to obtain J = (41I"me/cI'h') exp( -bt)[l-exp( -CI V)]. (8) This is the limiting form of Stratton'sl (14) for zero temperature. It is to be noted that the constants, if; of Holm, and bl and (I of Stratton, are all functions of applied voltage V. The justification for (5) is that the tunnel current is pre dominantly due to electrons whose Ex values are close to the Fermi level. With this in mind, Stratton's method of evaluating the integral can be interpreted as: (a) Replacing the true barrier potential ",(x) by a rectangular barrier whose height "'r is so determined that -{ (X21 J <{'r -(X2l -Xll)-I ) XlI ['" (x) ]ldx J'; (9) that is, <{'r is the square of the average value of [<('(x)]!. (b) Substituting (9) for ",(x) in (2), and approximating the radical in (2) by the first term of its binomial expansion; namely, [",(xH1/- Ex]l== ("'r+1/- Ex)l== (<('r)!+!('1-Ex)/("'r)'. (10) (c) Finally, carrying out the integrations of (1) and (2). This procedure gives exactly the same expression as (8) and the same coefficient for bl, but a somewhat different coefficient for CI. Since the hi term dominates, this interpretation is essentially valid. It can be inferred that the approximations used by Holm and Stratton are basically the same, although this is not obvious at first glance. Holm approximates only the barrier shape. Stratton approximates, in addition, the dominating range of energy of the tunneling electrons, and his approximation of barrier shape is generally superior to Holm's. Because of the additional approximation, Stratton's equations, for some special cases, are not as accurate as Holm'S, although the numerical discrepancy, from an experimental viewpoint, is not too significant. From (8), Stratton derives the low-voltage current density for a symmetric barrier as J = (811"me/clo'h') exp( -blo) exp( -b12V') sinh(!clOeV), (11) where the b's and c's are coefficients of bl and CI, as power series ill V; namely, h=blo-b u V +bl, V' CI =CIO-CIJ V +C12 V'. (12) The factor e in the hyperbolic sine term is missing in the original equation of Stratton. The dimension of V is his (6) seems to be taken as electron-volts, while, in his (24), the dimension is taken as volts, A similar expression can be derived from (4), above for low voltages; namely, J _ 3''''0 [-411"S(2m",o)!] [1I"S€'(2m)lV'] -211"s'h exp h exp 8h<{,o' . h[7rs.(2m)W] Xsm h(",o)' ' (13) where 'Po is the initial height of the trapezoidal barrier and s is the separation between electrodes. Since a trapezoidal barrier is assumed, XI=O and X2=S. Equations (11) and (13) are of exactly the same functional form. It is of interest to compare the numerical coefficients. For a trapezoidal barrier, the coefficients in (11) become blo = 4".s (2m 'Po)l/h, bJ2= -".e's(2m)!/6h'Pol, CIO= 27rs(2m)!/h( 'Po)l. (14) All coefficients in ell) are the same as the corresponding coeffi cients in (13), except that the numerical factor! is not present in (11), and the 1 factor in the second exponent of (13) is, in (11), l. Thus, (13) gives a zero bias conductance! times as large as (11). As V tends to zero, (13) gives u=J /V = 3.' (2m 'Po)!/2sh' exp[ -hs(2m'Po)l/h], (15) while (11) gives u= (.'(2m"'0)'/sh' exp[ -hs(2m"'o)i/h], (16) which is the same as the result of Sommerfeld and Bethe. 6 For small voltages, the choice between (3) and (9) for determin ing barrier height is essentially immaterial; any numerical dis crepancy is due to the further approximation introduced by (10). Therefore, (13) may be expected to give a somewhat more accu rate approximation for low voltages, although (11) seems prefera ble for more general cases. By exploiting the separate merits of Stratton's and Holm's approximations, the basic integral can be evaluated as follows: (a) Replace any arbitrary barrier potential 'P(x) by a rectangu lar barrier of height "', and width dX, with the values (17) and 'Pr= {~Xi:'1 ["'(x)]ldX}'. (9) (b) Using (17) and (9), integrals of both (1) and (2) can be evaluated to be J =3E(2m)!/A' dxh'{ 'Pr exp[ -A'('Pr)!] -('Pr+'V) exp[-A'('Pr+,V)']l, (18) where A'=4,..dx(2m)!/h. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.120.209 On: Wed, 03 Dec 2014 03:59:232492 COMMUNICATIONS Equation (18) has the same appearance as the result of Simmons,3 the difference being that Simmons uses", as defined in (3) above, instead of 'Pr as given by (9). JR. Stratton, J. Phys. Chern. Solids 23,1177 (1962). , R. Holm, J. App!. Phys. 22, 569 (1951). 3]. G. Simmons, HA Generalized Formula for the Electric Tunnel Effect Between Similar Electrodes Separated by a Thin Insulating Film," J. App!. Phys. (to be published) . • E. L. Murphy and R. H. Good, Jr., Phys. Rev. 102, 1464 (1956). • A. Sommerfeld and H. Bethe, Handbuch der Physik von Geiger und Scheel (Julius Springer-Verlag, Berlin, 1933). Low-Temperature Internal Friction in Nylon-4 K. D. LAWSON Bennington College, Bennington, Vermont A~D J. A. SAUER AND A. E. \\'OODWARD Physics Department, The Pennsylvania State University University Park, Pennsylvania (Received 25 March 1963) IT is well established that many linear polymers with flexible chain segments-such as polyethylene, some polyesters, and various polyamides-show a maximum of internal friction in the range 120° to 1700K with dependence upon the test frequency. This transition, generally referred to as the 'Y transition, is believed due to the reorientational motions of a relatively small number of chain segments. The minimum number of flexible chain units involved, or needed for this relaxation process, has not been definitely deter mined. From measurements on various polymers,' from studies of the restrictive effects of radiation-induced crosslinking in poly ethylene,' and from observations made on various ethylene copoly mers,3 it is conjectured that the minimum number involved is from 3 to 5, A materia! which should enable this minimum number to be more precisely defined is po!ypyrrolidine (ny!on-4) because its structure [-(CH2),-CO-NH- ] provides just 3 flexible CH, units situated between rather rigid hydrogen-bonded amide groups. Internal friction measurements were made over the temperature range from 100° to 3000K by means of a torsional pendulum ap paratus. The polymer specimen, 9.50 cm long, 1.565 cm wide and 0.00686 cm thick, was cut from a larger film prepared by casting from a 2% solution of nylon-4 in formic acid. Before being placed in the testing apparatus the specimen was annealed in a vacuum oven at 135°C for 24 h. During a test run, the specimen was kept E "' U IIJ 0 8 .::! <II <II 0 ...J ...J « u Z « %: u IIJ :2 ~-----------=.O .006 .004 ,002 100 200 TEMPERATURE OK S! 2 2.0 " ... ~ 1.5 -:; 0.4 300 ,2 !. ... § III :3 ~ !a m FIG. 1. Mechanical10ss and shear modulus vs absolute temperature in oylon-4. under dry nitrogen at 13 to 50 }.I, and the average heating rate maintained at about O.4°C/min. The internal friction of the speci men was obtained from optical measurements of the decay in amplitude after a small initial angular displacement. For the displacements used, the observed losses were independent of amplitude. The dynamic shear modulus was determined from the specimen size and the frequency of oscillation. The data are presented in Fig. 1 in terms of the mechanical loss and the shear modulus vs the absolute temperature. There is an internal friction maximum at about 127°K (0.31 cps), accom panied by a modulus dispersion. A second-loss peak, of much lower strength, is centered at about 213°K (0.295 cps). These results give definite indication that the low-temperature 'Y-relaxation process can occur in linear polymers with as few as 3 consecutive methylene groups present in the main polymer chain. A corresponding low-temperature relaxation can also be produced by side-chain motions but in this case only 2 units seem to be re quired since the transition has been observed in polybutene.4 The higher-temperature internal friction maximum near 213°K is analogous to the /3-relaxation process that has been observed in other polyamides. It is believed to arise from motion of amide and adjacent methylene units in the amorphous regions6 or from motion of absorbed water molecules,6 and hence should not depend primarily on the number of consecutive methylene groups. This peak has also been observed by Kawaguchi' for a frequency of 160 cps at about 230°K. We thank Dr. F. A. Bovey for supplying the polypryrrolidine. Our gratitude is given, also, to the NSF for a summer fellowship (KDL) and to the John Simon Guggenheim Memorial Foundation for a fellowship (AEW) in which tenure this work was completed. * Work supported in part by AEC Contract AT (30-1)-1858 and by NSF Grants G-14143 and GP-685. I A. H. Willbourn, Trans. Faraday Soc. 54,717 (1958). 'N. Fuschillo and J. A. Sauer, J. Ap])!. Phys. 28, 1073 (1957). 3 F. P. Reding, J. A. Faucher, and R. D. Whitman, J. Polymer Sci. 57, 483 (1962). 'A. E. Woodward, J. A. Sauer, and R. A. Wall, J. Chern. Phys. 30, 854 (1959). 'A. E. Woodward, J. M. Crissman, and J. A. Sauer, J. Polymer Sci. 44, 23 (1960). 6 K. H. IIlers, Makromo!' Chern. 38, 168 (1960). 7 T. Kawaguchi, J. App!. Polymer Sci. 2, 56 (1959). Interference between the Infrared Beams from Opposite Ends of a GaAs Laser A. E. MICHEL AND E. J. WALKER IBM Thomas J. Watson Research Center, Yorktown Heights, New York (Received 18 February 1963) THE interference between light beams from a lasing GaAs diode demonstrates in a simple and direct manner that the Iigth beams coming from opposite ends of a GaAs laser are spatially coherent and bear a fixed-phase relationship with each other. The interference fringe positions from a number of diodes agree well with those expected from theory. The interference of beams from opposite ends of a laser was first carried out by Kisliuk and Walsh! for a ruby laser. The smaller physical size and larger beam spread of the GaAs laser made it possible to use a simpler experimental arrangement. The sche matic diagram, Fig. 1 shows that only a single mirror M was needed to superpose portions of the two beams. The diodes were rectangular parallelepipeds approximately 100 }.I in cross section and 500 to 800 }.I long with optically flat ends and roughened sides. 2 A set of concentric interference rings centered about the shadow of the diode is predicted; the ring positions for small angles, reduce to fln2=2A[1/R+l/(2D+L)]n+C, (1) where fin is the angular position of the nth maximum, A is the wavelength and the distances R, D, and L are indicated in Fig. 1. Figure 2 is an enlargement of the interference pattern for R=5 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.120.209 On: Wed, 03 Dec 2014 03:59:23
1.1713827.pdf	Energy Dependence of Proton Irradiation Damage in Silicon W. Rosenzweig, F. M. Smits, and W. L. Brown Citation: Journal of Applied Physics 35, 2707 (1964); doi: 10.1063/1.1713827 View online: http://dx.doi.org/10.1063/1.1713827 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of proton energy on damage generation in irradiated silicon J. Appl. Phys. 107, 084903 (2010); 10.1063/1.3371714 Mediumenergy proton irradiation of fullerene films: polymerization, damage and magnetism AIP Conf. Proc. 786, 61 (2005); 10.1063/1.2103821 Quenching of benzene fluorescence in pulsed proton irradiation: Dependence on proton energy J. Chem. Phys. 67, 2793 (1977); 10.1063/1.435196 Energy Dependence of Neutron Damage in Silicon J. Appl. Phys. 38, 204 (1967); 10.1063/1.1708956 TEMPERATURE AND ILLUMINATION DEPENDENCE OF IRRADIATION DAMAGE IN SILICON Appl. Phys. Lett. 2, 235 (1963); 10.1063/1.1753750 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.51.150.161 On: Sun, 30 Nov 2014 22:33:39ACOUSTOELECTRIC EFFECT 2707 the velocity-dependent forces. Using (AS) we find Combining these results, we find f F. aj. ! Re v-·--d3v m* av NoeEdc-jucx Ho/e -! Re[p.(E.+qqCu/iew)+j.x H./e] and = -(l/m)! Re[p8(E 8+qqCu/iew) +j.xH./e] (A9) = (m/eT){jdc+! ReNseu(m/1Il)}. (All) J' Fele iJIde V_·~-d3V m* av This is the same as (1.5) when p. and H. are expressed in terms of i. and E •. White;' used a somewhat similar method to derive an expression for the acoustoelectric current in the absence of a magnetic field. However, he considered a one-dimensional model, which has limited validity. =-(l/m)[-NoeEdc+jdcxHo/e]' (AlO) JOURNAL OF APPLIED PHYSICS VOLUME 35. NUMBER 9 SEPTEMBER 1964 Energy Dependence of Proton Irradiation Damage in Silicon W. ROSENZWEIG, F. M. SMITS, AND W. L. BROWN Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey (Received 6 March 1964) The energy dependence of radiation damage in silicon for proton energies in the range 1.35 to 130 MeV has been measured by observing the degradation of the bulk minority carrier diffusion length in silicon solar cells. Variahility in proton flux determination at four different accelerators was minimized by employing pre bombarded solar cells with known minority carrier diffusion lengths as calibrated solid-state ionization l!hambers. Where beam intensity measurement comparisons with Faraday cups could be made, agreement to better than 5% was obtained. The quantity characterizing the damage rate is the rate of change of the inverse square diffusion length with flux K -=d(1/D)/dif>. The 1-f)-cm p-type silicon degraded, on the average at a rate six times less rapid than 1-Q-cm n type, independent of energy. Room temperature annealing gave 30% to 50% decrease in K whenever the diffusion length was measured during and after irradiation. The energy variation of K agrees with the variation predicted by Rutherford scattering below 8 MeV, but decreases less rapidly at higher energies. The measured diffusion lengths increased with excess carrier density n from 2% per decade at n = 109cm-' to 20% per decade at n = 101'cm-'. The reported results, obtained at low excess carrier density, can be used to predict solar cell degradation under conditions of outer space illumination if the appropriate excess carrier density is used. Failure to take into account the diffusion length variation will result in an underestimate of the solar cell output of less than 7%. INTRODUCTION THE energy dependence of the rate of lifetime degradation in l-Q·cm p-type silicon for proton energies in the range from 1.35 to 130 MeV has been measured by observing the degradation of the bulk minority carrier diffusion length in silicon solar cells. Such results are important in assessing the damage to solar cells on satellites operating in the Van Allen belt. As expected, for the energy range covered, the lifetime degradation per proton decreases monotonically with increasing proton energy. However, significant devia tions of the energy dependence from the predictions of a simple theoretical model were observed. EXPERIMENTAL PROCEDURE Changes in diffusion length can be observed in a con venient way by means of a silicon solar cell. This stems * Present address: Sandia Corporation, Albuquerque, New Mexico. from the fact that the shallow-diffused junction collects excess carriers which are generated by the radiation during bombardment primarily from the bulk. A meas urement of the radiation induced short-circuit current thus yields a direct determination of the minority carrier diffusion length as the bombardment progresses.1•2 Moreover, the excess carrier density produced by this excitation is sufficiently low so that the effects of vari ation of diffusion length with excess carrier density are negligible (see below and Fig. 5). For particle radiation, such as protons and electrons, an absolute diffusion length measurement is obtained by a determination of the ratio of the radiation-induced solar cell short circuit current density to the incident radiation current density divided by the average specific ionization of the incident particles.2 For heavy particles, the specific ionization can be determined from published 1 J. J. Loferski and P. Rappaport, Phys. Rev. 111, 432 (1958). 2 W. Rosenzweig, Bell System Tech. J. 41, 1573 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.51.150.161 On: Sun, 30 Nov 2014 22:33:392708 ROSENZWEIG, SMITS, AND BROWN data on the rate of energy loss divided by the well known value of 3.6 eV to produce a hole-electron pair in silicon. For electrons, the average specific ionization must be measured experimentally. The procedure for doing this is described in Ref. 2. It consists of measuring, by means of a solar cell, the relative ionization, as a function of depth in silicon, for monoenergetic electrons over their entire depth of penetration. The resultant curve is then normalized to give the total amount of ionization which the incident electron is known to be able to produce. The technique of measuring diffusion length with a low-intensity, 1-MeV electron beam has been developed to the point at which ±5% accuracy has been attained. Diffusion lengths measured in this way have been found to be in good agreement with proton beam measure ments at energies of 1.35 to 4.5, 16.8, and 130 MeV. This allows one to use, with confidence, solar cells which have been calibrated with I-MeV electrons as proton beam intensity monitors in those circumstances where the particle energy is known but in which an intensity measurement cannot be made readily with Faraday cups. Such a procedure was followed throughout the proton. bombardment study to be described here. At each pro ton energy at least two solar cells were exposed simul taneously to fluxes of equal intensity. One of the cells was heavily prebombarded so that its change of diffusion length was negligible during the additional bombard ment. The others were test cells for which the diffusion length degradation was being determined. Electrical connection was made to each of the cells by means of a pressure contact and the radiation-induced short circuit current was monitored during the bombardment. The effects of beam intensity fluctuations were mini mized by measuring the integral of the short circuit current over a short time interval which corresponded to a fixed amount of incident charge (ordinarily the order of 1010 protons per cm2). Postbombardment meas urements of the diffusion length by means of the electron beam were made at intervals for up to one month after the bombardment. In order to cover the desired energy range, bombard ments were carried out at four locations: Princeton University cyclotron, Harvard University cyclotron, Naval Research Laboratory (NRL 5-MeV Van de Graaff), and McGill University cyclotron. At Harvard and McGill (incident proton beam energy of 130 and 96 MeV, respectively), the cells were arranged in face to-face pairs along the axis of a stack of aluminum ab sorbers so that exposures could be obtained at various energies. The cells were untinned (having only thin evaporated contacts) so that the absorber thickness preceding any of the cells could be kept uniform and closely controlled. The error due to uncertainty in ab sorber thickness was negligible. One cell of each pair was a prebombarded monitor and the other the test speci men. In the face-to-face arrangement the active regions of both cells are exposed to protons of essentially identi cal flux and energy. At Princeton and NRL the cells ,vere exposed to a broad beam obtained by scattering with thin gold foils. Four cells were mounted in a plane perpendicular to the proton beam axis in a region of beam uniformity of better then 5%. At least one of the four cells was a pre bombarded monitor. The beam energy at Princeton, after scattering, was 16.8 MeV. Exposures at lower energies were obtained by placing aluminum absorbers over pairs of ceUs. At NRL the primary beam energy could be varied from two to five MeV which resulted in bombarding energies (after scattering) of 1.35 to 4.65 MeV. On the basis of simple theory it is expected that the diffusion length degrades according to the equation: 1/D=(1/Lo2)+K<I>, (1) in which L is the diffusion length after bombardment flux <I> for a material in which the initial diffusion length is Lo. Equation (1) is a statement of the hypothesis that the carrier recombination rate is the sum of two rates; the first results from recombination through centers which are present initially and the second from recom bination through centers which are introduced by the radiation and whose concentration is proportional to the exposure. The coefficient of proportiolilality K (we will refer to it as the damage coefficient) is a measure of the relative damage rate for the given condition of ir radiation and type of material. In the present study, the variation of K with proton energy was measured. Most of the solar cells used were produced by the Western Electric Company from 1.0 to l.5-Q· em p-type pulled crystals and have shallow phosphorus-diffused junctions. Some other cells were used as will be indicated below. INITIAL VALUE 1010 PRINCETON-4 BLOCK NO.3 •• PROTON ENERGY. 16.' MEV o a PROTON ENERGY. 6.5MEV PROTON FLUX ( ... ", ® FIG. L Diffusion length vs bombardment flux. Curves 1 and 2 are sample cell and monitor cell at 16.8 MeV. Curves 3 and 4 are sample cell and monitor cell at 6.5 MeV. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.51.150.161 On: Sun, 30 Nov 2014 22:33:39ENE R G Y DE P E ~ DEN CEO F PRO TON I R R A D I AT ION DAM AGE 1;,\ S i 2709 RESULTS An example of the degradation in diffusion length with bombardment flux is shown in Fig. 1. In this plot the ordinate is the instantaneous diffusion length and the abscissa is the cumulative proton flux. A Faraday cup was used as a monitor of the flux. The initial.dif fusion lengths of all four cells shown were determmed by the 1-:\1eV electron beam technique.2 It is evident that the diffusion lengths of the two heavily prebom barded solar cells, shown in curves 2 and 4, remained unchanged during the experiment as expected. The cell of curve 2 was exposed to the full proton energy (16.8 MeV). The cell of curve 4 was shielded by an aluminum absorber of 343 mg/ cm2 and thus was exposed to protons with a mean energy of 6.5 MeV. The energy variation of the protons due to straggling in the absorber is ap proximately ±O.4 MeV. This energy vari.ation intro duces a negligible error in the results as wlll be shown below. Although the Faraday cup was used as a flux monitor the absolute value of the flux at each of the two proton energies was determined from the monitor cell response and the knowledge of the monitor cell dif fusion length and the rate of energy loss of the protons. The points on curves 1 and 3 of Fig. 1 show the dif fusion length degradation of initially unirradiated cells exposed to the same proton fluxes and energies as the monitors of curves 2 and 4, respectively. The curves fitted to these points are given by Eq. (1). Values of K are determined from the fitted curves. There is a ten dency for the data points to fall more slowly than the theoretical curve. Such a tendency is absent in the case of bombardment damage by electrons. This may be due to more significant changes in carrier concentration, for corresponding changes in diffusion length, for protons than for electrons. 10 --------, Me GilL Ep =96 MEV ___ STACK NO.3 ---STACK NO.2 MONITORS --SPECIFIC IONIZATION 0.IOL---..J2.5----J5.0----~15:----------::10 Thickness of aluminum (g/cm') FIG. 2. Relative ionization vs thickness of aluminum absorber for incident protons of 96 MeV. IO-!5 ----~-- ~ --_. __ . -------- ------- ---._----, K • HARVARD + McGill • PRINCETON o NRL lo-eLI ------I~O -----7.IO:!,;OC------;;;;1000 PROTON ENERGY (MEV) Frc. 3. Damage coefficient l\ V8 proton energy. In thick absorber stack experiments, such as at Harvard and McGill, there is a question of beam broad ening due to multiple scattering of the protons. At McGill the beam was sufficiently broad (Gaussian in tensity distribution with a width at half-maximum of 6.3 cm) that the flux density decrease was less than about 10%. This situation is illustrated by a series of measurements the results of which are plotted in Fig. 2. The dots represent measurements obtained with a single monitor cell placed behind various thicknesses of alu minum absorber (stack # 3). The crosses are the relative monitor cell responses for one of the experimental stacks. The dashed line is relative specific ionization as a func tion of depth for a 96-MeV proton and agrees with the experimental curve to the extent that beam spreading and energy straggling can be neglected. At Harvard, the incident beam was narrow (2.5 cm diameter colli mation) and the flux density changed by almost a factor of two between the first pair of cells in the stack and the last. By using cell pairs, this variation is directly meas ured at each position. The good agreement between the results at the two accelerators (see Fig. 3 and below) illustrates the reliability of the method. The damage coefficient K was evaluated for all runs and is plotted for the n-on-p cells as a function of proton energy in Fig. 3. The experimental results at the four accelerators are separately identified. For the purpose of eliminating some of the variability due to differences in material, corresponding halves of the same solar cell were irradiated at different energies. Measured points for such pairs are identified by connecting lines in Fig. 3. Relative K values for one set of 1-Q'cm p-on-n cells in the energy range from 16.8 to 130 MeV were greater by a factor of 6.2±2, independent of energy. Postirradiation measurements of diffusion length by means of the 1-MeV electron beam indicated a room [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.51.150.161 On: Sun, 30 Nov 2014 22:33:392710 ROSENZWEIG, SMITS, AND BROWN O'.~ '", 6 ~~ t-- '~ 0'+ D~ ". • + ~ " +, " ':" , , , " , " " 7 , -, " a HARVARD " + McGILL o PRINCETON· o NRL e 10 100 1000 PROTON ENERGY (MEV) FIG. 4. Damage coefficient, K, following a room temperature anneal vs proton energy. temperature annealing of the damage. Increases of dif fusion length by as much as 25% over a period of several days have been noted. The diffusion length measure ments obtained approximately two weeks after bomb ardment are used to obtain a plot of K versus proton energy as shown in Fig. 4. The solid curve represents a fit (by eye) of the points in Fig. 3. The dashed lines will be discussed below. The postirradiation measurements of the diffusion length revealed a gradual variation of the diffusion length with electron beam intensity. Figure 5 shows some typical plots of the measured diffusion length as a function of excess carrier density. The diffusion length is calculated from the formula2 L=SeJ8c/J• and the excess carrier density from n=J8CL/qD, (2) (3) in which L is the diffusion length, n is the excess carrier density, J. is the electron beam current density, J sc is the specific ionization per incident electron, q is the electronic charge, and D is the minority carrier dif fusion coefficient. DISCUSSION A notable feature of the experimental results is the presence of an energy range, between about 8 and 40 MeV, in which the damage coefficient remains almost unchanged. If Rutherford scattering is assumed to be primarily responsible for elastic scattering between the incident proton and the silicon nuclei bound in the lattice, then the frequency of collisions in which the silicon atom receives an energy greater than some dis placement threshold energy (small compared to the maximum energy it can receive in a single collision) falls off inversely as the proton energy. Thus the rate of production of primary vacancy-interstitial pairs also falls off with the inverse proton energy. Some of the struck atoms have enough energy to produce secondary displacements slowing the fall-off with energy by a logarithmic factor.3 The dashed lines in Fig. 4 have the predicted energy dependence. The experimental damage coefficients appear to follow such a variation below 8 MeV and again above 40 MeV. The two dashed lines are separated by a factor of four in K value. Such a large factor is completely outside the range of experi mental error or material variability. The question arises as to whether these unexpected results might be due to the production of secondary particles, e.g., neutrons, protons, or alphas, as the proton beam is slowed down in the absorber. The relative number of these particles is expected to be small. But even if they are present in significant number, the spurious charged particles affect the measured results only to second order since the flux determination is based on an ionization measurement and the variation of the ionization with particle type and energy follows the Rutherford scattering law. The possibility of neutron damage was examined during the McGill experiment by placing solar cells in the aluminum stack just beyond the range of the protons. The damage to these cells was less than four per cent of the damage to the cells in the proton beam. The influence of secondary particles is thus found to be quite small and cannot give rise to the observed departure from the Rutherford scattering predictions. The influence of nuclear interactions on the scattering cross sections has been examined by Baiker, Flicker, and Vilms.4 They apply the optical model theory of the nucleus and find that higher energy impacts occur more frequently than expected on the basis of Rutherford scattering alone. Only a portion of this energy is used up in the production of additional displacements; the 100 ----,------ ,-f-- 80 PROTON ENERGIES _ o 94.3MEV ~60 e ~ )6,8 MEV '" 4.65 MEV I -r---: I ~20 r----~~- Ui Dc I r12 - ~ i!' ~ 1:1l;:==::::::;;;:::J;:===..---=-"T- --=± - 6 10' 1010 101\ 1012 EXCESS CARRIER DENSITY (eM") FIG. 5. Diffusion length vs excess minority carrier density. 3 G. J. Dines and G. H. Vineyard, Radiation Effects in Solids (Interscience Publishers, Inc., New York, 1957). 4 J. A. Baicker, H. Flicker, and J. Vilms, Appl. Phys. Letters 2, 104 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.51.150.161 On: Sun, 30 Nov 2014 22:33:39ENERGY ])EPE~DE~CE OF PROTON IRRADIATION DAMAGE IN Si 2711 rest results in ionization. Even if it is assumed that all of the energy goes into displacement production, the resultant calculated displacement density vs proton energy curve does not deviate from the simple inverse energy variation by as much as the curve shown in Fig.4. It is conceivable that more than one type of defect is being produced with differing energy-dependent pro duction rates. Thus, for example, the enhancement of the more energetic collisions due to nuclear interactions might give rise preferentially to defects whose energy level (s) and capture cross sections are more effective in degrading lifetime. An enhancement of this effectiveness would help to bring the computations by Baiker et at., into better agreement with the present experiment. Another interesting effect is the increase in diffusion length with increasing excess minority carrier density. Shockley-Read single level recombination theory5 pre dicts such an effect when the capture cross section for minority carriers is much larger than that for majority carriers. In this case it is possible to achieve a variation of ditTusion length with the square root of the excess carrier density. The variation observed here is much less rapid. It appears to be similar to what one might expect for a distribution of levels which progressively fill with electrons as the quasi-Fermi level moves toward the conduction band. The observation of the excess carrier density depen dence of the diffusion length has previously been re ported by Denney and his co-workers.6 It has further been suggested that the observed departure of the energy variation of the damage rate from the simple Rutherford law predictions might be entirely due to an energy de pendence of this nonlinear effect. The results in Fig. 5 show that this is not possible. The nonlinearity of the four curves is very nearly independent of proton energy.7 • W. Shockley and W. T. Read, Phys. Rev. 87, 835.(1952). 6 J. M. Denney et a1. STL Report 8653-6017-KU-OOO, and 8653- 6026-KU-OOO, Contract No. NAS 5-1851. 7 J. M. Denney and co-workers have also observed the energy independence of the nonlinearity; however, their findings on the energy variation of the damage rate do not agree with ours. Thus the energy variation of K computed from this data is very nearly independent of excess carrier density although the absolute values of K do depend on this quantity. The diffusion length degradation results reported here can be used to predict solar cell bombardment damage for cells whose parameter changes have been correlated with diffusion length (see, e.g., Ref. 8). It will, however, be necessary to introduce a correction factor for the K values plotted in Fig. 4 based on an average bulk gener ation rate for solar illumination at a particular bomb ardment level as compared to the value at about 1017 cm-3 sec! used in the diffusion length determinations. An upper limit to the magnitude of this correction can be determined from Fig. 5 if it is assumed that all the current generated under solar illumination is due to carriers generated uniformly in the bulk. For the least damaged cell in Fig. 5 this would give a value L""S2 J.l under "solar illumination" as compared to 1.=35 f.I for the low-level excitation used to evaluate K in Figs. 3 and 4. This implies that the effective K values are a factor of no more than 2.3 times less than given in the plots, for this degree of damage. It is to be noted, moreover, that the correction becomes smaller as the degree of damage increases. Studies of silicon solar cell characteristics8 show that this magnitude correction in K implies an underestimate in solar cell power output of, at most, 7%. ACKNOWLEDGMENTS The authors greatly appreciate the assistance by J. A. O'Sullivan and W. M. Augustyniak, of Bell Telephone Laboratories, and J. Weller of the Naval Research Laboratory. The courtesy and cooperation of Dr. A. Kohler, at Harvard, Professor R. Sherr and A. Emann, at Princeton, K. Dunning, at NRL, and Professor Bell and R. Mills, at McGill, are also gratefully acknowledged. 8 W. Rosenzweig, H. K. Gummel, and F. M. Smits, Bell System Tech. J. 42, 399 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.51.150.161 On: Sun, 30 Nov 2014 22:33:39
1.1931131.pdf	Shallow Donor Thermionic Emitter John K. Gorman Citation: Journal of Applied Physics 33, 3170 (1962); doi: 10.1063/1.1931131 View online: http://dx.doi.org/10.1063/1.1931131 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermionic Performance of Rhenium Emitters AIP Conf. Proc. 746, 910 (2005); 10.1063/1.1867215 Carbon Nanotubes as Thermionic Emitters AIP Conf. Proc. 699, 773 (2004); 10.1063/1.1649642 Low emittance thermionic electron guns AIP Conf. Proc. 184, 1532 (1989); 10.1063/1.38018 Thorium Sulfide as a Thermionic Emitter J. Appl. Phys. 21, 1193 (1950); 10.1063/1.1699564 Zirconium Carbide as a Thermionic Emitter J. Appl. Phys. 20, 886 (1949); 10.1063/1.1698554 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:4931 ;0 C. A. PI.I:'iT .I:-.ID I\". A. SIBI.EY less than tht: observed values. Again we may suppose that the precipitate s observed by etching of the crystal faces do not correspond to precipitates existing at bram:h points of dislocations. However, the fact that the ~RL crystals did not develop any peaks in the scatter ing curves at devilled qucn.::h temperatures, where the ~catte ring power approaches that of the untreated Harshaw samples, suggests that either the dislocation lines in the NRL crystals are much more n.onuniform or t hat the individual scattering units are ditTerent than those in the Harshaw crystals. For example, scattering by gasl'QUS impurities trapped at dislocations is a dis t inn p05sibility and the particular electronic state of s:;uch impurities determines the effective polarizability of the scattering unit. It would be very interesting to set: if crystals grown by the Kyropoulo s technique have different types of dislocation arrays than those grown by the Stockbarger method. [v. SUMMARY A dillerencc in scauering properties of KCI crystals was observed for Harshaw crystals and grown at i\RL. The Harshaw crystals had a orientation dependence which was not exhibited N R L samples, and thermal treatments produced ent effects in these specimens. The dislocation densities of all the samples essentially the same, and the difference in SGltt" ... properties can be ascribed to difference s in the rel~ul,,,;, of the dislocation networks, ditTerences in defect centrations at dislocations, or differences in defect which would give a difference in elfcctive pola for the :;cattering units. ACKNOWLEDG MENTS The authors are indebted to Dr. W. H. Vaughan the two :,\RL crystals used in this work. JOUR:-':AL OF APPL IED PHY SIC~ VOLU),IE JJ. XUMBER [I Shallow Donor Thermionic Emitter* JOIII' K. GORMAr-:' Sperry GY"':lfope COIl/puny, Greu/ Neck, Xf"'':' rurk (Received April 16. 1(62) A possible approach to the developmt:nt of a low work function thermionic emitter involves the introduction of shallow donor states into a maLrix having a low electron affinity. It is the intent of this puper to c.'q;lnre the concept of a shallow donor emitter from both a generall)()inl of view and with specific application to barium oxide as the host lauice. The prohlem is t1iscusSt.'t1 in terms uf the single-donor model with consideration given the various characteristics which would be required for a practical matrix additive system. Although the eit.'CLron affinity of BaO has been ('stim,lted to lit: as low as 0.6 eV, the ordinary oxygen vacancy donor pre..:;cnt is a dt:ep level with an ionization energy of about 1.4 eV; it yields a work function of l..t to 1.5 eV at tOOOoK. By comparison, a work function of O.s.~ eV would be expecled for shallow dunor BaO at this tt:mpcrature . The substitution of a I. [NTRODUCTION THE sa.t uratcd thermionic emission. available from a solid can gent'rally be approximated by the Rkhardson-Dushman expression, r ~ \201"e-·l'"r, (I) in whkh I is the t'mission density in Aj em:! obtainable at an ab::iolutc temperature T. The work function 4> is characteristic of the emitter and will generally be some· what dependent upon the temperaturc. If tP is expressed in electron volts. the Boltzmann constant k bt'comes 8.63X 10-" eV!dcg. .. This wMk was :o;upporled hy the Romc .\ir Uc\'cl(Jplll clll Center .. \ir Research and Dt'vc1opmcl1t Command , Griflis Air Furce Base, undt'r Contract 1\0 . .-\F30(602)2495 . tripositive rare-earth ion for a Baz+ ion in the 1all li'CC~C~i'~:::~:::::J as a possihle mechanism for the incorporat ion of an ir center. Some speculation is otTeretl concerning the ionizatioa. energy of this type of {bnor as well as the associated 3ctivatiOD proctss. An attempt was made to ohserve the donor behavior of several rare-earth ion additives in BaD and also srO by,'"dYiJoI i the temperature dependence of the effective work lu,>el,on. were secured for La, Cd, ~d. Er, and Eu in BaD, and Eu in SrO at analytical concentrations of from 0.01 to 0.05 While no lowering of the work function is reported, it is no definitive interpretation of a negative result can be certain other experimental information becomes available, larly the solubility and oxidation stale of the additive ion in matrix crystal. There arc sevt'ral !"5implifying features II·)\;~::~~';. Eq. (0. In particular one has negleeled: (I) quant lim-mechanical retltl"lion of electrons; field dt'pendence of the work function arising from Schottky effect and, in semiconductors, field tion, and (3) the variation of 4> from one 10 another on a polynystalline surface (patch effect).' The spec.:ific influences of these effects on emission been discussed by Herring and Nichols,· and re(:entl'ot by Hensley:! and a detailed analysis is not ,",,,,,nted the present context. It will suffice to point out Eq. (\) may be inlerpreled as a definition of an ] C. Herrin g" and M. H. :'-lichuls, Kcvs. ~1 odem Phys. 21, 185 (1949). ~ E. O. Hensley, J .. "\ppi. Phys. 32, 301 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:49SHALLOIV IJO~OR THER:VIIO .'1IC EMITTER 3171 ffedi\"(; work function tPr: derived from 1 and ;5t approximation CPt; may be identified \"ith a 't' ble ay:r3ge of the true work function, the ~i:g ddined as the position of the Fermi level to th~ \"anlllm level. work fUllction of an II-type semiconductor may ~ -..i>res,en ( "" ", (2) x is thl' electron, affinity and f.L the ~lectronic potenli;d refernng to the conductIOn band the t.'!lt'rgy zero. The electron affinity is the differ'en", between t he vacuum level and the ..,auctIOn band edge and in the absence of adsorbed layers may be regarded as an int rinsic properly host cry..-l:d. The chemical potential of the elec is the difference between the Fermi level and the CQDduction band t.:dge and determines the density of cooduction ekct rons, I/, If the Fermi level lies morc that thoUt 4kT below the conduction bnnd, It is approxi ... ted by (3) iDwbich Sr i:-; Lhl: elTective density of states ptr em:l in me conduct ion h:lno. The latter is given by Y,,= 2 (2-n-IIIkT,' Ii') I, (4) m is the diecLi ve mass of an elect ron near the of the conduction band, and the other symbols usual meaning. The electronic chemical is generally determined by the numbers and donors and acceptor levels which lie in the Li.>rbi,ldcn band, subject to the condition that the bulk solid be electric:llly neutral. The "lOrk function of n-type emitter may frequently be expressed by the ISJ'gJ,,,ltJnc)rapprox:imation. It is assumed in this model IJ is ddt.:rmined by a single nondegenerate dOllor and secondly, that there are no acceptors present. electro-neutrality condition then implies simply It is equal to the number of ionized donors. An . expression for" and hence cf> by Eq" (2) may be only when the donors are either ~Jightly . or almost completely ionized. Demars sligltlly iOlfized: at-most (om pletely ionized: (6) the~t: t.:xprt:ssions E .. is the ionization energy of the and .Yd their number/cm3 irrespective of the state l" loni,oation. It is apparent from these equations that a work function is associated with a low value for X a high density of donor levels relatively close to the ~-","u' ;"ic m band edge. Although Ei docs not appear ~"F'UCitlyi n Eq. (6), the assumption that there is a high . r,l1ion of ionized donors in the absence of accel_r I!; laillamount to assuming a low E,.. At tcmpera-tures and donor densities of practical interest, Eq. (5) would be applicable to Ei values of the order of an electron volt while Eq" (6) would apply to values approximating a few 11eV. There is evidence to indicate that certain solids possess X values ''''hich are considerably lower than the lowest work functions observed for practical emitters. Specific instances are considered below. Consequently, it would appear that a reasonable approach to the developm ent of a 10\v work function emitter would be a consideration of the possibility of incorporating im purity donor levels of low E, into such a host crystal. In other words, an attempt might be made to approach a value similar to X by increasing the electron population in the conduction band. As indicated above, this is equivalent to raising both the Fermi level and f.l. 2. THE SHALLOW DONOR "Cnder certain favorable conditions the binding energy between the donor electron and the remaining positive core may be reduced to a few MeV through polarization of the surrounding lattice. Donors of this type are referred to as "shallow" and are almost completely ionized even at room tempera ture. Shallow donors are more commonly observed in silicon and germanium, and a considerable body of literature concerning their occurrence in these matrices has been compiled. True shallow states arc reasonably described by the effective mass theory.3 Briefly, the ground state is de~cribed by a wave function, 1/I(r) of the form4 (7) where F(r) is a nonperiodic envelope function cen tralized about the donor core and u(r) is a Bloch func tion near the conduction band edge; F(r) is itself a solution to the ctTective mass Schrodingt:r equation. In the simple isot.ropic case the latter is equivalent to the wave equation for the hydrogen atom in which (1) the free electron mass is replaced by an clTective electron mass which eliminates the periodic crystal potential from the Hamiltoni an, and (2) the Coulombic potential due to the core is reduced by the bulk static dielectric constant of t.he matrix K. The envelope functions are, in f<.lct, expanded hydrogcnic functions; the correspond ing Bohr radii may be of the order of 10 A" Thus, F(r) modulates 'It(r) over a number of lattice cells in the vicinity of the donor core. The principal consequence of t his is that. F.i may be greatly rcdu{'cd if K is large, being given by E,= (E,/K') (m/m), (8) where l!.h is the ionization energy of a hydrogen atom (13.6 eV) and m is the free electron mass. It is apparent t.hat in addition to a low electron 3 \V. Kohn, Solid State Phys. 5, 25S (l95i). ~ This strictly holds for a single conduction band minimum (at the origin) in k space. In the general case, a linear combination of terms like (7), one for eac:h eauivalent minimum , is required. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:493172 J 0 II N K. GO R M .\ :'i aflinity a suitable matrix ~hould havt: a high dielectric constant and a low effective mass to promote shallow donor format ion. One would also require that the matrix have favorable vacuum properties, that is, a low vapor pressure and thermal stability at practical operating temperaturt:s. The nature of the impurities to be con sidered as potentially useful donors would naturally depend upon the particular matrix. A favorable solu bility is of primary importan ce to achieve meaningful donor ("ollrcntrations. l. THE ELECTRON AFFINITY ft is unfortunate that there are v(:ry fL.w value:; uf X rcporlt:d in the literature. This is not ~urprising, how· ever, since there is no singularly direct and reliable method for procuring such data. By measuring the electrical condunancc of porous samples, Hensley!.·6 estimated X to be about 0.6 eV for BaO, srO, and e"o. The analysis was based on the concept I hat at higher temperatures the conductance is dt:termined by e1ec· trons in the pores of the sample, while at lower tem peratures the ordinary bulk conductivity predominate s. This is because the larger density of dt:ctrons in the conduction band as compared to the pores, the ratio being about eX/kT, is offset: at elevated temperatur es by t he higher mobility afforded by the pores. Experi mentally, X is derived from an estimate of the It:mpc·ra ture at which the 1\\'0 conductivities become equal, plus an estimate of the average dimension of the pores. Hensley also included thermoelt'ctric power measure ments to characterize the "transition" temperature. Thton:tical estimates of X for the alkaline earth oxides arc also available. Values of X for several ionic compound s were obtained by 'Wright 7 using a Born-type cycle originated by ~fott.8 The maximum value which appearcd possible for the alkaline earth oxides increased from 0.5 to 1.0 eV in the order Ea, Sr, Ca, Yig, and Be although values approaching zero were not ruled out. The maximum values, at least, are in reasonable agree· ment with Hen~ley's estimates. The uncertainty is associated with a corresponding uncertainty in the width of the conduction band, which generally precludes a wider application of this metho(J.9 With certain Limitation s the electron aOinity may also be derived from stparate photoemission and photo· conductive measurements. This is exemplified by the work of Spicer10 on several alkali antimonides. The photoemission threshold for these materials corrc::;ponds 6 E. B. Hcnsley, Report on Fifteenth Annual Physical Elec tronics Conference, ':\fassachusctts Institute of Technology, Cambridge, Massachusetts, 1955, p. 18. IE. n. Hensley, J. App!. Phys. 23, 1122 (1952). 7 D. A. Wright, Proc. Phys. Soc. 60, 13 (194.8). a:-.l. F. Mott, Trans. Faraday Soc. 34, 500 (t93H). • Earlier calculations by Moll su,ggcst that X for the alkali halides are less than a volt. 10 W. E. Spicer, Report on Sevcnteenth Annual Physical Elct: t rOllics Conference, :"o.1assachusctts Institute of Tcdlllulogy I Cambridge, Massachusetts, 1957, p. 151. " R. H. Plumlee, RCA Rev. 17, 23t (1956). to the excitation of dectrons from the top of the band. The energy of a threshold photon Eo may forc be identified with the difference between this and the vacuum level. The widlh of the forbidden EG may similarly be estimated from the thre,;ho,ld photoconduction, The electron affinity is the difference x=Eo-Ec. Estimates of X ranging from 0.5 to 2.3 eV are by Spicer. The optical values for Eo and Eo l'ver, differ somewhal from the thermal le']Ulilb,riu values owing to the Franck-Condon orinc:inllp F.o-E(; will not coincide preciscly with the thermal interest in thermioni c emission, the di,;cr,ep;ln':y ably being larger for more polar crystals. In ad,iition' this, the general applicability of the technique is to substances which exhibit. reasonably photoprocesses. 4. APPLICATION TO BaO The low electron afl:inities and refractory ouoH,;;:; the alkaline eart.h oxides recommend these sullst,.1IIi as possible host crystals for a shallow donor Barium oxide, in particular, has the relatively dielectric constant of 34. Assuming m/mR:::.l, implies [Eq. (8)J that a shallow donor in BaO have an E; of about 0.012 eV. Using Hensley's x= 0.6 eV one can employ E'l. (6) to estimate function of BaO containing shallow donors. At and Nd= 10"/cm', cp is computed to be 0.83 should be noted that a high concentration of required to maintain a low work function at ture~ of practical interest. At low donor c~rc;:;;:: q, increases rapidly with temperature and approaches a value characteristic of the material. A donor density of 101i/cm3 would co;rrespo to an impurity conccntration of about 0.05 However, this is still probably not sufficiently produce a significant degradation of x. The single-donor model described is based on the electron distribution which would exist in bulk of the matrix and does not consider surface effects which might make their own contrib,rtI to cp, These effects generally fall into two categories; adsorbed dipolar layers which modify X directly, (2) filled surface state,. The laller change the value of JJ. by producing a raising or lowering of structure (but not Ihe Fermi level) in a comperu;ati space-charge layer ncar the surface. Each effect result in an increase or decrease in q, depending upon orientation of the dipolar layer in the first case or wh(·ther electrons or holes are trapped in the ~talcs in the second. Since both ctTcl:ls primarily ad:.;orbed impuritie s, it seems likely that in a V;tt.:uum ~ystem and at higher temperature s their Ilucncc would be :::mall for BaO.12 There is, at I~ E. B. Hensley (private communi cation). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:49S HAL LOll· D 0 :'\ 0 R TilE R :\1 I 0 :'\ ICE :\1 ITT E R 31 i3 . 'vc t:\"idcll(t: that these surfan' contribution s art' . ~ant in IbO :.lnel they will not be discussed furt her. ~. of inlen.:sl to compare the tJ> calculated for shallow It 15 BaO wit h that expected for ordinary ddeet BaO. doDO~po rtal1l defect donor in BaO is belitved 10 caB o of an Oxygen .vacan~y c.ont~illin g two electrons" It is ~pd onor havlllg an 1Olllzat.1011 energy of 1..1 cV,;' and . therefore ~lightly ionized under ordinary circum • (fS. Applying Eq. (5) </> is found to be 1.41 tVal t(#)0K whell Yt/= 1019 (see reference 13). The difference ween Ihi~ and the ~hallow donor value is 0.58 tV :ch at lOOooK is equivalent to an emission ratio of about SO). Tn the context of practical cathodes, the ..,niwde ?f the emission density m~y , of cQu.rsc, bt'Come limlled by other factors (heatIng, sparkmg, etc.). Conve:r:-ely, the emission available from oefect ,.0 at IOOOoK would be obtained at about 6200K from the shallow donor system. -S. THE SUBSTITUTIONAL IONIC DONOR A method by which donor centers may be int roduced iltoan ionic solid is through the ~ubstitution of some of the host cal ions by impurity ions of greater positive e.14-!7 The formal charge associated with the anpurity occupied site will simply be the ditlerence IIrtween the charges of the impurity and host ions. Thus, in barium oxide if a BaH ion is replaced by a tripositivt il)l1 M3+ the resultant center has a formal (barge of +e with resped to the ideal latlin:, and can potentially trap an electron to form a neutral donor. emately, it is possible to derive such a center by lacing the host anion by one of lower negative charge, .hich will also re~ult in a formal positive charge. Considering specifically a tripositive substitutional rmpurity in BaO, it would appear t hat a likely choice might bl.:: seltcted from among the rare-earth group of ions. The lat1er combine two essential properties: (1) They form relatively refractory oxides and would prc SIlIJlably b~ 1hermally stable in th~ BaO matrix, and (2) they po~sess large ionic radii which increases 1he ibility of their replacing the large BaH ion 10 form 1 substitutional solution. A ,large ionic raelitls is also 'dtsirablc to promote shallow donor format.ion through &reater interaction of the donor electron with surround ing lattice. The sub!,litut.ion of a number of tripo ~ili\·e ions in laO requires that some additional charge compensaling chan i~1ll occur to maintain electrical neutrality. This y Come about by the incorporation of an equivalent mber of BaH vacancies, each of which would have a I Act.unl defect donor densities are expected to he much lower II Ihls with correspondingly higher work functions. R. A. Smith, Scmicolldllc/ ors (Cambridge l;niversilv Press 11 ror~. t(59), p. 64. . , lie. \\·ag:ner, J. Chem. Phys. 18, 62 (1950). /. A. Kroger and H.]. Vink, Physica 20, 950 (t954). ~. A. Kroger nnd H. J. Vink, Solid Stilte Physics, edited b~· '~~3tz and D. Turnbull (Academic Press Tnc., New Yurk, 1956), . ,p. 307. formal charge of -2e. Thus, OJle \',h:allcy wuuld be: includeo for every two ions substituted. Charge com pensation by this process has been called I he Koch· \\'agner mechani sm. If; It is important to recognize that the simple tripositive ion in the laaice is equivalent to an ionized donor which has lost its electron to a barium \"acancy acceptor. This can be easily understood by visualizing the /leulral barium vacancy acceptor. The latter would consist of the simple vacancy associated \\lilh two holes, these being essentially missing electrons on the oxygen ions coordinated about the vacancy. If the holes are tilled with electron~, the simple dinegative vacancy remains. The fact that t he donor cledrons occupy acceptor levels means that th<:y arc not avail able to the conduction band; the acceptors thus produce a lowering of the Ft'rmi level with the resull that the vacancy-rich structure would not be a good emitter. Activation of this structure would necessitate a re moval of these vacancies so as to replenish the donor levels with electrons. Charge compensation involving the substitution of dedrons for cation vacancies has been referred to as the Verwey-Selwood mechanism16 and is tantamount to a chemical reduction of the system. If E,. for the impurity donor should be low, a large fraction of these electrons would 1hen reside in the conduction band. There is cvidenct: which does, in fact, indicate that activation of the ordinary (Ba,Sr)O cathode romaining defect donors involves the removal of barium vacancy acceptors. The only dilTerence from the situation de scribed above is that the electrons art: returned to oxygen vacancies rather t han impurity centers. Accord ing to Hensley and Okumural.'! barium vacancies are initially generated in equal numbers at. the time of conversion. The barium vacancy acceptors have an appreciably higher mobility than the oxygen vat'ancies and diffuse to the coating-ba:::e metal interface where they are eliminated. This presumably involves their being filled with free barium released at' the interfare by the rt'ducing action of ba~e metal impurities on the coating. The density of oxygen vacancy donors i5 believed to remain essentially COJ1stant during lhe activation. This interpretation of the proct'~s is based upon the observed agreell1tnt bet ween availablt: mobility data for barium vacanciesl:! and the mobility measured for whateve r :-;peries determine s the cathode activity. The diJTusion of the activit.y was studitd by observing the profile of the emission from a coating on a pure (passive) platinum ribbon using a small probe anode. A narrow tab of active nickd was incorporated at the center of the ribbon to produce a peaked initial distribution of the activity. 1l is perfectly reasonable to expect that the removal of vacancies from impurity doped BaO would proceed in an analogous manner provided that an active base nickel is employed. Even 18 E. B. Hensley and K. Okumura , Bull. Am. Phys. Soc. 5. 69 (1961 ) . l~ R. \V. Reddington, Phys. Rev. 87, tM6 (1952). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:4931 i~ .I 0 11:\ K. (; 0 R \1 .\ :\ in the absence of Lase metal activators, the sy~tt:m might hcromc activated through loss of molecular oxygen to the surroundin g vacuum. If shallow donors arc formt'd this can be formulated as (to) where ,. c represents a BaH vacancy, e-a conduction election, a.nd S a. pair of empty surface sites resulting from tht: transport to the ::iurfacc of Vr and the vacancy remaining from the removal of 0-.14 A Born cycle analysis of this process has been made21) which indicates that it is endothermi c to the extent of about 0.5 ('Y. Qualitatively, this suggests the activation by loss of oxygen is at least thermodyn amica.lly feasible. Con sequently, the likelihood of activation by base metal reducing agents is expected to be even greater. It should be appreciated, neverthele ss, that a practical cathode is a dynamic system constantly interacting with its environment and may be far removed from equilibrium. Thu~, thermodynami c computations are of limited application. An actual cathode is likely to be continu ously subjected to activating (reducing) processes and simultaneously to poisoning (oxidizing) processes. An a.ctivated condition is simply characterized by a favor able balance of the two which, according to the above discussion, would maintain a low density of acceptors. Assuming that the impurity ion does enter the lattice to form a donor, there still rtmains the question of the magnitude of Ej• Although the etTcctive mass theory provides a fair description of shallow states once their cxistenct! is established, it cannot predict a priori that it particular impurity will form a shallow level in a given matrix. An approximation method, based on techniqut'~ described by Rt'iss~!l and Kaus,22 was used to estimate the possibility that a shaJlow donor level might be derived from a substitutional rare-earlh ion in BaO.2O \Vhile the computation was made specifically for Gd3+ ion, the result is not expected to differ significantly for the other 41 rare earths. The GdH ion occupies a 8a2+ vat.:ancy, the latter being approxima ted as a spherical hole of effective radius R in the surrounding dielet'tric. Inside tht! hole the potential experienced by the extra associated donor electron is assumed to be that of the isolated tripositive core in tiaC1l0 plus a. constant contribution from the surrounding lattice. The variable part of the con ~ potential is described by the Thomas-fermi-Dirac function23 which accounts for screening of the nucleus by other core electrons. 1n the region exterior to R the potential is simply that due to an effective charge of +e reduced by the bulk dielectric constant K that is, V = eJ Kr. If R were to become quite large the wave function of the donor electron would approach the 4/ :!O "Shallow Donor Emission Cathode Study," First Technical Note, Sperry Report No. NA-8250-8278--1 [Contract i\'o. AF 30 (602) 2495). ~1 H. Reiss, J. Chern. Phvs. 25, 681 (1956). ~ P. E. Kaus, Phys. Rev. 109, 1944 (lQ58). 2a R. l.atter, Phys. Rev. 99, 510 (1955). function in an i~olate(1 Gd:1+ ion, designaled '" level). For R approaching l.l"ro the ~tate woul(11 described by lhe dielectric 4/ funclion y" that hypothetical function for an atom in which the dielectric is imagined to extend to the nUcleus. If largt: tht! dielectric function will constitute a state. Since the actual state in question should be when ~ between the~e extremes, one attempts to stnlct a wave function for the actual donor state a linear combination of the two: The a's are constants which tix the relative "'I and lh; their ratio will depend upon R. The energy which can be associated with the co,np.JSiI. will be the best approximation to the true energy was computed for various values of R conventional variation method. For R less than 3.7 A the analysis indicated a shallow ";=0.008 eV. A sharp transition to deep donor is calculated for larger values of R. If it can be that the appropriate value for R in BaO is not ditTerent from the Pauling radius of the m;;oo; •• ion, which is 1.35 A., the donor elect ron would ~hallow level. 6, EFFECTIVE WORK FUNCTION PLOTS Expt:rimcntal e\'idence of shallow donor behavior several rare-eart.h ions in BaO and also SrO was through direct thermionic tmission m('aSllnemPn'ls saturated emission from each system was determi"ed various temperatures and the da.ta ploUed as work function vs temperature.::! The . function was derived from Eq. (1). A factor of included on the right-hand ,ide as an apparent coefficient associated with the porosit.y of the If the effective work function plot is identified temperature dependence of the Fermi level, values for Ei and the donor density may be Oem'MInI an application of the single-dono r approximatio n. slight curvature predicted by the theoretical ex I""''' [Eqs. (5) and (6)J is ordinarily masked by the ~catt(:ring among the experimental points. In practice, a. straight line is drawn through a live group of points which is interpreted as the to the median point of the group. The slope of a is used to obtain a value for iVd. For slightly donors a is derived by differentiating Eq. (5) respect to T:24 a= (-k, 2)ln (J\'d/1Y ,)+tk. The intercept at T=O is cquallo the Richardson function <PH defmed by <P f: = <P ,,+a T. ~4 The slightly ionized approximati on is used unless there evidence that the donon; aTC shallow. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:49S H .\ I. I. U \I. L> U :\ U I{ THE I{ \I [ () " [c r: ,[ [ T T E R 31;5 (1-1 ) [11 whidl r: n1:.ty be cakulatcu..ln tht::--l' t'.\:prl" . ..;~ion :-;, fill I T(orrl' ~pond to the median point. \" :Ul{ . ' 7. EXPERIMENTAL The lllihod t.: :'~lmp!c:-; \\Tre prepared by decom(Jo sil ion I h' (arbon:,! l'~ on planar but tons of 1 nco 21,1 nickel " 1 l . all r. Th\.' (arbl)llatts Wl'ft; dcpo:;utd on the but tons by 1~;roprt'(ip i\alio ll from saturated ~olutions of tht: ,,' . b··d TI bi .lrbonall' III t'.xces~ car Dille aCi. le rarC-L'art h irn~)uriIY to he coprecipitatt.'d with tht: ~arillm (or strontium) c:lrbonate was added as the nitrate at a ncentralion of abollt 5X 10-6 moles/liter. The prt: ~Opitalion W;\:-i cHeclc::l through the e.il'clro~ytic release of hnlrogl'n at lhe OIckei surface whICh reused t hl' pH and' hl'nel' (he (arbonate ion (Onc('lll rat ion in the "iriniIY of 1 he :,urface. The particles were transported to Ihl' :,urfan by dect rophoresis to form a dcnst: uni form coaling. Coating weights of about one mg/em:! OWC an acta of 0.65 <:tn~ were obtained by thi!:i Lt'ch ni(IUl'. The 10lal concentration of the rart:-t'arth ion in Ihl' t'tJaling \\'a~ determined ~pectrog raphi cally, typical r;liu6 ranging irom 0.01 to o.ns mole %. Spectro ;copically pure barium (or strontium ) carbonate ob tainl1:1 from Johnson-lVlatthey and Co. was used in the preparation of the hath. The ran:-earth nitratc's wen: prrpart:d from oxides of 99.90/0 purity. Each sample was converted, proce ~~cd: and ~tudied in a c!t:mouillablt: tt'st diode slnu:tur e.~'f) A knife-edge sral bl'tWt:t:ll tht: pt'rmancntly mounted envelope and the rtmovablt: base assembly permitted the interchange of r;llhode !'amples. Experimental diode ~t rtl(:tures previously u~t'd at Sperry were t he basis for this design.:!" ;\fter an inil ial evacuation by a c.:hareoal-liquid nit rogen cryogl.'nic pump, t he system wa~ continuou sly exhausted by a titanium ion pump. This arrangem ent eliminated an~ .' pos~ibility of contamination by pump oils; the ulumalt pre:-sure was less than 8X 10-9 Torr. The u:,c of dcrtrocltpo!'iltd sarnple~ also eliminated tht' problem of contamination by binder decomposition products. To funhl'r m:.intain a dean ~vs1em, a fresh nickel allodt: \\":1:-in:'t:rll.'d in the :-'Inh:tu~e ('aeh lime Ihe samplt wa..: changed. Pos!"iblc evaporation of the coating onlo tilt' anodl: wa~ prevented by placing a nickd shield bet wtell [hI; anode and cathodt> during proct:"sing at higher I~mperature s. The shield was manipulat td by an ex lt~llal permanl'nt magnet. .rhe perveance of the diode was estimated to be ~:' X 10<' .vvt, corresponding to a space-charge emis ;lOn dCllsity of 1.9X 10-3 A 'em'! for a standard anode v,," 'Igc f 9 \. S' I· I' . '0 . "mct on y saturate{ emlSSlon currents are . fl' mealllng U In the computation of 4>E, the maximum ~ nsity measured was not premitted to exceed ~}{, W..; Ollhuis, A.STM Symposium on Cleaning oj Electronic rtr;:~c Cl)~lpon{'n ts and yIatcrial, SpC'cial Puhlication 246 .)'J), p. 116. olle tcnth of thi~ valm'. This elTct:tivt'iv diminatnl thl.: inthlt'nn: I)f ~pan' (,lIargc.:! l'~ing: l Kcithkv ~T()dd ·)10 de('(roml'ler tht: practical nlngl' of rurrl'-nt nH:a~Url'- 111tnl was from auout 10-111 to 10-4 A. The t:orrespond· ing tel~p 7rature interval was naturally dependent upon the aCl1vlly of thl' particular cathode. r~ually a range from about 150° 10 500°C was covered. The tempera I lIrt' wa:.; measurl'd to wit hin ±2° by means of aPt Pt Rh thermocouple :-:.pot-welded 10 till' un(kr ~ide of Ihe nickel button. After conversion of Ihe carbonates 10 the oxides, the temperature of Ihe coating \vas rai~t:d to a maximum valuc of about 1050°(' for 11 fe\\' minult:s 10 promote any pos:;;ible solulion proC('s~ bltweell tht' matrix oxide and the additive oxide. This was followed by an activation period of about ~ hour al 8oooe. These temperatur es were nen'ssarily arbitrary since little is kno\vn about the systems in que~tion. Howe\'er, at temperature s in {'XCCSS of 1050°C e\'aporation of HaO ill 1'GCUO becomes objenionable. Emission data for I.:ach plot were l'ol Il.:cted on a point-by-poinl ba~is proceeding at first. to progressively higher temperatures and then n:tracing 10 lower temperature s after reaching the maximum cur rent. Csing this lechnique it was possible to observe any "hy:o:teresis" effects which might be introduced at higher tl.:l11peraLur es, Several plots of this type wt're prepared for ~ach sample. Prior to the recording of data for each individu al plot, the sample wa~ heated for .,\ hour at 8IX)oC for rl'activation. Heating for longer periods or at temperatures up to 900°(' did not yield any ~i~nifi callt change in emission. 8. RESULTS AND DISCUSSION Thl' rl'sults of t.ht' thermionic emis:sion measurements for the v,nious ~ample s art: summarized in Table f. The various entries for the Rirhard:o:on work function, t.he empirical ionizat ion energy, et c., represt'nt a veragt's derived from plots reAecting a higher state of al,tivation of the ~alTIple. Although a decay to lowt:r ~tates of artivation was also observed for several of tht samples, particularly as a "hysteresis" efflTI, thi:-was takt:n as evidence for an increase in the acceptor conn:ntration in an.:oroance with Ihe interpretation given abovt'. Con sl'qut'nlly, thl'se data wen: not considt'rnl signil'icant in krms of tilt: single-donor modt:!. Tht:' averagt: devialion among the valuc :-; ior cPu for a givt:1l ::;umple is abo in duoed a~ an indit:ation of typical reproducibilitv. It is evident that no lowtring of the work function has been tffected by Ihe addition of the various rare-earth ions, typical (lveragt's for cPH approximating 1"+-1.6 eV. The corresponding empirical Eo' values are all 1.6 eV or greater which is at least 0.2 tV greater than the value expected for the defect donor in BaO, The results for the SrO systems were characterized by relatively pro nounced hysteresis effects and mu~t be regarded as \('ss significant than the BaO data, especially since duplicate samples wert: not studied in t he former case. The efi'er live work function at a typiral operating temperature, [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:493176 J 0 II :\ K. GOR:\Ic\'i T.\IlLE 1. Summary of thermioni c emission data. ~~--- --- ~ ~--- ~---~--~~~-- --- ~---~ -----~. Sample .\dditive Additive cone. .\v. cbH . \ \'. f~i :\v.(I' .\v . no. ~la trix ion (mole percent) (cY) (cY) (eV~deg-1 ) ---~-~ --~~ - I BaO nOIlC 2 BaD none 3 BaO La 0.025 -I BaO La 0.025 5 BaO Cd 0.035 (, BaO Cd 0.035 7 HaO ~d 0.010 8 BaO Xd 0.010 9 BaO Er 0~018 10 BaO Er 0.018 II BaO Eu 0~02S 12 RaO Eu 0.025 Ll SrO none 1-1 S,O Eu 0.018 15 S,O Sm 0.050 x Onl~ ' one value availahle. say 7000K, can be obtained from the respective t:ntry- for a by means of Eq, (13), Allhough the apparent donor densities are not listed, they may al:-;o be derived from a through Eq. (12), using a nominal value of 8X 1019 for ,\Te• These densities aTe found to be about 1016 or 1017/cm3, this being considerably smaller than the densities which would correspond to the additive concentration in the doped systems (assuming rompittc solubility). Only in the case of sample 11, Eu in BaO, did the average .V" (3.8X 1018/cm3) approach the total additive concentration (S.9X 101~/cm3). The fact that the differences between the CPR averages for duplicate samples are in general comparabl e wit h the variations among the averages of diJJerent systems casts doubt on the experimental significan ce of these variation s. for reasons which are outlined below it appears probable that the data arc simply character istic of the pure T~aO or SrO matrices in somewhat different states of activation. This would naturally imply that the empirical ionization energy would not reflect the donor behavior of the additive. A definitive interpretation of a negalive n:sult is precluded by the uncertain status of several important factors. It is of interest therefore to consider the various circumstances which would yield such a result, particularly as they relate to the general model. If the impurity ion is to produce a decn'ase in the work function , the following condition s must clearly obtain: (1) a substitutional solid solution must be formed which contains a reasonably high concentration of the additivt:; (2) all acceptors must be eliminated, and (3) the levels associated with the substitutional ion, if not shallow, must at least be somewhat closer to the conduction band than the defect donors. These con· ditions are probably interdependent to some degree. The solubility requirement is an obvious one. If the solubLiity were negligible, all of the additive ion would exist as the separate trioxide phase and the emission would be characteristi c of the pure (defect) alkaline ('ar1h oxide phase. Even if the thermodynami c potential 1.37 1.6 •. IXIO 0.03 1.35 1.6 J~6 om 1.39 l.i 3.8 0.01 1.60 2.1 -I ~ I 0~02 1.5i 2.0 3.5 O~ 10 1.-1-2 1.7 3.8 0.0.1 1.-1-7 1.8 2.8 0.01 tAt \.7 3.6 0,06 Ul 1.7 3.5 Om 1 .. 3 1.7 H a 1.50 2.1 2.0 o.m U8 1.9 2.5 om 1.25 I.. 6.5 0.03 1.82 2.5 -U a 1.47 1.8 6.7 0.01 for solid solution formation exists, phase equilibri, may not have been attained at the highest temlpe,.. reached during processing (tOSO°C)t6; that is, kinetic barrier for the transport of the additive ion be quite large. If a substitutional solution is H" 'mleo.~ position of the associated donor levels will determli whether or not tP is lowered. Consider first a true level or one \vhich at least has an I!.; lower than 1.-1-eV of the oxygen vacancy donor. Activation of system might be expected to proceed less readily t he defect system since t he process would transfer of electrons from eliminated acceptors to levels which may lie appreciably higher in the This would clearly lower the negative free energy over-all reduction. A ~econd possibility is, th,erefOllI,l that a high \vork function may reflect an i'nc,)mJpI activation rather than the high Ri erroncously from the effcclive \vork function plot. If this actually t he case the difficulty might be cir,curnv,,,, I hrough t hc U!:ie of base nickels of higher reducing tial.t7 The equivalence of a low \vork function to a reducing pott:ntial ha~ , in fact, been emphasized Plumlee. II The alternate possibility remains that the level associated with the substitutional tripositive may be even deeper than the l.-l-eV defect level. In event, a high work function would naturally still expected even though thc elimination of acceptors might be enhanced. It is likely that neutral donor would simply consist of the sulbstitutilll ion reduced to its 2+ oxidation state. Thc added tron would then exist in a level characteristic of rare-earth core which would not be very much turbed from the levd in the frec ion. Since the radius of the dipositive ion will be appreciably than that of the triposi1 ive ion, their solubilities ~6 Sample 4. (La in BaO), however was prcconverted in a gen atmosphere, during which a maximllm temperature of t400°C (for one-half hour) was rcached. :7 Magnesium and silicon constitute the principal a" cti,'ato~. the Inca 225 alloy used. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:49S II ALL 0 II' I> 0 :\ () R T II f: R :11 1 () :\ 1 C 1·: \1 ITT E R 31 i7 Ie dilten..:nt. This i~ dCl11oll :-itrated by the \'"ork of q~l' and Ba~lk ~~" {'oncc.rnin~ the ~xidation :-itatL'. of J Ilium ion 1Il 1 he alkahne l'arth oXIdes. By ohservlng ",ro ", I l I 'I l" the ~pcctral emlsS1~ n c.:~ItC~ )y u trano ~t .r'H r~lIon tbt'dipo~itin : and tr~pO~ltl\,C lnll~ ('ould be dl~tll1g~Il ~},ll·.d . the yariou .... mutrlCl':-i. 1t was concluded thaI. l:..u·l ...... IS ~urt'd only 10 .th~ extent th~1I t!lt rt.: .... ultc'll1l.Eu ~+.('aJ1 bf~laiJilizcd by It~ Inl'orporatJOIl mlo the lattice. \\ hen ttlt'limit of ~olubility is Tt';u.:ht'd no fUrl her reduction (J('ftlr:-'. TIl(' r~ldius of Eu~! i..-dose ... ! t () t h<l t oi Sr~+ (1.12 tIld I.I:).\, Tl'spCC1in:ly) and, ('o!lsequt.:ntly, till' fl' duflion i:-i found to oeem to the gn;'lIt':·;t t'xl('l1\ ill Sr(), In ~pilL' of the fact that the Hat) matri :-.: afford ... tlw Jdvantagl' oi a higher II idl'!"t rir ('on~1 a III , llw Iargt'r ionic radiu ... of the lla:!" jim (1.35 . ) i ... a pOI)rt'r ma t ('11 1 I) the radii (citIH;r 2+ or 3+) of tht rarc-carth ion ... gelll'f Illy, ini<:rring a Jcs~ favorabll' ~oJlIhil it~, fanor. 11 mi~h t brn:a:,oned tha.t a simpkr approa('h to a ~ub::-:litutiol1al donor would be to add a dipositin' ion, that is, the BeUtral donor diredly. The rliHirulty hen: i~ that thc \'try t'xi~tenrc of a stahle 2+ oxidation ... tatl· oUhidt: of the matrix in itself suggtsls Ihat the resultant h'\'l,1 will be dl'cp. Of t he tift tTIl ra rt.;-t'art h elt:ll1ellts, till' ('xist t'llre daqua si-~tahlt.: 2+ oxidation stall' has bl'cn l'stah!i!'htc1 .Iy for t:UTopium, yt Il'rbium, thulium) and ::;amarium. :!!1 «thi:st:, thermionic da!;:l werl' strured only for europ iun in BaO and Sr() and samariulll in SrO. As.<;uming that: arti\'ation prooucc ..-a tillin:.! of the additivl: donor len'l::; it is ~till po:-.sihlc that neutr,IL uygcn vacancies may :;till ht· induded in the lattin' to lorm a two·donor sv~txm. HO\\'l'\'l'r, t IH' vacann' donor .. would only aIled 'tilt' Ft·rmi kvel appreriablY if tht: additivl: In'e!s Wl'rt: either d('cp or in low concentration. In this caSl' the effect in' work funclion plot ~hould he chardl'kristic of PUrl' lbO. If tht· actdilivt' l('\'el wen' ~low , any \';[cancy donor, prest'llt should ht: e~~l'l1- tially inl'rt. 9, CONCLUSION It i:-'lpparent thaI all aoequ:1tt· intcrprt:talion of the pre~tnt data would nert~s itak a ciarilic<ttioll of :::oml' of the ~ints discussed in the pre\'iou::: section. Additional ~nmenta l information would be required, partiru- Y Conrt:rning Iht' solu bilit \' and oxidation :,-tale of I hI..' ~e-tarth ion in the matrix.l·niortuna tc\),) the number • exPt . . ~ental method~ wll1l'il can be brought to hear (J9~;. ~I. jafll.' and E. Banks. J. Elcdwchl.'l11. Soc. 102. Sl~ . ~. . '~:i\ Sp{'~ld im; ami :\. II. Daan~·. Til!' Narc F,ur/ft.. (Jphn . O!lS, IlIf .. Xcw \·or1-. 19611. p. 11. on the problem j ... limited by (1) the low umn:nlratioll of the additivl' and (2) the fact that. the sample must be studit:d undcr \'iriually till: same environmental l:ondi tions which obtain in the te~l diode :;tructurc. The above mentioned work of Jafft.· ann Bank ... suggests that tilt' observation of tluore~(xnre might prove informativc . \\,hilt-BaO and SrU arc primarily of ionic character, shallow dOllor stalt:s art' mon..; ('ommon ly oh;erved in matriu .':-i which are kss polar and havt· apprcl'iably loweT IXlIld gaps. '1'hi..-aspect of thl' problem dt:StT\T .... furtlwr \,un..-idcralioll. ThrTl' an:, nc\'crtlll'll' s..-, a few Tepo[b of shallow substitutional ionic donors. including (dh ill ('dS:1\1 and hydrugen in /'nO.:iI,:\:! The laltel' ~y:-i1t'1l1 i..-e..-pt'l'ially signiticant since tht' hand gaps uf Ba() and Znn do not differ greatly (-lA <lnd 3 tY, respectively), The hyurogen is believed to combinl' with an oxide ion on a normal site to form tIw spt,cics OH-a:; which. ba:-;ed on Hall conducti vity mca~urenU'nt s, has all ionization energy of O.fl.,J. c\'y To the exlt'nt that a less polar matrix may tend tn promott' thl' formation of shallo\\' donors, greater ('on sideratioll should be gin'n 10 ::-:urh matt'rial ", a~ possible hlJ~t latlict'i' in the ~mittTr ('untl'xt. Thl' ~parsity of (':-,:pt'rimtntal t,\,id(!nn' preclude s any gt'll('ra!ization ~ regarding tht.; compatibility of a 10\\' ell'(,tron amnity and a low band gap. It might ht.' mentioned. how('ver, that tht' photoelt'ctr ic work of Spict::rUJ indicated all electron aninilY of 0.6 t.:\' and a band gap of 1,.1. e\' for ("s: .. Sb. Ex1l·n .-:ioll of shallow donor tmi.,,~i()n rt'~l'arrh 1(, les~ polar ll1ilteri:.\I .. would, in any ca.se, entail addiliona l preliminary ml·;J:-illft:ment ..-of x. pl)~~ibly Ihrou~h 1l1)I)ii ration of tht: techniqul's elL-snibed abon·. ACKNOWLEDGMENTS Tht' author is illdd.llt'd to i'rofc ~:-;ur E. B. l-lcn..-!t:y uf tlH' rniycr:-;ity of \Ii..-~ouri and to Dr. C. C. \\'ang of thl' Spt.'rry (;~'ro sful)t' Company for their many valuable ;o;uggrslions during thl' ('our:;l' of this work. Thallk:-art al~o due 10 Dr. 1.. Hulmbo e and Dr. 1<. \Y. Olthuis of Spl'rry ior their coml1l<:nt~ in n:vil'wing the manuscript. 30 F .. \. Kroger, II. J. \'ink. and Van den BOlltngaard, Z. Physik Cnell1. 8203. I (19S-lI. 3: D. G. Thomas nnd j. j. Lander, J. ('hcm. Ph .... s. 25, IUC! (19S6 t. .!~ .\. R. Hutson. Repurt on :-:;('\,cll1\.'(·nth :\nnuaJ Ph~'~ica[ Electronic:; Conicrcncc. :\Ias:>athusctts Institute of Tt·chnol og:~·. Caml.rid/{(·. :\Iassachusl'tts. 19:;;, p. if>. J,l It has u(,(,1l suggested I,y Plumlee (refl'rl.'nl:e II! that a similar ,;pccit's constiLUtt·S till' dd('ct donor in t[l(' (Ha, Sr 10 cathode . 31Thc ckC'lron atlinit\· of ZnO. hm\·{'\·cr. ha~ l'l·l.'n estimaled to tit· 3 !O 4-(,\'. (rct'{"H:ncc·;.. . [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.177.236.98 On: Mon, 15 Dec 2014 03:48:49
1.1728416.pdf	Properties of HighResistivity Gallium Arsenide Compensated with Diffused Copper Joseph Blanc, Richard H. Bube, and Harold E. MacDonald Citation: Journal of Applied Physics 32, 1666 (1961); doi: 10.1063/1.1728416 View online: http://dx.doi.org/10.1063/1.1728416 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Compensation assessment in ‘‘undoped’’ highresistivity GaAs J. Appl. Phys. 66, 256 (1989); 10.1063/1.343866 Diffusion of chromium in gallium arsenide J. Appl. Phys. 59, 2398 (1986); 10.1063/1.336341 Properties of a highresistivity layer in epitaxially grown gallium arsenide film Appl. Phys. Lett. 22, 446 (1973); 10.1063/1.1654706 Electrical Properties of HighResistivity NickelDoped Silicon J. Appl. Phys. 41, 2644 (1970); 10.1063/1.1659275 Diffusion of Tin in Gallium Arsenide J. Appl. Phys. 32, 1180 (1961); 10.1063/1.1736193 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53JOURNAL OF APPLIED PH YSICS VOLUME 32, NUMBER 9 SEPTEMBER, 1961 Properties of High-Resistivity Gallium. Arsenide Compensated with Diffused Copper* JOSEPH BLANC, RICHARD H. BUBE, AND HAROLD E. MACDoNALD RCA Laboratories, Radio Corporation of America, Princeton, New Jersey (Received November 30, 1960) Low-resistivity n-type GaAs crystals with silicon donors are compensated with diffused copper to produce high-resistivity crystals in a manner which is amenable to semiquantitative description in terms of a simple thermodynamic mode!. The high-resistivity GaAs :Cu crystals are subjected to photoelectronic analysis, including room temperature Hall and photo-Hall measurements, to obtain information about the effects of deep-lying imperfections on the properties of the initial n-type GaAs. In addition to three deep d?nors previously reported, five acceptors are revealed. A 0.42-ev acceptor level, when compensated, pro vides a long electron lifetime resulting in high n-type photosensitivity at low temperatures. Evidence for eff~t~ ?n the electron mobility is obtained for compensated deep donor levels, important mainly in high resistivity n-type material, and for compensated acceptors lying 0.22 ev above the valence band important mainly at low temperatures. ' INTRODUCTION GALLIUM arsenide is a heteropolar III-V semi conductor with a band gap of about 1.4 ev at 300°K. It thus occupies a position intermediate between the lower band gap group IV elemental semiconductors, e.g., Ge and Si, and the higher band gap II-VI photo conductors, e.g., CdS and other chalcogenides of Zn or Cd. When the resistivity of GaAs is low because of the incorporation of suitable imperfections, its elec trical properties resemble, qualitatively at least, those of Ge and Si. When the resistivity is much higher (the intrinsic resistivity at 3000K is of the order of 10L 109 ohm cm), behavior is found which is very similar to that encountered in II-VI compounds. A previous in vestigationl of high-resistivity crystals of n-type GaAs produced by normal growth processes has indicated the presence of donor levels lying about 0.5, 0.6, and 0.7 ev below the bottom of the conduction band. When these centers are compensated, they act as electron traps and can be so detected. Strong evidence was also found for a density of shallow trapping centers with an activation energy of about 0.2 ev. The present investigation started with low-resistivity n-type GaAs (the donor impurity presumably being Si), and had for its aim the reproducible compensation of this material to resistivities greater than 103 ohm cm, so that experimental techniques suitable for the high resistivity range might be usefully applied. In addition to revealing the properties of imperfections, both chemical and structural, in the compensated material, it was hoped that a comparison of properties before and after compensation would yield some insight into the effects of deep-lying imperfections on the properties of the initial low-resistivity n-type GaAs. Copper was chosen as the compensating agent in this investigation because it is the most thoroughly studied acceptor in GaAs and offers several distinctive * The research reported in this paper was sponsored by the Air Research and Development Command, United States Air Force, under contract. Some of these results were reported in preliminary form at the Prague Conference on Semiconductor Physics, 1960. 1 R. H. Bube, J. App!. Phys. 31, 315 (1960). advantages. Fuller and Whelan2 have studied the solu bility of copper in GaAs as a function of temperature, and have shown that copper diffuses rapidly at rela tively low temperatures. The acceptor ionization energy of 0.14 ev has been established by Meyerhofer and by Whelan and Fuller,4 who also showed that each copper atom acts as a singly ionized acceptor in n-type GaAs. Thus, a thermodynamic characterization of Cu in GaAs is possible, and diffusion experiments can be made on a reasonable time scale. This paper describes both the phenomena involved in the compensation process as well as the results of photoelectronic analysis of the resultant high-resistivity GaAs: Cu crystals. To standard techniquesl involving photoconductivity, spectral response, infrared quench ing, and thermally stimulated current, have been added measurements of Hall effect and photo-Hall effect at room temperature. The location of five acceptor levels is determined, together with the behavior of these various centers relative to the photoconductivity pro cess. The possible correlations between the existence of various levels and the mobility of the initial material are explored. EXPERIMENTAL Preparation of Crystals All 14 samples of GaAs reported on in this investiga tion were monocrystalline slices cut from ingots grown by a conventional Bridgman technique.6 The crystals are listed in Table I in order of increasing electron con centration at 300°K. This ranges from 1.0XlOl6 cm-3 to 5.0X1Q17 cm-3; the electron mobility (also given in Table I) varies from 2700 to 5600 cm2jv sec at 3000K, 2 C. S. Fuller and J. M. Whelan, J. Phys. Chern. Solids 6, 173 (1958). 3 D. Meyerhofer, Prague Conference on Semiconductor Physics, 1960; F. D. Rosi, D. Meyerhofer, and R. V. Jensen, J. App!. Phys. 31, 1105 (1960). 4 J. M. Whelan and C. S. Fuller, J. App!. Phys. 31, 1507 (1960). • L. R. Weisberg, F. D. Rosi, and P. G. Herkart, in Properties of Elemental and Compound Semiconductors (Interscience Pub lishers, Inc., New York, 1959), Vo!' Vof Metallurgical Society Conferences, pp. 25-65. 1666 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53HI G H -RES 1ST I V I T Y G a A s COM PEN SAT ED WIT H D IFF USE D C u 1667 and from 2100 to 13300 cm2/v sec at 78°K. In any given sample, the electron concentration did not de crease by more than 10% in going from 3000 to 78°Kj semiquantitative spectrochemical analysis, where avail able, is in accord with the assumption that the electron concentration is nearly equal to the Si impurity con centration. This supports the hypothesis· that Si is the principal shallow donor in these samples, and that it is not highly compensated. The crystals listed in Table I fall into three categories. First, crystal 10G-32 was used as a reference exhibiting the "normal" Cu acceptor level with 0.14-ev ionization energy; after diffusion of Cu at 650°C, as described below, the crystal was p type with hole concentration of 9X1016 cm-3 and hole mobility of 270 cm2/v sec at 300°C, and had been previously measured by conven tional Hall techniques by Meyerhofer. Second, there are the four crystals designated as the 631 series; these crystals had low initial electron concentrations within a factor of 1016 cm-3 and high ratios of electron mobility at 78°K to that at 300°K. Cu was diffused into the crystals of the 631 series at 500°C. Third, there are the other nine crystals listed in Table I, which were annealed at successively higher temperatures, as de scribed below, until their resistivity exceeded 103 ohmcm. Compensation by Cu was achieved through the fol lowing set of procedures. The low-resistivity n-type GaAs crystals, with dimensions of approximately lX2X6 mm3 (volume of about 0.01 cm3) were electro plated with 1017 atoms of Cu or greater. One or more samples were then placed in quartz vials, evacuated to a pressure of 10-6 mm Hg and sealed. The samples were annealed in a simple controlled furnace for a period of 16 hr, and then were quenched by manually plunging the vials into a water bath. The nine specimens on which a detailed study of the compensation phenomenon was made as a function of increasing initial electron concentration are listed in Table II. These crystals were first annealed at 575°C; those samples which still had a resistivity below 103 TABLE I. Initial properties of GaAs. Jln Jln Diffusion nat 300oK, (300 OK) (78°K) temperature Crystal cm-3 cm2jv sec cm2jv sec °C 631-14 1.0X1016 5000 12600 500 10G-32 l.lX1()l6 4300 9300 650 631-9 1.2X1016 5600 13300 500 631-8 1.3XIOl6 5200 12400 500 631-1 1.9xl016 3700 6900 500 GAJ-18-2a 3.8X1016 4100 4800 575 ES44 4.1XI016 4900 7300 575 GAJ-18-2b 5.8XI016 5000 6400 575 GAJ-18M 7.3X1616 4600 5700 575 1OG40-4 8.8X101s 4200 4800 600 ES36 1.2X 1017 4400 4900 650 ES37 1.3 X 1017 4300 4700 650 ES 41 3.4XI017 3200 650 GAJ-18F 5.0X1017 2700 2100 750 TABLE II. Electrical properties of GaAs after diffusion at various temperatures. Initial electron concentra- Crystal tion. em-3 GAJ 18-2a ES44 GAJ 18-2b GAJ IBM ES36 ES 37 ES 27 ES41 GAJ 18F 3.8 X 1011 4.1 XI01$ 5.8 XI018 7.3XH)l1 1.2 XIO·7 1.3 XI017 2.2 XI017 3.4 X 10'7 5.0XI0 11 Electron concentration, em-a t after diffusion at: 575'C 650·C 7 SO·C p > Ill' ohm-em p>Ul' ohm-em p >103 ohm-em p>l03 ohm-em 1.1 XI0lf 5.0 X 1010 1.9 XI011 3.4XH)l1 4.3 X1017 p>l03ohm-em p >10' ohm-em p>l03ohm.em p>l03 ohm-em 2.1 X 1017 p >103 ohm-em ohm cm (n type) underwent the same treatment at 650°C. One sample was repeatedly annealed at 700° and 750°C before the resistivity increased to above 103 ohm cm. The details of the compensation process are summarized in Table II, and the final annealing temperatures before the beginning of photoelectronic analysis are given in Table I as well.6 A separate set of control experiments showed that similar crystals an nealed at the same temperatures, but in the absence of Cu did not become compensated. Spectrochemical analysis of two of the crystals for Cu content by Whitaker confirms that the density of Cu incorporated is within 25% of the initial electron concentration, and that therefore the high-resistivity behavior is indeed due to a close compensation of a shallow donor by the diffused Cu. Measurements All measurements were made on crystals with ohmic indium contacts, melted onto the crystals in vacuum at about 350°C. Photoelectronic measurements as a function of temperature, light intensity, wavelength, etc., were made in an atmosphere of dry helium in a suitable cryostat. Measurements of Hall effect and photo-Hall effect were made using the apparatus de scribed in a previous publication,7 in which a Cary 31-31V vibrating-reed electrometer is used for measure ment of both voltage drop in the crystal and of the Hall voltage. A conventional6-contact arrangement was used, measurements being made with both directions of applied voltage and both directions of magnetic field. Excitation by monochromatic radiation was obtained from a Bausch & Lomb grating monochromator, or with interference filters where indicated. Since the spec tral response (see Fig. 8) of photoconductivity in these GaAs crystals is limited almost entirely to volume absorbed light, i.e., to light with wavelength greater than the absorption edge, some measurements were made with a broad band of excitation derived from a No. 1497 microscope lamp operated at 6 v. The crystal 6 Four more samples were diffused, which, after diffusion, were highly inhomogeneous as indicated by Hall measurements; longer annealing times did not remove the inhomogeneities. By the same criteria, the 14 samples reported on here were homogeneous, although not necessarily on a microscopic scale. 7 R. H. BubeandH. E. MacDonald, Phys. Rev. 121,473(1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:531668 BLANC, BUBE, AND MAcDONALD T ABLE III. Summary of photoelectronic data on GaAs: Cu crystals. Majority Thermally stimulated Dark Dark conductivity carrier conductivity activation energy, lifetime, Activation energy current data 300oK, ev }lsec from tJ.i vs T, ev Trap depth, Trap density Crystal mho/cm High T Low T 3000K 900K High Ta Low Ta ev cm-3 631-1 10-8 0.66 0.42 0.04 8 0.09 1 0.26-0.43 1013 0.59 2X1016 631-8 10-8 0.58 0.37 0.13 0.003 0.26 0.53,0.69 1018 631-9 10-7 0.28 0.15 0.1 0.004 0.13 0.56 3X1016 ES 41 10-6 0.42 0.25 0.2 310 0.34 ES36A 10-6 0.45 0.05 1600 0.30 b 0.23 2X1014 lOG 40-4 3XI0-6 0.45 0.14 1700 0.37 j 0.25 2X1014 GAJ-18-2a 3XlO-6 0.40 0.2 200 0.21 0.26-D.32 4X1015 GAJ-18-2b 3XIQ-6 0.40 0.04 2 0.32 0.28-0.39 3XI014 GAJ-18M 5XlO-s 0.43 0.3 43 0.34 0.33 2X1012 631-14 5XIQ-s 0.43 0.12 0.2 0.002 0.28 0.03 ES 37 10-5 0.36 0.25 4 0.07 0.25 0.08,0.04 GAJ-18F 3XI0-s 0.35 0.22 3 0.007 0.10 ES 36 3XI0-4 0.21 3 0.06 0.09,0.03 ES44 8XlO-4 0.17 2 0.002 0.11,0.05 lOG32 10-1 0.12 46 0.04 0.09 a Photocurrent increasing with increasing temperature. b Photocurrent shows sudden temperature Quenching with increasing temperature as described in the text. Equivalent activation energy for sensitizing centers is 0.41 ev. Confirmed also by infrared quenching data. itself provided the effective short-wavelength cutoff of this band, and a water filter was normally used to give a long-wavelength cutoff at 1 f.L to exclude optical quench ing effects, as described in this paper. The use of this source had the advantage of providing a considerably higher exciting intensity; wherever corroborating checks were made with volume-absorbed monochromatic radiation, no differences, except that of intensity, were observed between the two types of excitation. In agree ment with these findings, the nature of the effects ob served and the magnitudes of the imperfection-level densities, calculated on the basis of the total volume of the crystal, were such as to confirm the contention that bulk properties were measured. The intensity of excitation was varied in a controlled manner by the use of calibrated wire-mesh neutral filters, accurate over 7 orders of magnitude. RESULTS Dark Conductivity The GaAs: eu crystals of this investigation are listed in order of increasing room temperature dark conduc tivity in Table 111.8 These conductivities cover the range from 10-8 to 10-1 mho/em. With the exception of crystal 631-1, the room temperature conductivity of all crystals is p type, as will be discussed in more detail under the section of Hall measurements. The dark conductivity of all these crystals was meas ured as a function of temperature within the range be- 8 Crystal ES 36 underwent photoelectronic analysis and then had new contacts applied for Hall measurements. In the course of this heating procedure, the dark conductivity dropped from 3XI0-4 mho/cm p type to 10-6 mho/cm p type, probably due to the precipitation of copper. The crystal with conductivity of 10-6 mho/cm has been called ES 36A and was subsequently sub jected to both Hall and photo electronic measurements. tween 90° and 4000K, the lower limit actually used being set by the lower limit to current detectability which was 10-11 amp. About half the crystals showed only a single slope in a plot of log conductivity vs l/T, sometimes over as many as 7 orders of magnitude of conductivity. The rest of the crystals showed one slope at higher temperatures and a smaller slope at lower temperatures. In every case, a slope observed first only at lower temperatures in higher-resistivity crystals, was observed over the whole range or at higher tempera tures in lower-resistivity crystals. Since the crystals involved were partially compen sated p type, with N A acceptors partially compensated by N D donors, in the range where p«N D,9 Ej=E+kTln[ND/(NA-ND)], (1) where Ej is the height of the Fermi level above the top of the valence band and E is the activation energy of the uncompensated acceptor centers. Equation (1) holds over any range where conductivity is associated pre dominantly with only one type of acceptor. Since the slope of a plot of log conductivity vs l/T is given by Slope= (T/k) (dEf/dT)- (Ejlk), (2) the slope is equal to -E/ k. This analysis assumes as a first approximation that the temperature dependence of the density of states cancels the temperature depend ence of the mobility. Even if the temperature depend ence of mobility departs appreciably from such an ap proximation, corrections of only a few hundredths of a volt in the given activation energies would be involved. If, for example, the mobility were independent of tem perature, the true value of E would be about 10% less than the value given by the slope. It is possible that 9 R. H. Bube. J. Chern. Phys. 23, 18 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53HIGH-RESISTIVITY GaAs COMPENSATED WITH DIFFUSED Cu 1669 IO',--..,---r.;;::----,---,-----,---,.--n 10."2 7 8 9 FIG. 1. Dark conductivity as a function of temperature for representative GaAs: eu crystals. The data are taken from a continuous recording, and experimental points have been therefore omitted. some of the observed variations in activation energy for a given imperfection level are traceable to variations in temperature dependence of mobility. Typical curves of dark conductivity as a function of temperature for a number of the crystals are given in Fig. 1. The dark conductivity activation energies are summarized in the third column of Table III and are graphically represented as a function of room tempera ture dark conductivity in Fig. 2. The results can be interpreted in terms of four acceptor levels located at about 0.42, 0.34, 0.22, and 0.13 ev above the top of the valence band. In addition, evidence is obtained in the two highest-resistivity crystals of donor levels lying ~.o.7'-----'----;---'----r---"'---.-----' ~ '" II: ~0..6 '" z o ~Q5 > 5 "'0.4 ~ :> ~ gO.3 o ~ UQ2 DONOR ENERGIES ACCEPTOR ENERGIES ------:-~~L0.42eV(tO'O05eV) • (2) ----------~-.! -O.34ev(tO.OI5ev) • o 0 ------- --r -O.22ev(:!:O.OIev) '" . ~ ~ ____ ~ _______ ~~~O~~~ o 0.1 ,":_,...---'c,.,---..J.-::z---'-~--L..::r--""""'''''--'''''''''''----' 10. 10 10. 10.- 10- 10.- 10.- 10' DARK CONDUCTIVITY AT 3OO0K,(o.hm-cmr' FIG. 2. Summary of the occurrence of various dark conductivity activation energies as a function of the dark conductivity of the GaAs: eu crystals. Solid dots imply that the activation energy is found either over the whole temperature range or at higher temperature; open dots imply that the activation energy is found only at lower temperatures. The activation energies plotted are those given in Table III. 0.6 and 0.7 ev below the bottom of the conduction band in agreement with past resultsl; the detailed rea sons for this interpretation are given in the section on Hall measurements. Photoconductivity vs Temperature In general, the crystals listed in Table III can be divided into two oppositely behaving groups as far as the temperature dependence of photoconductivity is concerned. Crystals with room temperature dark con ductivity less than 5 X 10-6 mho/ cm show a rapid in crease in photosensitivity upon cooling; crystals with room temperature dark conductivity greater than 5 X 10-6 mho/ cm show a rapid decrease in photosensi tivity upon cooling. This behavior is reflected in the values of the majority carrier lifetime given in the fourth column of Table III. The majority carrier lifetime is calculated from the equation 3r-r-,-..,-,.-,--,-n ) r'OG40_4 ," ,.:' z w a: a 3r=;:=:::;::~~~, ~::;=~;:::;:::;::::;:~ ~,o. r-Es 36A 021 &: 10 ,0. 23456789 (3) FIG. 3. Typical variation of photocurrent with temperature for GaAs: eu crystals with a room temperature dark conductivity less than SXlO-6 mho/em. where L is the interelectrode spacing, G is the photo conductivity gain (i.e., the number of charge carriers between the electrodes for each photon absorbed), p. is the carrier mobility, and V is the applied voltage. The use of the concept of majority carrier lifetime im plies simply that the photoconductivity is dominated by carriers of one type. Presumably the minority car riers will be captured first, and then recombination with majority carriers will take place at a later time. Naturally, the majority carrier lifetime under photo excitation need not refer to the same carriers as are the majority carriers in the dark; only Hall effect meas urements can identify the majority carriers under any particular condition of photoexcitation. The identity of the majority carriers is not specifically required for the calculation of a lifetime by Eq. (3), if one is willing to assume a reasonable value of mobility to obtain an order-of-magnitude figure for the lifetime. In those cases [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:531670 BLANC, BUBE, AND MACDONALD for which actual Hall measurements were not available, a mobility of 1()3 cm2/v sec was assumed to give such order-of-magnitude lifetime values. An inspection of Table III shows that most of the crystals with con ductivity less than 5XlO-6 mho/cm are 1()2 to 1()4 times more sensitive at 900K than at 300oK; crystals with conductivity greater than 5XlO-6 mho/cm, however, although as much as 1()2 times more sensitive than the higher-resistivity crystals at 300°K, are 1()2 to 1()3 times less sensitive at 900K than at 300°K. Figure 3 gives a number of examples of those crystals for which photosensitivity increases with decreasing temperature. The increase occurs abruptly and over a narrow temperature range. Figure 4 shows the depend- ~IO :::: <I ::t.. ~I z LLI a: a: '1 :> 10 v ~ ifl' E SENSITIZING = 0.41 ev SP. 6.103 Sn ~~2--~3---4L-~5---6L-~1---8L-~9--~10-J ~T' oK'I. 10'3 FIG. 4. Dependence of photocurrent on temperature for crystal GAJ-18M for two different light intensities. From the temperature breakpoint with increasing temperature, and its dependence on light intensity, the hole ionization energy for sensitizing centers is calculated to be 0.41 ev, and the ratio of hole capture cross section to electron capture cross section is calculated to be 6X loa. ence of photocurrent on temperature for crystal GAJ IBM for two different intensities of excitation. Low temperature Hall measurements by Whelan10 on such materials have shown that the photoconductivity is n type. The behavior is therefore identical with that found in such materials as CdS and CdSe, where ther mal quenching of photoconductivity is associated with the thermal freeing of holes from centers (called "sen sitizing centers") with a much larger capture cross sec tion for holes than for electrons. Such centers have an effective negative charge; photoexcited holes are readily captured, but the subsequent probability for capture of a photoexcited electron is small. Thus, the majority carrier lifetime is long and the photosensitivity is high. If this process is analyzed in terms of a rate equation analysis previously applied to such materials as CdS and CdSe,1l it is possible to determine both the location of the energy level of these sensitizing centers and the 10 J. M. Whelan (private communication). 11 R. H. Bube, J. Phys. Chern. Solids 1, 234 (1957). 10 .. I :t. ~ z ~ 2 3 4 a:: alO 5 l: Q., 4 5 6 2 10 631-14 10 ,lsev ~ev 3456189103456789 I/T:K~I x 10· 3 FIG. 5. Typical variation of photocurrent with temperature for GaAs:Cu crystals with a room temperature dark conductivity greater than 5X 10-6 mho/cm. ratio of the capture cross sections from the data of Fig. 4. If this is done, it is found that the sensitizing centers lie 0.41 ev above the top of the valence band, and the capture cross section of these centers for holes is 6 X 103 times larger than their subsequent cross section for electrons. As is indicated by the graphs of Figs. 3 and 4, the photocurrent in these crystals also rises exponentially with increasing temperature beyond a minimum follow ing temperature quenching. The activation energies corresponding to this exponential rise in photocurrent are summarized in the left portion of the fifth column of Table III. These energies may be divided into three groups with the following average values: 0.33±O.01 ev (5 crystals), O.25±O.01 ev (4 crystals), and O.11±O.02 ev (2 crystals). The temperature dependence of crystals with dark conductivity at room temperature in excess of 5XIo-6 mho/cm is illustrated by the examples given in Fig. 5 and summarized in the right portion of the fifth column of Table III. The photosensitivity decreases exponen tially with decreasing temperature, usually with one or both of two characteristic slopes: O.09±O.OO4 ev (5 crystals), or O.04±O.OO4 ev (4 crystals). Obvious exceptions to the separation of the crystals into two groups depending on their room temperature 10 .. :t. ,..,. Z LLI ~I :> u ~ 0 :I: Q. °3 4 6 ~~--+-~--~--8~-J903~~4~~--~6--~~8~~9 ¥T,·K-1x 10-! FIG. 6. Variation of photocurrent with temperature for GaAs: Cu crystals 631-8 and 631-9, indicating the rapid decrease of sensitivity observed with these crystals at low temperature. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53HI G H -RES 1ST I V IT Y G a A s COM PEN SAT ED WIT H D IFF USE D C u 1671 dark conductivity are 631-8 and 631-9, which have a smaller majority carrier lifetime at 90° than at 300°K. The actual dependence of photocurrent on temperature for these crystals is shown in Fig. 6. It is seen that they are like other high-resistivity crystals in that the photo sensitivity rises rapidly with decreasing temperature. They are unlike other high-resistivity crystals in that the photosensitivity falls off again sharply at low tempera tures.12 The curves of Fig. 6 are measured at high light intensity and indicate about equivalent photocurrent at 300° and 9OoK. The majority carrier lifetimes, how ever, are calculated for low light intensities, and the difference between sensitivity at 3000K and 900K for these crystals becomes much greater at low light in tensities, as shown in Fig. 7. Thus, these two crystals > 8 ~u5t :l .... Z \oJ II:: II:: ~IO ~ Q.. 0-631-8 >-631-9 16--1O~'''''1 ",---..I.---lI-.l.--L..-I.IO-l'---l-....J102 LIGHT INTENSITY FIG. 7. Variation of photocurrent with light intensity for GaAs: Cu crystals 631-8 and 631-9 at90° and300oK. The difference in sensitivity is much greater between the two temperatures for low-intensity excitation than for high. do fit the general patterns of high-resistivity crystals except for an additional unidentified process which re duces sensitivity at low temperatures. Spectral Response and Optical Quenching The absorption edge of GaAs at 900K is located at about 8400 A, corresponding to a band gap of about 1.47 ev. Spectral response curves are given in Fig. 8 12 Over an appropriate range of light intensities and applied voltages at 90oK, an oscillating photocurrent was found in crystal 631-8, in the presence of steady illumination and a normal de voltage. The phenomenon is essentially identical with that re ported for CdS [So H. Liebson, J. Electrochem. Soc. 102, 529 (1955); R. H. Bube and L. A. Barton, RCA Rev. 20, 564 (1959)] and for ZnSe [R. H. Bube and E. L. Lind, Phys. Rev. no, 1040 (1958)]. The oscillations are characterized by a slow buildup, often with accurately reproduced complex structure, followed by a sudden decrease. Buildup and relaxation of space-charge effects seems the most likely mechanism for such behavior. FIG. 8. Typical spec tral response curves at 900K for GaAs: Cu crys tals with high sensitivity (ES 41) and with low sensitivity (GAJ-18F). The curves are corrected for equal photon inci dence at each wave length. The rapid de crease in sensitivity at wavelengths above 10000 A for crystal ES 41 is the result of optical quenching. 900K 10 I~.~~~~~~~~~~~ 4000 6000 8000 10 000 12000 14 000 WAVELENGTH. A for two crystals of GaAs: eu; crystal ES 41 is one that has high sensitivity at low temperatures; crystal GAJ- 18F is one that has low sensitivity at low temperatures. Both curves indicate that the photosensitivity for sur face-absorbed light (in the spectral range where the absorption constant is very large) is very low. The sur faces of the measured crystals were subjected to abrasive blasting before attachment of electrodes, and the result of this treatment is apparently a high surface recombina tion velocity as is the case in Ge. The photosensitivity rises sharply as the exciting wavelength exceeds that of the absorption edge and volume excitation becomes predominant. Beyond the edge, the response stays high over quite a range during which excitation from imper fections dominates. The curve for insensitive crystal GAJ-18F is similar to those reported for high-resistivity 100 , .1, .0'..' I. .1 _ 90 111 BIAS' O.1JLA '\ ao 1 li70 ~IOJLA t-z \oJ 560 l- I z I ~so- II:: / t \oJ I IL 40 I \ lOl-I , 20 I 10 I - 0 I I I I I I 0.4 0.5 0.6 0.7 0.8 0.. 1.0 1.1 1.2 PHOTON ENERGY,ev FIG. 9. Typical optical quenching spectra for GaAs: Cu crystal GAJ-i8-2a, measured at 9OoK, for two different values of the bias current generated by 8580 A. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:531672 BLANC, BUBE, AND MACDONALD 0.1 234567891011 yT,oK" x 163 FIG. 10. Temperature dependence of optical quenching, measured in terms of the ratio of photocurrent without an H20 filter to exclude the quenching radiation to the photocurrent with an H20 filter; for crystal 631-1. n-type crystals! : an appreciable fraction of the maximum response is still found when the exciting energy is as low as 0.9 ev. The response for sensitive crystals, however, shows a sharp decrease as the exciting photon energy decreases below 1.2 ev, and is down by a factor of about 1()4 at 1.1 ev. This sharp decrease occurs as optical quenching by photons with energy less than about 1.2 ev exceeds the excitation by these same photons. Such optical quenching occurs when the light raises electrons from the valence band to the sensitizing centers, thus freeing holes which subsequently are captured at sites where recombination with free electrons is probable. The pro cess of optical quenching was investigated by exciting a bias photocurrent by 1.4S-ev light through an inter ference filter, and measuring the quenching effect as a function of wavelength of a secondary monochromatic radiation. The results are shown in Fig. 9 for two differ ent intensities of the bias excitation. The quenching effect was extremely strong and only by going to high bias-excitation intensity was it possible to detect some shape in the curve apart from simply total quenching. The monochromator available for the quenching ex periments did not permit coverage of the whole range, but a reasonable extrapolation of the spectrum indicates an optical quenching energy of about 0.4 ev. The de crease in percent quenching on the high-energy side is not the result of a decrease in actual quenching, but rather of an increase in excitation by these wavelengths which obliterates the quenching effect. The temperature dependence of the optical quenching effect is pictured in Fig. 10 for crystal 631-1 to demon strate that the optical quenching disappears as thermal quenching sets in, as is evident by comparison with the data for crystals 631-1 in Fig. 3. The data of Fig. 10 are given in terms of the photocurrent excited by a broad-band excitation spectrum from an incandescent lamp without interposing a water filter to that obtained when a water filter is interposed. The interposition of the water filter eliminates those wavelengths which cause optical quenching. Finally, at the high-tempera ture end, when optical quenching is no longer present, the current is higher without the water filter because of the reflection and scattering induced by the filter. Thermally Stimulated Currents The crystals of Table III fell into four categories as fa: as the possibility of measuring significant thermally stImulated currents was concerned. First, there were the crystals with room temperature dark conductivity greater than 5 X 10-6 mho/ cm which had relatively high dark currents even at low temperatures and very short free carrier lifetimes at low temperatures; for these crystals no measurements of thermally stimulated cur re?ts were possible. Second, there were the crystals WIth room temperature dark conductivity less than 5XlO-6 mho/cm which had high sensitivity at low temperatures, but still fairly high dark currents at higher temperatures; for these crystals only measure ments of thermally stimulated current at low tempera tures due to shallow traps were possible. Third, there were crystals 631-8 and 631-9, for which the low tem perature sensitivity was too low to permit detection of shallow traps, but for which deep traps could be de tected because of low dark conductivity at higher temperatures. Fourth, there was crystal 631-1 which combine.d the properties of high sensi tivi ty at low temper atures WI th low dark conductivi ty at higher temperatures for which both shallow and deep traps could be meas~ ured. These limitations on detection must be considered when interpreting the summary of thermally stimulated current data given in the sixth column of Table III. There is no evidence that the same basic pattern of a low density of shallow traps in the range 0.2 to 0.4 ev, and a higher density of deeper traps in the range 0.5 to 0.7 ev, is not characteristic of all the crystals of Table III. Figure 11 shows the thermally stimulated current curves for three of the crystals for which only shallow tr~ps . we.re ~etectable. All of the curves are complex WIth mdIcatIOns of considerable structure; perhaps as many as four different trap depths are indicated by the three curves. On the reasonable assumption that the ther:nally. stimulated curr.ent is contributed by majority carr~ers, . I.e:, those carners for which the majority carner lIfetIme has been determined, and that these majority carriers are electrons, as indicated by Hall effect on the photoconductivity in this temperature GAJ-IS- 2b GAJ-ISM 0.0002 0.0002 0.01 o '--'L-;!::::-'--~-'---~-.J L-'-:o-1:!:;60.-'---~140::-'----'!'2""0 lL-.J L-'-~--,-"--,---"~-.J TEMPERATURE. ·C FI~. 11. Thermally stimulated current associated with a low ~e,!-slty of shallow traps, typical of crystals with high photosensi tIvity at low temperature. Note structure in the curves. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53HI G H -RES 1ST I V I T Y G a A s COM PEN SAT ED WIT H D IFF USE D C u 1673 range,1° the trap depth may be calculated: Etrop= E/n=kT In(Nc/n), (4) where Ejn is the distance of the steady-state electron Fermi level below the bottom of the conduction band, T is the temperature of the thermally stimulated current maximum, Nc is the effective density of states in the conduction band, and n is the density of free carriers at the thermally stimulated current maximum (calcu lated in this case from the measured conductivity as suming an electron mobility of loa cm2/v sec). The den sity of traps of a given depth can be determined from the area A under that portion of the curve Nt=A/evG, (5) where Nt is the trap density, e is the electronic charge, v is the volume of the crystal, and G is the gain calculated for steady-state excitation of a photocurrent with magnitude equal to that of the average thermally stimulated current. FIG. 12. Thermallystimu lated current associated with a high density of deep traps, measurable only in the highest resistivity crys tals. The traps found are the same as those previously reported for high-resistivity n-type crystals. 0.09'r----------, 0.08 0.01 o 100 TEMPERATURE,'C The thermally stimulated current associated with deep traps in crystals 631-8 and 631-9 is shown in Fig. 12. These curves are essentially identical with the curves associated with deep traps reported for high resistivity n-type crystals previously.l The happy combination of circumstances which per mits both shallow and deep traps to be detected in crystal 631-1 gives rise to the curves of Fig. 13. The shallow traps again exhibit marked structure. In all the crystals, the density of shallow traps is relatively low, varying from about 1012 to 1015 cm-a; the density of deep traps is considerably higher and of the same order as reported for high-resistivity n-type crystals 1: about 1016 to 1018 cm-s. No evidence is found for the high density of shallow traps with depth about 0.2 ev 0.0007 :;: 0.0006 ~> -'8 i _0.0005 -' ...... lI>::L ~ .,.:0.0004 ..JZ .. '" :;;:E 0.0003 ~a I-0.0002 0.0001 200 FIG. 13. Thermally stimulated current for GaAs: eu crystal 631-1 showing both the small density of shallow traps and the higher density of deep traps. or less (probably hole traps lying above the valence band) and densities in the 1017 to 1018 cm-3 range, char acterized by a single thermally stimulated current peak without structure, which was found so consistently in the previously described! high-resistivity n-type crystals (see reference 19). The detailed structure of the low-temperature ther mally stimulated current curve for crystal 631-1 stimulated an effort to measure this portion of the curve with increased resolution. The result of such an effort is given in Fig. 14. The fine structure is completely reproducible from run to run. Hall and Photo-Hall Measurements Hall-effect measurements were made in the dark and under high-intensity broad-band excitation ("-'6 X 103 ft-c) at room temperature on ten of the GaAs:Cu crystals, as listed in Table IV. Any differences between -164 -156, '149 -141 -134 -1,6 -HS -til -103 'TEMPERATURE, 'C FIG. 14. The result of using higher resolution in the measurement of the shallow traps of Fig. 13. The data of Fig. 14 were measured at a heating rate of 0.29 deg/sec instead of 0.43 deg/sec used in Fig. 13; in addition, the data of Fig. 14 were recorded on an ex panded temperature scale by a factor of about three. The detailed structure (11 peaks or indications of peaks are easily discernible) is completely reproducible. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:531674 BLANC, BUBE, AND MAcDONALD TABLE IV. Summary of photo-Hall data on GaAs:Cu crystals. Conductivity Hall mobility, mho/em cm2/v sec Crystal Dark Light Dark Light dp/dn 631-1 3X10-7 6X10- 6 -65 -2220 631-8 9X10-7 9X10-6 +90 -39 60 ES36A 10-6 6X1O-6 +173 -232 26 631-9 2X10-6 10-· +3.7 -312 14 1O-G-40-4 5X1O-6 3X10-· +106 +127 GAJ-18M 6X10-6 2X10-- +103 +124 GAJ-18-2b 6X10-6 3X1O-· +146 +96 230 631-14 10--5X1O- 6 +183 +225 ES44 10-' 2X10-' +30 +36 10-G-32 4XlO-1 +267 the dark conductivities listed in Tables III and IV for the same crystal arose either as the result of heating to affix the Hall electrodes, or, in the case of the higher resistivity crystals, as the result of not waiting a long time for the conductivity to drop to its equilibrium dark value after photoexcitation. For crystals with a dark conductivity greater than 5XIO-6 mho/cm, the Hall mobility is positive in both dark and light and relatively independent of photoexcitation. For crystals with dark conductivity less than 5XIO-6 mho/cm, the Hall mobility either changes from positive to negative under photoexcitation, or, in the case of crystal 631-1, increases rapidly with photoexcitation while remaining negative over the whole range. Figure 15 shows the dependence of Hall mobility on the crystal conductivity (as varied by photoexcitation) for crystals 631-8, ES 36A, and 631-9, all of which show conversion from p type to n type with photoexcitation, and for crystal 631-1, which shows a strong dependence of Hall mobility on intensity of photoexcitation. When both carriers are contributing to the conduc- n ~---.- I t ;/ I l n --" --/+ ..... 631-9 ",. 1 0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 CONOUCTIVITY.(ohm-cm) 'x 10-6 FIG. 15. Variation of Hall mobility with conductivity, as varied by increasing photoexcitation, for those crystals of high-resistivity GaAs: eu showing a conversion from p type to n type under photoexcitation. See Table IV. tivity, the Hall constant R is given by pJl.p2-nJl.n2 e(pJl.i-nJl.n2) R= ~ e(pJl.p+nJl.n)2 0-2 where for the sake of simplicity we have assumed a correlation factor of unity between the Hall mobility Jl.H and the microscopic mobility. If values for either Jl.n or Jl.P and the value of the mobility ratio Jl.n/Jl.p=b are known, it is possible to solve the equations for R and 0-simultaneously to obtain both nand p as a func tion of conductivity (or of light intensity) from such data as are given in Fig. 15. If po=a/eJl.p and no=o-/CJl.n, then the result may be expressed either as po(1-Jl.H/ jJ.p) (7a) n b(l +b) p=po-nb, (7b) or as b2no(1 +Jl.ll/ jJ.n) (7c) p= (1+b) n=no-p/b. (7d) Using Eqs. (7), assuming b= to, values of p and n as a function of light intensity were calculated for those crystals in which two-carrier conductivity effects under photoexcitation were indicated. If I).p and I).n are defined as the differences between the values of p and n, re spectively, for high intensity excitation and no excita tion, the effect of the photoexcitation on the con ductivity can be evaluated as in the last column of Table IV. In those cases where the Hall coefficient is positive in the dark, such an analysis indicates that photoexcitation creates to to to2 free holes for each electron; the photoconductivity may therefore be said to be tip type" even though photoexcitation causes a change in the sign of the Hall coefficient. The variation of mobility in n-type crystal 631-1 with photoexcitation intensity cannot be attributed to a two-carrier conductivity effect, i.e., the very low value of Hall mobility in the dark is a real electron mobility and not the result of the participation of both electrons and holes in the conductivity. A useful criterion that must be met by true two-carrier conductivity under thermal equilibrium is that the values of nand p calculated from Eqs. (7) must be consistent with the value of the np product given by (8) where me is the effective electron mass, mh is the effec tive hole mass, and EG is the band gap. For GaAs at 300"K, using me=O.07 mo, mh=O.5 mo, and EG=1.4 ev, np=2Xto12 cm-6• This means that for the criterion of zero Hall mobility to be met, i.e., pJl.p2=nJl.n2, and the criterion of np = 2 X 1012 cm-3 to be met with jJ.n = 2 X 103 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53HIGH-RESISTIVITY GaAs COMPENSATED WITH DIFFUSED Cu 1675 cm2/v sec and b= 10 then n= 105 cm-3 p= 107 cm-3 , , , and a-=3.5XlO-10 mho/em. If an attempt is made to account for the low dark mobility of crystal 631-1 by a two-carrier conductivity model, the values of nand p resulting from Eqs. (7) have a product np=7X1017 cm-6• It may therefore be concluded that the large increase in electron mobility with photoexcitation intensity in crystal 631-1 is the result of a change in the scattering, i.e., a change in the occupancy of charged scattering centers, as a result of the motion of the steady state Fermi level with photoexcitation. Such a change in occupancy involves the deep compensated-donor trapping centers. Similar measurements of the varia tion of Hall mobility with photoexcitation have been made on a number of high-resistivity n-type GaAs crystals grown without any deliberately added impurity. All of these crystals show a marked increase in Hall mobility with increasing light intensity at fixed tem perature. The effects are therefore quite comparable to those reported for similar experiments with CdS.7 If the data are analyzed in terms of a model based on point scatterers, large scattering cross sections of the order of 10-11 to 10-10 cm2 are calculated. An investiga tion of this problem by itself is to be reported later. DISCUSSION Analysis of the Compensation Process An inspection of Tables I and II reveals considerable regularity in the compensation phenomena. Crystals with initial electron concentration of 1016 cm-3 became high resistivity after diffusion of Cu at 500°C; those with electron concentration in the range of 4 to 7 X 1016 after diffusion at 575°C; those in the range of 1 to 3 X 1017 after diffusion at 650°C. The one sample with initial electron concentration of 5 X 1017 cm-3 required a dif fusion temperature of 750°C before becoming high resistivity. This regularity suggests an inherent correla tion between the density of donors present and the solubility of Cu at any given temperature.13 By examination of a simple model in which only a shallow donor is present in the initial material, one can calculate both the enhancement of Cu solubility due to the presence of shallow donors and the carrier con centration at room temperature if none of the dissolved Cu precipitates in the quenching process. The results 13 There have been no experiments to date performed on these cry~tals to prove that diffusion at the respective temperatures for penods of 16 hr is sufficient to give compositional equilibrium. However, the data presented above indicate in two ways that equi~ibrium has in fact been reached. In crystal 631-8, a trap denSIty of the order of 1018 cm-3 was calculated on the assumption that the whole volume of the sample was hig}I resistivity· if the sample were not high resistivity throughout, it would be ne~essary to conclude the presence of a still higher trap density, which seems unreasonable. Also, the fact that the diffusions at 500°C with ma terial of 1016 cm-3 yielded high-resistivity material is in accord with the equilibrium calculations outlined here. Since these rema~ks apply to diffusion at 500°C, a fortiori the diffusions at hIgher. temperatures almost certainly yield equilibrium concentratlOns. '" E u , u z I o Z 10'6 575 7.1 ,lOiS 7.5 , lOiS 650 1.6", 10'6 3.3 ,10'6 750 4.7 , 10'6 1.6 x 10'1 ------------- FIG. 16. Variation of carrier concentration with initial donor concentration after copper diffusion at three representative temperatures. of an extension of this model by Reiss et al.14 yield as a close approximation Ncu (9) where N Cu is the solubility of Cu in the presence of N D shallow donors, N cuo is the solubility of Cu in the absence of other impurities, and ni is the intrinsic electron concentration. Since both ni and N cuo are functions of temperature, N Cu is also an implicit func tion of temperature. If ni and N cuo are known, it is pos sible to calculate N Cu and the difference (N D-N cu) which indicates the electrical type of the material and the carrier concentration after the quench to room temperature. The difference (N D-N cu) as calculated using Eq. (9) may be either positive or negative, i.e., the resulting crystals may be n type or p type depending on conditions. In order to carry out these calculations, the Cu solubility in the absence of impurities was determined by extrapolating the data of Fuller and Whelan2 to the relevant temperatures, and the intrinsic carrier concen tration was inferred from the high-temperature Hall data of Folberth and Weiss.!' The results of these calcu lations for three temperatures are shown in Fig. 16, where the parameters entering the computation are also indicated. For crystals containing about 1 to 2X1016 shallow donors, diffusion at 575°C results in a difference IN D -N Cu I of 1015 cm-3 or less, which shows that in 14 H. Reiss, C. S. Fuller, and F. J. Morin, Bell System Tech. J. 35,535 (1956). 16 o. G. Folberth and H. Weiss, Z. Naturforsch. lOa, 618 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:531676 BLANC, BUBE, AND MACDONALD this range of initial electron concentrations, the mater ial is expected to be fairly high resistivity after quench ing. Similarly, for diffusions at 650°C, the initial donor concentration calculated for high-resistivity resultant crystals ranges from about 8 to 9X 1016 cm-3• For dif fusions at 750°C, the computations yield 5 to 6X 1017 cm-3 for this range. These calculations are in good semi quantitative agreement with the experimental results of Table II. There are, however, certain objections which may be raised concerning the quantitative accuracy of Eq. (9) even for the simple case considered, quite apart from inaccuracies in the values of n, and N cuo used. The first of such objections is that Si, which is the main shallow donor in these samples of GaAs, is known to act amphoterically.16 A second difficulty is that Eq. (9) does not contain the acceptor ionization energy, which corresponds to the implicit assumption that all Cu atoms are ionized. It is difficult to calculate precisely what these effects will be, but both will tend to lessen the enhancement of Cu solubility. Order of magnitude estimates indicate that the amphoteric behavior of Si will not influence the results at concentrations less than 3X1017 cm-3, and that the ionization energy of Cu will not become important unless Ncu>2Ncuo. Of more importance, for present purposes, are ques tions dealing with the behavior of deep donors and ac ceptors in the initial material, the presence of which may not be directly revealed by Hall measurements on the initial material. The enhancement of solubility in the presence of such impurities is a complicated function of concentration and ionization energy, but, once again qualitatively, the presence of deep donors will tend to enhance the solubility of Cu, and the presence of deep acceptors to repress itP It is noteworthy that the presence of deep donors or acceptors will be revealed when the difference (N D-N cu) is less than the concentration of deep levels, since under that condition the Fermi level (in extrinsic material) is controlled by the deep levels. The variation of dark conductivity with temperature would then be controlled by ionization energies of the deeper levels and not by either Cu or Si. The main compensation effect is nevertheless the enhancement of Cu solubility by the shallow donor, although this effect by itself can clearly not account for a compensation, for example, of 1017 electrons before diffusion to 1011 cm-3 after dif fusion, an implication that the Cu and donor concentra tions are equal to one part per million. That effects not ascribable to the shallow donors do indeed occur can be seen from a careful examination of 16 J. M. Whelan, J. D. Struthers, and J. A. Ditzenberger, Prague Conference on Semiconductor Physics, 1960. 17 There is no particular difficulty in formally writing down equations relevant to the cases considered above. The expressions so obtained however are quite cumbersome, e.g., taking only Si amphotericism into account yields a biquadratic equation. Under the present circumstances, it hardly seems worthwhile to attempt numerical solutions to these equations. Tables I and II. The most striking example is a com parison between crystals ES 36 and ES 37, which had initially identical electron concentrations and mobilities. The two samples, which were annealed at the same time and in the same vial, differed quantitatively after diffusion and quenching. After diffusion at 575°C, ES 36 had an electron concentration of 1.1X1016 cm-3, while ES 37 had an electron concentration a factor of 5 higher. This difference in behavior persisted for diffusion at 650°C, after which ES 36 had a conductivity of 3XIo-4 mho/em, while ES 37 had a conductivity lower by a factor of 30. It is clear that these differences must be due to imperfections other than the shallow donor (if equilibrium was reached in each case), although it is not possible at this stage to decide which of the two crystals was "purer." It may be concluded, therefore, that although the compensation phenomenon can be explained to first order by a simple compensation mechanism, the re sulting high resistivities are the result of the presence of deep levels in the initial material. Nature of Imperfections in GaAs: Cu Crystals The presence of five levels located in the lower half of the forbidden gap of GaAs are indicated by the meas urements of the present investigation. The height of these various levels above the valence band and the the various techniques used in their determination are summarized in Table V. Four of the levels are identified as acceptor levels by their effect on the p-type conduc tivity. The deepest of these levels, with hole ionization energy of about 0.42 ev, is also identified as the sensitiz ing center for n-type photoconductivity at low tempera tures in crystals with room temperature dark conduc tivity less than 5 X 10-6 mho/ cm. The data indicate that this center acts as a sensitizing center only when it is compensated; photoexcited holes are captured which then have a small probability of recombining with free electrons. It is only when the Fermi level lies high enough, i.e., when the conductivity is low enough, that a sufficient proportion of these centers are compensated (occupied by electrons in dark thermal equilibrium) to give rise to the low-temperature increase in sensi- TABLE V. Summary of activation energies in GaAs: eu crystals. Phenomenon Activation energy, ev p-type conductivity 0.42 0.34 0.22 0.13 vs temperature Thermal quenching of 0.41 photoconductivity Optical quenching of 0.4 photoconductivity Photoconductivity vs 0.33 0.25 0.11 temperature at higher temperatures Photoconductivity vs 0.09 0.04 temperature at lower temperatures [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53HI G H -RES 1ST I V I T Y G a A s COM PEN SAT E D WIT H D IFF USE D C u 1677 TABLE VI. Estimates of imperfection center densities." Occupancy Lower limit Upper limit Sample Dark p, cm-' Ej, ev EA,ev by holes Nv, cm-' NA, cm-S NA, cm-' 10 G 32 9.1X1015 0.18 0.12 0.15 1.1 X 10'6 lOIS 2XI016 ES 44 3.0XlO14 0.27 0.17 0.035 4X1016 4X 1014 4X1016 631-14 4.1 X 10" 0.44 0.43 0.57 1016 1012 2X1016 GAJ-18-2b 2.6X 1011 0.45 0.40 0.21 6XI016 4XI011 3X1017 GAJ-18M 3.4X 1011 0.45 0.40 0.21 7XI016 4XlO11 4XlO17 10 G 40-4 2.8X 1011 0.45 0.45 0.67 9XI016 3XI0u 1017 ES36A 5.5X 1010 0.49 0.45 0.29 1017 6X101o 4X1017 • Dark p. cm-' from Hall effect at 300oK. Ef-height of Fermi level above the top of the valence band calculated from value of p. EA--activation energy for dark conductivity; see Table 1. NJ>-density of donors in initial n-type GaAs before copper diffusion. Lower limit to N A. the density of acceptors. calculated on the basis of a single acceptor level without any compensation being present.; see text. Upper limit to N A calculated on the basis of a single acceptor level with compensation present; see text. tivity. It is significant in this connection that the divid ing line between the photoconductivity characteristics of the crystals of Table III at 5XlO-6 mho/cm corre sponds to a location of the Fermi level about 0.47 ev above the top of the valence band; under these condi tions, about! of the 0.42-ev centers are compensated. As the Fermi level drops with increasing p-type con ductivity, the sensitizing ability of these centers disap pears. It is significant that the whole photoconductivity characteristics of crystal ES 36 were altered when its p-type conductivity was lowered to become crystal ES 36A as described in reference 8 of this paper. It is interesting to note that the hole ionization energy for sensitizing centers in Cu-compensated photoconducting CdS, CdSe, and GaAs is approximately the same frac tion of the total forbidden gap: 1.0 ev out of 2.4 ev in CdS, 0.6 ev out of 1.7 ev in CdSe, and 0.42 ev out of 1.4 ev in GaAs. The other four levels listed in Table V act as recombi nation centers for photoconductivity, i.e., photo sensitivity decreases as the length of time a photoexcited hole stays captured by these centers. Hall-effect meas urements show that for data on these centers obtained from the variation of photocurrent with temperature at higher temperatures, p-type photoconductivity is involved. For data obtained from the variation of photocurrent with temperature at lower temperatures in insensitive crystals, it is not currently known what type the photoconductivity is, but it may very well also be p type. The level located at about 0.13 ev above the top of the valence band is the level normally associated with Cu as an acceptor in GaAs. It is clear that in all of the crys tals of Table III, except reference crystal lOG32, these normal Cu levels are completely ionized at room tem perature and the dark conductivity is supplied by deeper levels. Identification of the other levels listed in Table V is uncertain. Whelan18 has mentioned a shallow ac ceptor level with ionization energy of 0.02 ev which might be related to the shallow level located at 0.04 ev 18 J. M. Whelan, paper in Semiconductors, edited by N. B. Han nay (Reinhold Publishing Corporation, New York, 1959), p. 154. in this investigation; Whelan and Fulletl suggested identification of this level with a Ga vacancy or a Ga vacancy-impurity complex.19 The two most resistive crystals, 631-1 and 631-8, show dark conductivity activation energies of 0.66 ev and 0.58 ev at higher temperatures. In view of the Hall and photo-Hall data of Table IV, it is believed that these crystals are p type in the dark at lower tempera tures where the conductivity is characterized by activa tion energies of 0.42 and 0.37 ev, and that they become n type in the dark with increasing temperature, room temperature being about the transition point. Crystal 631-8 is still p type in the dark, whereas crystal 631-1 is n type in the dark. It therefore seems most reasonable to ascribe these dark conductivity activation energies of about 0.6 and 0.7 ev to deep donors; likewise to ascribe the deep traps of 0.5, 0.6, and 0.7 ev to compensated deep donors, as in the previous investigation.! It is evident that temperature measurements of Hall and photo-Hall effect on all of these crystals are required for a full and unambiguous interpretation; such an in vestigation is intended. An effort has been made to obtain rough limits on the density of several of the levels summarized in Table V where sufficient data were abailable to permit it. A summary of such data is given in Table VI. The concen tration of holes in the dark is obtained from Hall data. The lower limit to the density of acceptor levels N A was obtained on the basis of a simple model of a single acceptor lying EA ev above the top of the valence band, ignoring the effects of any other levels which might 19 By a combination of photoconductivity and photomagneto electric measurements, A. Amith (private communication) has found levels at 0.65 ev below the conduction band and 0.23 ev above the valence band in high-resistivity n-type GaAs without added impurity, and levels at 0.5 ev above the valence band in low-resistivity p-type GaAs prepared by Cu diffusion into high resistivity n-type material. In our own work we have found con ductivity activation energies of about 0.5 ev when Cu was diffused into high-resistivity n-type GaAs; high densities of the 0.2-and O.5-ev trapping centers were also formed in such a diffusion process. In high-resistivity crystals grown without added impuri ties, but under excess As pressure, the photosensitivity increases with decreasing temperature according to an activation energy of 0.09±0.OO4 ev (5 crystals). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:531678 BLANC, BUBE, AND MACDONALD be present in the forbidden gap. In such a model, the density of free holes p is given by p= :v exp(-EA/kT) {[1+4:: eXP(EA/kT)r- 1}, (10) which generally simplifies to where Nv is the effective density of states in the valence band. When N A«Nv, p=N A. The upper limit to the density of acceptor levels N A was calculated assuming a single acceptor level appreciably compensated so that occupancy by holes NA-ND-p 1 NAt exp[(E/-E A)/kTJ+1' (12) where N D is the density of donors in the initial material, assumed given by the second· column of Table I, and E/ is the height of the Fermi level above the top of the valence band, calculated from p=Nvexp(-E//kT). (13) It is probable that the true density is closer to that of the upper limit. There remains finally consideration of the shallow trapping levels, present in densities between 1012 and 1015 cm-3, which are probably electron traps lying be tween 0.2 and 0.4 ev below the conduction band. In order to be detected, these shallow traps require a crystal with high sensitivity at low temperatures. This may explain why they were not detected in the pre vious investigation of high-resistivity n-type GaAs.l These shallow traps are characterized by considerable structure which may be due to the presence of excited states or to a type of spin-orbit interaction; it is a de tailed sharp structure not hitherto observed in thermally stimulated current data on any other materials to the best of our knowledge. CONCLUSIONS The compensation of n-type GaAs by diffused Cu can be semiquantitatively understood in terms of a simple thermodynamic model involving only the pres ence of shallow donors in the initial material. The direc tion which departures from this idealized model take is indicated. The levels revealed by the present investigation are identical with the donor and electron trapping (com pensated donor) levels located at 0.5, 0.6, and 0.7 ev below the bottom of the conduction band, as previously reported,! and in addition are comprised of acceptor levels lying 0.42,0.34,0.22,0.13, and 0.04 ev above the valence band. The 0.42-ev centers, when compensated, provide a long electron lifetime and hence give rise to high n-type photosensitivity. Such centers appear to be negatively charged centers under thermal equilibrium, with a cross section for capture of holes 6 X 103 times larger than the cross section for the subsequent capture of electrons. When the Fermi level lies above these levels, they cap ture photoexcited holes, reduce the hole lifetime, but increase the electron lifetime. When the Fermi level lies below these levels, the room temperature p-type photoconductivity increases as the lifetime of holes be comes longer. These levels must be predominantly associated with the copper compensation process of n-type GaAs because no evidence is found of high photo sensitivity at low temperatures, associated with capture of photoexcited holes with an activation energy of 0.42 ev in high-resistivity n-type material grown without intentionally added impurities. A residual concentra tion of such centers in n-type material, however, can reduce the hole lifetime if the conductivity is high enough so that the hole demarcation level lies near the 0.42-ev levels (coincidence of the demarcation level and the 0.42-ev levels occurs for a room temperature con ductivity of about 1.5 mho/cm). Hall data reported here and found on other high resistivity n-type GaAs crystals grown without in tentionally added impurities, indicate that the Hall mobility is sensitive to photoexcitation in n-type ma terials. There is strong evidence, therefore, that deep donors can playa role in the scattering process. Their effect is most pronounced in high-resistivity n-type material, where they exist in the compensated charged state, which can be removed by raising the Fermi level by photoexcitation. In low-resistivity n-type material, they are present in the uncompensated, uncharged state and make only a much smaller contribution to the scattering. Although the effect of photoexcitation on Hall mo bility seems much less in p-type material than in n type, according to Table IV, this is only apparent and not real. Since the change in scattering is given by dif ferences in 1/ J.L, the absolute changes in scattering for the smaller values of J.L in p-type material are of the same order of magnitude as for the larger values of J.L in n-type material. The 631 series of GaAs: Cu crystals is characterized by a high ratio of electron mobility at 78°K to that at 300°K. The absence of the 0.22-ev acceptor from this series is outstanding. In crystal 631-14, for example, the conductivity activation energy does not shift from the 0.42-ev level at high temperatures to the 0.25-ev level at low temperatures, as in several other crystals, but shifts directly from the 0.42-ev level to the 0.13-ev level. Similarly, crystal 631-9 shows the 0.13-ev level at low temperatures under conductivity conditions much lower than would be expected in comparison with the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53HI G H -RES 1ST I V IT Y G a A s COM PEN SAT ED WIT H D IFF USE D C u 1679 other crystals (see Fig. 1). There is reason, therefore, to propose that the O.22-ev center, when present as a compensated acceptor, is an efficient scatterer, particu larly at low temperatures. The marked absence of the high density of O.2-ev trapping levels found so consistently in high-resistivity n-type GaAs grown without intentionally added im purityl (see also reference 19) suggests that these levels are associated with an imperfection which is not present in the final GaAs: Cu crystals. ACKNOWLEDGMENTS The authors are indebted to L. R. Weisberg for many stimulating and helpful discussions, and to E. J. Stofko for the preparation and mounting of the crystals for measurement. JOURNAL OF APPLIED PHYSICS VOLUME 32. NUMBER 9 SEPTEMBER. 196! Flash Method of Detennining Thennal Diffusivity, Heat Capacity, and Thennal Conductivity W. J. PARKER, R. J. JENKINS, c. P. BUTLER, AND G. L. ABBOTT u. S. Naval Radiological Defense Laboratory, San Francisco 24, California (Received September 29, 1960) A flash method of measuring the thermal diffusivity, heat capacity, and thermal conductivity is described for the first time. A high-intensity short-duration light pulse is absorbed in the front surface of a thermally insulated specimen a few millimeters thick coated with camphor black, and the resulting temperature history of the rear surface is measured by a thermocouple and recorded with an oscilloscope and camera. The thermal diffusivity is determined by the shape of the temperature versus time curve at the rear surface, the heat capacity by the maximum temperature indicated by the thermocouple, and the thermal conduc tivity by the product of the heat capacity, thermal diffusivity, and the density. These three thermal prop erties are determined for copper, silver, iron, nickel, aluminum, tin, zinc, and some alloys at 22°C and 135°C and compared with previously reported values. INTRODUCTION THERE has been a renewed interest in developing new methods of determining the thermal con ductivity and the thermal diffusivity of materials in recent years. This is largely a result of the rapid ad vances of materials technology and the many new applications of materials at elevated temperatures. There are a number of presently existing steady-state and non-steady-state methods of measuring these parameters. However, there is some dissatisfaction with the length of time required to make reliable measure ments, and in some cases, the large sample sizes required by these techniques impose intolerable limitations. The difficulty of extending these methods to high tempera tures has proven to be a stumbling block in high temperature technology. The heat flow equation can be solved for a wide variety of boundary conditions, and these solutions can often generate values of the thermal properties. How ever, inability to satisfy the boundary conditions has led to difficulties in some of the classical techniques. Two of these difficulties are caused by surface heat losses and thermal contact resistance between the specimen and its associated heat sources and sinks. The problem of thermal contact resistance has been virtually eliminated in some recent thermal diffusivity determinations by * This research was sponsored by the Wright Air Development Division of the Air Research and Development Command, U. S. Air Force, under contract. thermally insulating the specimen and introducing the heat by an arc image furnace. A system of this type has been described by Butler and Inn! in which the thermal diffusivity is expressed in terms of the differences be tween the temperature versus time curves taken by thermocouples located at two points along a thermally insulated rod continuously irradiated at the front ~ur face by a carbon arc. It has been suggested2 that the Angstrom method, which utilizes a periodic front surface temperature variation for diffusivity measurements, can also be adapted to the arc image furnace. It is necessary to make these two types of determinations in a vacuum chamber in order to eliminate convective heat losses. However, above lOoo°C the radiation losses create a problem of considerable magnitude. The technique described in this report utilizes a flash tube to eliminate the problem of the thermal contact resistance, while the heat losses are minimized by mak ing the measurements in a time short enough so that very little cooling can take place. Although this method has only been tested for metals in the vicinity of room temperature, there is no reason to believe that measure- 1 C. P. Butler and E. C. Y. Inn in Thermodynamic and Transport Properties of Gases, Liquids and Solids (American Society of Mechanical Engineers, 29 W. 39th St., New York, New York, 1959). 2 A. Hirschman, W. L. Derksen, and T. 1. Monahan, "A Proposed Method for Measuring Thermal Diffusivity at Elevated Tem peratures," Armed Forces Special Weapons Project Report, AFSWP-1145, Material Laboratory, New York Naval Shipyard, April, 1959. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.112.236.84 On: Fri, 05 Dec 2014 22:45:53
1.1754005.pdf	EFFECT OF SURFACE SCATTERING ON ELECTRON MOBILITY IN AN INVERSION LAYER ON pTYPE SILICON F. Fang and S. Triebwasser Citation: Applied Physics Letters 4, 145 (1964); doi: 10.1063/1.1754005 View online: http://dx.doi.org/10.1063/1.1754005 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/4/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Weak antilocalization effect in high-mobility two-dimensional electron gas in an inversion layer on p-type HgCdTe Appl. Phys. Lett. 99, 042103 (2011); 10.1063/1.3615303 Scattering of electrons in silicon inversion layers by remote surface roughness J. Appl. Phys. 94, 392 (2003); 10.1063/1.1577227 Band Structure Investigation on p-Type Silicon Inversion Layers by Piezoresistance and Mobility Measurements J. Vac. Sci. Technol. 9, 759 (1972); 10.1116/1.1317774 Mobility Anisotropy and Piezoresistance in Silicon pType Inversion Layers J. Appl. Phys. 39, 1923 (1968); 10.1063/1.1656464 EFFECT OF SURFACE STATES ON ELECTRON MOBILITY IN SILICON SURFACEINVERSION LAYERS Appl. Phys. Lett. 9, 344 (1966); 10.1063/1.1754779 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:25:34Volume 4, Number 8 APPLIED PHYSICS LETTERS 15 April 1964 for luminescence such as that observed in electro luminescent but not coherently oscillating GaAs.2 At room temperature some superlinearity is ob served in the edT e-diode light output. Irradiation of these diodes with intense white light produces an open circuit photovoltage of 1.35 V at 77° and 0.7 V at 3000K. At 3000K the wavelength dependence of the short circuit photo voltaic current shows extrinsic peaks at 8500 and o 9000 A; the latter vanishes at lower temperatures. The above results show that (in the case of one II-VI material) efficient p-n junction electrolumi nescence is obtainable, not, unfortunately, at present without the concurrence of contact difficulties which apparently prohibit the observation of laser action. The high quantum efficiency tends to establish CdTe as a direct gap material. 10 The authors thank the following for their sub stantial contributions to this work: M. 1. Nathan and R. S. Levitt who made some of the optical and photometric measurements; S. p. Keller who gave support and technical advice; and W. N. Hammer and]. A. Kucza who contributed much original and creative effort in making measurements, fabricating devices, and preparing materials. 1The research herein reported is part of Project DEFENDER under the joint sponsorship of the Advanced Research Projects Agency, the Office of Naval Research, and the Department of Defense. 2R. ]. Keyes and T. M. Quist, Proc. ERE 50, 1822 (1962). 3K. Weiser and R. S. Levitt, Appl. Phys. Letters 2, 178 (1963). 41. Melngailis, Appl. Phys. Letters 2, 176 (1963). 5G• Cheroff, C. Lanza, and S. Triebwasser, Rev. Sci. Instr. 34, 10, 1138 (1963). 6D• de Nobel, thesis, University of Leiden, 1958. 7F• Morehead and G. Mandel (to be published). Sp. W. Davis and T. S. Shilliday, Phys. Rev. 118, 1020 (1960). 9H• W. Leverenz, Luminescence in Solids (John Wiley & Sons, Inc., New York, 1945). lOB. Segall, M. R. Lorenz, and R. E. Halsted, Phys. Rev. 129, 2471 (1963). EFFECT OF SURFACE SCATTERING ON ELECTRON MOBILITY IN AN INVERSION LAYER ON p-TYPE SILICON (transverse electric field effect on carrier mobil ity; E/T) In a surface channel conductivity controlled device such as the in sulating gate field effect transistor 1 shown in Fig. I, the transverse electric field (normal to the insulator-semiconductor interface) in the channel is in general large enough to warrant con sideration of the effect of surface scattering of carriers at the semiconductor-insulator interface. F or example, assuming a constant transverse field, Schrieffer2 was able to show for the case of large fields and diffuse surface scattering such that the surface scattering time is much smaller than the bulk scattering time, that the effective mobility is inversely proportional to the transverse field. This Letter presents some observations which indicate the dramatic change of carrier mobility caused by the transverse field as shown by small signal transconductance and channel conductance measure ments. F. Fang and S. Triebwasser IBM Watson Research Center Yorktown Heights, New York (Received 25 February 1964; in final form 30 March 1964) The source-drain conductance, GSD' IS gIven by GSD = (W/L)NQJ1 , (1) where N = the surface density of mobile electrons, W /L is the width to length ratio of the channel, Q the charge on the electron, and J1 the electron mobility.3 The transconductance of the device, Gm, is defined by (2) Figure 2(a) shows G~ 1 plotted as a function of V for such a device. On the same curve is shown thge gate to source-drain capacitance at 1 Mc/ sec. G m was independent of frequency from 20 cps to 10 Mcps and no observable relaxation for a pulsed gate voltage. The capaCItance was also frequency 145 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:25:34Volume 4, Number 8 APPLIED PHYSICS LETTERS 15 April 1964 n-TYPE FIELD-INDUCED INVERSION LAYER GATE OXIDE METAL Fig. 1. Insulated gate field effect transistor. independent over a large frequency range. The region of interest is that over which an n-type surface channel exists to which the n-type source and drain make Ohmic contact. Clearly G m varies as (V + V )-1 where V represents a kind of built-in g ox ox voltage which is discussed below. From Eq. (2), GSD should then vary as log (V + Va) as is verified in Fig. 2(b). The data contafned in Fig. 2(a) and 2(b) are from independent measurements. The lack of frequency dependence and relaxation of G m and C strongly suggests that there is a negligible number of active surface states at the interface in a highly inverted layer. To good ap proximation N is given by for a highly inverted layer where Cox is the oxide capacitance per unit area.4 At V = -Vox' the depletion layer is completely formed,g further motion of the Fermi level with respect to the conduction band at the Si-Si02 interface being minimal at higher voltages.5 Consider that in the region of interest the field at the Si02-Si interface is 'V 105 V/cm. An electron with energy (kT) would be brought to rest in 25 A o in such a field. The bulk mean free path is >200 A. If the interface can be represented as a thermal diffuse scatterer, then a classical electron 0 on the average would be contained in a sheet 25 A wide. Since the de Broglie wavelength of such a particle is 'VIOO A, Boltzmann statistics applied to con ventional three dimensional band states are inap propriate, but rather it would be more correct to assume, for the purpose of a spatial population density calculation, the electron sees some sort of average potential. F or a first approximation it is assumed that the density of free carriers is uniform over the charge sheet. If one now writes the equation of motion in one dimension of a charged 146 particle of energy U 0 in the self-consistent field generated by a uniform charge sheet, a rather simple result is obtained for the scattering time: where E = 47TC (V + V )/E = (47TqN)/E (5) o ox ox g s s IS the field in Si at the Si-Si02 interface in the spirit of approximation (3), and E 1 the residual field just beyond the free carrier sheet is given by where Es is the dielectric constant of Si, X 1 is the width of the depletion region, and V 1 is defined by Eq. (6). Equation (4) obtains when Eo » E l' 6 a condition which is satisfied in the region of interest. Now 11 is found from Ils = qT/m* = (2U jm)'!l (2/E) log (2EjE 1)' (7) which excep.t for the logarithmic to Schrieffer's Eq. (11) and (16). finds factor is similar Accordingly, one x (E/27T) log [(Vox + Vg)/(V /2)J. (8) The slope of the line in Fig. 2(b) gives an ex perimental value of 4.3 x 10 -4 mho for the coefficient of the logarithmic term. If kT /2 is assumed as a reasonable value for U 0' the theoretical value 6 C 5 en en 10 ~ ::Ii 4 J: 2: 0 on 3 >" g + x 'f 2 :Y '" 2 0 1L-4---L-----~ __ ~ -12 -8 -4 0 4 8 12 16 20 0 1000 2000 VGATE-SOURCE (VOLTS) Gso(/Lmho) (0) (b) Fig. 2. (a) Gm -1 vs Vg and Cg(SD) vs Vg• (b) GSD vs log (Vg + Vox), This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:25:34Volume 4, Number 8 APPLIED PHYSICS LETTERS 15 April 1964 would be 19.2 x 10-4 mho. The GSD = 0 intercept from Fig. 2(b) yields a value of (V /2) of 0.42 V as compared with a theoretical value of 0.61 V, calculated from E,q. (6). The effective electron mobility at V -3.5 V and 0 are 450 and 200 cm 2 V-I sec-I, r~spectively. These results demonstrate clearly the effect of surface scattering on mobility in an inversion layer on the surface of p-type Si. A simple one dimensional model involving an assumption of monoenergetic electrons yields theoretical results in surprisingly good agreement with the experimental data.. Although arguments based on quantization of momentum normal to the Si-SiO interface can be offered to justify 2 f" I the model in part, the arguments are not suf lClent y consistent to be convincing. We acknowledge helpful discussions with D. C. Mattis and P. J. Price. IThis device in the form shown in Fig. 1 was first described by K. Kahng and M. M. Atalla at the IRE·AIEE Solid State Device Res. Coni., Pittsburgh, Pa., June, 1960. The original idea is contained in W. Shockley and G. L. Pearson, Phys. Rev. 74,232 (1948). 2J. R. SchrieHer, Phys. Rev. 97,641 (1955). 3See, for example, P. K. Weimer, Proc. IRE 50, 1462 (1962). • 4This has been directly verified recently by A. B. F owlet and F. F. Fang in a field effect surface Hall measurement (to be published). SFor a more complete discussion of the behavior of the SiO ·Si structure see L. M. Terman, Solid·Slate Electron. 2 5, 285 (1962), and R. Lindner, Bell System Tech. J. 41, 803 (1962). 6The complete result is given by 7== 2mUo . COSh-I(Eo) tE~ -Ei}q2 E 1 which reduces to Eq. (5) for Eo» E l' EFFECT OF FAST-NEUTRON-INDUCED DEFECTS ON THE CURRENT CARRYING BEHAVIOR OF SUPERCONDUCTING Nh3Sn 1 (flux jumping; 4.20J<; critical current density increased; E) Previous work 2 has shown that neutron-induced defects are capable of changing the superconducting properties of Nb 3Sn. Neutron bombardment increased the shielding ability (magnetization) of thin cylindri cal Nb 3Sn samples. The magnetization data can be used to calculate the critical current density, J , c as shown by Kim and others. 3 The magnetization data demonstrated that J c was increased by neutron bombardment. However, the induced damage so increased the i!lstability of cylinders that extensive "flux jumping" occurred during the measurements. This phenomenon, although permitting qualitative evidence for the increase in J c' made it impossible to obtain quantitative data on the field dependence of J c' In particular, no information was obtained about the constants a and B of the Lorentz force o model of Kim and others 3 which shows that (1) where H is the external ma'gnetic field. Previous work 4,5 has shown good agreement between magnetization determination of J c and direct measure ments in the strip geometry, and also excellent G. W. Cullen and R. L. Novak RCA Laboratories, Princeton, New Jersey (Received 2 March 1964) agreement between the Lorentz force model and the field dependence of J c for both longitudinal and transverse fields. The following experiments were carried out therefore to permit measurements that would produce definitive data on the increases in current density after irradiation. The specimens were Nb 3Sn strips that had been formed by vapor deposition 6 on a ceramic sub strate. 7 The strips were 'V 1 cm long with a cross sectional area of 2 x 10-5 cm 2. The critical current was measured at 4.2'X as a function of the external magnetic field. The samples were irradiated at 50'1:: using the RCA facilities at the Industrial Reactor Laboratories. The specimens were cadmium-shielded to eliminate any effects produced by thermal neutrons. The fast neutron flux was measured utilizmg the 58Ni (n,p) 58Co reaction which has an effective threshold near 5 MeV. The measurements were repeated after the specimens had remained at room temperature for about two weeks, when the artificially induced radioactivity had decayed to a point where the specimens could be readily mounted and measured. 147 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:25:34
1.1840957.pdf	Determination of Centrifugal Distortion Coefficients of AsymmetricTop Molecules James K. G. Watson Citation: J. Chem. Phys. 46, 1935 (1967); doi: 10.1063/1.1840957 View online: http://dx.doi.org/10.1063/1.1840957 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v46/i5 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 46, NUMBER 5 1 MARCH 1967 Determination of Centrifugal Distortion Coefficients of Asymmetric-Top Molecules JAMES K. G. WATSON Department of Chemistry, The University, Reading, Berkshire, England (Received 16 September 1966) The rotational Hamiltonian of an asymmetric-top molecule in a given vibrational state~ obtained by the usual vibrational perturbation treatment, contains more parameters than can be ?etermmed from ~he observed energy levels. This Hamiltonian is therefore transformed by.means of ~ umtary transfor.matlOn to a reduced Hamiltonian which is suitable for fitting to observed energies. The umtary transformatIOn can be chosen so that the reduced Hamiltonian has the following properties: (i) It is totally symmetric in the point group D2, regardless of the symmetry of the molecule; (ii) It contains only (n+ 1) inde~~ndent ter~s of total degree n in the components of the total angular momentum, for each e,:en value .of n; (m) Its mll:tnx elements in a symmetric-top basis satisfy the selection rule AK = 0, ±2. This paper IS ~onc:rned ma.mly with the possibility of carrying out this reduction in general. However, the reduce~ Ha~ltoman des~nb~d above contains one less quartic coefficient than has been used previously, and this particular case IS diS cussed in more detail. I. INTRODUCTION THE centrifugal distortion coefficients of polyatomic molecules can be related to the vibrational potential constants, and can therefore be used as data in the evaluation of the latter. It was originally shown by Wilsonl that the quartic distortion coefficients TafJ",/6 introduced by Wilson and Howard2 depend to a good approximation only on the harmonic force constants. The validity of this theory has been strikingly confirmed recently by the determination of the vibrational fre quencies of F20 from an analysis of its pure rotation spectrum,8 and similar results have been obtained for other small molecules. Some of the higher-order co efficients, which depend on the anharmonic potential constants, have been discussed in recent papers by Chung and Parker.4 Most theoretical treatments of this problem have been devoted to the calculation of the distortion co efficients from the potential constants, assuming the latter to be known. Rather less attention has been paid to the important practical question of the determina tion of the distortion coefficients from the observed rotational energy levels of the molecule in its various vibrational states. The principal exception is a paper by Kivelson and Wilson,5 in which the quartic cen trifugal terms were treated as a perturbation of the rigid-rotor Hamiltonian and the first-order corrections were expressed in a convenient form. Successful determinations of the distortion co efficients of planar asymmetric-top molecules have J E. B. Wilson, Jr., J. Chern. phys. 4, 526 (1936); 5, 617 (1937) . 2 E. B. Wilson, Jr., and J. B. Howard, J. Chern. phys. 4,260 (1936). 8 L. Pierce, N. Di Cianni, and R. H. Jackson, J. Chern. Phys. 38, 730 (1963). , K. T. Chung and P. M. Parker, J. Chern. Phys. 38, 8 (1963); 43,3865,3869 (1965). I D. Kivelson and E. B. Wilson, Jr., J. Chern. Phys. 20, 1575 (1952). been made but for such molecules there are only four independe~t quartic coefficients,6 rather than the six which appear in Kivelson and Wilson's equation. There have therefore been very few attempts to apply Kivelson and Wilson's equation in its complete form, which is only required for nonplanar molecules. How ever, Dreizler, DendI, and Rudolph7 have found that in treating centrifugal distortion in the nonplanar molecules dimethyl sulfoxide and dimethyl sulfide they were unable to obtain determinate values for the coefficients, although the observed energy levels could be fitted accurately. The present paper is devoted to a general considera tion of the determination of centrifugal distortion co efficients of asymmetric-top molecules from observed energy levels. Use is made of the fact that the eigen values of the Hamiltonian are unaltered when it is sub jected to a unitary transformation. Thus, ~hose uni tary transformations whose effects are e~Ulval:nt to merely changing the values of the coeffiCients III the Hamiltonian lead to indeterminacies in the coefficients, since the sets of coefficients before and after transfor mation are equally consistent with the given set of eigenvalues. It is found that previous treatments of the quartic and all higher coefficients are indeterminate, unless they are constrained by additional relations such as those used for planar molecules. One consequence, which has already been reported briefly,S is that one of the terms can be eliminated from Kivelson and Wilson's equation, which therefore contains only five determinable distortion coefficients. On the basis of the present results, it is possible to understand in detail the form of the indeterminacy found by Dreizler et al.,7 as will be described in another paper. 6 (a) J. M. Dowling, J. Mol. Spectry. 6, 550 (1961); (b) T. Oka and Y. Morino, J. Phys. Soc. Japan 16, 1235 (1961). 7 H. Dreizler and G. DendI, Z. Naturforsch. 20a, 30 (1965); H. Dreizler and H. D. Rudolph, ibid. 20a, 749 (1965). 8 J. K. G. Watson, J. Chern. Phys.45, 13QO (1966). 1935 Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1936 JAMES K. G. WATSON II. METHOD The problem described in Sec. I is studied in the following way. The rotational Hamiltonian of an asym metric-top molecule in a given vibrational level is as sumed in a general form as a power series in the com ponents of the total angular momentum (Sec. III). This is then transformed by a unitary operator which is a function only of the total-angular-momentum vector and of various parameters (Sec. IV), to give a transformed Hamiltonian which is again a power series. The eigenvalues of the transformed Hamiltonian are therefore identical to those of the original Hamil tonian, but its coefficients depend on the parameters in the unitary transformation. Because these parameters are arbitrary, it follows that the only combinations of the coefficients in the Hamiltonian which can be de termined from the energy levels are those which are obtained by eliminating the parameters of the unitary transformation. The number of determinable combina tions is therefore just the number of coefficients which contribute independently to the Hamiltonian, minus the number of parameters which contribute independ ently to the unitary transformation. However, since both numbers are in principle infinite, this statement is not very meaningful as it stands. It becomes mean ingful when we consider the orders of magnitude of the various coefficients involved (see Secs. III and IV), since this allows us to associate particular degrees of freedom in the unitary transformation with particular terms in the Hamiltonian. It is then fairly easy to find the number of determinable combinations of the co efficients of the terms of each degree in the Hamiltonian. To actually carry out the transformation in detail would be extremely laborious, and has not been done. The main objects of the present paper are to find the number of determinable combinations of coefficients as described above and, by a suitable choice of the unitary transformation, to transform the Hamiltonian to a reduced Hamiltonian which contains only de terminable combinations of the coefficients. The form of the reduced Hamiltonian (see Sec. VI) can be de cided on the basis of order-of-magnitude arguments. This reduced Hamiltonian can therefore be used for the empirical fitting of rotational energy levels, so as to avoid the indeterminacies which would other wise occur. At the same time, the coefficients in the re duced Hamiltonian will ultimately be interpretable in terms of the potential constants of the molecule. The idea of the reduced Hamiltonian can be illus trated by a familiar example. If we neglect all higher terms, the rotational levels of a given nondegenerate vibrational state are the eigenvalues of the Hamiltonian Hrot=! L ilaplaI/3, (1) a./3 where the elements of the constant symmetric matrix jIaf3 are the effective values, for the particular vibra-tional state, of Wilson and Howard's2 tensor J.!afj; Ja is a component of the total-angular-momentum vector; and a, {3 are summed over x, y, z. Since the axes are usually chosen to be the principal inertial axes of the equilibrium configuration, they are not in general the principal axes of jIafj' However, since we can reduce jIaf3 to diagonal form by a simple rotation of axes, it is obvious that the rotational energies can depend only on the principal values of the matrix jIa(J. In fact, a ro tation of axes is just a particular type of unitary trans formation (see Sec. V). Thus, in order to fit the ob served rotational energies in this approximation, it is sufficient to use the reduced Hamiltonian (2) where X, Y, Z are the principal values of !ila(J' The number of coefficients (three) in the reduced Hamil tonian is just the difference between the number of independent elements (six) of jIa/3 and the number of degrees of freedom (three) in a rotation of axes. It should, however, be noted that HR is not completely equivalent to Hrot since the two have different eigen functions, and the difference would have to be taken into account in an accurate treatment of rotational intensities. This result may seem very trivial. However, when it is generalized to the terms of higher degree, interest ing new results are obtained. III. ROTATIONAL HAMILTONIAN Let us suppose that the usual vibrational perturba tion treatment has been performed for a general asym metric-top molecule, without vibrational degeneracies or resonances, so that the calculation of the rotational levels of a particular vibrational level has been reduced to finding the eigenvalues of a rotational Hamiltonian whose coefficients are appropriate to the vibrational state in question. The only remaining dynamical vari ables are the components of the total angular mo mentum, and it is assumed that the Hamiltonian is expressed as a power series in them. These components, in units of ft, are denoted by Jx, JII, J.; they satisfy the commutation relations appropriate to the components in moving or "molecule fixed" axes. These commutation relations can obviously be used to alter any expression involving the angular momenta, in a way which is equivalent to changing the coefficients of the various terms. In the case of the Hamiltonian this leads to certain indeterminacies among the co efficients; for instance, the Taf3a/3 (ar£{3) are not sepa rable from the TaaIJf3 and the principal rotational con stants.5 In fact, a general power-series expression of the type we are considering contains no more inde- Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsCENTRIFUGAL DISTORTION COEFFICIENTS 1937 pendent terms in quantum mechanics than it does in classical mechanics. To see this, consider the follow ing expression for the rotational Hamiltonian: H= t hpqr(J"pJllq:+J:Jllq"l) , (4) P.q.""O which contains one independent term for each com bination of powers of J"" III, J., i.e., the classical number of terms; the expression in parentheses is chosen in the manner shown because it is convenient that it should be Hermitian. The hpqr are constant coefficients. If now we have a quantum product of p factors J"" q factors JII, and r factors J., in any order, then from the commutation rules it differs from t(J"PJyq,r+J:Jyq"l) by terms of lower degree in the components of J. These latter terms in tum differ from similar expressions out of (4) by terms of yet lower degree in J, and so on. By carrying through this procedure one can therefore express any term of the quantum-mechanical Hamil tonian in the form (4). It follows therefore that the rotational Hamiltonian may be assumed without loss of generality to be in the form (4), which is referred to here as the standard form. For instance, the Hamil tonian (1) is in standard form as long as jIa{J is sym metric; and similarly Kivelson and Wilson's5 equation (5) is in standard form. This choice of the Hamiltonian in standard form avoids the indeterminacies of the type affecting Ta(Ja{J, referred to above. The vibrational perturbation treatment can be per formed so as to preserve the Hermitian property of the Hamiltonian. Since the expression in parentheses in (4) is Hermitian, it follows that the coefficients hpqr in the standard form are all real. A second property of the Hamiltonian, which can also be preserved in the perturbation treatment, is its invariance under the operation of time reversal, i.e., reversal of all mo menta accompanied by complex conjugation of all coefficients. When applied to the standard form (4), this means that the coefficients hpqr are real for even values of n = p + q + r and purely imaginary for odd values of n=p+q+r. The two properties used in con junction therefore imply that the coefficients of terms with odd values of n vanish, and that the coefficients of terms with even values of n are real. We are also interested in the number of terms of different types in (4). The total number of terms of given n is just the number of partitions of n into three parts, including zero as a possible part. If p is chosen, then the possible choices of q are 0, 1, "', n -p, numbering (n -p + 1); the value of r is then fixed as n -p -q. Thus, the number of partitions is " L (n+l-p) =!(n+l) (n+2). p=O The various terms in (4) can be classified according to the symmetry species of the point group D2, using the fact that J"" JII, J. belong to the species B"" BII, B. TABLE I. The number and species of terms in the standard form of the Hamiltonian (4).- D2 species p q T Number of terms A e e e Hm+l) (m+2) B", e 0 0 } Bv 0 e 0 !m(m+l) each B. 0 0 e Total (2m+1) (m+1) • p+q+r=2m. for fixed m .• is even, 0 is odd. respectively.9 The species of each term depends only on the parities of p, q, r and is shown in Table I. The use fulness of this classification will be found to rest on the fact that the commutation relations (3) are invariant to the operations of D2• It is not assumed that the mole cule has any symmetry, so that, in general, terms of all species are present initially in the Hamiltonian. The number of terms of different species is obtained as follows. For p+q+r=2m, the number of partitions of the type eee (e is even) is just the number of un restricted partitions of m, namely Hm+l) (m+2). The number of partitions of the type eoo (0 is odd) follows from the fact that if p is taken as 2s, there are m-s choices of q; thus, the number is t (m-s) =!m(m+l). .-0 These results are collected in Table 1. A. Orders of Magnitude It is found in practice that the power series (4) for the rotational Hamiltonian converges rapidly for small values of the total angular momentum. In other words the coefficients hpqr for different values of n = p+q+; are well separated in order of magnitude. It will sim plify the discussion of the transformations to be made if these orders of magnitude are taken in a more specific form. An examination of Dunham's formulas for a diatomic molecule1o in relation to Schiff's discussion of the Born-Oppenheimer treatmentll suggests that the following choice is physically reasonable: (5) In this equation K is a small parameter which may be regarded as the ratio of a nuclear displacement for low vibrati~nal quantum numbe:s to a typical bond length, and T.IS the energy of a typIcal valence-shell electronic 9 The subscript on the B-species symbol refers to the C2 axis for which the character is + 1. This notation is more convenient here than the conventional one; the correspondence is B.=Ba• Bu=B2. B.=B\. 10 J. L. Dunham, Phys. Rev. 41. 721 (1932). The formulas are quoted in C. H. Townes and A. L. Schawlow Microwave Spectroscopy (McGraw-Hill Book Co., New York' 1955) pp. 10-11. ' , 11 L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Co., New York, 1949), p. 288. Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1938 JAMES K. G. WATSON TABLE II. The number and species of terms in the standard form (9) of S2m-1.a D2 species p q , Number of terms A 0 0 0 im(m-l) Bz 0 e e } B" e 0 e lm(m+l) each B. e e 0 Total m(2m+l) a p+q+r=2m-!, for fixed m. e is even, 0 is odd. transition. The ratio of the coefficients for successive even values of n is then of order 0. This scheme of ordering is used in the following way. When the Hamiltonian is subjected to a general unitary transformation, all the coefficients in the Hamiltonian are changed to some extent. However, the change is not significant if it is of small magnitude relative to the coefficient itself, and in such cases the coefficient will be regarded as "determinable." On the other hand, if the change is of the same magnitude as the coefficient itself, the coefficient is "indeterminable." When a coefficient is indeterminable in this way, the parameters in the unitary transformation could be chosen to elim inate the corresponding term from the Hamiltonian altogether. However, all terms whose coefficients are individually indeterminable cannot be eliminated simultaneously, because certain coefficients of this type occur in determinable combinations. The reduced Hamiltonian is obtained by choosing the unitary trans formation so as to leave only these determinable com binations of coefficients. It follows from these remarks that the results are not dependent on the quantitative validity of (5), which is used mainly as a general qualitative guide.12 IV. UNITARY TRANSFORMATION The rotational Hamiltonian, as obtained from the vibrational perturbation treatment, has been assumed in a general form in Sec. III. We now consider the pos sibility of subjecting the Hamiltonian to a unitary transformation in such a way that this general form is unaltered but the individual coefficients are changed. Since the eigenvalues are unaffected by a unitary trans formation, it will follow that only those combinations of the coefficients which remain invariant during a unitary transformation can be determined from the observed eigenvalues. If U is some unitary operator (ut = U-l), then the transformed Hamiltonian H can be written H=U-IHU. (6) In discussing the transformation later, we suppose that a set of unitary transformations are applied suc- 12 I am indebted to Dr. Takeshi Oka for pointing out the rele vance of the Born-Oppenheimer treatment in discussions of orders of magnitude. cessively, which is equivalent to expressing U as a product and considering each factor separately. The following remarks then apply to each factor of U. It follows directly from (6) that H is Hermitian if H is Hermitian. Since we want H to be purely a func tion of J:z, JII, J" we choose U to be such a function. If U is also chosen to be invariant to time reversal, then H will be invariant if H is invariant. Now, the most convenient form for a unitary operator is U= exp(iS); (7) the unitary condition then merely requires that S should be Hermitian. The invariance of U under time reversal requires that S should change sign:_These two requirements imply that, when S is expressed in a standard form similar to (4), it has real coefficients and contains terms of odd p+q+r only. The reason for taking S in standard form is that two expressions can only be regarded as distinct if their standard forms are different; the choice therefore avoids a possible am biguity in S. On the basis of this result we now introduce the factorized form of U: U= exp(iS 1) exp(iSa) exp(iS5)···, (8) where SrI contains only terms with p+q+r=n: Sn= 2: Spqr(J.,PJ,/J,r+J.rJIIV.p) , (9) p+q+r=n where spIlr is real. From (8) it follows that ut = U-l= ••• exp( -iS5) exp( -iSs) exp( -iS1) (10) so that we can apply the transformations of different n successively. Note that, because_the .. different.S,. do not in general commute, exp(iS 1) exp(iSa) exp(iS5)··· is not in general equal to exp[i(SI+SS+S5+···)J. The number and symmetry classifications of the terms in S" can be obtained in the same way as for the Hamiltonian (Sec. III). The results are given in Table II. We now introduce the notation H2m+2= exp( -iS2m+l)H2m exp(iS 2m+1), (11) where m takes the values 0, 1, 2, ••• and Ho is the Hamiltonian H of (4). Then H2= exp( -iS1)Ho exp(iS1), H4= exp( -iSs)H2 exp(iSs) , ••• , Hoo=H. When H2m has been reduced to standard form, it can be written as H2m= !: hpqr(2m)(J.,pJIIQ.r+J,rJIIV,l). (12) (p-f-qtr even) Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsCENTRIFUGAL DISTORTION COEFFICIENTS 1939 A. Orders of Magnitude It is necessary now to choose the orders of magnitude of the coefficients Spqr in such a way that the rapid convergence of the Hamiltonian is unaffected, i.e., so that (5) applies equally to fl. The required orders of magnitude can be obtained in the following way. It is easily seen that H2m+2 is obtained from H2m by replacing Ja by exp( -iS2m+l)Ja exp(iS 2m+l) (a=x, y, z). Expansion of the exponentials gives exp( -iS2m+l)Ja exp(iS2m+l) _ ~ .t [[ ••• [[J a, S2m+l], S2m+l], ••• ], S2m+l] -~1, , t~O t! (13) where the term in t contains the tth commutator; this result is readily obtained by induction. Since S2'"7l+l is of degree 2m+ 1 in J and the degree is reduced by one for each commutation, the term in t is of degree 1 + (2m+1)t -t=1 + 2mt in the components of J. Thus, except for the case m=O, the effect of the unitary transformation is to replace J a by a power series whose first term is J a. It is neces sary that this series should converge as rapidly as the Hamiltonian. Successive terms in this series differ by 2m in the power of J, and the ratio of their coefficients is of the order of magnitude of the coefficients in S2m+l; on the other hand the ratio of the coefficients of terms in the Hamiltonian which differ by 2m in the power of J is 1(4m. The coefficients in S2m+l must therefore be of order 1(4m, so that (14) Writing H2m symbolically as TeCI(4J2+1(8J4+ ••• ) and the transformation (13) as J~ J +0mJ2mH+~J4mH+ ... , we see that H2m+2=H2m+O(Te 0m+4) and therefore that (15) hpqr(2m+2) = hpqr(2m) +0 (T.,,4m+4) • (16) (It should be noted here that the 0 symbol means "of order not greater than" and not "of order equal to.") Now, hpqr(2m)'r::5T.,,2(p+q+r), so that hpqr(2m) is not altered significantly if 4m+4> 2(p+q+r), i.e., if 2m?;:p+q+r. Thus, the coefficients of the terms in the Hamiltonian with p+q+r=n are only altered significantly by the transformations in (11) with m< tn. It follows from the discussion at the end of Sec. III that in considering the determinability of the coefficients in the Hamil-tonian up to the terms of degree n it is sufficient to take the unitary transformations (11) as far as to obtain Hn. v. TRANSFORMATION OF HAMILTONIAN Let us now consider the successive transformations described in Sec. IV. The first transformation is exceptional in that it is equivalent to replacing J", by a linear combination of J"" J'll J. rather than by a power series. In fact, if we write Sl in the form - Sl=we·J, (17) where e is a unit vector, then exp(iSl) is the unitary operator corresponding to a rotation ofaxes13 through an angle w about the line e. The three parameters SlOO = twe"" SOlO = twell, S001 = twe. [compare (17) with (9)J introduce three degrees of freedom into H2• From the results at the end of Sec. IV the coefficients of the quadratic terms are not changed significantly by the later transformations, so that this is the only transformation that affects the determina bility of the quadratic coefficients. The situation is in fact just as was described in Sec. II-the energy levels depend only upon the principal values of the quadratic coefficients, which are the three determinable combina tions of coefficients, and the three parameters in Sl can be chosen to bring the quadratic coefficients to diag onal form. With this choice of Sl the quadratic terms in H2 are in the reduced rigid-rotor form HR of (2); comparison with (12) shows that this corresponds to (18) (19) The X, V, Z notation is used for the principal rota tional constants rather than the conventional A, B, C because no particular order of the constants is implied. During this transformation the coefficients hpqr of the later terms in the Hamiltonian are changed to their new values hpqr(2), which are the values relative to the principal axes of the quadratic terms. The later transformations in (11) produce minor changes in the quadratic coefficients (see for example Sec. VIII); for simplicity, these changes are ignored in the following discussion, and the quadratic terms are taken in the form (2) with X, V, Z assumed constant. From (2) and the commutation relations (3) one can derive the following result, which is useful in the 13 E. C. Kemble, The Fundamental Principles of Quantum Me chanics (Dover Publications, Inc., New York, 1958), p. 307. Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1940 JAMES K. G. WATSON later parts of this section: i[HR, J.,PJIIV:+J:J"V,,P] =2p(Z-Y) (J",P-lJ"'l-HJ.r+l+J.r+lJ,,'lt-lJ,,,P-l) +2q(X -Z) (J",P+lJ"lj-lJ.r+l+J.r+ lJ"HJ",I>H) +2r(Y -X) (J",P+lJ"'lt- lJrl+J.r-lJ,,'lt-lJ,,,I>H) Let us consider first the B" species, which is typical of the three B species. Putting m=2 in Tables I and II, we find that there are three quartic terms in H of Species B"" and three terms in Sa of Species B",. The corresponding coefficients are hoal' hOl3, h211 and S120, SlO2, Saoo. Equation (22) then gives + terms of lower degree in J. (20) h0l3(4) = h01a(2) +2 (Z-Y)SI02' The argument at the end of Sec. IV shows that the second transformation is associated with the deter minabilityof the quartic coefficients in the Hamiltonian. From Table I there are 15 independent quartic terms in the Hamiltonian, while from Table II there are 10 independent terms and therefore 10 variable param eters in Sa. Thus, if all these terms in Sa affect the Hamiltonian independently, then the number of de terminable combinations of the quartic coefficients ob tained will be 15-10=5. From (5) and (14) the quartic coefficients are of order KST., and the coefficients in Sa are of order 0. When H4 is expanded (21) The only terms of order K8T. in H4-H2 come from i[HR, Sa], since the coefficients in HR are the only ones of H2 which are-of order 0T •. The significant changes between the quartic coefficients hpqr(2) and hpqr(4) are therefore produced by the quartic terms of i[HR, Sa]. Using the relation (20) together with the standard forms (9) and (12), we can derive these changes to be hpq-r(2m) = hpqr(2m-2) +2 (p+l) (Z-Y)Sp+l.q-I,r-l +2 (q+ 1) (X -Z) Sp-l,'lt-l,r-l +2(r+l) (Y -X)Sp-I,q---I,r+I, (22) where p+q+r=2m=4 in the present case; in this equation terms of smaller magnitude, resulting from later terms in (21), have been ignored. When any of p, q, r in (22) are zero, the right hand side apparently contains S coefficients with negative subscripts, which have not previously been introduced; the presence of such coefficients is merely a consequence of writing (22) in a general form, and can be remedied by taking them to be identically equal to zero. Now the angular-momentum commutation rules (3) are invariant under the operations of the point group D2, and HR is totally symmetric. Thus, the terms in i[HE, Sa] are of the same symmetry species as the cor responding terms in Sa. This result can also be seen by examining the different possible combinations of parities of p, q, r in (20). We can therefore discuss the determinability of the coefficients of the quartic terms of the different symmetry species separately. +4(Y -X)SI02. (23) The three coefficients hoal (4), hola(4), h211 (4) are therefore independent functions of S12O, S102, S300 and by a suitable choice of the latter they could be made to take any arbitrary values, subject only to order-of-magnitude restrictions. The only exception arises when Z = Y or, more generally, when Z-Y is very small, i.e., for an accidental symmetric top. For the moment it is as sumed that X, Y, Z are sufficiently different from each other that such exceptional cases can be ignored. This difficulty is discussed in Sec. VII. The coefficients hoa1(2l, h013(2), h211(2) are therefore in determinable and the transformation should be chosen to eliminate the corresponding terms from the reduced Hamiltonian. This is achieved by putting (24) in (23), which can then be solved to give definite values of S120, Sl02, S300, so that these three degrees of freedom in the unitary transformation are removed. The same result holds for the B" and B. terms. We therefore conclude that the nontotalIy symmetric quartic terms can be eliminated from the reduced Hamiltonian. It should be noted that it has not been assumed that the molecule has any symmetry at all. This result is similar to one obtained more directly by Kivelson and Wilson,5 namely that these nontotalIy symmetric terms do not contribute to the energy in first order. The new feature which has emerged is that these terms can be transformed completely into the terms of higher degree in the Hamiltonian, and therefore that the effects of these terms in second or higher order are indistinguishable from the effects of higher degree terms in the Hamiltonian. Thus, any determinable combinations of coefficients which contain these co efficients will also contain the coefficients of some at least of the higher-degree terms. Turning now to the A -species terms, we see from Tables I and II that there are six quartic terms in H and one term in Sa. The former have coefficients h400, ho40, hOO4, h022, ~02, ~o, and the latter has coefficient S111. The general rule (22) then shows that h400(4), h04Q(4), how(4) differ insignificantly from h400(2), h04Q(2), ho04(2), Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsCENTRIFUGAL DISTORTION COEFFICIENTS 1941 respectively, whereas for the others we have h022(4) = h022(2) +2 (Z -Y)SllI, ~02(4) =~02(2)+2(X -Z)Slll, h220(4) = h220(2) +2 (Y -X) S111. (25) Since each of h022(4) , h202(4), h220(4) is affected by this degree of freedom in the unitary transformation, each of them is indeterminable individually. We can, how ever, eliminate the parameter SllI from (25) in two ways, which give two determinable combinations of these coefficients. The most symmetrical choice is ~(4)+h202(4)+h<nO(4) =h022(2)+~OZ(2)+~20(Z) (26) and Xh022(4)+ Yh202(4)+Zh<nO{4) =Xh022(2)+ Y~02(2)+Zh220(Z). (27) The five determinable combinations of the quartic co efficients are, therefore, h4IYP) , h040(Z), ho04(2), ~2(Z)+ hzoz(Z)+h<no(Z) and Xh022(2)+Y~02(2)+Zh<nO(2), since none of these is affected significantly by this transformation. The parameter SllI is now chosen so that only five independent quartic terms are left in the reduced Hamiltonian. The way in which this may be done is somewhat arbitrary, but it is convenient to make a choice which simplifies the calculation of the eigen values of the reduced Hamiltonian. Such a choice is suggested in Sec. VI. Once the choice has been made, we will have chosen all the coefficients in Sa in a definite way, and can then perform the complete transforma tion from Hz to H4• The details of the transformation are not considered here, but it should be noted that the coefficients hpq,(4), which originally depended on the indefinite parameters in Sa, are now definite quan tities which could be related to the original hpq, of (4). C. H2m=exp(-iS 2m-1)H 2m-2 exp(iS2m-1) We now suppose that the first (m-1) transforma tions have been performed, choosing the parameters in Sl, S3, ••• , S2m-3 in some suitable way, and we con sider the mth transformation. According to the result at the end of Sec. IV, this transformation is associated with the determinability of the coefficients of the terms of total degree 2m in the Hamiltonian. From Table I there are (m+1) (2m+1) such terms, while from Table II there are m (2m+ 1) terms in S2m-1. Thus, if the latter produce independent changes in the Hamiltonian, the number of determinable combinations of coefficients will be (m+l) (2m+l) -m(2m+l) =2m+l, and there will be only 2m+ 1 independent terms of degree 2m in the reduced Hamiltonian. With appropriate changes, the discussion at the beginning of Pt. B. of this section applies here also, and the significant changes in the coefficients of the terms of degree 2m are again given by (22). We con-sider again the terms of the different symmetry species in D2 separa tel y. For the B", species, Tables I and II give the same number !m(m+1) of terms in the Hamiltonian and in S2m-1. It may be expected, therefore, that the co efficients of the B", terms are all indeterminable and that these terms can all be eliminated from H2m, i.e., that we can satisfy the equations with (pqr) = (eoo) , (28) by a suitable choice of the coefficients of the B", terms i~ S2m-1. To see that this is true, we consider the equa tlOns. (22). and (28) for successive even values of p, startmg With p=O. For p=O, (22) and (28) give 0=hOq/2m-2)+2(Z-Y)Sl,q-1,r-l, (29) where q and r are both odd and q+r=2m. These m equations can be solved immediately for the m quan tities Sl,g-l,,-l, as long as Z ¢ Y . We then have for p = 2 0=~qt"(Zm-Z)+6(Z -Y)S3,q-1,r-1 +2(q+1) (X -Z)Sl,q-J-1,r-1 +2(r+1) (Y-X)Sl,H,r+l, (30) where q and r are again both odd, and now q+r = 2(m-1). Using the solutions of (29) in the last two terms of (30), we can solve the (m-l) equations (30) for the (m-l) quantities S3,q-1,,-1. It is obvious from the form of (22) that this procedure can be continued up to and including the final equation for S2m-1,O,O, which corresponds to p=2m-2, q=r=1 in (28). Thus (except in the case Z = Y) Eqs. (28) can always be satisfied and the B", terms of degree 2m can be eliminated from H2m. The same result obviously applies to the BII and B. terms. We therefore reach the conclusion that all the non totally symmetric terms of degree 2m can be removed from H2m. Since this has been established for general m, it follows that all the terms which are non totally symmetric in D2 can be removed from the re duced Hamiltonian. This holds for a molecule of any symmetry. The (2m+l) determinable combinations of coeffi cients of the terms of degree 2m must therefore all belong to the A-species terms, as may be confirmed from the appropriate rows of Tables I and II. The form of these determinable combinations is not dis cussed in this paper, but in Sec. VI it is shown that the transformation can be chosen so that there are only (2m+1) independent terms of degree 2m in the re duced Hamiltonian. That result therefore proves that there are only (2m+l) determinable combinations of coefficien ts. VI. REDUCED HAMILTONIAN In Sec. V it was shown that the terms which are non totally symmetric in D2 can be eliminated com- Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1942 JAMES K. G. WATSON pletely from the reduced Hamiltonian, and the equa tions governing the reduction of the totally symmetric terms were given. The method of carrying out this final reduction is not unique, but it is desirable to choose a method which simplifies the calculation of the rota tional energy levels. There are two general methods of calculating the energy levels. In the first, the matrix of the Hamiltonian in a basis of symmetric-top wavefunctions is set up and diagonalized exactly. This method is straightforward in principle, but requires a large computer. In the second method, the centrifugal terms are treated as a per turbation on the rigid-rotor terms, and the centrifugal corrections are evaluated by means of perturbation theory. The latter method was used by Kivelson and Wilson5 to obtain the first-order corrections due to the quartic terms, and could be extended to include the second-order corrections from the quartic terms and the first-order corrections from the sextic terms, and so on. In this section it is shown that the reduced Hamil tonian can be chosen so that its matrix elements in a symmetric-top basis satisfy the selection rule ilK = 0, ±2. This form is probably the most convenient for the first method of calculation, because special methods have been developed for diagonalizing matrices of this type. It seems likely however that a different reduction would be more convenient for the perturbation method of calculation. The general problem of the perturba tion method is not discussed here. We note first from Table I that, since p, q, r are all even in the A -species terms, these terms are functions of J",2, Jy2, J,2 only. If we take the z axis as the sym metric-top axis, then the matrix elements of J,2 and (J",2+J y2) are purely diagonal, while J,l? and Jy2 in dividually have matrix elements with !lK =0, ±2. Let us now consider the general transformation of Sec. V.C and the general equation (22). We start with the terms with r=O and take them in order of in creasing r. For r=O, Eq. (22) reduces to hp,q,o(2m) = hp,q,o(2m-2) +2 (Y -X) Sp-l,q-l,l, (31) where p and q are both even and p+q = 2m. If either q or p is zero, (31) becomes h2m,o,O(2m) = h2m,0 ,o(2m-2) or ho ,2m ,o(2m) = ho,2m ,O(2m-2) (32) and so the two coefficients h2m,o,o(2m) and hO,2m,O(2m) do not depend significantly on the transformation and cannot be chosen to have arbitrary values, On the other hand, for the other (m -1) combinations of p and q in (31), the coefficients hp,q,o(2m) can be chosen to have arbitrary values and (31) can be solved for Sp-l,q-l,l, as long as X ~ Y. The desired choice of the coefficients is obtained in the following way. The ex-pression [h2m,O,o(2m)J",2+ho,2m,o(2m)Jln(J",2+Jll)m-l +(J",2+J,l)m-I[h 2m,o,o(2m)J",2+h o,2m,o(2m)J,n (33) is a polynomial of degree 2m in J", and Ju whose only nonzero matrix elements have!lK =0, ±2. This expres sion is not in the standard form (4), but may be re arranged to bring it into standard form. Because of the noncommutativity of J", and JII the rearrangement introduces terms of lower total degree in J"" Iu, and I.; but, since the coefficients of these additional terms are of smaller magnitude than the normal coefficients of such terms in the Hamiltonian, these terms can be ignored on order-of-magnitude grounds. When the re arrangement is carried out, we obtain the standard form of (33) in which the coefficients of J",2m and J,}m are 2~m,o,o(2m) and 2ho,2m,o(2m) and the coefficients of the remaining terms are functions of h2m,o,o(2m) and ho,2m,o(2m). Therefore, if these functions are used for hp,q,o(2m) in (31) and the Sp-l,q-1,l are chosen to satisfy (31), the terms with r=O will be brought to the desired form, For r=2, Eq. (22) becomes hp,q,2(2m) = hp,q,2(2m-2) +2 (p+ 1) (Z -Y)Sp-t1,q-l,1 +2(q+1) (X -Z)Sp-1,q+l,l+6(Y -X)Sp-l,q_1,3, (34) where p and q are both even and p+q=2m-2. The S coefficients with the final SUbscript 1 are now known quantities, so that (34) governs the possibility of choosing hp,q,2(2m) in some suitable way by an appro priate choice of Sp-1,q-l,3, When either p or q is zero, the term in Sp-1,q-1,3 drops out and therefore the two co efficients ~_2,O,2(2m) and hO,2m_2,2(2m) are fixed and cannot be chosen arbitrarily. We now construct the expression [~m_2,o,2(2m)J",2+ho,2m_2,2(2m)Iu2J (J",2+J l) m-2J.2 +I,2(J:I,2+Iu2)m-2[~m--2,o,2(2m)J.,2+ho,2m_2,2(2m)JII2J, (35) which has matrix elements with ilK =0, ±2 only. The coefficients in the standard form of (35) are functions of h2m_2,O,2(2m) and hO,2m_2,2(2m), and when these functions are used as the values of hp,q,2(2m) in the (m-2) equa tions (34) with p~O, q~O, then Eqs. (34) can be solved for the Sp-1,q-1,3. (Again the only exception arises for X = Y.) In this way the terms with r=2 can be brought to the desired form. The procedure described for r=O and r=2 can ob viously be continued successively to higher even values of r. The last value of r which need be considered is r=2m-4, since the terms with higher values of rare already in the desired form. In that case, Eq. (22) provides one new equation, for the coefficient S1,1,2m--3, which may be solved in the same way; and this com pletes the choice of all the S coefficients, because the Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsCENTRIFUGAL DISTORTION COEFFICIENTS 1943 three subscripts of the s coefficients must all be odd (see A species in Table II). It has therefore been shown that the unitary trans formation can be chosen so that the reduced Hamil tonian consists of a sum of expressions of the type ex emplified by (33) and (35), whose only matrix ele ments in a representation which diagonalises J. have AK =0, ±2. Once this result is established, it is pos sible to write down immediately the reduced Hamil tonian in a more convenient form. This is obtained by putting JX2=![J2-J.2+ (J",2-Jy2)], Jy2=![JLJ.2- (JxLJy2)]. (36) The reduced Hamiltonian then contains only terms which are either independent of (Jx2-Jy2) or linear in (Jx2-Jy2). Since J and J.2 are purely diagonal in the representation chosen, the former set of terms have matrix elements with AK =0 while the latter set of terms have matrix elements with AK =±2. If the terms of like degree in J2 and J.2 are collected together, the reduced Hamiltonian takes the form Hred=[!(X+Y)J2+{Z_!(X+Y) }Jl-AJ(J2)2 -AJKJ2J.2-A KJ.4+HJ(J2)3+H JK(]2)2J.2 +HKJJ2J.4+HKJ.6+ ••• ]+[(Ji-Ji) a(X -Y) -!5JJ2_!5KJ.2+1/J(J2)2+1/JK]2J.2+ 11KJ.4+ ••• } +{HX -Y) -!5J]2-!5KJ.2+ 11J(J2)2+1/JK]2J.2 +1/KJ.4+ ••• } (Jx2-Jy2)], (37) where the terms up to and including the sex tic terms have been written explicitly. The matrix elements of the first bracket are diagonal in K, and those of the second bracket have AK =±2. The second bracket is written in the form shown becauseJ.2 does not commute with Jx2-J1I2. In (37) the coefficients are as follows: X, Y, Z are the effective principal rotational con stants; AJ, AJK, AK, !5J, !5K are the quartic distortion co efficien ts; HJ, HJK, HKJ, HK, 1/J, l1JK, 1/K are the sextic distor tion coefficients. The relation of the above quartic distortion coefficients to the T a~'Y8 of Wilson and Howard2 is discussed in Sec. VIII. The meaning of the above sextic coefficients in terms of the hpqT coefficients of (4) is not discussed here, and they must be regarded for the moment as empirical parameters. It may be remarked however , , that they depend not only on the sextic coefficients of the original Hamiltonian (4) but also on the quartic coefficients, because the sextic terms are affected by the transformation which was used to reduce the quartic terms. VII. DISCUSSION OF PREVIOUS SECTIONS It is convenient at this point to summarize the re sults of the previous sections. The rotational Hamiltonian for a given vibrational level has been assumed in a general form, and this general form has been reduced to a new form which has the same eigenvalues but fewer parameters. This procedure removes the indeterminacies in the co efficients which arise either from the possibility of using the commutation rules or from the possibility of performing a unitary transformation on the Hamil tonian. Any remaining indeterminacy must have some other source. The reduction is carried out in two stages: (i) Rearrangement to standard form. The "general" Hamiltonian contains various terms which differ in the order of the angular-momentum operators J J J x, y, z but .have the same ~otal power of each of Jx, Jy, J •. In realIty, however, thIS expression is no more general than the special form (4), called the standard form to which it can be rearranged with some use of the co~mutation rules. The extra coefficients which appear in the "gen eral" form therefore contain indeterminacies which are removed by using (4). It should be stressed that the standard form is completely equivalent to the "general" form, in that it has exactly the same eigenvalues and eigenfunctions. (ii) Unitary transformation to reduced form. The standard form still contains indeterminacies associated with t~e possibility of carrying out a unitary trans format~on. The parameters in a general unitary trans formatlOn have therefore been chosen to eliminate as many terms as possible from the transformed Hamil tonian, thus giving the reduced Hamiltonian. The reduced Hamiltonian has exactly the same eigenvalues as ~he original Hamiltonian, but it is not completely eqUlvalent to it because its eigenfunctions are related to thos~ of the .original Hamiltonian by a unitary trans formatlOn. This fact would have to be taken into ac count in an accurate treatment of rotational intensities and it follows conversely that the intensities contai~ fur~her information not contained in the energy levels. It IS doubtful, however, whether the use of the in tensities would be a practicable proposition. The main concern here has been with the possibility of carrying out the reduction in a particular way. Order of-magnitude considerations have been used to sim plify the discussion. From these it follows that the re duced ~amiltonian can be chosen to have the following propertles: (i) It is totally symmetric in the point group D2, regardless of the symmetry of the molecule' (ii) it contains only (n+1) independe~t terms of total degree n in the components of the total angular momentum, for each even value of n; Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1944 JAMES K. G. WATSON (iii) its matrix elements in a symmetric-top basis satisfy the selection rule IlK =0, ±2. This form is particularly convenient for calculations by matrix diagonalization, and it is desirable that the coefficients appearing in it should be related to the vibrational potential function. The significance of the quartic coefficients is discussed in Sec. VIII. The role of the reduced Hamiltonian may be clarified by a vibrational analogy. In the absence of resonances, the vibrational term values of an asymmetric-top molecule are given by G(v) = L Wi(Vi+!) + LL Xii(Vi+!) (Vi+!) i ''";?i + LLLYijk(Vi+!) (Vj+!) (Vk+!) +.... (38) ,'";?,'";?k The coefficients Wi, Xiii Yijk, ••• appearing in this ex pression are the determinable combinations of the vibrational potential constants. They provide the only information on the vibrational potential energy that can be derived from the vibrational energy levels of a single isotope. There are in general, however, con siderably fewer parameters appearing in (38) than in the expression for the vibrational potential energy, so that the extra parameters in the latter must be re garded as not determinable on the basis of the above data. The reduced rotational Hamiltonian performs a similar function to (38) in the fitting of observed energy levels. The necessary complication in the rota tional problem is that one cannot give an explicit ex pression for the energy levels, and it is necessary to solve a secular equation. It may also be noted that a common feature of the reduced rotational Hamiltonian and of (38) is that they are no more complicated for the most unsymmetrical molecule than for the most symmetrical molecule. (We consider here only asym metric tops, so that the highest symmetry is D2h.) This is because the number of determinable coefficients is not greater than the number of coefficients which are nonzero by symmetry, even for the highest sym metry. Failure of the Reduction Just as the vibrational expression (38) breaks down in cases of resonance, so does the reduction of the rota tional Hamiltonian break down in certain cases, as was pointed out in Sec. VI. These anomalous cases arise in the present treatment when the molecule is acci dentally a symmetric top. More precisely, the break down occurs when the difference between two of the rotational constants is of the same order of magnitude as the centrifugal constants. Since we are only con sidering molecules which are not symmetric tops by symmetry, the likelihood of this occurring in practice is rather small. In any case, the breakdown is only partial if the molecule is not accidentally a spheri cal top. If we consider the case X~y¢Z, then the A-species terms could be reduced in the way described in Sec. VI by taking the symmetric-top axis along the X or Y axis. The terms of Species B", and BII could also be eliminated as described in Sec. V. The method there fore only really fails for the elimination of the B. terms, and this is associated with the fact that S2m-l contains B. terms such as J.2m-l which commute with HR of (2). On the other hand, the reduction as described breaks down completely for an accidental spherical top, because all the unitary transformations then commute with HR. These anomalous cases would therefore re quire special consideration to see how the indetermina cies associated with the unitary transformations arise. A more serious case of failure arises when the Hamil tonian (4) does not converge rapidly. This occurs when two vibrational levels are nearly degenerate and in Coriolis or Fermi interaction. It is then impossible to treat the vibrational levels separately as has been done here. A similar failure occurs for "nonrigid" mole cules, for which the concept of small vibrations about equilibrium, which is the basis of the normal vibrational perturbation treatment, no longer applies. vm. DETERMINATION OF QUARTIC COEFFICIENTS In practice, the result of most immediate importance obtained in the previous sections is that there are only five independent quartic terms in the reduced Hamil tonian, instead of the six terms of Species A in the standard form (4). The rest of this paper is devoted to an examination of this particular case in more detail. It is necessary to note first that the quartic terms of the reduced Hamiltonian contain only those quartic terms which contribute to the rotational energy in first order. The remaining quartic terms in the standard form (4) contribute in second or higher order, but the results of Sec. V show that their effects are entirely equivalent to the effects of higher-power terms in the Hamiltonian, and their coefficients are experimentally inseparable from those of the higher-power terms, unless data other than the rotational energies of one isotope is used. The situation is very similar to that arising in the discussion of anharmonic effects on vibrational energies (Sec. VII), where the second-order effects of cubic potential terms and the first-order effects of quartic potential terms are treated together in the con stants Xiii which also contain contributions from the vibrational angular momentum; the cubic and quartic potential constants can only be separated by the use of vibration-rotation interaction constants or of isotopic data. In the present section we ignore the higher-power terms in the Hamiltonian and we are therefore pri marily concerned with the first-order effects of the quartic terms. From the point of view of orders of magnitude (Sec. III), we are interested in the rota tional energies correct to order ~T •. A second point to note is that we must take account of the small changes in the effective principal rota- Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsCENTRIFUGAL DISTORTION COEFFICIENTS 1945 tional constants referred to at the end of Sec. V.A. These changes were ignored previously because they do not alter the values of the constants significantly, but their contribution to the energies is of the same order as the centrifugal contribution for low values of J and must be taken into account when we consider explicit expressions for the energies. It is sufficient here to have the constants correct to order K8T •• A. Hamiltonian and Transformation For an orthorhombic molecule, the rotational Hamil tonian as obtained from the vibrational perturbation treatment contains only terms of Species A in D2 because of molecular symmetry. For molecules of lower symmetry there are, in general, also terms which are non totally symmetric in D2• In the latter case we as sume that the non totally symmetric terms in the operators S1 and Sa have been chosen as outlined in Sec. V so as to remove the non totally symmetric quad ratic and quartic terms from the Hamiltonian. Thus, in either case we are left with only the A-species terms in the standard form (4). We can therefore concentrate on the one-parameter problem of the reduction of the A -species quartic terms. To facilitate comparison with other work we revert to the notation of Kivelson and Wilson5 for the quartic coefficients in the standard form of the Hamiltonian. If all higher terms are ignored, the latter becomes H=XJ",2+YJi+ZJ.2+1 LT'aap(J.l,H{32, (39) a.fJ where T'aapp=T'PPaa and a and [:3 are summed independ ently over x, y, z. The six T' aapp are related to the un primed T coefficients of Wilson and Howard2 as follows: The effective rotational constants X, Y, Z are equiva lent to Kivelson and Wilson's [:3, ,,(, a, respectively, and contain the correction terms given in their Eq. (34). All the coefficients in (39) are assumed to have values appropriate to the vibrational level being considered. The relation with the previous notation in (4) is T'",== 8h400, T' IIYZZ = 4ho22, T'yyyy=8h o40, T' xx .. = 4h202, T' .... = 8ho04, T'",,,,W= 4h220• (40) The unitary operator by which (39) is to be trans formed is exp(iS'a), where S's is the single term of Species A in Sa; namely S'a=Sl11(JJy!.+J Jy!",), ( 41) in which S111 is of order of magnitude 0, i.e., the ratio of a quartic centrifugal coefficient to one of the prin cipal rotational constants. To terms of order ~T. we therefore have exp( -is'a)H exp(iS'a)~H+i[HR, S'a}=·f1. (42) Now, retaining the terms of lower degree neglected in (20), we have i[HR, S'aJ=2S11d (Y -X) (J",2J1l+J1I2J",2+2J.2) +(Z-Y) (J1I2J.2+JN1I2+2J,,?) +(X -Z) (J.2J",2+J",2J.2+2J1I2) I and, therefore, if f1 is expressed as f1 =XJ",2+Y'JII2+ZJ.2+1 L f'aafj(J.la2Jl, we find a.p X=X+4(Z-Y)S111, Y' = Y +4(X -Z)S111, Z =Z+4(Y -X)S111, i' aaaa=r' aaaa, f'1IYZZ = T'1Iy .. +8(Z -Y)Slll, f'""" •• = T'",,,, •• +8 (X -Z)S111, (43) (44) f'''''''W=T'''''''lIy+8(Y -X)S111' (45) As far as the quartic coefficients are concerned, this is just the result given in (25) with the change of no tation in (40). The parameter S111 can be eliminated from the nine equations (45) to give eight quantities which have the same values for H or fl. Since the energies cannot de pend on S111, they therefore depend only on these eight quantities, which are therefore the determinable com binations of coefficients. The eight quantities can be taken as x=X -!T'w •• , ID = Y -!T' """'" 8 = Z -!T' "''''W' , , , T zux, T yyy,,, T sa:z, Tl = r' yyzz +r' xxzz +T' xxw, T2 = X T'I/Y"+ Y T' """ •• + ZT' """w, (46) where some smaller terms have been ignored in T2. It may be noted that T1 and T2 have the same values for any permutation of the axes. It is convenient now to convert to Nielsen's con stants DJ, DJK, DK, OJ, Rs, and Re, whose relation to the T'S is given in Kivelson and Wilson's Appendix.5 With the tilde used to identify coefficients corresponding to fl, Eq. (45) leads to DJ=DJ+!(X -Y)S111, DJK=DJK-3(X -Y)S111; DK=DK+(5/2) (X-Y)S111, ~J=OJ; R5 = Rs+!(X+ Y-2Z) S111, R6=R6+1(X -Y)Slll. (47) Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1946 JAMES K. G. WATSON When the matrix elements of i1 are calculated in a symmetric-top basis, the matrix elements with ilK =±4 are proportional to Ra. Thus, if S1ll is chosen so that R6=0, then i1 will be in the reduced form described in Sec. VI, with only IlK =0, ±2 matrix elements. We therefore take stants in the reduced Hamiltonian i1 are x =X -[16Rr,(Z -Y) / (X -Y)], Y=Y+[16R 6(Z-X)/(X-Y)J, Z=Z+16Rr,. (53) Slll= -4Rr,/(X -Y), and the coefficients in the reduced form become (48) It can be shown that these equations are equivalent to X =x-2IlJ-Il JK+2oJ+2oK, DJ=DJ- 2Ra, DK = DK -10Rr" DJK = DJK+ 12R6; BJ=oJ; R5=Rr,+[2(2Z-X -Y)R6/(X -Y)]; Ra=O. (49) Since the five nonvanishing coefficients with tildes in (49) are the quantities which are determined experi mentally, it is convenient to adopt a simpler notation for them. OJ is unchanged and is therefore retained, and the others are represented by capital and small deltas, as follows: (50) The coefficient OK is chosen in this way since it pro vides a closer analogy with OJ [see Eq. (55)]. Using (49) and the expressions in Kivelson and Wilson's Appendix,6 one finds ~=-Hr'",,,,,,,,,,+r'I/lI1IlI)' +!(r'IfIf .. -r'",,,, •• +r'''''''1fIf(2Z-X -Y)/(X -Y)}. (51) It is a curious fact that the value of ilK is invariant to a permutation of the axes. The combinations of the r'aaf3f3 occurring in (51) can be expressed in terms of 1"1 and r2 of (46); the expressions for IlJK and ilK are obvious, while OK = [ir'= (X -Z) / (X -Y)] +[ir'I/lI1IlI(Y -Z)/(X -Y)] -[iT1(X+Y)/(X-Y)] + [ir2/ (X -Y)]. (52) From (45) and (48), the effective rotational con-Y =ID-2IlJ-IlJK-2o J-2oK, Z=.8-2IlJ, (54) where X, ID, .8 are given in (46). Thus, as expected, the coefficients in the reduced Hamiltonian can be expressed in terms of the eight determinable comb ina tionsin (46). [In (52) the difference between X, Y,Z and X, ID, .8 is not significant to the present accuracy.] By some manipulation, the reduced Hamiltonian can be brought to the form i1 =[!(X+Y)J2+{Z-iCX+Y) }J.2-IlJ(J2)2 -IlJK]2J.2_IlKJ.4] +[(J",LJi) {t(X-Y) -OJJ2-oKJ.2} +{t(X-Y) -OJJL oKJ.2} X (J",2_J"i)], (55) which is the same as (37) with the sextic terms neg lected. This result shows the analogy between OJ and OK. If the quadratic terms of (55) are written as Ho=XJ,?+Y J,l+ZJ.2 =iCX+Y)J2+{Z-iCX+Y) }J,2 +iCX-Y) (Jz2-J,l), (56) then Jz2-J,l can be eliminated from (55) and (56) to give i1 =Ho-dJ(J2)2-dJK]2J.2-dKJ.4_dwJHo]2 -!dWK(HoJ.2+J.2Ho), (57) in which dJ=IlJ-{2oJ(X+Y)/(X-Y) }, dJK=IlJK-{20K(X+Y)/(X-Y)} -{2oJ(2Z-X-Y)/(X-Y) }, dK=IlK-{2oK(2Z-X-Y)/(X-Y) }, dWJ=40J/(X-Y) , dWK =40K/ (X -Y). (58) Equation (57) can be used as the basis of an ap proximate treatment of the effects of centrifugal dis tortion, similar to that of Kivelson and Wilson," in which the quartic terms are regarded as a perturbation on the unperturbed Hamiltonian Ho and the first-order perturbation corrections are evaluated. If Wo is the Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsCENTRIFUGAL DISTORTION COEFFICIENTS 1947 appropriate eigenvalue of Ho, it is readily found that the approximate eigenvalue of il is W = (il)= Wo-dJJ2(J+ 1) 2-dJKJ(J+ 1) (1.2) -dK(J.4)-d WJWol(J+l) -dwKWo(1.2), (59) where J is the total-angular-momentum quantum number. The angle brackets denote average values. A more direct derivation of this result has already been reported briefly.s The method of the previous report makes it possible to express the average values of all the quartic terms, such as (J.4), in terms of those of the quadratic terms. The averages are described more fully in the appendix at the end of this paper. Equation (59) gives the form of the perturbation expression which is most closely related to the reduced Hamiltonian (55). However, for the practical fitting of rotational energy levels, the necessity of evaluating (J.4) is an unnecessary complication. By using Eq. (4) of Ref. 8 to eliminate the term in (J.4), or by choosing S111 = {dK(X -Y)/12(Z-X) (Z-Y)} -{4R6/(X-Y)} (60) in the present section, one can derive the alternative first-order expression W =wo-PwW02-PWJwol(J + 1) -pJJ2(J + 1) 2 -pwKWo(1.2)-PJxl(J+l) (J.2), (61) in which wo is the energy of a rigid rotor with rota tional constants X'=X- {16R6(Z-Y)/(X -Y)} +{dK(X-Y)/3(Z-X) }, Y'=Y+{16~(Z-X)/(X-Y) } -{dK(X -Y)/3(Z-Y)}, Z' =Z+16R o-{dK(X -y)2/3(Z-X) (Z-Y)}, (62) and the P coefficients are given by PW= -dx/3(Z-X) (Z-Y), PWJ=dwJ-(X+Y)pw, PJ=dJ+XYpw, pWK=dwK-2(2Z-X-Y)pw, pJK =dJK+2 (XZ+YZ-2XY)pw. (63) The difference between X', Y', Z' of (62) and X, f, Z of (53) must be taken into account in comparing results obtained by the use of the two equations (59) and (61). To facilitate the use of available tables, (61) can con veniently be rewritten in terms of Ray's reduced energy14 E(K) and its derivative E' = dE/ dK = (Jb2), 14H. C. Allen, Jr., and P. C. Cross, Molecular Vib-Rotors (John Wiley & Sons, Inc., New York, 1963), Chaps. 2 and 3 and Appendix IV. where K is the asymmetry parameter K= (2B -A -C) / (A -C). With a IIr representation, i.e., (x, y, z) == (c, a, b), respectively, in the definition of the centrifugal con stants (51), the result is W =Wo-l:.EP- (l:.JK+2K5J+4l:. E)J(J+l)E' where -(l:.J-l:.E)J2(J+l)2+25 JJ(J+l)E +2(5K+2Kl:.E)EE', (64) B. Sum Rules The sum rules for the asymmetric-top rotational energies, with inclusion of the quartic centrifugal terms, have recently been given by Allen and O1son15 and in a more symmetrical form by Fraley and Rao.16 Examination of Fraley and Rao's equations shows that these rules provide a method of determining seven combinations of constants, plus one check, when suf ficient data are available. These combinations of con stan ts can be taken as x, ID, .8, r' xxxx, r'IIUYY, r' •••• , and rl of (46). Thus, this method of determination does not contradict the previous statements about the de terminability of the quartic distortion coefficients. C. Planar Molecules As was noted in the Introduction, successful de terminations of the quartic centrifugal constants have been made for some planar molecules. For such mole cules it is customary to use certain relations between the ra{J'Y~' which were given for the general planar molecule by Dowling and by Oka and Morino,6 although strictly speaking these relations only hold for the equilibrium values of the ra{J'Y~' These relations are equivalent to two relations among the r' aa{J{J, which may be expressed in the form r'aace=!A 2C2[ (r'aaaa/ A4) -(r'bbbb/ B4) + (r'eece/C4)], r'bbee=!B2C2[ -(r' aaaa/ A4) + (r'bbbb/ B4) + (r'eece/C4)], (66) in which all the constants have their equilibrium values. The four independent r' aa{J{J for a planar molecule may therefore be taken as r'aaaa, r'bbbb, r'eece, and r'aabb. Now, Eqs. (45) show that the r'aaaa are determinate quantities and that the principal rotational constants are, to all intents and purposes, also determinate quan ti ties. Thus, the use of (66) makes r' aaee and r' bbee de- 16 H. C. Allen, Jr., and W. B. Olson, J. Chern. Phys. 37, 212 (1962). It should be noted that in this reference, and in W. B. Olson and H. C. Allen, Jr. O. Res. Nat!. Bur. Std. A67, 359 (1963)J, the (1,3) matrix elements of the O± submatrices are given as E1,B instead of the correct E1•a±E1._I• This does not affect any conclusion of either paper. 16 P. E. Fraley and K. N. Rao, J. Mol. Spec try. 19, 131 (1966). Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1948 JAMES K. G. WATSON terminate quantities, and acts as a constraint on the unitary transformation. When the conditions (66) are applied, the problem therefore becomes well determined and values for the four independent distortion con stants can be obtained. The success of these treatments can be understood on this basis. The two relations (66) between the six r'aapp lead to one relation between the five distortion constants in (46). With some use of the planarity condition AB = (A + B) C, the required relation is found to be (67) which in fact involves only three of the distortion con stants in (46). From (67) can be derived correspond ing relations between the distortion constants in (51), etc. As mentioned above, these relations apply strictly only to the equilibrium values of the constants. It may be that the higher-order effects found for some planar molecules, suchs as F20, are to some extent spurious, being due to departures from these relations. It would be interesting therefore to have an unconstrained fit of a planar molecule to the five distortion constants in (46), to test how well (67) is satisfied. ACKNOWLEDGMENTS I would like to thank Professor 1. M. Mills for his comments on the first draft of the manuscript, which helped me to eliminate several obscurities. This work was performed during the tenure of a research fellow ship from Imperial Chemical Industries, Ltd., and I wish to express my gratitude to them. APPENDIX: AVERAGES OF QUARTIC TERMS As mentioned in Sec. VIII, the method of a previous reportS makes it possible to express the values of all quartic terms in the angular momenta, averaged over rigid-rotor eigenfunctions, in terms of the average values of the quadratic terms. This Appendix gives more detailed formulas for these averages. The rigid-rotor Hamiltonian is taken as HR in (2) and its eigenvalue is denoted by WR• When (2) is averaged over the eigenfunction it gives If small changes in the rotational constants are treated as a perturbation, it follows from first-order perturba tion theory that (JZ2) =awR/ax, (J,l) =awR/ay, (J.2)=aw R/az, (A2) and these equations can be used for calculating (Jz2), (J,l), (J.2). Since J2 is conserved, averaging of J2 gives (A3) so that, from (Al) and (A3), (Jz2) = {WR-YJ(J+l)+(Y-Z) (Jhl/(X -V), (J,l) = {WR-XJ(J+l) +(X -Z) (J.2)}/(Y -X). (A4) Thus, it is sufficient to calculate (J.2) by differentia tion, once WR has been calculated for a given level. For the sake o(symmetry, the following equations are expressed in terms of (Ji), (JII2), (J.2). They can be modified as desired by use of the above equations. The average values of quartic terms~which are non totally symmetric in D2 vanish by symmetry, so that only the totally symmetric terms are of interest. Let us first consider terms of the type Ja2Jp2 for a ~ /3. We note first that (AS) For instance, from we have +(Y -Z) (J.4-J.4) = WR(J.2)- (J.2)WR=0. (A7) Thus, (Jz2J.2) = (IN,,2), and (AS) follows by cyclic interchanges. It is easily seen that Since the wavefunctions are eigenfunctions of HR-ZP, the average value of the left-hand side is (HR-ZJ2)X (J.2). By use of (Al) and (A3), the average value of (A8) gives (X -Z) {(J,,2J.2)_(J,,2)(J.2)}+(Y -Z) {(JII2J.2) _(JII2)(J.2)} =0. (A9) By cyclic interchanges, two further equations can be obtained from (A9); however, the three are not in dependent because their sum is an identity. An additional relation can be obtained by the aver aging of (43). The average value of the left-hand side vanishes, and by using (AS) the result can be written (X -Y) {(J,,2JII2)-(J:x,2)(J,l>I +(Y -Z) {(J1I2J.2) -(JII2)(J.2)1+(Z-X) {(JNi) -(J .2) (J:z;2) 1+311 =0, (AIO) Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsCENTRIFUGAL DISTORTION COEFFICIENTS 1949 where II=i[(X -V) {(J",2)(J1I2)+(J.2)} +(Y -Z) {(J,l)(J.2)+(J",2)} +(Z-X) {(J.2)(J.,2)+(J1I2)}]. (All) The quantity II is a function of the quadratic averages only, and is invariant under a cyclic interchange. From the three equations of the type (A9) the first three terms in (AlO) are equal, so that one obtains (J1I2J.2)= (JN1I2)= (J1I2) (J.2)-{II/(Y -Z)}, (IN,,,2)= (J",2J.2)= (J.2)(J",2)- {II/(Z-X)}, (J",2J1I2)= (J1I2J",2)= (J",2)(J1I2)-{II/(X -Y) }, (A12) The averages of the type (Ja4) can now be obtained as follows. By the averaging of (HR-YJ2) (HR-ZJ2) = (X -Y) (X -Z)J",' +(X -V) (Y -Z)J,Hi+(Z-Y) (X _Z)JN",2 -(Y-Z)2JN,}, (A13) THE JOURNAL OF CHEMICAL PHYSICS one obtains, by using (Al) and (A3), (X -Y) (X -Z) { (J",4)-(J",2)2} +(X -Y) (Y -Z) {(JH1I2)- (J",2) (Jy2) } +(Z-Y) (X -Z) {(IN,,,2)_(J.2)(J,,,2)} -(Y _Z)2{ (JNy2)-(J.2) (J1I2) } =0. (A14) Substitution from (A12) therefore gives (Jx4) = (Jz2)2+{ll(Y -Z)/(X -V) (X -Z)}, (Jy4) = (JII2)2+{ll(Z-X)/(Y -Z) (Y -X)}, (J.4) = (J.2)2+{ll(X -Y)/(Z-X) (Z-Y)}, (A1S) where the second and third equations are obtained by cyclic interchanges. Equations (A12) and (A1S) there fore give the desired expressions for the quartic averages in terms of the quadratic averages. By taking (x, y,'z) == (a, b, c) and putting X = 1, Y =K, Z = -1, they can be expressed in terms of the asymmetry parameter K. If one substitutes from (A4) into the third equation of (A1S), taking account of the change of notation from X, Y, Z to (3, ,,(, ex, respectively, one obtains Eq. (4) of a previous communication. s VOLUME 46, NUMBER 5 1 MARCH 1967 Electron Spin Resonance Absorption Spectra of COa-and COa3-Molecule-Ions in Irradiated Single-Crystal Calcite* R. A. SERWAyt Illinois Institute of Technology and IIT Research Institute, Chicago, Illinois AND S. A. MARSHALL Argonne National Laboratory, Argonne, Illinois (Received 1 November 1966) Single crystals of x-and -y-irradiated calcite reveal a number of paramagnetic defect centers, two of which have been tentatively identified as the COa-and coaa-molecule-ions. The ESR absorption spectrum of what is believed to be the COa-molecule-ion is found to have symmetry about the calcite [l11J direction with spin-Hamilton parameters given by gJl=2.0051, g.L=2.0162, AII=13.1 Oe, and A.L=9.4 Oe. The corresponding parameters for what is believed to be the COa3-molecule-ion which also exhibits symmetry about the crystal [111J direction are given by gl!=2.0013, g.l=2.0031, AII=171.22 Oe, and A.L=111.33 Oe. The parameters of these two spectra are discussed and compared with those reported for the isoelectronic NOs and NOa2-species. Optical measurements reveal two absorption bands, one at 6500 A and another at 4850 A. The longer-wavelength band exhibits anisotropy and is found to have temperature-dependent decay characteristics which are similar to those of the COa-molecule-ion ESR spectrum. An activation energy of 0.12 eV is obtained for this center. Thermal decay data suggest that the shorter-wavelength band is not associated with either of these paramagnetic species. INTRODUCTION DEFECT centers produced in various inorganic single crystals by ionizing radiations may some times be studied by electron spin resonance (ESR) and * This work was performed under the joint auspices of the U.S. Air Force and the U.S. Atomic Energy Commission. t Submitted by R. A. Serway to the faculty of the Illinois Institute of Technology as part of a Doctoral dissertation. optical absorption spectroscopy. When this is the case, information such as symmetry and polarization, un paired electron spin density on nonzero spin nuclei, temperature-dependent rates of formation and decay as well as concentration of species may be determined. The ESR absorption spectra of x-and "{-irradiated single-crystal calcite (CaCOa) have been studied from 4.2°K to room temperature. In crystals subjected to Downloaded 04 Sep 2013 to 128.148.252.35. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions
1.1714087.pdf	Complex vs Band Formation in Perovskite Oxides J. B. Goodenough and P. M. Raccah Citation: Journal of Applied Physics 36, 1031 (1965); doi: 10.1063/1.1714087 View online: http://dx.doi.org/10.1063/1.1714087 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/36/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Formation of charge-transfer-complex in organic:metal oxides systems Appl. Phys. Lett. 102, 163304 (2013); 10.1063/1.4802923 Formation of transition layers at metal/perovskite oxide interfaces showing resistive switching behaviors J. Appl. Phys. 110, 053707 (2011); 10.1063/1.3631821 Thermochemistry of complex perovskites AIP Conf. Proc. 535, 288 (2000); 10.1063/1.1324466 Ferroelectric fatigue in perovskite oxides Appl. Phys. Lett. 67, 1426 (1995); 10.1063/1.114515 Covalency Criterion for Localized vs Collective Electrons in Oxides with the Perovskite Structure J. Appl. Phys. 37, 1415 (1966); 10.1063/1.1708496 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 91.182.86.89 On: Wed, 09 Apr 2014 13:08:13JOURNAL OF APPLIED PHYSICS VOLUME J6, NO.3 (TWO PARTS-PART 2) MARCH 1965 Complex vs Band Formation in Perovskite Oxides J. B. GOODENOUGH AND P. M. RACCAH Lincoln Laboratory,* Massachusetts Institute of Technology, Lexington, Massachusetts It is argued that there is a critical cation-anion covalent mixing parameter Ac such that ligand-field theory is appropriate for A<Ac, but band theory must be used for A>Ac. This provides, therefore, a criterion for distinguishing metallic vs magnetic compounds in those structures, like perovskite, where cation-cation interactions are negligible. It is also argued that A.>Ao can be anticipated where the cations are in a low-spin state. The fact that LaNi03 contains low-spin NillI and exhibits no Jahn-Teller distortion suggested that >-'>Ao in this compound. Metallic conductivity from -200° to 300°C and Pauli paramagnetism from 4° to 3()()OK seem to confirm this suggestion. Where A"" A., there is the possibility of a phase change in which A < Ao in some directions, A> Ao in others. LaCo03 seems to illustrate this situation. It undergoes a transition at 12100K, the cobalt ordering into alternate (111) planes of high-spin Co3+ and planes containing low spin COllI. Below 4000K the latter planes contain only COllI ions. The magnetic C03+ ions couple anti ferromagnetically via Co3+-"diamagnetic COIIlO. complex"-Co3+ superexchange to give TN",,900K. THE theory of magnetism lacks adequate criteria for predicting whether a material will contain spon taneous atomic moments at low temperature. Investi gations of two criteria have been initiated previously: (1) the density of states at and just above the Fermi level of a nonmagnetic host doped with a transition metal atom,l and (2) a critical cation-cation separation Re.2 In this paper we investigate another: (3) the critical covalent mixing parameter Ac for cationic d with anionic sand p states. Figure 1(a) shows two transition-metal cations separated by a distance R. MoW has argued for a critical separation Re in solids below which the elec trons in overlapping f. and f., orbitals are metallic, above which they are magnetic. His argument essen tially implies that the overlap integral has a critical magnitude ~c such that for ~.= (f.,j.,) <~c the ligand field theory is applicable, for ~.> ~e a band theory must be used. Figure 1 (b) shows two transition-metal cations with an anion intermediary, as is found in the perovskite structure. In the theory of 1800 superexchange,4 the one-electron wave functions 1/I=y/O)+y/1) contain un perturbed, ligand-field functions of the form 1/1'<0) = N.(f.+A.4>.) and 1/I,,(0)=N,,(j..,+A,,¢ .. ). The perturba tion terms are of the form 1/;.(1)= 'L.,(b .. ,jU)1/;.,(O), where baa' = €.~. is the off-diagonal transfer integral connecting 1/;.(0) and 1/;.,(0) and U is the Coulomb energy required to move an electron from 1/1.(0) to 1/;.,(0). Again, the overlap integral has a critical magnitude ~e, and by definition U~ and the ligand-field perturbation pro cedure explodes as ~. increases to ~e.5 Since f. and f.' do not overlap appreciably, ~.=N.2A.2 and ~ .. = * Operated with support from the U. S. Air Force. 1 A. M. Clogston, B. T. Matthias, M. Peter, H. J. Williams, E. Corenzwit, and R. C. Sherwood, Phys. Rev. 125, 541 (1962). 2 D. B. Rogers, R. J. Arnott, A. Wold, and J. B. Goodenough. J. Phys. Chern. Solids 24,347 (1963). 3 N. F. Mott, Can. J. Phys. 34, 1356 (1956). 4 P. W. Anderson, Phys. Rev. 115, 2 (1959). Anderson used orthogonalized ligand-field orbitals. 6 J. B. Goodenough, Transition Metal Compounds, in Informal Proceedings Buhl International Conference on Materials, edited by E. R. Schatz (Gordon and Breach, New York, 1964), p. 65. N ,,2A,,2. It follows that there must be a critical covalent mixing parameter Ao such that for A<Ac ligand-field theory is applicable, for A> Ac band theory must be used. Covalent mixing destabilizes the cationic d orbitals by an amount ~E=E;A2, where E;';::::!,EM-EI is the differ ence in Madelung and ionization energies for the effective charges on the ions. In ligand-field theory, the principal contribution to the cubic-field splitting at an octahedral-site cation is E;(A.2-A,..2). If the transition~metal cation is in a low-spin state, this splitting is larger than the intra-atomic exchange splitting ~ex, or A.2> A,,2+ (~exjE;). Since the geometry of the perovskite structure makes A"jA. a fairly large fraction, it is reasonable to anticipate a A.> Ac and the consequent formation of IT* band states wherever cations of a perovskite are in a low-spin state. In the perovskite LaNiOs, the NiIlI cations are in a low-spin state. Also, there is no static Jahn-Teller distortion associated with the single eg electron and no evidence, from neutron diffraction data, of magnetic order down to "'-'lOoK.6 This suggested to us that A.> Ac in LaNiOa, so that the partially filled eg orbitals are transformed into partially filled IT* band states. The conductivity and susceptibility were investigated to check this. In the case of cation-cation interactions, it is known5 that where R';::::!,Ro, crystallographic distortions may occur in which cation sublattices are shifted toward one another to make some R < Rc and some R> Re. Similarly, distortions in which cation and anion sub lattices are shifted toward one another to make some A<Ao and some A>Ac can be expected where A';::::!,Ae. The high ferroelectric Curie temperature in BaTiOs has been interpreted,7 for example, as the result of a A,..';::::!, Ac. Several years ago magnetic data for the system Lal_",Sr",Co03-<l suggested8 that the perovskite LaCoOa orders at low temperature into low-spin COlli and high spin COH on alternate (111) cobalt planes. This im- 6 W. C. Koehler and E. O. Wollan, J. Phys. Chern. Solids 2, 100 (1957). 7 J. B. Goodenough (to be published). 8 J. B. Goodenough, J. Phys. Chern. Solids 6, 287 (1958). 1031 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 91.182.86.89 On: Wed, 09 Apr 2014 13:08:131032 J. R. GOODENOUGH AND P. M. RACCAH (0 ) (b) (3+) (c) FIG. 1. Schematic angular dependence of u and 11" ionic d orbitals for ~wo ca.tions separated (a) by a distance R, (b) by an inter venmg amon, and (c) by an intervening anion that is shifted toward one so as to make X.<X, and X.'>X" plies that A.~Ac and that at low temperatures the oxygen sublattice moves toward the COIll planes (COIll has a considerably smaller ionic radius because it has no eg electrons) and away from the Co3+ (111) planes to make A.(II!»Ac and A.(3+)<Ao, as shown in Fig. l(c). Each COIll forms a COIl106 "complex" that is isolated from the other "complexes" by Co3+ ions. Therefore, the eg orbitals of the COIll ions are trans formed into molecular orbitals of a "complex" rather than into crystalline 0'* band states. To further test this idea, LaCo03 was investigated. Although some of the experimental results were anticipated,9 our inter pretation is quite different. (1) LaNi03. This rhombohedral perovskite, prepared by reacting the oxalates at 800°C under pure oxygen is metallic from -200° to +300°C and exhibits a lo~ (Xg= 4.2°X 10-6 emu) temperature-independent sus ceptibility between 4° and 300°K. Since conductivity is primarily via d electrons in transition-metal oxides these data strongly support the existence of a partiall~ filled 0'* band, or A.>Ac. Note that there is no spon- gR. R. Heikes, R. C. Miller, and R. Mazelsky Physica 30 1600 (1964). " taneous magnetization of the band states even though the 0'* band is orbitally degenerate and must be narrow. (2) LaCoOa. This perovskite was prepared as pre viously reported.Io Below "-'4000K, it is a semiconduc tor, exhibits a susceptibility maximum at 900K, and has a Jleff~3.85JLB corresponding to half of the cations in a low-spin state and g~2.4. A g> 2.0 is anticipated for Co3+. In the range 400° < T < l2100K the conductivity increases much more rapidly with temperature than in the low-temperature region, and above 12100K it seems to become metallic. The susceptibility and lattice parameter show anomalies in the range 400< T< 700°K.9 With DTA and high-temperature x rays we were able to identify a first-order rhombohedralp rhombohedral phase change at Tt= l2100K and to exclude any phase change between 900K and Tt, par ticularly in the range 400° < T < 700°K. TGA showed no change in oxygen content on passing through Tt• The rhombohedral unit cell contains two molecules, dis tinguishing alternate (111) planes of co bal t ions. These properties can be accounted for as follows: Below 4OO0K, trivalent cobalt orders into alternate (111) planes of low-spin COIll and high-spin Co3+, and for T<90oK the magnetic C03+ ions are coupled anti ferromagnetically via Co3+-"diamagnetic COIll com plex"-Co3+ superexchange. That such long-range super exchange can give TN"-'90oK has already been estab lished experimentally by Blasse.II It is assumed that A.(3+)~Ac-'Y and A.(III)~Ac+'Y, so that X2=A/3+)A.(III)~Ao2_'Y2 and the compound is a semiconductor with activation energy q=q('Y), where q-+O as 1'-+0. In the range 400< T<l2100K, the subarray of (111) planes containing COIll consists of a statistical distribution of COIll and Co3+. This causes a dS/dT>O, where S is the average spin per cation. It also causes a d'Y/dT<O, because the average cobalt radius on the CoIll-containing array increases with temperature to make A/Ill) decrease and A.<3+) increase toward a common value Ac. Since dq/dT<O, the conductivity increases very sharply with T in this temperature interval. For T> 12100K, all the cobalt are Co3+ and A~Ao. The details of this model will be reported elsewhere. It is concluded that the covalent mixing parameter Ac is a useful criterion for magnetic vs metallic states and for interpreting "complex" vs 0'* band formation. 10 A. Wold, B. Post, and E. Banks, J. Am. Chern. Soc. 79, 6365 (1957) . • • 11 G. Bl~~se, "Su: les composes oxygenes des elements de trans- 1tiO.n a. I etat solide," Colloque International Du C.N.R.S., Umverslty of Bordeaux (24--27 September 1964). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 91.182.86.89 On: Wed, 09 Apr 2014 13:08:13
1.1713657.pdf	Electron Tunneling through Asymmetric Films of Thermally Grown Al2O3 S. R. Pollack and C. E. Morris Citation: Journal of Applied Physics 35, 1503 (1964); doi: 10.1063/1.1713657 View online: http://dx.doi.org/10.1063/1.1713657 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Tunneling through an epitaxial oxide film: Al2O3 on NiAl(110) J. Vac. Sci. Technol. B 12, 2122 (1994); 10.1116/1.587721 Tunnel Current through a Thin Al2O3 Film with Nonuniform Thickness J. Appl. Phys. 42, 2981 (1971); 10.1063/1.1660658 HotElectron Transfer through ThinFilm Al–Al2O3 Triodes J. Appl. Phys. 37, 66 (1966); 10.1063/1.1707893 Photocurrents Through Thin Films of Al2O3 J. Appl. Phys. 36, 796 (1965); 10.1063/1.1714221 Tunneling Through Asymmetric Barriers J. Appl. Phys. 35, 3283 (1964); 10.1063/1.1713211 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:51JOURNAL OF APPLIED PHYSICS VOLUME 35, NUMBER 5 MAY 1964 Electron Tunneling through Asymmetric Films of Thermally Grown AhO 3 S. R. POLLACK AND C. E. MORRIS UNIVAC, Di!/ision Sperry Rand C()yporation, Bt1~e Bell, Pennsyll!ania (Received 2 December 1963) , The curren t through Al-AhOa-metal structures was studied in detail and a description based upon electron tunneling is presented. It is shown that: the trapezoidal energy barrier model of Simmons adequately accounts for the details of the current-voltage characteristic over 9 current decades; there is a built-in Voltage across a thermally grown oxide film of 0.92 V, in agreement with Mott's theory; the vacuum work function of the counterelectrode determines the barrier height at the oxide-counterelectrode interface; and the electron affinity of A1203 is 1.58 eV. INTRODUCTION THE quantum-mechanical process of tunneling has been suggested by many authors as the predom inant electron transfer mechanism through extremely thin insulating films. Theoretical analysesH of this phenomenon have yielded current-voltage (J-l/) char acteristics which relate the tunnel current to insulator film properties, such as electronic band structure and thickness, and to the properties of the counterelectrode. This paper presents the data obtained in a detailed study of electron tunneling through thermally grown films of aluminum oxide, and suggests a model which is consistent with the data and also accounts for the results described by previous workers. The techniques of measurement are also described in detail since, as we shall show, these details can influence the conclusions drawn from the data. The work is described in three parts. The first de scribes the experimental details including sample prepa ration and measurement techniques. The oxide model is then presented along with the data. This is followed by a discussion which relates some of the published electrochemical data on thermally grown aluminum oxide to the model described here. EXPERIMENTAL Sample Preparation All of the samples described in this paper are of the type AI-Al 20.-metaL They were prepared by first vacuum depositing 99.999% Al strips onto appropriately cleaned microscope slides. The system was maintained during the deposition at a pressure of approximately 8X 10-6 Torr, as measured at the substrate. The average Al deposition rate was 100 to 200 A/sec and the thickness of the Ai film was approximately 3000 A. The oxide film was grown on the Al by exposing it to room air at some temperature T, for a time t. T varied from 23°C to 350°C and t ranged from 1 to 1300 h. The deposition rate, thickness, and purity of the counterelectrode metal varied with the material used as shown in Table I. The substrate temperature was always reduced to room 1 R. Holm, ]. AppJ. Phys. 22, 569 (1951). 2 R. Stratton, J. Phy!;. Chem. Solids 23,1177 (1962). 3 J. G. Simmons, 1-App!. Phys. 34, 2581 (1963). temperature prior to the deposition of the counter electrodes. The active area of the samples was typically 10-2 cruz. A SiO film approximately 1000 A thick was used to cover the edge where the counterelectrode passed over the Al film on which the oxide was grown. This was done for two reasons. First, the edge contributed a significant portion of the current for samples containing an oxide TABLE I. COUllterelectrode parameters. Counter- Deposition electrode rate Thickness metal Purity {%) (A/sec) (!.) At 99.999 100-200 3000 Au 99.99 100-150 2000 Bi 99.9999 20-50 2000 Mg 99.97 20-50 2000 Ni 99.999 2-5 500 Pb 99,9999 100-150 4000 Cu 99.9999 100-150 4000 grown at 23°C. Figure 1 shows the current for two AI-A1 20,,-Cu samples, oxidized at 23°C for 20 min, which are identical except for the presence of the SiO film. The higher current in the sample without the SiO covered edge can be attributed to a linear J-V contribu tion from that edge. That is, the difference between the two curves is the J-V characteristic of a resistor which can be thought of as being in parallel with the u; ~ if! > IOr------r------r------r----~r-----~ 1.0 (HI-o NO SiO OVER EOG~ • WITH SiO OVER EOGE •• •• _ -0 • • .' • '/'" 00 o<f' .' . / • . /0 • • /R:4f1. /,<f O,OI~--~- _ _:":-4--_l:_-_--J'-- __ _! M~ ~ ~ w m ~ J (A/cm2) FIG. 1. Linear contribution to current due to edge emission through a 23"C oxide. 1503 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:511504 S. R. POLLACK ANI) C. E. MORRIS y INPUT ~ MOSELY L-----~---..,i\pUT o. Bridge Circuit for J-V Measurement UNBLANKING PULSE U b. Pulse Circuit tor J-V Meosurement MODEL 20 1-MEGOHM X -Y RECOROER FIG. 2. Circuits used for obtaining "fast" J -V data. sample and of magnitude 4 njmm of edge. The presence of edge emission was further demonstrated by preparing two sets of samples, one with constant active area but with varying edge length, and the other with varying area and constant edge length. In both cases the current at 1 V correlated with edge length rather than the active area of the sample. A set of samples was also made using Pb as a counterelectrode, and similar edge effects were noted. Although the edge current decreased rapidly as the oxidation temperature increased, it was felt that the SiO film would obviate the problem com pletely. The second reason for the SiO film was that it permitted data collection at higher voltages without edge breakdown. Measurement Technique It was realized quite early in the experiment that the application of a few volts across the oxide insulator resulted in electric fields in excess of the field required to cause Al ion migration through the insulator at room temperature. The effects of this ion motion on the J-V characteristic are described in some detail later. At this time we simply state that the sample changes when a voltage is applied across it. This has led to the classifi cation of data as being either "fast" or "slow" depend ing upon whether the data were taken in a time which was short or long, respectively, compared to the time during which a sample was observed to change when under the influence of an applied voltage. Where such changes are observed it is clear that an explanation of the J-V characteristic of a sample in terms of the details of an energy-barrier model is meaningful only for "fast" data. Furthermore it was found that fast data could not be taken over the entire voltage range at room temperature, because of limitations in our instrumentation. However, selected voltage ranges could be observed and these are described below. On the other hand it was possible to obtain fast data at 77°K so that, unless otherwise specified, only fast data at 77°K are reported here. Three separate test circuits were employed to take data on each sample. For sample currents less than 10-7 A at 77 oK, the change in current with time at a fixed voltage was always Jess than 2%. Therefore all current measurements below 10-7 A could be taken using a dc applied voltage. A Keithley Model 610A electrometer was used as a voltmeter and ammeter. For sample currents between 10-7 A and 10-5 A the data were taken by placing the sample across one arm of a balanced Wheatstone bridge as shown in Fig. 2(a). The bridge output voltage is directly proportional to the current through the sample. The duration of the triangular input voltage could be varied between 2 and 60 sec and its amplitude between 0.1 and 5.0 V. The J-V characteristics were plotted directly by an X -Y recorder. If the repetition of two triangular pulses of the same amplitude and duration produced the same trace, the data were classified as fast. This was accom plished at 77°K by limiting the pulse duration to less than 5 sec. Fast data in this current range were not taken at room temperature because the short pulse duration required was less than the response time of the recorder. When the sample current exceeded 10-5 A a pulse technique was used to obtain "fast" data. The pulse circuit is shown in Fig. 2 (b). The input pulse length was adjusted for each measurement so as to exceed the capacitive rise time of the sample, and the unblanking pulse was applied to the cathode-ray tube of an X -Y oscilloscope after the transient capacitive current spike. In this way one obtains a point on the X -Y oscilloscope with the X and Y coordinates indicating the voltage and current, respectively. In general the input pulse lengths VACUUM I I ABSORBED VACUUM X ELECTRON AFFINITY ~COUNTER ELECTRODE + ~ OXYGEN 10NS-+ + ~, SURFACE STATE """""..,..,4,..,.:;;p.:-=-=-~- -=:-=-.::::-.: -+ -L ---~ -= ",,"i'/77:h-; I -""'iii "/ FERMI ENERGY ENERGY ELEcLLRON At X n-TYPE TRANSITION REGION I BAND I GAP I I I I I s 1--( TUNNEL ) I THICKNESS I I BARRIER I OXIDE COUNTER ELECTRODE (METALI FIG. 3. Electronic energy-barrier diagram of AI-AbO. electrode structure. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:51ELECTRON TlTN:-.JELING THROllGH ASYMMETRIC FILMS OF AI203 1505 ranged from 200 /.Isec to 5 J.lsec as the curren t varied from 10-5 A to 10-1 A. It was necessary to correct for the series resistance of the films when working in the higher current range (e.g.> 10-2 A), and all data shown have been so corrected. This technique could also be used to obtain fast data even at room temperature. There was approximately one-half of a decade. ov~r lap in the current measurement ranges of the ClrcUlts so that there was no scaling of data. That is, the fit of the data obtained from one circuit to another was always checked and found to be smooth as long as the data were "fast." RESULTS A. Proposed Model The electronic energy barrier proposed here is shown schematically in Fig. 3. This model is similar to that suggested by others4•5 for this and related metal-oxide structures' however there are important differences in detail, and they will be pointed out. The oxide fil?: is comprised of two regions, a semiconducting tranSItIOn region and a barrier-type oxide region. The .thickn~ss of the barrier oxide region and the electromc barner heights, CPl and CP2, are the parameters determ~ed by tunneling. The details of the band structure m the transition region can not be determined from the data presented here; however, the nonvanishing barrier 0 ~s indicated.5 A sharp oxide-counterelectrode interface IS shown in Fig. 3. Handy6 has shown that the counter electrode metal does penetrate into the oxide for 2 to 5 A so that the interface shown is actually the interface at the penetration depth. Geppert1 has pointed out that a rectangular or a trapezoidal potential barrier is a good approximation in a thin insulating film if one also includes the image correction. We therefore replace the actual barrier of Fig. 3 by an idealized barrier shown by the solid curve Al TRANSITION REGION BARRIER OXIDE -1+ COUNTER 0-------1 r----- -OELECTRODE l30·' V FIG. 4. Idealized trapezoidal potential barrier, showing barrier shapes in a biased and unbiased condition. ---- 4 J. C. Fisher and 1. Giaever, J. App\. Phys. 32, 172 (1961). • G. T. Advani, J. G. Gottling, and M. S. Osman, Proc. IRE 50, 1530 (1962). 6 R. M. Handy, Phys. Rev. 126, 1968 (1962). 7 D. V. Geppert, J. App!. Phys. 34, 490 (1963). Al I I I I I I I I I I I I I I I I I \ I v TRANSITION REGION BARRIER OXIDE COUNTER ELECTp.CnE FIG. 5. Schematic representation of the effect of the transition region on the wavefunctio?: No~e smaller amplitude and ther~f?re smaller tunneling probabIlIty III the presence of the transItion region. in Fig. 4. If 0 is small ($10-1 eV) the transition regio~ behaves like an n-type semiconductor and the apph cation of an external electric field results in most of the field appearing across the barrier region. An applied voltage V therefore produces the barrier potential shown by the dashed curve in Fig. 4 .. When the Al electrode' on which the oxide is grown is biased nega tively or positively the observed current is labeled 11 or 12, respectively. The tunnel currents, II and 12 have been calculated by Simmons3 for a trapezoidal barrier, i.e. for the case 0=0 in Fig. 4. He includes a hyperbolic image correction which is a better approximation at higher voltages than the symmetric parabola used by Holm.! The analysis of the data presented here is based upon the calculation of Simmons. Disagreement with his calculation can be anticipated because O~O, and this disagreement can be described in the following way. A nonzero (but small) 0 obviously has a negligible effect upon his results for 12 for applied voltages V>o. The effect upon II, however, is not negligible. When the temperature T is the order of o/k, where k is Boltzmann's constant, the electron concentration in the conduction band of the oxide transition region is large and approximate agree ment with Simmons should result. However, as T decreases, a decrease in 11 results, and this decrease continues until the number of electrons in the Fermi tail with energy greater than 0 is extremely small, i.e., when T-:;,o/3k. At these temperatures the electrons are tunneling essentially from the metal rather than from the conduction band of the transition region, thereby resulting in a decrease in the tunneling probability, as described schematically in Fig. 5. The temperature dependence is noW considerably weaker than at higher temperatures since the supply function is no longer decreasing with temperature. The voltage de pendence of the current should not be affected by the temperature. B. Data Figures 6-9 show the I-V characteristics at 77°K for typical samples oxidized at different temperatures. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:511506 S. R. POLLACK AND C. E. MORRIS 4 ;;; ... SAMPLE NO. 127-4-2 65' OXIDE (10 DAYS) --JIo THEORETICAL {S'I7.51 4>,'1.6eV -J2. THEORETICAL K' 8 fl4> '0.geV <53 2! . ~ J,. EXPERIMENTAL {77'K ° J2. EXPERIMENTAL > 1~'-;-9;---10~-8;---IOL_7;---10~-6;---IOL-5'--':-;--IOL_'-3 -IOL-;;-2 -IOL_.-, -IO-';O~-'tO+.l FIG. 6. J-V characteristics taken at 77°K of a 65°C oxide with Al electrodes shown together with the theoretical curves calculated from Simmons' equations at T=ooK. The general shape of the curves was stable with time and in most cases the data could be reproduced many days later. The structure in J2 at approximately 1.6 V for all the samples is a result of the passage of the <PI barrier below· the Fermi level at an applied voltage V=<pI/e. This initiates the start of the Fowler-Nord heim8 region which is characterized by a stronger dependence of J upon V. A similar break in JI is also observed at a voltage of approximately 2.5 V. In like fashion this occurs when the <P2 barrier passes below the > 6r--r--.---,---.---r---.--,---,---.---. 4 2 SAMPLE NO. 96-5-11 180' OXIDE (23 DAYS) --J,. THEORETICAL {S' 251 4>, = I.hV -J2. THEORETICAL K'8 fl4>'0.9IV 4 J,. EXPERIMENTAL{n' K o J2. EXPERIMENTAL ~4 1~'-;-9;---10~-8;--!l.10'-,-7~-10~-6:---IO~-5;---IOL_4;---IOL-3'--10~-2;---IOL-':---IO-';;O--'tO+' . J (A/em2) FIG. 7. J-V characteristics taken at 77 oK of a lS0°C oxide with Al electrodes shown together with the theoretical curves calcu lated from Simmons' equations at T=ooK. Fermi level. In Figs. 6--9 theoretical curves for J I and h are shown together with the barrier parameters. These curves were obtained by computer calculations based on Simmons' equations using a dielectric constant of 8. In order to obtain the fit shown here, the effective active area of the sample was adjusted to values between 10% and 0.1% of the geometrical area. This value of the effective area was consistent with the results of pulsed breakdown measurements which indicated that although the breakdown was uniformly distributed over the entire active area, only 0.1% to 10% of the 8 R. H. Fowler and L. Nordheim, Proc. Roy. Soc. (London) A119, 173 (1928). area actually broke down. In all cases J2 is described completely by the tunnel theory whereas the experi mental values of J I are less than the theoretical values. This is discussed more completely in Sec. C below. Data from 70 samples oxidized at various tempera tures were analyzed using Simmons' equations. It was found that <PI = 1.58 eV ±0.03 eV, <P2= 2.50 eV ±0.06 eVand (<p2-<PI)=0.92 eV ±0.07 eV for samples oxi dized in the temperature range 400°> T~ 65°C. Over this range of oxidation temperature there were no con- 6 5 SAMPLE NO. 91-5-3 260' OXIDE (55 DAYS] --J" THEORETICAL {S' 301 4>, '1.6 IV -J2, tHEORETICAL K ' B fl4>' 0.9 IV ~ J •• EXPERIMENTAL{n' K J2, EXPERIMENTAL a J" EXPERIMENTAL{300' K Jz. EXPERIMENTAL ~ 4 O~~~~~~~~~-L~~~~ __ ~ __ ~~ 10-9 to-8 10-7 10-6 10-5 10-4 10-3 to-2 10-' 100 tO+1 , J (A/em2) FIG. 8. J-V characteristics taken at 77°K of a 260°C oxide with Al electrodes shown together with the theoretical curves calcu lated from Simmons' equations at T=ooK. sistent differences in the values for the barrier heights. These barrier height parameters differ from the values quoted by Meyerhoffer and Ochs9; however, the average barrier is in good agreement. This is due to an error in their theoretical analysis for asymmetrical barriers. A comparison of the J-V curves in Figs. 8 and 9 indi cates that the 4-h, 360°C oxide appears to have tunnel ing properties similar to the 55-day, 260°C oxide. Oxidation for still longer times at 360°C resulted in an mcrease in current density for oxidation times up to 4 SAMPLE NO. BHI-5 360' OXIDE (4 DAYS) --J •• THEORETICAL { S' 30 1 4>, 'l.hV -J2. THEORETICAL K ' B flt/> '0.9 IV 4 J,. EXPERIMENTAL{77' K o Jz. EXPERIMENTAL O~~~~~~~~_~~~~~-L~~~ 10-1 to-8 10-7 10-6 10-5 10-4 10-3 10-2 10-· 100 IO+! J (A/em2) FIG. 9. J-V characteristics taken at 77°K of a 360°C oxide with AI electrodes shown together with the theoretical curves calcu lated from Simmons' equations at T=ooK. 9 D. Meyerhoffer and S. A. Ochs, J. App!. Phys. 34, 2535 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:51E LEe T R 0:'\ T l' !\ :\ ELI :'\ G T H R 0 {T G II AS Y 1\1 1\1 E T RIC F 1 L 1\1 S 0 F .\ 12 03 150i (a) (b) hr;. 10. Electron-ditIraction pattern of (a) 360°C oxide, 10 days (b) 3600C oxide, 55 days. "d" values ot 2.78, 2.375, 1.97, 1.50, and 1.392 X corresponding to the 220, 311, ·100, 511, and 440 reflections, respectively, indicate ,-alumina. Also evident is the greater crystal line size in the 55-day sample. 10 days followed by a decrease for longer oxidation times. It was felt that this unusual result may have been due to the presence of a crystalline phase which occurs in AlzOo, and subsequent electron diffraction studies revealed the presence of '}'-alumina for samples oxidized for over 3 days (see Fig. 10). Although the value for the barrier heights do not appear to be altered by the phase transition, anomalous tunneling behavior does result. The thicknesses of the samples were also determined from the theoretical curves and are shown in Fig. 11. The thickness indicated by capacitance measurements assuming a dielectric constant of 8, was larger than the tunneling measured value by a factor of 1.5 to 2.0. This is typical of what others have observed and is related to both the anomalous capacitance effectlO and to the fact that a tunneling measurement favors the smallest thickness whereas the capacitance indicates approxi mately the average thickness. For example the capaci tance of the sample oxidized at 260°C in Fig. 11 was 0.016,uF, which, for k= 8 yields a thickness of 45 A, in agreement with the corrected values of Hunter and Fowle!l (see Discussion). On the other hand the thick ness determined by tunneling is 30 A; however, the III C. .-\. ]'dead, PhI'S. Rev. Letters 6,545 (1961). 11 l\I. S. Hunter and P. fowle, ]. Electrochem. Soc. 103, 482 (1956). effective area of the sample was only 10% of the actual area. If one assumes that the remaining 90% of the sample is 45 A, then the capacitance measurement is altered by only 5% by the small tunnel regions and the larger thickness is indicated. The J-V curves of samples oxidized at room tempera ture were not as consistent as the curves of the higher temperature samples. Figure 12 shows several different J-V curves for samples oxidized at 23°C. It is apparent from these curves that the oxide film formed at room ;;; 70,---,---,---,---,---,---,---,---,---,--, 60 50 HUNTER AND FOWLE (CORRECTED) ;;; 40 '" '" z ~ 30 :J: >-TUNNELING THICKNESS °0~~5~0--~iO~O--~i5~0--~20~O--~25~0--~30~0--~35~O--4~0~0--7.45~0--~500 OXIDATION TEMPERATURE (OC) FIG. 11. Comparison of thicknesses as measured by tunneling and anodizing methods. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:511508 S. R. POLLACK AND C. E. MORRIS FIG. 12. J-V characteristics taken at 77°K of several 23°C oxide samples. temperature does not reach an equilibrium state in air for many hours. This is evidenced by the decreasing current with age and by the absence of distinct barrier formation. Samples oxidized for up to 36 days were somewhat more stable and showed barrier heights of approximately 1.8 eV for CP1 and 2.7 eV for 4>2 with a thickness of 17 A. These values for 4>1 and 4>2 are con sistently larger than those obtained from samples oxi dized at temperatures greater than 23°C. However, since long term stability in the 1-V curve was not ob served in these samples these values of 4>1 and 4>2 were not used to determine the barrier heights quoted above. C. Temperature Effects The effect of temperature upon II and J2 was studied. It was mentioned previously that fast data at or near room temperature could not be obtained over all current ranges. It is possible, however, to obtain large current high-voltage data using pulse techniques, and the typical effects of temperature can be seen in Fig. 8 for an all Al sample oxidized at 260°C. Note that the change in h between 300° and 77°K is small whereas II increases and actually becomes larger than 12 when the 2.5 2.0 1.5 1.0 0.5 VOLTS J (x to-1 AMP) 56 o 48 40 32 24 16 16 24 S2 40 48 56 0.5 AI-AI201-AI 260'C OXIDE liD DAY OXIDATION) 2.0 2.5 Fro. 13. Effect of temperature upon hand h at low voltages. 4.5 2.5 Al-AIZOS-Au 26O"C OXIDE (55 DAY OXIDATION TIME) • JI {3000K .0 Jz .. JI {nOK .. J2 2.0,-::-~--.....I...:;-----'-;-----'-;;-------' 10-5 10-4 10-3 10-2 10-1 J (A/cm2) FIG. 14. Effect of temperature upon JI and .T2 at high voltages for a sample with a gold counterelectrode. applied voltage V> 2.9 V. The crossover voltage is in excellent agreement with the value predicted by Sim mons for structures with CPl=1.6 eV and (cp2-4>I)=0.9 e V as can be seen from the theoretical curves in Fig. 8. This indicates that when the conduction band popula tion is large, as it is, for example, at room temperature for 0<0.1 eV, the model shown in Fig. 4 can be treated as trapezoidal, in which case tunneling theory accurately describes the I-V characteristic. For voltages greater than the crossover voltage, or for current densities greater than 1.0 A/cm2, the voltage dependence of the current density becomes weaker than the theoretical values shown in Fig. 8 above 3 V. This may be due to space-charge effects in the presence of high trap densities.12 The temperature dependence at lower voltages can be seen in Fig. 13. The I-V traces were taken with a 2-sec-duration pulse, so that the room temperature curve, strictly speaking, is not "fast." The I-V curve produced by a second pulse however did not differ from the first curve by more than 10%. The near constant values of 12 and the decreased values of II with decrease in temperature were typical and agree with the qualita tive discussion of the effects of 0~0 on the tunnel current. The results of Advani et al.5 indicate an ex ponential temperature dependence for II, with a ther mal activation energy of the order of 10-1 V and is probably equal to 0, defined here. When metals other than Al are used as counter electrodes the same temperature effects are obtained, i.e., J2 is relatively temperature independent whereas 11 decreases with decreasing temperature. This can be seen by the pulse data in Fig. 14 for an AI-AbOrAu sample. A reverse in recitification at 4.1 V occurs for this sample at room temperature and agrees with Simmon's equations with 4>1=1.9 eV and (4)2-4>1)= 2 eV. These are the barrier parameters obtained for an Al-AI 20a-Au structure. 12 D. V. Geppert, J. App!. Phys. 33, 2993 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:51E LEe T RON TUN N ELI N G T H R 0 UGH AS Y M MET RIC F I L M S 0 F A 12 03 1509 D. Electrode Effects The effect of different metals as counter electrodes was also studied. Figure 15 shows typical I-V curves for several samples oxidized for 36 days at 180°C, and with the indicated metals deposited on the same alu minum oxide film. All of the metals studied are not shown for sake of clarity. The values of CPI and CP2 were obtained from these curves and the barrier asymmetry (CP2-CPl) minus the built-in barrier asymmetry (~CPAI) in the all aluminum sample is plotted against the work function of the counter electrode material in Fig. 16. The theoretical curve is the curve one would obtain if CP2 were directly proportional to the work function of the counterelectrode. Although the effect of the work func tion is slightly larger than expected, the data in Fig. 16 are consistent with the model presented here. This indi cates that the barrier portion of the oxide is at the oxide-air interface since a conducting region between the oxide and the counterelectrode would not vield the complete work-function effect of Fig. 16. • For the case of Mg the curves are more nearly sym metric although 12 is still greater than II, indicating that CPI <CP2. This is as expected considering that the difference in work function between Al and Mg is not sufficient to overcome the built-in voltage of 0.92 V. These results are similar to those obtained by Miles and Smithl3 on plasma anodized samples of AhOa, however, the process of plasma anodization appears to produce a symmetric oxide. The atom-size effect described by Handy6 was also observed for these samples. For example, Fig. 15 shows that the low-voltage resistivity with a Ni counter electrode was less than with AI, whereas with Bi it was greater. However, the extent to which the atom size determined the resistivity was not as great on the thicker oxide films (oxidation temperature 180°C as compared to Handy's samples at 23°C) for electrode materials with atom size less than that of AI. 4 SAMPLE No. 96-5 180· C OXIDE 123 DAYS) J, J2 E~~~m~E Mg A\ FIG. 15. Dependence of tunnel currents on the counter electrode material. 13 J. L. Miles and P. H. Smith, Presented at Spring Meeting Electrochemical Society, Pittsburgh, pennsylvania, April 1963 Extended Abstract No. 83, Electric Insulation Division. ' > ~6 '" g 3 a: .... u '" L;j 2 a: Ni ~ OXIDATION TEMPERATURE OF ALL SAMPLES AT 1800 C ~I o u ~0~.8'-~-0~.6,--job4~-~0".2--~0--~0~.2~~0~4--~0~.6--~078--~1.0~~1.2 t.'CE -t.'AL leV) . FIG. 16. Obse~ved built in asymmetry (~<I>CE) minus the built III asymmetry With an Al counterelectrode (~<I>At> vs the vacuum work function of the counterelectrode. E. Voltage-Induced Transients The application of a constant de voltage across a sample at room temperature resulted in a current which changed mona tonically for a time tv until it reached some final value. This change can be described as fol lows. A sample is first maintained in a zero bias condi tion at toom temperature for at least 20 min. Biasing t?e A~ electrode, on which the oxide was grown, nega tIve (I.e. It), the curve 1 of Fig. 17 is taken with an X -Y recorder using a triangular pulse 3 sec in duration. At 1.6 V, however, the voltage is held constant and the current grows until it reaches curve 2 after which time the change in current is negligible. The voltage is then reduced in 3 sec and 2 is produced. If the process is repeated after a wait of at least 20 min this time . ' reversmg the polarity, the curves 3 and 4 are obtained for 12 where, however, the current is observed to decrease at constant voltage. This effect was first reported by Fisher and Giaver.4 Upon a further exami- 1.8 1.6 1.4 1.2 1.0 V (VOLTS) CURRENT (ARB. UNITS) 35 0 0.8 0.6 0.4 0.2 30 25 20 15 10 5 0.2 0.4 to 15 20 25 30 35 1.6 1.8 J2 FIG. 17. Change in tunnel current with time at constant voltage. For discussion see text. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:511510 S. R. POLLACK AN J) C. E. MORRIS _ --~--- 2 -----::--..::::---:-- 3 - ---.....", INCREASE IN (]" WITH J1 BIAS LEGEND: (o,b) o. .. NO BIAS CONDITION 2. INSTANTANEOUS CONDITION 3. FINAL BARRIER PROFILE o BIAS I I I DECREASE IN (]" +J2 BIAS : WITH J2 BIAS ________ J b. I I I I I I o BIAS : +J, BIAS L ________ _ :FIG. 18. Change in barrier parameters produced by the variation in spatial distribution of ions in an applied field. nation, the following characteristics of this effect have been noted. If a fast measurement of 11 is taken im mediately after curve 4, one obtains 5, and if 12 is measured immediately after 2 one obtains 6. This indi cates that the transient change in the sample increases or decreases the tunnel current in one polarity and also increases or decreases, respectively, the tunnel current in the other polarity. If the terminal voltage (in this case 1.6 V) is not so high as to cause permanent damage to the sample the cycle can be repeated with, for ex ample, the current on curve 5, growing back to curve 2 while at. a constant. volt.age of 1.6 V. After approxi mately 20 min at 300° K and zero volts all "memory" of previous cycling is gone. The time tv was generally 3 to 5 min at 300oK. At lower temperatures tv increased and the magnitude of t.he current change (i.e., f11 in Fig. 17) at constant voltage decreased. At 77°K the samples would generally break down under dc bias before a large enough voltage to observe these effects could be reached. It is also characteristic of this effect that a given state of the sample can be quenched-in at 77 oK. For example, if one measures 11 along curve 1 of Fig. 17, allows the current to increase to curve 2, ~and then quenches the sample at 77°K, subsequent measure ments of 11 and 12 yield curves indicating a larger current at a given voltage than would ordinarily be obtained in a sample quenched from a state in which no voltage-induced transient had taken place. Likewise a smaller current is obtained if the sample is quenched after the transition from curve 3 to 4 at constant volt age. These larger or smaller currents can be observed for an indefinite period of time at low temperatures. Similar transient effects are also obtained on samples with counterelectrodes of Pb, Cu, and Au. These transient effects are probably related to the ionic transient effects observed14-16 on amorphous oxide films except that instead of observing ionic conductance directly, it is the effect of the ionic motion on the barrier parameters and therefore on the tunnel current that is observed. The following three considerations demonstrate that the transient currents are not ionic. The first is that the increased current state quenched in at low temperatures persists for both polarities; the second is that the direct contribution of an ionic current should cause the current to increase in both polarities, contrary to what is observed; and the third is that a sample was maintained with a voltage across it for 34 days such that 100 times the weight of the counter electrode would have crossed the oxide if only 10% of the current had been ionic. No noticeable change in the counterelectrode was observed. Therefore these tran sients are a secondary effect probably produced by the variations in the density and spatial distribution of ions in the applied field. For example, if the positive "sur face" charge density at the barrier-semiconducting transition layer interface adjusts in t.he presence of an applied field so as to decrease the total electric field in the oxide, then a change in 4>1 results as shown in Fig. 18. Such a change qualitatively accounts for the transients observed if the creation of the positive sur face charge can be characterized by a thermal activa tion energy. A quantitative description of this phe nomenon may provide an interesting method for study ing ionic conduction parameters in very thin oxide films. As a result of the voltage-induced transients, dc or slow data of 11 and 12 appear to be such that 11>h up to approximately 1.5 V. When the data are taken at 77 oK where these transients do not take place, the data indicate that 11< 12 and an apparent reversal in the direction of recitification with temperature occurs. If, however, fast data are taken at room temperature, then 11 <h as it is at 77°K and no reversal in rectification takes place. This can be seen in Fig. 17 where the fast curves, 1 and 3, for 11 and 12, respectively, indicate 11 <h, whereas the slow data of curves 2 and 4 indicate 11>12. We believe that this is the cause of the dc measured reverse in rectification in thermally oxidized aluminum samples reported in the literature. Care must also be taken to avoid quenching the sample at 77°K immediately after taking slow data at room temperature since the transient state persists at the low temperature. F. Aging Effect on 1-V Characteristics We have observed the effects of aging upon the tunnel characteristics and have obtained results similar to Handy for samples oxidized up to a few hours. If, however, samples are oxidized for extended periods of 14 c. P. Bean, J. C. Fisher, and D. A. Vermilyea, Phys. Rev. 101, 551 (1956). 15 J. F. Dewald, J. Phys. Chern. Solids 2, 55 (1957). 16 D. A. Vermilyea, J. Erectrochem. Soc. 104,427 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:51ELECTRON TFNNELING THROFGH ASYMMETRIC FILMS OF A120, 1511 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10° 10+1 10+2 J (A/em2) FIG. 19. Effect of age on the J-V characteristics of Al-AbOa-Al and AI-AbO a-Au structures. time at temperatures above 3()()OK, the change in the tunnel current with age is greatly reduced. Figure 19 shows J-V data taken 35 days apart for a sample oxidized at 260°C for 55 days. The J-V characteristic of the sample with an Al counterelectrode is stable above 0.5 V whereas with Au, the characteristic changes with age. This change comes about by an increase in 1/>1 and a small decrease in 1/>2 resul ting in a decreased value of the built-in field. Such a change is reasonable since the oxide was in equilibrium in a field of 0.92 V/26 A=3.6X106 V/cm prior to depositing the counter electrode, whereas after depositing the Au electrode, the field is increased to 1.95 V/26A=7.50X106 V/cm. Since the anodization field for Al is approximately 7.2X106 V/cm, it is not suprising to find that doubling the field in that range results in a nonequilibrium oxide in which ionic rearrangement takes place. This explana tion also explains why Handy always observed that samples with Cu, Ni, Ag, Au, and Sn counterelectrodes have an increasing tunnel resistivity since all of these materials have work functions greater than AI. Although this effect may not be the only one operative in produc ing long-term changes in tunnel characteristics it appears to playa significant role when the oxide has otherwise obtained equilibrium. DISCUSSION The salient features of the model presented here are the n-type transition region between the parent alu minum and the insulating oxide, and the built-in asym metrical electronic energy barrier that exists even for an all aluminum sample. A transition region, as indi cated in Fig. 3, has been proposed and, as suggested by Geppert,1 such transition regions can have a marked effect on the J-V tunneling characteristic. The thick ness of this region and its identification as the amorphous film described by Eley and Wilkinson17 can not be 17 D. D. Eley and P. R. Wilkinson, Proc. Roy. Soc. (London) A254, 327 (1959). determined from the results of this work. A diffuse interface region is indicated however by the tempera ture dependence of J1 and by the low value of 1/>1 (1.58 eV). On the other hand, the assumption of a sharp oxide counterelectrode interface yields excellent agreement between theory and experiment for J2, even though a penetration of the counterelectrode into the oxide does occur. The built-in field in the oxide inhibits cation diffusion from the counterelectrode so only atomic diffusion takes place, and, as Handy observed, this is influenced most strongly by the relative size of the diffusing atom and the interstice in AbOa. As a result the penetration, which extends for only 2 to 5 A, can be looked at simply as a decrease in oxide thickness rather than as a "thick" transition region. This is also indicated by the correlation of the work function of the counterelectrode with the built-in asymmetry (Fig. 16). The agreement between experiment and theory as indicated by the accurate fit of the J-V characteristic over 7 to 9 current decades, and by the occurrence of the Jr-J2 crossover at the predicted voltage, demon strates the validity of the rectangular or, more appro priately, the trapezoidal energy-barrier model in ex tremely thin films. The accurate determination of the barrier heights ~1-1/>2 for the all aluminum system indicates that the built-in voltage predicted by MoWS in his theory of formation of protective oxide films does exist and is equal to 0.92 V.19 Since the 1/>2 barrier height is proportional to the counterelectrode work function, one can obtain the electron affinity X for the oxide. X will be given by the differences between the counterelectrode vacuum work function and the 1/>2 barrier height. Using the preferred value for I/>Al of 4.08 eV and 2.50 eV for 1/>2 we obtain x= 1.58 eV. Heil20 has determined X by a different method and obtains x=2 eV. It was suggested by Keller and Edwards21 that the thermally grown oxide film on Al consists of a compact barrier-type region plus a conducting region the thick ness of which is determined primarily by the presence of water vapor. Hunter and Fowlell have measured the barrier thickness and found that it is determined by the temperature of oxidation and that the net effect of water vapor on the barrier oxide is to increase the time it takes to reach its maximum thickness. Their results for barrier thickness are shown in Fig. 11, together with the values determined by tunneling. The curve marked H-F (corrected) is their thickness curve, corrected for the built-in voltage of 0.92 described in this paper. If one assumes that this voltage is unaltered by the liquid 18 N. F. Mott, Trans. Faraday Soc. 43, 429 (1947); also see N. Cabrera and N. F. Mott, Rept. Progr. Phys. 12, 163 (1948- 1949). 19 For a further discussion of the built-in voltage see S. R. Pollack and C. E. Morris, J. Electrochem. Soc. (to be published) and Solid State Comm. 2 (Jan. 1964). ' 20 H. Hei!, Bull. Am. Phys. Soc. 7, 327 (1962). 2\ F. Keller and J. D. Edwards, Metals Progr. 54, 198 (1948). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:511512 S. R. POLLACK AND C. E. MORRIS anodizing bath then it must be added to the applied anodic voltage when calculating the thickness by the method of Hunter and Fowle. The tunnel thickness is considerably less than the corrected values for the bar rier thickness. Although a smaller thickness is to be expected from such measurements, an explanation for the large discrepancy is still lacking. Hunter and Fowle also indicate a very rapid rate of growth of the barrier film until it reaches its terminal thickness. The time inferred from their data is approxi mately 20 min in dry O2 or air. We have also observed a very rapid barrier oxide growth, however the details of the energy barrier require considerably longer times, of the order of days, to reach equilibrium. Since these details are influenced by the final distribution of Al ions throughout the oxide it is not surprising to find long-term changes taking place. A thickness measure- JOURNAL OF APPLIED PHYSICS ment alone can not detect such changes whereas the tunneling probability and therefore the tunnel current, is very sensitive to the details of the energy barrier. The model suggested here can also be used to inter pret the photoresponse data of Lucovsky et al.22 on similar metal-oxide-metal structures. ACKNOWLEDGMENTS We would like to thank N. Goldberg and H. Callen for many helpful discussions. We also thank J. Simmons for making a copy of his work available to us prior to publication. The electron diffraction studies by J. Comer and K. Caroll were invaluable. The assistance of P. Kornreich is also gratefully acknowledged. 22 G. Lucovsky, C. J. Repper, and M. E. Lasser, Bull. Am. Phys. Soc. 7,399 (1962); and J. Appl, Phys. (to be published). VOLUME 35. NUMBER 5 MAY 1964 Electrical Effects due to the F Center in the Potassium Halides J. N. MAYcoCK* RIAS, Martin Company, Baltimore, Maryland (Received 30 September 1963; in final form 6 January 1964) The electrical conductivity of the additively colored potassium halides has been investigated as a function of color density and temperature. These studies show that the crystals have a conductivity lower than that of the pure crystals for temperatures outside the intrinsic range. The experimental thermal dissociation energy of an F center is compared with the theoretical values. INTRODUCTION WHEN single crystals of the alkali halides are heated in an alkali metal vapor atmosphere it is possible to produce crystals containing only F centers. This color center being electrically neutral will not directly affect the electrical properties of the host material. However, as the balance between anion and cation lattice vacancies has been disturbed it should be possible to observe some difference in the electrical properties between the pure host crystal and the colored crystal. Several authorsl-6 have examined the conductivity changes of irradiated alkali halide single crystals. U~- * This work was supported by the U. S. Air Force Office of Scientific Research of the Office of Aerospace Research, under Contract No. AF49(638)-1017. 1 R. Smoluchowski, Report of the Bristol Conference on Defects in Crystalline Solids, 1954 (Physical Society, London, 1955), p.252. 2 F. Seitz, Rev. Mod. Phys. 26, 7 (1954). a K. Kobayashi, Phys. Rev. 102, 348 (1956). 4 H. S. Ingham and R. Smoluchowski, Phys. Rev. 117, 1207 (1960). 6 R. W. Christy and E. Fukushima, Phys. Rev. 118, 1222 (1960). 6 P. Berge and G. Blanc, Bull. Soc. Franc. Mineral. Crist. 83, 257 (1960). fortunately this method of coloring produces the F cen ter plus numerous other centers of both electron and hole origin. It is therefore desirable to produce crystals containing only the F center. This can be done by either additive coloration or electrolytic coloration. Crystals additively colored were first quantitatively investigated by PohF who investigated the migration of F centers by applying a dc field to an additively colored crystal at various temperatures. By this technique he was able to devise a value for the mobility of an F center. Re cently Jain and Sootha,8 and Krasnopevtsev9 have renewed interest in the effect of F centers on the elec trical conductivity of potassium chloride and bromide single crystals. This work indicated that the additively colored crystals behaved qualitatively the same as the irradiated material with respect to their electrical prop erties. Unfortunately this work did nothing to clarify the processes taking place within the crystal, so the present work was initiated to determine the conduction processes taking place when an electric field is applied 7 R. W. Pohl, Proc. Phys. Soc. (London) 49 (extra part), 3 (1937). 8 S. C. Jain and G. D. Sootha, Nature 193, 566 (1962). 9 V. V. Krasnopevtsev, Fiz. Tverd. Tela 4, 1807 (1962) [English transl.: Soviet Phys.-Solid State 4,1327 (1963)]. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.127.238.233 On: Fri, 21 Nov 2014 18:12:51
1.1736064.pdf	Operation of TunnelEmission Devices C. A. Mead Citation: J. Appl. Phys. 32, 646 (1961); doi: 10.1063/1.1736064 View online: http://dx.doi.org/10.1063/1.1736064 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v32/i4 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 03 Aug 2013 to 128.171.57.189. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 32, NUMBER 4 APRIL, 1961 Operation of Tunnel-Emission Devices C. A. MEAD California Institute of Technology, Pasadena, California (Received February 15, 1960) The operation of a new class of devices employing the principle of tunnel emission is discussed. It is shown that a controlled electron source may be obtained with the use of a metal-insulator-metal diode structure where the second metal layer is very thin. A triode geometry may be secured by the addition of an additional insulator and a metal collector layer. Limitations on the operating frequency, current density, and current transfer ratio of such devices are discussed. Experimental results on diode and triode are dis cussed. Experimental results on diode and triode structures which employ several materials are presented. Successful triodes and vacuum emitters have been realized with the use of AhO, insulating films. Experi ments using Ta20s are described, and the results are discussed. TUNNEL EMISSION TUNNEL emission is the phenomenon occurring at a metal-insulator interface when a high electric field is present within the insulator. 1 The phenomenon is most easily studied with reference to a diode structure consisting of two metal plates separated by a thin in sulating layer, a potential being applied between the two metal plates. When the field is increased to a suffi ciently high value, electrons in the metal impinging upon the interface may "tunnel" through the insulator forbidden region into the conduction band. The mecha nism by which this tunneling occurs is shown sche matically in Fig. 1. The wave function for a stream of electrons near the Fermi level in the metal traveling to the right is a sine wave as shown. The insulator for bidden region does not permit propagating wave solu tions. The problem is very much like that of an electro magnetic wave in a waveguide beyond cutoff, yielding exponentially damped solutions. In the conduction band, propagating solutions are again possible and the wavelength decreases as energy is gained from the electric field. Upon entering the left-hand metal, the FIG. 1. Energy band structur of metal-insulator-metal diode with applied electric field showing wave function of tunneling electron (schematic). 1 C. A. Mead, Proc. Inst. Radio Engrs. 48, 359, 1478 (1960). wavelength abruptly decreases still farther because of the metal-insulator work function. An excellent survey of the theoretical work done on this problem and a complete list of references has been given by Chynoweth.2 In general, a solution to the problem gives a current-voltage characteristic of the form 1 (E 2 -= --) exp(-Eo/E), 10 Eo (1) where 1 and E are the current density and electric field, respectively. In the expression given by Chynoweth, and 4¢!(2m)! Eo ""'"---- 3hq 10= 2qcf>2m /9h7r2, where cf> is the metal-insulator work function, m* is the effective mass of the electron, and q is the charge of the electron. The conditions given for the validity of these expressions are that the electron image force be not too strong and that the energy gap of the insulator be large compared with the metal-insulator work func tion. Also unstated is the condition that the applied voltage be greater than the work function, another way of stating that the electrons are tunneling into the con duction band of the insulator and not directly into the second metal. Typical values for cf> are of the order of 1 ev, making Eo nearly 108 v/cm and 10 of the order of 1010 amp/cm2• A plot of the v-amp characteristic is shown in Fig. 2. It can be seen that the current density increases ex tremely rapidly with increasing electric field. In most cases the electric field required for significant current density is many times that required for avalanche break down in the bulk insulator. Such breakdown, however, requires a large number of electronic mean free paths and in the present case is averted by making the in sulating layer very thin, i.e., less than one mean free path. The energy distribution of the tunneling is plotted against the electron wave number in Fig. 3. It can be 2 A. G. Chynoweth, Progr. in Semiconductors 4, 97 (1959). 646 Downloaded 03 Aug 2013 to 128.171.57.189. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsOPERATION OF TUNNEL-EMISSION DEVICES 647 seen that the electrons are concentrated very close to the Fermi level (k/kj= 1). TUNNEL-EMISSION AMPLIFIER A very significant feature of the tunnel-emission process is that it constitutes a controlled source of majority carriers. Suppose we make the right-hand metal layer thin compared with an electronic mean free path in the metal. A typical electron tunneling from the left-hand metal will now pass through the thin metal layer and out through the surface. Such a device may in principle be operated at very high current densities and may well constitute the most practical high current density "cathode" for many conventional and micro wave tube applications. In order for the electrons to J I(T' 1(:r7~_--'-r __ ---r---.--....,-- ..... ---' o .2 .3 .4 E FIG. 2. Theoretical v-amp characteristic of diode as given by Eq. (1). .6 appear in the vacuum, the electron energy (correspond ing to the voltage applied between the metal layers) must be greater than the right-hand metal-vacuum work function. A triode structure may also be con structed by adding another insulating layer to the right of the thin metal region and then a third metal layer, the purpose of which is to collect electrons emitted from the surface of the thin metal layer. The energy band representation of such a structure is shown in Fig. 4. The device thus formed is similar to a transistor, and the same terminology is applied to the metal layers. Three major areas which should be investigated with respect to this device concern (a) frequency limitations, (b) current density and area limitations, and (c) current Relative Tunnel Current 0r-__ ~~.2~5·3===At=:.:5:::.6~~1~~.8~~.9 __ ~I.O~ 1.00 .98 k/k, .96 .94 .92 .90 FIG. 3. Momentum distribution of tunnel electrons (normalized to Fermi momentum). transfer ratio limitations. These areas will now be con sidered in detail. Frequency Limitations Since the actual emitter-base tunneling takes place in an extremely short time, we should expect the major limitations on the gain bandwidth to be input capaci tance and base-collector transit time. The capacitance limitation is very similar to that of an ordinary vacuum tube. A high-frequency figure of merit M may be defined as follows: M=1/RC, where R is the incremental common base input resist ance and C is the emitter-base capacitance. This figure of merit is independent of the area of the device unless the current density is not uniform, a condition which will be discussed shortly. From Eq. (1) we may evaluate the incremental input resistance, assuming E«Eo: R=dE/AJE o, where A is the area of the device and d is the emitter base insulator thickness. The capacitance is that of a plane-parallel capacitor: C=eA/d. The figure of merit may thus be written as FIG. 4. Energy band representation of tunnel emission triode (sche matic). M=JEo/eE, (2) Downloaded 03 Aug 2013 to 128.171.57.189. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions648 C. A. MEAD and may be uniquely evaluated in terms of the normal ized current density. It can be seen that the figure of merit is very nearly proportional to the current density, and the desirability of operating at relatively high cur rent densities is quite obvious. Base-collector transit time may become a problem if the collector insulating region is made too thick. Two cases will be considered: (1) Collector insulating layer thick compared with electronic mean free path. In this case we may define an electron mobility p, in the insulator. The transit time t may then be expressed in terms of the collector base voltage Vcb: (3) It should be pointed out that as the collector is made thicker the collector base capacitance is reduced, thus making possible higher gains at frequencies approaching the figure of merit. However, the transit time rapidly becomes important as d is increased. (2) Collector insulating layer thin compared with electronic mean free path. In this case the transit time is determined only by the electronic velocity and in sulator thickness, and for all reasonable thicknesses will be extremely short. Current Density and Area Limitations Equation (2) shows clearly the desirability of opera tion at the highest possible current density (or total current for a given area if the distribution of current is nonuniform). One limitation on the current density is that of space charge in the base-collector insulator. The space-charge limited value of current density is given by E V! J=2.33X10-6_.-ampjcm2, (4) EO d2 provided the film is thin compared with a mean free path. For film thicknesses of the order of the mean free path, the value will be somewhat smaller than indicated by this expression. In general, it is necessary to make the collector insulator region thin enough to prevent space-charge limitations at the highest current density to be encountered. Another rather serious limitation of the effective cur rent density is that of the self-bias effect. Since it is not possible to make the current transfer ratio of the device exactly unity, some current will be required to flow I Emitt., Insulator Bose Insulator Collector 2b I FIG. 5. Model for clllculating the effects of lateral base current. laterally in the thin metal base region. If the emitter collector current transfer ratio a < 1, the lateral voltage drop resulting from this current decreases the emitter base electric field near the center of the device and reduces the current density there. By this mechanism, current is effectively confined to a small strip along the edge of the emitter. The situation is very similar to that encountered in the junction transistor.3 The "character istic length" with which the current density decreases will now be determined. A two-dimensional structure is envisioned, a cross section of which is shown in Fig. 5. The lateral (x-directed) base current j per unit length of the structure is given by j(x) = IX (1-a)J(x)dx. o (5) The lateral voltage drop from the edge of the emitter vex) caused by this lateral current flowing through the base sheet resistance R. is -v(x)= IX j(x)R.dx. (6) This voltage in turn affects the emitter-base electric field Veb-v(x) E=--- d (7) which in turn controls the total current density by Eq. (1). Substituting Eq. (7) into Eq. (1) and assuming v«Veb, we arrive at the approximate result, J(x)=J(O) exp(-Eo vex»). E Veb (8) The three equations (5), (6), and (8) must now be solved simultaneously for J(x). Fortunately, the equa tions are identical in form to those encountered in a similar calculation for junction transistors, and the solution has been found4: J(x)=J(b) sec2[(b-x)jsJ, (9) where 2Veb(EjEo) S2=----- (1-a)R.J(b) (10) The constant s has the dimensions of length and may be thought of as the "characteristic crowding distance," or distance from the edge of the emitter where the cur rent density has fallen appreciably. It is quite clear that a heavy penalty in performance will be paid if the x dimension of the unit is large compared with this distance. Current Transfer Ratio Limitations The fraction of emitter current which actually reache s the collector will be referred to as the device curren t 3 N. H. Fletcher, Proc. I. R. E. 43, 551 (1955). 4 C. A. Mead, Solid State Electronics 1, 211, (1960). Downloaded 03 Aug 2013 to 128.171.57.189. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsOPERATION OF TUNNEL-EMISSIOi'\ DEVICES 649 gain a. This parameter is of major importance in the application of the device. As we have seen in the pre ceding section, higher values of a permit operation with less self-bias crowding. Also, it may be desired to operate the device in the common emitter connection where the current gain is a very sensitive function of a. Limitations on the current transfer ratio stem chiefly from two sources: traps in the insulators and base-insulator inter faces, and "collisions" in the base region and first in sulating layer. It may be thought that electrons tunnel ing from the valence band of the emitter-base insulator into the base region would also constitute an important source of base current. This would be true if the insulator forbidden band were centered upon the metal Fermi level. The problem is very much like that of the emitter efficiency of a transistor, which is low if the Fermi levels in the two regions are equidistant from the center of the semiconductor forbidden regions. The problem is solved by moving the Fermi level in the emitter region nearer the edge of the band (by increasing the doping). Simi larly, by making the metal-insulator work function less than half the forbidden gap, the base current tunneling from the valence band may be made much less than the emitter current. In the discussion which follows we will neglect base current from this source. Traps in the insulating layers may be avoided by using insulating layers of high purity and good crystal structure. Traps at the interfaces may be more difficult to eliminate. It is anticipated that investigations in this area will prove to be a large part of the development of devices of this type. As electrons traverse the thin base region, some will suffer "collisions" and lose enough x-directed energy that they are not able to surmount the work function into the vacuum or second insulator. It has already been stated that the mean free path in the first insulator should be large compared with the thickness of the layer. For reasonable current gain, the second metal "base" layer must also be thin compared with the mean free path I. It should be noted that this mean free path is a very different thing from that normally referred to in connection with the conductivity of the metal. Very little is known about the behavior of "hot" electrons with energies of only a few ev in a metal. However, two very significant experiments have recently been re ported. It may be inferred from work done by Thomas5 that the mean free path for electrons in potassium varies with electron energy as shown in Fig. 6. He attributes the very rapid decline in mean free path around 3 ev to the plasma resonance of the metal. The striking thing about his result is the very long mean free path at energies less than the plasma resonance energy. Since it is possible to make quite continuous metal layers under 100 A thick, such a film should in principle be capable of meeting the requirements of a control element for tunnel emission devices. Similar results indicating 6 H: Thomas, Z. Physik 147, 395 (1958). 1000~--------------------------, 500 100 234 ELECTRON ENERGY (tv) FIG. 6. Mean free path of electrons in potassium as a function of energy above Fermi level (after Thomas). very long electron paths have also been reported for copper.6 Another possible mechanism by which electrons may be lost in the base is the reflection of electronic wave functions from the metal-insulator interface. This prob lem has been dealt with7 in connection with metal vacuum interfaces which should exhibit similar charac teristics. The result of such an investigation is that for electron energies in which we are interested, the reflec tion coefficient is very small compared with unity. A note here is in order concerning the choice of base thickness. If it may be assumed for the moment that all electrons are lost because of collisions in the base and that the collector multiplication factor is unity, the current gain may be written a=exp(-d/I). An approximate expression for the base-sheet resistance is given by8 P( 4L) R8~-1+--, d 'lrd where p is the bulk resistivity of the base material and L is the conductivity mean free path in the metal. For a given geometry and set of requirements on the device, d must be selected for a compromise between self-bias crowding and optimum a. If the electron mean free path in the first insulator is not long compared with the thickness, electrons may suffer collisions and lose suffi cient x-directed momentum that they are not able to surmount the work function into the vacuum or second 6 R. Williams and R. H. Bube, J. App!. Phys. 31,968 (1960). 7 L. A. MacColl, Phys. Rev. 56, 699 (1939). 8 L. Holland, Vacuum Deposition of Thin Films (John Wiley & Sons, Inc., New York, 1956), pp. 236, 347. Downloaded 03 Aug 2013 to 128.171.57.189. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions650 C. A. MEAD I (amp) " 8 "'(]) '" 80 4.5 5 5.5 6 6.5 V (volt) FIG. 7. Experimental v-amp characteristics of three similar AI-AbOa-AI diodes. 7 insulator. For this reason it is desirable to make the first insulator thin compared with the conduction band mean free path. It should be noted in this connection that experimental information is available for only a very few semiconductors. Finally, it should be pointed out that devices with a plurality of thin base layers are possible just as are vacuum tubes with several grids. Such arrangements may be found desirable for various applications as the state of the art advances. EXPERIMENTAL RESULTS Triode Structures The first experimental tunnel-emission diodes were fabricated from aluminum because of the ease with which thin oxide films of known thickness may be formed on the surface by anodizing. 8 Initially, aluminum was evaporated on a glass substrate and anodized to the desired oxide thickness in a dilute ammonium citrate AI ColieClor FIG. 8. Cross section of experimental tunnel-emission triode. solution. l\fore aluminum was evaporated through a mask on the surface of the oxide in the form of circular dots approximately 0.2 mm in diam. Additional diodes were prepared in a similar manner on the electropolished surface of an aluminium single-crystal substrate. The v-amp characteristics of three typical diodes anodized at 5 v (corresponding to approximately 70 A) is shown in Fig. 7. Early triodes were prepared as shown in Fig. 8. Aluminum was evaporated on a glass substrate in the form of a stripe approximately 5 mm wide, and was anodized to the desired oxide thickness (50-100 A). To avoid field concentrations at the edges, silicon monoxide was evaporated over all but a 1-mm stripe in the center. Thin aluminum base layer stripes approximately 1 mm wide were evaporated through a mask which allowed them to extend to the left so that contact could be made. The sheet resistance of the film was monitored during deposition, and was controlled to a value of approxi mately 10 ohms per square. Since the films began to show conductivity at greater than 100 kohm/square it is felt: that at the thickness used, a reasonably uni form film was obtained. Judging from interferometer measurements and sheet resistance calculations, the film AI. / .... : .. :.:.~ .... ·"AI..O, .. > I . \,-' :_.' '...;...' . ':.....:' __ _ ~~~--~ Al 7 I I I I I / / / / I / / / / / / / / GLASS SUBSTRATE FIG. 9. Cross section of experimental vacuum emitter. was estimated to be approximately 300 A thick. A very thin film of silicon monoxide (also of the order of 100 A thick) was then evaporated over the central part of the assembly, and finally thick aluminum collector stripes were evaporated in registry with the base stripes but extending to the right. Contact to all regions was made by means of pure indium solder, no difficulties being encountered even with the very thin base films. At current levels of a few JLamp, units constructed by the technique just described showed current transfer ratios up to approximately 0.1. Emission into a Vacuum In order to study tunnel emission into a vacuum, diodes were constructed as shown in Fig. 9. Aluminum was first evaporated on a glass substrate. Circular areas approximately 0.1 mm in diam, which were to serve as the active area of the device, were masked by a photo resist process. All the remaining aluminum was anodized to 200 v (approximately 2500-A oxide thickness). The resist was then removed and the active areas were anodized to approximately l00-A oxide thickness. A very thin (10 ohms per square) aluminum film was then Downloaded 03 Aug 2013 to 128.171.57.189. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsOPERATION OF TUNNEL-EMISSIOi-J DEVICES 651 evaporated through a mask in the form of a rectangle which covered the active area completely and extended onto the thick oxide to provide a contact area. Contact was made by the use of indium solder. This technique allowed several dozen of the devices to be fabricated at once and minimized frustration caused by the destruc tion of one device. The entire assembly was mounted in a vacuum facing an anode plate spaced approximately 1 mm. The emitter-base characteristic was observed on a v-amp curve tracer, while the average anode current was monitored by a sensitive oscilloscope. For some devices, current transfer ratios of the order of 0.01 have been observed; however, many are much lower and individual samples vary widely. The transfer ratio of one particular vacuum emitter is shown in Fig. 10 as a function of emitter current. The transfer ratio invariably increases rather rapidly with emitter current. The de- O~--~----~--;----r---;----~--~ o 3 4 1. (ma) 6 FIG. 10. Current transfer ratio of vacuum emitter as function of emitter current (emitter-base voltage was approximately 7 v). crease in transfer ratio at low currents is thought to be caused by traps in the nearly amorphous insulator and at the insulator-metal interface. AhOa Problem In all of the experiments described thus far, the cur rent obtained before the device was destroyed was quite low. The v-amp characteristics of the tunneling were sometimes quite noisy and erratic. It has been suggested that such difficulties are caused by the presence of hydroxide in the anodic A1203 film.9 Some work has been done at various laboratories on thermally grown oxide films; however, one would not like to give up the controllability of the anodic process and the desirable property of producing a film in which the electric field is very uniform over the entire surface. For these rea sons, a film was sought which would be very stable 9 K. R. Shoulders (private communication). 10" ..---------------------------------, 77'K 10.1 .. ! 10" 10·' 10-10 10-11 0 6 10 12 V (volt) FIG. 11. Experimental v-amp characteristic of Ta-Ta20.-Au diode at nOK [solid line is a fit of Eq. (l)J. chemically but which could be formed by anodic tech niques. These requirements were met by tantalum oxide. Results with Tantalum Tantalum diodes were constructed in the same manner as the aluminum diodes already described. Since tantalum is very difficult to evaporate, the diode dots used were either gold or aluminum. For a given anodiz ing voltage, tunneling was found to occur at a consider ably lower voltage than for the aluminum units. These diodes have been found to be remarkably stable, and currents of nearly an amp have been observed before destruction. 4 V 1.,1t 2 /1 o 0 ~ 0 /1 0 0 0 4 ma /140 m a t::. t::. c,. 0 t::. 0 0 0 °OL------10-0------~20~0-------30~0------4-0-0--------500 T (OK) FIG. 12. Experimental temperature dependence of Ta-Ta2-0.-Au diode. Downloaded 03 Aug 2013 to 128.171.57.189. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions652 C. A. MEAD A typical v-amp characteristic of one of these diodes is shown in Fig. 11. The solid line is the theoretical curve of Eq. (1) where the effective area and metal insulator work function have been adjusted to make the slope fit at the upper end of the curve. Such a procedure seems quite artificial, since the areas obtained are of the order of 10-4 of the true area. It was speculated that perhaps emission was occurring only at localized very small areas. To test this hypothesis, diodes were con structed as shown in Fig. 9 with a very thin gold film as the front electrode. The v-amp characteristic was monitored on a curve tracer and the diode was observed through a microscope. As the thin gold film was heated by the tunneling electrons, it eventually melted and presumably formed very small globules on the surface, resulting in a marked black color. In every case this effect started at the center of the diode and grew larger until it covered the whole active area. During this process no singularities were observable which could be attributed to high current points. The v-amp character istic showed no change except that resulting from the change in area until the dark area reached the outer edge of the active area, at which time the diode became an open circuit. From this result it may be concluded that there were no macroscopic singularities in the tunnel emission current. However, the tunneling may proceed by means of impurities in the film, which could be considered microscopic singularities. Effect of Crystal Orientation One would expect the metal-insulator work function to be a function of crystal face, as in normal field emis sion. Diodes were built on ordinary rolled tantalum sheet (Fansteel capacitor grade), on sheet recrystallized at approximately 2800°C in argon [diodes made on (111) faces and also on crystal boundaries], and on sputtered tantalum films furnished .by N. Schwartz at the Bell Telephone Laboratories. When anodized at the same voltage, the reproducibility between diodes was approxi mately 5% in voltage, and within this tolerance no measurable difference in the diode characteristics was ob served. Additional diodes were made on different crystal faces of niobium with the same result. Temperature Dependence The voltage necessary for a given tunnel current is a reasonably sensitive function of temperature, as shown in Fig. 12. It is believed that this temperature depend ence is caused by a corresponding change in the metal insulator work function. Although no direct evidence is available on this point, it has been shown1o that a very similar temperature dependence of the tunnel voltage in thin germanium p-n junctions is attributable to the change in band gap with temperature. 10 A. G. Chynoweth, Phys. Rev. 118,425 (1960). Tantalum Triode Experiments Both triodes and vacuum emission diodes have been constructed with the use of tantalum in a manner similar to that discussed for aluminum. Some triodes showed feeble transfer characteristics but were not very re producible. Tantalum diodes similar to that shown in Fig. 9, with thin aluminum front films, were tested for tunnel emission into a vacuum. Emitter currents up to 100 ma were used, and the anode meter was sensitive to 10-9 amp. From this experiment it was concluded tha t the transfer ratio, if any, was less than 10-7• Since the front film was aluminum of the same thickness as that used in the aluminum oxide experiments, it is' highly unlikely that all the electrons are being lost in the metal film. It is believed that the mean free path in the tantalum oxide film is sufficiently short that essen tially all tunneling electrons suffer at least one collision from the time they enter the conduction band until they reach the metal. This prevents them from overcoming the aluminum-vacuum work function even if they suc cessfully negotiate the metal film. However, in a triode structure the base metal-collector insulator work func tion is presumably much lower than the corresponding vacuum work function, and electrons may pass over even after losing some of their energy. In summary, anodically grown tantalum oxide films are chemically very stable and show interesting tunneling character istics; however, the electronic mean free path appears to be so short that they are essentially useless for triodes or vacuum emitters. CONCLUSIONS It should be emphasized that the work reported here is certainly in its very early stages. The most serious limitations at present are our almost total lack of knowledge of the pertinent properties of materials, both metals and insulators, and the great need for suitable techniques for fabricating the desired structures. The effects of such basic parameters as crystal structure and orientation, metal-insulator interface structure, and impurities in the various layers are all totally unknown. Many basic questions are brought to mind which have not yet been given even superficial consideration. Hence the results given here must be treated as preliminary. Nonetheless, the feasibility of a new class of devices operating on the principle of tunnel emission has been demonstrated and hence this brief report is given in the hope that it will aid other investigators in the field. ACKNOWLEDGMENTS The author would like to thank his colleagues throughout the industry for their many helpful com ments and suggestions. A great deal of apparatus and many of the experimental units were constructed by H. M. Simpson. The work was supported in part by a generous grant from the International Telephone and Telegraph Corporation. Downloaded 03 Aug 2013 to 128.171.57.189. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.1705112.pdf	PhaseIntegral Approximation in Momentum Space and the Bound States of an Atom Martin C. Gutzwiller Citation: J. Math. Phys. 8, 1979 (1967); doi: 10.1063/1.1705112 View online: http://dx.doi.org/10.1063/1.1705112 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v8/i10 Published by the AIP Publishing LLC. Additional information on J. Math. Phys. Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsJOURNAL OF MATHEMATICAL PHYSICS VOL U M E 8, N U M B E R 10 OCTOBER 1967 Phase-Integral Approximation in Momentum Space and the Bound States of an Atom MARTIN C. GUTZWILLER IBM Watson Laboratory, Columbia University, New York, New York 10025 (Received 13 February 1967) The phase integral approximation of the Green's function in momentum space is investigated for an electron of negative energy (corresponding to a bound state) which moves in a spherically symmetric potential. If the propagator rather than the wavefunction is considered, all classical orbits enter into the formulas, rather than only the ones which satisfy certain quantum conditions, and the separation of variables can be avoided. The distinction between classically accessible and classically inaccessible regions does not arise in momentum space, because any two momenta can be connected by a classical trajectory of given negative energy for a typical atomic potential. Three approaches are discussed: the Fourier transform of the phase integral approximation in coordinate space, the approximate solution of Schrodinger's equation in momentum space by a WKB ansatz, and taking the limit of small Planck's quantum in the Feynman-type functional integral which was recently proposed by Garrod for the energy momentum representation. In particular, the last procedure is used to obtain the phase jumps of 1T/2 which occur every time neighboring classical trajectories cross one another. These extra phase factors are directly related to the signature of the second variation for the action function, and provide a physical application of Morse's calculus of variation in the large. The phase integral approximation in momentum space is then applied to the Coulomb potential. The location of the poles on the negative energy axis gives the Bohr formula for the bound-state energies, and the residues of the approximate Green's function are shown to yield all the exact wavefunctions for the bound states of the hydrogen atom. I. INTRODUCTION of classical orbits for which there is no simple and THE present investigation was undertaken with general description. Second, at a given negative the ultimate goal of finding analytic (as op-energy, E < 0, any two momenta p' and p" can be posed to numerical), approximate expressions for connected by a classical trajectory in the case of a single electron wavefunctions of bound states in typical atomic or molecular potential. But two posi atoms or simple molecules. The phase-integral ap-tions, q' and q", can be connected by a classical proximation, sometimes called the WKB method, trajectory only if they lie both in the region where provides such expressions. However, it turns out the potential energy V(q) is smaller than the total that a somewhat unusual approach working in energy E. The propagator F(P" p' E) will, therefore, momentum space is more appropriate than the well-be approximated by an expression P(P" p' E) with known form involving Hamilton's action function smoother behavior than the propagator G(q" q' E) in coordinate space. Actually, we construct a phase- whose approximation G(q" q' E) has some rather integral approximation for the propagator, or Green's artificial singularities around V(q') = E or V(q") = E. function, F(P" p' E), in terms of the initial momentum Third, it appears that the common procedure of p', the final momentum p", and the energy E. The separating the variables in a problem of spherical singularities of F along the negative E axis give the symmetry has an adverse effect upon the phase approximate wavefunctions. This procedure is tested integral approximation. The well-known difficulty in for the Coulomb potential, where it is found to yield obtaining Bohr's formula for the hydrogen levels the exact wavefunctions for all the bound states. vanishes entirely if we construct either P or G in three Although this last result seems better than expected, dimensions without bothering to separate variables. there are good reasons to believe that the present The general formula for P(P" p' E) is easily written scheme is indeed more efficient than the usual ones, down, but its derivation does not satisfy a mathe at least in the case of bound states for typical atomic matician's requirement for rigor. Even the phase potentials. First, the connection between classical integral approximation K(q" q' t) for the propagator and quantum mechanics is much simpler for the K(q" q' t) from position q' to position q" in the given propagator than for the individual wavefunctions. time t has not yet been established with the desirable The construction of the approximate propagator degree of accuracy and generality for singular po requires the knowledge of all the cla~sical paths tentials such as the Coulomb potential, although K which go from the initial to the final point, whereas is certainly better understood than (J which in turn is an approximate wavefunction requires a special class better known than P. The author found it helpful to 1979 Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1980 MARTIN C. GUTZWILLER arrive at E by several different methods. The emphasis of this report is, therefore, not on the applications for which the method was originally devised, but on the more basic problems concerned with the phase integral approximation. In particular, it is necessary to obtain E directly from the path-integral expression for F which was recently discovered by Garrodl as a generalization of the Feynman integral for K. Apart from the very difficult question of justifying Feynman type integrals and deriving their limits for vanishing Planck's quantum, certain results from the classical calculus of variation are needed, in particular the character of the second variation. As far as K is con cerned, these results have been obtained by Morse,2 but the class of problems arising from E and G (which one may legitimately call isoperimetric) has apparently not been considered as yet and their solution is only conjectured for the purpose at hand. The discussion of the various topics is presented in the following manner. Section IIA summarizes some of the well-known results about the propagator K(q"t",q't'), and proposes a general formula for the limit Ii ---+ 0 when t" -t' is arbitrarily long. The crucial phase jumps at a focal point are related to Morse's theory of the second variation, which in turn arises quite naturally if one goes to vanishing Ii in Feynman's path integral for K. Section lIB dis cusses the same ideas for the Green's function G(q" q' E), although our mathematical background in this instance is much poorer. Three different ways to obtain the limit of G for small Ii are presented, by taking the Fourier transform of K, by solving the inhomogeneous Schrodinger equation, and by letting Ii vanish in Garrod's path-integral expression. A second variation is again needed, except that the variational quantity is not covered by Morse's theory, and certain conjectures have to be made. Section IIC carries the arguments over into the study of F(p" p' E), in particular the three methods for going to the limit Ii ---+ O. The investigation of Garrod's path integral and the study of the second variation for the action integral are now particularly important, because Schrodinger's equation is not local anymore, and the phase jumps cannot be obtained in the customary manner. Since the formulas for t~e limits G(q" q' E) and F(p" p' E) as Ii vanishes are completely analogous, 1 C. Garrod, Rev. Mod. Phys:38, 483 (1966). • M. Morse, The Calculus of Variations in the Large (American Mathematical Society, Providence, Rhode Island, 1935). For more contemporary presentations, cr. J. Milnor, Morse Theory (Princeton University Press, Princeton, New Jersey, 1962); H. M. Edwards, Ann. Math., 2nd Ser. 80,22 (1964); S. Smale, J. Math. Mech. 14,1049 (1965). any detailed calculations can be carried out in either case. The more familiar G is chosen in Sec. IlIA to exhibit the simplifications due to a spherically sym metric potential. Section IIIB establishes the same result by performing explicitly the limiting process in Garrod's path integral for G(q" q' E); this feat has only been possible for a spherically symmetric poten tial, although the formula for G is believed to be valid more generally. The various results are listed in Sec. IIIC for E, as they are needed for the Coulomb problem. The Kepler orbits in momentum space are discussed in Sec. IVA. Since they are circles, their geometry is much easier to understand than in coordinate space, and simplifies all explicit calculations. The phase integral approximation E is worked out in Sec. IVB on this basis. In particular the phase jumps at focal points are obtained, and compared with those of another famous problem, the linear oscillator. The resulting approximate Green's function is shown in Sec. IVC to have poles at the negative values of E in agreement with Bohr's formula. The residues are worked out and are compared with the residues in the exact Green's function which has recently been established by various authors. The complete agree ment confirms our original expectation that bound states are best described by the phase-integral method in momentum space. n. GENERAL FORMULAS A. Time and Space Coordinates Consider a simple physical system without spin, e.g., an electron in a given elect~omagnetic field. Its coordinates are given by a vector q, and its momentum by a vector p. In case the components of q or p have to be specified, they are indicated by an upper index, such as qi or pi. The propagation function K(q" t", q't') for this system depends on the initial coordinates q' and time t' , as well as the final coordinates q" and time t" > t'. K is found from the requirements that ili(aK/at") -Hop(P" q" t)K = 0, (1) lim K(q"t", q't') = b(q" -q'), (2) t"-+t' where Hop(p q t) is the Hamiltonian operator. Hop is obtained formally from the classical Hamiltonian H(Pq t) if P is replaced by the operator -ilia/aq. Planck's quantum divided by 271 is written as Ii. Equation (1) is SchrOdinger's equation, and the initial condition (2) appears quite naturally if one tries to solve the initial value problem for (1). After a suggestion by Dirac, it was demonstrated Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1981 by Feynman that K can be represented as an integral over all possible trajectories from q' at t' to q" at t" in the following manner. Let the time interval from t' to t" be subdivided into N subintervals by inserting t1, t2, ••• , tN-I, and define a discrete path from q' to q" by inserting the intermediate points ql, q2, ... , qN-I. An action integral RN along this path is given by RN = f(tn -tn_1)L(qn -qn-l, qn' tn)' (3) 1 I tn -tn_1 where q' = qo, t' = to, q" = qN' and t" = tN. Also, we have introduced the classical Lagrangian ~ ·oH L(tjq t) = £., i -. -H, i Op' (4) where the momenta pi are eliminated on the right-hand side with the help of the relation tji = dqi /dt = oR/ Opi. Feynman's formula is then given by K = lim IT [ m J! N ... rIO 1 27Tili(tn -tn_I) X f d3ql ... f d3qN_l exp [iRN/Ii]. (5) For definiteness we have assumed a 3-dimensional q-space, and a particle of mass m. The physical content of (5) is discussed in a recent monograph by Feynman and Hibbs.3 The constant in front of the (N -I)-fold integration has been chosen mainly to obtain the relation f d3qK(q"t", qt)K(qt, q't') = K(q"t", q't'). (6) Nelson4 has recently discussed Feynman's formula as an analytic continuation of Wiener's formulaS for Brownian motion, but we would like to start directly from (5). Pauli6 investigated the limit of K for small time intervals t" -t'. The result can be written in terms of the action integral R(q"t", q't') = t'~(tj q t) dt, (7) Jt' calculated along the classical trajectory which carries the particle from q' at time t' to q" at time t". The approximate value K is given by K(q"t", q't') = (27Tili)-!(DR)! exp [iR(q"t", q't')/Ii], (8) where DR is the determinant of the mixed derivatives DR = (_1)3 det I (o2R)/(oq'oq") I. (9) 8 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hili Book Company, Inc., New York, 1965). 4 E. Nelson, J. Math. Phys. 5, 332 (1964). '·N. Wiener, J. Math. Phys. 2, 131 (1923). 6 W. Pauli, Ausgewahlte Kapitel aus der Feldquantisierung, Lecture Notes, Zurich, 1951. Since the initial momentum p' along the classical trajectory is given by p' = -oR/oq', one can interpret DR as the Jacobian o(P')/o(q") between the range d3p' of initial momenta and the volume d3q" covered by the endpoints. The validity of (8) has been established by Choquard7 for potentials without singularities. But even for the Coulomb potential, one has always at least two classical trajectories connecting any given pair of points q' and q" in a given time t" -t'. For a short time interval t" -t', one trajectory follows quite closely the straight line from q' to q", whereas the other trajectory heads first for the center of attrac tion, then turns around it in a sharp twist, and goes to the final point following an almost radial path again. The formula (8) for K will, therefore, not be sufficient for a typical. atomic potential. Actually, the singularity (2) in K at t" = t' follows from (8) if we evaluate R for the direct path from q' to q". Formula (8) remains presumably valid for sufficiently small Iq" -q'l if it is applied only to the direct tra jectory, since the contribution from the indirect trajectory would remain finite. Pauli showed that K as given by (8) satisfies Schrodinger's Eq. (I) up to a remainder which is proportional to li2. It is, therefore, reasonable to expect that the limit of K, as Ii goes to zero, has an appearance very much like (8). Feynman's formula (5) shows that there is a contribution to the limit of vanishing Ii from every path q' = qo, ql' ... , qN = q' for which RN is stationary. Thus we expect in general a sum of terms like (8), one for each classical tra jectory from q' at t' to q" at t". The continuity of the result requires that each term in this sum takes the exact form (8) as q" approaches q' along the direct path while t" -t' is sufficiently small. As t" increases from t', and q" runs along a given classical trajectory, the amplitude (DR)! becomes infinite every time q" passes a focal point. A detailed examina tion of Schrodinger's equation in its neighborhood shows that (8) remains valid even beyond the focal point, if we take the amplitude (IDR!)! and insert a special phase factor exp (-i7T/2) for every reduction by I in the rank of the Jacobian o(q")/o(p') = l/DR at the focal point. Thus, we obtain K(q"t", q't') = (27Tili)-! ! (DR)! classical paths X exp [i: + Phases], (10) as the limit of K(q" t", q't') for vanishing Ii. 7 Ph. Choquard, Helv. Phys. Acta 28,89 (1955). Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1982 MARTIN C. GUTZWILLER This last expression was obtained by solving Eq. (1) up to terms in Ii!, by imposing the initial condition (2), and by forcing the result to be continuous. The relation (6) can be checked for (10) if the integral over q is computed by the stationary phase method. However, it would help our understanding of similar approximations for G(q" q' E) and F(P" p' E) if the expression (10) could be directly derived from Feynman's integral (5). We shall indicate the necessary steps, although we realize that there are many gaps to be filled before any mathematical rigor can be claimed. Let N be large enough so that the particular classi cal path of interest can be adequately described by a sequence q' = iio, iiI' ... , iiN = q" corresponding to the times t' = to, t1, ... , t N = t". The approximate action RN given by (3) is stationary for q1 = iiI' ... , qN-1 = iiN-1' Ifwewriteqi = iii + 15qiforj = 1,'" , N -1, we find RN = R(q"t", q't') + t !,R;,15q j15q, + .. " (11) il where the omitted terms are of order either (15q)3 or lIN. Since 15qj is a vector, the matrix Ril has more elements than are actually suggested by (11). The integrations over q1, ... ,qN-1 are easily p!»"formed if the exponent in (5) is replaced by the two terms in (11). The matrix Ri! has to be transformed to principal axes, so that one gets 3N -3 Fresnel inte grals. Thus we find an amplitude equal to (2'ITIi)i(3N-8) . (ldetRjll)-l and a phase factor exp [iRlli + (3N -3)i'IT14 -iM'lT12], where M is the number of negative eigenvalues of Ri!' In order to show agreement with (10) we have to establish that and that M equals the number of focal points between q' and q", each counted according to the rank of D"il. The proof of (12) can be accomplished without difficulty in the case of a spherically symmetrical potential because Ril simplifies greatly in polar co ordinates and its determinant can be evaluated by writing out the appropriate recursion formulas (cf. Gel'fand and Yaglom,8 as well as Montro119). Unfor tunately, such a procedure has not been successful in • I. M. Gel'fand and A. M. Yaglom. J. Math. Phys. 1,48 (1960). • E. W. Montroll, Commun. Pure Appl. Math. S, 415 (1952). the case of a nonseparable potential. On the other hand, the relation between M and the focal points is a well-known result of the calculus of variation in the large, as worked out by Morse2 in a classic mono graph. Morse's results can, therefore, be interpreted physically in terms of the extra phase which a wave loses at a caustic due to its spilling over into the classically forbidden region. The results of Morse have not received any atten tion in the textbooks of classical mechanics. Yet, in every course there is at least one student to ask whether, indeed, the integral S L dt becomes minimal along the classical trajectory. If the answer might have seemed unimportant because there has been no physical application for it so far, it is all the more interesting to find such an application in the transition from classical to quantum mechanics. The fact that f L dt becomes minimal for a sufficiently short path gives Morse's theory a simplicity which will not be matched by the later examples of a second variation (cf. Secs. lIB and 1IC). B. Energy and Space Coordinates In order to describe stationary states of a physical system, one has to know the propagator at constant energy. We assume from now on that H is independent of t, so that K depends only on the difference t" -t'. The Green's function G(q" q' E) is defined as G(q" q' E) = -!-("" dtK(q"t, q'O) exp [iEt] , (13) Iii Jo Ii where E is in the upper half of a complex E plane. The homogeneous differential equation (1) and the initial condition (2) are now combined into the inhomoge neous equation [E -Hop(P"q")]G(q"q'E) = 15(q" -q'). (14) If the homogeneous equation [E -Hop(pq)]tp = 0 has no acceptable solution for an interval of real values of E, then Eq. (14) has a solution which is, moreover, symmetric in q' and q". Green's function G can then be continued analytically into the lower half of the complex E plane by putting G(q" q' E) = [G(q' q" E)]. (15) Thus, the behavior of G along the real E axis is directly related to the existence of solutions for the homogeneous equation which corresponds to (14). The details of this relation are discussed in any modern textbook on Green's functions. The expression for G in terms of an integral over all paths from q' to q" has only been discovered very recently by Garrod.l The crucial step is to consider Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1983 all possible paths in "phase space" rather than co ordinate space only. Thus one introduces a sequence of coordinates q' = qo, ql' ... ,qs = q" (as before), but in addition a sequence of momenta PI' Pi' ... , P.v-l. A path in "phase space" is described by the combined sequence q' = qo, Pi' ql, Pi' ... , q.V-l' P.v-l' q.v = q", and the mean energy & along this path is defined by t .v-I p2 1 S & = -L -1!. + -L V(q,,). (16) N I 2m N 0 For simplicity's sake the Hamiltonian has been as sumed to consist only of the usual kinetic energy r/2m and the potential energy V(q). Eut, both a relativistic kinetic energy and a vector potential could equally well have been included. The Green's function now becomes fN-1 f,Y-1 G = lim (27Tn)-3.V IT d3q" IT d3Pn N~oo 1 I x exp U S.vJ /(E -E), (17) where Ss is the action along the path qo, PI ' ql' ... , PS-!, q-y in phase space, N S.v = LPn-l(q" -q,,-l)· (18) 1 The few formal steps from (5) to (17) are explained in Appendix A, because our definition of G differs slightly from Garrod's. There are three ways to:.finding the approximation G of G for small n. First, one can simply insert (10) into (13) and evaluate the integral over t by the station ary phase method. Second, the inhomogeneous wave equation (14) can be solved in the limit of vanishing n. Third, the limit of (17) can be found as n goes to zero. The first method is the most straigbtforward and is carried out in Appendix B. Its result is expressed in terms of the classical action S(q" q' E) =iq"p dq, q' (19) evaluated along the classical path which leads from q' to q" at the given energy R(P q) = E. The phase integral approximation G becomes G(q" q' E) = -~ L (IDsD~ 2 7T n classical paths x exp [~ + Phases]' (20) where the determinant Ds now contains not only the second mixed derivatives with respect to q' and q", but also with respect to E, a2s a2s oq'oq" oq'oE Ds= (21) Actually, the element 02S/0P might just as well be replaced by 0, because the 3 x 3 determinant I 02s/aq'aq" I vanishes. The phases in (20) are the same as in (10), except when a2R/at2 = -oE/ot < 0, i.e., a higher-energy orbit leads to a longer transit time. The interpretation of Ds can be made as follows: Consider the family of classical trajectories which leave q' with the initial momentum P' in the neigh borhood d3p' = dO' dE; their endpoints lie in a neighborhood d3q" = dO" dt of q"; Ds is then the Jacobian dO'/dO". The phases in (20) are again -i7T/2 times the reduct\on in rank of the 2 x 2 matrix associated with dO" /dO' at a focal point. The second method for obtaining G has been studied extensively, e.g., by Avila and Keller,lo in the case where E -V(q) is positive and bounded for all q. This situation corresponds to the scattering of particles by a potential without singularities, whereas we are interested in particles which are trapped in a singular potential such as the Coulomb potential. Nevertheless, the general considerations are similar; in particular, the discussion of caustics, as in the work of Ludwig,ll can be taken over directly. But one will not have an infinity of trajectories from q' to q" if E -V(q) is bounded and 'positive. Kohn and Sham12 have obtained (j in one dimension with the help of the well-known expression for G in terms of a Wronskiaa. Their method has not been generalized to more than one dimension; formula (20) leads exactly to their result. The singularity of (j for a small distance Iq" -q'l can be obtained directly from the inhomogeneous equation (14). It is found that up to terms in Iq" -q'12 (j( " 'E)"-' _ m q q = 27Tn2Iq" -q'l x exp {i Iq" -q'l [2m(E -V(q»]l/n}, (22) where q = l(q' + q"). This expression corresponds to limiting the expansion (20) to the shortest trajectory from q' to q", and evaluating S(q" q' E) in powers of Iq" -q'l. The approximation (22) for (j is completely equivalent to the Thomas-Fermi approximation, 10 c. s. S. Avila and 1. B. Keller, Comm. Pure Appl. Math. 16, 363 (1963). 11 D. Ludwig, Comm. Pure Appl. Math. 19,215 (1966). 12 W. Kohn and L. J. Sham, Phys. Rev. 137, A1697 (1965). Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1984 MARTIN C. GUTZWILLER which appears usually as the Fourier transform S d3(q" -q') exp [-p(q" -q')/Ii] of (22), namely GTF(PqE) = [E -(p2/2m) -V(q)]-I; (23) cf. Baraff and Borowitz.13 The third method seems to be the most interesting because it leads to a new viewpoint in classical mechanics and to some new problems for the calculus of variations in the large. It is natural to perform the integrations in (17) in two steps. First, one integrates over the variables Pn and qn on a hypersurface of constant average energy E, as given by (16). Second, & is integrated from -00 to + 00. As Ii goes to zero, one is, therefore, faced with finding the stationary pathq' = qo,p!, ql, ... ,PN-!, qN = q" for SN under the subsidiary condition (16). In the limit of large N, one has to solve the "isoperimetric" problem: Find the curves p(t), q(t) in phase space for which S pdq is stationary, given the endpoints q' and q", as well as the average energy E = f H(Pq) dt/(t" -t'). The Euler equations of this problem are the Hamilton equations of motion, but the usual variational principle at constant energy demands that S P dq be stationary for given endpoints q' and q", while H(Pq) = E at each point p(t), q(t); cf. Whittaker.14 Garrod1 noticed this novel variational principle. For the purpose of finding G, one has to go one step further, since the second variation of S N is needed. Let N again be large enough to describe a particular classical path in phase space by a sequence q' = ijo, p!, ijI, ... , PN-! ;ijN = q" taken at equal time inter vals. With qn = ijn + flqn and Pn = fin + flpn, where n = }, 1, ... , N -t, one expands in powers of flqn and flpn, SN = S(q"q'E) + bIS + b2S + ... 1 it" E = H(pq) dt + flIE + b2E + ... , (t" -I') t' (24) where the omitted terms are either of order l/N or of third order in flqn and flpn. The classical path is stationary if the condition (25) is identically fulfilled in all bqn and flpn for a param eter T such as to satisfy (16). The second variation of the exponent in (17) subject to (16) becomes [fl2S]& = [fl2S -Tfl2E]61&=O, (26) i.e., the variables flqn and flpn in the quadratic form 13 G. E. Baraff and S. Borowitz, Phys. Rev. 121, 1704 (1961). U E. T. Whittaker, A Treatise on the ANl{ytical Dynamics of Particles and Rigid Bodies (Cambridge University Press, Cambridge, England, 1937), 4th ed., p. 247. b2S -Tb2E are subject to the linear constraint bIE = O. The further steps in the integration over bqn and flpn, with (26) inserted into the exponent of (17), are straightforward. One finds an amplitude (2rrli)3N-2 times A(q" q' E) dE, where A contains the determinant of the matrix associated with (26) and a Jacobian, because the integration uses internal coordinates for (26) in addition to E, rather than tlqn and flpn. The phase factor is simply exp [is(q'' q' E)/Ii -iMrr/4], where M can be called the index of the classical trajectory. M is equal to the number of negative eigenvalues in (26) minus the number of positive ones. In the case of a spherically symmetric potential, it will be shown in Sec. IIIB that the amplitude A(q" q' E) equals (IDs!)!, where E is replaced bye. As in the case of K and Eq. (12), we have not been able to show this identity in the more general case of an arbitrary potential, but we shall assume it henceforth. The index M starts out with a: value 2 for the most direct tra jectory from q' to a nearby endpoint q". As can be observed from the sign of Ds, the index M changes at every focal point. We conjecture that M increases at every focal point by twice as much as the rank of dO"/dO' is reduced. There does not seem to exist a simple relationship between S L dt on one hand, and S p dq at constant average energy on the other, al though the equations of motion for stationary trajectories are identical. Since the amplitude A(q" q' E) does not depend on Ii, the main variation in the integral over E comes from the phase factor exp [is(q'' q' E)/Ii -iMrr/4]. Therefore, A is pulled out of the integral with E replaced by E, and S(q" q' E) is expanded around E to first power in E -E. The remaining integral becomes f+OO ~ eiH&-E)/1i = {-2rri for I> 0, -00 E -E 0 for I < 0, (27) where t = oS(q"q' E)/oE is the transit time for the particle to go from q' to q". The denominator in (17) automatically limits the contributions from the various paths in phase space to the ones which Corre spond to going forward in time, provided the imagi nary part of E is positive. Thus we find again the approximation (20), but this time on the basis of Garrod's formula (17). C. Energy and Momentwu The propagator F(p" P' E) for a particle to start out with a momentum p' and end up with a mo mentum P" while propagating with the energy E, is Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1985 defined by F(p" p' E) = (27T1i)-Sf dSq"f dSq'G(q" q' E) X exp [i(p' q' -p" q")/ Ii]. (28) The inhomogeneous Schrodinger equation (14) be comes [E -Hop(p"q")]F (p" p' E) = b(p" -p'), (29) where the Hamilton operator is the integral operator H . F = (p,,2/2m)F(p" p' E) + f dSpV(p"p)F(p p' E). (30) V(P" p) is the Fourier transform of the potential V(q), i.e., V(p" p') = (27T1i)-Sf dSq V(q) exp [-i(p" -p')qj Ii]. (31) The path integral expression (17) can directly be inserted into (28) to yield the formula F == lim (27T Ii)-SN-Sf IT dSqnf It dSPn N~oo 0 l X exp [-~ TN]/(E -&), (32) where TN is the action along the path p' = p_! ' qo, Pl' ql,···, PN-!, qN, PN+! = P" in phase space given by N TN = l qn(Pn+! -Pn-!), (33) ° and the average energy & is given by the same formula (16). The semiclassical or WKB method has been used occasionally in momentum space. KohnlD describes the motion of electrons in a solid in this manner. Goldman et al.16 discuss the transformation between WKB wavefunctions in coordinate and in momentum space for one dimension. Schiller17 writes the equa tions for the phase and the amplitude in a time dependent situation. But none of these authors has investigated the Green's function F in the semi classical approximation, nor were they interested in bound states, even for a spherically symmetric poten tial. There are again the three ways to finding the approximation F of F for small Ii which were discussed in the preceding section. 15 w. Kohn, Proc. Phys. Soc. (London) 72, 1147 (1958), cr. also E. I. Blount, Phys. Rev. 116, 1636 (1962). 18 I. I. Goldman, V. D. Krivchenko, V. I. Kogan, and V. M. Galitskii, Problems in Quantum Mechanics (Academic Press Inc. New York, 1960), pp. 11 and 92. 17 R. Schiller, Phys. Rev. 115,1100 and 1109 (1962). The first method consists in applying the Fourier transform (28) to the formula (20) for G, and evaluat ing the integral by the stationary phase method. The procedure corresponds very closely to the calculations in Appendix B. The result involves the classical action T(p" p' E) = i~"q dp, (34) calculated along the classical path in momentum space which leads from P' to p" at the given energy H(Pq) = E. The phase-integral approximation F is given, in complete analogy to (20), by F(p" p' E) = -~ l (IDTI)! 27T1i classical paths X exp [ -i~ + Phases} (35) where the 4 x 4 determinant DT contains again the second mixed derivatives of T with respect to P' and P" as well as E, (PT o2T -- op'op" op'oE uT = cPT 02T (36) oEop" OE2 Again, the element 02T/o£2 may be replaced by zero, because the 3 x 3 determinant 102 T/ op' op" I vanishes. The determinant (36) has a completely analogous interpretation to (21), in terms of the family of classical trajectories which go from p' into the neigh borhood of p" at the given energy E. Presumably the phases in (35) are similarly related to the caustics which are generated by this family of trajectories in momentum space. But it is important to realize that the two families, one in coordinate space and the other in momentum space, are not simply the same set of curves in different representations. This fact becomes especially apparent if one studies the char acter of the focal points along the classical trajectory. Thus, a Kepler orbit in coordinate space has two singly counting focal points followed by the doubly counting starting point, whereas in momentum space there is one doubly counting focal point followed by the doubly counting starting point to which all trajec tories of the family return. The second method of deriving F consists in using a trial solution of the type B(p" p' E)exp [-iT(P" p' E)/Ii] in order to solve the inhomogeneous Schrodinger equation (29) to first order in Ii. The potential-energy term in (30) is evaluated in Appendix C with the help of the stationary phase method. The Hamiltonian Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1986 MARTIN C. GUTZWILLER operating on B exp [-iT/ii] becomes HopF = exp [i;] . {[~: + V(q")]B +'Ii[dVdB B d2 V d2T] } I dq" dp" + "2 dq"dq" . dp"dp" +"', (37) where q" = dT/dp", and the terms in the last line are sums over the components of p" and q". The re mainder is of order 1i2. If this expression is inserted into the left-hand side of (29), it is not evident at all which terms in (37) are to be matched by the r5(p" -p') on the right-hand side of (29). Obviously this in homogeneous term in (29) determines the amplitude B, exactly as the r5(q" -q') on the right-hand side of (14) determines the amplitude A of G, whose behavior for smalllq" -q'l is expressed in (22). Upon closer examination, the following is found. The terms p"2/2m + V(q") in the first line of (37) are equal to E, provided T is the appropriate action function (34). The square bracket in the second line of (37) vanishes if B is proportional to (DT)! and p" =;1= p'. The behavior of (DT)! as p" approaches p' can be most easily investigated if one starts with the Thomas-Fermi approximation S ~ Iq" -q'l . {2m[E -V{l(q' + q"»]}! as in (22), and examines the transformation into momentum space, p" = dS/dq" and p' = -dS/dq'. The Jacobian of this transformation is given in the limit of q' = q" by the expression m4v/2m{E -v(q» with Vn V12 VIS VI V21 V22 V2S V2 '11= - (38) VSI VS2 Vss Vs VI V2 Va 0 where Vii = d2V/dqidqi and Vi = dV/dqi. The value of -(27T/i2)-I(IDTI)!exp (-iT/ii) for small Ip" -p'l is obtained as 1 IdV/dql [ i" , ] -27T/i2 IP" -p'I' (v)!-exp -h Ip -pl' Iql , (39) where q is chosen on the surface 2m[E -V(q)] = (p" + p')2/4 such that the direction of -dV/dq coin cides with the direction of p" -p'. Since the square bracket in the second line of (37) contains only first derivatives, it will not lead to a singularity r5(p" -p') if we insert (39). This is, in fact, what one has to expect, since the factor 1/(27T/i2) in (39), together with the factor iii in (37), yields a term of order /i-I, whereas the right-hand side of (29) is of order IiO. The inhomoge neous term in (29) is, therefore, not generated by the formula (35) for F [as the b(q" -q') in (14) is generated by the formula (20) for G]; it would come about only by going to the next term in the expansion (37) for H opF. If the singularity at p" = p' is to be included explicitly in an approximation for F, one would have to write r5(p" -p') V(p"p') + ----~~~~----- E -p2/2m (E -p"2/2m)(E -p,2/2m) - 21/i2 L (IDTI)~exp [-i T + Phases], 7T classical paths Ii (40) where the first two terms are obtained from an ex pansion of F in powers of V(p"p'). For Coulomb-like potentials, these terms are of order IiO and /i-I. _ The preceding discussion shows that, contrary to G(q" q' E), the Green's function F(P" p' E) is not easily obtained by solving the inhomogeneous Schrodinger equation. It seems very hard to get higher-order terms in the expansion (37) for HopF. Also, the behavior of F near a caustic and the extra phase factor cannot be determined from (29), because Schrodinger's equation is an integral equation in momentum space. The expansion of G(q" q' E) near a caustic, however, is based on finding solutions to Schrodinger's equation which are only valid in a small neighborhood. If F is derived directly from the path integral formula (32), i.e., by the third method, the procedure is absolutely identical with the derivation of G from (17). The discussion at the end of the preceding section can be repeated exactly with p and q, as well as T and S, interchanged. A detailed examination of (17) or (32) in the limit of small /i appears, therefore, quite worthwhile. III. SPHERICALLY SYMMETRIC POTENTIAL A. Approximate Green's Function in Polar Coordinates The classical orbit going from q' to q" lies in the plane which is determined by q', q", and the center of force at the origin. The action S(q" q' E) depends only on the absolute values r' and r" of q" and q', and on the angle cp between q" and q'. By straightforward calcula tion, one finds for the determinant (21) that Sr'r" Sr'''''' Sr'E Ds= S'" S""r" S"""''' S""E (41) "2,,,2 sin cp SEr" SE",,, SEE Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1987 q/ and rp" are the polar angles of q' and q" in the plane of the orbit. The indices on S indicate the derivatives of S with respect to these quantities. The determinant in (41) is obtained by finding the orbit corresponding to r', r", rp, and E in a plane. The equations of motion can be solved by quadra tures if we know the angular momentum M of the orbit. It is, therefore, advisable to use M as a third parameter, besides r' and r", rather than rp. The connection between rp and M follows immediately if we combine the two conservation laws for angular momentum and for energy, mr2 drp/dt = M, (42) (dr/dt)2 + r2(drp/dt)2 = 2m[E -VCr)]. (43) rp increases always if M > 0, even if r sometimes increases and sometimes decreases. Therefore, the integrand in f.r" M dr rp = rp" -rp' = r r2[2m{E -VCr) -M2/2mr2)]l (44) has to be interpreted as making positive contributions, even if r is made to run back and forth between certain maximum and minimum values, r max and r min' before reaching the limits of integration, r' and r". In the same sense we find that S =f.r " dr 2m[E -VCr)] . (45) r' [2m{E -VCr) -M2/2mr2)]l If the derivatives with respect to rp' and rp" in (41) are now expressed as derivatives with respect to M, one obtains finally M m D -' 1 s -r,2r,,2 sin rp [2m(E -VCr') -M2/2mr,2)] X m (Orp)-l. (46) [2m(E -V(r") -M2/2mr"2)]l oM The last factor can be expressed formally as an integral over r with the help of (44), namely orp -f.r"dr 2m[E -VCr)] (47) oM -r' r2[2m(E -VCr) -M2/2mr2)]! . It is important to notice certain special cases of (46). If rp tends to zero while M tends to a nonvanishing limit, the approximation (22) is obtained after in serting (45) and (46) into (20). If r" approaches either rmax or rmln' where E -VCr) -M2/2mr2 vanishes, the amplitude DB stays finite. Formally, this comes about becaur.e the integral (47) diverges while the denominator in (46) vanishes. Physically, it means simply that the orbits do not crowd one another at their point of greatest or smallest distance from the origin. However, D. becomes infinite wher ever orp/oM = O. A plot ofr" vs rp reveals immediately a caustic for the family of classical trajectories in the same plane which leave q' with different angular momenta M. Similarly, the vanishing of sin rp in the denominators of (41) and (46) indicates a focal point for the family of trajectories which leave q' in different planes, but with the same absolute value of angular momentum. Each occurrence contributes a phase -i7T/2 to the formula (20). These two types of focal points can coincide, such as in the Coulomb potential where all trajectories of a given energy E return to the initial point q', independently of the direction or the magnitude of their angular momentum. The formulas (45) and (46) can be inserted into (21) in order to yield G(q" q' E), provided one can solve Eq. (44) so as to find the angular momentum M in terms of the distances " and ,", and the angle rp. For the Coulomb potential this problem should not be too hard to treat explicitly. But we shall not go into these details here, because a more interesting example of the same calculation is given in the last three sections. B. Garrod-Feynman Integral in the Limit of Small Ii The phase-integral approximation in coordinate space at a given energy E can be obtained from the results in the preceding section in the case of a spherically symmetric potential. The same formulas are gotten directly from the path-integral expression (17) for G(q" q' E) by going to the limit of small Ii. This second derivation is important because it can be used equally well to find the limit of small Ii for the path-integral expression (32) of F(P" p' E). It also yields the phase jumps at the focal points and gives new insights into the Garrod-Feynman integrals, (17) and (32). The first task is to rewrite (17) as well as the original Feynman formula (5) in polar coordinates. Edwards and Gulyaev18 have discussed this trans formation for K(q" t", q't') in the case of a free particle. But their arguments are greatly simplified for our purpose by the following remarks. The propagator K satisfies Schrodinger's Eq. (1), exactly as the transition probability in Brownian motion satisfies the Fokker Planck equation (cf. Wang and Uhlenbeck19). The 18 S. F. Edwards and Y. V. Gulyaev, Proc. Roy. Soc. (London) 279,229 (1964). 19 M. C. Wang and G. E. Uhlenbeck. Rev. Mod. Phys. 17, 323 (1945). Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1988 MARTIN C. GUTZWILLER only quantItIes of importance are, therefore, the various momenta of K in the limit of vanishing t" -t: For a nonrelativistic particle of mass m in a potential V(q), one has the relations lim(t" -t'r1{fK(q"t", q't') d3q" -I} t"-+t' = V(q')jili, (48) lim(t" -n-1f(q;" -q;')K(q"t", q't')d3q" = 0, (49) t"-+t' lim(t" -t')-lf(q;" -q;')(ql" -q!')K d3q" t"-+t' = (ilijm)oj!' (50) It can be shown, by straightforward computation, that these relations are satisfied, not only by the kernel [mj27Tili(t" -t')]i x exp i{(q" -q')2j2m(t" -t') -(t" -t')V(q')}jli in Cartesian coordinates, but also by the expression 3 [27Tili(; -t')r x exp -.! [(r" -r'l + r'r"«(J" _ (J')2 . { 1 Ii 2m(t" -t') + r'r" sin (J' sin (J"(T" -T')2] -(t" -t')V(q') + ~(! + 1 )(t" -t')}, (51) 2mr'r" 4 4 sin (J' sin (J" where we have used polar coordinates by putting ql = r sin (J cos T, q2 = r sin (J sin T, q3 = r cos (J for both the initial and final points. The symmetric oc currence of the single and the double primed co ordinates in the first part of the exponent is essential in order to guarantee the relations (49) and (50). The last term in the exponent looks like an additional potential, and has to be inserted if (48) is to be satisfied. The expression (51) is now used to generate the propagator K, i.e., the action function RN in (5) is written in polar coordinates as N {I 2 RN = I (tn -tn-I) 2 ( )2 [(r n -r n-l) I m tn -tn_l + rnrn_I«(Jn -(In_I)2 + rnrn-l sin (In sin (In_l 2 1i2 (Tn -Tn-I) ] -V(qn) + 8 mrnrn_l into Green's function G in the manner of Appendix A. Three momenta sn-i' Ln-t, M n-t are inserted between the coordinate triples (r n-l, (In_l, Tn-I) and (r n' en' Tn)· The average energy & is now defined by 1 s-![ L 2 _ 1i2j4 6 = --I S! + ........:.::.n_----'_ 2mN t r n+V n-i M! -1i2j4 ] 1 N + . . + - ! V(qn), (53) r n+V n-} sm (In+t SID (In-t N 0 instead of the Cartesian formula (16). Green's func tion is given by G = lim (27T1i)-3Nf'Ir drn d(Jn dTn .. "'l-rJ) 1 N-t x f If dSn dLn dMn' (r~r~ sin (Jo sin (IN)-i x exp [~SN ]/(E -E), (54) and the action S N along the path in phase space by N SN =! [sn-t(rn -rn-l) + Ln-t«(Jn -(In-l) I + M n-t( Tn -Tn-I)], (55) instead of the Cartesian formulas (17) and (18). Sn is naturally associated with the projection of Pn onto the direction of qn' whereas M nand Ln correspond to the components of the angular momentum parallel and perpendicular to the z axis. The third task is to apply the procedure at the end of Sec. 2 to the energy 6 and the actiqn S.v given by (53) and (55). The equations of motion for the classical trajectory follow from (25), and are the following: N(r n -r n-l) = TSn_tjm, N«(Jn -(In-l) = TLn_tjmr nr n-l' N( Tn -Tn-I) = TM n_tjmr nr n-l sin (In sin 0n-l; (56) o { L2 ! _ 1i4 N(sn_! -sn+t) = T;-VCr n) + n- urn 2mr nr n-l L2 1 -1i2j4 + n+1f 2mrnrn+1 M~_! -1i2j4 (57) Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1989 The further calculations are greatly simplified if the coordinate system is chosen such that 0' = 00 = 177' = ON = 0" because this implies that On = t77' and Ln = o. Also, we find that all Mn are equal to some constant M. The additional terms ;'2/4, which appear in the last two Eqs. (57), can be neglected compared to the classical quantities M nand Ln. In the limit of M'T -1 0 NmrOr1 -1 M'T C + 1) 1 2Nmr1 ro r2 0 1 M'T infinite N, the remaining Eqs. (56) and (57) can be reduced to (42) and (43) with the help of (53). In order to compute the second variation (26), we notice that the subsidiary condition 61[; = 0 does not involve the variations 6Lj, 601, 6L!,···, tJ() N-1 , 6LN_j, and that the quadratic form 62S -'T62[; does not couple them to the other variations. The quadratic form (26) decays, therefore, into a sum, of which the first term has the matrix 0 0 0 0 -1 0 1 Nmr1r2 (58) 2 0 0 -1 0 0 0 in terms of the variations (59) The normalization in (59) with the help of M has been chosen such as to make the matrix (58) dimensionless. The eigenvalues and the determinant of (58) will be discussed in Appendix D. This part of the second N M'T C + 1) 0 2Nmr2 r1 r3 1 M'T Nmr2ra variation (26) can be fully understood without any difficulty. The remainder can be simplified if we integrate immediately over the variations 6rpn and then over all but one of the variations 6M n. This provides a factor (277';,)N-1 to the integral (54) and reduces the second variation (26) because all 6rpn have been eliminated and all 6M n have been replaced by a single one 6M. Thus, the second part of the quadratic form (26) can be represented by the matrix 2M'T 2M'T L 'T 0 --- 0 --- I Nmrn_1rn Nmr~ Nmr: 0 'T/Nm -1 0 0 2M'T -1 -rVll/N 'TV12/N --- I Nmr~ 2 0 0 1 'T/Nm -1 (60) 2M-r 0 'TV21/N -1 'T V22/N --- Nmr~ Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1990 MARTIN C. GUTZWILLER in terms of the variations (tJM, tJs!, tJr1, tJsi, tJr2," .). (61) The quantities Vj I are defined as 02 {S N-! M2 } f'jl=-- ~V(rn)+L . or jorl 0 ! 2mr n_!r n+! (62) The variation tJM can be eliminated from the quadratic form (60) with the help of the subsidiary condition tJ1t: = 0, i.e., x M N-! N-l tJM ~ + L Sn tJsn + L Vnbr" = 0, (63) mrnrn_l ! m 1 where f'n is the same kind of derivative as (62). The eigenvalues and the determinant of the resulting matrix are obtained in Appendix E. With the help of (27) the formula (54) for G is, therefore, reduced to G = -27Ti . eiSllifIT donf'fi dLn exp..! [(58)] ror~27TIi)N 1 ! Ii . (27TlirN-lf IT dr nf "n dSn (OM) 1 ! oE rj,'! X exp..! [(60) with (63)]. (64) Ii The derivative oMjoE at constant rj and Sl is obtained from (53) with E = E after neglecting the 1i2j4 terms and setting Ln = 0, On = !7T, as well as Mn = M. Compared to the exponentials, the variation of oMjoE with rj and SI is slow, so that it can be evalu ated for the classical trajectory and pulled out of the integral. If we insert the results of Appendixes D and E into (64), we find the formula (20) with the ampli tude given by (46). This completes the discussion of the Garrod-Feynman integral for small Ii in the case of a spherically symmetric potential. C. Polar Coordinates in Momentum Space For a spherically symmetric potential, the classical trajectory in momentum space lies again in the plan which is spanned by the initial and the final momen tum. The action T(p" pi E) depends only on the absolute values pi and p" of pi and p", and on the angle 'Y} between pi and p". The determinant (36) in the amplitude of F(P" pi E) is now given by Tp'p" Tp'~" Tp'E DT = T~ T~,p" T~,~" T~'E (65) p'2p"2 sin 'fJ TEp" TE~" TEE where 'Y}' and 'Y}" are the polar angles of pi and p" in the plane of the classical trajectory. The indices on T indicate the derivatives of T with respect to these quantities. The determinant in (65) is found from the orbit which corresponds to pi, p", 'Y}, and E in a plane. The variables which are conjugate to p and 'Y} are the projection 0 of the position vector q onto the direction of motion and the angular momentum M. The radial distance r is given by r2 = (OTjop)2 + p-2(oTjo'Y})2 = 02 + M2jp2 (66) so that the Hamilton-Jacobi equation becomes p2j2m + V[(OTjop)2 + p-2(oTjo'Y})2]! = E. (67) A more familiar-looking equation is obtained by introducing the inverse r(V) of V(r). Such an inverse exists for the typical potentials where the force of attraction increases monotonically as the distance from the center decreases. The new equation (OTjop)2 + p-2(oTjo'Y})2 -r2(E -p2j2m) = 0 (68) looks like an ordinary Hamilton-Jacobi equation of a fictitious particle with polar coordinates p and 'Y}, at zero energy, in a radial potential given by -t . r2(E -p2/2m). As p increases indefinitely, this radial potential van ishes; but since the energy is zero, the very large values of p become accessible. For p = 0, the potential has the value -tr2(E) where r(E) is the maximum distance of the real particle with the energy E < O. In analogy to the formulas (44) through (47), one has now " I LP" M dp 'f} = 'f} -'f} = p' p2[r2(E _ p2/2m) _ M2/p2]!' (69) (70) (71) (72) All remarks concerning the critical points in coordi nate space apply again to (71) with respect to momen tum space. F can be computed according to (35). But, as explained earlier, it is of great interest to arrive at this result directly as the limit of small Ii of the Garrod-Feynman integral (32) for a spherically symmetric potential. Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1991 The first task is again to rewrite (32) in polar coordi nates. The detailed correspondence between coordinate and momentum space is maintained by the coordinate transformation p! = Pn sin 'n cos 'YIn' p! = Pn sin 'n sin 'YIn , p! = Pn cos 'n' for half-integer n, and (73) I ( -. r + Ln -r) -M n '-qn = an SID ., -:: cos., cOS'YI - _ . SID 'Yj, P P . SID' qn = an SID ., + -:: cos., SID 'YI + _ . COS 'YI, 2 ( --:--y Ln -r) . _ M n - P P' SID' 3 -Y Ln-.-y qn = an cos., --:-SID." (74) P for integer n, where p = (Pn-lPn+l)!' cos, = (cos 'n-l cos 'n+l)!' sin, = (sin 'n-! sin 'n+l)!' and ij = H'YIn-! + l1n+!)' The integral (32) is then trans formed into F = lim (27T1i)-3N-3 N-+"" f N fN -l x IJ dan dLn dMn f} dPn d'YIn d'n X (p,2p,,2 sin " sin ''')-! exp [-(i/Ii)TN]/(E -B), (75) where the action TN along the path in phase space is given by N TN = 2 [aiPn+l -Pn-l) + Li'n+l -'n-l) o + Mn('YIn+! -'YIn-!)] + [terms at least quadratic in (Pn+! -Pn-!), an+l -'n-!), ('YIn+! -'YIn-!)], (76) and the average energy e can be written as 1 N-! 2 1 N e = -2 Pn + -2 V(/qnD (77) N ! 2m N 0 with /qn/2 = a; + L! + .M!. Pn-lPn+! Pn-!Pn+l SID 'n-! SID 'n+l -(a~ + L! )<1-cos('n+l- 'n-!»' Pn-!Pn+! (78) One.would like to get rid of the last terms in (76) and (78). Obviously, they can not simply be neglected, since even in the expression (54) for G additional terms, -1i2/8mr n+lr n-! and -1i2/8mr n+lr n-! sin On+! sin 0n-l' had to be inserted into e. The arguments of the previ ous section are not applicable because F satisfies an integral equation, rather than a Fokker-Planck-like Schrodinger equation. More than just the zero, first, and second moment of the propagator for small times are needed in momentum space. It is not clear whether simple formulas like (53), (54), (55) can be found for F in polar coordinates as N --00. As Ii goes to zero, however, the variations in the coordinate differences (Pn+! -Pn-!), (an -an_I), etc., become small. It is sufficient to keep only the first parts of (76) and (78). The correspondence be tween the formulas (53) through (55) for G and the formulas (75) through (78) for F is complete again. VCr n) in (53) is replaced by p!/2m in (77), and the kinetic energy term, [s! + .. ']/2m, in (53) is replaced by the potential energy, V([a! + ... ]l), in (77). In order to apply the arguments of the previous section to the discussion of (75) in the limit of vanishing Ii, they have to be sufficiently general so as to include a kinetic energy which is not simply the square of the momentum. The Appendices D and E treat this gen eral case, and are, therefore, immediately applicable to the formulas (75) through (78), after the prelimi nary steps corresponding to the formulas (56) through (64) for G have been completed. In this manner we are ultimately again lead to the expression (35) for F with the expression (71) for DT and the phase jumps at focal points which were dis cussed earlier. It is evident from the arguments in the Appendices D and E that the rotational invariance has been used extensively, so that the limit of small Ii, in the Garrod-Feynman integral has been estab lished only for potentials of spherical symmetry. IV. PHASE-INTEGRAL APPROXIMATION FOR THE COULOMB PROBLEM IN MOMENTUM SPACE A. Classical Kepler Orbits in Momentum Space The orbits in momentum space can be obtained in a straightforward manner if one computes the integral (69) with the Coulomb potential VCr) = -e2/r. (79) It seems, however, more appealing to describe these orbits in a geometric manner, particularly because they turn out to be so simple. Starting from the trajectory in coordinate space r = (M2/me2)(1 + £ cos <p)-l, (80) £ = [1 + 2M2E/me4]!, (81) Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1992 MARTIN C. GUTZWILLER one gets immediately the momenta dql dql dcp me2 . PI = m -= m - . -= - - sm cp, dt dcp dt M dq2 dq2 dcp me2 P2 = m -= m - . -= -(E: + cos cp). dt dcp dt M (82) (The Cartesian components of P and q are called (PI' P2) and (ql' q2) in this section.] If cp is eliminated between the last two equations, we get the equation of a circle in momentum space P~ + [P2 -(me2/M)E:]2 = (me2/M)2, (83) with the radius me2/ M and the center at a distance E:me2/M from the origin. All orbits in momentum space intersect a circle of radius ( -2mE)! around the origin at diametrically opposite points, as can be recognized from the solution PI = ± ( -2mE)! and P2 = 0 of (83). Conversely, for any circle in momen tum space which intersects the circle of radius (-2mE)! around the origin, we can find a value M between 0 and (-me4/2E)! such that its radius is given by me2/ M and the distance of its center from the origin by (2mE + m2e4/M2)!. Let us now find the locus of the centers 9f all such circles which pass through a given point, say (p,O), for a given energy E < O. Suppose that one such circle goes through the point [( -2mE)! cos oc, (-2mE)! sin oc]. Its center (PI' P2) lies, therefore, on the bisectrix given by (PI -p)2 + p~ = (PI -(-2mE)! cos OC)2 + (P2 -(-2mE)! sin OC)2, as well as on the straight line through the origin and perpendicular to the direction (cos oc, sin oc), i.e., PI cos oc + P2 sin oc = O. If we eliminate oc from these two equations, we find PI = Hp + (2mE/p)]. (84) The locus of the centers of all orbits through (p, 0) is the straight line perpendicular to (p, 0) at a distance Hp + (2mE/p)] from the origin. For p < (-2mE)! the quantity (84) is negative so that the origin lies on the same side of the locus as the point (p, 0). It is now easy to find the center of the orbit in momentum space which passes through two given momenta, p' and pH. We have only to intersect the two loci for the centers of the circles through p' and through pH. Since these loci are straight lines, there is exactly one intersection. We find, therefore, the important statement that: For given E < 0 there is exactly one classical orbit in momentum space which connects a given initial momentum p' with a given final momentum p". The exception to this statement arises in the special case where p' and p" are "opposite" each other with respect to the circle of radius ( -2mE)! around the origin, i.e., p" = 2mEp'//p'/2. In that case all orbits through p' go also through p". This last configuration is of particular interest because it turns out that all the classical trajectories starting from a momentum p' intersect one another at the "opposite" momentum, and nowhere else. Again this situation is much simpler than for the Coulomb potential in coordinate space where all the classical trajectories of a given energy E < 0 starting from a position q' touch one another along a caustic. In momentum space this caustic has seemingly contracted into a point. The action function (34) can be obtained from (80) and (82) by writing f"'u d d T = -(r cos cp --.El + r sin cp J!1) dcp cP' dcp dcp = i~ul +LEdc:s cp = (-;;)!(U" -u'), (85) where u is the "eccentric anomaly" which is given by u = 2 arctan [(1 -E)/(l + E)]! tan cp/2. (86) The "true anomaly" cp is measured from the point of closest approach, the perihelion. B. Phase Integral Approximation In order to Hnd explicit expressions for the approx imate Green's function F(p" P' E) as given by (35), one has to find a relation between the polar angle rJ in momentum space and the angular momentum M which occurs in the formulas (70) and (71). In terms of the quantity P = ![ Ipl _ (-2mE)!] 2 (-2mE)! Ipl = ![ p -(-2mE)l] , (87) 2 (-2mE)! p one finds after some obvious algebra that M = (me4/2E)! sin rJ[P"2 -2P'P" cos rJ + p'2 + sin% rJ]-!. (88) The determinant DT in (35) is then obtained from (71) as me8 D ---------------~--------~---- T -_2Ep,2p,,2( -E + p'2/2m)( -E + p,,2/2m) X (p,,2 -2P' p" cos rJ + p,2 + sin2 rJ). (89) Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1993 The denominator vanishes only if P" ->-P' at the same time as 'Y) ->-0, or if P" ->--P' at the same time as 'Y) ->-TT. The latter case corresponds to p" being "opposite" to p'. For the action function T, we find from (86) that T= (_ ~;)t [P"2 -2P' P" cos 'Y) + p,2 + sin2 'Y)]! x arctan "---------"--------'-=- (P' P" + cos 'Y) (90) The arctan as well as the root in its argument are uniquely defined in the range (0, TT) for 0 < p' < 00, and 0 < 'Y) < TT. As anyone of the three independent variables in (90) reaches the end of its domain, there is always a natural definition for T to preserve its continuity. It suffices to construct the corresponding classical trajectory which does not go through the "opposite" momentum in order to find the correct value of T. The action T from a momentum p' to its opposite is always given by TT( -me4j2E)!. As we follow the trajectory through the opposite momentum to a final momentum p", the total action accumulated is given by T= (-~~)! { [P,,2 _ 2P'P" cos 'Y) + p,2 + sin2 'Y)]!} x 2TT -arctan ; P'P" + cos'Y) (91) 'Y) is again the angle between p' and p" measured as in the case of (90) and restricted to the interval o < 'Y) < TT. If we follow the orbit any further, the total action can be obtained from (90) or (91) by adding as many times 2TT( -me4j2E)! as full orbits have been completed. In order to apply the formula (35), we have to determine the extra phase factors which come from the critical points along the classical trajectory. For the Kepler orbits in momentum space the two kinds of critical points discussed in Sec. 4 coincide, since all trajectories of given energy E leaving a given momen tum p' meet again at the opposite momentum what ever the direction or the absolute value of their angular momentum. A factor exp ( -iTT) = -1 is picked up for ea~h traversal of such a doubly critical point. The same factor-enters into (35) when the trajectory goes through the initial point p' again. Since both (90) and (91) are expressed in terms of the angle 'Y) which is defined by p' • p" = p' p" cos 'Y), it seems appropriate to use this scalar product in (89), (90), and (91) rather than 17. It should be noticed that the amplitude (DT)! stays the same for all the trajectories which go from p' to p", because according to (36) only the derivatives of T with respect to p' and p" are needed, whereas, the actions along different trajectories from p' to p" differ .only by multiples of 2TT( -me'j2E)t and possibly a sign. The summation over all trajectories reduces to the geometric series of the powers of exp [2TTi( -me4j2EIi2)!]. After some rearranging we can finally write for F(P" p' E) the expression -TTIi2(p,2 _ 2mE)(p"2 -2mE) Ip" 0-p'l [(p,2 _ 2mE)(p"2 _ 2mE) + 2mE(p" _ p')2]! . sm 2 --arctan sm TT --. . { (-me4)! [(p,2 -2mE)(p,,2 -2mE) + 2mE(p" -p'}2J!}/' (-me4)! 2Eli2 -2mE(p" -p')2 2Eli2 (92) The arctan varies between 0, when p' and p" are opposite each other, and iTT, when p' = p". Therefore, the amplitude of (92) becomes infinite as p" approaches p', but it stays finite as p" goes through the opposite momentum of p'. The only singularity in (92) is the one which was described by the formula (39). The above results for the approximate Green's function of the Coulomb problem in momentum space is so simple because the caustics have shrunk to points. The corresponding function G(q" q' E) in coordinate space is expected to be more complicated, although it will have the same denominator. In this connection, the three-dimensional harmonic oscil lator is of interest, because it combines the features of the Coulomb problem in momentum as well as in coordinate space. To each initial point, q' or p' corre sponds an "opposite" point, -q' or -p', where all trajectories through q' or p' meet again. But in between, the trajectories belonging to one plane touch one another along an envelope. There will be effec tively a total of six critical points for each full oscillator orbit, whereas there are only four critical points for each Kepler orbit, whether in momentum or in co ordinate space. The quantum condition of Bohr and Sommerfeld requires, therefore, half-integer quantum numbers for the three-dimensional oscillator (with a minimum of I); but for the Coulomb problem, the quantum numbers are integer, as shown above. Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1994 MARTIN C. GUTZWILLER C. Comparison with the Exact Green's Function The main purpose for studying the phase-integral approximation in momentum space was to find wave functions for bound states. The formula (92) is, therefore, of greatest interest for negative values of the energy. We shall put -2mE = y2 with a real y > 0, whenever the Green's function is examined along the negative E axis. The expression under the root in (92) can then be written as (p'2 + y2)(p"2 + y2) _ y2(p" _ p')2 = p'2p"2 + y4 + 2y2p' p" cos 'Yj, which vanishes if and only if p' p" = y2 and cos 'Yj = -1. The only singularities in (92) along the negative real E axis arise from the zeros of the denominator. The corresponding poles are at (93) where n is a positive integer. The Bohr formula is obtained without any gimmickry. It would have resulted with similar ease from G(q" q' E). The residues at the poles (93) can be most conven iently expressed in terms of the Bohr momentum Yn = me2/nli and the angular variable [(p'2 + y2)(p"2 + y2) _ y2(p" -p')2J! {J = 2 arctan . y2(p" _ p')2 The residue at the pole (93) is found to be (_1)n+1(8n/7T2)y!(p,2 + y~)-2(p"2 + y~)-2 (94) X (sin n{Jn/sin (In). (95) The quotient sin n{3/sin {3 is a rational function of the momenta p' and p", as is seen immediately if sin n{3 is expanded in terms of sin {J and cos {3 and the elementary formulas sin (2 arctan IX) = 21X/(1 + 1X2), cos (2 arctan IX) = (1 -1X2)/(1 + 1X2) (96) are used together with (94). As an example let us put n = 1 in (95), which gives (8/7T2)y~(p'2 + y~)-2(p"2 + y~)-2. (97) According to Bethe and Salpeter, 20 this expression is just the product of the two normalized Is functions of the hydrogen spectrum, with variable p'2 and p"2, respectively. Similarly, we obtain for n = 2 from (95) and (96) (32/ 7T~y~(p'2 + y~r3(p"2 + y~)-3 X {(p,2 _ y~)(p"2 -y~) + 4y~(p', p")}. (98) 10 H. A. Bethe and E. E. Salpeter, "Quantum Mechanics of One and Two-electron Systems," in Encyclopedia of PhYSics (Springer Verlag, Berlin, 19S7), p. 12S. If we expand the scalar product (p', p") = p~p; + p~p~ + p~p; , we are left with four terms in the braces. Together with the factors in front, each term is again a product of normalized hydrogen wavefunctions, the first term in the braces providing the 2s function and the last three terms the three 2p functions. Instead of comparing further the residues (95) with the known hydrogen wavefunctions in momentum space, it is more efficient to compare directly the approximate Green's function (92) with the exact one. The latter has been obtained in closed form by Okub021 and has recently been discussed by other authors.22 Along the negative real E axis we can write F(" , E) = _ 2m!5(p" -p') P P ,2 + 2 P Y 4m2e2 27T21i(p'2 + y2)(p"2 + y2)(p" _ p,)2 m e y drrmes/1!Y 8 3 4 l'" -7T21i2(p'2 + y2)(p"2 + y2) 1 "'''' X [(' _ 1)2(p'2 + y2j(p"2 + y2) + 4'y2(p" _ p,)2]-I. (99) The first two terms are obtained from the integral equation (29) by a formal expansion of F in powers of the potential. The last term is formally of the same power in Ii as the approximation (35) for the Green's function which led us earlier to consider the expression (40) as being possibly superior to (35), especially for smallip" -p'l. The last term in (99) can be regarded as a Mellin transform with me2/liy as the new variable instead of ,. Since the function of , differs from zero only in the interval from 1 to 00 where it can also be expressed in powers of 11', the integrltl over' presents no great difficulties. Thus, we can write the last term in (99) for 1 > me2/liy as m e y drrme2/IIY 8 34 f'" 7T21i2(p'2 + y2)2(p"2 + y2)2 1 "'''' "'(_l)n+1 sinn{3 x~ '--n=1 ,n+1 sin {J 8m3e4y =----------~-------7T21i2(p'2 + y2)2(p"2 + y2)2 '" (_1)n+1 sin n{J X ~ '--. (100) n=1(me2/liy) -n sin {3 The poles of (99) in the left-hand part of the complex E plane are correctly given by (100), but the expan sion converges only on the negative real E axis and 21 S. Okubo and D. Feldman, Phys. Rev. 117,292 (1960). 22 L. Hostler, J. Math. Phys. 5, 123S (1964); J. J. Schwinger, J. Math. Phys. 5, 1606 (1964). Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1995 is, therefore, useless as a representation of the last term in (99). Indeed, the angle f3 becomes complex for values of E off the negative real axis, so that sin nf3 becomes exponentially larger for increasing n. The poles of (99) and their residues are shown by (100) to be the same as (93) and (95). The phase integral approximation is thus shown to yield exactly all the bound states of the hydrogen atom. Although we have thereby achieved the main goal of this paper, it may be of interest to discuss just a few points which are concerned with the approxi mate Green's function F for positive values of the energy. If we insert -2mE = y2e-iro into (92) and let w grow from 0 to TT, we get the analytic continua tion of (92) from the negative to the positive real axis through the upper half-plane. In order to go from some point E' on the negative E axis to a point E" on the positive E axis, we can either first adjust y for w = 0 and then let w grow to TT, or we can first let w go from 0 to TT and then adjust y2. These two procedures give the same purely imaginary result, whatever the vectors p' and p". A well-defined dis continuity across the positive real axis is obtained between the results ofthe analytic continuation through the upper and through the low'er half of the complex E plane. The formula (92) for F has all the attributes of a well-behaved Green's function, which is all the more surprising because, for a given positive energy, certain parts of momentum space are classically inaccessible. It would be interesting to compare the discontinu ities across the positive E axis for the phase-integral approximation F with those of the exact Green's function (99). This seems almost more difficult than the comparison of the bound states, because the latter form a countable set, whereas the former form a continuum which depends on the three vari ables p', p", and cos ",. This problem is, therefore, not examined at this time, although it may be of interest for the discussion of scattering problems and virtual bound states. The original goal still seems of greater importance, i.e., the extension of the phase integral method to bound states in general spherically symmetric potentials and eventually to simple molec ular potentials such as in the diatomic molecules. The successful treatment of the hydrogen atom which was presented in this in,vestigation, constitutes a crucial first step in this direction. ACKNOWLEDGMENTS The author would like to express his gratitude to Dr. W. Schlup for many interesting discussions, and to Dr. F. Odeh and Dr. B. Weiss for enlightening remarks about the existence and the meaning of Morse theory. APPENDIX A The starting point is the identity (~)i exp [im(q" -q')2] 2TTilit 2lit = _1_ fd3p exp..i [P(q" -q') -t It], (2TTIi)3 Ii 2m (At) which can be used in the Feynman integral (5) between any two consecutive points q i and q H1 , if the Lagrangian has the classical form L = mcr/2 -V(q). If one calls P Hi the momentum between q i and q i+l , the expression (5) becomes K = lim (2TTIi)-3Nj IT d3qn jn d3Pn N~oo 1 l X exp!.. [SN -(t" -t')E], (A2) Ii with the abbreviations SN and E as defined by (16) and (18). The time intervals t i+l -t i have all been chosen equal in order to simplify the definition of E. The Fourier transform (13) immediately yields the expression (17) for G(q" q' E), if the integration over t from 0 to 00 is interpreted as a Laplace integral where E has a small positive imaginary part. APPENDIX B If (8) is inserted into (13) the exponent becomes R(q"t, q'O) + Et apart from the factor i/Ii. For given values of q", q', and E this exponent is stationary for to defined by the equation , -oR/at = E. (BI) The exponent is now expanded in powers of t -to, so that R(q" t, q'O) + Et = R(q" to, q'O) + Eto + Ht -to)2(02R/ot2) lto + .. '. (B2) The factor (DR)' in (8) is assumed to vary slowly and can, therefore, be evaluated at t = to without any further corrections. The integration over t is elemen tary and gives G(q" q' E) = -(1/2TTIi2)(DR)!(~2; IJ-! x exp [is(q'' q E)/ Ii], (B3) S(q" q' E) = R(q" to, q'O) + Eto, (B4) where to is to be eliminated with the help of (81). The second derivatives of R have to be written in Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1996 MARTIN C. GUTZWILLER terms of the derivatives of S. One finds that etc. One can write, therefore, the following sequence of equations: ~_~_~. 02S /02S oq"oq' -oq"oq' oq"oE oq'oE oE2' (BS) Id3PV(P" p) . B(p p' E) exp [_ ~ T(p p' E)] o2R = _ (02S)-1 ot2 . OE2 ' (B6) which leads immediately to the wave amplitude (Ds)l in (20). The 3 X 3 determinant lo2Sjoq"oq'l vanishes, be cause the equation H(oSjoq" , q") = E can be differ entiated with respect to q', which gives three linear homogeneous equations in the quantities oHjop" with the matrix o2Sj oq' oq". An interesting special case of this remark arises in one dimension, where one has the well-known formula S(q" q' E) = f"[2m(E -V(q»]l dq = S(q" E) -Seq' E) + const. (B7) The expression (20) becomes, therefore, after ad justing the normalization _1_ ! ei'const/lI . ( 02S )1 21T1i classical paths oEoq" X eiS"/lI( a2s )1 e-iS'/lI. (B8) oEoq' The constant in (B7) which reappears in the exponent of (B8) is different for each classical path, depending on the number of cycles in the path. If these details are properly considered, one arrives at the formulas of Kohn and Sham,12 APPENDIX C The main problem in obtaining the expansion (37) for the Hamiltonian in momentum space comes from the potential energy. The discussion is, therefore, limited to evaluating the potential term in (30), if the trial solution B(P" p' E) exp [-iT(P" p' E)jli] is in serted. We shall present first a short and rather formal argument, and then attempt to give a more rigorous, although lengthy, proof. The inverse Fourier transform of (31) is given by V(q) = I V(p" p') exp [~(P" -P,)q] d3p", (C1) and the derivatives of V(q) are written as av = 2 IV(p" p') . (p" -p') oq Ii X exp [~ (p" -P,)q] d3p", (C2) = B(p" p' E) exp [ -~ T(p" p' E)] (C3) d3pV(p" p) P P exp~ [_ T(p P 'E) + T(p"p 'E)] I B( 'E) . B(p" p' E) Ii = B(p") exp [~ T(p") ] f d3pV(p" p) X {B(P) exp.!. [_ T(p) + T(p") + (p -p") OT]} B(p") Ii op" X exp [~ (p" -P)q,,} where q" = +oTjop" and the argument p' as well as E has not been written anymore in the last line. The next step is the questionable one, since it consists in simply expanding the terms inside {} in powers of p -p". Thus, one obtains { } _ 1 + _1_ ( _ ") oB -B(p") p Pop" i ( ")( ") o2T + 21i p -P P -P op"op" + ... , (C4) where the neglected terms would all contribute to the order li2 and higher. The expansion (37) for the Hamiltonian follows immediately with the help of (C2) and if we assume that V(P" p') depends only on the differences p" -p'. If the expansion (C4) is carried further, higher terms in (37) are obtained without difficulty. A more careful procedure consists in applying Parseval's theorem to the integral in the second line of (C3). Apart from the factor R(P" p' E) exp [( -ijli)T(p" p' E)], the potential-energy term in the Hamiltonian becomes Id3 V() 1 Id3 B(p) qq' (21T1i)3 P B(p") i X exp/i [-T(p) + T(p") -(p -p")q]. (C5) The factor which multiplies V(q) can be regarded as a density function a(q) which weighs the various contributions of the potential V(q). One finds, indeed, that S a(q) d3q = 1 whatever R(p) and T(P). It is, therefore, reasonable to study a(q) , its main peak and its spread, particularly in the limit of small Ii. Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1997 Jones and Kline23 have investigated the asymptotic expansion of multiple integrals by the method of stationary phase. The method used in this Appendix differs somewhat and treats only interior critical points, although boundary points would have to be examined in a more complete theory. Also, we assume analyticity for the functions B(P) and T(P). Given q, the exponent in the integrand of (C5) becomes stationary for a value p = p which is obtained from the equation aTjap = q. (C6) If we assume p to be a regular point of T, a real linear transformation (C7) can be found, such that we can write the expansion T{p) -T{p) + (p -p)q = ! 2, £ipi2 + t 2,fJ;zmpip!pm i 11m +"2\ 2, fJ;lmnpip!pmpn + .. '. (C8) ilmn The coefficients fJ are symmetric in all their indices, and £; = + I or -1. By purely algebraic manipula tions, we can define coefficients Y for a further expansion pi = pi + 2, Yilmp!pm + 2, Yilmnplpmpn + ... , 1m 1m" (C9) such that we have to all orders in p the equality T{p) -T(p) + (p -p)q = ! 2, £iP,2. (CIO) i If we require that the y's are symmetric in all their indices, they are uniquely determined in terms of the fJ's, namely £iY:llm = -lfJ:J!m, (CIl) (CI2) These relations increase rapidly in complexity. The variable of integration in (C5) is now changed from p to p. The Jacobian a(p)ja(p) as well as B(P) can be expressed in terms of IX, fJ, and Y, and then expanded in powers of p. The integration over p is 28 D. S. Jones and M. J. Kline, J. Math. & Phys. 37, 1 (1958). trivial because of (ClO) and yields after some obvious manipulations 1 11 eiEk1T/4 det IIX (p-) I (27TIi)i B{p") k il X exp J. [-T(p) + T(p") -{p -p")q] Ii {_iii aB . B{p) --2 ~£I£m a _, IXilfJ!mm JIm p iii a2B . _ [ 1 + -2 ~£m a-:la-! IXimIXlm + zIiB(p) - -2,£i£lfJiill ,1m p p Y il + l4 .2,£j£Z£m(3fJiilfJlmm + 2fJ:llmfJ;zm)] + ... }. '1m (C13) The whole expression is to be considered as a distri bution function a(q) , with p related to q through (C6). The result (C13) is now multiplied by V(q) and integrated over q, again by the stationary phase method with T(p) -T(P") + (p -p")q as the rapidly varying phase. This phase is stationary at p = p" or q = aT/ap", provided the determinant of the second derivatives of T with respect to p does not vanish at p = p". This requirement makes it also possible to use p as variable of integration rather than q. There fore, we can also multiply (C13) with V( +aTjap)' det I a2Tjapapi , replace q in the exponent by aT/ap, and integrate over p. The phase T(P) -T(P") -(p -p") . aTj ap is now treated with respect to p -p" exactly as the phase T(P) -T(P) -(p -p)' aTjap was treated with re spect to p -p. There are some minor, though obvious, modifications because the latter phase is not the same function of p -p as the former of p -p". All the slowly varying quantities in (C13) have to be expanded in powers of p -p", although this is not necessary for the terms which are already of order Ii. Finally, we can express the coefficients fJ in terms of the derivatives of T with respect to p, and use such relations as a2T 2, IXimIXln ~ = £mbmn, il ap ap (C14) in order to express everything in term~ of T and its derivatives. All the complicated terms in (C13) are cancelled out, and one is left with the relatively simple expansion (37). Whereas the derivation of (C13) can be made sufficiently rigorous, provided we include a discussion of the boundary points, the further integration over Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions1998 MARTIN C. GUTZWILLER q or p using (Cl3) may be much harder to justify in view of its complicated structure. APPENDIX D In order to find the eigenvalues and the determinant of the matrix (58), we first add a term -AT/N in the diagonal elements. Let Un be the determinant which results from (58) after all rows and columns beyond n have been eliminated. The following recursion formulas are then obtained: Un =!:...[ M - AJUn-l- Un-I' N mrn+l'n-l (Dl) Un = - + - A U n-l -U n-l , T[ M M J N 2mr n+lr n 2mr nr n-l (D2) for half-integer and for integer n, respectively. The initial values are (D4) An alternating sign has to be 'eliminated before going to the limit of large N. Therefore, we define Un = (_l)n-!U n for half-integer n, Un = (_l)n-IUn for integer n. (D5) In the limit of large N with T remaining constant, the recursion formulas (Dl) and (D2) become dW/dt = -[(M/mr2) -A]U, (D6) dU/dt = [(M/mr2) -A] W, (D7) where W = Un for half-integer n, and U = Un for integer n. The initial conditions (D3) and (D4) reduce to W(O) = 0, U(O) = -1. (D8) The discrete parameter n has been replaced by the continuous parameter t = nT/N. The consecutive values of the radial distance r n are assumed to lie very close to corresponding values ret) along the classical orbit. Because of (42), the solution of the initial-value problem (D6), (D7), and (D8) can be written im mediately in terms of the polar angle fP along the orbit, W = sin (fP -At), U = -cos (fP -At). (D9) If N is sufficiently large, we find, therefore, the following approximate value for the determinant of (58), det 1(58)1 = (_1)N-l sin [(fP" -fP/) -A(t" -t')]. (DlO) The eigenvalues are, therefore, given by A = (V7T -fP" + fP/)/(t" -t'), where '/I is any integer, positive or negative, larger than -N and smaller than N. APPENDIX E The matrix (58) for the variations bLn and bOn was easy to discuss because its determinant could be evaluated explicitly for large N even after including a term -AT/N in the diagonal. In the case of the matrix (60) with the subsidiary condition (63), such a direct procedure can again be devised; but it is important not to specialize the particular form of the Hamiltonian at an early stage, because the general features might easily be lost in the arithmetic. Also, the treatment of the phase-integral approximation in momentum space is equivalent to the treatment in coordinate space, only if the kinetic energy is allowed to be a more general function of momentum than the usual p2/2m. Such a generalization would auto matically include a relativistic Hamiltonian. We shall assume, therefore, that the kinetic energy is an arbitrary function D(p) of p = Ipl and the potential energy an arbitrary function VCr) of r = Iql, so that H(pq) = D(p) + VCr). In terms of the momenta Sn' Ln, and M n' as well as the coordinates rn, On' fPn' we have in the limit Ii -+ 0, P2n = in + L~ + M! . 0 ·0' r n_!r n+l r n_!r n+! sm n-! sm n-l (El) 1 N-! 1 N e = -I D(Pn) + -I VCr n) N ! N 0 (E2) instead of (53). The formulas (77) and (78) are ob tained from (El) and (E2) by the formal replacements D +---+ V, p +---+ r, S +---+ (1, 0 +---+" fP +---+ 1], provided we neglect the last term in (78). A discussion of the second variation of S.v as given by (55) with a constant e as given by (E2) includes, therefore, a discussion of the second variation of TN as given by (76) with out the quadratic terms with a constant e as given by (77). Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissionsBOUND STATES OF AN ATOM 1999 The equations of motion (56) and (57) as well as the matrix (58) are hardly affected by the new kinetic energy, and can be treated exactly as before. The matrix of the second variations in (~M, ~s!, ~rl' ~s!, ~r 2, ••• ) is most easily written out in terms of the ~JeMM ~JeM! ~Je~l1 function N-! N Je = 2 D[s! + M2/r n_!r n_t]t + 2 V(r n). (E3) ! 0 Let lower indices on Je indicate the corresponding derivatives, half-integers for Sn and integers for r n. Instead of (60), one now has the matrix ~JeM! ~JeM2 ~JeMt ~JeH -(1 -~Jeh) 0 0 ~JeMl -(1 -~Jelt) ~Jell (1 + ~Jel!) ~Je12 1 ~JeM! 0 (1 + ~Jeh) ~JeH -(1 -~Jeh) (E4) 2 ~JeM2 0 ~Je21 where ~ = TIN. The subsidiary condition (63) becomes N-! N-l ~M . Je M + 2 Jen~Sn + 2 Jen~r n = o. (E5) t 1 A comparison with (60) shows that a number of new off-diagonal terms arise from the more general energy (E2). The 2N X 2N matrix (E4) is reduced to a 2N -:-1 x 2N -1 matrix in (~St, ~rl' ~s!, ~r2' ... , ~SN_!) by eliminating the variation ~M with the help of (E5). The second variation which goes into the exponent of (64) has, therefore (apart from a factor -t), the matrix C;j + ~JeMM[(ai + bi)(oj + bj) -bibj], (E6) where C ij is the matrix (E4) without the first row and the first column. The quantities aj and bj are given by the derivatives of Je, aj = Jej/JeM, bj = -JeMj/Je MM. (E7) The signature of the matrix (E6), i.e., the difference between the number of positive and the number of negative eigenvalues, as well as its determinant, have to be found. Let r n be the determinant which is obtained from (E6) after all rows and columns beyond the index n have been deleted. According to a theorem from linear algebra, cf., Bocher, 24 the signature equals the sum over sign (r n-tr n) from 11 = t to 11 = N -t with ro = 1. The determinant r n can be written in terms of the determinants Ck!, which are obtained from the matrix ciJ by deleting all rows and columns before k and beyond I. In terms of Ck! = (-1 )k(2k-1)+1(2!-1) Ck!, one 2& M. B6cher, Introduction to Higher Algebra (The Macmillan Company, New York, 1907), p. 147. -(1 -~Je2!) ~Je22 finds after some straightforward algebra that (_l)n(2n- Or n = Cln -2~JeMM X 2 (aioj + aibj + b;aj)Ch-iCi+ln i<j<n { } = aibjbka! + biajbka! + biajakb! + aibjakbl -2aiajbkbl -2b;bjakal• Certain terms of equal indices have been neglected because their contribution is only of order l/N or smaller. Since the elements of (cij) differ from zero only if they are close to the diagonal, one can easily derive recursion formulas for Ck!. In the limit of large N, one obtains ordinary linear differential equations in the parameter t = m/ N = n~, provided the sequence of momenta Sn and distances r n approximates the classical trajectory s(t), r(t) in phase space. With the Hamiltonian H(sr) = D(S2 + M2/r2)! + V(r), (E9) this trajectory satisfies the equations of motion ds/dt = -oH/or, dr/dt = oH/os. (EIO) The "initial" values are r(O) = r', r(T) = rH, and the angular momentum M is determined such that lTd oH iTd MDp " , t-= t--=g; -g;, o aM 0 pr2 (Ell) with p = (S2 + M2/r2)!; e.g., let Ckl = Wif I is half integer, and Ckl = U if I is integer. In terms of t" = Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions2000 MARTIN C. GUTZWILLER IT/N, the recursion formulas with respect to I become dW = a2H U + a2H W, dt" as2 asar dU a2H a2H -= - -W --U. (EI2) dt" ar2 asar These are the Jacobi equations for the so-called assessory problem; cf. Caratheodory.25 Solutions to (El2) can be constructed if a family of solutions s(t, ft) and r(t, ft) to (ElO) is known which depends on some parameter /1. One finds immediately that U = as/aftlt~t", W = ar/aftlt~t". (E13) The initial conditions for U and W depend on k. One finds for half-integer k that U(t') = I and W(t') = 0, whereas U(t') = 0 and W(t') = -I forinteger k, with t' = k'T/N. Since the angular momentum M is con sidered a constant in (ElO) and (EI2), the only parameter left to yield a family of solutions is the energy E. The function U and W can be written as integrals over the classical trajectory in the following manner: in the case of half-integer k, the second Eq. (ElO) is integrated by writing t" -t' =fr " dr: (E14) r' H. where s in H. is assumed to be eliminated with the help of H(sr) = E. The above equation is then differ entiated with respect to E at constant t", so that one obtains the relation 0-ar" I . -.L _f.r" dr H as I (E15) -aE t" H~ r' H; '8 aE r' where ar' /aE was assumed to vanish in accordance with the initial condition W(t') = O. The derivative as/aE follows from H(sr) = E, so that ar" I = H';fr"dr HB8 = -H'; t"dt(1-). (E16) aEt" r' H! Jt' Hs. The lower indices always designate partial derivatives, whereas the prime or double prime indicate the time at which the quantity is to be evaluated. If the expres sion (E16) is inserted for W into the first Eq. (EI2), the corresponding function U is obtained. After adjusting the result to the initial condition U(t') = I, one finds that U = H~ + H'H"It"(J.-) H" • r , H ' B t s. s W = -H~H'; r:"dt(-.t.) Jt H. s (E17) 1& C. CarathCodory. Variationsrechnung und partielle Differential gleichungen erster Ordnung (B. G. Teubner. Leipzig. 1935). p. 260. for half-integer k. Similarly, it follows that W = -H; + H'H" t"dt(~) H; r 8JI' Hr / U = -H;H; r:"dt(l..) JI Hr r (EIS) for integer k. The integrals in (E17) diverge when t passes a classical turning point where dr/dt = H. = O. But a close examination of U and W as t" approaches such a time, shows that these functions approach well defined, finite values and can be continued in a natural fashion without discontinuities. The same is true for (ElS). With the help of identity it" I It" I I I dt(-) = dt(-) +---, (EI9) t' Hr r t' H. • H~H; H;H; the integrals (E IS) can be written like the integrals (E 17), and vice versa. Also, this identity shows how to avoid a divergent integral in the neighborhood of a turning point. The calculation of r n from (ES) presents no difficulties in the limit of large N. The coefficients aj and bi are written for half integer j as ajro.-J HBdtl iNH1I1dt, bj ro.-J -HM• dt I iN H1I1M dt, and for integer j as airo.-J Hrdtl iNHllfdt, bjro.-J -HMrdtl iNH 1J,IMdt. (E20) (E2l) It seems advisable to obtain first the sums of the type 1 (i <j)C!i_!a i, 1 (i <j)ajCi+!n, etc., with the help of (E 17) and (E IS), then the spms of the type 1 (i < j < k < i)ajCi+!k-!ak, etc., and finally the complete sums as they occur in (ES). The various successive integrations can always be combined and simplified, although the procedure is very tedious and one suspects that there must be some shortcuts to avoid these lengthy computations. The result is (-l}-1V-1rN = H~H;,,{fVHMM dt -iN (:~). dt} x [(t N -to) I iN H.l1 dt r. (E22) The integrals in braces can be shown to equal a(IPN -IPo)/aM at constant ro and rN, whereas the ratio (tN -to)/J~ Hlll dt is equal to aM/aE, i.e., the change in angular momentum which is necessary to accommodate a change in average energy while keep ing the same orbit s(t) and r(t). The integral (64) is combined with (D 1 0) and (E22) to give (20) with (46). Downloaded 16 Jun 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions
1.1709856.pdf	Atomic Mating of Germanium Surfaces D. Haneman, W. D. Roots, and J. T. P. Grant Citation: Journal of Applied Physics 38, 2203 (1967); doi: 10.1063/1.1709856 View online: http://dx.doi.org/10.1063/1.1709856 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/38/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Atomic imaging and modeling of H2O2(g) surface passivation, functionalization, and atomic layer deposition nucleation on the Ge(100) surface J. Chem. Phys. 140, 204708 (2014); 10.1063/1.4878496 Atomic imaging of nucleation of trimethylaluminum on clean and H2O functionalized Ge(100) surfaces J. Chem. Phys. 135, 054705 (2011); 10.1063/1.3621672 Characterization of the “clean-up” of the oxidized Ge(100) surface by atomic layer deposition Appl. Phys. Lett. 95, 212902 (2009); 10.1063/1.3268449 Electronic effects induced by single hydrogen atoms on the Ge(001) surface J. Chem. Phys. 128, 244707 (2008); 10.1063/1.2938091 Ion implanted nanostructures on Ge(111) surfaces observed by atomic force microscopy J. Vac. Sci. Technol. B 15, 809 (1997); 10.1116/1.589414 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:41JOURNAL OF APPLIED PHYSICS VOLUME 38, NUMBER 5 APRIL 1967 Atomic Mating of Germanium Surfaces D. HANEMAN,* W. D. ROOTS, AND J. T. P. GRANT School of Physics, University of New South Wales, Australia (Received 6 June 1966; in final form 21 September 1966) Single crystals of germanium have been partially split in ultrahigh vacuum (~1O--9 Torr), and the surfaces of the split recontacted with high precision. Initially, an internal n-~ structure appears about the mated split in n-type specimens, indicating that the surfaces, although in intimate, possibly atomic, contact retain their new surface structures sufficiently to trap electrons. Subsequently, portion of the junction disap~ pears to an extent which varies from specimen to specimen. This is interpreted as due to the occurrence of atomic bonding across portion of the contacted region, and suggests that cleaved and annealed surface structures are not drastically different from ideal surfaces. When irradiated with a light spot the remaining junction region causes a photovoltage to appear which reverses in sign as the spot traverses the mated region. Due to the narrowness of the latter, applications as light detectors of high positional sensitivity are indicated. The p-type specimens show no junctions when surfaces are mated, confirming that clean cleaved surfaces remain p type. Specimens (n or p) containing surfaces mated in air show no photovoltage. Specimens containing surfaces mated above 200°C, the cleaved-to-annealed surface transition tempera ture, appear to have higher photovoltages than those mated at room temperature. INTRODUCTION IN this work, information about the properties of clean semiconductor surfaces is obtained by a new technique. Surfaces are created by cleavage in ultra high vacuum and replaced by methods which can in principle effect atom-on-atom-recontact precision. The properties of the mated region then give information about the nature of the original surfaces. A point of particular interest is whether it is possible to obtain a "perfect" join. If, as proposed by some,l·2 the arrangements of atoms on the clean surface are drastically different from the undistorted termination of the bulk lattice at a (111) plane, then two clean annealed surfaces, created initially by cleavage in ultra high vacuum would not rebond perfectly if replaced with atom-on-atom precision. If however the surface atoms are only slightly shifted from the normal po sitions, as proposed by one of us earlier,3.4 then in principle perfect rebonding might be possible if each surface atom were brought back into contact with the atom that was its neighbour prior to cleavage. Experiments have been carried out for germanium. This material exhibits brittle cleavage, preferably along (111) planes. Historically, much work on recontacting surfaces has been done with mica and similar lamellar materials. s Although these can be cleaved in vacuum so that comparatively large (",,0.5 cm2) flat areas are obtained, different kinds of atom can be exposed, and it is likely that some sideways displacement may have taken place before the cleaved strips are recontacted. Apart from strength of adhesion and surface energy, little further information has been obtained. As will * Visiting Professor, Brown University, Providence, R.I. 1 J. J. Lander and J. Morrison, J. App!. Phys. 34, 1403 (1963). 2 R. Seiwatz, Surface Sci. 2, 473 (1964). 3 D. Haneman, Phys. Rev. 121, 1093 (1961). 4 N. R. Hansen and D. Haneman, Surface Sci. 2, 566 (1964). 5 P. J. Bryant, L. H. Taylor, and P. L. Gutshall, Transactions of the 10th N ationaJ Vacuum Symposium (The Macmillan Co., New York, 1963), p. 21. be shown below, flat cleavages are actually a disad vantage when attempting atom-on-atom recontact. Work on pressing metal surfaces into intimate contact results in types of welding due to plastic flow.6 It is important to our experiment to avoid such plasticity effects, if the properties of virgin clean surfaces are to be obtained. Germanium at room temperature shows no plastic flow. At a stress of 6 kg/mm2 the dislocation velocity is about 5X 10-s cm-sec1 at 400°C,7 and by extrapolation would be of order 10-18 cm-sec1 at room temperature, which is completely negligible. METHOD The technique of obtaining and replacing surfaces with high precision was based on creating a small controlled split in a block-shaped germanium single crystal. The surfaces of the split were the ones to be studied, and the recontact was obtained by closing the crack. To help understand the mode of coming together of the surfaces, preliminary studies of the topography of cleaved (111) surfaces were made by taper sectioning samples.8 These taper sections (Ref. 8 and Fig. 1) showed that the surfaces exhibited tear marks which were steps exhibiting various orientations and of heights up to t-t }J, and more. In the mating experiments the crystals were therefore split so that the widest separations of the cleavage surfaces, namely at the mouth of the crack, were less than the average heights of the steps. Referring to Fig. 2, the jaw opening 20 may be shown, from standard elasticity equations,9 to be 20= (8L2/61/2) [V(t3E)]1/2, where L is the crack length, 2t is the specimen thickness, 6 F. P. Bowden, Proceedings of The Symposium on Adhesion and Cohesion (Elsevier Publishing Co., Amsterdam, 1962), p. 121. 7 A. R. Chaudhuri, J. R. Patel, and L. G. Rubin, J. App!. Phys. 33,2736 (1962). 8 D. Haneman and E. N. Pugh, J. App!. Phys. 34,2269 (1963). 9 J. J. Gilman, J. App\. Phys. 31, 2208 (1960). 2203 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:412204 HAKEMAN, ROOTS, AND GRANT FIG. 1. Taper section of cleaved germanium surface, showing profiles of steps. Taper ratio 10: 1. Magnification X 180. As reproduced X 40. E is the Young's modulus (14.0X1011 dyn/cm2) and A is the surface energy (1060 erg/cm2 by the measure ments of JaccodinelO). The theory of the method by which the surface-energy value was obtained is not above criticism, but the value is at least approximate and appears only as a square root. For a crack length of 1 mm, the jaw opening is then calculated as 0.36 J.I.. Thus, splits of length 0.5 mm or less were used. In this way the steps and other surface irregularities acted as guides or keys which did not fully disengage. On allowing the crack to close, the surfaces were therefore guided back by a complex pattern of keys which would in principle result in perfect repositioning of the sur faces. The presence of large-sideways forces could con ceivably disturb this by causing pressures against the step sides. This possibility was minimised by having a split relatively small in length with respect to the bulk crystal, and by using a finely polished point-shaped wedge to separate the surfaces. Another possibility is that small particles might chip out from the surfaces on splitting and lodge in the crack preventing proper closure. Careful microscopic observation of dozens of pairs of cleaved surfaces revealed perfect male and female parts. No missing piece was detected. The above evidence thus suggested that the technique was capable in principle of obtaining surface recontact that was perfect to an atomic scale. Evidence of a more direct nature was obtained from electrical measure ments described later. Attempts to produce partial splits in germanium were at first only rarely successful, as reported by others,9,lO due to a strong tendency for the splits to leave the central (111) planes (these are the cleavage planes) and divert out through the sides of the specimen. Since the experiments had to be performed in ultrahigh vacuum to prevent surface contamination, it was es sential to have a reliable method. After a considerable number of trials of various methods and shapes of specimen, the following arrangement was found to be reliable in producing partial splits, although in some cases the splits were not as small «0.5 mm in length) as required. The apparatus is illustrated in Fig. 3. 10 R. J. Jaccodine, J. Electrochem. Soc. 110, 524 (1963). The specimens are of square section, 1. 75 mm wide and 6 mm high with a small groove at the top of depth ! mm parallel to (111) planes. The base of the groove is scratched as a last step after etching. From 4 to 6 ohmic electrical contacts are provided around the top as indicated. The crystal is split by forcing a finely polished guided tungsten wedge vertically into the center of the groove at a very slow rate, of order 1 J.I./sec. This is achieved by a set of motion reduction levers which transmit pressure to the top of the wedge at approx. 1/100th the original rate, the original pres sure being exerted by a magnetically operated screw or by a Varian linear motion feedthrough. The specimen is supported by steel blocks which exert some side pressure. In addition pressure is applied to the top of the accurately squared crystal by a block carrying a small clearance hole for the wedge. The pressures may be adjusted in vacuo. The crystal is insulated by thin mica sheets. In practice, it has been found that the surfaces can be held apart after splitting by the friction between the top sides of the crystal and the mica covered top block. The wedge can thus be withdrawn and the surfaces allowed to close by releasing the top pressure. Both the side-and top-blocks carry heaters which allow the experiments to be performed at temper atures up to 350°C. SPLITTING PROCEDURE A. Room Temperature High power optical microscopy both during (X40) and after (X 400) splitting reveals no trace of cracks of the type referred to in these experiments. (Longer straight cracks, of approximately 1.5 mm and more, are visible as thin black lines. The short cracks can be made visible by appropriate etching, Fig. 4.) The onset and presence of the split is detected by an increase in resistance across contacts whose joining path is reduced in section by the occurrence of the split. An emf bridge is used to detect the voltages between the various sets of contacts, supplied with a current which is kept constant at 1 mA by a simple circuit. (Larger currents cause detectable heating.) The bridge is balanced, and the pressure slowly applied to the splitting wedge until a slight kick in the bridge null detector (galvanometer) indicates that a split has occurred. This kick is easily separated from the small slow resistance change caused ,~I~---------+------~~ gfoc-L .j FIG. 2. Schematic diagram of opening of split in a specimen by forces applied at jaws. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:41ATOMIC MATING OF GERMANIUM SURFACES 2205 FIG. 3. (a) Diagram of crystal in splitting jig. Adjustable horizontal and vertical compressions can be applied. Pointed splitting wedge applies pressure through hole in top block. (b) Photograph of crystal in position. Magnification X6.S. As re produced X3.2S. (c) Photograph of entire unit. Force is applied from outside vacuum chamber (not shown) onto set of motion reduction levers mounted at top, which transmit force to splitting wedge. Magnification X 1.4. As reproduced XO.7. by the application of pressure. If required, further splitting is induced. The wedge can then be withdrawn, the crack being held open by friction between the crystal tops and the top block. The latter can be (b) (c) FIG. 4. (a) Photograph of face of specimen after splitting, showing base of prepared groove. Magnification X21S. As re produced ~X100. (b) After etching in "Billig" etch. (c) After heavier etch. A scratch is shown for comparing the effect of etch. N ate the fine crack lines around base of groove. These were pro duced deliberately by rough grinding of base of groove. Under normal preparation of groove none are visible. Note also the small crack line. Such lines are occasionally noticed after etching. They are produced bv an internal fork of the crack meeting the surface here. High power microscop.\· of end of etch line shows no dislocation pits. released at any time, including after heating. With experiments conducted in a baked-out steel tank pumped by a 75-liter/sec getter ion pump, barely any pressure change in the usual background of 5 X 10--10 to 10--9 Torr takes place during these operations. In the usual case of four contacts, six pairs of re sistances could be measured all of which were influenced by the crack to an extent depending on its shape and their position relative to it. In a typical case, the re sistance across two contacts on either side the split might increase 6% while that across two on one side of it ("parallel" contacts) might increase by only 0.2%. To test the effects of subsequent heating, it was es sential that the characteristics of the contacts did not change. Hence, the crystals were cycled through high temperatures in vacuum prior to splitting to ensure that the contacts were unaffected by heat treatment. In addition the resistances across the two pairs of "parallel" contacts, which were almost unaffected by the split, provided another check as to the stability of the contacts during heating after splitting. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:412206 HANEMAN, ROOTS, AND GRANT B. Above Room Temperature The crystals were sometimes split while the assembly was held at 210DC or above (the temperature region in which the cleaved surface structure changes to the different annealed surface structure). This caused some difficulties due to the sensitivity of the specimen re sistances to slight temperature changes. An alundum wedge was used instead of the tungsten wedge to minimize heat conduction changes. It was always neces sary to make measurements with the emf supply polarity reversed as well, to allow for the emf generated by the crystal due to even 1 DC temperature difference between the contacts. Ge has a comparatively high thermoelectric power of about 0.5 m V ;eC, which can be appreciable relative to the resistive emf developed across the crystal by the 1-mA supply, since the re sistance at 210DC is of order only a few ohms. However, with such precautions satisfactory measurements could be made. The temperature of the specimen was in all cases measured by the change in resistance using a calibration obtained previously against a contacting thermocouple in a uniform vacuum oven. BEHAVIOR ON RECONTACTING SURFACES IN VACUUM In: all, over 20 successful high-vacuum splits were obtained. (1) p-type. In the case of p-type specimens (0.04 [2·cm) the extra resistance caused by the split dis appeared completely (to reproducibility accuracy of 1 part in 5(0) on allowing the split to close. This oc curred provided the initial split length was less than about 0.5 mm. For longer splits it was never found possible to regain the original resist~nce. Th.ese t;ia~s confirmed the theory in the precedmg sectIOn; It IS essential to prevent the cleavage steps disengaging if perfect recontact is to be attempted, otherwise the cross-sectional area after recontact is less than the corresponding area before splitting. As an example of the sensitivity of this criterion, a p-type specimen (No. 56) was partially split to a depth of 0.5 mm. The original resistance across a set of contacts HC was 28.66 units and increased to 29.68 units on splitting. On recontact it recovered to 28.66 units. The split was then reopened and lengthened to 0.8 mm by careful insertion of the wedge in the groove, HC increasing to 30.39 while the split was open. Sub sequent recontact only restored HC to 29.09 units. The results below therefore all refer to crystals with initial split lengths less than 0.5 mm, found to be a safe upper limit.. . (2) n-type. In the case of n-type speCImens qUIte different behavior was observed. On closure the extra resistance caused by the split only disappeared in part, even for the shortest splits. Furthermore, the reduction in resistance was not always instantaneous, sometimes taking place over a period of minutes or even hours. On heating, the residual extra resistance decreased and at about 130DC disappeared entirely to within the reliability of measurement (approx. 0.2%). However, it recovered completely on cooling, i.e., the curve was completely reversible. . .. These phenomena are indicated schematically m FIg. S (a), and results for a particular specimen in Fig. 5(b) . The explanation of these effects is that, as is well known, a surface potential barrier forms at the surfaces exposed by the split. This potential barrier is p type for both n-and p-type cleaved surfaces. On recontact, the barrier does not disappear over all the contacted region, hence forming an n-p-n blocking layer in n-type specimens, as shown in Fig. 6. On heating, electrons are excited thermally in eventually sufficient numbers to swamp the effects of charges trapped at impurities .or inhomogeneities (contacted surfaces), and the matenal becomes intrinsic throughout. At this stage the blocking layer becomes ineffective, and electrons pass freely across the contacted region. For comparison, the emitter-to-collector resistance of a commercial n-p-n alloy transistor was measured as a function of temper- RESISTANCE (a) r--~_--+_~_-_-~~A~~~~L ______ "-. ____ _ RESI.5TANCE r AFTER SPUT RES'STANC BE""CRE SPLit AutoHea( ---.l 20·e. 120"C 220'C lOO"C TEMPERATURE (b) Gel?' 1·30hm.cm n I~pe ,~ -----_."--- Of,!-, -50~~".~~"-"~'--':250 TEMPERAlI..A£ ·C FIG. 5. (a) Schematic diagram of behavior of resistance across n-type specimen after splitting. Many specimen~ show some auto-healing. The remaini~g extra resist3;nce. dlsappears on heating but returns on coohng, due to npn JunctIOn centered at split. (b) Results for a particular specimen. RAG denotes re sistance between contacts A and G, (b.s.) and (a.s.) refer to before splitting and after splitti,ng: Ratios rather th~ absolute values are plotted.in order to ehmm3;te effec~s of particular (re producible) behaVior of contacts dunng heatmg. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:41ATOMIC MATING OF GERMANIUM SURFACES 2207 ature, being somewhat analogous to the n-p-n region at the split. As shown in Fig. 7, the resistance of both structures disappears at about 130°C. On cooling, the barrier reappears, and hence, the resistance rises again. (3) Photoscanning. Further evidence concerning the n-p-n structure was derived from scanning the region in which the split was presumed to have occurred with a fine ("'@-J.!-wide) light spot. This caused a photo voltage to appear across the contacts. The voltage reversed sign when the spot crossed the split, as shown in Fig. 8. By noting the point of reversal, the position of the split could be identified, and its entire course traced on the crystal side by suitable scanning with the light spot. No photovoltages (detection threshold 1 J.!V) were found for p-type specimens. This is in accord with the absence of a detectable internal potential barrier as found from the resistance measurements. (4) Crystals that were split and recontacted above 200°C, the reported transition temperaturell for cleaved- ~rmlle;.~ -___ Valence band FIG. 6. Schematic diagram of energy bands (n-type specimen) in vicinity of mated but unhealed surfaces. This is similar to schemes postulated for medium-angle grain boundaries. The sharp peak in the bands shown on this scale has probably a complex form on a larger scale. to-heated surface configurations, showed (in the 3 cases) rather larger photovoltages than for similar mated splits produced at room temperature. (5) Crystals (n-type) split in air showed some re duction in resistance on closure. However, no photo voltages were detected from such specimens, presum ably due to the presence of a trapped air layer, sand wiched between the sides of the split. (6) The remaining split could be revealed by etching with CP4 or ferricyanide etch [8g K3Fe(CN)6+12g KOH+100 ml H20]. Table I lists results for a number of representative specimens. DISCUSSION OF VACUUM MATING Intimacy of Recontact As described above, the extra resistance caused by the split disappeared entirely in p-type specimens at 11 J. J. Lander, G. W. Gobeli, and J. Morrison, J. Appl. Phys. 34,2298 (1963). t Auto HNl ~H. FGD t. '0 --, .. G. J7' ,., -.j -'1 ,., 10 II) ~ 10 100 120 '100 160 180 200 ao aeo 260 ZSO TffiF£RATLRE ·C FIG. 7. Comparison of temperature behavior of resistance across mated unhealed split vs that of resistance between n-type regions of commercial n-p-n-alloy transistor (adjusted ordinate scale). Note effects of p-type layer disappear at similar tempera tures (120D-130DC)_ room temperature, and also in n-type specimens when measured at temperatures above 130°C, where charge barriers were inoperative. This showed that the cross sectional area of recontact was the same (to within 0.2%) as before splitting. From tunnelling theory, the transmission of electrons through a physical gap as small as 5 A and of potential height as low as 0.1 eV would only be 10%. Thus, even such small gaps would act almost like open circuits, and would prevent com plete recovery of the resistance. Since complete recovery was observed, it is concluded that the surfaces were in intimate contact, as expected from the technique used of guiding the surfaces back by means of their own cleavage steps which had not been allowed to disengage. -lOll -so ~ o Ei -so 15 I CL -100 o 1500 2000 DISTANCE ().1) Split and recontacted Split and recontacted at 21)0C at 22'C FIG. 8. Plots of photovoltage across contacts sited on either side of remanent split in n-type specimens as function of position of 50-I'-wide light spot. Sharp reversal of photovoltage occurs as spot crosses split. The distance between the peaks is same as size of light spot, down to smallest spots tried (151')' Zero on graph refers to edge of specimen. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:412208 HAN"EMAN, ROOTS, ANI> GRANT TABLE I. Characteristics of representative mated specimens. ------------~-.-- Auto heal. I nitial split Increase (%) in resistance (%) change Photovoltagea Environment of contacts across in resist. (/-LV) Pressure Temp. Front (Torr) (OC) Crystal Type 53 n 10-9 23 14 54 n 10-9 23 10.7 39 n 2! X 10-8 210 16 2!X10-8 23 1 760 22 5.8 56 p 3 n • Tungsten filament source, 80-/-L-wide light spot. The difference in electrical behavior of p-type and n-type germanium upon mating is important. Since a p-type surface forms on clean, cleaved nor p germanium (these results confirm similar conclusions by Gobeli and Allen12), n-type germanium is much more sensitive than p-type to the presence of internal mated clean surfaces, by virtue of the opposite-conductivity-type region they introduce. After the mating processes have ceased, there remains an n-p-n region about the mated split in n-type germanium. This is due to electrons trapped at the contacted region. This region consists of atoms which had, on the former surfaces, formed some surface arrangement and had presumably not rebonded properly with their opposites on recontact. However, immediately upon closing the split, there was some spontaneous reduction of the resistance (Fig. 5). This effect, referred to as "auto healing", varies in magnitude between specimens, from 0% to 70% in trials so far. It is this healing behavior which is of particular interest. Auto Healing The fact that part of the open split resistance dis appears on recontact suggests that some of the blocking n-p-n barrier, composed initially of two separate n-p surface regions coming into contact, has disappeared on contact. The variation found between specimens in the size of the effect is due to the difficulty of producing identical cracks in specimens. The initiation and sub sequent progress of a split is extremely sensitive to the microscopic configuration at the points where maximum stress is applied, and reproducibility at this level has been found difficult to achieve. The cases of largest spontaneous disappearance of the resistance, or "auto healing", are believed to corre spond to conditions of best atomic recontact having been achieved. In this type of experiment the most positive result is the most significant, as indicating what can be achieved when conditions which are difficult to control are fortuitously most favorable. 12 G. W. Gobeli and F. G. Allen, Surface Sci. 2, 402 (1964). ------ ------ Rear L. side R. side Front Rear Front Rear ----------- ------ 13.5 2.7 0 -4.6 -4.3 36 40 -2.3 30 25 2.5 4 very small 120 140 2.7 0.8 0.7 -1 -2.7 0 0 7.2 -0.74 -0.5 0 0 While no result has yet been achieved in which the extra split resistance (n-type specimens) disappears completely, the fact that in several cases a large re duction has been observed seems noteworthy. It seems difficult to account for this other than by postulating a high degree of bonding between the mated surfaces. The degree of atomic disorder at the interfaces where auto healing has taken place must be less than that required to form blocking junctions. It is possible to form estimates of the degree of such disorder. Evidence about this is derived from work on twin boundaries in Ge, work on intimate heterojunctions between Ge and materials of similar structure and lattice constant, and work on low-and medium-angle grain boundaries in Ge. Effect of Related Defect Regions on Conduction Measurements on (111) twin boundaries were re ported by Billig and Ridout.13 By placing a collector probe near an incident light spot on a Ge surface, no change in signal gradient was detected as the collector passed across the boundary, indicating absence of a significant potential barrier. Such a boundary of course represents a relatively mild disturbance to the lattice potential and absence of large concentrations of trapped carriers at the twin boundary is not surprising. Conduction across Ge-Si (111) heterojunctions was studied bv Oldham and Milnes.14 The lattice constants a are 5.65754 A for Ge and 5.43072 A for Si. One can regard the mismatch as being accommodated by pure edge dislocations, whose spacing in this case would be 155 A. The current-voltage characteristics of the Ge-Si junctions could not be explained simply by the dis continuities in the conduction and valence band edges consequent on the different electron affinities and band gaps in the two materials. In particular, both n-n and p-p Ge-Si heterojunctions showed barriers of 0.4-0.6 eV, suggesting that the Fermi level was fixed near the center of the Si band by the presence of a layer of interface states. However in the case of Ge-GaAs 13 E. Billig and M. S. Ridout, Nature 173, 496 (1954). 14 W. G. Oldham and A. G. Milnes, Solid-State Electron. 7, 153 (1964). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:41ATOMIC MATIXG OF GERMANIUM SURFACES 2209 heterojunctions15 (a for GaAs is 5.6534 A) the mismatch is mucp less, the corresponding dislocation spacing being ~166 A. Work on these showed that normal Ge-GaAs heterojunctions appeared to have no detectable «5X 101O/cm2) interface-state densities. These results suggest that a sheet of disturbance such as a heterojunction shows interface states that detect ably interfere with the conduction if the spacing be tween mismatch centers is about 150 A, but not if it is of order 8000 A. Further information comes from the considerable work that has been done on grain-boundary junctions by Matare,t6 Mueller17 and others. A low and medium angle « 25°) grain boundary can be described as an array of edge dislocations. Theoretical predictions by Read and Shockley18 about the grain-boundary energy on the basis of such a dislocation model were confirmed by measurements on tricrystals by Wagner and Chalmers,19 Grain boundaries are now known to have a strong p-type character, high carrier trapping proper ties, and high photoelectric sensitivity among other special properties.16.17 A grain boundary structure in n-type Ge behaves like an n-p-n junction, and tran sistors have been based on this property. It is generally agreed that electrons are trapped at dangling bonds connected with the boundary. There are clearly some similarities in the behavior of grain boundaries and mated surfaces. It is of interest to determine the lowest angle of misfit between two Ge lattices, which still leads to the properties described above. Although systematic work has not been done for a range of slowly varying angles, sufficient results for our purposes are available. For very small angle of misfit boundaries, or lineage boun daries, the arrays of dislocations have been observed by etch pits. Recombination of holes and electrons at boundaries corresponding to approximately 1 min of arc misfit has been observed.20 The distance D between the dislocations is given by D= (a/2 sinO), where 0 is the misfit angle (assuming no twist), and a is the lattice constant. In this case, D is approximately 1 J.I.. A distinct p-type layer does not appear to be formed for these very small angles. Much more pro nounced effects, including strong p-type character have been observed for boundaries with misfit or tilt angles between 1° and 25°. The photo response for 1°-tilt 16 L. Esaki, W. E. Howard, and J. Heer, Surface Sci. 2, 127 (1964). 16 H. F. Matare, Report on 24th Annual Conference of Physical Electronics (MIT Press, Cambridge, Mass., 1964). 17 R. K. Mueller, J. App!. Phys. 32, 635, 640 (1961). 18 W. T. Read and W. Shockley, Phys. Rev. 78, 275 (1950). 19 R. S. Wagner and B. Chalmers, J. App!. Phys. 31, 581 (1960). 20 F. L. Vogel, W. T. Read, and L. C. Loven, Phys. Rev. 94, 1791 (1954). FIG. 9. Schematic appearance of base of an open split on atomic scale. There is a gradual transition from conditions where atoms are bonded to one where spacing is too large for bonding. On closing split, conditions for rebonding would be favourable if drastic rearrangements of atoms on surfaces of split had not taken place. boundaries was measured bv Lindemann and Mueller21 [axis of relative rotation for -the two sides was a common (100) direction, the mean boundary plane being (110) or (100)]. These boundaries had a capacitance (10-20 pF /mm2) which was sufficient for their light-power detection ability to be as good as for boundaries be tween lattices of up to 25° tilt. For 1°-tilt, the dis location spacing D is approximately 170 A. Thus, we know that there are sufficient charges trapped for an effective barrier to be formed for a dislocation spacing of 170 A in Ge, but for a spacing of 1/.1. a proper blocking junction is not formed. These figures are in accord with the heterojunction results discussed above (D= 150 A gives appreciable interface states, D= 8000 A does not). Regarding the interface disorder as equivalent to dis locations, we may conclude that the spacing between these must be greater than approximately 200 A in order for no effective barrier to be observed in n-type Ge at room temperature. From the above discussion it appears that the degree of perfection of bonding between the mated surfaces that is required for no blocking junction to appear is such that, compared with a set of dislocations, the spacing of such a set is greater than about 200 A. This is a state of high perfection. This result is not unreasonable when one considers the conditions at the base of a split (Fig. 9). There would be a gradual transition from regions of bulk bonding to regions of separation. As the two sides of the split are allowed to return (under slight positive pressure) the atoms at the very "bottom" of the split could rebond with comparative ease, followed by the next set, and so on. In principle this could cause the split to heal entirely if the atoms on the split surfaces had not formed some arrangement drastically different from normal. In practice, mechanical perfection, com plete absence of disturbing stresses, and minimal con tamination would all be required. In fact, such per fection of conditions was not achieved and the degree of observed heal for macroscopically identicalmechani cal starting conditions and vacua of order 10-9 Torr, varied from 0% to 70%. In some cases the healing was immediate, in others it occurred over periods of minutes or, rarely, hours. Since the split region that remains 21 W. W. Lindemann and R. K. Mueller, J. App!. Phys. 31, 1746 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:412210 HANEMAN, ROOTS, AND GRANT after healing, being detected by photovoltage and etch ing methods, is observed to extend to the top of the specimen, one concludes that it is indeed the lower end, extending to the former "bottom" of the opened split, which healed, A direct proof of this is not yet possible. As mentioned above, the separation of surfaces in this region while open is much less than the jaw opening of i-! p., and not visible under optical microscopy. After healing, one can only be sure of the region that has not healed. Hence, one can not see the base of the split while open, or positively identify it after healing. Care ful optical microscopy of the etch line of the remaining split simply shows a thin black line which becomes gradually fainter and disappears. Heavy etching causes it to end abruptly (see Fig. 4). There is no indication whatever of dislocation etch pits associated with this or any other part of the split. Light spot scans of the end of the split for photovoltage appear to show that the junction ends in the vicinity of the disappearance of the etch line, but the discrimination of this technique is not good due to the diffusion lengths (sizeable fraction of 1 mm) of the optically generated carriers. Other types of scanning may be more useful. Heat treatment of surfaces mated at room temper ature, up to 300°C, does not lead to any permanent change in the remaining split resistance, nor does ex posure of the structure to air, nor attempts to apply strong side forces, in vacuo or in air. In some cases, samples after exposur,e to air were immersed in oil (no effect), and subjected to hydrostatic pressures of 3000 atm, without a permanent effect. The above tests all suggest that after healing has taken place it is difficult to permanently extend the healed region. The conditions for healing are delicate. Whatever the cause that prevents the split, although closed, from healing completely (e.g., the presence of a slight sideways force which would be more effective at the upper end of the split), this cause is not overcome by the methods mentioned. This is not surprising as some of the methods are perforce gross compared to the finely balanced conditions required. The experiments described give information about clean Ge surfaces obtained by cleavage at room temper ature. The structure and other properties of such surfaces are altered when they are annealed above a critical temperature, approximately 200°C for Ge.H Most theoretical and experimental work regarding structures has been concerned with the annealed sur faces which yield LEED patterns similar to those ob tained by the common surface cleaning methods of ion bombardment and annealing. Hence, attempts were made to obtain information about annealed surfaces by the mating technique. MATING OF HEATED SURFACES Several methods were used. Two crystals were split and mated at 210°C, one split at 210°C and mated at room temperature, and also one split at room temper-ature and not allowed to close until heated to 220°C. The splitting while "hot" required very even temper ature conditions to minimize the relatively large thermo electric emf's mentioned earlier, and the use of a polished alundum rather than tungsten wedge to minimize tem perature changes due to conduction losses while apply ing pressure. It was necessary to hold the crystal open (pressure now ",,5 X 10-8 Torr) while the 12 emf bridge readings were taken (about 5 min) as a reference. Since closing the crystal at this temperature immedi ately restored the original resistances across all contacts, the degree of healing could not be determined till the crystal was cooled to room temperature and the various resistance ratios compared. All these temperature stabi lization procedures were time consuming. No healing was found for three crystals tried, the resistance ratios when cold being the same as when the crystal was open at high temperature. In the case of one crystal some healing appeared to have taken place. In all cases the photovoltage developed across the splits by a standard light spot seemed significantly larger (see Fig. 8) than for specimens split and mated at room temperature. We believe the paucity of mating so far to be more an instrumental than fundamental result, probably due to conditions in the limited number of samples tried being imperfect. The side and vertical pressures on the crystal were controlled by springs set in vacuo to certain tensions, and these pressures applied to the crystal were not affected by thermal expansions of supporting jig parts. These arrangements were made in order to ensure that the external conditions while hot were very similar to those for the crystal during the room temperature splits. The properties of these crystals after mating were not permanently affected by reheat ing, air exposure or application of pressure. As in the case of room temperature specimens the surfaces were known to be in intimate contact because of the dis appearance of the junction above 130°C. The higher photovoltages referred to above seem significant, and could indicate a higher potential barrier, and or different carrier recombination rates at the mated region than for crystals split and mated at room temperature. Many measurements of barrier properties of grain-boundary structures have been made,t6.17 and they suggest that similar methods might be useful for mated surface barriers. Unfortunately, the bulk Ge below the mated split acts as an electrical shorting path. Attempts have been made to remove this by etching and fine-sand blasting. However, the bottom of the split in the interior of the specimen is usually not a straight line. Damage to the mated region while seeking to delineate the bottom frequently results. Successful contacts to the mated surface barrier itself are also necessary for further work on barrier properties. At this stage one may say that the properties of crystals mated at 210°C seem to be different from those mated at room temperature, but healing takes place in both cases. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:41ATOMIC MATING OF GERMANIUM SURFACES 2211 SURFACE MODELS Measurements by LEED on (111) surfaces of Ge obtained by cleavage in ultrahigh vacuum have been reported by Farnsworth et al.,22 and by Lander, Gobeli and MorrisonY The latter alone reported a structur~ with a rectangular unit mesh which changed over after a few minutes annealing at about 200°C to a structure similar to that obtained on surfaces cleaned by ion bombardment and annealing. A model was proposed to account for the data, although sufficient intensity measurements for a detailed check were not reported. As mentioned earlier, quite different surface models2-4 have been shown to be capable of accounting for the detailed LEED intensity data reported by Lander and ~orrisonl for annealed Ge surfaces. This uncertainty IS due to the lack of precise knowledge of the scattering processes affecting low-energy electrons in the surface layers. An attempt to fit the less extensive cleaved surface intensity data by a variation of the rumpled surface modep,4 has been made by Miller in this lab oratory. The model assumes that (12) rows of surface layer atoms [indexed with respect to the rhombic unit ce~l of an undis torted ( 111 ) plane] are al terna tel y raIsed and lowered with respect to their "normal" heights. To ~ccount for the reported intensity data, an asyn:metry m the (01) direction is required, and is prOVIded by assuming that the lowered surface layer atoms are moved slightly in the (01) direction towards the layer 2 atoms. A degree of fit to the LEED data is possible with this model as well as the one proposed by Lander and Morrison. Under the circumstances a detailed discussion of the possible arrangements of atoms on cleaved Ge surfaces is not useful at this stage. However, the present work supplies a further constraint on proposed surface struc tures, additional to those from measurements such as LEED,1l,12 photoemission,12 electron paramagnetic reso nance,23 and surface conductivity.24 The main conclusions refer to the requirements for the arrangements of atoms on clean Ge faces. The observed autohealing could take place if the surface atoms possessed a structure that is obtained from the normal one by only minor atom shifts.3 A schematic diagram for the case of contacted cleaved surfaces prior to healing is shown in Fig. 10. The atoms on one face are shown displaced sideways with respect to those on the opposite face. The distance between two such opposite atoms must be such that the force between them is sufficient to pull them into a proper bonding state if healing is to take place. The potential between two atoms is attractive at separations larger than 2ll H. E. ~arnsworth, J. B. Marsh, and J. Toots, Proceedings of the Internattonal Conference on Semiconductors Exeter (Institute of Physics and the Physical Society, London, '1962), p, 836. 23 M. F. Chung and D. Haneman, J. App!. Phys. 37, 1879 (1966). 24 D. E. Aspnes and P. Handler, Surface Sci. 4, 353 (1966). FIG. 10. Schematic appearance of atoms on contacted cleaved surfac:s. Co~t~rnination on the surfaces (open for 5 min at "-'10-9 Torr! IS n.eghglble. ~toms on one surface are shown displaced. The relative Sideways IDlsfit between atoms on opposite sides of split must. not be too large « 1 A) for re bonding to take place. Re bondmg would not be generally possible if migrations of atoms had occurred on surfaces. normal and attempts to set limits on the interatom spacing can be made if the potential is known. Un fortunately, sufficiently precise information is not avail able. Inform~tion can be obtained from the following ap proach. Smce there are several equivalent directions in the plane, the atoms on one surface may shift laterally, as mentioned previously, in a direction differ ent from that in which their opposite neighbors have moved. To enable rebonding, the shifts must therefore be less than-about half the interatom sRacing in the plane of the surface, i.e., less than about 2 A. It appears, therefore, from considerations such as these, that shifts of up to about an angstrom might be possible. Note added in proof. The above discussion has referred to the limits for possible lateral displacements of atoms on reconstructed cleaved surfaces. In the case of an nealed surfaces, the model of small vertical displace n:ents of surface atoms, proposed by one of us pre vlOusly,3 appears to be fully compatible with the o~served autohealing. When the surfaces are replaced WIth the proper precision, (occurring particularly near the base region), corresponding atoms are opposite each other and thus in position to reform their bonds. The activation energy for healing is not known but forma tion of a bulk structure would be expected to result in a lowering of energy and thus be favored. Surface arrangements involving drastic changes1,2 from the simple bulk termination of the lattice, would appear to be incapable of accounting for the auto healing observed in these experiments. DEVICE POSSIBILITIES The internal n-p-n structure that appears at un healed vacuum contacted surfaces has properties that suggest possible device applications. The sensitive elec tron trapping region, which causes the p-type depletion layers, is extremely thin. Estimates of the depth of distortion from a clean Ge surface1.4 yield 5-9 A, so that the mate~ surface barrier region itself is probably less than 20 A thick. It is thus even thinner than a [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Sun, 21 Dec 2014 12:35:412212 HANEMAN, ROOTS, AND GRANT grain boundary junction region, and, like the latter, suggests applications as a light sensor with extreme positional sensitivity. At present, the photovoltages obtained (Fig. 8) have not been as large as from grain-boundary structures but are readily amplified. Note added in proof. Recently, Palmberg and Peria [Surface Sci. 6, 57 (1967) ] have concluded on the basis of LEED, alkali adsorption and work function measurements, that the rumpled surface modeI,a in JOURNAL OF APPLIED PHYSICS 8X2 unit cell form, is alone compatible with their data on (111) germanium surfaces. ACKNOWLEDGMENTS This work was supported by a grant from W.D. and R.O. Wills (Australia) Ltd. to whom appreciation is extended. Dr. A. Ewald of Sydney University provided· facilities, and gave assistance for the tests at high hydrostatic pressures. VOLUME 38, NUMBER 5 APRIL 1967 Self-Locking of Modes in Lasers* H. STATZ AND G. A. DEMARS Raytheon Research Division, Waltham, M assachuse/ts AND C. L. TANG School of Electrical Engineering, Cornell University, Ithaca, New York (Received 31 October 1966) We investigated phase-locking effects between longitudinal modes in la~ers. In order to show the general trend to be expected for a large number of oscillating modes, we treat three-, four-, and five-mode oscillations. The expected phases depend in a complicated manner upon the relaxation times TI and T2 of the medium, on the degree of inhomogeneous broadening, the mode separation and location of the medium in the cavity. Simple formation of sharp output spikes at the fundamental frequency are expected where crystals like ruby or YAG are placed near the edges of the cavity. Sharp spikes at twice the fundamental frequency are expected when these solids are placed in the center of a cavity. Certain filters, when placed near the edge or center of the cavity are expected to came similar locking effects. Gases and solids are expected to act quite differently. The calculations are based on the maximum-emission principle. This principle will be discussed in a later pUblication. Some experimental results are also presented. I. INTRODUCTION IN a previous paperl it has been pointed out that the various simultaneously oscillating modes in general are not independent of each other, but there are mechanisms in the laser medium which tend to intro duce definite phase relationships. In the meantime, very dramatic results of mode locking have been obtained by Stetser and DeMaria.2 In this case the principal interaction between the modes occurs in a saturable absorber. The mode-locking mechanisms and observations in general refer to the many simultaneously oscillating longitudinal modes. These modes are essentially equidistant in frequency with a separation given by c/2L, where c is the velocity of light and L is the optical length between mirrors. Most workers in the field understand normally by mode locking that all the various modes have the same phase and that their electric fields are described by exp i (wot+nflwt) , where Wo is the oscillating frequency of one mode and flw is * Supported in part by Air Force Cambridge Research Labora tory, L. G. Hanscom Field, Bedford, Mass. I H. Statz and C. L. Tang, J. App!. Phys. 36, 3923 (1965). 2 D. A. Stetser and ]. A. DeMaria, App!. Phys. Letters 9, 118 (1966) . the mode spacing. The quantity n is an integer and may be considered the mode number. By adding up all these electric fields one obtains an output pattern consisting of pulses with a repetition period r = 271'/ flw and a pulse width approximately given by flr=27r/(n maxflw), where nmax is the number of oscillating modes. When a large number of modes is oscillating, such as in a glass laser, pulse lengths approaching 10-13 sec can be obtained2 with correspondingly high peak powers. In general, not as many simultaneously oscillating modes are found in lasers. First of all, the linewidth sets an upper limit to the number of oscillating modes and, in addition, other factors cause a limitation and a selection of the· oscillating modes. For example, the spatial competition between modes for the inverted population in homo geneously broadened lines causes a limitation of the oscillating modes.3 Also, depending upon the location of the crystal in the cavity,4 certain additional mode selection rules become operative. For inhomogeneously broadened gas-laser transitions, holes are being eaten into the lines, and longitudinal modes with a spacing smaller than the width of the spectral hole are prevented 3 C. L. Tang, H. Statz, and G. A. DeMars,]. App!. Phys. 34, 2289 (1963). 4 V. Evtuhov, App!. Phys. Letters 6, 141 (1965). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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1.1714372.pdf	Electron Emission, Electroluminescence, and VoltageControlled Negative Resistance in Al–Al2O3–Au Diodes T. W. Hickmott Citation: Journal of Applied Physics 36, 1885 (1965); doi: 10.1063/1.1714372 View online: http://dx.doi.org/10.1063/1.1714372 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/36/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Characteristics of electron emission of Al-Al2O3-Ti/Au diode with a new double-layer insulator J. Vac. Sci. Technol. B 32, 062204 (2014); 10.1116/1.4900632 Surface plasmon polariton-assisted electron emission and voltage-controlled negative resistance of Al – Al 2 O 3 – Au diodes J. Appl. Phys. 107, 093714 (2010); 10.1063/1.3407510 Voltage-controlled negative resistance and electroluminescent spectra of Al – Al 2 O 3 – Au diodes J. Appl. Phys. 106, 103719 (2009); 10.1063/1.3262619 Temperature dependence of voltage-controlled negative resistance and electroluminescence in Al – Al 2 O 3 – Au diodes J. Appl. Phys. 104, 103704 (2008); 10.1063/1.3021092 Anomalous voltagecontrolled negative resistance in Ptinsulatorconductive polymerAu junctions Appl. Phys. Lett. 47, 724 (1985); 10.1063/1.96016 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:12X-RAY SPECTRA OF THE LIGHTER ELEME~TS 1885 decrease in atomic number accounts for other changes in spectral intensity. Thus in Table I the increase in relative intensities of the generally minor I and 'YJ lines for the elements between copper and titanium is not due to an increase in their absolute intensities but to the progressive weakening of the a and {31 lines resulting from the diminishing number of electrons in the 3d levels. Potassium and chlorine have no 3d electrons and their spectra are reduced to a single 1+'YJ line. Similarly, the fall in intensities of the '/'1 and {3z lines in passing from silver to rhodium and the JOURNAL OF APPLIED PHYSICS inability to find them for yttrium and lighter elements is associated with a depopulation of the 4d levels. No satisfactory explanation can be given of the low efficiencies of the rhodium and yttrium spectra. It is possible that the preparation of rhodium was impure since its K spectrum was also weak compared to those of its neighbors, silver and ruthenium. An impurity could not, however, be found from a routine x-ray spectral analysis. Impurity does not explain the poor emission of the yttrium sample whose K spectrum was in all respects normal. VOLUME 36. NUMBER 6 JUNE 1965 Electron Emission, Electroluminescence, and Voltage-Controlled Negative Resistance in Al-Al 203-Au Diodes T. W. HrCKMOTT General Electric Research Laboratory, Schenectady, New York (Received 6 October 1964; in final form 17 February 1965) The temperature dependence of the conductivity of Al-Ab03-Au diodes that exhibit voltage-controlled negative resistance (VCNR) in their current-voltage (I-V) characteristics, as well as electron emission and electroluminescence from such diodes, have been studied. Electron emission into vacuum and electro luminescence are both characterized by a steep increase in intensity for diode voltages greater than 1.8 V. Electron emission exhibits a second rise when the diode voltage exceeds the work function of the metal facing vacuum; electroluminescence, in contrast, is quenched when the diode voltage exceeds about 4V. The resistance of Al-A]'03-Au diodes is independent of temperature down to 3°K if V m, the voltage for maxi mum current in the 1-V characteristic, is not exceeded. If the full 1-V characteristic is traced out as tem perature is decreased, diode resistance increases, VCNR in the I-V characteristic disappears, and electron emission into vacuum from the diode disappears. The attenuation length for electrons emitted into vacuum through the gold films of Al-A]'03-Au diodes is ",200 A, independent of diode voltage; the attenuation length in the oxide is greater than 200 A. Retarding potential measurements of the normal energy component of emitted electrons, and electroluminescence of diodes, show that some electrons gain energies in the oxide film that are higher than the applied voltage. The maximum excess energy gained is 4.1 V. Electrolumines cence occurs from spots on Al-AJ.03-Au diodes. The spectrum covers the visible range with peaks of higher intensity at 1.8, 2.3, and 4.0 V. The experimental data are used to derive values of the parameters of a pro posed model of VC~R in metaHnsulator-metal diodes. INTRODUCTION ON the basis of measurements of the potential distribution in metal-oxide-metal-oxide--metal structures (triodes)1 a qualitative model has been pro posed for the establishment of conductivity and the phenomena associated with voltage-controlled negative resistance (VCNR) in the current-voltage (1-V) charac teristics of metal-insulator-metal sandwiches.1-12 The 1 T. W. Hickmott, J. App!. Phys. 35, 2679 (1964). 2 T. W. Hickmott, J. App!. Phys. 33, 2669 (1962). 3 T. W. Hickmott, J. App!. Phys. 34, 1569 (1963). 4 T. W. Hickmott, J. App!. Phys. 35, 2118 (1964). 6 G. S. Kreynina, L. N. Selivanov, and T. I. Shumskaia, Radio Eng. Elec. Phys. 5, 8, 219 (1960). 6 G. S. Kreynina, Radio Eng. Elec. Phys. 7, 166 (1962). 7 G. S. Kreynina, Radio Eng. Elec. Phys. 7, 1949 (1962). 8 S. R. Pollack, W. O. Freitag, and C. E. Morris, Electrochem. Tech. 1, 96 (1963). 9 P. H. Nielsen and N. M. Bashara, IEEE Trans. Electron Devices EDll, 243 (1964). 10 H. Kanter and W. A. Feibelman, J. App!. Phys. 33, (1962). 11 H. T. Mann, J. App!. Phys. 35, 2173 (1964). 12 R. A. Cola, J. G. Simmons, and R. R. Verderber, NAECON Proc. (1964). establishment of conductivity and VCNR in triodes is accompanied by the formation of two distinct regions in the insulator.! In the bulk of the insulator, conductivity is Ohmic, potential drops are small, and are determined primarily by the magnitude of the current through the triode. A high-field region, approximately 120-150 A thick, also forms within triodes, usually near the negative electrode. The potential in this region increases monotonically with voltage applied to triodes; processes occurring in the high-field region determine VCNR in the I-V characteristic, as well as electron emission and electroluminescence which occur from metal-insulator metal diodes or triodes. The I-V characteristics of metal-insulator-metal diodes are identical to the characteristics of triodes made with the same insulator and metal electrodes. In the present paper, detailed measurements are reported of the temperature dependence of conductivity of AI Ab03-AU diodes, and of electron emission and electro luminescence from such diodes, in an effort to under- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:121886 T. W. HICKMOTT ;;: ~10-2 Iz loU II: II: :::> u loU C ~ 10-3 AI-AI203-Au DECREASING TEMPERATURE 153' 6 9 10 Vf (VOLTS) FIG. 1. Temperature dependence of the I-V characteristic of an AI-AbO a-Au diode for decreasing temperature. Oxide thick ness, 500 A; gold thickness, 110 A. Au+, Al-. stand and elucidate some of the processes occurring in the high-field region of these structures. EXPERIMENTAL Experimental procedures used in making metal insulator-metal sandwiches have been described.2,4 Sample areas were generally 10 mm2• Oxide thicknesses were ",,500 A and were determined from capacitance measurements using a value of 8 for the dielectric constant. Gold evaporations were done from a tungsten helix using a weighed amount of gold and a standard evaporation configuration. Gold thicknesses were meas ured to ±5% by determining the characteristic x-ray fluorescent emission of gold.Ia A few results on AI SiO-Au triodes are presented. Their preparation has been described.I Current-voltage characteristics of diodes and elec tron emission into vacuum were measured simultane ously. Emission currents into vacuum from diodes were collected by a gold-plated stainless-steel collector at +22 V and measured by a Keithley model 415 mi croammeter. Emitter-collector distances were 1 cm. Both electron emission Ie and current through the oxide film sandwich II were displayed as functions of the diode potential V I on Moseley 135 X -Y recorders. Four-probe measurements of potential were made to 13 H. A. Liebhafsky, H. G. Pfeiffer, E. H. Winslow, and P. D. Zemany, X-ray Absorption and Emission in Analytical Chemistry (John Wiley & Sons, Inc., New York, 1960), p. 153. minimize errors due to the potential drop in the evapo rated metal electrodes. Such errors are particularly large in diodes carrying high currents or when very thin metal counterelectrodes having high resistance are used. Even with four-probe measurements, errors of a few tenths of a volt in locating the voltage for maximum current V m can occur when high currents are carried by the diode. Conductivity of AI-AbO a-Au diodes was developed by applying potentials in vacuum.2 A continuously pumped metal cryostat immersed in liquid nitrogen was used to measure the temperature dependence of the I-V characteristics and electron emission of Al-AhOa-Au diodes. Sample temperatures were measured by a copper-constantan thermocouple attached by indium solder to the glass substrate behind the diode to be measured. Temperatures given are those just before the voltage across the diode was raised. In some diodes, temperature rises up to 150°C were measured when maximum diode currents were several hundred milliamperes. The large temperature rise when large diodes currents were dissipated means that diode temperatures were not known during high-current measurements. At low temperatures, and when the diode current was small, this problem was not serious. TEMPERATURE DEPENDENCE OF DIODE CONDUCTANCE AND ELECTRON EMISSION Two distinct modes of temperature dependence of I-V characteristics were observed for AI-AhOa-Au diodes after forming of VCNR, depending on whether the voltage for maximum current V m was exceeded in tracing out the 1-V characteristic or whether the applied voltage was kept below V m' Conductivity and I-V characteristics of the diodes were independen t of temper ature, within 10%, from room temperature down to 2°K, if the applied voltage did not exceed V m, if con ductivity was established at room temperature, and if the diode resistance was less than a few thousand ohms. Injection of charge carriers from the metal into the insu lator is not thermally activated in a diode with fully de veloped conductivity. If, on the other hand, the diode was cycled to lOV and back, tracing out the full I-V characteristic, the temperature dependence of conductivity was markedly different as shown in Fig. 1 for a typical AI-AhOa-Au diode. As temperature was lowered, the peak current and minimum current decreased gradually although the voltage for maximum current remained nearly constant, and the shape of the I-V curves also remained nearly constant. (In Fig. 1, V m is shifted below 2.8 V because high currents produced IR drops in the gold leads.) At some temperature, which was generally around 210°- 2200K but varied from diode to diode, negative resis tance was traced out for increasing voltage but no nega tive resistance appeared when the voltage was decreased, as in the 213°K curve in Fig. 1. If temperature was decreased further, this small residual current decreased [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:12PHENOME:-JA IN AI-AI 20a-Au DIODES 1887 slightly and then reached a nearly constant value. As temperature was raised back to room temperature, the diode redeveloped conductivity, VCNR and a maxi mum current nearly as high as at room temperature, at about 160oK, much lower than the temperature at which VCNR disappeared for decreasing temperature. Above about 170oK, as the diode tempera ture was raised, the peak current 1m and the I-V characteristics re mained almost independent of temperature. Al-Si0-Au diodes exhibit identical temperature dependence. Electron emission from oxide films that exhibit VCNR develops at the same time as conductivity is developed in the oxide; if conductivity cannot be developed, no electron emission into vacuum is observed. Two features are characteristic of emission into vacuum of electrons from metal-oxide-metal sandwiches, with oxide films greater than 100 A thick, which show VCNR in their I-V characteristics.3 Emission currents are first detected above the noise when about 1.8 V are applied to the film, and a second increase in total current occurs at applied potentials just above the work function of the metal which faces vacuum. The similarity of emission from diodes with different insulating oxides and some of the evidence that emitted electrons pass through the metal film rather than through holes in the metal film have been discussed previously.3 The qualitative fea tures of electron emission are also independent of the metal electrode facing vacuum; emission is greater from electrode metals with lower work functions. Electron emission into vacuum from diodes also de pends on temperature when the full characteristic is traced out. As temperature was lowered, the ratio of current emitted into vacuum to current through the film Ie/If decreased, as shown in Fig. 2, where Ie/If as a function of voltage is plotted for the same diode whose I-V characteristics are shown in Fig. 1. This is a convenient way to plot data to compare emission from different diodes because of pronounced nonlinearity of current through the diode, because of variations of diode current from diode to diode that depend on the way con ductivity has been developed, and because of variations of diode current for different runs on the same diode. Qualitative features of emission were the same as at room temperature until VCNR vanished; Ie/If de creased more rapidly than If did. When the temperature was reached at which VCNR was no longer observed, electron emission was not found when the voltage across the diode was less than 6 V. VCNR in the 1-V character istic and low-voltage electron emission into vacuum disappeared together. As the diode temperature was decreased further, current through the diode decreased very little but the fraction of current emitted into vacuum at higher diode voltages decreased rapidly until, at some low temperature, 155°K for the diode in Fig. 2, no further electron emission into vacuum could be detected above noise. Electron emission remained below noise as the temperature was further lowered. Thus, for decreasing temperature, the fraction of electrons emitted AI-A1203-Au DECREASING TEMPERATURE 10-1 FIG. 2. Temperature dependence of the ratio of vacuum emission current to diode current at different applied voltages for the Al-AbOa-Au diode of Fig. 1. into vacuum at a given diode potential dropped steeply, as shown in Fig. 3. Emission of electrons below 5 V depends on the same processes· as VCNR in the diode. If VCNR is eliminated from the I-V characteristic, no low-voltage electrons are emitted into vacuum, regard less of temperature. As diode temperature was raised, film conductivity and electron emission both developed to magnitudes characteristic of diodes at room temper ature, for temperatures greater than about 160°K. For increasing temperature, the fraction of electrons emitted into vacuum at a given diode potential remained nearly constant over a temperature range above 1700K in which the peak diode current was close to that found at room temperature. If V m was not exceeded as the temperature of an AI-AhOa-Au diode decreased, Ie/If at constant voltage decreased by approximately 2 as the diode temperature was reduced to 220°K. Ie/If then remained constant below 2200K where VCNR would not be observed if the full 1-V characteristic were traced out. If the resistance of metal-oxide-metal diodes is es tablished at room temperature, diode resistance meas ured at very small voltages is nearly independent of temperature. However, if resistance is established at a low temperature by exceeding V m in the I-V character istics, a pronounced temperature dependence of diode resistance is found. A maximum room-temperature resistance was established in several diodes by raising the diode voltage to 10 V and turning the voltage off quickly. The diode temperature was then lowered to [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:121888 T. W. HICKMOTT Ie iI NEGATIVE RESISTANCE VANISHES I 10" 16~O--~18~O--~2~OO~~2~20~~2~40~~26~O--~28~O--~300 TEMPERATURE (OK) FIG. 3. Variation of the ratio of vacuum emission current to diode current at constant diode voltage as the temperature of the diode of Fig. 1 was decreased. about 90°K. If the voltage applied at 900K was kept between 1.8 and 2.8 V, or less than V m, conductivity developed as it would at room temperature although the final conductivity reached was smaller than would develop at room temperature. Diode conductivity which develops for V < V m will develop at very low tempera tures as well as at room temperature. Raising V above V mat 900K traced out VCNR; the resulting high diode resistance, generally higher than could be developed at room temperature, was maintained when the diode voltage was reduced to zero. The change of resistance with increasing temperature was then measured using a small constant current (1-10 J,LA) through the diode. The potential drop across the diode was less than 0.3 V, so that the diode resistance was nearly Ohmic and was not changed by the applied voltage. In Fig. 4, the resistance of four diodes with approxi mately the same initial resistance, developed at 900K, is plotted as a function of increasing diode temperature. The detailed behavior is a function of previous diode history, of the rate of temperature rise, and of the manner in which diode resistance has been established, and has not been studied in detail. However, all the diodes showed a steep decrease in resistance between 100° and 2400K, the temperature region in which VCNR vanishes when the full I-V characteristic is traced out and in which electron emission vanishes. If diode resis tance is established at low temperatures and at diode voltages greater than V m, some state is produced in the oxide which will contribute to increased conductivity of the diode as it is warmed up. Alternatively, if I-V characteristics are traced out as temperature is lowered, the state produced by V> V m will reduce the number of electrons which would otherwise be emitted into vacuum, and will reduce Ie/Ir, as in Fig. 2. The presence of a; high field in AI-Ab03-Au diodes was deduced from measurements of the potential dis tribution in AI-Ab03-AI-Ab03-Au triodes.l The tem perature dependence of the I-V characteristics of triodes is qualitatively the same as that of AI-Ab03-Au diodes and is one of the principal reasons for extending the model to diodes. In Fig. S(a),I-V curves at different temperatures are plotted for the AI-Si0--Au triode whose I-V characteristics were shown in Figs. 2-4 of Ref. 1. As with diodes, the shape of the curves remained nearly constant, the peak current decreased, and finally VCNR was no longer traced out. In Fig. Sea) the po tential within the triode was concentrated between the Al grid and the Al cathode at all temperatures; as is typical of triodes, the potential between the Au plate and the Al grid was small, proportional to the triode current, 'and determined by processes occurring in the high-field region between grid and cathode. The current density between plate and grid when conductivity is Ohmic is Jpg=Jpc= (nJ,L)pgeFpg= (nJ,L)pgeVpg/dpg, (1) where J pc is the current density through the triode, F pg is the field between plate and grid, npg is the number of ~ ~ 60 ill :§ tl 50 z ~ (f) ~ 40 w '" co 030 20 10 80 320 FIG. 4. Variation of resistance of four AI-At.Oa-Au diodes with increasing temperature after establishing high diode resistance at 9OoK. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:12PHENOMENA IN AI-AI20s-Au DIODES 1889 charge carriers/cm3 in the plate-grid region, J.l.pg is the carrier mobility in this region in cm2 /V -sec, V pg is the plate-grid potential, dpg is the plate-grid separation, and e is the electron charge. The quantity (nJ.l.)pg, di rectly proportional to Ipc/Vpg, was calculated under the assumption of uniform conductivity over the diode area. It was nearly constant over the whole range of triode voltages at each temperature as shown in Fig. 5(b). Between 2960 and 212°K, the temperature range in which VCNR was observed in the triode, (nJ.l.)pg decreased by only a factor of 3 while the triode current decreased by more than 100. When VCNR vanished, (nJ.l.)pg dropped somewhat more, corresponding to a higher fraction of the total potential drop between plate and grid. If conduction is through singularities that occupy only a small fraction of the total diode area, (nJ.l.)pg is actually much larger but its behavior with temperature remains unchanged. Electron emission into vacuum from triodes showed identical quantitative behavior as from diodes when temperature was decreased. The fraction of the total current emitted from triodes was smaller than for diodes, as would be expected if the Al grid layer attenu ated electron emission. As triode temperature decreased, low-voltage emission vanished when VCNR was no longer traced out; further decrease in temperature re sulted in the disappearance of Ie at about the same temperature as for diodes. The decrease in film con ductivity and the appearance of a low-temperature t! E, u~ ~~ -... g: ;} .!!. AI-SiO -AI-5,0 -Au .1296,i .3212'K 4 ~ o 264'K '" 181'K o 234 'K 0 • • • o· 0 0 0 • • • • • 0 0 0 0 0 i ° • • 0 0 0 0 1014 8 8 8 8 ° ~ • 0 0 • 0 i ° it ° 0 0 I!! • • • • • • • • • • • • • • '" • • • '" '" '" '" '" '" '" '" '" '" '" '" '" '" '" '" '" 1O"L--+-+-+----j4-+-~------J;____t_-+-----' Vpc I VOLTS) FIG. 5_ (a) Temperature dependence of the I-V characteristic of an AI-SiO-AI-SiO-Au triode for decreasing temperature. (b) Variation of the dependence of (np.)pI on plate-cathode voltage for the same triode with decreasing temperature. FIG. 6. The ratio of vacuum emission current to diode current at constant values of the applied voltage for AI-AI,Os-Au diodes with varying gold thickness. Oxide thickness, 500 A. state in the oxide, produced by the field and resulting in the reduction of electron emission, occur because of processes in the grid-cathode region of triodes where the bulk of the potential drop remains concentrated at low temperatures. ELECTRON ATTENUATION IN GOLD FILMS AND THE ENERGIES OF EMITTED ELECTRONS The transmission of hot electrons through thin gold films has recently been studied theoretically and experi mentally.10.14-21 Triode measurements show that elec trons are accelerated in a region about 120-150 A thick near the cathode of a sandwich that shows VCNR, are attenuated by passage through the insulator and through the metal electrode facing vacuum, and a small fraction is then emitted into vacuum. An attenuation length L for electrons passing through metal films of thickness d can be defined by the relation Ie/If=k(V,I,r/»exp( -d/ L), (2) 14 C. A. Mead, Phys. Rev. Letters 8, 56 (1962); 9, 46 (1962) . 15 C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. E. LaBate, Phys. Rev. 127, 2006 (1962) . 16 H. Kanter, J. App!. Phys. 34,3629 (1963). 17 S. M. Sze, Solid-State Electron. 7, 509 (1964). 18 R. E. Collins and L. W. Davies, App!. Phys. Letters 2, 213 (1963); Solid-State Electron 7, 445 (1964). 19 J. J. Quinn, Phys. Rev. 126, 1453 (1964); App!. Phys. Letters 2, 167 (1963). 20 R. Stuart, F. Wooten, and W. E. Spicer, Phys. Rev. 135, A495 (1964). 21 K. Motizuki and M. Sparks, J. Phys. Soc. Japan, 19, 486 (1964). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:121890 T. W. HICKMOTT 400 300 V> 200 i :"! 100 _1 Ie • (.!.t. ) L If -If d~o e APPLIED VOLTAGE . 1 1 FIG. 7. Dependence of the attenuation length for electrons in gold films on the voltage applied to an AI-AbO a-Au diode. Oxide thickness, 500 A. where k is a function of the applied potential V, the oxide thickness l, and the work function cp of the metal facing vacuum. Since electrons are scattered in the insulator, k(V,l,cp) may reflect details of the band struc ture of the insulator. Some evidence for such an effect was found for Ta-Ta 206-Au diodes in which a sharp drop in electron emission into vacuum was observed at an electron energy corresponding to the bandgap of Ta206.3 Electron emission during the warming up of diodes from low temperature, after full conductivity was de veloped at about 170oK, was measured in order to de termine L. For each diode, at least 10 measurements of Ie/If were made as the film warmed up. For most values of gold thickness at least two different diodes were used. The difficulties in using a wide range of gold film thick ness to study the attenuation length of electrons have been discussed by Kanter and Feibelman.1o The high resistance of very thin films means that the potential drop across the width of the sample can be large; for thick films, emission into vacuum is dominated by thin spots in the gold film. The former problem was mini mized by using 5-mm gold strips and bringing indium contacts close to the edge of the aluminum strip. The thinnest film used for quantitative measurements, 110 A thick, had a resistance of 9.0 n from one side of the aluminum strip to the other. Resistances of the other gold films ranged between 2.5 and 0.40 Q. Figure 6 shows log Ie/If as a function of gold thick ness at different values of diode voltage. The spread in la/If at any given temperature masked any variation due to changing temperature which may have been present. The vertical lines show the range of Ie/If for each sample. An attenuation length has been derived for each value of applied voltage by drawing the best average line through the points; the uncertainty in L was obtained by drawing lines of maximum slope and minimum slope through the same points. Only gold thicknesses less than 800 A were considered since the points at 1100 A were generally high, as if emission were determined primarily by thin spots. The results are shown in Fig. 7 where L is plotted as a function of volt age applied to the diode. L is nearly constant over the whole range of diode voltages at which emission was observed, possibly rising at high voltages and at low voltages. In Fig. 8, (Ie/lf)d=o= k(V,l,cp) is plotted show ing the steep dependence of emission on applied po tential and the rise in emission above 5 V. Between 5 and 8 V, log k(V,I,cp) ex: V. The major variation in emis sion is found in the pre-exponential factor. There is no apparent change in I e/ If which corresponds to the onset of negative resistance in the 1-V characteristics at 2.8 V. There is, however, an inflection in k(V,l,cp) at 8.2 V which seems to be outside the experimental error of the points and may be associated with electron transitions from the conduction band to the valence band of the insulator, just as was found in Ta205.3 A fundamental question with regard to electron emission from oxide sandwiches, and one that is also AI-AI,O,-Au (::L APPLIED VOLTAGE FIG. 8. Dependence on applied voltage of the pre-exponential factor for the attenuation of electrons emitted into vacuum from Al-AbOa-Au diodes. Oxide thickness, 500 A. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:12PHENOMENA IN AI-AI203-Au DIODES 1891 central to understanding negative resistance, is the means by which electrons receive sufficient energy within the oxide to escape into vacuum when the ap plied potential is only 2 V, since the work function of gold is 4.7 V. Some insight into this problem is provided by using plane-parallel geometry to measure the normal component of the energy of emitted electrons by means of retarding potentials,lo A 100X 100 mesh accelerating grid at a potential of + 7 V was placed between col lector and emitter. Both collector and grid were gold plated stainless steel. A small magnetic field parallel to the direction of emission served to collimate the electrons. For each value of collector potential, the 1-V characteristic of the diode was traced out and electron current to the collector Ie was measured as a function of the potential applied to the diode. In Fig. 9, lei If is plotted as a function of collector-emitter potential Ve for different values of potential across the diode for a sandwich with 500 A of oxide and a gold film 400 A thick. A total of 32 I-V traces were made to obtain the retarding potential curve. The constancy of Iellf above zero collector potential, within the usual variation found, indicates that the contact potential between the acceler ating grid and the collector was small. The accelerating grid was necessary because indium solder contacts to the gold film, needed for proper measurements of the 1-V characteristics of the diodes, resulted in contact potentials that distorted the fields at low collector po- o 95V ~..-::-.""lA...."--. 8 5V o • • • 6.SV ••• Ie " " 11 " S.SV • • 4.5V 10-8 3.5V 3.0V •• 10-1~');-8 --'..l---:c!----'--'j_ 4,--l-ll.-.---!'-2'--"---*O-"--:;:++2 -'---:;c+ Jr-4 --"---;c;+ 6 COLLECTOR POTENTIAL (VOLTS) FIG. 9. Retarding potential measurements. Dependence of the ratio of vacuum emission current to the collector to current through an Al-AJ,03-Au diode on collector potential at constant values of diode potential. Oxide thickness, 500 1; gold thickness, 4001. AI-AI,O,-Au 5 VI (VOLTS) FIG. 10. Ratio of vacuum emission current to diode current at different applied voltages for Al-AJ,O.-Au diodes with varying oxide thickness. Gold thickness, 225 .A. tentials. One source of error that was not eliminated by use of an accelerating grid is the spread in energy due to the resistive drop across the gold film. For the diode of Fig. 9, the maximum potential across the gold film was 0.2 V at V = V m' Above 4 V the resistive drop was less than 0.1 V. Noise and irregularities in emission make the data spread too much to permit taking of derivatives and determining the relative number emitted in each energy range. However, certain qualitative conclusions can be reached. A measurable fraction of the emitted electrons have energies that approach the maximum applied voltage over the whole range of diode voltages for which emission is above noise. For these high-energy electrons, a 1-V increase in applied potential increases the maxi mum energy of emitted electrons by about 0.9 V, for example, at Ie/If= 10-9 in Fig. 9. For applied voltages greater than the work function of gold, cfJAu=4.7 V, there is an inflection in the curve and a large fraction of the electrons appear at lower energy than for low applied voltages. The inflection appears approximately at a collector potential Ve such that Vf+ Vc= 5V, where Vf is the applied potential. Thus two distinct groups of emitted electrons appear, low-energy and high-energy. The current due to high-energy electrons is about 10-8 of the total diode current and is nearly constant for the whole range of diode voltages. The fraction of low-energy electrons emitted increases rapidly as the applied voltage exceeds the work func tion of gold. The largest fraction of the electrons, which [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:121892 T. W. HICKMOTT IP21 PHOTOMULTIPLIER 460m~tl.54'V 10-' .. 2.70 2.82 . .!'j~ ... 440m~ {2.61.V --' ~ 10-7 g >-"I 2.82 .... 2.94 "'-in E Z ... 395m~ {2.96.V 0 ~ <0 --' 3.14 ... .. 1Il 3.36 .... "'-I(T' -1 ~ E 0 !i;l ... ... ~ ... -j --' cl 10-7 ~ t; ... "'- ;oj ... ~ .... ~ ~ > 10-' ~ "'-E ~ 10-7 ~ .. !il 10-' L-..L..-.JJ...--L----''--.l.-...l.----L--L_L....l.-..L---.J 2 3 4 5 6 7 8 9 10 II V, (VOlTS) FIG. 11. Dependence of relative electroluminescent intensity at different wavelengths on voltage applied to an AI-AI203-Au diode. Oxide thickness, 550 A; gold thickness, 200 A. determines the potential between plate and grid in triodes, is readily scattered within the oxide, has very low mobility, and does not have an energy large enough to escape into vacuum. The steep rise in emission when the applied voltage exceeds the work function of gold shows that, at high voltages, the largest fraction of emitted electrons comes through the metal. The characteristic electron emission at voltages less than rpAu from sandwiches that show VCNR has been discussed as if the electrons were also coming through the gold film facing vacuum. If even some of the electrons are emitted through the metal fIlm, there must be some mechanism within the insu lator by which they gain energy. Alternatively, emission may be through pinholes in the gold film. The electron affinity of insulators is much less than the work function of metals22; thus, electrons accelerated through the oxide would need less energy to escape directly from the insulator than to escape through the metal. The best evidencethat electrons are emitted through the metal at low voltages comes from the dependence of energy of emitted electrons on gold thickness. If electron emission occurred through pinholes, their energy for any diode voltage would be determined by the height of the oxide-vacuum barrier and by the energy distribution of electrons in the insulator, and should be independent of gold thickness. Increasing the gold thickness would cut down the intensity of electron emission at low energies by reducing the number and area of pinholes; it should not affect the maximum 22 N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals (Oxford University Press, London, 1948), 2nd ed. energy of the emitted electrons. Thickening the gold film reduces the maximum energy of emitted electrons, and also reduces their intensity, for all values of diode voltage. We therefore conclude that some electrons gain enough energy in the insulator to exceed the work function of gold when applied voltages are greater than about 1.8 V. Pinholes remain a possible, but unknown, source of emitted electrons, but a significant fraction of electrons emitted at low diode voltages comes through the metal film . An estimate of the magnitude of the excess energy which electrons gain in the insulator can be obtained from retarding potential measurements in Fig. 9. The minimum energy electrons need to gain is 4.7 V, the work function of gold.23 The maximum collector po tential Vern to reduce electron emission to noise measures the maximum energy the electrons have gained, but depends on the noise level of the circuit. To derive a value of excess energy Ee, the value of Vern has been taken as the value that reduces Ie/If to SXlO-lO• Ee is defined by (3) where Vf is the diode voltage. The value of Ee varies between 4.1 and 2.9 V; it is constant and equal to 4.1 V for V(:S:4.S V. Ee decreases at high diode voltages where electron energy loss in the gold film may be greater. Thus anomalous emission below 4.7 V is due to the small fraction of electrons that have been accelerated to high energies in the high-field region of the oxide near the cathode of the diode. MeadI4 and Kanter and FeibelmanIo,I6 have used geometries similar to that of the present work in meas uring L in gold, but have used thinner oxide films in which tunneling was believed to be the dominant con duction mechanism. The values of attenuation lengths in Fig. 7 are higher than the value of 100 A reported by Mead for 7-V tunneling electrons or the value of 60 A reported by Kanter and Feibelman. Possible reasons for these discrepancies are differences in gold film structure, differences in the energy distribution of electrons entering the gold film, or differences in scat tering at the metal-oxide interface just before electrons enter the metal film. The latter seems particularly likely to be important since, as low-temperature measurements of I-V characteristics show, the metal-oxide barrier is low in diodes that show VCNR. On the other hand, for tunneling from metal to metal, or from metal to the conduction band of an insulator, to be a dominant con duction mechanism in insulator sandwiches requires a metal-insulator barrier of 1 V or more. The value of L is appreciably lower than the value of 740 A found by photoelectric emission measurements for electrons with energies less than 1 V above the Fermi level of gold.I5 In contrast to the measurements of Sze, Moll, and SuganoI7 who found a decrease in attenuation from 700 to 70 A as the electron energy increased from 1 to 5 e V 23 J. C. Riviere. Proc. Phys. Soc. London B70, 676 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:12PHENOMENA IN Al-AI 20a-Au DIODES 1893 above the Fermi level, L in Fig. 7 is nearly constant in energy. ELECTRON ATTENUATION BY AbOa Discrepancies arise in the measurement of attenu ation lengths for emitted electrons by the oxide film when tunneling structures are usedIO,14,16,IS,24 or when cathodes of the Malter type26-ao are studied. The electron attenuation length in AbOa has been reported as 5 Ns and as 24 N° in very thin tunneling structures. On the other hand, Malter-type cathodes are generally several thousand angstroms thick and yet give significant elec tron emission. Figure 10 shows the dependence of Ie/If on diode voltage for diodes with different oxide thicknesses and a gold thickness of 225 A. Vertical lines on the curves indicate the noise in the electron emission; the greater the thickness of the oxide, the greater the noise, par ticularly at higher diode voltages. The maximum voltage which could be applied to the thinnest oxide film was limited by destructive breakdown and shorting of the film at about 5 V. Careful measurements of the attenu ation of electrons by AbOa films such as have been made for Au films, have not been carried out. Figure 10 shows that the electron attenuation length in AhOa is at least 200 A and probably longer. Electron emission processes from diodes that exhibit VCNR appear to be more closely related to emission from Malter cathodes than from tunneling structures. ELECTROLUMINESCENCE OF Al-AlzOa-Au DIODES Electroluminescence in AhOa films during anodi zation or when immersed in electrolyte has been studied by many investigatorsal-aa. The brightness and wave length of luminescence depend sensitively on impurities in the oxide but mechanisms of electroluminescencehave not been well established. Electroluminescence has also been reported in AI-AhOa-Au and AI-SiQ-Au diodes in which VCNR was established in the I-V character istic.l,lo Examination of the spectral distribution of emitted light using glass filters and a photomultiplier showed the existence of high-energy electroluminescence 24 W. Haas and R. Johannes, Brit. J. App!. Phys. 14, 287 (1963). 26 L. Malter, Phys. Rev. 50, 48 (1936); see also, K. G. McKay, Advances in Electronics (Academic Press Inc., New York, 1948), Vol. 1, p. 1. 26 D. Dobischek, H. Jacobs, and J. Freely, Phys. Rev. 91, 804 (1953). 27 N. Y. Basalayeva, T. M. Yekimenko, M. I. Yelinson, D. V. Zernov, Y. V. Savitskaya, and A. A. Yasnopol'skaya, Radio Eng. Elec. Phys. 6, 1541 (1962). 28 M. M. Vuldynskii, Zh. Tekhn. Fiz. 20, 1306 (1950). 2' M. I. Elinson and D. V. Zernov, Radio Eng. Elec. Phys. 2, 1, 112 (1957). 30 R. Johannes, K. Ramanathan, P. Cholet, and W. Haas, IEEE Trans. Electron Devices ED 10, 258 (1963). 31 See H. F. Ivey, Electroluminescence and Related Effects, Supplement No.1 of Advances in Electronics and Electron Physics (Academic Press Inc., New York, 1963), p. 161 for references. 32 J. Wesolowski, M. Jachimowski, and R. Dragon, Acta Phys. Po!. 20, 303 (1961). 33 L. Lewowska and B. Sujak, Acta Phys. Pol. 23, 13 (1963). .:j~ ~ ~ in z ~ ~ !z tt 1:3 z ;;; :3 51 ~ !oJ 2!: ~ ii1 I P21 PHOTOMUlTIPLIER 10" ..., ~ bi ~ ..., ... on 10" -j ~ ... :ll ~ ... ~ Q Z ... :e ... 10"'--7-'-t--;-~-T5 ~6-!;-7 --:!:8--;9~-;';;10---!":---' V, (VOLTS) 10" ..., FIG. 12. Dependence of relative electroluminescent intensity at different wavelengths on voltage applied to an Al-AIzOa-Au diode. Oxide thickness, 550 A; gold thickness, 200 A. in AI-SiO-Au diodes; some of the emitted light had energy greater than the voltage applied to the diode. Electroluminescence is observed with either diode po larity rather than when Al is anodic as in the AI-AhOa electrolyte system. An AI-AhOa-Au diode of 10 mm2 area with oxide thickness of 550 A and gold thickness of 200 A was used for more careful measurements of the spectral distri bution of electroluminescent light. A 1P21 photomulti plier, operated at 900 V, which was sensitive between 300 and 700 mJl, was used to measure light output. Combinations of Coming glass filters were used to obtain relatively narrow bandpass filters that covered the spectral range to which the photomultiplier was sensitive. The only exception was a filter for the range between 300 and 360 mJl. An evaporated silver film, about 600 A thick, provided a bandpass filter with about 20% transmission in this spectral range.a4 The trans mission of each combination of filters used was measured with a Cary model 14 spectrophotometer. In Figs. 11 and 12, the ratio of photomultiplier current Ip to current through the oxide film If is plotted as a function of diode voltage when different filters were used. The relative electroluminescent intensity has been corrected for variations in the percent transmission of the filters, for variations in the sensitivity of the 1P21 photomultiplier, and for the small variations in the transmission of the gold film with wavelength.a5 These corrections are contained in the quantity C. In obtaining Figs. 11 and 12, repeated tracings of the I-V 34 H. R. Philipp suggested the use of a silver filter. 36 The spectral dependence of the transmission of thin gold films was provided byR. H. Doremus. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:121894 T. W. HTCKMOTT (a) (b) FIG. 13. Photograph of electroluminescence from the AI AhOz-Au diode used in Figs. 11 and 12. Bright spots in (b) show luminescent areas. characteristic were necessary. J -V characteristics were reproducible to about ± 10% in taking the data in these figures. Several traces were made with each filter and the particular trace shown in Fig. 11 or Fig. 12 was representative of measurements at each wavelength. The wavelength that labels each of the curves was the wavelength for maximum transmission of the filter; the width of the filter between wavelengths of zero trans mission is shown in terms of energy in the brackets. Certain features are common to all the curves in Figs. 11 and 12. No light emission at any wavelength was detected below 1.6-1.8 V, the same voltage at which electron emission into vacuum rises above noise. Above the threshold voltage, electroluminescence in creased extremely rapidly; the relative intensity more than doubled for each 0.1-V increase. The peak in the relative intensity was at about 3.5 V, followed by a marked quenching at about 4 V and a rather gradual rise above 5 V. Both the relative and absolute intensity of electroluminescence decreased rapidly above 3.5 V since the diode current also decreased rapidly. The de crease in the relative intensity above 4 V contrasts strongly with electron emission from similar diodes which increases steeply above 5 V, as in Fig. 1. Superimposed on the broad spectrum of luminescence are three emission bands of high relative intensity. The strongest of these is the peak at 504 m,u whose maximum relative intensity is 20 times higher than that at 475 m,u and 10 times higher than that at 545 m,u. The second peak is at 320 m,u. The relative intensity of this curve is about 10 times higher than that of the line at 360 m,u, although measurement of the intensity of the line at 320 m,u is more uncertain since it is close to the cutoff wavelength of the photomultiplier at 300 m,u. The third maximum in relative intensity of electrolumines cence lies near 720 m,u. Thus there is evidence for emis sion bands of high relative electroluminescent intensity between 3.4 and 4.5 V, between 2.2 and 2.6 V, and be tween 1.4 and 1.8 V, these being the energy bandpasses of the corresponding filters. As with AI-SiO-Au diodes, electroluminescence at low voltages is characterized by energies greater than the applied voltage. Electroluminescence from AI-Ab03-Au diodes occurs from bright spots scattered at random on the diode. After measuring the spectral distribution of electro luminescence, the diode of Figs. 11, 12 was photographed in order to show the distribution of emitted light. In Fig. 13(a), half of the 1O-mm2 diode is shown at X35 magnification using Polaroid type 57 film. Ab03 formed by anodization in molten bisulfate eutectic often shows growth patterns on the oxide and these are clearly visible. After photographing the diode, the J -V chara~ teristic from 0 to 10 V was traced 30 times while the camera shutter was open. Repeated tracings were necessary in order to have sufficient intensity to photo graph light spots. Bright spots in Fig. 13(b) show lumi nescent regions of the diode. About forty spots appeared on the diode. The majority of the spots were stable during repeated tracings of the J-V characteristic although a few spots would appear or disappear after a small number of traces. Such spots would not be photo graphed. Every luminescent spot on the diode was associated with a visible dark spot or flaw on the diode, although there were many flaws which had no lumines cence associated with them. An attempt was made to correlate luminescent spots and regions of high electron emission by using a phos phor screen to view electron emission. Emission from this particular diode was from spots at high voltages. Below 5 V, electron emission was less spotty but was not uniform over the diode area. Some correspondence between flares of luminescent intensity and increased electron emission was observed, but regions of maximum electron emission did not generally coincide with lumi nescent spots. The question of whether conduction is primarily through a small number of flaws in the insu lator or more widely distributed in area remains to be resolved, although it is probable that conduction is not uniform over the diode area. The maximum average current density for the diode of Fig. 13 was about 5 A/cm2• When all conduction was through luminescent spots, the current density in the conducting area was about 104 A/cm2 since the area of each spot was about [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:12PHENOMENA IN AI-AI 20s-Au DIODES 1895 10--6 cm2• Careful correlation of measurements of elec tron emission, electroluminescence, and destructive di electric breakdown of diodes might provide a conclusive answer. It is clear that luminescence occurs at singu larities in the oxide; however, the processes that control luminescence are controlled by diode voltages and indi vidual bright spots increase and decrease their bright ness together. N arrow-bandpass dielectric interference filters ob tained from Bausch & Lomb were used to examine more closely the relative electroluminescent intensity of the spectral region covered by the bandpass filter at 504 m~ in Fig. 12. The maximum electroluminescent intensity lay between 530 and 570 m~, corresponding to electron transitions of 2.2 to 2.3 eV. DISCUSSION Table I shows values of the energy level differences in Al20a required for the model discussed in Ref. 1, and deduced from experimental measurements on AI AhO"Au diodes. The third column gives the experi mental evidence from which the number is derived. Conduction processes in the insulator can be discussed in connection with Fig. 14, which is a schematic repre sentation of the model at different diode voltages. Thicknesses of the high-field and Ohmic regions of the insulator are not in correct proportion. The bandgap of insulators, in general, is poorly de termined. From measurements of the optical absorption edge, the bandgap of a-AhOa is greater than 8 va6; no value has been reported for anodized AhOa although a value of 6.5 V was suggested to account for electrolytic rectification.a7 A bandgap of 8.2 V, derived from the decrease in electron emission which occurs at that voltage in Fig. 8, is reasonable but has no other experi mental confirmation. One experiment suggests that Ee-El = 4.1 eV. Al-AbOa-Au diodes, as prepared, have very high resistance. Conductivity and VCNR are developed by application of voltage to the diode. For impure AbOa diodes, such as those anodized in fused bisulfate eutectic, the voltage to develop conductivity is approximately constant, independent of the thickness of the oxide, and is 4.1 V.4 In Fig. 14, this voltage would be associated with the ionization of impurity atoms in the midbandgap. According to the simple theory of the barrier height at a metal-insulator interface, Ee-Ef=rPmi=rPm-X, where rPm is the metal work function and X is the electron affinity of the oxide.22 rPAI=4.20 eV and rPAu=4.70 eV.2a Electron affinities of insulators are not well determined; for MgO, which behaves similarly to AbOa in Malter effect cathodes, x< 1 eva8 and X may be closer to 0.1 eV.a9 This would suggest that Ec-EF=3.2 to 4.1 eV 38 A. Gilles, J. Phys. Rad. 13,247 (1952). 37 A. W. Smith, Can. J. Phys. 35, 1151 (1957). 38 J. R. Stevenson and E. B. Hensley, J. App\. Phys. 32, 166 (1961). 39 A. B. Laponsky, Rept. 23 MIT Conference on Physical Electronics (1~63), p. 152. (01 o VOLTS Ec VA~M Ec ~ At OXIDE fAU~ Au ffi El z EF E .... Fz 0 e: EH ------- EH ~ Ev Ev (cl 4 VOLTS Ec VACUUM EF Ec ~ OXIDE fAu~ Au ffi El z 0 EF~ ~ .... EH Ev ., (bl 6 Ec 4 At 2 EF 0 EH -2 Ev -4 (dl Ec EF 4 2 0 -2 -4 OXIDE 2 VOLTS VACUUM Eo Au ,---,E:.::l.J--J EF EV VACUUM Ec Au 'Au EF Ey FIG. 14. Schematic diagram of the potential distribution at dif ferent voltages applied to an AI-AI.Os-Au diode which exhibits VCNR in its I-V characteristic. at the Al-Al 20a interface, which is close to the value found. Values for the barrier height at the AI-AbOa interface have been derived from measuring]- V charac teristics of Al-Al20a-metal diodes and analyzing the data in terms of tunnel emission or of Schottky emis sion. Values of the barrier height which have been re ported are 0.35,40 0.72,410.74,42 0.78,431.58,441.64,451.8,18 2.2,46 and 2.0-2.5 eV,47 for AbOa films prepared by anodization or by thermal oxidation. The variety of experimental values indicates the problems in comparing experiment and theory; it is noteworthy that the bar riers are all low when one considers values of work functions and electron affinities. One possible explan ation for the discrepancy between different values is that conduction occurs through impurities in the insu lating oxide film rather than through the true conduc tion band of the insulator. The impurity distribution, in turn, would be determined by details of preparation of the insulator and would be expected to vary from laboratory to laboratory. As the voltage across an Al-Al 20a-Au diode is raised, three different phenomena occur at about 1.8 V. Elec troluminescence and electron emission into vacuum both 40 G. T. Advani, J. G. Gottling, and M. S. Osman, Proc. IRE 50, 1530 (1962). 41 M. Hacskaylo, J. App\. Phys. 35, 2943 (1964). 42 P. R. Emtage and W. Tantraporn, Phys. Rev. Letters 8,267 (1962). 43 T. E. Hartman and J. S. Chivian, Phys. Rev. 134, A1094 (1964). . 44 S. R. Pollack and C. E. Morris, J. App!. Phys. 35, 1503 (1964). 46 T. E. Hartman, J. App\. Phys. 35, 3283 (1964). 46 J. Nakai and T. Miyazaki, Jap. J. App\. Phys. 3, 677 (1964). 47 D. Meyerhofer and S. A. Ochs, J. App\. Phys. 34, 2535 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:121896 T. W. HICKMOTT TABLE 1. Energy levels in Ah03. Energy Energy Experimental source levels (eV) Eo=Ec-Ey 8.2 (1) Electron emission Ec-EI 4.1 (1) Development of conductivity AI-AhOs-Au diodes in E1-EH 2.3 (1) Electroluminescence (2) tJ21/dV2 becomes negative EH-Ey 1.8 (1) Appearance of electron emission and electroluminescence (2) Voltage to redevelop conductivity (3) Electroluminescence E[-Ey 4.1 (1) Electroluminescence (2) Excess energy of electrons emitted into vacuum rise steeply out of noise. In Ref. 1, these two phenomena have been associated with the formation of mobile holes in the valence band. Neutralization of holes by elec trons from the impurity band can result in a radiative transition or in excitation of an electron from an im purity band to the conduction band. According to Fig. 14(b), the mobile hole is formed by transition of an electron from the valence band to the hole level when the voltage exceeds 1.8 V. The conductivity of an AI AI20a-Au diode can be reduced to very Jow values by exceeding V m, going into the negative resistance region of the J-V characteristic, and suddenly turning off the diode voltage. The low conductivity thus established is stable until the diode voltage exceeds about 1.8 V; as the diode voltage exceeds 1.8 V, conductivity re develops until the maximum diode conductivity is reached. Increased diode conductivity is associated with an increase in the number of ionized states in the impurity band capable of contributing to conductivity. It is possible that the number of these states increases because of electron transitions into the valence band or hole band from the impurity band. As the voltage across an AI-AhOa-Au diode exceeds about 2.3 V, the second derivative of the J-V charac teristic becomes negative.2 In Fig. 14, this is associated with the onset of neutralization of sites in the impurity band by electrons from the hole band EH, which results in a decrease of conductivity. The voltage for maximum current, 2.8 V, is the voltage at which the decrease in conductivity due to neutralization of impurity centers exceeds the increase due to their formation. The excess energy of emitted electrons, particularly those emitted at diode voltages less than the work function of gold, is about 4.1 V and is taken to be the energy difference between the impurity band and the valence band. For electrons in the impurity band to gain this energy and be emitted, Er-Ev>Ec-EI; otherwise electrons would not be excited into the conduction band but would be trapped in the insulator. Maxima in the elec troluminescence at wavelengths corresponding to about 1.8, 2.3, and 4.0 eV also support the assignment of levels in Fig. 14 and Table I. The energy level scheme in Fig. 14 is consistent with experimental data but is obviously speculative. Several questions remain unanswered. In Figs. 11 and 12, a sharp quenching of electroluminescence occurs at about 4.0 V, and affects electroluminescence at all wave lengths. This could be due to a decrease in the number of electrons in the impurity band capable of making radiative transitions, to a decrease in the number of sites in the hole or valence band capable of receiving electrons, or to a reduction in the probability of an electroluminescent transition. It is not clear which is the controlling factor. The reduction in electron emis sion into vacuum with decreasing temperature when the complete I-V characteristic is traced out, shown in Fig. 1, appears to be related to the change in diode resistance when it is warmed up, as shown in Fig. 4, since both phenomena occur in the same temperature range. The development of conductivity at low temper atures depends on both temperature and field; the impurity band of Fig. 14 may have energy levels sepa rated by relatively small energies. It has been shown that V m in metal-oxide-gold diodes depends on the dielectric constant of the insulating oxide; the higher the dielectric constant, the lower V m is.2 If the model of Ref. 1 is correct, this would imply that the separation of Er and EH depends on the dielectric constant and that the two levels are connected. The nature of their dependence remains to be explained. Finally, the ques tion of whether Fig. 14 applies only to singular areas of the diode, whether all the diode current passes through those spots that are electroluminescent, or whether a larger fraction of the diode area is involved in conduc tion, needs to be investigated. ACKNOWLEDGMENTS The author is indebted to F. S. Ham for many helpful discussions as well as for a critical review of the manuscript. The photomultipliers and filters used to study electroluminescence were kindly provided by D. T. F. Marple. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.203.227.62 On: Fri, 28 Nov 2014 20:39:12
1.1713823.pdf	Potential Distribution and Negative Resistance in Thin Oxide Films T. W. Hickmott Citation: Journal of Applied Physics 35, 2679 (1964); doi: 10.1063/1.1713823 View online: http://dx.doi.org/10.1063/1.1713823 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in SWITCHING AND NEGATIVE RESISTANCE IN THIN FILMS OF NICKEL OXIDE Appl. Phys. Lett. 16, 40 (1970); 10.1063/1.1653024 A Reply to Comments on the Paper ``Potential Distribution and Negative Resistance in Thin Oxide Films'' J. Appl. Phys. 37, 1928 (1966); 10.1063/1.1708626 Comment on the Paper ``Potential Distribution and Negative Resistance in Thin Oxide Films'' J. Appl. Phys. 36, 2329 (1965); 10.1063/1.1714482 Impurity Conduction and Negative Resistance in Thin Oxide Films J. Appl. Phys. 35, 2118 (1964); 10.1063/1.1702801 Negative Resistance in Thin Anodic Oxide Films J. Appl. Phys. 34, 711 (1963); 10.1063/1.1729342 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:53JOURNAL OF APPLIED PHYSICS VOLUME 35, NUMBER 9 SEPTEMBER 1964 Potential Distribution and Negative Resistance in Thin Oxide Films T. W. HICKMOTT General Electric Research Laboratory, Schenectady, New Yark (Received 17 February 1964) AI-SiO-AI-SiO-Au triodes with SiO thicknesses between 150 and 500 A have been used to measure the po tential distribution in thin oxide films before, during, and after the development of voltage-controlled nega tive resistance (VCNR) in the current-voltage characteristics. Development of VCNR in the triode is accompanied by the establishment of a high-field region about 120 A in thickness near the negative electrode. If triode potentials are reversed after developing conductivity, VCNR is still found in the current-voltage (I-V) characteristic of the triode but the potential distribution in the triode is only slightly changed. VCl'\R in the /-V characteristic is a high-field phenomenon but it does not depend on field emission of electrons from the metal electrodes. Conductivity in the bulk of the insulator is Ohmic with electron mobilities ",1O-L1O-2 cm'/V-sec. The behavior of AI-SiO-Au diodes is identical to that of triodes. Electroluminescence of AI-SiO Au diodes. which appears when conductivity is developed, is characterized by a steep rise in intensity at 1.i-i V, the voltage at which electron emission into vacuum from such diodes is first detected. Both electro luminescence and electron emission provide evidence for high-energy processes in the oxide film. A phe nomenological model of conductivity and voltage-controlled negative resistance in thin oxide films is de veloped in which impurity conduction is the most important conduction mechanism. VOLTAC;E-CONTROLLED negative resistance has been observed in the current-voltage character istics of metal-oxide-metal diodes in which the oxide thickness varies between about 100 and 20000 A.1-9 Insulators for which the effect has been observed include Ab03, SiO, Ta205, I':r02, Ti02, MgO-Ab03, and MgO. The voltage V m for maximum current through such diodes depends on the dielect ric constant of the oxide and, to a lesser extent, on the metals which form the electrodes of the diode.1.9 However, V m is independent of the thickness of the oxide. For heavily doped or im pure insulators, establishment of conductivity and nega tive resistance by application of voltage to the diode also depends primarily on diode voltage and is independent of oxide thickness.9 Although diode voltage controls the current-voltage characteristics, the insulating films are so thin that the fields are high for small applied voltage. Determination of potential distributions in the oxide film before, during, and after the establishment of diode conductivity is of primary importance for understanding the conduction mechanisms. It has been found that negative resistance can be developed in triode structures, metal-oxide-metal oxide-metal sandwiches.1 The current-voltage charac teristics of such triodes are identical to those of diodes containing the same oxide. The central metal film can be used as a probe to measure the potential distribution within the oxide when conductivity is developed be t ween top and bottom electrodes. Information about J T. W. Hickmott, J. App!. Phys. 33, 2669 (1962). 2 T. W. Hickmott, J. App!. Phys. 34, 1569 (1963). 3 H. Kanter and W. A. Feibelman, J. App!. l'hys. 33, 3580 (1962). 4 G. S. Kreynina, L. N. Selivanov, and T. 1. Shumskaia, Radio Eng. Elec. Phys. 5, 8, 219 (1960). , G. S. Kreynina, Radio Eng. Elec. Phys. 7, 166 (1962). fi G. S. Kreynina, Radio Eng. Elec. Phys. 7, 1949 (1962). 7 S. R. Pollack, W. O. Freitag, and C. E. Morris, Electrochem. Tech. 1,96 (1963). 8 S. R. Pollack, J. App!. Phys. 34, 877 (1963). 9 T. W. Hickmott, J. Appl. Phys. 35, 2118 (1964). fields in the insulator obtained in this way provides a basis for a qualitative model of conductivity and nega tive resistance in metal-insulator-metal diodes. EXPERIMENTAL One triode configuration that has been used to study the potential distribution and field during forming of conductivity and negative resistance of oxide films is shown schematically in Fig. 1. Samples were formed by evaporating the base aluminum strip (cathode), evapo ra ting silicon monoxide of desired thickness, evaporating an aluminum layer "-' 150 A thick (grid), evaporating a second silicon monoxide layer, and then evaporating a 350-A gold layer (plate) on top. These electrode desig nations, plate, cathode, and grid, will be used regard less of the polarity of the triode voltage. The aluminum cathode and grid had a layer of ~20 A of oxide form prior to evaporation of SiO. Silicon monoxide films of FIG. 1. Prepara tion of AI-SiO-AI SiO-Au triodes and the circuit for meas uring their electrical characteristics. ~~~ EVAPORATE EVAPORATE EVAPORATE ALUMINUM SiO ALUMINUI/ CATHODE GRID ~ ... ~ ..• I"': ". ~ , . EVAPORATE EVAPORATE SiO GOLD PlATE 2679 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:532680 T. \\. HIe K MOT T desired thickness were deposited by evaporation of 10 mesh vacuum-degassed SiO from a molybdenum boat a tara te of '"V 40 AI sec and at pressures of '" 1 X 10-5 Torr. SiO depositions were monitored by a quartz crystal deposition thickness monitor. By varying rela tive thicknesses of insulating layers, the position of the grid between plate and cathode could be varied. Triode areas were approximately 4 mm2• There was no direct path from cathode to plate. In a second type of triode, the insulating layer between cathode and grid was pro duced by anodizing the cathode layer in fused KHS0 4- NH4HS04 eutectic to a desired thickness.I A relatively thick aluminum layer was then evaporated and anodized in boric acid-sodium borate solution to an oxide thick ness of 230 A. The thickness of the aluminum grid after anodizing was then unknown, but the grid had electrical continuity. A gold electrode was then evapo rated on top of the triode. Results obtained on the potential distribution of the two types of triodes, using different insulators and different geometrical configu rations, were qualitatively similar. Since triodes using SiO were better characterized, results obtained with them will be discussed in some detail. The circuit used for determining the potential distri bution in triodes is also shown schematically in Fig. 1. Three-probe potential measurements were made with triodes as in Fig. 1; plate and cathode resistances were low enough that errors due to resistive drops in the metal films were small. Potentials were measured with a grounded Keithley 610A electrometer and with a I ! 1. 1 ~ 10 5 10 '" E! :; 4 8 4~ 0 ~ ~ ,,~ ~ l< l< ~ !pc/ \ > 69 !pc( v. > -6 :; 3 I \ 3:; ~ rJ " .. I .. I ~ ~ >~ -/V9C /' ~4 /V9C \ 2 4 I /' I ./ ·.-v~9 /, ./ /'VP9 /( /' v/ ,/ :.< 36 .f. ..2 12 ~ 12 30 10 30 10 V> V> ~ 524 !:; 0 ~ 8 8;: '2 ~ ~ ;18 ~6 6 ::: > > 4 Vpe (VOLTS) FIG. 2. The potential distribution in triodes at different stages of the development of plate-cathode conductivity. Note scale changes for each of the curves. Keithley 600A battery operated electrometer which did not require a ground connection. Both electrometers had 1014_Q input impedance. Voltages were provided by a battery and voltage divider network, or by a program able power supply capable of delivering one ampere; X Y recorders were used to record all electrical quantities. THE POTENTIAL DISTRIBUTION IN TRIODES Conductivity was developed in the triode by applying voltage between plate and cathode V pc with the gold plate positive. Some of the characteristic features of the potential distribution at different stages of forming are illustrated in Figs. 2 to 4 in which currents and poten tials for repeated tracings of the current-voltage charac- 50r ~ §:f £1. 30 u; ~ 40 24 ~ o - '0 > ... .:; 12 12~ 18 ; K K > . ~ 8.'0: 12 :r '- ~ 12 12 ~ 2.4 2.0 '58 1.65 2 ~ ~ 1.21 teristics of a typical triode are illustrated. The triode had 150 A of insulation between cathode and grid, and 450 A between grid and plate. The numbers of each curve designate its number in sequence of developing conductivity. The particular curves are chosen to illu strate salient features of the potential distribution. Unless otherwise indicated, all I-V (current-voltage) curves are for increasing triode voltage; generally, triode currents are smaller for decreasing voltage than for increasing voltage. I pc and V pc symbolize current aml voltage when the plate is positive; Iep and Vep are used when the cathode is positive. Scale factors vary from curve to curve in Figs. 2-4. In the newly made triode, resistance between all pairs of electrodes was very high and leakage currents were [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:53POTENTIAL DISTRIBUTIOK IN THIN OXIDE FILMS 2681 less than 10-10 A for 1 V across the triode. With the circuit of Fig. 1, a spontaneous potential appeared be tween grid and plate, and between grid and cathode, which could be as large as 1 V but may have been of an instrumental nature. This potential disappeared after resistance of the triode was reduced during development of conductivity. Conductivity first appeared in the tri ode at about 4.6 V. In Fig. 2(a), as Vpc was increased, I pc showed negative resistance and the potent~al divid~d nearly evenly between Vpg and VgC, approxImately m the ratios of their resistances. Although V pg rose sharply and V gc dropped as I pc increased at 4.6 V, this voltage change was not permanent; in Fig. 2 (b), the potential in the triode divided between V po and Vue in nearly the same ratio as in Fig. 2(a). The most characteristic feature of the potential dis tribution in a triode as negative resistance becomes fully developed is that nearly the whole potential d~op ap pears at the negative electrode, between gnd and cathode. In Fig. 2(b), as I pc increased steeply between 5.5 and 6.0 V, V pg and V go fluctuated erratically. At 6.0 V, I pc climbed steeply, V pg dropped, and V gc becan:e nearly equal to V pc. This was a perman<:nt change. 1I1 the potential distribution. As long as a smgle polarIty was applied to the triode, the potential drop remained primarily between grid and cathode. In Figs. 2(c) :,nd 2(d), and Fig. 3(a), I pc and V po are shown as functlOns of V pc after the negative resistance characteristic was fully developed by increasing the triode voltage to 10 V. V was a small fraction of the applied voltage V pc and w~; determined by I pc j I pc in turn was determined by processes happening between grid and cathode where the primary potential drop occurred. ConduCtance and negative resistance can be de veloped between plate and cathode without developing significant conductance to the grid. For Fig. 2(c), resist ance between plate and grid and between grid and cathode was greater then 200000 n if measured by ap plying voltage directly to the grid. Further tracings of the current-·voltage curves resulted in conductivity de veloping to the grid, and for Fig. 2 (d), Rpg = Rye = 1250. n if measured directly to the grid. Rpo= V po/ I pc= 2 Q III both Figs. 2(c) and 2(d), which is much less than Rpg measured between plate and grid. The effective resistance of the triode appears to be determined at the metal-insulator interface. Potential mea surements on triodes before conductivity had been developed to the grid, as in Figs. 2(a) to 2(c), are some what uncertain since the metal-insulator contact of the grid may be non-Ohmic, or Ohmic but with ahighresist ance. They are shown to illustrate the sequence of behavior of a typical triode. In some cases, conductivity was developed to the grid by applying potential di rectly between plate and grid, but this was not done with this triode. Developing conductivity between either plate and grid or cathode and grid developed conduc tivity between the grid and the other electrode and E 40 2.0 32 1.6 '" :; 0 1.22: ~ lop 0.8 Q4 4 6 Vep (VOLTSl 20 25 10 Vep iVOLTSl FIG. 4. The potential distribution in triodes ~t .different stages of the development of plate-cathode conductiVity. Note scale changes for each of the curves. usually changed V pg/ I pc very little. Figures 2(~) . to Fig. 3 (a) are typical of the development of conductlVlty and negative resistance when only one polarity has been applied to the triode. Figure 3(a) shows the current voltage characteristic that had developed just before polarity was reversed across the triode at curve 20. Reversal of polarity of the voltage applied to the triode making cathode positive and plate negative, re sulted'in a decrease of lcp, a broadening of the peak in the current-voltage characteristic, and a shifting of V m to ""3.5 V where it did not coincide with the maximum of Vgp at about 4.5 V. lcp and Vgp were b?th more erratic and noisier than with the original polarIty. How ever negative resistance was still found, as shown in :Figs: 3(b) and 3(c) and the primary potential drop re mained between grid and cathode. Vgp was higher than for the initial polarity and it did not follow Iepas closely. The mechanism responsible for negative resistance does not depend on having a high field at the negative electrode of the triode; field emission from the negative electrode does not determine current-voltage characteristics or negative resistance. With further development of the current voltage characteristic by repeating tracings with cathode positive and plate negative, Vop increased and the. trace shown in Fig. 3(d) was obtained. Vop had a maXimum value of 1.77 V for Vcp=4.3 V. Potentials in the triode can shift when polarity is reversed though the process may be a slow one and is not essential for negative resistance. Although Vgp increased markedly, it ap proximately followed I cp, while V cg remained a mono tonically increasing function of V cpo [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:532682 T. W. HICKMOTT Restoration of the original polarity with plate positive and cathode negative restored the high value of I m and again reduced V po to relatively low values, as shown in Fig. 4 (a). V po was proportional to I pc for V pc < 4 V but above 4 V, V pg became somewhat erratic and the ratio I pc/ V pa dropped steadily, instead of remaining constant as in Figs. 2(c) and 2(d), and Fig. 3(a). When polarity was again reversed, making cathode positive and plate negative, the traces in Fig. 4(b) were ob served. Initially, most of the potential drop remained between cathode and grid. At Vcp=2.4 V, Vea dropped abruptly and became proportional to I cpo .4 sudden shift oj potential jrom one portion oj the oxide to the other occurred. As long as this polarity was maintained, the potential drop was primarily between grid and plate. VCg was very small, rather erratic, and roughly pro portional to lep as shown in Fig. 4(c). In curve 45, which is not shown, the original polarity was restored again, and Veo remained small and pro portional to I pc' In Fig. 4(d), run 46 is shown. For in creasing voltage, V co and I pc were proportional; the high-field region of the insulator was at the anode instead of the cathode. When the voltage was decreased, the potential abruptly shifted to the grid-cathode region at V pe= 4.6 V, and V Pa once again became proportional to I pc' Certain features illustrated in Figs. 2 to 4 are typical of the potential distribution in triodes that show nega tive resistance between plate and cathode. The general features are the same whether SiO or Ab03 is the insu lator of the triode. (1) Conductivity between electrodes in newly made triodes is low. Potentials applied between plate and cathode divide between the grid-cathode and plate-grid regions in the ratio of resistances of the two regions. When 1=.4 exp(BV), as is the case for many of the oxide films before negative resistance is established, the apparent resistances of the films are voltage dependent. This may be reflected in the voltage division between the two portions of the oxide film as V pc is increased. For very low I pc, negative resistance may be found when voltage division of this kind is present, after the initial development of conductivity. (2) Development of conductivity and negative re sistance between plate and cathode is followed by markedly nonlinear potential distributions in which nearly all the potential drop occurs at the negative electrode, between grid and cathode. This region is characterized by extremely high fields. Twelve to thir teen volts can be applied to triodes before destructive breakdown occurs. For the triode in Fig. 2, F ge= 8X 106 V/cm at Vpe=12V, and the actual field may be even higher. Likewise, in diodes with 120-A insulating films, negative resistance can be established and character istics are similar to diodes with greater insulator thick ness. The critical processes determining conductivity and negative resistance occur in a region of the oxide that is less than 120 A thick. (3) Development of conductivity to plateandcathode does not necessarily result in conductivity to the grid. Grid conductivity may be developed separately or it may develop during establishment of plate conductivity. (4) Negative resistance and high conductivity are still found in triodes when the voltage is reversed, although the potential drop remains in the grid-cathode region. High fields are not required at the electron emit ting surface to have either high currents or negative resistance. (5) The high-field region within the triode can shift from the cathode-grid region to the plate-grid region, in which case the potential between grid and cathode will be determined by lcp. ~egative resistance in thin insulating films is accompanied by a high field somewhere within the insulator. If the sandwich structure has been cycled repeatedly with one polarity, the high-field region will generally be at the negative electrode but it may be located almost anywhere in the insulator. The high field region may shift within the insulator without its shift being detectable in the I-V characteristics. Com paring Fig. 2(d) and Fig. 4(d), it would be difficult to tell from I pc-V pc curves that the primary potential drop was between grid and cathode in the first case and be tween grid and plate in the second. (6) A characteristic stage in the development of con ductivity is illustrated in Figs. 2(c) and 2(d). Vpo is proportional to I pc for all, or nearly all, values of V pc, and the plate-grid region has Ohmic conductivity. I pc in turn, is determined by processes occurring in the grid-cathode region. The current density between plate and grid when conductivity is Ohmic is ] pg= ] pc= (nj.J.)poeFpy= (nj.J.) pfieV po/ dp!I' (1) where npfl is the number of charge carriers/emS, j.J.po is the carrier mobility in cm2/V-sec, V PO is the plate-grid potential, dpo is the plate-grid separation, and e is the electron charge. (nj.J.)po can be measured for triodes with different plate-grid distances by using current-voltage curves when conductivity is developed to the same ex tent as in Figs. 2(c) and 2(d). In Table I, dpg, dge, and d pc are given for seven triodes with SiO insulation. These values were derived from measurements of capacitance on the triodes, assuming a dielectric constant for SiO of 6. This value of the dielectric constant may be high since the dielectric constant of SiO depends on evapo- TABLE 1. Charge carrier concentration and mobility in triodes. dpy due dpc (nf.L)pg n1,o (10-6 (10-' (10-6 1014/cm- f.Loc J.LP!/ 1016/ Triode em) em) em) V-sec cm'/V-sec cm'/V-sec cm3 1 4.4 1.5 5.9 4 6.5XlO-s 1.5XlO-2 2.7 2 4.1 2.2 6.1 4.5 5.7XlO-s 1.3 X 10-2 3.5 3 2.8 3.7 6.2 1.8 3.7XlO-s 3.2XlO-3 5.6 4a 1.9 4.4 5.3 1.5 3.4XlO-s 1.3 X 10-3 12 4b 1.9 4.0 6.0 2.5 4.3XlO-5 3.5XlO-3 7.1 Sa 1.0 4.5 5.1 0.55 5.8XlO-s 4.4XlO-4 13 5b 0.9 5.0 5.5 0.98 2.4X10-5 8.0XlO-4 12 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:53POTE:.JTIAL DISTRIBUTI01\ 11\ THI:.J OXIDE FILMS 2683 ration rate.1O Development of conductivity of sample 1 is illustrated in Figs. 2-4. In Fig. 5, (n}J.) pg is plotted as a function of V pc for selected experimental curves that are comparable to Fig. 2(d), in which Ipc/V pg is con stant, at least for 0< V pc< 6 V. The constancy of (JI}J.) P!l for individual samples and the order of magnitude constancy from sample to sample indicates that the jlotential in the region between plate and grid is linear and conductivity is Ohmic. In a number of samples, (ilP.) I'll decreased steeply at higher voltages. This may be due to a penetration of the grid-cathode poten t ial drop into the plate-grid region with a consequent increase of V pg and an apparent reduction of (np.) pg. In Fig. 4(d), during run 46, (11}J.)"" for V pc> Vm was ~.2(H)13/cm-V-sec). When V w was decreased during run 46, the potential drop abruptly shifted to the grid cathode region; for run 47, Vpg was proportional to Ipc for Vpc< Vm. (np.) po, determined for run 47, was 7.9 (1Q13/cm-V-sec). The equality of (np.)"c and (11P.)py when both are proportional to I pc is evidence that the conduction mechanisms in th'e two portions of the tri odes are the same, and that 11 and}J. are nearly the same in both regions. Electron emission into vacuum from triodes shows the same characteristic features as have been reported for AI-SiOAu diodes.2 In Fig. 6, the ratio of electron emission into vacuum Ie to current through the triode I pc is plot ted for three triodes. The triode numbers cor respond to Table 1. A st.eep rise in electron emission 10' I-TRIODE # • -I 0-2 • - 3 0-40 • -4b .. -5a A -5b • • • • • Q o o A<>8 00 A • 0 " " •• • • Q - lo'2~.-.L.-;-----'--t-.....l..-F6 -"'--t-"'-----.i;,O----' Vpc (VOLTS) FIG. 5. Dependence of (n/l,)po on plate-cathode voltage for triodes with different plate-grid separations. ---- \0 D. R. York, J. Electrochem. Soc. 110,271 (1963). --TRIODE I --TRIODE 2 -----TRIDDE Sb ____ f Ie BELOW NOISE 5 6 Vpc (VOLTS) FIG, 6. Electron emission into vacuum from three AI-SiO-AI SiO-Au triodes with different grid-cathode spacings. Gold thickness, 350 A. around 2.5 V was followed by a leveling off or decrease in the ratio of Ie/lpc and by a second rise in emission above the work function of gold. In general, the fraction of electrons that were emitted from the triode was smaller than for diodes, the voltage at which emission first appeared was higher, and the noise in the emitted current was greater than for diodes. However, the quali tative emission characteristics remained unchanged. A qualitative picture of t.he potential distribution in triodes with well developed conductivity and for an arbitrary plate-cathode potential is shown in Fig. 7. The positions of the Fermi levels of cathode, grid, and CATHODE + + DISTANCE (1) FIG. 7. Schematic diagram of the potential distribution in triodes after development of plate-cathorle conductivity. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:532684 T. \V. HIe K MOT T plate are well defined when a voltage is applied but potentials in the insulator are only schematic. No energy band relations are indicated within the insulator nor is the nature of the metal-insulator interface considered. The maximum field in the insulator is probably greater than the average field which is shown. The processes that determine conduction and negative resistance are shown as concentrated in a region of less than 120 A near the cathode after forming of the current-voltage characteristic. This would occur if the triode were cycled several times with the plate positive. Electrons gain energy in the high-field region, but most charge carriers would be collected by the grid. To maintain current continuity, electrons are injected from the grid into the bulk of the oxide where they diffuse and scatter to such an extent that the concept of a mobility for the charge carriers is valid for the bulk of the oxide film. Some of the electrons, accelerated between grid and cathode, scattered in passing through grid and oxide, and then collected by the gold film, may also possess enough energy to escape into vacuum through the gold. The fraction that is emitted is a relatively small fraction of the total number of charge carriers that determine the plate-grid potential. In Fig. 7, the potential drop is shown between grid and cathode. The occurrence of voltage-controlled negative resistance (VCNR) does not depend on the potential drop being in this particular region of the oxide, as shown by Fig. 4(c) or Fig. 4(d). However, VCNR which may occur with either polarity of the triode does depend on having a high-field region somewhere in the insulator. In Table I, values of (np,)py for V < V m are given for triodes in which the plate-grid separation has been varied. Their near constancy as well as the dependence of V py on 1 pc show further that plate-grid resistivity is Ohmic. No independent values of mobility of charge carriers in oxide films have been reported but it is possi ble to derive an approximate value. The current density through an ideal space-charge-limited diode is given by (2) where K is the dielectric constant of the insulator, P,o is the carrier mobility, V is the applied voltage, and d is the diode thickness. Equation (2) is derived on the as sumption of an Ohmic metal-insulator contact and no trapping effects within the insula torY If trapping effects reduce the fraction of charge injected into the diode that passes through the insulator, Eq. (2) is modified by substituting an effective mobility P,e for the true mobility, but the same current-voltage relationship is retained. Provided that V < V m, the current through a diode or triode with fully developed conductivity is proportional to V2.! We therefore assume that the triodes in Table I act as ideal SCL diodes in order to derive a value of mobility from Eq. (2). This, however, is the value of mobility in the grid-cathode region where II A. Rose, Phys. Rev. 97, 1538 (1955). triode current is determined. Table I shows p,yc for each of the triodes derived under the assumption that the triode behaves as an ideal SCL diode. The total thick ness of the triode has been used to derive p,yc. From the requirement of current continuity through the triode, (np,)ycFac= (np,)yeVucldyc = (1lp,)pyV py/dpy= (1lp,)pyFpy. (3) The potential drop between grid and cathode is 50-100 times that between plate and grid. Assume 1lpy=llyc' Some support for this assumption comes from curve 46 [Fig. 4(d)] and curve 47 which were discussed before, in which it was found that (1lp,)py= (1lp,)yC when conduc tion in the two regions of the triode is compared under conditions where the potential drop is determined by 1 pc. p,py and npo can then be derived separately and are shown in Table 1. The approximate nature of the values of 1lpg and JIpy is apparent, but other values of these quantities are not available; they may be correct within an order of magnitude. The relative constancy of the values for a number of triodes is at least encouraging. The low values of the mobilities are apparent. In the discussion of the behavior of triode structures, the designations of cathode, grid, and plate have been used for the three electrodes, by analogy with a vacuum triode. One requirement for a three-layer active device is that a large fraction of the charge carriers can pass through and be controlled by the grid. The existence of electron emission from triodes as in Fig. 7 shows that a small fraction of the electrons are transmitted through the grid. The exact fraction transmitted is unknown. However, the great majority of the charge carriers, and particularly the majority which determine the potential within the oxide, are not transmitted through the grid. Instead, the grid acts primarily as a region of low resist ance connecting two regions of higher resistance. This is shown by Fig. 3(c) and Fig. 4(d) in which the high field region in the insulator is at the anode instead of at the cathode. In such a case electrons would not gain enough energy between the negative electrode and the grid to be accelerated through the grid. Careful exami nation of the effect of grid thickness on electron emission into vacuum from triodes offers a method of determining the electron attenuation length in metal films. Thus measurements of potentials in triodes provide some information on mobilities and charge carrier den sities in the Ohmic region of the insulator between plate and grid. It is surprising that the grid does not have a greater effect, but current-voltage relationships and de tailed behavior of diodes are identical to triodes. The qualitative model of Fig. 7 should be applicable to diodes if the grid region is eliminated. ELECTROLUMINESCENCE OF OXIDE SANDWICHES Kanter and Feibelman3 reported light emission and scintillations from AI-AbOa-Au diodes that showed negative resistance. Results on Al-SiO-Au diodes con- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:53POTENTIAL DISTRIBUTIOl\' I:,{ THIN OXIDE FILMS 2685 firm these observations. When conductivity and nega tive resistance are developed by the application of forming potentials, a number of bright spots appear on typical diodes. These bright spots tend to be fairly stable through several tracings of the current-voltage characteristic though they may suddenly shift or dis appear. Bright spots were frequently associated with visible flaws, blisters, or other gross structural defects of the oxide films. An RCA 1P21 photomultiplier tube, operated at 950 V, was used with optical filters to study light emission somewhat more quantitatively. The response of the tube extended from about 300 to 700 mJL. In Fig. 8, the ratio of photomultiplier current I p to current through a 1 mm2 AI-SiO-Au diode If is plotted as a function of diode voltage. In curve 8a no filter was used; the spectral response of the filters used in curves 8b to 8d is shown at the bottom of the figure. Both visually and with the photomultiplier, no light emission was visible below 1.8 V, the same voltage at which electron emission into vacuum from AI-SiO-Au diodes was just detectable.2 At that diode voltage the light intensity increased extremely rapidly, as did elec tron emission. Visible light was emitted from 10-15 small spots. However, in contrast to electron emission into vacuum, the intensity of emitted light dropped steeply when V m exceeded at 2.9 V. The photomultiplier showed a second peak of high relative intensity at about 3.8 V, and then dropped to a lower level before rising to nearly the highest value of relative intensity at 10 V. Spectral distribution measurements using filters pro vided further evidence for the existence of high-energy processes in the oxide film. In curve 8c, the filter trans mitted a narrow band of energies between 2.6 and 3.4 eV; light emission first appeared at 1.9 V, indicating that electron transitions with an energy at least 0.7 V greater than V f were occurring. Likewise, in curve 8d, about 0.6 V separated the appearance of light and the cutoff energy of the filter. Most of the light emission at the second peak of curve 8a at 3.8 V, has energy be tween 3.2 and 4.0 V as in curve 8d. At high voltages, light from the diode had a broad spectral distribution. Thus both electron emission into vacuum and the radia tion of visible light due to electroluminescence of oxide sandwiches provide evidence for processes occurring in the oxide that can impart energies to the charge carriers that are significantly higher than the applied potentials. Electroluminenscence at low voltages has been re ported in ZnS/2 CdS/3 and ZnSel4 in which light with energy greater than the applied voltage has been observed. Mechanisms proposed in these works have involved hole injection across p-n junctions in the compounds and similar phenomena may occur in metal insulator-metal diodes that exhibit negative resistance. 12 W. A. Thornton, Phys. Rev. 116, 893 (1959). 13 R. C. Jaklevic, D. K. Donald, J. Lambe, and W. C. Vassell, App!. Phys. Letters 2,7 (1963). 14 M. Aven and D. A. Cusano, J. App!. Phys. 35, 606 (1964). Ip if 123456789'10 II VI (VOLTS) .___---- .. ENERGY (.VI FIG. ~. Light emissio~ f~om an AI-SiO-Au diode. Dependence of the :atlo of ph~tomult~pher c~rrent to diode current on voltage applied to the dIOde, llsmg optical filters with different passbands. DISCUSSION Measurements of the potential distribution in triodes provide a basis for a model of conduction and negative resistance in thin insulating films. Establishment of conductivity in diode or triode results in the localization ?f the potential drop in a fairly narrow region of the msulator, a region which is then characterized by high fields when potentials less than the band gap of the insulator are applied. Processes occurring in this high field region determine the current-voltage character istics, negative resistance, electron emission, and electro luminescence. Ridleyl5 has shown that a voltage controlled differential negative resistance which is a bulk property of solids is accompanied by the formation of domains of high field. This type of current-voltage characteristic has been reported in germanium,t6 GaAs 17 and CdS. IS Evidence has been reported for field conce~ trations in bulk samples such as have been found in the present work in thin oxide films. High-field domains in semiconductors are characterized by slow motion across the sample and subsequent reformation of the high-field domainp,18 Slow drift and reformation of high-field domains has not been observed in thin oxide films. This may be due to boundary effects which are important for very thin films. Negative resistance in thin oxide fi1n:s differs also in that voltages necessary for negative resIstance are much smaller than are necessary in bulk samples of CdS, Ge, or GaAs. In addition to the presence of field concentration in 15 B. K. Ridley, Proc. Phys. Soc. (London) 82 954 (1963) 15 B. K. Ridley and R. G. Pratt, Phys. Lett~rs 4, 300 (1963). 17 A. Barraud, Compt. Rend. 256, 3632 (1963). .18 K .. W. BOer, Festkorperprobleme, edited by F. Sauter (Fred erick Vleweg und Sohn, Braunschweig, 1962), Vol. 1, p. 38. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:532686 T. W. HICK1IOTT the insula tor, certain other experimental obserya liolls are salient to a model of conduction and negative resist ance in thin oxide films.!,2,9 (1) Initially, diodes have very high resistance wit~ currents that are exponentially dependent on V or V'. (2) Forming of conductivity of the oxide by appli cation of a potential can depend on either voltage or on field, depending on the nature and purity of the oxide; the purer the oxide, the higher the fields that are neces sary to develop conductivity. If the oxide impurity concentration is too low, dielectric breakdown of the oxide film occurs before forming of conductivity. In addition metals used as electrodes in the diode deter- , mine the ease of establishment of conductivity and the final magnitudes of the currents. (3) Electroluminescence and electron emission appear simultaneously with the initial development of diode conductivity and not before. Both rise steeply above noise at the same voltage, about 1.8 V for AI-SiO-Au diodes, independently of the magnitude of the diode curren t or thickness of the oxide. (4) For a diode with fully developed conductivity, la V2 for V < V m, a characteristic of space-charge-limited currents in insulators. (5) The conductivity of a diode is independent of temperature at least down to 3°K, provided that V m is not exceeded, indicating that the barrier at the metal oxide interface is very low and conduction is not ther mally activated. (6) The shape of the current-voltage characteristic, particularly in the negative resistance region, is nearly independent of temperature although 1", decreases when the full current-voltage characteristic is traced out as temperature is lowered.! Reduction of conductivity in going from peak to valley is very fast, < 10-6 sec!; re establishment of conductivity with decreasing voltage has a time constant of seconds. (7) Negative resistance in diodes or triodes is found with either polarity although the potential in the insu lator does not shift readily when polarity is reversed. (8) Forming of oxide conductivity produces a per manent change in the oxide; the original high resistivity is not recovered. Likewise, when diode conductivity is reduced by going into the negative resistance region of the current-voltage characteristic, a semipermanent change in the oxide occurs. If the potential is reduced rapidly, conductivity of the diode will be characteristic of the high voltage and will remain low until redeveloped by applying a potential greater than about 1.8 V to the diode. Diode conductivity can be restored to any value be tween the minimum value found when diode voltage is turned off and the maximum value that is developed for the applied voltage equal to V m by applying voltages such that 1.8< V < V m.1 Four distinct conduction phenomena in metal-oxide-metal diodes lllust be explained by any model. First is the forming of conductivity, the development of con ductivity by the application of voltage to the diode. Second is the nature of conduction for V < V m after development of negative resistance in the current voltage characteristic. Third is the reduction of diode conductivity by the application of voltages greater then V"" and its subsequent redevelopment when the diode voltage is decreased. Finally, some explanation must be given of processes by which electrons gain enough energy in the insulator to be emitted into vacuum at low voltages and also to produce electro luminescence in diodes. The phenomenological model of impurity conduction which is developed in the rest of the discussion is plausible but incomplete. It is specu lativebut is offered as a framework in which to fit many of the complex experimental observations. The simple model of a metal-insulator contact has been discussed Illany times.19 The barrier to the passage of electrons from the metal into the insulator is given by ¢>lIti=¢>"'-X, where ¢>", is the work fUllction of the metal and X is the electron affinity of the insulator. Electron affinities of insulators are not well known but seem generally to be in the range of 1 V or less so ¢>mi for an ideal metal-insulator contact should be 3.5 to 4 V. Experimental studies of tunneling between metal films separated by very thin insulators have been in terpreted in terms of such a simple model with barrier heights between 0.7 and 2.5 V being obtained by dif ferent workers.3,2o,2l The validity of a band model for conduction in amorphous insulators with low carrier mobility, as in oxide films, is at best dubious. However, it is a convenient way to express energy relations in the system and a band picture will be used for this reason. Frenkej22 pointed out that the prebreakdown currents and dielectric strengths of amorphous materials were nearly the same as those in crystalline solids. He treated an amorphous solid as an assemblage of isolated atoms and derived a model of these phenomena which did not depend on the band structure of the material. A band gap and an electron affinity can be defined re gardless of whether conduction is described by a collec tive electron model.!9 Previous models for the forming of conductivity in oxide filmsl,7 have suggested that positive charge is formed in the insulator by application of a high field. The positive charge is concentrated at the negative electrode and its primary effect is to reduce the barrier ¢>mi to a low value, permitting electrons to flow readily from the metal into the insulator. Such a model is not consistent with the results on triodes nor with the inde pendence of negative resistance on polarity of the ap- 19 N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals (Oxford University Press, London, 1948), 2nd ed. 20 J. G. Simmons and G. J. Unterkofier, J. Appl. Phys. 34, 1828 (1963). 21 D. Meyerhofer and S. A. Ochs, J. Appl. Phys. 34, 2535 (1963). 22 J. Frenkel, Tech. Phys. USSR 5, 685 (1938). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:53POTEN"TIAL DISTRIBUTION" IN THIN OXIDE FIL;VIS 2687 plied voltage. Impurities in the oxide are important both ill establishing conductivity and in determining the magnitude of currents which can be developed.9 Con duction through an impurity band in the insulator, located close to the Fermi level of the metals and half way between conduction and valence band of the insu lator, would satisfy many of the experimental obser vations since impurity conduction is characterized by low carrier mobility and small temperature depen dence.23 A possible model for establishment of conduc tivity and for negative resistance in diodes is shown in Fig. ? The high-field region is indicated schematically as bemg near the cathode in Fig. 9, but could be in some other region of the insulator. In the unformed diode, a large number of immobile neutral impurity centers are distributed throughout the oxide with energy approximately midway between va lence band and conduction band. The nature and number of these depend on the insulator and its method of preparatioll; they may be due to foreign atoms in the Int t ice or to structural faults. As voltage across the diode is increased, two primary contributions to the leakage currents will be Schottky emission from the cathode24 and Schottky emission from impurity centers.25,26 Both give an exponential dependence of current on voltage and the latter will leave i?nized impurity centers in the insulator. As the poten tial, or field, reaches a critical value, the number of ionized impurity centers in the insulator becomes large enough that an impurity band forms in the insulator. Impurity ionization occurs nearly uniformly through the whole oxide thickness. In addition, modification of the metal-oxide interface in such a way that electrons pass readily into the insulator, without thermal acti vation, is a critical step in forming which will depend on the metal electrodes. The impurity band lies well below the conduction band of the insulator. If its lo cation in energy before forming varies by more than a few tenths of a volt from the midband-gap EG/2 con ductivity will form with great difficulty or not ~t all. For impurity conduction, the impurity concentration at the metal-oxide interface does not have to be high enough to reduce rf>mi by several volts as has been sug gested before.1,7 The magnitude of current depends on the number of impurity sites, on their separation since only a fraction of impurity sites will be ionized and on matching at the metal-oxide interface. Cond~ction is by electrons of low mobility hopping from site to site indicated by (1) in Fig. 9, and the impurities contri~ buting to conductivity are distributed nearly uniformly throughout the insulator. Conduction in the low-field region of the insulator is also by hopping from impurity site to impurity site. The mobile charge carriers are : N. F. Mott and W. D. Twose, Advan. Phys. 10, 107 (1961). P. R. Emtage and W. Tantraporn, Phys. Rev. Letters 8 267 (1962). ' 2. J. Frenkel, Phys. Rev. 54, 647 (1938). 26 D. A. Vermilyea, Acta Met. 2, 346 (1954). UNFORMED DIODE ~------Ec ALUMINUM tmi OXIDE GOLD __ Ey FORMED DIODE r+--I-'--::'~- Ec . FIG. 9. Schematic diagram of the establishment of conductiv Ity, and conduction processes, in the high-field region of an Al oxide-Au diode. electrons injected at the electrodes, but conductivity is controlled by positive impurity centers. Below V"" current is space-charge limited. However, space-charge-limited currents alone cannot account for negative resistance, nor for electrolumines cence and electron emission. An immobile hole level E/I has therefore been postulated in the forbidden gap of the insulator between the impurity band and the valence band. As the diode voltage is raised, immobile hole levels are filled by electrons tunneling from the valence band or by impact ionization from a few electrons accelerated to energies greater than EH-Ev, indicated by (2) in Fig. 9, leaving behind a mobile hole in the valence band. This hole moves toward the cathode under the influence of the high field (3). Between its point of formation and the cathode, the hole is neutralized by an electron from the impurity states, (4) or possibly directly by the metal, imparting its recombination energy to another electron in an impurity state, which is then excited into the true conduction band of the insulator, approximately Ea/2 V above the Fermi level of the base metal and the energy level of the impurity conduction band. (5) Elec trons in the conduction band, in turn, are accelerated in the high field by the full potential across the diode (6) and some receive enough energy to appear in vacuum. The contribution of electrons in the conduction band to determining the potential between plate and grid in a triode is negligible compared to the contribution of low-mobility electrons in impurity states. The ones which escape are only slightly scattered and lose little en ergy passing through the insulator or metal. Alterna tively, electrons can be captured by transition to the im purity band of theinsulator (7) giving light emission with [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:532688 T. W. HICKMOTT energy greater than the applied voltage. Electrolumi nescence may also be due to radiative transitions be tween E[, HH, and Ev. For voltages greater than Ec-E[, electrons may also tunnel from the impurity band into the conduction band where they can be rapidly accelerated and contribute to the emitted cur rent. Thus the mechanism by which electrons gain energy and are emitted into vacuum at low diode voltages is electron excitation from the impurity band into the conduction band by some kind of recombination process. Geppert27 has suggested the possibility of a similar cumulative avalanche process in metal-insula tor metal structures, in which electrons are excited from the valence band to the conduction band rather than from an impurity band. Electrons excited from an im purity band into the conduction band should be charac terized by a nearly constant excess energy equal ap proximately to Eu/2. Those excited from the valence band should have excess energy approximately equal to EG, and measurement of the energy of electrons emitted into vacuum from metal-insulator-metal diodes should differentiate between the two possibilities. Negative resistance in the current-voltage character istic is determined by some mechanism which depends on a high field within the insulator, which is independent of temperature, which has a time constant less than a microsecond, and which reduces the number of impurity centers in the insulator that can contribute to electronic conductivity. The most probable such mechanism would seem to be tunneling of electrons within the insulator, either from the valence band or from a second impurity level, into the impurity band through which conduction occurs, resulting in neutralization of impurity sites. Neutralization from the hole levels introduced to ac count for energetic processes of electron emission and electroluminescence is indicated as (8) in Fig. 9. The negative resistance region of the current-voltage charac teristic above 2.8 V represents a competitive reaction between the neutralization of positive impurity centers in the oxide by tunneling of electrons from the hole levels, which reduces conductivity, and creation of im purity centers by some ionization process, which in creases conductivity. As the diode voltage is increased, neutralization, being an exponential function of field or voltage, becomes more and more important; the number of impurity centers that are ionized and capable of con tributing to conduction decreases. These neutralization processes occur in a relatively small high-field region of the insulator; in the bulk of the insulating film, fields are small and the number of charge carriers as well as their mobility remains nearly constant. The slow process which determines the speed at which a complete current-voltage cycle can be traced out is the refor mation of conducting impurity centers in the insulator 27 D. V. Geppert and B. V. Dore, Stanford Research Institute, Technical Report ASD-TDR-63-672 under Contract No. AF 33 (657)-8721. when the voltage is decreased; neutralization by high field processes is rapid. The proposed mechanism, while tentative, explains many of the experimental phe nomena. Other processes involving multiple collisions and energy transfer in a high-field region of the insulator might also be sufficient to raise electrons to an energy high enough to escape into vacuum or to give off visible light by transition to trapping states in the insulator. In addition, other energy levels or bands of energy levels may occur in the forbidden gap which can contrib ute to excitation and neutralization of charge carriers. No ionic conduction mechanisms have been invoked in trying to explain the experimental results, although either the forming of conductivity or negative resistance might involve ionic conduction. Several observations seem to preclude ionic mechanisms. Establishment of negative resistance and potential shifts in triodes are difficult to explain if ionic motion is necessary. Anodi zation of aluminum, a process that depends on ion motion, requires fields in the oxide of 5-lOX 106 V /cm.28 Development of initial conductivity in impure AbO~ films occurs at a nearly constant voltage with average field strengths varying from 3 X 106 V! cm in the thinnest films to 4.5 X 105 V / em in the thickest film, much less than that needed for anodization.9 While it is true that triode measurements show a field concentration once conductivity is developed, this does not occur until after initial development of conductivity. The time con stants for forming of conductivity and for reducing conductivity in tracing out negative resistance are much too short to involve ionic motion in the insulator. Study of voltage transients in formation of anodic films has shown that the times for ion motion in high fields are seconds.29 Conductivity of metal-oxide-metal diodes can initially be developed with millisecond voltage pulses and once conductivity is formed, the resistance of diodes can be either increased or decreased with micro second pulses.l These are time constants of electronic processes rather than of ionic processes. Although ionic mechanisms cannot be eliminated completely, particu larly in redeveloping conductivity as diode voltage is reduced, they do not seem to be of primary importance for negative resistance. Voltage-controlled negative resistance in the current- TABLE II. Dependence of V m on insulator band gap and dielectric constant. Insulator Vm (V) Vm2 (VJ2 RG (eV) AI-SiO-Au 2.9-3.1 8.4-9.5 AI-AI,O,-Au 2.8-2.9 7.9-8.4 (8.4) Ta-Ta2O,,-All 2.2 4.8 4.6 Zr-Zr02-All 2.1 4.4 (4.3) Ti-Ti02-Au 1.7 2.9 3.0 28 L. Young, Anodic Oxide Films (Academic Press Inc., New York, 1961). '" D. A. Vermilyea, J. Electrochem. Soc. 104,427 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:53POTEKTIAL DISTRIBGTION IN THIl\ OXIDE FILMS 2689 voltage characteristics of metal-oxide-metal diodes is found quite generally for different oxides.1.6 For oxides with higher dielectric constant than SiO, V m generally occurs at lower voltages, and a correlation between V m and K! has been reported.l Band gaps for anodic oxide films are not well known; for AIz03, Eo> 8 V, for Ta205, a value of 4.6 e V has been found30 while a value of 3.0 has been reported for Ti(h31 In Table II, a correlation between V m2 and Ee is shown. For AIz03, Ee is derived from electron emission measurements.32 For Zr02 and SiO, Eo is not accurately known but a steep drop in electron emission from Zr-Zr02-Au diodes occurs at 4.3 30 L. Apker and E. A. Taft, Phys. Rev. 88, 58 (1952). 31 R. H. Bube, Photocond'uctivity of Solids (John Wiley & Sons, Inc., New York, 1960), p. 233. 32 T. W. Hickmott (to be published). V, just as it does for Ta205 diodes at 4.6 V.2 The higher the dielectric constant, the lower V m and Eo are, and the empirical relations V",2= 1O.3-0.18K(V)2 can be derived from Table II. If the model of Fig. 9 is correct, the impurity levels and hole levels are closely connected and their separation is determined by the dielectric constant of the insulator. ACKNOWLEDGMENT It is a pleasure to acknowledge many stimulating conversations with F. S. Ham. D. MacKellar kindly provided facilities for SiO evaporation. JOURNAL OF APPLIED PHYSICS VOLUME 35, NUMBER 9 SEPTEMBER 1964 Pressure Theory of the Thermoelectric and Photovoltaic Effects MILTON GREEN Burroughs Corporation, Defense and SPace Group, Paoli Research Laboratory, Paoli, Pennsylvania (Received 7 June 1963; in final form 16 April 1964) The theory is based upon the hypothesis that free charge carriers--electrons and holes-and phonolls exert pressures inside a solid. Gradients of such pressures exert motive forces on the carriers. On this basis, the hole cu rren t density / p, in the absence of a magnetic field, is assumed to be / p=upE-/lpgrad P p-/lp",gradP "', where Up, /lp, and P p are, respectively, the conductivity, mobility, and pressure of holes; /lop", is the inter action mobility between holes and phonons; P", is phonon pressure; and R is the electrostatic field. A similar expression is obtained for electrons by exchanging the subscript p for n. (The two mobilities associated with electrons, however, are negative.) The theory is applied to the nondegenerate semiconductor, with the assumption that the equation of the ideal gas law applies. (Thus, Pp= pkT, Pn=nkT, where k is the Boltzmann constant, T is temperature Kel vin, and p and n are concentrations of holes and electrons, respectively.) It is also assumed- for small cur rents-that deviation from the equilibrium pressures can be neglected. Assumptions concerning the phonon effect are quite general; the contribution from this source to the hole current density I"~ is given by /"", = -up(kT /e)op grad In T, where eis magnitude of electronic charge. The dimensionless quantity op, the phonon-dragging coefficient for holes (a temperature- and material-dependent parameter), is not amenable to calculation by the theory, in its present form, and must be determined experimentally. Again, a similar expression exists for electrons. I. INTRODUCTION IN this paper, thermoelectricity and voltaic photo electricity are treated mainly from a field theory approach. By this is meant that the problem is dealt with in terms of such electrical point-to-point parame ters of a circuit as electric fields, conductivity, charge carrier concentrations,"" mobilities, space charge, and however, is considered completely as a field theory. On the other hand, there is an abundance of literature on the statistical approach to thermoelectricity. Herring5 has collected a fairly large bibliography, as has Price.2 In behalf of the field theory treatment, it can be said that the fundamental physical processes involved are more easily understood,6 since the concepts are con crete, simpler, and also more familiar. current density. r,... Theoretical treatises involving, in part, such an ap proach as taken here have appeared.1-4 None of these, 1 F. W. G. Rose, E. Billig, and J. E. Parrott, J. Electron. Control 3,481 (1957). 2 P. J. Price, Phil. Mag. 46, 1252 (1955). 3 P. J. Price, Phys. Rev. 104, 1223, 1245 (1956). I J. Tauc, Phys. Rev. 95, 1394 (1954); Rev. Mod. Phys. 29, 30XJ19S7). The mathematical formulation of the flow equations, taken up in Sec. II, begins with the usual forces that act upon charge carriers-namely, electrostatic po- 5 C. Herring, Phys. Rev. 96, 1163 (1954). 6 Rose et at. 1 state, "The usual theoretical treatment of this effect (thermoelectricity) involves statistical techniques which do not readily lend themselves to a clear exposition of the subject." [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Wed, 26 Nov 2014 05:26:53
1.1777057.pdf	Band Structure of HgSe and HgSe–HgTe Alloys T. C. Harman and A. J. Strauss Citation: Journal of Applied Physics 32, 2265 (1961); doi: 10.1063/1.1777057 View online: http://dx.doi.org/10.1063/1.1777057 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High temperature structural studies of HgS and HgSe quantum dots Appl. Phys. Lett. 83, 4011 (2003); 10.1063/1.1625433 Effect of Fe on the carrier instability in HgSe Appl. Phys. Lett. 47, 407 (1985); 10.1063/1.96127 On the anomalous phonon mode behavior in HgSe J. Appl. Phys. 56, 2541 (1984); 10.1063/1.334319 Some Properties of HgSe–HgTe Solid Solutions J. Appl. Phys. 32, 2254 (1961); 10.1063/1.1777054 Angular Dependence of Magnetoresistance in HgSe J. Appl. Phys. 32, 1800 (1961); 10.1063/1.1728459 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.153.184.170 On: Thu, 27 Nov 2014 09:25:24ELECTRICAL AND OPTICAL PROPERTIES OF ZnSe 2265 reflection at 3.22 ev and half-width very roughly 0.04 ev. From the energy difference for the two absorptions at room temperature as evaluated from Fig. 4, one obtains O.4S± 0.04 ev for the splitting of the valence band at r due to spin-orbit interaction. We suggest that the absorption peaks in Fig. 4 at 4.75 and 5.10 ev are due to transitions between the two valence band states and the conduction band at LCi.e., k= 271"/ a(-U,~)]' The valence band splitting at L due to spin-orbit interaction is theoretically expected to be about i of that at r. Within the experimental error the separation of the two peaks, 0.35±0.08 ev, is in accord with this prediction. One may speculate that the absorption peak at 6.4 ev is due to transitions between the valence and con duction bands in the vicinity of XCi.e., k= 271"/ a (1,0,0)]' This ordering of the different direct-transition band gaps suggested for ZnSe is the same as that suggested for Ge by Phillips.28 The relative magnitude of the absorption at 4.7S and 6.4 ev and the width of the peak at 6.4 ev are comparable to the corresponding structure observed in GeP We believe that interpretation of the structure above 6.4 ev, as well as confirmation of the assignment of the lower energy structure, will be greatly aided by improved calculations of the band structure of II-VI compounds, and by more accurate optical data. ACKNOWLEDGMENTS We wish to thank H. R. Philipp and E. A. Taft for assistance in obtaining the reflectance data in the 3.8 to 14.5 ev energy range, and for advice on the Kronig Kramers inversion calculations, and Henry Ehrenreich for stimulating discussions. 28 J. C. Phillips, J. Phys. Chern. Solids 12, 208 (1960). JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 32, NO. 10 OCTOBER, 1961 Band Structure of HgSe and HgSe-HgTe Alloys T. C. HARMAN AND A. J. STRAUSS Lincoln Laboratory,* Massachusetts Institute of Technology, Lexington 73, Jfassachusetts A detailed analysis of Hall coefficient data obtained at temperatures between 77° and 3500K has been made for HgSe and HgSeo.5Teo.5 samples containing excess donor concentrations up to 1019 em-a. On the basis of previous magnetoresistance, Seebeck coefficient, and reflectivity data, a spherically symmetric non quadratic conduction band exhibiting the E(k) dependence described by Kane was adopted in making the analysis. Calculations based on a conventional two-band model failed to give quantitative agreement with experiment, but good agreement was obtained on the basis of a model in which the conduction band and one valence band overlap in energy. Therefore the materials are semimetals rather than semiconductors. The best fit to the data was obtained with an overlap energy of 0.07 ev for both HgSe and HgSeO.5Teo.5, with hole density-of-states masses of 0.17 /no and 0.30 /no, respectively. With increasing carrier concentration, the optical absorption edge for heavily doped HgSe exhibits a shift to higher energies which is characteristic of n-type materials with low electron effective masses. Qualitatively, the optical data are consistent with a semimetal band model rather than with a semiconductor model, since the interband absorption edge ap parently occurs at photon energies less than the Fermi energy. INTRODUCTION THE II-VI compounds HgSe and HgTe, which crystallize in the zinc-blende structure, form a continuous series of pseudo binary solid solutions. Values of 0.01-0.02 ev for the energy gap of HgTe have been obtained by several authors1-s who analyzed the variation of Hall coefficient with temperature on the basis of a simple two-band semiconductor model. Zhuze6 has reported the energy gap of HgSe to be 0.12 * Operated with support from the U. S. Army, Navy, and Air Force. 1 I. M. Tsidilkovski, Zhur. Tekh. Fiz. (USSR) 27,1744 (1957). 2 T. C. Harman, M. J. Logan, and H. L. Goering, J. Phys. Chern. Solids 7, 228 (1958). 3 R. O. Carlson, Phys. Rev. 111, 476 (1958). 4 J. Black, S. M. Ku, and H. T. Minden, J. Electrochem. Soc. 105, 723 (1958). 5 W. D. Lawson, S. Nielsen, E. H. Putley, and A. S. Young, J. Phys. Chern. Solids 9, 325 (1959). 6 V. P. Zhuze, Zhur. Tekh. Fiz. (USSR) 25, 2079 (1955). ev, without specifying the nature of the experimental data or the band model employed, while Goodman7 has predicted a gap of 0.7 ev for HgSe on theoretical grounds. No energy gap values for HgSe-HgTe alloys have been reported. In the present investigation, the variation of Hall coefficient and resistivity with temperature has been determined experimentally for samples of HgSe and the following alloys: HgSeo.75Teo.25, HgSeo.5Teo.5, and HgSeO.25TeO.75' A detailed analysis has been made of the Hall coefficient data for HgSe and HgSeO.6Teo.5 samples varying widely in net donor concentration. It has not been possible to explain these data on the basis of a simple two-band model. Good agreement with experi ment is obtained, however, with a band model in which the conduction band and one valence band overlap 7 C. H. L. Goodman, Proc. Phys. Soc. (London) B67, 258 (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.153.184.170 On: Thu, 27 Nov 2014 09:25:242266 'r. C. H ARM A NAN D A. J. S T R A U S S E FIG. 1. Schematic «k) diagram for the band structure model adopted for HgSe and HgSeO.5TeO .•. k in energy, as shown schematically in Fig. 1. Data on the infrared absorption of HgSe are also consistent with this model. Thus, these two materials are semi metals rather than semiconductors. This result suggests that HgTe may also be a semimetal and that previous analyses of its electrical properties should be recon sidered from this point of view. EXPERIMENTAL PROCEDURE Ingots of HgSe and HgSe-HgTe alloys were prepared by a modified Bridgman method. Commercial high purity elements (99.999+%) were placed in a quartz tube tapered to a point at one end. The tube was sealed off under vacuum and heated in a horizontal two-zone resistance furnace. After reaction was com plete, the furnace was rotated into the vertical position, and the melt was frozen directionally by lowering the tube out of the furnace at the rate of about 4 mm/hr. The vapor pressure of mercury over the melt was kept constant during crystallization by controlling the tem perature of the upper zone of the furnace. Single crystals of undoped HgSe up to 2.5 em in diameter and 10 cm in length were obtained by this method, while doped HgSe ingots and alloy ingots were generally composed of large grains. Where the electrical proper ties of single crystal and polycrystalline samples could be compared, they were found to be the same within experimental error. Undoped ingots of HgSe were n type, probably due to the presence of excess mercury.s The minimum donor concentration which could be ob tained was about 1 X 1017 cm-3. Donor concentrations up to 3X 1019 cm-3 were obtained by doping with aluminum, but no acceptor impurity could be found. In the HgSeo.5Teo.5 alloy, donor concentrations up to 7X lOIS cm-3 and acceptor concentrations up to 3X 1019 cm-3 were obtained by doping with aluminum and copper, respectively. 8 The stoichiometry of HgSe, HgTe, and their alloys, as well as the electrical behavior of various impurities in these materials, will be discussed in a subsequent publication. The resistivities and Hali coefficients of parallelepiped samples cut from HgSe and alloy ingots were measured by conventional dc potentiometric methods. The mag netic field used for the Hall measurements was approxi mately 6000 gauss. Measurements at room temperature and liquid nitrogen temperature were made with pres sure contacts, while indium-soldered contacts were used for measurements at liquid helium temperature and for those made as a function of temperature. In making the latter measurements, the sample was first cooled to either liquid nitrogen or liquid helium temperature and then allowed to warm up slowly while data were taken automatically with a recording potentiometer. Infrared reflection and transmission data used to calculate optical absorption coefficients were obtained with Perkin Elmer model 221 (double beam) and model 12C (single beam) spectrophotometers, respec tively. The samples were etched before measurement in order to remove the work damage produced by grinding and polishing. EXPERIMENTAL DETERMINATION OF FREE ELECTRON CONCENTRATIONS Experimental values of the free-electron concentra tion (n) in all samples for which n~ p, the free-hole concentration, were calculated from the measured Hall coefficients (RH) according to the usual expression for one-carrier conduction: n= -1/ RHec. This expression is found to be applicable to such samples on the basis of data which show that the ratio of electron mobility to hole mobility (b= Il-n/ Il-p) is of the order of 100 in the HgSe-HgTe system. For HgSeO.5TeO.5, a comparison between the Hall mobilities of extrinsic n-type and p-type samples at 4.2°K gives b= 85, while analysis of the temperature dependence of RH for a sample con taining an excess acceptor concentration of 3.4X lOIS cm-3 gives b=1.1X10 2. This analysis utilizes the ex pression Rmax/Rext= (1-b)2/4b, where Rmax is the maximum negative value of RH and Rext is the value of RH in the extrinsic range. In deriving this expression, it is assumed that only one species of electrons and one species of holes make appreciable contributions to the conductivity, and also that the rate of change of b with temperature is small compared to the rate of change of the intrinsic concentration with temperature. No addi tional assumptions concerning the details of band structure, statistics, or scattering mechanism are re quired. In the case of HgTe, the same analysis of RH as a function of temperature gives b= 100,2,3 but no comparison between the Hall mobilities of extrinsic n-type samples is possible, since extrinsic n-type ma terial cannot be prepared. s Neither type of data on the mobility ratio of HgSe can be obtained, since extrinsic p-type material cannot be prepared. S It seems reason able, however, to assume that the mobility ratio in HgSe is also of the order of 100, particularly since the Hall mobilities of electrons in HgSe, HgTe, and their alloys are the same to within 25%. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.153.184.170 On: Thu, 27 Nov 2014 09:25:24BAKD STRUCTURE OF HgSe AND HgSe~HgTe ALLOYS 2267 For a semiconductor with b»l and n~ p, the exact expression for RH reduces to RH= -An/nee, where An is a parameter whose value depends on degree of de generacy, scattering mechanism, the product of carrier mobility and magnetic field, and band structure. Since analysis of galvanomagnetic and thermomagnetic data for HgSe indicates that the value of A n is very nearly unity/ n is very nearly equal to -l/RHee when n~p. THEORETICAL CALCULATION OF FREE ELECTRON CONCENTRATIONS The energy-band model adopted in the theoretical calculations is shown schematically in Fig. 1. Free electrons are present in the conduction band C, as in the simple two-band model, but free holes are present only in valence band V 2, not in valence band V I as in the two-band model. In order to calculate n as a func tion of temperature and donor concentration, theoreti cal expressions for nand p were first obtained in terms of the band parameters of bands C and V 2, respectively. These expressions were derived on the basis of the general equation for the concentration of free carriers in a band: (1) where /0 is the equilibrium distribution function and k is the wave vector. The properties of the conduction band of HgSe re quired to evaluate n according to Eq. (1) are known with considerable accuracy from previous investiga tions. Measurements in this laboratory of magneto resistance as a function of the angle between current and magnetic field indicate that the band is spherically symmetric,9 although it should be noted that magneto resistance results described by Rodot and RodotlO are inconsistent with spherical symmetry. B,oth Seebeck coefficient datal! and infrared reflectivity measure mentsl2 show that the electron energy relative to the bottom of the conduction band (€) does not exhibit a simple quadratic dependence on wave vector (k), so that the effective mass is not independent of energy. The data are found to be consistent with the expression given by Kanel3 for e (k) : e= -elll/2+ (e012+8J>2k2/e)!j2, (2) where €YI is the energy gap between the conduction band and valence band VI, as shown in Fig. 1, and P is a matrix element defined by Kane.ls When the inte gral f /r/J3k in Eq. (1) is evaluated on the basis of this expression, using Fermi-Dirac statistics, the equation 9 T. C. Harman and A. J. Strauss (to be published). 10 M. Rodot and H. Rodot, Compt. rend. 250, 1447 (1960). 11 T. C. Harman, Bull Am. Phys. Soc. 5, 152 (1960). 12 G. B. Wright, A. J. Strauss, and T. C. Harman, Bull. Am. Phys. Soc. 6, 155 (1961). 13 E. O. Kane, J. Phys. Chern. Solids 1, 249 (1957). obtained for the free-electron concentration is 3 (3)!(kT)3J'" X!(X+cJ>I)t(2X+cJ>I) n=--- dX, 471'2 2 Pol +exp(X -fJ) (3) where X=t/kT, cJ>(if!iil€Y1/kT, and fJ=EF/kT; EF is the Fermi level relative to the bottom of the conduction band. The following values of P and EOI for the conduction band of HgSe have been used in evaluating Eq. (3): P=9XIO-s ev-cm, EOI=O.l ev. These are the values which yield theoretical results for the Seebeck coeffici ents in quantitative agreement with the experimental datal!; somewhat different values of the band param eters are derived on the basis of the reflectivity data. It is of interest that P has essentially the same value for HgSe as for the lII-V compounds lnSh, InAs, GaSb, InP, and GaAs.H Seebeck and reflectivity data for HgSeO.5 Teo,5, al though not as extensive as those for HgSe, appear to be consistent with the Kane model of the conduction band. Therefore, Eq. (3) has been used in the theoretical calculations for the alloy as well as for HgSe. The data indicate that, for electrons of a given energy, the effec tive mass is considerably greater in HgSeo.5Teo.5 than in HgSe. The band parameter values adopted for the alloy are: P=9XlO-8 ev cm (the same as for HgSe), egl=0.2 ev. For electrons at the bottom of the conduc tion band, the effective mass calculated from these values is 0.014 mo, compared with the corresponding mass of 0.007 mo for HgSe. The fact that the conduction band in HgSe and HgSeo.5Teo.5 has the form given by Kane indicates that a valence band VI is present at an energy EUI below the bottom of the conduction band, as shown in Fig. 1. No other information concerning the valence bands of HgSe or HgSeO.5TeO.5 is available from pre vious investigations. As stated above, the additional valence band V2 shown in Fig. 1 has been included in the band structure in order to account for the data obtained in the present investigation, since these data could not be explained quantitatively by a structure containing only bands C and V I. In obtaining a theoretical expression for p analogous to Eq. (3) for n, it was assumed that valence band V2 is a simple parabolic band, in which hole energy exhibits a quadratic dependence on wave vector. Evaluation of Eq. (1) for such a band, using Fermi-Dirac statistics, gives the well-known expression p=47r(2kT/h2)i(mp)iF,,[ -(fJ+cJ>2)], (4) where mp is the density-of-states effective mass for holes, Fi is the Fermi integral, fJ is the reduced Fermi level defined as in Eq. (3), and cJ>2=€Y2/kT; Eg2 is the energy separation between bands C and V 2, as shown in Fig. 1. Equation (4) is valid regardless of the number 14H. E._Ehrenreich, Phys.~Rev. 120, 1951 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.153.184.170 On: Thu, 27 Nov 2014 09:25:242268 T. C. HARMAN AAfD A. J. STRAUSS ,.. ,.. " ) " 3.0 ~~-r~_~~~~~~~-~~ __ ~XIOI7 2.0 , 1.0 0 f017 12 If 10 .. "300 } • "300/"n EXPERIMENTAL - 9 _ THEORETICAL B 1: lOtS "77 (cm-3) 7-£o 6 ~ FIG. 2. Free-electron concentration at 3000K (naoo) and free electron concentration ratio (naoo/n77) versus free-electron con centration at nOK (nn) for HgSe. of valleys associated with band V 2 and is applicable in the presence of degenerate valence bands. The evaluation of EY2 and mp * for HgSe and HgSeO.6Teo.6, which formed an important part of the present investigation, was accomplished by substituting trial values of these parameters into Eq. (4), calculating n as a function of temperature and donor concentration according to the method now being described, and selecting those values which gave the best fit to the experimental data. The value of EY2 obtained in this manner is -0.07 ev for both HgSe and HgSeo.6Teo.5' The values of mp * for HgSe and HgSeo.5Teo.5 are 0.17 mo and 0.30 mo, respectively. By using the band parameter values listed for HgSe and HgSeo.6Teo.6, theoretical values of nand p for a specified temperature can be calculated from Eqs. (3) and (4), respectively, provided that the Fermi level is also specified. Since the present calculations were re stricted to cases in which n~ p, the values of Ef adopted were those which led to values of nand p satisfying the relationship n=p+N D, (5) where LV D is the excess donor concentration. This rela tionship is applicable because the excess donors are fully ionized over the entire temperature range, as shown by the fact that for all samples the Hall coeffi cient increased to a constant value when the tempera ture was reduced sufficiently. Complete ionization is predicted theoretically, since for electron effective masses as low as those in HgSe and HgSeo.6Teo.6 the ionization energy of shallow donors is expected to vanish at concentrations much lower than those studied in the present investigation, due to the overlap of con duction band and donor wave functions. In order to obtain theoretical values suitable for comparison with experimental data, two sets of calcu lations were made on the basis of Eqs. (3), (4), and (5). In one case, n was evaluated as a function of N D for certain fixed temperatures, using a desk calculator while in the other the variation of n with T for certai~ fixed donor concentrations was calculated with an IBM 709 computer. All band parameter values were taken to be independent of temperature. RESULTS AND DISCUSSION Experimental data for free-electron concentrations at 300° and nOK in HgSe samples varying widely in net donor concentration are shown in Fig. 2, where nsoo and nSOO/1t77 are plotted against 1t77. The qualitative features of the data may be explained in terms of the relationship of nsoo and n77 to N D, in the same manner as if HgSe were an n-type semiconductor. Samples containing sufficiently high donor concentrations are extrinsic at both temperatures, with N D»P; according to Eq. (5), nsoo= n77= N D, and nsoo/ n77= 1. As N D is decreased below 1 X 1018 cm-S, psoo-which is equal to the concentration of intrinsic electrons promoted from the valence band to the conduction band-becomes appreciable compared to N D. Therefore, nsoo becomes greater than N D. On the other hand, n77 remains equal to N D until N D is reduced to values considerably less than 1 X 1018 cm-s, since the intrinsic carrier concentra tion at nOK is lower than at 300oK. Therefore nn decreases more rapidly than nsoo with decreasing N D and nsoo/ nn increases as nn decreases. ' Although the general features of the data in Fig. 2 are consistent with a conventional semiconductor model in which there is a positive energy gap between the valence and conduction bands, calculations based on such a model failed to give quantitative agreement with experiment. Calculations based on the semimetal (?verlapping band) model of Fig. 1 did give quantita tIve agreement with the experimental data, however. The results of these calculations are shown as theoreti cal curves in Fig. 2. In order to obtain the curves, n;;oo and n77 were first calculated as functions of N D as shown in Fig. 3. As stated previously, the band' pa rameter values used in the calculations were fOl = 0.1 ev, E1I2= -0.07 ev, and mp=0.17 mo. Each of the points o 0.2 0.4 0.6 0.8 1.0 ND cm-3 FIG. 3. Theoretical dependence of free-electron concentration (n) at 300°, 7r, and OOK on net donor concentration (N D) for,HgSe: [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.153.184.170 On: Thu, 27 Nov 2014 09:25:24BAND STRUCTURE OF HgSe AND HgSe-HgTe ALLOYS 2269 used in drawing the curves of Fig. 2 was then obtained by comparing n77 for a given N D with n300 for the same ND. In addition to the results for 300° and 77°K, Fig. 3 shows the theoretical variation of n with N D at OaK. Whereas the intrinsic free electron concentration eni) is zero for a semiconductor at OaK, ni calculated ac cording to the semimetal model has the rather large value of 1.0X 1017 cm-3 at OaK. As the temperature is increased, ni changes very slowly, increasing to only 1.2X 1017 cm-3 at nOK and to 3.6X 1017 cm-3 at 3000K. According to this result, none of the HgSe samples studied in the present investigation was in the intrinsic range even at room temperature. Experimental results for HgSeo.5Teo.5 at 300° and 77°K are shown in Fig. 4. These results are more com plex than those for HgSe, since they include data for samples containing excess acceptors as well as for those containing excess donors. For samples containing suffi ciently high acceptor concentrations, the free-electron concentrations cannot be obtained from the measured Hall coefficients, since RH is given by the expression for mixed conduction rather than by the single carrier expression -1/nec. Therefore, the data in Fig. 4 are presented in terms of R77/R300 and -1/R77ec; these coordinates are equivalent to nSOO/n77 and nn, respec tively, for samples in which n?-p. The experimental curve in Fig. 4 consists of two branches. The part of the upper branch for values of -l/Rnec greater than 2.8X 1017 cm-3 includes data for samples in which n?-p. The variation of n30o/nn ( = R77/ R300) observed in this region occurs for the same reasons described previously for the case of HgSe, and the theoretical curve shown was calculated in the same way as the corresponding curve for HgSe shown in Fig. 2. As stated above, the band parameter values adopted for HgSeo.5Teo .• in order to obtain quantita tive agreement with the experimental data were fOl =0.2 ev, f02= -0.07 ev, and mp=0.30 mo. The values of ni calculated for these parameters are 2.8X 1017 cm-3 at 77°K and 7.8X 1017 cm-3 at 3000K. -.. -EXPERIMENTAL -THEORETICAL I ~ • .1. _ ----.1-_._.1 __ L I -L ! j 1 tOl8 tlR77 ec (cm-') FIG. 4. Variation of Hall coefficient ratio (R,,/ Raoo) with -l/R"ec for HgSeo.5Teo .•. 10'9 102 8 • HgS. } S EXPERIMENT • Hg .0.5TeO.5 6 -THEORY • • 4 I' " 8 " fi 2 "i a: I 10 8 6 o 2 4 6 8 10 12 103/PK FIG. 5. Dependence of Hall coefficient (RH) on 11T for two samples of HgSe and two samples of HgSeo.5Teo.5. The theoretical part of the curve in Fig. 4 ends at the point where the net donor concentration becomes zero due to the compensation of donor and acceptor impurities and consequently n77=ni. The remainder of the curve is traversed as the net acceptor concentration (N A) is increased. Initially, R77 increases with increasing N A, both because of the decrease in nn and because of the onset of mixed conduction. Therefore, -1/ R77eC decreases, and the experimental points fall along the upper branch of the curve. In this region, the increase in Rn causes an increase in R77/ R300, since the samples remain very nearly intrinsic at 3000K and R300 there fore remains essentially constant. When N A increases sufficiently, R77 begins to decrease toward zero as a result of mixed conduction. Therefore, -1/ Rnec in creases, and the points fall along the lower branch of the curve. In this region, R300 is greater than Ri at 3000K, so that for a given value of Rn the value of R77/R300 is lower than for the upper branch of the curve. In addition to the Hall coefficient data obtained at 300° and nOK for a large number of samples, RH was measured for several samples as a function of tempera ture between 77°K and about 350°K. The data for two samples of HgSe and two samples of HgSeo .• Teo.5 are shown in Fig. 5. The three theoretical curves shown were calculated in the manner described above, using the same band parameters used to calculate the theo retical curves of Figs. 2, 3, and 4. The agreement be tween theory and experiment is seen to be quite satis factory. No attempt was made to calculate a theoretical curve for the fourth sample, which contained sufficient excess acceptors to be in the mixed conduction region over the whole temperature range. Optical absorption coefficients measured at 3000K for three samples of HgSe with free-electron concentra tions from 1.5 X 1018 cm-3 to 1.8 X 1019 cm-3 are shown as a function of wavelength in Fig. 6. The minima in the curves result from the simultaneous occurrence of two absorption processes, one of which increases with in- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.153.184.170 On: Thu, 27 Nov 2014 09:25:242270 T. C. HARMAN AND A. ]. STRAUSS 2 396 103 8 ";" 6 E u 0 4 ·396 2 ·36A .. 37A-1 102L-____ -L ______ ~ __ ~~L_L_ ____ ~ 1 2 4 6 8 10 20 A(fL) FIG. 6. Absorption coefficient (a) versus wavelength (X) for samples of HgSe with carrier concentrations of 1.5 X lOIS cm-3 (37A-l), 5.0XlOIS cm-3 (36A), and 1.8XI019 cm-3 (39B). Each vertical arrow indicates the wavelength which corresponds to the Fermi energy for the designated sample. creasing photon energy while the other decreases. The former process is presumably an interband absorption, while the latter is presumably free-carrier absorption. The usual method of obtaining corrected data for the interband absorption by subtracting out the free-carrier absorption could not be applied, since the latter does not follow a simple power law in the spectral region in vestigated, and therefore could not be extrapolated accurately to shorter wavelengths. Consequently, no attempt was made to analyze the data in a quantitative fashion. The qualitative features are of considerable interest, however. The absorption edge for interband transitions ex hibits a marked shift to higher energies with increas ing free-electron concentration. Such an increase in energy is also observed for n-type InSb,!5 InAs,t6 and CdSnAs 2P·18 It occurs in materials with low electron effective masses because the Fermi level increases ap preciably with increasing concentration; as the lower states in the conduction band become filled, photons of higher energy are required to promote electrons from 10 M. Tanenbaum and H. B. Briggs, Phys. Rev. 91. 1561 (1953). 16 R. M. Talley and F. Stern, J. Electronics 1, 186 (1955). 17 A. J. Strauss and A. J. Rosenberg, J. Phys. Chern. Solids 17, 278 (1961). IS W. G. Spitzer, J. H. Wernick, and R. Wolfe, Solid-State Electronics 2, 96 (1961). the valence band to higher unoccupied states in the conduction band.l9 The shift of the absorption edge in HgSe thus supports the conclusion that this edge is associated with promotion of electrons into the con duction band. In terms of the present investigation, it is even more significant that intense interband absorption (0:"-' 103 cm-3) occurs in HgSe at photon energies considerably less than the Fermi energy. In order to illustrate this fact, a vertical arrow has been placed in Fig. 6 at the wavelength corresponding to the Fermi level calculated from Eq. (3) for each sample investigated. In each case, the absorption edge occurs at wavelengths significantly greater than the one indicated by the arrow. Qualita tively, this observation is consistent with the semimetal band model proposed for HgSe in the present investiga tion, rather than with a semiconductor band model. If phonon absorption or emission is neglected, on the basis of the semimetal model the minimum photon energy required to promote an electron into the con duction band would be less than the Fermi energy by the value of the overlap energy (€Y2) between the valence and conduction bands. The semiconductor model, on the other hand, requires a minimum energy equal to the Fermi energy plus the energy gap. A de tailed experimental and theoretical analysis of the shape of the absorption edge would be required before accept ing the optical data as convincing evidence for the semimetal model. CONCLUSION Detailed analysis of Hall coefficient data for HgSe and HgSeo .• Teo .• leads to the conclusion that these materials are semimetals, in which the valence and con duction bands overlap by approximately 0.07 ev, rather than semiconductors. Optical absorption data for HgSe are consistent with this conclusion. ACKNOWLEDGMENTS The authors are grateful to A. E. Paladino for his assistance in preparing the materials and making many of the electrical measurements, to Mrs. M. C. Plonko for making the optical measurements, and to S. Hilsen rath and Mrs. N. B. Rawson for performing most of the theoretical calculations. They are also pleased to acknowledge the helpful comments of Dr. G. B. Wright and Dr. J. M. Honig. 19 E. Burstein, Phys. Rev. 93, 632 (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.153.184.170 On: Thu, 27 Nov 2014 09:25:24
1.3047264.pdf	Nuclear physics: A report on the Paris conference Michael Danos Citation: Physics Today 18, 3, 44 (1965); doi: 10.1063/1.3047264 View online: http://dx.doi.org/10.1063/1.3047264 View Table of Contents: http://physicstoday.scitation.org/toc/pto/18/3 Published by the American Institute of PhysicsNuclear Physics A report on the PARIS CONFERENCE By Michael Danos Being an affair organized on the occasion of the thirtieth anniversary of the discovery of artificial radioactivity by Frederick and Irene Joliot-Curie, the Paris Conference on Nuclear Physics had, and was supposed to have, aspects of both a conference and of a show. Several factors contributed to its eminent success as a conference. In chronological order of impact, the first of these was the ele- vated spirits induced, at least for non-Parisians, just by the magic of Paris. The second was the excellence of the facilities and of the organization. The third was the high quality of the papers and, in general, of their presentation. The last, but in the end the most important, factor was the impression of the vitality, one even may say re- birth, of nuclear physics as a field. Right after the consolidation of quantum me- chanics and after the discovery of the neutron, nuclear physics became the forefront of physics. One of the subjects of this field was the problem of the nuclear force. About fifteen years ago this child of nuclear physics "ran away from home", changed its name to elementary-particle physics and took all the glamour and excitement with it, leaving behind a semistagnant array of disjointed subjects: the different nuclear effects were de- scribed by different models not having too much in common with each other. Nuclei seemed to be too complicated to be treated rigorously but too small for statistical mechanics to apply. Soon thereafter, following the introduction of the shell model and, later, of the many-body tech- niques, nuclear physics began slowly to emerge from the doldrums. Its character gradually changed, becoming more and more "fundamen- tal"; a growing number of effects could be under- stood from "first principles". In this way the dif- ferent subjects began to merge and to form a com- mon field. The grand, over-all impression gen- erated by the conference was that nuclear physics has rejoined the first ranks in the quest for the unknown, that connections to neighboring fields have been established to their mutual benefit, that the fragmentation has been replaced by an all-inclusive description. The closest of the related fields is, naturally, elementary-particle physics. Awhole review paper was devoted to the interac- tions between nuclear physics and elementary- particle physics (Rapporteur: Van Hove, CERN), and throughout the conference there were many papers where these and other interconnections were ostensibly present. The conference was organized to take place in two parts. During the first three days perhaps half of the contributed papers were presented by their authors. The fourth day, a Sunday, was free to give the participants an opportunity to do sightseeing in the closer vicinity of Paris and to deprive the rapporteurs of the excuse of lack of time for the preparation of their presentations. The final three days were taken up by the review lectures of the rapporteurs, the last of which was the traditional summary talk; it was given by Peierls (Oxford) . This by now well-tested ar- rangement again turned out to be very successful and satisfactory. I did not even hear any grum- bling from the many authors who had no chance to present their contributions in person. In reporting on the physics discussed at the conference, I shall attempt to indicate the present status of nuclear physics as it emerged from the papers presented at the conference, drawing mostly, but not exclusively, on the summary pa- pers given by the rapporteurs. (The names in parentheses are also mostly those of the rap- porteurs.) For complete coverage the reader should consult the conference Proceedings* In this survey, the different subjects will be arranged according to their "fundamentality". The sense in which this is meant is the following. As in any but the simplest systems, an exact de- scription of nuclei is both unattainable and un- desirable. One can gain physical insight only after suitable simplifications have been carried out. These simplifications can have the character of *Thc Proceedings of the International Conference on Nuclear Physics are published in two volumes. Volume 2, containing contributed papers, has already been issued. Volume 1, in- cluding the reviews of the rapporteurs and discussions, is expected to appear this month. They are published by the Centre National de la Recherche Scientifique, Service des Publications, 15 quai Anatole France, Paris 7, France. 44 • MARCH 1965 • PHYSICS TODAYThe Conference on Nuclear Physics was or- ganized under the sponsorship of UNESCO and the International Union of Pure and Applied Physics. It was held July 2-8, 1964, in the UNESCO Palace in Paris. Dr. Danos is a physicist in the Radiation Physics Division of the National Bureau of Standards. approximations or of models. The former are evidently more desirable than the latter. Actually, one can judge a scientific field by the preponder- ance of one over the other: in a very young field one has almost only models; as it matures, more and more models change into approximations to a "fundamental" theory. The approximations also provide the links to neighboring fields; a model usually starts out as an ad hoc invention to ex- plain a certain more-or-less narrow set of observa- tions and, in general, is meaningless outside of the particular narrow field. Only when the con- nection of the model with a "fundamental" the- ory, with "first principles", has been established (i.e., after it has developed to the point that it represents a certain well-defined approximation to the fundamental theory) , can it be used to connect with neighboring fields. (Peierls) To begin with the most fundamental subject discussed at the conference, I would like to report on the interdisciplinary field par excellence, viz., the weak interactions. It has always led a semi- autonomous life, belonging neither to elementary- particle physics nor to nuclear physics but con- tributing greatly to both fields. It has the habit of coming up with surprises. This time the subject was the universality of the four-fermion interac- tion, or—depending on one's point of view—parity impurities of nuclear states. It was always clear that a certain contribution to nuclear forces has to come from virtual /J-decays in which an electron and a neutrino are exchanged between two nu- cleons. However, this would be a second order process and thus of completely undetectable order of magnitude. On the other hand, a process where the lepton bracket in, say (nOp) (vOe) is replaced by a nucleon bracket to give (nOp) (pOn) , which is a nonleptonic, AS = 0 interaction, would be of first order and would yield effects wjiich, under favorable conditions, could be detected. As fre- quently is the case, two independent groups, sep- arated by almost half an earth's revolution (to be precise, by 145.7° longitude) have performed experiments to test for the parity impurity in- duced by such a first-order process; the Caltech group used circularly polarized photons, while theMoscow group used polarized neutrons. Both groups found an effect in agreement with the order of magnitude of a first-order process, but in both experiments the remaining uncertainties were still so large as to leave skeptics unconvinced about the reality of the effect. Perhaps one should pay more attention to the skeptics, particularly in the field of weak interactions: the object of excitement of not so long ago, the intermediate heavy boson, is retreating into the distance of ever-increasing mass, perhaps to leave behind only its name, W, with no substance to cling to—a name which, with a little fantasy, has a faint opti- cal resemblance to the grin of the Cheshire cat. At present the boson is already beyond 1.3 GeV. (Nataf, Orsay) The list of the specifically nuclear fundamental subjects begins with the nucleon two-body force. The quality of the data has by now improved so much that one has to begin to take into account the departures from charge independence in the description of the nuclear forces. For example, the range of the force resulting from a one-pion exchange is different for charged and for neutral mesons because of their different mass; in the one-pion-exchange contribution to the p-p and n-n interaction, only neutral mesons can participate, while both neutral and charged mesons can be exchanged in the p-n interaction. The over-all fit to the experimental data of the phase shifts gen- erated by potentials is already quite good, in par- ticular for the p-p forces. The p-n data are im- proving from day to day. At present, those po. tentials still give the best chi square in the fits to the data which have the least restrictions im- posed on them by theoretical considerations. Ob- viously the description is ahead of the understand- ing, a rather universal state of affairs in many instances in nuclear physics, and because of the implicit challenges quite satisfactory for a theorist, although perhaps not as satisfactory for an experi- mentalist. (Amati, CERN and Palermo) Information on the nuclear forces has come from a field from which not many expected it to come: nuclear-matter calculations. That this event could take place at all is a consequence of the funda- mental character of the many-body calculations: in principle, at least, they are exact. In practice, approximations have to be made to render the problem tractable. The question always concerns the quality of the approximations, and the quality has continued to improve over the last years. When carried out with "realistic" forces—i.e. those which reproduce the two-nucleon scattering data—it seems that these calculations have de- PHYSICS TODAY • MARCH 1965 . 45termined a feature of the two-nucleon force which until now has not been measured in high-energy experiments because of their insufficient accuracy. The feature is the repulsive core. A hard core, either an infinite or a finite "square tower", keeps the nucleons too far apart, gives nuclear matter too much saturation. A soft core of the theo- retically much more pleasing Yukawa form allows the particles to come closer together and to ex- perience the attractive potential to a larger degree. This adds several MeV to the nucleon binding energy in nuclear matter. (Bethe, Cornell) On the question of nuclear many-body forces, field theory is quite impotent beyond indicating that they should exist. Unfortunately, it also has not yet been possible to extract any definite in- formation from experimental data. (Peierls) The next fundamental subject, already alluded to above, is the nuclear-matter calculations. In his review, Bethe reported on Bethe, concerning a typical many-body effect; namely, the contribu- tion of the three-hole graphs. They form the most complicated set of graphs yielding a con- tribution to the binding energy, which is pro- portional to the square of the density. As so often is the case in many-body problems, the usual per- turbation treatment does not yield a convergent series. (As an aside, the convergence of the : Brueckner-Goldstone treatment is also in doubt.) By a tour de force, Bethe summed all three-hole graphs, finding that they contribute something of the order of a few MeV to the binding energy. This then would combine with the effect of the soft core to increase the binding energy of nuc- clear matter from about 8 MeV per nucleon to the vicinity of the experimental value, which is around 16 MeV per nucleon. A very important fundamental quantity is the two-body correlation function in nuclei. It be- gins to be accessible to experiment. Results have begun to appear from inelastic electron scattering, an experimental technique which has become feasi- ble with the advent of electron linear accelerators. As the experience with this new tool grows, highly significant results can be expected. (Bishop, Orsay) The last really fundamental subject concerns the very light nuclei. These systems are so small that perhaps it will be almost possible to obtain a solution of their Schrodinger equations in the not-too-distant future. Contrary to previous re- ports, the exotic nuclei like 4H and 5H do not seem to exist. On the other hand, the a-particle has acquired structure in that several bumps asso- ciated with an intermediate excited a-particle ap- pear in diverse reaction cross sections. One ofthem is evidently associated with a resonance: the t-p scattering phase shift goes through 90° at an excitation energy of the a-particle of about 20 MeV. The meaning of the other bumps is still unclear at this time. (Wilson, Harvard) This ends the list of fundamental subjects and brings up the models. In general terms, three ingredients determine the quality of a model: (1) the zero order approximation, (2) the residual interactions, and (3) the selection of the most important contributions of (2) . The ingredient (2) can alternatively be called "the connection with fundamental theory". (Bloch, Saclay) In the terminology of Peierls, given above, a model be- comes an approximation when the ingredients (2) and (3) are worked out. Here also, the models will be mentioned in the order determined by their closeness to fundamental theory. The foremost of the nuclear models, the shell model, is well along the road towards becoming an approximation; it is practically there. The model-ingredients (1) [H1 = %( (7\ + Vt)] and (2) [H.2 — %jVij — SjJ7,-] are quite well known, and (3) is being explored. As put in the language of the trade, shell-model calculations have to be car- ried out using realistic nuclear forces. Attempts along these lines using the Scott-Moszkowski cut- off procedure have given quite promising results in calculations for light nuclei. (Brown, Nordita) The stepsister of the shell model, the optical model, lags behind in the approach to becoming an approximation. It by now reproduces the ex- perimental results very excellently, including po- larization data. The optical model thus has in- gredient (1) of very high quality. However, even (2) is not well in hand: the rapporteur (Hodgson, Oxford) issued a plea for a derivation of the optical-model parameters from fundamental the- ory. As it stands, improvements are achieved by retreat from the model towards the shell model in that the absorptive part is at least partially re- placed by explicit introduction of nucleon varia- bles in the target nucleus. One such calculation is known by the name "doorway states". It con- cerns the single-particle excitations of the target nucleus which can be reached by two-body colli- sions involving the incoming particle and a parti- cle of the target nucleus. Even more complicated reaction cross sections involving incoming deuter- ons have been treated this way. These calculations have been astonishingly successful when clone in a "realistic" way, i.e., when allowing interactions of finite range between a target nucleon and an incoming nucleon, and when describing the nu- cleons by optical-model wave functions. One may 46 • MARCH 1965 • PHYSICS TODAYRIDL EXPANSIBLE PROJECT-MATCHED INSTRUMENTS NEW MULTICHANNEL ANALYZER SERIES JOINS RIDL'S EXPAN- SIBLE PROJECT-MATCHED FAMILY OF INSTRUMENTS. Like all RIDL's Expansible Project-Matched Instruments, Series 34-27 Scientific Analyzer Systems offer unmatched versatility to meet any specific require- ment. <JThe heart of Series 34-27 multichannel analyzers is a 400-channel common-memory unit which accepts a variety of interchangeable amplifiers, ADC's, and outputs. €J Build your own system specifically adapted to your present work, whatever it may be, including pulse height analysis, Mbssbauer effect studies, or NMR experiments. Expand it to match the needs of future work by merely changing one or more plug-in modules. NUC;R-B-219 Want more data on RIDL's new Series 34-27 Scientific Analyzer Systems? Ask your Nuclear-Chicago sales engineer or write for information. •::-What is Expansible Project-Matched Instrumentation (EPMI)? It's much more than four words to the physical scientist. It is arecognition of his varying and individualistic work and of hisneed for equally versatile and creative tools. One example of theway RIDL fills this need is the well received Designer Series™of counting modules. Now Series 34-27 Scientific Analyzer Sys-tems expand the concept of matching the instrumentation to theproject. Other important EPMI benefits are reliability, capability,availability, information, service, and the ability to speedily andeconomically adapt an existing system to future needs.RIDL DIVISION 4509 West North Ave., Melrose Park, III. 60160 NUCLEAR-CHICAGO CORPORATION In Europe: Donker Curtiusstraat 7 Amsterdam W, The Netherlands PHYSICS TODAY MARCH 1965 47predict that the optical model itself will always remain in the status of a model in the sense that a treatment of ingredients (2) and (3) would dis- solve the model. In our journey away from fundamentals we now reach the very important (albeit rather model-like) models, the diverse collective models describing the low-energy nuclear spectra. BCS, particularly when eliminating the indeterminacy of the particle number, is very good in (1) ; (2) and, in particular, (3) are hard to come by, im- provements are quite intractable. Similarly collec- tive variables lead to excellent descriptions of the spectra—i.e., again (1) is extremely good; (2) is weak. Bloch (Saclay) even calls the absence of a fundamental derivation of the collective varia- ables "the main gap in nuclear-structure theory". The need for such a "fundamental" description manifests itself in all cases where both "single- particle" and "collective" features appear simul- taneously, where an interplay of single-particle and collective aspects takes place. An obvious example is the Nilsson model. Another example is 16O, where the "mysterious" 0+ level at 6.6 MeV has revealed itself to be the head of a rotational band; in other words, it is a second, deformed "ground state". (Brown, Nordita) A further ex- ample concerns reactions: an incoming particle may not excite a single-particle state (doorway state) but may excite a collective state instead, even assuming that only two-body forces exist in nature. (Bohr, Copenhagen) The reader may have noticed that in the dis- cussion the reaction theories have not been men- tioned separately. This has not been an over- sight; after all, reactions are nothing but nuclear states belonging to the continuum rather than to the discrete spectrum, described in a time- dependent formulation. They thus should be con- sidered together with nuclear structure and not as a separate subject. (Bloch, Saclay) I now would like to discuss briefly some mis- cellaneous subjects reported at the conference. There was Flerov's (Dubna) report on heavy-ion reactions. When trying to produce new high-Z elements, one finds that one's efforts are stymied by a quite unexpected phenomenon: the spon- taneous-fission probabilities in many nuclei are greater by extremely large factors than the rates extrapolated from the known cases; factors of 10lr> are not uncommon. In his talk, Flerov mentioned that the element 104 had not yet been definitely identified. [In the meantime, press reports have indicated that Flerov finally was successful in dem- onstrating this element.]It now appears quite certain- that the experi- mentally observed energy gap in nuclei is not present in nuclear matter but is a consequence of the finite size of nuclei. The absence of the energy gap in infinite systems is qualitatively as- sociated with the fact that for the most important interactions, namely, the head-on collisions of particles near the Fermi surface, which correspond to about 150-200-MeV laboratory energy, the phase I shift is experimentally very small, a feature re- produced by realistic nuclear forces. In finite sys- tems, the preponderance of these interactions seems to be sufficiently weakened to allow a gap to appear, the magnitude of which decreases with increasing size of the system. This effect corre- sponds to experimental observations. (Bethe, Cor- nell) A new semiempirical approach to nuclear many-body calculations was proposed by Migdal (Moscow). He suggests using theoretical considera- tions to determine the form of the equations and fitting parameters by comparison with experiment, rather than calculating them from the nuclear forces, which, in principle, could be done. With some approximations, he, in fact, reduced the number of parameters to one, and applied his analysis to several nuclear problems. One may expect that this method will correlate many ex- perimental data in a semiquantitative manner. Concerning the low-energy levels, the rotational spectra are very regular up to states of very high angular momentum, while the situation with re- spect to vibralional levels already is quite con- fused at two-phonon excitations. Part of the con- fusion may result from the closeness of their energy to the edge of the energy gap above which the density of the "single-particle" states increases drastically. Baranger (Pittsburgh) formulated it this way: "Deformed nuclei are much more de- formed than spherical nuclei are spherical." An effect of the sudden appearance of deforma- tions at N = 88 has now been detected by mass spectroscopy: the trend of the binding energy of two neutrons as a function of the neutron number has a discontinuity in the slope at N — 88. This is the strongest support so far for the often-expressed hypothesis that both a deformed and a spherical regime exists in all nuclei; it just so happens that the dependence of the energy on the neutron num- ber is different for these two states so that the deformed state becomes the lower one at N = 88. (Kerman, MIT) The inevitable has happened: dispersion-rela- tion techniques have been applied to the study of nuclear reactions. In this approach no distinc- 48 MARCH 1965 • PHYSICS TODAYSCIPP SERIES ANALYZERS Expandability feature, exclusive with Victoreen SCIPP series, enables user to expand SCIPP 400 single-parameter pulse height analyzers to 1600-channel single-parameter or 1600-channel multiparameter operation as his requirements dictate. —for the first time — there is no need to buy an analyzer with a built-in obsolescence factor. New Victoreen SCIPP series analyzers are designed to grow as your needs grow. SCIPP 400A can perform precision pulse height analysis.. .yet can be expanded readily to SCIPP-1600TP for multiparameter operation ... or can perform other analyses. Other important features of SCIPP series analyzers: 8MC digitizing rate; silicon semi-conductors used throughout. Full details on request. Meantime check condensed specification data below. 8MC DIGITIZING RATE • PATCH PROGRAMMABLE 400,1600,10,000 words (word length 106, decimal). Patch program can be changed to meet virtually any requirement. Built-in address and data register. Random parallel access. Serial and parallel outputs. On-line data handling.Memory subgrouping, transfer and overlap capability. Digital factor-of-two display of mem- ory live, static, static/live. Single or dual ADC units available — parallel transfer from ADC address register to memory address register. Analog function of pulse input ADC's. High-speed readout and readin.TlULLAMdrW // VVICTOREEN w A DIVISION OF THE VICTOREEN INSTRUMENT COMPANY 5B57 West 95th Street • Oak Lawn, Illinois, U.S.A. ,This unique HAEFELY Accelerator is the heart of the world's largest electron microscope at Toulouse, France. Five years of successful operation without a single breakdown. 1.5MeV Cockroft-Walton type Stability 1:100.000 Emile Haefely & Co. Ltd. Basel/Switzerland For more information, contact the U.S. representative: B. FREUDENBERG INC. 50 Rockefeller Plaza New York, N. Y. 10020 Tel: 2)2 PI 7-9130tion is made between elementary particles and composite particles, and the aim is to establish connections between different reactions involving a set of particles, say, a proton, a triton, lnB and 12B. A difficulty of principle, not counting mathe- -matical difficulties, is the existence of unphysical energies (energies which cannot be realized in ex- periments; e.g., a center-of-mass energy of Mpc2 + M,,c2— 10 MeV in proton-neutron scattering). Nevertheless, information on the scattering ampli- tude at these energies is needed in the formal- ism; thus, not all input data can be determined from experiment. The astonishing fact is that it has been possible to collect essentially all of the indeterminacy of the problem into one parameter which can be adjusted from the data of one reac- tion. Then one can make predictions concerning other reactions, and, as far as tested, the procedure seems to work. (Shapiro, Moscow) Before closing, I should like to report on some technicalities which contributed greatly to the suc- cess of the conference. The meeting had on the order of a thousand participants. It had the feel of a small, intimate conference. This incredible feat was the result of the smooth and carefully prepared organization and depended decisively on the excellent facilities. The meetings took place in the UNESCO Palace where all seats were equipped with functioning earphones and where even in the plenary sessions there was desk space for about three-fourths of the participants. Thus, even at the more remote corners of the hall, the participants experienced no crowding, were acous- tically just three meters away from the speaker, and could follow the proceedings without strain. The opportunity provided by the conference was utilized by Rosenfeld (Copenhagen) to try to do something about the paper explosion and the consequence of it, the information implosion; because of the first, people may in despair give up reading altogether. It seems that nothing feasible was proposed to cure the evil at the root, viz., to stem the torrent of papers. However, an inter- national committee was organized to set up, among other things, a list of key words ostensibly to facilitate literature search by computers. This list is supposed to be as small as possible, but sufficiently large so that a line or two of key words chosen from the list should suffice to specify the contents of the article quite precisely. What I like about the proposal is that this line, printed between title and abstract, should allow the reader to skip both of these and to reduce the reading time of journals to the time it takes to page through them. I hope something comes of it. 50 MARCH 1965 PHYSICS TODAY
1.1725379.pdf	Statistical Model Including Angular Momentum Conservation for Abnormal Rotation of OH* Split from Water Tadao Horie and Takashi Kasuga Citation: The Journal of Chemical Physics 40, 1683 (1964); doi: 10.1063/1.1725379 View online: http://dx.doi.org/10.1063/1.1725379 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/40/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Rotational transfer, an angular momentum model J. Chem. Phys. 98, 4586 (1993); 10.1063/1.465020 Angular momentum conservation demonstration Phys. Teach. 27, 561 (1989); 10.1119/1.2342870 NOTES: Conservation of Angular Momentum in a Rotating Fluid Phys. Teach. 12, 493 (1974); 10.1119/1.2350516 The Conservation of Angular Momentum Am. J. Phys. 33, 345 (1965); 10.1119/1.1971501 Statistical Interpretation for Abnormal Rotation of OH* Split from H2O2 by Electron Impact J. Chem. Phys. 31, 783 (1959); 10.1063/1.1730462 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 07:48:32OPTICAL PUMPING AND CHEMICAL REACTIONS 1683 are to be studied by optical pumping techniques, only these gases will be useful. In turn, these very stable gases will have to be decomposed by photolysis in the vacuum uv or by particle bombardment. Optical pump ing may still be very useful in interpreting a limited but important class of reactions. THE JOURNAL OF CHEMICAL PHYSICS ACKNOWLEDGMENTS One of us (Richard Bersohn) is indebted to Professor A. Kastler of Paris for a stimulating stay at his labora tory. Financial support for this work was received from the U.S. Air Force and the U.S. Atomic Energy Com mission. VOLUME 40, NUMBER 6 15 MARCH 1964 Statistical Model Including Angular Momentum Conservation for Abnormal Rotation of OH* Split from Water TADAO HORIE AND TAKASHI KASUGA Faculty of Science, Osaka University, Osaka, Japan (Received 23 October 1963) In order to give an account of extreme, non thermal populations of the rotational states of the 22:+ level of OH split by electron impact from water, a proposal is made for a statistical model which includes the law of angular momentum conservation within it. The interaction volume is taken to be double-walled and spher ically symmetric with respect to the center of gravity of the whole system which consists of OH* and H. It is shown that abnormal populations quite similar to typical ones thus far experimentally observed can be derived from the model proposed with the aid of the Franck-Condon principle and a predissociation of OH. I. INTRODUCTION AFTER the announcement of Oldenberg in 1934, .ft several kinds of nonequilibrium populations have so far been reported of the rotational ,tates of the ex cited OH* (2~+) radicals produced in various types of discharges through water vapor. The mechanism, how ever, of free radicals production is so complicated especially in discharge plasmas, that any theoretical treatment has not yet been devoted to those abnormal phenomena. Such being the case, it may go without saying that experimental conditions should be made as simple as possible. Along this line, two measurements have been done; one in the U.S. National Bureau of Standards by a microwave discharge through water vapor ex tremely diluted with rare gas,! and the other in Osaka University by a crossed-beam technique with water molecular jet and electron beam.2 Both of them have revealed similar populations of abnormally rotating OH* radicals. It is most likely that there exist two groups of OH* split by electron impact from H20. One includes highly rotating radicals at the rotational temperature of 14000oK, and the other slowly rotat ing ones at the room temperature. In addition, strange to say, the abnormally rotating radicals are strikingly predominant in number over the thermally rotating ones. Here we proposed a statistical model including the law of angular momentum conservation, which will lead us to an over-all distribution of rotational popu- 1 H. P. Broida and W. R. Kane, Phys. Rev. 89, 1053 (1953). 2 T. Horie, T. Nagura, and M. Otsuka, Phys. Rev. 104, 547 (1956); J. Phys. Soc. Japan 11, 1157 (1956). lations without mlssmg the abnormal feature men tioned above. After having been vertically raised by electron impact up to an electronically excited level, the H20 molecule splits into Hand OH. Without going into detail on the transient behavior of the dis sociation, an assumption is made as follows. The excess energy possessed by the excited water molecule is dis tributed in an at-random fashion among the transla tional and rotational degrees of freedom of Hand OH before the fragments happen to separate from each other far beyond a certain distance (which is a little longer than twice the bond length of OH). 2. ANGULAR MOMENTUM CONSERVATION AND INTERACTION VOLUME Before excitation takes place, the water molecule has an angular momentum C around its center of gravity, which here will be assumed fixed in space. The molecule is raised by some means up to a certain electronically excited state, which is supposed to have an excess energy E, and then splits into free radicals. The angular momentum of the whole system should be conserved throughout the course of splitting. This conservation is expressed by Dirac's delta function as o(rxp+RxP+N-C), where N is angular momen tum vector of OH, and where p, P and r, R are linear momentum vectors and positional vectors of Hand OH, respectively. Similarly, the energy and linear momentum conservations are also given by using delta functions as usual. If the center of gravity of the whole system is taken as the origin of coordinate, the phase integral 1>(E)dE for the OH radical to have rotational energies E to This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 07:48:321684 T. HORIE AND T. KASUGA t L c FIG. 1. Dependence of the limits of integration with respect to L upon N or (21.)t. E+dE is defined as <1>( E) = const f d3r f d3R f d3p f d3P f dOd¢ f dP8dP4> Xo[E-(p2/2m) -(p2/2M) -(N2/2I) J Xo(p+P)o(mr+MR) Xo(rxp+RxP+N-C)O[E-(N2/2I)J, (1) where 0, cp are the spherical polar coordinates of the molecular axis of OH* with respect to the rectangular axis fixed in space, and P8, P4> the conjugate momenta, and where m is mass of H, M mass of OH, I moment of inertia of OH. The integrals over Rand Pin Eq. (1) are immedi ately evaluated, since in general (2) and we have <I>(E) =constf dJr fd3P fdOdcp f dP8dP4> Xo[E-(p2/2,u) -(N2/2I) J Xo(L+N-C)O[E-(N2/2I)J, (3) where ,u is the reduced mass of m and M, and L is (m/,u)rxp. The integrals with respect to all the other variables except for rand L are also found step by step with the aid of Eq. (2) and the method of change of variables, and we have the following integral with no more delta function, Referring to the integral over r, a tentative assump tion is made of the interaction volume in which H and OH* exchange energies with each other. It is bounded externally by a sphere around the origin of radius Ro and internally by a sphere of radius ro, where Ro is a little larger than twice the bond length of OH, roo We then have <1>( E) = const[ CE!(~-E)]{ f dL[2Io( E-E) -DJi -fdL[2io(E-E) -DJ!}, (5) where 10 is (m/,u)2,uR 02, io (m/,u)2,ur02, and the latter nearly equals I. The interval of the first integral in Eq. (5), for instance, is determined as follows. Owing to the law of angular momentum conservation, the three vectors, L, N, and C should make a triangle. Accordingly, when N::; C, L takes values ranging from C -N to N +C. On the other hand, when N?:. C, L varies from N -C to N + C. In addition, the integrand should be real. As a result, the interval depends upon N, or (2IE)l, as summarized in Fig. 1. The range of integra tion with respect to L is shown by hatching parallel to the ordinate for (2lE)!::;C, by dots for C::; (2Ie)l::; (2I El) I, and by hatching parallel to the abscissa for (2Iel)!::; (2Ie)!::; (2Ie2)1. Finally, we have <I>(e) =const(1/Ce 1) X {Io[sin-lA+A (1-A2)!-sin- lB-B(1-B2)!J -io[sin-la+a(1-a2)Lsin-lb-b(1-b2)lJ}, (6) where A stands for Lmax/[2Io(E-e) Jt, B for Lmin/[2Io(E-e) J!, and a, b for the same expressions in which 10 is replaced by io, and where Lmax and Lmin are the upper and lower limits of the interval of the integral over L. 3. NUMERICAL CALCULATION In order to make numerical calculations by the use of Eq. (6), it is convenient for e to be replaced by rotational quantum number K as e= (1/2l) (h/271')2K(K+l). Similarly, E and C will also be replaced by effective rotational quantum numbers KE and Kc defined as E= (1/2I) (h/271') 2KE(KE+ 1) , and C= (h/271') [Kc(Kc+l) Jt. Then, <I>(e)de is reduced to a function of K, <I>(K; 1', Kc, KE), containing three parameters. Among these, I'is (io/Io)l, and depends upon the radii of the inter action volume, ro and Ro. It seems plausible to assume that Ro is a little larger than 2ro, and therefore l' will tentatively be fixed at 0.45. The second parameter Kc will also be fixed at 2 in the following manner. The water mole<:;ule at th~ This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 07:48:32ABNORMAL ROTATION OF OH 1685 ground state has three axes of rotation. The distribu tion function of rotational energies is given by dudvdw exp( -l/kT) [(lj2iu)u2 + (1/2Iv)v2+ (1/2Iw)w2], where Iu, Iv, Iw and u, v, ware moments of inertia and angular momenta with respect to the three axes. According to the mathematical study of Okamoto,3 the above function can be replaced in a good approxi mation by (/dC exp( -ljkT) (C2/2Im), where (/= U2+V2+W2, and 1m is the harmonic mean of I", Iv, and I w. Here the temperature T depends upon the experimental condition. In the crossed-beam experi ment, the molecular beam of water emerges from a reservoir maintained at the room temperature. On the other hand, Eq. (6) has the factor l/C. Accordingly, <I>(K; Kc, KE) depends upon Kc mainly through the factor (2Kc+ 1) exp[ -0.08Kc(Kc+ 1)], which shows a maximum at the Kc value of 2. We are now ready to make numerical calculations of <I>(K; KE) in dependence on the third parameter KE• Figure 2 shows some of the calculated populations of the OH rotational states, where the value of KE is attached to each curve. By the way, the expression in the first bracket of Eq. (6), for instance, has a geo metrical meaning as the area surrounded by a unit circle and two parallel straight lines distant in the same direction from the center by A and B. Both A and B vary with K or ~, and at a certain value of K corresponding to ~1 indicated in Fig. 1 A becomes unity. Above this value of K, the A line remains in tangential contact with the circle, while the B line moves toward the A line with an increase in K, and finally both coincide with each other. Such a geometrical behavior makes it easier to see the numerical calcula tion. 4. COMPARISON WITH EXPERIMENT Everyone of the relative population curves obtained above shows an edge at the rotational quantum num- 0·10 0·05 o 5 10 15 20 25 30 Rotational Quantum Number, K FIG. 2. Calculated distributions of rotational populations in dependence on KE. Here K c is fixed at 2, and'Y at 0.45. The area under the curve is normalized to unity. 3 M. Okamoto, Osaka Math. J. 13, 1 (1961); Tokei Danwakai, 6,1 (1962) (in Japanese). 5 L ..' > ':;1 a; 0:: 048121620 32 RotQtional Guantum Number -+-- FIG. 3. Calculated relative rotational populations of OH* (2~+) split from H20. (a) Lyman-alpha photon dissociation of H20. (b) Electron-impact dissociation of H20. The dotted line indicates deviation due to predissociation of OH. ber of 2. It is simply because the parameter Kc was fixed at 2. In actuality, however, the distribution of the parent molecules with respect to Kc is approxi mately proportional to the factor, as mentioned in the last section, of the Maxwell-Boltzmann distribution law. The edge is readily rounded off by taking it into account. Tanaka, Carrington, and Broida have recently re ported relative popUlations of OH observed in emission from photon dissociation of water.4 The radiation source they used had a large amount of emission at the Lyman alpha (1215-A) line, which does correspond to the KE value of about 24. This effective rotational quantum number must be composed of two parts, 22 and 2, where the former corresponds to the energy available directly from Lyman-alpha and the latter the rota tional energy initially possessed by the parent mole cule. The calculated curve for KE = 24 is shown in Fig. 3(a). This curve exhibits an extremely abnormal popUlation around K = 20. Quite the same character istic feature is also recognized by a pen recorder figure in a recent private communication from Broida to one of the authors. In case of electron-impact excitation, the exciting energy is not so monoenergetic, contrary to photon excitation. Accordingly, E is expected to have a rather broad distribution, as has often been pointed out else where.6 According to the Franck-Condon principle, the distribution of E may be determined by reflecting the position probability functions associated with the nor mal vibrations of the ground state of water onto the potential energy surface of the excited state, when both of the states are well informed. In such a case, it is necessary to take the weighted mean of the popu lation curves presented in Fig. 2 for a wide range of values of the parameter KE• For simplicity, the excess energy will be assumed to have a Gaussian distribution symmetric with respect 4 I. Tanaka, T. Carrington, and H. P. Broida, J. Chern. Phys. 35, 750 (1961). iF. H. Field and J. L. Franklin, Electron Impact Phenomena (Academic Press Ltd., London, 1961), p. 59. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 07:48:321686 T. HORIE AND T. KASUGA to a certain value of KE• If KE is estimated at 28, and the half-width at 12, the relative population presented by the solid-line curve in Fig. 3, (b) is obtained. Here the dotted curve indicates a deviation due to the pre dissociation of OH which takes place beyond K = 24.1,6 The curve (b) followed by the dotted branch seems quite similar to the result of the electron-impact ex periment which has been reported in detail several years before.2 From the results thus far obtained, it is most likely that the rotational distribution for elec- 6 A. G. Gaydon and H. G. Wolfhard, Proc. Roy. Soc. (London) A208,63 (1951). THE JOURNAL OF CHEMICAL PHYSICS tron impact can be derived from that for photon excitation with the aid of the Franck-Condon principle and the predissociation effect. In conclusion, so far as the characteristic features are concerned, the statistical model proposed here has led us to both types of the abnormal populations ob served in the photon excitation and in the electron impact experiment, notwithstanding that they look strikingly different from each other. ACKNOWLEDGMENT The authors are grateful to Professor H. P. Broida for helpful and stimulating informations. VOLUME 40, NUMBER 6 IS MARCH 1964 Analytical Expressions for the Hartree-Fock Potential of Neutral Atoms and for the Corresponding Scattering Factors for X Rays and Electrons* T. G. STRAND AND R. A. BONHAMt Chemistry Department, Indiana University, Bloomington, Indiana 47405 (Received 14 August 1963) Approximate analytical expressions for the Hartree-Fock potential of neutral atoms to Z =36 have been obtained by fitting the radial electron density with an analytical expression by least squares. The expression for the radial density corresponds to the following form of the screening factor: Zp(r) 2 m --= ~a'Yiexp(-aAir)+r ~b'Yiexp(_bAir), Z i-I j-1 where m=2 for Z=2 to Z=18 and m=3 for Z=19 to Z=36. The corresponding expressions for the mean radius, the mean square radius, the diamagnetic susceptibility, and the atomic scattering factors for x rays, and for electrons according to the first Born approximation are given. The accuracy of the approximate expressions is discussed in relation to results obtained by numerical calculations from the Hartree-Fock wavefunctions for the atoms. 1. INTRODUCTION IN a previous paper,' approximate analytical expres sions for the Thomas-Fermi-Dirac screening factor for neutral atoms were given along with the correspond ing expressions for the radical electron density and the scattering factors for x rays and electrons. A bibliog raphy of analytical electron screening functions and analytical expressions for scattering factors has also been given in this paper. In the present work, approximate analytical expres sions for the Hartree-Fock (HF) screening factors have been determined for neutral atoms to Z=36, including extrapolated values of the parameters for scandium (element). The corresponding expressions for the radial densities, and the scattering factors for x rays and electrons are given for all of these atoms. * Contribution Number 1171 from the Chemical Laboratories of Indiana University. t We wish to thank the Air Force Office of Scientific Research for financial support of this work. 1 R. A. Bonham and T. G. Strand, J. Chern. Phys. 39, 2200 (1963) . 2. ANALYTICAL FORMULAS The radial electron density of neutral HF atoms could be accurately represented by the expression cor responding to the following form of the screening fac tor, Z~.(r)/Z: Zp(r)/Z= La1'i exp( _aAir)+r Lb1'j exp( _bAjr), (1) i i where r is the radial distance, Zp(r) the effective nu clear charge for the potential, Z the atomic number, and the a1' i, aAi, b1' it and bA/s are parameters to be determined for each atom. For r= 0, the following condition for the a1'{S is obtained: (2) The expression (1) has previously been used by Ibers2 to obtain analytical expressions for the Hartree (H) or HF potential of the atoms Z=9, 18, 74, and 80. The electrostatic potential, Zer-1[Zp(r)/Z], and the 2 J. A. Ibers and J. A. Hoerni, Acta Cryst. 7, 405 (1954). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Mon, 22 Dec 2014 07:48:32
1.1729123.pdf	Modulation of Carrier Surface Lifetime and the Time Constants of Surface States in Si G. C. Alexanderakis and G. C. Dousmanis Citation: Journal of Applied Physics 34, 3077 (1963); doi: 10.1063/1.1729123 View online: http://dx.doi.org/10.1063/1.1729123 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Interplay of bulk and surface properties for steady-state measurements of minority carrier lifetimes J. Appl. Phys. 111, 123703 (2012); 10.1063/1.4729258 Effects of high carrier densities on phonon and carrier lifetimes in Si by time-resolved anti-Stokes Raman scattering Appl. Phys. Lett. 90, 252104 (2007); 10.1063/1.2749728 Steady-state and time-resolved photoconductivity measurements of minority carrier lifetime in ZnTe J. Appl. Phys. 86, 6599 (1999); 10.1063/1.371629 A contactless method for determination of carrier lifetime, surface recombination velocity, and diffusion constant in semiconductors J. Appl. Phys. 63, 1977 (1988); 10.1063/1.341097 Determination of effective surface recombination velocity and minoritycarrier lifetime in highefficiency Si solar cells J. Appl. Phys. 54, 238 (1983); 10.1063/1.331693 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Tue, 02 Dec 2014 08:59:19MICROPLASMA BREAKDOWN IN GERMANIUM 3077 that the slow pulses at low temperature and the shorter ones at room temperature originate in the same centers. The slow pulses frequently observed at liquid N 2 temperatures are found to be constant in amplitude at a constant voltage, and random in length. Individual centers, however, may have pulse patterns in which the pulses vary in average duration and in average repeti tion rate. This is attributed to variations in carrier generation rates in the vicinity of the centers. Pulse data obtained over the microplasma instability region show the slope of the volt-ampere characteristic to be approximately linear and coextensive with the static characteristic. The high resistance of the diode in breakdown, as shown by this slope, suggests that there is a current limiting element in series with the break down region which we believe to be the spreading resist ance arising from the small size (about 1000 A in diameter) of the center. A further consequence of the presence of centers is found in multiplication experiments made by photo injection of carriers. We find a sudden decrease in the magnitude of the measured value of the multiplication as each center breaks down. Microplasma breakdown voltage is found to have a positive temperature dependence which tends to in crease with base resistivity, though it actually varies considerably from center to center, even within the same junction. The temperature coefficient of break down, however, appears to be constant, 0.0021 (0C)-l at -196°C, independent of base resistivity and of in dividual center characteristics. It has been suggested that suitable diodes can be used as cryogenic thermom eters capable of reading to better than ±O.Ol°C at -253°C and that other diodes have possible applica tions as rapid photoactivated switches with a very high ratio of open to closed resistance. JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 10 OCTOBER 1963 Modulation of Carrier Surface Lifetime and the Time Constants of Surface States in Si* G. C. ALEXANDRAKIS Nuclear Research Center "Democritos," Athens, Greece AND G. c. DOUSMANISt Center for the Advanced Study of Physics and the Philosophy of Science and Nuclear Research Center "Democritos," Athens, Greece (Received 5 April 1963) The effects of ac fields on the surface potential (IPs) and recombination velocity (s) (or surface lifetime) of carriers in Si surfaces have been studied by means of a method used earlier in Ge. The field is applied normally to the "back surface" of large area p-n junctions and its effects on-the surface are detected by means of changes in the reverse saturation current of the diode. The study yielded the following results: The range of the equlibrium values of IPs at 3000K is about ±0.5 V. Mainly one recombination energy level is found at ±16 kT from mid-gap. The effectiveness of the modulation of s is a measure of surface response. The data are compared with frequency response curves derived for some specific distributions of time con stants for the surface states. The response to applied fields is much larger in the higher frequency range (5 to 50 kc/sec) rather than at 10-1000 cps. The time constants for the Si surface states involved in this behavior are "" 10-4 sec, one to two orders to magnitude shorter than in Ge, and this may account for the difficulties encountered earlier in modulating IPs in Si at the low frequencies used in Ge. These surface states, that inhibit modulation at low frequencies, may be the usual states on the outside of the oxide ("slow" states), part of the "fast" states at the semiconductor-oxide interface, or may be spread in the space-charge region. I. INTRODUCTION THE time constants of surface states in semicon ductors can be measured by determining changes in surface properties induced by ac fields applied nor mally to the surface. The extent of such changes, in surface parameters such as conductivity or surface re- * Work performed under the auspices of the Greek Atomic Energy Commission. t Present address: RCA Laboratories, Princeton, New Jersey. combination velocity (surface lifetime), is a function of frequency. In the present work, measurements of changes of surface recombination velocity have been made over an appreciable range of frequencies. From these data the time constants of the Si surface states are determined. In addition, the range of the equilib rium values of the surface potential and the structure of the energy levels of the surface states that give rise to surface recombination have been studied. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Tue, 02 Dec 2014 08:59:193078 G. C. ALEXANDRAKIS AND G. C. DOUSMANIS EXCESS ELECTRONS / AT SURFACE C-BAND ~= "FAST" SURFACE STATES <1</ob(NEG~ --------V----r-~r'~--------~------E, }<..<I~(PO~ . SPACE-CHARGE REGION V-BAND IONIZED ACCEPTORS OXIDE· LAYER FIG. 1. Energy bands at a semiconductor surface for the case of p-type bulk materials with an n-type surface. "Fast" states, located at the semiconductor-oxide interface act as electron-hole recombination centers. The charge in the "slow" surface states, on the outside of the oxide layer, establishes the equilibrium value of the surface potential <{". The carrier densities are given everywhere by n=ni expq<{,/kT, where <{' varies between <Pb deep in the bulk and 'P, at the surface edge of the space-charge region. II. EXPERIMENTAL METHOD AND RESULTS ON CHANGES IN SURFACE POTENTIAL AND RECOMBINATION VELOCITY The experimental technique used is the same as the one applied earlier to Ge1j2 and is only briefly de scribed here. The ac electric field is applied normally to the "back" surface of large Si diodes. The thickness of the Si wafer is smaller than the carrier diffusion length. The reverse saturation current 18 of such diodes is determined mostly by the carrier's surface, rather than by the bulk lifetime. The ac fields modulate the surface potential 'P} (Fig. 1). The surface recombination velocity s, being a function of 'P8, is also modulated. Changes in s produce a change in the reverse saturation current 18 which is applied to the vertical input of an oscilloscope. The ac field simultaneously drives the scope horizontal and one obtains directly patterns of I, vs E or, qualitatively, of s vs the surface potential 'Ps since changes in I, are proportional to changes in 10TO 600 V 0) / +----l--+--+ .... V FIG. 2. Circuit for measurement of the effects of applied fields on the surface recombination veloc ity. The change in the saturation cur rent I, of the re verse-biassed p-n junction is plotted on the oscilloscope directly as a func tion of the field applied to the sur face. 1 J. E. Thomas, Jr., and R. H. Rediher, Phys. Rev. 101, 984 (1956). 2 G. C. Dousmanis, Phys. Rev. 112,369 (1958). 3 G. D. Watkins in Progress in Semiconductors, edited by A. F. Gibson (Heywood and Company Ltd., London, 1960), Vol. 5, p. 1. See this review article for further references on this subject; Also, J. T. Law in Semiconductors, edited by N. B. Hannay (Reinhold Publishing Corporation, New York, 1959), p. 676. s, and 'P. is a monotonic (although not a linear) func tion of the applied field E. The circuit used for measurements is very simple and is shown in Fig. 2. The semiconductor diode, and the mica and electrode assembly is placed in an airtight chamber so that the surface can be exposed to various gases at nearly atmospheric pressures. The additional RC circuit shown is used to correct a displacement voltage drop that can obscure the effect of E on S.2 Theoretical patterns of s vs 'P. for a single level for the surface recombination states are shown in Fig. 3. s in Fig. 3 is given by the formula3: CURVE A B C (E,-Ei) / KT +22 OR -22 +16 OR -16 +8 OR -8 FIG. 3. Calculated curves of s vs <{'. for a single surface recom bination state. E,-E; is the energy of the state in relation to the middle of the forbidden gap. The curves apply to the case of equal cross sections for capture of holes and electrons (C p = Cn). In the experiments one observes oscilloscope patterns similar to portions bac or b'a'c', depending on whether the surface is n or p type. The thicker portions of graph B indicate data obtained with 5 \1-cm p-type Si (Sec. II). Figure 3 shows s for three different values of the energy Et-Ei (measured from mid gap) for the "fast" state that is responsible for surface recombination. C p and en are the probabilities (= cross section x thermal velocity) for capture of holes and electrons, respectively. 'Pb is the "bulk" potential denoting the distance of the Fermi level from mid gap and 'P. is the surface poten tial. IV t is the number of states per cm2 and k, T have their usual meaning. If the applied field is small, the pattern observed on the oscilloscope is a straight line and its slope indicates the type of surface (p, n, or intrinsic if the maximum of s is observed) at equilibrium. If the frequency of the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Tue, 02 Dec 2014 08:59:19MODCLATIO~ OF CARRIER SURF .. \CE LIFETIME I~ Si 3079 E- n-TYPE cPa p-TYPE FIG. 4. A pattern J., vs applied field in n-type Ge. Qualitatively the curve is a portion of one of the s vs .p, curves of Fig. 3. The equilibrium value of the surface potential is positive (n-type surface). The field varies it over the n+ side and over the intrinsic range towards the p side exhibiting the maximum of surface recombination. applied field is high in comparison with the time con stants of the slow surface states, then the field modu lates very effectively the surface potential around the equilibrium value, indicated by an "operating point" a or a' in Fig. 3 (hence the surface recombination velocity). In this case, the slow states do not have sufficient time to change their charge during the ac cycle. If the frequency is decreased then the charge of the slow states changes and shields the space-charge region from the external field. The amplitude of the vertical scope deflection for given field magnitude as a function of frequency is an indication of the effective ness of modulation. It. has been used before in this manner to determine the time constants of the slow surface states in Ge.2 A similar technique was used previously to measure the same Ge time constants by noting the field effect on the surface conductivity.4 The curve of s vs 'Po is bell shaped with s being large for small (in general) values of 'P" and rather small at extreme values of the potential (strongly p-or n-type surface.) Figures 4 and 5 show results obtained with germanium.2 As in earlier work,1.2 the ac field frequency E- n -TYPE INTRINSIC p-TYPE cPa FIG. 5. Another pattern of J, vs 1, in n-type Ge. The suriace in the absence of the field is close to intrinsic. The abscissa can be changed from E to .p" in which case the graph fits a curve such as graph c of Fig. 3. 4 R. H. Kingston and A. L. McWhorter. Phys. Rev. 103. 534 (1956). is 50-100 cps. The pattern in Fig. 4 is surface recom bination vs applied field (or induced charge). The curve is only qualitatively similar to the curves of s vs surface potential,I.2 since 'Ps is not a linear but only a monotonic function of E. In the Ce patterns (Figs. 4 and 5) the applied field induces large changes in surface potential, changing it from Il type to intrinsic and over to the p side of the s vs 'P, curve (Fig. 3). Similar patterns are observed in p-type Si. The results are shown in Figs. 6 and 7. (Because the Si diodes are p type, one ob serves a minimum on the oscilloscope instead of the maximum seen on Figs. 4 and 5 with n-type Ce. Figures 6 and 7 are the reverse of the ones seen on the scope.) The Si and Ge specimens are treated with the usual CP4 etch. In dry air one obtains in Si a p-type surface, as indicated from the slope of the patterns in 6(b) and i(b). Addition of H20 vapor, or I'."2+H20, changes the surface from p to n type, as in the case of Ge surfaces.2.:l,5 QINDUCED -. QINDUCEO- (0)N2+H2O Ib) ROOM AIR In-TYPE SURFACE) Ip-TYPE SURFACE) Ie) INTERMEDIATE BETWEEN (0) AND Ib) I INTRINSIC SURFACE) FiG. 6. Patterns of f s vs induced charge Q (or applied field) in p-type Si with an ambient atmosphere of N,+H 20 (a), of room air (b), and intermediate helween the two (c). The observed patterns of s vs H can be changed to those of s vs 'P,.2 Changes in I, are proportional to changes in s, but the change of the values of the abscissa from E (0 'Ps is more involved.2 One uses the fact that at the maximum of s the value of the poten tial equals (!)Xln(CpC,,) where Cp, C" are the capture probabilities for hoks and electrons. At the maximum 1".,=0, if one assumes that Cp=C". The measured value of the induced charge, then, at s maximum is equal and opposite to the surface charge at equilibrium. This, from the known curves2 of Q vs 'Ps, yields the equilib rium value of the surface charge. Every point on the J~ scale then is changed from Q to the corresponding 'P., by adding to the induced Q the equilibrium value of the surface charge, and reading from the published curves of charge vs potentiaF the corresponding value of 'Ps. More details on this method will be found in Ref. 2. 5 R. H. Kingston, J. .\ppl. Phys. 27, 101 (19.16). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Tue, 02 Dec 2014 08:59:193080 G. C. ALEXANDRAKIS AND G. C. DOUSMANIS 1./ ,~~ C/em" - C/em" - 1 1 1 1 1 I -1.411011 0 +1.4110" -1.411011 0 +1.4110" FlG. 7. Patterns of I. vs E in 5 ll-cm p-type Si, similar to those of Fig. 6. The values of the induced charge is obtained from the measured values of applied voltage and the capacitance. Using the method, the results on Fig. 7 are plotted on the s vs CPs graph in Fig. 3, and one sees that they indicate an energy level for the surface recombination states at ± 15.7 kT from the middle of the forbidden gap. This agrees well with results published earlier.2,6 The error in qCPs/kT and (Et-Ei)/kT introduced by the assumption C p= Cn is t In(C p/Cn). For C p com parable to Cn this is negligible, but if Cp/Cn,,-,104 (states reported by Rupprecht),7 t In(Cp/Cn) =4.6 which is 23% of the total range of qCPs/kT(±20). A self-consistent method of dealing with the case of arbitrary values of Cp/Cn is described in detail in Ref. 2. In the present work the error introduced by this is not expected to be larger than the uncertainties from other sources (induced charge measurements, specimen resistivity, etc.) The above method overestimates the values of CPs (and that of the energy level distance from mid gap) because it neglects charge trapped in the fast surface states. Although this charge can be a substantial frac tion of the total induced change, it appears that it can be neglected for our present purposes without apprecia ble errors in CPs and Et-Ei• For large bending of the bands, the surface potential is a logarithmic function of the charge, so that CPs is not substantially affected even if an appreciable fraction of the charge goes to the fast states. Thus in Ge this method2 yielded values of CPs that are in fair agreement with results of other methods3 (e.g., conductivity measurements) which if anything, underestimate CPs because of neglecting changes in surface mobility. From Figs. 3 and 7, then, the range of (qcp.,) observed in Si is in the range of ±20 kT or ±SOO mY. Another type of pattern, observed earlier in Ge,2 has also been found in Si: s as a function of increasing field increases at first, then flattens, and then increases again. Two energy levels for the fast states could account for such behavior. Figure 8 shows patterns showing step-line portions obtained for various combinations of surface state parameters. Each curve is a superposition of the 6 H. Statz, L. Davis Jr., and G. A. de Mars, Phys. Rev. 98, 540 (1955); H. Statz, G. A. de Mars, L. Davis Jr., and A. Adams Jr., Phys. Rev. 101, 1272 (1959); 106, 455 (1957). 7 G. Rupprecht, J. Phys. Chem. Solids 14, 208 (1960). single-level curves of Eq. (1), but state 2 is assumed to be twice as effective as state 1 for surface recombina tion, The calculated curves of Fig. 8 show that one can indeed obtain patterns of the observed type from a superposition of single-level curves. The slope of the patterns of I. vs Q changes sub stantially when the Si surface is exposed to light. A similar behavior has been reported earlier in Ge.1,2 It has been attributed to: (a) The shift of the equilibrium CPs (Shift of the "operating point" a or a' in Fig. 3) when carriers are injected by illumination. The injected carriers always tend to flatten the bands, i.e., reduce the absolute value of 'Ps. (b) At high minority carrier injection ills is no longer proportional to ils. We add that, even at moderate values of iln/no (iln=optically injected minority carrier density, no=minority carrier density in bulk at equilibrium), s is no longer given by Eq. (1).8 That is, one no longer moves along the s vs CPs curves of Fig. 3. Equation (1) is obtained from the Shockley-Read recombination model9,lO by neglect ing terms in the recombination rate that are propor tional to (iln)2. Theory predicts8 that the surface re combination velocity does depend substantially on injection and this would be one of the main causes of the change in ills upon illumination. III. FREQUENCY RESPONSE Studies of time effects (and the frequency responses) in semiconductor surfaces have appeared in the litera- 100 80 I/) "J60 ? t-< uj40 DO 20 CURVE A B C o 'tfs Ki" B (Etl -E.q J.n ~ (Et2-E.1 In ££.!. KT Cn2 KT Cn2 2 -12 -10 12 10 -12 -2 12 2 -4 -6 4 FIG. 8. Calculated curves of s vs '1'. for two recombination states, one of which (level 2) is twice as effective for surface recombination. The surface state parameters for each curve are shown above. As in the case of Ge(2) one observes in Si oscillo scope patterns similar to the step-like structure of the. cu~ves above, indicating more than one level for surface recombmation. 8 G. C. Dousmanis, J. App!. Phys. 30, 180 (1959). 9 W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952). 10 D. T. Stevenson and R. J. Keyes, Physica 20, 1041 (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Tue, 02 Dec 2014 08:59:19MOD U L A T ION 0 F CAR R I E R SUR F ACE L I F E TIM E INS i J081 ture.3.4.11-14 In field-effect work,4 a fraction of the charge induced on the semiconductor can be considered as leaking through the oxide to the slow states. This fraction is a function of frequency: If the frequency of the ac field is very high in comparison to the time constants of the slow surface states, then these states have no time to change their charge during the ac cycle. All the induced charge changes appear in the space charge region (and the fast-surface states, in cluding the recombination states). If the frequency of the field is decreased then the slow states start re sponding to the field changes and part of the induced charge enters the slow state. As a result less charge appears in the space-charge region, and the surface potential does not vary as much as in the previous case. Consequently, the change in s and the corresponding change in the observable Is is smaller. Then, for a given magnitude of induced charge (or applied field) the vertical scope deflection (dI.) is a measure of the change in s or the effectiveness of the induced charge in modulating 'Ps and s. This effectiveness is smaller at lower frequencies where the slow-surface state tends to "shield" the space-charge region from the applied field. In evaluating the frequency response, one associates 4 an RC circuit associated with the slow-surface states (time constant T). It is the leakage of charge from the space-charge region to the outside of the oxide layer that gives rise to the shielding effect by the slow states at low frequencies. The surface response to the ac field is high when most (or all) the induced charge appears in the space-charge region. For a single time constant of the slow states the relative response (S) of the sur face to the ac field is given by4 S= jwT/(1+ jwT). (2) The limiting values for S are unity (large w), and zero (w=O) as one expects from the physical model. S is also zero when R=O. If all the slow states do not have the same time constant one sums or integrates over a distribution assuming that the effects of states with different con stants are additive. Let geT) be the density of states per unit state time. Then one has for the differential response: dS=g(T)[jwT/(1+ jwT)]dT, (3) and lT2 jwT [fT. J-1 s= geT) . dT. g(T)dT. Tl 1+ JwT T1 (4) 11 C. G. B. Garrett, Phys. Rev. 107,478 (1957). 12 J. N. Zemel and J. O. Varela, J. Phys. Chern. Solids 14, 142 (1960). 13 F. Berz, J. Phys. Chern. Solids 23, 1795 (1962). "D. H. Lindley and P. C. Banbury, J. Phys. Chern. Solids 14,200 (1960). FIG. 9. Theoretical patterns of surface response vs frequency for different types of distribution for the time constants of the surface states (see text). The response for all three curves is nor malized to unity at 105 cps. Tl and T2 denote the lower and upper limits of the range over which the distribution of time constants ex tends. We evaluate S for three types of distribution: Single time constant, g(T)=o(T-To), (Sa) g(T) = constant, (Sb) geT) = constant/To (Sc) From (4) we obtain the magnitude of S for Sa, Sb, and Sc, respectively: S(W)={[1 tan-l (WT2) -tan-l (WTl)J2 w(T2-T1) + 1 In2(1 +W2T22)}! 4w2(T2-Tl)2 1+wT12 ' Sew) 1 [(tan-1wT2-tan- 1wTl)2 In (T2/T1) +i In2(1 +W2T 22)Ji. 1+wT 12 (6a) (6b) (6c) The type of response one obtained from distributions of time constants given by 6 (a), (b), and (c) are shown in Fig. 9. In the Ge conductivity measurements of Kingston and McWhorter4 the response given by g(T)=const./T [6 (c)] agreed with the data. In the response of surface recombination in Ge the same dis tribution was found, with some indication that a dis tribution with g(T)= Constant [6 (b)] was also present.2 The present data on Si are shown in Fig. 10. One notes first that the time constants involved are 1 to 2 orders of magnitude shorter than in Ge. (In Fig. 9 the Ge data2-4 would fall mostly on the left of all three theoretical curves shown and the Si data mostly on the right.) The Si response (Fig. to) is still rising at 30 kc/sec so that one does not know at what frequency [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Tue, 02 Dec 2014 08:59:193082 G. C. ALEXANDRAKIS AND G. C. DOUSMANIS '" U> z :r U> .... '" ... > !i ..J .... '" SCALE FOR C,D.E.F 0.1 0.08 0.06 0.04 0.02 A'G'(p·2.S ll-cm) C. D' SHp' S D-em) a. Sl(p'O.OIIl-cm) E. F, S. (p·O.OI D-cml SCALE FOR A,a 1.0 0.1 0.6 0.4 t I ~ 5 II! FIG. 10. Data on the response of the Si surface to an applied ac field as a function of frequency, and comparison with similar data in Ge. One sees that the Si surface does not respond ap preciably to ac fields in the range of 10-1000 cps, where the Ge surface because of longer time constants, shows a significant respon;e.2•4 The measured response is taken as unity at 3 X 104 cps. The ordinate scale on the right applies to curves A and B, and that on the left applies to the other four curves. the response flattens. And, since the slow states re spond to such high frequencies, the results may be complicated by effects arising from the "fast" states whose time constants are in the f.l.sec range. The Si data indicate that the rise of the response with the frequency is more abrupt than in Ge. This would indicate at least a strong admixture of a portion of states whose density is either constant with T [6 (b)J or is a delta function of a single constant (Fig. 9). The present data cannot discriminate between a dis tribution of the constant type [6 (b) J and an inter mediate one between a constant and one whose density varies as liT [6 (b) and (c)]. Evidence for such an intermediate distribution, as noted above, was reported in Ge.2 That the surface potential and the recombination velocity are much more effectively modulated at 10 kclsec rather than, say, at 100 cps is also demonstrated more directly by the following observation: with a low voltage applied to the electrode at the low frequency one observes on the scope a straight line that indicates a small variation of sand <P. about their zero-field value. This same voltage, at 10 kclsec, is sufficient to sweep out a large portion of the s vs <P. curve from the n to the p-side so that it shows not only curvature (Fig. 3) but the entire region of maximum s. In semiconductor field-effect work, either in meas urements of surface conductivity or recombination it was consistently found that Si surfaces required, at low frequencies, considerably larger applied fields for the surface parameters to yield changes comparable to those in Ge. Besides differences in the state density in the two materials, the difference in the time constants of the slow states has to be considered: the slow states shield the space charge region from the field at the fre quencies of 50-1000 cps that were used before. Such frequencies are high in comparison to the slow-state time constants in Ge, but not in comparison to the Si constants. Since the difficulty of modulating the Si surface at low frequencies is connected with the values of its time constants, and not with field strength, it would appear that modulation by other means, such as photo created carrier injectionl' would also be less effective than in Ge at low frequencies. Aside from surface states, one may equally well assign this behavior to states that exist in the Si space charge region with time constants in the 10-3 to 10-4 sec range. Such states at frequencies higher than 1()4 cps would not have time to change their charge and would thereby allow effective modulation of the surface potential. At low frequencies they would follow the field variations and such modulation would not be effective. The data do not allow discrimination as to whether the short time constants observed are to be connected to the "slow" states of earlier literature (on the exterior side of the oxide layer) or with states that are spread in the space-charge region. IV. CONCLUSION The modulation of the surface recombination velocity (or carrier surface lifetime) in Si surfaces by an applied ac field yields, as in Ge, information on the type of surface (p, n, or intrinsic) one obtains with a given surface treatment and ambient atmosphere. Also in formation on the ranges of surface potential and the values of the energy levels for the surface states that are responsible for surface recombination. The values of the surface potential and energies for the recombina tion states in Si are in fair agreement with results ob tained by other techniques.3.6.7.l6.17 Study of the am plitude of the effect of the field as a function of fre quency shows that short time constants (in the 10-3 to 10-'-sec range) do not allow effective surface modu lation at low frequencies. These time constants are one to two orders of magnitude smaller than those found by the same method in germanium. They may be associated either with the "slow" states on the exterior side of that oxide layer, the "fast" states at the semi conductor-oxide interace or with states distributed in the semiconductor space-charge region. ACKNOWLEDGMENTS G. C. Dousmanis wishes to express his indebtedness to the Royal Hellenic Foundation for a grant that made this work possible, and his appreciation for the hospitality of the Foundation and of the Greek Atomic Energy Commission. The authors also take much pleasure in thanking K. Laskaris, Director of the Electronics Division, for his continuous interest and several helpful discussions. 1& E. O. Johnson, Phys. Rev. 111, 153 (1958). 16 H. V. Harten, J. Phys. Chern. Solids 14, 220 (1960). 17 D. Gerlich, J. Phys. Chern. Solids 23, 837 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Tue, 02 Dec 2014 08:59:19
1.1713826.pdf	Acoustoelectric Effect S. G. Eckstein Citation: Journal of Applied Physics 35, 2702 (1964); doi: 10.1063/1.1713826 View online: http://dx.doi.org/10.1063/1.1713826 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in An anomalous acoustoelectric effect Phys. Today 54, 9 (2001); 10.1063/1.4796234 Acoustoelectric effect in piezodielectric semiconductor layered structure J. Appl. Phys. 44, 3034 (1973); 10.1063/1.1662702 Acoustoelectric Effects in Indium Antimonide J. Appl. Phys. 42, 2041 (1971); 10.1063/1.1660484 BrillouinScattering Studies of the Acoustoelectric Effect J. Acoust. Soc. Am. 49, 1037 (1971); 10.1121/1.1912447 MagneticField Dependence of the Acoustoelectric Effect J. Appl. Phys. 34, 510 (1963); 10.1063/1.1729303 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.97.125.60 On: Wed, 03 Dec 2014 14:37:24JOURNAL OF APPLIED PHYSICS VOLUME 35, NUMBER 9 SEPTEMBER 1964 Acoustoelectric Effect* S. G. ECKSTEIN Argonne National Laboratory, Argonne, Illinois (Received 3 February 1964) . C:eneral exp.ressiot;s for }~e a~oustoel:ctr!c current in the presence of external electric and magnetic fields ~la\e heen denved. 1 he Welllrcich. r~lahon IS shown to be valid in materials in which carriers of one sign only Me ~)resent, to the extent that collisIOn-drag effects may be neglected. The Weinreich relation is not valid for seml.metals. However, in ?igh magnetic fields, the acoustoelectric current and field in semimetals are pro l:ortlOnal to the attenuah~n. The acoustoelectric field is then in the direction qX H, where q is the wave \ ect~r, ~nd H t?~ ma~netJc field. Th~ acoustoelectric current reinforces the original direct current under amplifYlllg conrlitlOns, III agreement With ohservations of anomalous magnetoresistance in Bi. INTRODUCTION WHE~ a sou~d. wave propagates through a ma- terial contammg conduction electrons, its mo mentum, as well as its energy, is attenuated by the electrons. The momentum attenuation acts as a dc force, causing the electrons to drift in the direction of the force. If there is a closed circuit in this direction a direct current will be produced. This is the acousto~ electric current; it is proportional to the sound in tens~ty, since the momentum attenuation is itself pro portlOnal to the sound intensity. If, on the other hand, the circuit is open, the drifting electrons produce a space charge whose electric field cancels the dc force due to the sound wave momentum attenuation. This back electric field is the acoustoelectric field. The acoustoelectric effect was first predicted by Parmenter,! and was identified by Weinreich2 as due to the momentum attenuation of the sound wave. The acoustoelectric field has been observed bv Weinreich Sanders, and White3 in n-type Ge; and ~ the acousto~ electric current has been observed by Wang4 and White" ~CdS. - The acoustoelectric current has also been observed indirectly in a group of interesting experiments. In these experiments the resistance of CdS6,7 and the mag netoresistance of Bi8 are observed to change when the drift velocity of the charge carriers exceeds the velocity of sound. Hutson9 has suggested the following inter pretation of the anomalous resistance: When the drift velo~ities exceed the velocity of sound, sound is ampli fied mstead of attenuated; and in fact, if no external sound wave is present, noise is amplified.8,10 The noise amplification causes an acoustoelectric current which , , * Based on work performed under the auspices of the U S. Atomic Energy Commission. . 1 R. H. Parmenter, Phys. Rev. 89 990 (1953). : G. We~nre~ch, Phys. Rev. 107,317 (1957). G. Welllrelch, T. M. Sanders and H. G. White Phys Rev 114, 33 (1959). ' ,. • : W. c. \,:,a!'g, Phys. Rev. Letters 9, 4-1-3 (1962). . n. L. White (to be published). 6 R. W. Smith, Phys. Rev. Letters 9, 87 (1962). 1 J. H. M~Fee, J. Appl. Phys. 34, 1548 (1963). • L. Esakl, Phys. Rev. Letters 8, 4 (1962). • A. R. Hutson, Phys. Rev. Letters 9,296 (1962). 10 A. R. Hutson, J. H. McFee, and D. L. White, Phys. Rev. Letters 7, 237 (1961). in the case of CdS opposes the original direct current; hence the apparent resistance is increased. On the other hand, the magnetoresistance of Bi is observed to de crease, and therefore, its acoustoelectric current should reinforce the original direct current. However, the acoustoelectric current has not been observed in Bi nor has it been calculated theoretically, so that the ~rgu ments of Hutson are less convincing in this case. The existing literature of the acoustoelectric effect is somewhat ambiguous with respect to the definition of the acoustoelectric field. The theoretical treatments2,1l,12 define this field as the electric field equivalent to the dc forces acting upon the electrons due to the sound wave. Thus, Weinreich pointed out, the rate of loss of mo mentum from the sound wave (which is equal to the rate of energy loss divided by the velocity of sound) is a dc force in the direction of propagation of sound and is equivalent to an effective electric field, which he defined to be the acoustoelectric field, This argument provided a relation (known as the Weinreich relation) between the attenuation of sound and the acousto electric field. The physically observable field is the back field which opposes carrier drift, and ensures that no current flows under open circuit conditions. It is this field which we prefer to call the acoustoelectric field. The situation is analogous to the Hall effect. In the Hall effect the force due to the magnetic field is J' x Hlc' this for~e is . " of course, eqUIvalent to an effective electric field jxHINec, where N is the number density of carriers and e their charge. However, the Hall field which is th~ physically observable field, opposes this effective field and is equal to -j x HI N ec. In the acoustoelectric case' the analogous fields may differ by more than a sign, fo; the acoustoelectric field need not oppose the dc forces if the current may flow in the direction of the forces whereas in the Hall case the current is always perpen~ dicular to the magnetic field forces. In a proper treatment of the acoustoelectric effect the acoustoelectric field should be introduced from th~ start in the equation of motion for the charge carriers, in analog.\' with the treatment of the Hall etTect. It i~ Il N. Mikoshiha, J. Appl. Phys. 34, 510 (1963). 12 H. Spector, J. Appl. Phys. 34, 3628 (1963). 2702 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.97.125.60 On: Wed, 03 Dec 2014 14:37:24ACOUSTOELECTRIC EFFECT 2703 especially obviolls that this must be done if semimetals are considered. If both the electrons and the holes attenuate the sound, then the forces acting upon them are in the same direction, namely that of the propaga tion of sound. The effective electric field acting upon the electrons is then oppositely directed to that acting upon the holes. This is, of course, in contradiction to the idea of a physically observable acoustoelectric field acting upon both electrons and holes. J n Sec. 1 the theory of the acoustoelectric effect will be formulated in a manner directly applicable to semimetals. External electric and magnetic fields will be taken into account explicitly, so that the theory will be valid for amplifying conditions. Expressions will be found for the acoustoelectric field and current in terms of the attenuation, for various experimental conditions. The status of the Weinreich relation will also be examined, and it will be found that the collision drag effect modifies the Weinreich relation. The acoustoelectric effect in semi metals will be ex amined in Sec. 2. The expression for the acoustoelectric current in the presence of high magnetic fields will show that this current does indeed reinforce the original direct current under amplifying conditions, as suggested by Hutson.9 Therefore, the magnetoresistance of Bi is expected to decrease for drift velocities greater than the velocity of sound, in agreement with the experi mental observation of Esaki.8 1. THEORY OF THE ACOUSTOELECTRIC EFFECT In the steady state, the net rate of momentum gain by a system of charge carriers is zero. Therefore, the sum of forces acting on the charge carriers, namely, applied field forces plus forces due to collisions, must be zero. Consider a system of No electrons of effective mass m* per unit volume. In the presence of a sound wave the electron density is N(r,t)= No+Ns(r,t), where Ns(r,t) varies, like the sound field, as exp[i(q·r-wt)]. In this expression q and ware the wave vector and frequency of the sound. Before collision, the average electron momentum is m(v), where (v) is the average electron velocity; after collision the electron momentum relaxes to an isotropic distribution about the velocity of the local moving lattice; that is, if the lattice has a local velocity u(r,t)=uo exp[i(q·r-wt)], the average electron momentum after collision is given by mu(r,t), where m is the actual electron mass. This is a result of the collision-drag effect, which is treated in detail by Holstein.13 Thus, the electrons gain momentum in collisions at a rate N (mu-m(v»)/ T= Fcoll, (1.1) whl're T is the relaxation time. Equation (1.1) is an l'xpression for t hl' forces acting upon the electrons due to collisions. 13 T. Holstein, Phys. Rev. 113, 479 (1959). The total applied field forces are given by pE+j x H/c-NqqCu/iw= Fappl , (1.2) where p is the charge density (p= -Ne), j the current, C the deformation potential tensor, and E and Hare the total electric and magnetic fields. (-qqCu/iw is the deformation potential force.) In the steady state the sum of forces FcolI+Fappl vanishes when integrated over the entire crystal. There fore, the dc part of FcolI+F appl must itself vanish. Let us separate the current, density, and fields into dc parts, and parts which vary in space and time as the sound wave. Thus, j=jdc+j,,; E=Edc+E.,; H=Ho+H.,; p=Po+Ps. In these expressions, E, and Hs are the self-consistent fields which accompany the sound wave; and j8, ps, E. and H. all vary as exp[i(q .r-wl)], Ho is the external magnetic field, and p = -N e. When these definitions are used, we find that the de parts of the forces are given by: Fcolldc=! Re(N,mu/T) + (m/eT)jdc (1.1') F appldc= POEdc+jdc X Hole +! Re[p,(E.+qqCu/iew)+i.xHs/c]' (1.2') The expression for F appldc may be simplified by using Maxwell's equations and the equation of con tinuity. First, Maxwell's equations give a relationship between E. and H.: H.= (c/v.)qXE s, (1.3) where VB is the velocity of sound, and q a unit vector in the direction of propagation of sound. The equation of continuity yields a relation between ps and is, namely, P.= (j.·q)/v s• (1.4) When (1.3) and (1.4) are substituted III (1.2'), the following expression is found: Fappldc= -NoeEdc+jdcxHo/c +qr! ReL· (E8+qqCu/iew)]/v s• (1.2") When the sum F eolldc+ F ,,"ppide is set equal to zero, the following relation between the direct current and field is found: jdc+ (eT/mC)jde x Ho =UOEdc- (eT/mvs)q! Rei.· (E.+qqCu/iew) -(m/m)! ReN.eu, (1.5) where uO=Noe2T/m. We will rewrite this relation in a more compact form: j,lc+w,T(jol(.XI7() = CTIl(Ed,+ t), (1.5') where wcT=eHOT/mc, and flo is a unit vector in the direction of the external magnetic field. The vector t. which is proportional to the sound intensity (and is [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.97.125.60 On: Wed, 03 Dec 2014 14:37:242704 S. G. ECKSTEIN essentially the effective field discussed in the introduc tion), is given by t=q! Rej,,' (Es+qqCu/iew)/- (Noevs) -(m/m)! ReNseu/lTo. (1.6) This result may also be obtained by integration of the first moment of the Boltzmann equation. This second derivation is given in the Appendix. It is of interest to find the relation between the effective iield t and the attenuation a. This relation may be found by substituting the expression for the attenuationl4,15 a=!S-1 Re{j,,· (E8+qqCu/iew) + [mu/erJ [js+ (m/m)iVoeuJ} (1.7) (where S is the power density of the sound wave) in Eq. (1.6). The result is: t= (-qaS/Noev,,)+ (m/m) (l/lTovs) X{t Requ·[j.,+(m/m).iVoeuJ+t Reuq·j,}. (1.8) The last two terms on the right-hand side of Eq. (1.8) are due to collision-drag effects: one term comes from the collision-drag terms in the expression for the attenu ation of sound (1.7); and the other is a result of the current due to the motion of the lattice -lVeu(m/m). The term proportional to [j,,+ (m/m)iVoeuJ tends to zero at low frequencies (i.e., when charge quasineu trality holds) and is negligible except at the very highest ultrasonic frequencies. The remaining term is finite even in the limit of zero frequency. This appar ently finite effect of collision-drag forces in the limit of charge quasineutrality is due to the fact that the dc ionic current15 t ReNsioneu(m/m) has been omitted from the total current in (1.5').16 The density N"ion may be found from the continuity equation for ions: If the ionic direct current is added to the electronic current of (1.5'), and a zero magnetic field is assumed, the expression for the total dc current becomes (1.10) where ttotal= (-qaS/Noev s)+ (m/m) (1/lTov,,) X{t Re[qu+uqJ[js+(m/m)NueuJ}. (1.11) Thus, as expected, collision-drag terms have a finite effect only when quasineutrality does not hold. Equations (1.8) and (1.11) show that the Weinreich relation, Noef,= -qaS/v s is valid only to the extent that collision-drag terms may be neglected. This is in 14 M. H. Cohen, M. J. Harrison, and W. A. Harrison, Phys. Rey. 117,937 (1960). 1& A discussion of the origin of the factors (m/1I1), which appear in the expressions for the attenuation and the ionic current, is giyen in Ref. 11. 16 This dc ionic current is not necessarily fictitious, even for a clamped crystal, since it does not require ion transport. disagreement with the results of Mikoshiball and Spectorl2 who report that the Weinreich relation is always valid. The derivation of these authors depends upon the introduction of fictitious collision-drag forces. Since Eq. (1.5) may be derived, as in the Appendix, by integrating the first moment of the Boltzmann equation, and since this identical Boltzmann equation is assumed to be valid in Refs. 11 and 12, the introduction of the fictitious collision-drag forces is inconsistent. In piezoelectric semiconductors and semiconductors with large deformation potentials, the attenuation is quadratic in the piezoelectric tensor (or deformation potential), whereas the collision-drag terms are at most linear in these quantities. Therefore, in these cases the collision-drag terms may be neglected in comparison with the attenuation term in the acoustoelectric field, and the Weinreich relation is an excellent approxima tion, even at very high frequencies. For these materials, the ionic direct current is negligible by comparison with the electronic direct current, and, therefore, Eq. (1.5') describes the entire direct current. For this case, Eq. (1.5') may be used, together with the appropriate boundary conditions, to find the acoustoelectric field and current. These are, of course, those parts of Edc and jdc which vary as the sound intensity. In the absence of the sound wave, let jdc=jO, and Edc= Eo+ Eu, where Eo is the applied iield and Elf the Hall field. Then Eq. (1.5') gives (1.12) In a typical observation of the acoustoelectric effect, either the potential drop across the sample or the current may be held constant when the external sound wave is applied. [If both current and potential are allowed to vary, the situation becomes needlessly com plicated, and it is then impossible to solve Eq. (1.5') for both the acoustoelectric field and current.J Suppose that the current is held constant. Then lTo(Edc+ t) = jo+wcr(joXHo) = lTo(Eo+ Ell) and consequently: Ric= Eo+ Ef{-t. Thus, under constant current conditions, j"e=O; Ea,,= -f,. (1.13) (1.14) (1.15) This result is also valid for open circuit conditions, which is the special case, jo= 0, Eo= O. If the potential drop across the sample is kept con stant when the sound wave is applied, then Edc will have the value Eo in the direction of the closed circuit. Since the current will flow entirelv in this direction, the scalar product of Eq. (1.5') with a unit vector Eo in the direction of the closed circuit gives: hence (1.16) (1.17) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.97.125.60 On: Wed, 03 Dec 2014 14:37:24ACOUSTOELECTRIC EFFECT 2705 This expression may be substituted in Eq. (1.5') to find the electric field: Therefore, for constant potential conditions, the acous toelectric current and field are given by: in" = ITo (S . Eo)Eo, En,,= -{e-(e·Eo)[E o+ (Ell/Eo)]}. (1.19) Equation (1.17) may be regarded as the modification of Ohm's law when a sound wave is applied. For those cases in which the Weinreich relation holds, the direct current will be (1.20) where the drift velocity Vd is given by -eEOT/m. For amplifying conditions (q. v d) is positive (in fact, larger than v.,) and the attenuation Q: is negative. Hence, the direct current decreases when a sound wave is applied, under amplifying conditions. This is in agreement with the observations of anomalous resistance in CdS for drift velocities greater than the velocity of sound. ] t should be noted that the acoustoelectric field is always perpendicular to the current. For example, in piezoelectric semiconductors, if a constant potential drop across the crystal is maintained when the sound wave is switched on and there is no external magnetic fiele!, the acoustoelectric fiele! will be [see Eq. (1.19)J (1.21) This fiele! should be observed whenever the direction of propagation of sound does not coincide with the direc tion of the current. 2. ACOUSTOELECTRIC EFFECT IN SEMIMETALS ]f both electrons and holes are present in a material, and if recombination is neglected, the rate of change of de momentum of each charge carrier separately is zero; hence each carrier obeys an equation similar to (1.5): idee+ (WeT )eideeX11o= lToe(Ede+ ee), jctc"-(WeT )hioehX11o= o-oh(Ede+ en), (2.la) (2.1b) where the superscripts (and subscripts) e and 7z refer to electrons ane! holes, respectively. The vectors ee and en are given by: ee= -if! Re[isc. (Es+qqCeu/iew)]/(Noev s) It is very convenient to find a relation between jde ane! Edc of the form (1.5'), where e is replaced by some average effective field t. This relation may be found by solving (2.la) and (2.1b) for idee and ideh and adding the currents to find ide, The resulting equation may then be solved for Ede in terms of ide' The final result is: (WcT)e- (WrT)/c ~ jde+ (jdcXHo) 1 + (WrT)e(WrT),. where (JOc.+Uo" (WcT)eCWcTh ~ r'l (jdc·HO)no 1 + (WcT)e(WcT)h lToe+o-Oh _ ~-~-(Edc+8), (2.4) 1+ (WcT)e(WcT)h We have assumed in (2.4) and (2.5) that the equi librium density of electrons and holes are equal, i.e., Noe= 1Voh= No. Equation (2.4) is of the form (1.5'), where WeT 1S replaced by and 0-0 is replaced a= [ITOe+ITOh][1 + (WeT )e(WcT h]-l; ex cept that in this case, an extra term proportional to (ide' H o)H 0 is obtained. This extra term is zero for crossed electric and magnetic fields, and in that case, the results (1.15) and (1.19) hold for the acoustoelectric current and field, when appropriate substitutions are made. In the general case, when the extra term is non zero, the dc current and field in the absence of the sound wave are given by ju=aEo/[1-K(E o·Ho)2] (2.6a) (WcT)av(EoXHo)+K(Eo·Ho)[EoX (EoXHo)] Ric = Eo+----------------- 1-K(Eo·Ho)2 = Eo+En, (2.6b) where K= (WcT)e(WcTh[1+(WcT)e(wcTh]-I. When the sound wave 1S applied and a constant current is maintained, (2.7a) -(m/me)! ReNseeu/o-o', (2.2a) and eh=(H Re[j/'. (Es-qqChu/iew)]/ (Noev s) + (m/mh)! ReNsheu/o-oh. (2.2b) The total direct current is the sum of hole ane! electron currents: (2.3) If a constant potential drop is maintained, iae= a(e· Eo)Eo/[l- K(Eo' HO)2], Eae= -{ e-(e· Eo)[Eo+ (Ell/Eo)]}. (2.7b) (2.8a) (2.8b) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.97.125.60 On: Wed, 03 Dec 2014 14:37:242706 S. G. ECKSTEIX The deformation potential is very large in semi metals such as Bi, and therefore the collision-drag terms may be neglected in the expressions for the attenuation and the acoustoelectric current and field. Thus, if the power dissipated by the electrons is JVe, and the power dissipated by the holes is H\, then lVe=:i Rej.e· (Es+qqCeuliew), W,.,=! Reish. (Es-qqChu/iew), (2.9)a (2.9b) and the total power dissipated is lif = We+ Wit. The attenuation is given by the relation: a= W I.'i. (2.10) 1£ Eqs. (2.9) are substituted in (2.2) and collision-drag terms are neglected, it is found that Hence ~e= -qWel(Noev s), B,,=qWhl(Noev.,). (2.11a) (2.11b) (2.12) It is obvious from this expression that a Weinreich-like relation does not exist for semimetals. We shall examine the acoustoelectric effect for the case of high magnetic fields (WeT» 1) for crossed electric and magnetic fields. These are the conditions appro priate to the anomalous magnetoresistance experiments of Esaki.8 For high magnetic fields, the first term in the expression (2.12) for t may be neglected for it is of order (ljwcT) relative to the second term. Hence --(WcT).(WcT)nCXS. ft B= qX o. Noev 8[(wcT).+ (wcThJ (2.13) For a constant potential drop across the sample, the total current is given by: (2.14) where Vd= (cEoXflo)IHo is the drift velocity of both electrons and holes. In the expression for the drift velocity, we neglected EH as being of order (II WeT) rela tive to Eo; and of course, neglected E.c in comparison with Eo. lf the expression for the direct current (2.14) is com pared with (1.21), we see that it has exactly the same form, except that the acoustoelectric current is in the opposite direction. Thus, in semimetals in high magnetic fields, the acoustoelectric current reinforces the original direct current for amplifying conditions, in agreement with the experimental observation of anomalous mag netoresistance by Esaki. 8 APPENDIX. DERIVATION OF ACOUSTOELECTRIC EFFECT BY INTEGRATION OF BOLTZMANN EQUATION Consider the linearized Boltzmann equation: a f a f F a f f-feq -+v·_+--· -----. (Ai) at dr m (Jv T Here feq is the distribution to which the electrons relax in the presence of a sound wave. As result of the collision drag cHect, this distribution is found to bc: feq(r,v,t) = fo[mv-mu(r,t), EF(r,f)], (A2) where fo is the equilibrium Fermi distribution and Ep(r,t), the local Fermi level, is chosen to give the correct electron density.l4·ls The force F is given hy: F= e(Edc+vxHolc+Es+v x H.Jc +qqCu/iew) = Fdc+F" (A3) where Fdc is a de force and Fs varies as the sound wave. Let us separate the distribution function f into a dc part fIle and a part which varies as the sound wave f.: (A4) Then (AS) and (A6) This distribution f will contain additional ac terms which vary as the second and higher powers of the sound wave, but these will be neglected here. The Boltzmann equation may also be separated into a dc part and ac parts. The equation for the dc part will be, from (Ai), Multiply (A7) by v and integrate over velocity space. Then fvfe qd3v = Nu(mlm), hence Jvj.qdCd3v =! ReNsu(mlm). Integration by parts gives Equation (A8) holds because the integrated part is zero since f is zero for v= co; and aF;jav.=O even for [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.97.125.60 On: Wed, 03 Dec 2014 14:37:24ACOUSTOELECTRIC EFFECT 2707 the velocity-dependent forces. Using (AS) we find Combining these results, we find f F. aj. ! Re v-·--d3v m* av NoeEdc-jucx Ho/e -! Re[p.(E.+qqCu/iew)+j.x H./e] and = -(l/m)! Re[p8(E 8+qqCu/iew) +j.xH./e] (A9) = (m/eT){jdc+! ReNseu(m/1Il)}. (All) J' Fele iJIde V_·~-d3V m* av This is the same as (1.5) when p. and H. are expressed in terms of i. and E •. White;' used a somewhat similar method to derive an expression for the acoustoelectric current in the absence of a magnetic field. However, he considered a one-dimensional model, which has limited validity. =-(l/m)[-NoeEdc+jdcxHo/e]' (AlO) JOURNAL OF APPLIED PHYSICS VOLUME 35. NUMBER 9 SEPTEMBER 1964 Energy Dependence of Proton Irradiation Damage in Silicon W. ROSENZWEIG, F. M. SMITS, AND W. L. BROWN Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey (Received 6 March 1964) The energy dependence of radiation damage in silicon for proton energies in the range 1.35 to 130 MeV has been measured by observing the degradation of the bulk minority carrier diffusion length in silicon solar cells. Variahility in proton flux determination at four different accelerators was minimized by employing pre bombarded solar cells with known minority carrier diffusion lengths as calibrated solid-state ionization l!hambers. Where beam intensity measurement comparisons with Faraday cups could be made, agreement to better than 5% was obtained. The quantity characterizing the damage rate is the rate of change of the inverse square diffusion length with flux K -=d(1/D)/dif>. The 1-f)-cm p-type silicon degraded, on the average at a rate six times less rapid than 1-Q-cm n type, independent of energy. Room temperature annealing gave 30% to 50% decrease in K whenever the diffusion length was measured during and after irradiation. The energy variation of K agrees with the variation predicted by Rutherford scattering below 8 MeV, but decreases less rapidly at higher energies. The measured diffusion lengths increased with excess carrier density n from 2% per decade at n = 109cm-' to 20% per decade at n = 101'cm-'. The reported results, obtained at low excess carrier density, can be used to predict solar cell degradation under conditions of outer space illumination if the appropriate excess carrier density is used. Failure to take into account the diffusion length variation will result in an underestimate of the solar cell output of less than 7%. INTRODUCTION THE energy dependence of the rate of lifetime degradation in l-Q·cm p-type silicon for proton energies in the range from 1.35 to 130 MeV has been measured by observing the degradation of the bulk minority carrier diffusion length in silicon solar cells. Such results are important in assessing the damage to solar cells on satellites operating in the Van Allen belt. As expected, for the energy range covered, the lifetime degradation per proton decreases monotonically with increasing proton energy. However, significant devia tions of the energy dependence from the predictions of a simple theoretical model were observed. EXPERIMENTAL PROCEDURE Changes in diffusion length can be observed in a con venient way by means of a silicon solar cell. This stems * Present address: Sandia Corporation, Albuquerque, New Mexico. from the fact that the shallow-diffused junction collects excess carriers which are generated by the radiation during bombardment primarily from the bulk. A meas urement of the radiation induced short-circuit current thus yields a direct determination of the minority carrier diffusion length as the bombardment progresses.1•2 Moreover, the excess carrier density produced by this excitation is sufficiently low so that the effects of vari ation of diffusion length with excess carrier density are negligible (see below and Fig. 5). For particle radiation, such as protons and electrons, an absolute diffusion length measurement is obtained by a determination of the ratio of the radiation-induced solar cell short circuit current density to the incident radiation current density divided by the average specific ionization of the incident particles.2 For heavy particles, the specific ionization can be determined from published 1 J. J. Loferski and P. Rappaport, Phys. Rev. 111, 432 (1958). 2 W. Rosenzweig, Bell System Tech. J. 41, 1573 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.97.125.60 On: Wed, 03 Dec 2014 14:37:24
1.1702450.pdf	Electron Emission from Thin AlAl2O3Au Structures H. Kanter and W. A. Feibelman Citation: Journal of Applied Physics 33, 3580 (1962); doi: 10.1063/1.1702450 View online: http://dx.doi.org/10.1063/1.1702450 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Characteristics of electron emission of Al-Al2O3-Ti/Au diode with a new double-layer insulator J. Vac. Sci. Technol. B 32, 062204 (2014); 10.1116/1.4900632 Surface plasmon polariton enhanced electroluminescence and electron emission from electroformed Al-Al2O3-Ag diodes J. Appl. Phys. 112, 073717 (2012); 10.1063/1.4758289 Growth and electronic structure of Sm on thin Al2O3/Ni3Al(111) films J. Chem. Phys. 136, 154705 (2012); 10.1063/1.4704676 Electron transport mechanism in Al/Al2O3/nInTe/Bi thinfilm structures J. Appl. Phys. 64, 6379 (1988); 10.1063/1.342074 Electron Emission, Electroluminescence, and VoltageControlled Negative Resistance in Al–Al2O3–Au Diodes J. Appl. Phys. 36, 1885 (1965); 10.1063/1.1714372 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:06JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 12 DECEMBER 1962 Electron Emission from Thin Al-Ah03-Au Structures H. KANTER AND W. A. FElm:LMAN Westingholtse R.esearch Laboratories, Pittsburgll, Penn.~ytvltni(t (Received July 16, 1962) Emitter cells with AhO. film thicknesses between 67 and 150 A and Au film thicknesses between 200 and 300 A were operated at voltages up to 10 V. Typical I-V characteristics for the total cell currents are pre sented. The curves suggest tunnel emission through the barrier at the AI-AJzO. interface when the cell is operated for the first time and only small currents have been drawn. After passage of large currents, the /-1' characteristics become temperature dependent. The normal energy distribution of the emitted electrons is measured and a linear dependence of the average energy on cell voltages is established. The fractions of current emitted through the Au into the vacuum is determined as a function of the AbO" thickness. Using the attenuation lengths of gold reported recently by Mead, an attenuation length of about 24 A is deduced for electrons, which within the insulator have gained an energy of nearly 3 eV. The emitted current could be increased considerably by depositing a low work function material (Ba) at the gold surface. The maximum fraction of current emitted into the vacuum was 10-', at an emitted current density of nearly 5 rnA/em'. INTRODUCTION IN a recent article, Meadl described a cold cathode device consisting of a thin AbOs layer several tens of angstroms thick, which was sandwiched between two metal electrodes. If a sufficiently large potential is applied to these electrodes, a considerable current up to tens of amperes per cm2 can be drawn through the struc ture. The current is carried by electrons which have penetrated the negative metal-insulator barrier into the conduction band of the insulator, where, by virtue of the applied and the contact field, they can gain consider able energy. In case the positive metal electrode is made sufficiently thin, part of this current is able to penetrate the electrode and to escape into an adjacent materiap·2 or into the vacuum.! For the most efficient operation of such a device, the transfer ratio (T) or that fraction of the current which leaves the metal again should be made as large as possible. Therefore, the second metal elec trode should be as thin as possible. A limit, however, is imposed by the condition of sufficient conductivity, which is usually satisfied for metal films in the order of 200 A or more thick depending on the specific resis tivity, surface roughness, evaporation method, etc. Because of the minimum thickness requirement the electron attenuation within the metal is of decisive importance. On the basis of measurements of photo electron escape depths from potassium by Thomas,a the electron attenuation length can be hundreds of ang stroms in case the electron energy is less than a few volts (about 3 eV for potassium) above the Fermi level. For gold, Spitzer and collaborat ors4 found an attenu ation length of about 700 A for 0.8-eV electrons, while Mead" reported about 100 A for electrons with energies above 4.7 eV. (The former authors observed emission across a barrier into an adjacent semiconductor, the 1 C. A. Mead, J. Appl.Phys. 32, 646 (1961). 2 J. P. Spratt, R. F. Schwarz, and W. M. Kane, Phys. Rev. Letters 6, 341 (1961). " H. Thomas, Z. Physik 147, 395 (1951). 4 w. G. Spitzer, C. R. Crowell, and M. M. Atalla, Phys. Rev. Letters 8, 57 (1962). • C. A. Mead, Phys. Rev. Letters 8, S6 (1962). latter measured on electrons emitted into the vacuum.) For gold, therefore, a strong energy dependence of the electron attenuation is apparent and it appears possible to increase considerably the emission through gold films into the vacuum by lowering the surface barrier. It was the purpose of this work to gather some ex ploratory data on the emission characteristics of thin film Al-AbOa-Au structures, including the improve ment of the transfer ratio by lowering the work function of the exit surface. Gold was chosen because some data on attenuation lengths are available, and because it can easily be evaporated, is a good conductor, and forms rather stable surfaces. The Al-AbOs base was used because of the ease of formation and control of the AbOa layer by anodization, and because its physical characteristics were known to be appropriate through the work of previous investigators.6-s Transfer ratios were determined as a function of the Au and AhOa film thickness and the voltage applied to the structure. The energy distribution as well as the mean energy of the emerging electrons was determined. The effect of lower ing the work function was investigated by depositing layers of Ba on top of the gold film. Using the transfer ratios of Au by Mead, the data allowed one to roughly estimate the attenuation of low energy electrons in Al20a films. EXPERIMENTAL METHOD The electron emitting structures were prepared on a microscope slide as demonstrated in Fig. 1. After wash ing, rinsing with distilled water, and drying the slides, 12 Al stripes about 2.5 mm wide and 600 A thick were deposited by evaporation. (All evaporations were made in a vacuum of several 10--6 Torr.) Aluminum oxide layers were formed by anodization in a 3% ammonium citrate solution9 at voltages between 5 and 11 V. Assum- 6 J. C. Fisher and 1. Giaver, J. Appl. Phys. 32, 172 (1961). 7 J. T. Advani, M. S. Thesis, MIT (May 1961). 8 R. M. Handy, Phys. Rev. 126, 1968 (1962). • The current density in the anodization process was always kept below 400 "A/cm2 and the process was stopped when the current had decreased to 2 "A/em'. 13.7 A/V was taken as the thickness voltage relation. 3580 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:06E LEe T RON E MIS S ION FRO M T H I N A 1 - A 1 2 0 3 -A 1I S T Rue T U RES 3581 ,I" x 3" Microscope Slide \ \ACliVe Areas FIG. 1. Microscope slide with 12 diode samples. in~ a linear thickness-forming v?ltage relation, the thickness of the AbOa layers vaned between 67 and 150 A. Subsequently, the slides were washed and baked in air at 150°C for an hour to remove the moisture from the AbOa. In a second evaporation, an Au strip about 7 mm wide was deposited over the entire length of the slide, forming 12 samples, each of about 6 mm2 active area. The thickness of the gold films was determined from the light transmission, using data of a previous investigation by Feibelman.lO Contact to the films was made with silver paint. The experimental arrangement, operated in a de mountable vacuum system at 2X 10--° Torr consisted of a grid of 100X 100 mesh/in.2, which was placed about 4 mm apart from the slide, and the collector electrode which was mounted about 3 mm apart from the grid: Both grid and collector were gold-plated to insure uniform surface conditions. The electrical circuitry is sketched in Fig. 2. In order to measure total emission grid and collector were connected and kept 22 V positiv~ with respect to the grounded gold film. Retarding curves used to obtain the width of the energy distributions were determined with the grid at +22 V by measuring t?e collector current as a function of a retarding poten tIal. The actual shape of the energy distributions were determined by graphical differentiation of retarding curves obtained with the arrangement slightly modified: The collector was covered with soot to reduce reflection and a magnetio field was applied to avoid defocusing by the retarding field. It is evident that in such a plane parallel arrangement, one can only obtain the normal component of the energy distribution. The grid, as well as nonuniformities in the surface potential, imposes a limit in the energy resolution, which is believed to be in the order of 1/2 V. The grid-collector arrangement could be replaced by a semitransparent phosphor covered slide in order to demonstrate the emission characteristics of the samples. The phosphor was covered with a lOX 10 per in.2 mesh to control the phos phor surface potential, which was kept at 1 to 2 keY. The effect of lowering the work function of the gold film was studied in a glass envelope which contained the slide with the samples, a collector electrode and one or two Ba-channels, such as are generally used to getter vacuum tubes. The system was slightly baked out at 10 W. A. Feibelman, "Light Transmission vs Measured Thick ness Curves for Some Thin Films," Westinghouse Research Report 6O-8-1Q-39-R3. FIG. 2. Circuit dia gram of experimen tal setup. 1S0°C for several hours.ll After cooling to room tem perature and bringing the cold trap to liquid nitrogen temperature, the pressure dropped to about 10--8 Torr. Although the Ba-channels were outgassed during the bake, firing of the channels increased the pressure to more than 10--6 Torr, a value which is insufficient for clean surface studies. Therefore, the results with regard to improvement of transfer ratios by lowering the work function need not represent optimum values. RESULTS AND DISCUSSION a. General J-V Characteristics The samples were operated by gradually increasing the sample voltage from zero and observing the diode current Id and the emission current Ie. The latter could be measured down to about 10--11 A. A typical I d-V d and I e-V d characteristic is shown in Figs. 3 and 4. As can be seen, the curves are not reversible, but the diode current for smaller voltages increases with increase of the charge which has passed through the sample. After some operation time (typically 200 mA/cm2 for 30 sec), the I d-V d characteristic became nearly stable, with slowly increasing diode currents only at rather large current values, which eventually lead to the destruction of the cell. All emission measurements reported below were carried out when the cell had reached nearly stable characteristics. After completing measurements at moderate currents, the measurements were extended to larger currents even if the cell could not be considered stable any more. Destruction of the cell usually deter mined the maximum operation current (about 1/2 A/cm2). The emission current changed approximately the same way as the diode current did, so that the transfer ratio varied slowly with cell current. Diode currents (and emission currents, which will be discussed later) have been measured for samples with AbOa thicknesses between 67 and 150 A. Figure 5 shows the diode current for the initial increase of voltage on the sample. In order to unify the data, I d is plotted as a function of the average field in the sample as determined by the AbOa film thickness and the applied voltage. 11 The temperature was kept low to avoid possible destruction of the samples by diffusion of Al and Au into the AbO •. It was ob served that originally good samples showed very poor characteris tics after baking at 250°C for several hours. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:063582 H. KANTER AND W. A. FEIBELMAN IO-I..-----r----...-------,----, 10'2 6 o I. Run, Vollage Applied 1st TIme v 2. Run, Voltage Decreosed t. 3. Run, Voltage Increased Again 7 8 9 Diode Voltage, Vd• (V) FIG. 3. I d-V d characteristic of cell for initial operation and after drawing considerable current. (The dielectric constant of the Ah03 was chosen as €= 1, see below.) As can be seen, all data points fall within a narrow region and would even coincide, if one would assume that the thickness of the thinnest film is ..!' 10-9 E- ~ ::J U C o :~ 10'10 E w o I. Run, Voltage Applied 151 TIme • 2. Run, VOltage Decreased • 3. Run, Vollage Increased Again IO·I"6-------l.-----.lS.-----J9!,---l Diode Voltage, Vd (V) FIG. 4. 1.-Va characteristic of cell for initial operation and after drawing considerable cell current. actually somewhat thicker than determined by the forming voltage.12 The very steep rise of fd with Vd suggests that fd is a tunnel current, i.e., during the initial operation, the barrier at the Al-AbOa interface is sufficiently large to prohibit thermal emission over the barrier. Indeed, a theoretical curve (solid line in Fig. 5) according to the Fowler-Nordheim formula agrees very well with the experimental data. The Fowler-Nordheim formula describing the tunnel current between two metal layers separated by an insulator readsl f = f o' (;)2 exp( -Eo/E), with 1 the current density and E the electric field. Here Eo<:::-[4 <pI (2m)tJ/ (3ft· e), and 10<:::-2ecp'lm /9ft3r, where <p is the metal-insulator barrier height, m* the effective mass, and e the charge of the electron. The conditions for the validity of the formula are that the image force is not too strong and that the energy gap of the insulator is large compared with the metal-insulator barrier height. Furthermore, the applied voltage must be sufficiently high to have the electrons tunnel into the conduction band of the insulator and not directly into the second metal. The curve in Fig. 5 was obtained using m= mo and <p= 1.6 eV. For E the average field for the thin insulator using €= 1 was inserted, which is, of course, only a very crude approximation. Due to polarization effects, the field will presumably be larger at the boundaries than the average field strength. Since nothing can be said about m either, the stated barrier height of 1.6 eV is, at best, only a good guess. A test of the tunnel hypothesis would be the temperature inde pendence of the initial I-V characteristic. Therefore, the initial I-V curves have been determined for room temperature and liquid nitrogen temperature on samples with 150 A-AhOa thickness. At liquid nitrogen tem perature, the diodes behaved practically the same way as they did at room temperature. There was the same steep rise of 1 d upon initial increase of the voltage V d. After some operating time, the Id-Vd characteristic flattened out as has been observed at room temperature. However, the change of the initial I d-V d curves with temperature, which is not shown, was nonreproducible in as much as the curves were shifted to lower as well as larger field strengths, depending on the sample. While it is thus not possible to claim these data as proof for the tunnel hypothesis, they also do not contradict them. 12 It is known, that for AhO. thicknesses below 1OO-A deviations from the linear thickness-forming voltage might occur, which seems not to be surprising in view of.,thePfact that before anodiza· tion the Al is already covered with an AhO, layer several tens of angstroms thick. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:06E LEe T RON E 1\1 ISS ION F R () 1\[ T II I N A 1--A 120 3 --A 1I S T RITe T U RES 3SS3 • 67 A AI203 250A Au A 75A 2S0A o 100A 225A . . 10-2 -v 125 A 320A o ISOIi. 320A ~ 10-3 ....:l' C ~ :; U 10-4 .... "0 0 i:3 10-5 10-6L-_-'-_-.J.. __ .L...._-.L_-l __ ..L-_--' o 1.6 3.2 4.8 6.4 8.0 9.6 x 10· Field Strength, F (V/cm) Fw. 5. Initial diode current vs applied average field. The solid line is calculated by the Fowler-Nordheim formula with m=mo and 1"=1.6eV. More consistent measurements are necessary to clarify this question. If we assume the barrier height of 1.6 eV to be a reasonable figure, it is very unlikely that one would initially observe emission over the barrier (Schottky effect). Since the I-V characteristics change with passage of large currents through the insulator, a change of barrier height to lower values due to modifications within the insulator is very likely. Under this condition, thermal emission across the barrier might become possi ble. These thermal currents are, of course, temperature dependent. According to Schottky's theory of thermal emission over a barrier, the height of which is controlled by an applied field, the emission current isl3 where E is the field strength and T the temperature. A plot of log I vs Ei therefore should give straight lines, the slope of which is inversely proportional to the tem perature. In our experiments, a temperature dependence of the Battened out I-V characteristics has been indeed observed. Fig. 6 shows the log 1-Vi characteristics of a sample for room and liquid nitrogen temperature. An increase in the slope with decrease in temperature is apparent. However, if Schottky's theory would apply, 13 See, for instance, W. B. Nottingham, Encyclopedia of Physics, edited byS. Fliigge (Springer-Verlag, Berlin, 1956), Vol. 21, p. 1 if. ~ ." .... C ~ :; U .... "0 0 i:3 o Room Temperature o Liquid Nitrogen Temperature 10-3 10.4 10-5 10-6 OL-----.J..I--~--2L----~3-~xlo·3 A,(v1/cm Y) Fw. 6. Diode current after passage of some charge for room and liquid nitrogen temperature. one would expect the curve for liquid nitrogen to fall below that for room temperature. Since no quantitative agreement is reached at all, it must be concluded that other effects, i.e., space-charge effects, appreciably modify or even dominate the temperature dependence of the currents. It is apparent that in order to untangle the various physical processes involved in the I-V characteristics, a careful study of samples with repro ducible characteristics is mandatory.14 On certain cells, it could be noticed that after operat ing for some time at moderate currents and at voltages sufficient for emission, a decrease in voltage below about 4 V lead to breakdown of the cell with considerable increase of the cell current. This current was not stable but showed large fluctuations. Upon increase of the voltage, the current dropped again, and showed fewer breakdown peaks. At sufficiently large voltage, these peaks completely disappeared. A more or less reproduci ble I-V curve including the "breakdown" region is demonstrated in Fig. 7. "Breakdown" peaks in the cell current below 5-V cell voltage were accompanied by bursts of emitted electrons, such that the fluctuations in I d and Ie did correspond to each other. It is probable that this type of ]-V characteristic with a negative 14 In a recent communication by P. R. Emtage and W. Tantra port in Phys. Rev. Letters 8, 267 (1962), a Schottky temperature dependence of diode currents was reported, from which barrier heights between 0.5 and 1 V were deduced. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:0635S4 II. KANTER AND W. A. FEIBELl\IAN 10 - t "\ \ I \ \ ~ 225 A Au BreakdOwn Normal Mode I Mode .1 Lig~~~~ssion i1 Uliform Light Spots (Scintillations) Across Surface Forms After Some TIme of Operation at -I mA .0IO.':--"2:---4t---i:S;---*"a------.lIO>;--+.r--i.14 Diode Voltage Vd (V) FIG. 7. Id-Va characteristic with breakdown region for Vd$4 V. resistance region is related to that reported recently by Hickmott.1b b. Imaging of the Emitting Surface By placing a phosphor close to the sample in order to be able to view an image of the emitting surface, a check of the uniformity of emission was possible. For an Au film 300 A thick, the phosphor area corresponding to the active region of the sample was uniformly bright-the edge region not appearing brighter than the rest of the area. Imaging has not been attempted at larger Au film thicknesses, since films of good conductivity can be made less than 300 A thick without difficulty. It has been observed, however, that transfer ratios as a func tion of gold film thickness dropped continuously with thickness only for thicknesses below about 350 A. Beyond that thickness, the transfer ratio leveled off and stayed constant for larger thicknesses. It is assumed that the emission observed on thicker films originated from the edge region of the sample, where the film thickness tapers off to zero. The variation of our Au film thicknesses was thus limited to between about 200 and 300 A, the lower limit given by the condition of sufficient conductivity. In order to keep the voltage drop across the sample smaller than 0.1 V at a cell cur rent of 10 mA, the film resistance had to stay below 10 n. On cells exhibiting the above mentioned breakdown characteristics of the cell current, the emission peaks appeared as bright flashing spots randomly distributed 15 T. W. Hickmott, J. Appl. Phys. 33, 2669 (1962). in space and time over the uniformly excited part of the phosphor. With increase in cell voltages, the flashes disappeared, while the phosphor increased in brightness. With a decrease in voltage, the uniform excitation dis appeared while more and more scintillations became visible until the cell voltage became too small and no light emission from the phosphor was observed at all. c. Light Emission from the Samples At large cell currents, a very faint uniform lumines cence emission from the active cell area could be ob served with the dark-adapted eye. The light appeared to be of pale greenish color, but this may have been due to the gold film, which looks green by transmission. At moderate currents, the emission was hardly recognizable any more. The color and intensity of the luminescence was the same for forward or reverse polarity,16 On un stable cells, exhibiting bursts of emitted electrons at low voltages (breakdown region), scintillations randomly distributed over the cell area were readily observable in the darkened room. It is believed that the breakdown in our samples indicated by a bright spot leads to local destruction of the thin film in a method similar to that utilized in breakdown resistant capacitors, in which after breakdown one layer burns out over a sufficiently --------_, ___ --7V 14mA T~7.HO·S S.75V4mA T_S'10-5 10.7 S.5V 1.5mA T =4.3'10.5 ~ 6.25V.S5mA - ,..!' 10·e T33.1·IO·5 -i 6V.2SmA f: T =2.35'10.5 :; U c 5.75V.l7mA 0 10-9 T_1.I3·10-5 'in .!!? E w 5.5V.15mA T=3.1·IO·6 10'10 10.11 ~--'~..L-.I....L..-'-.J....L __ .L-_...I-_-1.._--J -3 -2 -I 0 I 2 3 4 Retarding Potential, VR , (V) FIG. 8. Retarding potential curves of emission current. Param eter is the cell voltage and cell current. The arrows on the curves determine the region in which the curves break off the saturation value and were obtained from a plot with a linear current scale. 16 The same effect has been reported by J. Wesolowski, M. Jackimowski, and R. Dragon, Acta Phys. Polon. 20,303 (1961), [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:06E L E C T R 0 l\' E!'vI ISS I 0 ~ FRO!'vI T H I N A 1 - A 1 2 0 3 -A u S T R U C T l' RES 3585 • 67A AI203 o 75A v 100A o 125A A 150A 250A Au 250A 225A 320A 320A Diode Polenliol. Vd (V) FIG. 9. Width of the energy distributions vs cell voltage Vd. large area, thus separating the breakdown spot from the rest of the structure. Therefore, a bright spot appears only for a very short time in our experiments. At larger operating voltages, the scintillations again disappeared, and the uniform luminosity became visible. The lumi nescence in AbOa films has been observed previously17 and is generally considered to be due to an electro luminescence process. d. Energy Distribution of Emitted Electrons Continuous electron emission into the vacuum was not observed for diode voltages less than 5 eV (mini mum observable emission current about 10-10 A/cm2). Beyond this voltage, the integral spectrum of the normal energy component of the emission current was measured for a variety of Al20a thickness, for cell voltages up to 10 V. A set of curves using a log Ie-V R scale for a sample with 100 A-AbOa and 225 A-Au (Fig. 8) is typical for a cell with stable characteristics. Inspection shows that the intersections of the curves with a certain small current ordinate value decrease about linearly with the cell voltage, suggesting a linear dependence of the width of the energy distribution on the cell voltage. In Fig. 9, the width of the energy distributions for various AbOa thicknesses is plotted as a function of cell voltage. The width is taken as the energy difference between that point of the abscissa (in Fig. 8), where the curves level off to a constant value (indicated by arrows a t V R "'" + 1 V) and the point where the curves have decreased to 10-11 A.18 Because the latter value appears 17 D. W. Mayer, Fifth National Conference on Tube Techniques, New York City, September 15, 1962; M. I. Elinson, G. F. Vasilev, and A. G. Zlidan, Radiotekhn. i Elektron. 4, 1718 (1959); I. Adams and T. R. AuCoin, IniernaJional Conference on Lumines cence, New York University, October 1961 (John Wiley & Sons, Inc., New York, 1962). IS Generally widths of distributions are defined between points representing a certain fraction of the maximum observable value. Since we are interested in the maximum observable energy, regardless of intensity, the above mentioned method of defining the width was adopted. to be a rather arbitrary cutoff point and because of the uncertainty in determining the point of level-off (even in a linear plot), the width can only be determined within certain limits. Nevertheless, the data show that the width decreases linearly with decrease in cell volt age, approaching zero between 3.9-and 4.9-V cell volt age depending on the Al20a film thickness. Further more, the absolute width approximates very closely the difference between applied voltage and the work func tion of the gold surface ('PAu"'-'4.7 V). As illustrated in Fig. 10, the width should extrapolate to zero for V d= 'PAu if electrons originated from near the Fermi surface and had entered the insulator conduction band by tunneling. In this case, the width of the energy dis tribution should follow the 450 line (dot-dash), indi cated in Fig. 9. The measured widths deviate from this line. Undoubtedly, part of the deviation results from variations in the surface potential. However, except for the thinnest film, the curves monotonically shift to the left and, thus, indicate larger widths at constant V d with decrease in oxide thickness. Therefore, the data suggest that the measured width can be larger than that expected for tunneling, by an amount which depends on the oxide thickness. The "excess" energy, as indicated by the shift towards the left in Fig. 9, is suggestive of Schottky-type emission over a barrier into the insulator. The decrease of the excess energy with increase of the oxide thickness reflects the attenuation within the oxide. A typical normal energy distribution of the emitted electrons is shown in Fig. 11. The curve was obtained by graphical differentiation of the also indicated retarding curve, taken from an x-y recorder. Except for the lack of more energetic electrons, the curve resembles very much the energy distribution as known from secondary electron emission from metals. The average normal en ergy of the emitted electrons, calculated by integrating the retarding curves, was found to increase linearly with the cell voltage V d and can be closely resembled by Ea(eV)=0.2 Vd(V)-0.3. While the above results apply to a stable cell, it is interesting to note that in unstable cells, where the emission current originates partly from breakdown spots, the energy of the emitted electrons is up to 2 eV larger than expected on the basis of Fig. 9. The more energetic electrons presumably have gained their excess --r------------------------- FIG. 10. Energy diagram of emitter structure. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:063SK6 II. KA:,\TER AKD W. A. FEIBELMAN -I -2 Retarding Potential (volt) FIG. 11. Normal energy distribution for electrons emitted from a cell operated at 7 V. energy in the "hot" mkroplasma of the breakdown discharge. e. Transfer Ratio The transfer ratio T for various AhOa and Au film thicknesses and for various diode voltages is tabulated in Table I. The transfer ratio was generally found to be between 10-6 and 10-4, depending on film thicknesses and voltages used.19 The order of magnitude of T agrees for the particular Au film thicknesses employed in our experiments, with those reported recently by Mead.5 (Mead used Be-BeO--Au structures, but did not state the BeO film thicknesses.) In order to evaluate T with regard to its dependence on film thickness, we extra polated all data to V d= 7 V. The transfer ratio for this voltage is listed in column 5 of Table 1. From our data alone, it is not possible to deduce the attenuation in Ali)a and Au separately. In the experi ments by Mead" mentioned above, however, the attenu ation was measured in Au for electrons, which after penetration of the Au still had sufficient energy to surmount the work function step (",4.7 eV) and to escape into the vacuum. An attenuation length of about 100 A was reported. Using this value, we can calculate from our data the transfer ratio for zero gold film thickness. 19 A slight dependence of T on operation time was observed when relatively large currents were drawn through the sample. T usually increased gradually with time and eventually went through a maximum. The change is believed to be due to heating of the sample. The quoted T values are those observed before a change with time had set in. Note added in proof. Recent measurements of the transfer ratio as a function of gold film thickness in our structures resulted in an attenuation length for Au near 60 A. This leads to larger values for T A120a than tabu lated in column 6 of Table 1. It does not, however, affect the essence of the following discussion. In agreement with Mead, transfer ratios for the insu lator alone are in the order 10-2 to 10-4 and in our case depend on the oxide thickness (column 6, Table I). Thus, most electrons have lost energy in the Al20a to values less than 4.7 eV above the Fermi level. In a plot of log T vs oxide thickness, which is not shown here, the data points, except for the thi.nnest film of 67 A, fall on a straight line, which intercepts the abscissa at T~5·1O-2. The small transfer ratio at zero oxide thick ness indicates that part of the low transfer ratio for the AbOg is apparently due to appreciable reflection of elec trons at the AhOa-Au interface. The attenuation length deduced from the slope of the Hne is >-=24 A and might be interpreted as an average range of electrons which have started at the bottom of the conduction band and have been accelerated to an energy of at least 4.7 eV above the Au Fermi level. The energy separation of the conduction band from the Al or Au Fermi level is roughly given by the barrier height at the Al-AlzOa interface. This barrier height was estimated to be 110t larger than 1.6 eV, even upon initial application of voltage to the diode. Thus, the electrons must have gained an energy of the order of 3 eV within the in sulator. TABLE 1. Transfer ratios for various film thicknesses and operating voltages. AhOa Au l'</(V) T 67 250 6.0 65XlO- s 4·10"-' 4.9.10-2 67 250 5.75 16 67 250 5.5 6.8 67 250 5.25 3.9 67 250 5.0 0.87 7$ 250 6.0 11 75 250 5.75 12 7$ 2SO 5.5 8.1 75 250 5.25 4.35 75 250 5.0 2.94 100 225 7.0 71 7.5·10-. 7.1.10-4 100 225 6.75 60 100 225 6.5 43 100 225 6.25 31 100 225 6.0 23.5 100 225 5.75 11.3 100 225 5.5 3.1 125 320 8.0 27 125 320 7.5 20 125 320 7.0 6 125 320 8.4 56 125 320 8 28.6 125 320 7.5 14.3 125 320 7.0 6.5 150 320 10,0 200 3·10-'6 7.4·10-" ISO 320 9.0 sn ISO 320 8,0 <) ISO 320 10,0 90 ISO 320 9.25 45 150 320 8.$ 24 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:06ELECTRON EMISSION FROM THIN AI-AI 203-Au STRUCTURES 3587 f. Improvement of the Transfer Ratio by Deposition of a Ba Layer on the Au Film Because of the probable increase in the attenuation lengt~ at energies smaller than abo~t 3 ey above. the Fermi level, as suggested by Thomas data, a con~lde: able improvement of T by lowering the work functIOn is verv likely. It appears to be advisable to use as exit metal a metal with a low work function, thus keeping the total film thickness to a minimum. For a metal with low work function we chose Ba (<p"-'2.5 eV), mainly because it is readily available in so-called getter channels used in vacuum tubes. We did not succeed in building a cell with a Ba film as a second electrode. Ba was de posited to a jilm thickness with a resistance of several hundred ohms per square on top of the Ah03. Under this condition, the total cell resistance became so low that the cell potential of more than about 4 to 5 y, necessary to observe emission, could not be applIed without immediate destruction of the cell by the large power dissipation. Therefore, we decided to ~se our conventional AI-AhOa-Au cells and to depOSit a Ba layer on top of the exit surface. Typical operating d~ta of a cell, before, during and after "activation" by firmg the Ba channel are presented in Table II. As mentioned before the maximum improvement of the transfer ratio could ~ot be maintained over a prolonged period of time because of the rather poor vacuum conditions, mainly caused by the insufficient degassing of the Ba channel before operation. Upon firing the Ba, a considera?le increase in the emission current was observed, which gradually leveled off with time. After s~opping ~he evaporation the emission decreased agam, reachmg after 1 to i min the values indicated by "steady" in Table II. The values, however, were actually still slowly decreasing. A typical maximum transfe.r ratio measured was T= 10-2 for V d= 7 V and an Au thickness of 225 A, which need not necessarily be considered an optimum value, because of the poor vacuum under which the experiments were carried out. The depression of the work function of the Au surface by deposition of a Ba layer was measured in separate experiments, carried out under comparable vacuum conditions in which a gold surface was bombarded by a slow electron beam. Upon deposition of a Ba layer on the Au-retarding electrode, beam retarding curves were usually shifted between 1.9 and 2.2 V. Thus, lowering the minimum escape energy of the electrons from about 4.7 eV (above the Fermi level) to about 2.7 eV con siderably increases the transfer ratio. Since the transfer ratio of the AlzOa-Au system with Ba at the surface (2.7-eV electrons) is T= 10-2 and thus larger than the transfer ratio which has been deduced for 4.7-eV elec Irons in AlzO;; alone (T"-' 10-3), it is apparent that a noticeably larger fraction of electrons with lower ener gies is available at the Ab03-Au interface. This fact is indicative of the rapid rise of the energy distribution Sample No.1 2 5 TABLE II. Improvement of transfer roatios by Ba deposition-H)O A-AJ,Oa; 225-A Au. 7 6 6.5 7 T(after Ba T(before) evaporation) 1(}-4 1(}-2 5.7·1(}-3 6.8.10-4 10-6 3.5.10-3 4·1(}-3 2.10- 3 10-' 1. 7.10-4 2·1(}-4 Remarks maximum during evaporation after 1 min after 2 min, steady maximum during first evaporation maximum during second evapora tion after 2 min, steady curve for electrons in the Ab03 with decrease in energy. In order to find the attenuation of these less energetic electrons in both Ab03 and Au, the effect of thickness variations of either material should be studied. In vestigations of this type are under way. CONCLUSION The principal results obtained by this work are the following: (1) The observed initial current-voltage character istics of the AI-Ab03-Au samples with AbOa film thicknesses between 67 and 150 A, agree with the theoretically expected characteristics for tunnel currents. (2) After large currents have been drawn through the insulator, the I-V characteristics change, generally to larger currents, especially at low voltages. These characteristics are temperature dependent. (3) As expected, emission into the vacuum does not set in before the Fermi level of the Al base is lifted near the vacuum level of the thin gold film. Upon further increase of the sample voltage, the average energy of the emitted electrons increases linearly with the applied voltage. The absolute width of the energy distribution can be slightly larger than the difference between the cell voltage and the work function potential of the Au. (4) The transfer ratio for thin Au films, defined as the ratio of emission current to diode current has been found to vary between 2X 10-4 and 10-6, depending on AbOa and Au film thickness and sample voltage. Using Mead's recent data on the attenuation of electrons in Au, an attenuation of about 24 A in AbOa was deduced for electrons which have been accelerated from the con duction band to more than 4.7 eV above the Fermi level, representing a gain in energy of roughly 3 eV. (5) It could be demonstrated that the transfer ratio can be improved considerably by lowering the work function of the Au film. Thus, as an example, an original transfer ratio, T= 10-\ was improved by deposition of Ba to as much as T= 10-2, corresponding to an emission current density of 5 mA/cm2• An improvement of T to [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:063588 H. KANTER AND W. A. FEIBELMAN values better than 10-2 appears possible through more refined surface techniques. It must be kept in mind that the experiments de scribed above served only to gain a general idea of what one can expect from an electron emission device. The quoted values are not considered at all optimum values. In order to start a more detailed investigation, it is desirable to use more stable and reproducible samples than have been used in this work. The AhOa insulator does not appear to be very suitable, at least when formed by anodization, a factor that has been observed by other investigators. Furthermore, it should be possi-ble to lower the effective work function much more than was possible in this work, either by using various low work function deposits of the alkali metals or alkali metal oxides, by more suitable combinations of thin metal films and deposits, or by the use of alloying tech niques as obtained by heating the films.2.J Transfer ratios better than 10-2 might very likely be possible. Another possibility might be the deposition of the second metal film in a mesh-like structure, such that part of the emitted electrons do not have to penetrate the metal film at all. 20 lu G. Shishkin and 1. L. Sokolskaya, Radiotekhn. i Elektron. 5, 1218 (1960). JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 12 DECEMBER 1962 Scattering of Plane Waves by a Cavity Ribbon in a Solid K. HARUMI Department of Applied Science, Tolwku University, Sendai, Japan (Received December 18, 1961) The scattering of plane compressional or shear waves by a cavity ribbon with width a in an elastic medium is computed by the use of the Mathieu functions when the displacement of the incident wave is in xy plane. The diffraction patterns for kca/2 = 1, 2, and 4, and for k. = 2kc are obtained; the diffraction patterns by a cavity ribbon are similar to those of the sound wave by a perfectly absorbing ribbon only when the incident compressional waves propagate to the direction normal to the plane of the ribbon. The scattering cross sections Q/a are of the order of (ka)3 in the Rayleigh case. These facts support the conclusion of the pre vious paper (rigid ribbon). I. INTRODUCTION THE scattering of a plane elastic wave in a solid by any obstacle has been studied by several authors,1-6 as referred to in the previous paper,7 here after this paper will be referred to as I. In order to obtain the property of the scattering of an elastic wave by a crack in a solid, the author treated the scattering of a plane compressional or shear wave incident on an infinitely long rigid ribbon as the first step. The present paper is the answer to this problem. The method which is used in this paper is similar to that of I, and the distributions-in-angle and the scatter ing cross sections are obtained for the incident compres sional and shear waves propagating along the positive y axis (Oo=7r/2). The Rayleigh case is considered and some expansions of the radial Mathieu functions used in the Rayleigh case are added in the Appendix. 1 K. Sezawa, Bull. Earthquake Res. lnst. Tokyo Univ. 3, 19 (1927); 4, 59 (1928). 2 K. Kato, Mem. Inst. Sci. Ind. Res. Osaka Univ. 9, 16 (1952). "C. F. Ying and R. TrueH, J. Appl. Phys. 27,1086 (1956). 4 R. M. White, J. Acoust. Soc. Am. 30, 771 (1958). f' N. G. Einspruch and R. TrueH, J. Acoust. Soc. Am. 32, 2H (1960). 6 N. G. Einspruch, E. J. WitterhoIt, and R. TrueH, J. Appl. Phys. 31, 806 (1960). 7 K. Harumi, J. App!. Phys. 32, 1488 (1961). We have heard8 that Skuridin9 treated the scattering by the cavity ribbon using the Kirchhoff's approxima tion (short wavelength limit), and obtained the diffraction patterns which show the sharp angular dependency, but unfortunately we have not seen this paper. II. INCIDENT WAVES AND SCATTERED WAVES In this paper, for simplicity, we confine ourselves to the case of the incident plane compressional or shear wave whose displacement is tangential to the xy plane which is perpendicular to the axis of the cavity ribbon of width a. The notations and the equations of this paper have much in common with I except that they lack the z component. In this paper, therefore, some of the equations are omitted throughout. As stated in I, the general solutions of the displace men t in two-space dimensions are expressed as the sum of the compressional part L and the shear part M,lO L= grad<I> (1) (2) 8 K. Bhagwanuin, Math. Rev. 17, 319 (1956). • G. A. Skuridin, Izv. Akad. Nauk SSSR Ser. Geofiz. 1955, 3. 10 P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), VA!. 2, p.1764. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Wed, 26 Nov 2014 10:17:06
1.1714260.pdf	Electrical Resistivity of Metals and Alloys Containing Localized Magnetic Moments A. J. Dekker Citation: Journal of Applied Physics 36, 906 (1965); doi: 10.1063/1.1714260 View online: http://dx.doi.org/10.1063/1.1714260 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/36/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Interplay between magnetic order and electrical resistivity in local moment manganites AIP Conf. Proc. 1461, 267 (2012); 10.1063/1.4736901 Magnetoresistance of antiferromagnetic metals with localized magnetic moments J. Appl. Phys. 61, 4393 (1987); 10.1063/1.338433 Superconductivity — A Probe of the Magnetic State of Local Moments in Metals AIP Conf. Proc. 34, 71 (1976); 10.1063/1.2946164 Local Moment Formation and Resistivity Minima in CoAl Alloys J. Appl. Phys. 40, 1476 (1969); 10.1063/1.1657728 Effect of Correlation on Occurrence of Local Magnetic Moments in Metals J. Appl. Phys. 40, 1103 (1969); 10.1063/1.1657544 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Sun, 21 Dec 2014 05:08:14JOURNAL OF APPLIED PHYSICS VOLUME 36, NO.3 (TWO PARTS-PART 2) MARCH 1965 Electrical Resistivity of Metals and Alloys Containing Localized Magnetic Moments A. J. DEKKER Institutefor Crystal Physics, University of Groningen, Holland In metals and alloys containing localized magnetic mo~ent~, an importa~t cont~ibu~ion. to t~e elec trical resistivity can be associated with disorder of the atonuc spm system. Thi~ con~nbution l~ ascn~ed to a scalar interaction between the spins of the conduction electrons and the atomIC Spll~S. '\ssummg a.Sl~ple form for this interaction, the experimental and theoretical sit~ation concerning the spm-dlsorder reslstiVlty is reviewed for pure metals, dilute alloys, and concentrated bmary alloys. INTRODUCTION THE advances made in the physics of metals and alloys during the last decade have resulted to a large extent from a few important !dea?, whi.ch ~ave stimulated experimental and theoretIcal mvestlgatIOns. Probably one of the most fruitful contributions in this respect is the detailed study by FriedeP and ~is gro,up concerning the perturbation produced by an ImpUrIty in a metal. In contrast to the view held previously, that the screening charge around an impurity decreases monotonically with distance, Friedel demonstrated that the screening charge has a long-range oscillatory character. He also derived a simple relation between the total screening charge and the phase shifts in the partial wave method, the Friedel sum-rule. This work provides the basis for much of our current understand ing of a variety of properties, including transport phenomena, Knight shift, positron annihilation, etc. Another fruitful idea, which has received a great deal of attention lately, relates to the exchange inter action between the conduction electrons and the lo calized electrons of incompletely filled inner atomic shells. The purpose of this paper is to review the progress that has been made in this area, at least inso far as it applies to the electrical resistivity caused by spin disorder. For a general survey of this topic we refer to Coles.2 The magnetic and transport properties of a ferro magnetic metal such as nickel can be discussed with some confidence in terms of the band theory, involving a broad 4s conduction band and a narrow 3d band, responsible for the magnetic properties. Thus, the electrical resistivity of nickel and some of its alloys3 can be understood as a consequence of the s-d scatter ing mechanism proposed by Mott.4 In its simplest form the band model cannot explain the occurrence , . of an antiferromagnetic arrangement, nor a CUrIe- Weiss law in the paramagnetic region, observed in certain transition metals and alloys. These properties evidently require a certain degree of localization of the magnetic moments and the introduction of ato~lic orbitals. The rather complicated problem of locahza tion of magnetic moments in the 3d (a~d similar) transition metals and alloys has been dIscussed by several authors.5-8 The situation with regard to the localization of magnetic moments is much clearer for the rare-earth metals than for the other transition elements and, therefore we shall be concerned almost exclusively with the 'former. The elements of the lanthanide group have the electron configuration (41)n6s25d, with the exception of Eu and Yb, which have (41)76s2 Il:nd (41) 146s2, respectively. As n increases, the relat~ve stability of the 41 levels is enhanced and they fll:ll m creasingly below the outer electrons, both energetlcally and radially. From neutron diffraction experiments9 one concludes that the root-mean-square radius of the 41 charge distribution is about 0.3 A, i.e., of the. order of 0.1 of the interatomic distance of ,,-,3.6 A m the metallic state. The relatively strong localization of the 41 electrons has important consequence: for the prop erties of the rare-earth metals. The dIrect exchange interaction between the 41 shells of neighboring atoms is far too smalllO to explain the relatively high magnetic ordering temperatures; in fact, our current unde~ standing of the properties of the rare-e.arth m~tals :s based on a long-range indirect exchange mteractIOn VIa the conduction electrons.ll•12 It explains many details of the sometimes complex types of magnetic ordering, including helicoidal spin arrangements.13-15 • The exchange interaction between the conductIOn electrons and the magnetic shells leads to spin-de pendent scattering and thus to a term in the electrical resistivity. After brief summaries of the indirect ex change mechanism and the method for calc~la~i~g the spin disorder resistivity, we discuss the resIstlvlty of 6 P. W. Anderson, Phys. Rev. 124, 41 (1961). 6 P. A. Wolff, Phys. Rev. 124, 1030 (1961). 7 J. Friedel, G. Leman, and S. Olszewski, J. Appl. Phys. 32, 325S (1961). .. 8 C. Kittel, Quantum Theory of Solids (John Wliey & Sons, Inc New York, 1963), Chap. 18. g W. C. Koehler and E. O. Wollan, Phys. Rev. 9, 1380 (1953). 1 See J. Friedel, Advan. Phys. 3, 446 (1954); Can. J. Phys. 10 R. Stuart and W. Marshall, Phys. Rev. 120, 353 (1960). 34, 1190 (1956); J. Phys. Radium 23, 692 (1962); and references 11 P. G. de Gennes, J. Phys. Radium 23, 510 (1962). in these papers. 12 Y. A. Rocher, Phil. Mag. Suppl. 11, 233 (1962). 2 B. R. Coles, Phil. Mag. Suppl. 7, 40 (1958). 13 M. K. Wilkinson, E. O. Wollan, W. C. Koehler, and J. W. 3 A. W. Overhauser and A. I. Schindler, J. Appl. Phys. 28, Cable, J. Appl. Phys. 32, 485S, 495S (1961). 554 (1957) 14 R. J. Elliott, Phys. Rev. 124, 346 (1961); Proc. Phys. Soc. 'N. F. Mott and H. Jones, Theory of the Properties of Metals (London) 84,63 (1964). 123 329 (1961) and Alloys (Clarendon Press, Oxford, England, 1936). 16 T. A. Kaplan, Phys. Rev., . 906 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Sun, 21 Dec 2014 05:08:14ELECTRICAL RESISTIVITY OF METALS 907 pure metals, dilute alloys, and concentrated alloys containing localized magnetic moments. EXCHANGE INTERACTION BETWEEN CONDUC TION ELECTRONS AND LOCALIZED SPINS The indirect coupling between localized atomic spins via the conduction electrons is closely related to an other important problem in metals, viz. the indirect coupling between nuclear spins via the hyperfme inter action. The latter problem has been studied in detail by Ruderman and Kittel.16 The existence of an exchange interaction between conduction electrons and an atomic spin has been proposed independently by Vonsovsk y17 and Zener,18 in analogy with the exchange interaction between the valence electrons and the electrons in an incompletely filled shell in a free atom. More recently, it has been studied in detail by several authors.l9-23 We shall assume here that the exchange interaction between a conduc tion electron of spin Se at r and an atom of spin Sn at Rn can be written as (1) where G is a quantity with the dimensions of energy times volume (see Kittel8). As a result of the spin orbit coupling, Sn is given by its projection on the total angular momentum In, so that Bn'=-2G(g-1)5(r-Rn)Se·Jn, (la) where g is the Lande factor. Consider now two atomic spins Sn and Sm embedded in a sea of free electrons. One can show that, as a result of the scattering produced by (1), the conduc tion electrons in the vicinity of Sn become polarized. In fact, if n+(r) and n_(r) represent the densities of electrons with magnetic quantum numbers mes= +! and mes= -! at a distance r from Sn Here, Z is the number of conduction electrons per atom, Q the atomic volume, EF the Fermi energy, kF the Fermi wave vector, mjn the total magnetic quan tum number of atom n, and F(x) = (x cosx-sinx)/x4 (3) is the oscillating Ruderman-Kittel function. As a result of this spin polarization, the spin Sm senses the presence of Sn and from a second-order perturbation 18 M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954). 17 S. Vonsovsky, J. Phys. (USSR) 10,468 (1946). 18 See C. Zener and R. R. Heikes, Rev. Mod. Phys. 25, 191 (1953) . 19T. Kasuya, Progr. Theor. Phys. (Kyoto) 16,-45 (1956). 20 K. Yosida, Phys. Rev. 106, 893 (1957). 21 S. H. Liu, Phys. Rev. 121, 451 (1960); 123, 470 (1961). 22 T. A. Kaplan and D. H. Lyons, Phys. Rev. 129, 2072 (1963). 23 A. Blandin, thesis, Paris (1961); A. Blandin and J. Friedel, J. Phys.Radium 20, 160 (1959). calculation one finds for the indirect coupling between the two spins, separated by a distance Rmn Bmnl! = [91rZ2G2(g-1)2/ EFQ2JF(2kFRmn)Jm·Jn. (4) For small values of 2kFRmn, F(x)~-1/3x, so that in that region the interaction is ferromagnetic; the first zero occurs for 2kFRmn=4.49. For two nearest neigh bors in a trivalent rare-earth metal, kF~1.4X 108 cm-1 and Rmn~3.6 A so that F(2kFRmn) ~-lO-3. For a discussion of various physical properties of the rare earth metals in terms of (4), we refer to de Gennesll and Rocher.l2 A discussion of the temperature de pendence of the wave vector of the magnetic order ob served in the heavy rare-earth metals has recently been given by Elliott and Wedgwood24; it involves superzone boundaries produced by the magnetic order via the in teraction (1). OUTLINE OF THE CALCULATION OF THE RESISTIVITY Since details of the calculation of the spin-disorder resistivity will be omitted in the following sections, we recall the general procedure here and apply it to a simple case. In these calculations we assume that the conduction electrons occupy a simple conduction band; their energy is given by E=h2k2/2m, where k is the wave vector and m an effective mass. The transport properties are discussed in terms of a relaxation time TF, where the subscript F refers to electrons with the Fermi energy. The reciprocal relaxation times corre sponding to the various scattering mechanisms are assumed to be additive. For N scattering centers per unit volume with a differential cross section A (0) per unit solid angle, the contribution to the reciprocal relaxation time is hkF ir • hkFN TF-1=-N A(O)(l- cosO)21rsmOdO=--A t. mom (5) Here, At is the total transport cross section per center. The contribution to the resistivity of these centers for a metal containing n conduction electrons of charge -e per unit volume is (6) As a simple example consider the spin-disorder re sistivity of a rare-earth metal in the temperature region T»Tc, where Tc marks the magnetic ordering temperature. Neglecting effects due to local order of the atomic spin system, we proceed as follows: Since the average atomic spin seen by the conduction elec trons vanishes, expression (1) immediately gives the perturbation produced by a single atom. In the Born approximation,25 the scattering of a conduction electron 24 R. J. Elliott and F. A. Wedgwood, Proc. Phys. Soc. (London) 84,63 (1964). 2i The Born approximation used in combination with the delta function potential is equivalent to s-wave scattering in the partial wave method (see Ref. 8, p. 361). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Sun, 21 Dec 2014 05:08:14908 A. J. DEKKER of spin S. from a state k to k' by a localized angular momentum J (chosen as origin) is governed by matrix elements of the form -2G(g-1) X (m • .', m/ I f exp[i(k-k') ·r]~(r-O)drS.·J I me" mj); (7) me., mj and me,', m/ are the magnetic quantum num bers in the initial and final states, respectively. The spatial integration yields a factor of unity and the only nonvanishing spin-matrix elements are (mes, mj! Se.J.! meB, mj)=m e8mj, (8) (me.±l, mj=Fl! Se±h' ! meB, mj) = [j(j+l)-mj(mj=Fl)]i, (9) where Se±= S,,±iS y, etc. The latter correspond to collisions with spin flip, whereas (8) leaves the mag netic quantum numbers unchanged. Now, the differential cross section corresponding to a particular process is equal to (m/211'fI,2)2 times the absolute square of the matrix element involved. For the processes without spin flip and with me.= ±!, this gives A±.±(8) = (m/211'fi,2)2G2(g-1)2m;. (10) For processes with spin flip we obtain, similarly, A±,=F(8) = (m/211'fI,2)2G2(g-1)2[j(j+l) -mj(mj±l)]. (11) As a consequence of the use of delta functions, the differential cross sections are independent of the scattering angle 8 and the corresponding total transport cross sections are obtained by multiplication with 4?r. In the paramagnetic state each of the (2j+l) possible values of mj is equally probable and from (10) and (11) one thus arrives at an average cross section per atom given by At= 4?r(m/211'fI,2)2G2(g-1)2j(j+ 1). (12) Combining this result with (6) one obtains the well known formula for the spin-disorder resistivity at high temperatures26-29 Paro= (311'Nm/2f1,ilEF)G2(g-1)2j(j+l), (13) where N represents the number of atoms per unit volume; it predicts a constant spin-disorder resistivity at high temperatures. We should remark that collisions involving spin flip are, in general, associated with the transfer of energy between the scattered electron and the atom involved; this will be the case if there is an internal or external magnetic field. In such cases, the right-hand side of (5) must be multiplied by 2/[1+ exp( -E/kT)], where E is the energy transferred to the electron in the collision.26 THE RESISTIVITY OF PURE RARE-EARTH METALS The electrical resistivity of polycrystalline samples of the heavy rare-earth metals has been measured by Colvin, Legvold, and Speddingll° between 1.3° and 320°K. As an example of the general behavior, we show in Fig. 1 the resistivity of terbium. One observes a sharp change in slope near the N eel temperature of 229°K and a slight increase in slope with increasing temperature near the ferromagnetic Curie point of 219°K. One assumes that the total resistivity is the sum of three contributions Po represents the residual resistivity due to impurities; pp(T) is due to phonon scattering, and PaCT) is the spin-disorder term. In accordance with (13) it is reasonable to assume that the slope of the nearly 125 115 105 .. 85 I C) .- :Ie ~65 ~ ::t: o 45 35 25 15 5 o 40 80 120 160 200 240 280 320 OK 26 T. van Peski-Tinbergen and A. J. Dekker, Physica 29, 917 FIG. 1. Electrical resistivity of terbium as a function of temper- (1963). ature, according to Colvin, Legvold, and Spedding.3O 27 T. Kasuya, Progr. Theor. Phys.16, 58 (1956); 22,227 (1959). 28 P. G. de Gennes and J. Friedel, J. Phys. Chern. Solids 4, 71 (1958). 30 R. V. Colvin, S. Legvold, and F. H. Spedding, Phys. Rev. 120, 21 R. Brout and H. Suhl, Phys. Rev. Letters 2, 387 (1959). 741 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Sun, 21 Dec 2014 05:08:14ELECTRICAL RESISTIVITY OF METALS 909 linear part in the paramagnetic region is caused solely by phonon scattering. In that case, an experimental value for PaN can be found by extrapolating the linear part in the paramagnetic region to T=O and sub tracting the residual resistivity Po. Values for PaN SO obtained are included in Table I for the elements Gd-Tm. Since these elements all have the same crystal structure, nearly the same lattice constants, and the same number of valence electrons, one expects them to have approximately the same exchange constant G. From a plot of PaN as a function of (g-1)2j(j+1), one finds indeed that the points fall reasonably well on a straight line.29-31 It is of interest to note that the indirect exchange interaction (4) between the atomic spins, combined with a molecular field model leads, for the paramagnetic Curie temperature Op, to the expression 3 Z2G2( 1)2'( '+1) kO = - 11" g-'J J "F(1.k R ) p EQ2 £...J FOn· F nr'O (15) Values for the sum appearing in (15) have been calcu lated for various structures12; for Z=3 and for an hcp lattice with c/a= 1.58, this sum is equal to -6.5XlO- 3. Employing (13) and (15) one can thus calculate m and G from experimental values of Pa", and Op. In fact, if mo represents the free electron mass, one obtains after putting in numerical values for the various quantities, and mo[ 9.6Pa", ]! G=m (g-1)2j(j+1) . (16) Here, P.", is expressed in JlQ-cm, Op in OK, and G in eV A3. The calculated values for m/mo and G are given in Table I; it is observed that these quantities are roughly constant for the heavy rare-earth metals. For temperatures below the magnetic ordering tem perature, the resistivity due to spin disorder has been calculated for a ferromagnetic metal on the basis of a molecular field model and assuming no orbital contribu tion to the magnetic moment.26-28 In this case, the con- TABLE I. The spin-disorder resistivity at high temperatures, P.", (in pn-cm), the paramagnetic Curie temperature 8", and the calculated values of m/mo and G for the heavy rare-earth metals. P.", 8,,(OK) m/mo G(eV A3) Gd 106.4 317 2.6 3.1 Tb 85.7 237 2.8 3.2 Dy 57.6 153 2.9 3.0 Ho 32.3 87 2.9 2.9 Er 23.6 42 4.2 2.2 Tm 14.9 20 5.7 1.9 31 A. R. Mackintosh and F. A. Smidt Jr., Phys. Letters l, 107 (1962). duction electrons see an average atomic spin 0' and one assumes that the perturbation which produces the scattering by an atom Sn at Rn is given by H'= -2GIl(r-Rn)S., (Sn-d). (17) This assumption is equivalent with a Nordheim ap proximation. In contrast to the band theory, this model leads to equal relaxation times for the "up" and "down" electrons; according to van Peski-Tin bergen and Dekker26 p.(T) = 3;e:: {S(S+1) -O'LO' tanh[2T~~~1) ]}, (18) where To is the ferromagnetic Curie temperature. In the corresponding formula given by de Gennes and Friedel,28 the last term in (18) is missing, because they neglected energy transfer between the conduction electrons and the spin system. Expression (18) re produces qualitatively the essential features of the resistivity of gadolinium (the only purely ferromagnetic metal of the heavy rare earths). At low temperatures, the molecular field model must be replaced by a spin-wave mode127,28,32; for a ferro magnetic metal this leads to a spin-disorder resistivity proportional to P. Unfortunately, it is rather difficult to obtain reliable experimental information in this region about Pa separately. The residual resistivity of the rare-earth metals is of the order of several JlQ-cm and it does not seem justified to consider this quantity as temperature independent. Attempts to fit the ex perimental data to a formula of the form p=cTn lead to n~, with sizeable variations.33 Mackintosh82 has taken into account magnetic anisotropy, which leads to a minimum energy ..:l required to excite a spin wave and to a resistivity proportional to P exp( -..:l/kT); this fits the data for Dy very well. Certain details of the p(T)-curves for polycrystalline samples show up in a more pronounced way in single crystals, particularly along the hexagonal axis.34-36 Normally, the resistivity along this axis rises as the temperature is lowered through the Neel point, exhibits a maximum at a lower temperature, and decreases suddenly at the ferromagnetic Curie point. These anomalies have been discussed by Elliott and Wedg wood37 in terms of the two types of ordering which 32 D. A. Goodings, J. App!. Phys. 34, 1370 (1963) j Phys. Rev. 13l, 542 (1963) j A. R. Mackintosh, Phys. Letters 4, 140 (1963). 33 F. A. Smidt, Jr., and A. H. Daane, J. Phys. Chern. Solids l4, 361 (1963). 34 S. Legvold, F. H. Spedding, and P. M. Hall, Phys. Rev. 117, 971 (1960). 35 S. Legvold, F. H. Spedding, and D. L. Strandberg, Phys. Rev. ll7, 2046 (1962). 36 S. Legvold, F. H. Spedding, and R. W. Green, Phys. Rev. Ill, 827 (1961). 37 R. J. Elliott and F. A. Wedgwood, Proc. Phys. Soc. (London) 81,846 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Sun, 21 Dec 2014 05:08:14910 A. J. DEKKER may occur together in the rare-earth metals: (Sn.)=M S cos(q·Rn+q,), (Snx)=M'S cos(q·R n) j (Snu)=M'S sin(q·Rn), (19) (20) where q is parallel to the c axis; M and M' determine the degree of order and are functions of T. This type of ordering produces an exchange field for the conduc tion electrons of lower symmetry than the lattice, introduces new boundaries in the Brillouin zone, and distort the Fermi surface. Taking this into account, Elliott and Wedgwood find that the calculated re sistivities exhibit the essential features of the experi mental curves. RESISTIVITY DUE TO LOCALIZED SPINS IN DILUTE ALLOYS It is well known that many dilute alloys of transition elements in a nonmagnetic host lattice exhibit resis tivity and magnetic anomalies at low temperatures.3S For example,39 a 0.1% solid solution of Mn in Cu shows a minimum near 18°K, followed by a maximum near 4 OK. The quantity (Pmax-Pmin) I Pmin is of the order of a few percent. Similar anomalies have been found recently in dilute alloys of Gd in Ag40 and y'41 Since the resistivity due to phonon scattering de creases monotonically with decreasing temperature, the experimental data imply that the impurity re sistivity itself must be temperature dependent. One possible explanation of the anomalies that has been suggested is contained in the "pair model," introduced by the author42,26 and by Brailsford and Overhauser43j this model found its logical conclusion in the more recent work of Beal,44 who introduced the long-range Ruderman- Kittel-Y osida indirect exchange interaction between the localized moments explicitly. The essential idea of this model is the following: Neglecting for the moment the ordinary Coulomb perturbation, the system of impurities produces a spin-dependent per turbation for a conduction electron of spin S. at r of the form n When one calculates the resistivity resulting from (21) 38 See, for example, G. J. van den Berg and J. de Nobel, J. Phys. Radium 23, 665 (1962) for references. 3D A. Kjekshus and W. B. Pearson, Can. J. Phys. 40, 98 (1962). 40 T. Sugawara, R. Soga, and 1. Yamese, J. Phys. Soc. Japan 19, 780 (1964). 41 T. Sugawara and 1. Yamese, J. Phys. Soc. Japan 18, 1101 (1963) . 42 A. J. Dekker, Physica 25, 1244 (1959); J. Phys. Radium 23, 702 (1962); see also Physica 24,697 (1958). 43 A. D. Brailsford and A. W. Overhauser, J. Phys. Chern. Solids 15, 140 (1960); 21, 127 (1961). 44 M. T. Beal, J. Phys. Chern. Solids 25, 543 (1964); thesis, Paris (1963). in the first Born approximation, one obtains terms of the form G2 f (Sn2)(1-cosO) sinOdO, (22) as well as terms which depend on the correlation be tween pairs of spins, f (SinqRnm) ) . G2 (Sn,Sm) 1+ (1-cosO smOdO, qRnm (23) where q= 2k sin!O. The factor containing sinqRnm re sults from the interference of waves scattered by the spins at Rn and Rm. The resistivity then contains a contribution from each pair which at temperatures above the magnetic ordering temperature TN is pro portional to G2S2( S + 1) 2[cos2kFRnmJ[cos2kFRnmJ> O. (24) 3kT (2kFRnm)3 (2kFRnm)2 The factors in square brackets are due to the Ruder man-Kittel coupling and the interference, respectively. Note that the coefficient of liT is always positive, no matter which pair one considers and that this contribu tion to the resistivity decreases with increasing T. When combined with the phonon contribution this can produce a minimum in the resistivity. At tempera tures below the magnetic ordering temperature TN (which is proportional to the concentration), the model leads to a resistivity which increases with increasing temperature, thus producing a maximum in the im purity resistivity near TN. The maximum would not be observed if TN falls below the experimental region, i.e., at very low concentrations. At high concentrations, the minimum would be masked by the phonon con tribution. If one neglects the phonon contribution, the model predicts that (Pmax-Pmin) I pmin is inde pendent of the concentration. From a comparison be tween the predictions of this model and a number' of experimental data, Beal44 concludes that the agree ment is satisfactory. More recently, an alternative explanation for the resistivity minimum has been proposed by Kondo.45 Since the temperature at which the minimum is ob served is rather insensitive to the concentration, and the relative depth of the minimum is independent of concentration, Kondo argues that the minimum cannot be due to a correlation between the localized spins. He assumes, in fact, that the explanation must involve scattering by independent localized spins alone. Since in the paramagnetic region the first Born approximation in this case leads to a temperature-independent re sistivity, he calculates the resistivity on the basis of (21) to the second Born approximation and arrives at a spin-disorder resistivity of the form ps= constx[1+(6ZGIE F) 10gT]. (25) 46 J. Kondo, Progr. Theoret. Phys. (Kyoto) 32,37 (1964). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Sun, 21 Dec 2014 05:08:14ELECTRICAL RESISTIVITY OF METALS 911 Kondo's theory thus predicts that a minimum in the total resistivity will be observed only if the exchange interaction G is negative; this conclusion requires further experimental confirmation. In the pair model, the occurrence of a minimum is independent of the sign of G. The predicted logarithmic temperature dependence is in good agreement with experimental data on iron in gold alloys46 and some other alloys. Kondo's theory predicts for the depth of the minimum (26) a prediction which is different from that given by the pair model. Also, the temperature at which the mini mum is observed is in Kondo's theory proportional to x!, whereas the pair model gives xl. Expression (26) is in good agreement with the experimental data ob tained by Knook47 for alloys of Fe in Cu. A further detailed comparison between theory and experiment is clearly desirable for dilute alloy systems. DISORDER RESISTIVITY IN BINARY ALLOYS In binary alloys AxBl-x containing localized magnetic moments, both atomic and spin disorder contribute to the electrical resistivity. Even at T=O, when the spin system has its maximum degree of order, there will be a contribution due to spin disorder. An interesting set of data on this subject has been obtained by Smidt and Daane,33 who measured the resistivity of the magnetic alloy systems Gd-Lu, Tb-Lu, and Gd-Er as a function of temperature and for various composi tions. Similar data are available for some Gd-Y alloys.48 For the Gd-Dy system, Bozorth and Suits49 have measured the resistivity at 4.2° and at 3200K for con centrations covering the whole range of compositions. Only recently has an attempt been made to discuss the disorder resistivity of such alloys in terms of a Nordheim model, extended to include spin-dependent scattering.50,51 So far, this has been worked out only for the simplest situations, viz., for the paramagnetic region and for T=O, assuming either ferromagnetic or antiferromagnetic ordering. We shall denote the total disorder resistivity by PdN for the paramagnetic region (complete spin disorder) and by PdQ for T=O. In the relevant literature the term "spin-disorder resistivity" usually refers to the quantity PdN -PdQ. Let the lattice of a binary alloy A"B1-x be divided into atomic cells of volume Q, and let the potential energy of an electron of spin S. be (Va-2GaS.· Sa) oCr-a) in an A cell, in a B cell. (27) (28) 46 D. K. C. MacDonald, W. B. Pearson, and 1. M. Templeton, Proc. Roy. Soc. (London) 266, 161 (1962). 47 B. Knook, thesis, Leiden (1962). 48 J. Hennephof, Phys. Letters 11,273 (1964). 49 R. M. Bozorth and J. C. Suits, J. Appl. Phys. 35, 1039 (1964). 60 A. J. Dekker, Phys. Letters 11,274 (1964). iii A. J. Dekker, Phys. Status Solidi 7, 241 (1964). Va and Vb represent the ordinary Coulomb potentials; delta functions are used only for simplicity. We assume complete atomic disorder. In the spirit of the Nordheim approximation, we assume that the scattering produced by each atom is determined by the difference between the actual and the average potential. In the region of complete spin disorder, the latter is o(r)[xV a+(1-x) Vb] and one obtains for the total disorder resistivit y51 J.1l'm PdN 2fie2EFQ[x(1-x)Vab2+xGa2(ga-1)2ja(ja+ 1) + (1-x)Gt,2(gb-1)2jb(jb+1)], (29) where Vab= Va-Vb. As expected, there is the usual Nordheim term in x(1-x) plus two terms correspond ing to the spin-dependent scattering by the A and B atoms. At T=O, the magnetic order in binary alloys is probably quite complicated because of the long range of the Ruderman-Kittel interaction. Consider, for example, a binary alloy containing one magnetic com ponent and assume that nearest neighbors are coupled ferromagnetic ally, next-nearest neighbors antiferro magnetically, and that all interactions of longer range can be neglected. At T=O, one would then expect ferromagnetic clusters coupled antiferromagnetically. The actual magnetic structure of these alloys is, of course, important, because it determines the average potential to be employed in the Nordheim approxima tion. For the moment, let us assume that at T=O the alloy consists of ferromagnetically ordered regions which are large compared to 1jkF. Within such a region the average potential seen by an electron with magnetic quantum number me8= ±t is then [xVa+ (1-x) Vb=FxGa(ga-1)ja =F(1-X)Gt,(gb-1)jb]o(r). (30) Assuming again that the perturbation which produces scattering by the two kinds of atoms is given by (27) or (28) minus expression (30), one obtains two differ ent relaxation times for the + and -electrons, 7F±-I= (J.1l'nj2fiE FQ)x(1-x) [Vab=FGa(gc 1)ja ±Gt,(gb-1)jb]2, (31) where Vab= Va-Vb. We note that in the analogous treatment for a pure metal below the Curie tempera ture leading to expression (18), the two relaxation times are equal, because in that case there is no Cou lomb scattering; in fact, (31) shows that if Vab=O, TF+=TF-. On the basis of (31) one obtains two possible expres sions for the disorder resistivity at T=O, depending on the relative values of the mean-free path of the con duction electrons ~% the size of the ferromagnetic [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Sun, 21 Dec 2014 05:08:14912 A. J. DEKKER regions L, and the spin depolarization length of the conduction electrons X •. A rough estimate of X. indi cates that this quantity is probably of the order of a millimeter and we shall assume therefore that X.»XF and X.» L. In that case one obtains for XF«L 37Nnx(1-x) [VabL {Ga(ga-1)ja-~(gb-1)jbI2J2 Pd~ 2e2hEFO Vab2+{Ga(ga-1)ja-~(gb-1)jbI2· (32) This expression results from (31) if one adds the con ductivities of the + and -electrons. However, if XF» L, one should average the scattering cross sections for the two types of electrons and one obtains 37Nnx(1-x) . . PdO= 2e2hEFO [Vab2+ {Ga(ga-1)Ja-~(gb-1)Jb}2J. (33) It should be remarked that expression (33) is also ob tained for an antifertomagnetic ground state. The dilemma presented by the choice between (32) and (33) also arises in the band theory of ferromagnetic metals62; in our case, however, the difference between (32) and (33) is a consequence of the interference between Coulomb and exchange scattering without spin flip. In fact, if Vab=O, (32) and (33) are identical. We now compare these results with experiment. In the first place, the data given by Smidt and Daane33 and Hennephof48 for the alloys containing only one magnetic component satisfy the predicted concentra tion dependence and one finds from the experiments 244x(1-x) +lOlx for GdxLul_" Pd<>o= 164x(1-x) +89x for Tb"Lul_", (34) 125x(1-x) +111x for Gd"Y1_" 324x(1-x) for Gd"Lul_" PdO= 264x(1-x) for Tb"Lul_". (35) 169x(1-x) for Gd"Y1_" The resistivities are expressed in }.IO-cm. Note that the observed values for Vab2 in (34) increase with increasing difference in the metallic radii of the components, as expected. From the numerical values obtained for Vab2 and G2 from the high-temperature data, one can calcu late the disorder resistivity at T=O by employing (32) or (33). Compared with the experimental values of the coefficients of x( 1-x) in (35), expression (32) gives 87, 34, 7, whereas (33) gives 323, 240, 211, re spectively. It is evident that the "small cluster" formula (33) gives the best agreement in all three cases. The rather large difference between theory and experiment for Gd"Yl-z may be due to the fact that the data cover 62 See, for example, J. M. Ziman, Electrons and Phonons (Clar endon Press, Oxford, England, 1960), p. 379. only the region 0.7 <x<1. A similar type of analysis applied to the data of Smidt and Daane33 for the alloys Gd-Er, containing two magnetic components, also shows good agreement with (33). If on the basis of the experimental evidence given above, we assume that Pd~ is given by (33), the "spin disorder resistivity" is given by Pd<>o-PdO= 2e:;;FO[xGa2(ga-1)2ja(ja+1) +(1-x)~2(gb-l)2jb(jb+1) -x(1-x) X {Ga(ga-1)ja-~(gb-1)jb)2]. (36) Note that Vab2 does not appear in this expression; it would be present if (32) had been used for Pd~. This formula differs from those employed by Weiss and Marotta63 and by Smidt and Daane33; these authors simply modified expression (13), derived for a pure metal, in order to obtain an expression for the spin disorder resistivity of an alloy. One can show61 that the procedure followed by Smidt and Daane is only equiva lent with (36) in the case of a binary alloy containing one magnetic heavy rare-earth component. CONCLUSION It appears that a scalar interaction between the spins of the conduction electrons and the localized atomic spins can explain semiquantitatively the most pronounced features of the resistivity of the pure heavy rare-earth metals. At low temperatures there is still some uncertainty concerning the relative im portance of various factors. It is evident that the spin wave spectrum will have a strong influence on the temperature dependence of the spin-disorder resistivity and further reliable experimental iI).formation on the electrical and magnetic properties in this region would be desirable. The influence of impurities should also be considered, since the residual resistivity of the "pure" metals is still of the order of some }.IO-cm. Efforts should be directed towards a reliable separation of the phonon resistivity from the total resistivity in this region. Finally, the whole problem of scattering of electrons by helicoidal spin arrangements and the influence of subzone boundaries produced by this kind of magnetic order deserves further study. In the field of dilute magnetic alloys, further accurate experimental information about the electrical and magnetic properties is desirable for a conclusive quan titative comparison between theory and experiment. The Nordheim approximation applied to binary rare-earth alloys has produced a paradox for the dis order resistivity at He temperatures which needs clarification. One would expect that magnetization of a sample at low temperatures would produce a strong reduction of the resistivity [determined by the differ ence between (33) and (32)]. 6a R. J. Weiss and A. S. Marotta, J. Phys. Chern. Solids 9,302 (1959) . [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Sun, 21 Dec 2014 05:08:14
1.1736062.pdf	Current Flow across Grain Boundaries in nType Germanium. I R. K. Mueller Citation: Journal of Applied Physics 32, 635 (1961); doi: 10.1063/1.1736062 View online: http://dx.doi.org/10.1063/1.1736062 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Diffusion of n-type dopants in germanium Appl. Phys. Rev. 1, 011301 (2014); 10.1063/1.4838215 Impact of field-enhanced band-traps-band tunneling on the dark current generation in germanium p - i - n photodetector Appl. Phys. Lett. 94, 223515 (2009); 10.1063/1.3151913 Photoinduced current transient spectroscopy of deep defects in n-type ultrapure germanium J. Appl. Phys. 86, 940 (1999); 10.1063/1.370828 Growth, optical, and electron transport studies across isotype n-GaAs/n-Ge heterojunctions J. Vac. Sci. Technol. B 17, 1003 (1999); 10.1116/1.590684 Current Flow across Grain Boundaries in nType Germanium. II J. Appl. Phys. 32, 640 (1961); 10.1063/1.1736063 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Wed, 24 Dec 2014 01:32:20MAG ~ E TIC V I S COS I T Y DUE TO SOL UTE AT 0 M P A IRS. I I . 635 magnetic viscosity is present in a certain temperature range the material is also sensitive to annealing in a magnetic field. In fact, the energy H J. associated with the magnetic viscosity must be, under certain condi tions, about equal to the anisotropy energy induced by annealing in a magnetic field. Also, the presence of wasp-waisted loops in the Rayleigh region (for loops traced with stabilized walls) is additional evidence that a magnetic viscosity is present. All the results obtained are in agreement with the theory given in Part I of this paper,! that magnetic viscosity (wbether resulting from interstitials in bcc structures or to the rotation of pairs), wasp-waisted loops in the Rayleigh region, squaring of the loops by annealing in a magnetic field, are all different aspects of the same phcnomenon---the dilIusion either of pairs of atoms in the alloys or of interstitial atoms in the bcc structures. As for the quantitative correspondence between the anisotropy energy deduced from the viscosity field, and the induced anisotropy energy Ku evaluated from the changes in the magnetization curves, the first value is much lower than the second. However, the values are within one order of magnitude and the discrepancy can be attributed in part to the difference in temperature of the two measurements and to the fact that in the measurement of HI it is difficult to cover the whole relaxation spectrum. JOURNAL OF APPLIED PHYSICS VOLUME 32. NUMBER 4 APRIL. 1961 Current Flow across Grain Boundaries in n-Type Germanium. I R. K. MUELLER Mechanical Division, General Jfiils, Inc., Minneapolis, Jfinnesota (Received September 6, 1960) A theory of the current flow across grain boundaries in n-type germanium is given. In the temperature range where carrier generation in the space charge region can be neglected and for donor concentrations in the bulk larger than l014/cm3, the current is carried essentially by electrons crossing the barrier, the zero bias conductance is independent of the donor concentration and is given by GQ=2.2·108Te-~o/kT. The apparent activation energy "'0 is directly related to the barrier height. The current for applied voltages which are large compared to kT!q fails to saturate. The deviation for saturation is related to the density of states in the boundary band. At sufficiently low temperatures the carrier generation in the space-charge region is the rate-determining process for the current flow across the boundary. 1. INTRODUCTION THE current flow across grain boundaries in n-type germanium was discussed several years ago by Taylor, Odell, and Fan.l The voltage-current relation derived in their paper does not agree with experimental results obtained in this Laboratory for a large number of carefully oriented bicrystals with a wide range of bound ary angles and impurity content which will be discussed in a following paper. It is therefore desirable to reex amine the theory of the current flow across grain bound aries. Taylor, Odell, and Fan treated the negatively charged boundary as a mathematical plane and assumed that the electron density is continuous across this bound ary. In the following model the capture rate of the boundary for electrons is taken into account, image force effects are considered, and the boundary is of finite width; the current to and across the boundary region is treated by analogy to thermionic emission, a concept which was first applied to semiconductor barriers by Torrey and Whitmer.2 For sufficiently low temperatures the current across the boundary is essentially determined by the 1 W. E. Taylor, N. H. Odell, and H. Y. Fan, Phys. Rev. 88, 867 (1952). 2 H. C. Torrey and C. A. Whitmer, Crvstal Rectifiers (McGraw Hill Book Company, Inc., New York, 1948). carrier generation and annihilation in the space-charge region. A description of the boundary model used and a dis cussion of the assumptions made precedes the analysis of the current flow across the boundary. 2. POTENTIAL BARRIER The shape of the potential barrier around the bound~ ary is determined by the nature of the surface states connected with the boundary. It is assumed that the surface states are distributed homogeneously over a plane boundary which restricts our results to bound aries with sufficiently high misfit angles. No detailed knowledge of the level structure of surface states is at present available; there are, however, experimental data which limit the choice of a boundary model. It is the purpose of this section to specify the characteristics of the barrier which enter into the calculations of the current flow across the boundary and to show that the assumptions made are compatible with experimental evidence. The observed conductance along grain boundaries3--5 3 A. G. Tweet, Phys. Rev. 99, 1182 (1955). 4 R. K. Mueller, J. Phys. Chern. Solids 8, 157 (1959). I) B. Reed, O. A. Weinreich, and H. F. Matare, Phys. Rev. 113, 454 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Wed, 24 Dec 2014 01:32:20636 R. K. MUELLER BOUNDARY REGION CONDUCTION BAND VALENCE BAND FIG. 1. Grain boundary barrier at equilibrium. which is practically temperature independent down to liquid helium temperatures6 implies a partially filled band of surface states. This band may, as suggested by Shockley,7 lay somewhere in the forbidden energy gap, or may be formed as suggested by Handler and Portnoy8 by a band of surface states perturbed out of the con duction band which overlaps with the valence band. We define the boundary region as the region into which the wave functions of the electrons and holes in the surface states extend appreciably. In Handler and Portnoy's modelS this would be the extension of the wave functions in the two-dimensional valence band corresponding to the lowest trough level. This is about 30 A for typical field values. No specific assumptions about the shape of the electronic potential in this region are necessary for our consideration. The boundary region contains a net negative charge which is compensated by the positive charge in the surrounding space-charge region. The observed net negative charge in the boundary region and the anoma lous Hall effect which indicates predominantly positive carriers for the sheet conductance 2 implies that the barrier region contains a large number of positive and negative carriers free to move parallel to the boundary. An electron which approaches the boundary region is therefore not only under the influence of the repulsive force due to the net negative charge on the boundary, but if it approaches the boundary within one Debye radius, it is also under the influence of an attractive image force which originates from the polarization of the boundary region. This image force9 reduces the po tential barrier and gives rise to two maxima in the electron potential outside the boundary region. If we assume, in order to estimate the position of these maxima, a linear potential outside the barrier region with an electric field F of about 106 v/cm, one finds for the distance of the potential maxima from the 6 H. F. Matare, J. App!. Phys. 30, 581 (1959). 7 W. Shockely, Phys. Rev. 91, 228 (1953). 8 P. Handler and W. M. Portnoy, Phys. Rev. 116, 516 (1959). 9 F. Seitz, The Modern Theory of Solids (McGraw-Hill Book Company, Inc., New York, 1940). boundary a value of the order of 100 A. The potential depression flrp against the maximum potential for a spatially fixed negative surface charge is given byI° (1) where q is the electronic charge and K is the dielectric constant. The existence of a relatively far-reaching attractive force around the boundary is indicated by the observed high capture rate for electrons crossing the boundary4 if one assumes a model for the capture process similar to that proposed by Lax for "giant traps."ll Since the barrier maxima occur about 100 A off the boundary, internal structures such as dislocation arrays12 with a spacing considerably smaller than 100 A should not affect the current flow appreciably. We define the "barrier height" rp (see Fig. 1) as (2) which is the energy difference measured from the Fermi energy EF to the edge of the conduction band Ecm at the barrier maxima. The temperature dependence of rp is assumed to be the same as the temperature depend ence of the energy gap, rp=rpo-cT (3) with c=4.4·1O-4cv;oKY This assumption is justified by the observed value of rp which is close to the gap energyI4.16 and the observed temperature independence of the boundary conductance.3-6 3. BARRIER AT EQUILIBRIUM The boundary represents a trapping site for elec trons.4 At equilibrium, owing to the principle of de tailed balancing, the number of electrons which are trapped per cm2 of boundary per second is equal to the number of electrons reemitted from the boundary per cm2/sec. In order to evaluate the rate at which electrons are trapped at the boundary we determine first the random current I r which crosses the barrier per unit area from either side under equlibrium conditions. If we neglect tunneling through the barrier, Iris given by I r= q(Nc/4)ve-¢/kT = q(N c/4)vec/k. e-¢olkT, (4) which is the well-known Richardson equation for thermionic emission. Nc=2.1015T~(cm-3)13 is the effec tive number of states in the conduction band. v= (8kT/Trm)i is the average thermal velocity, which 10 W. Shottky, Z. Physik 18, 63 (1923). 11 M. Lax, J. Phys. Chern. Solids 8, 66 (1959). 12 F. C. Frank, Pittsburgh Rept.; p. 150 (1950). Office of Naval Research (NAVEXOS-P-834). 13 H. Brooks, Advances in Electronics and Electron Phys. 7, 120 (1955). 14 R. K. Mueller, Rept. on Eighteenth Ann. Con£. Phys. Elec tronics, M.I.T. (1958). 15 R. K. Mueller, J. Appl. Phys. 30, 546 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Wed, 24 Dec 2014 01:32:20CURRENT FLOW IN n-TYPE GERMANIUM. I 637 becomes 1.4·106T! cm/sec if the effective mass m is assumed to be 0.2 electron masses. The barrier pene tration by tunneling can be accounted for by a correc tion factor16 in Eq. (4). For the temperature range con sidered here and for electrical fields of about 105 v/cm which are typical for grain boundary barriers, this factor is close to unity and shall be neglected. A certain fraction r of the current I T is reflected from the boundary region, and a certain fraction "I from the net current (l-r)I T is captured at the boundary. The reflection can be assumed to be small, and we shall in the following neglect it. This is reasonable in our ap proximation since the effect of the neglected barrier penetration by tunneling cannot be distinguished from a boundary region reflection and both factors tend to compensate each other. According to the principle of detailed balancing, the emission current I em from the boundary to either side at equilibrium is "II T' The coefficient "I has been de termined experimentally4 and ranges from 0.6 to 0.2 for different boundary angles. 4. CURRENT FLOW ACROSS BOUNDARY If a bias is applied between the two sides of the boundary, the barrier changes and a steady state is reached if the net rate U e of electrons captured at the boundary is equal to the net rate U h of holes captured in the potential well which the boundary barrier repre sents for the holes, i.e., when the charge in the boundary region and surrounding hole inversion layer is sta tionary. This condition, (5) cp+=Eem+- j+ cp-=Eem--j-CPB+= Eem +-jB CPB-=E em--jB (8) which describe the potential barrier for electrons from the positively and negatively biased sides of the sample to the boundary and from the boundary to the posi tively and negatively biased sides of the sample (see Fig. 2). Eem + and Eem -are the edges of the conduction band at the barrier maximum on the positively and negatively biased sides. Eem + and Eem -are different because under applied voltage the electrical field at the barrier top and therefore the image force depression is different on the positively and negatively biased sides. For moderate applied biases, however, this effect is small and we shall neglect it for the present considera tions. We neglect also the variation of the Fermi level in the boundary band, and discuss the influence of both effects in Sec. 5. With this approximation and in view of Eqs. (2), (6), and (7), we find CPB+=CPB-=CP cp+=cp+q(V -~V) cP-=cP-q~V. (9) The first of Eqs. (9) implies that the emISSIon current from the boundary is unaffected by the applied bias: Iem +=Iem -=Iem="II T' (10) The other two give for electron currents IT+ and Ir- from the positively and negatively biased sides to the boundary in view of Eq. (4): allows one to determine the difference between the IT+=ITe-q(V-t:.V)!kT quasi-Fermi levels in bulk and boundary states under IT-=ITeqt:.V!kT. (11) applied bias and therefrom the current across the boundary. The net rate of electron capture U e is We define three quasi-Fermi levels j-, j+, and jB. U (+ ) (+ ) (12) j d j+ q .=1' IT +IT--Iem +Iem- . -an are the Fermi energies in the bulk on the positively and negatively biased sides of the boundary, several hole diffusion lengths away from the boundary, where the material can be considered practically at equilibrium. jB is the quasi-Fermi level of the electrons in the boundary states. The difference between j-and j+ is proportional to the applied voltage V,17 4>- the difference j--j+=qV; j--jB=q~V (6) (7) has to be determined from the stationary-state condi tion Eq. (5). In order to describe the steady-state condition, we introduce the four quantities 16 A. Sommerfeld and H. Bethe, Handbudz der Physik (Springer Verlag, Berlin, 1933), 2nd ed., Vol. 24a. 17 W. Shockley, Electron and Holes in Semiconductors (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1950). f V VALENCE BAND FIG. 2. Grain~boundary barrier~under bias. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Wed, 24 Dec 2014 01:32:20638 R. K. MUELLER On introducing Eqs. (10) and (11), one finds gU e= 'YI r[eqAV/kT (e-qV/kT + 1) -2]. (13) The implicitly assumed voltage independence of the capture rate 'Y is consistent with the approximation Ecm+=Ecm- made previously. For the evaluation of the net rate of hole capture we neglect the carrier generation in the space-charge region. the temperature range where this is permissible is dis cussed in Sec. 6. Under this condition the hole current h+ from the positively biased side to the inversion layer and the hole current h-from the inversion layer to the negatively biased side can be determined in the same manner as for biased p-n junctions18 yielding I h+= I.(1_e- Q(V-AV)/kT) I h-= I. (eQAV/kT -1), which give for the net rate of hole capture U h : gUh=I h--Ih+=I.,[eqAv/kT(e-qv/kT + 1)-2J, (14) (15) where I. is the hole saturation current per unit area. Equations (13) and (15) show that the net rates of electron and hole capture are, under the assumption made, proportional to the same function of LiV and V. Since I. and 'YI r are voltage independent and unequal, the stationary-state condition, Eq. (5), requires that both capture rates vanish individually. This gives for Li V the relation (16) The total current across the boundary consists of an electron and a hole contribution. Since the net rates of hole and electron capture vanish individually, each contribution can be considered independently. The total electron current Ie across the boundary is Ir--(1-'Y)Ir+-Iem-. With Eqs. (10), (11), and (16), one finds Ie= (2-'Y)Ir tanhgV /2kT, and similarly for the total hole current, Ih=I. tanhgV/2kT. The total current across the boundary I = I e+ h is then I=2Ir(1-'Y/2+/3) tanhgV /2kT (17) with /3=Is/2Ir. The ratio /3 measures the relative im portance of the hole vs the electron contribution. With 200 J.lsec for the bulk lifetime and with the observed value of 0.71 ev for cpo given in Part II, (3 becomes 1.4 .1013/N d at room temperature. The hole contribution to the current across the boundary is therefore sig nificant only for very low-doped bicrystals. For suffi cien tly high-doped bicrystals (N d? 1014 cm-3), we can neglect /3 in Eq. (17), i.e., the total current across the boundary is carried essentially by electrons and the zero bias conductance Go becomes (1-'Y/2) (gIr/kT). If we introduce Ir from Eq. (4), we have Gn= (1-'Y/2)KTe-¢o/kT with K = (g2;Y cv/4k'J'2)eC/k= 2.2 .1OR(mho/ cm2°K). 18 W. Shockley, Bell System Tech. J. 28, 933 (1949). (18) 5. DEVIATION FROM SATURATION In Sec. 4 we have neglected the variation 0 is of the Fermi level in the boundary band and the variation ocp of the image force depression under applied bias. With these approximations we found that the current across the boundary Eq. (17) saturates for applied voltages large compared to kT/g with a saturation current Io=2Ir(1-'Y/2+/3). We shall now consider in a first order approximation the dependence of the current on OjB and ocp for applied biases large compared to kT/g. We limit our consideration here again to sufficiently high-doped samples and consider the electron contribu tion only. For applied biases large compared to kT / g, the voltage drop occurs essentially between the boundary and the positively biased side. The variation of the image force depression is therefore significant only on the positively biased side, and we have CPB+=cp-ois-ocp (19) CPB-=CP-OjB, which gives for the emission currents from the boundary in a first-order approximation Iem += I emf 1 +[( OjB+ OcJ»/k TJ} Iem -=Iem[1 + (ois/kT)J, where I em is the equilibrium value 'Y I r. (20) Since the current I r+ from the positively biased side to the boundary decreases exponentially with applied bias, we can neglect it in the present consideration. The total current I across the boundary is therefore Ir--Iem-. We determine Ir-from the steady-state condition U e= 0 which gives, according to Eq. (12), (21) where 1 is the capture rate under applied bias. If one assumes a capture process by single phonon interaction into shallow bound statesll (created by the image force), then the deviation of 1 from the equilibrium capture rate 'Y depends only on ocp and not on oj B: 1= 'Y(1-aocp/kT). (22) The assumed capture process implies that the low energy electrons are more readily captured than the high-energy electrons, and one can derive an upper limit for the parameter a by assuming that the capture rate for electrons with a kinetic energy at the barrier maximum between zero and ocp is unity for the un disturbed boundary. This leads directly to the rela tion 'Y~ocp/kT+1(1--ocp/kT) and consequently to O~a~ (l-'Y)h. With Eqs. (20)-(22) we find for the total current I 1= (2-'Y)Ir(1+ojB/kT+ jocp/kT). (23) The factor j is of the order of unity limited between 1/(2-'Y)~j~1h· [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Wed, 24 Dec 2014 01:32:20CURRENT FLOW IN n-TYPE GERMANIUM. 639 We define a saturation region conductance G.at as bias, and we find (I -Io)/V and introduce the dimensionless parameter ( / ) V oF~ qNd KF . (30) p=Gsat/GO=2(OjB/qV+ joc/>/qV). (24) Both OjB and oc/> depend on the variation of the elec trical field of on the positively biased side of the boundary. We shall now express both quantities in terms of of. The change of charge per cm2 oQ inside the barrier region due to a field variation of at the barrier top is KoF. Assuming complete degeneracy in the boundary band,8 we have on the other hand OQ=q~YBOjB, and therefore OjB= (K/qNB)oF, (25) where N B is the density of states in the boundary band per cm2/ev. The dependence Of the image force depression on the field strength F is given in Eq. (1). From there it follows for sufficiently small of (26) where F is the field at equilibrium. To obtain a relation between of and the applied voltage, we follow the calculation for the potential dis tribution in a space charge layer given by Kingston and Neustadter.19 It is assumed in their calculations that the distribution of states in the valence band is not affected by the presence of the high field in the space-charge region. This is, as Handler and Portnoy8 pointed out, not the case at least in the immediate neighborhood of the boundary. It can be expected, however, that the field around the barrier maxima which are about 100 A off the boundary in our model can be determined in a reasonable approximation by assuming the undisturbed distribution of states in the valence band. A straightforward calculation gives for the field F(V) as function of the applied bias F(V)= !~:T (np+N/(~k~V)) r, (27) where np is the hole concentration around the barrier maxima, V the applied voltage, and ~ the "built in potential" which is related to c/>, defined in Eq. (2), by ~=c/>/q- (kT/q) In (NclNd). (28) For applied voltages (29) the field varies practically linearly with the applied 19 R. H. Kingston and S. F. Neustadter, J. App!. Phys. 26, 210 (1955). On combining Eqs. (24)-(26) and (30), we find for p in the voltage range defined in Eq. (29) p=Nd{ (2/qN BF)+ j(q/KF)!}. (31) 6. GENERATION AND ANNIHILATION OF CARRIERS IN SPACE-CHARGE REGION In order to evaluate the temperature range for which the foregoing analysis is valid, we have to consider the effects of carrier generation and annihilation in the space-charge region. Since this process has an activation energy of the order of one-half the gap energy,20 whereas the current flow described by Eq. (18) has the activation energy c/>o which is dose to the gap energy, there exists a temperature range T< Tc for which the current due to space-charge generation dominates the current across the boundary. The space-charge generated current in p-n junctions has been treated comprehensively by Sah, Noyce and Shockley21; we can limit our discussion to the modifica tions necessary to adapt the results described in their paper to the grain boundary case. If a bias V is applied across the boundary, the p-type inversion layer around the boundary is biased in the forward direction against the negative side of the bulk with a bias ~ V and biased in the reverse direction with a bias (V -~ V) against the positively biased side. ~ V is defined as in Eq. (12) as jB-j-. The steady-state con dition implies that the corresponding currents I-(~ V) and 1+ (V -~ V) are equal. Equating these two currents gives a rather complex equation for ~ V which reduces for biases small compared to kT / q to ~ V = V /2. This leads to a zero bias conductance G8P equal to one-half the zero bias conductance of a p-n junction. On using the maximum value determined by Sah et at., which occurs for trapping centers with energy levels midway in the forbidden energy gap, one gets (32) where ni is the intrinsic carrier density; W the width of the depletion region; TO= (TpOTno)!, the mean of the limiting values of the lifetime of holes (T pO) and elec trons (T no) in strongly n-and p-type material. The temperature Tc for which Gsp is equal to Go defined in Eq. (1) depends on the donor density, the lifetime TO, and the barrier height c/>o. For typical values, TO~ 10-6 sec, N d= 1015 cm-3, and c/>0=0.71 ev, one finds Tc~200°K. 20 E. M. Pell, J. App!. Phys. 26, 658 (1955). 21 C. Sah, R. N. Noyce, and W. Shockley,~Proc. LR.E. 45, 1228 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Wed, 24 Dec 2014 01:32:20
1.1728409.pdf	Peltier Coefficient at High Current Levels John R. Reitz Citation: Journal of Applied Physics 32, 1623 (1961); doi: 10.1063/1.1728409 View online: http://dx.doi.org/10.1063/1.1728409 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The central role of the Peltier coefficient in thermoelectric cooling J. Appl. Phys. 115, 123517 (2014); 10.1063/1.4869776 Peltier current lead experiment and their applications for superconducting magnets Rev. Sci. Instrum. 75, 207 (2004); 10.1063/1.1633987 Apparatus for Measurement of Peltier Coefficients Rev. Sci. Instrum. 35, 1302 (1964); 10.1063/1.1718729 Experiments with Peltier Junctions Pulsed with High Transient Currents J. Appl. Phys. 34, 1806 (1963); 10.1063/1.1702684 Peltier Coefficient at a SolidLiquid Interface J. Appl. Phys. 31, 1690 (1960); 10.1063/1.1735923 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Fri, 19 Dec 2014 17:15:27LETTERS TO THE EDITOR 1623 In this experiment a 92-g sample of Pechiney graphite, having a density of 1.75±0.04 g/cm3, was impregnated by a solution impregnation technique with 1.61 g of uranium-235 (0.0332 g/cm3) in the form of UO •. The resulting UO. particle size is so small that virtually all of the fission products recoil into the graphite matrix. This sample, 25 mm in diameter and 100 mm long, was sealed in an aluminum container and had a thermocouple located at its center. It was irradiated to 1.3XlOl8 nvt (thermal) in a water-cooled loop in the BRI reactor at a nominal tempera ture of 32°C (maximum temperature 36°C). Post-irradiation calorimetric measurements indicated that 31.2±3.2 cal/g of stored energy had accumulated during exposure to 3 MWD/T. Neutron bombardment alone would have yielded approximately one calorie per gram of stored energy. For the calorimetric tests we employed the dipping technique,' wherein the sample is inserted into a massive furnace-calorimeter consisting of a copper cylinder, which provides a uniform axial surface temperature, surrounded by thermal insulation. Calorim eter measurements were made (1) before irradiation (for reference), (2) after irradiation to measure Wigner energy release, and (3) twice after the energy release to examine the reproduci bility of the experimental procedure. The furnace was maintained at its reference temperature 200°C, for several hours before the test was conducted. Among the four calorimeter tests, only the Wigner energy release measurement produced a result that was substantially different from the others, and in this case the maxi mum sample temperature exceeded the furnace temperature by 11SC. The stored energy in neutron bombarded material has been attributed to the trapping of atoms carbon displaced from their normal lattice positions, and one MWD/T of exposure results in 0.45 call g of energy storage, or 5 X 10-0 displaced atoms per carbon atom.l Thus one can estimate that each fission event produced 30000 displaced carbon atoms. This larger energy storage could also have resulted from (1) a larger number of displaced atoms; (2) a different process for energy storage because of the higher density of damage in the case of fission fragments, or (3) a different type of damage that is a consequence of the high charge of fission fragments. The energy storage process is currently being examined further. The results of the present and the further tests will be described more completely in another report. I G. R. Henning and J. E. Hove, Proc. Geneva ConI. 7, 666 (1955). • H. M. Finniston and J. P. Howe, Prog. Nuclear Energy 5, Ser. 2, 551 (1959). • R. J. Harrison, USAEC Report ORNL-I722 (1954). 4J. C. Ben and J. H. W. Simmons, USAEC Report TID·7565 (Part 1) (1959), p. 83. Peltier Coefficient at High Current Levels* JOHN R. REITZ Case Institute of Technology, Cleveland, Ohio (Received April 10, 1961) THE optimum thermoelectric materials are extrinsic semi conductors with carrier concentrationsl in the range 1019 electrons (or holes) per cm3• To maximize the performance of a thermocouple, one leg is made of p-type material, the other of n-type material. The junction between the two legs thus assumes the character of a p-n junction and is shown schematically in Fig. 1. To the left of the junction the current is carried primarily by holes, whereas to the right it is carried primarily by electrons. For the direction of current shown, hole-electron pairs must be thermally created, and the junction is a heat-absorbing junction. The thermal energy required to make a hole-electron pair is just the vertical projection of the transition arrow shown in the figure, and this is seen to be in accord with the usual definition of the Peltier coefficient' where s= (lie) (Ak-r/T). (1) (2) 1 c: o ~ -C) Q) Q) p-type • ~J • Fermi level n -type FIG. 1. An isothermal i unction between p-type and n-type semiconductors at low current density. This is a heat-absorbing iunction for direction of current shown. Here S is the Seebeck coefficient, T is the absolute temperature of the junction, k is Boltzmann's constant, and I (the chemical potential) measures the position of the Fermi level relative to the carrier band edge. A is a number which depends upon the precise mechanism of charge carrier scattering (A = 2 for scattering by acoustical phonons). The quantity AkT may be regarded as the average thermal energy of charge carriers emitted from the junction. Of course the transition shown in Fig. 1 does not occur; the junction is too wide. Electron-hole pairs are created by vertical transitions across the full energy gap after which the created carriers diffuse and drift in the isothermal junction region. The net thermal energy absorbed per pair is the same as indicated, but the transport of charge through the junction is impeded by the rectifying contact. The situation is improved by replacing the junction with a metal weld as shown in Fig. 2, but a single metal cannot in general eliminate the rectifying contacts at both surfaces. In Fig. 3 we show the establishment of ohmic contacts3 at both surfaces by using a low work function metal in contact with the n-type material and a high work function metal in contact with the p-type materiaL The importance of an ohmic contact for many semiconductor applications is well known, but its importance for high-current Peltier junctions has apparently not received adequate discussion. At appreciable current levels a rectifying contact is accompanied by a voltage drop across the contact; this may be interpreted in terms of a contact resistance, but actually it is a reduction in --'. electrons FIG. 2. An isothermal semiconductor-metal-semiconductor iunction at high current density. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Fri, 19 Dec 2014 17:15:271624 LETTERS TO THE EDITOR -J p-type -electrons ••• FIG. 3. An ohmic junction at low or high current density. Metal! has a low work function and metal 2 has a high work function. Peltier coefficient of the heat-absorbing junction and an increase in Peltier coefficient of the heat-producing junction. This state ment follows from an examination of Fig. 2 where, as usual, the voltage drop across the junction (A V) is depicted as a shift in Fermi level. The net thermal energy to create an electron-hole pair is thus reduced by an amount eA V. Note that at the heat absorbing junction, current flows through the rectifier in the "reverse" direction, but at the heat-producing junction its passage is in the "forward" direction. Thus the presence of a rectifying contact reduces the efficiency of the thermocouple as the current density is increased. Since current densities in power generating thermocouples may run as high as several hundred amperes per cm2, the effect appears to be important. The rectifying contact can probably not be eliminated com pletely in practical thermocouples. The choice of semiconductor for a thermoelectric application is controlled by its figure of merit, or materials parameter, and the choice of junction metal is limited by considerations of chemical compatibility and stability. The ohmic contact to the n-type material offers the most difficulty, particularly for high-temperature junctions. The harmful effects of the rectifying contact can be partially compensated, however, by changing the carrier concentration of the semiconductor. Thus, to gain back the appropriate Peltier coefficient at the heat absorbing junction, the doping level of the crystal should be decreased. The optimum concentration of charge carriers is therefore different at high current from its value at low current, and in some cases the difference would appear to be substantial. * Supported in part by the U. S. Atomic Energy Commission, and in part by the National Carbon Company, a division of Union Carbide Corporation. 1 C. Zener, Thermoelectricity, edited by P. H. Egli Uohn Wiley & Sons. Inc .. New York, 1960), p. 8. 'See, e.g., A. H. Wiison, The Theory of Metals (Cambridge University Press, New York, 1953), 2nd ed., p. 232. • A good discussion of ohmic contacts is given by L. V. Azaroff, Introduc tion to Solids (McGraw-Hill Book Company, Inc., New York, 1960), p. 343. Maser Action in Emerald F. E. GOODWIN Hughes Research Laboratories Malibu, California (Received March 23; in final form, May 24, 1961) THE paramagnetic resonance spectra of emerald(chromium doped beryl) have been reported by Geusic et al.' The stable physical properties and zero-field splitting at 53.6 kMc make this material especially attractive for use in solid-state-maser amplifiers in the millimeter-wave region. Bogle2 has shown that emerald may also be of importance at X band and L band because of favorable cross-relaxation effects. The spin Hamiltonian of emerald is identical to that for ruby, x=gtlS.B+D[ S.2_~]. except that the D factor is greater by 4.66. Therefore, the wealth of I I I ~ vp =5Bi4 KMC ~ I ~ O~------~------r\~------~ UJ ~ -26.81--------= o 2 HOC.KGAUSS FIG. 1. Energy-level diagram for emerald maser (e =90 deg). experimental and computed data available for ruby masers is useful in predicting the operation of emerald masers.3,4 This correspondence reports the successful operation of synthetic emeralds in a single-cavity reflection-type maser amplifier operating at 10 kMc. The c axis was oriented at 90 deg with respect to a magnetic field of 1900 gauss and a pump fre quency of 58.4 kMc was used (see Fig. 1). The particulars of operation are given in Table I. The expected magnetic Q(QM) can be approximated from QM= hAvm kTAvm 87r(p2) (n2-nl)F '" 7rNO(P2) (vp-2v.)F· For the values given in Table I, QM~180, which is in approximate agreement with the experiment value: QM ~ 2vm/G!B ~ 160. The cavity, with dimensions of 0.140XO.280XO.280 in., was constructed of copper and filled with three emerald slabs allowing a T E011 resonance. The c axis was in the plane of the crystal slabs and perpendicular to the magnetic field (see Fig. 2). Pump power was introduced into the cavity through the signal iris. Slabs of beryl were used as seeds on which to grow the emerald material. When the emerald had grown to a thickness of 0.050 in., the samples were removed. As is typical of early growth, these crystals exhibited a number of imperfections, as was evidenced by microscopic twinning and spontaneous nuclei. The filling factor of 80% was a result of these imperfections. The crystal TABLE I. Particulars of operation for the emerald maser. Signal frequency (P.) Pump frequency (pp) Magnetic field (Hdo) Orientation Hdo to c axis Temperature (T) Power gain (G) Bandwidth (D, 3 db) Voltage gain bandwidth Paramagnetic linewidth (L\.Pm) Filling factor (F) Magnetic Q (expected) Magnetic Q (measured) Concentration (ions/em') Average (squared) dipole moment of the maser transition (,,') 10.0 kMc 58.4 kMc 1900 gauss 90 degrees 4.2°K 16 db 20 Mc 126 Mc 300 Me 80% 180 160 2.5 X 101• 4 XI 0 -40 erg'/ gauss' axis exhibited a spread of 3 deg within the sample which caused a broadening of the paramagnetic resonance Iinewidth to 300 Mc for 8=90 deg and to 500 Mc for 8=55 deg, values five to eight times greater than that for ruby. The maser performance obtained indicates that the broadening of the spectral lines due to microscopic spreading of the c axis does not destroy the maser properties at 8=90 deg; however, preliminary attempts to achieve maser action at 9=55 deg were not successful. Crystals of a longer growth cycle are being synthe sized; it is expected that the later samples will be relatively free [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.251.14.35 On: Fri, 19 Dec 2014 17:15:27
1.1732608.pdf	Magnetic Susceptibility of the Cubic Sodium Tungsten Bronzes John D. Greiner, Howard R. Shanks, and Duane C. Wallace Citation: The Journal of Chemical Physics 36, 772 (1962); doi: 10.1063/1.1732608 View online: http://dx.doi.org/10.1063/1.1732608 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/36/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Low temperature magnetic susceptibility of uranium and rare earth tungsten oxide bronzes J. Chem. Phys. 61, 3920 (1974); 10.1063/1.1681684 Electrical Properties of Some Dilute Cubic Sodium Tungsten Bronzes J. Chem. Phys. 36, 87 (1962); 10.1063/1.1732323 Electrical Resistivity of Cubic Sodium Tungsten Bronze J. Chem. Phys. 35, 298 (1961); 10.1063/1.1731904 Sodium Diffusion in Sodium Tungsten Bronze J. Chem. Phys. 22, 266 (1954); 10.1063/1.1740049 The Magnetic Susceptibility of Tungsten Bronzes J. Chem. Phys. 18, 1296 (1950); 10.1063/1.1747930 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sat, 20 Dec 2014 18:20:55772 STERN, KAUDER, AND SPINDEL to be, within the estimated experimental error, inde pendent of the degree of complexing of the nitrate ion, as evidenced by the constancy of the measured fraction ation factors when the nitrate ion was in solutions in which the degree of complexing varied from -0 to ",75%. THE JOURNAL OF CHEMICAL PHYSICS ACKNOWLEDGMENTS We would like to thank Dr. E. U. Monse for his many helpful discussions and for his aid in preparing this manuscript. We would also like to thank Robert Green berger for his assistance in the laboratory. VOLUME 36, NUMBER 3 FEBRUARY I, 1962 Magnetic Susceptibility of the Cubic Sodium Tungsten Bronzes* JOHN D. GREINER, HOWARD R. SHANKS, AND DUANE C. WALLACEt Institute for Atomic Research and Departments of Physics and Chemistry, Iowa State University, Ames, Iowa (Received June 22, 1961) The sodium tungsten bronzes (Na."W0 3) in the cubic range 0.45<x<1.0 were found to have mass susceptibilities from 0.OO7X1o-e for x=0.49 to 0.053X1o-e for x=0.85. This feeble paramagnetism was found to be temperature independent from 70° to 3000K for three representative samples with x=0.49, 0.76, and 0.85. The susceptibility of WO, was determined to be -0.060X1o-e emu/g and is also tempera ture independent. Satisfactory agreement between calculated and observed susceptibilities was obtained with a model which assumes that the bronzes consist of a dispersion of sodium ions in a WO, lattice. The molar susceptibility, then, can be calculated from the equation XM=X (WO,) +xx (Na+) +x •. The term x. for the Pauli paramagnetism was obtained for two cases: (a) for nearly free electrons (m= 1.6m) , and (b) for the density of states taken from the literature data on the low temperature heat capacity of the bronzes. Best quantitative agreement was obtained between the calculated and the observed susceptibilities for case b and indicates a more rapid increase in the density of states than simply Et. INTRODUCTION THE cubic sodium tungsten bronzes have the chemical formula, Na.,wOa with 0.45<x<1.0, and crystallize with the perovskite structure.1 In the unit cell tungsten atoms are at the cube centers, oxygen atoms are at the face centers, and sodium atoms are distributed at the cube corners (when x= 1, all the corners will be occupied). These materials exhibit the metallic properties of luster and high electrical and thermal conductivities. The bronzes are of interest because the number of conduction electrons may be controlled through control of the sodium concentration. A number of investigations2-6 of the electronic proper ties of these materials have already been reported. These include a limited amount of data on the magnetic susceptibility. The present report describes a more complete investigation of the magnetic susceptibility of Contribution No. 1034. Work was performed in the Ames Laboratory of the U. S. Atomic Energy Commission. t Present address: Sandia Corporation, Albuquerque, New Mexico. 1 G.'Hiigg, Z. Physik. Chern. B29, 192 (1935). 2 L. D. Ellerbeck, H. R. Shanks, P. H. Sidles, and G. C. Daniel son, J. Chern. Phys. 35,298 (1961). I R. W. Vest, M. GrifIe1, and J. F. Smith, J. Chern. Phys. 28, 293 (1958). 'W. Gardner and G. C. Danielson, Phys. Rev. 93, 46 (1954). & F. Kupka and M. J. Sienko, J. Chern. Phys. 18, 1296 (1950). a P. M. Stubbin and D. P. Mellor, J. Roy. Soc. New South Willes 82,225 (1948). the cubic bronzes and proposes a model which accounts for the observed magnetic properties. EXPERIMENTAL PROCEDURE The sodium tungsten bronze samples used in this investigation were single crystals prepared by the electrolytic reduction of a fused mixture of sodium tungstate and tungsten trioxide. The sodium concen tration x was determined from x-ray measurements of the cubic lattice parameters ao through use of the relationshi p7 ao=0.0820x+3.7845 A. These crystals were shown to be electrically homogeneous by Ellerbeck et al.2 All of the bronzes which were measured were in the range of cubic symmetry and therefore isotropic in their magnetic behavior. The W03 measurements were made on powdered samples of purified tungstic acid anhydride (reagent grade) manufactured by the Fisher Scientific Company. The susceptibility measurements were made by the Faraday method.s Shaped pole pieces were used which provided a uniform force field that extended over some 10 cc thus making the force on samples placed within this region independent of sample geometry or position. The magnet current was supplied by an electronically regulated motor-generator which produced fields up to 7 B. W. Brown and E. Banks, J. Am. Chern. Soc. 76, 963 (1954). 8 See E. C. Stoner, Magnetism and Atomic Structure (E. P. p\ltton and Company, New York, 1926), p. 39. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sat, 20 Dec 2014 18:20:55MAG NET Ie SUS C E P T I B I LIT Y 0 F SOD I U M TUN G S TEN B RON Z E S 773 10 koe that were constant to better than t%. The constants of the apparatus were calibrated against pure water and triply distilled mercury and were checked periodically with a secondary platinum standard. The method of Honda and Owen9•10 was used to correct for ferromagnetic impurities, which in all cases were found to be less than 10 ppm. The force exerted on the sample by the field was measured on an enclosed analytical balance which permitted weighings to be made under vacuum. The restoring force and detection of the beam position were accomplished electrically from outside the system with a sensitivity of better than 0.02 dyne. Random errors in the force determinations were reduced by making two or more runs on each sample. The temperature measurements were made with the sample suspended freely in a copper tube that was surrounded by a vacuum-jacketed liquid-nitrogen cryostat. Heat transfer between the sample and the copper tube was provided by a helium exchange gas whose pressure was reduced at the time of measure ment to prevent convection currents. The temperature of the copper tube was lowered by transferring heat to TABLE I. Room temperature magnetic susceptibility of NazWO. and WOs. Mole Mass sus-Molar sus- Sample fraction of ceptibility, ceptibility, Sample mass sodium X XM No. (grams) (x) (emuXI0-6) (emu X 10-11) lUA 5.070 0.489 0.007 1.7 (T)-UIA 5.070 0.489 0.007 1.7 123A 2.274 0.596 0.014 3.4 124A 9.148 0.597 0.010 2.5 122A 2.280 0.640 0.026 6.4 184A 1.431 0.694 0.033 8.2 166A 1.935 0.764 0.031 7.7 (T)a 166A 1.935 0.764 0.030 7.5 21A 4.709 0.771 0.038 9.5 116A 4.660 0.793 0.047 11.8 125A 2.964 0.851 0.044 11.1 (H)b 125A 2.893 0.85 0.053 13.3 (T)a 125A 2.893 0.85 0.048 12.1 Vest d al. 4.652 0.887 0.039 9.8 WOa-l 1.645 -0.059 -13.7 WOa-2 2.325 -0.059 -13.7 (T)a WOa-33.063 -0.060 -13.9 • (T) Measurement made after sample returned to room temperature from liquid-nitrogen temperature. b (H) Measurement made after annealing the sample at 650°C for 24 hr. 9 K. Honda, Ann. Physik 32, 1048 (1910). 10 M. Owen, Ann. Physik 37,679 (1912). 0.08 0.06 0.04 jo.oz I ~ 0 ;<-0.02 -0D4 -0.06 -o.oe f- i50 " ~ I r ~ ~ 0 v I KlO 00 T I I2SA " .- 16610 iliA ~ - WO:.-3 ~ ..., I I -200 250 -~)() FIG. 1. The temperature dependence of the magnetic suscepti bility of NazWOa and WOa. the liquid-nitrogen reservoir through conduction along a cupro-nickel gradient tube. A noninductively wound manganin heater, placed between the sample and the gradient tube, was used to maintain the sample at any temperature up to 3oooK. A copper resistance thermometer monitored the heater i~put while a copper constantan thermocouple, located just below the sample, was used for temperature measurement. Thermocouple readings were made before removing the exchange gas to ensure thermal equilibrium. The reliability of this method of temperature measurement had previously been checked by comparing the tem perature measured with the apparatus thermocouple against a reference thermocouple placed in a dummy sample of mercury. Provisions were made for extending the low-temperature range to about 65°K by reducing the pressure over the liquid nitrogen with a high capacity pump. RESULTS The room-temperature data are given in Table I. The standard deviations obtained from least-squares treatment of the data gave a probable error for the mass susceptibility of the bronze samples of ±O.OO4X 10-6 emu/g. In the case of the WOg determinations, the introduction of a correction term for the Pyrex container raised this value to ±0.005XlO-6 emu/g. The temperature dependence of the susceptibility was measured on three representative bronze samples with x values of 0.85,0.76, and 0.49. The results, given in Fig. 1, show that the paramagnetism of each sample remains essentially constant over a wide temperature range. A small, reproducible increase with decreasing temperature was noted for sample l11A below 110oK, but investigation of this effect at temperatures lower than 69°K was not possible with the present cryostat . The diamagnetism of WOg was also measured from 300° to 107°K and was found to be invariant with temperature. The results of other investigations of the sodium tungsten bronzes and on WOg are given in Table II. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sat, 20 Dec 2014 18:20:55774 GREINER, SHANKS, AND WALLACE TABLE II. Results of other investigators. Mass susceptibility, X Investigator Specimen (emuX1~) Tilk and Klemm11 WOo -0.060 Conroy and Sienko12 WOo -0.090 Kupka and Sienk05 Nao.56<WO, 0.013 Nao.956WO. 0.057 Stubbin and Mellor Nao.6-o.7WO. 0.20- Nao.9l!WO. 0.43- NaQ.9W0 3 0.42" -Temperature independent from 300' to 4800K. The values of Kupka and Sienk05 and Tilk and Klemmll are in excellent agreement with the present investiga tion. In contrast, the magnitude of the values de termined by Stubbin and Mellor6 is greater by a factor of 10; however, the data of Stubbin and Mellor do corroborate that the susceptibilities are temperature independent. The mass susceptibility of WOs reported by Conroy and Sienko12 is -0.090X10-6 emu/g. Reference to their calibration check runs, however, gives values for H20 and NaCI that are approximately 0.02 X 1O-S emu/ g more diamagnetic than other reported values.1s.14 A correction of this amount applied to Conroy and Sienko's value would bring all the WOs data into agreement. A measurement was made on a sample of the original Nao.89WOS specimen used by Vest et al.s The value of the susceptibility at room temperature of this sample is included in Table I. CALCULATION OF SUSCEPTIBILITY The importance of the WOs octahedron as a basic unit in the structure of pure tungsten trioxide and of the alkali tungsten bronzes has been pointed out by Hagg and Magneli.15 For all of these materials the WOs lattice is composed of W06 octahedra which are bound together at the corners (oxygen atoms). The present model, then, considers the sodium tungsten bronzes (for 0.45 <x< 1.0) as a WOs lattice with sodium atoms randomly distributed on the interstitial (perovksite type) positions. [Note added in prooj. From a private communication with D. W. Lynch and R. G. Dorothy of Ames Laboratory, measurements of the optical properties of the cubic bronzes indicate 11 W. Tilk and W. Klemm, Z. anorg. u. allgem. Chern. 240, 355 (1939). J.ll L. E. Conroy and M. J. Sienko, J. Am. Chern. Soc. 74, 3520 (1952). Ia P. W. Selwood, Magnetochemistry (Interscience Publishers, Inc., New York, 1956), 2nd ed., p. 25. 14 W. Klemm, Z. anorg. u. allgem. Chern. 244, 391 (1940). 15 G. Hagg and A. Magneli, Revs. Pure and Appl. Chern. 4, 235 (1954). that the energy levels are changed very little from those in the insulator WOs]. It is assumed that the contribution of the WOslattice to the magnetic suscepti bility is independent of sodium concentration and is the same per mole as that of pure WOs. It is further assumed that in the bronze each sodium atom ionizes completely to contribute one nearly free electron to a conduction band; this is strongly suggested by the Hall effect measurements of Gardner and Danielson.4 On the basis of this model, the susceptibility per mole of Na",WOs should be independent of temperature and should be given by XM=X(WO S) +xx(Na+) +X.. (1) Figure 1 shows that the measured susceptibility values are independent of temperature over the range studied (from 69° to 3000K). From the present meas urements, X (WOs) = -13.9± 1.2 X 10-6 emu/mole. Ac cording to Brindley and Hoare,16 the susceptibility of Na+ is -6.1X10-s emu/mole. The susceptibility of the conduction electrons x. is computed below for two different cases. A. Case of Nearly Free Electrons If the effect of the lattice potential is taken into account by replacing the electron mass m by an effective mass m, the total density of electronic states g at the Fermi energy r is given by17 (2) 24,----,-.,.-,--,.--y----,--.- .... -::r---. 20 -4 -8 -160.\"i.O;--"~t;_-;;~_t.:--+.-+._--};-__:~~::__.....!1O ~ IlaWOs FIG. 2. Comparison between the results of the theoretical calculations and the measured room temperature susceptibilities for the sodium tungsten bronzes. 16 G. W. Brindley and F. E. Hoare, Proc. Phys. Soc. (London) 49,619 (1937). 17 See for example A. H. Wilson, The Theory of Metols (Cam bridge University Press, Cambridge, England, 1954), 2nd ed. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sat, 20 Dec 2014 18:20:55MAGNETIC SUSCEPTIBILITY OF SODIUM TUNGSTEN BRONZES 775 where h is Planck's constant and where g(r) is the number of states per mole per unit energy if n is the number of electrons per unit volume and V is the molar volume. For Na"WO a, (3) where ll() is the cubic lattice parameter and hence aoa is the volume of one unit cell. The molar susceptibility for this case is17 (4) where p. is the Bohr magneton and the second factor in brackets is just the correction for the diamagnetism of the conduction electrons. A reasonable fit to the experimental results can be obtained if m* is taken to be 1.6m. With m* = 1.6m, the susceptibility of the bronzes as computed from (1) and (4), with the help of (2) and (3), is shown by the dashed line in Fig. 2. B. Computation from Electronic Specific Heat Vest et al.s have measured the electronic specific heat, in the temperature range 10 to 4 OK, of five samples of Na"WOs (x values were 0.56, 0.65, 0.73, 0.81 and 0.89). The electronic specific heat C. was proportional to the temperature T for each sample: C.='Y(x) T per mole. (5) If the exchange and correlation forces between the conduction electrons are neglected, then 'Y is related to the density of states at the Fermi energy by17 (6) where k is the Boltzmann constant, and it is understood that g(r) depends on x. Now the susceptibility of the conduction electrons can be represented by (7) where Xp is the spin paramagnetism and Xd is the dia magnetic contribution. To zeroth order in kT/r the spin paramagnetism is17 (8) Thus XP was calculated from (8) for each of the five samples of Vest et al.3 with the aid of the measured values of 'Y(x) and the use of (6). The diamagnetic contribution was more difficult to compute. For this part, two assumptions were made, namely: (i) The electronic energy E was considered to be a monotonic increasing function of the magnitude of the electronic wave vector k; that is E=E(k), where k = I k I. This assumption is much less restrictive than the nearly free electron approximation, where E is proportional to k2• • (ii) It was further assumed that E(k) is independent of sodium concentration. Thus the "shape" of the conduction band was considered to be independent of x, with more conduction band states being occupied as more sodium atoms are added to the crystal. The use of these two assumptions is supported in large part by the fact that the resulting Xd was found to be never greater in magnitude than 8% of the XP for the corre sponding sample. Thus, the smallness of Xd justifies an approximate calculation of this contribution. Wilson18 has given for the most important contribu tion to the molar diamagnetism of the conduction electrons e2V f[a2 E a2 E (a2 E )21~jo Xd= 127rh2c2 ak,,2 akl-ak"aky jaEdk, (9) where the magnetic field has been taken in the kz direction, and where e is the electronic charge, c is the velocity of light, and jo is the Fermi distribution function. The integral in (9) is over k space, but the major contribution is at energies near the Fermi energy due to the factor ajo/aE. For the present case E is a function only of k= (ki+kl+k z2)i, and the integration can be done. The result to zeroth order in kT/r is -e2V(dE d2E) Xd= 9h2c2 dk +2k dk2 \' (10) Now since E increases monotonically with k [by assumption (i)], at T=O all states with k5:ko are occupied and all others are unoccupied, where ko = k (0. Thus x can be related to ko by the density of states in k space [which is 2/(211-)3, where the 2 is included to account for the spin degeneracy]: n=x/aos=[2/(21f)3]t1rk o3=kN31f2• (11) In differentiating this equation, the x dependence of ll() can be neglected to good approximation (this has been verified numerically by carrying out the calcula tions helow without introducing this simplification). Thus (12) It is now convenient to transform the derivatives of E with respect to k, which appeared in (10), to derivatives of r with respect to x. This can be done by observing that (13) or (14) After some manipUlation there results, with the aid of (12), (15) l8A. H. Wilson, Proc. Cambridge Phil. Soc. 49, 292 (1953). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sat, 20 Dec 2014 18:20:55776 GREINER, SHANKS, AND WALLACE Now the density of states per mole per unit energy range is defined as19 2 f dS g(t) = Nao3 (21r) 3 (dE/dk)' (16) where N is Avogadro's number and where the integral is over the Fermi surface. For the present case, (dE/dk) is constant over a surface of constant energy, and so N (dE/ dk )r(dko/ dx)' (17) where the last equality follows with the help of (12). There results finally, with reference to (14), d~/dx=N /g(r); (18) d2~ d[1] dx2=Ndx g(~) . (19) A graph of l/g(r) vs x was prepared from the electronic specific heat data. This curve was dif ferentiated graphically to give d2~/dx2, by (19), and Xd was then calculated from (15). The total susceptibility was then calculated from (1) for the five samples of Vest et al.3 and the results are shown in Fig. 2. The reported limits of error on the electronic specific heat measurements were used to obtain limits of error on the calculated electronic susceptibility. These calculated limits when added to the limits of error in the present meas~rement of the susceptibility of tungsten trioxide, gave limits for the susceptibility as calculated from t~e electronic specific heat. These limits are also shown III Fig. 2. DISCUSSION The results of the calculations are compared with the experimental values in Fig. 2. The two room temperature values of Kupka and Sienk05 are also shown in the figure. Kupka and Sienko also gave susceptibility values which were calculated from a "free electron" model, but these values did not agree with their measurements. From the comparison between calculated and 19 See N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys (Clarendon Press, Oxford, England, 1936), p.85. measured susceptibilities, two conclusions can be drawn, namely: (i) The model of the preceding section gives a good quantitative account of the magnetic properties of the bronzes for 0.45 < x < 1.0 as regards both the x de pendence and the temperature dependence of the susceptibility. At the same time the model is quali tatively consistent with results reported for other electronic properties such as Hall effect and electronic specific heat. (ii) As regards the trend of increasing susceptibility with increasing x, the calculation based on the elec tronic specific heat appears to be in better agreement with the measured results than does the nearly free electron calculation. Since the slope of the measured susceptibility vs x curve is greater than that given by a nearly free electron calculation, then, on the basis of the present model, the density of states must increase faster with energy than the Ei of a nearly free electron band. It is seen in Fig. 2 that the two points of highest x, as calculated from the electronic specific heat, do not agree with experimental results as well as do the lower-x ones. According to the specific heat data, the density of states at the Fermi energy begins to increase rapidly in the region x=0.7S.20 This increase may result from the Fermi surface approaching a zone boundary (a spherical Fermi surface would touch the zone boundary at x=p) and hence departure from spherical constant energy surfaces may be expected for higher energies. Since the calculation does not consider nonspherical energy surfaces, this could be the reason why the last two points do not agree as well as the others. ACKNOWLEDGMENTS The authors wish to express their appreciation to Dr. J. F. Smith and Dr. G. C. Danielson for their encouragement and for many helpful suggestions. We would also like to thank Dr. J. M. Keller for reviewing this work and for helpful criticisms. Acknowledgment is also made to Miss Mary Beeler and Miss Joyce Schoenbeck for their assistance in the numerical analysis of the experimental data. 20 Additional information concerning the structure of Nao.nWO. can be found in the neutron diffraction work of M. Atoji and R. E. Rundle, J. Chern. Phys. 32, 627 (1960). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sat, 20 Dec 2014 18:20:55
1.1729602.pdf	Role of Oxygen in Reducing Silicon Contamination of GaAs during Crystal Growth J. F. Woods and N. G. Ainslie Citation: Journal of Applied Physics 34, 1469 (1963); doi: 10.1063/1.1729602 View online: http://dx.doi.org/10.1063/1.1729602 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Periodic chains of gold nanoparticles and the role of oxygen during the growth of silicon nanowires Appl. Phys. Lett. 89, 173111 (2006); 10.1063/1.2364835 The role of crystalgrowth properties on silicon implant activation processes for GaAs J. Appl. Phys. 64, 1464 (1988); 10.1063/1.341818 Clustering mechanism during growth of GaAs on silicon J. Vac. Sci. Technol. B 6, 1137 (1988); 10.1116/1.584266 Oxygen contamination of Ge during thermal evaporation for Ohmic contacts to GaAs J. Vac. Sci. Technol. B 6, 582 (1988); 10.1116/1.584404 Incorporation of boron during the growth of GaAs single crystals Appl. Phys. Lett. 36, 989 (1980); 10.1063/1.91393 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 15 Oct 2014 14:05:38JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 5 MAY 1963 Role of Oxygen in Reducing Silicon Contamination of GaAs during Crystal Growth J. F. WOODS AND N. G. AINSLIE IBM Thomas J. Watson Research Center, Yorktown Heights, New York (Received 22 October 1962) GaAs grown in a horizontal Bridgman crystal growth apparatus to which oxygen has been added exhibits lower silicon content than that grown without oxygen. Material grown under oxygen additions of 10-20 Torr exhibits, at room temperature, carrier densities in the 2-4XI015 cm-3 range and mobilities between 7500-8650 cmz V-I sec-I. Silicon concentrations computed from the reaction 4Ga+Si0 2 -> 2Ga20+Si are compared with electrical determinations of donor densities and spectroscopic determinations of silicon concentrations with reasonably good agreement. It is concluded that suppression of Si02 dissociation at the walls of the silica reaction tube is the most important action of oxygen on GaAs properties although oxygen doping may playa role in the production of high resistivity GaAs. INTRODUCTION IT has recently been reported I that the electrical properties of gallium arsenide made by the hori zontal Bridgman technique are very sensitive to the amount of oxygen present in the system during crystal growth. It was found that by adding oxygen to the fused silica reaction tube in amounts varying from zero to ISS Torr at room temperature, the room-tem perature resistivity increased by about ten orders of magnitude, the room-temperature carrier density de creased by about the same amount, and the electron mobility passed through a maximum in the 10-20 Torr range. The room-temperature mobilities in this range of o~ygen pressure, the highest reported to date, are consIstently 7500-8500 cm2 V-I seci at room temper ature, and 20000-30000 cm2 V-I secI at 77 OK. The best crystal exhibited a room-temperature mo bility of 8650 cm2 V-I seci and a 77°K mobility of 30000 cm2 V-I secI. The effects that oxygen has upon the electrical properties of GaAs have been examined in light of more recent experimental work. It is the purpose of this paper to summarize the experimental findings to date, and to describe the mechanism by which oxygen probably changes the properties of GaAs in the ob served manner. EXPERIMENTAL RESULTS GaAs ingots weighing approximately 100 g were synthesized in a standard horizontal Bridgman crystal growth apparatus by reacting pure, vacuum decanted gallium with arsenic vapor. The arsenic pressure abov~ the GaAs was maintained througii the use of a con densed arsenic source kept at 6lO± lOoe; the arsenic always underwent a vacuum bake-out prior to sealing the fused-silica reaction tube. Since oxygen doping in hibited single-crystal growth, perhaps due to increased wetting of the boats, measurements were made on monocrystalline specimens taken from usually poly crystalline ingots. IN. G. Ainslie, S. E. Blum, and J. F. Woods J. Appl. Phys 33 2391 (1962). ,. , Figure 1 summarizes the variation of the room temperature electrical properties with pressure of the added oxygen. Despite the fact that oxygen doping probably occurs, the GaAs grown in the 10-20 Torr O2 range exhibits electrical properties that are character isti~ of ;elatively pure material. This is seen in Fig. 2, whIch gIves the temperature dependence of the mobility for GaAs grown in the 10-20 Torr range, and, for purposes of comparison, less pure GaAs grown under essentially zero oxygen pressure. Due to the predomi nance of lattice scattering over ionized impurity scatter ing, the GaAs given the oxygen treatment shows a rising mobility with decreasing temperature to about 77°K. Below 77°K the mobility decreases with de creasing temperature as impurity scattering becomes the important factor limiting mobility. At temperatures below 400K the Hall coefficient (Fig. 3), passes through a maximum as impurity con duction becomes dominant. The mobility in the im purity band at 4.2°K is much lower for the material g~own under 10-20 Torr O2 than for material grown WIthout oxygen, indicating higher purity. Figure 4 shows the relation between the mobility at 4.2°K and 16 17 16 q-15 § 14 ~ 13 iii z 12 w 011 a: ~IO a: ~ 9 u '" 6 0 ..J 7 :9000 u " " .. 8000 g N 7000 § ,.. ': 6000 ..J CD 0 ;:; 5000 CARRIER DENSITY o I 5' 10 100 6 7 - 6 j ,.. 4 ': > 3 ~ 2 ~ o '1 -2 a: ROOM TEMPERATURE PRESSURE OF ADDED OXYGEN (TORR) FIG. 1. Electrical properties of GaAs grown by horizontal Bridg ~!lI?-technique_ as a function of the pressure of oxygen added initially to the reaction tube. 1469 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 15 Oct 2014 14:05:381470 J. F. WOODS AND N. G. AINSLIE 40000r-------------.------~ 10000 8000 0 6000 " ~ 4000 .,- -0 > 1 >-f-:::; iD 0 ::E. 10 TEMPERATURE (OK J FIG. 2. Log plot of Hall mobility vs temperature for GaAs grown in the 10-20 Torr O2 range and for GaAs grown under zero oxygen pressure. the maximum mobility (800-17S0K). Since the former depends particularly on the density of shallow donors, whereas the latter depends on the density of various impurities and defects, a completely smooth variation is not to be expected. The trend of the data, however, clearly shows the effect of oxygen in improving the mobility by reducing the density of shallow donors. It has been postulated1 that oxygen causes these electrical effects principally by excluding the donor silicon from the GaAs through suppression of the Si02 dissociation reaction at the walls of the fused-silica reaction tube during crystal growth. The results of careful emission spectroscopic analyses furnish evidence to support this idea. Samples of GaAs ingots grown under three oxygen pressures were analyzed to have the following silicon densities: O2 pressure (Torr, room temp) 0 10 155 n (Si) cm-3 1.5 X 10'7 5 X 10'6 not detected. The silicon levels corresponding to 0 and 10 Torr com pare quite favorably with the measured room-tempera ture carrier densities of such material (see Fig. 1). In the discussion to follow further quantitative justi fication is given for the proposed role of oxygen in terms of the recent thermodynamic data reported by Cochran and Foster,2 and the energy level model for high-resistivity GaAs proposed by Blanc and Weisberg. s DISCUSSION Cochran and Foster2 give thermodynamic data as a function of temperature in the range 1050o-1600oK for the following reaction: 4Ga(in GaAs)+Si02 ~ 2Ga20(vapor) +Si(in GaAs). (1) 2 C. N. Cochran and L. M. Foster, J. Electrochem. Soc. 109, 144 (1962). a J. Blanc and L. R. Weisberg, Nature 192, 155 (1961). They also report gallium activities at 1081 ° and 1523 oK; by extrapolation it is possible to obtain values for gallium activity at any temperature of interest. The following expression relates the Ga20 pressure to the silicon activity in the melt: plGa 20]a[Si]=K(T), (2) where the symbols p and a represent pressure and chemical activity, respectively, and K(T) is the mass action constant mUltiplied by the fourth power of the gallium activity. Although Ga20 is not as stable as the Ga20s condensed phase, Ga20 nonetheless seems to form initially. Discussion of Ga20S formation has been deferred to subsection D. A. Absence of Oxygen According to reaction (1), the Si02 of the reaction vessel dissociates to form silicon and Ga20. If care is taken to exclude all oxygen from the reaction tube by SUbjecting the arsenic to a vacuum bakeout to remove As20S and by vacuum decanting the gallium to remove Ga20S, then, providing the Ga20 that forms by reaction of Ga with Si02 does not dissolve in the melt, the following expression relates p[Ga20] to the mole frac tion N[Si] of silicon in the GaAs melt. p[Ga20]=2pRT m(V m)(_1 __ )N[Si], (3) Vg MGaAs where p=density of the GaAs melt, V m=volume of melt, Vg=volume of J:\as in reaction tube, R=gas con stan t, M GnAs = average of the atomic weights of gallium and arsenic, and T m=mean gas temperature. Equation (3) is valid for the case in which N[Si] is very small relative to the atomic fractions of Ga and As. Solving Eqs. (2) and (3) simultaneously, assuming that a[Si] =N[Si], one calculates the equilibrium values of p[Ga 20] and N[Si]. These values are given in the I Z I&J c::; ii: " I&J o '" oJ oJ C I :z: 10 10 20 30 40 !50 60 70 80 238 I03/T FIG. 3. Hall coeffi cient vs reciprocal temperature for GaAs grown in the 10-20 Torr 02 range. Data in the 4.2°- 12.5°K range are not shown. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 15 Oct 2014 14:05:38R 0 LEO FOX Y G E N IN RED U C I N G S I LIe 0 NCO N TAM I NAT ION 1471 >t: ..J iii o 2 ~ :z: 104 2XI04 3XI04 HALL MOBILITY MAXIMUM (cm2/volt-sec) FIG. 4. Hall mobility at 4.2°K vs Hall-mobility maximum taken from curves of Hall mobility vs temperature: 0 grown under zero oxygen pressure;. vapor grown GaAs [from V. J. Lyons and V. J. Silvestri, J. Electrochem. Soc. 108, 177 C, Abstract 140 (1961)J; f::,. grown under various pressures of oxygen. following tabulation along with n[Si], the atomic density of silicon in the GaAs crystal. Melt tempera tures of 1510° and 1573°K are assumed since in all likelihood the actual melt temperatures in the hori zontal Bridgman runs lay between these two values. The calculations were made using p=6 g cm-3, Vo/V m = 15, T m= 1200oK, and a crystal-liquid segregation coefficient of unity: K(T) P[Ga20J N[SiJ n[Si] 15100K (max melting temperature) 1.05 X 10-11 atm2 1.71 Torr 2.06XlO-6 9.27x 1016 em-a 1573°K 5.00X 10-10 atm2 6.21 Torr 7.50XlO-6 3.38X 1017 em-a. The calculated values of n[Si] compare favorably with the spectroscopic analyses for silicon shown previously. B. Effect of Small Amounts of Added Oxygen We now examine what happens when oxygen is de liberately added to the reaction tube. If oxygen does not dissolve in the melt and if Ga203 formation does not occur, then it can be shown, by comparing the reactions for SiO and Si02 formation4,5 with reaction (1), that virtually all added oxygen combines to form Ga20. Therefore, when 10 Torr of oxygen at room tempera ture are added to the system, the Ga20 pressure during the run would be 80 Torr since each O2 molecule forms two Ga20 molecules, and T ".::: 1200 oK is a factor of four greater than room temperature. Substituting p[Ga20] = 80 Torr into (2), the following silicon levels are deduced: N[Si] n[Si] 15100K 9.48 X 10-10 4.27X1013 cm-3 1573°K 4.52XlO-8 2.03XlOlIi cm-3. 4 H. L. Schick, Chern. Rev. 60, 331 (1960). 6 H. F. Ramstead and F. D. Richardson, Trans. AIME 221, 1021 (1961). It is seen that the measured carrier density, 2-5X1015 (see Fig. 1), of material grown with this amount of added oxygen agrees well with the silicon level calcu lated for a melt temperature of 1573°K. Also, as with the case in which no oxygen is added, the calculated silicon density at 1573°K agrees roughly with the spectroscopic analysis reported above. If, however, it is assumed that only l of the added oxygen combines to form Ga20, and! of it dissolves in the GaAs melt, the silicon level calculated for a melt temperature of 15100K would be 64X4.27X 1013= 2.73 X 1015 cm-3, which is also in reasonable agreement with the measured carrier density and the spectroscopic analysis. In addition, the melt would contain 7.59X 1018 oxygen atoms cm-3 for the conditions of the present experiments in which V oIV m = 15. Since the actual melt temperature during an experiment lies between the extremes of 1510° and 1573°K, the amount of dis solved oxygen required for agreement between the calculated and observed silicon contents would there fore have to lie between 0 and! of the total amount of added oxygen. C. The Effect of Large Amounts of Added Oxygen For the GaAs grown under zero or low oxygen pressures, there is reasonable agreement between the calculated silicon densities, the measured carrier densi ties, and the results of emission spectrographic analyses for silicon. At higher oxygen pressures, however, the carrier density is no longer approximately equal to the silicon density. This is entirely reasonable and is due either to the effects of other impurities, or to defects in the crystal which become relatively more important as the Si content is reduced. Nonetheless, the silicon content calculated from the thermodynamic data can be related to the observed electrical properties of the GaAs through a simple energy level model. A model sufficient to explain these electrical data has been reported by Blanc and Weisberg3 and is shown in Fig. 5. Using their notation, N D is identified as the silicon density, N DD is a deep-donor density, and N A CONDUCTION BAND I -4 Ell-US6 -4.2 XIO T eV 1 VALENCE BAND FIG. 5. Energy level model used in text to describe electrical properties of GaAs. Energies are indicated from conduction band edge. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 15 Oct 2014 14:05:381472 J. F. WOODS AND N. G. AINSLIE is the density of acceptor impurities. Ed and ED are the activation energies of the deep donor and the silicon, respectively, and Eg is the forbidden energy gap. Since the material to be discussed is n type, the acceptor levels are taken to be completely ionized so their activation energy does not enter the equations to follow. When the electron gas in the conduction band is not degenerate, the electron concentration n is given by nl [ ED-E,]-1 n=-+N D l+gD exp'--- n kT (4) in which ni is the intrinsic carrier density; gd and gD are the degeneracy factors for the deep donors and the silicon, respectively; E, is the Fermi energy; and kT has its usual meaning. Equation (4) expresses the charge neutrality condition when the acceptors are completely ionized. Since, in the nondegenerate case, nand E, are related through the expression n=Nc exp( -E,/kT), (5) where N c is the effective density of states in the con duction band edge, Equation (4) is a quartic in n. In the GaAs in question, however, ED~O.OI eV and Ed~0.76 eV.6 Thus, at temperatures less than 7000K either exp[(ED-E, )/kT]«I, or exp[(Ed- E, )/kT]»I, or both. These conditions permit simplification of Eq. (4) for various temperature ranges and doping condi tions. Two cases are of particular interest here: In the first case in which N D is large (N D»N A), the Fermi level will be far above Ed and n will be much larger than ni at T < 700oK. In this case, the first and third terms on the right-hand side of Eq. (4) are negligible. Using Eq. (5), Eq. (4) reduces to NDNc exp( -ED/kT) n=--------- gDn+Nc exp(-ED/kT) (6) In the second case in which N D is not large relative to N A, the Fermi energy is far below ED. In this case the second term on the right-hand side of Eq. (4) reduces to N D. Using Eq. (5) again, the expression for n becomes NDDNcexp(-Ed/kT) nl n +--(NA-N D). (7) gdn+Nc exp( -Ed/kT) n When gd= 2, Eq. (7) becomes Blanc and Weisberg's3 Eq. (1). For the condition N A <N DD, N D <O.IN A, and 6 C. H. Gooch, C. Hilsum, and B. R. Holeman,]. App!. Phys. 32,12069 (1961). Measurements in this laboratory on GaAs grown under high oxygen pressures agree with the measurements reported in this reference. nNn«n, Eq. (7) approaches the limit (NDD )(Nc) n= NA -1 -; exp(-Ed/kT). (8) The carrier densities measured on samples grown in zero or low oxygen over-pressures are described by Eq. (6). These carrier densities range from 4X1017 cm-3 to 2X 1015 cm-3. Samples with carrier densities less than 1.5X1016 show distinct carrier "freeze-out", as Eq. (6) would predict, and exhibit impurity-band conduction at the lower temperatures. 7 At high oxygen pressures (80-155 Torr) the carrier densities are described by Eqs. (7) and (8) and lie in the 107-108 cm-3 range. This is the usual high resistivity GaAs described by workers in several laboratories. When GaAs is grown under intermediate oxygen pressures (30-80 Torr), results are not very repro ducible. There are, however, in this range of oxygen pressure some ingots which exhibit carrier densities that vary approximately as exp (-O.4/kT) and have values of 101L1013 cm-3 at room temperature. This last type of material, grown under intermediate oxy gen pressures, can also be described by Eq. (7). At these carrier densities nlJn is negligible, and N c Xexp(-Ed/kT)«n, so Eq. (7) becomes NDDNc exp(-Ed/kT) n= (NA-ND). (9) gdn If INA -N D I «n, then n2= (N DDNcI gd) exp( -Ed/kT). (10) Equation (10) yields an apparent activation energy of !Ed for n. The condition INA-NDI«n is a rather stringent one, and a lack of experimental reproduci bility is to be expected. Nonetheless, the O.4-eV slope has been observed in two ingots which were subse quently examined by optical absorption and photo conductivity measurements8 to see whether or not it was a real level. No signs of absorption near 0.4 e V could be found in either specimen, but a level was ob served at approximately 0.8 eV. It thus appears likely that these specimens do indeed exhibit behavior corre sponding to Eq. (10). It should be noted that all these various forms of behavior [Eqs. (6)-(10)J could in principle be ob served in a series of specimens by varying N D only; cV DD and N A may be constant from specimen to speci men. However, if N DD, N A, or both, increase with increasing oxygen pressure, the variation of carrier density n with silicon N D will be qualitatively the 7 The existence of impurity-band conduction would suggest an impurity density of the order of 1017 cm-3 on the basis of simple theory. However, the high mobilities (> 7000 em' V-I secl) are evidence against such high ionized impurity concentrations. 8 W. ]. Turner, A. E. Michel, and W. E. Reese (private com munication, to be published). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 15 Oct 2014 14:05:38R 0 LEO FOX Y G EN IN RED U C I N G S I LIe 0 NCO N TAM I NAT ION 1473 17 16 .. Ie 15 u " Z 14 o ;:: ~ 13 f-z IU ~ 12 o t.> ~ II 0: ft.> ~ 10 IU ... o 9 <.0 o -oJ 8 7 6 !!? '"' 2 Q " " co: co: z z NOO=1017 5C=~13======~14=======15~=:~-1~6-----17 LOG OF SHALLOW OONOR (SILICON) CONCENTRATION ,No (cm-') FIG. 6. Log plot of electron concentration vs shallow donor concentration at 3000K for model shown in Fig. 5. Equation (6) describes the extreme right-hand branch of each curve, Eq. (S) the extreme left-hand branch, with the steep intermediate region described by Eq. (9). same. Figure 6 shows a plot of n vs N D at 3000K for several values of N DD and N A •. In plotting Fig. 6 the following values have been used: n/=4.23X1012 cm-5, Eg= 1.56-4.2X 1O-4T eV, effective mass of electrons=0.072m e, effective mass of holes=0.5m e, Ncexp(-Ea/kT)=1.76X104 (Ea=O.S eV), gd=l, Ncexp(-Ev/kT)=3.3X1017 (ED=O.OI eV). Equation (6) describes the right-hand branch of each curve in Fig. 6, whereas Eq. (S) describes the left-hand branch. The intermediate region in which n undergoes very large changes with small changes of N D is de scribed by Eq. (9). As N D, identified with silicon, decreases in the region described by Eq. (6) due to chemical suppression by an increasing oxygen pressure, the mobility increases (see Fig. 1) indicating a decrease in the density of ionized scattering centers. As N D is further decreased to the region described by Eq. (9), in which NAro../N D, mobility would not be expected to continue to increase with decreasing N D since N A now comprises an appreciable fraction of the total number of ionized scattering centers in the crystal. Also, since n decreases sharply in this range of N D, the reduced screening of charged centers would be a factor to re-duce mobility. Furthermore N DD, although essentially un-ionized, may also limit the mobility through neutral impurity scattering. Thus it is reasonable to expect the mobility to stop rising as the oxygen pressure exceeds a certain value, and to decrease at higher pressures as seen in Fig. 1. Now, whether N DD and N A are constant with in creasing oxygen pressure, or increase proportionately, the final value of n at high oxygen pressures will be the same [see Eq. (S)]. Thus, from these sorts of data it is not clear whether or not N DD or N A are associated with oxygen; however, the reduction of the silicon concen tration lY D seems certain to be caused by the oxygen. Assuming that iV DD and l'{ A are independent of the oxygen over-pressure during crystal growth, though they may vary from ingot to ingot over some range, it is possible to deduce representative values from the measurements. From the observed fact that the carrier density decreases fairly smoothly with oxygen pressure to approximately 2 X 1016 cm-3, and then becomes er ratic over a considerable range of intermediate oxygen pressures, it may be deduced that N A is also approxi mately equal to 2 X 1015 cm-3• This is because such erratic behavior would occur in the region !N A <If D <2N A in which the electron density n drops precipitously with decreasing N D; in this range small changes in the relative values of N A and N D would yield enormous changes in n. Since 0.4 eV is the apparent activation energy of n ex hibited by the material grown under intermediate oxy gen pressures, where Eq. (10) seems to apply, and since slopes of about 0.7S eV have been obtained in high resis tivity material, Ed may be taken to be about O.S-aT, where a is the temperature coefficient of the energy level. FromEq. (10) N DD=gdr2/[e2RWc exp( -Ed/kT)] where r is the ratio of the Hall mobility to conductivity mobility, and R is the Hall coefficient. For an ingot grown under 50 Torr O2, N DD=4X1019gar2 exp[ -a/k], and for one grown under SO Torr O2, NDD=6X1020gar2 Xexp[ -a/k]' Since gdr2 exp[ -a/k] is not less than 0.01, these values of N DD represent quite high deep donor concentrations. Assuming, as before in the case of small oxygen additions (subsection B), that i of the added oxygen at high (50-155 Torr) levels dissolves in the melt, one calculates the following oxygen and silicon densities: Room temperature pressure of added O2, Torr n[O]cm-S n[Si](=ND) cm-S 15100K 1573°K 50 3.S4X 1019 1.09X 1014 5.19X 1015 SO 6.0SX1019 4.27X101S 2.03X1015 155 1.1SXI020 1.14 X 1013 5.41XlO14• Even though only 1 of the added oxygen remains in the gas phase to suppress the Si02 dissociation in ac cordance with reaction (1), it seems to be enough to depress the silicon concentration to the 1015 cm-3 level or lower, and the oxygen that dissolves in the melt would be sufficient to dope the GaAs to the rather high [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 15 Oct 2014 14:05:381474 J. F. WOODS AND N. G. AINSLIE IOO~--------~~--------~IO~O--------~150 ROOM TEMPERATURE PRESSURE OF ADDED OXYGEN (TORR) FIG. 7. Plot of log dissolved oxygen "n[O] and dissolved silicon n[Si] concentration vs initial pressure of added oxygen. 10lL1020 cm-3 range. Figure 7 gives the calculated oxygen and silicon densities as functions of the amount of oxygen added to the system. It is assumed that the amount of dissolved oxygen lies somewhere between 5% and 95% of the total amount added. The assumed temperature limits are 1S1OoK and 1S73°K. The figure makes it apparent that over wide ranges of oxygen uptake and temperature the oxygen content of the GaAs is consistently high (between 1018 cm-a and 1020 cm-a), whereas the silicon content is depressed to relatively low levels due to reaction (1). It must be pointed out that these experiments and calculations neither confirm nor deny the possibility that N DD is associated with dissolved oxygen. If N DD were to increase relative to N A as N D de creases, as would be the case if N DD were the oxygen content, then n would increase directly with oxygen pressure at high pressures. This would be a small effect relative to the other effects attributed to oxygen, and has not yet been observed. D. Ga20a Formation A baffling aspect of the oxygen-doping experiments is the apparent failure of the stable Ga203 condensed phase to form. Again using data reported by Cochran and Foster,2 it is found that the Ga20 could co-exist with Ga203 and GaAs by the reaction, 3 Ga20(vapor)+As 4(vapor) ~ 4 GaAs(condensed) + Ga203 (condensed) at the following temperatures and pressures: P[Ga20] 100 Torr 10 Torr 1 Torr T 15000K (1227°C) 14000K (1127°C) 1308°K (1035°C). Since the temperature of the reaction tube is not uni form, but rather decreases from the melt temperature at one end to ",600°C at the arsenic reservoir end, it would seem that the Ga20 pressure could never build up to pressures large enough to suppress the Si02 dissociation. Rather, the oxygen should be continu ously removed by Ga20a formation in the colder regions of the system. This apparently either does not occur at all, or does not occur fast enough to allow the silicon content of the melt to build up to undesirable levels corresponding to the low Ga20 pressures that would result. As suggested by Cochran and Foster,2 the rate at which Ga20a forms could be limited by the rate at which Ga20 vapor diffuses through the en veloping arsenic gas (kept at approximately 1 atm of pressure) to the cold zone of the reaction tube. Alterna tively, other kinetic factors such as GaAs or Ga20a nucleation in the cold zone may retard the loss of Ga20. One experiment is of particular interest in this con nection. Rather than doping with gaseous oxygen, a run was carried out in which solid Ga20a was placed in the reaction boat where GaAs was to be grown. The amount of oxygen contained in the' added Ga20a was equal to that which would have resulted from a 17-Torr gaseous oxygen addition and, indeed, the resulting GaAs had properties typical of material grown in the 10-20 Torr range: it exhibited a room-temperature mobility of 8090 cm2 V-I secl and an 86°K mobility of 27460 cm2 V-I secl. CONCLUSIONS (1) GaAs, exposed to various over-pressures of ox ygen during crystal growth by the horizontal Bridgman technique, exhibit carrier densities that correspond fairly well with silicon contents calculated from a re action in which the oxygen suppresses the dissociation of Si02 walls of the fused silica reaction tube. (2) For small oxygen additions (up to 20 Torr at room temperature when the gas volume is 15 times the melt volume) the principal function of oxygen seems to be to increase the apparent purity of the GaAs, as evidenced by much higher mobilities and lower carrier densities, by causing a decrease in the concentration of the shallow donor silicon. (3) For large oxygen additions (between 80 and 155 Torr at room temperature when the gas volume is 15 times the melt volume) the resulting GaAs becomes high resistivity, or semi-insulating, but still n type. This is probably due to a deep-donor level, present [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 15 Oct 2014 14:05:38ROLE OF OXYGEN IN REDUCING SILICON CONTAMINATION 1475 either independent of the oxygen or perhaps introduced by oxygen, which prevents the material from going p type when the silicon content is reduced below the concentration of acceptor impurities. ACKNOWLEDGMENTS Quantitative spectrographic analyses for silicon were performed under the supervision of W. Reuter. We JOURNAL OF APPLIED PHYSICS wish to thank W. Turner, A. Michel, and W. Reese for permission to report some of their results on optical absorption and photoconductivity phenomena in ad vance of publication. Special thanks are due also to R. Zimer for assistance in building the crystal growth equipment, and to J. Keller and G. Moran for carrying out the many elec trical measurements. VOLUME 34. NUMBER 5 MAY 1963 Effect of Low-Temperature Phase Changes on the Mechanical Properties of Alloys near Composition TiNi W. J. BUEHLER, J. V. GILFRICH, AND R. C. WILEY U. S. Naval Ordnance Laboratory, Silver Spring, Maryland (Received 24 August 1962; in final form 28 December 1962) X-ray diffraction and dilation studies have shown that alloys near the stoichiometric TiNi composition undergo transformation into the related phases ThNi and TiNia at low temperatures. The main factors controlling these phase transformations are alloy composition, temperature, and mode of plastic deforma tion. In plastic deformation, tensile or compressive stressing produced separate and unlike decomposition phases; this finding was dramatically demonstrated by unique temperature-sensitive dimensional changes in plastically deformed specimens. Changes of large magnitude in vibration damping have also been noted and appear related to variations in the phase equilibria of the system. X-RAY diffraction and related studies were made of the titanium-nickel system around the equiatomic compound TiNi to explain some unusual changes,I·2 in physical and mechanical properties of this material with small temperature changes near 65°C. Arc-cast samples were examined by x-ray diffraction at room tempera ture, while hot-rolled sheets (rolling temperature 700°C) of similar composition were examined at various tem peratures from 25° to lO00°C in a high temperature diffractometer. The phases present in the various samples are listed in Table I. The specimens were arc melted from "iodide" titanium (Brinell Hardness No. less than 85) and "carbonyl" nickel (99.99% pure) using a nonconsumable tungsten electrode and a water cooled copper hearth. The compositions listed are the nominal values as calculated from the raw materials, which were quite accurately weighed. The weights after melting showed no significant losses and so the nominal com positions are assumed reasonably correct. The x-ray patterns of the three phases Ti2Ni, TiNi, and TiNia can be readily distinguished3 and since the x-ray technique made use of a counter diffractometer, minor amounts of these phases could be detected quite readily (down to a 1 W. J. Buehler and R. C. Wiley, Am. Soc. Metals Trans. Quart. 55, 269 (1962). 2 J. V. Gilfrich, "X-ray Diffraction Studies on the TiNi System," in Proceedings 11th Annual Conference on Applications of X-ray Analysis, Denver Research Institute, 1962 (Plenum Press, Inc., New York, 1963). 3 M. Hansen, Constitution of Binary Alloys (McGraw-Hill Book Company, Inc., New York, 1958), p. 1052. few percent). The x-ray penetration of these samples by the molybdenum radiation was the order of 0.004 in. so the information was characteristic of the surface to this depth but not necessarily of the bulk of the material. For the arc-cast buttons, the surface was in the "as cast" condition and no surface preparation was used. The hot-rolled sheet was about 0.020 in. thick and both sides of representative samples were examined in the x-ray work. In all cases both sides of the sheet were found to be identical. TiNi, a CsCl-type body-centered cubic intermetallic compound, does exist in a stable or metastable form, at room temperature, either as a single phase or one com ponent of a two-phase system with the other phase either Ti2Ni or TiNia, depending on the actual composi tion around the equiatomic point. As the amount of Ni increases, the amount of TiNia increases, as reported by Margolin et al.4 However, at less than 54 wt % Ni, TiNi dissociates into Ti~i and TiNis, as reported by Duwez and TaylorS and by Poole and Hume-Rothery,6 but not as reported by Purdy and Parr,7 who claim a transfor mation of TiNi at room temperature into a previously unreported "7r" phase which they index as hexagonal, a=4.572 A, c=4.660 A, c/a= 1.02. The 54-wt % Ni t H. Margolin, E. Ence, and J. P. Nielsen, Trans. AIME 197, 243 (1953). Ii P. Duwez and J. L. Taylor, Trans. AIME 188, 1173 (1950). 8 D. M. Poole and W. Hume-Rothery, J. lnst. Metals 83,473 (1955). 7 G. R. Purdy and J. G. Parr, Trans. Met. Soc. AIME 221,636 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 15 Oct 2014 14:05:38
1.1735318.pdf	ElectronBombardment Induced Recombination Centers in Germanium J. J. Loferski and P. Rappaport Citation: Journal of Applied Physics 30, 1318 (1959); doi: 10.1063/1.1735318 View online: http://dx.doi.org/10.1063/1.1735318 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Color-center waveguides in low-energy electron-bombarded lithium fluoride Appl. Phys. Lett. 88, 261111 (2006); 10.1063/1.2218039 Analysis of multijunction solar cell degradation in space and irradiation induced recombination centers J. Appl. Phys. 93, 5080 (2003); 10.1063/1.1561999 A stabilized electronbombardment ion source Rev. Sci. Instrum. 45, 308 (1974); 10.1063/1.1686616 INFRARED BIREFRINGENCE OBSERVATIONS ON ELECTRONBOMBARDED SILICON Appl. Phys. Lett. 9, 355 (1966); 10.1063/1.1754610 ElectronBombardment Induced Recombination Centers in Germanium J. Appl. Phys. 30, 1181 (1959); 10.1063/1.1735289 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.120.209 On: Sat, 22 Nov 2014 05:39:261318 DISCUSSION conductivity, annealed and observed almost complete recovery of thermal conductivity in the region of the maximum. However, after standing for a couple of months, a remeasurement of thermal conductivity showed that the thermal conductivity had increased above its original value by an appreciable amount. It should be noted that the irradiation was sufficient to produce p-type ma terial after all of the activated Ge70 had decayed to gallium. One possible explanation of the enhancement is that the hole-phonon interactions have a much smaller effect on the conductivity. Transport Properties R. K. WILLARDSON G. K. Wertheim: The energy level structure of electron irradi ated Si is different for vacuum floating zone and quartz crucible grown crystals. The results agree with those obtained by G. D. Watkins using spin resonance techniques. Do you find a difference in energy level structure for neutron irradiated Si of the two types? R. K. Willardson: The energy level structure of neutron irradi ated Si is the same for n-type material of either type as far as we can ascertain. Recombination G. K. WERTHEIM H. Y. Fan: It seems to me that the Hall effect and the recom bination type of measurements are very sensitive, in some respects much too sensitive. You are likely to see energy levels or defects which are introduced to a very small extent but which are very effective in pinning down the Fermi level when the resistivity is high, or they are very effective for recombination when the capture cross section is large. For instance, in n-type Si irradiated with neutrons we see only two definite absorption bands, whereas all previous measurements of Hall effects indicated you might have a spread of levels. I think in such cases, if you want to spot the major levels, some measurements which are a little less sensitive, like optical absorption, perhaps should be made. G. K. Wertheim: I think I differ with you fundamentally because my feeling is that any level that you can see is of interest because it contains some information about the nature of the bombard ment damage. The mere fact that it is introduced in a small density does not make it less interesting, and perhaps this is a good argument for the use of lifetime measurements because, if the cross section is large, it provides a rather sensitive tool to get us something that we cannot see with optical means. Radiation Effects on Recombination in Germanium O. L. CURTIS, JR. H. Y. Fan: I would like to point out another factor in connec tion with the recombination type of measurements, that is, the surface effect. Photoconductivity does depend upon the carrier lifetime, and some previous work at Purdue by StOckman showed a distinct photoconductivity peak corresponding to some energy level toward the middle of the energy gap such as shown here at 0.32 ev in the case of 14-Mev neutron radiation. However, some subsequent measurements by Spear at Purdue showed that this effect was purely a surface effect. We are all aware that the trapping and the carrier recombination surface effects can be very important. So here is another thing that we must bear in mind. O. L. Curtis, Jr.: I believe that we do not have surface effects in these samples. These samples are about 7 or 8 millimeters in the smallest dimension; and, whereas you might well expect surface effects in small samples, even with fairly short lifetimes, still with post-irradiation lifetimes of the order of 20 microseconds or so it seems hardly possible that the surface can be playing an important role in our measurements. Now there is something to be borne in mind. That is, if you use, say, a white light to excite the carriers, you might excite the carriers only at the surface, essentially; and you might find predominantly surface effects, whereas you think because of the size of your specimen you should be eliminating them. For these measurements we used a germanium filter of the order of a half-millimeter in front of our specimens so that the carriers that are excited are excited fairly uniformly inside the specimen. J. J. Loferski: I would like to speak in defense of devices. There seems to be the feeling abroad that if one attaches to a piece of germanium anything other than a couple of ohmic contacts the measurements that one makes on that device are to be regarded at least with suspicion and perhaps to be ignored entirely. Now this is not true. Careful measurement made on properly made devices can, for instance, follow lifetime changes with an accuracy of 1% or better; and that is pretty difficult to do if you are measuring the lifetime directly. Usually plus or minus 10% is pretty good for direct lifetime measurements. Also, the great sensitivity that one gets on such pieces with other than only ohmic contact makes it possible to follow recom bination-center concentrations of the order of 1010 or even less per cm3 in germanium. P. Rappaport: We have tried to compare the results that one gets when measuring lifetime on a slab of germanium with just two ohmic contacts to those one gets from lifetime measurements on junction diodes, which is perhaps the simplest type of device. As Curtis suggested, we had difficulty with that experiment. We have in the past, however, had satisfaction from such devices. The difficulty is that, when using junction diodes to measure life time changes when one is concerned with these changes as a function of resistivity, there is another parameter that changes in the junction. It is the collection efficiency for the excess carriers that are induced in the semiconductor, and that is the thing that we have not been able to pin down well enough to be able to com pare the results with those obtained on bulk specimens. Electron-Bombardment Induced Recombination Centers in Germanium J. J. LOFERSKI AND P. RAPPAPORT O. L. Curtis, Jr.: Because of the possibility of multiple levels, it seems apparent that in order to know anything about the proper ties of recombination centers one must make lifetime measure ments both as a function of temperature and carrier concentra tion. The temperature dependence of p-type material shows that such an analysis as you have made in the p-type region is mean ingless. Our measurements on C060 gamma-irradiated, p-type material, mentioned in the previous paper, indicated a very similar dependence on carrier concentration to that you show for 1-Mev electron irradiation; but our observations of the dependence of lifetime on temperature reveal that recombination did not take place at the 0.26-ev level, rather that the occupation of a level in this region probably determined the number of upper levels available for recombination. One cannot safely determine energy level position solely on the basis of measurements as a function of carrier concentration. Magnetic Susceptibility and Electron Spin Resonance E. SoNDER H. Brooks: If your susceptibility data are interpreted on the basis of clustering, then perhaps it might mean that the clusters are considerably larger than we have been accustomed to thinking in the past, and that the flux necessary to produce overlap is con siderably less than 1018 to 10'9. G. Leibfried: The closed-shell repulsion in covalent materials is much smaller than in metals; this would cause the damage due to one fast neutron to be distributed over an area a factor of 5 to 10 larger. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.120.209 On: Sat, 22 Nov 2014 05:39:26DISCUSSION 1319 A. G. Tweet: This is a comment concerning interpretation of extremely strong temperature dependence of mobility. Under certain conditions such effects may be caused by inhomogeneities in samples rather than a large density of defects. For example, in material doped with only 1015 to 1016 nickel atoms per cm3, re sults that are uninterpretable have been obtained. Further in vestigation has shown that this is caused by an inhomogeneous distribution of impurity. Magnetic Susceptibility and Electron Spin Resonance-Experimental G. BEMSKI M. Niseno.fJ: We have found spin-resonance absorptions in many samples of neutron-irradiated silicon. Irradiation until the Fermi level was near the center of the forbidden gap produced a reproducible pattern, independent of the original donor or acceptor concentration. Under annealing, while the Fermi level remained at the center of the gap, two changes took place in the spin-reso nance pattern. However, even after annealing at 500°C two centers still remained, indicating that at least four types of cen ters are introduced by neutron irradiation at room temperature. We also found that the intensity of the lines found in these ma terials increased with irradiation (the range of flux used was 1011-10'9), indicating that these resonances are produced by a re distribution of electrons on defects rather than by electrons origi nating from impurities. In samples whose post-irradiation Fermi level was closer than 0.1 ev to the conduction band or 0.2 ev to the valence band, different types of spin-resonance absorption patterns were seen. Magnetic Susceptibility and Electron Spin Resonance-Experimental G. D. WATKINS L. Slifkin: I have a question about how the irradiation-induced vacancies get to the oxygen atoms. For the case of germanium, subtracting Logan's value for the creation energy of vacancies, which is a little over 2 ev, from the self-diffusion activation energy of less than 3 ev, one obtains a little less than 1 ev for the mobility energy. This gives, for the temperature range in which the A centers are produced, a frequency of vacancy motion of about 1 per day. It might be expected to be even less in silicon, which has a higher melting point and, presumably, a higher activation energy for vacancy motion. It is, therefore, a bit difficult to see how the vacancy can diffuse to the oxygen atom in silicon at such a low temperature. H. Brooks: Is it absolutely excluded that the oxygen moves? A. G. Tweet: Although oxygen is interstitial, it is bonded and, therefore, probably does not move. I refer you to work of Logan and Fuller. G. D. Watkins: Is it possible that in self-diffusion measurements one is dealing with aggregates of vacancies and that the only place where you really form a single vacancy is in the radiation experiment? H. Brooks: The binding energy of the vacancies would have to be pretty high. A. G. Tweet: If diffusion proceeded by both vacancies and divacancies, a break would appear in the activation energy curve for self-diffusion. The fact that no such break has been observed implies that diffusion is all going by divacancies with a binding energy of 3 ev or more, or by single vacancies. G. H. Vineyard: At least two things about these annealing processes deserve comment. In your Fig. 5 two stages are visible in the formation of the A center. No simple movement of one kind of defect is going to explain that. Second is the fact that on anneal ing to 3000K only 3% of the number of centers expected for room temperature irradiation remain. Perhaps you have an explanation. G. D. Watkins: I think Dr. Wertheim will be able to say some thing about the second of your comments. (G. K. Wertheim presented some results concerning the tem perature dependence of the introduction rate of the A center in silicon for electron bombardment. These results have been sub mitted for publication in The Physical Review.) J. Rothstein: An oxygen atom coupled to a vacancy should have a dipole moment. It is conceivable that this might be detected by measuring dielectric loss. Moreover, one might pick it up by measuring infrared absorption. It seems possible that if one applied a polarizing field one might actually get a Stark-splitting of the level. Nature of Bombardment Damage and Energy Levels in Semiconductors J. H. CRAWFORD, JR., AND J. W. CLELAND R. W. Balluffi: Would you expect a large range of sizes of the damage spikes? J. H. Crawford: Yes. A large range would be expected. This is the reason that two different sizes were shown in the figure; the large size is completely blocking, whereas the small size is not. What we have done here, in essence, is to arbitrarily split the damage into two groups. The first group is composed of isolated defects or small clusters which produce the effects of isolated energy levels, whereas the large groups are envisioned as affecting almost entirely those properties requiring current transport. W. L. Brown: I would like to comment on your p-type bombard ment results. These are in contrast to what we have seen with electron bombardment of specimens at liquid nitrogen tempera ture. While you observe an increase in hole concentration we observe only a decrease in hole concentration over the entire tem perature range when the bombardment occurs at 77°K. R. L. Cummerow: Our results with electron bombardment at room temperature in which n-type material is converted to p type seem to agree with your proposal of a limiting Fermi level .\*. The question I would like to ask is, does the mobility determina tion support your two-level scheme? That is to say, trapping with the Fermi level halfway between these two levels would require that mobility decrease with long bombardment, whereas if a single level and annealing were responsible for the limiting con ductivity. the mobility should show no further decrease. J. H. Crawford: We have insufficient data on the mobility in the impurity scattering range to decide this point. H. Y. Fan: Do I understand that you assume that the energy levels near the center of the gap that determine the behavior of converted material are produced at about the same rate as the 0.2-ev level below the conduction band? J. H. Crawford: Perhaps. The only way one can obtain a limiting Fermi level value is by the introduction of a filled level near 0.2 ev above the valence band and an empty state quite near the center of the gap at the same rate. H. Y. Fan: In the case of electron bombardment our results indicate that the level near the middle of the gap is produced in relatively low abundance. J. H. Crawford: For the gamma irradiation we know simply from the shape of the Hall coefficient vs temperature curve that there is a deep vacant state present in concentration equal to the 0.2-ev state below the conduction band. R. L. Cummerow: I have some evidence that there is a level in the center of the gap, but it is somewhat indirect and was obtained by analyzing the potential variation in the junction produced by electron bombardment. Precipitation, Quenching, and Dislocations A. G. TWE>;T W. L. Brown: It has been shown that electron irradiation altered the precipitation of lithium in a manner that would indicate that [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.120.209 On: Sat, 22 Nov 2014 05:39:26
1.1735317.pdf	Radiation Effects on Recombination in Germanium O. L. Curtis Jr. Citation: Journal of Applied Physics 30, 1318 (1959); doi: 10.1063/1.1735317 View online: http://dx.doi.org/10.1063/1.1735317 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhanced non-radiative recombination in the vicinity of plasma-etched side walls of luminescing Si/Ge-quantum dot structures Appl. Phys. Lett. 103, 161106 (2013); 10.1063/1.4825149 Radiation effects on the behavior of carbon and oxygen impurities and the role of Ge in Czochralski grown Si upon annealing J. Appl. Phys. 105, 123508 (2009); 10.1063/1.3148293 Effective superconfiguration temperature and the radiative properties of nonlocal thermodynamical equilibrium hot dense plasma Phys. Plasmas 12, 063302 (2005); 10.1063/1.1931109 Photoluminescence investigation of phononless radiative recombination and thermal-stability of germanium hut clusters on silicon(001) Appl. Phys. Lett. 79, 2261 (2001); 10.1063/1.1405148 Phononless radiative recombination of indirect excitons in a Si/Ge type-II quantum dot Appl. Phys. Lett. 71, 258 (1997); 10.1063/1.119514 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.55.97 On: Tue, 09 Dec 2014 05:31:481318 DISCUSSION conductivity, annealed and observed almost complete recovery of thermal conductivity in the region of the maximum. However, after standing for a couple of months, a remeasurement of thermal conductivity showed that the thermal conductivity had increased above its original value by an appreciable amount. It should be noted that the irradiation was sufficient to produce p-type ma terial after all of the activated Ge70 had decayed to gallium. One possible explanation of the enhancement is that the hole-phonon interactions have a much smaller effect on the conductivity. Transport Properties R. K. WILLARDSON G. K. Wertheim: The energy level structure of electron irradi ated Si is different for vacuum floating zone and quartz crucible grown crystals. The results agree with those obtained by G. D. Watkins using spin resonance techniques. Do you find a difference in energy level structure for neutron irradiated Si of the two types? R. K. Willardson: The energy level structure of neutron irradi ated Si is the same for n-type material of either type as far as we can ascertain. Recombination G. K. WERTHEIM H. Y. Fan: It seems to me that the Hall effect and the recom bination type of measurements are very sensitive, in some respects much too sensitive. You are likely to see energy levels or defects which are introduced to a very small extent but which are very effective in pinning down the Fermi level when the resistivity is high, or they are very effective for recombination when the capture cross section is large. For instance, in n-type Si irradiated with neutrons we see only two definite absorption bands, whereas all previous measurements of Hall effects indicated you might have a spread of levels. I think in such cases, if you want to spot the major levels, some measurements which are a little less sensitive, like optical absorption, perhaps should be made. G. K. Wertheim: I think I differ with you fundamentally because my feeling is that any level that you can see is of interest because it contains some information about the nature of the bombard ment damage. The mere fact that it is introduced in a small density does not make it less interesting, and perhaps this is a good argument for the use of lifetime measurements because, if the cross section is large, it provides a rather sensitive tool to get us something that we cannot see with optical means. Radiation Effects on Recombination in Germanium O. L. CURTIS, JR. H. Y. Fan: I would like to point out another factor in connec tion with the recombination type of measurements, that is, the surface effect. Photoconductivity does depend upon the carrier lifetime, and some previous work at Purdue by StOckman showed a distinct photoconductivity peak corresponding to some energy level toward the middle of the energy gap such as shown here at 0.32 ev in the case of 14-Mev neutron radiation. However, some subsequent measurements by Spear at Purdue showed that this effect was purely a surface effect. We are all aware that the trapping and the carrier recombination surface effects can be very important. So here is another thing that we must bear in mind. O. L. Curtis, Jr.: I believe that we do not have surface effects in these samples. These samples are about 7 or 8 millimeters in the smallest dimension; and, whereas you might well expect surface effects in small samples, even with fairly short lifetimes, still with post-irradiation lifetimes of the order of 20 microseconds or so it seems hardly possible that the surface can be playing an important role in our measurements. Now there is something to be borne in mind. That is, if you use, say, a white light to excite the carriers, you might excite the carriers only at the surface, essentially; and you might find predominantly surface effects, whereas you think because of the size of your specimen you should be eliminating them. For these measurements we used a germanium filter of the order of a half-millimeter in front of our specimens so that the carriers that are excited are excited fairly uniformly inside the specimen. J. J. Loferski: I would like to speak in defense of devices. There seems to be the feeling abroad that if one attaches to a piece of germanium anything other than a couple of ohmic contacts the measurements that one makes on that device are to be regarded at least with suspicion and perhaps to be ignored entirely. Now this is not true. Careful measurement made on properly made devices can, for instance, follow lifetime changes with an accuracy of 1% or better; and that is pretty difficult to do if you are measuring the lifetime directly. Usually plus or minus 10% is pretty good for direct lifetime measurements. Also, the great sensitivity that one gets on such pieces with other than only ohmic contact makes it possible to follow recom bination-center concentrations of the order of 1010 or even less per cm3 in germanium. P. Rappaport: We have tried to compare the results that one gets when measuring lifetime on a slab of germanium with just two ohmic contacts to those one gets from lifetime measurements on junction diodes, which is perhaps the simplest type of device. As Curtis suggested, we had difficulty with that experiment. We have in the past, however, had satisfaction from such devices. The difficulty is that, when using junction diodes to measure life time changes when one is concerned with these changes as a function of resistivity, there is another parameter that changes in the junction. It is the collection efficiency for the excess carriers that are induced in the semiconductor, and that is the thing that we have not been able to pin down well enough to be able to com pare the results with those obtained on bulk specimens. Electron-Bombardment Induced Recombination Centers in Germanium J. J. LOFERSKI AND P. RAPPAPORT O. L. Curtis, Jr.: Because of the possibility of multiple levels, it seems apparent that in order to know anything about the proper ties of recombination centers one must make lifetime measure ments both as a function of temperature and carrier concentra tion. The temperature dependence of p-type material shows that such an analysis as you have made in the p-type region is mean ingless. Our measurements on C060 gamma-irradiated, p-type material, mentioned in the previous paper, indicated a very similar dependence on carrier concentration to that you show for 1-Mev electron irradiation; but our observations of the dependence of lifetime on temperature reveal that recombination did not take place at the 0.26-ev level, rather that the occupation of a level in this region probably determined the number of upper levels available for recombination. One cannot safely determine energy level position solely on the basis of measurements as a function of carrier concentration. Magnetic Susceptibility and Electron Spin Resonance E. SoNDER H. Brooks: If your susceptibility data are interpreted on the basis of clustering, then perhaps it might mean that the clusters are considerably larger than we have been accustomed to thinking in the past, and that the flux necessary to produce overlap is con siderably less than 1018 to 10'9. G. Leibfried: The closed-shell repulsion in covalent materials is much smaller than in metals; this would cause the damage due to one fast neutron to be distributed over an area a factor of 5 to 10 larger. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.55.97 On: Tue, 09 Dec 2014 05:31:48DISCUSSION 1319 A. G. Tweet: This is a comment concerning interpretation of extremely strong temperature dependence of mobility. Under certain conditions such effects may be caused by inhomogeneities in samples rather than a large density of defects. For example, in material doped with only 1015 to 1016 nickel atoms per cm3, re sults that are uninterpretable have been obtained. Further in vestigation has shown that this is caused by an inhomogeneous distribution of impurity. Magnetic Susceptibility and Electron Spin Resonance-Experimental G. BEMSKI M. Niseno.fJ: We have found spin-resonance absorptions in many samples of neutron-irradiated silicon. Irradiation until the Fermi level was near the center of the forbidden gap produced a reproducible pattern, independent of the original donor or acceptor concentration. Under annealing, while the Fermi level remained at the center of the gap, two changes took place in the spin-reso nance pattern. However, even after annealing at 500°C two centers still remained, indicating that at least four types of cen ters are introduced by neutron irradiation at room temperature. We also found that the intensity of the lines found in these ma terials increased with irradiation (the range of flux used was 1011-10'9), indicating that these resonances are produced by a re distribution of electrons on defects rather than by electrons origi nating from impurities. In samples whose post-irradiation Fermi level was closer than 0.1 ev to the conduction band or 0.2 ev to the valence band, different types of spin-resonance absorption patterns were seen. Magnetic Susceptibility and Electron Spin Resonance-Experimental G. D. WATKINS L. Slifkin: I have a question about how the irradiation-induced vacancies get to the oxygen atoms. For the case of germanium, subtracting Logan's value for the creation energy of vacancies, which is a little over 2 ev, from the self-diffusion activation energy of less than 3 ev, one obtains a little less than 1 ev for the mobility energy. This gives, for the temperature range in which the A centers are produced, a frequency of vacancy motion of about 1 per day. It might be expected to be even less in silicon, which has a higher melting point and, presumably, a higher activation energy for vacancy motion. It is, therefore, a bit difficult to see how the vacancy can diffuse to the oxygen atom in silicon at such a low temperature. H. Brooks: Is it absolutely excluded that the oxygen moves? A. G. Tweet: Although oxygen is interstitial, it is bonded and, therefore, probably does not move. I refer you to work of Logan and Fuller. G. D. Watkins: Is it possible that in self-diffusion measurements one is dealing with aggregates of vacancies and that the only place where you really form a single vacancy is in the radiation experiment? H. Brooks: The binding energy of the vacancies would have to be pretty high. A. G. Tweet: If diffusion proceeded by both vacancies and divacancies, a break would appear in the activation energy curve for self-diffusion. The fact that no such break has been observed implies that diffusion is all going by divacancies with a binding energy of 3 ev or more, or by single vacancies. G. H. Vineyard: At least two things about these annealing processes deserve comment. In your Fig. 5 two stages are visible in the formation of the A center. No simple movement of one kind of defect is going to explain that. Second is the fact that on anneal ing to 3000K only 3% of the number of centers expected for room temperature irradiation remain. Perhaps you have an explanation. G. D. Watkins: I think Dr. Wertheim will be able to say some thing about the second of your comments. (G. K. Wertheim presented some results concerning the tem perature dependence of the introduction rate of the A center in silicon for electron bombardment. These results have been sub mitted for publication in The Physical Review.) J. Rothstein: An oxygen atom coupled to a vacancy should have a dipole moment. It is conceivable that this might be detected by measuring dielectric loss. Moreover, one might pick it up by measuring infrared absorption. It seems possible that if one applied a polarizing field one might actually get a Stark-splitting of the level. Nature of Bombardment Damage and Energy Levels in Semiconductors J. H. CRAWFORD, JR., AND J. W. CLELAND R. W. Balluffi: Would you expect a large range of sizes of the damage spikes? J. H. Crawford: Yes. A large range would be expected. This is the reason that two different sizes were shown in the figure; the large size is completely blocking, whereas the small size is not. What we have done here, in essence, is to arbitrarily split the damage into two groups. The first group is composed of isolated defects or small clusters which produce the effects of isolated energy levels, whereas the large groups are envisioned as affecting almost entirely those properties requiring current transport. W. L. Brown: I would like to comment on your p-type bombard ment results. These are in contrast to what we have seen with electron bombardment of specimens at liquid nitrogen tempera ture. While you observe an increase in hole concentration we observe only a decrease in hole concentration over the entire tem perature range when the bombardment occurs at 77°K. R. L. Cummerow: Our results with electron bombardment at room temperature in which n-type material is converted to p type seem to agree with your proposal of a limiting Fermi level .\*. The question I would like to ask is, does the mobility determina tion support your two-level scheme? That is to say, trapping with the Fermi level halfway between these two levels would require that mobility decrease with long bombardment, whereas if a single level and annealing were responsible for the limiting con ductivity. the mobility should show no further decrease. J. H. Crawford: We have insufficient data on the mobility in the impurity scattering range to decide this point. H. Y. Fan: Do I understand that you assume that the energy levels near the center of the gap that determine the behavior of converted material are produced at about the same rate as the 0.2-ev level below the conduction band? J. H. Crawford: Perhaps. The only way one can obtain a limiting Fermi level value is by the introduction of a filled level near 0.2 ev above the valence band and an empty state quite near the center of the gap at the same rate. H. Y. Fan: In the case of electron bombardment our results indicate that the level near the middle of the gap is produced in relatively low abundance. J. H. Crawford: For the gamma irradiation we know simply from the shape of the Hall coefficient vs temperature curve that there is a deep vacant state present in concentration equal to the 0.2-ev state below the conduction band. R. L. Cummerow: I have some evidence that there is a level in the center of the gap, but it is somewhat indirect and was obtained by analyzing the potential variation in the junction produced by electron bombardment. Precipitation, Quenching, and Dislocations A. G. TWE>;T W. L. Brown: It has been shown that electron irradiation altered the precipitation of lithium in a manner that would indicate that [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.55.97 On: Tue, 09 Dec 2014 05:31:48
1.1707893.pdf	HotElectron Transfer through ThinFilm Al–Al2O3 Triodes O. L. Nelson and D. E. Anderson Citation: Journal of Applied Physics 37, 66 (1966); doi: 10.1063/1.1707893 View online: http://dx.doi.org/10.1063/1.1707893 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/37/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Characteristics of electron emission of Al-Al2O3-Ti/Au diode with a new double-layer insulator J. Vac. Sci. Technol. B 32, 062204 (2014); 10.1116/1.4900632 Hot-electron degradation in hydrogenated amorphous-silicon-nitride thin-film diodes J. Appl. Phys. 89, 5491 (2001); 10.1063/1.1364652 Hotelectron impact excitation of ZnS:Tb alternatingcurrent thinfilm electroluminescent devices J. Appl. Phys. 78, 2101 (1995); 10.1063/1.360188 Experimental Evidence of HotElectron Transport through Thin Metal Films J. Appl. Phys. 43, 1830 (1972); 10.1063/1.1661404 HotElectron Transport in Al–Al2O3 Triodes Produced by Plasma Oxidation J. Appl. Phys. 39, 5104 (1968); 10.1063/1.1655931 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:53JOURNAL OF APPLIED PHYSICS VOLUME 37, NUMBER 1 JANUARY 1966 Hot-Electron Transfer through Thin-Film Al-Al 20a Triodes* O. L. NELSONt AND D. E. ANDERSON Physical Electronics Laboratory, University oj Minnesota., Minneapolis, Minnesota (Received 15 July 1965) Triode devices, consisting of Al-AI.03-AI-AI.0 3-Al films, were used to inject hot electrons into an oxide film. Transmission ratios were measured as a function of collection-oxide thickness, collection bias, and injection at 7r and 300oK. These data were compared with a model for hot-electron penetration in which electron-electron interactions in the metal were invoked, with ll. mean free path 1, ex: (E-E/)-2; once-scattered electrons were included in the collected fraction. Assuming an energy loss of 0.1 eV per interaction in the oxide (from optical absorption data), the comparison of the model and the experimental data yielded a mean free path between these interactions of approximately 12 A. The predicted transmission ratios agreed quite well with the experimental data. I. INTRODUCTION RESULTS pertaining to hot-electron interactions in anodized AhOs films have been reported which were obtained from measurements of electron emission into a vacuum from diode structures. Kantor and Feibelman1 measured the transmission dependence as a function of Au overlayer thickness and extrapolated to zero thickness for several AbOa thicknesses, from which an attenuation length of 24 A in the oxide was obtained. Collins and Davies,2 using a somewhat similar procedure incorporating the energy distribution of emitted elec trons, deduced an attenuation length of approximately 5 A and large energy-loss interactions. In the above investigations the oxide involved was subjected to high fields, and the determination of attenuation length was somewhat indirect. Thin-film metal-oxide tunnel triode devices have been fabricated and suggested as possible active circuit elements.3,4 Such devices also provide a convenient structure in which to investigate hot-electron inter actions, because the energy of the injected electrons and the position of collection can be systematically varied by adjustment of the various film thicknesses. In particular, the injection parameters can be main tained constant for several units while the collection oxide thickness is varied. Also the field across the collection oxide can be independently varied. The triode samples used in this investigation were fabricated with a common thin middle base metal and oxide for each set of seven triode units. Measurements of the transfer ratio were obtained as a function of collection-electrode bias for each of the collection-oxide thicknesses. A range of injection parameters was obtained by using the several samples. * Work supported by the Aeronautical Systems Division, Air Force Systems Command, United States Air Force, under Contract No. AF 33(657)-10475. t Present address: 3M Company, Central Research Labora tories, St. Paul, Minnesota. 1 H. Kantor and W. A. Feibelman, J. Appl. Phys. 33, 3580 (1962) . 2 R. E. Collins and L. W. Davies, Solid-State Electron. 7, 445 (1964). 3 C. A. Mead, J. Appl. Phys. 32, 646 (1961). 4 G. T. Advani, J. G. Gottling, and M. S. Osman, Proc. lnst. Radio En~s. 50, 1530 (1962). 66 An analysis of the results requires a knowledge of the potential barrier presented by the collection oxide as a function of bias and thickness. This information was obtained from internal photoemission in thin-film Al-AI203-AI diode units," and used to interpret the triode results in terms of the variation of transfer ratio as a function of barrier height and collection-oxide thickness for a fixed barrier height. These results are then compared with models for electron transmission through a triode device. II. SAMPLE FABRICATION AND MEASUREMENT The triode samples were fabricated on glass sub strates which had been cleaned in detergent and H20 and then fire-polished. As illustrated in Fig. 1, seven Al tabs O.1SXO.OS in. were vacuum-deposited part way across the substrate. The evaporation was accomplished from a high-purity Ai filament supported by braided W wires in a glass vacuum system, using a Ti sputter pump at pressures less than 10--6 Torr. Each tab was then anodized in 3% ammonium tartrate, pH 5.5, using an Al cathode, to form an oxide film of the desired thickness. The anodizing voltage was maintained for 5 min and the resulting thickness was taken as 13 A/V. 6 The thin base-metal film was vacuum-deposited to overlay the ends of the tabs and the thickness was monitored using a quartz-crystal resonance frequency shift monitor. 7 ,8 This layer was then anodized as FIRE-POLl 0 METAL BASE FILM. ANODIZED ~~~Zt5~BSTRATE • FIG. 1. Geometry of thin-film triode samples. Electrode overlap area for each triode nominally O.05XO.05 in. ---- 5 O. L. Nelson and D. E. Anderson, Bull. Am. Phys. Soc. 11, 389 (1965); O. L. Nelson and D. E. Anderson (to be published). • G. Hass, J. Opt. Soc. Am. 39, 532 (1949). 7 S. J. Lins and H. S. Kukuk, Vacuum Symposium Transac tions (Pergamon Press Ltd., London, 1960), p. 333. 8 G. Sauerbrey, Z. Physik 155, 206 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:53HOT··ELECTRON TRANSFER THROUGH AI-AI,O. TRIODES 67 described above, except that the voltage was main tained for 15 min to minimize effects of higher film resistance, to form an oxide film common to all units. Finally, seven tabs were deposited to overlay the ends of the first tabs. These and the common oxide then provide nearly identical injection sources for the seven units. For notation purposes the bottom tabs are designated electrodes I, the upper tabs electrodes III, and the middle layer as electrode II or base. The common area of overlap was nominally O.05XO.05 in., but the actual areas of the injection diode, collection electrode-base overlap, and common overlap of the three films were measured using a microscope so the currents could be expressed as current per unit area. Current-transfer measurements were obtained using the circuitry shown in Fig. 2. The currents were measured using battery-operated Keithley electrom eters and the outputs were recorded as indicated. The collection current could be measured to 10-10 A with a meter volt-drop of less than 10 m V; the collection bias was provided by a battery and precision potentiometer with a dial. The collection circuit and sample were housed in a metal case to provide electrical shielding. Only one point in the circuit, the power-supply con nection to the base layer, was grounded. The program mable regulated power supply was driven by a motor operated potentiometer to provide a constant rate of change of voltage. Data were obtained by fixing the collection bias and driving the injection current to the desired maximum, then back to zero. III. EXPERIMENTAL RESULTS Twelve samples were fabricated using the geometry shown in Fig. 1. The film thicknesses are tabulated in Table 1. Entries appear only for those triode areas from which complete data were obtained. Some of the omitted areas had initially shorted oxides or ones which shorted early during measurement. Incomplete measurements were obtained from some other samples. Also included in Table I are the tunnel (injection) biases VB-III = -Veb required to inject 1 mA/cm2 from electrode III. Figure 3 shows a current-voltage characteristic for a typical injection-diode section. This agrees well with the tunnel-current analysis,9 with an effective injection barrier height of 1.50 eV. This effective barrier height, derived from a Fowler-Nordheim plot, is not the actual maximum barrier presented by the injection x-v RECORDER TIME BASE RECORDER FIG. 2. Triode measurement circuit. 9 R. H. Fowler and L. Nordheim, Proc. Roy. Soc. (London) A1l9, 173 (1928). 90 eo 70 60 "50 z o j::: Id 30 .., z 20 10 AI-AI-20, #5R, 300'K, SWEEP 17 II °OL-----LI----~2~--~3----~4~--~5--~ INJECTION VOLTAGE FIG. 3. Typical current-voltage characteristic for an injection diode, sample 20, 300 oK. Injection !ror,n elect~ode.III.of tri~de unit 5, sweep number 17. Arrows mdlcate directIOn III which recorder plot was obtained. Sweep rate 0.2 V /sec. oxide; from photoemissive studies· the true barrier maximum was approximately 2.0 eV. Figure 4 shows an example of the ratio of collection current to injection current vs injection current, ob tained from the current-time records, from a triode unit. The injection diode section was that shown in Fig.3. Data for two temperatures, 3000 and 77°K, are shown, with collection-electrode bias as a parameter. The numbers refer to the order in which the data were obtained. Slight aging and hysteresis effects were observed, and the current ratio shows a slight depend ence on injection current. No apparent correlation with other parameters was discovered for these effects. The first two are fairly commonly observed in thin-film tunnel devices, and the dependence of current ratio on injection current may result from current-density variations produced by resistive volt drops in the thin-metal films. These data are typical of those obtained for the other units listed in Table I. Data were obtained for injection from electrodes I as well as from electrodes III, but further discussion is limited to the latter case. :For further comparison the injection-current density was fixed by using the measured geometrical areas and the transfer ratio is then defined as the ratio of collected current density at electrode I to injected current density from electrode III. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:5368 O. L. NELSON AND D. E. ANDERSON TABLE I. Thicknesses of!AI~3 and Al films of tunnel-current triodes. Electrode I-base Base Base- VIII-base oxide thickness, X metal" electrode III for Ie III= 1.0 Sample #1 2 345 6 7 (X) oxide, (A) mA/cm2, V 13 39 52 65h 65 78 210 52 2.9 14 78 200 65 3.8 15 59 78 215 46 3.0 17 59 65 320 52 3.0 18 52 2 78 310 65 3.8 19 39 52 65 78 91 325 46 2.7 20 65 78 91 300 78 {4.7 (77°K) 4.8 (3000K) r7 (77°K) 24 39 39 52 65 78 260 52 2.8 (3000K) at 0.2 mA/cm2 .. The Al thickness was determined by total weight of Al evaporated and solid-angle arguments, and by using a quartz-crystal resonance-frequency monitor. The values are believed to be accurate to within 10%. The metal converted by anodizing was accounted for. b Although this film was formed to 59 11.. its tunnel-current-voltage characteristics suggest the oxide is 65 11.. Possibly connection was made to it during formation of the next film. The effect of collection bias on the current ratio can be obtained from these data. Figure 5 shows the transfer ratio vs collection bias for 1.0 mA/cm2 injection-current density from the several triode areas of samples 18, 19, and 20 at nOK. Figure 6 shows similar data from samples 20 and 24 at 300° and n°K. These data were obtained from many triode units with a fairly wide range of injection parameters and collection-oxide thicknesses. In spite of this the plots are very similar in shape, with some exceptions, and show a nearly ex ponential dependence of transfer ratio on collection bias. These data can be compared in terms of another parameter, collection-oxide thickness. Figure 7 shows the'transfer ratio for fixed injection-current density and at zero collection bias, vs the collection-oxide thickness for the several samples. There is scatter in these data, but the general dependence is quite similar for all samples. Also included are the transfer ratios for AI-Al-24, 3000K with bias values of + 1 V and -1 V. These are quite similar to those obtained for zero bias. From these data, it appears that a simple empirical rela tionship can describe the results quite well; a (V c,X 0) o· Ie/Ie I AI-AI-20,300 K o -5 10 -. 10 1,5 21 .. IZ via • II • • II • 19 It .. 20 II ... r ... 0.8 OV -0.2· • • ·-0.4 II I. 1-0.6 II • r-0.8 81 8. • • I-LOV AI-AI-20.77 K _5 -0.6V ~------6 ... 0.4 1()5~1~_~:;;;:1'=:::==;;~ %2 V ";:"~"'f""'I"ir--------OV OV -7 19' -f4i·-,....------0.2 ~9 ..... _------ 0.4 8 -------- 0.6 10 -----~--;------, .... O~-I'" 10-'" CURRENT. Ie., AM PS FlO. 4. Current transfer ratio Ie/I. vs injection current Ie as a function of collection biasYl2. Sample 20, triode unit 5, at 77° and 300cK. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:53HOT-ELECTRON TRANSFER THROUGH AI AI20a TRIODES 69 5 ~ ~. 2 c:f"AI-AI-20, ~ "'3 10'" ~ ~ 5 z ", I!: .... z 2 ! -t .10 ~ 5 oJ 8 tl 10 ~---~0~.6--~-0~.4~---0~.2-----0~---0~.2----0~.4----0~.6 "12' COLLECTION BIAS, VOLTS FIG. 5. Transfer ratio vs collection bias at 1.0 mA/cm! injection current density. Samples 18, 19, and 20 at nOR. cc exp[eV./'Y- Xo/B], where'Y is approximately 0.7 eV, o~16 A, and V. and Xo are the collector bias and oxide thickness, respectively. The electron-energy diagram characterizing the triode is shown schematically in Fig. 8. The interpreta tion of the transfer parameters of the triode in terms of a hot-electron model requires a detailed knowledge of the potential barrier profile presented by the collection oxide, including the dependence on bias, oxide thick ness, and temperature. These questions motivated an investigation of in ternal photoemission across the oxide of thin-film Al-Al20rmetal diodes, which were fabricated using the same techniques as described above. An analysis of the results showed5 a potential barrier profile which could be described quite accurately by the metal-insulator contact theory, modified by the image effect. Using Simmons' approximationlO for the image correction, this has the form (for a symmetric AI-AhOrAI diode) ¢(x, V) =<1>0-(eV .,x/Xo)-[aXo2/x(x-XII)], where V. is the applied collection voltage, Xo is the oxide thickness in angstroms, x is distance in angstroms measured from the base metal, and a is a parameter dependent on Xo and K, the relative dielectric constant, given by 5.75 (KXO)-l in electron volts. The high frequency value of K was chosen. <1>0 is a constant found experimentally to be slightly dependent on Xo, increas ing by 1/260 eV/A. If these parameters are inserted, 10 J. G. Si=ons, J. Appl. Phys. 34,1793 (1963). the barrier for 3000K can be approximated by Xo 2.12Xo ----,eV. x(x-Xo) Vi is an effective internal bias representing the asym metry, which for the units measured appeared to be approximately +0.2 eV, higher at the deposited elec trode side. If this is included, the constant should be adjusted appropriately to give the same barrier height at zero applied bias. At 77°K, <1>0 was approximately 0.2 eV greater and Vi appeared to be nearly zero . The transfer-ratio data are now analyzed in terms of this potential profile. Samples 20 and 24 are chosen as representative of the triode results. The barrier height maximum can be calculated for a given oxide thickness, applied bias, and temperature. This was done, and Fig. 9 shows the transfer ratio vs the barrier height maximum, ¢max, from units of these samples. Two choices of ¢max were used for the 3000K data, one with zero internal bias and one with an internal bias of +0.2 eV to account for the slight asymmetry. These plots should represent the integral of the energy distri bution of the collected electrons, modified by inter actions and collection factors in the collection oxide. An examination of Fig. 9 shows that the dependence -of transfer ratio on oxide thickness is not merely in the 10 tllO -I 10 101'-~--~--~----~----~----~--~~ -1.2 -0.8 -0.4 0 0.4 0.8 1.2 VIZ. COLLECTION BIAS. VOLTS FIG. 6. Transfer ratio vs collection bias at constant injection current density. Sample 20 measured at 3000K and 1.0 mA/cm!' sample 24 at 300° and nOK, 0.2 rnA/em!. • [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:5370 O. L. NELSON AND D. E. ANDERSON IO~'r-----------------------------------' , , \ AI-AI-24,3Od'.,A 0.2 mA/Cfff', "'...... \ ......... V 'tiV "', , "', \ "'" '\. ':'... ,--..... 6 ~ ...... "'Ie ...... " IO~L--L----~~~~--~-~~--~~ 40 50 60 70 80 90 BASE-COLLECTION ELECTROOE OXIOE THICKNESS, X., A FIG. 7. Transfer ratio vs Al20a thickness at zero collection bias, 1.0 mA/cm2 injection-current density (except 0.2 mA/cm2 for sample 24). Samples 13, 18, 19, 20, and 24 at 300° and/or 77°R. Data for sample 24 also shown at + 1 V and .,..-1 V bias. change in barrier height. At a given barrier height, the transfer ratio decreases with increasing oxide thickness. Similar dependence is seen at both 77° and 300oK, with higher transfer ratios at 77°K for a given barrier height. This is shown more explicitly in Fig. 10, where trans fer ratios at constant barrier height are plotted as a function of oxide thickness. For convenience we have chosen the zero bias value of CPmax for 78' A thickness in this case; this corresponds to 2.2 eV at 3000K and 2.4 eV at 77°K. The results presented in Fig. 10 suggest a nearly exponential dependence of transfer ratio on collection- METAL!.) INJECTION OXIDE -XM !!...Q n •• ( IE ,IE. I METAL(c) X-O FIG. 8. Schematic electron-energy diagram for triode analysis. All energies measured with respect to the bottom of the conduction band of the base metal. oxide thickness. The dashed lines a and b of Fig. 10 show exponential characteristic lengths of 13 and 20 A. The interpretation of these data require the con struction of a physical model. Two basic approaches are possible; either the collected electrons penetrate through the thin base metal or they flow through pinholes in the metal. Results from early samples with different geom etry and with known "pinholes" resulting from im proper film registry demonstrated that a proper choice of pinhole size and density could yield results qualita tively similar to these, but also demonstrated a strong dependence on the "pinhole" dimensions. The results presented above show consistency from unit to unit on each sample and also between samples fabricated at different times. This regularity would be 10 ... 010 ~ -. 10 1.8 2.0 2.2 '" 2.4 't'_, .v 2.6 2.8 3.0 FIG. 9. Transfer ratio at constant injection-current density vs calculated collection·oxide barrier-height maximum. Dependence of <Pm." on thickness, temperature, and bias has been included. expected if the electrons penetrate the metal film, but seems somewhat surprising from a mechanism relying upon random pinhole defects. These arguments are, of course, not conclusive, and in fact some of the effects observed from individual triode units may best be described by the pinhole assumption. Nevertheless, it is implicitly assumed in the next section that the col, lected electrons have penetrated through the metal film, and it is shown that quantitative agreement between the model and experiment can be obtained. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:53HOT-ELECTRON TRANSFER THROUGH AI-A120, TRIODES 71 IV. A POSSIBLE MODEL FOR THE TRIODE DEVICE: COMPARISON WITH EXPERIMENTAL DATA A. Interactions in the Base Metal with a Simple Collection Barrier The thin-film tunnel triode consists of five regions: emitting metal, tunnel barrier oxide, very thin metal base, collection oxide, and finally collection metal. These were shown schematically in Fig. 8 where the distances and energy levels were defined, and where electron distributions were also sketched. The electrons which tunnel from the emitter arrive in the middle (base) metal and have a certain probability of traversing it and continuing through the collection oxide to reach the collector. This probability, called the transfer ratio, should be a function of the tunnel-injection energy €i, the base metal and collection-oxide thicknesses XM and X 0, the electron range in metal and oxide 1M and 10, the oxide potential barrier <I> (x, Vc), and the collection bias V •. Consider first the injected beam of electrons. These electrons are assumed to tunnel from energies near the Fermi energy in the emitter and are incident on the middle metal with an energy distribution determined by the tunnelling probability and insulator interactions. The half-width of the total energy distribution of 10-4 -I 10 . a Ae -X./20A b Se -X.1I3A C/X. • d O/X ••• -X./30A data adjusted to equivalent barrier height by optical results IO'-~40~--~50~-----~50~--'ro~--~8~O----~90~.~ BASE-COLLECTION ELECTRODE OXIDE THICKBS, X., A FIG. 10. Transfer ratio at constant CPm.x vs collection-oxide thickness. <Pma" was chosen as the zero-bias value for a 78-A oxide, as determined photoelectrically. Samples 20 and 24 at 300° and 77°K. Calculateda(X o} for several models shown as dashed lines. electrons which tunnel from an idealized metal at low temperatures into vacuum (image forces neglected) is given in terms of tunnel-equation parameters.!l If the experimental parameters obtained from the triode devices are used in this relationship, a half-width of approximately 0.05 eV is predicted, much less than the apparent width seen from the transfer-ratio data. This observation must be incorporated into this analysis. Collins and Davies2 assumed that the spread in energy of hot electrons emitted into vacuum was a result of many strong interactions in the tunnel barrier oxide. They assumed also that only electrons which did not interact in the metal overlayer could be collected. It is difficult to explain such large energy-loss inter actions in an insulator (although there is evidence that they may occur in Ta205).12 Further, it is demon strated that such interactions are not necessary to explain the spread in energy if once-scattered electrons in the. metal which are still energetically capable of escape are included among those collected. We now assume that the tunnelling electrons interact nearly elastically in the oxide. Even if the direction of these electrons were completely randomized, the inci dent beam would be narrowed to a small cone upon entering the metal, as is discussed later. For the present, then, the incident beam will be approximated as mono energetic and normally incident on the base-metal film. This approximation is slightly relaxed when collection of that fraction of electrons which suffers no interactions is discussed. Finally, this approximation is re-examined in Sec. IVC. In the base metal these electrons will interact most strongly with the conduction electrons. An electron electron mean free path le(~) is assumed which is a function of the incident electron energy. The prob ability that an incident electron suffers an inelastic collision between x and x+dx is thus pc=exp[ -(XM+x)/I.( ~i)Jdx/le( ~i)' These electrons can be viewed as a supply function for further propagation. Since they lose energy to con duction electrons because of the interaction, we re quire the conditional probability p(el Ei)dE that an electron of energy ei is scattered to an energy between E and e+dE given that it suffers a collision. Berglund and Spicer13 have considered this problem under the assumption that the scattering matrix is a constant, independent of the various k vectors of the electrons involved. Their result is where 1l R. D. Young, Phys. Rev. 113, 110 (1959). 12 C. A. Mead, Phys. Rev. 128, 2088 (1962). 1& C. N. Berglund and W. E. Spicer, Phys. Rev. 136, A1030 and At044 (1964). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:5372 O. L. NELSON AND D. E. ANDERSON and P.(f,fi)df= p(f)[1-feE)] t 211" 1 M.12 Jo h Xp (71)f(T/)p(T/+ E;-£)[1-f(71+E,;- e)]d71dE, with p( E) = the density of electron states, f( E) = Fenni partition function, 1 M.I = scattering matrix, 71 = energy of target electron before interaction. Assuming the density of states is given by p( E) = DEt, D= 411" (2m)i/h3, which is the fonn for the free-electron approximation for a metal, this can be integrated over 71. To obtain a more tractable expression, p. was expanded in powers of (E;-E)/EF, and peE;) in powers of (E,;-EF)/EF. In this fonn, the energy distribution is P.(E;,E)dE/P(Ei) ~2(E/ E;)I(E';-E)/(E,;-EF)2dE, forDS E,;-E< E,;-EF«EF. As a consequence of the expansions, this conditional probability no longer has a unity integral over all E. However, the nonnalizing factor would change by less than 8% for typical energies. The probability that the electron which has energy E was the target electron rather than the incident one is accounted for simply by multiplication by two in this approximation. The total probability of a collision, P( Ei), is effectively the reciprocal of the lifetime of an electron with energy E,;. Thus the mean free path can be defined as 1.( Ei) = v/ P, where v is the electron velocity, or group velocity of the wavefunctions. Assuming the free-electron velocity, 1.( E';)= const. ENP(E,;)-::::.L/(E.'; EF-l)2, where L is a constant involving the interaction matrix. Quinn14 calculated an approximate mean free path for Al of 1000 A for E,;-EF= 1 eV, with approximately the above energy dependence, for (E,;-EF)«EF. Calcula tions made by Sparks and Motizuki15 using the Y aka wa potential approach agreed with Quinn's expression for the low-energy, high-density limit. The Fenni energy for Al has been calculated by Segall16 to be approxi mately 12 eV, in agreement with soft x-ray experi ments,17 so we set L= 1000/144 A. Now we must consider the problems of collection or subsequent interactions for these once-scattered elec trons. The distribution of these electrons is of the fonn et(E;-E)dE. For example, if E,;-E~2(EB- EF), where EB is the energy barrier for escape, roughly 25% of these electrons are energetically capable of escape. If these are assigned an average energy of HEB-EF)+EF, of the order of 7% of these will be energetically capable of escape after a second interaction. Further, their average collection cone will be smaller than for the once-scat tered electrons. Thus we assume that electrons which suffer:more than one collision are no longer capable of escape. The probability that an electron with energy be- 14 J. J. Quinn, Phys. Rev. 126, 1453 (1962). 16 M. Sparks and K. Motizuki, J. Phys. Soc. (Japan) 19,486 (1964). 1~ B. Segall, Phys. Rev. 124, 1797 (1961). 17 F. Seitz, The Modern Theory of Solids (McGraw-Hill Book Company Inc., New York, 1940), p. 436. tween E and E+de at x will reach the metal-collection oxide interface without suffering a second collision is exp[x/l.(E) cosO], where () is the direction in which the electron is moving relative to the x axis. We assume that the electrons were scattered isotropically in direc tion, which was implicit in the assumption of constant scattering matrix. By combining the probabilities for each step of the process, the probability that an electron initially in jected normally into the base metal will be transmitted to the base-metal-collection-oxide interface with energy between E and E+dE after suffering one electron electron interaction can be detennined. Multiplying by the number of incident electrons per second No, the number transmitted per second to the oxide interface at x= 0 after suffering one interaction is nlc(ei,E,l.,cosO,X M)dEd(cos(}) (Ei- ~) =4N O(E/ Ei)! ded(cosO) (Ei-EF)2 X /0 e[-(X M+X) Il'('i)le[XII'(')COS91~. -XM l.(ED Now we must consider the base-metal-collection oxide interface. As previously discussed, the potential profile presented by the oxide is a function of distance, not defined at the interfaces (as there presented it went to -00, which is not realistic). The image-modified pro file must be truncated in some manner. We shall assume for the present that the profile can be represented by a step function from 0 to EB at the interface. The inter pretation of EB will be discussed later for several assumed models for interactions in the oxide. However, for the simplest assumption of no energy losses in the oxide and a mean free path for elastic scattering much longer than the oxide thickness, EB will just correspond to the potential-barrier maximum ¢,u",,+ EF' Let us now find the fraction of incident electrons which can just enter the oxide by traversing the inter face potential EB. In the previous expression cosO enters in one exponential argument. The range of cosO will be from 1 to (EB/E)'2::(EB/Ei)t, assuming again a free electron-energy-momentum relationship. Theseenergies are measured from the metal conduction-band edge, and E~12 eV for AI, so for the present application (EB/ Ei)c~d4/16, yielding a minimum cosO of about 0.93. The cos() modifications of the collection path are there fore ignored, and cos(} will be approximated by 1 in the exponential term. Integration with _respect to x then yields the number of electrons per second arriving at the metal-oxide interface (x= 0) ; nIc( Ei,e,I.,X M)ded(cos(}) (Ei-E) XM =4No(ejEi)t,-- (Ei-EF)21.(Ei) [e-XMfl6(,l_e-XMlle('il] X ded(cos(}). XM/l.(Ei)- XM!l.(E) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:53HOT-ELECTRON TRANSFER THROUGH AI-AhO. TRIODES 73 Substitution for l., multiplication by a transmission 800,..-----------------_ factor T assumed18 to be t, and integration over cosO from 1 to (fBI e)! yields the number which can just traverse a potential barrier EB and arrive at x=O+; nlC(e;,E,L,X M,EB)dE e[-(XMI L) (.t<F-])'J_ e(-(XMIL) ('i!<F-l)2j X---------------------- for Ei~ E~ fB. The electrons which arrive at x=O having suffered no collisions in the metal, must also be considered. We had assumed these to be normally incident on the metal because of the focusing effect of the metal. When they enter the collection oxide, however, a large portion of their kinetic energy will be lost to provide the potential energy required by the barrier. Thus any spreading of the beam in the metal as a result of elastic collisions (e.g., electron-phonon) will cause some fraction of these electrons to be reflected at the oxide interface. Further, the assumption of a mono energetic beam will be questionable when considering this collection. We thus assign a collection factor C to the nonscattered electrons which may be a function of all the parameters already considered, as well as the energy distribution of the tunneling electrons and the electron-phonon interaction range in the metal. The fraction of electrons which can enter the oxide will then be given by +2/" [(e/Ei)L (EB/E,)l] <a X {exp[ _:M (e!EF-l)2] The dependence of O!, on the various parameters is obscured by the complexity of the expression. To obtain a pictorial impression of this dependence nlc(E)/No WaS calculated for selected values of the parameters. Some 18 The plane-wave solution for wave function transmission over a narrow square potential barrier typical of those encountered here showed rapid oscillation of T with energy between unity and an increasing lower.envelope. The arithmetic average of these bounds increased from 0.5 to 0.7 from ('-Ea)=O to (E-EB)=3 eV. 600 CURVES (I b C XII 3001 400A 3001 EI-E" s.v 5eV 3.V ....!!!..lCIO· IN. 275 23 33600 '>400 . .-2 .. -z ... J -200 o 2.0 E-E" •• v 4.0 15.0 FIG. U. Once-scattered electron distributions vs energy cal~ culated from. the model for selected parameters: EF= 12 eV, L=l000jl44A, EB-EF=2.0, 2.2, and 2.5 eV. of these are shown in Fig. 11 for choices of parameters applicable to the experimental devices. Also listed are the calculated values of no/2N 0, the nonscattered electron contribution for C= 1. Graphically, this con~ tribution would be represented by an incident-energy distribution sharply peaked at E;, with integrated area of the cited values. Curves of this type were graphically integrated for the following choices of parameters: L= 1000/144 A; Ep= 12 eV; E.= 17 eV; and XM=300 and 400 A. The results were plotted as a function of barrier height EB on Fig. 12. Curves a and c contain only the once scattered contribution (C=O) and curve b contains the full nonscattered contribution (C= 1). In the special case of no interactions in the collection oxide which we are now assuming, these curves corre spond to the predicted transfer ratio. eB-EF is simply the potential-barrier maximum c/>max presented by the oxide, which would change with applied bias as was discussed previously. The parameters chosen for the calculated curves should be pertinent to the experi mental triode AI-Al-20, :IF 5, and the data from this unit are presented on the figure. The calculated c/>max vs bias relation for a 7S-A oxide film was used and c/>max for zero bias was chosen as 2.2 eV. Comparisons of these data with the calculated curves show a difference in magnitude and in detailed structure, but the general barrier-height dependence is similar. Another point of comparison of this model with the data is the oxide-thickness dependence of the trans fer ratio. This model would predict no dependence [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:5374 O. L. NELSON AND D. E. ANDERSON 10 -4 10-' -. 10 0 ~ cr cr III u. II) ! .... I07L--L ____ ~ ____ ~--~~--~~--~~--~. 1.8 2.0 2.2 2.4 2.6 2.8 3.0 E.-(iF eV r FIG. 12. Calculated transfer ratio vs barrier height for several approximations and parameter choices. Experimental data from Al-Al-20, # 5 (78-A collection oxide, 3000K) shown for com parison. (a) n'(EB), incident once-scattered fraction. XM=300 A, EI-EF=5 eV, C=O. (b) n'(EB), XM=300 1, Ei-EF=5 eV, C=1. (c) ni(EB), XM=400 A, E'-EF=5 eV, C=O. (d) [Curve a] 'e-Xmax/l O X, Xo= 78 A. (e) [Curve a} (1/26) 'e-Xmax/30 X, Xo=78A. on oxide thickness after the experimental barrier height versus thickness is corrected for as in Fig. 10. The data on this figure do not agree with this prediction. We must conclude then that the model assuming no inter action in the oxide is too simple. B. Model Incorporating Interactions in the Collection Oxide We shall now consider in more detail interactions in the collection-oxide film. Referring again to Fig. 8, the oxide presents a potential barrier c/>(x). The collection metal at x=Xo is biased to aid or retard collection of electrons across the oxide. We have presumed in dis cussing the injection oxide that the impinging electron interactions in the oxide are of an elastic nature, i.e., electron-phonon interactions with large momentum (direction) changes but relatively small energy loss. This type of process is very difficult to treat in the collection oxide if the distance between interactions is short. The electrons may still be collected after a number of interactions which is large for a trajectory following technique, but small for a diffusion type of analysis. Also, for a large number of interactions the small energy losses associated with each one should no longer be neglected, and thus the details of the barrier shape become important. Application of bias can further complicate the situation. One limiting case, that of negligible energy loss and long mean free path, has already been discussed. Another limiting case for interactions in the oxide would be the assumption of large energy losses for each interaction. (This is, incidentally, incompatible with the previous assumptions concerning the incident tunnelling beam, but its implications can now be more fully assessed.) Then electrons which suffered an inter action before reaching the position of the barrier maxi mum would be returned by the small retarding field j the rest would be collected. For a symmetric barrier with zero bias, c/>max would occur at Xo/2 and a plot of transfer ratio vs oxide thick ness would then be of the form exp ( - X 0/21eo). This is compatible with the data of Fig. 10 if leo, the average x distance between these strong interactions, is chosen between 6 and 10 A. However, experimental data of a vs X 0 for + 1 V, OV and -IV bias had comparable slopes as seen in Fig. 7, while the predicted slopes would be widely different. For example, xmax(+IV)~.2Xo and Xmax( -1 V)~0.8X 0 so the predicted apparent slopes would be 50, 20, and 12 A for + 1 V, 0, and -IV, respectively. Under the assumption that the electrons drift against a slightly retarding field until they cross the barrier maximum, an indication of the bias dependence is given by curve d of Fig. 12, where curve a was used as the basis. EB is still interpreted as the barrier maximum. Xmax is the calculated position of the barrier-profile maximum as a function of bias for the theoretical image-modified c/>(V). As was previously concluded, this model does not agree with the experimental data. A third model for the interactions in the oxide can be formulated assuming a fairly large number of lossless interactions. Consider first the case of no external bias and assume for the moment the potential profile is a constant value EB. We shall assume an energy-independ ent range r between collisions, with probability dis tribution per) and an isotropic direction distribution .. Then we will define an average distance 10 which an electron moves in either the + or -direction between each interaction given by 10= (x+)= t 111'" cos8rp(r)drd(cosO) Here lp is the mean free path j we thus examine the simpler random-walk problem with fixed x-directed increments of +10 or -10, The electrons are all injected in the +x direction so they can be assumed to originate one 10 unit into the oxide. The collection electrode is Xo/lo units away. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:53HOT-ELECTRON TRANSFER THROUGH AI-AI20a TRIODES 75 Both the 0 and X 0/10 positions are assumed absorbing, and we want the probability of absorption at Xo/lo. This problem is discussed by Feller19 as the "ruin" problem and the solution, for equal probabilities of + or -jumps and starting from the first position, is p (collection) = 10/ X o. This thickness dependence, (con stant/Xo), is shown as Curve c on Fig. 10. Note that the-shape is independent of to (but does not hold unless Xo/lo»l) and does not agree very well with the experi mental data. The dependence of transfer ratio on barrier height would be just the calculated ai, multiplied by lo/Xo for the assumed constant barrier. For the image modified barrier profile, electrons could be scattered into or out of the collection cone of escape. In general EB would be replaced by ¢(x)+ Ep, and the solution would become very complex. The expected number of jumps before absorption is also discussed by Feller. For equal-jump probabilities for the present problem, starting from the first position (x= 10), the expected number of jumps is (Xo/lo-l). If Xo/lo is large then, even though the energy loss per collision is small, the total loss should not be ignored. Typically the energy exchange in electron-phonon interactions is of the order of a few hundredths of an electron volt. Harris and Piper20,21 have measured the optical parameters of thin-film AbOa in the infrared region and found an absorption edge at approximately 0.1 eV. If it is assumed that the incident electrons can excite this absorption mechanism (possibly optical phonons), then the total energy loss affecting collection could be approximated by 0.1 eV times the expected number of interactions before they passed the barrier maximum at xmax• For a symmetric barrier at zero bias, this would be 0.1 (Xo/2Io) eV. The effect of this would be analogous to an apparent increase of the barrier height by this amount. From the calculated fraction of electrons which just enter the oxide vs EB shown in Fig. 12, Curve a, it is found that for the parameters used there ai( EB) ex: exp[ -EB/0.45]. The first-order effect of these small energy losses as a function of oxide thickness would then be to multiply the impinging fraction of electrons by exp[ -!(Xo/lo) (0.1/0.45)]. To appraise the validity of this estimate it will be used to improve the agreement between the experi mental data of Fig. 10 and the calculated Curve c. It is seen that fair agreement would obtain if an exponential multiplicative term decreasing by a factor t from Xo=40 A to Xo=80 A were included. Thus (40/1oH!:::: In(O.25) or ZrE:;!;3 A and the apparent thickness dependence of the energy-loss term is exp[-Xo/30 A]. The apparent thickness dependence can now be used with the calculated barrier maximum position as a 19 W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley & Sons, Inc., New York, 1957), Vol I, 2nd ed., Chapter XIV. 2l) L. Harris and J. Piper, J. Opt. Soc. Am. 52,223 (1962). 21 L. Harris, J. Opt. Soc. Am. 45, 27 (1955). function of bias. The resultant transfer-ratio variation with bias is shown as Curve e of Fig. 12, where Curve a was again used as the basis. Although the general slope of this curve is in good agreement with the data, the detailed shape is not. It may be pointed out, how ever, that both tf>,nax(V) and xma,,(V) for this plot are calculated and in particular do not account for the transition regions between metal and oxide which were discussed in regard to the photo threshold data. Inclu sion of this effect would probably tend to smooth the energy dependence of Curve e. Further, no effect from the variation of EB with distance, and hence change of collection cone across the oxide, has been included. Several types of interactions of hot electrons with the oxide have been considered. Although these were not exhaustive and invoked several approximations, the last one predicts results in good agreement with the data. It is also physically realistic in that it incorporates small-energy-loss interactions such as expected for electron-phonon exchanges. The effective mean free path it provides, 1~12 A, is within the expected magni tude22 for a material such as amorphous AI20a• One further point of agreement can be presented. The magnitude of the transfer ratio predicted from this model for a triode with the dimensions of AI-Al-20, # 5 at zero bias is ai(EB= 14.2 eV) . lo/Xoe-XO! 6()rv6 X 10-6• If all of the nonscattered electrons are included, a""'9X 10-6• The value obtained experimentally was 8X 10-6• Further, if the transfer ratio vs X 0 data of Fig. 10 are extrapolated back to Xo=O, the intercept is approximately 10-a. The value of ai(EB= 14.2 eV), pre sumed to be the transfer ratio in the absence of inter actions in the oxide, is 6X1D-4. This excellent quantitative agreement may be some what fortuitous, but does indicate internal consistency and prediction of values of the same magnitude as the experimental data. C. Effect of Oxide Interactions on Injection Assumptions The model which evolved in the preceding section incorporated electron interactions in the collection oxide and, in fact, assumed an energy loss of 0.1 e V per interaction. The injected tunnelling electrotl beam was, however, assumed to suffer negligible interactions in the injection oxide. This apparent inconsistency will now be discussed. One method of accomplishing this would be to proceed through the construction again, with a distributed in jection source. Since a number of assumptions and approximations were made which limit the quantitative 22 An energy loss of approximately 0.1 eV per collision has been incorporated into an analysis of secondary electron escape in MgO and effective ranges of the order of 5 A have been deduced. See, for example, W. S. Khokley and K. M. van Vliet, Phys. Rev. 128, 1123 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:5376 O. L. NELSON AND D. E. ANDERSON accuracy of the calculation, this is justified only if the resulting changes are of appreciable magnitude. It will be argued that at least for the case of the example followed in the previous section the change will be small. Since the collection factor for nonscattered electrons was not determined, the analysis is applicable for in jection energies large enough so this contribution is small. For this case, the electrons tunnel through the injection-oxide barrier and appear somewhere in the conduction band of the oxide. Although the description of tunnelling electrons is somewhat nebulous, for the specific example discussed above (e,-EF= 5 eV), these electrons first appear some distance into the injection oxide, subjected to a strong field. For an oxide of 80 A with a 2.2-eV barrier, the electrons may be in the oxide conduction band a distance of 45 A before reaching the metal. With the postulated mean free path of 12 A, x-directed range23 of 3 A, they may then suffer of the order of 5-10 interactions before injection into the metal. For the postulated energy loss of 0.1 eV per interaction, the energy spread would then be of the order of 0.5 to 1 eV. Consideration of the once-scattered electron energy distribution curves of Fig. 11 and the expression from which they came will indicate that a lower injection energy will primarily eliminate the high-energy tail. Since this tail is a small contribution to the total inte gral, the number of collected once-scattered electrons should not change appreciably. The nonscattered con tribution would be increased, but we have not con sidered this in detail. To illustrate this, let us consider the particular example used previously. For Et-EF=5.0 eV, XM=300 A, EB= 2.2 eV, the once-scattered contribution to the incident transfer ratio is 5.7XID-5 for a simple barrier as seen from Curve a, Fig. 12. The total nonscattered contribution would be 2.75XID-5 if all were collected. Now for E.-EF=4.5 eV, the once-scattered contribu tion would be 5.4XID-5 and the nonscattered contribu tion 5XIo-5. It is thus seen that the change of injection energy does not change the results appreciably for this example. We see then that the model could be made self-consistent with respect to interactions in the oxides and the results, for the choice of parameters of interest here, would be approximately the same. tt The forward focusing of the high field would probably tend to increase the mean +x-directed distance between collisions, resulting in fewer collisions before reaching the base-metal film. V. CONCLUSIONS The thin-film triode devices provided information concerning hot-electron penetration of the oxide films. These results depend strongly on hot-electron inter actions in the thin base-metal film. Since experimental data are minimal for such interactions in Al films, theoretical analyses were utilized. Further, the data from these devices could be dominated by small atypical areas (such as pinholes or areas of defects) and so must be viewed with some caution. This concern is somewhat allayed by the self-consistency of the data and the regular dependence on the various parameters. The analysis of these data, utilizing the potential barrier profile obtained from the photoelectric measure ments, showed strong evidence of interactions of hot electrons in the oxide. The exact nature of these could not be deduced, but large-energy-Ioss interactions are not consistent with the data. A model was developed which included electrons which had been scattered once in the metal. A mono energetic electron beam incident on the thin base metal film via tunnelling was assumed for simplicity of the calculations, but a small energy spread would not affect the results appreciably, for the range of parameters used for comparison with the data. Only electron-electron interactions were considered in the metal and a mean-free path was assumed of the form l.a:. (E-EF)-2. The once-scattered electrons were as sumed to have an isotropic direction distribution and those with x-directed energy in excess of a potential barrier EB at the interface could enter the oxide. A col lection factor was assigned to the nonscattered electrons which reached the interface. Under the assumptions of this model, an energy loss of 0.1 eV per collision in the oxide (in agreement with an optical absorption edge) was assigned. Using a random-walk approximation in the oxide, a value for the average distance between interactions was ob tained by comparison with the data. This was approxi mately 12 A, which, of course, depends to some extent on the assumptions of the model and the chosen values of calculated parameters. ACKNOWLEDGMENTS We wish to thank R. Sodoma who constructed the deposition masks and fabricated many of the samples, and E. D. Savoye for many helpful discussions. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sun, 21 Dec 2014 22:40:53
1.1777165.pdf	Absorption Edge in Degenerate pType GaAs I. Kudman and T. Seidel Citation: Journal of Applied Physics 33, 771 (1962); doi: 10.1063/1.1777165 View online: http://dx.doi.org/10.1063/1.1777165 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Optical absorption edge of semi-insulating GaAs and InP at high temperatures Appl. Phys. Lett. 70, 3540 (1997); 10.1063/1.119226 Experimental observation of a minority electron mobility enhancement in degenerately doped ptype GaAs Appl. Phys. Lett. 63, 536 (1993); 10.1063/1.109997 Modulation effects near the GaAs absorption edge J. Appl. Phys. 68, 6388 (1990); 10.1063/1.346891 ABSORPTION EDGE MEASUREMENTS IN COMPENSATED GaAs Appl. Phys. Lett. 5, 37 (1964); 10.1063/1.1754037 Optical Absorption Edge in GaAs and Its Dependence on Electric Field J. Appl. Phys. 32, 2136 (1961); 10.1063/1.1777031 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 00:25:48Journal of Applied Physics Volume 33, Number 3 March,1962 Absorption Edge in Degenerate p-Type GaAst I. KUDMAN AND T. SEIDEL Radio Corporation of America, Semiconductor and Materials Division, Somerville, New Jersey (Received August 11, 1961) Infrared absorption for p-type degenerate GaAs is studied at room temperature for various hole concen trations. At high absorption coefficients, a Burstein-like shift is observed for samples doped above 1019 jcm3; this shift is interpreted as a decrease in the valence band electron p3pulation. A direct transition analysis was made on 1017 jcm3 material, yielding an energy gap of 1.39±0.02 ev. The free carrier absorption was extrapolated to shorter wavelengths and subtracted from the data. The resulting absorption edges extend to energies beyond the fundamental edge and reveal the presence of an added absorption mechanism. I. INTRODUCTION DETAILED infrared studies on GaAs have been mainly restricted to wavelengths longer than the fundamental absorption edge. In this region, Braun steinl and Spitzer and Whelan,2 respectively, have studied p-and n-type GaAs. With regard to the "ab sorption edge," there is a lack of data for absorption coefficients greater than 103 em-I. This paper describes a study of the absorption edge to 1.4X 104 em-I. The main feature of this study is the effect of p-type de generacy on the shape and height of the absorption edge. Insofar as studies of other degenerate semiconductors are concerned, there is an abundance of literature, notably on InSb3 and Ge.4 These authors have observed shifts in the edge both toward shorter and toward longer wavelengths as the doping increases. For high values of the absorption coefficients a (in the order of 2-5X1()3), the edge of p-type GaAs shifts to shorter wavelengths, while for lower values of the absorption coefficients, there is a shift to longer wavelengths. The wor~ described was performed under the sponsorship of th.e .~lectron.1C Technology Laboratory, Wright Air Development DIVISIon, AIr Research and Development Command, United States Air Force. t Presented in part at the April Meeting of the American Physical Society in Washington, 1961; Bull. Am. Phys. Soc. 6 312 (1961). ' I R. Braunstein, J. Phys. Chern. Solids 8, 280 (1959). 2 W. Spitzer and J. Whelan, Phys. Rev. 114, 59 (1959). 3 G. Gobeli and H. Y. Fan, Phys. Rev. 119, 613 (1960). 4 J. Pankove, Phys. Rev. Letters 4, 454 (1960). II. EXPERIMENTAL TECHNIQUE Single-crystal specimens of GaAs with various Zn densities were used for the transmission measurements. These slices were taken from ingots grown by the hori zontal Bridgman technique. Hall measurements were made on single specimens adjacent to those used for transmission data. Measurements have shown that the carrier densities of the Hall sample and its adjacent mate do not differ by more than 10%. The thickness of the sample was measured interferometrically; the final thicknesses were of the order of 3 J.t. A Perkin-Elmer model 112 spectrometer with NaCI optics was used for these measurements. The absorption coefficient was computed from the standard equation which includes the effect of multiple internal reflections in the sample. The reflectivity was assumed to be 0.31 and independent of doping. This assumption was sup ported by measurements of the transmission at the same wavelengths for various thicknesses at p = 6 X 1019 cm-3• III. DATA AND RESULTS Figure 1 shows the room-temperature absorption as a function of photon energy as calculated from the transmission measurements. The resolution and carrier densities are indicated. Two absorption mechanisms are predominant-free carrier absorption and absorption due to the band-to-band transitions. The dashed portion of the curves corresponds to resolution twice that indicated in Fig. 2. Experimental points are not included because confusion would result in the regions 771 Copyright © 1962 by the American Institute of Physics. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 00:25:48772 I. KUDMAI\' AND T. SEIDEL Z 4 o ;::: c.. 0:: Sl CO -< 3 10 4 2 1010 • t5X 1017 -1.IX10'9 • 2.6XIO" • 6.0XI()t9 • lOX IQ20 FIG. 1. Absorption coefficient as a function of photon energy of p-type GaAs at 300oK. Doping (em-a) is indicated. Resolution is shown for solid lines; dashed lines are for data with twice this resolution. of crossover. In the region above 1.4 ev, a Burstein-like shift in the absorption does occur.5 For meaningful analysis, however, the free carrier absorption must be subtracted from the edge. Results of this subtraction are shown in Fig. 2. The values for the subtracted ab sorption below 1.4 ev are still large for samples with p~10l9, but the data have been cut off at the interception with the edge of the pure sample. The figure illustrates the true extent of the Burstein-like shift. The one sample which has been extended may be considered typical of the behavior for all the samples doped above 10l9/cm3 for wavelengths just longer than those corre sponding to 1.4 ev. This absorption which extends beyond the fundamental edge of the pure sample suggests another absorption mechanism. IV. ANALYSIS Previous work6 indicates that the fundamental ab sorption may be due to direct transitions, where the band structure model has conduction band minimum and valence band maximum at k= (000). The measure ments on the 1017 sample are consistent with this hy pothesis. Figure 3 shows an analysis for direct transi tions with a matrix element independent of photon energy; values of Q' between 9000 cm-1 and 13 000 cm-1 have been used. The squares are for a smaller area sample and have not been weighed as much. The threshold energy is 1.39±O.02 ev. These data are used as a basis for the interpretation of the remainder of the results. The absorption for direct transitions and for a 5 E. Burstein, Phys. Rev. 93, 632 (1954). 6 H. Ehrenreich, Phys. Rev. 120, 1951 (1960). full valence band may be written as follows: (1) where mrl is the reduced effective mass for the heavy hole-and-conduction band and 11: r2 is for the light-hole and-conduction band, hvo is the threshold energy, and K is a constant. This expression includes the fact that the valence band is degenerate. With the measurements from the direct transition analysis in Fig. 3 and the effective masses quoted by Ehrenreich,6 it is possible to evaluate the constant K. The absorption for the heavily doped samples must include the probability for occupation of an electron in the valence band. For a degenerate valence band there will be two values of Hand k (corresponding to the photon energy and elec tron momenta) for which a transition of the same energy hI' may occur. The absorption then becomes where fr and h are the Fermi probabilities for occupa tion in the heavy-and light-hole bands, respectively. The constants K, mrl, mr2, and hvo have been evaluated, and the absorption may be computed as a function of the photon energy with the doping p as a parameter. The Fermi energies have been evaluated for a de generate valence band and constant effective masses. The valence-band structure is assumed to be the same as that of Ge with different m*'s; the inversion sym metry effects are ignored. Equation (2) may be com puted numerically for a given photon energy. A com parison of calculated results for p= 1019 and 3X 1019 cm-3 with experimental data is given in Fig. 4. The spirit of the calculation requires the theory and data to be 5 'E 5 FIG. 2. Absorption edge with extrapolated free carrier absorption subtracted. Dashed portion shows how the absorption extends beyond the pure sample's edge. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 00:25:48:\ B S 0 R l' T I 0:\ E [) G E r:\ D E G E :\' ERA T E P -T Y l' EGa A s 773 identical for the 1017 sample. The agreement for the 1019 and 3XlOI9 cases may be considered to be satis factory. There is somewhat less agreement for the 6X 1019 and 1020 cases. In view of the approximations made, however, the fit is considered to support the hypothesis of a Burstein shift at room temperature. Figure 2 showed that the absorption in the heavily doped samples typically extended to the low-energy side of the fundamental edge of the pure sample. For these doping levels, one can conceive of indirect transi tions from large k values in the valence band to the bottom of the conduction band. However, an analysis along these lines on the 1020 sample required the exist ence of a 0.02 ev phonon for k values less than 20% of the way to the edge of the Brillouin zone. Because this requirement is incompatible with the phonon spectrum in GaAs, the analysis was ruled out. In addition, liquid nitrogen data on the 1020 sample were not consistent with results expected for indirect transition. This be havior, therefore, reveals the importance of considering the impurity band effects. V. DISCUSSION AND CONCLUSIONS The infrared transmission reveals a tilting of the fundamental absorption edge for heavy doping at room temperature. These data are interpreted here as being due to a decrease in the valence band electron popula tion; good numerical comparisons have been made in support of this hypothesis. Because there is not an exact agreement, however, it is appropriate to review all the approximations made in the theory. The following assumptions were made: (1) Effective masses are 4.0 3.5 'E ~ 20 ~ " ~ 15 1.0 .5 1.36 / / / / / hll (ev) FIG. 3. Direct transition analysis on 1Ol7/cm3 p-type GaAs for absorption greater than 9000 em-I. Threshold energy is 1.39±0.02 ev. Square points are for a second sample. 5 'E FIG. 4. Comparison of the direct transition absorption calcu lated from Eq. (2), with the experimental values of the absorption for p= 1019 and 3X 1019 cm-3• constant. (2) Hall coefficient = 1/ pe. (3) The inversion symmetry effects peculiar to zinc-blende structures are unimportant. (4) The density of states is not affected by doping and the analysis can be made on a compara tive basis. (5) The threshold energy is unaffected by doping. Modification of the analysis by (1), (3), and (5) would improve the agreement, while (2) would increase the disagreement. However, some of the parameters which are required for such a modification are not available, and it would be inconsistent to consider one of these corrections without the others. Even anisotropy may prove to be important; this consideration is of the same order of complexity as inversion symmetry. The absorption on the low-energy side of the pure sample's absorption edge is best explained by invoking absorp tion from the impurity band. This explanation requires the existence of a nonzero momentum matrix element connecting the impurity band electrons and the s orbitals of the conduction band. A new density of states, corresponding to impurity-valence-band states, must be handled with caution.7 ACKNOWLEDGMENTS The authors thank R. Braunstein, H. Ehrenreich, T. Kinsel, J. Pankove, and W. Spitzer for their dis cussions and suggestions, P. Del Priore for making the thickness measurements with the interferometer micro scope, and P. Vohl for growing the crystals. 7 R. H. Parmenter, Phys. Rev. 97, 587 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 00:25:48
1.1777177.pdf	LowNoise Beams from Tunnel Cathodes G. Wade, R. J. Briggs, and L. Lesensky Citation: Journal of Applied Physics 33, 836 (1962); doi: 10.1063/1.1777177 View online: http://dx.doi.org/10.1063/1.1777177 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Lownoise noise J. Acoust. Soc. Am. 78, 100 (1985); 10.1121/1.392571 Lownoise 115GHz receiver using superconducting tunnel junctions Appl. Phys. Lett. 43, 786 (1983); 10.1063/1.94455 Lownoise selfpumped Josephson tunnel junction amplifier Appl. Phys. Lett. 39, 650 (1981); 10.1063/1.92841 Lownoise tunnel junction dc SQUID’s Appl. Phys. Lett. 38, 723 (1981); 10.1063/1.92472 Arrays of superconducting tunnel junctions as lownoise 10GHz mixers Appl. Phys. Lett. 34, 711 (1979); 10.1063/1.90615 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 12:14:38JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 3 MARCH, 1962 Low-Noise Beams from Tunnel Cathodes G. WADE AND R. J. BRIGGS Spencer Laboratory, Raytheon Company, Burlington, M~assachusetts AND L. LESENSKY RAD Division, Avco Corporation, Wilmington, Massachusetts (Received August 18, 1961) The tunnel cathode consists of a metal-insulator-metal sandwich in which the electrons tunnel through the insulator materia!. This paper presents an analysis of the noise associated with the space-charge waves of a beam emitted from such a cathode. The beam noise temperature for a refrigerated tunnel cathode is shown to be 27300, where 0 is the value in volts of a built-in energy window for the emitted electrons. The value of 0 is adjustable by means of a dc potentia!. Assuming a beam noise temperature of 30oK, the current density is calculated for a variety of cathode parameters. A discussion is presented of the significance of the parameters and of the difficulties which would be encountered in constructing such cathodes for low noise. INTRODUCTION MICROWAVE noise temperatures as low as about 2500K have been measured on electron beams used in traveling-wave tubes.1.2 These noise tempera tures, involving the longitudinal fluctuations in space charge waves, were measured on beams emitted from thermionic cathodes. Ten years ago, the lowest noise temperatures attainable were an order of magnitude higher than the above figure. Success in lowering the noise temperature has resulted from operating on the beam in the region just beyond the cathode. Specifically, in such a tube the beam is made to flow through an extended low-velocity region where the voltage is less than a few tenths of a volt.3-6 If instead of this operation the beam is accelerated rapidly as it leaves the cathode, and if the beam voltage beyond that region is every where greater than a few volts, the noise temperature will not be lower than the cathode temperature, ordinarily around HOOoK. Low noise can also be obtained from beams in which the coupling is to cyclotron waves rather than to space charge waves.7 For cyclotron waves, the noise tempera ture is a measure of the transverse fluctuations and can be reduced by the application of large magnetic fields. In the above cases, the noise reduction is due funda mentally to what happens to the electrons at the cathode surface and in the vacuum beyond. Recently, there has been work on non thermionic emission involving 1 B. P. lsraelsen, E. W. Kinaman, and D. A. Watkins, "Develop ment of ultra-low-noise traveling-wave amplifiers at Watkins Johnson Company," Proc. Symposium on the Application of Low Noise Receivers to Radar and Allied Equipment, Lexington, Massachusetts, 1960. 2 O. Hodowanec and H. J. Wolkstein, "An ultra-low-noise wide-band traveling-wave tube," presented at Electron Devices Meeting, Washington, D. C., 1960. 3 A. W. Siegman, D. A. Watkins, and H. C. Hsieh, J. App!. Phys. 28, 1138 (1957). 4 M. R. Currie and D. C. Forster, Proc. lnst. Radio Engrs. 46, 570 (1958). 5 M. R. Currie, Proc. lnst. Radio Engrs. 46, 911 (1958). 6 M. R. Currie and D. C. Forster, J.App!' Phys. 30, 94 (1959). 7 R. Adler and G. Wade, J. App!. Phys. 31, 1201 (1960). operation on the electrons within the cathode material itself.8-13 The noise characteristics of such beams are inherently different from the characteristics of therm ionic beams. This paper analyzes the noise associated with the space-charge waves of the beam for a type of emission capable in principle of giving rise to very low temperature. In addition to noise temperature, the corresponding density of the emitted current is calcu lated. A discussion is presented of the significance of the parameters involved in producing low noise and of the difficulties which would be encountered in con structing such cathodes. IDEALIZED MODEL FOR TUNNEL CATHODES The cathode treated here provides for electron tunneling through insulator material and is called the tunnel cathode. This section describes the cathode and presents an idealized model for the emission mechanism involved. The next sections analyze the model in terms of the noise temperature and the current density. The analysis concerns a slightly modified version of what has been proposed by Mead and Geppert.11-13 The modification was made solely in the interest of low noise and may well be disadvantageous from other standpoints. Geppert suggested the possibility of low noise from this general type of emission,12 and his suggestion has served as the impetus for the present investigation. Even though the noise reduction in the present scheme is not precisely the same as that de scribed by Geppert, the analysis does have general application to his scheme. The tunnel cathode involves a metal-insulator-metal sandwich with provision for electron tunneling through 8 J. A. Burton, Phys. Rev. 108, 1342 (1957). 9 A. M. Skellett, B. G. Firth, and D. W. Mayer, "Some properties of the MgO electron primary emitter," Proceedings of the Fourth National Conference on Tube Techniques, September, 1958 (New York University Press, New York, 1959). 10 R. E. Simon and W. E. Spicer, J. App!. Phys. 31, 1505 (1960). 11 C. A. Mead, Proc. lnst. Radio Engrs. 48, 359, 1478 (1960). 12 D. V. Geppert, Proc. lnst. Radio Engrs. 48, 1644 (1960). 13 C. A. Mead, J. App!. Phys. 32, 646 (1961). 836 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 12:14:38LOW-NOISE BEAMS FROM TUN:'-JEL CATHODES 837 METAL SUBSTRATE FERMI SEA ELECTRON DENSITY IN METAL SUBSTRATE VACUUM LEVEL ANODE POTENTIAL I FIG. 1. Diagram of tunnel cathode and corresponding ideali~ed energy band picture for zero degrees. absolute. The dashe? lme indicates a modification in the potential profile correspondmg to the energy distribution used in the analysis. the insulator material, as illustrated in Fig. 1. The upper part of the figure shows a diagram of the sandwich (not to scale). One battery is connected from the metal substrate to the metal film and a second battery from the metal film to the anode. The lower part of the figure ~hows an idealized energy-band picture and indicates the electron density in the metal substrate for a tem perature of absolute zero. The emitted electrons tunnel from the metal substrate into the metal film and then pass over the cathode surface barrier into the vacuum. In Fig. 1, we have assumed that the energy gap in the forbidden region is somewhat more than twice the height of the true work function in the metal film. (In the following discussion, we will assume somewhat arbitrarily that the metal-insulator work function is approximately equal to one-half the insulator energy gap.) This permits Vb to be adjusted in such a way as to give the potential profile shown in the figure. The opposite side of the potential barrier across the insulator is at the left boundary of the metal film for all electrons in the metal substrate regardless of their energies. Hence, the tunneling electrons tunnel completely through the insulator and arrive in the metal film without loss of energy. The most energetic of these electrons have energies above that of the vacuum level. Note that the above description would not be valid if the energy gap were less than twice the height of the work function. Under this circumstance, if Vb were adjusted so that the most energetic electrons in the metal substrate would have energies above the potential of the vacuum level, the opposite side of the potential barrier for these electrons would be in the insulator and not at the surface of the metal film. Hence, these electrons in tunneling through the barrier would emerge into the insulator material rather than into the metal film. Collisions in the insulator material might be detrimental in reducing the electron energies to below that of the vacuum level. It is to avoid this situation that we assume an energy gap of more than double the work function. If the metal film is sufficiently thin, there will be small probability of the electrons losing their energies by collisions in traveling through the film. Several workers have reported recent experiments which imply that the mean free paths of electrons in metals may be . . h f d 14-16 A slgmficantly greater than ereto ore suppose . film thickness of several hundred angstroms may be sufficiently thin for the above purpose. If the electrons are not involved substantially in collisions, many elec trons will arrive at the cathode surface with energies above the vacuum level, and hence these electrons will be emitted into the vacuum. The velocity distribution in emISSIOn of this kind obviously differs greatly from the half-~ax:-vell!an distribution in thermionic emission. The dlstnbutlOn and, more particularly, the spread of ve~ocities play significant roles in determining the magmt~de of the beam noise. In general, the greater the veloCIty spread, the greater the beam noise. Since tunneling is the emission mechanism for this cathode, high temperature is not essential in its operation. The velocity spread is lowest when the temperature is reduced to zero degrees absolute. We will assume absolute zero in the following calculations on noise temperature and current density. We will discuss the effects of finite temperature in the Discussion section of the paper. For absolute zero, the velocity spread has an upper limit imposed by the Fermi level and a lower limit imposed by the vacuum level. The magnitude of the spread can be adjusted by vary ing Vb. CALCULATION OF NOISE TEMPERATURE The noise temperature Tn of an electron beam is a figure of merit relating to the minimum quantit~ of noise which would appear at the output of an amplIfier using the beam in its amplifying process. In microw~ve amplifiers, the main source of noise is the beam nOIse. Under the assumption that all other amplifier sources of noise are negligible, we can identify the noise tem perature of the beam with that of the amplifier itself. For any amplifier, the noise temperature is equal to the temperature of the input match for which the amplifier noise output is just double the value corresponding to the match being at OaK. Noise temperature is related to noise figure as indicated in the following expression: 14 H. Thomas, Z. Physik 147, 395 (1958). 15 R. Williams and H. R. Bube, J. Appl. Phys. 31, 968 (1960). 16 J. P. Spratt, R. F. Schwarz, and W. M. Kane, Phys. Rev. Letters 6, 341 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 12:14:38838 WADE, BRIGGS, AND LESE:-JSKY Consider an electron beam in which the emitted elec trons are accelerated by strong electrostatic fields immediately after emerging from the cathode surface. The accelerating fields are of sufficient strength to prevent the accumulation of space charge in the cathode-anode region. Under these circumstances, the noise temperature of the beam is independent of any subsequent lossless region through which the beam might pass, as long as the mean velocity of the beam is large compared with its velocity spread. If the emission process is thermionic, the above conditions apply to temperature-limited operation. Associated with the space-charge waves of a beam emitted in this fashion, there are two uncorrelated noise sources. One is due to current fluctuation at the cathode and the other is due to velocity fluctuation there. The electrons emerge from the cathode in completely random fashion, and the current fluctuation is called shot noise. For this situation, it can be shown that the noise temperature is given by17 (1) where m=electronic mass, k=Boltzmann's constant, (V02)av= the mean square velocity component normal to the anode plane of the electrons passing the anode plane, and (fJv2),w= the mean square deviation in the velocity component normal to the anode plane, or the variance, of the electrons passing the anode plane. [The energy distribution for thermionic emission is very nearly half-Maxwellian. If we use this fact to calculate (V02)av and (fJv2).w, we find from Eq. (1) that the noise temperature Tn is simply the cathode temperature.] In the emission process being considered in the present paper, the current fluctuation and the velocity fluctuation at the cathode are uncorrelated. The elec trons are emitted in random fashion giving rise to shot noise current fluctuation. Since, in these respects, tunnel-cathode emission is like temperature-limited thermionic emission, Eq. (1) is also valid in the present analysis. In the tunnel cathode, electron tunneling through the insulator is similar to the tunneling which occurs in field emission.I8 If we modify the potential profile shown by the solid line in Fig. 1, we can use the above similarity to obtain the energy distribution function for the emitted electrons. The modification is given by the dashed line in the figure. As far as the electrons having sufficient energy for emission are concerned, the modi fication is obviously very slight. For these electrons, the energy distribution in the metal film is the same as that in the vacuum. The approximate energy distribu tion for field emission at zero absolute temperature is 17 R. W. DeGrasse and G. Wade, Proc. Inst. Radio Engrs. 44, 1048 (1956). See especially Eqs. (3), (4), and (8). 18 R. H. Fowler and L. Nordheim, Proc. Roy. Soc. (London) 119, 173 (1928). given by19 Pt(w)=At(wo-w) exp[ -(wo-w)/e,B] for wo-efJ<w<wo (2) =0 otherwise, where Pt(w)dw is the number of electrons passing the anode plane per unit time having kinetic energy com ponents normal to the cathode and anode planes between wand w+dw; w is the electron kinetic energy component at the anode normal to the cathode and anode planes (i.e., mv2/2); At is independent of w; Wo is an energy given by (eVo+eV b-eljJ) in mks units;,B is a voltage given numerically by 0.97 X lO-IOE/ (ljJ)!; E is the accelerating field at the metal substrate surface in volts per meter; ljJ is one-half the insulator energy gap in volts; and fJ is a voltage defined in Fig. 1. The nature of the energy distribution function is made clear by reference to Fig. 2. The figure shows the points in velocity space for the electrons in the metal substrate at zero absolute temperature. All the points are contained within a sphere whose radius is deter mined by the energy range of the Fermi sea. Assume that the z direction is perpendicular to the cathode and anode planes. In the velocity space, we can imagine a plane perpendicular to the z-directed velocity axis intersecting the sphere at a distance (27JfJ)! from its surface. Only those electrons whose points are contained within the portion of the sphere to the right of the plane are capable of being emitted after tunneling. Following emission, the electrons are accelerated in the z direction by fields from the battery of potential Vo. As previously stated, the energy distribution of these electrons, as they arrive at the anode plane, is given by Eq. (2). From Eq. (2), we can calculate (V02)av and (fJv2),w as follows: i'" vz2Pt(w)dw 2wo ( 'V(2)av (3) i'" Pt(w)dw m and 1 (efJ)2 (4) 36 mwo Here we have assumed that (e{3/wo) and o/{3 are much less than unity. Using these expressions in Eq. (1), we obtain for the beam-noise temperature (5) As illustrated in Fig. 1, fJ is the difference between Vb and the true work function of the metal at the cathode 19 See for example, G. Richter, Z. Physik 119, 406 (1942), or N. S. Mott and 1. N. Sneddon, Wave Mechanics and its Applica tions (Clarendon Press, Oxford, England, 1948). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 12:14:38LOW-'JOTSE BEAMS FROM TUNNEL CATHODES 839 FIG. 2. Points in velocity space for the electrons in the metal substrate at OOK. The shaded region is the origin of the tunneling elec trons which are emitted. Vy V. Vz surface. By adjusting V b, in principle, 0 can be made to have any value. For 0 equal to a tenth of a volt, the calculated noise temperature is close to the best obtainable by the techniques previously mentioned which operate on the beam in the region beyond the cathode surface. To obtain noise temperatures as low as in parametric amplifiers and masers, 0 would have to be about a hundredth of a volt or lower. It is interesting to note that a triangular energy distribution [such as the one given by Eq. (2) with the exponential factor omittedJ also leads to Eq. (5) under the assumption of small relative velocity spread. Hence, the energy distribution expressed by (2) is essentially triangular over the energy range of interest. CALCULATION OF CURRENT DENSITY The current density available from a tunnel cathode can be calculated by integrating the energy distribution function given by Eq. (2) as follows: 1= fv~~eo Pt(w)dw=l o[l-e-(olf3) (1+0/i3)J, (6) where 10= A t(ef3)2. From the above, it is clear that the constant 10 is just the total current density which tunnels from the metal substrate into the metal film, and in the case of a triangular barrier, it is given by20 10=~ E exp[-~ (2me)! cplJ 8d cp 3 fz E E2 [ cp' = 1.55XHJ-6-; exp -6.86X109 ~J (7) in mks units. In the present case, the barrier is not actually tri angular in shape as given by the dashed line in Fig. 1, but has·the truncated shape given by the solid line of Fig. 1. Since the values of 0 in which we are interested are in the vicinity of 0.01 ev, the above expression for the emitted current density [Eq. (6)J should be an excellent approximation in the present case. The total 20 A. G. Chynoweth, Progr. in Semiconductors 4; 97 (1959). TABLE I. Values of the emitted current density for various values of the insulator thickness and energy gap; Tn=30oK. -----._--- </> t ( volts X 10' ) f3 ( amps) ( _amps) (volts) (A) E meter (volts) Jo meter2 J meter2 1.5 20 3/4 0.06 2.9XIO· 2.3XIO' 2 15 4/3 0.09 8XIO' 5 XIO' 20 I 0.07 3XIQ3 30 30 2/3 0.046 8 X 10-' I. 7 XlO-3 40 1/2 0.035 2.7 XIO-' 9.4XIO-s 3 15 2 0.10 4 X 10' 1.7XI02 20 3/2 0.084 57 0.4 30 I 0.056 1.5 XlO-' 2.3 XlO-' 4 15 8/3 0.13 3XIQ3 13 20 2 0.10 2 1 XIO-2 current density which tunnels from the metal substrate should be somewhat greater than the value of 10 given in Eq. (7), since the tunneling distance is less for the lower energy electrons. Assume the following values for half the insulator energy gap and for the insulator thickness: cp= 1.5 ev 1=20 A. Then E=cp/t=0.75X 109 volts/meter is the maximum electric field which can be applied across the insulator without allowing electrons to tunnel into the insulator conduction band. The values of 10 and /3 are /3=0.06 ev, and 10=2.9X104 amps/sq meter. A noise temperature of approximately 300K corre sponds to a value of 0=0.01 v. The emitted current density for this value of 0 is 1=2.3XI02 amps/sq meter. This value of current density would give a beam current of 50 jJ.a if emitted from a cathode of 20 mils diameter. Table I demonstrates the dependence of the emitted current density on the film thickness and the insulator energy gap. In all cases, it is assumed that the electric field is as large as possible without allowing electrons to tunnel into the insulator conduction band. The emitted current density is computed for the case of the noise temperature being at 30oK. The current density calculations presented here neglect traps in the insulator and reflections at the metal-vacuum interface.21 It should also be mentioned that the values of tunneling current density computed here for a given insulator energy gap and insulator thickness may be several orders of magnitude too small, according to recent experiments.22 Some authors have attributed this discrepancy to a reduced effective mass for the electron in the insulator material, which is not accounted for in the theory,22 Another possible explana tion is a smoothing of the potential profile due to an image force effect or to the fact that the boundary 21 C. A. Mead, J. App!. Phys. 32, 646 (1961). 22 J. C. Fisher and I. Giaever, J. App!. Phys. 32, 172 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 12:14:38840 WADE, BRIGGS, i\~D LESENSKY between the insulating film and the metal substrate is graded and irregular rather than sharp and smooth. It would seem that the image force alone could not account for the discrepancy since the dielectric constant of the insulator is so large. In any event, this may allow for somewhat larger film thicknesses to be used than would be indicated by our computations. DISCUSSION The results of the previous calculations show that it is theoretically possible to obtain low-noise electron beams of reasonable current densities from tunnel cathodes. The beam noise temperature, as given by Eq. (5), is a linear function of the voltage 0 for small values of o. The emitted current density, however, is a quadratic function of 0 as can be shown by expanding Eq. (6) as follows: j""'t(O/{3)2jo, o«{3. (8) Clearly, this places a lower limit on the value of noise temperature which can be achieved simultaneously with a useful emitted current density. In all of the above calculations, the cathode was assumed to be at zero degrees absolute temperature. For finite temperature, the velocity spread and hence the beam noise will be higher. The emitted current density will also be higher. The effect of finite cathode temperature on beam noise temperature can be seen qualitatively by assuming that the distribution shown in Fig. 1 is modified to include a Boltzmann tail. Tunneling probability is greater for the electrons with higher energy in the tail than for those with lower energy. A larger proportion of the higher energy electrons will be emitted than of the lower energy electrons. The corresponding tail for the emitted electrons will have a fatter appearance than does the Boltzmann tail in the metal substrate. Hence the noise temperature of the emitted beam will be greater than the cathode temperature. Geppert and Barnes have recently made a calculation of noise temperature in volving only the Boltzmann tail.23 This is the extreme opposite of the case considered here for zero tempera ture. In a sample calculation, the above authors show that for room temperature operation, the beam noise temperature is only slightly higher than room tempera ture.23 For extremely low noise, the cathode must be refrigerated and 0 must be a small fraction of a volt. Therefore, variations in 0 over the cross section of the cathode must be much less than a fraction of a volt or spotty emission will result. Hence, very stringent requirements are placed on the allowable variations in the vacuum work function over the cross section of the metal film. 23 D. V. Geppert and C. W. Barnes, Jr., "The equivalent noise temperature of the tunnel cathode" (to be published in the Transactions of the Professional Group on Electron Devices). A closely related problem is the necessary uniformity of the insulator thickness over the cross section. Since the total thickness of the insulating film is a small number of atomic layers, a variation in the thickness of only one or two atomic layers would result in extremely nonuniform emission. (The critical effect of variations in the insulator thickness on the uniformity of emission is seen from the fact that the current density is an exponential function of the thickness.) Another important limitation of the tunnel cathode is based on the leakage current, that is, the lateral current flow due to the electrons which tunnel through the insulator with energies below the vacuum level. This current must pass through the battery supplying the potential V b. The thickness of the metal film must be small compared to a mean free path in order to allow for electron transmission into the vacuum. The resist ance it offers to the leakage current is large, and the resultant potential drop along the film will cause reduced and nonuniform emission. As possible solutions to this problem one might consider: (1) A metallic grid on the cathode surface which would be thick enough to establish a more uniform potential along the cathode surface. (2) A superconducting metal film. In the second case, the flow of current laterally across the metal film is accomplished by the superconducting electrons only, since no electric field is required to cause a finite current flow. The hot electrons injected into the metal film by the tunneling process will give up their excess energy to the lattice through collisions, and then will drop down into the superconducting state. The energy lost by these electrons will boil off some of the liquid helium bath, which then must be replenished. Since the amount of energy absorbed by the boiling of liquid helium is approximately 3 joules/ cc, a tunnel cathode which supplies an emitted current of 50 J.La would boil off about 20 cc of liquid helium per hour in continuous operation.24 This is small compared to the amount of helium which would be lost from the cooler because of heat conduction and radiation alone. Another limitation in the tunnel cathode is break down in the insulator. If the work function of the outer metal film is large, the applied potential V b necessary to lower the vacuum level below the Fermi level of the metal substrate may cause insulator breakdown. A judicious choice of insulator material would be necessary to circumvent this difficulty. CONCLUSIONS This paper has presented an analysis of the noise properties and available current density of the tunnel cathode. Electron emitters using quantum-mechanical 24 In making this calculation, it was arbitrarily assumed that the leakage current was approximately twice the value of Jo given by Eq. (7). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 12:14:38LO\\/-NOISE BEAt-.IS FROM TUNNEL CATHODES 841 tunneling are capable in principle of giving low-noise behavior. Lnder a set of idealized conditions described above, the noise temperature is shown to be Tn=2730 0, where 0 is the energy window for transmitted electrons expressed in volts. Obviously, there will be many difficulties encountered in any attempt to realize low-noise emission from tunnel cathodes. To list a few: (1) Materials must be found with the appropriate energy band structure. (2) The work function of the metal film must be extremely uniform over the cross section. (3) The insulator thickness must be uniform within very close tolerances. Because of the unsolved technological problems, this work must be considered as being in its preliminary stages. ACKNOWLEDGMENTS The authors wish to acknowledge valuable discussions with Dr. S. Aisenberg, Dr. W. Feist, and Dr. S. Wolsky of the Raytheon Company. JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 3 MARCH, 1962 Thermoelectric Properties of Bismuth-Antimony Alloys G. E. SMITH AND R. WOLFE Bell Telephone Laboratories, Incorporated, Murray Hill, New Jeresy (Received August 7, 1961) The thermoelectric figure of merit (z), resistivity, and Seebeck coefficient have been measured between 20° and 3000K on single crystals of several alloys in the range from 1% to 40% antimony in bismuth. These materials are semimetals (0 to 5% antimony) or small energy gap intrinsic semiconductors (5 to 40% antimony) and all are n type. The Seebeck coefficients and figures of merit are anisotropic, the larger values being those measured parallel to the threefold symmetry axis. In the 12% antimony alloy the larger z rises from l.OX 1O-3;OK at 3000K to a maximum of 5.2X1o--';oK at 80cK and falls rapidly at lower temperatures. All of the alloys between 3% and 16% antimony have a maximum z near 5X 10--";oK at a temperature between 700K and 100°K, The 5% antimony alloy has the highest z at room temperature INTRODUCTION THE electrical properties of single crystals of bismuth-antimony alloys have been investigated recently by Jain.! He found that alloys containing less than five atomic percent antimony have overlapping valence and conduction bands and are therefore semi metals. In alloys containing between 5% and 40% antimony, the resistivities increase as the temperature is lowered below lOOoK and he deduced that these materials are intrinsic semiconductors with small energy gaps (EgS;0.014 ev). The thermoelectric properties of bismuth-antimony alloys were first measured almost 50 years ago.2 In all of the early work on these materials, polycrystal line specimens of unknown purity were used. In recent measurements on pure single crystal bismuth, Chandrasekhar3 studied the anisotropy of the Seebeck 1 A. L. Jain, Phys. Rev. 114, 1518 (1959); see also, S. Tanuma, J. Phys. Soc. Japan 14, 1246 (1959). 2 G. Gehlhoff and F. Neumeier, Verhandl. deut. physik. Ges. 15,876, 1069 (1913). 'B. S. Chandrasekhar, J. Phys. Chern. Solids 11, 268 (1959). (z=1.8X1O-Sj"K). In this material, the Seebeck coefficient is practically constant (S= -1l0±10 j.lV rK) between 77° and 3000K and the ratio of the thermal to electrical conductivities is close to the theoretical Wiedemann-Franz ratio above 100oK. As a result, z is inversely proportional to the absolute temperature (zT=0.52±0.OS) between 100° and 300oK. In the 12% antimony alloy, S rises from -110j.lvj"K at 3000K to -220j.lvrK at 20oK. A specimen of this material, doped with 0.01% lead, is p type below 42°K. A qualitative explanation of these results is given in terms of mixed conduction by electrons and holes having properties similar to those in pure bismuth. The use of these alloys (and semimetals in general) in thermoelectric refrigeration at low temperatures is discussed. coefficient. At room temperature, the Seebeck coefficient measured parallel to the threefold symmetry axis is twice as large as that measured in a direction perpendic ular to this axis (SII= -103 ~vrK; Sl= -51 ~v;oK). In the present investigation single crystals of bismuth antimony alloys were investigated and a similar anisotropy was found. This anisotropy in S is reflected in the thermoelectric figure of merit z defined by Z= S2/ KP, where K is the thermal conductivity and p is the electrical resistivity. The figure of merit determines the usefulness of any material in thermoelectric applications.4,5 The bismuth antimony alloys were considered to be the best materials for thermoelectric refrigeration6 and power generation7 until 1954 when the properties of semiconductors such as bismuth telluride were investigated.8 Th~ semi- 4 A. F. roffe, Semiconductor Thermoelements and Thermoelectric Conling (Infosearch, London, 1957). • H. J. Goldsmid, Applications oj Thermoelectricitv (John Wiley & Sons, Inc., New York, 1960). - 6 W. C. White, Elec. Eng. 70,589 (1951). 7 M. Telkes, J. Appl. Phys. 25, 165 (1954). 8 H. J. Goldsmid and R. W. Douglas, Brit. J. Appl. Phys. 5, 386 (1954) j H. J. Goldsmid, J. Electronics 1, 218 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 12:14:38
1.1735287.pdf	Recombination Properties of Bombardment Defects in Semiconductors G. K. Wertheim Citation: Journal of Applied Physics 30, 1166 (1959); doi: 10.1063/1.1735287 View online: http://dx.doi.org/10.1063/1.1735287 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Highly nonlinear defect-induced carrier recombination rates in semiconductors J. Appl. Phys. 114, 144502 (2013); 10.1063/1.4824065 Communication: Non-radiative recombination via conical intersection at a semiconductor defect J. Chem. Phys. 139, 081102 (2013); 10.1063/1.4819784 Effect of ion bombardment and annealing on the electrical properties of hydrogenated amorphous silicon metal–semiconductor–metal structures J. Appl. Phys. 97, 023519 (2005); 10.1063/1.1834710 New method for complete electrical characterization of recombination properties of traps in semiconductors J. Appl. Phys. 57, 4645 (1985); 10.1063/1.335501 Abstract: Properties of interfacial defects in III–V compound semiconductors J. Vac. Sci. Technol. 13, 37 (1976); 10.1116/1.568895 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:02JOURNAL OF APPLIED PHYSICS VOLUME 30, NUMBER 8 AUGUST, 1959 Recombination Properties of Bombardment Defects in Semiconductors* G. K. WERTHEIM Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey The theory of recombination via defects having energy levels in the forbidden gap is reviewed. Emphasis is given to those aspects which complicate interpretation of lifetime data, such as the inherent difference between steady state and transient measurements, large-signal behavior, competing recombination mecha nisms, trapping, the possible existence of strongly temperature-dependent cross sections, and the properties of multilevel defects. A summary of the known recombination properties of bombardment-produced defects is given. INTRODUCTION CRYSTALLINE defects give rise to energy levels in the forbidden gap of semiconductors. The location of such defects is usually determined from measurements of the Hall effect taken over a range of temperature. When such measurements are combined with those of conductivity the state of charge of the defect can sometimes be inferred as well. Additional information can be obtained from measurements of the lifetime of nonequilibrium carrier concentrations, which yield capture cross sections of defects for minority carriers, and under some circumstances for majority carriers. Since the magnitude and temperature depend ence of these cross sections are related to the state of charge of the capturing defect, lifetime measurements can help to define the nature of crystalline imperfec tions. The basic aspects of the recombination of excess carriers in semiconductors are now familiar and have been recently treated in a number of review articles.1-4 Further details on many of the subjects which will be mentioned only briefly here can be found in these articles. There are four chief recombination mecha nisms: (1) recombination via levels or states in the forbidden gap, (2) recombination via surface states, (3) recombination by the Auger effect, and (4) band to-band recombination in which the excess energy is radiated as a photon, sometimes accompanied by phonons. For the study of crystalline defects, the first mechanism is the most important, since the measure ments are here directly related to the properties of the defects. We will consequently confine ourselves almost entirely to this subject. The second mechanism, recombination via surface states, often adds complications which may mask the volume effect, especially since the surface recombination velocity may depend on the bombardment of the speci men. This field has been explored to only a slight extent although changes in device parameters ascribable * This work was supported in part by the Wright Air Develop ment Center of the U. S. Air Force. 1 E. S. Rittner, Proceedings of the Conference on Photoconductivity, Atlantic City, 1954 (John Wiley & Sons, Inc., New York, 1956). 2 A. Hoffman, Halbleiter Problerne II (Friedrich Vieweg und Sohn, Braunschweig, 1955). 3 G. Bemski, Proc. Inst. Radio Engrs. 46, 990 (1958). 4 P. Aigrain, Nuovo cimento 7, Supp!. 2, 724 (1958). to surface changes are well known. Recombination via the Auger effect6 has recently been invoked to explain the lifetime in InSb which is inherently short (10-7 sec at room temperature). Here radiation effects on lifetime have not been studied, probably because the inherently short lifetime makes measurements difficult, and also because the preparation of high purity material with controlled lifetime is not very far advanced. Radiative band-to-band recombination6 has been studied in many semiconductors. The change in the fraction of carriers recombining by this mechanism could be used as an index of the imperfections introduced by bombard ment.7 This, however, does not appear to be a powerful tool. Theory of Recombination via Defect Levels Recombination via levels in the forbidden gap was first discussed by Hall and by Shockley and Read. 8 The general result of their analysis is that the net capture rates of a given defect for electrons and holes are given by U n=Cn[N°on- (no+nl+on)oN], U p=Cp[N-op+ (po+Pr+op)oN]' ( 1) The terms are defined in Table 1. To discuss a par ticular recombination process we then need solutions to a set of coupled differential equations (don/dt)=gn-Un, (dop/dt)=gp-U p, (2) subject to the neutrality condition op-on=oN. (3) Relatively simple solutions are obtained only in re stricted cases; a general solution has not been obtained and would be of dubious value because of its complexity. We will consider steady state and transient solutions separately. In either case the desired solution is the free time of a minority carrier, i.e., the time spent by an excess minority carrier in the minority band before 6 P. T. Landsberg and A. R. Beattie, Proceedings of the Inter national Conference on Semiconductors, Rochester, 1958; J. Phys. Chern. Solids 8, 73 (1959). 6 W. van Roosbroeck and W. Shockley, Phys. Rev. 94, 1558 (1954). 7 R. Braunstein, Phys. Rev. 9, 1892 (1955). 8 R. N. Hall, Phys. Rev. 87, 387 (1952); W. Shockley and W. T. Read, ibid. 87, 835 (1952). 1166 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:02R E COM BIN A T ION PRO PER TIE S 0 F B 0 MBA R D MEN T DE FEe T S 1167 TABLE 1. Definition of symbols. Un, Up-net capture rates of a defect for electrons and holes. Cn, Cp-the capture constants for electrons and holes equal to (UnVn) and (upvp). U n, up-capture cross sections for electrons and holes. Vn, vp-thermal velocities of electrons and holes. lin, lip-deviations from the thermal equilibrium carrier concen trations, no and po. N°, N~-the thermal equilibrium density of empty and filled defect states, N°+N~=N.· liN-deviation from the thermal equilibrium defect popu lation. nl, PI-carrier concentrations when the Fermi level is at the defect level. gn, gp-the volume rate of generation of electrons and holes by external means. it is annihilated. In the steady state case this is taken to be the excess minority carrier concentration divided by the net capture rate which is equal to the generation rate. This is not the lifetime obtained from a measure ment of the diffusion length. It should also be noted that the free time of excess majority carriers is in general not equal to that of minority carriers. The minority carrier lifetime is given by where a= {TnoN- in p-type TpoN° in n-type. If the recombination center density is sufficiently small this reduces to the familiar form TO TpO(nO+nl)+TnO(PO+Pl) no+po (5) which also applies generally to the diffusion length. If the recombination center density is sufficiently small and the injection level sufficiently large so that the excess hole and electron densities are equal, we obtain for the large signal case T T pO(nO+nl+on) + TnO(po+pl+on) no+po+on (6) In the transient case9-11 the solution is the character istic time of the decay of the excess carrier concentra tion. If the density of injected carriers is sufficiently small so that the differential equations are linear this is the time constant of an exponential. The result then 9 E. 1. Adirovich and G. M. Goureau, Soviet Phys.-"Doklady" 1,306 (1956); Doklady Akad. Nauk. S.S.S.R. 108,417 (1956). 10 D. J. Sandiford, Phys. Rev. 105,524 (1957). 11 G. K. Wertheim, Phys. Rev. 109, 1086 (1958). is similar to Eq. (4), T Tpo(N°+nO+nl)+ TnO(N-+PO+Pl) no+po+NON-/N (7) which for small recombination center densities becomes identical with the steady state solution, Eq. (5). An approximate solution valid only in the region of transition from small to large signal has also been givenY For very large deviations from equilibrium the decay is not exponential and is given by the following equation12 : on=ono exp( -t ) TnO+TpO X (1 + (no+po)/ono)l- l'O/('nO+'pol] (8) 1 + (no+po)/on In the intermediate region solutions in closed form have not been obtained. In certain cases, such as the one which arises when the recombination level and the Fermi level are in the same half of the energy gap, it may be experimentally impossible to realize an injection level small enough to operate in the linear region. This comes about since the injection level in n type should be small compared to (N-+PO+Pl). Some machine calculations applicable to this situation have recently been reported.13 Detailed discussions and extensions of various aspects of these equations have been given in a number of publicationsY-17 An extension to degenerate semi conductors has been given by Rose.ls The idealization made in the foregoing discussion that only a single species of defect contributes to recombination is seldom met in real situations. The multitude of levels usually found in bombarded material should serve as adequate warning that single level solutions may not be applicable. The independent multilevel steady state case has been discussed by Kalashnikov and Okada19 and the transient case by WertheimY In both cases it turns out that it is not proper to add recombination rates of the individual species of defects unless certain restrictive conditions are met. These are that the concentrations and the cross sections for carrier capture of both defects be such that no appreciable fraction of the injected carriers is trapped. 12 G. M. Goureau, Zhur. Eksptl. i Teoret. Fiz. 33, 158 (1957); Soviet Phys. JETP 6, 123 (1958). 13 K. C. Nomura and J. S. Blakemore (to be published). 14 W. Shockley, Proc. lnst. Radio Engrs. 46, 973 (1958). 15 Lashkarev, Rashba, Romanov, and Demidenko, Zhur. Tekh. Fiz. 28, 1853 (1958). 16 P. T. Landsberg, Proc. Phys. Soc. (London) 1370,282 (1957). 17 D. H. Clarke, ]. Electronics and Control 3, 375 (1957). 18 F. W. G. Rose, Proc. Phys. Soc. (London) 71, 699 (1958). 19 S. G. Kalashnikov, Zhur. Tekh. Fiz. 26, 241 (1956); J. Okada, J. Phys. Soc. Japan 12, 1338 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:021168 G. K. WERTHEIM z '\ CRYSTAL 904 ~~ROL kAMPLJ 1 700°C 3 . .)' V' I .,.-,... ..... 750°C --_.----L-t=i /--w :?: i= 5 w u. :::i Z 10-7 5 2 / 80~ -.--I -~ . V ".. / I -.... ~ 3 4 5 6 7 8 9 10¥ToK FIG. 1. Lifetime in a multilevel system (p-type germanium containing nickel). The case where one of the defects communicates only with the minority carrier band, i.e., acts as a minority carrier trap, has been treated by Haynes and Hornbeck20 and others.u·21 This case is particularly applicable to bombarded germanium at low temperature. Another case, not usually met in the elemental semiconductors, is that discussed by Rose22 which deals with continuous energy level spectra. More important than the case of multiple independ ent levels is that where two or more levels belong to the same defect, i.e., they correspond to successive stages of ionization, rather than to the ground and excited states of one electronic level. This case applies to most of the well known chemical recombination centers in germanium and silicon, and also to radiation damage in germanium. The most important chemical recombination centers in germanium, copper, and nickel, have three and two levels, respectively, while the most important one in silicon, namely gold, has two. The equilibrium statistics are easily generalized for such defects,23-26 but the nonequilibrium recombi nation case leads to considerable difficulty. The general situation has recently been discussed by Sah and Shockley.26 The examination of experimental results in terms of this picture has just begun. An illustration may be drawn from the study of chemical impurities in germanium. One of the most thoroughly studied systems, nickel in germanium, 2fi J. R. Haynes and J. Hornbeck, Phys. Rev. 97, 311 (1955); 100, 606 (1955) . .. H. Y. Fan, Phys. Rev. 92, 1424 (1953); Fan, Navon, and Gebbie, Physica 20, 855 (1954). 22 A. Rose, Phys. Rev. 97, 322 (1955). 23 P. T. Landsberg, Proc. Phys. Soc. (London) B69, 1056 (1956). 24 W. Shockley and J. T. Last, Phys. Rev. 107,392 (1957). 25 V. E. Khartsiev, Zhur. Tekh. Fiz.28, 1651 (1958). 26 Chin-Tang Sah and W. Shockley, Phys. Rev. 109, 1103 (1958); M. Bernard, J. Electronics and ControlS, 15 (1958). offers an instructive example of recombination through two levels belonging to the same atom. These levels correspond to successive added electrons. In p type the temperature dependence of lifetime was found to have the behavior shown in Fig. 1.27 This behavior cannot be realized in terms of single level recombination statistics, unless one of the captured constants has an exponential temperature dependence with an activation energy of 0.2 ev. In terms of the double level model of the simplest kind, where only small changes from the thermal equilibrium concentration are allowed, this behavior follows directly from the fact that a level exists only if the atom is in either of the two states of charge adjoining that level. Specifically an atom exhib1ting two levels exists in three states of charge Fig. 2. The lower level exists only if the atom is in either of the lower two states of charge, and the upper only if the atom is in either of the two higher states. The levels of nickel in germanium fall as shown in Fig. 2. In p type the Fermi level may pass through the lower level as the temperature is changed, producing large changes of lifetime by modulating the density of the upper level. In a situation such as this, the lifetime equation may be written where we have assumed that electron capture will be the time limiting step in recombination via both the lower and upper level. Figure 3 shows measurements of lifetime in another p-type specimen, together with a fit made using experimentally measured concentrations of the various charge states of nickel and substituting in the foregoing equation, assuming that the capture constants are independent of temperature. This assumption seems to be borne out by the fit obtained. Capture Cross Sections The usefulness of lifetime measurements depends ultimately on our ability to relate the measured magni tude and temperature dependence of a cross section to NICKEL LEVELS & CHARGE :o/:WfW$~W://@ Ec I I 10.31 ev -2 I I ® t -1 CD , : 0.22 ev NEUTRAL 1 I Ev 0;:?;:»~f;;/~;/;/>~:?j( /', FIG. 2. The level scheme and states of charge of a multilevel impurity. 27 G. K. Wertheim, Bull. Am. Phys. Soc. Ser. II, 4, 27 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:02R E COM BIN A T ION PRO PER TIE S 0 F B 0 MBA R D MEN T D E FEe T S 1169 values derived from a model of a crystalline defect. There are three types of capture processes, those in volving capture by a neutral defect, those involving capture by a defect whose charge is opposite to that of the carrier, and those where the charge is the same. Recombination always involves one Coulomb-attractive process and either a neutral or a repulsive one. The Coulomb-attractive capture process is seldom observed in conventional lifetime measurements since the other, slower process usually determines the lifetime. A theoretical treatment of the capture process has been given only for the Coulomb-attractive case where the capture process depends on the Coulomb potential and not on the details of the electronic structure of the defect. The cross section for this process is very Jarge at low temperature and decreases rapidly as the temperature increases. Characteristic values observed experimentally are 10-13 cm2 at 78°K and 10-16 cm2 at 300°K. In the neutral case there is no long-range attractive potential so that cross section should be related to the characteristic atomic dimension. The capture constant may well be independent of tempera ture. Characteristic values obtained experimentally at room temperature are in the vicinity of 10-16 cm2• They do not exhibit a strong temperature dependence. In the repulsive case, cross sections smaller than 10-16 cm2 are to be expected. These may exhibit an exponential temperature dependence of the form exp( -E/kT) where E may be as large as a few tenths of an electron volt. This behavior could arise from a potential barrier surrounding the defect which can be overcome only by carriers which have an energy greater than that corre sponding to the band edge. The experimental evidence supporting this model has recently been challenged. Experimentally cross sections greater than 10-16 cm2 have been observed for repulsive defects. It is apparent that the distinction among the three capture processes cannot be made on the basis of room temperature cross sections alone, since values ranging from 10-15 to 10-16 cm2 have been obtained in all cases. At low temperature, 78°K, the results should be less ambiguous. These facts indicate that the temperature dependence of the cross section is an important param eter which can help to determine the charge of the recombination defect. No attempt has so far been made to use this parameter in the study of bombard ment damage. EXPERIMENTAL Three types of measurement are usually made: (1) lifetime as a function of defect density at fixed temper ature, (2) lifetime as a function of temperature at fixed defect density, and (3) lifetime as a function of carrier concentration at fixed temperature and defect density. The first two methods have the advantage that con siderable information can be obtained from a single specimen. In addition, measurements made at fixed temperature avoid the complication which may arise 8 6 4 2 I/) o Z 810-6 W V) ~ w :E i= w u. :::i 8 6 4 2 4 2 J I I CRYSTAL 532-8 / /72 \ \ / 7i"'-7 IT; ~ /; fr-( 1 1 r ! T--+-7j 72 /./ ,p-': ~ 3 456 103/ToK 7 8 FIG. 3. Comparison of measured lifetime with that computed from experimentally obtained charge state of the impurity. from temperature-dependent capture constants and from the anneal of the damage introduced. On the other hand, since energy levels can be determined only if a range of Fermi level positions is examined, the third method is in some ways the most advantageous. However, if this method is used, the assumption must be made that all the specimens are sufficiently similar that the rate of introduction of damage will be the same and, more important, that there are no Fermi level dependent annealing processes. There are a large number of methods for measuring lifetimes. These have been discussed in a recent review article.3 We would only repeat that transient and steady state methods may give different answers when the defect concentration is high [see Eqs. (4) and (7)]. Steady state methods using the diffusion of carriers have the advantage that they measure the free time of minority carriers directly. Included among these methods are simple diffusion length measurements from an injecting source to a collector and the PME effect but not the PME-PC null method, which is ver; sensitive to small trapping effects. Transient methods, utilizing injection by a pulse of light or radiation, do not readily distinguish between trapping and lifetime effects, although auxiliary experiments can usually establish the presence of trapping. REVIEW OF THE LITERATURE Most of the work dealing with bombardment effects on lifetime in semiconductors has been concentrated on germanium, probably because this material is the one most readily available in controlled purity, and because its bombardment induced levels are better known than [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:021170 G. K. WERTHEIM INTERSTITIAL VACANCY /////(//,:{//////(//////////////,///, E I C lO.20eV y X I 10.18 ev 0.07ev 10.02 ev : 'f _EV FIG. 4. Energy level scheme of neu· tron bombardment damage in germa nium. those of any other semiconductor. It has been found that the defects introduced by heavy particle bombard ment of germanium differ significantly from the defects produced by electrons or gamma rays, although certain similarities in the two cases have also been found. Heavy particles such as fast neutrons, protons, deu terons, or alpha particles, will produce clustered defects or disordered regions, while electrons and gamma rays may be expected to produce vacancy-interstitial pairs or, under some conditions, isolated vacancies and inter~ stitials. We will therefore discuss the two types of bombardment separately. I. Germanium (a) Heavy Particle Bombardment The energy level scheme produced by fast neutron28 bombardment at room temperature is shown in Fig. 4. Recombination in n-type germanium has been studied both as a function of defect density and temperature.29 The results are compatible with recombination via the level 0.23 ev below the conduction band. These studies were made using uniform bars of a variety of resis tivities and injection at a surface barrier contact. The hole-capture cross section of the defect was found to be 3Xl0-15 cm2 (revised value). An analogous study using the base region of a pnp transistor and pile neutron bombardment has been reported by Messenger and Spratt.30 Their results are in reasonable agreement with those of Curtis and co-workers. The recombination level was found to be 0.23 ev below the conduction band and the hole-capture cross section 1 X 10-15 cm2• In additi~n they give a value for the electron capture cross sectIOn of 4XlO-I5 cm2• The difference between th~ two report~d hole-capture cross sections may have arIsen from dIfferent ways of estimating the defect concentration and the neutron flux, and is probably not significant. The similarity in magnitude of the hole and electron-capture cross sections tends to rule out 28 Cleland, Crawford, and Pigg, Phys. Rev. 98 1742 (1955)' 99, 1170 (1955). " (1;5~~rtis, Cleland, Crawford, and Pigg, J. Appl. Phys. 28, 1161 30 G. C. Messenger and J. P. Spratt, Proc. lnst. Radio Engrs 46, 1038 (1958). . the assignment of the O.23-ev level to a transition between a singly and doubly charged state. Measurement after bombardment with neutrons of higher energy (14 Mev) have been reported by Vavilov and co-workers.31 They did not locate the defect level but give a cross section for hole capture of 1 X 10-15 cm~ based on theoretical, computed defect densities. These may not be applicable if the recombination level is not one of major defect levels. :Moreover, the cross section represents only a lower limit since the occupancy of the defect is not known. Similar measurements have been reported by Curtis and Cleland,32 using 14.5-Mev neutrons. They conclude that recombination proceeds via a level close to the middle of the gap which exhibits acceptor nature. The density of this defect was not determined. Results in p-type germanium are much more compli cated. Some of the difficulty may arise from known annealing processes which take place readily between room temperature33 and 800K after some types of bombardment. We will consider only the high temper ature region where trapping is not significant. In this region measurements have been reported by Curtis et al.34 The behavior here indicates that the level 0.23 ev below the conduction band is still the dominant recombination center, but it appears that it is the second level of a defect which has its first level in the lower half of the gap. Under these circumstances the passage of the Fermi level through the lower level will strongly modify the lifetime in a manner similar to that discussed for nickel in germanium above. In particular the 0.23-ev level does not exist when the Fermi level is more than a few kT below the lower level, so that the lifetime in p type at low temperature is much longer than would be computed on the basis of the 0.23-level alone. (A fuller account of this work appears elsewhere in this issue.) The assumption of a double level defect leads to good agreement between the cross section for hole capture determined in n-and p-type material. A study of the effects of deuteron bombardment on life~ime35 has not yielded cross sections or energy levels whIch may be compared to those given in the foregoing. The. gene.ral ~greement among the various papers dealmg WIth pIle neutron bombardment is gratifying. The results indicate that the recombination center is identical with the defect which controls carrier concen tration in n-type germanium. (b) Electron and Gamma Bombardment Bombardment of germanium with cobalt-60 gamma rays (1.17 and 1.33 Mev) has yielded the following 31 Vavil?v, Spitsyn, Smirnov, and Chukichev, Zhur. Eksptl. Tegret. FIZ. 32,.702 (1957); Soviet Physics JETP. 5, 579 (1957). O. L. CurtIs, Jr., and J. W. Cleland, Bull. Am. Phys. Soc. Ser. II, 4, 47 (1959). 33 G. ~. Gobeli, Phys. Rev. 112, 732 (1958). : Curtl~, Clel~nd, and Crawford, J. App!. Phys. 29,1722 (1958). Hashlgutchl, Matsuura, and Ishino J. Phys Soc Japan 12, 1351 (1957). ,. . [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:02RECOMBINATION PROPERTIES OF BOMBARDMENT DEFECTS 1171 energy level scheme36 (Fig. 5). One may expect that electron bombardment will produce the same defects, since the displacement of germanium atoms is due to the intermediate Compton electrons and photoelectrons produced by the gamma rays. The maximum energy of the Compton electrons is 0.817 and 0.964 Mev, respectively. The contribution of photoelectrons is small. In n type the dominant recombination level was found to be located 0.20 ev below the conduction band,34 close to the level found after pile neutron bombardment, but the cross section for hole capture in this case was only 4X 10-16 cmz, smaller by a factor of eight than that found after neutron bombardment. This is sur prising, since the existence of the same level in the two cases suggests that the damage configuration is identical. Experiments in p type again indicate that the recombi nation level is the second level of a defect whose first level is in the lower half of the gap, giving further support to the notion that a similar defect is involved in both cases. The difference in cross sections may be due to clustering of defects in the neutron case. Electron bombardment effects in 1l type also have been reported by Smirnov and Vavilov.37 They used 0.55-and 0.75-Mev electrons, and obtained cross sections of 5 X 10-17 and 1 X 10-16 cm2, respectively. The energy level of the defect was not determined, but we may assume that it is identical with that found above. However, since the Fermi level position in these samples is not given, it is not possible to deduce whether a correction for the filling of the defect should be applied to obtain the true cross section. The values quoted would then represent only lower limits. A more detailed study of electron bombardment effects has been reported by Rappaport and Loferski,38 using 1-Mev electrons, and by Baruch,39 using a 2.0-Mev electrons. In both cases germanium samples with a wide range of resistivities were measured after the same amount of bombardment. The lifetimes were measured at room temperatures. No specific results are given in reference 38. Baruch found the recombination level to be 0.18 ev below the conduction band, with a hold capture cross section of 1.6X 10-15 cm2 and an electron capture cross section of 1.6X 10-16 cm2• The electron capture cross section is based on limited data. The agreement among these papers is satisfactory in insofar as the recombination level is found to be located about 0.20 ev below the conduction band. The cross sections are in less satisfactory agreement. The reported value ranges from 1.0XlO-16 to 16.0XI0-16 cm2• Some of this difference may again be due to incomplete 36 Cleland, Crawford, and Holmes, Phys. Rev. 102, 722 (1956). 87 L. S. Smirnov and V. S. Vavilov, Zhur. Tekh. Fiz. 27, 427 (1957); Soviet Phys. Tech. Phys. 2, 387 (1957). 3S P. Rappaport and J. J. Loferski, Bull. Am. Phys. Soc. Ser. II, 3, 141 (1958). 39 P. Baruch, Proceedings of the International Conference on Semiconductors, Rochester, 1958; }. Phys. Chem. Solids 8, 153 (1959). FIG. 5. Energy level scheme of gamma bom bardment damage in germanium. INTERSTITIAL ~NA% I 10.20 ev ! X I 10.26 ev I I Ec W~~Ev knowledge of the defect density in the actual specimens under consideration. In addition to the high temperature recombination behavior of bombardment defects, the low temperature trapping aspects have also been investigated. Shulman40 has reported hole traps located 0.28 ev above the valence band in1l-type electron bombarded germanium. The cross section of these traps for holes was reported to be 6.0X lO-16 cm2; the cross section is of the acti vation energy type with AB .... -'O.OS ev. Other studies41 have shown hole trapping below 22SoK with a trap located 0.11 ev above the valence band. In p type the trapping behavior depends on the bombarding temper ature and annealing of the crystal. Samples electron bombarded at 800K exhibit traps which anneal rapidly at temperatures above 22SoK. These traps are appar ently located somewhat below the middle of the gap. In general it is clear that trapping dominates the recombination processes in bombarded germanium at temperatures below 200°K. II. Silicon The extent of work on recombination in bombarded silicon is very much smaller than that in germanium. Possible reasons for this have been suggested in the foregoing. (a) Neutron Bombardme1lt of Silicon Recombination in silicon bombarded with neutrons from a fission plate has shown that the dominant recombination level is located close to the middle of the energy gap.42 (This is in sharp contrast to the electron bombardment results discussed in the follow ing.) The recombination process shows a strong de pendence on the injection level, suggesting that the recombination level is not discrete. A detailed analysis has not been given. The effect of pile neutrons on transistors has also been analyzed to obtain a measure 4Q R. G. Shulman, Phys. Rev. 102, 1451 (1956). 41 G. K. Wertheim (unpublished); see also W. L. Brown, J. AppL Phys. 30, 1320 (1959), this issue. 42 G. K. Wertheim, Phys. Rev. 111, 1500 (1958). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:021172 G. K. WERTHEIM .. I I I :0.4 ev I t IO.27eV 0.05peV I __ l--_ ~~~EV FIG. 6. Energy level scheme of electron bom bardment damage in pulled silicon containing oxygen. of the radiation e£fect.43 Cross sections were not obtained but the lifetime has been expressed as a function of flux as 3.0X 1Osq,-1 in n type and as 2.0X 106(p-1 in p type. The lifetimes obtained from these equations are greater by factors of eight and five in nand p type than those of reference 42. The differences may arise from differ ences in neutron dosimetry or from differences in the neutron spectrum. (b) Electron Bombardment Hall effect and conductivi ty measurements in electron bombarded silicon have shown three levels well within the forbidden gap44-46 (Fig. 6). These are responsible for carrier concentration changes in material of moder ate resistivity and, on the basis of the recombination statistics, should have the dominant effect on carrier lifetime. Other levels close to the band edges, found in some experiments,46 should have little or no effect on recombination. In n-type material the recombination behavior is entirely in accord with the single level model. A good fit to the data is obtained using temper ature independent hole-and electron-capture constants for the level 0.27 ev above the valence band.11·44 The room temperature cross sections were found to be 8.0X 10-13 cm2 and 9.5 X 10-1• cm2 for holes and elec trons, respectively. The hole-capture cross section is extremely large, and suggests that capture must take place to a Coulomb-attractive defect. This indicates that the defect is an acceptor. On the other hand, carrier removal experiments indicate that this level has no effect on carrier concentration when the Fermi level is above it. This in turn means that the damage site is over-all neutral. A consistent picture is obtained on the assumption that the damage site contains two defects, having opposite unit charge when the Fermi 43 G. C. Messenger, Proceedings of the Brussels Conference, 1958 . .. G. K. Wertheim, Phys. Rev. 105, 1730 (1957). 4b G. K. Wertheim, Phys. Rev. 110, 1272 (1958). 46 D. E. Hill, thesis, Purdue University (unpublished); Bull. Am. Phys. Ser. II, 3, 142 (1958). level is above the 0.27-ev level. The magnitude of the Coulomb-attractive cross section suggests that the two defects must be separated by a distance of perhaps SO A. In p type the behavior of the lifetime is in accord with recombination through the level 0.16 ev below the conduction band. In this case the cross sections were both found to be approximately 2X 10-16 cm2• Since one of the two must be a Coulomb-attractive capture cross section which may be expected to be larger, we are led to the conclusion that the damage site must contain a pair of close-spaced defects of opposite charge. Certain difficulties which remain in this picture have been discussed,4''' The anneal of these lifetime effects has been studied by Bemski and Augustyniak.47 Recombination via the level 0.45 ev below the conduc tion band has not been observed. III. Other Investigations A number of other studies concerned with recombi nation in bombarded material have also been reported in recent years. Loferski and Rappaport48 have used the short circuit current of a diode under bombardment to determine the threshold for the production of damage. In the case of a step junction the short circuit current is a direct measure of the diffusion length near the junction, and consequently a measure of the lifetime. The thresholds in germanium and silicon were found to be 14.5 ev and 12.9 ev, respectively. No information about the location or cross sections of the defects produced was given. Electron bombardment has also been used to reduce the carrier storage time in switching diodes,49,oo and to increase the turn-on current of pnpn cross points. S1 In both cases the desired effect arises from the reduction of lifetime in a region into which carriers are injected. Bombardment effects in transistors ascribable to changes in lifetime in the base region have also been reported.52.5& IV. Other Semiconductors Studies of the effects of bombardment on lifetime in other semiconductors have not yet been reported. As a matter of fact the systematic study of recombination properties in these substances is itself not far advanced. In the III-V compounds where the production of high purity single crystal material is more advanced than in any but the elemental semiconductors, long lifetimes have not yet been achieved. In the large gap II-VI compounds the emphasis has been on luminescent 47 G. Bemski and W. M. Augustnyiak, Phys. Rev. 108, 645 (1957). 48 J. J. Loferski and P. Rappaport, Phys. Rev. 98, 1861 (1955); 111,432 (1958). 49 Miller, Bewig, and Salzberg, J. AppJ. Phys. 27, 1524 (1956). 50 R. Gorton, Nature 179, 864 (1957). bl G. Backenstoss (unpublished). 52 Florida, Holt, and Stephen, Nature 173, 397 (1954). 1i3]. W. Easley, Proc. lnst. Radio Engrs. WESCON (1958); J. J. Loferski, J. App\. Phys. 29, 35 (1958). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:02RECOMBINATION PROPERTIES OF BOMBARDMENT DEFECTS 1173 TABLE II. A summary of recombination properties of bombardment defects in germanium and silicon. Author(s) Sample type Curtis, Cleland, Crawford, and Pigg& n-type Ge Messenger and Sprattb n-type Ge Vavilov, Spitsyn, Smirnov, and Chukichev' n-type Ge Curtis and Clelandd n-type Ge Curtis, Cleland, and Crawford" n-type Ge p-type Ge Smirnov and Vavilovf n-type Ge P. Baruchg n-and p-type Ge G. K. Wertheimh n-type Si p-type Si G. C. Messengeri n-type Si p-type Si G. K. Wertheimj n-type Si p-type Si J. W. Easleyk n-type Si • See reference 29. c See reference 31. • See reference 34. b See reference 30. d See reference 32. f See reference 37. rather than conductive processes. These usually reflect trapping rather than recombination processes. Radia tion effects on luminescence in these substances are known. v. Other Imperfections Dislocations are the only other nonchemical defects whose lifetime effect has been thoroughly studied. The major fraction of the work in this field has been done in germanium. A review of this work has recently been given by Haasen and Seeger. 54 Good agreement has been obtained by two entirely different approaches. 55. 56 The major difference between these and an earlier paper can now be attributed to difficulties in the measurement of the dislocation density. An interesting extension of radiation damage study is possible in view of the suggestion that the defects produced by the annealing of a crystal supersaturated with copper or nickel are vacancies."7 Vacancies can also be produced by quenching from temperatures near 64 P. Haasen and A. Seeger, Halbleiter Probleme, IV (Friedrich Vieweg und Sohn, Braunschweig, 1958), pp. 68. 66 J. P. McKelvey, Phys. Rev. 106,910 (1957). 66 G. K. Wertheim and G. L. Pearson, Phys. Rev. 107, 694 (1957). 67 P. Penning, Philips Research Repts. 13, 17 (1958). Cross section Irradiation Defect level X1016cmll pile neutrons E,-E=0.23 40 (up) and Co'" gammas E,-E=0.23 5 (up) pile neutrons E,-E=0.23 10 (up) 40 (un) 14-Mev neutrons not determined 10 (up) 14.5-Mev neutrons E-Ev=0.32 [up!un=300] pile neutrons E,-E=0.20 30 (up) and CoSO gammas Ec-E=0.20 4 (up) pile neutrons multilevel behavior and CoSO gammas Ec-E=0.2 0.55 Mev not determined 0.5 (up) 0.75-Mevelectrons not determined 1.0 (up) 2-Mevelectrons Ec-E=0.18 16(up) 1.6 (u,.) fission plate neutrons middle of gap [r=3.9X101>I>-'] r=4.3XlOl>I>-1 pile neutrons not determined [r=3X 101>I>-IJ r = 2 X 101>I>-1 0.7-Mevelectrons E-Ev=0.27 8000 (up) 95 (unl 0.7-Mevelectrons E,-E=0.16 18(up) 19(un) fission plate neutrons not determined [r=5.7X105q,-1] g See reference 39 . i See reference 43. k See reference 53. h See reference 42. j See reference 45. the melting point58 and by plastic deformation. 59 A comparison between the defects produced in these three ways with those produced by bombardment may show which of the bombardment levels are to be assigned to an isolated vacancy. Some thoughts along this line have been put forward by Seeger,SO and a number of other experiments are possible. CONCLUSIONS It is apparent from the preceding as well as from the summary in Table II, that our understanding of the recombination effects of radiation damage in germanium and silicon has made considerable progress. Difficulties usually arise when it proves impossible to associate the recombination level with one otherwise identified. This situation is similar to our experience with chemical im purities where proper interpretation of lifetime data became possible only after the level scheme had been established by other means; the difficulties are not sur prising in view of the luxuriant complexity Df the multi level recombination problem. The conceptually simple 68 R. A. Logan, Phys. Rev. 101, 1455 (1956). 69 A. G. Tweet, Phys. Rev. 99, 1245 (1955). 66 A. Seeger, Proceedings of the Brussels Conference, 1958 (to be published). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:021174 G. K. WERTHEIM equations for the net capture rates of individual defects lead to differential equations which have useful solutions only in the simplest cases. One well-known solution is the single level case, but unfortunately most known recombination centers are not of this type. Multilevel defects give rise to an entirely new class of effects which may readily be confused with temperature dependent capture constants and may suggest erroneous energy levels. Determinations of cross sections can usually be made with confidence only if both the level scheme and the charge state of the defect are known from Hall or conductivity measurements. The lack of consistency that has been noted among various measurements of similar systems can arise in a variety of ways. In the case of neutron damage the chief source of discrepancy may well lie in the neutron dosimetry since the total integrated flux is often used to compute the defect density using introduction rates measured elsewhere or computed from theory. It appears desirable to determine the defect density as directly as possible, when meaningful cross sections are needed. Under certain favorable conditions it can be obtained from lifetime measurements alone.ll When this is not possible Hall measurements are called for. Complications may also arise from radiation annealing. 61 61 Mac Kay, Klontz, and Gobeli, Phys. Rev. Letters 2, 146 (1959). A possible mechanism here is that energetic carriers give up energy to the damage configuration, facilitating rearrangement. Additional complications may be due to the failure of the reciprocity law62; i.e., the amount of damage may depend not only on the integrated flux, but also on the rate at which it is administered. Finally, the suggestion has also been put forth that annealing and rearrangement of the microscopic defect structure may depend on the electronic state of charge of the defect, that is to say on the Fermi leve1.28•63 If this is correct it may not be proper to assume that a given bombard ment will necessarily produce the same defects in an n and a p-type crystal. Some of these intriguing ideas may be amenable to investigation using the recombination process. The principal achievement of the papers discussed here has been to show that bombardment defects have measurable, reproducible recombination properties. Cross sections and energy levels have been established in a number of cases. Little has been done so far to use this information to establish the detailed structure of bombardment defects, which is, after all, the central problem in the study of radiation effects. 62 J. W. Mac Kay (private communication); W. L. Brown (private communication). 63 W. L. Brown, J. App\. Phys. 30, 1320 (1959), this issue. JOuRNAL OF APPLIED PHYSICS VOLUME 30. NUMBER 8 AUG us T. 1 9 5 9 Radiation Effects on Recombination in Germanium ORLIE L. CURTIS, JR. Solid State Division, Oak Ridge National Laboratory,* Oak Ridge, Tennessee The properties of recombination centers in germanium are obtained on the basis of lifetime data in con junction with other information available. For recombination centers introduced by C060 gamma rays and fission neutrons, the recombination energy level position is placed at 0.20 ev below the conduction band. The room temperature hole-capture cross sections resulting are 1.1 X 10-16 em' and 6X 10-16 ern' for C060 gamma ray and fission neutron irradiation, respectively. For the case of 14-Mev neutron irradiation the energy level is located 0.32 ev above the valence band. The room temperature hole and electron cross sections are ",6 X 10-1• em' and 2.2X 10-17 em', respectively. The capture probabilities are assumed to be independent of temperature except for the case of gamma irradiation, for which there is apparently a fairly strong variation corresponding to a change in the activation energy of 0.07 ev. The selection of the values given above is not entirely unique. The assumptions made in their determination are discussed. The values given are directly applicable only in the case of n-type material, the situation in p-type material being more complex. I. INTRODUCTION MINORITY carrier lifetime measurements are being used to an increasing extent as a sensitive de tector of radiation damage. That lifetime is very sensi tive to radiation was recognized as early as 1953 when measurements were made on diodes irradiated in the Oak Ridge graphite reactor.! Lifetime changes in * Oak Ridge National Laboratory is operated by Union Carbide Corporation for the U. S. Atomic Energy Commission. 1 B. R. Gossick (personal communication). transistors as well as bulk samples were reported at about the same time by others,2 but the low initial life times made fairly large irradiation necessary for meas urable effects. Several investigators have used the properties of devices to study the effect of irradiation on lifetime. For instance, some of the earlier measurements depended upon relating the lifetime to the short-circuit current of the photovoltaic effect,3 and the reverse 'Florida, Holt, and Stephen, Nature 173, 397 (1954). 3 J. J. Loferski and P. Rappaport, Phys. Rev. 98, 1861 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Thu, 28 Aug 2014 14:30:02
1.1704989.pdf	On the Propagation of Gravitational Fields in Matter Peter Szekeres Citation: Journal of Mathematical Physics 7, 751 (1966); doi: 10.1063/1.1704989 View online: http://dx.doi.org/10.1063/1.1704989 View Table of Contents: http://aip.scitation.org/toc/jmp/7/4 Published by the American Institute of PhysicsJOURNAL OF MATHEMATICAL PHYSICS VOLUME 7, NUMBER 4 APRIL 1966 On the Propagation of Gravitational Fields in Matter PETER SZEKERES Center for Radiophysics and Space Research, Cornell University, Ithaca, New York (Received 14 September 1965) A purely covariant treatment is made of those solutions of the Einstein field equations which rep resent pure gravitational radiation propagating in fluid and electromagnetic media. The analysis involves a discussion of the full Bianchi identities in carefully selected tetrad frames. In this way the interaction between the gravitational field and the medium is transferred to a coupling between a preferred frame for the gravitational field and one for the matter field. The gravitational radiation no longer propagates along shear-free null geodesics, as it does in vacuum, and the shear and ray curva ture of the propagation vector are shown to depend directly on the properties of the medium. Some new solutions of the field equations, representing transverse gravitational waves propagating in an electromagnetic field, are exhibited and discussed in some detail. It is shown that no such solutions exist, at least in simple cases, for perfect fluids. Finally, the treatment presented here is compared with the more usual electromagnetic treatment, and it is shown why the theories require basically different approaches. 1. INTRODUCTION A CONSIDERABLE amount has been written about the propagation of gravitational radia tion in empty space.1 These investigations rely heavily on the study of what are called algebraically special gravitational fields, which correspond phys ically to the case of "pure" radiation. The principal result is the theorem of Goldberg and Sachs (1962): A vacuum metric is algebraically special if and only if it admits a shear-free null geodesic congruence. Although it is possible to considerably relax the vacuum conditions2 it is by no means true that the theorem holds in general. This paper deals with the question; what happens to the Goldberg-Sachs theo rem when there are perfect fluids or electromagnetic fields present? The answer to this question should furnish clues to the following problems: (a) the inter action of gravitational fields with matter, (b) the generation of gravitational waves in physically real istic sources, (c) the establishment of criteria for the presence of gravitational radiation in matter, (d) a new function theory for nonvacuum gravita tional fields. The analysis rests upon the decomposition of the curvature tensor into the trace-free Weyl tensor and a sum of terms arising from the Ricci tensor: Rabed = Cabed + gal.Rdlb + Ralegdlb -jRgalegdlb.3 (1.1) 1 See, for example, F. A. E. Pirani, "Gravitational Radia tion", article in Gravitation, an Introduction to Current Re search, edited by L. Witten (John Wiley & Sons, Inc., New York, 1962). 2 W. Kundt and A. Thompson, Compt. Rend. Acad. Sci. Paris 254, 4257 (1962). 8 Square brackets denote antisymmetrization, A1abl = [1/2!](Aab -Aba). Round brackets denote symmetrization. On account of the Einstein field equations (1.2) the Ricci terms in (1.1) can be equated with the presence of matter. The Weyl tensor, having all the symmetries of a vacuum Riemann tensor, is to be thought of as representing the free gravitational field. At any point of space-time the Ricci tensor and Weyl tensor are completely independent, but in a region they are connected through the differential Bianchi identities, which can be written in the fol lowing form4 : Cabed:d = Rela:bl -figelaR.bl' (1.3) The remarkable resemblance that (1.3) bears to Maxwell's equations leads to the suggestion that the Bianchi identities represent the interaction between the gravitational and matter fields. The right-hand side J abc of (1.3) is to be regarded as a matter current; it satisfies a "conservation equation" Jabe:e = 0, (1.4) analogous to the conservation equation of electrody namics r.a = o. The matter current represents that part of the source which interacts with the free gravitational field. Those parts of the matter which do not contribute to J abc are called gravitationally inert; the propagation of the free gravitational field is in no way dependent 4 W. Kundt and M. Triimper, Akad. Wiss. Mainz. No. 12 (1962). 751 752 PETER SZEKERES upon them. There is nothing corresponding to this in electrodynamics where, by Maxwell's equations, the electromagnetic field determines the complete charge-current distribution. The difference between the two cases can be expressed by saying that photon telescopes can be used to explore the universe com pletely with regard to its electric charges, but a graviton telescope may fail to detect the presence of matter in certain states. In Sec. 2 the Bianchi identities (1.3) are considered when there is a perfect fluid present and the Weyl tensor is algebraically special. It is found that the gravitational field propagates along a null direction whose shear and refraction (as measured by the curvature of the rays) is determined completely by the dynamical and kinematical properties of the fluid. Futhermore the fluid decomposes into separate parts which interact independently with the Petrov type-N, type-III and type-D components of the gravitational field. In Sec. 3 a similar analysis is carried out for electromagnetic fields. In this case it is found that the shear and refraction of the gravi tational field depend on the optical properties of the electromagnetic field. Some exact solutions with a Petrov type N gravi tational wave propagating along shear-free null geo desics in a nonnull electromagnetic field are exhibited in Sec. 4. In Sec. 5 it is shown that Petrov type N solutions cannot exist in a perfect fluid if the fluid I:>ressure vanishes. Without the condition p = 0 the problem remains unsolved, but it is pointed out that "almost perfect" fluid solutions of Petrov type N may exist. In conclusion the physical significance of the analysis is discussed, with particular emphasis on its relation with electromagnetic theory. 2. GRAVITATIONAL FIELDS IN PERFECT FLUIDS (i) Dynamics and Kinematics of Fluids For a perfect fluid the energy-stress tensor takes the form (2.1) where The kinematics of the fluid are studied by breaking up the covariant derivative of the 4-velocity in the following way: (2.2) where 8 = ua;a, Wab = hlachbldUc;a, and With respect to a Fermi propagated frame, Wab and U ab are respectively the rates of rotation and shear of neighbouring particles of the fluid5 ; 8 is the rate of expansion of the timelike congruence. We define shear and rotation scalars u and W by From the field equations (1.2), we obtain the Ricci tensor Rab = -(p + P.)UaUb + !(p -P.)gab' (2.3) and the contracted Bianchi identities result in equa tions of motion for the fluid, Ji. + (p. + p)8 = 0, habp,b + (p. + p)ua = O. (2.4a) (2.4b) The full Bianchi identities (1.3) yield, on substituting (2.3),' -(p. + P)(WabUc -UlaWblc + UlaUblc)' (2.5) The right-hand side of this equation is the matter current Jab. discussed in Sec. 1. Equations (2.4) only involve 8, Ua, (J. and habp.b; we say that these quantities constitute the inert part of the fluid since they are not connected with the propagation of the free gravitational field. Jab. involves essentially the shear and rotation of the fluid, and the spatial gradi ent of the density; these constitute the gravitationally active part of the fluid, the part that can be found by observing the propagation of the free gravitational field. (ll) Algebra of the Weyl Tensor In order to study the Weyl tensor it is convenient to set up a quasi-orthonormal tetrad of null vectors ka, ma, ta, la satisfying kama = tala = 1, kaka = mama = tata (2.6) Introducing three self-dual bivectors (2.7) 6 J. Ehlers, Akad. Wiss. Mainz. No. 11 (1961). ON THE PROPAGATION OF GRAVIT ATION AL FIELDS IN MATTER 753 we can decompose the Weyl tensor into tetrad com ponentsG Oabed + iO~bed = 01 Vab Ved + 02(VabMed + Mab Ved) + 03(MabMed + Uab Ved + VabUed) + OiUabM ed + MabUed) + OSUabU ed, where 0* -l(_)f 01; abed -2 g Eabi; cd' (2.8) (2.8) The various terms in (2.8) have the following phys ical interpretations7 : the 01 term represents a trans verse wave component in the ka direction, the O2 term a longitudinal wave component, and the 03 term a "Coulomb" component. The 04 and 05 terms represent longitudinal and transverse com ponents in the ma direction. (iii) Optics of Null Congruences The principal optical properties of a null con gruence having ka as tangent can be studied from the tetrad components of the complex vector (2.9) Lb is determined up to a phase e18 , since ta may be subjected to transformations of the form We shall call La the optical vector of the null con gruence; its tetrad components are 'Y = Lbkb = 'Y(1) + i'Y(2) , 12 = Lbmb , (2.10) 'Y vanishes if and only if ka is geodesic; it measures the ray curvature or the departure from geodicity in the rays. Consequently we may think of it as representing the refraction of the null congruence. U is called the shear, (J the expansion, W the twist, and 12 the angular velocity or rotation of the null con gruence.6 (iv) Propagation of the Gravitational Field Consider now an algebraically special Weyl tensor. This means that there exists a null vector ka, such that C4 = Cs = 0 in (2.8). The Weyl tensor is of Petrov type N if C2 = C3 = 0 for this ka, of Petrov type III if C3 = 0, and of Petrov type II or D if C3 ~ O. A simple calculation from (2.8) with these specializations yields the following relations: • R. Sachs, Proc. Roy. Soc. (London) A264, 309 (1961). 7 P. Szekeres, J. Math. Phys. 6, 1387 (1965). In Petrov type N uabV"Cabed;d = C1(l''Y -k'u), in Petrov type III vabOabed;d = 2C2(le'Y -keu), and in Petrov type II or D vabV"Cabed;d = 3Ca(l''Y -k·u). (2.11) (2.12) (2.13) When there is a fluid present with streamlines ua, we normalize ka to make kaua = -1, and defining Sa = habkb (whence sasa = 1, saua = 0) we can choose the null vector ma such that (2.14) Substituting the Bianchi identities (2.5) into Eqs. (2.11), (2.12), and (2.13) we find the following ex pressions for the shear and refraction of the prin cipal null congruence ka (denoted here by Uo and 'Yo to distinguish them from the fluid quantities): In Petrov type N 301'Yo = !(~.ata -3(~ + P)(Wob + UOb)tOSb), (2.15) 3C1Uo = i(~.osa -3(~ + P)(Wab + uOb)lOtb), in Petrov type III 3Cao = !(~.osa + 3(~ + P)(Wab + UOb)rt), 3C2Uo = -H~.or + 3(~ + P)(Wob + UObWSb), in Petrov types II or D (2.16) 3C3'Yo = -~.blb -(~ + p)(3Wabrsb -uablbse), (2.17) 3C3Uo = (~ + p)Ube~br. ka is called the principal null direction of the gravita tional field; the field is to be regarded as propagating along this direction. Equations (2.15), (2.16), and (2.17) show that the shear and refraction of the principal null direction of an algebraically special gravitational field are determined by the tetrad components of the spacelike density gradient, the rotation and the shear of the fluid. If the Weyl tensor is of Petrov type N we have C2 = C3 = 0, and the right-hand sides of equations (2.16) and (2.17) must vanish. It follows then from (2.15) that4 (~ + P)Uab = C1uo(3sasb -hab) , (2.18) ~ + P)Wab = 2s[o(lbICl'Y0 + tb1C1'YO), (2.19) 754 PETER SZEKERES h\p..a = 3(CI'Yolb + CdOtb + O'OSb)' (2.20) Hence the optical shear and the refraction are directly proportional to the shear and the rotation of the fluid: v2 !CI'Yo! = (p. + p)w, v'3 !CIO'o! = (p. + p)O'. (2.21) (2.22) From (2.22) we see that 0'0 is real if and only if CI is real; this means that the principal axes of the optical shear coincide with the polarization axes of the transverse gravitational field (the axes ta , r which make CI real). Equation (2.18) shows that the fluid shear has a principal axis in the ray direction Sa and is degenerate in the transverse (ta, 1a) plane. From (2.19) and (2.21) it is seen that the refraction of the wave is determined by the rotation of the fluid. The axis of rotation of the fluid must lie in the trans verse plane of the wave; if it coincides with one of the polarization directions (CI real and 'Y~l) or 'Y~2} = 0) then the wave is reflected at right angles to it, whereas if it is at 450 to the polarization di rections the wave is deflected in the direction of the rotation axis (Fig. 1). For a type-III Weyl tensor the right-hand side of Eq. (2.17) must vanish, since Ca = O. Hence we have 3C2O'o = -(p. + P)O'belbse, (2.23) and ka is shear-free if and only if sa (the longitudinal wave direction according to an observer traveling with the fluid) is a principal axis of the fluid shear (Fig. 2). Equation (2.16) can be split up into real and imaginary parts 6C2'Y~1l = p..asa -!(p. + P)O'beSbS" , (2.24) FIG. 1. Propagation of a transverse gravitational wave (type N) in a perfect fluid. The central ellipsoid represents the shear of the fluid streamlines. The broken lines denote graviton paths. They are deflected from the geodesic by a vector da which makes an angle t/> = 28 ± iT with the rotation axis CJf', where 8 is the angle CJf' makes with one of the polariza tion axes of the plane wave. The magnitUde of this deflection is proportional to the angular velocity", of the fluid. A circular cross section of gravitons is transformed into an ellipse, by an amount proportional to the fluid shear <T in the direction of wave propagation. FIG. 2. Propagation of a longitudinal wave (type III) in a perfect fluid. The circle of gravitons is transformed into an ellipse, by an amount depending on the angle 8 between the principal fluid shear axis and the direction of wave propagation 8a• The deflection out of the plane of the polariza tion is proportional to cos t/>, the angle between 8a and the rotation axis ",a. where Hence the ray is only left undeflected in a direction orthogonal to its longitudinal plane of polarization [the (sa, ea) plane] if the axis of rotation of the fluid is orthogonal to the ray direction. The refraction in its own plane is determined by the components in the ray direction of the density gradient and fluid shear. It is unaffected by any rotation the fluid may have about 8a as axis. Equations (2.15), (2.16), and (2.17) suggest that not only can the matter be split up into gravita tionally inert and active parts, but the active part J abc can be further split up into separate parts inter acting with the transverse wave component, the longitudinal wave component and the Coulomb part of the field. For example, the shear tensor can be split up as a sum of three terms: and From Eqs. (2.18), (2.23), and (2.17) it appears that for an algebraically special field with principal null vector ka = sa + ua, the first term interacts with the shear of the type-III component, and the last with the Coulomb component. This splitting off is really the essence of Kundt and Thompson's state ment of the Goldberg-Sachs theorem:2 Any two of the following imply the third: (A) Cabed is algebraically special with ka for principaZ null vector. (B) ka is shear-free and geodesic. (C) VabV"Cabed;d = 0 vabCabed;d = 0 for Petrov type III uabV"Cabc/d = 0 for Petrov type N. ON THE PROPAGA TION OF GRA VIT A TION AL FIELDS IN MA TTE R 755 From Eqs. (2.11-(2.13) it is clear that (A), (B) => (C), and (A), (C) => (B). The proof that (B), (C) => (A) is less trivial. 3. INTERACTION OF GRAVITATIONAL AND ELECTROMAGNETIC FIELDS An electromagnetic field is represented by a skew symmetric tensor Fab satisfying Maxwell's equations (Fab + iF:b);b = o. (3.1) The energy-stress tensor is given by Tab = FaiFb i -!gabFijFij = -Rob. (3.2) (i) Null Field The electromagnetic field is said to be null if there exists a null vector such that (Fab + iF:b)ka = 0, from which it follows that the Maxwell tensor can be written in the form (3.3) where the conventions of Sec. 2 are adopted. Max well's equations (3.1) now imply that ka;bkbr = 0 and ko;blalb = 0, ka is shear-free and geodesic. From the field equations (3.2) we have and the Bianchi identities (1.3) can be written as Cab.d;d = R.1a;bl = -tck.k1a;bl + k.;lbkal), (3.4) whence From the Goldberg-Sachs-Kundt-Thompson theo rem quoted at the end of Sec. 2, it follows that the gravitational field must be algebraically special with ka as principal null direction. This result is what we might expect intuitively-the gravitational field as sociated with a pure radiation electromagnetic field consists of pure gravitational radiation. If the Weyl tensor is of Petrov type N, we can contract (3.4) with nb and find that o = k. ;bl"tb = Z = () + 1M. Hence the expansion and twist must vanish if the Weyl tensor represents a pure transverse gravita tional wave. All solutions of the field equations representing this situation have been found by Kundt.8 8 W. Kundt, Physik, 163,77 (1961). (ii) Non-null Field The Maxwell tensor has the form Fab + iF:b = A(2Plaqbl + 2f1arbl), (3.5) where Pa, qa are the principal null vectors of the electromagnetic field.9 Pa, qQ, Ta, fa form a quasi orthonormal null tetrad (we call it the electromag netic frame). A is the (complex) electromagnetic amplitude or field strength. Maxwell's equations (3.1) can now be regarded as expressing the gradient of the field amplitude in terms of optical parameters of the principal null directions: !(In AL = _Z(vl qa -z(alpa + n(p)ra + n(a)fa, (3.6) where z(v) = L~v)rb, z(a) = Lialfb , n(vl = Liv) qb, n(al = Lia'pb, Lip) , Lia) being the optical vectors pa and qa, The field equations (3.2) result in Rab = IA 12 (2P(aqb) -!gab). (3.7) On substituting into the Bianchi identities we can carry out a similar analysis to that for a fluid medium. There are two cases to be distinguished: (a) The gravitational field is algebraically special and its principal null vector ka coincides with one of the null vectors Pa or qa of the electromagnetic field. The two fields shall be called aligned in this case; it has been shown by Kundt and Triimper4 that ka must be shear-free and geodesic. (b) The gravitational and electromagnetic fields are nonaligned; that is, ka does not coincide with either pa or qa. It is possible to scale these null vectors such that kapa = -kaqa = -1. By a spacelike rotation ra ~ e,era we can achieve that ka = Pa -qa + ra + fa. The null tetrad for the gravitational field can be completed by choosing ma = H-Pa + qa + ra + fa), ta = !(P. + qa + fa -ra). This normalization amounts to a coupling of the 9 J. L. Synge, Relativity, the Special Theory (North Holland Publishing Company, Amsterdam, 1956). 756 PETER SZEKERES gravitational and electromagnetic frames, so as best to view the interaction. Substituting (3.7) into the right-hand side of the Bianchi identities (1.3), and using the identities (2.11)-(2.13) and Maxwell's equations in the form (3.6), we arrive at the following relations: For a type N Weyl tensor C{'f = -[A [2 (L~p) + L~Q»ma , CIU = ! [A [2 (L~P) + L~Q»t" , for a type III Weyl tensor 2C2'Y = [A [2 (L~p) + L~Q»)l" , 2C2u = ! [A [2 (L~p) + L~Q»ka, for a type II or D Weyl tensor 3C3'Y = 4 [A [2 «L~p) + L~Q»m· + (L~P) -L~Q» r) , 3Cau = [A[2 (-(L~p) + L~Q»t· + (L~p) -L~Q»ka). (3.8) (3.9) (3.10) Hence with this choice of tetrads, the interaction between an algebraically special gravitational field and a nonaligned electromagnetic field is completely determined by the tetrad components in the gravita tional frame of the sum and difference of the two optical vectors of the electromagnetic field. If the W eyl tensor is of Petrov type N then the right-hand sides of (3.9) and (3.10) must vanish; if the principal null vector k. of the gravitational field is to be shear free and geodesic it is clear that the sum of the op tical vectors, L!p) + L!Q), must vanish. Exact solu tions representing this situation are discussed in the next section. 4. EXACT ELECTROMAGNETIC SOLUTIONS (i) Null Solutions In the light of the preceding analysis it would be interesting to exhibit some exact solutions repre senting gravitational waves propagating through various media. As a first example there exist the metrics of Kundt8 representing a type N gravita tional field having u = (J = w = n = 0 (planefronted waves with parallel rays), accompanied by a plane electromagnetic wave, dl = !(dx2 + dy2) - 2 du dr + 2U du2, where U = U(x, y, u) satisfied The coordinates are those introduced by Robinson and TrautmanlO in which Xl = u hypersurfaces const are null The vectors k. = U,a, are tangent to the family of null geodesics lying in the hypersurfaces, and x2 = r is chosen as an affine parameter along these geodesics. The coordinates x3 = x and x· = y label the geodesics on each surface U = const. (li) Nonaligned Nonnull Solutions There also exist solutions of the field equations with a nonnull electromagnetic field and which are of Petrov type N. To find these solutions we use the relations obtained from the Bianchi identities in the previous section and put these into the N ew man-Penrose formalismll to obtain further simplifica tions. Finally we set up Robinson-Trautman co ordinates and use the methods of Newman, Tambur ino and Unti12 •13 to obtain the exact solutions. The procedure is long and cumbersome, but fairly straightforward. The final result is the following metric: di = ! cos2 ttr(dx2 + dy2) - 4 du dr -2T(2r + K -I sin 2ttr) du dx + 4K -2(2T2 sin2 Kr -2e2 .. -rK aKj au) du2, (4.1) where T = T(U, x) = eU coth (e"x + feu»~, K = K(U, x) = g(u)e" sinh (eUx + feu»~, g(u) and feu) are arbitrary functions of u. This metric is of Petrov type N with principal null vector pointing along k. ex: u,. = (1, 0, 0, 0). k. is geodesic, shear-free and twist-free, but it will have an expansion and a rotation. The Ricci tensor turns out to be where p. = (lKe-", -r(!e-" aKjau + T2e-" tan Kr + TsecKr), e-"'TtanKr + seCKr, 0), and t = _po + (0, -2rT sec ttr, 2 sec Kr, 0). 10 1. Robinson and A. Trautman, Phys. Rev. Letters 4, 431 (1960). 11 E. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962). 12 E. Newman and L. Tamburino, J. Math. Phys. 3, 902 (1962). 13 E. Newman, L. Tamburino, and T. Unti, J. Ma.th. Phys. 4, 915 (1963). ON THE PROPAGATION OF GRAVITATIONAL FIELDS IN MATTER 757 p. and t are a pair of null vectors satisfying paqG = l. By (3.7) the metric can be considered as repre senting a transverse gravitational wave propagating along shear-free null geodesics through a nonnull electromagnetic field. The principal null vectors of this field are p. and q., neither of which are aligned with the gravitational wave kG, and the electromag netic field strength is A = 2eu • The electromagnetic field has the odd character that it is not a wave field (since it is not null-the electric and magnetic fields are nowhere equal and perpendicular) yet its amplitude propagates with the velocity of light. It may be thought of as a "quasi wave" field. For a timelike observer the passage of the field will appear like an electromagnetic sheet whose strength rises (or diminishes) exponentially without limit. We may calculate the strength C1 of the gravitational wave in the frame (kG, mG, ta , ta ) determined from the normalizations of Sec. 3. It is = !g(u)eU sinh (eUx + feu»~ tan Kr. Thus the arbitrary function g(u) measures the strength of the gravitational wave, which is seen to be quite independent of the electromagnetic field strength A. The function feu) is merely a phase function on the wave hypersurfaces u-const, which can be set to zero by a coordinate transformation It is interesting that C1 has singularities at r (n + !)'Ir,,-1. These are real singularities of the manifold, and there is no way of avoiding them. Another way in which these singularities show up is in the expansion of the gravitational propagation vector ka = U,a' When there is no electromagnetic field we have that 0 = kG;. satisfies dO/dr = ~, so that 0= I/r and the waves are spherical, emanating from a source at r = O. With the electromagnetic field present the equation becomes modified to read dO/dr = ~ +l, so that o = "tanKr. fold to the region _!'Ir,,-1 < r < !'Ir,,-1 it will be incomplete. (iii) Aligned Nonnull Solutions The metric (4.1) is by no means the most general one representing a pure transverse gravitational wave in a nonnull electromagnetic field. It is not even the most general one with shear-free geodesic propagation vector ka' The analysis in the Penrose Newman formalism makes it clear that the electro magnetic field strength A may be variable over the hypersurfaces u = const. However it must be con stant along the tangents ka if these are to be shear free and geodesic: A,aka == aA/ar = O. The full integration of the field equations in this more general case is considerably more complicated, and a closed form for the metric has not been found. The metric (4.1) represents the case of a type-N wave in a nonaligned electromagnetic field. There exist further solutions representing a type-N wave in an aligned field. As pointed out in Sec. 3 (ii) (a), the principal null vector kG is shear-free and geodesic. For Petrov type N it turns out furthermore that ka has vanishing expansion, twist and angular mo mentum (that is, it is a p.p. wave), and the elec tromagnetic field amplitude A is constant. This makes the Newman-Penrose field equations fairly straightforward to integrate. The result is ds2 = !P-2(dx2 + dy2) - 2 du dr -P-2(X du dx + Y du dy) + {U -! IAI2 r2 + !P-2(X2 + y2) I du2 (4.2) where P = P(u, x, y) satisfies p2\12 In P = ! IA 12 = const. U(x, y, u) satisfies \12U = _p2, and z = X + iY = feu, z) -4 aU /az, where z = x + iy, \12 == a2/ax2 + a2/ay2, and f is an arbitrary analytic function of z. This metric is of Petrov type N with propagation vector pointing along kG = u,. = (1, 0, 0, 0). The Ricci tensor has the form (4.3) The waves are infinitely divergent at the points where r = (n + !)'Ir,,-1. If we choose to restrict the mani- ma = (-1, -11A12 r2 + U, X, Y). 758 PETER SZEKERES kQ and mG are the principal null vectors of the elec tromagnetic field. The null vector is neither shear free nor geodesic. Completing the tetrad with the vectors tG , lB, where tG = (0,0, P, iP), we find for the shear and refraction of m G "I = ma;btBmb = -2P au/az. In this frame the gravitational field strength CI can be calculated; -8 a(p2 au/az) _ ap2 _ X ap2 _ Yap2 az au ax ay -P2(IAI2 r + 4 az/az). We see that the field strength varies along the geo desics of propagation: CI.BkB = acI/ar = 2 -p2 IAI2. If the null vector mG has vanishing shear, it is clear we cannot use the metric (4.2) since p2 = O. This situation is represented by the metric ds2 = !P-2(dx2 + dy2) -2 du dr + 2(U -1lA 12 r2) du2 , (4.4) where p2V2lnP2 = ! IAI2, V2U = O. The Ricci tensor is again of the form (4.3) but with mB = (-1, -1 IAI2 r2 + u, 0, 0). In this case mG is shear-free, but it is still not geodesic. The gravita tional field strength is given now by CI = -8 a(p2 au/az)/az, and is constant along the kB geodesics, aC 1/ ar = O. (iv) A Conformally Flat Solution The metrics (4.2), (4.3) are all the metrics rep resenting a pure transverse gravitational wave prop agating through an aligned nonnull electromagnetic field. From the metric (4.4) we can obtain an in teresting case if we put U = O. mB is now geodesic, "I = 0, but also CI = O. This means that the Weyl tensor vanishes, and there is no free gravitational field at all. That is, the metric ds2 = !P-2(dx2 + dy2) -2 du dv -! IA 12 r2 du2 , where V2lnP2 = 0 represents a conformally flat space, with a nonnull electromagnetic field present. 5. EXACT FLUID SOLUTIONS The question we now investigate is whether there exist any Petrov type N solutions of the field equa tions with a perfect fluid. A partial answer has been given by Kundt and Trfunper,4 who show that no solutions exist if w = 0 (w = angular velocity of fluid). By Eq. (2.21) this is seen to be equivalent to the statement that no Petrov type N solutions with perfect fluids exist in which the waves are prop agated along null geodesics ("I = 0). However, the case p = J1 + A(t), where t = const are the hypersurfaces to which the uB are orthogonal (they exist on account of the pos tulate w = 0), eludes the Kundt-Trfunper analysis. They discard this case as unphysical since it is usual to have p < iJ1. This is not totally convincing, however, since J1 might be almost constant on the hypersurfaces t = const, and A (t) chosen in such a way as to have p < iJ1 satisfied everywhere. There appears to be no straightforward way of eliminating this case, and it must remain an open question whether there exists solutions of Petrov type N with p = J1 + ACt)· The more general case w ~ 0 is much harder to analyze since the fluid streamlines are no longer hypersurface-orthogonal and it is not possible to set up suitable Gaussian coordinates. We have man aged to deal with the case p = 0, where by (2.4b) the streamlines are geodesic, uB = O. The result, proved in the Appendix, is the following: No solutions of Petrov type N with incoherent matter (p = 0) exist. While the question of the existence of type N solutions is still not decided, we see from the above results that such solutions, if they exist, must be of a complexity considerably exceeding that of any fluid solutions that have been found to date. To conclude this discussion, we give a simple argument to show that locally there can be a fluid present in a Petrov type N metric. Consider a conformal transformation of the metric, The Ricci tensor transforms as flBb = RBb + 2Ua;b -2uaub + (2ucuC + UC ,.)gab, where Ua = U,B' The Weyl tensor remains invariant ON THE PRO P A GAT ION 0 F G R A V I TAT ION A L FIE L D SIN MAT T E R 759 so that the Petrov type of the metric is unchanged by the conformal transformation. If we consider gab to be the metric tensor for a vacuum solution Rab = 0, and let u be a solution of the partial dif ferential equation then where Using the field equations in the new space flab -!1l0ab = -Tab, where 1l = llabOab = 6( 0 -l)e -2", we find that Tab = -20"ab + (40/3 -1)e-2"hab (5.1) + (3 -2 0)e-2UUaUb , (5.2) where ua = e"ua is a timelike unit vector in the gab space, and hab = dab + UaUb. Thus we have generated a perfect fluid solution from the vacuum if we can find a solution of Eq. (5.1) with O"ab = O. We cannot find such a solution if the initial metric is of Petrov type N, since the fluid streamlines would be hyper surface-orthogonal (w = 0), contradicting the result of Kundt and Trumper. However it is clear that at any point of the manifold it is possible to find a solution having O"ab = 0 at that point. In this way we can generate a "local fluid." But as we depart from this point we will have O"ab ~ 0, and aniso tropies will appear in the energy tensor. It is not inconceivable that we might find a solution in which 0" remains small relative to 0 at least for a sizable region of the manifold, and in this region we will have an "almost-perfect" fluid. We can obtain an upper bound for the size of the region in which Tab remains physical. From (5.2) it is seen that the density and mean pressure are given by It = e-2"(3 -20), P = e-2U(40/3 -1). Hence, if It and p are both to be positive we must have ! S (j <!. Furthermore 0 should be much closer to the lower value than the higher, else the pressure dominates the density. Now we can use the Ricci identities Contracting over a and c and using the vacuum condition Rab = 0, we find on further contracting with ub that o == ao/au = _20"2 -102. If initially at u = Uo, 0 = ! + E, we will have ao/au < - 136' hence 0 can only remain > ! until a time Ul = Uo + 16 E/3, after which the pressure becomes nega tive. 6. RELATION TO ELECTROMAGNETIC THEORY The results obtained in this paper for the prop agation of gravitational waves in matter have a strangely unfamiliar ring when we try to compare them with the usual electromagnetic treatment. For example, the" refraction" discussed here is nothing like the refraction of electromagnetic waves, for there is no slowing down of the waves-there is merely a deflection from the straightest, the geo desic, path-while the other feature of the inter action, the shear of the waves, is something never discussed in electromagnetic theory. It is not hard to see where the difference between the two theories lies. We could treat the electromagnetic field in a similar way, discussing the Maxwell equations Fab _·a .b -J , and obtaining a departure from geodicity and a shear in the electromagnetic wave coupled to the current vector l. But this treatment would be entirely wrong if applied, say, to light passing through a slab of glass. In this case the interesting features occur at the atomic scale, where the cur rent l becomes extremely complicated. When we smooth out all these tiny currents we have l = 0, so that the field should propagate as though there was no matter present at all, Fab •b = o. But at the atomic level there is the creation of a large number of oscillating dipole moments which produce their own field, out of phase with this freely propagating field in just such a way as to produce a total transmitted wave traveling with a speed less than that of light in vacuum. Feynman14 14 R. Feynman, Lectures on Physics, Vols. I and II (Addi son-Wesley Publishing Company, Inc., Reading, Massachu setts, 1963). 760 PETER SZEKERES has recently given a very clear and beautiful treat ment of just this problem. There are several reasons why such a discussion would not be applicable to the gravitational case. In the first place general relativity is a continuum theory and is only valid at that scale where we can regard the matter as smoothed out into a highly regular fluid. It is very difficult to see how one could treat a system of discrete particles in the theory. This feature arises again and again, its most famous instance perhaps occurring in cosmology where the whole galactic population is smeared out into a continuum. Secondly, the principle of equivalence demands that all masses respond equally to the gravitational field, with the result that no dipole moments are created in the matter. It is true that .quadrupole moments may occur, but there is still another point to bear in mind here. It is only on the astronomical scale that matter is held together by purely gravitational forces; on the terrestrial scale it is the much larger electromagnetic forces that are important. A comparable situation in the elec tromagnetic theory would be if the atoms were held together not by the electric forces but by some field which was stronger by a factor of about 1040 (even the nuclear forces pale into insignificance here). In such a case the induced dipole moments would be weaker by a corresponding factor, and the usual phenomenon of refraction would never be observed. Our analysis of refraction would then have to follow lines similar to those discussed in this paper. The above discussion raises some inevitable que ries. If large-scale gravitational waves arise, or have arisen at a more chaotic epoch of the universe, how do these propagate through the galactic system? The analysis should now follow the more familiar electromagnetic treatment, with induced quadrupole moments in the galaxies replacing atomic dipole moments. At the other end of the scale, we may ask how very short wavelength gravitational radiation (of atomic dimensions) would propagate in ordinary matter. Again, the electromagnetic treatment should be the one to adopt. ACKNOWLEDGMENTS The author wishes to express his gratitude to Dr. F. A. E. Pirani for his invaluable help and continued encouragement in this work, and to Dr. M. Trtimper for reading some parts of this work and making several illuminating remarks. This work was supported partly by a Commonwealth Scholar-ship held at Kings College, London, and partly by Contract AFOSR 49(638)-1527. APPENDIX: PETROV TYPE-N SOLUTIONS WITH INCOHERENT MATTER Consider a fluid with p = O. From Eqs. (2.4a, b) we have Jl = -p.O, 'Ita = O. Let us assume w ~ O. If the Weyl tensor is of Petrov type N with principal null vector ka = Ua + Sa, we have from (2.21) that "{ ~ 0 (ka is not geodesic). Take r a the unit vector pointing along "( la + -yta, and qa the unit vector pointing along i("{la --yta). Ua, Sa, ra, and qa form an orthonormal tetrad. From Eqs. (2.17) to (2.22) and (2.2) we have Ua;b = 2wslarbJ + 3U(SaSb -ihab) + iOhab (A1) and P.,b = P.(3wrb + V3 USb + flub)' (A2) If we put these into the current conservation equa tion (1.4) we get W = -i",(20 + V3 U) (A3) and (A4) Consider the Ricci identities Using (1.1) and the field equations (2.3) this may be rewritten in terms of the Weyl tensor C\edUa = 2Ub;ldeJ + iP.UldgeJb' (A5) Contracting over band c, and a further contraction with ud results in the well-known Raychaudhuri equation o = -ip. + 2",~ -2u~ -if. (A6) Using the fact that C\ed is of Petrov type N, (A7) results in and The last equation together with (A4) gives that Sd = rd = O. (A9) ON THE PROP AGATION OF GRA VIT ATION AL FIELDS IN MATTER 761 Using the Weyl tensor symmetry Ca[bedl = 0, and the fact that /L,a is a gradient in (A2) /L, [a:bl = 0, we find, using (A7), that That is, o = Sa:btSb = ka:btkb = i('l -1~/V2 hi. Hence ,),(1),),(2) = 0, that is, either ')'(1) or ,),(2) is zero, which means that qa and r a coincide with the polari zation directions of the transverse wave. This means that we can write the Weyl tensor as Cabcd = 2C(k[arblk[crdl -k[aqblk[cqdl)' By (A5), (A6), and (AS) we find (AlO) Now, V2 h'l = ka:brakb, and by (2.21) it follows that Now (All) From (A5) and (A7) it follows that the last term vanishes, while the second term can be written as wsa:crarc -1(0 + 2v3 (j)sa:.rasc. If we now differentiate (All) along ua we find using (A3), that W2Sa:brarc = Iw2(v3 (j -0) -/Lv3 (j. A final differentiation along ua of this equation re sults in /LW2 = O. Hence w = 0 and our theorem is proved, since by (A10) this means C = 0 and the Weyl tensor vanishes.
1.3047156.pdf	The physics of liquids …a conference report Joseph L. Hunter and Edward F. Carome Citation: Physics Today 18, 1, 67 (1965); doi: 10.1063/1.3047156 View online: http://dx.doi.org/10.1063/1.3047156 View Table of Contents: http://physicstoday.scitation.org/toc/pto/18/1 Published by the American Institute of Physicsthe physics of LIQUIDSa conference report By Joseph L. Hunter and Edward F. Carome Under the sponsorship of the National Science Foundation, the Physics Department of John Car- roll University played host to about sixty research- ers in the field of the physics of liquids at a four- day conference from June 1 through June 4, 1964. Although the meeting was built around a nucleus of prepared talks, several of the participants pre- ferred the more informal approach of participat- ing from the floor in all of the talks. Among the latter were Daniele Sette of the University of Rome, Henry S. Frank of the University of Pitts- burgh, Martin Greenspan of the National Bureau of Standards, and Robert T. Beyer of Brown Uni- versity. As a matter of fact, most of those de- livering talks preferred to think of themselves as discussion leaders rather than lecturers. Successive half-day sessions were devoted to the following: viscosity and viscoelasticity; x-ray and neutron diffraction by liquids; nuclear magnetic resonance effects and positron annihilation in liquids; dielectric and ultrasonic relaxation in liquids; and liquid chemistry. A full day was de- voted to general liquid theory. The first day was given over substantially to general liquid theory, handled by Peter Gray, of The University, Newcastle-on-Tyne, and by Herbert S. Green, of the University of Adelaide. Their task was to give to the assembled group, composed mainly of experimentalists, some notion of the newest approach and emphasis in liquid theory. Gray opened the seminar with the state- ment that the statistical mechanical theory of liquids may be conveniently divided into two parts, equilibrium and nonequilibrium theory, and made the point that the two are, at the present time, qualitatively different. The equilibrium theory is formally exact, and well-defined mathe- matical approximations are introduced to obtain Joseph L. Hunter of John Carrol] University was the di- rector and Edward F. Carome of John Carroll University and Ernest Yeager of Western Reserve University were co- directors of the conference reported here. Included also in the planning were William Cramer of the Office of Naval Research and Theodore A. Litovitz of Catholic University.numerical results. On the other hand, the non- equilibrium theory requires approximations of a physical or intuitive nature. For the equilibrium case, Gray described the fundamental problem: the expression of thermo- dynamic quantities in terms of the molecular po- tential and the radial distribution function. The radial distribution function involves a power series in the density in which the coefficients are irreduc- ible cluster integrals (or diagrams) . Representa- tion of the density requires a summation over the various types of diagrams. Different closed integral equations are produced, depending on the approxi- mation used in the summation, the most promi- nent being the Yvon-Born-Green, the hyper-netted chain, and the Percus-Yevick. Agreement is gen- erally good at low densities, but becomes pro- gressively worse at higher densities. In nonequilibrium theory, Gray discussed formu- lae for the viscosity, thermal conductivity, and dif- fusion coefficients. These are obtained again in terms of the pair potential and radial distribution functions by solving the kinetic equations to first order in the velocity, temperature, and concentra- tion gradients. The agreement of the numerical values so obtained with experiment is reasonably good and strongly correlated with that of the thermodynamic functions. An important outcome of these calculations is that the theory is entirely unequivocal as to the existence of a bulk vis- cosity; in the case of liquid argon, for example, its calculated value varies between one and three times the shear viscosity at different temperatures and densities. In his talk, Green chose to describe in some detail a modern problem in the statistical me- chanics of equilibrium processes, and one in non- equilibrium processes. For equilibrium processes, he chose a modification of the Monte Carlo method as an illustration. A set of M (M~25) particles is started from a random configuration in a box with periodic boundary conditions. Each particle is visited in turn; the particle is either left in its position P or displaced to a randomly chosen neighboring point P', according to whether a ran- PHYSICS TODAY JANUARY 1965 • 67dom number between zero and one exceeds or is less than a function of the potential energy of the two particles (which function may also vary be- tween zero and one) . Favored distributions result from this process, and Green described the re- sults of his work in applying the results to electro- lytes. Two intriguing results are that pairs of op- posite charges predominate at temperatures below 104/A', where K is the dielectric constant, and that electric waves may result from an initial non- equilibrium ensemble. Green also described, in some detail, a method of dealing with irreversible processes in which the evaluation of the all-important autocorrelation functions is reduced to the solution of a (comparatively) well-known hierarchy of integral equations for the few-particle distributions. The method is based on the formalisms of Kubo, Mori, M. S. Green, and H. S. Green. Heretofore, such equations were, practically speaking, unsolvable. However, with the help of advanced computational techniques and improved approximations of the hyper-netted chain type, there is now hope that they will finally yield. Green estimated that, within the next few years, they will allow transport co- efficients for liquids to be evaluated with an ac- curacy similar to that obtained for dynamic variables. In order to lighten the load on the various participants during the first day of the conference, several other talks were interspersed between por- tions of the theoretical ones presented by Gray and Green. In one of these, Carome discussed the results of several experiments on laser-induced acoustic effects performed by his research group at John Carroll University. Intense plane-wave acoustic impulses have been generated in an opti- cally absorbing liquid layer using the defocused beam from a Q-spoiled ruby laser. He indicated that such signals might be of use in studying re- laxing liquids. The focused beam from a similar laser also has been used to generate wideband ultrasonic and hypersonic waves in liquids and solids, and acoustic signals in excess of two kilo- megacycles have been propagated and detected acoustically in various liquids. Though stimulated Brillouin scattering is probably the source of some of the observed signals, it appears that sources such as dielectric breakdown also are active. Jacek Jarzynski of the American University de- scribed his ultrasonic measurements in the alloys of molten metals. Although there were many in- teresting points in regard to these experiments, perhaps the most interesting was the variation of volume viscosity, particularly its variation withthe percentage of different alloy materials. Jarzyn- ski gave results for potassium, sodium, silver, tin, and several other pure metals and combinations of these metals in alloys. A tin-silver alloy is a case in point. For large percentages of tin, the alloy manifests a large volume viscosity (a ratio of volume-to-shear viscosity of about five to one). However, as the percentage of silver is increased, the ratio of volume-to-shear viscosity falls rapidly, becoming less than one and approaching zero asymptotically, although Jarzynski did not actually reach the zero value experimentally. The morning of the second day was devoted to viscosity and viscoelasticity, primarily (but not entirely) from the experimental approach of ul- trasonic propagation. As was the case with Gray and Green in general liquid theory, this was a cooperative endeavor of Joseph L. Hunter of John Carroll University, Theodore A. Litovitz of the Catholic University, and John Lamb of the Uni- versity of Glasgow. Hunter laid the groundwork for present-day theories of viscoelasticity, starting with the theory of the ideal liquid, with its single elastic constant. He then showed that this ap- proach may first be generalized by the introduc- tion of a second constant; if this is a viscous con- stant, one has the theory of viscous liquids; if it is a second elastic constant (the shear elasticity) one has solid elasticity theory. He discussed the partial generalizations possible, and arrived at the one unique to present-day viscoelastic theory, defining the various viscoelastic moduli. He also gave the historical background to the concept of the bulk viscosity. He concluded with a description of most recent measurements which enable the evaluation of the viscoelastic constants. Litovitz used the viscoelastic constants, in partic- ular the relaxational compressional modulus, to introduce a well-knit theory of viscosity in which free volume figures vary prominently. He indicated that the relaxational moduli and the free volume should be closely related, although this relation has not been evident until recently because of the comparatively little information on the moduli. Litovitz reviewed the best-regarded recent theories of viscosity, pointing out the strengths and weak- nesses of each; he then showed that a theory which he proposed fits experimental values better than the existing theories. Very briefly this theory takes into account the facts that a molecule not only must have the strength to break a bond, but it must also have a space to go to if it is to succeed in breaking it. Whereas Litovitz was primarily interested in the compressional modulus, Lamb's talk was devoted 68 JANUARY 1965 PHYSICS TODAYThe new RIDL Nanolyzer* offers two principal advantages over other presently available multichannel an- alyzers: speed and accuracy. It can accept, analyze, and store data seven to fifty times faster than con- ventional analyzers. It will also store at rates of over 200,000 counts per second with a minimum of distor- tion when Nal(TI) detectors are used. These benefits are obtained by using several totally new circuit concepts, chief among which are those described at the right. Of equal importance are the Nanolyzer's convenient operation and the ease of servicing. Its com- pact packaging has been achieved by using printed circuit boards with Silicon Nand Arithmetic Package (SNAP-LOGIC) encapsulated cir- cuits. This also contributes to easy maintenance. The Nanolyzer can be used with all standard readout devices. For more information, please consult your RIDL sales engineer or write for your copy of our 24-page Nanolyzer Brochure.THIN-FILM MEMORY Plug-in circuit board with 128-word thin-film memory plane. The high-speed, thin-film memory planes used for data storage are the heart of the Nanolyzer. The memory is word-organized with 256 words of 24 bits each. It can store 10s (minus 1) bits of information in each memory location in a 1, 2, 4, 8 BCD code. The memory is located on two identical plug-in circuit boards for simplicity and serviceability. Each plug-in board contains, in addition to a 128- word memory plane, all of the select matrix and memory drivers for that memory plane and preamplifiers for each of the 24-bit outputs. 100-MC ADC The 100-mc Analog-to-Digital Con- verter uses a radically new technique for pulse-height analysis. By ac- curately determining when the pulse to be analyzed arrives, a sample of the pulse that is linear and inde- pendent of pulse amplitude can be used for analysis. Besides eliminat- ing the annoying linearity problems associated with peak detection, this technique reduces analyzer "open time" to a minimum and provides a time base for subsequent analysis.PILE-UP REJECTION 100K CIRCUIT OUT CIRCUIT IN 0 20 40 60 80 100 120 CHANNEL NUMBER Cs137 spectra accumulated at high counting rates with and without pile- up rejection circuit. When the ADC is operated at count- ing rates exceeding 100,000 counts per second, spectrum distortion due to pile-up will severely limit the use- fulness of the accumulated data. Therefore, a pile-up rejection circuit has been incorporated in the ADC to minimize spectrum distortion due to pile-up. Pile-up rejection is accom- plished by inspection logic which inspects the ADC baseline just be- fore the pulse arrives and rejects the incoming pulse from analysis if the baseline is not at ground due to pile-up or baseline shift. The chart above compares a typical distor- tion-free curve obtained with the Nanolyzer using pile-up rejection with a curve of poorer resolution obtained from a conventional ana- lyzer without pile-up rejection. ANTIWALK DISCRIMINATOR The ADC input circuits use pulse- sampling techniques for accurate data analysis at extremely high counting rates. Difficulties of deter- mining the peak are eliminated by the built-in ANTIWALK discrim- inator whose output occurs at the input pulse crossover. The discrimi- nator output determines the start of the sampling period, ensuring a "true" sample independent of pulse height.Time resolution of 5to10 nano- seconds, independent of discrimi- nator level, is typical for this circuit. •Registered trademark of RIDL NUC:R-4-26<S RIDLRADIATION INSTRUMENT DEVELOPMENT LABORATORY A DIVISION OF NUCLEAR-CHICAGO CORPORATION 4509 West North Ave., Melrose Park, III. 60160 In Europe: Donker Curtiusstraat 7 Amsterdam W, The Netherlands Scientists and engineers interested in challenging career opportunities are invited to contact our personnel director. PHYSICS TODAY JANUARY 1965to measurement of the shear modulus. He de- scribed the values of this modulus obtained in a number of silicone liquids. These are mixtures of linear polysiloxanes of varying molecular weights. The shearing modulus is measured by reflecting the shear waves at a quartz-silicon interlace. It is possible, though very difficult, to measure the phase, as well as the magnitude, of the shear re- flection coefficient. Lamb was able to obtain the angle of the shear reflection coefficient up to a frequency of 70 megacycles, which was sufficient for the purpose of the experiment. The afternoon of the second day was devoted to diffraction by liquids: George W. Brady of Bell Telephone Laboratories spoke on the diffraction of x rays and P. A. Eglestaff of the British Atomic Energy Research Establishment (Harwell) spoke on the diffraction of neutrons. Brady discussed the fundamentals of the theory of large and small angle diffraction, and then de- scribed the major experimental techniques and the major difficulties involved in diffraction by liquids. He chose a very interesting representative analy- sis: the structure of FeCl3 in acid and neutral solution. A striking finding was that the coordina- tion was octahedral in neutral concentrated solu- tion, whereas in acid the solute turned into a polymeric form of alternating tetrahedral and oc- tahedral units. In addition to the geometry of the basic units, x-ray diffraction has also been found useful as a clue to clustering in the critical region. From this viewpoint, Brady discussed various forms of correlation found in the solution C7F1C — C8H10. Eglestaff discussed the scattering of slow neutrons in terms of the probability of exchanging energy between the neutrons and the system when a mo- mentum transfer takes place. From this probability function a Fourier transform is obtained, part of which is connected with the neutron-scattering pattern. The behavior of the scattering gives qualitative information about the asymptotic be- havior of the atoms; for instance, for one special condition it gives information about the way in which slow diffusion processes in normal liquids take place. Two extremes of behavior may be distinguished, which Eglestaff termed Lorentzian and Gaussian. The former is characterized by slowly fluctuating interactions with other atoms, and the latter by rapid fluctuations. These in- fluence the line shape of the scattering distribu- tions. Thus one gets quantitative information of the velocities of motion involved and the dis- tances between collisions. Also, the scattering func- tion may be related to other correlation functions. There is a case in which the scattering is propor-tional to the velocity-correlation function, and this provides a means of determining the number of degrees of freedom for particular modes of mo- tion: e.g., modes leading to diffusion. In another instance, the scattering function can be related to the properties of sound-wave propagation in the system. In general, one must beware of adopting too naive a concept of the relation between neu- tron diffraction and liquid structure as such. J. G. Powles of Queens College, University of London, began his talk on nuclear magnetic reso- nance in liquids by saying quite emphatically that he was mainly concerned with using nuclear- magnetic-resonance techniques in studying the rate and the nature of molecular motion in liquids, and not in elucidating the structure of the mole- cules themselves. He stated that nmr is an ideal tool for this, since the measured quantity, nu- clear magnetization, is very sensitive to molecular motion but has quite a negligible reaction on mo- lecular motion. Only a very restricted part of the molecular motion is "seen" by nmr. (Measure- ments in benzene have clearly demonstrated that the nuclear magnets see only the low-frequency Fourier components.) However, in spite of this, nmr is quite successful in evaluating correlation times of molecular motion. In iso-butyl bromide, correlation times varying with temperature over the range from 10~2 to 10"11 sec have been de- duced, and are confirmed by the more direct measurement of dielectric relaxation, which de- pends on a closely related correlation function. Powles then described interesting findings by nmr with respect to the degree of difference between solid-liquid and liquid-vapor close to their critical points. Basically, these indicate that the micros- copic difference between phases is not as marked as macroscopic properties would suggest. Powles also mentioned the advantage of nmr because of the possibility of varying parameters such as pressure, temperature, and composition with relative ease because of the relatively re- mote contact between the sample and the meas- uring device. Powles had been interrupted several times in his talk by those defending rival interpretations, or otherwise displeased by his forthright approach. Intransigent to the end, he concluded with the hope that data from nmr and other related meth- ods would "save us from the present unhealthy and empirical recourse to the discussion of dubious concepts such as activation energy and microvis- cosity and so on". The ensuing discussion was noisy. Leonard A. Roellig of Wayne State University 70 JANUARY 1965 PHYSICS TODAYGS 2O3 PRECISION GIMBAL SUSPENSION Precise, quick, repeatable positioning of optical and mechanical elements* The magnetic coupling system and precision micrometer drive of the GS 203 deliver precise, quick, repeatable rotations. • True independent rotation about vertical and horizontal axes. • Index and vernier define an ab- solute and resettable angular posi- tion of the mounted element. Compact, rugged construction in anodized aluminum and stainless steel, using only four moving parts, assures trouble-free performance. Applications Gas laser resonators; Mach-Zehn- der and other optical interfero- meters; external mirror solid state laser systems; Q-Switch shutter mounts; general purpose laboratory positioners for mirrors, beam split- ters, and other optical or mechani- cal components. Specifications Resolution Resettability Total Angular Motion Aperture for Normally Incident Beam Maximum OD of1 second 6 seconds 15 degrees 1.75 inches Mounted Element 2.00 inchesFor applications requiring even greater resolution, the GS 253, equipped with micro-motion dif- ferential screw devices, reduces the rate of gimbal rotation by a factor of 20. Mounting A removable, flat, aluminum base plate is easily drilled to fit a wide variety of mounting systems. Price and Availability You can purchase a GS 203 Pre- cision Gimbal Suspension for $145.00. Shipped from stock, FOB Ithaca, New York. The GS 253 is $295.00. Accessories GS 001 Adaptor Set for mounting elements less than 2 inches in diameter, (give mirror diameter and thickness). $12.50 GS 002 Height Adaptor matches mirror heights when one of a pair of GS 203's is mounted on a TR 112 Translation Stage. $10.50 GS 003 Element Rotator allows continuous rotation of a 1.4 inch diameter element. $180.00Industrial and University Research Laboratories Using The GS 203 Aerospace Corp. - American Optical - Bell Telephone Labs. - Boeing - Cal. Tech. - Columbia - Cornell - Corning Glass Works - Dalmo Victor - Eastman Kodak - EG&G - Electro-Optical Systems - Ford - General Electric - Griffiss AFB - Hughes - IBM - Johns Hopkins - Laser Inc. - Laser Systems Center - Lockheed • Lincoln Labs. - Martin - MIT - Michigan State - Naval Ord. Lab. - Naval Research Lab. - North American - Ohio State - Perkin-Elmer - Philco - Picatinny - RCA - Raytheon - STL - Sperry Rand - Stan- ford - Tech. Ops. - TRG - Univ. Dayton - Univ. Maryland - Univ. Michigan - Univ. Penn. - Univ. Roch. - U.S. Army Elec. Command - U.S. Navy Elec. Lab. Catalog & Other Products We also manufacture precision translation stages, micro-motion differential screw devices, gas laser resonators, and optical inter- ferometers. For full product infor- mation or to receive a copy of our complete product catalog, write or phone: P.O. Box 81, Ithaca, N. Y. 14851 Telephone 607-272-3265 LANSING RESEARCH CORPORATION See us at the 1965 13th Annual Physics Show—Statler Hilton Hotel, New York City—January 27-30, 1965. Booth 117 PHYSICS TODAY • JANUARY 1965 . 71discussed the relation between positron annihila- tion and many properties of liquids which are of fundamental interest. Briefly, the characteristics of positrons are influenced by their environment. These characteristics include the lifetimes of the free positron, and singlet and triplet positronium; the rate of formation of positronium, and the rates of the two-gamma and three-gamma annihilation modes of positronium. All these are found to de- pend sensitively upon the physical state, molecular composition, pressure, temperature, and other pa- rameters of a liquid. Roellig discussed the various experimental methods employed in positron an- nihilation, and gave experimental results in liquid metals, cryogenic liquids, and conventional liquids. He also described some very recent measurements of his own in superfluid helium and in teflon. He concluded by describing in general how posi- tron annihilation would be employed in three im- portant cases: the Fermi surfaces of liquid metals, solid-liquid phase changes, and microscopic density changes in fluids. Robert Cole of Brown University, in his talk on dielectric polarization and relaxation in liquids, began with the following points which he con- sidered important for a present-day understanding of dielectric measurements: (1) the approximate character of such quantities as polarizability and permanent dipole moments as molecular constants; (2) the approximate validity of the Lorentz field for nonpolar liquids; (3) the failure of the Lorentz field for polar liquids; (4) the need in some cases of considering quadrupole interaction fields and energies. He mentioned that the Kirkwood theory of static dielectric constants, modified to treat in- duced moments, consistently has given good semi- quantitative results when used to study local equilibrium correlations of a representative dipole and its neighbors. It accounts quite well for the large dielectric constants and temperature coeffi- cients of HF, HCN, and the alcohols. He then described an extension of Kirkwood's equilib- rium theory by Kubo, Glarum, and himself which relates dielectric relaxation to the time-dependent correlation function of a dipole with itself and its local environment. This theory indicates that one should not expect major differences between macroscopic and microscopic functions. Cole gave examples leading to single and multiple relaxation times. He also surveyed what he considered the most interesting recent developments in the field of dielectric relaxation. He compared the simple behavior of the aliphatic alcohols with the non-exponential relaxation functions of the glycols and the alkali halides. Finally he noted that broad relaxation spectra need not imply distributions of relaxation times; they may rather result from co- operative processes which are intrinsically non- exponential in time. George McDuffie of Catholic University traced out the relations among the findings in the three fields of measurements which may be represented as ultrasonic, dielectric, and nuclear magnetic. In each case, information is gained with regard to relaxation processes in liquids. Static viscosity measurements serve to supplement these. If one restricts one's attention to associated liquids, cer- tain similarities in the behavior of the static viscosity and the characteristic times for ultra- sonic, dielectric, and nuclear magnetic processes become evident. As an example, the temperature dependence is similar and shows a non-Arrhenius behavior. Similarities can also be noted with re- spect to pressure dependence and effects of im- purity molecules. Again, the activation enthalpy is nearly the same for all four processes. The dielectric relaxation time tends to be larger than the ultrasonic relaxation time and the nmr cor- relation time; in one group of liquids it is only slightly larger (2.5-5:1), but in another group it is considerably larger (100:1). There is an in- teresting observation with respect to distribution of times: for the case in which the dielectric re- laxation time is about equal to the other times, a distribution of dielectric relaxation times is re- quired, but where the dielectric time is much larger than the others a single time suffices. Richard E. Nettleton of the Bureau of Stand- ards began his presentation "The Phenomenology of Liquid Transport" by stressing the tradi- tional aim of irreversible thermodynamics: that of providing a unified way of regarding constitutive relations. Take the problem of writing the most general stress-strain relation for a viscoelastic ma- terial, or that of determining relations among the elastic constants. The Onsager reciprocity relation may be applied here. Why, then, have not the relations among phenomenological coefficients which may be obtained from Onsager's theorem (and the Gibbs entropy equation) found much ap- plication to the numerical evaluation of these coefficients? The reason appears to be that there are other more direct means available; as an ex- ample, for chemical reactions, the kinetic coeffi- cients are all readily calculable from the model. However, there are problems in which some, but not all, kinetic coefficients are calculable from 72 JANUARY 1965 PHYSICS TODAYNEW FROM EG«G...Announcing 3 additions to EG&G's M1OO Modular Counting System. The M1OO provides nanosecond time resolutions for high speed data handling and reduction, and operates at continuous and periodic rates in excess of 100 megacycles per second. 150 MC SCALING SPARK CHAMBER HIGH SPEED LINEAR GATING HV PUtSERDUAL LINEAR GATE GAIl INPUTGATE "OM SIOO PRESCALER 150 megacycles. Resolves random input signals with a pulse-to-pulse resolution of 6.5 nanoseconds. En- ables resolution of high speed inputs with low speed sealer. Accepts stan- dard EG&G M100 logic input pulses of —700 millivolts. Scales by a factor of 8. Delivers a 5 volt negative out- put pulse when driving a high im- pedance or —700 millivolts when driving 50 ohms. Front panel 1-2- 4 readout. Price: $420. Delivery from stock.HV100 PULSER Designed for pulsing spark chambers and spark gaps. Delivers up to 5,000 volts into 50 ohms with 1 nano- second risetime. Can be driven at rates up to 300 pulses per second with total delay as low as 40 nano- seconds. When operating at full out- put, the HV100 will not induce any response whatsoever from other M100 modules operating within the same manifold. Total time jitter: ±1 nanosecond. Price: $495. Delivery from stock.LG100 LINEAR GATE Passes signals up to 200 mega- cycles. Can be gated up to 75 mega- cycles. Gate opening and closing time less than 2V2 nanoseconds. Operated by —700 millivolt logic pulse. Range: ±1 volt with 1% lin- earity. Pass and gating circuits com- pletely direct-coupled. Opening and closing gating transients are each zero integral. Two independent gates packaged in one module. Price: $450. Delivery from stock. For detailed technical information, write EG&G's Salem Laboratory, 35 Congress Street, Salem, Massachusetts 01971, or call 617-745-3200. These new modules will be demonstrated at the Physics Show — Booth #64, 65. EDGERTON, GERMESHAUSEN & GRIER, INC. BOSTON • LAS VEGAS • SANTA BARBARA PHYSICS TODAY JANUARY 1965 73a model, and here the Onsager reciprocity relation may be useful. Take the irreversible approach to the steady states of heat conduction and diffusion. Over times of about 10-13 sec, Fourier's and Fuchs' laws may be augmented by inertial terms pro- portional to the time derivatives of the heat and particle flows. In this case, the generalized law of heat conduction can be obtained from a Debye- wave model of thermal propagation, but the terms associated with thermal diffusion can best be obtained from Onsager reciprocity, which obvi- ates the need for doubtful assumptions about mo- lecular motion in a fluid. Specifically, theoretical expressions may be obtained for the thermal con- ductivity and the thermal diffusion coefficient. Shirley V. King of Birkbeck College, London, one of the coworkers of J. D. Bernal of the Uni- versity of London in the geometrical approach to liquid structure, then presented a film on an aspect of close-packed spheres. Bernal's group has done some exceedingly interesting work on the statistics of close-packed spheres, and the film showed a case in which a shallow pan was filled with several layers of ball bearings and then agitated in a random manner. For whatever rea- son (and it is a matter of some controversy) structure began to emerge in the assemblage in the form of an area having definite crystalline form. This area then enlarged as the random agitation continued. Miss King did not have time to discuss this phenomenon sufficiently to give any kind of a complete explanation. But it is to be noted that it is not necessary to conclude that it illustrates order emerging almost miraculously from disorder by random agitation. One of the findings of Bernal in his studies of close-packed spheres is the distinction between "heaps" and "piles", the heap representing a disorderly arrange- ment, and the pile an orderly arrangement, of many objects thrown together. In the experimen- tal case in question, the random agitation may be said to have enabled a "heap" to become a pile, it being presumed that the more regular packing is encouraged by agitation. The author is admittedly on dangerous ground here, but this very rough explanation has been attempted, since close-packed sphere theory is a particularly fun- damental, as well as fascinating, field of physics. Ernest Yeager of Western Reserve University be- gan the chemistry session with a very fundamental discussion of the various effects associated with the propagation of ultrasonic waves through elec- trolytic solutions. In recent years, the relaxational absorption observed for many electrolytes has been explained quantitatively in terms of specific chemi-cal processes including ionization, hydrolysis, ionic association, and even rearrangement of solvent molecules bound to ionic associates. Often several processes are perturbed simultaneously from equi- librium by the sound waves, and the interpreta- tion of the resulting complex relaxation spectra requires considerable insight into the nature and coupling of the processes. Dr. Yeager reviewed the normal reaction coordinate approach to the de- scription of the relaxation spectra of coupled processes. Some ions of low charge density depress rather than increase the ultrasonic absorption. An explanation for this depression was proposed on the basis of the Hall two-state model for the structure of water. The principal effect responsi- ble for the depression of the absorption is believed to be a decrease in the energies of activation for the interconversion of the two (or more) structures. Gordon Atkinson of the University of Maryland next spoke on ultrasonic absorption in electrolytes. He stated that the present theories of electrolytic solutions, based on the Debye model of rigid spheres in a continuum solvent, are inadequate, and that one is forced to consider specific solvent effects. One of the promising techniques for the examination of such effects is ultrasonic absorp- tion. He gave a brief description of application of ultrasonic findings to interpretation of the dy- namics of electrolytic systems, in particular varie- ties of relaxation mechanisms which were useful in interpretation. He examined ultrasonic ab- sorption results in MnSO4 solutions in detail, and found a consistent interpretation in terms of a three-step association process in a manner first proposed by Eigen. Frank T. Gucker, of the University of Indiana, traced the relationship of various thermodynamic properties of solutions. Among those included in the discussion were the molar enthalpy, molar heat capacity, volume, compressibility, and free energy. The changes of these quantities with concentration and other parameters of the solu- tion are important to an understanding of the fundamental theory of solutions. If a very accurate density measurement and a very accurate measure- ment of the velocity of sound can be made, the compressibility can be computed to the same ac- curacy, and valuable information concerning the other parameters can be gained, particularly if temperature and pressure variation is also em- ployed. Gucker described two velocity-determin- ing systems of very great precision. As an example of the precision, fifteen measurements of the veloc- ity of sound in water at a temperature of 35°C showed a standard deviation of 0.002 percent. 74 . JANUARY 1965 PHYSICS TODAY