title
stringlengths 9
24
| content
stringlengths 0
1.29M
|
|---|---|
PhysRevB.75.205431.pdf | Anomalous Coulomb diamonds and power-law behavior sensitive to back-gate voltages in carbon
nanoscale peapod quantum dots
J. Mizubayashi,1,2J. Haruyama,1,2I. Takesue,1,2T. Okazaki,3,2H. Shinohara,4,2Y. Harada,5,2and Y. Awano5,2
1Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, Japan
2JST-CREST, 4-1-8 Hon-machi, Kawaguchi, Saitama 332-0012, Japan
3National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8565, Japan
4Nagoya University, Furo-cho, Chigusa, Nagoya 464-8602, Japan
5Fujitsu Laboratory, 10-1 Wakamiya, Morinosato, Atsugi, Kanagawa 243-0197, Japan
/H20849Received 21 February 2006; revised manuscript received 14 March 2007; published 21 May 2007 /H20850
We report anomalous charging effect of single electrons /H20849Coulomb diamonds /H20850observed in carbon nanoscale
peapod quantum dots that encapsulate a series of C 60molecules. We find that behaviors of diamonds are
anomalously sensitive to back-gate voltages /H20849Vbg/H20850, exhibiting two evidently different Vbgregions and a large
polarity on Vbg. In particular, we find only a sequence of one large diamond followed by three smaller ones
existing around ground state. Magnetic-field dependence indicates the presence of shell filling by spin singletto doubly degenerate electronic levels for these. The encapsulated-C
60molecules indirectly affect this shell
filling at low Vbgpossibly via nearly free electrons. In contrast, they act as individual quantum dots coupled in
series in high Vbgregion. It directly contributes to highly overlapped very large diamonds. Moreover, we report
power-law behaviors on conductance versus energy relationships observed in the same carbon nanoscalepeapods. We find that the values of powers are also highly sensitive to applied V
bgwith three different regions
and anomalously high at high source-drain voltages. Because the power laws are found at voltages, which arethe nearest outside of the above-mentioned fourfold Coulomb diamonds, correlation of the anomalous powerswith orbital-related Tomonaga-Luttinger liquid is discussed.
DOI: 10.1103/PhysRevB.75.205431 PACS number /H20849s/H20850: 73.63. /H11002b, 71.10.Pm, 73.22.Lp, 73.23. /H11002b
I. INTRODUCTION
Carbon nanoscale peapods, which are single-walled car-
bon nanotubes /H20849SWNTs /H20850encapsulating a series of fullerenes
such as C 60,C70, and Gd @C82/H20849C82encapsulating Gd /H20850mol-
ecules in their inner space,1,2have recently attracted consid-
erable attention. This is because their unique nanostructuresare expected to yield exotic electronic states, quantum charge/H20849spin/H20850transports, and one-dimensional /H208491D/H20850quantum phe-
nomena. There are, however, still a few reliable reports thatexperimentally reported such electronic states and quantumphenomena.
From theoretical viewpoints, in C
60@/H20849n,n/H20850peapods that
are armchair-type SWNTs encapsulating C 60molecules, it
has been predicted that electrons that were transferred fromthe SWNT accumulated in the space between the C
60mol-
ecules and SWNTs, forming the so-called nearly free-electron /H20849NFE /H20850states.
3Hybridization of these NFE states
with the /H9266and/H9268orbitals of C 60molecules introduced four
asymmetric subbands including the approximately doublydegenerate ground states in the C
60@/H2084910,10 /H20850peapod in con-
tradiction to the two subbands in conventional SWNTs.3,4
Measurements of semiconductive peapods encapsulating
a series of Gd @C82by a scanning tunnel microscope re-
vealed that a conduction band was periodically modulatedaround Gd @C
82in a real space due to the hybridization of
orbitals between the SWNT and Gd @C82.5Moreover, elec-
trical measurements of peapods encapsulating C 60and
Gd@C82indicated the possibility of the presence of variable
range hopping.6References 3–6at least suggested the pres-
ence of charge transfer and orbital hybridization between theencapsulated fullerenes and SWNTs.On the other hand, it is well known that SWNTs are
within a 1D ballistic charge transport regime and exhibit avariety of quantum effects, such as quantized energy levels,Tomonaga-Luttinger liquids /H20849TLLs /H20850, and shell /H20849orbital /H20850fill-
ing/H20849atomiclike behaviors /H20850as quantum dots.
7–10For instance,
when carbon nanotubes /H20849CNs/H20850act as quantum dots, electron
can be placed on the quantized electronic levels /H20849orbitals or
shells /H20850in the dots one by one via single-electron charging
effect. This effect has caused shell filling in CN quantumdots,
7–10such as even-odd effect, shell filling in two spin-
degenerate electronic states, and Kondo effect. In particular,two different types of shell filling have been experimentallyreported, that is, antiparallel spins /H20849spin singlet /H20850and parallel
spins /H20849spin triplet /H20850. Reference 22proposed a theoretical
model that explained the results by taking into considerationseveral families of single-electron states and Coulomb repul-sion. Our present system is analogous to this model.
Moreover, the behavior of TLLs, which is a collective
phenomenon arising from electron-electron interaction in 1Dconductors, has been identified by observing power laws inrelationships of conductance vs energy in CNs.
12–15The re-
ported correlation exponent g, which denotes the strength of
an electron-electron interaction, was as low as /H110110.2. This
implied the presence of a strong repulsive Coulomb interac-tion existing in CNs. How such phenomena are affected byencapsulating a series of fullerenes, however, has not yetbeen investigated in any carbon nanoscale peapods to date.
For present study, we report finding anomalous behaviors
of Coulomb diamonds observed in carbon nanoscale peapodsquantum dots that encapsulate a series of C
60molecules.
First, we find that behaviors of diamonds are anomalouslysensitive to back-gate voltages /H20849V
bg/H20850, exhibiting two evi-PHYSICAL REVIEW B 75, 205431 /H208492007 /H20850
1098-0121/2007/75 /H2084920/H20850/205431 /H208497/H20850 ©2007 The American Physical Society 205431-1dently different Vbgregions /H20851i.e.,/H208491/H20850small diamond region
for −1.7 V /H11021Vbg/H11021+1.7 V, and /H208492/H20850large diamond region for
Vbg/H11021−1.7 V and +1.7 V /H11021Vbg/H20852and a large polarity on Vbg.
In particular, we find only a sequence of one large diamondfollowed by three smaller ones existing only around groundstate in + V
bgregion. Magnetic-field dependence indicates the
presence of shell filling to doubly degenerate electronic lev-els by spin singlet for these. The size of diamond indicatesthat this is independent of the encapsulated-C
60molecules,
basically in this low Vbgregion. However, they might indi-
rectly affect this shell filling via nearly free electrons, whichaccumulate on the space between C
60’s and SWNT by elec-
tron transfer from the SWNT. In contrast, we find that theencapsulated-C
60molecules directly contribute to overlapped
very large diamonds by acting as individual quantum dotscoupled in series in high V
bgregion.
Next, we report finding the presence of power laws in
conductance vs energy relationships, which is also highlysensitive to V
bgshowing three Vbgregions in different power-
law behaviors /H20851i.e.,/H208491/H20850Vbg/H110210.8 V, /H208492/H208500.8 V/H11021Vbg/H110211.7 V,
and/H208493/H208501.7 V/H11021Vbg/H20852. Anomalously high values of powers /H20849
/H110228/H20850are also found for Vbg’s/H110220.8 V in high Vsdregions. The
power laws are found at voltages, which are the nearest out-
side of the above-mentioned fourfold Coulomb diamonds.Because these fourfold diamonds mean single-electron fillingto doubly degenerate orbitals in the peapod quantum dot,correlation of the anomalous powers with orbital-relatedTLL states is discussed.
II. ANOMALOUS COULOMB DIAMONDS SENSITIVE TO
BACK-GATE VOLTAGES AND SHELL-FILLING
EFFECT
For the present study, field-effect transistors /H20849FETs /H20850, using
peapods encapsulating C 60molecules as the channel, were
fabricated. A scanning electron microscopy /H20849SEM /H20850top view
indicated that the FETs included two bundles of peapods2as
the channels. The number of peapods included in one bundlewas estimated to be approximately 20 from atomic force mi-croscopy /H20849AFM /H20850, and transmission electron microscopy
/H20849TEM /H20850observations. Since change in the observed differen-
tial conductance was largely independent of the change inV
bg, metallic transport in the present peapod was
confirmed.11
A. Anomalously Vbg-sensitive Coulomb diamonds
The measurement results by single-electron spectroscopy
are shown in Fig. 1:/H20849a/H20850for −4 V /H11021Vbg/H11021+4 V and /H20849b/H20850ex-
pansion of /H20849a/H20850for 0 V /H11021Vbg/H11021+2 V. Figure 1/H20849a/H20850indicates
that/H208491/H20850the sizes of Coulomb diamonds are highly sensitive
to applied Vbgand /H208492/H20850they have significant polarities on
applied Vbg/H20849i.e., asymmetric features between + Vbgand − Vbg
ranges /H20850. From the first viewpoint, the sizes of diamonds can
be evidently classified to the following two Vbgregions: /H208491/H20850
small diamond region for −1.7 V /H11021Vbg/H11021+1.7 V, and /H208492/H20850
large diamond region for Vbg/H11021−1.7 V and +1.7 V /H11021Vbg.I n
the first Vbgregion, the sizes of diamonds are smaller than
/H1101110 mV except for those appearing at Vbg=+1 V /H20851noted asn=4 in Fig. 1/H20849b/H20850/H20852, whereas in the second Vbgregion, they are
approximately 20 mV except for a few diamonds at Vbg/H11021
−1.7 V and larger than 40 mV at Vbg/H11022+1.7 V.
From the second viewpoint, the sizes of diamonds are
very asymmetric as mentioned above. Moreover, periodicityof diamonds is quite different between − V
bgand + Vbgranges
in the first Vbgregion. A sequence of closed one large dia-
mond /H20849noted as n=4/H20850followed by three closed smaller ones
/H20851noted as n=1–3 in Fig. 1/H20849b/H20850/H20852is observable for 0 V /H11021Vbg
/H11021+1.7 V, whereas they cannot be observed for −1.7 V
/H11021Vbg/H110210 V. Although some similar-size diamonds are ob-
servable for −0.5 V /H11021Vbg/H110210 V, they are unclosed and other
diamonds for −1.7 V /H11021Vbg/H11021−0.5 V become much smaller
asVbgvalue decreases. In the second Vbgregion, the dia-
monds are highly overlapped and cannot be separated. Inparticular, the overlapping is very significant for +1.7 V/H11021V
bg.
It is a well-known fact that conventional CN quantum
dots can place many electrons on those many quantized elec-tronic levels one by one via single-electron tunneling andthat shell-filling effects are mostly independent of V
bg, be-
cause the shape of dots is independent of Vbg.Vbgjust
changes the positions of chemical potentials in the conven-tional CN quantum dots, unlike most cases of semiconductorquantum dots. Only the case of impurity- or defect-inducedmany small quantum dots, which were connected in series,showed diamonds with inhomogeneous sizes, resulting instochastic Coulomb diamonds. However, no such behaviorshave been observed in our empty SWNT quantum dots /H20849i.e.,
without C
60’s/H20850. Therefore, the above-mentioned anomalously
high Vbg-sensitive Coulomb diamonds indicate strong asso-
ciation with contribution of the encapsulated-C 60molecules.
Here, the value of Uc=e2/2Ceff/H20849the single-electron charg-
ing energy of the system; Ceffis the effective capacitance for
single charging effect21/H20850should be approximately 6–10 meV
in the case of empty and high-quality SWNTs quantum dots/H20849i.e., without C
60’s, defects, and impurities /H20850, if one follows
the expectations based on only the length of present carbonpeapod of 500 nm following a previous study of SWNTbundles /H20849e.g., a U
cof/H1101125 meV for a tube length of
100–200 nm.7The values of Uc/H20849/H1102110 meV /H20850in the first Vbg
region mentioned above are in approximately good agree-
ment with this estimation of Uc=6–10 meV. This strongly
indicates no contribution of the encapsulated-C 60molecules
at least to CeffofUcand only the SWNT acts as a quantum
dot. Hence, single electrons flow only through the SWNT inthis low V
bgrange around 0 V and cannot flow into the C 60
molecules.
B. Shell filling to doubly degenerated electronic levels at
low +Vbg
In particular, the period of Coulomb diamonds observed
for 0 V /H11021Vbg/H11021+1.7 V, as shown in Fig. 1/H20849b/H20850, can be inter-
preted as the four diamonds, a sequence of one large dia-mond /H20849noted as n=4/H20850followed by three smaller ones /H20849noted
asn=1–3 /H20850, as mentioned above. This sequence indicates the
possible presence of shell-filling effect with the doubly de-generate electronic levels only at ground states, based onMIZUBAYASHI et al. PHYSICAL REVIEW B 75, 205431 /H208492007 /H20850
205431-2previous reports of the fourfold diamonds in SWNT /H20849Ref. 9/H20850
and multiwalled CN /H20849MWNT /H20850quantum dots.10However, the
observed sequence of these diamonds is only one set for0V/H11021V
bg/H11021+1.7 V, because appearance of much larger dia-
monds obstructs the observation of further sets of such se-quence. This result is very anomalous as compared withthose periodically observed over wide ranges of V
bgin Refs.
9and10In conventional CN quantum dots, such one-set
degenerate levels cannot be found, because individual non-degenerate electronic level is formed only from quantizationof two subbands existing in bulk of a SWNT, while only insome cases, all levels are doubly degenerate, such as thefourfold diamonds, as observed in Refs. 9and10.
Hence, in order to confirm the presence of shell-filling
effects and doubly degenerate levels for Fig. 1/H20849b/H20850, we have
investigated the V
bgshift of the linear-response conductance
peaks /H20851i.e., shown by arrows in Fig. 1/H20849b/H20850/H20852as a function of
magnetic field Bperpendicular to the tube axis. The result,
Fig.2/H20849a/H20850, reveals that adjacent peaks shift in opposite direc-
tions. This is a behavior of spin singlet state whose spinsalternate as S=0→1/2→0... and exist on the same orbital
state, unlike a spin triplet state formed by Hund’s rule. InFig. 2/H20849b/H20850, we plot additional energy, which was deduced
from the separation of adjacent peaks involving electrons onthe same orbital in Fig. 2/H20849a/H20850/H20849i.e., peaks 1 and 2, and peaks 3
and 4 /H20850, as a function of magnetic field B. A dashed line
shows the result of the best fit of the data to U
c+gL/H9262BB,
010
-2020
-10
0 0.5 1.0 1.5
Vbg[mV]
sd[mV]
010
-2020
-10
0
V
sdn=1n=1 n=2n=3 n=4n=4 n=5
(b)(a)
FIG. 1. /H20849Color online /H20850/H20849a/H20850Coulomb diamonds observed in a carbon nanoscale peapod quantum dot at T=1.5 K. The zaxis is the
differential conductance with the magnitudes of which are indicated on the right side. Dotted lines at Vbg= ±1.7 V indicate boundaries for
two different Vbgregions for small and large diamonds. /H20849b/H20850Expansion of Coulomb diamonds /H20849red regions surrounded by the dotted lines /H20850for
0V/H11021Vbg/H11021+1.7 V in /H20849a/H20850.nindicates the number of electrons confined in peapod quantum dots for each diamond. The dotted diamonds are
guides to the eye. Arrows mean the Vbgpoints for Fig. 2.
Magnetic field [T]012340.00.51.0
1234
(a)
Magnetic field [T]PeakPosition:Vbg[V]
01234Addition energy [meV]
0.000.250.50
from 1,2from 3,4
(b)
Magnetic field [T]Addition Energy [meV]
FIG. 2. /H20849a/H20850Vbgshift of conductance peak positions /H20851atVbg
=0.11, 0.26, 0.53, and 0.77 around Vsd=0 V shown by arrows in
Fig.1/H20849b/H20850/H20852in Fig. 1/H20849b/H20850as a function of magnetic field B./H20849b/H20850Addi-
tional energy obtained from each peak pair in Fig. 2/H20849a/H20850versus B.ANOMALOUS COULOMB DIAMONDS AND POWER-LAW … PHYSICAL REVIEW B 75, 205431 /H208492007 /H20850
205431-3where /H9262Bis the Bohr magneton and gLis the Lande factor,
and gives gL=1.96. This value of gLis approximately con-
sistent with those mentioned in Ref. 10. Therefore, we con-
clude that Fig. 1/H20849b/H20850indicates the presence of doubly degen-
erate electronic levels existing only at ground states and thepresence of a shell filling to such levels by spin singlet.
Because of the absence of interaction between the C
60
molecules and the SWNT as mentioned above, these doublydegenerate electronic levels are not directly associated withthe encapsulated-C
60molecules. Therefore, this does not cor-
respond to the model predicted in Ref. 22. These, however,
may be indirectly associated with the C 60molecules via
NFEs and doubly degenerate subbands due to the NFE statesas follows.
Reference 3predicted that the doubly degenerate sub-
bands, which exist only at the ground states, originated fromthe hybridization of orbitals of the encapsulated-C
60mol-
ecules and NFE states in bulk of C 60@/H2084910,10 /H20850peapod as
mentioned in the Introduction. In contrast, in the case of
conventional quantum dot structure, this degeneration issolved and the subbands are quantized to many electroniclevels /H20849shells /H20850. However, because the channel length of
present peapod FET is as long as 500 nm, the level spacing/H9004E=h
/H9263F/2L/H20849Land/H9263Fare the tube length and Fermi veloc-
ity, respectively /H20850should be as small as /H110113 meV even in
nondegenerate levels. In such a case, the doubly degeneratesubbands around the ground state may still remain, resultingin formation of approximately double-degenerate electroniclevels around V
bg=0 V, which was observed in Fig. 1/H20849b/H20850,
even in the quantum dot.
In contrast, this shell-filling effect was not observable for
−1.7 V /H11021Vbg/H110210 V as mentioned above. Moreover, the sizes
of unclosed diamonds are inhomogeneous without specifiedperiodicities in this V
bgrange. These behaviors are analogous
to stochastic Coulomb diamonds, which have been reportedin impurity- or defect-induced small quantum dots connectedin series, although the number of quantum dots is very smallin the present case because the size of diamonds is as smallas less than 10 mV.
These significantly different behaviors of diamonds for
±V
bgrange evidently support the correlation of the doubly
degenerate levels and shell filling with NFE states mentionedabove. This is because NEF states can be formed by electrontransfer from SWNT to the space between the encapsulatedC
60’s and the SWNT as mentioned in the Introduction3and
applying + Vbginduces this electron transfer as well as the
single-electron injection from source electrode. In contrast,applying − V
bgobstructs these electron transfers, resulting in
the above-mentioned stochastic diamonds in the low − Vbg
range. This also implies a possibility of the presence of some
defect-induced small quantum dots in the SWNT for yieldingthese stochastic diamonds, because no electron transfer fromthe SWNT exists and, thus, such electrons in the SWNT maybe injected into the small dots.
C. Interplay between encapsulated-C 60molecules and SWNTs
at high Vbg
On the other hand, the sizes of diamonds in the second
Vbgregion are much larger than those in the first Vbgregionshowing heavy overlapping and, thus, this means the values
ofCeffare smaller than those in the first Vbgregion. This
implies disappearance of the shell filling and the connectionof a few dots mentioned above in the first ± V
bgregions. In
conventional Coulomb diamonds, such large and uncloseddiamonds have been interpreted by a series connection ofdefect- or impurity-induced many small dots. However, be-cause such large diamonds have not been found in our emptySWNT quantum dots as mentioned earlier, they do not cor-respond to the present case. This indicates a possibility thatindividual encapsulated-C
60molecules behave as individual
small quantum dots and they are electrostatically coupled toeach other, because applying high + V
bgstrongly induces the
above-mentioned electron transfer from SWNT to the spacebetween C
60’s and SWNT for yielding NFEs. The induced
transfer results in excess accumulation of the NEFs and theexcess NFEs can make single-electron tunneling into indi-vidual C
60molecules. In such a case, electron will flow only
through the encapsulated-C 60molecules connected in series
under applied Vsd, because C 60molecules are fully encapsu-
lated into the inner space of the SWNT and neighboringC
60’s face each other via tunnel junction in most parts of the
present peapods.
If there is no such a coupling and encapsulated-C 60mol-
ecules act as independent quantum dots, our system corre-sponds to the model proposed in Ref. 22. This will lead to
shell filling by parallel spins /H20849spin triplet /H20850, because of the
presence of Tomonaga-Luttinger liquid /H20849Coulomb repulsion /H20850
as mentioned in the next section.
The values of U
c’s of /H1101120 meV for Vbg/H11021−1.7 V and
/H1101140 meV for +1.7 V /H11021Vbgin the second Vbgregion are at
least two and four times larger than those for the first Vbg
region /H20849/H1102110 meV /H20850. Hence, the values of the total capaci-
tance of C 60molecules, which are coupled with SWNTs, can
be estimated to be the value of 1/3 capacitance of SWNTs/H20849C
SW/H20850for +1.7 V /H11021Vbg. In contrast, it can be estimated to be
the same value as /H20849CSW/H20850forVbg/H11021−1.7 V. This indicates the
presence of different origins to yield effective capacitance
CeffforUcfor ± Vbg. As mentioned above, electron transfer
from SWNT to the C 60’s could not occur in the − Vbgrange.
Hence, this large value of Ceffmay be attributed to single
charging effect of defect or impurity-induced small quantumdots as well as that for −1.7 V /H11021V
bg/H110210 V, although large
−Vbgvalue induces charging effect, resulting in the larger
diamonds.
In conclusion, our argument implies that the anomalously
high Vbg-sensitive Coulomb diamonds are attributed to the
following: /H208491/H20850shell filling to doubly degenerate electronic
levels associated with NFE states in low + Vbgregion with no
direct interaction between the encapsulated-C 60molecules
and the SWNT and /H208492/H20850single-electron injection from the
SWNT into the C 60molecules, which act as individual quan-
tum dots coupled in series, in high + Vbgrange. Moreover, we
argued that diamonds in − Vbgregion originate from defect-
or impurity-induced small quantum dots. In order to furtherclarify these arguments, more investigation is expected, suchas by changing the number of encapsulated-C
60molecules.MIZUBAYASHI et al. PHYSICAL REVIEW B 75, 205431 /H208492007 /H20850
205431-4III. ANOMALOUS POWER-LAW BEHA VIORS AND
ORBITAL-RELATED TOMONAGA-LUTTINGER LIQUID
A. Anomalous power-law behaviors
Figure 3shows the double-logarithmic plot of differential
conductance divided by T/H9251as a function of eV/kBTmeasured
atVbg= +0.4 V for three different temperatures. All data col-
lapse on a single universal value showing saturation ateV/k
BT/H11021hvF/L. These results are qualitatively consistent
with those in previous reports of TLL states in CNs.13
Figure 4shows the relationships of differential conduc-
tance /H20849dIsd/dVsd/H20850toVsdon doubly logarithmic scales for one
−Vbgand three + Vbgregions. In the − Vbgregion, any differ-
ential conductance did not follow a linear relationship, asshown in Fig. 4/H20849a/H20850. On the contrary, saliently linear relation-
ships with different
/H9251values are observable in the + Vbgre-
gion, although the Vsdregions with exhibiting power laws are
narrow at Vsd’s/H1102210 mV in Figs. 4/H20849c/H20850and4/H20849d/H20850/H20849e.g., half
decade /H20850. The behaviors are classified into the following three
Vbgregions; /H208491/H20850Vbg/H110210.8 V, the linearities with 1.6 /H11021/H9251/H110212
are observable only at Vsd/H110210.01 V /H20851Fig. 4/H20849b/H20850/H20852;/H208492/H208500.8 V
/H11021Vbg/H110211.7 V; two linear relationships with different /H9251
ranges /H20849i.e.,/H9251=2–3 and /H9251=8–10 for Vsd/H110210.01 V and Vsd
/H110220.02 V, respectively /H20850are observable /H20851Fig. 4/H20849c/H20850/H20852, and 3.
1.7 V/H11021Vbg, the linearities with /H9251=10–12 are observable
only at Vsd/H110220.01 V /H20851Fig.4/H20849d/H20850/H20852.
The summary of values of /H9251observed in all the Vbgregion
included in Figs. 4/H20849b/H20850–4/H20849d/H20850is shown in Fig. 5. The differ-
ences in tendencies of /H9251among the three regions are appar-
ent in this figure. Moreover, the values of /H9251observed in
empty SWNTs are also shown in this figure. All the valuesare less than 1, which is consistent with previous reports ofTLLs in SWNTs. This implies that Fig. 4is unique to pea-
pods.
B. Correlation with possibly orbital-related Tomonaga-
Luttinger liquid
The presence of power laws has been discussed as evi-
dence for TLLs in CNs,12–15as mentioned in the Introduc-T=1 . 5K
T=5K
T=1 0K
(a)Vbg=+0.4VGT-α[arb. Units]
eV/kBT
FIG. 3. The double-logarithmic plot of differential conductance
/H20849G=dIsd/dVsd/H20850divided by T/H9251as a function of eV/kBTmeasured at
Vbg= +0.4 V for three different temperatures.(a)
Vsd [V]0.0001 0.01 0.1e-11e-10e-9e-8e-7e-6e-5
Vbg=1.0V
Vbg=1.2V
Vbg=1.4V11010010000
0.1
0.01 dI/dV [nS]1000
Vsd[V]0.0001 0.01 0. 1e-11e-10e-9e-8e-7e-6e-5
Vbg=1.8V
Vbg=2.0V
Vbg=2.4V
Vbg=2.8V
Vbg=4.0V
Vbg=4.2V
11010010000
0.1
0.01 dI/dV [nS]1000Vsd[V]0.0001 0.01 0.1e-11e-10e-9e-8e-7e-6e-5
Vbg=-3.8V
Vbg=-3.0V
Vbg=-0.2V11010010000
0.1
0.01 dI/dV [nS]1000
Vsd[V]0.0001 0.01 0.1e-11e-9e-8e-7e-6e-5
Vbg=0.2V
Vbg=0.4V
Vbg=0.8V11010010000
0.1
0.01 dI/dV [nS]1000
0.001 0.001
0.001 0.001(c)(b)
(d)(a)
FIG. 4. /H20849Color online /H20850Relationships of dIsd/dVsdto source-drain
voltage /H20849eVsd/k/H11271T=1.5 K measured /H20850on doubly logarithmic scales
for four different Vbgregions; /H20849a/H20850Vbg/H110210V , /H20849b/H208500V/H11021Vbg/H110210.8 V,
/H20849c/H208500.8 V/H11021Vbg/H110211.7 V, and /H20849d/H208501.7 V/H11021Vbg. These power laws pri-
marily appear just at the nearest-outside regions of fourfold Cou-lomb diamonds in Fig. 1/H20849b/H20850. Only the power law in the low V
sd
region at Vbg=1 V appears inside of the large Coulomb diamond in
fourfold diamonds. The liner lines were obtained from accurate datafitting including measurement points as many as possible and val-ues of power
/H9251were exactly estimated.
Vbg(V)012345 6Power α
02468101214Vsd<10mV
Vsd>10mV
××××××××××
Vbg[V]The values of power α
×SWNTs12 3
FIG. 5. /H20849Color online /H20850Dependence of power /H9251on different Vbg
values, estimated from Figs. 4/H20849b/H20850–4/H20849d/H20850, in both the present peapod
and the empty SWNT quantum dots. Three different Vbgregions
/H20851/H208491/H20850Vbg/H110210.8 V, /H208492/H208500.8 V/H11021Vbg/H110211.7 V, and /H208493/H208501.7 V/H11021Vbgcor-
responding to Figs. 4/H20849b/H20850–4/H20849d/H20850, respectively /H20852separated by dotted
lines are evident. Several values of /H9251were added in addition to Fig.
4/H20849d/H20850only in region 3.ANOMALOUS COULOMB DIAMONDS AND POWER-LAW … PHYSICAL REVIEW B 75, 205431 /H208492007 /H20850
205431-5tion. The values of /H9251were very sensitive to the boundary
conditions between the metal electrodes and CNs,13namely,
the tunneling density of state, such as /H9251bulk=/H110110.3 and /H9251end
=2/H9251bulkfor the tunneling from a Au electrode to the bulk and
to the end of CNs within the large-channel number TLLstates, respectively.
13The formulas of /H9251for each tunneling
were also given by /H9251bulk=/H20849g−1+g−2/H20850/8 and /H9251end=/H20849g−1
−1/H20850/4. However, it should be noted that even the maximum
value of /H9251reported in previous CNs to date is approximately
1.25, except for Refs. 14and17. Therefore, we imply that
the/H9251values of 1.6–12 observed in Figs. 3and4are anoma-
lously large in comparison with the /H9251values reported thus
far in conventional TLLs.14The junction structures in this
study, in which the ends of the peapod bundles were placedunder a Au electrode, should have shown a maximum
/H9251endof
only /H110110.6. In fact, the empty SWNTs have exhibited /H9251=
/H110110.8 even at the maximum case, as explained for Fig. 5
above.
Here, it should be noticed that the power laws, as shown
in Figs. 4/H20849b/H20850–4/H20849d/H20850, were found at the Vbg’s in the nearest-
outside voltage regions of the Coulomb diamonds, as shownin Fig. 1of Sec. II. Region 1 /H20849V
bg/H110210.8 V /H20850for Fig. 4/H20849b/H20850
mostly agrees with the Vbgregion including three small dia-
monds /H20851n=1,2,3 in Fig. 1/H20849b/H20850/H20852, whereas region 2 /H208490.8 V
/H11021Vbg/H110211.7 V /H20850for Fig. 4/H20849c/H20850agrees with the Vbgregion in-
cluding one large diamond and one small diamond /H20851n=4, 5
in Fig. 1/H20849b/H20850/H20852. Region 3 /H20849Vbg/H110211.7 V /H20850for Fig. 4/H20849d/H20850agrees
with the Vbgregion, showing very large diamonds without
shell filling. Moreover, no shell-filling effect was observed in−V
bgregion in Fig. 1/H20849a/H20850. This agrees with the presence of no
power laws in Fig. 4/H20849a/H20850.
Each Coulomb diamond region for this shell /H20849orbital /H20850-
filling region meant the presence of electrons, which wereplaced one by one on electron orbitals via single-electroncharging effect, in the peapod quantum dot. Hence, the ob-served power laws sensitive to V
bgare strongly associated
with the number of such electrons /H20849n/H20850in the peapod quantum
dot and the number of /H20849partially /H20850occupied electronic levels
/H20849N/H20850; i.e., n=1,2,3 /H20849N=1,2 /H20850forVbg/H110210.8 V and n=4,5 /H20849N
=2,3 /H20850for 0.8 V /H11021Vbg/H110211.7 V.
The correlation of power laws, the values of /H9251, and TLL
states with the electronic-level filling effect /H20849i.e., orbital-
filling effect /H20850in CNs have not yet been reported in previous
studies. Only a single study,13however, predicted that a
small gand large /H9251could be obtained from the large Nin
peapods. The theory predicted g=/H208491+2Nvq//H9266/H6036vF/H20850−1/2for
armchair CNs, where Nandvqare the number of /H20849partially /H20850
occupied symmetric subbands with degenerate Fermi vectorwaves and the same bandwidth, and the electron-electroninteraction matrix element, respectively. If the subbands areasymmetric and each of them crosses the Fermi level onlyonce, Ncan be replaced by N/2. This holds true for the
subbands of the C
60@/H2084910,10 /H20850peapod in this study.
We quantitatively examine the validity of this theory for
the present measurement results by replacing Nto the num-
ber of /H20849partially /H20850occupied electronic levels /H20849orbitals /H20850and us-
ing the same value of vq. The value of g=0.135 is obtained
from/H9251end=/H20849g−1−1/H20850/4/H20849Ref. 13/H20850using/H9251=1.6 that was ob-
served in region 1 /H20849N=1,2 /H20850in Figs. 4/H20849b/H20850and5. The value ofvqcan be estimated by substituting these values of g
=0.135 and N=2 in g=/H208491+2/H20849N/2/H20850vq//H9266/H6036vF/H20850−1/2.18Then, g
=0.11 and g=0.099 are, respectively, obtained for N=3 and
N=4 by substituting this estimated value of vqing=/H208491
+2/H20849N/2/H20850vq//H9266/H6036vF/H20850−1/2. The value of g=0.11 obtained for N
=3 is in approximately good agreement with g=0.082 esti-
mated from /H9251end=/H20849g−1−1/H20850/4 by using /H9251=2.8 that was ob-
served in the portion of region 2 with low Vsdvalues in Figs.
4/H20849c/H20850and5.
On the other hand, this gvalue is irrelevant to /H9251=8–10
that is observed in the portion of region 2 with high Vsd
values in Figs. 4/H20849c/H20850and5. Moreover, the gvalue of 0.099
leads to /H9251=2.96 for N=4, which is significantly less than the
values of /H9251/H1102210 observed in region 3 at high Vsdvalues in
Figs. 4/H20849d/H20850and5. These indicate that different values of vq
should be used for the case of higher Vsd. Because strength of
electron-electron interaction varies from low to high Vsd’s,
this is reasonable. When different values of vqare used for
large N,/H9251=10 /H20849forN=3/H20850and 12 /H20849forN=4/H20850could be ob-
tained from the values of 2 vq//H9266/H6036vF=1160 and 1250, re-
spectively, g=/H208491+2/H20849N/2/H20850vq//H9266/H6036vF/H20850−1/2, and/H9251end=/H20849g−1−1/H20850/4.
Consequently, the theory18is quantitatively relevant when
N=2 and 3 /H20849at lower Vsd/H20850under the same value of vqand
N=3/H20849at higher Vsd/H20850and 4 under the larger values of Vq. This
indicates that the presence of two power laws observed inFig.2/H20849c/H20850is attributed to change in
vqdue to increase in Vsd.
Therefore, we conclude that the power laws with large valuesof
/H9251/H208491.6/H11021/H9251/H1102112/H20850can be attributed to the TLL via the occu-
pied doubly degenerated electronic levels, which are located
near the ground states unique to the peapod quantum dots.
However, the high Vsdregion for power laws in region 2
does not locate at the nearest outside of Coulomb diamonds.In addition, power laws observed in region 3 are not consis-tent with no shell-filling effect reported in. /H20849Ref. 16/H20850These
may indicate that such power laws are not associated withorbital-related TLLs. Therefore, further investigation is re-quired to reconfirm relevance of the values of
vqand com-
prehensive understanding of these power-law behaviors.
Hence, other interpretation should be discussed described
as follows. If the capacitance of peapods is /H11011600 times
smaller than those in MWNTs due to the presence of C 60
molecules electrostatically coupled with the SWNT in seriesand the value of Nis as large as 10–20 like those in MWNTs,
the large-channel TLL model coupled with external electro-magnetic environment shown for MWNTs /H20849Refs. 13,19, and
20/H20850may explain the
/H9251=8–12, because /H9251in conventional
MWNTs is given by 2 R/RQ=2/H20849L/C/H208501/2/RQ/H110150.44, where L
is the kinetic inductance given by RQ/2NvF/H20849/H110151n H //H9262m/H20850,C
is the external electrostatic capacitance /H20849/H1101530 aF/ /H9262m/H20850, and
RQ/H20849=h/e2/H20850is the quantum resistance. This may correspond
to the high values of /H9251in region 3, because we reported that
the encapsulated-C 60molecules were not electrostatically
coupled with the SWNT at Vbg/H11021+1.7 V /H20849i.e., in regions
1 and 2 /H20850, while the coupling occurred at Vbg/H11022+1.7 V
/H20849region 3 /H20850.
IV . CONCLUSION
In conclusion, we reported anomalous behaviors of Cou-
lomb diamonds observed in carbon nanoscale peapod quan-MIZUBAYASHI et al. PHYSICAL REVIEW B 75, 205431 /H208492007 /H20850
205431-6tum dots that encapsulated a series of C 60molecules. We
found that behaviors of diamonds were anomalously sensi-tive to V
bg, exhibiting two evidently different Vbgregions
/H20851i.e., /H208491/H20850small diamond region for −1.7 V /H11021Vbg/H11021+1.7 V,
and/H208492/H20850large diamond region for Vbg/H11021−1.7 V and +1.7 V
/H11021Vbg/H20852and a large polarity on Vbg. In particular, we found
only a sequence of one large diamond followed by threesmaller ones existing around ground state. Magnetic-field de-pendence indicated the presence of shell filling to doublydegenerate electronic levels by spin singlet state for these.The size of diamond indicated that this was independent ofthe encapsulated-C
60molecules. However, they might indi-
rectly affect this shell filling via NFEs, which accumulate onthe space between C
60’s and SWNT by electron transfer from
the SWNT. In contrast, we found that the encapsulated-C 60
molecules directly contributed to overlapped very large dia-monds by acting as individual quantum dots coupled in se-ries in high V
bgregion.Moreover, we reported the power-law behaviors on con-
ductance vs energy relationships observed in the same car-bon nanoscale peapods. We found that the values of powers
/H9251were highly sensitive to Vbgalso showing three different
regions /H20851/H208491/H20850Vbg/H11021−1.7 V, /H208492/H20850−1.7 V /H11021Vbg/H11021+1.7 V,
/H208493/H20850+1.7 V /H11021Vbg/H20852and anomalously high /H20849/H9251/H110228/H20850at high Vsd
voltages. Because the power laws were found at the nearest-
outside voltages of the above-mentioned fourfold Coulombdiamonds, the correlation of the anomalous powers withorbital-related TLL states was discussed. Further investiga-tion is required in order to develop a comprehensive under-standing of these phenomena /H20849e.g., changing the number of
encapsulated-C
60molecules /H20850.
ACKNOWLEDGMENTS
We acknowledge T. Nakanishi, S. Tarucha, W. Izumida, P.
E. Lindelof, and M. Thorwart for fruitful discussions.
1B. W. Smith et al. , Nature /H20849London /H20850393, 323 /H208491998 /H20850.
2M. Yudasaka, S. Iijima et al. , Chem. Phys. Lett. 380,4 2 /H208492003 /H20850.
3S. Okada, S. Saito, and A. Oshiyama, Phys. Rev. Lett. 86, 3835
/H208492001 /H20850.
4Y.-G. Yoon and S. G. Louie, Appl. Phys. Lett. 83, 5217 /H208492003 /H20850.
5J. Lee, T. Okazaki, H. Shinohara, Y. Kuk et al. , Nature /H20849London /H20850
415, 1006 /H208492002 /H20850.
6K. Hirahara, K. Suenaga, S. Bandow, H. Kato, T. Okazaki, H.
Shinohara, and S. Iijima, Phys. Rev. Lett. 85, 5384 /H208492000 /H20850.
7D. H. Cobden, M. Bockrath, P. L. McEuen, A. G. Rinzler, and R.
E. Smalley, Phys. Rev. Lett. 81, 681 /H208491998 /H20850.
8J. Nygard et al. , Nature /H20849London /H20850408, 342 /H208492000 /H20850.
9W. Liang, M. Bockrath, and H. Park, Phys. Rev. Lett. 88,.
126801 /H208492002 /H20850.
10M. R. Buitelaar, A. Bachtold, T. Nussbaumer, M. Iqbal, and C.
Schonenberger, Phys. Rev. Lett. 88, 156801 /H208492002 /H20850.
11The metallic behavior and the tube diameter /H20849/H110111.6 nm /H20850con-
firmed by TEM indicate the possibility of a C 60@/H2084910,10 /H20850as
used in Ref. 3Based on this, Gmaxwas estimated to be /H1101140
/H11003/H208514/H208492e2/h/H20850/H11015640/H20852/H9262S for our FET. Since the actually total
Gmaxobserved here was, however, as low as /H1101110/H9262S, a large
contact resistance /H20849of the order of M /H9024/H20850at the electrode and/or
peapods interface is estimated, thus resulting in peapod quantumdots.
12M. Bockrath et al. , Nature /H20849London /H20850397, 598 /H208491999 /H20850.
13A. Bachtold, M. de Jonge, K. Grove-Rasmussen, P. L. McEuen,M. Buitelaar, and C. Schonenberger, Phys. Rev. Lett. 87,
166801 /H208492001 /H20850.
14H. W. Ch. Postma, C. Dekker et al. , Science 293,7 6/H208492001 /H20850; they
reported an /H9251=1.66, and interpreted by correlated sequential
tunneling /H20849Ref. 17/H20850. Since they, however, integrated G0over the
entire Vbgregions, their /H9251value cannot be compared with our
results. In addition, some possibilities for the origin of powerlaws other than TLLs have been discussed, e.g., a Coulomb
blockade strongly coupled with its external electromagnetic en-vironment and a phenomenon related to 1D localization /H20849Refs.
16,19, and 20/H20850. However,
/H9251is too small in these models.
15H. Ishi, H. Kataura et al. , Nature /H20849London /H20850426, 540 /H208492003 /H20850.
16R. Egger and A. O. Gogolin, Phys. Rev. Lett. 87, 066401 /H208492001 /H20850.
17M. Thorwart, M. Grifoni, G. Cuniberti, H. W. C. Postma, and C.
Dekker, Phys. Rev. Lett. 89, 196402 /H208492002 /H20850.
18W. Que, Phys. Rev. B 66, 193405 /H208492002 /H20850.
19G.-L. Ingold and Yu. V. Nazarov, in Single Charge Tunneling ,
edited by H. Grabert and M. H. Devoret /H20849Plenum, New York,
1992 /H20850, Vol. 294, p. 21.
20J. Haruyama, I. Takesue, T. Hasegawa, and Y. Sato, Phys. Rev. B
63, 073406 /H208492001 /H20850.
21G.-L. Ingold and Yu. V. Nazarov, in Single Charge Tunneling ,
Nato ASI, Series B: Physics, edited by H. Gravert and M. H.Devoret /H20849Plenum, New York, 1992 /H20850, Vol. 294.
22Y. Oreg, K. Byczuk, and B. I. Halperin, Phys. Rev. Lett. 85, 365
/H208492000 /H20850.ANOMALOUS COULOMB DIAMONDS AND POWER-LAW … PHYSICAL REVIEW B 75, 205431 /H208492007 /H20850
205431-7 |
PhysRevB.102.214437.pdf | PHYSICAL REVIEW B 102, 214437 (2020)
Editors’ Suggestion
Fractons from polarons
John Sous1,2,*and Michael Pretko3
1Department of Physics T42, Technische Universität München, Garching, 85747, Germany
2Department of Physics & Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z3
3Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA
(Received 24 April 2019; revised 7 December 2020; accepted 8 December 2020; published 28 December 2020)
Fractons are a type of emergent quasiparticle that cannot move freely in isolation, but can easily move in bound
pairs. Similar phenomenology is found in boson-affected hopping models, encountered in the study of polaronsystems and hole-doped Ising antiferromagnets, in which motion of a particle requires the creation or absorptionof background bosonic excitations. In such models, individual low-energy quasiparticles cannot move freely,while bound pairs have drastically increased mobility. We show that boson-affected hopping models can providea natural realization of fractons, either approximately or exactly, depending on the details of the system. We firstconsider a generic one-dimensional boson-affected hopping model, in which we show that single particles moveonly at sixth order in perturbation theory, while motion of bound states occurs at second order, allowing for abroad parameter regime exhibiting approximate fracton phenomenology. We explicitly map the model onto afracton Hamiltonian featuring conservation of dipole moment via integrating out the mediating bosons. We thenconsider a special type of boson-affected hopping models with mutual hard-core repulsion between particlesand bosons, experimentally accessible in hole-doped mixed-dimensional Ising antiferromagnets, in which thehole motion is one dimensional in an otherwise two-dimensional antiferromagnetic background. We show thatthis system exhibits perfect fracton behavior to all orders in perturbation theory. We suggest diagnostic signatures
of fractonic behavior, opening a door to use already existing experimental tools to study their unusual physics,such as universal gravitation and restricted thermalization. As an example, gravitational attraction manifestsas phase separation of holes in doped antiferromagnets. In studying these models, we identify simple effectiveone-dimensional microscopic Hamiltonians featuring perfect fractonic behavior, paving the way to future studieson fracton physics in lower dimensions, where a wealth of numerical and analytical tools already exist. In theseHamiltonians we identify pair-hopping interactions as the mechanism of dipole motion, and argue that this mayprovide a connection to topological edge states in boundary fractonic systems.
DOI: 10.1103/PhysRevB.102.214437
I. INTRODUCTION
Strongly interacting quantum many-body systems can exist
in phases of matter featuring a wide range of exotic phe-nomena, including topologically protected edge currents andquantum error-correcting properties [ 1,2]. Perhaps most sur-
prisingly, such phases can exhibit exotic new quasiparticlesresulting from fractionalization of the fundamental particlescomprising the system. For example, celebrated fractionalquantum Hall systems host quasiparticles carrying only afraction of the charge of an individual electron [ 3]. These
fractionalized quasiparticles typically also feature anyonicstatistics, partway between ordinary bosons and fermions.Further types of fractionalization are also possible such asthe separation of the spin and charge of the constituentelectrons [ 4–7].
While the notion of fractionalized quasiparticles has been
understood for several decades, theoretical work over the pastfew years has uncovered a new more unusual type of quasi-particle. Certain quantum phases of matter are known to host
*Present address: Department of Physics, Columbia University,
New York, New York 10027, USA.“fracton” quasiparticles, characterized by an exotic set of mo-
bility restrictions. Specifically, a fracton is a quasiparticle thatdoes not have the ability to move by itself. Rather, fractonscan only move by coming together to form certain mobilebound states, see Fig. 1. This restriction on mobility is usually
encoded in the system in the form of higher-moment charge
conservation laws, such as the conservation of dipole moment
that often arises as a consequence of an emergent symmetrictensor gauge field [ 8–12].
These unusual fracton quasiparticles have attracted intense
interest due to their fundamental importance as a new for-mulation of exotic gauge field theories [ 8–10] and for their
practical applications. Most notably, fractons present a novelroute to the possible construction of self-correcting quantummemories, providing a platform for quantum information stor-age and processing [ 13–15]. Fracton physics has also shed
new light on many seemingly disconnected areas of theoret-ical physics, including explaining the restricted mobility oftopological crystalline defects [ 16–20] and providing a new
mechanism for many-body localization in the absence of dis-order [ 21–26]. Fractons have also drawn a bevy of unexpected
connections with topological order [ 27–30], quantum Hall
physics [ 10,31], deconfined quantum criticality [ 32], gravita-
tion [ 33], and holography [ 34]. We refer the reader to Ref. [ 35]
2469-9950/2020/102(21)/214437(18) 214437-1 ©2020 American Physical SocietyJOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
FIG. 1. Fractons from polarons. Fractons and dipoles are
schematically represented in (a): A fracton of charge Q cannot move,but a bound state of two fractons (typically of opposite charge)
can, in accordance with dipole conservation. Boson-affected hopping
models, used to study certain polaron systems, are represented in(b) where a particle (light-blue dot) can only move by creating
a boson (orange square) at its departure site or by absorbing one
at its arrival site. The particle-bath coupling leads to formation ofpolarons: a particle dressed by a bosonic cloud (cloudy light-red
oval). A single particle (upper panel), in forming a polaron, becomes
localized by string excitations, while two particles (lower panel) be-come bound (forming a bipolaron) and can move via boson-mediated
pair-hopping interactions. This behavior is exact in one-dimensional
systems with mutual hard-core repulsion between the particles andbosons, and is approximate in absence of the constraint. By stagger-
ing charge, see Sec. III C, we identify a single polaron as a fracton,
and a bipolaron as a dipole.
for a review of fractons, and to selected literature [ 36–54]
for details.
Fractons were first proposed in the context of three-
dimensional quantum spin-liquid models [ 13,21,27,28], and
were later shown to be realized as topological lattice defectsof ordinary two-dimensional crystals [ 16]. Fracton models can
in principle be engineered directly using Majorana islands[45]. But, there is still an important need for concrete, physi-
cally accessible systems realizing fracton physics. It would beparticularly useful to construct models that are realizable inone dimension, as opposed to previous models that are stableonly in higher dimensions. Such a fracton system could thenbe studied using the wide variety of analytic and numericaltechniques available for studying one-dimensional models.
Towards this end, we here show that fracton physics can be
realized in a class of models featuring boson-affected hopping(depicted in Fig. 1), which can be found in both one- and
higher-dimensional systems, see [ 55] for a concise demon-
stration. Systems that are described by such models includehole-doped antiferromagnets in two or three dimensions andpolaronic systems in any dimension. Such systems can exhibit
either exact or approximate fracton behavior, depending onthe specific details. In these models, quasiparticles can onlyhop via the creation or absorption of background bosonicexcitations. For example, in a two- or three-dimensional Isingantiferromagnet, a hole can only move to a neighboring siteat the expense of creating energetically costly spin misalign-ments (i.e., magnons) [ 56–60], as illustrated in Fig. 2. As such,
there is no “bare” nearest-neighbor hopping term for the holes.Rather, a hole can only move through very weak beyond-nearest-neighbor hopping processes of a perfectly orientedspin (up arrow in Fig. 2) or through a complicated sequence
of nearest-neighbor boson-mediated hoppings [ 56].
As such, the hole acquires a finite effective mass only at
sixth and higher orders in perturbation theory in the nearest-neighbor limit [ 56]. In contrast, a bound state of two holes
[61–63] is capable of moving through a much simpler second-
order process. As a consequence, a broad parameter regime
exists where the individual holes are effectively immobile [ 64]
compared to a “dipole” of two holes, which has an enormouslysmaller effective mass, thereby providing an approximate re-alization of fractons. This intuition gained from the Isingantiferromagnet holds much more generally. In a wide class ofboson-affected hopping models, bound pairs of particles, i.e.,bipolarons, have significantly enhanced mobility compared to
the individual particles, in close analogy with the physics of
fractons. While this fracton behavior is generically approxi-mate, we will find a special class of boson-affected hoppingmodels that exhibit true fracton behavior, valid to all orders inperturbation theory.
In this work we establish a more precise relationship
between fracton physics and boson-affected hopping mod-els, such as those encountered in the study of polaronsand hole-doped Ising antiferromagnets [ 55]. We begin by
briefly reviewing the physics of fractons, such as their higher-
moment conservation laws. We also write a one-dimensional
lattice Hamiltonian governed by such a conservation law.This Hamiltonian features only pair-wise hopping, withoutsingle-body hopping terms, and manifestly exhibits the frac-ton phenomenon. In a conventional system of particles on alattice, such a Hamiltonian with no single-particle hoppingwould be enormously fine tuned and is likely hard to en-gineer. However, we show that the fractonic properties ofthis Hamiltonian can be realized naturally (albeit approxi-mately) in boson-affected hopping systems. To this end we
FIG. 2. Fractons from hole-doped Ising antiferromagnets. Motion of a hole in an Ising antiferromagnetic background is impeded due to
the creation of energetically costly spin misalignments, i.e., magnons, panel (a). While not perfectly immobile (due to high-order Trugmanloops), a single hole is drastically less mobile than a bound state of two holes, which moves comparatively much easier, panel (b). In the
mixed-dimensional limit of the holes moving along a line in the two-dimensional system, the magnetic polaron (i.e., the hole dressed by
magnons) is perfectly localized, as the bosonic strings restrict the hole to its original site, and Trugman loops are absent.
214437-2FRACTONS FROM POLARONS PHYSICAL REVIEW B 102, 214437 (2020)
work with a generic boson-affected hopping Hamiltonian that
can describe a variety of physical systems. We show that,upon perturbatively integrating out the bosons mediating thehopping, one naturally obtains a fracton Hamiltonian throughfive orders of perturbation theory. We then move on to a par-ticular quasi-one-dimensional boson-affected hopping model,describing holes restricted to move along one dimension ofa two-dimensional system, which can be readily realized inultracold atoms. We show that a hard-core constraint foundin this model eliminates single-particle hopping to all ordersin perturbation theory, resulting in perfect fracton physics
exhibiting exact conservation of dipole moment. We note that
the fracton realizations we discuss here are both ungauged and
of type-I models, known to host mobile bound states.
In this sense, systems with boson-affected hopping serve as
a natural playground for explicitly studying fractonic physics,allowing us to study the dynamics of fractons and dipolesas well as some of their established phenomenology, includ-ing their restrictions on thermalization and their gravitationalbehavior. We investigate to what extent this phenomenologysurvives in approximate fracton systems, where the mobilityconstraints are weakly violated. Specifically, we consider howthe thermalization and gravitational properties associated withfractons are altered in the presence of a small fracton mobility,finding that both survive to an extent, providing useful ex-perimental diagnostics of fracton physics. In fact, gravitationmanifests as phase separation of holes doped in antiferromag-nets, in agreement with previous studies. Besides providingan experimentally accessible platform for fractons, boson-affected hopping systems also come with a well-establishedliterature, on topics ranging from polarons to antiferromag-netism, from which we hope important insights may be drawnfor better understanding the physics of fractons. Our work alsoopens the door for application of powerful one-dimensionalanalytic and numerical techniques to fracton systems.
II. THE PHYSICS OF FRACTONS
The essential physics of fractons is governed by perfectly
correlated hopping, in which a particle can only move if thereis corresponding motion of a second particle (or group ofparticles). As such, in the absence of other particles withwhich to correlate its motion, a single fracton is strictlyimmobile. Mathematically, this structure is neatly encodedin the language of higher-moment conservation laws, suchas conservation of dipole moment, which severely restrictthe motion of charges. These conservation laws can arise,for example, in the context of symmetric tensor gauge theo-ries, which describe a variety of fracton systems [ 8,9,11,12].
To illustrate the main principle, consider a tensor versionof Maxwell theory, with a rank-2 symmetric tensor electricfield E
ijwith a charge density ρdefined via a generalized
Gauss’s law:
∂i∂jEij=ρ (1)
(where all indices refer to spatial coordinates and repeated
indices are summed over). As compared with a conventionalGauss’s law, this equation is notable for the presence of anunusual extra conservation law. As in conventional electro-magnetism, the total charge in the system is encoded in anappropriate electric flux through the boundary, which indi-
cates that the total charge only changes through flux of chargethrough the boundary. For a closed system we can thereforeconclude that the total charge is constant:
/integraldisplay
d
dxρ=const. (2)
Unlike conventional Maxwell theory, however, the presence of
two derivatives in Gauss’s law allows us to conclude that thetotal dipole moment in the system/integraltext
d
dx(ρ/vectorx) is also encoded
as an electric flux through the boundary. For a closed systemwe can then conclude that the total dipole moment is alsoconserved:/integraldisplay
d
dx(ρ/vectorx)=const. (3)
This extra conservation law severely restricts the motion of
the charges of the theory. An individual charge is not capableof moving at all, since motion in any direction would changethe total dipole moment of the system. In contrast, a dipolarbound state of two equal and opposite charges is free to movein any direction, provided it maintains its dipole moment, asindicated in Fig. 1. These arguments also extend to other types
of tensor gauge theories, which can exhibit conservation ofeven higher charge moments. In this work, however, we focuson fracton systems exhibiting only conservation of charge anddipole moment.
To date, the study of fractons has focused mainly on three-
dimensional spin liquid models, as well as elasticity theory intwo and three spatial dimensions. In contrast, there has beenlittle investigation into stable realizations of fracton physicsin one dimension. Putting aside concerns of stability, one canconstruct a one-dimensional Hamiltonian exhibiting fractonphysics by simply demanding the conservation of both chargeand dipole moment, a task we accomplish in this work, seeSecs. IIIandIV. However, the absence of one-body hopping
matrix elements, necessary for dipole conservation, is not anatural feature of ordinary systems of particles. In this workwe show how Hamiltonians of this type can naturally arise inboth approximate and exact ways in the context of systemswith boson-affected hopping.
To construct a system with the desired charge and dipole
conservation laws, it is simplest to consider two species ofhard-core particles, created (destroyed) by f
†
σ(fσ), which
we regard as carrying opposite charges, σ=±. These par-
ticles can have either bosonic or fermionic statistics, withlittle effect on the subsequent analysis (though we will laterspecialize to the case of fermions). The dipole conserva-tion law forbids single-particle hopping. The lowest-orderdipole-conserving hopping process corresponds to motionof the smallest dipole, a bound state of a positive anda negative charge separated by one lattice site, from onepair of sites to the next, as illustrated in Fig. 1(see also
Fig. 5). An effective Hamiltonian consistent with conser-
vation laws thus takes the form H=−/epsilon1
0/summationtext
i,σf†
i,σfi,σ−
t/summationtext
i(f†
i+1,σf†
i+2,−σ+f†
i−1,σf†
i,−σ)fi+1,−σfi,σ. By design, this
Hamiltonian exhibits fracton phenomenology, with mobiledipoles and stationary charges. While this Hamiltonian is ex-plicitly fractonic, the absence of single-body hopping matrixelements is not a natural feature of typical systems. Such a
214437-3JOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
restriction on free single-body motion, however, does arise in
the context of boson-affected hopping, to which we turn next.
III. APPROXIMATE FRACTON BEHA VIOR FROM
BOSON-AFFECTED HOPPING
A. The model
With an understanding of fractons in hand, we now turn
our attention to what ap r i o r i would seem like a completely
disconnected area of physics. We consider models where aset of quasiparticles fcan only move via interaction with an
auxiliary set of bosons b. Specifically, any process that hops
anfparticle from one site to the next necessarily creates
or absorbs a boson b. We refer to this type of motion as
boson-affected hopping, which we will see leads to fractonbehavior for the fparticles. The fparticles can in principle
have either bosonic or fermionic statistics. In this paper wewill mostly focus on the case where fis fermionic, to match
with the properties of the most common physical realizations,though a system with bosonic fparticles would have largely
similar behavior. From here on, therefore, we refer to the
fparticles as the fermions and the bparticles as the bosons.
We have already described in the Introduction how boson-
affected hopping can arise in a hole-doped antiferromagnetin two or higher dimensions, since motion of a single holerequires the creation of energetically costly spin misalign-ments, i.e., magnons. Before moving on to an analysis ofsuch a model, it is also useful to consider how this type ofmotion arises in a very different physical context, namely thestudy of polarons. The concept of a polaron most commonlydescribes the motion of an electron in a polarizable crystal,in which the ions adjust their positions in order to screen thecharge of itinerant electrons, as seen in Fig. 1. Dragging such
a screening cloud of ions around the crystal leads to a dramaticincrease in the effective mass of the electron bound in thecloud of phonons, the polaron [ 65]. Since the motion of a
particle (electron or hole, etc.) in a crystal requires significantrearrangement of the background, one can effectively describethe dynamics of the particle as conditioned upon the creationor annihilation of distortions in the medium. Subsequently,one more generally finds polaronic effects in a variety ofsystems ranging from particles in ordered phases [ 66–69]t o
impurities in ultracold gases [ 70,71]. As an example, this
approach was pioneered by Edwards [ 72] in his eponymous
model, which was used to study a variety of quantum phenom-ena in boson-affected systems [ 73–78]. The Edwards model
presumes that a carrier moves by creating and /or annihilating
excitations in the background that can be parametrized asbosons. One can see that this model captures various featuresof the Holstein [ 79] and Peierls polaron [ 80,81] physics, mag-
netic polarons [ 82,83], and the Falicov-Kimball model [ 84].
We note that the phonon-induced modulation of the electronhopping in solids due to out-of-phase lattice distortions isdescribed, to linear order, by more elaborate models, such asthe Peierls model [ 85–87] (also known as the Su-Schrieffer-
Heeger model [ 88,89]). Certain features of the boson-affected
hopping models we discuss here carry over to the Peierlsmodel, such as the pair hopping of bound pairs of fermionsknown as bipolarons [ 90]. However, other features, such as theheavily suppressed mobility of single particles, do not always
carry over to Peierls polarons [ 80].
With these physical contexts in mind, we now abstract to
a generic boson-affected hopping model. We first consider amodel with only a single species of fermion f, which will
demonstrate the central idea in the simplest context. Later,we will extend the analysis to multispecies models, which isimportant for certain contexts and will make a connection witheven simpler fracton models. We will only explicitly analyzeone-dimensional models, though much of the same physicswill carry over immediately to higher-dimensional systems.
Before focusing on purely boson-affected hopping, we first
write down a model that has boson-affected andconventional
hopping processes for the fermions. In one dimension, such a
Hamiltonian, can be written as
H=− tf/summationdisplay
/angbracketlefti,j/angbracketrightf†
ifj+g/summationdisplay
/angbracketlefti,j/angbracketrightf†
ifj(b†
j+bi)
−μ/summationdisplay
if†
ifi+ωb/summationdisplay
ib†
ibi. (4)
The first term represents unassisted hopping of the fermions,
while the second represents boson-affected hopping. The last
two terms are the chemical potential of the fermions and theenergy cost to create a boson, respectively. We genericallytake the bosons to be gapped, as in the cases of opticalphonons and Ising magnons. Note that we have not giventhe bosons any dynamics (i.e., any bhopping terms) on the
grounds that, in typical physical realizations, the bosons areeffectively static variables compared with the fermions. Forexample, in the context of polarons, this corresponds to thestatement that ions move enormously slower than electrons.It is also worth noting that we have included only terms cor-responding to boson creation on the departure site and bosonabsorption on the arrival site of the fermions, which is a typ-ical situation in certain systems, such as the two-dimensionalantiferromagnet of Fig. 2and the Edwards model. In the
Peierls model [ 85–89] describing the linear coupling of
electron hopping to the lattice ∼(f
†
ifj+H.c.)(X i−Xj),
where X i∝b†
i+bi, the particle can move by creating or
annihilating bosons at both the arrival and departure sites.
Generically, the Hamiltonian of Eq. ( 4) has fully mobile
particles, as a consequence of the bare hopping tf, with the
boson coupling simply serving to modulate the effective mass.However, in certain physical situations, such as hole-dopedIsing antiferromagnets, all hopping processes necessarily in-volve the creation or absorption of bosonic defects, such that
t
fis rigorously zero. In other cases, tfis not strictly zero, but it
can often be negligibly small in strongly coupled systems. Aswe will soon see, single-particle mobility is generated at sixthorder in perturbation theory in this model anyway. Thus, aslong as t
fis small compared to these sixth-order corrections,
it will have no effect on the subsequent analysis. Whether tfis
rigorously zero or simply negligibly small, we will drop thisterm for the moment, writing the effective Hamiltonian as
H=g/summationdisplay
/angbracketlefti,j/angbracketrightf†
ifj(b†
j+bi)−μ/summationdisplay
if†
ifi+ωb/summationdisplay
ib†
ibi.(5)
While this Hamiltonian cannot be solved exactly for arbi-
trary density of fparticles, it can be usefully analyzed via
214437-4FRACTONS FROM POLARONS PHYSICAL REVIEW B 102, 214437 (2020)
perturbation theory expanding around the solvable point g=
0, at which all particles are trivially stationary.
B. Dynamics
1. Single-particle dynamics
We first consider the single-particle sector, satisfying/summationtext
if†
ifi=1 (i.e., one particle in the infinite system), of
the Hamiltonian in Eq. ( 5). We can now find the effective
Hamiltonian within this sector by performing a perturba-tive calculation, effectively integrating out the bbosons.
The details of this perturbative calculation can be found inAppendix A. To gain physical intuition, it is instructive to
inspect the behavior at the second order. We find, to secondorder in g, that the effective single-particle Hamiltonian takes
the form
h
1=−/epsilon10/summationdisplay
if†
ifi, (6)
which contains only an on-site energy /epsilon10=2g2/ωb, char-
acterizing the polaron formation energy. (Note that thisrenormalization energy is half that obtained in the Peierlsmodel of electron-phonon coupling [ 80].) Importantly, how-
ever, the Hamiltonian does not contain any hopping processesfor the fermions, so at this level of perturbation theory, theparticles are strictly locked in place, behaving as fractons,see Fig. 3. While the single-particle Hamiltonian of Eq. ( 6)
was calculated explicitly only to second order, it is easy tosee pictorially that a process moving a single particle ap-pears only at sixth order in perturbation theory [ 76,91], see
Fig.4. Such processes will lead to a single-particle dispersion
of order ( g
6/ω5
b) cos(2 k). Through fifth order in perturbation
theory, however, the single-particle Hamiltonian will featureno hopping terms, and the particles will behave as fractons.The polaron, which is approximately localized by the costlystring excitations, thus physically realizes a fracton.
2. Two-particle dynamics
We now turn our attention to the two-particle sector of
the theory, satisfying/summationtext
if†
ifi=2 (i.e., two particles in the
infinite system), which will feature nontrivial dynamics ata much lower order in perturbation theory. Two particlesbecome bound by moving together through a second-orderprocess in which one particle hops and emits a boson, whichis immediately absorbed by the other particle hopping in thesame direction, as seen in Fig. 5. We again refer the reader to
Appendix Afor technical details of the perturbative calcula-
tion. To second order in g, we obtain the effective Hamiltonian
for the two-particle sector as
h
2=−/epsilon10/summationdisplay
if†
ifi−t/summationdisplay
i(f†
i+1f†
i+2+f†
i−1f†
i)fi+1fi
+J/summationdisplay
if†
ifif†
i+1fi+1, (7)
where /epsilon10=2t=J=2g2/ωb. The first term is the same on-
site energy seen in the single-particle Hamiltonian, while thefinal term is an interaction energy between nearest-neighborpairs. Meanwhile, the second term represents a pair-hoppingterm, moving a pair of particles at sites iandi+1t oe i t h e rFIG. 3. Polaron (fracton) and bipolaron (dipole) dispersion and
effective masses. We consider an exemplary case of g=0.4a n d
ωb=1 in the one-dimensional boson-affected hopping model with
tf=0. To facilitate comparison we have shifted the energy of each
quasiparticle band such that the zero of energy coincides with theband minimum at the center of the Brillouin zone. Note that for two-
particle states K=k
1+k2,w h e r e kis the single-particle momentum.
The polaron dispersion is shown for two cases: (1) in presence ofa mutual hard-core constraint between the particles and the bosons
(dashed line), and (2) in absence of the constraint (dotted curve).
In (1) the polaron dispersion is perfectly flat providing an exactrealization of a fracton. In (2) the polaron acquires a finite bandwidth
through a sixth-order process, i.e., E
P(k)=−2(g6/ωb5)c o s ( 2 k). In
the regime considered, this polaron remains much heavier than thebipolaron, whose dispersion is shown only up to second order,
E
BP(K)=−2(g2/ωb)c o s ( K) (solid curve). This is further demon-
strated in the inset, where we show the inverse of the masses ofthe polaron (diamond symbol) and bipolaron (circle symbol) in the
unconstrained model. In this sense, and through staggering charge
(Sec. III C), bipolarons realize dipoles, approximately in the absence
of the constraint, and exactly in its presence (in one dimension).
sites i+1 and i+2 or sites i−1 and i. This behavior of
the two-particle state mimics that of a dipole. Importantly,however, there are still no single-particle hopping terms. Tothis order in perturbation theory, we can therefore concludethat single particles are immobile, while bound states of twoparticles can move freely, with a dispersion of order −tcos(K)
(where K=k
1+k2is the momentum of the two-polaron
bound state) demonstrated in Fig. 3(see details of calculation
in Appendix B), which is perfectly in line with the expected
behavior of fracton systems.
Before moving on, we note that the final Jterm of
the Hamiltonian is a repulsive interaction, arising from thefact that a particle cannot hop to a neighboring site thatis already occupied [ 92,93]. Nearest-neighbor particle pairs
thereby miss out on a portion of the binding energy that wouldhave been obtained from virtual hopping processes to neigh-boring sites. Importantly, however, it should be noted thatthis “repulsion” will not destabilize the two-particle boundstate. Since the individual particles have no mobility in theHamiltonian, they will not be able to lower their energy bymoving apart. In other words, while well-separated particleswould have a lower energy than a nearest neighbor pair, there
214437-5JOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
FIG. 4. Weak single-particle mobility and approximate fracton behavior. A schematic representation of a sixth-order process giving rise
to single-particle mobility. The particle (light-blue dot) moves by creating a string of bosonic excitations (orange-red squares), which it then
“cleans up” by retracing its steps. Through this process the particle moves two sites apart and thus acquires a −2(g6/ωb5)c o s ( 2 k) dispersion.
As we explain in the main text, pairs of particles have a more enhanced mobility that already manifests at earlier orders, see Fig. 5. This thus
presents a case of approximately fractonic behavior.
are no matrix elements in the Hamiltonian that can take the
system between these two configurations. As such, the Jterm
of the Hamiltonian merely serves to raise the energy of thetwo-particle bound state.
It is also important to point out that the behavior of po-
larons and bipolarons in such boson-affected hopping models(e.g., the Edwards model) departs from the usual view thatpolaronic and bipolaronic quasiparticles must be associatedwith mass enhancement. Indeed, here we find that single po-larons experience pronounced mass enhancement. However,bipolarons do not and are light in comparison. For morediscussions about light polaronic quasiparticles, we refer thereader to Refs. [ 80,81,90] studying Peierls polarons and bipo-
larons, which are both shown to be light at strong coupling.These results challenge the standard view of polaronic massenhancement known for Holstein and Fröhlich models.
C. Identification of conservation laws
The phenomenology of immobile particles forming mobile
bound states matches perfectly with the properties of fracton
systems, but to make the connection more precise, we should
identify the conserved quantities in the effective Hamiltonianobtained via perturbation theory, to see how they relate to typi-cal conservation laws in fracton systems, such as conservationof charge and dipole moment. The effective Hamiltonian ofEq. ( 7) manifestly obeys conservation of charge (i.e., parti-
cle number),/summationtext
in(f)
i=const. However, making a connection
with conservation of dipole moment is slightly trickier. Themobile excitations of the theory are bound states of two iden-
tical particles, not opposite charges, and the motion of such a
bound state does not conserve the naively defined dipole mo-
ment D=/summationtext
in(f)
ixi. However, there is a simple workaround
that allows us to obtain a dipolar conservation law. We cansimply define a new “charge density” as follows:
n/prime
i=n(f)
iexp/parenleftbigg
iπ/summationdisplay
j<in(f)
j/parenrightbigg
, (8)
which staggers the sign of charges from one particle to
the next. This definition automatically ensures that the mo-bile bound states consist of two particles of opposite n
/prime
charge. In other words, the mobile bound states are true
dipoles of the new charge density. The pair-hopping tpro-
cesses of the Hamiltonian can move a dipole from onepair of sites to the next, but the magnitude of the dipoleis always left unchanged. Since the on-site and interactionenergies (i.e., /epsilon1
0andJterms) of the Hamiltonian do not
change the charge configuration of a state, we can thereforeconclude that our effective Hamiltonian of Eq. ( 7) exhibits
conservation of dipole moment, with respect to the staggeredcharge density:
D
/prime=/summationdisplay
in/prime
ixi=const., (9)
which can explicitly be checked to commute with the Hamil-
tonian [ D/prime,h2]=0. While this version of dipole conservation
has a slightly nonlocal form in terms of the original fermions,
due to the definition of n/primein terms of a semi-infinite string,
it is every bit as effective at restricting their motion. The twodensities nandn
/primediffer only by a sign, and any motion of a
single fermion will change the value of D/prime, so the immobility
of single particles can be understood as a direct consequenceof this emergent conservation law. This conservation lawholds up to sixth order in perturbation theory, at which pointit is violated by the generation of single-particle hopping.Thus, this generic boson-affected hopping model gives rise toapproximate fracton behavior, over a wide parameter regimebetween second and sixth orders of perturbation theory. In
FIG. 5. Dipole-conserving two-particle dynamics. A two-particle bound state moves through boson-mediated pair-hopping interactions.
Here we schematically show a second-order process that involves the exchange of a single boson (orange-red square) between the two particles
(light-blue dots). One particle hops first, leaving behind a boson at the site between the two particles. The second particle then hops to the sitepreviously occupied by the first particle by absorbing the boson. In this way, the bound state moves over by a single lattice spacing acquiring
a−2(g
2/ωb)c o s ( K) dispersion, where K=k1+k2. Since the relative distance between the two staggered charges (see Sec. III C) remains
invariant, the two-particle state provides an ideal realization of a dipole.
214437-6FRACTONS FROM POLARONS PHYSICAL REVIEW B 102, 214437 (2020)
a later section we will investigate the properties of such ap-
proximate fracton systems and establish to what extent typicalfracton phenomenology survives.
IV . EXACT FRACTON BEHA VIOR FROM
FERMION-BOSON HARD-CORE REPULSION IN LOWER
DIMENSIONS
A. The model
While the previous model featured only approximate frac-
ton behavior, it is possible to realize fractons exactly througha small modification to the model system. We now imposea mutual hard-core constraint between the fermions and thebbosons, such that any site can host at most one total ex-
citation. This can be implemented at the Hamiltonian level,for example, by adding a repulsive term between bosons andfermions as
H=g/summationdisplay
/angbracketlefti,j/angbracketrightf†
ifj(b†
j+bi)−μ/summationdisplay
if†
ifi+ωb/summationdisplay
ib†
ibi
+U/summationdisplay
if†
ifib†
ibi, (10)
then taking the U→∞ limit. In other words, we project
all states of the form f†
ib†
i|0/angbracketrightout of the Hilbert space. We
discuss below how this constraint can be physically realizedin a simple way in antiferromagnets. For now, let us work outthe physical consequences of this constrained Hilbert space.
B. Dynamics
The first notable consequence of the mutual hard-core
constraint between bosons and fermions is that all of the“backtracking” higher-order processes contributing to single-particle motion, such as seen in Fig. 4, are forbidden. A
fermion can only backtrack by reabsorbing a bparticle that
it just emitted, perfectly retracing its steps. The constraint-enforced, continuing motion of the particle in a certain onedirection only involves first the creation and then subsequentannihilation of bosonic strings, effectively localizing the par-ticle to its original position, while the “cleaning up” typeof motion seen in Fig. 4is no longer possible. As such, a
single particle is now immobile to allorders in perturbation
theory, leading to perfect fracton behavior. The single-particleHamiltonian then takes the exact form
h
1=−/epsilon10/summationdisplay
if†
ifi, (11)
where /epsilon10must be determined order by order in g.
Importantly, while the hard-core constraint makes single
particles perfectly immobile, it still permits mobility of two-particle bound states, which thereby play the role of mobiledipoles of fractons. This can easily be seen from the fact thatthe second-order process of Fig. 5does not involve states with
fermions and bosons on the same site at any point. To sec-ond order in perturbation theory, the two-particle Hamiltoniantakes the same form seen earlier:
h
2=−/epsilon10/summationdisplay
if†
ifi−t/summationdisplay
i(f†
i+1f†
i+2+f†
i−1f†
i)fi+1fi
+J/summationdisplay
if†
ifif†
i+1fi+1. (12)At this level we see that the mutual hard-core constraint has
no significant effect on dipole dynamics.
Making use of the hard-core constraint, we can determine
the dynamics of particles to even higher orders in the g/ωb
expansion. As noted earlier, the single-particle Hamiltonian
has a trivial on-site form to all orders of perturbation theory,with only the prefactor /epsilon1
0being renormalized order by order.
In contrast, there will be some noteworthy changes to the two-particle Hamiltonian. To fourth order in perturbation theory,we find that the two-particle Hamiltonian takes the form
h
2=−/epsilon10/summationdisplay
inf
i+Jz1/summationdisplay
inf
inf
i+1+Jz2/summationdisplay
inf
inf
i+2
−t1/summationdisplay
i(f†
i+1f†
i+2+f†
i−1f†
i)fi+1fi
+t2/summationdisplay
i(f†
i+2f†
i+3+f†
i−2f†
i−1)fi+1fi. (13)
We refer the reader to Appendix Afor calculational details.
The first three terms represent an on-site energy and interac-tions between nearest neighbors and next-nearest neighbors.The second line is the same pair-hopping interaction we sawat second order, moving a pair of particles over by one site.The last term is also a pair-hopping interaction that moves apair of particles over by two lattice sites. This new term alsomanifestly conserves the dipole moment D
/prime=/summationtext
in/prime
ixiof the
staggered charge density.
More generally, let us schematically consider the form of
h2to any order in perturbation theory. It is clear that, by
repeated emission of bosons by one fermion and repeatedabsorption by the other, a nearest-neighbor bound state ofparticles can hop to any location. As such, the exact Hamil-tonian, to all orders in perturbation theory, will contain termshopping a pair by arbitrary distances. Furthermore, the con-straint forbids terms that change the net separation betweenthe two fermions, as those require backtracking. It is alsoeasy to check that there are no processes that can move apair of particles that are not nearest neighbors, as there are nosufficiently long-ranged terms in the Hamiltonian. As such,we conclude that the exact two-particle Hamiltonian takes theschematic form:
h
2=−/epsilon10/summationdisplay
inf
i+/summationdisplay
i,δJzδnf
inf
i+δ
−/summationdisplay
i,δtδ(f†
i+δf†
i+δ+1+f†
i−δf†
i−δ+1)fi+1fi,(14)
where /epsilon10is determined by the order of the expansion, Jzδis
a density-density interaction between particles δsites apart,
andtδis the matrix element for hopping a pair by δsites.
Both decay rapidly as a function of δ. All terms of this form
manifestly conserve the dipole moment D/prime, which we can then
rigorously conclude is conserved to all orders, i.e., [ D/prime,h2]i s
exactly zero.
C. Experimental realization: Hole-doped mixed-dimensional
Ising antiferromagnets
We have shown that a one-dimensional boson-affected
hopping model supplemented by a mutual hard-core
214437-7JOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
FIG. 6. Trugman loops in two-dimensional boson-affected hopping systems. Sixth-order processes, known as Trugman loops, involving
the creation of a string of bosons (orange-red squares) around closed loops by the particle (light-blue dot), give rise to single-particle mobility,
even in the presence of mutual hard-core repulsion between the particles and bosons. If, however, the particles are restricted to move only
along one direction in a two-dimensional system, as in hole-doped mixed-dimensional Ising antiferromagnets, closed loops are absent and the
system is perfectly fractonic.
constraint between fermions and bosons exhibits perfect frac-
ton behavior, with strictly immobile individual particles andmobile two-particle bound states. For this to be meaningful,however, it is important to establish an experimentally real-
izable physical context described by such a model. To this
end we first note that such a mutual hard-core constraint isnaturally found in the context of hole-doped antiferromag-nets, since there is no meaningful way that a hole can existon the same site as a misaligned spin. In one dimension, ahole doped in an antiferromagnet exhibits spin-charge sep-aration and magnons formed as misaligned spin “defects”
can only exist in higher dimensions. In other words, holes in
antiferromagnets only exhibit boson-affected hopping in di-mensions higher than one. Our proof of exact fracton behaviorwas, however, carried out only in one dimension. Indeed, intwo (and higher) dimensions, there are sixth-order processesknown as Trugman loops leading to single-particle mobility(see Fig. 6), so fracton behavior is once again approximate in
these higher-dimensional systems, even when supplemented
by a hard-core constraint.
Fortunately, however, there is an intermediate situation be-
tween one- and two-dimensional systems that ideally servesto realize perfect fracton behavior. In so-called “mixed-dimensional” Ising antiferromagnets, the hole motion isrestricted to a line in a two-dimensional antiferromagnet [ 94].
On one hand, spin-charge separation cannot occur as the holemoves through the lattice unlike in the one-dimensional case.On the other, the holes cannot move in loops and thus cannotacquire a finite effective mass [ 56], so they behave as fractons
to all orders of perturbation theory (the flat band in Fig. 3).
We refer the reader to Ref. [ 55] for a detailed construction of
fractons in hole-doped antiferromagnets. Here we summarizethe main idea.
The mixed-dimensional Ising antiferromagnet can be
potentially realized in promising experiments with Rydberg-atom arrays [ 95–97], trapped ions [ 98,99], and polar
molecules [ 100,101], and perhaps with ultracold atoms in op-
tical lattices [ 102–104]. By experimentally tuning to the limit
of nearest-neighbor Ising interactions, the parent undoped sys-tem of such a two-dimensional square lattice is described byan effective Hamiltonian H
Ising=J/summationtext
/angbracketlefti,j/angbracketrightSz
iSz
j. The groundstate of this Hamiltonian is a classical Néel state |/Psi1GS/angbracketright=
/Pi1i∈Ac†
i,↑/Pi1j∈Bc†
j,↓|0/angbracketrightwith all spins on sublattice Aup, and all
spins on sublattice Bdown. Here the c†
↑/↓creates a particle
(fermion) with spin ↑/↓. Individual holes doped into this
system can move through the hopping of the particles carryingthe spin, which can be taken to a very good approximation,to be nearest neighbor. As outlined in Ref. [ 94], by applying
a strong gradient potential V(y) along the ydirection taken
to be one of the principal axes of the square lattice, the holeis forced to move only along the xdirection. We expect that
mixed-dimensional antiferromagnets can also be engineeredin solid-state devices.
The motion of the hole occurs as a result of the hopping
of a particle whose spin becomes either perfectly oriented ordisoriented with respect to the sublattice it belongs to. Wecan thus regard the disoriented spin as a bosonic defect or amagnon with a creation operator:
d
†
i=/braceleftbigg
σ−
i,ifi∈A,
σ+
i,ifi∈B,(15)
where σ±is the spin-1 /2 raising /lowering Pauli matrix. We
define the hole operator as
h†
i,↓=ci,↑,ifi∈A, (16)
h†
i,↑=ci,↓,ifi∈B. (17)
The only way a hole can move is through the creation of
a defect, which represents the displaced particle now with amisaligned spin orientation. Implementing these processes weobtain a Hamiltonian [ 91]
H=−t/summationdisplay
/angbracketlefti,j/angbracketright,σ[h†
j,σhi,σ(d†
i+dj)+H.c.]+HIsing.(18)
Here/angbracketleft·/angbracketrightrefers to nearest neighbors, and the Hamiltonian
respects a no-double occupancy constraint such that each sitehas either a hole or a spin:/summationtext
σh†
i,σhi,σ+d†
idi+did†
i=1.
Notice that this model resembles that of Eq. ( 10). Here the
holes take the role of the ffermions, and the magnons that
of the bbosons. We observe that this Hamiltonian naturally
respects the mutual fermion-boson hard-core constraint as ahole can only exist at a site empty of a spin. Furthermore,
214437-8FRACTONS FROM POLARONS PHYSICAL REVIEW B 102, 214437 (2020)
in both models each site can host at most one boson. This is
easy to see for the Ising antiferromagnet, for which a magnoncorresponds to a spin flip of a spin-1 /2 particle. For the model
Eq. ( 10), while such a condition is not formally enforced, we
note that due to the absence of backtracking, it is impossible tocreate more than one boson per site. Finally, we note that thebosonic excitations in the hole-doped Ising antiferromagnetcannot be described by a term such as ω
d/summationtext
id†
idias bosonic
defects nearby cost less energy to create than those fartheraway. This, however, poses no qualitative difference on thephysics in the limits we consider here [ 91]. We can easily
see perfect fracton and dipole behavior in this system upto all orders in perturbation theory, Eq. ( 14). This physics
of the quasi-one-dimensional model extends to two dimen-sions in the low-energy limit corresponding to dynamicaltimescales shorter than the inverse of the sixth-order Trugmanloop, promoting ideal fracton behavior to a broad parame-ter regime in the more widely accessible two-dimensionalantiferromagnets.
In contrast to the f-bmodel considered above, the holes
come in two spin flavors. Therefore, one can regard thehole’s spin as a fracton’s charge degree of freedom, thenconservation of dipole moment is automatic with no need forstaggering charge [ 55].
Besides the mixed-dimensional limit antiferromagnet, new
avenues of research on approximately fractonic quasiparticlesexist in two-dimensional Ising antiferromagnets. Next-nearestneighbor (spatially diagonal) hopping presents a compli-cation since it promotes single-particle mobility. Couplingone-dimensional spin chains or moiré engineering to re-alize rectangular antiferromagnets with diagonal distancessufficiently larger than those of square lattices presents onepossible solution. Dysprosium phosphases and dysprosiumaluminum garnets serve as good avenues to approximatelyrealize our model in two dimension and in which a t
/prime
term is very likely absent [ 105]. Using cold-atom quan-
tum simulators of mixtures of bosonic and fermionic statesactivated optically to realize the fermion-boson model or Ryd-berg simulators of hole-doped Ising antiferromagnets provideanother alternative.
The realization of fracton physics in polaronic systems is
the central result of our work. Indeed, experiments studyingmagnetic polarons already show indications of the frac-ton phenomenon, including their restricted mobility andthe string-mediated binding of dipoles [ 106–110]. In the
next section we suggest sharp diagnostics that will helpefforts targeting the observation of the exotic features offractons.
V . DIAGNOSTICS AND PHENOMENOLOGY
In the previous sections we have seen how systems with
boson-affected hopping can give rise to fracton physics. Wenow describe ways to diagnose the presence of fracton physicsin such systems, such as correlation function signatures andtypical fracton phenomenology, such as restricted thermaliza-tion and gravitational behavior. In many cases, the fractonbehavior of these models is only approximate, breaking downat some high order in perturbation theory. We therefore alsostudy to what extent typical fracton phenomenology survives
in approximate fracton systems.
A. Pair correlation function
One immediate consequence of the fracton behavior of
boson-affected hopping models is the fact that, within thetwo-particle sector, the distance between those particles afterintegrating out the bosons never changes. This is true regard-less of whether we consider a nearest-neighbor pair, which isfree to move around the system, or a stationary configurationwith greater separation between the particles. This perfectlocking of the two particles with each other will have a clearmanifestation in the density-density correlation functions ofthe system. For example, let us consider the real-space boson-integrated density-density correlation function
C(d)=Tr
b/angbracketleftBigg
1
N/summationdisplay
iˆniˆni+d/angbracketrightBigg
(19)
(which is independent of ifor a translationally invariant sys-
tem). Here Tr bis a trace over the bparticles. Since, for any
given two-particle state, the particles are separated by a con-stant distance D(the dipole moment), this correlation function
will be nonzero only for d=D, such that
C(d)∼δ(d−D). (20)
In contrast, a two-particle state in a system without dipole
conservation would feature a more generic distribution of thiscorrelation function, without such a sharp peak. This corre-lation function may prove a useful tool for detecting fractonbehavior in both experimental and numerical contexts.
For contexts in which fracton behavior is approximate, not
exact, the density-density correlation function will no longerbe a strict delta function at a fixed dipole moment. For dipoleswith D>1 (i.e., with particles separated by more than one
lattice site) in the boson-affected models we consider, therewill no longer be any reason for the two-particle configurationto have any particular fixed dipole moment in the presenceof single-particle mobility, due to absence of long-range pair-hopping interactions, see Eq. ( 14). As such, the correlation
function in the approximate case would eventually flatten outcompletely. For D=1 dipoles, however, it is still energeti-
cally favorable for two particles to form a dipole, due to thegain in kinetic energy in the bound state. While a D=1 state
does not have its dipole preserved precisely, we still expectmost of the wave function to have its weight in the D=1
sector as long as the single-particle hopping is sufficientlysmall. We then expect that the system will have a rounded,but still prominent, peak in C(d) near d=1. This behavior
can be interpreted in terms of an effective attraction betweenthe particles of the system, a topic to which we turn next.
B. Gravitational behavior
A hallmark feature of fracton systems is the presence of a
universal attractive force between the fractons, which has a di-rect interpretation as an emergent gravitational force [ 33,34].
This attraction arises as a simple consequence of the fact thatfractons are more mobile (i.e., have a smaller effective mass)in the vicinity of other fractons. As a toy model to illustrate
214437-9JOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
the principle, consider a particle with an effective mass m(r),
where ris the distance away from a second particle in the
system, which we regard as fixed. Neglecting any interparticleinteractions, the energy of one particle can be written as
E=
1
2m(r)v2. (21)
In terms of the constant energy E, the velocity of the particle
is then given by
v=/radicalBigg
2E
m(r). (22)
Since the effective mass of a particle decreases at small r,t h e
result is an increase in velocity when particles are close, and
a corresponding decrease in velocity as particles move away,
amounting to an effective attraction. Note that this attractionholds whether the fracton behavior is exact [ m(r)→∞ as
r→∞ ] or is simply approximate.
We can also generalize this toy analysis to include the ef-
fects of an intrinsic short-range repulsion, as we found earlierin the Hamiltonian of Eq. ( 7). In this case, the velocity of a
particle can be written as
v=/radicalBigg
2[E−V(r)]
m(r), (23)
where V(r) is some short-range repulsive potential, such as
V(r)=V0e−r/afor lattice scale a. Let us take the decrease
inm(r) with rto be short range, which is the generic case
[33]. We model the approximate fracton behavior by allowing
for a small single-particle hopping parameter t0, such that
the total effective hopping takes the form t=t0+ηe−r/afor
some parameter η, which sets the energy scale of the dynamics
of the fracton. Taking advantage of the fact that tsets the scale
of the inverse effective mass, we can write the velocity profileof the particle as
v∼/radicalbigg
2t0(E−V0e−r/a)/parenleftBig
1+η
t0e−r/a/parenrightBig
≈/radicalbig
2t0E/bracketleftbigg
1+1
2/parenleftBigη
t0−V0
E/parenrightBig
e−r/a/bracketrightbigg
, (24)
where the last line is an approximation for rbigger than
a few lattice spacings (at long distances). Importantly, notethatt
0/lessmuchηin our situation of interest. Specifically, in the
boson-affected model we have been considering, we haveη∼g
2/ωb, while t0∼g6/ω5
b. As such, we can conclude that
η/t0∼(g/ωb)−4/greatermuchV0/Efor small gat fixed ωb, since V0/E
scales as g2/ωb. We can therefore identify the force between
particles as attractive for almost all states. (Note that thiscondition will fail for states with sufficiently small E,b u t
small- Estates are those whose fractons are far apart such
that they do not pass close to each other near r=0, and
so they do not interact anyway.) In this way we see thateven systems with only approximate fracton behavior willexhibit a near-universal attraction between particles, whichwe therefore expect will cluster together in the system. Thisclustering of holes will result in their phase separation [ 55],
consistent with previous studies on hole-doped antiferromag-nets [ 111–113], providing a smoking-gun signature of fracton
behavior. (We remark that in certain physical contexts, suchas electron systems, the short-range gravitational attraction
between particles will compete with a long-range Coulombrepulsion (which may though be screened). This will still leadto clustering effects at short distances, which may then beovertaken by emulsion physics at longer scales [ 31].)
C. Restricted thermalization
Another notable feature of fracton systems is the fact that
they tend to reach thermal equilibrium very slowly, if at all.It has been shown that three-dimensional fracton systemsapproach equilibrium logarithmically slowly, in a manifes-tation of “asymptotic many-body localization” [ 22]. Even
more dramatically, certain one-dimensional fracton systemscan fail to reach equilibrium at all, forever maintaining amemory of their initial conditions [ 23]. Specifically, a system
initialized with a fracton at a specific location will foreverremember the position of that fracton. A proposed explanationis based on the properties of random walks in one dimension,in which particles always eventually return to their origin. Assuch, when a fracton moves by emitting a dipole, that dipoleshould eventually return to the fracton and be reabsorbed.We expect similar localization of fermions in boson-affectedhopping models exhibiting perfect fracton behavior, such asthe mixed-dimensional Ising antiferromagnet, at least undersimilar initial conditions. When a fermion moves, it is ac-companied by the creation of a string of bosons. Even if theboson has weak dynamics, it will likely return to the vicinityof the fermion and be reabsorbed, preserving localization ofthe fermion, at least approximately. We leave a more rigor-ous analysis of these ideas to future work. This localization,which occurs even in the absence of disorder, could pro-vide a key signature in diagnosing fracton physics in thesesystems.
For approximate fracton behavior, single-particle mobility
will eventually cause systems to relax to thermal equilibrium.However, the relaxation rate will be highly dependent on theinitial conditions, since two-particle bound states can dispersetheir energy around the system much more effectively thansingle particles can. To see this, let us consider a systemof particles moving at a characteristic velocity v. When the
system has only O(1) particles, the thermalization time will
be limited by τ∼L/v, i.e., the time it takes a particle (fracton
or dipole) to travel ballistically across the system length L.
At larger densities, the thermalization time will be limitedby diffusive processes τ∼(L
2/Dv), where Dis the diffusion
constant of the system, set by the density. In either case, therelaxation time depends inversely on the characteristic veloc-ityv, which behaves roughly as v∼√
kT/mfor particles
of effective mass m. Since two-particle bound states have a
mass m2/lessmuchm1, where m1is the single particle mass, we see
that such dipolar states will have a much lower thermalizationtime. Such a parametric difference in relaxation times betweenstates initialized with isolated particles versus two-particlebound states is a clear indication of the presence of approx-imate fracton physics.
Finally, we note that, due to the large separation in scale
of the effective masses of fractons and dipoles, there isthe possibility that dipoles may also end up localized (orat least slow to thermalize), by a mechanism akin to the
214437-10FRACTONS FROM POLARONS PHYSICAL REVIEW B 102, 214437 (2020)
quantum disentangled liquid [ 114], as we discuss further in
the next section.
VI. EXTENSIONS: MULTISPECIES SYSTEMS AND FINITE
FRACTON DENSITIES
We have now established a robust connection between the
physics of fractons and systems with boson-affected hopping,such as polarons. In this section we discuss various ways inwhich our analysis can be usefully extended. We also outlineseveral directions for future topics of investigation relevant tothese ideas.
A. Multispecies systems
In the previous sections we have considered boson-affected
hopping models featuring only a single species of fermion,which is appropriate to certain physical contexts. In other situ-ations, however, the fermions can come with an internal flavor,such as the spin degree of freedom. It is therefore important towork out how the preceding analysis extends to multispeciesboson-affected hopping models. Also, as we will see, certaintypes of multispecies systems will lead to realizations of morefamiliar fracton systems with a simpler set of conservationlaws. For concreteness, we restrict our attention to two-speciesmodels, though models with a larger number of species couldalso be considered without much difficulty.
We begin by considering the most straightforward mul-
tispecies generalization of the previously studied boson-affected hopping model, which we write as
H=g/summationdisplay
/angbracketlefti,j/angbracketright,σf†
i,σfj,σ(b†
j+bi)+ωb/summationdisplay
ib†
ibi−μ/summationdisplay
i,σf†
i,σfi,σ,
(25)
where iandjare site indices and σruns over the two species,
which we label as +and−. Once again we can perturbatively
eliminate the bosons to derive an effective Hamiltonian forthe fermions. Carrying through the same perturbative cal-culation as in the single-species case, through second orderin perturbation theory, the effective Hamiltonian within thesingle-particle sector takes the form
h
1=−/epsilon10/summationdisplay
i,σf†
iσfiσ, (26)
with/epsilon10=2g2/ωb. At this level, the two species behave
completely independently, both described by Hamiltonianswithout hopping terms. As in the single-species case, thefirst processes contributing to single-particle mobility occur atsixth order in perturbation theory, just as in Fig. 4. Below this
order, both species of fermions behave as fracton excitations.
Similarly, we can calculate the effective Hamiltonian
within the two-particle sector, which, to second order in per-turbation theory, takes the form
h
2=−/epsilon10/summationdisplay
i,σf†
i,σfi,σ−t/summationdisplay
i,σ,σ/prime(f†
i+1,σf†
i+2,σ/prime+f†
i−1,σf†
i,σ/prime)
×fi+1,σ/primefi,σ+Jz/summationdisplay
i,σf†
i,σfi,σf†
i+1,σfi+1,σ
+Jxy/summationdisplay
i,σf†
i,−σf†
i+1,σfi+1,−σfi,σ, (27)where /epsilon10=2t=Jz=Jxy=2g2/ωb.T h e/epsilon10term and Jzterm
are on-site energies and nearest-neighbor interactions, re-spectively. The tterm represents pair hopping that moves
two particles (either of the same or opposite species) in thesame direction by one lattice site. The final J
xyexchange
term allows two particles of opposite species on neighbor-ing sites to switch places by exchanging a boson. It iseasy to see that, if we define the total fermion density n
i=/summationtext
σf†
i,σfi,σ, then this quantity obeys the same conservation
law as in the single-species case,/summationtext
in/prime
ixi=const., where
n/prime
i=niexp(iπ/summationtext
j<inj).
While the above model has fracton behavior up to sixth
order in perturbation theory, with an emergent dipole con-servation law, it still has the same sort of mildly nonlocalform as in the single-species case, due to the string/summationtext
j<inj
used to define the sign of the charge. In light of this, it is a
useful exercise to construct a boson-affected hopping modelwith a purely local conservation law, regardless of whetheror not the model is realistic. To this end, we will constructa model that directly maps the two species of fermions ontopositive and negative charges, with ρ
i=ni+−ni−, rather than
relying on a nonlocal sign structure. However, if we wantthe dipole moment of this charge density P=/summationtext
iρixito be
conserved, then we need to slightly change our model fromthe simplest multispecies model of Eq. ( 25). First of all, the
J
xyterm of h2, which exchanges the position of a +and
−charge, obviously violates Pdipole conservation. Thank-
fully this term can be eliminated by introducing a mutualhard-core repulsion between positive and negative charges,which prevents such a switch. Another violation of dipoleconservation comes from the σ=σ
/primepiece of the tterm,
which moves two particles of equal charge in the same di-rection. This issue can also be overcome by giving a senseof directionality to the particles. Specifically, we stipulate thata positive charge can only move right by emitting a bosonand left by absorbing a boson, whereas a negative charge canonly move right by absorbing a boson and left by emitting aboson. These changes can be implemented by writing a newHamiltonian as
H
/prime=g/summationdisplay
i(f†
i+1,+fi,+b†
i+f†
i,+fi+1,+bi
+f†
i+1,−fi,−bi+1+f†
i,−fi+1,−b†
i+1)
+ωb/summationdisplay
ib†
ibi−μ/summationdisplay
i,σf†
i,σfi,σ+U/summationdisplay
if†
i,+fi,+f†
i,−fi,−,
(28)
where we take the U→∞ limit. In this limit the only
dynamical processes allowed by this Hamiltonian involve mo-tion of dipoles with a ( ρ,−ρ) charge configuration, while
bound states of the same charge (if present) cannot move.As such, this model exhibits the local conservation of dipolemoment/summationtext
iρixi=const. This demonstrates that, as a proof
of principle, completely local higher-moment conservationlaws can be realized in boson-affected hopping systems. Weleave the construction of more realistic models of this form tofuture work.
214437-11JOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
B. Finite densities of fractons
In this work we have studied boson-affected hopping mod-
els, focusing on the one- and two-particle sectors. Such ananalysis was sufficient for demonstrating the (often approx-imate) immobility of fractons, as well as the mobile natureof dipoles. However, it is both important and interesting toconsider what happens in sectors with a larger particle num-ber, especially with a finite density of particles. Some aspectsof the finite-density problem can be understood as simpleextensions of our work, while other questions require a morecomplicated analysis that we leave to future work.
As a first order of business, it is important to note that
the immobility of an isolated fracton is not affected by thepresence of other particles far away from the fracton underconsideration. In general, the mobility of a fracton is only af-fected by particles on nearby sites, and in the particular modelunder consideration, is only determined by the presence orabsence of a nearest-neighbor particle. A particle more than asingle lattice spacing away cannot give rise to fracton mobilityat any order in perturbation theory. Similarly, a dipole cannotbe prevented from moving by far-away particles. A dipole is
only blocked from motion when it comes directly into contact
with another particle, due to ordinary hard-core repulsion. As-suming that boson-mediated interactions are predominantly ofthe two-body type, we expect the mobility of an individualfracton or dipole to be effectively unchanged when there is afinite density of other particles in the system.
While the behavior of individual particles is roughly un-
changed at finite densities, there are plenty of interestingquestions to ask regarding the interactions between particles.For example, a finite-density system will generically have a fi-nite concentration of both dipoles, with a relatively light mass,and isolated fractons, with an infinite (or at least extremelyheavy) mass. This sort of situation is precisely what is con-sidered in the study of “quantum disentangled liquids” [ 114],
which is a fluid made from one very heavy and one very lightspecies of particles. It has been argued that, while the heavyparticles reach thermal equilibrium, the light particles havenonergodic behavior due to being localized in the effectivedisordered landscape created by the heavy particles. Applyingthis logic to the present situation, we can speculate that thereare certain initial conditions for which the fractons reachthermal equilibrium, while dipoles are effectively localizedon the fractons. It remains to be seen whether or not thereis a more general relationship between fracton physics andquantum disentangled liquids.
More generally, there is important work to be done in
analyzing the effective Hamiltonians for one-dimensionalfracton systems. For example, we have shown that certainboson-affected hopping systems give rise to the followingHamiltonian:
H=−/epsilon1
o/summationdisplay
if†
ifi−t/summationdisplay
i(f†
i+1f†
i+2+f†
i−1f†
i)fi+1fi
+J/summationdisplay
if†
ifif†
i+1fi+1, (29)
which explicitly describes the dynamics of fractons (with
staggered charge) and dipoles in one spatial dimension. Onecan also obtain a more conventional fracton theory with nocharge staggering by working with a two-species model of
the form
H=−/epsilon1
0/summationdisplay
i,σf†
i,σfi,σ
−t/summationdisplay
i,σ(f†
i+1,σf†
i+2,−σ+f†
i−1,σf†
i,−σ)fi+1,−σfi,σ
+/summationdisplay
i,σ,σ/primeVσ,σ/primef†
i,σfi,σf†
i+1,σ/primefi+1,σ/prime
+U/summationdisplay
inf
i,σnf
i,−σ, (30)
with U→∞ limit. Here σandσ/primerun over two possible
values. What sort of phases do these Hamiltonians host, andare those phases described by a simple mean-field approach?Does the gravitational clustering of fractons play an importantrole in the phase diagram of this model, such as described inRef. [ 31]? These are important questions left to the future.
We also note that a Hamiltonian of the form Eq. ( 29)
is related to the model considered in Ref. [ 115]. There it
was shown that a strong pair-hopping term may lead to angapless phase [ 116] with properties very different from a
phase dominated by single-particle hopping. The boundariesbetween such a pair-hopping phase at its two edges and thesingle-particle-hopping one host a Majorana mode. This in-
dicates that, at finite fermion densities and when sandwiched
between systems dominated by single-particle hopping, thisfracton model Eq. ( 29) should give rise to topological physics.
For example, consider a one-dimensional polaronic systemcharacterized by strong electron-phonon coupling, such thatthe system is dominated by pair hopping, as discussed above.If such a system is then coupled at its boundaries to an-other one with weak electron-phonon coupling, dominated bysingle-particle hopping, the analysis of Ref. [ 115] reveals that
the resulting boundary must host a robust Majorana mode.At least in this specific instance we therefore conclude thatthe boundaries between a fractonic phase and a nonfractonicphase lead to topologically protected behavior. It is also ofgreat interest to investigate these ideas in the two-speciesfracton model of Eq. ( 30).
It is at present unclear whether there is any deeper con-
nection between fracton physics and topological boundarymodes. It is therefore an important task to find approximateor exact solutions to one-dimensional fracton Hamiltonians,which may yield important insights into the phases of frac-tonic matter and their connection with more conventionalfermion theories.
VII. CONCLUSIONS
In this work we have shown that boson-affected hopping
models, which arise in the study of electron-phonon andmagnetic (holes doped into antiferromagnets) polarons, pro-vide a physical realization of fractons, in either an exact orapproximate way, depending on the details of the specificmodel considered. In these models, individual quasiparticlesare either perfectly or nearly immobile, since single-particlemotion requires the creation of costly bosonic excitations,while bound states of these quasiparticles exhibit no such
214437-12FRACTONS FROM POLARONS PHYSICAL REVIEW B 102, 214437 (2020)
FIG. 7. Feynman diagrams for one- and two-boson processes. A
solid line with an arrow represents the particle, and a wiggly line
represents the boson. Diagrams with crossed boson lines such asthe last diagram in the top panel vanish in the constrained model,
leading to a self-consistent solution similar in spirit to the Born
approach [ 55].
mobility restrictions, closely mirroring the physics of frac-
tons. More concretely, we have shown how boson-affectedhopping models can be mapped explicitly onto a fractonHamiltonian with dipole conservation via perturbatively inte-grating out the mediating bosons. This effective Hamiltoniancontains only pair-hopping terms, corresponding to the mo-tion of dipoles, while all single-particle hopping elements areprecisely zero. This mapping generically holds through sixthorders in perturbation theory, so there is a wide parameterregime in which these systems exhibit approximate fractonbehavior.
We have also shown how to obtain exact fracton behavior
in a one-dimensional boson-affected hopping model by im-posing a mutual hard-core constraint between the fermionsand bosons. Such a constraint is naturally realized in hole-doped mixed-dimensional Ising antiferromagnets, in whichholes in a two-dimensional antiferromagnet are confined toa one-dimensional subspace. In these systems, dipole momentis conserved to all orders in perturbation theory, giving rise toperfect fracton behavior.
Our work identifies boson-affected hopping systems as a
new platform for studying the physics of fractons, present-ing potential for observing their exotic phenomenology inexperiments on polarons, such as [ 106–110]. When fracton
physics is only approximate, we can can regard the systemto correspond to a small perturbation away from a fractonphase, in an appropriate sense. Using this approach we haveshown that many of the reach phenomenology of fractons sur-vive in approximately fractonic setups. We predict polaronswill feature the universal short-range attraction character-istic of fracton systems, arising from a lowered effectivemass in the presence of other fractons. We find that thisuniversal attraction survives in systems featuring only approx-imate fracton behavior, provided the violation is sufficientlyweak. This attraction leaves its mark in phase separation ofholes in doped antiferromagnets. We have also predicted thatboson-affected hopping systems will be slow to reach thermalequilibrium, and we have estimated the corresponding ther-malization time. We conjectured that one-dimensional modelswith perfect fracton behavior, such as the mixed-dimensionalIsing models, will exhibit true localization, even at finitetemperature.
Our results open the door for a productive exchange of
ideas between a range of previously distant fields, such asfractons and polarons, and there are many interesting ques-
tions that remain to be answered. We expect that the manypowerful numerical and analytic tools available for one-dimensional systems may be productively used to study theeffective fracton Hamiltonians arising in boson-affected hop-ping models. How do these models behave at finite densities?Is there some precise connection that can be made withthe physics of quantum disentangled liquids? Will the pair-hopping interactions identified in our effective fracton modelslead to topological edge states? Are there results from the the-ory of polarons that elucidate new fracton phenomenology?These and many other questions can now be formulated inlight of the connections drawn by our work.
ACKNOWLEDGMENTS
We acknowledge useful conversations with Mona Berciu,
Jonathan Ruhman, Anton Andreev, and Pablo Sala. J.S. ac-knowledges the hospitality of the Stewart Blusson QuantumMatter Institute at the University of British Columbia. J.S.acknowledges support from the Deutscher AkademischerAustauschdienst (DAAD) short-term grant and the Natu-ral Sciences and Engineering Research Council of Canada(NSERC). M.P. acknowledges support from the Air ForceOffice of Scientific Research under Award No. FA9550-17-1-0183.
APPENDIX A: PERTURBATIVE CALCULATIONS
We here detail the perturbative calculations described in the
main text, which allows us to obtain effective Hamiltoniansfor the ffermions in the one- and two-particle sectors by
integrating out the bbosons. We consider the simpler case of
a single species of fermions. We rewrite the Hamiltonian ofEq. ( 5) (with μ=0) as
H=H
0+V, (A1)
where we take our unperturbed Hamiltonian as
H0=ωb/summationdisplay
ib†
ibi, (A2)
which has all particles trivially localized. The perturbing in-
teraction takes the form
V=g/summationdisplay
/angbracketlefti,j/angbracketrightf†
ifj(b†
j+bi). (A3)
We perform an expansion in g/ωbintegrating out the
bosons at the given order of perturbation theory. It is obviousthat odd orders in the expansion give zero correction. In sec-ond order, the calculation amounts to studying the effects ofa single mediating boson, while in fourth order it correspondsto the effects of coupling to two bosons. See Fig. 7for the set
of diagrams considered in the calculation.
1. Unconstrained model
We first study the unconstrained model with both fandb
particles allowed on the same site.
214437-13JOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
a. Single-particle Hamiltonian
Using standard perturbation theory techniques [ 117], the
effective single-particle Hamiltonian can be written as
h2nd
1=PV1−P
E0−H0VP, (A4)
where Pis the projector onto states with zero bbosons, E0
is the unperturbed energy, and h2nd
1is evaluated for single-
particle states. To evaluate VPon single-particle states in the
Hilbert space, note that Pprojects out all states with nonzero
bosons, and we are left with
Vf†
i|0/angbracketright=g(f†
i+1+f†
i−1)b†
i|0/angbracketright. (A5)
The 1 −Psimply returns the same state. Acting with H−1
0on
this intermediate state simply returns the same state with aneigenvalue ω
−1
b. Acting with PVthen gives 2 f†
i|0/angbracketright. Putting
everything together we have
h2nd
1=−2g2
ωb/summationdisplay
if†
ifi, (A6)
which features only an on-site energy, without any single-
particle hopping terms. As discussed in the main text,single-particle hopping only appears at sixth order in pertur-bation theory, as seen in Fig. 4.
b. Two-particle Hamiltonian
As in the single-particle case, the two-particle effective
Hamiltonian takes the form [ 117]
h2nd
2=PV1−P
E0−H0VP (A7)
evaluated on the two-particle states of the Hilbert space. We
begin by evaluating on states where the two particles areseparated by one lattice site. The projector Peliminates all
states with nonzero bosons, and we are left with
PV1−P
E0−H0(Vf†
if†
i+1|0/angbracketright)
=−g2
ωdPV/braceleftBig
f†
i−1b†
if†
i+1+fib†
i+1f†
i+2/bracerightBig
|0/angbracketright
=−g2
ωd/braceleftBig
2f†
if†
i+1+(f†
i+1f†
i+2+f†
i−1f†
i)/bracerightBig
|0/angbracketright
=−2g2
ωdf†
if†
i+1|0/angbracketright−g2
ωd(f†
i+1f†
i+2+f†
i−1f†
i)|0/angbracketright
≡−/epsilon10f†
if†
i+1|0/angbracketright−t(f†
i+1f†
i+2+f†
i−1f†
i)|0/angbracketright. (A8)
The first term represents the polaron formation energy for two
particles −2/epsilon10, pushed up by a nearest-neighbor repulsion
J=/epsilon10, while the second term is a pair-hopping interaction
mediated by the bosons.
We can also evaluate h2nd
2for the states f†
if†
i+nforn>1.
In this case the particles are sufficiently far apart that nosingle-boson process can allow interaction between them.This simply generates the polaron renormalization of the en-ergy, as expected, but no extra interactions.Putting all these pieces together, we arrive at an effective
two-particle Hamiltonian of the form
h
2nd
2=−/epsilon10/summationdisplay
if†
ifi−t/summationdisplay
i(f†
i+1f†
i+2+f†
i−1f†
i)fi+1fi
+J/summationdisplay
if†
ifif†
i+1fi+1, (A9)
which is Eq. ( 7) of the main text. This Hamiltonian features
only two-body hopping processes, while single-particle mo-tion is absent (up to sixth order in perturbation theory), leadingto approximate conservation of dipole moment and fractonphenomenology, as discussed earlier.
2. Model with mutual hard-core constraint
As described in the main text, we can eliminate all contri-
butions to single-particle mobility, to all orders in perturbationtheory, by imposing a mutual hard-core constraint betweenthe fermions and the bosons of the theory that forbids thecleaning-up backtracking motion of a single fermion, such asthat of Fig. 4. While this constraint makes single particles
fully immobile, it still permits the mobility of two-particlebound states. Indeed, since the second-order perturbation the-ory analysis in the previous subsubsection did not involve anystates violating the mutual hard-core condition, h
2is identical
with or without this condition, up to second order. However,the hard-core condition allows a simplified analysis of higher-order corrections to this Hamiltonian, which we now calculateto fourth order.
a. Single-particle Hamiltonian
The fourth-order correction to the effective single-particle
Hamiltonian is given by [ 117]
h4th
1=PV1−P
E0−H0V1−P
E0−H0V1−P
E0−H0VP
−1
2/bracketleftBigg
PV/parenleftbigg1−P
E0−H0/parenrightbigg2
VP V1−P
E0−H0VP
+PV1−P
E0−H0VP V/parenleftbigg1−P
E0−H0/parenrightbigg2
VP/bracketrightBigg
.(A10)
Evaluating h4th
1on the single-particle states, we find
h4th
1=3g4
ωb3/summationdisplay
if†
ifi, (A11)
which reflects two types of polaronic renormalization pro-
cesses: f†
i|0/angbracketrightV−→b†
if†
i±1|0/angbracketrightV−→f†
i|0/angbracketrightV−→b†
if†
i±1|0/angbracketrightV−→f†
i|0/angbracketright
and f†
i|0/angbracketrightV−→b†
if†
i±1|0/angbracketrightV−→b†
ib†
i±1f†
i±2|0/angbracketrightV−→b†
if†
i±1|0/angbracketrightV−→
f†
i|0/angbracketright. The calculation formula, Eq. ( A10), keeps track of
these different contributions. Note that the second type ofcontributions corresponds to processes in which the fermioncreates and subsequently absorbs longer two-site stringsof bosons.
214437-14FRACTONS FROM POLARONS PHYSICAL REVIEW B 102, 214437 (2020)
Summarizing, we see that at the fourth order in perturba-
tion theory the polaron formation energy is
/epsilon10=2g2
ωb−3g4
ωb3. (A12)
b. Two-particle Hamiltonian
We now turn to the fourth-order correction to the effective
two-particle Hamiltonian given by [ 117]
h4th
2=PV1−P
E0−H0V1−P
E0−H0V1−P
E0−H0VP
−1
2/bracketleftBigg
PV/parenleftbigg1−P
E0−H0/parenrightbigg2
VP V1−P
E0−H0VP
+PV1−P
E0−H0VP V/parenleftbigg1−P
E0−H0/parenrightbigg2
VP/bracketrightBigg
.(A13)
A lengthy calculation of the action of h4th
2on all two-particle
states results in
h4th
2=3g4
ωb3/summationdisplay
inf
i−3g4
ωb3/summationdisplay
inf
inf
i+1+3g4
ωb3/summationdisplay
inf
inf
i+2
+2g4
ωb3/summationdisplay
i(f†
i+1f†
i+2+f†
i−1f†
i)fi+1fi
+1
2g4
ωb3/summationdisplay
i(f†
i+2f†
i+3+f†
i−2f†
i−1)fi+1fi. (A14)
The second term is a nearest-neighbor interaction that coun-
teracts the polaronic renormalization with an origin similarto that discussed in the last section. Note that longer-rangenext-nearest-neighbor density-density and pair-hopping in-teractions appear first at this order. The dipole-conservinglong-range pair hopping occur as a result of a process inwhich, for example, one fermion leaves a two-site string ofbosons for its partner to absorb, allowing the pair to moveover by two sites.
Putting everything together we obtain
h
2=h2nd
2+h4th
2
=−/epsilon10/summationdisplay
inf
i+Jz1/summationdisplay
inf
inf
i+1+Jz2/summationdisplay
inf
inf
i+2
−t1/summationdisplay
i(f†
i+1f†
i+2+f†
i−1f†
i)fi+1fi
+t2/summationdisplay
i(f†
i+2f†
i+3+f†
i−2f†
i−1)fi+1fi, (A15)
which is the effective Hamiltonian of Eq. ( 13) and the coeffi-
cients that appeared at the second order are now renormalizedby additive factors ∝g
4/ωb3at this fourth order.
Crucially, as we discuss in the main text, the mutual hard-
core constraint forbids cleaning-up backtracking motion, andthus always ensures bosons are created one at a site in stringconfigurations. This ensures perfect polaron immobility, but
does not affect the string-mediated dipole-conserving bipo-laron mobility.
From this we infer the behavior to all orders of perturbation
theory in Eq. ( 14) of the main text. We expect the physics to
hold generally even in the limit of large number of bosons, aseach site will still host at most a single boson and the stringconfigurations will just simply become longer.
APPENDIX B: EQUATION OF MOTION FOR DIPOLES
In a Hamiltonian exhibiting fracton physics, such as that
of Eq. ( 7), the individual fermions are localized (forming
polarons) and have no dispersion. However, bound states ofpairs of fermions, i.e., bipolarons, have a nontrivial disper-sion. We wish to find the equation of motion [ 118] for such
bound states. We do so by solving for the Green’s functionG(ω)=
1
ω+iη−H|η→0+. We first take advantage of the identity
G(ω)[ω+iη−H]=1, evaluating its expectation value in a
set of normalized basis states. To this end we define momen-tum states as
|K,n/angbracketright=1
√
N/summationdisplay
ieiK(Ri+n/2)f†
if†
i+n|0/angbracketright, (B1)
with n>0, and K=k1+k2is the dipole (bipolaron) momen-
tum. We derive the equations of motion by evaluating
/angbracketleftK,1|G(ω)[ω+iη−h2]|K,n/angbracketright
=g(ω,K,n)−/angbracketleftK,1|G(ω)h2|K,n/angbracketright=δn,1,(B2)
where we have defined g(ω,K,n)=/angbracketleftK,1|G(ω)|K,n/angbracketright.T o
proceed, we compute the action on the states |K,n/angbracketrightof the
Hamiltonian h2, working out term by term:
−/epsilon10/summationdisplay
if†
ifi|K,n/angbracketright=− 2/epsilon10|K,n/angbracketright, (B3)
−t/summationdisplay
i(f†
i+1f†
i+2+f†
i−1f†
i)fi+1fi|K,n/angbracketright
=− 2tcos(K)|K,1/angbracketrightδn,1, (B4)
and
J/summationdisplay
if†
ifif†
i+1fi+1|K,n/angbracketright=J|K,1/angbracketrightδn,1. (B5)
Since |K,1/angbracketrightis an eigenstate of each of the terms in h2,i ti s
also an eigenstate of h2, and we can immediately identify the
dipole Green’s function g(ω,K,1) from Eq. ( B2)a s
g(ω,K,1)=1
ω+iη+2/epsilon10−J+2tcos(K). (B6)
The two-polaron bound state energy corresponds to the pole
ofg(ω,K,1), and we find the dipole (bipolaron) dispersion:
EBP(K)=−2/epsilon10+J−2tcos(K), (B7)
as stated in the main text.
[1] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and
S. Das Sarma, Non-Abelian anyons and topologicalquantum computation, Rev. Mod. Phys. 80, 1083
(2008) .
214437-15JOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
[2] S. J. Devitt, W. J. Munro, and K. Nemoto, Quantum er-
ror correction for beginners, Rep. Prog. Phys. 76, 076001
(2013) .
[3] T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers,
Quantum Hall physics: Hierarchies and conformal field theorytechniques, Rev. Mod. Phys. 89, 025005 (2017) .
[4] P. W. Anderson, The resonating valence bond state in La
2CuO 4
and superconductivity, Science 235, 1196 (1987) .
[5] S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, Topology of
the resonating valence-bond state: Solitons and high- Tcsuper-
conductivity, Phys. Rev. B 35, 8865(R) (1987) .
[6] M. Berciu and S. John, Quantum dynamics of charged and
neutral magnetic solitons: Spin-charge separation in the one-dimensional Hubbard model, Phys. Rev. B 61, 10015 (2000) .
[7] E. Demler, C. Nayak, H.-Y . Kee, Y . B. Kim, and T. Senthil,
Fractionalization patterns in strongly correlated electron sys-tems: Spin-charge separation and beyond, Phys. Rev. B 65,
155103 (2002) .
[8] M. Pretko, Subdimensional particle structure of higher rank
U(1) spin liquids, Phys. Rev. B 95, 115139 (2017) .
[9] M. Pretko, Generalized electromagnetism of subdimensional
particles, Phys. Rev. B 96, 035119 (2017) .
[10] M. Pretko, Higher-spin Witten effect and two-dimensional
fracton phases, Phys. Rev. B 96, 125151 (2017) .
[11] H. Ma, M. Hermele, and X. Chen, Fracton topological order
from Higgs and partial confinement mechanisms of rank-twogauge theory, Phys. Rev. B 98, 035111 (2018) .
[12] D. Bulmash and M. Barkeshli, The Higgs mechanism in
higher-rank symmetric U(1) gauge theories, Phys. Rev. B 97,
235112 (2018) .
[13] J. Haah, Local stabilizer codes in three dimensions without
string logical operators, Phys. Rev. A 83, 042330 (2011) .
[14] S. Bravyi and J. Haah, Quantum Self-Correction in the 3D
Cubic Code Model, Phys. Rev. Lett. 111, 200501 (2013) .
[15] B. Terhal, Quantum error correction for quantum memories,
Rev. Mod. Phys. 87, 307 (2015) .
[16] M. Pretko and L. Radzihovsky, Fracton-Elasticity Duality,
P h y s .R e v .L e t t . 120, 195301 (2018) .
[17] A. Gromov, Chiral Topological Elasticity and Fracton Order,
P h y s .R e v .L e t t . 122, 076403 (2019) .
[18] S. Pai and M. Pretko, Fractonic line excitations: An inroad
from three-dimensional elasticity theory, Phys. Rev. B 97,
235102 (2018) .
[19] M. Pretko and L. Radzihovsky, Symmetry Enriched Fracton
Phases from Supersolid Duality, Phys. Rev. Lett. 121, 235301
(2018) .
[20] A. Kumar and A. C. Potter, Symmetry-enforced fractonicity
and two-dimensional quantum crystal melting, P h y s .R e v .B
100, 045119 (2019) .
[21] C. Chamon, Quantum Glassiness in Strongly Correlated Clean
Systems: An Example of Topological Overprotection, Phys.
Rev. Lett. 94, 040402 (2005) .
[22] A. Prem, J. Haah, and R. Nandkishore, Glassy quantum dy-
namics in translation invariant fracton models, Phys. Rev. B
95, 155133 (2017) .
[23] S. Pai, M. Pretko, and R. M. Nandkishore, Localization in
Fractonic Random Circuits, Phys. Rev. X 9, 021003 (2019) .
[24] S. Pai and M. Pretko, Dynamical Scar States in Driven Fracton
Systems, Phys. Rev. Lett. 123, 136401 (2019) .[25] P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F.
Pollmann, Ergodicity-Breaking Arising from Hilbert SpaceFragmentation in Dipole-Conserving Hamiltonians, Phys.
Rev. X 10, 011047 (2020) .
[26] V . Khemani, M. Hermele, and R. Nandkishore, Localization
from Hilbert space shattering: From theory to physical real-izations, P h y s .R e v .B 101, 174204 (2020) .
[27] S. Vijay, J. Haah, and L. Fu, A new kind of topological
quantum order: A dimensional hierarchy of quasiparticles builtfrom stationary excitations, P h y s .R e v .B 92, 235136 (2015) .
[28] S. Vijay, J. Haah, and L. Fu, Fracton topological order, gen-
eralized lattice gauge theory and duality, P h y s .R e v .B 94,
235157 (2016) .
[29] W. Shirley, K. Slagle, Z. Wang, and X. Chen, Fracton Models
on General Three-Dimensional Manifolds, Phys. Rev. X 8,
031051 (2018) .
[30] S. Pai and M. Hermele, Fracton fusion and statistics, Phys.
Rev. B 100, 195136 (2019) .
[31] A. Prem, M. Pretko, and R. Nandkishore, Emergent phases of
fractonic matter, P h y s .R e v .B 97, 085116 (2018) .
[32] H. Ma and M. Pretko, Higher rank deconfined quantum crit-
icality and the exciton Bose condensate, Phys. Rev. B 98,
125105 (2018) .
[33] M. Pretko, Emergent gravity of fractons: Mach’s principle
revisited, P h y s .R e v .D 96, 024051 (2017) .
[34] H. Yan, Fracton topological order and holography, Phys. Rev.
B99, 155126 (2019) .
[35] R. M. Nandkishore and M. Hermele, Fractons, Annu. Rev.
Condens. Matter Phys. 10, 295 (2019) .
[36] K. Slagle and Y . B. Kim, Quantum field theory of X-cube frac-
ton topological order and robust degeneracy from geometry,Phys. Rev. B 96, 195139 (2017) .
[37] H. Ma, E. Lake, X. Chen, and M. Hermele, Fracton topological
order via coupled layers,
P h y s .R e v .B 95, 245126 (2017) .
[38] S. Vijay, Isotropic layer construction and phase diagram for
fracton topological phases, arXiv:1701.00762 .
[39] H. Ma, A. T. Schmitz, S. A. Parameswaran, M. Hermele,
and R. M. Nandkishore, Topological entanglement entropy offracton stabilizer codes, P h y s .R e v .B 97, 125101 (2018) .
[40] A. T. Schmitz, H. Ma, R. M. Nandkishore, and S. A.
Parameswaran, Recoverable information and emergent con-servation laws in fracton stabilizer codes, P h y s .R e v .B 97,
134426 (2018) .
[41] A. Prem, S. Vijay, Y .-Z. Chou, M. Pretko, and R. M.
Nandkishore, Pinch point singularities of tensor spin liquids,Phys. Rev. B 98, 165140 (2018) .
[42] T. Devakul, Y . You, F. J. Burnell, and S. L. Sondhi, Fractal
symmetric phases of matter, SciPost Phys. 6, 007 (2019) .
[43] D. J. Williamson, Z. Bi, and M. Cheng, Fractonic matter
in symmetry-enriched U(1) gauge theory, Phys. Rev. B 100,
125150 (2019) .
[44] A. Gromov, Towards Classification of Fracton Phases: The
Multipole Algebra, P h y s .R e v .X 9, 031035 (2019) .
[45] Y . You and F. von Oppen, Majorana quantum Lego, a route
towards fracton matter, P h y s .R e v .R e s . 1, 013011 (2019) .
[46] A. T. Schmitz, Gauge structures: From stabilizer codes to
continuum models, Ann. Phys. 410, 167927 (2019) .
[47] A. Prem, S.-J. Huang, H. Song, and M. Hermele, Cage-Net
Fracton Models, P h y s .R e v .X 9, 021010 (2019) .
214437-16FRACTONS FROM POLARONS PHYSICAL REVIEW B 102, 214437 (2020)
[48] H. Song, A. Prem, S.-J. Huang, and M. A. Martin-Delgado,
Twisted fracton models in three dimensions, Phys. Rev. B 99,
155118 (2019) .
[49] Y . You, T. Devakul, F. J. Burnell, and S. L. Sondhi, Symmetric
fracton matter: Twisted and enriched, Ann. Phys. 416, 168140
(2020) .
[50] K. Slagle, D. Aasen, and D. Williamson, Foliated field theory
and string-membrane-net condensation picture of fracton or-der,SciPost Phys. 6, 043 (2019) .
[51] H. Yan, O. Benton, L. D. C. Jaubert, and N. Shannon, Rank-2
U(1) Spin Liquid on the Breathing Pyrochlore Lattice, Phys.
Rev. Lett. 124, 127203 (2020) .
[52] Y . You, Non-Abelian defects in fracton phase of matter, Phys.
Rev. B 100, 075148 (2019) .
[53] A. Dua, D. J. Williamson, J. Haah, and M. Cheng, Compactify-
ing fracton stabilizer models, Phys. Rev. B 99, 245135 (2019) .
[54] T. Wang, W. Shirley, and X. Chen, Foliated fracton order in
the Majorana checkerboard model, Phys. Rev. B 100, 085127
(2019) .
[55] J. Sous and M. Pretko, Fractons from frustration in hole-doped
antiferromagnets, npj Quantum Mater. 5, 81 (2020) .
[56] S. A. Trugman, Interaction of holes in a Hubbard antiferro-
magnet and high-temperature superconductivity, Phys. Rev. B
37, 1597 (1988) .
[57] B. I. Shraiman and E. D. Siggia, Mobile Vacancies in a Quan-
tum Heisenberg Antiferromagnet, Phys. Rev. Lett. 61, 467
(1988) .
[58] C. L. Kane, P. A. Lee, and N. Read, Motion of a single hole in
a quantum antiferromagnet, Phys. Rev. B 39, 6880 (1989) .
[59] S. Sachdev, Hole motion in a quantum Néel state, Phys. Rev.
B39, 12232 (1989) .
[60] A. L. Chernyshev and P. W. Leung, Holes in the t-J
zmodel: A
diagrammatic study, Phys. Rev. B 60, 1592 (1999) .
[61] B. I. Shraiman and E. D. Siggia, Two-Particle Excitations in
Antiferromagnetic Insulators, Phys. Rev. Lett. 60, 740 (1988) .
[62] Yu. A. Dimashko, The t-Jzmodel: Two holes pairing by means
of one string, Physica C: Supercond. 206, 393 (1993) .
[63] L. Vidmar and J. Bon ˇca, Two holes in the t-Jmodel form a
bound state for any nonzero J/t,J. Supercond. Nov. Magn. 26,
2641 (2013) .
[ 6 4 ]K .B i e n i a s z ,P .W r z o s e k ,A .M .O l é s ,a n dK .W o h l f e l d ,F r o m
“weak” to “strong” electron localization in a Mott insulator,SciPost Phys. 7, 066 (2019) .
[65] A. S. Alexandrov and N. F. Mott, Polarons & Bipolarons
(World Scientific, Singapore, 1995).
[66] F. C. Zhang and T. M. Rice, Effective Hamiltonian for the
superconducting Cu oxides, Phys. Rev. B 37, 3759 (1988) .
[67] H. Eskes and G. A. Sawatzky, Tendency Towards Local Spin
Compensation of Holes in the High- T
cCopper Compounds,
P h y s .R e v .L e t t . 61, 1415 (1988) .
[68] M. Berciu, I. Elfimov, and G. A. Sawatzky, Electronic po-
larons and bipolarons in iron-based superconductors: The roleof anions, P h y s .R e v .B 79, 214507 (2009) .
[69] F. Trousselet, P. Horsch, A. M. Ole ´s, and W.-L. You, Hole
propagation in the Kitaev-Heisenberg model: From quasipar-ticles in quantum Néel states to non-Fermi liquid in the Kitaevphase, Phys. Rev. B 90, 024404 (2014) .
[70] P. Massignan, M. Zaccanti, and G. M. Bruun, Polarons,
dressed molecules and itinerant ferromagnetism in ultracoldfermi gases, Rep. Prog. Phys. 77, 034401 (2014) .[71] R. Schmidt, M. Knap, D. A. Ivanov, J.-S. You, M. Cetina,
and E. Demler, Universal many-body response of heavy im-purities coupled to a Fermi sea, Rep. Prog. Phys. 81, 024401
(2018) .
[72] D. M. Edwards, A quantum phase transition in a model with
boson-controlled hopping. Physica B: Condensed Matter 378–
380, 133 (2006) .
[73] A. Alvermann, D. M. Edwards, and H. Fehske, Boson-
Controlled Quantum Transport, Phys. Rev. Lett. 98, 056602
(2007) .
[74] G. Wellein, H. Fehske, A. Alvermann, and D. M. Edwards,
Correlation-Induced Metal Insulator Transition in a Two-Channel Fermion-Boson Model, Phys. Rev. Lett. 101, 136402
(2008) .
[75] S. Ejima, G. Hager, and H. Fehske, Quantum Phase Transition
in a 1D Transport Model with Boson-Affected Hopping: Lut-tinger Liquid Versus Charge-Density-Wave Behavior, Phys.
Rev. Lett. 102, 106404 (2009) .
[76] M. Berciu and H. Fehske, Momentum average approximation
for models with boson-modulated hopping: Role of closedloops in the dynamical generation of a finite quasiparticlemass, Phys. Rev. B 82, 085116 (2010) .
[77] S. Ejima, S. Sykora, K. W. Becker, and H. Fehske, Phase
separation in the Edwards model, Phys. Rev. B 86, 155149
(2012) .
[78] D.-N. Cho, J. van den Brink, H. Fehske, K. W. Becker, and
S. Sykora, Unconventional superconductivity and interactioninduced Fermi surface reconstruction in the two-dimensionalEdwards model, Sci. Rep. 6, 22548 (2016) .
[79] T. Holstein, Studies of polaron motion: Part II. The “small”
polaron, Ann. Phys. 8, 343 (1959) .
[80] D. J. J. Marchand, G. De Filippis, V . Cataudella, M. Berciu,
N. Nagaosa, N. V . Prokof’ev, A. S. Mishchenko, and P. C. E.Stamp, Sharp Transition for Single Polarons in the One-Dimensional Su-Schrieffer-Heeger Model, Phys. Rev. Lett.
105, 266605 (2010) .
[81] J. Sous, Peierls bipolarons and localization in solid-state and
molecular systems, Ph.D. thesis, The University of BritishColumbia, 2018.
[82] G. Martinez and P. Horsch, Spin polarons in the t-Jmodel,
Phys. Rev. B 44, 317 (1991)
.
[83] F. Grusdt, M. Kánasz-Nagy, A. Bohrdt, C. S. Chiu, G. Ji, M.
Greiner, D. Greif, and E. Demler, Parton Theory of MagneticPolarons: Mesonic Resonances and Signatures in Dynamics,Phys. Rev. X 8, 011046 (2018) .
[84] L. M. Falicov and J. C. Kimball, Simple Model for
Semiconductor-Metal Transitions: SmB
6and Transition-
Metal Oxides, Phys. Rev. Lett. 22, 997 (1969) .
[85] S. Bariši ´c, J. Labbé, and J. Friedel, Tight Binding and
Transition-Metal Superconductivity, Phys. Rev. Lett. 25, 919
(1970) .
[86] S. Bariši ´c, Rigid-atom electron-phonon coupling in the tight-
binding approximation. I, Phys. Rev. B 5, 932 (1972) .
[87] S. Bariši ´c, Self-consistent electron-phonon coupling in the
tight-binding approximation. II, P h y s .R e v .B 5, 941 (1972) .
[88] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in Poly-
acetylene, Phys. Rev. Lett. 42, 1698 (1979) .
[89] A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su,
Solitons in conducting polymers, Rev. Mod. Phys. 60, 781
(1988) .
214437-17JOHN SOUS AND MICHAEL PRETKO PHYSICAL REVIEW B 102, 214437 (2020)
[90] J. Sous, M. Chakraborty, R. V . Krems, and M. Berciu, Light
Bipolarons Stabilized by Peierls Electron-Phonon Coupling,P h y s .R e v .L e t t . 121, 247001 (2018) .
[91] M. Berciu and H. Fehske, Aharonov-Bohm interference
for a hole in a two-dimensional Ising antiferromagnet ina transverse magnetic field, P h y s .R e v .B 84, 165104
(2011) .
[92] J. Sous, M. Chakraborty, C. P. J. Adolphs, R. V . Krems, and
M. Berciu, Phonon-mediated repulsion, sharp transitions and(quasi)self-trapping in the extended Peierls-Hubbard model,Sci. Rep. 7, 1169 (2017) .
[93] J. Sous, M. Berciu, and R. V . Krems, Bipolarons bound
by repulsive phonon-mediated interactions, P h y s .R e v .A 96,
063619 (2017) .
[94] F. Grusdt, Z. Zhu, T. Shi, and E. Demler, Meson forma-
tion in mixed-dimensional t-Jmodels, SciPost Phys. 5, 057
(2018) .
[95] H. Labuhn, D. Barredo, S. Ravets, S. de Leseleuc, T. Macri,
T. Lahaye, and A. Browaeys, Tunable two-dimensional arraysof single Rydberg atoms for realizing quantum Ising models,Nature (London) 534, 667 (2016) .
[96] J. Zeiher, R. van Bijnen, P. Schausz, S. Hild, J.-y. Choi,
T. Pohl, I. Bloch, and C. Gross, Many-body interferome-try of a Rydberg-dressed spin lattice, Nat. Phys. 12, 1095
(2016) .
[97] J. Zeiher, J.-y. Choi, A. Rubio-Abadal, T. Pohl, R. van Bijnen,
I. Bloch, and C. Gross, Coherent Many-Body Spin Dynamicsin a Long-Range Interacting Ising Chain, Phys. Rev. X 7,
041063 (2017) .
[98] D. Porras and J. I. Cirac, Effective Quantum Spin Systems with
Trapped Ions, Phys. Rev. Lett. 92, 207901 (2004) .
[99] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K.
Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, En-gineered two-dimensional Ising interactions in a trapped-ionquantum simulator with hundreds of spins, Nature (London)
484, 489 (2012) .
[100] R. Barnett, D. Petrov, M. Lukin, and E. Demler, Quantum
Magnetism with Multicomponent Dipolar Molecules in anOptical Lattice, P h y s .R e v .L e t t . 96, 190401 (2006) .
[101] A. V . Gorshkov, S. R. Manmana, G. Chen, J. Ye, E. Demler,
M. D. Lukin, and A. M. Rey, Tunable Superfluidity and Quan-tum Magnetism with Ultracold Polar Molecules, Phys. Rev.
Lett. 107, 115301 (2011) .
[102] M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L.
Pollet, I. Bloch, and C. Gross, Spin- and density-resolved mi-croscopy of antiferromagnetic correlations in Fermi-Hubbardchains, Science 353, 1257 (2016) .
[103] L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan,
H. Zhang, E. Khatami, N. Trivedi, T. Paiva, M. Rigol, andM. W. Zwierlein, Observation of spatial charge and spin cor-relations in the 2D Fermi-Hubbard model, Science 353, 1260
(2016) .
[104] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanasz-
Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M.Greiner, A cold-atom Fermi-Hubbard antiferromagnet, Nature
(London) 545, 462 (2017) .
[105] W. P. Wolf, The Ising model and real magnetic materials,
Brazilian J. Phys. 30, 794 (2000) .
[106] T. Dahm, V . Hinkov, S. V . Borisenko, A. A. Kordyuk, V . B.
Zabolotnyy, J. Fink, B. Büchner, D. J. Scalapino, W. Hanke,and B. Keimer, Strength of the spin-fluctuation-mediated pair-ing interaction in a high-temperature superconductor, Nat.
Phys. 5, 217 (2009) .
[107] S. Dal Conte, L. Vidmar, D. Golež, M. Mierzejewski, G.
Soavi, S. Peli, F. Banfi, G. Ferrini, R. Comin, B. M. Ludbrook,L. Chauviere, N. D. Zhigadlo, H. Eisaki, M. Greven, S. Lupi,A. Damascelli, D. Brida, M. Capone, J. Bon ˇca, G. Cerullo, and
C. Giannetti, Snapshots of the retarded interaction of chargecarriers with ultrafast fluctuations in cuprates, Nat. Phys. 11,
421 (2015) .
[108] T. A. Hilker, G. Salomon, F. Grusdt, A. Omran, M. Boll, E.
Demler, I. Bloch, and C. Gross, Revealing hidden antifer-romagnetic correlations in doped Hubbard chains via stringcorrelators, Science 357, 484 (2017) .
[109] C. S. Chiu, G. Ji, A. Bohrdt, M. Xu, M. Knap, E. Demler, F.
Grusdt, M. Greiner, and D. Greif, String patterns in the dopedHubbard model, Science 365, 251 (2019) .
[110] J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A. Hilker,
E. Demler, G. Salomon, I. Bloch, and C. Gross, Imagingmagnetic polarons in the doped Fermi-Hubbard model, Nature
(London) 572, 358 (2019) .
[111] V . J. Emery, S. A. Kivelson, and H. Q. Lin, Phase
Separation in the t-JModel, P h y s .R e v .L e t t . 64, 475
(1990) .
[112] M. Marder, N. Papanicolaou, and G. C. Psaltakis, Phase sepa-
r a t i o ni na t-Jmodel, P h y s .R e v .B 41, 6920 (1990) .
[113] C. D. Batista and G. Ortiz, Quantum Phase Diagram of the t-J
z
Chain Model, P h y s .R e v .L e t t . 85, 4755 (2000) .
[114] T. Grover and M. P. A. Fisher, Quantum disentangled liquids,
J. Stat. Mech. (2014) P10010 .
[115] J. Ruhman and E. Altman, Topological degeneracy and pairing
in a one-dimensional gas of spinless fermions, Phys. Rev. B
96, 085133 (2017) .
[116] The strong-pairing phase is gapless in the charge sector, but
gapped to single fermion excitations.
[117] M. Takahashi, Half-filled Hubbard model at low temperature,
J. Phys. C: Solid State Phys. 10, 1289 (1977) .
[118] M. Berciu, Few-Particle Green’s Functions for Strongly Cor-
related Systems on Infinite Lattices, P h y s .R e v .L e t t . 107,
246403 (2011) .
214437-18 |
PhysRevB.95.045422.pdf | PHYSICAL REVIEW B 95, 045422 (2017)
Thermoelectric transport in monolayer phosphorene
Moslem Zare,1Babak Zare Rameshti,1,*Farnood G. Ghamsari,1,2and Reza Asgari1,3
1School of Physics, Institute for Research in Fundamental Sciences (IPM), 19395-5531, Tehran, Iran
2Department of Physics, Kharazmi University, 15719-14911, Tehran, Iran
3School of Nano Science, Institute for Research in Fundamental Sciences (IPM), 19395-5531, Tehran, Iran
(Received 25 August 2016; revised manuscript received 29 November 2016; published 24 January 2017)
We apply the generalized Boltzmann theory to describe thermoelectric transport properties of monolayer
phosphorene in the presence of short- and long-range charged impurity interactions. First, we propose a low-energy Hamiltonian to explore the accurate electronic band structure of phosphorene in comparison with thoseresults obtained by density-functional simulations. We explain the effect of the coupling between the conductionand valence bands on the thermoelectric properties. We show that the electric conductivity of phosphorene ishighly anisotropic, while the Seebeck coefficient and figure of merit, without being influenced via either thepresence or absence of the coupling term, are nearly isotropic. Furthermore, we demonstrate that the conductivityfor the ntype of doping is more influenced by the coupling term than that of the ptype. Along with thermopower
sign change, profound thermoelectric effects can be achieved.
DOI: 10.1103/PhysRevB.95.045422
I. INTRODUCTION
Thermoelectric materials, based on a fundamental inter-
play between their electronic and thermal properties, haveattracted much interest for application in energy conversiondevices [ 1–11]. The efficiency of the thermoelectric devices
is quantified by a dimensionless figure of merit ZT, which
relates the Seebeck coefficient (thermopower) to the thermalconductivity. The small thermal conductivity and relativelyhigh thermopower and electrical conductivity are requiredfor high efficiency thermoelectric materials. Even if theSeebeck coefficient becomes large, a heat current inevitablyaccompanies a temperature gradient and thus makes a tradeoff.The main stream to prevail this issue is based on othermaterials with high power factor, such as doped narrow-gapsemiconductors [ 4–6], or on nanostructuring, such as PbTe
(1.5n m )/Pb
0.927Eu0.073Te (45 nm) multiple quantum well
[1,2] and Bi 2Te3[3]. It is well understood that this efficiency
improvement is due to the sharp peaked electronic density
of states (DOS) in low-dimensional materials [ 1,12], which
is the optimal way toward high thermoelectric efficiency[13]. Low dimensional systems could have dramatically larger
ZTvalues than the corresponding bulk materials because of
decreased thermal conductivity caused by phonon boundaryscattering and improved power factors on account of quantumconfinement. Although the efficiency is largely enhancedvia dimensionality reduction, however, it typically affectselectronic properties of conventional materials. Large efforts inimproving thermoelectric performance target energy filtering,which provides a way to increase the Seebeck coefficient byintroducing a strongly energy-dependent scattering mecha-nism [ 13–16]. Recent advances in fabrication technologies
have made exploring two-dimensional materials possible forthermoelectric applications [ 7–11].
Recently, isolated two-dimensional black phosphorus (BP),
known as phosphorene with a puckered structure, receivedtremendous interest owing to its extraordinary electronic and
*b.zare.r@ipm.iroptical properties in engineering applications [ 17–20]. The
optical and transport properties of monolayer of BP exhibitstrong in-plane anisotropy as bulk BP for two distinct zigzagand armchair directions. These anisotropic features mostlyoriginate from anisotropic bands, like silicon [ 21]. A nearly di-
rect band gap of BP increases with decreasing number of layersfrom 0 .3e Vi nb u l kt o0 .8e V<E
g<2 eV for a monolayer
[22–29]. According to theoretical predictions, phosphorene
has a high carrier mobility of around 1000 cm2V−1s−1[30]
and a high on/off ratio of 104in phosphorene field-effect
transistor at room temperature [ 31]. Besides the bulk BP, it has
also predicted that the phosphorene may have unique potentialthermoelectric applications [ 32–40].
In this paper, we first propose an accurate low-energy
model Hamiltonian protected all needed symmetries andcompare that with those that appeared in literature. Then, weinvestigate the electronic contribution to the thermoelectrictransport of the monolayer of phosphorene. We consider aphosphorene sheet in diffusive transport regime when thermalgradients and bias voltages are applied to the system. Thegeneralized Boltzmann transport equation is applied to obtainthe conductivity, Seebeck coefficient, and the figure of merit.Moreover, the diffusive transport coefficients are calculated byconsidering a short-range potential and a long-range charge-charge Coulomb potential with a Thomas-Fermi screeningas the source of scattering. Our calculations show thatalthough the electrical conductivity of phosphorene is highlyanisotropic, the Seebeck coefficient and the correspondingfigure of merit are nearly isotropic. The figure of merit, whichis a measure of thermoelectric efficiency, reaches to ∼1.2a t
low temperatures, irrespective of the underlying scatteringmechanisms. We also investigate the effect of the interbandcoupling term on thermoelectric transport coefficients. Theseresults propose that a monolayer of phosphorene could be apromising material for the thermoelectric applications.
This paper is organized as follows. In Sec. II, we first
introduce the system and then explain the method whichis used to calculate the conductivity and thermoelectriccoefficients using the generalized Boltzmann method. InSec. III, we present and describe our numerical results for the
2469-9950/2017/95(4)/045422(9) 045422-1 ©2017 American Physical SocietyZARE, RAMESHTI, GHAMSARI, AND ASGARI PHYSICAL REVIEW B 95, 045422 (2017)
conductivity and thermoelectric coefficients for phosphorene.
Finally, we conclude and summarize our main results inSec. IV.
II. MODEL AND BASIC FORMALISM
A. Hamiltonian of monolayer phosphorene
Phosphorene has an orthorhombic puckered structure. The
lattice constants of the conventional unit cell, considering fouratoms per unit cell in x(armchair) and y(zigzag) directions,
are respectively a
x=4.63˚A and ay=3.3˚A. Notice that the
primitive unit cell’s lattice constants are a0
x/y=ax/y/2. The
spin degeneracy of the system is gs=2 and possesses no
valley degeneracy.
We consider a monolayer of phosphorene at low tempera-
ture. The electronic band structure of phosphorene has beencalculated based on
V ASP package density-functional theory
[41]. The V ASP package provides the first conduction band
in the vicinity of the /Gamma1point, which is the exact position of
the conduction band minimum. However, the V ASP package
suggests a slightly indirect band gap with its actual valancemaximum occurring along the /Gamma1-Yhigh symmetry line [ 42].
Having ignored this slight shift, we can write down a low-energy model Hamiltonian. For this purpose, the electronicband structure basically could be described by a four-bandmodel in the tight-binding model, however, it can be expressedby a two-band model owing to the C
2hpoint group invariance.
Expanding the tight-binding model [ 25,43] around the /Gamma1point,
one obtains the low-energy k·pmodel of phosphorene [ 44]
as,
Heff=/parenleftBigg
Ec+ηck2
x+νck2
y γkx+αk2
x+βk2
y
γkx+αk2
x+βk2
y Ev−ηvk2
x−νvk2
y/parenrightBigg
(1)
in the conduction and valence band basis. Parameter αis
usually ignored because of the existence of the linear leadingorder term γk
x. Odd crosses terms of momentum components
in the dispersion relation due to simultaneously nonzero γ
andβbreak the time reversal (TR) symmetry. Within the
L¨owdin partitioning procedure the γkxterm comes from
the unperturbed Hamiltonian [ 44] and must be valid. We
obtain the TR invariant low-energy Hamiltonian of monolayerphosphorene as
H
eff=/parenleftBigg
Ec+ηck2
x+νck2
y γkx
γkx Ev−ηvk2
x−νvk2
y/parenrightBigg
,(2)
where Ec(v)is the band edge at the /Gamma1point with direct
energy gap Eg=Ec−Ev, and the off-diagonal γkxelement
is the interband coupling term with the real parameter γ.
Other parameters can be extracted from the knowledge ofDFT results [ 41] where we have E
g=0.912 eV , ηc=0.008,
ηv=0.038 in units of eV nm2,νc=0.030 and νv=0.005
in units of eV nm2which implies that effective masses
have values mcx=0.146,mvx=0.131,mcy=1.240 and
mvy=7.857 in units of the electron bare mass m0. Notice
that the hole mass along the zigzag ( ydirection) is much
(almost 10 times) greater than that along the armchair ( x
direction) which induces strong in-plane anisotropy. The onlyparameter which remains to identify is the γ, and we find
γ=0.480 eV nm, by fitting the low-energy dispersion ofFIG. 1. Phosphorene band energy dispersion along the Y-/Gamma1-X
direction in the Brillouin zone. The conduction and valence bandsare compared with those theoretical works in Refs. [ 45]a n d[ 44]a n d
with simulation results obtained within DFT-V ASP (black curves) in
Ref. [ 41].
the model Hamiltonian to that obtained by DFT-V ASP results.
Notice that due to the time-reversal symmetry, the off-diagonalterm includes only γk
xand other terms like βk2
ymight be
zero.
As discussed in Ref. [ 30] based on the symmetry of the
system, it is necessary to have a finite value of the νvalbeit
it is zero in Ref. [ 44]. Remarkably, due to the time-reversal
symmetry βpossibly being zero, however, it is finite in the
other parameterized Hamiltonian.
We demonstrate the band dispersion of the conduction
(upper panel) and valence bands (lower panel) in Fig. 1where
we compare our results with theoretical works in Refs. [ 44] and
[45] and V ASP-DFT simulations [ 41]. All parameters are given
in Table I. In the vicinity of the /Gamma1point, all discussed models
capture the physics of the low energy along one directionof the momentum. In the 2D case, the isofrequency profilesare obtained by horizontally cutting the dispersion surfaceseparately calculated by means of a plane wave model. Weillustrate an isofrequency contour surface in the kspace to
explore their symmetries for E
s(/vectorkF)=0.05–0.5 eV with a step
of 0.05 eV in both the electron and hole doped cases in Fig. 2.
The first and second rows (a) and (b) refer to the Fermi surfaceswith parameters used in Refs. [ 45] and [ 44], respectively.
As we stated before, the mass values in Ref. [ 45] are not
entirely suitable for phosphorene, although the isofrequencycounter Fermi surfaces are quite like elliptic structure forE
s(/vectorkF)<0.5 eV and predict that the interband coupling term
can be ignored. This is also the case in the dispersion relationstructure of proposed low-energy Hamiltonian in Ref. [ 25].
045422-2THERMOELECTRIC TRANSPORT IN MONOLAYER PHOSPHORENE PHYSICAL REVIEW B 95, 045422 (2017)
TABLE I. The effective band masses, mcx,mcy,mvx,a n dmvy(in units of the electron bare mass m0), gap energy Eg(in units of eV), and
γ=ϑa0
x/π(in units of eV nm) where ϑ=4 or 6.85, β=θ(a0
x/π)2(in units of eV nm2)w h e r e θ=2o r7 , ηs=η0/msx−γ2/Eg(in units of
eV nm2)a n dν(in units of eV nm2) based of theoretical works in Refs. [ 45]a n d[ 44]. The values of ϑandθdiffer in those references. In this
work we use parameters presented in Ref. [ 41], the first row.
Eg mcx mvx mcy mvy γβ η c ηv νc νv
Present 0.912 0.146 0.131 1.240 7.857 0.480 0 0.008 0.038 0.030 0.005
Ref. [ 45] 2.00 0.15 0.15 0.70 1.00 0.2839 0.0101 0.2137 0.2137 0.0544 0.0381
Ref. [ 44] 0.70 0.1128 0.1080 1.5123 ∞ 0.4862 0.0353 0 0.0151 0.0252 0
On the other hand, the counter plot of Ref. [ 44] breaks the
time-reveal symmetry even at low electron or hole density. Thisis basically based on the extra off-diagonal terms that appearedin the low energy of their model Hamiltonian. Finally, wepresent the isofrequency counter surface in the kspace based
on our parameters and importantly the shape of the Fermisurfaces in our model are almost an elliptic shape especiallyat low charge density. Our results predict that the interbandcoupling term plays a role.
(a)
(b)
(c)
FIG. 2. Isofrequency contour surface in the kspace at zero
temperature for Es(/vectorkF)=0.05–0.5 eV with a step of 0.05 eV in
both the electron and hole doped cases. The first and second rows (a)
and (b) refer to parameters used in Refs. [ 45]a n d[ 44], respectively.
The last row, (c) plots are based on our model Hamiltonian.By diagonalizing the Hamiltonian Eq. ( 2), we end up with
two energy bands given by
Eτ=1
2/bracketleftbig
Hc+Hv+τ/radicalBig
4H2cv+(Hc−Hv)2/bracketrightbig
(3)
with Hc=Ec+ηck2
x+νck2
y,Hv=Ev−ηvk2
x−νvk2
y,
Hcv=γkx, and τ=± 1 denotes the conduction (valence)
band. The corresponding eigenvector reads
/Psi1c(v)=1/radicalbig
1+|χc(v)|2/parenleftbigχc(v)
1/parenrightbig
, (4)
where χc(v)=[Hc−Hv+τ/radicalbig
4H2cv+(Hc−Hv)2]/2Hcv.
Furthermore, having calculated the band energy dispersion
given by Eq. ( 3), thexandycomponents of the velocity can
be calculated as
vx=kx/bracketleftbigg
ηc−ηv+τ2γ2+(Hc−Hv)(ηc+ηv)/radicalbig
4H2cv+(Hc−Hv)2/bracketrightbigg
(5)
vy=ky/bracketleftbigg
νc−νv+τ(Hc−Hv)(νc+νv)/radicalbig
4H2cv+(Hc−Hv)2/bracketrightbigg
. (6)
B. Anisotropic transport framework
In this section we use the generalized semiclassical Boltz-
mann formalism for an anisotropic system to establish thetransport coefficients in the diffusive regime. In particular, wetake into account two important cases of short-range (SR)impurities (e.g., defects or neutral adatoms) with Dirac deltapotential and long-range (LR) Coulomb impurities in ourinvestigation. The thermoelectric properties of phosphorenein the presence of both the electric field and the temperaturegradient will be found.
In the diffusive regime, the transport coefficients can be
obtained from the charge current and the energy flux density.More details are provided in Appendix. The nonequilibriumdistribution function in the presence of driving forces is neededto calculate the current densities. For this purpose, we take theBoltzmann equation up to a linear order in the presence ofthermoelectric fields. The collision integral is given by
/parenleftbiggdf
dt/parenrightbigg
coll.=/integraldisplayd2k/prime
(2π)2w(k,k/prime)[f(k,E,T)−f(k/prime,E,T)],(7)
where w(k,k/prime) is the scattering rate from state kto state k/prime
which needs to be specified according to the microscopic
origin of the scattering mechanisms. As the relaxation timeapproximation provides an inadequate explanation for the fullaspects of the anisotropic features of the transport properties,
045422-3ZARE, RAMESHTI, GHAMSARI, AND ASGARI PHYSICAL REVIEW B 95, 045422 (2017)
an exact integral equation approach might be implemented
[46–48]. The scattering w(k,k/prime) rates using the Fermi golden
rule within the lowest order of the Born approximation aregiven by
w(k,k
/prime)=2π
/planckover2pi1nimp|/angbracketleftk/prime|ˆV|k/angbracketright|2δ(εk−εk/prime), (8)
where nimpis the areal density of randomly distributed
scatterers and ˆVk−k/primeis the Fourier transformation of the
interaction potential between an electron and a single impurity.The short-ranged impurities are approximated with a zero-range hard-core potential ˆV
k−k/prime=V0. On the other hand,
the long-ranged Coulombic interaction owing to the chargedimpurities is screened by other electrons of the system likethe Thomas-Fermi approach. The generalized conductivityσ(ε;θ,θ
/prime)i sg i v e nb y
σ(ε;θ,θ/prime)=e2/integraldisplayd2k
(2π)2δ(ε−ε(k))v2(φ)
×[a(φ) cosθ+b(φ)s i nθ] cos(θ−ξ(φ))(9)
withθ=θ/prime=0f o r σxxandθ=θ/prime=π/2f o r σyy.W e
concentrate on low enough temperatures where only electrons
contribute effectively in thermal transport and disregard
phonon contribution.
III. NUMERICAL RESULTS AND DISCUSSION
In this section our numerical results for the thermoelectric
transport in phosphorene are presented. We investigate theelectrical conductivity, Seebeck coefficient ( S), and its corre-
sponding figure of merit ZT, considering both the SR and LR
potentials. It should be noted that we set T∼20 K in all calcu-
lated quantities. Moreover, n
imp.=1010cm−2corresponding
to the chemical potential approximately μ∼10−4eV is used
for the impurity concentration of both short-range and long-range potentials to ensure that the diluteness criteria is satisfied.It is worthwhile to mention that there are essential criteria forutilizing the Boltzmann equation. These criteria are listed asfollows. Particles might interact via binary collisions, impuritydensity is low in terms of the charge carriers, an external fieldmight has long-range wavelength, and all collisions are elasticand involve only uncorrelated particles.
Figure 3shows the variation of the electrical conductivity
of phosphorene versus the chemical potential μin the
presence of short-range impurity interaction with V
k−k/prime=
V0=1000 eV ˚A2[49], in the zigzag, σyy, and armchair, σxx,
directions. The influence of the interband coupling term γis
also indicated. While at the presence of the coupling term γthe
electrical conductivity in the armchair direction σxxis greater
than the conductivity in the zigzag direction σyy, for both n-
andp-doped regimes, however the conductivity in the armchair
direction is smaller than that of the zigzag direction σxx<σyy
for the n-doped regime. Intriguingly, the conductivity in the
armchair direction σxxis more influenced by the inclusion
of the coupling term γand enhanced significantly. In both
doping regimes, the coupling term does not alter notably theconductivity in the zigzag direction σ
yy. All these behaviors
are the characteristics of the dispersion of monolayer phos-phorene, as indicated in Fig. 4. In the absence of the couplingFIG. 3. The conductivity of monolayer phosphorene as a function
of the chemical potential μat the presence of short-range impurity
potential along the zigzag, σyy, and armchair, σxx, directions. The
effect of the coupling term γis also shown. Note that the chemical
potential is measured from the middle of the gap value.
termγ=0, phosphorene dispersion relation reduces to two
separate ovals for the conduction and valence bands. In order tounderstand aforementioned features, we use the intuition basedon the Drude formula with an effective mass tensor, σ∼1/m
∗,
of the transport. Around the /Gamma1point, the components of
the effective mass for γ/negationslash=0a r e m−1
c,xx=0.333>m−1
c,yy=
0.06145, and m−1
v,yy=− 0.00969 /lessmuchm−1
v,xx=− 0.393, while
for the γ=0a r em−1
c,xx=0.01637 <m−1
c,yy=0.061452, and
m−1
v,xx=− 0.07613 >m−1
v,yy=− 0.00969. It is worth noting
that unlike the graphene [ 50], the conductivity of phosphorene
has an explicit energy dependence when only short-rangescatterers are present.
Due to the fact that the long-range charge-impurity
Coulomb interactions are mostly the dominant scatterersin samples, we also consider the Coulomb interaction. Tothis end, we use an interaction potential including staticThomas-Fermi screening, as is commonly used for a 2Delectron gas [ 52] to account partially for screening, as
V
k−k/prime=2πe2/(ε(|k−k/prime|+qTF)) where qTF=2πe2N(μ)/ε
is the Thomas-Fermi screening vector with the density of statesof the system, N(μ). We use the dielectric constant of the
common substrate SiO
2which is about ε=2.45. In Fig. 5,t h e
conductivity of phosphorene as a function of doping is plottedin the presence of LR potentials, for both the zigzag, σ
yy, and
armchair, σxx, directions. The overall energy dependence is
the same as SR interactions which indicated that despite thedetails of scattering phenomena, the dispersion of phosphoreneplays a main role in the conductivity. However, there is a cleardiscrepancy between two scattering mechanisms for the roleof the coupling parameter γ. Interestingly, the conductivity in
the zigzag direction σ
yyis more affected by the coupling term
than the σxx, in contrary to SR interactions where σxxis more
altered by the coupling term. On the other hand, while theσ
xxis enhanced by the coupling for SR potentials, here σyyis
suppressed due to the coupling term. We should mention thatat very low temperatures the variation of thermal conductivitywill be similar to the charge conductivity K≈(π
2/3)kBTσ.
045422-4THERMOELECTRIC TRANSPORT IN MONOLAYER PHOSPHORENE PHYSICAL REVIEW B 95, 045422 (2017)
(a)
(b)
FIG. 4. The band structure of phosphorene in the first Brillouin
zone, indicated by the gray square plane. Dispersion of phosphoreneis depicted in the planes of X-/Gamma1-XandY-/Gamma1-Y. The corresponding
constant-energy contour plots in the Brillouin zone are shown in both
n-a n dp-doped cases. The influence of the coupling term γis also
indicated in panels (a) and (b).
Figure 6shows the anisotropy ratio calculated using the
SR and LR potentials. First of all, as seen in the figure,the ratio is significantly large specially at low charge carrierdensity. The curves are monotonic in terms of the chemicalpotential and find that the ratio slightly changes with the typeof impurity. In the inset, we show the anisotropic ratio ofthe hole doped case. We also compare our numerical resultswith those obtained by Liu et al. ,[51] in the case that d=0,
the distance between charged impurity with phosphorene,in which they computed the mobility within the Boltzmanntransport equation under detailed balance condition togetherwith the anisotropy in momentum. As seen in the figure, thereis a discrepancy between our fully self-consistent method withthe approximated relaxation time result. This predicts that theBoltzmann transport equation with the anisotropic momentumFIG. 5. The conductivity of monolayer phosphorene as a function
of doping μat the presence of long-range charged impurity potential
for the zigzag, σyy, and armchair, σxx, directions. The effect of
coupling term γis also shown.
cannot provide a full description of the transport properties in
phosphorene, except at very low doping regime.
The variation of the Seebeck coefficients (thermopower) S,
as a more feasible quantity in real experiments, with doping atthe presence of SR and LR potentials is obtained as shown inFig. 7. When the Fermi level lies in the valence band thermally
activated holes, which move along the same direction as thetemperature gradient owing to the positive charge, it results ina positive thermopower, however thermally activated electronsin the conduction band lead to a negative thermopower.Moreover, the figure of merit attains its maximum value aroundthe chemical potential as can be seen in Fig. 8where the ZT
is depicted as a function of the carrier density n
2Dfor both
scatterers. The figure of merit becomes large where the power
FIG. 6. The anisotropy ratio of the conductivity, σxx/σyy,o f
monolayer phosphorene as a function of electron doping μat the
presence of short- and long-range impurity potentials. Inset: the ratio
of the conductivity as a function of the hole chemical potential. Solid
line refers to data calculated in Ref. [ 51] for long-ranged impurity
potential at d=0.
045422-5ZARE, RAMESHTI, GHAMSARI, AND ASGARI PHYSICAL REVIEW B 95, 045422 (2017)
FIG. 7. Seebeck coefficient of phosphorene as a function of
carrier density n2Dat the presence of short- and long-range Coulomb
potentials. The effect of the coupling term γis also depicted. Despite
theγ, the Seebeck coefficients are nearly isotropic for both directions.
factor S2σis very strong while the thermal transport Kis
not. Our results reveal that, in contrast to highly anisotropicelectrical and thermal conductivities, the Seebeck coefficientand the thermoelectric figures of merit are nearly isotropic,consistent with the prior work [ 34]. In fact, for the charge
carrier density less than about 1 ×10
14cm−2, both σand its
derivative with respect to the energy have approximately thesame anisotropic behavior, consequently it leads to a nearlyisotropic behavior of the Seebeck coefficient. Interestingly,in spite of underlying scattering mechanisms, thermopowerand its corresponding figure of merit are not affected by thecoupling term γ, on the contrary the charge conductivity.
In Fig. 9, the figures of merit as a function of doping are
plotted at T=300 K in the presence of LR potentials, for
both the zigzag and armchair directions. Notice that in thiscase, we calculate Eq. ( A7) in the Appendix numerically.
FIG. 8. The variation of corresponding figures of merit are
depicted as a function of the carrier density at the presence of short-
and long-range charge-impurity potentials. The effect of the couplingtermγis also depicted. Figures of merit are also nearly isotropic.FIG. 9. The variation of figures of merit along the armchair
and zigzag directions as a function of the carrier density at the
presence of long-range charge-impurity potential at room temperature
T=300 K. The effect of only electron contribution in the thermal
conductivity Kelis illustrated by symbols while the full effect of
the electron and phonon contributions in the thermal conductivity
Kel+Kph(which is ∼13 and ∼30 Wm−1K−1along armchair and
zigzag, respectively) are shown by solid and dashed lines.
The contribution of the phonon in the thermal conductivity
Kph∼20–40 Wm−1K−1[34,53–55] can only affect the figure
of merit, and the thermopower is not altered by the presenceof the phonon. At high temperatures, the phonon becomesimportant but it only results in the overall decline of the figuresof merit, without affecting their qualitative behavior.
IV . CONCLUSION
In conclusion, the thermoelectric transport in phosphorene
in the presence of short- and long-ranged charged impuritypotentials is studied using the generalized semiclassicalBoltzmann approach for anisotropic systems. The chargeconductivity, which is slightly different for n- and p-doped
cases mostly owing to the unique dispersion of phosphorene,is found to be highly anisotropic, while the Seebeck coefficientand the corresponding figure of merit, without being affectedeither by type of scatterers or the presence/absence of couplingterm, are nearly isotropic. Intriguingly, the conductivity forn-doped cases is more influenced by the coupling term, albeit
in a dissimilar manner for different scatterers, than p-doped
cases. Furthermore, it is shown that thermopower changes signdue to the conversion of electrons to holes and vice versa atthe edge of the bands. We also reveal that phosphorene couldbe a very promising material for thermoelectric studies andapplications.
Since several works on thermodynamics in 2D crystalline
material systems are available, a proper comparison with thoseresults seems to be in order. Recent investigations, basedon DFT calculations, showed that the ZTvalue of BP can
only reach 0.22 at room temperature, while for monolayerphosphorene it reaches to 1.78 [ 36]. It has been argued that
applying strain is a practical way to enhance the thermoelectricefficiency of BP, and the largest ZTvalue of 0.87 can be
045422-6THERMOELECTRIC TRANSPORT IN MONOLAYER PHOSPHORENE PHYSICAL REVIEW B 95, 045422 (2017)
TABLE II. Reported values of the Seebeck coefficients and
their corresponding figures of merit for monolayer of phosphorene,
silicene, graphene, and MoS 2systems.
T[K] S[μVK−1] ZT
Phosphorene [ 36] 300 3000 1.78
Phosphorene [ 34] 300, 500 2000, 2800 1.5, 3.8
Phosphorene [ 57] 300 500, 600 1.65, 2.12
Phosphorene [ 56] 300, 500 450, 500 0.1, 0.14
Phosphorene [ 32] 300 1400 up to 6.5
Silicene [ 58] 300–600 up to 858 2.8–4.9
Graphene 300 up to 80 0.79–1
[7,8,59,60]
Graphene [ 61] 150, 300 up to 60, 120
Graphene [ 62]T <40, 300 up to 12, 50
MoS 2[10] 300, 500 up to 1050.5, 1
Present 20 ∼175 ∼1.2
achieved [ 35]. Although the ZTvalues obtained for BP are
very small to compete with typical thermoelectric materials,e.g., Bi
2Te3, Fei et al. using first principle simulations showed
that both the electrical and thermal conductance of monolayerphosphorene are highly anisotropic and the ZTvalue is greater
than 1.0 at room temperature and can attain up to 2.5 at500 K [ 34], due to the optimal ratio of conductances with
orthogonally preferred conducting directions. Zhang et al.
demonstrated, based on first principle calculations, that theZTvalue for phosphorene nanoribbons can achieve up to 6.4
at room temperature [ 32]. Liao et al. implying first principle
calculations reported ZTvalues of 0.1, 0.14 at 300, 500 K,
respectively, for p-doped samples [ 56]. Lv et al. , based on
the semiclassical Boltzmann equation and DFT calculations,showed that ZTvalue of phosphorene at room temperature by
strain can reach up to 1.65 [ 57].
Pan et al. , using the nonequilibrium Green’s function
method and molecular dynamics simulations, predicted thattheZTvalue of zigzag silicene nanoribbon can achieve up
to 4.9 [ 58]. Experimentally, the thermopower of graphene has
been varied from 20 to 90 μVK
−1while the temperature
is changed from 10 to 300 K [ 8]. Wei et al. , experimentally
achieved up to 50 μVK−1for the thermopower of graphene at a
temperature range of 11–255 K [ 7]. Checkelsky et al. reported
measurement of thermopower in graphene that reaches up to ∼
100μVK−1at room temperature [ 59]. By means of atomistic
simulation, Mazzamuto et al. predicted that the thermopower
of graphene nanoribbons attain the value of 300 μVK−1[60].
The investigation based on self-consistent Born approximationpredicted the value of 0.4 μVK
−2for the thermoelectric power
of graphene S/T, at the presence of charged impurity scat-
terers [ 61]. Bao et al., by presenting a balance-equation-based
theoretical examination of thermoelectric power in graphene,found that Schanges from 1 to 50 μVK
−1as temperature
goes from 10 to 300 K [ 62]. Buscema et al. , by scanning
photocurrent microscopy, have observed a thermopower ashigh as 10
5μVK−1for a single-layer MoS 2, which is tunable
via an external electric field [ 10]. Finally, the predicted and
measured values of the Seebeck coefficient and ZTof 2D
materials are listed in Table II.ACKNOWLEDGMENT
This work was partially supported by Iran Science Elites
Federation.
APPENDIX
In the diffusive regime, the transport coefficients can be
obtained from the following expression for the charge currentjand energy flux density j
q,
/bracketleftbigg
j
jq/bracketrightbigg
=/integraldisplayd2k
(2π)2/bracketleftbigg
−e
ε(k)−μ/bracketrightbigg
v(k)f(k), (A1)
where v(k) is the semiclassical velocity of the carriers which
is related to the energy dispersion εkthrough v=(1//planckover2pi1)∇kεk.
The nonequilibrium distribution function f(k) describes the
evolution of the charge distribution in the presence of thermo-electric forces. In the linear response theory we seek a solutionof Eq. ( 6) in the form of
f(k,E,T)−f
0=Ex∂Exf+Ey∂Eyf
+∇Tx∂∇Txf+∇Ty∂∇Tyf+··· (A2)
by parameterizing E,k, and vasE=E(cosθ,sinθ),k=
k(cosφ,sinφ), and v(k)=v(φ)(cosξ(φ),sinξ(φ)), respec-
tively; we end up for nonequilibrium distribution functionwith,
f(θ,α)−f
0=[A(φ) cosθ+B(φ)s i nθ)]E
+[C(φ) cosθ+D(φ)s i nθ]∇T(A3)
where, A(φ)=∂Exf,B(φ)=∂Eyf,C(φ)=∂∇Txf, and
D(φ)=∂∇Tyf. By invoking Eq. ( A3) into Eq. ( 7) we obtain
the following set of linear integral equations [ 46–48]
cosζ(φ)=¯w(φ)a(φ)−/integraldisplay
dφ/primev(φ/prime)
v(φ)w(φ,φ/prime)a(φ/prime),(A4)
sinζ(φ)=¯w(φ)b(φ)−/integraldisplay
dφ/primev(φ/prime)
v(φ)w(φ,φ/prime)b(φ/prime),(A5)
with similar relations for c(φ) and d(φ). Here
w(φ,φ/prime)=(2π)−1/integraltext
k/primedk/primew(k,k/prime) and ¯w(φ)=/integraltext
dφ/primew(φ,φ/prime).
Also, the quantities A(φ)=−ev(φ)[−∂εf0]a(φ),
B(φ)=−ev(φ)[−∂εf0]b(φ),C(φ)=v(φ)(ε−μ
T)[−∂εf0]c(φ),
andD(φ)=v(φ)(ε−μ
T)[−∂εf0]d(φ) are defined. Inserting
solutions of Eqs. ( A4) and ( A5) into Eq. ( A3) yields the exact
solution of the Boltzmann equation up to the linear order in E
and∇T.
By invoking the expression for f(θ,φ) into Eq. ( A1)f o rt h e
charge and heat currents, the response matrix, which relatesthe resulting generalized currents to the driving forces, canbe expressed in terms of some kinetic coefficients L
αas the
following,
/parenleftbigg
j
jq/parenrightbigg
=/parenleftbigg
L0−L1/eT
L1/e−L2/e2T/parenrightbigg/parenleftbigg
E
−∇T/parenrightbigg
. (A6)
Diagonal response elements explain the electrical σand
thermal Kconductivities, and the two off-diagonal thermo-
electric coefficients are related to each other through the
045422-7ZARE, RAMESHTI, GHAMSARI, AND ASGARI PHYSICAL REVIEW B 95, 045422 (2017)
Onsager relation. The Seebeck coefficient (thermopower)
S=−1
eT(L0)−1·L1describes the voltage generation due to
the temperature gradient while Peltier coefficient /Pi1=TS
accounts for the heat current induction due to the chargecurrent, respectively. The figure of merit, which is the abilityof a material to efficiently produce thermoelectric power,is described by a dimensionless quantity denoted by ZT=σS2
KT. All of the coefficients obey the relation
Lα(θ,θ/prime)=/integraldisplay
dε/bracketleftbigg−∂f0
∂ε/bracketrightbigg
(ε−μ)ασ(ε;θ,θ/prime). (A7)
All of the thermoelectric properties described by Lαcan be
found by calculating the generalized conductivity.
[1] L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B 47,12727
(1993 ).
[ 2 ] L .D .H i c k s ,T .C .H a r m a n ,X .S u n ,a n dM .S .D r e s s e l h a u s , Phys.
Rev. B 53,R10493(R) (1996 ).
[3] R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn,
Nature (London) 413,597 (2001 ).
[4] R. Arita, K. Kuroki, K. Held, A. V . Lukoyanov, S. Skornyakov,
and V . I. Anisimov, Phys. Rev. B 78,115121 (2008 ).
[5] N. Hamada, T. Imai, and H. Funashima, J. Phys.: Condens.
Matter 19,365221 (2007 ).
[6] J. M. O. Zide, D. Vashaee, Z. X. Bian, G. Zeng, J. E. Bowers, A.
Shakouri, and A. C. Gossard, P h y s .R e v .B 74,205335 (2006 ).
[7] P. Wei, W. Bao, Y . Pu, C. N. Lau, and J. Shi, Phys. Rev. Lett.
102,166808 (2009 ).
[8] Y . M. Zuev, W. Chang, and P. Kim, Phys. Rev. Lett. 102,096807
(2009 ).
[9] T. Kato, S. Usui, and T. Yamamoto, Jpn. J. Appl. Phys. 52,
06GD05 (2013 ).
[10] M. Buscema, M. Barkelid, V . Zwiller, H. S. J. van der Zant,
G. A. Steele, and A. Castellanos-Gomez, Nano Lett. 13,358
(2013 ).
[11] S. Konabe and T. Yamamoto, P h y s .R e v .B 90,075430 (2014 ).
[12] D. Bilc, S. D. Mahanti, K. F. Hsu, E. Quarez, R. Pcionek,
and M. G. Kanatzidis, P h y s .R e v .L e t t . 93,146403 (2004 ).
[13] G. D. Mahan and J. O. Sofo, Proc. Natl. Acad. Sci. USA 93,
7436 (1996 ).
[14] N. Neophytou and H. Kosina, P h y s .R e v .B 83,245305 (2011 );
J. Appl. Phys. 112,024305 (2012 ).
[15] V . M. Fomin and P. Kratzer, P h y s .R e v .B 82,045318 (2010 ).
[16] X. Zianni, Appl. Phys. Lett. 97,233106 (2010 ).
[17] H. Liu, Y . Du, Y . Deng, and P. D. Ye, C h e m .S o c .R e v . 44,2732
(2015 ).
[18] S. Das, M. Demarteau, and A. Roelofs, ACS Nano 8,11730
(2014 ).
[19] M. V . Kamalakar, B. N. Madhushankar, A. Dankert, and S. P.
Dash, Small 11,2209 (2015 ).
[20] W. Lu, H. Nan, J. Hong, Y . Chen, C. Zhu, Z. Liang, X. Ma, Z.
Ni, C. Jin, and Z. Zhang, Nano Res. 7,853 (2014 ).
[21] T. Markussen, A.-P. Jauho, and M. Brandbyge, Nano Lett. 8,
3771 (2008 ).
[22] Y . Takao and A. Morita, Physica B+C 105,93(1981 ).
[23] R. W. Keyes, Phys. Rev. 92,580 (1953 ).
[24] Y . Du, C. Ouyang, S. Shi, and M. Lei, J. Appl. Phys. 107,093718
(2010 ).
[25] A. N. Rudenko and M. I. Katsnelson, P h y s .R e v .B 89,201408(R)
(2014 ).
[26] S. Zhang, J. Yang, R. Xu, F. Wang, W. Li, M. Ghufran, Y .-W.
Zhang, Z. Yu, G. Zhang, Q. Qin, and Y . Lu, Acs Nano 8,9590
(2014 ).[27] L. Liang, J. Wang, W. Lin, B. G. Sumpter, V . Meunier, and M.
Pan, Nano Lett. 14,6400 (2014 ).
[28] D. C ¸ akır, H. Sahin, and F. M. Peeters, P h y s .R e v .B 90,205421
(2014 ).
[29] V . Tran, R. Soklaski, Y . Liang, and L. Yang, P h y s .R e v .B 89,
235319 (2014 ).
[30] L. Li, Y . Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H.
Chen, and Y . Zhang, Nat. Nanotechnol. 9,372 (2014 ).
[31] H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tomanek, and
P. D. Ye, ACS Nano 8,4033 (2014 ).
[32] J. Zhang, H. J. Liu, L. Cheng, J. Wei, J. H. Liang, D. D. Fan, J.
Shi, X. F. Tang, and Q. J. Zhang, Sci. Rep. 4,6452 (2014 ).
[33] W. Xu, L. Zhu, Y . Cai, G. Zhang, and B. Li, J. Appl. Phys. 117,
214308 (2015 ).
[34] R. Fei, A. Faghaninia, R. Soklaski, J.-A. Yan, C. Lo, and L.
Yang, Nano Lett. 14,6393 (2014 ).
[35] G. Qin, Q.-B. Yan, Z. Qin, S.-Y . Yue, H.-J. Cui, Q.-R. Zheng,
and G. Su, Sci. Rep. 4,6946 (2014 ).
[36] H. Y . Lv, W. J. Lu, D. F. Shao, and Y . P. Sun, arXiv:1404.5171 .
[37] E. Flores, J. R. Ares, A. Castellanos-Gomez, M. Barawi, I. J.
Ferrer, and C. S ´anchez, Appl. Phys. Lett. 106,022102 (2015 ).
[38] J.-W. Jiang, Nanotechnology 26,055701 (2015 ).
[39] Y . Cai, Q. Ke, G. Zhang, Y . P. Feng, V . B. Shenoy, and Y .-W.
Zhang, Adv. Funct. Mater. 25,2343 (2015 ).
[40] Z.-Y . Ong, Y . Cai, G. Zhang, and Y .-W. Zhang, J. Phys. Chem.
C118,25272 (2014 ).
[41] M. Elahi, K. Khaliji, S. M. Tabatabaei, M. Pourfath, and R.
Asgari, Phys. Rev. B 91,115412 (2015 ).
[42] P. Li and I. Appelbaum, P h y s .R e v .B 90,115439 (2014 ).
[43] M. Ezawa, New J. Phys. 16,115004 (2014 ).
[44] A. S. Rodin, A. Carvalho, and A. H. Castro Neto, Phys. Rev.
Lett.112,176801 (2014 ).
[45] T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno,
and F. Guinea, P h y s .R e v .L e t t . 113,106802 (2014 ).
[46] B. Z. Rameshti, R. Asgari, P h y s .R e v .B 94,205401 (2016 ).
[47] K. V ´yborn ´y, A. A. Kovalev, J. Sinova, and T. Jungwirth, Phys.
Rev. B 79,045427 (2009 ).
[48] A. Faridi, R. Asgari, and A. Langari, Phys. Rev. B 93,235306
(2016 ).
[49] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod.
Phys. 83,407 (2011 ).
[50] B. Z. Rameshti and A. G. Moghaddam, P h y s .R e v .B 91,155407
(2015 ).
[51] Y . Liu, T. Low, and P. P. Ruden, Phys. Rev. B 93,165402 (2016 ).
[52] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54,437
(1982 ).
[53] G. Qin, Q.-B. Yan, Z. Qin, S.-Y . Yue, M. Hu, and G. Su, Phys.
Chem. Chem. Phys. 17,4854 (2015 ).
[54] T.-H. Liu and C.-C. Chang, Nanoscale ,7,10648 (2015 ).
045422-8THERMOELECTRIC TRANSPORT IN MONOLAYER PHOSPHORENE PHYSICAL REVIEW B 95, 045422 (2017)
[55] Y . Hong, J. Zhang, X. Huangc, and X. C. Zeng, Nanoscale 7,
18716 (2015 ).
[56] B. L. Liao, J. W. Zhou, B. Qiu, M. S. Dresselhaus, and G. Chen,
Phys. Rev. B 91,235419 (2015 ).
[57] H. Y . Lv, W. J. Lu, D. F. Shao, and Y . P. Sun, Phys. Rev. B 90,
085433 (2014 ).
[58] L. Pan, H. J. Liu, X. J. Tan, H. Y . Lv, J. Shi, X. F. Tang, and G.
Zeng, Phys. Chem. Chem. Phys. 14,13588 (2012 ).[59] J. G. Checkelsky and N. P. Ong, P h y s .R e v .B 80,081413
(2009 ).
[60] F. Mazzamuto, V . Hung Nguyen, Y . Apertet, C. Ca ¨e r ,C .C h a s s a t ,
J. Saint-Martin, and P. Dollfus, Phys. Rev. B 83,235426 (2011 ).
[61] X.-Z. Yan, Y . Romiah, and C. S. Ting, Phys. Rev. B 80,165423
(2009 ).
[62] W. S. Bao, S. Y . Liu, and X. L. Lei, J. Phys.: Condens. Matter
22,315502 (2010 ).
045422-9 |
PhysRevB.101.115401.pdf | PHYSICAL REVIEW B 101, 115401 (2020)
Conductance quantization and shot noise of a double-layer quantum point contact
D. Terasawa ,1,*S. Norimoto ,2T. Arakawa,2,3M. Ferrier,2,4A. Fukuda,1K. Kobayashi ,2,5and Y . Hirayama6
1Department of Physics, Hyogo College of Medicine, Nishinomiya 663-8501, Japan
2Graduate School of Science, Department of Physics, Osaka University, Toyonaka 560-0043, Japan
3Center for Spintronics Research Network, Osaka University, Toyonaka, Osaka 560-8531, Japan
4Laboratoire de Physique des Solides, CNRS, Université Paris-Sud, Université Paris Saclay, 91405 Orsay Cedex, France
5Institute for Physics of Intelligence and Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan
6Graduate School of Science and CSIS, Tohoku University, Sendai 980-8578, Japan
(Received 10 April 2019; revised manuscript received 24 January 2020; accepted 9 February 2020;
published 2 March 2020)
The conductance quantization and shot noise below the first conductance plateau G0=2e2/hare measured
in a quantum point contact fabricated in a GaAs /AlGaAs tunnel-coupled double quantum well. From the
conductance measurement, we observe a clear quantized conductance plateau at 0 .5G0and a small minimum
in the transconductance at 0 .7G0. Spectroscopic transconductance measurement reveals three maxima inside
the first diamond, thus suggesting three minima in the dispersion relation for electric subbands. Shot noisemeasurement shows that the Fano factor behavior is consistent with this observation. We propose a model thatrelates these features to a wave-number directional split subband due to a strong Rashba spin-orbit interactionthat is induced by the center barrier potential gradient of the double-layer sample.
DOI: 10.1103/PhysRevB.101.115401
I. INTRODUCTION
In quantum point contacts (QPCs) on two-dimensional
electron gas (2DEG) systems, nanometer-scale confinementembodies a quantum ballistic transport analogous to the trans-verse modes of optical waveguides. The transverse modesor subbands are well separated in energy; thus, the conduc-tance through a QPC becomes quantized in a unit of G
0=
2e2/h[1–3], where hdenotes Planck’s constant, eis the
elementary charge, and the coefficient 2 expresses the spindegeneracy that is understood using the Landauer-Büttikerformalism [ 4–6]. Although many theoretical studies suggested
the lifted spin degeneracy state (0 .5G
0plateau) at zero mag-
netic field [ 7–11], this degeneracy is typically not resolved.
Instead, a small plateau appears at 0 .7G0[3], and it has
attracted considerable interest (for a review, see [ 12]). The
Landauer-Büttiker model has been tested by measuring shotnoise, i.e., the discrete noise of the charge that is carriedby particles in the probabilistic scattering process [ 13–19],
in this system. Previous shot noise measurements for QPCson 2DEGs have contributed significantly to the elucidationof basic physics and complemented the conductance results[20–27]. Furthermore, in addition to their fundamental phys-
ical importance, semiconductor nanostructures with a QPCoffer electronic devices that can manipulate electron chargesand spins; thus, they are feasible for spintronic devices [ 28,29]
and quantum computation [ 30]. In particular, a QPC on a
tunnel-coupled double layer (coupled quantum wire) is acandidate for implementing a qubit [ 31–33]. Hitherto, several
studies [ 34–40] have been conducted that resolved the coupled
*terasawa@hyo-med.ac.jpwave-function modes of double-layer systems, and the ob-
tained information is useful for quantum engineering. Theresolution of spin degeneracy and the generation of spincurrents with only electrical controls, such as using spin-orbitinteractions (SOIs) [ 27,41–47], remain to be addressed in
future studies. In addition, the shot noise for tunnel-coupledQPCs should be measured, because additional degrees offreedom are expected to affect many-body interactions in thenonequilibrium regime [ 48].
In this study, we fabricated a QPC in a double-layer 2DEG
of a GaAs /AlGaAs double quantum well (DQW) sample, and
we investigated the conductance quantization in this double-layer QPC system. Here, we report the shot noise results whenthe conductance is below the first conductance plateau, G
0.
Previously, researchers have reported the coexistence of 0 .5G0
and∼0.7G0plateaus [ 27,37,49–51]. Using a high mobility
and low electron density double-layer sample, we observeda clear conductance plateau at 0 .5G
0, and transconductance
minima at 0.5 and ≈0.7G0at zero magnetic field and the
lowest temperature available for the dilution refrigerator usedin this experiment. Energy spectroscopy reveals a rich struc-ture of subband edge (SBE) lines with three maxima insidethe first SBE diamond, between the 0 .5G
0andG0plateau
region. They are dependent on the magnitude and direction ofthe magnetic fields, and consistent with the horizontal (in thewave-number direction) subband splitting model discussedherein. From the shot noise measurement, the Fano factor F,
i.e., the current noise normalized to the noise of Poissoniantransmission statistics, exhibits reductions at 0 .5G
0andG0,
and a small reduction at 0 .7G0. In addition, we observe a
difference in Fwith regard to the positive and negative biases
that further suggests an SOI dispersion with Zeeman splitting.We hypothesize that this splitting is caused by the Rashba SOI
2469-9950/2020/101(11)/115401(13) 115401-1 ©2020 American Physical SocietyD. TERASAW A et al. PHYSICAL REVIEW B 101, 115401 (2020)
2.85 MHzAmp.Vsd
Vg
Digitize rS
DGate
Gatex
y
xzBilayer QPC1.5 K 20 mK
20 nF
QPC
100 nm
500 nmfront layerback layer
FIG. 1. Schematics of the sample and current noise measure-
ment setup. The sample is placed upside down on the cold finger ofthe mixing chamber, as shown in the horizontal view in the bottom
panel. Right inset: scanning electron microscopy image of the split
gates.
[52] that is induced by a strong potential gradient of the center
barrier and the high mobility of the sample. This study wouldinvoke further investigations for spin-related physics and aquasiparticle’s charge in the double-layer system.
The remainder of this paper is structured as follows:
In Sec. II, we describe the sample of this experiment
(Sec. II A), the experimental setups for conductance measure-
ments (Sec. II B), and the shot noise measurements (Sec. II C).
In Sec. III, we present the experimental results on the conduc-
tance measurement (Sec. III A ) and shot noise measurement
(Sec. III B ). A discussion is presented in Sec. IV. After calcu-
lating the wave functions in the DQW at the QPC (Sec. IV A ),
we discuss the effect of the SOI for the conductance and shotnoise (Sec. IV B ). We present the conclusions in Sec. V,a s
well as a brief mention of future perspectives.
II. EXPERIMENT
A. Sample preparation
The sample used in this study was fabricated on a DQW
heterostructure grown by molecular beam epitaxy on a GaAs(100) surface in the NTT Basic Research Laboratories. Thewafer comprises two 20-nm-wide GaAs quantum wells sep-arated by a 3-nm-wide AlAs barrier layer; thus, the center-to-center distance disd=23 nm. The DQW was located
600 nm below the surface, and it was doped from both sidesusing 1 ×10
12cm−2Siδ-dopings 200 nm away from both
layers. The energy gap between the DQW symmetric andantisymmetric states /Delta1
SASwas measured to be 0.29 meV
through the analysis of Shubnikov–de Haas (SdH) oscillationat low magnetic fields (see Appendix A). The total electron
density is 1 .20×10
11cm−2, with 0 .64×1011cm−2in the
symmetric state and 0 .56×1011cm−2in the antisymmetric
state. The sample was processed in a shape of a standardHall bar of width 50 μm and four voltage probes separated
by 180 μm( s e eF i g . 1). Two of the probes were used in
this experiment. Ohmic contacts were created using AuGe /Ni
metals. They were contacted with both layers simultaneously.Subsequently, a pair of split gates of width 500 nm and length
100 nm was created, under which a coupled double-layer QPCwas formed. The scanning electron microscopy image of thesplit gates is shown in Fig. 1. In this setup, the conductance
and current noise are the results of the transport measurementthrough this QPC. The low-temperature electron mobility isas high as ≈2.5×10
6cm2/(V s), given the low electron
density in the DQW. This value provides a mean free pathof≈14μm and a momentum relaxation time of ≈95 ps
from the Drude model. The sample was mounted on the coldfinger of the mixing chamber of a dilution refrigerator witha base temperature of 20 mK. We determine the x-,y-, and
z-directions with regard to the current flow direction through
the QPC and the 2DEG plane: the x-direction is perpendicular
to the current and in-plane to the 2DEG; the y-direction is
parallel along the current and in-plane to the 2DEG; thez-direction is perpendicular to the 2DEG. Magnetic fields
B=(B
x,By,Bz) were applied using a vector magnet, with
maximum fields of Bx=3,By=1, and Bz=8T .W eu s e
B=|B|as the magnitude of the total magnetic fields; thus,
B=0 T represents Bx=By=Bz=0T .
B. Conductance measurement
We measured the two-terminal differential conductance
G=dIsd/dVsd(IsdandVsddenote the source-drain current
and voltage, respectively) and the transconductance dG/dVg
(Vgdenotes the gate voltage applied to the split gates) simul-
taneously, using two lock-in amplifiers. First, Gwas mea-
sured using a standard lock-in technique with a frequencyof 387 Hz and an amplitude of V
ac
sd=10μV rms; simulta-
neously, a small ac gate modulation Vac
g=4m Vr m sw a s
applied through the second lock-in amplifier with a frequencyof 13 Hz. The output signal of the first lock-in amplifier,which includes the ac modulation signal from V
ac
g, was input
to the second lock-in amplifier, whose ac modulation wasreferenced by itself. This method allows us to measure thetransconductance directly; therefore, it is sensitive enoughto detect a small change in the transconductance. A dc gatevoltage V
dc
gwas also applied to the sample; thus, the total
voltage applied to the split gate VgisVg=Vdc
g+Vac
g.I n
addition, a dc voltage VS
sdwas applied to the source to cancel
the voltage arising from the Seebeck effect because the drainwas grounded at the mixing chamber, and dc voltage V
dc
sdwas
applied to the source electrode. Thus, the total voltage appliedto the source V
sdwasVsd=Vac
sd+Vdc
sd−VS
sd. For practical use
in graphs and image plots, we ignored the ac component of Vg
andVsd.
C. Shot noise measurement
The current noise, i.e., the current fluctuation around its
average, was measured at 300 mK following Refs. [ 53–55].
The voltage fluctuation generated in the parallel circuit of thesample and a 2.85-MHz LCresonator was measured as an
output signal of a homemade cryogenic amplifier [ 54]a ta1K
pot and a room-temperature amplifier, as shown schematicallyin Fig. 1. Subsequently, the time-domain noise signal acquired
by a digitizer was converted to a power spectrum through fastFourier transform (FFT). The current spectral density S
Iwas
115401-2CONDUCTANCE QUANTIZATION AND SHOT NOISE OF A … PHYSICAL REVIEW B 101, 115401 (2020)
TABLE I. Typical values of parameters for noise measurement.
AZ 0(/Omega1) C(pF) Sout
V(V2/Hz) Sout
I(A2/Hz)
8.7×1056.1×1041.0×1021.3×10−196.0×10−28
obtained by fitting the resonance peak P0that was described
as a function of the sample differential resistance Rd=1/Gat
a finite Vsd,
P0=A/bracketleftBigg
Sout
V+/parenleftbiggZ0Rd
Z0+Rd/parenrightbigg2/parenleftbig
Sout
I+SI/parenrightbig/bracketrightBigg
, (1)
where Adenotes the total gain of the cold and room-
temperature amplifiers, Z0denotes the impedance of the LC
resonance circuit, and Sout
VandSout
Idenote the current and
voltage noise of the amplifier, respectively. After a series ofcareful calibration procedures, we obtained the parameters asshown in Eq. ( 1). Their typical values are tabulated in Table I.
For a finite temperature, S
Iis described by the following
equation [ 18]:
SI=2F
Rd/bracketleftbigg
eVcoth/parenleftbiggeV
2kBTe/parenrightbigg
−2kBTe/bracketrightbigg
+4kBTe
Rd,(2)
where Tedenotes the electron temperature and Fdenotes the
Fano factor. For high bias region ( |eV|>2kBTe), the equation
above becomes simpler; SIbehaves linearly on /angbracketleftIsd/angbracketrightas
SI=2eF/angbracketleftIsd/angbracketright. (3)
We evaluated the Fano factor using this simpler form as it
yielded more reliable values [ 55].
III. RESULTS
A. Results of the conductance measurement
Figure 2(a) shows Gas a function of Vgat 2 K and 35 mK.
Reflecting the property of double-layer systems at 2 K, G
drops twice at Vg≈− 1.0 and−1.5 V (indicated by the down-
ward arrows), corresponding to the depletion of the front andback 2DEGs under the split gate, respectively. Then at 35 mK,several conductance plateaus are observed for V
g<−2.8V
1.5
1.0
0.5
0.0G (mS)
-3 -2 -1 0
Vg (V) 2K
35mK1.5
1.0
0.5
0.0G (2e2/h)
-3.15 -3.10 -3.05
Vg (V)dG/dVg (a.u.)α
βγ(a) (b)
FIG. 2. (a) Gas a function of Vgat 2 K and 35 mK at zero
magnetic field B=0T .( b ) G(left axis, in units of G0=2e2/h)a n d
dG/dVg(right axis, in arbitrary unit) as a function of Vg.before the channel is pinched off at Vg=− 3.14 V. Figure 2(b)
shows detailed structures of GanddG/dVgforG<1.5G0.
The resistances of the leads and at the contacts are subtractedaccordingly. We observe a clear 0 .5G
0plateau in Gand a local
minimum in the dG/dVgwith a small plateau around 0 .7G0
(indicated by the upper arrow). The simultaneous observation
of these two features for B=0 T has been reported in several
experiments [ 27,37,49,51,56]. To the best of our knowledge,
however, this has never been observed in a double-layersystem before. To supplement the explanation, unlike thetypical so-called “0.7 anomaly” in that a relatively highertemperature is required to observe a plateaulike feature [ 3],
this minimum in dG/dV
gis clearly observed at extremely low
temperatures such as T/lessorequalslant35 mK, indicating that it originates
in a ground state. In addition, a 0.7 plateau is evolved into aclear 0.5 plateau by changing the electron density [ 8,36,37],
or by increasing the in-plane magnetic field parallel to thechannel [ 3]. Therefore, the concurrent observation of 0.5 and
0.7 plateaus is rather unusual. Physically, the peaks observedindG/dV
gimply that the Fermi energy crosses the SBEs. In
Fig. 2(b), three peaks are shown between the G=0 and G0
regions, suggesting that the Fermi energy crosses three SBEs
in this region. We name these three peaks α,β, andγfrom
low to high Vg.
Subsequently, the energy spectroscopy for the channel un-
der the double-layer QPC was measured. Subband spacings oftransverse modes at the QPC are observed in a spectroscopicmeasurement by controlling the Fermi energy E
Fthrough
Vgand the chemical potentials between the source and drain
/Delta1μ sd=μs−μd=eVsd. Figure 3(a) shows the image plot of
dG/dVgas a function of VsdandVg. The dark regions represent
lowdG/dVg; therefore, these regions indicate plateau regions
in the conductance, whereas the brighter regions representhigh dG/dV
g, indicating that a Fermi energy passes through
an SBE. It should be noted that the pinch-off voltage is differ-ent from that in Fig. 2due probably to unexpectedly localized
electric charges. As compared to ordinary monolayer QPCcases [ 57–61], or even several tunnel-coupled double-layer
QPC cases [ 35,38,39], the data reveal a rich SBE structure,
particularly inside the first (lowest) SBE diamond [see alsoFig. 3(b), which is an enlarged image plot of Fig. 3(a) around
the first SBE structure]. In Figs. 3(a) and 3(b),w ed r a w
the SBE lines by connecting the maxima in dG/dV
gon the
image plot with the primary integer series in solid lines. Thefirst large diamond appears from V
g/similarequal− 2.8 V and closes at
/similarequal−2.7 V, with a width of approximately 1.5 mV . As is well
known, this width is to determine the subband spacing in theQPC. The electrostatic potential at the narrow constriction canbe described as a saddle point model [ 62–64] given by
V(x,y)=V
0−1
2m∗ω2
yy2+1
2m∗ω2
xx2, (4)
where V0is the electrostatic potential at the saddle, and the
confinement potential curvatures are expressed in terms of theharmonic oscillation frequencies ω
xandωy. It should be noted
that our coordinate is different from that used in Ref. [ 63], in
which the propagation direction is x. The subband spacing in
this diamond corresponds to ¯ hωx=0.75 meV. The observed
diamond shapes resemble slightly crushed rhombuses as com-pared to those in previous reports (e.g., [ 58]). Subsequently,
we focus on the small structures by drawing split SBE lines in
115401-3D. TERASAW A et al. PHYSICAL REVIEW B 101, 115401 (2020)
-2.8-2.7 Vg (V)
-0.8 -0.4 0.0 0.4 0.8
Vsd (mV)
100 80 60 40 20 0dG/dVg (a.u.)
αβγ
12G (2e2/h)
-0.8 -0.4 0.0 0.4 0.8
Vsd (mV)
-2.8-2.7-2.6-2.5-2.4Vg (V)
-0.8 -0.4 0.0 0.4 0.8
Vsd (mV)1234
0.25 0.25 0.5
60 40 20 0
dG/dVg (a.u.)
αβγ(a)
(b) (c)
1.01.52.0
0.5
FIG. 3. (a) Image plot of dG/dVgas a function of VsdandVgatT=20 mK and B=0 T with primary SBE lines (solid lines) and with full
SBE splitting lines (dash-dotted lines and a broken line), which were drawn based on the dG/dVgmaxima. The numbers express the plateau
values in the units of G0=2e2/h. (b) Enlarged image plot of dG/dVgwith contours of Gin the units of G0(indicated by the slanted numbers
near the right axis). The line profile at the dashed yellow line is shown later in Fig. 5(d).( c )Gin units of G0as a function of Vsdfor various Vg.
thedG/dVgresult, using dash-dotted lines and a broken line.
An enlarged image plot focusing on the structure in the firstdiamond is shown in Fig. 3(b). From this experimental result,
we observe three split SBE lines corresponding to the threepeaks observed in Fig. 2(b) (α,β, andγ) for the first-integer
SBE. We will demonstrate that this SBE splitting is supportedby the in-plane magnetic fields dependence of dG/dV
g.F i g -
ure3(c) shows the Gprofiles in units of G0as a function of
Vsd. As shown, the conductance is asymmetric with respect
to the positive and negative sides of Vsd. This asymmetry in
Gis large below G<G0. As an example of the asymmetric
behavior, we show a line profile of GatVg=− 2.766 V [the
horizontal broken yellow line in Fig. 3(b)] with a red curve in
Fig. 3(c). This asymmetric behavior was observed previously
[58] and explained in terms of self-gating effects. However, by
analyzing the results of the shot noise measurements, whichwill be presented in Sec. III B , we inferred that this asymmetry
has an intrinsic physical origin.
As we have explained in Sec. II, the in-plane components
of the magnetic field, B
xandBy, can be applied to the QPC
independently. Figure 4shows the image plots of dG/dVg
as functions of (a) VgandBxand (b) VgandBy.A s Bxis
increased with By=0 T (fixed), each SBE except for the
lowest SBE [marked with αin Fig. 4(a)] separates into two,
then the upper branches move upward. Even the SBE betweenthe 0.5 and 1 plateaus decouples into two (marked with βand
γ). Therefore, the SBE under the G
0plateau splits into three,
which is consistent with the observed SBE lines in Fig. 3.T h e
other SBEs show a Zeeman splitting similar to the cases ofmonolayer QPCs [ 65–67]a sB
xincreases. It is remarkable that
the SBE splitting starts at approximately Bx=1 T. However,
as shown in Fig. 4(b), the SBEs indicate no clear dependences
onBybelow 1 T; instead, they decrease slightly, particularly
for higher SBEs. The lowest SBE shows no dependence ofBxandBy. In addition, no clear onset of the second subladder
(antisymmetric wave-function series) occurs for both in-planefields below G<5G
0, contrary to the previous double-layer
QPC data [ 34,35,38].
Figures 5(a)–5(c) show the image plots of dG/dVgfor
Bx=1.0, 2.0 ( By=0 T), and By=1.0T( Bx=0 T), re-
spectively. As Bxincreases, the structure in the first diamond
(indicated by the white circles, SBE lines of βandγin Figs. 3
and 4) shows an interesting change. The lower broad peak
separates into two peaks gradually, in contrast to the upperpeak that becomes a clear single peak. This is demonstratedin Fig. 5(b) (B
x=2.0 T) as we indicate with three white
arrows. Meanwhile, at By=1.0 T, each of the lower and
upper peak smears out and becomes a broad peak. In Fig. 5(d),
3.0 2.0 1.0 0.0
Bx (T) -2.8-2.7-2.6-2.5-2.4Vg (V)
0.512345
1.0 0.0
By (T)-2.8-2.7-2.6-2.5Vg (V)
50403020100dG/dVg (a.u.)
0.51234(a)
(b)
αβγ
FIG. 4. Image plots of dG/dVgas a function of (a) BxandVgat
By=0T ,a n d( b ) ByandVgatBx=0T .
115401-4CONDUCTANCE QUANTIZATION AND SHOT NOISE OF A … PHYSICAL REVIEW B 101, 115401 (2020)
-2.8-2.7Vg (V)
-0.5 0.0 0.5
Vsd (mV)12
-0.5 0.0 0.5
Vsd (mV)12
-2.8-2.7
-0.5 0.0 0.5
Vsd (mV)12
dG/dVg (a.u.)
1.0 0.5 0.0 -0.5
Vsd (mV)By = 1.0 T
Bx = 2.9 T
Bx = 2.0 T
Bx = 1.0 T
B = 0 T(a)
Bx = 1.0 T (b)
Bx = 2.0 T (c)
By = 1.0 T (d)
FIG. 5. Image plots of dG/dVgas a function of VsdandVgat (a) Bx=1.0, (b) Bx=2.0, and (c) By=1.0 T. Primary SBEs are indicated
by the solid white lines. The three white arrows in (b) indicate three peaks inside the first diamond. In addition, the yellow arrow in (b) showsthe Zeeman gap opening. (d) Line profiles of dG/dV
gat the lower peak in the first diamond [at the yellow broken lines in (a)–(c)] as a function
ofVsdforB=0,Bx=1.0,Bx=2.0,Bx=2.9, and By=1.0 T. For the Bx=2.9 T data, see Fig. 14(c) . Each trace is offset for clarity.
we plot the dG/dVgprofile of the lower peak at Vg=2.795 V
[indicated by yellow broken lines in Figs. 5(a)–5(c)]a tB=0,
Bx=1,Bx=2.0,Bx=2.9, and By=1T .A t B=0T ,a
small shoulder appears on the left side of the center peak (at
Vsd=0 mV). However, we observe two peaks at Bx=2.0 and
2.9 T clearly, and at Bx=1.0 T slightly. Thus, the observed
structure inside the first diamond shows a clear dependence onthe magnitude of B
x. Meanwhile, the higher SBEs in Fig. 5(b)
(indicated by the yellow horizontal arrow at Vg=− 2.705 V)
change differently; they exhibit a small diamond structure inaccordance with the Zeeman gap opening as B
xincreases (see
also Fig. 14in Appendix B).
In addition, we observe a result that is different from the
previous results of the 0.7 anomaly. Figure 6shows the image
plots of dG/dVgfor several temperatures from 100 to 600 mK.
Interestingly, the structure inside the first diamond smears outasTis increased, showing a broad vague peak at the center of
the diamond. Therefore, it is clear that the structure observedin this study originates from the band dispersion of the
-2.75-2.70-2.65-2.60Vg (V)
-1.0 -0.5 0.0 0.5 1.0
Vsd (mV)100 mK
-2.75-2.70-2.65-2.60Vg (V)
-1.0 -0.5 0.0 0.5 1.0
Vsd (mV)200 mK
-2.75-2.70-2.65-2.60Vg (V)
-1.0 -0.5 0.0 0.5 1.0
Vsd (mV)400 mK
-2.75-2.70-2.65-2.60Vg (V)
-1.0 -0.5 0.0 0.5 1.0
Vsd (mV)600 mK(a) (b)
(c) (d)
FIG. 6. Image plot of dG/dVgas a function of VsdandVgat
B=0Tf o r T=(a) 100, (b) 200, (c) 400, and (d) 600 mK.double-layer system. Conversely, the dG/dVgminimum for
the 0.5G0plateau is robust. Gforms a clear plateau at 0 .5G0;
after this plateau, it increases without forming additional clearplateaus.
B. Results of the shot noise measurement
To further obtain information on the phenomenon from
a different aspect, we performed shot noise measurements.Figure 7(a) shows Gas a function of V
gat 300 mK. The
overshoot observed at the 0 .5G0plateau is more prominent at
higher temperatures, resembling the one observed in [ 68]. We
attribute the appearance of this overshoot to a resonance modedue to the superimposed transmission and reflection on thelowest SBE at the QPC region. Figure 7(b) shows S
Ias a func-
tion of IsdforVg=− 2.88,−2.85, and −2.83 V. SIshows a
parabolic behavior for |eVsd|/lessorsimilar2kBT, and then shows a linear
dependence for |eVsd|/greaterorsimilar2kBT, which is a typical behavior of
the shot noise with crossover from thermal noise to shot noise.We observe an asymmetric dependence between the positiveand negative I
sdnear G=0.7G0, which was also observed
previously [ 22,58] and explained in terms of the self-gating
effect in QPC. However, this asymmetry in SIis observed
00.511.52G (2e2/h)
-2.90 -2.80 -2.70
Vg (V)3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0SI ( x 10-27 A2/Hz)
-4 0 4
Isd (nA)(a) (b)
FIG. 7. (a) Gas a function of VgatT=300 mK and B=0T
forVsd=0V .( b ) SIas a function of IsdforVg=− 2.88,−2.85, and
−2.83 V (from the bottom trace to the top). Each trace is offset for
clarity. The colors of the traces correspond to the colors of the arrows
in (a).
115401-5D. TERASAW A et al. PHYSICAL REVIEW B 101, 115401 (2020)
1.0
0.5
0.0F
-2.90 -2.85 -2.80 -2.75
Vg (V)012G (2e2/h) F-
F+G
1.0
0.5
0.0F
0 1 2
G (2e2/h) No Spin Splitting
Full Spin Splitting
F- F+(a)
(b)
FIG. 8. (a) F+andF−(left axis) and G(right axis) as a function
ofVgatB=0T .( b ) F+andF−as a function of G. The solid lines and
broken lines represent the theoretically expected values of the Fano
factor for no spin splitting and full spin splitting, respectively. Fano
factor reductions at G=0.5G0and 0.7G0are indicated by the upper
arrows.
only for −2.875/lessorequalslantVg/lessorequalslant−2.84 V (for 0 .5G0<G<G0), and
it does not occur in other Vgvalues, thus suggesting other
possibilities. Accordingly, the slope of SIis always higher
in the negative side of Isdfor 0.5G0<G<G0.A sw eh a v e
stated earlier, we derived the Fano factor from the slope ofS
IasF=SI/(2e/angbracketleftIsd/angbracketright). Due to the asymmetry between the
positive and negative Isdsides of the SI, we used the Fano
factor of the positive side F+and negative side F−, and we
plotted them as a function of Vg, as shown in Fig. 8(a). Further,
the zero bias ( Vsd=0) conductance Gis plotted on the right
axis in Fig. 8(a). Consistent with the SIresult, F−is larger than
F+between the 0 .5G0andG0plateaus.
In a noninteracting scattering process, theory predicts [ 18]
F=/summationtext
nTn(1−Tn)/summationtext
nTn, (5)
where Tndenotes the transmission probability of the nth
channel. We replot F+andF−as a function of Gin Fig. 8(b),
along with the theoretical value of Fwhen no spin splitting
(the solid lines) and full spin splitting (the broken lines)occur. Both F
+andF−are suppressed at G=G0and 2 G0,
thus implying the formation of a single perfect conductancechannel in the coupled DQW for the plateau region. Twoimportant features of F
+and F−observed are (i) a clear
suppression at G=0.5G0and a rapid increase after this
reduction as Gis decreased, and (ii) a small reduction at
G∼0.7G0[both reductions are indicated by the upper arrows1.0
0.5
0.0F+
0 0.5 1
G (2e2/h) B = 0 T
Bx = 1 T
1.0
0.5
0.0F-
0 0.5 1
G (2e2/h) B = 0 T
Bx = 1 T(a)
(b)
FIG. 9. (a) F+and (b) F−as a function of GforB=0Ta n d
Bx=1 T. The solid lines and broken lines represent the same as
those in Fig. 8(b).
in Figs. 8(a) and8(b)]. Regarding the first point, the decrease
in the Fano factor indicates that EFfinishes crossing an SBE.
After the suppression at 0 .5G0, the Fano factor is increased
even when the plateau of Gis established. Generally, the
increase in the Fano factor indicates that a new conductionchannel opens as Gincreases from G=0.5G
0. The second
point suggests that, as shown previously [ 22,23,26] regarding
the 0.7 anomaly, the existing channels’ transmission proba-bilities contribute unequally to the conductance. This smallreduction appears for both F
+andF−.T h e Fvalues are larger
than the theoretical values of Fat the conductance plateau
region. For the enhanced Fano factor, three possibilities canbe considered: electron heating [ 55], channel mixing, and
1/fnoise. However, the 1 /fnoise scarcely contributes to the
enhancement in this experiment due to the noise measurementtechnique using a high resonant frequency LCcircuit and
double-high electron mobility transistor amplifier [ 54].
Furthermore, we measured the shot noise in the presence of
in-plane magnetic fields. Figure 9shows F
+andF−against G
forB=0 and Bx=1 T. In the presence of in-plane magnetic
fields, the Fano factor increases. At Bx=1 T, the difference
between F+andF−becomes larger than the zero-field dif-
ference between the 0.5 and 1 plateau regions. As a notabledifference, F
−obeys the theoretical dependence well.
IV . DISCUSSION
In this section, first we summarize our observations before
presenting a discussion of the results. First, it is shown that
115401-6CONDUCTANCE QUANTIZATION AND SHOT NOISE OF A … PHYSICAL REVIEW B 101, 115401 (2020)
three maxima exist inside the first diamond for the dG/dVgre-
sult, especially in the presence of a large Bx.N e x t , G,dG/dVg,
andFexhibit an asymmetric dependence with respect to Vsd.
However, in our results, an apparent beginning of the second-layer SBE such as those observed in Refs. [ 34,35,38] is not
observed, contrary to expectation. We cannot completely denythe possible effects from double-layer wave-function mixingon the issues above. Thus, we must specify whether our obser-vation originated from double-layer wave functions. Hence,we conducted computer simulations using the
NEXTNANO
simulation software [ 69]. The simulation results do not sup-
port the formation of double-layer wave functions; thus, it isdifficult to explain the results solely based on double-layereffects. Having obtained the simulation results, we propose apossible explanation for the experimental results above usingthe spin effect, i.e., the SOI-modified dispersion relation inparticular.
A. Simulation results
Because the system contains two layers (front and back),
we must consider two subladders for the wave functions andconfinement potentials. We denote the wave function of thesystem as
/Psi1
l,m(x,y,z)=u(y)ψl,m(x,z)( 6 )
with direction yfor propagating modes, and directions xand
zfor lateral and vertical (quantum-well) confinement, respec-
tively. The envelope wave function can further be denoted as
ψl,m(x,z)=χm(x)φp
l(z), (7)
where χm(x) denotes the mth lateral mode and φp
l(z) denotes
thelth vertical wave function in the quantum well. For tunnel-
coupled vertical modes,
φp
l(z)=αϕf
l(z)+βeiθϕb
l(z),α2+β2=1, (8)
where ϕfandϕbdenote the wave function in the front and
back layers (subladder index), respectively, and θdenotes the
interlayer phase difference. The index puses S or AS: for p=
S,θ=0 for the symmetric bonding state, and for p=AS,
θ=πfor the antisymmetric bonding state.
To confirm the SBEs in the first diamond, the wave-
function energies at the QPC were simulated using the self-consistent Schrödinger-Poisson method with
NEXTNANO .W e
first performed a one-dimensional (1D) simulation in the z
direction with reference to the characteristics of the bulk, i.e.,the calculated /Delta1
SASandVgdependence of Gto determine the
simulation parameters (see Appendix C). Subsequently, we
proceeded with two-dimensional (2D) simulations in the xz
plane as a function of Vg. The SBE energies are calculated
as the eigenvalues of the quantized wave functions in the xz
plane under a lateral parabolic confinement potential. It isnoteworthy that although the 2D simulation did not considertheydirection, we assume that the y-directional eigenenergies
exhibit a qualitatively equivalent dependence on V
gin the
QPC region. Thus, the lateral potential and width are deter-mined based on the V
gvalue. Figure 10(a) shows the SBE
energies as a function of Vgat the center of the QPC region.
We found that the energy of the lowest antisymmetric wave10-1710-1510-1310-1110-910-710-510-3
|φ|2
640 620 600
z (nm)-20-1001020E (meV)
S1
Vg=-2.28AS18
6
4
2
0Band Edge (meV)
-2.32 -2.30 -2.28 -2.26 -2.24 -2.22 -2.20
Vg (V)AS2
AS1S6
S5
S4
S3
S2
S1
(a)
(b)
front
back
FIG. 10. (a) Vgdependence of the SBEs. S and AS denote
symmetric and antisymmetric wave functions, respectively, and the
number index represents the mth lateral mode. (b) |φ|2for S1 and
AS1 at Vg=− 2.28 V [the dotted green line on the simulation result
of (a)]. The black line represents the quantum-well potential V(z)f o r
this gate voltage value. The origin of the z-axis starts from the sample
surface.
function ( l=1,p=AS,m=1) was higher than that of the
fifth symmetric wave function ( l=1,p=S,m=5), because
the screening effect of the front layer was extremely strongto allow for the electrons to realize the antisymmetric wavefunction (hereafter, we denote the wave function using twoindexes, pandm, such as AS1, because lis always 1). In
Fig. 10(b) , we show the |φ|
2of the lowest symmetric wave
function (S1) and the antisymmetric wave function (AS1)at the first plateau region. The wave function shows a largeimbalance between the front and back layers, indicating anextremely weak coupling between the two layers under anapplied strong electric field of approximately ∼4V/(μm).
Hence, we expect electrons to exist primarily in the back layerand their wave function to permeate to the front layer; thus, thesystem behaves as a single-layer system with a large potentialgradient toward the front layer.
B. Possible explanation with SOI-induced split
dispersion relation
To explain the structure in the first diamond (indicated by
the white circles in Fig. 5), the following simple relationship
between the density of states (DOS) and conductivity can beuseful. As is well known, the ballistic electron transport in aQPC shows the conductance that changes stepwise depending
115401-7D. TERASAW A et al. PHYSICAL REVIEW B 101, 115401 (2020)
(b) (c)
Zeeman Rashba SOI Rashba SOI + Zeeman(a)
-Ez/2Ez/2E
ky
-ERE
-kR kR
ky
-ER - Ez/2-ER + Ez/2E
-kR kR
ky
FIG. 11. Dispersion relations for (a) Zeeman splitting, (b) Rashba SOI splitting, and (c) Rashba SOI plus Zeeman splitting.
on the number of subbands below the Fermi level. Each
subband carries the current
j=e2Vsdn(E)v(E), (9)
where n(E)=1
2π∂k
∂Edenotes the 1D unidirectional density
of states, and v=2π
h∂E
∂kdenotes the group velocity. There-
fore, cancellation between the DOS and the Fermi velocitycauses the conductance quantization. Equation ( 9) describes
the importance of the DOS, because the conductance is theresult of the integral of the current divided by the appliedvoltage. Experimentally, a sudden DOS change results in alarge conductance jump and a large transconductance peak.In our experiment, the brighter the SBE in the dG/dV
gplot,
the larger are the DOS changes. Therefore, we observedthree large DOS changes within the first diamond, as shownexplicitly in Fig. 5(b).
For the candidate of the threefold DOS change, we suggest
the dispersion relation that splits in the wave number kdi-
rection, such as the SOI-induced splitting [ 43,44,70] and the
in-plane magnetic-field-induced splitting for tunnel-coupleddouble-layer systems [ 71], because three minima appear in the
subbands. However, taking into account the simulation result,the possibility of realizing an in-plane magnetic-field-inducedsplitting is highly unlikely, because well-developed tunnel-coupled wave functions are a prerequisite for this to occur (wewill discuss this in detail later). Regarding the SOI in this case,the space inversion symmetry is expected to be maintained forthexandydirections, but broken for the zdirection. Thus, the
Rashba SOI [ 52] with regard to the potential gradient in the z
direction and the current in the ydirection ([0,0,∂V(z)/∂z]×
[0,k
y,0]/bardblBx)is expected. The Hamiltonian regarding the
Rashba SOI with this broken symmetry is
H=¯h2k2
y
2m∗−¯h2
4m∗2c2σx∂V(z)
∂zky (10)
=¯h2k2
y
2m∗+αRσxky, (11)
where V(z) denotes the potential function of the DQW, σx
denotes the xcomponent of the Pauli matrix, and αRis the
so-called Rashba parameter. From Eq. ( 11) above, we can
derive the dispersion relation with the Rashba SOI as
E/arrowparrleftright(ky)=¯h2k2
y
2m∗±αRky. (12)Then, the energy assumes a minimum value of
−¯h2k2
R/(2m∗)=−ERat ky=∓m∗αR
¯h2=∓kR. Further,
according to analysis [ 43,70], the k-directional split subbands
are mixed; consequently, the subbands repel and open agap into the upper and lower branches [see Fig. 11(b) ].
Importantly, the lower branch contains two minima and theupper branch contains one minimum, at which the up- anddown-spin DOSs are degenerated; hence, this SOI-modifieddispersion exhibits three large DOS changes. Furthermore, inthe presence of B
x,E q .( 12) is modified as follows:
E/arrowparrleftright(ky)=¯h2k2
y
2m∗±α/prime
Rky±1
2g∗μBBx, (13)
-0.10-0.050.000.050.10ΔVsd (mV)
3 2 1 0
Bx (T)30
20
10
0ΔVg (mV)
3 2 1 0
Bx (T)
-2.80-2.75-2.70-2.65Vg (V)
-0.8 -0.4 0.0 0.4 0.8
Vsd (mV)(a)
(b) (c)Zeeman
ΔVg+
ΔVg+ΔVg-
ΔVg-ΔVsd+
ΔVsd+ΔVsd-
ΔVsd-1.52
1Bx = 2.0 T
FIG. 12. (a) Enlarged image plot of dG/dVgatBx=2.0T .T w o
sets of Zeeman splitting SBE lines have been indicated. Plots of (b)
/Delta1Vsdand (c) /Delta1Vgas a function of Bx.
115401-8CONDUCTANCE QUANTIZATION AND SHOT NOISE OF A … PHYSICAL REVIEW B 101, 115401 (2020)
FIG. 13. (a) Gas a function of Bz. (b) SdH oscillation extracted
from (a). /Delta1Grepresents the conductance subtracted the background
conductance change. (c) FFT power spectrum of the data in (b).
where μBdenotes the Bohr magneton. The dispersion re-
lations of the Zeeman splitting, Rashba SOI splitting, andRashba SOI plus Zeeman splitting cases are illustrated inFig. 11. The Rashba parameter should be modified because of
an additional magnetic confinement potential created by B
x,
m∗ω2
Bxz2/2[35](ωBx=eBx/m∗)i nt h e yzplane, as follows:
α/prime
R=¯h2
4m∗2c2∂
∂z/bracketleftbigg
V(z)+1
2m∗ω2
Bxz2/bracketrightbigg
. (14)
Thus, the Rashba energy increases with the increase in Bx,
which is a magnetic field parallel to the Rashba SOI field.This indicates that the two minima in the dispersion curvesof Rashba SOI separate with the increase in B
x; further, the
crossing point and a side of a minimum separate vertically,whereas the other side approaches. As shown in Fig. 5,t h e
lower two maxima inside the first diamond separate as B
x
increases, and thus agree qualitatively to the behavior of
minima in the dispersion curves of Rashba SOI.We extract the positions of the lower two maxima as /Delta1Vsd+
and/Delta1Vsd−. In addition, the separation of the center maximum
and each lower maximum is extracted as /Delta1Vg+and/Delta1Vg−[see
Fig. 12(a) for graphical illustration]. Figures 12(b) and12(c)
show the /Delta1Vsdand the /Delta1Vgvalues, respectively, as a function
ofBx. Although /Delta1Vg−increases slightly, the overall changes
correspond well to the three points in the dispersion curvesof the Rashba SOI plus Zeeman splitting—the crossing pointand the two minima. Therefore, the three maxima observedinside the first diamond can be attributed to these points. Con-sidering that the Rashba SOI field is proportional to
∂V(z)
∂zpy,
the principle behind the observed SOI is simple: the strongpotential gradient and high mobility (or the large relaxationtime [ 72]) of the sample. In our opinion, the center barrier in
the DQW produces this strong potential gradient, as shown inthe potential profile V(z)i nF i g . 10(b) .
Furthermore, the shot noise results support the conjecture
above in that the SBE splitting originates from the SOI. Asshown in Fig. 9, the additional B
xincreases F−to theoretical
values. In addition, the difference between F−andF+becomes
larger at Bx=1 T. Given that Bxis in the same direction
as that of the effective Rashba magnetic field Beff, when the
current flows from the source to drain, Vsd>0 (hence the
electron momentum is in the opposite direction), a positive Bx
supports Beff. However, the situation is completely different
when Vsdis negative, because a positive Bxcancels Beffas
Beffis induced to the negative xdirection. Therefore, in the
presence of the positive Bx, the separation by the Rashba SOI
is enhanced for Vsd>0 and decreased for Vsd<0. Conse-
quently, Gis suppressed for Vsd>0 and hence F+, and vice
v e r s a .A ss h o w ni nF i g . 8, this anisotropic Fano factor is
observed at 0 T. This is attributed to the effective Zeemanenergy gμ
BBeff.
An alternative SOI-like dispersion splitting can be con-
sidered in a tunnel-coupled double-layer system. Accordingto Ref. [ 71], an in-plane field induces the subband splitting
in proportion to the magnitude of the in-plane field in the
-2.8-2.7-2.6-2.5Vg (V)
-0.5 0.0 0.5
Vsd (mV)123
-0.5 0.0 0.5
Vsd (mV)123
-0.5 0.0 0.5
Vsd (mV)12
-2.8-2.7-2.6-2.5
-0.5 0.0 0.5
Vsd (mV)12(a) Bx = 1.0 T (b) Bx = 2.0 T (c) Bx = 2.9 T
(d) By = 1.0 T
Zeeman Zeeman
FIG. 14. Overall view of image plots of dG/dVgas a function of VsdandVgat (a) Bx=1.0, (b) Bx=2.0, (c) Bx=2.9, and (d) By=1.0T .
The yellow arrows in (b) and (c) show subband openings due to the Zeeman splitting. (e) Line profiles of dG/dVgat the white arrows in
(a)–(d) as a function of VsdforB=0,Bx=1.0,Bx=2.9, and By=1.0 T. Each trace is offset for clarity.
115401-9D. TERASAW A et al. PHYSICAL REVIEW B 101, 115401 (2020)
direction perpendicular to the in-plane field for 2DEG sys-
tems. Thus, Bxsplits the subband in the kydirection as /Delta1ky=
d/[¯h/(eBx)]. However, the estimated separation for Bx=1T
is/Delta1ky=3.5×107m−1, thus yielding(¯h/Delta1ky)2
2m∗=0.69 meV.
Although the theory considers a double-layer 2DEG system,this value is significantly large, comparable to the observedfirst diamond splitting. Furthermore, we cannot explain thesmall split that is already observed at the zero magnetic field.In addition, a strong double-layer coupling is a prerequisitefor this splitting. As shown in Fig. 10(b) , the wave functions
in the lower subbands are the highly unbalanced bondingstate. Therefore, this cannot be the primary contribution to thehorizontal splitting.
Finally, we would like to briefly discuss the reentrant con-
ductance behavior that was observed in strong SOI systemsin previous studies [ 43–45]. In this study, a small reentrant
feature was confirmed as shown in Fig. 2(b), and in the
conductance data in Figs. 7and 8, although we interpreted
them as a resonant mode. However, these features are notapparent compared with those in Refs. [ 43–45]. We attribute
this to the band structure of the sample: the second lowestband exists immediately above the lowest band. This configu-ration suppresses “the helical gap” and obscures the reentrantbehavior.
V . CONCLUDING REMARKS AND PERSPECTIVES
Herein we have revealed the SBE lines as a consequence
of the wave-number direction subband splitting induced bya strong SOI. We have observed the coexistence of a 0 .5G
0
plateau and a structure at 0 .7G0in a double-layer QPC system.
The structure observed in the dG/dVgspectroscopy has re-
vealed three maxima corresponding to the three minima in thedispersion relation of the wave-number-directional subbandsplitting. We attribute this splitting to a strong SOI due tothe high potential gradient at the center barrier and the highmobility of the double-layer sample. The Fano factor obtainedfrom the shot noise measurement has indicated an asymmetrictransmission probability. This result further supports the SOI-modified dispersion model and the asymmetry observed in theconductance measurement. However, multiple unansweredquestions still exist that require theoretical considerations
and additional experiments. This experiment includes usefulinformation on spintronics and quantum engineering thatwould benefit applications. In particular, a strong SOI in aGaAs/AlGaAs sample invokes spintronic applications in this
well-developed platform. In addition, we intend to performshot noise measurements in the QHE region of this system inthe future.
ACKNOWLEDGMENTS
We are grateful to K. Muraki and T. Saku of the NTT basic
research laboratories and A. Sawada for providing us with ahigh mobility sample, and to M. Hashisaka and A. Ueda fortheir productive discussion. This work was supported by theJSPS KAKENHI (JP15K17680, JP15H05854, JP18H01815,JP19H05826, JP19H00656).
APPENDIX A: SHUBNIKOV–DE HAAS OSCILLATION
ANALYSIS
First, we measure the Shubnikov–de Haas (SdH) oscilla-
tion at zero bias ( Vsd=0) and zero split gate voltages ( Vg=0)
in low magnetic fields at the lowest temperature available inthis experiment to obtain the electron densities and tunnelcoupling strength between the layers. Figure 13(a) shows G
as a function of B
z. As a clear sign of the weak localization
effect [ 73], positive magnetoconductance is observed initially.
Subsequently, the difference in density between the symmetricstate and the antisymmetric state results in a beating of theSdH oscillations in G[74]. This beating is resolved into
two sharp peaks of a Fourier power spectrum from the fastFourier transform analysis of the 1 /B
zdependence of G,a s
shown in Fig. 13(c) by the arrows. The density ρcorrespond-
ing to each peak is, as we mentioned earlier, 0 .64×1011
and 0.56×1011cm−2from a well-known relation between
the SdH frequency f=/Delta1Bzandρ,ρ=2ef/h, and the
energy separation between the symmetric and antisymmet-ric states /Delta1
SASis/Delta1SAS=π¯h2(ρS−ρAS)/m∗=0.29 meV,
where m∗=0.067mein GaAs with medenoting the electron
rest mass, and ρSandρASdenote the electron density in the
symmetric and antisymmetric states, respectively.
0.6
0.4
0.2
0.0E (eV)
1500 1000 500 0
z (nm) z (nm)0.4
0.2
0.0
Density (1018/cm3)
Fermi energy
-15-10-505
E (meV)
660 640 620 600Fermi energy
|ϕAS|2
|ϕS|2
(a) (b)
FIG. 15. (a) Potential distribution for the z-direction of the sample (upper) and the electron density profile (lower). The two downward
arrows indicate the positions of δ-doping. (b) Probability densities for the lowest two energy wave functions (symmetric ϕSand antisymmetric
ϕASstate) at the DQW confinement for the zdirection.
115401-10CONDUCTANCE QUANTIZATION AND SHOT NOISE OF A … PHYSICAL REVIEW B 101, 115401 (2020)
TABLE II. Comparison of /Delta1SAS between experiment and
calculation.
Experiment Calculation
ρS(×1010/cm2) 6.4 6.2
ρAS(×1010/cm2) 5.6 5.5
/Delta1SAS(mV) 0.29 0.25
APPENDIX B: SUPPLEMENTAL dG/dVgDATA
Figures 14(a) –14(d) show the overall view of the image
plots of dG/dVgas a function of VsdandVgforBx=1.0,
2.0, and 2.9 T and By=1.0 T, respectively. For Bx=2.0 and
2.9 T, the spin degeneracy is resolved for higher SBEs; conse-quently, we observe a minimum (dark region) correspondingto the 2 .5G
0plateau (indicated by yellow arrows). From this
gap opening, the Zeeman splitting is ≈0.09 meV at Bx=
2.0 T. Compared to the bare g-factor of GaAs ( |g|=0.44),
the Zeeman energy, |g|μBB, at this in-plane magnetic field is
approximately twice that of the bare Zeeman splitting.
APPENDIX C: COMPUTER SIMULATION USING
NEXTNANO SOFTWARE
To estimate the double-layer effects on the conductance,
we must calculate the wave functions at the double-layerQPC under a strong electric field confinement. Hence, weused the electronic simulator software,
NEXTNANO [69]. To
supplement the main text, we provide the 1D simulationresults of the /Delta1
SAScalculation and the Vgdependence of the
wave functions. Figure 15(a) shows the potential profile for
thezdirection and the electron density profile. Due to our
careful design, two Si δ-doping positions, indicated by the
two downward arrows, render the DQW symmetric againstthezdirection successfully. Figure 15(b) shows the energy60
40
20
0Electron Density (109/cm2)
-1.5 -1.0 -0.5 0.0
Vg (V)-3-2-1012Eigen-Energy (meV)E2
E1
Dens2
Dens1
FIG. 16. 1D eigenenergies and electron densities for each layer
as a function of Vg. E and Dens represent eigenenergies and electron
densities, respectively; 1 and 2 correspond to the back layer and front
layer, respectively.
of symmetric and antisymmetric wave functions and their
probability density profiles at Vg=0 V. The tunnel gap, /Delta1SAS,
is calculated as 0.25 meV , which is extremely close to theexperimental value. We tabulate the measured and calculatedvalues of /Delta1
SASin Table II, along with the densities of the
lowest symmetric and antisymmetric wave functions.
Figure 16shows the calculated eigenenergies from the
1D simulation ( z-direction) for the lowest wave function and
the electron density for each layer as a function of Vg.A s
Vgincreases in the negative direction, the potential of the
front layer increases, the symmetric state electrons depopulatefrom the front layer, and the energy separation between thesymmetric and antisymmetric wave functions becomes larger.As shown in Fig. 2(a),Gdrops twice at the two downward
arrows. Although these two points represent two pinch-offpoints in the bulk 2DEGs of the front and back layers undertheμm-scale gate electrodes, we assume that the calculation
results above correspond to this Gbehavior.
[1] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G.
Williamson, L. P. Kouwenhoven, D. van der Marel, andC. T. Foxon, Quantized Conductance of Point Contacts ina Two-Dimensional Electron Gas, Phys. Rev. Lett. 60,848
(1988 ).
[2] D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H.
Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A.Ritchie, and G. A. C. Jones, One-dimensional transport andthe quantisation of the ballistic resistance, J. Phys. C 21,L209
(1988 ).
[3] K. J. Thomas, J. T. Nicholls, M. Y . Simmons, M. Pepper, D. R.
Mace, and D. A. Ritchie, Possible Spin Polarization in a One-Dimensional Electron Gas, Phys. Rev. Lett. 77,135(1996 ).
[4] R. Landauer, Spatial variation of currents and fields due to
localized scatterers in metallic conduction, I B MJ .R e s .D e v .
1,223(1957 ).
[5] M. Büttiker, Four-Terminal Phase-Coherent Conductance,
P h y s .R e v .L e t t . 57,1761 (1986 ).
[6] M. Büttiker, Y . Imry, R. Landauer, and S. Pinhas, Generalized
many-channel conductance formula with application to smallrings, Phys. Rev. B 31,6207 (1985 ).[7] H. Bruus, V . V . Cheianov, and K. Flensberg, The anomalous
0.5 and 0.7 conductance plateaus in quantum point contacts,Physica E 10,97(2001 ).
[8] D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane,
R. G. Clark, P. J. Stiles, R. G. Clark, A. R. Hamilton,J. L. O’Brien, N. E. Lumpkin, L. N. Pfeiffer, and K. W.West, Many-body spin-related phenomena in ultra low-disorder quantum wires, Phys. Rev. B 63,121311(R)
(2001 ).
[9] C.-K. Wang and K.-F. Berggren, Local spin polarization in
ballistic quantum point contacts, Phys. Rev. B 57,4552 (1998 ).
[10] S. Daul and R. M. Noack, Ferromagnetic transition and phase
diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping, Phys. Rev. B 58,2635 (1998 ).
[11] K. Yang, Ferromagnetic Transition in One-Dimensional
Itinerant Electron Systems, P h y s .R e v .L e t t . 93,066401
(2004 ).
[12] A. P. Micolich, What lurks below the last plateau: experimen-
tal studies of the 0 .7
×2e2/hconductance anomaly in one-
dimensional systems, J. Phys.: Condens. Matter 23,443201
(2011 ).
115401-11D. TERASAW A et al. PHYSICAL REVIEW B 101, 115401 (2020)
[13] M. Büttiker, Scattering Theory of Thermal and Excess
Noise in Open Conductors, Phys. Rev. Lett. 65,2901
(1990 ).
[14] M. Büttiker, Scattering theory of current and intensity noise
correlations in conductors and wave guides, P h y s .R e v .B 46,
12485 (1992 ).
[15] G. B. Lesovik, Excess quantum noise in 2d ballistic point
contacts, JETP Lett. 49, 592 (1989).
[16] B. Yurke and G. P. Kochanski, Momentum noise in vacuum
tunneling transducers, P h y s .R e v .B 41,8184 (1990 ).
[17] T. Martin and R. Landauer, Wave-packet approach to noise
in multichannel mesoscopic systems, P h y s .R e v .B 45,1742
(1992 ).
[18] Y . Blanter and M. Büttiker, Shot noise in mesoscopic conduc-
tors, Phys. Rep. 336,1(2000 ).
[19] K. Kobayashi, What can we learn from noise?–Mesoscopic
nonequilibrium statistical physics, Proc. Jpn. Acad., Ser. B 92,
204(2016 ).
[20] A. Kumar, L. Saminadayar, D. C. Glattli, Y . Jin, and B. Etienne,
Experimental Test of the Quantum Shot Noise Reduction The-ory,P h y s .R e v .L e t t . 76,2778 (1996 ).
[21] M. Reznikov, M. Heiblum, H. Shtrikman, and D. Mahalu,
Temporal Correlation of Electrons: Suppression of Shot Noisein a Ballistic Quantum Point Contact, P h y s .R e v .L e t t . 75,3340
(1995 ).
[22] P. Roche, J. Ségala, D. C. Glattli, J. T. Nicholls, M. Pepper,
A. C. Graham, K. J. Thomas, M. Y . Simmons, and D. A. Ritchie,Fano Factor Reduction on the 0.7 Conductance Structure of aBallistic One-Dimensional Wire, Phys. Rev. Lett. 93,116602
(2004 ).
[23] L. DiCarlo, Y . Zhang, D. T. McClure, D. J. Reilly, C. M.
Marcus, L. N. Pfeiffer, and K. W. West, Shot-Noise Signaturesof 0.7 Structure and Spin in a Quantum Point Contact, Phys.
Rev. Lett. 97,036810 (2006 ).
[24] G. Gershon, Y . Bomze, E. V . Sukhorukov, and M. Reznikov,
Detection of Non-Gaussian Fluctuations in a Quantum PointContact, Phys. Rev. Lett. 101,016803
(2008 ).
[25] M. Hashisaka, Y . Yamauchi, S. Nakamura, S. Kasai, T. Ono,
and K. Kobayashi, Bolometric detection of quantum shot noisein coupled mesoscopic systems, P h y s .R e v .B 78,241303(R)
(2008 ).
[26] S. Nakamura, M. Hashisaka, Y . Yamauchi, S. Kasai, T. Ono, and
K. Kobayashi, Conductance anomaly and Fano factor reductionin quantum point contacts, P h y s .R e v .B 79,201308(R) (2009 ).
[27] M. Kohda, S. Nakamura, Y . Nishihara, K. Kobayashi, T. Ono,
J. Ohe, Y . Tokura, T. Mineno, and J. Nitta, Spin-orbit inducedelectronic spin separation in semiconductor nanostructures,Nat. Commun. 3,1082 (2012 ).
[28] S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, S. von
Molnár, M. Roukes, A. Chtchelkanova, and D. Treger, Spin-tronics: A spin-based electronics vision for the future, Science
294,1488 (2001 ).
[29] D. D. Awschalom and M. E. Flatté, Challenges for semiconduc-
tor spintronics, Nat. Phys. 3,153(2007 ).
[30] C. H. Bennett and D. P. DiVincenzo, Quantum information and
computation, Nature (London) 404,247(2000 ).
[31] E. Bielejec, J. A. Seamons, J. L. Reno, and M. P. Lilly,
Tunneling and nonlinear transport in a vertically coupledGaAs/AlGaAs double quantum wire system, Appl. Phys. Lett.
86,083101 (2005 ).[32] A. Bertoni, P. Bordone, R. Brunetti, C. Jacoboni, and S.
Reggiani, Quantum Logic Gates Based on Coherent ElectronTransport in Quantum Wires, P h y s .R e v .L e t t . 84,5912 (2000 ).
[33] A. Ramamoorthy, J. P. Bird, and J. L. Reno, Using split-gate
structures to explore the implementation of a coupled-electron-waveguide qubit scheme, J. Phys.: Condens. Matter 19,276205
(2007 ).
[34] K. J. Thomas, J. T. Nicholls, M. Y . Simmons, W. R. Tribe,
A. G. Davies, and M. Pepper, Controlled wave-function mixingin strongly coupled one-dimensional wires, Phys. Rev. B 59,
12252 (1999 ).
[35] G. Salis, T. Heinzel, K. Ensslin, O. J. Homan, W. Bächtold, K.
Maranowski, and A. C. Gossard, Mode spectroscopy and levelcoupling in ballistic electron waveguides, P h y s .R e v .B 60,7756
(1999 ).
[36] K. J. Thomas, J. T. Nicholls, M. Pepper, W. R. Tribe, M. Y .
Simmons, and D. A. Ritchie, Spin properties of low-densityone-dimensional wires, P h y s .R e v .B 61,R13365(R) (2000 ).
[37] S. Nuttinck, K. Hashimoto, S. Miyashita, T. Saku, Y .
Yamamoto, and Y . Hirayama, Quantum point contacts in adensity-tunable two-dimensional electron gas, Jpn. J. Appl.
Phys. 39,L655 (2000 ).
[38] S. F. Fischer, G. Apetrii, U. Kunze, D. Schuh, and G. Abstreiter,
Energy spectroscopy of controlled coupled quantum-wirestates, Nat. Phys. 2,91(2006 ).
[39] L. W. Smith, W. K. Hew, K. J. Thomas, M. Pepper, I. Farrer,
D. Anderson, G. A. C. Jones, and D. A. Ritchie, Row cou-pling in an interacting quasi-one-dimensional quantum wireinvestigated using transport measurements, Phys. Rev. B 80,
041306(R) (2009 ).
[40] S. Ichinokura, H. Hatano, W. Izumida, K. Nagase, and Y .
Hirayama, Electrical control of tunnel coupling between ver-tically coupled quantum point contacts, Appl. Phys. Lett. 103,
062106 (2013 ).
[41] S. Datta and B. Das, Electronic analog of the electro-optic
modulator, Appl. Phys. Lett. 56,665(1990 ).
[42] F. Nichele, S. Hennel, P. Pietsch, W. Wegscheider, P. Stano,
P. Jacquod, T. Ihn, and K. Ensslin, Generation and Detectionof Spin Currents in Semiconductor Nanostructures with StrongSpin-Orbit Interaction, Phys. Rev. Lett. 114,206601 (2015 ).
[43] C. H. L. Quay, T. L. Hughes, J. A. Sulpizio, L. N. Pfeiffer,
K. W. Baldwin, K. W. West, D. Goldhaber-Gordon, and R. dePicciotto, Observation of a one-dimensional spin-orbit gap in aquantum wire, Nat. Phys. 6,336(2010 ).
[44] J. Kammhuber, M. C. Cassidy, F. Pei, M. P. Nowak, A. Vuik, O.
G ü l ,D .C a r ,S .R .P l i s s a r d ,E .P .A .M .B a k k e r s ,M .W i m m e r ,and L. P. Kouwenhoven, Conductance through a helical statein an indium antimonide nanowire, Nat. Commun. 8,478
(2017 ).
[45] S. Heedt, N. Traverso Ziani, F. Crépin, W. Prost, S.
Trellenkamp, J. Schubert, D. Grützmacher, B. Trauzettel, andT. Schäpers, Signatures of interaction-induced helical gaps innanowire quantum point contacts, Nat. Phys. 13,563(2017 ).
[46] A. Srinivasan, D. S. Miserev, K. L. Hudson, O. Klochan, K.
Muraki, Y . Hirayama, D. Reuter, A. D. Wieck, O. P. Sushkov,and A. R. Hamilton, Detection and Control of Spin-Orbit In-teractions in a GaAs Hole Quantum Point Contact, Phys. Rev.
Lett. 118,146801 (2017 ).
[47] T. Masuda, K. Sekine, K. Nagase, K. S. Wickramasinghe,
T. D. Mishima, M. B. Santos, and Y . Hirayama, Transport
115401-12CONDUCTANCE QUANTIZATION AND SHOT NOISE OF A … PHYSICAL REVIEW B 101, 115401 (2020)
characteristics of InSb trench-type in-plane gate quantum point
contact, Appl. Phys. Lett. 112,192103 (2018 ).
[48] M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. Delagrange,
R. Weil, R. Deblock, R. Sakano, A. Oguri, and K. Kobayashi,Universality of non-equilibrium fluctuations in strongly corre-lated quantum liquids, Nat. Phys. 12,230(2016 ).
[49] R. Crook, J. Prance, K. J. Thomas, S. J. Chorley, I. Farrer, D. A.
Ritchie, M. Pepper, and C. G. Smith, Conductance quantizationat a half-integer plateau in a symmetric GaAs quantum wire,Science 312,1359 (2006 ).
[50] P. Debray, S. M. S. Rahman, J. Wan, R. S. Newrock, M. Cahay,
A. T. Ngo, S. E. Ulloa, S. T. Herbert, M. Muhammad, and M.Johnson, All-electric quantum point contact spin-polarizer, Nat.
Nanotech. 4,759(2009 ).
[51] P. P. Das, A. Jones, M. Cahay, S. Kalita, S. S. Mal, N. S. Sterin,
T. R. Yadunath, M. Advaitha, and S. T. Herbert, Dependenceof the 0 .5×(2e
2/h) conductance plateau on the aspect ratio of
InAs quantum point contacts with in-plane side gates, J. Appl.
Phys. 121,083901 (2017 ).
[52] Y . Bychkov and E. Rashba, Properties of a 2d electron gas with
lifted spectral degeneracy, JETP Lett. 39, 78 (1984).
[53] Y . Nishihara, S. Nakamura, K. Kobayashi, T. Ono, M. Kohda,
and J. Nitta, Shot noise suppression in InGaAs /InGaAsP quan-
tum channels, Appl. Phys. Lett. 100,203111 (2012 ).
[54] T. Arakawa, Y . Nishihara, M. Maeda, S. Norimoto, and K.
Kobayashi, Cryogenic amplifier for shot noise measurement at20 mk, Appl. Phys. Lett. 103,172104 (2013 ).
[55] T. Muro, Y . Nishihara, S. Norimoto, M. Ferrier, T. Arakawa,
K. Kobayashi, T. Ihn, C. Rössler, K. Ensslin, C. Reichl, and W.Wegscheider, Finite shot noise and electron heating at quantizedconductance in high-mobility quantum point contacts, Phys.
Rev. B 93,195411 (2016 ).
[56] L. P. Rokhinson, L. N. Pfeiffer, and K. W. West, Spontaneous
Spin Polarization in Quantum Point Contacts, Phys. Rev. Lett.
96,156602 (2006 ).
[57] K. J. Thomas, J. T. Nicholls, N. J. Appleyard, M. Y . Simmons,
M. Pepper, D. R. Mace, W. R. Tribe, and D. A. Ritchie,Interaction effects in a one-dimensional constriction, Phys. Rev.
B58,4846 (1998 ).
[58] A. Kristensen, H. Bruus, A. E. Hansen, J. B. Jensen, P. E.
Lindelof, C. J. Marckmann, J. Nygård, C. B. Sørensen, F.Beuscher, A. Forchel, and M. Michel, Bias and temperaturedependence of the 0.7 conductance anomaly in quantum pointcontacts, Phys. Rev. B 62,10950 (2000 ).
[59] S. M. Cronenwett, H. J. Lynch, D. Goldhaber-Gordon, L. P.
Kouwenhoven, C. M. Marcus, K. Hirose, N. S. Wingreen, andV . Umansky, Low-Temperature Fate of the 0.7 Structure ina Point Contact: A Kondo-Like Correlated State in an Open
System, Phys. Rev. Lett. 88,226805 (2002 ).
[60] T.-M. Chen, A. C. Graham, M. Pepper, I. Farrer, D. Anderson,
G. A. C. Jones, and D. A. Ritchie, Direct observationof nonequilibrium spin population in quasi-one-dimensionalnanostructures, Nano Lett. 10,2330 (2010 ).
[61] C. Rössler, S. Baer, E. de Wiljes, P.-L. Ardelt, T. Ihn, K. Ensslin,
C. Reichl, and W. Wegscheider, Transport properties of cleanquantum point contacts, New J. Phys. 13,113006
(2011 ).
[62] L. I. Glazman, G. B. Lesovik, D. E. Khmel’nitskii, and R. I.
Shekhter, Reflectionless quantum transport and fundamentalballistic-resistance steps in microscopic constrictions, JETPLett. 48, 238 (1988).
[63] M. Büttiker, Quantized transmission of a saddle-point constric-
tion, P h y s .R e v .B 41,7906 (1990 ).
[64] G. B. Lesovik and I. A. Sadovskyy, Scattering matrix approach
to the description of quantum electron transport, Phys.-Usp. 54,
1007 (2011 ).
[65] A. C. Graham, M. Y . Simmons, D. A. Ritchie, and M.
Pepper, Anticrossing of Spin-Split Subbands in Quasi-One-Dimensional Wires, P h y s .R e v .L e t t . 100,226804 (2008 ).
[66] W. K. Hew, K. J. Thomas, M. Pepper, I. Farrer, D. Anderson,
G. A. C. Jones, and D. A. Ritchie, Spin-Incoherent Transport inQuantum Wires, P h y s .R e v .L e t t . 101,036801 (2008 ).
[67] T.-M. Chen, A. C. Graham, M. Pepper, I. Farrer, and D. A.
Ritchie, Bias-controlled spin polarization in quantum wires,Appl. Phys. Lett. 93,032102 (2008 ).
[68] F. Sfigakis, C. J. B. Ford, M. Pepper, M. Kataoka, D. A. Ritchie,
and M. Y . Simmons, Kondo Effect from a Tunable Bound StateWithin a Quantum Wire, P h y s .R e v .L e t t . 100,026807 (2008 ).
[69] https://www.nextnano.de .
[70] O. Goulko, F. Bauer, J. Heyder, and J. von Delft, Effect of
Spin-Orbit Interactions on the 0.7 Anomaly in Quantum PointContacts, Phys. Rev. Lett. 113,266402 (2014 ).
[71] S. K. Lyo, Transport and level anticrossing in strongly coupled
double quantum wells with in-plane magnetic fields, Phys. Rev.
B50,4965 (1994 ).
[72] V . Edelstein, Spin polarization of conduction electrons induced
by electric current in two-dimensional asymmetric electronsystems, Solid State Commun. 73,233(1990 ).
[73] G. Bergmann, Weak localization in thin films: a time-of-
flight experiment with conduction electrons, Phys. Rep. 107,1
(1984 ).
[74] G. S. Boebinger, A. Passner, L. N. Pfeiffer, and K. W. West,
Measurement of Fermi-surface distortion in double quantumwells from in-plane magnetic fields, P h y s .R e v .B 43,12673
(1991 ).
115401-13 |
PhysRevB.78.193404.pdf | Kohn anomaly and interplay of electron-electron and electron-phonon interactions in epitaxial
graphene
S. Y . Zhou,1,2D. A. Siegel,1,2A. V . Fedorov,3and A. Lanzara1,2
1Department of Physics, University of California–Berkeley, Berkeley, California 94720, USA
2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
3Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
/H20849Received 14 October 2008; published 20 November 2008 /H20850
The interplay of electron-phonon /H20849el-ph /H20850and electron-electron /H20849el-el /H20850interactions in epitaxial graphene is
studied by directly probing its electronic structure. We found a strong coupling of electrons to the soft part oftheA
1gphonon evident by a kink at 150 /H1100615 meV, while the coupling of electrons to another expected
phonon E2gat 195 meV can only be barely detected. The possible role of the el-el interaction to account for the
enhanced coupling of electrons to the A1gphonon, and the contribution of el-ph interaction to the linear
imaginary part of the self-energy at high binding energy are also discussed. Our results reveal the dominantrole of the A
1gphonon in the el-ph interaction in graphene and highlight the important interplay of el-el and
el-ph interactions in the self-energy of graphene.
DOI: 10.1103/PhysRevB.78.193404 PACS number /H20849s/H20850: 71.38. /H11002k, 71.20. /H11002b, 71.55.Ak, 71.27. /H11001a
Electron-phonon /H20849el-ph /H20850coupling is among the most im-
portant interactions since it is at the origin of a variety ofinteresting phenomena such as the hoppinglike charge trans-port in organic semiconductors,
1,2charge-density wave
formation,3metal-insulator transition, superconductivity,4
and ballistic transport.5,6The el-ph interaction is particularly
intriguing in graphitic materials, where the special electronicproperties of Dirac fermions and the interplay of el-ph andelectron-electron /H20849el-el /H20850interactions result in a wide range of
novel physics.
7Because of its peculiar pointlike Fermi sur-
face, which can be connected by the wave vectors of thephonons at /H9003andK,
8electron screening of the lattice vibra-
tions decreases dramatically around these two points, caus-ing two Kohn anomalies.
9Moreover, in the case of single-
layer graphene, the peculiar band structure also results in thebreakdown of the Born-Oppenheimer approximation,
10,11a
shift of the E2gphonon frequency as a function of carrier
concentration10–13and sample thickness,14and a predicted
anomalous phonon-induced self-energy15–17that deviates
from that of conventional metals.18
Despite the intense research effort, two key components
in understanding el-ph interaction—the phonon modes in-volved and the coupling strength of the interaction—havenot been resolved. Angle-resolved photoemission spectros-copy /H20849ARPES /H20850is an ideal tool in this respect as it directly
measures the renormalized electronic band structure of a ma-terial and therefore provides direct insights about many-bodyinteractions. In recent years ARPES has been successfullyused to detect the signature of the el-ph interaction in theelectronic spectra in the form of a kink in both graphite
19–21
and graphene.22,23However, not only is there a discrepancy
in the value of the observed coupling strength19–23with re-
spect to the theoretical predictions15–17but also consensus on
which and how many phonon modes are involved has beenmissing so far. Theoretically it was proposed that due to theKohn anomalies at /H9003andK, both the E
2g/H20849195 meV /H20850andA1g
/H20849165 meV /H20850phonons contribute to the el-ph interaction.15Ex-
perimentally, although a kink has been reported in the elec-tronic dispersion,
22,23the large uncertainty in the kink energymakes it difficult to distinguish which, if not both, phonons
are involved. Therefore a more detailed study with improveddata quality is needed to complete our understanding of theel-ph coupling in graphene and to provide key insights forthe el-ph coupling in other graphitic materials.
In this Brief Report we present a high-resolution ARPES
study of the el-ph interaction and its contribution to the elec-tron self-energy in epitaxial graphene. The greatly improveddata quality with reduced noise level has enabled us to naildown the kink energy in the electronic dispersion to150/H1100615 meV and to reveal additional fine structures in the
electron self-energy at 60 /H1100615 and 200 /H1100615 meV. More im-
portantly, the direct comparison between the electronic dis-persion measured here and the reported phonon-dispersionrelation
8has allowed us to identify the soft part of the A1g
phonon /H20849Kohn anomaly /H20850as the main scattering channel re-
sponsible for the ARPES kink and the fine structure at 200meV in the self-energy with the E
2gmode. The enhanced
coupling to the A1gmode with respect to the E2gmode to-
gether with the much larger experimental el-ph couplingstrength /H9261/H110150.14 as compared to the theoretical one is dis-
cussed in terms of Coulomb interactions. In addition, wereport on the linear imaginary part of the self-energy at highbinding energy with similar magnitude along various direc-tions, which reflects the contribution from both el-ph andel-el interactions. Our results point to the dominant role oftheA
1gphonon in the el-ph interaction in graphene and high-
light the important interplay of el-ph and el-el interactions inthe intriguing physics of Dirac fermions in graphene.
High-resolution ARPES data were taken on single-layer
epitaxial graphene at Beamline 12.0.1 /H20849Figs. 1–3/H20850and Beam-
line 7.0.1 /H20849Fig. 4/H20850of the Advanced Light Source /H20849ALS /H20850of
the Lawrence Berkeley National Laboratory with a total-energy resolution of 25 and 35 meV , respectively. Sampleswere grown on n-type SiC wafers as detailed elsewhere.
24,25
The samples were measured with 50 eV photon energy at a
temperature of 25 K and with vacuum better than 3.0/H1100310
−11Torr.
Figure 1/H20849a/H20850shows an ARPES intensity map taken throughPHYSICAL REVIEW B 78, 193404 /H208492008 /H20850
1098-0121/2008/78 /H2084919/H20850/193404 /H208494/H20850 ©2008 The American Physical Society 193404-1the Dirac point /H20849Kpoint /H20850. One can easily identify a charac-
teristic energy /H20849pointed to by a horizontal black arrow /H20850,
where the slope of the dispersion /H20849i.e., velocity /H20850changes and
the intensity suddenly decreases due to the disappearance ofcoherent peaks in the energy distribution curves /H20849EDCs /H20850at
high binding energy. These are typical signatures of electron-boson coupling, where the broadening of the spectra beyondthe kink energy is due to the onset of the bosonic modeself-energy.
19–23To identify the exact bosonic modes in-
volved in the coupling and the strength of the coupling, weextract the dispersion relation from the peak positions andthe Im /H9018from the peak width by fitting the momentum dis-
tribution curves /H20849MDCs /H20850. The high statistics of the data in
panel /H20849b/H20850allows us to nail down the kink position to
150/H1100615 meV.
25,26The extracted kink energy is also consis-tent with a sudden drop of the MDC width /H20851inset of panel
/H20849d/H20850/H20852and a change in the electron velocity /H20851panel /H20849c/H20850/H20852, both
occurring at −150 meV. From the renormalization of theelectron velocity we extract the el-ph coupling constant /H9261
=
vb/vF−1, where vb=1.0/H11003106m/s is the bare band veloc-
ity and vFis the renormalized Fermi velocity vF=0.87
E-EF(eV)
kx(Å-1)0.0
-0.2
1.7 1.80.0
-0.1
-0.2
1.70 1.75
kx(Å-1)(a) (b)
K M
-0.4(c)
(d)ReΣ(eV)
0.000.010.0201
-0.2 -0.1 0.0v (106m/s)
E-EF(eV)0.0 -0.3MDC width
E- EF(eV)
FIG. 1. /H20849Color online /H20850/H20849a/H20850ARPES data taken along KM direc-
tion /H20849solid line in the inset /H20850./H20849b/H20850Dispersion /H20849black curve /H20850extracted
by fitting the raw data. The dashed line is the fit using two straightlines with different slopes. Within 20 meV below E
F, the dispersion
is affected by the resolution, and therefore we fit the dispersion onlyin the range between −250 and −20 meV. The gray dotted line is aguide for the deviation of the low-energy dispersion from the ex-trapolation of the high-energy dispersion. /H20849c/H20850Extracted velocity as a
function of energy from dispersions plotted in panel /H20849b/H20850. The sym-
bols are the raw data and the black solid line is a guide for the eyes./H20849d/H20850Extracted real part of the self-energy as a function of energy
Re/H9018/H20849E/H20850=E/H20849k/H20850−
vbkFwhere E/H20849k/H20850is the measured dispersion and kF
is the Fermi wave vector. The inset shows the MDC width.
E-EF(meV) E-EF(eV)
(b)(a)
Γ M Γ K200
1500.0
-0.15
-0.4
-0.8
A1g∆q~K∆q~0
A1gE2g E2g
FIG. 2. /H20849Color online /H20850/H20849a/H20850Intensity maps for the neighboring K
andK/H11032points as functions of energy and momentum. The two ar-
rows show the el-ph interaction with the E2gphonon and A1gpho-
non, respectively. /H20849b/H20850Phonon dispersions for the A1gnear KandE2g
near/H9003/H20849Refs. 8and9/H20850.
E-EF(eV)
k(Å-1)0.0
-0.1
-0.2
0.02 Å-1
ReΣ(eV)
0.000.020.040.06
E-EF(eV)-0.2 -0.1 0.0(a) (b)
αβγδα β γ δ
αβγδ
FIG. 3. /H20849Color online /H20850/H20849a/H20850Dispersions along various directions
/H20849labeled as /H9251,/H9252,/H9253, and/H9254/H20850shown in the Brillouin zone in the inset.
The dashed lines are fits of the dispersion using two straight lineswith different slopes. The dotted line is a guide to the eyes for thedeviation at low binding energy. The horizontal gray shadow high-lights the kink energy. /H20849b/H20850Corresponding Re /H9018extracted by sub-
tracting the bare band dispersion from the measured dispersion. Thebare band dispersion is taken as a straight line connecting the dis-persions at E
Fand −250 meV.
/s48
/s45 /s49
/s45 /s50
/s45 /s51/s48 /s46 /s53 /s48 /s46 /s48 /s45 /s48 /s46 /s53 /s48 /s46 /s53 /s48 /s46 /s48 /s45 /s48 /s46 /s53 /s48 /s46 /s53 /s48 /s46 /s48 /s45 /s48 /s46 /s53/s69 /s45 /s69/s70/s40 /s101 /s86 /s41
/s107 /s45 /s75 /s40Å-1)/s77 /s68 /s67 /s119 /s105 /s100 /s116 /s104 /s40Å-1)
/s48 /s46 /s48/s48 /s46 /s49/s48 /s46 /s50
/s45 /s51 /s45 /s50 /s45 /s49 /s45 /s51 /s45 /s50 /s45 /s49 /s45 /s51 /s45 /s50 /s45 /s49
/s45 /s51 /s45 /s50 /s45 /s49 /s45 /s51 /s45 /s50 /s45 /s49 /s45 /s51 /s45 /s50 /s45 /s49
/s69 /s45 /s69 /s40 /s101 /s86 /s41/s75/s97
/s99/s98/s40 /s97 /s41 /s40 /s98 /s41 /s40 /s99 /s41/s75 Γ /s75 /s77
/s40 /s100 /s41 /s40 /s101 /s41 /s40 /s102 /s41/s73 /s109Σ( eV )
/s48 /s46 /s48/s48 /s46 /s50/s48 /s46 /s52/s40 /s103 /s41 /s40 /s104 /s41 /s40 /s105 /s41/s75 /s75
FIG. 4. /H20849Color online /H20850/H20849a/H20850–/H20849c/H20850Dispersions along various direc-
tions. The inset shows the constant energy map at −3 eV, where thetrigonal distortion can be clearly observed. The three lines in theinset label the three cuts shown in panels /H20849a/H20850–/H20849c/H20850./H20849d/H20850–/H20849f/H20850Extracted
MDC width as a function of energy for data shown in panels /H20849a/H20850–
/H20849c/H20850. The gray line is a guide for the transition from a more linear
behavior to a more quadratic behavior. /H20849g/H20850–/H20849i/H20850Imaginary part of the
self-energy Im /H9018=
1
2v·/H9004k, where vis the velocity at each energy
and/H9004kis the MDC width subtracted by the width at EF/H20851dotted
lines in panels /H20849d/H20850–/H20849f/H20850/H20852to take care of the impurity scattering and
finite experimental resolution.BRIEF REPORTS PHYSICAL REVIEW B 78, 193404 /H208492008 /H20850
193404-2/H11003106m/s. This gives an experimental /H9261=0.14, which is
almost an order of magnitude larger than the predicted valueof 0.02 for the A
1gphonon.15The identification of the kink at
150 meV with strength of 0.14 is also supported by recentscanning tunneling microscope /H20849STM /H20850measurements.
27
To check whether other phonon modes contribute to this
large value of /H9261, we show in panel /H20849d/H20850the real part of the
electron self-energy Re /H9018. In addition to the main peak at
−150 meV that dominates Re /H9018, two additional fine struc-
tures at /H11015−60 and /H11015−200 meV can also be resolved. The
existence of these fine structures indicates the involvementof other collective modes in the coupling. Since the areaunderneath Re /H9018is an indication of the coupling strength,
clearly these additional modes contribute only a small frac-tion to the total coupling constant and, hence, cannot be re-sponsible for the large discrepancy between the experimentaland theoretical /H9261.
To single out the allowed scattering processes, in Fig. 2
we compare the ARPES constant energy map at E
Fand at the
kink energy /H20851panel /H20849a/H20850/H20852with the predicted phonon dispersion
/H20851panel /H20849b/H20850/H20852.8Clearly the soft part of the A1gphonon near the
zone corner Kpoint is the only mode with the right energy
and momentum q/H11015/H20841/H9003K/H20841to scatter states separated in energy
by 150 meV /H20849kink energy /H20850from near the Kpoint to the K/H11032
point /H20849intervalley scattering /H20850, therefore being likely the
dominant source for the kink in the dispersion and the maxi-mum peak in the Re /H9018. Similarly, the E
2gphonon near the /H9003
point has the right energy /H20849195 meV /H20850and momentum q/H110150
to connect states between 200 meV and the Fermi energywithin the same Kpoint /H20849intravalley scattering /H20850and is re-
sponsible for the fine structure in Re /H9018at 200 meV . Since the
sample is slightly electron doped, the qvector for intravalley
and intervalley scattering is /H110154%/H20841/H9003K/H20841larger than q=/H20841/H9003K/H20841
and 0 but still in the proximity of the two Kohn anomalies.
9
Finally, the fine structure in Re /H9018at 60 meV is likely due to
coupling with an out-of-plane phonon as reported by STMstudies.
28
Figure 3compares the el-ph interaction along different
directions. A similar kink is present in the dispersion alongall the directions at the same energy of 150 /H1100615 meV /H20851see
gray region in panel /H20849a/H20850/H20852. Although similar additional fine
structures involving the two other phonon modes are alsoobserved in the self-energy in panel /H20849b/H20850, the most important
finding is that the A
1gphonon is still the dominant one.
Although on a qualitative level the data presented here are
in good agreement with theoretical calculations, on a quan-titative level there are two key differences: /H208491/H20850Experimen-
tally we found that the intervalley scattering with the A
1g
phonon is by far the dominant scattering source. This is in
contrast to theoretical prediction where both intervalley /H20849A1g
phonon /H20850and intravalley /H20849E2gphonon /H20850scatterings are treated
in an almost equal footing, although the latter is decreased byhalf.
9/H208492/H20850The experimental el-ph coupling strength of 0.14,
mostly accounted for by the A1gphonon /H20849as discussed in Fig.
1/H20850, is much larger than the predicted value of 0.02.15This
holds even if finite experimental resolution, which makes /H9261
twice as big, is taken into account.15Therefore, an additional
mechanism needs to be included to explain the observed en-hancement, by approximately a factor of 3, of the el-ph cou-pling strength. One likely candidate is through the interplaywith el-el interaction, as pointed out theoretically.
15,16More
specifically, it has been argued that this interplay gives rise,in the presence of a linear dispersion, to a linear imaginarypart of the self-energy Im /H9018in agreement with experimental
reports,
22,29and that the el-ph coupling contributes to 1/3 of
its total magnitude.16By extending this study to the entire
momentum region /H20849Fig.4/H20850, we have shown that this linearity
in Im /H9018survives with a similar magnitude throughout the
entire Dirac cone, even when the dispersion is not linearbecause of the trigonal distortions /H20851Figs. 4/H20849b/H20850and4/H20849c/H20850/H20852, and
is hence a general property of Dirac fermions. Panels /H20849a/H20850–/H20849c/H20850
show the ARPES intensity maps along three different direc-tions. From the /H9003K/H20851panel /H20849a/H20850/H20852toMK direction /H20851panel /H20849c/H20850/H20852,
both the extracted dispersion /H20851see solid black line in panels
/H20849a/H20850–/H20849c/H20850/H20852and MDC width between −1 and −3 eV change
from a linear to a quadratic behavior due to the trigonaldistortion /H20851see also inset of panel /H20849a/H20850/H20852.
30The corresponding
Im/H9018are shown in panels /H20849g/H20850–/H20849i/H20850. Clearly, even when the
dispersion is not linear /H20851panel /H20849c/H20850/H20852,I m/H9018still shows a linear
dependence with similar magnitude /H20851panel /H20849i/H20850/H20852, suggesting
that the overall contribution of the el-ph interaction to theself-energy of graphene is comparable along all directions, inline with the isotropic el-ph coupling reported in Fig. 3and
theoretical prediction.
31
These data clearly establish the importance of the inter-
play between el-ph and el-el interactions suggesting that thelatter might be responsible for the observed enhancement ofthe coupling strength. Indeed it has been argued that in thecompletely unscreened case, the el-el interaction can enhancethe coupling to the A
1gphonon near Kby up to a factor of 3,
leaving the coupling to the E2gphonon near /H9003almost
unaffected.32This picture reconciles the disagreement be-
tween the theoretical and experimental coupling strengthmeasured by ARPES and can also account for the large in-tensity ratio between the 2 Dand 2 D
/H11032peaks reported by Fer-
rariet al. ,33which is likely due to the enhanced renormaliza-
tion of the el-ph coupling of peak D. Finally it is interesting
to note that a similar enhanced coupling to phonons withnonzero wave vectors through el-el interaction also occurs inthe case of transition-metal dichalcogenides where the qua-siparticles are Dirac fermions, resulting as well in a linearIm/H9018.
34These similarities suggest that the physics here dis-
cussed is not only a property of graphene but a more generalproperty of Dirac materials.
35
In conclusion, we have reported on the strong interplay of
el-ph and el-el interactions in graphene. We identified thedominant role of the A
1gphonons at 150 meV in the el-ph
interaction along the various directions near the Kpoint. Al-
though the fine structures due to coupling with other phononmodes are observed, we show that the enhancement of thecoupling to the soft part of the A
1gmode is likely induced by
the interplay between el-el and el-ph interactions. This studydemonstrates the important role of this interplay in a Diracfermion system and highlights the importance of includingboth interactions in the self-energy of graphene.
We thank D.-H. Lee for useful discussions. This work was
supported by the National Science Foundation through GrantBRIEF REPORTS PHYSICAL REVIEW B 78, 193404 /H208492008 /H20850
193404-3No. DMR03-49361, by the Office of Science, Office of Basic
Energy Sciences, Division of Materials Sciences and Engi-neering of the U.S. Department of Energy under ContractNo. DEAC03-76SF00098, and by the Laboratory DirectedResearch and Development Program of Lawrence Berkeley
National Laboratory under the Department of Energy Con-tract No. DE-AC02-05CH11231. S. Y . Zhou thanks the Ad-vanced Light Source for financial support.
1V . Coropceanu, M. Malagoli, D. A. da Silva Filho, N. E. Gruhn,
T. G. Bill, and J. L. Bredas, Phys. Rev. Lett. 89, 275503 /H208492002 /H20850.
2H. Yamane, S. Nagamatsu, H. Fukagawa, S. Kera, R. Friedlein,
K. K. Okudaira, and N. Ueno, Phys. Rev. B 72, 153412 /H208492005 /H20850.
3G. Gruner, Rev. Mod. Phys. 60, 1129 /H208491988 /H20850.
4J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108,
1175 /H208491957 /H20850.
5A. Javey, J. Guo, Q. Qang, M. Lundstrom, and H. Dai, Nature
/H20849London /H20850424, 654 /H208492003 /H20850.
6Z. Yao, C. L. Kane, and C. Dekker, Phys. Rev. Lett. 84, 2941
/H208492000 /H20850.
7A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,
and A. K. Geim, arXiv:0709.1163v2, Rev. Mod. Phys. /H20849to be
published /H20850.
8J. Maultzsch, S. Reich, C. Thomsen, H. Requardt, and P. Orde-
jon, Phys. Rev. Lett. 92, 075501 /H208492004 /H20850.
9S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robert-
son, Phys. Rev. Lett. 93, 185503 /H208492004 /H20850.
10S. Pisana, M. Lazzeri, C. Casiraghi, K. S. Novoselov, A. K.
Geim, A. C. Ferrari, and F. Mauri, Nature Mater. 6, 198 /H208492007 /H20850.
11A. H. Castro Neto, Nature Mater. 6, 176 /H208492007 /H20850.
12J. Yan, Y . Zhang, P. Kim, and A. Pinczuk, Phys. Rev. Lett. 98,
166802 /H208492007 /H20850.
13A. H. Castro Neto and F. Guinea, Phys. Rev. B 75, 045404
/H208492007 /H20850.
14A. Gupta, G. Chen, P. Joshi, S. Tadigadapa, and P. C. Eklund,
Nano Lett. 6, 2667 /H208492006 /H20850.
15M. Calandra and F. Mauri, Phys. Rev. B 76, 205411 /H208492007 /H20850.
16C.-H. Park, F. Giustino, M. L. Cohen, and S. G. Louie, Phys.
Rev. Lett. 99, 086804 /H208492007 /H20850.
17W.-K. Tse and S. Das Sarma, Phys. Rev. Lett. 99, 236802
/H208492007 /H20850.
18N. W. Ashcroft and N. D. Mermin, Solid State Physics /H20849Thomas
Learning, United States, 1976 /H20850.
19S. Y . Zhou, G.-H. Gweon, and A. Lanzara, Ann. Phys. /H20849N.Y . /H20850
321, 1730 /H208492006 /H20850.
20K. Sugawara, T. Sato, S. Souma, T. Takahashi, and H. Suematsu,
Phys. Rev. Lett. 98, 036801 /H208492007 /H20850.21C. S. Leem, B. J. Kim, C. Kim, S. R. Park, T. Ohta, A. Bostwick,
E. Rotenberg, H. D. Kim, M. K. Kim, H. J. Choi, and C. Kim,Phys. Rev. Lett. 100, 016802 /H208492008 /H20850.
22A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg,
Nat. Phys. 3,3 6 /H208492006 /H20850.
23J. L. McChesney, A. Bostwick, T. Ohta, K. V . Emtsev, Th. Sey-
ller, K. Horn, and E. Rotenberg, arXiv:0705.3264 /H20849unpublished /H20850.
24C. Berger, Z. Song, T. Li, X. Li, A. Y . Ogbazghi, R. Feng, Z.
Dai, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. deHeer, J. Phys. Chem. B 108, 19912 /H208492004 /H20850.
25S. Y . Zhou, G.-H. Gweon, A. V . Fedorov, P. N. First, W. A. de
Heer, D.-H. Lee, F. Guinea, A. H. Castro Neto, and A. Lanzara,Nature Mater. 6, 770 /H208492007 /H20850.
26E. Rollings, G.-H. Gweon, S. Y . Zhou, B. S. Mun, J. L. Mc-
Chesney, B. S. Hussain, A. V . Fedorov, P. N. First, W. A. deHeer, and A. Lanzara, J. Phys. Chem. Solids 67, 2172 /H208492006 /H20850.
27Note that the kink energy is far away from the deviation near the
Dirac point /H20849/H11015−0.4 eV /H20850reported in epitaxial graphene /H20849Refs. 22
and35/H20850and is therefore unaffected by such deviation.
28G. Li, A. Luican, and E. Y . Andrei, arXiv:0803.4016 /H20849unpub-
lished /H20850.
29Y . B. Zhang, V . W. Brar, F. Wang, C. Girit, Y . Yayon, M. Pan-
lasigui, A. Zettl, and M. F. Crommie, Nat. Phys. 4, 627 /H208492008 /H20850.
30S. Xu, J. Cao, C. C. Miller, D. A. Mantell, R. J. D. Miller, and Y .
Gao, Phys. Rev. Lett. 76, 483 /H208491996 /H20850.
31S. Y . Zhou, G.-H. Gweon, J. Graf, A. V . Fedorov, C. D. Spataru,
R. D. Diehl, Y . Kopelevich, D.-H. Lee, S. G. Louie, and A.Lanzara, Nat. Phys. 2, 595 /H208492006 /H20850.
32C. H. Park, F. Giustino, J. L. McChesney, A. Bostwick, T. Ohta,
E. Rotenberg, M. L. Cohen, and S. G. Louie, Phys. Rev. B 77,
113410 /H208492008 /H20850.
33D. M. Basko and I. L. Aleiner, Phys. Rev. B 77, 041409 /H20849R/H20850
/H208492008 /H20850.
34A. C. Ferrari, J. C. Meyer, V . Scardaci, C. Casiraghi, M. Lazzeri,
F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, andA. K. Geim, Phys. Rev. Lett. 97, 187401 /H208492006 /H20850.
35A. H. Castro Neto, Phys. Rev. Lett. 86, 4382 /H208492001 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 78, 193404 /H208492008 /H20850
193404-4 |
PhysRevB.74.125118.pdf | Potential superhard osmium dinitride with fluorite and pyrite structure:
First-principles calculations
Chang-Zeng Fan *and Song-Yan Zeng
Department of Material Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
Li-Xin Li, Zai-Ji Zhan, Ri-Ping Liu, and Wen-Kui Wang
Key Laboratory of Metastable Material Science and Technology, Yanshan University, Qinhuangdao 066004, China
Ping Zhang
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Yu-Gui Yao
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
/H20849Received 14 November 2005; revised manuscript received 2 August 2006; published 28 September 2006 /H20850
We have performed systematic first-principles calculations on dicarbide, -nitride, -oxide, and -boride of
platinum and osmium with the fluorite structure. It is found that only PtN 2,O s N 2, and OsO 2are mechanically
stable. In particular, OsN 2has the highest bulk modulus of 360.7 GPa. Both the band structure and density of
states show that the new phase of OsN 2is metallic. The high bulk modulus is owing to the strong covalent
bonding between Os 5 dand N 2 pstates and the dense packed fluorite structure. In addition, the total-energy
calculation for pyrite structure has also been performed, which indicates its mechanical and energetic stabilitybut much lower bulk modulus compared to the fluorite structure.
DOI: 10.1103/PhysRevB.74.125118 PACS number /H20849s/H20850: 81.05.Zx, 62.20.Dc, 71.20.Be, 61.66.Fn
The search for hard materials compared to or even harder
than diamond, which has the highest measured hardness of96 GPa /H20849Ref. 1/H20850and bulk modulus of 443 GPa,
2has a long
history and has stimulated a variety of great achievements inhigh-pressure research.
3–6Consequently, many new super-
hard materials have been prepared by high-pressure tech-niques, especially after the laser-heated diamond-anvil cellswas invented. In general, two groups of materials are pow-erful candidates for superhard materials: /H20849i/H20850strong covalent
compounds formed by light elements, such as polymorphy ofC
3N4,7B6O,8and c-BC 2N.9/H20849ii/H20850Partially covalent heavy
transition metal boride, carbide, nitride, and oxide. RuO 2
/H20849Ref. 10/H20850and OsB 2/H20849Ref. 11/H20850are such examples. Theoreti-
cally, the nature of hardness has been extensively investi-gated and many new models have been proposed.
1,3,12–16For
the strong covalent materials, hardness can be directlyderived,
13–16while for some metallic transition metal-based
superhard materials, it is acknowledged that bulk modulus orshear modulus can measure the hardness in an indirectway.
1,12,17That is, materials with high bulk or shear modulus
are likely to be hard materials.
In the present paper, we focused on the bulk modulus,
mechanical, and energetic stability of osmium dinitride/H20849OsN
2/H20850with fluorite structure18by calculating the elastic
constants within the density functional based electronic
structure method.19We report that the proposed OsN 2com-
pound has very high value of bulk modulus /H20849360.7 GPa /H20850
which is even higher than that of OsO 2with the same struc-
ture /H20849347.5 GPa /H20850and is comparable with that of orthorhom-
bic OsB 2/H20851365–395 GPa /H20849Ref. 11/H20850/H20852.
All first-principles calculations were performed with the
CASTEP code.20The ultrasoft pseudopotential21was em-
ployed to describe the interaction between ions and elec-trons. Both the local-density approximation /H20849LDA /H20850/H20849Ref.22/H20850
and the generalized gradient approximation /H20849GGA /H20850/H20849Ref.23/H20850
were used to describe the exchange and correlation poten-tials. For the Brillouin-zone sampling, the Monkhorst-Pack/H20849MP/H20850scheme with a grid of 0.03 Å
−1was adopted.24The
plane-wave cutoff energy is chosen to be 550 eV for LDAand 500 eV for GGA calculations. For the self-consistentfield iterations, the convergence was assumed when /H20849i/H20850the
total energy difference between the last two cycles was lessthan 1 /H1100310
−6eV/atom; /H20849ii/H20850the maximal force on each atom
was below 0.006 eV Å−1; and /H20849iii/H20850the maximal atomic dis-
placement was below 2 /H1100310−4Å. We have tested that with
even more strict parameters the total energy can be con-verged within 0.002 eV/atom for all the systems studied. Af-ter getting the equilibrium geometry configuration, we ap-plied the so-called “stress-strain” method to obtain the elasticconstants in that the stress can be easily obtained within thedensity functional based electronic structure method.
25The
stress-strain relation can be described as
/H20849/H92681,/H92682,/H92683,/H92684,/H92685,/H92686/H20850=C/H20849/H92551,/H92552,/H92553,/H92554,/H92555,/H92556/H20850T. /H208491/H20850
For the cubic crystal, there are only three nonzero indepen-
dent symmetry elements /H20849c11,c12, and c44/H20850. Applying two
kinds of cubic distortions /H20849/H92551and/H92554/H20850along the crystallo-
graphic directions shown in Figs. 1/H20849a/H20850and1/H20849b/H20850, respectively,
can give stresses relating to these three elastic coefficients,yielding an efficient method for obtaining elastic constantsfor the cubic system. For the hexagonal crystal, there are fiveindependent symmetry elements /H20849c
11,c12,c13,c33, and c44/H20850.
In order to obtain the additional components c13and c33,
another monoclinic distortion /H20851/H92553, see Fig. 1/H20849c/H20850/H20852is needed.
For each strain, in our practical calculations, its value is var-PHYSICAL REVIEW B 74, 125118 /H208492006 /H20850
1098-0121/2006/74 /H2084912/H20850/125118 /H208496/H20850 ©2006 The American Physical Society 125118-1ied from −0.003 to +0.003 with a step of 0.0012, then each
of three elastic constants takes the arithmetic average valueof the six steps. The bulk modulus is obtained from the elas-tic constants by the relation B=/H20849c
11+2c12/H20850/3.
The lattice and elastic constants of diamond, pure plati-
num, hexagonal, and cubic pure osmium were calculated toverify the reliability of the present calculations. It is wellknown that LDA usually underestimates the lattice constantsand overestimates the elastic constants, while GGA overesti-mates the lattice constants and underestimates the elasticconstants.
26,27For this reason we adopted to use the average
of the LDA and GGA results as our theoretical estimates. Asshown in Table I, the theoretical average lattice constant and
bulk modulus of diamond are 3.546 Å and 446.5 GPa, whichagree well with the experimental values of 3.567 Å /H20849Ref.12/H20850
and 443 GPa,
2with an error of 0.59% and 0.79%, respec-
tively. For the cubic osmium and cubic platinum, the theo-retical bulk modulus values /H20849Pt: 287.2 GPa, Os: 418.8 GPa /H20850
are also in accordance with other theoretical results /H20851Pt:
279 GPa,
26Os: 417.1 GPa /H20849Ref. 28/H20850/H20852. With regard to thehexagonal Os, as shown in Table I, the calculated equilib-
rium lattice parameter c/ais 1.581, in good agreement with
the experimental value of 1.580.29,30The calculated bulk
modulus for hexagonal Os is 424.0 GPa, which is also inaccordance with previous theoretical results /H20851403 GPa,
31
429.2 GPa /H20849Ref. 28/H20850/H20852and experimental measurements
/H20851462±12 GPa,29411±6 GPa,32395±15 GPa /H20849Ref. 33/H20850/H20852.
Based on above-mentioned accordance, therefore, we believethat the plane-wave ultrasoft pseudopotential /H20849PW-PP /H20850
method we employed is reliable in investigating the me-chanical properties of osmium and platinum compounds.
Now we turn to fully study OsN
2with fluorite structure.
The results of lattice constant, elastic constants, and bulkmodulus of OsN
2are listed in Table II. For comparison, we
have in addition given a calculation on platinum dinitride/H20849PtN
2/H20850and osmium dioxide /H20849OsO 2/H20850, and the results are also
listed in Table II. Given the elastic constants and bulk modu-
lus, the shear modulus Gand the Young’s modulus Ecan be
deduced as follows: G=/H20849c11−c12+2c44/H20850/4,E=9BG//H208493B
TABLE I. The calculated equilibrium lattice parameters a/H20849Å/H20850, elastic constants cij/H20849GPa /H20850, bulk modulus B/H20849GPa /H20850, polycrystalline shear
modulus G/H20849GPa /H20850, Young’s modulus E/H20849GPa /H20850, and Poisson’s ratio /H9263of typical pure crystals. The boldface numbers represent the theoretical
estimations of bulk modulus by present calculations.
ac 11 c33 c44 c13 c12 BG E /H9263
Diamond LDA 3.525 1105.8 607.3 140.5 462.3 545.0 1173.8 0.08
GGA 3.566 1053.3 569.1 119.5 430.7 518.0 1109.3 0.07
Ave. 3.546 1079.6 588.2 130.0 446.5 531.5 1141.6 0.08
Expt. 3.567a443b
Pt/H20849fcc/H20850 LDA 3.921 /H208493.890c/H20850 391.1 82.3 279.0 316.4 /H20849320c/H20850 69.2 193.5 0.40
GGA 3.998 /H208493.967c/H20850 307.9 65.7 232.9 257.9 /H20849238c/H20850 51.6 145.1 0.41
Ave. 3.960 /H208493.928c/H20850 349.5 74.0 256.0 287.2 /H20849279c/H20850 60.4 169.3 0.41
Expt. 3.924d276e
Os/H20849fcc/H20850 LDA 3.798 /H208493.814f/H20850 686.9 361.3 323.1 444.4 /H20849441.3f/H20850 271.6 676.9 0.25
GGA 3.851 /H208493.851f/H20850 614.7 328.0 282.5 393.2 /H20849392.9f/H20850 247.1 612.9 0.24
Ave. 3.824 /H208493.841f/H20850 650.8 344.7 302.8 418.8 /H20849417.1f/H20850 259.4 644.9 0.25
Os/H20849hcp/H20850 LDA 2.712 808.7 888.6 271.2 264.7 243.7 449.0
GGA 2.750 /H208492.746a/H20850 730.1 798.3 246.9 230.5 209.8 398.9
Ave. 2.731 1538.8 843.5 259.1 247.6 226.8 424.0
Expt. 2.7313g395,g462h
aReference 12.
bReference 2.
cReferences 26and37.
dReference 47.eReference 48.
fReference 28.
gReference 33.
hReference 29.
FIG. 1. /H20849Color online /H20850The schematic of strain
types: /H20849a/H20850/H92551;/H20849b/H20850/H92554;/H20849c/H20850/H92553.FAN et al. PHYSICAL REVIEW B 74, 125118 /H208492006 /H20850
125118-2+G/H20850, and v=E//H208492G/H20850−1. These quantities are also shown in
Table II.
The key criteria for mechanical stability of a crystal is that
the strain energy must be positive,34which means in a hex-
agonal crystal that the elastic constants should satisfy thefollowing inequalities:
c
44/H110220,c11/H11022/H20841c12/H20841,/H20849c11+c12/H20850c33/H110222c132, /H208492/H20850
while for a cubic crystal,
c44/H110220,c11/H11022/H20841c12/H20841,c11+2c12/H110220. /H208493/H20850
It is straightforward to verify from Table Ithat the elastic
constants of the hexagonal osmium satisfy formula /H208492/H20850, im-
plying the stability of hcp Os, which is consistent with theexperimental observation. In the same manner, from our cal-culation results in Table II, one can find that PtN
2, OsN 2and
OsO 2with fluorite structure are also mechanically stable
since their elastic constants fit well in formula /H208493/H20850. The sta-
bility of these three crystals can also be confirmed by pro-viding the Poisson’s ratio, whose value is usually between −1and 0.5, corresponding to the lower and upper limit wherethe materials do not change their shapes. Note that thepresent result of bulk modulus of PtN
2is 295.2 GPa. The
previous full-potential linearized augmented plane wavescalculation gives 290 GPa.
26Remarkably, the two ap-
proaches agree well, suggesting again the reliability ofPW-PP method in exploring the structural properties of tran-sition metal compounds. On the other hand, we obtained thebulk modulus of OsO
2to be 347.5 GPa, which is /H1101113%
smaller than that obtained from the full-potential linearmuffin-tin orbital method.
35,36This difference may comefrom different density functional based electronic structure
method. It reveals in Table IIthat OsN 2has the highest bulk
modulus of 360.7 GPa in our series of calculations; thisvalue of OsN
2is much higher than that of other noble metal
dinitride /H20849347 GPa for IrN 2, 190 GPa for AgN 2, and
222 GPa for AuN 2/H20849Ref. 37/H20850. The shear modulus of OsN 2is
calculated to be 103.4 GPa, comparable with that of PtN 2
/H20849104.6 GPa /H20850as shown in Table II, but much smaller than
those of diamond /H20849531.5 GPa /H20850and OsO 2/H20849237.1 GPa /H20850. Thus
compared to diamond or OsO 2, OsN 2cannot withstand the
shear stress to a large extent. It is interesting to point out thatOsN
2was implicitly referred to in Ref. 37to be unstable or
a little bulk modulus.38
Furthermore, the electronic structure and chemical bond-
ing of OsN 2with fluorite structure are studied by calculating
its total charge density, Mulliken population, and density ofstate /H20849DOS /H20850. In Fig. 2, we plot the total electron density in a
/H208491¯10/H20850plane which cut through both the Os and N sites. The
bonding behavior of OsN 2can be effectively revealed by
analyzing the charge density data in real space /H9267/H20849r/H20850at three
types of crystalline symmetry points, as indicated by filled
circles, open squares, and open circles in Fig. 2. We found
that the charge density of these three kinds of points areabout 0.8, 0.3, and 0 eÅ
−3, respectively. Thus the charge
density maximum lies between Os and N atoms, indicatingformation of strong covalent bonding between them. Com-bining the fact that each N atom occupies the tetrahedralinterstitial formed by four Os atoms around it, it is not diffi-cult to understand that OsN
2has a low compressibility. Table
IIIshows bond Mulliken population analysis of OsN 2, OsO 2,
and PtN 2. It indicates that for these three kinds of materials
the bonding is formed between metal atom and nonmetalTABLE II. The calculated equilibrium lattice parameters a/H20849Å/H20850, elastic constants cij/H20849GPa /H20850, bulk modulus B/H20849GPa /H20850, polycrystalline shear
modulus G/H20849GPa /H20850, Young’s modulus E/H20849GPa /H20850, and Poisson’s ratio /H9263of some fluorite and pyrite crystals. The boldface numbers represent the
theoretical estimations of bulk modulus by present calculations.
ac 11 c44 c12 BG E /H9263
OsO 2/H20849fluorite /H20850 LDA 4.770 /H208494.763a/H20850 721.3 243.1 206.6 378.2 /H20849411,a392b/H20850 250.2 615.0 0.23
GGA 4.861 632.2 211.2 158.9 316.7 223.9 543.6 0.21
Ave. 4.816 676.8 227.0 182.8 347.5 237.1 579.3 0.22
PtN 2/H20849fluorite /H20850 LDA 4.943 /H208494.866c/H20850 499.9 87.4 232.6 321.7 /H20849316c/H20850 110.5 297.4 0.35
GGA 5.040 /H208494.958c/H20850 427.9 77.5 188.6 268.3 /H20849264c/H20850 98.6 263.5 0.34
Ave. 4.992 /H208494.912c/H20850 463.9 82.5 210.6 295.0 /H20849290c/H20850 104.6 280.5 0.35
PtN 2/H20849pyrite /H20850 GGA 4.874 /H208494.875d/H20850 689 129 102 297.8 /H20849278d/H20850 211.3 512.7 0.21
GGAe4.862 668 99 167 272 184 452 0.23
OsN 2/H20849fluorite /H20850 LDA 4.781 544.5 103.9 309.8 388.0 117.4 319.9 0.36
GGA 4.856 465.4 79.7 267.3 333.3 89.4 246.2 0.38
Ave. 4.819 505.0 91.8 288.6 360.7 103.4 283.1 0.37
OsN 2/H20849pyrite /H20850 GGA 4.925 523 107 213 316 131 345.3 0.32
aReference 35.
bReference 36.
cReferences 26and37.
dReference 42.
eReference 41.POTENTIAL SUPERHARD OSMIUM DINITRIDE WITH … PHYSICAL REVIEW B 74, 125118 /H208492006 /H20850
125118-3atom, while a weak bonding is formed between two non-
metal atoms. This is compatible to the analysis of the elec-tron density of OsN
2in Fig. 2. Table IIIalso lists the Mul-
liken atomic population analysis results, from which we cansee the total charge transfer from Os to N is 1.10, resultingOs in +1.10 charge state and N −0.55 charge state. There-fore, the chemical bonding between Os and N has some char-acter of ionicity. Table IIIshows that the transferred charge
in OsN
2is almost the same as that in OsO 2, and is more than
that in PtN 2. Thus we can make a conclusion that the charge
transfer effect is more influenced by the metal atom ratherthan the nonmetal atom. It is interesting to note that themechanical properties of OsN
2are also very similar to OsO 2
rather than PtN 2, as revealed in our above discussions.
The partial DOS is shown in Fig. 3; no energy gap near
the Fermi level is seen, indicating the metallic nature ofOsN
2. At the Fermi level the total DOS is 1.88 states/ev
formula units. It reveals that from −18 to −12 eV the statesare mainly N /H208492s/H20850states with a small contribution from Os
/H208495dand 6 p/H20850. The states above −9 eV mainly come from Os
5da n dN2 porbitals.
It was recently demonstrated in both theory and experi-
ment that the synthesized platinum nitride crystallized in py-rite structure.
39–42The pyrite structure /H20849space group number
205 /H20850, which was also observed in the silica recently,43is
cubic with 12 atoms per primitive cell. For pyrite PtN 2, boththe four Pt atoms and the midpoints of the four nitrogen pairs
arrange in fcc positions and result in a NaCl-type arrange-ment. In addition, each pair of nitrogen atoms aligns alongone of the /H20849111/H20850directions. Besides the lattice constant a, the
position /H20849u,u,u/H20850of N is the only free structural parameter of
pyrite PtN
2. Inspired by these advances, we have also per-
formed a series of ab initio total-energy calculations to find if
OsN 2favors the intriguing pyrite structure as platinum ni-
tride does. Figure 4/H20849a/H20850gives the energy of OsN 2with the
internal parameter uvaried from 0.23 to 0.40. The plot re-
veals that fluorite OsN 2lies at a local minimum, indicating
its metastable nature. The location of lowest total energy is at0.3614 for GGA calculations, corresponding to the latticeconstant of 4.9246 Å of the pyrite structure. The bond lengthof nitrogen pairs in pyrite OsN
2is found to be 1.365 Å, even
smaller than that of pyrite PtN 2/H208491.51 Å /H20850. The separation of
nitrogen pairs in the pyrite OsN 2and PtN 2is nearly the same
as that of a single-bonded cubic-gauche structure of N ob-served recently,
44,45which means that the nitrogen pairs
probably consist of some characteristics of covalent bonding.The calculated formation energies of pyrite and fluorite OsN
2
at an ambient pressure are 0.69 and 1.10 eV /H20849per formula
unit /H20850, which are smaller than those of pyrite PtN 2/H208490.72 eV /H20850
and fluorite PtN 2/H208493.5 eV /H20850, respectively. Figure 4/H20849b/H20850plots the
enthalpy vs pressure for the fluorite and pyrite structures of
TABLE III. The calculated atomic and bond Mulliken popula-
tion analysis of OsN 2,O s O 2, and PtN 2. NM1 and NM2 denote the
first and second nonmetal atoms, and M denotes the metal atom.
Atomic /H20849e/H20850 Bond /H20849e/H20850
NM1 NM2 M NM1-M NM2-M NM1-NM2
OsN 2−0.54 −0.54 1.09 1.25 1.25 −0.7
OsO 2−0.55 −0.55 1.10 0.98 0.98 −0.45
PtN 2−0.45 −0.45 0.9 1.08 1.08 −0.34
FIG. 2. /H20849Color online /H20850Total electron density of OsN 2at the
/H208491¯10/H20850plane. The density at three types symmetry points /H20849they are
labeled with filled circles, open squares, and open circles /H20850are ap-
proximately 0.8, 0.3, and 0 eÅ−3.
FIG. 3. /H20849Color online /H20850Partial densities of states of OsN 2.
FIG. 4. /H20849a/H20850Total energy of OsN 2versus the internal parameter u;
/H20849b/H20850Enthalpy vs pressure for the fluorite and pyrite structures of
OsN 2.FAN et al. PHYSICAL REVIEW B 74, 125118 /H208492006 /H20850
125118-4OsN 2. In the overall range of external pressure that we have
studied, the enthalpy of pyrite OsN 2is always lower than
that of fluorite OsN 2, implying that no first-order phase tran-
sition might occur at zero temperature between these twostructures. In addition, the elastic constants of pyrite OsN
2
and PtN 2are also calculated and listed in Table II, wherein
the results for PtN 2show good agreement with those of ex-
perimental and other available theoretical results. The calcu-lated elastic constants of pyrite OsN
2satisfy formula /H208493/H20850.
Therefore, it is also a mechanically stable crystal structurethough its bulk modulus is about 10% lower than that offluorite OsN
2.
In conclusion, the OsN 2with fluorite structure is first re-
ported to be mechanically stable and have a very high bulkmodulus of 360.7 GPa by the first-principles calculations.The electronic and chemical bonding properties have been
investigated, indicating that the bonding is a mixture of co-valent and ionic components. It is found that the electronicproperties of OsN
2are very similar to that of PtN 2with the
same structure. As a pyrite-type PtN 2and a orthorhombic-
type OsN 2have been very recently synthesized under high
pressure and high temperature conditions,41,46we expect that
the OsN 2as well as PtN 2with fluorite structure may be ex-
perimentally prepared in the future.
We are grateful to R. Yu for useful discussions. This work
was supported by NSFC /H20849Grant Nos. 10404035, 10534030,
50325103, and 10544004 /H20850and SKPBRC /H20849Grant No.
2005CB724400 /H20850.
*Corresponding author. Electronic address: chzfan@hit.edu.cn
1D. M. Teter, MRS Bull. 23,2 2 /H208491998 /H20850.
2C. Kittel, Introduction to Solid State Physics , 8th ed. /H20849Wiley, New
York, 1996 /H20850.
3A. Y. Liu and M. L. Cohen, Science 245, 841 /H208491989 /H20850.
4R. A. Andrievski, Int. J. Refract. Met. Hard Mater. 19, 447
/H208492001 /H20850.
5V. V. Brazhkin, A. G. Lyapin, and R. J. Hemley, Philos. Mag. A
82, 231 /H208492002 /H20850.
6P. F. Mcmillan, Nat. Mater. 1,1 9 /H208492002 /H20850.
7G.-M. Rignanese, J.-C. Charlier, and X. Gonze, Phys. Rev. B 66,
205416 /H208492002 /H20850.
8D. W. He, Y. S. Zhao, L. Daemen, J. Qian, T. D. Shen, and T. W.
Zerda, Appl. Phys. Lett. 82, 643 /H208492002 /H20850.
9Z. C. Pan, H. Sun, and C. F. Chen, Phys. Rev. B 70, 174115
/H208492004 /H20850.
10K. Benyahia, Z. Nabi, A. Tadjer, and A. Khalfi, Physica B 339,1
/H208492003 /H20850.
11R. W. Cumberland, M. B. Weinberger, J. J. Gilman, S. M. Clark,
S. H. Tolbert, and R. B. Caner, J. Am. Chem. Soc. 127, 7264
/H208492005 /H20850.
12M. I. Petrescu, Diamond Relat. Mater. 13, 1848 /H208492004 /H20850.
13F. M. Gao, J. L. He, E. D. Wu, S. M. Liu, D. L. Yu, D. C. Li, S.
Y. Zhang, and Y. J. Tian, Phys. Rev. Lett. 91, 015502 /H208492003 /H20850.
14J. He, E. Wu, H. Wang, R. Liu, and Y. J. Tian, Phys. Rev. Lett.
94, 015504 /H208492005 /H20850.
15A. Šim ůnek and J. Vacká ř, Phys. Rev. Lett. 96, 085501 /H208492006 /H20850.
16F. M. Gao, Phys. Rev. B 73, 132104 /H208492006 /H20850.
17M. Mattesini, R. Ahuja, and B. Johansson, Phys. Rev. B 68,
184108 /H208492003 /H20850.
18Practically, systematic calculations were performed on dicarbide,
-nitride, -oxide and -boride of platinum and osmium with thefluorite structure, respectively, but only found that PtN
2,O s N 2,
and OsO 2are mechanically stable. The unstable phases are not
shown here.
19P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 /H208491964 /H20850;W .
Kohn and L. J. Sham, Phys. Rev. 140, A1133 /H208491965 /H20850.
20M. D. Segall, P. L. D. Linda, M. J. Probert, C. J. Pickard, P. J.
Hasnip, S. J. Clark, and M. C. Payne, J. Phys.: Condens. Matter
14, 2717 /H208492002 /H20850.21D. Vanderbilt, Phys. Rev. B 41, 7892 /H208491990 /H20850. For the GGA cal-
culations, the pseudopotential generated by using the relatedPerdew-Burke-Ernzerhof type of exchange-correlation functions
should be used.
22D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 /H208491980 /H20850.
23J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 /H208491996 /H20850.
24H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 /H208491976 /H20850.
25O. H. Nielsen and R. M. Martin, Phys. Rev. B 32, 3792 /H208491985 /H20850.
26R. Yu and X. F. Zhang, Appl. Phys. Lett. 86, 121913 /H208492005 /H20850.
27C. Stampfl, W. Mannstadt, R. Asahi, and A. J. Freeman, Phys.
Rev. B 63, 155106 /H208492001 /H20850.
28B. R. Sahu and Leonard Kleinman, Phys. Rev. B 72, 113106
/H208492005 /H20850.
29H. Cynn, J. E. Klepeis, C. S. Yoo, and D. A. Young, Phys. Rev.
Lett. 88, 135701 /H208492002 /H20850.
30L. Fast, J. M. Wills, B. Johansson, and O. Eriksson, Phys. Rev. B
51, 17431 /H208491995 /H20850.
31J. C. Zheng, Phys. Rev. B 72, 052105 /H208492005 /H20850.
32F. Occelli, D. L. Farber, J. Badro, C. M. Aracne, D. M. Teter, M.
Hanfland, B. Canny, and B. Couzinet, Phys. Rev. Lett. 93,
095502 /H208492004 /H20850.
33T. Kenichi, Phys. Rev. B 70, 012101 /H208492004 /H20850.
34J. F. Nye, Physical Properties of Crystals /H20849Oxford University
Press, Oxford, 1985 /H20850.
35U. Lundin, L. Fast, L. Nordstrom, B. Johansson, J. M. Wills, and
O. Eriksson, Phys. Rev. B 57, 4979 /H208491998 /H20850.
36H. W. Hugosson, G. E. Grechnev, R. Ahuja, U. Helmersson, L.
Sa, and O. Eriksson, Phys. Rev. B 66, 174111 /H208492002 /H20850.
37R. Yu and X. F. Zhang, Phys. Rev. B 72, 054103 /H208492005 /H20850.
38R. Yu /H20849private communication /H20850. This inconsistency may come
from the intrinsic technique flaws in the code for obtaining theelastic constants in the
WIEN2K software package, in which the
implanted rhombohedral distortion is not volume conservative.When applying a non-volume-conservative strain, the expres-sion
/H9278elast=V0/2cij/H9255i/H9255jfor elastic energy is no longer accurate,
resulting in unexpected results in some special cases. This maybe the reason why the conclusion concerning the stability trendof fluorite OsN
2in Ref. 37is opposite to what we obtained in
the present work.POTENTIAL SUPERHARD OSMIUM DINITRIDE WITH … PHYSICAL REVIEW B 74, 125118 /H208492006 /H20850
125118-539E. Gregoryanz, C. Sanloup, M. Somayazulu, J. Badro, G. Flquet,
H. K. Mao, and R. J. Hemley, Nat. Mater. 3, 294 /H208492004 /H20850.
40R. Yu, Q. Zhang, and X. F. Zhang, Appl. Phys. Lett. 88, 051913
/H208492006 /H20850.
41J. C. Crowhurst, A. F. Goncharov, B. Sadigh, C. L. Evans, P. G.
Morrall, J. L. Ferreira, and A. J. Nelson, Science 311, 1275
/H208492006 /H20850.
42A. F. Young, J. A. Montoya, C. Sanloup, M. Lazzeri, E. Greg-
oryanz, and S. Scandolo, Phys. Rev. B 73, 153102 /H208492006 /H20850.
43Y. Kuwayama, K. Hirose, N. Sata, and Y. Ohish, Science 309,
923 /H208492005 /H20850.44M. I. Eremets, A. G. Gavriliuk, I. A. Trojan, D. A. Dzivenko, and
R. Boehler, Nat. Mater. l3, 558 /H208492004 /H20850.
45T. Zhang, S. Zhang, Q. Chen, and L. M. Peng, Phys. Rev. B 73,
094105 /H208492006 /H20850.
46A. F. Young, C. Sanloup, E. Gregoryanz, S. Scandolo, R. J. Hem-
ley, and H. K. Mao, Phys. Rev. Lett. 96, 155501 /H208492006 /H20850.
47W. B. Pearson, A Handbook of Lattice Spacing and Structures of
Metals and Alloys /H20849Pergamon, New York, 1958 /H20850.
48E. A. Brandes, Smithells Metal Reference Book , 6th ed. /H20849Butter-
worth, London, 1983 /H20850.FAN et al. PHYSICAL REVIEW B 74, 125118 /H208492006 /H20850
125118-6 |
PhysRevB.91.174408.pdf | PHYSICAL REVIEW B 91, 174408 (2015)
Local character of the highest antiferromagnetic temperature of Ce systems
in Sc-rich CeTi 1−xScxGe
J. G. Sereni, P. Pedrazzini, M. G ´omez Berisso, A. Chacoma, and S. Encina
Low Temperature Division, CAB-CNEA and Instituto Balseiro, 8400 San Carlos de Bariloche, Argentina
T. Gruner, N. Caroca-Canales, and C. Geibel
Max-Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany
(Received 13 February 2015; revised manuscript received 21 April 2015; published 7 May 2015)
The highest antiferromagnetic (AFM) temperature in Ce based compounds has been reported for CeScGe with
TN=47 K, but its local or itinerant nature has not been deeply investigated yet. In order to shed more light
into this unusually high ordering temperature we have investigated structural, magnetic, transport, and thermalproperties of CeTi
1−xScxGe alloys within the range of stability of the CeScSi-type structure: 0 .25/lessorequalslantx/lessorequalslant1.
Along this concentration range, this strongly anisotropic system presents a complex magnetic phase diagramwith a continuous modification of its magnetic behavior, from ferromagnetism for 0 .25/lessorequalslantx/lessorequalslant0.50 (with 7 K /lessorequalslant
T
C/lessorequalslant16 K) to AFM for 0 .60/lessorequalslantx/lessorequalslant1 (with 19 K /lessorequalslantTN/lessorequalslant47 K). The onset of the AFM phase is associated to
a metamagnetic transition with a critical field increasing from Hcr=0a tx≈0.55 to≈6Ta t x=1, coincident
with an increasing contribution of the first excited crystal electric field doublet. At a critical point xcr≈0.65 a
second transition appears at TL/lessorequalslantTN. In contrast to observations in itinerant systems like CeRh 2Si2or CeRh 3B2,
no evidences for significant hybridization of the 4 felectrons at large Sc contents were found. Therefore, the
exceptionally large TNof CeScGe can be attributed to an increasing Ruderman-Kittel-Kasuya-Yosida interaction
between Ce double layers as Sc content grows.
DOI: 10.1103/PhysRevB.91.174408 PACS number(s): 75 .20.Hr,71.27.+a,75.30.Kz,75.10.−b
I. INTRODUCTION
Long range magnetic order of cerium ions can be found
along four decades of temperature, from 1.6 mK in CMN [ 1]
up to 115 K in CeRh 3B2[2], though most known transitions
occur between a few degrees and Tord≈12 K. Different
temperature ranges of magnetic order are characterized bydifferent magnetic behaviors. While magnetic interactionsbetween localized Ce-4 fmoments are typically observed
forT
ord<12 K [ 3], in the cases where Tordexceeds that
temperature, two types of behaviors can be distinguished.The one with localized Ce-4 fmoments is related to the rare
cases of Ce binary compounds formed in cubic-bcc structure.This is the crystal structure with the highest symmetry thatallows one to have the /Gamma1
8quartet as the ground state.
However, its fourfold degeneracy is reduced by undergoingdifferent types of transitions mostly connected with structuralmodifications. Those compounds are CeZn ( T
N=30 K), CeTl
(TN=25.5K ) , C e M g ( TC=20 K), CeCd ( TC=16.5K ) ,
and CeAg ( TN=15.6K )[ 4].
The ternary Ce-based compounds belonging to the other
group show evidences of itinerant character [ 5]. Among them,
the outstanding case is the above mentioned CeRh 3B2that
shows the highest ordering temperature with Tord=115 K [ 2].
The following highest ordering temperature has been reportedfor CeScGe with T
ord=47 K [ 6], but the local or itinerant
nature of its magnetic state is under discussion. Within thisgroup, one of the most studied compounds is CeRh
2Si2, which
shows a Tord=36 K [ 7,8]. Also in this case evidences for
local and itinerant magnetic character were equally claimedby different authors [ 9–11].
Such ambiguity, that appears frequently in Ce compounds,
was identified as the local-itinerant dilemma of Ce-4 felec-
trons [ 12]. The eventual itinerant character of the 4 forbitalswas compared to the U-5 fbehavior [ 5], whose compounds
frequently show high T
ordvalues with clear itinerant char-
acteristics arising from the extended character of the 5 f
orbitals. In Ce-4 fsystems, however, itineracy is related to
some degree of hybridization with the conduction band and theconsequent weakening of the 4 feffective magnetic moment,
μ
eff. In terms of Doniach model [ 13], the question arises
whether there is an optimal combination of weak 4 f-band
hybridization that enhances the intersite RKKY interaction byincreasing the in-site 4 f-band exchange without reducing μ
eff
significantly.
In the case of CeScGe, it was formerly reported as a
ferromagnet (FM) [ 6] that usually implies local 4 fcharacter.
However, band-structure calculations [ 14], based on itinerant
character, suggested that FM and antiferromagnetic (AFM)
states compete in energy. A subsequent study on CeScSi
and CeScGe [ 15] recognized both compounds as AFM. Thus
CeScGe presents the best characteristics for the proposed
investigation.
A further aspect that has to be taken into account for a
proper knowledge of the ordered phase at those temperatures
is the role of the excited crystal-electric-field (CEF) levels.
This contribution can be properly evaluated by tracing Tord
from relatively low values, where only a doublet with local-4 f
character contributes to the magnetic ground state, to high
values where involvement of excited CEF levels becomes
very likely. These conditions are fulfilled by CeTi 1−xScxGe
alloys, that cover an extended range of transition temperatures
fromTord≈7K i n C e T i 0.75Sc0.25Ge to Tord≈47 K at the
stoichiometric limit CeScGe [ 6]. The lowest Tordvalue is
determined by the limit of stability of the CeScSi-type structure
(atx=0.25) on the Ti rich side. Below x≈0.15 this alloy
stabilizes in the CeFeSi-type structure.
1098-0121/2015/91(17)/174408(8) 174408-1 ©2015 American Physical SocietyJ. G. SERENI et al. PHYSICAL REVIEW B 91, 174408 (2015)
II. EXPERIMENTAL DETAILS AND RESULTS
Polycrystalline samples of CeTi 1−xScxGe with 0 .25/lessorequalslantx/lessorequalslant
1 were synthesized by arc melting under Ar atmosphere thenominal amounts of the constituents with purity above 99.99%.The samples were turned over and remelted several timesto ensure homogeneity. Then, the samples were placed ina tungsten boat wrapped with zirconium foil and annealedat 1200
oC for one week. The quality of the samples was
verified by means of x-ray powder-diffraction measurementsusing Cu- Kα
1radiation ( λ=1.54056 ˚A) in a Stoe-Stadip-MP
diffractometer. The pattern was indexed on the basis of thetetragonal CeScSi-type structure.
Specific heat was measured between 0.5 and 50 K using a
standard heat pulse technique in a semiadiabatic He
3calorime-
ter. The magnetic contribution Cmis obtained by subtracting
the phonon contribution extracted from LaTi 0.5Sc0.5Ge. Mag-
netization measurements were carried out using a QuantumDesign MPMS magnetometer operating between 2 and 300 K,and as a function of field up to 50 kOe. Electrical resistivity wasmeasured between 2 K and room temperature using a standardfour probe technique with an LR700 resistance bridge.
A. Structural properties
The CeTi 1−xScxGe system forms in two related crystal
structures: CeFeSi-type for low Sc content (up to x=0.15)
and CeScSi-type beyond x=0.23 [16]. In the latter, each
second Ce-double layer is shifted by (1 /2, 1/2) with a
rearrangement of Sc and Si atoms, becoming a body centeredtetragonal instead of primitive tetragonal of CeFeSi, seeFig.1, with the consequent doubling of the clattice parameter.
Sc concentration xdependencies of the unit cell volume
V(x) and the c/a ratio of tetragonal lattice parameters are
shown in Fig. 2. Both parameters show a linear dependence
within the expected experimental dispersion. The increase ofV(x) can be explained by the larger atomic volume of Sc
with respect to Ti, whereas the reduction of the c/a ratio
is due to the increase of the aparameter because cremains
practically unchanged. This indicates that an expansion in thebasal plane of the tetragonal structure occurs, whereas theinterlayer distances in the caxis direction are only slightly
affected.
B. Magnetic susceptibility
In Fig. 3, the inverse of the high temperature magnetic
susceptibility (1 /χ) is presented after subtracting a Pauli type
paramagnetic contribution χP. This temperature independent
contribution, already reported for CeScGe [ 17], is observed
to decrease from χP=0.9×10−3atx=0.4t o0 .47×
10−3emu/Oe mol at x=1, as shown in the inset of Fig. 3.T h e
1/χdependence on concentration (evaluated for T> 50 K)
indicates a decrease of Ce effective magnetic moment fromμ
eff(x)≈2.25μBatx=0.4t o≈2μBatx=0.7. Beyond that
concentration μeff(x) remains practically unchanged. Notably,
the paramagnetic temperature θP(x) is positive along the
full concentration range and even increases from θP≈8K
atx=0.4, up to 19 K at x=0.8. At that concentration it
slightly decreases down to 17 K at x=1 as shown in the
inset of Fig. 3.
CeCeSi
ScdCe-Sc
FIG. 1. (Color online) CeScSi-type structure, showing double Ce
layers (yellow open network) and ligands layers (blue full network).
Left side: dCe-Sc indicates Ce-Sc spacing. Right side: curved arrow
represents Sc mediated Ce-interlayers interaction.
The variation of χ(T) around the magnetic transition is
shown in Fig. 4(using a double logarithmic representation)
for alloys with x/greaterorequalslant0.60. The respective maxima of χ(T)
between 20 /lessorequalslantTN/lessorequalslant50 K indicate the AFM character of the
magnetic transition. The samples with x/lessorequalslant0.50 are included
280284288292296300
0.0 0.2 0.4 0.6 0.8 1.03.683.703.723.743.763.78Vol [A]
c / a
x [Sc conc]
FIG. 2. (Color online) Unit cell volume variation in
CeTi 1−xScxGe as a function of Sc content (left axis) and c/a
lattice parameters ratio (right axis). Straight lines are guides to
the eyes. The shaded area indicates the coexistence region of bothstructures.
174408-2LOCAL CHARACTER OF THE HIGHEST . . . PHYSICAL REVIEW B 91, 174408 (2015)
0 50 100 150 200 250 3000100200300400500600
0.4 0.6 0.8 1.00.40.60.81.0
T [K] 1 / (χT - χP ) [mol Oe / emu]χχχχχP [10-3emu /mol Oe]
x [Sc conc.]5101520
θP [K]θθθθ
FIG. 3. (Color online) Inverse high temperature magnetic sus-
ceptibility measured in a field of H=10 kOe, after subtracting
aP a u l i χPcontribution. Inset: χP(x) contribution (left axis) and
paramagnetic temperature θP(x) (right axis).
in the inset of Fig. 4using a more extended M(T) scale because
of their FM character. The typical M(T) dependence of a FM is
observed for x=0.25 and 0.3 alloys, whereas those with x/greaterorequalslant
0.4 show an incipient AFM component even in field cooling
procedure. The thermal dependence in the paramagnetic phaseis compared in the main figure with a general Curie-Weissfunction χ(T)=0.56/(T−15.5) emu/Oe mol (notice the
positive θ
P=15.5 K) to demonstrate the stability of the
paramagnetic phase along the full concentration range. Onthe ordered phase, χ(T< T
N) shows a smooth behavior with
the presence of a weak maximum at T=TLbarely detected
between 0 .65/lessorequalslantx/lessorequalslant0.80. This anomaly is better observed in
specific heat measurements and its origin is discussed in thecontext of the magnetic phase diagram.
Those different behaviors can be sorted into three ranges
of Sc concentration: (i) for x/lessorequalslant0.6t h e M(T) dependence
100.010.1
11 00.51.01.52.02.53.0
70χ [emu /Oe mol ]
T [K]
M [emu /mol ]
T [K] x= 0.23
x= 0.30
FIG. 4. (Color online) Low field ( H=100 Oe) magnetic sus-
ceptibility in a double logarithmic representation. Full curve is a
reference for the paramagnetic phase (see the text) and dashed curve
indicates the position of the second transition at T=TL.I n s e t M(T)
measurements for x/lessorequalslant0.50 samples.0.00.20.40.60.81.00 100 200 300
0.900.951.001.0510 100
05 0 1 0 0 1 5 0024 T [K]ρ / ρ300K
T [K]ρ(T) / ρ100K
0.40ΔR / R0 [%]
H [kOe]
FIG. 5. (Color online) (a) Electrical resistivity normalized at
300 K of all measured samples. Inset: detail of the logarithmic T
dependence of samples between 0 .23/lessorequalslantx/lessorequalslant0.4 normalized at 100 K
for better comparison. (b) Percent magnetoresistance increase up toH=160 K Oe.
indicates an increasing mixture of an AFM component in
the dominant FM in the GS, with the ordering temperatureincreasing from T
ord=7Ka tx=0.25 up to 16 K at x=0.60.
Atx=0.65 the χ(T) maximum displays two shoulders which
can be distinguished after a detailed analysis of the χ(T)
curvature, i.e., its second derivative ∂2χ/∂T2. This feature
reveals that the x=0.65 concentration is placed very close
to a critical point. (ii) Between 0 .65/lessorequalslantx/lessorequalslant0.8, a slight kink
appears at T=TL(x), below TN, as seen in Fig. 4. Within this
range of concentration, TNincreases from 19 K at x=0.65
up to 35 K at x=0.8, whereas TL(x) increases from 18 K up
to 26 K. (iii) Above x=0.9,TNkeeps growing from 38 K at
x=0.9u pt o4 7Ka t x=1, whereas TL(x) is hardly seen in
χ(T) measurements.
C. Electrical resistivity
The thermal dependence of the electrical resistivity ρ,
normalized at 300 K, is presented in Fig. 5(a) for samples
between 0 .30/lessorequalslantx/lessorequalslant1.0. The rough ρvalues at 300 K show a
dispersion of about 20% around a mean value of 220 μ/Omega1cm,
with a tendency to a maximum at x=0.8. That dispersion
174408-3J. G. SERENI et al. PHYSICAL REVIEW B 91, 174408 (2015)
20 40 6001234
01 0 2 0 3 0 4 00.00.20.40.60.8
ΔΔΔΔ δδδδ
ΔΔΔΔCm [J/mol K]
T [K]Cm / T [J/mol K2 ]
T [K]
FIG. 6. (Color online) Magnetic contribution to the specific heat
divided temperature within the 0 .25/lessorequalslantx/lessorequalslant0.65 range. Doted curve
indicates the 1 /Torddecreasing trend of the maximum below x=0.5.
Inset: fit of Cm(T) for sample x=0.25 at high temperature to evaluate
the CEF spectrum; see the text.
is attributed to random microcracks in the samples. The
continuous increase of the ρ(T) slopes indicates the weakening
of Kondo type scattering with growing Sc content. In the insetof that figure, a detail of the logarithmic dependence of sampleson the Ti-rich side (0 .23/lessorequalslantx/lessorequalslant0.60) normalized at 100 K
are collected to better analyze the role of the Kondo effect.A logarithmic increase of ρ(T) approaching the magnetic
transition from high temperature is observed at x=0.23
as an indication of Kondo scattering contribution. However,the intensity of this electronic scattering rapidly decreaseswith Sc concentration until it vanishes at x=0.60. At that
concentration, a resistivity upturn characteristic of the openingof a gap in the Fermi surface emerges. This anomaly developsupx=0.80 and then decreases again as the system reaches
its stoichiometric CeScGe limit.
The change of the ρ(T) dependence at T=T
ordis in
agreement with the transitions observed from magnetic mea-surements. In the FM region a downward kink in ρ(T)i so b -
served at T
ord, whereas a gap opening, usually related to AFM
order, occurs for x> 0.50. Beyond the critical concentration,
the width of the anomaly exceeds the temperature differencebetween T
NandTL. The weakening of the ρ(T) anomaly
approaching the stoichiometric limit can be attributed to thevariation of the Fermi level within the energy gap associatedto the AFM state. As Ti
4+atoms are replaced by Sc3+ones,
the chemical potential decreases and eventually approachesthe lower energy side of the gap. This simple picture has tobe considered within the complexity of the Fermi surface in asystem with many conduction electrons.
D. Specific heat
Specific heat results show several peculiar features along
the full concentration range in coincidence with the differentregimes observed in magnetic measurements. The resultsobtained on the 0 .25/lessorequalslantx/lessorequalslant0.65 alloys are presented in
Fig.6asC
m/T, where Cmindicates the magnetic contribution01 0 2 0 3 0 4 0 5 00.00.20.40.6Cm / T [J/ mol K2 ]
T [K]0.00.20.40.6
FIG. 7. (Color online) Magnetic contribution to the specific heat
in the 0 .65/lessorequalslantx/lessorequalslant0.80 range (left axis), and the 0 .90/lessorequalslantx/lessorequalslant1
samples shifted for clarity (right axis). Dashed curves represent thefits on the ordered phase of samples x=0.65 and 0.80 (see the text).
to the specific heat after phonon subtraction. Within the exper-
imental accuracy, the maxima of Cm(x)/Tcoincide with those
observed in M(T). A common feature along this concentration
range is the lack of a Cmjump at T=Tord. Instead, a broadened
transition is observed, followed by a tail in Cm/T(T> T ord)
that reveals a significant contribution of magnetic correlationsas precursors of the magnetic transition. This tail becomesmore extended in temperature with increasing T
ord(x)u pt o
x=0.50. Samples with x=0.60 and x=0.65 recover a
larger slope just above Tordtogether with a moderate increase of
theCm/Tmaximum. This change of tendency coincides with
the modification of the magnetic structure around x=0.50.
In this concentration range, the maximum of Cm/Tdecreases
as∝1/Tord. Although such a decrease is a consequence of
plotting Cm(T)/T, it reveals that Cm(Tord) remains almost
constant with decreasing Tordand does not extrapolate to zero
as one would expect in the case of a quantum critical point.
An increase of the low temperature curvature in Cm(T)/T
indicates a progressive opening of a gap of anisotropy ( /Gamma1)i n
the magnon spectrum with concentration. To evaluate that evo-lution, the low temperature C
m/Tdependence was analyzed
using the function Cm/T=γ0+B×T×exp(−/Gamma1/T )a s
shown in Fig. 7for samples with x=0.65 and 0.8. The values
174408-4LOCAL CHARACTER OF THE HIGHEST . . . PHYSICAL REVIEW B 91, 174408 (2015)
computed for those concentrations are /Gamma1(x=0.65)=7K
and/Gamma1(x=0.80)=15 K, respectively. Concerning the γ0(x)
dependence, it drops from γ0≈220 mJ mol−1K−2atx=0.25
down to ≈20 mJ mol−1K−2atx=0.60.
The main characteristic of the alloys with x> 0.65 is
the split of Cm/Tinto two maxima, in agreement with the
χ(T) results. While the lower transition (at T=TL)s h o w s
a cusplike anomaly between x=0.7 and 0.9, the upper one
(TN) is associated to a jump in Cm/T(TN). Such a split cannot
be distinguished in Cm(T) nor in M(T) measurements on
thex=0.65 sample. However, in preliminary thermopower
measurements a maximum and a marked shoulder are observedat 21 and 23 K, respectively.
Forx/greaterorequalslant0.9, a further change in C
m/T(T) is observed
as depicted in the lower part of Fig. 7. The broad shoulder
ofCm/T(T) in samples 0 .9/lessorequalslantx/lessorequalslant1 can be attributed to
the increase of the GS degeneracy from Neff=2t o4a s
the contribution of the first exited CEF doublet graduallyincreases. Also both transitions change their aspects because,while the one at T
Ltransforms into a steplike anomaly, that at
TNbecomes sharper and grows significantly. Notice that the
/Delta1Cm(TN)j u m pi n x=1i s≈16 J mol−1K−1, in between the
values predicted in a mean field approximation for a doublet(/Delta1C
m=1.5R=12.5Jm o l−1K−1) and a quartet ( /Delta1Cm=
2.2R≈18 J mol−1K−1)[18]. The small peak observed in x=
0.9a tT≈7 K can be attributed to an extrinsic contribution
of a small amount of Ce oxide.
E. Magnetization
TheM(H) hysteresis loops measured at T=1.8K o n
the alloys with x/lessorequalslant0.5 reveal the FM character of the
ordered phase [see Fig. 8(a)]. Measurements performed up
toH=50 K Oe are presented in Fig. 8(b) showing that at
high magnetic field M(x) increases up to a maximum value
forx=0.50. The saturation magnetization ( Msat), extracted
from Fig. 8(b) asM(x,H)=Msat×(1−a/H ), increases
from 1.04 μB/f.u. for x=0.30 up to 1.15 μB/f.u. for x=0.50.
Atx=0.6Msatdecreases to 1 μB/f.u., though this value is
extracted from a reduced fitting range. Similarly, the coercivefield increases from 1 .1KO ef o r x=0.30 up to 2.6 kOe for
x=0.50.
Atx=0.60, the spontaneous FM magnetization is re-
placed by a small susceptibility, indicating a transition toan AFM state. Coincidentally, a metamagnetic transition(MMT) appears with the critical field H
crincreasing with Sc
concentration up to our experimental limit of H=50 kOe
with an initial ratio of ∂H cr/∂x=2.2KO e /Sc%.
In contrast with M(H) measurements performed on stoi-
chiometric CeScGe [ 15,17], which report a MMT transition
around Hcr≈60 K Oe with a weak associated hysteresis, no
signal of MMT was observed in preliminary magnetoresis-tance measurements up to H=160 K Oe on the Sc rich
side. In Fig. 5(b) we show the monotonous increase of the
positive magnetoresistence of CeT
0.05Sc0.95Ge in percent units
/Delta1R/R 0, where /Delta1R=R(H)−R0withR0=R(H=0).
Similar behavior is observed for CeScGe, both measuredatT=4.6 K. This disappearance of the MMT at high
Sc concentration coincides with the decrease of the areaof hysteresis loop in M(H)a sH
cr(x) increases, shown in
Fig.8(b).0 1 02 03 04 05 00.00.20.40.60.81.0-0.6-0.30.00.30.6-10 -5 5 01 0M [μB / f.u.]
H [kOe]H [kOe]M [μB / f.u.]
x=0.3
x=0.4
x=0.5
FIG. 8. (Color online) Magnetization measurements at T=
1.8 K: (a) hysteresis loops centered at H=0f o rt h eF M0 .3/lessorequalslantx/lessorequalslant
0.5 alloys. (b) Magnetization curves up to H=50 kOe, showing a
metamagnetic transformation in samples with x/greaterorequalslant0.6.
III. DISCUSSION
A. Evaluation of the crystal electrical field splitting
In order to have a qualitative evaluation of the energy of the
excited CEF levels we have analyzed the Cm(T) dependence
of the x=0.25 sample which shows the lowest ordering tem-
perature. For that purpose we have fitted the experimental datataking into account that the CEF splits the sixfold Hund’s rulemultiplet, originated in the J=5/2 angular momentum of Ce,
into three Kramer’s doublets. Due to the eventual hybridization(i.e., Kondo effect) acting on the first excited level, thestandard Schottky anomaly ( C
Sch) may not describe the Cm(T)
dependence properly. Thus a standard approach to mimic abroadening of the levels was applied. For this procedure thefirst CEF exited doublet at /Delta1
1is described as a set of four
single levels equally distributed in energy around the nominalvalue of the represented broadened level. This method is onlyapplicable to weakly hybridized level (i.e., /Delta1
1/kB>TK).
To properly account for the total magnetic contribution tothe entropy, a second excited doublet (at /Delta1
2) was included
in the evaluation of the full CEF spectrum. Since /Delta12largely
exceeds the temperature range of our Cm(T) measurements, no
hybridization effects were taken into account for this analysis.
174408-5J. G. SERENI et al. PHYSICAL REVIEW B 91, 174408 (2015)
0 1 02 03 04 05 00.00.51.01.52.02.5Sm / Rln 2
T[ K ]
FIG. 9. (Color online) Thermal evolution of the magnetic contri-
bution to the entropy ( Sm) normalized by Rln 2.
The applied formula is
CSch(T)=/Sigma1iAi/bracketleftbigg/parenleftbigg/Delta1i
T/parenrightbigg/slashbigg
cosh/parenleftbigg/Delta1i
T/parenrightbigg/bracketrightbigg2
. (1)
Since the measured specific heat Cm(T) contains GS ( CGS)
andCSchcontributions, the proper fit has to be performed
adding both components: Cm=CGS(T)+CSch(T). In this
case,CGScorresponds to the high temperature tail of the
magnetic anomaly of the x=0.25 sample which can be
properly described as CGS∝1/T2.I nt h ei n s e to fF i g . 6the
result of this fit to the experimental data is shown, includingthe detail of the C
Schfunction. The computed parameters are
/Delta11≈35 K and /Delta12≈155 K, with an effective broadening of
the first CEF doublet evaluated as δ=12 K. These results
were checked with the evaluation of the magnetic entropyinvolved in this Schottky anomaly. This analysis cannot beapplied for higher Sc concentrations because the increaseofT
ordprogressively approaches /Delta11. The analysis of the
entropy collected at ≈50 K, see Fig. 9, excludes significant
changes expected in the CEF levels splitting. Nevertheless,the reduction of the electrical charges at the transition metalsites from Ti
4+[Ar4s23d2]t oS c3+[Ar4s23d1] may affect the
CEF levels eigenfunctions with the consequent effect on theanisotropy and the magnetic interaction between neighboringplanes.
B. Entropy
The analysis of the thermal evolution of the magnetic
contribution to the entropy Sm(T) is shown in Fig. 9normalized
by the value of a Kramer’s doublet level: Rln 2. The alloy x=
0.25 with the lowest ordering temperature ( TC=7 K) reaches
Sm=Rln 2 at T=11 K, slightly above TC, suggesting
that only the GS doublet contributes to the magnetic order.However, since in the paramagnetic phase S
m(T) increases
continuously (i.e., without showing any plateau around Rln 2)
one may infer that the low energy tail of the first excitedCEF level starts to contribute to the entropy at quite lowtemperature. This fact qualitatively confirms the CEF level0.2 0.4 0.6 0.8 1.001020304050T [K]
[Sc conc]
FIG. 10. (Color online) Sc concentration dependence of the mag-
netic phase diagram showing the transition temperatures extracted
from magnetic and thermal measurements (left axis) within the range
of the CeScSi-type structure.
spectrum extracted in the previous subsection from the fit of
Cm(T).
Although the contribution of the first excited doublet to
the ordered state may be marginal in the alloys with low Sccontent, it becomes significant for higher concentrations asS
m(Tord) increases with Tord(x). This feature becomes evident
forx/greaterorequalslant0.5 alloys because Sm(Tord) clearly exceeds Rln 2.
Notice that around this concentration the AFM sets on. Forx=1 it practically reaches the value corresponding to two
doublets, i.e., Rln 4, in agreement with the broad shoulder
observed in C
m(T)/Tand the value of /Delta1Cm(TN) (see Fig. 7).
The comparison of the entropy distribution above and
below Tord[i.e.,Sm(T< T C) andSm(T> T C), respectively]
provides a hint to figure out the effective dimensionality ofa magnetic system. According to theoretical predictions [ 19],
for two dimensional (2D) systems the entropy accumulatedbetween T=0 and T=T
ordis similar to that contained
in the tail of Cm(T> T ord). In this case, the samples with
0.25/lessorequalslantx/lessorequalslant0.5s h o w Sm(T< T C) and Sm(T> T C) values
close to the prediction for a simple square Ising lattice ofspins 1/2. Such is not the case for samples beyond the criticalconcentration (i.e., x/greaterorequalslant0.65) for which the shape of AFM
transition tends to the characteristic /Delta1C
m(TN) jump of a 3D
second order transition. Also this change occurs around themodification of the magnetic character from FM to AFM.
C. Magnetic phase diagram
The magnetic characteristics of this system are summarized
in the phase diagram presented in Fig. 10. The two most
relevant features are the FM to AFM change between 0 .50/lessorequalslant
x/lessorequalslant0.60. A critical point (CP) occurs at x≈0.65 where three
phases, paramagnetic, AFM I, and AFM II, converge accordingto the phase boundaries defined by respective anomaliesobserved in the specific heat. There, an intermediate phaseAFM II sets in between T
N(x) andTL(x). Along the full con-
centration range different regions were identified as follows.(i) Between 0 .25/lessorequalslantx/lessorequalslant0.50 a FM-GS, as determined by M
174408-6LOCAL CHARACTER OF THE HIGHEST . . . PHYSICAL REVIEW B 91, 174408 (2015)
vsHhysteresis loops of Fig. 6(a), shows an increasing Msat
that reaches its maximum value at x=0.50. (ii) A continuous
change from FM to AFM occurs between 0 .50<x< 0.60,
evidenced by the MMT with Hcrarising from zero. ρ(T)
measurements confirm this continuous change because thekink at T=T
Cprogressively develops a characteristic AFM
upturn (see Fig. 5) due to the opening of a gap in a fraction
of the Fermi surface. (iii) Around the CP ( x≈0.65) a weak
transition emerges at TL(x) below TNaccording to Cm(T) and
χ(T) results. Notably, TN(x) rises continuously up to TN=
47 K (at x=1) despite the weakening of the Ce effective
magnetic moment.
As it was mentioned in the Introduction, there are early
band-structure calculations [ 14] based on an itinerant character
of CeScGe, that requires one to have at least a moderatehybridization between Ce-4 fconduction states. Such ex-
change is typically manifested as an increase of ρ(T)a tl o w
temperature, due to Kondo scattering, and the density of statesreflected in γ
0. None of these effects are observed on the
Sc-rich limit. On the contrary, they appear on the Ti-rich sidedespite the FM behavior. Since ρ(T,x) and γ
0(x) indicate
a weakening of this interaction with Sc content, the highT
Nvalue observed in CeScGe cannot be attributed to any
hybridization mechanism but rather to an intrinsically strongRuderman-Kittel-Kasuya-Yosida (RKKY) interaction.
A general description of the magnetic evolution of this
system can be proposed by considering each Ce-double layeras a nearly 2D-FM unit that weakly interacts with neighboringlayers. The positive and even growing θ
P(x) values support an
increasing intra layer Ce-Ce neighbors FM interaction. Once
the amount of Sc-3 d1electrons becomes relevant (at x≈0.50)
theinter layers RKKY interaction takes over favoring an AFM
stacking of the double layers within an anisotropic but 3Dscenario. The thermal population of the first CEF excitedlevel also contributes to the FM-AFM change because of themodification of the involved moments.
Despite the ≈12% difference between Ti and Sc Gold-
schmit metallic radii ( r
Ti=1.46˚A and rSc=1.64˚A, re-
spectively), the Ce-T spacing ( dCe-T) only increases by
≈2% between CeTiGe ( dCe-Ti=3.45˚A[20]) and CeScGe
(dCe-Sc=3.53˚A[15]) evaluated within the same CeScSi
structure. This variation is in agreement with the slightchange ( ≈1%) observed on the caxis between x=0.30
andx=1. Within a rigid sphere picture, this d
Ce-Sc=
3.53˚A and the sum of Sc and Ce metallic radii ( rSc=
1.64˚A and rCe=1.86˚A) become comparable. There-
fore, a significant increase of the electronic overlap be-tween Sc-3 dand Ce-5 delectrons can be expected with
the consequent strengthening of the Ce- inter layers RKKY
interaction.
In this context one should notice that isotypic rare earth
(R) compounds of the RTGe family also present very highordering temperatures, like GdScGe (CeScSi type structure)
that orders at T
C=350 K [ 21], more than a factor 7 higher than
in CeScGe. Similar values are observed for GdTiGe (of CeFeSitype structure) with T
N=412 K [ 22]. These behaviors are in
clear contrast with itinerant cases like CeRh 2Si2and CeRh 3B2
[2,7], for which the TNratio between GdRh 2Si2and CeRh 2Si2
is only a factor 3.
IV . CONCLUSIONS
Apart from the significantly high ordering temperature
of CeScGe at TN=47 K, the main magnetic characteristics
shown by this system are the continuous change from FM toAFM-I order and the presence of a CP point (at x=0.65)
associated to an intermediate AFM-II phase. These modifica-tions of the magnetic ground state occur without affecting thelocal character of the Ce-4 forbital.
The layer type of the crystalline structure seems to play
a basic role because it favors the formation of FM sheetsinvolving two neighboring Ce planes. At low Sc content(0.25/lessorequalslantx/lessorequalslant0.50) the FM intra plane RKKY interaction,
resembling 2D-type order, dominates the scenario. The con-tinuous increase of the positive θ
Ptemperature indicates that
this Ce-Ce interaction even enhances with concentration.
After reaching the maximum of Msatatx=0.50, the system
develops a MMT transition from Hcr=0. Such a continuous
transition from a FM state to an AFM reveals the similar energyof these competing phases, that can be described as a smoothmodification from a FM stacking to an AFM stacking of FMlayers. This change in the inter layers RKKY interaction can
be driven by the variation of the electronic configuration of theintermediaries atoms Ti
4+and Sc3+.
The increasing inter layers interaction strengthens the 3D
character of the AFM phase evidenced by the Cm(T)j u m p
at the magnetic transition. The appearance of an intermediatephase AFM-II indicates that the magnetic order parameterchanges before reaching the ground state configuration. TheS
m(T,x) evolution confirms the first excited CEF level increas-
ing contribution as TNbecomes comparable to /Delta11. Therefore,
the record high ordering temperature of CeScGe has to beattributed to the convergence of different factors, such as the Cedouble-layer structure of FM character, the increasing RKKYinteraction with Sc-3 d
1concentration, and the vicinity of the
first CEF excited level. All these conditions are related withthe local characters of the Ce-4 forbitals.
ACKNOWLEDGMENTS
This work was partially supported by ANPCyT (Grant
No. 2010-1060) and SECyT-UNCuyo (Grant No. 06/C457).J.G.S., P.P., and M.G.B. acknowledge support as members ofCONICET, while S.E. acknowledges support in the form of ascholarship.
[1] R. A. Fisher, E. W. Hornung, G. E. Brodale,
and W. F. Giauque, J. Chem. Phys. 58,5584
(1973 ).
[2] S. K. Malik, R. Vijayaraghavan, and W. E. Wallace, J. Magn.
Magn. Mater. 37,303(1983 ).[3] See, for example, E. Bauer, Adv. Phys. 40,417(1991 ).
[4] See, for example, J. G. Sereni, in Handbook Phys Chem of Rare
Earths , edited by K. A. Gschneidner, Jr. and L. Eyring (Elsevier
Science Publishers, B. V ., 1991), V ol. 15, Chap. 98; J. Phys. Soc.
Jpn. 67,1767 (1998 ) and references therein.
174408-7J. G. SERENI et al. PHYSICAL REVIEW B 91, 174408 (2015)
[5] L. E. De Long, J. F. Huber, and K. S. Bedell, J. Magn. Magn.
Mater. 99,171(1991 ).
[6] P. C. Canfield, J. D. Thompson, and Z. Fisk, J. Appl. Phys. 70,
5992 (1991 ).
[7] C. Godard, L. C. Gupta, and M. F. Ravet-Krill, J. Less-Common
Met. 94,187(1983 ).
[8] See, for example, S. Kawarazaki, M. Sato, Y . Miyako, N.
Chigusa, K. Watanabe, N. Metoki, Y . Koike, and M. Nishi,Phys. Rev. B 61,4167 (2000 ).
[9] S. Kawarazaki, Y . Kobashi, J. A. Fernandez-Baca, S. Murayama,
Y.¨Onuki, and Y . Miyako, Physica B 206-207 ,298(1995 ).
[ 1 0 ]R .S e t t a i ,A .M i s a w a ,S .A r a k i ,M .K o s k i ,K .S i g i y a m a ,T .
Takeuchi, K. Kindo, Y . Haga, E. Yamamamoto, and Y . ¨Onuki,
J. Phys. Soc. Jpn. 66,2260 (1997 ).
[11] M. G ´omez Berisso, P. Pedrazzini, J. G. Sereni, O. Trovarelli,
C. Geibel, and F. Steglich, E u r .P h y s .J .B 30,343(2002 ).
[12] A. R. Mackintosh, Physica B 130,112(1985 ).
[13] S. Doniach, Physica B 91,231(1977 ).
[14] T. Shigeoka, M. Yokohama, M. Kosaka, Y . Uwatoko, M.
Furugen, S. Ishida, and S. Asano, Physica B 281-282 ,96(2000 ).[15] S. Singh, S. K. Dhar, C. Mitra, P. Paulose, P. Mainfrinetti, and
A. Palenzona, J. Phys.: Condens. Matter 13
,3753 (2001 ).
[16] T. Gruner, N. Caroca-Canales, O. Stocker, M. M. Koza, J. Sereni,
U. Burkhard, and C. Geibel (unpublished).
[17] Y . Uwatoko, T. Ishii, G. Oomi, H. Takahashi, N. M ˆori, S. Nimori,
G. Kido, J. L. Serrao, D. Mandrus, Z. Fisk, and J. D. Thompson,Physica B 237-238 ,207(1997 ).
[18] See, for example, P. H. Meijer, J. H. Colwell, and B. P. Shah,
Am. J. Phys. 41,332(1973 ).
[19] See, for example, C. Domb and A. R. Miedema, in Progress in
Low Temperature Physics , edited by C. J. Gorter (North Holland,
1964), V ol. IV , Chap. VI, p. 300.
[20] B. Chevalier, W. Hermes, E. Gaudin, and R. Pottgen, J. Phys.:
Condens. Matter 22,146003 (2010 ).
[21] S. Couillaud, E. Gaudin, V . Franco, A. Conde, R. P ¨ottgen, B.
Heying, U. Ch. Rodewald, B. Chevalier et al. ,Intermetallics 19,
1573 (2011 ).
[22] S. A. Nikitin, I. A. Tskhadadze, I. V . Telegina, A. V .
Morozkin, and Y . Seropegin, J. Magn. Magn. Mater. 182,375
(1998 ).
174408-8 |
PhysRevB.84.165110.pdf | PHYSICAL REVIEW B 84, 165110 (2011)
Signature of helium segregation in hydrogen-helium mixtures
Sebastien Hamel,1Miguel A. Morales,1,2and Eric Schwegler1
1Lawrence Livermore National Laboratory
2Physics Department, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
(Received 22 December 2010; revised manuscript received 19 July 2011; published 11 October 2011)
The conductivity and reflectivity of mixtures of hydrogen and helium under high pressure are calculated using
first-principles molecular dynamics and the Kubo-Greenwood formula. Hydrogen-helium mixtures have beenpredicted to undergo demixing below a certain critical temperature. The impact of phase segregation of heliumon the optical properties of the mixtures is explored. The change in reflectivity upon demixing is found to varywith frequency with larger variation at higher frequency.
DOI: 10.1103/PhysRevB.84.165110 PACS number(s): 78 .15.+e, 78.20.Bh, 96 .15.Nd, 96 .30.Kf
I. INTRODUCTION
The lack of data from either direct observation or experi-
ments leaves unanswered fundamental questions in planetaryscience. The hydrogen-helium mixtures account for mostof the mass in giant planets, and so understanding theirproperties is crucial to advancing our understanding of theinternal structure of these bodies. The use of theoretical databased on first-principles simulation of material properties tocomplement known data is an important development of recentplanetary models.
1–5
Recently, the thermodynamic properties of H-He mixtures
were investigated using first-principles methods,6,7in particu-
lar the conditions under which the demixing of helium (some-times referred to as helium rain) may occur. These studies pre-dict that a large fraction of the interior of Saturn is in a regimewhere the hydrogen-helium mixture should phase separate.This phase separation provides an additional source of heat thatmay explain the planet’s high luminosity considering its massand age.
8,9(Saturn and Jupiter are believed to have been formed
approximately at the same time as the sun.) In order to verifythis prediction, we need experimental confirmation of wherethe mixture will phase separate under extreme conditions.For now, the experimental evidence for this demixing isindirect, but the relevant conditions of temperature (thousandsof Kelvin) and pressure (a few Mbar) may soon be achieved inusing ramp-wave compression laser experiments. In this paper,based on first-principles calculations of electrical conductivity,we propose that reflectivity measurements carried out athigh frequency will show a clear signature of the heliumsegregation.
II. METHOD
We used configurations drawn from first-principles molec-
ular dynamics (FPMD) simulations of H-He mixtures atdifferent temperatures and densities to investigate the impactof pressure, temperature, and helium concentration on theelectrical (and thermal) conductivity of the mixtures. Weconsidered four temperatures: 4, 6, 8, and 10 kK, ninehelium concentrations (0%, 2%, 5%, 10%, 20%, 40%, 60%,80%, and 100%), and densities corresponding to pressuresranging from 200 to 2000 GPa. For pure hydrogen, we alsoconsidered a lower density (0.37 g /m
3) at 10 kK, which is
on the hydrogen principal Hugoniot, in order to validate ourapproach with previously published experimental and theoret-
ical data.10,11
For each temperature and density, the electronic conduc-
tivity is evaluated on 15 well-spaced configurations drawn
from the FPMD trajectory. The FPMD simulations performedin this work were based on Kohn-Sham density functionaltheory, using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional. Empty states were included with an
electronic temperature set to the ionic temperature. This
electronic temperature is used to determine the (possiblyfractional) occupation number of the orbitals according to aFermi distribution. To integrate the equation of motion duringthe dynamics we used a time step of 8 a.u.
Most of the FPMD simulations were performed with the
QBOX code. We used Born-Oppenheimer MD (BOMD)within the NVT ensemble (with a weakly coupledBerendsen thermostat), as implemented in the QBOXcode (http://eslab.uc- davis.edu/software/qbox). We used aHamann-type local pseudopotential with a core radius ofr
c=0.3 au to represent hydrogen and a Troullier-Martins-type
nonlocal pseudopotential with sandpchannels and rc=1.091
au to represent helium. A plane-wave energy cutoff of 90 Rywas used for r
s<1.10 and of 115 Ry for rs>1.10. We used
250 electrons. The Brillouin zone was sampled at the /Gamma1point.
To reduce systematic effects and to get accurate pressures,we added a correction to the EOS designed to correct for theplane-wave cutoff and the sampling of the Brillouin zone. To
compute this, we studied 15–20 configurations at each density
and composition by using a 4 ×4×4g r i do f kpoints with
a plane-wave cutoff of at least 300 Ry. The actual plane-wavecutoff used depended on density and was chosen to achievefull convergence in the energy and pressure. Averages wereaccumulated for at least 2000 time steps after having first equi-librated the system, using a suitable effective model and subse-
quently allowed 300–500 time steps of equilibration with DFT.
Additional simulations performed in this work, including
conductivity calculations as well as FPMD simulations ofmetastable mixtures reported below, were performed withthe Vienna ab initio Simulation Package (V ASP).
12VA S P
and QBOX use different pseudopotentials, and we verifiedthat pressures were consistent when large plane-wave energycutoffs were used. The V ASP simulations are also performed atthe PBE
13level of approximation to the exchange-correlation
functional with projector-augmented wave (PAW)14
165110-1 1098-0121/2011/84(16)/165110(7) ©2011 American Physical SocietyHAMEL, MORALES, AND SCHWEGLER PHYSICAL REVIEW B 84, 165110 (2011)
pseudopotentials to account for the core electrons.15
The additional simulations of the phase-separated system
performed with V ASP used 250 hydrogen atoms and 167helium atoms to reach a concentration of 40% He.
For each configuration drawn from the trajectories, we
used the gamma point electronic density from the MD toevaluate the set of Kohn-Sham orbitals at the (
1
4,1
4,1
4)kpoint
(again using Fermi broadening at the ionic temperature). Basedon these orbitals,
16we use the Kubo-Greenwood formula
to estimate the real component of the frequency-dependentconductivity:
11,17,18
σ(ω)=2πe2¯h2
3m2ω/Omega1/summationdisplay
kW(k)N/summationdisplay
j=1N/summationdisplay
i=13/summationdisplay
α=1[F(/epsilon1i,k)−F(/epsilon1j,k)]
×|/angbracketleft/Psi1j,k|∇α|/Psi1i,k/angbracketright|2δ(/epsilon1i,k−/epsilon1j,k−¯hω), (1)
where eandmare the electron charge and mass, /Omega1the cell
volume, αdenotes the x,y, and zdirections, and F(/epsilon1i,k)i st h e
occupation number. The sum over kpoints is performed using
only the (1
4,1
4,1
4)kpoint. This approximation was validated
u s i n ga4 ×4×4g r i do f kpoints for a few configurations. The
change in the DC conductivity with configurations was foundto be much larger that the change due to k-point sampling,
which was at most a few percent. The dc conductivity isobtained as the zero frequency limit of σ(ω) averaged over
the different configurations, while the imaginary componentof the conductivity is obtained by using the Kramers-Kronigtransform:
σ
2(ω)=−2
πP/integraldisplay∞
0σ(ν)ω
(ν2−ω2)dν, (2)
where Pdenotes the principal value of the integral. Using
the complex conductivity, we can get the complex dielectricfunction /epsilon1, the index of refraction n, the coefficient of
extinction k,and the reflectivity R:
/epsilon1
1(ω)=1−4π
ωσ2(ω);/epsilon12(ω)=4π
ωσ(ω), (3)
/epsilon1(ω)=/epsilon11(ω)+i/epsilon12(ω)=[n(ω)+ik(ω)]2, (4)
R(ω)=[1−n(ω)]2+k2(ω)
[1+n(ω)]2+k2(ω). (5)
III. RESULTS AND DISCUSSION
A. Hydrogen
In Fig. 1we plot the frequency-dependent electronic
conductivity obtained using Eq. ( 1) for hydrogen along a
isotherm of 10 000 K for densities ranging from low-density(LD) 0.37 g /cm
3to high-density (HD) 2.33 g /cm3.T h e
frequency dependence of the conductivity of hydrogen showsa peak at zero photon energy for all densities except the lowest,which has a peak at finite photon energy. This change in shapeof the frequency dependence has importance consequences forthe reflectivity, which is 0.5 for this 0.37 g /cm
3point, while
it is 0.7–0.8 for the higher densities. This and a relatively lowdc conductivity of hydrogen at a density of 0.37 g /cm
3are the
result of a pseudogap forming in the DOS. This change from ametal at HD to a semimetal at LD can be seen in the Electron0 1 02 03 04 0 50 60
Ener gy (eV)01000020000300004000050000Conductivity (ohm-1cm-1)1823 GPa ; 2.33 g/cc ; R = 0.84
1378 GPa ; 2.03 g/cc ; R = 0.83
1054 GPa ; 1.77 g/cc ; R = 0.81
633 GPa ; 1.38 g/cc ; R = 0.77
315 GPa ; 0.98 g/cc ; R = 0.73
46 GPa ; 0.37 g/cc ; R = 0.51
FIG. 1. (Color online) Frequency-dependent electronic conduc-
tivity of hydrogen at 10 000 K for densities ranging from 0.37 to2.33 g/cm
3. The corresponding reflectivity at 1064 nm is included in
the legend.
Localization Function (ELF)19reported in Fig. 2.T h ev o l u m e
enclosed by the silver isosurface (ELF =0.5: homogeneous
electron gas-like) but not the red one (ELF =0.75: more local-
ized) is much larger for the HD hydrogen, pointing to a moredelocalized charge density. For the LD hydrogen the ELF showa high degree of localization around pairs of hydrogen atoms.
B. Hydrogen-helium mixtures
For the densities and temperatures considered here, pure
helium exhibits a large band gap. At the highest density(4.62 g /cm
3) and temperature (10 000 K) calculated, we
obtained an average gap of 9.5 eV . At the lowest density(1.95 g /cm
3) and temperature (4000 K), the gap increases
to 13.8 eV . Even with these large gaps, the temperaturebroadening of the Fermi distribution allows for some smalloccupation of the conduction bands leading to a small conduc-tivity (1–10 ohm
−1cm−1) for helium. This is to be compared
with conductivities in the tens of thousands ohm−1cm−1for
hydrogen under similar density and temperature conditions.Hence, for all the H-He mixtures calculated in this workthe dc conductivity is essentially coming from the hydrogencomponent of the fluid. For all the concentrations, the dcconductivity was found to increase approximately linearly withpressure and decrease exponentially with the concentration of
FIG. 2. (Color online) ELF of hydrogen at 10 000K for densities
of 0.37 and 2.33 g /cm3. The silver isosurface is for a value of
0.5 (homogeneous electron gas-like), and the red is for 0.75 (more
localized). The white spheres denote the hydrogen atoms and have
the same absolute size to illustrate the change in density.
165110-2SIGNATURE OF HELIUM SEGREGATION IN HYDROGEN- ... PHYSICAL REVIEW B 84, 165110 (2011)
0 500 1000 1500 2000010 00020 00030 00040 000
Pressure GPaConductivity ohm1cm1
0 500 1000 1500 20000200040006000800010 00012 00014 000
Pressure GPaConductivity ohm1cm1
0 500 1000 1500 200002468101214
Pressure GPaConductivity ohm1cm10 500 1000 1500 2000010 00020 00030 00040 00050 00060 000
Pressure GPaConductivity ohm1cm1
(a)
(b)
(c)
(d)
FIG. 3. (Color online) DC conductivity vs pressure for 4 kK
(black), 6 kK (red), 8 kK (blue), and 10 kK (green): (a) Pure hydrogen,
(b) H-He mixture with 10% helium, (c) H-He mixture with 40%
helium, (d) pure helium. The standard deviation over the differentconfigurations is used as an estimate of the uncertainty. For this
system, this uncertainty is larger than the error coming from size
effects or from an incomplete sampling of kspace.
helium. Compared to the impact of helium concentration and
pressure on the dc conductivity, the impact of temperature israther small in the range considered here but tends to decreasethe dc conductivity (Fig. 3, Tables I–IV). The only exception
to this is pure helium, where, because of the presence of a gap,the conductivity increases with temperature.TABLE I. DC conductivity of H-He mixtures at 4 kK.
X Density (g /cm3)P(GPa) σDC(S/cm) /Delta1σ DC(S/cm)
0 0.98 241 .9 17190 1332
0 1.38 528 23610 12990 1.77 916 .5 38365 4012
0 2.03 1218 .7 53017 5115
0 2.33 1635 .4 50856 4355
0.02 1.01 239 .7 15213 853
0.02 1.42 522 .4 21341 1833
0.02 1.83 905 .8 34762 1682
0.1 1.15 231 .4 9848 725
0.1 1.62 499 .1 13811 1169
0.1 2.08 863 .6 24242 1859
0.1 2.38 1150 33798 3531
0.1 2.73 1544 .1 33207 2330
0.2 1.31 220 .2 6244 364
0.2 1.84 471 .8 8670 288
0.2 2.36 817 .3 16223 849
0.2 2.70 1087 .5 20681 2168
0.2 3.10 1463 .7 19510 1060
0.4 1.53 203 .9 3682 116
0.4 2.15 432 .4 5414 101
0.4 2.77 747 .2 7604 116
0.4 3.16 996 .1 9039 155
0.4 3.63 1341 .8 10275 256
1 1.95 160 .30 0
1 2.74 345 .80 0
1 3.52 605 .20 0
1 4.02 813 .50 0
1 4.62 1105 .90 0
For the 10 000 K isotherm, we obtained a good fit ( R2=
0.999) of the dc conductivity using
σHHe(P,X )=− 0.23+9201.41b1(X)+19.6767b2(X)P,
b1(X)=e−5.62559X−1.07678X2−9.69242X12, (6)
b2(X)=e−2.77475X−0.457604 X2−4.40873X5,
where Pis the pressure in GPa, X=NHe
NH+NHeis the helium
concentration, and σis in units of ohm−1cm−1(see Fig. 4).
The fit was constructed first by using a linear function forthe pressure dependence and then fitting the concentrationdependence of the slopes and intercepts. For the pressuredependence fit, the standard deviation of the conductivity wasused as a measure of uncertainty [i.e., the conductivity valuesare weighted by 1 /(/Delta1σ)
2]. There appears to be a plateau of
conductivity reached at higher pressure and lower temperaturefor pure hydrogen as well as mixtures with low concentrationof helium. Hence, the linear regression may not be the bestchoice, but considering the errors bars (given by the standarddeviation of the snapshots), going beyond a linear fit is notwarranted at this time. This possible plateau will need to beinvestigated at higher pressure. We note that this functionalform used for σ
HHe(P,X ) interpolates smoothly the FPMD
data but may not be reliable outside the calculated range ofdensities.
165110-3HAMEL, MORALES, AND SCHWEGLER PHYSICAL REVIEW B 84, 165110 (2011)
TABLE II. DC conductivity of H-He mixtures at 6 kK.
X Density (g /cm3)P(GPa) σDC(S/cm) /Delta1σ DC(S/cm)
0 0.98 267 16496 682
0 1.38 565 .1 22790 1226
0 1.77 964 .8 33942 2809
0 2.03 1276 .1 48410 6871
0 2.33 1701 48070 2211
0.02 1.01 264 .4 14978 871
0.02 1.42 560 .2 19993 1528
0.02 1.83 953 .8 31600 1833
0.1 1.15 253 .3 9980 537
0.1 1.62 533 .3 13563 761
0.1 2.08 908 .9 23149 1871
0.1 2.38 1201 .8 30836 2238
0.1 2.73 1604 .6 30726 2809
0.2 1.31 241 .7 6148 270
0.2 1.84 503 .5 8943 470
0.2 2.36 857 .9 15491 1390
0.2 2.70 1138 19231 14540.2 3.10 1518 .9 18613 1260
0.4 1.53 223 .7 3693 85
0.4 2.15 461 .2 5331 113
0.4 2.77 786 .7 7271 198
0.4 3.16 1045 .6 8697 283
0.4 3.63 1394 .5 9834 327
1 1.95 178 .90 0
1 2.74 372 0 0
1 3.52 641 .80 0
1 4.02 853 .40 0
1 4.62 1153 .90 0
With mixtures, it is often practical to consider mixing
models in order to interpolate between the pure components.One of these models is the pressure-matching linear mixingmodel (PMLM)
18defined as
σPMLM
HHe (P,X )=/bracketleftbiggVH(P)
VHHe(P,X )/bracketrightbigg
σH(P)
+/bracketleftbiggVHe(P)
VHHe(P,X )/bracketrightbigg
σHe(P), (7)
FIG. 4. (Color online) The symbols are the FPMD dc conductivity
with error bar taken from the standard deviation between configura-
tions. The lines are from a fit to the FPMD dc conductivities using
Eq. ( 6). The temperature is 10 000 K.TABLE III. DC conductivity of H-He mixtures at 8 kK.
X Density (g /cm3)P(GPa) σDC(S/cm) /Delta1σ DC(S/cm)
0 0.98 291 .6 15603 720
0 1.38 600 .4 22259 1164
0 1.77 1010 .1 32282 3003
0 2.03 1328 .8 42109 3609
0 2.33 1762 .4 46969 3175
0.02 1.01 289 .4 14622 672
0.02 1.42 594 .8 19628 723
0.02 1.83 998 .8 29713 2080
0.02 2.09 1314 .9 37884 2849
0.02 2.40 1742 .3 42006 2249
0.05 1.08 282 .8 12224 507
0.05 1.51 581 .3 16543 713
0.05 1.94 978 25822 1422
0.05 2.22 1285 .3 33117 2176
0.05 2.55 1707 .4 35367 2405
0.1 1.15 276 9858 373
0.1 1.62 564 .6 13772 634
0.1 2.08 948 .9 22495 939
0.1 2.38 1249 28047 1559
0.1 2.73 1660 .3 29147 1869
0.2 1.31 262 .3 6215 293
0.2 1.84 533 .1 8886 290
0.2 2.36 896 .1 14650 857
0.2 2.70 1181 18779 1234
0.2 3.10 1570 .7 19577 1172
0.4 1.53 241 .3 3728 97
0.4 2.15 488 .4 5289 140
0.4 2.77 823 .4 7244 154
0.4 3.16 1083 .5 8651 223
0.4 3.63 1443 .8 9708 219
1 1.95 194 .60 0
1 2.74 396 .50 0
1 3.52 672 .10 0
1 4.02 892 .71 0
1 4.62 1195 .51 0
where
VHHe(P,X )=(1−X)VH(P)+XV He(P). (8)
Here we consider this linear mixing model for two reasons.
First, our explicit simulation of mixtures of helium andhydrogen of different concentration can be used to access thequality of the PMLM model for the dc conductivity and forthe equation of state of the mixture.
Second, the PMLM model describes a system where
two components are completely isolated from each otherbut at equivalent temperature and pressure (it is a mixingof noninteracting species). When the two components arehomogeneously mixed, this is always an approximation to thereal system. But when the two components are phase separatedbut remain in mechanical equilibrium (i.e., they have the samepressure and temperatures), this model becomes a very goodapproximation of the real system. In fact, in this case, the
165110-4SIGNATURE OF HELIUM SEGREGATION IN HYDROGEN- ... PHYSICAL REVIEW B 84, 165110 (2011)
TABLE IV . DC conductivity of H-He mixtures at 10 kK.
X Density (g /cm3)P(GPa) σDC(S/cm) /Delta1σ DC(S/cm)
0 0.98 315 15265 1022
0 1.38 633 .2 21579 823
0 1.77 1053 .8 30972 1156
0 2.03 1377 .7 38064 3011
0 2.33 1823 .3 43639 2402
0.02 1.01 310 .5 14524 689
0.02 1.42 627 .3 19381 936
0.02 1.83 1043 28328 12340.02 2.09 1364 .5 35242 2441
0.02 2.40 1801 .4 40624 2005
0.05 1.08 305 .4 12023 442
0.05 1.51 613 .2 16540 662
0.05 1.94 1019 .8 25925 1551
0.05 2.22 1334 .3 31414 2061
0.05 2.55 1765 .2 34052 1097
0.1 1.15 297 9675 387
0.1 1.62 595 13283 4940.1 2.08 991 .6 21352 1001
0.1 2.38 1296 .5 27284 1879
0.1 2.73 1713 .5 28772 1721
0.2 1.31 282 6485 265
0.2 1.84 562 .2 8982 493
0.2 2.36 934 .2 14387 894
0.2 2.70 1224 17964 1280
0.2 3.10 1619 .4 19355 1101
0.4 1.53 260 3765 76
0.4 2.15 514 .7 5315 111
0.4 2.77 856 .7 7208 194
0.4 3.16 1124 .6 8619 377
0.4 3.63 1489 .7 9642 309
0.6 1.71 238 .887 714 168
0.6 2.40 477 .353 1337 374
0.6 3.09 794 .106 2089 371
0.6 3.53 1041 2661 693
0.6 4.06 1384 .34 3094 365
0.8 1.84 224 .014 154 88
0.8 2.59 447 .027 229 103
0.8 3.32 745 .506 546 272
0.8 3.80 977 .054 410 243
0.8 4.37 1304 .
88 876 372
1 1.95 207 .11 0
1 2.74 417 .53 1
1 3.52 702 .54 1
1 4.02 928 .88 2
1 4.62 1238 .49 2
PMLM model is an approximation only in that the impact of
the interface on the transport properties is neglected.
A good fit ( R2=0.998) of the FPMD pressure-
temperature-density data from which we can evaluate thevolume fractions in the linear mixing model is given by
ρ={a
H(T)+[aHe(T)−aH(T)]X}Pc(T), (9)where Xis the concentration in He. As a function of
temperature, the parameters are given by
aH(T)=0.109651 −7.07335 ×10−6T+2.37907 ×10−10T2,
aHe(T)=0.247818 −1.57444 ×10−5T+5.33961 ×10−10T2,
c(T)=0.422708 +9.48002 ×10−6T−2.6347×10−10T2,
with densities given in (g /cm3), temperatures in Kelvin and
pressures in GPa.
For a mixture with 40% helium at 1500 GPa, the calculated
conductivity is four times smaller than for pure hydrogen at thesame pressure. By comparison, a PMLM model based on purehelium and pure hydrogen would predict only a factor of tworeduction in conductivity. This also means that if the conditionsare such that the H-He system segregates into a pure He frac-tion and a pure H fraction under constant pressures (reachinga state well described by a linear mixing rule), the overall
500100015002000Pressure GPa
0.0
0.5
1.0Helium concentration0500010 000σPMLM FPMDScm
500
1000
1500
2000Pressure GPa
0.0
0.5
1.0 Helium concentration0204060σPMLM FPMD
σFPMD500100015002000Pressure GPa
0.0
0.5
1.0 Helium concentration020 00040 000σScm
FIG. 5. (Color online) (Top) The dc conductivity of the
H-He mixture given by PMLM model (pink) and calculated with
FPMD (blue). (Middle) Difference between the PMLM model andFPMD dc conductivity. The PMLM model corresponds to the
conductivity the completely demixed fluid. (Bottom) The relative
difference
σPMLM−σFPMD
σFPMD exhibits the largest difference at high helium
concentration.
165110-5HAMEL, MORALES, AND SCHWEGLER PHYSICAL REVIEW B 84, 165110 (2011)
0 500 1000 1500 2000
Pressure (GPa)00.20.40.60.81Reflectivity at 1064 nmHydrogen
2% Helium
5% Helium
10% Helium
20% Helium
40% Helium
60% Helium
80% Helium
Helium
FIG. 6. (Color online) Reflectivity at 1064 nm as a function of
pressure and helium concentration at a temperature of 10 000 K.
conductivity will increase dramatically. Such a demixing of
helium in metallic hydrogen is predicted for Saturn (and to alesser extend in Jupiter).
6,7Note that the increase is particularly
important for higher helium concentration (Fig. 5).
Whilethe dc conductivity is a crucial quantity for magnetic
fields and thermal profile model of planets, the quantitythat is most relevant for laser-driven shock-compressionexperiments is the reflectivity. This is what is measured withthe use of VISAR diagnostics.
20,21In Fig. 6we show the
FPMD reflectivity at a frequency of 1064 nm as a functionof pressure and helium concentration for the 10 000 Kisotherm. As for the DC conductivity, the reflectivity of H-Hemixtures increases with pressure and decreases with heliumconcentration.
In order to investigate the impact of helium segregation on
the reflectivity, we prepared a (40% He) mixture in both themixed and segregated phases, as well as the pure states under1500 GPa and 10 000 K conditions. The segregated phase isa metastable system that was prepared by first fixing the crosssection of a pure H and a pure He and varying the length of theboxes to reach the target pressure. Once the pure systems areequilibrated, the boxes are brought in contact with each other,and the stress at the interfaces is relieved by running MDon each subsystem, freezing the atomic position of the othersubsystem. Then the volume is fixed, and a MD is continuedwith all the atoms free to move. Under this large pressure,very little diffusion was observed at this point. Configurationsfrom this MD were drawn (Fig. 7), and we evaluated the
conductivity using Eq. ( 1) in the direction along the slab.
This arrangement comes very close to a pressure-matchinglinear mixing (the only difference is in the “thickness” of theinterface).
FIG. 7. (Color online) 40% He mixtures at 10 000 K, 1500 GPa,
and average density 3.63 g /cm3: (a) mixed phase, (b) segregated in
“slabs” with the conductivity calculated along the slab.0 1 02 03 04 0 50 60
Ener gy (eV)010000200003000040000Conductivity (S/cm)Hydrogen
Helium
Mixed X0.4
Slabs
Linear Mixing
3 comp LM
FIG. 8. (Color online) Signature of He segregation in the fre-
quency dependence of the conductivity.
The frequency-dependent conductivity σ(ω) for pure H,
pure He, the 40% He mixture (mixed and segregated) aswell as the prediction of the pressure-matching LM modelare shown in Fig. 8. One can clearly see the very different
features of the pure hydrogen and pure helium system withthe latter’s onset of conductivity around ∼10 eV photon
energy (gap energy). As they should, these features (H peakat 0 eV and He peak at 30 eV) are also seen in the LMmodel and in the “slab” σ(ω),which is very close to the
LM model, but are completely missing from the fully mixedsystem. Under these P-T conditions, the pure H and pureHe spectra have a isosbectic point at 17.675 eV , and anymixing models based only on the pure references are boundto pass through that point. That the mixed system does notpass through that point is clear evidence of an electronicallydifferent fluid from either of its components. Using a three-component LM model that includes the mixed spectra, wecan estimate the volume fraction ( ∼20%) needed to reproduce
the slab calculation and obtain an interface “thickness” of∼1.2˚A.
Using Eq. ( 5), we evaluate the frequency-dependent reflec-
tivity. As noted above, even though at very low frequencythe mixed 40% He system has half the conductivity of thesegregated system and a quarter of that of pure hydrogen, thelow-frequency part of the reflectivity of these systems is sim-ilar. Only with increasing photon energy do the reflectivitiesdiverge from one another (Fig. 9). Even though the difference
24 681 0 1 2 1 4 1618 20 22 24 2628 30
Energy (eV)00.20.40.60.81ReflectivityHydrogen
Helium
Mixed
Slab
Linear Mixing
FIG. 9. (Color online) Signature of He segregation in the fre-
quency dependence of the reflectivity.
165110-6SIGNATURE OF HELIUM SEGREGATION IN HYDROGEN- ... PHYSICAL REVIEW B 84, 165110 (2011)
TABLE V . Reflectivity for H-He mixtures at 10 000 K, 1500 GPa.
λ(nm) energy (eV) H He ×0.4 Mixed LM ×0.4 slab
1064 1 .166 0.83 0.08 0.64 0.76 0.74
532 2 .335 0.81 0.08 0.59 0.73 0.71
350 3 .545 0.79 0.08 0.53 0.70 0.67
248.25 5 .0 0.78 0.08 0.48 0.68 0.64
124.125 10 .0 0.79 0.09 0.38 0.61 0.54
between mixed and segregated is obvious around 10 eV , the
separation is significant around 3 eV . We report values forsome laser frequency used in VISAR diagnostics in Table V.
We propose that measuring an increase in reflectivity underconstant pressure (and temperature) conditions may be thesignature of the demixing of helium from the H-He mixture.Using a multichannel VISAR to identify a larger effect athigher frequency would be further indication that demixing isobserved.
ACKNOWLEDGMENT
This work was performed under the auspices of the U.S.
Department of Energy under contract DE-AC52-07NA27344.
1J. J. Fortney and W. B. Hubbard, Icarus 164, 228 (2003).
2J. J. Fortney and W. B. Hubbard, Astrophys. J. 608, 1039 (2004).
3J. J. Fortney and N. Nettelmann, Space Sci. Rev. 152, 423 (2010).
4N. Nettelmann, B. Holst, A. Kietzmann, M. French, R. Redmer,
and D. Blaschke, Astrophys. J. 683, 1217 (2008).
5J. V orberger, I. Tamblyn, B. Militzer, and S. A. Bonev, Phys. Rev.
B75, 024206 (2007).
6M. A. Morales, E. Schwegler, D. M. Ceperley, C. Pierleoni,
S. Hamel, and K. Caspersen, Proc. Natl. Acad. Sci. USA 106,
1324 (2009).
7W. Lorenzen, B. Holst, and R. Redmer, P h y s .R e v .L e t t . 102, 115701
(2009).
8D. J. Stevenson and E. E. Salpeter, Astrophys. J. Suppl. Ser. 35,
221 (1977).
9D. J. Stevenson and E. E. Salpeter, Astrophys. J. Suppl. Ser. 35,
239 (1977).
10P. M. Celliers, G. W. Collins, L. B. DaSilva, D. M. Gold, R. Cauble,R. J. Wallace, M. E. Foord, and B. A. Hammel, Phys. Rev. Lett. 84,
5564 (2000).
11B. Holst, R. Redmer, and M. P. Desjarlais, Phys. Rev. B 77, 184201
(2008).
12G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993); 49, 14251
(1994); G. Kresse and J. Furthm ¨uller, Comput. Mater. Sci. 6,1 5
(1996); Phys. Rev. B 54, 11169 (1996).13J. P. Perdew, K. Burke, and M. Ernzerhof, P h y s .R e v .L e t t . 77, 3865
(1996); 78, 1396 (1997).
14P. E. Bl ¨ochl, Phys. Rev. B 50, 17953 (1994); G. Kresse and
D. Joubert, ibid.59, 1758 (1999).
15We used a plane-wave energy cutoff of 500 eV for the conductivity
calculations. A few configurations were tested with a 1200 eVcutoff to verify that spectra were well converged with respect toplane-wave energy cutoff.
16In the conductivity calculation, we use a sufficient amount ofunoccupied orbitals to capture all excitations below 60 eV . Thisallowed us to verify that the Kramers-Kronig transform wasconverged for the frequency interval considered here.
17T. R. Mattsson and M. P. Desjarlais, Phys. Rev. Lett. 97, 017801
(2006).
18D. A. Horner, J. D. Kress, and L. A. Collins, P h y s .R e v .B 77,
064102 (2008).
19A. D. Becke and K. E. Edgecombe, J. Chem. Phys. 92, 5397
(1990); B. Silvi and A. Savin, Nature (London) 371, 683
(1994).
20D. G. Hicks, T. R. Boehly, P. M. Celliers, J. H. Eggert, S. J. Moon,D. D. Meyerhofer, and G. W. Collins, P h y s .R e v .B 79, 014112
(2009).
21P. M. Celliers, G. W. Collins, L. B. Da Silva, D. M. Gold, andR. Cauble, Appl. Phys. Lett. 73, 1320 (1998).
165110-7 |
PhysRevB.98.144435.pdf | PHYSICAL REVIEW B 98, 144435 (2018)
Exotic specific heat anomaly in the GdY system: A probable signature of the Lifshitz transition
A. Vl. Andrianov
Faculty of Physics, M.V . Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia
(Received 25 October 2017; revised manuscript received 26 September 2018; published 25 October 2018)
The order-order magnetic phase transition “helical antiferromagnet–ferromagnet” in the Gd xY1−xsystem
have been examined by means of the specific heat vs temperature. The observed anomaly in the specific heatdoes not follow the ordinary second-order transition scenario but resembles the exotic second-and-a-half–orderbehavior, according to the Ehrenfest’s classification of the phase transitions. Such a behavior was first predictedfor the Lifshitz electronic topological transition. We suggest that the observed anomaly is likely a product of theunderlying Lifshitz transition associated with the magnetic phase transition.
DOI: 10.1103/PhysRevB.98.144435
I. INTRODUCTION
The Lifshitz transition [ 1], also known as the electronic
topological transition—the qualitative change of the Fermisurface (FS) shape under pressure, content, or temperaturevariation—is one of the basic phenomena of the physics ofmetals. In terms of the density of states, it occurs when Fermilevel crosses the van Hove singularity, therefore providingcharacteristic anomalies in the physical properties of themetal.
In early studies the Lifshitz transition was considered as a
rather exotic phenomenon that occurs only when Fermi leveland the van Hove singularity appeared accidentally—or by
precise choice of the object—to be very close to each other.
Later it appeared to be more frequent; Lifshitz transitionshave been observed experimentally in Bi and its alloys, inAl, Zn, Cd, As, In, in Re and its solid solutions, in LiMgand CdMg systems, etc., mostly under pressure and/or content
variation [ 2–13]. The observations of the Lifshitz transition in
Ti [14,15] and WTe
2[16] were remarkable as manifestations
of the “real” thermodynamic phase transition that occurs intemperature, not in pressure.
The general thermodynamic feature of the Lifshitz tran-
sition in temperature is a characteristic additional term pro-portional to ( T−T
1)5/2in the free energy; T1hereafter
corresponds to this phase transition [ 1]. According to the
classical Ehrenfest classification [ 17] this transition deserves
the name “second-and-a-half–order” transition. Consequentlythis transition is associated with the characteristic asymmetricsquare-root anomaly on the specific heat temperature depen-dence c
p(T):/Delta1cp(T)∝(T−T1)1/2[1]. The case of the
finite temperature is studied in [ 18].
In Fig. 1we compare qualitative sketches of the cp(T)
dependencies in the cases of (a) the ordinary second-ordertransition (considering our object, the lower-symmetry phaseis situated at the higher temperatures, which is uncommonbut feasible) and (b) the second-and-a-half–order transition,following [ 1]. In the most general description the second-
order transition is reflected by a more or less broadened peakwhile the second-and-a-half–order transition is reflected by asort of kink. In other words, in the case of the second-and-a-half–order transition the transition temperature is associated
not with the local peak, but with the onset of the wide anomalyon the c
p(T) dependence, which is a specific signature of this
transition.
Being just an addition to the electronic specific heat of
the given metal, which is in turn a tiny fraction of the totalspecific heat typically dominated by phonon contribution,this anomaly was expected unlikely to be observable exper-imentally. It is noteworthy that while the term “second-and-a-half–order transition” is often used as a synonym to the
“Lifshitz transition,” the second-and-a-half–order transition in
the strict Ehrenfest’s sense [i.e., a square-root anomaly on thec
p(T) dependence] has not been observed, to the best of our
knowledge.
In [ 19] we have suggested that the “helical
antiferromagnet–ferromagnet” magnetic phase transitionin the GdY system, believed to arise from the underlyingLifshitz transition, demonstrates the characteristicsecond-and-a-half–order transition behavior. In this workwe are going to confirm this suggestion.
Heavy rare-earth hcp metals and their alloys with each
other and relative yttrium are examples of solids in which thegeometry of the Fermi surface directly determines the typeof magnetic structure via “magnetic nesting” first proposedfor chromium by Lomer [ 20] and for rare-earth metals by
Dzyaloshinski [ 21]( s e er e v i e w[ 22]). It is well established
that the various forms of complex periodic magnetic structuresevident in these metals (i.e., helical, sinusoidal, cycloid etc.)all correspond to one plausible geometry of the FS, while thesimple collinear ferromagnetic order is associated with thealternative shape of the FS (see Refs. [ 23–28] for details).
The phase transitions from the complex periodic mag-
netic structure to the simple collinear ferromagnetic phasewith temperature are typical of the heavy rare-earth metals.In the straightforward approach it means that all of thesetransitions should be associated with the second-and-a-half–order transition, i.e., the Lifshitz transition from one plausiblegeometry of the FS to the alternative one. Actually the mag-netic structures in these objects are typically subject to thestrong magnetic anisotropy and therefore these transitions are
2469-9950/2018/98(14)/144435(5) 144435-1 ©2018 American Physical SocietyA. VL. ANDRIANOV PHYSICAL REVIEW B 98, 144435 (2018)
(a)
(b)
FIG. 1. Qualitative sketch of the temperature behavior of the spe-
cific heat in the vicinity of the phase transition: (a) phase transitionatT
1is second order (considering our object, the lower-symmetry
phase is situated at the higher temperatures, which is uncommon
but feasible); (b) phase transition at T1is a second-and-a-half–order
transition, arising from the Lifshitz transition. Dashed line depicts a
background dependence. T1: temperature of the phase transition.
accompanied by drastic crystalline lattice distortions. As a re-
sult, these transitions in Tb, Dy, and their relatives are actuallyfirst order. However, Gd and its solutions with nonmagneticyttrium are an important exception because the magneticanisotropy in Gd is several orders of magnitude smaller thanin the other heavy rare-earth metals due to its zero orbitalmoment [ 22].
The magnetic structures and phase transitions in the
Gd
1−xYxsystem were studied thoroughly in [ 29–34]. Com-
positions with 0 .40>x> 0.32 do order magnetically into
a helical antiferromagnetic state at TNand then turn into a
collinear ferromagnetic state at T1and remain ferromagnetic
down to the lowest temperatures, which is typical for theheavy rare-earth metals. Nevertheless, it was clearly demon-s t r a t e di n[ 32] that the “helical antiferromagnet–ferromagnet”
transition in the Gd
66Y34system is definitely not first order
in contrast with the other heavy rare-earth metals’ solutions.In particular, no thermal hysteresis was observed on thesmooth temperature dependence of the magnetic helical wavevector that approaches zero at T
1[32]. The direct study
of the FS in this system by positron annihilation [ 34,35]
and angle-resolved photoemission spectroscopy [ 36,37] along
with ab initio calculations [ 26] confirmed the change in
the FS shape. Therefore, we suggested that the discussedmagnetic phase transition, with the Lifshitz transition in thebackground, may be actually second-and-a-half–order [ 19].
The expected asymmetric square-root anomaly, according tothe FS geometry, should be associated with the helical phase,i.e., located at T> T
1. The specific heat measurements were
performed to check this suggestion.
II. EXPERIMENTAL DETAILS
The textured samples chosen for this study were cut from
specimens of Gd 0.66Y0.34,G d 0.65Y0.35, and Gd 0.69Y0.31,p r e -
pared in Moscow Institute for Metals by O. D. Chistyakovfrom the components of 99.99% purity by arc melting andfollowing annealing under the same conditions for all threespecimens. (The specimens of Gd
0.66Y0.34and Gd 0.65Y0.35
w e r et h es a m ea su s e di n[ 19]; see detailed description and
characterization therein.)
The magnetic states of the samples were examined by
means of ac magnetic susceptibility temperature depen-dencies χ(T). The values of transition temperatures ob-
tained from these dependencies are 201 ±0.5, 202 ±0.5,
and 213 ±0.5 K for the magnetic ordering temperatures;
125±2, 152 ±2, and 190 ±2Kf o r T
1for Gd 0.65Y0.35,
Gd0.66Y0.34, and Gd 0.69Y0.31, respectively. These values match
the “temperature-content” phase diagram obtained in [ 31]
within ±0.3% content accuracy that proves the quality of the
samples. Therefore, the magnetic ordering temperatures forall three samples vary within ±3%; their Debye temperatures,
according to density, within ±1%; and their saturation mag-
netization values, according to content, within ±3%, rather
close to each other, while T
1values vary drastically. It means
that Gd 0.69Y0.31, ferromagnetic from the lowest temperatures
to 190 K, may serve as a reference object for the remainingtwo samples in this temperature range that includes the studiedphase transition at T
1in both of them.
The temperature dependencies of the specific heat were ob-
tained with a PPMS Quantum Design relaxation calorimeterin a temperature range 2–270 K on heating, with and withoutmagnetic field, using the same setup for all three samples. Thetemperature rise in a single measurement never exceeded 4 K.
III. RESULTS
The resulting cp(T) dependence for Gd 0.65Y0.35atH=0
is presented in Fig. 2(a); the dependencies for the remaining
two samples almost coincide with it on this scale. The peakatT
Nis clear while the anomaly at T1is weak. Obviously
there is no evidence of the first-order transition at T1. Elec-
tronic specific heat coefficients γatT→0a r e1 4 ±2, 17±
3, and 12 ±2m J/K2mol for Gd 0.65Y0.35,G d 0.66Y0.34, and
Gd0.69Y0.31respectively; approximately the same value.
The detailed cp(T) dependencies for H=0 and H=
5 kOe for all the samples in the vicinity of T1are presented in
Fig. 2(b). The accuracy of these dependencies, estimated by
data points spread, is about ±0.2[ J/mol K], and calorimeter
reported errors are about ±0.1[ J/mol K]. We assume that
H=5 kOe magnetic field suppresses completely the helical
antiferromagnetic phase in both the helically ordered samplesin favor of the ferromagnetic phase. The anomaly in c
p(T)
associated with the magnetic transition at T1is well resolved
at least for Gd 0.65Y0.35, while for Gd 0.66Y0.34it is at the edge
of the experimental accuracy. It is already clear that theseanomalies do not resemble a peak at T
1[Fig. 1(a)] and are
closeer to the second-and-a-half–order behavior [Fig. 1(b)].
144435-2EXOTIC SPECIFIC HEAT ANOMALY IN THE GdY … PHYSICAL REVIEW B 98, 144435 (2018)
(a)
(b)
FIG. 2. (a) Temperature dependence of the specific heat cp(T)i n
zero magnetic field for Gd 0.65Y0.35.T1=125±3 K is the tempera-
ture of the helical antiferromagnet–ferromagnet transition, obtained
independently by neutron scattering [ 29] and magnetic susceptibility
[19].TN=200±0.5 K is the magnetic ordering temperature. The
dependencies for Gd 0.65Y0.35,G d 0.66Y0.34,a n dG d 0.69Y0.31almost
coincide on this scale. Dotted rectangle corresponds to Fig. 2(b).( b )
Temperature dependencies of the specific heat cp(T) in the vicinity
ofT1for Gd 0.69Y0.31(H=0, solid squares), Gd 0.66Y0.34(circles,
solid for H=0 and open for H=5 kOe, shifted vertically for
clarity), and Gd 0.65Y0.35(triangles, solid for H=0 and open for
H=5 KOe, shifted vertically for clarity). Lines are guides for the
eye. Arrows mark T1, the temperature of the helical antiferromagnet–
ferromagnet transition. PM: paramagnetic; HAFM: helical antiferro-magnetic; FM: ferromagnetic phases.
Subtracting the cp(T) dependence for the reference object,
ferromagnetic Gd 0.69Y0.31, from dependencies for Gd 0.65Y0.35
and Gd 0.66Y0.34we obtain the resulting /Delta1cp(T) dependencies,
presented in Fig. 3. Interpolation of Gd 0.69Y0.31cp(T) de-
pendence for the subtraction was performed employing linearleast-square fit within ±5 K from the required temperature;
dependencies for Gd
0.65Y0.35and Gd 0.66Y0.34were used with-
out any preprocessing. We also present a fragment of thedependence for Gd
0.65Y0.35in the magnetic field H=5k O e
that suppresses the magnetic transition at T1. The presenceFIG. 3. Temperature dependencies of the specific heat /Delta1cp(T)
for Gd 0.65Y0.35atH=0 (solid triangles) and H=5 kOe (open
triangles, magnetic transition suppressed), and Gd 0.66Y0.34(solid cir-
cles) after subtracting the dependence for the reference Gd 0.69Y0.31.
Error bars are reported by the calorimeter. Dashed lines stay for
the linear background dependencies and for the square-root fits
associated with the presumed second-and-a-half–order transition,suggested to arise from the underlying Lifshitz transition. T
1are the
temperatures of this transition, obtained from the square-root fits.
HAFM: helical antiferromagnetic; FM: ferromagnetic phases.
of the anomaly at H=0 and its absence at H=5 kOe prove
that the observed anomaly is really associated with the studiedtransition.
IV . DISCUSSION
The anomaly associated with the transition is clearly re-
solved, and both the samples behave in a similar way. Itis worth mentioning that the very shape of the anomaly atH=0, even without physical background, is sufficient to
identify this transition as an Ehrenfest second-and-a-half–order. Indeed, the Ehrenfest order of transition is higher thantwo, because the additional specific heat at T→T
1tends
to zero, in contrast with the ordinary second-order behavior,Fig. 1(a). On the other hand, the clearly nonlinear and up-
arched anomaly corresponds to the order of transition lowerthan three. The square-root dependence predicted for thesecond-and-a-half–order transition fits reasonably the /Delta1c
p(T)
dependencies for both the samples, in agreement with theexpectations, Fig. 1(b). Hence we conclude that the studied
phase transition at T
1is likely a second-and-a-half–order
transition according to the Ehrenfest classification.
The relation of such a behavior with the underlying Lifshitz
transition, the only known second-and-a-half–order transitionso far, seems highly likely. On the other hand, the studiedtransition is definitely not a “genuine” Lifshitz transitiondescribed in [ 1] but presumably a complex combination of
Lifshitz transition and magnetic phase transition (see above).Moreover, the role of magnetoelastic phenomena is also im-portant [ 33] and cannot be neglected. Therefore, the studied
transition is apparently a product of the interplay of con-ductive, magnetic, and elastic subsystems in the vicinity of
144435-3A. VL. ANDRIANOV PHYSICAL REVIEW B 98, 144435 (2018)
Lifshitz transition. A reasonable suggestion is that this com-
plex transition preserves nevertheless the square-root anomalyin the specific heat characteristic of the genuine Lifshitztransition. The alternative is an assumption that the observedanomaly is not related with the Lifshitz transition at all andarises from some other phenomenon, which seems unlikely asno such alternate phenomena are known. Of course, these pre-liminary suggestions require theoretical consideration. Proba-bly the recent progress in the ab initio calculations [ 28] would
provide the full description of this combined Lifshitz-and-magnetic transition.
The probable alternative scenarios should also be consid-
ered. The approach that does not involve Lifshitz transitionshould examine temperature-dependent energies of the twocompeting magnetic phases, helical and ferromagnetic, underthe most general assumptions but without employing specificrelations characteristic of the Lifshitz transition. This prob-lem is destined for the phenomenological Landau analysis,which has been performed in [ 38] with special interest to the
GdY system. It revealed that the studied transition should besecond order (while introduced hexagonal in-plane magnetic
anisotropy makes it even first order). Therefore, we concludethat the observed non-second-order behavior arises from someextra phenomenon, and all the features point to the Lifshitztransition.
V . CONCLUSION
In summary, we find that the “order-order” magnetic
phase transition in temperature in the GdY system is asso-ciated with an exotic specific heat anomaly that identifiesthis transition as an Ehrenfest second-and-a-half–order. Here,Ehrenfest’s second-and-a-half–order transition was directlyobserved as a specific heat anomaly. There are plausiblereasons to associate this anomaly with the underlying Lifshitztransition.
ACKNOWLEDGMENTS
The author is grateful to O. D. Chistiakov for the provision
of the samples and to the WSBS team for their hospitality.
[1] I. M. Lifshitz, Zh. Eksp. Teor. Fiz. 38, 1569 (1960) [Sov. Phys.
JETP 11, 1130 (1960)].
[2] Yu. P. Gaidukov, N. P. Danilova, and M. B. Shcherbina-
Samoilova, Zh. Eksp. Teor. Fiz. 77, 2125 (1979) [Sov. Phys.
JETP 50, 1018 (1979)].
[3] N. B. Brandt, V . S. Egorov, M. Yu. Lavrenyuk, N. Ya. Minina,
a n dA .M .S a v i n ,Z h .E k s p .T e o r .F i z . 89, 2257 (1985) [Sov.
Phys. JETP 62, 1303 (1985)].
[4] D. R. Overcash, T. Davis, J. W. Cook, Jr., and M. J. Skove, Phys.
Rev. Lett. 46,287(1981 ).
[5] Yu. P. Gaidukov, N. P. Danilova, and M. B. Shcherbina-
Samoilova, Pis’ma Zh. Eksp. Teor. Fiz. 25, 509 (1977) [Sov.
Phys. JETP Lett. 25, 479 (1977)].
[6] S. L. Bud’ko, A. N. V oronovskii, A. G. Gapotchenko, and E. S.
Itskevich, Zh. Eksp. Teor. Fiz. 86, 778 (1984) [Sov. Phys. JETP
59, 454 (1984)].
[7] C. L. Watlington, J. W. Cook, Jr., and M. J. Skove, Phys. Rev.
B15,1370 (1977 ).
[8] J. E. Schirber and J. P. Van Dyke, Phys. Rev. Lett. 26,246
(1971 ).
[9] I. Ya. V olynskii, V . I. Makarov, and V . V . Gann, Zh.
Eksp. Teor. Fiz. 69, 1019 (1975) [Sov. Phys. JETP 42, 518
(1975)].
[10] C. W. Chu, T. F. Smith, and W. E. Gardner, Phys. Rev. B 1,214
(1970 ).
[11] A. N. Velikodnyi, N. V . Zavaritskii, T. A. Ignat’eva, and A. A.
Yurgens, Pis’ma Zh. Eksp. Teor. Fiz. 43, 597 (1986) [Sov. Phys.
JETP Lett. 43, 773 (1986)].
[12] V . S. Egorov and A. N. Fedorov, Zh. Eksp. Teor. Fiz. 85, 1647
(1983) [Sov. Phys. JETP 58, 959 (1983)].
[13] S. V . Varyukhin and V . S. Egorov, Pis’ma Zh. Eksp.
Teor. Fiz. 39, 510 (1984) [Sov. Phys. JETP Lett. 39, 621
(1984)].
[14] V . I. Nizhankovskii, M. I. Katsnelson, G. V . Peschanskikh, and
A. V . Trefilov, Pis’ma Zh. Eksp. Teor. Fiz. 59, 693 (1994) [Sov.
Phys. JETP Lett. 59, 733 (1994)].[15] P. Souvatzis, O. Eriksson, and M. I. Katsnelson, Phys. Rev. Lett.
99,015901 (2007 ).
[16] Y . Wu, N. H. Jo, M. Ochi, L. Huang, D. Mou, S. L. Budko,
P. C. Canfield, N. Trivedi, R. Arita, and A. Kaminski, Phys.
Rev. Lett. 115,166602 (2015 ).
[17] P. Ehrenfest, Communications from the Physical Laboratory of
the University of Leiden, Suppl. No. 75b, 1933.
[18] Ya. M. Blanter, M. I. Kaganov, A. V . Pantsulaia, and A. A.
Varlamov, Phys. Rep. 245,159(1994 ).
[19] A. Vl. Andrianov and O. A. Savelieva, Europhys. Lett. 82,
47012 (2008 ).
[20] W. M. Lomer, Proc. Phys. Soc. 80,489(1962 ).
[21] I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 47, 336 (1965) [Sov.
Phys. JETP 20, 223 (1965)].
[22] J. Jensen and A. R. Macintosh, Rare Earth Magnetism
(Clarendon, Oxford, 1991).
[23] S. B. Dugdale, H. M. Fretwell, M. A. Alam, G. Kontrym-
Sznajd, R. N. West, and S. Badrzadeh, Phys. Rev. Lett. 79,941
(1997 ).
[24] V . Thakor, J. B. Staunton, J. Poulter, S. Ostanin, B. Ginatempo,
and E. Bruno, P h y s .R e v .B 68,134412 (2003 ).
[25] L. Nordström and A. Mavromaras, Europhys. Lett. 49,775
(2000 ).
[26] I. D. Hughes, M. Däne, A. Ernst, W. Hergert, M. Lüders,
J. Poulter, J. B. Staunton, A. Svane, Z. Szotek, and W. M.Temmerman, Nature (London) 446,650(2007 ).
[27] A. Vl. Andrianov, O. A. Savel’eva, E. Bauer, and J. B. Staunton,
Phys. Rev. B 84,132401 (2011 ).
[28] E. Mendive-Tapia and J. B. Staunton, Phys. Rev. Lett. 118,
197202 (2017 ).
[29] S. Bates, S. B. Palmer, J. B. Sousa, G. J. McIntyre, D. Fort, S.
Legvold, B. J. Beaudry, and W. C. Koehler, Phys. Rev. Lett. 55,
2968 (1985 ).
[30] S. B. Palmer, S. Bates, G. J. McIntyre, J. B. Sousa, D.
Fort, and B. J. Beaudry, J. Magn. Magn. Mater. 54–57 ,519
(1986 ).
144435-4EXOTIC SPECIFIC HEAT ANOMALY IN THE GdY … PHYSICAL REVIEW B 98, 144435 (2018)
[31] R. J. Melville, S. B. Palmer, S. Bates, and G. J. McIntyre,
J. Magn. Magn. Mater. 116,267(1992 ).
[32] R. J. Melville, R. S. Eccleston, G. J. McIntyre, and S. B. Palmer,
J. Phys.: Condens. Matter 4,10045 (1992 ).
[33] S. B. Palmer, G. J. McIntyre, A. V . Andrianov, and R. J.
Melville, J. Magn. Magn. Mater. 177,1023 (1998 ).
[34] S. J. Crowe, S. B. Dugdale, Zs. Major, M. A. Alam, J. A. Duffy,
a n dS .B .P a l m e r , Europhys. Lett. 65,235(2004 ).[35] H. M. Fretwell, S. B. Dugdale, M. A. Alam, D. C. R. Hedley,
A. Rodriguez-Gonzalez, and S. B. Palmer, P h y s .R e v .L e t t . 82,
3867 (1999 ).
[36] K. M. Döbrich, A. Bostwick, E. Rotenberg, and G. Kaindl,
Phys. Rev. B 81,012401 (2010 ).
[37] K. M. Döbrich, A. Bostwick, J. L. McChesney, K. Rossnagel, E.
Rotenberg, and G. Kaindl, P h y s .R e v .L e t t . 104,246401 (2010 ).
[38] A. Michelson, Phys. Rev. B 16,585(1977 ).
144435-5 |
PhysRevB.97.085137.pdf | PHYSICAL REVIEW B 97, 085137 (2018)
Extremely large magnetoresistance and electronic structure of TmSb
Yi-Yan Wang,1Hongyun Zhang,2Xiao-Qin Lu,1Lin-Lin Sun,1Sheng Xu,1Zhong-Yi Lu,1Kai Liu,1
Shuyun Zhou,2,3and Tian-Long Xia1,*
1Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices,
Renmin University of China, Beijing 100872, People’s Republic of China
2State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University,
Beijing 100084, People’s Republic of China
3Collaborative Innovation Center of Quantum Matter, Beijing 100084, People’s Republic of China
(Received 11 November 2017; published 23 February 2018)
We report the magnetotransport properties and the electronic structure of TmSb. TmSb exhibits extremely large
transverse magnetoresistance and Shubnikov-de Haas (SdH) oscillation at low temperature and high magneticfield. Interestingly, the split of Fermi surfaces induced by the nonsymmetric spin-orbit interaction has beenobserved from SdH oscillation. The analysis of the angle-dependent SdH oscillation illustrates the contributionof each Fermi surface to the conductivity. The electronic structure revealed by angle-resolved photoemissionspectroscopy (ARPES) and first-principles calculations demonstrates a gap at the Xpoint and the absence of
band inversion. Combined with the trivial Berry phase extracted from SdH oscillation and the nearly equalconcentrations of electron and hole from Hall measurements, it is suggested that TmSb is a topologically trivialsemimetal and the observed XMR originates from the electron-hole compensation and high mobility.
DOI: 10.1103/PhysRevB.97.085137
I. INTRODUCTION
Recently, rare earth monopnictides LnX (Ln =La, Y , Ce, Nd
and X=Sb, Bi) have drawn much attention and been studied
widely [ 1–27]. In these materials, extremely large magnetore-
sistance (XMR) is a remarkable signature since conventionalnonmagnetic metals usually show a small magnetoresistance(MR) of only a few percent. XMR has also been observed inseveral other materials such as WTe
2[28–30] and (Nb/Ta)As 2
[31–35]. Several mechanisms have been proposed to explain
the origin of XMR, for example, magnetic field inducedmetal-to-insulator transition [ 2], the breaking of topological
protection [ 36], or the compensation of hole and electron
[8,14]. For a semimetal with topologically nontrivial electronic
structure, the topological protection suppresses backscatteringat zero magnetic field. The application of a field will break theprotection and result in XMR [ 36]. However, the nontrivial
topological state is not indispensable for the generation ofXMR since topologically trivial materials (such as LaSb [ 4],
YSb [ 17,19], and CeSb [ 23]) can also exhibit XMR. In fact,
XMR can be explained by the electron-hole compensationfrom the semiclassical two-band model [ 8,14]. In that case, the
balance between electron concentration and hole concentrationwill lead to unsaturated quadratic behavior of the MR, and thevalue of MR depends on the mobility of carriers.
The topological property of the LnX family is interesting. A
previous theoretical work [ 1] predicts that LaX (X =N, P, As,
Sb, Bi) are topological semimetals or topological insulators.Later ARPES experiments show that LaSb is a topologicallytrivial material without band inversion [ 4] while LaBi is a
*tlxia@ruc.edu.cntopological semimetal with multiple Dirac cones in the surface
band structure [ 12,15]. By drawing the topological phase
diagram of CeX (X =P, As, Sb, Bi) as a function of the
spin-orbit-coupling (SOC) effect, Kuroda et al. demonstrates
the topological phase transition from trivial to nontrivial withthe increase of the SOC effect [ 24]. Consequently, it is of
interest to explore the possible topological materials in othermembers of LnX with strong SOC effect.
TmSb is an isostructural compound to LaSb/LaBi. In this
work, we have grown the high quality single crystals of TmSband investigated the detailed magnetotransport properties andthe electronic structure. The transverse MR of TmSb reaches3.31×10
4% at 2.3 K & 14 T. The split of Fermi surfaces
(FSs) is found through the analysis of SdH oscillation, whichis attributed to the nonsymmetric spin-orbit interaction. Theangle-dependent MR are measured to clarify the contributionof each Fermi surface (FS) to the conductivity. In addition,the electronic structure of TmSb has been studied by ARPESexperiments and first-principles calculations. The trivial Berryphase and the absence of band inversion indicate that TmSbis a topologically trivial semimetal. The Hall measurementsreveal the compensation of carriers and the high mobility,which constitute the origin of the observed XMR.
II. EXPERIMENTAL METHODS AND CRYSTAL
STRUCTURE
Single crystals of TmSb were grown from Sb flux. Tm
and excess Sb were placed in a crucible with a ratio ofTm:Sb =1:6. Then the crucible was sealed into an evacuated
quartz tube and heated to 1150
◦C. After cooling to 750◦C
in 300 hours, the excess antimony flux was removed with acentrifuge. The elemental composition was checked by energy
2469-9950/2018/97(8)/085137(7) 085137-1 ©2018 American Physical SocietyWANG, ZHANG, LU, SUN, XU, LU, LIU, ZHOU, AND XIA PHYSICAL REVIEW B 97, 085137 (2018)
dispersive x-ray spectroscopy (EDS, Oxford X-Max 50). X-ray
diffraction (XRD) patterns of powder and single crystal werecollected from a Bruker D8 Advance x-ray diffractometerusing Cu K
αradiation. TOPAS-4.2 was employed for the
refinement. Resistivity measurements were performed on aQuantum Design physical property measurement system (QDPPMS-14T). ARPES measurements were taken at the Dream-line beamline of the Shanghai Synchrotron Radiation Facility(SSRF). The crystals were cleaved in situ along the ( 001 )
plane and measured at T ∼20 K with a working vacuum better
than 5 ×10
−11Torr. The first-principles calculations were
performed with the projector augmented wave (PAW) method[37,38] as implemented in the V ASP package [ 39]. For the
exchange-correlation functional, we adopted the generalizedgradient approximation (GGA) of Perdew-Burke-Ernzerhof(PBE) type [ 40]. The kinetic energy cutoff of the plane-wave
basis was set to be 250 eV . The Brillouin zone was sampledwith a 20 ×20×20k-point mesh and the Gaussian smearing
method with a width of 0.05 eV was used to broaden the Fermisurface. Both cell parameters and internal atomic positions
were fully relaxed until all forces became less than 0.01 eV /˚A.
The calculated lattice constant 6.131 ˚A of TmSb agrees well
with the experimental value 6.105 ˚A[41]. In the study of
electronic structure, the modified Becke-Johnson (MBJ) [ 42]
exchange potential at the meta-GGA level of the Jacobs ladderwas used and the SOC effect was included. For the calculationsof Fermi surfaces, the maximally localized Wannier functions(MLWF) method [ 43,44] was employed. TmSb crystallizes in
the NaCl-type structure as shown in the inset of Fig. 1(a).T h e
obtained TmSb crystals are in the shape of cubes. The singlecrystal XRD pattern indicates that the surface of the crystal isthe (00l) plane [Fig. 1(a)]. The powder XRD pattern of TmSb
crystals can be well refined as shown in Fig. 1(b). The refined
lattice parameter a(6.08(0) ˚A) is in good agreement with the
value in the Inorganic Crystal Structure Database (ICSD) [ 41].
III. RESULTS AND DISCUSSION
We have investigated the magnetotransport properties of
TmSb in detail. Figure 2(a) shows the temperature dependent
resistivity ρxx(T) under different magnetic fields. TmSb ex-
hibits metallic behavior under zero magnetic field. After apply-ing a moderate field, an upturn appears in ρ
xx(T) curve with the
temperature decreased. The upturn can be enhanced by increas-ing magnetic field. Similar behavior has also been observed inthe isostructural compounds LaSb/LaBi/YSb [ 2,8,9,17] and
other XMR materials (such as WTe
2[28], NbAs 2/TaAs 2[31]).
Especially in WTe 2, the upturn has been successfully explained
by the following of Kohler’s rule in high quality sampleswith low charge carrier density [ 29]. Resistivity plateau is
another phenomenon usually observed in XMR materials. Theresistivity plateau seems to be absent in TmSb. However,as seen in the ∂ρ/∂T curves [inset on the left of Fig. 2(a)]
derived from the main panel, a minimum at T
i∼5.6 K can be
obtained under different fields, indicating that the resistivityplateau starts to emerge. The resistivity plateau is suggested tooriginate from the temperature-insensitive resistivity at zerofield [ 7,14]. The inset on the right of Fig. 2(a) plots the
transverse MR of TmSb as a function of field. The MR follows
FIG. 1. (a) Single crystal XRD pattern of a TmSb crystal, showing
only the ( 00l) reflections. Inset: the crystal structure of TmSb. The
blue and red balls represent Tm and Sb, respectively. (b) Powder
XRD pattern of TmSb with refinement. Red circle and black solid linerepresent the data of experiment and the fit curve, respectively. The
difference plot is in blue. The pink vertical lines denote the positions
of Bragg peaks of TmSb. The inset is an image of TmSb single crystal.
B1.76(red solid line) and the value reaches 3 .31×104%a t
2.3 K & 14 T. Usually, in semimetals with perfect electron-hole compensation, the MR will exhibit quadratic behavior(MR∝B
2) and not be saturated. The index in TmSb deviates
from 2, indicating that the electron and hole in TmSb may beslightly imbalanced.
SdH oscillation has been observed at low temperature and
high field [Fig. 2(b)]. The oscillation becomes weaker with
the increase of temperature. After subtracting a smooth back-ground, the SdH oscillation amplitude /Delta1ρ
xx=ρxx−/angbracketleftρxx/angbracketright
can be obtained as shown in the inset of Fig. 2(b). Figure 2(c)
presents the fast Fourier transform (FFT) analysis of the SdHoscillation. Seven peaks (including three pairs of peaks andone single peak) are identified from the FFT spectra. Since theOnsager relation F=(φ
0/2π2)A=(¯h/2πe)Adescribes that
the frequency Fis proportional to the extremal cross-sectional
areaAof FS normal to the field, three pairs of peaks mean
the FSs split under the field. In fact, the split of FSs has alsobeen observed in de Haas-van Alphen (dHvA) type oscillationof TmSb [ 45]. TmSb is paramagnetic [ 46], and the large
magnetization is contributed by the local magnetic momentof Tm
3+ions which develops with the application of magnetic
field [ 45]. The spin degeneracy is lifted under the field and the
085137-2EXTREMELY LARGE MAGNETORESISTANCE AND … PHYSICAL REVIEW B 97, 085137 (2018)
FIG. 2. Magnetotransport properties of TmSb (Sample 1, RRR =70). (a) Temperature dependence of resistivity ρxx(T)a tB=0T ,6T ,1 0
T, 14 T. Inset on the left: ∂ρ/∂T as a function of temperature. Inset on the right: MR versus magnetic field B at 2.3 K. The MR follows B1.76,
which can be well fitted as shown in the red solid line. (b) Magnetic field dependence of resistivity ρxx(B) at different temperatures. Inset:
The amplitude of SdH oscillation plotted as a function of 1/B. (c) The FFT spectra of the corresponding oscillations. Inset: The projection of
the calculated FSs from the direction of kx. (d) Temperature dependent FFT amplitude of the frequencies. The solid lines are fittings using the
thermal factor in LK formula. (e) The fit (red solid line) of SdH oscillation at 2.3 K using the multiband LK formula. (f) FFT spectra of the SdH
oscillations with the change of θat 2.3 K. (g) The frequencies originating from electronlike FSs plotted as a function of the angle θ. The solid
lines are fits to the equation presented in the text. The inset on the left shows angle dependence of the frequencies originating from holelike
FSs. The inset on the right is a schematic diagram of the measurements.
nonsymmetric spin-orbit interaction is formed [ 47], resulting
in the split of FSs.
The inset of Fig. 2(c)shows the projection of the calculated
FSs on the ky-kzplane. In the current measurement ( I//x ,
B//z ), there are three kinds of electronlike FSs ( α/prime,α/prime/primeand
α/prime/prime/prime) based on the difference of extremal cross-sectional area.
The other two holelike FSs are denoted as βandγ, respectively.
However, only the frequencies from α/prime,β, andγare observed
in the FFT spectra [F( δ1) and F( δ2) come from the mixture of
the FSs α/primeandα/prime/prime, which will be discussed below]. The absence
of the frequencies from α/prime/primeandα/prime/prime/primeis understandable. Since the
field is parallel to the zaxis, the extremal cross-sectional area
ofα/prime/primeis close to that of β. So the frequencies from α/prime/primemix
with that from βand cannot be separated in the FFT spectra.
Rotating the field will change the extremal cross-sectional areaofα
/prime/primeand make its corresponding frequencies appear, which
has been proven by the angle-dependent MR (see below). Forthe elliptical FS α
/prime/prime/prime, a possible explanation is that the mobility
along the long axis is much smaller than the mobility along theshort axis. Such anisotropic mobility has been derived from thequantitative analysis in YSb/LaSb, where both the anisotropyand multiband nature are considered [ 7,20].
The amplitude of SdH oscillation is described by Lifshitz-
Kosevich (LK) formula:
/Delta1ρ∝λT
sinh(λT)e−λTDcos/bracketleftbigg
2π/parenleftbiggF
B−1
2+β+δ/parenrightbigg/bracketrightbigg
.(1)
In the formula, λ=(2π2kBm∗)/(¯heB).kBandm∗are the
Boltzmann constant and the effective mass of carrier, re-spectively. T
Dis the Dingle temperature, and 2 πβis the
Berry phase. δis a phase shift, with the value of δ=0 and
±1/8 for the 2D and 3D systems, respectively. Figure 2(d)shows the temperature dependence of FFT amplitude of the
corresponding frequencies. The data can be well fitted by thethermal factor R
T=(λT)/sinh(λT) in LK formula. The fitted
effective masses (see Table I) are comparable with that of
LaSb [ 3] and NdSb [ 25]. As for F( δ1) and F( δ2), the effective
masses are 0 .554meand 0.542me, respectively. Berry phase
is a way to roughly estimate the topological property of thematerials. Since the oscillation is multifrequency, we fit theoscillation pattern using multiband LK formula [Fig. 2(e)]t o
obtain the values of the Berry phase and Dingle temperature.As shown in Table I, the values of Berry phase are far away
from the nontrivial value π, suggesting that TmSb is possible
topologically trivial material.
Angle-dependent MR measurements are performed to fur-
ther understand the contribution of each FS. Figure 2(f)
shows the FFT spectra of SdH oscillations with rotatingt h efi e l di nt h e y-zplane. With the θchanging from 0
◦
to 90◦, the extremal cross-sectional area of α/primenormal to
the field increases while that of α/prime/primedecreases. As a result,
the frequencies from α/primeincreases, and the frequencies from
TABLE I. Parameters derived from SdH oscillation. F, oscillation
frequency; A, extremal cross-sectional area of FS normal to field; kF,
Fermi vector; m∗, effective mass; TD, Dingle temperature; 2 πβ, Berry
phase.
F(T) A ( ˚A−2)kF(˚A−1)m∗/meTD(K) 2 πβ
α/prime
1383.5 0.037 0.108 0.278 11.4 0.38 π+0.25π
α/prime
2428.6 0.041 0.114 0.264 10.5 −0.27π+0.25π
β1699.3 0.067 0.146 0.300 8.5 0.29 π−0.25π
β2795.2 0.076 0.155 0.345 5.8 0.29 π−0.25π
085137-3WANG, ZHANG, LU, SUN, XU, LU, LIU, ZHOU, AND XIA PHYSICAL REVIEW B 97, 085137 (2018)
FIG. 3. Fermi surface intensity plot and band dispersions along high-symmetry directions measured by ARPES. (a) Schematic of the first
and second 3D BZs with high symmetry points marked by red points. The purple area illustrates the k-space location of the red lines in (b),
which indicates the mapping area. (b) ARPES intensity plot of TmSb close to kz∼0 with hv=53 eV at T ∼20 K with high symmetry points
marked on it. (c), (e), (g) Photoemission intensity plots of cut1, cut2, and cut3 indicated in (b), respectively. (d), (f), (h) 2D curvature intensityplots of (c), (e), (g), respectively, and white open circles in (d) and (f) indicate half of the larger electron pockets at Xpoints.
α/prime/primecan be identified when θ=30◦before decreasing with
angle gradually. The angle-dependent frequencies from β
are nearly unchanged while the frequency from γvaries
slightly.
Figure 2(g) presents the angle dependence of the frequen-
cies. Two-dimensional FS is suggested to exist in LaSb sincethe frequency F(θ) follows F(0)/cos(θ−nπ/2) [2]. However,
the data in TmSb can’t be well fitted (not presented here) bythe above function. In fact, it is suggested to be a pseudo-two-dimensional characteristic of ellipsoidal FS [ 9], because the ex-
tremal cross-sectional area A=πab//radicalbig
sin2θ+(a2/b2)cos2θ
(aandbare the semimajor and semiminor axes of the
ellipsoid, respectively) can be approximated as πb2/cosθ
for small θvalues and a/greatermuchb. Reasonably, the equa-
tionF(θ)=F(0)//radicalbig
(b/a)2sin2(θ−nπ/2)+cos2(θ−nπ/2)
(n=0, 1 for α/prime
1(α/prime
2),α/prime/prime
1(α/prime/prime
2), respectively) is employed to
describe the angle-dependent frequencies and the experimentaldata can be well fitted as shown by the solid lines in Fig. 2(g).
The obtained a/bof FSα
/prime/prime
1(α/prime/prime
2) is 2.06 (2.07). Then the values
of F(α/prime/prime
1)=697.0 T and F( α/prime/prime
2)=793.8Ta t θ=0◦can be
derived, which are close to F( β1) and F( β2) as expected. Then
the frequency F( δ1) can be identified as F( α/prime
2)+F(α/prime/prime
2). Such
a frequency is the consequence of magnetic breakdown effect[48], which is caused by quantum tunneling of carriers between
the orbits on different FSs [ 49]. F(δ
2) is the split frequency
of F(δ1) under the field, which has a similar effective mass
as F(δ1). As shown in the inset of Fig. 2(g), with the change
ofθ, the frequencies from βare nearly unchanged while the
frequency from γvaries slightly. Such behaviors are expected
since the FS βis nearly spherical and the FS γis slightly
anisotropic. The behavior of angle-dependent frequencies isclearly related to the shape of FSs.ARPES measurements were performed to reveal the elec-
tronic structure of TmSb. TmSb crystalizes in a face-centeredcubic (FCC) structure. The first and second three-dimensionalBrillouin zones (BZs) are shown in Fig. 3(a). ARPES mea-
surements were performed at T ∼20 K at a photon energy
of 53 eV . Figure 3(b) shows the measured Fermi surface map
close to the k
z∼0 plane, which contains pockets at the /Gamma1and
Xpoints. To reveal their dispersions, we show in Figs. 3(c),
3(e)and3(g) cuts through these pockets as indicated by lines in
Fig. 3(b). To enhance the dispersing bands, the corresponding
curvature [ 50] plots are also shown in Figs. 3(d),3(f)and3(h).
The pockets at the /Gamma1point are clearly identified to be hole
pockets and labeled by βandγin Fig. 3(d). The ellipsoidlike
pocket at each Xpoint is labeled by αin Figs. 3(d),3(f)and
3(h), which is also seen in calculations in Fig. 4(b) with its long
axis along the /Gamma1-Xdirection. We note that there exit two hole
pockets in Figs. 3(e) and3(f)at theXpoint which are similar
to the two hole pockets βandγat the/Gamma1point in Figs. 3(c)and
3(d). Similar observation has been reported in LaSb [ 4,15],
LaBi [ 15], and YSb [ 19]. There are two possible scenarios
for explaining this: (1) kzbroadening; (2) band folding due
to new surface reconstruction. Because negligible change isobserved in the measured dispersion when changing photonenergies similar to that reported in LaBi and LaSb [ 15], we
think that k
zbroadening is a more likely scenario. This suggests
that those in Figs. 3(e) and3(f)may come from the kz∼0.5
plane (top or bottom of the Brillouin zone). The electronpocket labeled with α
∗in Figs. 3(d) and3(f)is also from this
kzbroadening. In all the cuts, an energy gap of ∼0.5e Vi s
observed at the Xpoint between the conduction band αand
the valence bands. The absence of band anticrossing along the/Gamma1-Xdirection indicates the topologically trivial characteristic
085137-4EXTREMELY LARGE MAGNETORESISTANCE AND … PHYSICAL REVIEW B 97, 085137 (2018)
FIG. 4. (a) Band structure along high-symmetry directions of the
Brillouin zone and (b) Fermi surfaces of TmSb calculated with the
MBJ potential and including the SOC effect. The Fermi level is set tozero.
of TmSb, which is similar to the case of LaSb and YSb
[4,17,19].
First-principles calculations have also been employed to
study the electronic structure of TmSb. As shown in Fig. 4(a),
the calculated band structure is quite consistent with thatobserved by ARPES. There are two hole bands ( βandγ)
and one electron band ( α) crossing the Fermi level. The gap
at the Xpoint is about 0.49 eV . Combined with the trivial
Berry phase obtained from SdH oscillation and the electronicstructure revealed by ARPES experiments and first-principlescalculations, TmSb is suggested to be a topologically trivialsemimetal. Figure 4(b) presents the calculated FSs of TmSb
with the SOC effect included. The colors of the FSs are in aone-to-one relationship with the corresponding bands crossingthe Fermi level. For the two hole pockets, βis nearly spherical,
butγhas a FS stretched in the {100}directions. The electron
pockets αare ellipsoidal and located at every Xpoint.
Since the topological trivial characteristic of TmSb has
been confirmed in the above discussion, the breaking oftopological protection is not suitable to explain the origin ofXMR in TmSb. Hall measurements are taken to achieve theinformation about carriers and to reveal the origin of XMRin TmSb. Figure 5(a) shows the field dependence of Hall
resistivity ρ
xy=[ρxy(+B)−ρxy(−B)]/2o fT m S b .T h e ρxy
curves are nonlinear, indicating that the electron and hole
coexist in TmSb. The Hall resistivity can be described by thesemiclassical two-band model:
ρ
xy=B
e/parenleftbig
nhμ2
h−neμ2
e/parenrightbig
+(nh−ne)(μhμe)2B2
(nhμh+neμe)2+(nh−ne)2(μhμe)2B2,(2)
where μh(μe) and nh(ne) are hole (electron) mobility and
hole (electron) concentration, respectively. As shown by thered solid lines in Fig. 5(a), the data can be well fitted by
the two-band model. Figure 5(b) presents the temperature
dependent carriers’ concentrations and mobility, which arederived from the fitting. The concentrations at 2.5 K areFIG. 5. (a) Magnetic field dependence of Hall resistivity at
different temperatures (Sample 2, RRR =24.1, MR 2.8K,14T=7.62×
103%). The red solid lines are the fits using the two-band model. (b)
The obtained carriers concentrations and mobility from the fits.
ne=9.43×1020cm−3andnh=8.75×1020cm−3.T h er a -
tionh/ne≈0.93 indicates the compensation of hole and
electron in TmSb. The values of mobility at 2.5 K ( μe=
4.32×103cm2V−1s−1,μh=3.24×103cm2V−1s−1)a r e
lower than those of LaSb/LaBi/YSb [ 3,8,17], which may be
caused by the lower quality of the sample used to measure theHall resistivity.
In the case of perfect electron-hole compensation ( n
e=
nh), the relation MR =μeμhB2can be derived from the field
dependent resistivity:
ρ(B)=(nhμh+neμe)+(nhμe+neμh)μhμeB2
e(nhμh+neμe)2+e(nh−ne)2(μhμe)2B2.(3)
Obviously, the MR will be unsaturated, and the value of MR
depends on the mobility of carriers. If the compensation isimperfect, the MR can be expressed as follows,
MR=η(μ
h+μe)2μhμeB2
(ημh+μe)2+(η−1)2(μhμe)2B2, (4)
where η=nh/ne(η/negationslash=1 but close to 1). The MR will deviate
from quadratic behavior slightly, and the feature of unsatu-ration retains when the value of ( η−1) is small. In TmSb,
the compensated carrier concentrations and high mobility aresuggested to be responsible for the XMR. Even the ratio η≈
0.93 indicates that the compensation is not very perfect, the
MR is still unsaturated and deviates from quadratic behaviorslightly as expected.
IV . SUMMARY
In summary, single crystals of TmSb are grown and the
magnetotransport properties have been investigated. Analysison the FFT spectra of the SdH oscillations observed at lowtemperature and high field clearly indicates the split of Fermisurfaces. The extracted trivial Berry phase from the fit of LKformula, combining with the electronic structures from ARPESmeasurements and first-principles calculations confirm thatTmSb is a semimetal with topologically trivial band structuresand nearly compensated concentrations of electron and hole.The XMR in TmSb is attributed to the electron-hole compen-sation and high mobility of carriers.
085137-5WANG, ZHANG, LU, SUN, XU, LU, LIU, ZHOU, AND XIA PHYSICAL REVIEW B 97, 085137 (2018)
ACKNOWLEDGMENTS
We thank Peng-Jie Guo for helpful discussions. This
work is supported by the National Natural Science Foun-dation of China (Grants No. 11574391, No. 11774424, No.11474356, and No. 91421304), the Fundamental ResearchFunds for the Central Universities, and the Research Fundsof Renmin University of China (Grants No. 14XNLQ07,
No. 14XNLQ03, and No. 16XNLQ01). Computational re-sources have been provided by the Physical Laboratory ofHigh Performance Computing at Renmin University of China.The Fermi surfaces were prepared with the XCRYSDENprogram [ 51].
[1] M. Zeng, C. Fang, G. Chang, Y .-A. Chen, T. Hsieh, A. Bansil,
H. Lin, and L. Fu, arXiv:1504.03492 .
[2] F. F. Tafti, Q. D. Gibson, S. K. Kushwaha, N. Haldolaarachchige,
a n dR .J .C a v a , Nat. Phys. 12,272(2016 ).
[3] F. F. Tafti, Q. Gibson, S. Kushwaha, J. W. Krizan, N.
Haldolaarachchige, and R. J. Cava, Proc. Natl. Acad. Sci. USA
113,E3475 (2016 ).
[4] L.-K. Zeng, R. Lou, D.-S. Wu, Q. N. Xu, P.-J. Guo, L.-Y . Kong,
Y .-G. Zhong, J.-Z. Ma, B.-B. Fu, P. Richard, P. Wang, G. T. Liu,L. Lu, Y .-B. Huang, C. Fang, S.-S. Sun, Q. Wang, L. Wang, Y .-G.Shi, H. M. Weng, H.-C. Lei, K. Liu, S.-C. Wang, T. Qian, J.-L.Luo, and H. Ding, Phys. Rev. Lett. 117,127204 (2016 ).
[5] W.-J. Ban, W.-T. Guo, J.-L. Luo, and N.-L. Wang, Chin. Phys.
Lett.34,077804 (2017 ).
[6] P.-J. Guo, H.-C. Yang, K. Liu, and Z.-Y . Lu, Phys. Rev. B 96,
081112 (2017 ).
[7] F. Han, J. Xu, A. S. Botana, Z. L. Xiao, Y . L. Wang, W. G. Yang,
D. Y . Chung, M. G. Kanatzidis, M. R. Norman, G. W. Crabtree,a n dW .K .K w o k , P h y s .R e v .B 96,125112 (2017 ).
[8] S. Sun, Q. Wang, P.-J. Guo, K. Liu, and H. Lei, New J. Phys. 18,
082002 (2016 ).
[9] N. Kumar, C. Shekhar, S.-C. Wu, I. Leermakers, O. Young, U.
Zeitler, B. Yan, and C. Felser, P h y s .R e v .B 93,241106 (2016 ).
[10] Y . Wu, T. Kong, L.-L. Wang, D. D. Johnson, D. Mou, L. Huang,
B. Schrunk, S. L. Bud’ko, P. C. Canfield, and A. Kaminski, Phys.
Rev. B 94,081108 (2016 ).
[11] R. Lou, B.-B. Fu, Q. N. Xu, P.-J. Guo, L.-Y . Kong, L.-K. Zeng,
J.-Z. Ma, P. Richard, C. Fang, Y .-B. Huang, S.-S. Sun, Q. Wang,L .W a n g ,Y . - G .S h i ,H .C .L e i ,K .L i u ,H .M .W e n g ,T .Q i a n ,H .Ding, and S.-C. Wang, P h y s .R e v .B 95,115140 (2017 ).
[12] J. Nayak, S.-C. Wu, N. Kumar, C. Shekhar, S. Singh, J. Fink,
E. E. Rienks, G. H. Fecher, S. S. Parkin, B. Yan et al. ,Nat.
Commun. 8,
13942 (2017 ).
[13] R. Singha, B. Satpati, and P. Mandal, Sci. Rep. 7,6321
(2017 ).
[14] P.-J. Guo, H.-C. Yang, B.-J. Zhang, K. Liu, and Z.-Y . Lu, Phys.
Rev. B 93,235142 (2016 ).
[15] X. H. Niu, D. F. Xu, Y . H. Bai, Q. Song, X. P. Shen, B. P. Xie,
Z. Sun, Y . B. Huang, D. C. Peets, and D. L. Feng, Phys. Rev. B
94,165163 (2016 ).
[16] N. Ghimire, A. Botana, D. Phelan, H. Zheng, and J. Mitchell, J.
Phys.: Condens. Matter 28,235601 (2016 ).
[17] Q.-H. Yu, Y .-Y . Wang, R. Lou, P.-J. Guo, S. Xu, K. Liu, S. Wang,
and T.-L. Xia, EPL119,17002 (2017 ).
[18] O. Pavlosiuk, P. Swatek, and P. Wi śniewski, Sci. Rep. 6,38691
(2016 ).
[19] J. He, C. Zhang, N. J. Ghimire, T. Liang, C. Jia, J. Jiang, S. Tang,
S. Chen, Y . He, S.-K. Mo, C. C. Hwang, M. Hashimoto, D. H.L u ,B .M o r i t z ,T .P .D e v e r e a u x ,Y .L .C h e n ,J .F .M i t c h e l l ,a n dZ.-X. Shen, Phys. Rev. Lett. 117,267201 (2016 ).[ 2 0 ] J .X u ,N .J .G h i m i r e ,J .S .J i a n g ,Z .L .X i a o ,A .S .B o t a n a ,Y .L .
Wang, Y . Hao, J. E. Pearson, and W. K. Kwok, P h y s .R e v .B 96,
075159 (2017 ).
[21] N. Alidoust, A. Alexandradinata, S.-Y . Xu, I. Belopolski, S. K.
Kushwaha, M. Zeng, M. Neupane, G. Bian, C. Liu, D. S. Sanchezet al. ,arXiv:1604.08571 .
[22] L. Ye, T. Suzuki, C. R. Wicker, and J. G. Checkelsky,
arXiv:1704.04226 .
[23] H. Oinuma, S. Souma, D. Takane, T. Nakamura, K. Nakayama, T.
Mitsuhashi, K. Horiba, H. Kumigashira, M. Yoshida, A. Ochiai,T. Takahashi, and T. Sato, P h y s .R e v .B 96,041120 (2017 ).
[24] K. Kuroda, M. Ochi, H. Suzuki, M. Hirayama, M. Nakayama,
R. Noguchi, C. Bareille, S. Akebi, S. Kunisada, T. Muro et al. ,
arXiv:1707.06500 .
[25] N. Wakeham, E. D. Bauer, M. Neupane, and F. Ronning, Phys.
Rev. B 93,205152 (2016 ).
[26] M. Neupane, M. M. Hosen, I. Belopolski, N. Wakeham, K.
Dimitri, N. Dhakal, J.-X. Zhu, M. Z. Hasan, E. D. Bauer, and F.Ronning, J. Phys.: Condes. Matter 28,23LT02 (2016 ).
[27] Y . Wu, Y . Lee, T. Kong, D. Mou, R. Jiang, L. Huang, S. L.
B u d ’ k o ,P .C .C a n fi e l d ,a n dA .K a m i n s k i , P h y s .R e v .B 96,
035134 (2017 ).
[28] M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop,
T. Liang, N. Haldolaarachchige, M. Hirschberger, N. P. Ong, andR. J. Cava, Nature (London) 514,205(2014 ).
[29] Y . L. Wang, L. R. Thoutam, Z. L. Xiao, J. Hu, S. Das, Z. Q. Mao,
J. Wei, R. Divan, A. Luican-Mayer, G. W. Crabtree, and W. K.Kwok, Phys. Rev. B 92,180402 (2015 ).
[30] J. Jiang, F. Tang, X. C. Pan, H. M. Liu, X. H. Niu, Y . X. Wang,
D. F. Xu, H. F. Yang, B. P. Xie, F. Q. Song, P. Dudin, T. K. Kim,M. Hoesch, P. K. Das, I. V obornik, X. G. Wan, and D. L. Feng,Phys. Rev. Lett. 115,166601 (2015 ).
[31] Y .-Y . Wang, Q.-H. Yu, P.-J. Guo, K. Liu, and T.-L. Xia, Phys.
Rev. B 94,041103 (2016 ).
[32] Z. Yuan, H. Lu, Y . Liu, J. Wang, and S. Jia, P h y s .R e v .B 93,
184405 (2016 ).
[33] D. Wu, J. Liao, W. Yi, X. Wang, P. Li, H. Weng, Y . Shi, Y . Li, J.
Luo, X. Dai et al. ,Appl. Phys. Lett. 108,042105 (2016 ).
[34] Y . Luo, R. McDonald, P. Rosa, B. Scott, N. Wakeham, N.
Ghimire, E. Bauer, J. Thompson, and F. Ronning, Sci. Rep. 6,
27294 (2016 ).
[35] B. Shen, X. Deng, G. Kotliar, and N. Ni, P h y s .R e v .B 93,195119
(2016 ).
[36] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. Cava, and N. Ong,
Nat. Mater. 14,280
(2015 ).
[37] P. E. Blöchl, P h y s .R e v .B 50,17953 (1994 ).
[38] G. Kresse and D. Joubert, P h y s .R e v .B 59,1758 (1999 ).
[39] G. Kresse and J. Hafner, Phys. Rev. B 47,558(1993 ); G. Kresse
and J. Furthmüller, Comput. Mater. Sci. 6,15(1996 );Phys. Rev.
B54,11169 (1996 ).
085137-6EXTREMELY LARGE MAGNETORESISTANCE AND … PHYSICAL REVIEW B 97, 085137 (2018)
[40] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 (1996 ).
[41] M. Abdusalyamova, A. Chuiko, A. Y . Golubkov, S. Popov, L.
Parfenova, A. Procofev, and I. Smirnov, J. Alloy. Compd. 205,
107(1994 ).
[42] A. D. Becke and E. R. Johnson, J. Chem. Phys. 124,221101
(2006 ); F. Tran and P. Blaha, P h y s .R e v .L e t t . 102,226401 (2009 ).
[43] N. Marzari and D. Vanderbilt, P h y s .R e v .B 56,12847 (1997 ).
[44] I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65,035109
(2001 ).
[45] S. Nimori, G. Kido, D. Li, and T. Suzuki, Physica B (Amsterdam)
211,148(1995 ).[46] B. R. Cooper and O. V ogt, P h y s .R e v .B 1,1211 (1970 ).
[47] Y . Ōnuki, A. Nakamura, T. Uejo, A. Teruya, M. Hedo, T.
Nakama, F. Honda, and H. Harima, J. Phys. Soc. Jpn. 83,061018
(2014 ).
[48] K. Kishigi, M. Nakano, K. Machida, and Y . Hori, J. Phys. Soc.
Jpn.64,3043 (1995 ).
[49] J. Hu, Y . L. Zhu, D. Graf, Z. J. Tang, J. Y . Liu, and Z. Q. Mao,
Phys. Rev. B 95,205134 (2017 ).
[50] P. Zhang, P. Richard, T. Qian, Y .-M. Xu, X.
Dai, and H. Ding, Rev. Sci. Instrum. 82,043712
(2011 ).
[51] A. Kokalj, Comput. Mater. Sci. 28,155(2003 ).
085137-7 |
PhysRevB.79.201204.pdf | Origin of the anatase to rutile conversion of metal-doped TiO 2
Sa Li and P. Jena
Department of Physics, Virginia Commonwealth University, Richmond, Virginia 23284-2000, USA
/H20849Received 27 January 2009; revised manuscript received 26 April 2009; published 13 May 2009 /H20850
Extensive calculations using density functional theory enable us to explain the origin of the surprising
room-temperature conversion of anatase to rutile phase of TiO 2when doped with Co and Ni, but not with Cu.
Contrary to earlier suggestion, neither high spin nor strain of the transition metals is found to be responsible forthis phase conversion. The driving mechanism, instead, is attributed to the increased interaction between Coand Ni atoms forming a linear chain in the rutile phase. We predict that Cr and Mn which have even largerspins than Co and Ni cannot induce this phase conversion.
DOI: 10.1103/PhysRevB.79.201204 PACS number /H20849s/H20850: 71.10.Hf, 61.72. /H11002y, 71.15.Nc, 75.50.Pp
Growing interest in the study of TiO 2semiconductor
stems from its current use in photovoltaics,1
electrochromics,2sensors,3and photocatalysis.4In the
ground state TiO 2exists in the anatase phase and undergoes
transition to the rutile phase at temperatures well in excess of600 °C.
5Since the optical and electrical properties of ana-
tase and rutile TiO 2are distinctly different, it is desirable to
be able to control phase content and conversion betweenthese two phases. In particular, it has been suggested that,due to the novel semiconducting properties of the anatase/rutile interface region, partial conversion from anatase torutile phase may render TiO
2with exciting photocatalytic
properties.6–8Consequently, there has been considerable
interest9–11in finding ways to induce this phase transition at
lower temperatures. In a recent paper, Gole et al.12reported
the surprising room-temperature phase conversion of anataseto rutile TiO
2by using transition-metal ions with highly un-
paired electron spins. They showed that Co, and to a lesserextent Ni, accomplishes this conversion, while it does notoccur when Cu is used as a dopant. The authors attributed theorigin of this phase conversion to the magnetic nature of Coand Ni.
In this Rapid Communication we show that Ni in TiO
2is
nonmagnetic and Co doping continues to enable the phaseconversion even after switching off the magnetic interaction.Furthermore, Cr and Mn whose atomic spins are larger thanthat of Co are also incapable of causing this phase transition.The origin of the phase conversion resulting from Co and Nidoping is found to be due to the increased interaction be-tween these atoms as they form a linear chain in the rutilestructure. Our studies also reveal some unusual magnetic be-havior of Cu, Fe, and Cr when doped in TiO
2: nonmagnetic
Cu couples ferromagnetically, ferromagnetic /H20849FM /H20850Fe
couples antiferromagnetically, and antiferromagnetic /H20849AFM /H20850
Cr couples ferromagnetically.
The relaxations in the lattice caused by the dopant at-
om /H20849s/H20850, the total energy, and the electronic structure were cal-
culated using Vienna ab initio simulation package /H20849V ASP /H20850
/H20849Ref. 13/H20850and the projector augmented wave /H20849PAW /H20850/H20849Ref. 14/H20850
method. The PAW generalized gradient approximation/H20849GGA /H20850/H20849Ref. 15/H20850potentials with the valence states 3 dand 4 s
for Ti, Cr, Mn, Fe, Co, Ni, and Cu and 2 sand 2 pfor O were
used. High precision calculations with a cutoff energy of400 eV for the plane-wave basis were performed. Inaddition, GGA+ Ucalculations with parameter values ofU=3.0 eV and J=0.87 eV /H20849Ref. 16/H20850for Cr, Mn, Fe, Co, Ni,
and Cu 3 delectrons were carried out. The geometries of the
above supercells /H20849ionic coordinates and c/aratio /H20850were op-
timized without any symmetry constraint. For sampling theirreducible wedge of the Brillouin zone we used k-point grids
of 6/H110036/H110034 and 4 /H110034/H110038 for the anatase and rutile tetrago-
nal supercells, respectively. In all calculations, self-consistency was achieved with a tolerance in the total energyof at least 0.1 meV . To study the magnetic properties ofmetal-doped TiO
2, we carried out spin-polarized calculations
including both ferromagnetic and antiferromagnetic configu-rations.
The anatase TiO
2has a tetragonal structure and belongs to
the space group I41 /amd /H20849141 /H20850. In the anatase structure,
each Ti atom is octahedrally bonded to six O atoms with fourO atoms lying at a distance of 1.94 Å from Ti while theother two are at 1.99 Å. TiO
2rutile structure belongs to the
space group P42 /mnm /H20849136 /H20850. In the rutile structure, each Ti
is also octahedrally coordinated with O atoms with four ofthem lying on the /H20849110 /H20850plane at a distance of 1.95 Å while
the other two lie along the /H20851110 /H20852direction at a distance of
1.98 Å. Our calculations show that the anatase phase is0.998 eV lower in energy than the rutile phase at 0 K.
We first discuss dopant concentration of 6.25% where one
Ti atom in the 48-atom supercell is replaced by Cr, Mn, Fe,Co, Ni, or Cu. At this concentration we found the anatasephase to be more stable than the rutile phase by 1.309, 0.795,0.672, 0.623, 0.742, and 0.574 eV for Cr, Mn, Fe, Co, Ni,and Cu dopants, respectively. The magnetic moments at theCr, Mn, Fe, Co, Ni, and Cu sites in the anatase phase are1.87, 2.65, 1.73, 0.73, 0, and 0.39
/H9262Band in the rutile phase
are 0.61, 2.62, 1.67, 0.67, 0, and 0.63 /H9262Brespectively.
GGA+ Ucalculations lead to slightly higher magnetic mo-
ment on each dopant atom. For example, the moment of Coin anatase TiO
2increases from 0.73 to 0.83 /H9262Bby means of
GGA+ Umethod. The magnetic behavior of Co, Mn, Fe, and
Ni-doped anatase TiO 2is similar to the findings of
Park et al.16The difference between numerical values of the
magnetic moments calculated by us and these authors16may
be due to the use of different muffin-tin radii.
The relative phase stability changes when the dopant con-
centration is increased to 12.5%. Note that for this concen-tration, we have chosen two different sites for the dopants inboth anatase and rutile structures /H20849Fig.1/H20850. We find that in the
anatase phase, Cr, Fe, Co, Ni, and Cu atoms prefer to occupyPHYSICAL REVIEW B 79, 201204 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
1098-0121/2009/79 /H2084920/H20850/201204 /H208494/H20850 ©2009 The American Physical Society 201204-1nearest-neighbor sites /H20849configuration A /H20850while Mn atoms pre-
fer to remain farther away /H20849configuration B /H20850. The M-Mdis-
tances are shown in Table I. When two Mn atoms are apart at
the distance of 4.85 Å, AFM configuration is only 7 meVmore favorable than FM configuration. In the rutile phase,however, all dopant atoms prefer to occupy nearest-neighborsites /H20849configuration C /H20850. In Table Iwe give the energy differ-
ence between the anatase and rutile phases of Ti
1−xMxO2
/H20849M=Cr, Mn, Fe, Co, Ni, and Cu; x=0.125 /H20850systems. We
define this energy difference as, /H9004E=E/H20849anatase /H20850−E/H20849rutile /H20850.
Here Eis the total energy of a given phase and negative
value for /H9004Emeans that the anatase phase is more stable
than the rutile phase. We see from Table Ithat Co and, to a
lesser extent, Ni-doped TiO 2favor the rutile phase as the
ground state while Cr, Mn, and Cu doping have no effect onthe phase conversion. GGA+ Ucalculations show similartrend, even though Ni-doped system slightly favors the ana-
tase structure. In Fe-doped TiO
2, Fe atoms couple antiferro-
magnetically, which is in agreement with recent findings thatsubstitution of Fe in anatase TiO
2does not introduce ferro-
magnetic ordering.17The anatase phase is preferred for Fe
doping, but the energy difference is small.
We now concentrate on the origin of the phase conversion
in Co and Ni and lack of it in Cu. We note that the magneticmoment of Co, Ni, and Cu free atoms are 3, 2, and 1
/H9262B,
respectively. In the bulk phase Co and Ni are ferromagneticwhile Cu is paramagnetic /H20849PM /H20850. However, when doped in
TiO
2, Co and Cu carry magnetic moment while Ni does not.
These features can be seen from the density of states /H20849DOS /H20850
in both the anatase and rutile phases. As an example, wepresent the DOS for 12.5% metal-doped rutile phase in Fig.2. In pristine rutile TiO
2, The lower valence band /H20849LVB /H20850is
dominated by O-2 sstates. The upper valence band /H20849UVB /H20850is
composed of O-2 pstates and Ti-3 dstates. The UVB has a
calculated bandwidth of 5.5 eV and is separated from theconduction band /H20849CB /H20850by a band gap of 1.7 eV . We see that
Co-doped TiO
2is a semimetal. It has a pseudo gap at the
Fermi level in the spin down states. Addition of Ni to TiO 2
not only introduces new states at the middle of the band gapbut also at the UVB. The delocalized Ni- dstates hybridize
strongly with O- pand Ti- dstates in the UVB, thus making
Ni-doped TiO
2paramagnetic.18Cu-doped TiO 2behaves
half-metallic, and the localized energy levels from the dopantdistribute just at the UVB. In the GGA+ Ucalculations, Co-
doped rutile TiO
2is still a semimetal, with spin-up 3 dva-
lence bands slight moving to higher energy. We see that themid-gap bands in Ni-doped system are broadened in theGGA+ Ucalculation. In contrast to the GGA results in Cu-
doped TiO
2, the Fermi level falls into the gap of the Cu 3 d
spin-up states when GGA+ Umethod is used.
It has been suggested that the phase conversion in Co and
Ni-doped TiO 2could have a magnetic origin. As mentioned
earlier, this cannot be the case at least for Ni since its mag-netic moment is zero. Equally important, Cu-doped TiO
2
film at a concentration of approximately 10 at. %has been
reported experimentally19to be ferromagnetic at room tem-
perature, yet it does not exhibit phase conversion. Our resultsshow that Cu at 6.25% concentration carries a magnetic mo-
TABLE I. The GGA optimized average M-Odistance /H20849Å/H20850,M-Mdistance /H20849Å/H20850, the magnetic moments /H20849/H9262B/H20850at each dopant site, the
preferred magnetic coupling, the energy difference /H20849eV/H20850between the anatase /H20849ana /H20850and rutile /H20849rut/H20850phases /H20849/H9004E=Eanatase −Erutile /H20850, and the
energy difference /H20849eV/H20850between two dopant configurations /H20849/H9254E=ED-EC/H20850for the dopant concentration of 12.5% shown in Fig. 1. The
GGA+ Ucalculated energy differences are shown to compare with GGA calculations.
DopantM-OM -M /H9262 Coupl. /H9004E
/H9254E
/H20849ED-EC/H20850 /H20849Ana. /H20850/H20849 Rut. /H20850/H20849 Ana. /H20850/H20849 Rut. /H20850/H20849 Ana. /H20850/H20849 Rut. /H20850/H20849 Ana. /H20850/H20849 Rut. /H20850/H20849 GGA /H20850/H20849Eana-Erut/H20850/H20849 GGA+ U/H20850/H20849Eana-Erut/H20850
Ti 1.96 1.96 3.04 2.95 0 0 PM PM −0.998 −0.998
Cr 1.91 1.92 2.90 2.95 1.95 2.02 FM FM −0.680 −0.636 0.223Mn 1.89 1.91 4.85 2.95 2.64 2.63 AFM AFM −0.472 −0.495 0.138Fe 1.89 1.90 2.83 2.94 1.93 1.85 AFM AFM −0.018 −0.057 0.578Co 1.88 1.89 2.81 2.94 0.66 0.65 FM FM 0.077 0.052 0.768Ni 1.88 1.89 2.88 2.95 0 0 PM PM 0.011 −0.046 0.453Cu 1.93 1.95 3.03 2.96 0 0.60 PM FM −0.154 −0.116 0.179
FIG. 1. /H20849Color online /H20850The crystal structures showing the dopant
sites for 12.5% concentration in anatase TiO 2/H20849configurations A and
B/H20850and rutile TiO 2/H20849configurations C and D /H20850.SA LI AND P. JENA PHYSICAL REVIEW B 79, 201204 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
201204-2ment of 0.39 and 0.63 /H9262Bin the anatase and rutile phases,
respectively. When Cu concentration increases to 12.5 %, the
moment on Cu in the anatase phase quenches to 0 /H9262Bbut
remains at 0.60 /H9262Bin the rutile phase. The coupling is fer-
romagnetic. We conclude that the magnetic signals seen ex-perimentally in the Cu-doped TiO
2can be directly assigned
to Cu substitution at the Ti site20when Cu concentration is
low. Oxygen vacancies can only enhance the moment on Cu.At 25% Cu concentration, rutile TiO
2is found to be nonmag-
netic which agrees with the calculations of Duhalde et al.19
However, contrary to the work of Li et al. ,21we found that
6.25% Cu-doped rutile TiO 2is magnetic. To understand the
source of this discrepancy we note that there are two PAWpotentials for Cu, one incorporates the 3 porbitals, while the
other does not. We repeated our calculations by including the3porbitals and found that when enough empty energy bands
are included, the moment on Cu is 0.43
/H9262B. Thus, we con-
clude that Cu-doped TiO 2is magnetic at low concentration,
but it is unable to cause the phase conversion. To further ruleout the possibility that magnetism is responsible for thephase conversion in Co-doped TiO
2, we repeated our calcu-
lation by switching off the spin polarization provision. Anonspin polarized calculation on Co-doped system for the12.5% concentration showed that the rutile structure is still0.172 eV lower in energy than the anatase structure.
The next possibility is that strain induced by dopants may
be responsible for the observed phase conversion. To illus-trate the role of strain, we note that the ionic radii of Co/H20849+2/H20850and Ni /H20849+2/H20850are, respectively, 0.74 and 0.72 Å whichare larger than the ionic radius of Ti /H20849+4/H20850, namely, 0.68 Å.
The ionic radius of Cu /H20849+2/H20850, on the other hand, is 0.69 Å
which is much closer to the Ti value. This is reflected in theoptimized average distance between Co-O, Ni-O, and Cu-Oin the doped sample given in Table I. Note that the distances
for the Co and Ni-doped systems are significantly smallerthan those of Ti-O distances both in the anatase and rutilephases. For Cu-doped systems, however, the distances aremuch closer to the value in pristine TiO
2. This may give the
impression that strain may be responsible for the phase con-version. To see if this indeed is the case, we repeated thecalculations for Co-doped TiO
2by not allowing the supercell
to relax. The rutile phase was again found to be 0.336 eVlower in energy than the anatase phase at the 12.5% concen-tration. Thus, phase conversion in Co doped cannot be in-duced by strain.
To resolve the mystery of the origin of phase transition,
we note that in the rutile phase the doped atoms form a linearchain at the 12.5% concentration, thus enabling the dopantatoms to interact more strongly with each other than they canin the anatase phase. To see if this interaction can play a role,we compare the energy difference between configurations Cand D /H20849see Fig. 1/H20850. We find that all the three dopants—
namely, Co, Ni, and Cu—prefer to form a linear chain struc-ture /H20849configuration C /H20850in the rutile phase. The energy differ-
ences between configurations C and D for Co, Ni, and Cudoping are, respectively, 0.768, 0.453, and 0.179 eV . Thisindicates that Co has the strongest preference to form a chainstructure. Note that if the dopant atoms prefer to remain iso-FIG. 2. /H20849Color online /H20850The total DOS for pristine and metal-doped rutile TiO 2/H20849configuration C /H20850:/H20849a/H20850undoped, /H20849b/H20850Co-doped, /H20849c/H20850
Ni-doped, and /H20849d/H20850Cu-doped rutile TiO 2structures.ORIGIN OF THE ANATASE TO RUTILE CONVERSION OF … PHYSICAL REVIEW B 79, 201204 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
201204-3lated /H20849configuration D /H20850in the rutile structure, the anatase to
rutile phase conversion will not take place. In Co-dopedTiO
2, the rutile phase /H20849configuration D /H20850is 0.691 eV higher in
energy than the anatase phase /H20849configuration A /H20850. Therefore,
Co and Ni doping tend to lower the energy of the dopedsystem by maximizing their interaction, and hence forming alinear chain as in the rutile phase.
To see if this indeed is the real mechanism, we carried out
systematic calculations on M-doped TiO
2using other 3 del-
ements, namely, Cr, Mn, and Fe. Among these Cr has thehighest spin moment in the 3 dseries, namely, 6
/H9262B, and Mn
atom has a spin moment of 5 /H9262B. However, due to its half-
filled 3 dand filled 4 sorbitals /H208493d54s2/H20850Mn atoms interact
weakly with each other and the cohesive energy of bulk Mnis the lowest among the 3 dtransition-metal atoms. Thus, one
would expect that Mn doping will result in the least energygain when Mn atoms form a linear chain in the rutile phaseand hence can hardly induce phase conversion in TiO
2. This
is exactly what our calculations yielded.
In Fig. 3we summarize the main results of this Rapid
Communication. For 6.25% dopant concentration, no phaseconversion is found for any of the dopants. However, for12.5% concentration Co and Ni induce phase conversionwhile Cr, Mn, and Cu do not. Fe doping seems to lie at thethreshold as the energy difference between the rutile andanatase phases is too small to predict its precise role in phaseconversion. Also plotted in Fig. 3is the energy difference
between configurations C and D. In configuration C the dop-ant atoms in the rutile phase form a linear chain /H20849clustering /H20850
while in configuration D, they do not cluster. The variation inthis energy is remarkably similar to that of the energy differ-ence between the anatase and rutile phases, giving clear in-dication that the increased interaction between Co atomsconfined to a linear chain in the rutile phase is responsiblefor the phase conversion.
In summary, we have studied systematically the relative
stability between the anatase to rutile phase in transition-metal /H20849M/H20850-doped TiO
2/H20849M=Cr, Mn, Fe, Co, Ni, and Cu /H20850and
explored the origin of the surprising room-temperature phaseconversion in Co and Ni-doped system. The phase conver-sion is neither induced by the high spin of the dopants nor bythe strain. Instead, it depends on the preferred configurationdopant atoms assume in the rutile phase. When the concen-tration reached 12.5%, the dopant atoms cluster to form a
linear chain in the rutile phase and the increased interactionbetween Co and Ni atoms in the linear chain configurationcompared to other dopants causes the observed phase con-version. Our calculations show that Co is the most suitabledopant due to its largest energy gain in the linear chain struc-ture. We predict that Mn and Cr doping will not be able tocause this phase conversion. It is difficult to predict the roleof Fe due to the minor energy difference between the anataseand the rutile phases. Experimental studies of Cr, Mn, andFe-doped TiO
2will help establish the accuracy of our pre-
diction.
This research used resources of the National Energy Re-
search Scientific Computing Center, which is supported bythe Office of Science of the U.S. Department of Energy un-der Contract No. DE-AC02-05CH11231. Partial support ofthis work by the Department of Energy is also acknowl-edged.
1B. O’Regan and M. Gratzel, Nature /H20849London /H20850353, 737 /H208491991 /H20850.
2U. Bach et al. , Adv. Mater. 14, 845 /H208492002 /H20850.
3L. D. Birkefeld et al. , J. Am. Ceram. Soc. 75, 2964 /H208491992 /H20850.
4A. Fujishima and K. Honda, Nature /H20849London /H20850238,3 7 /H208491972 /H20850.
5F. Dachille et al. , Am. Mineral. 53, 1929 /H208491968 /H20850.
6T. Ohno et al. , J. Catal. 203,8 2 /H208492001 /H20850.
7G. H. Li et al. , J. Mol. Catal. A: Chem. 275,3 0 /H208492007 /H20850.
8D. C. Hurum et al. , J. Phys. Chem. B 107, 4545 /H208492003 /H20850.
9Y . H. Zhang and A. Reller, Mater. Sci. Eng., C 19, 323 /H208492002 /H20850.
10S. Riyas et al. , Adv. Appl. Ceram. 106, 255 /H208492007 /H20850.
11D. J. Reidy et al. , J. Eur. Ceram. Soc. 26, 1527 /H208492006 /H20850.12J. L. Gole et al. , J. Phys. Chem. C 112, 1782 /H208492008 /H20850.
13G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 /H208491996 /H20850.
14P. E. Blochl, Phys. Rev. B 50, 17953 /H208491994 /H20850.
15Y . Wang and J. P. Perdew, Phys. Rev. B 44, 13298 /H208491991 /H20850.
16M. S. Park et al. , Phys. Rev. B 65, 161201 /H20849R/H20850/H208492002 /H20850.
17C. E. Rodriguez-Torres et al. , J. Phys.: Condens. Matter 20,
135210 /H208492008 /H20850.
18T. Umebayashi et al. , J. Phys. Chem. Solids 63, 1909 /H208492002 /H20850.
19S. Duhalde et al. , Phys. Rev. B 72, 161313 /H20849R/H20850/H208492005 /H20850.
20C. E. R. Torres et al. , Appl. Surf. Sci. 254, 365 /H208492007 /H20850.
21Q. K. Li et al. , Europhys. Lett. 81, 17004 /H208492008 /H20850.FIG. 3. /H20849Color online /H20850 The energy difference
/H20849/H9004E=Eanatase −Erutile /H20850between anatase and rutile for M-doped TiO 2
/H20849M=Cr, Mn, Fe, Co, Ni, and Cu /H20850with 6.25% and 12.5% dopant
concentration measured with respect to that of pristine TiO 2. The
energy of the rutile structure is set to be zero. The square line is theenergy difference /H20849
/H9254E=ED-EC/H20850between configurations C and D,
which correspond to the energy gain when dopants are confined tolinear chain structure in the rutile phase.SA LI AND P. JENA PHYSICAL REVIEW B 79, 201204 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
201204-4 |
PhysRevB.103.134504.pdf | PHYSICAL REVIEW B 103, 134504 (2021)
Superconductivity with and without glue and the role of the double-occupancy forbidding
constraint in the t-J-Vmodel
Luciano Zinni , Matías Bejas , and Andrés Greco
Facultad de Ciencias Exactas, Ingeniería y Agrimensura and Instituto de Física Rosario (UNR-CONICET). Avenida Pellegrini
250-2000 Rosario, Argentina
(Received 2 December 2020; revised 13 February 2021; accepted 22 March 2021; published 5 April 2021)
The occurrence of retarded (with glue) and unretarded (without glue) pairing is thoroughly discussed in
cuprates. We analyze some aspects of this problem in the context of the t-J-Vmodel in a large- Napproximation.
When 1 /Nrenormalizations are neglected the mean-field result is recovered, where the unretarded d-wave
superconducting pairing triggered by the spin-exchange interaction Jis obtained. However, the presence of a
nonnegligible nearest-neighbors Coulomb interaction V(q) kills superconductivity. If the non-double-occupancy
constraint and its fluctuations are considered, the situation changes drastically. In this case, V(q) is screened
making d-wave superconductivity very robust. In addition, we show that the early proposal for the presence of
an unretarded pairing contribution triggered by the spin-exchange interaction Jcan be discussed in this context.
DOI: 10.1103/PhysRevB.103.134504
I. INTRODUCTION
The origin of superconductivity in high- Tccuprates is un-
der an intense debate since its discovery in 1986. Not onlythe high value of the superconducting critical temperatureT
cis surprising, but these materials are also anisotropic and
the metallic and superconducting properties show a two-dimensional character. The phase diagram in the temperatureand doping plane of these materials shows unconventionalcharacteristics, as the dome-shaped behavior of T
cagainst
doping in the proximity to the antiferromagnetic insulatorand the pseudogap phase at low doping (see Ref. [ 1]f o ra
review). In addition to these features, the superconductinggap has d-wave symmetry [ 1,2]. All members of the cuprate
family share similar characteristics, suggesting the existenceof a universal physics.
Phenomenological theories where pairing is due to anti-
ferromagnetic fluctuations [ 3,4] were proposed for explaining
thed-wave symmetry of the superconducting gap in the
proximity to antiferromagnetism. In this scenario, the two-dimensional antiferromagnetic fluctuations play the role ofa retarded glue, as phonons in conventional low-temperaturesuperconductors.
In the early times, the Hubbard and the t-Jmodels were
recognized [ 5] as minimal microscopic models for cuprates.
The Hubbard model treated in the framework of a weakcoupling random phase approximation shows d-wave super-
conductivity [ 6], where the effective pairing interaction is
mediated by the dynamical spin susceptibility which actsas a pairing glue. In this approach, the nearest-neighborsCoulomb interaction, V(q)=2V[cos( q
x)+cos(qy)], which
is expected to be nonnegligible in cuprates [ 7], affects su-
perconductivity because it has a d-wave repulsive projection.
It is therefore important to understand why Tcremains large
even if a nearest-neighbors Coulomb interaction is present.The same effect of V(q) is expected in the antiferromagnetic
phenomenological theories [ 3,4].
The study of d-wave superconductivity and the role of
the nearest-neighbors Coulomb interaction in the Hubbardmodel is huge [ 8–11]. In Ref. [ 8] it was shown that d-wave
superconductivity in the Hubbard model is almost unaffectedbyV(q) if the strong coupling limit is properly treated. In
addition, since the two-dimensional Hubbard model reduces
to the t-Jmodel in the large- Ulimit [ 12], the question about
the role of V(q) on superconductivity is also of interest in
thet-Jmodel. On the other hand, retarded (with glue) and
unretarded (without glue) paring [ 13,14] is under discussion
in the t-Jmodel. While in Ref. [ 14] pairing was discussed
as composed by a retarded and an unretarded contributions,
in Ref. [ 13] only an unretarded pairing was considered as the
relevant one.
At the mean-field level, the t-Jmodel shows d-wave su-
perconductivity [ 15] arising from the unretarded exchange
interaction J[cos( q
x)+cos(qy)], where Jis the spin-exchange
coupling. Since the exchange interaction has the same form ofthe nearest-neighbors Coulomb interaction, the last one is, in
principle, detrimental to superconductivity. Using a large- N
approach on the t-J-Vmodel, in this paper we discuss super-
conductivity and the role of the nearest-neighbors Coulombinteraction V(q). When the local constraint that prohibits dou-
ble occupancy is not included superconductivity is stronglyaffected by V(q), even for Vof the order of J. Including the
constraint d-wave superconductivity is robust against V(q),
even for V/greatermuchJ. We also found that the leading contribution
to superconductivity is mainly provided by the unretardedexchange, however, this contribution is efficient only if theconstraint is properly included. In Sec. IIwe present a sum-
mary of the formalism, in Sec. IIIthe results and discussions,
and in Sec. IVthe conclusions.
2469-9950/2021/103(13)/134504(8) 134504-1 ©2021 American Physical SocietyZINNI, BEJAS, AND GRECO PHYSICAL REVIEW B 103, 134504 (2021)
II. MODEL AND SUMMARY OF THE FORMALISM
As a minimal model, we study the t-J-Vmodel on a square
lattice,
H=−/summationdisplay
/angbracketlefti,j/angbracketright,σtij˜c†
iσ˜cjσ+/summationdisplay
/angbracketlefti,j/angbracketrightJij/parenleftbigg
/vectorSi·/vectorSj−1
4ninj/parenrightbigg
+/summationdisplay
/angbracketlefti,j/angbracketrightVijninj, (1)
where ˜ c†
iσ(˜ciσ) is the creation (annihilation) operator of elec-
trons with spin σ(=↑,↓) in the Fock space without double
occupancy at any site, ni=/summationtext
σ˜c†
iσ˜ciσis the electron density
operator, /vectorSiis the spin operator. The hopping (spin exchange)
tij(Jij) takes the value t(J) between the first nearest-neighbors
sites. Vijis a nearest-neighbors Coulomb interaction with
strength V.
It is nontrivial to study the t-Jmodel because of the local
constraint that prohibits the double occupancy at any site. Inaddition, the operators involved in the t-Jmodel are Hub-
bard operators [ 16] which satisfy nonstandard commutation
rules. We employ here a large- Ntechnique based on a path
integral representation in terms of the Hubbard operators (seeRefs. [ 17,18] and references therein). In the large- Nscheme,
the number of spin components is extended from 2 to Nand
the physical quantities are computed in powers of 1 /N.I n
what follows the spin index σis called p.
In the framework of the large- Npath integral approach, the
t-Jmodel is mapped to an effective theory described in terms
of fermions, bosons, and their mutual interactions [ 18].
A. Fermions
We obtain a fermionic propagator [solid line in Fig. 1(a)]
G(0)
pp/prime(k,iωn)=δpp/prime
iωn−εk, (2)
with the electronic dispersion
εk=−2/parenleftbigg
tδ
2+/Delta1/parenrightbigg
[cos(kx)+cos(ky)]−μ. (3)
For a given doping δ, the chemical potential μand/Delta1are
determined self-consistently by solving
1−δ=2
Ns/summationdisplay
knF(εk), (4)
and
/Delta1=J
4Ns/summationdisplay
k[cos(kx)+cos(ky)]nF(εk), (5)
where nFis the Fermi function and Nsis the total number
of lattice sites. The momentum kis measured in units of the
inverse of the lattice constant. In Eq. ( 2)ωnis a fermionic
Matsubara frequency. The Green’s function G(0)
pp/prime(k,iωn)i s
O(1) in the context of the 1 /Nexpansion.
In the present formalism, the spin-exchange term or J-
term of the t-J-Vmodel [Eq. ( 1)] is treated by introducing
a bond-field variable that describes charge fluctuations onFIG. 1. (a) Summary of the Feynman rules. Solid line repre-
sents the fermionic propagator G(0)
pp/prime. Dashed line represents the
6×6 bosonic propagator D(0)
ab./Lambda1pp/prime
aand/Lambda1pp/prime
abrepresent the interac-
tion between two fermions and one and two bosons, respectively.(b) Diagrammatic representation of the Dyson equation. (c) The two
different contributions to the irreducible boson self-energy. (d) Ef-
fective interaction between fermions. Looking at the order of the
propagators and vertices we see that V
effisO(1/N), thus supercon-
ductivity arises at O(1/N) in this large- Nscheme.
the bond connecting nearest-neighbors sites along the x- and
y-directions. /Delta1is the static mean-field value of this bond field.
Although the electronic dispersion [Eq. ( 3)] looks like that in a
free electron system, the hopping integral tis renormalized by
doping δbecause of electron-correlation effects. In addition,
there is a contribution /Delta1which depends on J.
B. Bosons
We define a six-component bosonic field
δXa=(δR,δ λ , rx,ry,Ax,Ay), (6)
where δRdescribes the fluctuations of the number of holes at
a given site, thus it is related to on-site charge fluctuations,δλis the fluctuation of the Lagrange multiplier introduced to
enforce the constraint that prohibits the double occupancy ata given site, and r
xandry(AxandAy) describe fluctuations
of the real (imaginary) part of the bond field coming from theJ-term.
The 6 ×6 bare bosonic propagator associated with δX
a
[dashed line in Fig. 1(a)], connecting two generic components
134504-2SUPERCONDUCTIVITY WITH AND WITHOUT GLUE AND … PHYSICAL REVIEW B 103, 134504 (2021)
aandbis
/bracketleftbig
D(0)
ab(q,iνn)/bracketrightbig−1=N⎛
⎜⎜⎜⎜⎜⎜⎝δ2
2/parenleftbig
V−J
2/parenrightbig
[cos(qx)+cos(qy)]δ
20000
δ
200 0 0 0
004
J/Delta12000
00 04
J/Delta1200
00 0 04
J/Delta120
00 0 0 04
J/Delta12⎞
⎟⎟⎟⎟⎟⎟⎠, (7)
where qandν
nare the momentum and bosonic Matsubara frequencies, respectively. The factor Nshows that the 6 ×6 bosonic
propagator D(0)
abisO(1/N), and it is frequency independent. The element (1,1) in Eq. ( 7) carries the information of1
4Jijninjand
Vijninjof Eq. ( 1), while the information of Jij/vectorSi·/vectorSjis contained in the elements ( a,a) with a=3–6.
C. Interaction vertices
For computing quantities up to O(1/N) the present large- Nscheme leads to three-legs and four-legs vertices [Fig. 1(a)].
The three-legs vertex
/Lambda1pp/prime
a=(−1)/bracketleftBigg
i
2(ωn+ω/prime
n)+μ+2/Delta1/summationdisplay
ηcos/parenleftBig
kη−qη
2/parenrightBig
cosqη
2;1 ;−2/Delta1cos/parenleftBig
kx−qx
2/parenrightBig
−2/Delta1cos/parenleftBig
ky−qy
2/parenrightBig
;2/Delta1sin/parenleftBig
kx−qx
2/parenrightBig
;2/Delta1sin/parenleftBig
ky−qy
2/parenrightBig/bracketrightBigg
δpp/prime, (8)
where η=x,y, represents the interaction between two fermions and one boson.
The four-legs vertex /Lambda1pp/prime
abrepresents the interaction between two fermions and two bosons. /Lambda1pp/prime
abfulfills the symmetry of
/Lambda1pp/prime
ab=/Lambda1pp/prime
ba, and the only elements different from zero are
/Lambda1pp/prime
δRδR=/bracketleftBigg
i
2(ωn+ω/prime
n)+μ+/Delta1/summationdisplay
ηcos/parenleftbigg
kη−qη+q/prime
η
2/parenrightbigg/parenleftbigg
cosqη
2cosq/prime
η
2+cosqη+q/prime
η
2/parenrightbigg/bracketrightBigg
δpp/prime, (9)
/Lambda1pp/prime
δRδλ=1
2δpp/prime, (10)
/Lambda1pp/prime
δRrη=−/Delta1cos/parenleftbigg
kη−qη+q/prime
η
2/parenrightbigg
cosq/prime
η
2δpp/prime, (11)
and
/Lambda1pp/prime
δRAη=/Delta1sin/parenleftbigg
kη−qη+q/prime
η
2/parenrightbigg
cosq/prime
η
2δpp/prime. (12)
Each vertex conserves momentum and energy and it is O(1). For readability reasons we drop the frequencies and momenta
in the left-hand side of the definitions of the three- and four-legs vertices /Lambda1pp/prime
aand/Lambda1pp/prime
ab[see Fig. 1(a) for the frequency and
momentum dependence].
By using the propagators and vertices summarized in Fig. 1(a)we can draw Feynman diagrams as usual.
From the Dyson equation [Fig. 1(b)], the bosonic bare propagator D(0)
abis renormalized at 1 /Norder
[Dab(q,iνn)]−1=/bracketleftbig
D(0)
ab(q,iνn)/bracketrightbig−1−/Pi1ab(q,iνn), (13)
where the 6 ×6 boson self-energy matrix /Pi1ab[Fig. 1(c)]i s
/Pi1ab(q,iνn)=−N
Ns/summationdisplay
kha(k,q,εk−εk−q)nF(εk−q)−nF(εk)
iνn−εk+εk−qhb(k,q,εk−εk−q)−δa1δb1N
Ns/summationdisplay
kεk−εk−q
2nF(εk),(14)
with hagiven by
ha(k,q,ν)=/braceleftbigg2εk−q+ν+2μ
2+2/Delta1/bracketleftBig
cos/parenleftBig
kx−qx
2/parenrightBig
cos/parenleftBigqx
2/parenrightBig
+cos/parenleftBig
ky−qy
2/parenrightBig
cos/parenleftBigqy
2/parenrightBig/bracketrightBig
;1 ;
−2/Delta1cos/parenleftBig
kx−qx
2/parenrightBig
;−2/Delta1cos/parenleftBig
ky−qy
2/parenrightBig
;2/Delta1sin/parenleftBig
kx−qx
2/parenrightBig
;2/Delta1sin/parenleftBig
ky−qy
2/parenrightBig/bracerightbigg
. (15)
The vertices /Lambda1pp/prime
aand/Lambda1pp/prime
abnot only represent interactions
from the Hamiltonian Eq. ( 1), but, as they come from thepath integral, they contain also contributions from the alge-
bra of the Hubbard operators and the non-double-occupancy
134504-3ZINNI, BEJAS, AND GRECO PHYSICAL REVIEW B 103, 134504 (2021)
constraint, which introduce a frequency dependence. Due to
this frequency dependence, the computation of the first andsecond diagrams in Fig. 1(c) leads to finite and infinite con-
tributions. However, the ghost fields from the Jacobian inthe path integral give rise to terms that cancel exactly theseinfinities [ 17,18].
The 6 ×6 dressed bosonic propagator D
abcontains all
possible charge fluctuations of the t-Jmodel on the square
lattice, and all are treated on equal footing [ 19]. The large-
Napproach weakens the effective spin interaction compared
with the one associated with the charge degrees of freedom.D
abwith a,b=1,2 describes on-site charge fluctuations as-
sociated to δRandδλ. The presence of δλindicates that
the non-double-occupancy constraint and its fluctuations aretaken into account in the calculation. The element (1,1) ofD
abis related to the usual charge-charge correlation function
[17].D22andD12correspond to fluctuations associated with
the non-double-occupancy condition and correlations betweennon-double-occupancy condition and charge-density fluctua-tions, respectively. We call this case as on-site charge sectoror the 2 ×2 sector. If a,b=3–6, D
abdescribes bond-charge
fluctuations associated to rx,ry,Ax, and Ay. We call this case
as the bond-charge sector or the 4 ×4 sector. Dabalso con-
tains the mixing of both sectors, however, it was shown thatthe coupling between on-site and bond-charge fluctuations isnegligible [ 20]. If J=0t h e6 ×6D
abreduces to the 2 ×2
sector, and only on-site charge fluctuations are involved.
The superconducting effective interaction between
fermions, Veff(k,k/prime;ωn,ω/prime
n), can be calculated using the
diagram in Fig. 1(d), which shows that in the present theory
pairing is mediated by charge fluctuations contained inD
ab. Note that we can also draw a diagram containing two
vertices /Lambda1pp/prime
aband two bosonic propagators Dab, however, this
contribution is omitted because it is O(1/N2). The analytical
expression for the effective interaction is
Veff(k,k/prime;ωn,ω/prime
n)=/Lambda1aDab(k−k/prime,ωn−ω/prime
n)/Lambda1b,(16)
where /Lambda1aand/Lambda1bare the three-legs vertices from Eq. ( 8) with
p=p/prime.
We use a weak coupling approximation to compute the
effective couplings λiin the different pairing channels or ir-
reducible representations of the order parameter on the squarelattice, i[i=(d
x2−y2,dxy,s/prime,p)],
λi=1
(2π)2/integraltext
(dk/|vk|)/integraltext
(dk/prime/|vk/prime|)gi(k/prime)Veff(k/prime,k)gi(k)/integraltext
(dk/|vk|)gi(k)2,
(17)
where the functions gi(k) encode the different pairing
symmetries, gdx2−y2(k)=cos(kx)−cos(ky), gdxy(k)=
cos(kx) cos( ky),gs/prime(k)=cos(kx)+cos(ky), and gp(k)=
sin(kx).vkis the quasiparticle velocity at momentum k.T h e
integrations are restricted to the Fermi surface, i.e., kand
k/primerun over Fermi surface momenta and iωn=iω/prime
n=0.λi
measures the strength of the interaction between electrons
at the Fermi surface in a given symmetry channel i.I f
λi>0, electrons are repelled hence, superconductivityFIG. 2. The superconducting coupling λandλ(0)versus doping
δforV=0a n d V=0.3. Inset: The superconducting coupling λand
λ(0)versus Vforδ=0.25. For this δ,Vc∼1.9.
is only possible when λi<0. The critical temperatures
Tccan then be estimated using a BCS expression:
Tci=1.13ω0exp(−1/|λi|), where ω0is a suitable cutoff
frequency which encodes retardation effects. If λiis
negligible, superconductivity is unexpected, no matterthe value of ω
0. Although it is an approximation, the weak
coupling scheme gives a way to select, in principle, thedominant pairing channels from all different contributionsindependently of their retarded or unretarded nature.It was introduced in retarded (with glue) cases as theelectron-phonon one [ 21], where λis the dimensionless
coupling strength due to the electron-phonon interaction.This approach was also used for spin-fluctuation interactionin the context of cuprates [ 3]. The fact that we calculate
on the Fermi surface in Eq. ( 17) does not invalidate the
study of retarded interactions. Obtaining an accurate valueofT
crequires considering retardation effects in more detail,
but that is not our aim. We study the main tendencies tosuperconductivity and from where they arise by computingthe coupling strength λof each contribution.
III. RESULTS AND DISCUSSIONS
We chose J=0.3,T=0, and 0 /lessorequalslantV/lessmuchVc, where Vcis the
onset of the instability to a checkerboard charge density wave[18,22]. Energy is given in units of t. There is no tendency to
superconductivity, i.e., λ
i>0, for any pairing channel except
fordx2−y2forδ<0.5. Thus, in the following we focus only on
thedx2−y2channel. For simplicity we call λdx2−y2(dx2−y2-wave)
asλ(d-wave) in what follows.
Using the 6 ×6Dabin Eq. ( 16) we compute λas a function
ofδforV=0 and V=0.3. Figure 2shows that, although
V(q) has a repulsive d-wave projection, λis almost unaffected
byV. In addition, d-wave superconductivity enhances with
decreasing doping. This result is contrary to other resultsthat suggest that superconductivity is killed by Valready for
values of the order of J(Refs. [ 23–25]).
134504-4SUPERCONDUCTIVITY WITH AND WITHOUT GLUE AND … PHYSICAL REVIEW B 103, 134504 (2021)
Using the 6 ×6D(0)
abinstead of Dabin Eq. ( 16)Veffis given
by [26]
V(0)
eff(k,k/prime;ωn,ω/prime
n)=/parenleftbiggJ
2−V/parenrightbigg
[cos( kx−k/prime
x)+cos(ky−k/prime
y)]
+J
2[cos( kx−k/prime
x)+cos(ky−k/prime
y)].
(18)
Note that V(0)
effis frequency independent. The first term in
V(0)
effcontaining JandVcomes from the 2 ×2 sector of the
t-Jmodel. The second term originates from the 4 ×4 sector
which is proportional to J. We call attention that considering ˜ c
as usual fermions in Eq. ( 1)V(0)
effcan be recovered as a mean-
field approximation of the t-Jmodel.
Using V(0)
effthe corresponding λ(0)can be computed. In
contrast to λ, while superconductivity is robust for V=0,λ(0)
vanishes for V=0.3( s e eF i g . 2). These results show that,
among other effects discussed later, the renormalization ofD
(0)
abby the 6 ×6 boson self-energy /Pi1ab[Eq. ( 13)] screens
out the effect of V. The inset in Fig. 2shows λandλ(0)
versus Vforδ=0.25. These results for λindicate that super-
conductivity is mostly unaffected by the Coulomb interactioneven for V/greatermuchJwhen the full-dressed D
abbosonic propagator
is considered. On the other hand, for the case of the barepropagator D
(0)
abno superconductivity occurs for V>0.3, as
can be expected from Eq. ( 18).
One difference between λandλ(0)forV=0 is that while
λ(0)smoothly decreases with decreasing doping, λtends to
large negative values at δ∼0.13. This behavior for λcan be
explained in the context of the flux phase instability, whichoccurs at a critical doping δ
c∼0.13 for present parameters
[19]. See the Appendix for details about the flux phase. Since
λis calculated on the Fermi surface, i.e., ωn=ω/prime
n=0, when
approaching δcthe effective superconducting coupling λtunes
the instability and diverges.
It was shown that λ, which includes the bosonic self-energy
/Pi1ab, is robust against V, but such robustness is not present
in the case of λ(0). Next, we discuss which are the relevant
components of /Pi1abthat lead to the different behavior between
λandλ(0). Since the flux phase belong to the five to six sector
ofDab(see the Appendix), we calculate λincluding only the
2×2 sector /Pi111,/Pi112,/Pi122, and the flux sector /Pi155,/Pi156and
/Pi166in the Dyson equation [Eq. ( 13)], i.e., leaving the other
components of /Pi1abas zero. We call this λCh−FP. Figure 3(a)
shows λCh−FPforV=0 and V=0.3. For completeness, in
the figure we included the results of λfor the full 6 ×6 case
of Fig. 2. These results show that /Pi111,/Pi112,/Pi122,/Pi155,/Pi156,
and/Pi166are the most important components of the bosonic
self-energy /Pi1absince they capture the same λbehavior as
using the full 6 ×6/Pi1ab. It is important to note that the
inclusion of /Pi1abin the Dyson equation introduces a frequency
dependence in the dressed bosonic propagator Dab, i.e., the
effective interactions are retarded in contrast to the unretardedinteractions from the undressed D
(0)
ab. This point is important
for later discussions.
Next we analyze the influence of the 2 ×2 on-site-charge
and FP sectors separately.FIG. 3. (a) λCh−FPversus doping for V=0a n d V=0.3.λfrom
Fig.2is included for comparison. (b) λChforV=0a n d V=0.3, and
λJversus doping. (c) λFPandλJversus δ.I n s e t : λFPandλJversus δ
for/Gamma1=0.05.
Considering only /Pi111,/Pi112, and/Pi122in the Dyson equation
the effective paring interactions can be written as
V(Ch)
eff(k,k/prime;ωn,ω/prime
n)
=−2/Lambda11(δ−/Pi112)+/Lambda12
1/Pi122−/braceleftbigδ2
2(2V−J)Fk,k/prime−/Pi111/bracerightbig
(δ−/Pi112)2+/Pi122/braceleftbigδ2
2(2V−J)Fk,k/prime−/Pi111/bracerightbig
+J
2[cos( kx−k/prime
x)+cos(ky−k/prime
y)], (19)
where Fk,k/prime=cos ( kx−k/prime
x)+cos ( ky−k/prime
y). Using Eq. ( 19),
we compute λCh. Figure 3(b) shows results for λChversus
δforV=0 and V=0.3. It can be seen that λChis almost
unaffected by Vshowing that the Coulomb repulsion is indeed
screened by the /Pi1abcomponents that belong to the 2 ×2 on-
site charge sector. The second term on the right hand side ofEq. ( 19) is the same as in Eq. ( 18). We call λ
Jthe contribution
from this term and its behavior is shown in Fig. 3(b).T h e
fact that the three curves are nearly coincident give us the
134504-5ZINNI, BEJAS, AND GRECO PHYSICAL REVIEW B 103, 134504 (2021)
clue that the components of the 2 ×2 on-site charge sector of
/Pi1abscreen the first term of Eq. ( 18) and consequently, only
the effective ( J/2) [cos ( kx−k/prime
x)+cos ( ky−k/prime
y)] interaction
from the 4 ×4 sector survives.
When the t-Jmodel is treated at mean-field level, su-
perconductivity is expected to be triggered by the exchangeterm J(k−k
/prime). However, superconductivity is killed by a
small nearest-neighbors Coulomb interaction. When the non-double-occupancy constraint is treated properly, the effect ofthe Coulomb interaction is screened. Thus, present resultsshow a clear difference between a treatment of supercon-ductivity at the mean-field level and a treatment in strongcoupling. We think that our results support the early point ofview [ 13] that superconductivity in cuprates has a contribution
from the unretarded J(k−k
/prime) term, but we claim that for such
a pairing to be realized the non-double-occupancy constraintshould be treated beyond mean-field.
The screening effect from the 2 ×2 sector can be under-
stood as follows. The second contribution of the first term inEq. ( 19)i sm a i n l y s-wave and gives a negligible contribution
in the d-wave channel, i.e., this term is not relevant for our
analysis. The third term has the form of the screening of theCoulomb interaction from the usual RPA, because /Pi1
22is just
a simple bubble. It is important to remember that /Pi122arises
here from fluctuations of the Lagrange multiplier introducedto impose the constraint. Then, this contribution screens theJand Vterms (first term of V
(0)
eff)f r o mt h e2 ×2 sector,
while the J-term from the 4 ×4 sector (second term in V(0)
eff)
remains. In addition, the first contribution −2/Lambda11(δ−/Pi112),
which is independent of JandVhas a small repulsive d-wave
projection. Then, if V=J=0, i.e., only the 2 ×2 sector is
present, superconductivity is not expected to be mediated bycharge fluctuations.
It is important to mention that the doping dependence of
λ
ChandλJdoes not show the steep behavior near δcseen in
Fig.2forλ. This is due to the fact that we did not include /Pi155,
/Pi156, and/Pi166from the FP sector (see the Appendix). To under-
stand the influence of only these components on λwe take the
dressed bosonic propagator Daband compute Veffby project-
ingDabonto the FP eigenvector (0 ,0,0,0,1/√
2,−1/√
2)
(Ref. [ 19]). We obtain
V(FP)
eff(k,k/prime;ωn,ω/prime
n)
=−(/Lambda15−/Lambda16)2ReχFP(k−k/prime,iωn−iω/prime
n), (20)
where /Lambda15and/Lambda16are the fifth and sixth components of the
vertices in Eq. ( 8), and
χFP(q,iνn)=[(8/J)/Delta12−/Pi1FP(q,iνn)]−1, (21)
which is the flux phase susceptibility [ 19] and/Pi1FP(q,iνn)t h e
electronic polarizability given by
/Pi1FP(q,iνn)=−1
Ns/summationdisplay
kγ2
FP(q,k)nF(/epsilon1k+q)−nF(/epsilon1k)
/epsilon1k+q−/epsilon1k−iνn,
(22)
with the form factor γFP(q,k)=2/Delta1[sin( kx+qx/2)−
sin(ky+qy/2)]. For q=(π, π ) the form factor γFP(q,k)transforms as [cos( kx)−cos(ky)], i.e., the flux instability has
d-wave symmetry. χFP(q,iνn) plays the role of a bosonic
glue, as phonons in usual superconductors.
This projection isolates the FP sector and allows us to
check its effect on λ.λFPversus δ, where λFPis calculated
using V(FP)
eff, is shown in Fig. 3(c). While at large doping λFP
goes to zero, the curve shows the steep behavior approaching
δc.I nF i g . 3(c), we also plot λJversus δ. Comparing λFPwith
λJwe conclude that the flux phase enhances superconductivity
only near the quantum critical point at δcassociated with the
flux instability. This tendency to enhance superconductivitycan be seen as triggered by quantum critical fluctuations [ 27].
Figure 3shows that the total coupling strength λcan
be computed in a good approximation as the sum of λ
FP
andλJ, i.e., as coming from an effective pairing interaction
Veff∼V(FP)
eff+J(k−k/prime). While V(FP)
effis retarded, J(k−k/prime)
is unretarded. It is well known that when introducing a finitebroadening /Gamma1in the analytical continuation iν
n=ν+i/Gamma1in
Eq. ( 22), the flux phase is pushed to lower dopings [ 28]. In the
inset of Fig. 3(c)we show results for λFPandλJfor/Gamma1=0.05.
For this /Gamma1the flux-phase does not set down at a finite doping,
andJ(k−k/prime) is a good approximation for computing the total
coupling strength for all dopings.
The authors of Ref. [ 14] showed that the pairing strength
is composed by a retarded spin fluctuation contribution andan unretarded term J(k−k
/prime), and that the retarded pairing
dominates. In agreement with this work we also found an un-retarded J(k−k
/prime) contribution. As discussed in our paper the
large- Napproximation weakens spin fluctuations over charge
fluctuations, then we cannot rule out the presence of a retardedspin-fluctuation pairing. In Ref. [ 14] the Coulomb potential
V(q) was not included, which can kill the superconductivity
from the spin fluctuation term. However, we showed that theconstraint in the t-Jmodel, when included, screens V(q).
Then, we think that our paper and that of Ref. [ 14] are com-
plementary. If pairing in cuprates is mainly retarded or mainlyunretarded is an open discussion. Although one can expecta retarded pairing as in conventional superconductors, someexperiments suggest that pairing may certainly be unretarded[29,30].
IV . CONCLUSION
Using a large- Napproach on the microscopic t-J-Vmodel
we studied d-wave superconductivity and the role of a nearest-
neighbors Coulomb repulsion on it. In this approach, pairingis mediated by a bosonic propagator which contains on-site charge and bond-charge fluctuations, both treated at thesame footing in present formalism. When the bare bosonicpropagator is considered, superconductivity arises from theunretarded exchange term J(k−k
/prime). However, the presence
of the nearest-neighbors Coulomb repulsion V(q)i sd e t r i -
mental to superconductivity and cancels pairing for valuesofV∼J, suggesting a fragile d-wave superconductivity. The
situation changes drastically when the bosonic propagator isdressed by interactions. In this case, superconductivity be-comes almost unaffected by Vand remains robust even for
V/greatermuchJ. The inclusion of the non-double-occupancy constraint
and its fluctuations screens the effect of V(q), while a pair-
ing contribution from the J-term remains. In other words,
134504-6SUPERCONDUCTIVITY WITH AND WITHOUT GLUE AND … PHYSICAL REVIEW B 103, 134504 (2021)
the scenario for a possible unretarded (without glue) pairing
contribution triggered by Jemerges in strong coupling, i.e.,
only if the local constraint is considered properly.
Our results may be useful for the comparison with similar
calculations in the Hubbard and t-Jmodels. A robust d-wave
superconductivity against a nearest-neighbors Coulomb repul-sionV(q) requires the nondouble occupancy to be considered,
and at this level an unretarded pairing contribution is obtained.In the large- Ulimit, the Hubbard model is mapped to the t-J
model. Then it would be interesting to check the role of V(q)
on superconductivity and, in addition, to disentangle retardedand unretarded interactions from the obtained pairing in theHubbard model.
ACKNOWLEDGMENTS
The authors thank P. Bonetti, W. Metzner, D. Vilardi, and
H. Yamase for fruitful discussions. A.G. thanks the Max-Planck-Institute for Solid State Research in Stuttgart forhospitality and financial support.
APPENDIX: SOME CHARACTERISTICS AND
DISCUSSIONS ON THE FLUX PHASE
In this Appendix we briefly discuss the main characteris-
tics of the flux phase (FP) and its possible connection withthe physics of the pseudogap. As discussed in Ref. [ 19],
for present parameters ( J=0.3 and T=0) the flux phase
[18,31–34] occurs at δ=δ
c∼0.13, with a modulation vector
Qclose to ( π,π ), i.e., the FP breaks the translational symme-
try. In present large- Napproximation the FP occurs when one
eigenvalue of D−1
abis zero, and since the associated eigenvector
is of the form (0 ,0,0,0,1/√
2,−1/√
2), the flux instability
is located in the sector 5-6 of the 6 ×6m a t r i x Dab(Ref. [ 19]).
Forδ<δ cthe imaginary components AxandAyof the bond
field become finite. The commensurate FP is characterizedby the modulation vector q=(π, π ) and describes staggered
circulating currents. In the FP state a d-wave gap, similar to
the pseudogap in cuprates, opens, and Fermi pockets withlow intensity in the outer part are developed [ 35] instead of
a large Fermi surface. The FP is equivalent to the dCDWwhich was proposed phenomenologically for describing the
pseudogap [ 36].
The FP is a bond-charge instability. As discussed in Ref.
[19], besides the FP there are several kinds of bond-charge
fluctuations, and in principle all of them can lead to an in-stability depending on the model parameters. However, inthe context of the present large- Nmethod, for hole-doped
cuprates it was found that the flux instability is robust in arealistic-parameters regime. In contrast, for electron-dopedcuprates the leading bond-charge instability can occurs forthe real components r
xandryof the fluctuations of the bond
field [ 37].
Although the flux phase or dCDW is a candidate for de-
scribing the pseudogap, its existence in the t-Jand Hubbard
models is controversial. While some reports show the pres-ence of the flux instability or its fluctuations [ 38,39], others do
not [ 40]. The FP is also controversial from the experimental
point of view. While the authors of Refs. [ 36,41,42]s h o w
that a series of experiments in the pseudogap phase can bedescribed in the context of the FP, angle-resolved photoemis-sion spectroscopy (ARPES) experiments do not show pocketsbut Fermi arcs [ 43,44] which are considered as an indication
that translational symmetry is not broken in the pseudogap. InRefs. [ 45–47] the interaction between the flux-phase fluctua-
tions and carriers in the proximity to the flux-phase instabilityleads to a reasonable description of the Fermi arcs and Ramanscattering without the necessity of the translational-symmetrybreaking. Recently [ 48], it was proposed that the FP is a good
candidate for describing the pseudogap.
The connection between the FP and the antiferromag-
netism and its fluctuations, which lead to d-wave supercon-
ductivity [ 8,49], is an interesting point. The FP occurs at
much larger doping ( δ=0.13 in the present calculation) than
the onset of antiferromagnetism. Then, at the onset of theFP both phases may interact weakly while, with decreasingdoping approaching the antiferromagnetic-insulating phase,antiferromagnetism, and its fluctuations may lead against theFP. On the other hand, the FP develops staggered magneticmoments much weaker than those in the antiferromagneticphase [ 50] which, in principle, indicates that the FP and an-
tiferromagnetism are distinct phases. In spite of that, it wasclaimed that antiferromagnetism can be also understood in theframework of the flux phase [ 51].
[1] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J.
Zaanen, Nature 518, 179 (2015) .
[2] T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999) .
[3] P. Monthoux, A. V . Balatsky, and D. Pines, P h y s .R e v .L e t t . 67,
3448 (1991) .
[4] P. Monthoux and D. Pines, P h y s .R e v .B 47, 6069 (1993) .
[5] P. W. Anderson, Science 235, 1196 (1987) .
[6] N. Bulut and D. J. Scalapino, Phys. Rev. B 54, 14971
(1996) .
[7] L. F. Feiner, J. H. Jefferson, and R. Raimondi, Phys. Rev. Lett.
76, 4939 (1996) .
[8] D. Sénéchal, A. G. R. Day, V . Bouliane, and A. M. S. Tremblay,
P h y s .R e v .B 87, 075123 (2013) .[9] G. Esirgen, H.-B. Schüttler, and N. E. Bickers, P h y s .R e v .L e t t .
82, 1217 (1999) .
[10] C. Husemann and W. Metzner, Phys. Rev. B 86, 085113 (2012) .
[11] M. Jiang, U. R. Hähner, T. C. Schulthess, and T. A. Maier, Phys.
Rev. B 97, 184507 (2018) .
[12] K. A. Chao, J. Spałek, and A. M. Ole ´s,Phys. Rev. B 18, 3453
(1978) .
[13] P. W. Anderson, Science 316, 1705 (2007) .
[14] T. A. Maier, D. Poilblanc, and D. J. Scalapino, Phys. Rev. Lett.
100, 237001 (2008) .
[15] G. Baskaran, Z. Zou, and P. Anderson, Solid State Commun.
88, 853 (1993) .
[16] J. Hubbard, Proc. R. Soc. London A 276, 238 (1963) .
134504-7ZINNI, BEJAS, AND GRECO PHYSICAL REVIEW B 103, 134504 (2021)
[17] A. Foussats and A. Greco, Phys. Rev. B 65, 195107 (2002) .
[18] A. Foussats and A. Greco, Phys. Rev. B 70, 205123 (2004) .
[19] M. Bejas, A. Greco, and H. Yamase, Phys. Rev. B 86, 224509
(2012) .
[20] M. Bejas, H. Yamase, and A. Greco, Phys. Rev. B 96, 214513
(2017) .
[21] G. Rickayzen, Green’s Functions and Condensed Matter ,
1st ed. (Academic, New York, 1980).
[22] A. T. Hoang and P. Thalmeier, J. Phys.: Condens. Matter 14,
6639 (2002) .
[23] N. M. Plakida and V . S. Oudovenko, J. Exp. Theor. Phys. Lett.
119, 554 (2014) .
[24] R. Zeyher and A. Greco, Eur. Phys. J. B 6, 473 (1998) .
[25] S. Zhou and Z. Wang, P h y s .R e v .B 70, 020501(R) (2004) .
[26] A term proportional to D(0)
12was neglected because does not
contribute to the d-wave channel.
[27] A. Abanov, Y .-M. Wu, Y . Wang, and A. V . Chubukov, Phys.
Rev. B 99, 180506(R) (2019) .
[28] H. Yamase, M. Bejas, and A. Greco, Phys. Rev. B 99, 014513
(2019) .
[29] J. Lorenzana, B. Mansart, A. Mann, A. Odeh, M. Chergui, and
F. Carbone, Eur. Phys. J.: Spec. Top. 222, 1223 (2013) .
[ 3 0 ]S .R .P a r k ,Y .C a o ,Q .W a n g ,M .F u j i t a ,K .Y a m a d a ,S . - K .
Mo, D. S. Dessau, and D. Reznik, P h y s .R e v .B 88, 220503(R)
(2013) .
[31] I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774 (1988) .
[32] J. B. Marston and I. Affleck, Phys. Rev. B 39, 11538 (1989) .
[33] D. C. Morse and T. C. Lubensky, Phys. Rev. B 43, 10436
(1991) .
[34] E. Cappelluti and R. Zeyher, P h y s .R e v .B 59, 6475 (1999) .
[35] S. Chakravarty, C. Nayak, and S. Tewari, Phys. Rev. B 68,
100504(R) (2003) .[36] S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak,
Phys. Rev. B 63, 094503 (2001) .
[37] M. Bejas, A. Greco, and H. Yamase, New J. Phys. 16, 123002
(2014) .
[38] P. W. Leung, Phys. Rev. B 62, R6112 (2000) .
[39] X. Dong and E. Gull, P h y s .R e v .B 101, 195115
(2020) .
[40] A. Macridin, M. Jarrell, and T. Maier, P h y s .R e v .B 70, 113105
(2004) .
[41] C. Honerkamp and P. A. Lee, P h y s .R e v .L e t t . 92, 177002
(2004) .
[42] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78,1 7
(2006) .
[43] M. R. Norman, H. Ding, M. Randeria, J. C. Campuzano,
T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K.Kadowaki, P. Guptasarma, and D. G. Hinks, Nature 392, 157
(1998) .
[44] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys.
75, 473 (2003) .
[45] A. Greco, P h y s .R e v .L e t t . 103, 217001 (2009) .
[46] A. Greco and M. Bejas, P h y s .R e v .B 83, 212503 (2011) .
[47] A. Greco and M. Bejas, J. Phys.: Condens. Matter 26, 485701
(2014) .
[48] A. Gourgout, A. Ataei, M.-E. Boulanger, S. Badoux, S.
Thériault, D. Graf, J.-S. Zhou, S. Pyon, T. Takayama, H. Takagi,N. Doiron-Leyraud, L. Taillefer, arXiv:2012.10484 .
[49] D. Vilardi, C. Taranto, and W. Metzner, Phys. Rev. B 99,
104501 (2019) .
[50] T. C. Hsu, J. B. Marston, and I. Affleck, P h y s .R e v .B 43, 2866
(1991) .
[51] C.-M. Ho, V . N. Muthukumar, M. Ogata, and P. W. Anderson,
Phys. Rev. Lett. 86, 1626 (2001) .
134504-8 |
PhysRevB.101.245416.pdf | PHYSICAL REVIEW B 101, 245416 (2020)
Valley degree of freedom in ferromagnetic Janus monolayer H-VSSe and
the asymmetry-based tuning of the valleytronic properties
Chaobo Luo , Xiangyang Peng ,*Jinfeng Qu, and Jianxin Zhong
Hunan Key Laboratory for Micro-Nano Energy Materials and Devices, School of Physics and Optoelectronics,
Xiangtan University, Hunan 411105, People’s Republic of China
(Received 18 December 2019; revised manuscript received 29 May 2020; accepted 1 June 2020;
published 15 June 2020)
By using density-functional theory-based GW method, we studied the valley degree of freedom of Janus
monolayer VSSe. The GW corrections lead to a doubling of the band gap and change the band dispersionconsiderably, indicating significant many-body effects. VSSe is confirmed to be ferromagnetic, which breaksthe time-reversal symmetry and the odd parity of the Berry curvature in momentum space. The dissimilarmagnitudes of Berry curvatures of the inequivalent valleys give rise to appreciable anomalous Hall conductivity(AHC). The calculated valley optical response of VSSe exhibits a clear valley-selective circular dichroism. Theferromagnetism induces large valley-Zeeman splitting, making it possible to realize the selective valley excitationeven by unpolarized light. The Janus VSSe is more tunable by external fields because of symmetry breaking.
Due to the relief of time-reversal symmetry, the valley-Zeeman splitting can be continuously tuned by varyingthe magnetization direction. The loss of mirror symmetry in VSSe enables a bidirection modulation of the bandgap by changing the direction of electric field. The strain can linearly tune the valley gap in a considerable range.The Berry curvature and AHC can be effectively regulated in the external fields.
DOI: 10.1103/PhysRevB.101.245416
I. INTRODUCTION
Transition-metal dichalcogenides (TMDs) have become a
prominent family of two-dimensional materials. As an analogto graphene, it surpasses graphene in possessing direct bandgap in visible light range and strong spin-orbit coupling,making TMDs excellent candidates for optical, electronic,spintronic, and photovoltaic applications [ 1–5]. The well-
known representative members of TMDs are monolayer MX
2
(M=Mo, W; X=S, Se), in which there is a central M
sublayer sandwiched by two mirror-symmetric Xsublayers.
Due to the lack of inversion symmetry, the H- MX 2has a new
degree of freedom, i.e., valley, which is coupled with spindegree of freedom to exhibit extraordinary quantum effectssuch as valley-spin locking and valley-spin Hall effect [ 6–14].
Recently, vanadium dichalcogenides H-V X
2(X=S, Se, Te)
are arising as a distinguished group among the TMDs by beingsimultaneously semiconducting and ferromagnetic [ 15–18].
Two-dimensional ferromagnetic semiconductors are under in-tensive research for their superior potential in spintronics. Theintrinsic ferromagnetism is further coupled with the valleydegree of freedom in V X
2to make a ferrovalley material [ 15].
Monolayer MXX/primeis a Janus variant of TMDs, in which the
two chalcogen layers are different and hence the mirror sym-metry existing in MX
2is broken [ 19–21]. Ordered Janus MXX/prime
has been synthesized by modified chemical vapor deposition
(CVD) methods under careful control to avoid the formationof random alloys [ 19,22,23]. The out-of-plane asymmetry
between the XandX
/primelayers can significantly enhance the
*xiangyang_peng@xtu.edu.cnperpendicular piezoelectric effect. Janus V XX/prime, in particular
Janus monolayer H-VSSe (briefed as VSSe in the following),has also been studied recently [ 24,25]. It can be expected
that ordered VSSe can be grown by the similar modifiedCVD methods as mentioned above. It is found that VSSe isstrongly piezoelectric and multiferroic with strong ferroelas-ticity and ferromagnetism. Therefore, VSSe is expected toprovide a unique platform to explore the electronic, optical,magnetic, and valleytronic properties and their synergisticeffects. However, the coupling of the ferromagnetism andvalleytronic properties in VSSe has not been explored. It isstill to know how the effective magnetic field induced byferromagnetism would affect the Berry curvatures, the re-lated optical dichroism, and the anomalous Hall conductivity(AHC). The valley gaps have great effect on the electrical,optical, and valleytronic properties. The conventional density-functional theory calculations of vanadium dichalcogenidesup to now did not consider the many-body effects and henceusually significantly underestimated the energy gap, which inturn would compromise the calculations of Berry curvatures,photoluminescence spectrum, AHC, etc.. In this study, weinvestigated the combining effects of magnetic exchange fieldand valleytronic properties in VSSe based on more rigorousquasiparticle GW method. The Berry phase-related quantumphenomena were examined.
In symmetrically equivalent directions, the response of
the system to the external perturbations are identical. InMX
2, for instance, the electric fields along the two opposite
perpendicular directions, which are equivalent due to themirror symmetry between the two Xlayers, will have the
same effect. With the breaking of the time-reversal symmetry,inversion symmetry, and mirror symmetry, VSSe should be
2469-9950/2020/101(24)/245416(8) 245416-1 ©2020 American Physical SocietyLUO, PENG, QU, AND ZHONG PHYSICAL REVIEW B 101, 245416 (2020)
FIG. 1. (a) The top and side views of VSSe. The red, yellow, and green spheres represent the V , S, and Se atoms, respectively. The diamond
indicates the unit cell of 2D VSSe. (b) The phonon spectrum of VSSe. (c) The average electric potential along zdirection of VSSe.
more tunable in external fields. Therefore, we investigated
valley control in VSSe by applying electric field, strain, andchanging the magnetization to find the covariation of theelectronic, spin, and valley freedoms.
II. METHODS
The first-principles calculation is carried out by using
the density-functional theory (DFT) package V ASP [26,27].
The generalized gradient approximation functional in Perdew-Burke-Ernzerhof (PBE) form is used [ 28]. It is found that a
12×12×1/Gamma1-centered kmesh and a 400-eV plane-wave
truncation energy are sufficient to give convergent results.Spin-orbit coupling (SOC) is taken into account. Consid-ering the deficiency of DFT in estimating the band gapof semiconductors, we calculated the band structure usingGW0 approximation to include many-body effects [ 29]. The
monolayer VSSe is simulated by a slab in a supercell with avacuum layer thicker than 15 Å. Projector augmented wave(PAW) pseudopotential is utilized to describe the interatomicinteraction. The criterion for atomic relaxation is 0.001 eV /Å.
Since the slab of VSSe is asymmetric, the dipole correctionshave been taken into consideration [ 30]. The phonon spectrum
is calculated using
PHONOPY [31] to check the dynamical
stability of monolayer VSSe. Based on the calculated Blochstates, the Berry curvature of the selected band is calculatedthrough the Kubo formula [ 32]. The total Berry curvature,
anomalous Hall conductivity, and optical conductivity areobtained by
WANNIER 90 [33]. A fine kmesh of 36 ×36×1
is used in WANNIER interpolation. The dielectric function of
VSSe is derived from the calculated optical conductivity tostudy the optical properties.
III. RESULTS AND DISCUSSION
The Janus VSSe is composed of an upper S layer, a lower
Se layer, and a middle V layer, as shown in Fig. 1(a). Each V
atom has six nearest Se and S neighbors. After optimization ofthe lattice and the atomic positions, it is found that the latticeconstant of VSSe (3.26 Å) [ 24,25] is between that of VS
2(3.18 Å) [ 17] and VSe 2(3.34 Å) [ 16]. The V–Se and V–S bond
lengths in VSSe are almost the same as they are in VSe 2and
VS2, respectively. Since V–S bond is much shorter than V–Se
bond, it is evident that the mirror symmetry with respect to theV plane in VS
2and VSe 2is lost in VSSe [Fig. 1(a)]. There is
no imaginary frequency in the calculated phonon spectrum asFig. 1(b) shows, indicating that VSSe is dynamically stable.
By using molecular-dynamics simulations, we further foundthat VSSe keeps stable and well ordered at the finite tempera-tures of 300 and 500 K (Fig. S1 in the Supplemental Material[34]). The calculated average electrostatic potential along z
axis is quite asymmetric, as depicted in Fig. 1(c). The potential
in the vacuum region is flat after dipole corrections (Fig. S3 inthe Supplemental Material [ 34]). We have done Bader charge
analysis and found that each V atom loses 1 .382e, whereas
each Se and S atom obtains 0 .605eand 0.777e, respectively,
agreeing with the electronegativity order S >Se>V.
Recently, more and more two-dimensional (2D) ferromag-
nets, such as CrI
3[35], VSe 2[36], and Cr 2Ge2Te6[37],
have been found in experiments, in spite of the Mermin-Wagner theorem. It was assumed that the magnetic anisotropymay play a role to stabilize the long-range magnetic order[35,37,38], which is the case for V X
2and V XX/prime. Our calcula-
tions show that monolayer H-VS 2and VSe 2are ferromagnetic
(FM), in consistency with the previous results [ 15–18]. For
VSSe, it is found that ferromagnetic configuration is ener-getically lower than the antiferromagnetic (AFM) one. EachV atom contributes 1 μ
Bmagnetic moment. According to the
nearest-neighbor Heisenberg model [ 39], the Curie tempera-
ture can be estimated by 3 kBTc/2=(EAFM−EFM)/N, where
kBis the Boltzmann constant, Nis the number of magnetic
atoms in the supercell, and EAFMandEFMthe total energy of
AFM and FM configurations, respectively. We compared thetotal energies of VSSe in FM and various AFM configurations(see Fig. S2 and Table S1 of the Supplemental Material [ 34]).
It is found that E
FMis lower than EAFM by 0.216 eV , which
corresponds to a Curie temperature of 418 K ( N=4), which
is between that of VS 2and VSe 2[40]. Therefore, VSSe
can be ferromagnetic at room temperatures. V X2and V XX/prime
have strong SOC, which can lead to magnetic anisotropy.
245416-2V ALLEY DEGREE OF FREEDOM IN FERROMAGNETIC … PHYSICAL REVIEW B 101, 245416 (2020)
FIG. 2. The PBE (a), (b) and GW (c), (d) band structures of H-VSSe monolayer with (b), (d) and without (a), (c) SOC. The red lines and
blue dashed lines in (a), (c) correspond to spin-up and spin-down states. The red and green triangles in (b) and (d) denote the contribution
from the dz2anddx2−y2±idxyorbitals of V atom, respectively. The arrows between V±KandC±Kdenote the valley-selective optical transitions
induced by left and right circularly polarized light σ+andσ−.
We calculated the magnetic anisotropic energy of VSe 2and
VSSe and found that their easy axis is in the material plane,whose energy is 0.58 [ 41,42] and 0.37 meV lower than the
perpendicular energy for VSe
2and VSSe, respectively. The
magnetic orientation can be effectively tuned by externalfields [ 43–45] or magnetic substrates [ 46,47]. It is found that
when the orientation of the magnetization is perpendicularto the material plane, VSe
2will exhibit intriguing properties
of ferrovalley, which are under intensive exploration [ 13,15–
18]. To compare the ferrovalley properties of Janus VSSe
with those of VSe 2, the perpendicular magnetic orientation is
assumed (unless otherwise stated) in the following.
We first calculated the band structure of VSSe with spin
polarization but without SOC. It can be found that there is aDirac valley at each of the Kand−Kpoints. Both valence-
and conduction-band edges of the valleys are spin-up states,as shown by the two red lines closest to the valley gap inFig. 2(a). The bands of opposite spins are well separated,
which breaks the time-reversal symmetry relation E
↑(k)=
E↓(−k), where the arrows denote the spin directions. In the
calculated DFT-PBE band structure, the top of the valenceband is located at the /Gamma1point and the indirect band gap for
spin-up states is 0.529 eV . The two valleys are degenerate inenergy with identical valley gaps of 0.744 eV . After includingthe many-body effects by using GW approximation, the topof the valence bands is moved from /Gamma1to±Kpoints so that
the band structure has direct gaps between the spin-up bandsat the two valleys. As shown Fig. 2(c), the renormalized
band gap is 1.494 eV , almost twice as much as the PBEgap. The dispersion of the bands also changes apparently dueto GW corrections. The analysis of the Bloch states at theDirac valleys shows that the conduction- ( C) and valence-
(V) band edges near C
±KandV±Kare, respectively, dz2anddx2−y2±idxydominant states from V atom, and therefore,
their corresponding orbital magnetic moment along the z
direction μLisμL(C±K)≈0 and μL(V±K)≈±2μB, respec-
tively. μBis the Bohr magneton.
The band structures taking the SOC into account are shown
in Figs. 2(b)and2(d). One can see that CKandC−Kremain en-
ergetically close, whereas VKis appreciably higher than V−K,
thereby reducing the valley gap at K(/Delta1K) and increasing the
gap at −K(/Delta1−K). Hence, the valley degeneracy is broken and
an evident valley splitting /Delta1=/Delta1−K−/Delta1Kis induced, which
is manifested as two splitting peaks in the photoluminescencespectra. /Delta1has a DFT value of 80 meV . After GW corrections,
/Delta1is significantly increased to 179 meV . The large valley split-
ting can be ascribed to the ferromagnetism and the strong SOCin V atom between its orbital ( μ
L) and spin ( μS) magnetic
moments, which is proportional to μS·μL. We calculated the
mean value of spin of the states of the valence-band edgeand found that /angbracketleftˆσ
x/angbracketright≈/angbracketleft ˆσy/angbracketright≈0 and /angbracketleftˆσz/angbracketright≈1, where ˆ σis the
Pauli operator. Therefore, the spin of the valence-band edgealmost remains parallel in the upward direction, producingan effective magnetic field B
effacting on μLand inducing
an energy shift of μLBeff. As discussed above, the orbital
magnetic moment μL(C±K)≈0, and therefore the energy
shiftμL(C±K)Beff≈0 for conduction valley edges CKand
C−K. In contrast, μL(V±K)≈±2μB, leading to an up and
down energy shift of 2 μBBefffor valence-band edge states VK
andV−K, respectively. The valley gap /Delta1Kand/Delta1−Kare thus
reduced and enlarged by 2 μBBeff, respectively. As a result, the
valley splitting /Delta1=/Delta1−K−/Delta1K=4μBBeff=0.23BeffmeV,
being close to the experimentally found /Delta1dependence on
the external magnetic field Bfor MoS 2and MoSe 2,/Delta1=
0.22BmeV [ 48,49], where BeffandBare in the unit of tesla.
In experiments, an external magnetic field of tens of tesla
245416-3LUO, PENG, QU, AND ZHONG PHYSICAL REVIEW B 101, 245416 (2020)
FIG. 3. (a) The zcomponent of the total Berry curvatures of valence bands /Omega1z
total.( b )T h e zcomponent of the Berry curvatures of the
valence-band edge /Omega1z
Vand the conduction-band edge /Omega1z
C. (c) The anomalous Hall conductivity dependence on the Fermi level. The points of
C±K,V±K,andCMare labeled in the band structures in Fig. 2(d). (d) The anomalous Hall conductivity under 3% tensile strain.
can only induce several meV valley splitting [ 48,49], whereas
ferromagnetic VSSe has a very large valley splitting of 179meV , which means a B
eff≈800 T. Suppose one is to create
the same valley splitting (179 meV) in MoS 2and MoSe 2by
external magnetic field B, then Bshould be of similar magni-
tude (∼800 T). This implies that intrinsic ferromagnetism is
much more efficient in producing valley splitting.
We calculated the zcomponent of Berry curvatures /Omega1z
n(k)
of the highest valence ( V) and the lowest conduction ( C) bands
according to the Kubo formula [ 32],
/Omega1z
n(k)=/summationdisplay
m/negationslash=n2Im/angbracketleftψnk|ˆvx|ψmk/angbracketright/angbracketleftψmk|ˆvy|ψnk/angbracketright
[εm(k)−εn(k)]2, (1)
where ˆ vxand ˆvyare velocity operators along xandydirec-
tions, respectively. |/Psi1nk/angbracketrightis the calculated wave function of
the Bloch state of band n(=CorV)a tkpoint, and εnkis the
energy eigenvalue. For nonmagnetic TMDs, such as MoS 2,
the Berry curvature has an odd parity /Omega1z
n(k)=−/Omega1z
n(−k), as
dictated by time-reversal symmetry [ 50]. The calculated Berry
curvatures of VSSe are shown in Figs. 3(a) and3(b).I nt h e
intrinsic ferromagnetic field of VSSe, the signs of /Omega1z
n(K) and
/Omega1z
n(−K) remain opposite but the magnitudes become differ-
ent. From Figs. 2(d) and3(b), one can find that /Delta1K</Delta1 −K
but|/Omega1z
n(K)|>|/Omega1z
n(−K)|(n=CorV). The inverse relation
between the energy gap and the magnitude of Berry curvaturecan be understood from the Kubo formula [Eq. ( 1)], which in-
dicates that the largest contribution to |/Omega1
z
n(±K)|comes from
the conduction- and valence-band edges C±KandV±Kdue
to the inverse dependence on [ εC(±K)−εV(±K)]2, where
εC(±K)−εV(±K) is just the energy gap at ±K. Actually,
we found that the Berry curvatures satisfy /Omega1z
n(K)//Omega1z
n(−K)≈
−/Delta12
−K//Delta12
K. Interestingly, it is found that this relation still
holds during the tuning of the valley splitting by external fields(see below).
If an in-plane electric field E
/bardblis applied, an anomalous
transverse current occurs driven by the Berry curvature E/bardbl×
/Omega1z
n(k)[50]. In VSSe, Berry curvature is prominent only at
the valley edges. As discussed above, /Omega1z
n(K) and /Omega1z
n(−K)
have opposite sign and unequal magnitudes, inducing oppositedeflection of the valley carriers in transverse direction but
at different rate. As a result, a net charge accumulation willdevelop at one side and a transverse Hall voltage is built.The net charge comes from the same valley with the samespin, which means a simultaneous charge, spin, and valleypolarization. In MoS
2,/Omega1z
n(K)=−/Omega1z
n(−K) and hence the
valley carriers have the same transverse deflection rate inopposite direction, without inducing charge Hall voltage. Wecalculated the anomalous Hall conductivity σ
AH
αβ, which is ba-
sically determined by the summation of the Berry curvaturesof all occupied states over the Brillouin zone. It is found σ
AH
αβ
is always zero for MoS 2because the odd parity of Berry
curvature in MoS 2cancels its summation over k. In VSSe,
the odd parity is broken and the calculation yield a nonzeroσ
AH
αβ. When the Fermi level is between the top valence edges of
the two valleys VKandV−K, which corresponds to the case of
hole doping, the calculated maximum AHC σAH
αβis 29.0S/cm.
Between the conduction edges of the two valleys CKandC−K,
the maximum absolute value of σAH
αβis 7.9S/cm.
In the nonmagnetic TMDs, such as MoS 2, there is valley-
selective circular dichroism, by which one can selectivelyexcite valley carriers at Kor−Kby using light with opposite
circular polarization. To see the effect of ferromagnetismon the optical properties of VSSe, we calculated the inter-band transition matrix P
C,V
±(k)=/angbracketleft/Psi1Ck|ˆP±|/Psi1Vk/angbracketright, where ˆP±=
(ˆPx±ˆPy)/√
2 and ˆPis the momentum operator. The signs +
and – stand for left and right circular polarization, respec-tively. As shown in Fig. 4(a), we calculated the k-resolved
normalized circular polarization η(k)=|PC,V
+(k)|2−|PC,V
−(k)|2
|PC,V
+(k)|2+|PC,V
−(k)|2, and
found that ηis nearly ±1a t±Kand in their neighborhood, in-
dicating that left ( σ+) and right ( σ−) circularly polarized light
can only excite the Kand−Kvalley, respectively, and that the
valley-selective circular dichroism persists in the presence offerromagnetism. If the incident light is unpolarized, which isa superposition of the σ
+andσ−components, the K(−K)
valley only absorbs the σ+(σ−) component.
σ+andσ−peaks can be observed in the polarization-
resolved photoluminescence (PL) experiment. In nonmag-netic MoS
2, the two peaks overlap because of the valley
245416-4V ALLEY DEGREE OF FREEDOM IN FERROMAGNETIC … PHYSICAL REVIEW B 101, 245416 (2020)
FIG. 4. (a)The circular polarization η(k) of the optical transition between the valence- and conduction-band edges in the first Brillouin
zone. The hexagon represents the Brillouin zone. The high-symmetry points K,−K,and/Gamma1are labeled. The color scale on the right side
indicates the values of η(k) over the Brillouin zone. (b) The imaginary part ε2of dielectric function for left-handed light σ+, right-handed light
σ−, and unpolarized light σ.
degeneracy. If an external magnetic field is applied, it has been
observed in PL spectra that the σ+andσ−peaks are split at a
small rate about 0.22 meV/T for MoS 2and MoSe 2[48,49]. To
study the optical properties such as valley Zeeman splitting ofthe ferromagnetic VSSe, we calculated the imaginary part ofthe dielectric function, which is related to optical conductivityσ(ω)b yε
2(ω)=4πσ(ω)/ω. The optical conductivity is cal-
culated via WANNIER 90 with a dense k mesh of 36 ×36×1.
In Fig. 4(b), one can see that σ+andσ−peak positions
correspond to the valley gaps /Delta1Kand/Delta1−K, respectively.
The splitting between the peaks is 179 meV , which is thesame as the calculated valley splitting (GW). Because theσ
+andσ−peaks are well split, the excitation energies for
Kvalley EKand for −Kvalley E−Kdiffer considerably,
making it possible to realize the valley-selective excitation byunpolarized light σ. As known, σcan be decomposed into σ
+
andσ−components. When the energy of σisEK, only the σ+
component of σcan excite the Kvalley. However, σof energy
EKwill not excite the −Kvalley because of the significant
excitation energy mismatch between EKandE−K. Similarly,
the unpolarized light of energy E−Kcan only excite the −Kvalley. We also calculated the PL spectra of unpolarized light
σ, as shown in Fig. 4(b). It can be seen that there are still two
well-split peaks located at the same position as the σ+and
σ−peaks, which correspond to Kand−Kvalley excitations,
respectively. This is not possible for nonmagnetic TMDs sincethe two valleys are energetically degenerate and have the sameexcitation energy [ 7]. In this case, one has to use light of
the same energy but with opposite circular polarization toselectively excite the valley carriers.
The lack of time-reversal symmetry and mirror symmetry
made VSSe more tunable than nonmagnetic TMDs. We stud-ied the control of valley freedom by magnetization, electricfield, and strain. As discussed above, the valley splitting is4μ
BBeffprovided that the orbital magnetic moment (in z
direction) is parallel to the effective magnetic field Beff.I f
there is an angle θbetween Beffand the orbital magnetic mo-
ment [Fig. 5(a)], the valley splitting becomes 4 μBBeffcosθ.
Therefore, one can rotate the direction of magnetization totune the valley splitting. In calculation, the spin quantizationaxis can be aligned to any specified direction. We chose aplane as shown in the inset of Fig. 5(a) and change the spin
FIG. 5. The tuning of band structures (a), the valley gaps /Delta1Kand/Delta1−K(b), and the total Berry curvatures of the valence bands (c) by
changing the magnetization angles θbetween the magnetic moment and zdirection. The inset of (a) indicates the rotation angle and the
rotation plane of the magnetization direction. For the definition of xandzdirections, please refer to Fig. 1(a). The inset of (b) shows the
variation of valley splitting /Delta1(θ)=/Delta1−K(θ)−/Delta1K(θ) with respect to θ.
245416-5LUO, PENG, QU, AND ZHONG PHYSICAL REVIEW B 101, 245416 (2020)
FIG. 6. Electric-field tuning of band structures (a), the valley gaps /Delta1K(Ez)a n d/Delta1−K(Ez) (b), and the total Berry curvatures of the valence
bands (c). The inset in (b) shows the variation of valley splitting /Delta1(Ez)=/Delta1K(Ez)−/Delta1−K(Ez) with respect to Ez.
quantization axis within this plane. In this way, the mag-
netization direction dependent properties can be calculated.We studied the θ-dependent band structure and found that
the band gap /Delta1
Kand/Delta1−Kincrease and decrease at Kand
−Kvalleys, respectively, for 0◦/lessorequalslantθ/lessorequalslant180◦,a ss h o w ni n
Figs. 5(a) and5(b). Between 0◦and 90◦, the valley splitting
/Delta1=/Delta1−K−/Delta1Kdiminishes but remains positive. When Beff
is lying in the plane ( θ=90◦), the valley splitting vanishes.
With further rotation from 90◦to 180◦, one can find that valley
splitting becomes negative and the magnitude grows until itfinally reaches −4μ
BBeff. Therefore, the valley splitting can
be continuously tuned from 4 μBBeffto−4μBBeff.W ea l s o
studied the modulation of Berry curvature /Omega1zby changing the
magnetization direction. In Fig. 5(c), it can be seen that the
Berry curvature difference |/Omega1z(K)|−|/Omega1z(−K)|is largest and
zero in magnitude when Beffis perpendicular and parallel to
VSSe plane, respectively.
Application of electric field is an effective way to tune the
band structure and align the bands. We studied the electricresponse of VSSe by applying perpendicular electric fieldE
z. In VSSe, the upper S and the lower Se layers are not
mirror symmetric, and consequently the direction of Ezshouldmake a difference when it points up or down. The GW
band structures with Ez=0 and±0.7e V/Å are plotted in
Fig. 6(a). One can see that the band is pushed up in positive
Ezbut is pressed down when the Ez.turns negative. The valley
gaps/Delta1Kand/Delta1−Kvary first slowly and linearly with small Ez,
and change rapidly in the narrow range around ±0.4e V/Å,
as shown in Fig. 6(b). But, the valley splitting /Delta1=/Delta1−K−
/Delta1Kremains almost invariant. In VSSe, the valley gaps are
dependent on the direction of electric field, being enhancedin positive E
zbut reduced in negative Ez. In addition, the
magnitude of the gap variation rate also differs in oppositeelectric field. This is quite different from MX
2TMDs. Our
calculations show that, for instance, the valley gaps of mono-layer VSe
2always grow with |Ez|regardless of the direction of
Ezdue to the mirror symmetry between the two Se layers. We
also calculated the Berry curvatures /Omega1z(±K) under different
Ez, and found the electric field has marginal effect on /Omega1z(±K),
as shown in Fig. 6(c).
In heterostructures or at different temperatures, VSSe is
usually strained. We studied strain effect on the electronicstructure of VSSe by calculating the GW band structuresunder different in-plane strains, as shown in Fig. 7(a).I ti s
FIG. 7. Strain tuning of band structures (a) and the valley gaps /Delta1K(/epsilon1)a n d/Delta1−K(/epsilon1) (b) and the total Berry curvatures of the valence bands
(c). The inset in (b) shows the variation of valley splitting /Delta1(/epsilon1)=/Delta1K(/epsilon1)−/Delta1−K(/epsilon1) with respect to /epsilon1.
245416-6V ALLEY DEGREE OF FREEDOM IN FERROMAGNETIC … PHYSICAL REVIEW B 101, 245416 (2020)
found that the tensile strain will move the bands up and the
compressive one will press the bands down, with respect tothe vacuum level [ 51]. The strain tuning of the gaps is quite
effective. The magnitude of the gap modulation is as large as0.63 eV for /Delta1
Kand 0.55 eV for /Delta1−Kwhen the strain is varied
between ±3%. The band gaps /Delta1±Khave linear dependence on
strain, being enhanced by stretch and reduced by compression.The slopes of the two lines in Fig. 7(b) indicate that /Delta1
K
has a larger variation rate with respect to strain than /Delta1−K.
Hence, the valley splitting /Delta1=/Delta1−K−/Delta1Kincreases under
tensile strain and decreases under the compressive strain. Themaximum modulation of /Delta1is 80 meV for ±3% strain range.
The change of the valley Berry curvature is also considerable[Fig. 7(c)]. The tensile (compressive) strains tend to enhance
(reduce) Berry curvatures /Omega1
z(±K) and also their magnitude
difference |/Omega1z(K)|−|/Omega1z(−K)|=|/Omega1z(K)+/Omega1z(−K)|. Since
the AHC is determined by the summation of the Berry cur-vatures over the Brillouin zone, and the Berry curvatures arenonzero only around ±K, it is expected that the AHC will be
considerably enhanced under the tensile strain as a result ofthe increase of |/Omega1
z(K)+/Omega1z(−K)|. With respect to the AHC
of the nonstrained VSSe [Fig. 3(c)], the calculated AHC under
3% tensile strain [Fig. 3(d)] is almost doubled when the Fermi
level is moved to between VKandV−Kby hole doping, and
between CKandC−Kby electron doping. In this way, one can
adjust the transverse Hall voltage by strain.
IV . CONCLUSION
In summary, we have studied the valleytronic properties
of monolayer Janus VSSe and investigated the control ofvalley degree of freedom. VSSe is found to be a ferro-magnetic semiconductor with strong SOC. The inequivalentDirac valleys of VSSe have the same spin and are not en-ergetically degenerate due to the breaking of time-reversalsymmetry by its ferromagnetism. Compared with the DFTresults, the GW renormalized band gap and valley splitting
are almost doubled and the valence-band maximum at Г
point is pressed down considerably lower than those at ±K
points, showing strong many-body effects in VSSe. The parityrelations E
n(k)=En(−k) and/Omega1z
n(k)=−/Omega1z
n(k) are violated.
There is a sizable magnitude disparity between the Berrycurvatures at Kand−K, which satisfies /Omega1
z
n(K)//Omega1z
n(−K)≈
−/Delta12
−K//Delta12
K, and hence results in appreciable anomalous Hall
conductivity. The valley-selective circular dichroism persistsin ferromagnetism. The calculated optical spectrum featurestwo well-separated peaks and a scheme of valley-selectiveexcitation by unpolarized light is thus proposed. The breakingof symmetries in VSSe makes the valley freedom more tun-able. The valley splitting can be continuously modulated andeven reversed by rotating the magnetization vector. Unlike themirror-symmetric MX
2, in which the electric response does
not depend on the direction of Ez, VSSe can be bidirectionally
tuned when Ezchanges direction due to the mirror-asymmetric
Janus structure. Application of lateral strain can effectivelymodify the band structure. Compressive strains will shiftdown the bands, reduce the valley gap, and increase thevalley splitting appreciably, whereas the tensile strains actoppositely. Accordingly, the variation of Berry curvature isconsiderable. The AHC can be enhanced by strains and hencethe transverse Hall voltage can be effectively controlled bystrain.
ACKNOWLEDGMENTS
The authors acknowledge the support of the National
Natural Science Foundation of China (Grants No. 11874315and No. 11874316), the National Basic Research Programof China (Grant No. 2015CB921103), Innovative ResearchTeam in University (Grant No. IRT 17R91), and HunanProvincial Innovation Foundation For Postgraduate (GrantNo. CX20190472).
[1] W. Choi, N. Choudhary, G. H. Han, J. Park, D. Akinwande, and
Y . H. Lee, Recent development of two-dimensional transitionmetal dichalcogenides and their applications, Mater. Today 20,
116 (2017) .
[2] H. Yuan et al. , Generation and electric control of spin-
valley-coupled circular photogalvanic current in WSe
2,Nat.
Nanotechnol. 9, 851 (2014) .
[3] X. Xu, W. Yao, D. Xiao, and T. F. Heinz, Spin and pseudospins
in layered transition metal dichalcogenides, Nat. Phys. 10, 343
(2014) .
[4] G. B. Liu, D. Xiao, Y . Yao, X. Xu, and W. Yao, Electronic
structures and theoretical modelling of two-dimensional group-VIB transition metal dichalcogenides, Chem. Soc. Rev. 44,
2643 (2015) .
[5] T. Cao et al. , Valley-selective circular dichroism of monolayer
molybdenum disulphide, Nat. Commun. 3, 887 (2012) .
[6] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control of val-
ley polarization in monolayer MoS
2by optical helicity, Nat.
Nanotechnol. 7, 494 (2012) .
[7] J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L.
Seyler, W. Yao, and X. Xu, Valleytronics in 2D materials, Nat.
Rev. Mater. 1, 16055 (2016) .[8] T. Olsen and I. Souza, Valley Hall effect in disordered mono-
layer MoS 2from first principles, Phys. Rev. B 92, 125146
(2015) .
[ 9 ] A .S r i v a s t a v a ,M .S i d l e r ,A .V .A l l a i n ,D .S .L e m b k e ,A .
Kis, and A. Imamo ˘glu, Valley Zeeman effect in elementary
optical excitations of monolayer WSe 2,Nat. Phys. 11, 141
(2015) .
[10] D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, Coupled Spin
and Valley Physics in Monolayers of MoS 2and Other Group-VI
Dichalcogenides, P h y s .R e v .L e t t . 108, 196802 (2012) .
[11] W. Yao, D. Xiao, and Q. Niu, Valley-dependent optoelectronics
from inversion symmetry breaking, P h y s .R e v .B 77, 235406
(2008) .
[12] H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Valley
polarization in MoS 2monolayers by optical pumping, Nat.
Nanotechnol. 7, 490 (2012) .
[13] J. Zhou, Q. Sun, and P. Jena, Valley-Polarized Quantum
Anomalous Hall Effect in Ferrimagnetic Honeycomb Lattices,Phys. Rev. Lett. 119, 046403 (2017) .
[14] W. Feng, Y . Yao, W. Zhu, J. Zhou, W. Yao, and D. Xiao, Intrin-
sic spin Hall effect in monolayers of group-VI dichalcogenides:A first-principles study, P h y s .R e v .B 86, 165108 (2012) .
245416-7LUO, PENG, QU, AND ZHONG PHYSICAL REVIEW B 101, 245416 (2020)
[15] W. Y . Tong, S. J. Gong, X. Wan, and C. G. Duan, Concepts
of ferrovalley material and anomalous valley Hall effect, Nat.
Commun. 7, 13612 (2016) .
[16] J. Liu, W. J. Hou, C. Cheng, H. X. Fu, J. T. Sun, and S. Meng,
Intrinsic valley polarization of magnetic VSe 2monolayers,
J. Phys.: Condens. Matter 29, 255501 (2017) .
[17] H. L. Zhuang and R. G. Hennig, Stability and magnetism of
strongly correlated single-layer VS 2,P h y s .R e v .B 93, 054429
(2016) .
[18] F. Zhang, W. Mi, and X. Wang, Tunable valley and spin
splitting in 2H-VSe 2/BiFeO 3(111) triferroic heterostructures,
Nanoscale 11, 10329 (2019) .
[19] A. Y . Lu et al. , Janus monolayers of transition metal dichalco-
genides, Nat. Nanotechnol. 12, 744 (2017) .
[20] T. Hu, F. Jia, G. Zhao, J. Wu, A. Stroppa, and W. Ren, Intrinsic
and anisotropic Rashba spin splitting in Janus transition-metaldichalcogenide monolayers, P h y s .R e v .B 97, 235404 (2018) .
[21] F. Li, W. Wei, P. Zhao, B. Huang, and Y . Dai, Electronic and
optical properties of pristine and vertical and lateral heterostruc-tures of Janus MoSSe and WSSe, J. Phys. Chem. Lett. 8, 5959
(2017) .
[22] R. Li, Y . Cheng, and W. Huang, Recent progress of Janus 2D
transition metal chalcogenides: From theory to experiments,Small 14, 1802091 (2018) .
[23] J. Zhang et al. , Janus monolayer transition-metal dichalco-
genides, ACS Nano 11, 8192 (2017) .
[24] C. Zhang, Y . Nie, S. Sanvito, and A. Du, First-principles
prediction of a room-temperature ferromagnetic Janus VSSemonolayer with piezoelectricity, ferroelasticity, and large valleypolarization, Nano Lett. 19, 1366 (2019) .
[25] J. Yang, A. Wang, S. Zhang, J. Liu, Z. Zhong, and L. Chen, Co-
existence of piezoelectricity and magnetism in two-dimensionalvanadium dichalcogenides, Phys. Chem. Chem. Phys. 21, 132
(2018) .
[26] G. Kresse and J. Furthmüller, Efficient iterative schemes for
ab initio total-energy calculations using a plane-wave basis set,P h y s .R e v .B 54, 11169 (1996) .
[27] G. Kresse and J. Furthmüller, Efficiency of ab-initio total en-
ergy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996) .
[28] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient
Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996) .
[29] M. Shishkin and G. Kresse, Implementation and performance
of the frequency-dependent GW method within the PAW frame-work, P h y s .R e v .B 74, 035101 (2006) .
[30] L. Bengtsson, Dipole correction for surface supercell calcula-
tions, Phys. Rev. B 59, 12301 (1999) .
[31] A. Togo and I. Tanaka, First principles phonon calculations in
materials science, Scr. Mater. 108, 1 (2015) .
[32] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den
Nijs, Quantized Hall Conductance in a Two-Dimensional Peri-odic Potential, P h y s .R e v .L e t t . 49, 405 (1982) .
[33] A. A. Mostofi, J. R. Yates, G. Pizzi, Y .-S. Lee, I. Souza, D.
Vanderbilt, and N. Marzari, An updated version of
WANNIER 90:
A tool for obtaining maximally-localised Wannier functions,Comput. Phys. Commun. 185, 2309 (2014) .
[34] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.101.245416 for molecular dynamics simu-lation, energies of FM and AFM configurations, band structures
with and without dipole corrections, and formation energies ofVSSe and MoSSe.
[35] B. Huang et al. , Layer-dependent ferromagnetism in a van der
Waals crystal down to the monolayer limit, Nature (London)
546, 270 (2017) .
[36] M. Bonilla et al. , Strong room-temperature ferromagnetism
in VSe
2monolayers on van der Waals substrates, Nat.
Nanotechnol. 13, 289 (2018) .
[37] C. Gong et al. , Discovery of intrinsic ferromagnetism in two-
dimensional van der Waals crystals, Nature (London) 546, 265
(2017) .
[38] B. Shabbir, M. Nadeem, Z. Dai, M. S. Fuhrer, Q.-K. Xue, X.
Wang, and Q. Bao, Long range intrinsic ferromagnetism in twodimensional materials and dissipationless future technologies,Appl. Phys. Rev. 5, 041105 (2018) .
[39] J. Kudrnovský, I. Turek, V . Drchal, F. Máca, P. Weinberger, and
P. Bruno, Exchange interactions in III-V and group-IV dilutedmagnetic semiconductors, Phys. Rev. B 69, 115208 (2004) .
[40] H. Pan, Electronic and magnetic properties of vanadium
dichalcogenides monolayers tuned by hydrogenation, J. Phys.
Chem. C 118, 13248 (2014) .
[41] S. Feng and W. Mi, Strain and interlayer coupling tailored
magnetic properties and valley splitting in layered ferrovalley2H-VSe
2,Appl. Surf. Sci. 458, 191 (2018) .
[42] H.-R. Fuh, C.-R. Chang, Y .-K. Wang, R. F. Evans, R. W.
Chantrell, and H.-T. Jeng, New-type single-layer magneticsemiconductor in transition-metal dichalcogenides V X
2(X=
S, Se and Te), Sci. Rep. 6, 32625 (2016) .
[43] J. Qi, X. Li, Q. Niu, and J. Feng, Giant and tunable val-
ley degeneracy splitting in MoTe 2,Phys. Rev. B 92, 121403
(2015) .
[44] A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y . Lyanda-
Geller, and L. P. Rokhinson, Evidence for reversible control ofmagnetization in a ferromagnetic material by means of spin–orbit magnetic field, Nat. Phys. 5, 656 (2009) .
[45] N. Miyata and B. Argyle, Magnetocrystalline anisotropy of
single-crystal europium oxide, Phys. Rev. 157, 448 (1967) .
[46] C. Zhao et al. , Enhanced valley splitting in monolayer WSe
2
due to magnetic exchange field, Nat. Nanotechnol. 12, 757
(2017) .
[47] D. Zhong et al. , Van der Waals engineering of ferromagnetic
semiconductor heterostructures for spin and valleytronics, Sci.
Adv. 3, e1603113 (2017) .
[48] D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson, A.
Kormanyos, V . Zolyomi, J. Park, and D. C. Ralph, Breakingof Valley Degeneracy by Magnetic Field in Monolayer MoSe
2,
Phys. Rev. Lett. 114, 037401 (2015) .
[49] Y . Wu, C. Shen, Q. Tan, J. Shi, X. Liu, Z. Wu, J. Zhang, P.
Tan, and H. Zheng, Valley Zeeman splitting of monolayer MoS 2
probed by low-field magnetic circular dichroism spectroscopyat room temperature, Appl. Phys. Lett. 112, 153105 (2018) .
[50] D. Xiao, W. Yao, and Q. Niu, Valley-Contrasting Physics in
Graphene: Magnetic Moment and Topological Transport, Phys.
Rev. Lett. 99, 236809 (2007) .
[51] D. Zhang and S. Dong, Challenges in band alignment between
semiconducting materials: A case of rutile and anatase TiO
2,
Prog. Nat. Sci. 29, 277 (2019) .
245416-8 |
PhysRevB.83.035309.pdf | PHYSICAL REVIEW B 83, 035309 (2011)
Circular photogalvanic effect on topological insulator surfaces:
Berry-curvature-dependent response
Pavan Hosur
Department of Physics, University of California, Berkeley, California 94720, USA
(Received 1 July 2010; published 18 January 2011)
We study theoretically the optical response of the surface states of a topological insulator, especially the
generation of helicity-dependent direct current by circularly polarized light. Interestingly, the dominant current,due to an interband transition, is controlled by the Berry curvature of the surface bands. This extends theconnection between photocurrents and Berry curvature beyond the quasiclassical approximation, where it hasbeen shown to hold. Explicit expressions are derived for the (111) surface of the topological insulator Bi
2Se3,
where we find significant helicity-dependent photocurrents when the rotational symmetry of the surface is brokenby an in-plane magnetic field or a strain. Moreover, the dominant current grows linearly with time until a scatteringoccurs, which provides a means for determining the scattering time. The dc spin generated on the surface is alsodominated by a linear-in-time, Berry curvature-dependent contribution.
DOI: 10.1103/PhysRevB.83.035309 PACS number(s): 72 .40.+w, 72.25.Fe, 78 .68.+m, 72.10.Bg
I. INTRODUCTION
Topological insulators (TIs) have caught the eye of many
a condensed-matter physicist and materials scientist in recentyears. In very simple terms, these are materials that have an
insulating bulk but conducting surface states (SSs) that are
protected against disorder by time-reversal symmetry. Thereason for the tremendous amount of attention they havereceived is twofold. One, they have been predicted to exhibit
a number of exotic phenomena such as the magnetoelectric
effect,
1magnetic monopolelike behavior,2and the existence
of topologically protected Majorana modes3with potential ap-
plications for topological quantum computing.4Two, a number
of materials have already been theoretically predicted5–9and
experimentally found10–16to be in this fascinating phase.
In their simplest incarnation, the SSs of TIs correspond
to the dispersion of a single Dirac particle, which cannotbe realized in a purely two-dimensional band structure withtime-reversal invariance. This dispersion is endowed withthe property of spin-momentum locking, that is, for eachmomentum there is a unique spin direction of the electron.Since several materials were theoretically predicted to be inthis phase, most of the experimental focus on TIs so far hasbeen toward trying to directly observe these exotic SSs in realor momentum space, in tunneling,
10and photoemission11–16
experiments, respectively, and to establish their special topo-
logical nature. However, so far there has been a dearth ofexperiments that study the response of these materials toexternal perturbations, such as an external electromagneticfield.
In order to fill this gap, we calculate here the response of
TI surfaces to circularly polarized (CP) light. Since photons inCP light have a well-defined angular momentum, CP light can
couple to the spin of the surface electrons. Then, because of
the spin-momentum-locking feature of the SSs, this couplingcan result in dc transport that is sensitive to the helicity (right-versus left-circular polarization) of the incident light. Thisphenomenon is known as the circular photogalvanic effect
(CPGE). In this work, we derive general expressions for the
direct current on a TI surface as a result of the CPGE at normalincidence within a two-band model, and we estimate its sizefor the (111) surface of Bi
2Se3, an established TI, and find
it to be well within measurable limits. Since bulk Bi 2Se3has
inversion symmetry and the CPGE, which is a second-ordernonlinear effect, is forbidden for inversion symmetric systems,
this current can only come from the surface.
We find, remarkably, that the dominant contribution to the
current is controlled by the Berry curvature of the electron
bands and grows linearly with time . In practice, this growth is
cut off by a scattering event that resets the current to zero.At the microscopic level, this part of the current involvesthe absorption of a photon to promote an electron from thevalence to the conduction band. The total current contains twoother terms—both time-independent—one again involving aninterband transition and the other resulting from intrabanddynamics of electrons. However, for clean samples at lowtemperatures, the scattering or relaxation time is expectedto be large, and these contributions will be eclipsed by thelinear-in-time one. Hence, this experiment can also be used tomeasure the relaxation time for TI SSs.
Historically, the Berry curvature has been associated with
fascinating phenomena such as the anomalous Hall effect
17and
the integer quantum Hall effect,18and therefore it is exciting
that it appears in the response here. Its main implication hereis that it gives us a simple rule, in addition to the requirementof the right symmetries, for identifying the perturbations thatcan give a linear-in-time CPGE at normal incidence: we lookfor perturbations that result in a nonzero Berry curvature. Putanother way, we can identify perturbations that have the rightsymmetries but still do not give this current because the Berrycurvature vanishes for these perturbations. Importantly, forTI SSs, the requirement of a nonzero Berry curvature amountsto the simple physical condition that the spin direction of theelectrons have all three components nonzero. In other words,if the electron spin in the SSs is completely in-plane, the Berrycurvature is zero and no linear-in-time CPGE is expected. Thespins must somehow be tipped slightly out of the plane, asshown in Fig. 1(a), in order to get such a response. Thus, a
pure Dirac (linear) dispersion, for which the spins are planar,cannot give this response; deviations from linearity, such asthe hexagonal warping on the (111) surface of Bi
2Te3,19are
essential for tilting the spins out of the plane.
035309-1 1098-0121/2011/83(3)/035309(7) ©2011 American Physical SocietyPA V AN HOSUR PHYSICAL REVIEW B 83, 035309 (2011)
FIG. 1. (Color online) (a) Schematic illustration of preferential
absorption at one out of two points related by the reflection symmetryabout the yzplane. The short arrows denote the spin direction of
electrons in various states. At low energies, the spins are completely
in-plane. They acquire a small out-of-plane component at higherenergies. The dotted lines represent incoming photons of helicity
−1 (left-CP photons). These photons can only raise the/angbracketleftS
z/angbracketrightof an
electron, and thus are preferentially absorbed by electrons whose
/angbracketleftSz/angbracketright<0 in the valence band. The chemical potential μmust be
between the initial and final states for any absorption to occur.(b) Constant energy contours for the surface conduction band of
Bi
2Se3. Dark lines denote lower energy. Part (a) is drawn at py=0.
(c) Geometry of the experiment. Light is incident normally on the(111) surface of Bi
2Se3. The dotted lines represent the mirror plane
mabout which the lattice has a reflection symmetry. The current ja2(t)
(see text) is along ˆx.
CPGE has been observed in the past in GaAs,20SiGe,21
and HgTe /CdHgTe (Ref. 22) quantum wells—all systems
with strong spin-orbit coupling. The effect in these systemscan be understood within a four-band model consisting oftwo spin-orbit-split valence bands and two spin-degenerateconduction bands. In contrast, TI SSs can be faithfully treatedwithin a two-band model. The simplicity of the latter systemmakes it more convenient for theoretical study comparedto semiconductor quantum wells, and, hence, enables us todetermine a connection between the CPGE and the Berrycurvature. In general, if a surface has no rotational symmetryabout the surface normal, such a photocurrent is allowed.
Finally, we estimate the current on the (111) surface of
Bi
2Se3using an effective model for the SSs.19,23This model
captures the deviations from linearity of the SS dispersion dueto the threefold rotational symmetry of the (111) surface ofBi
2Se3. These deviations have been observed in photoemission
experiments on Bi 2Te3.11Similar deviations are expected for
Bi2Se3,23though they cannot be seen in the slightly smaller
momentum range compared to Bi 2Te3over which data are
currently available.24To get a direct current with CP light
at normal incidence, rotational symmetry about the surfacenormal needs to be broken. Based on the requirement ofnonzero Berry curvature, we propose to do this in two ways:(i) by applying an in-plane magnetic field and includingdeviations from linearity of the dispersion, and (ii) by applying
a strain.
With a magnetic field of 10 T (with a 1% strain) and
assuming a scattering time of 10 ps (the scattering time inGaAs is ∼1 ns over a wide range of temperatures;
25we use a
conservative estimate for Bi 2Se3here), we find that a current
density of ∼100 nA /mm (∼10 nA/mm) can be obtained due
to the CPGE with a 1 W laser. This value can be easilymeasured by current experimental techniques. Conversely, thescattering time, crucial for transport processes, for Bi
2Se3SSs
can be determined by measuring the current. In comparison,circular photogalvanic currents of a few nanoamperes per Wattof laser power have been measured in quantum wells of thesemiconductors GaAs,
20SiGe,21and HgTe /CdHgTe.22
A connection between the optical response of a sys-
tem and the Berry curvature of its bands has been pre-viously noted at low frequencies, where a semiclassicalmechanism involving the anomalous velocity of electronsin a single band explains it.
26,27Here, we show it for
interband transitions where no quasiclassical approximationis applicable. Instead, we calculate the quadratic responsefunction directly. A connection is still present that pointsto a deeper relation between the response functions and theBerry curvature.
This paper is organized as follows. In Sec. II, we state
the symmetry conditions under which a CPGE may occur.We present our results, both general as well as for Bi
2Se3
in particular, in Sec. III A and describe the microscopic
mechanism in Sec. III B . The calculation is described briefly
in Sec. III C and in detail in Appendix B. In Sec. IV,w eg i v e
our results for the optical injection of dc spin, and in Sec. V
we briefly discuss the situation where the rotational symmetryof the surface is broken by shining the light off-normally.
II. SYMMETRY CONSIDERATIONS FOR THE CPGE
In this section, we specify the symmetry conditions under
which one can get a CPGE on the surface of a TI. But first, letus briefly review the concept of the CPGE in general.
The dominant dc response of matter to an oscillating electric
field is, in general, quadratic in the electric field. When theresponse of interest is a current, the effect is known as thephotogalvanic effect. This current can be written as
j
α=ηαβγEβ(ω)Eγ(−ω), (1)
where Eα(t)=Eα(ω)eiωt+E∗
α(ω)e−iωtis the incident electric
field,E∗
α(ω)=Eα(−ω), and ηαβγis a third-rank tensor, which
has nonzero components only for systems that break inversionsymmetry, such as the surface of a crystal.
Forj
αto be real, one has ηαβγ=η∗
αγβ. Thus, the real
(imaginary) part of ηαβγis symmetric (antisymmetric) under
interchange of βandγ, and therefore describes a current that is
even (odd) under the transformation ω→−ω. Consequently,
jαcan be conveniently separated according to
jα=Sαβγ/parenleftbiggEβ(ω)E∗
γ(ω)+E∗
β(ω)Eγ(ω)
2/parenrightbigg
+iAαμ(E×E∗)μ,
(2)
where Sαβγis the symmetric part of ηαβγandAαμis a second-
rank pseudotensor composed of the antisymmetric part of ηαβγ.
035309-2CIRCULAR PHOTOGALV ANIC EFFECT ON TOPOLOGICAL ... PHYSICAL REVIEW B 83, 035309 (2011)
For CP light, E∝ˆx±iˆyifˆzis the propagation direction and
only the second term in Eq. ( 2) survives, and hence represents
the CPGE. This effect is odd in ω. On the other hand, the first
term, which is even in ω, represents the linear photogalvanic
effect as it is the only contribution for linearly polarized light.Since the transformation ω→−ω, or equivalently, E→E
∗,
reverses the helicity of CP light, that is, changes right-CPlight to left-CP light and vice versa, the CPGE is the helicity-dependent part of the photogalvanic effect.
The helicity of CP light is odd (i.e., right- and left-CP
light get interchanged) under mirror reflection about a planethat contains the incident beam, but invariant under arbitraryrotation about the direction of propagation. Let us considernormal incidence of CP light on a TI surface normal to thezaxis. Let us further assume that there is a mirror plane that
is they-zplane [see Fig. 1(c)]. Then, the only component of
direct current that reverses direction on switching the helicity isa current along the xaxis. If there is also rotation symmetry R
z
about the zaxis [such as the threefold rotation symmetry on the
(111) surface of Bi 2Se3], then no surface helicity-dependent
direct photocurrent is permitted. One needs to break thisrotation symmetry completely by applying, for example, andin-plane magnetic field, strain, etc., to obtain a nonvanishingcurrent.
III. HELICITY-DEPENDENT DIRECT PHOTOCURRENT
We now present our main results for the photocurrent and
estimate it for Bi 2Se3. After painting a simple microscopic
picture for the mechanism, we give a brief outline of the fullquantum-mechanical treatment of the phenomenon.
A. Results
A general two-band Hamiltonian (in the absence of the
incident light) can be written as
H=/summationdisplay
pc†
pHpcp=/summationdisplay
p|Ep|c†
pˆn(p).σcp (3)
up to a term proportional to the identity matrix, which is not
important for our main result, as it involves only interbandtransitions. Here ˆn(p) is a unit vector, σare the spin-Pauli
matrices, and c
p=(cp↑,cp↓)Tis the electron annihilation
operator spinor at momentum p. Clearly, this can capture
a Dirac dispersion, for example, with E(p)=±vFpand
ˆn(p)=vFˆz×p. It can also capture the SSs of Bi 2Se3in
the vicinity of the Dirac point, which includes deviationsbeyond the Dirac limit. We also assume the Hamiltonian has areflection symmetry mabout the yaxis, where ˆzis the surface
normal. Using the zero-temperature quadratic response theorydescribed in Sec. III C , we calculate the current due to the
CPGE and find that
/vectorj
CPGE (t)=[jna+ja1+ja2(t)]ˆx, (4)
where the subscripts a(na) stand for “absorptive” and “non-
absorptive,” respectively. The absorptive part of the responseinvolves a zero-momentum interband transition between a pairof levels separated by energy ¯ hω. These terms are only nonzero
when there is one occupied and one empty level. In thispart of the response, we find a term that is time-dependent,j
a2(t). In particular, this term grows linearly with the time
over which the electromagnetic perturbation is present, whichis allowed for a dc response. In reality, this linear growthis cut off by a decay process that equilibrates populations,and is characterized by a time constant τ. In clean samples
at sufficiently low temperatures, characterized by large τ,t h i s
contribution is expected to dominate the response, and hence itis the focus of our work. The other contributions are discussedin Appendix B. Conversely, because of the linear growth with
time, one can determine the lifetime of the excited states bymeasuring the photocurrent. This term is
j
a2(t)=−πe3¯hE2
0tsgn(ω)
4/summationdisplay
pδ(¯h|ω|−2|Ep|)vx(p)F(p)
(5)
where we have assumed that the chemical potential is in
between the two energy levels ±|Ep|connected by the optical
frequency ¯ hω, and that temperature can be neglected compared
to this energy scale. Here, vx(p)=∂|Ep|
∂pxis the conventional
velocity and F(p)=i/angbracketleft∂pxu(p)|∂pyu(p)/angbracketright+c.c., where |u(p)/angbracketright
is the conduction band Bloch state at momentum p,i st h e
Berry curvature of the conduction band at momentum p.F o r
the class of Hamiltonians ( 3) with which we are concerned,
the Berry curvature is given by (see Appendix A)
F(p)=ˆn·/parenleftbigg∂ˆn
∂px×∂ˆn
∂py/parenrightbigg
, (6)
which is the skyrmion density of the unit vector ˆnin
momentum space. Since ∂piˆn⊥ˆnfori=x,y,F(p)/negationslash=0 only
if all three components of ˆnare nonvanishing. For linearly
dispersing bands, ˆnhas only two nonzero components [e.g.,
Hp=pyσx−pxσy,ˆn∝(py,−px,0)]. Hence, corrections
beyond the pure Dirac dispersion are essential. Also, due tom, the Berry curvature satisfies F(p
x,py)=−F(−px,py).
Since in Eq. ( 5)w eh a v et h e xvelocity multiplying the
Berry curvature, which also transforms the same way, a finitecontribution is obtained upon doing the momentum sum.
We now calculate j
a2(t) for the threefold-symmetric (111)
surface of Bi 2Se3starting from the effective Hamiltonian19,23
H=vF(pxσy−pyσx)+λ
2(p3
++p3
−)σz, (7)
where vF∼5×105m/s( R e f . 6) andλ=50.1e V ˚A3.23A
spin-independent quadratic term has been dropped since itdoes not modify the answers for interband transitions, whichonly involve the energy difference between the bands.
To get a nonzero j
CPGE , the threefold rotational symmetry
must be broken. We propose to do this in two separate ways,which are detailed as follows.
1. Applying a magnetic field B in the x direction
This field has no orbital effect, and can be treated by adding
a Zeeman term
H/prime
Zeeman =−gxμBBσx, (8)
035309-3PA V AN HOSUR PHYSICAL REVIEW B 83, 035309 (2011)
where gxis the appropriate gfactor and μBis the Bohr
magneton, to the Hamiltonian ( 7). To lowest order in λandB,
we get
ja2(t)=3e3vFE2
0λ(gxμBB)2t
16 ¯h2ωA, (9)
where Ais the laser spot size. For gx=0.5,23and assuming
the experiment is done in a 10-T field with a continuous-wavelaser with ¯ hω=0.1 eV , which is less than the bulk band
gap of 0.35 eV ,
13A∼1m m2, a laser power of 1 W, and
the spin relaxation time t∼10 ps, we get a current density
of∼100 nA /mm, which is easily measurable by current
experimental techniques. Note that the expression ( 9)f o rja2(t)
contains the parameter λ, which measures the coupling to σz
in Eq. ( 7). Since /vectorB=Bˆxbreaks the rotation symmetry of
the surface completely, a naive symmetry analysis suggests,wrongly, that deviations from linearity, measured by λ, are not
needed to get j
a2(t).
2. Applying a strain along x
This can be modeled by adding a term
H/prime
strain=δλp3
xσz (10)
toHin Eq. ( 7). This gives
ja2(t)=3e3vF(δλ)E2
0ωt
27A (11)
to lowest order in λandδλ. For a 1% strain, δλ/λ=0.01,
and the same values for the other parameters as in Eq. ( 9), we
get a current density of ∼10 nA/mm. Equation ( 11) does not
contain λ; this is because δλalone both breaks the rotation
symmetry and tips the spins out of the xyplane.
B. Physical process
The appearance of the Berry curvature suggests a role
of the anomalous velocity in generating the current. Suchmechanisms have been discussed in the literature in the contextof the CPGE.
26,28However, those mechanisms only work
when the electric field changes slowly compared to the typicalscattering time. The SSs of Bi
2Se3probably have lifetimes
of tens of picoseconds, and thus we are in the opposite limitwhen ¯ hω=0.1 eV , which corresponds to a time scale 10
3
times shorter.
In this limit, the dc responses are a result of a preferential
absorption of the photon at one of the two momentum points foreach pair of points ( ±p
x,py) related by m, as shown in Fig. 1(a)
forpy=0. According to the surface Hamiltonian ( 7), the spin
vector S=σ
2¯hgets tipped out of the xyplane for states that lie
beyond the linear dispersion regime, but the direction of thetipping is opposite for ( p
x,py) and ( −px,py). Thus, photons
of helicity −1, which can only raise/angbracketleftSz/angbracketrightof an electron, are
preferentially absorbed by the electrons that have /angbracketleftSz/angbracketright<0
in the ground state. The response, then, is determined by theproperties of these electrons. Clearly, the process is helicity-dependent as reversing the helicity would cause electrons with/angbracketleftS
z/angbracketright>0 to absorb the light preferentially.
This is consistent with the requirement of a nonzero Berry
curvature, which essentially amounts to the spin direction ˆn
having to be a three-dimensional vector. In the linear limit,where H=vF(pxσy−pyσx), the spin is entirely in-plane,
and all the electrons absorb the incident light equally.
C. Calculation in brief
We now briefly outline the calculation of the helicity-
dependent photocurrent. The detailed calculation can be foundin Appendix B. Readers only interested in our results may wish
to skip this section.
1. The model
The Hamiltonian and relevant electric field (vector poten-
tial) perturbations for getting a direct current to second orderin the electric field of the incident photon are
H=|E
p|ˆn(p).σ, (12)
H/prime=jxAx(t)+jyAy(t), (13)
jα=∂H
∂pα, (14)
Ax(t)+iAy(t)=A0ei(ω−i/epsilon1)t, (15)
where Ais the vector potential, ˆzis assumed to be the surface
normal, and /epsilon1is a small positive number that ensures slow
switch-on of the light.
2. Quadratic response theory
In general, the current along xto all orders in the
perturbation H/primeis
/angbracketleftjx/angbracketright(t)=/angbracketleftbig
T∗/parenleftbig
ei/integraltextt
−∞dt/primeH/prime(t/prime)/parenrightbig
jx(t)T/parenleftbig
e−i/integraltextt
−∞dt/primeH/prime(t/prime)/parenrightbig/angbracketrightbig
, (16)
where T(T∗) denotes time-ordering (anti-time-ordering) and
O(t)=eiHtOe−iHt. Terms first order in H/primecannot give a
direct current. The contribution to the current from the second-order terms can be written as
/angbracketleftj
x/angbracketright(t)=/integraldisplayt
−∞dt/prime/integraldisplayt1
−∞dt/prime/prime/angbracketleft[[jx(t),H/prime(t/prime)],H/prime(t/prime/prime)]/angbracketright
=/integraldisplayt
−∞dt/prime/integraldisplayt1
−∞dt/prime/primeχxαβ(t,t/prime,t/prime/prime)Aα(t/prime)Aβ(t/prime/prime),(17)
where α,β∈{x,y},χxαβ(t,t/prime,t/prime/prime)=χxαβ(0,t/prime−t,t/prime/prime−t)=
/angbracketleft[[jx,jα(t/prime−t)],jβ(t/prime/prime−t)]/angbracketright≡χxαβ(t/prime−t,t/prime/prime−t) due to time
translational invariance, and the expectation value is overthe ground state, which has all states with E
p<(>) 0 filled
(empty). For Hamiltonians of the form of Eq. ( 12), the
expectation value of any traceless operator Oin the Fermi
sea ground state can be written as a trace,
/angbracketleftO/angbracketright=/summationdisplay
p1
2Tr/braceleftbigg/parenleftbigg
1−H
|Ep|/parenrightbigg
O/bracerightbigg
=−/summationdisplay
pTr(HO)
2|Ep|.(18)
This gives
χxαβ(t1,t2)=−/summationdisplay
pTr(H[[jx,jα(t1)],jβ(t2)])
2|Ep|. (19)
Equation ( 19) is the zero-temperature limit of the finite-
temperature expression for the quadratic susceptibility provenin Ref. 29.
035309-4CIRCULAR PHOTOGALV ANIC EFFECT ON TOPOLOGICAL ... PHYSICAL REVIEW B 83, 035309 (2011)
Because of the mirror symmetry m,χxαβ(t1,t2) is non-
vanishing only for α/negationslash=β. To get a direct current, we
retain only the nonoscillating part of Ax(t+ti)Ay(t+tj)=
A2
0
2e2/epsilon1t{sin[2ωt+ω(ti+tj)]−sin[ω(ti−tj)]}. Thus,
jdc
x(t)=A2
0e2/epsilon1t
4/integraldisplay0
−∞dt1/integraldisplayt1
−∞dt2{(χxxy−χxyx)(t1,t2)e/epsilon1(t1+t2)
×sinω(t2−t1)]}. (20)
3. The result
After carrying out the two time integrals, we get the
three currents mentioned in Eq. ( 4) .F o rc l e a ns a m p l e sa t
low temperatures, ja2(t), which grows linearly with time, is
expected to dominate. A general expression for this term is (inthe units e=¯h=v
F=1, where vFis the Fermi velocity)
ja2(t)=iA2
0πtsgn(ω)
2ω2/summationdisplay
pδ(|ω|−2|Ep|)
×Tr(Hjx)Tr(H[jx,jy]). (21)
Using Eqs. ( 12) and ( 14) and the Lie algebra of the Pauli
matrices, [ σi,σj]=2i/epsilon1ijkσk, where /epsilon1ijkis the antisymmetric
tensor, the above traces can be written as
Tr(Hjx)=2|Ep|vx(p), (22)
Tr(H[jx,jy])=4i|Ep|3ˆn·/parenleftbigg∂ˆn
∂px×∂ˆn
∂py/parenrightbigg
=4i|Ep|3F(p). (23)
Equations ( 21), (22), and ( 23) give our main result, Eq. ( 5).
IV . OPTICAL SPIN INJECTION
Having understood the microscopic mechanism underlying
the generation of the photocurrent ja2(t), we wonder, next,
whether such a population imbalance can lead to any otherhelicity-dependent macroscopic responses. Since each ab-sorbed photon flips the zcomponent of the spin of an electron,
a net/angbracketleftS
z/angbracketrightis expected to be generated on the surface. Such a
process of optical spin injection was discussed for thin filmsof topological insulators,
27without, however, recognizing the
role of the Berry curvature in the interband transition.
The calculation of /angbracketleftSz/angbracketrightis identical to that of jCPGE .T h e
total/angbracketleftSz/angbracketrightgenerated consists of the same three parts as jCPGE ,
and the dominant part is
Sz
a2(t)=−πe2E2
0¯htsgn(ω)
8/summationdisplay
pδ(¯h|ω|−2|Ep|)nz(p)F(p).
(24)
Szdoes not break the rotational symmetry of the surface,
so we calculate Sz
a2(t) directly for the threefold-symmetric
Hamiltonian ( 7) and obtain
Sz
a2(t)=e2E2
0(¯hω)3λ2t
210A. (25)
For the same values of all the parameters as for ja2(t),
we get Sz
a2(t)∼10¯h, which means only ten electron spins
are flipped over an area of ∼1m m2. This is a very smallnumber and cannot be measured by the current experimental
techniques. However, the result that the dominant spin injectedonto the surface is also controlled by the Berry curvatureis still theoretically interesting, as it points toward a deeperconnection between the Berry curvature of electron bands andthe helicity-dependent dc responses of systems with strongspin-orbit-coupled coupling.
V . CPGE AT OBLIQUE INCIDENCE
Experimentally, a very attractive way of breaking the rota-
tional symmetry of the surface is by performing the experimentwith obliquely incident light. Indeed, such experiments havealready been performed successfully on graphene at lowfrequencies.
30At the microscopic level, the effect there has
been attributed to photon drag, where the current arises as aresult of the in-plane component of the photon momentum q
/bardbl
getting transferred to the electrons in graphene. In general,
an analogous process is expected to contribute to the CPGEat high frequencies as well. We can estimate the size ofthe photon-drag effect on TI surfaces in the Dirac limit byconsidering a mechanism similar to the one described inSec. III B , that is, the electrons at ( ±p
x,py) absorb the incident
light unequally if the light is incident in the yzplane. Now,
no out-of-plane tipping of the spin is needed, because, ifone thinks of the helical photon as simply a spin-raisingor -lowering operator for spins parallel to its propagationdirection ˆz
/prime, the electrons at ( ±px,py) already have opposite
/angbracketleftSz/prime/angbracketright,and hence will absorb the light unequally. Thus, the
general expression for the current may contain only thosematerial parameters that appear in the pure Dirac dispersion.As before, it must be quadratic in the photon electric field,and must change sign when q
/bardblandωare both reversed,
since that corresponds to switching the photon helicity.Thus, to lowest order in q
/bardbl, the linear-in-time current, based
simply on symmetry and dimensional analysis, must be ofthe form
/vectorj
photon drag (t)∼e3E2
0v2
Fq/bardblt
¯h2ω2Aˆx. (26)
Forq/bardbl=c/ω,cbeing the speed of light in vacuum, and the
same values for all the other parameters as in Sec. III A ,w e
get a current density of ∼1μA/mm. This will dominate the
response at off-normal incidence, but can be suppressed bycareful alignment of the experimental setup. However, since aresponse might appear even in the pure Dirac limit in whichthe Berry curvature vanishes, the role of the Berry curvatureis not clear for this process.
In graphene, helicity-dependent direct photocurrents have
also been predicted by applying a dc bias.
31However, with a
dc bias across a TI surface and ordinary continuous lasers, wefind the current to be too low to be measurable.
VI. CONCLUSIONS
In summary, we studied the CPGE on the surface of a
TI at normal incidence, and applied the results to the (111)surface of Bi
2Se3. If the rotational symmetry of the TI surface
035309-5PA V AN HOSUR PHYSICAL REVIEW B 83, 035309 (2011)
is broken by applying an in-plane magnetic field or a strain,
we predict an experimentally measurable direct photocurrent.A striking feature of this current is that it depends on theBerry curvature of the electron bands. Such a dependence canbe understood intuitively as a result of the incident photonsgetting absorbed unequally by electrons of different momenta,and hence different average spins. The current grows linearlywith time until a decay process equilibrates populations, whichprovides a way of determining the excited-states lifetime.We also calculated the amount of dc helicity-dependentout-of-plane component of the electron spin generated. Thisdoes not require any rotational symmetry breaking; however,the numerical value is rather small with typical values ofthe parameters. Finally, we estimated the size of the CPGEdue to the photon-drag effect at oblique incidence assuminga differential absorption mechanism similar to the one dis-cussed for normal incidence, and found a rather large value.However, the role of the Berry curvature in this process wasunclear.
For future work, we wonder whether the Berry curvature
dependence of the helicity-dependent response to CP light sur-vives for three- and higher-band models. This is a practicallyrelevant question, as semiconductor quantum wells such asthose of GaAs, SiGe, and HgTe /CdHgTe demand a four-band
model for modeling the CPGE.
ACKNOWLEDGMENTS
We would like to thank Ashvin Vishwanath for enlightening
discussions, Joseph Orenstein for useful experimental inputs,and Ashvin Vishwanath and Yi Zhang for invaluable feedbackon the draft. This work was supported by the Office ofBasic Energy Sciences, Materials Sciences Division of theUS Department of Energy under Contract No. DE-AC02-05CH1123.
APPENDIX A: PROOF OF BERRY CURVATURE
EXPRESSION
Here we show that the Berry curvature defined for Bloch
electrons as
F(p)=i(/angbracketleft∂pxu|∂pyu/angbracketright−/angbracketleft∂pyu|∂pxu/angbracketright)( A 1 )
can be written as
F(p)=ˆn·(∂pxˆn×∂pyˆn)( A 2 )
for the band with energy |Ep|for Hamiltonians of the form
Hp=|Ep|ˆn(p)·σ.
At momentum p, the Bloch state |up/angbracketrightwith energy |Ep|is
defined as the state whose spin is along ˆn(p). Defining |↑/angbracketrightas
the state whose spin is along +ˆz,|up/angbracketrightis obtained by performing
the appropriate rotations,
|up/angbracketright=e−iσz
2φ(p)eiσy
2θ(p)|↑/angbracketright, (A3)
where θ(p) andφ(p) are the polar angles that define ˆn(p):
ˆn(p)=sinθ(p) cosφ(p)ˆx+sinθ(p)s i nφ(p)ˆy+cosθ(p)ˆz.
(A4)Substituting Eq. ( A3)i nE q .( A1), one gets
F(p)=sinθ(p)[∂pxθ(p)∂pyφ(p)−∂pxφ(p)∂pyθ(p)],(A5)
which, on using Eq. ( A4) and some algebra, reduces to the
required expression Eq. ( A2).
APPENDIX B: CURRENT CALCULATION FOR
THE CPGE
Here we explain the current calculation of Sec. III A in more
detail and also state results for the parts of the current that wechose not to focus on there.
As shown in Sec. III C , the relevant susceptibility is
χ
xαβ(t,t/prime,t/prime/prime)=−1
2/summationdisplay
pTr/parenleftbiggH
|Ep|[[jx(t),jα(t/prime)],jβ(t/prime/prime)]/parenrightbigg
=−/summationdisplay
p1
2|Ep|Tr(H[[jx,jα(t1)],jβ(t2)])
≡χxαβ(t1,t2), (B1)
where t1=t/prime−t,t2=t/prime/prime−t, and the nonvanishing compo-
nents of χxαβare those for which α/negationslash=β. The nonoscillating
part of the current, hence, is
/angbracketleftbig
jdc
x/angbracketrightbig
(t)=jCPGE (t)=A2
0e2/epsilon1t
4/integraldisplay0
−∞dt1/integraldisplayt1
−∞dt2[χxxy(t1,t2)
−χxyx(t1,t2)]e/epsilon1(t1+t2)sin[ω(t2−t1)]. (B2)
SincejCPGE (t) is an odd function of ω, it reverses on reversing
the polarization, as expected.
The traces in the susceptibility expressions are calculated
by introducing a complete set of states in place of the identityseveral times. Thus,
χ
xxy(t1,t2)=−/summationdisplay
p1
2|Ep|Tr(H[[jx,jx(t1)],jy(t2)])
=−1
2/summationdisplay
p/summationdisplay
nmlsgn(En){ei(Em−En)t2
×(ei(El−Em)t1−e−i(El−En)t1)XnlXlmYmn+c.c.},
(B3)
where Xnl=/angbracketleftn|jx|m/angbracketright, etc., and the subscript ponEphas
been dropped to enhance the readability. Similarly,
χxyx(t1,t2)=−/summationdisplay
p1
2EpTr(H[[jx,jy(t1)],jx(t2)])
=−1
2/summationdisplay
p/summationdisplay
nmlsgn(En){ei(Em−En)t2Xmn
×(ei(El−Em)t1XnlYlm−e−i(El−En)t1YnlXlm)
+c.c.}. (B4)
Substituting ( B3) and ( B4)i n( 20), we get
jCPGE (t)=A2
0e2/epsilon1t
4Re/integraldisplay0
−∞dt1/integraldisplayt1
−∞dt2e/epsilon1(t1+t2)
×sin[ω(t1−t2)]/summationdisplay
p,nmlsgn(En)ei(Em−En)t2
035309-6CIRCULAR PHOTOGALV ANIC EFFECT ON TOPOLOGICAL ... PHYSICAL REVIEW B 83, 035309 (2011)
×{(ei(El−Em)t1−e−i(El−En)t1)XnlXlmYmn
−Xmn(ei(El−Em)t1XnlYlm−e−i(El−En)t1YnlXlm)},
(B5)
where Re denotes the real part. Carrying out the the two time
integrations gives
jCPGE (t)=A2
0e2/epsilon1t
8Im/summationdisplay
p/summationdisplay
nmlsgn(En)
×/parenleftbigg1
Em−En+ω−i/epsilon1−1
Em−En−ω−i/epsilon1/parenrightbigg
×/braceleftbiggXnl(XlmYmn−YlmXmn)
El−En−2i/epsilon1
+Xlm(YmnXnl−XmnYnl)
El−Em+2i/epsilon1/bracerightbigg
, (B6)
where Im stands for the imaginary part. Using Im(1
/Omega1−i/epsilon1)=
πδ(/Omega1) and Re(1
/Omega1−i/epsilon1)=1
/Omega1in the limit /epsilon1→0, we get,after some algebra, jCPGE (t)=jna+ja1+ja2(t), where (Tr
denotes the trace)
jna=A2
0
16/summationdisplay
pω/parenleftbig
ω2−12E2
p/parenrightbig
i|Ep|3/parenleftbig
ω2−4E2p/parenrightbig2Tr(Hjx)Tr(H[jx,jy]) (B7)
comes from intraband processes and is constant in time,
ja1=−πA2
0sgn(ω)
32/summationdisplay
pδ(|ω|−2|Ep|)
E2p
×Tr(H[jx,[jx,jy]])( B 8 )
is a result of an interband transition absorption as indicated by
theδfunction in energy and is also constant in time, and
ja2(t)=iA2
0πtsgn(ω)
8/summationdisplay
pδ(|ω|−2|Ep|)
×Tr(Hjx)Tr(H[jx,jy])
E2p, (B9)
which also results from interband absorption and increases
linearly in time. The last term was the main focus of our work.
1A. M. Essin, J. E. Moore, and D. Vanderbilt, P h y s .R e v .L e t t . 102,
146805 (2009).
2X.-L. Qi, R. Li, J. Zang, and S. C. Zhang, Science 323, 1184
(2009).
3L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407
(2008).
4G. P. Collins, Sci. Am. 294, 57 (2006).
5L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).
6H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang,
Nat. Phys. 5, 438 (2009).
7B. Yan, C.-X. Liu, H.-J. Zhang, C.-Y . Yam, X.-L. Qi, T. Frauenheim,
and S.-C. Zhang, Europhys. Lett. 90, 37002 (2010).
8S. Chadov, X.-L. Qi, J. K ¨ubler, G. H. Fecher, C. Felser, and S.-C.
Zhang, Nat. Mater. 9, 541 (2010).
9B. Yan, H.-J. Zhang, C.-X. Liu, X.-L. Qi, T. Frauenheim, and S.-C.
Zhang, Phys. Rev. B 82, 161108(R) (2010).
10P. Roushan, J. Seo, C. V . Parker, Y . S. Hor, D. Hsieh, D. Qian,
A. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani, Nature
(London) 460, 1106 (2009).
11Y . 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).
12D. Hsieh, D. Qian, L. Wray, Y . Xia, Y . Hor, R. J. Cava, and M. Z.
Hasan, Nature (London) 452, 970 (2008).
13Y . 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, Nat. Phys. 5,
398 (2009).
14H. Lin, L. A. Wray, Y . Xia, S. Jia, R. J. Cava, A. Bansil, and M. Z.Hasan, Nat. Mater. 9, 546 (2010).
15H. Lin, R. S. Markiewicz, L. A. Wray, L. Fu, M. Z. Hasan, and
A. Bansil, e-print arXiv:1003.2615 .
16T. Sato, K. Segawa, H. Guo, K. Sugawara, S. Souma, T. Takahashi,
and Y . Ando, Phys. Rev. Lett. 105, 136802 (2010).
17F. D. M. Haldane, P h y s .R e v .L e t t . 93, 206602 (2004).18D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. denNijs,
Phys. Rev. Lett. 49, 405 (1982).
19L. Fu, P h y s .R e v .L e t t . 103, 266801 (2009).
20S. D. Ganichev, E. L. Ivchenko, S. N. Danilov, J. Eroms,
W. Wegscheider, D. Weiss, and W. Prettl, Phys. Rev. Lett. 86, 4358
(2001).
21S. D. Ganichev, F. P. Kalz, U. R ¨ossler, W. Prettl, E. L. Ivchenko,
V . V . Bel’kov, R. Neumann, K. Brunner, and G. Abstreiter, Mater.Res. Soc. Symp. Proc. 690, F3.11.1 (2002).
22B. Wittmann, S. N. Danilov, V . V . Bel’kov, S. A. Tarasenko, E. G.
Novik, H. Buhmann, C. Brune, L. W. Molenkamp, Z. D. Kvon,N. N. Mikhailov, S. A. Dvoretsky, N. Q. Vinh, A. F. G. van derMeer, B. Murdin, and S. D. Ganichev, Semicond. Sci. Technol. 25,
095005 (2010).
23C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang,Phys. Rev. B 82, 045122 (2010).
24D. 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).
25L. Munoz, E. Perez, L. Vina, and K. Ploog, Phys. Rev. B 51, 4247
(1995).
26E. Deyo, L. E. Golub, E. L. Ivchenko, and B. Spivak, e-printarXiv:0904.1917 .
27H. Z. Lu, W. Y . Shan, W. Yao, Q. Niu, and S. Q. Shen, Phys. Rev.
B81, 115407 (2010).
28J. E. Moore and J. Orenstein, Phys. Rev. Lett. 105, 026805 (2010).
29Paul N. Butcher, Nonlinear Optical Phenomena (Engineering
Experiment Station, Ohio State University, Columbus, 1965),Chap. 7, Eq. 7.25.
30J. Karch, P. Olbrich, M. Schmalzbauer, C. Brinsteiner, U. Wurst-bauer, M. M. Glazov, S. A. Tarasenko, E. L. Ivchenko, D. Weiss,J. Eroms, and S. D. Ganichev, e-print arXiv:1002.1047 .
31T. Oka and H. Aoki, Phys. Rev. B 79, 081406(R) (2009).
035309-7 |
PhysRevB.85.094509.pdf | PHYSICAL REVIEW B 85, 094509 (2012)
Three-dimensional electronic structure and interband nesting in the stoichiometric
superconductor LiFeAs
T. Hajiri,1,2,*T. Ito,1,3R. Niwa,1M. Matsunami,2,4B. H. Min,5Y . S. Kwon,5and S. Kimura2,4,†
1Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan
2UVSOR Facility, Institute for Molecular Science, Okazaki 444-8585, Japan
3Nagoya University Synchrotron Radiation Research Center, Nagoya University, Nagoya 464-8603, Japan
4School of Physical Sciences, The Graduate University for Advanced Studies (SOKENDAI), Okazaki 444-8585, Japan
5Department of Emerging Materials Science, DGIST, Daegu 711-873, Republic of Korea
(Received 20 December 2011; revised manuscript received 20 February 2012; published 19 March 2012)
We report the three-dimensional electronic structure of iron pnictide superconductor LiFeAs obtained by
polarization-dependent angle-resolved photoemission spectroscopy. The obtained orbital characters of eachFermi surface (FS) as well as the band dispersions were qualitatively consistent with those derived from localdensity approximation band calculations. It was found that FS nesting appears between a two-dimensional holeband at the zone center and an electron band at the zone corner with the same d
xyorbital character. A shadow
band attributed to ( π,π,π ) band folding was also observed without the spin-density-wave transition. This result
suggests that FS nesting between bands with the same orbital dxycharacter due to spin fluctuation plays an
important role in LiFeAs.
DOI: 10.1103/PhysRevB.85.094509 PACS number(s): 74 .25.Jb, 74.70.Xa, 79 .60.−i
I. INTRODUCTION
Recently discovered iron pnictide superconductors1have
two-dimensional (2D) Fe-As layers that are similar to theCu-O planes in high- T
csuperconducting cuprates. In high- Tc
cuprates, 2D magnetic interaction is important for the origin
of the high Tcbecause of their 2D electronic structure. Such
2D interaction as well as the 2D nesting condition in ironpnictides has long been a focus of discussions.
2,3However, the
crystal structures of iron pnictides are more three dimensional(3D) than those of cuprates. It is therefore important toclarify the 3D electronic structure of iron pnictides in order tounderstand the effective interaction of Cooper pair formation.Recently, Yoshida and co-workers pointed out the importanceof the 3D nesting condition, as well as its orbital characterin BaFe
2(As 1−xPx)2(122 type).4However, there is no further
evidence for the 3D nesting condition in other iron pnictideswith different crystal structures, such as LiFeAs (111 type)and others.
LiFeAs is an ideal system for investigation of unconven-
tional superconductivity as well as the orbital characters ofmulti-Fermi surfaces, because the undoped LiFeAs showsa superconductivity below the critical temperature T
c=
18 K without the antiferromagnetic (AFM)/spin-density-wave(SDW) and structural transitions, which is different from otheriron pnictides such as the 122 type. So far, a scenario forthe high T
cof the 122 type due to spin fluctuation5as well
as a SDW6has been proposed. The same superconducting
mechanism is expected in the case of LiFeAs. An AFMwave vector similar to that of other iron pnictides has, infact, been observed in polycrystalline LiFeAs by neutronscattering.
5The fundamental electronic structure has never
been clarified, however. For example, only two and one holepockets at the /Gamma1point were observed in angle-resolved pho-
toemission spectroscopy (ARPES)
7and de Haas–van Alphen
(dHvA)8experiments, respectively, despite the expectation
that there would be three hole pockets as a result of a bandcalculation.In this paper, we report the electronic structure as well as the
orbital characters of LiFeAs using polarization-dependent 3DARPES. The obtained band dispersions and orbital charactersare qualitatively in good agreement with those derived fromlocal density approximation (LDA) band calculations. Consid-ering a 3D nesting condition, we find that each 2D hole andelectron Fermi surface (FS) of d
xyorbital character is weakly
nested. This weak nesting suggests that ( π,π,π ) interband
scattering is important for the superconducting behavior ofLiFeAs.
II. EXPERIMENT
The 3D-ARPES experiments were performed on single
crystals of stoichiometry LiFeAs with a Tconset value of
19.7 K.9ARPES measurements were carried out at the
“SAMRAI” end station of the undulator beamline 7U ofUVSOR-II at the Institute for Molecular Science.
10In this
system, two linearly polarized lights perpendicular and parallelto the mirror plane (namely, SandPpolarizations), shown in
Fig. 1(a), can be irradiated to a sample without changing the
sample position. The total energy and angular resolution wereset at 6–15 meV and about 0 .17
◦, respectively. All of the
measurements were performed using in situ cleaved samples
atT=12 K in an ultrahigh vacuum of about 8 ×10−9Pa. The
photon energies corresponding to high-symmetry kzpoints
were determined using normal-emission ARPES. The innerpotential was determined to be 15.4 eV . The Fermi level ( E
F)
of the samples was calibrated with reference to that of gold.The obtained ARPES spectra were compared with an LDAband-structure calculation with spin-orbit coupling using the
WIEN2K code.11Figure 1(b) shows the calculated 3D FS, which
is consistent with the previous calculation.12
III. MATRIX ELEMENT CALCULATIONS
When linearly polarized light is used, the orbital symmetry
of electronic states can be determined using the dipole
094509-1 1098-0121/2012/85(9)/094509(6) ©2012 American Physical SocietyT. HAJIRI et al. PHYSICAL REVIEW B 85, 094509 (2012)
hν
Samplexyy
SS
Py
x
y
x
y
x
y
xy
xy
xy
xy
xy
xx
zEES
Analyzer slit50°
Mirror planedxzdx2-y2dyz
SP
Pdx2-y2 dxy(odd)
(odd)(even)
dxz dyzdxy
dz2dz2
&
(even)
30
SampleAnalyzerAnalyzer
Sample
°(a)
(b)(c)
(d)
xy
zxy
ΓZ
MAz
EP
FIG. 1. (Color online) (a) Experimental geometry of the polarization-dependent ARPES. (b) Calculated 3D FS. (c),(d) Initial-state electronic
orbital excited by polarized light at normal emission (zone center) and φ=30◦(zone boundary), respectively. The spatial symmetry of the 3 d
orbitals was oriented with respect to the mirror plane, and the orbitals were selectively excited with each polarization ( SorP) under normal
(c) and 30◦emission (d) configurations. The mirror plane including the direction of incidence and emission was defined along the xaxis, as
shown by the bold lines.
selection rule. The photoemission intensity can be expressed
asI∝/summationtext
i|/angbracketleftf|A·p|i/angbracketright|2δ(Ef−Ei−hν), where Aandp
are the vector potential of the electromagnetic field and themomentum operator, respectively, and |f/angbracketrightand|i/angbracketrightare the final-
and initial-state wave functions. In the case of normal emission,since the final state |f/angbracketrighthas an even symmetry with respect to
the mirror plane,
13,14the nonvanishing condition of the dipole
transition /angbracketleftf|A·p|i/angbracketrightis that the initial state |i/angbracketrightmust have the
same symmetry (even/odd) as the dipole operator A·p.F o r
example, if A·pis even (odd) corresponding to the P(S)
polarization, then the initial states with even (odd) symmetryshould be reflected in the ARPES spectra. In Table I,w e
summarize the polarization-dependent sensitivity for the Fe 3 d
orbitals along two high-symmetry directions, /Gamma1-Mand/Gamma1-X
lines. To extend the analysis of the polarization-dependentARPES for the sample rotation φaround the xaxis in
Fig. 1(a), we evaluated the angular dependence of the matrix
element using experimental geometry.
15For our experimental
geometry, the angle between the incident synchrotron radiationand the electron analyzer in the mirror plane is 50
◦[see the
TABLE I. The possibility to detect 3 dorbitals along /Gamma1-Mand
/Gamma1-Xhigh-symmetry directions.
Pol. dxydyzdxzdx2−y2dz2dxz+dyz√
2dxz−dyz√
2
/Gamma1-MS /circlecopyrt/circlecopyrt /circlecopyrt /circlecopyrt
P /circlecopyrt/circlecopyrt/circlecopyrt/circlecopyrt /circlecopyrt
/Gamma1-XS /circlecopyrt/circlecopyrt /circlecopyrt /circlecopyrt
P /circlecopyrt/circlecopyrt/circlecopyrt /circlecopyrt /circlecopyrtgeometry in Fig. 1(a) for details]. In Figs. 1(c) and 1(d),
we summarize the polarization-dependent sensitivity for theFe 3dorbitals at the zone center ( /Gamma1andZpoints; normal
emission) and at the zone corner ( MandApoints; φ≈30
◦),
respectively.
IV . RESULTS AND DISCUSSION
To clarify the band character, polarization-dependent 3D-
ARPES images were accumulated near high-symmetry points[Figs. 2(a) and 2(b)]. In each figure, the left (right) image
represents the energy bands excited by the S(P) polarized
light. We found that the FSs of LiFeAs mainly consist of threeand two hole pockets at the /Gamma1andZpoints, respectively, and
two electron pockets at the MandApoints. In all of the
images, the prominent features as indicated by the solid anddotted curves clearly change in a manner that is dependent onthe polarization. For the Spolarization in the left-hand panels,
two hole pockets (solid lines) are clearly visible around the /Gamma1
andZpoints, as well as an electron pocket with its bottom at
about 100 meV at the Mpoint and an electron pocket with its
bottom at about 40 meV at the Apoint. On the other hand,
for the Ppolarization in the right-hand panels, two (one) hole
pockets (solid lines) appear at the /Gamma1(Z) point as well as one
(two) electron pocket(s) with their bottom at about 40 meV attheMpoint (at about 40 and 100 meV at the Apoint).
To check the orbital character of the observed bands,
the experimental band dispersions compared with the bandcalculation renormalized by a factor of 1.6 were plotted asshown in Fig. 2(c). The renormalization factor was determined
by the fitting of the whole of the valence band; specifically
094509-2THREE-DIMENSIONAL ELECTRONIC STRUCTURE AND ... PHYSICAL REVIEW B 85, 094509 (2012)
|k////| (π / a) / a)
S pol. pol. S pol. pol. S pol. pol.
|k////| (π / a) / a)(b)(b)Γ Z M A0
200200100100P pol. pol. P pol. pol. P pol. pol. P pol. pol. S pol. pol.Binding Energy (meV)Binding Energy (meV) Binding Energy (meV)Binding Energy (meV)(a)(a)
Binding Energy (meV)Binding Energy (meV)(c)(c)Γ Z M A
AMZΓ0
2002001001000
100100
0
100100
0
100100
0
100100dxy/x2-y2
dxy/x2-y2P pol. pol. P pol. pol. P pol. pol. P pol. pol. S pol. pol.
hν = 2323 eVeV hν = 3535 eVeV hν = 3939 eVeV hν = 2626 eVeV hν = 1818 eVeVkz = 0 kz = π kz = π kz = 0 kz = -πS pol. pol. S pol. pol. S pol. pol. dyz
dyz
dxy dxz/yz
dxy dxz/yzdxz
0 0.50.5 0.50.5 0.50.5 0.50.5 0.50.5 0.50.5 0.50.5 0.50.5 0 0 0
0 0.50.5 0.50.5 0.50.5 0.50.5 0.50.5 0.50.5 0.50.5 0.50.5 0 0 0 0.50.5 0.50.5 0
FIG. 2. (Color online) (a) Polarization-dependent ARPES images at high-symmetry points. The left- and right-hand images in each panel
were obtained using SandPpolarized light, respectively. (b) Same as (a) but momentum-distribution curve’s (MDC’s) second-derivative
images. (c) Experimentally obtained band dispersions (bold solid lines) and their orbital characters. The calculated band dispersions (thin solidlines) renormalized by 1.6 are also plotted. See the text for details.
we made the best match of the hole bands with the top at
about 0.3 eV at the /Gamma1andZpoints. At the /Gamma1point ( φ=0◦),
three hole pockets are observed; the middle-sized (smallest)hole pocket is observed only with the S(P) polarization, while
the largest hole pocket is observed with both polarizations. Themiddle-sized hole pocket observed with the Spolarization
can be attributed to the d
yzorbital because the intensity of
thedyzorbital is expected to be larger than that of the dxy
orbital on the basis of the matrix element calculation.15In
the same manner, since the intensity of the dx2−y2orbital is
also smaller than that of the dxzorbital in the Ppolarization,
the smallest hole pocket can be attributed to the dxzorbital.
The outer hole pocket is attributed to a mixture of the dxy
orbital band with the Spolarization and the dx2−y2orbital band
with the Ppolarization. At the Zpoint, in a similar manner,
the middle-sized hole pocket is the dyzorbital and the outer
band observed in both polarizations is the combination of thed
xy/x2−y2orbitals. It should be noted that the middle-sized hole
pocket mainly observed with the Spolarization appears only
nearEFwhen the Ppolarized light is used. This suggests
that the tail of the peak (might be a quasiparticle peak)above E
F, which has the different orbital character from the
middle-sized hole pocket, is observed with the Ppolarization.
These band characters are consistent with those of the LDAband calculation. It should be noted that our ARPES datashow three hole pockets across E
Fat the /Gamma1point, which is
a different result from the two hole pockets obtained by theprevious ARPES studies.
7,16
For the MandApoints, the electron band observed with
theSpolarization (outer band) can be attributed to the dxy
orbital as shown in Figs. 1(c) and 1(d). On the other hand,
since the inner band is observed only when using the P
polarized light, it can be attributed to the dxzordyzorbitals.These band characters are also consistent with the band
calculation.
The outermost hole pockets ( dxy/x2−y2orbital) near the
/Gamma1andZpoints and the innermost pockets ( dxy) near the /Gamma1
point observed with the Ppolarization are in good agreement
with the renormalized band calculation. However, the wavevector of the middle-sized hole band across E
Fobserved
with the Spolarization is smaller than the calculated value.
Moreover, the bottoms of the electron pockets at the M
andApoints are located at lower binding energies than
those in the band calculation. These results suggest thatthe band (orbital)-dependent renormalization effect with thelarger renormalization factor should be included. Actually, theband-dependent renormalization factor ∼4 is suggested by a
previous ARPES study on the 122 type.
17
To clarify the orbital chararacter in detail, we measured the
ARPES image along the /Gamma1-Xdirection with the Ppolarization
in Fig. 3.A tt h e /Gamma1point, three hole pockets are observed.
Based on Table I, the outermost hole pocket can be attributed
to the dxyorbital because it shows two dimensionally (shown
later). Along the /Gamma1-Xdirection, since thedxz+dyz√
2anddxz−dyz√
2
orbitals have even and odd symmetries, respectively, both of
the inner and middle-sized hole pockets can be attributed to be
thedxz+dyz√
2orbital. If both of the inner and middle-sized hole
pockets are thedxz+dyz√
2orbital, they can be observed along the
/Gamma1-Mdirection with both polarizations in Figs. 2(a) and2(b) as
shown in Table I. But the inner and middle-sized hole pockets
are only observed by the PandSpolarizations, respectively.
Therefore the inner and middle-sized hole pockets can beattributed to be the d
xzanddyzorbitals, respectively.
For the Mpoint, the two electron pockets with their bottom
at about 40 and 100 meV are observed. The energy of these
094509-3T. HAJIRI et al. PHYSICAL REVIEW B 85, 094509 (2012)
00 1 1 00
200200100100Binding Energy (meV)Binding Ener gy (meV)
|k|k////|(π / a)/a )1 1Γ X X M X X
(hν=23eV)eV) (hν=2626eV)eV)k = 0 =0 k = 0=0
FIG. 3. (Color online) ARPES images near the /Gamma1andMpoints
along/Gamma1-Xdirection with the Ppolarization. The left- and right-hand
images in each panel are the intensity mapping and the MDC second-
derivative image, respectively. The solid lines are the prominent
features.
bottoms are consistent with those along the /Gamma1-Mdirection in
Fig. 2. Similar to the /Gamma1point, since the deeper electron pocket
is quasi-two dimensional (shown later), it can be attributed tothed
xyorbital. On the other hand, the deeper electron pocket is
considered to be thedxz+dyz√
2orbital. However, since the deeper
electron pocket is observed only by the Spolarization along
the/Gamma1-Mdirection in Figs. 2(a) and2(b), it can be attributed to
thedxzordyzorbitals.
Next, the 3D FS as well as the band dispersions are
discussed. Figure 4shows the photon energy dependence
of the FSs along the kzdirection; Spolarization was used
in the /Gamma1-Zplane to emphasize the dyzorbital hole pocket
that is strongly renormalized from the band calculation, andPpolarization in the M-Aplane to observe all electron
pockets. The orbital characters of the FSs determined by thepolarization-dependent ARPES are denoted. For the /Gamma1-Zline,
the smallest FS is not visible because the d
xyorbital cannot be
observed using the Spolarization. The dyzorbital hole FS near
the/Gamma1-Zline (dyzsolid line) has weak kzdependence and the
Fermi wave vector or the size of the FS is much smaller thanthat of the band calculation ( d
yzdotted line). The outermost
hole FS near the /Gamma1-Zline (dxy/x2−y2solid line), however, is
almost flat. On the other hand, near the M-Aline, the dxz/yz
ΓΓ MZ AS pol. P pol.
dxy/x2-y2 (h)
dyz(h)
dxz(h)
dxy(e)dxz/yz(e)
FIG. 4. (Color online) Out-of-plane FS at the /Gamma1MAZ plane
obtained by Spolarization ( /Gamma1-Zline) and Ppolarization ( M-Aline).
The experimentally obtained and calculated FSs are depicted by solid
and dashed lines, respectively.
0.8
0.80.4
0.40
0Binding Ene rgy (eV)
)Ve(
ygrenE
gnidniBΓΓ MΓ M
Z AZ (a)
(b)(c)
(d)A
Z A A
FIG. 5. (Color online) (a) ARPES spectra at the Zpoint. The solid
circles are values that are expected and the open squares are those
that are not expected by the band calculation. (b) ARPES image with
the band calculation at the Zpoint. (c) Schematic band dispersions
obtained from the experiments. The open squares used in (c) have the
same meaning as in (a) and (b). (d) Schematic figure of the 3D FS
nesting conditions. The solid and dashed lines indicate the originaland nested FSs, respectively. The bold arrow indicates the expected
nesting wave vector of ( π,π,π ).
orbital electron FS ( dxz/yz solid line) is very wavy or 3D like.
The size of the dxyorbital electron FS ( dxysolid line) at the
Mpoint is slightly larger than that of the band calculation ( dxy
dashed line).
A possible band nesting without magnetic ordering is
discussed. Figures 5(a) and 5(b) show the ARPES spectra
near the Zpoint in a wide energy range measured using
thePpolarized light. The calculated band dispersions with
the renormalization factor of 1.6 are also plotted. Besides theband dispersion corresponding to the calculated value [solidcircles in Fig. 5(a)], an undefined band dispersion denoted
by open squares can be observed. As shown in the schematicband dispersions in Fig. 5(c), the undefined band dispersion is
very similar to the band at the Mpoint. This finding strongly
suggests that the band dispersion at the Mpoint appears at the
Zpoint through band folding with the ( π,π,π ) wave vector
from the MtoZpoints.
Band folding with the same ( π,π,π ) wave vector has
been reported in 122-type iron pnictide
18accompanied by
structural and SDW/AFM transitions.19However, no evidence
of band nesting without structural and SDW/AFM transitionshas been reported in the 122 type. In a previous ARPESstudy on LiFeAs, no nesting between hole and electron FSshas been observed.
7However, the NMR study indicated
the existence of spin fluctuation in LiFeAs similar to thatin other SDW/AFM iron pnictides.
6The existence of the
stripe-type AFM order is also forecast by a first-principlescalculation.
20Moreover, in NaFeAs, which is, like LiFeAs,
094509-4THREE-DIMENSIONAL ELECTRONIC STRUCTURE AND ... PHYSICAL REVIEW B 85, 094509 (2012)
a 111-type iron pnictide, band folding due to magnetic
fluctuation emerges at a temperature more than 20 K abovethe SDW transition temperature.
21Therefore our observations
support the proposition that LiFeAs involves a strong spinfluctuation with neither a magnetic nor an SDW transition.This result suggests that the ground state of LiFeAs is locatedvery close to the magnetic transition.
According to a theory of the spin-fluctuation superconduc-
tivity mechanism, line nodes may appear when the pnictogenheight is small,
22,23due to the change in nesting conditions
caused by the disappearance of a hole FS of dxycharacter
around the zone center. This is because the disappearance of thed
xyFS induces weakening of the interband scattering between
the hole and electron FSs and more pronounced intrabandscattering between the electron and electron FSs. This leadsto the change in the symmetry of the superconducting gap andthe appearance of line nodes. From such a standpoint, sinceLiFeAs has a d
xyhole FS around the zone center and a large
pnictogen height,24this material does not seem to have line
nodes.
Since we were able to observe the dxyhole FS, we shall
discuss the FS nesting properties in the 3D momentum space.Considering the ( π,π,π ) FS nesting as shown in Fig. 5(d),
the inner and middle-sized hole FSs are very small and theirnesting with the electron FSs is therefore not clear. On theother hand, the outer hole FS shows weak nesting with the 2Delectron FS in the 3D momentum space. AFM spin fluctuationbetween hole and electron FSs in LiFeAs has been observedin a neutron-scattering experiment, suggesting that AFM spinfluctuations are the origin of the superconductivity.
5,25In our
results, these nesting FSs have the same dxyorbital character.
Hence the nesting between these 2D hole and electron FSswith a d
xyorbital character may make a dominant contribution
to the enhancement of the spin fluctuation.
As discussed so far, because of the existence of the dxyhole
FS, the ( π,π,π ) band folding, and the weak nesting between
the hole and electron FSs, interband scattering between thehole and electron FSs might be dominant. According to theabove-mentioned theory of the spin-fluctuation superconduc-tivity mechanism,
22,23thedxyFS affects the form of thesuperconducting gap as well as the superconducting instability.
If thedxyFS is located around the zone center, the interband
scattering between the hole and electron FSs is dominant.Consequently, no node will appear. These results suggest thatLiFeAs is a nodeless superconductor.
Very recently, dHvA measurements have shown that there
is only one hole FS around the zone center.
8This is different
from our result and the previous ARPES data.7They pointed
out that such discrepancy could be caused by the surface effectsof ARPES measurements. If there are the surface effects, 2Dband dispersions would be expected, but our data clearly show3D band dispersions. Hence our ARPES data represent theintrinsic bulk electronic structure.
V . CONCLUSION
In summary, we performed polarization-dependent 3D-
ARPES measurements to investigate the 3D electronic struc-ture of LiFeAs. The band dispersions as well as their orbitalcharacters were resolved, and band folding with the ( π,π,π )
wave vector was clearly observed. Moreover, we found weaknesting between the 2D hole and electron FSs with thesame d
xyorbital character, which may enhance the spin
fluctuation. These results suggest that LiFeAs is a nodelesssuperconductor and that the FS with the same orbital characternested by the spin fluctuation plays a role in the appearance ofsuperconductivity.
ACKNOWLEDGMENTS
The authors gratefully acknowledge H. Miyazaki for fruit-
ful discussions and M. Sakai for his technical assistance duringthe experiments. Part of this work was supported by the Use-of-UVSOR Facility Program (BL7U, 2010) of the Institutefor Molecular Science and by a Grant-in-Aid for ScientificResearch (B) from JSPS (Grant No. 22340107). The workat SKKU was partially supported by Basic Science ResearchProgram (Grant No. 2010-0007487) and the Mid-career Re-searcher Program (Grant No. 2010-0029136) through the NRFfunded by the Ministry of Education, Science and Technology.
*hajiri.tetsuya@f.mbox.nagoya-u.ac.jp
†kimura@ims.ac.jp
1Y . Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.
Chem. Soc. 130, 3296 (2008).
2I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev.
Lett.101, 057003 (2008).
3K. Terashima, Y . Sekiba, J. H. Bowen, K. Nakayama, T. Kawahara,
T .S a t o ,P .R i c h a r d ,Y - M .X u ,L .J .L i ,G .H .C a o ,Z - A .X u ,H. Ding, and T. Takahashi, Proc. Natl. Acad. Sci. USA 106, 7330
(2009).
4T. Yoshida, I. Nishi, S. Ideta, A. Fujimori, M. Kubota, K. Ono,S. Kasahara, T. Shibauchi, T. Terashima, Y . Matsuda, H. Ikeda, andR. Arita, P h y s .R e v .L e t t . 106, 117001 (2011).
5A. E. Taylor, M. J. Pitcher, R. A. Ewings, T. G. Perring, S. J. Clarke,
and A. T. Boothroyd, Phys. Rev. B 83, 220514(R) (2011).6L. Ma, J. Zhang, G. F. Chen, and W. Yu, P h y s .R e v .B 82, 180501(R)
(2010).
7S. V . Borisenko, V . B. Zabolotnyy, D. V . Evtushinsky, T. K. Kim,I. V . Morozov, A. N. Yaresko, A. A. Kordyuk, G. Behr,A. Vasiliev, R. Follath, and B. B ¨uchner, P h y s .R e v .L e t t . 105,
067002 (2010).
8C. Putzke, A. I. Coldea, I. Guillam ´on, D. Vignolles, A. McCollam,
D. LeBoeuf, M. D. Watson, I. I. Mazin, S. Kasahara, T. Terashima,T. Shibauchi, Y . Matsuda, and A. Carrington, Phys. Rev. Lett. 108,
047002 (2012).
9Y . J. Song, J. S. Ghim, B. H. Min, Y . S. Kwon, M. H. Jung, and J-S.Rhyee, Appl. Phys. Lett. 96, 212508 (2010).
10S. Kimura, T. Ito, M. Sakai, E. Nakamura, N. Kondo, T. Horigome,
K. Hayashi, M. Hosaka, M. Katoh, T. Goto, T. Ejima, and K. Soda,Rev. Sci. Instrum. 81, 053104 (2010).
094509-5T. HAJIRI et al. PHYSICAL REVIEW B 85, 094509 (2012)
11P. Blaha, K. Schwarz, P. Sorantin, and S. B. Trickey, Comput. Phys.
Commun. 59, 399 (1990).
12I. A. Nekrasov, Z. V . Pchelkina, and M. V . Sadovskii, JETP Lett.
88, 543 (2008).
13S. H ¨ufner, Photoelectron Spectroscopy (Springer, Berlin, 2003).
14R. Matzdorf, Surf. Sci. Rep. 30, 153 (1998).
15S. M. Goldberg, C. S. Fadley, and S. Kono, J. Electron Spectrosc.
Relat. Phenom. 21, 285 (1981).
16K. Umezawa, Y . Li, H. Miao, K. Nakayama, Z. H. Liu, P. Richard,
T. Sato, J. B. He, D. M. Wang, G. F. Chen, H. Ding, T. Takahashi,and S. C. Wang, P h y s .R e v .L e t t . 108, 037002 (2012).
17T. Sato, K. Nakayama, Y . Sekiba, P. Richard, Y .-M. Xu, S. Souma,
T. Takahashi, G. F. Chen, J. L. Luo, N. L. Wang, and H. Ding, Phys.
Rev. Lett. 103, 047002 (2009).
18M .Y i ,D .H .L u ,J .G .A n a l y t i s ,J .H .C h u ,S .K .M o ,R .H .
He, M. Hashimoto, R. G. Moore, I. I. Mazin, D. J. Singh,Z .H u s s a i n ,I .R .F i s h e r ,a n dZ .X .S h e n , P h y s .R e v .B 80, 174510
(2009).19Q. Huang, Y . Qiu, Wei Bao, M. A. Green, J. W. Lynn, Y . C.Gasparovic, T. Wu, G. Wu, and X. H. Chen, P h y s .R e v .L e t t . 101,
257003 (2008).
20Y .-F. Li and B.-G. Liu, E u r .P h y s .J .B 72, 153
(2009).
21C. He, Y . Zhang, B. P. Xie, X. F. Wang, L. X. Yang, B. Zhou,F. Chen, M. Arita, K. Shimada, H. Namatame, M. Taniguchi,X. H. Chen, J. P. Hu, and D. L. Feng, P h y s .R e v .L e t t . 105, 117002
(2010).
22K. Kuroki, H. Usui, S. Onari, R. Arita, and H. Aoki, P h y s .R e v .B
79, 224511 (2009).
23H. Ikeda, R. Arita, and J. Kunes, P h y s .R e v .B 81, 054502 (2010).
24K. Hashimoto, S. Kasahara, R. Katsumata, Y . Mizukami,
M. Yamashita, H. Ikeda, T. Terashima, A. Carrington, Y . Matsuda,and T. Shibauchi, Phys. Rev. Lett. 108, 047003 (2012).
25M. Wang, X. C. Wang, D. L. Abernathy, L. W. Harriger, H. Q.
Luo, Y . Zhao, J. W. Lynn, Q. Q. Liu, C. Q. Jin, C. Fang, J. Hu, andP. Dai, P h y s .R e v .B 83, 220515(R) (2011).
094509-6 |
PhysRevB.19.1270.pdf | PHYSICAL RKVIK% 8 VOLUME 19,NUMBER 2 15JANUARY1979
Simplemodelofhydrogen andlithium chemisorption onjelliumsubstrates
J.P.Muscat andD.M.Nuns
Department ofMathematics, Imperial College, LondonS.8'.7.,England
(Received 28September 1977)
Withinasimpleone-parameter quasianalytic model,weareabletoreproduce themainone-body features
offirst-principles calculations forhydrogen chemisorption onjelliumsurfaces. Theseinclude thequalitative
variation inthewidthandpositionoftheresonance peaksastheadatom-substrate separation isallowedto
change.Thegeneral variation ofthesefeatures withtheelctron densityofthesubstrate isalsoreproduced
withremarkable accuracy. Preliminary resultsforLiadsorption arealsopresented, andalsoseemtobe
compatible withthelimiteddatafromfirst-principles calculations. Nevertheless thelimitation ofthepresent-
modeltoanl=0solution meansthatitcannot include sp-hybridization effectswhichappeartobe
important. Thesimplicity ofthecurrent modelenablesasimplephysical interpretation ofthemechanism of
chemisorption onfree-electron-like substrates. Inaddition thereisastrongpossibility ofextension ofour
modeltosystems,ofgreaterpractical importance, forwhichafirst-principles calculation isasyetnot
possible.
I.INTRODUCTION
Thispaperhasarisenfromtheneedtounder-
stand, andtoplaceinthecontextofthewhole
theoryofchemisorption, aseriesofimportant
recentfirst-principles calculations forchemi-
sorption onfree-electron-like substrates. ''
Thesecalculations employtheremarkably suc-
cessfullocal-density-functional (LDF)approach
weremindourselves thatthisapproach seemsto
giveanexcellent approximation tothegroundstate
ofvarioussystems, itsapplication totheexcita-
tionspectrum being,however, ratherlessfirmly
based.Furthermore, theapproach isformally
similartotheHartree theory,itbeingthecase
thatoncetheself-consistent Hartree-like poten-
tialisestablished, solvingforthewavefunctions
andenergylevelsisonlyaone-body problem.
Thecalculations inquestion''alluniformly
employasastarting pointthejelliummodel,in
whichthesubstrate ion-core chargedensity.is
smeared outtoformauniform positiveback-
groundtruncated stepwise atthesurface. This
modelisoftenconsidered asausefulstarting
pointforconsidering thesurfaceproperties of
thecleansp-metal surfaces.'Inthechemisorp-
tioncalculations apointcharge, ofmagnitude
equaltothatofthenucleusoftheadatom inques-
tion,isaddedtothejelliumbackground atadis-
tancedfromthe"jellium edge,"atwhichtheback-
grounddropstozero.Themajornumerical task.
ofre-solving theLDFequations isthenunder-
taken.Thechemisorption energy, thedipolemo-
ment,andthechangeindensityofstates&N(c)
onadsorption areamongthequantities calculated.
Calculations existforH,0,Li,Cl,andSiad-
sorbates onjellium doneinthisway.''Calcula-
tionshavealsobeenmadeforHand0adsorbates
inwhichthesubstrate atomicstructure istakenintoaccount usingaperturbational calculation to
firstorderinthesubstrate ion-core pseudopo-
tentials.''Thesubstrate pseudopotentials are
foundtomodifyradically suchproperties asthe
equilibrium distance d(andhencethenatureof
substrate-adsorbate binding); thechemisorption
energy anddipolemoment beingalsomodified.
However, 4N(e)atagivendisnotusuallycor-
rectedforthepseudopotentials, andweshall
similarly ignorethiseffectinthefollowing. It
is,however, possible totakeintoaccount the
substrate pseudopotentials inanonperturbational
waybymeansofaclustercalculation, asdone
byHarris andPainter,'forexample.
Awordshouldbesaidontheexperimental im-
portance ofspmetalsassubstrates inchemi-
sorption, forwhichlesswell-established high-
qualityexperimental information appears tobe
available thanfortransition metalorsemiconduc-
torsubstrates. Forexample, onexposure of
aluminum tooxygenthereremains uncertainty
astowhether adsorption orabsorption ofthe
oxygentakesplace."'Thechemisorption ofhy-
drogen onsuchmetalsasAl,Mg,andthealkalis
seemsnottohavebeenestablished."Onthe
otherhand,hydrogen shouldchemisorb (ifitis
notabsorbed) onthelattersystems initsatomic
state.'Suchanadsorbed layerisstablewith
respecttodesorption iftheatomicchemisorption
energy4Eexceeds halftheH,dissociation energy,i.e.,exceeds2.3eV."For,say,Allow-index
faces,theabove-mentioned calculations suggest
AEisinsufficient forsuchstability,'"'but,
nevertheless, atsufficiently lowtemperatures
suchthatdiffusion isnegligible theadsorbed H
atomsshouldbeobservable inametastable state
asfoundforHonsomenoblemetals."
Inthealkaliadsorbates therathersimilar
properties ofvarioustransition metalsubstrates"
1270
SIMPLE MOBKL OFHYDROGEN ANDLITHIUM. ..
andthelargeradiusofthevalence-shell elec-
tronicstatessuggests thatthetransition metald
bands,specially whennarrow, mightriotbe.very
important inthesubstrate-adsorbate interaction.
Insuchacasethesubstrate spbandmightbe
dominant andafree-electron-like modelofthe
substrate—whosedelectrons aretreatedas
cores—mightbeusefulincalculating some
properties. Langhasrathersuccessfully dis-
cussedtheworkfunction changeinalkalichemi-
sorption ontransition metalsystems usinga
jellium modelforthesubstrate."
Oneaimofthepresent work,ofwhichshort
accounts havepreviously beenpublished,'""is
toaidinterpretation andunderstanding ofthe
jellium-based calculations forhydrogen onfree-
electron-like substrates. Inchemical terms, we
wouldliketoknowwhether thehydrogen istobe
regarded asadsorbed inessentially atomicform,
orwhether thereisarecognizable chemical bond.
Theformer viewseemssuggested byearlierwork
withinanapproximate LDFformalism,"wherean
atomiclike resonance wascalculated. Inviewof
theimproved treatments nowavailable,''which,
asweshallshow,suggest thecontrary view,the
resultsofRef.16willnotbefurtherdiscussed
here.
Weshallconcentrate onthechangeindensityof
statesEN(e)onchemisorption, andfollowtheus-
ualapproximation ofassuming themaineffectof
substrate pseudopotentials istogiveamorereal-
isticdwithout changing bN(e}atthatd.Were-
gardthisquantity ascontaining valuable informa-
tionastothenatureofthechemical bondinthe
LDFgroundstate.Itsrelationship withtheactual
excitation spectrum isregarded assecondary
here.Thepresent workisconfined tomonovalent
adsorbates, mainlyhydrogen butlithiumisalso
considered. Theapproach istostartbytrying
toidentify themainphysical elements inthe
adatom-substrate interaction, whichareincorpo-
ratedintoasimplemodelwhoseresultsarethen
compared withthefirstprinciples calculations.
Letusfirstoutlinethephysical ideas,taking
hydrogen astheexample. Ourstarting pointis
theAnderson modelofchemisorption withinthe
restricted Hartree-Fock formalism,'""Thishas
anumberofpointsincommon withLDFformal-
ism.Inthispicture onestartswithaself-con-
sistenteffective hydrogen 1slevel,E,«,which
foraslightly negatively charged adatomliesa
littleabovethemeanoftheionization andaffinity
levels—sayat6eVbelowvacuumlevel.The
changeindensityofstatesAN(c)duetobringing
uptheadatomisthen
(la)where
Ik}i,sabandstateofthesemi-infinite
metalwithenergy e~and
I1s}istheHvalence
orbital, Vbeingtheperturbation onbringing the
atomuptothesurface."A(e)istheHilbert
transform of&(e).
ItisseenfromEq.(1)thatif6issmallthere
isasharppeakorresonance in4Nwhichoccurs
atanene.rgyz„given bythesolution ofthetrans-
cendental equation
e—e,«—A(e)=0. (3)
InFig.1weillustrate thegraphical solution of
(3}.4(e)andA(e)havebeentakenforsimplicity
tohavetheform
g=c(1 ga)3~2
A=c(-,'c(3—2e)+[e(a—1)—8(-1—e)](e'-1)"'j,
2-.
"I,
h,n(E)/~~
I
I
1—I
l
I
l
I/
//
0LP
-l -5
FIG.1.Anderson modelapproach. (a)Functions6
andAforc=1(seeSec.I)withgraphical solution of
Eq.(3).(b)Quantity 4n(e)plottedvseatee&f=-0.2
forvarious valuesofc;curveslabeledbyc.wherethephaseshiftq(e)isgivenby"
q(e)=—tan'(a(e)/[e—c,«—&(c)]];
0&ad&v. (lb)
Thefunction4(e},centraltothetheory,isde-
finedby
&(e)=P1(lsIvl»I'&(e—e~} (»
1272 J.P.MUSCAT ANDD.M.NE%NS 19
where8isthestepfunction andcisascaling
constant. NotetheE''behavior of4atbottom
ofband,aswouldbeappropriate forthesurface
densityofstates. However, sinceanuppercutoff
isrequired, wehaveforsimplicity takenasym-
metric6,thoughthiswouldbemorenaturalfora
tight-binding system. Accordingly, theresults
obtained fromthismodelwillnotbeverymeaning-
fulforajelliumsurfacein,say, &~0.5.From
Fig.1itisseenthat&,liesbelowz,«.The
larger A(whichscaleswith6),is,thelarger
thislowering willbe;(a)and(b)inFig.1refer,
respectively, tosmallandlarge.&.Sincethe
matrixelement V,~,andhence6,isexpected to
increase astheatomapproaches thesurface, it
isthusexpected that&,milldropuniformly below
e,«astheatomapproaches thesurface; (a)and
(b)inFig.1canberegarded asappropriate for
largeandsmallatom-surface distance, respec-
tively.
Thewidthoftheresonance ish(s,).Thisreso-
nancewidthisthusexpected atfirsttoincrease
astheatomapproaches thesurface, butifq,gets
nearthebottomofthebanditwill,decrease again.
Finally,ifa,goesbelowthebandedge[whichis
actually thecaseinFig.1(b)]e,becomes alo-
calizedstate.
InFig.1(b)wealsoillustrate thephaseshift
q(c)through theproportional quantity dn(e)
=2q(e)/v. &n(c)isthechangeinnumber ofelec-
tronsboundbelowenergy &duetoaddingthehy-
drogen,itsderivative being&N(c)[seeEq.(la)j.
a,«ischosentolienearthebandcenter. The
curvec=0.08corresponds toalinesomewhat
steeper thancase(a)ofFig.1(a).Theshapeof
hn(e)ischaracteristic ofawell-defined atomic
resonance, showingarapidchange indn(e)asone
traverses throughit,from4nsmallat&=-0.5to
4nnearly2at&=0.Inthecasec=2.7onthe
otherhand,onehasaboundstatebelowbandand
alsoaboveit,corresponding tocase(b)ofFig.
1(a).InstudiesoftheAnderson modelofchemi-
sorption" itiswellestablished thatthesebound
statesarebonding andantibonding states,respec-
tively. Weexpectthesestatestogether tocon-
tributeAn=2electrons (sinceoneatomicorbital
ja)hasbeenaddedtothesystem), buteachstate
only4n=1;thus,thevalueof4nnearmid-
bandshouldcorrespond toaboutunity,corre-
sponding toabondingstate.Lookingatthec=2.7
curve, weseeithas4n=2forznearbottomof
theband,asitmustduetotheboundstate,but
thisvaluedropsrapidlytoabout1atmidband in
accordance withtheabovesimpleargument.
Asubtlesituation isfoundatvaluesofcbetween
theatomic andcovalent limits. Atc=0.65,an
atomiclike resonance, broader thanatc=0.08isfound,but4ntendstosaturate atabout1.2;this
resonance isnotpurelyatomicbuthas.some
bondingcharacter. Atc=1.2wehaveagaina
sharpresonance, justabovebottomofband.The
resonance itselfbinds1.4electrons, seeming
paradoxically moreatomiclike thanatc=0.65.
However, theresonance givesitselfawayon
lookingatbn(c)forlargere,whereafall-off
reminiscent ofthebondingstatecasec=2.7is
seen,whichshowstheresonance isreallya"vir-
tualbondingstate."
Tosummarize thediscussion oftheAnderson
model,itisexpected thatthereisanarrow
atomiclike resonance atE,«-6eVbelowvacuum
atlarged,whichbroadens andshiftsdownwith
decreasing d.Ifthecouplingisstrongenough, on
furtherdecreasing d,theresonance willnarrow
onapproaching thebottomofthebandandmay
thenappearasabondingstate.Consideration of
thearea&n(e)undertheresonances canfurther
beusedasadiagnosis forbondingoratomic
character. Furthermore, iff,«andE~lienear
thecenteroftheband,thenbonding-type reso-
nancesorboundstateswillbeoccupied (andanti-
bonding unoccupied) leadingtoanimportant con-
tribution tothebinding energyfromthesestates.
ThechangeindensityofstatesbN(c)accord-
ingtoLDFcalculations whenaprotonisintro-
ducedatdistance dfromthejellium edgeis
showninFig.2forr,=2and3.'Broadly, the
features justdescribed areindeedseen.At
largedthereisanarrowlevelat&,«=6eVbe-
lowvacuum lhvelforr,=2andr,=3(atd=~the
levelisfoundtobeat7eVbelowvacuum). In
bothcasesthislevelatfirstbroadens andshifts
downwithdecreasing d,thennarrows onapproach-
ingthebottomoftheband.Finally, itisknown
fromcalculations doneforaprotonwellinside
jellium thatwhenx,&1.9thereisaboundstate
justbelowthebottomoftheband."
Anattempttomakeamorequantitative applica-
tionoftheseideas,whichseemqualitatively cor-
rect,encounters difficulties incalculating &(e).
Thisappliesparticularly tothehigh-energy be-
haviorwhichisessential togetcorrectifthe
Hilberttransform A(e)istobecalculated.
Inthepresent work,theexplicitcalculation of
&(e)isavoided byusingtheKorringa-Kohn-
Rostoker (KKR)formalism."Inthismethod, the
potential ofthesystemisapproximated byone
whichisspherically symmetric withinasphereof
radiusRcentered ontheproton, whereas outside
thesphereitisthepotential oftheunperturbed
surface. Itisthenpossible toexpress AN(e)in
termsof(i)theone-electron Green's function of
theunperturbed surface and(ii)thesolution of
theSchrodinger equation ofangular momentum E
19 SIMPLE MODEL OFHYDROGEN ANDLITHIUM. .. 127$
EF—
0.4-
0
0-2
0-110
hN(E)au(a)
-2.0
20
(b)butsinceitismainlythebindingstrength ofthe
spherical potential whichisimportant, thisisnot
believed tobeserious.
Inthecaseofalkaliatomsonetendstobase
one'sthinking onthesituation fordatorexceed-
ingtheequilibrium distance fromthesurface. In
thisregionasinglealkaliatomislargelyionized,
andtheadatom-surface interaction isnotusually
considered strongenoughtoformaresonance
withsignificant bondingcharacter, thoughitmight
bestrongenoughtosignificantly hybridize the
adatomsandpvalenceorbitals.'Inthissitua-
tion,shiftsinresonance position originate in
shiftsinz,«fromsuchcausesastheimagepo-
tentialorotherscreening effectsratherthanfrom
A(&),"the"weak-coupling" caseofFig.1(a)being
probably nearthetruthinthephysical region.
Considerable interest nevertheless attaches tothe
resonance width4itself.Weshallassume that
theresonance involves the2s(inthecaseofLi)
atomicorbital, provided itisreasonably narrow
compared withtheenergyseparation ofthe2sand
2porbitals. Accordingly, weagainusetheKKR
modelasforhydrogen, modifying thesphere
radiusandinternal potential appropriately; in
thiscaseourguidetotheseis,respectively, the
Liatomicradiusandtheposition oftheresonance
inthefirstprinciples calculations, 'inanattempt
toreproduce correctly theshapeoftheresonance. .
-05
20I
I
40
hN(6)a.u"-15
FIG.2.Changeindensityofstates4N(q)dueto
chemisorption oftheHatomontheielliumsurface
according tofirst-principles calculations ofRef.5.
Curveslabeledbydistance ffofprotonfromjellium
edge.(a)x~=2;(b)r=3.
andenergy ainsidethesphere. Thephysical
ideasalreadyexpressed intheAnderson frame-
workcanreadilybeincorporated. Theinclusion
ofonlya1svalenceorbitalontheHatomgoes
overintheKKRmethodtoretaining onlythel=0
solution insidethesphere. Theideaofaconstant
self-consistent adatomvalence energy &,«leads
totheassumption thattheI=0bindingstrength
ofthepotential insidethespherebeindependent
ofdistance fromthesurface. Theradiusofthe
spherechosenisrathersmall(actually 1a.u.),
inordertominimize theregionofspaceinwhich
theSchrodinger equation isconstrained toanl=0
solution. Thisnecessitates takingasomewhat
deeperpotential insidethespherethanisrealistic,H.FORMALISM
LetH,denotetheHamiltonian ofthefreesur-
face,andvthepotential intheneighborhood of
theadsorbing atom. W'eshallassume thatvis
aspherically symmetric potential whichisnon-
zeroonlywithinasphereofradiusA.Ifwenow
denotetheeigenvalues ofthetotalHamiltonian H
including vbye„,andthoseofHobyq~,thenthe
difference indensityofstates(including spin)in-
troduced bytheperturbation visgivenby
AN(e)=2+5(e—I„)—2+5(E- e~).
n(4)
AN(e)=2v'Im—lnf(e).d&
Weshallnotgivethefullexpression forf(e)
withintheKKRmethod, butconfineourselves to
thecaseofimportance inthispaperwhereonly
l=0solutions totheSchrodinger equation insideIfwenowintroduce afunctionf(e)whichisanalyt-
iceverywhere exceptatafinitenumberofpoles
e,(theenergyeigenvalues ofH,),andwhosezeros
areate„,wecanshowbyuseofawell-known
theorem ofanalysis" thatEq.(2)canberewritten
intheform
1074 J.P.MUSCAT ANDD.M.NEWNS
thesphereareconsidered. Letthel=0solution
tothe'Schrodinger equation insidethesphereat
energysbep,(r),andG'(r,r')betheGreenfunc-
tionG'=(e—H,)'belonging totheunperturbed
Hamiltonian H,;thenachoiceforf(e)is
G(r,r')=lim dS dS'G2(r,r').(6b)
r=R-2( r=R-f
Heretheintegrations areoverthesurface ofthe
spheres ofradiusrandr'(r'&r). Itisseenthat
f(e)isindeedzeroattheperturbed eigenvalues
e„[seeE(1.(3.10)ofRef.21],andithaspolesat
theunperturbed eigenvalues &~duetothepres-
enceofthefunctions G'(r,r').
ThushN(c)isgiveninourapproximation by
~l(t(e)=2v-'im—lnG(r,r')
—,,G(r,r'&(,(~'&).(7&
Thegreatadvantage ofthismethodisthatitis
capable ofconsidering separately thesolutions
insidethespherethroughg„andthoseoutside
throughC',andthentomatchthematthesur-
faceofthesphere. WenoteherethattheKKR
method" isvariational withrespecttothee„and
thustoEN(e); infact,itminimizes themismatch
atthesurfaceofthesphere. Thedetermination
ofP,(r)presents noproblem; onesimplyhasto
solvetheSchrodinger equation forthechosen
potential v(r).Themaindifficulty isthusthe
calculation oftheGreenfunctionG'(r,r').For
afree-electron gaswithasurface,6'isgiven
bythegeneralexpression
G'(-,-)=Q' ~"('~"',(8)6—6p+zs
wherecapitallettersaredesigned torepresent
variables parallel tothesurface, andlower-case
letterstorepresent variables perpendicular to
thesurface, E„-=2(K2+k,')inatomicunits,ands
isaninfinitesimal positive (luantity. TheIt»'s
depend onthemodelofthesurfacechosen. We
haveadoptedafinite-barrier potential model-to
describe thesurface. Thebarrier ofheight Vo
istakenatz=so.Byconsiderations ofcharge
neutrality ofthesystem,ithasbeenshownthat
thejellium edgeissituated atadistancez,-from
thebarrier,"where+2-1sin'—
22—1
Herek,=(2V,)'~2andkzistheFermiwavevec-
tor.
Therearetwopossibilities forthewavefunc-
tions,"according towhetherk,isgreaterorless
than0o.For0„&k„theeigenvalue spectru~ is
nondegenerate andthewavefunctions aregivenby
4»,(z)=cos[k»(z-80)—5];z&8(&
5isaphaseshiftgivenbytant&=-q,/k,andq,
=(k,'—k2)'~2.Fork,&k,theeigenvalue spectrum
isdoublydegenerate, andthetwolinearly inde-
pendent wavefunctions aretakentobe
1'k-q'egg(ggo)+ ge~kg('0)'g(g2 k,+q,'
y(»(&)
ge((('(»-»&&.
,0,+q,'
(12)
q'—jp+qg gefqg(g-go) ~8)8q,'+k,
andq,'=(k,'—k;)''.
Thesewerechoseninsuchawaythatg"&rep-
resents theanalytic continuation ofE(ls.(10)for
k,&k,inwhichcase g»(2&istheonlypossible
'choicewhichislinearly independent of(rj(».
kgThesedefinethecontinuum completely. One
cannowconstruct theGreen's functions byin-
sertingthe(r&'sasgivenbyK(ls.(10—12)intoE(l.
(8).Inordertocalculate G,itisthennecessary
toexpandthe(I&'sintermsofspherical harmonics
aboutthespherecenter(heretakenastheorigin),
andthenkeeponlythel=0termintheexpansions.
Onenowhastosumoverk.Theintegration over
kisstraightforward although ratherlengthy and
is.detailed intheAppendix. Therearetwopossi-
bleexpressions forG(r,r')according towhether
thebarrieratzoliestotherightortotheleftof
thespherecenterattheorigin.Theseexpres-
sionsaregivenby
19 SIMPLE MODEL OFHYDROGEN ANDLITHIUM. .. 1275
[2x'—1—2g(x'—1)'']e'*dx);
/kps,&0, (14)
wherek=(2z)''v=[2(V,—z))''n=2k,z„andj,isthespherical Besselfunction oforderzero.
TheKKRformalism onlypermits evaluation of
Grigorously whenthespheredoesnotoverlap
thebarrier,i.e.,zp~Rorzp&-R.Inpractice,
continuity ofthewavefunctions forz«zpinEqs.
(10)-(12) shouldkeeptheresultsreasonable for
asmallamountofoverlap. Weshallinthefollow-
ingassume theresultsremainreasonable pro-
0.5R.
Beforegoingontocalculate 4N,letuslook
brieflyattheproperties ofG(r,r'),andfirstof
all,letuslookatthesimplelimitVp-.Inthis
caseitiseasilyshownbyrepeated integration by
partsthatEqs.(13)and(14)reduceinthelimit
kp~to
e-fur'~2&~gp4',(k-r), —j,(kr'); z,&0—(,)0x''2kzo
0;8&0
(15)
i.e.,theresults foundinourprevious workfor
aninfinite-barrier potential model."
Inthelimitzp-~,asinthelimitV,-O,one
obtains theresultfortheGreen's function inthe
caseofanimpurity inthebulk,ifonlythes
phaseshiftistakenintoaccount,"i.e.,Fornegative valuesofz„oneseesimmediately
thattheimaginary partofG(r,r')isvanishingly
smallatlowenergies duetotheexponential term .
in(14)whichisverysmallforz/V,«I.Athigh
energies thebehavior israthersimilartothat
foundforzp&0.
Thefunction p,(r)whichappears in(7)isthe
regular /=0solution oftheSchrodinger equation
inx&Ratenergy &.Thisfunction beingalways
real,itiseasilyshownthatEq.(7)reduces to
n,N(z)=2v'—dn
dE(18)
y,(z)=L,(z), (20)
L„(z)=, lnG(r,r')~„„,„.ImG(r,r')y,—Im(8/Br')G(r, x')
ReG(r,~')y,—Re(s/9~')G(r, ~')'
wherey,isthelogarithmic derivative of$0cal-
culatedatr=R,andtherealandimaginary parts
ofG(x,r')andtheirderivatives withrespectto
x'arealsotakenatr=x'=R.
Thepositions oftheresonances aregivenbythe
zerosofthedenominator ofEq.(19),whichoccur
when
G(~,r')=4~q,(kr)e-"-"'/~'. (16) Limiting resultsare
Inthelimitz,--~,Eq.(16)issuitablytrans-
formedtotakeaccountofthechangein"local
bottomofband"whichisinthiscaseatthevacu-
umlevel,i.e.,
G(r,r')=-4',(iver)e""'/z'.(17)
Letusnow1ookatthebehavior ofGatlowen-
ergiesforfinitez,&0.Thebulktermin(13)which
givesthebehavior atsmall Eoftheimaginary
partoftheGreen's function foransphaseshift
inthebulk[seeEq.(16)]iseliminated completely
byatermwhicharisesfromthepresence ofthe
surface(i.e.,whichdepends onz,),thusgiving
fortheimaginary partoftheGreen's function an
z''behavior atlowenergies. Orinotherwords,
theatomseesasurfacelike densityofstates(e'')
insteadofabulk-like densityofstates(z'~')..
Thehigh-energy behavior ofG(r,r')isnotquite
sosimple. G(r,x')isanoscillating function ofz.
Theamplitude ofthesuccessive oscillations de-
creases ratherslowlyasz'(21)
(2fe/)'~';z(0RL„=(1
R~——-ktankR e&0.(22a)
(22b)
WenoticethatEq.(19)strongly resembles the
equation forthephaseshiftinabulkimpurity scat-
teringproblem, towhichitreduces ifEq.(16)is
substituted forGinto(19),Asinbulkscattering
theory,asharpresonance canoccurifthede-
nominator of(19)goesthroughzero[i.e.,(20)is
satisfied] whenthenumerator issmall.Thisdoes
nothappenforE=0bulkresonances asthenumer-
atorhasthec.'i"behavior whichdoesnotvanish
rapidly enoughatsmalle,incontrast tothea'~'
behavior foran/=1resonance."
Inthepresent problem whenzo&R(spherelies
&276 J.P.MUSCAT ANDD.M.NEWNS
insidebarrier), asjustpointed outaboveforImG,
thenumerator of(19)behavesase'~'forsmallz,i.e.,likeanl=1phaseshift,sothatawell-defined
resonance canindeedexistevenwhenthesphere
isinsidethesurface. Whenz&R(sphere outside
barrier) thenumerator of(19)isattenuated expo-
nentially toanextentincreasing withdistance from
surface andmodulus ofenergybelowvacuum. Now
aresonance ofsharpness increasing withdistance
fromsurface canoccur.Thepresentformalism
thusshowsverysimply howaresonance canbe
developed whichisatoncesharpyetessentially
l=0-like insidethesphere.
Onenoticesherealsotheanalogy between our
expression for4VasgivenbyEqs.(18)and(19),
anditsequivalent intheAnderson formulation,
i.e.,Eq.(1).However, arigorous mapping ofone
expression ontotheotherprovesdifficult tocarry
outunambiguously exceptinthelimitwherethe
protonisfaroutside thesurface. Inthiscasewe
findthattheexpressions foraandAinEq.(1)
are
p(g)g-i~-~ e-2~(~os)
0(23a)
—1/2
a(e)=-z'A'—1——e'""os'(23b)«p Vp
where
(23c)
Here&,«istheenergyoftheboundstateofthe
wellatz,-~(spherefaroutsidesurface), given
bythesolution ofEq.(20)withL&(c)givenbyEq.
(21).
m.RESULTS
u(~)=-v, e". (24)
WetakeR=1a.u.;Infact,smallchanges inR
havelittleeffectontheresults. Rshouldnot,
however, betakentoolargetosatisfactorily use
aonephaseshiftmodel. Theonlyremaining pa-A.Hydrogen
Thecalculation ofgand4Ndepends ontheform
oftheperturbation v(r),whichmanifests itself
through thelogarithmic derivative y,(e).Infact,
p,(e)wasfoundtobeverylittledependent uponthe
actualshapeofthemodelpotentials chosen—we
havetriedsquare-well, Yukawa, andexponential
potentials—provided thattheseweretakentobe
equallystrongly binding. W'egivetheresultsfor
theformrameter isthusv„which waschosensoasto
giveaweakly boundstatefortheprotoninbulk
jelliumashasbeenfoundwhenr,&1.9infirst
principles calculations. Avalueofup=2.7a.u.
givestheboundstateat0.02a.u.belowthebottom
oftheband;thisisindependent ofx,sincezis
measured withrespecttobottomofband.Equa-
tion(24)implicitly assumes thatv(r)isunchanged
(relative tobottomofband)whenz,&R,i.e.,
sphereoutsidebarrier, anassumption requiring
justification. Nowmostofthebarrier potential
Vpisseenfromjelliumcalculations tobemade
upofexchange-correlation potential, especially
atlargex,.Evenatr,=2theelectrostatic con-
tribution isonlynP-0.25.a.u.,compared with
V,=0.6a.u.Itcouldbearguedthatforzp&Rwe
shouldincrease v(r)byhg(arathersmallcorrec-
tionforr,&3).However, wearguethatsincethe
relatively smallsphereradiuschosenexcludes a
considerable portionoftheattrative regionaround
theprotonwhichissampled bythe"1sorbital,"
weshouldhavesomecompensatory factorwhich
wearbitrarily selectbyneglecting hP.
Thefinitesquarebarrier modelhasbeenused
asanapproximation tothejellium onebyMahan."
ThefactthatMahanrequired theextreme value
Vp6ptofitjelliumofx,&3suggests thatthe
analogy between thefinitebarrier andjellium
models mightbebetteratlowerelectron densities
r,~3.Inthiscontext werecallthatinthepresent
workwedonotemployMahan's variational pro
ceduretodetermine V,butwesimplytaketheval-
ueofVpasthecalculated jelliumworkfunction
plusFermilevel.
Theresultsofourcalculation ofb,N(e)based
onthepotential (24)withV,=2.7a.u.areshown
inFigs.3and4andinTableI."Itisne'cessary
torecallthattheproblem ofsphereoverlap with
thebarrier prevents theinclusion ofresultsfrom
ourcalculation atdvaluesforwhich~s,~&0.5a.u.
inFig.3andTableI;inFig.4wehaveinterpo-
latedsoastosmoothly fillintheexcluded region.
Thereisastriking general agreement between our
results andthoseofthefirst-principles calcula-
tions.Ourresults giveafairlyclearly defined
resonance peakindNatmostd.Thedvariation
ofthemaximum inthisresonance isseenfrom
Fig.4tobeinveryfairgeneralagreement with
thatofthefirstprinciples calculations forr,
=2.3,and4.Furthermore, thewidthisnarrow
atlarged,thenincreases asddecreases, and
finallynarrows againastheboundstateisabout
toseparate justasinthefirstprinciple calcula-
tions.Quantitatively, Fig.3andTableIshow
thattheresonance widthisfoundtobeinfair
overallaccordwiththelatterexceptthatourcal-
culation cannotreproduce theawkward situation
19 SIMPLE MODEL OkHYDROGEN, A5DLITHIU'M. ~. 1277
0.6,
rs=2
30(1.90)—r,=4
3
(a.u)
-0.2—
0.2—
-10(090)
10 20h~{e) O.U.'I
30
03—
6(a.u.)
02— -06-2 0 2d(a.U.)
01—
-05(1.1)FIQ.4.Energyrelativetovacuumlevelofmaximum
inresonance peakofANasafunction ofdforthree~~
values. Brokencurves—Hef.5;fullcurves-present
work.Levelsatd=~shownatright(brokenline-aef.
5,fulllines—presentwork).Fermilevelsshownat
tople@.
10} I
20 30
QN(P)0,U."
FIG.3.AN(e)inFig.2,according topresentwork.
Curveslabeled byvaluesofdandbyAz(ez}inparen-
thesis (a)r,=2.;(b)r~=3.
inwhichthesphereisatthebarrier. Wetake
thisasindicating thatthephysical modelwhich
isthecentralpointofthispaperisphysically
correct.
Notwithstanding thegeneral agreement justmen-
tioned,letusindicateitslimitations. First,in
theprobably ratherunstable regionofnegative d
wheretheboundstateisabouttoseparate, our
resonances aretoonarrow[seeespecially ther,
=2caseinFig.3(a)].Second,theredoesseemto
beadiscernible tendency inTableIforourreso-
nancewidthtoreachitsmaximum atalargerval-
ueofdthanthatfoundinthefirstprinciples cal-culations; thismaypossiblyarisefromdiffer-
encesbetween thejellium andfinitebarrier mod-
els.Third,thechoice(24)leadstodifferent po-
tentialsv(r)relative tovacuum whenr,varies,so
theresonance position converges todifferent
"atomiclevels"atdinourmodel.Here,how-
ever,theresonance positions inthefirstpri~~ci-
plescalculations (seeFig.4)stillshowjustsuch
nonconvergent behavior atthelargestpositive d
valuesavailable, andfurthermore seemcuriously
tobeconverging toalevelmellabovethatgiven
fortheneutral8atominLDFapproximation. In
thisconnection thereisseeninourr,=2reso-
nanceposition inFig.3tobeananalogous small
one-body shiftupwards atd=3a.u.relative to
d=~."Theupwardshiftisclearlyseenanalyti-
callytooccurfromEqs.(3)and(23a)whenthe
levele,~~liesabove ~Vo.However, thiseffect
1278 J.P.MUSCAT ANDD.M.NEWNS 19
TABLEI.Resonance widthsandpositions. 20
s2
d'
bEr,
E
8
ag(e~)f-1.0-0.3
-15.20.00,71.1
-116'''-732.03.0-5.6
1.8
0.7
0.9~~~5.0
33~~~
p.~'~~
r—3~~~4425
4.04.22.51.2
14141619-15.7-14.0''-12.0—8,2-6.0-5.4
10
d
Erg
Er,
2Qg
2&2
.sq(~~)—1.0-8.6-8.8
0.4
0.4
1.1—0.5-8.3-8.2
0.9
0.8
1.10.00.5-7.8-7.1
-7.7
1.52.6
1611
r=41.p-6.1-6.2
2.0
F1
1.31.5-5.4
-4.1
1.7
3.0
1.03.0
-2.5
1.0
0.2 056/VO10
d
Erg
Er2
2Qg
2&2
sg(c„)-1.0-5.9
—6.0
0.3
0.1
1.2-0.5-5.6-5.9
0.5
0.3
1.20-5.2
-5.6
0.5
0.5
1.30.5
49
—5.0
0.7
0.9
1.21.01.53.0
-4.5
—4.4—1.9
0.7
1.6
1.21.2
0.2
seemstoosmalltoexplainthewholeoftheeffect
inthefirst-principles resultsjustalludedto.
Finally, therearisestheimportant question of
theareaunderthehN(e)curves, whichwillnow
bediscussed.
InFig.5weillustrate hn(e),asdefined inSec.
I,forr,=2(r,=2isverysimilarexceptfora
slightly lowerrelative position of&~).Curves
suchasthatford=1.1mayberegarded asillus-
tratingthephenomenon ofan"incomplete reso-
nance"."'"However, wegainfurther information
byusingtheAnderson-model discussion inSec.I.
Itisseenthat,apartfromthelackofacommon
intersection, curvesd=3.0,1.1,and-1.0inFig.
5haveastrongresemblence tocurvesc=0.08,
0.65,and1.2,respectively, ofFig.1(b).We,
therefore believeitisreasonable tointerpret
theminthesameway.Wecouldtherefore con-
cludethatatd=3.0wehaveanatomiclike reso-
nance,atd=1.1apartlybonding andpartly
atomiclike one,whereas atd=-1.0wehavea
virtual bondingstate.Itisremarkable thatFig.
5resembles socloselycurvesderivedfroma~dina.u.
Energyofmaximum inresonance fromRef.5(eV).
Energyofmaximum inresonance—present work(eV).
"Fullwidthathalf-maximum ofresonance fromRef.5
(eV).
Fullwidthathalf-maximum ofresonance—present
work(eV).
~According topresentwork.FIG.5.hn(e)vseatrs=2.Curves label.edbyvalues
ofd.
narrow-bandAnderson model(Fig.I).
Bycomparison ofFig.5withFig.1(b),it.also
seemsthate,«andalsoe~arereasonably near
thebandcenter. Inaccordance withthediscussion
inSec.I,thequantity hn(ez)shouldthenafforda
singlenumber givinginformation onthebonding
character. Ifhn(e„)isnearunitythereisanoc-
cupiedlocalized bondingstateoroccupiedreso-
nanceofstrongly bondingcharacter, withunoc-
cupiedanti-bonding states, implying adegreeof
covalency. Ontheotherhand,ifbn(c~)isnear
2or0,itimplies, respectively, afullorempty
atomicresonance. Figuresfordn(c~)areadded
toFig.3inbrackets andarefoundinTableI.
Itisseenthatatr,=2thereismoderate bonding
character [hn(ez)=1.36]atd=1.1,theJellium
equilibrium distance. Thedvalueisreduced on
theAlclose-packed surfaces,"'soalthough
bn(e~)ismoreorlessunchanged atasmaller
valueofdsuchas0.'|,weexpectfromFig.5that
theresonance hasmorebondingcharacter. Co-
valentcharacter increases furtherasindicated by
4n(e~)approaching unityatnegative d.Atlarge
d,hn(e~)approaches 2,corresponding toanoc-
cupiedatomicresonance. Similar behavior is
foundatr,=3exceptt".ttheatomicresonance is
unoccupied. Theimportant conclusion ofthisdis-
cussionisthatforapractical systemsuchasH
onAlthereissomething resembling acovalent
bondinthattheoccupied resonance hasstrongly
bondingcharacter.
Ofcourse, thenumerically calculated bnvalues
inFig.2arerequired tobestrictly unitybyself-
consistency. Oursdifferfromunityduetopro-
jection onthel=0subspace. Itisassumed here
thattheeffectsneglected byusaremainlyofthe
19 SIMPLEMODEL OF8YDROGEN AlVDLITHIUM. .. 1279
natureofscreening andthusinformation aboutthe
bondisbestderivedfromtheapproximate 4n.
However, onemaywonder whether thewholeof
suchanotabledisagreement inareahn(&z)be-
tween,saythecaser,=2,d=2inFigs.2(a)and
3(a)canbeexplained inthisway.
B.Lithium
Inthecaseoflithiumthereistheadditional
complication ofa1s'core.Weconsider onlythe
Livalence shellandthusincludethecoreby
meansofa"modelpotential."Forsimplicity,
wehavechosenasquare-well perturbing potential
ratherthan(24),sincethecoreshouldeliminate
theattractive centralregionfromthemodelpo-
tential. Thusourv(r)is
v(r)=-v,',r—R.
Arbitrary units
LW
lh
CF
I
6~a.ui
I
I0.3-I
I
I
I
T
I
I
I
I
I
0
FIG.6.AN(e)forLiadsorbed onjelliumatr,=2.
Brokencurve—Ref.2;fullcurve—presentwork.The
latterarelabeledbythevalueofRused.(25)
8shouldideallybetheatomicradiusofLi,but
asforHasomewhat smallerBandlargerv,was
usedtoreducetheregionofspaceconstrained to
anl=0solution. Awasvariedinordertodeter-
minetheroleitplaysinthecalculation. There-
maining parameter v,wasthenchosensuchthat
theposition oftheresonance coincides withthat
foundbyLangandWilliams intheirfirst-princi-
plescalculations' forLiadsorption onjellium(r,
=2).
TheresultsattheLangandWilliams equilibrium
distance of2.5a.u.fromthejelliumedge'"are
illustrated inFig.6,andcompared tothoseof
thefirst-principles calculations.'Thecurvesare
labeled bytheRvalues. AchangeinRisthusnot
seentoproduceadrasticvariation inthereso-
nancewidth.Thebestagreement withthecalcu-
lationofLangandWilliams wasobtainedforR
=2.5a.u.Incomparison, wenotethattheatomic
radiusofLiis2.9a.u.Thecorresponding value
ofv,wasgivenbyv,=0.45a.u.Thisparticularwellgivesanl=0boundstatesituatedatabout3
eVbelowthevacuumlevel,incomparison tothe
ionization energyof5.4eVforLi.Thereisthus
anupwardshiftof2.4eVofthatlevelattributable
totheimageforceorotherscreening effect.The
valueoftheresonance widthatthisvalueof8
=2.5a.u.isabout2.5eV.Thismakesitlarger
thanthe2s-2penergyseparation of1.85eVfor
Li,"whichnecessarily meansthatthesetwolevels
arestrongly hybridized—aneffectwhichhasbeen
predicted forCsadsorption ontransition metals."
Ourl=0calculation isthusinadequate andmust
beextended totakeaccount ofthel=1solution in-
sidethesphere.
Ifthedistance fromthejellium edgedisal-
lowedtovary(Randv,beingkeptfixedatR=2.5
andv,=0.45a.u.),onefindsthatwhereas thepo-
sitionoftheresonance hardlyvariesatall,the
widthchanges dramatically fromabout4.5eVat
d=2a.u.toabout1.5eVatd=3a.u.Oneshould
remarkatthisstagethattheformersituation is
poorlydescribed withinourmodel, notonlybe-
causeofthenecessity ofinclusion ofanl=1solu-
tioninsidethesphere, butalsobecause thereis
acertain amountofoverlap between thesphere
andthesurfacebarrier. (i.e.,z,=0.7a.u.).On
theotherhand,ourmodelshouldadequately de-
scribethecased=3a.u.
Inclusion ofthesubstrate ionicpseudopotentials
infirst-order perturbation theorywasshownto
leadtoareduction ofdtod=2.1a.u.,'foradsorp-
tionofLiinathreefoldsiteonAl(111). This
valueofdisstillmuchlargerthantheionicra-
diusofLi(1.2a.u.).
Itisalsoofinterest toconsider thecaseof
lower-electron-density substrates suchasthe
casex,=3,sincethisshouldrepresent reasonably
wellthespbandofAgorofNi(r,=3.08corre-
sponding to0.6selectrons perNiatom). Thepa-
rameters representing theadatomproperties are
takenfromabove(R=2.5a.u.,v0=0.45a.u.).A
newfeatureoftheresultsfor~,=3isthepres-
enceofadownward shiftoftheresonance position
asdisdecreased, whichwasabsentinthecase
x,=2.Thisdownward shiftmayberegarded as
abondingshiftandindicates thepossibility that
adsorption mightbelessionicfortheselower-
densitysubstrates.
IV.CONCLUSION
Wehavederivedavirtually analytic expression
forthechangeindensityofstateshN(e)whena
monovalent atomisininteraction withasurface
approximated bythefinitesquarebarrierpoten-
tial.Theatomisrestricted tohaveonlyans-like
valenceorbital; thisisachieved bykeeping only
1280 J.P.MUSCAT ANDD.M.NKWNS 19
thel=0solution inthemuffin-tin-KKB tec'hnique.
Themodelisappliedtohydrogen chemisorption
onjellium, forwhichelaborate self-consistent
firstprinciples calculations areavailable. The
potential insidethehydrogen muffintin,chosen
byconsideration ofthesituation, forhydrogen deep
insidejellium, iskeptindependent bothofdistance
dfromjellium edgeandofr,.Goodoverallagree-
mentwiththefirst-principles calculations wasthen
found,involving reproduction ofthedramatic vari-
ationofwidthandposition oftheresonances asa
function ofd,forx,=2,3,and4.Theassumed
potential seems, however, tobreakdownifthe
hydrogen istoofaroutsidethesurface. Weare
therefore abletoconfirmasimplechemical pic-
tureoftheresonances occurring inthefirst-prin-
ciplescalculations asessentially 1sincharacter.
Wearealsoabletoallotbondingcharacter tothe
resonances. Theyarefoundtobesubstantiallyatomicincharacter whentheprotonlieswellout-
sidethesurface, buttodevelop strongbonding
character neartheequilibrium distance.
Inapplying themodeltolithiumchemisorption
onjellium, againgoodresultswereobtainedfor
theessentially atomic-like 2sresonance bycom-
parison withfirst-principles calculations. Ex-
tensionofourmethodtotreatproblems notsofar
dealtwithinfirst-principles calculations, suchas
a/sorption ofthehigheralkalis,isunderway.
ACKNOW( LEDGMENTS
Wearegrateful toH.Hjelmberg andB.I.Lund-
qvistforproviding uswiththeirresultspriorto
publication. 'Oneofus(J.P.M.)wouldliketoac-
knowledge thesupport oftheScienceResearch
Council througharesearch assistantship.
APPENDIX: CALCULATION OFG{r,r')
~.&o)0
Theintegration overkmustbesplitintotwocontributions arisingfromthek,&k0andk,&k0regions.
Afterexpansion oftheplanewavesinspherical harmonics, andtruncation ofthisexpansion beyondthe
firstterm,wefind
G(rjr')=8m'g''.1+,'—1cos2k,ze+'1--$ sin2k,z,e(k,—k,)j,(k~)j,(6') 2k,'2k,k''~'.
+ZS k0k k
X/2
—;-1cos2k,z,e(k,—k,)
00
where8istheunitstepfunction.
Thecalculation oftheimaginary partofG(r,r')presents nodifficulty, anditisfoundthatImG(r,r)canbewritten intermsofasingleintegral whichinturncanbeevaluated numerically,
fk/00jmG(r,r')=4»kj,(kr)j,(kr')(1+—'''([sx'-1 —kx(»'—2)'i'e(x —1))cssnx
0'(A1)
+2x(1-x*)"'sinnxe(1 —x))C»).
WenowlookattherealpartofG(r,r')Aftercha.ngingtheorderofintegration, wefind(A2)
ReV(r,r')=2
~','"y'dy+k, tf[2x'—1—2x(x'—1)'~'e(x—1)]cos{).x
+2x(l-x')'~'sino[x6(l-x))dx 'yj",y'ydy."j(r)j(r')
e—y'/2(AS)
Thefirsttermcanbeintegrated analytically. Theintegration withrespecttoyinthesecondtermcanbe
written intermsofsineandcosineintegrals. Theseareregular functions everywhere inthecomplex
planeexceptforthesingularities oflogarithmic natureofthecosineintegrals."Onecanthen,byuseof
theresiduetheorem, rewritetheseintermsofsimpler integrals tofind
R»G(r,r')=4»j(kr)(kn(kr')—kj{kr')([kx'—1—2»(x'—()'&'e(x—1)]sinnx
0/A0
2(i.)ie(i.)cess.)C),(A4)
SIMPLE MODE LOFHYDROGEN ANDLITHIUM. .. I28I
wheren,isthespherical Neumann function oforderzero.Ifwenowmakeuseoftheidentity
f[2x'—1—2x(x'—1)'~'e(x—1)]cos(]x+2x(1-x')'~'sinnxe(1-x)]Cx=0,
0
wecanrewriteEqs.(A2)and(A4)intoequation(13)ofthetext.(A5)
o(0
Oncemoretherearetwocontributions toG(x,v')arisingfromthek,&k,andk,)k,regions. Thewave
functionfork,&k,isarealexponential; itcan,however, berewritten intermsofplanewaveswhichwe
thentreatinthemannerdescribed above. Inthiswaywefind
(AB)
where q=[k'—(k'+j(')]''
Theimaginary partofG(x,r')isthengivenby
[2x'&1—2x(1+x')' ']ensnxdx); E&V,.
k4',(in)j,(in")k, 2x(1x')'~-'e"dx;E&V„
ImG(x,x')= Elao
4',(ik~)j,(i~r')k,—+'LK'
0(A7)
Inordertocalculate therealpartofG(x,x'),wehavetochangetheorder.ofintegration asinthecaseof
0.Thereisasupplementary difficulty inthiscaseduetothefactthatwemustintegrate alongallof
therealaxisandpartoftheimaginary axisaswell.Aftersomemanipulation wefind
0'
4trj,(ttrx),+kj,(txx')-[k'x' —1—k'x(x'—1)''e(x—1)]e'dx);EeV,*
ReG(r,r')= ~/a,
-4lli„(t'n)(trn, (ixx')+kj (iev') (2x'+1—kx(x'+1)' ']sinnxdxj; E&V,.
i~/00(AB)
Equation (14)nowfollowsimmediately fromEqs.(A7}and(AB}.
~N.D.LangandA.B.Williams, Phys.Rev.Lett.34,
531(1975).
2N.D.LangandA.B.Williams, Phys.Rev.Lett.37,
212(1976).
3O.Qunnarsson, H.Hjelmberg, andB.I.Lundqvist,
Phys.Rev.Lett.37,292(1976);Surf.Sci.63,348
(1977).
K.Y.Yu,J.N.Miller,P.Chye,W.E.Spicer, N.D.
Land,andA.R.Williams, Phys.Rev.B14,1446
(1976).
H.Hjelmberg, O.Gmmarsson, andB.I.Lundqvist,
Surf.Sci.68,158(1977);H.Hjelmberg (private
communication).
6P.Hohenberg andW.Kohn,phys.Bev.136,B864
(1964);W.KohnandL.J.Sham,Phys.Bev.140,
A1133(1965).
¹D.Lang,SolidStatePhys.28,225(1973).J.Harris andQ.S.Painter, Phys.Bev.Iett.36,151
(1976).
9P.0,Qartland, Surf.Sci.62,183(1977);C.W.B.
Martinson, L.-Q.Petersson, S.A.Flodstrom, andS.B.M.Hagstrom, Proceedings oftheInternational
Symposium onPhotoemission, Noordwijk, 1976(un-
published), p.177.J.R.Anderson, inStructure ofMetallic Catalysts
(Academic, NewYork,1975).
~~D.O.Hayward andB.M.W.Trapnell, inChemisorp-
tion(Butterworths, London,1964).
~B.P.H.Gasser, Chem.Soc.Ann.Bep.LIX,7(1962).
N.D.Lang,Phys.Bev.B4,4234(1972).
~J'.P.Muscat andD.M.Newns,Phys.Lett.A60,348
(1977).J.P.Muscat andD.M.Newns,Phys.Lett.A61481
(1977).
6S.C.Ying, JikR.Smith, andW.Kohn,Phys.Bev.B
11,1483(1975).
~P.W.Anderson, Phys.Rev.124,41(1961).
T.B.Qrimley, Proc.Phys.Soc.London90,751(1967);
D.M.Edwards andD.M.Newns,Phys.Lett.A24,236
(1967);D.M.Newns,Phys.Bev.178,1123(1969).
~A.Blandin, Proceedings oftheInternational Schoolof
Physics EnricoI'ermi, Course37(Academic, New
1282 J.P.MUSCAT A50O.54.5E%NS
York,1967)-,p.393.
~DC.O.Almbladh, U.vonBarth,Z.D.Popovic, and
M.J.Stott,Phys.Rev.B14,2250(1976).
~W.KohnandN.Rostoker, Phys.Rev.94,llll(1954).
~~J.P.Muscat andD.M.Newns,J.Phys.C7,2630
(1974).
~3J.W.Qadzuk, Surf.Sci.6,133(1967);J.P.Muscat
andD.M.Newns, SolidStateCommun. 11,737(1972);
A.C.Hewson andD.M.Newns,Jpn.J.Appl.Phys.
Suppl.2,Pt.2,121(1974).
~4K.T.Whittaker andG. ¹Watson, inCourseofMod-
ensAnalysis {Cambridge U.P.,London,1965),Sec.
6.3.
~5J.Bardeen, Phys.Rev.49,653(1936).
A.J.Bennett andC.B.Duke,Phys.Rev.160,541
(1967).~~L.I.Schiff,inQuantum' Jlfechanics {McGraw-Hill-
Kogakusha, NewYork,1968),p.126.
~8J.Friedel, lecturenotes"D6phasages,"troisi6me
cycledePhysique desSolides, Orsay,1967(unpub-
lished).
G.D.Mahan,phys.Rev.B12,5585(1975).
'OThehorizontal scaleinPig.2ofRef.15wasinad-
vertently inerrorbyafactorof2;inaddition, thed=~leve],satr,=3andr,=4inFig.3ofthisrefer-
encewereslightly inerror.
3~J.B.Pendry,J.phys.C10,809(1977).
Atomic EnergyLevels, Natl.Bur.Stand.Circ.No.
467(U.S.GPO,Washington, D.C.,1949),Vol.I.
-3N.N.Lebedev, inSpecia/Punctions andNeirAppli-
cations(Dover, NewYork,1972),p.34.
|
PhysRevB.95.085310.pdf | PHYSICAL REVIEW B 95, 085310 (2017)
Thermal transport across metal silicide-silicon interfaces:
First-principles calculations and Green’s function transport simulations
Sridhar Sadasivam,1Ning Ye,2Joseph P. Feser,2James Charles,3,4Kai Miao,3,4Tillmann Kubis,3and Timothy S. Fisher1,*
1School of Mechanical Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA
2Department of Mechanical Engineering, University of Delaware, Newark, Delaware, 19716, USA
3Network for Computational Nanotechnology (NCN), Purdue University, West Lafayette, Indiana 47907, USA
4School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA
(Received 12 September 2016; revised manuscript received 12 November 2016; published 22 February 2017)
Heat transfer across metal-semiconductor interfaces involves multiple fundamental transport mechanisms
such as elastic and inelastic phonon scattering, and electron-phonon coupling within the metal and acrossthe interface. The relative contributions of these different transport mechanisms to the interface conductanceremains unclear in the current literature. In this work, we use a combination of first-principles calculations underthe density functional theory framework and heat transport simulations using the atomistic Green’s function(AGF) method to quantitatively predict the contribution of the different scattering mechanisms to the thermalinterface conductance of epitaxial CoSi
2-Si interfaces. An important development in the present work is the direct
computation of interfacial bonding from density functional perturbation theory (DFPT) and hence the avoidanceof commonly used “mixing rules” to obtain the cross-interface force constants from bulk material force constants.Another important algorithmic development is the integration of the recursive Green’s function (RGF) methodwith B ¨uttiker probe scattering that enables computationally efficient simulations of inelastic phonon scattering and
its contribution to the thermal interface conductance. First-principles calculations of electron-phonon couplingreveal that cross-interface energy transfer between metal electrons and atomic vibrations in the semiconductor ismediated by delocalized acoustic phonon modes that extend on both sides of the interface, and phonon modesthat are localized inside the semiconductor region of the interface exhibit negligible coupling with electrons inthe metal. We also provide a direct comparison between simulation predictions and experimental measurementsof thermal interface conductance of epitaxial CoSi
2-Si interfaces using the time-domain thermoreflectance
technique. Importantly, the experimental results, performed across a wide temperature range, only agree wellwith predictions that include all transport processes: elastic and inelastic phonon scattering, electron-phononcoupling in the metal, and electron-phonon coupling across the interface.
DOI: 10.1103/PhysRevB.95.085310
I. INTRODUCTION
Interfaces between heterogeneous materials provide a
plethora of possibilities for the design of devices with en-gineered electronic and optical properties. This work concernsthe study of heat transport across metal-semiconductor hetero-junctions that form a technologically important class of inter-faces used in electronic devices. The understanding of chargeand heat transport through metal contacts to semiconductorchannels is critical to ensure reliable operation of field effecttransistors that form the basic building block of high-powerelectronic devices. Understanding of thermal transport throughmetal-semiconductor interfaces is also important in the designof modern memory storage devices such as heat assisted mag-netic recording [ 1] and phase change memory [ 2]. Apart from
their technological relevance, metal-semiconductor interfacesalso provide a material system in which various physical
mechanisms of heat transport such as elastic interfacial phonon
scattering, inelastic phonon scattering, and electron-phononcoupling co-exist. In this work, we provide a rigorous modelingframework to understand the contribution of various interfacialscattering mechanisms to thermal transport across cobaltsilicide (CoSi
2)–silicon interfaces that are extensively used
in microelectronic devices [ 3].
*tsfisher@purdue.eduElastic scattering of phonons at an interface is the most
widely studied framework to understand and predict thermalinterface conductance at heterojunctions. Under the elastictransport framework, a phonon of energy ¯ hωincident from one
side of an interface is either transmitted across the interface orreflected back into the same material. For elastic interfacialtransport, the primary quantity of interest is the phonontransmission function that represents the probability that aphonon incident from one side of the interface transmits to theother side. Anharmonic or three-phonon scattering processestypically become important at room temperature and above,in which a phonon of energy ¯ hωincident on the interface
could transmit or reflect multiple phonons with appropriateenergies to ensure energy conservation. This mechanism hasbeen postulated to be important in acoustically mismatchedinterfaces such as Pb-diamond [ 4,5].
Electrons are the primary energy carriers in metals, while
phonons are dominant in intrinsic semiconductors. Henceelectron-phonon coupling can be another important energytransfer mechanism that affects thermal interface conductancein metal-semiconductor interfaces. Electron-phonon couplingwithin the metal provides an additional resistance to heattransfer [ 6]. However, electron-phonon coupling across an
interface, i.e., coupling between metal electrons and semi-conductor phonons, provides a parallel heat flow path inaddition to phonon-phonon heat transfer across the interface.Time domain thermoreflectance (TDTR) experiments in the
2469-9950/2017/95(8)/085310(15) 085310-1 ©2017 American Physical SocietySRIDHAR SADASIV AM et al. PHYSICAL REVIEW B 95, 085310 (2017)
FIG. 1. Schematic of various mechanisms involved in heat
transfer between the dominant energy carriers, i.e., electrons in the
metal and phonons in the semiconductor. Phonon-phonon energy
transfer across the interface could involve elastic and inelastic
interfacial scattering processes. Electron-phonon coupling could
involve coupling between electrons in the metal with phonons inthe metal and with phonons in the semiconductor.
literature [ 7,8] suggest that direct electron-phonon coupling
can contribute significantly to heat transport across metal-semiconductor interfaces, and models [ 9–14] have been devel-
oped to quantify its contribution. The different mechanismsof heat transport at a metal-semiconductor interface aresummarized in Fig. 1.
Simplified empirical models are commonly used to in-
terpret experimental thermal conductance data for metal-semiconductor interfaces. Elastic interfacial phonon scat-tering is commonly modeled using the acoustic [ 15] and
diffuse [ 16] mismatch models (AMM, DMM), which are
heuristic approaches applicable for smooth and rough inter-faces respectively. Also, simplifying assumptions such as theDebye approximation to phonon dispersion can compromisethe quantitative accuracy of such models. Even atomisticsimulation approaches for elastic interfacial thermal transportsuch as the atomistic Green’s function (AGF) method ofteninvolve empirical force constant models that can producesignificant discrepancies when compared to calculations thatemploy harmonic force constants obtained from ab initio ap-
proaches [ 17]. The contribution of inelastic phonon scattering
to thermal interface conductance has also been modeled ina simplified manner using heuristic extensions to the elasticdiffuse mismatch model [ 18]. The strength of electron-phonon
coupling is typically modeled using idealized approximationssuch as bulk metal deformation potentials [ 9,19], and such
approximations are expected to be inaccurate for the directcoupling of metal electrons with joint or interface phononmodes. Little work exists on rigorous first-principles determi-nation of the strength of coupling between electrons and jointinterface phonon modes at a metal-semiconductor interface.
Apart from the complexity of various thermal transport
mechanisms described above, the uncertainty in interfacialatomic structure has historically made direct comparisonsbetween simulations and experiments difficult. Much of theexisting experimental data [ 8,20,21] on thermal conductance
of metal-semiconductor interfaces involves materials withmismatched lattice constants, for which the interface atomicstructure is likely to be at least partially amorphous. Ex-
perimental studies that simultaneously characterize interfa-cial atomic structure along with interface conductance arescarce [ 22,23]. However, predictive atomistic transport sim-
ulations that involve first-principles approaches are typicallylimited to crystalline epitaxial interfaces because of theassociated computational tractability. This disparity betweensimulations and experimental studies often makes quantitativecomparisons challenging. To overcome this difficulty, wechoose to work with CoSi
2(metal)–Si (semiconductor) inter-
faces in the present work. Both CoSi 2and Si have FCC lattice
structures with similar lattice constants of 5.36 and 5.43 ˚A,
respectively. Measurements of thermal interface conductanceon CoSi
2(111)/Si (111) interface using the TDTR technique
have been reported in our recent work [ 24], and the interface
has been verified to be epitaxial and smooth using TEMimaging (see Ref. [ 24] for a TEM image of the interface). We
use the same experimental data to compare with the presentsimulation predictions on a lattice-matched CoSi
2(111)/Si
(111) interface; the interfacial atomic configuration was alsochosen to match with the atomic configuration of samplesused in the experiment (see Sec. II Bfor details of the various
interfacial atomic configurations). The close correspondencebetween the atomic structures used in the present work andthe experimental data reported in Ref. [ 24] enables a direct
comparison between simulations and experiments.
Although the primary focus of the present work is the study
of thermal transport across metal-semiconductor interfaces,the methods developed and reported here are also expected tobe useful for a broad class of problems that use the nonequilib-rium Green’s function (NEGF) method for atomistic transportsimulations. From a methodology standpoint, we report aframework that combines first-principles calculations of in-teratomic force constants with the atomistic Green’s functionmethod and evaluate the validity of the “mixing rule” thatis commonly used to approximate interfacial bonding at aheterojunction. The conventional AGF method that is suitablefor elastic phonon transport is extended to include anharmonicphonon scattering using a B ¨uttiker probe approach [ 25]. Since
the B ¨uttiker probe approach is not directly compatible with
the conventional recursive Green’s function (RGF) method(see Sec. II C 1 ), we develop a modification that enables the use
of the RGF method in simulations that involve B ¨uttiker probe
scattering. The new RGF algorithm enables computationallyefficient simulations of phonon-phonon scattering using theB¨uttiker probe approach and is expected to be applicable for
a wide range of problems that require efficient representationof dephasing processes under the NEGF framework. Ab initio
calculations of electron-phonon coupling are also integratedinto the AGF transport simulations.
Apart from the development of new methods, the present
work also provides useful insights into the physics of ther-mal transport across metal-semiconductor junctions. Rigor-ous first-principles calculations indicate that elastic phonontransport underpredicts the experimental data over a widetemperature range. Analysis of the cross-interface heat fluxaccumulation function provides useful insights on the mi-croscopic mechanisms responsible for increased interfaceconductance due to anharmonic phonon scattering in the bulkmaterials forming the interface. First-principles calculations
085310-2THERMAL TRANSPORT ACROSS METAL SILICIDE- . . . PHYSICAL REVIEW B 95, 085310 (2017)
of electron-phonon coupling on an interface supercell along
with a detailed analysis of the contribution from differentkinds of phonon modes to the Eliashberg function revealthat delocalized phonon modes mediate cross-interface energytransfer between metal electrons and the semiconductor lattice.We also obtain an effective length scale of electron-phononinteraction in the semiconductor by comparing simulationpredictions with experimental data and evaluate the accuracyof prior approximations to the length scale of joint or interfacephonon modes.
II. METHODS
A. First-principles calculations
All first-principles calculations in this paper were per-
formed under the framework of density functional theory(DFT) using a plane-wave basis set as implemented in the
QUANTUM ESPRESSO suite of codes [ 26]. Rappe-Rabe-Kaxiras-
Joannopoulos (RRKJ) ultrasoft pseudopotentials were used forboth Co and Si atoms, and the exchange correlation energy wasapproximated under the generalized gradient approximation(GGA) using the Perdew-Burke-Ernzerhof (PBE) functionalform. Three sets of first-principles calculations are performedfor the results reported in this paper; these involve calculationson bulk Si (six atom nonprimitive unit cell along [111]direction), bulk strained CoSi
2(nine atom unit cell along
[111] direction) where a tensile strain is applied along the(111) plane, and a Si (111)-CoSi
2(111) interface supercell.
The relaxed lattice constants of bulk Si and bulk CoSi 2are
5.44 and 5.36 ˚A, respectively. For all simulations considered
in this paper, a tensile strain of 1.5% is applied on CoSi 2along
the (111) plane to match the lattice constants of Si and CoSi 2.
Table Ishows the cutoff energies and k-point grids used for
DFT calculations on all three systems. Structural relaxation iscarried out to reduce the Hellmann-Feynman forces on every
atom below 10
−3eV˚A−1. A full stress relaxation is carried
out for bulk Si while the stresses on bulk strained CoSi 2and
the interface supercell are relaxed only along the transportdirection. CoSi
2is stretched along the in-plane direction to
match its lattice with Si, and hence the in-plane stresses arenot relaxed.
Phonons are analyzed using density functional perturbation
theory (DFPT) where the dynamical matrices are obtained ona4×4×3q-point grid for bulk Si (six atom unit cell along
the [111] direction), bulk strained CoSi
2(nine atom unit cell
along the [111] direction), and a 4 ×4×1q-point grid for the
interface supercell. The real-space interatomic force constants(IFCs) needed for Green’s function transport calculations areobtained from a Fourier transform of the dynamical matrices.B. Coherent phonon transport using the
atomistic Green’s function method
The terms “coherent” and “ballistic” phonon transport
are used interchangeably in the present manuscript andrefer to simulations performed under the conventional AGFframework. These simulations do not model phonon dephasing(cf. coherent) and inelastic phonon scattering (cf. ballistic).However, elastic interfacial scattering, i.e., reflection andtransmission of phonon waves at the interface is included inthis framework. The next section describes modifications tothe conventional AGF approach to model phonon dephasingand inelastic phonon scattering. The details of the AGF methodare available in prior reports [ 27,28], and a brief description is
provided in this section. The basic conceptual framework forthe AGF method involves a “device” region that is connected tosemi-infinite “leads”. The device Green’s function Gis given
by
G(ω)=(ω
2I−Hd−/Sigma11−/Sigma12)−1, (1)
where Hdis the force constant matrix corresponding to the
device region, and /Sigma11,/Sigma12are the contact self-energies. The
contact self-energies are obtained from the surface Green’sfunctions g
1,g2as follows:
/Sigma11=τ1g1τ†
1,/Sigma1 2=τ2g2τ†
2, (2)
where τ1,τ2represent the force constant matrices for in-
teraction between atoms in the device region and the semi-infinite contacts. g
1andg2are the surface Green’s functions
of the contacts that are obtained using the Sancho-Rubiomethod [ 29,30]. The phonon transmission function across the
device is obtained from the Caroli formula:
T(ω)=Tr(/Gamma1
1G/Gamma1 2G†); (3)
where /Gamma11,2=i[/Sigma11,2−/Sigma1†
1,2] denotes the imaginary part of
the contact self-energies and physically represents the phonon“escape rate” [ 28] from the device into the respective contacts.
In all the above expressions, the dependence of the Green’sfunction, contact self-energy, and the transmission functionon the transverse phonon wave vector q
||(for structures
with periodicity in the transverse or in-plane direction)and frequency ωis implicitly assumed. After obtaining the
transmission function, the thermal interface conductance G
Q
can be obtained using the Landauer formula:
GQ=/summationdisplay
q||1
2π/integraldisplay∞
0¯hωT(ω,q||)∂fo
BE
∂Tdω, (4)
where fo
BEis the equilibrium Bose-Einstein distribution func-
tion.
TABLE I. Parameters used for DFT, DFPT calculations on bulk Si, bulk strained CoSi 2, and the Si-CoSi 2interface supercell.
Parameter Bulk Si Bulk strained CoSi 2 Interface supercell
Kinetic energy cutoff (eV) 680 820 820
Charge density cutoff (eV) 6800 8200 8200Electron k-point grid 12 ×12×91 6 ×16×12 16 ×16×1
Phonon q-point grid 4 ×4×34 ×4×34 ×4×1
085310-3SRIDHAR SADASIV AM et al. PHYSICAL REVIEW B 95, 085310 (2017)
C. Inelastic scattering using B ¨uttiker probe approach
The AGF formulation presented in the previous sec-
tion is applicable only for elastic phonon transport, i.e.,anharmonic scattering mechanisms such as umklapp scatteringare not considered. The formulation for extension of AGF toinclude anharmonic phonon scattering has been developed inRef. [ 31]; however, the approach is computationally expensive,
and we are not aware of its application to study phonontransport through realistic three-dimensional crystals. Theauthors recently proposed a phenomenological B ¨uttiker probe
approach to model anharmonic phonon scattering withinthe AGF method [ 25]. The approach is an extension to phonons
of the widely used B ¨uttiker probe method to model inelastic
electron scattering processes in the NEGF framework [ 32–34].
Although the method is heuristic, it provides a computationallyefficient alternative to the self-consistent Born approximation(SCBA) [ 35] that is not phenomenological but is computation-
ally intensive in both memory and time. The essence of theB¨uttiker probe approach involves attaching fictitious contact
probes to every atom in the device, and the temperatures ofthese fictitious contacts are then iteratively solved to ensureenergy conservation in the device region. The B ¨uttiker probes
contribute an additional self-energy to the device Green’sfunction (in addition to the self-energies due to the realcontacts):
G=(ω
2I−Hd−/Sigma11−/Sigma12−/Sigma1BP)−1. (5)
In the present formulation, the B ¨uttiker probe self-energy
is assumed to be a diagonal matrix whose diagonal elementsare of the form
/Sigma1
BP(j,p)(ω)=−i2ω
τ(ω), (6)
where /Sigma1BP(j,p)denotes the B ¨uttiker probe self-energy at atom
jand vibrational direction p(x,y,z). Similar to the matrices
/Gamma11,/Gamma12, we also define /Gamma1BP=i(/Sigma1BP−/Sigma1†
BP) that represents the
imaginary part of the B ¨uttiker probe self-energy. τ(ω) denotes
the frequency dependent scattering time due to umklappscattering and is assumed to be of the form τ
−1(ω)=Aω2
for both Si and CoSi 2. Quadratic frequency dependence of
the umklapp scattering rate has been used in prior studiesinvolving the BTE [ 36] and Landauer approach [ 37,38].
The parameter Ais chosen by fitting (see Ref. [ 39]) to the
experimental thermal conductivity of bulk Si (at differenttemperatures) and bulk CoSi
2(at room temperature). Due to
lack of experimental data on the temperature dependence ofthe lattice thermal conductivity, the scattering parameter A
is assumed to be independent of temperature for CoSi
2.T h i s
assumption is acceptable because the thermal conductivity ofCoSi
2is dominated by electrons, and the interface conductance
is found to exhibit a weak dependence on the lattice thermalconductivity of CoSi
2(see Ref. [ 39]).
Recursive Green’s function method for efficient solution of
B¨uttiker probe temperatures
B¨uttiker probes offer a heuristic but efficient method to
implement scattering in NEGF simulations. However, thepopular recursive Green’s function (RGF) method [ 40] that
avoids full inversion in the calculation of the retarded Green’sfunction Gand the lesser Green’s function G
n=G(/Gamma11+
/Gamma12+/Gamma1BP)G†is not compatible with B ¨uttiker probes. This
incompatibility can be understood from the following equationused to enforce heat current conservation in each B ¨uttiker
probe i:
Q
i=/summationdisplay
j/summationdisplay
q||/integraldisplay∞
0¯hω
2πTr(/Gamma1iG/Gamma1jG†)/bracketleftbig
fo
BE(ω,Ti)
−fo
BE(ω,Tj)/bracketrightbig
dω=0, (7)
where the summation in the variable jruns over all other
B¨uttiker probes ( i/negationslash=j) and the contacts. The computation
of the transmission function matrix Tr( /Gamma1iG/Gamma1jG†) between
every pair of B ¨uttiker probes requires calculation of the full
Green’s function matrix G. The equation for charge current
conservation in electronic transport is similar to the foregoingequation for heat current conservation [ 32,34]. Hence prior
implementations [ 25,32,34]o ft h eB ¨uttiker probe formalism
have employed direct matrix inversion instead of the RGFmethod to calculate the full device Green’s function matrix.
Equation ( 7) enforces the condition that the total integrated
energy flux in each B ¨uttiker probe is zero, i.e., inelastic scat-
tering between different energy levels is allowed. Alternativeimplementations of the B ¨uttiker probe approach invoke energy
flux conservation at each phonon frequency instead of the totalintegrated energy flux over all phonon frequencies [ 32,33].
Such an approach is suitable only for elastic dephasing and ishence not adopted in this work.
Equation ( 7) can be cast in a slightly different form as
Q
i=/summationdisplay
q||/integraldisplay∞
0¯hω
2πTr/parenleftbig
/Sigma1in
iA−/Gamma1iGn/parenrightbig
dω=0, (8)
where /Sigma1in
i=fo
BE(ω,Ti)/Gamma1iandA=i(G−G†) denotes the
spectral function. In the above equation, the matrices /Sigma1in
i
and/Gamma1iare block-diagonal. Hence, only the block-diagonals
ofAandGnneed to be computed, and this can be done using
the RGF algorithm. However, the computation of /Sigma1in
iandGn
require knowledge of the B ¨uttiker probe temperatures. Hence
the use of RGF with B ¨uttiker probes requires that /Sigma1in
iandGn
are recalculated during every Newton iteration of the solution
for B ¨uttiker probe temperatures. Although this step is not
needed in a conventional B ¨uttiker probe implementation with
full matrix inversion, the computational advantage of RGFover full inversion far outweighs the computational expense ofrepeated RGF calculations in every Newton iteration. Also, thememory required to store and invert the full Green’s functionmatrix can become prohibitively large with increasing devicelength.
Anantram et al. [40] provide a detailed discussion of the
RGF methodology for computation of the block diagonals ofGandG
n. Here, we provide only the modifications needed to
combine RGF with the B ¨uttiker probe formalism. The RGF
algorithm for computation of the block-diagonal elementsof the retarded Green’s function Gremains unchanged.
However, the RGF algorithm for computation of G
nrequires
modification to also calculate the derivative of the diagonalelements of G
nwith respect to the B ¨uttiker probe temperatures.
We also assume that all B ¨uttiker probes within a RGF “block”
have the same temperature, i.e., the number of B ¨uttiker probe
085310-4THERMAL TRANSPORT ACROSS METAL SILICIDE- . . . PHYSICAL REVIEW B 95, 085310 (2017)
temperatures that need to be solved is equal to the number of
blocks in the device region.
The equation for left-connected gnLis given by [ 40]
gnLi+1
i+1,i+1=gLi+1
i+1,i+1/parenleftbig
/Sigma1in
i+1,i+1+σin
i+1,i+1/parenrightbig
gLi+1†
i+1,i+1,(9)
where σin
i+1,i+1=Bi+1,ignLi
i,iB†
i,i+1,B=(ω2I−Hd−/Sigma11−
/Sigma12−/Sigma1BP). Our terminology follows that of ref. [ 40] where
gLdenotes the left-connected retarded Green’s function, and
/Sigma1indenotes the lesser self-energy. The derivative of gnLi+1
i+1,i+1
with respect to the B ¨uttiker probe temperature Tjis given by
∂gnLi+1
i+1,i+1
∂Tj
=⎧
⎪⎪⎨
⎪⎪⎩gLi+1
i+1,i+1Bi+1,i∂gnLi
i,i
∂TjB†
i,i+1gLi+1†
i+1,i+1, ifj<i+1,
gLi+1
i+1,i+1/Gamma1BP,i+1gLi+1†
i+1,i+1∂fo
BE(ω,T)
∂T/vextendsingle/vextendsingle
Tj,ifj=i+1,
0, ifj>i+1.
(10)
The equation for Gn
i,iis given by
Gn
i,i=gnLi
i,i+gLi
i,i/parenleftbig
Bi,i+1Gn
i+1,i+1B†
i+1,i/parenrightbig
g†Li
i,i
−/parenleftbig
gnLi
i,iB†
i,i+1G†
i+1,i+Gi,i+1Bi+1,ignLi
i,i/parenrightbig
.(11)
The derivative of Gn
i,iwith respect to B ¨uttiker probe tem-
peratures can be computed using the derivatives of the leftconnected Green’s function g
nLcomputed in Eq. ( 10):
∂Gn
i,i
∂Tj=∂gnLi
i,i
∂Tj+gLi
i,i/parenleftbigg
Bi,i+1∂Gn
i+1,i+1
∂TjB†
i+1,i/parenrightbigg
g†Li
i,i
−/parenleftBigg
∂gnLi
i,i
∂TjB†
i,i+1G†
i+1,i+Gi,i+1Bi+1,i∂gnLi
i,i
∂Tj/parenrightBigg
.(12)
Overall, the RGF algorithm for Gnneeds to be modified to
compute the derivatives of Gnwith respect to the B ¨uttiker
probe temperatures. The new RGF algorithm’s commutationinvolves the following steps:
(1)g
nL1
11=gL1
11/Sigma1in
11gL1†
11,∂gnL1
11
∂T1=gL1
11/Gamma1BP, 1gL1†
11∂fo
BE(ω,T)
∂T|
T1,
∂gnL1
11
∂Tj=0(j> 1).
(2) For i=1,2,..., N −1 and j=1,2,..., N , compute
Eqs. ( 9) and ( 10).
(3)Gn
NN=gnLN
NN,∂Gn
NN
∂Tj=∂gnLN
NN
∂Tjforj=1,2,..., N .
(4) For q=N−1,N−2,..., 1 and j=1,2,..., N ,
compute Eqs. ( 11) and ( 12).
The algorithm proceeds as follows:(1) Start with an initial guess for the B ¨uttiker probe
temperatures.
(2) For each phonon frequency, compute G
R
ii,Gn
ii, and∂Gn
ii
∂Tj
using the RGF algorithm described above.
(3) Compute energy current densities in each B ¨uttiker
probe using Eq. ( 8).
(4) Compute the Jacobian matrix whose ( i,j)thelement is
given by the following equation:
Ji,j=/summationdisplay
q||/integraldisplay∞
0¯hω
2πTr/parenleftBigg
/Gamma1iAii∂fo
BE
∂T/vextendsingle/vextendsingle/vextendsingle/vextendsingle
Tjδij−/Gamma1i∂Gn
ii
∂Tj/parenrightBigg
dω, (13)
where δijis the Kronecker delta function.(5) Update the temperature of B ¨uttiker probes using the
Newton equation:
Tnew=Told−J−1f. (14)
(6) If /bardblTnew−Told/bardbl>/epsilon1, go back to step 1 with the new
guess for B ¨uttiker probe temperatures as Tnew.
An alternative to the Newton-Raphson method is the secant
method in which the exact Jacobian needs to be computedonly in the first iteration. For further iterations, the Jacobiancould be updated using the Broyden’s update formula [ 41], and
this method was also found to give satisfactory convergence.With the secant method, the derivative of the lesser Green’sfunction with respect to B ¨uttiker probe temperatures need only
be computed in the first iteration. For the remaining iterations,the traditional RGF function is sufficient. For large devicelengths, the computation and storage of the full Jacobian matrixcan become prohibitively expensive; an alternative approachinvolves approximation of the Jacobian by a sparse blockdiagonal matrix. Such an approximate Jacobian was also foundto lead to convergence, however, with an increased number ofiterations compared to the exact Jacobian. The approximateJacobian provides a memory-time tradeoff as storage of thesparse block diagonal matrix involves lesser memory but thecomputational time increases relative to the Newton Raphsonscheme with exact Jacobian. All the results presented in thispaper involve the secant method in which the exact Jacobianis computed in the first iteration and the Broyden’s updateformula is used for further iterations.
Figure 2shows a comparison of the computational times for
AGF simulations of bulk silicon with B ¨uttiker probe scattering
using direct inversion and the RGF algorithm described above.The computational times were obtained using
MATLAB , and
both direct inversion and the RGF methods were parallelizedover transverse wave vectors. The computational time for fullmatrix inversion increases rapidly with device length [matrixinversion scales as O(n
3
||n3
z), where n||,nzdenote the number of
atoms per slab and the number of slabs in the transport directionrespectively], while that for the RGF algorithm proposed aboveshows a more gradual scaling with device length [RGF scalesasO(n
3
||nz)].
8 10 12 14 16 18 20020040060080010001200
Length of device (nm)Time (minutes)
Full Inversion
RGF
FIG. 2. Comparison of computational times for different device
lengths obtained using direct inversion and the RGF algorithm.
085310-5SRIDHAR SADASIV AM et al. PHYSICAL REVIEW B 95, 085310 (2017)
Apart from the computational time improvement, another
important advantage of the RGF method over full inversion isthe reduced memory needed to store and invert full Green’sfunction matrices. The device sizes considered in the presentwork (see Sec. IV) are computationally intractable with direct
matrix inversion. Hence the extension of the RGF algorithm toB¨uttiker probes is expected to be critical for application of the
B¨uttiker probe method to realistic device sizes. Also, the results
in Fig. 2confirm that the computational expense of repeated
AGF calculations of G
n(ω;q||) for each RGF iteration is far
less than the computational expense for a single calculationof the full retarded Green’s function of the device G(ω;q
||)
through direct inversion.
D. Fourier diffusion of electrons coupled with phonons
Electrons are the primary heat carriers in metals, and they
transfer energy to phonons near the interface between metaland semiconductor. Intrinsic Si is the semiconductor of interestin this paper, and hence the contribution of electrons in Si tothermal transport is neglected. We neglect any cross-interfaceelectron transport through the CoSi
2-Si interface and consider
diffusive transport of electrons in CoSi 2. Electrons are included
in the AGF simulation within the framework of a two-temperature model that is commonly used to interpret ultrafastlaser experimental data and is also used to model energytransfer between electron and phonon subsystems withinthe Eliashberg function framework. The primary assumptioninvolved in the definition of a local electron and phonontemperature is the existence of electron-electron and phonon-phonon collisions that enable local equilibrium separatelywithin electron and phonon subsystems. The electron andphonon subsystems exchange energy through electron-phononcoupling that is expressed in terms of the Eliashberg function.The steady-state Fourier diffusion equation for electrons in themetal ( x> 0) is given by
k
ed2Te(x)
dx2+Qep(x)=0, (15)
where Qep(x) denotes the volumetric heat source term due to
coupling between electrons and phonons and is expressed interms of the Eliashberg function α
2F(ω) as derived by Allen
in Ref. [ 42]:
Qep(x)=2πD(Ef)/integraldisplay∞
0(¯hω)2α2F(ω)/bracketleftbig
fo
BE(Tp(x))
−fo
BE(Te(x))/bracketrightbig
dω. (16)
In the foregoing equation Te(x),Tp(x) denote the local electron
and lattice temperatures respectively at location x, andD(Ef)
is the electronic density of states at the Fermi energy. The aboveterm can be included as a source term in the B ¨uttiker probe at
location x, i.e., the energy current conservation equation for
thei
thB¨uttiker probe in the metal is given by
/summationdisplay
q||/integraldisplay∞
0¯hω
2πTr(/Sigma1in
iA−/Gamma1iGn)dω+Qep,i/Delta1xi=0,(17)
where /Delta1xiis the length that the B ¨uttiker probe occupies
along the transport direction. In a phonon-only simulation,the total energy current in each B ¨uttiker probe is set tozero to ensure energy flux conservation [see Eq. ( 8)], i.e.,
B¨uttiker probes redistribute the energy, but with no net transfer
of energy between electrons and phonons via the fictitiousB¨uttiker contacts. When electrons are included in the transport
calculation, the total energy current in each B ¨uttiker probe is
set to the electron-phonon energy exchange given by Eq. ( 16).
The local lattice temperature T
pin Eq. ( 16) is obtained by
equating the local phonon energy density to the product ofa local Bose-Einstein distribution at temperature T
pand the
local phonon density of states:
/summationdisplay
q||/integraldisplay∞
0ω2Gn(ω;q||)dω=/summationdisplay
q||/integraldisplay∞
0ω2A(ω;q||)fo
BE(ω,Tp)dω.
(18)
The above equation makes use of the following expressions for
the local phonon number density ρ(ω) and the local phonon
DOSD(ω) in terms of the lesser Green’s function Gn(ω) and
the spectral function A(ω), respectively,
ρ(ω)=/summationdisplay
q||ωGn(ω;q||)
π,D (ω)=/summationdisplay
q||ωA(ω;q||)
π.(19)
Equations ( 15), (17), and ( 18) constitute a set of coupled
nonlinear equations that are solved iteratively to obtain theelectron temperature, the B ¨uttiker probe temperature, and
the local device temperatures. Similar to the methodologyfor B ¨uttiker probe temperatures in Sec. II C 1 , the Newton-
Raphson method is used for the solution of the above equationand details of the algorithm are provided in Ref. [ 39].
III. COHERENT PHONON TRANSPORT
This section contains results for the phonon transmission
function and thermal interface conductance of Si-CoSi 2
interface from ballistic phonon transport calculations (i.e.,B¨uttiker probe scattering turned off). Different interfacial
atomic configurations are possible for the Si-CoSi
2interface
depending on the coordination number of the Co atom closestto the interface (possible values of 5, 7, and 8) and the relativecrystal orientation between the (111) surfaces of Si and CoSi
2.
The “A” type orientation occurs when the Si-CoSi 2stacking is
continuous while the “B” orientation occurs when the CoSi 2
crystal is rotated by 180◦about the [111] direction. Previous
first-principles calculations in the literature [ 43,44] indicate
that the 8A and 8B configurations have the lowest interfacialenergies and are hence the most probable interfacial atomicstructures. Both configurations are considered for the presentphonon transport calculations using AGF.
While bulk IFCs are quite commonly obtained from first-
principles calculations, little work exists on the use of rigorousDFPT calculations to determine the force constants betweenatoms belonging to different materials across a heterogeneousinterface. Force constants (in AGF) and interatomic potentials(in molecular dynamics) between atoms belonging to differentbulk materials are commonly represented using simplifyingapproximations without rigorous calculations of the actualstrength of interfacial bonding [ 45,46]. In the present work,
both bulk and cross-interface IFCs needed for AGF transportsimulations are determined entirely from DFPT calculations.
085310-6THERMAL TRANSPORT ACROSS METAL SILICIDE- . . . PHYSICAL REVIEW B 95, 085310 (2017)
(a)
(b)
FIG. 3. Atomic structures of Si-CoSi 2interface supercells used in
DFPT calculations (two unit cells shown along the in-plane direction
for clarity). The red dotted boxes indicate the region around the
interface for which IFCs are extracted from the interface supercellcalculation. (a) 8A configuration. (b) 8B configuration.
Figures 3(a) and3(b) shows supercells of the 8A and 8B
interfacial atomic configurations respectively. Each supercellcontains two interfaces because of periodic boundary condi-tions. Although DFPT calculations are performed on a finiteinterface supercell, transport simulations are performed ona single Si-CoSi
2interface formed by semi-infinite Si and
CoSi 2crystals. The red dotted boxes enclose atoms around the
interface for which the force constants are obtained from theinterface supercell DFPT calculation. In the atomic structureconsidered for transport calculations, the IFCs for atomsoutside the red dotted box are assumed to equal the bulk IFCsof Si (left of the box) and CoSi
2(right of the box). Results that
illustrate the convergence of cross-interface force constantswith respect to the size of interface supercell are provided inRef. [ 39].
Enforcement of the acoustic sum rules is an important
consideration when IFCs obtained from DFPT calculationsare used in thermal transport simulations. Acoustic sum rulesconstitute a set of translational invariance conditions on theIFCs to ensure that long wavelength acoustic modes of a crystalhave zero vibrational frequency:
/summationdisplay
jHiα,jβ=0, (20)
where i,jdenote atom indices while α,βdenote the directions.
The spatial range of inter-atomic interactions is artificiallytruncated in DFPT by the finite q-point grid used in the
calculations. Although the neglected long-range interactionsare typically small, this procedure results in small violationsof the translational invariance conditions. Hence the raw forceconstants obtained from DFPT do not satisfy the acoustic sumrules exactly and need to be enforced as a post-processingstep on the IFCs [ 47]. Common DFT codes such as Quantum
Espresso automatically enforce translational invariance for theIFCs of the crystal on which DFPT calculations are performed.However, in the present calculations, the IFCs obtained forbulk Si and bulk CoSi
2are combined with that obtained for the
interface supercell. Hence the IFCs for Si and CoSi 2unit cellsnearest to the interface will require modifications to ensure
that the acoustic sum rules are satisfied. In the present work,the diagonal blocks of the force constant matrix are modifiedto ensure that Eq. ( 20) is satisfied.
Cross-interface force constants between heterogeneous
materials are commonly approximated using simplifying as-sumptions due to the computational complexity of performingdirect DFPT calculations on an interface supercell with alarge number of atoms. If the materials on both sides ofthe interface have the same lattice structure such as Si-Geinterfaces, a common approximation is to assume the sameforce constants for both materials with the assumption thatinterfacial scattering is primarily affected by the change inatomic mass across the interface [ 17]. Other approximations
include the use of empirical corrections to obtain the cross-interface force constants from the bulk force constants [ 45].
Another common approximation involves the use of mixingrules to obtain the strength of cross-interface interactions froman average of the bulk material parameters [ 48].
In order to evaluate the validity of a simple averaging
approximation for the cross-interface force constants, Fig. 4(a)
compares the phonon transmission function at normal inci-dence ( q
||=0) when the cross interface IFCs are assumed to
be a simple arithmetic average of the bulk IFCs and when thecross-interface IFCs are obtained from DFPT on the interfacesupercell shown in Fig. 3(a). In the averaging approach, the
cross-interface Co-Si and Si-Si IFCs are obtained by averagingthe interactions in bulk Si and bulk CoSi
2. The averaging
approximation is found to over-estimate the transmissionfunction for most of the frequency range except at very lowfrequencies or long wavelengths [see inset in Fig. 4(a)] where
the predictions from both the average and DFPT IFCs convergeto the acoustic mismatch limit. Long-wavelength phonons areinsensitive to the local details of interfacial bonding, andhence the transmission function at low phonon frequenciesis expected to be insensitive to the exact interfacial forceconstants. However, rigorous predictions of cross-interfaceforce constants are necessary for accurate prediction of trans-mission at higher frequencies that are expected to dominatephonon transport at room temperature and beyond. The thermalinterface conductance computed directly from Eq. ( 4) includes
contributions from the ballistic contact conductances at thecontact-device interfaces in addition to the conductance ofthe Si-CoSi
2interface in the middle of the device region. To
obtain the conductance of the Si-CoSi 2interface alone, we
use the following expression to subtract the ballistic contactresistances from the total resistance obtained from Eq. ( 4)[17]:
G
/prime
Q(T)=GQ(T)
1−1
2/parenleftbigGQ(T)
GQ,Si(T)+GQ(T)
GQ,CoSi2(T)/parenrightbig, (21)
where GQ(T) is the interface conductance computed from
Eq. ( 4), and G/prime
Q(T) is the thermal interface conductance of a
single Si-CoSi 2interface and plotted in Fig. 4(b).GQ,Si(T) and
GQ,CoSi 2(T) are the ballistic conductances of homogeneous Si
and CoSi 2slabs, respectively.
The use of a simple arithmetic average for cross-interface
IFCs over-estimates the thermal interface conductance [seeFig. 4(b)] by almost 70% at room temperature, and the
errors increase at higher temperatures. Hence the prediction
085310-7SRIDHAR SADASIV AM et al. PHYSICAL REVIEW B 95, 085310 (2017)
FIG. 4. Results from ballistic phonon transport calculations for
aS i - C o S i 2interface. (a) Phonon transmission function at normal
incidence computed using average and DFPT force constants for
the 8A interface. The inset shows the same graph for small phonon
frequencies or long wavelengths. (b) Thermal interface conductance
for 8A and 8B Si-CoSi 2interfaces.
of phonon thermal interface conductance for temperatures
beyond a few tens of degree Kelvin requires the rigorous pre-diction of interfacial bonding strength, and simple averagingapproximations are not expected to be quantitatively accurate.Similar conclusions on the overestimation of interface conduc-tance due to simple approximations that neglect local changesin the force field near a heterogeneous interface were found inRef. [ 49].
IV . EFFECT OF ANHARMONIC SCATTERING ON
THERMAL INTERFACE CONDUCTANCE
In this section, the effect of anharmonic phonon scattering
in Si and CoSi 2on the thermal interface conductance is
presented. This section contrasts with results in the previoussection for which phonon transport in Si and CoSi
2wereassumed to be ballistic. The B ¨uttiker probe scattering rates
for both Si and CoSi 2were assumed to be of the form
τ−1(ω)=Aω2, and the parameter Awas fitted to obtain the
bulk thermal conductivity of Si and CoSi 2(see Ref. [ 39]). Since
Si has a relatively high phonon thermal conductivity comparedto CoSi
2, this circumstance corresponds to a low scattering
rate on the Si side of the interface and a high scattering rate inCoSi
2. To understand the effect of bulk scattering rates on the
interface conductance, we also performed simulations in whichthe scattering rate in CoSi
2is reduced by a factor of 100, while
the Si scattering rate is maintained the same (case A) and thescattering rate in Si is increased by a factor of 100 while thatin CoSi
2is maintained the same (case C). Case B corresponds
to the nominal scattering rate in both Si and CoSi 2, i.e., the
scattering rate that reproduces the bulk thermal conductivityof Si and CoSi
2. The present simulations assume that all the
B¨uttiker probes on the Si and CoSi 2sides of the interface
have scattering rates of bulk Si and bulk CoSi 2, respectively.
However, the local anharmonicity near the Si-CoSi 2interface
is likely to differ from the bulk anharmonicities of Si andCoSi
2. Future work on determining the change in umklapp
scattering rates near a heterogeneous interface is needed toimprove the present model.
Figures 5(a)–5(c) show the local device temperature profile
corresponding to all three cases and the associated thermalinterface conductance. A temperature difference of 10 Kis applied across the leads in all cases. The conductanceis enhanced with increase in the bulk scattering rates ofthe materials comprising the interface. As expected fromconventional scattering theory, the bulk material conductanceshowever decrease with increased scattering rates [observe theprogressive rise in temperature drops within Si and CoSi
2
in Figs. 5(a)–5(c)]. The foregoing results suggest that the
inclusion of inelastic phonon scattering in the AGF simulationsproduces contrasting effects on the interface and bulk materialconductances.
To elucidate the microscopic mechanisms responsible for
the enhancement in interface conductance with inelasticscattering, Figs. 5(d)–5(f) show the spectral variation of heat
flux from the Si and CoSi
2contacts. For case A with low
scattering on both sides of the interface, the spectral heatfluxes from the two contacts follow each other, suggestingthat “vertical transport”, i.e., mixing between different energylevels is insignificant. However, higher scattering rates resultin a shift of the frequencies at which the spectral heatflux is a maximum. Also, the maximum allowed phononfrequency in strained CoSi
2is about 7 ×1013rad s−1, while
that in Si is close to 1014rad s−1. Hence the phonons in
Si between 7 ×1013and 1014rad s−1do not contribute to
cross-interface heat transport in a ballistic simulation [seeFig. 4(a)]. Inelastic scattering enables phonon scattering into
the high-energy optical modes of Si whose contribution isenhanced with a rise in the scattering rates of Si and CoSi
2.
The elevated participation of high-energy optical phonons inSi can also be observed in Figs. 5(g)–5(i) where phonons
with frequencies larger than 7 ×10
13rad s−1contribute 8%,
10%, and 18% of the total energy flux in Si for casesA, B, and C, respectively. Although the spectral heat fluxshows spatial variation, the total energy flux integrated overall phonon frequencies is independent of position. Similar
085310-8THERMAL TRANSPORT ACROSS METAL SILICIDE- . . . PHYSICAL REVIEW B 95, 085310 (2017)
−30 −20 −10 0 10 20 30300302304306308310
Distance (nm)Temperature (K) SiCoSi2GQ = 310 MW/m2/KCase A
(a)−30 −20 −10 0 10 20 30300302304306308310
Distance (nm)Temperature (K)Si CoSi2GQ = 520 MW/m2/KCase B
(b)−30 −20 −10 0 10 20 30300302304306308310
Distance (nm)Temperature (K)Case C
GQ = 850 MW/m2/K
SiCoSi2
(c)
0 2 4 6 8 10
x 101300.20.40.60.81x 10−4
Phonon frequency (rad/s)J(ω) (W/m2/rad/s)Case ASi contact
CoSi2 contact
(d)0 2 4 6 8 10
x 101301234x 10−5
Phonon frequency (rad/s)J(ω) (W/m2/rad/s)Case BSi contact
CoSi2 contact
(e)0 2 4 6 8 10
x 101300.511.52x 10−5
Phonon frequency (rad/s)J(ω) (W/m2/rad/s)Case CSi contact
CoSi2 contact
(f)
0 2 4 6 8 10
x 101300.20.40.60.81
Phonon frequency (rad/s)Heat flux accumulation92%
Case A
Si contact
CoSi2 contact
(g)0 2 4 6 8 10
x 101300.20.40.60.81
Phonon frequency (rad/s)Heat flux accumulation90%
Case B
Si contact
CoSi2 contact
(h)0 2 4 6 8 10
x 101300.20.40.60.81
Phonon frequency (rad/s)Heat flux accumulation82%
Case C
Si contact
CoSi2 contact
(i)
FIG. 5. (a)–(c) Device temperature profile for cases A, B, and C, where case B corresponds to nominal scattering rates in Si and CoSi 2,
while cases A and C correspond to artificially decreased and increased scattering rates, respectively. The magenta lines correspond to linear
fits of the temperature profiles on either side of the interface. (d)–(f) Spectral variation of the energy flux from Si and CoSi 2contacts for cases
A, B, and C, respectively. (g)–(i) Accumulation of energy flux in the Si and CoSi 2contacts with respect to phonon frequency for cases A, B,
and C, respectively.
conclusions on the enhancement of thermal interface conduc-
tance due to inelastic scattering have been reported in priorwork [ 4,5,50].
V . EFFECT OF ELECTRON-PHONON COUPLING ON
THERMAL INTERFACE CONDUCTANCE
A. First-principles calculations
The results from first-principles calculations of electron-
phonon coupling, both in bulk strained CoSi 2and Si-CoSi 2
interface supercells, are reported in this section. The phononlinewidth γ
qpdue to electron-phonon scattering is givenby [42,51]
γqp=2πω qp/summationdisplay
νν/prime/integraldisplayd3k
/Omega1BZ/vextendsingle/vextendsinglegqp
kν,k+qν/prime/vextendsingle/vextendsingle2δ(Ekν−Ef)
×δ(Ek+qν/prime−Ef), (22)
where gqp
kν,k+qν/primeis the electron-phonon coupling matrix ele-
ment for scattering of an electron with energy Ekin band νby
a phonon of energy ¯ hωqpinto a state with energy Ek+qin band
ν/prime. The above expression is valid at low temperatures when
electron-phonon scattering is restricted to a narrow energywindow around the Fermi surface. The phonon linewidth canbe used to compute the spectral Eliashberg function α
2F(ω)
085310-9SRIDHAR SADASIV AM et al. PHYSICAL REVIEW B 95, 085310 (2017)
which quantifies the strength of electron-phonon coupling:
α2F(ω)=1
2πD(Ef)/summationdisplay
q,pγqp
¯hωqpδ(ω−ωqp). (23)
The spectral Eliashberg function can be used to obtain
an effective volumetric electron-phonon coupling coefficientGep:
Gep=2πD(Ef)/integraldisplay∞
0(¯hω)2α2F(ω)∂fo
BE
∂Tdω. (24)
To understand the spatial variation of electron-phonon cou-
pling across a semiconductor-metal interface, we also define alocal Eliashberg function α
2Fl(ω)a s[ 52]
α2F(ω)=1
2πD(Ef)/summationdisplay
q,pγqp
¯hωqpδ(ω−ωqp)
=1
2πD(Ef)/summationdisplay
q,pγqp
¯hωqpδ(ω−ωqp)/summationdisplay
l/summationdisplay
m=x,y,zφqp,lmφ∗
qp,lm/parenleftBigg/summationdisplay
l/summationdisplay
m=x,y,zφqp,lmφ∗
qp,lm=1/parenrightBigg
=1
2πD(Ef)/summationdisplay
l/summationdisplay
m=x,y,z/summationdisplay
q,pγqp
¯hωqpδ(ω−ωqp)φqp,lφ∗
qp,l=/summationdisplay
lα2Fl(ω), (25)
where φqpdenotes the phonon eigenvector, the index lruns
over all the atoms in the unit cell, and the index mrepresents
the vibrational degrees of freedom ( x,y,z) for each atom.
The local Eliashberg function is then used to define a localvolumetric electron-phonon coupling coefficient G
ep,l:
Gep,l=/parenleftbigg
2πD(Ef)/integraldisplay∞
0(¯hω)2α2Fl(ω)∂fo
BE
∂Tdω/parenrightbiggVunit cell
Vl,
(26)
where the additional factor Vunit cell/Vlensures that Gep,lis
the local volumetric coupling coefficient around atom lthat
occupies a volume Vl.
The Eliashberg functions of bulk strained CoSi 2and the
interface supercell calculated from DFPT are provided in theRef. [ 39] along with a discussion on convergence with respect
tok-point grid and smearing parameters. Equations ( 22)
and ( 23) are appropriate for bulk materials in which transla-
tional periodicity is assumed in the scattering matrix elementsg
qp
kν,k+qν/prime. These equations also apply for the Si-CoSi 2inter-
face supercells because these supercells represent Si-CoSi 2
superlattices with periodicities of the order of a few nm. TheEliashberg function for the supercells physically representsthe coupling between electron and phonon modes of theSi-CoSi
2superlattice. To ensure that the electron-phonon
coupling coefficients obtained from DFT/DFPT calculationson superlattices are transferrable to transport simulations ofa single Si-CoSi
2interface, we performed calculations on
a series of Si-CoSi 2supercells with varying Si and CoSi 2
slab thicknesses. Figure 6(a) shows the spatial variation of
the electron-phonon coupling coefficient for three interfacesupercells (SC) of the 8B configuration with different lengthsof the Si and CoSi
2slabs forming the interface. The local
coupling coefficient on the CoSi 2side of the interface is
averaged over one Co and two Si atoms to remove atomisticfluctuations in the local coupling coefficient. The electron-phonon coupling coefficients for bulk strained CoSi
2and bulk
intrinsic Si are also shown in Fig. 6(a). The electron-phonon
coupling in intrinsic bulk Si is zero since the Fermi levellies in the middle of the band gap, and the delta functions
around the Fermi surface in Eq. ( 22) are zero. An important
observation from Fig. 6(a) is the appearance of a nonzero
coupling coefficient on the Si side of the interface. Also, themagnitude of this coupling is approximately constant in Sibeyond two atomic layers from the interface for all threesupercells considered in Fig. 6(a). The convergence of the
plateau on the Si side of the interface for supercells SC2 andSC3 in Fig. 6(a) indicates that the electronic wave functions
of CoSi
2are sufficiently localized within the CoSi 2slab and
do not tunnel across the Si slabs of the superlattice. TheSi-CoSi
2interface forms a Schottky barrier with a p-type
Schottky barrier height of 0.2 eV . Hence the local electronicDOS at the Fermi level decays rapidly in Si away from theSi-CoSi
2interface [see Fig. 6(b)]. However, such a two-order-
of-magnitude decay in local electronic DOS does not result in acommensurate reduction in the local electron-phonon couplingcoefficient shown in Fig. 6(a).
This unusual result can be understood by considering the
relative contributions of different types of phonon modes of theSi-CoSi
2interface supercell to the overall Eliashberg function.
The different phonon modes of the interface supercell areclassified into four types based on the spatial localization of thephonon eigenvector corresponding to the mode. The presentapproach is analogous to the classification of interface modesin Ref. [ 53]. The interface supercell [SC 2 in Fig. 6(a)]s h o w n
in Fig. 7(a) is decomposed into three regions consisting of Si,
CoSi
2, and interfacial atoms. The criteria for classification of
a phonon mode φqpare defined as follows:
φqp=⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩Si mode, if
/bardblφqp,Si/bardbl
/bardblφqp,tot/bardbl>0.85,
CoSi 2mode, if/bardblφqp,CoSi2/bardbl
/bardblφqp,tot/bardbl>0.85,
interfacial mode, if/bardblφqp,int/bardbl
/bardblφqp,tot/bardbl>0.85,
delocalized mode, if none of the above,(27)
where /bardblφqp,Si/bardbl,/bardblφqp,CoSi 2/bardbl, and/bardblφqp,int/bardbldenote the norm of
the phonon eigenvector within the Si, CoSi 2and interfacial
regions respectively in Fig. 7(a)./bardblφqp,tot/bardbldenotes the norm
085310-10THERMAL TRANSPORT ACROSS METAL SILICIDE- . . . PHYSICAL REVIEW B 95, 085310 (2017)
−4 −3 −2 −1 0 1 200.511.522.53x 1017
Distance along supercell (nm)Gep (W/m3/K)SiCoSi2
InterfaceGep for bulk
strained CoSi2
Gep for
bulk Si SC1
SC2 SC3
−2 −1 0 1 210−410−310−210−1100
Distance along supercell (nm)LDOS at Ef (arb. units)Si CoSi2
interface
location
(a) (b)
FIG. 6. (a) Spatial variation of the electron-phonon coupling coefficient Gepacross a Si-CoSi 2interface for different supercell lengths. The
coupling coefficient in bulk strained CoSi 2and bulk Si are also shown for comparison. (b) Spatial variation of the local electron DOS at the
Fermi energy across the Si-CoSi 2structure shown in Fig. 3(c).
of the phonon eigenvector of the entire interface supercell and
is normalized to unity. The choice of spatial extent of theinterfacial region and the value 0.85 in Eq. ( 27) are arbitrary
and used only to provide a physical understanding of themechanism of cross-interface coupling between electrons inmetal and phonons in the semiconductor.
The contribution of the different types of phonon modes
to the total phonon DOS and the total Eliashberg functionof the interface supercell are shown in Figs. 7(b) and 7(c),
respectively. Figure 7(b) indicates that phonon modes in the
frequency range of ω=(8−10)×10
13rad s−1are localized
in the Si region of the interface. These high-frequencymodes correspond to optical modes of Si and are above
the maximum allowed phonon frequency of bulk CoSi 2.
Although the optical modes of Si contribute to phononDOS of the interface supercell, their contribution to theEliashberg function shown in Fig. 7(c) is negligible. This result
demonstrates that modes localized in Si do not couple withmetal electrons. However, the significant volumetric couplingcoefficient on the Si side of the interface in Fig. 6(a) can
be attributed to delocalized modes whose vibrational energyis distributed across Si and CoSi
2atoms of the interface
supercell. Metal electrons transfer energy to delocalizedphonon modes whose vibrational patterns dictate that a portion
0 2 4 6 8 10 12
x 101300.10.20.30.40.50.6
ω (rad/s)Phonon DOS (arb.units)Si modesinterfacial
modes delocalized
modes CoSi2 modes
0 2 4 6 8 10 12
x 101300.050.10.150.20.25
ω (rad/s)α2F(ω)CoSi2 modes
Si
modesinterfacial
modes delocalized
modes
(a)
(b)( c)
FIG. 7. (a) Partitioning of different regions in the Si-CoSi 2interface supercell used for the classification of phonon modes. (b) and (c)
Contribution of Si, CoSi 2, interfacial, and delocalized phonon modes to the total DOS (b) and Eliashberg function (c) of the Si-CoSi 2interface
supercell.
085310-11SRIDHAR SADASIV AM et al. PHYSICAL REVIEW B 95, 085310 (2017)
of the energy is transferred to silicon atoms across the
interface.
Hence our results suggest that energy exchange between
electrons in metal and atomic vibrations in the semiconductoris manifested primarily by the coupling between electrons anddelocalized interface modes whose vibrational energy is dis-tributed across Si and CoSi
2atoms. An important implication
of this result is that strength of direct electron-phonon couplingis intimately tied to the strength of interfacial bonding andthe phonon-phonon conductance across the interface. For aninterface with weak or van der Waals bonding, the contributionof such delocalized modes to phonon DOS is expected to bemuch smaller, and the phonon modes will be localized oneither side of the interface.
Figure 7(c) also suggests that the mechanism of energy
transfer from metal electrons to phonons in Si is primarilymediated by acoustic delocalized phonon modes and thecontribution from coupling between electrons and opticalmodes of Si is negligible. This result contrasts with electron-phonon coupling in bulk Si where the contributions fromacoustic and optical modes are similar in magnitude (seeSec. VIin Ref. [ 39]). In the interface supercell considered
here, optical modes of Si are localized to the Si side of theinterface where the electron DOS at Fermi level is very small[see Fig. 6(b)]. The acoustic modes in Si are delocalized
with the acoustic modes of CoSi
2, leading to their stronger
coupling with electrons in CoSi 2. Figure 7(c)also indicates that
coupling between metal electrons and CoSi 2optical phonon
modes contributes significantly to the Eliashberg function ofthe interface supercell. However, such coupling is localizedwithin the metal and contributes little to energy transfer acrossthe interface. Localized interfacial modes, i.e., modes withvibrational energy localized to a few atomic layers around theinterface are observed to contribute to the Eliashberg functionin a small frequency range ω=(7–8)×10
13rad s−1.
B. Effect of electron-phonon coupling on
thermal interface conductance
In this section, results from first-principles calculations
of electron-phonon coupling are incorporated into the AGFtransport simulations. The details of the approach are describedin Sec. II D and Ref. [ 39]. The primary difference between
the results presented here and those in Sec. IVis the
presence of nonzero energy fluxes in the B ¨uttiker probes
to represent the energy exchanged between electrons andphonons. Hence the simulation results presented in this sectioninclude contributions from both anharmonic phonon scatteringand electron-phonon coupling.
We consider first the case where electrons exchange energy
only through the B ¨uttiker probes in the metal, i.e., no
direct coupling between metal electrons and semiconductorphonons. The Eliashberg function of bulk strained CoSi
2is
used to calculate the energy exchange term Qepin Eq. ( 16).
Figure 8(a) shows a typical electron and lattice temperature
profile obtained from such a simulation along with the heatfluxes from the various B ¨uttiker probes in Si and CoSi
2[see
Fig. 8(c)]. The heat fluxes in all the B ¨uttiker probes on the
Si side of the interface are zero while the heat fluxes in theB¨uttiker probes of CoSi
2decrease away from the interface.This decay in the electron-phonon energy transfer away from
the interface is a consequence of the equilibrium betweenelectrons and phonons away from the interfacial region [see atemperature profile in Fig. 8(a)].
Also, comparison of the lattice temperature profiles in
Fig. 5(b) with that in Fig. 8(a) shows that for the same
scattering rates and applied temperature difference across theSi and CoSi
2contacts, the lattice temperature drop in CoSi 2
is reduced when electrons are included in the simulation. Thereduced lattice temperature drop in CoSi
2is a consequence of
electrons in metal providing a parallel heat flow path with lowerresistance compared to phonons ( κ
e,CoSi 2=46 W m−1K−1and
κp,CoSi 2=4.9Wm−1K−1). Hence a significant fraction of
energy in CoSi 2is carried by electrons that transfer energy
to the lattice near the metal-semiconductor interface.
The present simulation is conceptually similar to the
analytical model developed by Majumdar and Reddy [ 6]w h o
suggested that electron-phonon coupling within the metaleffectively provides a resistance in series with the phonon-phonon resistance across the interface. Hence the interfaceconductance in Fig. 8(a) is smaller than the phonon-only
conductance in Fig. 5(b). Majumdar and Reddy’s model for the
effective conductance with electron-phonon coupling is givenby
G
Q=/radicalbigGepκp
1+√
Gepκp
Gpp, (28)
where Gepis the effective electron-phonon coupling effi-
cient in the metal, κpis the lattice thermal conductivity of
the metal, and Gppis the phonon interfacial conductance.
The electron-phonon coupling coefficient in bulk CoSi 2is
3.1×1017Wm−3K−1[see Fig. 6(a)],κp=4.9Wm−1K−1,
andGpp=5.2×108Wm−2K−1[see Fig. 5(b)]. Substituting
these values in Eq. ( 28), we obtain GQ=365 MW m−2K−1,
which is close to the value from the simulation in Fig. 8(a).
Although the temperature profiles presented so far in Figs. 5
and 8involve conditions near room temperature, similar
simulations were also performed at temperatures of 100, 150,200, and 250 K to obtain the temperature dependence ofinterface conductance. At each temperature, the B ¨uttiker probe
scattering rate in Si was changed to match the bulk thermalconductivity corresponding to that temperature (see Ref. [ 39]).
Figure 9shows a comparison of simulation predictions with
experimental measurements using the time-domain thermore-flectance (TDTR) technique [ 24].
The simulation predictions using the various models are
presented to provide a quantitative understanding of thecontributions from each heat transfer mechanism to thethermal interface conductance. Ballistic AGF simulationswith only coherent interface scattering (black solid curvedenoted by “A” in Fig. 9) underpredict the thermal interface
conductance for all temperatures with a 33% difference atroom temperature. Also, an elastic transport model doesnot capture the temperature dependence of the interfaceconductance. Experimental data suggests that the thermalinterface conductance increases by 37% from 150 K to roomtemperature; however, the AGF simulation predicts a modest15% increase in interface conductance for the same change intemperature. The stronger dependence of the experimental data
085310-12THERMAL TRANSPORT ACROSS METAL SILICIDE- . . . PHYSICAL REVIEW B 95, 085310 (2017)
−30 −20 −10 0 10 20 30300302304306308310312
Distance (nm)Temperature (K)CoSi2SiTe
TpGQ = 359 MW/m2/K
−30 −20 −10 0 10 20 30300302304306308310312
Distance (nm)Temperature (K)GQ = 437 MW/m2/K
TpTe
Si CoSi2
−30 −20 −10 0 10 20 300246810x 108
Distance (nm)BP energy flux (W/m2)
Si CoSi2due to cross−interface
e−ph couplingno direct
coupling
with direct
coupling (1.9 nm)(a)
(c)(b)
FIG. 8. (a) Electron and lattice temperature profile across Si-CoSi 2interface with electron-phonon coupling inside the metal region only.
(b) Electron and lattice temperature profile across Si-CoSi 2interface with electron-phonon coupling inside the metal region and in two unit
cells of Si closest to the interface. In both (a) and (b), the red line corresponds to a linear fit of the lattice temperature profile in Si and the
green line corresponds to a linear fit of the electron temperature profile in CoSi 2away from the interface region. (c) Heat flux distribution
in the B ¨uttiker probes across the Si-CoSi 2interface corresponding to the temperature profiles in (a) and (b). For the simulation with direct
electron-phonon coupling, the first B ¨uttiker probe in Si closest to the interface has a nonzero energy flux.
on temperature suggests the importance of inelastic scattering
processes in cross-interface energy transport. The inclusionof inelastic phonon scattering (magenta curve with circlesdenoted by “B” in Fig. 9) in the AGF simulations increases
the interface conductance by about 80% at room temperature,and the simulation predictions are closer to experimental data.However, if electrons in metal are also considered in thesimulation with electron-phonon coupling limited to the metalregion only (red curve with hexagrams denoted by “C” inFig. 9), the thermal interface conductance decreases by about
30% at room temperature, and the simulation under-predictsthe experimental data. We note that this simulation considersthe contributions from both anharmonic phonon scattering andelectron-phonon coupling within the metal.
The DFPT calculations of electron-phonon coupling pre-
sented in the previous section do not consider anharmonicityof phonon modes in the interface supercell. In a single Si-CoSi
2
interface with semi-infinite Si and CoSi 2slabs on either side,
the interface phonon modes will be localized around theinterface. The spatial extent of these modes will depend onthe anharmonic interaction strength with bulk Si and bulkCoSi
2modes. The local electron-phonon coupling coefficient
Gepis expected to equal the bulk values for Si and CoSi 2beyond the spatial extent of these interface modes. Differentapproximations for the extent of joint or interface phononmodes have been proposed in the literature. Huberman andOverhauser [ 9] proposed that the joint modes extend to a
distance equal to the bulk mean free path of the materialsforming the interface. For Si, the average phonon meanfree path is of the order of 40 nm and the use of thislength predicts a large contribution to thermal transport fromcross-interface electron-phonon coupling [ 52]. Results from
application of the analytical model developed by Hubermanand Overhauser to the present Si-CoSi
2interface is discussed
in the Ref. [ 39]. More recently, Lu et al. [54] argued that the
extent of interfacial phonon modes should equal the distanceover which the temperature profile obtained in moleculardynamics simulations is nonlinear. This length is typicallyof the order of 1–2 nm, and this model predicts a much smallercontribution of cross-interface electron-phonon coupling tointerface conductance. In the present work, we obtain anapproximate estimate of this length by fitting the simulationpredictions to experimental data.
With the assumption that cross-interface electron-phonon
coupling is responsible for the difference between experimen-tal data and the simulation results represented by the red curve
085310-13SRIDHAR SADASIV AM et al. PHYSICAL REVIEW B 95, 085310 (2017)
0 100 200 300 4000100200300400500600
Temperature (K)G (MW/m2/K)
AExperimentB
D
C
FIG. 9. Comparison of simulation predictions with experimental
data (blue squares with error bars). A (black solid curve): phonon-
only simulation with elastic interface scattering. B (magenta circles):phonon-only simulation with anharmonic phonon scattering in both
Si and CoSi
2. C (red hexagrams): electrons and phonons considered in
the simulation with electron-phonon energy transfer inside the metalregion only. D (green diamonds): electrons and phonons considered
in the simulation with electron-phonon energy transfer included in
two (1.9 nm) unit cells of Si closest to the interface.
in Fig. 9, we use the coupling coefficient on the Si side of
the interface [see Fig. 6(a)] to model energy transfer between
electrons in metal and the semiconductor lattice. Curve “D” inFig.9represents the thermal interface conductance obtained by
coupling electrons in metal with two unit cells of Si closest tothe interface along the transport direction. Direct coupling withtwo unit cells of Si, which represents a length of approximately1.9 nm, is found to be sufficient to obtain a close match withexperimental data at various temperatures. The close matchto experimental data suggests that the extent of joint interfacemodes in Si is much smaller than the bulk mean free pathof Si. The small spatial extent of joint modes is likely dueto the increased anharmonicity of interfacial phonon modesas compared to the bulk phonon modes. Similar conclusionsregarding increased anharmonicity of the interfacial regionare discussed in Ref. [ 55] by computing the anharmonic
contribution to the potential energy of interfacial atoms inSi/Ge interfaces. The temperature profile corresponding tothe simulation with direct electron-phonon coupling [seeFig. 8(b)] is similar to that obtained from the simulation
with electron-phonon coupling only in the metal region [seeFig. 8(a)]. However, the nonzero energy flux in the B ¨uttiker
probe closest to the interface in Si [see Fig. 8(c)] is indicative
of direct electron-phonon energy transfer, and this effect con-tributes to the enhancement in thermal interface conductance.VI. CONCLUSIONS
This work reports first-principles calculations of phonons
and electron-phonon coupling at a Si-CoSi 2interface and
compares simulation predictions of thermal interface con-ductance to experimental measurements using the TDTRtechnique. TEM imaging of the Si-CoSi
2interface confirms
the epitaxial nature of the interface and thus enables aquantitative comparison between simulation and experiment.From a methodology standpoint, important contributions fromthe present work include the development of computationallyefficient methods to include inelastic phonon scattering in aGreen’s function transport simulation and the incorporation ofresults from first-principles calculations of electron-phononcoupling into the AGF framework. We also evaluate thevalidity of the “mixing rule”, a heuristic approximation tointerfacial bonding at heterojunctions, using comparisons toresults obtained from rigorous first-principles calculationsof interfacial bonding, and find that simple averaging ofinterfacial force constants can result in errors of approximately100% in thermal interface conductance at room temperature.
Elastic scattering of phonons at an interface is the most
widely used framework to understand and predict the ther-mal interface conductance of heterojunctions, but the needto include inelastic phonon and coupled electron-phononprocesses has become apparent, largely due to the lack ofagreement between models and experiments. The presentwork provides a rigorous evaluation of the contributionsfrom various transport processes for a Si-CoSi
2interface.
Importantly, the experimental results, performed across awide temperature range, only agree well with predictionsthat include all transport processes: elastic and inelasticphonon scattering, electron-phonon coupling only in the metal,and electron-phonon coupling across interface. The relativecontributions of the various transport mechanisms wouldhowever be specific to the metal-semiconductor interface.For example, the extent of joint phonon modes is expectedto be strongly sensitive to the strength of bonding at theinterface (e.g., van der Waals versus covalent bonding). Also,the polarity of interfacial bonds could have a significantimpact on the strength of direct electron-phonon coupling.An interesting possibility for future work would involve asystematic study of the effect of interfacial bonding parameterson the relative contributions from the various cross-interfacethermal transport mechanisms.
ACKNOWLEDGMENTS
S.S. acknowledges financial support from the Office of
Naval Research (Award No. N000141211006) and Helen andMarvin Adelberg fellowship from the School of MechanicalEngineering at Purdue University.
[1] M. H. Kryder, E. C. Gage, T. W. McDaniel, W. A. Challener,
R. E. Rottmayer, G. Ju, Y .-T. Hsia, and M. F. Erden, Proc. IEEE
96,1810 (2008 ).
[2] J. P. Reifenberg, D. L. Kencke, and K. E. Goodson, IEEE
Electron Device Lett. 29,1112 (2008 ).[3] S. P. Murarka, Intermetallics 3,173(1995 ).
[ 4 ] Y .A .K o s e v i c h , P h y s .R e v .B 52,1017 (1995 ).
[5] G. T. Hohensee, R. Wilson, and D. G. Cahill, Nat. Commun. 6
6578 (2015 ).
[6] A. Majumdar and P. Reddy, Appl. Phys. Lett. 84,4768 (2004 ).
085310-14THERMAL TRANSPORT ACROSS METAL SILICIDE- . . . PHYSICAL REVIEW B 95, 085310 (2017)
[7] P. E. Hopkins, J. L. Kassebaum, and P. M. Norris, J. Appl. Phys.
105,023710 (2009 ).
[8] L. Guo, S. L. Hodson, T. S. Fisher, and X. Xu, J. Heat Transfer
134,042402 (2012 ).
[9] M. L. Huberman and A. W. Overhauser, Phys. Rev. B 50,2865
(1994 ).
[10] A. Sergeev, Physica B: Condens. Matter 263-264 ,217(1999 ).
[11] G. Mahan, P h y s .R e v .B 79,075408 (2009 ).
[12] J. Ren and J.-X. Zhu, P h y s .R e v .B 87,241412 (2013 ).
[13] J. Lombard, F. Detcheverry, and S. Merabia, J. Phys.: Condens.
Matter 27,015007 (2015 ).
[14] T. Lu, J. Zhou, T. Nakayama, R. Yang, and B. Li, P h y s .R e v .B
93,085433 (2016 ).
[15] W. Little, Can. J. Phys. 37,334(1959 ).
[16] E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61,605(1989 ).
[17] Z. Tian, K. Esfarjani, and G. Chen, Phys. Rev. B 86,235304
(2012 ).
[18] P. E. Hopkins, J. Appl. Phys. 106,013528 (2009 ).
[19] A. V . Sergeev, P h y s .R e v .B 58,R10199(R) (1998 ).
[20] R. J. Stoner and H. J. Maris, Phys. Rev. B 48,16373 (1993 ).
[21] R. J. Stevens, A. N. Smith, and P. M. Norris, J. Heat Transfer
127,315(2005 ).
[22] R. M. Costescu, M. A. Wall, and D. G. Cahill, Phys. Rev. B 67,
054302 (2003 ).
[23] R. B. Wilson, B. A. Apgar, W.-P. Hsieh, L. W. Martin, and
D. G. Cahill, Phys. Rev. B 91,115414 (2015 ).
[24] N. Ye, J. P. Feser, S. Sadasivam, T. S. Fisher, T. Wang, C. Ni,
and A. Janotti, P h y s .R e v .B 95,085430 (2017 ).
[25] K. Miao, S. Sadasivam, J. Charles, G. Klimeck, T. S. Fisher, and
T. Kubis, Appl. Phys. Lett. 108,113107 (2016 ).
[26] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C.
Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Daboet al. ,J. Phys.: Condens. Matter 21,395502 (2009 ).
[27] W. Zhang, T. Fisher, and N. Mingo, Numer. Heat Transfer, Part B
51,333(2007 ).
[28] S. Sadasivam, Y . Che, Z. Huang, L. Chen, S. Kumar, and T. S.
Fisher, Annu. Rev. Heat Transfer 17,89(2014 ).
[29] M. L. Sancho, J. L. Sancho, J. L. Sancho, and J. Rubio, J. Phys.
F15,851(1985 ).
[30] F. Guinea, C. Tejedor, F. Flores, and E. Louis, P h y s .R e v .B 28,
4397 (1983
).[31] N. Mingo, P h y s .R e v .B 74,125402 (2006 ).
[32] R. Venugopal, M. Paulsson, S. Goasguen, S. Datta, and M.
Lundstrom, J. Appl. Phys. 93,5613 (2003 ).
[33] J. Maassen, F. Zahid, and H. Guo, P h y s .R e v .B 80,125423
(2009 ).
[34] A. Afzalian, J. Appl. Phys. 110,094517 (2011 ).
[35] M. Luisier, P h y s .R e v .B 86,245407 (2012 ).
[36] D. Singh, J. Y . Murthy, and T. S. Fisher, J. Heat Transfer 133,
122401 (2011 ).
[37] N. Mingo, P h y s .R e v .B 68,113308 (2003 ).
[38] C. Jeong, S. Datta, and M. Lundstrom, J. Appl. Phys. 111,
093708 (2012 ).
[39] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.95.085310 for details of numerical models,
methods, and convergence related to electron-phonon coupling.
[40] M. Anantram, M. S. Lundstrom, and D. E. Nikonov, Proc. IEEE
96,1511 (2008 ).
[41] C. G. Broyden, Math. Comput. 19,577(1965 ).
[42] P. B. Allen, Phys. Rev. Lett. 59,1460 (1987 ).
[43] R. Stadler, D. V ogtenhuber, and R. Podloucky, Phys. Rev. B 60,
17112 (1999 ).
[44] M. Wardle, J. Goss, P. Briddon, and R. Jones, Phys. Status
Solidi A 202,883(2005 ).
[45] Z. Huang, T. Fisher, and J. Murthy, J. Appl. Phys. 109,074305
(2011 ).
[46] Z.-Y . Ong and E. Pop, Phys. Rev. B 81,155408 (2010 ).
[47] N. Mingo, D. A. Stewart, D. A. Broido, and D. Srivastava, Phys.
Rev. B 77,033418 (2008 ).
[48] M. N. Luckyanova, J. Garg, K. Esfarjani, A. Jandl, M. T. Bulsara,
A. J. Schmidt, A. J. Minnich, S. Chen, M. S. Dresselhaus, Z.Ren, E. A. Fitzgerald, and G. Chen, Science 338,936(2012 ).
[49] X. Gu, X. Li, and R. Yang, Phys. Rev. B 91,205313 (2015 ).
[50] E. S. Landry and A. J. H. McGaughey, P h y s .R e v .B 80,165304
(2009 ).
[51] R. Bauer, A. Schmid, P. Pavone, and D. Strauch, Phys. Rev. B
57,11276 (1998 ).
[52] S. Sadasivam, U. V . Waghmare, and T. S. Fisher, J. Appl. Phys.
117,134502 (2015 ).
[53] K. Gordiz and A. Henry, J. Appl. Phys. 119,015101 (2016 ).
[54] Z. Lu, Y . Wang, and X. Ruan, P h y s .R e v .B 93,064302 (2016 ).
[55] K. Gordiz and A. Henry, Sci. Rep. 6,23139 (2016 ).
085310-15 |
PhysRevB.85.235433.pdf | PHYSICAL REVIEW B 85, 235433 (2012)
Crossed Andreev reflection in quantum wires with strong spin-orbit interaction
Koji Sato,1Daniel Loss,2and Yaroslav Tserkovnyak1
1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
2Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
(Received 28 September 2011; published 15 June 2012)
We theoretically study tunneling of Cooper pairs from an s-wave superconductor into two semiconductor
quantum wires with strong spin-orbit interaction under magnetic field, which approximate helical Luttingerliquids. The entanglement of electrons within a Cooper pair can be detected at low temperatures by the electriccurrent cross correlations in the wires. By controlling the relative orientation of the wires, either lithographically ormechanically, on the substrate, these current correlations can be tuned, as dictated by the initial spin entanglement.This proposal of a spin-to-charge readout of quantum correlations is alternative to a recently proposed utilizationof the quantum spin Hall insulator.
DOI: 10.1103/PhysRevB.85.235433 PACS number(s): 71 .10.Pm, 72 .25.Hg, 73 .63.Nm, 74 .78.Na
I. INTRODUCTION
One of the key features and resources of quantum me-
chanics is entanglement, particularly in the particle spinsector, which has been an enticing subject since the Einstein-Podolsky-Rosen thought experiment
1and, more recently,
fueled by the modern proposals for spin-based quantuminformation processing and computation.
2–4In order to use
an entangled pair of electrons for quantum informationtechnology in a scalable semiconductor setting, it is essentialto have a solid-state system that can separate the entangledelectrons over appreciable distance. Detecting electron spinentanglement is possible via bunching or antibunching cor-relations in beam splitters
5and transport through Coulomb-
blockaded quantum dots forming a Josephson junction.6A
conceptual headway came with a proposal to spatially separatespin-singlet Cooper pairs (CP’s) injected from an s-wave
superconductor via crossed Andreev reflection (CAR)
7in
a quantum dot setup8and in a normal-metal fork.9Later,
more elaborate considerations for an s-wave superconductor in
junction with quantum wires,10,11quantum beam splitter,12and
quantum dots13have been put forward. CAR is essential in all
these proposals, and it has been experimentally manifested inthe negative nonlocal differential resistance in the system of asuperconductor in junction with normal metal.
14CP splitter
experiments have been recently performed with quantumdots
15and carbon nanotubes.16As another form of CP splitter,
we theoretically proposed the system with a superconductorstraddling a strip of two-dimensional quantum spin Hallinsulator (QSHI),
17to inject a CP into its gapless edge states.
Utilizing the helical Luttinger-liquid character of the QSHIedges (where each electron moves in the opposite direction toits time-reversed Kramers partner with opposite spin), the spinentanglement can be converted into nonlocal charge-currentcross correlations.
In this paper, we consider CP injection into quantum
wires with strong spin-orbit interaction (SOI), such as self-doped (and possibly backgated, to control their electrondensity) InAs nanowires. If only SOI is considered, the spindegeneracy at the /Gamma1point ( k=0) is preserved because of
the time-reversal symmetry. However, this degeneracy canbe lifted by external magnetic field (facilitated in InAs bya large g factor of electrons and generally enhanced byelectron-electron interactions
18). When the chemical potential
is set in the corresponding gap at the /Gamma1point, gapless
states which propagate in the opposite directions with almostopposite spins can be realized at the Fermi points. Note thatsuch a system can closely resemble the helical edge state of theQSHI.
19We consider an s-wave superconductor connected to
a pair of such semiconductor wires in the regime where two CPelectrons split into different wires, in the presence of electron-electron repulsion. Effective spin-quantization axes for theleft- and right-moving electrons injected into the Fermi points
of the two wires are tilted|in one wire relative to the other|by
their geometric misalignment. Such tilt affects the current crosscorrelations in the wires in a way that is similar to the tunablebreaking of the inversion symmetry discussed in Ref. 17.
When temperatures and voltage bias between the super-
conductor and the wires are smaller than the superconductorgap/Delta1, single-particle injection into the wires is suppressed.
In this regime, transport is dominated by the CP tunneling.This process, however, is exponentially suppressed if thedistance between the wires exceeds the coherence length of
a CP and algebraically on the scale of the Fermi wavelength
in the superconductor (depending sensitively on its spatialdimensionality),
10posing a potentially serious constraint on
the interwire separation. Very importantly, furthermore, ifthe applied voltage and temperature are smaller than /Delta1,t h e
parasitic tunneling of two CP electrons into the same wireis suppressed with a power law that is governed by the
Luttinger-liquid correlations.
10In this work, we thus focus
on the regime where a CP splits ejecting electrons into thedifferent wires. There is a time lag of ∼/Delta1
−1between two such
tunneling events, the longer it is the weaker the Luttinger-liquidsuppression of the same-wire CP tunneling. However, whentwo electrons are forced to split and enter different wires atlow energies, the leading-order tunneling rates are independent
of this time delay (neglecting any interwire interactions).
10
Therefore, we consider a simplified model with equal-time CP
injection of two electrons into two different wires.11
This paper is organized as follows. In Sec. II, we introduce
the Hamiltonian for the quantum wire with spin-orbit couplingand magnetic field, and discuss tunneling matrix elements.We consider both Rashba and Dresselhaus SOI with the wireoriented in the xyplane under the magnetic field in the
235433-1 1098-0121/2012/85(23)/235433(7) ©2012 American Physical SocietyKOJI SATO, DANIEL LOSS, AND YAROSLA V TSERKOVNYAK PHYSICAL REVIEW B 85, 235433 (2012)
zdirection. In Sec. III, we calculate the noise spectrum of the
currents in the wires with Keldysh formalism. (Details of thecomputation are relegated to the appendices.) Final remarks onpossible extensions of our theory and experimental feasibilityare provided in Sec. IV.
II. HAMILTONIAN FOR QUANTUM WIRES
For the wires, Rashba and Dresselhaus SOI in combination
with the Zeeman splitting are considered. Lateral confinementin the wire governs subbands, of which we suppose (atsufficiently low temperature and appropriate backgate bias)only the lowest one is occupied, whose Kramers pairs are splitby the lack of both time-reversal and inversion symmetries. Inthis system, the one-dimensional effective Hamiltonian for awire oriented along the xaxis is given by
20,21
H0=¯h2k2
2m∗+αkˆσy+βkˆσx−ξˆσz, (1)
where m∗is the effective mass of electron, α(β) is the strength
of the Rashba (Dresselhaus) SOI, and kis the electron wave
number. The Dresselhaus part is for the case when a zinc-blende heterostructure is grown in the [001] crystallographicdirection, while the wire is oriented in the [100] direction.
22
2ξ=gμBBis the Zeeman energy gap at k=0, with magnetic
fieldBapplied along the zaxis, g is the g factor, and μBthe
Bohr magneton. ˆσ=(ˆσx,ˆσy,ˆσz) are Pauli matrices. Other wire
and magnetic field orientations are discussed in Sec. IV.
Defining the k-dependent effective field R(k)=(βk,αk,
−ξ), the Hamiltonian can be written as
H0=¯h2k2
2m∗+R(k)·ˆσ, (2)
and the eigenspinors are found by rotating spinors
such that R(k)·ˆσ|χ±(k)/angbracketright=±R(k)|χ±(k)/angbracketright, where R(k)=/radicalbig
k2(α2+β2)+ξ2. The subscripts ±here label spin up/down
along R(k). The energy eigenstates are thus given by ψ±(k)=
χ±(k)eikx, with energy /epsilon1±(k)=¯h2k2/2m∗±R(k). The upper
and lower ( /epsilon1+and/epsilon1−) bands are sketched in Fig. 1. When the
kF kFμ
∋
FIG. 1. (Color online) Single-particle electron dispersion with
Rashba and Dresselhaus SOI. Zeeman splitting 2 ξis induced at k=0
by a magnetic field in the zdirection, and the chemical potential is set
in this gap. One-dimensional effective theory is then linearized near
±kF, which define respectively the right- and left-moving electron
branches.FIG. 2. (Color online) S-wave superconductor bridging two
identical wires. The lower wire is rotated by angle θwith respect
the upper wire. The superconductor is biased by Vwith respect to
the wires.
chemical potential μis set within the gap, we can linearize
the remaining left and right-moving /epsilon1−branches within a
Luttinger-liquid picture. This requires ( eV,k BT)/lessmuchξand
electron-electron interactions that are not strong enough tohybridize the /epsilon1
±bands. On the other hand, we require the mag-
netic field to be weak enough on the scale set by Hcof the su-
perconductor (which can be enhanced in mesoscopic structuresup to the paramagnetically-limited value of order /Delta1/μ
B.23)
Inversion asymmetry between the two wires is introduced
by tilting the lower wire (which is otherwise defined along thesame crystallographic axis), which rotates the spin quantiza-tion axis at each Fermi point of the /epsilon1
−band. The upper wire is
along the xaxis, whereas we suppose the lower wire is placed
in the xyplane at an angle θwith respect to the xaxis, as
shown in Fig. 2. (This may in practice be realized by growing
both wires parallel to each other on an unstrained crystal, andthen distorting the crystal in the xyplane to effectively tilt the
wires; depending on the interwire separation, a finite θmay
not require a large strain, whose additional effect on the SOIis neglected.) The SOI in the lower wire is thus given by
H
SO=αk(cosθˆσy−sinθˆσx)+βk(cosθˆσx+sinθˆσy).(3)
Reflecting this rotation of the lower wire, the effective fields
for the upper ( u) and lower ( d) wires are given by
R(u)(k)≡R(k,θ=0)=(βk,αk, −ξ),
R(d)(k)≡R(k,θ)
=[k(−αsinθ+βcosθ),k(αcosθ+βsinθ),−ξ].
(4)
The corresponding Fermi-point eigenspinors are |χ(u,d)
r,l/angbracketright≡
|χ(u,d)(±kF)/angbracketright, which solve
R(n)
r,l·ˆσ/vextendsingle/vextendsingleχ(n)
r,l/angbracketrightbig
=−R(n)
r,l/vextendsingle/vextendsingleχ(n)
r,l/angbracketrightbig
(5)
forR(u,d)
r,l≡R(u,d)(±kF). We will assume electronic correla-
tions are not strong enough to significantly affect these Fermi-point spinors. Anticipating tunneling of electrons with well-defined spins from the superconductor into the Fermi pointsof our wires, we can effectively decompose the fermionic field
235433-2CROSSED ANDREEV REFLECTION IN QUANTUM WIRES ... PHYSICAL REVIEW B 85, 235433 (2012)
operators ψ(n)
σ(σ=↑,↓) in terms of the right (left) movers
ψ(n)
r,lin the nth wire as21
ψ(n)
σ=/angbracketleftbig
χ(n)
r|σ/angbracketrightbig
ψ(n)
r+/angbracketleftbig
χ(n)
l|σ/angbracketrightbig
ψ(n)
l. (6)
The full wire Hamiltonian (1)is bosonized24near the Fermi
points to give an essentially helical (so long as the Zeemantermξis weak) Luttinger-liquid:
25
H0=v/summationdisplay
n=u,d/integraldisplaydx
2π/bracketleftbigg1
g(∂xφ(n))2+g(∂xθ(n))2/bracketrightbigg
, (7)
where φ(n),θ(n)=(φ(n)
r±φ(n)
l)/2 obey commutation relations
[θ(n)(x),φ(m)(y)]=(iπ/2)sgn( x−y)δnm.φ(n)
r,lparametrize
fermionic operators as ψ(n)
r,l∝e±iφ(n)
r,l.
The tunneling Hamiltonian, which describes nonlocal
injection of the spin singlet CP from an s-wave superconductor
into the two quantum wires is given by11
HT=Te−2eVt[ψ(u)
↑ψ(d)
↓(0)−ψ(u)
↓ψ(d)
↑(0)]+H.c.(8)
In this model, two electrons from a singlet CP split and
tunnel simultaneously into the upper and lower wires at theirrespective origins. Vis the voltage applied between the super-
conductor and the wires, which is set to be smaller than /Delta1to
preclude quasiparticle excitations. Expanding spin-dependent
operators ψ(n)
↑/↓in terms of the chiral modes pertinent to the
w i r e sa si nE q . (6), we can rewire the tunneling Hamiltonian
(8)as
ψ(u)
↑ψ(d)
↓−ψ(u)
↓ψ(d)
↑=/summationdisplay
μ,ν=r,lKμνψ(u)
μψ(d)
ν, (9)
where Kμνare the complex-valued expansion coefficients,
which are given by
Kμν=/angbracketleftbig
χ(u)
μ/vextendsingle/vextendsingle↑/angbracketrightbig/angbracketleftbig
χ(d)
ν/vextendsingle/vextendsingle↓/angbracketrightbig
−/angbracketleftbig
χ(u)
μ/vextendsingle/vextendsingle↓/angbracketrightbig/angbracketleftbig
χ(d)
ν|↑/angbracketrightbig
.(10)
In Section III, the current-current correlations are cal-
culated, which depend on |Kμν|2(reflecting spin-rotational
symmetry of a singlet CP):
|Kμν|2=1−/vextendsingle/vextendsingle/angbracketleftbig
χ(u)
μ/vextendsingle/vextendsingleχ(d)
ν/angbracketrightbig/vextendsingle/vextendsingle2=1
2/parenleftbig
1−ˆR(u)
μ·ˆR(d)
ν/parenrightbig
.(11)
Here, ˆR(n)
μ=R(n)
μ/R(n)
μ, and |Kμν|2can be evaluated using
Eq. (4).R(n)
μ=√
k2
F(α2+β2)+ξ2,f o rn=u,landμ=
±, independent of the orientation of the wire or elec-
tron chirality. Furthermore, since R(u)
μ·R(d)
ν=μνk2
F(α2+
β2) cosθ+ξ2(where μandν=± respectively for r,l), we
find that |K++|2=|K−−|2and|K+−|2=|K−+|2. Lumping
Zeeman and SOI energies into a dimensionless parameter
λ=ξ/kF/radicalbig
α2+β2, we finally arrive at a rather simple
expression for Eq. (11):
|Kμν|2=1−μνcosθ
2(1+λ2). (12)
III. NOISE SPECTRUM
In this section, the current-current correlations at the four
end points of the two wires in Fig. 2are considered. Thesymmetrized noise spectrum,
Sij(ω)=Sji(−ω)=/integraldisplay∞
−∞dteiωt/angbracketleft{δIi(t),δIj(0)}/angbracketright,(13)
is calculated using Keldysh formalism.17,26HereδIi(t)=
Ii(t)−/angbracketleftIi(t)/angbracketrightare the current fluctuations, ilabeling four
outgoing channels in the wires ( i=1, upper right; 2 upper
left; 3, lower left; and 4, lower right branches) (see Fig. 2). We
henceforth bosonize the current operators (with details of thecomputation provided in Appendix A), finally obtaining the
following expressions for the noise spectra at zero frequency(ω=0):
S
13=S31=S24=S42=eI(1+g2cosθ)≡S+,
S14=S41=S23=S32=eI(1−g2cosθ)≡S−,
(14)
S11=S22=S33=S44=eI(1+g2),
S12=S21=S34=S43=eI(1−g2).
Here, Iis the average current flowing though each of the
four branches, which is given by Eq. (A7) . This current
vanishes in the limit λ/greatermuch1, when both wires become fully
spin polarized thus blocking the CP tunneling. Notice thatthe magnetic field did not scramble the helical structure ofthe interwire cross correlations, which are the same [apartfrom the overall suppression by (1 +λ
2)] as the case of the
time-reversal symmetric QSHI.17This is one of the key results
of this paper.
The interwire cross-correlation spectra (14) are
given by
S±(θ,λ)∝1±g2cosθ
1+λ2, (15)
which are modified from those in Ref. 17only by the
magnetic-field suppression factor of (1 +λ2)−1.I nR e f . 17,
the angle θdependence for the CP injection into the helical
edge states of a QSHI is due to a tunable asymmetry betweentwo edges (induced by a local application of strain or gatevoltage to an otherwise inversion-symmetric system). Here, θ
dependence comes from the mechanical rotation of the lowerwire by the angle θ. Notice that the definitions for S
+andS−
are interchanged here in comparison to Ref. 17. This is because
the quantum wires considered here do not have the inversionsymmetry of helical edge states on the opposite sides of a QSHIstrip. Despite this fundamental difference, we can clearly seethe same structure in the CP noise cross correlations for boththe present quantum-wire system and the helical QSHI edges.According to Eq. (15), we can extract the Luttinger-liquid
interaction parameter g(which is typically
27g∼0.1−1i n
semiconducting wires) from the interwire cross correlations:g
2cosθ=(S+−S−)/(S++S−). While in Ref. 17the angle
θis a parameter that may not be precisely known, in the
present setup the rotation angle θof the lower wire can
be experimentally well defined, so that gcan be found by
measuring S+andS−for an arbitrary value of θthat is away
fromπ/2.
IV . CONCLUSION AND DISCUSSION
The backscattering caused by disorder in the wire scrambles
the ballistic transport and smears the correlations pertaining
235433-3KOJI SATO, DANIEL LOSS, AND YAROSLA V TSERKOVNYAK PHYSICAL REVIEW B 85, 235433 (2012)
to entangled electron pairs, which could hinder practical
implementation of our proposal. The nonlocal charge crosscorrelations thus have to be detected on the length scalesshorter than the mean free path. Partitioning of electronsaccordingly to their spin is the key feature for our result, whichwas obtained by setting the chemical potential in the gap toprobe into a single band. If it is set otherwise, for instance, inthe region of multiple bands, this feature is lost.
In the discussion so far, we were considering only one
specific crystallographic orientation of the wires. Namely,the heterostructure growth is in the [001] crystallographicdirection and each wire is defined (e.g., electrostatically) alongthe [100] direction. However, while the Rashba SOI is rota-tionally invariant around the normal axis, the Dresselhaus SOIis sensitive to the wire orientation on a crystal’s surface.
22,28
Suppose that with the same crystal growth direction of [001],
the wire is defined at an angle θDfrom the [100] direction. In
this case, the Dresselhaus SOI part of the Hamiltonian is givenby
22
HD=βk[cos(2 θD)ˆσx−sin(2θD)ˆσy]. (16)
This crystallographic orientation and the associated Hamil-
tonian are now chosen for the upper wire, with ourcoordinate system still placed (as in Fig. 2) with the
xdirection collinear with the wire. The corresponding
effective-field vector is then R
(u)(k)=[βkcos(2θD),αk−
βksin(2θD),−ξ]. Since the lower wire is rotated in the
xyplane by the angle θwith respect to the upper
wire, R(l)obtained by the corresponding rotation on R(u)
is given by R(l)=[−αksinθ+βkcos(2θD−θ),αkcosθ−
βksin(2θD−θ),−ξ]. The absolute value of R(u,l)is modified
byθD:R(u,l)=/radicalbig
k2[α2+β2−2αβsin(2θD)]+ξ2. Both the
direction and the magnitude of R(u,l)are thus modified,
affecting Kμνin Eq. (11). We still have |K++|2=|K−−|2
and|K+−|2=|K−+|2according to Eq. (11). In fact, the
modification of |Kμν|2can be absorbed by redefining λ
entering Eq. (12)asλ=ξ/kF/radicalbig
α2+β2−2αβsin(2θD), with
all subsequent relations for the noise spectra unmodified. Inparticular, apart from the modified geometric spin factor λ,
which suppresses the overall strength of the CAR, S
±in
Eq.(15) remain the same. This means we can choose any wire
orientation on the crystal surface without altering the essenceof the noise cross correlations. One special point is θ
D=π/4
when α=β(orθD=−π/4 when α=−β), corresponding
to the “persistent spin helix”,20,29where λblows up and the
CAR is fully blocked (reflecting exact cancellation of the SOIterms).
Let us also comment on a possible triplet pairing of the
injected electrons, e.g., if the two terms in the tunneling Hamil-
tonian (8)acquire a relative phase difference: e
iδ/2ψ(u)
↑ψ(d)
↓−
e−iδ/2ψ(u)
↓ψ(d)
↑. We can rewrite it as cos( δ/2)(ψ(u)
↑ψ(d)
↓+
ψ(u)
↓ψ(d)
↑)−isin(δ/2)(ψ(u)
↑ψ(d)
↓+ψ(u)
↓ψ(d)
↑). The correspond-
ing modification of |Kμν|2in Eq. (12) can be accounted for
by the replacement θ→θ−δ, with the same δshift of θ
appearing in the noise expressions. Interestingly, the phasedifference in the tunneling terms has the same effect on thecurrent correlations as a mechanical rotation of the wires. Sucha triplet component in tunneling can be effectively induced bytunneling away from the Fermi points at finite temperature
and/or voltage, and artificially enhanced in more complextunneling setups.
30
Another concern to be mentioned is that, if the supercon-
ductor is in a slab shape, the perpendicular critical field isreduced. This issue can be mitigated by applying an in-planemagnetic field. For the case of a thin-film superconductor,the critical field is further enhanced (up to its paramagneticlimit
23) when the magnetic penetration depth is greater than
its thickness. Rin Eq. (4)needs to be modified accordingly.
Since the magnetic-field and SOI contributions to Rare not
generally perpendicular to each other anymore, the resultingenergy band is not symmetric as in Fig. 1. In turn, |K
μν|2in
Eq.(12)and the formula in Eq. (14)acquire certain corrections.
In the limit of λ/lessmuch1, the corrections are small, however, and
we recover the same noise behavior as in Eq. (15).I nt h e
strong-field limit, λ/greaterorsimilar1, on the other hand, a more careful
analysis would be warranted.
Now let us return to Eq. (15) to see the feasibility of this
theory in an experiment. A very large magnetic splitting (onthe scale of the SOI) in the wires, λ/greatermuch1, blocks Andreev
reflection,
31when the Fermi level is inside the /Gamma1-point gap.
The SOI is large in the InAs-based heterostructures and wires,where the Rashba parameter is α/lessorsimilar10
−11eV m (being tunable
by electrostatic gating),32β/lessmuchα, and g factor is ≈15. For
electron densities in the range of 10–100 μm−1, this gives for
the magnetic field B∼0.1–1 T corresponding to λ∼1. Both
αand g factor can be considerably lower (both up to two orders
of magnitude) in InGaAs-based heterostructures, which canmake also α∼β,
28while the corresponding magnetic field
range remains roughly the same. This gives us a favorableoperational bound for the magnetic field, which opens the/Gamma1-point gap without compromising the strength of the CAR,
while also not exceeding the paramagnetically-limited criticalfield (with T
c/greaterorsimilar1 K). Taking everything into account, this
means the experiments can be done at temperatures closet o1K .
ACKNOWLEDGMENTS
This work was supported by the Alfred P. Sloan Foundation,
the NSF under Grant No. DMR-0840965 (Y .T.), and by theSwiss NSF and DARPA QuEST (D.L.).
APPENDIX A: A VERAGE CURRENT AND NOISE
SPECTRUM
In this Appendix, we evaluate the average current in each
wire and current-current correlations, Eq. (13). Tunneling of
Cooper pairs is treated perturbatively with the Keldysh for-malism. Using Luttinger-liquid bosonization formalism,
24,25
the fermionic field is expressed as
ψ(n)
μ(x)=1√
2πaeiμ[kFx+φ(n)
μ(x)],
where ais the short-distance cutoff. The Klein factor is omitted
here as it has no effect on the final results derived below.
235433-4CROSSED ANDREEV REFLECTION IN QUANTUM WIRES ... PHYSICAL REVIEW B 85, 235433 (2012)
μ=r,l=± labels the left- and right-moving branches. In this
bosonized representation, the tunneling Hamiltonian, Eq. (8),
becomes
HT=T
2πae−iω0t/summationdisplay
μ,ν=r,lKμνei[μφ(u)
μ(0)+νφ(d)
ν(0)]+H.c..
Here,ω0=2eV/¯his the Josephson frequency corresponding
to the bias Vapplied between the superconductor and the
wires, and Kμνis given in Eq. (9).We define the current to be positive as it flows away from the
superconductor. The bosonized current operators at distancex/greatermuchafrom the superconductor, along branches 1 through 4
in Fig. 2,a r eg i v e nb y
24
I1,2=±I(u)(±x)=±e(vg/π)∂xθ(u)(±x),
I4,3=±I(d)(±x)=±e(vg/π)∂xθ(d)(±x).
The average current and the noise spectrum are given by26,33
I(n)(x,t)=1
2/summationdisplay
η/angbracketleftbig
Tce−i
¯h/integraltext
cdt/prime/primeHT(t/prime/prime)I(n)(x,t,η )/angbracketrightbig
,
S(nm)(x,t;x/prime,t/prime)≈/summationdisplay
η/angbracketleftbig
Tce−i
¯h/integraltext
cdt/prime/primeHT(t/prime/prime)I(n)(x,t,η )I(m)(x/prime,t/prime,−η)/angbracketrightbig
,
respectively, where Tcis the Keldysh contour-ordering operator, η=± labels the upper (lower) branch of the Keldysh contour
for the field operators, and the time evolution of the operators on the right-hand side is given in the interaction picture. Since tothe leading order in tunneling the average current Iis proportional to |T|
2, we have dropped the /angbracketleftIi(t)/angbracketright/angbracketleftIj(0)/angbracketrightterm in the noise
spectrum, which is of order |T|4. When calculating the noise spectrum, it is convenient to exponentiate the operator θ(n)in the
following way:
∂xθ(n)(x,t)=∂x(−i∂λ)eiλθ(n)(x,t)/vextendsingle/vextendsingle
λ=0.
Up to the second order in T, we finally find
I(n)(x,t)=−sgn(x)|T|2evg
4πa2h2/summationdisplay
μ,ν,ε
η,η 1,η2|Kμν|2η1η2∂x(−i∂λ)/integraldisplay∞
−∞dt1/integraldisplay∞
−∞dt2e−εiω 0(t1−t2)
×/angbracketleftbig
Tceiλθ(n)(x,t,η )eiε[μφ(u)
μ(0,t1,η1)+νφ(d)
ν(0,t1,η1)]e−iε[μφ(u)
μ(0,t2,η2)+νφ(d)
ν(0,t2,η2)]/angbracketrightbig/vextendsingle/vextendsingle
λ=0, (A1)
S(nm)(x,t;x/prime,t/prime)=−sgn(x)sgn(x/prime)1
2/parenleftbiggevg|T|
πah/parenrightbigg2/summationdisplay
μ,ν,ε
η,η 1,η2|Kμν|2η1η2∂x∂x/prime∂λ1∂λ2/integraldisplay∞
−∞dt1/integraldisplay∞
−∞dt2e−iεω 0(t1−t2)
×/angbracketleftbig
Tceiλ1θ(n)(x,t,η )e−iλ2θ(m)(x/prime,t/prime,−η)eiε[μφ(u)
μ(0,t1,η1)+νφ(d)
ν(0,t1,η1)]e−iε[μφ(u)
μ(0,t2,η2)+νφ(d)
ν(0,t2,η2)]/angbracketrightbig/vextendsingle/vextendsingle
λ1,λ2=0. (A2)
Here,ε=± corresponds to the annihilation (creation) part of HTandhis the Planck’s constant. The above expression is then
evaluated using standard bosonic operator identities,24giving
I(n)(x,t)=−isgn(x)evg|T|2
4πa2h2/summationdisplay
μ,ν,ε
η,η 1,η2ε|Kμν|2η1η2/integraldisplay∞
−∞dt1/integraldisplay∞
−∞dt2e−εiω 0(t1−t2)
×/summationdisplay
κ(δn,uδμ,κ+δn,dδν,κ)Pη1η2(t1−t2)[Qκ,ηη 1(x,t−t1)−Qκ,ηη 2(x,t−t2)], (A3)
S(nm)(x,t;x/prime,t/prime)=sgn(x)sgn(x/prime)1
2/parenleftbiggevg|T|
πah/parenrightbigg2/summationdisplay
μ,ν,ε
η,η 1,η2|Kμν|2η1η2/integraldisplay∞
−∞dt1/integraldisplay∞
−∞dt2e−iεω 0(t1−t2)/summationdisplay
λ,κLnm,λκ
×Pη1η2(t1−t2)[Qλ,ηη 1(x,t−t1)−Qλ,ηη 2(x,t−t2)][Qκ,−ηη1(x/prime,t/prime−t1)−Qκ,−ηη2(x/prime,t/prime−t2)],(A4)
where Lnm,λκ=(δu,nδμ,λ+δd,nδν,λ)(δu,mδμ,κ+δd,mδν,κ) and Pηη/prime(t) and Qμ,ηη/prime(x,t) are expressed in terms of the Green’s
functions of φ(n)(x,t) andθ(n)(x,t):
Pηη/prime(t)=/productdisplay
n=u,dexp/bracketleftbig
G(n)φφ
ηη/prime(0,t)+G(n)θθ
ηη/prime(0,t)/bracketrightbig
,Q μ,ηη/prime(x,t)=μ∂xG(n)θφ
ηη/prime(x,t)+∂xG(n)θθ
ηη/prime(x,t). (A5)
235433-5KOJI SATO, DANIEL LOSS, AND YAROSLA V TSERKOVNYAK PHYSICAL REVIEW B 85, 235433 (2012)
Here,24
G(n)φφ
ηη/prime(x,t)=/angbracketleftbig
Tcφ(n)(x,t,η )φ(n)(0,0,η/prime)−φ(n)(0,0)2/angbracketrightbig
=−g
4/summationdisplay
r=±ln[1+iDηη/prime(t)(vt−rx)/a],
G(n)θθ
ηη/prime(x,t)=/angbracketleftbig
Tcθ(n)
η(x,t)θ(n)
η/prime(0,0)−θ(n)(0,0)2/angbracketrightbig
=−1
4g/summationdisplay
r=±ln[1+iDηη/prime(t)(vt−rx))/a],
G(n)φθ
ηη/prime(x,t)=/angbracketleftbig
Tcφ(n)
η(x,t)θ(n)
η/prime(0,0)/angbracketrightbig
=G(n)θφ
ηη/prime(x,t)=/angbracketleftbig
Tcθ(n)
η(x,t)φ(n)
η/prime(0,0)/angbracketrightbig
=−1
4/summationdisplay
r=±rln[1+iDηη/prime(t)(vt−rx)/a],(A6)
where Dηη/prime(t)=/Theta1(ηη/prime)sgn(η/primet)+/Theta1(−ηη/prime)sgn(η/prime). We finally obtain the current as
I(n)(x)=isgn(x)evg
4πa2h2|T|2/summationdisplay
μ,ν,κ|Kμν|2(δn,uδμ,κ+δn,dδν,κ)[Qκ,++(x)−Qκ,+−(x)+Qκ,−+(x)−Qκ,−−(x)]
×[P+−(ω0)−P+−(−ω0)−P−+(ω0)+P−+(−ω0)]=sgn(ω0)1
1+λ22πe|T|2
v2h2/Gamma1(2γ+2)/parenleftbigg|ω0|a
v/parenrightbigg2γ
|ω0|,(A7)
where Pηη/prime(ω) andQμ,ηη/prime(x)≡Qμ,ηη/prime(x,ω=0) denotes the Fourier transform and γ=(g+g−1−2)/2.Kμνhere is taken from
Eq.(12). Similarly for the noise spectrum, we get:
S(nm)(x,x/prime,ω=0)=−sgn(x)sgn(x/prime)1
2/parenleftbiggevg|T|
πah/parenrightbigg2/summationdisplay
μν|Kμν|2/summationdisplay
λ,κLnm,λκ [P−+(ω0)+P−+(−ω0)+P+−(ω0)+P+−(−ω0)]
×{[Qλ,++(x)−Qλ,+−(x)][Qκ,−+(x/prime)−Qκ,−−(x/prime)]+[Qλ,−+(x)−Qλ,−−(x)][Qκ,++(x/prime)−Qκ,+−(x/prime)]}
=eI[δn,mF1(x,x/prime)+δn,−mF2(x,x/prime)], (A8)
where
F1(x,x/prime)=1+g2sgn(x)sgn(x/prime),(for n = m) ,F 2(x,x/prime)=1−g2cosθsgn(x)sgn(x/prime),(forn/negationslash=m),
andIis given by the absolute value of the current in Eq. (A7) .
APPENDIX B: RELEV ANT INTEGRALS
The evaluation of the average current and the noise
spectrum in Eqs. (A7) and(A8) reduces to finding Pηη/prime(ω)
andQμ,ηη/prime(x,ω=0)=Qμ,ηη/prime(x), which are the Fourier trans-
forms of the functions defined in Eq. (A5) . Using the Green’s
functions in Eq. (A6) , we find
Pηη/prime(t)=1
[1+iDηη/prime(t)vt/a ]2γ+2,
Qμ,ηη/prime(x,t)=i/summationdisplay
r=±r+μ
4agDηη/prime(t)
1+iDηη/prime(t)(vt−rx)/a.
P−+(ω) can be evaluated by noting the integral:
/integraldisplay∞
−∞dteiωt
(δ+it)ν=2π
/Gamma1(ν)|ω|ν−1e−|ω|δ/Theta1(ω).
Furthermore, we see P+−(ω)=P−+(−ω), and
the expression appearing in Eqs. (A7) and (A8)become
P−+(ω0)−P−+(−ω0)−P+−(ω0)+P+−(−ω0)
=4π
/Gamma1(2γ+2)/parenleftbigga
v/parenrightbigg2γ+2
|ω0|2γ+1e−|ω0|a/vsgn(ω0),
and
P−+(ω0)+P−+(−ω0)+P+−(ω0)+P+−(−ω0)
=4π
/Gamma1(2γ+2)/parenleftbigga
v/parenrightbigg2γ+2
|ω0|2γ+1e−|ω0|a/v.
To be internally consistent with the low-energy Luttinger-
liquid description, we henceforth drop the factor e−|ω0|a/v.T h e
remaining relevant terms entering Eqs. (A7) and(A8) are
Qμ,++(x)−Qμ,+−(x)=Qμ,−+(x)−Qμ,−−(x)
=i/summationdisplay
rr+μg
4g/integraldisplay∞
0dt/bracketleftbigg1
a+i(vt−rx)+1
a−i(vt−rx)/bracketrightbigg
=i/summationdisplay
rr+μg
2gv/bracketleftbiggπ
2−tan−1/parenleftbigg
−rx
a/parenrightbigg/bracketrightbigg
≈iπ
2vg[μg+sgn(x)] .
1A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
2D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
3D. P. DiVincenzo and D. Loss, J. Magn. Magn. Mater. 200, 202
(1999).4R. A. ˙Zak, B. R ¨othlisberger, S. Chesi, and D. Loss, Riv. Nuovo
Cimento Soc. Ital. Fis. 33, 345 (2010).
5G. Burkard, D. Loss, and E. V . Sukhorukov, P h y s .R e v .B 61,
R16303 (2000).
235433-6CROSSED ANDREEV REFLECTION IN QUANTUM WIRES ... PHYSICAL REVIEW B 85, 235433 (2012)
6M . - S .C h o i ,C .B r u d e r ,a n dD .L o s s , Phys. Rev. B 62, 13569
(2000).
7G. Deutscher and D. Feinberg, Appl. Phys. Lett. 76, 487 (2000).
8P. Recher, E. V . Sukhorukov, and D. Loss, Phys. Rev. B 63, 165314
(2001).
9G. B. Lesovik, T. Martin, and G. Blatter, Eur. Phys. J. B 24, 287
(2001).
10P. Recher and D. Loss, P h y s .R e v .B 65, 165327 (2002).
11C .B e n a ,S .V i s h v e s h w a r a ,L .B a l e n t s ,a n dM .P .A .F i s h e r , Phys.
Rev. Lett. 89, 037901 (2002).
12P. Samuelsson, E. V . Sukhorukov, and M. B ¨uttiker, P h y s .R e v .B
70, 115330 (2004).
13O. Sauret, D. Feinberg, and T. Martin, Phys. Rev. B 70, 245313
(2004).
14D. Beckmann, H. B. Weber, and H. v. L ¨ohneysen, P h y s .R e v .L e t t .
93, 197003 (2004); S. Russo, M. Kroug, T. M. Klapwijk, and A. F.
Morpurgo, ibid. 95, 027002 (2005); J. Wei and V . Chandrasekhar,
Nat. Phys. 6, 494 (2010).
15L. Hofstetter, S. Csonka, J. Nyg ˚ard, and C. Sch ¨onenberger, Nature
(London) 461, 960 (2009).
16L. G. Herrmann, F. Portier, P. Roche, A. L. Yeyati, T. Kontos, and
C. Strunk, P h y s .R e v .L e t t . 104, 026801 (2010).
17K. Sato, D. Loss, and Y . Tserkovnyak, Phys. Rev. Lett. 105, 226401
(2010).
18B. Braunecker, G. I. Japaridze, J. Klinovaja, and D. Loss, Phys.
Rev. B 82, 045127 (2010).
19C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005); 95,
146802 (2005); B. A. Bernevig and S.-C. Zhang, ibid. 96, 106802
(2006).
20J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801
(2003).
21J. Sun, S. Gangadharaiah, and O. A. Starykh, P h y s .R e v .L e t t . 98,
126408 (2007); S. Gangadharaiah, J. Sun, and O. A. Starykh, Phys.
Rev. B 78, 054436 (2008).22M. Scheid, M. Kohda, Y . Kunihashi, K. Richter, and J. Nitta,
Phys. Rev. Lett. 101, 266401 (2008); M. Wang, K. Chang,
L. G. Wang, N. Dai, and F. M. Peeters, Nanotechnology 20, 365202
(2009).
23P. M. Tedrow, R. Meservey, and B. B. Schwartz, Phys. Rev. Lett.
24, 1004 (1970).
24T. Giamarchi, Quantum Physics in One Dimension (Oxford Uni-
versity Press, Oxford, 2004).
25A. Str ¨om and H. Johannesson, Phys. Rev. Lett. 102, 096806
(2009).
26C. de C. Chamon, D. E. Freed, and X. G. Wen, Phys. Rev. B 53,
4033 (1996).
27O. M. Auslaender, A. Yacoby, R. de Picciotto, K. W. Baldwin,L. N. Pfeiffer, and K. W. West, Science 295, 825 (2002);
O. M. Auslaender, H. Steinberg, A. Yacoby, Y . Tserkovnyak, B. I.
Halperin, K. W. Baldwin, L. N. Pfeiffer, and K. W. West, ibid. 308,
88 (2005).
28L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch ¨o n ,a n dK .E n s s l i n ,
Nat. Phys. 3, 650 (2007).
29B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev. Lett. 97,
236601 (2006).
30P. Virtanen and P. Recher, P h y s .R e v .B 85, 035310 (2012).
31M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. Lett. 74, 1657
(1995).
32J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett.
78, 1335 (1997); T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi,
ibid. 89, 046801 (2002); S. Dhara, H. S. Solanki, V . Singh,
A. Narayanan, P. Chaudhari, M. Gokhale, A. Bhattacharya,and M. M. Deshmukh, P h y s .R e v .B 79, 121311 (2009);
S. Est ´evez Hern ´andez, M. Akabori, K. Sladek, C. V olk, S. Alagha,
H. Hardtdegen, M. G. Pala, N. Demarina, D. Gr ¨utzmacher, and
T. Sch ¨apers, ibid. 82, 235303 (2010).
33A. Cr ´epieux, R. Guyon, P. Devillard, and T. Martin, Phys. Rev. B
67, 205408 (2003).
235433-7 |
PhysRevB.75.035111.pdf | Facilitated gapless conduction through DNA molecules
Juyeon Yi *
Department of Physics, Pusan National University, Busan 609-735, Korea
Beom Jun Kim†
Department of Physics, BK21 Physics Research Division, and Institute of Basic Science, Sungkyunkwan University,
Suwon 440-746, Korea
/H20849Received 8 June 2006; revised manuscript received 13 October 2006; published 9 January 2007 /H20850
We examine the effect of counterions on coherent electronic conduction through DNA molecules. Evoking
the fluctuation nature of counterions, we consider that a randomness is brought into the molecular energy levelsthrough electrostatic interaction among base charges and counterions. It is demonstrated that the conductanceis greatly suppressed, and in addition, a remarkable reduction of the band gap width occurs, in the presence ofionic fluctuation. The current-voltage characteristics are also evaluated, which reveal that the current curvechanges its type from semiconductorlike at weak randomness to Ohmic at strong randomness. Within thepicture regarding the molecule as a disordered wire, the conductance variance is evaluated to reproduce thewell-known universal relation to the conductance average. We propose that an experimental confirmation of theuniversal relation can be good evidence for the validity of our model.
DOI: 10.1103/PhysRevB.75.035111 PACS number /H20849s/H20850: 85.65. /H11001h, 87.14.Gg, 73.23. /H11002b
The question of whether DNA conducts or not has ac-
quired considerable attention in recent years. The initiativewas fueled by molecular electronics taking advantage of self-assembly wherein molecules are spontaneously formed in ahighly ordered structure. If electron transport through DNAcould be performed in a similar fashion to semiconductors,we then have genuine conducting nanosized wires with sav-ing expenditure for manufacturing. Initial experiments wereperformed to measure the electric conductivity of DNA mol-ecules, and the revealed properties were diverse, from super-conducting to insulating.
1,2At the same time, theoretical at-
tempts were made to acquire quantitative data on electronicband structure,
3and model the system, grasping the essential
factors governing the electric conductivity of the molecules.4
Knowing that there exist /H9266orbitals in the bases of nucle-
otides densely stacked along the helix, a consensus has beenreached that the orbital overlap paves the way for the chargemigration.
By far, many interesting issues have been addressed, in-
cluding polaron transport,
5sequence dependence,6and con-
tact effects.7Much effort has been especially put into studies
to explore the effect of counterions and water molecules onthe conductivity of DNA. Let us briefly mention the relatedexperimental observations, with special interest on the bandgap: In the current-voltage /H20849I-V/H20850characteristics of DNA in
ambient solution /H20849solvation water with counterions /H20850, no band
gap was found.
8–11Intriguingly, the recent study by Gutiérrez
et al. has shown that the electron coupling to a dissipative
environment /H20849e.g., solution /H20850induces electronic states within
the band gap, leading to the Ohmic behavior.12This is in a
sharp contrast to the experimental findings in Refs. 2and13,
where I-Vcurves obtained for poly /H20849G/H20850-poly /H20849C/H20850in vacuum
manifest the conductance zero plateau indicating the bandgap size of a few eV .
The above mentioned existing experimental studies
2,8–11,13
clearly suggest the key role of the counterions in conducting
behavior of DNA, i.e., counterions facilitate the gapless con-duction. Aiming to describe the feature, a simple model
Hamiltonian of the tight-binding chain can be
H0=/H20858
i/H9280ici†ci+t/H20858
i/H20849ci†ci+1+ci+1†ci/H20850, /H208491/H20850
where /H9280iis the molecular orbital energy, ci/H20849ci†/H20850is a fermion
annihilation /H20849creation /H20850operator at the ith base, and tis the
hopping strength between consecutive bases. Here the pres-ence of counterions condensed onto the negatively chargedphosphate group naturally affects the orbital energy
/H9280i, which
can be regarded as an effective electrostatic interaction be-tween an electron and condensed charges at the ith base. We
note that ions in water are mobile to diffuse along the helix:Given the diffusion constant of ions D=10
−6cm2/sec, ions
can diffuse from base to base in a few nanoseconds. Besides,the absorption/desorption process of ions occurs repeatedly.These imply that
/H9280i’s are not uniform across the system and
over time, thus having spatio-temporal randomness. A closeresemblance is noteworthy here, between molecules in solu-tion and disordered wires. However, the latter disorder iswell-known to yield a profound effect of the Anderson local-ization. In this regard, the experimental observation of theconductivity enhancement in the disordered DNA moleculesappears to be counterintuitive. From this reasoning, it is clearthat the simple tight-binding Hamiltonian /H208491/H20850cannot capture
the essential physics behind the conduction through DNA.Furthermore, in the limiting case when the orbital energy
/H9280i
is uniform and thus H0possesses the translational symmetry,
a simple application of the Fourier transformation results inenergy levels, E/H20849k/H20850=
/H9280−2tcosk, having no gap. Conse-
quently, the Hamiltonian /H208491/H20850fails to incorporate the experi-
mental fact that ordered DNA molecule /H20849/H9280i=/H9280/H20850exhibits an
I-Vcharacteristics with the big-gap semiconducting nature,
signaling again a deficiency in H0.
The purpose of the present work is to explore the under-
lying physics of the gapless or gapful conductance of DNAPHYSICAL REVIEW B 75, 035111 /H208492007 /H20850
1098-0121/2007/75 /H208493/H20850/035111 /H208495/H20850 ©2007 The American Physical Society 035111-1molecules with focus on the role of counterions in water. In
order to achieve this, we describe the effect of counterions asthe random orbital energy
/H9280iwith the Coulomb repulsion
interaction included in the model Hamiltonian. The conduc-tance is then evaluated through the use of the Green’s func-tion method and the the Landauer-Büttiker formalism is em-ployed to compute the I-Vcharacteristics.
14In good
agreement with existing experimental studies, we find thatthe randomness suppresses not only the transmission ampli-tude but also the gap width, and that in the absence of therandomness the system supports the charge ordering with thegapful I-Vcharacteristics. In fact, our picture where the mol-
ecules in solutions are regarded to be disordered wires, drawsan issue of great interest. It is well known that the conduc-tance of disordered mesoscopic systems sensitively dependson disorder configuration, and becomes a sample-specificquantity, leading to the prominent conductance fluctuation/H20849CF/H20850.
15,16While traditional CF has been obtained in an en-
semble of systems, CF is supposed to occur intrinsically in amolecule because in solution the variety of disorder configu-ration can be generated in correspondence with ion fluctua-tion, forming an equivalent disorder ensemble but in a singlemolecule. This conjecture can be readily confirmed by ex-periment. To the end, we evaluate the relation, independentof system details, between conductance variance and itsaverage.
We start from our model Hamiltonian
H=H
0+HU,
HU=U/H20858
i/H20873ci†ci−1
2/H20874/H20873ci+1†ci+1−1
2/H20874, /H208492/H20850
where H0is given by Eq. /H208491/H20850andHUrepresents the repul-
sion between nearest bases. The short-ranged interaction canbe considered to be the screened Coulomb repulsion betweenspinless fermions in a tight-binding version. To take full ac-count of the long-range correlation would be more desirableif a nontrivial role of the interaction nature is unveiled.
Including spin variables, H
Uhas an additional on-site re-
pulsion for the occupation of spin-up and spin-down par-ticles, and the subtraction factor 1/2 is replaced by 1. Thiswould be critically dealt with for understanding spin-associated phenomena such as the proximity-inducedsuperconductivity
1and antiferromagnetic signature in the de-
crease of paramagnetism,17which are, however, beyond the
scope of our work. Furthermore, since within our scheme,the on-site repulsion simply yields the renormalization of therepulsion parameter,
18let us retain the spinless particles in-
teracting via the short-ranged repulsion.
Although there exist a number of molecular levels, we are
only interested in the two bands near the Fermi level, i.e.,HOMO /H20849highest occupied molecular orbital /H20850and LUMO
/H20849lowest unoccupied molecular orbial /H20850split by the gap be-
tween them. In H
U, the factor 1/2 is subtracted from the
charge density operator to guarantee the particle-hole invari-ance for the half-filled system in the presence of the repul-sion. When U→/H11009, the Coulomb repulsion H
Udominates
and accordingly the ground-state configuration is character-ized by the charge distribution of alternating occupied and
unoccupied sites, i.e, /H20855ci†ci/H20856gs=1 and /H20855ci+1†ci+1/H20856gs=0/H20849or vice
versa /H20850with the quantum mechanical expectation value /H20855¯/H20856gs
computed for the ground state. Even when Uis finite,19,20
charges tend to support an order which can be measured by
/H9004defined as
/H20855ci†ci/H20856gs/H110131/2 +/H9004/H20849−1/H20850i. /H208493/H20850
When the system is perfectly ordered in the charge configu-
ration we have /H9004=1/2, whereas /H9004=0 in the perfectly disor-
dered case with /H20855ci†ci/H20856gs=1/2.
We next split ni/H11013ci†ciinto two parts, the expectation
value and fluctuation around it /H20849denoted by : ¯:/H20850, i.e., ci†ci
=1/2+ /H9004/H20849−1/H20850i+:ni: and make an ansatz that the product of
fluctuations : ni::ni+1: is negligibly small, which in turn leads
us to the effective Hamiltonian
Heff=H0+U/H20858
i/H208512/H9004/H20849−1/H20850i+1ci†ci+/H90042/H20852. /H208494/H20850
When the homogeneously sequenced molecule in vacuum
with/H9280i=/H9280is considered, the above Hamiltonian is readily
diagonalized to give the two subbands,
E±/H20849k/H20850=/H9280+U/H90042±2/H20881t2cos2k+U2/H90042, /H208495/H20850
having band gaps at k=±/H9266/2 of the width 4 U/H9004. It should be
noted that the Coulomb repulsion introduced in the modelprovides a source for the band gap, which is exactly whatwas lacking in the tight-binding Hamiltonian H
0. In our for-
mulation, /H9004needs to be determined in a self-consistent man-
ner via the so-called gap equation /H11509Etot//H11509/H9004=0, which de-
scribes the condition that the true values of /H9004should
correspond to the minimum of the total energy of the system.Here, the total energy is written as E
tot
=/H20858k,/H9251=±f/H20851E/H9251/H20849k/H20850/H20852E/H9251/H20849k/H20850, where f/H20849E/H20850=1/ /H20851e/H9252/H20849E−/H9262/H20850+1/H20852is the
Fermi-Dirac distribution function with /H9262being the chemical
potential. From the experiment measuring currents throughpoly /H20849dG/H20850-poly /H20849dC/H20850in vacuum,
2the gap width is estimated to
be a few eV , indicating the repulsion strength Uin our ap-
proach is also an order of eV . Such a big gap implies that noelectrons can smear into the upper band, for which it isstraightforward to see /H9004/H110151/2 as expected. This implies that
the charge ordering indeed persists without randomness inour model. In fact, in Ref. 21the effective Hamiltonian was
introduced in the same spirit as in the present work to renderthe origin of the semiconducting behavior of the homopoly-mer. Different from the original DNA model where the lad-der topology was used,
21we here consider a DNA molecule
as a one-dimensional chain. One possible drawback of thissimplification is that it might underestimate the conductancegap since the interstrand interaction is expected to amplifythe potential alteration and thus enlarge the gap. However,since this simplification of DNA as a one-dimensional chaincannot change the existence of a relatively large band gap,and since our motivation here is to examine the conductionproperty of DNA in a qualitative way, we believe that theneglect of the interstrand interaction can be justified.
We now examine in detail how the electric conduction
property of DNA is altered in the presence of counterions inJUYEON YI AND BEOM JUN KIM PHYSICAL REVIEW B 75, 035111 /H208492007 /H20850
035111-2solution. As mentioned above, counterions absorbed or des-
orbed on and off the phosphate group give rise to the orbitalenergy shift through electrostatic interaction with base
charges. In more detail, we first write
/H9280i=/H9280/H208491−ZNi/H20849c/H20850/H20850, where
/H9280measures the scale of the orbital energy, Zis the valency of
a single counterion, and Ni/H20849c/H20850is the number of counterions
adsorbed on the ith phosphate. For instance, when a single
/H20849Ni/H20849c/H20850=1/H20850ion of Na+/H20849Z=1/H20850is adsorbed onto a phosphate
group carrying the charge − e/H20849e/H110220/H20850, the phosphate becomes
neutral, and therefore, net ionic strength vanishes leading to
/H9280i=0 at the base. In reality, the absorption/desorption of
counterions on the phosphate is not a static but a dynamicprocess occurring all over along the DNA helix. Accordingly,
it is very reasonable to treat N
i/H20849c/H20850as a fluctuating random
variable, which can effectively be included as randomness inthe on-site energy
/H9280i. Although the effective interaction
strength /H9280has above been assumed to be uniform over dif-
ferent bases, this cannot be a drawback of our formulation ofthe model since the fluctuating counterionic adsorption pro-cess will eventually nullify the difference in the bases.
In the present work, we assume that
/H9280iis a uniformly
distributed random variable in /H20851−W,W/H20852to incorporate the
charge neutrality condition /H20855/H9280i/H20856=/H9280/H208491−Z/H20855Ni/H20849c/H20850/H20856/H20850=0 with /H20855¯/H20856
being the disorder average. The variance of /H9280iis related to
the strength of counterion number fluctuation, /H9254/H9280/H11013/H20881/H20855/H9280i2/H20856
=/H9280/H9254N/H20849c/H20850. Although the explicit expression for the temperature
dependence of /H9254Ni/H20849c/H20850is not taken into consideration, there
exists an implied notion for the number fluctuation strength.Since the number fluctuation arises from the entropic gain
coupled to temperature,
/H9254Ni/H20849c/H20850would be more dominant for
ions with higher valency at a higher temperature.
Transport properties of a molecule is traced according to
the Landauer formula,14where the dimensionless conduc-
tance T/H20849E/H20850/H20849or the transmission /H20850is given by
T/H20849E/H20850=T r /H20853/H9003LGr/H9003RGa/H20854, /H208496/H20850
with the retarded Green function, Gr/H20849E/H20850=/H20851/H20849E+i0+/H20850I−Heff
−/H9018/H20852−1. The self-energy correction /H9018=/H9018L+/H9018Ris related to
the coupling strengths /H9003Land/H9003Rto the left /H20849L/H20850and the right
/H20849R/H20850metallic leads, via /H9003L/H20849R/H20850=−2 Im /H9018L/H20849R/H20850. We also assume
the self-energy is energy independent, which is valid for
wide-band leads mostly used in experiments. The I-Vchar-
acteristics are obtained from T/H20849E/H20850via
I=2e
h/H20885
−/H11009/H11009
dE/H20851f/H20849E−/H9262/H5129/H20850−f/H20849E−/H9262r/H20850/H20852T/H20849E/H20850, /H208497/H20850
where /H9262/H5129and/H9262rare the chemical potential of the left and the
right leads, respectively, the difference of which is controlledby the applied voltage bias V:
/H9262/H5129−/H9262r=eV. For a given ran-
dom configuration of /H9280iat disorder strength W, we first com-
pute/H9004self-consistently from the minimum condition of Etot,
and then compute the dimensionless conductance T/H20849E/H20850,
which is then averaged over 4 /H11003103disorder configurations.
Figure 1displays the numerically evaluated transmission
andI-Vcharacteristics for various W. At a weak randomness,
the transmission curve exhibits a gap, the width of which isabout 2 eV at W=0. This is indeed consistent with the ex-
periments where I-Vcharacteristics of homogeneously se-
quenced DNA in vacuum, corresponding to W=0 in our
model, were observed to exhibit a semiconductorlikebehavior.
2,13AsWis increased, i.e., as the randomness be-
comes more significant, it is seen in Fig. 1that the gap width
gets narrower. When the randomness becomes extremelystrong, e.g., W=2, the Ohmic behavior shows up without a
gap. This is again in perfect qualitative agreement with ex-perimental evidence
8–11that DNA molecules in ambient so-
lution show gapless current curves.
From the numerical result of the gapless conduction at
strong disorder W/H110152, we examine the strength of the coun-
terion fluctuation /H9254N/H20849c/H20850. In reality, especially at high tempera-
tures, Ni/H20849c/H20850is expected to be better described as Gaussian
random variables. Although we have used the uniform distri-bution for the random variable
/H9280i, it is legitimate to estimate
the relation between the numerical parameter Wand the ac-
tual strength of the number fluctuation of counterions bysimply comparing the variances of the two, yielding
W/
/H9254N/H20849c/H20850=/H208813Z/H9280.22The interaction strength /H9280is roughly esti-
mated without screening effects as
/H9280=e2//H208494/H9266/H92550a/H20850=kBTR/H20849/H5129B/a/H20850/H20849/H9255w//H92550/H20850, /H208498/H20850
with the Bjerrum length /H5129B=e2/4/H9266/H9255w/H110157 Å in water, /H92550/H20849w/H20850
being the dielectric constant of vacuum /H20849water /H20850, and kBTR
being the thermal energy at room temperature. The Bjerrum
length defines the length scale at which the electrostatic in-teraction between two unit charges becomes comparable tothe thermal energy at room temperature, and /H5129
B/H110157Å i n
water. Taking distance between bases and phosphate as a
/H110155 Å, one gets the estimation for the interaction strength
between base charges and the counterions as /H9280/H110152.8/H20849eV/H20850.
This gives /H9254N/H20849c/H20850/H110151//H9280/H110150.36 for the gapless conduction at
W/H110152. It seems fair to conclude that ion fluctuation required
for the gapless conduction is significant and yet in realizablerange. Here we would like to mention results from thedensity-functional calculation on the effects of counterions.
23
By considering a chain of sodium counterions, it was shown
FIG. 1. /H20849Color online /H20850The left panel displays the averaged
transmission for the molecule of 16 bases for various randomness/H20849t=1 has been used /H20850. As the randomness increases, the gap width
that amounts to 2 eV for W=0 gets reduced and eventually vanishes
for strong randomness, e.g, W=2. In the right panel, corresponding
current-voltage characteristics are shown, demonstrating thecurrent-zero plateaux for weak disorder and the Ohmic behavior forW=2. /H20849The energy parameters here are in units of eV . /H20850FACILITATED GAPLESS CONDUCTION THROUGH DNA … PHYSICAL REVIEW B 75, 035111 /H208492007 /H20850
035111-3that counterions alter the on-site potentials of the bases:
When a single sodium atom is replaced by H 3O counterion,
HOMO and LUMO of the base nearest to H 3O is shifted in
relative to that of the base nearby sodium ions. Furthermore,they have shown that nonuniform distribution of ions causedthe conductance reduction. This also goes well with our re-sult /H20849see the left panel of Fig. 1displaying the conductance
suppression by increasing W/H20850. Though a direct comparison is
not feasible, the results suggest that our consideration of thecounterions affecting the orbital energy through the electro-static interaction is reasonable.
Not only the gap width, but also the transmission ampli-
tude is suppressed as Wis increased, as can be seen in Fig. 1.
We display the transmission amplitude and the gap width inFig. 2, in the upper and the lower panels, respectively. The
finite-size effect in the localization phenomenon is reflectedas the exponential decay of the conductance as a function ofthe system size N, i.e, /H20855T/H20856/H11011exp/H20849−N/
/H9264/H20850with the localization
length scale /H9264, depending on disorder strength. As shown in
the upper panel of Fig. 2, where the logarithm of the maxi-
mum amplitude of the averaged transmission is presented,our system indeed belongs to the case displaying
/H9264/H110111/W.
This goes well with the observation of the insulating behav-ior of molecules longer than 40 nm.
24Consequently, for a
long molecule, despite the gap reduction, randomness is dis-advantageous to electric conduction for small voltage bias.
Up to now, by employing a picture regarding the mol-
ecules in solution as disordered wires, the gapless and thegapful I-Vcharacteristics observed in the experiments are
successfully found to depend on the randomness originatingfrom counterion fluctuation in solution. The validity of thepicture can be verified by experiments. In fact, other promi-nent effects of randomness have been demonstrated as theconductance fluctuations. For an ensemble of disorderedwires of a finite size in a diffusive regime /H20849T/H110221/H20850,i ti s
known that conductance variance is universal such as
/H9254T
=/H20881/H20855T2/H20856−/H20855T/H208562/H110111. When the disorder strength is increased,
/H9254Tdecreases below the universal value and is shown to be
related to /H20855T/H20856.16Interestingly, this universal behavior is also
reproduced within our model, as presented in Fig. 3. Sinceour model is restricted to be a single-mode one-dimensional
chain, for which T/H333551, the diffusive regime is not reachable,
neither is the universal value of /H9254T. Nonetheless, the scaling
law /H20855T/H20856/H11011/H20849/H9254T/H208502for small T, and ln /H20855T/H20856/H11011/H20855lnT/H20856for relatively
large Tare clearly observed. Since these relations do not
depend on the system details, it is readily measurable in ex-periments. Having in mind that molecules are disordered, itis obviously expected to have conductivity fluctuations ofmolecules with strong fluctuations in their energy levels.Nonetheless, in studying conduction properties of DNA, thishas been less recognized and proper analysis of experimentaldata has been left aside. Our results not only render an originof the gapless conductivity of DNA in solution, but also drawattention to the mesoscopic nature of DNA wires and theirconductance fluctuations.
In summary, we have investigated the solvation effects on
the conduction properties of DNA molecules. By having paidattention to the fact that the ions are mobile along the helixin solution, and continue adsorption and desorption, thenumber of counterions are considered as random variables.Hence, the on-site potential shift due to the electrostatic in-teraction among base charges and condensed ions forms arandom profile over the bases. It has been shown that therandomness suppresses the conduction but also causes a sig-nificant reduction of the gap width, facilitating the gaplessconduction. Especially upon a close resemblance betweenthe DNA molecules in solution and one-dimensional disor-dered wires, we have proposed that the picture could be ex-perimentally substantiated through observing conductancefluctuations and validating if they obey the characteristicscaling laws, such as /H20849
/H9254T/H208502/H11011/H20855T/H20856for small T, and /H20855lnT/H20856
/H11011ln/H20855T/H20856for large T. Finally, a few brief remarks are made on
numerical results based on efficient methods such as density-
functional theory /H20849DFT /H20850and molecular dynamics combined
with DFT, where the density of states /H20849DOS /H20850of DNA were
obtained. When a dry DNA is considered, the band gap be-tween HOMO and LUMO levels was clearly seen.
3,23,25
Compared with experiments, this is consistent with the re-
sults in Refs. 2and13, and with our results for W=0. How-
ever, there still exists a controversy over the effects of coun-terions and water molecules. It was found that conductionchannel from the solvents
23,26,27was introduced in between
FIG. 2. The disorder effects on the gap width and transmission
amplitude: The upper panel displays the logarithm of maximumtransmissions obtained for N=16,32,48 and scaled with the system
size. It indicates that the transmission amplitudes exponentially fallsoff as the system size and the randomness increase. In the lowerpanel, the gap parameter evaluated for the different system sizes isshown to vanish for strong disorder.
FIG. 3. Universal relations between /H20855T/H20856and/H9254/H20855T/H20856for small
transmission amplitude. Points were obtained for N=16 with W
=0.5 /H20849/H11003/H20850andW=1/H20849/H11569/H20850, for N=32 with W=0.5 /H20849/H17039/H20850andW=1/H20849/L50098/H20850,
and for N=48 with W=0.5 /H20849/H17005/H20850andW=1/H20849/H17010/H20850. The inset shows a
linear relation between /H20855lnT/H20856and ln /H20855T/H20856for those system sizes with
W=0.25.JUYEON YI AND BEOM JUN KIM PHYSICAL REVIEW B 75, 035111 /H208492007 /H20850
035111-4the HOMO and LUMO levels, effectively reducing the band
gap, but still its width remained a few eV . On the other hand,Westerhoff et al. found the gap even larger when DNA is in
solution.
28Even apart from the discrepancy among the theo-
retical results, it appears that those are not compatible withthe observations.
8–11In those studies, a snapshot of ionic
configuration—where for example, every base accompaniesNa
+—was considered rather than taking account of a number
of possible ionic distribution. This implies that for giving asalient account of gapless conduction in experiments and
pursuing further related issues, sufficient counting of ionicfluctuation must be essential.
This work was supported by Korea Research Foundation
Grant funded by the Korean Government /H20849MOEHRD /H20850by
Grant No. KRF-2004-005-C00044 /H20849J.Y ./H20850and No. KRF-2005-
005-J11903 /H20849B.J.K. /H20850, and by Korean Science and Engineer-
ing Foundation by Grant No. R04-2004-000-10031 /H20849J.Y ./H20850.
*Electronic address: jyi@pusan.ac.kr
†Electronic address: beomjun@skku.edu
1A. Kasumov, M. Kociak, S. Gu’eron, B. Reulet, V . V olkov, D.
Klinov, and H. Bouchiat, Science 291, 280 /H208492001 /H20850;P .J .d e
Pablo, F. Moreno-Herrero, J. Colchero, J. Gómez-Herrero, P.Herrero, A. Baró, P. Ordejón, J. M. Soler, and E. Artacho, Phys.Rev. Lett. 85, 4992 /H208492000 /H20850.
2D. Porath, A. Bezryadin, S. de Vries, and C. Dekker, Nature
/H20849London /H20850403, 635 /H208492000 /H20850.
3J. P. Lewis, P. Ordejon, and O. F. Sankey, Phys. Rev. B 55, 6880
/H208491997 /H20850; R. Di Felice, A. Calzolari, E. Molinari, and A. Garbesi,
ibid. 65, 045104 /H208492001 /H20850; M. Hjort and S. Stafström, Phys. Rev.
Lett. 87, 228101 /H208492001 /H20850; P. Maragakis, R. L. Barnett, E. Kax-
iras, M. Elstner, and T. Frauenheim, Phys. Rev. B 66,
241104 /H20849R/H20850/H208492002 /H20850.
4J. F. Feng and S. J. Xiong, Phys. Rev. E 66, 021908 /H208492002 /H20850;G .
Cuniberti, L. Craco, D. Porath, and C. Dekker, Phys. Rev. B 65,
241314 /H20849R/H20850/H208492002 /H20850.
5R. Bruinsma, G. Grüner, M. R. D’Orsogna, and J. Rudnick, Phys.
Rev. Lett. 85, 4393 /H208492000 /H20850;Y o o et al. ,ibid. 87, 198102 /H208492001 /H20850;
S. Komineas, G. Kalosakas, and A. R. Bishop, Phys. Rev. E 65,
061905 /H208492002 /H20850; S. S. Alexandre, E. Artacho, J. M. Soler, and H.
Chacham, Phys. Rev. Lett. 91, 108105 /H208492003 /H20850.
6R. A. Caetano and P. A. Schulz, Phys. Rev. Lett. 95, 126601
/H208492005 /H20850.
7E. Maciá, F. Triozon, and S. Roche, Phys. Rev. B 71, 113106
/H208492005 /H20850.
8B. Xu, P. Zhang, X. Li, and N. Tao, Nano Lett. 4, 1105 /H208492004 /H20850.
9L. Cai, H. Tabata, and T. Kawai, Appl. Phys. Lett. 77, 3105
/H208492000 /H20850; J. Gu, L. Cai, S. Tanaka, Y . Otsuka, and H. Tabata, J.
Appl. Phys. 92, 2816 /H208492002 /H20850;D .H .H a et al. , Chem. Phys. Lett.
355, 405 /H208492002 /H20850;J .S .H w a n g et al. , Appl. Phys. Lett. 81, 1134
/H208492002 /H20850.
10P. Tran, B. Alavi, and G. Gruner, Phys. Rev. Lett. 85, 1564
/H208492000 /H20850.
11A. Bonincontro, G. Gareri, A. Giansanti, and F. Pedone, Phys.
Rev. A 38, 6446 /H208491988 /H20850.
12R. Gutiérrez, S. Mandal, and G. Cuniberti, Phys. Rev. B 71,
235116 /H208492005 /H20850; Nano Lett. 5, 1093 /H208492005 /H20850.
13M. S. Xu, S. Tsukamoto, S. Ishida, M. Kitamura, Y . Arakawa, R.
G. Endres, and M. Shimoda, Appl. Phys. Lett. 87, 083902
/H208492005 /H20850.
14R. Landauer, IBM J. Res. Dev. 1, 223 /H208491957 /H20850; J. Math. Phys. 37,
5259 /H208491996 /H20850; D. S. Fisher and P. A. Lee, Phys. Rev. B 23,
R6851 /H208491981 /H20850; S. Datta, Electronic Transport in Mesoscopic
Systems /H20849Cambridge University Press, Cambridge, 1999 /H20850.
15P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 /H208491985 /H20850;Y . Imry, Europhys. Lett. 1, 249 /H208491986 /H20850; P. A. Lee, A. D. Stone,
and H. Fukuyama, Phys. Rev. B 35, 1039 /H208491987 /H20850.
16A. D. Stone, K. A. Muttalib, and J.-L. Pichard, in Anderson Lo-
calization , edited by T. Ando and H. Fukuyama /H20849Springer-
Verlag, Berlin, 1988 /H20850; N. Giordano, Phys. Rev. B 38, 4746
/H208491988 /H20850; B. L. Al’tshuler, V . E. Kravtsov, and I. V . Lerner, Sov.
Phys. JETP 64, 1352 /H208491986 /H20850; M. Kemp, A. Roitberg, V . Mujica,
T. Wanta, and M. A. Ratner, J. Phys.: Condens. Matter 100,
8349 /H208491996 /H20850.
17O. Schiemann, N. J. Turro, and J. K. Barton, J. Phys. Chem. B
104, 7214 /H208492000 /H20850.
18When spin-1/2 particles are considered, the short-ranged repul-
sion among them can be written as HU=/H20858i/H20851U0/H20849ni↑−1/2 /H20850/H20849ni↓
−1/2 /H20850+U1/H20849ni−1/H20850/H20849ni+1−1/H20850/H20852with ni=ni↑+ni↓andni/H9268=ci/H9268†ci/H9268.I ti s
straightforward to see that apart from irrelevant constants, HU
=/H20858i/H20851/H20849U0/2/H20850/H20849ni−1/H208502+U1/H20849ni−1/H20850/H20849ni+1−1/H20850/H20852. Under the strong re-
pulsion, the ansatz for the density alternation is still valid, ni
=1+/H9004/H20849−1/H20850i+:n:. Applying the same scheme as used for the
spinless particles, we have exactly the same form of the effectiveHamiltonian given by Eq. /H208494/H20850but with U=U
1−U0/2/H20849Ref. 19/H20850.
19For the repulsion parameters, see Table 2 in E. B. Starikov, Phi-
los. Mag. Lett. 83, 699 /H208492003 /H20850, which were obtained for the
dimers of base pairs. The parameter values are shown to dependon the molecular form and base species, leading the wide rangeofU, roughly 1.3–2.5 eV.
20Considering the interaction to be the screened Coulomb repul-
sion, we make a rough estimation for the value of Usuch as
U/H11015/H20849e2/4/H9266/H92550r/H20850e−/H9260rwith rand/H9260−1being the interbase distance
and the screening length, respectively. Taking r/H110153.4 Å and
/H9260a/H110151, we obtain U/H110151.5 eV.
21J. Yi, Phys. Rev. B 68, 193103 /H208492003 /H20850.
22B.-Y . Ha and A. J. Liu, Phys. Rev. E 60, 803 /H208491999 /H20850.
23Ch. Adessi, S. Walch, and M. P. Anantram, Phys. Rev. B 67,
081405 /H20849R/H20850/H208492003 /H20850; Ch. Adessi and M. P. Anantram, Appl. Phys.
Lett. 82, 2353 /H208492003 /H20850.
24A. J. Storm, J. van Noort, S. de Vries, and C. Dekker, Appl. Phys.
Lett. 79, 3881 /H208492001 /H20850.
25Y . Ye, R. Chen, A. Martinez, P. Otto, and J. Ladik, Physica B
279, 246 /H208492000 /H20850; A. K. Bakhshi, P. Otto, J. Ladik, and M. Seel,
Chem. Phys. 108, 215 /H208491986 /H20850.
26F. L. Gervasio, P. Carloni, and M. Parrinello, Phys. Rev. Lett. 89,
108102 /H208492002 /H20850.
27A. Hübsch, R. G. Endres, D. L. Cox, and R. R. P. Singh, Phys.
Rev. Lett. 94, 178102 /H208492005 /H20850.
28L. M. Westerhoff and K. M. Merz, Jr., J. Mol. Graphics Modell.
24, 440 /H208492005 /H20850.FACILITATED GAPLESS CONDUCTION THROUGH DNA … PHYSICAL REVIEW B 75, 035111 /H208492007 /H20850
035111-5 |
PhysRevB.86.125139.pdf | PHYSICAL REVIEW B 86, 125139 (2012)
Pseudopotential-based first-principles approach to the magneto-optical Kerr effect: From metals
to the inclusion of local fields and excitonic effects
Davide Sangalli,1,2Andrea Marini,3and Alberto Debernardi1
1MDM Lab, IMM, Consiglio Nazionale delle Ricerche, Via C. Olivetti, 2 I-20864 Agrate Brianza, Italy
2European Theoretical Spectroscopy Facilities (ETSF)
3Istituto di Struttura della Materia of the National Research Council, Via Salaria Km 29.3, I-00016 Monterotondo Stazione, Italy
(Received 10 May 2012; revised manuscript received 2 August 2012; published 27 September 2012)
We propose a first-principles scheme for the description of the magneto-optical Kerr effect within density-
functional theory (DFT). Though the computation of Kerr parameters is often done within DFT, starting from theconductivity or the dielectric tensor, there is no formal justification to this choice. As a first step, using as referencematerials iron, cobalt, and nickel, we show that pseudopotential based calculations give accurate predictions.Then we derive a formal expression for the full dielectric tensor in terms of the density-density correlationfunction. The derived equation is exact in systems with an electronic gap, with the possible exception of Cherninsulators, and whenever the time-reversal symmetry holds and can be used as a starting point for the inclusionof local fields and excitonic effects within time-dependent DFT for such systems. In case of metals instead weshow that, starting from the density-density correlation function, the term which describes the anomalous Halleffect is neglected, giving a wrong conductivity.
DOI: 10.1103/PhysRevB.86.125139 PACS number(s): 71 .15.−m, 71.45.Gm, 78 .20.Bh
I. INTRODUCTION
The magneto-optical Kerr effect (MOKE) consists in the
rotation of the polarization plane of light reflected from thesurface of a magnetic material. It was discovered in 1877 byJohn Kerr
1,2while he was examining the light reflected from
a polished electromagnet pole. Very recently it became the
object of an intense experimental investigation, mainly for tworeasons. First one can exploit this effect to read suitably mag-netically stored information using optical means in modernhigh-density data storage technology.
3–5Second, the MOKE
can be used as a powerful probe in many fields of research suchas microscopy for domain observation, surface magnetism, and
magnetic interlayer coupling in multilayers.
4,6–9It can also be
used to observe plasma resonance effects in thin layers andstructural and magnetic anisotropies.
10–12
The microscopic origin of the Kerr effect is a combined
action of the spin-orbit coupling (SOC) and the net spin po-larization of the material.
13Indeed the existence of a nonzero
magnetization in the ground state is due to a spontaneoussymmetry breaking of the system. This symmetry breakingis transferred, through the SOC, to the spatial part of thewave functions so that ψ
+Lz(x) is different from ψ−Lz(x).
Accordingly the absorption is different for light with rightand left circular polarization.
The problem has been addressed in the literature and ab
initio calculations, based on density-functional theory (DFT),
are available for transition metals like Fe, Co, and Ni
13–21and,
recently, for other materials such as full-Heusler films andMn-doped GaAs.
22–26
The Kerr parameters are commonly obtained from the
Kubo formula27for the optical conductivity tensor using
the single-particle Kohn-Sham (KS) wave functions. This isequivalent to the computation of the dielectric constant at therandom-phase approximation (RPA) without the inclusion oflocal fields (LFs) and exchange-correlation (xc) corrections.We will refer to this approach as the independent particles RPA(IP-RPA) scheme.An alternative approach, based on the Luttinger’s
formula,
22has been proposed in the literature,23–25starting
from the current-current correlation function, χ
jj.A l s oi nt h i s
case the KS wave functions are used and LF and xc effects are
not considered.
In all these works the Kerr parameters are computed within
the DFT framework starting from the dielectric tensor, ε(ω),
or, which is the same, from the conductivity, σ(ω). However
while the diagonal terms of the dielectric tensor, i.e. εii(ω),
can be expressed in terms of the density-density correlationfunction, χ
0
ρρ(orχρρif LF and xc effects are included), to the
best of our knowledge, no such expression has been derivedfor the off-diagonal terms, i.e., ε
ij(ω) with i/negationslash=j. The latter
can be obtained only starting from χ0
jj, which however is not
expressed as a functional of the density. In this case the sole
formal justification to the use of KS quantities to construct χ0
jj
is that a posteriori the approach gives good results and that
KS wave functions can be regarded as a good approximationto quasi-particle wave functions.
In the present work instead we derive an expression for
the full dielectric tensor in terms of χ
0
ρρand so for the
construction of the Kerr parameters within a density basedapproach. Besides a formal justification to the use of the DFTapproach, the result we propose can be regarded as a startingpoint to go beyond the RPA-IP scheme to include LF andxc effects. Indeed while the description of transition metalswithin the RPA-IP approach is reasonable, for semiconductorsimportant deviations are expected, in particular when excitons(magnetic excitons) exist. Magnetic semiconductors are in factmaterials of great interest, especially in view of spin electronics(spintronics) applications, and the MOKE can be a valuabletool for the investigation of their properties.
28,29
In the present work we limit our discussion to the polar
geometry, that is, when the propagation direction of the photon(thezaxis) and the magnetization of the system are both
perpendicular to the surface of the sample ( xy). Experimen-
125139-1 1098-0121/2012/86(12)/125139(8) ©2012 American Physical SocietySANGALLI, MARINI, AND DEBERNARDI PHYSICAL REVIEW B 86, 125139 (2012)
tally this is the most studied geometry, and it is also the one
which in general gives the largest MOKE signal. To supportour theoretical derivation with numerical results we haveimplemented in the plane waves and pseudopotentials basedcode
YAMBO30the computation of the Kerr parameters. We
have tested our implementation in transition metals for whichwell assessed all electron calculations and experimental resultsare available. For these materials the IP-RPA approximation issufficient.
Thus in Sec. IIwe show results on bulk iron, cobalt, and
nickel in order to validate our pseudopotential based approach,at the IP-RPA level.
Then we discuss how to construct a formal equation for the
dielectric tensor starting from χ
0
ρρin Sec. III. This approach
is compared with the one based on χ0
jj. The two differ by a
term which, in general, is zero in systems with an electronic
gap or whenever the pure time-reversal symmetry holds.This term describes the anomalous Hall conductivity (AHC).Taking iron as a reference system we show that, neglectingthis term, a large error is induced in the computation of theoff-diagonal conductivity in metals. However the anomalousHall conductivity is zero in systems with an electronic gap,
31
thus our approach is exact for dielectrics. We then show howLF and xc effects can be included replacing χ
0
ρρwithχρρ.
The result is a scheme, in principle exact, to compute the Kerrparameters within time-dependent DFT (TDDFT).
II. MOKE PARAMETERS WITH THE IP-RPA
APPROXIMATION
A. Theoretical background
The description of the MOKE can be obtained in terms
of the dielectric function ε(ω) or equivalently of the optical
conductivity σ(ω). The two are related by the equation
σ(ω)=ω
4πi(ε(ω)−1), (1)
where the dielectric tensor at the IP-RPA level can be
constructed from the (paramagnetic) χ0
jj, according to the
equation32
εα,β(ω)=/parenleftbigg
1−4πe2n
mω2/parenrightbigg
δα,β−4πe2
ω2χ0
jαjβ(0,ω).(2)
Hereeis the electron charge, mis the electron mass, /Omega1is the
unit cell volume, αlabels the Cartesian axis, and n=Nel//Omega1
is the number of electrons per unit volume. The IP responsefunction is
χ
0
jαjβ(q,ω)
=/summationdisplay
cv/integraldisplayd3k
(2π)3/bracketleftbig
χ0
jαjβ(q,ω)/bracketrightbig
cvkq
=1
m2/summationdisplay
cv/integraldisplayd3k
(2π)3/bracketleftbigg/parenleftbig
pα
cvk(q)/parenrightbig∗pβ
cvk(q)fvk(1−fck−q)
¯hω−(/epsilon1ck−q−/epsilon1vk)+iη
−/parenleftbig
pα
vck(q)/parenrightbig∗pβ
vck(q)fvk−q(1−fck)
¯hω+(/epsilon1ck−/epsilon1vk−q)+iη/bracketrightbigg
, (3)
where the first term on the right-hand side is the resonant part,
while the second is the antiresonant one. c(v) are conduction(valence) band indices, fikare the occupation factors, /epsilon1i(k)
is the electronic energy at k, andpα
cvk(q) is the expectation
value of the momentum operator /angbracketleftck−q|ˆpα|vk/angbracketrightw h i c hi no u r
pseudopotential based scheme must be computed as33
/angbracketleftck−q|/parenleftbigg
ˆpα−im
¯h[ˆxα,ˆVNL]/parenrightbigg
|vk/angbracketright, (4)
where ˆxαis theαcomponent of the position operator and VNL
is the nonlocal part of the pseudopotential. The infinitesimal
ηfactor implies that the electromagnetic field is adiabatically
turned on at t=− ∞ but can also be viewed as a finite lifetime
broadening which accounts for scattering process and finiteexperimental resolution. In the present work we use η(ω)=
0.3e V+0.03¯hωto mimic an experimental resolution which
decreases linearly with energy as in Ref. 14.
B. MOKE spectra for transition metals
As a first step we check how a pseudopotentials based
approach performs in the description of the Kerr parameters,as it is common wisdom
14that all electron calculations are
needed to describe the wave function in the core region, wherethe SOC is mostly effective, and thus to evaluate the MOKE.
We start from a ground-state DFT calculation for bulk
iron, cobalt, and nickel with the
ABINIT code,34using norm-
conserving Hartwigsen, Goedecker, and Hutter (HGH)35
pseudopotentials and including the SOC; we found out thatit is crucial, for a correct description of the MOKE, to havethe SOC effect included both in the pseudo-Hamiltonian andin the construction of the pseudopotential.
Bulk iron is studied in its bcc phase with the experimental
cell parameter a=2.87˚A, an energy cutoff of 65 Ha, and a
k-points sampling of the Brillouin zone (BZ) 14 ×14×14.
Bulk cobalt is studied in its fcc phase with the experimentalcell parameter a=3.55˚A, an energy cutoff of 55 Ha, and a
k-points sampling of the BZ 8 ×8×8. Finally bulk nickel is
studied in its fcc phase with the experimental cell parametera=3.52˚A, an energy cutoff of 65 Ha, and a k-points sampling
of the BZ zone 14 ×14×14. Semicore electrons are also
included in the pseudopotentials for all systems, as we foundthe density of states to be poorly described using the HGHpseudopotentials with only valence electrons, constructedfrom the parameters of Ref. 35.
Then we compute the dielectric function with the
YAMBO
code30according to a modified version of Eq. (2):
εα,β(ω)=/parenleftbigg
1+4πe2
ω2χ0
jαjβ(0,0)/parenrightbigg
δα,β−4πe2
ω2χ0
jαjβ(0,ω),
(5)
where the diamagnetic term has been replaced by the zero-
frequency value of χjj. Indeed in cold semiconductors the
diamagnetic term must be exactly balanced by χ0
jαjβ(0,0), due
to the effective-mass sum rule,36–38but this balance is slowly
converging with the number of cvstates included and the use
ofχ0
jαjβ(0,0) speeds up the convergence.39In metals instead
the difference between the diamagnetic term and the zero-
frequency value of χjjgives the Drude term, which, with this
choice, is set to zero. However it has been shown40that in
practice an ultrafine sampling of the BZ would be needed to
125139-2PSEUDOPOTENTIAL-BASED FIRST-PRINCIPLES ... PHYSICAL REVIEW B 86, 125139 (2012)
24681012Re[σxx(ω)]
05101520253035ω Im[σxx(ω)]
24681012[1015 s-1]
05101520253035
[1030 s-2]
024 6 8
[eV]24681012
024 6 81 005101520253035Iron bcc
Nickel fccCobalt fcc(a) (b)
(c) (d)
(e) (f)
FIG. 1. (Color online) Plot of σxx(ω)f o r( a )a n d( b )b u l kb c c
iron, (c) and (d) bulk fcc cobalt, and (e) and (f) bulk fcc nickel.
The continuous (red) lines are the results from the preset work. Thedashed lines are all electron results from Ref. 14. The symbols are
experimental measurements. (a) and (b) Ref. 41. (c) and (d) Ref. 42.
(e) and (f) Filled circles, Ref. 42. Empty squares, Ref. 43. Filled
triangles, Ref. 44. Filled squares, Ref. 41. Empty triangles, Ref. 45.
compute this difference. Hence it is preferable to include it
with a semiclassical model as described in Ref. 40. The Drude
term is included only in the computation of the diagonal part ofε
(ω). The optical conductivity is finally constructed from Eq.
(1). Here for the Drude term we used the same parameters of the
reference all electron calculation, i.e., ωP=(4.9+i1.8π)e V
for iron, ωP=(8.3+i2π) eV for cobalt, and ωP=(7.5+
i2.24π) eV for nickel.
The results for the optical conductivity are plotted in Figs. 1
and 2. For all systems there is a systematic blue-shift of
the theoretical peaks against the experimental data. This isa known problem of the LDA, due to the self-interaction error,which tends to delocalize the dorbitals and accordingly gives
wrong eigenvalues. The same consideration also explains theoverestimation of the intensity, as delocalization increasesthe orbitals overlap and thus the intensity of the dipolescomputed to construct the dielectric function. However a goodagreement is found with the reference all electron calculations.The diagonal component, σ
xx(ω), is commonly computed
from pseudopotential based calculations, provided that the-0.8-0.6-0.4-0.200.20.40.60.8ω Re[σxy(ω)]
-0.4-0.200.20.40.60.81ω Im[σxy(ω)]
-0.8-0.6-0.4-0.200.20.40.60.8[1030 s-1]
-0.4-0.200.20.40.60.81
[1030 s-2]
024 6 8
[eV]-0.8-0.6-0.4-0.200.20.40.60.8
024 6 81 0-0.4-0.200.20.40.60.81Iron bcc
Nickel fccCobalt fcc(a) (b)
(c) (d)
(e) (f)
FIG. 2. (Color online) Plot of σxy(ω) for (a) and (b) bulk bcc
iron, (c) and (d) bulk fcc cobalt, and (e) and (f) bulk fcc nickel.The continuous (red) lines are the results from the preset work. The
dashed lines are all electron results from Ref. 14. The symbols are
experimental measurements. (a) and (b) Filled circles, Ref. 46; empty
squares, Ref. 47. (c) and (d) Ref. 42. (e) and (f) Filled circles, Ref. 42;
empty squares, Ref. 48.
dipoles are constructed as in Eq. (3). For the off-diagonal
component, σxy(ω), instead, it has been reported that it must
be computed using all electron wave functions,14because
it depends crucially on the correction to the wave functiondue to the SOC term in the Hamiltonian. However, in ourresults, the differences in the diagonal and the off-diagonalparts of the optical conductivity compared to the referenceall electron calculations are of the same order. Hence wecan conclude that pseudopotentials based calculation can beused to compute the Kerr parameters with the same levelof confidence of absorption spectra, which depend only onthe diagonal part of the optical conductivity. We mentionthat, in the literature, pseudo wave function have been usedto construct the off-diagonal conductivity at ω=0i nt h e
computation of the anomalous Hall conductivity.
49–51Also
in this case a good agreement with the all electron calculationswas found.
Thus we finally compute the complex Kerr parameters
according to the equation
/Psi1
K(ω)=θK(ω)+iγK(ω)=−εxy
(εxx−1)√εxx,( 6 )
125139-3SANGALLI, MARINI, AND DEBERNARDI PHYSICAL REVIEW B 86, 125139 (2012)
-0.8-0.6-0.4-0.200.2θK(ω)
-0.4-0.200.20.4γK(ω)
-0.8-0.6-0.4-0.200.2[deg]
-0.4-0.200.20.4
[deg]
024 6 8
[eV]-0.8-0.6-0.4-0.200.2
024 6 81 0-0.4-0.200.20.4Iron bcc
Nickel fccCobalt fcc(a) (b)
(c) (d)
(e) (f)
FIG. 3. (Color online) Plot of the Kerr parameters /Psi1K(ω)=θK+
iγKfor (a) and (b) bulk bcc iron, (c) and (d) bulk fcc cobalt, and (e)
and (f) bulk fcc nickel. The continuous (red) lines are the results
from the preset work. The dashed lines are all electron results from
Ref. 14. The symbols are experimental measurements. (a) and (b)
Filled circles, Ref. 46; empty diamonds, Ref. 47; empty triangles,
Ref. 52; stars, Ref. 53; filled squares, Ref. 54. (c) and (d) Filled
circles, Ref. 42; empty triangles, Ref. 55. (e) and (f) Filled circles,
Ref. 42; empty triangles, Ref. 47; empty squares, Ref. 46.
which is the standard expression for the polar geometry in
the small angles limit. Here the photon propagates along thezdirection and describes a linearly polarized wave with the
electric field along the xdirection. Results are reported in
Fig.3. The blue-shift of the theoretical results is still present,
while the overestimation of the dipoles in both σ
xx(ω) and
σxy(ω) is compensated and the intensity of the MOKE signal
is closer to the experimental data than for the case of the opticalconductivity.
To conclude this section, we have shown that for Fe, Co,
and Ni the Kerr results computed from our pseudopotentialapproach are in good agreement with the results obtained fromall electron calculations.
III. BEYOND THE IP-RPA APPROXIMATION
A. A density based approach
In the previous section we have constructed the Kerr
parameters starting from the KS wave functions using Eq. (2),as it is commonly done in the literature. However the use of
the KS wave function to construct χ0
jjis not formally justified.
Moreover the inclusion of LF and xc effects within a density
based approach in Eq. (2)is not straightforward.
However, to describe the MOKE only the long wavelength
term, i.e., q=0, of the dielectric function is needed, where
the distinction between longitudinal and transverse fieldsdisappears. In this limit the diagonal part of the dielectric tensorcan be constructed from χ
0
ρρ, which, at finite q, describes only
the longitudinal term of the dielectric tensor. This approachis formally justified within a density based approach andmoreover would allow a straightforward inclusion of LF andxc effects within the TDDFT scheme. It is then tempting totry to construct the full dielectric tensor at q=0 from χ
0
ρρand
use the result to go beyond the IP-RPA scheme.
Here we provide a heuristic derivation where only lon-
gitudinal fields are considered, as our final goal is to take theq→0 limit. We will prove a posteriori that the result is correct
for systems with an electronic gap or, more generally, whenthe pure time-reversal symmetry exists, and we will discussin detail the difference between the derived equation andEq.(2).
We consider a nonuniform system. The dielectric function
is defined as
E
ext(q,ω)=ε(qq/prime,ω)Etot(q/prime,ω). (7)
Assuming that only longitudinal fields exist, Eq. (7)can be
written in terms of the potentials
Vext(q,ω)=ˆqε(qq/prime,ω)ˆq/primeVtot(q/prime,ω), (8)
where the two are related by the equation
Vext(q,ω)=Vtot(q,ω)−Vind(q,ω)( 9 )
=Vtot(q,ω)−4πe2
q2χ0
ρρ(q,q/prime,ω)Vtot(q/prime,ω).(10)
Inserting Eq. (10) into Eq. (8)and taking the limit q→
q/prime→0 we can define a generalization of the relation that
holds between εααandχρρ:32
εαβ(ω)=δαβ−lim
qα,qβ→04πe2
q2χ0
ρρ(qα,qβ,ω). (11)
In order to compare Eqs. (11) and(2)we first notice that
the latter is divergent for ω→0. After some algebra Eq. (2)
can be rewritten as36
εα,β(ω)=Aαβ
ω2+Bαβ
ω+δαβ
+/summationdisplay
cv/integraldisplayd3k
(2π)34πe2¯h2
(/epsilon1ck−/epsilon1vk)2/bracketleftbig
χ0
jαjβ(0,ω)/bracketrightbig
cvk0.
(12)
Aαβdescribes the contribution from the electrons at the
Fermi surface, i.e., the Drude term, and is zero in coldsemiconductors, when there are not partially filled bands. Thisterm is also included in Eq. (11)in theq→0 limit as discussed
in Ref. 40. Once the ω
−2has been isolated using the relation
xα
cvk=−i¯hpα
cvk/[m(/epsilon1ck−/epsilon1vk)] in the last term of Eq. (12)
125139-4PSEUDOPOTENTIAL-BASED FIRST-PRINCIPLES ... PHYSICAL REVIEW B 86, 125139 (2012)
together with
lim
q,q/prime→0χ0
ρρ(q,q/prime,ω)
=lim
q,q/prime→0/summationdisplay
cv/integraldisplayd3k
(2π)3/bracketleftbigg(iq·x∗
cvk)(iq/prime·xcvk)fvk(1−fck−q)
¯hω−(/epsilon1ck−q−/epsilon1vk)+iη
−(iq·x∗
vck)(iq/prime·xvck)fvk−1(1−fck)
¯hω+(/epsilon1ck−/epsilon1vk−q)+iη/bracketrightbigg
, (13)
we obtain the remaining part of Eq. (11).
B. The anomalous Hall effect
Hence term Bαβis not included in Eq. (11). It can be
explicitly written, at the RPA-IP level, as
Bαβ=¯he2
2π2m2/summationdisplay
uw/integraldisplay
d3k(fuk−fwk)/parenleftbig
pα
wuk/parenrightbig∗pβ
wuk
(/epsilon1wk−/epsilon1uk)2.(14)
This can be shown to be zero when the time-reversal symmetry
holds36or in any case when α=βinverting the mute indices
uandwin the second term on the right-hand side. In
MOKE experiments however the time reversal is broken bythe existence of a ground-state magnetization and by the SOCterm in the Hamiltonian and for the construction of /Psi1
K(ω)w e
need the terms α/negationslash=β.
ThusBαβcan differ from zero. In the following, we briefly
discuss its physical meaning. To fix the ideas we chose α=x
andβ=y. It can be easily proven that the the Bxycoefficient
is (apart from being a trivial factor arising from the relationbetween ε
andσ) the intrinsic anomalous Hall conductivity
(AHC), which is responsible for the anomalous Hall effect inmagnetic metals.
In fact, according to Ref. 56the AHC reads
σ
AHC
xy=−e2
¯h/summationdisplay
u/integraldisplayd3k
(2π)3fuk/Omega1z
u(k). (15)
That is, σAHCcan be expressed as a BZ integral of the Berry
curvature of the uband,/Omega1z
u(k) (summed over all the occupied
states). The latter quantity can be written in terms of theingredients of Eq. (14) as
56
/Omega1z
u(k)=−¯h2
m2/summationdisplay
w,w/negationslash=u2Im/parenleftbig
px
uwkpy
wuk/parenrightbig
(/epsilon1uk−/epsilon1wk)2. (16)
After some straightforward algebra, one can easily prove
that the AHC can be expressed as
σAHC
xy=Bxy
4πi, (17)
which provides the relations between Band the AHC. In the
case of magnetic metals, our expression constitutes an alter-native approach to compute σ
AHCwith respect to the methods
based on the computation of the Berry phase.51In the case of
insulators instead σAHChas been recognized as a topological
invariant,57also called Chern number, which can take only
integer values. Thus, in the dielectric, Bxycan be nonzero
only in the so-called Chern insulator, hypothetical materialsshowing a quantum Hall effect without external magnetic field.In practice for all the presently known dielectrics Eq. (11) can
be considered exact.024 6 8
[eV]-1.5-1.0-0.50.00.51.0Re[σxy] [103 (Ω*cm)-1]
024 6 8-1.0-0.50.00.51.01.5
ω Im[σxy] [1030 s-2]-3-2-101ω Re[σxy] [1030 s-2]
00.511.52
Im[Δεxy] Re[Δεxy]σAH=665 ( Ωcm)-1x100(a) (b)
(c) (d)
FIG. 4. (Color online) Off-diagonal element of the conductivity
tensor, σxy(ω), of iron computed starting from Eq. (11), dot-dashed
(green) line, and from Eq. (2), continuous (red) line. The dashed lines
are all electron results from Ref. 14. The symbols are experimental
measurements: filled circles, Ref. 46; empty squares, Ref. 47. (a) and
(b)σxy(ω) computed with the smearing used in the present work. (c)
and (d) σxy(ω) evaluated at η=0.05 eV . The difference between the
dot-dashed (green) line and the continuous (red) line is represented
with a thin (black) line. Inset: Difference in terms of the dielectricfunction.
Also in this case we have tested, at the IP-RPA level,
the effect of Bαβon bulk iron, comparing the conductivity
computed starting from either Eqs. (2)or(11).I nF i g s . 4(a)
and4(b), we show the error induced in the computation of the
off-diagonal conductance on bulk iron.
To clarify the relation between this difference and the AHC
also numerically we have considered, in Figs. 4(c)and4(d),t h e
plot of the conductivity at small smearing, i.e., η=0.05 eV,
as Eqs. (2)and(11) are equal only in the limit η→0.58From
Fig.4(c)it is clear that the difference of the two gives a constant
value, as expected theoretically, apart from the region ω/similarequal0,
where the 1 /ω2term makes Eq. (2)numerically unstable. We
can thus extract the σAHC
xy=665 (/Omega1cm)−1, which is not so
far from the theoretically computed value 750 .8(/Omega1cm)−1of
Ref. 56. The difference is likely due to the sampling of the
BZ. As for the case of the Drude term, the anomalous Hallconductivity depends on the contribution from the electrons
125139-5SANGALLI, MARINI, AND DEBERNARDI PHYSICAL REVIEW B 86, 125139 (2012)
at the Fermi surface and thus a very fine sampling of the BZ
should be needed, which is beyond the scope of the presentwork. Also we see in Fig. 4(d) that at small ηthe difference
between the imaginary parts of the conductivity computedwith the two approaches goes to zero [Fig. 4(b)] as expected
from the theoretical derivation. Finally in the insets we havealso represented the differences at the level of the dielectricfunction which are Im[ /Delta1ε]∝1/ωand Re[ /Delta1ε]∝δ(ω), thus
respecting the Krames-Kronig relations.
C. Inclusion of local fields and excitonic effects
The generalization of Eq. (2)to include LF and xc effects is
nontrivial and needs a careful distinction between longitudinaland transverse induced fields. The result, derived in Ref. 59,
is to replace the IP χ
0
jjwith the one constructed from the
analytical part of the electron-hole (eh) propagator L(12,34)
solution of the modified Bethe-Salpeter equation:
L(12,34)=L0(12,34)+L0(12,1/prime2/prime)[v(1/prime,3/prime)δ(1/prime,2/prime)δ(3/prime,4/prime)
−iW(1,2)δ(1/prime,3/prime)δ(2/prime,4/prime)]L(3/prime4/prime,34), (18)
with 1 representing spatial, time, and spin coordinates: 1 =
(x1,t1,σ1). The long-range part of the exchange interaction
v(1,2)=δ(t1−t2)/|x1−x2|between the electron and the
hole is truncated with the substitution v→vwhere in
reciprocal space
vG(q)=/braceleftbigg4πe2/|q+G|2ifG/negationslash=0,
0i f G=0.. (19)
From the electron-hole propagator the χ
jjandχ
ρρare
constructed with the relations32
χρρ(1,2)=−i¯hL(1,2; 1+,2+), (20)
χ
jj(1,2)=−i¯h−¯h2
4m2[(∇1−∇/prime
1)(∇2−∇/prime
2)L(1,2; 1/prime2/prime)]1/prime=1+,2/prime=2+,
(21)
with 1+=limτ→0(x1,t1+τ,σ 1) The result is then
εα,β(ω)=/parenleftbigg
1−4πe2n
mω2/parenrightbigg
δα,β−4πe2
ω2χjαjβ(0,ω). (22)
A possible strategy to remain within a density based
formalism,60starting from Eq. (2), could be to use the
unphysical LTDDFTreplacing iW(1,2)δ(1/prime,3/prime)δ(2/prime,4/prime) with
fxc(1,2)δ(1/prime,2/prime)δ(3/prime,4/prime)i nE q . (18). However this is not
formally justified and at least a current based formalism shouldbe used, i.e., current DFT.
61–63Indeed for the description of
absorption spectra, the diagonal part only of the dielectricfunction is commonly constructed from
χρρto include LF and
xc effects64starting from εii(ω)[χ0
ρρ]. A similar derivation can
be used to include LF and xc effects replacing χ0
ρρwithχρρ
in Eq. (11). In this case however the Dyson equation for the
response function should be written for a nonhomogeneoussystem assuming, as we did in the IP case, that transversefields can be neglected as we are looking for the q→0 limit.
The result is
ε
αβ(ω)=δαβ−lim
qα,qβ→04πe2
q2χρρ(qα,qβ,ω), (23)which, according to the discussion of the previous sections,
should hold when the time-reversal symmetry exists or forsystems with an electronic gap.
31,65Equation (23) must then
be compared with Eq. (22).I fχ
jjandχρρare constructed from
the same Lusing Eqs. (20) and(21) the two are diagonalized
by the same vectors in cvspace, AI
cvk; here Iis the index of
the excitation, which now can be a mixture of electron-holepairs. In this case, inserting the vectors A
I
cvkin the equations,
the two approaches will differ by the term
Bαβ=¯he2
2π2m2/summationdisplay
I/summationdisplay
uw/integraldisplay
d3k(fuk−fwk)
×/parenleftbig
AI
wukpα
wuk/parenrightbig∗AI
wukpβ
wuk
(¯hωI)2, (24)
which defines a generalization of the anomalous Hall effect.
HereωIare the poles of χρρ. In common metals usually we
haveAI
cvk=δI,(cv)I[i.e., each vector AIis different from zero
only for a specific transition cv=(cv)I] and ¯hωI=/epsilon1ck−/epsilon1vk,
thus Eq. (24) reduces to Eq. (14).
However if one remains within a pure DFT approach, then
the vectors which diagonalize χρρ,AI,TDDFT
cvk do not, in general,
diagonalize Land thus χ
jj. In this case Eqs. (23) and(22)
could also differ by a term proportional to AI,TDDFT
cvk−AI
cvk.
This term must be zero for α=β, while its relevance in the
caseα/negationslash=βand its eventual physical meaning are left under
study.
IV . CONCLUSIONS
We have proposed a scheme to compute the magneto-
optical Kerr effect in magnetic semiconductors. The schemehas two main novelties. First, it is based on pseudopotentialscalculations. This is the most widely used approach to describeextended systems and we have shown that pseudo-wave-functions can be used to obtain the Kerr parameters. The resultswe find are comparable with all electron calculations, providedthat the spin-orbit interaction is correctly accounted for in theconstruction of the pseudopotential.
Second, we have discussed the inclusion of local-field and
excitonic effects in the computation of the MOKE. We haveshown that two strategies can be used: (i) the Bethe-Salpeterequation, through the result derived in Ref. 59, but also,
in almost any case of interest, (ii) an approach based ontime-dependent density-functional theory and in general on thedensity-density correlation function through the result derivedin the present manuscript.
ACKNOWLEDGMENTS
This work was partially funded by the Cariplo Founda-
tion through the Oxides for Spin Electronic Applications(OSEA) project (No. 2009-2552). D. Sangalli would liketo acknowledge G. Onida and the European TheoreticalSpectroscopy Facility (ETSF)
66Milan node, for the oppor-
tunity of running simulations on the ETSF-Milano (ETSFMI)cluster, and P. Salvestrini for technical support on the clus-ter. We also acknowledge computational resources providedby the Consorzio Interuniversitario per le Applicazioni di
125139-6PSEUDOPOTENTIAL-BASED FIRST-PRINCIPLES ... PHYSICAL REVIEW B 86, 125139 (2012)
Supercalcolo Per Universit ´a e Ricerca (CASPUR) within
the project Magnetic Oxides for Spin Electronics (MOSE).Finally D. Sangalli and A. Debernardi would like to thank
R. Colnaghi for technical support.
1J. Kerr, Philos. Mag. 3, 321 (1877).
2J. Kerr, Philos. Mag. 5, 161 (1878).
3G. A. Bertero and R. Sinclair, J. Magn. Magn. Mater. 134, 173
(1994).
4T. K. Hatwar, Y . S. Tyan, and C. F. Bruker, J. Appl. Phys. 81, 3839
(1997).
5R. Alcaraz de la Osa, J. M. Saiz, F. Moreno, P. Vavassori, andA. Berger, Phys. Rev. B 85, 064414 (2012).
6W. R. Bennett, W. Schwarzacher, and W. F. Egelhoff, Phys. Rev.
Lett. 65, 3169 (1990).
7Y . Suzuki, T. Katayama, P. Bruno, S. Yuasa, and E. Tamura, Phys.
Rev. Lett. 80, 5200 (1998).
8C. Zinoni, A. Vanhaverbeke, P. Eib, G. Salis, and R. Allenspach,
P h y s .R e v .L e t t . 107, 207204 (2011).
9A. L. Balk, M. E. Nowakowski, M. J. Wilson, D. W. Rench,
P. Schiffer, D. D. Awschalom, and N. Samarth, P h y s .R e v .L e t t .
107, 077205 (2011).
10R. Q. Wu and A. J. Freeman, J. Magn. Magn. Mater. 200, 498
(1999).
11W. B. Zeper, F. J. A. M. Greidanus, P. F. Garcia, and C. R. Fincher,J. Appl. Phys. 65, 4971 (1989).
12D. Weller, H. Br ¨andle, G. Gorman, C.-J. Lin, and H. Notarys, Appl.
Phys. Lett. 61, 2726 (1992).
13G. Y . Guo and H. Ebert, P h y s .R e v .B 51, 12633 (1995).
14A. Delin, O. Eriksson, B. Johansson, S. Auluck, and J. M. Wills,
P h y s .R e v .B 60, 14105 (1999).
15P. M. Oppeneer, T. Maurer, J. Sticht, and J. K ¨ubler, P h y s .R e v .B
45, 10924 (1992).
16P. M. Oppeneer, T. Kraft, and H. Eschrig, Phys. Rev. B 52, 3577
(1995).
17M. Y . Kim, A. J. Freeman, and R. Wu, Phys. Rev. B 59, 9432
(1999).
18J. M. Luttinger, in Mathematical Methods in Solid State and
Superfluid Theory ,e d i t e db yR .C .C l a r ka n dG .H .D e r r i c k( O l i v e r
and Boyd, Edinburg, 1967), Chap. 4, p. 157.
19A. Vernes, L. Szunyogh, and P. Weinberger, P h y s .R e v .B 65, 144448
(2002).
20A. Vernes and P. Weinberger, Phys. Rev. B 70, 134411
(2004).
21A. Vernes, I. Reichl, P. Weinberger, L. Szunyogh, and C. Sommers,P h y s .R e v .B 70, 195407 (2004).
22F. Ricci, S. Picozzi, A. Continenza, F. D’Orazio, F. Lucari,
K. Westerholt, M. Y . Kim, and A. J. Freeman, Phys. Rev. B 76,
014425 (2007).
23A. Stroppa, S. Picozzi, A. Continenza, M. Y . Kim, and A. J.Freeman, P h y s .R e v .B 77, 035208 (2008).
24L. Uba, S. Uba, L. P. Germash, L. V . Bekenov, and V . N. Antonov,
P h y s .R e v .B 85, 125124 (2012).
25F. Ricci, F. D’Orazio, A. Continenza, F. Lucari, and A. J. Freeman,
P h y s .R e v .B 83, 224421 (2011).
26F. Haidu, M. Fronk, O. D. Gordan, C. Scarlat, G. Salvan, and D. R.
T. Zahn, Phys. Rev. B 84, 195203 (2011).
27R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).28G. Acbas, M.-H. Kim, M. Cukr, V . Nov `a k ,M .A .S c a r p u l l a ,O .D .
Dubon, T. Jungwirth, Jairo Sinova, and J. Cerne, Phys. Rev. Lett.
103, 137201 (2009).
29C. Sun, J. Kono, A. Imambekov, and E. Morosan, P h y s .R e v .B 84,
224402 (2011).
30A. Marini, C. Hogan, M. Gr ¨uning, and D. Varsano, Comput. Phys.
Commun. 180, 1392 (2009).
31With the exception of the so-called Chern insulator.
32G. Strinati, Rivista del Nuovo Cimento 11, 1 (1988).
33A. J. Read and R. J. Needs, P h y s .R e v .B 44, 13071 (1991).
34X. Gonze et al. ,Comput. Phys. Commun. 180, 2582 (2009).
35C. Hartwigsen, S. Goedecker, and J. Hutter, Phys. Rev. B 58, 3641
(1998).
36J. E. Sipe and E. Ghahramani, P h y s .R e v .B 48, 11705 (1993).
37K. S. Virk and J. E. Sipe, Phys. Rev. B 76, 035213 (2007).
38G. F. Giuliani and G. Vignale, Quantum Theory of the Electron
Fluid (Cambridge University Press, New York, 2005), Chap. 3.4.2.
39The inclusion of a big number of cvexcitations would be very
problematic for the description of local fields and excitoniceffects where a matrix whose dimensions are N
cvxNcvmust be
diagonalized.
40A. Marini, G. Onida, and R. Del Sole, P h y s .R e v .B 64, 195125
(2001).
41P. B. Johnson and R. W. Christy, P h y s .R e v .B 9, 5056
(1974).
42K. Nakajima, H. Sawada, T. Katayama, and T. Miyazaki, Phys. Rev.
B54, 15950 (1996).
43M. Shiga and G. P. Pells, J. Phys. C 2, 1847 (1969).
44M. Ph. Stoll, Solid State Commun. 8, 1207 (1971).
45H. Ehereinreich, H. R. Philipp, and D. J. Olencha, Phys. Rev. 131,
2469 (1963).
46P. G. Van Engen, Ph.D. thesis, Technical University Delft, 1983.
47G. S. Krinchik and V . A. Artemjev, J. Appl. Phys. 39, 1276
(1968).
48J. L. Erskine, Physica B +C89, 83 (1977).
49J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Phys. Rev. 75,
195121 (2007).
50X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Phys. Rev. B 74,
195118 (2006).
51X. Wang, D. Vanderbilt, J. R. Yates, and I. Souza, Phys. Rev. B 76,
195109 (2007).
52T. Katayama, H. Awano, and Y . Nishihara, J. Phys. Soc. Jpn. 55,
2539 (1986).
53S. Visnovsky, R. Krishnan, M. Nylt, and P. Prosser (unpublished).
54T. Kawagoe and T. Mizoguchi, J. Magn. Magn. Mater. 113, 187
(1992).
55D. Weller, G. R. Harp, R. F. C. Farrow, A. Cebollada, and J. Sticht,Phys. Rev. Lett. 72, 2097 (1994).
56Y . Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth,
D. S. Wang, E. Wang, and Q. Niu, Phys. Rev. Lett. 92, 037204
(2004).
57D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs,Phys. Rev. Lett. 49, 405 (1982).
125139-7SANGALLI, MARINI, AND DEBERNARDI PHYSICAL REVIEW B 86, 125139 (2012)
58In particular the expression ( ω+iη)−1is used in the implementa-
tion which in the limit η→0 gives lim η→01
ω+iη=1
ω−iπδ(ω).
59R. Del Sole and E. Fiorino, Phys. Rev. B 29, 4631 (1984).
60For completeness we also mention that an alternative approach to
the inclusion of LF and xc effects within TDDFT can be based onthe real-time propagation approach.
67
61G. Vignale and M. Rasolt, P h y s .R e v .L e t t . 59, 2360 (1987).
62G. Vignale, P h y s .R e v .B 70, 201102(R) (2004).
63S. H. Abedinpour, G. Vignale, and I. V . Tokatly, Phys. Rev. B 81,
125123 (2010).
64The analytic part only of the electron-hole propagator is used tobetter describe the long-range part of the coulomb interaction inextended systems. Indeed, within the dipole approximation, it is alsopossible to express ε
−1(ω)[χρρ]. This is equivalent to ε(ω)[χρρ].The substitution ε−1(ω)[χρρ]→ε−1(ω)[χ0
ρρ] leads to the IP ap-
proximation while the replacement ε(ω)[χρρ]→ε(ω)[χ0
ρρ] leads
to the IP-RPA approximation where the long-range contributionof the Coulomb interaction is described at the RPA level, whilemicroscopic fields are described at the IP level, i.e., are neglected.
65We are presently doing simulations on magnetic semiconductors,namely, EuS and EuO, to check the inclusion of LF and xc effectson the Kerr parameters.
66A. Y . Matsuura, N. Thrupp, X. Gonze, Y . Pouillon, G. Bruant,and G. Onida, Comput. Sci. Eng. 14, 22 (2012) http://www.etsf.it ;
http://www.etsf.eu .
67D. Varsano, L. A. Espinosa-Leal, X. Andrade, M. A. L. Marques,
R. Di Felice, and A. Rubio, PhysChemChemPhys 11, 4481
(2009).
125139-8 |
PhysRevB.98.121108.pdf | PHYSICAL REVIEW B 98, 121108(R) (2018)
Rapid Communications
Giant planar Hall effect in the Dirac semimetal ZrTe 5−δ
P. Li, C. H. Zhang, J. W. Zhang, Y . Wen, and X. X. Zhang*
King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering Division (PSE),
Thuwal 23955-6900, Saudi Arabia
(Received 3 March 2018; revised manuscript received 26 June 2018; published 24 September 2018)
Recently, giant planar Hall effect originating from chiral anomaly has been predicted in nonmagnetic
Dirac/Weyl semimetals. ZrTe 5is considered to be an intriguing Dirac semimetal at the boundary of weak
topological insulators and strong topological insulators, although this claim still remains controversial. Here,we report the observation in ZrTe
5−δof the giant planar Hall resistivity that shows two different magnetic-field
dependences as predicted by theory and a maximum at the Lifshitz transition temperature. We found thatthe giant planar Hall resistivity fades out with decreasing the thickness of ZrTe
5−δnanoplates, which may
be ascribed to the vanishing of the 3D nature of the samples. In addition, we have observed a nontrivialBerry phase, chiral-anomaly-induced negative longitudinal magnetoresistance, and a giant in-plane anisotropicmagnetoresistance in these ZrTe
5−δnanoplates. All the experimental observations demonstrated coherently that
ZrTe 5−δis a Dirac semimetal.
DOI: 10.1103/PhysRevB.98.121108
Dirac and Weyl semimetals as a new type of quantum
materials have drawn tremendous attention recently for thenovel physics and potential applications [ 1–8]. One of the
most intriguing magnetotransport properties in these materi-als is the chiral-anomaly-induced negative magnetoresistance(NMR) [ 5,9,10]. Although NMR is considered a signature
of Dirac or Weyl semimetals, several extrinsic factors, suchas current jetting and conductance fluctuation, may also leadto NMR [ 11–14]. Very recently, giant planar Hall effect
(PHE) was predicted theoretically to appear in Dirac andWeyl semimetals [ 15,16], where PHE refers to the transverse
voltage when a magnetic field is applied in plane with the elec-trical current. PHE is a weak magnetic field effect and usuallyobserved in ferromagnetic materials where it originates fromthe spin-orbit coupling [ 17–19]. Different from ferromagnetic
material, the PHE in topological Dirac and Weyl semimetalswas theoretically demonstrated to originate from nontrivialBerry phase and chiral anomaly [ 15,16,20–24]. The angular
dependence of PHE resistivity and longitudinal anisotropicmagnetoresistance in Dirac and Weyl semimetals can be de-scribed as [ 15,16]
ρ
xy=−/Delta1ρchiralsinϕcosϕ, (1)
ρxx=ρ⊥−/Delta1ρchiralcos2ϕ, (2)
where/Delta1ρchiral=ρ⊥−ρ//is resistivity anisotropy induced by
chiral anomaly and Berry phase, ρ⊥andρ//are the resistivity
for a magnetic field applied transversely and parallel to thecurrent in the current plane, respectively.
The monolayer ZrTe
5was initially predicted and
demonstrated to be a 2D quantum spin Hall insulator [ 25–27].
*xixiang.zhang@kaust.edu.saAlthough angle-resolved photoemission spectroscopy
(ARPES) experiments showed directly that a bulk ZrTe 5is a
weak topological insulator (TI) [ 28], it was also demonstrated
that bulk ZrTe 5is a 3D Dirac semimetal by the observation
of chiral anomaly in magnetotransport [ 29] and nontrivial
Berry phase [ 30,31], and by magnetoinfrared spectroscopy
[32]. More interestingly, it has been shown that the Dirac
semimetal states will appear in ZrTe 5at the boundary of
strong and weak TIs [ 33–35]. Therefore, the nature of layered
transition-metal chalcogenide ZrTe 5is still under hot debate
and more evidence is needed to uncover its Dirac semimetalstate.
In this Rapid Communication, we report the observa-
tion of chiral-anomaly-induced planar Hall effect in ZrTe
5−δ
nanoplates. The nontrivial Berry phase ( ∼π)i nZ r T e 5−δ
was obtained from the Shubnikov–de Haas (SdH) oscillation
data. The chiral-anomaly-induced giant planar Hall resistivityreached the maximum value of 254.0 μ/Omega1cm (14 T) near
the resistivity anomaly temperature ∼150 K, which strongly
supports the existence of the Dirac point in ZrTe
5−δ.
ZrTe 5−δsingle crystals grown by iodine-assisted vapor
transport technique were obtained from HQ Graphene. Thethin plates of ZrTe
5−δwere exfoliated mechanically onto the
SiO 2(280-nm) /Si substrates. The electrodes were patterned
by standard e-beam lithography and deposited by e-beamevaporation. The magnetotransport measurements were car-ried out on a physical property measurement system (Dyna-cool system, Quantum Design Inc.) by standard lock-in meth-
ods (see Supplemental Material [ 36]). The crystal structure
of ZrTe
5−δwas imaged using monochromated Cs-corrected
high-resolution scanning transmission electron microscopy.The ionic gating on ZrTe
5−δdevices was carried out using
LiClO 4+Polyethylene oxide (PEO) +methanol as the solid
electrolyte [ 37,38].
Figure 1(a) shows the optical image of a typical exfoliated
ZrTe 5−δmicroribbon device and measurement configuration
2469-9950/2018/98(12)/121108(7) 121108-1 ©2018 American Physical SocietyLI, ZHANG, ZHANG, WEN, AND ZHANG PHYSICAL REVIEW B 98, 121108(R) (2018)
FIG. 1. (a) Optical image of ZrTe 5−δdevices. The scale bar is 30 μm. (b) HAADF image of ZrTe 5−δsample in abplane. The dashed
rectangle gives the unit cell of ZrTe 5−δinabplane. The green and purple dots represent Zr and Te atoms, respectively. The scale bar is 1 nm.
(c) Temperature dependence of resistivity of ZrTe 5−δunder different magnetic fields. (d) Hall resistivity of ZrTe 5−δas a function of magnetic
field at selected temperatures. The inset gives ρxy−Tcurves ( B=5T ) .
of PHE ( ρxy) and anisotropic magnetoresistance (AMR) ( ρxx)
[36]. The high angle area dark-field (HAADF) image of
ZrTe 5−δ(abplane) is shown in Fig. 1(b). The unit cell of
ZrTe 5−δina−bplane is indicated by the white dashed
rectangle, in which Zr and Te atoms are highlighted with,respectively, green and purple dots for clarity. The latticeconstant along bis identified to be ∼14.5 Å. According to
the previous report, ZrTe
5will transform from strong TIs to
weak TIs with increasing the lattice constant blarger than
14.46 Å, whereas the Dirac semimetal states will appear atthe boundary of the transformation [ 33,34]. Therefore, the
lattice constant of our sample supports the existence of Diracsemimetal state.
Figure 1(c) shows the typical temperature dependence of
the longitudinal resistance of ZrTe
5−δunder magnetic fields of
zero and 14 T ( I//a, B//b). A pronounced resistance anomaly,
a metal-insulator transition (MIT), appeared at ∼135 K in
the zero-field curve, which agrees well with previous results[38,39]. Notably, the transition temperature is slightly in-
creased to ∼150 K as the magnetic field is increased to 14 T.
This MIT transition was ascribed to the consequence of theLifshitz transition [ 40] that was accompanied with the change
of the dominated carrier type, from ntype (low temperatures)
toptype (high temperatures) [ 28]. It was reported recently
that ZrTe
5single crystals grown by chemical vapor transport
are of Te deficiency (ZrTe 5−δ) and that the MIT near 130 Kshould be correlated to the bipolar conduction [ 41]. The
metallic behavior at low temperatures should be the criticalrequirement for a Dirac/Weyl semimetal, which was observedin our samples. The single crystals grown by flux method haveaZ r T e
5stoichiometry with a MIT temperature below 5 K.
More importantly, it is demonstrated that the stoichiometricZrTe
5is a narrow-gap semiconductor instead of a semimetal
[41]. Elemental analysis results show that our sample is
ZrTe 5−δ(δ=0.22, ZrTe 4.78) with Te vacancies, consistent
with that reported [ 41]. Not surprisingly, we also observed a
sign change in Hall resistance near 135 K accompanied by thechange of dominated carrier from n-t optype, as shown in
Fig. 1(d) and its inset [ 36]. The anomalous Hall effect at low
magnetic fields in Fig. 1(d) was observed in previous reports
[38,40,42], which were interpreted within the framework of
Berry curvature generated by Weyl node in ZrTe
5[42].
To explore the physics of the quantum magnetotransport
underlying the experimental data [Fig. 2(a)], we calculated
dρ/dB from the transverse magnetoresistance measured with
magnetic field applied perpendicular to the current plane ( B//b
axis) and in the current plane to reveal the SdH oscillations.The SdH oscillations are clearly seen in the curves of dρ/dB
as a function of B
−1for different temperatures [Fig. 2(b)].
A small shoulder at B=5.3 T, indicated by the arrow, can
be ascribed to the effect of spin splitting [ 30]. To extract the
Berry phase, we plotted the Landau fan diagram in Fig. 2(c).
121108-2GIANT PLANAR HALL EFFECT IN THE DIRAC … PHYSICAL REVIEW B 98, 121108(R) (2018)
FIG. 2. (a) Magnetic-field-dependent magnetoresistance ratio of ZrTe 5−δwith B//b. (b) SdH oscillation amplitude as function of B−1
at different temperatures. The arrow indicates the spin splitting at high fields. (c) Landau fan diagram of ZrTe 5−δ. (d) Longitudinal
magnetoresistance with B//aat different temperatures.
According to the Lifshitz-Onsager quantization rule, BF/B=
n−γ+δ, where n,BF, andγare, respectively, the Landau
index, oscillation frequency, and the Onsager phase factor,γ=1/2−φ
B/2π.δis an additional phase shift whose value
is within the range of ±1/8 and depends on the degree of the
dimensionality of the Fermi surfaces. The Berry phase φBcan
then be obtained from the intercept of the Landau fan curvein Fig. 2(c). To avoid the effect of spin splitting at high fields
on the determination of the Berry phase [ 6], we only linearly
fitted the Landau index with n> 3 as a function of B
−1.F r o m
the intercept obtained from the linear fitting, we find that anontrivial Berry phase φ
B=1.07π±0.02π[36] and that this
value agrees very well with that reported previously.
This nontrivial Berry phase motivated us to further explore
the chiral-anomaly-induced NMR and other exotic transportproperties in our ZrTe
5−δsamples. Figure 2(d) shows the
longitudinal MR at different temperatures measured withB//I//aaxis in the current plane. In addition to the pro-
nounced SdH oscillations, the negative MR was clearly seenat low temperatures ( T<40 K) under strong magnetic fields.
Based on the geometry of our devices, we can exclude thecontribution of current jetting effect to the observed NMR[11,12,36].
The nontrivial Berry phase and NMR support strongly the
existence of a topological semimetal state in ZrTe
5−δand sug-
gest that a giant PHE could be observed in our sample ZrTe 5−δ
as a signature of topological semimetal state [ 15,16,20–23].Figures 3(a)–3(c) show the angular dependence of the pla-
nar Hall resistance Rplanar
xy of ZrTe 5−δmeasured at different
temperatures, where the sample is rotating in acplane. Ap-
parently, the data of ρplanar
xy measured at 2 and 200 K and
under both fields do not follow the sin2 ϕ(ϕ: angle between
BandIinacplane) dependence [Eq. ( 1)] thoroughly, but the
data measured at 150 K indeed can be described by a sin2 ϕ
angular dependence. Another interesting feature is that the
main peak in ρplanar
xy−ϕcurves for B=14 T shifts gradually
from∼310° below 150 K to ∼130° above 150 K. Based on
the data in Fig. 1(d), we understand that the dominant carrier
changes from n-t optype at about 135 K. This asymmetric
angular dependence of ρplanar
xy could be due to the contribution
of a normal Hall effect that arose from the perpendicular
component of the applied magnetic field. This perpendicularcomponent could be easily caused by the small misalignmentof the sample surface ( acplane) with respect to the magnetic
field, as shown in Fig. 3(d).
To separate the normal Hall contribution from ρ
planar
xy ,w e
calculated the average of the measured ρplanar
xy−ϕunder −14
and 14 T, as the green curves show in Figs. 3(a)–3(c). Typ-
ically, we found that ρplanar
xy becomes much more symmetric
[Fig. 3(e)] and shows twofold feature over 360°. However, the
data cannot be directly described by Eq. ( 1) due to a resistivity
shift of 150 μ/Omega1cm away from zero. This resistivity shift
should arise from the longitudinal misalignment during the
121108-3LI, ZHANG, ZHANG, WEN, AND ZHANG PHYSICAL REVIEW B 98, 121108(R) (2018)
FIG. 3. Measured planar Hall resistivity ρplanar
xy of ZrTe 5−δ(t=200 nm) under the opposite magnetic fields (14 and −14 T). (a) T=2K ;
(b)T=150 K; (c) T=200 K. (d) The schematic gives the measurement configuration with the misalignment that includes out-of-plane
magnetic-field component. The yellow circle gives the ideal rotation of magnetic field during the measurement of PHE. The blue circlerepresents the real ratio of magnetic field with misalignment, which will give the contribution of Hall effect in measured ρ
planar
xy. (e) Typical
fitting of averaged ρplanar
xy and the fitting curves ( T=200 K). (f) Typical fitting of angular-dependent ρplanar
xy with three contributions: intrinsic
PHE, Hall effect, and longitudinal resistivity offset.
fabrication of Hall bar device. This small longitudinal resis-
tivity contribution can be subtracted by fitting the averageddata to Eq. ( 3):
ρ
planar
xy=−/Delta1ρchiral
xy sinϕcosϕ+a/Delta1ρchiral
xycos2ϕ+b. (3)
The first term is the intrinsic PHE originating from chiral
anomaly; the second and third terms are, respectively, thein-plane AMR and longitudinal resistance offset caused bythe misalignment of Hall bar. However, compared to the stan-dard longitudinal anisotropic magnetoresistance observed inthese devices [Fig. 4(a)], the angular-dependent longitudinal
misalignment-resistance is quite small, evidenced by the smallresistance shift in Hall effect [Fig. 1(d)]. After taking the
longitudinal Hall bar misalignment-resistance binto account,
theρ
planar
xy can be well fitted by Eq. ( 3), as the red line
shown in Fig. 3(e). To exclude the artifacts in the analysis
of the raw data, we also fitted measured ρplanar
xy directly with
three contributions, twofold PHE, onefold Hall effect, andresistance offset due to the misalignment of Hall bar, as shown
in Fig. 3(f). We found that the amplitude of the intrinsic ρplanar
xy
obtained from above two fitting strategies are identical, 67.00
±0.60μ/Omega1cm in Fig. 3(e) and 66.60 ±0.60μ/Omega1cm in
Fig. 3(f), which suggests that both methods result in nearly
the same intrinsic ρplanar
xy .
121108-4GIANT PLANAR HALL EFFECT IN THE DIRAC … PHYSICAL REVIEW B 98, 121108(R) (2018)
FIG. 4. (a) Measured in-plane giant anisotropic magnetoresistance of ZrTe 5−δat different temperatures ( B=14 T). (b) Intrinsic planar
Hall resistivity ρchiral
xy as a function of rotation angle at different magnetic fields ( T=150 K). (c) The magnetic-field dependence of chiral-
anomaly-induced planar Hall resistivity ρchiral
xy. It can be divided into weak and strong magnetic-field regions. The red line gives the fitting by
Eq. ( 5) for weak tendency of saturation at strong magnetic fields. The error bar comes from the fitting by Eq. ( 3). (d) The intrinsic planar Hall
resistivity ρchiral
xy as a function of temperature ( B=14 T) with various ZrTe 5−δnanoplate thicknesses. (e) The ionic gating effect on planar Hall
effect ( t=58 nm). The maximum temperature in ρchiral
xy−T(B=14 T) curves corresponds to the MIT temperatures as the gating voltage
increases from 0 to 2.5 V .
To further confirm our argument, we measured the
longitudinal AMR [AMR =(Rϕ−R⊥)/R⊥] at different
temperatures as shown in Fig. 4(a). For comparison, the
angular-dependent intrinsic planar Hall resistivity ρchiral
xy mea-
sured under different magnetic fields is shown in Fig. 4(b).
The exact 45° phase difference between the angular-
dependent twofold AMR and intrinsic PHE agrees well with
the theoretical predictions and Eqs. ( 1) and ( 2)[15,16,36].
Generally, the anisotropic magnetoresistance in conventionalmagnetic materials is quite small and is caused by the spin-orbit coupling [ 17]. The giant AMR in this study reaches as
high as −43% at 2 K and 14 T, which is believed to be closely
related to the giant PHE induced by chiral anomaly [ 21,22].
It should also be noted that ρ
chiral
xy increases monotonically to
254.0±2.0μ/Omega1cm as the magnetic field is increased to 14 T.This giant PHE observed in our ZrTe 5−δdevice is about four
orders of magnitude higher than that observed in conventionalferromagnetic metals [ 19].
To gain a deeper insight into the physical mechanism
underlying the giant PHE, we plotted typical ρ
chiral
xy−B
curve for t=67-nm-thick device in Fig. 4(c) [36]. Appar-
ently, ρchiral
xy does not follow a simple linear or quadratic
dependence on the magnetic field Bas the MR and Hall
effect observed in conventional ferromagnetic materials[17,18]. Instead, it shows roughly two different field de-
pendences: for low fields ( B<3T ) , ρ
chiral
xy depends on B2;
with increasing the field further, B> 6T ,ρchiral
xy shows a
weak tendency of saturation. Actually, it has been theo-retically predicted that the chiral-anomaly-induced planarHall effect can be divided into different magnetic field
121108-5LI, ZHANG, ZHANG, WEN, AND ZHANG PHYSICAL REVIEW B 98, 121108(R) (2018)
regions [ 15]:
La/greatermuchLc,ρchiral
xy∝/parenleftbiggLc
La/parenrightbigg2
∝B2; weak magnetic −field region (4)
La/lessmuchLc,ρchiral
xy∝1
σ/parenleftbigg
1−2La
Lx/parenrightbigg
forLa<Lx<L2
c
La; strong magnetic field region (5)
La/lessmuchLc,ρchiral
xy∝1
σ/parenleftbigg
1−L2
a
L2c/parenrightbigg
forLx>L2
c
La; strong magnetic field region (6)
where σ,Lx,La, and Lcare, respectively, conductivity,
sample length, magnetic length ( La∝B−1), and chiral charge
diffusion length [ 15]. The weak tendency of saturation at
strong magnetic field will follow a −B−1[Eq. ( 5)] or−B−2
[Eq. ( 6)] relation, depending on the sample length Lx[15].
The red line in Fig. 4(c) gives that the fitting by Eq. ( 5)s h o w s
the typical weak tendency of saturation at high magneticfields. The consistency between our observation and theoryis the strong evidence of the existence of topological state inour samples.
To understand the behavior of ρ
chiral
xy in more detail, we
investigated the dependence of the giant planar Hall effecton the thickness of ZrTe
5−δnanoplates [ 36]. Figure 4(d)
shows the amplitude of the planar Hall resistivity with variousZrTe
5nanoplate thicknesses as a function of the temperature
under a magnetic field of 14 T, in which the values ofplanar Hall resistivity are taken from the angular dependenceofρ
chiral
xy measured at different temperatures. Interestingly, a
maximum intrinsic planar Hall resistivity ρchiral
xy appears in
most of samples except for two ultrathin ZrTe 5−δdevices
(t=25 and 30 nm). Particularly important, very strong and
well-defined peaks occur in the temperature range of 110 ∼
150 K in the vicinity of the corresponding MIT temperatures.This peak behavior of ρ
chiral
xy (T) is different from the mono-
tonic decrease of ρchiral
xy with increasing temperature observed
in some Dirac/Weyl semimetals [ 20–23]. An ARPES study
showed that the Fermi level in ZrTe 5−δwill sweep from
electronlike band to holelike band across MIT temperature[28], although no definite evidence can be found to support
the existence of the Dirac cone in the ARPES spectra [ 28].
However, our observation of giant PHE and its interestingbehavior in Fig. 4(d) will be a piece of strong evidence to
support the existence of Dirac fermions in ZrTe
5−δ, since
the giant planar Hall effect is caused by the chiral anomalyand Berry phase of carriers near the Dirac point [ 15,16].
When the Fermi level in ZrTe
5−δis sweeping across the Dirac
point near MIT temperature, the chiral-anomaly-induced in-trinsic ρ
chiral
xy will be enhanced [Fig. 4(d)]. Nevertheless, in
other well-known Dirac/Weyl semimetals, such as Cd 3As2,
GdPtBi, and WTe 2[20–23], their Dirac point will deviate
from Fermi level further as the temperature increases [ 43,44].Therefore, a monotonic decrease of ρchiral
xy with increasing
temperature is observed in these Dirac/Weyl semimetals. An-other striking feature in Fig. 4(d) is the monotonic decrease of
planar Hall resistivity as the device thickness decreases. Thisis a typical characteristic and further indication that ZrTe
5−δ
is a 3D Dirac/Weyl semimetal.
To further consolidate our argument, we investigated the
effect of ionic gating on planar Hall effect in ZrTe 5−δ.G i v e n
the relatively high carrier density in ZrTe 5−δ,i ti sa l m o s t
impossible to tune the carrier density in thick devices (e.g.,t=174 nm) [ 36]. We indeed observed the shift of MIT tem-
perature from 120 K ( V
g=0 V) to 340 K ( Vg=3.0 V) under
the ionic gating in a thin device ( t=30 nm), in agreement
with previous gating experiment [ 38]. Nevertheless, limited
by the relatively weak planar Hall effect in thin ZrTe 5−δ,i t
is hard to investigate the gating effect on planar Hall effect[36]. Therefore, we carefully examined the gating effect on a
device with moderate thickness ( t=58 nm), as displayed in
Fig. 4(e). By applying V
g=1.5 V, the transition temperature
inρ−Tcurve shifted from 118 K down to 106 K. When the
gating voltage was increased to 2.5 V , the transition tempera-ture was increased to 146 K. This nonmonotonic dependenceagrees well with previous report [ 38]. More important, the
maximum in the temperature-dependent planar Hall resistivityoccurs at the same temperature where the MIT transitionappears in the corresponding ρ−Tcurve, which strongly
indicates that the planar Hall effect in ZrTe
5−δis a direct
consequence of chiral anomaly related to the Dirac/Weylpoint.
To conclude, we have observed a giant planar Hall effect in
ZrTe
5−δ, accompanied with a nontrivial Berry phase, negative
longitudinal magnetoresistance, and a giant anisotropic mag-netoresistance, which demonstrated fully that ZrTe
5−δwith
the MIT temperature near 130 K is indeed a Dirac semimetal.
We thank Prof. Feng Liu at University of Utah for very
helpful discussions. The research reported in this publicationwas supported by King Abdullah University of Science andTechnology (KAUST). P.L. acknowledges the financial sup-port of Grant No. CRF-2015-SENSORS-2709 (KAUST).
P.L. and C.Z. contributed equally to this work.
[1] Z. Liu et al. ,Science 343,864(2014 ).
[2] S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. Pesin, and
A. Vishwanath, P h y s .R e v .X 4,031035 (2014 ).[ 3 ] B .Q .L v et al. ,P h y s .R e v .X 5,031013 (2015 ).
[4] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X.
Dai, and B. A. Bernevig, Nature (London) 527,495(2015 ).
121108-6GIANT PLANAR HALL EFFECT IN THE DIRAC … PHYSICAL REVIEW B 98, 121108(R) (2018)
[5] J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M.
Hirschberger, W. Wang, R. Cava, and N. Ong, Science 350,413
(2015 ).
[6] P. J. Moll, N. L. Nair, T. Helm, A. C. Potter, I. Kimchi, A.
Vishwanath, and J. G. Analytis, Nature (London) 535,266
(2016 ).
[7] A. C. Potter, I. Kimchi, and A. Vishwanath, Nat. Commun. 5,
5161 (2014 ).
[8] P. Li et al. ,Nat. Commun. 8,2150 (2017 ).
[9] H. B. Nielsen and M. Ninomiya, Phys. Lett. B 130,389(1983 ).
[10] X. Huang et al. ,P h y s .R e v .X 5,031023 (2015 ).
[11] R. Dos Reis, M. Ajeesh, N. Kumar, F. Arnold, C. Shekhar, M.
Naumann, M. Schmidt, M. Nicklas, and E. Hassinger, New J.
Phys. 18,085006 (2016 ).
[12] F. Arnold et al. ,Nat. Commun. 7,11615 (2016 ).
[13] P. Goswami, J. H. Pixley, and S. Das Sarma, Phys. Rev. B 92,
075205 (2015 ).
[14] T. Schumann, M. Goyal, D. A. Kealhofer, and S. Stemmer,
P h y s .R e v .B 95,241113 (2017 ).
[15] A. A. Burkov, Phys. Rev. B 96,041110 (2017 ).
[16] S. Nandy, G. Sharma, A. Taraphder, and S. Tewari, Phys. Rev.
Lett. 119,176804 (2017 ).
[17] T. McGuire and R. Potter, IEEE Trans. Magn. 11,1018 (1975 ).
[18] H. X. Tang, R. K. Kawakami, D. D. Awschalom, and M. L.
Roukes, P h y s .R e v .L e t t . 90,107201 (2003 ).
[19] S. Kokado, M. Tsunoda, K. Harigaya, and A. Sakuma, J. Phys.
Soc. Jpn. 81,024705 (2012 ).
[20] N. Kumar, C. Felser, and C. Shekhar, Phys. Rev. B 98,041103
(2018 ).
[21] H. Li, H. Wang, H. He, J. Wang, and S.-Q. Shen, Phys. Rev. B
97,201110 (2018 ).
[22] M. Wu et al. ,arXiv:1710.01855 .
[23] Y . Wang, J. Gong, D. Liang, M. Ge, J. Wang, W. Zhu, and
C. Zhang, arXiv:1801.05929 .[24] S. Liang, J. Lin, S. Kushwaha, R. Cava, and N. Ong, Phys. Rev.
X8,031002 (2018 ).
[25] H. Weng, X. Dai, and Z. Fang, P h y s .R e v .X 4,011002 (2014 ).
[26] X.-B. Li et al. ,Phys. Rev. Lett. 116,176803 (2016 ).
[27] R. Wu et al. ,Phys. Rev. X 6,021017 (2016 ).
[28] Y . Zhang et al. ,Nat. Commun. 8,15512 (2017 ).
[29] Q. Li et al. ,Nat. Phys. 12,550(2016 ).
[30] Y . Liu et al. ,Nat. Commun. 7,12516 (2016 ).
[31] X. Yuan et al. ,NPG Asia Mater. 8,e325 (2016 ).
[32] R. Y . Chen, Z. G. Chen, X.-Y . Song, J. A. Schneeloch, G. D. Gu,
F. Wang, and N. L. Wang, P h y s .R e v .L e t t . 115,176404 (2015 ).
[33] G. Manzoni et al. ,Phys. Rev. Lett. 117,237601 (2016 ).
[34] Z. Fan, Q.-F. Liang, Y . Chen, S.-H. Yao, and J. Zhou, Sci. Rep.
7,45667 (2017 ).
[35] J. L. Zhang et al. ,P h y s .R e v .L e t t . 118,206601 (2017 ).
[36] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.98.121108 for measurement configuration,
Raman spectra, thickness determination by atomic force mi-croscope, quantum oscillation for Berry phase, exclusion ofcurrent jetting, magnetic-field dependence of planar Hall resis-tivity at different temperatures, temperature dependence of re-sistance anisotropy, ionic gating effect on PHE, magnetic-field-dependent planar Hall resistivity with different thicknesses, andmore AMR and PHE data.
[37] Y . J. Yu et al. ,Nat. Nanotechnol. 10,270(2015 ).
[38] J. Niu et al. ,Phys. Rev. B 95,035420 (2017 ).
[39] G. Zheng et al. ,P h y s .R e v .B 93,
115414 (2016 ).
[40] H. Chi, C. Zhang, G. Gu, D. E. Kharzeev, X. Dai, and Q. Li,
New J. Phys. 19,015005 (2017 ).
[41] P. Shahi et al. ,P h y s .R e v .X 8,021055 (2018 ).
[42] T. Liang et al. ,Nat. Phys. 14,451(2018 ).
[43] S. Borisenko, Q. Gibson, D. Evtushinsky, V . Zabolotnyy, B.
Büchner, and R. J. Cava, P h y s .R e v .L e t t . 113,027603 (2014 ).
[44] C. Wang et al. ,P h y s .R e v .B 94,241119 (2016 ).
121108-7 |
PhysRevB.99.115307.pdf | PHYSICAL REVIEW B 99, 115307 (2019)
Mechanism of ultrafast spin-polarization switching in nanostructures
V . N. Mantsevich,1I. V . Rozhansky,2,3N. S. Maslova,1P. I. Arseyev,4,5N. S. Averkiev,2,3and E. Lähderanta3
1Lomonosov Moscow State University, Quantum Technology Center, Physics Department, 119991 Moscow, Russia
2Ioffe Institute, St. Petersburg, 194021, Russia
3Lappeenranta University of Technology, FI-53851 Lappeenranta, Finland
4P .N. Lebedev Physical Institute RAS, 119991 Moscow, Russia
5Russia National Research University Higher School of Economics, 119991 Moscow, Russia
(Received 16 November 2018; revised manuscript received 21 February 2019; published 11 March 2019)
We consider time-dependent processes in the optically excited hybrid system formed by a quantum well
(QW) coupled to a remote spin-split correlated bound state. The spin-dependent tunneling from the QW tothe bound state results in the nonequilibrium electron spin polarization in the QW. The Coulomb correlationsat the bound state enhance the spin polarization in the QW. We propose a mechanism for ultrafast switchingof the spin polarization in the QW by tuning the laser pulse frequency between the bound state spin sublevels.Mn-doped core /multishell nanoplatelets and hybrid bound state-semiconductor heterostructures are suggested
as promising candidates to prove the predicted effect experimentally. The obtained results open a possibility forspin polarization control in nanoscale systems.
DOI: 10.1103/PhysRevB.99.115307
I. INTRODUCTION
As power consumption in modern electronics becomes one
of the central problems, utilization of electron spin is verypromising for spintronic devices and information processing[1–3]. It requires precise manipulation /switching of spin po-
larization [ 4,5]. Generation and detection of spin-polarized
currents is the key problem in the spintronic devices [ 6–13].
There is a growing interest in semiconductor spin lasers, inwhich the spin polarized carriers are injected by circularlypolarized light or by electrical injection [ 14–16]. The spin
lasers demonstrate threshold reduction [ 17,18] and gain in
polarization degree for spin to optical polarization conversion[19,20]. Perhaps the greatest potential of spin lasers is ultrafast
spin and polarization dynamics [ 15]. Spin-polarized light-
emitting diodes also proved promising for the spin injection,a pure circular polarization of the electroluminescence atroom temperature with no external magnetic field has beendemonstrated [ 21]. A great progress has been made in sta-
tionary spin transport in magnetic [ 22–25] and nonmagnetic
tunnel junctions in the presence of spin-orbit and exchange
interactions [ 26,27] and in quantum dot (QD) systems [ 28–30]
in magnetic field.
For charge and spin control in small devices time-
dependent effects and transient processes are essential[9,31–36]. Thus, time evolution of spin and charge config-
urations in correlated low-dimensional systems is of greatinterest both from fundamental and technological points ofview. Time-dependent characteristics also provide an impor-tant information about the properties of nanoscale systems.Nowadays, there are various experimental methods (polariza-tion photoluminescence (PL), magneto-optical Faraday /Kerr
effect, etc.) for time-resolved detection of the spin polarization[1,37].One of the most perspective ideas of controlling spin
polarization is based on the carriers spatial separation. Hybridbound state-semiconductor heterostructures formed by a QWand bound states (such as magnetic impurity ions) separatedby a thin spacer seems to be good candidates to realize thisidea [ 38–41]. Usually, spin-polarized carriers are injected
from the bound state (ferromagnetic δlayer) into the semi-
conductor QW; the magnetic properties and spin polarizationof the carriers can be controlled via the spacer thickness,shape, and the δ-layer parameters [ 42–45]. This method al-
lows us to obtain spin polarization, however, with no controlon its time evolution. Another mechanism of dynamic spinpolarization of electrons due to the spin-dependent tunnelingfrom a semiconductor QW into the bound state spin-split byexchange interaction was proposed and realized experimen-tally [ 46,47]. Linearly polarized laser pulse creates nonpo-
larized electrons in the QW. Spin-dependent tunneling intothe bound state results in accumulation of the electron spinpolarization in the QW detected by the circular polarization ofthe PL.
In this paper we analyze the dynamic spin polariza-
tion theoretically. We extend the formalism considered inRefs. [ 48–54]. It was first applied to the correlated QD
coupled to a reservoir to describe the nonstationary spinpolarized currents due to the time evolution of a magneticmoment in the QD under applied bias and external magneticfield. In Refs. [ 48
,49] the spin generation due to the spin-
dependent tunneling in the hybrid QW-bound state system
was explained. In the present paper we generalize the non-stationary approach to the case when initial nonequilibriumcarriers distribution in the QW is tuned by a laser pulse. Weshow that the spin polarization in QW and circular polarizedPL change their signs when the pulse frequency is tuned tomatch one of the bound state spin sublevels. Our results open
2469-9950/2019/99(11)/115307(6) 115307-1 ©2019 American Physical SocietyV . N. MANTSEVICH et al. PHYSICAL REVIEW B 99, 115307 (2019)
the possibility to control the sign of electrons spin polarization
in the QW.
II. THEORETICAL MODEL
We consider nonstationary processes in the system formed
by the QW coupled to a spin-split correlated bound statewith the energy ε
1separated from the QW by a tunnel
barrier [see Fig. 1(a)]. At the initial time QW is optically
excited with linearly polarized light generating unpolarizednonequilibrium electrons with energies ε
k, where kis the in-
plane vector. The barrier is characterized by the tunneling rate/Gamma1
wi. The electron-hole recombination processes in the QW
are described by the relaxation rate γw; a separate relaxation
channel at the bound state allows electrons to disappear withthe rate γ
i. The suggested model gives the possibility to
analyze dynamic spin injection processes caused by the spin-dependent tunneling between the QW and bound state consid-ering exactly high order correlation functions for the boundstate electrons. The Hamiltonian of the system consists of theQW part, the bound state part, which includes the Hubbardterm corresponding to the on-site Coulomb repulsion, and thetunneling part describing electrons transfer between the QWand the bound state:
ˆH=/summationdisplay
σ,kεkˆc+
kσˆckσ+/summationdisplay
σ/parenleftbig
ε1ˆnσ
1+Uˆnσ
1ˆn−σ
1/parenrightbig
+/summationdisplay
kσtk(ˆc+
kσˆc1σ+ˆc+
1σˆckσ). (1)
Here index klabels continuous spectrum states in the QW;
tkis the tunneling amplitude between a QW state kand the
bound state. The bound state energy level ε1can be split by
an exchange interaction or a weak external magnetic fieldinto two spin sublevels: ε
σ=ε1+σ/Delta1, where σ=±1/2i s
the electron spin projection and /Delta1is the splitting energy.
Operators ˆ c+
kσ(ˆckσ) are the creation (annihilation) operators
for the QW states. ˆ nσ
1=ˆc+
1σˆc1σis the bound state electron
occupation number, where operator ˆ c1σdestroys the electron
with spin projection σ.Uis the on-site Coulomb repulsion for
the double occupation of the bound state.
We neglect the tunneling of holes between the QW and
the bound state as it is usually less efficient than for elec-trons due to the difference in the effective mass. The holes
(a) (b)
FIG. 1. (a) Scheme of the model structure under external laser
excitation with frequency ω. (b) Schematic energy diagram showing
the bound state split energy levels, initial and equilibrium electrons
distribution in the QW.contribution to the resulting nonequilibrium spin polarization
is negligible as their spin relaxation is much faster than for theelectrons [ 46].
III. NONSTATIONARY ELECTRONIC TRANSPORT
FORMALISM
Let us further consider ¯ h=1 and e=1 elsewhere and
assume the low temperature regime. The equations of motionfor the electron operators products ˆ n
σ
1,ˆnσ
1k=ˆc+
1σˆckσ, and
ˆnσ
k/primek=ˆc+
k/primeσˆckσcan be written as:
i∂ˆnσ
1
∂t=−/summationdisplay
k,σtk·/parenleftbig
ˆnσ
k1−ˆnσ
1k/parenrightbig
, (2)
i∂ˆnσ
1k
∂t=−/parenleftbig
εσ
1−εk/parenrightbig
·ˆnσ
1k−U·ˆn−σ
1ˆnσ
1k+tk·/parenleftbig
ˆnσ
1−ˆnσ
k/parenrightbig
−/summationdisplay
k/prime/negationslash=ktk/prime·ˆnσ
k/primek, (3)
i∂ˆnσ
k/primek
∂t=− (εk/prime−εk)·ˆnσ
k/primek−tk/prime·ˆnσ
1k+tk·ˆnσ
k/prime1.(4)
Substituting the solution of Eq. ( 4) into Eq. ( 3) reveals the
relaxation term i/Gamma1wiˆnσ
1kdue to the tunneling with the rate
/Gamma1wi=πν0t2
kdetermined by the unperturbed density of states
ν0and the tunneling amplitude tk[48]. Further we assume
ν0constant for 2D electrons and tk. independent of k.T h e
equations for the bound state occupation numbers n±σ
1are
further obtained by averaging operator equations ( 2)–(4) and
by decoupling QW electrons occupation numbers from thebound state occupation numbers [ 48]. Within the decoupling
procedure the operators ˆ n
σ
kare replaced with the distribution
function fσ
k. Assuming that equilibrium state corresponds
to the empty bound state and equilibrium Fermi distribu-tion of electrons in the QW the following equations can beobtained [ 48]:
∂n
σ
1
∂t=−2·/Gamma1wi·Iσ
k−γi·nσ
1,
∂fσ
k
∂t=2·/Gamma1wi·Jσ
k−γw·/parenleftbig
fσ
k−f0
k/parenrightbig
, (5)
where
Iσ
k=nσ
1−/parenleftbig
1−n−σ
1/parenrightbig
·/Xi1(εσ)−n−σ
1·/Xi1(εσ+U)
Jσ
k=1
ν0π·/bracketleftbigg/parenleftbig
1−n−σ
1/parenrightbig/parenleftbig
nσ
1−fσ
k/parenrightbig
·ϒ
(εσ−εk)2+ϒ2
+n−σ
1/parenleftbig
nσ
1−fσ
k/parenrightbig
ϒ
(εσ+U−εk)2+ϒ2/bracketrightBigg
(6)
and QW occupation function /Xi1(x) with x=εσ,εσ+Ureads:
/Xi1(x)=/integraldisplay
dεk·fσ
k(εk)·1
πϒ
(x−εk)2+ϒ2. (7)
In Eqs. ( 5) we have introduced relaxation rates γwandγide-
scribing relaxation processes in the QW and at the bound state,respectively. The relaxation rate ϒ=/Gamma1
wi+γiis introduced to
describe correctly the bound state structure; it accounts for itsbroadening due to both tunneling and relaxation [ 48].
In the absence of the tunnel coupling the electrons in the
QW are described by Fermi distribution f
0
kwith a chemical
115307-2MECHANISM OF ULTRAFAST SPIN-POLARIZATION … PHYSICAL REVIEW B 99, 115307 (2019)
potential μ0and a temperature T0. Let us consider an optical
excitation by a short laser pulse with Gaussian spectral distri-bution. By tuning the laser wavelength the peak of the excitednonequilibrium electron distribution in the QW could be putin a resonance with one of the bound state spin sublevels asshown in Fig. 1(b). For the bound state the initial conditions
are:n
σ
1=n−σ
1=0. The tunneling of the QW electrons into
the bound state leads to the renormalization of the stationarydistribution function in the QW. Solving Eqs. ( 5)–(7)i nt h e
stationary case (∂nσ
1
∂t=∂ˆnσ
k
∂t=0) one can get stationary bound
state occupation numbers:
nσst
1=/Phi1(εσ)−/Delta1/Phi1σ·/Phi1(ε−σ)
1−/Delta1/Phi1σ·/Delta1/Phi1−σ, (8)
where
/Phi1(εσ)=2/Gamma1wi
2/Gamma1wi+γi·/Xi1(εσ),
/Delta1/Phi1σ=/Phi1(εσ)−/Phi1(εσ+U). (9)
Functions /Phi1(εσ) are determined with stationary distribution
functions fst
k, which can be found from Eqs. ( 5)–(7). Solution
of Eqs. ( 5)–(7) in the stationary case reveals the presence
of residual spin polarization for electrons in the QW for/Gamma1
wi/γw/lessmuch1. The spin polarization given by ρs=N↑−N↓,
where Nσ=/integraltext
fσ(εk)dεk, results in the circular polarization
of the PL from the QW:
PPL∼N↑−N↓
N↑+N↓. (10)
The polarization degree PPLis proportional to the spin polar-
ization of the electrons in the QW. The coefficient dependson the radiative recombination details, in particular, on theoccupation of heavy and light hole subbands [ 2]. In our model
spin polarization of the electrons in the QW appears due to thetunnel leakage. So, for the considered effect we neglect theinfluence of the valence band structure on the QW electronsspin polarization. This assumption is valid for not too wideQWs with separated heavy and light holes subbands.
IV . RESULTS AND DISCUSSION
Kinetics of the photoexcited electrons in the QW is char-
acterized by the recombination processes with a typical timeγ
−1
wand tunneling between the QW and the bound state with a
time/Gamma1−1
wi. Figure 2shows the kinetics of the spin polarization
in the QW. At a small time the spin-dependent tunneling leadsto a linear increase of electron spin polarization [ 48,49], it fur-
ther approaches its stationary value given by the equilibriumcarries distribution in the QW.
The Coulomb correlations strongly influence the carriers
dynamics when the carrier lifetime at the bound state exceedsthe tunneling time, which is in its turn smaller than therelaxation time in the QW:
γ
−1
i/greatermuch/Gamma1−1
wi;/Gamma1−1
wi/lessorequalslantγ−1
w. (11)
The role of Coulomb correlations and bound state energy
in the nonequilibrium spin polarization of the photoexcitedelectrons is shown in Fig. 2. We assume the initial Fermi
distribution of the photoexcited electrons in the QW witha chemical potential μ
∗. In the following calculations weFIG. 2. Time evolution of the spin polarization in the QW. Solid
black ( U=0) and red ( U=1) lines: ε↑=0.35,ε↓=0.25,μ∗=0.3.
Dashed blue ( U=0) and green ( U=1) lines: ε↑=0.2,ε↓=
0.1,μ∗=0.15. Parameters are μ0=0,W=2,γi=0.005,/Gamma1wi=
6γi,γw=5γi.
neglect thermal broadening of the distribution and consider
the energy relaxation via the hole recombination processes.The thermal broadening and energy relaxation via electron-phonon interaction affects the tunneling rate. However, it isless important than the electron-hole recombination, as thelatter directly changes the number of carriers in the QW.The calculations were performed following the Eqs. ( 5)–(7).
Two main effects can be clearly seen in Fig. 2. Firstly, the
presence of Coulomb correlations increases spin polarization.Secondly, spin polarization of photoexcited electrons (and,thus PL circular polarization) is sensitive to the relative posi-tion of the equilibrium distribution chemical potential μ
0=0
and the bound state spin split levels. It substantially increaseswhen the bound state energy levels are located closer to theequilibrium chemical potential μ
0.
Our theory predicts an effect, the ultrafast switching of spin
polarization of electrons in a QW with a laser pulse. By tuningthe excitation laser frequency the nonequilibrium distributionmaximum of the excited electrons can be shifted along theenergy scale matching one of the spin-split bound state energylevels [see Fig. 1(b)].
Figure 3shows calculation results for the Gaussian energy
distribution of photoexcited electrons in the QW. Conse-quently, the tunneling of the electrons with the correspondingspin projection into the bound state becomes more effectiveand causes spin polarization of the resident electrons in theQW. Changing the maximum of the electron energy distribu-tion between the two bound state spin sublevels results in theswitching of the spin polarization and, subsequently the rever-sal of the circular PL polarization sign [Fig. 3(a)]. This finding
opens a possibility to generate spin polarized train pulseswith opposite polarization as illustrated in Fig. 3(b). Here the
nonpolarized electrons in the QW are generated by laser trainpulses with the peak of the laser spectrum alternating betweenthe two spin sublevels of the bound state. Consequently, spinpolarization of the electrons in the QW changes its sign fromone pulse to another. We do not consider the process ofthe nonequilibrium electrons generation assuming they arecreated instantly by the laser pulse. The increasing part of
115307-3V . N. MANTSEVICH et al. PHYSICAL REVIEW B 99, 115307 (2019)
6 9 12 15(a)
(b)
FIG. 3. (a) Time evolution of the spin polarization in the
QW. Dashed lines: ω=ε↑=0.35. Solid lines: ω=ε↓=0.25.
(b) Switching of the spin polarization sign by tuning the laserfrequency between ω=ε
↑=0.35 and ω=ε↓=0.25. Parameters
areμ0=0,U=1,W=2,γi=0.005,/Gamma1wi=6γi.
the spin polarization response pulse indicated by the solid
lines in Fig. 3(b) is determined by the spin inertia time [ 55],
which is in our system the inverse tunneling rate /Gamma1−1
wi.T h ef u l l
decay of the spin polarization is due to spin relaxation; thistime is assumed to be the longest on the problem timescale(∼10 ns [ 46]). The spin relaxation is not accounted for in our
theory, so the decay of the spin polarization pulse is shownschematically by the dashed line representing an exponentwith the characteristic time t≈γ
i. So, Fig. 3(b) describes the
case of the the interval between the laser pulses exceedingspin relaxation time. The red line illustrates the case of avery fast recombination in the QW exceeding the tunnelingrate. The amplitude of the polarization appears to be smallas the carriers in the QW relax to equilibrium faster than thepolarization develops.
The requirements to observe the predicted spin polar-
ization switching are (i) possibility to create nonequilib-rium distribution of photoexcited electrons and (ii) inversetunneling rate should not exceed the time of the electrondistribution thermalization. Promising candidates for exper-imental observation are, for example, Mn-doped colloidalcore/multishell nanoplatelets [ 56,57] or semiconductor het-
erostructures formed by several QWs separated by the barrierswith different width and height [ 46,47]. In the latter case one
of the QWs is doped with Mn. In both cases the pump-probetechnique (with both pump and probe laser pulses linearly
polarized) should be used to reveal the PL circular polariza-tion. Linearly polarized pump pulse creates nonequilibriumdistribution of the electrons in the core of the nanoplatelet(for the first configuration) or in the QW without Mn dopantatoms (for the second configuration). The spin-dependenttunneling of the electrons to the Mn impurity states leads tothe spin polarization of the electrons in the core or QW andcan be detected by the emergent polarization of the probepulse.
To observe the polarization sign switching the inequalities
Eqs. ( 11) should be fulfilled. The radiative recombination rate
in the QW is of the order of 1 ns. The bound state relaxationtimeγ
−1
iis determined by recombination processes; its value
can vary from 1 ns down to 10 ps [ 49,58]. The tunneling rate
extracted from experiments on dynamic spin injection in semi-conductor heterostructures [ 46,47,59,60]/Gamma1
wi∼1–100 ps−1
depending on the barrier thickness [ 61], which can be also
tuned by an external bias. Typical relaxation rate of thenonequilibrium electron distribution excited by the laser pulseis about a few picoseconds, for example in CdSe thin films itwas reported to be 1–5 ps [ 62]. So, inequalities Eqs. ( 11) can
be realized experimentally.
Another important parameter is the magnitude of the bound
state spin splitting. As the typical width of the nonequilibriumelectron distribution excited by the 1 ps laser pulse is about0.5 meV , for the effective spin-dependent tunneling the spinsplitting should exceed this value. The appropriate values ofthe spin splitting are typical for the semiconductor superstruc-tures with magnetic impurities due to exchange interaction.For example, in (Ga,Mn)As heterostructures the magnitude ofthe exchange induced splitting is about 2.5 meV [ 47,58]. It
can be also tuned by an external magnetic field.
V . CONCLUSION
We considered time-dependent processes in the system
formed by a QW coupled to a remote spin-split correlatedbound state. It was shown that Coulomb interaction at thebound state leads to the significant increase of the spinpolarization in the QW. The residual spin depends on theequilibrium Fermi level in the QW. We propose a mecha-nism for ultrafast switching of the spin polarization in theQW. As energy distribution of photoexcited electrons dependson the excitation laser pulse spectrum, the spin-dependenttunneling efficiency can be controlled by matching of thelaser pulse frequency to the spin split bound state energy.The time-dependent electron spin polarization in the QW and,consequently, the circular polarized PL could reverse the signdepending on the laser pulse frequency. We suggested possi-ble hybrid low-dimensional structures as candidates to probethe predicted effect and discussed the conditions necessaryfor the experimental observation. We believe that these resultsopen a possibility for spin polarization control in nanoscalesystems.
ACKNOWLEDGMENTS
We thank E. A. Zhukov for fruitful discussions and ac-
knowledge the financial support from the Russian Foundation
115307-4MECHANISM OF ULTRAFAST SPIN-POLARIZATION … PHYSICAL REVIEW B 99, 115307 (2019)
for Basic Research (RFBR) Grants No. 18-02-00668, 18-02-
00052, and from Russian Science Foundation project 17-12-01182 (theoretical model). The work was also supported by
the Academy of Finland Grant No. 318500.
[1] D. Awschalom, D. Loss, and N. Samarth, Semiconductor
Spintronics and Quantum Computation, Nanoscience andTechnology (Springer, Berlin, 2002).
[2] I. Žuti ´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76,323
(2004 ).
[3] D. A. B. Miller, Proc. IEEE 97,1166 (2009 ).
[4] H. Dery, P. Dalal, L. Cywinski, and L. J. Sham, Nature
(London) 447,573(2007 ).
[5] H. Dery, Y . Song, P. Li, and I. Žuti ´c,Appl. Phys. Lett. 99,
082502 (2011 ).
[6] G. A. Prinz, Science 282,1660 (1998 ).
[7] R. Crook, J. Prance, K. J. Thomas, S. J. Chorley, I. Farrer,
D. A. Ritchie, M. Pepper, and C. G. Smith, Science 312,1359
(2006 ).
[8] P. Chuang, S.-C. Ho, L. W. Smith, F. Sfigakis, M. Pepper, C.-H.
Chen, J.-C. Fan, J. P. Griffiths, I. Farrer, H. E. Beere, G. A. C.Jones, D. A. Ritchie, and T.-M. Chen, Nat. Nanotechnol. 10,35
(2014 ).
[9] V . N. Mantsevich, N. S. Maslova, and P. I. Arseyev, JETP Lett.
97,352(2013 ).
[10] D. V . Averin and J. P. Pekola, Phys. Rev. Lett. 101,066801
(2008 ).
[11] A. Bayat, C. E. Creffield, J. H. Jefferson, M. Pepper, and
S. Bose, Semicond. Sci. Technol. 30,105025 (2015 ).
[12] M. D. Shulman, O. E. Dial, S. P. Harvey, H. Bluhm,
V . Umansky, and A. Yacoby, Science 336,202(2012 ).
[13] E. P. Blair and C. S. Lent, J. Appl. Phys. 113,124302 (2013 ).
[14] J. Lee, R. Oszwałdowski, C. Gøthgen, and I. Žuti ´c,Phys. Rev.
B85,045314 (2012 ).
[15] P. E. Faria Junior, G. Xu, J. Lee, N. C. Gerhardt, G. M. Sipahi,
and I. Žuti ´c,Phys. Rev. B 92,075311 (2015 ).
[16] M. Lindemann, T. Pusch, R. Michalzik, N. C. Gerhardt, and
M. R. Hofmann, Appl. Phys. Lett. 108,042404 (2016 ).
[17] J. Rudolph, D. Hägele, H. M. Gibbs, G. Khitrova, and
M. Oestreich, Appl. Phys. Lett. 82,4516 (2003 ).
[18] C. Gothgen, R. Oszwaldowski, A. Petrou, and I. Žuti ´c,
Appl. Phys. Lett. 93,042513 (2008 ).
[19] J. Lee, W. Falls, R. Oszwaldowski, and I. Žuti ´c,Appl. Phys.
Lett.97,041116 (2010 ).
[20] D. Saha, D. Basu, and P. Bhattacharya, P h y s .R e v .B 82,205309
(2010 ).
[21] N. Nishizawa, K. Nishibayashi, and H. Munekata, Proc. Natl.
Acad. Sci. USA 114,1783 (2017 ).
[22] E. Y . Tsymbal, O. N. Mryasov, and P. R. LeClair, J. Phys.:
Condens. Matter 15,R109 (2003 ).
[23] H. J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier,
H.-P. Schönherr, and K. H. Ploog, Phys. Rev. Lett. 87,016601
(2001 ).
[24] Y . Ohno, D. K. Young, B. Beschoten, F. Matsukura, H. Ohno,
and D. D. Awschalom, Nature (London) 402,790(1999 ).
[25] R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt,
A. Waag, and L. W. Molenkamp, Nature (London) 402,787
(1999 ).
[26] V . I. Perel’, S. A. Tarasenko, I. N. Yassievich, S. D. Ganichev,
V . V . Bel’kov, and W. Prettl, Phys. Rev. B 67,201304 (2003 ).[27] M. M. Glazov, P. S. Alekseev, M. A. Odnoblyudov, V . M.
Chistyakov, S. A. Tarasenko, and I. N. Yassievich, Phys. Rev. B
71,155313 (2005 ).
[28] P. Recher, E. V . Sukhorukov, and D. Loss, P h y s .R e v .L e t t . 85,
1962 (2000 ).
[29] R. M. Potok, J. A. Folk, C. M. Marcus, V . Umansky, M. Hanson,
and A. C. Gossard, P h y s .R e v .L e t t . 91,016802 (2003 ).
[30] J. A. Andrade and P. S. Cornaglia, Phys. Rev. B 94,235112
(2016 ).
[31] I. Bar-Joseph and S. A. Gurvitz, Phys. Rev. B 44,3332 (1991 ).
[32] S. A. Gurvitz and M. S. Marinov, P h y s .R e v .A 40,2166 (1989 ).
[33] V . Mantsevich, N. Maslova, and P. Arseyev, Solid State
Commun. 152,1545 (2012 ).
[34] C. A. Stafford and N. S. Wingreen, Phys. Rev. Lett. 76,1916
(1996 ).
[35] B. L. Hazelzet, M. R. Wegewijs, T. H. Stoof, and Y . V . Nazarov,
Phys. Rev. B 63,165313 (2001 ).
[36] E. Cota, R. Aguado, and G. Platero, Phys. Rev. Lett. 94,107202
(2005 ).
[37] Spin Physics in Semiconductors , edited by M. I. Dyakonov
(Springer, Berlin, 2008).
[38] A. M. Nazmul, T. Amemiya, Y . Shuto, S. Sugahara, and
M. Tanaka, P h y s .R e v .L e t t . 95,017201 (2005 ).
[39] B. A. Aronzon, M. V . Kovalchuk, E. M. Pashaev, M. A. Chuev,
V . V . Kvardakov, I. A. Subbotin, V . V . Rylkov, M. A. Pankov,I. A. Likhachev, B. N. Zvonkov, Y . A. Danilov, O. V . Vihrova,A. V . Lashkul, and R. Laiho, J. Phys.: Condens. Matter 20,
145207 (2008 ).
[40] S. V . Zaitsev, M. V . Dorokhin, A. S. Brichkin, O. V . Vikhrova,
Y . A. Danilov, B. N. Zvonkov, and V . D. Kulakovskii, JETP
Lett.90,658(2010 ).
[41] B. P. Zakharchenya and V . L. Korenev, Phys. Usp. 48,603
(2005 ).
[42] I. Žuti ´c, J. Fabian, and S. C. Erwin, P h y s .R e v .L e t t . 97,026602
(2006 ).
[43] E. Dias Cabral, M. A. Boselli, R. Oszwałdowski, I. Zutic, and
I. C. da Cunha Lima, P h y s .R e v .B 84,085315 (2011 ).
[44] C. Song, M. Sperl, M. Utz, M. Ciorga, G. Woltersdorf,
D. Schuh, D. Bougeard, C. H. Back, and D. Weiss, Phys. Rev.
Lett.107,056601 (2011 ).
[45] L. D. Anh, P. N. Hai, Y . Kasahara, Y . Iwasa, and M. Tanaka,
Phys. Rev. B 92,161201 (2015 ).
[46] V . Korenev, I. Akimov, S. Zaitsev, V . Sapega, L. Langer,
D. Yakovlev, Y . A. Danilov, and M. Bayer, Nat. Commun. 3,
959(2012 ).
[47] I. Akimov, V . L. Korenev, V . F. Sapega, L. Langer, S. V . Zaitsev,
Y . A. Danilov, D. R. Yakovlev, and M. Bayer, Phys. Status
Solidi B 251,1663 (2014 ).
[48] N. S. Maslova, I. V . Rozhansky, V . N. Mantsevich, P. I. Arseyev,
N. S. Averkiev, and E. Lähderanta, P h y s .R e v .B 97,195445
(2018 ).
[49] I. V . Rozhansky, K. S. Denisov, N. S. Averkiev, I. A. Akimov,
and E. Lähderanta, Phys. Rev. B 92,125428 (2015 ).
[50] N. S. Maslova, V . N. Mantsevich, and P. I. Arseyev, JETP Lett.
105,260(2017 ).
115307-5V . N. MANTSEVICH et al. PHYSICAL REVIEW B 99, 115307 (2019)
[51] N. Maslova, P. Arseyev, and V . Mantsevich, Solid State
Commun. 252,78(2017 ).
[52] V . Mantsevich, N. Maslova, and P. Arseyev, Physica E
(Amsterdam) 93,224(2017 ).
[53] V . Mantsevich, N. Maslova, and P. Arseyev, J. Magn. Magn.
Mater. 456,194(2018 ).
[54] N. S. Maslova, V . N. Mantsevich, and P. I. Arseyev, JETP Lett.
108,485(2018 ).
[55] E. A. Zhukov, E. Kirstein, D. S. Smirnov, D. R. Yakovlev,
M. M. Glazov, D. Reuter, A. D. Wieck, M. Bayer, andA. Greilich, P h y s .R e v .B 98,121304 (2018 ).
[56] J. R. Murphy, S. Delikanli, T. Scrace, P. Zhang, T. Norden,
T. Thomay, A. N. Cartwright, H. V . Demir, and A. Petrou,Appl. Phys. Lett. 108,242406 (2016 ).[57] S. Delikanli, M. Z. Akgul, J. R. Murphy, B. Barman, Y . Tsai,
T. Scrace, P. Zhang, B. Bozok, P. L. Hernández-Martínez,J. Christodoulides, A. N. Cartwright, A. Petrou, and H. V .Demir, ACS Nano 9,12473 (2015 ).
[58] K. S. Denisov, I. V . Rozhansky, N. S. Averkiev, and E.
Lähderanta, Semiconductors 51,43(2017 ).
[59] S. Amaha, W. Izumida, T. Hatano, S. Teraoka, S. Tarucha, J. A.
Gupta, and D. G. Austing, Phys. Rev. Lett. 110,016803 (2013 ).
[60] J. Fransson, P h y s .R e v .B 69,201304 (2004 ).
[61] K. Kayanuma, E. Shirado, M. Debnath, I. Souma, S.
Permogorov, and Y . Oka, Physica E (Amsterdam) 10,295
(2001 ).
[62] D. E. Logan, M. P. Eastwood, and M. A. Tusch, J. Phys.:
Condens. Matter 10,
2673 (1998 ).
115307-6 |
PhysRevB.97.214415.pdf | PHYSICAL REVIEW B 97, 214415 (2018)
Calculating the transport properties of magnetic materials from first principles including thermal
and alloy disorder, noncollinearity, and spin-orbit coupling
Anton A. Starikov, Yi Liu,*Zhe Yuan,*,†and Paul J. Kelly‡
Faculty of Science and Technology and MESA+Institute for Nanotechnology, University of Twente,
P .O. Box 217, 7500 AE Enschede, The Netherlands
(Received 20 February 2018; revised manuscript received 24 May 2018; published 13 June 2018)
A density functional theory based two-terminal scattering formalism that includes spin-orbit coupling and spin
noncollinearity is described. An implementation using tight-binding muffin-tin orbitals combined with extensiveuse of sparse matrix techniques allows a wide variety of inhomogeneous structures to be flexibly modelledwith various types of disorder including temperature induced lattice and spin disorder. The methodology isillustrated with calculations of the temperature dependent resistivity and magnetization damping for the importantsubstitutional disordered magnetic alloy permalloy (Py), Ni
80Fe20. Comparison of calculated results with recent
experimental measurements of the damping (including its temperature dependence) indicates that the scatteringapproach captures the most important contributions to this important property.
DOI: 10.1103/PhysRevB.97.214415
I. INTRODUCTION
As long as device dimensions were much larger than the
spin-flip diffusion length of the constituent materials, the effectof the electron spin on transport properties went largely unde-tected. When attention focused on magnetic materials in thinfilm and multilayer form, new properties such as interface mag-netic anisotropy and oscillatory exchange coupling emerged,culminating in the discovery of giant magnetoresistance [ 1,2]
(GMR) almost 30 years ago [ 3]. This heralded the emergence of
the field of spintronics [ 4], which exploits the spin of electrons
in addition to the charge used in conventional electronics,triggering a flood of new discoveries including tunnelingmagnetoresistance (TMR) [ 5,6], spin-transfer torque (STT)
[7–9], the spin Hall effect [ 10,11], the spin Seebeck effect,
etc. [ 3,12] Spin-dependent electron transport manifests itself
on microscopic length scales in magnetically inhomogeneoussystems such as magnetic bilayers, multilayers, and magnetictextures where interface and finite size effects are dominant. Asimportant as the fundamental physics of spin-dependent trans-port are the applications that spintronics makes possible. TheGMR effect allowed magnetic read heads to be miniaturizedand led to an explosion in the density of data that could be storedon a hard disk. The TMR effect in magnetic tunnel junctions(MTJs) forms the basis for new forms of nonvolatile storage,magnetic random access memories (MRAM); MTJs are alsoused as sensor elements in read heads. STT makes it possibleto write information in MRAMs more efficiently leading toSTT-RAMs [ 13,14] or to make microwave frequency STT os-
cillators (STOs) where the injected spin forces a magnetization
*Present address: The Center for Advanced Quantum Studies and
Department of Physics, Beijing Normal University, 100875 Beijing,China.
†Corresponding author: zyuan@bnu.edu.cn
‡Corresponding author: P.J.Kelly@utwente.nlto precess with gigahertz (GHz) frequency [ 8,15,16]. Passage
of a spin-polarized current can also cause a domain wall tomove, which is the principle behind a form of shift registercalled “racetrack memory” [ 17,18].
The search for new and improved kinds of magnetic storage
provided another focus of attention in the field of spintronics:magnetization dynamics in response to external fields andcurrents in nanoscale systems [ 19]. The physics of such devices
involves two major contributions: (i) spin-dependent scatteringof electrons in bulk materials and at interfaces, and (ii) spin-non-conserving scattering of electrons because of spin-orbitcoupling (SOC) when spin is no longer a good quantumnumber, or because of magnetic disorder. A breakdown ofspin conservation is essential for spin-relaxation processes thatare described with material dependent time and length scales,conventionally the Gilbert damping parameter αand the spin-
flip diffusion length l
sf, respectively. Predicting and controlling
these properties is very important for understanding anddesigning new spintronic devices leading to numerous exper-imental [ 20–27] and theoretical [ 28–31] material-dependent
studies on the subject. The development of a new theoreticalframework [ 32] for calculating magnetization damping and
its implementation in the framework of density functionaltheory [ 33–37] has motivated systematic reinvestigation of the
damping in alloys [ 38,39] and of the temperature dependence
of damping in permalloy [ 40] allowing quantitative confronta-
tion of theory and experiment without invoking adjustableparameters such as the relaxation time in the torque correlationmethod (TCM) [ 29–31,41].
In this paper, we describe in detail a method we recently
used to calculate the resistivity ρ, the spin-flip diffusion
length (SDL), and the Gilbert damping parameter for Ni
1−xFex
substitutional alloys [ 33], the resistivity and damping for the
itinerant ferromagnets Fe, Co, and Ni with thermal disorder[34], the resistance [ 42] and anisotropic damping [ 43]o f
magnetic domain walls, the nonadiabatic STTs in ballistic
2469-9950/2018/97(21)/214415(23) 214415-1 ©2018 American Physical SocietyANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
S
R
L
FIG. 1. A sketch of a two-terminal configuration for the scattering
problem. A gray scattering region ( S) is sandwiched between left ( L)
and right ( R) semi-infinite leads that have translational symmetry. A
current flows along the direction of the zaxis.
systems [ 44], interface-enhanced damping [ 45], thermal dis-
order effects in transport [ 46], and a novel interface spin Hall
effect [ 47]. It extends earlier work [ 48–50] by including SOC
and noncollinearity.
Central to the method is the scattering formalism [ 51]f o rt h e
conductance of a two-terminal device [ 52]. The system under
investigation is attached to reservoirs by semi-infinite leads thatsupport well defined scattering states (Fig. 1). For crystalline
leads, these states are right- and left-propagating Bloch statesthat are incident upon the scattering region and either reflectedfrom it or transmitted through it. The probability amplitude thattheνth right-propagating state incident from the left lead with
spinσ
/primeis scattered into the μth right-propagating state with
spinσin the right lead defines the transmission matrix element
tσσ/prime
μν. The crystal momenta and band indices of the scattering
states are labeled by μorν. Similarly, the reflection matrix rσσ/prime
μν
can be defined for right-propagating states that are reflected
back into left-propagating states in the left lead. Denoting leadsas left ( L) and right ( R) for a two-terminal system, we have two
transmission matrices t
LRandtRL, for electrons coming from
left- and right leads, and, similarly, two reflection matrices rLL
andrRR. Together, they form the scattering matrix
S=/parenleftbiggrLL tRL
tLR rRR/parenrightbigg
(1)
that contains all the information needed to study a number of
important physical properties of the system. The best knownsuch property is the conductance Gthat can be expressed
according to the Landauer-Büttiker formalism as [ 52,53]
G=e
2
hTr{tt†}. (2)
The scattering formalism is not restricted to calculations of the
conductance but can provide us with useful information aboutspin-dynamics and spin-relaxation processes in the scatteringregion. In particular, it can be used to calculate the Gilbertdamping parameter αand the spin-flip diffusion length l
sf
[33,45,46].
The present study is based upon a first-principles tight-
binding (TB) linearized-muffin-tin-orbital (LMTO) imple-mentation of the scattering formalism. TB-LMTOs form aminimal basis set [ 54–56] that allows us to construct a highly
efficient computational method, especially when combinedwith sparse-matrix techniques [ 57,58]. In combination with
the local spin-density approximation (LSDA) from densityfunctional theory (DFT), it allows us to study physical systemseither entirely ab initio , i.e., without introducing any free
parameters or in the case of finite temperature transport, aminimal number thereof. We extend earlier work [ 49,50]b y
introducing noncollinear magnetism [ 59] and spin-orbit inter-
action using the Pauli-Schrödinger Hamiltonian. We general-ize the wave-function matching (WMF) technique introducedby Ando [ 51] to eliminate the need for a principal layer
decomposition and dispense altogether with partitioning thescattering region into layers. Compared to the widely usedrecursive Green’s function method, the interaction range in theleads and scattering region is arbitrarily long without loss of nu-merical stability and optimal use can be made of sparse-matrixsolvers which greatly improves the computational efficiency.Not having to divide the scattering geometry into “blocks” or“layers” results in greater flexibility in applications to disorder.
We illustrate how this framework can be used to investigate
spin-dependent transport in the diffusive regime and spin-relaxation phenomena using recent developments in scatteringtheory and spin dynamics [ 32]. In Sec. II, we outline the
technical details of the method and illustrate it by applyingit to the Ni
80Fe20alloy permalloy in Sec. III. Some technical
details of the SOC implementation with LMTOs are given inAppendix A. Appendix Bcontains a brief discussion of some
limitations of the method as well as numerical tests about theexchange-correlation functional in the LSDA and the basis setof TB-LMTOs.
II. FORMALISM
To solve the scattering problem for the infinite system
depicted in Fig. 1, we need to solve the single-particle
Schrödinger equation
(H−EI)/Psi1=0, (3)
at some specified energy E, usually the Fermi energy. We as-
sume that the ground-state charge and spin-densities and Kohn-Sham potentials for all atoms in the system have already beencalculated self-consistently. Here, /Psi1is a vector of coefficients
/Psi1
iwhen the wave function /Psi1is expanded in some localized
orbital basis ( i≡Rlmσ for the MTOs we will use, where R
is an atom site index and lmσ have their conventional orbital
angular momentum and spin meaning; see Appendix A1).H
is the Hamiltonian matrix in the localized orbital basis and asummation over iis implied in ( 3). Its sparsity is determined
by the range of the localized orbitals, which is minimal forTB-MTOs. The system in Fig. 1is infinite so that the dimen-
sions of the Hamiltonian Hand unit matrix Iin Eq. ( 3) are both
infinite. By applying the “wave-function matching” method[51], the semi-infinite leads with full translational symmetry
can be replaced with appropriate boundary conditions in theform of energy-dependent embedding potentials on the bound-ary layers. This reduces the problem to a finite size and resultsin a two-stage process for calculating the scattering matrix. Inthe first stage, to be discussed in Sec. II A, eigenmodes u
mof
the leads are calculated by solving the Schrödinger equationfor each of the leads in turn taking translational symmetry intoaccount. By calculating their wave vectors k
mand velocities
vm, the eigenmodes can be classified as being either left-going
um(−) or right-going um(+). They form a basis in which to
expand any left- and right-going waves in the leads and their
214415-2CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
transformation under a layer translation in the leads is easily
calculated by using a generalization of Bloch’s theorem forcomplex k[51].
In the second stage, discussed in Sec. II B, these solutions
from the first stage can be used to construct the energy-dependent boundary conditions for the Hamiltonian in thescattering region, which can be a slab of a random alloy, asingle interface, a multilayered structure, a tunnel junction,a slab of thermally disordered material, etc. One then has tosolve a system of linear equations (LEQs) with the originalHamiltonian modified by incorporating the boundary condi-tions in the role of a coefficient matrix to obtain the wavefunctions /Psi1that provide all information about the scattering
in the system and can be used to calculate the transmissionand reflection probability amplitudes, t
μνandrμν, and more.
As a result of choosing a localized basis to minimize thehopping range, the Hamiltonian matrix is very sparse. Thiscan be exploited by using efficient numerical methods suchas incomplete LU-factorization (which takes into account thesparsity of the matrix) to solve the LEQs. The solution scaleslinearly with the extent of the scattering region in the transportdirection.
Once the scattering matrix is known, we can extract the
resistivity and Gilbert damping parameter as discussed inSec. II C. The scattering formalism will be presented in its
general form not depending on details of the underlyingbasis set and Hamiltonian; the most relevant aspects of the
LMTOs used in the current implementation are sketched in
Appendix A1. In Sec. II D, we discuss ways of modeling
different kinds of disorder using large supercells transverseto the transport direction.
A. Eigenstates of ideal leads
We make use of an assumed two-dimensional (2D) trans-
lational symmetry in the plane perpendicular to the transportdirection to characterize states in this and the next section witha lateral wave vector k
/bardblin the corresponding two-dimensional
Brillouin zone (2D BZ). All variables therefore have an implicitdependence on k
/bardblthat will be suppressed for simplicity. When
we refer to the number of atoms (orbitals) in a layer, we referto the finite number of atoms (orbitals) in a translational unitcell.
Because the ideal leads have translational symmetry in the
transport direction, they can be decomposed into an infinitenumber of translationally invariant layers. When the hoppingrange of the Hamiltonian of this system is greater than thecorresponding periodicity, i.e., when hopping to layers beyondthe nearest neighboring layers is not negligible, the usualapproach would be to increase the layer thickness until onlyhopping between neighboring layers occurs; these are calledprincipal layers . In general, the principal layer procedure
results in increased computational cost and decreased accuracy.To remedy this, we formulate the WFM method for arbitraryhopping range between layers, thus generalizing previousformulations of the WFM method [ 49–51,60].
We start with an ideal wire with translational symmetry
(Fig. 2), in which every layer contains N
Oatom centered
orbitals and is coupled to some number ( N) of layers to its
left and right. Then the Schrödinger equation for the ith layer
FIG. 2. Hamiltonian matrix of an ideal quantum wire partitioned
into slices determined by the translational symmetry of the leads.
Hi≡Hi,iis the on-layer term of the Hamiltonian, Bl≡Hi,i+land
B†
l≡Hi,i−ldescribe hopping to the lth and −lth neighboring layers,
respectively.
is given by
(EI−Hi)/Psi1i+N/summationdisplay
l=1(Bl/Psi1i+l+B†
l/Psi1i−l)=0, (4)
where Hi≡Hi,iis the on-layer term of the Hamiltonian, Bl≡
Hi,i+landB†
l≡Hi,i−ldescribe hopping to the lth and −lth
neighboring layers respectively, and /Psi1i+lis the wave function
on the lth neighboring layer. Taking into account translational
symmetry in the periodic crystal, the wave function on anyarbitrary layer lis related to the wave function on layer l−1
by a generalized Bloch factor λas
/Psi1
l=λ/Psi1l−1. (5)
Combining ( 4) and ( 5) leads to the generalized eigenvalue
problem of rank 2 ×N×NO:
(EI−H0)/Psi10+N−1/summationdisplay
l=1(Bl/Psi1l+B†
l/Psi1−l)+B†
N/Psi1−N
=−λBN/Psi1N−1, (6a)
/Psi1l=λ/Psi1l−1∀l∈[−N+1,N−1], (6b)
where, without loss of generality, i=0 has been assumed.
Nontrivial solutions of ( 6) can be separated into two classes.
The first class consists of N×NOleft-going waves, and
the second class of N×NOright-going waves. Each class
can contain both propagating Bloch waves and nonpropa-gating, evanescent waves. The corresponding Bloch factorsare denoted by λ(+) andλ(−). Of these N×N
Osolutions,
onlyNOBloch factors λ(+) correspond to the translation of
right-propagating waves to a neighboring layer, the rest of theλ(+) factors describe translations to more distant layers and do
not provide any additional information; thus we have only N
O
unique translation factors among the λ(+). Similarly, for left-
propagating waves, there are only NOunique translations in the
set ofλ(−) factors. By using only the NOorbitals belonging to
thel=0 layer with eigenvectors from ( 6) corresponding to the
set of unique translation factors λ(±), we construct normalized
eigenvectors um(±)(≡/Psi1l=0,mwhere lis the layer index and m
is the mode index) and use these to form the NO×NOmatrices
[51]
U(±)=(u1(±)···uNO(±)). (7)
Any arbitrary wave function on the l=0 layer can then
be represented as a linear combination of left- and right-
214415-3ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
FIG. 3. Geometry of the finite scattering problem comprising left ( L), and right ( R) leads sandwiching the scattering ( S)r e g i o nt h a ti s
augmented by a finite number Nof lead layers chosen to be sufficiently large so that there is no hopping from the scattering region proper to
the (white) lead layers.
propagating waves
/Psi1=/Psi1(+)+/Psi1(−), (8)
and any left- or right-propagating wave can be expanded in
terms of the eigenstates of the lead as
/Psi1(±)=U(±)C(±), (9)
where Cμ(±) is a vector of coefficients. We define the NO×
NOdiagonal eigenvalue matrices by
/Lambda1(±)=δnmλm(±). (10)
Using the Bloch condition ( 5) and following Ando’s original
procedure [ 49–51,61], we define the translation matrices
F(±)=U(±)/Lambda1(±)U−1(±). (11)
The translation of the wave function on layer iover an arbitrary
number of layers lis then given by
/Psi1i+l=Fl(+)/Psi1i(+)+Fl(−)/Psi1i(−), (12)
allowing us to construct the full solution for the entire lead.
B. The scattering problem
The scattering region Sis now inserted between the left
and right leads. It is important to emphasize that we donot define a layered structure inside the scattering region.Regardless of its size or contents, the scattering region actsas one large metalayer. The resulting problem is infinite butby making use of the translational symmetry in the leads andthe solutions obtained in the previous section, the leads can beincorporated in the scattering problem in the form of boundaryconditions imposed in the lead layers adjoining the scatteringregion. The system is partitioned as shown in Fig. 3where the
infinite scattering geometry is truncated to include only N+1
(translationally invariant) lead layers on the left and right inaddition to the original (disordered) scattering region where
Nis the hopping range (in terms of number of layers) in the
Hamiltonian describing the leads.
Specifically, we assume that the N+1 lead layers attached
to the original scattering region on the left side are indexed as−N,..., 0. Then the wave function in the layer in the left lead
with index −(N+Q), where Q> 0, i.e., the wave function in
the white layers on the left in Fig. 3can be related to the wave
function /Psi1
L,−N=/Psi1L,−N(+)+/Psi1L,−N(−), a superposition of
left- and right-propagating waves, as
/Psi1L,−N−Q=/Psi1L,−N−Q(+)+/Psi1L,−N−Q(−)
=F−Q
L(+)/Psi1L,−N(+)+F−Q
L(−)/Psi1L,−N(−)
=/bracketleftbig
F−Q
L(+)−F−Q
L(−)/bracketrightbig
/Psi1L,−N(+)
+F−Q
L(−)/Psi1L,−N, (13)
allowing us to express an arbitrary /Psi1L,−N−Qin terms of /Psi1L,−N
and/Psi1L,−N(+). The set of Schrödinger equations for layers
−N,..., 0 become
(EI−HL,n)/Psi1L,n+−n/summationdisplay
l=1BL,l/Psi1L,n+l+BLS,n/Psi1S
+N/summationdisplay
l=1B†
L,l/Psi1L,n−l=0∀n∈[−N,0], (14)
where BLS,nis the coupling between the lead layer on the
left with index n(in the truncated transport geometry) to
the (original) scattering region S,BL,lis the hopping to the
lth next layer in the left lead, and /Psi1Sis the wave function
in the scattering region. By splitting the summation/summationtextN
l=1=/summationtextN+n
l=1+/summationtextN
l=N+n+1in the last term on the left-hand side (lhs)
and using ( 13) to eliminate /Psi1L,n(∀n<−N), Eq. ( 14) can be
transformed to
(EI−HL,n)/Psi1L,n+−n/summationdisplay
l=1BL,l/Psi1L,n+l+BLS,n/Psi1S+/parenleftbig
1−δN,−n/parenrightbigN+n/summationdisplay
l=1B†
L,l/Psi1L,n−l+N/summationdisplay
l=N+n+1B†
L,lF−l+N+n
L (−)/Psi1L,−N
=−N/summationdisplay
l=N+n+1B†
L,l/bracketleftbig
F−l+N+n
L (+)−F−l+N+n
L (−)/bracketrightbig
/Psi1L,−N(+)∀n∈[−N,0]. (15)
This effectively acts as a boundary condition for the left-hand side of ( 15) and removes a direct dependence on the wave functions
to the left of the −Nth layer.
214415-4CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
Injection of electrons from the left electrode leads to only right-propagating waves on the right-hand side of the scattering
region and the wave function in the layer with index N+Qcan be related to the wave function in the rightmost layer ( N) within
the truncated transport geometry as
/Psi1R,N+Q=FQ
R(+)/Psi1R,N, (16)
allowing /Psi1R,N+Qto be eliminated from the Schrödinger equation
(EI−HR,n)/Psi1R,n+n/summationdisplay
l=1B†
R,l/Psi1R,n−l+B†
RS,n/Psi1S+(1−δN,n)N−n/summationdisplay
l=1BR,l/Psi1R,n+l+N/summationdisplay
l=N−n+1BR,lFl−N+n
R (+)/Psi1R,N=0
∀n∈[0,N], (17)
where B†
RS,nis the coupling between the scattering region
and the lead layer with index n(in the truncated transport
geometry), and BR,lis the hopping to the l-th next layer in
the right lead.
Combining the sets of Eqs. ( 15) and ( 17) with the
Schrödinger equation for the scattering region
(EI−HS)/Psi1S+N−1/summationdisplay
l=0[B†
LS,l/Psi1L,−l+BRS,l/Psi1R,l]=0 (18)
results in a set of inhomogeneous LEQs. By assuming that
/Psi1L,−N(+)=UL(+) and solving the LEQs with multiple right-
hand sides in one go, we can obtain the wave functions in thesystem for electrons that are injected in all possible propagatingmodes of the left lead (i.e., modes with |λ|=1).
Bloch states incident from the left and propagating to the
right are scattered by the breaking of translation symmetry intoleft-going states on the left-hand side and right-going states onthe right-hand side as
/Psi1
L,−N(−)=UL(−)˜r, (19a)
/Psi1R,N(+)=/Psi1R,N=UR(+)˜t. (19b)
Once we know the set of wave functions /Psi1for all incoming
states from the left lead, we can calculate the elements of thematrices ˜r=˜r
μνwith dimension ML×ML, where MLis the
number of propagating modes in the left lead, and of ˜t=˜tμν
with dimensions MR×ML,
˜r=U−1
L(−)[/Psi1L,−N−UL(+)], (20a)
˜t=U−1
R(+)/Psi1R,N. (20b)
The elements of the physical reflection and transmission
probability amplitude matrices can be found by normalizingwith respect to the currents:
r
μν=/radicalBigg
vL,μ(−)
vL,ν(+)˜rμν;tμν=/radicalBigg
vR,μ(+)
vL,ν(+)˜tμν, (21)
where vR,L(±) are the group velocities of the eigenmodes in the
left and right leads, which are determined using the expressionsderived in Appendix A2:
v
ν(±)=2a
¯hN/summationdisplay
n=1nIm/bracketleftbig
λn
ν(±)u†
ν(±)Bnuν(±)/bracketrightbig
. (22)When SOC is included, spin is no longer a good quantum
number and separating the equation of motion for differentspinors is not possible. Nevertheless, it will be convenient forthe purposes of analyzing our results to decompose the matricesr
μνandtμνinto the spin projections rσσ/prime
μνandtσσ/prime
μν.T h i si s
discussed in Appendix A3.
When only nearest neighbor hopping is allowed ( N=1),
the expressions in Secs. II A andII Breduce to expressions
known from earlier work [ 49–51,61]. For arbitrary values
ofN, they represent a generalized WFM technique that is
similar to the widely used recursive Green’s function method.The advantage is that proper treatment of sparsity of theresulting LEQs allows us to use efficient sparse-matrix LEQsolvers, which drastically improves computational efficiency.Additionally, departure from the recursive Green’s functionmethod allows us to describe the scattering region withoutintroducing “blocks” or “layers.” This eliminates numericalissues and simplifies application of the method in complicated,incommensurable systems. The equivalence of the WFMmethod with the Kubo-Greenwood formalism in the linear-response regime was shown earlier [ 61].
C. Extracting material-specific parameters from
the scattering matrix
Once we know the scattering matrix ( 1) consisting of the
rσσ/prime
μνandtσσ/prime
μνmatrices calculated from the left and right-hand
sides, we can use this to extract various bulk (and interface)parameters currently of interest in the field of spintronics.We focus here on the bulk resistivity and Gilbert dampingof the important Ni
80Fe20ferromagnetic alloy, permalloy, as
illustrative examples.
1. Resistivity
The total resistance of a diffusive conductor (e.g., alloy)
of length Lsandwiched between two identical ideal (ballistic)
leads can be expressed as
1/G=1/G Sh+R, (23)
where Gis the total conductance of the system and GSh=
(2e2/h)Nis the Sharvin conductance of each lead with
Nconductance channels per spin. R, the resistance of the
scattering region corrected for the finite conductance of theballistic leads [ 49,62], has two contributions:
R(L)=2R
i+Rb(L), (24)
214415-5ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
where Riis the resistance of a single alloy |lead interface, and
Rb(L) is the bulk resistance of an alloy layer of thickness L.
For a sufficiently thick alloy layer, Ohmic behavior is recoveredwhenR
b(L)≈ρL, where ρis the bulk resistivity.
In materials whose SOC is weak, the transport of electrons is
found to be well described by considering currents of spin-upand spin-down electrons separately. For a stack of materialscomprising ferromagnetic (FM) and nonmagnetic (NM) met-als, the resistance of each spin channel is obtained by addingresistances in series, the two spin channels are then added inparallel according to the “two-current series-resistor” (2CSR)model [ 63–65]. When a ferromagnetic alloy is sandwiched
between nonmagnetic leads, each spin species sees two spin-dependent interface resistances R
σ
iand a spin-dependent bulk
term:Rσ(L)=2Rσ
i+ρσL. The total resistance that results
from adding these terms in parallel can be written as
R(L)=2Ri(β2−2βγ+1)
1−γ2+ρL
+4R2
i(β2−1)(β−γ)2
(γ2−1)[(γ2−1)ρL+(β2−1)2Ri],(25)
in terms of the total interface resistance Rigiven by
1
Ri=1
R↑
i+1
R↓
i, (26)
the corresponding interface spin asymmetry γ=(R↓
i−
R↑
i)/(R↑
i+R↓
i), the total resistivity of the alloy ρgiven by
1
ρ=1
ρ↑+1
ρ↓, (27)
and the corresponding bulk spin asymmetry β=(ρ↓−
ρ↑)/(ρ↓+ρ↑).
When the SOC can no longer be considered weak, eight
parameters are used to describe the resistance of a diffusiveNM|FM|NM system [ 66]. Two parameters are required to
describe the NM metal, a resistivity ρ
NMand a spin-flip
diffusion length lNM. Three parameters are required to describe
the FM metal: a resistivity ρσ
FMfor each spin channel as well
as a spin-flip diffusion length lFM. And three parameters are
required to describe the interface; an interface resistance Rσ
i
for each spin channel and an interface spin-flip scattering
parameter δ(also called the spin memory loss parameter)
[67–69]. The nontrivial evaluation of all of these parameters
would go beyond the present task of illustrating the use ofthe scattering formalism and will be the subject of a separatepublication [ 70]. For the purpose of extracting a resistivity from
a series of calculations of R(L), we will use ( 25) in the form
R(L)=a+ρL+b/(ρL+c). (28)
For sufficiently thick slabs where the third term in ( 28) van-
ishes, ρwill be extracted from the slope of R(L). Otherwise,
all four independent parameters will be used to perform the fit.Both approaches will be examined in Sec. III B.
2. Gilbert damping
The magnetization dynamics of ferromagnets is commonly
described using the phenomenological Landau-Lifshitz-Gilbert equation
dM
dt=−γM×Heff+M×/bracketleftbigg/tildewideG(M)
γM2s·dM
dt/bracketrightbigg
, (29)
where Ms=|M|is the saturation magnetization, /tildewideG(M)i s
the Gilbert damping parameter (that is in general a tensor),and the gyromagnetic ratio γ=gμ
B/¯his expressed in terms
of the Bohr magneton μBand the Landé gfactor, which is
approximately 2 for itinerant ferromagnets. For a monodomainferromagnetic layer sandwiched between nonmagnetic leads,NM|FM|NM, the energy dissipation due to Gilbert damping is
dE
dt=/integraldisplay
Vd3rd
dt(Heff·M)
=/integraldisplay
Vd3rHeff·dM
dt=1
γ2dm
dt·/tildewideG(M)·dm
dt,(30)
where m=M/Msis the unit vector of the magnetization
direction for the macrospin mode. By equating this energyloss to the energy flow into the leads [ 71] associated with “spin
pumping” [ 72],
IPump
E=¯h
4πTr/braceleftbiggdS
dtdS†
dt/bracerightbigg
=¯h
4πTr/braceleftbiggdS
dmdm
dtdS†
dmdm
dt/bracerightbigg
,
(31)
the elements of the tensor /tildewideGwere expressed in terms of the
scattering matrix [ 32]
/tildewideGij(m)=γ2¯h
4πRe/braceleftbigg
Tr/bracketleftbigg∂S
∂mi∂S†
∂mj/bracketrightbigg/bracerightbigg
. (32)
Physically, energy is transferred from the slowly varying
spin degrees of freedom to the electronic orbital degrees offreedom where it is rapidly lost to the lattice (phonon degreesof freedom). Our calculations focus on the role of elasticscattering as the rate-limiting first step.
To calculate the Gilbert damping tensor /tildewideG
ijusing ( 32),
we need to numerically differentiate the scattering matriceswith respect to the magnetization orientation m. Expressing
this orientation in spherical coordinates ( θ,φ) with the polar
angle θ=0 corresponding to the equilibrium magnetization
direction m, we vary the magnetization direction about θ=0
to calculate the 2 ×2 damping tensor in a plane orthogonal to
m. Specifically, ∂S/∂m
iin (32) can be replaced by ∂S/∂ei,
where eiare components of the Cartesian basis vectors in
the plane orthogonal to mandφ0defines the orientation
of the coordinate system in this plane. Then the derivativesof the scattering matrix can be approximated as
∂S
∂e1≈S(/Delta1θ,φ 0)−S(/Delta1θ,φ 0+π)
2/Delta1θ, (33a)
∂S
∂e2≈S(/Delta1θ,φ 0+π/2)−S(/Delta1θ,φ 0+3π/2)
2/Delta1θ,(33b)
where /Delta1θis a small variation of the polar angle. Substitution
of (33)i n t o( 32) yields four elements of the 3 ×3 damping
tensor for any particular orientation mof the magnetization.
For cubic substitutional alloys, the damping can be assumedto be isotropic (see Appendix A4), so we limit ourselves to
differentiating about a single orientation mand our primary
214415-6CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
interest will be in the diagonal elements of /tildewideG=/tildewideGii. When the
damping is enhanced by FM |NM interfaces [ 22,72–74], the to-
tal damping of a ferromagnetic slab of thickness Lsandwiched
between leads can be written /tildewideG(L)=/tildewideGif+/tildewideGb(L), where
/tildewideGifis the interface damping enhancement and we express the
bulk damping in terms of the dimensionless Gilbert dampingparameter αas
/tildewideG
b(L)=αγM s(L)=αγμ sAL, (34)
where μsis the magnetization density and Ais the cross
section.
D. Modelling disorder
The structures used in spintronics studies are typically
stacked layers of magnetic and nonmagnetic materials thatexhibit various types of disorder. The magnetic materialsthemselves are frequently magnetic alloys like permalloy thatare intrinsically “chemically” disordered and are chosen tohave desirable magnetic properties. Even when two materials(like Fe and Cr) have the same crystal structure (bcc) and areclosely lattice matched, it is not possible to exclude intermixingat an interface and becomes desirable to be able to model it.
Other important material combinations such as permalloy
and Pt have a large lattice mismatch. For thin layers, this canbe accommodated by straining either or both layers (pseudo-morphic growth) but above a critical thickness these will relaxto their preferred structures with or without the formation ofmisfit dislocations. Since fully relaxed interface structures canonly be modelled using lateral supercells [ 45,47], we apply
the supercell approach to all forms of disorder studied in thiswork.
Lastly, many experiments are performed at room and el-
evated temperatures making it desirable to take into accounttemperature induced lattice and spin disorder. Our approachwill be to model such thermal disorder in large lateral su-percells. By doing so, we will be able to make contactwith a large body of experiments that have been interpretedwith phenomenological models [ 65] that assume a diffusive
transport regime.
1. Chemical disorder (random alloys)
To study bulk alloys and interface mixing, we can cal-
culate atomic sphere (AS) potentials self-consistently usingthe coherent-potential approximation (CPA) implemented withMTOs [ 75,76] or with periodic supercells [ 54–56]. Transport
calculations are performed with large lateral M×Nsupercells
in which atomic sites are randomly populated with AS poten-tials subject to the constraint imposed by the stoichiometry ofthe targeted experimental system. This can be done either byenforcing the desired stoichiometry layer by layer or globally.
For example, to simulate a (001) simple cubic A
25B75
random alloy using an 8 ×8 lateral supercell, we can randomly
assign 16 out of the 64 sites to Aatoms and the rest to B
atoms maintaining the 25:75 stoichiometry in every layer assketched in Fig. 4. In the second case, we assign elements Aand
Brandomly throughout the complete slab of material that is
chemically disordered. In both cases, configuration averagingis carried out by repeating the scattering calculations for anumber of different realisations of random disorder.
FIG. 4. Illustration of the configuration for a supercell with
chemical disorder. Two arbitrary layers in an 8 ×8 supercell are
shown for an A25B75alloy. Atoms of type Aare black and atoms
of type Bare grey.
2. Positional (thermal) disorder
Thermal lattice disorder (or other kinds of positional dis-
order) can be modelled in lateral supercells by displacing
atoms in the scattering region from their equilibrium positions,denoted R
i, by a randomized displacement vector uifor each
atomic site resulting in the new set of atomic coordinates /hatwideRi=
Ri+ui(Fig. 5). The scattering matrices are then calculated
for a number of such disordered configurations and the resultsaveraged. The main physical approximation which is invoked
is the adiabatic approximation. Though formally problematic
for a metal with a gapless spectrum, within the frameworkof the lowest order variational approximation (LOV A) to theBoltzmann equation, the adiabatic approximation is foundto describe the thermal and electrical transport properties oftransition metals very well [ 77].
Different approaches to generating u
iare possible, ranging
from ab initio molecular dynamics, through first-principles
lattice dynamics [ 46], to parameterized Gaussian disorder
[34,46]. The latter and simplest approach, which is adopted
here, is based on the harmonic approximation whereby theenergy cost of displacing atoms is quadratic in their displace-ments. Components of u
iare then distributed normally with a
root-mean-square (rms) displacement /Delta1that can be chosen
in different ways. It can be related to the temperature and
extracted from experiment within the Debye model. Or it can be
chosen to reproduce an experimental temperature-dependentresistivity. As the particular method of generating displace-ments does not affect the implementation of the transportmethod, we will not concern ourselves overly with the rela-tionship between the displacements {u
i}and the temperature
in this paper.
FIG. 5. Schematic of frozen thermal lattice disorder. Atoms (gray
balls) on an ideal lattice (left) are displaced from their equilibrium
positions by random vectors uito form a static configuration (right)
for the electronic scattering calculation.
214415-7ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
3. Noncollinear configurations and magnetic disorder
For collinear ferromagnets (or antiferromagnets), thermal
spin disorder can be modelled by rotating magnetic moments inthe scattering region away from their equilibrium orientations[34] (Fig. 6). To lowest order in the polar angles describing this
orientation, the energy varies quadratically and temperature-
induced spin disorder can be modelled with Gaussian disorder.
In a magnetic domain wall separating domains in which themagnetization is collinear, the magnetization rotates continu-ously from one preferred orientation to the other. Addressingboth these problems, spin disorder and domain walls, requiresan implementation of the scattering formalism whereby atomson different sites can have differently oriented magnetization
directions. This is simply achieved in the atomic spheres
approximation (ASA) [ 78–80].
We assume that the spin quantization axis σ
zof the collinear
system and the spatial zaxis are collinear with the direction of
transport. We then rotate the local magnetic moment (exchangepotential) on an arbitrary site (atomic sphere) by rotating allspin-dependent atomic parameters, which are 2 ×2 operators
in spin-space [such as the potential function P
α(A5)o rS O C
parameters ( A13)], using the rotation operator
/hatwideRsi=exp/parenleftbiggiσyθi
2/parenrightbigg
exp/parenleftbiggiσzφi
2/parenrightbigg
, (35)
where θiandφiare polar and azimuthal angles in the AS
on site /hatwideRi, while leaving spin-independent operators like the
structure constant matrix unchanged. The LMTO Hamiltonian
(A18) can then be constructed in the usual way using matrix
operations on the modified operators.
By using a suitable distribution of spinor-rotation angles,
we can simulate thermal disorder or ordered structures likedomain walls [ 34,42–44]. For thermal disorder, we can choose
φ
ito be random while assuming a Gaussian distribution for
θiwith an rms rotation /Delta1/Theta1 related to the temperature using
some model, e.g., to reproduce an experimental temperature
dependent resistivity [ 34] or magnetization [ 46]. Alternatively,
we can calculate the interatomic exchange interactions fromfirst principles and use these to determine the magnon dis-persion relations. By occupying the magnon modes for somechosen temperature, random sets of φ
iandθican be generated
and used as input to a scattering calculation in a frozen-
magnon approximation [ 46]. Details of how {θi,φi}depend
on temperature fall outside the scope of this paper.
FIG. 6. Schematic of frozen thermal spin disorder. Atomic mag-
netic moments (arrows) of a ferromagnetically ordered system (left)
are tilted by random polar angles θi(and azimuthal angles φi, not
shown) to form a static spin-disordered configuration (right) for theelectronic scattering calculation.Although the above scheme is only applied to magnetization
fluctuations about the global quantization axis in this paper, wehave applied it to nontrivial noncollinear magnetizations suchas spin spirals and magnetic domain walls in references [ 42–
44]. By explicitly taking the spatially varying magnetization
into account, we calculated the domain wall resistance [ 42],
the enhancement of the Gilbert damping by noncollinearity[43], and the nonadiabatic STT parameter [ 44] in domain walls
with different profiles. It could equally well be applied to studytransport properties in spin glasses [ 81] or amorphous magnets
[82] where the Gilbert damping is generally an anisotropic
tensor depending on the symmetry [ 83,84], as demonstrated
for magnetic domain walls [ 43].
III. CALCULATIONS
The scattering calculations are carried out in two distinct
steps. In the first step, semirelativistic [ 85] AS potentials
are calculated self-consistently for the atoms in the structurewe are interested in, starting with the calculation of “bulk”potentials for the left and right leads. AS potentials can becalculated self-consistently for the scattering region using thesurface Green’s function (SGF) method [ 86] or a supercell
approach with a conventional “bulk” band structure code[54–56]. For substitutional random alloys, this is done very
efficiently by combining the SGF method with the CPA [ 86].
In the second step, the WFM method outlined in Sec. IIis
used to calculate the scattering matrix for the fully relativisticPauli-Schrödinger Hamiltonian using a TB-MTO basis; fordetails see Appendix A1. In this step, the scattering states in
the left and right leads are first determined following Sec. II A
and then the scattering problem is solved according to Sec. II B.
Although the calculations are entirely ab initio in the
sense that the computational scheme does not contain anyfree parameters, the results do depend on the numericalimplementation, which is necessarily approximate. In thissection, we discuss a number of relevant issues and illustratesome potential difficulties for a Cu |Py|Cu system consisting
of a length Lof permalloy sandwiched between copper leads.
Both fcc materials are chosen to have a (111) orientation inthe transport direction. The lattice constant of permalloy is
taken to be a
Py=3.5412 ˚A according to Vegard’s law. This
is slightly smaller than the experimental lattice constant of
Cu,aCu=3.614˚A. As we will be interested only in the bulk
properties of the alloy, we will choose the lattice constant ofthe copper leads to be equal to that of permalloy and ignore the(small) modifications introduced into the electronic structureof Cu that may result.
Should the lattice mismatch be important, as is the case
when modeling the interface properties of an A |B interface
between materials A and B, the lattice constant ratio a
A/aB
can be approximated by the ratio of two integers NAandNB
such that NAaA∼NBaB. When the A and B lattices are chosen
to be aligned, this can result in unfavorably large values of NA
andNB. By dropping the alignment condition, more flexibility
can be achieved by searching for lattice vectors in each latticewhose lengths match; in general, this will require rotating thelattices with respect to one another. This approach made itpossible to study fully relaxed interfaces between permalloyand the nonmagnetic materials Cu, Pd, Ta, and Pt [ 45,47].
214415-8CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
A. Modelling diffusive transport with supercells
The (lateral) supercell approach allows us to flexibly model
interfaces between materials with different crystal structuresand lattice constants by imposing a degree of lattice periodicityin order to be able to use Bloch’s theorem. A substitutionalalloy, or a crystalline material at finite temperature has, how-ever, no translational symmetry and the supercell approachis formally only correct in the thermodynamic limit. Justas practical experience has shown that periodic boundaryconditions can be very effectively used to model symmetry-breaking surfaces, interfaces, impurities, etc., with very smallsupercells, we will see that we can model diffusive transportwith lateral supercells of very modest size.
1./Gamma1-point calculations with large lateral supercells
By assuming lattice periodicity parallel to the interface, the
wave functions can be characterized by a wave vector k/bardblin
a 2D BZ no matter how large the period might be. In thethermodynamic limit, this BZ becomes vanishingly small, theband dispersion becomes negligible and neither BZ samplingnor configuration averaging over configurations of disordershould be necessary. We explore this limit in Fig. 7where the
dependence of the resistance of a Cu |Py|Cu system is shown
as a function of the Py slab thickness Lfor 5×5, 15×15, and
25×25 supercells without SOC, i.e., examining the majority
and minority spin subsystems separately and neglecting theeffects of periodicity entirely by using only k
/bardbl=(0,0)≡/Gamma1.
The first feature we observe in Fig. 7is a roughly two-order-
of-magnitude difference between the resistances of majorityand minority spins. Such a difference is qualitatively consistentwith what is known about the mean-free paths of electronsin the two spin-channels [ 87]. For minority spins, this is
extremely short with reported values in the range 4–8 ˚A, while
for majority spins, it is much larger, in the range 50–200 ˚A
[88,89]. We can understand this [ 43] in terms of the energy
bands that were calculated for fcc Fe and Ni using the ASpotentials calculated self-consistently for permalloy with theCPA [ 86,90] ,s h o w ni nF i g . 8. At the Fermi energy, the
majority-spin bands for Ni and Fe are almost identical so that
0.050.1Rma
01020
Rmin5×5
0.650.70.75Rmaj
01020
Rmin15×15
0 10 200.60.70.8
L [nm]Rmaj
0 10 2001020
L [nm]
Rmin25×25×
FIG. 7. Resistance in f /Omega1m2of a Cu |Py|Cu structure as a function
of the thickness Lof Py for /Gamma1-point calculations for 5 ×5 (top), 15 ×
15 (middle), and 25 ×25 (bottom) lateral supercells, for majority
(left column) and minority (right column) spins, with 10 randomconfigurations of chemical disorder per thickness.-9-6-303E-EF (eV)Majority Spin Minority Spin
X L-9-6-303E-EF (eV)
X LNi Ni
Fe Fe
FIG. 8. Band structures calculated with the Ni and Fe AS po-
tentials and Fermi energy that were calculated self-consistently for
Ni80Fe20using the coherent potential approximation. The same AS
radii were used for Ni and Fe.
in a disordered alloy the majority-spin electrons see essentially
the same potentials on all lattice sites and are only very weaklyscattered by the randomly distributed Ni and Fe potentials. In
contrast, the minority-spin bands are quite different for Ni and
Fe, which can be understood in terms of the different exchangesplittings; the magnetic moments calculated for Ni and Fein permalloy in the CPA are 0.63 and 2 .61μ
B, respectively.
The random distribution of Ni and Fe potentials in permalloythen leads to strong scattering of minority-spin electrons intransport. This picture is consistent with previous calculationsof the resistivity and Bloch spectral function of permalloy[91–93].
As we approach the diffusive limit, we expect to see a
linear dependence of the resistance on the slab thickness.The minority spins clearly exhibit this behavior even for thesmallest supercell size studied, N×N=5×5. Increasing N
only reduces the spread between results for different configu-rations of alloy disorder. For the majority spins, the situation israther different: the resistance depends nonlinearly on Lwith
notable oscillations that we attribute to constructive and de-structive interference of electron waves in the Fabry-Perot-likeCu|Py|Cu cavity. Only for thick Py or large supercells do the
oscillations vanish, restoring the expected linear dependenceof the resistance for L> 10 nm.
In the case of a 5 ×5 supercell, with only five layers of Py
(∼1 nm, the smallest Py slab thickness Lin Fig. 7), the system
size is already comparable to the mean free path of electrons inthe minority-spin channel, and we are in the diffusive limit. Forthe majority-spin channel, the lateral dimensions of the slabonly begin to approach the reported mean free path [ 88,89]
for a 25 ×25 supercell and even for such large supercells
the resistance only shows diffusive (linear) behavior when thelength of the slab is larger than the mean free path.
Once the system is large enough to approach the diffusive
limit and the length dependence of the resistance becomeslinear, a resistivity ρ
/Gamma1can be determined from the slope of
214415-9ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
00.511.52ρmaj
0 0.05 0.1 0.15 0.2100110120130
1/N (SC size is N ×N)ρmin
FIG. 9. Resistivity in μ/Omega1cm of Py as a function of the supercell
size for /Gamma1-point calculations: for majority (top) and minority (bottom)
spins. The “error bars” measure the spread on averaging different
configurations of alloy disorder.
Rversus L. The dependence of ρ/Gamma1on the supercell size N
is shown in Fig. 9. One can see that in the minority-spin
case the resistivity is converged with a negligible error bar toρ
/Gamma1
min∼105μ/Omega1cm when the linear dimensions of the supercell
are much larger than the mean free path. For the majority spincase, as discussed above, we are not yet in the diffusive limitand quantum (interference) effects are still observable. Forsufficiently large values of L, we extract resistivity values of
orderρ
/Gamma1
maj∼0.5μ/Omega1cm. Though the “error bar” that results
from configuration averaging is small, there is still a strongdependence on the size of the lateral supercell.
2. Small supercell with integration over 2D BZ
Although /Gamma1-point calculations with a large supercell might
be the most direct way of simulating a diffusive medium, thecomputational cost is very high. We therefore study the effect ofimproving the sampling of the wave functions by making use ofBloch’s theorem. For an N×Nlateral supercell, we calculate
the transmission using a Q×Qset of k
/bardblpoints in the 2D BZ
associated with the supercell. The total transmission is given bysummation of partial transmissions. The same effect could beobtained for a QN×QNsupercell with /Gamma1point sampling only
(and with disorder in an N×Nunit cell artificially repeated
Q×Qtimes).
The resistance of a Py slab is shown in Fig. 10for a 5 ×5
supercell as a function of the thickness Lfor different Q×Q
samplings of the 2D BZ. As discussed in the previous section,for a 5 ×5 supercell, especially for majority spins, we were far
from the diffusive limit and the /Gamma1point picture was dominated
by interference effects; these are still visible in the top leftsubplot in Fig. 10when only 4 ×4 k points are used for
the BZ integration. Nevertheless, even though the resistancedisplays oscillatory behavior for any individual kpoint, these
oscillations average out with increasing BZ sampling densityresulting in a substantially linear dependence visible in thebottom left subplot of Fig. 10for a sampling of 256 ×256
k-points. For minority spins, the picture is simpler, as wealready approach the diffusive limit for a 5 ×5 supercell and0.70.750.8Rmaj
01020
Rmin4×4
0.70.750.8Rmaj
01020
Rmin48×48
0 10 200.70.750.8
L [nm]Rmaj
0 10 2001020
L [nm]
Rmin256×256
FIG. 10. Resistance in f /Omega1m2o faP ys l a bi naC u |Py|Cu scattering
geometry as a function of the slab thickness Lfor a 5 ×5 supercell
with different k-point samplings of the Brillouin zone. The normalized
area of the 2D element used in the BZ summation is defined as
/Delta12k||=ABZ/Q2where ABZis the area of the downfolded supercell
BZ. Results are shown for Q=4 (top), 48 (middle), and 256
(bottom) for majority (left) and minority (right) spins, with six random
configurations of chemical disorder for every value of L.
therefore even a quite small BZ sampling results in a very
linear dependence. Once the sampling is dense enough andwe observe linear diffusive behavior, we can extract valuesof the Py resistivity from the slope of R(L). In the limit that
L→0, the resistance does not vanish because of the interface
resistance and the finite (Sharvin) conductance of the idealleads.
To investigate the convergence of this procedure and com-
pare with the /Gamma1-only limit, we plot in Fig. 11the resistivity
calculated for 5 ×5 and 15 ×15 supercells as a function of the
equivalent BZ sampling NQ×NQ together with the resultsρ ρ
FIG. 11. Resistivity in μ/Omega1cm of Py as a function of equivalent
k-point sampling ( N×Q) in which the downfolded 2D BZ with area
ABZfor an N×Nsupercell is sampled with BZ element /Delta12k||=
ABZ/Q2for 5×5 (dash-dotted blue line) and 15 ×15 (solid red line)
supercells. For comparison, the results of /Gamma1-only calculations ( Q=
1) with variable supercell size are also shown (dashed black line).
Results for majority and minority spins are shown in the upper andlower panels, respectively. The converged values ρ
maj=0.57μ/Omega1cm
andρmin=105–109 μ/Omega1cm are in very good agreement with the
calculated values in the literature [ 91] around 0.6 and 100 μ/Omega1cm,
respectively.
214415-10CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
for the /Gamma1-only calculations for an N×Nsupercell from Fig. 9.
For majority-spin electrons, /Gamma1-only calculations are dominated
by interference effects and far from convergence so an ac-ceptable estimate of the resistivity could not be made. Whenthe BZ sampling is increased, then the results for both 5 ×5
and 15 ×15 supercells converge rapidly to the same value of
0.57±0.01μ/Omega1cm for majority spin electrons, suggesting that
this is the true calculated (albeit experimentally unobservablebecause SOC has been omitted) resistivity. For minority spins,the BZ-integrated 15 ×15 supercell result provides us with a
converged value of 105 ±1μ/Omega1cm, which is similar to the
/Gamma1-only result with a large supercell. The converged value for
a5×5 supercell is 4% larger (109 ±1μ/Omega1cm), which can be
attributed to the error that results from the limited averagingof configuration space possible with a limited supercell size;for a mean-free-path of 1 nm, a volume containing only5×5×5 atoms is sampled by a minority-spin electron before
undergoing a collision. For our present purposes, this erroris not big enough to justify the greater expense associatedwith larger supercells and we will limit ourselves to the 5 ×5
supercell with 32 ×32 2D BZ sampling throughout the rest of
this paper. This is something which should be born in mindwhen comparing to experiment where calculations shouldbe explicitly tested for convergence with respect to lateralsupercell size as well as BZ sampling. Additional tests of otheraspects of the numerical implementation of the method can befound in Appendix B1.
B. Resistivity calculations with SOC
Figure 12shows the thickness dependence of the Py layer
resistance where SOC was included with the magnetizationperpendicular to the current direction ( R
⊥) and parallel to it
02 0 40 60 80 100
L [nm]012345R (f m2)R||
R
||=2.70±0.02 cm
exp=4.2 4.8 cm (Poly.)=2.15±0.01 cm
FIG. 12. Resistance calculated for Cu |Py|Cu using a 5 ×5s u -
percell with SOC as a function of the layer thickness Lwith the
magnetization parallel to ( R/bardbl: dots, solid line) and perpendicular to
(R⊥: squares, dashed line) the current direction. Calculated results
are shown by symbols while the lines are fits using the 2CSRmodel. The fitted resistivities are nearly a factor of two smaller
than the experimental values in the range 4.2–4.8 μ/Omega1cm measured
for polycrystalline samples at low temperature [ 95–98], where the
presence of grain boundaries can increase the resistivity [ 99,100].(R/bardbl). What is most striking about these results compared to
those without SOC in Fig. 10is the nonlinearity of R(L). When
a current of electrons is injected from the Cu lead on the leftinto disordered Py, they need not scatter immediately at theinterface but do so on a length scale measured in terms of theelastic mean free path. However, we do not see any evidencefor such an effect in the absence of SOC (Fig. 10) and there is
no good reason why SOC should greatly alter this. When weinclude SOC, spin is no longer a good quantum number andthe unpolarized current must adapt to the finite polarizationof Py. This it does asymptotically on a length scale given by
l
Py
sf, the spin-flip diffusion length in Py, which was calculated
in Ref. [ 33]t ob e ∼5.5 nm in good agreement with values
determined experimentally [ 69,94]. However, in Fig. 12,R
appears to be varying nonlinearly on a length scale much larger
than this value of lPy
sf. To understand why, we must return to
calculations for Py without SOC.
Within the 2CSR model, the resistances of the individual
spin channels are first determined and then added in parallelto determine the total. These individual spin resistances areshown in Fig. 13for the same system as studied in Fig. 12but
with the SOC switched off. The majority and minority spinresistances are perfectly linear (except for a small mean-free-path effect showing up for a Py thickness smaller than 2 nm inthe majority spin channel). The total resistance shown in thebottom panel exhibits a curvature that is absent in the individualspin channels and in addition, the slope is about a factor offour smaller than with SOC included. We can understand thecurvature from ( 25)o rb y( 28); we only expect to observe
the linear behavior characteristic of Ohm’s law when the thirdterm on the right-hand side of these equations is negligiblecompared to the other terms (or vanishes). This only happenswhenρL/greatermuchR
i(orβ=γ).
It is instructive to try and extract a value for the resis-
tivity from Fig. 13(c) , while pretending not to know the
0.60.70.80.9R↑ fΩ m2
010203040R↓ fΩ m2
0 5 10 15 20 25 30 350.511.5R fΩ m2a)
b)
c)
FIG. 13. Resistance calculated for Cu |Py|Cu for a 5 ×5 supercell
without SOC as a function of the Py slab thickness Lfor (a) majority-
spin electrons, (b) minority-spin electrons and (c) total. The multiplesymbols for a given length are results for different configurations of
disorder. Linear least-square fits for L> 20 nm are shown by solid
lines, a nonlinear least squares fit for total resistance is representedby a dashed line.
214415-11ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
slopes of ρmaj=0.57±0.01μ/Omega1cm and ρmin=109μ/Omega1cm
from Figs. 13(a) and13(b) , which when combined result in
ρnonrel=(ρ−1
maj+ρ−1
min)−1=0.567±0.009μ/Omega1cm. We can do
this either by fitting the expression ( 28) to the calculated data
or by studying systems so long that the contribution of the bulkresistivity to the total dominates the interface terms and can beextracted from the slope of R(L). A linear fit of R(L)f o rL>
20 nm yields a total resistivity of 1 .08±0.07μ/Omega1cm, which
is almost twice as large as the true value, ρ
nonrel, demonstrating
that the nonlinear contribution from the interface terms in(28) is still not negligible even though R(L) looks reasonably
linear. Estimating the resistivity with higher accuracy usingthis approach requires calculations for much longer systems.
T h ea l t e r n a t i v ei st ofi t R(L)u s i n g( 28). To ensure a
stable fitting procedure, we use an iteratively re-weighted leastsquares algorithm with bisquare weights [ 101]. In this case the
estimated total resistivity is 0 .53±0.05μ/Omega1cm. Though this
is in much better agreement with ρ
nonrel, an additional error of
7% has nevertheless been introduced and the error bar itselfhas increased by a factor of 5.
Returning now to Fig. 12, we find that the R(L) data with
SOC shown as symbols can be fitted very well using ( 28)
(solid and dashed lines) yielding resistivity values of ρ
/bardbl=
2.70±0.02μ/Omega1cm and ρ⊥=2.15±0.01μ/Omega1cm. The av-
erage resistivity ¯ ρ=(ρ/bardbl+2ρ⊥)/3=2.33±0.02μ/Omega1cm is a
factor four larger [ 91,99] thanρnonrel=0.567±0.009μ/Omega1cm.
This compares reasonably well with CPA calculations per-
formed within the Kubo-Greenwood formalism [ 92,93,99]b u t
is almost a factor of two smaller than experimental valuesin the range 4 .2–4.8μ/Omega1cm measured for polycrystalline
samples [ 95–98]. Note that the resistivity can be significantly
enhanced by the grain boundaries [ 100]. The magnetoresis-
tance anisotropy value we estimate is ( ρ
/bardbl−ρ⊥)/¯ρ×100% =
24%±1%, which compares reasonably with experimental
values in the range 16%–18% [ 95–97] and previous theoretical
estimates [ 92,93,99].
Our calculations confirm the overall picture that in spite
of its smallness for 3 dmaterials, SOC plays an essential role
in determining the transport properties of alloys when thereis a very large difference between the resistivities of majorityand minority spins in the absence of SOC. When temperature-induced lattice and spin disorder are included below, the bulkresistivity will increase and the curvature seen for low valuesofLdecreases; it will turn out that low-temperature Py is
peculiarly difficult to describe accurately in our real-spaceapproach because of the large mismatch between the two spinchannels.
1. Influence of the Interface
Determining an alloy resistivity from calculations that
include SOC using a 2CSR model that neglects it entirely isunsatisfactory. The 2CSR model does identify an importantissue however, namely, the essential role played by interfaces inthe scattering formalism. Clearly, the interface is obscuring thecharacter of the bulk property we want to study by introducinga number of extraneous effects: mean-free-path effects at theinterface, spin-dependent interface resistances, and interfaceand bulk spin flipping required to bring the unpolarized currentinjected from the Cu lead into equilibrium with the spin-polarized current in Py. Since the asymptotic resistivity should-9-6-303E-EF (eV)Majority Spin Minority Spin
X L-10-50510E-EF (eV)
X LVCA Fe20Ni80VCA Fe20Ni80
HMF Cu HMF Cu
FIG. 14. (Top) Self-consistent virtual-crystal approximation band
structure for a nuclear charge ZVCA=(1−x)ZNi+xZ Fe.A tt h e
Fermi level, the majority-spin states (lhs) are free-electron-like,the minority-spin states have mainly 3 dcharacter (rhs). (Bottom)
Majority (lhs) and minority (rhs) spin band structures of Cu in which
a replusive constant potential of 1 Ry has been added to the minorityspin potential, the effect being to remove all minority-spin states from
the Fermi energy.
be independent of the leads, ideally, we would choose lead
materials that are perfectly matched to the properties of thealloy we are studying with the same current polarization andminimal interface resistance. However, we are constrained inour choice of lead material to choose something with fulllattice periodicity. We examine a number of possibilities inthis section.
We could choose leads to be ordered alloys with the
same chemical composition as Py. However, for an arbitraryNi
1−xFexchemical composition this might require using im-
possibly large unit cells. Instead, we adopt the virtual-crystalapproximation (VCA) [ 102] and construct artificial atoms with
nuclear charge Z
VCA=(1−x)ZNi+xZFeand a correspond-
ing number of neutralizing valence electrons. A self-consistentcalculation with this procedure for an fcc lattice with the samelattice constant as Py results in a magnetic moment of 1 .06μ
B;
the corresponding bands are shown in Fig. 14(top row). At
the Fermi level, we see that the majority-spin states (lhs) arefree-electron-like, while the minority-spin dstates [right-hand
side (rhs)] are partly occupied so we expect a better matchingof the electronic structures at the interface that should result ina smaller interface resistance. The result of calculating R
/bardbl(L)
using these VCA leads is shown in Fig. 15(black dots). The
curvature for low values of Lis strongly reduced compared to
Fig. 12and a resistivity value of ρ/bardbl=2.76±0.01μ/Omega1cm is
directly extracted from the linear dependence.
The procedure can be further refined by noting that the 1 /L
term in ( 25) vanishes if R↓
i→∞ , i.e.,γ=1. This situation
would correspond to using half-metallic ferromagnetic (HMF)[103] leads. There is no need to actually use a “real” HMF,
we can construct one from Cu leads by simply adding astrong ( ∼1 Ry) repulsive potential to the minority spin Cu
lead potentials to eliminate all minority spin states from the
214415-12CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
02 0 40 60 80 100
L [nm]0123456R|| (f m2)
2 HMF Cu Leads
1 HMF Cu Lead
VCA Fe20Ni80 Leads
FIG. 15. Resistance calculated with SOC as a function of the
Py slab thickness L. The black dots are calculated using the VCA
Ni80Fe20leads, while the red (blue) dots are obtained using Cu leads
with one (both) of the Cu leads replaced by the artificial HMF Cu.
The solid lines are linear fits to the calculated values.
vicinity of the Fermi level (Fig. 14, lower panels). In this way,
only majority spins are injected into Py whose low-temperaturepolarization we calculated to be β∼0.9[46]. If we do this
for the left-hand lead, we obtain the results shown in Fig. 15
as red dots; constructing both leads in this way, we obtainthe results shown as blue dots. In both cases, we achievenearly ideal Ohmic behavior. We attribute the small deviationsfrom linearity for small values of Lto spin flipping on a
length scale of l
Py
sfas the fully polarized injected electrons
equilibrate to β∼0.9. The slopes are essentially identical
within the error bars of the calculation, consistent with theslope obtained with VCA leads with ρ∼2.8μ/Omega1cm and only
slightly larger than the value we extracted with unpolarized Culeads, ρ∼2.7μ/Omega1cm.
All theoretical estimates of the permalloy resistivity are
below the reported experimental range ¯ ρ=4.2–4.8μ/Omega1cm
[95–98]. The discrepancy has been explained by noting that
theoretical calculations have been carried out for monocrys-talline, monodomain permalloy, while experimental observa-tions are made on polycrystalline samples [ 99]. In the latter
case, additional scattering on the grain boundaries increasesthe resistivity. We also have no information about the domainstructure of the experimental samples, but one can expect thatfor multidomain samples the resistivity should further increasedue to the additional scattering involved.
C. Gilbert damping
The Gilbert damping can be calculated by numerically
differentiating the scattering matrices with respect to themagnetization orientation as formulated in Sec. II C 2 .T h e
value of /tildewideGresulting from the differentiation may depend on
the choice of the finite but small value of /Delta1θchosen for the
numerical procedure. An example is shown in Fig. 16for Py
in the Cu |Py|Cu system. One can see that numerically /tildewideGis
very stable and does not depend on the choice of /Delta1θfor
variation of the polar angle over a large range, indicating linear10−510−410−310−210−10.0960.0980.10.1020.104G/(γ μsA) [nm]
Δθ[π]
FIG. 16. Gilbert damping of a 11.2-nm-thick layer of Ni 80Fe20
alloy as a function of /Delta1θ, the finite difference of the polar angle
used for numerical differentiation. Calculations are performed usinga5×5 supercell.
dependence of the scattering matrix on small variations of the
magnetization orientation.
When the thickness of the Py layer is increased, the total
damping of the system grows proportionally, as anticipated in(34) and demonstrated in Fig. 17. Moreover, the strict linearity
and negligible variation for different configurations of disorderindicate that the Gilbert damping is very insensitive to detailsof how the random chemical disorder in the alloy is modelled.The value of αextracted from the slope of /tildewideG(L)/(γμ
sA)
is 0.0046±0.0002, which falls at the lower end of a range
of values, 0 .004–0 .013, reported in the literature for mea-
surements at room temperature [ 22,24–27,39,40,73,105–114].
Very recently, measurements were carried out as a function oftemperature from room temperature (RT) to low temperatures,decomposing the damping into bulk and interface contributions[40]. These yield a value for the bulk damping of 0 .0048±
0.0003 at 5 K in remarkably good agreement with the value
calculated above (that has been confirmed by subsequentcoherent potential approximation calculations [ 36,37] within
the error bars of the calculations).
At room temperature, Zhao et al. reported a value of α=
0.0055±0.0003 that is at the low end of the 0 .004–0 .013
range measured previously. An even more recent RT study ofthe Ni
1−xFexalloys as a function of xreported [ 39] values of
αin essentially perfect agreement over the full concentration
range with the values calculated by Starikov et al. forT=0
[33]. Schoen et al. attributed the better agreement they obtained
with theory to corrections they made (i) for interface damping
0 5 10 15 20 2500.10.2
L [nm]G/(γμsA) [nm]
αcalc= 0.0046 ± 0.0002
αexp= 0.0048 ± 0.0003 (T= 5 K)
FIG. 17. Total damping of the Py slab as a function of its thickness
L[104]. The error bars, which measure the spread over different
configurations of disorder for every thickness, are less than the markersize in this plot and thus nearly invisible. The damping extracted via a
linear fitting (blue solid line), α
calc=0.0046±0.0002, is in excellent
agreement with the experimental value reported at low temperature[40].
214415-13ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
00.050.10.150.2α
SOC multiplier
1 2 3 4 5 600.20.4α1/2
SOC multiplier
FIG. 18. Scaling of Gilbert damping with the SOC strength. The
dots correspond to the calculated results, and the solid line shows thequadratic fit. The quadratic dependence of αon the SOC strength is
in agreement with a recent CPA calculation for Os-doped Py [ 36].
enhancement [ 22,23,73,115] and (ii) for radiative damping. For
Py, they reported an RT value of α=0.0050. Taken together
with the Zhao et al. result, α=0.0055±0.0003, this would
seem to indicate a minimal temperature dependence of themagnetization damping that is striking in view of a factor fiveincrease in the resistivity of Py over the same temperaturerange. To elucidate this (and the apparent disagreement be-tween the T=0 calculations of Starikov et al. [33] and the
older room-temperature experiments), we undertook a studyof the effect of thermally induced lattice and spin disorder onthe resistivity and damping that will be the subject of the nextsection.
The ingredients contributing to the magnetization damping
in the calculations are disorder and SOC; omitting either willlead to a vanishing bulk damping. In the next section, we willexamine what happens when we “tune” the amount of disorder.Before doing this, we investigate how the damping depends onthe magnitude of the SOC term in ( A12) when we scale it
with a parameter λ:H
so→λH so. From the results shown in
Fig.18for Py, it is clear that the damping scales quadratically
with the strength of the SOC. This is the scaling expected forthe strong interband scattering limit of the torque correlationmodel. Though only strictly applicable to ordered solids withwell-defined band structures [ 29–31,41], the strong interband
scattering limit is the appropriate limit for a disordered alloywhere momentum is not well defined.
D. Thermal lattice and spin disorder
To resolve the discrepancy between room-temperature mea-
surements [ 22,24–27,39,73,105–114] and T=0 calculated
values of αfor the Ni 1−xFexalloy system [ 33] and because we
are aware of only a single low-temperature measurement [ 40],
we extend to Py the method introduced in Ref. [ 34] to study
the temperature dependence of damping in Fe, Co, and Ni.Finite temperatures lead to displacements of the atoms fromtheir equilibrium positions (lattice disorder) and to rotationsof atomic magnetic moments away from their equilibriumorientations. A correct theoretical description of temperature4681012|| [ cm]
0 0.01 0.02 0.03 0.044.44.64.85.0 [ 10 ]
/a0[103]
FIG. 19. Resistivity ρ/bardbl(top) and damping parameter (bottom) in
Ni80Fe20alloy as a function of the rms displacement of atoms from
their equilibrium positions (in units of the lattice constant a0).
effects in solids would begin with the fundamental lattice and
spin excitations (phonons and magnons, respectively) and theoccupancy of these excitations [ 46] but this lies outside the
scope of the present paper. Instead, we apply a simpler schemeof uncorrelated Gaussian atomic and spin displacements [ 34]
to study the effects of thermal lattice and spin disorder on theresistivity and damping of Py.
We describe lattice disorder in terms of independent random
displacements u
iof the NSatoms in the scattering region
labeled ifrom their equilibrium positions Rii.e., we describe
the lattice as a collection of independent harmonic oscillators,see Fig. 5. The displacements u
iare distributed normally with
rms displacement /Delta1=√/summationtext
iu2
i/NS. As shown in Fig. 19,
increasing /Delta1leads to increased scattering and increased
resistivity. For /Delta1/a 0=0.029 corresponding to a resistivity
of 8.2μ/Omega1cm, the resistance R(L)o ft h eC u |Py|Cu system
(unpolarized Cu leads) is shown as a function of Lin Fig. 20.
The increased bulk resistivity leads to a substantial decreaseof the curvature observed in Fig. 12underlining the peculiar
difficulties presented by low-temperature Py.
While an rms displacement of 4% (in units of the lattice
parameter a
0) is enough to cause an almost fourfold increase in
the resistivity, the damping increases by only ∼5%. Figure 19
therefore indicates that while the resistivity might dependstrongly on additional structural sources of scattering in Pysuch as dislocations, grain boundaries, etc., the damping isexpected to be insensitive to this additional disorder. Theresistivity estimated theoretically for crystalline Py at zerotemperature can be expected to be lower than the valuesdetermined experimentally for polycrystalline samples.
The weak dependence of αon lattice disorder is consistent
with the quadratic scaling of the damping with the SOCstrength expected in the strong interband scattering limit ofthe torque correlation model. The electronic structure of Pystrongly resembles that of clean Ni in sofar as the majorityspindband is filled (seen clearly in the VCA, Fig. 14); it
is known from experiment [ 20] and TCM calculations in the
strong interband scattering limit [ 29,31,41] that the damping
214415-14CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
02 0 40 60 80 100
L [nm]0246810R|| (f m2)/a0=0
/a0=0.029
FIG. 20. Resistance calculated for Cu |Py|Cu with lattice disorder
corresponding to /Delta1/a 0=0.029 in a 5 ×5 supercell and including
SOC as a function of the layer thickness L. The magnetization is
parallel to the current direction. The results without lattice disorder
are included for comparison. The solid lines illustrate the linear fits
to the calculated values.
parameter of Ni depends only weakly on the relaxation time
in this limit.
Reverting to the ideal crystal structure, we next introduce a
certain level of spin disorder, as sketched in Fig. 6, by tilting the
atomic moments randomly from their equilibrium orientationsthrough angles θ
ithat are assumed to be distributed normally
with a rms tilt angle /Delta1/Theta1=/radicalBig/summationtext
iθ2
i/NS. The results plotted
in Fig. 21show that the resistivity depends strongly on spin
disorder (suggesting that measured resistivity values shouldalso depend on the domain structure of the samples); for therange of /Delta1/Theta1 shown, it increases by a factor of almost ten.
Compared to the lattice disorder case, the damping parameteralso increases more strongly, by almost 20%. A major factor inthis increase is, however, the reduced effective magnetizationdensity μ
sin (34)a s/Delta1/Theta1increases. Note that these calculations
assume that the external magnetic field is negligibly weak.
102030|| [ cm]
0 0.05 0.1 0.15 0.24.55.05.5 [ 10 ]
[][103]
FIG. 21. Resistivity (top) and damping parameter (bottom) in
Ni80Fe20alloy as a function of the rms rotation /Delta1/Theta1 of the atomic
magnetic moments and macrospin.The qualitative picture sketched above can be improved by
introducing quantitative measures for the rms displacementsand rotations in terms of the temperature T. In the Debye
model [ 116], the mean-square displacement of the ith atom
from equilibrium /angbracketleftu
2
i/angbracketrightdepends on Tas
/angbracketleftbig
u2
i(T)/angbracketrightbig
=/Delta12=3¯h2T
mk/Theta12
D/bracketleftbigg
/Phi1/parenleftbigg/Theta1D
T/parenrightbigg
+/Theta1D
4T/bracketrightbigg
, (36)
where /Theta1Dis the Debye temperature and /Phi1(x) is the Debye
integral function
/Phi1(x)=1
x/integraldisplayx
0y
ey−1dy. (37)
Expanding the integrand in ( 37) as a power series in yleads to
/Phi1(x)=1−x
4+x2
36+...so that in the high-temperature limit
where x< 1,/Phi1(x)/similarequal1−x/4 and ( 36) reduces to the classical
statistics [ 116]
/Delta12=3¯h2T
mk/Theta12
D. (38)
In the low-temperature limit T/lessmuch/Theta1D(x/greatermuch1), zero point
motion (zpm) is dominant and /Delta12=3¯h2
4mk/Theta1 D. Because it does
not contribute to give a low-temperature resistivity, we neglectthe zpm at T=0 but keep the complete Debye integral ( 37)
for the finite temperature calculation.
To map spin disorder onto temperature, we can use a cubic
spline to interpolate the experimental magnetization [ 117,118]
at an arbitrary temperature [ 46] or fit the experimental data
using some empirical analytical function [ 119]. We assume
that the magnetization at finite temperature can be defined interms of the mean value of the cosine of the tilting angle θ:
M(T)=M
0/angbracketleftcosθ/angbracketright. (39)
For atomic tilting angles θidistributed according to the Fisher
distribution [ 120] with probability density
P(θ,κ)=κsinθeκcosθ
2s i n h κ, (40)
the expectation value of the zprojection of the local magne-
tization can be expressed via the distribution parameter κas
/angbracketleftcosθ/angbracketright=cothκ−1/κ, which is just the Brillouin function
for large Jandκ∼1/kBT. This results in a transcendental
equation which relates temperature and magnetic momentdistribution:
M(T)
M(0)=cothκ−1
κ. (41)
The azimuthal angle φin (35) defining the orientation of the
projection of the magnetic moment in the xyplane is assumed
to be distributed uniformly in [0 ,2π], which is reasonable for
an isotropic material like Py.
The thermally induced random field satisfies the fluctua-
tion dissipation theorem, i.e., the time-averaged correlationfunction of the random field is proportional to the Gilbertdamping and depends on the temperature [ 121–123]. In our
modeling of the spin disorder, we do not explicitly involvethe random field but generate several snapshots of magneticconfigurations which are essentially “independent” of oneanother. So the implementation using the random distribution
214415-15ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
010203040 ( cm)
0 100 200 300 400 500
Temperature (K)4.04.55.05.56.0 (10-3)
Zhao
Schoen
Calc.Calc.
Expt.
FIG. 22. Resistivity (top) and bulk damping (bottom) of Ni 80Fe20
as a function of the temperature. The red symbols are the calculated
results (the red line is a guide to the eye). The asterisks (black)
in the resistivity plot correspond to tabulated average experimental
values from [ 124]. The asterisks (green) in the damping plot are the
data of Zhao et al. [40]. The blue diamond is a room-temperature
measurement corrected for spin pumping and radiative contributions[39].
of the atomic magnetic moments ( 40) does not violate the
fluctuation dissipation theorem.
The temperature dependent resistivity and damping calcu-
lated for Py using /Theta1D=450Kand the experimental mag-
netization [ 118] are compared with experiment in Fig. 22.
For temperatures around room temperature, the model ofindependent harmonic oscillators describes phonon motionreasonably well and the phonon spectrum of transition metalsis quite similar to the spectrum described by the Debye model.The agreement between the calculated and experimentallyobserved values of ρ(T)[124] and of α(T)[40] is remarkably
good. The results for the damping confirm an earlier reportof an, at best, weak temperature dependence of α[125]. A
recent room-temperature measurement [ 39] of the damping
containing corrections for spin pumping and radiative effectsis in almost perfect agreement with our RT result. We note thatthe measurements of Zhao et al. [40] were corrected for spin
pumping but not for radiative effects.
IV . CONCLUSION
We have developed a method to calculate the scattering
matrix including SOC and noncollinearity and illustrated ithere with calculations of the resistivity and Gilbert dampingof permalloy. The very efficient implementation with tight-binding muffin tin orbitals allows it to be applied to a widerange of materials and systems. In addition to zero-temperaturecalculations for ideal disordered alloys [ 33], the method can
be used to model temperature-induced disorder [ 34,46], for
systems such as interfaces that are not periodic [ 45,47], or
for noncollinear systems [ 42,43]. Our results for disordered
alloys are in agreement with or have been confirmed by otherestablished theoretical methods like CPA. Where comparisoncan be made, our results are in good agreement with experiment
so they can be used to predict material parameters.
ACKNOWLEDGMENTS
We acknowledge many useful discussions with Rien Wes-
selink and Kriti Gupta. This work was financially supportedby the “Nederlandse Organisatie voor Wetenschappelijk On-derzoek.” (NWO) through the research programme of “Sticht-ing voor Fundamenteel Onderzoek der Materie,” (FOM) andthrough the use of supercomputer facilities of NWO “ExacteWetenschappen” (Physical Sciences). It was also supported byEU Contract No. IST-033749 DynaMax, EU FP7 Contract No.NMP3-SL-2009-233513 MONAMI, ICT Grant No. 257159MACALO, the Royal Netherlands Academy of Arts and Sci-ences (KNAW) and “NanoNed,” a nanotechnology programmeof the Dutch Ministry of Economic Affairs.
APPENDIX A: COMPUTATIONAL DETAILS
1. Linearized Muffin-Tin Orbitals (LMTOs) and
the Pauli-Schrödinger Hamiltonian
The practical implementation of the WFM method pre-
sented in this paper is based on LMTOs and the ASA thatare described in detail elsewhere [ 54–56]. Here, we will
focus only on those aspects that are important for the presentversion of the scattering formalism. An earlier, nonrelativisiticversion of the method [ 49,50] can be referred to for additional
details regarding the use of muffin-tin orbitals in transportcalculations.
In the ASA, muffin-tin spheres are expanded to fill the vol-
ume of the solid [ 126]. Considering only the l=0, spherically
symmetric component of the potential inside the resulting ASlocated at R, a solution φ
Rl(E,r R) of the radial Schrödinger
equation can be determined numerically for energy Eand
angular momentum lresulting in the partial wave [ 56]
φRL(E,rR)≡φRl(E,r R)Ylm(ˆrR), (A1)
with rR≡r−R,L≡lmlabels the angular momentum, and
Ylmis a spherical (or real, cubic) harmonic. ˆrRdenotes a unit
vector and rR≡|r−R|. Later, we can drop the explicit R
dependence where it does not give rise to ambiguity. In termsof the logarithmic derivative D
lofφl(E,r)a tr≡s(the AS
radius),
Dl(E,s)≡rφ/prime
l(E,r)
φl(E,r)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
r=s≡sφ/prime
l(E,s)
φl(E,s), (A2)
we can define the energy-dependent potential function
P0
l(E)=2(2l+1)/parenleftbiggw
s/parenrightbigg2l+1Dl(E)+l+1
Dl(E)−l, (A3)
which in the ASA depends only on the potential and is
independent of the crystal structure. When there is more thanone atom type, wis an average AS radius. In terms of the
“canonical” structure constant matrix S
0
R/primeL/prime,RL[56,126], which
depends only on the positions of the ions, we can define theHermitian matrix at E=E
νas
h0(Eν)=−/bracketleftbig˙P0(Eν)/bracketrightbig−1/2/bracketleftbig
P0(Eν)−S0/bracketrightbig/bracketleftbig˙P0(Eν)/bracketrightbig−1/2,(A4)
214415-16CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
where P0
RLis an ( m-independent) element of the diagonal
matrix P0and ˙P0is the energy-derivative of P0.I nt h i s
expression only S0is a nondiagonal matrix. To first order in
E−Eν,h0(Eν) is the Hamiltonian in the ASA [ 54–56]. The
problem presented by the long range [ 126] of the structure
constant matrix S0is resolved by introducing a generalized
representation characterised by a set of l-dependent screening
parameters αland defining the so-called “screened” structure
constants Sαand potential functions Pαdefined by
Pα=P0/bracketleftbig
1−αP0/bracketrightbig−1, (A5)
Sα=S0/bracketleftbig
1−αS0/bracketrightbig−1, (A6)
where αis a diagonal matrix with m-independent elements
αl. For a suitable choice of screening parameters, the range
ofSαis essentially limited to the first- and second-nearest
neighbors for close-packed structures [ 54–56]. In the screened
representation, the two-center tight-binding matrix becomes
hα(Eν)=−/bracketleftbig˙Pα(Eν)/bracketrightbig−1/2/bracketleftbig
Pα(Eν)−Sα/bracketrightbig/bracketleftbig˙Pα(Eν)/bracketrightbig−1/2.
(A7)
The energy-independent, linearized (at Eν) muffin-tin or-
bitals for the AS located at Rare defined [ 56]a s
/vextendsingle/vextendsingleχα
RL(Eν)/angbracketrightbig
=|φRL(Eν)/angbracketright+/vextendsingle/vextendsingle˙φα
R/primeL/prime(Eν)/angbracketrightbig
hα
R/primeL/prime,RL(Eν),(A8)
where
/vextendsingle/vextendsingle˙φα
R/primeL/prime(Eν)/angbracketrightbig
=1
Nα
l(Eν)∂/bracketleftbig
|φR/primeL/prime(E)/angbracketrightNα
l(E)/bracketrightbig
∂E/vextendsingle/vextendsingle/vextendsingle/vextendsingle
E=Eν,(A9)
with the normalization function
Nα
l(Eν)=/bracketleftbig
(s/2)˙Pα
l(Eν)/bracketrightbig1/2. (A10)
For simplicity, we will later assume that the orbitals are
constructed at Eνand omit any explicit energy dependence.
Now we can construct the energy independent Hamiltonianmatrix correct to second order in E−E
ν:
/angbracketleftbig
χα/vextendsingle/vextendsingleH−EνI/vextendsingle/vextendsingleχα/angbracketrightbig
=hα+hαoαhα, (A11)
where oα=/angbracketleftφ|˙φα/angbracketright= ˙Nα/Nαis the so-called overlap matrix.
Equation ( A11) shows that the hopping range of the LMTO
Hamiltonian is double the hopping range of the screenedstructure constant matrix, defined by the three-center integralh
αoαhα. In a transport calculation dominated by what happens
at the Fermi energy EF, we can choose Eν=EFand the
second (three-center) term in Eq. ( A11) can be omitted.
We include the spin-orbit interaction in a perturbative way
by adding a Pauli term to the Hamiltonian,
Hso=1
c2rdV(r)
drˆL·ˆS. (A12)
In the LMTO basis set, the matrix elements of Hsoare given
by
/angbracketleftχα|Hso|χα/angbracketright=γ1+γ2hα+hαγ+
2+hαγ3hα,where γ1,γ2, andγ3are spin-orbit parameters for one-, two-,
and three-center terms,
γ1=/angbracketleftbig
φ/vextendsingle/vextendsingle1
c2rdV(r)
drˆL·ˆS/vextendsingle/vextendsingleφ/angbracketrightbig
=K⊗ξ, (A13a)
γ2=/angbracketleftbig
φ/vextendsingle/vextendsingle1
c2rdV(r)
drˆL·ˆS/vextendsingle/vextendsingle˙φα/angbracketrightbig
=K⊗˙ξα,(A13b)
γ3=/angbracketleftbig˙φα/vextendsingle/vextendsingle1
c2rdV(r)
drˆL·ˆS/vextendsingle/vextendsingle˙φα/angbracketrightbig
=K⊗¨ξα,(A13c)
and we introduce a matrix of coefficients
Klmσ,l/primem/primeσ/prime=/angbracketleftbig
lmσ/vextendsingle/vextendsingleˆL·ˆS/vextendsingle/vextendsinglel/primem/primeσ/prime/angbracketrightbig
, (A14)
and a set of SOC potential parameters:
ξlσσ/prime=/angbracketleftbig
φlσ(r)/vextendsingle/vextendsingle1
c2rdVσσ/prime(r)
dr/vextendsingle/vextendsingleφlσ/prime(r)/angbracketrightbig
, (A15a)
˙ξα
lσσ/prime=/angbracketleftbig
φlσ(r)/vextendsingle/vextendsingle1
c2rdVσσ/prime(r)
dr/vextendsingle/vextendsingle˙φα
lσ/prime(r)/angbracketrightbig
, (A15b)
¨ξα
lσσ/prime=/angbracketleftbig˙φα
lσ(r)/vextendsingle/vextendsingle1
c2rdVσσ/prime(r)
dr/vextendsingle/vextendsingle˙φα
lσ/prime(r)/angbracketrightbig
. (A15c)
The expressions for ˙ξand¨ξcan be slightly reworked by taking
(A9) into account:
˙ξα
lσσ/prime=˙ξlσσ/prime+ξlσσ/primeoα
lσ/prime, (A16a)
¨ξα
lσσ/prime=¨ξlσσ/prime+˙ξlσσ/prime/parenleftbig
oα
lσ+oα
lσ/prime/parenrightbig
+ξlσσ/primeoα
lσoα
lσ/prime,(A16b)
where
˙ξlσσ/prime=/angbracketleftbig
φlσ(r)/vextendsingle/vextendsingle1
c2rdVσσ/prime(r)
dr/vextendsingle/vextendsingle˙φlσ/prime(r)/angbracketrightbig
, (A17a)
¨ξlσσ/prime=/angbracketleftbig˙φlσ(r)/vextendsingle/vextendsingle1
c2rdVσσ/prime(r)
dr/vextendsingle/vextendsingle˙φl/primeσ/prime(r)/angbracketrightbig
.(A17b)
The parameters ξ,˙ξ, and ¨ξcan be obtained by radial integration
over the AS. Off-diagonal (in spin-space) elements of Vσσ/prime(r)
are assumed to be the average of potentials for different spins[127].
Thus the complete Pauli-Schrödinger Hamiltonian will be
H
so=/bracketleftbig
EνI+γ1/bracketrightbig
+/bracketleftbig
hα+γ2hα+hαγ+
2/bracketrightbig
+/bracketleftbig
hα(oα+γ3)hα/bracketrightbig
, (A18)
where we have grouped one-, two-, and three-center terms.
Even when E=Eν, omitting the three-center term in ( A18)
is formally less readily justified than when SOC is neglectedbecause it introduces longer-range hopping in the Hamiltonianmatrix. The practical impact on the resistivity and damping ofneglecting the three-center terms is examined in Fig. 23where
no significant effect can be seen while the computational costis reduced by some 70%.
2. Velocities
In this section, we derive the expression ( 22) for the group
velocities of eigenmodes in the ideal wire for the generalizedWFM framework. The derivations are similar to previous work[60,61]. The vectors u
mof (7) are solutions of the polynomial
214415-17ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
1.01.52.02.5R|| (f m2)
w.o. three-center terms
with three-center terms
0 5 10 15 20 25 30
L [nm]0.050.100.150.20G/(sA) (nm)(a)
(b)
FIG. 23. Effect of the three-center terms in Hsoon (a) the total
resistance and (b) the total damping of Ni 80Fe20atT=0. Red open
circles include three-center terms, black filled circles omit them.
equation of order 2 N:
λNHum+N/summationdisplay
n=1/parenleftbig
λN+n
mBnum+λN−n
mB†num/parenrightbig
=0. (A19)
Left multiplying by u†
mand differentiating with respect to
energy leads to
d
dE/bracketleftBigg
λN
mu†
mHum+N/summationdisplay
n=1/parenleftbig
λN+n
mu†
mBnum+λN−n
mu†
mB†num/parenrightbig/bracketrightBigg
=λN
m+2ıλN−1
mdλm
dEN/summationdisplay
n=1nIm/parenleftbig
λn
mu†
mBnum/parenrightbig
=0,(A20)
which yield the following expression for dE/dλ m:
dE
dλm=−2ı
λmN/summationdisplay
n=1nIm/parenleftbig
λn
mu†
mBnum/parenrightbig
. (A21)
For propagating states, λm=eikma,kmis real, and ais the
thickness of the periodic lead layer so
dkm
dE=1
ıaλmdλm
dE. (A22)
The standard definition of group velocity is υm=1
¯hdE
dk, there-
fore, substituting Eq. ( A21) and ( A22), we obtain
υm=ıaλm
¯hdE
dλm=2a
¯hN/summationdisplay
n=1nIm/parenleftbig
λn
mu†
mBnum/parenrightbig
. (A23)
3. Spin-projections of the scattering matrix
For convenience of interpretation it is useful to decompose
the transmission and reflection matrices into spin-projectedones. Although spin is not a valid quantum number when SOCis included, we can still characterize states in the leads as stateswhich have a distinctive spin projection onto the σ
zaxis (σz=
±1
2). For leads consisting of light nonmagnetic metals, this is
a reasonable approximation and can be achieved in practiceas follows: for every pair of spin-degenerate lead eigenmodes
u1,u2, we can construct a new pair of orthogonal eigenmodes
u/prime
1,u/prime
2by taking a linear superposition of the original modes
/bracketleftbiggu/prime
1
u/prime
2/bracketrightbigg
=/bracketleftbigga11a12
a21a22/bracketrightbigg
×/bracketleftbiggu1
u2/bracketrightbigg
(A24)
and choosing the coefficients aijto maximize the σz=+1
2
component of u/prime
1and the σz=−1
2component of u/prime
2. We denote
these new states as uσ+anduσ−. This basis set transformation
allows us to to operate with reasonably well defined spin-projected scattering matrices. For example, the matrix r
σσ/prime
μν
describes reflection from the νσ/primestates into the μσstates in
the same lead.
4. Reduction of the Gilbert damping tensor to a scalar
For a crystal with cubic symmetry and a collinear mag-
netization, the damping torque can in general be written asτ=m×(α·˙m). If the magnetization mis taken to be along
thezaxis, then to leading order in the transverse components
of the magnetization ( m
x,my/lessmuchmz∼1), the damping torque
can be written in Cartesian coordinates as
τx=−mz(αyx˙mx+αyy˙my), (A25a)
τy=mz(αxx˙mx+αxy˙my). (A25b)
Without loss of generality, we choose the momentary direction
of magnetization precession to be along the xaxis, i.e., ˙m=
˙mˆxand|mz|=1, as sketched in Fig. 24(a) . Then τx=−αyx˙m
andτy=αxx˙m. Keeping the system otherwise unchanged, we
rotate the coordinate axes through 90◦clockwise about the z
axis as shown in Fig. 24(b) . In this case, we can write the
damping torque τx/prime=−αyy˙mandτy/prime=αxy˙m. Comparing the
components in Figs. 24(a) and24(b) , we find αxx=αyyand
αxy=−αyx.
If we reverse the magnetization in Fig. 24(a) but keep ˙m
unchanged, the damping torque should be reversed as plottedin Fig. 24(c) . If we then rotate the system 180
◦about the x
axis, it reproduces the configuration of Fig. 24(a) except that
the component τxis inverted. As a consequence, τx=−αyx˙m
must be zero indicating that the off-diagonal elements αyx=
−αxy=0. In the same manner, it can be proved that in a
collinear magnetic system with cubic symmetry, the Gilbertdamping tensor reduces to a scalar, α=α1, where 1is the
3×3 unit matrix.
τx m
x y
τy
τx m
x y
τy
m=z m=−z
τx m
xy
τym=z
(a) (c) (d)
τ′y m
′x ′y τ′xm=z (b)
FIG. 24. Geometry of damping torque exerted on magnetization.
See text for detailed analysis.
214415-18CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
APPENDIX B: LIMITATIONS
A number of factors limit the application of the present
method. First and foremost, memory considerations limit thesize of system that can be addressed. A calculation for a singleconfiguration of the longest scattering region shown in Fig. 12
and containing about 15 000 =5×5×600 atoms requires
about one hour on a supercomputer node with 32 cores and256 GB memory, where the calculation parallelized perfectlyover the two-dimensional 32 ×32k-point summation. The
maximum length of ∼105 nm places an upper limit on,
e.g., the spin-flip diffusion length that could be studied. Forlarger systems, the calculations need to be performed with amultithreading sparse matrix solver or simply with extendedmemory. For metals, the computing time scales linearly withthe length Lof the scattering region and quadratically with the
size of the lateral supercell. For a lateral supercell containingMatoms, a calculation for a single k-point scales as M
3.T h e
BZ size scales as 1 /M, so for constant sampling density of
reciprocal space, the scaling goes as M2L. The upper limit of
lateral supercell size with SOC is in practice about 10 ×10 and
30 nm long; this scattering region also contains about 15 000atoms.
A second important limitation is the ASA. This conven-
tional description of the potential in combination with the MTOscheme is usually very good for close-packed systems; opensystems can frequently be reasonably well modelled by fillingspace with artificial “empty atomic spheres” [ 56,128], i.e.,
without nuclei inside. For lower symmetry structures, wherethe spherically symmetric potentials around the nuclei are notsufficient to characterize the real Kohn-Sham potential, theASA breaks down. In these cases, the reliability and accuracyof transport calculations as currently implemented with MTOand ASA are limited by the ASA description of the Kohn-Shampotentials. Andersen has suggested ways of circumventing thislimitation without sacrificing the efficiency of the ASA [ 129].
The ultimate limitation is the DFT itself, or rather the
approximation to the exchange-correlation potential, the func-tional derivative of the exchange-correlation (XC) energy, thathas to be chosen. Some of the uncertainties that result fromdifferent choices are discussed briefly next.
1. Additional numerical tests
Although the formalism described in this paper is parameter
free, the practical implementation requires approximating theXC energy of DFT [ 130] as in the local density approximation
[131] where electron gas data have been parameterized by vari-TABLE I. Dependence of the atomic magnetic moment μs,
nonrelativistic resistivities ρmajandρmin, the relativistic resistivity
ρ/bardbl, and the Gilbert damping parameter for Py, on the basis set and
choice of exchange-correlation potential: von Barth-Hedin (vBH)[132], Perdew-Zunger (PZ) [ 133], and V osko-Wilk-Nusair (VWN)
[134]. Calculations are performed with a 5 ×5 supercell and a
k-point sampling grid of 32 ×32 (equivalent to 160 ×160 for a 1 ×1
primitive interface cell). Resistivities are given in μ/Omega1cm and the
magnetic moment is in Bohr magneton μ
Bper atom.
XC/Basis μs ρmaj ρmin ρ/bardbl α(10−3)
vBH/spd 1.025 0 .57±0.01 109 ±12.7±0.14.6±0.2
vBH/spdf 1.001 0 .67±0.01 101 ±12.6±0.14.3±0.2
PZ/spd 1.010 0 .92±0.01 108 ±13.1±0.14.7±0.2
VWN/ spd 1.022 0 .60±0.01 107 ±12.8±0.14.5±0.2
ous workers [ 132–134]. The numerical values of the resistivity
and damping parameter will depend on the parametrizationchosen. Our implementation with TB-MTOs requires truncat-ing the orbital angular momentum expansion at some maxi-mum value of the orbital angular momentum and this will alsoinfluence the numerical results. For all calculations presentedin this paper, the von Barth-Hedin (vBH) XC potential andanspd basis set were used. To demonstrate the uncertainty
resulting from the somewhat arbitrary choice of XC potentialand basis sets, we show the results of calculations for Cu |Py|Cu
with different choices of potentials and spdorspdf basis set
in Table I.
The quantity most dependent on these choices is the permal-
loy majority-spin resistivity in the absence of SOC. The veryweak scattering of majority spins makes this very sensitive tosmall details of the electronic structure, which in turn dependstrongly on the exchange splitting. Once SOC is included, themean-free path is reduced and the sensitivity of the resistivityand, especially, of the Gilbert-damping parameter to these“technical” details becomes acceptable.
Various forms of the XC potentials have been implemented
in computer programs and examined for different physicaland chemical quantities [ 135]. For the transport properties of
magnetic metals and alloys that we study in this paper, there isno clear evidence to show that one XC potential is better thanthe others. The test for the basis set using the vBH XC potentialalso indicates that the minimal spd basis is good enough for
convergence, as initially proposed by Lambrecht and Andersen[136]. The slight difference in the calculated resistivity and
Gilbert damping can be regarded as the “uncertainty” arisingfrom an arbitrary choice of XC potential.
[1] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F.
Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas,Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Super-lattices, Phys. Rev. Lett. 61,2472 (1988 ).
[2] G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, En-
hanced magnetoresistance in layered magnetic structures withantiferromaganetic interlayer exchange, Phys. Rev. B 39,4828
(1989 ).[3] See the collections of articles in Ultrathin Magnetic Structures
I-IV,e d i t e db yJ .A .C .B l a n da n dB .H e i n r i c h( S p r i n g e r - V e r l a g ,
Berlin, 1994–2005); in Spin Current , edited by S. Maekawa,
S. O. Valenzuela, E. Saitoh, and T. Kimura, SemiconductorScience and Technology V ol. 17 (Oxford Science Publications,Oxford, 2012).
[ 4 ]S .D .B a d e ra n dS .S .P .P a r k i n ,S p i n t r o n i c s , Annu. Rev.
Condens. Matter Phys. 1,71(2010 ).
214415-19ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
[5] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey,
Large Magnetoresistance at Room Temperature in Ferromag-netic Thin Film Tunnel Junctions, Phys. Rev. Lett. 74,3273
(1995 ).
[6] S. Yuasa and D. D. Djayaprawira, Giant tunnel magnetoresis-
tance in magnetic tunnel junctions with a crystalline MgO(001)barrier, J. Phys. D: Appl. Phys. 40,R337 (2007 ).
[7] J. C. Slonczewski, Current-driven excitation of magnetic mul-
tilayers, J. Magn. Magn. Mater. 159,L1(1996 ).
[8] L. Berger, Emission of spin waves by a magnetic multilayer
traversed by a current, Phys. Rev. B 54,9353 (1996 ).
[9] D. C. Ralph and M. D. Stiles, Spin transfer torques, J. Magn.
Magn. Mater. 320,1190 (2008 ).
[10] Axel Hoffmann, Spin Hall effects in metals, IEEE Trans. Magn.
49,5172 (2013 ).
[11] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and J.
Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87,1213 (2015 ).
[12] R. L. Stamps, S. Breitkreutz, J. ˚Akerman, A. V . Chumak, Y .
Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen, S. A. Majetich,M. Kläui, I. L. Prejbeanu, B. Dieny, N. M. Dempsey, and B.Hillebrands, The 2014 magnetism roadmap, J. Phys. D: Appl.
Phys. 47,333001 (2014 ).
[13] J. ˚Akerman, Toward a universal memory, Science 308,508
(2005 ).
[14] M. H Kryder and C. S. Kim, After hard drives—what comes
next? IEEE Trans. Magn. 45,3406 (2009 ).
[15] A. Ruotolo, V . Cros, B. Georges, A. Dussaux, J. Grollier, C.
Deranlot, R. Guillemet, K. Bouzehouane, S. Fusil, and A. Fert,
Phase-locking of magnetic vortices mediated by antivortices,Nat. Nanotechnol. 4,528(2009
).
[16] A. Slavin, Spin-torque oscillators get in phase, Nat. Nanotech-
nol.4,479(2009 ).
[17] S. S. P. Parkin, M. Hayashi, and L. Thomas, Magnetic domain-
wall racetrack memory, Science 320,190(2008 ).
[18] L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin, Dynamics
of magnetic domain walls under their own inertia, Science 330,
1810 (2010 ).
[19] Y . Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin,
Nonlocal magnetization dynamics in ferromagnetic nanostruc-tures, Rev. Mod. Phys. 77,1375 (2005 ).
[20] S. M. Bhagat and P. Lubitz, Temperature variation of ferromag-
netic relaxation in the 3 dtransition metals, Phys. Rev. B 10,
179(1974 ).
[21] B. Heinrich, D. J. Meredith, and J. F. Cochran, Wave number
and temperature-dependent Landau-Lifshitz damping in nickel,J. Appl. Phys. 50,7726 (1979 ).
[22] S. Mizukami, Y . Ando, and T. Miyazaki, Ferromagnetic reso-
nance linewidth for NM/80NiFe/NM films (NM =Cu, Ta, Pd
and Pt), J. Magn. Magn. Mater. 226–230 ,1640 (2001 ).
[23] S. Mizukami, Y . Ando, and T. Miyazaki, Magnetic relaxation of
normal-metal (NM)/80NiFe/NM films, J. Magn. Magn. Mater.
239,42(2002 ).
[24] L. Lagae, R. Wirix-Speetjens, W. Eyckmans, S. Borghs, and
J. de Boeck, Increased Gilbert damping in spin valves andmagnetic tunnel junctions, J. Magn. Magn. Mater. 286,291
(2005 ).
[25] J. O. Rantschler, B. B. Maranville, J. J. Mallett, P. Chen,
R. D. McMichael, and W. F. Egelhoff, Damping at nor-mal metal/permalloy interfaces, IEEE Trans. Magn. 41,3523
(2005 ).[26] N. Inaba, H. Asanuma, S. Igarashi, S. Mori, F. Kirino, K. Koike,
and H. Morita, Damping Constants of Ni-Fe and Ni-Co AlloyThin Films, IEEE Trans. Magn. 42,2372
(2006 ).
[27] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y . Ando, A.
Sakuma, and T. Miyazaki, Magnetic damping in ferromagneticthin films, Jpn. J. Appl. Phys. 45,3889 (2006 ).
[28] V . Kamberský, J. F. Cochran, and J. M. Rudd, Anisotropic
low-temperature FMR linewidth in nickel and the theory of‘anomalous’ damping, J. Magn. Magn. Mater. 104,2089
(1992 ).
[29] K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Identification of
the Dominant Precession-Damping Mechanism in Fe, Co, andNi by First-Principles Calculations, P h y s .R e v .L e t t . 99,027204
(2007 ).
[30] K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Spin-orbit
precession damping in transition metal ferromagnets, J. Appl.
Phys. 103,07D303 (2008 ).
[31] V Kamberský, Spin-orbital Gilbert damping in common mag-
netic metals, Phys. Rev. B 76,134416 (2007 ).
[32] A. Brataas, Y . Tserkovnyak, and G. E. W Bauer, Scattering
Theory of Gilbert Damping, Phys. Rev. Lett. 101,037207
(2008 ); Magnetization dissipation in ferromagnets from scat-
tering theory, Phys. Rev. B 84,054416 (2011 ).
[33] A. A. Starikov, P. J. Kelly, A. Brataas, Y . Tserkovnyak, and
G. E. W. Bauer, Unified First-Principles Study of GilbertDamping, Spin-Flip Diffusion and Resistivity in TransitionMetal Alloys, P h y s .R e v .L e t t . 105,236601 (2010 ).
[34] Y . Liu, A. A. Starikov, Z. Yuan, and P. J. Kelly, First-principles
calculations of magnetization relaxation in pure Fe, Co, and Niwith frozen thermal lattice disorder, Phys. Rev. B 84,014412
(2011 ).
[35] H. Ebert, S. Mankovsky, D. Ködderitzsch, and P. J. Kelly, Ab-
Initio Calculation of the Gilbert Damping Parameter Via LinearResponse Formalism, P h y s .R e v .L e t t . 107,066603 (2011 ).
[36] S. Mankovsky, D. Ködderitzsch, G. Woltersdorf, and H. Ebert,
First-principles calculation of the Gilbert damping parametervia the linear response formalism with application to mag-netic transition metals and alloys, P h y s .R e v .B 87,014430
(2013
).
[37] I. Turek, J. Kudrnovský, and V . Drchal, Nonlocal torque
operators in ab initio theory of the Gilbert damping in random
ferromagnetic alloys, P h y s .R e v .B 92,214407 (2015 ).
[38] M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva,
H. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw, Ultra-low magnetic damping of a metallic ferromagnet, Nat. Phys.
12,839(2016 ).
[39] M. A. W. Schoen, J. Lucassen, H. T. Nembach, B. Koopmans,
T. J. Silva, C. H. Back, and J. M. Shaw, Magnetic propertiesof ultrathin 3 dtransition-metal binary alloys. II. Experimental
verification of quantitative theories of damping and spin pump-ing,P h y s .R e v .B 95,134411 (2017 ).
[40] Y . Zhao, Q. Song, S.-H. Yang, T. Su, W. Yuan, S. S. P. Parkin,
J. Shi, and W. Han, Experimental investigation of temperature-dependent Gilbert damping in permalloy thin films, Sci. Rep.
6,22890 (2016 ).
[41] V . Kamberský, On ferromagnetic resonance damping in metals,
Czech. J. Phys. 26,1366 (1976 ).
[ 4 2 ] Z .Y u a n ,Y .L i u ,A .A .S t a r i k o v ,P .J .K e l l y ,a n dA .B r a t a a s ,S p i n -
Orbit-Coupling-Induced Domain-Wall Resistance in DiffusiveFerromagnets, P h y s .R e v .L e t t . 109,267201 (2012 ).
214415-20CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
[43] Z. Yuan, K. M. D. Hals, Y . Liu, A. A. Starikov, A. Brataas, and
P. J. Kelly, Gilbert Damping in Noncollinear Ferromagnets,P h y s .R e v .L e t t . 113,266603 (2014 ).
[44] Z. Yuan and P. J. Kelly, Spin-orbit-coupling induced torque
in ballistic domain walls: Equivalence of charge-pumping andnonequilibrium magnetization formalisms, P h y s .R e v .B 93,
224415 (2016 ).
[45] Y . Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, and P. J.
Kelly, Interface Enhancement of Gilbert Damping from FirstPrinciples, P h y s .R e v .L e t t . 113,207202 (2014 ).
[46] Y . Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, M. van
Schilfgaarde, and P. J. Kelly, Direct method for calculatingtemperature-dependent transport properties, P h y s .R e v .B 91,
220405(R) (2015 ).
[47] L. Wang, R. J. H. Wesselink, Y . Liu, Z. Yuan, K. Xia, and
P. J. Kelly, Giant Room Temperature Interface Spin Halland Inverse Spin Hall Effects, P h y s .R e v .L e t t . 116,196602
(2016 ).
[48] K. Xia, P. J. Kelly, G. E. W. Bauer, I. Turek, J. Kudrnovský,
and V . Drchal, Interface resistance of disordered magneticmultilayers, P h y s .R e v .B 63,064407 (2001 ).
[49] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and G. E. W.
Bauer, First-principles scattering matrices for spin-transport,P h y s .R e v .B 73,064420 (2006 ).
[50] M. Zwierzycki, P. A. Khomyakov, A. A. Starikov, K. Xia, M.
Talanana, P. X. Xu, V . M. Karpan, I. Marushchenko, I. Turek,G. E. W. Bauer, G. Brocks, and P. J. Kelly, Calculating
scattering matrices by wave function matching, Phys. Status
Solidi B 245,623(2008 ).
[51] T. Ando, Quantum point contacts in magnetic fields, Phys. Rev.
B44,8017 (1991 ).
[52] S. Datta, Electronic Transport in Mesoscopic Systems
(Cambridge University Press, Cambridge, 1995).
[53] Y . Imry, Introduction to Mesoscopic Physics , 2nd ed. (Oxford
University Press, Oxford, 2002).
[54] O. K. Andersen and O. Jepsen, Explicit, First-Principles Tight-
Binding Theory, P h y s .R e v .L e t t . 53,2571 (1984 ).
[55] O. K. Andersen, O. Jepsen, and D. Glötzel, Canonical de-
scription of the band structures of metals in, in Highlights
of Condensed Matter Theory
,e d i t e db yF .B a s s a n i ,F .F u m i ,
and M. P. Tosi, International School of Physics “EnricoFermi,” Varenna, Italy (North-Holland, Amsterdam, 1985),pp. 59–176.
[56] O. K. Andersen, Z. Pawlowska, and O. Jepsen, Illustration
of the linear-muffin-tin-orbital tight-binding representation:Compact orbitals and charge density in Si, Phys. Rev. B 34,
5253 (1986 ).
[57] P. R. Amestoy, I. S. Duff, J. Koster, and J. Y . L’Excellent,
A fully asynchronous multifrontal solver using distributeddynamic scheduling, SIAM J. Matrix Anal. Appl. 23,15
(2001 ).
[58] P. R. Amestoy, A. Guermouche, J. Y . L’Excellent, and S. Pralet,
Hybrid scheduling for the parallel solution of linear systems,Parallel Computing 32,136(2006 ).
[59] S. Wang, Y . Xu, and K. Xia, First-principles study of spin-
transfer torques in layered systems with noncollinear magneti-zation, P h y s .R e v .B 77,184430 (2008 ).
[60] P. A. Khomyakov and G. Brocks, Real-space finite-difference
method for conductance calculations, P h y s .R e v .B 70,195402
(2004 ).[61] P. A. Khomyakov, G. Brocks, V . Karpan, M. Zwierzycki, and
P. J. Kelly, Conductance calculations for quantum wires andinterfaces: mode matching and Green functions, Phys. Rev. B
72,035450 (2005 ).
[62] Kees M. Schep, Jeroen B. A. N. van Hoof, Paul J. Kelly, Gerrit
E. W. Bauer, and John E. Inglesfield, Interface resistances ofmagnetic multilayers, P h y s .R e v .B 56,10805 (1997 ).
[63] S. F. Zhang and P. M. Levy, Conductivity perpendicular to the
plane of multilayered structures, J. Appl. Phys. 69,4786 (1991 ).
[64] S.-F. Lee, W.P. Pratt, Jr., Q. Yang, P. Holody, R. Loloee, P. A.
Schroeder, and J. Bass, Two-channel analysis of CPP-MR datafor Ag/Co and AgSn/Co multilayers, J. Magn. Magn. Mater.
118,L1(1993 ).
[65] T. Valet and A. Fert, Theory of the perpendicular magnetore-
sistance in magnetic multilayers, Phys. Rev. B 48,7099 (1993 ).
[66] J. Bass, CPP magnetoresistance of magnetic multilayers: A
critical review, J. Magn. Magn. Mater. 408,244(2016 ).
[67] A. Fert and S.-F. Lee, Theory of the bipolar spin switch,
Phys.
Rev. B 53,6554 (1996 ).
[68] K. Eid, D. Portner, J. A. Borchers, R. Loloee, M. A. Darwish,
M .T s o i ,R .D .S l a t e r ,K .V .O ’ D o n o v a n ,H .K u r t ,W .P .P r a t t ,Jr., and J. Bass, Absence of mean-free-path effects in thecurrent-perpendicular-to-plane magnetoresistance of magneticmultilayers, P h y s .R e v .B 65,054424 (2002 ).
[69] J. Bass and W. P. Pratt, Jr., Spin-diffusion lengths in metals
and alloys, and spin-flipping at metal/metal interfaces: anexperimentalist’s critical review, J. Phys.: Condens. Matter 19,
183201 (2007 ).
[70] K. Gupta, R. J. H. Wesselink, Z. Yuan, A. N. Other, S. W.
Els, and P. J. Kelly, Calculating the temperature dependenceof interface transport parameters from first-principles: Py |Pt
(unpublished).
[71] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, Optimal
Quantum Pumps, P h y s .R e v .L e t t . 87,236601 (2001 ); M.
Moskalets and M. Büttiker, Dissipation and noise in adiabaticquantum pumps, P h y s .R e v .B 66,035306 (2002 ); Floquet
scattering theory of quantum pumps, 66,205320 (2002 ).
[72] Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Enhanced
Gilbert Damping in Thin Ferromagnetic Films, P h y s .R e v .L e t t .
88,117601 (2002 ); Spin pumping and magnetization dynamics
in metallic multilayers, P h y s .R e v .B 66,224403 (2002 ).
[73] S. Mizukami, Y . Ando, and T. Miyazaki, The Study on Ferro-
magnetic Resonance Linewidth for NM/80NiFe/NM (NM =
Cu, Ta, Pd and Pt) Films, Jpn. J. Appl. Phys. 40,580(2001 ).
[74] M. Zwierzycki, Y . Tserkovnyak, P. J. Kelly, A. Brataas, and
G. E. W. Bauer, First-principles study of magnetization relax-ation enhancement and spin transfer in thin magnetic films,Phys. Rev. B 71,064420 (2005 ).
[75] J. Kudrnovský and V . Drchal, Electronic structure of random
alloys by the linear band-structure methods, P h y s .R e v .B 41,
7515 (1990 ).
[76] J. Kudrnovský, V . Drchal, C. Blaas, P. Weinberger, I. Turek, and
P. Bruno, Ab-initio theory of the CPP magnetoconductance,Czech. J. Phys. 49,1583 (1999 ).
[77] S. Y . Savrasov and D. Y . Savrasov, Electron-phonon interac-
tions and related physical properties of metals from linear-response theory, Phys. Rev. B 54,16487 (1996 ).
[78] L. M. Sandratskii and P. G. Guletskii, Symmetrised method for
the calculation of the band structure of noncollinear magnets,J .P h y s .F :M e t .P h y s . 16,L43(1986 ).
214415-21ANTON A. STARIKOV , YI LIU, ZHE YUAN, AND PAUL J. KELLY PHYSICAL REVIEW B 97, 214415 (2018)
[79] J. Kübler, K.-H. Hock, J. Sticht, and A. R. Williams, Density
functional theory of non-collinear magnetism, J. Phys. F: Met.
Phys. 18,469(1988 ).
[80] L. M. Sandratskii, Noncollinear magnetism in itinerant-
electron systems: Theory and applications, Adv. Phys. 47,91
(1998 ).
[81] Y . Niimi, M. Kimata, Y . Omori, B. Gu, T. Ziman, S. Maekawa,
A. Fert, and Y . Otani, Strong Suppression of the Spin Hall Effectin the Spin Glass State, P h y s .R e v .L e t t . 115,196602 (2015 ).
[82] D. Wesenberg, T. Liu, D. Balzar, M. Wu, and B. L. Zink, Long-
distance spin transport in a disordered magnetic insulator, Nat.
Phys. 13,987(2017 ).
[83] K. M. D. Hals and A. Brataas, Spin-orbit torques and
anisotropic magnetization damping in skyrmion crystals, Phys.
Rev. B 89,064426 (2014 ).
[84] Y . Tserkovnyak and H. Ochoa, Generalized boundary condi-
tions for spin transfer, P h y s .R e v .B 96,100402 (2017 ).
[85] D. D. Koelling and B. N. Harmon, A technique for relativistic
spin-polarised calculations, J. Phys. C: Solid State Phys. 10,
3107 (1977 ).
[86] I. Turek, V . Drchal, J. Kudrnovský, M. Šob, and P. Weinberger,
Electronic Structure of Disordered Alloys, Surfaces and Inter-faces (Kluwer, Boston-London-Dordrecht, 1997).
[87] B. Dieny, Giant magnetoresistance in spin-valve multilayers,
J. Magn. Magn. Mater. 136,335(1994 ).
[88] B. Dieny, Quantitative interpretation of giant magnetore-
sistance properties of permalloy-based spin-valve structure,
Europhys. Lett. 17,261(1992 ).
[89] B. A. Gurney, V . S. Speriosu, J.-P. Nozieres, H. Lefakis,
D. R. Wilhoit, and O. U. Need, Direct Measurement of Spin-Dependent Conduction-Electron Mean Free Paths in Ferromag-netic Metals, Phys. Rev. Lett. 71,4023 (1993 ).
[90] P. Soven, Coherent-potential model of substitutional disordered
alloys, Phys. Rev. 156
,809(1967 ).
[91] J. Banhart, H. Ebert, and A. Vernes, Applicability of the
two-current model for systems with strongly spin-dependentdisorder, P h y s .R e v .B 56,10165 (1997 ).
[92] S. Khmelevskyi, K. Palotás, L. Szunyogh, and P. Weinberger,
Ab initio calculation of the anisotropic magnetoresistance in
Ni
1−cFecbulk alloys, P h y s .R e v .B 68,012402 (2003 ).
[93] I. Turek, J. Kudrnovský, and V . Drchal, Ab initio theory
of galvanomagnetic phenomena in ferromagnetic metals anddisordered alloys, Phys. Rev. B 86,014405 (2012 ).
[94] J. Bass and W. P. Pratt, Jr., Current-perpendicular (CPP)
magnetoresistance in magnetic metallic multilayers, J. Magn.
Magn. Mater. 200,274(1999 ).
[95] J. Smit, Magnetoresistance of ferromagnetic metals and alloys
at low temperatures, Physica 17,612(1951 ).
[96] T. R. McGuire and R. I. Potter, Anisotropic magnetoresistance
in ferromagnetic 3 dalloys, IEEE Trans. Magn. 11,1018
(1975 ).
[97] O. Jaoul, I. A. Campbell, and A. Fert, Spontaneous resis-
tivity anisotropy in Ni alloys, J. Magn. Magn. Mater. 5,23
(1977 ).
[98] M. C. Cadeville and B. Loegel, On the transport properties in
concentrated Ni-Fe alloys at low temperatures, J. Phys. F: Met.
Phys. 3,L115 (1973 ).
[99] J. Banhart and H. Ebert, First-principles theory of spontaneous-
resistance anisotropy and spontaneous Hall effect in disorderedferromagnetic alloys, Europhys. Lett. 32,517(1995 ).[100] B.-H. Zhou, Y . Xu, S. Wang, G. Zhou, and K. Xia, An ab initio
investigation on boundary resistance for magnetic grains, Solid
State Commun. 150,1422 (2010 ).
[101] W. Dumouchel and F. O’Brien, Integrating a robust option into
a multiple regression computing environment, in Computing
and Graphics in Statistics , edited by A. Buja and P. A. Tukey
(Springer-Verlag New York, Inc., New York, NY , USA, 1991),pp. 41–48.
[102] L. Nordheim, Electron theory of metals I, Annalen der Physik
401,607(1931 ); On the electron theory of metals II,
401,641
(1931 ).
[103] R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J.
Buschow, New Class of Materials: Half-Metallic Ferromag-nets, Phys. Rev. Lett. 50,2024 (1983 ).
[104] The finite intercept at L=0 in Fig. 17,/tildewideG
i≡/tildewideG(L=0) corre-
sponds to an interface damping enhancement that was observedfor thin ferromagnetic films [ 137–139]a n dN M |Py|NM layers
[22,23,115]. It was qualitatively explained in terms of how a
precessing magnetization “pumps” the transverse componentof magnetization through the interface [ 72]. The material
parameter that describes this “spin pumping” is the “mixingconductance” introduced to describe spin dependent transportbetween non-collinearly aligned elements of a magnetic cir-cuit [ 140]. To describe the interface damping enhancement
quantitatively, diffusive scattering and spin-flip scattering inthe NM layer have to be described as well as interface spin-flipscattering and modification of the mixing conductance by SOC
[45].
[105] C. E. Patton, Z. Frait, and C. H. Wilts, Frequency dependence
of the parallel and perpendicular ferromagnetic resonancelinewidth in permalloy films, 2-36 Ghz, J. Appl. Phys. 46,5002
(1975 ).
[106] W. Bailey, P. Kabos, F. Mancoff, and S. Russek, Control of
magnetization dynamics in Ni
81Fe19thin films through the use
of rare-earth dopants, IEEE Trans. Magn. 37,1749 (2001 ).
[107] J. P. Nibarger, R. Lopusnik, Z. Celinski, and T. J. Silva, Varia-
tion of magnetization and the Landé gfactor with thickness in
Ni-Fe films, Appl. Phys. Lett. 83,93(2003 ).
[108] S. Ingvarsson, G. Xiao, S. S. P. Parkin, and R. H. Koch, Tunable
magnetization damping in transition metal ternary alloys, Appl.
Phys. Lett. 85,4995 (2004 ).
[109] H. Nakamura, Y . Ando, S. Mizukami, and H.
Kubota, Measurement of magnetization precession forNM/Ni
80Fe20/NM (NM= Cu and Pt) using time-resolved Kerr
Effect, Jpn. J. Appl. Phys. 43,L787 (2004 ).
[110] R. Bonin, M. L. Schneider, T. J. Silva, and J. P. Nibarger,
Dependence of magnetization dynamics on magnetostrictionin NiFe alloys, J. Appl. Phys. 98,123904 (2005 ).
[111] K. Kobayashi, N. Inaba, N. Fujita, Y . Sudo, T. Tanaka, M.
Ohtake, M. Futamoto, and F. Kirino, Damping constants forpermalloy single-crystal thin films, IEEE Trans. Magn. 45,
2541 (2009 ).
[112] J. M. Shaw, T. J. Silva, M. L. Schneider, and R. D. McMichael,
Spin dynamics and mode structure in nanomagnet arrays:Effects of size and thickness on linewidth and damping, Phys.
Rev. B 79,184404 (2009 ).
[113] C. Luo, Z. Feng, Y . Fu, W. Zhang, P. K. J. Wong, Z. X. Kou, Y .
Zhai, H. F. Ding, M. Farle, J. Du, and H. R. Zhai, Enhancementof magnetization damping coefficient of permalloy thin filmswith dilute Nd dopants, Phys. Rev. B 89,184412 (2014 ).
214415-22CALCULATING THE TRANSPORT PROPERTIES OF … PHYSICAL REVIEW B 97, 214415 (2018)
[114] Y . Yin, F. Pan, M. Ahlberg, M. Ranjbar, P. Dürrenfeld, A.
Houshang, M. Haidar, L. Bergqvist, Y . Zhai, R. K. Dumas,A. Delin, and J. ˚Akerman, Tunable permalloy-based films for
magnonic devices, P h y s .R e v .B 92,024427 (2015 ).
[115] S. Ingvarsson, L. Ritchie, X. Y . Liu, Gang Xiao, J. C. Slon-
czewski, P. L. Trouilloud, and R. H. Koch, Role of electronscattering in the magnetization relaxation of thin Ni
81Fe19films,
P h y s .R e v .B 66,214416 (2002 ).
[116] B. T. M. Willis and A. W. Pryor, Thermal Vibrations in
Crystallography (Cambridge University Press, London, New
York, 1975).
[117] R. J. Wakelin and E. L. Yates, A Study of the order-disorder
transformation in iron-nickel alloys in the region FeNi 3,Proc.
Phys. Soc. B 66,221(1953 ).
[118] R. Weber and P. E. Tannenwald, Second-order exchange in-
teractions from spin wave resonance, J. Phys. Chem. Solids
24,1357 (1963 ); Long-range exchange interactions from spin-
wave resonance, Phys. Rev. 140,A498 (1965 ).
[119] M. D. Kuz’min, Shape of Temperature Dependence of Sponta-
neous Magnetization of Ferromagnets: Quantitative Analysis,P h y s .R e v .L e t t . 94,107204 (2005 ).
[120] R. Fisher, Dispersion on a sphere, Proc. R. Soc. London A 217,
295(1953 ).
[121] J. Foros, A. Brataas, Y . Tserkovnyak, and G. E. W. Bauer,
Current-induced noise and damping in nonuniform ferromag-nets, P h y s .R e v .B 78,140402 (2008 ).
[122] J. Foros, A. Brataas, G. E. W. Bauer, and Y . Tserkovnyak,
Noise and dissipation in magnetoelectronic nanostructures,
P h y s .R e v .B 79,214407 (2009 ).
[123] J. Xiao, G. E. W Bauer, K.-C. Uchida, E. Saitoh, and S.
Maekawa, Theory of magnon-driven spin Seebeck effect, Phys.
Rev. B 81,214418 (2010 ).
[124] C. Y . Ho, M. W. Ackerman, K. Y . Wu, T. N. Havill, R. H.
Bogaard, R. A. Matula, S. G. Oh, and H. M. James, Electricalresistivity of ten selected binary alloy systems, J. Phys. Chem.
Ref. Data 12,183(1983 ).
[125] G. Counil, T. Devolder, J.-V Kim, P. Crozat, C. Chappert,
S. Zoll, and R. Fournel, Temperature dependences of theresistivity and the ferromagnetic resonance linewidth inpermalloy thin films, IEEE Trans. Magn. 42,3323 (2006 ).
[126] O. K. Andersen, Linear methods in band theory, P h y s .R e v .B
12,3060 (1975 ).
[127] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, First-
principles calculation of the magnetocrystalline anisotropyenergy of iron, cobalt and nickel, Phys. Rev. B
41,11919
(1990 ).
[128] D. Glötzel, B. Segall, and O. K. Andersen, Self-consistent
electronic structure of Si, Ge and Diamond by the LMTO-ASAmethod, Solid State Commun. 36,403(1980 ).
[129] M. Zwierzycki and O. K. Andersen, The overlapping muffin-tin
approximation, Acta Phys. Pol. A 115,64(2009 ).
[130] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys.
Rev. 136,B864 (1964 ).
[131] W. Kohn and L. J. Sham, Self-consistent equations including
exchange and correlation effects, Phys. Rev. 140,A1133
(1965 ).
[132] U. von Barth and L. Hedin, A local exchange-correlation
potential for the spin-polarized case: I, J. Phys. C: Solid State
Phys. 5,1629 (1972 ).
[133] J. P. Perdew and A. Zunger, Self-interaction correction to
density-functional approximations for many-electron systems,Phys. Rev. B 23,5048 (1981 ).
[134] S. H. V osko, L. Wilk, and M. Nusair, Accurate spin-dependent
electron liquid correlation energies calculations: a criticalanalysis, Can. J. Phys. 58,1200 (1980 ).
[135] R. M. Martin, Electronic Structure: Basic Theory and Practical
Methods (Cambridge University Press, Cambridge, United
Kingdom, 2004), pp. 42–44.
[136] W. R. L. Lambrecht and O. K. Andersen, Minimal basis sets
in the linear muffin-tin orbital method: Application to thediamond-structure crystals C, Si, and Ge, P h y s .R e v .B 34,
2439 (1986 ).
[137] B. Heinrich, K. B. Urquhart, A. S. Arrott, J. F. Cochran, K.
Myrtle, and S. T. Purcell, Ferromagnetic-Resonance Study ofUltrathin Bcc Fe(100) Films Grown Epitaxially on Fcc Ag(100)Substrates, Phys. Rev. Lett. 59,1756 (1987 ).
[138] W. Platow, A. N. Anisimov, G. L. Dunifer, M. Farle, and
K. Baberschke, Correlations between ferromagnetic-resonancelinewidths and sample quality in the study of metallic ultrathinfilms, P h y s .R e v .B 58,5611 (1998 ).
[139] R. Urban, G. Woltersdorf, and B. Heinrich, Gilbert Damping
in Single and Multilayer ultrathin films: Role of interfaces innonlocal spin dynamics, Phys. Rev. Lett. 87,217204 (2001 ).
[140] A. Brataas, Y . V . Nazarov, and G. E. W. Bauer, Finite-
Element Theory of Transport in Ferromagnetic-Normal MetalSystems, P h y s .R e v .L e t t . 84,2481 (2000 ); Spin-transport
in multi-terminal normal metal-ferromagnetic systems withnon-collinear magnetizations, Eur. Phys. J. B 22,99(2001 ).
214415-23 |
PhysRevB.72.075117.pdf | Calculation of pair correlations in a high-density electron gas: Constraints for effective
interparticle potentials
R. Díez Muiño,1,2I. Nagy,3,2and P. M. Echenique4,2,1
1Unidad de Física de Materiales, Centro Mixto CSIC-UPV/EHU, P . Manuel de Lardizabal 3, 20018 San Sebastián, Spain
2Donostia International Physics Center, P . Manuel de Lardizabal 4, 20018 San Sebastián, Spain
3Department of Theoretical Physics, Institute of Physics, Technical University of Budapest, H-1521 Budapest, Hungary
4Departamento de Física de Materiales, Facultad de Químicas, UPV/EHU, Apartado 1072, 20080 San Sebastián, Spain
/H20849Received 18 January 2005; revised manuscript received 6 May 2005; published 11 August 2005 /H20850
The pair-correlation function g/H20849r/H20850of an interacting, unpolarized electron gas is modelled by using the
geminal representation and effective potentials for the two-electron relative motion. We put forward a gener-alization of the impurity-related Fumi theorem, in order to obtain the scattering-mediated part of the kineticenergy change. Considering the high-density limit, where a perturbatively exact expression for the ground-stateenergy is available, a rigorous constraint on the effective interactions in the even and odd channels is deduced.The constraint is implemented by using physically motivated one-parametric potentials in these channels.
DOI: 10.1103/PhysRevB.72.075117 PACS number /H20849s/H20850: 71.10.Ca, 71.45.Gm
I. INTRODUCTION
Modern density functional theory /H20849DFT /H20850has had a major
impact on electronic structure calculations.1By far, the most
popular approximation for this theory is the local density
approximation /H20849LDA /H20850for the exchange-correlation energy.
The success of the resulting simple computational schemerests, physically, on a rigorous relation between theexchange-correlation energy and the pair-correlationfunction.
2More precisely, the normalization condition for the
exchange-correlation hole continues to hold when the LDAis made for the exact density depletion around an electron.Due to the nature of LDA, the finer microscopical details ofthe underlying pair-correlation problem of an interactingelectron gas are, however, not transparent in this scheme.
A local functional for the exchange-correlation energy,
which depends on the local density of occupied states, wasintroduced recently.
3The proposed local parametrization also
rests on the homogeneous electron gas model and uses thedecomposition of its exchange-correlation hole in scatteringstates of different relative energies. In this framework, thekeyquantity is the energy change e
xc/H20849k,n0/H20850of an electron
pair with relative momentum kin a system of density n0,
when the antisymmetrization is imposed and the interactionbetween particles is switched on.
3
The idea of pair-approximation to realize the important
short-range correlation in the model system directly by usingeffective /H20849screened /H20850interactions in a two-body Schrödinger
equation for the spatial part of two-electron wave functions/H20849geminals /H20850has attracted a considerable interest.
4–12Based on
the effective potential concept, and focusing on the importantpair-correlation function g/H20849r/H20850, the description of the standard
many-body system can be simplified by transferring a part of
complexities from the total wave function to a model Hamil-tonian. This intuitive treatment of complexities via effectivepotentials for particle-interaction is quite similar to the oneused in nuclear physics.
13
A combination of the above ideas is appealing and physi-
cally transparent. An approach based on geminals can beuseful to discuss the details of the energy change exc/H20849k,n0/H20850
/H11013ex/H20849k,n0/H20850+ec/H20849k,n0/H20850. The main goal of the present paper is
to describe this quantity in the high density limit. It is this
limit in which exact results, in first order of the coupling
constant /H9251/H11013e2, exist for the on-top value /H20851g/H20849r=0/H20850/H20852of the
pair-correlation function and the potential /H20849exchange /H20850energy
/H20851/H9255x/H20849n0/H20850/H20852per particle.
These exact results, should provide natural constraints for
describing the g/H20849r/H20850andec/H20849k,n0/H20850quantities via geminals and
effective interparticle interactions. Similarly as in recent at-
tempts to formulate density matrix functional theory,14,15the
prototype many-body system can play here a crucial role inunderstanding the physical details of dynamical correlation.
The embedded-pair approximation
4–12rests, basically, on
the scattering aspects of correlated motions. Therefore, simi-larities and differences in comparison with well-known re-sults obtained for the screening problem of a charged-impurity are conceptually important.
The rest of the paper is organized as follows. Section II
contains a brief summary of the Hartree-Fock approximationin order to show the links to an effective potential approach.In this section, our basic constraint for the kinetic energychange at the high-density limit is formulated using the scat-tering approach for geminals. The explicit results, obtainedby implementing the approach via model potentials, aregiven there as well. Section III is devoted to a short summaryand discussion. We shall use Hartree atomic units in the pa-per unless otherwise stated.
II. METHOD AND RESULTS
The antisymmetry of the state-function under permutation
of identical fermions leads to a special correlation betweenthe positions of two such particles whose spins are parallel,even if there is nointeraction between the particles. The
system is described by a single Slater determinant of mo-mentum eigenstates, i.e., plane waves. In this noninteractingcase one has,
5in a partial-wave representation for the gemi-PHYSICAL REVIEW B 72, 075117 /H208492005 /H20850
1098-0121/2005/72 /H208497/H20850/075117 /H208497/H20850/$23.00 ©2005 The American Physical Society 075117-1nals, the following expression for the so-called ideal /H20851g0/H20849r/H20850/H20852
pair distribution function:
g0/H20849r/H20850=3
2/H20858
odd l
l=1/H11009
/H208492l+1/H20850/H20855jl2/H20849kr/H20850/H20856+1
2/H20858
even l
l=0/H11009
/H208492l+1/H20850/H20855jl2/H20849kr/H20850/H20856.
/H208491/H20850
The averages in Eq. /H208491/H20850, denoted by /H20855¯/H20856, are obtained by
weighting over the normalized, ideal distribution function
P0/H20849k/H20850of the relative momenta13,16
P0/H20849k/H20850=2 4k2
kF3/H208751−3
2k
kF+1
2/H20873k
kF/H208743/H20876, /H208492/H20850
where kF=/H208493/H92662n0/H208501/3is the Fermi momentum.
The pair distribution functions are averages for the mov-
ing particles. An electron at r=0 is not fixed; the reference
point moves with the electron. In a geminal-based represen-tation the relative motion is apparent. With g
0/H20849r/H20850the electron
charge density and the hole-density are − n0g0/H20849r/H20850and
n0/H20851g0/H20849r/H20850−1/H20852, respectively. The hole-density is normalized to
−1, and thus one may consider an electron with its hole as a
quasiparticle.17We stress the point, that this depletion is pre-
scribed solely by the Pauli’s exclusion principle; the Hartree-
Fock hole is not the result of an electrostatic repulsion.
The perturbative potential energy, the exchange energy
per particle /H9255x/H20849n0/H20850, is given by
/H9255x/H20849n0/H20850=1
2/H20885
0/H11009
4/H9266r21
rn0/H20851g0/H20849r/H20850−1/H20852dr. /H208493/H20850
By employing standard results from trigonometry
/H20858
odd l
l=1/H11009
/H208492l+1/H20850jl2/H20849kr/H20850=1
2/H208731−sin 2 kr
2kr/H20874, /H208494/H20850
/H20858
even l
l=0/H11009
/H208492l+1/H20850jl2/H20849kr/H20850=1
2/H208731+sin 2 kr
2kr/H20874, /H208495/H20850
we can, after a straightforward manipulation, rewrite Eq. /H208493/H20850
as a function of the relative momentum
/H9255x/H20849rs/H20850=n0
2/H20885
0kF
dk P0/H20849k/H20850ex/H20849k/H20850. /H208496/H20850
In this equation 2 ex/H20849k/H20850=−4/H9266//H208492k/H208502, and the Wigner-Seitz pa-
rameter rsis defined from n0=3/ /H208494/H9266rs3/H20850. At the perturbative
first-order there is no other contribution to the ideal kinetic
energy /H9255kin0/H20849rs/H20850=/H208493/10 /H20850kF2.
To obtain the same form for /H9255x/H20849rs/H20850from the more
conventional18Wigner-Seitz expression
/H9255x/H20849rs/H20850=−1
n0/H20885d3k1d3k2
/H208492/H9266/H208506nk10 4/H9266
/H20849k1−k2/H208502nk20, /H208497/H20850
one should use the k=/H20849k1−k2/H20850/2 and s=/H20849k1+k2/H20850/2 vari-
ables and perform integrations over these variables under theconstraint19of/H20841s+k/H20841/H33355kF.I nE q . /H208497/H20850nki0refers /H20849i=1,2 /H20850to
ideal Fermi distribution functions.
The Hartree-Fock wave functions for the homogeneous
system are plane waves, thus the above approximation asso-ciates interacting particles with a wave function of noninter-acting ones and results in the ideal g
0/H20849r/H20850. Of course, the
interaction among particles causes their positions to be cor-
related, even if there was no symmetrization postulate. Con-ventionally, the term dynamical correlation is used to namethis effect.
Clearly, it is possible to arrange g/H20849r/H20850/H11013g
0/H20849r/H20850+/H9004g/H20849r/H20850so
that the normalization condition
4/H9266/H20885
0/H11009
drr2n0/H9004g/H20849r/H20850=0 , /H208498/H20850
is still obeyed, yet the potential energy from Eq. /H208493/H20850is lower
/H20849i.e., larger negative value /H20850than the one found in the Hartree-
Fock approximation. This arrangement will usually costsome kinetic energy change /H20849/H9255
kin/H20850, so that one cannot just
adjust /H9004g/H20849r/H20850to maximize the potential energy gain /H20849/H9255pot/H20850
alone. Furthermore, these changes are at least second order
in the coupling constant /H9251and, in addition, they should sat-
isfy the virial theorem.20,21
The pair approximation via effective potentials rests5on
the following mathematical observation. The jl/H20849kr/H20850partial
waves, which are used to the expansion for geminals in Eq.
/H208491/H20850, are scatteringlike /H20849free /H20850solutions of the simple radial
Schrödinger equation,
/H20873T+l/H20849l+1/H20850
2/H9262r2/H20874jl/H20849kr/H20850=k2
2/H9262jl/H20849kr/H20850. /H208499/H20850
In this equation, T=/H20851−1/ /H208492/H9262r/H20850/H20852/H20851/H20849d2/dr2/H20850r/H20852, and the reduced
mass is /H9262=1/2.
The intuitive treatment, i.e., the direct approach for an
interacting homogeneous system,4–12consists of replacing
the noninteracting functions jl/H20849kr/H20850by interaction-based func-
tions Rl/H20849k,r/H20850in Eq. /H208491/H20850and using the P0/H20849k/H20850distribution of
Eq. /H208492/H20850. The symmetry-preserving Rl/H20849k,r/H20850functions are scat-
tering solutions of the
/H20873T+V±/H20849r/H20850+l/H20849l+1/H20850
2/H9262r2/H20874Rl±/H20849k,r/H20850=k2
2/H9262Rl±/H20849k,r/H20850, /H2084910/H20850
Schrödinger equation. Here V±/H20849r/H20850=/H20849/H9251/r/H20850f±/H20849r/H20850are screened
potentials in even /H20849/H11001/H20850and odd /H20849/H11002/H20850channels.8This choice
for the effective potentials is motivated, mainly, by the ap-parent role of parity in Eq. /H208491/H20850.
By using for R
0+/H20849k,r=0/H20850the Lippmann-Schwinger integral
equation22and taking the perturbative substitution /H20851j0/H20849kr/H11032/H20850
→R0+/H20849k,r/H11032/H20850/H20852in its kernel, one gets
R0+/H20849k,0/H20850=1−/H9262
k/H20885
0/H11009
dr/H11032V+/H20849r/H11032/H20850sin/H208492kr/H11032/H20850. /H2084911/H20850
Based on the rigorous23,24high-density expression for the
on-top pair-function 2 g/H20849r=0,rs/H20850=1−/H9252rswith/H9252=0.732, it is
easy to show via Eq. /H2084911/H20850that this form is guaranteed by
using a one-parametric screened potential with screening pa-MUIÑO, NAGY , AND ECHENIQUE PHYSICAL REVIEW B 72, 075117 /H208492005 /H20850
075117-2rameter /H11008kF. A bare Coulomb potential gives, from the per-
turbative Lippmann-Schwinger equation, the 2 g/H20849r=0,rs/H20850=1
−6/H9266//H208495kF/H20850expression. This is clearly an overestimation for
/H9252.
The rigorous result was obtained by many-body perturba-
tion theory /H20849up to second order in the coupling,23for the
energy /H20850via the static structure function, and by a special
double-perturbation approach24for/H9004g/H20849r/H20850. This method re-
sults in an everywhere nonpositive /H9004g/H20849r/H20850thus the normaliza-
tion condition for the pair correlation function is not satis-
fied. The effect leads to overcorrelation.24In the present
work, we shall use the exact on-top value for g/H208490,rs/H20850as a
constraint on V+/H20849r/H20850.
Now, let us turn to the exciting problem of characterizing
thekdependence of the energy change ec/H20849k,n0/H20850to the key
quantity of a local functional3based on exc/H20849k,n0/H20850/H11013ex/H20849k/H20850
+ec/H20849k,n0/H20850. This is the input to the equivalent of Eq. /H208496/H20850for
/H9255xc/H20849rs/H20850to describe a ground-state characteristic of a homoge-
neous, interacting system.
Motivated by the above discussion on the effect of a re-
arrangement via /H9004g/H20849r/H20850, we shall apply the following decom-
position: ec/H20849k,n0/H20850/H11013ekin/H20849k,n0/H20850+epot/H20849k,n0/H20850. The term describ-
ing the potential-energy change is related /H20849in the investigated
high-density limit /H20850to an e2-order change /H20851/H9004g/H20849r/H20850/H11008e2/H20852in the
pair function and to the switching-on of the true Coulomb
interaction. This term is, already, at least second order in thecoupling constant.
The only remaining term which still may scale linearly
with the coupling
/H9251/H11013e2in the effective pair approximation
for the high density electron gas with certain screened inter-
particle interactions, is ekin/H20849k,n0/H20850. Obviously, the correct
second-order scaling20,21of the kinetic energy change can
prescribe a nontrivial constraint on the geminal-based ap-proach for this system.
In the applied scattering description for /H9004g/H20849r/H20850and
/H9255
kin/H20849n0/H20850, first we outline how one can rewrite the important
normalization condition, Eq. /H208498/H20850, in terms of eigenphase
shifts /H9254l±/H20849k/H20850of even and odd channels. The standard
expression8,25on the volume integral
/H20885d3r/H20851/H20841Rl±/H20849k,r/H20850/H208412−/H20841jl/H20849kr/H20850/H208412/H20852=2/H9266
k2d
dk/H9254l±/H20849k/H20850, /H2084912/H20850
provides the desired link. By changing the order of kandr
integrations in Eq. /H208498/H20850, after employing Eqs. /H208491/H20850and /H208492/H20850, one
can easily arrive at the following constraint:
/H9004n−+/H9004n+=0 , /H2084913/H20850
for which the channel terms are as follows:
/H9004n−
n0=3
2/H20858
odd l
l=1/H11009
/H208492l+1/H20850/H208832/H9266
k2d
dk/H9254l−/H20849k/H20850/H20884,/H9004n+
n0=1
2/H20858
even l
l=0/H11009
/H208492l+1/H20850/H208832/H9266
k2d
dk/H9254l+/H20849k/H20850/H20884. /H2084914/H20850
In these expressions, the averages over P0/H20849k/H20850are denoted, as
before, by /H20855¯/H20856. Further simplifications can be made by em-
ploying partial-integrations in making the prescribed aver-ages.
For comparison, in the screening problem of a static em-
bedded charge /H20849Z/H20850, say antiproton, in a paramagnetic gas,
one has the following Friedel sum on the total induced elec-
tron charge /H20849/H9004n/H20850:
/H9004n=/H20858
l=0/H11009
/H208492l+1/H208502
/H208492/H9266/H208503/H20885d3knk02/H9266
k2d
dk/H9254l/H20849k/H20850, /H2084915/H20850
as a normalization condition, /H9004n=Zat consistency.
The Friedel condition ensures the charge neutrality of the
entire /H20849grand-canonical /H20850system in the presence of the
charged impurity. It does not say where the excess electronsare located. This comes from self-consistent-field approxi-mations, where the Poisson equation makes the necessarypotential-density connection.
Due to the volume-integral in Eq. /H2084912/H20850, there is some
similarity between the present and the impurity problems.The essential physical difference is related to the fact that the
Hartree-Fock hole is already properly normalized in the pair-correlation problem. The interaction results only in a rear-rangement via /H9004g/H20849r/H20850=g/H20849r/H20850−g
0/H20849r/H20850.
For two particles with relative momentum kinteracting
via a common potential it is well known26,27that the interac-
tion energy, which is the shift of the energy levels of thesystem produced by the potential, is given by
/H9004E
k=−2/H9266
/H9262k/H20858
l=0/H11009
/H208492l+1/H20850/H9254l/H20849k/H20850. /H2084916/H20850
In the case of an embedded impurity /H20849/H9262=1/H20850, an average
of/H9004Ekover a Fermi distribution function gives the total
energy change in the system.28This change is related to the
number /H20849Z/H20850of excess electrons and their energetic redistri-
bution in the field of the external charge.18The kinetic-
energy change in the grand-canonical system is ZkF2/2, due
to the excess electrons. Now we discuss how to obtain theseobvious physical results via Eq. /H2084916/H20850and an additional term
based on phase shifts.
Let us use, for simplicity, Eq. /H2084910/H20850with a common poten-
tial and Eq. /H208499/H20850for the interaction-free case. Multiplying Eqs.
/H208499/H20850and /H2084910/H20850byj
l/H20849kr/H20850andRl/H20849k,r/H20850, respectively, and substract-
ing the resulting noninteracting forms from the interacting
ones, we perform volume integrations on both sides of theobtained result. The kinetic-energy change e
kin/H20849l,k/H20850in the l
channel is given by
ekin/H20849l,k/H20850=2/H9266
/H9262k/H9254l/H20849k/H20850+k2
2/H92622/H9266
k2d
dk/H9254l/H20849k/H20850, /H2084917/H20850
which contains now, as a first term, the corresponding /H20849nega-
tive /H20850interaction-energy term from Eq. /H2084916/H20850on the right-hand
side. The second term is based on Eq. /H2084912/H20850.CALCULATION OF PAIR CORRELATIONS IN A HIGH- … PHYSICAL REVIEW B 72, 075117 /H208492005 /H20850
075117-3Summing over land integrating, as in Eq. /H2084915/H20850, over a
Fermi distribution function /H20849using/H9262=1/H20850one gets
/H9255kin/H20849n0/H20850=2
/H9266/H20858
l=0/H11009
/H208492l+1/H20850/H20885
0kF
dkd
dk/H20873k2
2/H9254l/H20849k/H20850/H20874. /H2084918/H20850
The above physical statement on the /H9255kin/H20849n0/H20850=ZkF2/2 value in
the impurity-related problem is verified, as Eqs. /H2084915/H20850and
/H2084918/H20850nicely show.
The modification via Eq. /H2084917/H20850to our geminal-based prob-
lem is straightforward. According to the relative-spin struc-ture we have
e
kin/H20849k,n0/H20850=ekin−/H20849k,n0/H20850+ekin+/H20849k,n0/H20850, /H2084919/H20850
where the channel contributions are
k2ekin−/H20849k,n0/H20850
2/H9266=3
2/H20858
odd l
l=1/H11009
/H208492l+1/H20850d
dk/H20873k2
2/H9262/H9254l−/H20849k/H20850/H20874,
k2ekin+/H20849k,n0/H20850
2/H9266=1
2/H20858
even l
l=0/H11009
/H208492l+1/H20850d
dk/H20873k2
2/H9262/H9254l+/H20849k/H20850/H20874. /H2084920/H20850
Finally, we perform an average over the P0/H20849k/H20850distribution
function to obtain a general expression for the important
quantity, the kinetic-energy change
/H9255kin/H20849rs/H20850=n0
2/H20885
0kF
dk P0/H20849k/H20850ekin/H20849k,n0/H20850. /H2084921/H20850
Thus, our main goal, i.e., to express these changes by scat-
tering characteristics is formulated.
The important high-density limit is treated via the first-
order Born /H20849B/H20850approximation. This corresponds to the use22
of the perturbative form for the phase shifts
/H9254l±/H20849k/H20850=− /H208492/H9262k/H20850/H20885
0/H11009
dr r2V±/H20849r/H20850jl2/H20849kr/H20850. /H2084922/H20850
To arrive at the perturbative, first order in e2, equivalents of
Eqs. /H2084913/H20850,/H2084914/H20850,/H2084919/H20850, and /H2084920/H20850, we shall use the above ex-
pression for the channel phase shifts.
The high-density constraint, /H9255kinB/H20849rs→0/H20850=0, obtained in
the Born approximation, could help to avoid
overcorrelation,24a possible drawback of modelling.15This
averaging to zero in first approximation resembles to thetreatment used in nuclear physics to eliminate /H20849in first order /H20850
tensor forces in the saturation problem.
13
A simple inspection of Eqs. /H2084914/H20850and /H2084920/H20850together with
Eq. /H2084922/H20850and Eqs. /H208494/H20850and /H208495/H20850shows, via order changes in
integrations and summations, that in the remaining kintegra-
tion the F±/H208490;2k/H20850/H11013V±/H208490/H20850±V±/H208492k/H20850forms will appear in our
constraints at the investigated perturbative limit. Here V±/H20849q/H20850
is the Fourier transform of the effective V±/H20849r/H20850. For example,
the equivalent of Eq. /H2084920/H20850in the Born limit is
k2ekin−/H20849k,n0/H20850=−3
8d
dk/H20851k3F−/H208490;2k/H20850/H20852,k2ekin+/H20849k,n0/H20850=−1
8d
dk/H20851k3F+/H208490;2k/H20850/H20852. /H2084923/H20850
For further analysis, one needs a suitable potential. As we
mentioned at Eq. /H2084910/H20850a parity-conserving approximation and
the electronic cusp suggest a screened, but Coulombic at r
=0, potential, V±/H20849r/H20850=/H208491/r/H20850f±/H20849r/H20850. The above-mentioned im-
portant role of V±/H20849q=0/H20850also gives an orientation in illustra-
tive modelling via the deduced, normalization and energetic
constraints.
Based on these remarks and previous experiences,10we
implement the high-density constraints using the following29
simple potential:
V±/H20849r/H20850=e2
re−/H9261±rcos/H20849/H9261±r/H20850. /H2084924/H20850
The required Fourier transform of this potential is
V±/H20849q/H20850=4/H9266e2q2
q4+4/H9261±4. /H2084925/H20850
We shall use in our problem the /H9261+=0.766 kFvalue which
gives10via Eq. /H2084911/H20850the constraining, 2 g/H208490,rs→0/H20850=1
−0.732 rs, exact23,24asymptotic form of the pair-correlation
function at the origin in the high density limit.
Notice, that at r=0 only the l=0 partial wave gives con-
tribution. With the above scaling for screening, we havecomputed the g/H208490,r
s/H20850function for different rsparameters.
The obtained results are exhibited in Fig. 1. A calculation
based on an approximation in ladder theory,30results of a
numerical solution of an effective Euler-Lagrangeequation,
31and the exact high-density expression are shown
in Fig. 1 as well. It is especially difficult to obtain an accu-rate value of g/H208490,r
s/H20850using quantum Monte Carlo methods
due to the absence of zero-variance property and arduous
numerics.32The ladder-based approximation gives the
2g/H208490,rs→0/H20850=1−0.663 rslimiting form. The agreement be-
tween the different results is quite reasonable.
After the above fixing of /H9261+, we have only the /H9261−as free
parameter but two constraints, Eqs. /H2084913/H20850to the norm and /H2084921/H20850
with/H9255kinB/H20849rs/H20850=0. Therefore, if these conditions could give
similar values for /H9261−an acceptable consistency will be
achieved. Remarkably, it turns out that this is indeed thecase. From the normalization constraint we get /H9261
−=1.58/H9261+
while from the /H9255B/H20849rs→0/H20850=0 energetic constraint we obtain a
/H9261−=1.70/H9261+value. Notice, that the similar scaling of a screen-
ing parameter with kFwas found earlier33by investigating
the density-density response function as a solution of thecorresponding Bethe-Salpeter equation within a Hartree-Fock-type theory.
In our parity-conserving approximation with a two para-
metric, i.e., minimal, model for effective interactions wehave/H9261
−/H11022/H9261 +. Therefore, in addition to the obvious effects
due to odd or even summations in l, we expect smaller
changes for /H9004g↑↑/H20849r/H20850than for /H9004g↑↓/H20849r/H20850under the actions of
V+/H20849r/H20850andV−/H20849r/H20850; these changes are determined mainly by the
short-range parts of effective interactions. The additive terms
to 2/H9004g/H20849r/H20850are given byMUIÑO, NAGY , AND ECHENIQUE PHYSICAL REVIEW B 72, 075117 /H208492005 /H20850
075117-4/H9004g↑↑/H20849r/H20850=2/H20855L−/H20849k,r/H20850/H20856 /H20849 26/H20850
for the parallel-spin component, and
/H9004g↑↓/H20849r/H20850=/H20855L−/H20849k,r/H20850/H20856+/H20855L+/H20849k,r/H20850/H20856 /H20849 27/H20850
for the antiparallel-spin component of the total change. The
L±/H20849k,r/H20850function to kaveraging, have the form of
L±/H20849k,r/H20850=/H20858
l±
/H208492l+1/H20850/H20851/H20841Rl±/H20849k,r/H20850/H208412−/H20841jl/H20849kr/H20850/H208412/H20852, /H2084928/H20850
where the /H11006refer to even /H20849/H11001/H20850and odd /H20849/H11002/H20850inlsummations,
respectively.
In Fig. 2 we plot our first-order /H9004g/H20849r/H20850/rsfunction, as a
function of the x=/H20849r/rs/H20850dimensionless variable and taking
the Born limit, i.e., rs→0. At small xvalues we have, by
construction, the exact limiting value, −0.732/2, for thisfunction. Note that our method is based on the normalizationof/H9004g/H20849r/H20850; its components are not constrained separately.
These components, from Eqs. /H2084926/H20850and /H2084927/H20850, are also exhib-
ited in Fig. 2 via dotted and dashed curves, respectively. Theinset shows the above curves around the zero value in anenhanced scale, in order to demonstrate the changes in sign.
As we mentioned earlier, the special double-perturbation
approach results in an everywhere nonpositive /H9004g/H20849r/H20850and anovercorrelation
24because the normalization condition is not
satisfied. This approach yields, with common potentials forboth the antiparallel and parallel cases, a very similar /H9004g
↑↓/H20849r/H20850
function for the x/H110211 range; see Fig. 1 of Ref. 24. The short-
range effect of Coulombic repulsion is, in both methods, thesame.
On the other hand, the corresponding /H9004g
↑↑/H20849r/H20850function is
quite different; see Fig. 2 of Ref. 24. It is zero, of course, at
x=0, but this function has a minimum value of −0.2 at about
x=1, and is nonpositive everywhere. The observed differ-
ence might be related to the not-normalized nature of theapproach and, partly, to the fact that in our case we havestronger screening in odd channels than in even channels.Clearly, in the present normalized method the deviation ofthe exchange-hole from the Hartree-Fock form is very mod-erate in first order of the true physical coupling.
Now we turn to the presentation of the obtained results for
kinetic energy changes which constitute the main motivationto the present work in the high-density, Born limit, in which/H9255
B/H20849rs→0/H20850=0 at first order of the coupling. The first-order
kF2ekin±/H20849k/H20850//H208492/H9266/H20850changes of Eq. /H2084923/H20850and their sum, via Eq.
FIG. 1. Pair-correlation function at the origin g/H208490,rs/H20850as a func-
tion of the electronic density parameter rs/H20849in atomic units /H20850. Results
obtained in this work are plotted with a solid line. The dotted linerefers to the ladder approximation and the dashed line to the exactresult in the high density limit. Solid triangles are the results of anumerical solution of an effective Euler-Lagrange equation.
FIG. 2. Difference in the pair-correlation function with respect
to the Hartree-Fock case /H9004g/H20849r/H20850=g/H20849r/H20850−g0/H20849r/H20850in the high-density per-
turbative limit, and as a function of the distance r. Both /H9004g/H20849r/H20850and
rare divided by rs. The solid line is the full /H9004g/H20849r/H20850, and the dashed
and dotted lines show the antiparallel /H9004g↑↓/H20849r/H20850and parallel /H9004g↑↑/H20849r/H20850
contributions to it, respectively. The inset provides a zoom of the/H9004g/H20849r/H20850/H110150 region, to remark the non-negative values of both the
parallel and antiparallel contributions to /H9004g/H20849r/H20850. All quantities in
atomic units.CALCULATION OF PAIR CORRELATIONS IN A HIGH- … PHYSICAL REVIEW B 72, 075117 /H208492005 /H20850
075117-5/H2084919/H20850, are plotted in Fig. 3, as a function of the dimensionless
u=k/kFvariable.
The formal similarity of Eq. /H208496/H20850and /H2084921/H20850suggest to use
thekF2ex/H20849k/H20850//H208492/H9266/H20850=−1/ /H208492u/H208502function as a reference, to un-
derstand in a quantitative way the total kinetic energy
change, kF2ekin/H20849k/H20850//H208492/H9266/H20850. At small relative momenta, this os-
cillating quantity tends to zero. This limit corresponds to the
“comoving” case, kinematically. The net energy change has amaximum value at about the biggest /H20849u=1/H20850relative mo-
menta, the “head-on” case, at which it almost cancels the
exchange contribution.
The solid curve crosses the zero value at intermediate, u
/H11229/H208491/2 /H20850, relative momenta. Here appears the nontrivial role
of oscillating potentials, via the parity weighting, as the com-parison of the dashed and dotted curves clearly shows. These
curves rest on the ± V
±/H208492k/H20850forms, because V±/H20849k=0/H20850=0 in the
present model. Notice that the parity weighting is prescribed
solely by the Pauli exclusion principle; we are not consider-ing a symmetry-broken state in our geminal approximation.Finally, in the formal u→/H11009limit one gets the
/H20841e
x/H20849k/H20850/ekin/H20849k/H20850/H20841=2 ratio.
III. SUMMARY
In conclusion, motivated by physically transparent ideas,
the feasibility of expressing the kinetic energy change of anelectron pair with given relative momentum, when the anti-symmetrization is imposed and the interaction is switchedon, has been investigated. For the high-density electron gas,where some exact results for strongly related characteristicsof the system are available, transparent and useful constraintson the effective pair potentials is deduced.
The parity-conserving implementation, via parametric
model potentials, gives a consistent physical picture on therearrangement driven by interparticle interactions. Beyondthis perturbative limit, further numerical implementations areneeded to establish a practical and well-constrained input tolocal functionals based on the concept of local density ofoccupied states.
Beyond the high-density Born approximation, a minimi-
zation of the scattering mediated energy change could be awell-motivated constraint for a variational treatment. Such atreatment may have
34superiority over conventional
coupling-constant integration with an approximate potentialenergy.
Notice, that a recently proposed extension
35of the
geminal-based idea for inhomogeneous, atomic systems restsalso on the minimization of the relative electron-electron ki-netic energy. A natural combination of results based on theseattempts may provide a deeper understanding of the fine mi-croscopical details of correlated electron motions, which areusually hidden in standard DFT.
ACKNOWLEDGMENTS
The authors are thankful for useful discussions with J.
Soler and J.I. Juaristi. The work of one of the authors /H20849I.N. /H20850
has been supported partly by the OTKA /H20849Grants Nos.
T046868 and T049571 /H20850. The authors acknowledge financial
support by the Basque Departamento de Educación, Univer-sidades e Investigación, the University of the Basque Coun-try UPV/EHU /H20849Grant No. 9/UPV 00206.215-13639/2001 /H20850,
and the Spanish Ministerio de Educación y Ciencia /H20849Grant
No. FIS2004-06490-C03-02 /H20850.
1W. Kohn, Rev. Mod. Phys. 71, 1253 /H208491998 /H20850.
2O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274
/H208491976 /H20850.
3J. M. Soler, Phys. Rev. B 69, 195101 /H208492004 /H20850.
4A. W. Overhauser, Can. J. Phys. 73, 683 /H208491995 /H20850.
5P. Gori-Giorgi and J. P. Perdew, Phys. Rev. B 64, 155102 /H208492001 /H20850.6B. Davoudi, M. Polini, R. Asgari, and M. P. Tosi, Phys. Rev. B
66, 075110 /H208492002 /H20850.
7I. Nagy, J. I. Juaristi, R. Díez Muiño, and P. M. Echenique, Phys.
Rev. B 67, 073102 /H208492003 /H20850.
8P. Ziesche, Phys. Rev. B 67, 233102 /H208492003 /H20850; Phys. Status Solidi
B241, 3544 /H208492004 /H20850.
FIG. 3. Kinetic-energy change ekin/H20849k,n0/H20850as a function of the
dimensionless quantity k/kF/H20849solid line /H20850. The odd channel ekin−/H20849k,n0/H20850
/H20849dotted line /H20850and even channel ekin+/H20849k,n0/H20850/H20849dashed line /H20850contributions
toekin/H20849k,n0/H20850are shown in the plot as well. The energies ekin/H20849k,n0/H20850
are multiplied by kF2//H208492/H9266/H20850. All quantities in atomic units.MUIÑO, NAGY , AND ECHENIQUE PHYSICAL REVIEW B 72, 075117 /H208492005 /H20850
075117-69B. Davoudi, R. Asgari, M. Polini, and M. P. Tosi, Phys. Rev. B
68, 155112 /H208492003 /H20850.
10I. Nagy, R. Díez Muiño, J. I. Juaristi, and P. M. Echenique, Phys.
Rev. B 69, 233105 /H208492004 /H20850.
11P. Ziesche and F. Tasnádi, Ann. Phys. 13, 232 /H208492004 /H20850.
12M. Corona, P. Gori-Giorgi, and J. P. Perdew, Phys. Rev. B 69,
045108 /H208492004 /H20850.
13K. A. Brueckner, Phys. Rev. 96, 508 /H208491954 /H20850.
14G. Csányi and T. A. Arias, Phys. Rev. B 61, 7348 /H208492000 /H20850.
15A. Beste and R. J. Bartlett, J. Chem. Phys. 120, 8395 /H208492004 /H20850.
16N. H. March, W. H. Young, and S. Sampanthar, The Many-Body
Problem in Quantum Mechanics /H20849Cambridge University Press,
Cambridge, 1967 /H20850.
17P. Fulde, Electron Correlation in Molecules and Solids /H20849Springer-
Verlag, Berlin, 1991 /H20850.
18G. D. Mahan, Many-Particle Physics /H20849Plenum, New York, 1981 /H20850.
19L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1
/H20849Butterworth-Heinemann, Oxford, 1980 /H20850.
20R. A. Ferrell, Phys. Rev. Lett. 1, 443 /H208491958 /H20850.
21N. H. March, Phys. Rev. 110, 604 /H208491958 /H20850.22C. J. Joachain, Quantum Collision Theory /H20849North-Holland, New
York, 1975 /H20850.
23J. C. Kimball, Phys. Rev. B 14, 2371 /H208491976 /H20850.
24V . A. Rassolov, J. A. Pople, and M. A. Ratner, Phys. Rev. B 59,
15625 /H208491999 /H20850.
25I. Nagy and A. Bergara, Nucl. Instrum. Methods Phys. Res. B
115,5 8 /H208491996 /H20850.
26C. J. Pethick and G. M. Carneiro, Phys. Rev. A 7, 304 /H208491973 /H20850.
27S. J. J. M. F. Kokkelmans, G. V . Shlyapnikov, and C. Salomon,
Phys. Rev. A 69, 031602 /H20849R/H20850/H208492004 /H20850.
28F. G. Fumi, Philos. Mag. 46, 1007 /H208491955 /H20850.
29I. Nagy and A. Bergara, J. Phys.: Condens. Matter 11, 3943
/H208491999 /H20850.
30H. Yasuhara, Solid State Commun. 11, 1481 /H208491972 /H20850.
31A. Kallio and J. Piilo, Phys. Rev. Lett. 77, 4237 /H208491996 /H20850.
32W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev.
Mod. Phys. 73,3 3 /H208492001 /H20850.
33G. E. Engel, Phys. Rev. Lett. 78, 3515 /H208491997 /H20850.
34A. W. Overhauser and J. T. Tsai, Phys. Rev. B 13, 607 /H208491976 /H20850.
35P. Gori-Giorgi and A. Savin, Phys. Rev. A 71, 032513 /H208492005 /H20850.CALCULATION OF PAIR CORRELATIONS IN A HIGH- … PHYSICAL REVIEW B 72, 075117 /H208492005 /H20850
075117-7 |
PhysRevB.101.035419.pdf | PHYSICAL REVIEW B 101, 035419 (2020)
Intrinsic resistance peaks in AB-stacked multilayer graphene with odd number of layers
Tomoaki Nakasuga,1Taiki Hirahara,1Kota Horii,1Ryoya Ebisuoka,1Shingo Tajima,1Kenji Watanabe,2
Takashi Taniguchi,2and Ryuta Yagi1,*
1Graduate School of Advanced Sciences of Matter (AdSM), Hiroshima University, Higashi-Hiroshima 739–8530, Japan
2National Institute for Materials Sciences (NIMS), Tsukuba 305-0044, Japan
(Received 30 May 2019; revised manuscript received 9 December 2019; published 21 January 2020)
We studied the band structure of AB-stacked multilayer graphene with odd numbers of layers by conducting
experiments to measure resistance ridge structures which were recently found to appear in the plot of theresistance with respect to a carrier density and a perpendicular electric flux density. The resistance ridgeswere found to exhibit qualitatively different structure depending on the parity of the number of layers, whichdetermines the presence or absence of a monolayerlike band. In the perpendicular electric field, pairs of nearlyflat bands (or heavy mass band) are formed at the bottoms of bilayerlike band because of the formation ofthe energy gap, and result in the split resistance ridge structures in the even numbers of layers. However, amonolayerlike band, which is present in AB-stacked graphene with odd numbers of layers, hybridizes with thebilayerlike bands; number of nearly flat bands, and thus, the number of resistance ridges, reduced as comparedwith the case of AB-stacked graphene with even numbers of layers. The mixing also opened an energy gap at thebottom of the monolayerlike band. The resistance ridge provides detailed information on the dispersion relationin multilayer graphene.
DOI: 10.1103/PhysRevB.101.035419
I. INTRODUCTION
Since the discovery of massless Dirac fermions in
graphene, a number of scientific investigations [ 1–3]h a v e
tried to elucidate their physical property and potential applica-bility. The electronic properties of graphene depend stronglyon the crystallographic structure. While monolayer graphenehas a single band with a massless dispersion relation [ 4–6],
bilayer graphene has a massive dispersion relation [ 4,7–10].
As the number of layers increases, many different stackingstructures become possible, each of which is expected to havea particular band structure. In particular, AB-stacked grapheneshows regularity in the evolution of its band structure. 2 N
(N: integer) layer graphene has Nbilayerlike band(s), and
2N+1 layer graphene has Nbilayerlike band(s) and a mono-
layerlike band, as shown in Fig. 1(a) [8,11–16]. Detailed
band calculations have suggested that the band structure ofmultilayer graphene is much more complicated than thoseshown in Fig. 1(a). The band structure of graphene in the
high-energy regime has been studied in optical spectroscopicexperiments [ 17–23]. Moreover, while the low-energy band
structure, which affects transport phenomena, has not beenfully revealed by optical measurements, the advent of tech-niques for making high-quality graphene samples has madeit possible to probe the low-energy band structure by usingShubnikov–de Haas oscillations [ 4,24–34].
Recently, high-quality multilayer graphene was found to
show intrinsic resistance peaks in its carrier density de-pendence [ 30,32,35]. Detailed measurements using graphene
samples with top- and bottom-gate electrodes have uncovered
*yagi@hiroshima-u.ac.jpintrinsic resistance ridges structure specific to the band struc-ture of AB-stacked four-layer [ 30,32] and six-layer graphene
[35]. These intrinsic resistance ridges are considered to be
a promising means of probing the band structure of two-dimensional materials. The ridges (or peaks) are related to thetopological changes in the Fermi surface [ 30]; the complicated
dispersion relations of multilayer graphene are tunable witha perpendicular electric field, and the resultant shape of theFermi surface (energy contour of the dispersion relations)varies depending on the chemical potential. The ridges inAB-stacked graphene with even number of layers grapheneare principally due to two reasons [ 32,35]. One is the nearly
flat band structure created at the bottoms of the bilayerlikeband by the perpendicular electric field; the field opens a bandgap at the bottoms of these bands, and makes them approxi-mately flat. This results in the heavy band mass [ 7,10,36–39].
Conspicuous split ridges appear on the map of resistivityas a function of carrier density and perpendicular electricflux density. The other reason is formation of mini-Diraccones [ 40], which are created by the perpendicular electric
field [ 30,32,35]. Trigonal warping locally closes the energy
gap created by the perpendicular electric field, and thereby,mini-Dirac cones are created. Mini-Dirac points appear assharp resistance ridges. In AB-stacked four-layer graphene,they appear at the charge-neutrality point [ 30,32], while in
AB-stacked six-layer graphene, they appear not only at thecharge-neutrality point but also at nonzero carrier densities[35]. In this paper we will examine the intrinsic resistance
peak structure of AB-stacked multilayer graphene with oddnumbers of layers. We found that the resistance ridges arequalitatively different from graphene with even number oflayers. We will show that this difference originates from thedispersion of the bilayerlike band which is hybridized with
2469-9950/2020/101(3)/035419(12) 035419-1 ©2020 American Physical SocietyTOMOAKI NAKASUGA et al. PHYSICAL REVIEW B 101, 035419 (2020)
(b)(a)
3L
(d)Energy
Wave
vector
BN
TGG
TG
BGSiO2hBN G(V)
10
-10
(V)1001000 (Ohm)
-50 50 04L 5L 6L 7L
(c)
FIG. 1. (a) Simplified dispersion relations for AB-stacked multilayer graphene. Graphene with an odd number of layers consists of bilayer
band(s) and a monolayer band. Graphene with an even number of layers consists of bilayer band(s). (b) Optical micrograph of encapsulated
graphene sample with top and bottom-gate electrodes (top). The bar is 10 μm. (c) Illustration of vertical structure of encapsulated graphene
in the effective sample area. G means graphene. TG means the top-gate electrode, and BG means the conducting Si substrate. (d) Back-gate
voltage ( Vb) dependence of resistivity of AB-stacked five-layer graphene sample for different top-gate voltages ( Vt).Vtwas varied between
−10 and 10 V in 2-V steps.
the monolayerlike band. The bottoms of the bilayerlike bands
form particular structure that is qualitatively different fromthat of the even numbers of layers.
II. EXPERIMENTAL
Our samples consisted of high-quality graphene flakes
encapsulated with thin flakes of h-BN, and equipped with
top- and bottom-gate electrodes, as shown schematically inFigs. 1(a) and 1(b). The graphene flakes were prepared by
mechanical exfoliation of high-quality Kish graphite crystals.Thin h-BN flakes were also prepared using a similar method.
A stack consisting of graphene layers and h-BN layers was
formed by using the transfer technique described in Ref. [ 41]
and Ref. [ 24]. The graphene sample was formed on a Si
substrate (covered with SiO
2) which was heavily doped and
remained conducting at low temperature. The substrate servedas the bottom gate electrode. The top gate was formed bytransferring graphene with a few layers onto the top of theencapsulated graphene. The resulting stack, consisting of thegraphene and h-BN flakes, was patterned into a Hall bar by
using reactive ion etching with a low-pressure mixture of CF
4
and O 2gas. Therefore, the top gate electrode and the effective
sample area had an exact geometry [Figs. 1(b) and 1(c)].
Electrical contact with graphene was attained by using theedge-contact technique reported in Ref. [ 41]. Electrical con-
tact with top gate electrode was carefully made at a pointwhere the graphene sample to be measured was not under-neath the h-BN, so as not to make a direct connection with the
graphene to be measured.
The number of layers and the stacking of the graphene
were verified by various methods. We identified their effecton the characteristic Landau-level structures [ 16], which are
specific to a particular number of layers and stackings (see theAppendix).
III. RESULTS AND DISCUSSION
A. AB-stacked five-layer graphene
AB-stacked five-layer graphene is a typical example of
odd-layer multilayer graphene with the multiple bilayerlikebands. It is expected to have two bilayerlike bands and amonolayer band [ 8,11–13,15,16,42,43]. Figure 1(d) shows
the back-gate voltage ( V
b) dependence of the resistivity for
different top-gate voltages ( Vt), which was measured at T=
4.2 K. The mobility ( μ) was calculated with the simple
formula μ=1/|ntoteρ|using data for the Vbdependence of
resistivity ( ρ) with Vt=0. It was about 1 .9×105cm2/Vs
in the electron regime, and 1 .1×105cm2/Vs in the hole
regime at large carrier densities. Here, ntotis the total carrier
density. It is clear that varying the top gate changed theoverall shape of the resistance traces with respect to V
b.T h e
data with Vt=0 show conspicuous double-peak structures
whose peak resistivities are approximately the same. Thesestructures would originate from the bottoms of the bilayerlikebands [ 30,32]. With increasing |V
tg|, the shapes of the traces
change into ones with a main peak and a small side peak. Theresistivity of the main peak increases with |V
tg|, until it satu-
rates and then slightly decreases for large |Vtg|. This behavior
is reminiscent of that of AB-stacked four-layer [ 30,32] and
six-layer graphene [ 35] and is strikingly different from the
behavior of bilayer [ 44–47] or AB-stacked trilayer graphene
[47–51]. Bilayer graphene shows insulating behavior as
the top-gate voltage increases, while trilayer graphene shows
035419-2INTRINSIC RESISTANCE PEAKS IN AB-STACKED … PHYSICAL REVIEW B 101, 035419 (2020)
02000(Ohm)
02000(a)
(b) (c)
(Ohm) (arb. units)
1
0
-1
FIG. 2. Top and bottom-gate voltage dependence of resistivity in AB-stacked five-layer graphene. (a) Map of resistivity as a function of Vt
andVb.T=4.2K . B=0 T. (b) Map of resistivity as a function of ntotandD⊥. (c) Similar map for dρ/dntot.
the opposite behavior; the resistivity of the peaks appearing
near the charge-neutrality point decreases with increasing top-gate voltage.
To investigate the above-mentioned properties further, we
measured the top- and bottom-gate voltage dependence of theresistivity in detail. The results are summarized in the mapof resistivity with respect to V
tandVb, as shown in Fig 2(a).
Resistivity peaks appear as ridges. Salient resistance ridgeson a linear line from the upper left to lower right satisfy thecondition of charge neutrality, which corresponds to the largepeaks in Fig 2(d) for|V
t|>0. In addition, side peaks which
are parabolic in shape are discernible in the figure. Becausethe peak structure would result from a variation in the disper-sion relation arising from the perpendicular electric field asin AB-stacked four- and six-layer graphene [ 30,32,35], we re-
plotted the map as a function of total carrier density ( n
tot) and
electric flux density ( D⊥) perpendicular to the graphene. The
total carrier density can be calculated by summing the carrierdensities induced by the top and bottom-gate voltages as
n
tot=[Ct(Vt−Vt0)+Cb(Vb−Vb0)]/(e). (1)
Here, CtandCbare the specific capacitances of the top and
bottom-gate electrodes, respectively. Vt0andVb0represent the
shift in gate voltage due to carrier doping associated with thetop and bottom-gate electrodes. The effect of the perpendicu-lar electric field can be estimated using electric flux density in-duced by the top and bottom-gate voltages, which is given by
D
⊥=[Ct(Vt−Vt0)–Cb(Vb−Vb0)]/2. (2)
From the charge-neutrality condition in Fig. 2(a), one
can estimate the ratio of the capacitances ( =Vt/Vb)t ob e
about 3.8. The specific capacitances were calculated fromthe Landau-level structure measured under the condition,D
⊥=0, to be Ct=395aF/μm2and Cb=104aF/μm2.Figures 2(b) and 2(c) show maps of ρand dρ/dntotas a
function of ntotandD⊥. In these figures, parabolic ridges are
discernible for both the electron and hole regimes. Similarbut more complicated parabolic ridge structures were alsoobserved in AB-stacked 4 [ 30,32] and six-layer graphene
[33,35].
The dispersion relations in the absence and presence of
perpendicular electric fields were numerically calculated toexamine the relation between the band structure and the resis-tance ridge structure in the five-layer graphene. The calcula-tion was based on the effective mass approximation and theSlonczewski-Weiss-McClure (SWMcC) parameters [ 52–54]
of graphite were used. Screening of induced carriers wastaken into account by using the distribution of induced carriersin graphene. We assumed that carriers in each layer decayexponentially with a decay length of λ, which is roughly con-
sistent with the results of the Thomas-Fermi approximation[55]. In the calculation we took λ=0.45 nm, approximately
the same value expected from a self-consistent calculation ofthe screening length [ 56], and approximately the same as the
experimental value obtained from Landau-level structures inmultilayer graphene [ 16]. Figure 3(a) shows the dispersion re-
lations of the AB-stacked five-layer graphene numerically cal-culated for different values of |D
⊥|. The dispersion relations
for AB-stacked four-layer graphene are shown in Fig. 3(b)
for comparison. For the five-layer graphene, there are twosets of bilayerlike band and a monolayerlike band, which arecomplicatedly hybridized near E=0[34] [more complicated
than what is shown in Fig. 1(a)]. Applying a perpendicular
electric field opens energy gaps., i.e., differences in energybetween the bottoms of the bands. The gaps increase withincreasing |D
⊥|; thereby, the dispersion relations look rather
simplified. For convenience, we labeled the band as αe−γh,
and the bottoms of the bands a, b, c,b/prime, and a/prime. Bands γeand
γhin the five-layer graphene are monolayerlike bands. The
035419-3TOMOAKI NAKASUGA et al. PHYSICAL REVIEW B 101, 035419 (2020)
FIG. 3. Band structure of multilayer graphene in perpendicular electric field. (a) Dispersion relations in AB-stacked five-layer graphene.
D⊥was varied from 0 to 0 .802×10−7and 4.81×10−7cm−2As. (b) Similar results for AB-stacked four-layer graphene. Bands are labeled αe,
βe,γe,αh,βh,a n dγh. The characteristic points in the bands are labeled a−canda/prime-b/prime. The right inset shows the definition of the Slonczewski-
Weiss-McClure (SWMcC) parameters. The SWMcC parameters of graphite were used for the calculations ( γ0=3.19 eV, γ1=0.39 eV,
γ2=− 0.02 eV, γ3=0.3e V ,γ4=0.044 eV, γ5=0.038 eV, and /Delta1p=0.037 eV).
remaining bands are principally bilayerlike bands, as in the
AB-stacked four-layer graphene, but they differ significantlybetween the four- and five-layer cases. In particular, theenergy gap between α
eandβeand the one between αhandβh
are significantly larger in the four-layer graphene than in the
five-layer graphene. It can be seen that the structures of bandα
eandαhin the five-layer graphene are more complicated
than those in the four-layer graphene; this difference wouldoriginate from the hybridization with the monolayerlike bandin the five-layer graphene.
The difference in the band structure results in particular
resistance ridge structures. The characteristic band positionsare closely related to the resistance ridges. We have calculatedthe semiclassical resistivity based on the Boltzmann equa-tion with the constant relaxation-time approximation. (Thecalculation is similar to the one performed on AB-stackedfour-layer graphene [ 30]). We took into account possible
energy broadening due to scattering. Figure 4(a) compares
the experimental and calculated maps of dρ/dn
totplotted as a
function of ntotandD⊥. It can be seen that the calculations ap-
proximately reproduced the experimental result. Conspicuousridge structures are labeled with the characteristic positions ofthe band structure in Fig. 3(a). Ridge cstems from the mini-
Dirac cones formed at the charge-neutrality point. Ridges b
andb
/primeare for the bottoms of the bilayerlike bands βeandβh.
As for positions aanda/prime, which correspond to the bottoms
of the monolayerlike bands, structures hardly appeared in theexperimental results, possibly because the variation in theconductivity was rather smaller than at the other characteristicpositions in the bands. As shown in Fig. 4(b), the structures
are barely visible in the simulation with reduced energybroadening.
Now let us discuss the differences between the five-layer
and four-layer cases. The resistance ridges of the five-layergraphene, which are parabolic in shape, are qualitativelydifferent from those of the four-layer graphene. The ridges inthe four-layer case show clear splitting with increasing |D
⊥|
[30,32]. This is due to formation of an energy gap between the
bilayerlike bands, as shown in Fig. 3(b). On the other hand, in
the five-layer graphene, the bottom of βealmost touches αe,
and the bottom of βhhas approximately the same energy as the
local bottom of αh.This qualitatively different band structure
results in the five-layer graphene not having any split ridgestructures for the bilayerlike bands.
Although the four-layer and five-layer graphene have sig-
nificantly different electronic band structures, they have sim-ilar resistance ridges that appear at n
tot=0f o r|D⊥|above
∼0.5×10−7cm−2As. In both cases, this is because the ridge
originates from the formation of mini-Dirac cones [ 30,35,40]
near E=0f o rl a r g e |D⊥|, as can be seen in Figs. 3(a) and
3(b). In the five-layer case, three sets of mini-Dirac cones
are created at different wave numbers in kspace. Among
them, the two located at kx/negationslash=0[ s e eF i g . 3(a)] arise from
the bilayerlike band because of trigonal warping. They areboth threefold degenerate in a valley ( KorK
/prime). The other
set of mini-Dirac cones, which are located at kx=0, appar-
ently originate from the monolayerlike band. The mini-Diraccone structure in the four-layer case is strikingly different.
035419-4INTRINSIC RESISTANCE PEAKS IN AB-STACKED … PHYSICAL REVIEW B 101, 035419 (2020)
-55
-5
5 -55
-5
5bc
bc b’ b’
bc b’a’ a
-55
-5
5(arb. units)
1
0
-1
(arb. units)
1
0
-1
(a)
(b)
FIG. 4. Resistance ridges and characteristic band points in AB-stacked five-layer graphene. (a) Map of dρ/dntotas a function of ntotand
D⊥.The left panel shows results from the experiment, and the right panel is the numerical calculation with /Gamma1=3 meV . Resistance ridges b,
c,andb/primecorrespond to the positions in the dispersion relation in Fig. 3(a). The areas surrounded by the red lines indicate the measured area in
the experiment. (b) Similar plot for numerical simulation with /Gamma1=1 meV. Resistance ridges originating from the monolayerlike band ( aand
a/prime) are discernible at large values of ntot.
There are large mini-Dirac cones (in positive kx) and small
mini-Dirac conelike structures (in negative kx) which have
gaps. The cones and the small conelike structure are boththreefold degenerate in the KandK
/primevalley. In the both the
four- and five-layer cases, perpendicular electric field resultedin complicated massive bands changing into linear bands nearthe charge-neutrality point, and thereby, the resistance ridgesnear the n
tot=0 appeared.
The simulation with reduced energy broadening reveals
the resistance ridges associated with the monolayer band forthe bottoms of the monolayer bands γ
eandγh[Fig. 4(b)].
However, they are hardly visible in the experimental data. Insufficiently large perpendicular electric fields, energy gaps arecreated for the monolayerlike band because of hybridizationwith bilayerlike bands [Fig. 3(a)]. Although the monolayerlike
band has a nonzero band mass near the bottoms of the bands,the mass is much smaller than those for the bottoms of thebilayerlike bands. This would make the resistance ridges forthe bottoms of γ
eandγhhard.B. AB-stacked seven-layer graphene
AB-stacked seven-layer graphene, which has three sets
of bilayerlike bands and a monolayerlike band, also showscharacteristic resistance ridges for odd numbers of layers. Westudied the intrinsic resistance peaks of the seven-layer samplethat had a similar structure to that of the five-layer sample. Themobility at a large carrier density was μ=6.9×10
4cm2/Vs
in the electron regime and 5 .0×104cm2/Vs in the hole
regime. Figure 5(a) shows a map of resistivity as a function
ofVbandVt, which was measured at T=4.2 K. The ratio
of the specific capacitance was estimated to be Ct/Cb=3.98.
Ct=446aF/μm2andCb=112aF/μm2. Figure 5(b) is a
replot as a function of ntotandD⊥. The resistance ridges are
distinct from those of the four-layer [ 30,33], five-layer, and
the six-layer cases [ 32,35].
Although the six-and the seven-layer graphene have more
complicated band structures than those of four- and five-layer graphene, they show characteristic differences in theband structure reflecting the even-odd layer number effect.
400
0400
0
FIG. 5. Intrinsic resistance ridges for AB-stacked seven-layer graphene (a) Map of ρas a function of VbandVt.T=4.2K . B=0T .
(b) Replot as a function of ntotandD⊥.
035419-5TOMOAKI NAKASUGA et al. PHYSICAL REVIEW B 101, 035419 (2020)
FIG. 6. Band structure of multilayer graphene in perpendicular electric field. (a) Dispersion relations of AB-stacked seven-layer graphene.
From left to right, D⊥was varied from 0 to 0 .802×10−7and 4.81×10−7cm−2As . (b) Dispersion relations of AB-stacked six-layer graphene.
Bands are labeled αe,βe,γe,δe,αh,βh,γh,a n dδe. The characteristic points in the bands are labeled with a–danda/prime-c/prime.
Figure 6(a) shows the numerically calculated dispersion re-
lation of the seven-layer graphene for some values of |D⊥|,
while Fig. 6(b) shows those for the six-layer case for compar-
ison. The SWMcC parameters of graphite, and λ=0.45 nm
were used in the calculation. Bands are labeled αe,βe,γe,δe,
αh,βh,γh, andδh. Characteristic points in the band diagram
are labeled a–danda/prime–c/prime. The dispersions for the seven-layer
graphene under a perpendicular electric field are much morecomplicated than those of the six-layer graphene because ofhybridization of the bilayerlike bands with a monolayerlikeband, as was seen earlier for the cases of the four- and five-layer graphene. In the six-layer graphene, the application ofa perpendicular electric field opens energy gaps between thebilayerlike bands, and the dispersion relations are nearly flatnear the bottoms of each band. On the other hand, no suchflat dispersion relations form in the seven-layer graphene. Thebottoms γ
eandβe(γhandβh) nearly make contact with the
small energy gaps. The structures apparently originate fromhybridization with the monolayerlike band. On the other hand,for large |E|, one can see that bands δ
eandδh, which originate
from the monolayerlike band, have rather simple shapes.
To see the correspondence of the intrinsic resistance ridges
to the dispersion relations, we compared the experimentalresults with the numerically calculated resistivities (Fig. 7).
It is clear that the theoretical results approximately explainthe experimental results. Resistance ridges appear at the corre-sponding positions in the band structures. First, let us examinethe ridges appearing in the vicinity of n
tot=0.One can recog-
nize the resistance ridge near the charge-neutrality conditionas in the four-, five-, and six-layer graphene. Comparing theexperimental results with those of the band calculation, it can
be seen that mini-Dirac cones are created at points din the
vicinity of the charge-neutrality point for large |D
⊥|.I nt h e
AB-stacked seven-layer graphene, the dispersion relations at|D
⊥|=0 show a semimetallic band structure; the electron
and hole bands overlap near E=0. Applying a perpendicular
electric field created mini-Dirac cones, from which conspicu-ous ridges formed.
Next, we turn to the other resistance ridges. The bottoms
of the bilayer bands bandb
/prime[Fig. 6(a)] appear as resistance
ridges in Fig. 7. Apparently, there are no split ridge structures
arising from the bottoms of bilayerlike bands, as in the five-layer case. Unlike the five-layer case, conspicuous arisingmini-Dirac cones [indicated by candc
/primein Fig. 6(a)] are visible
as in the six-layer case [ 35], at carrier densities different from
charge neutrality. In addition, the experimental results do haveclear ridge structures for the monolayerlike band aanda
/prime,a s
in the five-layer case; the lack should again be due to relativelysmall carrier density and light band mass.
IV . DISCUSSION
First, we address the evolution of the resistance ridge
structure in AB-stacked multilayer graphene with increas-ing number of layers. The numerically calculated resistanceridge structures for four to seven layers are summarized inFig. 8. (The calculation for the four-layer case is reported in
Ref. [ 30].) It is clear that the resistance ridges (peaks) appear
at different positions in the diagram: The ridge structures havea specific pattern depending on the number of layers. One can
035419-6INTRINSIC RESISTANCE PEAKS IN AB-STACKED … PHYSICAL REVIEW B 101, 035419 (2020)
5-55
-5 5-55
-5(arb. units)
1
0
-1
(arb. units)
1
0
-1
Experiment Calculation
FIG. 7. Resistance ridges and characteristic band points in AB-stacked seven-layer graphene. Map of dρ/dntotas a function of ntotandD⊥.
The left panel shows experimental results, and the right panel is a calculation with energy broadening /Gamma1=3m e V . b, c, d,b/prime,a n d c/primecorrespond
to the positions in the dispersion relation. The areas surrounded by the red lines indicate the measured area in the experiment.
thus determine the number of layers and stacking by using the
diagram. Resistance ridges due to bilayer bands show splittingin AB-stacked graphene with even numbers of layers, whilethe splitting is absent from AB-stacked graphene with oddnumbers of layers. In addition, the resistance ridges due tothe monolayerlike band in the graphene with the odd numbersof layers are rather small.
On the other hand, the mini-Dirac points form relatively
strong peaks compared with the bottoms of the bands. Forexample, ridge structures at n
tot=0 appear regardless of the
number of layers. The six- and seven-layer graphene showrelatively strong peaks at the mini-Dirac points (MDP) at
nonzero carrier densities.
On the ridges formed at ntot=0, the resistivity tends to
increase with increasing |D⊥|, but it saturates (and slightly
decreases in some cases) at large |D⊥|. The early graphene
research reported that bi- and trilayer graphene had differentresponses to a perpendicular electric field: Bilayer graphenebecomes insulating because the energy gap opens [ 45,46,57],
while trilayer graphene becomes more metallic [ 48–50,57]
(i.e., its resistivity decreases). However, this sort of behaviordoes not persist in graphene consisting of more layers, as
4L 5L 6L
b b b bb
mb bm b bbMDPMDP MDP MDP MDP7L
mmMDPMDP MDPb b(arb. units)
(arb. units)5
-5
5
-5
5 -55 -5 5 -5
5 -55 -5
5 -5 5 -55 -51
0
-1
01
FIG. 8. Evolution of resistance ridge structure in AB-stacked multilayer graphene. Numerically calculated maps of dρ/dntot(upper panels)
andρ(lower panels) are plotted against ntotandD⊥. From left to right, the number of layers are 4, 5, 6, and 7. bstands for the ridge structure due
to bilayerlike bands, and mstands for that due to monolayerlike bands. MDP stands for the resistance ridge structure arising from mini-Dirac
points. /Gamma1=1 meV, and the SWMcC parameter of graphite was used for these calculations.
035419-7TOMOAKI NAKASUGA et al. PHYSICAL REVIEW B 101, 035419 (2020)
(a)
(b)
(c)
FIG. 9. Dispersion relation and conductance. Schematic drawings of ntotdependence of electrical conductance ( σ) for a parabolic band (a),
a nearly flat band (b), and a Dirac cone (c). The left panels are cases of a single band, and the right panels are cases with another band with a
larger carrier density.
shown in previous work [ 30,32,35] and this study. The be-
havior is consistent with the formation of mini-Dirac cones inthe vicinity of the charge-neutrality point. AB-stacked five- toseven-layer graphene (and possibly the four-layer graphene)are semimetallic near the charge-neutrality point in the ab-sence of a perpendicular electric field. Electrons and holesare compensated, so that there would be considerably manycarriers that contribute to the conductance. The formation ofmini-Dirac cones tends to decrease the number of carriers.The absence of insulating behavior can be understood fromthe minimum conductivity of monolayer graphene at thecharge-neutrality point. Theory predicts a minimum conduc-tivity of about e
2/¯hat the Dirac point [ 4,6,58]. Although
a Dirac point is difficult to realize in an actual experimentbecause of inhomogeneity [ 59–62], many experiments have
shown that there is a minimum conductivity, whose value isnot universal.
Next we describe intuitive understanding of the resistance
ridges. Figures 9(a)–9(c) show n
totdependence of conduc-
tance ( σ) for different dispersion relations. Let us consider
the Drude conductivity, σ=ne2τ/m, where, mis a band mass
andτis a relaxation time which we assume to be constant for
simplicity. First, we examine a case of a single band. For aband with parabolic dispersion relation (i.e., mis a constant),
σincreases with n
totmonotonically if energy Eis swept from
E1toE2[the left panel of Fig. 9(a)]; no conspicuous ridge
structure appears as expected. If there is a nearly flat bandat the bottom as shown in the left panel of Fig. 9(b),mnear
the bottom is much heavier than higher energies. dσ/dn
totis smaller near ntot=0 than larger values of ntot,a n dak i n k
structure would appear at the carrier density where the nearlyflat band is filled out. Although the Drude formula of σis
invalid because of m=0 in case of Dirac cone, σis still
given by σ=neμ.I fμdoes not vary largely [the left panel
of Fig. 9(c)], a V-shaped structure will appear in the n
tot
dependence of σ, as one can often see in monolayer graphene.
If there is an extra band which does not have large variationsin its mass within the region between E=E
1andE2,ntot
dependence of σdoes not change qualitatively as shown in
the right panels in Figs. 9(a)–9(c). The variation of σfor
mini-Dirac cones is sharper than that for the nearly flat band.This is because the nearly flat band requires much largercarrier density to fill out than the carrier density to pass theDirac point. Ridge structure in Fig. 8would be qualitatively
explained by combining these simple patterns according to theband structure.
Finally, we comment on the method of probing the band
structure from the resistance ridges. As in cases of the Ra-man G
/primeband spectra shape [ 16,63–66] and the Landau fan
diagrams [ 4,24–34], the resistance ridge structure can be used
to identify its number of layers and stacking, by comparingwith the ridge structures for known number of layers andstacking. For materials with unknown band structure, onemight also estimate the number of bands and formation ofenergy gap via perpendicular electric field from the resistanceridge structures. However, to obtain more information, oneneeds to compare the experimental resistance ridge structureswith the results from band calculations. As has been described
035419-8INTRINSIC RESISTANCE PEAKS IN AB-STACKED … PHYSICAL REVIEW B 101, 035419 (2020)
in this paper, small structures in the dispersion relation, but
important for transport phenomena, can be clearly detected.Moreover, band parameters (SWMcC parameters in the caseof graphene), and carrier screening length λcan be estimated
from the ridge if it is compared with the band calculation. Fortwo-dimensional materials other than graphene, this kind ofparameter can possibly be determined.
The resistance ridge has advantages over the Shubnikov–de
Haas effect: the ridges directly reflect the dispersion relationswhile, with the S–dH effect, one can detect electronic states inmagnetic field (i.e., Landau levels), which are totally differentfrom those at the zero magnetic field. The resistance ridgecould be used for a method to detect low-energy dispersionrelations in various two-dimensional materials.
V . SUMMARY AND CONCLUDING REMARKS
Intrinsic resistance ridge structures of AB-stacked five- and
seven- layer graphene, which appear as a function of carrierdensity and perpendicular electric field, were studied togetherwith the band structure by using an encapsulated graphenedevice equipped with top and bottom-gate electrodes. Wefound that the intrinsic resistance peaks (ridges) in multilayergraphene with an odd number of layers are strikingly differentfrom the graphene with an even number of layers: Onlygraphene with an even number of layers show split ridges dueto the formation of nearly flat bands. This difference resultsfrom hybridization of the bilayerlike band with the mono-layerlike band in the graphene with odd number of layers.Thus, these results show that the resistance ridges can beused to probe the electronic band structure of two-dimensional
materials.
ACKNOWLEDGMENT
This work was supported by KAHENHI Grant No.
25107003 from MEXT Japan.
APPENDIX
1. Determination of number of layers and stacking
The number of layers and their stacking were determined
by combined use of atomic force microscopy (AFM) andRaman spectroscopy. In particular, the number of layers andstacking were determined after calibrating the relation be-tween the Raman spectral shape and the number of layersof graphene determined by AFM. The spectral shape ofthe ABA stacking showed a systematic evolution [ 16,63–66]
that was considerably different from that of ABC stacking[63,65–69]. The details are described in Ref. [ 16]. We also
used the Landau-level structures which can be deduced fromthe Shubnikov–de Haas oscillations in the low-temperaturemagnetoresistance. The Landau-level structures reflect theelectronic band structure of graphene directly, meaning that itis one of the most reliable methods to determine the number oflayers and stacking. A map of magnetoresistance with respectto the carrier density and magnetic field (Landau fan diagram)reveals graphene’s detailed low-energy band structure that isspecific to the number of layers and stacking. The number oflayers and stacking of the measured samples were verified by
07
-5 5 -5 5
-5 51814 -14 -22
07
07B(T)
B(T) B(T)10000
1000
100
10(Ohm)
0101000
100
(arb. units)(a) (b)
(c)
DOS
(arb. units )
01-2
-2 18 -22
07 B(T)14 18 -2
-50 50
E(meV )(d)
K
K’
14
FIG. 10. Landau-level structure in AB-stacked five-layer graphene. (a) Map of longitudinal resistivity ρxxin AB-stacked five-layer
graphene. D⊥=0c m−2As.T=4.2 K. Numbers show filling factors for some energy gaps. (b) Map of dρxx/dntot. Red bars indicate
Landau levels for the monolayerlike band, which appear as a beating of the magnetoresistance oscillations. (c) Numerically calculated energy
eigenvalues for AB-stacked five-layer graphene. Red and black lines show data for KandK/primepoints, respectively. The SWMcC parameters of
graphite were used for this calculation. (d) Map of numerically calculated density of states (DOS).
035419-9TOMOAKI NAKASUGA et al. PHYSICAL REVIEW B 101, 035419 (2020)
-6
-5 507-5 507-6 (a) (b)
(c) (d)
-5 507
0101000
100
(arb. Units)
DOS
(arb. units )
01
07
-50 50E(meV)K
K’
(Ohm)
1000
100
10
1
-6
FIG. 11. Landau-level structure in AB-stacked seven-layer graphene (a) Map of longitudinal resistivity of AB-stacked seven-layer
graphene. D⊥=0c m−2As.T=4.2 K. Numbers show filling factors for some energy gaps. (b) Map of dρxx/dntot. (c) Numerically calculated
energy eigenvalues. Red and black lines show data for KandK/primepoints, respectively. The SWMcC parameters of graphite were used. (d) Map
of numerically calculated DOS.
referring a list of fan diagrams for AB-stacked graphene with
known numbers of layers [ 16].
2. Landau-level structure in AB-stacked 5-layer graphene
The AB-stacked five-layer graphene sample showed
Shubnikov–de Haas oscillations in the magnetoresistancewhich was measured at T=4.2 K. Figures 10(a) and10(b)
show maps of the longitudinal resistivity ( ρ
xx) and its deriva-
tive with respect to the magnetic field ( dρxx/dB), plotted
as a function of magnetic field Band carrier density ntot.
Here, ntotwas varied by controlling the top and bottom-gate
voltages so as to satisfy the condition D⊥=0. The stripes
are Landau levels for particular bands with particular Landauindices. The observed Landau-level structure near the charge-neutrality point is approximately the same as that in theprevious report for AB-stacked five-layer graphene, whichwas measured from a sample with a single gate electrode[16]. This confirms that our sample was identified as AB-
stacked five-layer graphene, because Landau-level structure isthe fingerprint of the electronic band structure of graphene.
The overall structure of the Landau levels can be ap-
proximately explained by a numerical calculation based onthe effective mass approximation. Figure 10(c) shows energy
eigenvalues calculated for the Slonczewski-Weiss-McClureparameters which are approximately the same as those ofgraphite, and Fig. 10(d) is the calculated density of states.
Although refining the SWMcC parameters would give a betterfitting to the experiment, energy gaps with ν=− 2,14,18 are
clearly visible in the experimental data. The filling factor forthe gaps satisfies the relation 4( N+1/2) with integer N,a sin the monolayer graphene. In addition, the Landau levels for
the monolayerlike band are visible in Fig. 8(b) (indicated by
the red bars).
The energy gap with ν=− 2 characterizes the AB-stacked
five-layer graphene. It occurs near the charge-neutrality pointabove a few tesla and appears between the zero-mode Landaulevels; no Landau-level crossings occur for the larger mag-netic field. Similar characteristic energy-gap structures appearin AB-stacked multilayer graphene with more layers, and onecan identify the number of layers by using the filling factor ofthe gap. The gap occurs at ν=0 in the case of AB-stacked
graphene [ 30,32,34], while it appears at ν=4 in AB-stacked
six layer [ 35]. To be shown later, in the seven layer, it appears
atν=6.
3. Landau-level structure in AB-stacked seven-layer graphene
Figures 11(a) and11(b) show maps of RxxanddRxx/dBas
a function of ntotandB. Highly complicated beatings of the
Shubnikov–de Haas oscillations can be seen. The energy gapsand the Landau-level crossing near the charge-neutrality pointapproximately reproduce the fan diagram measured for single-gated graphene samples [ 16], which confirms that our sample
is the AB-stacked seven-layer graphene. Conspicuous energygaps appear at ν=− 6. Figure 11(c) shows the numerically
calculated Landau-level spectra for the SWMcC parametersof graphite, while Fig. 11(d) is a map of the corresponding
density of states. The calculation approximately accountsfor the overall Landau-level structures and positions of theconspicuous energy gaps. In particular, the energy gap at
035419-10INTRINSIC RESISTANCE PEAKS IN AB-STACKED … PHYSICAL REVIEW B 101, 035419 (2020)
ν=− 6 is visible between the zero-mode Landau levels of
the bilayerlike band.
4. Calculation of dispersion relation and Landau levels
The dispersion relations at zero magnetic field were cal-
culated using the Hamiltonian for the effective mass approxi-mation which is based on the tight-binding model [ 5,8,34,42].
Landau levels were numerically calculated by expanding thewave functions with Landau functions [ 14,34,70–72] and
evaluating the eigenvalues of the Hamiltonian. The density ofstates was calculated by assuming that each Landau level hada carrier density of degeneracy multiple eB/h[34].
The electrostatic potential due to the perpendicular electric
field was calculated by taking the screening of each layer intoaccount. Multilayer graphene is atomically thin, as are othertwo-dimensional materials, so that an externally applied per-pendicular electric field is expected to penetrate the graphenebut to be shielded layer by layer [ 16,22,55,56,73–78]. The
internal electric field significantly changes the electrostaticpotential for each layer in the graphene and affects the bandstructure [ 16,56]. Here, we used the same method as in
Ref. [ 16], where it was assumed that the external electric
field diminishes exponentially with the screening length λ,
which is a fitting parameter to be experimentally determined.We estimated it to be about 0.43 in our previous work on
the Landau-level structure in AB-stacked multilayer graphenein which we measured samples with a single gate electrode[16,35]. The resistance ridges observed in the present experi-
ment were best explained for λ∼0.45 nm. Here, we assumed
the dielectric constant in the graphene to be ε/ε
0=2.0.
5. Calculation of conductivity at zero magnetic field
The Drude conductivity was calculated by using the nu-
merically calculated dispersion relations. The resistivity wasthen determined by taking the reciprocal of the conductiv-ity. A constant relaxation time was assumed. For a smallelectric field E
xapplied in the x-direction, the solution of
the Boltzmann equations is simply approximated at low tem-perature by shifting the wave number ( k) of all the existing
electrons by −eE
xτ. Thus, the conductivity is proportional to
the sum of the group velocities for all of the filled electronicstates. To make a comparison with the experiment, we tookenergy broadening of the distribution function into account;this would possibly arise for various reasons, e.g., scattering,inhomogeneity, etc. We assumed that the derivative of thedistribution function with respect to energy is simply a Gaussfunction with a standard deviation, /Gamma1/√
2. The details of the
distribution function would not change the important featureof the simulation.
[1] A. K. Geim and K. S. Novoselov, Nat. Mater. 6,183(2007 ).
[2] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod.
Phys. 83,407(2011 ).
[3] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,
and A. K. Geim, Rev. Mod. Phys. 81,109(2009 ).
[4] K. S. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, M. I.
Katsnelson, I. V . Grigorieva, S. V . Dubonos, and A. A. Firsov,Nature (London) 438,197(2005 ).
[5] P. R. Wallace, Phys. Rev. 71,622(1947 ).
[6] N. H. Shon and T. Ando, J. Phys. Soc. Jpn. 67,2421 (1998 ).
[7] E. McCann, D. S. L. Abergel, and V . I. Fal’ko, Solid State
Commun .143,110(2007 ).
[8] B. Partoens and F. M. Peeters, P h y s .R e v .B 75,193402 (2007 ).
[9] M. Koshino and T. Ando, P h y s .R e v .B 73,245403 (2006 ).
[10] E. McCann, P h y s .R e v .B 74,161403(R) (2006 ).
[11] M. Koshino and T. Ando, P h y s .R e v .B 76,085425 (2007 ).
[12] M. Koshino and T. Ando, Solid State Commun .149,1123
(2009 ).
[13] M. Nakamura and L. Hirasawa, Phys. Rev. B 77,045429
(2008 ).
[14] H. K. Min and A. H. MacDonald, Phys. Rev. B 77,155416
(2008 ).
[15] S. Latil and L. Henrard, P h y s .R e v .L e t t . 97,036803 (2006 ).
[16] R. Yagi, T. Hirahara, R. Ebisuoka, T. Nakasuga, S. Tajima, K.
Wtatanabe, and T. Taniguchi, Sci. Rep. 8,13018 (2018 ).
[17] M. Sprinkle, D. Siegel, Y . Hu, J. Hicks, A. Tejeda, A. Taleb-
Ibrahimi, P. LeFevre, F. Bertran, S. Vizzini, H. Enriquez, S.Chiang, P. Soukiassian, C. Berger, W. A. deHeer, A. Lanzara,and E. H. Conrad, P h y s .R e v .L e t t . 103,226803 (2009 ).
[18] A. B. Kuzmenko, I. Crassee, D. van Der Marel, P. Blake, and
K. S. Novoselov, Phys. Rev. B 80,165406 (2009 ).[19] K. F. Mak, C. H. Lui, J. Shan, and T. F. Heinz, Phys. Rev. Lett.
102,256405 (2009 ).
[20] L. M. Zhang, Z. Q. Li, D. N. Basov, M. M. Fogler, Z. Hao, and
M. C. Martin, P h y s .R e v .B 78,235408 (2008 ).
[21] T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg,
Science 313,951(2006 ).
[22] T. Ohta, A. Bostwick, J. L. McChesney, T. Seyller, K. Horn,
and E. Rotenberg, Phys. Rev. Lett. 98,206802 (2007 ).
[23] C. Coletti, S. Forti, A. Principi, K. V . Emtsev, A. A. Zakharov,
K. M. Daniels, B. K. Daas, M. V . S. Chandrashekhar, T. Ouisse,and D. Chaussende, A. H. MacDonald, M. Polini, and U. Starke,Phys. Rev. B 88,155439 (2013 ).
[24] T. Taychatanapat, K. Watanabe, T. Taniguchi, and P. Jarillo-
Herrero, Nat. Phys. 7,621(2011
).
[25] L. C. Campos, T. Taychatanapat, M. Serbyn, K. Surakitbovorn,
K. Watanabe, T. Taniguchi, D. A. Abanin, and P. Jarillo-Herrero, P h y s .R e v .L e t t . 117,066601 (2016 ).
[26] Y . Asakawa, S. Masubuchi, N. Inoue, S. Morikawa, K.
Watanabe, T. Taniguchi, and T. Machida, P h y s .R e v .L e t t . 119,
186802 (2017 ).
[27] A. Kumar, W. Escoffier, J. M. Poumirol, C. Faugeras, D. P.
Arovas, M. M. Fogler, F. Guinea, S. Roche, M. Goiran, and B.Raquet, P h y s .R e v .L e t t . 107,126806 (2011 ).
[28] W. Z. Bao, Z. Zhao, H. Zhang, G. Liu, P. Kratz, L. Jing, J.
Velasco, D. Smirnov, and C. N. Lau, P h y s .R e v .L e t t . 105,
246601 (2010 ).
[29] C.-H. Ho, S.-J. Tsai, R.-B. Chen, Y .-H. Chiu, and M.-F. Lin, J.
Nanosci. Nanotechnol. 11,4938 (2011 ).
[30] Y . Shi, S. Che, K. Zhou, S. Ge, Z. Pi, T. Espiritu, T. Taniguchi,
K. Watanabe, Y . Barlas, R. Lake, and C. N. Lau, P h y s .R e v .L e t t .
120,096802 (2018 ).
035419-11TOMOAKI NAKASUGA et al. PHYSICAL REVIEW B 101, 035419 (2020)
[31] Z. Wu, Y . Han, J. Lin, W. Zhu, M. He, S. Xu, X. Chen, H. Lu,
W. Ye, T. Han, Y . Wu, G. Long, J. Shen, R. Huang, L. Wang,Y . He, Y . Cai, R. Lortz, D. Su, and N. Wang, P h y s .R e v .B 92,
075408 (2015 ).
[32] T. Hirahara, R. Ebisuoka, T. Oka, T. Nakasuga, S. Tajima, K.
Watanabe, T. Taniguchi, and R. Yagi, Sci. Rep. 8,13992 (2018 ).
[33] T. Hirahara, R. Ebisuoka, K. Watanabe, T. Taniguchi, and R.
Yagi, J. Phys. Conf. Ser. 969,012150 (2018 ).
[34] M. Koshino and E. McCann, P h y s .R e v .B 83,165443 (2011 ).
[35] T. Nakasuga, S. Tajima, T. Hirahara, R. Ebisuoka, T. Oka, K.
Watanabe, T. Taniguchi, and R. Yagi, Phys. Rev. B 99,085404
(2019 ).
[36] A. A. Avetisyan, B. Partoens, and F. M. Peeters, P h y s .R e v .B
80,195401 (2009 ).
[37] A. A. Avetisyan, B. Partoens, and F. M. Peeters, P h y s .R e v .B
79,035421 (2009 ).
[ 3 8 ]E .V .C a s t r o ,K .S .N o v o s e l o v ,S .V .M o r o z o v ,N .M .R .P e r e s ,
J. M. B. Lopes dos Santos, J. Nilsson, F. Guinea, A. K. Geim,and A. H. Castro Neto, Phys. Rev. Lett. 99,216802 (2007 ).
[39] Y . B. Zhang, T. T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl,
M. F. Crommie, Y . R. Shen, and F. Wang, Nature (London) 459,
820(2009 ).
[40] T. Morimoto and M. Koshino, Phys. Rev. B 87,085424 (2013 ).
[41] L. Wang, I. Meric, P. Y . Huang, Q. Gao, Y . Gao, H. Tran, T.
Taniguchi, K. Watanabe, L. M. Campos, and D. A. Muller,Science 342,614(2013 ).
[42] B. Partoens and F. M. Peeters, P h y s .R e v .B 74,075404 (2006
).
[43] F. Guinea, A. H. Castro, and N. M. R. Peres, Solid State
Commun. 143,116(2007 ).
[44] T. Taychatanapat and P. Jarillo-Herrero, Phys. Rev. Lett. 105,
166601 (2010 ).
[45] J. Yan and M.S. Fuhrer, Nano Lett .10,4521 (2010 ).
[46] H. Miyazaki, K. Tsukagoshi, A. Kanda, M. Otani, and S.
Okada, Nano Lett .10,3888 (2010 ).
[47] W. J. Zhu, D. Neumayer, V . Perebeinos, and P. Avouris,
Nano Lett. 10,3572 (2010 ).
[48] M. F. Craciun, S. Russo, M. Yamamoto, J. B. Oostinga, A. F.
Morpurgo, and S. Tarucha, Nat. Nanotechnol. 4,383(2009 ).
[49] S. H. Jhang, M. F. Craciun, S. Schmidmeier, S. Tokumitsu, S.
Russo, M. Yamamoto, Y . Skourski, J. Wosnitza, S. Tarucha, J.Eroms, and C. Strunk, Phys. Rev. B 84,161408(R) (2011 ).
[50] K. Zou, F. Zhang, C. Capp, A. H. MacDonald, and J. Zhu, Nano
Lett.13,369(2013 ).
[51] A. A. Avetisyan, B. Partoens, and F. M. Peeters, P h y s .R e v .B
81,115432 (2010 ).
[52] J. C. Slonczewski and P. R. Weiss, Phys. Rev. 109,272(1958 ).
[53] J. W. McClure, Phys. Rev. 108,612(1957 ).[54] M. S. Dresselhaus and G. Dresselhaus, Adv. Phys. 30,
139
(1981 ).
[55] F. Guinea, P h y s .R e v .B 75,235433 (2007 ).
[56] M. Koshino, P h y s .R e v .B 81,125304 (2010 ).
[57] M. Koshino and E. McCann, Phys. Rev. B 79,125443 (2009 ).
[58] Y . S. Zheng and T. Ando, Phys. Rev. B 65,245420 (2002 ).
[59] S. Adam, E. H. Hwang, V . M. Galitski, and S. Das Sarma, Proc.
Natl. Acad. Sci. USA (PNAS) 104,18392 (2007 ).
[60] Y . W. Tan, Y . Zhang, K. Bolotin, Y . Zhao, S. Adam, E. H.
Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, Phys. Rev.
Lett. 99,246803 (2007 ).
[61] J. G. Checkelsky, L. Li, and N. P. Ong, Phys. Rev. Lett. 100,
206801 (2008 ).
[62] R. Yagi, S. Fukada, H. Kobara, N. Ogita, and M. Udagawa,
Physica 42,673(2010 ).
[63] L. M. Malard, M. A. Pimenta, G. Dresselhaus, and M. S.
Dresselhaus, Phys. Rep. 473,51(2009 ).
[64] A. C. Ferrari, J. C. Meyer, V . Scardaci, C. Casiraghi, M. Lazzeri,
F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, andA. K. Geim, P h y s .R e v .L e t t . 97,187401 (2006 ).
[65] C. H. Lui, Z. Q. Li, Z. Y . Chen, P. V . Klimov, L. E. Brus, and
T. F. Heinz, Nano Lett. 11,164(2011 ).
[66] T. A. Nguyen, J. U. Lee, D. Yoon, and H. Cheong, Sci. Rep. 4
,
4630 (2014 ).
[67] C. X. Cong, T. Yu, K. Sato, J. Z. Shang, R. Saito, G. F.
Dresselhaus, and M. S. Dresselhaus, ACS Nano 5,8760 (2011 ).
[68] Y . F. Hao, Y . Y . Wang, L. Wang, Z. H. Ni, Z. Q. Wang, R. Wang,
C. K. Koo, Z. X. Shen, and J. T. L. Thong, Small 6,195(2010 ).
[69] D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C.
Hierold, and L. Wirtz, Nano Lett. 7,238(2007 ).
[70] S. J. Yuan, R. Roldan, and M. I. Katsnelson, Phys. Rev. B 84,
125455 (2011 ).
[71] E. McCann and V . I. Fal’ko, Phys. Rev. Lett. 96,086805 (2006 ).
[72] M. Koshino and E. McCann, Phys. Rev. B 87,045420 (2013 ).
[73] P. B. Visscher and L. M. Falicov, P h y s .R e v .B 3,2541 (1971 ).
[74] R. van Gelderen, R. Olsen, and C. M. Smith, Phys. Rev. B 88,
115414 (2013 ).
[75] L. Pietronero, S. Strassler, H. R. Zeller, and M. J. Rice,
Phys. Rev. Lett. 41,763(1978 ).
[76] D. P. DiVincenzo and E. J. Mele, Phys. Rev. B 29,1685
(1984 ).
[77] N. J. Lee, J. W. Yoo, Y . J. Choi, C. J. Kang, D. Y . Jeon, D. C.
Kim, S. Seo, and H. J. Chung, Appl. Phys. Lett. 95,222107
(2009 ).
[78] H. Miyazaki, S. Odaka, T. Sato, S. Tanaka, H. Goto, A. Kanda,
K. Tsukagoshi, Y . Ootuka, and Y . Aoyagi, Appl. Phys. Express
1,034007 (2008 ).
035419-12 |
PhysRevB.91.045125.pdf | PHYSICAL REVIEW B 91, 045125 (2015)
Chemical tuning of electrical transport in Ti 1-xPtxSe2-y
Justin S. Chen,1Jiakui K. Wang,1Scott V . Carr,1Sven C. V ogel,2Olivier Gourdon,2Pengcheng Dai,1and E. Morosan1
1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
2Los Alamos Neutron Science Center, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received 3 October 2014; revised manuscript received 16 December 2014; published 20 January 2015)
The structural and transport properties of polycrystalline Ti 1-xPtxSe2-y(x/lessorequalslant0.13,y/lessorequalslant0.2) are studied,
revealing highly tunable electrical properties, spanning nearly ten orders of magnitude in scaled resistivity.Using x-ray and neutron diffraction, Pt is found to dope on the Ti site. In the absence of Pt doping (for x=0),
Se deficiency ( y> 0) increases the metallic character of TiSe
2, while a large increase of the low-temperature
resistivity is favored by a lack of Se deficiency ( y=0) and increasing amounts of doped Pt ( x> 0). The chemical
tuning of the resistivity in Ti 1-xPtxSe2-ywith Se deficiency andPt doping results in a metal-to-insulator transition.
Simultaneous Pt doping and Se deficiency ( x,y > 0) confirms the competition between the two opposing trends
in electrical transport, with the main outcome being the suppression of the charge density wave transition below2Kf o r y=2x=0.18. Band structure calculations on a subset of Ti
1-xPtxSe2-ycompositions are in line with the
experimental observations.
DOI: 10.1103/PhysRevB.91.045125 PACS number(s): 71 .30.+h,71.20.Nr,71.45.Lr
I. INTRODUCTION
Layered crystal structures with van der Waals gaps lend
themselves to great tuning versatility. In particular, the layeredtransition-metal dichalcogenides (LTMDs) are a family of two-dimensional compounds, with electrical transport propertiesranging from metals to insulators, mostly with a charge densitywave (CDW) ground state [ 1]. Chemical modifications of these
systems include intercalation (in the van der Waals gaps) orsubstitution with magnetic or nonmagnetic elements, greatlyenhancing the range of physical properties observed in theLTMDs, to include unconventional superconductivity [ 1–7],
magnetism [ 7–10], or metal-insulator transitions. [ 3,11].
The layered crystal structure of the hexagonal LTMDs has
prompted studies analogous to those in hexagonal graphene,
particularly aiming at the fabrication of electronic devices.
MoS
2in particular has attracted recent interest, as its properties
can also be tuned through a variety of methods such asapplication of strain [ 12] or electric fields [ 13]. The CDW
state has been exploited in several LTMDs, mainly TiTe
2
[14] and TaS 2[15], to produce devices with nonlinear I-V
characteristics. There also was recent interest in TiSe 2,
with new fabrication techniques using mechanical exfoliation
[16] and chemical vapor deposition [ 17] developed for the
controlled fabrication of thin layers of TiSe 2.T h eL T M D sh a v e
a variety of applications for studying the physics of competingphases as well as for their practical use as two-dimensionaldevices.
Here, we report the effects of chemical tuning on the
transport properties of TiSe
2. A combination of Pt doping
on the Ti sublattice, together with Se deficiency, in the seriesTi
1-xPtxSe2-yresults in the scaled resistivity changing by nearly
ten orders of magnitude. In particular, Se deficiency alone(x=0,y> 0) completely suppresses the CDW transition and
imparts a metallic character to the y=0.2 system, resulting
in a drop in the scaled low- Tresistivity by nearly two orders
of magnitude. By contrast, Pt doping with no Se deficiency(x>0,y=0) hardly changes the CDW transition temper-
ature, but reveals insulating behavior with (i) an increasinghigh-temperature ( T> T
CDW)g a pEgand (ii) an increase ofmore than eight orders of magnitude of the resistivity scaled
at 300 K for x/lessorequalslant0.13. A combination of Se deficiency and
Pt doping ( y=2x,x> 0) shows a suppression of the CDW
slower than in the former case ( x=0), but considerably faster
than in the latter ( y=0), while the resulting high- Tresistivity
(T> T CDW) is metallic. The large range of resistivity values
covered in TiSe 2by a single (chemical) tuning parameter
renders this system a potential candidate for further explorationof transport properties adequate for applications. Furthermore,the increase of the room-temperature transport gap with x
makes this system ideal for transport gap engineering fordevices. We argue that the transport properties are driven by thechanging chemical potential with xandy, with band structure
calculations qualitatively supporting this scenario. Theoreticalcalculations similar to these may aid in the future choice ofchemical tuning parameters to control the targeted propertiesof TiSe
2.
II. METHODS
Polycrystalline samples of Ti 1-xPtxSe2-ywere synthesized
for various xandyvalues using solid state reaction. Stoichio-
metric amounts of Pt, Ti, and Se powders were sealed undervacuum in a silica tube. Reaction temperatures up to 1000
◦C
were necessary to eliminate impurity phases such as Pt andPtSe
2, but this high temperature caused some Se evaporation.
The mass difference before and after heating was compensatedfor by the addition of Se to the mixture, which was then heatedat 650
◦C first in powder form, and then in pellet form, for
several days, with an intermediate grinding. The full procedurewas carried out up to two times. A control TiSe
2sample was
synthesized following the same protocol, and the Ti:Se =1:2
stoichiometry was confirmed by the resistivity peak height[18]. It appears that the PtSe
2-TiSe 2solubility limit is reached
in the region of 0 .13<x< 0.25, since samples made with
x/greaterorequalslant0.25 show significant amounts of both binary phases.
Phase determination of the samples was done with powder
x-ray diffraction (XRD) using a Rigaku D/max ULTIMA IIdiffractometer with a Cu Kαradiation source. Rietveld anal-
ysis was performed using the
GSAS /EXPGUI suite of programs
1098-0121/2015/91(4)/045125(7) 045125-1 ©2015 American Physical SocietyCHEN, W ANG, CARR, VOGEL, GOURDON, DAI, AND MOROSAN PHYSICAL REVIEW B 91, 045125 (2015)
[19]. For consistency, Si powder was used as standard in
all XRD powder measurements. Neutron powder diffraction(NPD) was performed at room temperature, for ( x,y)=(0,0),
(0.02,0.04), (0.05,0), (0.09,0.18), and (0.09,0), using the time-of-flight HIPPO instrument at the Los Alamos Neutron ScienceCenter (LANSCE). For the analysis, high-resolution data weretaken from the backscattering bank at a nominal diffractionangle of 144 .447
◦. Electrical transport measurements were
performed down to T=2 K with a Quantum Design physical
properties measurement system using a standard four-probetechnique. Band structure calculations were performed withthe full-potential linearized augmented-plane-wave methodimplemented in the
WIEN2K package [ 20]. A 10 ×10×10
k-point grid was used, together with the −6.0 Ry separation
energy between the core and valence states. A plain densityfunctional theory (DFT) calculation with a localized exchangeinteraction did not reproduce the small 0.15 eV band gap inTiSe
2that was observed in scanning tunneling spectroscopy
data [ 21]. Therefore, a screened hybrid functional Yukawa
screened – Perdew-Burke-Ernzerhof (YS-PBE0) [ 22]w a s
used as the exchange-correlation potential, reproducing theband gap as with the GW calculation [ 21]. Pt-doped andSe-deficient sample calculations were performed by construct-
ing supercells with a unit cell of (2 a,2a,c) with one Ti atom
replaced by Pt or one Se atom removed from the supercell.
III. DATA AND ANALYSIS
TiSe 2has rather unusual properties among the many known
LTMDs. This compound displays a commensurate CDW, withlong-debated properties of the normal state because of thesmall indirect gap in TiSe
2. It was difficult to unambiguously
distinguish between a positive (semiconductor) [ 21,23,24] and
a negative (semimetal) [ 25,26] gap, with the most recent
scanning tunneling spectroscopy studies favoring a small-band-gap semiconductor [ 21]. The CDW transition around
T
CDW=220 K is marked by a broad peak in resistivity, without
a preceding incommensurate CDW state as is the case in manyLTMDs [ 2,4,6,26]. Unlike the case in many other LTMDs,
t h eC D Ws t a t ei nT i S e
2does not result from Fermi surface
nesting, but has been claimed to originate from an excitonicinsulator state [ 2,23,25]. Most intriguing, TiSe
2does not have
a superconducting state above 0.4 K [ 4], as is the case in many
LTMDs, but a superconducting state can be induced either by
0.00 0.05 0.10
xi nT i1-xPtxSe2-ysquares - from XRD
triangles - from NPD(e) (d) y=2 x y=0 x=0 (c)
0.20 0.15 0.10 0.053.5353.5403.5453.5503.555
yi nT i1-xPtxSe2-yÅ
0.00 0.05 0.105.985.996.006.01c(Å)2468 1 0 1 2 1 4(b) NPDIntensity (arb. units)
Q(Å-1)x = 0.05
y=0
10 20 30 40 50 60 70 80 90Ti1-xPtxSe2-yIntensity (arb. units)
2θ(degrees)(a) XRD
x = 0.05
y=0
FIG. 1. (Color online) (a) Ti 1-xPtxSe2-yXRD pattern for ( x,y)=(0.05,0), with calculated peak positions marked by black (top) vertical
lines. Green (bottom) vertical lines mark the Si standard peak positions. (b) Normalized Ti 1-xPtxSe2-yNPD pattern for ( x,y)=(0.05,0).
The refined lattice parameters from XRD (squares) and NPD (triangles) for Ti 1-xPtxSe2-yare shown for (c) x=0 (no Pt), (d) y=0( n oS e
deficiency), and (e) y=2x(Pt doping and Se deficiency). The refined lattice parameters from NPD were calibrated with those from XRD for
(x,y)=(0,0).
045125-2CHEMICAL TUNING OF ELECTRICAL TRANSPORT IN Ti . . . PHYSICAL REVIEW B 91, 045125 (2015)
0 50 100 150 200 250 300100101102103104ρ(mΩcm)
T( K )y=0
y=0 . 2
y=0
y=0 . 1Ti1-xPtxSe2-y
x=0x=0.05
FIG. 2. (Color online) The temperature-dependent resistivity for
Ti1-xPtxSe2-ywithx=0a n dx=0.05 with no Se deficiency ( y=0,
solid lines) and with Se deficiency ( y> 0, dashed lines).
the intercalation of transition metals M=Cu or Pd [ 4,6]o r
by pressure [ 27].
The transition metals Mknown to intercalate in TiSe 2do
not form stable MSe2compounds isostructural with TiSe 2.
However, PtSe 2and TiSe 2are known isostructural LTMDs
[28]. It is therefore not surprising that Pt is found to dope in
place of Ti rather than intercalate between the layers [ 28–31].
Structural studies with XRD and NPD indeed confirm that thisis the case in all Ti
1-xPtxSe2-ysamples presented here. As an
example, the refined XRD and NPD data for Ti 0.95Pt0.05Se2
are shown in Fig. 1. The XRD refinement confirms the P3m1
space group for Ti 1-xPtxSe2-yfor all reported ( x,y) values.
The lattice parameters aandcare plotted in Figs. 1(c)–1(e),both as a function of y[x=0, open symbols, increase
to the left, Fig. 1(c)], and as a function of x[fory=0,
full symbols, Fig. 1(d), and for y=2x, half-full symbols,
Fig. 1(e)]. Without doping ( x=0, open symbols), the lattice
parameter a(c) increases (decreases) for Se deficiency close
toy=0.2 [Fig. 1(c)]. A similar effect occurs with doping and
Se deficiency, when y=2x[Fig. 1(e)]. Surprisingly, it would
appear that the amount of Pt is irrelevant from a structuralperspective, given that the change in both aandcfory=2x
is the same as when the composition is changed only by Sedeficiency y, for the same values of y[Figs. 1(c) and 1(e)].
This is also consistent with the fact that the lattice parametersremain virtually unchanged with doping alone ( y=0) for x
values up to 0.13 [Fig. 1(d)].
To determine the location of the Pt atoms, several structure
models were used for the NPD refinement. From GeneralStructure Analysis System (GSAS) refinements, a model withPt and Ti sharing the same site appears to be the most accurate.Attempts to employ an intercalation model (with the Pt atomslocated in the van der Waals gaps) consistently led to valuesfor Se occupancy and atomic displacements U
isothat were
nonphysical. Se occupancy increased to ∼1.5, indicating 50%
more Se atoms occupying a site than is physically possible. TheU
isofor Pt in the van der Waals gap was 0.6–0.8, an order of
magnitude larger than the typical Uisovalues. This represented
an implausible increase from room temperature Uisovalues in
crystals, usually on the order of 0.01, with higher values of0.1–0.2 for loosely bound atoms in organic molecules [ 32].
Therefore, it is concluded that Pt is not located in the van der
Waals gap, but is only partially substituting on the Ti sites.
The effects of Se deficiency ( y/greaterorequalslant0) versus Pt doping
(x/greaterorequalslant0) on the transport properties are first illustrated by
the change in the absolute resistivity values in Fig. 2.A t
0 50 100 150 200 250 30010-1101103105107
0 50 100 150 200 250 30010-1100101102103
0 50 100 150 200 250 30010-1100101102103
0 100 2000.60.70.80.90 100 2000.50.60.70.80.9 x=0
y=2 x(c)(a)Ti1-xPtxSe2-yρ/ρ(300K)
T( K )0
0.02
0.05
0.07
0.09
0.13y=0(b)
x=
0.02 0.05 0.07 0.09ρ/ρ(300K)
T( K )x=0 0.1 0.2ρ/ρ(300K)
T( K )y=
ρ/ρ(300K)
T( K )ρ/ρ(300K)
T( K )
FIG. 3. (Color online) Temperature-dependent resistivity scaled at 300 K, ρ/ρ(300 K), for Ti 1-xPtxSe2-yfor (a) x=0 (no Pt), (b) y=0
(no Se deficiency), and (c) y=2x(Pt and Se deficiency). The insets of (a) and (c) show details of the resistivity of the metallic phases.
045125-3CHEN, W ANG, CARR, VOGEL, GOURDON, DAI, AND MOROSAN PHYSICAL REVIEW B 91, 045125 (2015)
T=300 K, the resistivity ρappears to increase by one or two
orders of magnitude from the y=0 samples (solid lines) to
they> 0 samples (dashed lines). Furthermore, the qualitative
temperature dependence of the resistivity is substantivelydifferent for the two sets of samples. The Se-deficient samples(dashed lines) display a decreasing resistivity with decreasingtemperature, indicative of a trend towards metallicity. Sedeficiency has been shown to suppress the height of the peakbelow the CDW transition in TiSe
2-ysingle crystals [ 18], but
without moving the CDW transition itself. The differencebetween the reported resistivity of Se-deficient single crystalsand that of the polycrystalline samples in the current study ismost likely in the actual amount of Se deficiency, which maydiffer from the nominal amounts y.
A comparison of the resistivity data, scaled at T=300 K,
for various xandyvalues for Ti
1-xPtxSe2-yis shown in
Fig. 3. With no Pt doping [ x=0, Fig. 3(a)], the resistivity
displays metallic behavior when y=0.2, with only small
changes to resistivity for y/lessorequalslant0.1. As seen in the inset and
below in Fig. 4(a), as the resistivity displays metallic behavior,
it appears that the CDW transition has been suppressed tobelow 2 K for this composition. In the case of Pt dopingwith no Se deficiency [ y=0, Fig. 3(b)], the scaled resistivity
increases by up to eight orders of magnitude as xincreases
up tox=0.13. Quantitatively, this can be described with an
exponent αintroduced as
ρ(300 K) /ρ(6 K)=10
−α. (1)
TheT=6 K was used for defining αsince, at lower
temperatures, the resistance values for the most insulatingsample ( x=0.13) surpassed the instrument limit for these
measurements. The exponent αdetermined using Eq. ( 1)
is maximum, α=7.2, for x=0.13. However, it is readily
apparent from Fig. 3(b) that this is an underestimate for α, since
ρis likely to still increase rapidly below 6 K. It is noteworthy
that the CDW transition appears minimally affected by Ptdoping without Se deficiency, as will be shown below by theresistivity derivative plots (Fig. 4).
In the case of y=2x[Fig. 3(c)], the low-temperature scaled
resistivity first shows a slight increase for x=0.02 (circle),
slower than that for the analogous xvalue with no Se deficiency
[Fig. 3(b)]. However, for x/greaterorequalslant0.05 the resistivity becomes
metallic at high temperatures, while the CDW persists up tox=0.05, albeit at decreasing temperatures. A competition
appears to exist between Pt doping, which drives the systemtowards an insulating state, and the Se deficiency, whichsuppresses the CDW and induces a poor metallic state. TheCDW transition is more evident in the resistivity derivativedρ/dT (Fig. 4), with T
CDW defined as the temperature where a
drop in the derivative occurs on cooling [ 6]. As mentioned
above, the CDW is completely suppressed in the ( x,y)=
(0,0.2) sample [pentagon, Fig. 4(a)] or partially suppressed for
(x,2x) with x/greaterorequalslant0.05 [Fig. 4(c)]. All other compositions, and
in particular those with no Se deficiency [Fig. 4(b)] display
a CDW transition at the same temperature TCDW≈220 K,
marked by a vertical dashed line in Fig. 4. However, the CDW
transition temperature is suppressed by nearly one order ofmagnitude for x=0.05 in Ti
0.05Pt0.95Se1.9[up triangle, inset,
Fig. 4(c)], down to ∼25 K.
0 50 100 150 200 250 300
02 0 4 0 6 0(c)(a)
0
0.1
0.2TCDW
Ti1-xPtxSe2-y
(b)y=
y=2 xx=
T( K )x=0dρ/dT (arb. units)y=0
0
0.02
0.05
0.07
0.09
0.13
FIG. 4. (Color online) Temperature derivatives of the resistivity
dρ/dT for Ti 1-xPtxSe2-yfor (a) x=0 (no Pt), (b) y=0( n oS e
deficiency), and (c) y=2x(Pt and Se deficiency). The inset in (c) is
the low-temperature region for y=2x.
For the insulating samples ([ y=0, Fig. 3(b)], the band gap
Egabove TCDW was estimated using [ 33]
ρ∝e−Eg/2kBT. (2)
The gap Egas a function of x(Fig. 5) is determined as the
slope of the linear fits for ln ρvs 1/T(inset) above T=250 K.
The gap Egincreases linearly with xand reaches a maximum
valueEg=125 meV at x=0.13.
In order to understand how tuning xandyproduces the
semiconducting and metallic states of Ti 1-xPtxSe2-y, hybrid
functional DFT calculations are performed for ( x,y)=(0,0),
(0.25,0), and (0,0.25). A density of states (DOS) plot isshown in Fig. 6. The calculations reveal E
g=0.2e Vf o r
TiSe 2, which agrees with the previous DFT calculations using
the GW approximation [ 34,35] and close to some of the
recent experimental estimates [ 2,25,36]. Pt doping alone,
045125-4CHEMICAL TUNING OF ELECTRICAL TRANSPORT IN Ti . . . PHYSICAL REVIEW B 91, 045125 (2015)
0.00 0.05 0.10 0.15 0.20020406080100120140
3.5 4.0 4.5 5.012345Eg(meV)
xTi1-xPtxSe2-y
y=0
TCDWρ/ρ (300K)
1/T(10-3K-1)x=0.02
0.05
0.07
0.09
0.13
FIG. 5. (Color online) Band gap Egas a function of Pt concen-
tration xin Ti 1-xPtxSe2-y. Inset: Linear fits of ln ρvs 1/Tabove 250 K
were used to determine Eg.
(x,y)=(0.25,0), increases the gap to Eg=0.5 eV , a trend
consistent with the observed linear change in Eg.
The valence band is between −6 eV and 0 eV for
(x,y)=(0,0), and widens with Pt doping by about 1 eV
at (x,y)=(0.25,0). This widening is possibly due to the
smaller electronegativity difference between Pt and Se (0.35)compared to that between Ti and Se (1.01) [ 37]. However, Se
deficiency in ( x,y)=(0,0.25) does not broaden the valence
band, but instead shifts it down by 1 eV . The Ti 3 delectron
band shifts to the Fermi level, resulting in a finite DOS for(x,y)=(0,0.25), albeit with a small value (local minimum)
atE
F. The Se deficiency can be understood as analogous to
electron doping, as fewer Ti electrons are transferred to Sesites for ( x,y)=(0,0.25) compared to ( x,y)=(0,0).
The Ioffe-Regel limit specifies the maximum resistivity
of a metal. The limit is estimated from the Drude transportequation ρ=ne
2τ/m , using the Fermi velocity (assuming a
spherical Fermi surface) for estimating the mean scatteringtimeτ, with the Fermi surface taking up the entirety of the
first Brillouin zone to estimate the charge density n, and using
-7 -6 -5 -4 -3 -2 -1 0 1 2 30246810
-1.0 -0.5 0.0 0.5 1.001234
x=0 ,y=0
x=0 . 2 5 ,y=0
x=0 ,y=0 . 2 5DOS (states/eV)
Energy (eV)Ti1-xPtxSe2-y
DOS (states/eV)
Energy (eV)
FIG. 6. (Color online) Hybrid functional DFT calculations for
(x,y)=(0,0), (0.25,0), and (0,0.25). Inset: The details near the Fermi
energy reveal a small but finite DOS only for the ( x,y)=(0,0.25)
sample.0.00 0.05 0.10 0.150.00.10.2ρ(6K)
α =yi nT i1-xPtxSe2-y
xi nT i1-xPtxSe2-y-1
0
12
3
4
5
6
7
8CDWTi1-xPtxSe2-yρ(300K)=1 0- α
FIG. 7. (Color online) x-ycontour plot of the resistivity exponent
α(see text). Symbols: actual data determined from ρ(T) data; dashed
line: possible boundary for the CDW state.
the electron’s rest properties for the values of eandm.T h e
large increase in resistivity for the Se-deficient samples, whichviolates the Ioffe-Regel limit of 500 μ/Omega1cm (calculated for a
nearly full first Brillouin zone with a typical lattice spacing of≈4˚A[38]) cannot be explained by the small DOS alone. The
large resistivity could be indicative of other sources of electronscattering such as from underlying CDW correlations [ 39,40],
or of the electrons becoming strongly localized, as is the casein some bad metals [ 41].
The overall effects of Pt doping and Se deficiency are
summarized in the ( x,y) contour plot of the resistivity exponent
α[Eq. ( 1)] in Fig. 7. It is readily apparent that the two chemical
control parameters xandyhave drastically different effects on
the transport properties of TiSe
2. First, for y=0,αincreases
linearly with xup toα≈8f o rx=0.13. The opposite trend,
albeit much slower, is revealed by Se deficiency without Ptdoping ( x=0), where αdecreases with ytoα≈− 1f o r
y=0.2. Along the line y=2x, the competition between
the two chemical control parameters xandyresults in a
nonmonotonic change in α.F o rl o w x, Pt doping wins over
Se deficiency, as the x=0.02 sample becomes slightly more
semiconducting than that with x=0. Further increase in xin
the case of y=2xresults in decreasing αand a metallic state
towards high x. The case of y=2xdelineates a metallic range
above, with increasing insulating character upon approachingthexaxis at high x. Interestingly, the CDW state seems to
be less sensitive to the change in the electrical resistivity. Thedashed line in Fig. 7indicates that the CDW state persists
for most semiconducting samples. The nearly divergent ρ
for (x,y)=(0.13,0) makes it very difficult to still determine
the signature of the CDW, while in the ( x,2x) samples with
x
/lessorequalslant0.07, downturns in dρ/dT close to the lowest measured
T[inset, Fig. 4(c)] suggest that a CDW transition might still
occur just below 2 K. This would be consistent with the trendobserved up to x=0.05, where T
CDW has been suppressed
down to 25 K.
045125-5CHEN, W ANG, CARR, VOGEL, GOURDON, DAI, AND MOROSAN PHYSICAL REVIEW B 91, 045125 (2015)
IV . DISCUSSION AND CONCLUSIONS
In Ti 1-xPtxSe2-y,t h exandychemical control parameters
drive changes from metallic to semiconducting behavior,simultaneous with changes of the aandclattice parameters.
Interestingly, aincreases and cdecreases with increasing yin
the undoped samples [ x=0i nF i g . 1(c)]. The accompanying
trend towards metallicity [Fig. 3(a)] might be associated
with increasing interlayer correlations, as the correspondinglattice parameter cbecomes smaller. By contrast, changing
xwith no Se deficiency [ y=0i nF i g . 1(d)]l e a v e s aandc
virtually unchanged, while the transport properties are greatlyaffected [Fig. 3(b)]. In conjunction with the band structure
calculations, these observations indicate that Se deficiencyshifts the chemical potential of the system towards the Ti3dbands and increases, at least crystallographically, the
three-dimensional (3D) character of these compounds as a
increases and cdecreases. Previously, Cu and Pd intercalation
in TiSe
2also led to an increase of the alattice parameter,
resulting in metallic (and eventually superconducting)behavior [ 4,6]. However, the clattice parameters for the
intercalated superconducting compounds increased, incontrast to the effect in the current Pt-doped samples. A morethorough exploration of the ( x,y) phase space is necessary to
fully determine the correlations between the electronic andstructural properties of TiSe
2. It is of note, however, that even
TiSe 2intercalation with transition metals that did not yield
a low-temperature superconducting state [ 42] resulted in an
expected decrease of the interlayer spacing c.
Previous experimental studies on PtSe 2indicated that this
dichalcogenide compound was a semiconductor with Eg=
100 meV [ 31], while TiSe 2was likely a semiconductor with
as m a l l( ≈20 meV) band gap Eg[2,21,23–25]. Considering
that Ti and Pt are chemically and electronically substantivelydifferent, it is not entirely unexpected that the change inthe gap value with xis nonmonotonic. Pt doping of only
13% in TiSe
2[Fig. 3(b)] appears to increase the gap beyond
that of pure PtSe 2, suggesting a higher maximum gap in
Ti1-xPtxSe2-y, if higher compositions were not precluded by the
solubility of the two end compounds. Interestingly, the CDW isobserved to persist in the semiconducting state in Ti 1-xPtxSe2-y
(y=0), as seen in Fig. 3(b). This was also the case in
Pd-intercalated TiSe 2[6]. However, in the latter case, much
smaller scaled resistivity values [ ρ/ρ(300 K) /lessorequalslant103] favored
a superconducting state for high enough Pd composition.However, given that the gap in TiSe
2is very small, there is
an ongoing debate over its exact value [ 2,18,21,23–26,36],
which is irrelevant for the current discussion.
Engineering a transport gap in TiSe 2may allow transistors
and other electronic devices to be made based on TiSe 2
[12,13,43]. While the band gap in the bulk is smaller
than the ideal 1 eV for some applications (photovoltaiccells [ 44]), the band gap of dichalcogenides was raised by
0.5 eV for MoS
2once the samples were exfoliated to a
single monolayer [ 43]. Additionally, for any practical device
using the CDW state, the CDW would have to exist atroom temperature [ 16]. Previous work showed that T
CDW
could also be raised in thin layer samples of TiSe 2[16].
Therefore a comparison of the properties of bulk and exfoliatedTi
1-xPtxSe2-ymay lead to such a desirable band gap Eg
andTCDW increase. Such experiments are currently under
way. Furthermore, tuning of the chemical parameters xand
ymakes Ti 1-xPtxSe2-yideal for studies of the fundamental
physics of the CDW state affected by both the band gap anddimensionality.
ACKNOWLEDGMENTS
E.M. and J.S.C. acknowledge support from the DOD
PECASE. The neutron scattering work at Rice was supportedby the U.S. DOE, BES, Contract No. DE-SC0012311 (P.D.).Part of the work is supported by the Robert A. WelchFoundation Grant No. C-1839 (P.D.). This work has benefitedfrom Lujan Neutron Scattering Center at LANSCE, whichis funded by the U.S. Department of Energy’s Office of BasicEnergy Sciences. Los Alamos National Laboratory is operatedby Los Alamos National Security LLC under DOE ContractNo. DEAC52-06NA25396.
[ 1 ] J .A .W i l s o na n dA .D .Y o f f e , Adv. Phys. 18,193(1969 ).
[2] K. Rossnagel, J. Phys.: Condens. Matter 23,213001
(2011 ).
[3] R. Ang, Y . Miyata, E. Ieki, K. Nakayama, T. Sato, Y . Liu, W. J.
Lu, Y . P. Sun, and T. Takahashi, Phys. Rev. B 88,115145 (2013 ).
[4] E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G. Bos,
Y . Onose, T. Klimczuik, A. P. Ramirez, N. P. Ong, and R. J.Cava, Nat. Phys. 2,544(2006 ).
[5] K. E. Wagner, E. Morosan, Y . S. Hor, J. Tao, Y . Zhu, T. Sanders,
T. M. McQueen, H. W. Zandbergen, A. J. Wiliams, D. V . West,a n dR .J .C a v a , P h y s .R e v .B 78,104520 (2008 ).
[6] E. Morosan, K. E. Wagner, L. L. Zhao, Y . Hor, A. J. Wiliams,
J. Tao, Y . Zhu, and R. J. Cava, P h y s .R e v .B 81,094524 (2010 ).
[7] J. J. Yang, Y . J. Choi, Y . S. Oh, A. Hogan, Y . Horibe, K. Kim,
B. I. Min, and S. W. Cheong, Phys. Rev. Lett. 108,116402
(2012 ).[8] E. Morosan, H. W. Zandbergen, L. Li, M. Lee, J. G. Checkelsky,
M. Heinrich, T. Siegrist, N. P. Ong, and R. J. Cava, Phys. Rev.
B75,104401 (2007 ).
[9] V . G. Pleschov, N. V . Baranov, A. N. Titov, K. Inoue, M. I.
Bartashevich, and T. Goto, J. Alloys Compd. 320,13(2001 ).
[10] M. Koyano, M. Suezawa, H. Watanabe, and M. Inoue, J. Phys.
Soc. Jpn. 63,1114 (1994 ).
[11] B. Sipos, A. F. Kusmartseva, A. Akrap, H. Berger, L. Forro, and
E. Tutis, Nat. Mater. 7,960(2008 ).
[12] H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund, S. T.
Pantelides, and K. I. Bolotin, Nano Lett.
13,3626 (2013 ).
[13] H. Qiu, L. J. Pan, Z. N. Yao, J. J. Li, Y . Shi, and X. Wang, Appl.
Phys. Lett. 100,123104 (2012 ).
[14] J. Khan, C. Nolen, D. Teweldebrhan, D. Wickramaratne,
R. Lake, and A. Balandin, Appl. Phys. Lett. 100,043109
(2012 ).
045125-6CHEMICAL TUNING OF ELECTRICAL TRANSPORT IN Ti . . . PHYSICAL REVIEW B 91, 045125 (2015)
[15] J. Renteria, R. Samnakay, C. Jiang, T. R. Pope, P. Goli, Z. Yan,
D. Wickramaratne, T. T. Salguero, A. G. Khitun, R. K. Lake,and A. A. Balandin, J. Appl. Phys. 115,034305 (2014 ).
[16] P. Goli, J. Khan, D. Wickramaratne, R. K. Lake, and A. A.
Balandin, Nano Lett. 12,5941 (2012 ).
[17] S. L. Benjamin, C. K. de Groot, C. Gurnani, A. L. Hector,
R. Huang, K. Ignatyev, W. Levason, S. J. Pearce, F. Thomas,and G. Reid, Chem. Mater. 25,4719 (2013 ).
[18] F. J. DiSalvo, D. E. Moncton, and J. V . Waszczak, Phys. Rev. B
14,4321 (1976 ).
[19] A. C. Larson and R. B. V on Dreele, Los Alamos National
Laboratory Report LAUR No. 86-748, 2000 (unpublished).
[20] P. K. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz,
Computer code
WIEN 2K,http://www.wien2k.at
[21] M. Cazzaniga, H. Cercellier, M. Holzmann, C. Monney, P. Aebi,
G. Onida, and V . Olevano, Phys. Rev. B 85,195111 (2012 ).
[22] F. Tran and P. Blaha, Phys. Rev. B 83,235118 (2011 ).
[23] M. M. May, C. Brabetz, C. Janowitz, and R. Manzke, Phys. Rev.
Lett.107,176405 (2011 ).
[24] T. E. Kidd, T. Miller, M. Y . Chou, and T. C. Chiang, Phys. Rev.
Lett.88,226402 (2002 ).
[25] H. Cercellier, C. Monney, F. Clerc, C. Battaglia, L. Despont,
M. G. Garnier, H. Beck, P. Aebi, L. Patthey, H. Berger, andL. Forro, Phys. Rev. Lett. 99,146403 (2007 ).
[26] G. Li, W. Z. Hu, D. Qian, D. Hsieh, M. Z. Hasan, E. Morosan,
R. J. Cava, and N. L. Wang, Phys. Rev. Lett. 99,027404
(2007 ).
[27] A. F. Kusmartseva, B. Sipos, H. Berger, L. Forr ´o, and E. Tuti ˇs,
Phys. Rev. Lett. 103,236401 (2009 ).[28] R. M. A. Lieth and J. C. J. M. Terhell, Preparation and
Crystal Growth of Materials with Layered Structures (D. Reidel
Publishing Company, Dordrecht, 1977).
[29] F. Gronvold, H. Haraldsen, and A. Kjekshus, Acta Chem. Scand.
14,1879 (1960 ).
[30] S. Furuseth, K. Selte, and A. Kjekshus, Acta Chem. Scand. 19,
257(1965 ).
[31] F. Hulliger, J. Phys. Chem. Solids 26,639(1965 ).
[32] W. Massa, Crystal Structure Determination , 2nd ed. (Springer-
Verlag Berlin Heidelberg, New York, 2004).
[33] N. W. Ashcroft and N. D. Mermin,
Solid State Physics
(Brooks/Cole, Boston, MA, 1976).
[34] L. Hedin, Phys. Rev. 139,A796 (1965 ).
[35] G. Strinati, H. J. Mattausch, and W. Hanke, P h y s .R e v .B 25,
2867 (1982 ).
[36] J. C. E. Rasch, T. Stemmler, B. Muller, L. Dudy, and R. Manzke,
Phys. Rev. Lett. 101,237602 (2008 ).
[37] CRC Handbook of Chemistry and Physics , 84th ed., edited by
D. R. Lide (CRC Press, Boca Raton, FL, 2003), Sec. 9.
[38] M. Gurvitch, P h y s .R e v .B 24,7404 (1981 ).
[39] M. Naito and S. Tanaka, J. Phys. Soc. Jpn. 51,219(1982 ).
[40] R. M. Fernandes, E. Abrahams, and J. Schmalian, Phys. Rev.
Lett.107,217002 (2011 ).
[41] J. Merino and R. H. McKenzie, Phys. Rev. B 61,7996 (2000 ).
[42] E. Morosan (unpublished).[43] Y . Yoon, K. Ganapathi, and S. Salahuddin, Nano Lett. 11,3768
(2011 ).
[44] M. Moustafa, T. Zandt, C. Janowitz, and R. Manzke, Phys. Rev.
B80,035206 (2009 ).
045125-7 |
PhysRevB.76.085315.pdf | Effects of photon and thermal coupling mechanisms on the characteristics of self-assembled
InAs/GaAs quantum dot lasers
C. Y. Jin, *H. Y. Liu, K. M. Groom, Q. Jiang, and M. Hopkinson
Department of Electronic and Electrical Engineering, EPSRC National Center for III-V Technologies, University of Sheffield,
Sheffield S1 3JD, United Kingdom
T. J. Badcock, R. J. Royce, and D. J. Mowbray
Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
/H20849Received 23 April 2007; published 9 August 2007 /H20850
The relative contributions of the photon and thermal coupling mechanisms to the behavior of self-assembled
InAs/GaAs quantum dot lasers are studied. A theoretical model, which takes into account a photon couplingprocess between the ground and first excited states of different sized dots, is proposed to fully explain thetemperature dependence of the threshold current density /H20849J
th/H20850of both undoped and p-doped lasers. The simu-
lation results suggest that the carrier distribution between the different energy states in a dot is modulated bythe intradot thermal excitation of carriers. This process, when combined with the photon coupling mechanism,can account for the negative characteristic temperature /H20849T
0/H20850appearing in different temperature ranges for
undoped and p-doped devices. Thermal coupling, which involves thermal carrier escape and recapture among
different dots, has also been studied. Below threshold, thermal coupling is found to be significant but isweakened as threshold is approached because of the decreased carrier lifetime. Near and above threshold, thephoton coupling mechanism is important and can be used to model the different temperature behaviors of thelasing spectra observed experimentally for the undoped and p-doped lasers.
DOI: 10.1103/PhysRevB.76.085315 PACS number /H20849s/H20850: 42.55.Px, 73.63.Kv
I. INTRODUCTION
In the early 1980s, a three-dimensional confinement struc-
ture, which was initially named a “quantum box”, was theo-retically proposed to eliminate the temperature dependenceof the lasing threshold in semiconductor devices.
1,2Since the
1990s, the use of strain-induced self-assembled quantumdots /H20849QDs /H20850has made it possible to form defect-free, three-
dimensional confinement structures suitable for applicationin semiconductor lasers.
3The first demonstration of a QD
laser based on self-assembled QDs was reported by Kirs-taedter et al. in 1994.
4Following initial results, the perfor-
mance of QD lasers was improved considerably, with a sig-nificantly lower threshold than for quantum well lasersachieved by many groups.
5–7However, due to the existence
of several nonideal factors, for example, finite-potentialbarriers,
5excited QD states /H20849ES/H20850,8and inhomogeneous
broadening of the optical transitions,9the threshold current
density /H20849Jth/H20850of present self-assembled QD lasers exhibits
characteristic temperatures /H20849T0/H20850departing significantly from
the infinite value predicted for an ideal QD laser.1
A negative T0/H20849a decreasing threshold current with in-
creasing temperature /H20850has been typically observed in un-
doped QD lasers at low temperatures /H20849/H33355200 K /H20850and has been
explained by a thermal coupling model /H20849or thermal redistri-
bution model /H20850involving a transition from a nonequilibrium
to an equilibrium carrier distribution within the QDs of theensemble.
9,10Evidence for this carrier thermal coupling pro-
cess has been provided by photoluminescence /H20849PL/H20850spectra
exhibiting two unusual behaviors with increasing tempera-ture: /H208491/H20850a spectral linewidth narrowing
11,12and /H208492/H20850a wave-
length redshift in addition to the normal redshift arising fromthe band gap shrinkage.
13,14Additional evidence for spectrallinewidth narrowing has been obtained from the observation
of thermal broadening phenomenon in laser emission spectraat low temperatures /H20849/H33355200 K /H20850.
15By comparing the
temperature-dependent threshold current and laser emission
spectra of two QD lasers with different barrier heights, fur-
ther evidence has been obtained supporting the thermal cou-pling model.
16Theoretical analysis of this behavior has also
been undertaken based on two major models: the masterequation model
17,18and the rate equation model.19However,
although the thermal coupling model is able to successfullyexplain the negative T
0, the mechanism responsible for the
low-temperature broadening of the laser emission spectra isstill the subject of much debate. A possible problem with thethermal coupling model is whether interdot carrier transfer issufficiently fast in comparison with the short carrier lifetimenear threshold.
20,21Moreover, the additional wavelength red-
shift phenomenon has only been observed in PL and has notyet been reported for laser emission spectra.
To improve the T
0value at and above room temperature
/H20849RT/H20850in 1.3 /H9262m emitting QD lasers, p-type modulation dop-
ing has been incorporated with the aim of reducing the ef-fects of hole excitation out of the lasing state.
22,23A very
high or even infinite T0has been demonstrated using this
approach.24–26Further improvement of the room-temperature
performance of 1.3 /H9262m emitting QD lasers was reported
recently27,28using a combination of p-type modulation dop-
ing and high-growth-temperature GaAs spacer layers placedbetween the QD layers.
29,30In this and other work,25an in-
finite or negative T0has been reported, extending to tempera-
tures as high as /H1101150 °C.
While the reduction of hole excitation out of the lasing
state should improve T0, it cannot explain an infinite or nega-
tive value, and hence other possible mechanisms have beenPHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
1098-0121/2007/76 /H208498/H20850/085315 /H2084912/H20850 ©2007 The American Physical Society 085315-1considered. Experimental results from a measurement of the
turn-on delay time and the unamplified spontaneous emissionhave been interpreted in terms of a decrease in both the non-radiative Auger and radiative recombination of the QDs, andthis has been proposed as the process responsible for theinfinite or negative T
0around RT.25,31However, possible
physical mechanisms for these effects are unclear. Alterna-tively, it has been proposed that the decrease of the thresholdcurrent density results from an interdot carrier thermal redis-tribution, which still occurs around RT in p-doped structures
because the positive charged QDs, a result of the extrinsicholes, increase the confinement energy for electrons. Re-cently, a thermal redistribution delayed to /H11011270 K in a
p-type QD laser has been observed by measuring the gain
spectra at the transparency point
32and by the behavior of
low-injection current EL spectra.33These results have been
interpreted as evidence for the translation of the thermal car-rier redistribution induced negative T
0region to higher tem-
perature in p-type-doped devices. However, the thermal-
carrier redistribution process is a one-way effect withincreasing temperature /H20849the thermal redistribution can only
increase with increasing temperature /H20850and hence cannot ex-
plain the observation of a positive T
0inp-doped QD lasers at
low temperature, a region where a negative T0exists in un-
doped QD lasers. In addition, our previous work suggestedthat a thermal-carrier redistribution process cannot solely ex-plain the behavior of the temperature-dependent PL and thelaser emission spectra.
28To fully explain the different tem-
perature behaviors of p-doped and undoped QD lasers, a
photon coupling model has been developed.28
The purpose of this paper is to study the relative contri-
butions of the thermal coupling mechanism /H20849TCM /H20850and the
photon coupling mechanism /H20849PCM /H20850to the behavior of self-
assembled QD lasers. A theoretical model which includesboth the TCM and PCM is developed to explain the variousexperimental observations reported for QD lasers. The line-width and emission wavelength of the temperature-dependent optical spectra, both below and above threshold,have been investigated experimentally and simulated usingour model. Via a detailed comparison between p-doped and
undoped devices, we find that although the TCM model canexplain some of the experimental behavior, the PCM pro-vides a better description of the full range of experimentaldata. Our main conclusion is that the TCM is dominant forsubthreshold behavior, whereas the PCM makes a major con-tribution to both the threshold performance and lasing behav-ior.
The paper is organized as follows: We first describe the
samples and the experimental details in Sec. II. In Sec. III,we present the theory developed to describe the temperaturebehavior of QD lasers. In Sec. IV, we discuss the relativecontributions of the TCM and PCM. Finally, we summarizeour conclusions in Sec. V.
II. EXPERIMENTAL DETAILS
Both p-type modulation doped and undoped QD laser
structures were grown by molecular beam epitaxy in an Ox-ford Instruments V90H system. The p-type modulationdoped device consists of five layers of InAs QDs, each
grown within an 8 nm In
0.15Ga0.85As quantum well to give a
dot-in-a-well /H20849DWELL /H20850structure.5Each DWELL is sepa-
rated by 50 nm GaAs spacer layers. These dot and well pa-rameters have been shown to optimize the optical propertiesof the InAs QDs.
34In the GaAs spacer layer, 6 nm of GaAs
is doped with Be at a level 1.0 /H110031018cm−3, with the doped
layer separated from the DWELL by 9 nm of undoped GaAs.This doping density results in approximately 15 acceptorsper QD. An identical structure but with undoped spacer lay-ers was grown for comparison. The growth temperature was510 °C for the In-containing layers. Following the InAs QDsand the InGaAs well, the initial 15 nm of the GaAs spacerlayer /H20849SPL /H20850was deposited at 510 °C, following which the
temperature was increased to 580 °C for the remainder ofthe GaAs SPL. This low-to-high growth temperature step iscritical for the growth of high quality structures without pro-ducing a shift in the wavelength from the required1.3
/H9262m.29,35The active region was grown at the center of an
undoped 150 nm GaAs/AlGaAs waveguide with n-type
lower and p-type upper cladding layers consisting of 1.5 /H9262m
Al0.4Ga0.6As deposited at 620 °C. A heavily doped 300 nm
p+-GaAs contact layer completed the growth.
Shallow ridge waveguide lasers, as shown in Fig. 1, were
fabricated by a SiCl 4inductively coupled plasma technique,
with etching below the p-doped AlGaAs cladding layer. La-
ser cavity lengths were 3 mm with as-cleaved facets. Lasercharacteristics were measured under pulsed injection/H208495
/H9262s,10 kHz /H20850from 60 to 380 K. The heavily doped p+top
layer was etched off in PL test structures as it is strongly
absorbing at the emission wavelength of the QDs.
The temperature dependence of Jthfor both the p-doped
and undoped lasers is plotted in Fig. 2. The inset shows laser
emission spectra recorded at RT for an injection current 1.1times the threshold current. The peak wavelengths of thelaser emission are 1289 and 1302 nm for the undoped and
p-doped devices, respectively. For the undoped QD laser, the
threshold current density decreases with increasing tempera-ture between 130 and 220 K, resulting in a negative T
0.10
Above /H11011220 K, Jthincreases gradually with increasing tem-
perature, with a more rapid increase occurring above 320 K.
FIG. 1. A schematic diagram of the p-type modulation doped
QD laser structure.JIN et al. PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-2For the p-doped laser, the threshold current density initially
increases gradually up to 200 K. Above 200 K, a negative T0
occurs between 220 and 320 K /H20849−50–50 °C /H20850. This is fol-
lowed by an abrupt increase above 320 K. A threshold cur-
rent density of 48 A cm−2is achieved at RT for the p-doped
structure.
III. THEORY
A. Thermal coupling mechanism and photon coupling
mechanism
Figure 3shows a schematic diagram of both the TCM and
PCM in self-assembled QD lasers. In the thermal couplingprocess, carriers in smaller dots, which are assumed to havehigher energy levels, are thermally excited into the barriersand are captured into larger dots which are generally thoseinvolved with the lasing. These additional carriers may in-crease the peak intensity of the optical spectra and hence themaximum gain for a given injection level. The concentrationof carriers into one subset of QDs at high temperatures, com-pared to a uniform population at low temperatures, has beensuggested as an explanation for the decrease of J
thin QD
lasers as the temperature is increased.
Thermal coupling is a nonideal process resulting from the
finite-potential barriers and the inhomogeneous broadeningof the optical transitions. If the gain spectrum of a QD laserwas dominated by the TCM, as shown in the left part of Fig.3, three physical effects should be observed with increasing
temperature and hence coupling: /H208491/H20850a spectral linewidth nar-
rowing, /H208492/H20850a spectral intensity increase, and /H208493/H20850a redshift of
the emission maximum in addition to the normal redshiftarising from band gap shrinkage. Thermal coupling is amonotonic process and can only increase with increasingtemperature.
In the photon coupling process, we first assume that pho-
tons generated by the QDs can be either amplified or ab-sorbed by transitions from both the ground state /H20849GS/H20850in
similar sized dots and the excited state /H20849ES/H20850of larger dots;
this process requires the presence of a distribution of QD
sizes, which is unique to self-assembled quantum dot sys-tems in that the optical properties comprise an inhomoge-neous distribution of discrete states. Absorption or amplifi-cation occurs between GS and ES transitions, which areseparated by less than the homogeneous broadening, /H6036/H9003
h/H9263.36
Consequently, the total modal gain is the sum of the GS and
ES gains at the relevant energy.
To re-emphasize the main physics of this model, the ES
from larger dots which lie within the homogeneous broaden-ing of the energy of the GS lasing transition contributes tothe gain or loss at the lasing wavelength. This effect initiallyappears insufficient to affect the threshold current density.However, the GS gain usually saturates at about one-third ofits possible maximum value around RT.
37At any tempera-
ture, the threshold gain will be close to this value, while theES gain can vary over a wide range, from almost the maxi-mum ES absorption a
maxto the threshold gain gthas the
thermal-carrier population of this state changes. Because themaximum ES absorption a
maxis typically of order twice the
maximum gain of the GS, the contribution from the ES tran-sition at the lasing energy can be significant. As the energeticspacing of the hole levels is believed to be of the order of afew meV, compared to many tens of meV for the electrons, itis the thermal excitation of holes which is expected to makethe major contribution to this process.
38The right-hand side
picture in Fig. 3is plotted with parameters used in the fol-
lowing numerical simulations and with a maximum ES ab-sorption at the lasing wavelength,
/H9251ES. It is seen that with
these parameters, a significant contribution from the ES oc-curs at the lasing wavelength.
The PCM is a nonideal process resulting from the exis-
tence of the ES levels and the inhomogeneous broadening ofthe discrete optical transitions. If the gain spectra of a QDlaser were dominated by the PCM, as shown in the righthand side picture in Fig. 3, two effects should be observed
with increased coupling, namely, /H208491/H20850a spectral intensity de-
crease, which is due to the ES absorption at the lasing wave-length, and /H208492/H20850a wavelength redshift, which is due to the
asymmetry of the ES distribution around the lasing wave-length. For the PCM, either a coupling increase or decreaseis possible with increasing temperature, reflecting either in-creased ES occupation by carrier excitation from the GS orFIG. 2. Measured threshold current densities for both the
p-doped and undoped lasers as a function of temperature. The inset
shows laser emission spectra measured for a current I=Ith/H110031.1 at
RT for both devices.
FIG. 3. Schematic diagrams of the thermal coupling mechanism
/H20849TCM /H20850and photon coupling mechanism /H20849PCM /H20850.EFFECTS OF PHOTON AND THERMAL COUPLING … PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-3decreased ES occupation due to carrier escape to the barriers.
When the coupling decrease in the PCM presents with in-creasing temperature, a spectral intensity increase and wave-length blueshift should occur.
Within both models, a negative T
0, indicating a decrease
ofJth, should be reflected by an increase in the optical spec-
tral intensity at the lasing wavelength. If we compare themain features of the TCM and PCM, a negative T
0indicates
a coupling increase in the TCM, which should be accompa-nied by a wavelength redshift. In contrast, a negative T
0re-
quires a coupling decrease within the PCM, and this shouldbe accompanied by a wavelength blueshift. Therefore, from astudy of the temperature-dependent peak wavelength of thelaser emission, it should be possible to deduce the relativeimportance of the TCM and PCM.
An additional means by which the relative contributions
of the TCM and PCM can be investigated is via their tem-perature dependence over a wide temperature range. The
TCM is a monotonic process which can only result in anenhanced effect with increasing temperature. Therefore, thismechanism can only account for a negative T
0and not a
positive T0. When using the TCM to explain the negative T0
in both p-doped and undoped devices, a significant delay in
the critical temperature for the onset of the thermal redistri-bution of carriers needs to be assumed for the p-doped de-
vices. In addition, the TCM cannot explain the positive T
0
observed at low temperatures in p-doped devices /H20849see Fig.
2/H20850. Such behavior requires the introduction of an entirely
different process, for example, increased Augerrecombination.
25However, at present, this is an entirely phe-
nomenal explanation,32with the physical mechanism respon-
sible for an increasing Auger recombination unknown.
The PCM, however, can either be enhanced or reduced
with increasing temperature. Hence, by appropriately apply-ing this model, both the negative T
0and the positive T0be-
haviors of a p-type modulation doped laser can be explained.
In analysis the temperature behavior of the PCM, we intro-duce a second assumption that the thermal excitation of holesto higher QD energy levels occurs at lower temperatures thanfor electrons. This is a consequence of the more closelyspaced hole levels. It is assumed that there are two criticaltemperatures, T
hand Te, at which hole and electron thermal
excitation to higher energy levels starts to occur. The secondassumption requires that T
h/H11021Te.
For p-doped lasers, a large number of holes are released
into the hole states of the QDs. Below Th, the hole ES is fully
occupied39and hence the ES absorption /H9251ESin Fig. 3is
blocked. Between Thand Te, the hole occupancy of the ES
decreases due to thermal excitation to higher hole levels. Asa result, state blocking of the ES absorption decreases andhence there is an increase in J
th, which causes a positive T0
below Te. Above Te, due to the thermal excitation of elec-
trons into the ES, absorption by the ES transition is increas-ingly blocked, resulting in a decreasing J
thand a negative T0
value at RT. This decrease in Jthcontinues until carrier exci-
tation to states in the barrier, followed by recombination inthe barriers, and becomes significant, resulting in the ob-served abrupt increase in J
th. In contrast, for the undoped
laser, below Th, the electron and hole ESs are essentially
unoccupied, so there is a strong ES absorption at the lasingwavelength. Above Th, the ES absorption is gradually
blocked by the thermal excitation of holes from the GS, re-sulting in a decrease of J
th/H20849there is no parasitic recombina-
tion via the ES because there are no electrons in this state /H20850
and the observed negative T0at low temperatures. Above Te,
the ES absorption continues to be blocked, but recombina-tion via the ES is now possible, resulting in a weakening ofthe decrease of J
th. Eventually, excitation to the barriers be-
comes significant and Jthstarts to increase. Hence, the PCM
is able to predict the occurrence of a negative T0over very
different temperature ranges for undoped and p-doped QD
lasers.
The PCM can also explain reports of a decrease in both
the nonradiative Auger recombination31and radiative
recombination25with increasing temperature via the pre-
dicted decrease of the GS occupancy. In addition, the in-crease of the nonradiative Auger recombination at low tem-peratures, which has been reported in some studies of
p-doped lasers,
25can be explained by the GS carrier density
increase required to keep a constant gthwhen the ES absorp-
tion increases in the p-doped device.
B. Rate equations
To theoretically investigate the relationship between the
TCM and PCM, a theoretical model which can describe boththe mechanisms has been developed. In Ref. 28, we de-
scribed a rate equation model to study the PCM assuming anequal distribution of carriers within different QDs. In thiswork, we have further divided the QDs into subgroups. If welabel different QD subgroups with j=1,2,..., J, assuming an
equal distribution of carriers within each QD subgroup, therate equations for electrons can be written as follows:
/H11509nr
/H11509t=I
qVa−/H20858
jVdotsj
Va/H20849Rcapj−Rescj/H20850
−/H20858
i=w,b/H20873Vi
Vani
/H9270i+L
Va/H20858
/H9263/H9003igiS/H20874, /H208491/H20850
/H11509nESj
/H11509t=/H20849Rcapj−Rescj/H20850−/H20849Rrelj−Rexj/H20850−nESj
/H9270ESj−L
Vdotsj/H20858
/H9263/H9003dotsgESjS,
/H208492/H20850
/H11509nGSj
/H11509t=/H20849Rrelj−Rexj/H20850−nGSj
/H9270GSj−L
Vdotsj/H20858
/H9271/H9003dotsgGSjS, /H208493/H20850
where nr=/H20858i=w,b/H20849Vi/Va/H20850nirepresents the normalized summa-
tion of electron densities in the well and barrier; nESjand nGSj
are the electron densities in the ground and excited states; Iis
the injected current, /H9270b,/H9270w,/H9270ESj, and /H9270GSjare the carrier life-
times in the different regions; Gb,Gw, andGdotsare the opti-
cal confinement factors; gb,gw,gGSj, and gESjare the material
gains of the barriers, wells, and ground and excited states;
Va,Vb,Vw,Vdotsjare the volumes of the active region, barri-
ers, wells, and the jthsubgroup of dots; Lis the cavity length;
and Srepresents the summation of the forward and backward
propagating light densities: /H20849S++S−/H20850.S±obey the propagation
rate equations:JIN et al. PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-4/H11509S±
vg/H11509t±/H11509S±
/H11509z=GcS±+Rsp, /H208494/H20850
where Gc=/H20858j/H20858i=b,w,GS,ES/H9003igij−/H9251is the coupled optical modal
gain, which describes the photon coupling process betweenthe GS and first ES. We simply assume the same opticallosses in different energy regions.
The second assumption of the PCM model is that the
thermal excitation of holes to higher QD energy levels oc-curs at lower temperatures than that for electrons. In thenumerical model, the hole distribution in the valence band isassumed to obey the Fermi-Dirac statistics above 80 K,whereas the electron distribution in the conduction band isdetermined by dynamical processes as described by the car-rier rate equations /H208491/H20850–/H208493/H20850. Charge neutrality is ensured by
the following equation, which must be satisfied in each re-gion of the device:
n=
/H20858
i=w,b,dots/H20873Vi
Va/H20874ni=p−pA, /H208495/H20850
where Viand niare the volume and electron densities for the
barriers, wells, and dots, and pAis the density of acceptors.
We simply assume that all holes released from the Be impu-rities are captured into the dots, well, and barrier regions.
C. Gain and spontaneous emission
The modal gain of the QDs is described by40
gij/H20849/H9263/H20850=1
hv/H9273i
V0/H9266e2/H6036NL
cnr/H92550m02/H20885
−/H11009/H11009
/H20841MB/H208412/H20841Mcv/H208412/H20849fic,j+fiv,j−1/H20850
/H11003Gij/H20849E/H11032/H20850Li/H20849h/H9263,E/H11032/H20850dE/H11032,i=GS,ES, /H208496/H20850
where /H9273i=2 and 4 give the degeneracy of the ground and
excited states, respectively, V0is the single dot volume, NLis
the dot layer number, nris the refractive index, Mcvis the
wave function overlap, MBis the Bloch matrix element, fc
and fvare the occupation factors of the electron and hole,
Gij/H20849E/H20850is the Gaussian distribution function given by
Gij/H20849E/H20850=1
/H208812/H9266/H9268 iexp/H20875−/H20849E−Eij/H208502
2/H9268i2/H20876,i=GS,ES, /H208497/H20850
where /H9268iis the width of the Gaussian distribution, and
Li/H20849h/H9263,E/H20850is the Lorentzian line shape given by
Li/H20849h/H9263,E/H20850=/H6036/H9003 cv,i//H9266
/H20849E−h/H9263/H208502+/H20849/H6036/H9003 cv/H208502,i=GS,ES, /H208498/H20850
where /H9003cv,iis the carrier polarization dephasing rate.
The modal gains of the InGaAs wells and GaAs barriers
are calculated as
gi/H20849/H9263/H20850=/H9003iai/H9267i/H20849fic+fiv−1/H20850,i=w,b, /H208499/H20850
where aiis the gain coefficient and /H9267iis the reduce density of
states.
The spontaneous emission coupled into different wave-
lengths in Eq. /H208494/H20850can be defined as41Rsp/H20849/H9263/H20850=/H20858
j/H20858
i=b,w,GS,ESEsti,j/H20849v/H20850/H9004/H9263, /H2084910/H20850
where /H9004/H9271is the frequency separation between the cavity
modes and Estis the rate per unit length of stimulated emis-
sion, which can defined as follows:
Esti,j/H20849/H9263/H20850=gij/H20849/H9263/H20850ficfiv
fic+fiv−1,i=w,b,GS,ES. /H2084911/H20850
D. Carrier transfer process
Carrier transfer processes are defined in Fig. 4. The rate of
carrier capture from the InGaAs wells into the dots is de-scribed by
42
Rcapj=nw/H208491−fESc,j/H20850
/H9270cap, /H2084912/H20850
where fESis defined as fESj=nESjV0//H9273ES. The rate of carrier
escape from the dots into the InGaAs wells is described by
Rescj=nESj/H208491−fw0/H20850
/H9270esc, /H2084913/H20850
where /H9270escis defined as /H9270esc=/H9270capexp(/H20849Ew−EESj/H20850/kT),43with
Ew−EESjdenoting the energy difference between excited
states and the InGaAs well band edge. The rate of carrierrelaxation from the excited states to the ground state is givenby
R
relj=nESj/H208491−fGSj/H20850
/H9270rel, /H2084914/H20850
where fGSjis defined as fGSj=nGSjV0//H9273GS. The rate of carrier
excitation from the ground state into the excited states isgiven by
R
exj=nGSj/H208491−fESj/H20850
/H9270ex, /H2084915/H20850
FIG. 4. A schematic diagram of electron transport processes in
the conduction band of a quantum dot laser.EFFECTS OF PHOTON AND THERMAL COUPLING … PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-5where /H9270exis defined as /H9270ex=/H9270relexp(/H20849EESj−EGSj/H20850/kT), with
EESj−EGSjdenoting the energy difference between the ground
state and excited states.
E. Bimodal grouping of dot distributions and simulation
results
An obvious method for grouping the dots is to divide
them into subgroups based on their emission wavelength.36
However, the main behavior difference, which can distin-guish between the TCM and PCM, is an opposite wavelengthshift of the emission in a negative T
0temperature region. The
wavelength shift is likely to be very small, about 10 nm overthe full temperature range. To simulate the wavelength shiftaccurately using relatively small temperature steps, for ex-ample 20 K, a huge number of dot groups would need to betaken into account. This would be very computer intensiveand so simplifications have to be found.
In this work, we divide the dots into two groups, each
with a Gaussian distribution of energy levels. This impliesJ=2. These two groups reflect a bimodal distribution of QDs,
which has been observed in many of our samples and inother reports.
35,44Based on an atomic force microscopy
analysis of our samples, we have assumed that a second dis-tribution of QDs exists, which contains /H1101110% –30% of the
total number of QDs and represents QDs with a slightlysmaller size. Although this bimodal grouping of the dot dis-tribution may somewhat underestimate thermal effects withinthe same dot subgroup, it can successfully describe the con-tinuous change of the spectral wavelength with increasingtemperature while significantly saving on computationaltime.
The parameters used in the simulations presented in this
paper are as follows. The dot density is
/H9267dots=4.3
/H110031010cm−2, the height of the dots is hdots=6 nm, and the
width of the dots is wdots=15 nm. The cavity length is L
=3 mm, the width W=10/H9262m, and the thickness d=0.4/H9262m.
The width of the In 0.15Ga0.85As well is dw=8 nm, the width
of the p-doped layer dp=6 nm, and the distance between
these two layers dwp=9 nm. Optical confinement for the bar-
riers is Gb=0.06, for the wells Gw=0.01, and for the dots
Gdots=0.0008. The QD layer number is NL=5. The facet re-
flectivity is R1=R2=0.3, optical loss /H9251=2 cm−1, and refrac-
tive index nr=3.3. The homogeneous broadening is assumed
to have a Lorentzian form with /H6036Gcv=6 and 12 meV for the
GS and ES, respectively. Within the models used, it is foundthat the lasing threshold is relatively insensitive to the homo-geneous broadening. Hence a constant, temperature-insensitive homogeneous broadening is assumed, althoughexperimentally, it has been shown that the homogeneousbroadening does increase steadily over the temperature rangeconsidered in the simulations.
45,46A Gaussian inhomoge-
neous broadening is assumed with values /H9268GS=16 meV and
/H9268ES=30 meV for the GS and ES, respectively. The same
broadening is used for both subsets of QDs. The density ofacceptors for the p-doped device is p
A=1.5/H110031018cm−3.W e
assume that all holes from the Be impurities are ionized into
the barriers. The band gap of the GaAs barriers is Egb
=1.424 eV, and the band gap of the In 0.15Ga0.85As well is1.19 eV. In the simulation, only the electron ground state and
the first excited state are included, whereas three hole levelsare included. The separations of the electron and hole statesare taken as 53 and 15 meV, respectively. The separation ofthe central energies of the two bimodal QD subsets is takenas 34 meV. It is assumed that 20% of the dots are in thesecond smaller size subset.
Figure 5shows the results of the simulation of the tem-
perature dependence of J
thfor both the p-doped and undoped
devices. These were obtained using the theoretical approachdiscussed above. A negative T
0is predicted between 220 and
320 K for the p-doped laser. In addition, a negative T0is
predicted for the undoped laser below 200 K. The results ofthe simulation show good agreement with the experimentalresults presented in Fig. 2. The inset of Fig. 5shows the
results of a simulation performed without the photon cou-pling process, obtained by simply ignoring the ES gain in therate equation model. Only a very weak negative T
0behavior
is predicted in the temperature range from 140 to 220 K forthep-doped device, and 80–120 K for the undoped device.
This negative T
0results entirely from the TCM and is sig-
nificantly weaker than the experimentally observed behavior.In addition, the low temperature positive T
0behavior ob-
served for the p-doped device is only reproduced when the
PCM is included in the simulations.
The temperature dependence of the peak GS gain, nor-
malized to the threshold gain, for an injection current I=Ith
/H110031.1 is plotted in Fig. 6. The total optical gain, which is the
sum of the GS gain and the ES gain at the lasing wavelength,equals the threshold gain at an injection level just abovethreshold. The normalized ES gain at the lasing wavelengthwill, therefore, be 1 minus the normalized peak gain of theGS /H20849a small wavelength shift of the peak gain is neglected
here /H20850. For the undoped device, the ES contribution at the
lasing wavelength gives a strong absorption of −0.26 g
that
80 K /H20849hence the normalized GS gain has a value greater than
1/H20850, but with increasing temperature, it increases and eventu-
ally saturates to a positive value /H110150.16 gthabove 200 K. For
thep-doped device, the ES gain at the lasing wavelength isFIG. 5. Simulated temperature dependence of the threshold cur-
rent densities for both the p-doped and undoped lasers. The inset
shows the results of the simulation without the inclusion of thephoton coupling process.JIN et al. PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-6zero at 80 K, corresponding to full Pauli blocking of the
transition. With increasing temperature, the ES gain de-creases to an absorption value of −0.1 g
that 200 K, followed
by an increase and saturation to a positive value /H110150.16
/H11003gthabove 350 K. These calculations confirm the first as-
sumption of the photon coupling model that the ES gain atthe lasing wavelength can make a significant contribution tothe total gain and can hence have a significant effect on theperformance of the laser device.
The electron and hole occupancies of the ES for both
devices are calculated as a function of temperature in Fig.7/H20849a/H20850for an injection current I=I
th/H110031.1. For the p-doped la-
ser, below 200 K, the hole occupation of the ES decreasesdue to intradot thermal excitation to higher confined states.This gives an increase in the ES absorption, as shown in Fig.6, and hence a positive T
0. Above 200 K, the electron occu-
pation of the ES increases due to intradot thermal excitationfrom the GS to the ES. This gives a decrease in the ESabsorption and hence a negative T
0near RT. In contrast, for
the undoped laser, below 200 K, there is a strong increase inthe electron occupation of the ES. As a result, a decrease ofthe ES absorption and a negative T
0occur. The hole occupa-
tion level fluctuates very weakly in the undoped device be-cause an equilibrium carrier distribution /H20849Fermi-Dirac statis-
tics/H20850is assumed; this may slightly underestimate the holes’
behavior at very low temperature /H20849/H11011100 K /H20850. Figure 7reveals
that, compared to the undoped device, the intradot thermal
excitation of electrons into the ES is delayed in the p-doped
device and this results in the negative T
0appearing in a dif-
ferent temperature range for the two devices. In addition, theintradot thermal excitation of holes into the ES accounts fora positive T
0appearing at very low temperatures in the
p-doped device.
To investigate the reason why the excitation of electrons
in the p-doped laser is delayed to higher temperatures, the
electron and hole occupancies of the GS for both devices areplotted as a function of temperature in Fig. 7/H20849b/H20850. When the
carrier occupation factors for both holes and electrons aresummed, a similar behavior to that exhibited by the gain, asshown in Fig. 6, is obtained. However, the electron occupa-
tion factor is lower in the p-doped device over the tempera-
ture range below 300 K. This is because the extrinsic holesin the QDs /H20849see the hole occupation factor /H20850result in fewer
electrons being required to reach g
th. This reduced electron
occupation decreases the excitation rate over the same tem-perature range because the intradot excitation of electrons isproportional to the electron density in the GS according to
Eq. /H2084915/H20850. Therefore, the onset for electron excitation out of
the GS is shifted to higher temperatures. In addition thesimulations show that the total GS occupation factor for bothelectrons and holes is approximately 1.35 for both devicesover the temperature range studied. Referring to the expres-sion for the gain in Eq. /H208496/H20850, the factor /H20849f
c+fv−1/H20850/H11015 0.35 at
threshold, which is near 1/3 of its maximum possible value
/H20849=1.0 /H20850. Hence, the GS threshold gain is approximately 1/3 of
the maximum gain, in agreement with the experimental re-
port in Ref. 37. This finding also explains why the ES ab-
sorption is significant in the simulations shown in Fig. 6.
From Figs. 5–7, it can be seen that the value of Th/H20849the
temperature where significant hole excitation out of the GSstarts to occur /H20850is less than 80 K for both the undoped and
p-doped QD lasers; this is below the regime where the nega-
tive and positive T
0behavior, start to occur for the undopedFIG. 6. Simulated maximum gain of the GS transition as a func-
tion of temperature. The gain is calculated for an injection currentofI
th/H110031.1 and is normalized to the threshold gain.
FIG. 7. Simulations of /H20849a/H20850the ES and /H20849b/H20850the GS carrier occu-
pancies, for an injection current I=Ith/H110031.1, as a function of tem-
perature for both the undoped and p-doped QD lasers.EFFECTS OF PHOTON AND THERMAL COUPLING … PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-7and p-doped device, respectively /H20849see Fig. 2/H20850.Tehas a value
near 200 K for the p-doped device and is about 100 K for
the undoped device, where the occupation factor exceeds acertain value /H20849/H110110.2/H20850. These results indicate that the photon
coupling process, modulated by the delay of the intradot ex-
citation, is the main reason why the negative T
0region ap-
pears at very different temperatures for the two devices. Theresults described in this section suggest that the PCM can beapplied to explain the temperature dependence of J
thin both
p-doped and undoped QD lasers.
IV. FURTHER RESULTS AND DISCUSSIONS
A theoretical model and numerical simulations have been
applied to explain the very different temperature-dependentJ
thbehaviors of both undoped and p-doped QD lasers. Quan-
titative changes in the spectral intensity, which are respon-sible for the negative or positive T
0, have been analyzed. The
PCM has been shown to provide a more complete descrip-tion than the TCM in explaining the temperature dependenceofJ
thover the full experimental temperature range. However,
two other processes, spectral linewidth narrowing and awavelength redshift, both of which have been demonstratedas experimental support for the TCM,
11–14have not yet been
considered. In this section, these phenomena are studied,both below and above threshold, to further test the predic-tions of the TCM and PCM.
A. Spectral linewidth narrowing
The full width at half maximum /H20849FWHM /H20850of the PL spec-
tra for the undoped and p-doped structures is plotted in Fig.
8as a function of temperature. The FWHM of the p-doped
structure is 2–4 meV larger than that of undoped structure,but the temperature range over which there is a rapidly de-creasing FWHM is similar in both structures, from100 to 180 K. Above this temperature range, the FWHM ofthep-doped structure fluctuates near 35 meV, whereas the
FWHM of the undoped structures slowly increases from32 to 34.5 meV, between 180 and 300 K. Although theminimum of the PL linewidth is shifted to higher temperaturein the p-doped device, the behavior is not fully consistent
with the TCM. In particular, when compared to the tempera-ture dependence of J
thin Fig. 2, the PL linewidth of the
p-doped device reaches a minimum at a temperature
/H20849/H11011260 K /H20850below the temperature minimum of Jth
/H20849/H11011310 K /H20850. In addition, at low temperatures between 100 and
180 K, although both devices exhibit a PL linewidth narrow-
ing, their threshold current densities change in opposite di-rections. These experimental observations do not support acomplete quantitative agreement between the TCM and tem-perature behavior of the PL linewidth. To further study theseprocesses, the temperature dependence of the laser emissionspectra was studied. The inset of Fig. 8shows typical laser
emission spectra recorded for an injection current density of1.1 times J
th. The lasing spectra exhibit a gradual narrowing
with increasing temperature. This narrowing is similar forboth devices, which suggests a similar temperature depen-dence of the homogeneous linewidth.
Hence, neither the PL nor the lasing spectra provide con-
clusive evidence for the TCM occurring in different tempera-ture regions for the undoped and p-doped lasers. Moreover,
although the broadening of the laser emission spectrum ofboth devices is observed at low temperatures, the linewidthof the p-doped laser is narrower than that of the undoped
laser, whereas the PL linewidth of the p-doped structure is
broader. This difference may indicate that the spectral broad-ening mechanisms are intrinsically different in PL and lasing.As suggested above, linewidth narrowing of the PL with in-creasing temperature results from a carrier thermal redistri-bution between QDs, whereas the linewidth of the laseremission spectra is determined by the homogeneous broad-ening of the optical gain. It should also be noted that thelasing spectra of the p-doped device are much narrower than
those of the undoped device for temperatures above 77 K,indicating a larger homogeneous linewidth for the p-doped
device.
B. Thermal redistribution with different injections
As discussed above, the broadening of the lasing spectra
at low temperatures may result from the homogeneousbroadening of the optical gain, rather than effects related tothe TCM. One possible reason for this is that the contributionfrom the TCM is reduced as the injection current approachesthreshold. To study this behavior, the thermal redistributionof carriers between the two QD subsets for different injectionlevels has been calculated. A thermal coupling factor K
thermal
is introduced to describe the degree of TCM, as shown in
Fig.9. For a bimodal distribution of dots,
Kthermal =nGS2
nGS1. /H2084916/H20850
The dotted line in Fig. 9indicates Kthermal =1 and corre-
sponds to an equal distribution of carriers within the twosubsets of dots.FIG. 8. PL FWHM for both the p-doped and undoped QD lasers
as a function of temperature. The inset shows lasing spectra of the
p-doped and undoped QD lasers for a number of different tempera-
tures. The lasing spectra were recorded for a current I/H20849T/H20850=Ith/H20849T/H20850
/H110031.1.JIN et al. PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-8Figure 9shows the temperature dependence of the ther-
mal coupling factor for the p-doped QD laser calculated for
different levels of injection current from I=1 mA to Ith
/H110031.1. For a current of 1 mA, Kthermal decreases from 1.0 at
100 K to a minimum value of 0.48 at 280 K; this reflects athermal redistribution process between the two subsets ofQDs. Almost half of the carriers in the second, smaller subset
of dots are thermally excited into the barrier and some ofthese are recaptured into the first subset of larger dots. How-ever, with increasing current, the thermal redistribution pro-cess weakens significantly, with a minimum K
thermal of 0.9
achieved at 250 K for a current of Ith/H110031.1. This value for
currents just above the threshold suggests that the TCM isless significant at and above threshold in the p-doped device.
In Fig. 10, the thermal coupling factor for the undoped QD
laser is plotted. At a current of 1 mA, K
thermal decreases from
1.0 at 100 K to a minimum value of 0.52 at 220 K. Again,the strength of the thermal redistribution process is reducedsignificantly for a current of I
th/H110031.1. The reason for this
decreasing strength of the thermal redistribution process asthe current approaches threshold is because the faster carrierrecombination rate reduces the fraction of carriers able toescape from the subset of smaller QDs before radiative re-combination occurs.
In the low-temperature regime below 200 K, it can be
seen that the thermal coupling factor can reach a valuegreater than 1.0. A value greater than 1 reflects strong carrierspectral hole burning, which occurs at low temperatures near100 K due to blocked carrier excitations, as described in Eq./H2084915/H20850by the exponential factor 1/exp /H20849/H20849E
ES−EGS/H20850/kT/H20850with a
very small value of kT. This blocked carrier excitation may
be responsible for a slower polarization dephasing rate, aswell as a broader laser emission spectrum, which has beenpreviously discussed in relation to the homogeneous broad-ening model.
20The results of Figs. 9and10indicate that
while the TCM is an important process for injection levelsbelow threshold, its importance is less at and above thresh-old.
A comparison of Figs. 9and10indicates that the critical
temperature of the TCM is different in the p-doped and un-doped devices. At an injection current of 1 mA, the critical
temperature is 280–300 K in the p-doped device and 220 K
in the undoped device. These temperatures agree very wellwith experimental studies of a TCM delayed by /H1101170 K in
the EL results of Refs. 32and33, as well as the PL data in
the present work. The main reason for the delayed criticaltemperature predicted by the present simulations is due to adelayed electron escape process in the p-doped device. From
Eq. /H2084913/H20850, it is found that a slower electron escape process at
a given temperature requires a smaller ES occupancy. TheES electron occupancy is lower in the p-doped device com-
pared to the undoped device at injection levels far belowthreshold, because a faster spontaneous recombination rateoccurs due to the presence of the extrinsic holes. It should benoted that this delayed critical temperature is mainly affectedby the thermal escape process described in Eq. /H2084913/H20850, whereas
the delayed intradot thermal excitation in Fig. 7/H20849a/H20850is domi-
nated by the thermal excitation process described in Eq. /H2084915/H20850.
However, the reason for these two kinds of delay is similar,both of which result from reduced electron occupation fac-tors in the QD states of the p-doped device.
In addition to the processes included in the present simu-
lations, an increased electron confinement has been proposedas a mechanism for a delayed TCM in p-type devices.
25This
arises from the Coulomb attraction by the extrinsic holes.However, it is difficult to evaluate by how much the electronconfinement will be altered by the extrinsic holes, and in thepresent simulations, the same electron and hole confinementpotentials are assumed for both devices. Including a deeperelectron confinement potential in the simulation results, theTCM minimum shifts to a higher temperature, as expected.However, in the absence of an accurate knowledge of themagnitude of the increased confinement potential that resultsfrom the extrinsic holes, this modified potential has not beenincluded in the simulations presented in this paper.
With increasing current injection, the critical temperature
at which the minimum K
thermal occurs is reduced, for ex-
ample, to 220 K at threshold for the p-doped device. The
same phenomenon can be observed in the undoped deviceFIG. 10. The temperature dependence of the thermal coupling
factor Kthermal for the undoped QD laser and for different injection
levels.FIG. 9. The temperature dependence of the thermal coupling
factor K thermal for the p-doped QD laser and different current injec-
tion levels. The inset shows the definition of Kthermal .EFFECTS OF PHOTON AND THERMAL COUPLING … PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-9where the critical temperature changes from 220 to 160 K as
the injection current increases from 1 mA to threshold. TheK
thermal minima at threshold, predicted by the simulations
plotted in Figs. 9and10, occur at the same temperature as
the minima in Jthsimulated by only considering the TCM
/H20849see inset of Fig. 5/H20850. It can hence be concluded that the sig-
nificance of the TCM is weakened at high-injection currentsand on its own appears unable to fully account for the nega-tive T
0observed in both p-doped and undoped lasers.
C. Wavelength shift
Finally, the predictions of the TCM and PCM for the tem-
perature dependence of the emission wavelength are consid-ered. Low-current EL spectra and lasing spectra /H20849EL spectra
above threshold /H20850were measured for comparison with the
simulations. Figure 11shows the measured temperature de-
pendence of the EL emission for injection currents of I
=1 mA and I
th/H110031.1 for both devices. The gross shift of the
emission is caused by the thermal band gap shrinkage, whichis not known accurately enough to reliably remove it fromthe experimental data. Hence, it is necessary to look forsmall effects against this background shift via a comparisonof the emission at the two different injection currents. Rela-tive to the EL at 1 mA, the lasing emission exhibits a blue-shift below 200 K for the undoped device and above 220 Kfor the p-doped device. These are the temperature regions
where the negative T
0is observed for both devices. This
behavior agrees with the predictions of the PCM as discussedabove. The detailed behavior of the emission in the negativeT
0regions is shown in the insets of Fig. 11for both devices.
Relative to the low-injection current EL, the lasing emissionexhibits a blueshift of 10 meV for the undoped device and5 meV for the p-doped device.
Because the temperature dependence of the QD band gap
is not accurately known, it is not easy to simulate the emis-sion behavior shown in Fig. 11. However, if only the wave-
length difference between the undoped and p-doped devices
is considered, the effects of the temperature-dependent bandgap can be neglected if it is assumed that this is the same forboth devices. Figure 12shows the measured temperature de-
pendence of the emission difference between the undopedand p-doped devices for a range of injection currents. The
peak emission is taken as the maximum of the whole spectra,which contains contributions from both subsets of dots; thisis particularly significant in the low-injection spectra. Theinset shows a schematic of the quantity plotted in the mainpart of the figure. Below 200 K, the lasing emission is domi-nated by the photon coupling process, which gives a blue-shift in the undoped device and a redshift in the p-doped
device. As a result, the wavelength separation between thedevices increases with increasing temperature. In contrast,the low-current behavior is dominated by a thermal redistri-bution process, which gives a redshift for both the undopedand p-doped devices. However, the thermal redistribution
process is delayed in the p-doped device by /H1101170 K as dis-
cussed above. This delay results in the separation betweenthe two devices decreasing above /H11011170 K, and this separa-
tion does not return to its pre-170-K value until the thermalredistribution process is completed in the p-doped device.
The results plotted in Fig. 12reveal a very different behavior
for the lasing emission and subthreshold EL below 200 K.
Figure 12also contains the results of a simulation based
on the theoretical model described above and assuming abimodal distribution of dots. Assuming a wavelength separa-tion of 2 nm at 80 K between the lasing spectra of the un-doped and p-doped devices, the simulation results describe
well the experimental results for above threshold injection.The simulated lasing wavelength difference increases by11 nm between 80 and 220 K. Experimentally, the increaseis 10 nm between 80 and 190 K. For low-current injection,the simulated and experimental values are 2 and 4 nm, re-spectively. The differences between the experimental andsimulated results may result from an underestimation of thethermal-carrier redistribution between the subgroups of QDs.As in other simulations discussed above, the results pre-sented in Fig. 12indicate that the effect of the TCM is sig-
nificant at low-injection current but that the PCM is neededto fully explain the wavelength behavior above threshold.FIG. 11. The temperature dependence of the emission wave-
length for two injection currents and for both the p-doped and un-
doped QD lasers.FIG. 12. Temperature dependence of the difference between the
peak wavelengths of the undoped and p-doped QD lasers for both
low-current injection and injection above threshold.JIN et al. PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-10V. CONCLUSION
In summary, a theoretical model accounting for both ther-
mal coupling and photon coupling processes in self-assembled QD lasers has been presented. The presence of anegative T
0region where a spectral blueshift coexists in both
undoped and p-doped QD lasers can be explained by a pho-
ton coupling process between the GS and the first ES in dotsof different sizes. The intradot thermal excitation of electronsis delayed in a p-type device, a result of a reduced electronoccupation of the GS. This behavior when combined with the
PCM can account for the very different temperature depen-dence of J
thobserved in the two types of device. Below
threshold, the TCM results in a spectral linewidth narrowingand a redshift of the emission for both undoped and p-doped
QD devices with increasing temperature. This behavior isweakened as the injection level is increased. Near and abovethreshold, the PCM makes an increased contribution to thebehavior of the lasers.
*c.jin@sheffield.ac.uk
1Y. Arakawa and H. Sakaki, Appl. Phys. Lett. 40, 939 /H208491982 /H20850.
2M. Asada, Y. Miyamoto, and Y. Suematsu, IEEE J. Quantum
Electron. QE-22 , 1915 /H208491986 /H20850.
3D. Leonard, M. Krishnamurthy, C. M. Reaves, S. P. Denbaars,
and P. M. Petroff, Appl. Phys. Lett. 63, 3203 /H208491993 /H20850.
4N. Kirstaedter, N. N. Ledentsov, M. Grundmann, D. Bimberg, V.
M. Ustinov, S. S. Ruvinov, M. V. Maximov, P. S. Kop’ev, Z. I.Alferov, U. Richter, P. Werner, U. Gosele, and J. Heydenreich,Electron. Lett. 30, 1416 /H208491994 /H20850.
5G. T. Liu, A. Stintz, H. Li, K. J. Malloy, and L. F. Lester, Elec-
tron. Lett. 35, 1163 /H208491999 /H20850.
6I. R. Sellers, H. Y. Liu, K. M. Groom, D. T. Childs, D. Robbins,
T. J. Badcock, M. Hopkinson, D. J. Mowbray, and M. S.Skolnick, Electron. Lett. 40, 1412 /H208492004 /H20850.
7H. Y. Liu, D. T. Childs, T. J. Badcock, K. M. Groom, I. R. Sellers,
M. Hopkinson, R. A. Hogg, D. J. Robbins, D. J. Mowbray, andM. S. Skolnick, IEEE Photonics Technol. Lett. 17, 1139 /H208492005 /H20850.
8O. B. Schekin, G. Park, D. L. Huffaker, and D. G. Deppe, Appl.
Phys. Lett. 77, 466 /H208492000 /H20850.
9L. V. Asryan and R. A. Suris, Semicond. Sci. Technol. 11, 554
/H208491996 /H20850.
10A. E. Zhukov, V. M. Ustinov, A. Y. Egorov, A. R. Kovsh, A. F.
Tsatsulnikov, N. N. Ledentsov, S. V. Zaitsev, N. Y. Geordeev, P.S. Kopev, and Z. I. Alferov, Jpn. J. Appl. Phys., Part 1 36, 4216
/H208491997 /H20850.
11D. I. Lubyshev, P. P. González-Borrero, E. Marega, Jr., E. Petit-
prez, N. La Scala, Jr., and P. Basmaji, Appl. Phys. Lett. 68, 205
/H208491996 /H20850.
12Z. Y. Xu, Z. D. Lu, X. P. Yang, Z. L. Yuan, B. Z. Zheng, J. Z. Xu,
W. K. Ge, Y. Wang, J. Wang, and L. L. Chang, Phys. Rev. B 54,
11528 /H208491996 /H20850.
13L. Brusaferri, S. Sanguinetti, E. Grilli, M. Guzzi, A. Bignazzi, F.
Bogani, L. Carraresi, M. Colocci, A. Bosacchi, P. Frigeri, and S.Franchi, Appl. Phys. Lett. 69, 3354 /H208491996 /H20850.
14S. Sanguinetti, M. Henini, M. Grassi Alessi, M. Capizzi, P.
Frigeri, and S. Franchi, Phys. Rev. B 60, 8276 /H208491999 /H20850.
15A. Patane, A. Polimeni, M. Henini, L. Eaves, P. C. Main, and G.
Hill, J. Appl. Phys. 85, 625 /H208491999 /H20850.
16K. M. Groom, A. I. Tartakovskii, D. J. Mowbray, M. S. Skolnick,
P. M. Smowton, M. Hopkinson, and G. Hill, Appl. Phys. Lett.
81,1/H208492002 /H20850.
17M. Grundmann and D. Bimberg, Jpn. J. Appl. Phys., Part 1 36,
4181 /H208491997 /H20850.
18M. Grundmann, O. Stier, S. Bognar, C. Ribbat, F. Heinrichsdorff,and D. Bimberg, Phys. Status Solidi A 178, 255 /H208492000 /H20850.
19H. Huang and D. G. Deppe, IEEE J. Quantum Electron. 37, 691
/H208492001 /H20850.
20M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Saka-
moto, Phys. Rev. B 61, 7595 /H208492000 /H20850.
21M. Sugawara, K. Mukai, and Y. Nakata, Appl. Phys. Lett. 74,
1561 /H208491999 /H20850.
22O. B. Shchekin and D. G. Deppe, Appl. Phys. Lett. 80, 3277
/H208492002 /H20850.
23O. B. Shchekin, J. Ahn, and D. G. Deppe, Electron. Lett. 38, 712
/H208492002 /H20850.
24S. Fathpour, Z. Mi, P. Bhattacharya, A. R. Kovsh, S. S. Mikhrin,
I. L. Krestnikov, A. V. Kozhukhov, and N. N. Ledentsov, Appl.Phys. Lett. 85, 5164 /H208492004 /H20850.
25I. P. Marko, N. F. Masse, S. J. Sweeney, A. D. Andreev, A. R.
Adams, N. Hatori, and M. Sugawara, Appl. Phys. Lett. 87,
211114 /H208492005 /H20850.
26H. Y. Liu, S. L. Liew, T. Badcock, D. J. Mowbray, M. S.
Skolnick, S. K. Ray, T. L. Choi, K. M. Groom, B. Stevens, F.Hasbullah, C. Y. Jin, M. Hopkinson, and R. A. Hogg, Appl.Phys. Lett. 89, 073113 /H208492006 /H20850.
27T. J. Badcock, H. Y. Liu, K. M. Groom, C. Y. Jin, M. Gutiérrez,
M. Hopkinson, D. J. Mowbray, and M. S. Skolnick, Electron.Lett. 42, 922 /H208492006 /H20850.
28C. Y. Jin, T. J. Badcock, H. Y. Liu, K. M. Groom, R. J. Royce, D.
J. Mowbray, and Mark Hopkinson, IEEE J. Quantum Electron.
42, 1259 /H208492006 /H20850.
29H. Y. Liu, I. R. Sellers, T. J. Badcock, D. J. Mowbray, M. S.
Skolnick, K. M. Groom, M. Gutiérrez, M. Hopkinson, J. S. Ng,J. P. R. David, and R. Beanland, Appl. Phys. Lett. 85, 704
/H208492004 /H20850.
30C. Y. Jin, H. Y. Liu, T. J. Badcock, K. M. Groom, D. J. Mowbray,
and M. Hopkinson, IEE Proc.: Optoelectron. 153, 280 /H208492006 /H20850.
31S. Ghosh, P. Bhattacharya, E. Stoner, J. Singh, H. Jiang, S. Nut-
tinck, and J. Laskar, Appl. Phys. Lett. 79, 722 /H208492001 /H20850.
32I. C. Sandall, P. M. Smowton, J. D. Thomson, T. Badcock, D. J.
Mowbray, H.-Y. Liu, and M. Hopkinson, Appl. Phys. Lett. 89,
151118 /H208492006 /H20850.
33T. J. Badcock, R. J. Royce, D. J. Mowbray, M. S. Skolnick, H. Y.
Liu, M. Hopkinson, K. M. Groom, and Q. Jiang, Appl. Phys.Lett. 90, 111102 /H208492007 /H20850.
34H. Y. Liu, M. Hopkinson, C. N. Harrison, M. J. Steer, R. Frith, I.
R. Sellers, D. J. Mowbray, and M. S. Skolnick, J. Appl. Phys.
93, 2931 /H208492003 /H20850.
35H. Y. Liu, I. R. Sellers, M. Gutierrez, K. M. Groom, W. M.EFFECTS OF PHOTON AND THERMAL COUPLING … PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-11Soong, M. Hopkinson, J. P. R. David, R. Beanland, T. J. Bad-
cock, D. J. Mowbray, and M. S. Skolnick, J. Appl. Phys. 96,
1988 /H208492004 /H20850.
36M. Sugawara, N. Hatori, H. Ebe, M. Ishida, and Y. Arakawa, J.
Appl. Phys. 97, 043523 /H208492005 /H20850.
37S. Osborne, P. Blood, P. Smowton, J. Lutti, Y. C. Xin, A. Stintz,
D. Huffaker, and L. F. Lester, IEEE J. Quantum Electron. 40,
1639 /H208492004 /H20850.
38G. Park, O. B. Shchekin, and D. G. Deppe, IEEE J. Quantum
Electron. 36, 1065 /H208492000 /H20850.
39D. G. Deppe, H. Huang, and O. B. Shchekin, IEEE J. Quantum
Electron. 38, 1587 /H208492002 /H20850.
40D. Bimberg, N. Kirstaedter, N. N. Ledentsov, Zh. I. Alferov, P. S.
Kop’ev, and V. M. Ustinov, IEEE J. Quantum Electron. 3, 196
/H208491997 /H20850.41C. Y. Jin, Y. Z. Huang, L. J. Yu, and S. L. Deng, IEEE J. Quantum
Electron. 40, 513 /H208492004 /H20850.
42T. W. Berg and J. Mørk, IEEE J. Quantum Electron. 40, 1527
/H208492004 /H20850.
43M. A. Lampert and P. Mark, Current Injection in Solid /H20849Aca-
demic, New York, 1970 /H20850.
44Y. C. Zhang, C. J. Huang, F. Q. Liu, B. Xu, J. Wu, Y. H. Chen, D.
Ding, W. H. Jiang, X. L. Ye, and Z. G. Wang, J. Appl. Phys. 90,
1973 /H208492001 /H20850.
45P. Borri, W. Langbein, J. Mørk, J. M. Hvam, F. Heinrichsdorff,
M.-H. Mao, and D. Bimberg, Phys. Rev. B 60, 7784 /H208491999 /H20850.
46P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D.
Ouyang, and D. Bimberg, IEEE J. Quantum Electron. 8, 984
/H208492002 /H20850.JIN et al. PHYSICAL REVIEW B 76, 085315 /H208492007 /H20850
085315-12 |
PhysRevB.71.165104.pdf | Quantum many-body dynamics in a Lagrangian frame: I. Equations of motion
and conservation laws
I. V. Tokatly1,2,*
1Lerhrstuhl für Theoretische Festkörperphysik, Universität Erlangen-Nürnberg, Staudtstrasse 7/B2, 91058 Erlangen, Germany
2Moscow Institute of Electronic Technology, Zelenograd, 124498 Russia
sReceived 16 August 2004; published 4 April 2005 d
We formulate equations of motion and conservation laws for a quantum many-body system in a co-moving
Lagrangian reference frame. It is shown that generalized inertia forces in the co-moving frame are described by
Green’s deformation tensor gmnsj,tdand a skew-symmetric vorticity tensor F˜mnsj,td, where jin the Lagrang-
ian coordinate. Equations of motion are equivalent to those for a quantum many-body system in a space with
time-dependent metric gmnsj,tdin the presence of an effective magnetic field F˜mnsj,td.To illustrate the general
formalism we apply it to the proof of the harmonic potential theorem. As another example of application weconsider a fast long wavelength dynamics of a Fermi system in the dynamic Hartree approximation. In thiscase the kinetic equation in the Lagrangian frame can be solved explicitly. This allows us to formulate thedescription of a Fermi gas in terms of an effective nonlinear elasticity theory. We also discuss a relation of ourresults to time-dependent density functional theory.
DOI: 10.1103/PhysRevB.71.165104 PACS number ssd: 71.10. 2w, 05.30. 2d, 47.10. 1g, 02.40. 2k
I. INTRODUCTION
Lagrangian and Eulerian formulations of fluid mechanics
are known as two alternative ways to describe dynamics ofcontinuum media.
1The more common Eulerian sor spatial d
formulation considers basic collective variables, such as den-sitynsx,tdor current jsx,tddistributions, as functions of
space-time coordinates xandt.
1,2This corresponds to the
description of a system from the standard point of view of anobserver in a fixed laboratory reference frame. Central no-tions of Lagrangian sor material ddescription are the trajec-
tories of infinitesimal fluid elements. Every small element ofa fluid can be uniquely labeled by its initial position
jthat
plays a role of independent, so called Lagrangian, coordi-nate. Lagrangian description represents the dynamics of con-tinuum media as it is seen by a local observer, moving witha flow. In the last decades the Lagrangian method attracts anincreasing attention as a powerful tool for studying nonlineardynamics of compressible media with numerous applicationsin cosmology, plasma physics, physics of semiconductors,etc.sfor a recent comprehensive review see Ref. 3 d. Recently
we have shown that the Lagrangian coordinate naturally ap-pears in time-dependent density functional theory sTDDFT d,
where it plays a role of a basic variable for a nonadiabaticexchange correlation potential.
4It is also interesting to note a
relation of Lagrangian fluid dynamics to noncommutativegeometry and noncommuting gauge fields.
5
Commonly Lagrangian and Eulerian descriptions are con-
sidered as inherent ingredients of the classical continuummechanics. In fact, they offer two alternative techniques forsolving the equations of classical hydrodynamics. However,the main idea of Lagrangian method, which is the descriptionof dynamics using co-moving coordinates, is clearly muchmore general and universal. In the present paper we formu-late microscopic equations of many-body dynamics in theco-moving Lagrangian reference frame. The transformationto the Lagrangian frame corresponds to an explicit separationof the convective motion of particles. This is a natural gen-
eralization of the common separation of the center-of massmotion in homogeneous many-body systems. The separationof the center-of-mass motion also plays an important role inthe theory of harmonically trapped systems. For the har-monic inhomogeneity the convective motion can be sepa-rated by the transformation to a global accelerated reference
frame, which is a key step in the proof of the harmonicpotential theorem
6–8sHPT d. In fact, the proof of HPT can be
viewed as the simplest application of the Lagrangian descrip-tion to quantum dynamics. In the case of a general inhomo-geneous flow the separation of convective “center-of-mass”motion leads to an appearance of inhomogeneous inertiaforces in the equations for the relative motion. We show thatthese forces can be uniquely described by the symmetric de-formation tensor g
mnsj,tdand a skew-symmetric vorticity
tensorF˜mnsj,td. The deformation tensor enters equations of
many-body dynamics as an effective time-dependent metric,
while the vorticity tensor plays a role of an effective mag-netic field.
Agreat advantage of the Lagrangian description of many-
body dynamics is that in the co-moving frame both the den-sity of particles and the current density become the exactintegrals of motion.The current density is zero in every pointof Lagrangian
j-space, while the particles’ density distribu-
tion preserves its initial form. These “conservation laws” areguaranteed by a fine local compensation of inertia forces,external forces, and the force of internal stresses. The aboveforce balance follows the local momentum conservation lawsthe exact microscopic Navier-Stokes equation dafter the
transformation to the Lagrangian frame. We explicitly dem-onstrate that the exact internal stress force takes a form of acovariant divergence of a symmetric second-rank stress ten-sor. As a byproduct of our formalism we obtain a micro-scopic representation for the local stress tensor in a generalquantum many-body system.PHYSICAL REVIEW B 71, 165104 s2005 d
1098-0121/2005/71 s16d/165104 s13d/$23.00 ©2005 The American Physical Society 165104-1The concept of quantum stress has been introduced by
Schrödinger in 1927.9Over the last two decades there has
been a growing interest in understanding properties of quan-tum systems, such as molecules or solids, in terms of thestress density ssee, for example, Refs. 10–13 and references
therein d. A derivation of a microscopic expression for the
kinetic part of the stress tensor in quantum many-body sys-tem causes no problem. This simple generalization of theone-particle result has been obtained in the classical paper byMartin and Schwinger.
14However, the derivation of the mi-
croscopic form for the interaction related stress tensor turnedout to be not that simple.
11,14–18In this paper we present two
alternative derivations of the symmetric form for the stressdensity, which has been obtained by Puff and Gillis in Ref.17. In particular, we show that this form is consistent withthe definition of the stress tensor via the variational deriva-tive of the energy with respect to the metric tensor.
The structure of the paper is the following. In Sec. II we
consider the standard Eulerian form of the conservation lawsin a quantum many-body system. In this section we alsopresent a compact derivation of the microscopic expressionfor the exact stress tensor. Section III is devoted to the for-mulation of quantum many-body theory in the co-movingLagrangian frame. In Sec. III A the key notions of Lagrang-ian coordinate and of the deformation tensor are formallydefined. The derivation of the equations of motion in an ar-bitrary local noninertial reference frame is presented in Sec.III B. Here we also derive the form of transformed many-body Hamiltonian and discuss the physical meaning of gen-eralized inertia forces. In Sec. III C we derive local conser-vation laws, and present a complete formulation of themany-body problem in the Lagrangian frame. It is shownthat this problem corresponds to the solution of the equationsof motion for the relative motion, supplemented by the localforce balance equation. The force balance equation plays arole of an additional gauge condition that fixes the referenceframe. Section IVcontains simple examples of application ofthe general theory. In Sec. IV A we interpret the harmonicpotential theorem
6in terms of dynamics in the Lagrangian
frame. In Sec. IV B we apply the general formalism to thestudy of semiclassical collisionless dynamics of a Fermi gas,and shortly discuss a connection of our approach to TDDFT.It is shown that in the regime of a fast long wavelengthevolution the kinetic equation in the Lagrangian frame canbe solved explicitly. In this case the behavior of the system isdescribed by an effective nonlinear continuum mechanics,which, after the transformation to the laboratory frame, re-duces to the generalized collisionless hydrodynamics ofRefs. 19 and 20. In Sec. V we summarize our results.
II. CONSERVATION LAWS INTHE LABORATORY
REFERENCE FRAME: DEFINITION OF THE STRESS
TENSOR
In this paper we consider a system of Ninteracting par-
ticles in the presence of a time-dependent external potentialU
extsx,td. The corresponding Hamiltonian takes the follow-
ing standard form:
H=Tˆ+Wˆ+Uˆ, s1dTˆ=−Edxc†sxd„2
2mcsxd, s2d
Wˆ=1
2Edxdx8wsux−x8udc†sxdc†sx8dcsx8dcsxd,s3d
Uˆ=EdxUextsx,tdc†sxdcsxd, s4d
wherewsxdis the interaction potential, and c†andcare the
field operators, which satisfy proper commutation relations
fc†sxd,csx8dg±=dsx−x8d. s5d
The upper slower dsign in Eq. s5dcorresponds to fermions
sbosons d, and fA,Bg±=AB±BA. Using Hamiltonian of Eqs.
s1d–s4dwe obtain Heisenberg equations of motion for
c-operators,
i]
]tcsxd=−„2
2mcsxd+Uextcsxd+Edx8wsux
−x8udc†sx8dcsx8dcsxd. s6d
Equation s6dallows to derive equations of motion for any
physical observable as well as for any correlation function.The most important of these equations are the local conser-vation laws or balance equations, which should be satisfiedfor an arbitrary evolution of the system. Below we concen-trate on conservation laws for the number of particles and formomentum. These local conservation laws follow the equa-tions of motion for the density, nsx,td, and for the current,
jsx,td, respectively. Computing the time derivative of the
density operator,
nˆsx,td=
c†sx,tdcsx,td, s7d
we obtain the continuity equation that is the local balance
equation for the number of particles,
]n
]t+]jm
]xm=0, s8d
wherensx,td=knˆsx,tdland
jmsx,td=kjˆmsx,tdl=−i
2mKc†]c
]xm−]c†
]xmcL. s9d
Here the angle brackets stand for averaging with the exact
many-body density matrix. Similarly using Eq. s6dwe derive
the equation of motion for the current, Eq. s9dssee, for ex-
ample, Refs. 14 and 18 d,
m]jm
]t+Fmkin+Fmint+n]
]xmUext=0, s10d
Equation s10dhas a meaning of the local force balance equa-
tion in the fixed laboratory reference frame. Vectors Fmkinand
Fmintin Eq. s10dcorrespond to the forces, which are related to
the kinetic and the interaction effects, respectively,
Fmkin=]
]xn1
2mK]c†
]xm]c
]xn+]c†
]xn]c
]xm−dmn
2„2nˆL,s11dI. V. TOKATLY PHYSICAL REVIEW B 71, 165104 s2005 d
165104-2Fmint=Edx8]wsux−x8ud
]xmr2sx,x8d. s12d
In Eq. s12dwe introduced the notation r2sx,x8d
=kc†sxdc†sx8dcsx8dcsxdlfor the two-particle density matrix.
Obviously, the last term on the left-hand side in Eq. s10dis
the force produced by the external potential. The kineticforce of Eq. s11dhas a form of a divergence of a symmetric
second rank tensor. This automatically implies vanishing in-
tegral kinetic force, eF
mkinsx,tddx=0. The Newton’s third law
requires that the force Fmintof Eq. s12dshould obey the same
property, which is however by far not obvious. In fact, thepossibility to represent Eq. s12din a divergence form has
been a subject of a long discussion in the literature.
11,14,16–18
An elegant symmetric representation of the stress tensor hasbeen presented sunfortunately without derivation dby Puff
and Gillis in Ref. 17. Since this representation is of primaryimportance for our paper, below we give a compact deriva-tion of the Puff and Gillis result.
The symmetry of the function
r2sx,x8dwith respect to the
permutation of coordinates allows us to transform vector Fmint,
Eq.s12d, as follows:
Fmintsxd=Edx8]wsux−x8ud
]xmr2sx,x8d
=1
2Edx8]wsux8ud
]x8mfr2sx−x8,xd+r2sx,x−x8dg
=−1
2Edx8]wsux8ud
]x8mfr2sx+x8,xd−r2sx,x−x8dg
=−1
2Edx8sex8n]n−1d]wsux8ud
]x8mr2sx,x−x8d,
where ]n=]/]xn. Using an obvious operator identity
ex8„−1=E
01
x8„elx8„dl
we arrive at the following final representation for the local
forceFmint:
Fmintsxd=]
]xnWmnsxd, s13d
whereWmnsxdis a stress tensor, which is responsible for the
contribution of interparticle interaction to the force balance21
Wmnsxd=−1
2Edx8x8mx8n
ux8u]wsux8ud
]ux8u
3E
01
r2sx+lx8,x−s1−ldx8ddl. s14d
In the next section we will show that parameter lin Eq. s14d
has a deep geometric meaning. It can be associated to thenatural parameter for a geodesic sstraight line in the present
casedthat connects two interacting particles ssee also the
Appendix d.Equations s11d,s13d, and s14dshow that the net internal
force,F
mkin+Fmint, is representable in a form of divergence of a
symmetric second-rank tensor Pmn. Tensor Pmndescribes lo-
cal internal stresses in the fluid. A contribution of the con-vective motion of particles to this tensor is known exactly.
2.
It is equal to the macroscopic momentum flow tensor,mn
vmvn, wherev=j/nis the fluid’s velocity. It is convenient
to separate this contribution explicitly and rewrite the con-servation laws of Eqs. s8dands10din the following familiar
form:
D
tn+n]
]xmvm=0, s15d
mnDtvm+]
]xnPmn+n]
]xmUext=0, s16d
whereDt=s]/]td+v„is the convective derivative and Pmn
=Tmn+Wmnis the exact stress tensor, which contains the ki-
netic,Tmn, and the interaction, Wmn, contributions. The inter-
action stress tensor, Wmn, is given by Eq. s14d, while the
kinetic part, Tmn, is defined as follows:
Tmn=1
2mKsqˆmcd†qˆnc+sqˆncd†qˆmc−1
2dmn„2nˆL,s17d
whereqˆ=−i„−mvis the operator of “relative” momentum
which accounts for the above-mentioned separation of themacroscopic convective motion.
Equations s15dands16dform a basis for a hydrodynamic
description of a nonequilibrim many-body system. Accord-ing to the Runge–Gross mapping theorem of TDDFT
22the
exact many-body wave function/density matrix sfor given
initial conditions dis a unique functional of velocity vsx,td.
Therefore the stress tensor Pmnis also a functional of v.
Hence Eqs. s15dands16dcan be viewed as a formally closed
system of equations that completely determine the dynamicsof collective variables nsx,tdandvsx,td. These dynamics are
governed by the external force, n
]mUext, and by the force of
internal stress, ]nPmn. Since the convective motion has been
explicitly separated from the stress tensor, only the relativemotion of particles contributes to P
mn. A particular form of
Pmnshould be determined from the solution of a many-body
problem in a reference frame moving with the “center-of-mass” velocity vsx,td. In the rest of the present paper we
derive equations of many-body dynamics in this co-moving
frame and present simple illustrative examples of their solu-tions.
III. QUANTUM DYNAMICS IN THE LAGRANGIAN
FRAME
A. Definition of the Lagrangian reference frame
Co-moving or Lagrangian frame is a local noninertial ref-
erence frame which moves with the velocity vsx,tdof the
fluid. Formally the transformation to the Lagrangian frame
corresponds to a nonlinear change of variables x=xsj,td,
which maps old coordinates xto new coordinates j. For a
given velocity distribution the transformation function,QUANTUM MANY-BODY DYNAMICS IN …I.… PHYSICAL REVIEW B 71, 165104 s2005 d
165104-3xsj,td, is defined by the following initial value problem:
]xsj,td
]t=vsxsj,td,td,xsj,0d=j. s18d
If the function vsx,tdis continuous and satisfies the Lipschitz
condition in x, there exists a unique solution to the first order
differential equation of Eq. s18d.23Therefore, under the
above conditions on the velocity distribution, the map: x
!jis unique and invertible.
Physically the function xsj,tdcorresponds to the trajec-
tory of an infinitesimally small fluid element. Every fluid
element sand therefore every trajectory dis uniquely labeled
by the element’s initial position—the Lagrangian coordinate
j. The inverse function j=jsx,td, which determines the
transformation from the Lagrangian to the laboratory refer-
ence frame, recovers the initial position of a fluid elementthat at instant tarrives at the point x. The nonlinear transfor-
mation of coordinates, x=xs
j,td, induces a change of metric
sdxd2=gmndjmdjn,gmn=]xa
]jm]xa
]jn. s19d
In classical continuum mechanics the symmetric second rank
tensorgmnsj,td, Eq. s19d, is known as Green’s deformation
tensor.1This tensor is normally used to characterize a de-
formed state of a system within the Lagrangian description.The corresponding contravariant tensor, g
mn, is defined as
follows:
gmagan=dnm,gmn=]jm
]xa]jn
]xa. s20d
Since the deformation tensor gmnhas a meaning of the metric
tensor in the Lagrangian j-space, it should play a key role in
the description of many-body dynamics. It is quite natural toexpect that the general equations of motion in the Lagrangianframe should reduce to those in a space with time-dependentmetricg
mnsj,td. Below we confirm this intuitive expectation
by explicit calculations.
B. Equations of motion in a local noninertial reference frame
In this section we derive quantum equations of motion in
a general noninertial snot necessarily Lagrangian dreference
frame. The frame is defined by its velocity vsx,td, which
enters the trajectory equation of Eq. s18d, and thus provides a
unique and invertible map, x!j. As a first step in the deri-
vation we perform a nonlinear transformation of coordinates,x=xs
j,td, in the equation of motion, Eq. s6d, and in the
commutation relations of Eq. s5d. Straightforward calcula-
tions lead to the result,
i]
]tcsjd=S−1
2m1
˛g]
]jm˛ggmn]
]jn
+iv˜msj,td]
]jm+Uextsj,tdDcsjd
+Edj8wslj,j8dc†sj8dcsj8dcsjd, s21dwhereUextsj,td=Uextsxsj,td,td, and field operators csj,td
satisfy the following equal-time commutation relations:
fc†sjd,csj8dg±=1
˛gdsj−j8d. s22d
To shorten the notations in Eq. s21dwe omitted the explicit
time dependence in the argument of c-operators. The first
term in the large parentheses in Eq. s21dis the Laplace op-
erator in a reference frame with metrics gmnssee, for ex-
ample, Ref. 24 d, while the second term comes from the trans-
formation of the time derivative in Eq. s6d. This term is
proportional to vector v˜msj,tdthat is the vector of velocity,
transformed to a new frame,
v˜msj,td=]jm
]xnvnsxsj,td,td. s23d
The interparticle distance, lj,j8, in the argument of the inter-
action potential in Eq. s21dequals to a length of geodesic
that connects points jand j8. Geodesic, zj,j8sld, param-
etrized by a parameter ls0,l,1d, is a solution to the
following equation:24
z¨msld+Gabmszdz˙asldz˙bsld=0, s24d
wherez˙=]z/]l, and Gabmis the affine connection
Gabm=1
2gmnS]gna
]jb+]gnb
]ja−]gab
]jnD. s25d
Equation s24dshould be solved with boundary conditions
zs0d=j,zs1d=j8. It is convenient to parametrize geodesics
by a natural parameter, which is chosen in such a way that an
absolute value of the “velocity,” uz˙u=˛gmnz˙mz˙n, becomes in-
dependent of lalong the curve zsld. For this parametrization
the length lj,j8, which enters Eq. s21d, is simply equal to uz˙u
at any point on the geodesic,
lj,j8=E
01˛gmnszdz˙msldz˙nslddl=˛gmnz˙mz˙n. s26d
Equation s21dis the equation of motion for the operator
csj,td=csxsj,td,td. Due to the Jacobian factor 1/ ˛gin the
commutation relations of Eq. s22d, the quantity csj,tdcannot
be interpreted as an operator for annihilation of a particle in
a given point of j-space. In particular, the operator nˆsjd
=c†sjdcsjddoes not correspond to the density operator in
the new frame. By definition the density is a number of par-
ticles per unit volume that is changed under a volume non-preserving coordinate transformation. Therefore it is naturalto define the physical field operators and the density operatoras follows:
c˜sjd=g1/4csjd,c˜†sjd=g1/4c†sjd, s27d
n˜ˆsjd=c˜†sjdc˜sjd=˛gc†sjdcsjd, s28d
which automatically accounts for the proper change of a unit
volume in the deformed reference frame. Obviously the re-
defined field operators c˜sjdsatisfy the common commuta-
tions relations,I. V. TOKATLY PHYSICAL REVIEW B 71, 165104 s2005 d
165104-4fc˜†sjd,c˜sj8dg±=dsj−j8d. s29d
The renormalization of c-operators, Eq. s27d, is equivalent to
the corresponding multiplicative redefinition of the many-body wave function. This redefinition is aimed to preservethe common probabilistic interpretation and the standardform of the normalization conditions in the new referenceframe ssimilar arguments were suggested by Podolsky
25in
early days of quantum mechanics d.
Let us show that the renormalization of field operators,
Eq.s27d, also simplifies the form of the equations of motion.
First we note that the differential operator on the right-handside in Eq. s21dsfirst two terms in the square brackets dcan
be rearranged as follows:
−1
2m1
˛g]
]jm˛ggmn]
]jn+iv˜m]
]jm
=1
˛gKˆm˛gKˆm
2m−mv˜mv˜m
2−i1
2˛gS]
]jm˛gv˜mD,s30d
where we introduced an operator of “kinematic” momentum
in the noninertial reference frame,
Kˆm=−i]
]jm−mv˜m. s31d
sRaising and lowering of tensor indices are performed ac-
cording to the standard rules, i.e., v˜m=gmnv˜norKˆm=gmnKˆn.d
Using the equation of trajectory xsj,td, Eq. s18d, and the
definition of metric tensor gmn, Eq. s19d, one can prove the
following identity:
g−1/4]g1/4
]t=1
4]lng
]t=1
2˛gS]
]jm˛gv˜mD. s32d
The quantity on the right-hand side in Eq. s32dcoincides
with the last term on the right-hand side in Eq. s30d. Hence
the sum of the corresponding term in the equation of motion,Eq.s21d, and of the time derivative of
creduces to the fol-
lowing compact form:
]c
]t+1
2˛gS]
]jm˛gv˜mDc=g−1/4]g1/4c
]t=g−1/4]c˜
]t.s33d
Substituting Eq. s30dinto Eq. s21dand using Eq. s33d,w e
obtain the final equation of motion for the renormalized field
operator c˜sj,td
i]c˜sjd
]t=Sg−1/4Kˆm˛gKˆm
2mg−1/4+Uext−mv˜mv˜m
2Dc˜sjd
+Edj8wslj,j8dc˜†sj8dc˜sj8dc˜sjd. s34d
Equation s34dallows us to recover a form of the transformed
Hamiltonian H˜fc˜†,c˜g, which, together with the commutation
relations of Eq. s29d, determines the dynamics of the system,
H˜=T˜ˆ+W˜ˆ+U˜ˆ, s35dT˜ˆ=Edj˛gsKˆmg−1/4c˜d†gmn
2msKˆng−1/4c˜d, s36d
W˜ˆ=1
2Edjdj8wslj,j8dc˜†sjdc˜†sj8dc˜sj8dc˜sjd,s37d
U˜ˆ=EdjSUext−mv˜mv˜m
2Dc˜†c˜. s38d
Equations s34d–s38drepresent the main results of this sec-
tion. Equation s34dis the Heisenberg equation of motion for
the physical field operator, while Eqs. s35d–s38destablish the
rules for the transformation of the many-body Hamiltonianto an arbitrary local noninertial reference frame.
Formally the Hamiltonian of Eqs. s35d–s38ddescribes a
system of quantum particles in the presence of an effectivevector potential mv
˜sj,tdand an additional effective scalar
potential mv˜2/2. The particles live in a space with the time-
dependent metric gmnsj,tdand interact via pairwise potential
which depends on the length of a geodesic connecting pair of
particles.Additional “potentials” and a nontrivial metric ten-sor are responsible for generalized inertia forces exerted on aparticle in a general noninertial reference frame. To get atransparent physical understanding of these forces it is in-structive to look on dynamics in the semiclassical approxi-mation. Since the most important inertial contributions enteronly quadratic parts of the Hamiltonian fEqs. s36dands38dg,
we neglect for a moment the interaction, and consider anequation of motion for the Wigner function,
f
˜psj,td=Ee−iphKc˜†Sj+h
2,tDc˜Sj−h
2,tDLdj
in a gas of noninteracting particles. In the semiclassical limit
the Wigner function satisfies the following kinetic equation:
]f˜p
]t+]H˜sp,jd
]p]f˜p
]j−]H˜sp,jd
]j]f˜p
]p=0, s39d
whereH˜sp,jdis the semiclassical Hamiltonian function,
which corresponds the noninteracting part of Eq. s35d,
H˜sp,jd=gmn
2mspm−mv˜mdspn−mv˜nd+Uext−mv˜mv˜m
2.
s40d
Substituting Eq. s40dinto Eq. s39dwe get the result
]f˜p
]t+gmn
mspm−mv˜md]f˜p
]jn−S]gab
]jnpapb
2m−]v˜a
]jnpa
+]Uext
]jnD]f˜p
]pn=0. s41d
Inertia forces do not explicitly show up in Eq. s41d. The
reason is that Eq. s41dis the equation for the function f˜p
which depends on the canonical momentum p. The physical
velocity of a particle in the new reference frame is propor-
tional to the kinematic momentum K=p−mv˜si.e.,]H˜/]pmQUANTUM MANY-BODY DYNAMICS IN …I.… PHYSICAL REVIEW B 71, 165104 s2005 d
165104-5=Km/md. Therefore it is more natural physically to consider
Kas an independent variable in the kinetic equation. The
distribution function of the kinematic momentum, f˜
K8sj,td,
can be introduced as follows:
f˜
K8sj,td=f˜K+mv˜sj,td. s42d
Performing the corresponding change of variables in Eq. s41d
we obtain the final semiclassical equation of motion for the
distribution function f˜
K8sj,tdin the local noninertial refer-
ence frame,
]f˜
K8
]t+Kn
m]f˜
K8
]jn−Fm]v˜n
]t+KmF˜mn−]gab
]jnKaKb
2m
+]
]jnSUext−mv˜mv˜m
2DG]f˜
K8
]Kn=0, s43d
where a skew-symmetric second rank tensor F˜mnis defined
as follows:
F˜mn=]v˜m
]jn−]v˜n
]jm. s44d
Since tensor F˜mnvanishes for an irrotational flow, we name it
the vorticity tensor.26In the next section Eq. s43dwill be
applied to the derivation of generalized collisionlesshydrodynamics.
19,20
The expression in the square brackets in Eq. s43dcontains
all inertia forces. These are all the terms except for the ex-ternal force,
]nUext. The first term in the square brackets is
the linear acceleration force, while the last term is related tothe kinetic energy of a moving frame. In a particular case ofa homogeneously rotating frame the last term is responsiblefor the centrifugal force. The second and the third terms inthe square brackets correspond to inertia forces that dependon a velocity of a particular particle. The second term is theclassical Coriolis force. This force is proportional to theskew-symmetric vorticity tensor, which defines a local angu-lar velocity of the reference frame. The third, bilinear inparticle’s momentum term is less common. The correspond-ing inertia force makes a free particle to move along a geo-desic in a local noninertial frame. Indeed, the third term inthe square brackets in Eq. s43dcan be rewritten as follows:
1
2m]gab
]jnKaKb=1
mgnmGabmKaKb, s45d
where we have used Eq. s25d, which relates the affine con-
nection Gabmto the metric tensor gmn. The right-hand side of
Eq.s45dis easily recognized as a covariant component of the
forceintheequationofgeodesic fsee,forexample,Eq. s24dg.
C. Conserving quantities and balance equations
1. The continuity equation
The first problem we address in this section is a proper
definition of the current operator, j˜mˆ, in a general noninertial
reference frame. The easiest way to establish a form of j˜mˆisto derive the equation of motion for the density operator
n˜ˆsj,td=c˜†sj,tdc˜sj,td. Using Eq. s34dto compute the time
derivative of the density operator we find that the desired
equation indeed reduces to the common form of the continu-ity equation,
]n˜ˆ
]t+]j˜mˆ
]jm=0, s46d
if we define the current operator, j˜mˆsj,td, as follows:
j˜mˆ=gmnF−i
2mSc˜†]c˜
]jn−]c˜†
]jnc˜D−v˜nc˜†c˜G. s47d
The standard form of the continuity equation, Eq. s46d,
should be considered as one more justification for the redefi-nition of field operators, Eq. s27d. We would like to outline a
very natural form of the current operator, Eq. s47d. Despite
thepresenceoftheJacobianfactors s˛gorg1/4dintheHamil-
tonian, they completely vanish in Eq. s47dfas well as in the
definition of the density operator of Eq. s28dg.
From this point we restrict ourselves to the Lagrangian
frame, which is the local reference frame, moving with thevelocityvof the fluid. In this special case the continuity
equation admits a very simple solution. Let us calculate the
expectation value of the current operator j˜mˆ, Eq. s47d. This
can be done, for example, by transforming the right-handside in Eq. s47dback to the laboratory frame, and by using
Eq.s9dtogether with the definition of the velocity, v=j/n.
The result takes an extremely simple form
j˜msj,td=kj˜mˆsj,tdl=0. s48d
Thus the current density is exactly zero in every point of the
Lagrangian j-space.This is of course not surprising, since an
observer in the co-moving frame should not see any current.Combining Eq. s48dand the continuity equation of Eq. s46d
we arrive at the conclusion that the density n
˜sj,tdis inde-
pendent of time
n˜sj,td=n˜sj,0d=n0sjd, s49d
wheren0sxdis the initial density distribution. Therefore in
the Lagrangian frame not only the number of particles Nis
an integral of motion, but the density itself is also a conserv-ing quantity. Evolution of the density in the laboratory framecan be calculated with the following formula fsee Eq. s28dg:
nsx,td=n
˜sjsx,td,td
˛gsjsx,td,td=n0sjsx,tdd
˛gsjsx,td,td. s50d
Equation s50dis, in fact, the explicit solution to the continu-
ity equation of Eq. s8d, which defines the density nsx,tdas a
functional of velocity vsx,td.
Equations s48dands49ddemonstrate the main advantage
of the Lagrangian frame for the description of many-bodydynamics. In this very special reference frame the inertiaforces are adjusted to get exactly zero current density andtherefore to keep the density of particles fixed during theI. V. TOKATLY PHYSICAL REVIEW B 71, 165104 s2005 d
165104-6whole evolution of the system. Equation s48dcan be used to
construct a complete many-body theory in the co-movingframe. The frame’s velocity venters the equation of motion,
Eq. s34d, as an external parameter. Imposing the local
“gauge” condition of Eq. s48dwe specify the reference frame
and thus get the complete theory with all quantities definedby the initial conditions.
2. Local force balance in the Lagrangian frame
Let us turn to the local momentum conservation law. In
the laboratory reference frame it is given by Eq. s16dfor,
equivalently, by Eq. s10dg. Since in the Lagrangian frame the
current density is zero, the local momentum conservationlaw should reduce to the zero force condition—the inertiaforces should exactly compensate the external force and theforce of internal stresses. Below we derive an explicit formof this balance equation by the direct transformation of Eq.s16dto the Lagrangian coordinates
j. First we express the
vector of velocity vand the stress tensor Pmnin terms of the
corresponding quantities, v˜andP˜
nm, in the Lagrangian frame
vm=]jb
]xmv˜bsj,td, s51d
Pmn=]jb
]xm]xn
]jgP˜
bgsj,td. s52d
Equation s51dfollows the definition of v˜m, Eq. s23d, while in
Eq. s52dwe adopted the standard tensor transformation
rules.24Substituting Eqs. s51dands52dinto Eq. s16d, trans-
forming the derivatives, and multiplying the result by
]xm/]ja, we obtain the following equation:
mn]xm
]ja]
]tS]jb
]xmv˜bD+]xm
]ja]jd
]xn]
]jdS]jb
]xm]xn
]jgP˜
bgD+n]Uext
]ja=0.
s53d
The first term in Eq. s53dcan be further simplified as fol-
lows:
]xm
]ja]
]t]jb
]xmv˜b=]v˜a
]t−v˜b]jb
]xm]vm
]ja=]v˜a
]t−1
2]v˜bv˜b
]ja.
s54d
In the sequence of transformations in Eq. s54dwe used an
obvious identity s]xm/]jads]jb/]xmd=dab, the trajectory
equation of Eq. s18d, and the definition of v˜m, Eq. s23d.A
similar simplification of the second term in Eq. s53dis even
more straightforward. One only needs to apply the chain rulefor the calculation of the spatial derivative and take into ac-count the following explicit representation for affine connec-tion ssee, for example, Ref. 24 d:
G
abm=]jm
]xg]2xg
]ja]jb. s55d
As a result we get a very natural expression for the second
term in Eq. s53d,]xm
]ja]jd
]xn]
]jdS]jb
]xm]xn
]jgP˜
bgD=P˜
a;bb, s56d
where the semicolon is used to denote the covariant deriva-
tive. The covariant divergence of the stress tensor in Eq. s56d
is defined as follows:24,27
P˜
m;nn=]P˜
mn
]jn+GnanP˜
ma−GmanP˜
na=1
˛g]˛gP˜
mn
]jn−1
2]gab
]jmP˜ab.
s57d
Substitution of Eqs. s54dands56dinto Eq. s53dleads to the
final form of the force balance equation in the Lagrangianframe
n
˜Fm]v˜m
]t+]
]jmSUext−mv˜nv˜n
2DG+˛gP˜
m;nn=0. s58d
Adirect comparison of the force term in the kinetic equation
of Eq. s43dand the term in the square brackets in Eq. s58d
shows that the latter is exactly the sum of the external forceand two inertia forces that are independent of particle’s mo-mentum. These three forces are balanced by the force ofinternal stresses fthe second term in Eq. s58dg. The net force,
exerted on every fluid element in the Lagrangian space, iszero, which results in a zero current density and a stationaryparticles’density distribution. It should be noted that the restof inertia forces sthose, which are different for different par-
ticles in a fluid element dimplicitly present in the kinetic part
of the stress tensor P
˜mn. To see this more clearly we need to
derive an explicit microscopic representation for this tensor.
3. Stress tensor in the Lagrangian frame
In general both kinetic, T˜mn, and interaction, W˜mn, contri-
butions to the total stress tensor, P˜mn=T˜mn+W˜mn, can be
found using the common transformations rules24
T˜mnsj,td=]xa
]jm]xb
]jnTabsxsj,td,td, s59d
W˜mnsj,td=]xa
]jm]xb
]jnWabsxsj,td,td. s60d
Here stress tensors, Tabsx,tdandWabsx,td, in the laboratory
reference frame are given by Eqs. s17dands14d, respectively.
We shall however follow another route, which takes full ad-vantage of geometric ideas we develop in this paper. Thetransformed many-body Hamiltonian of Eqs. s35d–s38d,i sa n
explicit functional of the metric tensor g
mn.Therefore we can
find the required stress tensor by computing the variationalderivative of the energy with respect to the metric tensor.More precisely, we make use of the fact that under a smallvariation of the metric, the variations of the kinetic energy,Eq.s36d, and of the energy of interparticle interaction, Eq.
s37d, are related to tensors T
˜mnandW˜mn, respectively,
dkT˜ˆl=Edj˛g
2dgmnT˜mn=−Edj˛g
2dgmnT˜mn,s61dQUANTUM MANY-BODY DYNAMICS IN …I.… PHYSICAL REVIEW B 71, 165104 s2005 d
165104-7dkW˜ˆl=Edj˛g
2dgmnW˜mn=−Edj˛g
2dgmnW˜mn.s62d
Such a definition of the stress tensors is closely related to the
common definition of the energy-momentum tensor in gen-eral relativity ssee, for example, Ref. 27 d. The main advan-
tage of this definition is that it automatically gives a symmet-ric form of the stress/energy-momentum tensors. Recently avery similar approach has been used to derive a microscopicexpression for the stress tensor in the equilibrium quantummany-body system within the local density approximation.
13
Reference 13 also contains a general discussion of the abovegeometric definition of the stress tensors in the context ofnonrelativistic quantum mechanics.
The variation of the Hamiltonian should be taken at con-
stant
c˜-variables ssince they satisfy the equations of
motion d13,27and at constant velocity v˜. The kinetic energy
operatorT˜ˆofs36dcontains the metric tensor only in a form
ofgmn,˛gandg−1/4. Noting that
d˛g=−1
2˛ggmndgmn,dg−1/4=1
4g−1/4gmndgmn,
we can easily compute the variation of T˜ˆand then identify
the kinetic stress tensor using Eq. s61d. The result of the
calculations takes the form
T˜mnsj,td=1
2mKsKˆmg−1/4c˜d†sKˆng−1/4c˜d
+sKˆng−1/4c˜d†sKˆmg−1/4c˜d
−1
2gmn1
˛g]
]ja˛ggab]
]jbc˜†c˜
˛gL. s63d
If we evaluate the right-hand side in Eq. s63dfor Euclidian
metric,gmn=dmn, we immediately recover Eq. s17d. There-
fore the commonly used symmetric form of the kinetic stresstensorT
mn, Eq. s17d, is in exact correspondence with the
geometric definition of Eq. s61d.
Calculation of the variation dW˜ˆ, Eq. s62d, is a little bit
more involved. Since the interaction Hamiltonian of Eq. s37d
depends on gmnonly via the length of geodesic, we have
dkW˜ˆl=1
2Edhdh8dlh,h8]wslh,h8d
]lh,h8r˜2sh,h8d,s64d
where r˜2sh,h8d=kc˜†shdc˜†sh8dc˜sh8dc˜shdl. The next step is
to compute the variation of the functional lh,h8fgmng, Eq.
s26d. Let lin Eq. s26dbe a natural parameter for a geodesic
in the space with “unperturbed” metric gmnsnot the “full”
metricgmn+dgmnd. For this parametrization the variation of
lh,h8fgmngtakes the formdlh,h8=1
2lh,h8E
01
dldgmnfzsldgz˙msldz˙nsld
=EdjE
01
dldfj−zsldgdgmnsjdz˙msldz˙nsld
2lh,h8,
s65d
wherezsld=zh,h8sldis the geodesic which connects points
handh8. Substituting Eq. s65dinto Eq. s64d, and using the
definition of Eq. s62dwe get the following representation for
the interaction part of the stress tensor:
W˜mnsj,td=−1
2˛gE
01
dlEdhdh8dfj−zh,h8sldg
3z˙h,h8msldz˙h,h8nsld
lh,h8]wslh,h8d
]lh,h8r˜2sh,h8d.s66d
Let us evaluate the right-hand side of Eq. s66dat the Euclid-
ean metric gmn=dmn. In this case lh,h8=uh−h8uwhile the
geodesic sparametrized by the natural parameter dis a straight
line
zh,h8sld=h+sh8−hdl.
The above expressions for lh,h8andzh,h8sldshould be sub-
stituted into Eq. s66d. Introducing a new variable j8=h8−h,
and removing the delta-function by the integration over h,
we obtain the result that exactly coincides with Eq. s14d.
Therefore the symmetric representation of Eq. s14d, which
has been obtained in the preceding section by somewhat ar-tificial manipulations, has a clear geometric meaning. In par-ticular, the internal parameter lin Eq. s14dis the natural
parameter for a geodesic connecting two interacting par-ticles.
Equations s63dand s66dare the principal results of the
present section. They define explicit microscopic representa-tions for the stress tensors in a local noninertial referenceframe.
The zero force condition of Eq. s58dwithP
˜mnof Eqs. s63d
and s66dis equivalent to the requirement of zero current
density, Eq. s48d. Hence we can use Eq. s58das an alternative
“gauge” condition to fix the velocity parameter v˜sj,td, enter-
ing many-body equations of motion, Eq. s34d.
IV. EXAMPLES AND APPLICATIONS
A. The harmonic potential theorem
As a first simple example of application of our general
formalism, we consider many-body dynamics in the presenceof the following external potential:
U
extsx,td=1
2mvmnxmxn+Emstdxm, s67d
where vmnis a constant tensor and Emstdis a time-dependent
vector fwithout loss of generality we can set Ems0d=0g. The
initial value problem with the external potential of Eq. s67dis
exactly solvable, which is known as the harmonic potentialtheorem sHPT d.
6It is also known that HPT is related to theI. V. TOKATLY PHYSICAL REVIEW B 71, 165104 s2005 d
165104-8covariance of the time-dependent Schödinger equation under
the transformation to a global accelerated reference frame.7,8
Therefore our formulation of the many-body problem shouldbe perfectly suited to the demonstration of HPT.
Within the present Lagrangian formulation one needs to
find a self-consistent solution to the many-body equation ofmotion, Eq. s34d, and to the force balance equation, Eq. s58d.
Let us assume that velocity vsx,td, which defines fvia Eq.
s18dgthe motion of the reference frame, is a function of t
only,vsx,td=Vstd. In this case the trajectory of a fluid ele-
ment takes a form
xs
j,td=j+Rstd, s68d
whereRstdis a solution to the following Cauchy problem:
]Rstd
]t=Vstd,Rs0d=0.
Clearly, if our anzatz, vsx,td=Vstd, is a self-consistent solu-
tion, then Rstdshould correspond to the center-of-mass co-
ordinate. Using Eq. s68dwe get the following results for the
metric tensor gmn, the velocity v˜sj,td, and the effective po-
tential, which enter Eqs. s34dands58d,
gmn=dmn,v˜sj,td=Vstd, s69d
Uextsxsj,td,td−mv˜nv˜n
2=1
2mvmnjmjn−Lstd+jmfmvmnRnstd
+Emstdg. s70d
Function Lstdin Eq. s70dis the classical Lagrangian for a
particle moving in the harmonic potential of Eq. s67d,
Lstd=m
2V2std−1
2mvmnRmstdRnstd−EmstdRmstd.
The equation of motion, Eq. s34d, and the force balance
equation, Eq. s58d, simplify, respectively, as follows:
i]c˜8
]t=S−„j2
2m+1
2mvmnjmjnDc˜8
+Edj8wsuj−j8udn˜ˆsj8dc˜8sjd
+jmSm]Vmstd
]t+mvmnRnstd+EmstdDc˜8,s71d
m]Vmstd
]t+mvmnRnstd+Emstd
+S1
n0sjd]
]jnP˜mnsj,td+mvmnjnD=0, s72d
where we also performed a gauge transformation, c˜8sj,td
=c˜sj,tdexpf−imVj+ie0tLdt8g, which corresponds to the
transformation from the canonical to the kinematic momen-
tum.
Let, initially, the system be prepared in a stationary state
sor in arbitrary mixture of stationary states d. This means thatatt=0 the stationary force balance equation is fulfilled,
1
n0sjd]
]jnP˜mnsj,0d+mvmnjn=0. s73d
If at allt.0 the center-of-mass coordinate Rstdsatisfies the
classical equation of motion,
m]2Rmstd
]t2+mvmnRnstd+Emstd=0, s74d
then both the equation of motion, Eq. s71d, and the force
balance equation, Eq. s72d, preserve their initial sstationary d
form. Therefore the many-body system in the co-movingframe remains in the initial stationary state, in particular,
P˜mnsj,td=P˜mnsj,0d. This statement is the essence of HPT.6
Within the present formulation of many-body dynamics it
appears quite naturally. In fact, HPT is a built-in property ofour self-consistent approach. This actually means that anyapproximate treatment of a self-consistent system of Eqs.s34dands58dshould automatically satisfy HPT.
B. Geometric formulation of generalized hydrodynamics:
nonlinear elasticity of a collisionless Fermi gas
The HPT type of motion provides an extremely simple
example of many-body dynamics without any deformation of
local fluid elements sgmnHPT=dmnd. In this section we apply our
approach to a much more general situation with a nontrivial
dynamics of a fluid. Namely, we consider a semiclassicaldynamics of an interacting Fermi system in the time-dependent Hartree approximation. The problem reduces tothe self-consistent solution of a semiclassical collisionlesskinetic equation fsee Eq. s43dg,
]f˜
K8
]t+Kn
m]f˜
K8
]jn−Fm]v˜n
]t+KmF˜mn−]gab
]jnKaKb
2m
+]
]jnSU−mv˜mv˜m
2DG]f˜
K8
]Kn=0, s75d
and a force balance equation fsee Eq. s58dg,
m]v˜m
]t+]
]jmSU−mv˜nv˜n
2D+˛g
n0P˜
m;nn=0. s76d
HereU=Uextsj,td+UHsj,tdis a sum of the external potential
and the Hartree potential,
UHsj,td=Ewslj,j8dn0sj8ddj8.
Since the interaction effects are already included son the
mean field level din the self-consistent potential, only the
kinetic part of the stress tensor contributes to Eq. s76d, i.e.,
P˜mn=m−1oKKmKnf˜
K8/˛g. The last expression for P˜mnis a
plain semiclassical limit of the general kinetic stress tensor,Eq.s63d.
The problem of solving Eqs. s75dands76dcan be refor-
mulated as follows. Let us substitute the sum of the inertiaand the external forces from the balance equation, Eq. s76d,QUANTUM MANY-BODY DYNAMICS IN …I.… PHYSICAL REVIEW B 71, 165104 s2005 d
165104-9into the kinetic equation of Eq. s75d. After the substitution
the potential Uin Eq. s75dcancels out and the kinetic equa-
tion reduces to the following universal form:
]f˜
K8
]t+Kn
m]f˜
K8
]jn−SKmF˜mn−]gab
]jnKaKb
2m−˛g
n0P˜
m;nnD]f˜
K8
]Kn=0,
s77d
where the stress tensor P˜mnis defined as follows:
P˜mnsj,td=1
˛go
KKmKn
mf˜
K8sj,td. s78d
Equations s77dand s78dconstitute a closed set, which is
structurally similar to the system of Vlasov equations with aself-consistent force. The skew-symmetric vorticity tensor,
F˜mnsj,td, and the symmetric deformation tensor, gmnsj,td,
enter Eqs. s77dands78das external parameters, which gov-
ern the evolution of the system. Hence these equations define
a distribution function, f˜
K8sj,td, as a unique functional of F˜mn
andgmn, provided the initial condition, f˜
K8sj,0d=f˜
Ks0dsjd,i s
given. Equation s78ddetermines the stress tensor as a univer-
salsi.e., independent of external potential dfunctional of F˜mn
andgmn,
P˜mn=P˜mnfF˜mn,gmngsj,td. s79d
The vorticity and the deformation tensors contain nine inde-
pendent scalar functions sthree from F˜mnand six from gmnd
which completely describe a deformed state of a system.1
Hence Eq. s79dplays a role of a generalized “equation of
state” which relates the stress tensor to the deformation. It isworth mentioning that the existence of such an equation ofstate is a direct consequence of Runge-Gross mappingtheorem
22in TDDFT.
Substituting the functional of Eq. s79dinto Eq. s76dwe
obtain a hydrodynamic equation of motion which determinesthe evolution of velocity for a given external potential.Therefore the description of many-body dynamics consists oftwo separate problems. The first one corresponds to the uni-versal kinetic problem of Eqs. s77dand s78d. By solving
these equations we find the stress tensor functional, Eq. s79d
sthe generalized equation of state d. The second problem is to
compute the velocity and density distributions by solving theclosed set of hydrodynamics equations, Eqs. s18dands76d.
The universal kinetic problem of Eqs. s77dands78d, can
be solved explicitly in the case of a fast long wavelengthdynamics, i.e., if the deformation tensor is a fast function oftime, but slowly changes in space. More precisely, we as-sume that the characteristic length scale, L, of the deforma-
tion inhomogeneity is much larger than
tu, where tis the
time scale of a dynamical process and uis the characteristic
velocity of a particle. This situation is, for example, commonin Coulomb systems where the plasma frequency determinesthe characteristic time scale of dynamics, while the corre-sponding spatial variations of the density can be arbitraryslow. Let us estimate different terms in Eq. s77dunder the
above assumption. The first and the second terms on theright-hand side of Eq. s77dare of the order of 1/
tandu/L,respectively. Both terms in the second line in of Eq. s77dalso
give a contribution ,u/L, while the term, related to the Co-
riolis force, is proportional to F˜,v˜T/L, where v˜Tis a rota-
tional sor transverse dcomponent of the velocity. According
to the force balance equation of Eq. s76d, for any physical
velocity the transverse part of the linear acceleration is com-
pensated by the transverse part of the vector s˛g/n0dP˜
m;nn.
Hence v˜Tshould be proportional to tu2/L. This means that
the contribution of Coriolis force to the kinetic equation is ofthe order of
tu2/L2. Therefore, to the leading order in the
small parameter g=tu/L!1, only the first term in Eq. s77d
gives a nonvanishing contribution. Thus the universal prob-lem of Eqs. s77dand s78dreduces to the following trivial
equation:
]
]tf˜
K8sj,td=0. s80d
Equation s80dshows that for a fast, small-gradient evolution
the distribution function in the Lagrangian frame preserves
its initial form, f˜
K8sj,td=f˜
Ks0dsjd. In this respect the dynamics
remind the HPT type of motion. However the evolution of
the velocity is by far not trivial. Below we consider a systemwhich evolves from the equilibrium state. Substituting theequilibrium distribution function into Eq. s78dwe get the
stress tensor functional,
P
˜mnsj,td=dmn
˛gsj,tdP0sjd, s81d
which is proportional to the initial equilibrium pressure,
P0sjd. The last step is to substitute the nonadiabatic “equa-
tion of state,” Eq. s81d, into the force balance equation of Eq.
s76d. This results in the following “hydrodynamic” equation
of motion:
mn0]v˜m
]t+n0]
]jmSU−mv˜nv˜n
2C+]gmnP0
]jn+1
2P0]gaa
]jm=0,
s82d
where we used the definition of the covariant divergence, Eq.
s57d, to compute the stress force, ˛gP˜
m;nn,i nE q . s76d.W e
would like to outline that n0sjdandP0sjdin Eq. s82dare the
time independent initial density and pressure, respectively.
Equations s82dands18dconstitute a closed set of continuum
mechanics equations which describe a long wavelength dy-namics of a Fermi gas in the time-dependent Hartree ap-proximation. Since the stress force in Eq. s82ddepends only
on the deformation tensor, g
mn, it is natural to interpret Eqs.
s82dands18das a nonlinear elasticity theory of a Fermi gas.
In the case of small deformations this theory reduces to thestandard linear elasticity theory with a nonzero shear modu-lus. Indeed, in the linear regime Eq. s18dtakes the form
]usj,td
]t=vsj,td, s83d
whereu=x−jis the displacement vector. The deformation
tensor reduces to the common linearized expression,I. V. TOKATLY PHYSICAL REVIEW B 71, 165104 s2005 d
165104-10gmn=dmn−]um
]jn−]un
]jm. s84d
Assuming for simplicity that the unperturbed state is homo-
geneous, and substituting Eqs. s83dands84dinto Eq. s82d,
we get the following equation of motion for the displacementvector:
mn
0]2um
]t2−]smn
]jn+n0]dU
]jm=0. s85d
The linearized stress tensor smntakes the standard elastic
form
smn=dmnK]ua
]ja+mS]um
]jn−]un
]jm−dmn2
3]ua
]jaD,s86d
whereK=5
3P0andm=P0are the bulk modulus and the shear
modulus of a Fermi gas, respectively.19,20,28
The full nonlinear set of equations, Eqs. s82dands18d,i s
equivalent to the generalized collisionless hydrodynamicsderived in Refs. 19 and 20 ssee also Ref. 29 d. In fact, Eqs.
s82dands18dand the generalized hydrodynamics of Refs. 19
and 20 correspond to the same theory in the Lagrangian andEulerian formulations, respectively. An advantage of thepresent Lagrangian formulation is the explicit form of the
stress tensor P
˜mn, Eq. s81d. The Lagrangian point of view
also gives a very clear microscopic picture of the fast colli-sionless dynamics. This is a kind of evolution of a many-body system with almost time-independent distribution ofparticles inside every moving and deforming fluid element.Using the nonadiabatic equation of state in the Lagrangianframe, Eq. s81d, we can easily recover the corresponding
expression for the stress tensor P
mnsx,tdin the laboratory
frame,
Pmnsx,td=]ja
]xm]jb
]xnP˜absjsx,td,td=g¯mnsx,td˛g¯sx,tdP0sjsx,tdd.
s87d
Hereg¯mnsx,tdis Cauchy’s deformation tensor,1
g¯mnsx,td=]ja
]xm]ja
]xn,g¯=1/g. s88d
One can check that Pmnsx,tdof Eq. s87dis a solution to the
equation for the stress tensor derived in Refs. 19 and 20. The
stress tensor Pmnsx,td, Eq. s87d, enters the force balance
equation in the laboratory frame, Eq. s16d. This is a hydro-
dynamic equation of motion in the Eulerian description. It is
quite natural that Pmnsx,tddepends on g¯mnsx,tdsince
Cauchy’s tensor is a common characteristics of deformations
in the Eulerian picture.
In contrast to the stress tensor in the Lagrangian frame,
Eq.s81d, the stress tensor of Eq. s87dis a highly nonlocal
function. It is proportional to the pressure P0at the initial
position of a fluid element which is currently at xfxis an
independent variable in Eq. s16dg. The locality of the stress
force in Eq. s82dis a key property of the present Lagrangian
formulation of nonadiabatic continuum mechanics of a Fermigas. This formulation should be much more convenient forapplications to particular nonlinear problems.
V. CONCLUSION
We applied the idea of the Lagrangian description in con-
tinuum mechanics to the theory of nonequilibrium quantummany-body systems. Reformulation of the microscopicmany-body theory in terms of Lagrangian coordinates corre-sponds to the transformation to the local noninertial refer-ence frame moving with the flow sthe co-moving Lagrangian
frame d.This transformation allows to separate the convective
motion of particles, which is a direct generalization of thecommon separation of the center-of-mass motion in homo-geneous systems. The motion of particles in the Lagrangianframe is influenced by the external forces and by generalizedinertia forces. We have shown that the inertia forces can bedescribed in purely geometric terms of Green’s deformation
tensorg
mnand the skew-symmetric vorticity tensor F˜mn. Ten-
sorsgmnandF˜mnenter equations of motion as an effective
metric tensor and an effective magnetic field, respectively.Our results demonstrate a close relation of the many-bodydynamics in Lagrangian frame to the quantum dynamics oncurved manifolds.
We also derived local conservation laws for the number of
particles and for momentum in the Lagrangian frame, andpresented closed microscopic expressions for the stress ten-sor and for the corresponding stress force. The local momen-tum conservation law in the Lagrangian frame reduces to azero force condition. The inertia forces exactly compensatethe external force and the stress force in every point of theLagrangian
j-space. The net force, exerted on every fluid
element, is exactly zero, which results in zero current densityand a time-independent density distribution. This property isthe main advantage of the Lagrangian description. It suggestsone of the most promising application of our formalism,which is a new reformulation of TDDFT in a form similar tothe static theory. Indeed the main practical problem of TD-DFT is an inevitable strong nonlocality of exchange correla-tion potentials.
4,7,8The physical reason for this is just the
nonadiabatic motion of fluid elements.When time is flowing,new and new fluid elements arrive at a given point x, and
bring an information about surrounding space, producing theabove nonlocality. Using our reformulation of the many-body theory as a basis for TDDFT one can completely re-move the very source on the nonlocality, which is of extremepractical importance. In this paper we did not touch thesequestions since it required an extended special consideration.A detailed formulation of TDDFT in the Lagrangian framewill be presented in the next paper of this series.
30
In this paper we also considered two illustrative examples
of application. The most interesting of them is the descrip-tion of a nonlinear semiclassical dynamics of a collisionlessFermi gas. We have shown that the full problem can be sepa-rated into two independent parts. The first one is the solutionof a universal kinetic problem, which defines the stress ten-sor as a universal functional of g
mnandFmn. This stress ten-
sor is used as an input for the second “hydrodynamic” part ofthe problem, determining the dynamics of the velocity vec-tor. This separation of the initial many-body problem can beQUANTUM MANY-BODY DYNAMICS IN …I.… PHYSICAL REVIEW B 71, 165104 s2005 d
165104-11viewed as a particular realization of TDDFT in the hydrody-
namic formulation.4,22In the case of a fast long wavelength
dynamics ssimilar to that for plasma oscillations d, the univer-
sal kinetic problem can be solved explicitly. The solution isextremely simple—the Wigner function in the Lagrangianframe is time independent. The corresponding “hydrody-namic” problem also can be formulated in the explicit form.It reduces to a closed nonlinear elasticity theory of a Fermigas. This elasticity theory is, in fact, the Lagrangian formu-lation of the generalized hydrodynamics derived in Refs. 19and 20. The generalized hydrodynamics proved to be usefulin the description of a small-amplitude collective dynamicsof an electron gas. It gives the correct sconsistent with the
kinetic treatment ddispersion of plasma waves in homoge-
neous systems
19,20and recovers the exact dispersion of the
edge modes in a confined geometry.31The results for the
standing plasma waves in a parabolically trapped electrongas are also quite reasonable.
32,33The Lagrangian “elasticity
theory” of a Fermi gas, derived in Sec. IV B is structurallymuch more simple than the Eulerian formulation of Refs. 19and 20. Therefore, we believe that it should provide a goodbasis for the description of nonlinear dynamical effects ininhomogeneous many-electron systems.
ACKNOWLEDGMENT
This work was supported by the Deutsche Forschungsge-
meinschaft under Grant No. PA 516/2-3.
APPENDIX: THE DIVERGENCE REPRESENTATION OF
THE INTERACTION STRESS FORCE
By definition the infraction stress force Fˆintis the commu-
tator of the current operator jˆand the interaction Hamiltonian
Wˆ,
Fˆintsxd=mfjˆsxd,Wˆg−. sA1d
In the coordinate representation operators jˆandWˆare de-
fined as follows:
jˆsxd=o
ifpˆi,dsx−xidg+, sA2dWˆ=1
2o
i,jwsxi,xjd, sA3d
whereiandjlabel particles. Calculating the commutator of
Eq.sA1dwe get the force in the following form:
Fˆintsxd=1
2o
i,jSdsx−xid]wsxi,xjd
]xi+dsx−xjd]wsxi,xjd
]xjD.
sA4d
If the interaction potential satisfies the Newton’s third law,
]wsxi,xjd
]xi=−]wsxi,xjd
]xj, sA5d
Eq.sA4dtakes the form
Fˆintsxd=1
2o
i,jfdsx−xid−dsx−xjdg]wsxi,xjd
]xi.sA6d
The difference of delta-functions in Eq. sA6dcan be trans-
formed as follows:
dsx−xid−dsx−xjd=s1−esxi−xjds]/]xdddsx−xid
=−sxi−xjd]
]xE
01
dlelsxi−xjds]/]xddsx−xid
=−]
]xsxi−xjdE
01
dldfx−xi−lsxj−xidg.sA7d
Inserting Eq. sA7dinto Eq. sA6dwe get the final representa-
tion for the interaction stress force,
Fˆ
mintsxd=]
]xnWˆmnsxd, sA8d
whereWˆmnsxdis the interaction stress tensor operator,
Wˆmnsxd=−1
2o
i,jE
01
dldfx−xi−lsxj−xidg
3sxin−xjnd]wsxi,xjd
]xim. sA9d
Thel-integration in Eq. sA9dis along the line that connects
two interacting particles.
*Electronic address: ilya.tokatly@physik.uni-erlangen.de
1Physical Acoustics , edited by W. P. Masson sAcademic, New
York, 1964 d, Vol. I, Part A.
2L. D. Landau and E. M. Lifshitz, Mechanics of fluids, Course of
Theoretical Physics , 2nd ed. sPergamon, New York, 1987 d, Vol.
6.
3H. Schamel, Phys. Rep. 392, 279 s2004 d.
4I. V. Tokatly and O. Pankratov, Phys. Rev. B 67, 201103 sRd
s2003 d.
5R. Jackiw, S. Y. Pi, and A. P. Polychronakos, Ann. Phys. sN.Y.d301, 157 s2002 d.
6J. F. Dobson, Phys. Rev. Lett. 73, 2244 s1994 d.
7G. Vignale, Phys. Rev. Lett. 74, 3233 s1995 d.
8G. Vignale, Phys. Lett. A 209, 206 s1995 d.
9E. Schrödinger, Ann. Phys. sLeipzig d82, 265 s1927 d.
10L. J. Bartolotti and R. G. Parr, J. Chem. Phys. 72, 1593 s1980 d.
11O. H. Nielsen and R. M. Martin, Phys. Rev. B 32, 3780 s1985 d.
12A. Filippetti and V. Fiorentini, Phys. Rev. B 61, 8433 s2000 d.
13C. L. Rogers and A. M. Rappe, Phys. Rev. B 65, 224117 s2002 d.
14P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 s1959 d.I. V. TOKATLY PHYSICAL REVIEW B 71, 165104 s2005 d
165104-1215L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics
sBenjamin, New York, 1962 d.
16A. Kugler, Z. Phys. 198, 236 s1967 d.
17R. D. Puff and N. S. Gillis, Ann. Phys. sN.Y.d46, 364 s1968 d.
18D. N. Zubarev, Noequilibrium Statistical Thermodynamics sCon-
sultants Bureau, New York, 1974 d.
19I. V. Tokatly and O. Pankratov, Phys. Rev. B 60, 15550 s1999 d.
20I. V. Tokatly and O. Pankratov, Phys. Rev. B 62, 2759 s2000 d.
21In the Appendix we show that the interaction force is represent-
able in a divergence form of Eq. s13dif the interaction potential,
wsx,x8d, satisfies the condition ]xwsx,x8d=−]x8wsx,x8dthat is
the formal expression for the Newton’s third law. If wsx,x8d
=wsux−x8ud, the corresponding stress tensor is explicitly sym-
metric.
22E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 s1984 d.
23G. A. Korn and T. M. Korn, Mathematical Handbook for Scien-
tists and Engineers , 2nd. ed. sMcGraw-Hill, New York, 1968 d.
24B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, ModernGeometry—Methods and Applications sSpringer-Verlag, New
York, 1984 d, Vol. 1.
25B. Podolsky, Phys. Rev. 32, 812 s1928 d.
26In classical continuum mechanics a similar tensor is sometimes
called the spin tensor sRef. 1 d. Since in the present quantum
context this term is somewhat misleading, we prefer to use the
term “vorticity” to name the tensor F˜mn.
27L. D. Landau and E. M. Lifshitz, The classical theory of fields,
Course of Theoretical Physics , 4th ed. sPergamon, Oxford,
1975 d, Vol. 2.
28S. Conti and G. Vignale, Phys. Rev. B 60, 7966 s1999 d.
29G. S. Atwal and N. W. Ashcroft, Phys. Rev. B 65, 115109 s2002 d.
30I. V. Tokatly, Phys. Rev. B 71, 165105 s2005 d.
31I. V. Tokatly and O. Pankratov, Phys. Rev. B 66, 153103 s2002 d.
32J. F. Dobson and H. M. Le, J. Mol. Struct.: THEOCHEM 501–
502, 327 s2000 d.
33J. F. Dobson and H. M. Le, Phys. Rev. B 66, 075301 s2002 d.QUANTUM MANY-BODY DYNAMICS IN …I.… PHYSICAL REVIEW B 71, 165104 s2005 d
165104-13 |
PhysRevB.73.113406.pdf | Reduction of activation energy barrier of Stone-Wales transformation in endohedral
metallofullerenes
Woon Ih Choi,1Gunn Kim,1,2Seungwu Han,3and Jisoon Ihm1,*
1School of Physics, Seoul National University, Seoul 151-747, Korea
2Department of Physics, North Carolina State University, North Carolina 27695-7518, USA
3Department of Physics, Ewha Womans University, Seoul 120-750, Korea
/H20849Received 16 July 2005; published 28 March 2006 /H20850
Using ab initio calculations, we examine effects of encapsulated metal atoms inside a C 60molecule on the
activation energy barrier for the Stone-Wales transformation. The encapsulated metal atoms we study are K,Ca, and La which nominally donate one, two, and three electrons to the C
60cage, respectively. We find that
isomerization of the endohedral metallofullerene via the Stone-Wales transformation can occur more easilythan that of the empty fullerene owing to the charge transfer. When K, Ca, and La atoms are encapsulatedinside the fullerene, the activation energy barriers are lowered by 0.30, 0.55, and 0.80 eV, respectively com-pared with that of empty C
60/H208497.16 eV /H20850. The lower activation energy barrier of the Stone-Wales transformation
implies the higher probability of isomerization and coalescence of metallofullerenes, which require a series ofStone-Wales transformations.
DOI: 10.1103/PhysRevB.73.113406 PACS number /H20849s/H20850: 61.48. /H11001c, 81.05.Tp, 34.10. /H11001x
Since its discovery,1the fullerene /H20849C60/H20850has been exten-
sively studied both theoretically and experimentally because
of its potential applications in the fields of nanomaterials andbiomedical science. Although the fullerene molecule usuallyhas the highest icosahedral symmetry allowed by sixty con-stituent carbon atoms, transformations to other isomers of
lower symmetry are still possible. One of the plausible pro-cesses for isomerization of the fullerene is the so-calledStone-Wales /H20849SW/H20850or “pyracylene” rearrangement, which is
the 90° rotation of two carbon atoms with respect to themidpoint of the bond.
2The SW transformation is also used to
describe the structural changes of sp2-bonded carbon
nanosystems.3For example, it has been proposed that the
fusion process of fullerenes or carbon nanotubes may occurthrough a sequence of such a rearrangement.
4–6One problem
in this proposal is that the activation energy barrier of theSW transformation is /H110117 eV,
7due to the breaking of two /H9268
bonds, which is too high to overcome at the temperature in
merging processes /H208491000–1500 °C /H20850.8,9
There are several factors that can affect the energy barrier
in the SW transformation. Eggen et al. found that the activa-
tion barrier for the SW rearrangement was substantially re-duced in the presence of a loosely bound carbon atom lo-cated preferentially in the region of paired pentagons,
10
explaining the growth of fullerenes in terms of autocatalysisby a carbon atom. On the other hand, the energy barrier ofthe SW transformation could change through certain kindsof endohedral doping into the fullerene cage. While Slaninaet al. reported the catalytic effect of various kinds of ele-
ments or ions in the bowl-shaped C
34H12,11relatively few
studies have been devoted to the SW transformation of theendohedral metallofullerene. In this paper, we investigate ef-fects of encapsulated metal atoms on the activation energybarrier in the SW transformation of the M@C
60molecule,
where M=K, Ca, or La atoms. The nominal charge transfers
from metal atoms to the fullerene will be 1–3 electrons, cor-responding to valence states of K
+,C a2+, and La3+. We findthat isomerization of the endohedral metallofullerene via the
SW transformation can occur more easily than that of thebare fullerene owing to the charge transfer from the encap-sulated metal atoms to the fullerene. It is consistent withexperiments that metallofullerenes are much easier to fuse
than empty fullerenes in peapods.
12,13
In this work, we perform ab initio pseudopotential calcu-
lations using the plane wave basis set14within the local den-
sity approximation for the exchange-correlation effect. Theultrasoft pseudopotential
15is adopted with the cutoff energy
of 30 Ry. The supercell size is 29 /H1100329/H1100329aB3, where aBis
the Bohr radius, which is large enough to exclude undesir-able interactions between supercells. For La, we treat 11 va-lence electrons /H208495s
25p66s25d1/H20850explicitly since the 5 sand 5 p
orbitals have chemical interactions with the fullerene
molecule.16The scalar relativistic effects important in heavy
elements such as La are taken into account. The atomic po-sitions are relaxed until the total root-mean-square force ofthe atoms becomes less than 0.014 eV/Å. We use thenudged-elastic-band /H20849NEB /H20850method
17to calculate the activa-
tion energy barrier in the SW transformation of the fullerene.Thirteen replicas are chosen including the initial and finalconfigurations to construct an elastic band. We use the climb-ing image technique to locate the transition state /H20849TS/H20850
precisely.
18
We start with optimizing the La position on a threefold
symmetry axis 1.5 Å away from the center of the cage andobtain the stable position of the La atom in C
60. For K @C60
and Ca @C60, the starting positions for optimization are at the
center19and on a fivefold axis 1.5 Å away from the center of
the cage,20respectively. After relaxation, we confirm that the
relaxed positions are local minima rather than saddle pointsby perturbing the final position of the endohedral dopant.
The relaxed geometry shows that the K atom prefers to be
at the center of C
60. On the other hand, the stable positions of
the La and Ca atoms are off-centered and the distances fromthe La and Ca atom to the nearest C atom are 2.5 Å andPHYSICAL REVIEW B 73, 113406 /H208492006 /H20850
1098-0121/2006/73 /H2084911/H20850/113406 /H208494/H20850/$23.00 ©2006 The American Physical Society 113406-13.1 Å, respectively. Upon encapsulation of the La atom, the
threefold degeneracy of the t1ustates is split into 2+1 since
the off-centered La atom breaks the icosahedral symmetry ofan isolated C
60.21Almost three electrons are transferred from
the La atom to the doubly degenerate states /H20849fourfold degen-
eracy including the spin degrees of freedom /H20850of C 60.21,22The
twofold degeneracy is split again by occupation of the de-generate states with an odd number of electrons.
In Fig. 1, the SW transformation of an empty fullerene is
shown with a schematic diagram of the potential energy. Inthe TS, the rotating carbon dimer /H20851shown in black in Fig.
1/H20849b/H20850/H20852has a bond length of 1.23 Å, with 1.40 Å bonds to each
of the two nearest carbon atoms for C
60. This result is in
agreement with previous studies.7,23To confirm that the en-
ergy maximum point along the pathway is actually a transi-tion state with the Hessian index of one, we analyze theHessian matrix at the highest point. Instead of the fullphonon-mode analysis whose computational load is veryheavy, we look into a 6 /H110036 Hessian matrix in a configura-
tional subspace constructed by two rotating carbon atoms.We find that only one of the six eigenvalues of the matrix isnagative, i.e., the Hessian index of one, and this demon-strates that the NEB method reliably identifies the transitionstate. Regarding the SW transformation of metallofullereneswith off-centered dopants, we rotate the C-C dimer which isthe closest to the metal atom /H20849Ca and La /H20850.
Table I lists the computed activation barriers /H20849E
a/H20850of the
SW transformations in various fullerenes and a graphene
sheet. The activation barrier in the SW transformation ofthe fullerene /H208497.16 eV /H20850is lower than that of the graphene
sheet /H208499.2 eV, the highest among the systems we study /H20850. The
barrier reduction in fullerenes is due to the curvature effectassociated with the deviation from the sp
2bonding.6When
K, Ca, and La atoms are encapsulated inside the fullerene,the activation energy barriers are lowered by 0.30, 0.55, and0.80 eV, respectively, compared with that of bare C
60.
To determine the major effect of the encapsulated metal
atom on the lowering of the activation barrier, we performcalculations for fullerenes charged with one or two electrons.
The results for C
603−are not presented because the third elec-tron is found to be unbound. When a fullerene is charged, the
activation barrier becomes lower than that of the neutralcage, similar to the endohedral metallofullerenes. The barrierreduction is approximately proportional to the number of do-
nated electrons. E
ain the SW transformation of C60−/H20849C602−/H20850is
almost identical to that of K @C60/H20849Ca@C60/H20850. Therefore, one
can conclude that the electron donation of the incorporated
metal atom lowers the barrier and that the chemical bondingbetween the metal atom and C
60is less important.
The reaction energy /H20849Er=Eproduct −Ereactant /H20850shows the
same trend as Eain Table I. The C2v-symmetry C 60with two
pairs of adjacent pentagons obtained by the SW transforma-tion is 1.59 eV higher in energy than C
60with the
Ih-symmetry. This value is in good agreement with other re-
sults found in the literature.24By contrast, for La @C60,Eris
0.3 eV, which is much smaller than that of C 60/H208491.59 eV /H20850.I t
implies that an isomer of the metallofullerene which has ad-
jacent pentagons can exist with higher probability comparedto empty C
60. In the case of K @C60which has an interme-
diate value of Er, the K atom at the center of the cage does
not move during the process of the SW transformation. Bycontrast, the La atom in the fullerene moves by 1.0 Å towardthe paired pentagons during the dimer rotation. The displace-ment of the La atom induces a large elongation of the C
60
cage compared with the empty C 60molecule. The geometry
of an empty fullerene is governed by the isolated-pentagonrule /H20849IPR/H20850,
25,26which means that the structure is most stable
when all pentagons are surrounded by five hexagons. TheIPR-violating cages such as Sc
2@C66and Sc 3N@C68, how-
ever, are stabilized by the electron transfer between the en-capsulated metal atoms and the carbon cage, which signifi-cantly decreases the strain energy caused by pairedpentagons and thus stabilizes the fullerene.
27,28Owing to the
108° bond angles of the pentagon, the apex atoms in thepaired pentagon regions of the empty fullerene have a hy-bridization very close to sp
3and exhibit a dangling-bondlike
character.29For metallofullerenes, on the other hand, the
electron donation from the metal atom weakens thedangling-bondlike character and stabilizes the IPR-violatingisomers.
To clarify the role of the encapsulated metal atom, we
investigate the electronic structures of C
60and La @C60in
detail. The inspection of the projected density of states/H20849PDOS /H20850onto the 6 sand 5 dorbitals of La for each reactant,
FIG. 1. Schematic energy diagram and the atomic configuration
along the reaction coordinate in the Stone-Wales transformation ofempty C
60./H20849a/H20850Energy profile along the reaction coordinate. The
dots represent the energies of the replicas in the SW transformationin the NEB method. /H20849b/H20850Atomic rearrangement from the I
hisomer to
theC2visomer of the empty C 60molecule. Here, the rotating carbon
dimer is shown in black.TABLE I. Activation barrier /H20849Ea/H20850and reaction energy /H20849Er/H20850of the
SW transformation in various sp2-bonded carbon systems. All en-
ergy values are in eV.
System Ea Er
C60 7.16 1.59
K@C60 6.85 1.07
Ca@C60 6.61 0.59
La@C60 6.36 0.30
C60−6.87 1.12
C602−6.59 0.67
C42H16/H20849for graphene /H20850 9.20 3.07BRIEF REPORTS PHYSICAL REVIEW B 73, 113406 /H208492006 /H20850
113406-2transition state, and product indicates that the 6 sand 5 den-
ergy levels of the La atom mainly lie above the Fermi level.It means that almost three electrons transfer from La to thefullerene in La @C
60. In Fig. 2 /H20849a/H20850, the highest occupied mo-
lecular orbital /H20849HOMO /H20850at −0.2 eV /H20849indicated by the arrow /H20850
corresponds to the dangling-bond-originated state as illus-trated in Fig. 2 /H20849b/H20850. This state is above the HOMO of C
60/H20849i.e.,
the fivefold degenerate states originating from the HOMO ofempty C
60which are split off now /H20850. The energy level which
has the dangling bond character in La @C60is at −0.7 eV
/H20849HOMO−2 /H20850as shown in Fig. 2 /H20849c/H20850by the arrow and the wave
function in Fig. 2 /H20849d/H20850. Here one can observe three electrons
coming from the La atom occupy the upper one and halflevels which are the lowest unoccupied molecular orbital/H20849LUMO /H20850and LUMO+1 levels in the transition state of C
60.
The overall shape of the two wave functions /H20851Figs. 2 /H20849b/H20850and
2/H20849d/H20850/H20852is similar except for small changes due to the perturba-
tion of the La atom.
Figures 3 /H20849c/H20850and 3 /H20849d/H20850show the PDOS onto 2 porbitals of
two carbon atoms /H20851labeled as A,BandA/H11032,B/H11032in Figs. 3 /H20849a/H20850
and 3 /H20849b/H20850/H20852which have the dangling bonds in the transition
state. In the case of empty C 60, the PDOS of AandBare
exactly same. For La @C60, on the other hand, the charge
density of nonbonding states at site A/H11032/H20849between −1 and
0e V /H20850is not identical to the density at site B/H11032. This difference
arises from the hybridization between the La atom and the Catom at site B. In the TS configuration of empty C
60, the
HOMO and LUMO are both localized states within thehemisphere containing two dangling atoms while the HOMO
and LUMO in the TS of La @C60are delocalized on the
whole cage. Consequently, the electron transfer and hybrid-ization between the fullerene and metal atom reduce the ac-tivation energy barrier of the SW transformation in the en-dohedral metallofullerene.
In summary, we have studied the effect of encapsulated
metal atoms inside a C
60molecule on the activation energy
barrier of the SW transformation. The metal atoms in ourstudy are K, Ca, and La, which donate one, two, and threeelectrons to the C
60cage, respectively. We have found that
isomerization of the endohedral metallofullerene by the SWtransformation can occur more easily than that of the emptyfullerene owing to the charge transfer. The reduction of theactivation energy barrier by the electron transfer may alsoexplain the fact that metallofullerenes are much easier tocoalesce than empty fullerenes inside nanotubes.
We thank H. Shinohara for valuable discussions. This
work is supported by the CNNC of Sungkyunkwan Univer-sity, the KRF Grant No. KRF-2005-070-C00041, the Sam-sung SDI-SNU Display Innovation Program, and the MOSTthrough the NSTP Grant No. M1-0213-04-001. S. Han ac-knowledges support by the KRF /H20849Grant No. KRF-2004-005-
C00057 /H20850. The computations were performed at the Super-
computing Center of KISTI through the SupercomputingApplication Support Program using the
PWSCF code.30
FIG. 2. /H20849Color online /H20850Density of states /H20849DOS /H20850for C 60and
La@C60in the transition state /H20851/H20849a/H20850and /H20849c/H20850, respectively /H20852and the
isodensity surface plots of particular states for C 60and La @C60/H20851/H20849b/H20850
and /H20849d/H20850/H20852. The energy levels of the wave functions in /H20849b/H20850and /H20849d/H20850are
indicated by arrows in /H20849a/H20850and /H20849c/H20850, respectively. The isodensity
value is 0.0014 e/Å3and the surface is color-coded according to
the sign of the wave functions.
FIG. 3. /H20849Color online /H20850Atomic structures of the transition state
configuration for /H20849a/H20850C60and /H20849b/H20850La@C60, and PDOS of two atoms
/H20851labeled A,BandA/H11032,B/H11032in/H20849a/H20850and /H20849b/H20850, respectively /H20852which have
dangling bonds in the transition state, for /H20849c/H20850C60and /H20849d/H20850La@C60.
The defect states due to the dangling bonds or distorted bondingconfigurations are indicated by arrows. The red solid and blue dot-ted lines are the PDOS onto 2 porbitals of A,A
/H11032andB,B/H11032atoms,
respectively.BRIEF REPORTS PHYSICAL REVIEW B 73, 113406 /H208492006 /H20850
113406-3*Corresponding author. Electronic address: jihm@snu.ac.kr
1H. W. Croat, J. R. Heath, S. C. O’Brien, R. F. Curl, and R. E.
Smalley, Nature /H20849London /H20850318, 162 /H208491985 /H20850.
2A. J. Stone and D. J. Wales, Chem. Phys. Lett. 128, 501 /H208491986 /H20850.
3F. Diederich, R. Ettl, Y. Rubin, R. L. Whetten, R. Beck, M. Al-
varez, S. Anz, D. Sensharma, F. Wudl, K. C. Khemani, and A.Koch, Science 252, 548 /H208491991 /H20850.
4S. Han, M. Yoon, S. Berber, N. Park, E. Osawa, J. Ihm, and D.
Tomanek, Phys. Rev. B 70, 113402 /H208492004 /H20850.
5Y. Zhao, B. I. Yakobson, and R. E. Smalley, Phys. Rev. Lett. 88,
185501 /H208492002 /H20850.
6M. Yoon, S. Han, G. Kim, S. Lee, S. Berber, E. Osawa, J. Ihm,
M. Terrones, F. Banhart, J.-C. Charlier, N. Grobert, H. Terrones,P. M. Ajayan, and D. Tománek, Phys. Rev. Lett. 92, 075504
/H208492004 /H20850.
7H. F. Bettinger, B. I. Yakobson, and G. E. Scuseria, J. Am. Chem.
Soc. 125, 5572 /H208492003 /H20850.
8B. W. Smith and D. E. Luzzi, Chem. Phys. Lett. 321, 169 /H208492000 /H20850.
9S. Bandow, M. Takizawa, K. Hirahara, M. Yudasaka, and S.
Iijima, Chem. Phys. Lett. 337,4 8 /H208492001 /H20850.
10B. R. Eggen, M. I. Heggie, G. Jungnickel, C. D. Latham, R.
Jones, and P. R. Briddon, Science 272,8 7 /H208491996 /H20850.
11Z. Slanina, X. Zhao, F. Uhlik, M. Ozawa, and E. Osawa, J. Or-
ganomet. Chem. 599,5 7 /H208492000 /H20850.
12T. Okazaki, K. Suenaga, K. Hirahara, S. Bandow, S. Iijima, and
H. Shinohara, J. Am. Chem. Soc. 123, 9673 /H208492001 /H20850.
13H. Shinohara /H20849private communication /H20850.
14J. Ihm, A. Zunger, and M. L. Cohen, J. Phys. C 12, 4409 /H208491979 /H20850.
15D. Vanderbilt, Phys. Rev. B 41, R7892 /H208491990 /H20850.
16K. Laasonen, W. Andreoni, and M. Parrinello, Science 258, 1916/H208491992 /H20850.
17Y.-G. Yoon, M. S. C. Mazzoni, and S. G. Louie, Appl. Phys. Lett.
83, 5217 /H208492003 /H20850.
18L. S. Wang, J. M. Alford, Y. Chai, M. Diener, J. Zhang, S. M.
McClure, T. Guo, G. E. Scuseria, and R. E. Smalley, Chem.Phys. Lett. 207, 354 /H208491993 /H20850.
19G. Henkelman and H. Jonsson, J. Chem. Phys. 113, 9978 /H208492000 /H20850.
20G. Henkelman, B. P. Uberuaga, and H. Jonsson, J. Chem. Phys.
113, 9901 /H208492000 /H20850.
21S. Han, J. Korean Phys. Soc. 44, 889 /H208492004 /H20850.
22Jing Lu, Xinwei Zhang, and Xiangeng Zhao, Chem. Phys. Lett.
332,5 1 /H208492000 /H20850.
23Y. Kumeda and D. J. Wales, Chem. Phys. Lett. 374, 125 /H208492003 /H20850.
24B. R. Eggen, C. D. Lantham, M. I. Heggie, B. Jones, and P. R.
Briddon, Synth. Met. 77, 165 /H208491996 /H20850.
25H. Kroto, Nature /H20849London /H20850329, 529 /H208491987 /H20850.
26T. G. Schmalz, W. A. Seitz, D. J. Klein, and G. E. Hite, J. Am.
Chem. Soc. 110, 1113 /H208491988 /H20850.
27C. R. Wang, T. Kai, T. Tomiyama, T. Yoshida, Y. Kobayashi, E.
Nishibori, M. Takata, M. Sakata, and H. Shinohara, Nature/H20849London /H20850408, 426 /H208492000 /H20850.
28S. Stevenso, P. W. Fowler, T. Heine, J. C. Duchamp, G. Rice, T.
Glass, K. Harich, E. Hajdu, R. Bible, and H. C. Dorn, Nature/H20849London /H20850408, 427 /H208492000 /H20850.
29E. Kaxiras, in Atomic and Electronic Structure of Solids /H20849Cam-
bridge University Press, Cambridge, 2003 /H20850.
30S. Baroni, A. Dal Corso, S. de Gironcoli, and P. Giannozzi, http://
www.pwscf.orgBRIEF REPORTS PHYSICAL REVIEW B 73, 113406 /H208492006 /H20850
113406-4 |
PhysRevB.102.121403.pdf | PHYSICAL REVIEW B 102, 121403(R) (2020)
Rapid Communications
Tunable and dual-broadband giant enhancement of second-harmonic and third-harmonic
generation in an optimized graphene-insulator-graphene metasurface
Jian Wei You and Nicolae C. Panoiu
Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom
(Received 8 May 2020; revised 26 August 2020; accepted 28 August 2020; published 10 September 2020)
We demonstrate a scheme to dramatically enhance both the second- and third-harmonic generation (SHG,
THG) in a graphene-insulator-graphene metasurface. The key underlying feature of our approach is the existenceof a double-resonance phenomenon, namely, the metasurface is designed to possess fundamental plasmonresonances at both the fundamental frequency and the higher harmonic. This dual resonant field enhancement,combined with a favorable spatial overlap of the optical near fields, lead to the increase of the THG and SHGby∼10
9and∼106, respectively. We also demonstrate that by tuning the Fermi energy of the graphene gratings
the dual-resonance property can be locked in over a remarkably broad spectral range of ∼20 THz, which is more
than three orders of magnitude larger than the spectral tunability achievable in metal-based plasmonic systems.Importantly, the enhanced nonlinear frequency generation process can be readily switched in the same systembetween the second and third harmonic. This type of graphene metasurface could open up new avenues towardsthe development of novel ultracompact and multifrequency active photonic nanodevices.
DOI: 10.1103/PhysRevB.102.121403
The first successful isolation of graphene from graphite [ 1]
via mechanical exfoliation has opened up a rapidly growingfield of research [ 2–5], primarily due to the unique and re-
markable properties of this new two-dimensional material.In its early stages research on graphene focused on its elec-tronic and mechanical properties, but it was soon realizedthat key optical properties, such as extreme optical near-fieldconfinement induced by the excitation of surface-plasmonpolaritons (SPPs) [ 6–11], tunability of the optical response
via gate voltage and chemical doping [ 12–14], and low losses
at high carrier densities [ 15,16], could transform graphene
into a promising and versatile material platform for a broadarray of optoelectronic applications. To this end, photonicdevices based on graphene, including diffractive elements, op-tical sensors, topological photonic devices, and photovoltaicand photoresistive devices [ 17–32], have already been demon-
strated.
In addition to advances in exploiting the linear physics
of graphene, its nonlinear optical properties could playan equally important role in key applications. Due to itscentrosymmetric nature, the leading nonvanishing nonlinearoptical interactions in graphene are of third-order type, suchas third-harmonic generation (THG) and Kerr effect. In partic-ular, it has been demonstrated that the strength of third-ordernonlinear optical interactions in graphene is several ordersof magnitude larger than in typical semiconductors [ 33–36].
More importantly, these nonlinear optical interactions canbe further enhanced upon resonant excitation of SPPs ingraphene structures, which leads to a number of excitingapplications [ 18–20,37–46], including frequency mixing [ 18],
photodetectors [ 25], generation of spatial solitons [ 43,44],
physical systems with tunable Dirac points [ 45], and Ander-
son light localization at the nanoscale [ 46].Although the second-harmonic generation (SHG) is gen-
erally forbidden in a freestanding graphene sheet, it isnevertheless permitted in two main configurations. First, SHGdoes arise in graphene nanostructures from nonlocal ef-
fects [ 36,47,48], namely, when nonlinear sources of SHG are
magnetic dipoles and electric quandrupoles. Second, by plac-ing a graphene sheet on a substrate, the inversion symmetryof the system is broken and SHG due to local nonlin-
ear polarization (electric dipoles) can occur [ 49–54]. Under
these conditions, the effective second-order susceptibility ofgraphene can be several orders of magnitude larger thanthat of semiconductors widely used in nonlinear optics, e.g.,GaAs [ 55].
In this Rapid Communication, we introduce a graphene-
insulator-graphene (GIG) optical structure with several uniqueoptical properties that cannot be achieved with metal-basedplasmonic nanostructures. In particular, the graphene meta-surface is designed to produce a tunable and dual-broadbandenhancement of both SHG and THG, by ∼10
6and∼109,
respectively, and the enhanced nonlinear frequency generationprocess can be readily switched between the second harmonic(SH) and third harmonic (TH). The giant enhancement ofthese two most ubiquitous nonlinear optical interactions isrealized by ensuring that the GIG structure possesses first-
order plasmon resonances at both the fundamental frequency
(FF) and higher harmonics (HHs), namely, SH and TH. Re-markably, we demonstrate that by tuning the Fermi energyof the graphene gratings the dual-resonance property can belocked in over a broad spectral range of ∼20 THz, which to
date is perhaps the largest spectral tunability of a resonantnonlinear optical interaction reported in a plasmonic system.For the sake of generality, for SHG, we consider both the casesof a nonlocal nonlinear polarization, which corresponds to
2469-9950/2020/102(12)/121403(6) 121403-1 ©2020 American Physical SocietyJIAN WEI YOU AND NICOLAE C. PANOIU PHYSICAL REVIEW B 102, 121403(R) (2020)
FIG. 1. (a) Schematic of a tunable GIG nanoresonator consisting
of GNRs with different widths placed at the opposite facets of an in-
sulator slab. (b) Illustration of physical mechanisms of enhancement
of SHG and THG in the GIG metasurface.
graphene structures in a stacked configuration embedded in a
background medium [ 56–58], and the case of a local nonlinear
polarization, when graphene is placed on a substrate.
The proposed periodic GIG structure is depicted in
Fig. 1(a). Its unit cell consists of two graphene nanoribbons
(GNRs) placed at opposite facets of an (insulator) dielectricspacer. Electrodes are placed in contact with the GNRs, whichallows one to tune their Fermi level. A TM-polarized planewave with frequency ω
0is incident from above onto the
GIG structure. As SPPs of GNRs are geometry dependent,their frequency can be set by properly choosing the width ofthe ribbons. Using this feature, the widths of the GNRs arechosen in such a way that the bottom and top GNRs havefirst-order SPP resonances at both the FF, ω
FF=ω0, and HH
(ωNL=2ω0for SHG and ωNL=3ω0for THG), respectively,
as per Fig. 1(b). In addition, the nonlinear optical response
of the GIG structure can be further optimized by requiringthat the bottom GNRs have higher-order plasmons at the HH,too [ 40]. Importantly, this nonlinear optical device can be
used to enhance both the SHG and THG by simply varyingthe Fermi level in the top GNRs, so as the frequency of itsfirst-order SPP is switched between 2 ω
0and 3ω0.
There are two key mechanisms that contribute to the re-
markably large enhancement of the nonlinear optical responseof the GIG structure, namely, by several orders of magnitudeas compared to that of a graphene sheet. The first one, indi-cated by path /circlecopyrt
1in Fig. 1(b), requires that the bottom GNR
has a first-order plasmon at ω0and a higher-order SPP at
ωNL[40]. Then, the field at ω0incident onto the bottom GNRs
generates a strong field on these GNRs at ω0, via the resonant
excitation of first-order SPPs, and, subsequently, higher-orderSPPs are resonantly generated by the nonlinear polarization inthese same bottom GNRs.
We now introduce a much more efficient mechanism con-
tributing to the enhancement of the nonlinear response of theGIG structure. It is schematically indicated by path /circlecopyrt
2in
Fig. 1(b) and relies on the fact that the top GNRs possessfirst-order SPPs at the HH. This mechanism can be described
as follows: the enhanced optical field due to the excitationof first-order SPPs on the bottom GNRs induces on the topGNRs a strong nonlinear polarization at the HH via near-fieldinteraction. This, in turn, resonantly excites first-order SPPson the top GNRs. Additionally, first-order SPPs on the topGNRs (at HH) are also directly generated via optical near-fieldcoupling with higher-order SPPs of the bottom GNRs.
In the final stage of the nonlinear optical interaction
between the incoming light and the GIG structure, the higher-order SPPs on the bottom GNRs and the first-order SPPson the top GNRs couple to the radiative modes to generatea strong signal at the HH. In fact, this GIG system acts asa nonlinear Yagi-Uda nanoantenna [ 59]: the bottom and top
GNRs are the driver at ω
0and the director at ωNL, respectively.
To illustrate these ideas, we considered a metasurface with
the periods of the bottom and top graphene gratings of /Lambda11=
200 nm and /Lambda12=100 nm, respectively. The widths w1and
w2of the GNRs and the thickness, h, of the spacer are de-
signed so as to achieve a double-resonance effect. We assumethat the spacer is made of polyethylene, which has relative per-mittivity of /epsilon1
s=2.28 and is practically lossless at midinfrared
frequencies [ 60]. The linear and nonlinear optical responses of
this GIG structure have been studied using an in-house devel-oped code based on the generalized-source finite-differencetime-domain (GS-FDTD) method; for details on the numeri-cal approach see the Supplemental Material (SM) [ 61].
In this method, the linear properties of graphene are mod-
eled using a linear surface optical conductivity [ 62],
σ
s=e2kBTτ
π¯h2ω/bracketleftbiggEF
kBT+2l n ( e−EF/kBT+1)/bracketrightbigg
+ie2
4π¯hlnξ−iω
ξ+iω.
(1)
Here, EF,T, andτare the Fermi energy, temperature, and re-
laxation time, respectively, ω=1−iωτ, andξ=2|EF|τ/¯h.
The nonlinear optical response is described by nonlinear
surface current densities determined by second- and third-order nonlinear surface susceptibilities [ 35,36,48–53]. In the
case of THG, the third-order surface current density ofgraphene is expressed as
J
(3)(/Omega13,ω)=σ(3)
s(/Omega13;ω)...E(ω)E(ω)E(ω), (2)
where /Omega13=3ωis the frequency at the TH and σ(3)
sis the
third-order nonlinear surface optical susceptibility. It is de-scribed by a single scalar function, σ
(3)
s, via the relation
σ(3)
s,ijk l=σ(3)
s(δijδkl+δikδjl+δilδjk)/3[35,36], with δijbeing
the Kronecker delta. Furthermore, in the case of SHG arisingfrom a local nonlinear polarization, the second-order nonlin-ear surface current density can be written as
J
(2)(/Omega12,ω)=σ(2)
s(/Omega12;ω):E(ω)E(ω), (3)
where /Omega12=2ωis the frequency at the SH and σ(2)
sis
the second-order nonlinear surface optical susceptibility.Symmetry considerations based on the fact that graphenebelongs to the D
6hsymmetry group lead to the conclusion
that the tensor σ(2)
s(/Omega12;ω) has three independent nonzero
components, σ(2)
s,⊥⊥⊥,σ(2)
s,/bardbl/bardbl⊥=σ(2)
s,/bardbl⊥/bardbl, and σ(2)
s,⊥/bardbl/bardbl, where the
symbols ⊥and/bardblrefer to the directions perpendicular onto
and parallel to the plane of graphene, respectively. The values
121403-2TUNABLE AND DUAL-BROADBAND GIANT ENHANCEMENT … PHYSICAL REVIEW B 102, 121403(R) (2020)
FIG. 2. (a) Wavelength dependence of the absorption, A,r e -
flectance, R, and transmittance, T. (b)–(d) Spatial profile of the
dominant component of the electric field, |Ex|, at the FF, determined
for the first three SPP resonances, respectively. (e) Dispersion mapof absorption spectra vs width of the bottom GNRs. Yellow, blue,
and green lines correspond to λ
(1)
FF,λ(1)
FF/2, and λ(1)
FF/3, respectively,
where λ(1)
FFis the width-dependent wavelength of the first-order SPP.
of these parameters used in this study are σ(2)
s,⊥⊥⊥=− 9.71i×
10−16Am V−2,σ(2)
s,/bardbl/bardbl⊥=σ(2)
s,/bardbl⊥/bardbl=− 2.56i×10−16Am V−2,
andσ(2)
s,⊥/bardbl/bardbl=− 2.09i×10−16Am V−1[50,53]. Note that,
as demonstrated in the SM, the qualitative conclusions ofour study do not change if instead of a local second-ordernonlinear response of graphene one considers a nonlocal one.
To characterize the linear optical response of the GIG
structure, we first calculated the absorption, A, transmittance,
T, and reflectance, R, corresponding to the bottom graphene
grating with geometrical parameters given in the inset ofFig. 2(a), and with E
F=0.4e V ,τ=0.2 ps, and T=300 K.
The results of these calculations are summarized in Fig. 2(a).
It can be seen that the absorption spectrum possesses a seriesof resonances, which are due to the excitation of SPPs onthe GNRs. The field distributions of the first three SPPs aregiven in Figs. 2(b)–2(d), respectively. They show that the local
optical field is strongly enhanced and confined around GNRs,with the largest field enhancement observed for the first-orderSPP. Moreover, the results presented in Fig. 2(a) show that
the absorption and reflectance spectra have resonances at thesame wavelengths, a feature that is particularly useful for theoptimization of the GIG structure.
A convenient procedure for designing a graphene grating
in which a double-SPP-resonance phenomenon occurs is il-lustrated by the dispersion map of the absorption at the FF,presented in Fig. 2(e). The bands in this map, which show the
FIG. 3. (a) Dispersion map of the top graphene grating. The ma-
genta line shows the width-dependent wavelength of the first-order
SPP. (b) Dependence of absorption spectra of the GIG structure onh. Red and green lines correspond to λ
(1)
FFandλ(1)
FF/3, respectively,
where λ(1)
FFis the thickness-dependent wavelength of the first-order
SPP. (c) Absorption spectra of the optimized top and bottom gratingsas well as that of the GIG structure, determined for the optimal
thickness h=40 nm for which the GIG structure possesses a double
resonance at frequencies ω
0(λ=15.9μm) and 3 ω0(λ=5.3μm).
width-dependent resonance wavelengths of SPPs of different
order, suggest that it is possible to choose the width w1in such
a way that a pair of SPPs exist at the FF and HH. Thus, if w1=
132 nm, a double resonance exists at the FF and SH, i.e., at(λ
FF,λSH=λFF/2), with λSH=λ(P0)=6.04μm, whereas
ifw1=173 nm, a double resonance exists at the FF and TH,
i.e., at ( λFF,λTH=λFF/3), with λTH=λ(P1)=5.25μm.
A drawback of the scheme we just described is that the
plasmon at the HH is a higher-order plasmon and therefore it
is less efficiently excited. In order to overcome this limitationand further enhance the nonlinear optical response of thedevice, another graphene grating is placed onto the spacer.The width w
2of the GNRs of this top grating can be freely
chosen. As such, it is chosen in such a way that at the HH (SHor TH) first-order plasmons exist in these GNRs. For example,
as illustrated in Fig. 3(a), when w
2=27 nm the wavelength
of the first-order plasmon of the GNRs of the top grating is
equal to λ(P1). Therefore, we expect that when w1=173 nm,
w2=27 nm, and h=40 nm the GIG structure possesses first-
order plasmons at both the FF and TH. This property is
verified by the dispersion map of the absorption in the GIGstructure, plotted in Fig. 3(b). This map shows that indeed the
GIG structure has first-order plasmons at λ
FF=15.9μm and
λTH=5.3μm, predominantly localized at the bottom and top
gratings, respectively. Note that due to the optical couplingbetween the top and bottom gratings, the double-resonancephenomenon in the decoupled bottom grating appears at a pairof wavelengths slightly blueshifted as compared to those inthe optimized GIG structure.
121403-3JIAN WEI YOU AND NICOLAE C. PANOIU PHYSICAL REVIEW B 102, 121403(R) (2020)
FIG. 4. (a) Spectra of THG for a graphene sheet, optimized bot-
tom grating, and optimized GIG structure. (b)–(d) Spatial profileof the dominant component of the electric field, |E
x|,a tt h eT H ,
determined for the resonances marked by /circlecopyrt1,/circlecopyrt2,a n d/circlecopyrt3in panel
(a), respectively.
The optical coupling between the two gratings leads to sev-
eral additional interesting phenomena, as per Fig. 3(b).F i r s t ,
the resonance wavelengths of SPPs vary with the thickness h,
especially at small values of hfor which there is a stronger
coupling. Second, for h/lessorsimilar40 nm, the resonance wavelengths
of the first-order plasmon of the top grating and the third-orderplasmon of the bottom grating are no longer equal, so thatone expects a smaller enhancement of the THG. On the otherhand, if his too large, the electric field at the FF in the bottom
grating can no longer excite the first-order plasmon at the THin the top grating, which also leads to decreased enhancementof the THG. Therefore, the optimum value of his∼40 nm.
Note also that for 21 nm <h<27 nm, the second-order plas-
mon in the GIG structure is almost completely suppressed, aphenomenon explained by the fact that the system has a boundstate in the continuum for h/similarequal24 nm [ 73].
These conclusions are further validated by the absorption
spectra presented in Fig. 3(c), where we compare the ab-
sorption in the bottom grating optimized to possess a doubleresonance at λ
FF=15.75μm and λTH=λFF/3=5.25μm,
the absorption in the top grating designed to possess a funda-mental plasmon at the TH wavelength, λ
TH=5.25μm, and
the absorption in the optimized GIG structure. These spectrashow that by adding the top grating the absorption at the TH isenhanced by more than 12 times, which suggests that the localoptical field and implicitly the nonlinear optical response ofthe GIG structure can be significantly enhanced.
To quantify the enhancement of the THG in our GIG struc-
ture, we computed the THG spectra for a graphene sheet, thebottom grating optimized to possess a double resonance atλ
FF=15.75μm and λTH=λFF/3=5.25μm, and the opti-
mized GIG structure, the results being compared in Fig. 4(a).
These spectra show that, as compared to the graphene sheet,the THG in the optimized bottom grating is enhanced by ∼10
5
when the FF coincides with that of the first-order plasmon of
the bottom GNRs. Under the same excitation conditions, anadditional 21 times enhancement is observed in the GIG struc-ture. These results are explained by the spatial profiles of theamplitude of the dominant component of the TH electric field,E
x, presented in Figs. 4(b) and 4(c). Thus, in the optimized
bottom grating, at the TH, a third-order plasmon is excited,
FIG. 5. (a) Absorption spectra of the optimized GIG structure
vs the Fermi energy of the two graphene gratings. Red and bluelines correspond to λ
(1)
FFandλ(1)
FF/3, respectively, where λ(1)
FFis the
Fermi-energy-dependent wavelength of the first-order SPP. Inset:
profile of the TH electric field, |Ex|, determined for EF=0.3e V .
(b) The same as in (a), but calculated for the case when only EFin
the top grating varies and EF=0.4 eV in the bottom grating. Inset:
profile of the SH electric field, |Ex|, determined for EF=0.3e V .
(c) Absorption spectra of a GIG structure optimized to enhance
SHG ( EF=0.2 eV) and THG ( EF=0.4 eV). (d) Spectra of SHG
determined for a graphene sheet placed on a polymer substrate, thebottom grating, and the optimized GIG structure.
whereas in the optimized GIG structure both a first-order
plasmon of the top grating and a third-order plasmon of thebottom grating are generated. Importantly, it can be seen thatw h e nt h eF F[ 3 ω
0in Fig. 4(a)] is equal to that of the first-order
plasmon of the top grating and the third-order plasmon of thebottom grating the THG is enhanced by ∼10
9, as compared to
the case of a graphene sheet.
A particularly important property of the proposed GIG
structure is the broadband nonlinearity enhancement at theHH, achievable by tuning the Fermi energy in the two grat-ings. The reason for this unique property is revealed by thedispersion map of the absorption of the optimized GIG struc-ture, presented in Fig. 5(a). Thus, it is clear from this figure
that the ratio between the wavelengths of the first-order SPPof the bottom GNRs on the one hand, and third-order SPPsof the bottom GNRs and first-order SPPs of the top GNRs onthe other hand, remains constant as the Fermi energy varies.
121403-4TUNABLE AND DUAL-BROADBAND GIANT ENHANCEMENT … PHYSICAL REVIEW B 102, 121403(R) (2020)
Consequently, the double-resonance property is precisely pre-
served as the Fermi energy varies. More specifically, as shownin Fig. 5(a), when the Fermi energy is varied from 0.2 to
1 eV , the resonance wavelength of the first-order plasmon ofbottom GNRs, and implicitly the operating wavelength at theFF, varies from 25 to 10 μm.
Another remarkable property of our proposed GIG struc-
ture is that it can enhance both the THG and SHG.Specifically, this functionality can be realized by tuning theFermi energy only in the top grating, such that the resonancewavelength of first-order SPPs of the GNRs in this grating isshifted from the TH to the SH. This is demonstrated by theabsorption map of the GIG structure presented in Fig. 5(b).
Thus, this figure shows two types of plasmon bands, namely,flatbands corresponding to SPPs in the bottom grating, whichobviously do not depend on E
Fin the top grating, and plas-
mon bands associated to the top grating, whose resonancewavelength depends on E
F. In particular, it can be seen that
whereas the resonance wavelength on the first-order SPPs ofthe bottom grating remains constant, λ
FF=15.75μm, the
resonance wavelength of first-order SPPs of the top gratingvaries from λ(P
4)=λFF/3=5.25μmt oλ(P3)=λFF/2=
7.875μm when EFis tuned from 0.4 to 0.2 eV , respectively
[see also Fig. 5(c)].
The strong enhancement of the SHG of the GIG struc-
ture, achieved for EF=0.2 eV, is clearly demonstrated by the
plots presented in Fig. 5(d), where we show the SHG spectracorresponding to a graphene sheet placed on the polymer sub-
strate, the bottom grating, and the combined GIG structure,determined for E
Ffor which the top GNRs have first-order
SPPs at the SH. As in the case of the TH, one can see thatstrongly enhanced SHG can be achieved in the optimized GIGstructure. In particular, at resonance, the SHG in the bottomgrating is ∼10
5larger than in the case of a graphene sheet,
whereas a further order of magnitude enhancement is achievedin the optimized GIG structure.
To conclude, a highly engineered GIG metasurface for
enhancement of SH and TH is studied in this Rapid Commu-nication. We demonstrate that it can be used to achieve tunableand dual-broadband enhancement of both nonlinear opticalinteractions, a property originating from the fact that oursystem possesses tunable double resonances. In practice, thisnonlinearity enhancement can be further improved by stack-ing several GIG units together to construct a 3D graphenemetamaterial [ 56]. This new type of graphene structures could
open up new research directions towards the development ofnovel ultracompact and multifrequency active photonic nan-odevices.
The authors acknowledge the use of UCL Legion High Per-
formance Computing Facility (Legion@UCL), and associatedsupport services, in the completion of this work. This workwas supported by European Research Council (ERC), GrantAgreement No. ERC-2014-CoG-648328.
[1] K. S. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, Y .
Zhang, S. V . Dubonos, I. V . Grigorieva, and A. A. Firsov,Science 306, 666 (2004) .
[2] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D.
Mayou, T. Li, J. Hass, A. N. Marchenkov, and E. H. Conrad,Science 312, 1191 (2006) .
[3] A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007) .
[4] A. A. Balandin, S. Ghosh, W. Bao, T. Calizo, D. Teweldebrhan,
F. Miao, and C. N. Lau, Nano Lett. 8, 902 (2008) .
[5] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,
and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009) .
[6] F. H. L. Koppens, D. E. Chang, and F. J. G. de Abajo, Nano
Lett. 11, 3370 (2011) .
[7] J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri,
F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera,P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R.Hillenbrand, and F. H. L. Koppens, Nature (London) 487,7 7
(2012) .
[8] Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M.
Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez,M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, andD. N. Basov, Nature (London) 487, 82 (2012) .
[ 9 ] A .N .G r i g o r e n k o ,M .P o l i n i ,a n dK .S .N o v o s e l o v , Nat.
Photonics 6, 749 (2012) .
[10] Y . V . Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy,
Int. J. Mod. Phys. B 27, 1341001 (2013) .
[11] F. J. G. de Abajo, ACS Photonics 1, 135 (2014) .
[12] M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju,
F. Wang, and X. Zhang, Nature (London) 474, 64 (2011) .[13] M. F. Craciun, S. Russo, M. Yamamoto, and S. Tarucha,
Nano Today 6, 42 (2011) .
[14] V . W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A.
Atwater, Nano Lett. 13, 2541 (2013) .
[15] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S.
Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard,and J. Hone, Nat. Nanotechnol. 5, 722 (2010) .
[16] T. Low and P. Avouris, ACS Nano 8, 1086 (2014) .
[17] Y . M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer,
H. Y . Chiu, A. Grill, and P. Avouris, Science 327, 662
(2010) .
[18] F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat.
Photonics 4
, 611 (2010) .
[19] Q. Bao and K. P. Loh, ACS Nano 6, 3677 (2012) .
[20] J. W. You, S. R. Bongu, Q. Bao, and N. C. Panoiu,
Nanophotonics 8, 63 (2018) .
[21] N. Yousefi, X. Lu, M. Elimelech, and N. Tufenkji, Nat.
Nanotechnol. 14, 107 (2019) .
[22] D. Jin, T. Christensen, M. Soljacic, N. X. Fang, L. Lu, and
X. Zhang, Phys. Rev. Lett. 118, 245301 (2017) .
[23] D. Pan, R. Yu, H. Xu, and F. J. G. de Abajo, Nat. Commun. 8,
1243 (2017) .
[24] J. W. You, Z. Lan, Q. Bao, and N. C. Panoiu, IEEE J. Sel. Top.
Quantum Electron. 26, 4600308 (2020) .
[25] F. Xia, T. Mueller, Y . M. Lin, A. Valdes-Garcia, and P. Avouris,
Nat. Nanotechnol. 4, 839 (2009) .
[26] F. Schedin, E. Lidorikis, A. Lombardo, V . G. Kravets, A. K.
Geim, A. N. Grigorenko, K. S. Novoselov, and A. C. Ferrari,ACS Nano 4, 5617 (2010) .
121403-5JIAN WEI YOU AND NICOLAE C. PANOIU PHYSICAL REVIEW B 102, 121403(R) (2020)
[27] N. Papasimakis, Z. Luo, Z. X. Shen, F. de Angelis, E. di
Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, Opt. Express
18, 8353 (2010) .
[ 2 8 ]T .T .L v ,Y .X .L i ,H .F .M a ,Z .Z h u ,Z .P .L i ,C .Y .G u a n ,
J. H. Shi, H. Zhang, and T. J. Cui, Sci. Rep. 6, 23186
(2016) .
[29] S. Kim, M. S. Jang, V . M. Brar, K. W. Mauser, L. Kim, and
H. A. Atwater, Nano Lett. 18, 971 (2018) .
[30] J. Shi, Z. Li, D. K. Sang, Y . Xiang, J. Li, S. Zhang, and
H. Zhang, J. Mater. Chem. C 6, 1291 (2018) .
[31] H. Hu, X. Yang, X. Guo, K. Khaliji, S. R. Biswas, F. J. G. de
Abajo, T. Low, Z. Sun, and Q. Dai, Nat. Commun. 10, 1131
(2019) .
[32] Y . Zhang, C.-K. Lim, Z. Dai, G. Yu, J. W. Haus, H. Zhang, and
P. N. Prasad, Phys. Rep. 795, 1 (2019) .
[33] E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A.
Mikhailov, Phys. Rev. Lett. 105, 097401 (2010) .
[34] S. Y . Hong, J. I. Dadap, N. Petrone, P. C. Yeh, J. Hone, and
R. M. Osgood, Jr., Phys. Rev. X 3, 021014 (2013) .
[35] J. L. Cheng, N. Vermeulen, and J. E. Sipe, New J. Phys. 16,
053014 (2014) .
[36] J. D. Cox, I. Silveiro, and F. J. G. de Abajo, ACS Nano 10, 1995
(2016) .
[37] D. A. Smirnova, A. V . Gorbach, I. V . Iorsh, I. V . Shadrivov, and
Y . S. Kivshar, P h y s .R e v .B 88, 045443 (2013) .
[38] M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and
M. Scalora, Opt. Lett. 38, 3550 (2013) .
[39] M. Weismann and N. C. Panoiu, Phys. Rev. B 94, 035435
(2016) .
[40] J. W. You, J. You, M. Weismann, and N. C. Panoiu, Philos.
Trans. R. Soc. A 375, 20160313 (2017) .
[41] G. Soavi, G. Wang, H. Rostami, D. G. Purdie, D. De Fazio,
T. Ma, B. Luo, J. Wang, A. K. Ott, D. Yoon, S. A. Bourelle,J. E. Muench, I. Goykhman, S. Dal Conte, M. Celebrano,A. Tomadin, M. Polini, G. Cerullo, and A. C. Ferrari,Nat. Nanotechnol. 13, 583 (2018) .
[42] J. W. You, Z. Lan, and N. C. Panoiu, Sci. Adv. 6, eaaz3910
(2020) .
[43] M. L. Nesterov, J. B. Abad, A. Y . Nikitin, F. J. G. Vidal, and
L. M. Moreno, Laser Photon. Rev. 7
, L7 (2013) .
[44] D. A. Smirnova, R. E. Noskov, L. A. Smirnov, and Y . S.
Kivshar, P h y s .R e v .B 91, 075409 (2015) .
[45] H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu,
P h y s .R e v .B 91, 201402(R) (2015) .
[46] H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye,
Sci. Rep. 5, 15585 (2015) .
[47] D. Smirnova and Y . S. Kivshar, P h y s .R e v .B 90, 165433
(2014) .
[48] M. T. Manzoni, I. Silveiro, F. J. G. Abajo, and D. E. Chang,
New J. Phys. 17, 083031 (2015) .[49] J. J. Dean and H. M. van Driel, Appl. Phys. Lett. 95, 261910
(2009) .
[50] J. J. Dean and H. M. van Driel, P h y s .R e v .B 82, 125411 (2010) .
[51] M. Glazov, JETP Lett. 93, 366 (2011) .
[52] S. A. Mikhailov, Phys. Rev. B 84, 045432 (2011) .
[53] Y . Q. An, J. E. Rowe, D. B. Dougherty, J. U. Lee, and A. C.
Diebold, Phys. Rev. B 89, 115310 (2014) .
[54] T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry,
Nat. Phys. 12, 124 (2016) .
[55] R. W. Boyd, Nonlinear Optics , 3rd ed. (Academic, New York,
2008).
[56] H. Yan, X. Li, B. Chandra, G. Tulevski, Y . Wu, M. Freitag, W.
Zhu, P. Avouris, and F. Xia, Nat. Nanotechnol. 7, 330 (2012) .
[57] I. V . Iorsh, I. S. Mukhin, I. V . Shadrivov, P. A. Belov, and Y . S.
Kivshar, P h y s .R e v .B 87, 075416 (2013) .
[58] D. A. Smirnova, I. V . Shadrivov, A. I. Smirnov, and Y . S.
Kivshar, Laser Photonics Rev. 8, 291 (2014) .
[59] L. Novotny and V . H. Niek, Nat. Photonics 5, 83 (2011) .
[60] Y . Wang, Y . Abe, Y . Matsuura, M. Miyagi, and H. Uyama,
Appl. Opt. 37, 7091 (1998) .
[61] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.102.121403 for details of the derivation and
validation of the GS-FDTD numerical method for simulationof linear and nonlinear optical properties of two-dimensionalmaterials. It includes Refs. [ 34–36
,39,40,48,55,62–72].
[62] G. W. Hanson, J. Appl. Phys. 103, 064302 (2008) .
[63] J. W. You, E. Threlfall, D. F. G. Gallagher, and N. C. Panoiu,
J. Opt. Soc. Am. B 35, 2754 (2018) .
[64] A. Taflove and S. C. Hagness, Computational Electrodynamics:
The Finite-Difference Time-Domain Method , 3rd ed. (Artech
House, Norwood, MA, 2005).
[65] R. M. Joseph and A. Taflove, IEEE Trans. Antennas Propag. 45,
364 (1997) .
[66] Advances in FDTD Computational Electrodynamics: Photonics
and Nanotechnology ,e d i t e db yA .T a fl o v e ,A .O s k o o i ,a n dS .G .
Johnson (Artech House, Norwood, MA, 2013).
[67] D. W. Wang, W. S. Zhao, X. Q. Gu, W. C. Chen, and W. Y . Yin,
IEEE Trans. Nanotechnol. 14, 250 (2015) .
[68] J. W. You, S. R. Tan, and T. J. Cui, IEEE Trans. Microwave
Theory Tech. 62, 2849 (2014) .
[69] J. W. You, H. G. Wang, J. F. Zhang, Y . Li, W. Z. Cui, and T. J.
Cui, IEEE Trans. Electron Devices 62, 1327 (2015) .
[70] J. W. You, H. G. Wang, J. F. Zhang, S. R. Tan, and T. J. Cui,
IEEE Trans. Electron Devices 61, 1546 (2014) .
[71] J. W. You, H. G. Wang, J. F. Zhang, S. R. Tan, and T. J. Cui,
IEEE Microwave Wireless Compon. Lett. 24, 730 (2014) .
[72] J. M. Jin, The Finite Element Method in Electromagnetics
(Wiley, New York, 2015).
[73] D. C. Marinica, A. G. Borisov, and S. V . Shabanov, Phys. Rev.
Lett. 100, 183902 (2008) .
121403-6 |
PhysRevB.78.174302.pdf | Influence of lattice heating time on femtosecond laser-induced strain waves in InSb
F. S. Krasniqi, *S. L. Johnson, P. Beaud, M. Kaiser, D. Grolimund, and G. Ingold
Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
/H20849Received 29 May 2008; published 6 November 2008 /H20850
Femtosecond laser-induced strain waves in InSb are studied by means of time-resolved x-ray diffraction. The
temporal evolution of the measured x-ray diffracted intensity reveals that the lattice dynamics depends on thetime scale of energy transfer from excited carriers to the lattice. A framework that accounts for this energy-transfer time /H20849lattice heating time /H20850is presented and applied to model the fluence dependence of the transient
x-ray diffraction data. In this model the initial strain wave dynamics depends crucially on the lattice heatingtime, which decreases with increasing fluence.
DOI: 10.1103/PhysRevB.78.174302 PACS number /H20849s/H20850: 62.30. /H11001d, 63.20. /H11002e
I. INTRODUCTION
Above band-gap excitation of a semiconductor crystal
with a subpicosecond laser pulse with pulse energies justbelow the damage threshold puts the crystal into a highlystressed state. Stress is typically relieved by lattice expansionthat starts at the crystal surface and subsequently drives atraveling expansion and compression strain wave into thebulk at the longitudinal speed of sound.
1,2The time-
dependent strain wave can be thought as a superposition ofcoherent phonons with wave vectors centered approximatelyabout the inverse of laser penetration depth.
3These transient
coherent lattice dynamics have been studied in a variety ofmaterials both by observing the resultant changes in opticalproperties /H20849reflectivity /H20850
1,4,5and by using x-ray
diffraction.2,6–8In contrast to optical reflectivity measure-
ments, time-resolved x-ray diffraction can directly observethe small shifts in interatomic distance associated with astrain wave as it propagates into the bulk of the crystal. Thesensitivity of x-ray diffraction to coherent lattice dynamicshas been demonstrated in many experiments in both Braggand Laue geometries.
9Generally, these studies are useful in
deducing the acoustic properties of the excited crystal andtesting models of electron-phonon coupling.
3,4,7
A simple model that describes the generation of a strain
wave in a laser-excited solid was proposed by Thomsen et
al.1They solved the elastic equations assuming that thermal
stress is instantaneously generated in an absorbing solid.This model has successfully been applied to explain the re-sults of both optical scattering and x-ray diffractionmeasurements.
1,2,9In principle, one might expect that the
model predicts reasonably the structure of the strain wave attimes well after carrier-lattice thermalization. During thethermalization process, on the other hand, the assumption ofinstantaneous heating may overestimate the strain. For in-stance, in polar semiconductors such as InSb, carrier-latticeenergy exchange is thought to be mediated by longitudinal-optical /H20849LO/H20850phonons.
10In these systems, the excess energy
of photoexcited carriers is first transferred to small momen-tum LO phonons, which subsequently decay to acousticphonons due to the anharmonicity of the crystal potential.The intrinsic lifetime of LO phonons is expected to governthe carrier-lattice thermalization dynamics
10,11and thus the
thermal component of the strain evolution. Intervalley scat-tering of carriers, significant for dense electron-hole plasmas
and high carrier energies, may further influence the thermal-ization time scale by reheating the carriers and reducing thecooling rate, whereas deformation-potential and piezoelectricscatterings with acoustic phonons are important for carrierexcess energies smaller than the LO phonon energy.
10,12
In the present paper we have employed time-resolved
x-ray diffraction using the femtosecond “slicing” source atthe Swiss Light Source /H20849SLS /H20850/H20849Ref. 13/H20850to investigate the
fluence dependence of the lattice heating time in laser-excited InSb. A model for the laser-excited strain waves isdeveloped assuming that energy transfer from photoexcitedcarriers to the lattice is mediated mainly by LO phonons. Wegive an analytical expression for the laser-induced strainwave that takes into account the energy-transfer time fromthe excited electrons to the lattice. This model is an exten-sion of the Thomsen model; it produces Thomsen-type strainprofiles in the limit of instantaneous heating /H20849i.e., when the
lattice heating time tends to zero /H20850. The x-ray diffraction from
a laser strained crystal is calculated using the Takagi-Taupindynamical theory for the depth-dependent strain gradients.
14
We find that the lattice heating time does affect the timeevolution of the x-ray diffracted intensity during carrier-lattice thermalization.
The paper is organized as follows. In Sec. II we briefly
describe the experimental setup. Based on results from pre-vious work on lattice heating and strain wave generation,Sec. III discusses the effect of laser heating on the strainwave. Here we present a model that can be used to calculatestrain wave profiles, including the lattice heating time. InSec. IV we calculate the x-ray diffraction from a strainedcrystal. In Sec. V we present and discuss our experimentalresults in the framework of models presented in two previoussections. Finally, in Sec. VI we present the conclusions ofour work.
II. EXPERIMENTAL SETUP
The experiment was performed at the SLS using a tunable
femtosecond undulator hard x-ray source. Short 140 /H1100630 fs
x-ray pulses are generated using the electron-beam slicingtechnique
13,15at a repetition rate of 1 kHz. Two mirrors focus
the x-ray beam onto the sample. The first is a horizontallymounted grazing incidence toroidal mirror placed 15 m be-PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
1098-0121/2008/78 /H2084917/H20850/174302 /H2084911/H20850 ©2008 The American Physical Society 174302-1fore the sample that both collimates the beam vertically and
brings the x rays to a horizontal focus of 250 /H9262m at the
sample. The second mirror is an elliptically bent grazing in-cidence optic positioned 43 cm before the sample. This mir-ror focuses the beam vertically to less than 10
/H9262m. Between
this last mirror and the sample, a double multilayer/H20849Mo /B
4C, 25 Å period /H20850monochromator selects an x-ray en-
ergy of 5.9 keV with a bandwidth of 1.2%. At a variable timerelative to the arrival of the x-ray pulses, a p-polarized fem-
tosecond laser pump pulse /H20851/H9261=800 nm, full width at half
maximum /H20849FWHM /H20850=120 fs /H20852excites the sample with a 16°
incidence angle, as measured from the crystal surface.
The x-ray diffraction measurements were performed on an
asymmetrically cut InSb single crystal /H20849500
/H9262m thick /H20850as
drawn in Fig. 1. The surface of the crystal was cut to an
angle /H9251=15.5° from the /H20849111 /H20850lattice planes. At 5.9 keV
x-ray energy the /H20849111 /H20850Bragg angle is 16.3°. The grazing
angle of incidence of x-rays with respect to the surface was
/H9258i=/H9258B−/H9251=0.8°, and the angle of the exiting x-ray beam with
respect to the crystal surface was /H9258e=/H9258B+/H9251=31.8°. The sig-
nal was measured by an avalanche photodiode detector.
The asymmetric Bragg geometry provides a better match
between the penetration depths of laser and x rays than doesa symmetric diffraction geometry. At grazing incidenceangles below 1° at these energies in InSb, photoabsorption ofx rays limits the penetration into the crystal. Under this ap-proximation, the x-ray penetration depth can be written as
/H9256x=1
/H9262/H20875sin/H20849/H9258B−/H9251/H20850
1+ /H20841b/H20841/H20876, /H208491/H20850
where /H9262is the linear absorption coefficient and bis the
asymmetry parameter defined as b=−sin /H20849/H9258B−/H9251/H20850/sin/H20849/H9258B+/H9251/H20850.
For x rays at 5.9 keV, /H9262/H110153.144/H11003105m−1/H20849Ref.16/H20850and Eq.
/H208491/H20850gives a penetration depth /H9256x/H1101543 nm.
The laser spot size on the sample was approximately
seven times larger than x-ray spot size to ensure probing of ahomogeneously excited area. The loss in temporal resolutiondue to the size of the x-ray beam and the angle between thepump and x-ray beams was less than 90 fs.III. GENERATION OF STRAIN WA VES
The fundamental interactions which occur during and fol-
lowing the absorption of subpicosecond, above-band-gap-energy laser pulses have been described by many authors.
17
In this section we give a brief summary of the processes thatare relevant to our work and the necessary considerations toincorporate the effects of lattice heating. The main objectiveof this section is to develop an expression for the latticetemperature that will be presented in Sec. III A which wewill use in Sec. III B to model strain waves and subsequentlyin the analysis and discussion of the experimental results.
A. Lattice heating
When a semiconductor crystal is excited with visible or
near-infrared photons of energy /H6036/H9275larger than the energy
gapEg, electrons are excited out of valence-band /H20849VB /H20850states
into conduction-band /H20849CB/H20850states. Neglecting recombination
and diffusion during excitation, the evolution of the carrierdensity profile during and immediately after the laser pulsecan be described by the following partial differential equa-tions for the carrier density N
e/H20849z,t/H20850and the pump intensity
I/H20849z,t/H20850/H20849Ref. 18/H20850:
/H11509
/H11509zI/H20849z,t/H20850=− /H20851/H92510+/H9251fc/H20849z,t/H20850+/H9252TPAI/H20849z,t/H20850/H20852I/H20849z,t/H20850/H20849 2/H20850
and
/H11509
/H11509tNe/H20849z,t/H20850=/H20875/H92510+1
2/H9252TPAI/H20849z,t/H20850/H20876I/H20849z,t/H20850
/H6036/H9275. /H208493/H20850
Here, zdenotes the spatial coordinate perpendicular to the
surface, /H92510and/H9252TPAaccount for linear and two-photon ab-
sorption /H20849for InSb at /H9261las=800 nm, /H92510/H110151.087/H11003107m−1
and/H9252TPA /H110158/H1100310−5m/GW /H20850,19,20and intraband free-carrier
absorption is calculated using the Drude model for the di-
electric constant21/H9255/H20849Ne/H20850,/H9251fc/H11011/H92552//H208512/H20881/H92551+/H20881/H20849/H925512+/H925522/H20850/H20852, where
/H92551/H20849Ne/H20850and/H92552/H20849Ne/H20850are the real and imaginary parts of /H9255/H20849Ne/H20850,
respectively. These equations can be solved using boundary
conditions I/H20849z=0,t/H20850=I0e−t2//H927002and N/H20849z,t=0/H20850=N0, where N0
/H110152/H110031016cm−3is the equilibrium carrier concentration at
T=300 K.
Following Ref. 22the evolution of the carrier density at
later times can be described by the differential equation
/H11509Ne
/H11509t=Da/H115092Ne
/H11509z2−CaNe2. /H208494/H20850
Here, the first term on the right-hand side describes the car-
rier diffusion characterized by the ambipolar diffusion coef-ficient D
athat depends in general on the carrier density and
temperature.23,24For InSb under high carrier density /H20849on the
order of 1020cm−3/H20850and excess energy /H20849about 1 eV /H20850condi-
tions Da/H1101540 cm2/s has been deduced from optical reflec-
tivity measurements.22For carrier densities on the order of
1021cm−3band-gap renormalization23,24slows down the dif-
fusion; in the case where this effect is ignored, the carrierdensity in the excitation depth is not more than 20% largerbecause of the large Auger recombination rate at these den-/c113B
d/c97/c113i/c113e
/c122laser
x-rays
/c106
/c113Bx-ray s
crystal
FIG. 1. Scheme of the asymmetric Bragg geometry. The grazing
angle of incidence of x rays with respect to the crystal surface isdenoted by
/H9258i;/H9251is the asymmetry angle, /H9258Bis the Bragg angle, /H9258e
is the exit angle, /H9272is the angle between the laser beam and the
crystal surface, and /H9256is the laser penetration depth.KRASNIQI et al. PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-2sities. Auger recombination characterized by the recombina-
tion coefficient Cais described by the second term. In highly
excited InSb, due to the screening of the Coulomb potentialthat mediates the carrier-carrier interaction, an Auger recom-bination rate with quadratic dependence on N
eis shown to be
more successful, with Ca/H110151.5/H1100310−9cm3/s.22Other re-
combination mechanisms, such as radiative recombination,are negligible for time scales /H110211n s /H20849Ref.11/H20850. This equation
does not distinguish between the /H9003andLvalleys in the CB.
However, by using the model described in Ref. 25/H20849including
diffusion in the /H9003valley with ambipolar diffusion coefficient
D
aand recombination in both /H9003andLvalleys with the same
Auger recombination coefficient Ca/H20850and assuming an inter-
valley scattering time /H9003→Lcomparable to the laser-pulse
duration, L→/H9003scattering time of 1 ps, and optical phonon
emission time of 150 fs, the time evolution of the carrierdensity differs not much from that computed using Eq. /H208494/H20850,
with discrepancies smaller than 30% at times /H110215 ps,
whereas at later times the discrepancy is larger but the valuesof carrier density are of the same order magnitude. Variationof the above-mentioned scattering times by /H1101140%increases
the discrepancy by /H1101114%.
A monochromatic laser pulse excites electron-hole pairs
at specific points in the band structure determined by thecondition E
c/H20849k/H20850−Ev/H20849k/H20850=/H6036/H9275, where Ec,Ev, and /H6036/H9275are the
CB, VB, and laser photon energies, respectively. The bigdifference in the curvatures of the CB and VB /H20849Ref. 27/H20850at
/H6036
/H9275/H110151.55 eV implies that most of the excess energies /H9004Ec
/H20849c=e,hfor electrons and holes, respectively /H20850reside initially
in the electrons rather than in the holes.50Although the ex-
cited carriers initially have a nonthermal distribution, scatter-ing processes such as carrier-carrier /H20849and to lesser extend
carrier-phonon /H20850scattering thermalize electrons and holes in a
very short time scale on the order of 100 fs.
11,28Under the
assumption that most of the photon energies are translatedinto the kinetic energy of the electrons, we may estimate theelectron temperature T
e0as
Te0/H110152/H9004Ee
3kB, /H208495/H20850
where kBis the Boltzmann constant.
It is generally recognized that in semiconductors of polar
character, carriers lose their excess energy primarily by emit-ting LO phonons via Fröhlich interaction.
11,29,30This interac-
tion favors optical phonons near the center of the Brillouinzone /H20849BZ/H20850since the rate matrix elements are proportional to
1/q
2, where qis the phonon wave vector.31If excitation of
carriers takes place in the /H9003valley, the photoexcited carrier
density is high /H20849/H114071020cm−3/H20850and the energy of the photo-
excited carriers is larger than that of the side valley mini-mums /H20849Land/or X/H20850, the electrons can scatter quickly /H20849on the
order of 100 fs /H20850to these valleys.
51The electrons in the side
valleys return back slowly /H20849on a time scale /H110221p s /H20850to the /H9003
valley.12,25In this case they may act to reheat the electrons in
the/H9003valley and slow the lattice heating.
In our analysis we will assume that the transfer of energy
from electrons to LO phonons, characterized by a character-istic time
/H9270op, will lead to an optical phonon population in
excess of the equilibrium value.11Fröhlich interaction favorsBZ center phonons, however the maximum wave vector qmax
of the LO phonons that interact with the electrons depends
on the excess energy of the electrons and the CB curvatureand can be on the order of 10
7cm−1.52The LO phonons
decay into acoustic phonons through anharmonic interactionwith a characteristic time constant
/H9270ap.10,32This later interac-
tion can be considered to bring the optical phonons into equi-librium with other lattice phonon modes.
11,32Quasiequilib-
rium distributions for all three systems /H20849electrons, LO
phonons, and the lattice /H20850are assumed to be established so
that the electrons, optical phonons, and lattice have time-dependent temperatures T
e,TLO, and Tl. The temporal and
spatial evolutions of the system composed of electrons, LOphonons, and the lattice can be described by a set of threecoupled nonlinear partial differential equations
C
e/H11509Te
/H11509t=/H11509
/H11509z/H20875/H20849ECB+2kBTe/H20850/H20873Da/H11509Ne
/H11509z/H20874/H20876+/H11509
/H11509z/H20873Ke/H11509Te
/H11509z/H20874
−Ce/H20873Te−TLO
/H9270op/H20874+/H20873Eg+3
2kBTeHe/H20874CANe2, /H208496/H20850
CLO/H11509TLO
/H11509t=Ce/H20873Te−TLO
/H9270op/H20874−CLO/H20873TLO−Tl
/H9270ap/H20874, /H208497/H20850
and
Cl/H11509Tl
/H11509t=CLO/H20873TLO−Tl
/H9270ap/H20874, /H208498/H20850
where ECBis the CB edge, Ceis the electronic heat capacity,
CLOis the LO phonon heat capacity, Clis the lattice heat
capacity, and Keis the electronic thermal conductivity.33The
electronic heat capacity is taken as
Ce=/H11509
/H11509Te/H208733
2NekBTeHe/H20874, /H208499/H20850
where Heis a degeneracy factor which depends on the elec-
tron energy and temperature.53The LO phonon heat capacity
is calculated by assuming that the phonon occupation num-ber depends only on E
LOand TLO, where ELOis the LO
phonon energy near the BZ zone center /H20851for InSb, ELO
/H110150.024 eV /H20849Ref.34/H20850/H20852; this implies a LO phonon heat capac-
ity that is similar to the Einstein model of heat capacityincluding only the LO phonon branch.
54In Eqs. /H208496/H20850and /H208498/H20850
the cooling of electrons due to the deformation-potentialscattering with acoustic phonons is not taken into accountsince in polar semiconductors this cooling mechanism isthought to become significant for carrier energies less thanE
LO.10Following Ref. 35, the energy-loss rate of electrons to
acoustic phonons through deformation-potential scattering ismore than a factor of 10 smaller than that needed to increasethe lattice temperatures within 10–15 ps to the values ob-served in the experiment. Reference 35describes the relax-
ation of electrons in a metal through deformation-potentialscattering. It assumes a parabolic band and chemical poten-tial equal to the Fermi energy which, depending on N
eand
Te, is typically larger than the quasi-Fermi levels in semicon-
ductors. In this case the Fermi distribution function feis
larger than that of a semiconductor, however since the pho-INFLUENCE OF LATTICE HEATING TIME ON … PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-3non emission rate is /H11008fe/H20849Ee−/H6036/H9275AP/H20850/H208511−fe/H20849Ee/H20850/H20852, with /H6036/H9275AP
being the acoustic phonon energy /H20849typically, /H1102110 meV /H20850, the
difference in chemical potentials does not have a large im-pact on the phonon emission rate.
The left-hand side /H20849lhs/H20850of Eq. /H208496/H20850represents the rate of
change of the energy density of the electron system. The firsttwo terms on the right-hand side /H20849rhs/H20850of Eq. /H208496/H20850represent the
rate of change of the electronic energy density due to diffu-sion, derived from the relaxation-time approximation of theBoltzmann equation.
33,36The third term describes the rate of
energy density transfer from electrons to the LO phonons.The emission of LO phonons takes place within hundreds offemtoseconds, but the decay of phonon population has beenobserved to be several picoseconds;
11,30this decay time is
long enough that a large nonequilibrium phonon populationis created within 1–2 ps after the excitation. Based on thisobservation we assume that the term describing the rate ofenergy transfer to the LO phonon system is of this form. The
fourth term describes the rate of energy density given to theelectron system by Auger heating. In every Auger recombi-nation event, the recombination energy /H11011E
g+/H208493/2/H20850kBTeHeis
transferred to another electron in the CB.37The energy den-
sity given to the electron system is written as a product of therecombination energy and Auger recombination rate.
In Eq. /H208497/H20850the lhs describes the rate of change of the
energy density of the LO phonon population; the first termon the rhs represents the rate of energy density LO phononsgain from the electrons and the second term represents decayinto acoustic phonons. The LO phonons have a small disper-sion and thus small group velocity. Taking a dispersion of 4meV over /H9004q=
/H9266/a, where a=6.479 Å is the lattice constant
/H20849Ref.34/H20850, the group velocity vLO/H110151.25/H11003105cm /s. Assum-
ing a LO phonon lifetime /H9270ap=9 ps, we obtain a propagation
length vLO/H9270ap/H1101511 nm, which is about 1/8 of the laser pen-
etration depth. We can conclude that LO phonons are trappedin the photoexcited region and do not propagate significantly.
In Eq. /H208498/H20850, the lhs describes the rate of change of the
energy density of the lattice, whereas the rhs represents therate of energy density which the lattice gains from opticalphonons. Diffusive thermal transport during the first few pi-coseconds /H20849/H1135115 ps /H20850is neglected because the estimated dif-
fusion length is about 1/7 of the laser penetration depth.
By solving Eqs. /H208492/H20850and /H208493/H20850we obtain the initial density
profile for Eq. /H208494/H20850/H20849see Fig. 2/H20850. The solution of Eq. /H208494/H20850is used
then in Eqs. /H208496/H20850–/H208498/H20850. The initial electron temperature is taken
from Eq. /H208495/H20850with/H9004E
e=/H6036/H9275−Eg/H110151.38 eV and the initial LO
phonon and lattice temperature are 300 K. The crystal sur-face is assumed to be impermeable for carriers and carrierenergy /H20849i.e., at z=0,
/H11509Ne//H11509z=0, and /H11509Te//H11509z=0 for all times /H20850.
DaandECBare taken as constant and the degeneracy factor
He=1.55Figure 3shows the evolution of electron, phonon,
and lattice temperatures for a laser fluence of 5 mJ /cm2,
/H9270op=2 ps, and /H9270ap=6 ps. The peak in electron temperature is
caused by Auger recombination that heats the electron sys-tem through the term /H11011T
e/H20849cf. Eq. /H208496/H20850/H20850. The LO phonon
temperature increases over a time /H11011/H9270opto a maximum value
of about 1100 K. Although this maximum value of TLOex-
ceeds the lattice melting temperature Tm=820 K, the rela-
tively high average frequency of LO phonons means that theactual magnitude of the average mean-square displacementof atoms is considerably lower than it would be if the lattice
reached this temperature. We estimate using the equipartitiontheorem that this temperature corresponds to a mean-squareamplitude of atomic vibrations that is 3.9% of the nearest-neighbor distance /H208492.793 Å /H20850well below the Lindemann
melting criterion /H20849/H1101110% of the nearest-neighbor
distance /H20850.
38,39The lattice temperature, on the other hand, in-
creases to a maximum value of about 500 K.
To avoid the computational difficulty of fitting the data to
a strain wave arising from the thermal stress derived from anexact solution of Eqs. /H208496/H20850–/H208498/H20850, we follow Ref. 3and observe
that the general shape of the lattice temperature is well de-scribed by an exponential function of the form
T
l/H20849t/H20850=T0+/H9004Tl/H208491−e−t//H9270/H20850. /H2084910/H20850
The rise time /H9270, hereafter referred to as lattice heating time,
depends strongly on the phonon decay time /H9270ap. Figure 4FIG. 2. Evolution of carrier density computed using Eq. /H208494/H20850.
Inset: Spatial distribution of carrier density at the end of laser pulsecomputed using Eqs. /H208492/H20850and /H208493/H20850compared to the analytic expres-
sion N
maxe−z//H9256. This distribution is considered as an initial condition
for Eq. /H208494/H20850. The initial condition for Eq. /H208493/H20850is the equilibrium car-
rier concentration at room temperature /H208492/H110031016cm−3/H20850. The tem-
poral profile of the laser intensity is a Gaussian of 120 fs FWHM.
FIG. 3. The dependence of electron, optical phonon and lattice
temperatures computed using Eqs. /H208496/H20850–/H208498/H20850.KRASNIQI et al. PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-4compares lattice temperatures computed using Eq. /H208498/H20850corre-
sponding to optical phonon decay times /H9270apo f3a n d8p s
with profiles of functional form /H2084910/H20850with rise times /H9270of 3
and 8 ps, respectively.
B. Strain waves
The transfer of energy to the lattice leaves the lattice in a
highly stressed state, which is eventually relieved by under-going thermal expansion. Following excitation, the latticemoves toward the new equilibrium state /H20849corresponding to an
expanded state /H20850but overshoots and coherently oscillates over
a range of frequencies, giving rise to the coherent acousticpulse, a strain wave.
3
Thomsen et al.1presented a thermoelastic model of strain
which describes the generation and propagation of a laser-induced coherent strain pulse, hereafter referred to as theThomsen model. In this model, an ultrafast laser pulse isabsorbed and deposits a significant amount of energy nearthe crystal surface. If the electron-phonon relaxation time isextremely fast /H20849i.e., the lattice is heated instantaneously /H20850, this
absorption will generate an instantaneous thermal stress
/H9268th=−3/H9252B/H9004Tl/H20849z/H20850, /H2084911/H20850
with
/H9004Tl/H20849z/H20850=/H208491−R/H20850F
/H9256Clexp/H20873−z
/H9256/H20874, /H2084912/H20850
where /H9252is the linear-expansion coefficient, Bis the bulk
modulus, Ris the reflectivity of the sample, Fis the laser
fluence, Clis the lattice heat capacity, and /H9256is the laser
penetration depth. For InSb, /H9252=4.7/H1100310−6/H20849Ref. 40/H20850,B
=46 GPa /H20849Ref. 26/H20850,Cl=0.832 /H11003106/H20849Ref. 40/H20850, and /H9256
/H1101592 nm /H20851at 800 nm /H20849Ref. 19/H20850/H20852.
With thermal stress of the form in Eq. /H2084911/H20850which assumes
an instantaneous conversion of the laser energy to heat, thelaser-induced strain wave
/H9257/H20849z,t/H20850is1/H9257/H20849z,t/H20850=3/H208491−R/H20850F/H9252B
/H9267v2/H9256Cl/H20877exp /H20849−z//H9256/H20850/H208751−1
2exp /H20849−vt//H9256/H20850/H20876
−1
2exp /H20849−/H20841z−vt/H20841//H9256/H20850sgn /H20849z−vt/H20850/H20878, /H2084913/H20850
where /H9267is the mass density and vis the longitudinal sound
velocity. For InSb, /H9267=5770 kg /m3/H20849Ref. 26/H20850and v
=3900 m /s.6Equation /H2084913/H20850represents a lattice strain wave.
The lattice strain profile is shown in Fig. 5for six different
times. It is seen that the resulting lattice motion correspondsto an acoustic pulse propagating into the solid at the velocityof sound. The pulse consists of a region of expansion /H20849or
positive strain /H20850followed by a region of compression /H20849nega-
tive strain /H20850.
In a more realistic model, one needs to consider the fact
that the excitation energy initially placed into the carrier sys-tem is not transferred instantaneously to the lattice. To incor-porate the time needed for excitation energy to be transferredto the lattice, the thermal stress is written in the form
/H9268th/H20849z,t/H20850=−3/H9252B/H20851Tl/H20849z,t/H20850−T0/H20852, /H2084914/H20850
where Tl/H20849z,t/H20850is the lattice temperature and T0=300 K.
Using the arguments in Sec. III A that the lattice tempera-
ture increase over the energy-transfer time is almost /H20849with
the largest discrepancy not more than 5% /H20850of the functional
form given by Eq. /H2084910/H20850, the thermal stress is written in the
form
/H9268th/H20849z,t/H20850=−3/H9252B/H208511 − exp /H20849−t//H9270/H20850/H20852Tleqexp /H20849−z//H9256/H20850, /H2084915/H20850
where Tleqis the lattice temperature when the equilibrium
with carriers is reached. Here, heat conduction is neglected.The shape of the strain wave is determined by the valueD
l//H20849/H9256v/H20850, where Dlis the thermal diffusion coefficient. For
InSb, Dl=0.16 cm2/s and Dl//H20849/H9256v/H20850=0.04, and according to
Ref. 1, the neglect of heat conduction makes only a small
change in the wave shape. However, the magnitude of thestrain /H20849especially in the incoherent part /H20850near the target sur-
face will be larger than that where heat conduction is con-FIG. 4. Approximation of calculated lattice temperature profiles
/H20849circles: /H9270ap=3 ps, squares: /H9270ap=8 ps /H20850with a function of the form
/H2084910/H20850for/H9270=3 and 8 ps, respectively.FIG. 5. Laser-generated strain waves at times t=2, 6, 10, 20, 30,
and 40 ps after excitation. The strain waves are calculated using Eq./H2084913/H20850for InSb at a laser fluence of 3 mJ /cm
2.INFLUENCE OF LATTICE HEATING TIME ON … PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-5sidered because the laser-deposited heat is trapped in the
excitation region. The difference in magnitudes of the strainwith and without consideration of heat conduction dependsupon the time delay /H9004tbetween the pump /H20849laser /H20850and the
probe /H20849x rays /H20850. For /H9004t/H1102140 ps, this difference will be less
than 12%, while in time range 40–80 ps, up to 27%. For/H9004t/H11022500 ps, heat conduction has a large effect on the shape
of the strain wave.
4
Such a functional form of the thermal stress was first used
in Ref. 3with/H9270representing the phenomenological electron-
phonon coupling time. There, an analytic expression for thecoherent part of the strain wave has been given. In this work
/H9270represents the LO phonon decay time /H20849cf. Fig. 4/H20850, and with
thermal stress of the form given by Eq. /H2084915/H20850the full strain
wave /H20849coherent+incoherent parts /H20850is obtained which in the
limit/H9270→0 reproduces the Thomsen strain wave. By using
Eq. /H2084915/H20850and Eqs. /H208494/H20850–/H208496/H20850in Ref. 1one can obtain an ana-
lytical expression for the laser-induced displacements u/H20849z,t/H20850
and subsequently for the strain /H9257/H20849z,t/H20850=/H11509u/H20849z,t/H20850//H11509z. The solu-
tions are56
/H20849i/H20850z/H11021vt
u/H20849z,t/H20850=G1/H20849z,t/H20850+1
2vG2/H20849z,t/H20850,
G1/H20849z,t/H20850=A1/H208511−e−v//H9256/H20849t−z/v/H20850/H20852/H208511−e−1 //H9270/H20849t−z/v/H20850/H20852,
G2/H20849z,t/H20850=A2/H92562e−2z//H9256
v/H20877/H20849e2z//H9256−1/H20850
/H9256−v/H9270/H20851/H9256/H208491−e/H20849z−vt/H20850//H9256/H20850
+v/H9270/H20849e/H20849z−vt/H20850//H20849v/H9270/H20850−1/H20850/H20852+/H20849ez//H9256−1/H208502/H20878+A2/H92562e−2z//H9256
v
/H11003v/H9270e−t//H9270/H208532v/H9270ez//H9256−ez//H20849v/H9270/H20850/H20851/H9256+v/H9270+e2z//H9256/H20849v/H9270−/H9256/H20850/H20852/H20854
/H20849v2/H92702−/H92562/H20850,
/H2084916/H20850
/H20849ii/H20850z/H11022vt
u/H20849z,t/H20850=A2/H92562e−z//H9256/H20877cosh /H20849vt//H9256/H20850−1
v2
+/H9270/H20851v/H9270e−t//H9270−v/H9270cosh /H20849vt//H9256/H20850+/H9256sinh /H20849vt//H9256/H20850/H20852
v/H20849v2/H92702−/H92562/H20850 /H20878,
/H2084917/H20850
where
A1=−3/H9252B/H9256
v2/H9267/H9004Tleq, /H2084918/H20850
A2=3/H9252B
/H9256/H9267/H9004Tleq. /H2084919/H20850
If we use /H9004Teq=/H208491−R/H20850F//H20849/H9256Cl/H20850in Eqs. /H2084918/H20850and /H2084919/H20850and take
the limit /H9270→0, we retrieve the traditional Thomsen strain
profile. Calculated strain profiles using Eqs. /H2084916/H20850–/H2084919/H20850are
shown in Fig. 6. The effect of the heating time /H9270is evident.
We see that the strain near the surface /H20849z=0/H20850increases over
the time /H9270, in contrast to Fig. 5where it is almost timeindependent, and the boundary between the expansive and
compressive regions is smoothed. Figure 7shows the strain
wave profile 20 ps after excitation for different heating times
/H9270.
IV . X-RAY DIFFRACTION IN LASER-EXCITED
CRYSTALS
X-ray diffraction in the presence of laser-induced strain is
calculated using the Takagi-Taupin /H20849TT/H20850dynamical theory
for the depth-dependent strain gradients.14This theory has
been successfully applied to model x-ray diffraction fromcoherent acoustic phonons.
2,3,6,7,11,41Within this theory the
differential equation for the ratio of the complex field ampli-
tudes of the diffracted x-rays Dhand incident x-rays D0can
be written as14
d/H9264
dz=/H9266i
/H9011/H20875/H92642−2S/H20849/H9253h/H20850/H9252/H9004/H9258/H9264−/H20841/H9253h/H20841
/H9253h/H20876, /H2084920/H20850
whereFIG. 6. Laser-generated strain waves at times t=2, 6, 10, 20, 30,
and 40 ps after excitation using a heating time /H9270=10 ps. The strain
waves are calculated using Eqs. /H2084916/H20850–/H2084919/H20850for InSb at laser fluence
of 3 mJ /cm2.
FIG. 7. Laser-generated strain wave 20 ps after excitation using
heating times /H9270=0.05, 4, 6, 8, and 10 ps. The strain waves are
calculated using Eqs. /H2084916/H20850–/H2084919/H20850for InSb at laser fluence of
3m J /cm2.KRASNIQI et al. PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-6/H9264=1
S/H20849P/H20850/H20881/H20841/H9253h/H20841
/H92530/H9273h¯
/H20881/H9273h/H9273h¯Dh
D0, /H2084921/H20850
/H9011=/H9261/H20881/H92530/H20841/H9253h/H20841
/H20841P/H20841/H20881/H9273h/H9273h¯, /H2084922/H20850
/H9252/H9004/H9258=/H20881/H92530//H20841/H9253h/H20841
/H20841P/H20841/H20881/H9273h/H9273h¯/H20875/H20849/H9004/H9258+C/H9257/H20850sin 2/H9258B−1
2/H92730/H20873/H9253h
/H92530−1/H20874/H20876,
/H2084923/H20850
C= cos2/H9251tan/H9258B+ sin/H9251cos/H9251, /H2084924/H20850
and
/H9273h=r0/H92612Fh
/H9266V. /H2084925/H20850
Here/H9004/H9258=/H9258−/H9258B,/H92530/H20849h/H20850are the direction cosines of the inci-
dent /H20849diffracted /H20850beam, /H9261is the x-ray wavelength, /H9257/H20849z,t/H20850is
the strain, Fhis the structure factor, r0is the classical elec-
tron radius /H20849r0=2.818 /H1100310−15m/H20850,Vis the volume of the
unit cell, S/H20849x/H20850is the sign function, and Pis the polarization
factor. In this experiment the x rays are spolarized and P
=1. Equation /H2084920/H20850is solved analytically /H20849see the Appendix /H20850
under the condition that deep in the crystal /H20849z=zm/H20850where the
strain vanishes /H9264/H20849zm/H20850=/H9264p, where /H9264pis the value of /H9264for a
perfect crystal.14,42From Eq. /H2084921/H20850one can find Dh/D0and
finally the rocking curve of the crystal
R=/H20841/H9253h/H20841
/H92530/H20879Dh/H20849z=0/H20850
D0/H20849z=0/H20850/H208792
. /H2084926/H20850
Figure 8compares the unperturbed rocking curves predicted
by Eq. /H2084926/H20850to the rocking curve of a thick crystal predictedby the dynamical theory of x-ray diffraction under grazing
incidence conditions.43The rocking curve computed by the
grazing incidence x-ray diffraction theory compares well tothat using TT theory indicating that the negligence of x-rayreflection does not have any impact on the rocking curve.
The laser-induced strain wave
/H9257/H20849z,t/H20850is a superposition of
coherent acoustic phonons with wave vectors qapproxi-
mately centered about the inverse of the laser penetrationdepth.
6,41Due to the coherent acoustic phonons of wave vec-
torqthe time-resolved x-ray diffraction intensity will oscil-
late with an angular frequency /H9275given by41
/H9275/H11015vq/H11015v/H9004/H9258B/H20841Gh/H20841
tan/H9258Bcos/H9251+ sin/H9251, /H2084927/H20850
where /H9004/H9258Bis the angular position off the Bragg peak and Gh
is the reciprocal-lattice vector. Equation /H2084927/H20850indicates that
by varying /H9004/H9258Bdifferent phonon modes with frequencies
/H9275/H20849/H9004/H9258B/H20850can be observed. Figure 9shows the normalized
time-dependent x-ray diffracted intensity for /H9004/H9258B=0.06° and
0.15°. The time evolution of the diffracted intensity has ba-sically two origins. First, the reduction of the intensity is dueto a shift of the rocking curve. Second, this reduction issuperimposed by an interference of x-rays originating fromtwo sources: /H20849a/H20850the bulk x-ray diffraction and /H20849b/H20850the diffrac-
tion from the strain wave. The diffraction from the strainwave leads to the oscillations in the time-dependent dif-fracted intensity with periods T=2
/H9266//H9275=48 and 18 ps, re-
spectively, which are comparable to those predicted by Eq./H2084927/H20850, which are 49 and 20 ps. The visibility of the temporal
oscillations is reduced with increasing bandwidth of the xrays.
41The inset of Fig. 9shows the effect of the x-ray band-
width on the time evolution of the diffracted intensity. Theoscillation amplitude of the time-dependent diffracted inten-FIG. 8. Comparison of the unperturbed rocking curve for InSb
/H208495.9 keV, 111 reflection, /H9251=15.5° /H20850calculated using the Takagi-
Taupin theory and the thick crystal rocking curve /H20851grazing incidence
x-ray diffraction /H20849GID-D /H20850/H20852and x-ray reflection /H20849GID-R /H20850calculated
using the theory of x-ray diffraction under grazing incidenceconditions.FIG. 9. Time evolution of the normalized diffracted intensity
calculated using Eqs. /H2084920/H20850–/H2084926/H20850at two different positions on the
rocking curve, /H9004/H9258B=0.06° and 0.15°. The laser-induced strain is
calculated using Eqs. /H2084916/H20850–/H2084919/H20850with/H9270=5 ps and Tleq=240 K. In-
set: The effect of the x-ray bandwidth on the time evolution of thediffracted intensity. The bandwidth of the beam is taken into ac-count by convolving the calculated rocking curves by a Gaussianfunction of FWHM given by Eq. /H2084928/H20850.INFLUENCE OF LATTICE HEATING TIME ON … PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-7sity decreases with the increase in the bandwidth of the x
rays.
V . EXPERIMENTAL RESULTS AND DISCUSSION
In this section we describe the effects of carrier-lattice
thermalization on laser-induced strain waves. In the experi-ment, we have measured the evolution of x-ray diffractionintensity as a function of laser fluence below the damagethreshold of 10–11 mJ /cm
2.6,44,57Figure 10shows the time-
dependent diffracted intensity measured at +0.06° from theunperturbed Bragg peak with fluences of/H208492.8/H110060.6/H20850mJ /cm
2, /H208495.6/H110061.2/H20850mJ /cm2, and
/H208498.4/H110061.8/H20850mJ /cm2. Time zero has been determined by us-
ing x-ray diffraction to probe laser-induced coherent opticalphonons in bulk bismuth.
13The effective time resolution is
195/H1100625 fs FWHM including time drifts of 70 fs measured
for several days. This enables multishot data accumulationduring extended consecutive time scans.
The fluence-dependent effects are immediately apparent.
First, the value of the intensity minimum decreases with in-creasing fluence. Second, at the lowest fluence/H20849/H110112.8 mJ /cm
2/H20850over the first 15 ps following the laser ex-
citation, the rate of decrease in the diffracted signal is slowerthan at higher fluences /H20849see the arrow identified with /H9004/H20850.
In order to extract physical information from the data pre-
sented in Fig. 10, we use the model presented in Sec. III. The
x-ray diffraction is simulated by using Eqs. /H2084920/H20850–/H2084926/H20850. The
bandwidth of the double multilayer monochromator /H9004E/Eis
taken into account by convolving the calculated rockingcurves by a Gaussian function of FWHM
/H9004
/H9258BW=/H20879/H9004E
E/H20879tan/H9258. /H2084928/H20850
Oscillations in the time-dependent diffraction intensities
shown in Fig. 10are washed out due to the large bandwidth
of the x rays. The contribution of the Debye-Waller factor inthe diffracted intensity is neglected. Over the temperature
range up to 600 K and for the mean-square displacement upto 8% of the nearest-neighbor distance the decrease in thediffracted intensity due to the Debye-Waller effect for the111 reflection is /H113514%. Figure 11shows the comparison be-
tween the simulated and measured rocking curves for theunperturbed /H20849zero strain /H20850and perturbed crystals. Once the
unperturbed rocking curve is reproduced, we simulate rock-ing curves with strain profiles computed using Eqs. /H2084916/H20850–/H2084919/H20850
by changing only the fluence Fand the heating time
/H9270,
whereas the other parameters are held fixed. The discrepan-cies in the small-angle side /H20849/H1101126%/H20850are due to the neglect of
heat conduction in the calculation of the strain wave. In thiscase the strain will be larger than that where heat conductionis considered because the laser-deposited heat is trapped inthe excitation region; the rocking curve is shifted more in thesmall-angle side and is broader than that with heat conduc-tion included. Since the small-angle side of the rocking curveis more sensitive to thermal expansion that is proportional tothe lattice temperature, the discrepancies due to the neglect
of heat conduction are pronounced more there. During thefirst 20 ps following the laser excitation when the effect ofthe lattice heating time is large, the discrepancies on thepositive angle side of the rocking curve are smaller than 4%whereas on the negative angle side up to 7%. For time delaysbetween 20 and 70 ps the discrepancies due to the neglect ofheat conduction are about 5% in the positive angle side andup to 22% on the negative /H20849small /H20850angle side.
As shown in Fig. 12, the simulations predict a faster in-
tensity drop for
/H9270=0. In the instantaneous heating limit theFIG. 10. Measured time-dependent normalized diffracted inten-
sities from InSb with step size of 670 fs for laser fluences 2.8, 5.6,and 8.4 mJ /cm
2. The time scans are measured at 5.9 keV
/H208492.101 Å /H20850, 111 reflection, /H9251/H1101515.5°.FIG. 11. Comparison between the simulated and the measured
rocking curves. The calculated rocking curves are convolved with aGaussian function corresponding to /H9004E/E/H110151.1%. In the perturbed
crystal case, the rocking curve is measured at a laser fluence ofabout 7 mJ /cm
2and time delay /H9004t/H1101575 ps. In the simulations we
have used F=7 mJ /cm2,/H9004t=75 ps, and /H9270=0 ps. Simulation A
does not take into account the effect of heat conduction. SimulationB accounts for heat conduction by assuming 26% lower surfacetemperature and 19% larger penetration depth. These values wereestimated by solving Eq. /H208498/H20850with heat conduction term D
l/H115092Tl//H11509z2
/H20849where Dlis the thermal diffusion coefficient /H20850included in the rhs,
initial condition Tl/H20849z,t=0/H20850=T0exp /H20849−z//H9256/H20850, and boundary condition
/H11509Tl/H20849z=0,t/H20850//H11509z=0.KRASNIQI et al. PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-8strain is initially large /H20849cf. Fig. 5/H20850. As the strain wave propa-
gates inside the crystal, the x rays probe a large-amplitudestrained region with a thickness that increases with the speedof sound. On the other hand, assuming a finite lattice heatingtime
/H9270, the strain which is initially zero increases during the
time/H9270as shown in Fig. 6. In this case the x rays probe a
strained region which increases with the speed of sound, butwhose amplitude increases with time. This gives rise to aslower drop in the x-ray intensity compared to the instanta-neous heating limit.
Experimental results are compared to simulations in Figs.
13and14. In the simulations, the lattice temperature increase
in the crystal surface predicted by Eq. /H208498/H20850for fluences 2.8,
5.6, and 8.4 mJ /cm
2is 158, 240, and 295 K, respectively.
The expected excitation densities at these fluences are 1.1/H1100310
21cm−3, 2.3/H110031021cm−3, and 3.3 /H110031021cm−3. In Fig.
13the measured diffracted intensity for the fluence of
2.8 mJ /cm2is compared to simulations assuming instanta-neous heating of the lattice, i.e., /H9270=0 ps. During the first
15–18 ps, the instantaneous heating time predicts a muchfaster drop in intensity than the measured one. Assuming alattice heating time
/H9270=11/H110064 ps, simulations predict a slow
decrease in the diffracted intensity during this time in agree-
ment with the experimental results /H20849see Fig. 14/H20850.58For higher
fluences of 5.6 and 8.4 mJ /cm2, a good agreement between
the experiment and calculations is found for /H9270=5/H110062 and
4/H110061.5 ps, respectively. For time delays /H1102220 ps the mea-
sured diffracted intensity is about 4% larger than that pre-dicted by the simulations. This discrepancy is mainly due tothe omission of heat conduction in the calculation of thestrain profiles that results in larger strains and thus smallerdiffracted intensities.
Reproduction of the slow drop of the diffracted intensity
by including the heating time in the strain wave suggests thatthis effect is a signature of the phonon dynamics subsequentto carrier energy relaxation as discussed in Sec. III A, sincethe lattice heating time is largely dependent on the opticalphonon decay time /H20849cf. Fig. 4/H20850. As the population of LO
phonons increases via energy transfer from the carriers, theatoms undergo anharmonic motion. This anharmonicitycouples the LO phonons to acoustic phonons. In a similarway, Chin et al.
11described the delay on the onset of the
time-dependent diffracted intensity from InSb. At a carrierdensity above 10
21cm−3, they were able to describe their
observed effect by assumin ga2p sL O emission time and 7
ps acoustic phonon generation time /H20849i.e., LO phonon decay
time /H20850. Since the heating time /H9270depends strongly on the pho-
non decay time, the decrease in the lattice heating time withincreasing laser fluence suggests that the LO phonon decaytime decreases with increasing laser fluence /H20849i.e., with in-
creasing excitation energy /H20850. A similar effect has been ob-
served by time-resolved Raman studies of phonon lifetimesin GaN which is a polar semiconductor like InSb. Tsen et
al.
46observed that the phonon lifetime of GaN decreases
from 2.5 to 0.35 ps when the e-hdensity increases from 1016
to 2/H110031019cm−3. Although GaN, which is a polar and direct-
gap semiconductor, differs from InSb in terms of band-gapFIG. 12. Calculated time-dependent diffracted intensity /H20849at
+0.06° relative to the Bragg peak /H20850assuming a strain history calcu-
lated using Eqs. /H2084916/H20850–/H2084919/H20850, with heating times /H9270=0, 5, and 10 ps.
FIG. 13. Comparison of the measured time-dependent normal-
ized diffracted intensity /H20849solid line /H20850to simulations assuming strain
history given by using Eqs. /H2084916/H20850–/H2084919/H20850with/H9270=0 ps /H20849dotted line /H20850.FIG. 14. Comparison of measured time-dependent normalized
diffracted intensities shown in Fig. 10to simulations assuming
strain history given by using Eqs. /H2084916/H20850–/H2084919/H20850.INFLUENCE OF LATTICE HEATING TIME ON … PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-9energy and band curvatures, the observations of Tsen et al.46
support the fact that the LO phonon decay depends strongly
on the electron density. Intervalley scattering may also influ-ence the carrier density dependence of the lattice heatingtime, but this should have the effect of increasing the latticeheating time as the fluence increases, which runs counter toour observations.
The matrix elements of the anharmonic decay rate depend
on the phonon occupation number
47which increases with
increasing LO phonon temperature and thus with increasinglaser fluence. This indicates that LO phonon decay rate in-creases /H20849i.e., the phonon lifetime decreases /H20850with increasing
laser fluence /H20849i.e., with increasing electron density /H20850which is
in accordance with our observations. However, Matulionis
48
suggested that the dependence of LO phonon decay time oncarrier density can be explained by assuming that plasmonsare involved in LO phonon disintegration /H20849LO phonons can
emit plasmons in the decay process /H20850. Therefore, a more rig-
orous approach that considers the decay of LO phonons viaboth acoustic phonons and plasmons is needed to provide aplausible answer regarding the dependence of the LO pho-non lifetime on the carrier density or laser fluence. To ourknowledge, the mechanism for the decay of LO phonons viaacoustic phonons and plasmons has not been worked out sofar.
VI. CONCLUSIONS
In conclusion, we have presented a study of laser-induced
strain waves in InSb over an intermediate fluence range/H20849/H110113–10 mJ /cm
2/H20850. We have presented a model that predicts
spatiotemporal evolution of strain waves during the latticeheating time. In the instantaneous heating limit the modelpredicts Thomsen-type strain waves. In the framework ofthis model and the Takagi-Taupin dynamical theory for thedepth-dependent strain gradients we have studied the fluencedependence of the transient x-ray diffraction signal. The netresult of this analysis is that the temporal evolution of thediffracted signal indicates that the lattice heating time de-creases with increasing fluence. This implies that the lifetime
of optical phonons decreases as the excitation energy in-creases, similar to previous observations in other polar,direct-band-gap semiconductors.
46Although the lifetime of
LO phonons in general is dominated by their decay into apair of acoustic phonons, the density dependence of the LOphonon decay time is not fully understood
46and requires
further investigations.
ACKNOWLEDGMENTS
The authors gratefully acknowledge financial support by
the European Union in Framework No. I3-JRP3. We thank P.Sondhauss for bringing Ref. 14to our attention.
APPENDIX
Analytical solution of Eq. /H2084920/H20850is
/H9264/H20849z/H20850=s/H92640+/H20849B/H92640+C/H20850tan/H20851s/H20849z−z0/H20850/H20852
s−/H20849A/H92640+B/H20850tan/H20851s/H20849z−z0/H20850/H20852, /H20849A1/H20850
where
/H9264/H20849z0/H20850=/H92640, /H20849A2/H20850
A=/H9266i
/H9011, /H20849A3/H20850
B=−/H9266i
/H9011S/H20849/H9253h/H20850/H9252/H9004/H9258, /H20849A4/H20850
C=−/H9266i
/H9011/H20841/H9253h/H20841
/H9253h, /H20849A5/H20850
and
s=/H20881AC−B2. /H20849A6/H20850
Hence the value of /H9264at the depth zcan be calculated by
knowing its value at z=z0.
*Present address: Max Planck Advanced Study Group at CFEL/
DESY, Notkestr. 85, 22607 Hamburg, Germany.faton.krasniqi@asg.mpg.de
1C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, Phys. Rev. B
34, 4129 /H208491986 /H20850.
2Ch. Rose-Petruck, R. Jimenez, T. Guo, A. Cavalleri, C. W. Sid-
ers, F. Ráksi, J. A. Squier, B. C. Walker, K. R. Wilson, and C. P.J. Barty, Nature /H20849London /H20850398, 310 /H208491999 /H20850.
3A. M. Lindenberg, Ph.D. thesis, University of California, 2001.
4G. Tas and H. J. Maris, Phys. Rev. B 49, 15046 /H208491994 /H20850.
5O. B. Wright, Phys. Rev. B 49, 9985 /H208491994 /H20850.
6A. M. Lindenberg, I. Kang, S. L. Johnson, T. Missalla, P. A.
Heimann, Z. Chang, J. Larsson, P. H. Bucksbaum, H. C.Kapteyn, H. A. Padmore, R. W. Lee, J. S. Wark, and R. W.Falcone, Phys. Rev. Lett. 84, 111 /H208492000 /H20850.
7D. A. Reis, M. F. DeCamp, P. H. Bucksbaum, R. Clarke, E.Dufresne, M. Hertlein, R. Merlin, R. Falcone, H. Kapteyn, M.
M. Murnane, J. Larsson, Th. Missalla, and J. S. Wark, Phys.Rev. Lett. 86, 3072 /H208492001 /H20850.
8M. F. DeCamp, D. A. Reis, A. Cavalieri, P. H. Bucksbaum, R.
Clarke, R. Merlin, E. M. Dufresne, D. A. Arms, A. M. Linden-berg, A. G. MacPhee, Z. Chang, B. Lings, J. S. Wark, and S.Fahy, Phys. Rev. Lett. 91, 165502 /H208492003 /H20850.
9See, for example, M. F. DeCamp, D. A. Reis, D. M. Fritz, P. H.
Bucksbaum, E. M. Dufresne, and R. Clarke, J. Synchrotron Ra-diat. 12, 177 /H208492005 /H20850, and references therein.
10J. Shah, Ultrafast Spectroscopyof Semiconductors and Semicon-
ductor Nanostructures /H20849Springer, Berlin, 1996 /H20850.
11A. H. Chin, R. W. Schoenlein, T. E. Glover, P. Balling, W. P.
Leemans, and C. V. Shank, Phys. Rev. Lett. 83, 336 /H208491999 /H20850.
12J. Shah, B. Deveaud, T. C. Damen, W. T. Tsang, A. C. Gossard,
and P. Lugli, Phys. Rev. Lett. 59, 2222 /H208491987 /H20850.KRASNIQI et al. PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-1013P. Beaud, S. L. Johnson, A. Streun, R. Abela, D. Abramsohn, D.
Grolimund, F. Krasniqi, T. Schmidt, V. Schlott, and G. Ingold,Phys. Rev. Lett. 99, 174801 /H208492007 /H20850.
14J. Gronkowski, Phys. Rep. 206,1 /H208491991 /H20850.
15R. W. Schoenlein, S. Chattopadhyay, H. H. W. Chong, T. E.
Glover, P. A. Heimann, C. V. Shank, A. A. Zholents, and M. S.Zolotorev, Science 287, 2237 /H208492000 /H20850.
16C. T. Chantler, J. Phys. Chem. Ref. Data 29, 597 /H208492000 /H20850.
17See, for example, S. K. Sundaram and E. Mazur, Nature Mater.
1, 217 /H208492002 /H20850, and references therein.
18K. Sokolowski-Tinten and D. von der Linde, Phys. Rev. B 61,
2643 /H208492000 /H20850.
19D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 /H208491983 /H20850.
20H. S. Brandi and C. B. de Araújo, J. Phys. C 16, 5929 /H208491983 /H20850.
21N. W. Ashcroft and N. D. Mermin, Solid State Physics /H20849Saun-
ders, Philadelphia, 1976 /H20850.
22P. M. Fauchet, Phys. Status Solidi B 110,K 1 1 /H208491982 /H20850.
23H. M. van Driel and J. F. Young, J. Phys. C 15, L31 /H208491982 /H20850.
24J. F. Young and H. M. van Driel, Phys. Rev. B 26, 2147 /H208491982 /H20850.
25C. J. Stanton and D. W. Bailey, Phys. Rev. B 45, 8369 /H208491992 /H20850.
26Semiconductors: Physics of Group IV Elements and III-V Com-
pounds , Landolt-Börnstein, New Series, Group III: Crystal and
Solid State Physics Vol. 17, Pt. A, edited by O. Madelung/H20849Springer, Berlin, 1982 /H20850.
27C. V. De Alvarez, J. P. Walter, R. W. Boyd, and M. L. Cohen, J.
Phys. Chem. Solids 34, 337 /H208491973 /H20850.
28Th. Elsaesser, J. Shah, L. Rota, and P. Lugli, Phys. Rev. Lett. 66,
1757 /H208491991 /H20850.
29J. Shah, Solid-State Electron. 21,4 3 /H208491978 /H20850.
30D. von der Linde, J. Kuhl, and H. Klingenberg, Phys. Rev. Lett.
44, 1505 /H208491980 /H20850.
31S. S. Prabhu, A. S. Vengurlekar, S. K. Roy, and J. Shah, Phys.
Rev. B 51, 14233 /H208491995 /H20850.
32H. M. van Driel, Phys. Rev. B 19, 5928 /H208491979 /H20850.
33J. R. Drabble and H. J. Goldsmid, Thermal Conduction in Semi-
conductors /H20849Pergamon, New York, 1961 /H20850.
34D. L. Price, J. M. Rowe, and R. M. Nicklow, Phys. Rev. B 3,
1268 /H208491971 /H20850.
35M. I. Kaganov, I. M. Lifshitz, and L. V. Tanatarov, Sov. Phys.
JETP 4, 173 /H208491957 /H20850.
36H. M. van Driel, Phys. Rev. B 35, 8166 /H208491987 /H20850.
37M. C. Downer and C. V. Shank, Phys. Rev. Lett. 56, 761 /H208491986 /H20850.
38F. A. Lindemann, Phys. Z. 11, 609 /H208491910 /H20850.
39J. F. Vetelino, S. P. Gaur, and S. S. Mitra, Phys. Rev. B 5, 2360
/H208491972 /H20850.
40CRC Handbook of Chemistry and Physics , edited by D. E. Lide,
87th ed. /H20849CRC, Boca Raton, 2006–2007 /H20850.
41J. Larsson, A. Allen, P. H. Bucksbaum, R. W. Falcone, A. Lin-
denberg, G. Naylor, T. Misalla, D. A. Reis, K. Scheidt, A.Sjögren, P. Sondhauss, M. Wulf, and J. S. Wark, Appl. Phys. A:Mater. Sci. Process. 75, 467 /H208492002 /H20850.
42C. R. Wie, T. A. Tombrello, and T. Vreeland, Jr., J. Appl. Phys.
59, 3743 /H208491986 /H20850.
43F. Rieutord, Acta Crystallogr., Sect. A: Found. Crystallogr. 46,526 /H208491990 /H20850.
44A. Rousse, C. Rischel, S. Fourmaux, I. Uschmann, S. Sebban, G.
Grillon, Ph. Balcou, E. Förster, J. P. Geindre, P. Audebert, J. C.Gauthier, and D. Hulin, Nature /H20849London /H20850410,6 5 /H208492001 /H20850.
45W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flan-
nery, Numerical Recipes in C /H11001/H11001 /H20849Cambridge University Press,
Cambridge, 2002 /H20850.
46K. T. Tsen, J. G. Kiang, D. K. Ferry, and H. Morkoç, Appl. Phys.
Lett. 89, 112111 /H208492006 /H20850.
47M. A. Stroscio and M. Dutta, Phonons in Nanostructures /H20849Cam-
bridge University Press, Cambridge, 2001 /H20850.
48A. Matulionis, Phys. Status Solidi A 203, 2313 /H208492006 /H20850.
49A. D. Polyanin, Handbook of Linear Partial Differential Equa-
tions for Engineers and Scientists /H20849Chapman and Hall, London/
CRC, Boca Raton, 2002 /H20850.
50In the case of parabolic band approximation, /H9004Ee=/H20849/H6036/H9275
−Eg/H20850//H208491+me/mh/H20850and/H9004Eh=−/H20849me/mh/H20850/H9004Ee, where meandmhare
the effective masses for electrons and holes.
51In the case of InSb, a laser pulse /H20849/H6036/H9275/H110151.55 eV /H20850of fluence just
below the melting threshold /H20849/H1101110 mJ cm−2/H20850creates an electron
density on the order of 1021cm−3. Because of the limited states
in the /H9003valley /H20849/H113511020cm−3/H20850, an intervalley scattering time
/H9270/H9003→L/H20849or/H9270/H9003→X/H20850smaller than /H20849or comparable to /H20850the laser-pulse
duration is needed to clear out the phase space for further tran-sitions.
52Energy and momentum conservations impose constrains on the
maximum wave vector qmaxof the LO phonons that interact with
the electrons. Conservation of energy states that Ec/H20849ki/H20850−Ec/H20849kf/H20850
=/H6036/H9275LOwhereas conservation of momentum, qmax=ki−kf, with
kiandkfstanding for initial and final states of the electron in the
kspace. For InSb, with Ec/H20849ki/H20850/H110151.33 eV and /H6036/H9275LO
/H110150.024 eV, qmax/H110153/H11003107cm−1.
53For parabolic bands Cecan be calculated following chapter 2 in
Ref. 21, with He=F3/2/H20849/H9257e/H20850/F1/2/H20849/H9257e/H20850, where /H9257e=/H20849/H9262e
−ECB/H20850//H20849kBTe/H20850,/H9262eis the quasi-Fermi level for the electrons, and
Fj/H20849/H9257/H20850is the Fermi-Dirac integral of the order j.
54Chapter 23, Eq. /H2084923.29 /H20850of Ref. 21.
55For parabolic bands, kBTe/H110111 eV and Neon the order of
1021cm−3,He/H110151.2.
56The resulting differential equation for the displacement u/H20849z,t/H20850is
of the form /H115092u//H11509t2=c1/H115092u//H11509z2+g/H20849c2,z,t/H20850, where gis an arbi-
trary function that does not depend on u, and c1and c2
/H11013/H20853c2a,c2b,.../H20854are constants. An analytical solution of this type of
equation can be found in Ref. 49.
57A possible surface damage has been ruled out by comparing the
unperturbed rocking curves before and after laser excitation.Visible sample damage /H20849very faint mark /H20850was first observed at a
laser fluence of about 11 mJ /cm
2but there was no measurable
loss in diffraction efficiency. The surface damage was consider-able at about 14 mJ /cm
2with about 10% unpumped diffraction
loss over 30 min.
58The optimal /H9270and its uncertainty /H9004/H9270are determined by minimiz-
ing the chi square with respect to /H9270and/H9004/H9258B/H20849Ref. 45/H20850.INFLUENCE OF LATTICE HEATING TIME ON … PHYSICAL REVIEW B 78, 174302 /H208492008 /H20850
174302-11 |
PhysRevB.88.035442.pdf | PHYSICAL REVIEW B 88, 035442 (2013)
Quantum confinement and Coulomb blockade in isolated nanodiamond crystallites
Asaf Bolker,1,2Cecile Saguy,2Moshe Tordjman,2and Rafi Kalish1,2
1Physics Department, Technion–Israel Institute of Technology, Haifa, Israel
2Solid State Institute, Technion–Israel Institute of Technology, Haifa, Israel
(Received 7 March 2013; published 29 July 2013)
We present direct experimental evidence of quantum confinement effects in single isolated nanodiamonds
by scanning tunneling spectroscopy. For grains smaller than 4.5 nm, the band gap was found to increase
with decreasing nanodiamond size and a well-defined, evenly spaced, 12-peak structure was observed on the
conduction band side of the conductance curves. We attribute these peaks to the Coulomb blockade effect,reflecting the 12-fold degeneracy of the first electron-energy level in the confined nanodiamond. The presentresults shed light on the size dependence of the electronic properties of single nanodiamonds and are of majorimportance for future nanodiamond-based applications.
DOI: 10.1103/PhysRevB.88.035442 PACS number(s): 73 .22.−f, 07.79.Fc, 81 .05.ug
I. INTRODUCTION
Nanodiamonds (ND) have unique mechanical, electrical,
and optical properties.1,2They are chemically inert, biocom-
patible, and their surfaces can be modified by applying variousterminations.
3Nanodiamonds containing nitrogen vacancy
(NV) color centers find applications in optics,4biotechnology,5
nanomedicine,6,7and quantum computing.8Moreover, the
production and the control of diamond nanocrystals with sizesas small as a few nanometers opens up the possibility torealize new diamond-based quantum electronic devices basedon Coulomb blockade and quantum size effects. As quantumsize effects are expected to strongly influence the electronicand optical properties of semiconductors, it is important toknow the grain size at which the transition from bulk propertiesto quantum confinement effects occurs. Up until now, variousexperimental and theoretical results were reported regardingthe size limit at which quantum effects, in diamond, set in.Due to the extremely small value of the exciton Bohr radiusin diamond, r=1.57 nm,
9quantum size effects are expected
to set in for very small size diamond nanocrystals. In fact,theoretical works have predicted that quantum effects occurfor NDs with sizes smaller than 3 nm (Ref. 10)o re v e n1n m ,
11
and that the band gap of ND with diameter between 1 and
1.5 nm is predicted to be smaller than that of bulk diamond.11,12
With regard to experimental works, some publications report
on the transition from bulk properties to quantum size effectsto set in already for ND grains as large as 27 nm, whereasin other works no confinement is observed in diamonds assmall as 4 nm.
10–15All these experimental results regarding
the electronic properties were obtained from ND layers or
from ND clusters ; such measurements are far from being
ideal for determining the true electrical properties of single
nanoparticles, as the band gap of quantum dots (QD) withinarrays were found to be reduced and the discrete energylevels to broaden with respect to the corresponding isolatedQDs. These effects were related to electron delocalization andcoupling between the electronic wave functions of neighboringQDs.
16,17So far, no result has been reported on single isolated
diamond nanocrystals. Hence a detailed study on various sizesingle ND crystals is required to experimentally determinewhen the quantum size effects set in and how the electronicproperties vary as a function of the grain size. Scanningtunneling spectroscopy (STS) provides a direct method to
study the local electronic properties of single nanocrystals andquantum dots. This technique allows local measurements, onthe nanoscale, of tunneling conductance spectra yielding directinformation on the density of states. This technique has beenwidely used on metallic and semiconductor quantum dots.
18–21
Here we report on the dependence of the electronic
properties of single diamond nanocrystals on size, varyingbetween 2.5 and 13 nm, measured by scanning tunnelingmicroscopy (STM) and spectroscopy (STS) at 30 K. Theinterpretation of the STS results is based on the capacitivemodel describing single-electron tunneling in metallic islandsexpanded to semiconductor quantum dots.
22,23We find that
quantum effects set in for NDs smaller than 4.5 nm. We showthat for those small grains, the band gap widens asymmetricallywith decreasing ND size. Furthermore, we find a 12-peakstructure related to Coulomb blockade effect on the conductionband side of the conductance curves. These results allow us toextract the size dependence of the band gap and single-electroncharging energy and to confirm the 12-fold degeneracy of thefirst level in the conduction band.
II. EXPERIMENT
The NDs measured in this work originate from a com-
mercially available high-purity and high-homogeneity NDpowder produced by laser synthesis by Ray Techniques Ltd.
24
The NDs were first annealed at 450◦Cf o r1hi na i r25to
remove the graphitic component. The ND powder was furthercleaned and dispersed using the procedure of Ref. 26.T h i s
includes the following stages: (i) acid cleaning in a mixtureof sulfuric and nitric acid and reflux for 5 days at 70
◦C;
(ii) repeated centrifugation to separate the nanodiamonds fromthe liquid; (iii) washing with deionized water in an ultrasonicbath for 1 h and then reflux with NaOH (0.1 M,8m l )f o r
1 h at 90
◦C followed by further rinsing in deionized water
in an ultrasonic bath for 1 h. The powder purity of the NDpowder thus obtained was subjected to x-ray photoelectronspectroscopy (XPS) and electron-energy-loss spectroscopy(EELS) measurements yielding information about impuritycontent and carbon bonding of the material. The XPS spectrumdid not reveal the presence of any impurities and EELS data
035442-1 1098-0121/2013/88(3)/035442(6) ©2013 American Physical SocietyBOLKER, SAGUY , TORDJMAN, AND KALISH PHYSICAL REVIEW B 88, 035442 (2013)
FIG. 1. (Color online) (a) STM image of the surface showing NDs. (b) STM image of 6 nm diameter (as estimated from its height) isolated
ND clearly showing the hexagonal structure of this specific grain.
clearly showed energy loss peaks due to diamond bulk and
diamond surface plasmons.
Two samples (A) and (B) were prepared by depositing the
NDs from a methanol ND suspension by sonication ontoa highly n-doped silicon wafer overgrown with a 0.9-nm
(±0.2 nm) native oxide layer, as determined by XPS. Sample
(B) was further subjected to a short ∼2-min chemical vapor
deposition (CVD) diamond overgrowth in order to increasethe ND size and to strengthen the ND attachment to thesubstrate. The crystallite sizes, as determined from scanningtunneling microscopy (STM) for samples (A) and (B) varyfrom 2.5 to 5 nm and from 5 to 15 nm, respectively. TheSTM/STS measurements were performed in an ultrahigh-vacuum variable temperature STM Omicron system using atungsten tip. The samples were heated, in situ , to 300
◦Cf o r
6 h before the measurements. Differential conductance dI/dV
curves were directly obtained by using a lock-in amplifieroperated at a modulation voltage of 5 mV and a time constantof 20 ms.
Figure 1(a) shows a typical topography image of NDs
placed on the surface, and Fig. 1(b), of an isolated single
6-nm-high ND. The images were obtained at a bias voltage(V
bias) of 7 V applied to the sample and a set point current ( ISP)
of 20 pA on sample (B). Because of the inherent properties ofthe tip, the nanodiamond size was determined, as is commonlydone, by their height and not by their lateral size. The height ofthe nanodiamond shown in Fig. 1(b) is 6 nm. Facets are clearly
observable for this specific nanodiamond. However, smallerNDs appear to have a round shape, as predicted in Ref. 11and
probably also due to convolution effects with the tip shape. Aroughness of about 1 nm was determined for the substrate bycontact atomic force microscopy (AFM) topographic images(not shown).
III. RESULTS AND DISCUSSION
Figure 2shows the dI/dV curves measured at 30 K
between −4 and +4 V at a bias voltage of 5 V and a
current set point of 2.5 nA for five different single NDs ofsize ranging between 2.5 and 13 nm. Typically several tens ofI(V) anddI/dV curves were recorded for each ND and the
results were averaged. At least three different nanoparticles ofeach size were measured, yielding reproducible results for thesame size ND. The forbidden gap values were extracted fromFig. 3for each grain by taking the difference between the first
positive and negative voltage points at which the measured
FIG. 2. (Color online) dI/dV curves as a function of the voltage
drop between the tip and the ND, ( V2=η×Vbias) for different grain
sizes: 13 nm (brown line), 7.5 nm (green line), 4.1 nm (blue line),
3.2 nm (red line), and 2.5 nm (black line).
035442-2QUANTUM CONFINEMENT AND COULOMB BLOCKADE IN ... PHYSICAL REVIEW B 88, 035442 (2013)
FIG. 3. (Color online) (a) Measurement configuration. (b) Equivalent circuit describing the DBTJ structure according to the capacitive
model.
dI/dV values exceeded the background noise caused by
thermal fluctuations. As seen in Fig. 2the conductance curves
and heights of peaks are lower for the negative bias thanfor the positive one, reflecting a higher effective barrier fortunneling at negative bias. In order to display the dI/dV
curves measured at the negative bias clearly, for NDs smallerthen 4.5 nm, their scales have been increased sixfold inFig. 2. The following clear features should be noticed from the
conductance curves of Fig. 2: (i) No peaks are observed within
the energy gap for all measured crystallites. Since the STSmeasurements probe the electronic properties of the outermostsurface, the fact that the band gaps for all nanodiamondsmeasured are larger or equal to that of bulk diamond andno states are noticeable within the measured band gaps clearlyproves that the nanodiamonds studied in the present workare not embedded in a substantial graphitic shell. (ii) Themeasured band gap for grains with diameters between 6 and13 nm is unchanged and is equal to 5.38 eV , in agreement withresults of previous spectroscopic measurements performed onbulk diamond at 30 K.
27In addition the measured energy
gap is symmetric around the tip Fermi level at zero biasvoltage ( E
F=0 eV). Hence NDs larger than 6 nm behave
like infinitely large bulk diamond. (iii) Twelve very distinct,evenly spaced peaks appear on the conduction band side forgrains smaller than 4.5 nm. The 13th peak is separated fromthe 12th peak by a larger gap. The average energy spacingbetween the 12 peaks observed for grains smaller than 4.5 nmincreases with decreasing ND size, from 56 meV for the4.5-nm grain to 100 meV for the 2.5-nm grain. (iv) The bandgap for grains smaller than 4.5 nm increases with decreasing
ND size, reaching a value as high as 5.73 eV for the 2.5-nmgrain.
For the analysis of the present results a double barrier
tunneling junction (DBTJ) configuration between a ND graincoupled to two metallic electrodes was assumed [see Figs. 3(a)
and 3(b)]. The first tunneling barrier consists of the native
silicon oxide layer separating the n
+silicon substrate and
the ND grain, while the vacuum between the tip and thegrain serves as the second barrier. The voltages V
1(between
the ND and the Si substrate) and V2(between the tip
and the ND) can be estimated from the capacitive modeldescribing single-electron tunneling in metallic islands andwas expanded to semiconductor quantum dots by Niquetet al.
23The parameters used for junctions 1 and 2 are the
barrier capacitances C1(sphere-plane capacitance28) andC2
(sphere-sphere capacitance29), the tunneling resistances are
R1andR2and tunneling rates are ( /Gamma11,/Gamma12), where R1∝
1//Gamma11, andR2∝1//Gamma12. The fraction of the total bias falling
on the tip/sample is given by η=V2/Vbias=C1/(C1+C2).
In the present case, the ND/substrate junction parameters(C
1and/Gamma11) are fixed by the sample structure, whereas the
tip/ND junction parameters ( C2and/Gamma12) can be experimentally
varied by changing Vbiasor the tunneling current set point,
thereby changing the tip/ND distance. This distance wasestimated from I(Z) measurements on single NDs to be 0.8
nm for V
bias=5 V and ISP=2.5 nA, while the tip diameter,
extracted by deconvolution of the scanned image using the
SPIP software,30was estimated to be 1.2 nm. Using these, the
calculated C1values were found to be larger by at least an
order of magnitude than the C2values ( C2=5.57×10−20F
vsC1=7.5×10−19F for a ND of diameter 2.5 nm), leading to
η/greaterorequalslant0.9 for all the measured ND grains. Hence more than 90%
of the tip-sample bias voltage falls on the tip/ND tunnelingjunction. The conduction band empty states and the valenceband filled states are probed by applying positive or negativebias voltages, respectively, to the sample. The dI/dV curves
are presented as a function of the actual tip/sample voltageηV
bias.
We propose, as will be shown below, that the 12 peaks no-
ticed in the conductance curves are caused by single-electroncharging effects, i.e., the Coulomb blockade. Generally, threeconditions must be met for the existence of observable single-electron charging effects, and all three hold for the presentcase: (i) The single-electron charging energy must be largerthan the thermal energy k
BTwhich for 30 K is 2.5 meV ,
significantly lower than the lowest measured average spacingenergy between the peaks. (ii) The tunneling resistance ofthe two junctions should be larger than the resistance quantato prevent tunneling event overlap and quantum fluctuationsfrom masking the Coulomb blockade. In the present case thetunneling resistance, R
T(RT=/Delta1V//Delta1I ), extracted from the
I-Vcurve step structure of the various ND grains ranging
from 0.5 to 1.5 G /Omega1is well above the resistance quanta, h/e2=
25.813 K /Omega1. (iii) The rate of electron tunneling between the
tip and the ND, /Gamma12, must be comparable to or larger than the
tunneling rate of electrons between the ND and the substrate,
035442-3BOLKER, SAGUY , TORDJMAN, AND KALISH PHYSICAL REVIEW B 88, 035442 (2013)
FIG. 4. (Color online) dI/dV curves as a function of the bias
voltage measured on a 4.1-nm grain at two different set point
parameters. The dI/dV curve taken at larger tip-sample separation
(red curve) was amplified 10 ×as compared to that taken at closer
tip-sample separation (black curve). In addition an offset on the y
axis between the two curves was added for clarity. /Sigma1andUmark the
polarization and charging energy, respectively. The inset shows the
I-Vcurves corresponding to the dI/dV curves.
/Gamma11(/Gamma12>/Gamma1 1), thus minimizing the charge leakage. In a STS
experiment this rate can be controlled by changing the distancebetween the tip and the ND grain. By increasing the tip/NDdistance, the tip/ND tunneling rate is expected to decrease(/Gamma1
2</Gamma1 1), thus removing the Coulomb blockade effects.31
Figure 4shows two dI/dV curves obtained for the same grain
at different tip-ND distances, one (black line) corresponding tothe set point for Coulomb blockade ( V
gap=5V ,ISP=2.5n A )
and the other (red line) corresponding to a higher tip-graindistance ( V
gap=7V ,ISP=1 nA). In addition the insert of
Fig. 4shows the corresponding I-Vcurves, confirming the
fact that the I-Vtaken at lower tip-sample distance (black)
exhibits higher conductance. The change in ηcaused by the
increase of the tip/ND distance has been taken into accountso that real level spacings are shown in Fig. 4. The curve
obtained for the Coulomb blockade set point conditions clearlyshow evenly spaced peaks at positive voltage bias. These areabsent in the curve measured for the larger tip/sample distance,indicating the disappearance of the single-electron chargingeffects. Therefore the peaks in the red curve must be relatedto tunneling to the ND discrete energy levels. Indeed, theenergy separation between the 12th and 13th peaks, /Delta1E
12,13
(black curve), differs from the equal spacing between the
first 12 peaks due to Coulomb blockade. This separation,/Delta1E
12,13, corresponds to the energy difference between the
first and second electron-energy levels of the ND plus thesingle-electron charging energy, E
12,13=U+/Delta11,2.
Figure 5shows a comparison between experimental and
calculated single-electron charging energy as a function ofND grain size. The experimental values represent the averagepeak separation measured between the first 12 peaks (the errorbars represent the standard deviation of these values). Thecalculated values of the charging energy Uwere deduced
fromU=e
2/CT, based on the capacitive model,23where,
CTis the total capacitance of the tip-ND-substrate, CT=FIG. 5. (Color online) Measured charging energy values, as a
function of grain size (black squares). The red line shows the charging
energy values calculated according to U=e2/C Twith oxide layer
thickness of 0.9 nm and tip-sample separation of 0.8 nm.
C1+Cdot+C2,,Cdotbeing the dot self-capacitance. As seen
in Fig. 5the calculated and measured charging energy values
both increase with decreasing the ND size. It was shown inRef. 23that the capacitive model for metallic QDs can be
applied with a good degree of accuracy to semiconductor QDswhen the dielectric constant of the nanocrystal is much higherthan that of the material on which the grain rests. In the presentcase, however, the dielectric constants of SiO
2and diamond
are relatively close, 3.9 and 5.7, respectively; hence theexperimental values are expected to follow the trend predictedby the capacitive model for metallic QDs with a moderatematch.
21,23
The fact that 12 clear peaks are seen in all positive dI/dV
data for grains smaller than 4.5 nm (see Fig. 2) confirms
the fact that the conduction band minimum of diamondis 12-fold degenerated due to spin and diamond six-valleydegeneracy. Hence, the first electron-energy level is expectedto be 12-fold degenerated. The first level may be further splitas the size of QD is reduced as predicted for Si QDs withdiameter below 3 nm (Ref. 32) (no similar calculations were
performed for nanodiamonds). This splitting was calculatedto be, at most, of the order of a few meV for Si QDs. Theenergetic resolution of our STS measurements at 30 K is 9meV [roughly 3 kT (Ref. 33)] and therefore even if a similar
splitting would exist in small NDs, our STS system couldnot measure it. However, since the conductance curves weremeasured under Coulomb blockade conditions the Coulombinteraction lifts the conduction band minimum degeneracy.
21
Hence, the evenly separated 12 peaks measured here for theNDs smaller than 4 nm must be attributed to the lifting ofthe first electron-energy level’s degeneration in the weaklyconfined ND.
Figure 6shows the size dependence of the band gaps
extracted from the measurement using E
gap=η/Delta1Ve –2/Sigma1,
(/Delta1V is the measured zero current voltage in the dI/dV pre-
sentation, and /Sigma1=U/2 is the grain polarization energy23,28).
The error bars represent the combined errors for the estimation
035442-4QUANTUM CONFINEMENT AND COULOMB BLOCKADE IN ... PHYSICAL REVIEW B 88, 035442 (2013)
FIG. 6. (Color online) Energy band gap as a function of nanodi-
amond grain size (red squares). The errors bars are derived from the
error in the measured single-electron charging energy Ua n di nt h e
estimation of η. The dotted horizontal line marks the Egvalue of bulk
diamond at 30 K.
of the polarization energy /Sigma1,o fη, and of the temperature line
broadening. While the larger grains (13, 7.5, and 6 nm) donot show any clear change in the measured energy gap, beingthe bulk value at 30 K,
27the grains with diameter smaller
than 4.5 nm do show an increase in the measured band gap,increasing by up to ∼300 meV above the bulk band gap for
the 2.5-nm grain.
The fact that the widening of the band gap is asymmetric,
noticeable on the valence band side, increasing with decreasinggrain size, can be explained as being due to the n
+silicon
substrate as seen in Fig. 7. This figure shows a schematic
representation of the tip/nanodiamond/ n+silicon substrate
indicating the relevant energies. Since the nanodiamonds ofsize up to 4.5 nm originating from sample (A) did not undergofurther CVD growth, they exhibit positive electron affinitydue to oxygen termination caused by extensive acid cleaningof the powder. At zero bias conditions the tip Fermi level isaligned with the Fermi level of the n-type silicon substrate,
which is closer to the conduction band minimum, resulting inthe measured asymmetry. Thus, as the band gap increases forsmaller nanodiamond grains the valance band is pushed further
FIG. 7. (Color online) Schematic model of the DBTJ energy band
structure. From left to right, the n-type silicon substrate with a band
gap of Eg=1.1 eV and an electron affinity χ=4.05 eV , the native
SiO 2layer with Eg=9 eV (marked in blue) and χ=1 eV , the
oxygen-terminated nanodiamond crystal with Eg/greaterorequalslant5.4 eV and χ=
1.7 eV , and the tungsten tip with a work function of φ=4.5 eV .
down leading to the observed asymmetry in the measured
spectrum.
IV . CONCLUSIONS
In summary, quantum size effects as well as the Coulomb
blockade were clearly observed in single NDs smaller than4.5 nm by STS measurements. In contrast, NDs of sizes largerthan 6 nm were shown to behave like bulk diamond. Theseresults disagree with previous experimental works performedon 4-nm nanodiamond clusters showing no band gap increase.In our case, unlike ND clusters or arrays, the electronicproperties of single isolated nanodiamonds are not affectedby neighboring nanodiamonds. Hence the results reportedin this work provide valuable information on the electronicproperties of single isolated nanodiamonds which may be usedfor future research and potential applications of NDs in opticaland electronic quantum devices.
ACKNOWLEDGMENTS
The authors thank A. Zunger, G. Bahir, and O. Millo for
fruitful discussions.
1I. Aharonovich, A. D. Greentree, and S. Prawer, Nat. Photonics 5,
397 (2011).
2V . N. Mochalin, O. Shenderova, D. Ho, and Y . Gogotsi, Nat.
Nanotechnol. 7, 11 (2012).
3A. Krueger and D. Lang, Adv. Funct. Mater. 22, 890 (2012).
4A. Cuche, A. Drezet, Y . Sonnefraud, O. Faklaris, F. Treussart, J.-F.
Roch, and S. Huant, Opt. Express 17, 19969 (2009).
5Y .-R. Chang, H.-Y . Lee, K. Chen, C.-C. Chang, D.-S. Tsai, C.-C.
Fu, T.-S. Lim, Y .-K. Tzeng, C.-Y . Fang, C.-C. Han, H.-C. Chang,and W. Fann, Nat. Nanotechnol. 3, 284 (2008).
6Z.-Y . Lien, T.-C. Hsu, K.-K. Liu, W.-S. Liao, K.-C. Hwang, and
J.-I. Chao, Biomaterials 33, 6172 (2012).
7X.-Q. Zhang, M. Chen, R. Lam, X. Xu, E. Osawa, and D. Ho, ACS
Nano 3, 2609 (2009).8A. D. Greentree, B. A. Fairchild, F. M. Hossain, and S. Prawer,
Mater. Today 11, 22 (2008).
9M. Cardona, T. Ruf, and J. Serrano, P h y s .R e v .L e t t . 86, 3923
(2001).
10A. S. Barnard, Cryst. Growth Des. 9, 4860 (2009).
11J.-Y . Raty, G. Galli, C. Bostedt, T. W. van Buuren, and L. J.
Terminello, Phys. Rev. Lett. 90, 037401 (2003).
12N. D. Drummond, A. J. Williamson, R. J. Needs, and G. Galli, Phys.
Rev. Lett. 95, 096801 (2005).
13Y . K. Chang, H. H. Hsieh, W. F. Pong, M.-H. Tsai, F. Z. Chien,
P. K. Tseng, L. C. Chen, T. Y . Wang, K. H. Chen, D. M. Bhusari,J. R. Yang, and S. T. Lin, P h y s .R e v .L e t t . 82, 5377 (1999).
14A. A. Fokin and P. R. Schreiner, Mol. Phys. 107, 823
(2009).
035442-5BOLKER, SAGUY , TORDJMAN, AND KALISH PHYSICAL REVIEW B 88, 035442 (2013)
15T. M. Willey, C. Bostedt, T. van Buuren, J. E. Dahl, S. G. Liu,
R. M. K. Carlson, L. J. Terminello, and T. M ¨oller, Phys. Rev. Lett.
95, 113401 (2005).
16P. Liljeroth, K. Overgaag, A. Urbieta, B. Grandidier, S. G. Hickey,
and D. Vanmaekelbergh, P h y s .R e v .L e t t . 97, 096803 (2006).
17D. Steiner, A. Aharoni, U. Banin, and O. Millo, Nano Lett. 6, 2201
(2006).
18U. Banin, Y . Cao, D. Katz, and O. Millo, Nature 400, 542
(1999).
19E. Bar-Sadeh, Y . Goldstein, C. Zhang, H. Deng, B. Abeles, andO. Millo, P h y s .R e v .B 50, 8961 (1994).
20P. Liljeroth, P. A. Zeijlmans van Emmichoven, S. G. Hickey,
H. Weller, B. Grandidier, G. Allan, and D. Vanmaekelbergh, Phys.
Rev. Lett. 95, 086801 (2005).
21B. Zaknoon, G. Bahir, C. Saguy, R. Edrei, A. Hoffman,
R. A. Rao, R. Muralidhar, and K.-M. Chang, Nano Lett. 8, 1689
(2008).
22Single Charge Tunneling: Coulomb Blockade Phenomena in Nanos-tructures , edited by H. Grabert and M. H. Devoret (Plenum Press,
New York, 1992).23Y . M. Niquet, C. Delerue, G. Allan, and M. Lannoo, Phys. Rev. B
65, 165334 (2002).
24RayND: Pure Nanodiamond Powder, http://www.nanodiamond.
co.il/Products.htm .
25A. E. Aleksenskiy, E. D. Eydelman, and A. Y . Vul, Nanosci.
Nanotechnol. Lett. 3, 68 (2011).
26C. Bradac, T. Gaebel, N. Naidoo, M. J. Sellars, J. Twamley,
L. J. Brown, A. S. Barnard, T. Plakhotnik, A. V . Zvyagin, andJ. R. Rabeau, Nat Nano 5, 345 (2010).
27R. P¨assler, Phys. Status Solidi B 216, 975 (1999).
28D. K. Ferry, S. M. Goodnick, and J. P. Bird, Transport in
Nanostructures (Cambridge University Press, Cambridge, 2009).
29D. Sarid, Exploring Scanning Probe Microscopy with MATHEMAT-
ICA (Wiley-VCH, Weinheim, Germany, 2007).
30J. F. Jørgensen, Scanning Probe Image Processor SPIPTM,V e r s i o n6
(Image Metrology, Denmark, 2012), http://www.imagemet.com/ .
31U. Banin and O. Millo, Annu. Rev. Phys. Chem. 54, 465 (2003).
32A. Franceschetti and A. Zunger, P h y s .R e v .B 62, 2614 (2000).
33A. Bolker, C. Saguy, M. Tordjman, L. Gan, and R. Kalish, Phys.
Rev. B 83, 155434 (2011).
035442-6 |
PhysRevB.77.195302.pdf | Coulomb effects in open quantum dots within the random-phase approximation
V . Moldoveanu1and B. Tanatar2
1National Institute of Materials Physics, P .O. Box MG-7, 077125 Bucharest-Magurele, Romania
2Department of Physics, Bilkent University, Bilkent, 06800 Ankara, Turkey
/H20849Received 26 October 2007; revised manuscript received 7 April 2008; published 2 May 2008 /H20850
The effect of electron-electron interactions on coherent transport in quantum dot systems is theoretically
investigated by adapting the well-known random-phase approximation /H20849RPA /H20850to the nonequilibrium Green–
Keldysh formalism for open mesoscopic systems. The contour-ordered polarization operator is computed interms of the Green functions of the noninteracting system. We apply the proposed RPA-Keldysh scheme forstudying Coulomb-modified Fano lines and dephasing effects in interferometers with side-coupled many-leveldots. Our method allows us to treat on equal footing the decoherence induced by the intradot interaction andthat by the Coulomb coupling to a nearby system. In the case of a single interferometer, we show that theintradot Coulomb interaction leads to a reduction of the Fano line amplitude. From the analysis of the inter-action self-energy, it follows that this effect originates in inelastic scattering processes in which electron-holepairs are involved. The interplay between the interdot and the intradot interactions in decoherence is discussedfor two nearby identical T-shaped interferometers. We also show that the intradot interaction does not preventthe observation of controlled dephasing due to a nearby charge detector, as long as the latter is subjected to asufficiently large bias.
DOI: 10.1103/PhysRevB.77.195302 PACS number /H20849s/H20850: 73.23.Hk, 85.35.Ds, 85.35.Be, 73.21.La
I. INTRODUCTION
Understanding the role of Coulomb interactions in trans-
port phenomena at nanoscale has become an important taskfor an accurate description of the underlying physics, espe-cially in the context of mesoscopic interferometry
1and semi-
conductor spintronics.2The reason is twofold. On one hand,
the electron-electron interactions in highly confined systemssuch as quantum dot arrays are responsible for nontrivialeffects that are appealing from the applications point of view/H20849Coulomb blockade,
3–6Kondo correlated transport,7charge
sensing,8,9etc. /H20850On the other hand, it was theoretically
predicted10–12that the coherent features of transport are dam-
aged by the inelastic processes due to the Coulomb interac-tion of the system with its environment. This statement wasconfirmed later on in the experiment of Buks et al.
13More
precisely, it was reported that the Aharonov–Bohm oscilla-tions in a ring with an embedded quantum dot are partiallyreduced when a quantum point constriction subjected to afinite bias is placed near the quantum dot. Since the proper-ties of the constriction and of the dot are easily tunable, thisdecoherence process is also called controlled dephasing; itopened the way to indirect measurement techniques of quan-tum interference in mesoscopic systems. In contrast, the hy-perfine interaction between the electronic and nuclear spinssets undesired limits for solid-state implementation of quan-tum computation algorithms.
14The problem of decoherence
induced by intradot interactions was theoretically addressedby Sivan et al.
15and by Altshuler et al.16
Since electron-electron interactions are a built-in feature
of semiconductor nanostructures, considerable experimentalefforts nowadays are focused on designing suitable quantumdot-based devices allowing the “reading” of interference ef-fects and the coherent manipulation of electrons while keep-ing the losses due to decoherence negligible. From the theo-retical point of view, the description of quantum transport ininteracting systems is not straightforward because one has to
essentially deal with a many-body problem that leaves noroom for an exact treatment except for very few simple mod-els. One crucial point then is to choose appropriate approxi-mation schemes for the Coulomb interaction in order to cap-ture subtle effects that play an important role in coherent orincoherent transport.
In this work, we propose a treatment of the Coulomb
interaction based on the random-phase approximation /H20849RPA /H20850
and on the Keldysh formalism, which we find useful in thestudy of dephasing in mesoscopic interferometers. Both theRPA and the Keldysh approaches are well established formaltools that lead to important progress in the description oftwo-dimensional electron gas properties and of the mesos-copic transport phenomena. In spite of this fact, the possibil-ity of combining them for studying open interacting systemsdriven by a finite bias has not been explored yet. In order toset the general context for our approach, we give in the fol-lowing a brief account of the Keldysh formalism and of itsmain applications. The main idea behind the nonequilibriumGreen’s function formalism is to compute all relevant quan-tities of a system coupled to several biased leads by using theequilibrium state of the noninteracting disconnected
system.
17–19The coupling to the leads plays the role of the
perturbation and is usually adiabatically switched. Then, astandard derivation leads to a closed formula for the currentin terms of nonequilibrium Green functions. In the interact-ing case, the difficult and technical problem that remains tobe solved is the calculation of these functions.
One approach is based on the equation-of-motion /H20849EOM /H20850
method that was initiated by Zubarev
20and is extensively
used by many authors in the study of Anderson and KondoHamiltonians /H20849see Refs. 21and22and references therein /H20850.
The usual strategy is to factorize the thermal averages ofproducts of four creation /H20849annihilation /H20850operators in order to
close the otherwise infinite chain of equations for higher-order Green functions /H20849see, for example, Refs. 23and24/H20850.PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
1098-0121/2008/77 /H2084919/H20850/195302 /H2084911/H20850 ©2008 The American Physical Society 195302-1One exactly solvable Fano–Kondo Hamiltonian within the
EOM method is presented in a recent work.25
Another way to approximately compute interacting Green
functions is to perform a perturbative expansion with respectto the interaction strength and to write down expressions forthe first and second-order contributions to the interactionself-energy. This procedure successfully describes the con-trolled dephasing in mesoscopic interferometers Coulombcoupled to charge detectors.
26,27This is because the second-
order diagram for the interaction self-energy is the electron-hole bubble, which already takes into account inelastic pro-cesses that induce decoherence in the system. It should bementioned that in this approach, a single-level quantum dotis considered and, therefore, the intradot interaction effectcannot be captured. König et al.
28argued that a single-
particle approximation oversimplifies the role of the interac-tion and therefore cannot capture decoherence effects due tointradot Coulomb repulsion. Clarifying the role of interdot
interactions in decoherence as well as the interplay betweenintradot and interdot interactions in controlled dephasingconstitutes another motivation for considering the RPA-Keldysh approach to steady-state transport in interactingquantum dot systems.
We mention that Faleev et al.
29performed a self-
consistent RPA calculation of the equilibrium Green func-tions and interaction self-energies for a homogeneous two-dimensional electron gas in the Kadanoff–Baym framework.Our approach is similar, the difference being that we con-sider open systems subjected to a finite bias and that the
numerical simulations are done for lattice models. In a recentwork, Wulf et al.
30calculated the admittance of one-
dimensional open systems subjected to an additional ac biassuperimposed to the source-drain bias. The steady-state re-gime of the system /H20849that is, in the absence of the ac bias /H20850is
described within the Landauer formalism and the random-phase approximation is used to estimate the density changeinduced by the ac bias. Here, we study steady-state transportand focus on decoherence effects due to electron-electroninteractions. We present two applications that are relevant tothe dephasing problem in mesoscopic interferometers. Thefirst model system we consider is a many-level one-dimensional quantum dot side-coupled to a single channellead /H20849the so-called T-shaped interferometer /H20850.
These systems attracted considerable attention since the
observation of the Fano interference in the experiment ofKobayashi et al.
31Johnson et al.9also reported different
transport regimes of a quantum dot coupled to a single con-ducting channel: pure Coulomb peaks and charge sensingeffect at very weak coupling to the channel or Coulomb-modified Fano lines at moderate coupling. The charge sens-ing effect allows measurements of Coulomb blockade withnoninvasive voltage probes and requires a theoretical de-scription beyond the orthodox picture of the Coulombblockade.
32,33
The role of electron-electron interactions on the transport
properties of side-coupled quantum dots has been theoreti-cally investigated especially in the context of the Fano–Kondo effect.
34Solving this problem requires nonperturba-
tive techniques /H20849such as the slave-boson mean-field theory or
the renormalization group method /H20850for dealing with the on-site /H20849Hubbard /H20850interaction at the quantum dot, which leads to
the strongly correlated Kondo state.35–37Numerical or ana-
lytical results have been obtained for single-level quantumdots only. On the other hand, Orellana et al.
38considered a
one-dimensional side-coupled array of noninteracting quan-tum dots and found that the transport properties /H20849resonances
or antiresonances /H20850depend on the number of sites in the array
/H20849odd or even /H20850. The conductance is computed by using a re-
cursive formula for the retarded Green function. Neverthe-less, to our best knowledge, no calculation of the Fano inter-ference for interacting many-level side-coupled dots hasbeen done yet.
The paper is organized as follows: In Sec. II, we describe
the method in a rather general form, in the sense that we donot specialize to the tight-binding Hamiltonian of a givenstructure. All we assume is that the electron-electron interac-tions are present only in some central region and not in theleads, as it is usually done in the Keldysh formulation ofelectronic transport. The spin degrees of freedom and theKondo problem are not considered in this work. For the roleof spin-flip effects in dephasing, we refer to the works ofKönig and Gefen
39and of Silva and Levit.40In Sec. III, we
show that the Fano interference is reduced when the Cou-lomb interactions inside the dot are taken into account. In thesecond half of Sec. III, we take two nearby T-shaped inter-ferometers and investigate in detail their coherence proper-ties in the presence of interdot and intradot interactions. Theeffect of a charge detector placed near the side-coupled quan-tum dot is also discussed. Finally, Sec. IV is devoted to con-clusions.
II. FORMALISM
In any theoretical approach to interacting quantum trans-
port, one starts with a formal tool to write down a formulafor the current through the considered system, in terms of theinteracting quantities. The explicit results are then obtainedby using approximation schemes for the interaction effects.Here, we use the nonequilibrium Keldysh formalism forelectronic transport and the random-phase approximation forthe Coulomb interaction. In view of the numerical imple-mentation, we shall work with tight-binding Hamiltonians.The system configuration is typical to the Keldysh approach:a central region /H20849C/H20850is coupled to noninteracting semi-infinite
leads via a time-dependent switching
/H9273/H20849t/H20850. An adiabatic cou-
pling is tacitly assumed in most of the theoreticalcalculations,
17,41which means that /H9273/H20849t/H20850vanishes in the re-
mote past, and the steady-state current is computed in thelong-time limit. Actually, recent rigorous results show thatthe steady-state current does not depend on the way in whichthis coupling is achieved.
42Transient current calculations
within the Keldysh formalism for noninteracting dots that aresuddenly coupled to biased leads were also performedrecently.
43
We use the index /H9253for the leads and di†/H20849di/H20850is the pair of
creation /H20849annihilation /H20850operators corresponding to the ith site
of the lead. We also denote by al†/H20849al/H20850the creation /H20849annihila-
tion /H20850operators on the lth site of the lattice describing the
central region. Then, the system Hamiltonian quite generallyreadsV . MOLDOVEANU AND B. TANATAR PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-2H/H20849t/H20850=Hcen+Hleads+/H9273/H20849t/H20850/H20849Htun+Hint/H20850,
Hcen=/H20858
l,m/H33528C/H20849/H9255l/H9254lm+tlm/H20850al†am,
Hleads=tL/H20858
/H9253/H20858
/H11021i,j/H11022/H33528L/H9253di†dj,
Htun=/H20858
i/H33528L/H9253/H20858
l/H33528C/H20849Vil/H9253di†al+ H.c. /H20850,
Hint=U
2/H20858
l/HS11005mnˆlnˆm
/H20841rl−rm/H20841. /H208491/H20850
In Eq. /H208491/H20850,Vil/H9253is the hopping coefficient between the corre-
sponding sites of the lead /H9253and of the central region. For
simplicity, we take Vil/H9253to be real and nonvanishing only if i,l
are nearest neighbors. The last term in the Hamiltonian is
written in terms of the on-site number operator nˆl=al†aland
describes the electron-electron interaction between chargeslocalized in different sites of the central region. The interac-tion strength is characterized by the parameter Uandr
lde-
notes the position of the lth site. tLis the hopping energy on
leads, /H11021,/H11022denotes nearest neighbor summation and for
simplicity the on-site energy of the leads is taken equal tozero.
Note that
/H9273/H20849t/H20850switches both the coupling to the leads and
the Coulomb interaction, which means that in our calcula-tion, the initial correlations are neglected.
18,19In the long-
time limit when the system achieves a steady state, this ap-proximation is permitted. Finally, t
lmare nearest neighbor
hopping parameters inside the central region and the on-siteenergies /H9255
lmay include a constant gate potential Vg.
The standard application of the Keldysh machinery leads
to the following preliminary formula for the current throughthe lead
/H9251in the steady state of the system:
J/H9251=e
/H9266/H6036/H20858
i/H33528L/H9251,m/H33528C/H20885
=/H11009/H11009
dERe/H20851Vmi/H9251Gim/H11021/H20849E/H20850/H20852, /H208492/H20850
where Gmi/H11021/H20849E/H20850is the Fourier transform of the lesser Green
function Gmi/H11021/H20849t,t/H11032/H20850=i/H20855ai†/H20849t/H20850dm/H20849t/H20850/H20856. Note that the operators are
written in the Heisenberg picture with respect to the totalHamiltonian and that we assumed the steady-state regime sothat the Green function depends only on time differences. Atthis point, one has to express the mixed index Green functionaccording to the Langreth rules,
18
Gli/H11021=/H20858
/H9253/H20858
j/H33528L/H9253/H20858
m/H33528C/H20849GlmRVmj/H9253gji/H11021+Glm/H11021Vmj/H9253gjiA/H20850, /H208493/H20850
where gA,/H11021are the advanced and lesser Green functions of
the semi-infinite leads that are known /H20849see, for example, Ref.
44/H20850.
Substituting their expressions into Eq. /H208492/H20850, one ends up
with the following /H20849we omit the energy dependence for the
simplicity of writing /H20850:J/H9251=ie
/H6036/H20885
−2tL2tL
dETr/H20851/H9003/H9251/H20849GR−GA/H20850f/H9251+G/H11021/H20852. /H208494/H20850
In Eq. /H208494/H20850, the Green functions are to be understood as ma-
trices and the trace means a sum over all sites of the centralregion. f
/H9251is the Fermi function of the lead /H9251and/H9003/H9251is a
matrix linewidth, which is essentially given by the density of
states in the lead, /H9267/H20849E/H20850=/H9258/H20849/H20841E/H20841−2tL/H20850/H208814tL2−E2/2tL, and by the
hopping constant between the lead /H9251and the central region
/H20851/H9258/H20849x/H20850is the step function /H20852,
/H9003lm/H9251/H20849E/H20850=2/H9266Vil/H9251Vjm/H9251/H9267/H20849E/H20850. /H208495/H20850
Equation /H208494/H20850was obtained for the first time by Jauho et al.41
and has been widely used in transport calculations for both
interacting and noninteracting structures. In the noninteract-ing case, the perturbation comes only from the coupling tothe leads, whose self-energy is known,
/H9018
L,lmR/H20849E/H20850=Vli/H9251Vmj/H9251ImgijR/H20849E/H20850, /H208496/H20850
/H9018L,lm/H11021/H20849E/H20850=2/H9266iVli/H9251Vmj/H9251gij/H11021/H20849E/H20850. /H208497/H20850
When the Coulomb interaction is taken into account, the
main technical task is to compute, within appropriate ap-proximations, the interacting self-energy /H9018
Ithat should then
be plugged into the Dyson and Keldysh equations,
GR=G0R+G0R/H20849/H9018LR+/H9018IR/H20850GR, /H208498/H20850
G/H11021=GR/H20849/H9018L/H11021+/H9018I/H11021/H20850GA, /H208499/H20850
where G0R,/H11021are Green functions of the noninteracting discon-
nected system. Using the known identity /H20849see Ref. 45/H20850GR
−GA=2iGRIm/H20849/H9018LR+/H9018IR/H20850GAand the Keldysh equation, one
obtains
J/H9251=e
h/H20885
−2tL2tL
dETr/H20851/H9003/H9251GR/H9003/H9252GA/H20849f/H9251−f/H9252/H20850−/H9003/H9251GRIm/H20849/H9018I/H11021
+2f/H9251/H9018IR/H20850GA/H20852. /H2084910/H20850
This is an alternative form for the current that is particularly
useful for emphasizing the limitations of the Landauer for-mula when applied to interacting systems.
One notices at once that the first term in the current has a
Landauer form in spite of the fact that the Green functionsappearing there are interacting quantities. The second term isgiven by the imaginary part of the interaction self-energy. Inour previous work,
27we used the above formula to investi-
gate the controlled dephasing in single-dot Aharonov–Bohminterferometers Coulomb coupled to a charge detector. Theself-energy was computed by a perturbative approach up tothe second order in the interaction strength.
Here, we propose an alternative method to compute the
interaction self-energy based on the random-phase approxi-mation. The starting point of the RPA scheme is to constructthe polarization operator /H9016. In the non-self-consistent ver-
sion of the RPA that we implement here, the polarizationoperator is built from the noninteracting Green functions ofCOULOMB EFECTS IN OPEN QUANTUM DOTS WITHIN … PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-3the coupled system. We denote these functions by Geffand
compute them by the Dyson equation /H20849in the Keldysh space /H20850
with respect to the self-energy of the leads,
Geff=G0+G0/H9018LGeff. /H2084911/H20850
Since the Green functions we deal with are contour or-
dered, the polarization operator /H9016has also lesser and greater
components, besides retarded and advanced ones /H20849k,lare
sites from the central region /H20850,
/H9016kl/H20849t1,t2/H20850=−Geff,kl/H20849t1,t2/H20850Geff,lk/H20849t2,t1/H20850. /H2084912/H20850
Using the rules for diagrammatic expansion of the Keldysh–
Green function, one is led to direct and exchange terms de-fined by the following RPA self-energies /H20849the corresponding
diagrams are given in Fig. 1/H20850:
/H9018˜kl/H20849t1,t2/H20850=iVkl/H20849t1,t2/H20850Geff,kl/H20849t1,t2/H20850, /H2084913/H20850
/H9018˜˜
kk/H20849t1,t2/H20850=−i/H20858
lVkl/H20849t1,t2/H20850Geff,ll/H20849t2,t2/H20850, /H2084914/H20850
where k,ldenote sites from the central region and time ar-
guments run along the two-branch Keldysh contour. Vis the
screened potential that obeys the Dyson equation with re-spect to the polarization operator /H9016,
V/H20849t
1,t2/H20850=V0/H20849t1,t2/H20850+/H20885dt/H20885dt/H11032V0/H20849t1,t/H20850/H9016/H20849t,t/H11032/H20850V/H20849t/H11032,t2/H20850.
The above integrals are along the Keldysh contour and we
introduced the instantaneous bare Coulomb potential,
V0,kl/H20849t,t2/H20850=U
/H20841rk−rl/H20841/H9254K/H20849t1−t2/H20850, /H2084915/H20850
where the delta function is defined on the Keldysh contour
/H20849see, for example, Ref. 19/H20850,
/H9254K/H20849t1−t2/H20850=/H9254/H20849t1−t2/H20850/H92703,/H92703=/H2087310
0− 1/H20874. /H2084916/H20850
Note that on each branch of the Keldysh contour K, the Cou-
lomb interaction is instantaneous and also that it does notcouple different branches on the contour. Using again theLangreth rules, one obtains explicit expressions for lesserand retarded quantities while the integrals are to be per-formed on individual pieces of the Keldysh contour. Theretarded polarization is computed via the Kramers–König re-lation and then the two basic equations for the polarization
operator become
/H9016
kl/H11021,/H11022/H20849E/H20850=−1
2i/H20885dE/H11032Geff,kl/H11021,/H11022/H20849E/H11032/H20850Geff,lk/H11022,/H11021/H20849E/H11032−E/H20850, /H2084917/H20850
/H9016R/H20849E/H20850=i
2/H9266/H20885dE/H11032/H9016/H11022/H20849E/H11032/H20850−/H9016/H11021/H20849E/H11032/H20850
E−E/H11032+i0. /H2084918/H20850
Noticing that the integration range in Eq. /H2084917/H20850is restricted to
/H20851−2tL,2tL/H20852and that Green function is nonvanishing only if
/H20841E/H11032−E/H20841/H110212tL, it follows that E/H33528/H20851−4tL,4tL/H20852. At the next step,
one has to compute the RPA interaction according to theDyson and Keldysh equations,
V
R/H20849E/H20850=V0+V0/H9016R/H20849E/H20850VR/H20849E/H20850, /H2084919/H20850
V/H11021,/H11022/H20849E/H20850=VR/H20849E/H20850/H9016/H11021,/H11022/H20849E/H20850VA/H20849E/H20850, /H2084920/H20850
where in Eq. /H2084920/H20850we have used the property V0/H11021=0. The
self-energies are then given by the following set of equations/H20849the argument Ecovers the interval /H20851−2t
L,2tL/H20852, as required
by the integral in the current formula /H20850:
/H9018/H11021,/H11022/H20849E/H20850=/H9018˜/H11021,/H11022/H20849E/H20850+/H9018˜˜/H11021,/H11022/H20849E/H20850, /H2084921/H20850
/H9018˜
kl/H11021,/H11022/H20849E/H20850=i
2/H9266/H20885dE/H11032Vkl/H11022,/H11021/H20849E/H11032/H20850Geff,kl/H11021,/H11022/H20849E−E/H11032/H20850, /H2084922/H20850
/H9018˜˜
kk/H11021,/H11022/H20849E/H20850=−i
2/H9266/H20885dE/H11032/H20858
lGeff,ll/H11021,/H11022/H20849E/H11032/H20850Vkl/H11022,/H11021/H20849E/H20850, /H2084923/H20850
/H9018R/H20849E/H20850=i
2/H9266/H20885dE/H11032/H9018/H11022/H20849E/H11032/H20850−/H9018/H11021/H20849E/H11032/H20850
E−E/H11032+i0. /H2084924/H20850
A particular feature of the RPA-Keldysh scheme is that
the usual first order, direct, and exchange diagrams are notrecovered when the RPA potential is introduced in the ex-
pressions for /H9018˜and/H9018˜˜. The reason for this is the following:
the self-energies contain the lesser and greater componentsV
/H11021,/H11022, which are given by the Keldysh equation whose first
term is of order 2 in the bare Coulomb potential. This isdifferent from the equilibrium version of RPA, wherein theself-energy contains the retarded component of the screenedpotential whose Dyson expansion starts with the first-orderterm. Therefore, we have to add by hand the first-order dia-grams in the final result for the retarded self-energy,
/H9018
˜
klR/H20849E/H20850=i
2/H9266/H20885dE/H11032Geff,kl/H11021/H20849E−E/H11032/H20850V0,kl/H20849E/H11032/H20850, /H2084925/H20850
/H9018˜˜
kkR/H20849E/H20850=−i
2/H9266/H20885dE/H11032/H20858
l/HS11005kGeff,ll/H11021/H20849E/H11032/H20850V0,kl. /H2084926/H20850
Equations /H2084925/H20850and /H2084926/H20850were obtained by again using the
diagrammatic expansion and the Langreth rules. Note that inEq. /H2084926/H20850, the integrals are actually decoupled and that
Im/H9018˜˜
kkR=0. Also, /H9018˜Ris off-diagonal because Vkk=0 by defi-kl k
FIG. 1. The two types of diagrams contributing to the interac-
tion self-energy. The solid lines represent noninteracting Greenfunctions G
effcalculated in the presence of the leads and the wiggly
line is the RPA potential.V . MOLDOVEANU AND B. TANATAR PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-4nition. Another useful quantity is the occupation number of
the system, which is computed as usual from the lesserGreen function /H20849iruns over all sites of the quantum dot /H20850,
N=−i
2/H9266/H20858
i/H20885
−2tL2tL
dEGii/H11021/H20849E/H20850=/H20885
−2tL2tL
dEN /H20849E/H20850, /H2084927/H20850
where N/H20849E/H20850is the density of states.
We end this section with some comments about the ex-
pected range of validity for the RPA approach presentedhere. As pointed out by Henrickson et al. ,
44first-order self-
consistent calculations break down when the interactionstrength exceeds the hopping constant on leads. This is be-cause in this range, elementary excitations are not capturedby the perturbative approach. In a very recent work,
46the
density of states for interacting electrons in graphene wascalculated within the RPA and no plasmonic excitations werereported. In the numerical simulations presented in Sec. III,the interaction strength is always much smaller than the hop-ping constant of the leads.
III. APPLICATION TO DEPHASING IN T-SHAPED
INTERFEROMETERS
In this section, we use the above RPA-Keldysh scheme to
study the transport properties of a quantum wire with a side-coupled quantum dot. The specific system we shall study is aone-dimensional quantum dot having two sites, one of whichis coupled to a single channel lead. In order to compare theeffects of interdot and intradot interactions, we consider alsotwo such interferometers that are Coulomb coupled whenplaced close to one another /H20849see the sketch in Fig. 2/H20850. The
on-site energies of the dots are denoted by /H9255
m/H20849upper inter-
ferometer /H20850and/H9261m/H20849lower interferometer /H20850,m=1,2, and the
hopping constant between the dots and the leads is denotedby
/H9270. The sites of the leads where the dots are coupled are
characterized by the on-site energy /H92550and/H92610. The Hamil-
tonian of the system then reads as
Hcen=/H20858
m=1,2/H20851/H20849/H9255m+Vg/H20850am†am+/H20849/H9261m+Vg/H11032/H20850bm†bm/H20852+tD/H20849a1†a2
+b1†b2+ H.c. /H20850+/H9270/H20849a0†a1+b0†b1+ H.c. /H20850
Hint=U
2/H20858
l,m,l/HS11005mnˆlnˆm
rl−rm, /H2084928/H20850Htun=tLa0†/H20849d0/H9251+d0/H9252/H20850+tLb0†/H20849d0/H9254+d0/H9253/H20850+ H.c. /H2084929/H20850
The annihilation and creation operators for the upper and
lower interferometer are denoted by am,am†andbm,bm†. Also,
we use the notations a0,a0†andb0,b0†for the operators asso-
ciated with the two sites on the leads where the dots areattached. The hopping constant between the sites of the dotsist
D, while tLis the hopping energy between the leads and
the central region. 0 /H9263,/H9263=/H9251,..,/H9253is the first site of the lead nu
that is attached to the central region. We take /H9255m=/H9261mto be
the energy reference. VgandVg/H11032simulate gate potentials ap-
plied on the dots. The last term in the Hamiltonian containsboth the interdot and the intradot interactions between thetwo dots. H
tunis the tunneling term between the leads and the
system. We can also include the interaction between the dotand the neighboring site of the lead but this is not essentialfor our discussion. The bias, the energy, the hopping con-stants on the leads, the coupling and interaction strengths,and the gate potential will be expressed in terms of the hop-ping energy t
Dof the dots, which is chosen as the energy
unit. The hopping energy on leads is tL=2tD, leading to a
bandwidth of the leads W=8tD/H20849recall that the spectrum of
the semi-infinite one-dimensional lead is /H20851−2tL,2tL/H20852/H20850. The
numerical simulations were performed in the very low-temperature regime kT=10
−4.
A finite bias is applied on the leads, i.e., V=/H9262/H9251−/H9262/H9252and
V/H11032=/H9262/H9253−/H9262/H9254, where /H9262/H9251,...,/H9262/H9254are the chemical potential of
the semi-infinite leads. We apply the bias in a symmetric waywith respect to zero, that is,
/H9262/H9251,/H9252=/H11006V/2 and /H9262/H9253,/H9254=/H11006V/H11032/2.
All the curves we present below were obtained by using asuitable grid for the energy range in the integrals in order toobtain stable results. Typically, one needs 1500 points in therange /H20851−4t
L:4tL/H20852. The main care here is to take properly into
account the very sharp peaks of the Green functions of thenoninteracting system.
We first look at the role of the intradot interaction and
consider only one interferometer. In Figs. 3/H20849a/H20850and3/H20849b/H20850, one
finds the first and the second Fano line shapes of the currentas a function of the gate potential V
g. The bias is fixed /H20849V
=0.2 /H20850and for the interaction strength we choose U
=0.1,0.2,0.3. We present the two peaks in separate plots inorder to better discern the dephasing effect. The Fano patternof the current as a function of the gate potential applied onthe lateral dot originates in the interference between elec-tronic waves freely passing through the wire /H20849forming the
so-called background signal /H20850and waves that are scattered at
least once at the side-coupled dot /H20849the resonant contribution /H20850.
It is clear that in the interacting case, both the amplitude andthe shape of the asymmetric Fano line change. A small re-duction of the peak is seen but the main differences appear inthe region of the Fano dip. At U=0.1, the dip is pushed
above the noninteracting one, but it almost disappears at U
=0.2. When further increasing the interaction to U=0.3, the
first Fano dip is recovered and a local maximum develops onits left side. In contrast, the second dip is even more dam-aged. Below, we shall discuss these features in more detail.Figure 3/H20849c/H20850shows the density of states in the dot as a func-
tion of energy and gate potential in the case U=0.2. It is
clear that at resonances /H20849i.e., at V
g/H11011−1 and Vg/H110111/H20850, the dotG
G
GG
GGαβ
γ δεεε
0120
1
2
λλλ
FIG. 2. Schematic of two T-shaped interferometers. The nota-
tions are explained in the text.COULOMB EFECTS IN OPEN QUANTUM DOTS WITHIN … PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-5loses one electron as the localized states enter the bias win-
dow /H20851−0.1:0.1 /H20852. This is why the two traces in Fig. 3/H20849c/H20850dis-
appear at energies E/H110220.1. We have also performed numeri-
cal simulations for three- and four-site quantum dots andqualitatively obtained the same results.
We emphasize that a similar dephasing effect was ob-
tained in our previous work,
27but there the effect was en-
tirely due to the Coulomb interaction between a single-sitedot embedded in an Aharonov–Bohm ring and a nearby de-tector. Here, it is the intradot interaction that leads to deco-herence.We extend our analysis by showing in Figs. 4/H20849a/H20850and4/H20849b/H20850
the current through the interferometer for fixed bias V=0.1
and different signs of the Fano parameter. More precisely,the Fano parameter qis positive /H20849i.e., the dip is located to the
left side of the peak /H20850if the on-site energy of the contact site
/H9255
0=−0.75 and negative if /H92550=0.75. We plot also the separate
contributions of each term in the current formula /H20851Eq. /H2084910/H20850/H20852.
The second term gives just a “bump” around the Fano reso-nance and we shall denote this contribution by J
ias it is
entirely due to the electron-electron interaction. The asym-metric shape of the resonance is given by the first term,which is proportional to the difference of the Fermi func-tions. Since in the noninteracting case this term is related tothe Landauer formula for the conductance, we use the nota-tionJ
c.
In both Figs. 4/H20849a/H20850and4/H20849b/H20850, a reduction of the Fano line
amplitude is noticed in the presence of electron-electron in-teraction. The line shapes move to the right, which is essen-tially due to a Hartree-type shift from the interaction self-energy. Another observation is that the second Fanoresonance is less affected by the Coulomb interaction, itsshift being also smaller than in the case of the first Fano line.This happens because there is more charge in the side-coupled quantum dot before the first resonant tunneling. It iswell known that at resonance the occupation number Nof the00.050.10.150.2
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Current
Gate potential
00.050.10.150.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4Current
Gate potential
0102030405060
EnergyGate potential
-1.5 -1 -0.5 0 0.5 1 1.5-2-1.5-1-0.500.511.52(b)(a)
(c)
FIG. 3. /H20849Color online /H20850/H20849a/H20850The first and /H20849b/H20850the second Fano line
shapes of the current through the interferometer as a function of thegate potential for different values of the interaction strength: fullline, U=0.3; dashed line, U=0.2; dotted line, U=0.1; long-dashed
line, U=0.0. /H20849c/H20850The density of states in the dot as a function of
energy and gate potential /H20849see the comments in the text /H20850. Other
parameters: V=0.2,
/H9270=0.35, /H9255=−0.75, and tL=1.00.050.10.150.2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Current
Gate potential
00.050.10.150.2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Current
Gate potential (b)(a)
FIG. 4. /H20849Color online /H20850/H20849a/H20850The contribution of the two terms in
the current formula for different signs of the Fano parameter /H20849a/H20850q
/H110210 and /H20849b/H20850q/H110220. Full line, the total current; dashed line, the Lan-
dauer current Jc; long-dashed line, the correction Ji; dotted line, the
noninteracting Fano line. Other parameters: U=0.2 V=0.2, /H9270
=0.35, and tL=1.V . MOLDOVEANU AND B. TANATAR PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-6dot decreases by 1 over a range that roughly equals the reso-
nance linewidth. Now, clearly, the first Fano line correspondsto the transition 2 →1 and the second one develops as the
quantum dot is emptied i.e., 1 →0/H20851see also the density of
states given in Fig. 3/H20849c/H20850/H20852. It is therefore understandable that
the Coulomb effects are weaker on the second resonance.
The above comments apply to both Figs. 4/H20849a/H20850and4/H20849b/H20850.
Now we discuss the details of dephasing. When q/H110210/H20851Fig.
4/H20849a/H20850/H20852, the suppression of the Fano line is mainly due to the
enhanced value of the dip at U=0.2. The reason is evident
when we look at the two contributions to the current. On onehand, J
calready displays a dip that is higher than the nonin-
teracting one, and on the other hand, the correction Jiadds to
the final value of the dip. Note that the Fano peaks are notdrastically affected and that contribution of J
idecreases on
the second resonance.
Turning to Fig. 4/H20849b/H20850in which q/H110220, we observe that the
suppression of the first Fano interference is symmetric, in the
sense that the dip is enhanced and the peak diminishes. It isalso interesting to mention that the contribution due to J
i
differently affects the two types of interference /H20849constructive
or destructive /H20850. In contrast to Fig. 4/H20849a/H20850where the maximum
ofJiis located below the Fano dip, in Fig. 4/H20849b/H20850this point is
rather below the Fano peak. As a consequence, the total peakis higher than J
cand the Fano dip is lower than the one in
Fig. 4/H20849a/H20850. Nevertheless, when comparing to the noninteract-
ing Fano line, we see that a dephasing still exists. For thesecond line shape, the constructive interference /H20849i.e., the
Fano peak /H20850is reduced and the destructive one is not changed.
The correction is again present but it does not change eitherthe peak or the dip and affects rather the middle of the Fanoline.
The above discussion suggests that in the presence of an
intradot interaction, both constructive and destructive Fanointerference are affected and that their sensitivity depends onthe sign of the Fano parameter, that is, on the order in whichthe two types of interference are experienced by the system.Ifq/H110210, the destructive interference appears first and the
interaction effects are predominant. If q/H110220, the Fano peak
amplitude reduces and the Fano dip is less affected.
Now, we discuss the behavior of the interaction self-
energy, which will shed some light on the main processesthat induce decoherence in the system. For this, we have toconsider the various matrix elements of /H9018
I.
In Fig. 5/H20849a/H20850, we give the imaginary part of the retarded
self-energy at the first site as a function of energy and gatepotential /H20851for a better visibility of Fig. 5/H20849a/H20850, we actually plot
−Im/H9018
I,11/H20852. It is evident that at a very small temperature, it
suffices to restrict the energy range to /H20851−0.1:0.1 /H20852, which
equals the bias window /H20849we take /H9262/H9251=0.1 and /H9262/H9252=−0.1 /H20850. One
observes that the main contribution to the imaginary partcorresponds to gate potentials that are located around the tworesonances. We remark that the maxima of the self-energyhave rather equal heights but they are not aligned in energy.Actually, the maximum around the second resonance is notcentered in the bias window. A similar behavior is obtainedfor −Im /H9018
I,22. The imaginary part of the off-diagonal element
/H9018I,12plotted in Fig. 5/H20849b/H20850is much smaller than the diagonal
counterpart. This is expected because for long-range poten-tials, as it is the case here, the exchange diagrams can beneglected with respect to the direct contribution.
47Note that
Im/H9018I,12is both positive and negative.
The imaginary part of the retarded self-energy is related to
the inverse of the quasiparticle lifetime. In our case, there aretwo contributions to the resonance width: one comes from
the leads’ self-energy /H9018
LRand is roughly on the order of
O/H20849/H9270/H208502and the other one is entirely due to the Coulomb re-
pulsion. From the diagrams that correspond to Im /H9018I,11,i t
follows then that the dephasing in the upper interferometer,say, is mainly due to the interaction with at least oneelectron-hole pair that is excited in either one of the twosubsystems. We recall that the creation and destruction of theelectron-hole pairs are inelastic scattering processes.
In what concerns the real part of the interaction self-
energy, it is responsible for the shift of the resonance and themain contribution is given by the Hartree diagram /H20851see Eq.
/H2084926/H20850/H20852. This diagram contains the on-site occupation number
that is constant except at resonance when electrons escapefrom the dot to the side-coupled leads. This behavior is eas-ily checked in Fig. 6/H20849a/H20850. Note that this term is energy inde-
pendent since the involved scattering process is elastic. Forcompleteness, we show in Fig. 6/H20849b/H20850the real part of the ex-
change self-energy /H9018
I,12. Again, it is smaller than Re /H9018I,22.
In the following, we investigate the transport properties of
two identical T-shaped interferometers that are mutuallycoupled via the Coulomb interaction between the side-coupled dots. If not otherwise stated, all interactions /H20849inter-
dot and intradot /H20850are taken into account. We take the same
lead-dot coupling strength on both systems. When the inter-ferometers have the same set of parameters, their Fano lineshapes coincide. In Figs. 7/H20849a/H20850and7/H20849b/H20850, we compare the sec-0.050.0 40.030.020.010
-0.4
-0.2
0
0.2
0.4-1.5-1-0.500.511.50.05
0.04
0.03
0.02
0.01
0Self-energy
Energy
Gate potentialSelf-energy
-5- 2x1 0-5-1.5 x 10-5- 1x1 0-6- 5x1 00-65x1 0-51x1 0
-0.4
-0.2
0
0.2
0.4-1.5-1-0.500.511.5-5-1.5 x 10-5- 1x1 0-6- 5x1 00-65x1 0Self-energy
Energy
Gate potentialSelf-energy
(b)(a)
FIG. 5. /H20849Color online /H20850/H20849a/H20850−Im/H9018I,11Ras a function of energy and
gate potential. The important contribution comes from energies in-side the bias window /H20851−0.1:0,1 /H20852and for gate potentials at which
the Fano resonances appear. /H20849b/H20850Im/H9018
I,12Ras a function of energy and
gate potential. Other parameters: U=0.2, Vsd=0.2, /H9270=0.35, and tL
=1.COULOMB EFECTS IN OPEN QUANTUM DOTS WITHIN … PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-7ond Fano resonance in this case to the similar curve from
Figs. 4/H20849a/H20850and 4/H20849b/H20850. The dephasing is now given by both
intradot and interdot Coulomb interactions. Overall, the Fanolines are similar to the ones obtained for the single interfer-ometer but several differences appear: /H20849i/H20850The Fano lines of
the Coulomb-coupled interferometers are shifted even fur-ther to the right when compared to the single interferometercase. /H20849ii/H20850The constructive interference is more suppressed
than in the single interferometer case. The reason for this isthat when both systems are in the constructive regime, alarge current passes through them and this amplifies thecharge sensing effect.
It is a known fact, both experimentally and theoretically,
that for two Coulomb-coupled systems, the dephasing effectincreases at higher values of the bias.
13,27We have checked
this feature for the two T-shaped interferometers. Below, wepresent numerical results that emphasize a more interestingeffect, namely, the enhancement of dephasing when twolev-
els participate in the quantum interference, at a fixed andrather low bias. From the experimental point of view, thissituation is met when bigger dots are used, which lead to asmaller level spacing, but also for double dots having a smallinterdot coupling. This is the situation we simulate in Figs.8/H20849a/H20850and8/H20849b/H20850by taking the hopping parameter between the
two sites t
D=0.25 /H20849the density of states in this case shows
that there are two levels that enter the bias window when thegate potential is varied /H20850. We take /H9255
0=−/H92610so that the two
systems show line shapes with Fano parameters of differentsign. Figure 8/H20849a/H20850shows the noninteracting Fano lines. Each
system exhibits only one Fano line, and by comparing Fig.8/H20849a/H20850to Fig. 7, one infers that in the two-level case, an addi-
tional shoulder appears in the middle of the Fano line. This isassociated with the entrance of the second level inside thebias window. We notice that the amplitude of the resultingFano line is not twice as large as the one shown in Fig. 7,
which means that the contributions of the two levels to thecurrent do no simply add. This suggests that a more compli-cated interference takes place in the system. Actually, eachlevel causes an interference with the background signal, butthe nature of this interference can be different /H20849purely con-
structive, purely destructive, or intermediate /H20850. When
electron-electron interactions are included in the calculation,a clear reduction of the constructive interference appears inthe current through the lower interferometer. The additionalshoulder is more difficult to discern. The upper interferom-eter shows, in turn, a Fano line whose dip is damaged. Webelieve that this dephasing effect for Coulomb-coupledT-shaped interferometers should be easily observed in ex-periments. One only has to compare the Fano line shapes ofa single interferometer and of the double interferometer. Letus stress again that this effect does not require a large bias.
The analysis we made so far shows that both intradot and
interdot Coulomb interactions cause a reduction in the Fanointerference. Since the intradot interaction is bigger than theinterdot repulsion, an important point would be to check ifby placing a quantum dot near an interacting interferometerthe controlled dephasing effects can still be discerned. Tothis end, we have performed numerical simulations for theinterferometer with a two-site side-coupled dot, which is0.020.060.10.1 40.180.22
-0.4-0.200.20.4-1.5-1-0.500.511.50.020.060.10.140.180.22Self-energy
Energy Gate potentialSelf-energy
-0.0 3-0.02-0.0100.010.020.03
-0.4-0.200.20.4-1-0.500.51-0.03-0.02-0.0100.010.020.03Self-energy
EnergyGate potentialSelf-energy
(b)(a)
FIG. 6. /H20849Color online /H20850/H20849a/H20850Re/H9018I,11Rand /H20849b/H20850Re/H9018I,12Ras a function
of energy and gate potential for the single interferometer. Otherparameters: U=0.2, V=0.2,
/H9270=0.35, and tL=1.00.050.10.150.2
0.6 0.8 1 1.2 1.4Current
Gate potential
00.050.10.150.2
0.6 0.8 1 1.2 1.4Current
Gate potential (b)(a)
FIG. 7. /H20849Color online /H20850The cumulative effect of the interdot and
intradot interaction can be noticed in the current through the upperinterferometer /H20849full line /H20850when comparing to the current for a single
interferometer /H20849dashed line /H20850. The dotted line represents the nonin-
teracting Fano line. The parameters are as in Fig. 4.V . MOLDOVEANU AND B. TANATAR PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-8Coulomb coupled to an additional single-site dot attached to
biased leads. Due to the charge sensing effect, one expects tosee changes in the current through the second dot when theFano resonance develops in the interferometer. Conversely,the Fano line itself should be modified as the electrons are“detected” by the nearby quantum dot. In Fig. 9/H20849a/H20850, we plot
the current through the interferometer as a function of thegate potential for two values of the bias applied on the de-tector. For comparison, we also show the current in the ab-sence of the detector /H20849the dotted line /H20850. It is clear that at bias
V=1.0, the amplitude of the Fano lines is reduced, both from
the peak and the dip. We remark also that at V=2.0, it is only
the Fano peak that decreases. Figure 9/H20849b/H20850confirms that the
single-site quantum dot detects the passage of electronsthrough the side-coupled dot. Away from resonances, thecurrent does not depend on V
g. This result suggests that con-
trolled dephasing can be also put into evidence for T-shapeinterferometers.
One of the advantages of the Keldysh formalism is that it
allows one to investigate the nonlinear transport regime, thatis, to study the dependence of the current on the applied bias.We show in Fig. 10/H20849a/H20850the behavior of the current through a
T-shaped structure with four side-coupled sites as a functionof the applied bias for different values of the interdot inter-action. The gate potential on the side-coupled sites is fixed to
V
g=0 and also the on-site energy of the contact site is /H92800
=0. The bias was varied in a symmetric way as follows: westart with a negative bias V=−4 by choosing
/H9262/H9251=−2 and
/H9262/H9252=2. Then, we simultaneously increase the chemical poten-
tial of the left lead and decrease the chemical potential of theright lead until the bias changes sign and reaches the finalvalue V=4.
ForU=0, the current displays the well-known steplike
structure. The jumps between two steps correspond to achange in the number of states located inside the bias win-dow. In the absence of the electron-electron interactions, thespectrum of the central region is symmetric with respect tozero. Consequently, the states whose energies differ just by asign simultaneously align to the positive /H20849negative /H20850chemical
potential of the leads, and at each passage between currentsteps, two more states enter or leave the bias window. In theinteracting case, one notices the appearance of additionalsteps /H20851see, for example, the step around V=2 shown in the
inset of Fig. 10/H20849a/H20850/H20852. This happens because the Coulomb in-
teraction pushes up the spectrum breaking its symmetry andthen two levels cannot enter or leave the bias window simul-00.050.10.150.2
-1.5 -1 -0.5 0 0.5 1 1.5Current
Gate potential
00.050.10.150.2
-1.5 -1 -0.5 0 0.5 1 1.5Current
Gate potential (b)(a)
FIG. 8. /H20849Color online /H20850The two-level Fano effect for two
T-shaped interferometers. By setting the hopping constant inside thedots to t
D=0.25, two levels can be brought inside the bias window
of the leads by varying the gate potential. /H20849a/H20850Noninteracting case
U=0.0 and /H20849b/H20850interacting case U=0.2. The bias is V=V/H11032=0.2. The
full line is the current through the upper interferometer; the dashedline represents the current through the lower interferometer.0.060.080.10.120.140.16
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Current
Gate potential
0.80.810.820.830.840.850.860.870.880.890.9
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Current
Gate potential (b)(a)
FIG. 9. /H20849Color online /H20850/H20849a/H20850The Fano interference in a T-shaped
interferometer is further reduced when a nearby charge detector issubjected to a finite bias V
/H11032. Full line, V/H11032=2; dashed line, V/H11032=1.
The dotted line represents the interacting Fano line in the absenceof the detector. /H20849b/H20850The current across the detector as a function of
the gate potential applied on the dot. The charge sensing effect leadsto changes in the detector current around each Fano resonance inthe side-coupled dot. Other parameters: U=0.2, V=0.1,
/H9270=0.35,
andtL=1.COULOMB EFECTS IN OPEN QUANTUM DOTS WITHIN … PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-9taneously. Clearly, the length of the steps increases with the
interaction strength, a feature that was also noticed by Hen-rickson et al.
44/H20849see Fig. 5 of Ref. 44/H20850within a self-consistent
approach to nonlinear transport in interacting quantum dots.Note that the interaction does not affect the symmetry of thecurrent curve with respect to the bias.
The dependence of the occupation number in the dot as a
function of bias is shown in Fig. 10/H20849b/H20850and offers a better
understating of the changes induced in the current curves by
the Coulomb interaction. If the bias window covers the entirespectrum of the system, the occupation number N/H110112.5,
which corresponds to nearly half-filling. The highest energylevel is the first one left above the bias window as the bias
window shrinks, while the lowest energy level is still activefor transport, as it is being pushed upward by the interaction.In this regime, the occupation number decreases. Then, thelowest level passes below the bias window and it can be fully
occupied, which leads to an increase of the charge accumu-lated in the dot and a decrease in the current. Note that whenthe interaction increases and the bias V=0, the occupation
number goes below 2.5 because the energy of the middlelevel is positive. We have also checked the current conserva-tion /H20849i.e., the identity J
/H9251=−J/H9252/H20850. The results presented aboveare consistent with previous self-consistent calculations of
Henrickson et al.44and therefore show the reliability of our
method. On the other hand, the RPA approach taken here isable to capture nontrivial effects due to the inelastic effectsthat cannot be reproduced by mean-field approximations.
We end with a discussion about the possible improve-
ments of the present method. It is clear that one could per-form self-consistent calculations by defining the polarizationoperator in terms of interacting Green functions. This proce-dure leads to longer times in the numerical simulations espe-cially for large number of sites. Also, the self-consistencycondition should be carefully checked at any value of therelevant parameters /H20849interaction strength or the tunneling
constant between the dot and the lead /H20850. In the self-consistent
scheme, the self-energies will be directly related to the inter-acting Green functions and the position of the poles is ex-pected to be slightly different from the noninteracting casedue to the Hartree shift. Nevertheless, for the few-level sys-tem we are considering here, this does not lead to qualitativechanges in the numerical results.
IV. CONCLUSIONS
We have implemented the random-phase approximation
in the framework of the nonequilibrium Keldysh Green’sfunction formulation for electronic transport in many-levelquantum dots. The starting point is the polarization operator,which in the present approach is built from noninteractingGreen functions. The calculation of interaction self-energytakes into account all scattering processes that involveelectron-hole pairs and also the contribution of the exchangediagram. This approach has therefore a clear advantage overthe second-order perturbation theory in the interactionstrength used previously in Ref. 27and could also be used as
an alternative to the equation-of-motion approach or to themean-field approximation.
As a first application of this method, we have considered
the interplay between the intradot and interdot interactions inelectronic transport in Coulomb-coupled T-shaped interfer-ometers. For a single interferometer, the numerical calcula-tions show that the intradot electron-electron interaction it-self suppresses the quantum interference, even in the lowbias regime. The various contributions to the interaction self-energy were analyzed as well as the dependence on the biasand gate potential.
In the presence of a second T-shaped interferometer or of
a charge detector coupled to leads, further dephasing appearsdue to the charge sensing effect. We show that the dephasingincreases when the Fano interference implies two levels ofthe dot, which are coupled to the continuum. The high tun-ability of side-coupled quantum dots should allow the obser-vation of our theoretical predictions in future experiments.
ACKNOWLEDGMENTS
V .M. acknowledges financial support by TUBITAK-
BIDEB and by CEEX Grant No. D11-45/2005. B.T. is sup-ported by TUBITAK /H20849Grant No. 106T052 /H20850and TUBA.-0.8-0.6-0.4-0.200.20.40.60.8
-4 -3 -2 -1 0 1 2 3 4Current
Bias0.20.250.30.350.40.450.50.550.6
1 1.5 2 2.5 3
22.12.22.32.42.52.62.7
-4 -3 -2 -1 0 1 2 3 4Occupation number
Bias(b)(a)
FIG. 10. /H20849Color online /H20850/H20849a/H20850Current vs bias for different interac-
tion strengths: dotted line, U=0; full line, U=0.15; dashed line,
U=0.25. The interaction leads to the formation of additional steps
when compared to the noninteracting case. The inset shows theformation of a new step in the bias range /H208511:3/H20852./H20849b/H20850The occupation
number of the dot as a function of bias for different interactionstrengths. Full line, U=0.15; dashed line, U=0.05; dotted line, U
=0.01. Other parameters:
/H9270=1 and tL=0.5.V . MOLDOVEANU AND B. TANATAR PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-101Y . Imry, Introduction to Mesoscopic Physics /H20849Oxford University
Press, Oxford, 2002 /H20850.
2Semiconductor Spintronics and Quantum Computation , edited
by D. D. Awschalom, D. Loss, and N. Samarth /H20849Springer, Berlin,
2002 /H20850.
3R. I. Shechter, Zh. Eksp. Teor. Fiz. 63, 1410 /H208491972 /H20850/H20851Sov. Phys.
JETP 36, 747 /H208491973 /H20850/H20852.
4E. Ben-Jacob and Y . Gefen, Phys. Lett. 108A , 289 /H208491985 /H20850.
5D. V . Averin and K. K. Likharev, J. Low Temp. Phys. 62, 345
/H208491986 /H20850.
6H. van Houten, C. W. J. Beenakker, and A. A. M. Staring, in
Single Charge Tunneling , NATO Advanced Studies Institutes,
Series B: Physics V ol. 294, edited by H. Grabert and M. H.Devoret /H20849Plenum, New York, 1992 /H20850.
7D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-
Magder, U. Meirav, and M. Kastner, Nature /H20849London /H20850391, 156
/H208491998 /H20850.
8M. Field, C. G. Smith, M. Pepper, D. A. Ritchie, J. E. F. Frost,
G. A. C. Jones, and D. G. Hasko, Phys. Rev. Lett. 70, 1311
/H208491993 /H20850.
9A. C. Johnson, C. M. Marcus, M. P. Hanson, and A. C. Gossard,
Phys. Rev. Lett. 93, 106803 /H208492004 /H20850.
10B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitskii, J. Phys. C
15, 7367 /H208491982 /H20850.
11A. Stern, Y . Aharonov, and Y . Imry, Phys. Rev. A 41, 3436
/H208491990 /H20850.
12S. A. Gurvitz, arXiv:quant-ph/9607029 /H20849unpublished /H20850; Phys.
Rev. B 57, 6602 /H208491998 /H20850.
13E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V . Umansky,
Nature /H20849London /H20850391, 871 /H208491998 /H20850.
14T. Fujisawa, D. G. Austing, Y . Tokura, Y . Hirayama, and S.
Tarucha, J. Phys.: Condens. Matter 15, R1395 /H208492003 /H20850.
15U. Sivan, Y . Imry, and A. G. Aronov, Europhys. Lett. 28,1 1 5
/H208491994 /H20850.
16B. L. Altshuler, Y . Gefen, A. Kamenev, and L. S. Levitov, Phys.
Rev. Lett. 78, 2803 /H208491997 /H20850.
17C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J.
Phys. C 4, 916 /H208491971 /H20850.
18H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and
Optics of Semiconductors /H20849Springer, Berlin, 1996 /H20850.
19M. Wagner, Phys. Rev. B 44, 6104 /H208491991 /H20850.
20D. N. Zubarev, Sov. Phys. Usp. 3, 320 /H208491960 /H20850.
21C. Lacroix, J. Phys. F: Met. Phys. 11, 2389 /H208491981 /H20850.
22V . Kashcheyevs, A. Aharony, and O. Entin-Wohlman, Phys. Rev.
B73, 125338 /H208492006 /H20850.23Y . Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. 66,
3048 /H208491991 /H20850.
24B. R. Bulka and P. Stefanski, Phys. Rev. Lett. 86, 5128 /H208492001 /H20850.
25I. V . Dinu, M. Ţolea, and A. Aldea, Phys. Rev. B 76, 113302
/H208492007 /H20850.
26A. Silva and S. Levit, Phys. Rev. B 63, 201309 /H20849R/H20850/H208492001 /H20850.
27V . Moldoveanu, M. Ţolea, and B. Tanatar, Phys. Rev. B 75,
045309 /H208492007 /H20850.
28J. König, Y . Gefen, and A. Silva, Phys. Rev. Lett. 94, 179701
/H208492005 /H20850.
29S. V . Faleev and M. I. Stockman, Phys. Rev. B 66, 085318
/H208492002 /H20850;62, 16707 /H208492000 /H20850.
30U. Wulf, P. N. Racec, and E. R. Racec, Phys. Rev. B 75, 075320
/H208492007 /H20850.
31K. Kobayashi, H. Aikawa, A. Sano, S. Katsumoto, and Y . Iye,
Phys. Rev. B 70, 035319 /H208492004 /H20850.
32G. Hackenbroich, W. D. Heiss, and H. A. Weidenmüller, Phys.
Rev. Lett. 79, 127 /H208491997 /H20850.
33R. Berkovits, F. von Oppen, and Y . Gefen, Phys. Rev. Lett. 94,
076802 /H208492005 /H20850.
34M. Sato, H. Aikawa, K. Kobayashi, S. Katsumoto, and Y . Iye,
Phys. Rev. Lett. 95, 066801 /H208492005 /H20850.
35Tae-Suk Kim and S. Hershfield, Phys. Rev. B 63, 245326
/H208492001 /H20850.
36R. Franco, M. S. Figueira, and E. V . Anda, Phys. Rev. B 67,
155301 /H208492003 /H20850.
37P. S. Cornaglia and D. R. Grempel, Phys. Rev. B 71, 075305
/H208492005 /H20850.
38P. A. Orellana, F. Dominguez-Adame, I. Gómez, and M. L.
Ladrón de Guevara, Phys. Rev. B 67, 085321 /H208492003 /H20850.
39J. König and Y . Gefen, Phys. Rev. Lett. 86, 3855 /H208492001 /H20850.
40A. Silva and S. Levit, Europhys. Lett. 62, 103 /H208492003 /H20850.
41A.-P. Jauho, N. S. Wingreen, and Y . Meir, Phys. Rev. B 50, 5528
/H208491994 /H20850.
42H. D. Cornean, H. Neidhardt, and V . A. Zagrebnov,
arXiv:0708.3931 /H20849unpublished /H20850.
43V . Moldoveanu, V . Gudmundsson, and A. Manolescu, Phys. Rev.
B76, 085330 /H208492007 /H20850.
44L. E. Henrickson, A. J. Glick, G. W. Bryant, and D. F. Barbe,
Phys. Rev. B 50, 4482 /H208491994 /H20850.
45H. M. Pastawski, L. E. F. Foa Torres, and E. Medina, Chem.
Phys. 281, 257 /H208492002 /H20850.
46Xin-Zhong Yan and C. S. Ting, Phys. Rev. B 76, 155401 /H208492007 /H20850.
47A. L. Fetter and J. D. Walecka, Quantum Theory of Many-
Particle Systems /H20849McGraw-Hill, New York, 1971 /H20850.COULOMB EFECTS IN OPEN QUANTUM DOTS WITHIN … PHYSICAL REVIEW B 77, 195302 /H208492008 /H20850
195302-11 |
PhysRevB.97.155434.pdf | PHYSICAL REVIEW B 97, 155434 (2018)
Quantum ring with the Rashba spin-orbit interaction in the regime of strong light-matter coupling
V . K. Kozin,1,2I. V . Iorsh,2O. V . Kibis,3,1,*and I. A. Shelykh1,2
1Science Institute, University of Iceland, Dunhagi 3, IS-107 Reykjavik, Iceland
2ITMO University, Saint Petersburg 197101, Russia
3Department of Applied and Theoretical Physics, Novosibirsk State Technical University, Karl Marx Avenue 20, Novosibirsk 630073, Russia
(Received 19 February 2018; revised manuscript received 5 April 2018; published 26 April 2018)
We developed the theory of electronic properties of semiconductor quantum rings with the Rashba spin-orbit
interaction irradiated by an off-resonant high-frequency electromagnetic field (dressing field). Within the Floquettheory of periodically driven quantum systems, it is demonstrated that the dressing field drastically modifiesall electronic characteristics of the rings, including spin-orbit coupling, effective electron mass, and opticalresponse. In particular, the present effect paves the way to controlling the spin polarization of electrons with lightin prospective ring-shaped spintronic devices.
DOI: 10.1103/PhysRevB.97.155434
I. INTRODUCTION
The rapidly developing field of spintronics deals with spin-
related phenomena in mesoscopic transport [ 1–5]. Generally,
the spins of individual carriers can be controlled either byapplication of an external magnetic field or via a changeof the strength of the spin-orbit interaction (SOI) in thesystem. The second approach forms the basis of so-callednonmagnetic spintronics, which has attracted an enormousamount of interest in the scientific community. In particular,two mechanisms of the SOI are relevant for semiconductorstructures: the Dresselhaus SOI [ 6], which was caused by the
inversion asymmetry of the crystal lattice, and the Rashba SOI[7–10], which originated from the inversion asymmetry of the
structure as a whole. The latter mechanism is of specific interestfor spintronic applications since it becomes dominant in con-ventionally used InAs/GaSb-, AlSb/InAs-, and GaAs/GaAlAs-based nanostructures [ 11–13], and it can be easily tuned by an
external gate voltage [ 14–16]. Recently, the alternative way of
tuning SOI by purely optical methods was developed [ 17–19].
It is based on the regime of strong light-matter couplingwhen the system “electron +electromagnetic field” cannot be
divided into weakly interacting optical and electronic subsys-tems. As a consequence, the hybrid electron-field object—theso-called “electron dressed by electromagnetic field” (dressedelectron)—appears as an elementary quasiparticle [ 20,21]. The
physical properties of dressed electrons can differ sufficientlyfrom their “bare” counterparts, as was demonstrated for a widevariety of condensed-matter structures, including bulk semi-conductors [ 22–24], quantum wells [ 25–29], quantum rings
[30–35], graphene [ 36–44], topological insulators [ 45], etc.
From the viewpoint of spintronic applications, it is cruciallyimportant that the SOI strength can be modified by laserirradiation [ 19] since this allows direct optical tuning of the
spin relaxation time in a two-dimensional (2D) electron gas
*oleg.kibis@nstu.ru[17], and therefore it paves the way to optically controlled
spintronic devices [ 18].
Although the first ferromagnetic spintronic device (the
Datta-Das spin transistor [ 46,47]) has been realized exper-
imentally, its technological production remains challengingdue to the difficulties with the efficient spin injection fromferromagnetic contacts. Therefore, the design of nonmagneticspintronic devices, which do not require the presence offerromagnetic elements, is still an actual problem. As a possibleway to solve the problem, it was proposed to use semiconductorquantum rings (QRs) with the Rashba SOI, which inducesthe phase shift between spin waves propagating in the clock-wise and counterclockwise directions. In turn, this results inthe large conductance modulation due to the interference oft h es p i nw a v e s[ 48]. As a consequence, the physical basis
of various QR-based nonferromagnetic spintronic devices—including spin transistors, spin filters, and quantum splitters—appears [ 49–64]. In the aforementioned previous studies on
the subject, the spin properties of QRs were assumed to becontrolled by gate voltage. As to the optical methods of thespin control of QRs, they have escaped attention up to now.The present theoretical research aims partially to fill this gapin the spintronics of QRs.
The paper is organized as follows. In Sec. II, we derived the
effective Hamiltonian of the irradiated QR with the Rashbaspin-orbit interaction within the Floquet theory of periodicallydriven quantum systems. In Sec. III, the elaborated theory
is applied to analyze spin and optical characteristics of theirradiated QR. Section IVcontains our conclusions.
II. MODEL
To describe an irradiated QR (see Fig. 1), we have to
start from the Hamiltonian describing an irradiated two-dimensional (2D) electron system with the Rashba spin-orbitinteraction [ 17]
ˆH
2D=(ˆp−eA)2
2m+α[σx(ˆpy−eAy)−σy(ˆpx−eAx)],
(1)
2469-9950/2018/97(15)/155434(6) 155434-1 ©2018 American Physical SocietyKOZIN, IORSH, KIBIS, AND SHELYKH PHYSICAL REVIEW B 97, 155434 (2018)
EM wave
Ξ
E0
Relectron
QRxzy
FIG. 1. Sketch of the system under consideration: The quantum
ring (QR) with the radius Rirradiated by a linearly polarized
electromagnetic (EM) wave with the electric-field amplitude E0.
The electron spin (the dark blue arrow) is directed along the local
quantization axis (the dashed blue line) with the spin angle ξ.
where ˆp=(ˆpx,ˆpy) is the operator of electron momentum, m
is the effective electron mass, eis the electron charge, αis
the Rashba spin-orbit coupling constant, σx,y,z are the Pauli
matrices, A=(Ax,Ay)=([E0/ω]cosωt,0) is the vector po-
tential of a linearly polarized electromagnetic wave (dressingfield) in the 2D plane, E
0is the electric field amplitude of the
wave, and ωis the wave frequency, which is assumed to be
far from resonant electron frequencies. Applying the standardapproach [ 66] to transform the 2D electron system into the
one-dimensional (1D) ring-shaped one, we arrive from the 2DHamiltonian ( 1) at the Hamiltonian of an irradiated QR,
ˆH
QR=ˆH/prime+/bracketleftBigg2/summationdisplay
n=1ˆVneinωt+H.c./bracketrightBigg
, (2)
where the stationary part,
ˆH/prime=ˆl2
z
2mR2+α
R/bracketleftBig
σρˆlz−i¯hσϕ
2/bracketrightBig
+e2E2
0
4mω2, (3)
is the Hamiltonian of the unperturbed QR up to a field-induced
constant shift of energy,
ˆV1=eE0
2mRω/parenleftBig
sinϕˆlz−i¯hcosϕ
2/parenrightBig
+αeE 0
2ωσy, (4)
ˆV2=e2E2
0
8mω2, (5)
is the periodic part with the two harmonics originated from the
irradiation, Ris the QR radius, ˆlz=−i¯h∂/ ∂ϕ is the operator
of angular momentum along the zaxis,ϕis the polar angle
of an electron in the QR, and σρ=cosϕσx+sinϕσyand
σϕ=− sinϕσx+cosϕσyare the Pauli matrices written in
polar coordinates. Applying the conventional Floquet-Magnusapproach [ 67–69] to renormalize the Hamiltonian of an irradi-
ated QR and restricting the consideration by the leading termsin the high-frequency limit, we can reduce the time-dependentHamiltonian ( 2) to the effective time-independent one,
ˆH=ˆH
/prime+2/summationdisplay
n=1[ˆVn,ˆV†
n]
¯hnω+2/summationdisplay
n=1[[ˆVn,ˆH/prime],ˆV†
n]+H.c.
2(¯hnω)2.(6)
Substituting Eqs. ( 3)–(5) into Eq. ( 6), one can rewrite the
effective Hamiltonian ( 6)a s
ˆH=ˆH0+ˆV, (7)where
ˆH0=ˆl2
z
2m∗R2+α
R/bracketleftBig
σρˆlz−i¯hσϕ
2/bracketrightBig
−/parenleftbiggeE0α
Rω2/parenrightbigg2ˆlzσz
m¯h
+e2E2
0
4mω2+1
2m/parenleftbigg¯heE 0
4mR2ω2/parenrightbigg2
, (8)
ˆV=/bracketleftbigg3
16γ2
1cos 2ϕ−γ2
1γ2/parenleftbigg
γ2
2−1
4/parenrightbigg
iσxsinϕ/bracketrightbigg¯h2
2mR2
+/bracketleftbiggiγ2
1sin 2ϕ
2−2γ2
1γ2/parenleftbigg
γ2
2−1
4/parenrightbigg
σxcosϕ/bracketrightbigg¯hˆlz
2mR2
+γ2
1cos 2ϕ
8mR2ˆl2
z, (9)
where
m∗=m
1+3(eE0/2mRω2)2(10)
is the effective electron mass renormalized by the irradiation,
γ1=|e|E0/(mRω2) is the dimensionless parameter describing
the strength of electron-field coupling, and γ2=mRα/ ¯his
the dimensionless parameter describing the strength of Rashbaspin-orbit coupling. As expected, the Hamiltonian ( 7) exactly
coincides with the Hamiltonian of an unirradiated QR [ 66]i n
the absence of the field ( E
0=0).
It should be noted that all effects that originated from
the direct spin interaction with the magnetic component ofthe dressing field (particularly, the Zeeman effect and theAharonov-Bohm effect) are relativistically negligible since theamplitude of magnetic induction of the field, B
0=E0/c,i s
very small for reasonable field intensities. Therefore, they areomitted in the developed theory. We also neglected effects thatarose from overlying electronic modes, assuming the typicaldistance between transverse electronic minibands (tens of meVfor state-of-the-art QRs [ 65]) to be sufficiently larger than the
photon and electron energies under consideration.
III. RESULTS AND DISCUSSION
To consider the Schrödinger problem with the effective
Hamiltonian ( 7), let us start from its part ( 8). Two exact
eigenstates of the Hamiltonian ( 8) can be written as
/Psi11(ϕ)=eijzϕ/parenleftbiggcos(ξ/2)e−iϕ/2
−sin(ξ/2)eiϕ/2/parenrightbigg
(11)
and
/Psi12(ϕ)=eijzϕ/parenleftbiggsin(ξ/2)e−iϕ/2
cos(ξ/2)eiϕ/2/parenrightbigg
, (12)
where
ξ=arctan/bracketleftbigg2m∗Rα/ ¯h
2(m∗/m)(eE0α/ω2¯h)2+1/bracketrightbigg
(13)
is the angle between the local spin quantization axis and the
zaxis (see Fig. 1). It follows from single-valuedness of the
eigenstates, /Psi11,2(ϕ)=/Psi11,2(ϕ+2π), that the zcomponent of
total angular momentum of the electron, jz, must satisfy the
condition, jz=λn+1/2, where n=0,1,2,... is the orbital
quantum number corresponding to the electron rotation in the
155434-2QUANTUM RING WITH THE RASHBA SPIN-ORBIT … PHYSICAL REVIEW B 97, 155434 (2018)
QR, and the sign λ=± describes the direction of the rotation
(counterclockwise/clockwise). Omitting constant terms thatonly shift the zero energy, one can write the electron energyspectrum of the eigenstates ( 11) and ( 12)a s
ε
s
λn=¯h2
2m∗R2/parenleftbigg
λn+1
2/parenrightbigg2
+¯h2
2m∗R2/vextendsingle/vextendsingle/vextendsingle/vextendsingleλn+1
2/vextendsingle/vextendsingle/vextendsingle/vextendsingles
×/radicaltp/radicalvertex/radicalvertex/radicalbt/bracketleftBigg
2/parenleftbiggm∗
m/parenrightbigg/parenleftbiggeE0α
¯hω2/parenrightbigg2
+1/bracketrightBigg2
+/bracketleftbigg2αm∗R
¯h/bracketrightbigg2
,(14)
where s=± 1 is the quantum number describing the spin
direction along the local quantization axis (see Fig. 1), and the
spins=+ 1 corresponds to the greater energy ( 14). Within
the conventional notation [ 70] based on the three quantum
numbers, |n,λ,s/angbracketright, the eigenstates ( 11) and ( 12) can be written
as
|n,+,−1/angbracketright=einϕ/parenleftbiggcos(ξ/2)
−sin(ξ/2)eiϕ/parenrightbigg
, (15)
|n,+,+1/angbracketright=einϕ/parenleftbiggsin(ξ/2)
cos(ξ/2)eiϕ/parenrightbigg
, (16)
|n,−,+1/angbracketright=e−inϕ/parenleftbiggcos(ξ/2)
−sin(ξ/2)eiϕ/parenrightbigg
, (17)
|n,−,−1/angbracketright=e−inϕ/parenleftbiggsin(ξ/2)
cos(ξ/2)eiϕ/parenrightbigg
(18)
forn=1,2,3,... and
|0,+,−1/angbracketright=/parenleftbiggcos(ξ/2)
−sin(ξ/2)eiϕ/parenrightbigg
, (19)
|0,+,+1/angbracketright=/parenleftbiggsin(ξ/2)
cos(ξ/2)eiϕ/parenrightbigg
(20)
forn=0. It follows from Eq. ( 14), in particular, that εs
−n=
εs
n−1. This means that the states |n,−,s/angbracketrightand|n−1,+,s/angbracketrightare
degenerated.
The eigenstates and eigenenergies ( 11)–(20) can be easily
verified by direct substitution into the Schrödinger equa-tion with the Hamiltonian ( 8). However, the total effective
Hamiltonian ( 7) consists of the two parts, including both the
discussed Hamiltonian ˆH
0and the term ˆV. Therefore, we have
to analyze the effect of the term ˆVon the found solutions of
the Schrödinger problem with the Hamiltonian ˆH0.I tf o l l o w s
from Eqs. ( 9) and ( 15)–(20) that/angbracketleftn/prime,λ/prime,s/prime|ˆV|n,λ,s/angbracketright∼δλ/primeλfor
n,n/prime/greaterorequalslant1. Thus, the term ˆVdoes not split the degenerate states
|n,−,s/angbracketrightand|n−1,+,s/angbracketright. It should be noted also that the
discussed regime of strong light-matter coupling is convention-ally defined as a light-induced renormalization of electronicproperties without the light absorption by electrons (see, e.g.,the discussion in Ref. [ 29]). Particularly, the main absorption
mechanism for semiconductor structures dressed by an off-resonant electromagnetic field—the collisional absorption ofthe field by conduction electrons—can be neglected if ωτ/greatermuch1,
where τis the electron relaxation time [ 28]. Therefore, we have
to consider the case of high frequencies, ω, when the condition
γ
1/lessmuch1 can take place. It follows from this that the discussed
term ˆV∼γ2
1can be considered as a weak perturbation for a
broad range of QR parameters. In particular, the conventionalFIG. 2. Electronic characteristics of InGaAs-based QR (the elec-
tron effective mass is m=0.045m0, the Rashba coupling constant
isα=104m/s, and the QR radius is R=200 nm) irradiated
by a dressing field with the frequency ω=1.6×1012rad/s: (a)
Dependence of the spin angle, ξ, on the irradiation intensity, I;
(b) dependence of the first nine electron energy levels, εs
λn,o nt h e
irradiation intensity, I, for the counterclockwise electron rotation in
the ring ( λ=+ ), where the dashed and solid lines correspond to
different spin orientations ( s=± 1).
criterion of perturbation theory,
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftn
/prime,λ/prime,s/prime|ˆV|n,λ,s/angbracketright
εs/prime
λ/primen/prime−εs
λn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessmuch1, (21)
can be satisfied for the first tens of energy levels ( 14)i n
the typical case of InGaAs-based QRs with the effectivemassm=0.045m
e, radius R≈200 nm, and the Rashba cou-
pling constant α≈104m/s. As a consequence, the effective
Hamiltonian ( 7) can be reduced to the simplified Hamiltonian
(8). Correspondingly, the found eigenstates and eigenenergies
(11)–(20) can be applied to describe electronic properties of
the irradiated QR.
It follows from the Hamiltonian ( 8) that the irradiation of
the QR results in two main effects: First, it renormalizes theelectron effective mass ( 10), and second it leads to the unusual
spin-orbit coupling ∼l
zσzdescribed by the third term of the
Hamiltonian ( 8). In turn, these effects lead to the dependencies
of the spin angle ( 13) and the energy levels ( 14)o nt h e
irradiation intensity, which are plotted in Fig. 2. It follows
from Fig. 2(a) that the irradiation has a very strongly effect on
155434-3KOZIN, IORSH, KIBIS, AND SHELYKH PHYSICAL REVIEW B 97, 155434 (2018)
the spin angle ( 13). Namely, the relatively weak irradiation can
decrease the angle to tens of percent of its initial value in theunirradiated QR, ξ
0=arctan (2 αmR/ ¯h). Since the modulation
of spin orientation by various external actions lies in the core ofmodern spintronics [ 1–4], the found strong dependence of the
spin polarization on the irradiation can be used, particularly inprospective ring-shaped spintronic devices operated by light.It follows from Fig. 2(b) that the irradiation also strongly
effects the energy of the electron levels in the QR and theirspin splitting. Such a light-induced modification of the energyspectrum ( 14) can manifest itself, particularly in the optical
measurements discussed below.
Let us consider a QR irradiated by a two-mode electro-
magnetic wave consisting of a strong dressing field (whichrenormalizes the energy spectrum of electrons according towhat was mentioned earlier) and a relatively weak probefield with the frequency /Omega1(which can detect the discussed
renormalization of the energy spectrum). The optical spectrumof absorption of the probe field can be obtained with use of theconventional Kubo formalism [ 71]. Within this approach, the
longitudinal conductivity describing the response of the QR tothe probe field polarized along the j=x,yaxis reads
σ
jj=/summationdisplay
n,λ,s
n/prime,λ/prime,s/prime/bracketleftbig
f/parenleftbig
εs/prime
λ/primen/prime/parenrightbig
−f/parenleftbig
εs
λn/parenrightbig/bracketrightbig
|/angbracketleftn/prime,λ/prime,s/prime|ˆvj|n,λ,s/angbracketright|2
/parenleftbig
εs/prime
λ/primen/prime−εs
λn/parenrightbig/parenleftbig
εs/prime
λ/primen/prime−εs
λn+¯h/Omega1+i/Gamma1/parenrightbig¯he2
iπR2,
(22)
where f(ε) is the Fermi-Dirac distribution function, ˆ vj=
ˆpj/mis the velocity operator, and /Gamma1=¯h/τ is the broadening
of energy levels depending on the electron relaxation time,τ. It should be noted that the used spin index, s=± 1,
describes the spin projection on the local quantization axis(see the dashed line in Fig. 1), which depends on the electron
location in the QR and, correspondingly, on the direction ofthe vector of electron velocity. As a consequence, the matrixof the velocity operator in Eq. ( 22),/angbracketleftn
/prime,λ/prime,s/prime|ˆvj|n,λ,s/angbracketright, is not
diagonal in this spin index. In particular, direct calculation re-sults in /angbracketleftn
/prime,±,s/prime|ˆvj|n,±,s/angbracketright∼(δn−n/prime,1+δn−n/prime,−1) and/angbracketleftn/prime,∓
,s/prime|ˆvj|n,±,s/angbracketright∼δn+n/prime,1. As a consequence, the probe field can
induce electron transitions between the electron states withmutually opposite local spin directions. Substituting Eqs. ( 14)–
(20) into Eq. ( 22), one can calculate the sought-after absorption
spectrum of the probe field (see Fig. 3), which is represented
by the real part of the conductivity, Re( σ
jj). In the absence of
the dressing field, the absorption spectrum of the QR plottedin Fig. 3(a) consists of the three peaks corresponding to the
following electron transitions: |5,+,+1/angbracketright→| 4,+,+1/angbracketright,
|7,+,−1/angbracketright→| 6,+,+1/angbracketright, and|7,+,−1/angbracketright→| 6,+,−1/angbracketright
(peak 1); |6,+,+1/angbracketright→| 5,+,+1/angbracketright,|8,+,−1/angbracketright→| 7,+
,1/angbracketright, and|8,+,−1/angbracketright→| 7,+,−1/angbracketright(peak 2); |
7,+,+1/angbracketright→
|6,+,+1/angbracketrightand|9,+,−1/angbracketright→| 8,+,−1/angbracketright(peak 3).
The evolution of this spectrum under the influence of the
dressing field is presented in Figs. 3(b)–3(d). In the absence
of the dressing field, the highest peak 3 originates from thetransitions |6,+,+1/angbracketright→| 5,+,+1/angbracketrightand|8,+,−1/angbracketright→
|7,+,−1/angbracketrightsince the chosen Fermi energy, μ=1 meV , lies
in the middle between the corresponding levels [see Fig. 2(b)].
Since the dressing field increases the distance between theenergy levels ( 14), it shifts the peaks to the right and deformsFIG. 3. Absorption spectra of the probe field with the frequency
/Omega1for the InGaAs-based QR (the electron effective mass is m=
0.045m0, the Rashba coupling constant is α=104m/s, the electron
relaxation time is τ=70 ps, the temperature is T=5K ,t h eF e r m i
energy is μ=1 meV , and the QR radius is R=200 nm) irradiated by
a dressing field with the frequency ω=1.6×1012rad/s and different
irradiation intensities, I.
them [see Figs. 3(b)–3(d)]. It should be noted that the shape of
the spectrum at the irradiation intensity I=1000 W /cm2is
very similar to case of an unirradiated QR [compare Figs. 3(a)
and3(d)]. Physically, the similarity appears since the Fermi
energy, μ=1 meV , lies at this intensity again in the middle
between the corresponding levels [see Fig. 2(b)]. However, the
highest peak in this case arises from the peak 1 in Fig. 3(a),
and therefore it corresponds to the transitions |5,+,+1/angbracketright→
|4,+,+1/angbracketright,|7,+,−1/angbracketright→| 6,+,+1/angbracketright, and|7,+,−1/angbracketright→
|6,+,−1/angbracketright. Finalizing the discussion, let us formulate how the
dressing field parameters should be chosen in experiments. Itfollows from the Hamiltonian ( 8) that the absolute value of the
light-induced renormalization of all electronic characteristicsis proportional to the squared ratio E
0/ω2. Therefore, we
have to keep this ratio not too small to observe the discussedrenormalization experimentally for reasonable dressing fieldamplitudes, E
0. This is why the dressing field frequency, ω,i n
Figs. 2and3is chosen to be in the THz range.
IV . CONCLUSIONS
In conclusion, we demonstrated that the key electronic
characteristics of QRs with the Rashba spin-orbit interaction —the structure of electron energy levels and the spin polarizationof electrons — strongly depend on an off-resonant irradiation.In particular, the modification of both electron effective massand spin-orbit coupling appears. It is shown that the irradiation-induced renormalization of the electron energy spectrum canbe observed in state-of-the-art optical experiments, whereasthe light sensitivity of the spin orientation can be exploited inprospective spintronic devices operated by light.
ACKNOWLEDGMENTS
The work was partially supported by the RISE Program
(project CoExAN), the Russian Foundation for Basic Research
155434-4QUANTUM RING WITH THE RASHBA SPIN-ORBIT … PHYSICAL REVIEW B 97, 155434 (2018)
(Project No. 17-02-00053), Rannis project 163082-051,
Ministry of Education and Science of Russian Federation(Projects No. 3.4573.2017/6.7, No. 3.2614.2017/4.6,No. 3.1365.2017/4.6, No. 3.8884.2017/8.9, and No.
14.Y26.31.0015), and Government of Russian Federation(Grant No. 08-08).
[1] D. D. Awschalom, D. Loss, and N. Samarth, Semiconduc-
tor Spintronics and Quantum Computation (Springer-Verlag,
Berlin, 2002).
[2] I. Zutic, J. Fabian, and S. D. Sarma, Spintronics: Fundamentals
and applications, Rev. Mod. Phys. 76,323(2004 ).
[3] D. D. Awschalom and M. E. Flatté, Challenges for semiconduc-
tor spintronics, Nat. Phys. 3,153(2007 ).
[4] M. Cahay, Spin transistors: Closer to an all-electric device, Nat.
Nanotechnol. 10,21(2015 ).
[5] V . Szaszko-Bogar, F. M. Peeters, and P. Földi, Oscillating spin-
orbit interaction in two-dimensional superlattices: Sharp trans-mission resonances and time-dependent spin-polarized currents,Phys. Rev. B 91,235311 (2015 ).
[6] G. Dresselhaus, Spin-orbit coupling effects in zinc blende
structures, Phys. Rev. 100,580(1955 ).
[7] E. I. Rashba, Properties of semiconductors with an extremum
loop. 1. Cyclotron and combinational resonance in a magneticfield perpendicular to the plane of the loop, Sov. Phys. SolidState 2, 1109 (1960).
[8] Y . A. Bychkov and E. I. Rashba, Oscillatory effects and the
magnetic susceptibility of carriers in inversion layers, J. Phys. C
17,6039 (1984 ).
[9] D. Bercioux and P. Lucignano, Quantum transport in Rashba
spin-orbit materials: A review, Rep. Prog. Phys. 78,106001
(2015 ).
[10] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional
Electron and Hole Systems (Springer-Verlag, Berlin, 2003).
[11] B. Miller, D. M. Zumbuhl, C. M. Marcus, Y . B. Lyanda-Geller,
D. Goldhaber-Gordon, K. Campman, and A. C. Gossard, Gate-Controlled Spin-Orbit Quantum Interference Effects in LateralTransport, Phys. Rev. Lett. 90,076807 (2003 ).
[12] A. Studenikin, P. T. Coleridge, N. Ahmed, P. Poole and A.
Sachrajda, Experimental study of weak antilocalization effectsin a high-mobility In
xGa1−xAs/InP quantum well, P h y s .R e v .B
68,035317 (2003 ).
[13] A. Ghosh, C. J. B. Ford, M. Pepper, H. E. Beere, and D. A.
Ritchie, Possible Evidence of a Spontaneous Spin Polarizationin Mesoscopic Two-Dimensional Electron Systems, Phys. Rev.
Lett.92,116601 (2004 ).
[14] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki,
Gate Control of Spin-Orbit Interaction in an InvertedIn
0.53Ga0.47As/In0.52Al0.48As Heterostructure, P h y s .R e v .L e t t .
78,1335 (1997 ).
[15] G. Engels, J. Lange, Th. Schäpers, and H. Lüth, Experimental
and theoretical approach to spin splitting in modulation-dopedIn
xGa1−xAs/InP quantum wells for B →0,Phys. Rev. B 55,
R1958 (1997 ).
[16] J. P. Heida, B. J. van Wees, J. J. Kuipers, T. M. Klapwijk, and
G. Borghs, Spin-orbit interaction in a two-dimensional electrongas in a InAs/AlSb quantum well with gate-controlled electrondensity, P h y s .R e v .B 57,11911 (1998 ).
[17] A. A. Pervishko, O. V . Kibis, S. Morina, and I. A. Shelykh,
Control of spin dynamics in a two-dimensional electron gas byelectromagnetic dressing, Phys. Rev. B 92,205403 (2015 ).[18] A. S. Sheremet, O. V . Kibis, A. V . Kavokin, and I. A. Shelykh,
Datta-and-Das spin transistor controlled by a high-frequencyelectromagnetic field, P h y s .R e v .B 93,165307 (2016 ).
[19] D. Yudin and I. A. Shelykh, Two-dimensional electron gas in
the regime of strong light-matter coupling: Dynamical conduc-tivity and all-optical measurements of Rashba and Dresselhauscoupling, Phys. Rev. B 94,161404(R) (2016 ).
[20] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-
Photon Interactions: Basic Processes and Applications (Wiley,
Chichester, 1998).
[21] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge
University Press, Cambridge, 2001).
[22] S. P. Goreslavskii and V . F. Elesin, Electric properties of a
semiconductor in the field of a strong electromagnetic wave,JETP Lett. 10, 316 (1969).
[23] Q. T. Vu, H. Haug, O. D. Mucke, T. Tritschler, M. Wegener,
G. Khitrova, and H. M. Gibbs, Light-Induced Gaps in Semicon-ductor Band-To-Band Transitions, Phys. Rev. Lett. 92,217403
(2004 ).
[24] Q. T. Vu and H. Haug, Detection of light-induced band gaps
by ultrafast femtosecond pump and probe spectroscopy, Phys.
Rev. B 71,035305 (2005 ).
[25] A. Mysyrowicz, D. Hulin, A. Antonetti, A. Migus, W. T.
Masselink, and H. Morkoc, “Dressed Excitons” in a Multiple-Quantum-Well Structure: Evidence for an Optical Stark Effectwith Femtosecond Response Time, P h y s .R e v .L e t t . 56,2748
(1986 ).
[26] M. Wagner, H. Schneider, D. Stehr, S. Winnerl, A. M. Andrews,
S. Schartner, G. Strasser, and M. Helm, Observation of theIntraexciton Autler-Townes Effect in GaAs/AlGaAs Semicon-ductor Quantum Wells, P h y s .R e v .L e t t . 105,167401 (2010 ).
[27] M. Teich, M. Wagner, H. Schneider, and M. Helm, Semiconduc-
tor quantum well excitons in strong, narrowband terahertz fields,New J. Phys. 15
,065007 (2013 ).
[28] O. V . Kibis, How to suppress the backscattering of conduction
electrons? Europhys. Lett. 107,57003 (2014 ).
[29] S. Morina, O. V . Kibis, A. A. Pervishko, and I. A. Shelykh,
Transport properties of a two-dimensional electron gas dressedby light, P h y s .R e v .B 91,155312 (2015 ).
[30] O. V . Kibis, Dissipationless Electron Transport in Photon-
Dressed Nanostructures, P h y s .R e v .L e t t . 107,106802
(2011 ).
[31] F. K. Joibari, Y . M. Blanter, and G. E. W. Bauer, Light-induced
spin polarizations in quantum rings, Phys. Rev. B 90,155301
(2014 ).
[32] H. Sigurdsson, O. V . Kibis, and I. A. Shelykh, Optically induced
Aharonov-Bohm effect in mesoscopic rings, Phys. Rev. B 90,
235413 (2014 ).
[33] O. V . Kibis, H. Sigurdsson, and I. A. Shelykh, Aharonov-Bohm
effect for excitons in a semiconductor quantum ring dressed bycircularly polarized light, Phys. Rev. B 91,235308 (2015 ).
[34] M. Hasan, I. V . Iorsh, O. V . Kibis, and I. A. Shelykh, Optically
controlled periodical chain of quantum rings, Phys. Rev. B 93,
125401 (2016 ).
155434-5KOZIN, IORSH, KIBIS, AND SHELYKH PHYSICAL REVIEW B 97, 155434 (2018)
[35] V . K. Kozin, I. V . Iorsh, O. V . Kibis, and I. A. Shelykh, Periodic
array of quantum rings strongly coupled to circularly polarizedlight as a topological insulator, Phys. Rev. B 97,035416 (2018 ).
[36] F. J. Lopez-Rodriguez and G. G. Naumis, Analytic solution
for electrons and holes in graphene under electromagneticwaves: Gap appearance and nonlinear effects, Phys. Rev. B 78,
201406(R) (2008 ).
[37] T. Oka and H. Aoki, Photovoltaic Hall effect in graphene, Phys.
Rev. B 79,081406(R) (2009 ).
[38] O. V . Kibis, Metal-insulator transition in graphene induced by
circularly polarized photons, P h y s .R e v .B 81,165433 (2010 ).
[39] T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Transport
properties of nonequilibrium systems under the application oflight: Photoinduced quantum Hall insulators without Landaulevels, P h y s .R e v .B 84,235108 (2011 ).
[40] G. Usaj, P. M. Perez-Piskunow, L. E. F. Foa Torres, and C. A.
Balseiro, Irradiated graphene as a tunable Floquet topologicalinsulator, Phys. Rev. B 90,115423 (2014 ).
[41] K. Kristinsson, O. V . Kibis, S. Morina, and I. A. Shelykh, Control
of electronic transport in graphene by electromagnetic dressing,Sci. Rep. 6,20082 (2016 ).
[42] O. V . Kibis, S. Morina, K. Dini, and I. A. Shelykh, Magneto-
electronic properties of graphene dressed by a high-frequency
field, P h y s .R e v .B 93,115420 (2016 ).
[43] O. V . Kibis, K. Dini, I. V . Iorsh, and I. A. Shelykh, All-optical
band engineering of gapped Dirac materials, Phys. Rev. B 95,
125401 (2017 ).
[44] I. V . Iorsh, K. Dini, O. V . Kibis, and I. A. Shelykh, Optically
induced Lifshitz transition in bilayer graphene, Phys. Rev. B 96,
155432 (2017 ).
[45] N. H. Lindner, G. Refael, and V . Galitski, Floquet topological
insulator in semiconductor quantum wells, Nat. Phys. 7,490
(2011 ).
[46] S. Datta and B. Das, Electronic analog of the electro-optic
modulator,
Appl. Phys. Lett. 56,665(1990 ).
[47] H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and M.
Johnson, Control of spin precession in a spin-injected field effecttransistor, Science 325,1515 (2009 ).
[48] T. Bergsten, T. Kobayashi, Y . Sekine, and J. Nitta, Experimental
Demonstration of the Time Reversal Aharonov-Casher Effect,Phys. Rev. Lett. 97,196803 (2006 ).
[49] J. Nitta, F. E. Meijer, and H. Takayanagi, Spin-interference
device, Appl. Phys. Lett. 75,695(1999 ).
[50] J. Nitta and T. Koga, Rashba spin-orbit interaction and its
applications to spin-interference effect and spin-filter device,J. Supercond. 16,689(2003 ).
[51] D. Frustaglia and K. Richter, Spin interference effects in ring
conductors subject to Rashba coupling, P h y s .R e v .B 69,235310
(2004 ).
[52] I. A. Shelykh, N. T. Bagraev, N. G. Galkin, and L. E.
Klyachkin, Interplay of h/e andh/2eoscillations in gate-
controlled Aharonov-Bohm rings, Phys. Rev. B 71,113311
(2005 ).
[53] A. G. Aronov and Y . B. Lyanda-Geller, Spin-Orbit Berry Phase
in Conducting Rings, P h y s .R e v .L e t t . 70,343(1993 ).[54] M. Popp, D. Frustaglia, and K. Richter, Spin filter effects in
mesoscopic ring structures, Nanotechnology 14,347(2003 ).
[55] A. A. Kiselev and K. W. Kim, T-shaped spin filter with a ring
resonator, J. Appl. Phys. 94,4001 (2003 ).
[56] I. A. Shelykh, N. G. Galkin, and N. T. Bagraev, Quantum splitter
controlled by Rashba spin-orbit coupling, Phys. Rev. B 72,
235316 (2005
).
[57] B. Mòlnar, F. M. Peeters, and P. Vasilopoulos, Spin-dependent
magnetotransport through a ring due to spin-orbit interaction,Phys. Rev. B 69,155335 (2004 ).
[58] P. Földi, B. Mòlnar, M. G. Benedict, and F. M. Peeters, Spintronic
single-qubit gate based on a quantum ring with spin-orbitinteraction, Phys. Rev. B 71,033309 (2005 ).
[59] P. Földi, O. Kálmán, M. G. Benedict, and F. M. Peeters, Quantum
rings as electron spin beam splitters, Phys. Rev. B 73,155325
(2006 ).
[60] R. Citro, F. Romeo, and M. Marinaro, Zero-conductance reso-
nances and spin filtering effects in ring conductors subject toRashba coupling, Phys. Rev. B 74,115329 (2006 ).
[61] N. Hatano, R. Shirasaki, and H. Nakamura, Non-Abelian gauge
field theory of the spin-orbit interaction and a perfect spin filter,Phys. Rev. A 75,032107 (2007 ).
[62] S. Matityahu, A. Aharony, O. Entin-Wohlman, and S.
Katsumoto, Robustness of spin filtering against current leakagein a Rashba-Dresselhaus-Aharonov-Bohm interferometer, Phys.
Rev. B 87,205438 (2013 ).
[63] S. Matityahu, A. Aharony, O. Entin-Wohlman, and S.
Tarucha, Spin filtering in a Rashba-Dresselhaus-Aharonov-Bohm double-dot interferometer, New J. Phys. 15,125017
(2013 ).
[64] S. Matityahu, A. Aharony, O. Entin-Wohlman, and C. A.
Balseiro, Spin filtering in all-electrical three-terminal interfer-ometers, Phys. Rev. B 95,085411 (2017 ).
[65] A. Lorke, J. R. Luyken, A. O. Govorov, J. P. Kot-
thaus, J. M. Garcia, and P. M. Petroff, Spectroscopy ofNanoscopic Semiconductor Rings, P h y s .R e v .L e t t . 84,2223
(2000 ).
[66] F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, One-
dimensional ring in the presence of Rashba spin-orbit interac-tion: Derivation of the correct Hamiltonian, P h y s .R e v .B 66,
033107 (2002 ).
[67] F. Casas, J. A. Oteo, and J. Ros, Floquet theory: Exponential
perturbative treatment, J. Phys. A 34,3379 (2001 ).
[68] N. GoldMan and J. Dalibard, Periodically Driven Quantum
Systems: Effective Hamiltonians and Engineered Gauge Fields,
Phys. Rev. X 4,031027 (2014 ).
[69] A. Eckardt and E. Anisimovas, High-frequency approximation
for periodically driven quantum systems from a Floquet-spaceperspective, New J. Phys. 17,093039 (2015 ).
[70] F. Nagasawa, D. Frustaglia, H. Saarikoski, K. Richter, and J.
Nitta, Control of the spin geometric phase in a semiconductorquantum ring, Nat. Commun. 4,2526 (2013 ).
[71] H. Bruuc and K. Flensberg, Many-Body Quantum Theory in
Condensed Matter Physics (Oxford University Press, Oxford,
2001).
155434-6 |
PhysRevB.72.075446.pdf | Metal-induced gap states in epitaxial organic-insulator/metal interfaces
Manabu Kiguchi,1,*Ryotaro Arita,2,†Genki Yoshikawa,3Yoshiaki Tanida,4Susumu Ikeda,1Shiro Entani,1Ikuyo Nakai,3
Hiroshi Kondoh,3Toshiaki Ohta,3Koichiro Saiki,1,3and Hideo Aoki2
1Department of Complexity Science and Engineering, University of Tokyo, Kashiwa, Chiba 277-8561, Japan
2Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan
3Department of Chemistry, University of Tokyo, Hongo, Tokyo 113-0033, Japan
4Fujitsu Laboratories Ltd., Atsugi, Kanagawa 243-0197, Japan
/H20849Received 11 January 2005; revised manuscript received 2 June 2005; published 31 August 2005 /H20850
We have shown, both experimentally and theoretically, that the metal-induced gap states /H20849MIGS /H20850can exist in
epitaxially grown organic insulator/metal interfaces. The experiment is done for alkane/Cu /H20849001/H20850with an
element-selective near edge x-ray absorption fine structure /H20849NEXAFS /H20850, which exhibits a prepeak indicative of
MIGS. An ab initio electronic structure calculation supports the existence of the MIGS. When the Cu substrate
is replaced with Ni, an interface magnetism may be possible with a carrier doping.
DOI: 10.1103/PhysRevB.72.075446 PACS number /H20849s/H20850: 73.20. /H11002r, 71.15.Mb, 73.40.Ns
I. INTRODUCTION
While there are mounting interests in the nature of hetero-
interfaces, organic crystal/metal interfaces are especially in-triguing due to their diverse possibilities not found in inor-ganic counterparts, which may facilitate in controlling anddesigning properties of interfaces. Electronic structures oforganic/metal interfaces are important from a technological
point of view as well, since the performance of devicesshould strongly depend on the electronic structure at the in-terface.
Recent years have in fact witnessed several interesting
results on organic-crystal/metal interfaces.
1One crucial fac-
tor in such organic-insulator/inorganic-metal heterointerfacesis the energy level alignment at the interface, which is stillfar from being fully understood. The band alignment, mea-sured by ultraviolet photoelectron spectroscopy /H20849UPS /H20850and
Kelvin probe, has been discussed in terms of various effects,such as electron transfer, image effect, modification of thesurface dipole at metal surface, chemical interaction and in-terfacial states, etc.
2However, there is no generally accepted
picture for organic/metal interfaces, which is contrasted withinorganic-semiconductor/metal interfaces where the bandalignment is known to be governed by the interface statescalled the metal-induced gap states /H20849MIGS /H20850.
3So a study of
MIGS at the organic/metal interfaces should be imperative.
The notion of MIGS was first introduced for
semiconductor/metal junctions in discussing the Schottkybarrier.
3MIGS are roughly free-electron-like metal wave
functions penetrating into the semiconducting side, wherethe penetration depth is inversely proportional to the bandgap in the conventional band picture. So a usual wisdomdictates that MIGS would be far too thin in insulator/metalinterfaces to be relevant. The present authors succeeded infabricating epitaxial alkali-halide/metal interfaces, which hasenabled us to obtain unambiguous evidences especially fromthe near edge x-ray absorption fine structure /H20849NEXAFS /H20850that
MIGS are in fact formed at the inorganic-insulator/metalinterfaces.
4–6One reason for the successful detection of
MIGS is the following: while signals from the interface areobscured by the signal from the substrate in conventionalexperimental methods, because the signals in these methods
involve significant contributions from the substrate, NEX-AFS, based on the x-ray absorption of the atom, providesinformation on the alkali halide with influences from thesubstrate being negligible.
While alkali halides are typical inorganic insulators, inter-
face states for organic-insulator/metal interfaces are a totallydifferent issue, since chemical bonds in organic insulators arecovalent as opposed to ionic bonds in inorganic, ionic insu-lators, and it is a fundamental question to ask whether theformation of MIGS at insulator/metal interfaces are universalenough to accommodate organic insulators. Experimentally,interface states in atomically well-defined organic-insulator/metal interfaces have yet to be observed to our knowledge
7
despite their prime importance. One obvious reason for thisis difficulties in fabricating atomically well-defined organic-insulator/metal interfaces and in detecting signals from theinterface.
4However, recent developments in the molecular
beam epitaxy /H20849MBE /H20850technique have made it possible to pre-
pare various types of heterointerfaces.8,9Some organic films
are in fact begun to be epitxially grown on metal substratesin a layer-by-layer fashion with MBE.
10
Given this background, the purpose of the present paper is
to examine the interface states in an atomically well-definedorganic insulator /H20849an alkane; C
44H90/H20850grown on Cu substrate.
We have found, with the element-selective NEXAFS, a pre-peak indicative of metal-induced gap states. At the same timewe have performed an ab initio electronic structure calcula-
tion. The theoretical result shows that /H20849i/H20850MIGS do exist at
the organic insulator/metal interface, and /H20849ii/H20850when we re-
place Cu with Ni, in which narrow 3 dbands rather than wide
4sbands dominate the electronic properties around the Fermi
energy, an interface magnetism is predicted to be possiblewith a carrier doping.
II. EXPERIMENT
The experiments were performed with a UHV chamber at
the soft x-ray beam line BL-7A in the Photon Factory in theInstitute of Materials Structure Science, Japan. We have em-PHYSICAL REVIEW B 72, 075446 /H208492005 /H20850
1098-0121/2005/72 /H208497/H20850/075446 /H208495/H20850/$23.00 ©2005 The American Physical Society 075446-1ployed an alkane /H20849CnH2n+2with n=44; tetratetracontane or
TTC /H20850as a simplest hydrocarbon, where the properties of this
series of molecules are known to be similar except for avariation in the band gap with n. A clean Cu /H20849001/H20850surface
was prepared by repeated cycles of Ar sputtering and anneal-ing at 900 K. TTC was then evaporated on Cu /H20849001/H20850with the
substrate temperature of 300 K with a Knudsen cell. Real-time observation of crystallinity and orientation of the filmwas probed by reflection high energy electron diffraction/H20849RHEED /H20850with a microchannel plate. Half-order streaks ap-
peared only along the /H20851110/H20852azimuth direction fo r a 1 mono-
layer /H20849ML/H20850thick TTC/Cu /H20849001/H20850. This indicates that the TTC
film epitaxially grows on Cu /H20849001/H20850with its molecular axis
parallel to /H20851110/H20852azimuth of the Cu substrates in a layer-by-
layer fashion.
10Carbon K-edge NEXAFS spectra were then
obtained in-situ by the partial electron yield method with amicrochannel plate. Since electrons with energies about200 eV are detected, the probing depth is estimated to be0.8 nm, which is deep enough to study the electronic struc-ture for the flat-lying 1 ML thick TTC film. Energy resolu-tion was 0.3 eV at C K-edge. The NEXAFS spectra were
taken at 30°, 55°, 90° x-ray incident angles from the surfaceparallel. We can qualitatively know the anisotropy of theorbital by changing the incident angles.
11
Figure 1 shows the NEXAFS fo ra1M LT T Co n
Cu/H20849001/H20850, as compared with the result for a bulk /H20849multilayer /H20850
TTC film. In C K-edge NEXAFS for n-alkane, there are
many /H20849at least five /H20850resonances, and all the resonances have
not been assigned up to now.12So we only discuss the as-
signed /H9268*/H20849C-C /H20850and/H9268*/H20849C-H /H20850resonances. For a bulk TTC, a
broad peak at about 293 eV is assigned t oaC1 s→/H9268*/H20849C-
C/H20850resonance, while the peak at 288 eV t oaC1 s→/H9268*/H20849C-
H/H20850.12The/H9268*/H20849C-C /H20850peak whose transition moment is parallel
to the C-C-C plane is most enhanced at the normal x-ray
incidence, which confirms that the TTC grows on Cu /H20849001/H20850
with its molecular axis parallel to the substrate. On the otherhand, the
/H9268*/H20849C-H /H20850peak, whose transition moment perpen-dicular to the C-C-C plane, splits into two. This should be
because one-half of the hydrogen atoms in the organic mol-ecule touch the substrate in the lying-down configuration asdepicted in the inset of Fig. 2, so that the /H20849C-H /H20850state splits
due to the interaction between TTC and Cu.
12
More importantly, a pronounced prepeak is seen to appear
below the bulk edge onset. Because the adsorption energy foralkanes on metal surfaces is small /H20849/H1101110 kJ/mol/CH
2chain /H20850,
the chemical nature of the interface should be a typical phy-sisorption state with a weak molecule-surface interaction, ascontrasted with a chemisorption with chemical bonds formedat the interface. The prepeak should originate from the prox-imity to a metal rather than from chemical bonds. So weassign the prepeak as coming from the MIGS. At the presentstage, we cannot completely rule out other possible originsof the prepeak such as the final-state interactions. The finalstate interaction between electrons and electron-vacanciescan distort or produce additional peak in NEXAFS.
13How-
ever, these effects would be tiny when the interatomic inter-action is weak.
14So the assignment of the prepeak to the
MIGS should be reasonable.
The right panel of Fig. 1 is a blowup of the absorption
edge for 1 ML TTC/Cu /H20849001/H20850. We can see that the prepeak is
greater for a grazing x-ray incidence than for the normalincidence. This indicates that the MIGS wave functions areoriented in the surface normal direction.
11
III.AB INITIO CALCULATION
Let us now move on to the first-principles /H20849density func-
tional theory /H20850electronic-structure calculation for the hetero-
FIG. 1. /H20849Color online /H20850Experimental result for the C K-edge
NEXAFS spectra in 1 ML C 44H90/H20849TTC /H20850films grown on Cu /H20849001/H20850
for the x-ray incidence angle /H9258varied over 30°, 55°, 90°. We also
display for comparison the result for a multilayer TTC /H20849bulk /H20850on
Cu/H20849001/H20850in black /H20849which is almost /H9258independent, here for /H9258=90° /H20850.
All the spectra are normalized by their edge jump. Right-hand panelis a blowup of the prepeaks, obtained by subtracting the bulk/H20849multilayer /H20850spectrum which has no structures at the edge onset.
FIG. 2. /H20849Color online /H20850Theoretical result for the band structure
of 1 ML polyethylene/Cu /H20849001/H20850. Top left-hand inset is the atomic
configuration considered here, where Cu is in red, C is in gray, andH is in white. Bottom left-hand side is a contour plot of the localdensity of states /H20849LDOS /H20850as a function of the growth direction zand
energy /H20849color coded /H20850, while the bottom right-hand side is the LDOS
atE
F−0.125 eV /H20849occupied /H20850and EF+0.125 eV /H20849unoccupied /H20850inte-
grated over the xyplane, as a function of z.KIGUCHI et al. PHYSICAL REVIEW B 72, 075446 /H208492005 /H20850
075446-2interface to examine how the above experimental result fits
the theoretical picture. Quite recently Morikawa et al.15have
performed a first principles calculation for interfaces be-tween alkane and various metals such as Cu, where thechange in the work function and a softening of the CHstretching mode were studied. Here we explore the local den-sity of states, and also predict what will happen when wereplace the substrate with a ferromagnetic d-band metal such
as Ni, for which the spin-density functional theory is em-ployed. The density of unoccupied states of Ni should bemuch greater than that of Cu, since Ni is ferromagnetic withthe major part of the minority-spin 3 dband sitting above the
Fermi energy. So, naively, the intensity of MIGS fororganic/Ni is expected to be much stronger than that fororganic/Cu, which has motivated us to study Ni.
A penalty for doing a first-principles band calculation is,
given the complexity of the system, we must replace thealkane with a finite nwith polyethylene, assumed to be infi-
nitely long. This reduces the size of the unit cell, enabling usto perform the spin density functional study. We have also
assumed that polyethylene chains are close-packed, while anexperiment
10for TTC indicates that the real packing has half
this density. However, the formation of the MIGS is in gen-eral dominated basically by the band gap of the insulator andthe charge transfer from the insulator to the metal /H20849or vice
versa /H20850.
5So, since the work function and the band gap of
polyethylene should not drastically depend on whether themolecules are close packed or not, we can expect that thepresent calculation should capture essential features of theelectronic structure of the interface.
We adopt the exchange-correlation functional introduced
by Perdew et al. ,
16and employ ultrasoft pseudopotentials in
separable forms.17,18The cutoff energy of the plane-wave
expansion for the wave function is taken to be 42.25 Ry. Theatomic configurations /H20849inset of Fig. 2 /H20850and the corresponding
electronic ground states are obtained with the conjugate gra-dient scheme.
19
Figure 2 shows the band structure along with the local
density of states /H20849LDOS /H20850for polyethylene/Cu. The LDOS at
Eis calculated as /H20858i/H20841/H9278i/H20849x,y,z/H20850/H208412with the summation taken
over the eigenstates having energies E−0.125 eV /H11021Ei/H11021E
+0.125 eV. The number of sampled kpoints is eight with the
Monkhorst-Pack method for the integration over the Bril-louin zone,
20where the bands are fitted to sinusoidal forms
and the tetrahedron method is employed. We can see in theresult that LDOS at E
Fhas peaks at the carbon sites, which
indicates MIGS are formed at the polyethylene/Cu interface.The wave function contour /H20849inset of Fig. 2 /H20850confirms this in
real space. The MIGS wave functions are seen to be orientedin the surface normal direction, which also agrees with theabove experimental result.
Ni substrate : We move on to polyethylene/Ni in Fig. 3,
where we must look at the majority- and minority-spin com-ponents separately, since the Ni substrate is spin polarized.We can see that the local density of states around E
Fsignifi-
cantly differs between the majority and minority spins, in-cluding those at the carbon sites. In the energy-resolvedLDOS /H20849color-coded panel in Fig. 3 /H20850this is seen to come from
a difference in the positions of MIGS in the occupied /H20849E
/H11021E
F/H20850side. To be more precise, there is a finite exchangesplitting for the MIGS in the polyethylene/Ni interface, al-
though the splitting is smaller than the splitting within the Nisubstrate. However, both the majority- and minority-spinMIGS lie below E
F/H20849with the latter lying just below EF/H20850. This
implies that the organic crystal, although lying on a ferro-magnetic substrate, is not spin-polarized. Figure 4 comparesfor Ni and Cu substrates the total /H20849i.e., sum of the majority
and minority spin /H20850LDOS /H20849which is relevant to the
NEXAFS /H20850. In accord with the above, the density of unoccu-
pied MIGS states is similar between polyethylene/Ni and
FIG. 3. /H20849Color online /H20850A plot similar to Fig. 2 for 1 ML
polyethylene/Ni /H20849001/H20850, where the spin density functional theory is
adopted. Green /H20849black /H20850lines in the band structure represent the
majority /H20849minority /H20850spin.
FIG. 4. /H20849Color online /H20850Theoretical results for the local density of
occupied and unoccupied states at EFare compared for
polyethylene/Ni and polyethylene/Cu.METAL-INDUCED GAP STATES IN EPITAXIAL … PHYSICAL REVIEW B 72, 075446 /H208492005 /H20850
075446-3polyethylene/Cu, while the density of occupied MIGS states
differs between them because the 3 dband resides just below
EFfor Ni. The theoretical prediction agrees with our prelimi-
nary NEXAFS result, a probe for unoccupied states, foroctane /H20849C
8H18/H20850on Ni /H20849111/H20850and Cu /H20849111/H20850substrates. The inten-
sity of prepeak is similar between the two systems, with the
intensity normalized by the edge jump being 0.27 for octane/Ni/H20849111/H20850against 0.30 for octane/Cu /H20849111/H20850.
We can summarize the band scheme in Fig. 5, which
schematically depicts the energy regions for MIGS in theorganic/Cu and organic/Ni interfaces. The MIGS band forthe majority spin in polyethylene/Ni lies below E
F,s ot h e
density of states in the unoccupied side is small, while thedensity of occupied states is large. By contrast, E
Fruns right
through the MIGS band when the substrate is Cu, but thedensity of MIGS is relatively low due to a low density ofstates of the Cu 4 sband. This results in little difference be-
tween organic/Ni and organic/Cu as far as the density ofunoccupied MIGS is concerned, so an experimental methodthat detects occupied states, such as resonant photoemissionor x-ray emission spectroscopy, will probe the difference.
While the spin-unpolarized MIGS on a ferromagnetic
substrate is a bit of a disappointment, the above picture natu-rally leads us to the following theoretical proposal. In theenergy diagram in Fig. 5, we can make the MIGS spin-polarized if we can introduce carriers into MIGS by, e.g.,chemical doping. Namely, while normally the majority andminority spins are /H20849almost /H20850fully occupied in organic/Ni, the
doped carriers will be accommodated in the minority-spinband as indicated by a dashed line in Fig. 5, so that weshould end up with polarized organic molecules. However,whether the argument based on the rigid-band picture is validis subtle, which should be a future problem.
IV. CONCLUSIONS
We have obtained a clear evidence that MIGS are formed
at atomically well-defined organic-insulator/metal interfacesby using NEXAFS. Ab initio electronic structure calculation
supports the existence of MIGS at the organic-insulator/metal interface. Theoretical result indicates that the densityof unoccupied MIGS states is similar betweenpolyethylene/Ni and polyethylene/Cu, while the density ofoccupied MIGS states should be different, and that a dopingwill make the organic molecule spin-polarized on Ni if anappropriate doping can be done.
ACKNOWLEDGMENTS
Calculations were performed with TAPP /H20849Tokyo ab-initio
program package /H20850, for which RA received technical advices
from Y . Suwa. SR8000 in ISSP, University of Tokyo, wasused for the numerical calculations. This work was supportedin part by creative scientific research Grant No. 14GS0207and a special coordination fund from the Japanese Ministryof Education.
*Present address: Department of Chemistry, Hokkaido University,
Sapporo 060-0810, Japan.
†Present address: Max-Planck-Institut für Festköoperforschung,
Stuttgart, Germany.
1S. Lukas, G. Witte, and Ch. Wöll, Phys. Rev. Lett. 88, 028301
/H208492002 /H20850.
2H. Ishi, K. Sugiyama, E. Ito, and K. Seki, Adv. Mater. /H20849Wein-
heim, Ger. /H2085011, 605 /H208491999 /H20850.
3J. Tersoff, Phys. Rev. Lett. 52, 465 /H208491984 /H20850.
4M. Kiguchi, R. Arita, G. Yoshikawa, Y . Tanida, M. Katayama, K.
Saiki, A. Koma, and H. Aoki, Phys. Rev. Lett. 90, 196803
/H208492003 /H20850.
5R. Arita, Y . Tanida, K. Kuroki, and H. Aoki, Phys. Rev. B 69,
115424 /H208492004 /H20850.
6M. Kiguchi, G. Yoshikawa, S. Ikeda, and K. Saiki, Phys. Rev. B
71, 153401 /H208492005 /H20850.
7For organic molecule/metal systems /H20851octane/Cu /H20849110/H20850/H20852, the inter-
face states were found by H. Öström, L. Triguero, M. Nyberg,H. Ogasawara, L. G. M. Pettersson, and A. Nilsson, Phys. Rev.Lett. 91, 046102 /H208492003 /H20850. However, ordered structures are not
formed there, which cannot be described with the band picture.
8M. Kiguchi, S. Entani, K. Saiki, T. Goto, and A. Koma, Phys.
Rev. B 68, 115402 /H208492003 /H20850.
9M. Kiguchi, S. Entani, K. Saiki, and G. Yoshikawa, Appl. Phys.
Lett. 84, 3444 /H208492004 /H20850.
10Y . Hosoi, Y . Sakurai, M. Yamamoto, H. Ishii, Y . Ouchi, and K.
Seki, Surf. Sci. 515, 157 /H208492002 /H20850.
11The intensity of the NEXAFS peak is I=cos2/H9258, where /H9258is the
angle between the electric vector of x ray and the transitionmoment. While the transition moment normal to the surfacetends to be more strongly detected with decreasing
/H9258, the true
grazing incidence is not essential for qualitatively discussing theanisotropy of MIGS. So we adopted 30° as a grazing incidenceangle.
12H. Kondo, F. Matsui, Y . Ehara, T. Yokoyama, and T. Ohta,
Langmuir 17, 8178 /H208492001 /H20850.
13J. L. Dehmer, A. F. Starace, U. Fano, J. Sugar, and J. W. Cooper,
Phys. Rev. Lett. 26, 1521 /H208491971 /H20850.
FIG. 5. /H20849Color online /H20850A schematic energy diagram for MIGS in
organic/Ni and in organic/Cu.KIGUCHI et al. PHYSICAL REVIEW B 72, 075446 /H208492005 /H20850
075446-414D. A. Muller, D. A. Shashkov, R. Benedek, L. H. Yang, J. Silcox,
and D. N. Seidman, Phys. Rev. Lett. 80, 4741 /H208491998 /H20850.
15Y . Morikawa, H. Ishii, and K. Seki, Phys. Rev. B 69, 041403 /H20849R/H20850
/H208492004 /H20850.
16J. P. Perdew, K. Burke, and Y . Wang, Phys. Rev. B 54, 16533
/H208491996 /H20850.17D. Vanderbilt, Phys. Rev. B 41, R7892 /H208491990 /H20850.
18K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt,
Phys. Rev. B 47, 10142 /H208491993 /H20850.
19J. Yamauchi, M. Tsukada, S. Watanabe, and O. Sugino, Phys.
Rev. B 54, 5586 /H208491996 /H20850.
20H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 /H208491976 /H20850.METAL-INDUCED GAP STATES IN EPITAXIAL … PHYSICAL REVIEW B 72, 075446 /H208492005 /H20850
075446-5 |
PhysRevB.80.052508.pdf | Suppression of the superconducting transition of RFeAsO 1−xFx(R=Tb, Dy, and Ho)
Jennifer A. Rodgers,1,2George B. S. Penny,1,2Andrea Marcinkova,1,2Jan-Willem G. Bos,1,2Dmitry A. Sokolov,1,3
Anna Kusmartseva,1,2Andrew D. Huxley,1,3and J. Paul Attfield1,2,*
1Centre for Science at Extreme Conditions, University of Edinburgh, King’ s Buildings, Mayfield Road,
Edinburgh EH9 3JZ, United Kingdom
2School of Chemistry, University of Edinburgh, Edinburgh EH9 3JJ, United Kingdom
3SUP A, School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
/H20849Received 31 July 2009; published 28 August 2009 /H20850
A suppression of superconductivity in the late rare-earth RFeAsO 1−xFxmaterials is reported. The maximum
critical temperature /H20849Tc/H20850decreases from 51 K for R=Tb to 36 K for HoFeAsO 0.9F0.1, which has been synthe-
sized under 10 GPa pressure. This suppression is driven by a decrease in the Fe-As-Fe angle below an optimumvalue of 110.6°, as the angle decreases linearly with unit-cell volume /H20849V/H20850across the RFeAsO
1−xFxseries. A
crossover in electronic structure around this optimum geometry is evidenced by a change in sign of thecompositional dT
c/dV, from negative values for previously reported large Rmaterials to positive for
HoFeAsO 0.9F0.1.
DOI: 10.1103/PhysRevB.80.052508 PACS number /H20849s/H20850: 74.62.Dh, 74.62.Bf, 74.70.Dd
Rare earth /H20849R/H20850oxypnictides RFeAsO /H20849Ref. 1/H20850were re-
cently discovered to superconduct when doped, with criticaltemperatures surpassed only by the high- T
ccuprates. Several
families of superconducting iron pnictides have subsequentlybeen discovered.
2These all have layered structures contain-
ing AsFeAs slabs with Fe tetrahedrally coordinated by As.The main types are the 1111 materials based on RFeAsO or
MFeAsF /H20849M=Ca,Sr,Ba /H20850, the 122 phases MFe
2As2, and the
111AFeAs /H20849A=Li,Na /H20850family. The related binaries Fe X/H20849X
=Se,Te /H20850are also superconducting.
The electron-doped 1111 materials RFeAsO 1−xFxand
RFeAsO 1−/H9254materials remain prominent as they have the
highest Tc’s, up to 56 K, and allow lattice and doping effects
to be investigated through variations in the R3+cation size
and the anion composition. A strong lattice effect is evidentat the start of the rare-earth series, as T
crises from 26 K for
LaFeAsO 1−xFxto 43 K under pressure,3,4and to a near-
constant maximum 50–56 K in the RFeAsO 1−xFxand
RFeAsO 1−/H9254series for R= P rt oG d ,5–10but whether lattice
effects ultimately enhance or suppress superconductivity forthe late R’s has been unclear. The late rare-earth
RFeAsO
1−xFxmaterials and the oxygen-deficient RFeAsO 1−/H9254
superconductors require high-pressure synthesis, leading to
significant challenges as single phase samples are difficult toprepare, and accurate analyses of cation stoichiometries andO and F contents are difficult. To investigate the effect of thelattice for later R, we have synthesized multiple samples of
RFeAsO
0.9F0.1/H20849R=Tb, Dy, and Ho /H20850under varying high-
pressure conditions. Here we report superconductivity inHoFeAsO
0.9F0.1for which the maximum Tcof 36 K is mark-
edly lower than in the previous Ranalogs. This is part of a
systematic suppression of superconductivity by the smaller,lateRcations. HoFeAsO
0.9F0.1also shows a reversal in the
sign of the compositional dTc/dV /H20849V=unit-cell volume /H20850
compared to the early Rmaterials, confirming that the de-
creasing Rsize has a significant effect on the bands contrib-
uting to the Fermi surface.
Polycrystalline ceramic RFeAsO 1−xFxsamples /H20849R=Tb,
Dy, and Ho /H20850were synthesized by a high-pressure method and
investigated by powder x-ray diffraction, magnetization, andconductivity measurements.11Initial results for RFeAsO 1−xFx
/H20849R=Tb and Dy /H20850were published elsewhere.12Both materials
were found to be superconducting with maximum Tc’s of 46
and 45 K, respectively. Little difference in superconductingproperties between samples with nominal compositions ofx=0.1 and 0.2 were observed, and the x=0.2 materials were
generally of lower phase purity, and so the x=0.1 composi-
tion was used in subsequent syntheses. The best samplestypically contain /H1101180%by mass of the superconducting
phase with residual nonsuperconducting R
2O3and RAs
phases also present. The sample purity and superconductingproperties are not sensitive to synthesis pressure over a rangethat moves to higher pressures as Rdecreases in size; R
=Tb and Dy superconductors were respectively prepared at7–10 and 8–12 GPa, heating at 1050–1100 °C. Repeatedsyntheses of TbFeAsO
1−xFxgave several samples with
higher Tc’s than the above value, the highest value is
Tc/H20849max /H20850=51 K /H20849Fig.1/H20850. Further DyFeAsO 1−xFxsamples did
not show higher transitions than before, so we conclude thatT
c/H20849max /H20850in this system is 45 K.
Tetragonal HoFeAsO 0.9F0.1was obtained from reactions
at 10 GPa pressure and the properties of six HoFeAsO 0.9F0.1
samples prepared under varying conditions are summarized
FIG. 1. Resistivity and /H20849inset /H20850susceptibility data for an optimum
sample of TbFeAsO 0.9F0.1, showing a sharp superconducting tran-
sition at Tc=51 K. The sample was prepared at 7 GPa and
1050 °C.PHYSICAL REVIEW B 80, 052508 /H208492009 /H20850
1098-0121/2009/80 /H208495/H20850/052508 /H208494/H20850 ©2009 The American Physical Society 052508-1in Table I. Crystal structure refinements and phase analysis
were carried out by fitting powder x-ray diffraction data /H20849Fig.
2/H20850.13Magnetization measurements demonstrate that all six
HoFeAsO 1−xFxsamples are bulk superconductors with Tc’s
of 29–36 K /H20849Fig. 3/H20850. Resistivities show smooth high-
temperature evolutions without apparent spin-density waveanomalies. The transitions to the zero resistance state havewidths of less than 4 K.
Although all of the samples in Table Ihave the same
starting composition, small variations in synthesis pressureand temperature result in a dispersion in xaround the nomi-
nal 0.1 value for the HoFeAsO
1−xFxphase and corresponding
variations in superconducting properties. Tcincreases to a
maximum value, Tc/H20849max /H20850, at the upper solubility limit of xin
RFeAsO 1−xFxsystems,7and this is consistent with the obser-
vation that the superconducting phases in samples 1, 3, and4, which are heated at high temperatures or for longer timesand so are likely to have a slightly lower F content, havelower T
c’s/H20849average 32.1 K /H20850than the other three samples,
made under nominally identical “optimum” conditions,which have average T
c=34.8 K. Sample 6 shows the highest
Tc=36.2 K and the lowest proportion of the HoFeAsO 1−xFx
phase and a correspondingly low diamagnetic volume frac-
tion. This demonstrates that the sample is at the upper limitof the superconducting composition range and so gives arealistic T
c/H20849max /H20850for the HoFeAsO 1−xFxsystem.Although the doping values xfor the high-pressure
RFeAsO 1−xFxsamples are not known precisely, comparing
ensembles of samples with similar phase purities made undersimilar conditions reveals a clear suppression of supercon-ductivity by lattice effects for heavier R. For example, all of
our TbFeAsO
1−xFxsuperconductors have higher Tc’s/H20849five
TbFeAsO 1−xFxsamples, Tc=45–51 K /H20850than all of the
HoFeAsO 1−xFxmaterials /H20849in Table I/H20850. The Tc/H20849max /H20850values of
51, 45, and 36 K for RFeAsO 1−xFxwith R=Tb, Dy, and Ho,
respectively, thus represent the trend correctly.
Figure 4shows a plot of the maximum critical tempera-
tures, Tc/H20849max /H20850, against unit-cell volume for many reported
RFeAsO 1−xFxandRFeAsO 1−/H9254systems and our above mate-
rials. Tc/H20849max /H20850rises slowly as cell volume decreases for R
=La to Pr and then shows a broad maximum, between R
=Pr and Tb in the RFeAsO 1−xFxmaterials, before falling
rapidly as Rchanges from Tb to Dy to Ho. This trend is not
seen in the reported RFeAsO 1−xsuperconductors, where
Tc/H20849max /H20850remains approximately constant,14,15apparently be-
cause they have larger cell volumes than their RFeAsO 1−xFx
analogs /H20849see Fig. 4/H20850.
The size of the R3+cation tunes the electronic properties
through variations in the geometry of the FeAs slab. A trendbetween the As-Fe-As /H20849or equivalent Fe-As-Fe /H20850angle and T
c
has been reported for the early Rmaterials.16The upper
panel of Fig. 4shows representative reported values for op-
timal RFeAsO 1−xFxsuperconductors including our R=Tb,
Dy, and Ho materials. This demonstrates that the angle de-creases monotonically with Rsize and so does not show a
universal correlation with T
c/H20849max /H20850. The Tc/H20849max /H20850variation in
theRFeAsO 1−xFxseries is described by a simple cos /H20849/H9278
−/H92780/H20850function, shown in Fig. 4, where the value of the As-
/X32/X30 /X34/X30 /X36/X30 /X38/X30 /X31/X30/X30/X49/X6E/X74/X65/X6E/X73/X69/X74/X79 /X28/X31/X30/X33/X63/X6F/X75/X6E/X74/X73/X29
/X32θ(/X64/X65/X67/X72/X65/X65/X73 /X29/X32/X34
/X31/X33
/X30
FIG. 2. Fitted x-ray diffraction profile for HoFeAsO 0.9F0.1
/H20849sample 5 /H20850at room temperature. The Bragg markers /H20849from top to
bottom /H20850are for the minority phases, Ho 2O3and HoAs, and for
HoFeAsO 0.9F0.1.-1.0-0.6-0.2χ'(1/4π)
40 30 20 10 0
T(K)1
23 456
(b) (a)
FIG. 3. Superconductivity measurements for HoFeAs 0.9F0.1;/H20849a/H20850
ac magnetic volume susceptibility for the six samples; /H20849b/H20850resistiv-
ities for samples 4 and 6.TABLE I. Synthesis conditions /H20849all samples were synthesized at 10 GPa /H20850, refined lattice parameters and
volume, Tc’s, mass fractions, and superconducting volume fractions for HoFeAsO 1−xFxsamples.
Sampletsynth
/H20849hr/H20850Tsynth
/H20849°C/H20850a
/H20849Å/H20850c
/H20849Å/H20850Vol
/H20849Å3/H20850Tc
/H20849K/H20850Mass frac.
/H20849%/H20850Diamag. frac.
/H20849%/H20850
1 2 1150 3.8246 /H208493/H208508.254 /H208491/H20850 120.74 /H208493/H2085029.3 75 70
2 2 1100 3.8272 /H208492/H208508.2649 /H208498/H20850121.06 /H208492/H2085033.0 74 85
3 1 1150 3.8258 /H208495/H208508.264 /H208492/H20850 120.96 /H208494/H2085033.2 73 76
4 3 1100 3.8282 /H208495/H208508.261 /H208492/H20850 121.07 /H208495/H2085033.7 84 74
5 2 1100 3.8282 /H208492/H208508.2654 /H208497/H20850121.13 /H208492/H2085035.2 81 57
6 2 1100 3.8297 /H208497/H208508.270 /H208492/H20850 121.30 /H208497/H2085036.2 58 46BRIEF REPORTS PHYSICAL REVIEW B 80, 052508 /H208492009 /H20850
052508-2Fe-As angle corresponding to the global maximum Tc,
/H9278max=110.6°, is close to the ideal 109.5° value for a regular
FeAs 4tetrahedron. All five of the Fe 3 dbands are partially
occupied and contribute to the Fermi surface of the iron ar-senide superconductors through hybridization with As 4 sand
4pstates.
17Decreasing the tetrahedral angle through 109.5°
marks the crossover from tetragonal compression to elonga-tion of the FeAs
4tetrahedra. In a crystal-field model, this
reverses the splittings of the t2anded-orbital sets and so a
significant crossover in the real electronic structure is likelyto occur near 109.5°.
Evidence for the above crossover also comes from a dis-
covered change in the sign of the compositional dT
c/dVnear
optimum doping in the RFeAsO 1−xFxsystems.18The unit-cell
parameters and volume for the six HoFeAsO 1−xFxsamples in
Table Ishow a positive correlation with Tc/H20849Fig.5/H20850, in con-
trast to early R=La /H20849Ref.19/H20850and Sm /H20849Ref.7/H20850analogs, where
lattice parameters and volume decease with increasing Tc.
The Tc,Vpoints for near-optimally doped R=La, Sm, and
HoRFeAsO 1−xFxsuperconductors are shown in Fig. 4to-
gether with the derived dTc/dVvalues. dTc/dVfor a single
RFeAsO 1−xFxsystem follows the overall trend in
dTc/H20849max /H20850/dVfor different R’s, changing from a negative
value at large R=La to a small positive slope at R=Ho.
The compositional dTc/dVfor a given RFeAsO 1−xFxsys-
tem reflects two competing effects of variations in the fluo-ride content xon the lattice volume. F
−is slightly smaller
than O2−so the anion substitution effect gives a negativecontribution to the compositional dTc/dV, independent of R.
The concomitant effect of doping electrons into the Fe d
bands tends to expand the lattice /H20849and increase Tc/H20850, but the
magnitude of this positive dTc/dVterm depends on the na-
ture of the bands at the Fermi surface. The observed shiftfrom negative to positive dT
c/dVasRchanges from La to
Ho shows that the decreasing size of the R3+cation leads to
significant changes in the Fermi surface, with volume-expanding /H20849antibonding /H20850bands more prominent for smaller
R. Calculations have confirmed that the electronic structure
near the Fermi level is sensitive to such small changes in the
Aszcoordinate /H20849equivalent to changing the Fe-As-Fe
angle /H20850.
20Small changes in the contributions of the dbands
are likely to be particularly important in a multigap scenariofor superconductivity, as evidenced in gap measurements ofTbFeAsO
0.9F0.1and other iron arsenide materials.21
In summary, our analysis of multiple samples of
RFeAsO 1−xFx/H20849R=Tb, Dy, and Ho /H20850superconductors demon-
strates that the maximum critical temperature falls from 51 KforR=Tb to 36 K for the previously unreported Ho analog.
Hence, the effect on the lattice of substituting smaller laterare earths in the RFeAsO
1−xFxlattice suppresses supercon-
ductivity. This lattice control appears to be through tuning ofthe interatomic angles in the FeAs layer, with the optimumangle being 110.6°, near the ideal tetrahedral value. Thecompositional dT
c/dVchanges sign around the optimum
angle evidencing significant changes in the Fermi surface. Itappears difficult to increase the critical temperatures above56 K in 1111 type iron arsenide materials through tuning
lattice effects, although the possibility of higher T
c’s in other
structure types remains open.
We acknowledge EPSRC, the Royal Society of Edinburgh
and the Leverhulme trust for support.FIG. 4. Variation in Fe-As-Fe angle /H9278/H20849upper panel /H20850and super-
conducting Tc/H20849lower panel /H20850with unit-cell volume for different
RFeAsO 1−xFx/H20849circles /H20850/H20849Refs. 19,22,5,7, and 12/H20850andRFeAsO 1−/H9254
/H20849triangles /H20850/H20849Refs. 14and15/H20850.Tc/H20849max /H20850points are shown as filled
symbols. The fit of equation Tc/H20849max /H20850=Tc/H20849max /H208500cosA/H20849/H9278−/H92780/H20850with
parameters Tc/H20849max /H208500=56 K, A=0.03, and /H92780=110.6° is also
shown. dTc/dVvalues are derived from the data for suboptimally
doped materials /H20849open symbols /H20850in the R=La /H20849Ref.19/H20850,S m /H20849Ref.7/H20850,
and Ho /H20849this Brief Report /H20850systems.FIG. 5. Variations in Tcwith the tetragonal unit-cell parameters
and volume for the six HoFeAsO 1−xFxsamples in Table I.BRIEF REPORTS PHYSICAL REVIEW B 80, 052508 /H208492009 /H20850
052508-3*Corresponding author; j.p.attfield@ed.ac.uk
1P. Quebe, L. J. Terbuchte, and W. Jeitschko, J. Alloys Compd.
302,7 0 /H208492000 /H20850.
2J. W. Lynn and P. Dai, Physica C 469, 469 /H208492009 /H20850.
3Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.
Chem. Soc. 130, 3296 /H208492008 /H20850.
4H. Takahashi, K. Igawa, K. Arii, Y. Kamihara, M. Hirano, and H.
Hosono, Nature /H20849London /H20850453, 376 /H208492008 /H20850.
5Z. A. Ren, J. Yang, W. Lu, W. Yi, G. C. Che, X. L. Dong, L. L.
Sun, and Z. X. Zhao, Mater. Res. Innovations 12, 105 /H208492008 /H20850.
6Z. A. Ren, J. Yang, W. Lu, W. Yi, X. L. Shen, Z. C. Li, G. C.
Che, X. L. Dong, L. L. Sun, F. Zhou, and Z. X. Zhao, EPL 82,
57002 /H208492008 /H20850.
7X. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and D. F. Fang,
Nature /H20849London /H20850453, 761 /H208492008 /H20850.
8Z. A. Ren, W. Lu, J. Yang, W. Yi, X. L. Shen, Z. C. Li, G. C.
Che, X. L. Dong, L. L. Sun, F. Zhou, and Z. X. Zhao, Chin.Phys. Lett. 25, 2215 /H208492008 /H20850.
9R. H. Liu, G. Wu, T. Wu, D. F. Fang, H. Chen, S. Y. Li, K. Liu,
Y. L. Xie, X. F. Wang, R. L. Yang, L. Ding, C. He, D. L. Feng,and X. H. Chen, Phys. Rev. Lett. 101, 087001 /H208492008 /H20850.
10P. Cheng, L. Fang, H. X. Yang, X. Zhu, G. Mu, H. Luo, Z. Wang,
and H. Wen, Sci. China, Ser. G 51, 719 /H208492008 /H20850.
11Samples were synthesized from stoichiometric amounts of RAs,
Fe2O3, FeF 2, and Fe, using a Walker multianvil module within a
1000 ton press. The products were dense, black, sintered poly-crystalline pellets. Powder x-ray diffraction data were collectedon a Bruker AXS D8 diffractometer using Cu K
/H92511radiation.
Data were recorded at 10 /H113492/H9258/H11349100° with a step size of 0.007°
for Rietveld analysis. ac magnetic susceptibility was measuredfrom 3 to 50 K with a field of 0.5 Oe oscillating at 117 Hz usinga Quantum Design SQUID magnetometer. Electrical resistivitywas measured by a four-probe method between 1.7 and 300 Kusing a Quantum Design physical property measurement systemand an APD cryogenics closed cycle refrigeration unit with anin-house built sample stage.
12J.-W. G. Bos, G. B. S. Penny, J. A. Rodgers, D. A. Sokolov, A.D. Huxley, and J. P. Attfield, Chem. Commun. 31, 3634 /H208492008 /H20850.
13HoFeAsO 0.9F0.1has a tetragonal structure /H20849space group
P4/nmm ; results from fit shown in Fig. 2; goodness of fit /H92732
=1.60, residuals; Rwp=3.94 %,Rp=3.02 %; cell parameters a
=3.8282 /H208492/H20850Å,c=8.2654 /H208497/H20850Å; atom positions /H20851x,y,z/H20852and iso-
tropic temperature /H20849U/H20850factors; Ho /H208511
4,1
4,0.1454 /H208494/H20850/H20852,
0.044 /H208492/H20850Å2;A s /H208511
4,1
4,0.6659 /H208495/H20850/H20852, 0.029 /H208492/H20850Å2;F e /H208493
4,1
4,1
2/H20850,
0.014 /H208492/H20850Å2;O , F /H208493
4,1
4,0/H20850, 0.26 /H208492/H20850Å2/H20850. The secondary Ho 2O3
phase is in a high-pressure B-type rare-earth oxide modification,
space group C2/m,a=13.841 /H208492/H20850Å, b=3.4984 /H208495/H20850Å, c
=8.608 /H208491/H20850Å,/H9252=100.08 /H208491/H20850°.
14K. Miyazawa, K. Kihou, P. M. Shirage, C. H. Lee, H. Kito, H.
Eisaki, and A. Iyo, J. Phys. Soc. Jpn. 78, 034712 /H208492009 /H20850.
15J. Yang, X. L. Shen, W. Lu, W. Yi, Z. C. Li, Z. A. Ren, G. C.
Che, X. L. Dong, L. L. Sun, F. Zhou, and Z. X. Zhao, New J.Phys. 11, 025005 /H208492009 /H20850.
16J. Zhao, Q. Huang, C. de la Cruz, S. Li, J. W. Lynn, Y. Chen, M.
A. Green, G. F. Chen, G. Li, Z. Li, J. L. Luo, N. L. Wang, andP. Dai, Nature Mater. 7, 953 /H208492008 /H20850.
17D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003 /H208492008 /H20850.
18The compositional dTc/dVquantifies the changes in Tcand unit-
cell volume Vdue to variations in doping level xat constant
/H20849atmospheric /H20850pressure, and is complementary to the pressure-
induced dTc/dVat constant x. Both derivatives are negative for
LaFeAsO 1−xFx, and we thus predict a positive pressure-induced
dTc/dV /H20849pressure suppression of superconductivity /H20850for
HoFeAsO 1−xFx.
19Q. Huang, J. Zhao, J. W. Lynn, G. F. Chen, J. L. Luo, N. L.
Wang, and P. Dai, Phys. Rev. B 78, 054529 /H208492008 /H20850.
20S. Lebègue, Z. P. Yin, and W. E. Pickett, New J. Phys. 11,
025004 /H208492009 /H20850.
21K. A. Yates, K. Morrison, J. A. Rodgers, G. B. S. Penny, J. W. G.
Bos, J. P. Attfield, and L. F. Cohen, New J. Phys. 11, 025015
/H208492009 /H20850.
22G. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong, P. Zheng, J.
L. Luo, and N. L. Wang, Phys. Rev. Lett. 100, 247002 /H208492008 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 80, 052508 /H208492009 /H20850
052508-4 |
PhysRevB.100.085415.pdf | PHYSICAL REVIEW B 100, 085415 (2019)
Ballistic thermoelectric properties of monolayer semiconducting transition
metal dichalcogenides and oxides
G. Özbal ,1R. T. Senger,1,2C. Sevik,3and H. Sevinçli4,*
1Department of Physics, Izmir Institute of Technology, 35430 Izmir, Turkey
2ICTP-ECAR Eurasian Center for Advanced Research, Izmir Institute of Technology, 35430 Izmir, Turkey
3Department of Mechanical Engineering, Faculty of Engineering, Eski¸ sehir Technical University, Eski¸ sehir, TR 26555, Turkey
4Department of Materials Science and Engineering, Izmir Institute of Technology, 35430 Izmir, Turkey
(Received 29 January 2019; revised manuscript received 17 May 2019; published 12 August 2019)
Combining first-principles calculations with Landauer-Büttiker formalism, ballistic thermoelectric transport
properties of semiconducting two-dimensional transition metal dichalcogenides (TMDs) and oxides (TMOs)(namely MX
2with M =C r ,M o ,W ,T i ,Z r ,H f ;X =O, S, Se, Te) are investigated in their 2H and 1T phases.
Having computed structural, as well as ballistic electronic and phononic transport properties for all structures,we report the thermoelectric properties of the semiconducting ones. We find that 2H phases of four of thestudied structures have very promising thermoelectric properties, unlike their 1T phases. The maximum roomtemperature p-type thermoelectric figure of merit ( ZT) of 1.57 is obtained for 2H-HfSe
2, which can be as high
as 3.30 at T=800 K. Additionally, 2H-ZrSe 2, 2H-ZrTe 2, and 2H-HfS 2have considerable ZTvalues (both n-
andp-type), that are above 1 at room temperature. The 1T phases of Zr and Hf-based oxides possess relatively
high power factors, however their high lattice thermal conductance values limit their ZTvalues to below 1 at
room temperature.
DOI: 10.1103/PhysRevB.100.085415
I. INTRODUCTION
Thermoelectric (TE) materials make it possible to drive
electric currents using temperature gradients, and converselycooling of a system just by using a voltage difference, namelythe Seebeck and Peltier effects, respectively. The performanceof TE conversion is quantified by the dimensionless figureof merit ZT, which includes strongly interrelated electronic
and thermal transport properties. Because of this interrelationamong Seebeck coefficient ( S), electrical conductance ( G),
and thermal conductance ( κ), significant enhancement of ZT
is an extremely difficult task. Therefore improvement of TEefficiency has been relatively slow, and typical ZTvalues do
not exceed 1 for bulk materials [ 1]. With the advances in
the production of low-dimensional structures, a new quest forhigh performance TE materials gained acceleration [ 2–7]. The
advent of atomically thin graphene provided a new platformto study transport and thermoelectric properties in two andone dimensions (2D and 1D) [ 8–10]. However, the absence
of an electronic band gap in 2D graphene and ultrahighthermal conductivity suppress its thermoelectric efficiency[11–15]. Still, there are numerous proposals to enhance the
TE performance of graphene [ 9,16–26]. A more recent family
of 2D materials, semiconducting TMDs and TMOs attractedattention due to the wide range of band gaps and lower latticethermal conductivities. One of the most detailed studies onstability, electronic, mechanical, and magnetic analysis ofsingle layer TMDs and TMOs belongs to Ataca et al. [27]. The
majority of the theoretical studies have been devoted towards
*haldunsevincli@iyte.edu.trMX 2(M=Mo,W; X =S,Se) monolayers [ 28–36], few layers
[37–42], hybrid nanoribbons [ 43,44], or heterostructures [ 45].
Phonon engineering [ 46], band structure engineering [ 47], and
strain engineering [ 48–52] approaches were also employed
extensively with the aim of improving thermoelectric perfor-mance of MX
2structures. In addition to Mo and W based
compounds, there are also a few studies Zr and Hf basedTMDs in their 1T phases [ 53–57]. However, a comprehensive
study on thermoelectric properties of pristine TMD /TMOs,
specifically the 2H phase of Ti, Zr, and Hf based structures,is still lacking. Here, we focus on expanding the library of2D TMD /TMO candidates starting with an investigation of
their ballistic properties. Electronic transport, thermal trans-port, and thermoelectric properties of 26 dynamically stablesemiconducting TMDs /TMOs are explored. Structural pa-
rameters are computed for obtaining accurate electronic bandstructures and vibrational spectra based on ab initio calcula-
tions. Thermoelectric coefficients are computed by combiningfirst-principles calculations and Landauer-Büttiker formalism.Also band gap corrections are performed using hybrid HSE06functionals when necessary.
II. METHODS
The geometrical optimization and electronic structure
calculations are performed using density functional theory(DFT) using plane-wave basis sets [ 58] by employing pro-
jector augmented wave (PAW) potentials [ 59]. The exchange-
correlation potential has been approximated by general-ized gradient approximation (GGA) using Perdew-Burke-Ernzerhof (PBE) functionals [ 60]. The plane-wave cutoff
energies are found to be in the range from 250 to 500 eV with
2469-9950/2019/100(8)/085415(10) 085415-1 ©2019 American Physical SocietyÖZBAL, SENGER, SEVIK, AND SEVINÇLI PHYSICAL REVIEW B 100, 085415 (2019)
convergence tests for each structure. The irreducible Brillouin
zone is sampled using the Monkhorst-Pack scheme with gridsizes of n×n×1(n=5–15) according to the convergence
tests [ 61]. The convergence thresholds for ionic and electronic
relaxations are set to 10
−3eV/Å and 10−6eV , respectively.
During the geometry optimization process, cell shape andvolume are preserved. The vacuum spacing is set to 15 Åto avoid any spurious interactions between layers. In orderto correct the band gap values Heyd-Scuseria-Ernzerhorf(HSE06) [ 62] hybrid functionals are used for selected five
MX
2structures, where 0.25 exact Hartree-Fock and 0.75 PBE
exchange mixing and the screening parameter of 0.2 Å−1are
used. The calculations are performed non-spin-polarized andspin-orbit interactions are not taken into consideration. Theinteratomic force constants (IFCs) are obtained by employingdensity functional perturbation theory (DFPT) [ 63]. Phonon
band structure and heat capacity calculations are performedby using the PHONOPY package [ 64].
The cohesive energy (E
c) per atom is computed as
Ecoh=(nXEX+nMEM−EMX 2)/(nX+nM). (1)
Here, nX(M)denotes the number of chalcogen (transition
metal) atoms in the unit cell. EX(M)is the energy of the
isolated single atoms, and EMX 2is the total energy of the MX 2
monolayer. In order to gain an understanding on bond charac-
teristics, charge transfer calculations are conducted by usingthe Bader method [ 65]. The percent ionic character (% IC)o f
metal and chalcogen /oxygen atoms can be calculated roughly
as [66]
%IC={1−exp[−0.25(X
A−XB)2]}×100 (2)
where XAand XBare electronegativities of the constituent
atoms.
Electronic transmission spectrum is given by the number
of transmission channels in the ballistic limit. Since eachstudied structure exhibits hexagonal symmetry, their trans-mission spectra are isotropic. Therefore both transmissionspectra and thermoelectric coefficients are given along onedirection. Dense k-point meshes 200 ×200×1 and 100 ×
100×1 are used in calculating transmission spectra using
PBE and HSE06 functionals, respectively.
Derivation of the electronic coefficients is performed by
using [ 67,68]
L
n(μ,T)=−2
h/integraldisplay
dEτel(E)(E−μ)n∂fFD(E,μ,T)
∂E,(3)
with nbeing an integer, τel(E) the electronic transmis-
sion spectrum, and fFD(E,μ,T) the Fermi-Dirac distribution
function at temperature Tand chemical potential μ.U s i n g
Ln, one can express the electrical conductance ( G), Seebeck
coefficient ( S), and the electrical part of the thermal con-
ductance ( κel)a sG=e2L0,S=(L1/L0)/eT, andκel=(L2−
L2
1/L0)/T, respectively.
Phonon thermal conductance is calculated using Landauer
formalism [ 69,70],
κph=1
2π/integraldisplay
dω¯hωτ ph(ω)∂fBE(ω,T)
∂T, (4)
where ωis the vibrational frequency, fBEstands for Bose-
Einstein distribution function, and τph(ω) is the phonon
FIG. 1. Crystal structures of TMDs and TMOs. The 2H phase
(a) and the 1T phase (b). The Brillouin zone, reciprocal lattice
vectors, and the high symmetry points are given in (c).
transmission spectrum obtained from counting phonon modes
with an average of 200 qpoints in the transverse direction.
After computing the electronic and phononic contributions tothe transport, the dimensionless thermoelectric figure of meritis obtained using
ZT=S
2GT/(κel+κph). (5)
III. STRUCTURAL AND ELECTRONIC PROPERTIES
MX 2monolayers consist of three atomic layers in the
sequence of X-M-X. These sublayers are arranged in thewell-known two phases (polymorphs): trigonal prismatic 2Hwhich is a member of P¯6m2(D
3h) symmetry group and
octahedral 1T which belongs to the P¯3m1(D3d). Schematic
representation of 2H and 1T phases from top and side viewsare shown in Figs. 1(a) and1(b), respectively, and their first
Brillouin zone is given in Fig. 1(c).
We first perform geometry optimizations and check the
dynamical stabilities of the TMD /TMO monolayers by
checking whether all vibrational frequencies are real andpositive. While all 2H structures group-VIB (Cr, Mo, W)TMDs /TMOs are dynamically stable semiconductors, their
1T phases are unstable. Their distorted 1T
dphases are dy-
namically stable, but all are metallic and therefore they arenot within the scope of this study. On the other hand, both2H and 1T phases of group IVB (Ti, Zr, Hf) TMDs /TMOs
are dynamically stable and semiconducting. The structuralparameters which determine the geometry are tabulated inTable I. The obtained parameters are in good agreement with
the literature [ 71].
For a given phase, d
MXincreases with increasing a.H o w -
ever, different phases (2H and 1T) of a given MX 2(ZrSe 2,
HfS 2, and HfSe 2) follow an opposite trend. For example,
the 1T phase of ZrSe 2has larger athan its 2H phase but
smaller dMXandh. For these structures, ais always larger
for the 1T phase, whereas dMXis reduced by ∼0.7% and h
is reduced by ∼6–7%. Bond angles θ1andθ2follow opposite
trends. Comparing two MX 2structures, one observes that the
structure with larger θ1has smaller θ2.A l s o θ1of the 1T phase
of a given structure is always larger than that in the 2H phase.
For compounds of the same phase, Ecohdecreases with
increasing aas expected. A comparison of 2H and 1T phases
of the same TMD reveal that Ecohis always larger for the 1T
085415-2BALLISTIC THERMOELECTRIC PROPERTIES OF … PHYSICAL REVIEW B 100, 085415 (2019)
TABLE I. Structural and electronic properties of semiconducting TMDs and TMOs, which are dynamically stable. The lattice parameter
of the unit cell ( a), bond lengths ( dMX), layer heights ( h), bond angles ( θ1,θ2), band gap ( EPBE
g), cohesive energy ( Ecoh), transferred charge to
X(ρM), charge received by X ( ρX), and the fractional ionic character (FIC), respectively. Bond lengths and angles are shown in Fig. 1and
electronic band structures are illustrated in Fig. S2 [ 93].
ad MX h θ1 θ2 EPBE
g Ecoh ρM ρX FIC
MX 2 Phase (Å) (Å) (Å) (deg) (deg) (eV) (eV /atom) ( e−)( e−)( % )
CrO 2 2H 2.63 1.91 2.32 74.77 86.96 0.37 5.34 1.48 −0.74 54.51
CrS 2 2H 3.04 2.29 2.94 79.82 83.26 0.93 4.17 1.00 −0.50 19.07
CrSe 2 2H 3.21 2.43 3.14 80.59 82.68 0.75 3.64 0.81 −0.41 17.97
CrTe 2 2H 3.48 2.64 3.41 80.63 82.66 0.53 3.09 0.56 −0.28 4.73
MoO 2 2H 2.83 2.05 2.47 74.18 87.39 0.91 6.24 1.67 −0.84 33.61
MoS 2 2H 3.18 2.41 3.13 80.77 82.55 1.67 5.14 1.07 −0.57 4.31
MoSe 2 2H 3.32 2.54 3.34 82.12 81.54 1.44 4.60 0.83 −0.42 3.73
MoTe 2 2H 3.55 2.73 3.61 82.74 81.07 1.08 4.04 0.52 −0.26 0.09
WO 2 2H 2.83 2.05 2.48 74.34 87.27 1.36 7.02 1.83 0.92 25.29
WS 2 2H 3.19 2.42 3.14 80.87 82.47 1.79 5.80 1.21 −0.61 1.20
WSe 2 2H 3.32 2.55 3.35 82.44 81.30 1.54 5.19 0.92 −0.46 0.90
WTe 2 2H 3.55 2.74 3.62 82.89 80.96 1.06 4.54 0.58 −0.29 1.68
TiS 2 2H 3.34 2.45 3.02 75.99 86.07 0.73 5.17 1.49 −0.75 23.69
TiSe 2 2H 3.49 2.58 3.24 77.61 84.89 0.60 4.68 1.39 −0.70 22.51
TiTe 2 2H 3.74 2.80 3.57 79.28 83.66 0.19 4.12 1.23 −0.61 7.54
ZrO 2 1T 3.28 2.12 1.93 79.05 100.95 4.44 7.71 2.54 −1.27 67.14
ZrS 2 1T 3.69 2.57 2.90 88.59 91.41 1.20 5.89 2.05 −1.02 32.34
ZrSe 2 2H 3.71 2.73 3.38 76.61 85.62 0.79 5.21 1.80 −0.90 31.07
1T 3.80 2.71 3.17 90.73 89.27 0.51 5.35 1.87 −0.94 31.07
ZrTe 2 2H 3.93 2.94 3.73 78.83 83.99 0.45 4.62 1.58 −0.79 13.78
HfO 2 1T 3.24 2.11 1.93 79.38 100.62 4.87 7.90 2.32 −1.16 68.17
HfS 2 2H 3.53 2.57 3.13 74.92 86.85 1.09 5.78 1.84 −0.92 33.61
1T 3.64 2.55 2.89 88.90 91.11 1.29 6.00 1.91 −0.95 33.61
HfSe 2 2H 3.67 2.70 3.36 76.78 85.50 0.88 5.25 1.68 −0.84 32.34
1T 3.76 2.68 3.14 90.92 89.08 0.60 5.42 1.76 −0.88 32.34
HfTe 2 2H 3.90 2.91 3.69 78.63 84.14 0.36 4.61 1.48 −0.74 14.79
phase. That is to say, 1T phases of studied Zr and Hf based
compounds are energetically more stable, in agreement withprevious studies [ 72–76]. According to Table I, TMOs exhibit
the highest ionic character in general, which is because ofthe largest charge transfer between the transition metal andthe oxygen atoms. 1T-HfO
2, which has the highest cohesive
energy and the widest electronic band gap shows the highestionic character, whereas 2H-MoTe
2possesses fractional cova-
lent character.
Electronic band diagrams of the investigated structures are
plotted in the Supplemental Material (see Fig. S2) [ 93]. The
2H phases of group-VIB dichalcogenides are direct band gapsemiconductors, whereas their oxides have indirect band gaps.Group-IVB dichalcogenides and oxides are all indirect semi-conductors. Electronic band gaps ranging between 0.19 eVand 4.87 eV are obtained with PBE functionals. HSE06 cal-culations are performed for materials which have a E
PBE
gless
than 0.5 eV . This is because the main effect of the hybrid func-tional to the band structure is to increase the band gap withchanging the band dispersions only slightly. When the bandgap of a structure is less than 10 k
BT, simultaneous contribu-
tion from the holes in the valence band and electrons in theconduction band suppresses the Seebeck coefficient [ 77–81].
The effect of hybrid functionals on electronic transport andthermoelectric properties will be discussed in more detaillater. The electronic band diagrams of 2H-CrO
2,2 H - T i T e 2,
2H-ZrTe 2, and 2H-HfTe 2within PBE +HSE06 functionals are
presented in Fig. 2. Band gap values are increased to 0.90,
0.97, 1.05, and 0.93 eV for CrO 2,T i T e 2,Z r T e 2, and HfTe 2,
respectively.
IV . VIBRATIONAL PROPERTIES
It is necessary to check whether imaginary or negative
frequencies exist in phonon dispersions to check the dy-namical stabilities of the structures. We note that dynami-cal stability is a necessary condition but not conclusive forexperimental realization. When both 2H and 1T phases ofMX
2monolayers are considered, 30 structures are found to
be dynamically stable. Four of these structures are excludedin this study as they are either metallic or semi-metallic.Phonon spectra of the remaining 26 structures are given inthe Supplemental Material (see Fig. S1) [ 93]. A cautionary
note is in order here. Smearing is a computational tool thatsmoothens the Fermi distribution function around the Fermienergy. It is a necessary ingredient in DFT calculations. Evengraphene can be found dynamically unstable if appropriatesmearing is not used. There does not exist a recipe fordetermining the smearing method or value. There are a fewmethods to implement smearing, most of them without a clear
085415-3ÖZBAL, SENGER, SEVIK, AND SEVINÇLI PHYSICAL REVIEW B 100, 085415 (2019)
CrO2
ZrTe2 HfTe2TiTe2
FIG. 2. Calculated electronic band structures of selected
2H-MX 2compounds; CrO 2, TiTe 2, ZrTe 2,H f T e 2b a s e do nP B E( r e d
solid line) and PBE +HSE (blue dashed line) functional. Fermi level
is set to zero for all subfigures.
physical meaning. Fermi-Dirac smearing, on the other hand,
is interpreted as electronic temperature. Being only applied onelectrons and not on ions, it should be not confused with realtemperature. Still, Fermi-Dirac smearing is used to identifytemperature dependent stabilization in certain cases [ 82–84].
In this work, we implement Fermi-Dirac smearing and scanpossible smearing values systematically.
For lower values of smearing ( σ=0.05 eV) 2H-TiS
2,
2H-ZrSe 2, and 2H-HfS 2out-of-plane ZA mode possess neg-
ative frequencies around the high symmetry points, whereas2H-HfSe
2has negative frequencies ony around the Kpoint.
When the smearing is increased ( σ=0.4 eV for 2H-HfSe 2
andσ=0.5 eV for 2H-TiS 2,2 H - Z r S e 2,2 H - H f S 2)a l l
phonons frequencies are found positive. These σvalues are
in the same range with those used in the literature [ 82,85,86].
The phonon band gap, which separates the acoustic modes
from the six optical branches, decreases with decreasingmass difference between the constituent elements of MX
2
compounds. The acoustic bandwidths become narrower withincreasing average mass, whereas all bands are pushed to-wards lower frequencies with increasing total mass of thecompounds. In previous studies, in which phonon-phononscattering was taken into account, the scatterings were limitedwhen a band gap is present. Therefore the absence of aphonon band gap was found helpful to reduce lattice thermalconductivity. Conversely, in the ballistic regime, the presenceof a phonon band gap reduces lattice thermal conductivity,TABLE II. Phonon thermal conductance values for various
temperatures.
κph(nW/K/nm)
MX 2 Phase 300 K 500 K 800 K
CrO 2 2H 2.09 2.49 2.66
CrS 2 2H 1.24 1.34 1.37
CrSe 2 2H 0.83 0.87 0.88
CrTe 2 2H 0.60 0.62 0.63
MoO 2 2H 1.63 1.89 2.00
MoS 2 2H 1.03 1.10 1.13
MoSe 2 2H 0.72 0.75 0.76
MoTe 2 2H 0.54 0.55 0.55
WO 2 2H 1.29 1.48 1.56
WS 2 2H 0.83 0.89 0.91
WSe 2 2H 0.66 0.68 0.68
WTe 2 2H 0.50 0.51 0.51
TiS 2 2H 0.95 1.00 1.02
TiSe 2 2H 0.95 0.99 1.00
TiTe 2 2H 0.70 0.72 0.73
ZrO 2 1T 1.45 1.71 1.81
ZrS 2 1T 0.83 0.87 0.89
ZrSe 2 2H 0.54 0.55 0.56
1T 0.71 0.72 0.73
ZrTe 2 2H 0.55 0.56 0.56
HfO 2 1T 1.28 1.50 1.60
HfS 2 2H 0.65 0.67 0.68
1T 0.71 0.74 0.75
HfSe 2 2H 0.51 0.51 0.52
1T 0.59 0.61 0.61
HfTe 2 2H 0.48 0.49 0.49
simply because there is no transmission within gap. Briefly,
both a wide phonon band gap and reduced phonon frequenciesdecrease thermal conductance and enhance the TE perfor-mance. Thermal conductance of the investigated TMD /TMOs
at various temperatures are listed in Table II.2 H - C r O
2is
composed of the lightest atoms in the group, hence has thelargest phonon thermal conductance at all temperatures. Astemperature increases thermal conductance of CrO
2increases
considerably. A similar trend appears for all the TMOs, whichis associated with the relatively higher ω
maxvalues, because
of oxygen being the lightest element in group VIA. As will bediscussed later, 2H phases of ZrSe
2,Z r T e 2,H f S 2, and HfSe 2
are found to be both n- and p-type promising thermoelectric
candidates. In order to clarify influence of thermal propertieson thermoelectric performance, vibrational spectra and trans-port properties are presented in Fig. 3, only for these four
materials.
In a previous study, which investigated the thermal conduc-
tivities of 2H group-VIB TMD family by using the Boltzmanntransport equation [ 87], it was found that the thermal con-
ductivities of sulfides (MS
2) and selenides (MSe 2) increase
as M changes from Cr to Mo, and from Mo to W, due tothe rapid increase in the phonon relaxation time. In contrastto this, we find that κ
phdecreases as M changes from Cr to
Mo to W at the ballistic limit, because of increasing atomicmasses. This inverse behavior is validated with the calculation
085415-4BALLISTIC THERMOELECTRIC PROPERTIES OF … PHYSICAL REVIEW B 100, 085415 (2019)
FIG. 3. Phonon dispersion relations, phonon transmission spec-
tra, and phonon thermal conductance as a function of temperature ofZrSe
2, ZrTe 2,H f S 2,a n dH f S e 2are shown, respectively.
of correlation between average atomic mass of the unit cell
and phonon thermal conductance, where %80, %78, and %77inverse correlations are found for 300 K, 500 K, and 800 K, re-spectively. In addition, it is found that the correlation betweenmandκ
phdoes not change at even higher temperatures.
We also note that in a recent theoretical study the bal-
listic thermal conductance value of MoS 2was reported as
1.06 nW /K for a sample having a width of 1.27 nm [ 88].
The corresponding thermal conductance per width value (0.84nW K
−1nm−1) is considerably less than our present result
(1.03 nW K−1nm−1). The disagreement is because we em-
ploy a fine sampling of the kpoints in the transmission
spectrum, whereas Cai et al. uses only the /Gamma1point. Hence
they find a stepwise transmission spectrum like in a one-dimensional system, which overestimates the contributionsfrom low energies.
FIG. 4. Heat capacities at various temperatures from 300 K to
1000 K are shown for the semiconducting compounds in the 2H
phase.
The vibrational heat capacity at constant volume is calcu-
lated using
Cν=kB/integraldisplay
dωρ(ω)p(ω,T), (6)
where ρis the phonon density of states, p(x)=−x2∂fBE/∂x,
fBE=1/(ex−1) being the Bose-Einstein distribution func-
tion, and x=¯hω/kBT.I nF i g s . 4and5the vibrational heat
capacities are plotted at T=300 K, 500 K, 800 K, and
1000 K. At lower temperatures, the heat capacity is dominatedby low frequency modes which possess low group veloc-ity and larger phonon density of states. Therefore, heaviercompounds, like Hf and Zr based structures, have higher heat
FIG. 5. Heat capacities at various temperatures from 300 K to
1000 K are shown for the semiconducting compounds in the 1T
phase.
085415-5ÖZBAL, SENGER, SEVIK, AND SEVINÇLI PHYSICAL REVIEW B 100, 085415 (2019)
capacities at 300 K. At increased temperatures, the differences
in the Cvtend to decrease. At 1000 K, the function p(x)i s
almost constant and equal to unity in the entire spectrum.The heat capacities approach the classical limit, which isproportional to the number of modes per unit cell.
V . THERMOELECTRIC PROPERTIES
According to the Mott formula [ 20,89,90]
S(T,μ)≈π2k2
BT
3edlnτ(E)
dE/vextendsingle/vextendsingle/vextendsingle/vextendsingle
μ, (7)
the logarithmic derivative of the electronic transmission deter-
mines the Seebeck coefficient at low temperatures. Namely,the abrupt changes in the transmission spectrum gives riseto large Seebeck coefficient and power factor. The structuresstudied in this work agree with this rule of thumb. The ther-moelectric coefficients, S,P, and ZT, for various temperatures
are tabulated in Table III. The chemical potential ( μ) is chosen
around the band edges where ZTis maximized. The differ-
ence between μat the valence band edge (conduction band
edge) and the μwhere p-type ZT(n-type ZT) is maximized
is crucial for determining the optimal doping levels of thesemiconductor (see Table S1) [ 93]. One observes that most
of the 2H group-VIB TMD /TMOs have relatively low ZTvalues compared to the 2H group-IVB TMD /TMOs. Notably,
oxide compounds from group VIB show considerably weakTE performance due to their low atomic masses and hencehighκ
ph. While p-type ZTvalues of the group-VIB TMOs
reach a maximum value around 0.11 at room temperature, thecorresponding values for n-type ZTcan be as high as 0.16.
There are various theoretical studies on the TE properties ofMX
2(M=Mo,W; X =S,Se) monolayers. In a previous work,
n-type ZTof the most studied MoS 2monolayer was pre-
dicted 0.04 by using the Boltzmann equation and equilibriummolecular dynamics (EMD) simulations [ 33]. Wickramaratne
et al. obtained different n-type ZT values (0.87 /1.35) by
adopting layer thickness dependent and constant κ
phvalues
in diffusive regime calculations. In another work, reportedZTvalues are overestimated compared to our findings for
well-studied MoS
2, MoSe 2,W S 2, and WSe 2in the frame of
ballistic transport [ 30]. In addition, previously reported p-
andn-type ZTvalues (0.58 /0.25) in the ballistic regime are
consistent with our results (0.47 /0.22) [ 34]. Also, there is an
agreement on the results of Huang et al. that p-type ZTof
MoS 2at room temperature and n-type ZTof WSe 2at high
temperatures are found to be higher than those of the MoSe 2
and WS 2[34].
Among all investigated compounds, ZrSe 2,H f S 2, and
HfSe 2are dynamically stable in both 2H and 1T phases.
TABLE III. p-a n d n-type Seebeck coefficient ( S), power factor ( P), and thermoelectric figure of merit ( ZT) at different temperatures based
on PBE calculations.
S(10−4V/K) P(10−3nW/K2nm) ZT
MX 2 Phase 300 K 500 K 800 K 300 K 500 K 800 K 300 K 500 K 800 K
CrO 2a2H 2.01 /−1.86 1.99 /−1.93 1.49 /−1.80 0.69 /1.15 0.89 /1.52 0.98 /1.92 0.10 /0.15 0.17 /0.27 0.23 /0.44
CrS 2 2H 2.04 /−1.99 2.20 /−2.23 2.47 /−2.43 1.57 /1.20 2.58 /1.66 3.50 /2.41 0.33 /0.27 0.74 /0.53 1.40 /1.02
CrSe 2 2H 2.13 /−1.97 2.35 /−2.39 2.65 /−2.66 1.30 /1.31 1.87 /1.75 2.74 /2.50 0.41 /0.40 0.83 /0.80 1.59 /1.49
CrTe 2 2H 2.08 /−2.24 2.48 /−2.40 2.46 /−2.43 1.35 /1.26 1.82 /1.77 2.76 /2.69 0.55 /0.54 1.08 /1.05 1.70 /1.59
MoO 2 2H 1.92 /−1.94 1.95 /−1.97 2.02 /−2.15 0.51 /0.82 0.66 /1.06 0.84 /1.35 0.09 /0.14 0.17 /0.26 0.30 /0.47
MoS 2 2H 2.05 /−2.07 2.37 /−2.13 2.52 /−2.52 1.89 /0.81 2.31 /1.17 2.88 /1.98 0.47 /0.22 0.86 /0.45 1.46 /0.97
MoSe 2 2H 2.08 /−2.00 2.27 /−2.45 2.57 /−2.79 1.02 /0.95 1.38 /1.68 1.95 /3.17 0.38 /0.35 0.74 /0.81 1.38 /1.91
MoTe 2 2H 2.18 /−2.11 2.36 /−2.48 2.71 /−2.99 1.06 /0.97 1.46 /1.62 1.98 /2.66 0.51 /0.46 1.00 /1.00 1.85 /2.21
WO 2 2H 2.01 /−1.88 2.01 /−2.10 2.06 /−2.14 0.50 /0.75 0.65 /0.96 0.82 /1.25 0.11 /0.16 0.21 /0.30 0.37 /0.54
WS 2 2H 2.08 /−2.00 2.28 /−2.12 2.58 /−2.62 1.41 /0.67 1.95 /1.00 2.38 /1.90 0.43 /0.22 0.86 /0.46 1.50 /1.08
WSe 2 2H 2.06 /−1.97 2.24 /−2.43 2.51 /−2.85 0.83 /0.86 1.09 /1.93 1.47 /3.43 0.34 /0.33 0.67 /0.92 1.21 /2.18
WTe 2 2H 1.99 /−2.07 2.37 /−2.33 2.59 /−2.82 0.82 /0.68 1.06 /0.98 1.42 /1.72 0.42 /0.36 0.83 /0.74 1.49 /1.57
TiS 2 2H 2.42 /−2.24 2.47 /−2.61 2.52 /−2.52 3.74 /4.43 3.98 /4.84 3.98 /5.29 0.98 /1.05 1.55 /1.79 2.14 /2.66
TiSe 2 2H 2.19 /−2.39 2.38 /−2.54 2.33 /−2.39 3.33 /3.43 3.68 /3.99 4.31 /4.64 0.86 /0.90 1.38 /1.51 1.81 /2.09
TiTe 2a2H 1.94 /−2.16 1.26 /−1.64 1.11 /−1.21 1.19 /2.78 1.39 /3.06 2.65 /2.86 0.40 /0.90 0.38 /0.87 0.38 /0.55
ZrO 2 1T 2.20 /−2.01 2.37 /−2.34 2.47 /−2.58 5.12 /3.35 5.56 /5.17 5.65 /6.25 0.86 /0.52 1.28 /1.08 1.87 /1.94
ZrS 2 1T 2.02 /−2.19 2.21 /−2.30 2.45 /−2.69 0.67 /1.85 0.88 /2.40 1.31 /2.78 0.23 /0.57 0.44 /1.04 0.87 /1.75
ZrSe 2 2H 2.38 /−2.43 2.63 −/2.76 2.75 /−2.87 3.36 /3.59 3.34 /4.00 3.29 /4.21 1.41 /1.42 2.19 /2.41 2.96 /3.61
ZrSe 2 1T 1.93 /−2.09 2.13 /−2.40 1.90 /−2.05 0.56 /1.80 0.72 /2.22 0.91 /2.71 0.22 /0.63 0.43 /1.16 0.65 /1.76
ZrTe 2a2H 2.26 /−2.35 2.65 /−2.50 2.11 /−2.02 2.56 /2.82 2.84 /3.21 4.10 /3.85 1.06 /1.18 1.73 /1.88 1.58 /1.67
HfO 2 1T 2.33 /−2.28 2.45 /−2.45 2.53 /−2.63 4.80 /2.67 5.28 /3.92 5.43 /5.30 0.92 /0.54 1.38 /0.99 2.00 /1.80
HfS 2 2H 2.38 /−2.26 2.62 /−2.62 2.70 /−2.83 3.85 /3.33 3.78 /3.59 3.59 /3.88 1.38 /1.17 2.11 /1.89 3.03 /2.92
HfS 2 1T 2.05 /−2.32 2.21 /−2.36 2.38 /−2.71 0.66 /1.87 0.85 /2.39 1.16 /2.69 0.26 /0.67 0.50 /1.19 0.92 /1.96
HfSe 2 2H 2.55 /−2.47 2.73 /−2.75 2.83 /−2.91 3.36 /2.74 3.35 /2.91 3.40 /3.19 1.57 /1.28 2.36 /2.04 3.30 /3.04
HfSe 2 1T 1.92 /−2.20 2.13 /−2.46 2.12 /−2.65 0.56 /1.84 0.73 /2.23 0.92 /2.61 0.26 /0.75 0.51 /1.35 0.84 /2.11
HfTe 2a2H 2.29 /−2.31 2.24 /−2.44 1.74 /−1.86 1.05 /2.41 1.52 /2.64 2.48 /3.09 0.56 /1.17 1.00 /1.75 0.91 /1.34
aElectronic transport and thermoelectric properties are performed based on HSE06 +PBE functional for these selected MX 2compounds.
p/n-type ZT values for various temperatures are listed in Table IV. In addition τel(E),S,PF,a n d ZTare demonstrated in Fig. S3 [ 93].
085415-6BALLISTIC THERMOELECTRIC PROPERTIES OF … PHYSICAL REVIEW B 100, 085415 (2019)
TABLE IV. p-a n d n-type ZTvalues at different temperatures
based on the HSE06 calculations.
ZT(p-/n-type)
MX 2 Phase 300 K 500 K 800 K
CrO 2 2H 0.09 /0.13 0.17 /0.23 0.30 /0.42
TiTe 2 2H 0.47 /0.81 0.90 /1.37 1.70 /2.18
ZrTe 2 2H 0.80 /1.08 1.49 /1.74 2.47 /2.67
HfTe 2 2H 0.52 /1.03 1.00 /1.71 1.76 /2.62
TiS 2 1T 0.11 /0.37 0.21 /0.70 0.38 /1.21
We predict substantial differences in their TE performances.
Both n-type and p-type ZTvalues of the 2H phases are
much larger than those of the 1T phase, and κphof the 1T
phases are always slightly higher than the 2H phases (seeTable II). The underlying reason lies mostly in their electronic
band structures. The frontier bands in the 2H phase are lessdispersive than in the 1T phase. The valence band maximumis almost flat, which leads to sharp changes in the DOS and theelectronic transmission spectrum and give rise to enhanced S,
P, and ZT. As a result, the p-type ZTvalues of the 2H phases
are five to six times higher than those of the 1T phases. In then-type ZT, the difference is not as dramatic as in the p-type
ZT, but those of 2H phases are considerably larger again. 1T
phases of ZrSe
2,H f S e 2, and HfS 2were previously predicted
to have promising ZTvalues, when phonon scatterings are
taken into account [ 53,56].
In order to quantify the role of κphonZT, we study their
correlation from the available data. The p-type ZTfor 2H
compounds is inversely correlated with κphas 55%, 60%,
and 59% at 300 K, 500 K, and 800 K, respectively. On theother hand, inverse correlation between n-type ZTandκ
ph
is slightly larger than that of p-type ZT. Inverse correlation
values of 57%, 62%, and 61% are obtained for the sametemperatures. These illustrate the role of κ
phin determining
the TE performance of the crystals considered.
The 2H phases of ZrSe 2,Z r T e 2HfS 2, and HfSe 2pro-
vide ZT values larger than 1 for both n- and p-type
carriers at room temperature. The electronic transmission,Seebeck coefficient, power factor, and TE figure of merit ofthese compounds are presented in Fig. 6. Although 2H-ZrSe
2
and 2H-ZrTe 2have almost the same thermal conductance
values, ZrTe 2has the lowest ZTcompared to other promising
compounds due to relatively smooth transmission spectra atthe valence band edge. It is also observed that PBE resultsyield a decreasing ZTfor ZrTe
2above 500 K. This stems from
the fact that the band gap of ZrTe 2as predicted from PBE
is not sufficient to support an efficient TE response at hightemperatures. We use hybrid functionals to correct the cal-culated band gap, which will be discussed separately below.Abrupt changes in the transmission spectra are observed at thevalence band edges of ZrSe
2,H f S 2, and HfSe 2(see Fig. 6).
Altough κphof 2H-HfTe 2is lower than that of 2H-HfSe 2,
itsp-type ZTis much lower than that of 2H-HfSe 2, which
is found to have the highest p-type ZTvalue (1.57) at room
temperature. The lower ZTvalue of 2H-HfTe 2is because
of its electronic transmission being smoother than that of2H-HfSe
2. In the case of n-type ZT, in addition to theseFIG. 6. Electronic transmission, Seebeck coefficient, power fac-
tor, and thermoelectric figure of merit are plotted around the Fermi
level for 2H-ZrSe 2, 2H-ZrTe 2,2 H - H f S 2, and 2H-HfSe 2.
four TMDs, for 2H-HfTe2 and 2H-TiS2, it exceeds 1 at room
temperature. It is worth mentioning that a remarkably high
p-type power factor is achieved for the 1T-ZrO 2and 1T-HfO 2
but their lattice thermal conductances are higher than those
of 2H-ZrSe 2,2 H - Z r T e 2,2 H - H f S 2, and 2H-HfSe 2by about a
factor of 2 or 3, thus their ZTvalues remain under 1 at 300 K.
In principle, TE response of these oxides can be enhanced byreducing κ
phwith phonon engineering.
Seebeck coefficient is reduced with simultaneous contribu-
tion of p- and n-type carriers. Accordingly, obtaining accurate
S,PF, and ZT values will mostly depend on electronic
band gap of material, especially at higher temperatures. Ifthe band gap of the material is smaller than about 10k
BT,S
is suppressed with increasing temperature as in the cases of
085415-7ÖZBAL, SENGER, SEVIK, AND SEVINÇLI PHYSICAL REVIEW B 100, 085415 (2019)
FIG. 7. Electronic band structure of 1T-TiS 2with PBE (solid
red) and hybrid HSE06 (dashed blue) functionals. Transition from
semimetallic to semiconducting phase occurs with the hybrid
functional.
2H-CrO 2,2 H - H f T e 2,2 H - T i T e 2, and 2H-ZrTe 2.T h e EPBE
gis
0.36 eV (0.45 eV) for 2H-HfTe 2(2H-ZrTe 2), and Sis sup-
pressed when temperature is above 300 K (500 K). Suppres-sion of Sreduces ZTwhen Tis above 500 K for 2H-HfTe
2,
because the increase in Gcompensates the decrease in Sat
lower temperatures. For 2H-CrO 2, a similar trend in Sis
observed, however ZTis not suppressed at higher tempera-
tures because Gincreases with T.2 H - T i T e 2has the narrowest
EHSE
g(0.19 eV) among the investigated TMDs. Therefore, the
decrease in SandZTappear above room temperature. τel(E),
S,PF, and ZTcalculated from hybrid-functional-corrected
band gaps are demonstrated in the Supplemental Material (seeFig. S3) [ 93]. Band gap correction using hybrid functionals
results in better ZTvalues for these MX
2compounds as seen
in Table IV.Besides the semiconducting TMD /TMOs, TE properties
of semimetallic 1T-TiS 2is also investigated. 1T-TiS 2is more
stable with a Ecoh=5.31 eV , which is higher than its 2H phase
for about 0.14 eV [ 91]. HSE06 correction exhibits a transition
from semimetallic to semiconducting behavior as shown inFig.7with a band gap of 0.62 eV , in agreement with previous
results [ 92]. It is clearly seen that 1T-TiS
2does not achieve
a high value of p-type ZT,b u t n-type ZTexceeds 1 when
temperature reaches 800 K (see Table IVand Fig. S3) [ 93].
We note that inclusion of spin-orbit coupling (SOC) should
be expected to give rise to quantitative changes in the thermo-electric coefficients of the structures. Lifting the spin degen-eracy, SOC can be expected to reduce the TE efficiency forstructures where the SOC is strong.
VI. CONCLUSION
We have investigated structural, electronic, vibrational, as
well as ballistic transport and thermoelectric properties of alarge family of TMDs /TMOs by using a combination of ab
initio and Landauer-Büttiker formalisms. We have identified
promising thermoelectric materials which possess high ZT
values close to or above 1 at room temperature. In particular,high p-type and n-type TE figure of merit are found for
2H-HfSe
2and 2H-ZrSe 2, respectively. Moreover, our cal-
culations reveal that two TMO monolayers, 1T-ZrO 2and
1T-HfO 2, can be promising p-type thermoelectric candidates
at room temperature.
ACKNOWLEDGMENTS
G.Ö., R.T.S., and H.S. acknowledge support from Sci-
entific and Technological Research Council of Turkey(TUBITAK) Grant No. 117F131. C.S. acknowledges thesupport from BAGEP Award of the Science Academy. Partof the numerical calculations are carried out at TUBITAKULAKBIM High Performance and Grid Computing Center.
[1] B. Poudel, Q. Hao, Y . Ma, Y . Lan, A. Minnich, B. Yu, X. Yan, D.
Wang, A. Muto, D. Vashaee, X. Chen, J. Liu, M. S. Dresselhaus,G. Chen, and Z. Ren, Science 320,634(2008 ).
[ 2 ]L .D .H i c k sa n dM .S .D r e s s e l h a u s , Phys. Rev. B 47,12727
(1993 ).
[ 3 ]L .D .H i c k sa n dM .S .D r e s s e l h a u s , Phys. Rev. B 47,16631
(1993 ).
[4] G. D. Mahan and H. B. Lyon, J. Appl. Phys. 76,1899 (1994 ).
[5] L. D. Hicks, T. C. Harman, X. Sun, and M. S. Dresselhaus,
P h y s .R e v .B 53,R10493 (1996 ).
[6] D. A. Broido and T. L. Reinecke, Appl. Phys. Lett. 67,100
(1995 ).
[7] M. Dresselhaus, G. Chen, M. Tang, R. Yang, H. Lee, D. Wang,
Z. Ren, J.-P. Fleurial, and P. Gogna, Adv. Mater. 19,1043
(2007 ).
[ 8 ]T .A .A m o l l o ,G .T .M o l a ,M .S .K .K i r u i ,a n dV .O .N y a m o r i ,
Crit. Rev. Solid State Mater. Sci. 43,133(2018 ).[9] P. Dollfus, V . H. Nguyen, and J. Saint-Martin, J. Phys.:
Condens. Matter 27,133204 (2015 ).
[10] Y . Xu, Z. Li, and W. Duan, Small 10,2182
(2014 ).
[11] Y . M. Zuev, W. Chang, and P. Kim, Phys. Rev. Lett. 102,096807
(2009 ).
[12] F. Ghahari, H.-Y . Xie, T. Taniguchi, K. Watanabe, M. S. Foster,
and P. Kim, P h y s .R e v .L e t t . 116,136802 (2016 ).
[13] A. A. Balandin,
Nat. Mater. 10,569(2011 ).
[14] J.-W. Jiang, J.-S. Wang, and B. Li, P h y s .R e v .B 79,205418
(2009 ).
[15] Z. Wang, R. Xie, C. T. Bui, D. Liu, X. Ni, B. Li, and J. T. L.
Thong, Nano Lett. 11,113(2011 ).
[16] Y . Anno, K. Takei, S. Akita, and T. Arie, Adv. Electron. Mater.
1,1500175 (2015 ).
[17] Y . Anno, Y . Imakita, K. Takei, S. Akita, and T. Arie, 2D
Materials 4,025019 (2017 ).
085415-8BALLISTIC THERMOELECTRIC PROPERTIES OF … PHYSICAL REVIEW B 100, 085415 (2019)
[18] R. D’Souza and S. Mukherjee, J. Appl. Phys. 124,124301
(2018 ).
[19] D. Dragoman and M. Dragoman, Appl. Phys. Lett. 91,203116
(2007 ).
[20] T. Gunst, T. Markussen, A.-P. Jauho, and M. Brandbyge, Phys.
Rev. B 84,155449 (2011 ).
[21] H. Sevinçli, C. Sevik, T. Ça ˘gin, and G. Cuniberti, Sci. Rep. 3,
1228 (2013 ).
[22] H. Sevinçli and G. Cuniberti, P h y s .R e v .B 81,113401 (2010 ).
[23] L. M. Sandonas, H. Sevinçli, R. Gutierrez, and G. Cuniberti,
Adv. Sci. 5,1700365 (2018 ).
[24] F. Mazzamuto, V . Hung Nguyen, Y . Apertet, C. Caër, C.
Chassat, J. Saint-Martin, and P. Dollfus, Phys. Rev. B 83,
235426 (2011 ).
[25] Y . Ouyang and J. Guo, Appl. Phys. Lett. 94,263107 (2009 ).
[26] K. Yang, Y . Chen, R. D’Agosta, Y . Xie, J. Zhong, and A. Rubio,
P h y s .R e v .B 86,045425 (2012 ).
[27] C. Ataca, H. Sahin, and S. Ciraci, J. Phys. Chem. C 116,8983
(2012 ).
[28] C. Adessi, S. Thebaud, R. Bouzerar, and G. Bouzerar, J. Phys.
Chem. C 121,12577 (2017 ).
[29] B. Ouyang, S. Chen, Y . Jing, T. Wei, S. Xiong, and D. Donadio,
J. Materiomics 4,329(2018 ).
[30] K.-X. Chen, X.-M. Wang, D.-C. Mo, and S.-S. Lyu, J. Phys.
Chem. C 119,26706 (2015 ).
[31] H. Babaei, J. M. Khodadadi, and S. Sinha, Appl. Phys. Lett.
105,193901 (2014 ).
[32] K. Nakamura, Jpn. J. Appl. Phys. 57,06HE04 (2018 ).
[33] Z. Jin, Q. Liao, H. Fang, Z. Liu, W. Liu, Z. Ding, T. Luo, and
N. Yang, Sci. Rep. 5,18342 (2015 ).
[34] W. Huang, H. Da, and G. Liang, J. Appl. Phys. 113,104304
(2013 ).
[35] G. Zhang and Y .-W. Zhang, J. Mater. Chem. C 5,7684 (2017 ).
[36] S. Kumar and U. Schwingenschlögl, Chem. Mater. 27,1278
(2015 ).
[37] A. Arab and Q. Li, Sci. Rep. 5,13706 (2015 ).
[38] D. Wickramaratne, F. Zahid, and R. K. Lake, J. Chem. Phys.
140,124710 (2014 ).
[39] M. Kayyalha, J. Maassen, M. Lundstrom, L. Shi, and Y . P. Chen,
J. Appl. Phys. 120,134305 (2016 ).
[40] K. Hippalgaonkar, Y . Wang, Y . Ye, D. Y . Qiu, H. Zhu, Y . Wang,
J. Moore, S. G. Louie, and X. Zhang, P h y s .R e v .B 95,115407
(2017 ).
[41] W. Huang, X. Luo, C. K. Gan, S. Y . Quek, and G. Liang, Phys.
Chem. Chem. Phys. 16,10866 (2014 ).
[42] R.-N. Wang, G.-Y . Dong, S.-F. Wang, G.-S. Fu, and J.-L. Wang,
Phys. Chem. Chem. Phys. 19,5797 (2017 ).
[43] Y . Ouyang, Y . Xie, Z. Zhang, Q. Peng, and Y . Chen, J. Appl.
Phys. 120,235109 (2016 ).
[44] Z. Zhang, Y . Xie, Q. Peng, and Y . Chen, Sci. Rep. 6,21639
(2016 ).
[45] T.-m. Wu, R.-x. Xu, X. Zheng, and W. Zhuang, C h i n .J .C h e m .
Phys. 29,445(2016 ).
[46] Y .-Y . Liu, Y .-J. Zeng, P.-Z. Jia, X.-H. Cao, X. Jiang, and K.-Q.
Chen, J. Phys.: Condens. Matter 30,275701 (2018 ).
[47] J. Hong, C. Lee, J.-S. Park, and J. H. Shim, Phys. Rev. B 93,
035445 (2016 ).
[48] W. Shen, D. Zou, G. Nie, and Y . Xu, Chin. Phys. B 26,117202
(2017 ).[49] Dimple, N. Jena, and A. D. Sarkar, J. Phys.: Condens. Matter
29,225501 (2017 ).
[50] S.-D. Guo, Comput. Mater. Sci. 123,8(2016 ).
[51] G. Zhang and Y .-W. Zhang, Mech. Mater. 91,382(2015 ).
[52] H. Y . Lv, W. J. Lu, D. F. Shao, H. Y . Lu, and Y . P. Sun, J. Mater.
Chem. C 4,4538 (2016 ).
[53] G. Ding, G. Y . Gao, Z. Huang, W. Zhang, and K. Yao,
Nanotechnology 27,375703 (2016 ).
[54] S.-D. Guo and J.-L. Wang, Semicond. Sci. Technol. 31,095011
(2016 ).
[55] Y .-X. Zhen, M. Yang, H. Zhang, G.-S. Fu, J.-L. Wang, S.-F.
Wang, and R.-N. Wang, Science Bulletin 62,1530 (2017 ).
[56] D. Qin, X.-J. Ge, G.-q. Ding, G.-y. Gao, and J.-T. Lü, RSC Adv.
7,47243 (2017 ).
[57] H. Y . Lv, W. J. Lu, X. Luo, H. Y . Lu, X. B. Zhu, and Y . P. Sun,
arXiv:1608.05464 .
[58] G. Kresse and J. Furthmüller, Phys. Rev. B 54,11169
(1996 ).
[59] G. Kresse and D. Joubert, P h y s .R e v .B 59,1758 (1999 ).
[60] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 (1996 ).
[61] H. J. Monkhorst and J. D. Pack, P h y s .R e v .B 13,5188 (1976 ).
[62] A. V . Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria,
J. Chem. Phys. 125,224106 (2006 ).
[63] S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev.
Mod. Phys. 73,515(2001 ).
[64] A. Togo and I. Tanaka, Scr. Mater. 108,1(2015 ).
[65] G. Henkelman, A. Arnaldsson, and H. Jónsson, Comput. Mater.
Sci.36,354(2006 ).
[66] W. Callister and D. Rethwisch, Materials Science and Engineer-
ing: An Introduction, 9th Edition: Ninth Edition (John Wiley and
Sons, Incorporated, New York, 2013).
[67] U. Sivan and Y . Imry, P h y s .R e v .B 33,551(1986 ).
[68] K. Esfarjani, M. Zebarjadi, and Y . Kawazoe, Phys. Rev. B 73,
085406 (2006 ).
[69] L. G. C. Rego and G. Kirczenow, Phys. Rev. Lett. 81,232
(1998 ).
[70] H. Sevinçli, S. Roche, G. Cuniberti, M. Brandbyge, R.
Gutierrez, and L. M. Sandonas, J. Phys.: Condens. Matter 31,
273003 (2019 ).
[71] F. A. Rasmussen and K. S. Thygesen, J. Phys. Chem. C 119,
13169 (2015 ).
[72] C. Gong, H. Zhang, W. Wang, L. Colombo, R. M. Wallace, and
K. Cho, Appl. Phys. Lett. 103,053513 (2013 ).
[73] P. Tsipas, D. Tsoutsou, S. Fragkos, R. Sant, C. Alvarez, H.
Okuno, G. Renaud, R. Alcotte, T. Baron, and A. Dimoulas, ACS
Nano 12,1696 (2018 ).
[74] S. Aminalragia-Giamini, J. Marquez-Velasco, P. Tsipas, D.
Tsoutsou, G. Renaud, and A. Dimoulas, 2D Materials 4,015001
(2017 ).
[75] M. J. Mleczko, C. Zhang, H. R. Lee, H.-H. Kuo, B. Magyari-
Köpe, R. G. Moore, Z.-X. Shen, I. R. Fisher, Y . Nishi, and E.Pop, Sci. Adv. 3,e1700481 (2017 ).
[76] C. Yan, C. Gong, P. Wangyang, J. Chu, K. Hu, C. Li, X. Wang,
X. Du, T. Zhai, Y . Li, and J. Xiong, Adv. Funct. Mater. 28,
1803305 (2018 ).
[77] G. Mahan, B. Sales, and J. Sharp, Phys. Today 50(3),42
(1997 ).
[78] J. O. Sofo and G. D. Mahan, P h y s .R e v .B 49,4565 (1994 ).
085415-9ÖZBAL, SENGER, SEVIK, AND SEVINÇLI PHYSICAL REVIEW B 100, 085415 (2019)
[79] R. P. Chasmar and R. Stratton, J. Electron. Control 7,52
(1959 ).
[80] G. S. Nolas, J. Sharp, and H. J. Goldsmid, Thermoelectrics:
Basic Principles and New Materials Development (Springer-
Verlag, Berlin, Heidelberg, 2001).
[81] A. M. Dehkordi, M. Zebarjadi, J. He, and T. M. Tritt, Mater.
Sci. Eng., R 97,1(2015 ).
[82] B. Singh, C.-H. Hsu, W.-F. Tsai, V . M. Pereira, and H. Lin,
P h y s .R e v .B 95,245136 (2017 ).
[83] D. L. Duong, M. Burghard, and J. C. Schön, P h y s .R e v .B 92,
245131 (2015 ).
[84] M. J. Wei, W. J. Lu, R. C. Xiao, H. Y . Lv, P. Tong, W. H. Song,
a n dY .P .S u n , P h y s .R e v .B 96,165404 (2017 ).
[85] C. Chen, B. Singh, H. Lin, and V . M. Pereira, Phys. Rev. Lett.
121,226602 (2018 ).[86] J.-A. Yan, M. A. D. Cruz, B. Cook, and K. Varga, Sci. Rep. 5,
16646 (2015 ).
[87] Z. Zhang, Y . Xie, Y . Ouyang, and Y . Chen, Int. J. Heat Mass
Transf. 108,417(2017 ).
[88] Y . Cai, J. Lan, G. Zhang, and Y .-W. Zhang, P h y s .R e v .B 89,
035438 (2014 ).
[89] M. Cutler and N. F. Mott, Phys. Rev. 181,1336 (1969 ).
[90] G. D. Guttman, E. Ben-Jacob, and D. J. Bergman, Phys. Rev. B
51,17758 (1995 ).
[91] C. Xu, P. A. Brown, and K. L. Shuford, RSC Adv. 5,83876
(2015 ).
[92] A. Samad, A. Shafique, and Y .-H. Shin, Nanotechnology 28,
175401 (2017 ).
[93] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.100.085415 for further figures and tables.
085415-10 |
PhysRevB.94.214508.pdf | PHYSICAL REVIEW B 94, 214508 (2016)
Hyperfine fields in the BaFe 2As2family and their relation to the magnetic moment
Gerald Derondeau,1,*J´an Min ´ar,1,2and Hubert Ebert1
1Department Chemie, Physikalische Chemie, Universit ¨at M ¨unchen, Butenandtstrasse 5-13, 81377 M ¨unchen, Germany
2New Technologies Research Center, University of West Bohemia, Pilsen, Czech Republic
(Received 29 September 2016; published 12 December 2016)
The hyperfine field Bhfand the magnetic properties of the BaFe 2As2family are studied using the fully
relativistic Dirac formalism for different types of substitution. The study covers electron doped Ba(Fe 1−xCox)2As2
and Ba(Fe 1−xNix)2As2, hole doped (Ba 1−xKx)Fe 2As2, and also isovalently doped Ba(Fe 1−xRux)2As2and
BaFe 2(As 1−xPx)2for a wide range of the concentration x. For the substituted compounds the hyperfine fields
show a very strong dependence on the dopant type and its concentration x. Relativistic contributions were found
to have a significantly stronger impact for the iron pnictides when compared to bulk Fe. As an important finding,we demonstrate that it is not sensible to relate the hyperfine field B
hfto the average magnetic moment μof the
compound, as it was done in earlier literature.
DOI: 10.1103/PhysRevB.94.214508
I. INTRODUCTION
Since the discovery of high-temperature superconductivity
in La(O 1−xFx)FeAs [ 1,2] the iron pnictides are currently one
of the most important prototype systems for unconventionalsuperconductivity. The mechanism of superconductivity ismore than likely connected to magnetic fluctuations [ 3–5],
which makes the magnetic behavior of the iron pnictides acrucial property [ 6,7]. Despite tremendous research over the
last years the complex magnetism of these compounds is stillnontrivial to explain and some problems remain unsolved.
For example, a discrepancy is observed concerning the
magnitude of the magnetic moment, depending on the chosenexperimental method. Neutron diffraction data predict forthe low-temperature phase of BaFe
2As2a total magnetic
moment of 0 .87μBper Fe from powder samples [ 8], while
from57Fe M ¨ossbauer spectroscopy [ 9,10] a value between
0.4μBand 0 .5μBwas estimated. One should note that
the magnetic moments in the iron pnictides are generallyconsidered to behave nearly itinerant [ 4,6,11–13], although
sometimes a localized picture might be more appropriate[14–16]. Furthermore, density functional theory (DFT) cal-
culations often overestimate the magnitude of the magneticmoments, ranging from approximately 1 .2μ
Bup to 2 .6μB
[7,11,17–19]. Thus, the magnetic moments are known to be
highly sensitive to the system and computational parameters,which makes estimations difficult and leads sometimes toseemingly contradicting reports [ 7,9,20–22]. Furthermore, the
importance of spin-orbit coupling for the iron pnictides wasonly recently stressed [ 23].
Nowadays, a lot of
57Fe M ¨ossbauer spectroscopy data are
available for the BaFe 2As2family with different types of
substitution and doping [ 10,24–27]. The previously mentioned
discrepancy between neutron diffraction and57Fe M ¨ossbauer
spectroscopy is often ascribed to possible nonzero contribu-tions of dorbitals to the hyperfine field with an opposite sign
to that of the Fermi contact field [ 24]. This would explain why
the suggested hyperfine proportionality constant Abetween
the experimentally measured hyperfine field B
expand the
*gerald.derondeau@cup.uni-muenchen.deunderlying magnetic moment μ(Fe) has a nonlinear behavior
and is in particular not comparable to the corresponding valuefor bulk Fe. This would imply that a more reliable estimationof magnetic moments based on
57Fe M ¨ossbauer spectroscopy
lies not between 0 .4μBand 0.5μBbut has a higher value.
Although such aspects were already suggested as a mostlikely explanation for this discrepancy [ 24], a quantitative
study of the theoretical hyperfine fields including relativisticcontributions is still missing [ 28].
To clarify this situation, we address in this paper the
antiferromagnetic state of the undoped mother compoundBaFe
2As2together with a large variety of different types
of substitution. These include electron doping in the caseof Ba(Fe
1−xCox)2As2and Ba(Fe 1−xNix)2As2, hole doping as
in (Ba 1−xKx)Fe 2As2, and also isovalently doped compounds
such as Ba(Fe 1−xRux)2As2and BaFe 2(As 1−xPx)2. To deal
adequately with substitutional systems the fully relativis-tic Korringa-Kohn-Rostoker-Green’s function (KKR-GF) ap-proach is used, which was already shown to be an appropriatetool to investigate various properties of the iron pnictides[29–31]. Chemical disorder due to substitution is dealt by
means of the coherent potential approximation (CPA), whicheffectively gives results comparable to the tedious average overmany supercell configurations and is much more reliable thanthe virtual crystal approximation (VCA) [ 29,32]. Application
of the CPA to the iron pnictides was already shown to bequite successful [ 29,30,33–35]. Using this approach, one can
not only investigate the type-resolved evolution of magneticmoments with composition, but also the doping dependence ofthe hyperfine fields. Furthermore, all contributions to the totalhyperfine field B
hfcan be separated, revealing the direct impact
of orbital non- s-electron parts within the fully relativistic
approach.
II. COMPUTATIONAL DETAILS
All calculations have been performed self-consistently
and fully relativistically within the four component Diracformalism, using the Munich
SPR-KKR program package
[36,37]. The crystal structure is based on the orthorhombic,
antiferromagnetic phase of BaFe 2As2in its experimentally
observed stripe spin state using a 4-Fe unit cell. This implies
2469-9950/2016/94(21)/214508(8) 214508-1 ©2016 American Physical SocietyGERALD DERONDEAU, J ´AN MIN ´AR, AND HUBERT EBERT PHYSICAL REVIEW B 94, 214508 (2016)
antiferromagnetically ordered chains along the aandcaxes
and ferromagnetically ordered chains along the baxis. With
spin-orbit coupling included within the relativistic approachthe orientation of the magnetic moments was chosen to be inplane along the aaxis, in line with experiment [ 20]. The lattice
parameters and As position zwhere chosen according to exper-
imental x-ray data [ 9]. To account for the influence of different
substitutions, a linear interpolation of the lattice parameterswith respect to the concentration xwas performed based on
Vegard’s law [ 38]. This interpolation was individually done
for each type of substitution, based on available experimentaldata [ 9,39–43]. More details on this procedure can be found
in previous publications [ 29,30]. The treatment of disorder
introduced by substitution is dealt with by means of the CPA.For the angular momentum expansion of the KKR Green’sfunction an upper limit /lscript
max=4 was used, i.e., s,p,d,f, and
gorbitals were included in the basis set, although contributions
to the hyperfine field of Fe from fandgorbitals are zero, as
one would expect. All DFT calculations used the local spin-density approximation (LSDA) exchange-correlation potentialwith the parametrization as given by V osko, Wilk, and Nusair[44]. The calculation and decomposition of the hyperfine field
B
hfis done in its fully relativistic form as discussed in detail
in Ref. [ 45].
III. RESULTS AND DISCUSSION
A. Undoped mother compound
The calculated total magnetic moment of Fe in the undoped
mother compound BaFe 2As2isμ(Fe)=0.73μB,a sw a s
already published in earlier work [ 30]. This moment splits into
a spin magnetic moment of μspin(Fe)=0.70μBand an orbital
magnetic moment of μorb(Fe)=0.03μB. Obviously, this is in
good agreement with experimental neutron diffraction data ofpure BaFe
2As2being 0 .87μB[8].
If the finite size of the atomic core is ignored, as usually
done, the fully relativistic approach described in Ref. [ 45] splits
Bhfinto five contributions. There are two contributions due to
theselectrons that are conventionally ascribed to the Fermi
contact interaction. The larger part is the core polarizationcontribution B
c
sthat was demonstrated in numerous studies
to be proportional to the local spin magnetic moment μspin
[46–48]. In addition, there is a s-electron contribution from
the valence band Bv
sthat is due to the polarization and
also dominantly due to the population mechanism [ 49]. For
systems with low symmetry there may be a spin dipolarcontribution to B
hffor the non- selectrons [ 45,50]. Apart from
p1/2states, states with higher angular momentum such as p
anddstates have zero probability density at the core and for
that reason do not contribute to Bhfvia the Fermi contact term.
If spin-orbit coupling is accounted for, as done here, thereis an additional contribution due to the spin-orbit inducedorbital magnetization [ 45,50]. As the orbital contribution is
in general dominating compared to the spin-dipolar one [ 45],
we use in the following the term orbital for the total fieldconnected with non- selectrons. Thus, for a transition metal,
the remaining three contributions are the orbital field B
c
nsof the
non-score states and the orbital fields Bv
pandBv
dof the valence
electrons with panddcharacter, respectively. One arrives forcc Fe
-35-30-25-20-15-10-50510
-35-30-25-20-15-10-50510B[T]
Bc
sBc
nsBv
sBv
pBv
dBhfBexpBhf(a) b
(b) Fe in BaFe 2As2
-8-7-6-5-4-3-2-1012
-8-7-6-5-4-3-2-1012B[T]
Bc
sBc
nsBv
sBv
pBv
dBhfBexpBhf
FIG. 1. Contributions to the hyperfine field Bhffor (a) bcc Fe
and for (b) Fe in antiferromagnetic BaFe 2As2. For comparison,
experimental values are shown as Bexp[9,10,51]./tildewideBhfis based on
Eq. ( 2) and includes an enhancement of the core polarization Bc
sof
25%.
the hyperfine field Bhfat the following decomposition [ 45]:
Bhf=Bc
s+Bc
ns+Bv
s+Bv
p+Bv
d. (1)
Figure 1(a) shows for bcc Fe numerical results for the
various contributions to the hyperfine field. As it is well known,B
hfof bcc Fe is dominated by its large core polarization
contribution Bc
s. This is enhanced by the field Bv
s, which
is also negative. All other contributions are much smallerand positive. Comparing the total calculated hyperfine fieldB
hf=−26.7 T with the corresponding experimental value
Bexp=−33.9 T, one finds the theoretical values too small
by about 25% [ 52]. This well known problem is primarily
to be ascribed to shortcomings of LSDA when dealing withthe core polarization cased by the spin polarization of thevalence electrons [ 53–55]. To cure this problem it is common
to enhance B
c
sby about 25% [ 53–56]. Using this empirical
approach one has for the enhanced hyperfine field /tildewideBhfthe
214508-2HYPERFINE FIELDS IN THE BaFe 2As2FAMILY . . . PHYSICAL REVIEW B 94, 214508 (2016)
TABLE I. Different contributions to the hyperfine field Bhffor
BaFe 2As2, depending on the magnetization direction axis. Consistent
with experiment is an orientation of magnetic moments along the a
axis, which was applied throughout this work.
Axis μspin(μB)Bc
s(T)Bc
ns(T)Bv
s(T)Bv
p(T)Bv
d(T)Bhf(T)
a 0.696 −7.37 0.035 1.48 0 .37 1 .86−3.62
b 0.698 −7.39 0.036 1.48 −0.34 2 .66−3.55
c 0.695 −7.36 0.035 1.48 −0.10−0.13−6.08
relation
/tildewideBhf=1.25Bc
s+Bc
ns+Bv
s+Bv
p+Bv
d. (2)
As can be seen in Fig. 1(a), this leads to /tildewideBhf=−32.9Tf o r
bcc Fe, in good agreement with experiment. Next, consider Fein the undoped mother compound BaFe
2As2as presented in
Fig. 1(b). Comparing the calculated Bhf=−3.62 T with the
experimental one Bexp=−5.47 T [ 9,10], the shortcomings
of LSDA are obviously the same as for bcc Fe, as onewould expect. However, the enhanced field /tildewideB
hf=−5.46 T
is in perfect agreement with experiment, confirming thetransferability of the enhancement factor in Eq. ( 2). Compared
with bcc Fe, the various contributions to /tildewideB
hfof Fe in BaFe 2As2
show two major differences. First, the sign of the valence band
s-electron contribution Bv
sis different, and, second, the spin-
orbit induced contribution of delectrons Bv
dis considerably
higher in the latter case. Both features lead to a very differentrelation between the enhanced hyperfine field /tildewideB
hfand the local
spin magnetic moment μspinfor the two systems. As /tildewideBhfof bcc
Fe is dominated by its enhanced core polarization contribution
/tildewideBc
s(/tildewideBhf//tildewideBc
s≈1.07), which is proportional to μspin,i ts e e m s
justified to assume that the experimental field Bexpreflects in a
one-to-one manner the local spin moment. For Fe in BaFe 2As2,
on the other hand, we find /tildewideBhf//tildewideBc
s≈0.59, i.e., the total field
/tildewideBhfcan by no means be used to monitor the local spin magnetic
moment of Fe.
It was found that this unexpected behavior of BaFe 2As2
compared to bcc Fe is mainly due to its in-plane orientationof magnetic moments along the aaxis. It was already
stressed that this magnetization direction is conform withexperiment [ 20] and can be theoretically described by the
applied inclusion of spin-orbit coupling. For comparison, weshow in Table Ithe components of the hyperfine field of
BaFe
2As2depending on the magnetization direction. It is
obvious that the contributions Bv
pandBv
ddepend strongly on
the chosen orientation, although the spin magnetic moment ofFeμ
spinand the other contributions to the hyperfine field only
marginally change. This has naturally an impact on the totalhyperfine field B
hfwhich thus depends on the magnetization
direction in BaFe 2As2.
B. Electron and hole doping
Having investigated the hyperfine field contributions of the
undoped BaFe 2As2including relativistic effects, an interesting
issue is their variation under different types of substitution inthe BaFe
2As2family.
Two examples of electron doping were investigated,
namely, Ba(Fe 1−xCox)2As2(Co-122) and Ba(Fe 1−xNix)2As2(a)
0.00.10.20.30.40.50.60.70.8
0 0.05 0.1 0.150.000.010.020.030.040.050.060.070.08µspin[µB]
µorb[µB]
xfor Ba(Fe 1−xCox)2As2µspin(Fe)
µorb(Fe)
µspin(Co)
µorb(Co)
µavg
(b)
-8-7-6-5-4-3-2-10123
0 0.05 0.1 0.150 0.025 0.05 0.075B[T]
xfor Ba(Fe 1−xCox)2As2x(Bexp)
Bc
s
Bv
s
Bc
ns
Bv
p
Bv
d
Bhf
Bhf
Bexp
FIG. 2. (a) Component-resolved magnetic moments for Co-122
depending on the concentration x. The left (right) scale refers to the
spin (orbital) magnetic moment. (b) Corresponding hyperfine field
contributions for Fe in Co-122. The experimental data Bexp(dashed
orange lines) [ 24] refer to the upper axis, with the upper and lower
axes for the concentration xchosen such that xcrit=xcrit,exp .
(Ni-122), with the corresponding data shown in Figs. 2
and3, respectively. Furthermore, one case of hole doping,
(Ba 1−xKx)Fe 2As2(K-122), has been considered (see Fig. 4).
In all cases, the magnetic moments of the components are pre-sented in Figs. 2(a),3(a), and 4(a), respectively, as a function
of the concentration. The magnetic moments for Co-122 inFig.2(a)were published before [ 30], and are reproduced here
to supply a reference for the hyperfine field and to allow fordirect comparison with other systems. The various figures givein a component-resolved manner the spin magnetic momentsμ
spin(left axis) and the orbital magnetic moments μorb(right
axis). The concentration dependent average of the systemwith composition Ba(Fe
1−xTMx)2As2is shown as μavg=
(1−x)[μspin(Fe)+μorb(Fe)]+x[μspin(TM)+μorb(TM)].
First consider the electron doped compounds Co-122
and Ni-122. Both systems show a similar decrease inμ
avguntil the breakdown of long range antiferromagnetic
(AFM) order at xcritis reached, with xcrit(Co-122) =0.125
andxcrit(Ni-122) =0.075, respectively. This is in reasonable
agreement with experiment, with the experimental xcrit,exp
being lower [ xcrit,exp (Co-122) ≈0.075 [ 57],xcrit,exp (Ni-122) ≈
0.0375 [ 58]]. Concerning the instability of the antiferromag-
netic order, the electronic structure calculations account for achange in the nesting condition due to a shift of the Fermi level
214508-3GERALD DERONDEAU, J ´AN MIN ´AR, AND HUBERT EBERT PHYSICAL REVIEW B 94, 214508 (2016)
(a)
-0.10.00.10.20.30.40.50.60.70.8
0 0.025 0.05 0.075 0.1-0.010.000.010.020.030.040.050.060.070.08µspin[µB]
µorb[µB]
xfor Ba(Fe 1−xNix)2As2µspin(Fe)
µorb(Fe)
µspin(Ni)
µorb(Ni)
µavg
(b)
-8-7-6-5-4-3-2-10123
0 0.025 0.05 0.075 0.10 0.025 0.05B[T]
xfor Ba(Fe 1−xNix)2As2x(Bexp)
Bc
s
Bv
s
Bc
ns
Bv
p
Bv
d
Bhf
Bhf
Bexp
FIG. 3. Same as for Fig. 2, but for Ni-122 with experimental data
f r o mR e f .[ 25].
due to doping but they do not explicitly account for fluctuating
magnetic moments or incommensurate spin-density waves[59]. This might explain the observed discrepancies between
x
critandxcrit,exp , implying that these aspects should be
accounted for in order to get better agreement.
In line with experiment, xcritfor Ni-122 is found to be only
half of Co-122. This had to be expected because of the formaldoubling of electron doping by Ni compared to Co substitutionof Fe. Another difference between these two compounds isthe lower Ni moment in Ni-122 compared to that of Co inCo-122. In this context one should also note that the rathersmall orbital moment of Ni has a different sign compared toits spin moment. The various hyperfine field contributions forFe in Co-122 and Ni-122 are shown in Figs. 2(b) and3(b),
respectively. The trends of the Fe magnetic moments and inthe hyperfine field contributions behave in a similar way. Thefigures show also experimental data for the hyperfine field B
exp
[24,25]. These has been plotted using a different scale for the
concentration xat the top of the figure that was chosen such
that theoretical and experimental critical concentrations agree(x
crit=xcrit,exp ). With the aforementioned enhancement of the
core polarization field by 25% and the rescaling of the xaxis,
one finds a very satisfying agreement for /tildewideBhfandBexpfor
Co-122 [Fig. 2(b)] as well as Ni-122 [Fig. 3(b)].
Next, the K-122 compound is discussed with its mag-
netic moments shown in Fig. 4(a) (see also Ref. [ 60]). A
breakdown of the AFM order is found from the calculationsatx
crit(K-122) =0.35, while a lower xcrit,exp (K-122) ≈0.25(a)
0.00.10.20.30.40.50.60.70.8
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.000.010.020.030.040.050.060.070.08µspin[µB]
µorb[µB]
xfor (Ba 1−xKx)Fe 2As2µspin(Fe)
µorb(Fe)
µavg
(b)
-8-7-6-5-4-3-2-10123
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40 0.05 0.1 0.15 0.2 0.25B[T]
xfor (Ba 1−xKx)Fe 2As2x(Bexp)
Bc
s
Bv
s
Bc
ns
Bv
p
Bv
d
Bhf
Bhf
Bexp
FIG. 4. Same as for Fig. 2, but for K-122 with experimental data
f r o mR e f .[ 10].
[40] is observed in experiment. It should be noted that the
substituted K does not have a noteworthy magnetic moment.As the Fe concentration does not change with substitutionon the Ba position, the average moment is therefore equalto the Fe moment, leading in this case to μ
avg=μ(Fe)=
μspin(Fe)+μorb(Fe). One can see that for K-122 the magnetic
moments change only marginally over a wide concentrationrange xand undergo a sharp drop for x> 0.25. The same
behavior can be seen in the hyperfine field contributions ofK-122, as shown in Fig. 4(b). Experimental data for B
exp
[10], referring again to the upper axis, are in good agreement
with the enhanced theoretical field /tildewideBhf. In particular, the
experimental Bexpis also nearly constant over a large range
of concentration, in variance to the electron doped systemsconsidered above.
C. Isovalent doping
The subsequently discussed Ba(Fe 1−xRux)2As2(Ru-122)
and BaFe 2(As 1−xPx)2(P-122) compounds are fundamentally
different from the systems considered above because of theisovalent doping. This means, in particular, that the VCAis inappropriate to deal with these systems in a meaningfulway. Still, a supercell approach could be applied to deal withthe substitution [ 61]. However, the large computational effort
makes theoretical work on these compounds rare and difficult.On the other hand, CPA-based approaches provide an efficientand powerful framework for this task.
214508-4HYPERFINE FIELDS IN THE BaFe 2As2FAMILY . . . PHYSICAL REVIEW B 94, 214508 (2016)
(a)
0.00.10.20.30.40.50.60.70.8
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.000.010.020.030.040.050.060.070.08µspin[µB]
µorb[µB]
xfor BaFe 2(As 1−xPx)2µspin(Fe)
µorb(Fe)
µavg
(b)
-8-7-6-5-4-3-2-10123
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4B[T]
xfor BaFe 2(As 1−xPx)2Bc
s
Bv
s
Bc
ns
Bv
p
Bv
d
Bhf
Bhf
FIG. 5. Same as for Fig. 2, but for P-122.
We show the component-resolved magnetic moments of
P-122 and Ru-122 in Figs. 5(a)and6(a), respectively. The first
point to note is that the calculations do not lead to a criticalconcentration x
critwithin the investigated regime of substitu-
tion, while on the experimental side one has xcrit,exp (Ru-122) ≈
xcrit,exp (P-122) ≈0.3[62,63]. Isovalent doping should in gen-
eral shift the Fermi level EFonly marginally, leading to an
unchanged nesting behavior. Thus, magnetic ordering maybe preserved as long as the substitutional limit x→1 has
a finite magnetic moment. In the case of electron or holedoping of BaFe
2As2the breakdown of magnetic order at a
critical concentration xcritcan be understood solely by the
nesting condition when the Fermi energy EFchanges due to
substitution. Note that also K-122 shows a finite xcritwith good
agreement to experiment, although the substitution happensnot on the Fe position but within the Ba layer. On the otherhand, isovalent substitution either within (Ru-122) or outside(P-122) the Fe layer cannot explain the magnetic breakdown bythe substitution alone. This indicates that other phenomena notaccounted for within the CPA mean field approach influencethe stability of the magnetic structure. In the literature, e.g.,magnetic dilution was discussed as the main driving force forthe magnetic breakdown in Ru-122 [ 64,65]. Although we find
a decrease in the magnetic moments due to the decrease inthe Fe content, it seems not sufficient to cause a breakdown ofthe magnetic order without further reasons. Spin fluctuationsand incommensurate spin-density waves can have an impacton the stability of the antiferromagnetic order, but also the(a)
0.00.10.20.30.40.50.60.70.80.9
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.000.010.020.030.040.050.060.070.080.09µspin[µB]
µorb[µB]
xfor Ba(Fe 1−xRux)2As2µspin(Fe)
µorb(Fe)
µspin(Ru)
µorb(Ru)
µavg
(b)
-9-8-7-6-5-4-3-2-10123
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4B[T]
xfor Ba(Fe 1−xRux)2As2Bc
s
Bv
s
Bc
ns
Bv
p
Bv
d
Bhf
Bhf
Bexp
FIG. 6. Same as for Fig. 2, but for Ru-122 with experimental data
f r o mR e f .[ 26].
emergence of a competing superconducting state might play a
role. In any case, it becomes obvious that the isovalently dopedcompounds of the BaFe
2As2family are even more difficult
to understand than the electron and hole doped variants.Nevertheless, LSDA-based calculations can reproduce thedecrease of the average magnetic moment μ
avgfor Ru-122
as well as for P-122, although the details of this reduction inthe magnetic moments are fundamentally different.
The magnetic moments and the hyperfine field contribu-
tions of Fe in P-122 shown in Fig. 5behave in a similar way
as those of K-122 (Fig. 4). In both cases the substitution takes
place outside the Fe layer; i.e., although the Fe concentrationdoes not change the total Fe moment, μ(Fe) does. The
hyperfine field contributions of Fe in P-122 vary again similarwith composition as the magnetic moments do. Of course,this has to be expected as the hyperfine field reflects themagnetization of the Fe atoms, which are the only magneticcomponents of these systems.
For Ru-122 the average moment μ
avgshown in Fig. 6(a)
decreases due to the increasing concentration of Ru whichhas a small induced magnetic moment of around μ(Ru)≈
0.07μ
B, independent on the concentration x. However, the
local Fe spin magnetic moment μspin(Fe) and orbital μorb(Fe)
magnetic moments surprisingly increase. This is a ratherunexpected finding as it was not observed so far withintheoretical investigations on the iron pnictides. Accordingly,the corresponding relation to the directly measurable hyperfine
214508-5GERALD DERONDEAU, J ´AN MIN ´AR, AND HUBERT EBERT PHYSICAL REVIEW B 94, 214508 (2016)
fieldBhfof Fe is of interest as it provides an element specific
probe of the magnetic properties. As can be seen in Fig. 6(b),
Bhfstays more or less constant over the whole investigated
regime of substitution, although μ(Fe) increases. This is due
to the fact that μspin(Fe) and μorb(Fe) simultaneously increase,
leading to a subsequent increase of the absolute values of Bc
s
andBv
d. Because the sign of both contributions is different,
their changes essentially compensate each other. This doesnot contradict with experimental findings of Reddy et al. [26]
depicted in Fig. 6(b) which show a more or less constant B
hf
for Ru concentrations x/lessorequalslant0.1. The rapid drop to lower Bhf
values for Ru-122 for x/greaterorequalslant0.2 is most likely connected to the
proximity to the critical concentration xcrit, which could not
be reproduced by our LSDA-based calculations.
In conclusion, a quite unexpected and interesting variation
of the magnetic moments and the hyperfine field with theconcentration xof the Ru-122 compound was found which
is consistent with experimental findings. This shows, inparticular, that Ru-122 and P-122 differ more from each otherwith respect to their magnetic properties, as one might expectfor two isovalently doped pnictides.
D. Relation to the magnetic moment
Finally, the results can be used to clarify the relation
between Bhfand the average magnetic moment μavg.I ti s
quite common to assume that the ratio Aavg
hf=−Bhf/μavgor
Ahf=−Bhf/μspin(Fe) is constant and use this value in order
to obtain the magnetic moments in related compounds fromthe Fe hyperfine fields. For example, Aavg
hf(Fe)=15 T/μB
was given for bulk Fe and Aavg
hf(Fe3+)=11 T/μBfor Fe3+
ions in Fe 2O3[25]. These values give for BaFe 2As2with
an experimental hyperfine field Bexp=−5.47 T a magnetic
moment μavg∼0.4–0.5μB[9,10]. Later on it was questioned
whether these ratios Aavg
hfare applicable to the iron pnictides
[24,25]. In addition, there is general work showing that a
scaling of Bhfwith the corresponding magnetic moment μavg
cannot be assumed ap r i o r i because Aavg
hfvaries strongly for
different materials [ 51]. This is in line with our results that
can be used to quantify Aavg
hf. Additionally, the assumption
of a constant ratio Aavg
hffor doped systems can be disproved,
supporting other work [ 24] which concludes that Bhfis indeed
not proportional to μavgfor BaFe 2As2-based substitutional
systems.
As stressed already, the core s-electron contribution Bc
s
is indeed proportional to μspin(Fe), which is quantified for
our calculations in Fig. 7(a), where we show the ratio Ac=
−Bc
s/μspin(Fe) for all investigated compounds depending on
the concentration x. Independent on x, we find the value
ofAcis nearly constant, 10.6 T /μB. This is in reasonable
agreement with earlier work of Lindgren and Sjøstrøm wherea value around 12.6 T /μ
Bwas calculated [ 48]. However, Bc
s
can obviously vary significantly from Bhf, as was extensively
shown in the literature.
At least for the undoped BaFe 2As2, the average mo-
ment equals the total Fe moment and is close to thespin magnetic moment of iron, μ
avg=μspin(Fe)+μorb(Fe)≈
μspin(Fe). Based on the calculations, one gets for BaFe 2As2
a ratio Ahf=−Bhf/μspin(Fe)=5.2T/μB, or based on the
enhanced hyperfine field /tildewideBhf, a ratio /tildewideAhf=7.8T/μB.T h i s(a)
9.09.510.010.511.011.512.0
0 0.1 0.2 0.3 0.4Ac[T/µB]
Dopant concentration xCo
Ni
K
Ru
P
(b)
3.54.04.55.05.56.06.5
0 0.1 0.2 0.3 0.4Ahf[T/µB]
Dopant concentration xCo
Ni
K
Ru
P
(c)
3.54.04.55.05.56.06.5
0 0.1 0.2 0.3 0.4Aavg
hf[T/µB]
Dopant concentration xCo
Ni
K
Ru
P
FIG. 7. (a) The ratio Ac=−Bc
s/μspin(Fe) is shown for all
investigated compounds, depending on the respective dopant and its
concentration x. The constant behavior shows a reasonable relation
to the magnetic moment μ. However, the same ratios are shown for
(b)Ahf=−Bhf/μspin(Fe) and for (c) Aavg
hf=−Bhf/μavg, having huge
deviations for an constant Ahfbehavior, depending on xa n do nt h e
chosen dopant.
is by a factor of 2–3 different from the ratio Aavg
hf(Fe) applied
in previous publications [ 9,10]. Consequently, the magnetic
moment of BaFe 2As2based on the measured hyperfine field
of 5.47 T should be not between 0 .4μBand 0.5μBbut
rather in the range between 0 .7μBand 1.0μB, which is in
better qualitative agreement with neutron diffraction, reporting
214508-6HYPERFINE FIELDS IN THE BaFe 2As2FAMILY . . . PHYSICAL REVIEW B 94, 214508 (2016)
0.87μB[8]. Nevertheless, one should keep in mind that this
is a qualitative estimation and it is clear from the literature[24,51] and from our work that an estimation of μ
avgbased on
Bhfshould be avoided as far as possible.
However, for the doped iron pnictides there is a sig-
nificant difference between μavgandμspin(Fe). Thus, the
relation between Bhfandμavgleads to an unpredictable,
nonlinear behavior of the ratio Aavg
hf. To quantify our claim
we plot the obtained values of Ahf=−Bhf/μspin(Fe) and
Aavg
hf=−Bhf/μavgdepending on the concentration xfor all
investigated compounds in Figs. 7(b) and7(c), respectively.
Already the ratio Ahf, which is coupled to the spin magnetic
moment of Fe, depends strongly on the respective dopant andon the concentration x. It becomes apparent that for such a
behavior no reasonable relation between B
hfandμspin(Fe)
is possible. This problem becomes even more obvious whenconsidering Aavg
hf. Here, the Ru-122 compound is interesting to
mention because Ahfdecreases with xwhile Aavg
hfincreases
with the concentration. This is due to the fact that theFe moment in Ru-122 increases while the average momentdecreases (see also Fig. 6). Thus, it can be crucially misleading
to relate B
hfto the average magnetic moment μavgin doped iron
pnictides. Consequently, the presented study clearly shows thatthe hyperfine fields B
hfof Fe obtained from57Fe M ¨ossbauer
spectroscopy are not suitable to make predictions about therespective magnetic moment μ
avgin doped iron pnictide
superconductors for different substitutions.
IV . SUMMARY
To summarize, this work presented a comprehensive the-
oretical study of the hyperfine fields in the iron pnictidesuperconductor family of BaFe
2As2with good agreement
with experiment. The CPA was applied to a variety ofcompounds, dealing accurately with the substitutional disorder
and accounting for all variants of doping. This includeselectron doped Ba(Fe
1−xCox)2As2and Ba(Fe 1−xNix)2As2,
hole doped (Ba 1−xKx)Fe 2As2, and also isovalently doped
Ba(Fe 1−xRux)2As2and BaFe 2(As 1−xPx)2. All systems were
investigated in their antiferromagnetic state which was used tostudy the magnetic moments depending on the concentration x
in detail. In order to get meaningful results the fully relativisticDirac formalism was applied, which ensured that all relativisticcontributions to B
hfwere accurately dealt with. Indeed, spin-
orbit induced contributions were found to have a significantlyhigher influence on Fe in BaFe
2As2, as found for bulk Fe.
In particular, the orientation of magnetic moments along the a
axis, consistent with experiment, plays a significant role for thehyperfine field. Consequently, we have quantified in detail whyit is not sensible to apply the bulk Fe ratio A
avg
hf(Fe)=15 T/μB
to the iron pnictides in order to obtain estimations for the
magnetic moment from57Fe M ¨ossbauer spectroscopy. As a
crude estimate, one might rather expect for undoped BaFe 2As2
ratios around 5.0–7 .5T/μB, leading to a magnetic moment of
roughly 0.7–1 .0μB, which is more consistent with neutron
diffraction reporting 0 .87μB[8]. However, it is best to avoid
such estimations, as was shown for the substituted iron pnictidesystems. Here, the behavior of Aavg
hfwith the concentration x
is clearly unpredictable and might lead to wrong conclusions.Thus, relating the hyperfine fields B
hfof Fe obtained via57Fe
M¨ossbauer spectroscopy with the magnetic moments should
be avoided for substituted iron pnictides.
ACKNOWLEDGMENT
We acknowledge financial support from the DFG Project
FOR 1346. J.M. acknowledges project CENTEM PLUS(LO1402) and VEDPMNF (CZ.02.1.01/15.003/358) of Czechministerium MSMT.
[1] Y . Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.
Chem. Soc. 130,3296 (2008 ).
[2] H. Takahashi, K. Igawa, K. Arii, Y . Kamihara, M. Hirano, and
H. Hosono, Nature (London) 453,376(2008 ).
[3] D. J. Singh, P h y s .R e v .B 78,094511 (2008 ).
[4] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys.
Rev. Lett. 101,057003 (2008 ).
[5] R. M. Fernandes, D. K. Pratt, W. Tian, J. Zarestky, A. Kreyssig,
S .N a n d i ,M .G .K i m ,A .T h a l e r ,N .N i ,P .C .C a n fi e l d ,R .J .McQueeney, J. Schmalian, and A. I. Goldman, P h y s .R e v .B 81,
140501 (2010 ).
[6] I. I. Mazin and M. D. Johannes, Nat. Phys. 5,141(2009 ).
[7] I. I. Mazin, M. D. Johannes, L. Boeri, K. Koepernik, and D. J.
Singh, P h y s .R e v .B 78,085104 (2008 ).
[8] Q. Huang, Y . Qiu, W. Bao, M. A. Green, J. W. Lynn, Y . C.
Gasparovic, T. Wu, G. Wu, and X. H. Chen, P h y s .R e v .L e t t .
101,257003 (2008 ).
[9] M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W.
Hermes, and R. P ¨ottgen, Phys. Rev. B 78,020503
(2008 ).
[10] M. Rotter, M. Tegel, I. Schellenberg, F. M. Schappacher, R.
P¨ottgen, J. Deisenhofer, A. G ¨unther, F. Schrettle, A. Loidl, and
D. Johrendt, New J. Phys. 11,025014 (2009 ).[11] A. N. Yaresko, G.-Q. Liu, V . N. Antonov, and O. K. Andersen,
Phys. Rev. B 79,144421 (
2009 ).
[12] I. Opahle, H. C. Kandpal, Y . Zhang, C. Gros, and R. Valent ´ı,
Phys. Rev. B 79,024509 (2009 ).
[13] J. Fink, S. Thirupathaiah, R. Ovsyannikov, H. A. D ¨urr, R.
Follath, Y . Huang, S. de Jong, M. S. Golden, Y .-Z. Zhang, H. O.Jeschke, R. Valent ´ı, C. Felser, S. Dastjani Farahani, M. Rotter,
and D. Johrendt, P h y s .R e v .B 79,155118 (2009 ).
[ 1 4 ] M .J .H a n ,Q .Y i n ,W .E .P i c k e t t ,a n dS .Y .S a v r a s o v , Phys. Rev.
Lett. 102,107003 (2009 ).
[15] L. X. Yang, Y . Zhang, H. W. Ou, J. F. Zhao, D. W. Shen, B.
Zhou, J. Wei, F. Chen, M. Xu, C. He, Y . Chen, Z. D. Wang,X. F. Wang, T. Wu, G. Wu, X. H. Chen, M. Arita, K. Shimada,M. Taniguchi, Z. Y . Lu, T. Xiang, and D. L. Feng, Phys. Rev.
Lett. 102,107002 (2009 ).
[16] L. Craco, M. S. Laad, S. Leoni, and H. Rosner, Phys. Rev. B 78,
134511 (2008 ).
[17] D. Kasinathan, A. Ormeci, K. Koch, U. Burkhardt, W. Schnelle,
A. Leithe-Jasper, and H. Rosner, New J. Phys. 11,025023
(2009 ).
[18] A. Sanna, F. Bernardini, G. Profeta, S. Sharma, J. K. Dewhurst,
A. Lucarelli, L. Degiorgi, E. K. U. Gross, and S. Massidda, Phys.
Rev. B 83,054502 (2011 ).
214508-7GERALD DERONDEAU, J ´AN MIN ´AR, AND HUBERT EBERT PHYSICAL REVIEW B 94, 214508 (2016)
[19] E. Akt ¨urk and S. Ciraci, Phys. Rev. B 79,184523 (2009 ).
[20] Y . Su, P. Link, A. Schneidewind, T. Wolf, P. Adelmann, Y . Xiao,
M. Meven, R. Mittal, M. Rotter, D. Johrendt, T. Brueckel, andM. Loewenhaupt, Phys. Rev. B 79,064504 (2009 ).
[21] H. Gretarsson, A. Lupascu, J. Kim, D. Casa, T. Gog, W. Wu,
S .R .J u l i a n ,Z .J .X u ,J .S .W e n ,G .D .G u ,R .H .Y u a n ,Z .G .Chen, N.-L. Wang, S. Khim, K. H. Kim, M. Ishikado, I. Jarrige,S. Shamoto, J.-H. Chu, I. R. Fisher, and Y .-J. Kim, Phys. Rev. B
84,100509 (2011 ).
[22] P. Vilmercati, A. Fedorov, F. Bondino, F. Offi, G. Panaccione,
P. Lacovig, L. Simonelli, M. A. McGuire, A. S. M. Sefat, D.Mandrus, B. C. Sales, T. Egami, W. Ku, and N. Mannella, Phys.
Rev. B 85,220503 (2012 ).
[23] S. V . Borisenko, D. V . Evtushinsky, Z.-H. Liu, I. Morozov, R.
Kappenberger, S. Wurmehl, B. B ¨uchner, A. N. Yaresko, T. K.
Kim, M. Hoesch, T. Wolf, and N. D. Zhigadlo, Nat. Phys. 12,
311(2016 ).
[24] P. Bonville, F. Rullier-Albenque, D. Colson, and A. Forget,
Europhys. Lett. 89,67008 (2010 ).
[25] I. Nowik, I. Felner, N. Ni, S. L. Bud’ko, and P. C. Canfield,
J. Phys.: Condens. Matter 22,355701 (2010 ).
[26] V . R. Reddy, A. Bharathi, A. Gupta, K. Sharma, S. Chandra, S.
Sharma, K. Vinod, and C. S. Sundar, J. Phys.: Condens. Matter
26,356002 (2014 ).
[27] J. M. Allred, K. M. Taddei, D. E. Bugaris, M. J. Krogstad, S.
H. Lapidus, D. Y . Chung, H. Claus, M. G. Kanatzidis, D. E.
Brown, J. Kang, R. M. Fernandes, I. Eremin, S. Rosenkranz, O.
Chmaissem, and R. Osborn, Nat. Phys. 12,493(2016 ).
[28] H. Pang, Y . Fang, and F. Li, Hyperfine Interact. 199,387
(2011 ).
[29] G. Derondeau, S. Polesya, S. Mankovsky, H. Ebert, and J. Min ´ar,
Phys. Rev. B 90,184509 (2014 ).
[30] G. Derondeau, J. Braun, H. Ebert, and J. Min ´ar,P h y s .R e v .B
93,144513 (2016 ).
[31] G. Derondeau, F. Bisti, J. Braun, V . A. Rogalev, M. Kobayashi,
M. Shi, T. Schmitt, J. Ma, H. Ding, H. Ebert, V . N. Strocov, andJ. Min ´ar,arXiv:1606.08977 .
[32] T. Berlijn, C.-H. Lin, W. Garber, and W. Ku, P h y s .R e v .L e t t .
108,207003 (2012 ).
[33] S. N. Khan and D. D. Johnson, P h y s .R e v .L e t t . 112,156401
(2014 ).
[34] S. N. Khan, A. Alam, and D. D. Johnson, Phys. Rev. B 89,
205121 (2014 ).
[35] A. Herbig, R. Heid, and J. Schmalian, Phys. Rev. B 94,094512
(2016 ).
[36] H. Ebert, D. K ¨odderitzsch, and J. Min ´ar,Rep. Prog. Phys. 74,
096501 (2011 ).
[37] H. Ebert et al. ,The Munich SPR-KKR Package , version 6.3,
http://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR , 2012.
[38] L. Vegard, Z. Phys. 5,17(1921 ).
[39] A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, D. J. Singh, and
D. Mandrus, P h y s .R e v .L e t t . 101,117004 (2008 ).
[40] M. Rotter, M. Pangerl, M. Tegel, and D. Johrendt, Angew. Chem.
Int. Ed. 47,7949 (
2008 ).[41] A. Thaler, N. Ni, A. Kracher, J. Q. Yan, S. L. Bud’ko, and P. C.
Canfield, P h y s .R e v .B 82,014534 (2010 ).
[42] M. Rotter, Ph.D. thesis, LMU M ¨unchen, 2010.
[43] M. Merz, P. Schweiss, P. Nagel, M.-J. Huang, R. Eder, T. Wolf,
H. von L ¨ohneysen, and S. Schuppler, J. Phys. Soc. Jpn. 85,
044707 (2016 ).
[44] S. H. V osko, L. Wilk, and M. Nusair, Can. J. Phys. 58,1200
(1980 ).
[45] M. Battocletti and H. Ebert, P h y s .R e v .B 64,094417 (2001 ).
[46] H. Akai, M. Akai, S. Bl ¨ugel, R. Zeller, and P. H. Dederichs,
J. Magn. Magn. Mater. 45,291(1984 ).
[47] M. E. Elzain, D. E. Ellis, and D. Guenzburger, P h y s .R e v .B 34,
1430 (1986 ).
[48] B. Lindgren and J. Sjostrom, J .P h y s .F :M e t .P h y s . 18,1563
(1988 ).
[49] H. Ebert, H. Winter, D. D. Johnson, and F. J. Pinski, Hyperfine
Interact. 51,925(1989 ).
[50] G. Y . Guo and H. Ebert, Phys. Rev. B 53,2492 (1996 ).
[51] S. Dubiel, J. Alloys Compd. 488,18(2009 ).
[52] H. Akai and T. Kotani, Hyperfine Interact. 120,3(1999 ).
[53] S. Bl ¨ugel, H. Akai, R. Zeller, and P. H. Dederichs, P h y s .R e v .B
35,3271 (1987 ).
[54] H. Ebert, P. Strange, and B. L. Gyorffy, J. Phys. (Paris) 49,31
(1988 ).
[55] D. Benea, O. Isnard, J. Min ´ar, H. Ebert, and V . Pop, J. Appl.
Phys. 109,083909 (2011 ).
[56] H. Ebert and H. Akai, Hyperfine Interact. 78,361(1993 ).
[57] C. Lester, J.-H. Chu, J. G. Analytis, S. C. Capelli, A. S. Erickson,
C. L. Condron, M. F. Toney, I. R. Fisher, and S. M. Hayden,Phys. Rev. B 79,144523 (2009 ).
[58] N. Ni, A. Thaler, J. Q. Yan, A. Kracher, E. Colombier, S. L.
Bud’ko, P. C. Canfield, and S. T. Hannahs, Phys. Rev. B 82,
024519 (2010 ).
[59] D. K. Pratt, M. G. Kim, A. Kreyssig, Y . B. Lee, G. S. Tucker,
A .T h a l e r ,W .T i a n ,J .L .Z a r e s t k y ,S .L .B u d ’ k o ,P .C .C a n fi e l d ,B. N. Harmon, A. I. Goldman, and R. J. McQueeney, Phys. Rev.
Lett. 106,257001 (2011 ).
[60] G. Derondeau, J. Min ´ar, S. Wimmer, and H. Ebert,
arXiv:1608.08077 .
[61] L. Wang, T. Berlijn, Y . Wang, C.-H. Lin, P. J. Hirschfeld, and
W. Ku, P h y s .R e v .L e t t . 110,037001 (2013 ).
[62] M. G. Kim, D. K. Pratt, G. E. Rustan, W. Tian, J. L. Zarestky,
A. Thaler, S. L. Bud’ko, P. C. Canfield, R. J. McQueeney, A.Kreyssig, and A. I. Goldman, Phys. Rev. B 83,054514 (2011 ).
[63] D. Hu, X. Lu, W. Zhang, H. Luo, S. Li, P. Wang, G. Chen, F.
Han, S. R. Banjara, A. Sapkota, A. Kreyssig, A. I. Goldman, Z.Yamani, C. Niedermayer, M. Skoulatos, R. Georgii, T. Keller, P.Wang, W. Yu, and P. Dai, Phys. Rev. Lett. 114,157002 (2015 ).
[64] R. S. Dhaka, C. Liu, R. M. Fernandes, R. Jiang, C. P. Strehlow,
T. Kondo, A. Thaler, J. Schmalian, S. L. Bud’ko, P. C. Canfield,and A. Kaminski, P h y s .R e v .L e t t . 107,267002
(2011 ).
[65] R. S. Dhaka, S. E. Hahn, E. Razzoli, R. Jiang, M. Shi,
B. N. Harmon, A. Thaler, S. L. Bud’ko, P. C. Canfield, andA. Kaminski, Phys. Rev. Lett. 110,067002 (2013 ).
214508-8 |
PhysRevB.97.085434.pdf | PHYSICAL REVIEW B 97, 085434 (2018)
Quantum thermodynamics for driven dissipative bosonic systems
Maicol A. Ochoa,1Natalya Zimbovskaya,2and Abraham Nitzan1,3
1Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
2Department of Physics and Electronics, University of Puerto Rico-Humacao, CUH Station, Humacao, Puerto Rico 00791, USA
3School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel
(Received 9 November 2017; revised manuscript received 26 January 2018; published 23 February 2018)
We investigate two prototypical dissipative bosonic systems under slow driving and arbitrary system-bath
coupling strength, recovering their dynamic evolution as well as the heat and work rates, and we verify thatthermodynamic laws are respected. Specifically, we look at the damped harmonic oscillator and the dampedtwo-level system. For the former, we study independently the slow time-dependent perturbation in the oscillatorfrequency and in the coupling strength. For the latter, we concentrate on the slow modulation of the energygap between the two levels. Importantly, we are able to find the entropy production rates for each case withoutexplicitly defining nonequilibrium extensions for the entropy functional. This analysis also permits the definitionof phenomenological friction coefficients in terms of structural properties of the system-bath composite.
DOI: 10.1103/PhysRevB.97.085434
I. INTRODUCTION
The formulation of thermodynamic concepts applicable
to molecular and nanoscale devices has recently motivatedintense research, as such systems provide a unique settingto study thermodynamic functions, heat transfer, power work,and dissipation at the nanoscale far from the thermodynamiclimit. The characteristics of these systems forbid the directapplication of traditional concepts from macroscopic statisticalthermodynamics, because fluctuations, thermal and quantum,as well as the system’s coupling to its environment canbe relevant for their complete description. In the quantumregime, dynamics [ 1], the broadening of energy levels, and
interference between different pathways can play importantroles and have been studied within the emerging field of quan-tum thermodynamics [ 2–9]. Models of quantum heat engines
that mimic macroscopic setups, for example, two-level Ottoengines that operate in two/four-stroke and continuous cycles[10–12], have been discussed, highlighting the role played
by quantum dissipation and friction [ 13–20] and providing
frameworks for analyzing efficiency and power in quantumheat engines [ 21–29]. Recently, a setting for the realization
of a four-stroke Otto engine with single trapped ions wastheoretically suggested [ 30,31] and experimentally achieved
[32]. Implications of quantum thermodynamics have also
been discussed in the framework of driven open quantumsystems as may be encountered in quantum pumps, where thedriving appears via a suitable time dependence of the system’sHamiltonian.
Such models are often discussed in the weak system-bath
coupling limit, where the thermodynamic functions associated
with the system of interest can be clearly identified. In contrast,
in the strong-coupling limit, one encounters difficulties partlystemming from the uncertainty about assigning the system-bath coupling to any part of the overall system, and alsobecause quantum mechanical broadening makes it difficult to
exactly characterize the system energy. A simple example is
the driven resonant level [ 33–39], where a single electroniclevel is coupled to a Fermi bath (or several such baths) while
its energy and/or coupling to the bath are modulated by an
external force. In the weak system-bath coupling regime,stochastic thermodynamics [ 40,41] and (for periodic driving)
Floquet theory [ 42] have been successfully used for describing
transport and thermodynamic implications of such driving
in a consistent form [ 36,43]. Strong system-bath coupling
[39,44–50] has proven more challenging (strong coupling
in nanothermoelectric devices is discussed in Ref. [ 51]). In
another context, the appearance of paradoxical behavior andanomalies in thermodynamic quantities [ 52,53] such as the
specific heat [ 54,55] has raised questions about the possibility
to achieve a consistent thermodynamic description of stronglycoupled quantum systems.
The driven resonant level model has been useful for un-
derstanding the implications of strong system-bath coupling
on the quantum thermodynamics of small systems. In this
paper we investigate the quantum thermodynamics of two
other prototypical systems operating in the strong-couplingregime and under slow driving—a driven harmonic oscillator
and a driven two-level system strongly coupled to their thermal
bosonic environments. The dynamics under the modulation
of system parameters in these models has been extensively
investigated before (see, e.g., Refs. [ 56,57]). Here, we aim
for a unified formulation that describes dynamic and thermo-
dynamic properties of these systems in the strongly coupled
regime, where the system-bath interaction can be of the same
order as the system energy itself. The first model, Eqs. ( 1)–(6)
below, a harmonic oscillator strongly coupled to a bosonic
bath, and driven by modulating in time its characteristic
energy (i.e., the oscillator’s frequency) or coupling to the bath,
may be applied to describe some physical systems such as
optomechanical heat engines [ 58,59], or molecules adsorbed
on insulator surfaces and subjected to mechanical stress. Inprevious theoretical studies, such models have been used to for-
mulate harmonic quantum Otto engines with time-dependent
frequency [ 30] as well as other quantum heat engines [ 60,61],
and have served to study the interplay between Markovian
2469-9950/2018/97(8)/085434(14) 085434-1 ©2018 American Physical SocietyOCHOA, ZIMBOVSKAYA, AND NITZAN PHYSICAL REVIEW B 97, 085434 (2018)
quantum master equations and Floquet theory [ 62] under
parametrically periodic driving. Indeed, the forced quantum
harmonic oscillator weakly coupled to a thermal bath (the
latter modeled as a set of two-level systems) was analyzed
using stochastic thermodynamics [ 63]. Recently, experimental
studies of the quantum thermodynamics of a two-dimensional
quantum harmonic oscillator having angular momentum were
reported [ 64].
The second model, Eqs. ( 50)–(53), a driven dissipative two-
level system in the strong-coupling regime, is similar to modelsused in quantum optics and quantum electrodynamics butdifferent from the familiar spin-boson model (the dynamics forthe latter was thoroughly described in, for example, Ref. [ 65]).
Previous studies using this model have concentrated on iden-tifying quantum signatures in the thermodynamic behavior ofsuch models in the weak-coupling regime [ 66]. A parametric
one-dimensional oscillator in a time-dependent potential hasbeen studied as a dissipative two-level system [ 67]. Stud-
ies under strong driving and non-Markovian dynamics [ 68]
stressing the nature of work and heat transfer, quantum jumpapproximations to the work statistics [ 69], and the dynamics
and thermodynamics near equilibrium [ 70] have been reported.
Notably, some experimental aspects associated with measuringwork and heat in a dissipative two-level quantum system, whereonly parts of the system and its environment are accessible tothe measurement, were analyzed [ 71].
In contrast to these studies, the present work does not
consider sudden adiabatic steps that uncouple the original
system from the surrounding baths, as such ideal steps maynot reproduce important aspects of their practical realization.Indeed, the operation of nanoengines often involves contin-uous variations such as the migration of chemically bondedmolecules on surfaces, plasmon-exciton couplings, opticallytrapped nanobeads, and optical tweezers. Our strategy closelyfollows the methodology adopted in Ref. [ 38] in the study of
the driven resonant level model, focusing on the dependenceof thermodynamic properties of the overall (system +bath)
system on the system parameters. This strategy permits us togo beyond descriptions based on perturbative treatments of thecoupling strength, such as used in most treatments of quantumBrownian motion (see, e.g., Refs. [ 56,57]). As a consequence
of the bosonic nature of the system under investigation, weare able to go beyond driving in the oscillator’s frequencyand consider in addition the time-dependent perturbations onthe coupling strength for the damped harmonic oscillator.Moreover, we identify quantum friction terms under finite-ratedriving for each case and we achieve a consistent dynamics aswell as thermodynamic characterization in each case.
In Sec. II, we study the damped harmonic oscillator exposed
to external perturbations that drive the oscillator frequency andthe coupling. Next, in Sec. III, we describe the thermodynamics
of the damped harmonic oscillator when the driving changesthe energy gap between levels. These results lead to thesubsequent discussion of quantum friction in Sec. IV.W e
summarize and conclude in Sec. V.
II. THE DRIVEN DAMPED HARMONIC OSCILLATOR
In this section we study a driven harmonic oscillator
coupled to a harmonic bath. The starting point is the standardHamiltonian (here and below we set ¯ h=1)
ˆH=ˆHS+ˆHB+ˆV, (1)
with
ˆHS=/Omega1ˆa†ˆa, (2)
ˆHB=/summationdisplay
mωmˆb†
mˆbm, (3)
ˆV=/summationdisplay
mumˆXˆYm, (4)
ˆX=ˆa+ˆa†, (5)
ˆYm=ˆb†
m+ˆbm, (6)
where ˆa(ˆa†) is the annihilation (creation) operator for the
primary boson of frequency /Omega1, coupled to a bath of bosonic
modes of frequencies ωm, coupling to the primary boson um,
and the corresponding annihilation (creation) operators ˆbm
(ˆb†
m). This bath is at thermal equilibrium with temperature
T=(kBβ)−1, where kBis the Boltzmann constant.
In describing the dynamics of this system, considerable
simplification is achieved by resorting to the rotating waveapproximation, keeping in Eq. ( 1) only coupling terms that
can conserve energy in low order. In this case, the dynamics isfully described by Green’s functions of the form /angbracketleftˆa(t)ˆa
†(t/prime)/angbracketright.
We define the nonequilibrium Green’s function in the Keldyshcontour,
G(τ
1,τ2)=−i/angbracketleftˆa(τ1)ˆa†(τ2)/angbracketrightc, (7)
and notice that the lesser projection G<at equal times provides
the reduced nonequilibrium density matrix for the primaryboson, i.e.,
ρ(t)=iG
<(t,t). (8)
In thermal equilibrium these functions are conveniently
described in frequency space. As in the driven resonancelevel model [ 35,38,39], the dynamics of the process under
study reflects the fact that upon driving, the system exploresdifferent regimes of bath population, the Fermi distribution inRefs. [ 38,39], and the Bose-Einstein distribution here. For sim-
plicity, we follow Refs. [ 38,39] in disregarding other effects,
in particular, those associated with the bath band structure byinvoking the wideband approximation. For static problems thisis justified under the assumption that /Gamma1is small enough so
that its frequency dependence is not explored within the widthof the spectral function A(ω). A necessary condition is that
the bath spectral region explored by the system is well aboveω=0 and well below any cutoff such as the environmental
Debye frequency ω
D, i.e., 0 /lessmuch/Omega1/lessmuchωDand/Gamma1/lessmuchωD.I f/Gamma1is
constant within this regime, the retarded projection Grand the
corresponding spectral density (density of modes projected onthe primary boson) A(ω)=−2I m [G
r(ω)] take the form (see
Appendix A)
Gr(ω)=1
ω−/Omega1+i(/Gamma1/2), (9)
A(ω)=/Gamma1
(ω−/Omega1)2+(/Gamma1/2)2, (10)
085434-2QUANTUM THERMODYNAMICS FOR DRIVEN DISSIPATIVE … PHYSICAL REVIEW B 97, 085434 (2018)
where [with g(ω) being the density of modes of the free bath]
/Gamma1(ω)=2π/summationdisplay
k|uk|2δ(ωk−ω) (11)
=/integraldisplay
dωkg(ω)|uk|2δ(ωk−ω) (12)
is assumed to be independent of ω. Under these assumptions,
the part of the free energy (the canonical potential) that dependson system parameters ( /Omega1and/Gamma1)i sg i v e nb y
F(/Omega1,/Gamma1)=1
β/integraldisplay∞
ωodω
2πA(ω)l n ( 1−e−βω). (13)
In Eq. ( 13),ωo>0 is the cutoff frequency introduced to
guarantee that the integral is finite and well defined. Theeffect of this lower cutoff on the rates that we evaluate in thissection is assessed in Appendix Band found to be irrelevant
for the present analysis as long as ω
ois smaller than other
characteristic energies of the system (i.e., 0 <ωo/lessmuch/Gamma1,/Omega1). In
the following, we omit the limits of integration when writingintegrals but we always keep in mind that a lower cutoff ω
o
has been set. The canonical potential F(/Omega1,/Gamma1) can be used
to determine the dependence on system parameters of allother thermodynamic functions relevant to our calculation (seeSec. II A).
The analysis in Secs. II AandII Bbelow is done under
this assumption. It is also possible that /Gamma1is small enough to
justify the wideband forms ( 9) and ( 10) of the Green’s and
spectral functions but is changing as /Omega1(t) explores different
regimes of the bath spectrum. This case can be treated byassuming that /Gamma1is independent of ωbut depends on /Omega1(t)( s e e
Sec. II C).
In what follows, we investigate the effect of driving either
on the frequency /Omega1or the couplings u
m(and consequently
/Gamma1), limiting our discussion to the case in which local driving
is slow compared with the relaxation rate that drives thesystem into equilibrium. Specifically, we consider that drivingin/Omega1is slow if the relation /Omega1
−1dt/Omega1/lessmuch/Gamma1holds, and also if
/Gamma1−1dt/Gamma1/lessmuch/Gamma1minis valid when the driving is in the coupling
terms uk, with /Gamma1mincorresponding to the minimum value
on/Gamma1achieved during modulation. Physically, we envision
that the strongly coupled composite system is embedded ina larger bath that determines the equilibrium temperature,and that we can follow the irreversible process of interestthrough measuring these rates. Under slow driving, workresults from the action of an external force that changesthe system parameter under consideration. The heat ratedeveloped as a result of this slow perturbation reflects en-tropy changes of the composite whose experimental measure-ment will entail the design of calorimeters encompassing thecomposite.
A. Driving the oscillator frequency
The extreme limit where /Omega1varies infinitely slowly with time
is referred to as the quasistatic limit, where there is a completetime-scale separation between the internal system dynamicsand the external driving. In this limit all equilibrium relation-ships remain valid, except that /Omega1(t) replaces the constant /Omega1.
In the wideband approximation, the retarded Green’s functionand corresponding spectral density, Eqs. ( 9) and ( 10), become
G
r(t,ω)=1
ω−/Omega1(t)+i(/Gamma1/2), (14)
A(t,ω)=/Gamma1
[ω−/Omega1(t)]2+(/Gamma1/2)2, (15)
the latter satisfying the following differential property,
∂
∂ωA(t,ω)=−∂
∂/Omega1A(t,ω). (16)
The canonical potential, Eq. ( 13) is given by
F(/Omega1,/Gamma1)=1
β/integraldisplay∞
ωodω
2πA(t,ω)l n ( 1−e−βω), (17)
and can be used to find the quasistatic entropy (as before,
we focus on the /Omega1-dependent part of this and all other
thermodynamic functions).
The equilibrium (quasistatic) energy E(0)for the composite
system (primary boson +bath) can be obtained from the canon-
ical potential Futilizing the expression E(0)=F+TS(0),
where S(0)represents the absolute entropy of the composite.
Using the canonical potential Fgiven by Eq. ( 13), we compute
the corresponding /Omega1-dependent contributions to all relevant
thermodynamic functions. Thus the entropy S(0)accepts the
form
S(0)(t)=kBβ2∂
∂βF=−kB/integraldisplaydω
2πA(t,ω){n(ω)l nn(ω)
−[1+n(ω)] ln[1 +n(ω)]}, (18)
where n(ω) is the Bose-Einstein distribution n(ω)=(eβω−
1)−1, and the quasistatic energy E(0)and heat capacity C(0)=
(∂/∂T )E(0),
E(0)(t)=F+TS(0)=/integraldisplaydω
2πA(t,ω)ωn(ω), (19)
C(0)(t)=kBβ2/integraldisplaydω
2πω2A(t,ω)n(ω)[1+n(ω)]. (20)
In Eqs. ( 18)–(20) the superscript (0) indicates that the cor-
responding quantity does not depend on the rate ˙/Omega1.I ti s
interesting to notice that these expressions for the equilibriumenergy E
(0)as well as the heat capacity C(0)suggest that an
extended subsystem that includes the primary boson and afraction of the coupling region will effectively describe thethermodynamics of the full system. To illustrate this point weagain focus on that part of the total (system +bath) energy that
depends on system parameters, and following the methodologyin Ref. [ 39], we extend the definition of the canonical potential
in Eq. ( 13) by introducing rescaling parameters which allow
for the computation of the independent contributions to thetotal system-bath energy from the primary boson part ˆH
S,t h e
harmonic bath ˆHB, and the coupling term ˆV(see Appendix C).
The resulting expressions read
/angbracketleftˆHS/angbracketright=/Omega1/integraldisplaydω
2πA(ω)n(ω), (21)
/angbracketleftˆV/angbracketright=2/integraldisplaydω
2πA(ω)(ω−/Omega1)n(ω), (22)
/angbracketleftˆHB/angbracketright=−1
2/angbracketleftˆV/angbracketright. (23)
085434-3OCHOA, ZIMBOVSKAYA, AND NITZAN PHYSICAL REVIEW B 97, 085434 (2018)
Consequently, E(0)=/angbracketleftˆHS/angbracketright+(1/2)/angbracketleftˆV/angbracketright, which suggests that
an effective system with Hamiltonian ˆHeff=ˆHS+(1/2)ˆV
defines the extended system. While this result may be appeal-ing, we stress that the occurrence of an effective Hamiltonianis neither needed in the present discussion of the equilib-rium thermodynamics nor in the subsequent extension to thenonequilibrium regime.
Equivalent expressions can be written in terms of rates. For
example, the rate of change of the internal energy Eis obtained
from Eq. ( 19)t ob e
˙E
(1)=˙/Omega1∂
∂/Omega1E(0), (24)
where the superscript indicates that this rate is linear in ˙/Omega1.
The reversible work associated with infinitesimal variations
in/Omega1must abide to the maximum work principle, therefore,
dW=d/Omega1∂ /Omega1F. Consequently, the reversible power for qua-
sistatic driving is
˙W(1)=˙/Omega1∂
∂/Omega1F=˙/Omega1/integraldisplaydω
2πA(t,ω)n(ω). (25)
This result indicates that a reversible work rate is proportional
to the equilibrium population /angbracketleftn/angbracketright=(2π)−1/integraltext
dωA (ω)n(ω)i n
the primary boson according to ˙W=˙/Omega1/angbracketleftn/angbracketright.
The quasistatic heat generated from an infinitesimal trans-
formation is proportional to the infinitesimal change in theentropy of the system as given by the differential dQ=
d/Omega1T∂
/Omega1S. Hence,
˙Q(1)=˙/Omega1T∂
∂/Omega1S(0)=˙/Omega1/integraldisplaydω
2πA(t,ω)ω∂n(ω)
∂ω. (26)
It is an immediate consequence from the definition of energy
for the composite system that the first law is satisfied. Indeed,
˙E(1)=˙F+T˙S(1)=˙W(1)+˙Q(1)can be easily verified. Obvi-
ously, all reversible changes in the composite system are firstorder in the driving rate ˙/Omega1.
Next, we extend our discussion to the variations that occur
at a small but finite speed, focusing on the nonequilibriumthermodynamics of the system. Following Ref. [ 38], we adopt
a dynamical approach based on the nonequilibrium Green’sfunctions formalism together with the gradient expansion ap-proximation. As outlined in Appendix D, this approach yields a
nonequilibrium correction to the boson distribution function asexperienced by the primary boson, n(ω)→φ
1(t,ω), that can
be obtained from the reduced density matrix of the primaryboson. The result reads
φ
1(t,ω)=n(ω)+˙/Omega1
2A(t,ω)∂
∂ωn(ω). (27)
Following Ref. [ 38], we define nonequilibrium rates in such
a way that in the limit of infinitely slow driving we recoverthe reversible quantities derived above. Nonequilibrium rateswill contain higher-order corrections in the driving rate ˙/Omega1and
we will introduce definitions that respect energy balance ateach order. In brief, our strategy consists of extending the ratesderived for the reversible case by substituting the Boltzmanndistribution n(ω) by the nonequilibrium distribution given
by Eq. ( 27). Thus, starting from Eq. ( 19), we postulate thefollowing form for the nonequilibrium energy,
E
(1)=/integraldisplaydω
2πA(t,ω)ωφ1(t,ω), (28)
such that E(1)=E(0)+(˙/Omega1/2)/integraltext
(dω/2π)ωA2∂ωn(ω). The
definition in Eq. ( 28) is consistent with the rate in Eq. ( 24)
up to first order in the modulation rate ˙/Omega1.1Likewise, the
nonequilibrium heat and work rates are obtained by extendingEqs. ( 25) and ( 26), that is,
˙W
(2)=˙/Omega1/integraldisplaydω
2πA(t,ω)φ1(t,ω)
=˙W(1)+(˙/Omega1)2
2/integraldisplaydω
2πA2∂
∂ωn(ω), (29)
˙Q(2)=˙/Omega1/integraldisplaydω
2πA(t,ω)ω∂φ1(t,ω)
∂ω
=˙Q(1)+(˙/Omega1)2
2/integraldisplaydω
2πAω∂
∂ω/parenleftbigg
A∂n(ω)
∂ω/parenrightbigg
. (30)
The superscript (2) in Eqs. ( 29) and ( 30) indicates that these
rates are exact up to second order in the driving rate /Omega1. These
definitions are consistent with the energy definition in Eq. ( 28)
for the system, and the identity ˙E(2)=˙W(2)+˙Q(2)holds.
Consider now the entropy production. In studying the
driven resonant electron level model, it was suggested that thenonequilibrium form for the entropy function can be obtainedfrom its equilibrium form by replacing the Fermi functionby the corresponding nonequilibrium distribution [ 38]. An
equivalent assumption would lead to an expression for theentropy given by Eq. ( 18) with n(ω) replaced by φ
1(ω)o f
Eq. ( 27). Such a strategy appears to fail in the systems
investigated here. Still, since our main concerns are variationsin the entropy, we can circumvent the actual definition ofa nonequilibrium entropy functional and consider the latterdirectly. Starting from Eq. ( 18) and the quasistatic evolution
derived from the differential dS
(0)=∂/Omega1Sd/Omega1 , we postulate that
a local variation in the nonequilibrium entropy functional maybe presented in a similar form, provided that n(ω) is replaced
byφ
1(t,ω)i n∂/Omega1S. This leads to
dS
dt=˙/Omega1∂S[φ1(ω)]
∂/Omega1, (31)
assumed correct to second order in ˙/Omega1, and consequently to
the following identity for the rate of entropy change to secondorder in ˙/Omega1,
TdS
(2)
dt=˙Q(1)−˙/Omega12
2/integraldisplaydω
2π/bracketleftbigg
A2∂
∂ωn(ω)
+Aω∂
∂ω/parenleftbigg
A∂
∂ωn(ω)/parenrightbigg/bracketrightbigg
. (32)
We identify the first term in the integral in Eq. ( 32) with the
extra power needed to vary /Omega1at a finite rate [as is indeed given
by Eq. ( 29)]. This term corresponds to the entropy production
1Define ˙E(1)=˙/Omega1∂/Omega1E(0),t h e nu s eE q .( 16) and integration by parts to
evaluate ∂/Omega1E(0). Upon substitution of n(ω)b yφ1(t,ω) in the resulting
expression one obtains ˙E(2).
085434-4QUANTUM THERMODYNAMICS FOR DRIVEN DISSIPATIVE … PHYSICAL REVIEW B 97, 085434 (2018)
FIG. 1. Heat, work, system energy, and entropy change rates upon
modulation of /Omega1as a function of the instant primary boson energy /Omega1.
Top (reversible rates): ˙W(1)(blue, solid), ˙Q(1)(yellow, dashed), and
˙E(1)(green, dotted). Not included ˙S(1)=T˙Q(1)as it is proportional to
Q(1). Bottom (nonequilibrium rates): ˙W(2)(blue, solid), ˙Q(2)(yellow,
dashed), and ˙E(2)−˙E(1)=T˙S(2)−˙Q(1)(green, dotted). Parameters
for this model are ˙/Omega1=2.5×10−2meV/fs,T=300 K, /Gamma1=5m e V .
caused by driving the system at such a finite rate. The second
integral in Eq. ( 32) is the second-order contribution to the heat
transferred to the external bath as follows from Eq. ( 30).
We close this section by considering a specific system in
Fig. 1where we modulate the primary boson energy at a
linear rate of 0.025 meV /fs from 15 meV to twice this value.
In the top panel in Fig. 1, reversible rates ˙Q(1)and ˙W(1)as
well as ˙E(1)are presented. The reversible entropy change rate
˙S(1)has not been included as this is a rescaled plot of ˙Q(1)
given by the temperature T[i.e., ˙S(1)=T˙Q(1)]. From this, we
notice that under quasistatic dynamics the work provided tothe system is quickly dissipated in the form of heat, leadingto a nearly vanishing energy change rate of the composite[˙Q
(1)∼− ˙W(1)]. The bottom panel in Fig. 1displays the
second-order contributions to the heat and work rates, as well asthe second-order contributions to the entropy change T(˙S
(2)−
˙S(1)) as given by Eq. ( 32). This illustrates that the terms in
the entropy change rate proportional to ˙/Omega12are positive and
carry the entropy production contribution due to finite ratedriving. In addition, from Eqs. ( 29), (30), and ( 32) we no-
tice that ˙E
(2)−˙E(1)=T˙S(2)−˙Q(1), which suggests that the
increased energy change rate in the system is associated withthe dissipation of energy under finite driving.
We conclude that the present approach to the dynamics
and quantum thermodynamics of the slowly driven dampedharmonic oscillator brings consistent results in the strong-coupling regime.
B. Driving the coupling strength
A different form for time-dependent perturbation appears
when we modulate the system-bath coupling strength whichis now characterized by the time-dependent parameter /Gamma1(t).
Again, if the driving rate is slow, we can assume that the systemchanges quasistatically and find the retarded Green’s functionby substitution of /Gamma1by/Gamma1(t)i nE q .( 9). As a result, we get
G
r(t,ω)=1
ω−/Omega1+i(/Gamma1(t)/2). (33)
Then, the spectral density of states is a time-dependent function
given by
A(t,ω)=/Gamma1(t)
(ω−/Omega1)2+[/Gamma1(t)/2]2, (34)
and the following relation between partial derivatives is satis-
fied,
∂
∂/Gamma1A(t,ω)=−∂
∂ωReGr(t,ω). (35)
The equilibrium thermodynamics of the system is again
derived from the canonical potential introduced by Eq. ( 13),
as well as from the equilibrium entropy given by Eq. ( 18).
While the driving is different from that considered above, themaximum work principle and the relation between reversibleheat and entropy still hold, thus the differential relations dW=
∂
/Gamma1Fd/Gamma1 anddQ=TdS=T∂/Gamma1Sd/Gamma1 remain valid. Therefore,
the adiabatic rates of changes in work and heat generated bythe reversible driving in the coupling strength can be presentedas follows,
˙W
(1)=˙/Gamma1∂
∂/Gamma1F=˙/Gamma1
/Gamma1/integraldisplaydω
2πA(t,ω)(ω−/Omega1)n(ω), (36)
˙Q(1)=T˙/Gamma1∂
∂/Gamma1S=˙/Gamma1
/Gamma1/integraldisplaydω
2πA(t,ω)(ω−/Omega1)ω∂n(ω)
∂ω.(37)
As before, the equilibrium relationship E(0)=F+TS(0)im-
plies that the first law ˙E(1)=˙W(1)+˙Q(1)is satisfied to this
order.
Beyond reversible driving, the nonequilibrium thermody-
namics is obtained after identifying the nonequilibrium formfor the distribution function, experienced by the primaryboson. As detailed in Appendix D, the nonequilibrium Green’s
functions technique and the gradient expansion approximationprovide the functional form for such a distribution,
φ
2(t,ω)=n(ω)−˙/Gamma1
2ReGr∂
∂ωn(ω). (38)
The resemblance in the structure of the distributions of
Eqs. ( 27) and ( 38) is evident, but they behave differently
when the frequency ωis close to /Omega1(see Fig. 2), since Aand
ReGrhave different symmetries around the primary boson
frequency: When ω=/Omega1,A(ω) takes its maximum value while
ReGrvanishes. Consequently, near /Omega1the absolute difference
|φ1−n|must reach its maximum while φ2−n=0, and
the dynamical behaviors associated with driving /Omega1and/Gamma1will
be different.
Repeating the considerations that lead to Eqs. ( 29) and ( 30),
we again obtain an expression for the rates in which the systemexchange work and heat due to /Gamma1variations up to order ˙/Gamma1
2by
replacing n(ω)b yφ2(ω) in the expressions for the reversible
085434-5OCHOA, ZIMBOVSKAYA, AND NITZAN PHYSICAL REVIEW B 97, 085434 (2018)
-8-6-4-2 0 2
0.46 0.48 0.5 0.52 0.54
ω (eV)φ1(ω) -n(ω)
φ2(ω) -n(ω)
FIG. 2. Difference between the nonequilibrium distributions for
the driven dissipative harmonic oscillator and the Bose-Einstein distri-
bution near the oscillator frequency /Omega1. The model under consideration
has as parameters /Omega1=0.5e V , /Gamma1=0.03 eV , T=300 K. In the
figure, we plot the difference φ1(ω)−n(ω) for a linear rate in /Omega1of
˙/Omega1=1m e V /fs (solid black) as well as the difference φ2(ω)−n(ω)
for a linear rate in /Gamma1of˙/Gamma1=1m e V /fs (dashed purple).
rates Eqs. ( 36) and ( 37),
˙W(2)=˙/Gamma1
/Gamma1/integraldisplaydω
2πA(t,ω)(ω−/Omega1)φ2(t,ω)
=˙W(1)−(˙/Gamma1)2
2/integraldisplaydω
2π(ReGr)2∂
∂ωn(ω), (39)
˙Q(2)=˙/Gamma1
/Gamma1/integraldisplaydω
2πA(t,ω)(ω−/Omega1)ω∂φ2(t,ω)
∂ω
=˙Q(1)−(˙/Gamma1)2
2/integraldisplaydω
2πReGrω∂
∂ω/parenleftbigg
ReGr∂n(ω)
∂ω/parenrightbigg
.
(40)
The time-dependent energy for the composite system is again
given by Eq. ( 28), this time with the nonequilibrium distribu-
tion given by Eq. ( 38). Consequently, energy conservation (the
first law) is established also at the second order in the drivingrate˙/Gamma1.
Finally, we verify that these rates are consistent with the
time derivative of the nonequilibrium entropy. While we donot introduce an explicit expression for this function, we canfind a suggestive form for its time derivative to second orderin˙/Gamma1by repeating the procedure that lead to Eq. ( 32), replacing
the function n(ω)i nt h e /Gamma1derivative of the entropy functional,
∂
/Gamma1S[n(ω)], by φ2(t,ω), leading to ˙S=˙/Gamma1∂/Gamma1S(φ2) correct to
second order, and hence
TdS(2)
dt=˙Q(1)−˙/Gamma12
2/integraldisplaydω
2π/bracketleftbigg
(ReGr)2∂n(ω)
∂ω
+ωReGr∂
∂ω/parenleftbigg
ReGr∂
∂ωn(ω)/parenrightbigg/bracketrightbigg
. (41)
Here, the first term in the integral corresponds to the entropy
production [see Eq. ( 39)] while the second one is the entropy
change due to heat transfer [see Eq. ( 40)]. Once more, wehave found a consistent dynamics as well as thermodynamic
description for the damped harmonic oscillator under slowdriving.
C. Including effects due to the bath band structure
In Secs. II A and II B we have neglected the effect of
variations in the density of relevant bath modes (modes withω∼/Omega1) upon variation of /Omega1. Here, we go one step beyond this
approximation and consider the situation in which the couplingparameter /Gamma1[Eq. ( 12)] varies due to bath band structure. We
still assume that /Gamma1depends on ωweakly enough ( ∂/Gamma1/∂ω /lessmuch1)
over the interval of modulation. In this case, we expect that thespectral function Acan be well described by the Lorentzian
A(t,ω)=/Gamma1(/Omega1(t))
(ω−/Omega1(t))2+(/Gamma1(/Omega1(t))/2)2, (42)
where we have included the functional dependence of /Gamma1on
the oscillator’s frequency /Omega1. The spectral function in Eq. ( 42)
satisfies the following identity,
∂
∂/Omega1A(t,ω)=−∂
∂ωA(t,ω)−∂/Gamma1
∂/Omega1∂
∂ωReGr(t,ω),(43)
which as in previous sections can be used to obtain the rates of
change in heat and work due to modulation in /Omega1.I nE q .( 43)
and below, we disregard derivatives of /Gamma1with respect to ω
since our considerations allow us to assume that this term isonly a function of /Omega1. The steps involved in the derivation of
energy fluxes have been illustrated above: Starting from thecanonical potential in Eq. ( 13), this time defined in terms of the
spectral function Ain Eq. ( 42), we obtain equilibrium entropy
and energy functionals in the corresponding forms given byEqs. ( 18) and ( 19) [with Agiven by Eq. ( 42)]. The reversible
work ˙W
(1)and heat rates ˙Q(1)are derived from the maximum
work principle and the fact that quasistatic heat due to aninfinitesimal transformation is proportional to the infinitesimalchange in the entropy of the system. As a consequence ofrelation ( 43), we find that the heat and work rates can each be
written in terms of two contributions: direct modulation in /Omega1as
well as a correction term, proportional to ∂/Gamma1/∂/Omega1 , originating
from the indirect modulation in /Gamma1. The explicit forms of the
reversible rates are proportional to ˙/Omega1and given by
˙W
(1)=˙/Omega1/integraldisplaydω
2πA(t,ω)n(ω)
+˙/Omega1
/Gamma1∂/Gamma1
∂/Omega1/integraldisplaydω
2πA(t,ω)(ω−/Omega1)n(ω), (44)
˙Q(1)=˙/Omega1/integraldisplaydω
2πA(t,ω)ω∂n(ω)
∂ω
+˙/Omega1
/Gamma1∂/Gamma1
∂/Omega1/integraldisplaydω
2πA(t,ω)(ω−/Omega1)ω∂n(ω)
∂ω. (45)
Beyond quasistatic dynamics and utilizing the results in
Appendix D, we find the nonequilibrium distribution function
˜φ(t,ω) valid to first order in ˙/Omega1,
˜φ(t,ω)=n(ω)+˙/Omega1(t)
2/parenleftbigg
A(t,ω)−∂/Gamma1
∂/Omega1ReGr/parenrightbigg∂
∂ωn(ω).(46)
085434-6QUANTUM THERMODYNAMICS FOR DRIVEN DISSIPATIVE … PHYSICAL REVIEW B 97, 085434 (2018)
Repeating the considerations that lead to Eqs. ( 29) and ( 30),
we once more obtain expressions for ˙W(2)and ˙Q(2). We notice
that the nonequilibrium rates up to second order in the drivingrate ˙/Omega1include corrections due to the bath structure that are
proportional to ( ∂/Gamma1/∂/Omega1 )
2,
˙W(2)=˙W(1)+(˙/Omega1)2
2/integraldisplaydω
2πA2∂
∂ωn(ω)
−(˙/Omega1)2
2/parenleftbigg∂/Gamma1
∂/Omega1/parenrightbigg2/integraldisplaydω
2π(ReGr)2∂
∂ωn(ω),(47)
˙Q(2)=˙Q(1)+(˙/Omega1)2
2/integraldisplaydω
2πAω∂
∂ω/parenleftbigg
A∂n(ω)
∂ω/parenrightbigg
−(˙/Omega1)2
2/parenleftbigg∂/Gamma1
∂/Omega1/parenrightbigg2/integraldisplaydω
2πReGrω∂
∂ω/parenleftbigg
ReGr∂n(ω)
∂ω/parenrightbigg
.
(48)
Finally, we remark that the entropy rate
TdS(2)
dt=˙Q(1)−˙/Omega12
2/integraldisplaydω
2π/bracketleftbigg
A2∂
∂ωn(ω)
+Aω∂
∂ω/parenleftbigg
A∂
∂ωn(ω)/parenrightbigg/bracketrightbigg
−˙/Omega12
2/parenleftbigg∂/Gamma1
∂/Omega1/parenrightbigg2/integraldisplaydω
2π/bracketleftbigg
(ReGr)2∂n(ω)
∂ω
+ωReGr∂
∂ω/parenleftbigg
ReGr∂
∂ωn(ω)/parenrightbigg/bracketrightbigg
(49)
also includes correction terms proportional to ( ∂/Gamma1/∂/Omega1 )2and
is consistent with the rates obtained in Eqs. ( 47) and ( 48).
III. THE DAMPED TWO-LEVEL SYSTEM
In this section, we consider a two-level molecule strongly
coupled with a thermal bath represented, as before, by a con-tinuum of harmonic modes. We will again disregard changesin the local bath band structure by adopting a widebandapproximation. The methods introduced in Sec. II Ccan be
implemented here if one needs to account for the effect ofsuch a structural change. In the Hilbert space of the molecule,each level is represented by a ket |i/angbracketright, with i∈{1,2}.T h e
Hamiltonian for the composite system is the sum of thefree Hamiltonian for the molecule ˆH
TLS, the harmonic bath
Hamiltonian ˆHB, and the coupling V,
ˆH=ˆHTLS+ˆHB+ˆV, (50)
ˆHTLS=ωLˆσz, (51)
ˆHB=/summationdisplay
kωkˆb†
kˆbk, (52)
ˆV=i1
2/summationdisplay
k(ukˆσ+ˆbk−u∗
kˆσ−ˆb†
k), (53)
where ˆ σz=(1/2)(|2/angbracketright/angbracketleft2|−|1/angbracketright/angbracketleft1|), ˆσ+=|2/angbracketright/angbracketleft1|, and ˆ σ−=
|1/angbracketright/angbracketleft2|. Here, ωLis the spacing between level energies, ωkarethe frequencies of the bath modes, and ukare the molecule-bath
coupling elements. A complete thermodynamic descriptionat equilibrium can be obtained from the free energy—thecanonical potential for the two-level system-bath compositesystem. The partition function and the free energy for thismodel are calculated in Appendix Efrom an approximate
description of the energy spectrum of the two-level systeminteracting with a finite but large bath. In the derivation weassume that the energy spacing between consecutive modes inthe bath is small and we take the limit of infinitesimal spacing.The result reads
F=1
β/integraldisplaydω
2πA(ω)l n ( 1−e−βω)−1
2/radicalBig
ω2
L+4η, (54)
where
η=lim
N→∞(1/4N)N/summationdisplay
k=1|uk|2, (55)
andA(ω) represents the spectral density. In standard models
for thermal baths,/summationtext
k|uk|2is constant and η→0a sN→
∞. Again, the equilibrium entropy functional is obtained by
differentiation of the canonical potential in Eq. ( 54) with
respect to the absolute temperature T. As a result, we arrive at
the following expression,
S(0)=−kB/integraldisplaydω
2πA(ω){n(ω)l n [n(ω)]
−[1+n(ω)] ln[1 +n(ω)]}. (56)
An approximate expression for the spectral density A(ω)i s
found using the nonequilibrium Green’s function (NEGF)technique in Appendix F. We get
A(ω)=/Gamma1S
2
(ω−ωL)2+(/Gamma1S/2)2. (57)
In this expression, S=−2/angbracketleftˆσz/angbracketrightis the difference in population
between the levels. The approximation employed to obtainEq. ( 57) assumes a factorization of a higher-order correlation
function in terms of lower-order ones, providing a simplesolution to the associated Dyson equation [see Eq. ( F4)]. We
notice that in the absence of population inversion, Sis positive.
If the change in ω
Ldue to driving is small relative to ωLitself,
we may disregard the dependence of SonωL.In this case, the
spectral function Adefined by Eq. ( 57) satisfies the equation
∂
∂ωA(ω)=−∂
∂ωLA(ω). (58)
This property of Ais used in the following computations of
work and heat rates.
The equilibrium energy functional E(0)=F+TS(0)can
be determined from Eqs. ( 54) and ( 56) and is given explicitly
by the expression
E(0)=/integraldisplaydω
2πA(ω)ωn(ω)−1
2/radicalBig
ω2
L+4η. (59)
Next, we introduce the quasistatic work and heat rates, utilizing
as in the previous section the maximum work principle andthe relation between entropy change and reversible heat. This
085434-7OCHOA, ZIMBOVSKAYA, AND NITZAN PHYSICAL REVIEW B 97, 085434 (2018)
leads to
˙W(1)=˙ωL∂
∂ωLF
=˙ωL/integraldisplaydω
2πA(ω)n(ω)−˙ωL
2ωL/radicalBig
ω2
L+4η,(60)
˙Q(1)=kB
β˙ωL∂
∂ωLS=˙ωL/integraldisplaydω
2πA(ω)ω∂n(ω)
∂ωL. (61)
The definition of the equilibrium energy E(0)and the fact
that the quasistatic energy variation is given by ˙E(1)=
˙ωL∂E(0)/∂ωLimply that energy balance (the first law) holds
for the rates derived in Eqs. ( 60) and ( 61), that is, ˙E(1)=
˙W(1)+˙Q(1).
It is interesting to compare the quasistatic evolutions of
this system and the damped harmonic oscillator consideredin Sec. II A. In the limit of large separation between levels,
S→1. Then Eqs. ( 26) and ( 61) yield identical expressions
for reversible heat rates provided that /Omega1is identified with ω
L.
The expressions for the reversible work flux in Eqs. ( 25) and
(60) appear different, however, this difference [which is also
reflected by the second term in Eq. ( 60)] just reflects the fact
the the ground state of the two-level system was chosen tobe−ω
L/2 [note that ηin Eq. ( 55) vanishes if ukis constant,
independent of the number of modes taken to model the bath].
As before, nonequilibrium effects appear in the next order
(2) in ˙ωLand explicit expressions can be derived following the
procedure used previously. First, we find the nonequilibriumdistribution function (see Appendix G),
φ
3(t,ω)=n(ω)+˙ωL
2S−1A(t,ω)∂
∂ωn(ω). (62)
Then we employ this function to compute the work and heat
nonequilibrium rates. For this purpose, we replace the Bose-Einstein distribution functions in expressions ( 60) and ( 61), by
φ
3(t,ω). The resulting nonequilibrium rates equal
˙W(2)=˙ωL/integraldisplaydω
2πA(ω)φ3(t,ω)−˙ωLωL
2/radicalBig
ω2
L+4η
=˙W(1)+(˙ωL)2
2S−1/integraldisplaydω
2πA2∂n(ω)
∂ω, (63)
˙Q(2)=˙ωL/integraldisplaydω
2πA(ω)ω∂φ3(ω)
∂ωL
=˙Q(1)+(˙ωL)2
2S−1/integraldisplaydω
2πAω∂
∂ω/bracketleftbigg
A∂
∂ωn(ω)/bracketrightbigg
.(64)
Also, making the same replacement [ n(ω)→φ3(ω)] in expres-
sion ( 59) for the energy functional, we can verify that energy
balance holds at second order in ˙ ωLfor the rates given by
Eqs. ( 63) and ( 64).
Similarly, the second-order contributions to the total
entropy rate ˙S(2), calculated from the differential dS=
(∂ωLS)dωL, permit a full identification of the entropy produc-
tion term. Indeed,
TdS(2)
dt=˙Q(1)−˙ωL
2S−1/integraldisplaydω
2π/bracketleftbigg
A2∂n(ω)
∂ω
+ωA∂
∂ω/parenleftbigg
A∂
∂ωn(ω)/parenrightbigg/bracketrightbigg
. (65)Here, the first integral on the right-hand side of Eq. ( 65)
corresponds to the rate of heat dissipated as entropy productionwhich is already identified by Eq. ( 63) as the nonequilibrium
work rate ˙W
(2), while the second integral is the heat flux
determined by Eq. ( 64). Thus we have achieved a complete and
consistent dynamic as well as thermodynamic representation ofthe damped two-level system under reversible and slow drivingof the energy gap ω
L.
Finally, note that (as expected) also the second-order terms
are the same as the damped harmonic oscillator in the limitS→1.
IV . FRICTION
Dissipation in a nanoscale engine due to its interactions with
the environment could be introduced in the equations of motionfor the system describing the time evolution of a physicalcoordinate by adding a phenomenological friction term. Theanalytic form for dissipative terms which may be ascribedto friction can be singled out from the detailed quantummechanical description of the dynamics of a particular opensystem. As known, friction is closely related to the powerdissipated in the system. It strongly depends on the system’sspeed. When the motion is infinity, slow friction approacheszero, and it increases as the system is speeding up [ 61]. In
Eqs. ( 29), (39), and ( 63) we have identified the power dissipated
under finite speed in the damped harmonic oscillator and in thedissipative two-level molecule subject to various drivings. Inorder to define a friction coefficient for each case, we have toassociate time perturbations with changes in certain externalcoordinates. Below, we use an example to show how theserelationships may be established.
In a recent experimental work [ 32], an Otto engine was
realized with a single trapped ion in a linear Paul trap with afunnel-shaped electrode geometry. The radial trap frequencyω
x,ywas observed to be decreasing in the axial zdirection as
ωx,y=ωo/slashbigg/parenleftbigg
1+z
rotanθ/parenrightbigg2
. (66)
This result suggests that the approximation, ωx,y=ωo(1−
2ztanθ/ro) may be employed for small θ.Thus, a displace-
ment along the zaxis in the trap induces a change of frequency
˙ωx,y=−2t a nθ˙z. This demonstrates that a linear relation
between the characteristic frequency of an atomic oscillatorand physical displacement is feasible. Thus, for the dampedharmonic oscillator, with the driven /Omega1(Sec. II A), we can
assume that ˙/Omega1=M
1˙z.
We may generalize this relationship and apply it to our
model. The dissipated power ˙W(2)is caused by a friction
force F1acting on the external coordinate zaccording to
˙W(2)=−F1˙z, with F1=−γ1˙z. Then, Eq. ( 29) leads to the
following form for the friction coefficient γ1,
γ1=−M2
1
2/integraldisplaydω
2πA2∂
∂ωn(ω). (67)
Similarly, if the rate of changes ˙/Gamma1in Eq. ( 39) and ˙ωLin Eq. ( 63)
could be related to some coordinate zvia˙/Gamma1=M2zand ˙ωL=
M3z, then the corresponding friction coefficients for motions
085434-8QUANTUM THERMODYNAMICS FOR DRIVEN DISSIPATIVE … PHYSICAL REVIEW B 97, 085434 (2018)
along these coordinates would be
γ2=−M2
2
2/integraldisplaydω
2π(ReGr)2∂
∂ωn(ω), (68)
γ3=−M2
3
2S−1/integraldisplaydω
2πA2∂n(ω)
∂ω. (69)
V . CONCLUSIONS
We have presented a systematic description of the dynamics
as well as the thermodynamics for a harmonic oscillator and atwo-level system coupled to a harmonic bath, both subject toslow driving rates. Our approach is an extension of the oneintroduced in Ref. [ 38]. The effects of driving are studied
within the nonequilibrium Green’s functions formalism and
the gradient expansion method. Our results are consistent withthe first and second laws of thermodynamics, yielding explicitexpressions for the work, heat, and entropy productions associ-ated with the driving process, valid for system-bath interactionsof arbitrary strengths. Similar to Ref. [ 38]( s e ea l s oR e f .[ 39])
we could identify, within the models studied, an effective
system Hamiltonian that accounts for system properties by
including half the system-bath interaction. Unlike Ref. [ 38],
a suggestive expression for the entropy production rate isobtained without the need to define the total entropy.
The formalism introduced in the present work can provide
a guideline for future thermodynamic treatments of stronglycoupled quantum nanoscale systems, and can be directly
applied to currently explored experimental setups such as the
realized optomechanical heat engine [ 58,59] or an approach of
a molecule to a metal surface.
ACKNOWLEDGMENTS
The research of A.N. is supported by the Israel-U.S. Bina-
tional Science Foundation, the German Research Foundation(DFG TH 820/11-1), the U.S. National Science Foundation(Grant No. CHE1665291), and the University of Pennsylvania.N.Z. acknowledges support from NSF-DMR-PREM 1523463.We thank Peter Hänggi and Peter Talkner for helpful discus-sions and critical comments.
APPENDIX A: RETARDED GREEN’S FUNCTION
FOR THE DAMPED HARMONIC OSCILLATOR Gr
Here, we derive Eq. ( 9). From the Hamiltonian given by
Eq. ( 1), we find that the Heisenberg equations of motion for ˆa
andˆbmare
id
dtˆa(t)=/Omega1ˆa+/summationdisplay
mumˆbm, (A1)
id
dtˆbm(t)=ωmˆbm+umˆa. (A2)
Next, we derive the equation of motion (EOM) for the Green’s
function defined in Eq. ( 7), in the Keldysh contour to later find
its retarded expression in frequency space. Indeed, utilizing
Eq. ( A1), we get
id
dτ1G(τ1,τ2)=δ(τ1,τ2)+/Omega1G(τ1,τ2)+/summationdisplay
umGma(τ1,τ2),
(A3)withGma(τ1,τ2)=−i/angbracketleftˆbm(τ1)ˆa†(τ2)/angbracketright.N o w ,w efi n dt h eE O M
forGma(τ1,τ2) utilizing Eq. ( A2), that is,/parenleftbigg
id
dτ1−ωm/parenrightbigg
Gma(τ1,τ2)=umG(τ1,τ2). (A4)
Forgm(τ1,τ2)=−i/angbracketleftˆbm(τ1)ˆb†
m(τ2)/angbracketright, the Green’s function that
solves the Dyson equation for a free boson (null self-energy),
we verify that the identity/parenleftbigg
id
dτ1−ωm/parenrightbigg
gm(τ1,τ2)=δ(τ1,τ2)( A 5 )
holds. The result described by Eq. ( A5) permits us to solve
Eq. ( A4),
Gma(τ1,τ2)=um/integraldisplay
dτ3gm(τ1,τ3)G(τ3,τ2). (A6)
Substituting Eq. ( A6)i n t o( A3), we obtain
id
dτ1G(τ1,τ2)=δ(τ1,τ2)+/Omega1G(τ1,τ2)
+/summationdisplay
|um|2/integraldisplay
dτ3gm(τ1,τ3)G(τ3,τ2).
(A7)
We now project onto the real line to derive the retarded form
Gr(t1,t2) of the Green’s function using Langreth rules. Then,
we define new variables s=t1−t2andt=(t1+t2)/2, such
that
i/parenleftbiggd
ds+1
2d
dt/parenrightbigg
Gr(t,s)=δ(s)+/parenleftbigg
/Omega1−i/Gamma1
2/parenrightbigg
Gr(t,s),(A8)
where we have adopted the wideband limit for the last term in
Eq. ( A8). We calculate the Fourier transform with respect to s
in Eq. ( A8) to get
Gr(t,ω)=/parenleftbigg
1−i
2d
dtGr(t,ω)/parenrightbigg/parenleftbigg1
ω−/Omega1+i(/Gamma1/2)/parenrightbigg
.(A9)
Thus the zeroth-order approximation for Gr(t,ω), correspond-
ing to the adiabatic limit, is obtained by disregarding the term
involving the derivative with respect to tin the right-hand side
of Eq. ( A9). The result is given in Eq. ( 9).
APPENDIX B: LOWER CUTOFF
FOR THE CANONICAL POTENTIAL
We introduce a cutoff frequency ωo=1/nsuch that
F(/Omega1,/Gamma1)=/integraldisplay∞
0dω
2πA(ω)l n ( 1−e−βω),
=/integraldisplay∞
ωodω
2πA(ω)l n ( 1−e−βω)
+/integraldisplayωo
0dω
2πA(ω)l n ( 1−e−βω), (B1)
We estimate∂
∂/Gamma1F(/Omega1,/Gamma1) to show that the terms below the lower
cutoff do not contribute to the rates ˙/Gamma1. In the region (0 ,ωo)w e
approximate ln[1 −e−βω]≈ln(βω). Then,
/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
∂/Gamma1/integraldisplayωo
0dω
2πA(ω)l n ( 1−e−βω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/lessorequalslant/integraldisplay
ωo
0dω
2π/Omega12
/Gamma1A(ω)|ln(βω)|
085434-9OCHOA, ZIMBOVSKAYA, AND NITZAN PHYSICAL REVIEW B 97, 085434 (2018)
=lim
n→∞/integraldisplayωo
1/(n+1)dω
2π/Omega12
/Gamma1A(ω)|ln(βω)|
/lessorequalslant−/Omega12
/Gamma1lim
n→∞ln/parenleftbiggβ
n+1/parenrightbigg/integraldisplayωo
1/(n+1)dω
2π1
(ω−/Omega1)2
/lessorequalslant−1
/Gamma1lim
n→∞ln/parenleftbiggβ
n+1/parenrightbigg/parenleftbigg1
n−1
n+1/parenrightbigg
→0.(B2)
APPENDIX C: EFFECTIVE HAMILTONIAN
FOR THE EXTENDED HARMONIC OSCILLATOR
In this Appendix we calculate the partial contributions to the
total energy of the dissipative harmonic oscillator utilizing themethod in Ref. [ 39]. In brief, we introduce rescaling parameters
(λ
S,λB,λV) in the Hamiltonian in Eq. ( 1) such that
ˆH(λS,λB,λV)=λSˆHS+λBˆHB+λVˆV. (C1)
This rescaling transfers to the spectral function Aas well as to
the canonical potential according to
A(λS,λB,λV)=λ−1
Bλ2
V/Gamma1
(ω−λS/Omega1)2+/parenleftbig
λ−1
Bλ2
V/Gamma1/parenrightbig2, (C2)
/Omega1(λS,λB,λV)=1
β/integraldisplay
A(λS,λB,λV)l n ( 1−e−βω).(C3)
With these definitions we can show that
∂
∂λSA(λS,1,1)=−/Omega1∂
∂ωA(λS,1,1), (C4)
∂
∂λBA(1,λB1)=λ−2
B/Gamma1∂
∂ωReGr(1,λB,1), (C5)
∂
∂λVA(1,1,λV)=−2λV/Gamma1∂
∂ωReGr(1,1,λV), (C6)
as well as
/angbracketleftˆHS/angbracketright=/Omega1/integraldisplaydω
2πA(ω)n(ω), (C7)
/angbracketleftˆHB/angbracketright=−/integraldisplaydω
2π(ω−/Omega1)A(ω)n(ω), (C8)
/angbracketleftˆV/angbracketright=2/integraldisplaydω
2π(ω−/Omega1)A(ω)n(ω). (C9)
Equations ( C7)–(C9) follow from the identity
/angbracketleftˆHi/angbracketright=−β∂
∂λi/Omega1(λi). (C10)
APPENDIX D: NONEQUILIBRIUM
DISTRIBUTION FUNCTIONS
Starting from the definition in Eq. ( 7), we can implement
the gradient expansion and keep only the terms up to first orderin energy and time derivatives. We then obtain
G
<(t,ω)=Gr(t,ω)/Sigma1<(t,ω)Ga(t,ω)
+i
2/bracketleftbigg
Gr(t,ω)∂Ga(t,ω)
∂t
−∂Gr(t,ω)
∂tGa(t,ω)/bracketrightbigg∂/Sigma1<(ω)
∂ω. (D1)Since
∂Gr
∂t=˙/Omega1(Gr)2,∂Ga
∂t=˙/Omega1(Ga)2, (D2)
GrGa=A(t,ω)
/Gamma1, (D3)
we get
iG<(t,ω)=An(ω)+˙/Omega1
2A2∂
∂ωn(ω), (D4)
where n(ω) is the Bose-Einstein distribution function. We
define the nonequilibrium distribution function φ1(t,ω)b yt h e
expression
iGr(t,ω)=A(t,ω)φ1(t,ω). (D5)
Consequently, φ1(t,ω) should be given by Eq. ( 27).
In Sec. II Bwe have studied the quantum thermodynamics
when driving affects the coupling strength. In this case andstarting from Eq. ( 33), we have
∂G
r
∂t=−i
2˙/Gamma1(Gr)2,∂Ga
∂t=i
2˙/Gamma1(Ga)2, (D6)
which after substitution in Eq. ( D1) lead to
iG<(t,ω)=An(ω)−˙/Gamma1
2AReGr∂
∂ωn(ω). (D7)
From this expression, we obtain the result for φ2(t,ω) given by
Eq. ( 38).
APPENDIX E: POTENTIAL FOR THE DAMPED
TWO-LEVEL SYSTEM
Here, we derive the expression for the canonical potential
for the dissipative two-level system discussed in Sec. III.W e
start by studying the Hamiltonian and the energy spectrum of atwo-level system coupled to a finite bath with Nnoninteracting
bosons. We assume that the frequency of boson mode kin
the bath is given by ω
k=k/Delta1ω (/Delta1ωis the inverse density
of modes, assumed constant), with k∈{0,..., N },/Delta1ω=
ωmax/N, andωmaxis an upper frequency cutoff defining the
bandwidth of the bath. Moreover, for each mode k, we consider
a finite number of phonons nk. Thus the bath is characterized
by the set of pararameters {N,/Delta1ω, {nk}}. System-bath coupling
is defined by the Hamiltonian in Eq. ( 53), which assumes the
rotating phase approximation. A basis for the composite system(TLS+finite bath) is obtained from the tensor product betweenthe energy eigenbasis for the two-level system and the diagonalbasis for the noninteracting bath: Denoting the two-level sys-tem eigenvectors by |l/angbracketright,l∈{1,2}, the basis for the composite
state is |l,{n
k|1/lessorequalslantk/lessorequalslantN}/angbracketright = |l/angbracketright⊗|n1/angbracketright⊗···⊗| nN/angbracketright.I nt h i s
basis and as a consequence of the interaction Hamiltonian inEq. ( 53), we find that
/angbracketleft1,n
1,..., n k+1,..., n N|ˆV|2,n1,..., n k,..., n N/angbracketright=−i
2uk,
(E1)
for all 1 /lessorequalslantk/lessorequalslantN, and also
/angbracketleftl,{nk}|ˆHTLS+ˆHB|l,{nk}/angbracketright =(−1)l
2ωL+N/summationdisplay
k=1ωknk.(E2)
085434-10QUANTUM THERMODYNAMICS FOR DRIVEN DISSIPATIVE … PHYSICAL REVIEW B 97, 085434 (2018)
LetεB=/summationtextN
k=1ωknkands=2+/summationtext
knk. Equations ( E1) and
(E2) indicate that the Hamiltonian acts on the state vec-
tor|l,{nk}/angbracketrightby preserving the total number s. In particular,
for a system in the initial state |2,{nk}/angbracketrightallowed transitionscouple relaxations at the two-level system (2 →1) with
excitations in a single mode in the bath ( nk→nk+1f o r
some k). Thus, in the subspace generated by the family of
kets,
{|2,{nk}/angbracketright,
|1,n1+1,{nk,k/negationslash=1}/angbracketright,
...,|1,{nk,k < j },nj+1,{nk,k > j }/angbracketright,
...,|1,{nk,k < N },nN+1/angbracketright}, (E3)
we find a matrix representation for the Hamiltonian Eq. ( 50), in terms of matrices AandB,
A=⎛
⎜⎜⎜⎜⎝−
ωL
2+εB 00 0
0 −ωL
2+ω1+εB 0 ... 0
00 −ωL
2+ω2+εB... 0
...............
00 0 ...−ωL
2+ωN+εB⎞
⎟⎟⎟⎟⎠, (E4)
B=⎛
⎜⎜⎜⎜⎜⎜⎜⎝ω
L−i
2u1−i
2u2...−i
2uN
i
2u1 00 ... 0
i
2u2 00 ... 0
......0......
i
2uN 00 ... 0⎞
⎟⎟⎟⎟⎟⎟⎟⎠, (E5)
such that
H({nk})=ˆHTLS+ˆHB+ˆV=A+B. (E6)
We emphasize that this is the representation of the Hamil-
tonian in the subspace defined by Eq. ( E3), which depends on
the initial set {nk}. We now investigate the partition function
/Xi1{nk}=Tr{exp[−βH({nk})]}by approximating the energy
eigenvalues in H({nk}) using Weyl’s matrix inequalities [ 72],
which we state next in our context. In brief, the eigenvaluesofAandBprovide lower and upper bounds for the energy
eigenvalues in H({n
k}) that depend on the inverse density of
bath modes /Delta1ω.
Since AandBare (N+1)-dimensional Hermitian matri-
ces, their eigenvalues, which we will denote by {αk}and{γk},
respectively, can be listed in decreasing order. Thus we write
αk=−ωL
2+ωN−k+εB, (E7)
with 0/lessorequalslantk/lessorequalslantNandω0=0, as well as
γ0=1
2/parenleftbig
ωL+/radicalBig
ω2
L+4η/parenrightbig
, (E8)
γN=1
2/parenleftbig
ωL−/radicalBig
ω2
L+4η/parenrightbig
, (E9)
γk=0 otherwise , (E10)
where we have introduced the parameter η=
(1/4N)/summationtext
k|uk|2. In order to obtain γi, we have noticed
that the characteristic polynomial p(γ)=det(B−γI) can be
evaluated by using the Laplace expansion theorem [ 73], andit is equal to
p(γ)=N/summationdisplay
l=1/bracketleftbigg
(ωL−γ)(−γ)−|ul|2
4/bracketrightbigg
(−γ)N−1(E11)
=N[(ωL−γ)γ+η](−γ)N−1. (E12)
If we denote by λkthe eigenvalues for the H({nk})i n
Eq. (E6) and they are listed in decreasing order, the eigenvalues
{αk},{γk}, and{λk}satisfy the following inequalities [ 72],
λk/lessorequalslantαj+γk−j(j/lessorequalslantk), (E13)
λk/greaterorequalslantαj+γk−j+N(j/greaterorequalslantk), (E14)
in particular, if k=j, then
λk/lessorequalslantαk+γ0, (E15)
λk/greaterorequalslantαk+γN. (E16)
Moreover, if j=k−1f r o mE q .( E13) we obtain
λk/lessorequalslantαk−1+γ1, (E17)
and if j=k+1 from Eq. ( E14)w eh a v e
λk/greaterorequalslantαk+1+γN−1. (E18)
From the inequalities in Eqs. ( E17) and ( E18), together with
the eigenvalues in Eqs. ( E7) and ( E10), we obtain upper and
lower bounds for λkwith 1/lessorequalslantk/lessorequalslantN−1,
−ωL
2+ωN−k−1+εB/lessorequalslantλk/lessorequalslant−ωL
2+ωN−k+1+εB,
(E19)
085434-11OCHOA, ZIMBOVSKAYA, AND NITZAN PHYSICAL REVIEW B 97, 085434 (2018)
and since ωk=k/Delta1ω ,E q .( E19) is equivalent to |λk−αk|/lessorequalslant
/Delta1ω. Consequently, for small /Delta1ω, we can approximate
λk=αk, (E20)
for 1/lessorequalslantk/lessorequalslantN−1. It remains to determine appropriate ap-
proximations for λ0andλN. For the former, considering
Eq. ( E15),
λ1/lessorequalslantλ0/lessorequalslantα0+γ0, (E21)
−ωL
2+ωN−1+εB/lessorequalslantλ0/lessorequalslant−ωL
2+ωN+εB+γ0,(E22)
as the ordering in {λk}dictates that λ1/lessorequalslantλ0. Since γ0can take
large values in the strong-coupling regime, in this case ourestimate will be
λ
0=−ωL
2+ωN−1+εB+C(/Delta1ω+γ0) (E23)
=α1+C(/Delta1ω+γ0), (E24)
where 0 /lessorequalslantC/lessorequalslant1 is a constant determined below. For the latter,
in view of Eq. ( E16), we find
αN+γN/lessorequalslantλN/lessorequalslantλN−1, (E25)
−ωL
N+εB+γN/lessorequalslantλN/lessorequalslant−ωL
2+ω1+εB, (E26)
which suggests that
λN=αN−1+C/prime(γN−/Delta1ω), (E27)
with 0/lessorequalslantC/prime/lessorequalslant1. Finally, we determine the constants Cand
C/primeby computing the trace for H({nk})i nE q .( E6). Indeed,
Tr{H({nk})}=Tr{A}+Tr{B} (E28)
=(1−N)ωL
2+(N+1)εB+/Delta1ωN(N+1)
2.
(E29)
On the other hand,
N/summationdisplay
k=0λk=α1+αN−1+N−1/summationdisplay
k=1αk
+C/prime(γN−/Delta1ω)+C(/Delta1ω+γ0) (E30)
=−(1+N)ωL
2+(N+1)εB
+/Delta1ωN(N+1)
2(E31)
+C/prime(γN−/Delta1ω)+C(/Delta1ω+γ0),(E32)
and if C/prime=C=1,/summationtextN
k=0λk=Tr{H(nk)}.
Next, we calculate the partition function for ˆH{nk},/Xi1{nk}=
Tr{exp(−βˆH{nk})},a s
/Xi1{nk}=e−βλ0+e−βλN+N−1/summationdisplay
k=1e−βλk(E33)
=e−βλ0−e−βα0+e−βλN−e−βαN+N/summationdisplay
k=0e−βαk
(E34)
=e−β(εB−ωL/2)R(ωmax,N), (E35)where we have introduced the function
R(ωmax,N)=e−βωmax(e−βγ0−1)
+e−βγN−1+N/summationdisplay
k=0e−βk/Delta1ω.(E36)
In order to recover the canonical partition function, we now
sum over all families {nk}. Letting Sbe such a collection, we
write ( μ=0)
/Xi1=/summationdisplay
{nk}∈S/Xi1{nk} (E37)
=⎛
⎝/summationdisplay
{nk}∈Se−β(εB−ωL/2)⎞
⎠R(ωmax,N). (E38)
We notice that
/summationdisplay
{nk}∈Se−β(εB−ωL/2)=eβωL/2/summationdisplay
{nk}∈SN/productdisplay
k=1e−βnkωk(E39)
=eβωL/2N/productdisplay
k=1∞/summationdisplay
n=0e−βnω k(E40)
=eβωL/2N/productdisplay
k=11
1−e−βωk, (E41)
and therefore
ln/Xi1=N/summationdisplay
k=1ln/bracketleftbiggeβωL/2
1−e−βωkR(ωmax,N)/bracketrightbigg
, (E42)
which in the thermodynamic limit leads to the integral form
ln/Xi1=/integraldisplayωmaxdω
2πA(ω)l n/bracketleftbiggeβωL/2e−βγN
1−e−βω/bracketrightbigg
. (E43)
Consequently, the final form for the canonical potential is
F=1
β/integraldisplaydω
2πA(ω)l n [ ( 1 −e−βω)e−β/Delta1/ 2], (E44)
with/Delta1=√ωL+4η, and that can be further simplified to the
form in Eq. ( 54).
APPENDIX F: SPECTRAL DENSITY FOR THE DAMPED
TWO-LEVEL SYSTEM
Consider the Green’s function
G(τ2,τ1)=−i/angbracketleftTcˆσ−(τ2)ˆσ+(τ1)/angbracketright. (F1)
The equation of motion for ˆ σ−is
id
dτ2ˆσ−(τ2)=ωL−2/summationdisplay
kVkˆSzˆak, (F2)
where Vk=iuk/2. Then, the equation of motion for the
Green’s function in Eq. ( F1)i s
id
dτ2G(τ2,τ1)=−2δ(τ2,τ1)/angbracketleftˆSz(τ1)/angbracketright+ωLG(τ2τ1)
−2/summationdisplay
kVk[−i/angbracketleftˆSz(τ2)ˆak(τ2)ˆσ+(τ1)/angbracketright].(F3)
085434-12QUANTUM THERMODYNAMICS FOR DRIVEN DISSIPATIVE … PHYSICAL REVIEW B 97, 085434 (2018)
In order to solve the EOM in Eq. ( F3) we approximate the
higher-order correlation function by the product
−i/angbracketleftˆSz(τ2)ˆak(τ2)ˆσ+(τ1)/angbracketright=/angbracketleft ˆSz(τ2)/angbracketright[−i/angbracketleftˆak(τ2)ˆσ+(τ1)/angbracketright].
(F4)
Such decoupling schemes were used in other contexts in
Refs. [ 74,75]. Following the same rationale as in Appendix A,
we find
−i/angbracketleftˆak(τ2)ˆσ+(τ1)/angbracketright=V∗
k/integraldisplay
dτ/primeuk(τ2,τ/prime)G(τ/prime,τ1), (F5)
which upon substitution in Eq. ( F3) leads to the expression
id
dτ2G(τ2,τ1)=δ(τ2,τ1)S(τ1)+ωLG(τ2τ1)
+S(τ2)/summationdisplay
k|Vk|2/integraldisplay
dτ/primeuk(τ2,τ/prime)G(τ/prime,τ1).
(F6)
This equation may be converted to the standard form of the
Dyson equation, by introducing the transformation ˜G(τ2,τ1)=
S−1/2(τ2)G(τ2,τ1)S−1/2(τ1)a ss h o w ni nR e f .[ 76]. As a result,
we find that in a stationary state,
Gr(ω)=S
(ω−ωL)+i/Gamma1S/2. (F7)
From this result, we obtain Eq. ( 57).APPENDIX G: NONEQUILIBRIUM DISTRIBUTION
G I V E NB YE Q .( 62)
Starting from the gradient expansion employed in Eq. ( D1),
which is valid for ˜G<introduced in Appendix F, and noticing
that
∂˜Gr
∂t=˙ωL(˜Gr)2,∂˜Ga
∂t=˙ωL(˜Ga)2, (G1)
we get
˜G<(t,ω)=˜Gr(t,ω)˜/Sigma1<(ω)˜Ga(t,ω)
+i˙ωL
2˜Gr(t,ω)˜Ga(t,ω)
×{˜Ga(t,ω)−˜Gr(t,ω)}∂
∂ω˜/Sigma1<(ω),(G2)
with Agiven by Eq. ( 57). From this result, we recover
G<(t,ω)=S1/2˜G<(t,ω)S1/2, and after some algebraic manip-
ulations we arrive at the expression
iG<(t,ω)=A(t,ω)/bracketleftbigg
n(ω)+˙ωL
2S−1A(t,ω)∂n(ω)
∂ω/bracketrightbigg
,(G3)
which brings the result for φ3(t,ω) given by Eq. ( 62).
[1] M. T. Mitchison, M. P. Woods, J. Prior, and M. Huber, New J.
Phys. 17,115013 (2015 ).
[2] J. Anders and M. Esposito, New J. Phys. 19,010201 (2017 ).
[3] M. Horodecki and J. Oppenheim, Nat. Commun. 4,2059 (2013 ).
[4] D. Gelbwaser-Klimovsky, W. Niedenzu, and G. Kurizki,
Adv. At. Mol. Opt. Phys. 64,329(2015 ).
[ 5 ] N .H .Y .N g ,L .M a n činska, C. Cirstoiu, J. Eisert, and S. Wehner,
New J. Phys. 17,085004 (2015 ).
[6] S. Vinjanampathy and J. Anders, Contemp. Phys. 57,545(2016 ).
[7] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk,
J. Phys. A 49,143001 (2016 ).
[8] M. Esposito, M. A. Ochoa, and M. Galperin, Phys. Rev. Lett.
114,080602 (2015 ).
[9] J. Millen and A. Xuereb, New J. Phys. 18,011002 (2016 ).
[10] R. Uzdin, A. Levy, and R. Kosloff, Phys. Rev. X 5,031044
(2015 ).
[11] R. Uzdin, A. Levy, and R. Kosloff, Entropy 18,124(2016 ).
[12] D. Gelbwaser-Klimovsky, A. Bylinskii, D. Gangloff, R. Islam,
A. Aspuru-Guzik, and V . Vuletic, arXiv:1705.11180 .
[13] G. Zolfagharkhani, A. Gaidarzhy, S.-B. Shim, R. L. Badzey, and
P. Mohanty, P h y s .R e v .B 72,224101 (2005 ).
[14] J. Wang, J. He, and Y . Xin, Phys. Scr. 75,227(2007 ).
[15] A. I. V olokitin and B. N. J. Persson, Phys. Rev. Lett. 106,094502
(2011 ).
[16] F. Intravaia, R. O. Behunin, and D. A. R. Dalvit, P h y s .R e v .A
89,050101 (2014 ).
[17] F. Plastina, A. Alecce, T. J. G. Apollaro, G. Falcone, G. Francica,
F. Galve, N. LoGullo, and R. Zambrini, P h y s .R e v .L e t t . 113,
260601 (2014 ).
[18] A. Alecce, F. Galve, N. L. Gullo, L. Dell’Anna, F. Plastina, and
R. Zambrini, New J. Phys. 17,075007 (2015 ).[19] N. Shiraishi, K. Saito, and H. Tasaki, Phys. Rev. Lett. 117,
190601 (2016 ).
[20] J. Klatt, M. B. Farias, D. A. R. Dalvit, and S. Y . Buhmann, Phys.
Rev. A 95,052510 (2017 ).
[21] F. Curzon and B. Ahlborn, A m .J .P h y s . 43,22(1975 ).
[22] B. Rutten, M. Esposito, and B. Cleuren, Phys. Rev. B 80,235122
(2009 ).
[23] M. Esposito, K. Lindenberg, and C. Van den Broeck, Phys. Rev.
Lett. 102,130602 (2009 ).
[24] U. Seifert, P h y s .R e v .L e t t . 106,020601 (2011 ).
[25] A. Levy, R. Alicki, and R. Kosloff, P h y s .R e v .E 85,061126
(2012 ).
[26] Y . Izumida and K. Okuda, Europhys. Lett. 97,10004 (2012 ).
[27] R. Uzdin and R. Kosloff, New J. Phys. 16,095003 (2014 ).
[28] R. S. Whitney, Phys. Rev. Lett. 112,130601 (2014 ).
[29] M. Bauer, K. Brandner, and U. Seifert, P h y s .R e v .E 93,042112
(2016 ).
[30] O. Abah, J. Rossnagel, G. Jacob, S. Deffner, F. Schmidt-
Kaler, K. Singer, and E. Lutz, P h y s .R e v .L e t t . 109,203006
(2012 ).
[31] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz,
Phys. Rev. Lett. 112,030602 (2014 ).
[32] J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F.
Schmidt-Kaler, and K. Singer, Science 352,325(2016 ).
[33] M. Esposito, K. Lindenberg, and C. Van den Broeck, Europhys.
Lett. 85,60010 (2009 ).
[34] D. M. Kennes and V . Meden, P h y s .R e v .B 87,075130 (2013 ).
[35] M. F. Ludovico, J. S. Lim, M. Moskalets, L. Arrachea, and D.
Sánchez, P h y s .R e v .B 89,161306 (2014 ).
[36] K. Proesmans, B. Cleuren, and C. Van den Broeck, J. Stat. Mech.:
Theory Exp. (2016 )023202 .
085434-13OCHOA, ZIMBOVSKAYA, AND NITZAN PHYSICAL REVIEW B 97, 085434 (2018)
[37] M. F. Ludovico, F. Battista, F. von Oppen, and L. Arrachea, Phys.
Rev. B 93,075136 (2016 ).
[38] A. Bruch, M. Thomas, S. Viola Kusminskiy, F. von Oppen, and
A. Nitzan, P h y s .R e v .B 93,115318 (2016 ).
[39] M. A. Ochoa, A. Bruch, and A. Nitzan, P h y s .R e v .B 94,035420
(2016 ).
[40] K. Sekimoto, Stochastic Energetics , V ol. 799 (Springer, Berlin,
2010).
[41] U. Seifert, Rep. Prog. Phys. 75,126001 (2012 ).
[42] S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406,379
(2005 ).
[43] G. B. Cuetara, A. Engel, and M. Esposito, New J. Phys. 17,
055002 (2015 ).
[44] M. Campisi, P. Talkner, and P. Hänggi, J. Phys. A 42,392002
(2009 ).
[45] A.-M. Daré and P. Lombardo, P h y s .R e v .B 93,035303 (2016 ).
[46] M. Esposito, M. A. Ochoa, and M. Galperin, Phys. Rev. B 92,
235440 (2015 ).
[47] M. Carrega, P. Solinas, M. Sassetti, and U. Weiss, Phys. Rev.
Lett. 116,240403 (2016 ).
[48] C. Jarzynski, Phys. Rev. X 7,011008 (2017 ).
[49] M. Perarnau-Llobet, H. Wilming, A. Riera, R. Gallego, and J.
Eisert, arXiv:1704.05864 .
[50] P. Strasberg and M. Esposito, P h y s .R e v .E 95,062101 (2017 ).
[51] G. Katz and R. Kosloff, Entropy 18,186(2016 ).
[52] M. Campisi, D. Zueco, and P. Talkner, Chem. Phys. 375,187
(2010 ).
[53] G. L. Ingold, E u r .P h y s .J .B 85,1(2012 ).
[54] P. Hänggi, G.-L. Ingold, and P. Talkner, New J. Phys. 10,115008
(2008 ).
[55] G.-L. Ingold, P. Hänggi, and P. Talkner, P h y s .R e v .E 79,061105
(2009 ).[56] H. Breuer and F. Petruccione, The Theory of Open Quantum
Systems (Oxford University Press, Oxford, U.K., 2002).
[57] M. Thorwart, P. Reimann, and P. Hänggi, Phys. Rev. E 62,5808
(2000 ).
[58] K. Zhang, F. Bariani, and P. Meystre, Phys. Rev. Lett. 112,
150602 (2014 ).
[59] A. Dechant, N. Kiesel, and E. Lutz, P h y s .R e v .L e t t . 114,183602
(2015 ).
[60] B. Lin and J. Chen, Phys. Rev. E 67,046105 (2003 ).
[61] Y . Rezek and R. Kosloff, New J. Phys. 8,83(2006 ).
[62] S. Kohler, T. Dittrich, and P. Hänggi, P h y s .R e v .E 55,300(1997 ).
[63] J. M. Horowitz, P h y s .R e v .E 85,031110 (2012 ).
[64] R. M. de Araújo, T. Häffner, R. Bernardi, D. Tasca, M. Lavery, M.
Padgett, A. Kanaan, L. Céleri, and P. Ribeiro, arXiv:1705.02990 .
[65] A. J. Leggett, S. Chakravarty, A. Dorsey, M. P. Fisher, A. Garg,
and W. Zwerger, Rev. Mod. Phys. 59,1(1987 ).
[66] A. Friedenberger and E. Lutz, P h y s .R e v .A 95,022101 (2017 ).
[67] C. Zerbe and P. Hänggi, Phys. Rev. E 52,1533 (1995 ).
[68] R. Schmidt, M. F. Carusela, J. P. Pekola, S. Suomela, and J.
Ankerhold, Phys. Rev. B 91,224303 (2015 ).
[69] F. W. J. Hekking and J. P. Pekola, P h y s .R e v .L e t t . 111,093602
(2013 ).
[70] V . Semin and F. Petruccione, Phys. Rev. A 90,052112 (2014 ).
[71] K. L. Viisanen, S. Suomela, S. Gasparinetti, O.-P. Saira, J.
Ankerhold, and J. P. Pekola, New J. Phys. 17,055014 (2015 ).
[72] R. Bhatia, Matrix Analysis , V ol. 169 (Springer, Berlin, 2013).
[73] M. Marcus and H. Minc, Introduction to Linear Algebra (Courier
Corporation, North Chelmsford, MA, 1965).
[74] T. J. Levy and E. Rabani, J. Chem. Phys. 138,164125 (2013 ).
[75] B. R. Bułka and T. Kostyrko, P h y s .R e v .B 70,205333 (2004 ).
[76] M. A. Ochoa, M. Galperin, and M. A. Ratner, J. Phys.: Condens.
Matter 26,455301 (2014 ).
085434-14 |
PhysRevB.78.054404.pdf | Crystallographic and magnetic structure of SrCoO 2.5brownmillerite:
Neutron study coupled with band-structure calculations
A. Muñoz,1,*C. de la Calle,2J. A. Alonso,2P. M. Botta,3V . Pardo,3D. Baldomir,3and J. Rivas3
1Departamento de Física Aplicada, EPS, Universidad Carlos III, Avenida Universidad 30, E-28911, Leganés-Madrid, Spain
2Instituto de Ciencia de los Materiales de Madrid, CSIC, E-28049, Cantoblanco-Madrid, Spain
3Departamento de Física Aplicada, Facultad de Física, Universidad de Santiago de Compostela, E-15782 Campus Sur s/n,
Santiago de Compostela, Spain
/H20849Received 20 December 2007; revised manuscript received 2 June 2008; published 5 August 2008 /H20850
A study of the crystallographic and magnetic structure of SrCoO 2.5with a brownmillerite-type structure has
been carried out from neutron powder-diffraction /H20849NPD /H20850measurements at temperatures ranging from 10 to 623
K, across the Néel temperature /H20849TN=537 K /H20850of this antiferromagnetic oxide. The study has been comple-
mented with differential scanning calorimeter /H20849DSC /H20850,dcsusceptibility and magnetization measurements. Al-
though the refinement of the crystal structure from NPD data is possible in the orthorhombic Pnma andIma2
space groups, the support of ab-initio band-structure calculations has allowed us to select, without ambiguity,
theIma2 space group as the ground state for SrCoO 2.5brownmillerite. In Ima2 the crystallographic structure
of SrCoO 2.5is described as layers of corner-sharing Co1O 6octahedra alternating along the aaxis with layers
of vertex-sharing Co2O 4tetrahedra, conforming chains running along the /H20851001 /H20852direction. The magnetic
structure below TN=537 K is G-type with the magnetic moments directed along the caxis. This magnetic
arrangement is stable from TNd o w nt o1 0K .A t T=10 K, the magnetic moment values for Co1 and Co2 atoms
are 3.12 /H2084913/H20850/H9262Band 2.88 /H2084914/H20850/H9262B, respectively, compatible with a Co2+L/H6018state, where L/H6018stands for a ligand hole.
The magnetic susceptibility curves show, below 200 K, a divergence of zero-field cooling and field coolingcurves, displaying broad maxima which are interpreted as due to the presence of ferromagnetic clustersembedded into an antiferromagnetic matrix. These inhomogeneities are inherent to the synthesis process, byquenching microcrystalline samples of SrCoO
3−xcomposition from high temperature, where cubic, ferromag-
netic perovskites have been identified by diffraction methods.
DOI: 10.1103/PhysRevB.78.054404 PACS number /H20849s/H20850: 75.25. /H11001z
I. INTRODUCTION
Among the nonstoichiometric ABO 3−/H9254perovskite oxides,
SrCoO 3−/H9254presents a rich phase diagram, exhibiting different
crystal structures as a function of the oxygen deficiency andalso depending even on the preparative conditions.
1–3Some
of the materials of this system have been recently receivingmuch attention since for certain compositions SrCoO
3−xdis-
plays a high oxygen mobility at room temperature, whatmakes it a very suitable compound for different technicalapplications such as gas sensing probes or oxygen mem-branes, with application in solid oxide fuel cells.
4,5
The nearly stoichiometric SrCoO 3phase can be prepared
under high oxygen pressure;2Bezdicka et al.6and Le Toquin
et al.7also proposed the synthesis of the fully stoichiometric
SrCoO 3using electrochemical oxidation of SrCoO 2.5/H20849or
Sr2Co2O5/H20850brownmillerite phase. Although SrCoO 2.5is meta-
stable at room temperature, the brownmillerite-type structurecan be stabilized by quenching SrCoO
3-/H9254from 1000 °C, in
air, into liquid nitrogen.1,8The quenching process procures
the ordering of the oxygen vacancies. On heating SrCoO 2.5
from room temperature, different intermediate rhombohedralphases appear and finally, at around 900 °C, the structurebecomes cubic.
9,10
The SrCoO 2.5brownmillerite is a superstructure of perov-
skite due to the long-range ordering of the 0.5 oxygen va-cancies per formula unit. Its crystal structure can be obtainedfrom the perovskite structure ABO
3by removing one third of
the oxygen atoms in every second /H20849h00/H20850layer of octahedra inan ordered way along the /H20851100/H20852cubic direction. Therefore,
the structure of SrCoO 2.5can be described as a stacking of
layers of CoO 4tetrahedra alternating with layers of CoO 6
octahedra along the aaxis, with the Sr atoms located in the
voids between the polyhedra. The lattice parameters of the
orthorhombic unit cell of SrCoO 2.5are related to the lattice
parameters of the cubic perovskite unit cell by ao/H110154ap,bo
/H11015/H208812ap, and co/H11015/H208812ap. The assignment of space group to
SrCoO 2.5has been controversial; in fact, A 2B2O5compounds
with brownmillerite structure have been described to crystal-lize in different orthorhombic space groups, Pnma /H20849no. 62 /H20850,
Ima2/H20849no. 46 /H20850andImma /H20849no. 74 /H20850, depending on the metal
composition and temperature; in some cases, as in Ca
2Fe2O5,
there has been some ambiguity in the assignment of thespace group due to the weakness of the h+k+l/HS110052nBragg
reflections.
11Let us point out that the different orthorhombic
space groups only imply slight differences in the arrange-ment of the BO
4tetrahedral chains. In a recent study,7the
Pnma space group has been proposed for SrCoO 2.5at room
temperature; according to the same study, the intermediateSrCoO
2.75and SrCoO 2.82phases crystallize in the cubic
Pm-3mand tetragonal I4/mmm space groups, respectively.
Magnetic measurements indicate that SrCoO 2.5/H20849Ref. 12/H20850
undergoes a long-range antiferromagnetic /H20849AFM /H20850order be-
lowTN=570 K. In this paper ab-initio band-structure calcu-
lations allowed us to identify the Ima2 space group as the
ground state for the crystal and magnetic structure ofSrCoO
2.5brownmillerite; thus we have investigated by neu-
tron powder-diffraction /H20849NPD /H20850the thermal evolution of thePHYSICAL REVIEW B 78, 054404 /H208492008 /H20850
1098-0121/2008/78 /H208495/H20850/054404 /H208498/H20850 ©2008 The American Physical Society 054404-1crystal and magnetic structures of SrCoO 2.5in the 10–623 K
temperature range, in complement with magnetization mea-surements. We also give an interpretation of the divergenceof the zero-field cooling /H20849ZFC /H20850and field cooling /H20849FC/H20850curves
at/H11011200 K, which is not due to any change in the magnetic
structure of brownmillerite but is originated by the presenceof ferromagnetic /H20849FM/H20850clusters in an AFM matrix.
II. EXPERIMENTAL
A. Sample preparation
SrCoO 2.5was obtained as a polycrystalline sample by a
citrate technique. Stoichiometric amounts of analytical gradeSr/H20849NO
3/H208502and Co /H20849NO 3/H208502·6H 2O were dissolved in a citric
acid concentrated aqueous solution. The so-formed organicresin was dried at 140 °C and slowly decomposed at tem-peratures up to 600 °C for 12 h. The sample was then heatedat 900 °C for 12 h in air. The brownmillerite-type structurewas obtained by quenching the sample from this temperaturein liquid N
2.
B. X-ray and neutron diffraction, magnetic measurements and
thermal analysis
The reaction products were characterized by x-ray diffrac-
tion /H20849XRD /H20850for phase identification and to assess phase pu-
rity. The characterization was performed using a Bruker-AXS D8 diffractometer /H2084940 kV , 30 mA /H20850in Bragg-Brentano
reflection geometry with CuK
/H9251radiation /H20849/H9261=1.5418 Å /H20850.
The NPD experiments were carried out at the Institut
Laue-Langevin, Grenoble /H20849France /H20850. Different patterns were
collected at 295 K /H20849RT/H20850, 373 K, 473 K and 623 K at the D1A
diffractometer with a /H9261=1.91 Å wavelength; above RT the
sample was contained in a quartz tube open to the air atmo-sphere, placed in a vanadium furnace. Additionally, a low-temperature NPD pattern was obtained at T=10 K with /H9261
=1.594 Å at the D2B diffractometer. The different NPD dia-grams were refined by using the Rietveld method
13with the
FULLPROF program.14The profile of the Bragg reflections
was simulated with a pseudo-V oigt function; the backgroundwas fitted either with a fifth-degree polynomial function/H20849NPD patterns collected at RT and T=10 K /H20850or by linear
interpolation from a high number of points of the back-ground.
The magnetic measurements of SrCoO
2.5were performed
in a sample property measuring system /H20849SPMS /H20850from Quan-
tum Design. The dcmagnetization was measured both in
ZFC and FC conditions. Three susceptibility curves wereobtained in the temperature interval 5 /H11021T/H11021320 K under a
magnetic field of 0.1, 0.5, and 1 kOe. Above RT, an addi-tional susceptibility curve was obtained in the 300 /H11021T
/H11021700 range under 1 kOe. Different magnetization curves
were measured at T=25 K and T=220 K for magnetic
fields ranging from-10 kOe to 10 kOe.
Thermal analysis was performed with a differential scan-
ning calorimeter /H20849DSC /H20850from Perkin Elmer. A small pellet of
SrCoO
2.5/H2084928 mg /H20850was heated from RT up to 720 K using a
heating rate of 20 K.min−1a n daN 2flowing atmosphere.C. Computational details
Full potential, all electron, electronic structure calcula-
tions based on the density functional theory /H20849DFT /H20850utilizing
the APW+lo method15were performed using the WIEN2K
software.16We have included the strong correlation effects to
deal with the Co delectrons by means of the LDA+U
scheme.17The nonorbitally dependent part of the exchange-
correlation functional was modeled using the Perdew-Burke-Ernzerhof generalized gradient approximation /H20849GGA /H20850
scheme.
18Spin-orbit effects have been introduced in a sec-
ond variational way using the scalar relativisticapproximation.
19We have converged all our calculations
with respect to the k-mesh and to Rmt/H11569Kmax,u pt o8 2
k-points in the irreducible wedge of the Brillouin zone: a
3/H110039/H110039 mesh and up to Rmt/H11569Kmax=7. Local orbitals were
added for a bigger flexibility in dealing with the semicorestates. Muffin-tin radii chosen were the following: 2.28 a.u.for Sr, 1.79 a.u. for Co, and 1.59 a.u. for O.
III. RESULTS
A. X-ray diffraction, thermal analysis and magnetic
measurements
SrCoO 2.5was obtained as a pure, well-crystallized brown-
millerite phase; Figure 1shows the XRD pattern, indexed in
an orthorhombic unit cell with a=15.7450 /H208495/H20850,b=5.5739 /H208492/H20850,
c=5.4697 /H208492/H20850Å. There are no traces of the competitive hex-
agonal phase with the same composition, which can be ob-tained by slow cooling of the same precursor oxides from900 °C in air.
The thermal evolution of the ZFC and FC magnetization
curves of SrCoO
2.5measured between 0.1 and 1 kOe are
compared in Fig. 2, in the temperature range 5 to 320 K. In
complement, a DSC thermogram measured above RT is dis-played in Fig. 3. An endothermic peak centered at 537 K is
observed. The inset shows the dcmagnetic susceptibility,
revealing a clear inflection about the same temperature. ThisFIG. 1. XRD pattern for SrCoO 2.5brownmillerite, indexed in an
orthorhombic unit cell with a=5.4579 /H208493/H20850,b=15.6388 /H2084910/H20850,c
=5.5643 /H208493/H20850Å, space group Ima2.MUÑOZ et al. PHYSICAL REVIEW B 78, 054404 /H208492008 /H20850
054404-2is in good agreement with the reported Néel temperature for
SrCoO 2.5,TN=570 K.12On decreasing the temperature be-
low 200 K /H20849Fig. 2/H20850the susceptibility of both ZFC curves
increase, displaying a broad maximum, the cusp of which isshifted to lower temperatures for higher applied fields. TheFC curves diverge from the corresponding ZFC curves below170 K /H20849H=0.1 kOe /H20850and 140 K /H20849H=1 kOe /H20850. Besides, at low
temperature, the FC curves show a thermal evolution char-acteristic of a FM ordering. This behavior can be clearlyobserved in the isothermal magnetization curves shown inFig. 4. At 25 K hysteresis is observed, although the remnant
magnetization is very small, 0.002
/H9262Bper formula. At 220 K
an almost linear response characteristic of an AFM materialis observed. This behavior already suggests the presence ofweak FM interactions, which are overimposed to the long-range antiferromagnetic ordering established below 537 K.
B. Crystallographic structure from neutron
powder-diffraction data
As it has been indicated in Sec. I, there was a serious
ambiguity related to the assignment of the orthorhombicspace group for SrCoO 2.5with brownmillerite structure;
therefore, both the orthorhombic space group Ima2 and
Pnma have been initially taken into consideration. The main
difference between both groups is that the inversion symme-try element is present in Pnma whereas Ima2 is a noncen-
trosymmetric space group. This fact determines the orienta-tion of the CoO
4tetrahedral chains along the longest lattice
parameter, aforIma2 and bforPnma ; as it can be seen in
Fig. 5/H20849a/H20850, for the space group Ima2 the octahedral chains
placed at x=0.25 and x=0.75 display the same orientation,
shifted by /H208491/2,1/2,1/2 /H20850. However, for the space group Pnma ,
the octahedral chain placed at y=0.25 exhibits a different
orientation from that placed at y=0.75 due to the fact that
both chains are related by the inversion symmetry element.FIG. 2. /H20849Color online /H20850Thermal evolution of the FC and ZFC dc
susceptibility obtained under different magnetic fields.
FIG. 3. DSC curve for SrCoO 2.5recorded above room tempera-
ture. The inset shows the variation of the magnetic susceptibility inthe same temperature range.FIG. 4. Isothermal magnetization curves at T=25 and 220 K.
a
bO2
Co1
O1
O3 Co2Sr
c
bO3
Co2O2
Sr(a) (b)
FIG. 5. /H20849Color online /H20850Views of the brownmillerite-type struc-
ture of SrCoO 2.5defined in the orthorhombic Ima2 space group: a /H20850
Layers of Co1O 6octahedra and Co2O 4tetrahedra alternate along
the /H20851100/H20852direction. b /H20850Chains of Co2O 4tetrahedra are running
along /H20851001/H20852direction.CRYSTALLOGRAPHIC AND MAGNETIC STRUCTURE OF … PHYSICAL REVIEW B 78, 054404 /H208492008 /H20850
054404-3Also, Fig. 5/H20849b/H20850illustrates the chains of Co2O 4tetrahedra,
linked through O3 oxygens, running along the caxis in the
Ima2 space group.
In a first trial, we performed the refinement of the crystal
structure from NPD data at T=10 K, at both space groups
Pnma andIma2. The Rietveld plots are displayed in Fig. 6,
showing in both cases a satisfactory agreement between theexperimental and calculated diagrams. The introduction ofthe magnetic structure was required to complete the fitting ofthe NPD patterns, as described below. Table Icontains the
main structural parameters in both space groups at 10 K. Thesubtle difference between the refinements in both spacegroups concerns the hklBragg reflections that verify the re-
lationship h+k+l=2n+1, which are forbidden for the space
group Ima2; however, these reflections are too weak to be
clearly observed and it is not possible to decide between bothspace groups from the quality of the crystal structure refine-ments.
AtT=10 K SrCoO
2.5is magnetically ordered; for the
space group Pnma the magnetic reflections are explained by
using the propagation vector k=0; in case that the space
group Ima2 is considered to define the crystallographic
structure, the magnetic peaks can be indexed by using thepropagation vector k=/H208491,1,1 /H20850. In both cases the magnetic
contribution to the calculated NPD patterns has been takeninto account. Again, the fit of the magnetic structures in both
space groups is equally possible, leading to similar Bragg R
factors, and describing a globally identical magnetic cou-pling of the Co spins. To solve this dilemma we have beenstrongly supported by the ab-initio calculations described be-
low, which clearly and unambiguously show a by far higherstability of the Ima2 crystal structure. Therefore the ortho-
rhombic Ima2 space group has been selected for the refine-
ment of the crystal and magnetic structures from the NPDpatterns acquired at 10, 295, 373, 473, and 623 K. TheTABLE I. Atomic positions, lattice parameters and discrepancy
factors for the refinement of the crystallographic structure ofSrCoO
2.5. In the orthorhombic space group Pnma /H2084910 K /H20850Sr, O1 and
O2 are at 8d /H20849x,y,z /H20850, Co1 at 4a /H208490,0,0 /H20850and CO2 and O3 at 4c
/H20849x,1/4,z /H20850. In the space group Ima2/H2084910 K and 295 K /H20850, Sr, O1 and O2
are at 8c /H20849x,y,z /H20850, Co1 at 4a /H208490,0,0 /H20850and Co2 and O3 at 4b /H208491/4,y,z /H20850.
AtT=295 K the magnetic RBragg factor is 9.3 %.
Lattice
parameters10 K
Pnma10 K
Ima2295 K
Ima2
a/H20849Å/H20850 5.4579 /H208493/H20850 15.6376 /H208497/H20850 15.7450 /H208495/H20850
b/H20849Å/H20850 15.6388 /H2084910/H20850 5.5644 /H208493/H20850 5.5739 /H208492/H20850
c/H20849Å/H20850 5.5643 /H208493/H20850 5.4580 /H208492/H20850 5.4697 /H208492/H20850
Vol/H20849Å3/H20850 474.94 /H208498/H20850 474.92 /H208494/H20850 480.03 /H208493/H20850
RP/H20849%/H20850 5.3 4.9 5.5
Rwp/H20849%/H20850 6.7 6.3 6.9
RBragg /H20849%/H20850 8.9 8.6 5.5
/H927321.5 1.3 1.7
Atom positionSr x 0.4981 /H2084913/H20850 0.1104 /H208491/H20850 0.11125 /H2084911/H20850
y 0.1102 /H208492/H20850 0.0111 /H208496/H20850 0.0085 /H208495/H20850
z 0.0107 /H208496/H20850 0.517 /H208493/H20850 0.503 /H208495/H20850
B/H20849Å/H20850 0.02 /H208492/H20850 0.02 /H208492/H20850 0.46 /H208495/H20850
Co1 x 0.000 0.000 0.000
y 0.000 0.000 0.000z 0.000 0.000 0.000B/H20849Å/H20850 0.02 /H208492/H20850 0.02 /H208492/H20850 0.41 /H2084914/H20850
Co2 x 0.020 /H208493/H20850 0.25 0.25
y 0.25 0.942 /H208492/H20850 0.938 /H208492/H20850
z 0.943 /H208492/H20850 0.045 /H208494/H20850 0.033 /H208496/H20850
B/H20849Å/H20850 0.02 /H20849
2/H20850 0.02 /H208492/H20850 0.41 /H2084914/H20850
O1 x 0.2575 /H2084912/H20850 0.9951 /H208493/H20850 0.9946 /H208493/H20850
y 0.9940 /H208493/H20850 0.2439 /H2084912/H20850 0.2534 /H2084912/H20850
z 0.2526 /H2084915/H20850 0.253 /H208493/H20850 0.250 /H208494/H20850
B/H20849Å/H20850 0.02 /H208492/H20850 0.02 /H208492/H20850 0.66 /H208496/H20850
O2 x -0.0013 /H2084915/H20850 0.1407 /H208492/H20850 0.1421 /H208492/H20850
y 0.1409 /H208492/H20850 0.0408 /H208495/H20850 0.0391 /H208496/H20850
z 0.0422 /H208495/H20850 0.018 /H208493/H20850 0.009 /H208495/H20850
B/H20849Å/H20850 0.02 /H208492/H20850 0.02 /H208492/H20850 0.79 /H208497/H20850
O3 x 0.6239 /H2084910/H20850 0.25 0.25
y 0.25 0.8705 /H2084910/H20850 0.8684 /H2084910/H20850
z 0.8694 /H2084911/H20850 0.641 /H208494/H20850 0.631 /H208495/H20850
B/H20849Å/H20850 0.02 /H208492/H20850 0.0/H208492/H20850 1.05 /H2084912/H20850FIG. 6. Comparison of the observed /H20849crosses /H20850, calculated /H20849solid
line/H20850and difference /H20849at the bottom /H20850NPD patterns at T=10 K; the
first and second rows of tick marks correspond, respectively, to thenuclear and magnetic reflections. /H20849a/H20850Pnma space group. /H20849b/H20850Ima2
space group.MUÑOZ et al. PHYSICAL REVIEW B 78, 054404 /H208492008 /H20850
054404-4atomic parameters and the most important bonding distances
and the bonding angles concerning the Co1O 6octahedra and
Co2O 4tetrahedra at 10 K and 295 K are listed in Tables I
andII. The evolution of the crystal structure above 295 K is
described elsewhere.10
C. Magnetic structure resolution details
Upon decreasing the temperature below TN=537 K new
reflections of magnetic origin appear on the NPD patterns, asshown in Fig. 7. As mentioned before, for the space group
Ima2 the new reflections of magnetic origin are indexed with
the propagation vector k=/H208491,1,1 /H20850. The possible magnetic
structures compatible with the crystal symmetry have beendetermined by following the method described by Bertaut.
20
The possible magnetic modes for k=/H208491,1,1 /H20850and for the 4 aand 4 bsites occupied by the Co atoms are given in Table III.
After checking the different solutions in the ordered region,
the best agreement with the experimental results is obtainedfor the magnetic modes corresponding to the /H9003
2irreducible
representation; although for this irreducible representationthe basis vectors for the atoms of the 4 bsite imply a possible
magnetic component along the ydirection; after the refine-
ment from the different NPD diagrams this m
ymagnetic
component turns to be negligible and, therefore, the mag-netic moments of the Co atoms in both sites are orientedalong the zdirection. The coupling of the magnetic moment
ism
1z=m2zfor the Co atoms of the 4 asite, and m3z=m4zfor
the Co atoms of the 4 bsite. The Co atoms which are ob-
tained by a /H208491/2,1/2,1/2 /H20850translation from the Co1 and Co2,
and Co3 and Co4 are antiferromagnetically coupled with re-spect to the initial Co atom. A plot of the magnetic arrange-ment of the Co spins is shown in Fig. 8; it can be described
as a G-type magnetic structure with AFM layers of CoO
6
octahedra alternating along the adirection with AFM layers
of CoO 4tetrahedra; the coupling of the octahedral and tetra-
hedral layers is also AFM. The thermal evolution of the or-dered magnetic moments at the octahedral and tetrahedralpositions is displayed in Fig. 9.A t T=10 K, the magnetic
moment values for Co1 and Co2 atoms are 3.12 /H2084913/H20850
/H9262Band
2.88 /H2084914/H20850/H9262B, respectively, for the 4 aand 4 bsites.
D. Electronic structure calculations
We have performed electronic structure calculations in the
two possible structures at low temperature /H20849T=10 K /H20850, withTABLE II. Bonding distances /H20849in Å /H20850and angles /H20849in degrees /H20850
corresponding to the CoO 6octahedron and CoO 4tetrahedron for
SrCoO 2.5, at 10 K and 295 K /H20849space group Ima2/H20850.
Bonding distances 10 K 295 K
Co1-O1 /H20849x2/H20850 1.938 /H2084914/H20850 1.970 /H2084910/H20850
Co1-O1 /H20849x2/H20850 1.963 /H2084913/H20850 1.939 /H2084910/H20850
Co1-O2 /H20849x2/H20850 2.214 /H208493/H20850 2.249 /H208493/H20850
/H20855Co1-O /H20856 2.038 /H2084910/H20850 2.053 /H208497/H20850
Co2-O2 /H20849x2/H20850 1.801 /H208495/H20850 1.794 /H208495/H20850
Co2-O3 2.25 /H208493/H20850 2.23 /H208494/H20850
Co2-O3 1.815 /H2084914/H20850 1.788 /H2084910/H20850
/H20855Co2-O /H20856 1.917 /H2084914/H20850 1.901 /H2084915/H20850
Bonding angles
Co1-O1-Co1 175.1 /H208496/H20850 175.0 /H208498/H20850
Co1-O2-Co2 156.2 /H208492/H20850 155.9 /H208492/H20850
Co2-O3-Co2 116.8 /H208499/H20850 117.4 /H208499/H20850TABLE III. Magnetic structures obtained for the space group
Ima2 and k=/H208491,1,1 /H20850. The atoms of the 4 asite are denoted as: Co1
/H208490,0,0 /H20850, Co2 /H208491/2,0,0 /H20850. For the 4 bsite the notation is: Co3 /H208491/4,y,z /H20850,
Co4 /H208493/4,−y,z/H20850.
4a 4b
Representation Co1 Co2 Co3 Co4
/H90031 /H208490,0,1 /H20850/H20849 0,0,−1 /H20850/H20849 1,0,0 /H20850/H20849 −1,0,0 /H20850
/H90032 /H208490,0,1 /H20850/H20849 0,0,1 /H20850/H20849 0,1,1 /H20850/H20849 0,−1,1 /H20850
/H90033 /H208491,1,0 /H20850/H20849 −1,1,0 /H20850/H20849 0,1,1 /H20850/H20849 0,1,−1 /H20850
/H90034 /H208491,1,0 /H20850/H20849 1,−1,0 /H20850/H20849 1,0,0 /H20850/H20849 1,0,0 /H20850
FIG. 7. /H20849Color online /H20850Thermal evolution of the NPD patterns
collected at T=373, 473, and 623 K with /H9261=1.91 Å. Only the
angular range around the /H208491,0,−1 /H20850+and/H208491,−1,0 /H20850+magnetic reflec-
tions is represented.cab1
2
4Co1 4a
Co2 4b3
FIG. 8. /H20849Color online /H20850A view of the magnetic structure of
SrCoO 2.5below TN=537 K. For sake of clarity, only the Co atoms
are represented.CRYSTALLOGRAPHIC AND MAGNETIC STRUCTURE OF … PHYSICAL REVIEW B 78, 054404 /H208492008 /H20850
054404-5space groups Pnma andIma2, based upon the atomic param-
eters listed in Tables IandII. We have always considered the
experimentally found AFM G-type structure, with the mag-
netization pointing along the caxis /H20849in the Ima2 setting /H20850. The
total energy in both possible structures for various values ofthe effective on-site Coulomb repulsion Uwas computed.
The absolute value of the total energies does not have anintrinsic physical sense, because it depends on many param-eters of the calculation /H20849number of local orbitals added, lin-
earization energies, k-points mesh, energy cutoff for the core
states, number of plane waves utilized, etc. /H20850, which all have
been previously converged to yield reliable results. The sameset of parameters was used for each space group with the aimto make the calculation independent from these variables.For this reason we show the difference between the totalenergies instead of the absolute energy values. For U=0, the
Ima2-based structure is more stable than the one based on
thePnma space group by 34 meV/Co. When strong correla-
tions are taken into account, the Ima2 structure gets stabi-
lized even further /H20849by 400 meV/Co when U=3 eV and by
550 meV/Co for U=5 eV /H20850. These results leave no doubt
about the much higher stability of the Ima2-based structure
with respect to the other possible space group.
We have also carried out the relaxation of the structure of
SrCoO
2.5within both space groups. For that sake, we have
started from our experimentally obtained lattice parametersand atomic positions and relaxed both the volume and theinternal atomic positions. The main conclusion remains un-changed, i.e., the Ima2 space group is more stable than the
Pnma . Our results show that the structure within the Pnma
space group relaxes to a volume 3% higher than experiment,whereas within the Ima2 space group, it relaxes to the ex-
perimental volume /H20849within less than 1% accuracy /H20850. The total
energy difference between both relaxed structures is approxi-mately 64 meV/Co within GGA and, when strong correlationeffects are considered by means of the LDA+U scheme, theenergy difference reduces to about 36 meV/Co for U
=5 eV, which is qualitatively equivalent to the results ob-tained with the experimental structures.IV. DISCUSSION
Ab-initio electronic structure calculations demonstrate
that the crystallographic structure of SrCoO 2.5is orthorhom-
bic,Ima2 space group. In principle, the orthorhombic Pnma
space group also seems to be compatible with the NPD data,but the total energy of this structure is considerably higher.The crystallographic structure of SrCoO
2.5can be, thus, de-
scribed in Ima2 as constituted by layers of Co2O 4tetrahedra
that alternate with layers of Co1O 6octahedra along the a
axis /H20851Fig. 5/H20849a/H20850/H20852.I nt h e Ima2 space group, the orientation of
Co2O 4tetrahedra placed at x=0.25 and x=0.75 is the same
/H20849with a shift of /H208491/2,1/2,1/2 /H20850due to the body centering of the
unit cell /H20850since they are not related by the inversion symme-
try element. The structure contains channels in the tetrahe-dral planes at x=0.25 and x=0.75 /H20851Figs. 5/H20849a/H20850and5/H20849b/H20850/H20852which
account for the appreciable oxygen chemical diffusioncoefficient
21,22and high oxygen permeability23of this mate-
rial.
NPD data showed that the magnetic structure is given by
the/H90032irreducible representation for both 4 aand 4 bsites;
after the refinement from NPD data the magnetic moments ofCo at both positions are arranged in a G-type magnetic struc-
ture. Unlike the former descriptions of the magnetic structureof SrCoO
2.5/H20849Ref. 12/H20850performed in the space group Icmm ,
where the spin direction could not be determined, we clearlyand unambiguously refined the spin direction along the caxis
from NPD data at different temperatures below T
N.I nt h e
magnetic structure the magnetic moments are antiferromag-netically coupled within each layer of octahedral units; in thesame way every Co atom within the tetrahedral layer is an-
tiferromagnetically coupled with its four nearest Co atoms.Across the adirection, the consecutive layers of CoO
6octa-
hedra placed at x=0 and x=1 /2 are ferromagnetically
coupled; the coupling between the layers of tetrahedral unitsplaced at x=1 /4 and x=3 /4 is also FM. In principle, the
arrangement of the magnetic moments can be understoodfrom the Goodenough-Kanamori rules.
24,25Within the layer
of octahedral units the superexchange path for the Co atomsis Co1-O1-Co1, with bonding angles very close to 180°; forinstance, at T=10 K this bonding angle is 175.1 /H208496/H20850°/H20849see
Table II/H20850. Therefore, the expected superexchange interaction
between half-occupied, e.g., orbitals is AFM. On the otherhand, the superexchange path connecting a Co1 atom of theoctahedral layer with the neighboring Co2 atom in the adja-cent tetrahedral layer is Co1-O2-Co2, of 156.2 /H208492/H20850°a t T
=10 K, also promoting the AFM interactions. With regard tothe coupling of the magnetic moments between the Co2 at-oms in the tetrahedral layer, the superexchange path is Co2-O3-Co2 and the bonding angle is far away from 180°; forinstance the bonding angle is 116.8 /H208499/H20850°a t T=10 K /H20851see
Table IIand Fig. 5/H20849b/H20850/H20852. If the Goodenough-Kanamori rules
are considered for the Co2-O3-Co2 superexchange path, thecoupling should very probably be FM; however, the AFMcoupling found between the nearest Co atoms of the tetrahe-dral units is indirectly determined from the coupling betweenthe adjacent layers of octahedra and tetrahedra: The interac-tion between neighboring Co1-Co1 atoms is strongly AFMas well as between Co1-Co2, which determines an AFM cou-pling between neighboring Co2-Co2 atoms within the tetra-hedral layer.FIG. 9. Thermal evolutions of the magnetic moments of the Co
atoms placed at the 4 aand 4 bsites.MUÑOZ et al. PHYSICAL REVIEW B 78, 054404 /H208492008 /H20850
054404-6AtT=10 K. the ordered magnetic moments for Co1 and
Co2 atoms are 3.12 /H2084913/H20850/H9262Band 2.88 /H2084914/H20850/H9262B, respectively,
which in both cases is considerably reduced from that ex-
pected for Co+3/H20849d6/H20850in high spin t2g4eg2configuration /H20849S=2/H20850.
The electronic structure calculations also explain the originof these experimentally observed magnetic moments for theCo atoms. The strong Co /H208493d/H20850—/H208492p/H20850hybridization in this
charge-transfer insulator leads to the formation of a ligandhole in the surrounding O ions. Hence, the Co atoms are in aCo
2+L/H6018state, where L/H6018stands for a ligand hole.26The magnetic
moment of the Co2+cations is only /H110113/H9262B/Co/H20849ideally cor-
responding to the d7configuration /H20850and a very large moment
is induced in the O ions, of approximately 0.4 /H9262B/Co. This
can be more clearly seen in Fig. 10, showing the density of
states /H20849DOS /H20850of both Co atoms and one O atom in the struc-
ture. It can be observed that the octahedral Co /H20849lowest panel /H20850
ions are in a t2g5eg2state, with the unoccupied t2gstate being at
1 eV above the Fermi level, separated by about 1 eV fromthe higher-lying unoccupied e
gstates. A similar Co2+L/H6018struc-
ture can be observed for the tetrahedrally-coordinated Coatom /H20849middle panel /H20850, by observing the t
2gdown-spin band
fully unoccupied /H20849eg4t2g3electronic configuration /H20850. The large
polarization of the O anions can be noticed in the upperpanel by observing the depletion of spin-down states close tothe Fermi level in the DOS curve of the O2 atom /H20849a similar
picture would be found for the other O atoms /H20850. The charge-
transfer character of the band gap can be noticed by observ-ing the large O pcharacter of the states close to the Fermi
level and the large Co dcharacter of the conduction band.
Another important point concerning the magnetic proper-
ties is the broad maximum observed in the ZFC magneticsusceptibility below 200 K /H20849Fig. 2/H20850, which is related to the
weak ferromagnetism observed in the isothermal magnetiza-tion curves /H20849Fig. 4/H20850. A similar behavior was previously de-
scribed for SrCoO
2.5by Le Toquin,27who proposed that the
weak FM component he observed /H208490.03/H9262B/H20850could be induced
by a FM interaction in the tetrahedral layers where the su-perexchange angle is close to 120°. However, the results ofthe present study of the magnetic structure of SrCoO
2.5, de-
fined in the Ima2 space group, do not allow the existence of
a weak ferromagnetism phenomenon for the propagationvector k=/H208491,1,1 /H20850.
28This can easily be understood as for
Ima2 the /H208491/2,1/2,1/2 /H20850lattice translation gives rise to an in-
version of the magnetic moment direction, in such a way thatfor each magnetic moment located at a particular /H20849x,y,z/H20850
position there is another antiparallel moment at /H20849x+1 /2,y
+1 /2,z+1 /2/H20850, which cancels any neat magnetization. In-
stead of a weak ferromagnetism effect, we propose that theanomalies observed in the magnetization curves are origi-nated by the presence of FM clusters of SrCoO
3−xembedded
in an AFM matrix of SrCoO 2.5. It is well known, since the
former investigations on the SrCoO 3−xsystem, that above
900 °C a cubic phase stabilizes, with the simple-cubic aris-totype perovskite structure, which can be identified by high-temperature in-situ XRD or NPD techniques.
9,10This cubic
phase has also claimed to be quenched from 1200 °C inliquid N
2/H20849Ref. 2/H20850. On the other hand, this cubic SrCoO 3−x
perovskite can be alternatively obtained under high pressure
conditions.29In all cases, the cubic SrCoO 3−xphase has been
reported to exhibit ferromagnetism below /H11011200 K, the
strength of the FM interactions depending on the oxygennonstoichiometry and, thus, on the Co
4+content, linearly
varying from TC=180 K for a sample with 50% Co4+to
TC=215 K for a perovskite with 90% Co4+.30We propose
that in the preparation process of brownmillerite, which isperformed by an out-of-equilibrium process by quenching aSrCoO
3-xsample in liquid N 2from 900 °C, clusters of the
high-temperature cubic oxygen-deficient phase are trappedand inhomogeneously distributed within the microcrystals.
These clusters are not detected or even detectable by dif-
fraction methods /H20849XRD or NPD /H20850, but their imprint is present
in the magnetic measurements. On the one hand, these FMregions are sufficiently isolated to be in the origin of theirreversibilities observed between the FC and ZFC curves.On the other hand, the observed reduction of the cusp tem-perature of the broad peak of the ZFC curves with the ap-plied measuring field, from T=70 K at H=0.1 kOe to T
=140 K at 1 kOe resembles the typical response describedfor superparamagnetic small-particles systems, whose block-ing temperature decreases for increasing applied magneticfields.
31In our system, this experimental observation consti-
tutes an additional support to the scenario of FM clusters inan AFM matrix, similar to that previously reported inBaCoO
3.32This is also coherent with the magnetization
curves /H20849Fig. 4/H20850, at temperatures below and above these sus-
ceptibility maxima. We suggest, therefore, that the weak FMsignal detected in the magnetization curves is not an intrinsicproperty of SrCoO
2.5brownmillerite, but a result of the pres-
ence of FM inhomogeneities. Of course, no anomaly is ob-served in the thermal evolution of the magnetic structure ofSrCoO
2.5across the FM TCof the clusters, since it is an
extrinsic phenomenon which does not affect the major AFMmatrix.
V. CONCLUSIONS
A study of the crystallographic and magnetic structure of
SrCoO 2.5from neutron powder diffraction in combination-8 -6 -4 -2 0 2 4 6-505
Energy (eV)Cooctd-505DOS (states/(eV spin))Cotetd-101
O2 p
FIG. 10. Density of states for SrCoO 2.5in the Ima2-based struc-
ture. The Co atoms are in a HS Co2+L/H6018state and the O atoms /H20849O2p
levels shown as a representative example /H20850are largely spin-
polarized. The upper /H20849lower /H20850panel represents up /H20849down /H20850spin states.CRYSTALLOGRAPHIC AND MAGNETIC STRUCTURE OF … PHYSICAL REVIEW B 78, 054404 /H208492008 /H20850
054404-7with ab-initio electronic structure calculations allowed us to
select the Ima2 space group, given the much higher stability
of the Ima2-based structure with respect to the other possible
space group /H20849Pnma /H20850. The magnetic structure is established
below TN=537 K, with a periodicity given by the propaga-
tion vector k=/H208491,1,1 /H20850. It is defined by an AFM G-type struc-
ture with the magnetic moments oriented along the cdirec-
tion. The magnetic structure can be described as AFM layersof CoO
6octahedra alternating along the adirection with
AFM layers of CoO 4tetrahedra; the coupling between the
octahedral and tetrahedral layers is AFM. The refined mag-netic moments at 10 K, close to 3
/H9262Bat both octahedral and
tetrahedral sites, are interpreted as arising from a Co2+L/H6018
state, where L/H6018stands for a ligand hole. We suggest that the
broad maximum and the divergence of FC and ZFC suscep-tibility curves observed below 200 K arise from the presence
of FM clusters of cubic SrCoO 3−xperovskite phases /H20849with
TC/H11011200 K /H20850embedded into the AFM matrix of SrCoO 2.5
brownmillerite.
ACKNOWLEDGMENTS
We thank the financial support of CICyT to Projects No.
MAT2007–60536 and MAT2006–10027, and we are gratefulto ILL for making all facilities available and the CESGA/H20849Centro de Supercomputación de Galicia /H20850for the computing
facilities. P.M.B. acknowledges Ministerio de Educación yCiencia /H20849Spain /H20850for the financial support /H20849J. de la Cierva
program /H20850.
*Corresponding author; angel.munoz@uc3m.es
1J. G. Grenier, S. Ghodbane, G. Demazeau, M. Pouchard, and P.
Hagenmuller, Mater. Res. Bull. 14, 831 /H208491979 /H20850.
2Y . Takeda, R. Kanno, T. Takada, O. Yamamoto, M. Takano, and
Y . Bando, Z. Anorg. Allg. Chem. 540, 259 /H208491986 /H20850.
3M. T. Anderson, J. T. Vaughey, and K. R. Poeppelmeier, Chem.
Mater. 5, 151 /H208491993 /H20850.
4V . V . Vashook, M. V . Zinkevich, and Y . G. Zonov, Solid State
Ionics 116, 129 /H208491999 /H20850.
5Z. P. Shao and S. M. Haile, Nature /H20849London /H20850431, 170 /H208492004 /H20850.
6P. Bezdicka, J. C. Wattiaux, J. C. Grenier, M. Pouchard, and P.
Hagenmuller, Z. Anorg. Allg. Chem. 619,7/H208491993 /H20850.
7R. Le Toquin, W. Paulus, A. Cousson, C. Prestipino, and C.
Lamberti, J. Am. Chem. Soc. 128, 13161 /H208492006 /H20850.
8T. Takeda and H. Watanabe, J. Phys. Soc. Jpn. 33, 973 /H208491972 /H20850.
9J. Rodríguez, J. M. González-Calbet, J. C. Grenier, J. Pannetier,
and M. Anne, Solid State Commun. 62, 231 /H208491987 /H20850.
10C. de la Calle, A. Aguadero, J. A. Alonso, M. T. Fernández-Díaz,
Solid State Sci. /H20849to be published /H20850.
11P. Berastegui, S. G. Eriksson, and S. Hull, Mater. Res. Bull. 34,
303 /H208491999 /H20850.
12T. Takeda, Y . Yamaguchi, and H. Watanabe, J. Phys. Soc. Jpn.
33, 970 /H208491972 /H20850.
13H. M. Rietveld, J. Appl. Crystallogr. 2,6 5 /H208491969 /H20850.
14J. Rodríguez-Carvajal, J. Phys. B 192,5 5 /H208491993 /H20850.
15E. Sjöstedt, L. Nördstom, and D. Singh, Solid State Commun.
114,1 5 /H208492000 /H20850.
16K. Schwarz and P. Blaha, Comput. Mater. Sci. 28, 259 /H208492003 /H20850.
17A. I. Liechtenstein, V . I. Anisimov, and J. Zaanen, Phys. Rev. B52, R5467 /H208491995 /H20850.
18J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 /H208491996 /H20850.
19D. Singh, Planewaves, Pseudopotentials and the LAPW Method
/H20849Kluwer, Dordrecht, 1994 /H20850.
20E. F. Bertaut, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr.,
Theor. Gen. Crystallogr. 24, 217 /H208491968 /H20850.
21M. V . Zinkevich and V . V . Vashook, Elektrokhimiya 28, 1800
/H208491992 /H20850.
22M. V . Zinkevich and V . V . Vashook, Vestsi Akad. Navuk Be-
larusi, Ser. Fiz.-Mat. Navuk 3,4 8 /H208491993 /H20850.
23H. Kruidhof, H. J. M. Bovwmeester, F. H. E. Doorn, and
A. J. Burggraaf, Solid State Ionics 63–65 , 816 /H208491993 /H20850.
24J. B. Goodenough, Phys. Rev. 100, 564 /H208491955 /H20850.
25J. Kanamori, J. Phys. Chem. Solids 10,8 7 /H208491959 /H20850.
26M. A. Korotin, S. Yu. Ezhov, I. V . Solovyev, V . I. Anisimov, D.
I. Khomskii, and G. A. Sawatzky, Phys. Rev. B 54, 5309 /H208491996 /H20850.
27R. Le Toquin, Ph.D. thesis, University of Rennes /H20849France /H20850, 2003.
28E. A. Turov, Physical Properties of Magnetically Ordered Crys-
tals /H20849Academic, New York, 1965 /H20850.
29H. Taguchi, M. Shimada, and M. Koizumu, J. Solid State Chem.
40,4 2 /H208491981 /H20850.
30H. Taguchi, M. Shimada, and M. Koizumu, J. Solid State Chem.
29, 221 /H208491979 /H20850.
31P. M. Botta, V . Pardo, D. Baldomir, C. de la Calle, J. A. Alonso,
and J. Rivas, Phys. Rev. B 74, 214415 /H208492006 /H20850.
32J. L. Dormann, D. Fiorani, and E. Tronc, Adv. Chem. Phys. 98,
283 /H208491997 /H20850.MUÑOZ et al. PHYSICAL REVIEW B 78, 054404 /H208492008 /H20850
054404-8 |
PhysRevB.82.060510.pdf | Antiferromagnetic ordering induced by paramagnetic depairing
in unconventional superconductors
Ryusuke Ikeda, Yuhki Hatakeyama, and Kazushi Aoyama
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
/H20849Received 16 June 2010; published 12 August 2010 /H20850
Antiferromagnetic /H20849AFM /H20850/H20849or spin-density wave /H20850quantum critical fluctuation enhanced just below Hc2/H208490/H20850
have been often observed in d-wave superconductors with a strong Pauli paramagnetic depairing /H20849PD/H20850includ-
ing CeCoIn 5. It is shown here that such a tendency of field-induced AFM ordering is a consequence of strong
PD and should appear particularly in superconductors with a gap node along the AFM modulation. Twophenomena seen in CeCoIn
5, the AFM order in the Fulde-Ferrell-Larkin-Ovchinnikov /H20849FFLO /H20850state and the
anomalous vortex lattice form factor in the high-field range below the FFLO state, are explained based on thispeculiar PD effect.
DOI: 10.1103/PhysRevB.82.060510 PACS number /H20849s/H20850: 74.70.Tx, 74.20.Mn, 74.25.Uv, 74.25.Dw
Recently, the heavy-fermion d-wave paired supercon-
ductor CeCoIn 5with strong paramagnetic depairing /H20849PD/H20850has
been thoroughly studied from the viewpoint of identifying itshigh-field and low-temperature /H20849HFLT /H20850phase near the zero-
temperature depairing field H
c2/H208490/H20850with a possible Fulde-
Ferrell-Larkin-Ovchinnikov /H20849FFLO /H20850state.1,2On the other
hand, this material also shows an antiferromagnetic /H20849AFM /H20850
or equivalently, a spin-density wave ordering3in the HFLT
phase and transport phenomena suggestive of an AFM quan-tum critical point /H20849QCP /H20850lying near H
c2/H208490/H20850.4,5A similar AFM
fluctuation enhanced near Hc2/H208490/H20850has also been detected in
other heavy-fermion superconductors6,7and cuprates,8all of
which seem to have strong PD and a d-wave pairing. A con-
ventional wisdom on this issue will be that, in zero field, thenonvanishing superconducting /H20849SC/H20850energy gap suppresses
AFM ordering and thus that the field-induced reduction inthe gap leads to a recovery of AFM fluctuation. However, itis difficult to explain, based on this picture, why an apparentQCP is realized not above but only just below H
c2/H208490/H20850/H20849Refs.
4and5/H20850in those materials. Rather, the fact that the field-
induced AFM ordering or QCP close to Hc2/H208490/H20850is commonly
seen in superconductors with a d-wave pairing and strong PD
suggests a common mechanism peculiar to superconductivityin finite fields and independent of electronic details of thosematerials such as the band structure.
In this Rapid Communication, we point out that, in nodal
d-wave superconductors, a field-induced enhancement of PD
tends to induce an AFM ordering just below H
c2/H208490/H20850. Al-
though relatively weak PD tends to be suppressed by thequasiparticle damping effect brought by the AFM fluctuation,strong PD rather favor coexistence of a d-wave superconduc-
tivity and an AFM order. Detailed mechanism of this field-induced enhancement of AFM fluctuation or ordering belowH
c2depends upon the relative orientation between the mo-
ment mof the expected AFM order and the applied field H:
inm/H20648Hcase, strong PD change the sign of the O /H20849m2/H20841/H9004/H208412/H20850
term in the free energy for any pairing symmetry, just likethat of its O /H20849/H20841/H9004/H20841
4/H20850term leading to the first-order Hc2
transition,2where m/H11013/H20841m/H20841and/H9004are order parameters of an
expected AFM phase and a spin-singlet SC one. In contrast,the field-induced AFM ordering in m/H11036H, which is possibly
satisfied in CeCoIn
5inH/H11036c,3is a peculiar event to thed-wave paired case with the momentum /H20849k/H20850-dependent gap
function wksatisfying wk=−wk+Qand tends to occur irre-
spective of the presence of the first-order Hc2transition,
where Qis the wave vector of the commensurate AFM
modulation and is /H20849/H9266,/H9266/H20850for the dx2−y2-wave pairing.9As a
consequences of this PD-induced magnetism, two striking
phenomena observed in CeCoIn 5, an AFM order3stabilized
by a FFLO spatial modulation and an anomalous flux densitydistribution in the vortex lattice,
10will be discussed.
First, we start from a BCS-type electronic Hamiltonian11
for a quasi-two-dimensional /H208492D/H20850material with /H9004andm
introduced as functionals. By treating /H9004at the mean-field
level, the free energy expressing the two possible orderingsis written in zero-field case as
F/H20849H=0/H20850=/H20858
r1
g/H20841/H9004/H20849r/H20850/H208412−Tln Tr c,c†,mexp /H20851−H/H9004m/T/H20852,
H/H9004m=/H20858
/H9251,/H9252=↑,↓/H20858
kcˆk,/H9251†ek/H9254/H9251,/H9252cˆk,/H9252+/H20858
q1
U/H20841m/H20849q/H20850/H208412
−/H20858
q/H20851m/H20849q/H20850·Sˆ†/H20849q/H20850−/H9004/H20849q/H20850/H9023ˆ†/H20849q/H20850+ H.c. /H20852, /H208491/H20850
where /H9023ˆ/H20849q/H20850=−/H20858kwkcˆ−k+q/2,↑cˆk+q/2,↓,Sˆ/H9263/H20849q/H20850=/H20858/H9251,/H9252/H20849/H9268/H9263/H20850/H9251,/H9252
/H11003/H20858kcˆk−q,/H9251†cˆk+Q,/H9252/2,cˆk,/H9251†creates a quasiparticle with spin in-
dex/H9251and momentum k,/H9268/H9263are the Pauli matrices, and the
positive parameters gandUare the attractive and repulsive
interaction strengths leading to the SC and AFM orderings,respectively. The dispersion e
k, measured from the chemical
potential, satisfies ek=−ek+Q+Tc/H9254I.12The dimensionless pa-
rameter /H9254Iis related to a small incommensurate part /H9254Qof
the AFM wave vector through the relation /H20841/H9254Q/H20841
/H11011/H20841/H9254I/H20841Tc//H208492EF/H20850/H20849/H11270/H20841Q/H20841/H20850, where EFis the Fermi energy. In the
case with a nonzero H, the Zeeman energy /H9262BH·/H20849/H9268/H20850/H9251,/H9252
needs to be added to ek/H9254/H9251,/H9252. Below, we focus on either the
case, m/H11036Horm/H20648Hby choosing the spin-quantization axis
along H. The orbital field effect will be included later.
To examine an interplay between the AFM and SC order-
ings below, let us consider the Gaussian AFM fluctuationtermF
min the free energy F,Fm=T/H20858/H9024ln det /H20851U−1/H9254q,q/H11032
−/H9273q,q/H11032/H20849/H9024/H20850/H20852, wherePHYSICAL REVIEW B 82, 060510 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS
1098-0121/2010/82 /H208496/H20850/060510 /H208494/H20850 ©2010 The American Physical Society 060510-1/H9273q,q/H11032/H20849/H9024/H20850=/H20885
0T−1
d/H9270/H20855T/H9270Sˆ
/H9263†/H20849q;/H9270/H20850Sˆ/H9263/H20849q/H11032;0/H20850/H20856ei/H9024/H9270/H208492/H20850
with a fixed /H9263andSˆ/H9263/H20849q;/H9270/H20850denotes Sˆ/H9263/H20849q/H20850at imaginary time /H9270.
For the moment, we focus on the Pauli limit with no orbitalfield effect and with uniform /H9004in which
/H9273q,q/H11032/H20849/H9024/H20850
=/H20851/H9273/H20849n/H20850/H20849q,/H9024/H20850+/H9273/H20849an/H20850/H20849q,/H9024/H20850/H20852/H9254q,q/H11032, and Fm=−T/H20858q,/H9024lnX/H20849q,/H9024/H20850,
where X−1/H20849q,/H9024/H20850=U−1−/H9273/H20849n/H20850/H20849q,/H9024/H20850−/H9273/H20849an/H20850/H20849q,/H9024/H20850, and /H9273/H20849n/H20850and
/H9273/H20849an/H20850are expressed by Figs. 1/H20849a/H20850and1/H20849b/H20850, respectively. The
/H9004-dependent part of /H9273/H20849n/H20850+/H9273/H20849an/H20850,/H9273s=/H9273/H20849n/H20850−/H9273/H20849n/H20850/H20841/H9004=0+/H9273/H20849an/H20850, has
been studied previously11inH=0 case. In the Pauli limit,
expressions of Fig. 1in nonzero Htake the form
/H9273/H20849n/H20850/H208490,/H9024/H20850=−T/H20858
p/H20858
/H9255,/H9268=/H110061d/H9255+/H9024,/H9268¯/H20849ep+Q/H20850d/H9255,/H9268/H20849ep/H20850
D/H9255+/H9024,/H9268¯/H20849p+Q/H20850D/H9255,/H9268/H20849p/H20850,
/H9273/H20849an/H20850/H208490,/H9024/H20850=T/H20858
p/H20858
/H9255,/H9268=/H110061/H20849−wpwp+Q/H20850/H20841/H9004/H208412
D/H9255+/H9024,/H9268¯/H20849p+Q/H20850D/H9255,/H9268/H20849p/H20850, /H208493/H20850
where /H9255/H20849/H9024/H20850is a fermion’s /H20849boson’s /H20850Matsubara frequency,
d/H9255,/H9268/H20849ep/H20850=i/H9255+/H9268/H9262BH+ep, and D/H9255,/H9268/H20849p/H20850=−d/H9255,/H9268/H20849ep/H20850d/H9255,/H9268/H20849−ep/H20850
+/H20841/H9004wp/H208412. Main features of /H9273/H20849n/H20850and/H9273/H20849an/H20850are seen in their
O/H20849/H20841/H9004/H208412/H20850terms. In zero field, /H9273s/H20849/H9004/H20850/H11013/H9273s/H208490,0 /H20850behaves like T−2
inT→0 limit and is negative so that the AFM ordering is
suppressed by superconductivity.11
To explain results in the case of strong PD, let us first
focus on m/H20648Hcase in which /H9268¯=/H9268. For a near-perfect nest-
ing, the O /H20849/H20841/H9004/H208412/H20850terms of /H9273/H20849n/H20850−/H9273/H20849n/H20850/H20841/H9004=0and/H9273/H20849an/H20850,a t /H20841q/H20841=/H9024
=0, take the same form as the coefficient of the quartic/H20851O/H20849/H20841/H9004/H20841
4/H20850/H20852term of the SC Ginzburg-Landau /H20849GL/H20850free energy
and thus, change their sign upon cooling.13Thus, /H9273s/H20849/H9004/H20850be-
comes positive for stronger PD, leading to a lower Fm, i.e.,
anenhancement of the AFM ordering in the SC phase. As
well as the corresponding PD-induced sign change in theO/H20849/H20841/H9004/H20841
4/H20850term which leads to the first-order Hc2transition,2the
PD-induced positive /H9273sis also unaffected by inclusion of the
orbital depairing.
Inm/H11036H, where /H9268¯=−/H9268, a different type of PD-induced
AFM ordering occurs in a d-wave pairing case with a gap
node along Q, where wp+Qwp/H110210: in this case, the O /H20849/H20841/H9004/H208412/H20850
term of /H9273/H20849n/H20850/H208490,0 /H20850remains negative and becomes
−N/H208490/H20850/H20841/H9004/H208412//H208512/H20849/H9262BH/H208502/H20852inT→0 limit with no PD-induced
sign change, where N/H208490/H20850is the density of states in the normal
state. Instead, the corresponding /H9273/H20849an/H20850/H208490,0 /H20850and thus, /H9273sare
divergent like N/H208490/H20850/H20851/H20841/H9004/H20841//H20849/H9262BH/H20850/H208522/H20841ln/H20851Max /H20849t,/H20841/H9254I/H20841/H20850/H20852/H20841inT→0
limit while keeping their positive signs, where t=T/Tc. This
divergence is unaffected by including the orbital depairing.Therefore, even in m/H11036H, the AFM order tends to occur
upon cooling in the dx2−y2-wave paired case. In contrast,
/H9273/H20849an/H20850/H208490,0 /H20850is also negative in the dxy-wave case satisfying
wpwp+Q/H110220 so that the AFM ordering is rather suppressed
with increasing H.
A possible AFM phase boundary in m/H20648c/H11036Hcase defined
as the temperature at which X0−1/H11013/H20851X/H208490,0 /H20850/H20852−1vanishes up to
O/H20849/H20841/H9004/H208412/H20850is shown in Fig. 2/H20849a/H20850together with the corresponding
/H9273s=0 curve, which is the upper bound of a PD-induced AFM
phase and shifts to higher temperature for larger /H20841/H9254I/H20841. Here,
the orbital depairing brought by the gauge field Ahas been
incorporated through the quasiparticle Green’s functionG
/H9255,/H9268/H20849r1,r2/H20850in real space with replacement2,15G/H9255,/H9268/H20849r1,r2/H20850
/H11013G/H9255,/H9268/H20849r1−r2/H20850exp /H20851ie/H20849/H6036c/H20850−1/H20848r2r1A·dl/H20852. In Fig. 2/H20849a/H20850, we have
used /H9251M/H20849ab/H20850/H20849Maki parameter in H/H11036c/H20850
/H110137.1/H9262BHc2,/H11036/H20849GL/H20850/H208490/H20850//H208492/H9266Tc/H20850=7.8 and the anisotropy /H9253
/H20849defined14from ek/H20850=4.5, where Hc2,/H11036/H20849GL/H20850/H208490/H20850=/H9253Hc2,/H20648/H20849GL/H20850/H208490/H20850
=/H9253/H6036c//H208492e/H926402/H20850is the Hc2/H208490/H20850value in H/H11036cdefined near Tc
with the coherence length /H92640. The AFM phase is lost in H
/H11022Hc2due to the discontinuous Hc2transition2int/H110210.215.
With a larger X0−1in the normal state, the uniform AFM
phase boundary in Fig. 2/H20849a/H20850is pushed down to t=0 to reduce
to an AFM-QCP.
The limitation to the O /H20849/H20841/H9004/H208412/H20850term of /H9273soverestimates the
AFM ordering. In fact, using the fullexpression of /H9273s, the
AFM region for /H9254I=0 is limited to an invisibly narrow region
in the vicinity of Hc2/H208490/H20850. Nevertheless, the PD-induced AF
order for nonzero /H20841/H9254I/H20841also follows from the full/H9273s/H20849/H9004/H20850:b y
substituting /H20841/H9004/H20841obtained from the gap equation into Eq. /H208493/H20850,
the lines on which the full /H9273s/H20849/H9004/H20850vanishes are obtained, in thep
p+Qp
p+Qp
pQ-
--(a) (b)
FIG. 1. Diagrams describing /H20849a/H20850/H9273/H20849n/H20850and /H20849b/H20850/H9273/H20849an/H20850, where a cross
denotes a particle-hole vertex on the AFM fluctuation, and doublesolid lines in /H20849a/H20850and /H20849b/H20850denote normal and anomalous Green’s
functions, respectively.0.25
0.2
0.15
0.1
0.05
0
3.09
0.00
0.40.6 0.81h0.850.90.9511.05h00.5δFmt
X0-1(a) (b)
FIG. 2. /H20849Color online /H20850Possible /H9273s=0 lines /H20849dashed curves /H20850int
vsh/H11013H/Hc2/H208490/H20850diagram in H/H11036c/H20648mfollowing /H20849a/H20850from the
O/H20849/H20841/H9004/H208412/H20850term of /H9273s/H20849/H9004/H20850for/H9254I=0 and /H20849b/H20850from the full /H9273s/H20849/H9004/H20850in the
Pauli limit for /H9254I=0.44 /H20851lower dashed /H20849green /H20850curve /H20852and 0.63
/H20851higher dashed /H20849blue /H20850one /H20852, respectively. With no AFM phase in
H/H11022Hc2/H20849T/H20850/H20849nearly vertical dotted curve /H20850, a dashed curve is the
upper bound of a possible uniform AFM phase for a fixed /H9254I.I n /H20849a/H20850,
the solid /H20849blue /H20850curve is the phase boundary of a uniform AFM
order which follows from the dashed curve and an assumed form ofX
0/H20841/H9004=0. The lower panel of /H20849a/H20850is the corresponding X0−1/H20849h/H20850att
=0.0075. In the FFLO state with a modulated /H20841/H9004/H20849/H9256/H20850/H20841, however, the
uniform AFM order parameter min the uniform /H9004case is trans-
formed to a modulated one m/H20849/H9256/H20850, and, due to this modulation, the
higher dashed /H20849blue /H20850curve in /H20849b/H20850is pushed up to the lower solid
/H20849blue /H20850one lying close to the second-order FFLO transition /H20849Ref. 14/H20850
/H20851higher solid /H20849red/H20850/H20852line. The lower panel of /H20849b/H20850is the normalized
/H9254Fm/H20849MF /H20850/H20849h/H20850att=0.075.IKEDA, HATAKEYAMA, AND AOYAMA PHYSICAL REVIEW B 82, 060510 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS
060510-2Pauli limit with uniform /H9004, as the dashed curves in Fig. 2/H20849b/H20850.
The/H9273s/H110220 region becoming wider with increasing /H20841/H9254I/H20841sug-
gests that, as in CeCoIn 5, the PD-induced AFM order near
Hc2/H208490/H20850should be incommensurate.3Similar results also fol-
low from the direct use of the tight-binding model.16
Interestingly, the AFM ordered region is expanded further
by the presence of the FFLO modulation /H9004/H20849/H9256/H20850
/H11013/H208812cos /H20849qLO/H9256/H20850, where /H9256is the component parallel to Hof the
coordinate. To demonstrate this, the gradient expansion15
will be applied to the O /H20849m2/H20850term of the mean-field expres-
sion of Fm. Up to the lowest order in the gradient, its
/H9004/H20849/H9256/H20850-dependent part simply becomes
/H9254Fm/H20849MF /H20850=/H20885d/H9256m/H20849/H9256/H20850/H20877−/H20849X0−1/H20850/H208492/H20850/H115092
/H11509/H92562−/H9273s/H20851/H9004/H20849/H9256/H20850/H20852/H20878m/H20849/H9256/H20850,/H208494/H20850
where /H20849X0−1/H20850/H208492/H20850=/H11509X−1/H20849k,0/H20850//H11509k2/H20841k=0, and we only have to ex-
amine its sign which, for uniform /H9004, corresponds to that of
−/H9273s. For simplicity, by using the qLOdata in Ref. 14,w efi n d
that/H9254Fm/H20849MF /H20850/H110210 below the lower /H20849blue /H20850solid curve in Fig.
2/H20849b/H20850, indicating that the FFLO order can coexist with the
AFM order in most temperature range. Close to the FFLOtransition /H20849red solid /H20850curve on which q
LO=0,m/H20849/H9256/H20850is found to
have the modulation /H11011sin/H20849qLO/H9256/H20850so that this AFM order is
continuously lost as the FFLO nodal planes goes away from
the system.17Oppositely, the form of m/H20849/H9256/H20850deep in the FFLO
state may not be examined properly in terms of the presentgradient expansion and will be reconsidered elsewhere.
Next, we examine the anomalous flux density distribution
in the vortex lattice of CeCoIn
5inH/H20648c/H20849Ref. 10/H20850as another
phenomenon suggestive of an AFM fluctuation enhanced justbelow H
c2. The flux distribution is measured by the form
factor /H20841F/H20841, which is the Fourier component of the longitudinal
magnetization Mc/H20849r/H20850/H11013/H20851Bc/H20849r/H20850−H/H20852//H208494/H9266/H20850at the shortest
reciprocal-lattice vector Kand implies the slope of the flux
density Bcin real space. Here, Mc/H20849r/H20850=/H20858K/HS110050Mc/H20849K/H20850eiK·ris
given by
Mc/H20849K/H20850=ic−1K−2/H20851K/H11003j/H20849K/H20850/H20852c−/H20879/H9254F
/H9254BPD/H20849−K/H20850/H20879
BPD=0,/H208495/H20850
where j/H20849K/H20850andBPD/H20849K/H20850are Fourier components of the or-
bital supercurrent density and a Zeeman field BPDimposed to
define Mc, respectively. It is straightforward to, according to
Ref. 2, obtain Mcin the weak-coupling model with no AFM
fluctuation by fully taking account of the PD and the orbitaldepairing. In our numerical calculations, the O /H20849/H20841/H9004/H20841
4/H20850contri-
butions to Mcare also incorporated. Nevertheless, it is in-
structive to first focus on its O /H20849/H20841/H9004/H208412/H20850terms. In H/H11270Hc2,/H20648/H20849GL/H20850/H208490/H20850,
the contribution to Mcfrom the spin part /H20849last term /H20850of Eq.
/H208495/H20850is given by
Mc/H20849K/H20850/H20849PD/H20850=/H9262BN/H208490/H20850
2/H9266T/H20841/H9004/H208412/H20849K/H20850Im/H9274/H208491/H20850/H208731
2+i/H9262BH
2/H9266T/H20874, /H208496/H20850
which is negative, where /H9274/H208491/H20850/H20849z/H20850is the first derivative of the
digamma function. Thus, the PD tends to enhance the mag-netic screening far from the vortex core. This is one of ori-gins of the field-induced increase in /H20841F/H20841. However, such an
enhancement of /H20841F/H20841in the weak-coupling model is much
weaker in the d-wave case than in the s-wave one,
18althoughit is the d-wave material CeCoIn 5, which has clearly shown
such an enhanced /H20841F/H20841.10This has motivated us to see how the
PD-induced AFM fluctuation is reflected in /H20841F/H20841. For this pur-
pose, let us first consider the m/H11036Hcase and start with the
Pauli limit again in which Mcis determined by the last term
of Eq. /H208495/H20850. By noting that this term can be obtained as the
first derivative of the free energy density with respect to
/H9262BH, it is found that the contribution to Mcfrom the Gauss-
ian AFM fluctuation is positive and proportional to
/H11011O/H20849/H20841/H9004/H20849r/H20850/H208412/H20850/H9262B2H/H20858qX/H20849q,0/H20850/T3, implying a suppressed
screening due to the AFM fluctuation, for weak PD /H20849/H9262BH
/H11270T/H20850, while it is negative and given by
−2/H9262BT/H20841/H9004/H20849r/H20850/H208412/H20849/H9262BH/H20850−3/H20841ln/H20851Max /H20849/H20841/H9254I/H20841,t/H20850/H20852/H20841/H20858/H9024/H20858qX/H20849q,/H9024/H20850, sug-
gesting an enhancement due to the AFM fluctuation of thescreening and thus, of /H20841F/H20841, for strong PD /H20849
/H9262BH/H11271T/H20850, respec-
tively.
In Fig. 3, we show hvs/H20841F/H208412curves obtained by incorpo-
rating the orbital depairing for both cases with and with nocontributions to Eq. /H208495/H20850from the AFM fluctuation through
F
m, where /H20841F/H20841was normalized by that in the GL region near
Tc. Based on the result in Fig. 2/H20849a/H20850, the familiar phenomeno-
logical form of X/H20849q,/H9024/H20850,N/H208490/H20850X/H20849q,/H9024/H20850=1 //H20851mN+/H20849/H20841q/H11003zˆ/H20841/H92640/H20849N/H20850/H208502
+/H92640/H20849N/H20850/H20841/H9024/H20841//H20841v/H20841/H20852was assumed in calculating the Fmcontribution
to Eq. /H208495/H20850, where mN=t+1− h/hQCPand/H92640/H20849N/H20850=0.6/H92640, and v2is
the squared Fermi velocity averaged over the Fermi surface.The remarkable peak just below H
c2of the red solid curve is
a reflection of the aforementioned growth of /H9273/H20849an/H20850inm/H11036H.
A similar /H20841F/H20841enhancement also occurs in m/H20648H/H20849dashed /H20850
curves with increasing H, reflecting the aforementioned sign
change in /H9273sdue to large PD. The favorable direction of the
moment min CeCoIn 5inH/H20648cis unclear at present. If, in
CeCoIn 5, the m/H11036Hcomponents are dominant in the AFM
fluctuation even in H/H20648c, the /H20841F/H20849t,h/H20850/H20841data10growing with
increasing field andon further cooling is a reflection of the
dx2−y2-wave pairing19accompanied by a strong AFM fluctua-
tion with Q/H20648/H20849/H9266,/H9266/H20850. With the same Q, the corresponding
/H20841F/H20849t,h/H20850/H20841in the s-wave case would become much lower than
the dotted ones, contrary to the trend in the weak-coupling0.55 0.65 0.75 0.85 0.9500.10.20.30.40.5
|F|2
h0.45
FIG. 3. /H20849Color online /H20850Normalized form factor /H20841F/H20849h/H20850/H208412curves in
H/H20648cobtained in the weak-coupling BCS model with no AFM fluc-
tuation /H20849dotted curves /H20850, in the case with AFM fluctuations with
m/H20648H/H20849dashed ones /H20850, and in the corresponding m/H11036Hcase /H20849solid
ones /H20850att=0.02 /H20851highest /H20849red/H20850curves /H20852, 0.04 /H20851middle /H20849blue /H20850/H20852, and
0.08 /H20851lowest /H20849green /H20850/H20852. Here, hQCP=0.942, /H9251M/H20849c/H20850=/H9251M/H20849ab/H20850//H9253=6.5, and
the typical quasiparticle damping rate /H92662T//H208534kF/H92640/H20849N/H20850/H20851mN/H20849t,h/H20850/H208521/2/H20854
due to 2D AFM fluctuation was used.ANTIFERROMAGNETIC ORDERING INDUCED BY … PHYSICAL REVIEW B 82, 060510 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS
060510-3results.18The curves in Fig. 3have been obtained by neglect-
ing the narrow FFLO region in CeCoIn 5inH/H20648c. Effects of
the FFLO modulation on Mcwill be reported elsewhere.
It has been argued in previous studies1,2,19that the anoma-
lous SC properties in CeCoIn 5in high fields are conse-
quences of strong PD. The present results indicate that theAFM fluctuation and order indicated in more recentmeasurements
3,10on this material just below Hc2/H208490/H20850also
stem from PD, and thus that its HFLT phase and the behav-iors suggestive of an AFM-QCP near H
c2/H208490/H20850commonly seen
ind-wave superconductors with strong PD,4–8including
CeCoIn 5, should be understood on the same footing. For in-
stance, the result in Fig. 2that the field-induced AFM order-
ing is discontinuously bounded by the first-order Hc2transi-
tion naturally explains why, in spite of the AFM ordering justbelow H
c2/H208490/H20850,3the transport properties in CeCoIn 5inH
/H11022Hc2have suggested a QCP notably below Hc2/H208490/H20850.4We
have also shown that, in dx2−y2-wave superconductors, the
AFM order is significantly enhanced by the FFLO periodic
modulation of the SC order parameter. In contrast, otherworks have assumed the presence of an attractive
/H9266-triplet
pairing channel as the only origin of the AFM ordering be-low H
c2.3,20However, the field-induced AFM transition in
Ref. 20is of first order in contrast to the observation1inCeCoIn 5. The present work has shown that an incommensu-
rate AFM ordering just below Hc2/H208490/H20850develops without as-
suming such an occurrence of other pairing state and dueonly to strong PD effect. In fact, the strange impurityeffects,
21arising even from a nonmagnetic doping, on the
HFLT phase of CeCoIn 5cannot be explained unless the
HFLT phase has an additional modulation of the SC orderparameter.
22
In conclusion, an AFM ordering or fluctuation enhanced
close to Hc2/H208490/H20850, often seen in unconventional superconduct-
ors, is a direct consequence of strong paramagnetic depairingand also of their d-wave pairing symmetry with a gap node
parallel to the AFM modulation. The AFM ordering en-hanced due to the FFLO modulation suggests that the HFLTphase in CeCoIn
5is a coexistent state of the AFM and FFLO
orders. Discussing the AFM-QCP issues in systems6,8with a
continuous behavior around Hc2based on the present theory
is straightforward and will be performed elsewhere togetherwith studies of a precise
/H9254Qorientation in CeCoIn 5based on
a more realistic electronic Hamiltonian.
This work was supported by Grant-in-Aid for Scientific
Research /H20849Grants No. 20102008 and No. 21540360 /H20850from
MEXT, Japan.
1A. Bianchi, R. Movshovich, C. Capan, P. G. Pagliuso, and J. L.
Sarrao, Phys. Rev. Lett. 91, 187004 /H208492003 /H20850.
2H. Adachi and R. Ikeda, Phys. Rev. B 68, 184510 /H208492003 /H20850.
3M. Kenzelmann, S. Gerber, N. Egetenmeyer, J. L. Gavilano, Th.
Strässle, A. D. Bianchi, E. Ressouche, R. Movshovich, E. D.Bauer, J. L. Sarrao, and J. D. Thompson, Phys. Rev. Lett. 104,
127001 /H208492010 /H20850.
4S. Singh, C. Capan, M. Nicklas, M. Rams, A. Gladun, H. Lee, J.
F. DiTusa, Z. Fisk, F. Steglich, and S. Wirth, Phys. Rev. Lett.
98, 057001 /H208492007 /H20850; F. Ronning, C. Capan, A. Bianchi, R.
Movshovich, A. Lacerda, M. F. Hundley, J. D. Thompson, P. G.Pagliuso, and J. L. Sarrao, Phys. Rev. B 71, 104528 /H208492005 /H20850.
5Y. Kasahara, Y. Nakajima, K. Izawa, Y. Matsuda, K. Behnia, H.
Shishido, R. Settai, and Y. Onuki, Phys. Rev. B 72, 214515 /H20849R/H20850
/H208492005 /H20850; K. Kumagai /H20849private communication /H20850.
6T. Park, F. Ronning, H. Q. Yuan, M. B. Salamon, R. Movshov-
ich, J. L. Sarrao, and J. D. Thompson, Nature /H20849London /H20850440,6 5
/H208492006 /H20850; T. Park, Y. Tokiwa, F. Ronning, H. Lee, E. Bauer, R.
Movshovich, and J. Thompson, arXiv:0910.2287 /H20849unpublished /H20850.
7F. Honda, M.-A. Measson, Y. Nakano, N. Yoshitani, E. Yama-
moto, Y. Haga, T. Takeuchi, H. Yamagami, K. Shimizu, R. Set-tai, and Y. Ōnuki, J. Phys. Soc. Jpn. 77, 043701 /H208492008 /H20850.
8T. Shibauchi, L. Krusin-Elbaum, M. Hasegawa, Y. Kasahara, R.
Okazaki, and Y. Matsuda, Proc. Natl. Acad. Sci. U.S.A. 105,
7120 /H208492008 /H20850.
9This assumption of 2D Qis a reasonable choice to describe the
PD-induced AFM order stabilized by a modulation parallel to agap node.10A. D. Bianchi, M. Kenzelmann, L. DeBeer-Schmitt, J. S. White,
E. M. Forgan, J. Mesot, M. Zolliker, J. Kohlbrecher, R.Movshovich, E. D. Bauer, J. L. Sarrao, Z. Fisk, C. Petrovi ć, and
M. R. Eskildsen, Science 319, 177 /H208492008 /H20850; M. R. Eskildsen
/H20849private communication /H20850.
11R. Konno and K. Ueda, Phys. Rev. B 40, 4329 /H208491989 /H20850; M. Kato
and K. Machida, ibid. 37, 1510 /H208491988 /H20850.
12A. B. Vorontsov, M. G. Vavilov, and A. V. Chubukov, Phys. Rev.
B79, 060508 /H20849R/H20850/H208492009 /H20850.
13K. Maki and T. Tsuneto, Prog. Theor. Phys. 31, 945 /H208491964 /H20850.
14R. Ikeda, Phys. Rev. B 76, 134504 /H208492007 /H20850.
15G. Eilenberger, Z. Phys. 190, 142 /H208491966 /H20850.
16Y. Hatakeyama and R. Ikeda, arXiv:1006.2465 , J. Phys.: Conf.
Ser. /H20849to be published /H20850.
17R. Ikeda, Phys. Rev. Lett. 102, 069703 /H208492009 /H20850; Y. Yanase and M.
Sigrist, J. Phys. Soc. Jpn. 78, 114715 /H208492009 /H20850.
18M. Ichioka and K. Machida, J. Phys.: Conf. Ser. 150, 052074
/H208492009 /H20850, and references therein.
19N. Hiasa and R. Ikeda, Phys. Rev. Lett. 101, 027001 /H208492008 /H20850.
20A. Aperis, G. Varelogiannis, and P. B. Littlewood, Phys. Rev.
Lett. 104, 216403 /H208492010 /H20850.The reduction in the Knight shift in
the HFLT phase stressed there is a reflection of the reduction/H20849Ref. 14/H20850in/H20841/H9004/H20841
2in the HFLT phase.
21Y. Tokiwa, R. Movshovich, F. Ronning, E. D. Bauer, P. Papin, A.
D. Bianchi, J. F. Rauscher, S. M. Kauzlarich, and Z. Fisk, Phys.
Rev. Lett. 101, 037001 /H208492008 /H20850.
22R. Ikeda, Phys. Rev. B 81, 060510 /H20849R/H20850/H208492010 /H20850.IKEDA, HATAKEYAMA, AND AOYAMA PHYSICAL REVIEW B 82, 060510 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS
060510-4 |
PhysRevB.88.064517.pdf | PHYSICAL REVIEW B 88, 064517 (2013)
Pressure dependence of superconductivity in simple cubic phosphorus
Kevin T. Chan, Brad D. Malone, and Marvin L. Cohen
Department of Physics, University of California, Berkeley, California 94720, USA
and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
(Received 26 April 2013; revised manuscript received 5 July 2013; published 30 August 2013)
The electronic structure and lattice dynamics for simple cubic (sc) P are calculated over the pressure range
0–70 GPa from first principles using the local-density approximation. The Rphonon mode is found to be unstable
below 20 GPa in the harmonic approximation, but may be stable down to a pressure less than 20 GPa whenanharmonicity is considered. The electron-phonon coupling is calculated for pressures above 20 GPa, and thesuperconducting transition temperature T
cis found to decrease with increasing pressure throughout this pressure
range. The result is in agreement with experimental results above 30 GPa. In contrast to experiment, no evidencefor a decrease in T
cwith decreasing pressure below 30 GPa is found. The structural transition from rhombohedral
A7 to sc is also investigated. An interesting two-step transition is found to occur theoretically which may have
relevance for the pressure dependence of Tc. Possible explanations for the discrepancy with experiment are
discussed.
DOI: 10.1103/PhysRevB.88.064517 PACS number(s): 74 .62.Fj, 74.70.Ad, 74 .25.Kc, 74 .20.Pq
I. INTRODUCTION
One hundred years after its discovery,1superconductivity
remains one of the most exciting subjects in condensed-matterphysics. In addition to the recent discoveries of unconventionalsuperconductivity in the iron-based superconductors,
2,3there
have been a large number of interesting developments in thesuperconductivity of simpler materials, including MgB
2,4,5
diamond,6cubic silicon doped with boron,7and the surprising
large superconducting transition temperature ( Tc)o f2 0Kf o r
lithium under pressure.8
A large number of other elemental solids have been found
to be superconducting, many in the past 20–30 years with theaid of much-improved diamond-anvil cells.
9The experimental
ability to vary the pressure over several hundred GPa hasallowed for superconductivity to be explored in high-pressurepolytypes not accessible at lower pressures, as well as the studyof the pressure dependence of T
cwithin single phases which
can vary because of underlying electronic and vibrationalchanges that occur as a material is compressed.
Phosphorus is semiconducting and takes on the orthorhom-
bicA17 structure at ambient pressure and temperature.
10,11
Under the application of pressure it transforms into the
semimetallic A7 structure at 4.5 GPa and to the metallic
simple cubic (sc) phase at 10 GPa,10,11which is the stable
form of phosphorus up to a pressure of 107 GPa.12Reports
of superconductivity in the sc phase are now over 40 yearsold.
13,14However, the maximal value and pressure dependence
ofTcremain the subject of much controversy in the literature.
Part of the confusion stems from the fact that the nature ofthe pressure variation of T
cdepends strongly on the path taken
in theP-Tdiagram.15,16While some differences can likely be
attributed to incomplete phase transformations along particularpaths, there is still discrepancy in the experimental results forthe pressure dependence of samples that are believed to befully transformed to the sc structure. Some experimental resultsindicate that T
cof the sc phase should be approximately 6 K and
vary only weakly on pressure.16Others report that Tcshould
exhibit a two-peak structure with the largest peak occurringnear 23 GPa with a transition temperature of ∼10 K.
17Themost recent results show only a single peak around 32 GPa
w i t ham a x i m a l Tcof 9.5 K.18
Unfortunately, the previous theoretical results have not
been able to resolve this discrepancy. Early estimates19of the
pressure variation of T cshow a single peaked structure with
the peak position near the second peak of the experimental
results of Ref. 17. These calculations rely on approximations
to the average phonon frequency entering into the McMillanequation for T
cand do not take into account the variation of the
phonon spectrum with pressure. Other estimates establish thatthe electron-phonon ( e-p) coupling strength is large enough to
roughly account for the range of the experimentally observedT
c, but, lacking a detailed description of the phonon spectra,
are unable to reliably establish the trend with increasing
pressure.20A more recent calculation predicts that Tcshould
instead be relatively constant over the full range of stabilityof the sc phase;
21this pressure variation is similar to the
experimental results reported in Ref. 16. These calculations,
which do not predict a drop in Tcat high pressures, are in
stark disagreement with the most recent experimental resultstaken at high pressure in Ref. 18. Another theoretical study
22
found a calculated Tcthat rose slightly from about 8.5 to 11 K
as the pressure increased from 10 to 35 GPa. This studyconsidered the change in phonon frequencies with pressure,but only estimated the Debye temperature from the calculatedbulk modulus.
In the present study we hope to shed light on the pressure
dependence of the transition temperature in simple cubic
phosphorus. Towards this aim, we have performed fully ab
initio calculations of the e-pcoupling strength as a function
of pressure over a wide range of the stability of the sc phase.In contrast to previous calculations, we calculate the lattice
dynamics at each pressure from first principles. We find that
at higher pressures, T
cdecreases with increasing pressure, in
good agreement with some experiments. However, we find
no evidence of the decrease in Tcas pressure is decreased at
lower pressures. Our harmonic phonon calculations indicate
that the sc structure has an instability in the Rphonon mode at
a pressure higher than observed in experiment. We therefore
calculate anharmonic phonon frequencies for the Rmode,
064517-1 1098-0121/2013/88(6)/064517(10) ©2013 American Physical SocietyKEVIN T. CHAN, BRAD D. MALONE, AND MARVIN L. COHEN PHYSICAL REVIEW B 88, 064517 (2013)
and find that the phonon modes are stable down to 5 GPa.
We also investigate the A7 to sc transition in P and find an
interesting two-step transition that may have implications for
the superconducting Tc.
II. METHODS
Thee-pcoupling formalism followed in the present study
is the same as in Ref. 23. For convenience, we reproduce
the relevant formulas here. The e-pmatrix element for the
scattering of an electron in band nat wave vector kto a
state in band mwith wave vector k+qby a phonon with
mode index νat wave vector qis
gν
mn(k,q)=/parenleftbigg¯h
2Mω qν/parenrightbigg1/2
/angbracketleftm,k+q|δqνVSCF|n,k/angbracketright.(1)
In this expression, |n,k/angbracketrightis the bare electronic Bloch state,
ωqνis the screened phonon frequency, Mis the ionic mass,
andδqνVSCFis the derivative of the self-consistent potential
with respect to a collective ionic displacement correspondingto phonon wave vector qand mode ν.
The phonon linewidth is given by
γ
qν=πωqν/summationdisplay
mn/summationdisplay
kwk/vextendsingle/vextendsinglegν
mn(k,q)/vextendsingle/vextendsingle2δ(/epsilon1m,k+q−/epsilon1F)
×δ(/epsilon1n,k−/epsilon1F), (2)
where wkis thek-point weight (normalized such that/summationtext
kwk=
2),/epsilon1n,kis the energy of the bare electronic Bloch state, and /epsilon1F
is the Fermi energy.
The sum over electron wave vectors kis performed on a
uniform grid over the whole Brillouin zone (BZ).
The phonon-mode-dependent coupling constant is given by
λqν=γqν
πN(/epsilon1F)ω2qν. (3)
In terms of the phonon linewidths, the Eliashberg spectral
function α2F(ω) can be written as24
α2F(ω)=1
2πN(/epsilon1F)/summationdisplay
qνwqγqν
ωqνδ(ω−ωqν). (4)
The sum over phonon wave vector qis performed on a uniform
grid over the irreducible BZ, with appropriate weights wq,
where/summationtext
qwq=1. In Eqs. (3)and(4),N(/epsilon1F) is the density of
states at /epsilon1Fper unit cell and per spin. The coupling constant λ
is given by the integral
λ=2/integraldisplay∞
0α2F(ω)
ωdω. (5)
Other important frequency moments of α2F(ω) are defined as
follows:
/angbracketleftω2/angbracketright=2
λ/integraldisplay∞
0ωα2F(ω)dω (6)
and
ωln=exp/parenleftbigg2
λ/integraldisplay∞
0lnωα2F(ω)
ωdω/parenrightbigg
. (7)We determine Tcusing the Allen-Dynes-modified McMil-
lan equation:25,26
Tc=ωln
1.2exp/parenleftbigg
−1.04(1+λ)
λ−μ∗(1+0.62λ)/parenrightbigg
, (8)
where μ∗is the Coulomb pseudopotential.27
First-principles electronic structure calculations are per-
formed with the QUANTUM -ESPRESSO code (QE)28within
density-functional theory29,30using the plane-wave pseu-
dopotential method31,32along with the local-density approx-
imation (LDA)33,34to the exchange-correlation energy. A
norm-conserving pseudopotential was constructed for P usingthe Troullier-Martins scheme
35as implemented in the APE
code.36The outermost 3 s23p3electrons are treated as valence
electrons and a nonlinear-core correction (NLCC) has beenadded.
37A plane-wave cutoff of 60 Ry for the valence
wave functions is used. Self-consistent charge density andtotal-energy calculations for sc P are performed using a30×30×30 Monkhorst-Pack
38(MP) kgrid with 0.03 Ry
Marzari-Vanderbilt (MV) smearing39for the occupations of
the electronic states. The sc lattice constant at a givenpressure is determined by relaxing the lattice constant until thecalculated pressure is within 0.05 GPa of the target pressure.
Electron-phonon calculations are performed for the sc
structure between 20 and 70 GPa. In order to converge thee-pquantities while maintaining a reasonable computational
cost, we use the electron-phonon Wannier (EPW) method,
40
as implemented in the EPW code,41to interpolate e-pmatrix
elements calculated on coarse electron and phonon grids tofinekandqgrids. Maximally localized Wannier functions
42,43
for use in the EPW method are generated using the WANNIER90
code.44
Bloch functions on an 8 ×8×8kgrid in the BZ are used to
generate Wannier functions for the lowest nine bands. Phononsin the harmonic approximation are calculated on an 8 ×8×8
qgrid in the BZ using density functional perturbation theory
45
as implemented in QE.
Matrix elements are interpolated onto a 100 ×100×100
/Gamma1-centered kgrid for the phonon linewidth calculations; the δ
functions of Eq. (2)are approximated by Gaussians of width
0.02 Ry. A 30 ×30×30/Gamma1-centered fine qgrid with a δ-
function smearing of 2 meV is used for the calculation ofα
2F(ω)[ E q . (4)]. With the chosen computational parameters,
λis estimated to be converged to less than 0.01.
In addition to e-pcalculations in the harmonic approx-
imation, we also estimate anharmonic phonon frequenciesfor the sc Rmode. With the frozen phonon method, the
anharmonic phonon modes can be determined by consideringatomic displacements with wave vector R, and solving the
Schr ¨odinger equation in a potential given by the total energy
versus atomic displacement in three dimensions. Such asolution for a general three-dimensional potential is difficult,so we estimate the phonon modes by fixing the phononpolarization along a given direction and considering the one-dimensional potential. We determine the phonon frequency asthe difference in energies of the ground and first excited statesin the one-dimensional potential. We consider polarizations inthe [111] and [100] directions.
064517-2PRESSURE DEPENDENCE OF SUPERCONDUCTIVITY IN ... PHYSICAL REVIEW B 88, 064517 (2013)
TheA7 to sc structural transition in P is studied using
variable-cell relaxation calculations; the method is similar tothat used in Ref. 23. Target pressures between 0 and 25 GPa are
considered. A 30 ×30×30 MP kgrid in the rhombohedral BZ
with 0.03 Ry MV smearing is used. Initial values of cos α=
0.536 and u=0.223, taken from experiment at 6.7 GPa,
11are
used for the starting structure for the relaxations. Structuresare relaxed until components of the forces on the atoms areless than 10
−4Ry/a.u. and the pressure is within 0.05 GPa of
the target pressure.
III. RESULTS
A. Electronic structure, phonons, and electron-phonon
coupling for sc P
The calculated volume and lattice constant as a function
of pressure are given in Table I. The calculated total energy
versus volume data are fit to a Birch-Murnaghan equation ofstate (EOS),
46giving V0=14.04˚A3,B0=131.6 GPa, and
B/prime
0=3.92 for the equilibrium volume, bulk modulus, and
the pressure derivative of the bulk modulus, respectively. Theequilibrium volume is somewhat smaller than that found fromfitting a Murnaghan or Birch-Murnaghan EOS to experimentaldata (15 .2˚A
3for Ref. 11and 15 .52˚A3for Ref. 12), and the
calculated B0somewhat larger than experiment (95 GPa for
Ref. 11and 70 .7G P af o rR e f . 12); our results are consistent
with the tendency of the LDA to overbind. Our EOS parametersare in good agreement with recent previous first-principlescalculations.
22,47–49Table Ialso shows, for each calculated
volume, the corresponding experimental pressure, Pexpt1 or
Pexpt2, obtained from the Murnaghan or Birch-Murnaghan
EOS with parameters from Ref. 11or12, respectively.
For example, a Murnaghan EOS with equilibrium volumeparameter V
0=15.2˚A3and bulk modulus parameter B0=
95 GPa from Ref. 11gives a pressure Pexpt1=8.0 GPa for a
volume V=14.06 ˚A3.
A previous calculation22indicates that the generalized-
gradient approximation (GGA) gives an equilibrium volumeand bulk modulus that is in better agreement with experimentalresults than the LDA for sc P, though not dramatically so.
TABLE I. Calculated pressure Pcalc, volume V, lattice constant
a, and corresponding experimental pressures Pexpt1andPexpt2for sc
P. The pressures Pexpt1 andPexpt2 are determined from the volume
by using the equation-of-state parameters given in Refs. 11and12,
respectively.
Pcalc(GPa) V(˚A3)a(˚A) Pexpt1(GPa) Pexpt2(GPa)
0 14.06 2.414 8.0 8.9
5 13.55 2.384 12.3 13.510 13.13 2.359 16.3 18.1
15 12.77 2.337 20.0 22.7
20 12.45 2.318 23.6 27.325 12.16 2.300 27.0 32.1
30 11.91 2.283 30.3 36.9
40 11.45 2.254 36.8 46.850 11.06 2.228 42.9 57.0
60 10.73 2.206 48.7 67.3
70 10.44 2.185 54.4 77.9FIG. 1. (Color online) Band structure of the simple cubic phase
of P for a number of pressures. The Fermi level is located at zeroenergy.
In the present work, we restrict ourselves to the LDA so
as to facilitate comparison with previous calculations ofsuperconductivity that use the LDA
19,20,22and because the
LDA has been used successfully to study superconductivity inother simple materials under pressure.
23,50–53Calculations of
superconductivity in sc P using the GGA could be interesting
but are beyond the scope of the present work.
The band structure of sc P as a function of pressure is shown
in Fig. 1. The largest changes near /epsilon1Fas a function of pressure
happen around the points RandMin the Brillouin zone.
The valence band at R, occupied for pressures below 15 GPa,
rises in energy as the structure is compressed and becomesunoccupied before a pressure of 20 GPa. The behavior at the M
point is the reverse, dropping lower in energy with increasingpressure. This behavior is in agreement with that found inprevious calculations,
19,20,22,47although the precise occupa-
tions of these specific bands at a given pressure differ slightlybetween calculations. At 0 GPa, N(/epsilon1
F) is 0.309 states /eV
atom (for both spins); it decreases slightly when pressureincreases to 20 GPa, then increases and levels off at higherpressures (Table II). This general trend also agrees with
previous calculations.
19,22Bands 2 and 3 do not change
dramatically with pressure near /epsilon1F.
TABLE II. Calculated frequency moments, N(/epsilon1F),/angbracketleftg2/angbracketright,λ,a n d
Tcfor sc P at various pressures. Equation (12) in the text is used to
determine the Coulomb pseudopotential μ∗as a function of pressure
in the calculation of Tc, with μ∗=0.18 atPcalc=25 GPa.
Pcalc ωln/angbracketleftω2/angbracketright1/2N(/epsilon1F) /angbracketleftg2/angbracketright Tc
(GPa) (K) (K) (states /eV atom) (eV /˚A)2λ (K)
20 418 449 0.292 60.61 0.795 10.26
25 435 468 0.296 63.05 0.776 9.64
30 444 482 0.301 65.32 0.771 9.39
40 456 506 0.303 68.81 0.739 8.2450 464 527 0.305 71.60 0.714 7.26
60 469 546 0.306 74.13 0.693 6.43
70 469 561 0.306 76.34 0.676 5.77
064517-3KEVIN T. CHAN, BRAD D. MALONE, AND MARVIN L. COHEN PHYSICAL REVIEW B 88, 064517 (2013)
FIG. 2. (Color online) Phonon dispersion for P in the simple cubic
phase for a number of pressures. The dynamical instability at the R
point at lower pressures corresponding to the distortion towards the
rhombohedral A7 structure can be seen.
The phonon dispersions are plotted in Fig. 2for several
pressures. The majority of the phonon frequencies are found toharden with increasing pressure. An exception is the transversebranch which shows a slight softening with increasing pressurealong the /Gamma1X,/Gamma1M, andXM directions. This softening agrees
with previous calculations and is consistent with an instabilityof the sc structure at pressures higher than those calculated inthe present study.
54The overall hardening of phonon modes
can also be seen in the phonon density of states F(ω), shown
in Fig. 4(top).
Below 20 GPa the lattice distortion corresponding to
the wave vector Ris found to have imaginary frequency,
suggesting that below 20 GPa, P is not stable in the sc structureand instead takes on the rhombohedral A7 structure. This result
differs from experiment,
11where the sc structure is stable down
to about 10 GPa (corresponding to a pressure of less than5 GPa in our calculations; see Table I). However, previous
calculations for As in the sc structure show that the Rphonon
mode is anharmonic near the transition to the A7 structure.
55
We therefore estimate the anharmonic phonon frequencies for
theRmode to determine if, theoretically, the sc structure might
be stable at these lower pressures. The results for polarizationin the [100] and [111] directions are given in Fig. 3for various
pressures.
At large pressures, anharmonic corrections are small. As
the pressure decreases, the anharmonicity of the Rmode
increases. The anharmonic phonon frequencies are real downto a theoretical pressure of 5 GPa and the Rmode does not go
completely soft. Therefore, the sc structure may remain stableto pressures lower than what is suggested by considering onlythe harmonic approximation. The transition to the A7 structure
is considered further in Sec. III B.
We now present results for e-pcoupling in sc P. We restrict
ourselves to the harmonic approximation, and therefore onlyconsider pressures from 20 to 70 GPa. The Eliashberg spectralfunction α
2F(ω) is shown in Fig. 4(bottom) for several
pressures. The shape of α2F(ω) is very similar to that of F(ω),
with somewhat enhanced weight at very low frequencies andFIG. 3. (Color online) Phonon frequencies for Rphonon mode
as a function of pressure. The anharmonic frequencies are estimatedfor polarizations along the [111] and [100] directions.
high frequencies, as compared to midrange frequencies. The
shift of α2F(ω) to higher frequencies as pressure is increased
follows the trend for F(ω) and leads to lower integrated λ
values. In the spectral function for 70 GPa one can observe thelarger coupling at low phonon energies relative to the curvesfor lower pressures. This enhanced coupling is related to thesoftening of the transverse modes that is seen in the phonondispersion in Fig. 2.
The wave-vector-resolved e-pcouplings λ
q=/summationtext
νλqνal-
low us to examine which modes contribute most strongly to theoverall e-pcoupling. Figure 5shows λ
q(top) and the nesting
function ξq(bottom) along several high-symmetry lines in the
BZ for several pressures. The nesting function is defined as
ξq=/summationdisplay
mn/summationdisplay
kwkδ(/epsilon1m,k+q−/epsilon1F)δ(/epsilon1n,k−/epsilon1F)( 9 )
FIG. 4. (Color online) Phonon density of states F(ω) (top),
Eliashberg spectral function α2F(ω) (bottom, solid), and integrated
λ(bottom, dashed) for sc P for a number of pressures.
064517-4PRESSURE DEPENDENCE OF SUPERCONDUCTIVITY IN ... PHYSICAL REVIEW B 88, 064517 (2013)
FIG. 5. (Color online) Electron-phonon coupling constant λq
(top) and nesting function ξq(bottom) along high-symmetry lines
in the Brillouin zone for sc P at various pressures. Values for λqatR
are 57, 6.0, and 2.8 for 20, 40, and 70 GPa, respectively.
[compare to Eq. (2)] and describes the phase space for
scattering across the Fermi surface. It can be seen that thecoupling at the softened mode Ris very strong, but that
there are also other strongly coupled regions throughout theBrillouin zone. The large coupling at RandXcan partially
be explained by large Fermi-surface nesting at these wavevectors. Wave vectors approximately midway between Rand
/Gamma1and between XandRshow large coupling and large
nesting as well. Interestingly, peaks in λ
qcan be seen at
specific wave vectors between /Gamma1andXandMand/Gamma1that
increase with pressure, contrary to the overall trend in λq.
Little or no corresponding enhancement of ξqcan be seen at
these wave vectors. However, these peaks correspond to wavevectors at which phonon softening occurs at 70 GPa (Fig. 2)
and are related to the aforementioned instability of the scstructure at higher pressures. These phonon modes contributeto the slightly enhanced coupling around 100 cm
−1seen in
theα2F(ω).
Frequency moments of α2F(ω) and the total e-pcoupling
parameter λ, along with N(/epsilon1F), are given in Table IIfor the
pressures calculated in this work. To facilitate comparison withprevious studies, we also include values for /angbracketleftg
2/angbracketright, the average
over the Fermi surface of the squared e-pmatrix elements [note
that/angbracketleftg2/angbracketrightdoes not include the factor (¯ h/2Mω qν)1/2present in
Eq.(1)]. These quantities are related by25
λ=N(/epsilon1F)/angbracketleftg2/angbracketright
M/angbracketleftω2/angbracketright. (10)
In Eq. (10),N(/epsilon1F) is for a single spin (i.e., one-half of the
value given in Table II, which is for both spins). Taking the
natural logarithm of the quantities in Eq. (10) and normalizing
to their values at 20 GPa, we have
lnλ
λ20=lnN(/epsilon1F)
N(/epsilon1F)20+ln/angbracketleftg2/angbracketright
/angbracketleftg2/angbracketright20+ln/angbracketleftω2/angbracketright20
/angbracketleftω2/angbracketright, (11)
where the subscript 20 denotes the value at 20 GPa. The results
as a function of pressure are plotted in Fig. 6.
ln
FIG. 6. (Color online) Trends in average phonon frequency /angbracketleftω2/angbracketright,
N(/epsilon1F), average e-pmatrix element /angbracketleftg2/angbracketright,a n de-pcoupling λas a
function of pressure. The subscript “20” denotes the value at 20 GPa.
We find that the contribution of N(/epsilon1F)t oλincreases
slightly from 20 to 30 GPa, and then remains almost constantas pressure increases. The contribution from the matrixelements does increase significantly with pressure. However,the contribution from increasing phonon frequencies is largerand results in a net decrease in λas pressure increases.
The magnitude and pressure trend of N(/epsilon1
F) agrees well
with previous studies.19,20,22The increase in average phonon
frequency with increasing pressure is also in agreementwith previous studies that consider this pressure trend.
20,22
However, our average phonon frequencies are significantly
larger in magnitude for similar pressures ( ∼450–500 K for
our calculations versus ∼350–400 K for other works). We
believe our calculations to be more accurate, since we havecalculated the full phonon dispersion from first principles,while previous studies used estimates. Likewise, the trendsin/angbracketleftg
2/angbracketrightwith pressure agree with previous studies, but the
magnitude of our values are roughly twice as large as thosein other works. The origin of this difference is unclear, butmay be related to the difference in calculational methods;previous studies used augmented plane-wave or muffin-tinorbital methods. Thus, the rough agreement in magnitude of
λbetween the present study and previous studies is somewhat
fortuitous.
Table IIalso shows T
ccalculated using the McMillan
equation [Eq. (8)]. The Coulomb pseudopotential parameter
μ∗is determined by using a modified Bennemann-Garland
relation56–58which relates μ∗toN(/epsilon1F):
μ∗=CN(/epsilon1F)
1+N(/epsilon1F). (12)
SinceN(/epsilon1F) varies little with pressure, μ∗is almost constant.
The constant Cis obtained by matching the calculated Tcto
that of experiment at one pressure. We use the experimentalresults for T
cof Karuzawa et al.18because they extend to
higher pressures than other studies.
Because, for a given pressure, the volume calculated using
the LDA differs from that of experiment, we have chosen
064517-5KEVIN T. CHAN, BRAD D. MALONE, AND MARVIN L. COHEN PHYSICAL REVIEW B 88, 064517 (2013)
FIG. 7. (Color online) Calculated Tcas a function of pressure
compared to the experimental results of Karuzawa et al. (Ref. 18). For
the calculations, pressures are shifted to match experiment, accordingto Table I.
to shift the calculated pressure Pcalcto an “experimental”
pressure Pexpt1 orPexpt2 (as given in Table Iand described
in Sec. III A ), for the purposes of comparison. This procedure
is equivalent to comparing theory and experiment at the samevolume . While this procedure is not rigorously justified, it
gave good agreement between experiment and theory forAs under pressure.
23A similar shift was used in studying
superconductivity in Al and Li under pressure.51We note that
if this procedure for shifting the pressure is not used, there areonly small quantitative differences in the results.
With this pressure shift, we find that setting μ
∗to 0.18
forPcalc=25 GPa allows us to match our results to the
experimental maximum Tcat around 32 GPa; hence C=0.79
in Eq. (12).
Aμ∗of 0.18 is somewhat larger than the generally
accepted value ( ∼0.13) for conventional superconductors.25
Some studies indicate that such a large μ∗is applicable
for Li,51,53,59so it is possible that the same is true for
P. Alternatively, one could question whether it is valid touse the Eliashberg/McMillan formalism with the Coulombpseudopotential or whether some alternate theory is required.
The calculated and experimental pressure dependence of
T
cis shown in Fig. 7. The calculations showing a decrease in
Tcwith increasing pressure are in good agreement with the
experimental data for pressures above 30 GPa (correspondingtoP
calc≈25–30 GPa). However, the calculated Tccontinues
to increase with decreasing pressure below 30 GPa; this resultis in disagreement with the experimental data, which shows adrop in T
cat lower pressures.
The qualitative behavior of Tc(P) above 30 GPa can be
understood by looking at Table IIand Eq. (8). While ωln
increases with pressure, λdecreases with pressure, so the
overall Tcdecreases with pressure. For the calculations, these
trends extend below 30 GPa. The disagreement betweencalculations and experiment raises questions about whetherother physical effects are occurring in experiment that are notaccounted for in the calculation.B.A7 to sc transition
The stability of P in the sc structure below a theoretical
pressure of 20 GPa was called into question by the soft R
phonon mode calculated in the harmonic approximation. Wetherefore consider whether the A7 structure is more stable than
the sc one in this pressure range. Similar theoretical studies oftheA7 to sc transition in As have been performed.
23,60–63Our
calculated structural parameters for A7 P—the rhombohedral
lattice constant arhom, rhombohedral angle α, and internal
parameter u, as well as nearest-neighbor ( d1) and next-nearest-
neighbor ( d2) distances—are given in Fig. 8.T h er e l a x e d
structure is sc for pressures of 20 GPa and above, as indicatedbyα=60
◦,u=0.25, and d1=d2. For pressures below
20 GPa, u< 0.25; this result is consistent with the imaginary
frequency for the Rmode found from the phonon calculations
in the harmonic approximation. In this pressure range, wecalculated the enthalpy for the sc phase and verified that it ishigher than the enthalpy for the relaxed A7 phase, showing
that the A7 phase is indeed favored.
Interestingly, the results indicate that there are two tran-
sitions that are well separated in pressure. Starting at lowpressures in the A7 structure, in the first transition αbecomes
close to 60
◦between 0 and 5 GPa, while uincreases only
incrementally and is far from the cubic value of 0 .25;α
remains close to 60◦above 5 GPa. In the second transition,
ureaches 0.25 at a pressure between 15 and 20 GPa. The
fact that a significant displacement uaway from 0.25 induces
only a small change in αfor pressures between 5 and 15 GPa is
surprising. By the symmetry of the sc structure, a displacementof the atoms with wave vector Rmust induce a rhombohedral
distortion of the unit cell. Our results do not violate symmetryconsiderations, as α/negationslash=60
◦below 20 GPa, but the fact that α
is so close to 60◦is unexpected.
In these calculations, the true A7 to sc transition occurs
between 15 and 20 GPa (above 20 GPa when shifted to thecorresponding experimental pressure), which is significantlyhigher than in experiment.
11Possible reasons for this discrep-
ancy are discussed in Sec. IV.
IV . DISCUSSION
A. Comparison of e-pcoupling in P and As
It is interesting to compare the case of P to that of As. Like
P, As has five valence electrons per atom and undergoes anA7 to sc structural transition as pressure is increased.
64,65The
experimentally measured Tcas a function of pressure for As
(Ref. 55) has a peak structure similar to several experimental
results for P.17,18The electronic and phononic structure for sc
As23is similar to that of P; in particular, the Fermi surfaces
for valence bands 2 and 3 (Fig. 1) have similar shapes.20,47,66
There are also important experimental differences between
P and As. The A7 to sc transition pressure for P is around
10 GPa at room temperature, which is significantly lower thanthat for As, which has been measured to be 24–32 GPa.
64,65
The pressure at which As reaches a peak in superconducting
Tcmatches the pressure of the A7 to sc transition; this
correspondence does not appear to be true for P. Furthermore,the maximum T
cfor P ( ∼10 K) is four times higher than that
for As (2.4 K).
064517-6PRESSURE DEPENDENCE OF SUPERCONDUCTIVITY IN ... PHYSICAL REVIEW B 88, 064517 (2013)
FIG. 8. (Color online) Lattice parameters (a) arhom,( b )α,a n d
(c)u, and (d) nearest-neighbor d1and next-nearest neighbor d2
distances for variable-cell relaxation calculations of P in the A7
structure with target pressures between 0 and 25 GPa.
A comparison can be made between the calculated fre-
quency moments, N(/epsilon1F), and λfor As in Ref. 23and for P in
the present work. We compare several quantities for sc As andP at pressures for which the calculated T
cis maximal for each
material. For As at a calculated pressure of 30 GPa, λ=0.50,
N(/epsilon1F)=0.290 states /eV atom, /angbracketleftω2/angbracketright1/2=284 K, and ωln=
253 K. The corresponding quantities for P at 20 GPa are givenin Table II. The atomic mass of P is 30.97, while that of Asis 74.92. With reference to Eq. (10), the P to As ratios of the
quantities λ,N(/epsilon1
F),/angbracketleftg2/angbracketright, and 1 /M/angbracketleftω2/angbracketrightare 1.59, 1.01, 1.63,
and 0.97, respectively. Thus, the difference in λbetween P and
As at these specific pressures is mostly due to differences in/angbracketleftg
2/angbracketright. Additionally, the P to As ratio of ωlnis 1.65.
We can conclude that the larger ωlnand larger matrix
elements contribute to a larger maximum Tcin P as compared
with As. One should note that the μ∗≈0.12 used in Ref. 23
is somewhat smaller than that needed in the present work(μ
∗≈0.18) to match Tcto experiment; the origin of this
difference is unclear.
Comparing the trends as a function of pressure, both P
and As in the sc structure have decreasing λwith increasing
pressure, due mainly to the increase in phonon frequencies.Both elements have increasing /angbracketleftg
2/angbracketrightwith increasing pressure
in the sc structure.
B. Comparison to experiment: Structure, e-pcoupling, and Tc
Our calculations can explain the decrease in Tcwith increas-
ing pressure above 30 GPa observed by Karuzawa et al.18in sc
P as coming from the increase in phonon frequencies, whichdecreases λ. A similar physical effect occurs in As.
23However,
the decrease in Tcwith decreasing pressure below 30 GPa
remains unexplained by our calculations. Nor do our resultsindicate a constant T
cwith pressure, as observed by Kawamura
et al. ,15,16nor any two-peak structure, as observed by Wittig
et al.17In addition, our calculated A7 to sc transition pressure
is significantly higher than what is found experimentally atroom temperature. In this section, we suggest several possibleexplanations for the discrepancy between experiment andcalculation which may serve as guide for future studies.
We consider the question of the structure of P as a function
of pressure. Experimentally, the A7 to sc transition pressure
is around 10 GPa at both room temperature
10,11and higher
temperatures,67,68and the pressure region of coexistence of
the two phases is fairly narrow. At lower temperatures thetransition pressure increases, and the region of coexistencebroadens (12–15.5 GPa at 21 K).
69–71Thus there is some
experimental uncertainty about the actual lowest free-energystructure at a given pressure in this range at low temperature.
Our calculations determine whether A7 or sc has the lowest
enthalpy structure at zero temperature at specified pressures.According to our calculations, the true A7 to sc transition,
whenαreaches 60
◦andureaches 0.25, occurs at a calculated
pressure above 15 GPa, corresponding to an experimentalpressure of about 20–23 GPa (see Table I). Such a pressure is
higher than any experimentally measured transition pressure.
Several possible explanations for the difference between
calculation and experiment are presented here. First, ourcalculations do not include the zero-point energy (ZPE) orfinite-temperature effects. If the fully anharmonic ZPE isincluded, the sc structure may in fact be stable below thecalculated pressure of 15 GPa, and possibly down to acalculated pressure of 5 GPa, corresponding to about 13 GPa inexperiment, for which our calculations show that αis still close
to 60
◦. Such a scenario would resolve much of the discrepancy
with experiment.
Another possibility is that an A7 structure, with αclose
to but not equal to 60◦, andu/negationslash=0.25 actually remains stable
064517-7KEVIN T. CHAN, BRAD D. MALONE, AND MARVIN L. COHEN PHYSICAL REVIEW B 88, 064517 (2013)
FIG. 9. (Color online) Simulated x-ray-diffraction spectra for
relaxed A7 and sc structures at various pressures. The 10.4 and 11.0
(hkl) diffraction lines discussed in the text are highlighted for clarity.
These lines appear to merge before the transition to the sc phase (withu=0.25) is completed.
to pressures higher than what is quoted in the experimental
works. Experimentally, the A7 to sc transition is determined
by observation of the merging of diffraction lines 10.4 and11.0 (hkl), which occur around 2 /Theta1≈24
◦, and the uparameter
is not determined accurately.11We simulate x-ray-diffraction
(XRD) spectra using the PLATON crystallographic tool72with
MoKαradiation and find that for our relaxed structures at
5–15 GPa, with αclose to 60◦andu/negationslash=0.25, the spectra
look very close to spectra for the sc structure (Fig. 9). In
particular, the merging of the 10.4 and 11.0 diffraction linesoccurs already at 5 GPa (see highlighted region in Fig. 9),
when the A7 structure is lower in energy and the uparameter
is still relatively far from its cubic value of 0.25. Experimentalbroadening and background may make the two structuresdifficult to distinguish with this method.
It is also possible that use of the GGA or another functional
for exchange and correlation, instead of the LDA, wouldgive better agreement with experiment for the structure asa function of pressure. However, in the case of As, the A7
to sc transition pressure is higher for the GGA than for the
LDA.
63If the same holds true for P, then the GGA will
be in further disagreement with experiment than the LDA.Experimental studies that determine uaccurately as a function
of pressure, and theoretical determination of the structureas a function of pressure, including ZPE and temperatureeffects and considering different functionals, would be usefulin resolving this issue.
We now turn to the question of why, experimentally, T
c
decreases with decreasing pressure below the peak. From a
theoretical perspective, this question cannot be answered with-out first resolving the structure at these pressures. However, wecan speculate based on the possible answers to the structural
question.
If the structure is indeed sc down to an experimental pres-
sure of ∼10 GPa, then the explanation of the T
cversus pressure
trend may require going beyond the approximations made inthe present study. It is possible that the full anharmonicity ofthe phonons must be considered, as was done for MgB
2.5,73,74
As shown in the present study, anharmonicity raises the R
phonon frequency significantly; if it raises the overall phononfrequencies throughout the BZ, it could reduce λ, leading to
al o w e r T
c.5Multiple-phonon processes leading to nonlinear
terms in the e-pcoupling might also be important.73,74Also,
inclusion of zero-point motion may have some effect on theelectronic structure and N(/epsilon1
F), which could affect λ.I ti sa l s o
possible that one may need to go beyond the isotropic approx-imation to Eliashberg theory
75assumed in the present work.
If the structure is in fact A7 at these pressures, then the
decrease in Tccould be explained by a decrease in N(/epsilon1F)a s
theA7 structure becomes more and more distorted from the
sc one as pressure is lowered. A similar mechanism occurs forAs and was elucidated in previous works.
23,55
Finally, we note an interesting observation that the two
structural transitions that we have found from our relaxationcalculations, occurring between 0 and 5 GPa and 15 and20 GPa, correspond well with the two T
cpeaks at 12 and
23 GPa observed by Wittig et al. ,17when the pressure shift
due to the LDA is accounted for. It would be interesting toinvestigate whether this observation has physical significance.
V . CONCLUSION
In this study, we calculate the e-pcoupling and Tcfor
phosphorus in the sc phase over the pressure range 20–70 GPa.Unlike prior theoretical results, our calculations explicitlytreat the pressure dependence of the lattice dynamics. Thedecrease in T
cabove 30 GPa is in good agreement with the
experimental results of Karuzawa et al.18and can be explained
as coming from the increase in phonon frequencies. Thedecrease in T
cas pressure is decreased below 30 GPa in sc
P remains unexplained, but we suggest several possible routesto understanding this puzzle. Calculations of the A7 and sc
structures in P reveal an interesting two-step transition whichmerits further study.
ACKNOWLEDGMENTS
This work was supported by National Science Foundation
Grant No. DMR10-1006184 and by the Director, Office ofScience, Office of Basic Energy Sciences, Materials Sciencesand Engineering Division, U.S. Department of Energy underContract No. DE-AC02-05CH11231. Computational supportwas provided by NSF through XSEDE resources at NICS andby DOE at Lawrence Berkeley National Laboratory’s NERSCfacility.
1H. Kamerlingh Onnes, Commun. Phys. Lab. Univ. Leiden 120b
(1911), reprinted in Proc. K. Ned. Akad. Wet. 13, 1274 (1911).
2Y . Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.
Chem. Soc. 130, 3296 (2008).3K. Ishida, Y . Nakai, and H. Hosono, J. Phys. Soc. Jpn. 78, 052001
(2009).
4J. Nagamatsu, N. Nakagawa, T. Muranaka, Y . Zenitani, andJ. Akimitsu, Nature (London) 410, 63 (2001).
064517-8PRESSURE DEPENDENCE OF SUPERCONDUCTIVITY IN ... PHYSICAL REVIEW B 88, 064517 (2013)
5H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G. Louie, Phys.
Rev. B 66, 020513 (2002).
6E .A .E k i m o v ,V .A .S i d o r o v ,E .D .B a u e r ,N .N .M e l ’ n i k ,N .J .
Curro, J. D. Thompson, and S. M. Stishov, Nature (London) 428,
542 (2004).
7E. Bustarret, C. Marcenat, P. Achatz, J. Ka ˇcmar ˇcik, F. L ´evy,
A. Huxley, L. Ort ´ega, E. Bourgeois, X. Blase, D. D ´ebarre, and
J. Boulmer, Nature (London) 444, 465 (2006).
8K. Shimizu, H. Ishikawa, D. Takao, T. Yagi, and K. Amaya, Nature
(London) 419, 597 (2002).
9C. Buzea and K. Robbie, Supercond. Sci. Technol. 18, R1 (2005).
10J. C. Jamieson, Science 139, 1291 (1963).
11T. Kikegawa and H. Iwasaki, Acta Crystallogr. Sect. B: Struct. Sci.
39, 158 (1983).
12Y . Akahama, M. Kobayashi, and H. Kawamura, P h y s .R e v .B 59,
8520 (1999).
13J. Wittig and B. T. Matthias, Science 160, 994 (1968).
14I. V . Berman and N. B. Brandt, Pis’ma Zh. Eksp Teor. Fiz. 7, 412
(1968) [JETP Lett. 7, 326 (1968)].
15H. Kawamura, I. Shirotani, and K. Tachikawa, Solid State
Commun. 49, 879 (1984).
16H. Kawamura, I. Shirotani, and K. Tachikawa, Solid State
Commun. 54, 775 (1985).
17J. Wittig, B. Bireckoven, and T. Weidlich, in Solid State Physics
Under Pressure , edited by S. Minomura (KTK Scientific, Tokyo,
1985), p. 127.
18M. Karuzawa, M. Ishizuka, and S. Endo, J. Phys.: Condens. Matter
14, 10759 (2002).
19M. Rajagopalan, M. Alouani, and N. E. Christensen, J. Low Temp.
Phys. 75, 1 (1989).
20M. Aoki, N. Suzuki, and K. Motizuki, J. Phys. Soc. Jpn. 56, 3253
(1987).
21H. Nagara, K. Mukose, T. Ishikawa, M. Geshi, and N. Suzuki,J. Phys.: Conf. Series 215, 012107 (2010).
22L. W. Nixon, Ph.D. thesis, George Mason University, 2010.
23K. T. Chan, B. D. Malone, and M. L. Cohen, P h y s .R e v .B 86,
094515 (2012).
24P. B. Allen, P h y s .R e v .B 6, 2577 (1972).
25W. L. McMillan, Phys. Rev. 167, 331 (1968).
26P. B. Allen and R. C. Dynes, Phys. Rev. B 12, 905 (1975).
27P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962).
28P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,
C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni,I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi,R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri,L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini,A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero,A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch,J. Phys.: Condens. Matter 21, 395502 (2009).
29P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
30W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
31J. Ihm, A. Zunger, and M. L. Cohen, J. Phys. C 12, 4409 (1979).
32M. L. Cohen, Phys. Scr. T1, 5 (1982).
33D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
34J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
35N. Troullier and J. L. Martins, P h y s .R e v .B 43, 1993 (1991).
36M. J. T. Oliveira and F. Nogueira, Comput. Phys. Commun. 178,
524 (2008).
37S. G. Louie, S. Froyen, and M. L. Cohen, P h y s .R e v .B 26, 1738
(1982).38H. J. Monkhorst and J. D. Pack, P h y s .R e v .B 13, 5188 (1976).
39N. Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne, Phys. Rev.
Lett. 82, 3296 (1999).
40F. Giustino, M. L. Cohen, and S. G. Louie, P h y s .R e v .B 76, 165108
(2007).
41J. Noffsinger, F. Giustino, B. D. Malone, C.-H. Park, S. G.Louie, and M. L. Cohen, Comput. Phys. Commun. 181, 2140
(2010).
42N. Marzari and D. Vanderbilt, P h y s .R e v .B 56, 12847 (1997).
43I. Souza, N. Marzari, and D. Vanderbilt, P h y s .R e v .B 65, 035109
(2001).
44A. A. Mostofi, J. R. Yates, Y .-S. Lee, I. Souza, D. Vanderbilt, andN. Marzari, Comput. Phys. Commun. 178, 685 (2008).
45S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev.
Mod. Phys. 73, 515 (2001).
46F. Birch, Phys. Rev. 71, 809 (1947).
47T. Sasaki, K. Shindo, K. Niizeki, and A. Morita, J. Phys. Soc. Jpn.
57, 978 (1988).
48A. Nishikawa, K. Niizeki, and K. Shindo, Phys. Status Solidi B
223, 189 (2001).
49R. Ahuja, Phys. Status Solidi B 235, 282 (2003).
50D. Kasinathan, J. Kune ˇs, A. Lazicki, H. Rosner, C. S. Yoo,
R. T. Scalettar, and W. E. Pickett, Phys. Rev. Lett. 96, 047004
(2006).
51G. Profeta, C. Franchini, N. N. Lathiotakis, A. Floris, A. Sanna,M. A. L. Marques, M. L ¨u d e r s ,S .M a s s i d d a ,E .K .U .G r o s s ,a n d
A. Continenza, P h y s .R e v .L e t t . 96, 047003 (2006).
52T. Bazhirov, J. Noffsinger, and M. L. Cohen, Phys. Rev. B 82,
184509 (2010).
53T. Bazhirov, J. Noffsinger, and M. L. Cohen, Phys. Rev. B 84,
125122 (2011).
54A. S. Mikhaylushkin, S. I. Simak, B. Johansson, andU. H ¨aussermann, P h y s .R e v .B 76, 092103 (2007).
55A. L. Chen, S. P. Lewis, Z. Su, P. Y . Yu, and M. L. Cohen, Phys.
Rev. B 46, 5523 (1992).
56K. H. Bennemann and J. W. Garland, Superconductivity in d- and
f- Band Metals, edited by H. C. Wolfe and D. H. Douglass, AIP
Conf. Proc. No. 4 (AIP, New York, 1972), p. 103.
57J. W. Garland and K. H. Bennemann, Superconductivity in d- and
f- Band Metals, AIP Conf. Proc. No. 4 (Ref. 56), p. 255.
58D. Papaconstantopoulos and B. Klein, Physica B +C107, 725
(1981).
59C. F. Richardson and N. W. Ashcroft, P h y s .R e v .B 55, 15130
(1997).
60C. R. da Silva and R. M. Wentzcovitch, Comput. Mater. Sci. 8, 219
(1997).
61M. Durandurdu, Phys. Rev. B 72, 073208 (2005).
62W. Feng, S. Cui, H. Hu, and H. Liu, Physica B: Condens. Matter
400, 22 (2007).
63P. Silas, J. R. Yates, and P. D. Haynes, P h y s .R e v .B 78, 174101
(2008).
64T. Kikegawa and H. Iwasaki, J. Phys. Soc. Jpn. 56, 3417
(1987).
65H. J. Beister, K. Str ¨ossner, and K. Syassen, P h y s .R e v .B 41, 5535
(1990).
66S. Shang, Y . Wang, H. Zhang, and Z.-K. Liu, Phys. Rev. B 76,
052301 (2007).
67H. Iwasaki, T. Kikegawa, T. Fujimura, S. Endo, Y . Akahama,T. Akai, O. Shimomura, S. Yamaoka, T. Yagi, S. Akimoto, andI. Shirotani, Physica B +C139–140 , 301 (1986).
064517-9KEVIN T. CHAN, BRAD D. MALONE, AND MARVIN L. COHEN PHYSICAL REVIEW B 88, 064517 (2013)
68T. Kikegawa, H. Iwasaki, T. Fujimura, S. Endo, Y . Akahama,
T. Akai, O. Shimomura, T. Yagi, S. Akimoto, and I. Shirotani,J. Appl. Crystallogr. 20, 406 (1987).
69I. Shirotani, H. Kawamura, K. Tsuji, K. Tsuburaya, O. Shimomura,
and K. Tachikawa, Bull. Chem. Soc. Jpn. 61, 211 (1988).
70I. Shirotani, K. Tsuji, M. Imai, H. Kawamura, O. Shimomura,
T. Kikegawa, and T. Nakajima, Phys. Lett. A 144, 102 (1990).
71I. Shirotani, J. Mikami, T. Adachi, Y . Katayama, K. Tsuji,
H. Kawamura, O. Shimomura, and T. Nakajima, Phys. Rev. B 50,
16274 (1994).72A. L. Spek, Acta Crystallogr. Sec. D 65, 148 (2009).
73T. Yildirim, O. G ¨ulseren, J. W. Lynn, C. M. Brown, T. J. Udovic,
Q. Huang, N. Rogado, K. A. Regan, M. A. Hayward, J. S. Slusky,T. He, M. K. Haas, P. Khalifah, K. Inumaru, and R. J. Cava, Phys.
Rev. Lett. 87, 037001 (2001).
74A. Y . Liu, I. I. Mazin, and J. Kortus, Phys. Rev. Lett. 87, 087005
(2001).
75P. B. Allen and B. Mitrovi ´c, in Solid State Physics , edited by
H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York,1983), V ol. 37, pp. 1–92.
064517-10 |
PhysRevB.104.045406.pdf | PHYSICAL REVIEW B 104, 045406 (2021)
Atomic manipulation of in-gap states in the β-Bi2Pd superconductor
Cristina Mier ,1,*Jiyoon Hwang,2,3,*Jinkyung Kim,2,3Yujeong Bae,2,3Fuyuki Nabeshima,4Yoshinori Imai,4,5
Atsutaka Maeda,4Nicolás Lorente,1,6,†Andreas Heinrich ,2,3,‡and Deung-Jang Choi1,6,7,§
1Centro de Física de Materiales, MPC (CSIC-UPV /EHU), 20018 Donostia-San Sebastián, Spain
2Center for Quantum Nanoscience, Institute for Basic Science, Seoul 03760, South Korea
3Department of Physics, Ewha Womans University, Seoul 03760, South Korea
4Department of Basic Science, University of Tokyo, Meguro, Tokyo 153-8902, Japan
5Department of Physics, Graduate School of Science, Tohoku University, Sendai, Miyagi 980-8578, Japan
6Donostia International Physics Center, 20018 Donostia-San Sebastián, Spain
7Ikerbasque, Basque Foundation for Science, 48013 Bilbao, Spain
(Received 6 May 2021; accepted 25 June 2021; published 6 July 2021)
Electronic states in the gap of a superconductor inherit intriguing many-body properties from the supercon-
ductor. Here, we create these in-gap states by manipulating Cr atomic chains on the β-Bi 2Pd superconductor.
We find that the topological properties of the in-gap states can greatly vary depending on the crafted spin chain.These systems make an ideal platform for nontrivial topological phases because of the large atom-superconductorinteractions and the existence of a large Rashba coupling at the Bi-terminated surface. We study two spin chains,one with atoms two lattice parameters apart and one with√
2 lattice parameters. Of these, only the second one
is in a topologically nontrivial phase, in agreement with the spin interactions for this geometry.
DOI: 10.1103/PhysRevB.104.045406
I. INTRODUCTION
The scanning tunneling microscope (STM) permits un-
precedented control at the atomic level [ 1]. Since the early
days of STMs, atoms have been moved, unveiling matter onthe atomic scale [ 2–6]. Atoms involve interactions that can
have a profound impact on the electronic properties of hostsubstrates; as such, designing atomic structures can lead tocreating new quantum states [ 7]. Magnetic atoms strongly
modify the low-energy electronic properties of superconduc-tors. This modification is due to the appearance of in-gap
states caused by the weakening of the Cooper-pair bind-ing. These in-gap states are usually called Yu-Shiba-Rusinov(YSR) states [ 8–10]. Recently, interest in in-gap states has
increased due to the suggestion of topological edge states ap-
pearing on chains of magnetic impurities on superconductors[11–18]. These zero-energy edge states imply the presence of
a topological superconducting phase. The zero-energy edgestates are Majorana bound states (MBSs) with nontrivial ex-change transformations. Braiding of MBSs is at the core ofcurrent proposals regarding topological quantum computation[19,20].
The STM has become a major tool in the study of MBSs
[14–17,21,22]. Indeed, its spectroscopic capabilities render it
unique for revealing in-gap states, granting access to unrivaledenergy and space resolutions. Recently, the spatial distribution
*These authors contributed equally to this work.
†nicolas.lorente@ehu.eus
‡heinrich.andreas@qns.science
§djchoi@dipc.orgof in-gap states was shown and used to infer new proper-ties of the states themselves [ 23–25]. The aforementioned
STM manipulation can be used to create atomically precisespin chains on superconductors [ 17,18,26]. The new in-gap
states evolve into bands and open gaps displaying new formsof superconductivity [ 11–13,27]. This evolution proves the
complexity of the induced electronic structure. Each addedimpurity locally creates a few states in the superconductinggap. As the number of impurities grows, the gap fillsup with
new quasiparticle states.
The study of impurity dimers illustrates the initial steps
of in-gap bands [ 28–33]. The quasiparticle states themselves
are difficult to describe. In the Bogoliubov–de Gennes ap-proximation, the quasiparticle states are taken as electron andhole superpositions despite violating particle-number conser-vation. Furthermore, the quasiparticle states are spin polarized[34,35], which has important implications for the way the
in-gap states hybridize [ 28]. In particular, the resulting states
reflect the spin ordering of the magnetic impurities [ 31].
However, recent work suggests that in the presence of strongRashba coupling, it is difficult to make a conclusion about theactual spin orientation of the impurities by studying the in-gapstates [ 32].
Here, we study atomic spin chains of Cr adsorbed on the
hollow sites of β-Bi
2Pd and grown along the two main surface
directions, /angbracketleft100/angbracketrightand/angbracketleft110/angbracketrightfor the Bi-terminated [001] sur-
face, using a home-built dilution fridge STM [ 36]. By doing
so, we are choosing two starkly different spin orientationsfor the chain ground state, as concluded in Ref. [ 31]. Dimers
along the /angbracketleft100/angbracketrightdirection with a Cr-Cr distance of two unit
cells (2 a, where ais the surface lattice parameter) present
antiferromagnetic (AF) coupling of their 5 μ
Bmagnetic
2469-9950/2021/104(4)/045406(10) 045406-1 ©2021 American Physical SocietyCRISTINA MIER et al. PHYSICAL REVIEW B 104, 045406 (2021)
moments [ 31]. Dimers along the /angbracketleft110/angbracketrightdirection are√
2a
apart, and they are instead ferromagnetically (FM) coupled.Here, we compare dimers, trimers, and tetramers of these twotypes of chains and conclude that the√
2a−/angbracketleft110/angbracketrightchains
are indeed FM coupled by comparing then with model cal-culations of spin chains solving the Bogoliubov–de Gennesequations [ 28,31]. As clearly seen in this work, the gap closes
rapidly for the√
2a−/angbracketleft110/angbracketrightchains; however, the 2 a−/angbracketleft100/angbracketright
chains maintain an almost constant gap for chains as long
as 12 Cr atoms. This difference has important implicationsfor the possibility of engineering topological phases on theβ-Bi
2Pd superconductor.
II. METHODS
A. Sample preparation and STM characterization
Theβ-Bi2Pd crystal was fabricated by the method written
in Ref. [ 37]. The chosen sample showed a Tcof 5.2 K. The
Bi-terminated surface of the β-Bi2Pd crystal was prepared
by cleavage in situ [31]. Cr atoms were deposited onto a
precooled β-Bi2Pd surface at a temperature T/lessorequalslant20 K to
have single isolated atoms. The experimental data were takenusing a home-built dilution fridge STM at T=30 mK and in
ultrahigh vacuum at the IBS Center for Quantum Nanoscience[36]. The very low temperature leads to a negligible thermal
smearing granting a resolution higher than the one obtainedby a superconducting tip [ 38–40]. We used a metallic PtIr tip
that permitted us to use the differential conductance dI/dVas
a direct measurement of the density of states of the substrate(refer to the Supplemental Material [ 41] for more details). The
conductance was measured using a lock-in amplifier with anAC modulation bias of 30 μV and a frequency of 330 Hz.
Lateral atomic manipulation was achieved by approaching
one side of a selected atom with the STM tip to reach junctionresistances on the order of a few tens of kilohms (typically, 3mV and tens of nanoamperes). Then the STM tip was laterallymoved to drag the atom to a desired position with the feedbackloop open.
B. Theory
We model the Cr spin chain in the dilute spin chain limit
[13] because density-functional-theory (DFT) calculations
show that no Cr dstates lie at the Fermi energy, preventing
charge transfer processes [ 31]. In this framework, we solve
a spin chain using Green’s functions for the superconductorin the Nambu basis set [ 42,43]. We add a Rashba term to
the Hamiltonian expressed in the local basis. The resultingdensity of states corresponds to the Bogoliubov–de Gennesstates of a BCS superconductor in the presence of an array ofclassical spins and subject to the strong Rashba interaction ofthe Bi-terminated surface.
The Fermi velocity entering the superconductor’s Green’s
function [ 42,43] is taken to be 0.15 (Hartree atomic units
¯h=m=e=1), and the Dynes parameter [ 44] control-
ling the width of the superconducting quasiparticle peaksis 0.05 meV . The small Dynes parameter leads to peaksin the density of states (DOS) sharper than the experi-mental ones but helps with the visualization of the evo-lution of in-gap states with the number of Cr atoms.β-Bi
2Pd is an s-wave superconductor that can be well
accounted for by a single gap [ 37,45]/Delta1=0.76 meV . For
the normal metal DOS, we use N=0.037/eV, which is 5
times larger than the corresponding Nfor a free-electron metal
with a Fermi velocity of 0.15 a.u., in order to capture the fiveelectrons of the Bi valence shell. The Hamiltonian taking intoaccount the superconductor is
ˆH
BCS=ξkτ3σ0+/Delta1τ2σ2, (1)
where σi(τi) are the Pauli matrices acting on the spin (par-
ticle) sectors, with σ0(τ0) being the 2 ×2 identity matrix
and the matrix product being a tensor one. ξkis the energy
from the Fermi level ( ξk=/epsilon1k−EF); the previous Hamilto-
nian is written in the four-dimensional Nambu basis: /Psi1=
(ˆc↑,ˆc↓,ˆc†
↑,ˆc†
↓)T.
To model the experimental system, we add the Hamilto-
nian describing the magnetic impurities [ 42,43]. To do this,
we change to a tight-binding basis, assuming a single, verycompact, atomic orbital per site. Additionally, the interactionswith the magnetic impurity are assumed to be strictly localizedto the site where the impurity is sitting [ 13]. The Hamiltonian
is then
ˆH=ˆH
BCS+ˆHimpurity =ˆHBCS+N/summationdisplay
j(Ujτ3σ0+Jj/vectorSj·/vectorα),(2)
with/vectorα=1+τ3
2/vectorσ+1−τ3
2σ2/vectorσσ2, where /vectorσis the spin operator
[9]. This Hamiltonian describes a BCS superconductor and
the interaction between its electrons and Nextra impurities.
The interaction contains an exchange coupling, with strengthJ
j, and a nonmagnetic potential scattering term Ujper im-
purity j. We will use the same impurity species, Cr, and
assume that the impurities are equivalent regardless of theiradsorption site and spin chain in order to study the sys-tem’s evolution with the number of atoms in the spin chains.
/vectorS
j=(Sj,x,Sj,y,Sj,z)=S(sinθjcosφj,sinθjsinφj,cosθj)i s
the spin of atom jconsidered to be a classical spin and hence
not an operator. The local term Ujdescribes a scalar potential
acting on the substrate’s electron. It is responsible for thepotential scattering term produced by the impurity. In the caseof a charged impurity, U
jis mainly given by the Coulomb
interaction between the total charge of the impurity and thecharge of the substrate’s electron. The potential scattering thatexplains the electron-hole asymmetry of the YSR bands istaken as U
j=5.5 eV . The values for the Kondo exchange
coupling Jjare about 2 eV , as estimated from fittings to single-
Cr YSR states [ 31].
The Hamiltonian is completed by a Rashba term:
ˆHRashba=iαR
2a/summationdisplay
i,j,α,β[ˆc†
i+1,j,α(σ2)α,βˆci,j,β
−ˆc†
i,j+1,α(σ1)α,βˆci,j,β+H.c.], (3)
where α,β are spin indexes. The interaction couples spins on
nearest-neighbor sites. The lattice parameter of the substrate isa, and the factor of 2 acomes from a finite-difference scheme
to obtain the above discretized version of the Rashba interac-tion. For the case of β-Bi
2Pd, we use a large Rashba coupling,
αR≈1.8 eV Å, which comes from our DFT calculations and
045406-2ATOMIC MANIPULATION OF IN-GAP STATES IN THE … PHYSICAL REVIEW B 104, 045406 (2021)
-3 -2 -1 0 1 2 30204060)Sn( Vd/Id
Sample bias (mV) Dimer
Trimer
Tetramer-3 -2 -1 0 1 2 30204060)Sn( Vd/Id
Sampe bias (mV) Dimer
Trimer
Tetramer(a) (c)
(b) (d)
FIG. 1. Chromium chains built on the β-Bi 2Pd surface by atomic manipulation. Topographic images of tetramer chains (a)√
2a−/angbracketleft110/angbracketright
and (b) 2 a−/angbracketleft100/angbracketrightunit cells apart (100 mV , 10 pA, 4 ×4n m2). The insets show the atomic geometry of the tetramer nanostructures. The
corresponding differential conductance is measured at the end atom (black dot) from the dimer to trimer to tetramer in tetramer chains (c)√
2
unit cells apart and (d) 2 unit cells apart. T=30 mK; AC modulation bias is 30 μV.
is in agreement with the couplings of Bi-terminated surfaces
[46].
The local or projected DOS (PDOS) is computed over
every local orbital iof the basis using
ρ(i,ω)=−1
πIm/bracketleftbig
G1,1
i,i(ω)+G4,4
i,i(−ω)/bracketrightbig
, (4)
where Gν,μ
iiis the resulting Green’s function evaluated on or-
bital ifor the Nambu components νandμby solving Dyson’s
equation:
ˆG=/bracketleftbigˆG−1
BCS−ˆHI/bracketrightbig−1, (5)
where ˆGBCSis the retarded Green’s operator for the BCS
Hamiltonian from Eq. ( 1) and ˆHI=ˆHimpurity +ˆHRashba .
The DFT calculations were performed using the V ASP
code [ 47]. The β-Bi2Pd slab was optimized using the
Perdew-Burke-Ernzerhof form of the generalized gradient ap-proximation [ 48], following the calculations of Ref. [ 31]. For
more details, see the Supplemental Material [ 41].
III. RESULTS AND DISCUSSION
The dI/dVover a single Cr adatom yields a single YSR
state given by peaks at V=±0.35 mV (see Refs. [ 31,41]).By lateral atomic manipulation, we place Cr atoms to cre-
ate linear√
2a−/angbracketleft110/angbracketrightor 2a−/angbracketleft100/angbracketrightchains. Figures 1(a)
and1(b) show constant-current images of the two tetramer
chains. The chain in Fig. 1(a)corresponds to the√
2a−/angbracketleft110/angbracketright
tetramer as depicted in the inset; the one in Fig. 1(b) is the
2a−/angbracketleft100/angbracketrighttetramer. As the chain is made larger, misplacing
a Cr atom becomes more common. Indeed, error-free√
2a−
/angbracketleft110/angbracketrightspaced nanostructures were difficult to obtain, while
2a−/angbracketleft100/angbracketrightchains are easier to manipulate. The reason lies
in the chemistry of the chains. For the more compact chains,the affinity of Cr atoms for certain conformations leads tononlinear arrangements. The less compact 2 a−/angbracketleft100/angbracketrightchain
is easier to fabricate by single-atom manipulation because theatoms do not approach each other as much, and hence, clusterformation is much less common.
Our DFT calculations yield a coherent picture with the
experiment. The Cr atoms are preferentially adsorbed on thehollow sites of the Bi-rich surface [ 31], and the Cr-Cr in-
teractions in the chains are mediated by a single Bi atomin the√
2a−/angbracketleft110/angbracketrightchains or a square of four Bi atoms in
the 2 a−/angbracketleft100/angbracketrightchains. Short 1 a−/angbracketleft100/angbracketrightchains can also be
obtained, but the structures easily become clusters due tothe Cr-Cr interaction. The√
2a−/angbracketleft110/angbracketrightdimer is 249 meV
less stable than the 1 a−/angbracketleft100/angbracketrightdimer. As a consequence,
045406-3CRISTINA MIER et al. PHYSICAL REVIEW B 104, 045406 (2021)
Sample bias (mV)(a) (b) (c)
Sample bias (mV)(d) (e) (f)
(g) (h) (i)
0.0204060
dI/dV signal (nS)ecnatsiD (Å) ecnatsiD (Å) ecnatsiD (Å)Dimer Trimer Tetramer
Hexamer Octamer Nonamer
Undecamer Dodecamer
Gap (meV)
Number of Atoms
FIG. 2. Differential conductance measured along Cr n2a−/angbracketleft100/angbracketrightchains with n=2i n( a ) , n=3i n( b ) , n=4i n( c ) , n=6i n( d ) , n=8
in (e), n=9 in (f), n=11 in (g), and n=12 in (h). The xaxis represents the sample bias; the yaxis displays the distances over the chain.
The color code gives the intensity of the differential conductance. The smallest gap in the system, defined as the distance between the lower
quasiparticle peak and the highest quasihole peak, is plotted in (i). In the absence of Cr atoms, the gap corresponds to 2 /Delta1,w h e r e /Delta1=0.76 meV
forβ-Bi 2Pd. The gap has been obtained at an edge atom or at the center of the spin chain.
shifting a single Cr atom towards another Cr to reach the
short√
2adistance likely produces a 1 a−/angbracketleft100/angbracketrightdimer. This
stacking error becomes more likely as the chain is manipu-lated more times to make it longer. The 2 a−/angbracketleft100/angbracketrightdimer
is only 30 meV less stable than the√
2a−/angbracketleft110/angbracketrightdimer.
But, still, the interactions between atoms for the larger Cr-Crdistance, 2 a−/angbracketleft100/angbracketrightchains, are weaker, resulting in easier
manipulation to build longer chains. Indeed, the bottom-upapproach of chain building is difficult on many other sub-strates [ 49]. Recent experiments showed long Mn chains built
in a similarly compact geometry but on a Nb(110) substrate,also giving rise to topological in-gap behavior [ 50,51].
Once the chains are built, the differential conductance
dI/dV, as a function of bias Vand surface position, is an
extraordinary probe of the electronic properties of the newsystems. Figures 1(c) and 1(d) show dI/dVspectra mea-
sured at T=30 mK for the dimer, trimer, and tetramer of√
2a−/angbracketleft110/angbracketrightand 2 a−/angbracketleft100/angbracketrighttypes, respectively. The dI/dV
spectra are taken at an edge atom [black dots in Figs. 1(a)
and1(b)]. The two sets [Figs. 1(c) and1(d)] are starkly in
contrast. Figure 1(c) clearly shows an in-gap state that is
shifting towards zero bias as the chain gets longer. With op-posite behavior, Fig. 1(d) shows no clear in-gap state and a
well-formed gap. Furthermore, the gap for the dimer is larger,but the trimer and tetramer show similar gaps, pointing at a
rapid stabilization of the gap with the chain size. The extrapeaks around ±1.5 mV on the outside of the gap appear
occasionally, depending on the tip. The in-gap states are notaffected by the appearance of these higher-energy structuresand hence by the actual configuration of the tip.
The in-gap states of the√
2a−/angbracketleft110/angbracketrightdimer agree well
with a model of two FM aligned spins. When the magneticmoments are coupled antiferromagnetically, the in-gap stateapproaches the individual Cr adatom YSR states [ 31], be-
havior that explains the apparent absence of YSR states inFig.1(d). The presence of YSR in-gap states can be revealed
by studying the spatial distribution of the differential con-ductance along the two types of chains. Figures 2,3, and 4
show the dI/dVin a color scale (bright yellow corresponds
to larger conductance, and dark blue represents zero conduc-tance) along the chain, with the yaxis (in angstrom) showing
the distances over the chain and the xaxis (in millivolts)
showing the STM junction’s bias.
A. 2 a−/angbracketleft100/angbracketrightspin chains
Figure 2shows the results for the 2 a−/angbracketleft100/angbracketrightspin chains.
As seen in Fig. 1(d), we find no obvious structure in the gap
in any of the studied chains. A closer look reveals atomic
045406-4ATOMIC MANIPULATION OF IN-GAP STATES IN THE … PHYSICAL REVIEW B 104, 045406 (2021)
(a)
(e) (f) (g) (h)(b) (c) (d)
FIG. 3. Differential conductance along a Cr 122a−/angbracketleft100/angbracketrightchain comparing (a) the experimental result and (b)–(h) the computed PDOSs
for different noncollinear spin arrangements. The first case, in (b), is for an antiferromagnetic (AF) arrangement of spins. The spins on
nearest-neighbor Cr atoms present a 180◦angle. In (c) the angle is 120◦, such that a period of three Cr atoms is needed to turn the spin along
the chain. The case in (d) corresponds to a period of four atoms or a 90◦configuration. For (e), the mutual angle is 72◦; for (f), the angle is
60◦. In (g), we plot period 10 or the 36◦spin angle, and finally, (h) corresponds to the ferromagnetic (FM) case. The best agreement with the
measured spectra in (a) corresponds to a noncollinear arrangement with spins forming 120◦.
modulations of the quasihole states that match the number
of atoms in the chains. The presence of YSR states can beinferred by the profile of the gap. The complete sequence ofchains from n=2t on=12 can be found in the Supplemental
Material [ 41]. All chains roughly show a smaller gap at the
edge atoms than at the center of the chain [see Fig. 2(i)].
In the first approximation, the gap is constant with chainlength. Beyond eight atoms, the chains show a smaller gapat the edge. However, the closing of the gap is very smalland almost constant for longer chains. These data indicate thatthe YSR states are not able to close the superconducting gap,
preventing any topological phase transition.
By computing the PDOS, Eq. ( 4), we can evaluate the
in-gap state spectra and compare them with the experimen-tal data. The 2 a−/angbracketleft100/angbracketrightdimer presents excellent agreement
between theory and experiment if no Rashba coupling is con-sidered and the dimer spins are coupled antiferromagnetically,as shown in Ref. [ 31]. In the present work, we have gone a step
further by including the spin-orbit Rashba coupling betweenelectronic spins, Eq. ( 3).
(a)
(d) (e) (f)(b) (c)
FIG. 4. Experimental differential conductance measured along the Cr adatoms of the√
2a−/angbracketleft110/angbracketright(a) dimer, (b) trimer, and (c) tetramer
chains. The corresponding calculations for the ferromagnetically coupled√
2a−/angbracketleft110/angbracketright(d) dimer, (e) trimer, and (f) tetramer chains. The color
code refers to the PDOS on the different sites of the tight-binding lattice, in this case the one corresponding to the Cr adatoms.
045406-5CRISTINA MIER et al. PHYSICAL REVIEW B 104, 045406 (2021)
(a)
(d)
(g) (h) (i)(e) (f)(b) (c)
FIG. 5. Topological phase transition induced by increasing the exchange coupling J. The three columns correspond to three different
values of the exchange coupling, (a) J=2.1e V ,( b ) J=2.3e V ,a n d( c ) J=2.5 eV , for the PDOS showing the quasiparticle states induced by
aC r 20√
2a−/angbracketleft110/angbracketrightchain. We see that the gap is virtually closed for J=2.3 eV and reopens for J=2.5 eV , displaying the MBS that indicates
the change in topological phase of the superconductor. (d)–(f) correspond to the respective values of Jand show the transversal spin density
/angbracketleftSx/angbracketrightalong the chain. We see that /angbracketleftSx/angbracketrightbecomes large and of opposite sign only at the two MBSs. Finally, (g)–(i) show the spin density /angbracketleftSz/angbracketrightof
the YSR states for the three different couplings. We find that the spin across the gap reverts when the TPT is achieved and the corresponding
MBSs have the same well-defined spin.
In the absence of Rashba coupling, the electronic spin is a
good quantum number, and the YSR states are spin polarized.The FM ordering between impurity spins leads to YSR statesthat are also FM ordered, allowing for extended in-gap states.However, AF ordering leads to localized in-gap states that donot disperse. The AF results are in good agreement with thespectra of Fig. 2(a).
When the Rashba coupling is added, the YSR states start
mixing between impurities that have opposite spins, leadingto splitting of the YSR states and to an important dispersionof the in-gap states [ 32]. As a consequence, the agreement
between theory and experiment worsens. Figure 3shows the
results for a Cr
122a−/angbracketleft100/angbracketrightspin chain. The experimental
spectra [Fig. 3(a)] do not match the states of the computed AF
configuration [Fig. 3(b)]. However, the agreement improves if
noncollinear configurations are used. The spectra in Fig. 3(c)
correspond to noncollinear Cr spins forming 120◦, which
leads to a spin spiral with a periodicity of three atoms. Thenoncollinearity compensates for the Rashba coupling mixing.The resulting YSR states do not disperse, leading to spectrathat compare favorably with the experimental one in Fig. 3(a).
Increasing the noncollinearity leads to smaller angles and
larger spiral periods. Figure 3shows how the gap fills up withstates as the Cr spin configuration approaches the FM ordering
[Fig. 3(h)].
DFT calculations with spin-orbit coupling yield the lowest
energy to the AF ordering, however. Further work is neededto clearly determine the spin configuration of the 2 a−/angbracketleft100/angbracketright
chains. To this end, spin measurements with a superconduct-ing tip are a recent promising technique [ 35].
B.√
2a−/angbracketleft110/angbracketrightspin chains
Figure 4presents the dI/dVmaps of the√
2a−/angbracketleft110/angbracketright
chains (top row) compared to model calculations of the PDOSon the surface sites (bottom row). Figure 4shows excellent
agreement between experiment and theory if the magneticmoments are ferromagnetically coupled, which is also in goodagreement with the results of Ref. [ 31]. The calculations for
the YSR structure confirm the FM ordering for Cr atomssitting along the√
2a−/angbracketleft110/angbracketrighthollow sites. Moreover, the
magnetic ordering is not altered by adding extra atoms to thedimer.
The data in Fig. 4permit us to have a clear picture of the
in-gap states for the√
2a−/angbracketleft110/angbracketrightchains. The dimer presents
two YSR bands, one closer to zero energy with a larger
045406-6ATOMIC MANIPULATION OF IN-GAP STATES IN THE … PHYSICAL REVIEW B 104, 045406 (2021)
(a)
(c) (d)(b)
FIG. 6. Majorana bound states in a 20-atom√
2a−/angbracketleft110/angbracketrightCr spin chain. (a) Color map plotting the PDOS as a function of energy and
distance. There is a clear state localized at the edges of the chain and at exactly zero energy. (b) PDOS at zero energy along the 20 Cr chain;
xaxis is the distance along the Cr chain. The localization of the PDOS to the edges at zero energy spans the four Cr edge atoms, and the
PDOS sharply falls beyond. The value of the PDOS between edges reduces as the chain length increases. (c) Color map (dark: negative, light:positive) showing that the transversal spin density /angbracketleftS
x/angbracketrightchanges sign with the edge, but (d) the spin density component /angbracketleftSz/angbracketrightis the same for both
edges. Moreover, these data can be correlated with a clear change in spin sign across the gap as the exchange-interaction value Jis increased,
which shows the closing and reopening of the gap into the topological phase. All these data signal the presence of a Majorana bound state in a20-atom√
2a−/angbracketleft110/angbracketrightCr spin chain.
density of states between the two Cr adatoms and one closer
to the quasiparticle continuum with a minimum between theatoms. Adding one more atom to form the trimer shifts thelowest-energy YSR state closer to zero but keeps its overallspatial distribution with a maximum PDOS on the centralatom. Furthermore, we find the second band closer to thequasiparticle continuum and, again, with a minimum PDOSover the central point of the chain. We also notice that as in thedimer case, the quasiparticle PDOS presents a reduction andan oscillation along the chain. Finally, the tetramer shifts bothbands closer to zero, although largely keeping their spatialdistributions. The PDOS at the quasiparticle edge presents thesame features as for the dimer and trimer.
In order to match the very fast experimental closing of the
gap with the chain length, the Kondo exchange coupling J
is increased from J=2.0 eV for the dimer, J=2.1e Vf o r
the trimer, and J=2.3 eV for the tetramer, respectively, in
Figs. 4(d)–4(f). This behavior can be rationalized by a pos-
sible geometrical and electronic rearrangement of the chainas the spin chain grows in size. The atoms place themselvesmore symmetrically and closer to the surface, leading toa larger hybridization with the substrate and thus to largercouplings.The MBSs appear naturally as soon as the exchange cou-
pling Jis larger than 2.3 eV . It is interesting to study how the
appearance of MBSs takes place as Jvaries. This is plotted
in Fig. 5. The panels are arranged in three columns. Each
column corresponds to a different value of J. The first one
hasJ=2.1 eV , the second one has J=2.3 eV , and the third
one has J=2.5 eV . The first row plots the PDOS along the
chain ( yaxis) as a function of the quasiparticle energy ( x
axis). We see the formation of YSR bands already for this20-atom chain. In the middle of the chain, there is a clear gapin the YSR structure. For small J, this gap is maintained all
along the chain; for the larger J, the gap is closed by an edge
state that is a MBS, as we shall briefly see. For J=2.3e V ,
we see that the lowest-energy bands are still separated by avery small gap, almost closing, and for J=2.5t h eg a pi s
well formed again. The closing and reopening of the gap area necessary condition to change to a topologically nontrivialsuperconducting band structure.
The second row is the transversal spin density component
/angbracketleftS
x/angbracketrightalong the chain for the same YSR state as above. We
see that the values are small and dispersed for J=2.1 and
2.3 eV . For J=2.3 eV the values of /angbracketleftSx/angbracketrightextend all over
the superconducting gap, giving the impression of many YSR
045406-7CRISTINA MIER et al. PHYSICAL REVIEW B 104, 045406 (2021)
(a)
(e) (f) (g) (h)(b) (c) (d)
FIG. 7. Cr n√
2a−/angbracketleft110/angbracketrightchains, with nfrom 5 to 20, for J=2.5 eV such that the superconductor is in the topological phase. The
zero-energy state moves away from the center of the chain to the borders as the chain is increased in size. At fairly low numbers, eight or even
seven atoms, the MBSs become clear, and a gap is formed at the center of the chain.
states closing the gap. But J=2.5 eV is very different. The
gap in /angbracketleftSx/angbracketrightis again clear, and very sharp values at just the edge
states appear and are of opposite sign. This is a clear signatureof a MBS [ 52].
The third row shows the spin /angbracketleftS
z/angbracketrightof the YSR states. From
the above data, we have evidence that a topological phasetransition (TPT) has taken place between J=2.1 eV and
J=2.5 eV , with J=2.3 eV being near the closing of the gap.
The spin shows it unambiguously. The YSR bands show oppo-site spin polarizations for their particle and hole components,which is clearly seen across the YSR gap. But the characterhas changed between J=2.1 eV and J=2.5 eV because
the spin polarization is the opposite one. This opposite spinpolarization is a clear hallmark of a TPT [ 53]. The edge states
show the same spin polarization as the MBSs [ 52].
The experimental data show that the gap is almost closed
for the tetramer Cr
4√
2a−/angbracketleft110/angbracketrightspin chain. Closing the gap
is a necessary condition for a TPT. Figure 5clearly shows
that the edge states for Jlarger than 2.3 eV are indeed MBSs
and that the TPT takes place somewhere close to 2.3 eV . The
change in YSR band character through the TPT is clearly seen
in the YSR spin polarization [ 53]; indeed, the spin inverts
across the transition.
Figure 6shows the calculation of a Cr 20√
2a−/angbracketleft110/angbracketrightspin
chain with J=2.5 eV . A clear spin-polarized edge state ap-
pears, with opposite transversal spin components /angbracketleftSx/angbracketrighton the
chain edges showing that indeed MBSs are formed [ 52]. The
number of atoms in the spin chain is decisive, clearly showing
MBSs. However, short chains may suffice to prove that indeedthe superconductor undergoes a TPT.
For a spin chain in the topological phase, the appearance
of the MBSs needs a certain minimum chain size becausethe MBSs have a certain extension and they overlap for smallchains. The consequence is that the zero-energy state becomes
localized in the center of the chain, and it is difficult to identify
the new superconducting phase as topological.
The behavior of MBSs with the chain’s length is shown
in Fig. 7for Cr
n√
2a−/angbracketleft110/angbracketrightchains, with nfrom 5 to 20.The parameters are the above ones with J=2.5 eV that cor-
respond to the topological phase. In the case of the pentamer,Fig. 7(a) clearly shows a closed gap. We find a zero-energy
state for Cr
5that looks very similar to the experimental (and
theoretical) one for Cr 4. The zero-energy state is clearly local-
ized in the center of the chain. As the chain length is increased,
the state localizes to the edges. At the same time there is an
excitation gap appearing in the center of the chain. For n=8
atoms, it is already possible to clearly differentiate the featuresof the well-formed MBSs even though the chain is still smalland the YSR states present a strong discrete nature. As thelength is increased, a clear MBS appears. These calculationsimply that Cr
n√
2a−/angbracketleft110/angbracketrightchains on β-Bi2Pd will clearly
show MBSs and topological features at fairly small chains.
Indeed, 20 atoms suffice to have an unambiguous topologicalspin chain.
IV . CONCLUSION
In summary, Cr n√
2a−/angbracketleft110/angbracketrightspin chains on β-Bi2Pd
show a fast closing of the superconducting gap as the numberof atoms in the chain increases. As few as four Cr atomssuffice to have in-gap states closing down the gap. We showedthat an eight-atom√
2a−/angbracketleft110/angbracketrightchain may already display all
features of MBSs. Our study revealed that the√
2a−/angbracketleft110/angbracketright
Cr spin chain shows ferromagnetic alignment of its spins. Thelarge magnetic moment of Cr plus a sizable Rashba couplingof theβ-Bi
2Pd surface lead to the topological phase transition.
Increasing the distance between Cr atoms leads to facile atommanipulation that translates in longer chains of Cr atoms onβ-Bi
2Pd but at the cost of not reaching a topological phase.
Indeed, our measurements show a persistent gap rather con-stant with chain length for Cr
n2a−/angbracketleft100/angbracketright, showing that this
type of chain will not induce a topological phase transitionon the β-Bi
2Pd superconductor. The topological character
of the Cr n√
2a−/angbracketleft110/angbracketrightspin chains reveals the presence of
Majorana bound states in our simulations for chains as shortas eight atoms.
045406-8ATOMIC MANIPULATION OF IN-GAP STATES IN THE … PHYSICAL REVIEW B 104, 045406 (2021)
ACKNOWLEDGMENTS
Financial support from the Spanish MICINN (Projects
RTI2018-097895-B-C44 and Excelencia EUR2020-112116),Eusko Jaurlaritza (Project PIBA_2020_1_0017), JSPS KAK-
ENHI (Grants No. JP18K03531 and No. JP19K14651), andthe Institute for Basic Science (Grant No. IBS-R027-D1) isgratefully acknowledged.
[1] P. Avouris, Acc. Chem. Res. 28, 95 (1995) .
[2] D. M. Eigler and E. K. Schweizer, Nature (London) 344, 524
(1990) .
[3] J. A. Stroscio and D. M. Eigler, Science 254, 1319 (1991) .
[4] G. Meyer, S. Zöphel, and K. H. Rieder, Appl. Phys. A 63, 557
(1996) .
[5] K. Morgenstern, N. Lorente, and K.-H. Rieder, Phys. Status
Solidi B 250, 1671 (2013) .
[6] S. Clair and D. G. de Oteyza, Chem. Rev. 119, 4717 (2019) .
[7] A. A. Khajetoorians, D. Wegner, A. F. Otte, and I. Swart, Nat.
Rev. Phys. 1, 703 (2019) .
[8] L. Yu, Acta Phys. Sin. 21, 75 (1965).
[9] H. Shiba, Prog. Theor. Phys. 40, 435 (1968) .
[10] A. I. Rusinov, Sov. JETP 9, 85 (1969).
[11] T.-P. Choy, J. M. Edge, A. R. Akhmerov, and C. W. J.
Beenakker, P h y s .R e v .B 84, 195442 (2011) .
[12] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani,
P h y s .R e v .B 88, 020407(R) (2013) .
[13] F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B 88,
155420 (2013) .
[14] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo,
A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science
346, 602 (2014) .
[15] M. Ruby, F. Pientka, Y . Peng, F. von Oppen, B. W. Heinrich,
and K. J. Franke, Phys. Rev. Lett. 115, 197204 (2015) .
[16] R. Pawlak, M. Kisiel, J. Klinovaja, T. Meier, S. Kawai, T.
Glatzel, D. Loss, and E. Meyer, npj Quantum Inf. 2, 16035
(2016) .
[17] H. Kim, A. Palacio-Morales, T. Posske, L. Rózsa, K. Palotás,
L. Szunyogh, M. Thorwart, and R. Wiesendanger, Sci. Adv. 4,
eaar5251 (2018) .
[18] D.-J. Choi, N. Lorente, J. Wiebe, K. von Bergmann, A. F.
Otte, and A. J. Heinrich, Rev. Mod. Phys. 91, 041001
(2019) .
[19] A. Kitaev, Ann. Phys. (NY) 303, 2 (2003) .
[20] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das
Sarma, Rev. Mod. Phys. 80, 1083 (2008) .
[21] A. Palacio-Morales, E. Mascot, S. Cocklin, H. Kim, S. Rachel,
D. K. Morr, and R. Wiesendanger, Sci. Adv. 5, eaav6600
(2019) .
[22] B. Jäck, Y . Xie, and A. Yazdani, arXiv:2103.13210 .
[23] G. C. Ménard, S. Guissart, C. Brun, S. Pons, V . S. Stolyarov, F.
Debontridder, M. V . Leclerc, E. Janod, L. Cario, D. Roditchev,P. Simon, and T. Cren, Nat. Phys. 11, 1013 (2015) .
[24] M. Ruby, Y . Peng, F. von Oppen, B. W. Heinrich, and K. J.
Franke, Phys. Rev. Lett. 117, 186801 (2016) .
[25] D.-J. Choi, C. Rubio-Verdú, J. de Bruijckere, M. M. Ugeda, N.
Lorente, and J. I. Pascual, Nat. Commun. 8, 15175 (2017) .
[26] A. Kamlapure, L. Cornils, J. Wiebe, and R. Wiesendanger, Nat.
Commun. 9, 3253 (2018) .
[27] Y . Peng, F. Pientka, L. I. Glazman, and F. von Oppen, Phys.
Rev. Lett. 114, 106801 (2015) .[28] M. E. Flatté and D. E. Reynolds, P h y s .R e v .B 61, 14810
(2000) .
[29] S. Kezilebieke, M. Dvorak, T. Ojanen, and P. Liljeroth, Nano
Lett.18, 2311 (2018) .
[30] M. Ruby, B. W. Heinrich, Y . Peng, F. von Oppen, and K. J.
Franke, Phys. Rev. Lett. 120, 156803 (2018) .
[31] D.-J. Choi, C. G. Fernández, E. Herrera, C. Rubio-Verdú, M. M.
Ugeda, I. Guillamón, H. Suderow, J. I. Pascual, and N. Lorente,Phys. Rev. Lett. 120, 167001 (2018) .
[32] P. Beck, L. Schneider, L. Rózsa, K. Palotás, A. Lászlóffy, L.
Szunyogh, J. Wiebe, and R. Wiesendanger, Nat. Commun. 12,
2040 (2021) .
[33] H. Ding, Y . Hu, M. T. Randeria, S. Hoffman, O. Deb, J.
Klinovaja, D. Loss, and A. Yazdani, Proc. Natl. Acad. Sci.
U.S.A. 118, e2024837118 (2021) .
[34] L. Cornils, A. Kamlapure, L. Zhou, S. Pradhan, A. A.
Khajetoorians, J. Fransson, J. Wiebe, and R. Wiesendanger,Phys. Rev. Lett. 119, 197002 (2017) .
[35] L. Schneider, P. Beck, J. Wiebe, and R. Wiesendanger, Sci. Adv.
7, eabd7302 (2021) .
[36] J. Kim, W.-J. Jang, T. H. Bui, D.-J. Choi, C. Wolf, F. Delgado,
D. Krylov, S. Lee, S. Yoon, C. P. Lutz, A. J. Heinrich, and Y .Bae, arXiv:2103.09582 .
[37] Y . Imai, F. Nabeshima, T. Yoshinaka, K. Miyatani, R. Kondo,
S. Komiya, I. Tsukada, and A. Maeda, J. Phys. Soc. Jpn. 81,
113708 (2012) .
[38] J. G. Rodrigo, H. Suderow, S. Vieira, E. Bascones, and F.
Guinea, J. Phys.: Condens. Matter 16, R1151 (2004) .
[39] S.-H. Ji, T. Zhang, Y .-S. Fu, X. Chen, X.-C. Ma, J. Li, W.-H.
Duan, J.-F. Jia, and Q.-K. Xue, Phys. Rev. Lett. 100, 226801
(2008) .
[40] C. Mier, B. Verlhac, L. Garnier, R. Robles, L. Limot, N.
Lorente, and D.-J. Choi, J. Phys. Chem. Lett. 12, 2983
(2021) .
[41] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.104.045406 for the experimental local
spectra of single Cr impurity, topography of Cr spin chains aswell as details on the DFT calculations, which includes Ref.[54].
[42] M. E. Flatté, P h y s .R e v .B 61, R14920 (2000) .
[43] M. E. Flatté and J. M. Byers, Phys. Rev. Lett. 78, 3761
(1997) .
[44] R. C. Dynes, V . Narayanamurti, and J. P. Garno, P h y s .R e v .L e t t .
41, 1509 (1978) .
[45] E. Herrera, I. Guillamón, J. A. Galvis, A. Correa, A. Fente,
R. F. Luccas, F. J. Mompean, M. Garcia-Hernandez, S. Vieira,J. P. Brison, and H. Suderow, Phys. Rev. B 92, 054507
(2015) .
[46] S. V . Eremeev, I. P. Rusinov, I. A. Nechaev, and E. V . Chulkov,
New J. Phys. 15, 075015 (2013) .
[47] G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6,1 5
(1996) .
045406-9CRISTINA MIER et al. PHYSICAL REVIEW B 104, 045406 (2021)
[48] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 (1996) .
[49] M. Steinbrecher, R. Rausch, K. T. That, J. Hermenau, A. A.
Khajetoorians, M. Potthoff, R. Wiesendanger, and J. Wiebe,Nat. Commun. 9, 2853 (2018) .
[50] L. Schneider, P. Beck, T. Posske, D. Crawford, E. Mascot, S.
Rachel, R. Wiesendanger, and J. Wiebe, Nat. Phys. (2021) , doi:
10.1038/s41567-021-01234-y .[51] L. Schneider, P. Beck, J. Neuhaus-Steinmetz, T. Posske, J.
Wiebe, and R. Wiesendanger, arXiv:2104.11503 .
[52] D. Sticlet, C. Bena, and P. Simon, P h y s .R e v .L e t t . 108, 096802
(2012) .
[53] M. Mashkoori, S. Pradhan, K. Björnson, J. Fransson, and A. M.
Black-Schaffer, P h y s .R e v .B 102, 104501 (2020) .
[54] M. Ruby, F. Pientka, Y . Peng, F. von Oppen, B. W. Heinrich,
and K. J. Franke, P h y s .R e v .L e t t . 115, 087001 (2015) .
045406-10 |
PhysRevB.81.085436.pdf | Bosonization of one-dimensional fermions out of equilibrium
D. B. Gutman,1,2,3Yuval Gefen,4and A. D. Mirlin5,2,3,6
1The Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
2Institut für Theorie der kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
3DFG Center for Functional Nanostructures, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
4Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
5Institut für Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
6Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia
/H20849Received 24 November 2009; published 24 February 2010 /H20850
Bosonization technique for one-dimensional fermions out of equilibrium is developed in the framework of
the Keldysh action formalism. We first demonstrate how this approach is implemented for free fermions andfor the problem of nonequilibrium Fermi edge singularity. We then employ the technique to study an interact-ing quantum wire attached to two electrodes with arbitrary energy distributions. The nonequilibrium electronGreen’s functions, which can be measured via tunneling spectroscopy technique and carry the informationabout energy distribution, zero-bias anomaly, and dephasing, are expressed in terms of functional determinantsof single-particle “counting” operators. The corresponding time-dependent scattering phase is found to beintrinsically related to “fractionalization” of electron-hole excitations in the tunneling process and at bound-aries with leads. Results are generalized to the case of spinful particles as well to Green’s functions at differentspatial points /H20849relevant to the problem of dephasing in Luttinger liquid interferometers /H20850. For double-step
distributions, the dephasing rates are oscillatory functions of the interaction strength.
DOI: 10.1103/PhysRevB.81.085436 PACS number /H20849s/H20850: 73.23./H11002b, 73.40.Gk, 73.50.Td
I. INTRODUCTION
One-dimensional /H208491D/H20850interacting fermionic systems
show remarkable physical properties. The electron-electroninteraction manifests itself in a particularly dramatic way in1D systems, inducing a strongly correlated electronic state—Luttinger liquid /H20849LL/H20850.
1–5A paradigmatic experimental real-
ization of quantum wires are carbon nanotubes,6for a recent
review, see Ref. 7. Further realizations encompass
semiconductor,8metallic,9and polymer nanowires,10as well
as quantum Hall edges.11,12
While equilibrium LL has been extensively explored,
there is currently a growing interest in nonequilibrium phe-nomena on nanoscale and, in particular, in nonequilibriumproperties of quantum wires. In a recent experiment,
13the
tunneling spectroscopy of a biased LL conductor has beenperformed /H20849see also a related work on carbon nanotube quan-
tum dots /H20850.
14A similar approach was used to study experi-
mentally nonequilibrium quantum Hall edges.15Quite gener-
ally, the tunneling spectroscopy technique allows one tomeasure the nonequilibrium Green’s functions G
/H11125/H20849/H9270/H20850. Analo-
gous experiments16have been carried out earlier in order to
study energy distribution function and inelastic relaxationprocesses in quasi-one-dimensional diffusive metallicsamples. The interpretation of the results for a metallicsample is based on the Fermi liquid theory, and, in particular,on a kinetic equation for a quasiparticle distribution function.In fact, even in that case, careful analysis requires taking intoaccount nonequilibrium dephasing processes,
17which lead to
additional broadening of the measured Fermi edge structuresin the tunneling current. In the case of strongly correlated,non-Fermi-liquid systems /H20849such as LL /H20850out of equilibrium,
the situation is much more complex. In this situation, notonly a quantitative theoretical analysis of G
/H11125, but even thevery notions of quasiparticle energy distribution and dephas-
ing, become highly nontrivial. The goal of the presentedwork is to construct a corresponding theory. To achieve thisgoal, we develop a formalism of nonequilibrium /H20849Keldysh /H20850
bosonization. While we consider systems of 1D interactingelectrons in this work, we expect that it will be an importantstep in understanding the properties of a broader class ofsystems—nonequilibrium quantum fluids in low dimensions.This includes, in particular, systems of cold atoms, with ei-ther fermionic or bosonic statistics.
The structure of the present paper is as follows. In Sec. II,
we discuss possible experimental realizations of a nonequi-
librium LL. In Sec. III, we develop a bosonization technique
for noninteracting electrons away from equilibrium. Workingwithin the Keldysh nonequilibrium formalism, we derive theaction of the bosonized theory. While this action is quadraticat equilibrium /H20849which is the essence of conventional
bosonization /H20850, it now includes arbitrary powers in the
bosonic fields. We demonstrate how this action can be usedto express the Green’s function of noninteracting fermions interms of a Fredholm functional determinant of a single-particle “counting” operator /H20849which is of Toeplitz type /H20850.W e
further discuss the relation between this problem and that ofcounting statistics. Specifically, our result is expressed interms of the determinant at the value of the phase /H20849“counting
field” /H20850/H9261=2
/H9266. On the other hand, the counting statistics at
this point is trivial, in view of charge quantization. We showthat the difference between the determinants used for ex-pressing the Green’s functions and those used for countingstatistics results from different continuations /H20849analytic vs pe-
riodic /H20850of the functional determinant beyond the nonanalyt-
icity point/H9261=
/H9266. In Sec. IV, we apply our technique to the
problem of Fermi edge singularity /H20849FES /H20850out of equilibrium.
We show that nonequilibrium FES Green’s function is ex-PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
1098-0121/2010/81 /H208498/H20850/085436 /H2084922/H20850 ©2010 The American Physical Society 085436-1pressed in terms of the same functional determinant but with
a shifted value of the argument, /H9261=2/H20849/H9266−/H92540/H20850, where/H92540is the
scattering phase on the core hole. Comparing our results forthis problem with those obtained earlier,
18we establish use-
ful identities between Fredholm determinants of counting op-erator at values of the counting field /H9261differing by 2
/H9266.I n
Sec.V, our formalism is extended to interacting fermions in a
quantum wire. First, we analyze the problem of tunnelingspectroscopy of a nonequilibrium LL in the case of spinlessfermions. We demonstrate that the nonequilibrium Green’sfunctions are expressed in terms of products of single-particle Fredholm determinants. The corresponding values ofthe counting fields are shown to be related to “fractionaliza-tion” of particle-hole excitations created during the tunnelingprocess, as well as at the boundaries with noninteractingleads. Our results for G
/H11125contain all information about
single-particle properties of the system, including tunnelingdensity of states, energy distribution, and dephasing. We find,in particular, that the dephasing rate oscillates as a functionof the interaction strength /H20849LL parameter K/H20850, vanishing at
certain values of K. At the end of the section, we generalize
the consideration to the case of spinful fermions, as well toGreen’s functions at different spatial points /H20849which is rel-
evant to the problem of dephasing in LL interferometers /H20850.
Section VIincludes a summary of our results as well as
prospects for future work.
Some of results of this work were presented in Ref. 19.
II. NONEQUILIBRIUM LUTTINGER LIQUID:
SETUPS
In this section, we specify the class of problems to be
considered and discuss possible experimental setups. We as-sume that electrons with distributions functions n
/H9257/H20849/H9280/H20850/H20849/H9257
=R,Llabels right and left movers /H20850are injected into a LL
wire from two noninteracting electrodes. It is convenient tomodel the electrodes as noninteracting 1D systems, so thatthe whole structure is a wire with spatially dependent inter-action that switches on near the points x=/H11006L/2; see Sec. V
for details.
It is worth noting that we assume the absence of electron
backscattering due to impurities inside the LL wire. Whenpresent in sufficient amount /H20849so that one can speak about a
disordered LL /H20850, such impurities strongly affect the electronic
properties of a LL wire. Specifically, they induce diffusivedynamics at sufficiently high temperature Tand localization
phenomena proliferating with lowering T/H20849Refs. 20–22/H20850,a s
well as inelastic processes.
23We also neglect the nonlinearity
of the electron dispersion whose influence on spectral andkinetic properties of 1D electrons was recently studied inRefs. 24and25.
We discuss now possible experimental realizations of the
problem. The simplest way to take the system out of equi-librium is to apply a voltage between two electrodes, so thatthe incoming distribution functions have different chemicalpotentials,
/H9262L−/H9262R=eV, but equal temperatures, TR=TL=T,
see, e.g., Ref. 26. However, in the case of a LL, this situation
is almost identical to the equilibrium one, in view of theabsence of electron backscattering. Indeed, the bosons re-main at equilibrium, so that the usual bosonization technique
/H20849within Matsubara formalism /H20850can be applied. The only non-
equilibrium effect will be a simple shift in the chemical po-tential of left movers as compared to that of the right movers.
A generalization of this setup that does yield a nontrivi-
ally nonequilibrium LL is shown in Fig. 1/H20849a/H20850. A long clean
LL is adiabatically coupled to two electrodes with differentpotentials,
/H9262L−/H9262R=eVand different temperatures TL,TR./H20849A
particularly interesting situation arises when one of tempera-tures is much larger than the other, e.g., T
L=0 and TRfinite,
so that nonequilibrium effects are most pronounced. /H20850This
model has been investigated in our previous works, Refs. 27
and28. While showing genuinely nonequilibrium effects /H20849in
particular, energy redistribution of electrons /H20850, this model,
when treated in the framework of Keldysh bosonization for-malism, is characterized by a Gaussian action. For this rea-son, we termed this setup “partially nonequilibrium” in Ref.27. We will verify in Sec. Vthat the results of the present
work /H20849pertaining to full nonequilibrium /H20850reduce to those ob-
tained earlier /H20849partial nonequilibrium /H20850in the case when both
n
RandnLare taken to be Fermi-Dirac functions.
The focus of this work is generic nonequilibrium situa-
tions, when at least one of the functions n/H9257is not of the
Fermi-Dirac form. Such situations naturally arise when elec-trons injected into a LL wire represent juxtaposition of par-ticles originating from reservoirs with different chemical po-tentials and mixed by impurity scattering. Two possiblerealizations of such devices are shown in Figs. 1/H20849b/H20850and1/H20849c/H20850.
In the first case, Fig. 1/H20849b/H20850, the mixture of left and right mov-
ers coming from reservoirs with
/H9262L/HS11005/H9262Ris caused by impu-
rities which are located in the noninteracting part of thewires.
29In the second setup, Fig. 1/H20849c/H20850, the LL wire is at-
tached to two thick metallic wires which are themselves bi-ased. We assume that those electrodes are diffusive but suf-ficiently short, so that energy equilibration there can beneglected. As a result, a double-step energy distribution is−V/2
TRV/2
TLa)
b)
−V/2
T
c)V/2
TLR
V/2 V/2
−V/2 −V/2V
V
Vtun
tun
tun
FIG. 1. /H20849Color online /H20850Schematic view of experimental setups
for tunneling spectroscopy of a LL out of equilibrium: /H20849a/H20850“partially
nonequilibrium” setup, with distribution functions n/H9257/H20849/H9280/H20850of Fermi-
Dirac form but with different temperatures; /H20849b/H20850,/H20849c/H20850“fully nonequi-
librium” setups characterized by double-step distribution functionsn
/H9257/H20849/H9280/H20850of electrons injected into the LL wire.GUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-2formed in the electrodes16and “injected” into the LL con-
ductor. Such double-step distributions are of particular inter-est for our problem, as they are of the “maximally nonequi-librium” form. The existence of multiple Fermi edges in thedistribution functions “injected” from the electrodes rendersthe electron-electron scattering processes,
17,30which govern
the nonequilibrium dephasing rate /H9270/H9278/H20849and thus the broaden-
ing of tunneling spectroscopy characteristics /H20850particularly
important.
The question of nonequilibrium dephasing induced by
electron-electron scattering is particularly intriguing in thecase of a 1D system. First, energy relaxation is absent in ahomogeneous LL system. Second, recent analysis of dephas-ing in the context of weak localization and Aharonov-Bohmoscillations has given qualitatively different results: whilethe weak-localization dephasing rate vanishes in the limit ofvanishing disorder,
22the Aharonov-Bohm dephasing rate is
finite in a clean LL.22,31In the case of a partially nonequilib-
rium setup the tunneling spectroscopy dephasing rate has aform similar to the equilibrium Aharonov-Bohm dephasingrate.
27,28As we show here, in the case of double-step distri-
butions dephasing acquires qualitatively distinct features; inparticular, the dephasing rate becomes an oscillatory functionof the interaction strength.
Having described the problems to be addressed, we turn
to the corresponding formalism. It is instructive to develop itfirst for the case of noninteracting fermions and then “turnon” the interaction.
III. FREE FERMIONS
In this section, we develop a bosonization formalism for
the case of free fermions out of equilibrium. Specifically, weconsider noninteracting fermions with a given distributionfunction n/H20849
/H9280/H20850and derive the corresponding bosonic action.
Using the latter, we calculate the fermionic Green’s function.Clearly, the Green’s function of noninteracting fermions istrivially obtained within the fermionic formalism. However,the results of this section are not just a complicated way tocalculate a simple quantity. Rather, they will play a crucialrole for developing the bosonic formalism for interactingsystems studied in the remainder of the paper.
A. Keldysh action: From fermions to bosons
Bosonization has been proved to be a very efficient tool
for tackling one-dimensional problems at equilibrium,1–5as it
maps a system of interacting fermions /H20849LL/H20850onto that of non-
interacting bosons. One can thus hope for similar advantagesof this approach for nonequilibrium problems as well. Thequestion though is whether the bosonization procedure canbe generalized to systems out of equilibrium? As we showbelow, the answer is affirmative, yet substantial modifica-tions are required.
Quite generally, operator bosonization procedure consists
of the following steps: /H20849i/H20850mapping between the Hilbert space
of fermions and bosons; /H20849ii/H20850construction of the bosonic
Hamiltonian H
Brepresenting the original fermionic Hamil-
tonian HFin terms of bosonic /H20849particle-hole /H20850excitations,i.e., density fields; /H20849iii/H20850expressing fermionic operators in
the bosonic language; /H20849iv/H20850calculation of observables
/H20849Green’s functions /H20850within the bosonized formalism by
averaging with respect to the many-body bosonic densitymatrix /H20849
/H9267B/H20850. Neither the Hilbert space nor the operators /H20849in-
cluding the Hamiltonian /H20850contain an information regarding a
state of the many-body system. Therefore, the first threesteps remain unchanged for a nonequilibrium situation. Themajor modifications occur in the step /H20849iv/H20850. Indeed, at equilib-
rium the fermionic density matrix is expressed through thecorresponding Hamiltonian as
/H9267F=exp /H20849−HF/T/H20850, implying
that the same relation holds in the bosonized theory, /H9267B
=exp /H20849−HB/T/H20850, which makes averaging with respect to /H9267B
straightforward. Out of equilibrium this is not so anymore: a
one-particle density matrix corresponding to a nonequilib-rium occupation n/H20849
/H9280/H20850of fermionic states translates into a
complicated density matrix of bosons, which does not allowthe application of Wick theorem. This poses a major diffi-culty in bosonizing fermionic problems away from equilib-rium and, as we see below, results in a non-Gaussian actionof the bosonized theory.
To construct the effective bosonic theory, we start with the
fermionic description. Within the LL model, the electronfield is decoupled into a sum of left- and right-moving terms,
/H9274/H20849x,t/H20850=/H9274R/H20849x,t/H20850eipFx+/H9274L/H20849x,t/H20850e−ipFx, /H208491/H20850
where pFis the Fermi momentum. The Hamiltonian of the
system reads
H0=−iv/H20885dx/H20849/H9274R†/H11509x/H9274R−/H9274L†/H11509x/H9274L/H20850, /H208492/H20850
where vis the electron velocity. The bosonic representation
for fermionic operators has the form1–5,32
/H9274/H9257/H20849x/H20850/H11229/H20873/H9011
2/H9266v/H208741/2
e/H9257ipFxei/H9278/H9257/H20849x/H20850, /H208493/H20850
where/H9011is an ultraviolet cutoff. The bosonic fields /H9278/H9257/H20849x/H20850
are related to the density of electrons /H20851given by/H9267/H9257/H20849x/H20850
=/H9274/H9257†/H20849x/H20850/H9274/H9257/H20849x/H20850in the fermionic language /H20852as
/H9267/H9257/H20849x/H20850=/H9257
2/H9266/H11509x/H9278/H9257, /H208494/H20850
and obey the commutation relations
/H20851/H9278R/H20849x/H20850,/H9278R/H20849x/H11032/H20850/H20852=− /H20851/H9278L/H20849x/H20850,/H9278L/H20849x/H11032/H20850/H20852=i/H9266sgn/H20849x−x/H11032/H20850./H208495/H20850
We use the convention that in formulas /H9257should be under-
stood as/H9257=/H110061 for right/left-moving electrons. The
bosonized Hamiltonian is expressed in terms of density fieldsin the following way:
H
0=/H9266v/H20885dx/H20849/H9267R2+/H9267L2/H20850. /H208496/H20850
We turn now to the Lagrangian formalism. Since we deal
with a nonequilibrium situation, the system is characterizedby an action defined on the Keldysh contour,
33BOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-3S0/H20851/H9274/H20852=/H20885
cdt/H20885dx/H20858
/H9257=R,L/H9274/H9257†i/H11509/H9257/H9274/H9257, /H208497/H20850
where/H9274,/H9274†/H20849t,x/H20850are fermionic fields, and /H11509R,L=/H11509t/H11006v/H11509x.T o
generate correlation functions, it is convenient to introduce asource term.
S
V/H20851/H9274/H20852=/H20885
cdt/H20885dxV/H9257/H20849x,t/H20850/H9274/H9257†/H20849x,t/H20850/H9274/H9257/H20849x,t/H20850. /H208498/H20850
The field components on the upper branch and lower are
denoted by + and −, respectively. It is convenient to performa rotation in Keldysh space,
33thus decomposing fields into
classical and quantum components /H20849the latter being denoted
by a bar /H20850,
V/H9257,V¯/H9257=/H20849V+,/H9257/H11006V−,/H9257/H20850//H208812, /H208499/H20850
/H9267/H9257,/H9267¯/H9257=/H20849/H9267+,/H9257/H11006/H9267−,/H9257/H20850//H208812, /H2084910/H20850
/H9274/H9257,/H9274¯/H9257=/H20849/H9274+,/H9257/H11006/H9274−,/H9257/H20850//H208812, /H2084911/H20850
/H9274/H9257†,/H9274¯
/H9257†=/H20849/H9274+,/H9257†/H11007/H9274−,/H9257†/H20850//H208812. /H2084912/H20850
In these notations, the density-correlation functions are en-
coded in the generating function
Z/H9257/H20851V/H9257,V¯/H9257/H20852=/H20855exp/H20853iV/H9257/H9267¯/H9257+iV¯/H9257/H9267/H9257/H20854/H20856S0. /H2084913/H20850
The calculation of the partition function can be performed in
either the fermionic or the bosonic description. In the fermi-onic language it can be readily done by evaluating a Gauss-ian integral over the Grassman variables,
Z
/H9257/H20851V,V¯/H20852= det /H208511+G/H92570/H20849/H92680V+/H92681V¯/H20850//H208812/H20852, /H2084914/H20850
where/H92680and/H92681are the unit matrix and the first Pauli matrix
in the Keldysh space, and G/H92570is the Keldysh Green’s func-
tion of free chiral fermions, which has the following matrixstructure:
G
/H92570=/H20873G/H92570rG/H92570K
0G/H92570a/H20874. /H2084915/H20850
Here G/H9257,0a,G/H9257,0r, and G/H9257,0Kare advanced, retarded and
Keldysh components,
G/H92570r,a/H20849/H9280,p/H20850=1 //H20849/H9280−/H9257vp/H11006i0/H20850; /H2084916/H20850
G/H92570K/H20849/H9280,p/H20850=/H208511−2 n/H9257/H20849/H9280/H20850/H20852/H20851G/H92570r/H20849/H9280,p/H20850−G/H92570a/H20849/H9280,p/H20850/H20852. /H2084917/H20850
We expand now the generating functional /H2084914/H20850in powers
of the source fields V/H9257,V¯/H9257. For higher-dimensional systems,
this would generate all terms of the type V/H9257nV¯
/H9257m. In 1D, the
situation is different. Specifically, in an equilibrium 1D sys-
tem only terms up to second order /H20849V/H9257V¯/H9257andV¯
/H92572/H20850are gener-
ated, which forms the basis of conventional bosonization.Out of equilibrium, this is not true anymore: terms of higherorders are generated as well, and the theory becomes non-Gaussian. What is crucial, however, is that all higher-orderterms are of the type V
¯
/H9257n, i.e., they do not depend on V/H9257.W e
will prove this statement in Secs. III B andIII C below.
The generating functional has thus the structure
Z/H9257/H20851V,V¯/H20852= exp/H20873iV/H9257/H9016/H9257aV¯/H9257+/H20858
n=2/H11009in
n!V¯
/H9257nSn,/H9257/H20874, /H2084918/H20850
where Sn,/H9257is the nth order irreducible vertex function,
Sn,/H9257/H20849x1,t1; ... ; xn,tn/H20850=−in/H20858
perm.TrKG/H92570/H20849x1,t1;xi2,ti2/H20850/H92681
/H208812
/H11003G/H92570/H20849xi2,ti2;xi3,ti3/H20850/H92681
/H208812/H11003¯
/H11003G/H92570/H20849xin,tin;x1,t1/H20850/H92681
/H208812. /H2084919/H20850
The multiplication in Eq. /H2084918/H20850and analogous formulas below
should be understood in the matrix sense with respect to thecoordinates,
V
/H9257/H9016/H9257aV¯/H9257=/H20885/H20851dx/H20852/H20851dt/H20852V/H9257/H20849x1,t1/H20850/H9016/H9257a/H20849x1,t1;x2,t2/H20850V¯/H9257/H20849x2,t2/H20850,
/H2084920/H20850
V¯
/H9257nSn,/H9257=/H20885/H20851dx/H20852/H20851dt/H20852V¯/H9257/H20849x1,t1/H20850¯V¯/H9257/H20849xn,tn/H20850
/H11003Sn,/H9257/H20849x1,t1; ... ; xn,tn/H20850, /H2084921/H20850
where /H20848/H20851dx/H20852/H20851dt/H20852implies integration over all spatial and time
coordinates. The summation in Eq. /H2084919/H20850goes over /H20849n−1/H20850!
permutations /H20853i2,i3,..., in/H20854of the set of indices /H208532,..., n/H20854/H20849la-
beling the space-time coordinates /H20850, and Tr Kdenotes the trace
over Keldysh indices. Clearly, after integration with V¯/H9257fields
in Eq. /H2084918/H20850all the /H20849n−1/H20850! terms of the sum in Eq. /H2084919/H20850yield
equal contributions, so that the total combinatorial factor is/H20849n−1/H20850!/n!=1 /n, as should be in the expansion of the loga-
rithm. We have chosen to define the vertex function in thesymmetrized form /H2084919/H20850/H20851and to introduce the corresponding
factor 1 //H20849n−1/H20850! in Eq. /H2084918/H20850/H20852, since S
n,/H9257/H20849x1,t1;...; xn,tn/H20850are
then equal to irreducible density-correlation functions/H20855/H20855
/H9267/H20849x1,t1/H20850¯/H9267/H20849xn,tn/H20850/H20856/H20856.
The quadratic part of the generating functional /H2084918/H20850is
determined by the polarization operator of fermions,
/H9016/H9257=/H208730/H9016/H9257a
/H9016/H9257r/H9016/H9257K/H20874, /H2084922/H20850
with the retarded, advanced, and Keldysh components given
by
/H9016/H9257r,a/H20849/H9275,q/H20850=1
2/H9266/H9257q
/H9257vq−/H9275/H11007i0, /H2084923/H20850
/H9016/H9257K/H20849/H9275,q/H20850=/H20851/H9016/H9257r/H20849/H9275,q/H20850−/H9016/H9257a/H20849/H9275,q/H20850/H20852B/H9257/H20849/H9275/H20850. /H2084924/H20850
Here, the functionGUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-4B/H9257/H20849/H9275/H20850=1
/H9275/H20885
−/H11009/H11009
d/H9280n/H9257/H20849/H9280/H20850/H208512−n/H9257/H20849/H9280−/H9275/H20850−n/H9257/H20849/H9280+/H9275/H20850/H20852,
/H2084925/H20850
governs the distribution function N/H9257/H20849/H9275/H20850of electron-hole ex-
citations moving with velocity vin direction/H9257,B/H9257/H20849/H9275/H20850=1
+2N/H9257/H20849/H9275/H20850. At equilibrium,
B/H9257/H20849/H9275/H20850=Beq/H20849/H9275/H20850=1+2 Neq/H20849/H9275/H20850= coth/H20873/H9275
2T/H20874, /H2084926/H20850
where Neq/H20849/H9275/H20850is the Bose distribution. By construction, the
second order density-correlation function S/H9257,n=2in Eq. /H2084918/H20850
is equal to the Keldysh component /H9016/H9257Kof the polarization
operator /H20849times − i/H20850.
In order to bosonize the theory, we should find a bosonic
counterpart of the action S0/H9257that reproduces the generating
functional /H2084918/H20850. According to Eq. /H2084913/H20850, we have
exp/H20849iS0/H9257/H20851/H9267¯/H9257,/H9267/H9257/H20852/H20850=/H20885DV/H9257DV¯/H9257Z/H9257/H20851V/H9257,V¯/H9257/H20852e−iV/H9257/H9267¯/H9257−iV¯
/H9257/H9267/H9257.
/H2084927/H20850
Substituting Eq. /H2084918/H20850into Eq. /H2084927/H20850, we obtain the bosonized
action
S0,/H9257/H20851/H9267/H9257,/H9267¯/H9257/H20852=−/H9267/H9257/H9016/H9257a−1/H9267¯/H9257−ilnZ/H9257/H20851/H9273¯/H9257/H20852. /H2084928/H20850
Here, Z/H9257/H20851/H9273¯/H9257/H20852/H11013Z/H9257/H20851/H9273/H9257=0,/H9273¯/H9257/H20852is a partition function /H2084918/H20850of
free fermions,
ilnZ/H9257/H20851/H9273¯/H9257/H20852=/H20858
n=2/H11009
in+1/H9273¯/H9257nSn,/H9257/n!, /H2084929/H20850
subject to the external quantum field
/H9273¯/H9257=/H9016/H9257a−1/H9267¯/H9257. /H2084930/H20850
The combined action of left- and right-moving electrons is
simply given by a sum of the corresponding chiral actions,
S0/H20851/H9267,/H9267¯/H20852=/H20858
/H9257S0/H9257/H20851/H9267/H9257,/H9267¯/H9257/H20852. /H2084931/H20850
Thus we have described a system of nonequilibrium free
fermions by a bosonic theory, Eq. /H2084931/H20850. In this approach in-
formation on the nonequilibrium state of the system is en-coded in the vertices /H20849S
n/H9257/H20850, schematically depicted in Fig. 2.
In Sec. III B, we discuss the status and implications of the
Dzyaloshinskii-Larkin theorem concerning these vertices.
B. Dzyaloshinskii-Larkin theorem
The appearance of higher-order /H20849n/H110222/H20850fermionic vertices
may seem to contradict the Dzyaloshinskii-Larkin theorem.34
The latter states that diagrams containing closed loops withmore than two fermionic lines vanish, i.e., the random phaseapproximation /H20849RPA /H20850is exact. Although the theorem was
formulated for the equilibrium case, its proof, given in Ref.34, ostensibly relies solely on particle conservation. Since
the latter remains valid out of equilibrium, one might expectthe theorem to hold under nonequilibrium conditions as well.To understand why Dzyaloshinskii-Larkin theorem is in fact
restricted to the equilibrium case only, and what its implica-tions for a nonequilibrium situation are, we carefully re-examine the arguments of Ref. 34.
One starts with the continuity equation for the chiral cur-
rent and density operators,
/H9275/H9267/H9257−/H9257qj/H9257=0 . /H2084932/H20850
Since within the LL model these operators are related to each
other through j/H9257=/H9257v/H9267/H9257, the continuity equation can be re-
written in terms of the density field only,
/H20849/H9275−/H9257vq/H20850/H9267/H9257=0 . /H2084933/H20850
As a consequence, correlation functions of densities satisfy
/H20849/H9275i−/H9257vqi/H20850/H20855/H9267/H9257/H20849/H92751,q1/H20850/H9267/H9257/H20849/H92752,q2/H20850¯/H9267/H9257/H20849/H9275n,qn/H20850/H20856=0 /H2084934/H20850
for any i=1,..., n. Therefore, the irreducible density-
correlation functions Sn/H9257/H20849/H92751,q1;/H92752,q2;.../H9275n,qn/H20850with n
/H110222 should be zero everywhere, except possibly for the mass
shell with respect to all arguments,35
Sn/H9257/H20849/H92751,q1;/H92752,q2; .../H9275n,qn/H20850
=/H9254/H20849/H92751−/H9257vq1/H20850/H9254/H20849/H92752−/H9257vq2/H20850¯/H9254/H20849/H9275n−/H9257vqn/H20850
/H11003S/H9257/H20849/H92751,/H92752, ...,/H9275n/H20850/H9254/H20849/H92751+¯+/H9275n/H20850. /H2084935/H20850
In the case n=2, the argument is not applicable in view of
the Schwinger anomaly, yielding the first term in the expo-nent on the right-hand side /H20849r.h.s. /H20850of Eq. /H2084918/H20850. When trans-
lated into the coordinate-time space, the mass-shell condition/H2084935/H20850implies that the correlation function depends in fact only
on the world line to which each of the points /H20849t
i,xi/H20850belongs
but not on the position of this point on the line:
Sn/H9257/H20849t1,x1; ... ; tn,xn/H20850/H11013/H20855 /H20855/H9267/H9257/H20849t1,x1/H20850¯/H9267/H9257/H20849tn,xn/H20850/H20856/H20856
=/H20855/H20855/H9267/H9257/H208490,/H92641/H20850¯/H9267/H9257/H208490,/H9264n/H20850/H20856/H20856, /H2084936/H20850
where/H9264i=x−/H9257vti. In the Keldysh formalism language, the
only nonzero irreducible density-correlation function in anyorder n/H110222 arises when one considers the correlator with allV
VV
V
V
V
VVV(a)
(c)(b)
S S
S423
FIG. 2. /H20849Color online /H20850Vacuum loops for free fermions in an
external field V¯. At equilibrium only S2is nonzero, according to the
Dzyaloshinskii-Larkin theorem. Away from equilibrium, all verticesappear. For details, see Sec. III B.BOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-5nfields being the classical components /H9267./H20849This follows from
the fact that the operators /H9267commute to a cnumber. /H20850These
correlation functions are the noise cumulants in the system.
The behavior of the correlation functions Sn/H9257on the “light
cone” /H2084935/H20850cannot be determined from particle conservation
law and requires an additional calculation. While at equilib-rium all S
n/H9257with n/H110222 do vanish /H20849which reconciles our
theory with the Larkin-Dzyaloshinskii theorem /H20850, out of equi-
librium they are in general nonzero. We consider this generalsituation in Sec. III C where we show that the bosonized
action can be presented in a compact form of a functionaldeterminant.
C. Bosonized action as functional determinant
As we have shown, the bosonic action, Eq. /H2084931/H20850,i se x -
pressed through the partition function Z/H20851V,V¯/H20852of free fermi-
ons in an external field V/H20849x,t/H20850defined on the Keldysh con-
tour. In one dimension the partition function can be cast in arelatively simple form. To achieve this, we first present thepartition function
Z/H20851V,V
¯/H20852=t r/H20853/H9267FSc/H20854. /H2084937/H20850
Here, Scis an evolution operator along Keldysh contour,
Z/H20851V,V¯/H20852= lim
t→/H11009tr/H20853/H9267Fe−iH/H20851V+/H20849−t/H20850/H20852/H9004te−iH/H20851V+/H20849−t+/H9004t/H20850/H20852/H9004t
/H11003¯/H11003e−iH/H20851V+/H20849t/H20850/H20852/H9004teiH/H20851V−/H20849t/H20850/H20852/H9004teiH/H20851V−/H20849t−/H9004t/H20850/H20852/H9004t
/H11003¯/H11003eiH/H20851V−/H20849−t/H20850/H20852/H9004t/H20854, /H2084938/H20850
and the trace is taken over the many-body fermionic Fock
space. Equation /H2084938/H20850can be further simplified by means of
the following identity:37
tr/H20853eH1eH2¯eHN/H20854= det /H208491+eh1eh2¯ehN/H20850. /H2084939/H20850
Here, hnis a matrix in the single-particle Hilbert space, and
Hn=/H20858
i,jhni,jai†aj /H2084940/H20850
is the corresponding operator quadratic in fermionic creation/
annihilation operators /H20849a†,a/H20850. The trace in the left-hand side
/H20849l.h.s. /H20850of Eq. /H2084939/H20850is taken in the many-body Fock space,
while the determinant on the r.h.s. is taken in the single-particle space.
Applying Eq. /H2084939/H20850in the continuum limit, we express the
partition function in the following form
Z
/H9257/H20851V/H9257,V¯/H9257/H20852= det /H208511−n/H9257+n/H9257U+,/H9257−1U−,/H9257/H20852. /H2084941/H20850
Here,
U+,/H9257/H20849t/H20850= T exp/H20873−i/H20885
0t
dth+,/H9257/H20874,
U−,/H9257−1/H20849t/H20850=T˜exp/H20873i/H20885
0t
dth−,/H9257/H20874 /H2084942/H20850
are evolution operators that correspond to the single-particle
Hamiltoniansh+,/H9257=−i/H9257v/H11509
/H11509x+V+/H20849x,t/H20850,
h−,/H9257=−i/H9257v/H11509
/H11509x+V−/H20849x,t/H20850. /H2084943/H20850
Thus the many-body problem of summing all vacuum loops
has been reduced to a calculation of a functional determinantof an operator in a single-particle Hilbert space. To simplifyit further, we analyze the properties of the evolution operatorUin one dimension. Its action on a wave function
/H9274/H20849x/H20850can
be described as
/H9274/H20849x,t/H20850= T exp/H20873−i/H20885
0t
dth+/H20874/H9274/H20849x,0/H20850, /H2084944/H20850
where/H9274/H20849x,0/H20850/H11013/H9274/H20849x/H20850. One can easily show that the resulting
wave function/H9274/H20849x,t/H20850satisfies the Schrödinger equation
i/H11509
/H11509t/H9274/H20849x,t/H20850=h+/H9274/H20849x,t/H20850. /H2084945/H20850
Solving Eq. /H2084945/H20850explicitly one finds
/H9274/H20849x,t/H20850=/H9274/H20849x−/H9257vt,0/H20850e−i/H208480td/H9270V+/H20851x+/H9257v/H20849/H9270−t/H20850,/H9270/H20852. /H2084946/H20850
Therefore, the action on a wave function of the evolution
operator forward and backward in time results in the phasefactor
/H20849U
−−1U+/H9274/H20850/H20849x/H20850=/H9274/H20849x/H20850e−i/H208480td/H9270/H20849V+−V−/H20850/H20849x+v/H9270,/H9270/H20850. /H2084947/H20850
Consequently, the partition function of the 1D fermions
can be cast as36
Z/H9257/H20851V/H9257,V¯/H9257/H20852=e−iV/H9257/H9016/H9257aV¯
/H9257/H9004/H9257/H20851/H9254/H9257/H20849t/H20850/H20852, /H2084948/H20850
where we introduced a determinant
/H9004/H9257/H20851/H9254/H9257/H20849t/H20850/H20852= det /H208511+/H20849e−i/H9254/H9257−1/H20850n/H9257/H20852, /H2084949/H20850
and
/H9254/H9257/H20849t/H20850=/H208812/H20885
−/H11009/H11009
d/H9270V¯/H9257/H20851/H9257v/H20849/H9270+t/H20850,/H9270/H20852/H20849 50/H20850
is the scattering phase accumulated by an electron moving
along a “light-cone” trajectory. Thus, according to Eq. /H2084948/H20850
the problem of summing up the vacuum loops is reduced toevaluation of a one-dimensional functional determinant /H2084949/H20850.
The determinant /H2084949/H20850is defined by the function
/H9254/H9257/H20849t/H20850in
the time space and n/H9257/H20849/H9280/H20850in the energy space, with /H9280andt
understood as canonically conjugate variables. It belongs tothe class of Fredholm determinants. For a specific case /H20849that
will be particularly important for us below /H20850when
/H9254/H9257/H20849t/H20850is
different from zero within a limited interval of time only, thedeterminant acquires the Toeplitz form. Such determinantshave been considered in the context of countingstatistics;
38,39see a more detailed discussion in Sec. III D .
It is also worth mentioning that there is a vast literature onthe connection of Fredholm determinants to quantum inte-grable models, classical integrable differential equationsGUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-6/H20849with soliton solutions /H20850, and free-fermion problems; we refer
the reader to Refs. 40–43and references therein.
At equilibrium the Taylor expansion of ln Zin/H9254termi-
nates at the second order /H20849Sn=0 for n/H110222/H20850, in agreement with
Dzyaloshinskii-Larkin theorem, Ref. 34. In that case the ac-
tion /H2084931/H20850is quadratic, reproducing the standard LL model.
Away from thermal equilibrium, high-order density correla-tions are finite.
38For this reason, we obtain a non-Gaussian
bosonized theory, despite the fact that the Hamiltonian /H208496/H20850is
quadratic. The higher-order terms Snwith n/H110222 appear in the
bosonic action due to a nondiagonal structure of the densitymatrix in the bosonic Fock space, which leads to a break-down of Wick theorem for the bosonic fields.
D. Green’s functions
We have thus shown that noninteracting fermions can be
equivalently described by the bosonic theory with the actiongiven by Eqs. /H2084931/H20850,/H2084928/H20850, and /H2084948/H20850. We apply now this for-
malism to calculate the free-fermion Green’s functions/H20849GFs /H20850,
G
/H9257/H11021/H20849x1,t1;x2,t2/H20850=i/H20855/H9274/H9257†/H20849x2,t2/H20850/H9274/H9257/H20849x1,t1/H20850/H20856,
G/H9257/H11022/H20849x1,t1;x2,t2/H20850=−i/H20855/H9274/H9257/H20849x1,t1/H20850/H9274/H9257†/H20849x2,t2/H20850/H20856. /H2084951/H20850
At equilibrium these GFs are related to the advanced and
retarded GFs via
G/H9257/H11022/H20849x,/H9280/H20850=/H20851G/H9257r/H20849x,/H9280/H20850−G/H9257a/H20849x,/H9280/H20850/H20852/H208511−n/H9257/H20849/H9280/H20850/H20852,
G/H9257/H11021/H20849x,/H9280/H20850=− /H20851G/H9257r/H20849x,/H9280/H20850−G/H9257a/H20849x,/H9280/H20850/H20852n/H9257/H20849/H9280/H20850. /H2084952/H20850
For free fermions, Eq. /H2084952/H20850is valid for an arbitrary distribu-
tion function n/H9257determining the filling of single-particle
states.
Due to Galilean invariance, the GFs depend only on /H9270/H9257
=t1−t2−/H9257/H20849x1−x2/H20850/v, so we may set x1=x2=vt2=0 in the ar-
gument of GF. Using Eqs. /H208493/H20850and /H2084951/H20850, we obtain
G0,/H9257/H11022/H20849/H9270/H9257/H20850=−i/H9011
2/H9266v/H20855TKei/H9278/H9257,−/H208490,/H9270/H9257/H20850e−i/H9278/H9257,+/H208490,0/H20850/H20856, /H2084953/H20850
and a similar result for the function G0,/H9257/H11021. At thermal equilib-
rium G0,/H9257/H11125can be readily calculated. A standard calculation
/H20849presented for completeness in Appendix A /H20850yields
G/H9257/H11125/H20849/H9270/H9257/H20850=/H11007i/H9011
2vT/H9270/H9257
sinh/H9266T/H9270/H92571
1/H11006i/H9011/H9270/H9257. /H2084954/H20850
Away from equilibrium the calculation of GFs, rather
simple within a fermionic framework, turns out to be quitecomplicated within a bosonic one. Nevertheless, this effortpays off, since the bosonic formalism will later allow us toextend the analysis to the interacting case.
Within the bosonic description, the GF can be represented
as a functional integral over the density fields. Since calcu-
lations of G
0,/H9257/H11022andG0,/H9257/H11021are quite similar to each other we
focus here onG0,/H9257/H11022/H20849/H9270/H9257/H20850=−i/H9011
2/H9266v/H20885D/H9267D/H9267¯eiS/H20851/H9267,/H9267¯/H20852
/H11003e/H20849i//H208812/H20850/H20851/H9278/H208490,/H9270/H9257/H20850−/H9278/H208490,0/H20850−/H9278¯/H208490,/H9270/H9257/H20850−/H9278¯/H208490,0/H20850/H20852. /H2084955/H20850
In a generic nonequilibrium situation, the bosonic action,
Eq. /H2084931/H20850, contains terms of all orders with no small param-
eter; the idea to proceed analytically in a controlled mannermay seem hopeless. This, however, is not the case: nonequi-librium bosonization is an efficient framework in which thefunctional integration can be performed exactly. Indeed, Z
/H9257
in Eq. /H2084928/H20850depends only on the quantum component /H9267¯,s o
that the action, Eq. /H2084931/H20850, is linear with respect to the classical
component/H9267of the density field. Hence, the integration with
respect to/H9267can be performed exactly
G0,/H9257/H11022/H20849/H9270/H20850=−i/H9011
2/H9266v/H20885D/H9267¯Z/H9257/H20851/H9273¯/H9257/H20852/H9254/H20849/H11509t/H9267¯+/H9257v/H11509x/H9267¯−j/H20850
/H11003e−/H20849i//H208812/H20850/H20851/H9278¯/H208490,/H9270/H20850+/H9278¯/H208490,0/H20850/H20852, /H2084956/H20850
where the source term is
j/H20849x,t/H20850=/H9254/H20849x/H20850/H20851/H9254/H20849t−/H9270/H20850−/H9254/H20849t/H20850/H20852//H208812. /H2084957/H20850
Resolving the/H9254function, we obtain an equation that deter-
mines the quantum component of the density field,
/H11509t/H9267¯/H9257+/H9257v/H11509x/H9267¯/H9257=j/H20849x,t/H20850. /H2084958/H20850
According to the structure of the first term in the action /H2084928/H20850,
we should look for the advanced solution of Eq. /H2084958/H20850which
is zero at times larger than those at which the source j/H20849x,t/H20850
acts. In other words, in the asymptotic regions /H20841x/H20841/H11022L/2 the
solution/H9267¯/H20849x,t/H20850should contain incoming waves only. Solving
Eq. /H2084958/H20850with this asymptotic conditions, we find the quan-
tum density component
/H9267¯/H9257/H20849x,t/H20850=/H9258/H20849−/H9257x/H20850
/H208812/H20853/H9254/H20849x−/H9257vt/H20850−/H9254/H20851x−/H9257v/H20849t−/H9270/H20850/H20852/H20854./H2084959/H20850
To find the Green’s function, we need to evaluate the factors
multiplying the delta-function in Eq. /H2084956/H20850, subjected to the
/H9254-function constraint. The most nontrivial factor /H20849which car-
ries the information about the distribution function /H20850is
Z/H9257/H20851/H9273¯/H9257/H20852, where/H9273¯/H9257is related to/H9267¯/H9257via Eq. /H2084930/H20850. According to
Eq. /H2084948/H20850,Z/H9257/H20851/H9273¯/H9257/H20852is expressed as a functional determinant of
the form /H2084949/H20850. We thus obtain
G0,/H9257/H11125/H20849/H9270/H20850=−1
2/H9266v1
/H9270/H11007i//H9011/H9004¯/H9257/H20851/H9254/H9257/H20849t/H20850/H20852. /H2084960/H20850
Here, we have denoted by /H9004¯/H9257the determinant normalized to
its value for zero-temperature equilibrium distribution, seeAppendix A. It is convenient to use this definition since thedeterminant/H9004
/H9257requires in fact an ultraviolet regularization.
On the other hand, the normalized determinant /H9004¯/H9257/H20849which is
equal to unity for the equilibrium, T=0 case /H20850is uniquely
defined. The prefactor in Eq. /H2084960/H20850that does not depend on
the distribution function is immediately determined from theequilibrium result.BOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-7According to Eqs. /H2084948/H20850and /H2084950/H20850, the mass-shell nature of
S/H9257nimplies that Z/H9257/H20851/H9273¯/H9257/H20852depends only on the world-line inte-
gral
/H9254/H9257/H20849t/H20850=/H208812/H20885
−/H11009/H11009
dt˜/H9273¯/H9257/H20849/H9257vt˜,t˜−t/H20850. /H2084961/H20850
Using Eq. /H2084930/H20850, we find an explicit solution for the “counting
field”/H9273¯/H9257,
/H9273¯/H9257/H20849x,t/H20850=2/H9266/H20875v/H9267¯/H9257/H20849x,t/H20850+/H9257/H20885
0x
dx˜/H9267¯/H9257/H20849x˜,t/H20850/H20876. /H2084962/H20850
Next, we calculate the value of /H9254/H20849t/H20850for our noninteracting
problem. Substitution of Eq. /H2084962/H20850into Eq. /H2084961/H20850allows us to
cast the result for the phases /H9254/H9257/H20849t/H20850into the following form:
/H9254/H9257/H20849t/H20850=−2/H9266/H208812/H9257lim
t˜→−/H11009/H20885
0/H9257v/H20849t˜+t/H20850
dx˜/H9267¯/H9257/H20849x˜,t˜/H20850. /H2084963/H20850
For the free-fermion problem the phase /H9254/H9257/H20849t/H20850=/H9261/H9275/H9270/H20849t,0/H20850
where
w/H9270/H20849t,t˜/H20850=/H9258/H20849t˜−t/H20850−/H9258/H20849t˜−t−/H9270/H20850/H20849 64/H20850
is a “window function” and /H9261=2/H9266. Thus, Z/H9257/H20851/H9273¯/H9257/H20852=/H9004¯/H9257/H9270/H208492/H9266/H20850,
where/H9004¯/H9257/H9270/H20849/H9261/H20850is the determinant /H2084949/H20850/H20849normalized to its T
=0 value /H20850for a rectangular pulse.
G0,/H9257/H11125/H20849/H9270/H20850=−1
2/H9266v/H9004¯/H9257/H9270/H208492/H9266/H20850
/H9270/H11007i//H9011. /H2084965/H20850
Determinants of the type /H2084949/H20850have appeared in a theory
of counting statistics.38,39Specifically, the generating func-
tion of current fluctuations /H9260/H20849/H9261/H20850=/H20858n=−/H11009/H11009ein/H9261pn/H20849where pnis
the probability of nelectrons being transferred through the
system in a given time window /H9270/H20850has the same structure as
/H9004/H9257/H9270/H20849/H9261/H20850. Taylor expansion of ln /H9260/H20849/H9261/H20850around/H9261=0 defines cu-
mulants of current fluctuations.
According to its definition, /H9260/H20849/H9261/H20850is 2/H9266periodic, which is a
manifestation of charge quantization that should show up inmeasurements of the transferred electric charge.
37–39,44–47
Thus,/H9260/H208492/H9266/H20850=1 is trivial. On the other hand, we have found
that the free-electron GF is determined by the nontrivialvalue of the functional determinant exactly at /H9261=2
/H9266. A res-
olution of this apparent paradox is as follows: the determi-nant/H9004
/H9257/H9270/H20849/H9261/H20850should be understood as an analytic function of
/H9261increasing from 0 to 2 /H9266. On the other hand, /H9260/H20849/H9261/H20850is
nonanalytic at the branching points /H9261=/H11006/H9266,/H110063/H9266,... T o
demonstrate this, it is instructive to consider the equilibriumcase that is treated in Appendix A. Then the expansion ofln/H9004
/H9257/H9270/H20849/H9261/H20850in/H9261is restricted to the /H92612term /H20849since RPA is
exact /H20850. It is easy to check that the /H9261=2/H9266point on this para-
bolic dependence correctly reproduces the fermion GF viaEqs. /H2084960/H20850and /H2084949/H20850. As to the counting statistics ln
/H9260/H20849/H9261/H20850,i ti s
quadratic only in the interval /H20851−/H9266,/H9266/H20852and is periodically con-
tinued beyond this interval, see Fig. 3.
The difference in the analytical properties of /H9260/H20849/H9261/H20850and
/H9004/H9257/H9270/H20849/H9261/H20850becomes especially transparent if one studies the
semiclassical /H20849long-/H9270/H20850limit,ln/H9004¯/H9257/H9270/H20849/H9261/H20850=/H9270
2/H9266/H20885
−/H11009/H11009
d/H9280/H20853ln/H208511+/H20849e−i/H9261−1/H20850n/H9257/H20849/H9280/H20850/H20852+i/H9261/H9258/H20849−/H9280/H20850/H20854.
/H2084966/H20850
For small positive /H9261the singularity of the integrand closest
to the real axis is located at /H9280=i/H20849/H9266−/H9261/H20850T, i.e., near/H9280=i/H9266T.A s
/H9261increases, the singularity moves toward the real axis,
crosses it at/H9261=/H9266and finally approaches /H9280=−i/H9266Tas/H9261
→2/H9266/H20849see inset of Fig. 3/H20850. The integral for ln /H9260/H20849/H9261/H20850is taken
along the real axis, resulting in nonanalyticity at /H9261=/H9266and in
zero value at/H9261=2/H9266. On the other hand, the contour of en-
ergy integration for ln /H9004¯/H9257/H9270/H20849/H9261/H20850with/H9261/H11022/H9266is deformed in the
complex plane to preserve analyticity, as shown in Fig. 3.
Specifically, the contour consists of the integration along thereal axis a part along the branch cut on the imaginary axis.The integration along the real axis yields
/H20885
−/H11009/H11009
d/H9280/H20875ln/H20873e/H9280/T+e−i/H9261
1+e/H9280/T/H20874+i/H9261/H9258/H20849−/H9280/H20850/H20876=−T/H9261˜2
2, /H2084967/H20850
where N/H11013/H20851/H9261/2/H9266/H20852,/H9261=/H9261˜+2/H9266N. The integration along the
branch cut of the logarithm yields − /H20849T/2/H20850/H20851/H208492/H9266N/H208502+4/H9266N/H9261˜/H20852,
resulting in the long- /H9270asymptotics
ln/H9004¯/H9257/H9270=−/H9270T/H92612/4/H9266. /H2084968/H20850
Substituting this in Eq. /H2084965/H20850, we correctly reproduce the
long-time asymptotics of the Green’s function G0/H11022at equilib-
rium, Eq. /H2084954/H20850.
Let us now turn to the nonequilibrium situation and con-
sider the double-step function
n/H9257/H20849/H9280/H20850=a/H9257n0/H20849/H9280−/H20850+/H208491−a/H9257/H20850n0/H20849/H9280+/H20850, /H2084969/H20850
where n0/H20849/H9280/H20850=/H9258/H20849−/H9280/H20850is the zero-temperature Fermi-Dirac dis-
tribution function, /H9280/H11006=/H9280−/H9262/H11006V/2, and 0/H11021a/H9257/H110211. The
value of/H9262is fixed by demanding that the total number of
electrons is the same as for the equilibrium distribution n0/H20849/H9280/H20850
/H20849which we use for normalization /H20850, yielding/H9280−=/H9280−/H208491−a/H20850eV,i
-iπ
πε
0 π −π −2π 2π
λln
ln
∆(
λ)κ(
λ)T
T
FIG. 3. /H20849Color online /H20850Analytic/H9004¯/H9257/H9270/H20849/H9261/H20850vs periodic/H9260/H20849/H9261/H20850continu-
ation of the functional determinant. The value of /H9004¯/H9257/H9270at/H9261=2/H9266
determines the free-electron GF, while ln /H9260/H208492/H9266/H20850=0 in view of
charge quantization. As an example, the equilibrium case is shown.Inset: contour of integration for the quasiclassical limit, Eq. /H2084966/H20850,o f
/H9004¯/H9257/H9270/H20849/H9261/H20850is deformed, since a singularity of the integrand crosses the
real axis at/H9261=/H9266.GUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-8/H9280+=/H9280+aeV. The distribution function in the time domain
n/H9257/H20849/H9270/H20850=/H20885
−/H11009/H11009d/H9280
2/H9266e−i/H9280/H9270+0/H9280n/H9257/H20849/H9280/H20850/H20849 70/H20850
can be straightforwardly calculated, and is given by a sum of
oscillating terms
n/H9257/H20849/H9270/H20850=/H208491−a/H9257/H20850eia/H9257eV/H9270n0/H20849/H9270/H20850+a/H9257e−ieV/H9270/H208491−a/H9257/H20850n0/H20849/H9270/H20850,/H2084971/H20850
where
n0/H20849/H9270/H20850=i
2/H92661
/H9270+i0/H2084972/H20850
is the T=0 Fermi-Dirac distribution function in time repre-
sentation.
On the other hand, we can find the time dependence of the
fermionic distribution function by using our nonequilibriumbosonization approach, leading to the identity /H2084965/H20850. In the
long-time limit, we need to evaluate the integral /H2084966/H20850, yield-
ing
ln/H9004¯/H9257/H9270/H20849/H9261/H20850/H11229eV/H9270
2/H9266/H20849ln/H208491−a/H9257+a/H9257e−i/H9261/H20850+a/H9257i/H9261/H20850. /H2084973/H20850
Analytically continuing in /H9261,w eg e t
ln/H9004¯/H9257/H9270/H208492/H9266/H20850/H11229ieV/H9270/H20877a/H9257−1 , a/H9257/H110221/2
a/H9257, a/H9257/H110211/2,/H20878 /H2084974/H20850
which reproduces the long-time time limit of the Green’s
function of free fermions with the distribution function /H2084971/H20850.
We have just demonstrated how the identity /H2084965/H20850works for a
double-step nonequilibrium distribution.
Equation /H2084965/H20850is a remarkable identity, as it connects two
seemingly unrelated objects: the distribution function of freefermions and a Fredholm determinant of the counting opera-tor. The value of /H9261=2
/H9266appearing in the bosonic representa-
tion of the free-fermion GF G0,/H9257/H20849/H9270/H20850has a clear physical
meaning: a fermion is a 2 /H9266soliton in the bosonic formalism.
IV. FERMI EDGE SINGULARITY
A natural question to ask is whether values of /H9004/H9257/H9270/H20849/H9261/H20850
away from/H9261=2/H9266are physically important. To see that this is
indeed the case, consider the Fermi edge singularity /H20849FES /H20850
problem. In this problem, an electron excited into the con-duction band, leaves behind a localized hole, resulting in ans-wave scattering phase shift,
/H92540, of the conducting
electrons.48In the mesoscopic context,49the FES manifests
itself in resonant tunneling experiments.50On a formal level,
it is described by the following Hamiltonian
H=/H20858
k/H9280kak†ak+E0b†b+/H20858
k,k/H11032Vk,k/H11032ak†ak/H11032bb†. /H2084975/H20850
While in the FES problem there is no interaction between
electrons in the conducting band, it has many features char-acteristic of genuine many-body physics. Historically, theFES problem was first solved by an exact summation of aninfinite diagrammatic series.
48Despite the fact that conven-tional experimental realizations of FES are three-
dimensional, the problem can be reduced /H20849due to the local
character of the interaction with the core hole /H20850to that of
one-dimensional chiral fermions. For this reason, bosoniza-tion technique can be effectively applied, leading to an alter-native and very elegant solution.
51
Away from equilibrium, the FES has been addressed in
Ref. 18where the canonical /H20849fermionic /H20850FES theory was
combined with the scattering matrix approach. Below, weapply the nonequilibrium bosonization technique to the sameproblem.
As mentioned above, the FES problem is effectively de-
scribed by chiral 1D electrons interacting with a core holethat is instantly “switched on.” As was shown in Ref. 51,
taking into account the core hole in the bosonization ap-proach amounts to replacement of e
i/H9278byei/H208491−/H92540//H9266/H20850/H9278in the
boson representation of the fermionic operator. Using Eqs./H208493/H20850and /H2084951/H20850, one gets
G
/H11022/H20849/H9270/H20850=−i/H9011
2/H9266v/H20855TKei/H208491−/H92540//H9266/H20850/H9278−/H208490,/H9270/H20850e−i/H208491−/H92540//H9266/H20850/H9278+/H208490,0/H20850/H20856/H2084976/H20850
and similarly for the function G/H11021. Within our nonequilibrium
formalism, this implies a replacement j→/H208491−/H92540//H9266/H20850jin Eq.
/H2084958/H20850. Performing the derivation as in the free-fermion case,
we thus obtain the nonequilibrium FES GF for electrons withan arbitrary distribution n/H20849
/H9280/H20850,
G/H11125/H20849/H9270/H20850=/H11007i/H9011/H9004¯/H9270/H208492/H9266−2/H92540/H20850/2/H9266v/H208491/H11006i/H9011/H9270/H20850/H208491−/H92540//H9266/H208502.
/H2084977/H20850
At equilibrium Eq. /H2084977/H20850can be further simplified /H20849see Appen-
dix A /H20850,
G/H11125/H20849/H9270/H20850=/H20873/H9266T/H9270
sinh/H9266T/H9270/H20874/H208491−/H92540//H9266/H208502/H11007i/H9011
2/H9266v/H208491/H11006i/H9011/H9270/H20850/H208491−/H92540//H9266/H208502,
/H2084978/H20850
reproducing the known results.48,51
For a double-step distribution, Eq. /H2084969/H20850, the long-time
limit is obtained as
/H9004/H9270/H208492/H9266−2/H92540/H20850/H11229e−/H9270/2/H9270/H9278/H208492/H92540/H20850, /H2084979/H20850
where the dephasing rate /H9270/H9278−1is given by
/H9270/H9278−1/H20849/H9261/H20850=−eV
2/H9266ln/H208731−4 a/H208491−a/H20850sin2/H9261
2/H20874. /H2084980/H20850
In the energy representation /H9270/H9278−1determines the broadening of
the split FES singularities. The same result for the broaden-ing of FES has been obtained by Abanin and Levitov in Ref.18within the fermionic framework. It is instructive to com-
pare their result with our analysis. In the bosonization tech-nique, we have expressed the GF of the FES problem interms of a functional determinant /H2084977/H20850. On the other hand,
within the fermionic approach
18the GF splits into a product
of an open line L/H20849/H9270/H20850/H20849i.e., single-particle Green’s function of
fermions in the presence of external time-dependent field /H20850
and closed loop eC/H20849i.e., vacuum loops of fermions in an
external field /H20850,BOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-9G/H11125/H20849/H9270/H20850=L/H11125/H20849/H9270/H20850eC, /H2084981/H20850
with the closed-loop part given by
eC=/H9004/H9270/H20849−2/H92540/H20850. /H2084982/H20850
This representation of the Green’s function is similar to the
functional bosonization approach,52–55that employs both fer-
mionic and bosonic variables. While functional and fullbosonization approaches yield equivalent results, this equiva-lence is highly nontrivial. Indeed, comparing Eq. /H2084981/H20850with
Eq. /H2084977/H20850and employing Eq. /H2084982/H20850, we establish the identity
/H11007i/H9011
2/H9266v/H208731/H11007i/H9270/H9011
1/H11006i/H9011/H9270/H20874/H208491−/H92540//H9266/H208502
/H9004/H9270/H208492/H9266−2/H92540/H20850=L/H11125/H20849/H9270/H20850/H9004/H9270/H20849−2/H92540/H20850
/H2084983/H20850
relating the functional determinants /H9004¯/H9270/H208492/H9266−2/H92540/H20850and
/H9004¯/H9270/H20849−2/H92540/H20850through the single-particle Green’s function L/H20849/H9270/H20850.
Since n/H9257is diagonal in energy space, while /H9254/H9257is diagonal
in time space, they do not commute, making the determinantnontrivial. It is worth noting that the functional determinants
/H9004¯/H9270/H20849/H9261/H20850for/H20841/H9261/H20841/H11021/H9266have been efficiently studied by numerical
means.56,57The identity /H2084983/H20850can be useful for the numerical
evaluation of/H9004¯/H9270/H20849/H9261/H20850at larger values of /H9261.
V. INTERACTING ELECTRONS
So far we have been dealing with noninteracting elec-
trons. Now, we focus on the main subject of this work:bosonization of interacting fermions, both for spinless andfor spinful cases. We begin by showing in Sec. VA how the
interaction can be incorporated into the nonequilibriumbosonization scheme developed above.
A. Keldysh action
For the problem of spinless interacting fermions, the
Hamiltonian reads
H=H0+Hee, /H2084984/H20850
where H0is given by Eq. /H208492/H20850and
Hee=1
2/H20885dxg/H20849x/H20850/H20851/H9267L/H20849x/H20850+/H9267R/H20849x/H20850/H208522, /H2084985/H20850
where g/H20849x/H20850is a spatially dependent interaction strength. To
model the coupling with noninteracting leads, we will as-sume that g/H20849x/H20850is constant within the interacting part of the
wire and “switches off” near the end points, x=/H11006L/2, see
Fig.4. This way of modeling leads was introduced in Refs.
58–60to study the conductance of a LL wire; it was also
exploited in Refs. 61and62to analyze the shot noise. In the
Lagrangian formulation, Eqs. /H2084984/H20850and /H2084985/H20850correspond to the
action
S/H20851
/H9274/H20852=S0/H20851/H9274/H20852+See/H20851/H9274/H20852,S/H20851/H9274/H20852=/H20885
cdt/H20885dx/H20858
/H9257/H20875/H9274/H9257†i/H11509/H9257/H9274/H9257−g/H20849x/H20850
2/H92672/H20849x,t/H20850/H20876,
where/H9267/H20849x/H20850=/H9267R/H20849x/H20850+/H9267L/H20849x/H20850. Decoupling the interaction term
via a bosonic field /H9272by means of a Hubbard-Stratonovich
transformation, we obtain the action
S/H20851/H9274,/H9272/H20852=/H20885
cdt/H20885dx/H20875/H20858
/H9257=R,L/H9274/H9257†/H20849i/H11509/H9257−/H9272/H20850/H9274/H9257+1
2/H9272g−1/H20849x/H20850/H9272/H20876.
/H2084986/H20850
The theory of fermions in an arbitrary field /H9272/H20849x,t/H20850/H20849on the
Keldysh contour /H20850can be bosonized using the results of Sec.
III. Introducing, as before, notations with /H20849without /H20850bar for
the quantum /H20849classical /H20850components, we obtain the action
S/H20851/H9272,/H9272¯,/H9267,/H9267¯/H20852=S0/H20851/H9267,/H9267¯/H20852+See/H20851/H9267,/H9267¯,/H9272,/H9272¯/H20852, /H2084987/H20850
where S0/H20851/H9267,/H9267¯/H20852is the bosonized action of nonequilibrium free
fermions, Eq. /H2084931/H20850and
See/H20851/H9267,/H9267¯,/H9272,/H9272¯/H20852=−/H20885dtdx /H20851/H9272/H9267¯+/H9272¯/H9267−/H9272g−1/H20849x/H20850/H9272¯/H20852./H2084988/H20850
Integrating out the auxiliary Hubbard-Stratonovich field /H9272,
we derive a theory written solely in terms of density fields,
S/H20851/H9267,/H9267¯/H20852=S0/H20851/H9267,/H9267¯/H20852−/H20885dtdxg /H20849x/H20850/H9267/H9267¯. /H2084989/H20850
Equation /H2084989/H20850constitutes a bosonic description for interact-
ing electrons out of equilibrium.
B. Tunneling spectroscopy of interacting
fermions, spinless case
We are now prepared to address the problem formulated
in the beginning of the paper: an interacting quantum wireout of equilibrium, Fig. 4. We will first calculate the GFs at
coinciding spatial points, which corresponds to tunnelingspectroscopy measurements. In Sec. VC, we will generalize
this analysis to GFs at different spatial points which are, inparticular, relevant to experiments on LL interferometers.
1. Tunneling into the interacting part of the wire
We consider GR/H11125/H20849/H9270/H20850for the tunneling point /H20849x=0/H20850located
inside the interacting part of the wire /H20849region II in Fig. 4/H20850;
generalization to tunneling into one of noninteracting leads/H20849regions I and III in Fig. 4/H20850is straightforward and will be
presented in Sec. VB3 .nR nL
K(x)II III I
LL
L/2 −L/2K=1
X
FIG. 4. /H20849Color online /H20850Schematic view of a LL conductor con-
nected to leads with two different incoming fermionic distributions.The LL interaction parameter K/H20849x/H20850is also shown; the dashed line
corresponds to its sharp variation in the boundaries.GUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-10Proceeding in the same way as for the noninteracting
case, we come to a representation of the GF in the form of an
integral over the density fields /H9267and/H9267¯, Eq. /H2084955/H20850. The only
difference as compared to the noninteracting case is that thebosonic action /H2084989/H20850now contains also the second term in-
duced by the interaction. Since this term is linear in the clas-sical component
/H9267/H9257, we can perform the integration over it in
the same way as we did in the noninteracting case. As aresult, we obtain equations satisfied by the quantum compo-
nents
/H9267¯/H9257of the density fields,
/H11509t/H9267¯R+/H11509x/H20875/H20873v+g
2/H9266/H20874/H9267¯R+g
2/H9266/H9267¯L/H20876=j,
/H11509t/H9267¯L−/H11509x/H20875/H20873v+g
2/H9266/H20874/H9267¯L+g
2/H9266/H9267¯R/H20876=0 , /H2084990/H20850
where the source term j/H20849x,t/H20850is defined by Eq. /H2084957/H20850. The
solution of Eq. /H2084990/H20850determines the phases /H9254/H9257/H20849t/H20850according to
Eqs. /H2084961/H20850and /H2084963/H20850. Remarkably, Eq. /H2084963/H20850expresses the phase
/H9254/H9257/H20849t/H20850affected by the electron-electron interaction, through
the asymptotic behavior of /H9267¯/H20849x,t/H20850in the noninteracting parts
of the wire /H20849regions I and III in Fig. 4/H20850. The phases/H9254/H9257/H20849t/H20850
determine the GFs via63
GR/H11125/H20849/H9270/H20850=/H11007i/H9011
2/H9266u/H9004¯R/H20851/H9254R/H20849t/H20850/H20852/H9004¯L/H20851/H9254L/H20849t/H20850/H20852
/H208491/H11006i/H9011/H9270/H208501+/H9253, /H2084991/H20850
where
/H9253=/H208491−K/H208502/2K, /H2084992/H20850
and
K=/H208491+g//H9266v/H20850−1 /2/H2084993/H20850
is the standard LL parameter in the interacting region.
To explicitly evaluate /H9254/H9257/H20849t/H20850for the structure of Fig. 4,i ti s
convenient to rewrite Eqs. /H2084990/H20850as a second-order differential
equation for the current
J¯=v/H20849/H9267¯R−/H9267¯L/H20850, /H2084994/H20850
/H20849/H92752+/H11509xu2/H20849x/H20850/H11509x/H20850J¯/H20849/H9275,x/H20850=0 , x/HS110050, /H2084995/H20850
where
u/H20849x/H20850=v/H208511+g/H20849x/H20850//H9266v/H208521/2=v
K/H20849x/H20850/H2084996/H20850
is a spatially dependent plasmon velocity. Reflection and
transmission of plasmons on both boundaries is characterized
by the coefficients r/H9257,t/H9257/H20849r/H92572+t/H92572=1/H20850; here the subscripts /H9257
refer to the boundaries between regions I/II and II/III. For
simplicity, we assume them to be constant over a character-istic frequency range
64/H9275/H11011/H9270−1. The scattering matrices on
the left and right boundaries have the form
SL=/H20873tL−rL
rLtL/H20874,SR=/H20873tRrR
−rRtR/H20874, /H2084997/H20850
where the first component corresponds to the left mover and
the second one to the right mover.Solution of Eq. /H2084995/H20850is quite straightforward. The bound-
ary points x=/H11006L/2 and the observation point x=0 divide
thexaxis into four regions /H20849I, II −,I I+, and III /H20850. In each of the
regions the function J¯/H20849/H9275,x/H20850satisfies the homogeneous wave
equation, with the velocity v/H20849in regions I and III /H20850oru/H20849in
regions II −and II +/H20850. The solution in each of the regions is
thus a sum of two waves propagating left and right. As dis-cussed after Eq. /H2084958/H20850, we need an advanced solution, which
imposes the condition that in the leads /H20849regions I and III /H20850
only incoming waves are present. There remain six coeffi-cients that are fixed by the boundary conditions at thesample/lead boundaries /H20851see Eq. /H2084997/H20850/H20852and by the matching
condition at the observation point /H20849x=0/H20850. The latter condition
is generated by the source term in Eq. /H2084990/H20850.
Solving Eq. /H2084995/H20850and using Eq. /H2084994/H20850, we find the quantum
density components
/H9267¯/H9257. In accordance with Eq. /H2084963/H20850the scat-
tering phases/H9254/H9257/H20849t/H20850are determined by the behavior of /H9267¯/H9257in
the asymptotic regions /H20849x/H11021−L/2 for/H9267¯Randx/H11022L/2 for/H9267¯L/H20850.
We find
/H9267¯R/H20849/H9275,x/H20850=/H208491+K/H20850tL
2/H208812Kveikx+i/H20849k−/H9260/H20850L/2
1−rRrLe−2i/H9260L/H208491−ei/H9275/H9270/H20850
/H11003/H208491−rRre−i/H9260L/H20850,x/H11021−L
2; /H2084998/H20850
/H9267¯L/H20849/H9275,x/H20850=/H208491+K/H20850tR
2/H208812Kve−ikx+i/H20849k−/H9260/H20850L/2
1−rRrLe−2i/H9260L/H208491−ei/H9275/H9270/H20850
/H11003/H20849r+rLe−i/H9260L/H20850,x/H11022L
2. /H2084999/H20850
Here, we use the notations k=/H9275/v,/H9260=/H9275/u, and r=/H208491
−K/H20850//H208491+K/H20850. Substituting this in Eq. /H2084963/H20850, we obtain/H9254/H9257/H20849t/H20850in
the form of a superposition of rectangular pulses,
/H9254/H9257/H20849t/H20850=/H20858
n=0/H11009
/H9254/H9257,nw/H9270/H20849t,tn/H20850, /H20849100 /H20850
where
tn=/H20849n+1 /2−1 /2K/H20850L/u /H20849101 /H20850
and
/H9254/H9257,2m=/H9266t−/H9257rLmrRm/H208491+/H9257K/H20850//H20881K,
/H9254/H9257,2m+1=−/H9266t−/H9257r/H9257m+1r−/H9257m/H208491−/H9257K/H20850//H20881K. /H20849102 /H20850
For the “partial equilibrium” state /H20849where nR/H20849t/H20850andnL/H20849t/H20850
are of Fermi-Dirac form but with different temperatures andchemical potentials /H20850the functional determinants are Gauss-
ian functions of phases, reproducing earlier results of func-tional bosonization.
28Indeed, using Eq. /H20849A10 /H20850,w efi n d
GR/H11125/H20849/H9270/H20850=/H11007i/H9011
2/H9266u1
/H208491/H11006i/H9011/H9270/H208501+/H9253
/H11003/H20873/H9266TR/H9270
sinh/H9266TR/H9270/H208741+/H9251/H20873/H9266TL/H9270
sinh/H9266TL/H9270/H20874/H9252
, /H20849103 /H20850
where the exponents 1+ /H9251and/H9252are given by the sumsBOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-111+/H9251/H11013/H20858
n=0/H11009/H20873/H9254R,n
2/H9266/H208742
,
/H9252/H11013/H20858
n=0/H11009/H20873/H9254L,n
2/H9266/H208742
. /H20849104 /H20850
Substituting here the results /H20849102/H20850for the phases /H9254/H9257,n,w e
obtain
1+/H9251=TL
1−RLRR/H208751+/H9253
2/H208491+RR/H20850/H20876,
/H9252=TR
1−RLRR/H20875RL+/H9253
2/H208491+RL/H20850/H20876, /H20849105 /H20850
in agreement with Ref. 28. Here T/H9257,R/H9257are plasmon trans-
mission and reflection probabilities on the left /H20849/H9257=L/H20850and
right /H20849/H9257=R/H20850boundaries, T/H9257=t/H92572,R/H9257=r/H92572,T/H9257+R/H9257=1. One
may check that, due to the sum rule
/H9251+/H9252=/H9253, /H20849106 /H20850
at thermal equilibrium /H20849TR=TL/H20850the GFs G/H11125are independent
of plasmon transmission/reflection amplitudes.
The phases/H9254/H9257/H20849t/H20850are shown in Fig. 5for two limits of
adiabatic /H20849r/H9257=0/H20850and sharp,
r/H9257=/H208491−K/H20850//H208491+K/H20850,
boundaries. Let us stress that when we speak here about
sharp boundaries, we mean that the extension of the contactregions is small compared to the characteristic plasmonwavelength /H11011u/
/H9275. It is assumed throughout the paper that
the structure is always smooth on the scale of the electronwavelength, so that no electron backscattering takes place.
In physical terms
/H9254/H9257/H20849t/H20850characterizes phase fluctuations in
the leads that arrive at the measurement point x=0 during the
time interval /H208510,/H9270/H20852. These fluctuations govern the dephasing
and the energy distribution of electrons encoded in the GFsG/H9257/H11125/H20849/H9270/H20850. Up to inversion of time, one can think of /H9254/H9257/H20849t/H20850as
describing the fractionalization of a phase pulse /H20849electron-
hole pair /H20850injected into the wire at point xduring the time
interval /H208510,/H9270/H20852. This is closely related to the physics of charge
fractionalization discussed earlier.31,60,65–68At the first step,
the pulse splits into two with relative amplitudes /H208491+K/H20850/2
and /H208491−K/H20850/2 carried by plasmons in opposite directions, cf.
Refs. 31,66, and 67. As each of these pulses reaches the
corresponding boundary, another fractionalization processtakes place: a part of the pulse is transmitted into a lead,while the rest is reflected. The reflected pulse reaches theother boundary, is again fractionalized there, etc. Let usstress an important difference between boundary fractional-ization of transmitted charge
60,68and that of dipole pulses
discussed here. While in the former case the boundaries canalways be thought of as sharp /H20849one is dealing with the small
q limit /H20850, in the present problem the way K/H20849x/H20850is turned on is
crucially important for reflection coefficients r
/H9257at/H9275/H11011/H9270−1.
For/H9270/H11270L/uthe coherence of plasmon scattering may be
neglected and the result splits into a product
/H9004¯/H9257/H20851/H9254/H9257/H20849t/H20850/H20852 /H11229/H20863
n=0/H11009
/H9004¯/H9257/H9270/H20849/H9254/H9257,n/H20850, /H20849107 /H20850
with each factor representing a contribution of a single phase
pulse/H9254/H9257,n/H20849t/H20850=/H9254/H9257,nw/H9270/H20849t,0/H20850.
We now apply our general results /H2084991/H20850and /H20849107/H20850to the
“full nonequilibrium” case, when n/H9257/H20849/H9280/H20850have a double-step
form, Eq. /H2084969/H20850. To obtain the exact form of the Green’s func-
tion G/H9257/H20849/H9270/H20850, one has to evaluate the Toeplitz determinants
numerically. Here, we restrict ourselves to the evaluation ofthe long-time asymptotics of G
/H9257/H20849/H9270/H20850that can be found ana-
lytically employing Eq. /H2084966/H20850and governs the broadening of
the split zero-bias-anomaly dips.27,28We focus on the adia-
batic limit when the distribution function remains unchangedand the broadening is solely due the nonequilibrium dephas-
ing rate,
27,281//H9270/H9278/H9257. We obtain
1//H9270/H9278R=1 //H9270/H9278RR+1 //H9270/H9278RL, /H20849108 /H20850
where 1 //H9270/H9278/H9257/H9257/H11032is the contribution to dephasing of the /H9257fer-
mions governed by the distribution of the /H9257/H11032fermions. These
dephasing rates are found to be
1//H9270/H9278R/H9257=−eV/H9257
2/H9266ln/H208731−4 a/H9257/H208491−a/H9257/H20850sin2/H9266/H208491+/H9257K/H20850
2/H20881K/H20874,
/H20849109 /H20850
see Fig. 6.
Two remarkable features of this result should be pointed
out. First, let us compare our results with the results of RPAapproximation. Consider the weak-interaction regime,
/H9253/H112701.
We then obtain
1//H9270/H9278RL/H11229/H9266/H9253eVLaL/H208491−aL/H20850, /H20849110 /H20850
1//H9270/H9278RR/H11229/H9266/H20849/H92532/8/H20850eVRaR/H208491−aR/H20850. /H20849111 /H20850
This should be contrasted with RPA which predicts equal
1//H9270/H9278RLand 1 //H9270/H9278RR, see Ref. 27. While 1 //H9270/H9278RLagrees with the
RPA result, 1 //H9270/H9278RRis parametrically smaller /H20849suppressed by-0.20.20.61δR/2π
t-0.400.4δL/2π
t-r2r2
r4
r3
r5-r4
-r-r3a)
b)c)
d)r(1+K)/2 K1/2
(1-K)/2 K1/2
FIG. 5. /H20849Color online /H20850Phases/H9254/H9257/H20849t/H20850entering Eq. /H2084991/H20850for the GFs
for sharp /H20851/H20849a/H20850,/H20849b/H20850;r=/H208491−K/H20850//H208491+K/H20850/H20852and adiabatic /H20851/H20849c/H20850,/H20849d/H20850/H20852
boundaries.GUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-12an extra factor of /H9253/H20850. The reason for this failure of RPA is
clear from our analysis. For a weak interaction the contribu-tions of R and L movers to G
Rare given by the functional
determinants/H9004/H9257/H9270/H20849/H9254/H9257/H20850with phases /H20849for adiabatic boundaries /H20850
/H9254L/H11229/H208491−K/H20850/H9266and/H9254R/H11229/H9266/H208491+K/H20850. While the contribution of
the small phase /H9254Lis captured correctly by RPA, a small- /H9254
expansion of ln /H9004R/H9270/H20849/H9254R/H20850fails for large/H9254R/H20849apart from equi-
librium and “partial equilibrium” where ln /H9004/H9257/H9270/H20849/H9254/H20850/H11008/H92542/H20850.
Another important observation is that for certain values of
the interaction parameter K/H20849different for/H9257=RandL/H20850the
dephasing rates 1 //H9270/H9278R/H9257vanish. This implies that, for these
values of K, the GF does not decay exponentially in time, so
that the power-law zero bias anomaly /H20849ZBA /H20850is not smeared.
The absence of dephasing indicates that for these values ofinteraction the system reduces in some sense to a noninter-acting model. As we are going to show, at these points thefunctional determinant can be calculated exactly.
2. Refermionization
The points of no-dephasing correspond to the value of the
phase/H9254/H20849argument of the functional determinant /H20850equal to
/H9254=2/H9266nwith an integer n. We will demonstrate that at these
points the functional determinant /H9004¯/H9270/H20849/H9254/H20850can be calculated
exactly by “refermionization.” The case /H9254=2/H9266corresponds
to the noninteracting /H20849K=1/H20850single-particle GF and has been
already analyzed in Sec. III. To study the case /H9254=4/H9266,w e
consider a two-fermion GF
G2=/H20855T/H9274†/H208491/H20850/H9274†/H208492/H20850/H9274/H208493/H20850/H9274/H208494/H20850/H20856. /H20849112 /H20850
We focus on the limit of merging points, t1=t2=0,t3=t4=/H9270;
x1,x2,x3,x4→x, which corresponds to simultaneous creation
and annihilation of two fermions, and thus should generate
/H9004¯/H9270/H208494/H9266/H20850. For noninteracting electrons the GF G2can be
readily calculated. Using Wick theorem, we find
G2=G/H208493,1/H20850G/H208494,2/H20850−G/H208494,1/H20850G/H208493,2/H20850. /H20849113 /H20850
If the spatial points strictly coincide, x1=x2=x3=x4=x, the
function G2vanishes. A finite result is obtained after splitting
the points by distances on the order of Fermi length, si
/H11011v//H9011. We thus findG2/H20849/H9270/H20850=1
2/H20849s1−s2/H20850/H20849s4−s3/H20850
/H11003/H20849/H11509s˜1−/H11509s˜2/H208502G/H20849/H9270,s˜1/H20850G/H20849/H9270,s˜2/H20850/H20841s˜1=s˜2=0, /H20849114 /H20850
where xi=x+si. In the bosonic description, this corresponds
to
G2/H20849/H9270/H20850=/H20873/H9011
2/H9266v/H208742
/H20855TKe2i/H9278/H20849/H9270/H20850−2i/H9278/H208490/H20850/H20856. /H20849115 /H20850
As was shown above, this correlation function can be evalu-
ated, with the result expressed in terms of a functional deter-minant,
G
2/H20849/H9270/H20850=/H20873/H9011
2/H9266v/H208742/H9004¯/H9270/H208494/H9266/H20850
/H20849/H9011/H9270/H208504, /H20849116 /H20850
where we used /H9270/H11271/H9011−1.
Comparing Eqs. /H20849114/H20850and /H20849116/H20850, we express/H9004¯/H9270/H208494/H9266/H20850
through the free-electron GFs,
/H9004¯/H9270/H208494/H9266/H20850=/H208492/H9266/H208502/H20849v/H9270/H208504/H20849/H11509s1−/H11509s2/H208502G/H20849s1,/H9270/H20850G/H20849s2,/H9270/H20850./H20849117 /H20850
The numerical coefficient /H208492/H9266/H208502was restored by comparison
with equilibrium case. For a double-step distribution func-tion, Eq. /H2084969/H20850, we find from Eq. /H20849117/H20850
/H9004¯/H9270/H208494/H9266/H20850=e2i/H20849a/H9257−1/H20850eV/H9270/H20873/H9266T/H9257/H9270
sinh/H9266T/H9257/H9270/H208742/H20875a/H9257/H20849a/H9257−1/H20850/H20849eV/H9270/H208502eieV/H9270
+/H20849a/H9257+/H208491−a/H9257/H20850eieV/H9270/H208502/H20873/H9266T/H9257/H9270
sinh/H9266T/H9257/H9270/H208742/H20876. /H20849118 /H20850
We see that/H9004¯/H9270/H208494/H9266/H20850shows oscillations in /H9270. At zero tempera-
ture there is no exponential damping. The absence of damp-ing is a manifestation of the vanishing dephasing rate, see thediscussion above. Another interesting property of the result/H20849118/H20850is the emergence of oscillations with three frequencies:
−2aeV,/H208491−2a/H20850eV, and /H208492−2a/H20850eV, implying three points of
singular behavior in the energy space. Let us recall that theinput double-step distribution had two such points: − aeVand
/H208491−a/H20850eV. With increasing interaction strength, the corre-
sponding twofold singularity gets progressively moresmeared /H20851see Eq. /H2084980/H20850/H20852, but then as
/H9254approaches/H9254=4/H9266,a
threefold singularity emerges at the new positions, see Fig. 7.
This procedure can be extended to a more general case of
/H9254=2/H9266n. Indeed, the simultaneous creation and annihilation
ofnnoninteracting fermions is described by
Gn/H20849/H9270/H20850=/H20855/H20851/H9274†/H20849/H9270/H20850/H20852n/H20851/H9274/H208490/H20850/H20852n/H20856, /H20849119 /H20850
where we again imply a point splitting on a distance of the
order of Fermi wavelength, i.e., /H11011v//H9011./H20849One can check that
the relation resulting from this consideration does not dependon details of the point-splitting procedure /H20850. The function G
n
can be expressed in terms of single-particle GFs as follows:0 0.2 0.4 0.6 0.8 1
K00.10.20.30.41/eVητφRη
RPARRRL
FIG. 6. /H20849Color online /H20850Dephasing rates 1 //H9270/H9278RRand 1 //H9270/H9278RLas func-
tion of LL parameter Kfor the adiabatic case and double-step dis-
tributions with aR=aL=1 /3.BOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-13Gn/H20849/H9270/H20850=Cnsn/H20849n−1/H20850/H20863
i/HS11005jn
/H20849/H11509si−/H11509sj/H20850G/H20849s1,/H9270/H20850
/H11003G/H20849s2,/H9270/H20850¯G/H20849sn,/H9270/H20850/H20841si=0. /H20849120 /H20850
Here, Cnare numerical coefficients on the order of unity. On
the other hand, in the bosonic framework we have
Gn/H20849/H9270/H20850=/H20873/H9011
2/H9266v/H20874n/H9004¯/H9270/H208492/H9266n/H20850
/H20849/H9011/H9270/H20850n2. /H20849121 /H20850
Demanding the equivalence of Eqs. /H20849120/H20850and /H20849121/H20850,w ee s -
tablish the identity
/H9004¯/H9270/H208492/H9266n/H20850=Cn/H20849v/H9270/H20850n2/H20863
i/HS11005jn
/H20849/H11509si−/H11509sj/H20850G/H20849s1,/H9270/H20850¯G/H20849sn,/H9270/H20850/H20841s=0,
/H20849122 /H20850
expressing the functional determinant /H9004¯/H9270/H20849/H9254/H20850through free fer-
mionic GFs G/H20849/H9270/H20850for/H9254=2/H9266n. The numerical coefficients Cn
can be restored by comparison with the known result for
/H9004¯/H9270/H208492/H9266n/H20850at equilibrium. The explicit form of /H9004¯/H9270/H20849/H9254=2/H9266n/H20850can
be readily found by substituting in Eq. /H20849122/H20850an explicit ex-
pression for the GF for a given distribution function.
3. Tunneling into noninteracting regions
Next, we discuss the tunneling spectroscopy for the non-
interacting parts of the wire. Let us focus on the right-moving electrons; the analysis of left-moving ones can be
done in the same way. For x1,x2/H11021−L/2/H20849region I in Fig. 4/H20850
the GF is the one of free fermions, as the right-moving par-ticles emerging from the left reservoir are not yet aware ofthe interacting region they are about to enter. The situation isless trivial for x
1,x2/H11022L/2/H20849region III /H20850. Indeed, while the
strength of the interaction in this region is zero, right-movingelectrons there have passed through the interacting part ofthe wire, which modifies their Green’s function. We will
show below that the GFs G
R/H11125/H20849x1,t1;x2,t2/H20850in the noninteract-
ing region satisfy Galilean invariance: they depend on /H20849x1
−vt1/H20850−/H20849x2−vt2/H20850only. For this reason, it is sufficient to con-
sider x1=x2to obtain the full information about the GF.
The evaluation of the GF is performed in the same way as
in the interacting region, yielding the result /H2084991/H20850. The phases
/H9254/H9257/H20849t/H20850are now given by
/H9254R/H20849t/H20850=/H20858
n=0/H11009
/H9254R,nw/H9270/H20849t,x1/v+2tn/H20850, /H20849123 /H20850
/H9254L/H20849t/H20850=2/H9266rRw/H9270/H20873t,x1−L
v/H20874+/H20858
n=0/H11009
/H9254L,nw/H9270/H20873t,x1
v+L
u+2tn/H20874,
/H20849124 /H20850
with the following amplitudes of rectangular pulses:
/H9254R,n=2/H9266tLtR/H20849rLrR/H20850n,
/H9254L,n=−2/H9266/H20849rLrR/H20850nrLtR. /H20849125 /H20850
In the case of smooth boundaries only one pulse is created,
/H9254R,0=2/H9266, reproducing the free fermion GF. Thus, in the adia-
batic case, the interaction has no influence on GFs in thenoninteracting parts of the wire, as expected. If the transitionbetween noninteracting and interacting parts of the wire isnot smooth, plasmon scattering takes place. This processleads to a redistribution of electrons over energies
28and thus
affects GFs in the noninteracting region.
C. Green’s functions at different points
and Aharonov-Bohm interferometry
So far we have discussed GFs at coinciding spatial points,
having in mind tunneling spectroscopy experiments. We nowconsider GFs at different spatial coordinates. Such GFs arerelevant to various physical quantities, in particular, in thecontext of Aharonov-Bohm interferometry. The similar prob-lem in the context of chiral edge state has been considered inRefs. 69and 70. Let us consider a four-terminal setup
formed by two quantum wires coupled by tunneling at twopoints, as schematically shown in Fig. 8. Each one of the
quantum wires is assumed to be a LL conductor connected totwo noninteracting electrodes with arbitrary /H20849in general, non-
equilibrium /H20850distribution functions, as shown in Fig. 4.W e
are interested in the Aharonov-Bohm effect, i.e., the depen-dence on the magnetic flux /H9021of the electric current flowing
from wire 1 into wire 2. Consider the situation where thetunnel coupling between the wires 1 and 2 is weak. We alsoassume that both arms of the AB-interferometer have equal-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
ε/V00.511-n
-2 -1 0 1 2
ε/V00.511.522.53y"
(b)(a)
FIG. 7. /H20849Color online /H20850Refermionization at a no-dephasing
point. Upper plot: double-step distribution function 1− n/H20849/H9280/H20850. Lower
plot: function y/H11033/H20849/H9280/V/H20850, where G/H92540=−/H9266/H11022/H20849/H9280/H20850=−i//H208492v/H20850/H20849V//H9011/H208503y/H20849/H9280/V/H20850is
the FES Green’s function /H2084977/H20850proportional to the determinant
/H9004¯/H9270/H208494/H9266/H20850with the argument /H9254=4/H9266being integer multiple of 2 /H9266. The
second derivative is plotted in order to emphasize singularities. Thearrow at
/H9280=0 denotes a delta-function contribution to y/H11033.GUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-14length dand tunneling occurs at points located inside the
interacting part of the wire. The flux dependent part of theelectric current is given by
I
/H9278=/H20841t12/H208412/H20885
−/H11009/H11009
dte−i/H9278/H20851G2/H11021/H20849d,t/H20850G1/H11022/H20849−d,−t/H20850
−G2/H11022/H20849d,t/H20850G1/H11021/H20849−d,−t/H20850/H20852+ h.c., /H20849126 /H20850
where the subscripts 1 and 2 label the wire and t12is the
tunneling matrix element between the wires. Separating theGF into left- and right-moving part, one gets
I
/H9278=/H20841t12/H208412/H20858
/H9257/H20885
−/H11009/H11009
dte−i/H9278/H20851G2,/H9257/H11021/H20849d,t/H20850G1,/H9257/H11022/H20849−d,−t/H20850
−G2,/H9257/H11022/H20849d,t/H20850G1,/H9257/H11021/H20849−d,−t/H20850/H20852+ h.c., /H20849127 /H20850
where we have neglected terms that oscillate fast with the
interferometer size d.
To analyze the GF G/H9257/H11125/H20849x1,x2,/H9270/H20850between two different
points of a wire, we proceed in the same way as in the casex
1=x2above. Integration over the classical component of the
density field leads to equations of motion for its quantumcomponent /H20849we choose
/H9257=Rfor definiteness /H20850,
/H20849−i/H9275+v/H11509x/H20850/H9267¯R,/H9275+/H11509x/H20873g
2/H9266/H9267¯/H9275/H20874=j/H20849/H9275,x;x1,x2,/H9270/H20850,
/H20849i/H9275+v/H11509x/H20850/H9267¯L,/H9275+/H11509x/H20873g
2/H9266/H9267¯/H9275/H20874=0 , /H20849128 /H20850
where we have used the /H20849/H9275,x/H20850representation; /H9267¯/H9275=/H9267¯R,/H9275
+/H9267¯L,/H9275. Equations /H20849128/H20850differ from the earlier Eq. /H2084990/H20850only
by the source term, which now reads
j/H20849/H9275,x;x1,x2,/H9270/H20850=1
/H208812/H20851/H9254/H20849x−x1/H20850ei/H9275/H9270−/H9254/H20849x−x2/H20850/H20852./H20849129 /H20850
Solving Eq. /H20849128/H20850, we find for x/H11021−L/2
/H9267¯R/H20849/H9275,x/H20850=/H208491+K/H20850tL
2/H208812Kveikx+i/H20849k−/H9260/H20850L/2
1−e−2i/H9260LrRrL/H20851e−i/H9260x2−ei/H20849/H9275/H9270−/H9260x1/H20850
−rRrei/H20849x2−L/H20850/H9260+rRrei/H9275/H9270+i/H20849x1−L/H20850/H9260/H20852. /H20849130 /H20850
Similarly, we find for x/H11022L/2/H9267¯L/H20849/H9275,x/H20850=/H208491+K/H20850tR
2/H208812Kve−ikx+i/H20849k−/H9260/H20850L/2
1−e−2i/H9260LrRrL/H20851rei/H9260x2−rei/H20849/H9275/H9270+/H9260x1/H20850
+rLei/H9275/H9270−/H20849x1+L/H20850/H9260−rLe−i/H20849x2+L/H20850/H9260/H20852. /H20849131 /H20850
Employing Eqs. /H2084963/H20850,/H20849130/H20850, and /H20849131/H20850, we obtain the fol-
lowing result for the GF:
GR/H11125/H20849x1,x2,/H9270/H20850=−1
2/H9266u/H20849/H11006i/H9011/H20850/H9253
/H11003/H9004¯R/H20851/H9254R/H20849t/H20850/H20852/H9004¯L/H20851/H9254L/H20849t/H20850/H20852
/H20873/H9270−x1−x2
u/H11007i
/H9011/H208741+/H9251/H20873/H9270+x1−x2
u/H11007i
/H9011/H20874/H9252.
/H20849132 /H20850
It is interesting to note that for spatially separated points the
scaling of GF with time /H20849and consequently with energy /H20850is
affected by plasmon scattering at the boundaries betweenwire and the leads. Surprisingly, even at equilibrium the GFinside interacting region is affected by the way interaction isturned on. For coinciding spatial points, the universal LLexponents, characteristic of an infinite wire, are restored dueto the sum rule /H20849106/H20850.
In the long wire limit, the functional determinant splits, as
before, into a product
/H9004
¯R/H20851/H9254R/H20849t/H20850/H20852 /H11229/H20863
n=0/H11009
/H9004¯R,/H9270−/H20849x1−x2/H20850/u/H20849/H9254R,2n/H20850/H9004¯R,/H9270+/H20849x1−x2/H20850/u/H20849/H9254R,2n+1/H20850.
/H20849133 /H20850
Here,/H9254/H9257,nare given by Eq. /H20849102/H20850. The calculation of /H9004¯Lis
performed in a similar way, yielding
/H9004¯L/H20851/H9254L/H20849t/H20850/H20852 /H11229/H20863
n=0/H11009
/H9004¯L,/H9270+/H20849x1−x2/H20850/u/H20849/H9254L,2n/H20850/H9004¯L,/H9270−/H20849x1−x2/H20850/u/H20849/H9254L,2n+1/H20850.
/H20849134 /H20850
We see that in the case of a GF at different spatial points,
the time argument of /H9004¯/H9257,/H9270/H20849/H9254/H9257/H20850/H20849determining the duration of
the pulses /H20850is replaced as compared to the case of x1=x2by
/H9270→/H9270/H11007/H9257x1−x2
u, /H20849135 /H20850
with the − /H20849+/H20850sign corresponding to even /H20849respectively, odd /H20850
pulses. It is easy to understand the reason for these alternat-ing signs. The even pulses are those that experience an evennumber of reflections, thus preserving their chirality, whilethe odd pulses experience an odd number of reflections andthus invert their chirality. We note that in the case when bothpoints are located in one of the noninteracting regions/H20849x
1,x2/H11022L/2o r x1,x2/H11021−L/2/H20850, the same consideration leads
to an analogous replacement of the time argument but withthe bare velocity
v,t12t12
Φ
12d d
FIG. 8. /H20849Color online /H20850Aharonov-Bohm setup in a four-terminal
geometry, with tunnel coupling /H20849dashed lines /H20850at two points. Both
interferometer arms have length d.BOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-15/H9270→/H9270−x1−x2
v/H20849136 /H20850
in the phases/H9254/H9257/H20849t/H20850, Eqs. /H20849123/H20850and /H20849124/H20850, entering/H9004¯/H9257,/H9270/H20849/H9254/H9257/H20850
and, correspondingly, in GFs. Since there is no fractionaliza-tion at the tunneling processes into a noninteracting region,only half of the pulses survives and no sign alternationarises.
Of particular interest is the value of GF at the interaction-
renormalized “light cone,” x
1−x2=/H11006ut. The value of the GF
at these points determines the integral for the interferencecurrent in Eq. /H20849127/H20850, see also Refs. 22and31. Let us con-
sider for simplicity the case of adiabatic barrier /H20849r
L=rR=0/H20850
when each of the products /H20849133/H20850and /H20849134/H20850reduces to the
first factor. Compared to the limit of coinciding spatialpoints, the duration of pulse in the functional determinanthas changed. The contribution associated with /H20849x
1−x2=/H9257ut/H20850
leads to a doubling of pulse duration in /H9004¯−/H9257, while/H9004¯/H9257has
disappeared altogether. As we see now, the dephasing rate
governing the exponential damping of the GF G/H9257/H11125atx1−x2
=/H9257/H11032u/H9270is
1//H9270/H9278/H20849/H9257/H11032/H20850AB;/H9257=2 //H9270/H9278/H9257,−/H9257/H11032, /H20849137 /H20850
where 1 //H9270/H9278/H9257/H9257/H11032are the partial dephasing rates for the tunneling
spectroscopy problem /H20849coinciding spatial points /H20850, as intro-
duced in Sec. VB1 . The dephasing rates /H20849137/H20850manifest
themselves in the interferometry measurements by inducingan exponential damping of the corresponding contributionsto the Aharonov-Bohm oscillations /H20849thus, the superscript
“AB” /H20850. In the limit of large interferometer size d, the contri-
bution with the lowest dephasing rate will dominate,
1/
/H9270/H9278AB= min
/H9257,/H9257/H11032=R,L2//H9270/H9278/H9257/H9257/H11032. /H20849138 /H20850
For double-step distributions, the dephasing rates /H20849137/H20850
are given /H20849up to a factor of 2 /H20850by Eq. /H20849109/H20850. With increasing
interaction the dephasing rate 1 //H9270/H9278ABbegins to oscillate as a
function of interaction parameter K, as illustrated /H20849for the
case of adiabatic contacts with leads /H20850in Fig. 6. This leads to
a remarkable prediction: the visibility of Aharonov-Bohmoscillations should be a strongly oscillating function of theinteraction strength.
D. Spinful Luttinger liquid
We now consider the problem of tunneling spectroscopy
for spinful electrons. The analysis is a straightforward exten-sion of the spinless case, analyzed in Sec. VB. We begin
with a fermionic Hamiltonian, which, in the spinful case, isgiven by
H=H
0+Hee, /H20849139 /H20850
H0=−iv/H20858
/H9268/H20849/H9274R,/H9268†/H11509x/H9274R,/H9268−/H9274L,/H9268†/H11509x/H9274L,/H9268/H20850, /H20849140 /H20850Hee=1
2/H20858
/H9257,/H9257/H11032;/H9268,/H9268/H11032/H20885dxg/H20849x/H20850/H9267/H9257,/H9268/H9267/H9257/H11032,/H9268/H11032. /H20849141 /H20850
where the index /H9268=↑,↓labels the spin projection. We now
switch to a Lagrangian description. To construct the free partof the action on the Keldysh contour, we repeat the stepsdescribed in detail in Sec. IIIand find
S
0=/H20858
/H9257=R,L;/H9268=↑,↓/H20849−/H9267¯/H9257,/H9268/H9016/H9257a−1/H9267/H9257,/H9268−ilnZ/H20851/H9273¯/H9257,/H9268/H20852/H20850,/H20849142 /H20850
where
/H9273¯/H9257,/H9268=/H9016/H9257a−1/H9267¯/H9257,/H9268. /H20849143 /H20850
The interacting part of the action reads
See=− /H20858
/H9257,/H9257/H11032,/H9268,/H9268/H11032/H20885dxg/H20849x/H20850/H9267/H9257,/H9268/H9267¯/H9257/H11032,/H9268/H11032. /H20849144 /H20850
To describe the tunneling spectroscopy measurements, we
need to find the single-particle GFs
G0,/H9257,/H9268/H11021/H20849x,t/H20850=i/H20855/H9274/H9257,/H9268†/H208490,0/H20850/H9274/H9257,/H9268/H20849x,t/H20850/H20856,
G0,/H9257,/H9268/H11022/H20849x,t/H20850=−i/H20855/H9274/H9257,/H9268/H20849x,t/H20850/H9274/H9257,/H9268†/H208490,0/H20850/H20856. /H20849145 /H20850
The fermionic operators are expressed in terms of bosonic
fields /H20849which now also carry the spin label /H20850in the usual way,
/H9274/H9257,/H9268/H20849x/H20850/H11229/H20873/H9011
2/H9266v/H208741/2
ei/H9257pFxei/H9278/H9257,/H9268/H20849x/H20850. /H20849146 /H20850
Substituting Eq. /H20849146/H20850into Eq. /H20849145/H20850, representing the GF as
a bosonic functional integral with the action S0+See, and per-
forming the integration over the classical component of thedensity field, we find the equation of motion satisfied by thequantum components of the field,
/H20849
/H11509t+v/H11509x/H20850/H9267¯R+/H11509xg
2/H9266/H20849/H9267¯R+/H9267¯L/H20850=j/H20849x,t/H20850,
/H20849−/H11509t+v/H11509x/H20850/H9267¯L+/H11509xg
2/H9266/H20849/H9267¯R+/H9267¯L/H20850=0 , /H20849147 /H20850
and
/H20849/H11509t+v/H11509x/H20850s¯R=j/H20849x,t/H20850,
/H20849−/H11509t+v/H11509x/H20850s¯L=0 . /H20849148 /H20850
Here, we have passed to new variables that describe the spin
and charge sectors of excitations,
/H9267¯R=/H9267¯R,↑+/H9267¯R,↓,/H9267¯L=/H9267¯L,↑+/H9267¯L,↓
s¯R=/H9267¯R,↑−/H9267¯R,↓,s¯L=/H9267¯L,↑−/H9267¯L,↓. /H20849149 /H20850
As one sees, the equations for the charge and spin degrees of
freedom are decoupled, which is a manifestation of spin-charge separation. The spin-density component obeys thesame equation as the density of free fermions, Eq. /H2084958/H20850.
Therefore, the spin sector is characterized by a LL parameterGUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-16Ks=1. As follows from Eq. /H20849148/H20850, the spin component s/H9257
propagates through the wire without any reflection.
To find the charge component, we define the charge den-
sity current
J¯=v/H20849/H9267¯R−/H9267¯L/H20850. /H20849150 /H20850
In terms of J¯, Eqs. /H20849147/H20850are reduced to a second order dif-
ferential equation,
/H20849/H92752+/H11509xuc2/H11509x/H20850J¯= 0 for x/HS110050, /H20849151 /H20850
where
uc2/H20849x/H20850=v2/Kc2/H20849x/H20850,
Kc=/H208731+2g
/H9266v/H20874−1 /2
. /H20849152 /H20850
Equation /H20849151/H20850coincides with Eq. /H2084995/H20850up to a different defi-
nition of the LL parameter. The interaction parameter /H9253=/H208491
−Kc/H208502/2Kcand the transmission and reflection amplitudes
are determined as for spinless fermions, with the replacementK→K
c.
The resulting expression for the Green’s function of spin-
ful fermions within the nonequilibrium bosonization ap-proach reads
G
R,↑/H11125/H20849/H9270/H20850=/H11007i/H9011
2/H9266/H20881uv/H20863/H9257,/H9268/H9004¯/H9257,/H9268/H20851/H9254/H9257,/H9268/H20849t/H20850/H20852
/H208491/H11006i/H9011/H9270/H208501+/H9253/2. /H20849153 /H20850
Here, we have assumed for generality that distribution func-
tions of spin-up and spin-down particles may be different.Therefore, the distribution function is labeled by two indices/H20849chirality and spin projection /H20850; these indices are inherited by
the functional determinant. The time-dependent phases of thespinful fermions
/H9254/H9257,/H9268/H20849t/H20850are expressed in terms of the scatter-
ing phases/H9254/H9257/H20849t/H20850of spinless fermions, Eq. /H20849100/H20850, in the fol-
lowing way:
/H9254L,↑/H20849t/H20850=/H9254L,↓/H20849t/H20850=1
2/H9254L/H20849t/H20850, /H20849154 /H20850
/H9254R,↑/H20849t/H20850=1
2/H20851/H9254R/H20849t/H20850+/H9254R0/H20849t/H20850/H20852, /H20849155 /H20850
/H9254R,↓/H20849t/H20850=1
2/H20851/H9254R/H20849t/H20850−/H9254R0/H20849t/H20850/H20852, /H20849156 /H20850
where/H9254R0/H20849t/H20850corresponds to noninteracting fermions and con-
sists of a single pulse with an amplitude 2 /H9266,/H9254R0/H20849t/H20850
=2/H9266w/H9270/H20849t,0/H20850.
We conclude that the inclusion of spin changes the scat-
tering phases in an essential way. This is most importantlyseen when considering the first pulse propagating to theright. Let us assume that there is no reflection at the bound-aries with noninteracting leads. The corresponding scatteringphases are each a superposition of the spin and chargemodes, see Eqs. /H20849155/H20850and /H20849156/H20850. Since the velocities of these
modes are different /H20849
vandu, respectively /H20850, then for suffi-ciently long wires, L/H20849v−1−u−1/H20850//H9270/H112711, the first pulse splits
into a charge and a spin parts. For a short wire, the spin pulseand the first charge pulse overlap. In this case one has to dealwith the general formula /H20849153/H20850and with time-dependent
phase containing both, spin and, charge contributions.Hence, if the wire is sufficiently short /H20849or, in other words, for
a given length of the wire the interaction is sufficientlyweak /H20850the spin-charge separation does not have enough time
to develop. For sufficiently long wires, spin-charge separa-tion does take place, in which case the respective determi-nants can be written as products of spin and charge contri-butions. This decomposition is not valid for short wires /H20849or,
for a given length of the wire, for sufficiently weak interac-tion /H20850. Note that at equilibrium there is significant simplifica-
tion. The GFs depend only on the sum of the scatteringphases squared, see Eq. /H20849103/H20850. Due to the sum rule /H20849106/H20850this
combination remains unchanged, regardless of whether spinand charge pulses overlap or not. Thus, at equilibrium onecan always think about these two modes separately and ad-ditively. Out of equilibrium, the dependence of the GF onscattering phases is more subtle, and the results for overlap-ping and separated pulses are different. Therefore, the spin-charge separation occurs in this case only for sufficientlylong wires. Focusing on this regime, we find
/H9004
¯R,↑/H20851/H9254R,↑/H20852=/H9004¯R,/H9270,↑/H20849/H9266/H20850/H20863
n=0/H11009
/H9004¯R,/H9270,↑/H20873/H9254R,n
2/H20874, /H20849157 /H20850
/H9004¯R,↓/H20851/H9254R,↓/H20852=/H9004¯R,/H9270,↓/H20849−/H9266/H20850/H20863
n=0/H11009
/H9004¯R,/H9270,↓/H20873/H9254R,n
2/H20874, /H20849158 /H20850
/H9004¯L,/H9268/H20851/H9254L,/H9268/H20852=/H20863
n=0/H11009
/H9004¯L,/H9270,/H9268/H20873/H9254L,n
2/H20874. /H20849159 /H20850
The first factor in each of Eqs. /H20849157/H20850and /H20849158/H20850originate
from the spin mode yielding the phase /H9266, i.e., a half of the
free-fermion phase value. The scattering phases of otherpulses /H20849originating from the charge mode /H20850have a half of
their values for spinless electrons.
Let us analyze this result. Consider the case of smooth
/H20849adiabatic /H20850contacts with leads, so that only one pulse passes
in each direction /H20849all
/H9254/H9257,nwith n/H113501 are zero /H20850. For the case of
partial equilibrium, the determinants can be evaluated explic-itly, yielding
/H20863
/H9268=↑,↓/H9004¯R,/H9268/H20851/H9254R,/H9268/H20849t/H20850/H20852=/H20873/H9266TR/H9270
sinh/H9266TR/H9270/H208741+/H9251/2
,
/H20863
/H9268=↑,↓/H9004¯L,/H9268/H20851/H9254L,/H9268/H20849t/H20850/H20852=/H20873/H9266TL/H9270
sinh/H9266TL/H9270/H20874/H9252/2
, /H20849160 /H20850
where/H9251,/H9252are given by Eq. /H20849105/H20850. For a double-step dis-
tribution function, a semiclassical limit of the determinants/H20849157/H20850–/H20849159/H20850can be readily evaluated. In the large-time limit
the behavior of the r.h.s. of Eqs. /H20849157/H20850–/H20849159/H20850is exponential,
yielding the partial decay ratesBOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-17/H9003RR=−eV
/H9266/H20877ln/H208511−4 aR/H208491−aR/H20850/H20852
+/H20858
n=0/H11009
ln/H208751−4 aR/H208491−aR/H20850sin2/H9254R,n
4/H20876/H20878,
/H9003RL=−eV
/H9266/H20858
n=0/H11009
ln/H208731−4 aL/H208491−aL/H20850sin2/H9254L,n
4/H20874, /H20849161 /H20850
and the total rate /H9003R=/H9003RR+/H9003RL. For simplicity, we have as-
sumed equal distributions for both spin components, a/H9257,↑
=a/H9257,↓/H11013a/H9257. Let us stress an important difference with the
spinless case. There, for smooth boundaries, the distributionfunction was not affected, and the decay rate /H20849inducing the
smearing of singularities in tunneling spectroscopy /H20850was
solely due to dephasing. In the spinful case, the situation isdifferent: independently of the shape of the boundary thespin-charge separation affects the distribution function ofelectrons. Indeed, imagine that we perform the tunnelingspectroscopy of right-movers in the right lead /H20849noninteract-
ing region III of Fig. 4/H20850for the case of adiabatic boundaries.
Then, the phases are
/H9254L,n=/H9254R,n/HS110050=0 and/H9254R,0=2/H9266.I nt h e
spinless case this implied that the distribution function re-mained unchanged. This is not so in the spinful situation,however: according to Eqs. /H20849157/H20850and /H20849158/H20850we get now a
product of four determinants with arguments /H11006
/H9266,
/H20851/H9004¯R,/H9270,↑/H20849/H9266/H20850/H208522/H9004¯R,/H9270,↓/H20849−/H9266/H20850/H9004¯R,/H9270,↓/H20849/H9266/H20850, /H20849162 /H20850
implying that the distribution function has changed. This ef-
fect remains finite even in the limit of vanishing interaction/H20849K→1/H20850as long as the long-wire condition, LT/H20849u
−1−v−1/H20850
/H112711, is satisfied. Returning to the spectroscopy of the inter-
acting region, we conclude that both effects—dephasing andchange in the distribution function—are necessarily presentin the spinful LL case and cannot be easily “disentangled.”The decay rates /H9003presented in Eq. /H20849161/H20850and in Fig. 9yield
the combined effect of interaction on the GF G
R/H11125/H20849/H9270/H20850and de-
termine the smearing of tunneling spectroscopy singularitiesin the energy space.
Finally, we discuss the extension of Eq. /H20849153/H20850to the case
of GFs at different points. In this case, we findG
R,↑/H11125/H20849x1,x2,/H9270/H20850=/H11007i/H9011
2/H9266/H20881uv/H9004¯R,↑,/H9270−x
v/H20849/H9266/H20850/H9004¯R,↓,/H9270−x
v/H20849−/H9266/H20850
/H11003/H20863n=0/H11009/H9004¯R,↑,/H9270nR/H20873/H9254R,n
2/H20874/H20863n=0/H11009/H9004¯R,↓,/H9270nR/H20873/H9254R,n
2/H20874
/H208751/H11006i/H9011/H20873/H9270−x
v/H20874/H208761/2/H208751/H11006i/H9011/H20873/H9270−x
u/H20874/H208761/2
/H11003/H20863n=0/H11009/H9004¯L,↑,/H9270nL/H20873/H9254L,n
2/H20874/H20863n=0/H11009/H9004¯L,↓,/H9270nL/H20873/H9254L,n
2/H20874
/H208751/H11006i/H9011/H20873/H9270−x
u/H20874/H20876/H9251/2/H208751/H11006i/H9011/H20873/H9270+x
u/H20874/H20876/H9252/2,
/H20849163 /H20850
where x=x1−x2and/H9270n/H9257=/H9270+/H9257/H20849−1/H20850n+1x/u. As for the spinless
case, Eq. /H20849132/H20850, the scaling of GFs with spatially separated
points is affected by plasmon scattering at the boundariesbetween the interacting regions and the leads even at equi-librium.
Before concluding this section, we point out a connection
between the spinful LL and the problem of the integer quan-tum Hall edge with two edge channels /H20849corresponding to two
Landau levels below the Fermi energy in the bulk /H20850. Nonequi-
librium properties and quantum coherence of such a systemare currently attracting large interest, in particular, in connec-tion with experiments on quantum Hall Mach-Zehnderinterferometers.
12In the quantum Hall setup the edge chan-
nel index plays a role of spin. The main difference is that,under conventional circumstances /H20849if one does not make spe-
cial efforts to couple counterpropagating edge modes /H20850, the
quantum Hall system is chiral: there are, say, only right-moving modes and no left-movers. This leads to a number ofessential simplifications: /H20849i/H20850the tunneling density of states
becomes trivial /H20849no ZBA /H20850;/H20849ii/H20850the charge fractionalization is
absent /H20849since plasmons can move only in one direction /H20850.
What remains is the charge-spin separation. This implies thatfollowing simplifications with functional determinants/H20849157/H20850–/H20849159/H20850the product of which determined the tunneling
spectroscopy Green’s function G
R,↑/H11125/H20849/H9270/H20850within our analysis.
First, the left-mover determinants /H20849159/H20850are now absent. Sec-
ond, out of the set of phases /H9254R,nonly the n=0 phase re-
mains, being equal to its free-fermion value, /H9254R,0=2/H9266. There-
fore, the product of determinants takes the form /H20849162/H20850/H20849that
has appeared above in the context of a GF in the noninter-acting region of a nonchiral spinful wire with adiabatic con-tacts /H20850. The analogous statement holds for the Green’s func-
tion with different spatial points, Eq. /H20849163/H20850. Specifically, for
the case of the two-channel chiral setup /H20849relevant in the
quantum Hall Mach-Zehnder interferometry context /H20850
12the
last fraction /H20849having L determinants in the numerator /H20850in Eq.
/H20849163/H20850disappears, while in the preceding-to-it fraction one
should keep only n=0 factor and set /H9254R,0=2/H9266. An equivalent
result was obtained by a different method in the recentwork.
690 0.2 0.4 0.6 0.8 1
K00.20.40.60.8ΓRη/eVη
RLRR
FIG. 9. /H20849Color online /H20850Decay rates/H9003RRand/H9003RL/H20849governing
smearing of tunneling spectroscopy singularities /H20850as functions of LL
parameter Kcfor spinful fermions. The adiabatic coupling to leads
and the double-step distributions with aR=aL=1 /3 are assumed.GUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-18VI. SUMMARY
Let us summarize the main results of this work, following
the flow of our presentation in the paper.
/H20849i/H20850We have developed a nonequilibrium bosonization ap-
proach and derived a bosonic theory describing the LL ofinteracting 1D electrons out of equilibrium. The theory ischaracterized by an action depending on density fields de-fined on the Keldysh time contour. In contrast to the equilib-rium case, this theory is not Gaussian, which is a manifesta-tion of the fact that the density matrix is nondiagonal in thebosonic Fock space. We have used this theory to calculatethe electronic GFs governing various observables.
/H20849ii/H20850We have first calculated the GF of noninteracting fer-
mions from our nonequilibrium bosonization approach. TheGF is expressed in terms of a functional determinant of theFredholm /H20849more specifically, Toeplitz /H20850type similar to those
that have earlier appeared in the context of counting statis-tics. The key difference is that in the case of counting statis-tics the determinant is nonanalytic and 2
/H9266periodic in the
counting field /H20849which reflects charge quantization /H20850, while in
our theory the determinant should be understood as an ana-lytically continued function. We have found that the free-fermion GF is described by the determinant exactly at thepoint 2
/H9266, which is related to the fact that in the bosonic
theory a fermion is represented by a 2 /H9266soliton.
/H20849iii/H20850We have next generalized the GF calculation to the
problem of nonequilibrium Fermi edge singularity describingexcitation of an electron into the conduction band within theprocess of photon absorption, accompanied by creation of acore hole. The result is obtained in terms of the same func-tional determinant as in the free-fermion case but the argu-ment is now shifted from 2
/H9266by twice the scattering phase on
the core hole.
/H20849iv/H20850We have then applied our formalism to the problem of
interacting 1D fermions. We have considered a model of aLL wire coupled to noninteracting 1D leads, with the inter-action strength “turned on” in specified fashion at the bound-ary between the wire and each of the leads. We have shownthat the electron GFs—which describe tunneling spectros-copy measurements—are again expressed in terms of Fred-holm determinants. The phases
/H9254/H9257/H20849t/H20850entering the expressions
for the corresponding operators have a physical interpreta-tion in terms of fractionalization processes taking place dur-ing the tunneling event, near the boundaries. If the charac-teristic energy scales for the tunneling spectroscopy are largecompared to the inverse flight time through the LL wire/H20849Thouless energy /H20850—which means that we are considering the
truly 1D /H20849rather than 0D /H20850regime—the functions
/H9254/H9257/H20849t/H20850repre-
sent a sequence of rectangular pulses separated by large in-tervals. As a result, the Fredholm determinant splits into aproduct of Toeplitz determinants of the same type as in thecases of noninteracting fermions and the Fermi edge singu-larity.
/H20849v/H20850We have analyzed the long-time asymptotics of the
determinant which yields the dephasing rate controlling thesmearing of LL tunneling singularities /H20849zero-bias anomaly /H20850.
The dephasing rate for the GF of electrons with
/H9257/H20849/H110061/H20850
chirality is a sum of two terms /H20858/H9257/H11032=/H1100611//H9270/H9278/H9257/H9257/H11032originatingfrom functional determinants which depend on the distribu-
tion function of left- /H20849/H9257/H11032=−1 /H20850and right- /H20849/H9257/H11032=1/H20850moving
electrons, respectively. For the case of double-step distribu-tions, there are two important findings:
/H20849a/H20850At weak interaction, comparing our exact results with
those of the RPA, we find that while 1 /
/H9270/H9278/H9257,−/H9257is correctly
obtained /H20849to leading order /H20850within RPA, the RPA result for
1//H9270/H9278/H9257,/H9257is parametrically wrong. This demonstrates that even
for a weak interaction a naive perturbative expansion /H20849lead-
ing to RPA /H20850may be parametrically incorrect in LL out of
equilibrium.
/H20849b/H20850Both 1 //H9270/H9278/H9257,/H11006/H9257are oscillatory functions of the interac-
tion strength /H20849or, equivalently, LL parameter K/H20850. Further-
more, each of them vanishes at certain values of K. At these
values the “counting phase” for the corresponding determi-nant becomes an integer multiple of 2
/H9266. We have calculated
the determinants at these no-dephasing points by a refermi-onization procedure.
/H20849vi/H20850We have generalized the above results to the case of a
GF with two different spatial arguments. When consideringthe value of the GF G
/H9257at its main peak, x1−x2=/H9257ut, the
dephasing rate is 2 //H9270/H9278/H9257,−/H9257, while 1 //H9270/H9278/H9257,/H9257does not contribute
/H20849and thus RPA is restored for weak interaction /H20850. The situation
is reversed for the value of the G/H9257at the other peak, x1−x2
=−/H9257ut, where the dephasing rate is 2 //H9270/H9278/H9257,/H9257. Such GFs /H20849with
x1−x2=/H11006/H9257ut/H20850enter the expression for the interference con-
tribution to current in an Aharonov-Bohm interferometerformed by two LLs coupled by tunneling at two points. Ourresults imply that the dephasing rate in such a nonequilib-rium LL interferometer /H20849and thus the visibility of Aharonov-
Bohm oscillations /H20850is an oscillatory function of the interac-
tion strength.
/H20849vii/H20850We have also considered the case of a spinful LL.
The general structure of the results for the GFs is similar,with the key difference being that now we encounter prod-ucts of determinants with phase arguments corresponding tothe spin and charge sectors. This is a manifestation of thespin-charge separation. One important consequence is thatthe temporal decay rate of the Green’s function /H20849and thus the
smearing of singularities in the tunneling spectroscopy /H20850re-
mains finite in the limit of vanishing interaction strength,assuming the limit of the large system size is taken first. Withincreasing interaction strength, the dephasing rate oscillates,similarly to the case of spinless fermions.
The nonequilibrium bosonization formalism developed in
this work has a variety of further applications. They include,in particular, counting statistics of charge transfer in an in-teracting 1D system away from equilibrium, analysis ofmany-body entanglement, quantum wires with several chan-nels, etc. Generalizations or modifications of our formalismshould be useful for a number of further prospective researchdirections, such as systems of cold atoms and fractionalquantum Hall edges away from equilibrium.
ACKNOWLEDGMENTS
We thank D. Abanin, D. Bagrets, L. Glazman, D. Ivanov,
L. Levitov, Y. Nazarov, P. Ostrovsky, E. Sukhorukov, and P.Wiegmann for useful discussions. Financial support byBOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-19German-Israeli Foundation, Einstein Minerva Center, U.S.-
Israel Binational Science Foundation, Israel Science Founda-tion, Minerva Foundation, SPP 1285 of the DeutscheForschungsgemeinschaft, EU project GEOMDISS, and Ros-nauka under Grant No. 02.740.11.5072 is gratefully ac-knowledged.
APPENDIX A: EQUILIBRIUM: GREEN’S FUNCTIONS G0Ò,
GÒVIA BOSONIZATION AND FREDHOLM
DETERMINANTS /H9004/H9270
GFs of free fermions at equilibrium can be readily found.
Since at equilibrium, the bosonic action is Gaussian, thefunctional integration over bosonic fields is straightforward.The fermionic GFs is thus expressed as
G
0/H11125/H20849/H9270/H20850=/H11007i/H9011
2/H9266veJ/H11125/H20849/H9270/H20850/H20849A1/H20850
in terms of the bosonic correlation functions
J/H11022/H20849/H9270/H20850=−1
2/H20855TK/H20851/H9278+,/H9257/H208490,0/H20850−/H9278−,/H9257/H208490,/H9270/H20850/H208522/H20856,
J/H11021/H20849/H9270/H20850=−1
2/H20855TK/H20851/H9278−,/H9257/H208490,0/H20850−/H9278+,/H9257/H208490,/H9270/H20850/H208522/H20856. /H20849A2/H20850
Explicitly calculating the correlations functions of the
bosonic fields, one finds
J/H11125/H20849/H9270/H20850=−/H20885
0/H11009d/H9275
/H9275e−/H9275//H9011/H20875/H208491 − cos/H9275/H9270/H20850coth/H9275
2T/H11006isin/H9275/H9270/H20876.
/H20849A3/H20850
Next, we calculate the integrals that appear on the r.h.s. of
Eq. /H20849A3/H20850. The integral with sin /H9275/H9270yields
/H20885
0/H11009d/H9275
/H9275e−/H9275//H9011sin/H9275/H9270= arctan/H9011/H9270. /H20849A4/H20850
The remaining integral can be split into two parts:
/H20885
0/H11009d/H9275
/H9275e−/H9275//H9011/H208491 − cos/H9275/H9270/H20850=1
2ln/H208491+/H90112/H92702/H20850/H20849 A5/H20850
and
/H20885
0/H11009d/H9275
/H9275e−/H9275//H9011/H20873coth/H9275
2T−1/H20874/H208491 − cos/H9275/H9270/H20850/H11229lnsinh/H9266T/H9270
/H9266T/H9270;
/H20849A6/H20850
in the second one, we have dropped the convergence factor,
e−/H9275//H9011, which is justified in view of T/H11270/H9011. Employing Eqs.
/H20849A4/H20850–/H20849A6/H20850, one gets
J/H11125/H20849/H9270/H20850=l n/H20873/H9266T/H9270
sinh/H9266T/H92701
1/H11006i/H9270/H9011/H20874. /H20849A7/H20850
Substituting Eq. /H20849A7/H20850into Eq. /H20849A1/H20850, one recovers the result
for the GF of free fermions, Eq. /H2084954/H20850.We proceed with GFs for FES at equilibrium,
G/H11125/H20849/H9270/H20850=/H11007i/H9011
2/H9266vexp/H20853/H208491−/H92540//H9266/H208502J/H11125/H20849/H9270/H20850/H20854. /H20849A8/H20850
Next, we relate the GF and the functional determinant /H9004/H9270/H20849/H9254/H20850.
At equilibrium, the latter can be evaluated as follows:
ln/H9004/H9270/H20849/H9254/H20850=−/H20873/H9254
2/H9266/H208742/H20885
0/H11009d/H9275
/H9275e−/H9275//H9011/H208491 − cos/H9275/H9270/H20850coth/H9275
2T.
/H20849A9/H20850
Using Eq. /H20849A4/H20850,w efi n d
/H9004/H9270/H20849/H9254/H20850=/H20873/H9266/H9270T
sinh/H9266/H9270T/H20874/H20849/H9254/2/H9266/H2085021
/H208491+/H92702/H90112/H20850/H208491/2/H20850/H20849/H9254/2/H9266/H208502.
/H20849A10 /H20850
Comparing Eq. /H20849A10 /H20850with Eq. /H20849A7/H20850, we establish the exact
relation /H20849including the proportionality factor /H20850between the
GF and the functional determinant,
G/H11125/H20849/H9270/H20850=/H11007i/H9011
2/H9266v/H208731/H11007i/H9011/H9270
1/H11006i/H9011/H9270/H20874/H208491/2/H20850/H208491−/H92540//H9266/H208502
/H9004/H9270/H208492/H9266−2/H92540/H20850.
/H20849A11 /H20850
We notice that the determinant /H9004/H9270/H20849/H9254/H20850as given by Eq. /H20849A10 /H20850
is a product of the temperature dependent and independentparts. It is convenient to normalize the result by its zerotemperature value,
/H9004
/H9270,T=0/H20849/H9254/H20850=1
/H208491+/H92702/H90112/H20850/H208491/2/H20850/H20849/H9254/2/H9266/H208502. /H20849A12 /H20850
We thus present Eq. /H20849A10 /H20850in the form
/H9004/H9270/H20849/H9254/H20850=/H9004¯/H9270/H20849/H9254/H20850
/H208491+/H92702/H90112/H20850/H208491/2/H20850/H20849/H9254/2/H9266/H208502, /H20849A13 /H20850
where/H9004¯/H9270/H20849/H9254/H20850is the normalized determinant,
/H9004¯/H9270/H20849/H9254/H20850=/H20873/H9266/H9270T
sinh/H9266/H9270T/H20874/H20849/H9254/2/H9266/H208502
. /H20849A14 /H20850
By construction, /H9004¯/H9270/H20849/H9254/H20850=1 for T=0. It turns out to be more
convenient to deal with the normalized determinant, since allultraviolet divergences /H20849/H9011-dependent factor /H20850are excluded
from this quantity.
APPENDIX B: HIGH-ORDER VERTICES FOR 1D
FERMIONS: DIAGRAMMATICS
In this appendix, we briefly sketch an explicit calculation
of third-order fermionic vertices by means of diagrammaticfermionic approach. Consider the third-order vertex shownin Fig. 2/H20849b/H20850,
S
3,/H9257/H208491,2,3 /H20850=/H20855TK/H9267/H9257/H208491/H20850/H9267/H9257/H208492/H20850/H9267/H9257/H208493/H20850/H20856
=/H20855/H92741,/H9257†/H92741,/H9257/H92742,/H9257†/H92742,/H9257/H92743,/H9257†/H92743,/H9257/H20856. /H20849B1/H20850GUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-20Here, the index i=1,2,3 includes the corresponding spatial
coordinate /H20849xi/H20850, time /H20849ti/H20850and Keldysh index sithat labels
upper and lower branches. Using Wick theorem, we find
−iS/H9257/H208491,2,3 /H20850=G/H9257/H208491,3/H20850G/H9257/H208493,2/H20850G/H9257/H208492,1/H20850
+G/H9257/H208491,2/H20850G/H9257/H208492,3/H20850G/H9257/H208493,1/H20850.
We choose first the following combination of Keldysh indi-
ces: s1=+, s2=+, s3=−. Using Eq. /H2084916/H20850and passing into
energy-momentum representation, we find
S3,/H9257/H20849/H92751,q1,+ ;/H92752,q2,+ ;/H92753,q3,−/H20850
=−i
v/H208492/H9266/H208502/H9254/H20849A1/H20850/H9254/H20849A2/H20850/H20885d/H9280
2/H9266n/H9257/H20849/H9280/H20850/H208511−n/H9257/H20849/H9280+/H92751+/H92752/H20850/H20852
/H11003/H208511−n/H9257/H20849/H9280+/H92751/H20850−n/H9257/H20849/H9280+/H92752/H20850/H20852. /H20849B2/H20850Here, Ai=/H9275i−/H9257vqiand/H92753=−/H92751−/H92752,q3=−q1−q2. There-
fore, the third-order correlation function is restricted to thelight-cone with respect to all its coordinates. At equilibriumthe integration over energy yields zero, making the correla-
tion function vanish. On the other hand, in the nonequilib-rium situation the result is in general nonzero. Repeating thecalculations for all possible choices of Keldysh indicess
1,s2,s3, we find that all third-order vertices are equal and
given by Eq. /H20849B2/H20850. When transformed from s=/H11006to quantum
and classical components, this means that the result is non-zero only when all indices are classical. Thus, the explicitdiagrammatic calculations of the third-order vertex confirmsthat /H20849i/H20850it is restricted to the light-cone, and /H20849ii/H20850only correla-
tions of classical fields are nonzero. We have checked thisalso for vertices of fourth order. These results are in fullagreement with the general treatment /H20849valid for vertices of all
orders /H20850performed in Sec. III.
1A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosoniza-
tion in Strongly Correlated Systems /H20849Cambridge University
Press, Cambridge, 1998 /H20850.
2M. Stone, Bosonization /H20849World Scientific, Singapore, 1994 /H20850.
3T. Giamarchi, Quantum Physics in One Dimension /H20849Clarendon
Press, Oxford, 2004 /H20850.
4D. L. Maslov, in Nanophysics: Coherence and Transport , edited
by H. Bouchiat, Y. Gefen, G. Montambaux, and J. Dalibard/H20849Elsevier, New York, 2005 /H20850,p .1 .
5J. von Delft and H. Schoeller, Ann. Phys. 7, 225 /H208491998 /H20850.
6M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley,
L. Balents, and P. L. McEuen, Nature /H20849London /H20850397, 598
/H208491999 /H20850; Z. Yao, H. W. Ch. Postma, L. Balents, and C. Dekker,
ibid. 402, 273 /H208491999 /H20850.
7C. Schönenberger, Semicond. Sci. Technol. 21,S 1 /H208492006 /H20850.
8O. M. Auslaender, A. Yacoby, R. de Picciotto, K. W. Baldwin, L.
N. Pfeiffer, and K. W. West, Science 295, 825 /H208492002 /H20850; E. Levy,
A. Tsukernik, M. Karpovski, A. Palevski, B. Dwir, E. Pelucchi,A. Rudra, E. Kapon, and Y. Oreg, Phys. Rev. Lett. 97, 196802
/H208492006 /H20850.
9E. Slot, M. A. Holst, H. S. J. van der Zant, and S. V. Zaitsev-
Zotov, Phys. Rev. Lett. 93, 176602 /H208492004 /H20850; L. Venkataraman, Y.
S. Hong, and P. Kim, ibid. 96, 076601 /H208492006 /H20850.
10A. N. Aleshin, H. J. Lee, Y. W. Park, and K. Akagi, Phys. Rev.
Lett. 93, 196601 /H208492004 /H20850; A. N. Aleshin, Adv. Mater. 18,1 7
/H208492006 /H20850.
11A. M. Chang, Rev. Mod. Phys. 75, 1449 /H208492003 /H20850; R. de Picciotto,
M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Ma-halu, Nature /H20849London /H20850389, 162 /H208491997 /H20850; M. Dolev, M. Heiblum,
V. Umansky, A. Stern, and D. Mahalu, ibid. 452, 829 /H208492008 /H20850;W .
Kang, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and K. W.West, ibid. 403,5 9 /H208492000 /H20850; M. Grayson, L. Steinke, D. Schuh,
M. Bichler, L. Hoeppel, J. Smet, K. v. Klitzing, D. K. Maude,and G. Abstreiter, Phys. Rev. B 76, 201304 /H20849R/H20850/H208492007 /H20850.
12Y. Ji, Y. C. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and H.
Shtrikman, Nature /H20849London /H20850422, 415 /H208492003 /H20850; I. Neder, M.
Heiblum, Y. Levinson, D. Mahalu, and V. Umansky, Phys. Rev.Lett. 96, 016804 /H208492006 /H20850; I. Neder, F. Marquardt, M. Heiblum,D. Mahalu, and V. Umansky, Nat. Phys. 3, 534 /H208492007 /H20850; E. Bieri,
M. Weiss, O. Goktas, M. Hauser, C. Schonenberger, and S.Oberholzer, Phys. Rev. B 79, 245324 /H208492009 /H20850.
13Y.-F. Chen, T. Dirks, G. Al-Zoubi, N. O. Birge, and N. Mason,
Phys. Rev. Lett. 102, 036804 /H208492009 /H20850.
14T. Dirks, Y.-F. Chen, N. O. Birge, and N. Mason, Appl. Phys.
Lett. 95, 192103 /H208492009 /H20850.
15C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna, D. Mailly,
and F. Pierre, Nat. Phys. 6,3 4 /H208492009 /H20850.
16H. Pothier, S. Gueron, N. O. Birge, D. Esteve, and M. H. De-
voret, Phys. Rev. Lett. 79, 3490 /H208491997 /H20850; A. Anthore, F. Pierre,
H. Pothier, and D. Esteve, ibid. 90, 076806 /H208492003 /H20850.
17D. B. Gutman, Y. Gefen, and A. D. Mirlin, Phys. Rev. Lett. 100,
086801 /H208492008 /H20850.
18D. A. Abanin and L. S. Levitov, Phys. Rev. Lett. 94, 186803
/H208492005 /H20850.
19D. B. Gutman, Y. Gefen, and A. Mirlin, arXiv:0906.4076 /H20849un-
published /H20850.
20W. Apel and T. M. Rice, Phys. Rev. B 26, 7063 /H208491982 /H20850.
21T. Giamarchi and H. J. Schulz, Phys. Rev. B 37, 325 /H208491988 /H20850.
22I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. Lett.
95, 046404 /H208492005 /H20850; Phys. Rev. B 75, 085421 /H208492007 /H20850;A .G .
Yashenkin, I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, ibid.
78, 205407 /H208492008 /H20850.
23D. A. Bagrets, I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov,
Semiconductors 42, 994 /H208492008 /H20850; D. A. Bagrets, I. V. Gornyi, and
D. G. Polyakov, Phys. Rev. B 80, 113403 /H208492009 /H20850.
24M. Khodas, M. Pustilnik, A. Kamenev, and L. I. Glazman, Phys.
Rev. B 76, 155402 /H208492007 /H20850; A. M. Lunde, K. Flensberg, and L. I.
Glazman, ibid. 75, 245418 /H208492007 /H20850; M. Pustilnik, M. Khodas, A.
Kamenev, and L. I. Glazman, Phys. Rev. Lett. 96, 196405
/H208492006 /H20850.
25J. Rech and K. A. Matveev, Phys. Rev. Lett. 100, 066407
/H208492008 /H20850; J. Rech, T. Micklitz, and K. A. Matveev, ibid. 102,
116402 /H208492009 /H20850.
26M. Trushin and A. L. Chudnovskiy, EPL 82, 17008 /H208492008 /H20850.
27D. B. Gutman, Y. Gefen, and A. D. Mirlin, Phys. Rev. Lett. 101,
126802 /H208492008 /H20850.BOSONIZATION OF ONE-DIMENSIONAL FERMIONS OUT … PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-2128D. B. Gutman, Y. Gefen, and A. D. Mirlin, Phys. Rev. B 80,
045106 /H208492009 /H20850.
29The tunneling spectroscopy problem in a nonequilibrium LL pro-
duced by an impurity located inside the interacting region wasconsidered in S. Ngo Dinh, Diploma thesis, Karslruhe, 2009; S.Ngo Dinh, D. A. Bagrets, and A. D. Mirlin, Phys. Rev. B 81,
081306 /H20849R/H20850/H208492010 /H20850.
30A. Altland and R. Egger, Phys. Rev. Lett. 102, 026805 /H208492009 /H20850.
31K. Le Hur, Phys. Rev. B 65, 233314 /H208492002 /H20850; Phys. Rev. Lett. 95,
076801 /H208492005 /H20850; Phys. Rev. B 74, 165104 /H208492006 /H20850.
32We omit Klein factors, as they are of no importance for our
problem.
33For review of the Keldysh technique see, e.g., J. Rammer and H.
Smith, Rev. Mod. Phys. 58, 323 /H208491986 /H20850; A. Kamenev, in Nano-
physics: Coherence and Transport , edited by H. Bouchiat, Y.
Gefen, G. Montambaux, and J. Dalibard /H20849Elsevier, New York,
2005 /H20850, p. 177; A. Kamenev and A. Levchenko, Adv. Phys. 58,
197 /H208492009 /H20850.
34I. E. Dzyaloshinskii and A. I. Larkin, Sov. Phys. JETP 38, 202
/H208491973 /H20850.
35Equation /H2084911/H20850of Ref. 34is different from our Eq. /H2084934/H20850in that the
former contains a product of kinematic factors /H20863i=1n/H20849/H9275i−/H9257vqi/H20850
corresponding to all nvertices. This would allow contributions
that are of mass-shell type with respect to just one of the verti-ces. However, one can in fact derive a much stronger equation,Eq. /H2084934/H20850, that imposes the mass-shell restriction with respect to
all vertices simultaneously.
36Due to the linearization of the electron spectrum, special care
needs to be taken in order to account for the anomalous contri-bution to the partition function Z/H20849related to the Schwinger
anomaly /H20850, as has been also mentioned in Sec. III B. This contri-
bution is given by the first term in the exponent on the r.h.s. ofEq. /H2084918/H20850. This term does not depend on the fermionic distribu-
tion and can be straightforwardly obtained in the standard dia-grammatic analysis performed at equilibrium. We restore it in
the final result for Z
/H9257/H20849V,V¯/H20850, Eq. /H2084948/H20850.
37I. Klich, in Quantum Noise in Mesoscopic Systems , edited by Yu.
V. Nazarov /H20849Kluwer, Dordrecht, 2003 /H20850.
38L. S. Levitov and G. B. Lesovik, JETP Lett. 58, 230 /H208491993 /H20850;L .
S. Levitov, H. W. Lee, and G. B. Lesovik, J. Math. Phys. 37,
4845 /H208491996 /H20850; D. A. Ivanov, H. W. Lee, and L. S. Levitov, Phys.
Rev. B 56, 6839 /H208491997 /H20850.
39B. A. Muzykantskii and Y. Adamov, Phys. Rev. B 68, 155304
/H208492003 /H20850; A. Shelankov and J. Rammer, EPL 63, 485 /H208492003 /H20850;K .
Schonhammer, Phys. Rev. B 75, 205329 /H208492007 /H20850; J. E. Avron, S.
Bachmann, G. M. Graf, and I. Klich, Commun. Math. Phys.
280, 807 /H208492008 /H20850; F. Hassler, M. V. Suslov, G. M. Graf, M. V.
Lebedev, G. B. Lesovik, and G. Blatter, Phys. Rev. B 78,
165330 /H208492008 /H20850.
40M. Jimbo, T. Miwa, Y. Môri, and M. Sato, Physica D 1,8 0
/H208491980 /H20850.
41A. G. Izergin and A. G. Pronko, Nucl. Phys. B 520, 594 /H208491998 /H20850.
42E. Bettelheim, A. G. Abanov, and P. Wiegmann, Phys. Rev. Lett.
97, 246402 /H208492006 /H20850.
43M. B. Zvonarev, V. V. Cheianov, and T. Giamarchi, J. Stat.
Mech.: Theory Exp. 2009 , P07035.
44M. Reznikov, R. de Picciotto, T. G. Griffiths, M. Heiblum, andV. Umansky, Nature /H20849London /H20850399, 238 /H208491999 /H20850; M. Reznikov,
M. Heiblum, H. Shtrikman, and D. Mahalu, Phys. Rev. Lett. 75,
3340 /H208491995 /H20850.
45L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, Phys. Rev.
Lett. 79, 2526 /H208491997 /H20850.
46A. H. Steinbach, J. M. Martinis, and M. H. Devoret, Phys. Rev.
Lett. 76, 3806 /H208491996 /H20850.
47R. J. Schoelkopf, P. J. Burke, A. A. Kozhevnikov, D. E. Prober,
and M. J. Rooks, Phys. Rev. Lett. 78, 3370 /H208491997 /H20850.
48P. Nozières and C. T. De Dominicis, Phys. Rev. 178, 1097
/H208491969 /H20850.
49K. A. Matveev and A. I. Larkin, Phys. Rev. B 46, 15337 /H208491992 /H20850.
50A. K. Geim, P. C. Main, N. La Scala, Jr., L. Eaves, T. J. Foster,
P. H. Beton, J. W. Sakai, F. W. Sheard, M. Henini, G. Hill, andM. A. Pate, Phys. Rev. Lett. 72, 2061 /H208491994 /H20850.
51K. D. Schotte and U. Schotte, Phys. Rev. 182, 479 /H208491969 /H20850.
52H. C. Fogedby, J. Phys. C 9, 3757 /H208491976 /H20850.
53D. K. Lee and Y. Chen, J. Phys. A 21, 4155 /H208491988 /H20850.
54C. M. Naon, M. C. von Reichenbach, and M. L. Trobo, Nucl.
Phys. B 435, 567 /H208491995 /H20850; C. M. Naon, M. J. Salvay, and M. L.
Trobo, Int. J. Mod. Phys. A 19, 4953 /H208492004 /H20850.
55I. V. Yurkevich, in Strongly Correlated Fermions and Bosons in
Low-Dimensional Disordered Systems , edited by I. V. Lerner et
al./H20849Kluwer, Dordrecht, 2002 /H20850, p. 69; A. Grishin, I. V. Yurkevich,
and I. V. Lerner, Phys. Rev. B 69, 165108 /H208492004 /H20850; I. V. Lerner
and I. V. Yurkevich, Nanophysics: Coherence and Transport
/H20849Elsevier, New York, 2005 /H20850, p. 109.
56I. Snyman and Y. V. Nazarov, Phys. Rev. Lett. 99, 096802
/H208492007 /H20850.
57I. Neder and F. Marquardt, New J. Phys. 9,1 1 2 /H208492007 /H20850.
58D. L. Maslov and M. Stone, Phys. Rev. B 52, R5539 /H208491995 /H20850.
59V. V. Ponomarenko, Phys. Rev. B 52, R8666 /H208491995 /H20850.
60I. Safi and H. J. Schulz, Phys. Rev. B 52, R17040 /H208491995 /H20850;59,
3040 /H208491999 /H20850.
61V. V. Ponomarenko, Phys. Rev. B 54, 10328 /H208491996 /H20850.
62B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev. Lett.
92, 226405 /H208492004 /H20850.
63The subscript/H9257in the notation for phases /H9254/H9257/H20849t/H20850corresponds to
the chirality of the corresponding component /H9267¯/H9257of the saddle-
point solution and to the chiral index of the Fredholm determi-
nant/H9004¯/H9257. Clearly, the saddle-point solution /H20849and thus/H9254/H9257/H20850depends
also on chirality of the Green’s function calculated. For definite-ness, we focus on G
Rand drop the additional index /H20849which will
be “R” in this case /H20850.
64Thereafter, we assume /H9270/H11270L/u.
65Y. Oreg and A. M. Finkelstein, Phys. Rev. Lett. 74, 3668 /H208491995 /H20850.
66I. Safi, Ann. Phys. 22, 463 /H208491997 /H20850; K.-V. Pham, M. Gabay, and
P. Lederer, Phys. Rev. B 61, 16397 /H208492000 /H20850.
67H. Steinberg, G. Barak, A. Yacoby, L. N. Pfeiffer, K. W. West, B.
I. Halperin, and K. Le Hur, Nat. Phys. 4,1 1 6 /H208492008 /H20850.
68E. Berg, Y. Oreg, E.-A. Kim, and F. von Oppen, Phys. Rev. Lett.
102, 236402 /H208492009 /H20850.
69I. P. Levkivskyi and E. V. Sukhorukov, Phys. Rev. Lett. 103,
036801 /H208492009 /H20850.
70J. T. Chalker, Y. Gefen, and M. Y. Veillette, Phys. Rev. B 76,
085320 /H208492007 /H20850.GUTMAN, GEFEN, AND MIRLIN PHYSICAL REVIEW B 81, 085436 /H208492010 /H20850
085436-22 |
PhysRevB.87.174523.pdf | PHYSICAL REVIEW B 87, 174523 (2013)
Effective interactions and fluctuation effects in spin-singlet superfluids
Andreas Eberlein and Walter Metzner
Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany
(Received 7 February 2013; published 28 May 2013)
We derive and evaluate one-loop functional flow equations for the effective interactions, self-energy, and gap
function in spin-singlet superfluids. The flow is generated by a fermionic frequency cutoff, which is supplementedby an external pairing field to treat divergencies associated with the Goldstone boson. To parametrize the singularmomentum and frequency dependencies of the effective interactions, the Nambu interaction vertex is decomposedin charge, magnetic, and normal and anomalous pairing channels. The one-loop flow solves reduced (mean-field)models for superfluidity exactly, and captures also important fluctuation effects. The Ward identity from chargeconservation is generally violated, but can be enforced by projecting the flow. Applying the general formalismto the two-dimensional attractive Hubbard model, we obtain detailed results on the momentum and frequencydependencies of the effective interactions for weak and moderate bare interactions. The gap is reduced byfluctuations, with a stronger reduction at weaker interactions, as expected.
DOI: 10.1103/PhysRevB.87.174523 PACS number(s): 05 .10.Cc, 71 .10.Fd, 74 .20.−z
I. INTRODUCTION
Numerous interacting Fermi systems undergo a phase
transition associated with spontaneous symmetry breakingat sufficiently low temperatures. Mean-field theory capturessalient features of the symmetry-broken phase such as long-range order, quasiparticle excitations, and collective modes.For example, the BCS wave function provides a surprisinglyfaithful qualitative description of the superfluid ground stateof an attractively interacting Fermi gas not only at weak butalso at strong coupling.
1,2Nevertheless, fluctuations often play
an important role, both above and below the energy scale forsymmetry breaking. At high energies, they renormalize theeffective interactions generating an instability of the normal(symmetric) state, which may enhance or reduce the scale forsymmetry breaking. At low energies, order parameter fluctu-ations usually suppress the order at least partially. Triggeredby the possibility of designing tunable attractively interactingFermi systems in cold atom traps, the issue of fluctuationeffects in fermionic superfluids has attracted renewed interest.
3
A framework to deal with fluctuation effects on all
energy scales is provided by the functional renormalizationgroup (fRG). This method provides a flexible source of newapproximation schemes for interacting Fermi systems,
4which
are obtained by truncating the exact functional flow for theeffective action as a function of a decreasing infrared cutoff/Lambda1.
5–7The common types of spontaneous symmetry breaking
such as superconductivity or magnetic order are associatedwith a divergence of the effective two-particle interaction ata finite scale /Lambda1
cin a specific momentum channel.8–10To
continue the flow below the scale /Lambda1c, an order parameter
describing the broken symmetry has to be introduced.
A natural procedure is to decouple the interaction by a
bosonic order parameter field, via a Hubbard-Stratonovichtransformation, and to study the coupled flow of the fermionicand order parameter fields. Thereby, order parameter fluctu-ations and also their interactions can be treated rather easily.This approach to symmetry breaking in the fRG framework hasbeen explored already in several works on antiferromagneticorder
11,12and superconductivity.13–18The bare microscopic
interaction can usually be decoupled by introducing a singleboson field. However, an effective interaction with only one
bosonic field corresponds to a strongly simplified represen-tation of the effective two-fermion interaction. Systems withcompeting instabilities corresponding to distinct order param-eters require the introduction of several bosonic fields.
19,20
Alternatively, one may explore purely fermionic flows in
the symmetry-broken phase. This can be done by adding
an infinitesimal symmetry-breaking term to the bare action,
which is promoted to a finite order parameter below thescale/Lambda1
c.21A simple one-loop truncation of the exact fRG
flow equation with self-energy feedback was shown to yieldanexact description of symmetry breaking for mean-field
models such as the reduced BCS model, although the effectivetwo-particle interactions diverge at the critical scale /Lambda1
c.21,22
A subsequent application to the two-dimensional attractive
Hubbard model showed that the same truncation, with a rathernaive parametrization of the effective two-particle vertex,yields surprisingly accurate results for the superconductinggap at weak coupling.
23However, the flow could be carried
out down to /Lambda1=0 only for a symmetry-breaking pairing field
/Delta10above a certain minimal value. At that value, a spurious
divergence of the two-particle vertex was found. Fortunately,
the minimal /Delta10was rather small, more than two orders of
magnitude below the size of the gap at the end of the flow, andin this sense close to the ideal case of an infinitesimal /Delta1
0.
In this paper, we further develop the fermionic fRG for
spin-singlet superfluids as a prototype for a broken contin-uous symmetry. We stay with the one-loop truncation used
previously, but we derive and apply a much more accurate
parametrization of the momentum and frequency dependenceof the flowing two-particle vertex, taking all singularities inthe particle-particle and particle-hole channels into account.We build on recent work on the structure of the Nambu two-particle vertex in a singlet superfluid,
24where constraints from
symmetries (especially spin-rotation invariance) were derived,
and insight into the singularities associated with superfluidity
was gained by analyzing the exact fRG flow of a mean-fieldmodel with charge and spin forward scattering in addition tothe reduced BCS interaction. Furthermore, a decomposition ofthe Nambu vertex in distinct interaction channels was derived,
174523-1 1098-0121/2013/87(17)/174523(22) ©2013 American Physical SocietyANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
extending the decomposition formulated by Husemann and
Salmhofer25for the normal state,26which will now be used
to separate regular from singular momentum and frequencydependencies. With an adequate parametrization of the vertexat hand, we can fully explore the performance of the one-looptruncated fermionic RG for symmetry breaking beyond mean-field models. Explicit results for the effective interactions, the
self-energy, and the gap function will be presented for the
two-dimensional Hubbard model with an attractive interactionas a prototypical case. The Ward identity relating the gap tothe vertex in the phase fluctuation channel (Goldstone mode)is not consistent with the truncated flow. The deviations aresmall at weak coupling, but they increase with the interactionstrength. This problem can be treated by projecting the flow on
the manifold of effective actions which respect the constraint
imposed by the Ward identity. We also analyze to what extenteffects of the Goldstone mode on other channels are capturedby the one-loop truncation.
The paper is structured as follows. In Sec. II, the basic
one-loop flow equations for the self-energy and the Nambutwo-particle vertex are written. Symmetry properties of theNambu vertex following from spin-rotation invariance anddiscrete symmetries are reviewed in Sec. III. The channel
decomposition for spin-singlet superfluids is derived inSec. IV, and the general structure of the flow equations is
discussed. In Sec. V, the random phase approximation is
revisited in the framework of the channel decomposed flowequations. The general formalism is applied to the attractiveHubbard model in Sec. VI, with results for the self-energy,
the gap function, and the effective interactions in all channels.Merits and shortcomings of the channel decomposed one-loopflow equations are summarized in the conclusions, Sec. VII.
II. TRUNCATED FLOW EQUATIONS
We analyze the superfluid ground state of attractively
interacting spin-1
2fermions. The system is specified by a bare
action of the form
S[ψ,¯ψ]=−/summationdisplay
k,σ[ik0−ξ(k)]¯ψkσψkσ+U[ψ,¯ψ], (1)
where ¯ψkσandψkσare Grassmann variables associated with
creation and annihilation operators, respectively. The variablek=(k
0,k) contains the Matsubara frequency k0in addition to
the momentum k, andσdenotes the spin orientation. ξ(k)=
/epsilon1(k)−μis the single-particle energy relative to the chemical
potential, and U[ψ,¯ψ] describes a spin-rotation invariant two-
particle interaction
U[ψ,¯ψ]=1
4/summationdisplay
ki,σi/bracketleftbig
U(k1,k2,k3,k4)δσ1σ4δσ2σ3
−U(k1,k2,k4,k3)δσ1σ3δσ2σ4/bracketrightbig¯ψk1σ1¯ψk2σ2ψk3σ3ψk4σ4.
(2)
Here and following, all temperature and volume factors are
incorporated in the summation symbols.
Our analysis is based on a truncation of the exact flow
equation4,5for the effective action /Gamma1/Lambda1[ψ,¯ψ], that is, the
generating functional for one-particle irreducible vertex func-tions in the presence of an infrared cutoff /Lambda1. The cutoff isimplemented by adding a regulator function to the inverse of
the bare propagator. The effective action /Gamma1
/Lambda1[ψ,¯ψ] interpolates
between the regularized bare action at the initial scale /Lambda10
and the final effective action /Gamma1[ψ,¯ψ] in the limit /Lambda1→0.
Spontaneous breaking of the U(1) charge symmetry in the
superfluid state can be treated by adding a small (ultimatelyinfinitesimal) symmetry-breaking field
δS[ψ,¯ψ]=/summationdisplay
k[/Delta10(k)¯ψ−k↓¯ψk↑+/Delta1∗
0(k)ψk↑ψ−k↓]( 3 )
to the bare action, which is then promoted to a finite order
parameter in the course of the flow.21
Expanding the exact functional flow equation for /Gamma1/Lambda1[ψ,¯ψ]
in powers of the source fields ψand ¯ψ, one obtains a hierarchy
of flow equations for the n-particle vertex functions.4We
truncate the hierarchy at the two-particle level, includinghowever self-energy corrections generated from contractionsof three-particle terms.
27This truncation was used in all pre-
vious fermionic fRG studies of symmetry breaking.21–24It is
exact for mean-field models. Our description of the truncationfollows closely the presentation in Ref. 24. However, we use
notations as in Ref. 4, where the regulator function is fully
included in the two-point vertex /Gamma1
(2)/Lambda1, and the sign convention
for/Gamma1(2)/Lambda1and the propagator G/Lambda1differs from that used in
Ref. 24.
In a superfluid state, it is convenient to use Nambu spinors
φksand ¯φksdefined as
¯φk+=¯ψk↑,φ k+=ψk↑,¯φk−=ψ−k↓,φ k−=¯ψ−k↓ (4)
instead of ψkσand ¯ψkσas a basis. The effective action as
a functional of the Nambu fields, truncated beyond quartic(two-particle) terms, has the form
/Gamma1
/Lambda1[φ,¯φ]=/Gamma1(0)/Lambda1−/summationdisplay
k/summationdisplay
s1,s2/Gamma1(2)/Lambda1
s1s2(k)¯φks1φks2
+1
4/summationdisplay
k1,...,k 4/summationdisplay
s1,...,s 4/Gamma1(4)/Lambda1
s1s2s3s4(k1,k2,k3,k4)
ׯφk1s1¯φk2s2φk3s3φk4s4, (5)
where /Gamma1(0)/Lambda1yields the grand canonical potential. For spin-
singlet pairing with unbroken spin-rotation invariance, onlyterms with an equal number of φand ¯φfields contribute.
The Nambu vertex /Gamma1
(4)/Lambda1
s1s2s3s4(k1,k2,k3,k4) is nonzero only for
k1+k2=k3+k4, due to translation invariance. The (scale-
dependent) Nambu propagator G/Lambda1is related to /Gamma1(2)/Lambda1by
(G/Lambda1)−1=/Gamma1(2)/Lambda1, and can be written as a 2 ×2m a t r i x
G/Lambda1(k)=/parenleftbiggG/Lambda1
++(k)G/Lambda1
+−(k)
G/Lambda1
−+(k)G/Lambda1
−−(k)/parenrightbigg
=/parenleftbiggG/Lambda1(k)F/Lambda1(k)
F∗/Lambda1(k)−G/Lambda1(−k)/parenrightbigg
.
(6)
The anomalous propagator F/Lambda1(k) is a symmetric function
ofk0and k. The Nambu self-energy /Sigma1/Lambda1is defined by
the Dyson equation ( G/Lambda1)−1=(G/Lambda1
0)−1−/Sigma1/Lambda1, where G/Lambda1
0is
the bare regularized propagator (in presence of /Delta10). In the
superfluid state it has the form
/Sigma1/Lambda1(k)=/parenleftbigg/Sigma1/Lambda1(k) /Delta10(k)−/Delta1/Lambda1(k)
/Delta1∗
0(k)−/Delta1∗/Lambda1(k)−/Sigma1/Lambda1(−k)/parenrightbigg
, (7)
174523-2EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
where /Sigma1/Lambda1(k) is the normal component of the self-energy and
/Delta1/Lambda1(k) is the (flowing) gap function.The gap function and the Nambu vertex are related by a
Ward identity following from global charge conservation (see,for example, Ref. 21)
/Delta1/Lambda1(k)−/Delta10(k)=/summationdisplay
k/prime/summationdisplay
s,s/prime/bracketleftbig
/Delta10(k/prime)G/Lambda1
s+(k/prime)G/Lambda1
−s/prime(k/prime)−/Delta1∗
0(k/prime)G/Lambda1
s−(k/prime)G/Lambda1
+s/prime(k/prime)/bracketrightbig
/Gamma1(4)/Lambda1
+s/primes−(k,k/prime,k/prime,k). (8)
The Ward identity implies that some components of the Nambu vertex diverge in case of spontaneous symmetry breaking ( /Delta1/Lambda1
finite for /Delta10→0), which is a manifestation of the massless Goldstone boson.
The flow equation for the Nambu self-energy is given by
d
d/Lambda1/Sigma1/Lambda1
s1s2(k)=/summationdisplay
k/prime/summationdisplay
s/prime
1,s/prime
2S/Lambda1
s/prime
2s/prime
1(k/prime)/Gamma1(4)/Lambda1
s1s/prime
1s/prime
2s2(k,k/prime,k/prime,k), (9)
where
S/Lambda1(k)=d
d/Lambda1G/Lambda1(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/Sigma1/Lambda1fixed=/bracketleftbig
1−G/Lambda1
0(k)/Sigma1/Lambda1(k)/bracketrightbig−1dG/Lambda1
0(k)
d/Lambda1/bracketleftbig
1−/Sigma1/Lambda1(k)G/Lambda1
0(k)/bracketrightbig−1(10)
is the so-called single-scale propagator. The truncated flow equation for the Nambu vertex (see Fig. 1) reads as
d
d/Lambda1/Gamma1(4)/Lambda1
s1s2s3s4(k1,k2,k3,k4)=/Pi1PH,d
s1s2s3s4(k1,k2,k3,k4)−/Pi1PH,cr
s1s2s3s4(k1,k2,k3,k4)−1
2/Pi1PP
s1s2s3s4(k1,k2,k3,k4), (11)
where
/Pi1PH,d
s1s2s3s4(k1,k2,k3,k4)=/summationdisplay
p,q/summationdisplay
s/prime
1,...,s/prime
4d
d/Lambda1/bracketleftBig
G/Lambda1
s/prime
1s/prime
2(p)G/Lambda1
s/prime
3s/prime
4(q)/bracketrightBig
/Gamma1(4)/Lambda1
s1s/prime
2s/prime
3s4(k1,p,q,k 4)/Gamma1(4)/Lambda1
s/prime
4s2s3s/prime
1(q,k 2,k3,p), (12)
/Pi1PH,cr
s1s2s3s4(k1,k2,k3,k4)=/summationdisplay
p,q/summationdisplay
s/prime
1,...,s/prime
4d
d/Lambda1/bracketleftBig
G/Lambda1
s/prime
1s/prime
2(p)G/Lambda1
s/prime
3s/prime
4(q)/bracketrightBig
/Gamma1(4)/Lambda1
s2s/prime
2s/prime
3s4(k2,p,q,k 4)/Gamma1(4)/Lambda1
s/prime
4s1s3s/prime
1(q,k 1,k3,p), (13)
/Pi1PP
s1s2s3s4(k1,k2,k3,k4)=/summationdisplay
p,q/summationdisplay
s/prime
1,...,s/prime
4d
d/Lambda1/bracketleftBig
G/Lambda1
s/prime
1s/prime
2(p)G/Lambda1
s/prime
3s/prime
4(q)/bracketrightBig
/Gamma1(4)/Lambda1
s1s2s/prime
3s/prime
1(k1,k2,q,p )/Gamma1(4)/Lambda1
s/prime
2s/prime
4s3s4(p,q,k 3,k4). (14)
The flow equation for the self-energy is exact (for an
exact /Gamma1(4)/Lambda1), while in the flow of /Gamma1(4)/Lambda1contributions from
/Gamma1(6)/Lambda1beyond self-energy feedback have been discarded.4,27
These discarded contributions are at least of order ( /Gamma1(4)/Lambda1)3,
and they involve overlapping loops leading to a reducedmomentum integration volume. The truncation is exact formean-field models with a reduced BCS and/or forward scat-tering interaction, although /Gamma1
(4)/Lambda1becomes large at the critical
scale.21,23,24Particle-particle terms in Nambu representation
contain particle-hole contributions in the original fermion basisand vice versa. In particular, the particle-particle contribution
FIG. 1. Feynman diagrams representing the flow equation for the
two-particle vertex. The dots denote differentiation of the propagatorproducts with respect to the scale /Lambda1.generating the Cooper instability is captured by the Nambu
particle-hole diagrams.
III. SYMMETRIES OF NAMBU VERTEX
The Nambu vertex /Gamma1(4)/Lambda1
s1s2s3s4(k1,k2,k3,k4) has 16 components
corresponding to the choices si=± fori=1,..., 4. Spin-
rotation invariance reduces the number of independent com-ponents of the Nambu vertex substantially. In Ref. 24,i tw a s
shown that the vertex can be parametrized by three functions ofk
1,k2,k3,k4, where k1+k2=k3+k4for translation invariant
systems. These functions are further constrained by discretesymmetries. In this section, we describe the spin-rotation
invariant form of the Nambu vertex as derived in Ref. 24.
In addition to the normal interaction, in the U(1) symmetry-
broken state there are also anomalous interactions correspond-
ing to operator products ¯ψ¯ψ¯ψ¯ψ+conjugate and ¯ψ¯ψ¯ψψ+
conjugate.
21,23Following Ref. 24, we write spin-rotation
invariant forms for the normal and anomalous interaction termsin the ψbasis, and then the corresponding expressions in
Nambu representation.
174523-3ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
A spin-rotation invariant normal interaction can always be
expressed as28
/Gamma1(2+2)[ψ,¯ψ]=1
4/summationdisplay
ki,σi/bracketleftbig
V(k1,k2,k3,k4)δσ1σ4δσ2σ3
−V(k1,k2,k4,k3)δσ1σ3δσ2σ4/bracketrightbig
ׯψk1σ1¯ψk2σ2ψk3σ3ψk4σ4. (15)
Here and in the remainder of this section, we suppress the
superscript /Lambda1for the scale dependence. One may also write
/Gamma1(2+2)[ψ,¯ψ] as a sum of a spin-singlet and a spin-triplet
component9
/Gamma1(2+2)[ψ,¯ψ]=1
4/summationdisplay
ki,σi/bracketleftbig
VS(k1,k2,k3,k4)Sσ1σ2σ3σ4
+VT(k1,k2,k3,k4)Tσ1σ2σ3σ4/bracketrightbig
ׯψk1σ1¯ψk2σ2ψk3σ3ψk4σ4, (16)
where Sσ1σ2σ3σ4=1
2(δσ1σ4δσ2σ3−δσ1σ3δσ2σ4),Tσ1σ2σ3σ4=
1
2(δσ1σ4δσ2σ3+δσ1σ3δσ2σ4), and
VS(k1,k2,k3,k4)=V(k1,k2,k3,k4)+V(k1,k2,k4,k3),(17)
VT(k1,k2,k3,k4)=V(k1,k2,k3,k4)−V(k1,k2,k4,k3).(18)
A spin-rotation invariant anomalous interaction with four
creation (or annihilation) operators can be written in the form24
/Gamma1(4+0)[ψ,¯ψ]=1
8/summationdisplay
ki/braceleftbig
WS(k1,k2,k3,k4)/parenleftbig¯ψk1↑¯ψk2↓
−¯ψk1↓¯ψk2↑/parenrightbig/parenleftbig¯ψk3↑¯ψk4↓−¯ψk3↓¯ψk4↑/parenrightbig
−WT(k1,k2,k3,k4)/bracketleftbig/parenleftbig¯ψk1↑¯ψk2↓+¯ψk1↓¯ψk2↑/parenrightbig
×/parenleftbig¯ψk3↑¯ψk4↓+¯ψk3↓¯ψk4↑/parenrightbig
−2/parenleftbig¯ψk1↑¯ψk2↑¯ψk3↓
ׯψk4↓+¯ψk1↓¯ψk2↓¯ψk3↑¯ψk4↑/parenrightbig/bracketrightbig
+conj./bracerightbig
.
(19)Conjugated terms denoted by “conj.” are obtained by reversing
the order of fields, replacing ¯ψkσbyψk∗σ, and complex
conjugation of the functions WS,T.
Finally, spin-rotation invariant anomalous interactions with
three creation and one annihilation operators, or vice versa,can be written as
24
/Gamma1(3+1)[ψ,¯ψ]=1
2/summationdisplay
ki/braceleftBigg
XS(k1,k2,k3,k4)/summationdisplay
σ¯ψk1σ/parenleftbig¯ψk2↑¯ψk3↓
−¯ψk2↓¯ψk3↑/parenrightbig
ψk4σ+XT(k1,k2,k3,k4)
×/bracketleftBigg/summationdisplay
σ/epsilon1σ¯ψk1σ/parenleftbig¯ψk2↑¯ψk3↓+¯ψk2↓¯ψk3↑/parenrightbig
×ψk4σ+2/parenleftbig¯ψk1↑¯ψk2↓¯ψk3↓ψk4↓
−¯ψk1↓¯ψk2↑¯ψk3↑ψk4↑/parenrightbig/bracketrightBigg
+conj./bracerightBigg
, (20)
where /epsilon1↑=1 and/epsilon1↓=− 1.
It is convenient to collect the 16 components of the Nambu
vertex /Gamma1(4)
s1s2s3s4i na4×4m a t r i x
/Gamma1(4)=⎛
⎜⎜⎜⎜⎝/Gamma1(4)
++++ /Gamma1(4)
++−+ /Gamma1(4)
+−++ /Gamma1(4)
+−−+
/Gamma1(4)
+++− /Gamma1(4)
++−− /Gamma1(4)
+−+− /Gamma1(4)
+−−−
/Gamma1(4)
−+++ /Gamma1(4)
−+−+ /Gamma1(4)
−−++ /Gamma1(4)
−−−+
/Gamma1(4)
−++− /Gamma1(4)
−+−− /Gamma1(4)
−−+− /Gamma1(4)
−−−−⎞
⎟⎟⎟⎟⎠. (21)
Rows in this matrix are labeled by s
1ands4, while columns are
labeled by s2ands3. With this convention, the Bethe-Salpeter
equation yielding the exact Nambu vertex in reduced (mean-field) models can be written as a matrix equation.
24Translating
the spin-rotation invariant structure of the various interactionterms to the Nambu representation, one obtains the Nambuvertex in the following form
29:
/Gamma1(4)(k1,k2,k3,k4)=⎛
⎜⎜⎜⎝VT(k1,k2,k3,k4) X(k1,k2,k3,k4) X∗(k∗
4,k∗
3,k∗
2,k∗
1)−V(k1,−k3,−k2,k4)
−X(k1,k2,k4,k3) W(k1,k2,k3,k4)V(k1,−k4,−k2,k3)X∗(k4,k3,k1,k2)
−X∗(k∗
4,k∗
3,k∗
2,k∗
1)V∗(k1,−k4,−k2,k3)W∗(k∗
4,k∗
3,k∗
2,k∗
1) X(k∗
1,k∗
2,k∗
3,k∗
4)
−V∗(k1,−k3,−k2,k4)−X∗(k4,k3,k2,k1)−X(k∗
1,k∗
2,k∗
3,k∗
4)VT∗(k1,k2,k3,k4)⎞
⎟⎟⎟⎠, (22)
where k∗=(−k0,k). The matrix elements WandXare related
to the anomalous (4 +0) and (3 +1) interactions, respectively:
W(k1,k2,k3,k4)
=WS(k1,−k4,−k3,k2)−WS(k1,−k3,−k4,k2)
+WT(k1,−k4,−k3,k2)−WT(k1,−k3,−k4,k2)
+2WT(k1,k2,−k3,−k4), (23)
X(k1,k2,k3,k4)=XS(k1,k2,−k3,k4)−XS(k2,k1,−k3,k4)
+XT(k1,k2,−k3,k4)−XT(k2,k1,−k3,k4)
+2XT(−k3,k2,k1,k4). (24)For translation invariant systems, the functions
V(k1,k2,k3,k4),W(k1,k2,k3,k4), and X(k1,k2,k3,k4)a r e
nonzero only if k1+k2=k3+k4, and can therefore be
parametrized by three energy and momentum variables.Discrete symmetries, such as time reversal and reflectioninvariance, and the antisymmetry under particle exchange fur-ther constrain the functions parametrizing the Nambu vertex.
24
IV . CHANNEL DECOMPOSITION
The two-particle vertex acquires a pronounced momentum
and frequency dependence in the course of the flow, whichbecomes even singular at the critical scale for sponta-neous symmetry breaking. A parametrization based on weak
174523-4EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
coupling power counting is not adequate in this situation.
Keeping the full dependence on the three independent mo-menta and frequencies is technically not feasible. The particle-particle and particle-hole contributions to the flow, Eq. (11),
depend strongly on certain linear combinations of momentaand frequencies, namely,
/Pi1
PH,d
s1s2s3s4(k1,k2,k3,k4):k3−k2,
/Pi1PH,cr
s1s2s3s4(k1,k2,k3,k4):k3−k1, (25)
/Pi1PP
s1s2s3s4(k1,k2,k3,k4):k1+k2.
This is because the poles of the contributing propagators
coalesce when the above combinations of momenta andfrequencies vanish or are situated at special nesting points(in case of nested Fermi surfaces). We therefore write thevertex as a sum of interaction channels, where each channelcarries one potentially singular momentum dependence, whichcan be parametrized accurately, while the dependence on theremaining two momentum variables is treated more crudely.This channel decomposition was introduced by Husemann andSalmhofer
25for the two-particle vertex in a normal metallic
state,26and extended by us for a superfluid state.24Most re-
cently, it was also formulated for an antiferromagnetic state.30
A. Interaction channels
Following Husemann and Salmhofer,25we write the normal
vertex in the form
/Gamma1(2+2)/Lambda1[ψ,¯ψ]=U[ψ,¯ψ]+1
2/summationdisplay
ki/primeC/Lambda1
k1+k4
2,k2+k3
2(k3−k2)
×/summationdisplay
σ,σ/prime¯ψk1σ¯ψk2σ/primeψk3σ/primeψk4σ
+1
2/summationdisplay
ki/primeM/Lambda1
k1+k4
2,k2+k3
2(k3−k2)
×/summationdisplay
σi/vectorτσ1σ4·/vectorτσ2σ3¯ψk1σ1¯ψk2σ2ψk3σ3ψk4σ4
+1
2/summationdisplay
ki/primeP/Lambda1
k1−k2
2,k4−k3
2(k1+k2)
×/summationdisplay
σ,σ/prime¯ψk1σ¯ψk2σ/primeψk3σ/primeψk4σ, (26)
where U[ψ,¯ψ] is the bare interaction, and the coupling
functions C/Lambda1,M/Lambda1, andP/Lambda1capture the “charge,” “magnetic”
(spin), and “pairing” channels, respectively. The matricescollected in /vectorτ=(τ
x,τy,τz) are the three Pauli matrices. The
prime at the sums over kiindicates momentum (and frequency)
conservation, k1+k2=k3+k4. The momentum argument in
brackets is the momentum transfer for the charge and magneticchannels, and the total momentum for the pairing channel.These are the variables for which a singular dependence isexpected. Comparing the ansatz (26) to the general spin-
rotation invariant form of the normal vertex (15), written in
terms of V
/Lambda1(k1,k2,k3,k4), one obtains the relation
V/Lambda1(k1,k2,k3,k4)
=U(k1,k2,k3,k4)+/bracketleftBig
C/Lambda1
k1+k4
2,k2+k3
2(k3−k2)+P/Lambda1
k1−k2
2,k4−k3
2(k1+k2)−M/Lambda1
k1+k4
2,k2+k3
2(k3−k2)
−2M/Lambda1
k1+k3
2,k2+k4
2(k1−k3)/bracketrightBig
δk1+k2,k3+k4. (27)
The flow equations for C/Lambda1,M/Lambda1, andP/Lambda1are obtained by
choosing a Nambu component involving the normal interaction
V/Lambda1, such as /Gamma1(4)/Lambda1
+−+− (k1,k2,k3,k4)=V/Lambda1(k1,−k4,−k2,k3), and
linking the flow of the various components to the Nambuparticle-particle and particle-hole terms such that momentain brackets correspond to the strong momentum dependenciesas in Eq. (25). One thus obtains
24
d
d/Lambda1C/Lambda1
kk/prime(q)=1
4/Pi1PP
+−+−/parenleftbigg
k+q
2,q
2−k,k/prime+q
2,q
2−k/prime/parenrightbigg
−/Pi1PH,cr
+−+−/parenleftbigg
k+q
2,−q
2−k/prime,k−q
2,q
2−k/prime/parenrightbigg
,
(28)
d
d/Lambda1M/Lambda1
kk/prime(q)=1
4/Pi1PP
+−+−/parenleftbigg
k+q
2,q
2−k,k/prime+q
2,q
2−k/prime/parenrightbigg
,
(29)
d
d/Lambda1P/Lambda1
kk/prime(q)=/Pi1PH,d
+−+−/parenleftbigg
k+q
2,k/prime−q
2,k/prime+q
2,k−q
2/parenrightbigg
.(30)
The pairing interaction can be split into a singlet and a triplet
component as
P/Lambda1
kk/prime(q)=PS,/Lambda1
kk/prime(q)+PT,/Lambda1
kk/prime(q), (31)
where PS,/Lambda1
kk/prime(q) is symmetric under sign changes of kandk/prime,
whilePT,/Lambda1
kk/prime(q) is antisymmetric.
For the anomalous (4 +0) interactions, the dependence
on the (total) momentum of the Cooper pairs contained in/Gamma1
(4+0)/Lambda1[ψ,¯ψ][ E q . (19)] is expected to become singular, which
is taken into account by the ansatz
Wν,/Lambda1(k1,k2,k3,k4)=Wν,/Lambda1
k1−k2
2,k4−k3
2(k1+k2)δk1+k2+k3+k4,0
(32)
forν=S,T. Equation (23) then yields
W/Lambda1(k1,k2,k3,k4)
=/bracketleftBig
WS,/Lambda1
k1+k4
2,k2+k3
2(k3−k2)−WS,/Lambda1
k1+k3
2,k2+k4
2(k1−k3)
+WT,/Lambda1
k1+k4
2,k2+k3
2(k3−k2)−WT,/Lambda1
k1+k3
2,k2+k4
2(k1−k3)
+2WT,/Lambda1
k1−k2
2,k3−k4
2(k1+k2)/bracketrightBig
δk1+k2,k3+k4. (33)
A Nambu vertex component capturing this interaction is
/Gamma1(4)/Lambda1
++−− (k1,k2,k3,k4). Matching again the strong momentum
dependencies in brackets with those of the particle-particleand particle-hole terms, one gets
24
d
d/Lambda1WS,/Lambda1
kk/prime(q)=/Pi1PH,d
++−−/parenleftbigg
k+q
2,k/prime−q
2,k/prime+q
2,k−q
2/parenrightbigg
−1
4/Pi1PP
++−−/parenleftbigg
k+q
2,q
2−k,q
2−k/prime,k/prime+q
2/parenrightbigg
,
(34)
174523-5ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
d
d/Lambda1WT,/Lambda1
kk/prime(q)=1
4/Pi1PP
++−−/parenleftbigg
k+q
2,q
2−k,q
2−k/prime,k/prime+q
2/parenrightbigg
.
(35)
The functions XS,/Lambda1(k1,k2,k3,k4) and XT,/Lambda1(k1,k2,k3,k4)
parametrizing /Gamma1(3+1)/Lambda1[ψ,¯ψ]i nE q . (20) are expected to depend
singularly on k2+k3, which is the total momentum of the
Cooper pair in /Gamma1(3+1)/Lambda1[ψ,¯ψ]. We therefore write
Xν,/Lambda1(k1,k2,k3,k4)=Xν,/Lambda1
k1+k4
2,k2−k3
2(k2+k3)δk1+k2+k3,k4(36)
forν=S,T. Equation (24) then yields
X/Lambda1(k1,k2,k3,k4)
=/bracketleftBig
XS,/Lambda1
k1+k4
2,k2+k3
2(k2−k3)−XS,/Lambda1
k2+k4
2,k1+k3
2(k1−k3)
+XT,/Lambda1
k1+k4
2,k2+k3
2(k2−k3)−XT,/Lambda1
k2+k4
2,k1+k3
2(k1−k3)
+2XT,/Lambda1
k4−k3
2,k2−k1
2(k1+k2)/bracketrightBig
δk1+k2,k3+k4. (37)
Anomalous (3 +1) interactions are contained in the Nambu
vertex component /Gamma1(4)/Lambda1
++−+ (k1,k2,k3,k4). Matching singular
momentum dependencies between the vertex on the left-handside and the particle-particle and particle-hole terms on theright-hand side of the flow equation yields
24
d
d/Lambda1XS,/Lambda1
kk/prime(q)=/Pi1PH,d
++−+/parenleftbigg
k−q
2,k/prime+q
2,k/prime−q
2,k+q
2/parenrightbigg
−1
4/Pi1PP
++−+/parenleftbigg
k/prime+q
2,q
2−k/prime,q
2−k,k+q
2/parenrightbigg
,
(38)
d
d/Lambda1XT,/Lambda1
kk/prime(q)=1
4/Pi1PP
++−+/parenleftbigg
k/prime+q
2,q
2−k/prime,q
2−k,k+q
2/parenrightbigg
.
(39)
Equations (28)–(30),(34),(35),(38), and (39) determine
the flow of the complete set of coupling functions describing
the Nambu vertex, that is, C/Lambda1
kk/prime(q),M/Lambda1
kk/prime(q),P/Lambda1
kk/prime(q),WS,/Lambda1
kk/prime(q),
WT,/Lambda1
kk/prime(q),XS,/Lambda1
kk/prime(q), and XT,/Lambda1
kk/prime(q), respectively. Note that the
above choice of Nambu components is not unique. Anycomponent containing V
/Lambda1,W/Lambda1, andX/Lambda1, respectively, could
have been chosen. The resulting equations for the functionsC
/Lambda1
kk/prime(q), etc., are equivalent.
Discrete symmetries and Osterwalder-Schrader positivity
(corresponding to Hermiticity) constrain the functions C/Lambda1
kk/prime(q),
etc., by relations analogous to those for the interactionfunctions presented in Sec. 3 of Ref. 24. The normal interaction
components obey
C
/Lambda1
kk/prime(q)=C/Lambda1
RkRk/prime(Rq)=C/Lambda1
kk/prime(−q)=C/Lambda1∗
−k,−k/prime(q), (40)
M/Lambda1
kk/prime(q)=M/Lambda1
RkRk/prime(Rq)=M/Lambda1
kk/prime(−q)=M/Lambda1∗
−k,−k/prime(q),(41)
P/Lambda1
kk/prime(q)=P/Lambda1
RkRk/prime(Rq)=P/Lambda1
k/primek(q)=P/Lambda1∗
−k/prime,−k(−q), (42)
where Rk=(k0,−k). Here, the first equation follows from
inversion symmetry, the second from inversion and time-reversal symmetry, and the third from inversion symmetry andpositivity. For the anomalous (4 +0) interaction, invariance
under spatial inversion and time reversal yield the relations
W
ν,/Lambda1
kk/prime(q)=Wν,/Lambda1
RkRk/prime(Rq)=Wν,/Lambda1∗
−k,−k/prime(−q) (43)FIG. 2. Decomposition of the Nambu vertex in bare interaction,
particle-hole channels, and particle-particle channel.
forν=S,T, and for the (3 +1) interactions
Xν,/Lambda1
kk/prime(q)=Xν,/Lambda1
RkRk/prime(Rq)=Xν,/Lambda1∗
−k,−k/prime(−q). (44)
The complete Nambu vertex can be written in the form (see
Fig. 2)
/Gamma1(4)/Lambda1
s1s2s3s4(k1,k2,k3,k4)=Us1s2s3s4(k1,k2,k3,k4)
+/bracketleftbigg
VPH,/Lambda1
s1s2s3s4/parenleftbiggk1+k4
2,k2+k3
2;k3−k2/parenrightbigg
−VPH,/Lambda1
s2s1s3s4/parenleftbiggk2+k4
2,k1+k3
2;k3−k1/parenrightbigg
+VPP,/Lambda1
s1s2s3s4/parenleftbiggk1−k2
2,k4−k3
2;k1+k2/parenrightbigg/bracketrightbigg
×δk1+k2,k3+k4, (45)
where the first term represents the Nambu components of the
bare interaction, while the other terms are generated by theparticle-hole and particle-particle contributions to the flow,that is,
d
d/Lambda1VPH,/Lambda1
s1s2s3s4/parenleftbiggk1+k4
2,k2+k3
2;k3−k2/parenrightbigg
=/Pi1PH,d
s1s2s3s4(k1,k2,k3,k4), (46)
d
d/Lambda1VPP,/Lambda1
s1s2s3s4/parenleftbiggk1−k2
2,k4−k3
2;k1+k2/parenrightbigg
=−1
2/Pi1PP
s1s2s3s4(k1,k2,k3,k4). (47)
The crossed particle-hole contribution yields the flow of VPH,/Lambda1
with indices 1 and 2 exchanged and a minus sign compared
to the direct contribution. Collecting terms with the variableq=k
3−k2in the channel decomposition, and writing the
components in matrix form as in Eq. (21), one obtains
VPH,/Lambda1(k,k/prime;q)
=⎛
⎜⎜⎜⎝K+,/Lambda1
kk/prime(q)X/Lambda1
kk/prime(−q)X/Lambda1
kk/prime(q)−K−,/Lambda1
k,−k/prime(−q)
X/Lambda1
k/primek(q)W/Lambda1
kk/prime(q)P/Lambda1
kk/prime(q)−X/Lambda1∗
k/primek(−q)
X/Lambda1
k/primek(−q)P/Lambda1∗
kk/prime(q)W/Lambda1∗
kk/prime(q)−X/Lambda1∗
k/primek(q)
−K−,/Lambda1∗
k,−k/prime(−q)−X/Lambda1∗
kk/prime(q)−X/Lambda1∗
kk/prime(−q)K+,/Lambda1∗
kk/prime(q)⎞
⎟⎟⎟⎠,
(48)
where
W/Lambda1
kk/prime(q)=WS,/Lambda1
kk/prime(q)+WT,/Lambda1
kk/prime(q), (49)
X/Lambda1
kk/prime(q)=XS,/Lambda1
kk/prime(q)+XT,/Lambda1
kk/prime(q), (50)
and
K±,/Lambda1
kk/prime(q)=C/Lambda1
kk/prime(q)±M/Lambda1
kk/prime(q). (51)
174523-6EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
Collecting terms with the variable q=k1+k2, one finds
VPP,/Lambda1(k,k/prime;q)
=2⎛
⎜⎜⎜⎜⎝P
T,/Lambda1
kk/prime(q)−XT,/Lambda1
k/primek(q)−XT,/Lambda1
kk/prime(q)M/Lambda1
k,k/prime(q)
XT,/Lambda1
−k/prime,k(q)−WT,/Lambda1
kk/prime(q)−M/Lambda1
k,−k/prime(q)−XT,/Lambda1∗
−k,k/prime(q)
XT,/Lambda1
−k,k/prime(q)−M/Lambda1
k,−k/prime(q)−WT,/Lambda1∗
kk/prime(q)−XT,/Lambda1∗
−k/prime,k(q)
M/Lambda1∗
kk/prime(q)XT,/Lambda1∗
kk/prime(q)XT,/Lambda1∗
k/primek(q)PT,/Lambda1∗
kk/prime(q)⎞
⎟⎟⎟⎟⎠.
(52)
Note that V
PH,/Lambda1captures the full information on the coupling
functions C/Lambda1
kk/prime(q), etc., characterizing the Nambu vertex.
By contrast, VPP,/Lambda1collects only magnetic and triplet pairing
components.
Forq=0, the matrix VPH,/Lambda1has the same structure as the
Nambu vertex for a mean-field model with reduced BCS andforward scattering interactions.
24Contributions with q/negationslash=0
correspond to fluctuations away from the zero momentum
Cooper and forward scattering channels.
It is convenient to use linear combinations of P/Lambda1andW/Lambda1
corresponding to amplitude and phase variables. For a real gap
function, these combinations are
A/Lambda1
kk/prime(q)=Re/bracketleftbig
P/Lambda1
kk/prime(q)+W/Lambda1
kk/prime(q)/bracketrightbig
, (53)
/Phi1/Lambda1
kk/prime(q)=Re/bracketleftbig
P/Lambda1
kk/prime(q)−W/Lambda1
kk/prime(q)/bracketrightbig
. (54)
Amplitude and phase variables for singlet and triplet compo-
nents can be defined by analogous linear combinations. NotethatP
/Lambda1
kk/prime(q) andW/Lambda1
kk/prime(q) are generally complex functions for
q0/negationslash=0, even for a real gap. For their real and imaginary parts,
we use the notation P/prime/Lambda1
kk/prime(q),W/prime/Lambda1
kk/prime(q) and P/prime/prime/Lambda1
kk/prime(q),W/prime/prime/Lambda1
kk/prime(q),
respectively. Instead of the representation (21), it can be
advantageous to use a Pauli matrix basis to represent theNambu vertex, as described in the Appendix.
B. Boson propagators and fermion-boson vertices
To achieve an efficient parametrization of the momentum
and frequency dependencies, the coupling functions arewritten in the form of boson-mediated interactions withbosonic propagators and fermion-boson vertices, as proposedby Husemann and Salmhofer.
25The bosonic propagators
capture the (potentially) singular dependence on the transfermomentum and frequency while the fermion-boson verticesdescribe the more regular remaining momentum and frequencydependencies. For example, the charge coupling function isdecomposed as
C
/Lambda1
kk/prime(q)=/summationdisplay
α,α/primeC/Lambda1
αα/prime(q)g/Lambda1
cα(k,q)g/Lambda1
cα/prime(k/prime,q), (55)
where the functions g/Lambda1
cα(k,q) provide a real orthonormal basis
set ofk-space functions, satisfying
/integraldisplay
dμ(k)g/Lambda1
cα(k,q)g/Lambda1
cα/prime(k,q)=δαα/prime (56)
with a suitable (not yet specified) k-space measure dμ(k).
Viewing C/Lambda1
kk/prime(q) as a boson-mediated interaction, the functions
C/Lambda1
αα/prime(q) can be interpreted as boson propagators and g/Lambda1
cα(k,q)
as fermion-boson vertices. Analogous decompositions areused for the magnetic and pairing coupling functions M
/Lambda1
kk/prime(q)andP/Lambda1
kk/prime(q), or the singlet/triplet components of the latter.
The anomalous (4 +0) coupling function W/Lambda1
kk/prime(q) can also
be decomposed in the form (55). Alternatively, one may
decompose the amplitude and phase coupling functions. Theanomalous (3 +1) coupling functions X
/Lambda1
kk/prime(q) require a more
general decomposition
X/Lambda1
kk/prime(q)=/summationdisplay
α,α/primeX/Lambda1
αα/prime(q)g/Lambda1
xα(k,q)˜g/Lambda1
xα/prime(k/prime,q), (57)
with two different sets of basis functions g/Lambda1
xαand ˜g/Lambda1
xα, since
thekandk/primedependencies of X/Lambda1
kk/prime(q) are generally different.
Summing over a complete set of basis functions, the abovedecomposition is exact. In practice, one has to approximatethe infinite sum by a finite number of terms, with a suitablechoice of boson propagators and fermion-boson vertices.
C. Classification of contributions to the flow
Inserting the channel decomposed Nambu vertex on the
right-hand side of the flow equation yields several contri-butions which can be distinguished by their topology whenrepresenting the coupling functions by boson-mediated inter-actions. For a graphical representation we use the symbolicnotation from Fig. 2, where bosons mediating interactions in
the (Nambu) particle-hole and particle-particle channels arerepresented by a wiggly and a double line, respectively. Allcontributions to the flow of the vertex are of second order in theinteraction. We discuss the different topologies for diagramswith two wiggly lines as examples. There are three distinctclasses, which we refer to as “propagator renormalization,”“vertex correction,” and “box contribution.” For the propagatorrenormalization (Fig. 3, left), the momenta of both bosonic
propagators coincide with the momentum transported throughthe fermionic bubble. Hence, a singularity in the bosonicpropagators generated by the bubble is amplified by feedbackof both propagators. For the vertex correction (Fig. 3, right),
the momentum of one of the bosonic propagators coincideswith the momentum of the fermionic (Nambu) particle-holepair. Potential singularities of the other bosonic propagatorare wiped out by the momentum integration. Note that atzero temperature all expected singularities of the vertexare integrable in two spatial dimensions, even the infraredsingularity associated with the Goldstone mode. For the boxcontribution (Fig. 4), singularities of the fermionic pair are not
amplified by singularities of the bosonic propagators which
FIG. 3. Examples for propagator renormalizaton (left) and vertex
correction (right). The variable pis integrated.
174523-7ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
FIG. 4. Example for a box diagram with integration variable p.
are both integrated. The contribution from the propagator
renormalization diagram thus dominates in the formation ofsingularities at special wave vectors q=k
3−k2. In mean-field
models with reduced interactions, it yields the complete flow,while vertex corrections and box contributions vanish.
The assignment of momenta in the channel decomposi-
tion was designed to deal with singularities generated bythe fermionic propagators. However, the box contributionexhibits another singularity generated by the singularity ofthe bosonic propagators at momentum zero. For k
1=k3(that
is,k=k/prime), the two bosonic propagators in Fig. 4carry the
same momentum variable. In the phase fluctuation channel(Goldstone mode), these propagators diverge quadratically atsmall momenta and frequencies (for /Delta1
0=0). The product
of two such singularities is not integrable in two dimensions.This problem can be treated by introducing a scale-dependentpairing field /Delta1
/Lambda1
0, which tends to zero continuously toward the
end of the flow. A finite pairing field regularizes divergencesin the Cooper channel (including the Goldstone mode), suchthat the right-hand side of the flow equations remains finite ateach finite scale, and the flow is integrable down to /Lambda1→0,
/Delta1
/Lambda1
0→0, as discussed in more detail in Sec. VI D4 . A scale
dependent /Delta1/Lambda1
0does not modify the structure of the flow
FIG. 5. Diagrammatic representation of contributions to the flow
ofVPH,/Lambda1. The dot denotes a /Lambda1derivative acting on the product of the
two fermionic propagators.
FIG. 6. Diagrammatic representation of contributions to the flow
ofVPP,/Lambda1.
equations, it merely yields additional contributions to the scale
derivative of the bare (Nambu) propagator G/Lambda1
0.
In addition to the contributions shown in Figs. 3and 4,
there are analogous contributions with wiggly lines replaced bydouble lines corresponding to the particle-particle channel and4-point vertices representing the bare interaction (as in Fig. 2),
including all possible mixtures of channels. The complete setof contributions to the flow of V
PH,/Lambda1andVPP,/Lambda1is shown in
Figs. 5and 6, respectively. Note that all diagrams are one-
particle irreducible, that is, they can not be cut by cutting asingle fermionic propagator line. Some of them can be cut bycutting an interaction line, but these lines do not correspondto particle propagators since the interaction is not representedby bosonic fields in our purely fermionic RG.
V . RANDOM PHASE APPROXIMATION
To gain insight into the singularity structure of the Nambu
vertex, it is instructive to consider the random phase approxi-mation (RPA) before analyzing the full set of flow equations.In the conventional formulation, the RPA corresponds to asummation of all (direct) particle-hole ladder contributionsto the Nambu vertex with bare interactions and mean-fieldpropagators. The self-energy is obtained from the usual mean-field equation, that is, self-consistent first-order perturbationtheory. In the channel decomposed functional RG frameworkderived in Sec. IV, the RPA is equivalent to the approximation
/Gamma1
(4)/Lambda1
s1s2s3s4(k,k/prime;q)=Us1s2s3s4(k,k/prime;q)+VPH,/Lambda1
s1s2s3s4(k,k/prime;q),(58)
that is, the crossed particle-hole and particle-particle channels
are discarded. Throughout this section we parametrize themomentum variables k
1,k2,k3,k4ask1=k+q
2,k2=k/prime−q
2,
k3=k/prime+q
2, and k4=k−q
2. The flow equation (46) for
VPH,/Lambda1can then be formally integrated to obtain an integral
equation which, expressed in term of /Gamma1(4)/Lambda1, reads as
/Gamma1(4)/Lambda1
s1s2s3s4(k,k/prime;q)=Us1s2s3s4(k,k/prime;q)
+/summationdisplay
p/summationdisplay
s/prime
iUs1s/prime
2s/prime
3s4(k,p;q)G/Lambda1
s/prime
1s/prime
2/parenleftbigg
p−q
2/parenrightbigg
×G/Lambda1
s/prime
3s/prime
4/parenleftbigg
p+q
2/parenrightbigg
/Gamma1(4)/Lambda1
s/prime
4s2s3s/prime
1(p,k/prime;q).(59)
174523-8EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
This is the familiar Bethe-Salpeter–type equation corre-
sponding to a summation of (Nambu) particle-hole ladders.Using this equation, the flow equation for the self-energy[Eq. (9)] can also be integrated, yielding the usual mean-field
equation
/Sigma1
/Lambda1
s1s2=/summationdisplay
k/prime/summationdisplay
s/prime
1,s/prime
2Us1s/prime
1s/prime
2s2(k,k/prime;0 )G/Lambda1
s/prime
2s/prime
1(k/prime). (60)
The integral equation (59) can be written in matrix form
such that Nambu index sums correspond to matrix products.In particular, choosing the Pauli matrix basis described inthe Appendix, one obtains
˜/Gamma1(4)/Lambda1(k,k/prime;q)=˜U(k,k/prime;q)+/summationdisplay
p˜U(k,p;q)
טL/Lambda1(p;q)˜/Gamma1(4)/Lambda1(p,k/prime;q), (61)
where the components of ˜L/Lambda1(p;q)a r eg i v e nb y
˜L/Lambda1
jj/prime(p;q)=1
2/summationdisplay
siτ(j)
s4s1τ(j/prime)
s3s2G/Lambda1s
2s4/parenleftbigg
p−q
2/parenrightbigg
G/Lambda1
s1s3/parenleftbigg
p+q
2/parenrightbigg
.
(62)
For a spin-rotation invariant system, the bare interaction
can be written in the form
U[ψ,¯ψ]=1
2/summationdisplay
k,k/prime,qC(0)
kk/prime(q)/summationdisplay
σ,σ/prime¯ψk+q/2,σ¯ψk/prime−q/2,σ/primeψk/prime+q/2,σ/primeψk−q/2,σ+1
2/summationdisplay
k,k/prime,qM(0)
kk/prime(q)/summationdisplay
σi/vectorτσ1σ4·/vectorτσ2σ3¯ψk+q/2,σ1¯ψk/prime−q/2,σ2
×ψk/prime+q/2,σ3ψk−q/2,σ4+1
2/summationdisplay
k,k/prime,qP(0)
kk/prime(q)/summationdisplay
σ,σ/prime¯ψq/2+k,σ¯ψq/2−k,σ/primeψq/2−k/prime,σ/primeψq/2+k/prime,σ, (63)
which is analogous to the decomposition of the fluctuation
contributions in Eq. (26). In the special case where the bare
coupling functions C(0)
kk/prime(q),M(0)
kk/prime(q), and P(0)
kk/prime(q) are nonzero
only for q=0, this becomes the reduced BCS and forward
scattering interaction of the model discussed in detail in
Ref. 24. In that case, the mean-field equation for the self-energy
is exact, and the Bethe-Salpeter equation yields the exact vertex/Gamma1
(4)/Lambda1
s1s2s3s4(k,k/prime;q=0).
For an explicit evaluation of the RPA vertex, we assume
separable interactions
C(0)
kk/prime(q)=C(0)(q)fc(k)fc(k/prime),
M(0)
kk/prime(q)=M(0)(q)fm(k)fm(k/prime), (64)
P(0)
kk/prime(q)=P(0)(q)fp(k)fp(k/prime),
with symmetric (under k/mapsto→−k) form factors, and a bare
gap function /Delta10(k)=/Delta10fp(k) with the same form factor as
the pairing interaction. The coupling functions contributing toV
PH,/Lambda1[see Eq. (48)] then also factorize:
C/Lambda1
kk/prime(q)=C/Lambda1(q)fc(k)fc(k/prime),
M/Lambda1
kk/prime(q)=M/Lambda1(q)fm(k)fm(k/prime),
P/Lambda1
kk/prime(q)=P/Lambda1(q)fp(k)fp(k/prime), (65)
W/Lambda1
kk/prime(q)=W/Lambda1(q)fp(k)fp(k/prime),
X/Lambda1
kk/prime(q)=X/Lambda1(q)fc(k)fp(k/prime),
and the gap function has the form
/Delta1/Lambda1(k)=/Delta1/Lambda1fp(k). (66)
The vertex assumes a particularly simple form in the Pauli
matrix basis, namely,
˜/Gamma1(4)/Lambda1(k,k/prime;q)=˜f(k)˜/Gamma1(4)/Lambda1(q)˜f(k/prime), (67)where ˜f(k) is the diagonal matrix
˜f(k)=diag[fm(k),fp(k),fp(k),fc(k)] (68)
and
˜/Gamma1(4)/Lambda1(q)=˜U(q)+˜VPH,/Lambda1(q) (69)
with
˜U(q)=⎛
⎜⎜⎜⎝2M(0)(q)0 0 0
0 P(0)(q)0 0
00 P(0)(q)0
00 0 2 C(0)(q)⎞
⎟⎟⎟⎠(70)
and
˜VPH,/Lambda1(q)=⎛
⎜⎜⎜⎜⎝2M/Lambda1(q)0 0 0
0 A/Lambda1(q)P/prime/prime/Lambda1(q)2X/prime/Lambda1(q)
0−P/prime/prime/Lambda1(q)/Phi1/Lambda1(q)−2X/prime/prime/Lambda1(q)
02 X/prime/Lambda1(q)2X/prime/prime/Lambda1(q)2C/Lambda1(q)⎞
⎟⎟⎟⎟⎠.
(71)
Here, A/Lambda1(q)=P/prime/Lambda1(q)+W/prime/Lambda1(q) and /Phi1/Lambda1(q)=P/prime/Lambda1(q)−
W/prime/Lambda1(q). Primes denote real parts and double primes imaginary
parts. At q0=0, all imaginary parts vanish. For q=0t h e
above matrix simplifies to the vertex previously obtained forthe reduced BCS and forward scattering model,
24in a slightly
different basis yielding some sign changes.
Inserting the factorized form of the vertex into the Bethe-
Salpeter equation (61), one obtains a linear algebraic equation
for˜/Gamma1(4)/Lambda1(q),
˜/Gamma1(4)/Lambda1(q)=˜U(q)+˜U(q)˜L/Lambda1(q)˜/Gamma1(4)/Lambda1(q), (72)
174523-9ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
where
˜L/Lambda1(q)=/summationdisplay
p˜f(p)˜L/Lambda1(p;q)˜f(p)=⎛
⎜⎜⎜⎝˜L/Lambda1
m(q)0 0 0
0 ˜L/Lambda1
a(q) ˜L/prime/prime/Lambda1
p(q) ˜L/prime/Lambda1
x(q)
0 −˜L/prime/prime/Lambda1
p(q) ˜L/Lambda1
φ(q)−˜L/prime/prime/Lambda1
x(q)
0 ˜L/prime/Lambda1
x(q) ˜L/prime/prime/Lambda1
x(q) ˜L/Lambda1
c(q)⎞
⎟⎟⎟⎠. (73)
The matrix elements ˜L/Lambda1
0jand ˜L/Lambda1
j0withj=1,2,3 vanish for symmetric form factors. The other matrix elements are given by
˜L/Lambda1
c(q)=/summationdisplay
p/bracketleftbigg
G/Lambda1/parenleftbigg
p−q
2/parenrightbigg
G/Lambda1/parenleftbigg
p+q
2/parenrightbigg
−F/Lambda1/parenleftbigg
p−q
2/parenrightbigg
F/Lambda1/parenleftbigg
p+q
2/parenrightbigg/bracketrightbigg
f2
c(p),
˜L/Lambda1
m(q)=/summationdisplay
p/bracketleftbigg
G/Lambda1/parenleftbigg
p−q
2/parenrightbigg
G/Lambda1/parenleftbigg
p+q
2/parenrightbigg
+F/Lambda1/parenleftbigg
p−q
2/parenrightbigg
F/Lambda1/parenleftbigg
p+q
2/parenrightbigg/bracketrightbigg
f2
m(p),
˜L/Lambda1
a(q)=−/summationdisplay
p/braceleftbigg
Re/bracketleftbigg
G/Lambda1/parenleftbigg
p+q
2/parenrightbigg
G/Lambda1/parenleftbigg
−p+q
2/parenrightbigg/bracketrightbigg
−F/Lambda1/parenleftbigg
p−q
2/parenrightbigg
F/Lambda1/parenleftbigg
p+q
2/parenrightbigg/bracerightbigg
f2
p(p),
˜L/Lambda1
φ(q)=−/summationdisplay
p/braceleftbigg
Re/bracketleftbigg
G/Lambda1/parenleftbigg
p+q
2/parenrightbigg
G/Lambda1/parenleftbigg
−p+q
2/parenrightbigg/bracketrightbigg
+F/Lambda1/parenleftbigg
p−q
2/parenrightbigg
F/Lambda1/parenleftbigg
p+q
2/parenrightbigg/bracerightbigg
f2
p(p), (74)
˜L/prime/prime/Lambda1
p(q)=−/summationdisplay
pIm/bracketleftbigg
G/Lambda1/parenleftbigg
p+q
2/parenrightbigg
G/Lambda1/parenleftbigg
−p+q
2/parenrightbigg/bracketrightbigg
f2
p(p),
˜L/prime/Lambda1
x(q)=2/summationdisplay
pRe/bracketleftbigg
G/Lambda1/parenleftbigg
p−q
2/parenrightbigg
F/Lambda1/parenleftbigg
p+q
2/parenrightbigg/bracketrightbigg
fc(p)fp(p),
˜L/prime/prime/Lambda1
x(q)=2/summationdisplay
pIm/bracketleftbigg
G/Lambda1/parenleftbigg
p−q
2/parenrightbigg
F/Lambda1/parenleftbigg
p+q
2/parenrightbigg/bracketrightbigg
fc(p)fp(p).
The system of linear equations (72) can be solved explic-
itly. The magnetic coupling function is decoupled from theothers, so that M
(0)(q)+M/Lambda1(q)={[M(0)(q)]−1−˜L/Lambda1
m(q)}−1.
Solving for the other coupling functions amounts to solving alinear 3 ×3 system. We do not write the explicit expressions
here, but discuss only the singularity structure of the couplingfunctions. Singularities arise because the determinant d
/Lambda1(q)=
det[1−˜U(q)˜L/Lambda1(q)] vanishes at q=0f o r/Delta10→0, if/Delta1/Lambda1(k)
remains finite, that is, in case of spontaneous symmetrybreaking. For q=0, the explicit solution for the coupling
functions and their behavior for /Delta1
0→0 was discussed in
detail in Ref. 24. For small q/negationslash=0, one can expand d(q)=
d/Lambda1=0(q)=d0+d1q2
0+d2q2+··· , where d0∝/Delta10for small
/Delta10, while d1andd2remain finite for /Delta10→0. Expanding
all coefficients to leading order in q0andq, one obtains the
singular coupling functions
/Phi1(q)∝−1
d0+d1q2
0+d2q2,
P/prime/prime(q)∝−q0
d0+d1q2
0+d2q2, (75)
X/prime/prime(q)∝−q0
d0+d1q2
0+d2q2
for/Lambda1=0. The other coupling functions C/Lambda1(q),M/Lambda1(q),A/Lambda1(q)
and the real part of X/Lambda1(q) remain finite for /Lambda1→0,/Delta10→0,
q→0.The divergence of the vertex in the phase fluctuation chan-
nel represented by the coupling function /Phi1/Lambda1(q) reflects the
Goldstone mode associated with the spontaneous breaking oftheU(1) symmetry. The Goldstone theorem, which guarantees
the existence of this mode, is obviously respected by the RPA.A less familiar interesting result of the above calculation isthe divergence of the (3 +1) interaction represented by the
coupling function X
/Lambda1(q). Atq=0 this interaction describes
pair annihilation (or creation) combined with a forwardscattering process.
VI. ATTRACTIVE HUBBARD MODEL
In this section, we compute the flow of the Nambu
vertex and the gap function for the two-dimensional attractiveHubbard model as a prototype of a spin-singlet superfluid. TheHubbard model describes interacting spin-
1
2lattice fermions
with the Hamiltonian
H=/summationdisplay
i,jtijc†
iσcjσ+U/summationdisplay
jnj↑nj↓, (76)
where c†
iσandciσare creation and annihilation operators
for fermions with spin orientation σon a lattice site i.F o r
the attractive Hubbard model, the interaction parameter U
is negative. The hopping matrix tijis usually short ranged.
We consider the case of nearest- and next-to-nearest-neighborhopping on a square lattice, with amplitudes −tand−t
/prime,
174523-10EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
respectively, yielding a dispersion relation of the form
/epsilon1(k)=− 2t(coskx+cosky)−4t/primecoskxcosky.(77)
The ground state of the attractive Hubbard model is a
spin-singlet s-wave superfluid at any filling factor.31For
t/prime=0 the superfluid order is degenerate with a charge density
wave at half-filling (only). The attractive Hubbard model hasbeen studied already in several works both at zero and finitetemperatures by resummed perturbation theory (mostly T
matrix),
32–35quantum Monte Carlo (QMC) methods,36–38and
dynamical mean-field theory.39,40
A. Regularization and counterterm
The renormalization group flow is governed by the scale
dependence of the regularized bare propagator, which wechoose to be of the following form:
/bracketleftbig
G/Lambda1
0(k)/bracketrightbig−1=/parenleftbiggik0−ξ(k)−δξ/Lambda1(k)+R/Lambda1(k0) /Delta10
/Delta10 ik0+ξ(k)+δξ/Lambda1(k)+R/Lambda1(k0)/parenrightbigg
, (78)
withξ(k)=/epsilon1(k)−μ. The regulator function
R/Lambda1(k0)=isgn(k0)/radicalBig
k2
0+/Lambda12−ik0 (79)
replaces frequencies k0with|k0|/lessmuch/Lambda1by sgn( k0)/Lambda1and thus
regularizes the Fermi-surface singularity of the bare fermionicpropagator. The (real) bare gap /Delta1
0induces symmetry breaking
and regularizes the Goldstone mode singularity forming in theeffective interaction below the critical scale /Lambda1
c. Instead of
linking the flow of /Delta10to the fermionic cutoff scale /Lambda1by
defining a /Lambda1-dependent /Delta1/Lambda1
0, we found it more convenient
to keep /Delta10fixed until /Lambda1has decreased to 0, and send /Delta10
to zero afterwards. The equations for the latter flow are
obtained simply by replacing /Lambda1-derivatives by derivatives with
respect to /Delta10. The counterterm δξ/Lambda1(k) is linked to the normal
component of the self-energy by the condition
d
d/Lambda1[δξ/Lambda1(k)+/Sigma1/Lambda1(0,k)]=0, (80)
such that the Fermi surface remains fixed during the flow.
Since there is a contribution proportional to ∂/Lambda1δξ/Lambda1to the
scale derivative of /Sigma1/Lambda1, solving Eq. (80) for∂/Lambda1δξ/Lambda1amounts to
solving a linear integral equation.
B. Parametrization
We now specify the approximate parametrization of the
self-energy and the interaction vertex. Due to the localbare interaction and the pairing instability occurring in thes-wave channel, the momentum dependence of the normal and
anomalous self-energy can be expected to be weak, at least atweak coupling. Perturbation theory
41and previous functional
RG calculations23showed that this is indeed the case. We
therefore discard the momentum dependence of the self-energy, keeping however the frequency dependence. The latter
is treated numerically by discretizing /Sigma1
/Lambda1(k0) and/Delta1/Lambda1(k0)o n
a suitable grid. The counterterm δξ/Lambda1is then also momentum
independent and can be interpreted as a shift of the chemicalpotential.
The interaction vertex is fully described by the coupling
functions C
/Lambda1
kk/prime(q), etc., introduced in Sec. IV, where singular
momentum and frequency dependencies have been isolated inone variable q. We now approximate these functions by thefollowing ansatz:
C
/Lambda1
kk/prime(q)=C/Lambda1(q)g/Lambda1
c(k0)g/Lambda1
c(k/prime
0),
M/Lambda1
kk/prime(q)=M/Lambda1(q)g/Lambda1
m(k0)g/Lambda1
m(k/prime
0),
A/Lambda1
kk/prime(q)=A/Lambda1(q)g/Lambda1
a(k0)g/Lambda1
a(k/prime
0),
/Phi1/Lambda1
kk/prime(q)=/Phi1/Lambda1(q)g/Lambda1
φ(k0)g/Lambda1
φ(k/prime
0), (81)
P/prime/prime/Lambda1
kk/prime(q)=P/prime/prime/Lambda1(q)g/Lambda1
φ(k0)g/Lambda1
φ(k/prime
0),
X/prime/Lambda1
kk/prime(q)=X/prime/Lambda1(q)g/Lambda1
c(k0)g/Lambda1
a(k/prime
0),
X/prime/prime/Lambda1
kk/prime(q)=X/prime/prime/Lambda1(q)g/Lambda1
c(k0)g/Lambda1
φ(k/prime
0).
The vertex thus assumes the form of a collection of boson-
mediated interactions with bosonic propagators coupled tothe fermions via fermion-boson vertices g
/Lambda1. The latter are
normalized to one at zero frequency ( k0=0). The momentum
dependence on kand k/primehas thus been neglected, and the
dependence on k0andk/prime
0has been factorized. For the attractive
Hubbard model, dependencies on kandk/primeare generated only
at order U3, and can thus be expected to be weak at least
at weak coupling. Neglecting the dependence on kandk/prime
implies a restriction to s-wave symmetry in charge, magnetic,
and pairing channels. As a consequence, all triplet components
vanish, such that A/Lambda1
kk/prime(q)=AS,/Lambda1
kk/prime(q),/Phi1/Lambda1
kk/prime(q)=/Phi1S,/Lambda1
kk/prime(q), and
X/Lambda1
kk/prime(q)=XS,/Lambda1
kk/prime(q), and in the matrix VPP,/Lambda1[Eq. (52)], only
four elements are nonzero. Compared to an exact decomposi-tion of the coupling functions as in Eqs. (55) and(57),t h es u m
over basis functions is replaced by just one term in the aboveansatz. Due to time-reversal and exchange symmetries, thereis no contribution to W
/prime/prime/Lambda1
kk/prime(q) of that form. We have allowed for
four distinct fermion-boson vertices g/Lambda1
c,g/Lambda1
m,g/Lambda1
a, andg/Lambda1
φ.T h e
factorization of the coupling functions is similar to the factor-ization (65) obtained for separable interactions in RPA. Instead
of parametrizing the fermion-boson vertices in the pairingchannel by a single function g
/Lambda1
p, we now distinguish between
g/Lambda1
aandg/Lambda1
φ. It turns out that they differ only slightly. The imag-
inary part of the pairing coupling function P/prime/prime/Lambda1
kk/prime(q) has little
impact on the flow. Instead of introducing another fermion-boson vertex for that quantity, we approximate its dependenceonk
0andk/prime
0byg/Lambda1
φ. The frequency dependence of the fermion-
boson vertices g/Lambda1(k0) is treated numerically by discretization.
The parametrization of the “boson propagators” C/Lambda1(q),
etc., requires special care to capture the singularities.We first consider the amplitude and phase channel. For
174523-11ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
small q, the functions A/Lambda1(q) and /Phi1/Lambda1(q) behave as
[A/Lambda1(q)]−1=−m/Lambda1
a−Z/Lambda1
aq2−¯Z/Lambda1
aq2
0+··· and [/Phi1/Lambda1(q)]−1=
−m/Lambda1
φ−Z/Lambda1
φq2−¯Z/Lambda1
φq2
0+··· , where m/Lambda1
φ→0f o r /Lambda1</Lambda1 c,
/Delta10→0. Actually, the regulator function can also generate
contributions of order |q0|, which disappear again as /Lambda1→0.
To deal with this technical complication, and to achieve anaccurate parametrization also at larger values of q
0andq,w e
parametrize A/Lambda1and/Phi1/Lambda1by two scale-dependent functions
[A/Lambda1(q)]−1=−m/Lambda1
a(q0)−e/Lambda1
a(q),
(82)
[/Phi1/Lambda1(q)]−1=−m/Lambda1
φ(q0)−e/Lambda1
φ(q),
where e/Lambda1
a(0)=e/Lambda1
φ(0)=0. The (discretized) momentum and
frequency dependencies of these functions are determinedfrom the flow. The above ansatz with functions of one ( q
0)
and two [ q=(qx,qy)] variables reduces the numerical effort
compared to a discretization of an arbitrary function ofq=(q
0,qx,qy). Tests within RPA indicate that it describes the
full functions sufficiently well. In particular, the behavior atsmallq
0and small qis captured correctly. The imaginary parts
P/prime/prime/Lambda1(q) andX/prime/prime/Lambda1(q) are odd functions of q0. This and the ex-
pected singularity structure [see Eq. (75)] motivate the ansatz
P/prime/prime/Lambda1(q)=−q0
m/Lambda1
p/prime/prime(q0)+e/Lambda1
p/prime/prime(q),
(83)
X/prime/prime/Lambda1(q)=−q0
m/Lambda1
x/prime/prime(q0)+e/Lambda1
x/prime/prime(q).
The parametrization of C/Lambda1(q),M/Lambda1(q), and X/prime/Lambda1(q)i s
slightly more complicated because at small qthese functions
can not be represented as a sum of a frequency- and amomentum-dependent function. We therefore distinguishthe cases |q|<q
maxand|q|>q maxwith a suitably chosen
qmax.F o r|q|>q maxwe make an additive ansatz analogous to
Eq.(82):
[C/Lambda1(q)]−1=−m/Lambda1
c(q0)−e/Lambda1
c(q),
[M/Lambda1(q)]−1=−m/Lambda1
m(q0)−e/Lambda1
m(q), (84)
[X/prime/Lambda1(q)]−1=−m/Lambda1
x/prime(q0)−e/Lambda1
x/prime(q).
For small q,t h e qdependence is increasingly isotropic, except
for the special case where the Fermi surface touches vanHove points (which we exclude). Hence, for |q|<q
maxwe
approximate the momentum dependence as isotropic,
C/Lambda1(q)=C/Lambda1(q0,|q|),M/Lambda1(q)=M/Lambda1(q0,|q|),
(85)
X/prime/Lambda1(q)=X/prime/Lambda1(q0,|q|),
reducing the number of variables again to two. To avoid a dis-
continuity at qmax, we connect the two regimes in momentum
space by a smooth partition of unity instead of step functions.
C. Flow equations
The flow equations for the scale-dependent functions
parametrizing the self-energy and interaction vertex are ob-tained by projecting the flow equations for the self-energy andthe coupling functions on the simplified ansatz. Dependencieson the fermionic momenta k,k
/primegenerated by the flow
are eliminated by a Fermi-surface average (but not the q
dependence, of course). This corresponds to keeping only theleading (in power counting) term in an expansion around the
Fermi surface, and averaging over the momentum dependencealong the Fermi surface, which is in line with the pure s-wave
ansatz for the interactions.
For the self-energy, we project on the momentum-
independent ansatz by averaging the flow Eq. (9)over the
Fermi surface as follows:
d
d/Lambda1/Sigma1/Lambda1(k0)=/angbracketleftrhs/Lambda1(k0,k)/angbracketrightk∈FS, (86)
where rhs/Lambda1(k0,k) stands for the right-hand side of the flow
equation (in Nambu matrix form), and /angbracketleft ···/angbracketright k∈FSdenotes a
Fermi-surface average. Momentum dependencies of the self-energy perpendicular to the Fermi surface are marginal inpower counting
42and lead to a (finite) renormalization of the
Fermi velocity. However, they are quantitatively small in theattractive Hubbard model, at least at weak coupling and awayfrom van Hove points, and have thus little influence on ourresults.
The flow equations for the coupling functions
C
/Lambda1
kk/prime(q),..., X/Lambda1
kk/prime(q) were derived in Sec. IV. Inserting
the ansatz for the interaction vertex on the right-hand side ofthese equations yields several terms which can be representedby Feynman diagrams of the form plotted in Fig. 5. We recall
that the pointlike vertex represents the bare (here Hubbard)interaction, the wiggly line any coupling function contributingtoV
PH,/Lambda1, and the double line coupling functions appearing in
VPP,/Lambda1[onlyM/Lambda1
kk/prime(q) in the absence of triplet terms]. Note that
the terms in Fig. 6are redundant since the complete set of
coupling functions is already captured by VPH,/Lambda1. The Nambu
index sums on the right-hand side of the flow equation for thecoupling functions can be transformed to a more convenientform by representing the vertex and the propagator productin the Pauli matrix basis defined in the Appendix and usedalready in Sec. V.
Some of the contributions, having the form of vertex
corrections or box diagrams, generate dependencies on kand
k
/primewhich are not allowed for in our ansatz. These dependencies
are projected out by a symmetrized Fermi-surface average.We discuss the procedure for the charge coupling function asa prototypical example, which can be extended directly to allother cases. The flow of the projected charge coupling functionis given by
d
d/Lambda1/bracketleftbig
C/Lambda1(q)g/Lambda1
c(k0)g/Lambda1
c(k/prime
0)/bracketrightbig
=/angbracketleftrhs/Lambda1(k,k/prime;q)/angbracketrightk±q
2,k/prime±q
2∈FS
≡rhs/Lambda1(k0,k/prime
0;q), (87)
where rhs/Lambda1(k,k/prime;q) denotes the complete right-hand side of
the flow equation for C/Lambda1
kk/prime[Eq. (28)], and
/angbracketleft ···/angbracketright k±q
2,k/prime±q
2∈FS=1
4/summationdisplay
/epsilon1,/epsilon1/prime=±1/angbracketleft ···/angbracketright k+/epsilon1q
2,k/prime+/epsilon1/primeq
2∈FS (88)
is a symmetrized Fermi-surface average. The latter averages
the four possible ways of integrating kand k/primeunder the
constraint that two of the four external momenta k±q
2and
k/prime±q
2of the vertex lie on the Fermi surface. For q=0,
corresponding to the forward scattering and Cooper channels,this becomes a Fermi-surface average with all four momentaon the Fermi surface. For q/negationslash=0, the set of momenta k
174523-12EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
satisfying both k+q
2∈FS and k−q
2∈FS is limited to few
points, except for special nesting vectors in case of a nestedFermi surface. The Fermi-surface average picks up the s-wave
component of the dominant processes near the Fermi surface.Indeed, the momentum dependence perpendicular to the Fermisurface is irrelevant in power counting.
42Note, however, that
we do not discard the dependence on q. That dependence
becomes important due to the formation of singularities, whichinvalidate the weak-coupling power counting.
The projection on the form factors in the channel decom-
position could also be carried out by integration over theentire Brillouin zone.
25,43However, for our simple ansatz with
only one (momentum-independent) form factor, it is better toapproximate the vertex by its Fermi-surface average instead ofa Brillouin zone average to capture the dominant contributionsat low energy scales. We checked this in some test cases byexplicit comparison of different projection procedures.
The flow equation for the bosonic propagator can be
extracted from Eq. (87) by setting k
0=k/prime
0=0. Since g/Lambda1
c(0)=
1 is independent of /Lambda1, one obtains
d
d/Lambda1C/Lambda1(q)=rhs/Lambda1(0,0;q). (89)
The functions m/Lambda1
c(q0) and e/Lambda1
c(q) parametrizing C/Lambda1(q)f o r
|q|>q maxare extracted by evaluating [ C/Lambda1(q)]−1at a fixed
momentum q∗as a function of q0, and at fixed frequency
q0=0 as a function of q, respectively. For q∗we choose a
momentum where C/Lambda1(q) is peaked, where it yields the largest
contribution. In the charge and magnetic channel, this happenstypically at finite momenta connecting antipodal Fermi points(2k
F-type momenta).
The product rule for differentiation applied to the left-hand
side of Eq. (87) atk/prime
0=0 yields the flow equation for the
fermion-boson vertex
d
d/Lambda1g/Lambda1
c(k0)=1
C/Lambda1(q)/bracketleftbig
rhs/Lambda1(k0,0;q)
−g/Lambda1
c(k0)rhs/Lambda1(0,0;q)/bracketrightbig
q=(0,q∗). (90)
The flow equations for the other coupling functions M/Lambda1
kk/prime(q),
etc., are projected on the ansatz in the same way. The flow ofthe fermion-boson vertices in the pairing channel g
/Lambda1
a(k0) and
g/Lambda1
φ(k0) is determined as in Eq. (90), with q∗=0.
The initial conditions for the flow at /Lambda10=∞ are as follows.
For the self-energy, counterterm, and gap function, the flowstarts at /Sigma1
/Lambda10=0,δξ/Lambda10=0, and /Delta1/Lambda10=/Delta10. The coupling
functions are initially zero, and the fermion-boson verticesare equal to one. Note that the coupling functions do notinclude the bare interaction. In a numerical evaluation, theflow starts at a large finite /Lambda1
0. The self-energy receives a
tadpole contribution of order one in the flow from /Lambda10=∞
to an arbitrarily large finite /Lambda10, yielding /Sigma1/Lambda10=U/2 with
corrections of order /Lambda1−1
0, and correspondingly δξ/Lambda10=−U/2.
The error of order /Lambda1−1
0made by starting the flow at a (large)
finite cutoff can be significantly reduced by using perturbativeresults at /Lambda1
0as initial conditions instead of the initial values
at/Lambda10=∞ . The coupling functions are then nonzero from the
beginning such that Eq. (90) is well defined at /Lambda10.
We conclude this section with a few words on numerical
aspects. More details can be found in Ref. 44. Momentum andfrequency dependencies were discretized on nonequidistant
grids such that the resolution is higher at smaller frequenciesand momenta. The positive frequency axis and radial momen-tum dependencies were discretized by around 30 points, andangular momentum dependencies by six angles per quadrantin the Brillouin zone. The functional flow equations werethus replaced by a system of around 2000 nonlinear ordinarydifferential equations with three-dimensional loop integralson the right-hand sides. The integrals were performed withan adaptive integration algorithm and the integration of theflow was performed with a third-order Runge-Kutta routine.Depending on parameters, the computation of a flow requiredbetween a day and a week on 20 CPU cores.
D. Results
We now present results for the effective interactions, the
normal self-energy, and the gap function as obtained from anumerical solution of the flow equations. Most of the numericalresults are obtained for a small fixed external pairing field /Delta1
0
chosen two to three orders of magnitude below the mean-field
gap/Delta1MF, but we also discuss some flows where /Delta10scales
toward zero after the fermionic cutoff has reached /Lambda1=0.
The Ward identity following from global charge conservationis enforced at zero frequency by projecting the flow, if notstated otherwise (for details, see Sec. VI D3 ). Bare interaction
strengths are chosen in the weak to moderate coupling range|U|/t=1–4. In the following, we set the nearest-neighbor
hopping amplitude t=1, that is, all quantities with dimension
of energy are in units of t.
1. Effective interactions
We begin with results for the coupling functions, which
describe the various effective interaction channels contributingto the the Nambu vertex. With our sign conventions, negativecoupling functions correspond to attractive effective interac-tions in the respective channel.
The flow of effective interactions in the pairing channel is
qualitatively similar to the one in RPA (see Sec. V). However,
the critical scale and the size of the coupling functions isreduced by fluctuations. Typical flows for the amplitude andphase couplings at q=0 are shown in Fig. 7for various
choices of the external pairing field /Delta1
0.F o rU=− 2, stable
flows without artificial singularities could be performed for ex-ternal pairing fields as small as three orders of magnitude belowthe size of the mean-field gap /Delta1
MF, with a final phase coupling
/Phi1/Lambda1=0(0) proportional to /Delta1−1
0within numerical accuracy, as
dictated by the Ward identity. The amplitude coupling A/Lambda1(0)
has a peak around /Lambda1c, whose size increases strongly upon
reducing /Delta10, while it reaches a finite value with a negligible
dependence on the external pairing field at the end of the flow.
The momentum and frequency dependence of A/Lambda1(q)
and/Phi1/Lambda1(q) around q=0 is shown for various choices of
/Lambda1in Fig. 8. For small momenta, the coupling functions
are isotropic functions of qwith a momentum dependence
proportional to q2, for both finite /Lambda1and/Lambda1=0. The frequency
dependence is linear for small q0at/Lambda1> 0, but essentially
quadratic for /Lambda1=0. The linear behavior at /Lambda1> 0 is caused
by the frequency-dependent regulator [Eq. (79)] and thus
disappears once the regulator has scaled to zero. The amplitude
174523-13ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
0 0.1 0.2
Λ−200−1000ΦΛ(0),AΛ(0)AΛ,Δ0=2·10−3
AΛ,Δ0=1 0−3
AΛ,Δ0=5·10−4
ΦΛ,Δ0=2·10−3
ΦΛ,Δ0=1 0−3
ΦΛ,Δ0=5·10−4
FIG. 7. (Color online) Scale dependence of the amplitude and
phase couplings A/Lambda1and/Phi1/Lambda1atq=0 for various choices of the
external pairing field /Delta10. The Hubbard model parameters are t/prime=
−0.1,U=− 2,n=0.5 (quarter-filling).
coupling A/Lambda1=0(q0,0) exhibits a tiny dip at q0=0. Overall,
the qualitative momentum and frequency dependencies ofthe coupling functions in the pairing channel do not deviatesignificantly from the behavior in RPA. This is also true forthe imaginary part of the pairing coupling P
/prime/prime/Lambda1(q) and the
imaginary part of the anomalous (3 +1) coupling X/prime/prime/Lambda1(q),
whose singular behavior at small momenta and frequencies iswell described by
X
/prime/prime/Lambda1=0(q)∝P/prime/prime/Lambda1=0(q)∝−q0
/Delta10+aq2
0+bq2, (91)
where a,bare positive constants.
The charge coupling function C/Lambda1(q) is generally negative at
all stages of the flow. It thus renormalizes the bare attraction inthe charge channel given by Uto an enhanced total attractive
interaction U+2C
/Lambda1(q). This effect is captured already inRPA. The enhancement is usually small. However, for densities
near half-filling and small values of t/prime, it becomes large
atq=(0,Q) with Q=(π,π). For t/prime=0 and half-filling,
U+2C/Lambda1(0,Q) is degenerate with the pairing interaction U+
P/Lambda1(0,0), reflecting the degeneracy of superfluidity with charge
density wave order due to a particle-hole symmetry in thisspecial case.
31In Fig. 9, we show the momentum dependence
of the charge coupling function in the static limit q0=0a t
the end of the flow ( /Lambda1=0) for two distinct choices of t/primeand
n. The function exhibits pronounced peaks at incommensurate
momenta situated at the Brillouin zone boundary. These peaksare present already in the bare polarization function (particle-hole bubble).
45They move toward ( π,π) and increase upon
approaching half-filling for t/prime=0. As a function of frequency,
the size of C/Lambda1(q) decays monotonically upon increasing |q0|.
The magnetic coupling function M/Lambda1(q) receives contribu-
tions beyond RPA which change its behavior qualitatively.In Fig. 10, we show its momentum dependence in the static
limitq
0=0 at the end of the flow for the same choices of t/prime
andnas in Fig. 9. The coupling function is negative in most
of the Brillouin zone, but it develops a pronounced positivepeak for small momenta q. This peak is a pure fluctuation
effect. In RPA, M
/Lambda1=0(0,0) vanishes due to the pairing gap.
Since the amplitude of the coupling function is small, thetotal interaction in the magnetic channel −U+2M
/Lambda1=0(q)
is dominated by the bare Hubbard interaction and remainspositive for all momenta. In Fig. 11, the frequency dependence
ofM
/Lambda1(q0,0) is shown at various stages of the flow. The
positive peak at q0=0 develops at and below the critical
scale/Lambda1cand is foreshadowed by a finite frequency peak for /Lambda1
near/Lambda1c. The flow is nonmonotonic and M/Lambda1(q0,0) exhibits
a sign change for small q0, but eventually M/Lambda1=0(q0,0)i s
positive for all frequencies. A similar sign change at finiteq
0and a pronounced finite frequency peak has been observed
0 0.025 0.050.075 0.1
qx/π−150−100−500AΛ(q0=0,qx,qy= 0), ΦΛ(q0=0,qx,qy=0 )
AΛ,Λ=0
AΛ≈ΦΛ,Λ≈2Λc
AΛ,Λ≈Λc
ΦΛ,Λ≈Λc
ΦΛ,Λ=0
0 0.05 0.10.15 0.2
q0−150−100−500AΛ(q0,q= 0), ΦΛ(q0,q=0 )
AΛ,Λ=0
AΛ≈ΦΛ,Λ≈2Λc
AΛ,Λ≈Λc
ΦΛ,Λ≈Λc
ΦΛ,Λ=0
FIG. 8. (Color online) Momentum dependence along the qxaxis (left) and frequency dependence (right) of the amplitude and phase
couplings A/Lambda1(q)a n d/Phi1/Lambda1(q) for small momenta and frequencies at various stages of the flow. The Hubbard model parameters are the same as
in Fig. 7and/Delta10=10−3.
174523-14EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
−0.6−0.4CΛ=0(q0=0,q)
0
0.5
1qx/π00.51
qy/π0
0.5
1qx/π−1.25−1−0.75−0.5CΛ=0(q0=0,q)
00.51
qy/π
FIG. 9. Momentum dependence of the static charge coupling function C/Lambda1=0(0,q) at the end of the flow, for t/prime=− 0.1,n=0.5( l e f t )a n d
t/prime=0,n=0.78 (right). The Hubbard interaction is U=− 2 in both cases.
previously in the charge coupling function for the repulsive
Hubbard model in the symmetric regime ( /Lambda1>/Lambda1 c).43,46
The real part of the anomalous (3 +1) coupling function
X/prime/Lambda1(q) is relatively small. Its singularity at the critical scale
/Lambda1cis considerably broadened by fluctuations (beyond RPA).
Nevertheless, its influence on the flow of the self-energyand the other coupling functions is important. Neglecting the(3+1) coupling would lead to artifacts such as nonmonotonic
flows of /Phi1
/Lambda1(0) even for small interactions U. While the
imaginary part of X/Lambda1(q) depends strongly on /Delta10for small
qand/Lambda1, the real part does not. In Fig. 12,w ep l o tt h e
momentum dependence of the static (3 +1) coupling function
at the end of the flow for the same choices of t/primeandn
as in Figs. 9and 10. Note that the imaginary part of the
static (3 +1) coupling function vanishes, that is, X/Lambda1(0,q)i s
real.
We now turn to the fermion-boson vertices, whose fre-
quency dependence is plotted in Fig. 13. Note that the vertices
are even functions of k0which are normalized to one at k0=0
by definition. The frequency dependence of the vertices is quiteweak. However, the frequency dependence of the vertices in
the pairing channel contributes significantly to the frequencydependence of the gap function /Delta1
/Lambda1(k0) and also to the flow of
/Phi1/Lambda1. The normal self-energy and the other coupling functions
are only weakly affected by the frequency dependence of thefermion-boson vertices. The magnetic vertex exhibits a smallpeak at low frequencies which develops at scales /Lambda1</Lambda1
cand
is therefore related to pairing fluctuations.
2. Normal self-energy and gap function
At weak to moderate interactions, the ground state of the
attractive Hubbard model is superfluid with Cooper pairsmade of weakly renormalized quasiparticles. Quasiparticlerenormalization occurs already at scales above the pairingscale/Lambda1
cand is described by the normal self-energy. The
momentum dependence of the self-energy is weak for thechoice of parameters considered in this work and we willpresent only results for the Fermi-surface average /Sigma1
/Lambda1(k0)=
/angbracketleft/Sigma1/Lambda1(k0,k)/angbracketrightk∈FS.I nF i g . 14, we show results for the imaginary
−0.500.5MΛ=0(q0=0,q)
0
0.5
1qx/π00.51
qy/π−0.500.5MΛ=0(q0=0,q)
0
0.5
1qx/π00.51
qy/π
FIG. 10. Momentum dependence of the static magnetic coupling function M/Lambda1=0(0,q) at the end of the flow, for t/prime=− 0.1,n=0.5( l e f t )
andt/prime=0,n=0.78 (right). The Hubbard interaction is U=− 2 in both cases.
174523-15ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
0 5 10
q0−0.4−0.200.2MΛ(q0,q=0)
Λ=0 .10
Λ=0 .20
Λ=0 .63
Λ=1 .28
0 0.5 1 1.5
q0−0.2500.250.5MΛ(q0,q=0)Λ=0 .0001
Λ=0 .024
Λ=0 .038
Λ=0 .050
Λ=0 .060
Λ=0 .078
Λ=0 .10
FIG. 11. (Color online) Frequency dependence of the magnetic coupling function M/Lambda1(q0,0) at various stages of the flow above (left) and
below (right) the critical scale for pairing. The model parameters are t/prime=− 0.1,U=− 2, and n=0.5.
part of /Sigma1/Lambda1(k0) as a function of frequency at the end of
the flow ( /Lambda1=0). We plot only the positive frequency axis
since Im /Sigma1/Lambda1(−k0)=− Im/Sigma1/Lambda1(k0). The real part (not plotted)
of/Sigma1/Lambda1(k0)i sa ne v e nf u n c t i o no f k0with a negative peak at
k0=0 that decays monotonically to the Hartree term Un/ 2
with increasing |k0|. The overall shape of the self-energy is the
same for all interaction strengths, only the size increases with|U|.
The slope of Im /Sigma1
/Lambda1(k0)a tk0=0 yields the quasiparticle
weight Z/Lambda1
fas
Z/Lambda1
f=/bracketleftbig
1−∂k0Im/Sigma1/Lambda1(k0)|k0=0/bracketrightbig−1. (92)
Zf=Z/Lambda1=0
f ranges from Zf=0.96 for U=− 1.5t oZf=
0.87 for U=− 3. Although the normal self-energy is fairly
small and the quasiparticle weight is only slightly suppressedfor small to moderate interactions, it has nevertheless asignificant impact on the size of the pairing gap.
F l o w so ft h eg a p /Delta1
/Lambda1(k0)a tk0=0 are shown in Fig. 15
for various choices of U. The small external pairing field /Delta10
increases to much larger gaps at scales near and below the
critical scale /Lambda1c, where A/Lambda1(0) has a peak. The edge of the gapflow at /Lambda1cbecomes sharper for smaller /Delta10. The gap at the end
of the flow assumes values close to /Lambda1c. The scale dependence
for/Lambda1</Lambda1 cobeys approximately
/Delta1/Lambda1(0)≈/radicalBig
/Lambda12c−/Lambda12, (93)
with increasing accuracy for smaller values of Uand/Delta10.I n
mean-field theory, this relation is exact for /Delta10→0, as one can
easily see by writing the gap equation in the presence of theinfrared regulator (79).F o rU=− 2 the gap flow lies almost
on top of the square root function (93) for small /Delta1
0, while for
stronger attractions deviations become visible. In particular,the final gap /Delta1
/Lambda1=0becomes clearly larger than /Lambda1c.
The flow in Fig. 15was obtained with frequency-dependent
effective boson propagators and fermion-boson vertices, and afrequency-dependent normal self-energy and gap as describedin Sec. VI B . Comparing with results obtained by discarding
the frequency dependence of some of these quantities, onefinds that only the frequency dependence of the bosonpropagators and of the imaginary part of the normal self-energyhave a substantial impact on the size of /Delta1
/Lambda1(0). The feedback
0
0.5
1qx/π−0.2−0.15−0.1−0.05XΛ=0(q0=0,q)
00.51
qy/π−0.4−0.20XΛ=0(q0=0,q)
0
0.5
1qx/π00.51
qy/π
FIG. 12. Momentum dependence of the static (3 +1) coupling function X/Lambda1=0(0,q) at the end of the flow, for t/prime=− 0.1,n=0.5( l e f t )a n d
t/prime=0,n=0.78 (right). The Hubbard interaction is U=− 2 in both cases.
174523-16EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
0 10 20
k011.11.2gΛ=0
a,gΛ=0
φ
gΛ=0
a
gΛ=0
φ0 0.5 111.05
0 10 20
k00.91gΛ=0
m,gΛ=0
c gΛ=0
c
gΛ=0
m
FIG. 13. (Color online) Frequency dependence of the fermion-boson vertices at the end of the flow ( /Lambda1=0). Left: amplitude and phase
vertices. Right: charge and magnetic vertices. The model parameters are t/prime=− 0.1,U=− 2, and n=0.5.
of the other frequency dependencies on the gap at k0=0i s
small.
The critical scale and the final gap are strongly reduced
compared to their mean-field values /Lambda1MF
cand/Delta1MF, respec-
tively, mostly due to fluctuations above the critical scale.
In Fig. 16, we plot the ratio /Delta1//Delta1 MFwith/Delta1=/Delta1/Lambda1=0(0)
as a function of Ufort/prime=− 0.1 and n=0.5. The lower
curve was obtained by a simplified static parametrization ofthe vertex and self-energy, where all frequency dependencieswere neglected. Notably, the reduction increases at weakerinteractions and does not extrapolate to one for U→0. This
is actually the expected behavior. In the weak coupling limitthe gap /Delta1has the same exponential Udependence /Delta1∝e
−b/|U|
with a (density-dependent) constant bas in mean-field theory.
However, the prefactor of the BCS mean-field formula isreduced by fluctuations, as first noted for the transitiontemperature in three-dimensional superconductors by Gorkovand Melik-Barkhudarov.
47The reduction factor in the weak
coupling limit can be computed by second-order perturbationtheory.
41,48,49For the parameters used in Fig. 16, one finds
/Delta1//Delta1 MF→0.3f o rU→0.44Both curves in Fig. 16should
tend to that value since the flow captures the perturbativecontributions. However, we can not reach the limit U→0
numerically. It is hard to compute the gap from a numerical
0 10 20 30 40
k0−0.15−0.1−0.050ImΣ(k0)
−UΔ0
1.51 0−4
2.01 0−3,n og(k0)
2.01 0−3
2.51 0−3
3.01 0−3
FIG. 14. (Color online) Frequency dependence of the imaginary
part of the normal self-energy for t/prime=− 0.1,n=0.5 (quarter-filling)
and various choices of Uat the end of the flow. The result labeled
as “no g(k0)” is obtained with constant (frequency-independent)
fermion-boson vertices.solution of the flow equations for smaller interaction strengths
than those shown since /Lambda1cand/Delta1decrease exponentially.
For strong attractions U, the attractive Hubbard model can
be mapped to a Heisenberg model in a uniform magneticfield.
31The gap ratio /Delta1//Delta1 MFthereby translates to the ratio
between the staggered magnetization msand the correspond-
ing classical result mcl
s. From numerical results for that ratio50
one can infer that the gap ratio in the strongly attractive
Hubbard model is 0 .6 at half-filling and even larger away from
half-filling. The observed increase of /Delta1//Delta1 MFwith increasing
|U|is therefore consistent with the expected trend. Similar
values for /Delta1//Delta1 MFbut with a less pronounced Udependence
have been obtained in an earlier fRG study with a simplerparametrization of the vertex.
23
We now discuss the frequency dependence of the gap
function. In Fig. 17,/Delta1/Lambda1=0(k0) is plotted as a function
of frequency for U=− 2,t/prime=− 0.1, and n=1
2. Results
obtained by computing the gap from a projected flow obeyingthe Ward identity at k
0=0 are compared to results where
the frequency dependence of the gap is computed directlyfrom the Ward identity ( /Delta1
WI), contrasting also calculations
with and without frequency-dependent fermion-boson verticesg(k
0). Note that the results discussed so far were all obtained
by enforcing the Ward identity only at k0=0. Unlike the
0 0.1 0.2 0.3 0.4
Λ00.10.20.30.4ΔΛ(k0=0 )−U Δ0
3.01·10−3
3.03·10−4
2.51·10−3
2.02·10−4
1.51·10−4
FIG. 15. (Color online) Scale dependence of the gap /Delta1/Lambda1(k0)a t
k0=0f o rt/prime=− 0.1,n=0.5 and various choices of Uand/Delta10.F o r
U=− 2a n d −3, the mean-field scale dependence/radicalbig
/Lambda12
c−/Lambda12, with
/Lambda1cdetermined from the peak of A/Lambda1(0), is shown for comparison.
174523-17ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
1 2 3 4
−U0.30.40.5Δ/ΔMFdynamical
static
FIG. 16. (Color online) Gap ratio /Delta1//Delta1 MFas a function of Uas
obtained from the flow with frequency-dependent (upper curve) and
static (lower curve) vertices and self-energies.
frequency dependence of the normal self-energy, the frequency
dependence of the gap is strongly affected by the frequency-dependent renormalization of the fermion-boson vertices.Neglecting it leads to a very weak or almost no (for /Delta1
WI)
frequency dependence. The gap /Delta1(k0) computed by projecting
the flow on the Ward identity at k0=0 exhibits a shallow
finite-frequency minimum, which is probably an artifact ofthe approximations associated with a (slight) violation of theWard identity at finite frequencies. /Delta1
WI(k0) has a minimum at
k0=0. A qualitatively similar frequency dependence of the
gap is also captured by the T-matrix approximation.35
3. Ward identity
For real gaps /Delta10and/Delta1/Lambda1the Ward identity (8)can be
simplified to
/Delta1/Lambda1(k)−/Delta10(k)
=−/summationdisplay
k/prime/Delta10(k/prime)[G/Lambda1(k/prime)G/Lambda1(−k/prime)+(F/Lambda1(k/prime))2]
×[V/Lambda1(k,−k,−k/prime,k/prime)−W/Lambda1(k,k/prime,k/prime,k)]. (94)
0 10 20
k00.10.110.120.13ΔΛ=0(k0)
withg(k0), ΔWI
withg(k0)without g(k0), ΔWI
without g(k0)0 0.5 10.10.11
FIG. 17. (Color online) Frequency dependence of the gap at the
end of the flow for various approximation schemes. The inset shows
the gap at small frequencies k0/lessorequalslant1. The model parameters are U=
−2,t/prime=− 0.1, and n=1
2.Expressing V/Lambda1andW/Lambda1by the coupling functions introduced
in the channel decomposition (Sec. IV), the identity can be
written as
/Delta1/Lambda1(k)=−/summationdisplay
k/prime/Delta10(k/prime)[G/Lambda1(k/prime)G/Lambda1(−k/prime)
+(F/Lambda1(k/prime))2]/Phi1/Lambda1
kk/prime(0)+O(/Delta10) (95)
for/Delta10→0. The first term on the right-hand side is of order
one since /Phi1/Lambda1
kk/prime(0)∝/Delta1−1
0for small /Delta10. With the approximate
parametrization for the Hubbard model described in Sec. VI B ,
the combination of interaction terms on the right-hand side ofEq.(94) can be written as
V
/Lambda1(k,−k,−k/prime,k/prime)−W/Lambda1(k,k/prime,k/prime,k)
=U+/Phi1/Lambda1(0)g/Lambda1
φ(k0)g/Lambda1
φ(k/prime
0)+1
2A/Lambda1(k/prime−k)/bracketleftbig
g/Lambda1
a(p0)/bracketrightbig2
−1
2/Phi1/Lambda1(k/prime−k)/bracketleftbig
g/Lambda1
φ(p0)/bracketrightbig2+C/Lambda1(k/prime−k)/bracketleftbig
g/Lambda1
c(p0)/bracketrightbig2
−3M/Lambda1(k/prime−k)/bracketleftbig
g/Lambda1
m(p0)/bracketrightbig2, (96)
where p0=(k0+k/prime
0)/2. For a small constant /Delta10and a
momentum-independent gap function /Delta1/Lambda1(k0), the Ward iden-
tity then assumes the form
/Delta1/Lambda1(k0)=−/summationdisplay
k/prime/Delta10[G/Lambda1(k/prime)G/Lambda1(−k/prime)+(F/Lambda1(k/prime))2]
×/Phi1/Lambda1(0)g/Lambda1
φ(k0)g/Lambda1
φ(k/prime
0)+O(/Delta10). (97)
The most important consequence of the Ward identity is the
divergence of the phase coupling /Phi1/Lambda1(0) in the limit /Delta10→0
for/Lambda1</Lambda1 c, reflecting the massless Goldstone boson asso-
ciated with spontaneous symmetry breaking. The truncatedflow equations do not obey the Ward identity exactly, and/Phi1
/Lambda1(0) deviates from the expected behavior ∝/Delta1−1
0for small
/Delta10. For small Uthe deviations are tiny. For example, for
t/prime=− 0.1,n=1
2, and U=− 2, the product /Delta10/Phi1/Lambda1=0(0) is
almost constant down to fairly small values of /Delta10, before
it increases and finally diverges at a finite /Delta10of the order
10−5, which is four orders of magnitude smaller than /Delta1/Lambda1=0.
The same behavior was observed already in more pronouncedform in Ref. 23.
The violation of the Ward identity can be quantified by
comparing the gap /Delta1
/Lambda1
RGcomputed from its flow equation to the
gap/Delta1/Lambda1
WIrequired by the Ward identity. The latter is computed
from Eq. (94) by inserting the coupling functions as determined
from the flow on the right-hand side. In Fig. 18,w ep l o t
the difference /Delta1/Lambda1
RG−/Delta1/Lambda1
WI, divided by /Delta1/Lambda1
RGU4, as a function
of the scale in units of /Lambda1c. One can see that the violation
builds up gradually at scales around /Lambda1c. The normalized
difference ( /Delta1/Lambda1
RG−/Delta1/Lambda1
WI)//Delta1/Lambda1
RGincreases rapidly from U=− 2
toU=− 3. For /Lambda1>/Lambda1 cit is roughly proportional to U4.F o r
/Lambda1</Lambda1 c, a pronounced /Delta10dependence appears. For much
smaller values of /Delta10than those shown, /Delta1/Lambda1
RG−/Delta1/Lambda1
WIcan
turn negative, which is related to an artificial divergence of/Phi1
/Lambda1at a finite /Delta10. We observed similar Udependencies
also for other hopping parameters and densities. On generalgrounds one would expect a violation of the Ward identityof order U
3at weak coupling, even if the one-loop flow was
carried out without additional approximations.27,44The above
results suggest that the violation sets in only at order U4,o r
contributions of order U3have very small prefactors.
174523-18EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
10−3 0.01 0.1 1 10
Λ/Λc010−30.002(ΔΛ
RG−ΔΛ
WI)/(ΔΛ
RGU4)
−U Δ0
2.01·10−3
2.02·10−3
2.52·10−3
3.03·10−3
FIG. 18. (Color online) Violation of the Ward identity as a
function of the scale /Lambda1fort/prime=− 0.1,n=1
2, and various values
ofUand/Delta10.T h eg a p /Delta1/Lambda1
RGdetermined from the flow equation is
compared to the gap /Delta1/Lambda1
WIdetermined from the Ward identity.
In Fig. 19,(/Delta1/Lambda1
RG−/Delta1/Lambda1
WI)//Delta1/Lambda1
RGis plotted as a function of /Lambda1
for a fixed set of parameters to compare the performance ofdifferent approximations. The graph labeled “dynamical” wasobtained by using the frequency-dependent parametrizationof the vertex and the self-energy as described in Sec. VI B .
The graph “no g(k
0)” was computed with constant (unity)
fermion-boson vertices, and the graph “static” by discardingall frequency dependencies. The lowest curve labeled “coord.proj.” was computed with the dynamical parametrization andthe Ward identity enforced by “coordinate projection” (asdescribed below). The latter obeys the Ward identity byconstruction, up to small discretization errors. Taking thefrequency dependencies into account obviously reduces theviolation of the Ward identity significantly.
Even for the most accurate parametrization of the vertex,
the Ward identity is not fulfilled by the truncated flow, asgenerally expected.
27A detailed discussion of this problem
in the case of superfluid order is provided in Ref. 44.T h e
deviations are small for weak interactions but increase rapidlywith|U|. Violating the Ward identity spoils the singular
infrared behavior of the coupling functions associated with
10−4 0.01 1 100
Λ00.050.10.15(ΔΛ
RG−ΔΛ
WI)/ΔΛ
RGstatic
nog(k0)
dynamical
coord. proj.
FIG. 19. (Color online) Violation of the Ward identity as a
function of /Lambda1forU=− 2,t/prime=− 0.1,n=1
2,a n d /Delta10=10−3.
Results from the flow equations with static and two distinct (withand without frequency-dependent fermion-boson vertices) dynamical
parametrizations of the vertex are compared to the result from the
Ward identity projected flow.the massless Goldstone boson for /Delta10→0. Even worse,
it leads to artificial singularities which prevent one fromcarrying out the flow down to /Lambda1→0 and /Delta1
0→0. In the
results presented in the preceding sections we have thereforeenforced the Ward identity by using a coordinate projectionprocedure, devised for the numerical solution of systems ofordinary differential equations with constraints.
51The flowing
quantities are thereby projected on the manifold spanned by theconstraint (Ward identity) in a way that the projected solutionstays as close to the solution of the flow equations as possible,while deviations from the constraint are damped exponentially.In practice, we have enforced the Ward identity only at zerofrequency ( k
0=0), to reduce the numerical effort.44This
has little effect on absolute values of results, but leads tothe slightly artificial frequency dependence of the gap at lowfrequencies discussed in Sec. VI D2 .
4./Delta10flow and singularities
We finally take a closer look at the singularities of the
vertex in the limit /Delta10→0. In particular, we complement
the numerical results for the Hubbard model by qualitativeanalytical estimates which are generally valid for fully gappedsinglet superfluids.
To this end, we assume that the fermionic cutoff has already
been removed ( /Lambda1→0), and we analyze the flow as a function
of a decreasing pairing field /Delta1
0.I nF i g . 20we show the flow
of/Delta1(0)=/Delta1/Lambda1=0(0),/Phi1(0)=/Phi1/Lambda1=0(0), and A(0)=A/Lambda1=0(0)
as a function of /Delta10, with an initial value /Delta10=0.005. Results
obtained for some fixed smaller values of /Delta10are shown
for comparison. The numerical computation of the /Lambda1flow
becomes increasingly difficult at smaller /Delta10. Furthermore,
there are systematic deviations between the results obtainedfrom the /Delta1
0flow and those computed at fixed /Delta10. These may
be related to divergencies in box diagrams for /Delta10→0, which
can and must be treated by a flow starting at an initially finite/Delta1
0, as we discuss in the following.
In a fully gapped superfluid, the fermionic propagator is
regularized by the pairing gap. However, the interaction vertexdevelops a singularity associated with the emergence of a
Goldstone boson. In particular, the phase coupling function
has the singular form
/Phi1(q)∝−1
/Delta10+aq2
0+bq2, (98)
for small q=(q0,q), where aandbare positive constants.
This singularity is dictated by the Ward identity. Relatedsingularities occur also for the imaginary parts of the pairingand anomalous (3 +1) coupling functions P
/prime/prime(q) andX/prime/prime(q),
respectively, but their impact is reduced by a numeratorproportional to q
0. The divergence of /Phi1(q)f o rq→0,/Delta10→0
is integrable in (2 +1) dimensions. Hence, self-energy and
vertex correction (Fig. 3) contributions involving integrals
over/Phi1(q) remain finite for /Delta10→0. However, box diagrams
(Fig. 4) yield contributions involving an integral over products
of two phase coupling functions
/summationdisplay
p∂/Delta10/bracketleftbig
Gs1s2(p−q/2)Gs3s4(p+q/2)/bracketrightbig
/Phi1(p−k)/Phi1(k/prime−p),
(99)
174523-19ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
0 0.0025 0.005
Δ(0)0.10.110.12Δ(k0=0 )
coord. proj.
no coord. proj.
coord. proj.
no coord. proj.
0 0.0025 0.005
Δ0−750−500−2500Φ(q=0 )
0 0.0025 0.005
Δ0−5.5−5−4.5A(q=0 )
FIG. 20. (Color online) /Delta10flows of /Delta1(0),/Phi1(0), and A(0) for an
initial value /Delta10=0.005. Results obtained for fixed smaller values of
/Delta10are shown for comparison (symbols). The model parameters are
t/prime=− 0.1,n=0.5,U=− 2.
where Gis the propagator and /Phi1the phase coupling function
for/Lambda1=0. The /Delta10derivative of the propagators is finite for
/Delta10→0, but for k=k/primethe singularities of the phase coupling
functions coalesce and the integral diverges as /Delta1−1/2
0.I ti s
therefore not possible to set /Delta10to zero before the fermionic
cutoff has been removed. The /Delta10flow is however well defined
and integrable.
Hence, the singularities associated with the Goldstone
mode do not lead to divergencies in other channels. In thisrespect, the one-loop flow analyzed in this work is qualitativelysimilar to the RPA. The fluctuation effects beyond RPA yieldonly finite renormalizations. On the other hand, it is knownfrom the theory of interacting bosons that the phase modedoes lead to a singular renormalization of the amplitudemode.
52In a renormalization group theory of fermionic
superfluids with auxiliary boson fields representing the orderparameter fluctuations, this effect appears already at one-loop
level.16The singular contributions involve scale derivatives
acting on the boson propagators. In the purely fermionicrenormalization group (without auxiliary bosons), analogoussingular contributions appear only at the two-loop level.
44
VII. CONCLUSION
We have analyzed ground-state properties of a spin-singlet
superfluid including fluctuations on all scales via a fermionicfunctional renormalization group flow in a formulation thatallows for symmetry breaking. The flow equations were trun-cated in a one-loop approximation with self-energy feedback.Spin-rotation invariance and discrete symmetries were fullyexploited to simplify the structure of the Nambu two-particlevertex. To parametrize the singular momentum and frequencydependencies of the effective interactions, the Nambu vertexwas decomposed in charge, magnetic, and various normal andanomalous pairing channels, which are all mutually coupledin the flow. We have shown that the channel decomposed one-loop flow equations are equivalent to the RPA for the vertexand to mean-field theory for the gap function, if only directNambu particle-hole contributions are taken into account.
53
The crossed particle-hole and the particle-particle (in Namburepresentation) contributions to the complete one-loop flowthus capture fluctuations beyond mean-field theory and RPA.
We have evaluated the flow equations for the two-
dimensional attractive Hubbard model as a prototype ofan interacting Fermi system with a spin-singlet superfluidground state. The dominance of s-wave terms in the effective
interactions in that model allows for a relatively simpleparametrization. The global U(1) Ward identity relating
the vertex to the gap function is violated by the one-looptruncation. The deviations are very small for a weak attraction,but increase rapidly for stronger interactions. To maintain thesingularity structure associated with the Goldstone boson, theflow was therefore projected on the Ward identity, analogouslyto evaluating a differential flow in the presence of a constraint.We have computed the effective interactions in the charge,magnetic, and pairing channels, including anomalous (3 +1)
interactions describing pair annihilation (or creation) com-bined with a one-particle scattering process. Unprecedentedcomprehensive results on the momentum and (imaginary)frequency dependencies of the effective interactions wereobtained and discussed. The singularities in the pairingchannels generated by the one-loop flow are qualitativelysimilar to the RPA, and are to a large extent fixed by the Wardidentity. The effective magnetic interaction develops a low-frequency small-momentum peak which is a pure fluctuationeffect. There are also significant quantitative fluctuation effectswhich are captured by the one-loop flow. In particular, the gapis strongly reduced compared to the mean-field value, witha stronger reduction at weaker interactions, as expected fromperturbative and numerical results. The expected divergenceof the superfluid amplitude mode in the low-energy limit is notcaptured by the one-loop truncation. This effect appears onlyat the two-loop level in the fermionic renormalization groupflow.
44
Aside from the channel decomposition of the vertex for
a system exhibiting spontaneous symmetry breaking, there
174523-20EFFECTIVE INTERACTIONS AND FLUCTUATION ... PHYSICAL REVIEW B 87, 174523 (2013)
are two other noteworthy technical upshots of our work,
which may be picked up in future calculations. First, wehave found that an accurate discretization of both momentumand frequency dependencies is computationally feasible
54and
has several advantages compared to the usual strategy of anansatz with a small number of scale-dependent coefficients. Inparticular, one avoids problems with momentum or frequencyderivatives which are necessary to extract the flow of suchcoefficients. Second, we have shown that a symmetry-breakingfield can be used as a convenient flow parameter, whichregularizes the flow at the critical scale and allows for acontrolled treatment of infrared divergences associated withthe Goldstone boson.
ACKNOWLEDGMENTS
We are grateful to J. Bauer, K.-U. Giering, N. Hasselmann,
T. Holder, C. Husemann, A. Katanin, B. Obert, and M.Salmhofer for valuable discussions.
APPENDIX: PAULI MATRIX BASIS
It is often convenient to represent the Nambu vertex in a
basis spanned by tensor products of Pauli matrices and the unitmatrix. The Pauli matrices τ
(1),τ(2),τ(3)and the unit matrix
τ(0)form a basis in the vector space of complex 2 ×2 matrices.
The tensor products τ(j)⊗τ(j/prime)form a basis in the space of
complex 4 ×4 matrices. The components of the Nambu vertex
in this basis are obtained as
˜/Gamma1(4)/Lambda1
jj/prime(k1,k2,k3,k4)=1
2/summationdisplay
siτ(j)
s4s1τ(j/prime)
s3s2/Gamma1(4)/Lambda1
s1s2s3s4(k1,k2,k3,k4).
(A1)
The inverse basis transformation is given by
/Gamma1(4)/Lambda1
s1s2s3s4(k1,k2,k3,k4)=1
2/summationdisplay
j,j/primeτ(j)
s1s4τ(j/prime)
s2s3˜/Gamma1(4)/Lambda1
jj/prime(k1,k2,k3,k4).
(A2)
The matrix formed by the components ˜/Gamma1(4)/Lambda1
jj/primeis denoted as
˜/Gamma1(4)/Lambda1. The tilde is used to distinguish this and other matrices
represented in the Pauli basis from matrices in the Nambuindex basis defined in Eq. (21).
The flow equations for the coupling functions parametrizing
the channel decomposed Nambu vertex can be derived mostconveniently in the Pauli matrix basis. Since the completeset of coupling functions is contained in the particle-holecontribution to the vertex, their flow is determined by theflow equation (46). Transformed to the Pauli matrix basis, the
equation reads as
d
d/Lambda1˜VPH,/Lambda1
jj/prime(k,k/prime;q)=/summationdisplay
p/summationdisplay
l,l/prime˜/Gamma1(4)/Lambda1
jl(k,p;q)∂/Lambda1˜L/Lambda1
ll/prime(p;q)
ט/Gamma1(4)/Lambda1
l/primej/prime(p,k/prime;q), (A3)
where
˜L/Lambda1
jj/prime(p;q)=1
2/summationdisplay
siτ(j)
s4s1τ(j/prime)
s3s2G/Lambda1s
2s4/parenleftbigg
p−q
2/parenrightbigg
G/Lambda1
s1s3/parenleftbigg
p+q
2/parenrightbigg
.
(A4)The decomposition (45) of the Nambu vertex can be written
in the Pauli matrix basis with momentum variables k1,4=
k±q/2 andk2,3=k/prime∓q/2a s
˜/Gamma1(4)/Lambda1
jj/prime(k,k/prime;q)=˜Ujj/prime(k,k/prime;q)+˜VPH,/Lambda1
jj/prime(k,k/prime;q)
−˜VPH/prime,/Lambda1
jj/prime/parenleftbiggk+k/prime−q
2,k+k/prime+q
2;k/prime−k/parenrightbigg
+˜VPP,/Lambda1
jj/prime/parenleftbiggk−k/prime+q
2,k−k/prime−q
2;k+k/prime/parenrightbigg
.
(A5)
Note that ˜VPH/prime,/Lambda1
jj/primeis defined by transforming VPH,/Lambda1
s2s1s3s4with the
first two Nambu indices exchanged to the Pauli matrix basis.The functions ˜L
/Lambda1
jj/prime(p;q) are given by products of normal and
anomalous propagators
L/Lambda1
00(p;q)=Re[G/Lambda1(p−)G/Lambda1(p+)]+F/Lambda1(p−)F/Lambda1(p+),
L/Lambda1
01(p;q)=iF/Lambda1(p−)ImG/Lambda1(p+)+iImG/Lambda1(p−)F/Lambda1(p+)
=L/Lambda1
10(p;q),
L/Lambda1
02(p;q)=iReG/Lambda1(p−)F/Lambda1(p+)−iF/Lambda1(p−)ReG/Lambda1(p+)
=−L/Lambda1
20(p;q),
L/Lambda1
03(p;q)=iIm[G/Lambda1(p−)G/Lambda1(p+)]=L/Lambda1
30(p;q), (A6)
L/Lambda1
11(p;q)=− Re[G/Lambda1(p−)G/Lambda1∗(p+)]+F/Lambda1(p−)F/Lambda1(p+),
L/Lambda1
22(p;q)=− Re[G/Lambda1(p−)G/Lambda1∗(p+)]−F/Lambda1(p−)F/Lambda1(p+),
L/Lambda1
33(p;q)=Re[G/Lambda1(p−)G/Lambda1(p+)]−F/Lambda1(p−)F/Lambda1(p+),
L/Lambda1
12(p;q)=Im[G/Lambda1(p−)G/Lambda1∗(p+)]=−L/Lambda1
21(p;q),
L/Lambda1
13(p;q)=ReG/Lambda1(p−)F/Lambda1(p+)+F/Lambda1(p−)ReG/Lambda1(p+)
=L/Lambda1
31(p;q),
L/Lambda1
23(p;q)=ImG/Lambda1(p−)F/Lambda1(p+)−F/Lambda1(p−)ImG/Lambda1(p+)
=−L/Lambda1
32(p;q),
where p+=p+q/2,p−=p−q/2.
The matrices representing the Nambu vertex in our approx-
imation for the Hubbard model are not all full, that is, severalmatrix elements vanish. More generally, for coupling functionswith a factorized momentum dependence and even parity formfactors the (direct) particle-hole contribution to the vertex hasthe form
˜V
PH,/Lambda1(k,k/prime;q)
=⎛
⎜⎜⎜⎜⎝2M/Lambda1
kk/prime(q)0 0 0
0 A/Lambda1
kk/prime(q)P/prime/prime/Lambda1
kk/prime(q)2 X/prime/Lambda1
kk/prime(q)
0 −P/prime/prime/Lambda1
kk/prime(q)/Phi1/Lambda1
kk/prime(q)−2X/prime/prime/Lambda1
kk/prime(q)
02 X/prime/Lambda1
kk/prime(q)2X/prime/prime/Lambda1
kk/prime(q)2C/Lambda1
kk/prime(q)⎞
⎟⎟⎟⎟⎠,
(A7)
and the particle-particle contribution ˜VPP,/Lambda1has only diagonal
elements given by the magnetic coupling function M/Lambda1
kk/prime(q).
However, the matrix elements of the crossed particle-holecontribution ˜V
PH/prime,/Lambda1(k,k/prime;q) are all nonzero and generally
given by linear combinations of several coupling functions.
174523-21ANDREAS EBERLEIN AND W ALTER METZNER PHYSICAL REVIEW B 87, 174523 (2013)
1D. M. Eagles, Phys. Rev. 186, 456 (1969).
2A. J. Leggett, in Modern Trends of Condensed Matter , edited by
A. Pekalski and J. Przystawa (Springer, Berlin, 1980).
3The BCS-BEC Crossover and the Unitary Fermi Gas ,e d i t e db y
W. Zwerger (Springer, Berlin, 2012).
4W. Metzner, M. Salmhofer, C. Honerkamp, V . Meden, andK. Sch ¨onhammer, Rev. Mod. Phys. 84, 299 (2012).
5C. Wetterich, Phys. Lett. B 301, 90 (1993).
6J. Berges, N. Tetradis, and C. Wetterich, Phys. Rep. 363, 223 (2002).
7P. Kopietz, L. Bartosch, and F. Sch ¨utz, Introduction to the
Functional Renormalization Group (Springer, Berlin, 2010).
8D. Zanchi and H. J. Schulz, P h y s .R e v .B 61, 13609 (2000).
9C. J. Halboth and W. Metzner, Phys. Rev. B 61, 7364 (2000).
10C. Honerkamp, M. Salmhofer, N. Furukawa, and T. M. Rice, Phys.
Rev. B 63, 035109 (2001).
11T. Baier, E. Bick, and C. Wetterich, P h y s .R e v .B 70, 125111 (2004).
12H. C. Krahl, S. Friederich, and C. Wetterich, Phys. Rev. B 80,
014436 (2009).
13M. C. Birse, B. Krippa, J. A. McGovern, and N. R. Walet, Phys.
Lett. B 605, 287 (2005).
14S. Diehl, H. Gies, J. M. Pawlowski, and C. Wetterich, P h y s .R e v .A
76, 021602(R) (2007).
15B. Krippa, E u r .P h y s .J .A 31, 734 (2007).
16P. Strack, R. Gersch, and W. Metzner, P h y s .R e v .B 78, 014522
(2008).
17S. Floerchinger, M. Scherer, S. Diehl, and C. Wetterich, Phys. Rev.
B78, 174528 (2008).
18L. Bartosch, P. Kopietz, and A. Ferraz, Phys. Rev. B 80, 104514
(2009).
19H. C. Krahl, J. A. M ¨uller, and C. Wetterich, Phys. Rev. B 79, 094526
(2009).
20S. Friederich, H. C. Krahl, and C. Wetterich, Phys. Rev. B 81,
235108 (2010); 83, 155125 (2011).
21M. Salmhofer, C. Honerkamp, W. Metzner, and O. Lauscher, Prog.
Theor. Phys. 112, 943 (2004).
22R. Gersch, C. Honerkamp, D. Rohe, and W. Metzner, Eur. Phys. J.
B48, 349 (2005).
23R. Gersch, C. Honerkamp, and W. Metzner, New J. Phys. 10, 045003
(2008).
24A. Eberlein and W. Metzner, Prog. Theor. Phys. 124, 471 (2010).
Note that fermionic propagators are defined with a different signconvention in that work.
25C. Husemann and M. Salmhofer, Phys. Rev. B 79, 195125
(2009).
26A similar channel decomposition was derived and applied also forthe single-impurity Anderson model by C. Karrasch, R. Hedden,R. Peters, T. Pruschke, K. Sch ¨onhammer, and V . Meden, J. Phys.:
Condens. Matter 20, 345205 (2008).
27A. A. Katanin, P h y s .R e v .B 70, 115109 (2004).
28M. Salmhofer and C. Honerkamp, Prog. Theor. Phys. 105,1
(2001).29In the matrix /Gamma1(4)presented in Ref. 24, Eq. (3.18), the order of the
first two momenta of the matrix elements /Gamma1(4)
−+++ and/Gamma1(4)
−−−+ was
erroneously reversed.
30S. A. Maier and C. Honerkamp, P h y s .R e v .B 86, 134404 (2012).
31For a review of the attractive Hubbard model, see R. Micnas,
J. Ranninger, and S. Robaszkiewicz, Rev. Mod. Phys. 62, 113
(1990).
32R. Fr `esard, B. Glaser, and P. W ¨olfle, J. Phys.: Condens. Matter 4,
8565 (1992).
33M. H. Pedersen, J. J. Rodr ´ıguez-N ´unez, H. Beck, T. Schneider, and
S. Schafroth, Z. Phys. B 103, 21 (1997).
34M. Letz and R. J. Gooding, J. Phys.: Condens. Matter 10, 6931
(1998).
35M. Keller, W. Metzner, and U. Schollw ¨ock, Phys. Rev. B 60, 3499
(1999).
36M. Randeria, N. Trivedi, A. Moreo, and R. T. Scalettar, Phys. Rev.
Lett. 69, 2001 (1992); N. Trivedi and M. Randeria, ibid. 75, 312
(1995).
37R. R. dos Santos, P h y s .R e v .B 50, 635 (1994).
38J. M. Singer, M. H. Pedersen, T. Schneider, H. Beck, and H.-G.
Matuttis, Phys. Rev. B 54, 1286 (1996).
39M. Keller, W. Metzner, and U. Schollw ¨ock, Phys. Rev. Lett. 86,
4612 (2001); J. Low Temp. Phys. 126, 961 (2002).
40M. Capone, C. Castellani, and M. Grilli, P h y s .R e v .L e t t . 88, 126403
(2002).
41A. Neumayr and W. Metzner, Phys. Rev. B 67, 035112 (2003).
42R. Shankar, Rev. Mod. Phys. 66, 129 (1994).
43C. Husemann, K.-U. Giering, and M. Salmhofer, Phys. Rev. B 85,
075121 (2012).
44A. Eberlein, Ph.D. thesis, University Stuttgart, 2013.
45See, for example, T. Holder and W. Metzner, P h y s .R e v .B 85,
165130 (2012).
46K.-U. Giering and M. Salmhofer, P h y s .R e v .B 86, 245122
(2012).
47L. P. Gorkov and T. K. Melik-Barkhudarov, J. Exp. Theor. Phys.40, 1452 (1961) [Sov. Phys.–JETP 13, 1018 (1961)].
48A. Georges and J. S. Yedidia, P h y s .R e v .B 43, 3475 (1991).
49A. Martin-Rodero and F. Flores, P h y s .R e v .B 45, 13008
(1992).
50See, for example, N. Trivedi and D. M. Ceperley, P h y s .R e v .B
40, 2737 (1989); A. L ¨uscher and A. M. L ¨auchli, ibid. 79, 195102
(2009).
51U. M. Ascher, H. Chin, and S. Reich, Numer. Math. 67, 131
(1994).
52See, for example, F. Pistolesi, C. Castellani, C. D. Castro, and G. C.Strinati, P h y s .R e v .B 69, 024513 (2004), and references therein.
53Recall that the Cooper (particle-particle) contributions are trans-
formed to particle-hole terms in Nambu representation.
54For flows in the symmetric regime a treatment of momentum andfrequency dependencies by discretization has been used already byGiering and Salmhofer (see Ref. 46).
174523-22 |
PhysRevB.73.184101.pdf | Density-functional calculations of the crystal structures and properties of CsCr 3O8
andACr3O8„A=In,Tl,Cu,Ag,Au …
R. Vidya, *P. Ravindran, A. Kjekshus, and H. Fjellvåg
Department of Chemistry, University of Oslo, Box 1033 Blindern, N-0315 Oslo, Norway
/H20849Received 26 October 2005; revised manuscript received 23 March 2006; published 1 May 2006 /H20850
Accurate ab initio density-functional calculations are performed to predict ground-state crystal structures
and to gain understanding of electronic structure and magnetic properties of CsCr 3O8and ACr3O8/H20849A
=In,Tl,Cu,Ag,Au /H20850. CsCr 3O8stabilizes in an orthorhombic /H20849prototype; Pnma /H20850structure in agreement with
experimental findings whereas the remaining compounds stabilize in the monoclinic KCr 3O8-type /H20849C2/m/H20850
structure. All compounds exhibit antiferromagnetic ordering in the ground state at 0 K. The electronic struc-tures are analyzed with the help of density-of-states, charge-density, and electron-localization-function plots.All compounds /H20849except InCr
3O8/H20850are found to be semiconductors /H20849insulators at 0 K /H20850with very small band gaps,
and Cr atoms in different environments consistently take different valence states.
DOI: 10.1103/PhysRevB.73.184101 PACS number /H20849s/H20850: 71.20. /H11002b, 78.20.Ci
I. INTRODUCTION
Oxide materials with transition metal constituents in
mixed-valence states have attracted much attention in recentyears since they exhibit exotic phenomena like colossal mag-netoresistance /H20849CMR /H20850and spin, charge, and orbital ordering.
1
Transition-metal compounds with mixed-valence Mn, Cu,
and Co are widely studied, whereas others with say, Cr andNi are less explored. Even though manganates have beenattracting much attention for mixed-valence behavior andcharge ordering features, mixed-valence states like V
4+and
V5+in/H9251-NaV 2O5,2,3Ti3+,T i4+in Ti 4O7,4Ni2+and Ni4+in
HoNiO 3/H20849Ref. 5 /H20850have also been reported. However, com-
pounds with Cr atoms in mixed-valence states are seldomobserved.
In an effort to gain understanding of mixed-valence Cr
compounds, we have earlier
6–8studied structural stability,
electronic structure, and magnetic properties for compoundswith the general formula ACr
3O8/H20849A=H,Li,Na,K,Rb as
well as Aabsent /H20850where Cr formally takes two different
valence states. In the first report6we used accurate density-
functional theory /H20849DFT /H20850calculations to analyze electronic,
magnetic, and bonding characteristics of ACr3O8/H20849Na, K, Rb /H20850
and later7we explored the ground-state structures of Cr 3O8
and LiCr 3O8which are of considerable interest as cathode
materials in rechargeable Li-ion batteries. In Ref. 8 wereport the findings for the highly hypothetical compoundHCr
3O8. In the present work we analyze the structural be-
havior, electronic structure, and magnetic properties of thecorresponding ACr
3O8phases with A=Cs, In, Tl, Cu, Ag,
and Au, with some special emphasis on CsCr 3O8. One of the
salient features of these compounds is that in addition tomixed-valence states, they have layered structures. Amongthe layered materials with CMR behavior, manganites likeLa
2−2xSr1+2xMn 2O7have been receiving attention recently as
they have strong anisotropy of electrical and magnetic trans-port, anomalous magnetoelastic properties, etc.
9Therefore
the studied compounds may also exhibit interesting physicalproperties.II. COMPUTATIONAL DETAILS
The results presented here are based on density-functional
calculations according to the projected-augmented plane-wave /H20849PAW /H20850/H20849Ref. 10 /H20850method as implemented in the V ASP
/H20849Vienna ab initio simulation package /H20850.
11In this approach the
valence orbitals are expanded as plane waves and the inter-actions between the core and valence electrons are describedby pseudopotentials. Since one of the main aims of thepresent work is to determine ground-state structures forACr
3O8, we have performed structural optimizations. We
considered around 12 different structure types as inputs foreach compound and evaluated the total energy for these teststructures. A detailed listing of all the considered input struc-tures are given elsewhere.
6,7The optimization of the atomic
geometry is performed via a conjugate-gradient minimiza-tion of the total energy, using Hellmann-Feynman forces onthe atoms and stresses in the unit cell. During the simula-tions, atomic coordinates and axial ratios are allowed to relaxfor different volumes of the unit cell. These parameters arechanged iteratively so that the sum of lattice energy and elec-tronic free energy converges to a minimum value. Conver-gence minimum with respect to atomic shifts is assumed tohave been attained when the energy difference between twosuccessive iterations is less than 10
−7eV per cell and the
forces acting on the atoms are less than 1 meV Å−1. The
structure with the lowest total energy is taken as the ground-state structure. The generalized gradient approximation/H20849GGA /H20850includes the effects of local gradients of the charge
density and generally gives better equilibrium structural pa-rameters than the local density approximation /H20849LDA /H20850. Hence
we have used GGA with the Perdew-Burke-Ernzerhof /H20851PBE
/H20849Ref. 12 /H20850/H20852functional to obtain the accurate exchange and
correlation energy for a particular atomic configuration. Wehave used 520 eV plane-wave energy cutoff for all the com-pounds. The calculations are carried out using 64 kpoints in
the irreducible Brillouin zone for the monoclinic C2/m
structure. We have used same energy cutoff and k-point den-
sity in all calculations. The above calculations are performedin paramagnetic /H20849P/H20850, ferromagnetic /H20849F/H20850, and antiferromag-PHYSICAL REVIEW B 73, 184101 /H208492006 /H20850
1098-0121/2006/73 /H2084918/H20850/184101 /H208498/H20850 ©2006 The American Physical Society 184101-1netic /H20849AF /H20850configurations. We have assumed a simple AF
structure as shown in Fig. 1 /H20849b/H20850of Ref. 6. The magnetic mo-
ments are estimated inside the atomic sphere radii. We havecalculated the total energy of the compounds as a function ofvolume for 10 different volumes, fitted the results to theso-called “universal equation of state,”
13and extracted the
bulk modulus /H20849B0/H20850.
III. RESULTS AND DISCUSSION
A. Structural optimization
Among the different atomic arrangements used as inputs
in the calculations for CsCr 3O8, the experimentally
established14structure /H20849prototype; Pnma /H20850and the
KCr 3O8-type /H20849C2/m/H20850variant came out with lower energies
/H20851Fig. 2 /H20849a/H20850/H20852than the other considered alternatives. Among
these the experimental arrangement has the lowest energyand represents the ground state. The structure /H20851Fig. 1 /H20849a/H20850/H20852
comprises three crystallographic types of Cr and six types ofO atoms which form Cr1O
6octahedra, Cr2O 4tetrahedra, and
Cr3O 4tetrahedra. These polyhedra are linked by corner shar-
ing to form layers extending parallel to the bcplane, sepa-
rated by Cs atoms which occupy interlayer positions. Thecalculated lattice parameters and atom positions for CsCr
3O8
are found to be in reasonable agreement with the availableexperimental values,
14except that ashows nearly 5.5% de-
viation between theory and experiment /H20849see Table I /H20850. Note
that the CsCr 3O8structure arranges the Cr–O polyhedral lay-
ers along the adirection. It is in this direction one also finds
the interlayer interactions that are governed by the weakervan der Waals type forces, which are not accounted properlyby the DFT calculations. This may be the reason for overes-timation of aand consequently for the somewhat too large
equilibrium volume /H20849remembering that GGA alone
15is likely
to overestimate volume by 2%–3% /H20850.
Among the considered structures for CuCr 3O8, our struc-
tural optimization shows that KCr 3O8-type /H20849C2/m/H20850,
LiCr 3O8-II-type, and ZnV 3O8-type /H20849Iba2/H20850structures are
present at the lower energy region in the total energy vs
volume curve. On the other hand, the other ACr3O8com-
pounds under consideration /H20849A=In,Tl,Ag,Au /H20850have lower
energies for KCr 3O8-type, LiCr 3O8-type, and LiV 3O8-type
/H20849P21/m/H20850structures /H20851see Figs. 2 /H20849b/H20850and 2 /H20849c/H20850; for the sake of
clarity higher-energy structures are left out /H20852. Hence the
ACr3O8compounds with A=In, Tl, Cu, Ag, and Au stabilize
in the KCr 3O8-type structure like the alkali-metal derivatives
of the family.6–8,16The KCr 3O8-type structure /H20851Fig. 1 /H20849b/H20850/H20852
consists of two types of Cr atoms arranged in fairly regularoctahedral and tetrahedral environments of O neighbors.These polyhedra form layers parallel to the a,bplane by
corner sharing and the Aatoms in interlayer positions. The
atomic arrangement in CsCr
3O8/H20849Pnma /H20850is similar to that in
the KCr 3O8-type structure, except that the orientation of half
of the tetrahedra of the former is different, and every secondlayer is rotated by 180° compared with the layers in theKCr
3O8-type structure. There is a significant distinction be-
tween the Cr-O distances in CrO 6octahedra and CrO 4tetra-
hedra in these structures, which immediately points to amixed-valence situation for Cr.Optimized structural parameters for the ground state of
theACr
3O8compounds are given in Table I. Except for a
rough linear relationship between the cell volume and theionic radius of A/H20849Fig. 3 /H20850there appears to be no clear cut
overall trend in the structural parameters for the ACr
3O8se-
ries as a whole /H20849despite the fact that all members except
CsCr 3O8are formally isostructural /H20850. Among the ACr3O8
compounds with A=In, Tl, Cu, Ag, and Au, only AgCr 3O8
and TlCr 3O8has been synthesized and characterized by mag-
netic measurements /H20849but structural determination has not
been attempted /H20850. Thus, the compounds with A=In, Cu, and
Au hitherto remain hypothetical. However, the negative val-ues of heat of formation indicate that these materials can besynthesized experimentally.
B. Magnetic properties
Cooperative magnetism with P, F, and AF configurations
was taken into account in the structural optimization calcu-
FIG. 1. /H20849Color online /H20850The optimized crystal structures of /H20849a/H20850
CsCr 3O8/H20849prototype; Pnma /H20850and /H20849b/H20850InCr 3O8/H20849KCr 3O8type; C2/m/H20850.VIDYA et al. PHYSICAL REVIEW B 73, 184101 /H208492006 /H20850
184101-2lations. As seen from Fig. 2 all the studied compounds sta-
bilize in the AF state. According to the measured17magnetic
susceptibility data, AgCr 3O8orders antiferromagnetically at
low temperatures; magnetic moment derived from the Curie-Weiss relationship gives
/H9262P=3.95±0.03 /H9262Bf.u.−1Calculated
magnetic moments for the studied compounds in theirground-state structures are listed in Table II. It is evident thatthe octahedral Cr1 site carries an appreciable magnetic mo-ment whereas the tetrahedral Cr2 site /H20849as well as Cr3 for
CsCr
3O8/H20850has an almost negligible magnetic moment /H20849more
significant but still small at the Cr2 and Cr3 sites inCsCr
3O8/H20850.
InCr 3O8represents a special case in that the magnetic
moment on Cr2 in the F state is appreciable /H208491.06/H9262B/H20850
whereas it follows the trend for the rest of the series for theAF state /H20849which is 0.20 eV lower in energy than the F state /H20850.
The delectrons in InCr 3O8may, e.g., be said to participate
more in magnetism than in bonding interaction as indicated
by the larger Cr2-O bond length in InCr 3O8/H208491.68 Å /H20850than
that in the other compounds /H20849e.g., 1.63 Å in TlCr 3O8/H20850,
whereas Cr1-O distance /H208491.91 Å /H20850is almost equal /H208491.92 Å in
TlCr 3O8/H20850. A general feature for the entire ACr3O8family is
that the oxygen atoms also possess small magnetic momentswhich originate from Cr dand O phybridization, and this
results in somewhat higher total moment values for the Fcases than the sum of the Cr moments. The magnetic mo-ments at O atoms range from 0.018 to 0.062
/H9262B. The mo-
ments at O1 and O3 are directed parallel to the majority-spinchannel of Cr atoms while the moments at O2 are directedparallel to the minority-spin channel of Cr atoms. The differ-TABLE I. Optimized ground-state structural parameters, bulk modulus /H20849B0/H20850, and heat of formation /H20849−/H9004H/H20850forACr3O8/H20849A=Cs, In, Tl, Cu,
Ag, and Au /H20850at 0 K. Except CsCr 3O8/H20849space group Pnma /H20850these compounds stabilize in KCr 3O8-type structure; space group C2/mwith A
in 2a/H20849000 /H20850and Cr1 in 2 c/H208490 0 1/2 /H20850positions.
Compound Unit cell /H20849Åo r° /H20850 Positional parameters B0/H20849GPa /H20850 −/H9004H/H20849kJ Mol−1/H20850
CsCr 3O8 a=16.8322 /H2084915.9570 /H20850aCs /H208494c/H20850: 0.2301, 1/4, 0.0150 /H208490.2137 ,1/4, 0.0064 /H20850 18.96 2189.38
b=5.5226 /H208495.5050 /H20850 Cr1 /H208494c/H20850: 0.4608, 1/4, 0.2575 /H208490.4524 ,1/4, 0.2407 /H20850
c=8.5903 /H208498.264 /H20850 Cr2 /H208494c/H20850: 0.6046, 1/4, 0.9516 /H208490.6104 ,1/4, 0.9581 /H20850
Cr3 /H208494c/H20850: 0.9198, 1/4, 0.8714 /H208490.9205 ,1/4, 0.8627 /H20850
O1 /H208494c/H20850: 0.8340, 1/4, 0.7831 /H208490.8370 ,1/4, 0.7619 /H20850
O2 /H208494c/H20850: 0.1799, 1/4, 0.4294 /H208490.1822 ,1/4, 0.4101 /H20850
O3 /H208494c/H20850: 0.0185, 1/4, 0.4515 /H208490.0110 ,1/4, 0.4699 /H20850
O4 /H208494c/H20850: 0.3996, 1/4, 0.4345 /H208490.3842 ,1/4, 0.4459 /H20850
O5 /H208498d/H20850: 0.3884, 0.9979, 0.1619 /H208490.3766, 0.9887, 0.1570 /H20850
O6 /H208498d/H20850: 0.0294, 0.0006, 0.1759 /H208490.0242, 0.0009, 0.1743 /H20850
InCr 3O8 a=8.4711 Cr2 /H208494i/H20850: 0.3576, 0, 0.2641 83.32 2005.53
b=5.6404 O1 /H208494i/H20850: 0.7771, 0, 0.5478
c=6.5323 O2 /H208494i/H20850: 0.7428, 0, 0.9571
/H9252=90.04 O3 /H208498j/H20850: 0.9721, 0.7614, 0.2741
TlCr 3O8 a=8.9606 Cr2 /H208494i/H20850: 0.3604, 0, 0.2942 27.91 2543.62
b=5.5066 O1 /H208494i/H20850: 0.7918, 0, 0.5729
c=7.7331 O2 /H208494i/H20850: 0.6834, 0, 0.9082
/H9252=92.84 O3 /H208498j/H20850: 0.9630, 0.7541, 0.3271
CuCr 3O8 a=8.3412 Cr2 /H208494i/H20850: 0.3462, 0, 0.2707 20.17 1862.75
b=5.5182 O1 /H208494i/H20850: 0.7727, 0, 0.5360
c=6.5480 O2 /H208494i/H20850: 0.7708, 0, 0.9434
/H9252=94.32 O3 /H208498j/H20850: 0.9631, 0.7577, 0.2797
AgCr 3O8 a=8.6410 Cr2 /H208494i/H20850: 0.3444, 0, 0.2848 23.34 1829.89
b=5.5211 O1 /H208494i/H20850: 0.7840, 0, 0.5525
c=7.1954 O2 /H208494i/H20850: 0.7473, 0, 0.9219
/H9252=93.80 O3 /H208498j/H20850: 0.9566, 0.7553, 0.3064
AuCr 3O8 a=8.7686 Cr2 /H208494i/H20850: 0.3412, 0, 0.2796 68.04 1706.14
b=5.4699 O1 /H208494i/H20850: 0.7860, 0, 0.5484
c=7.0013 O2 /H208494i/H20850: 0.7577, 0, 0.9254
/H9252=91.50 O2 /H208498j/H20850: 0.9543, 0.7551, 0.3006
aExperimental value from Ref. 14.DENSITY-FUNCTIONAL CALCULATIONS OF THE ¼ PHYSICAL REVIEW B 73, 184101 /H208492006 /H20850
184101-3ent size of the Cr magnetic moments is a clear indication of
different valence states. However, our prior experiences6–8
show that assignment of formal valence states for Cr fromtheir magnetic moments is not straightforward.C. Electronic structure
AllACr3O8compounds /H20849except InCr 3O8/H20850, exhibit insulat-
ing behavior at 0 K with small, but distinct band gaps /H20849Eg/H20850
FIG. 2. Calculated cell volume vs total energy for ACr3O8with /H20849a/H20850A=Cs, /H20849b/H20850A=In and Tl, and /H20849c/H20850A=Cu, Ag, and Au.VIDYA et al. PHYSICAL REVIEW B 73, 184101 /H208492006 /H20850
184101-4between the valence band /H20849VB /H20850and conduction band /H20849CB /H20850.
The Egvalues for the compounds under consideration are
included in Table II. InCr 3O8has a pseudogaplike feature at
the Fermi level /H20849EF/H20850. The presence of pseudogaplike features
in DOS is considered as favorable for stability, but this indi-
cator cannot be taken in support of a probable materializationof such a compound. In fact, compounds with monovalent Inare relatively uncommon /H20849for instance, only 114 compounds
with monovalent In are reported in the ICSD database
18
compared to 1072 compounds with In in the trivalent state /H20850
and a standard ionic radius for In+is not available. In order
to get insight into the occurrence of the higher magneticmoment at the Cr2 site in InCr
3O8than the other ACr3O8
compounds we have examined the total and site-projected
DOS for this phase /H20849not shown /H20850. Interestingly, this exercise
showed a half-metallic behavior with filled band in themajority-spin channel and a 2.6 eV energy gap in theminority-spin channel. The exchange-splitting energy for Cr1is around 1.5 eV whereas that for Cr2 is 0.8 eV. It is thelatter distinct exchange splitting which is responsible for thesizeable magnetic moment at the Cr2 site. It should be notedthat owing to the limitation of usual density-functional cal-culations transition metal oxides are often wrongly predicted
to be metal instead of insulator. This can be remedied bygoing beyond GGA such as LDA+ Umethod,
19self-
interaction corrected density-functional calculations20or dy-
namical mean field theory.21Additional calculations with
such approaches may clarify why InCr3O8 is metallic ratherthan insulator. Hence, provided InCr
3O8can be obtained its
properties appear to deserve a closer attention.
Figure 4 shows the total and site-projected DOS profiles
for CsCr 3O8together with CuCr 3O8as representative for the
ACr3O8compounds with KCr 3O8-type structure. The well-
localized states around −7 eV in Fig. 4 /H20849a/H20850originate from Cs
porbitals. Cr dand O pstates are present from −6 eV up to
EF. The energetic distribution of these DOS profiles show
some similarities to those of Cr dand O pDOS in CrO 2/H20849d2/H20850
and Cr 2O3/H20849Ref. 22 /H20850which extend from −8 to EF. Moreover,
CrO 2/H20849Cr in octahedral coordination /H20850and Cr 2O3/H20849Cr in
square-planar coordination /H20850have Cr- dstates closer to EFlike
in Cr1 /H20849from −0.9 eV to EF/H20850which has octahedral coordina-
tion. These states are attributed to the unpaired electrons int
2g-like orbitals which are responsible for the substantial
magnetic moment at Cr1. Since Cr2 and Cr3 have tetrahedralcoordination, their DOS profiles are significantly differentfrom that of the Cr1. The majority- and minority-spin chan-nels of Cr2 and Cr3 are more or less equally filled, resultingin the small magnetic moments. The average Cr1-O distanceis 1.94 Å whereas the average Cr2-O and Cr3-O distancesare 1.67 Å. This implies that Cr1 states participate more inmagnetic interaction than in bonding interaction and viceversa for Cr2 and Cr3. As Cr2 and Cr3 dstates participate
more in bonding interaction, they are localized and hence nosignificant states are seen closer to E
F./H20851It is worthwhile to
recall that in a similar compound KCr 3O8,6the integrated
crystal orbital Hamiltonian population /H20849COHP /H20850value for tet-
rahedral Cr is more than the octahedral Cr, implying stronger
bonding interaction in the former than in the latter. /H20852It is
interesting to note that Cr1, Cr2, Cr3, and O2 states havewell-defined sharp peaks around −6 eV. Among the threedifferent types of O, O2 is the one which places itself at theapex of Cr1O
6octahedra and Cr2O 4, Cr3O 4tetrahedra. Thus
O2 mediates the superexchange interaction between Cr1 andCr2/Cr3 which is manifested by the DOS peaks seen around−6 eV.
According to our calculations CuCr
3O8should be a semi-
conductor with a 0.41 eV band gap. The prominent states for
TABLE II. Calculated magnetic moment /H20849in/H9262Bper Cr atom /H20850forACr3O8in F and AF states. Total refers
to the total magnetic moment per formula unit. Band gap /H20849Eg/H20850is given in units of eV .
CompoundFA F
Eg Cr1 Cr2 Total Cr1 Cr2
CsCr 3O8a2.488 0.231 2.866 2.481 0.233 0.61
CuCr 3O8 2.608 0.231 2.888 2.533 0.018 0.41
AgCr 3O8 2.518 0.249 2.879 2.476 0.000 0.58
AuCr 3O8 2.422 0.226 2.854 2.476 0.011 0.45
InCr 3O8 2.750 1.063 4.776 2.579 0.673 psuedogap
TlCr 3O8 2.495 0.212 2.868 2.422 0.016 0.53
aMoment at Cr3 is 0.208 and 0.212 in F and AF states, respectively.
FIG. 3. Optimized equilibrium volume for ACr3O8/H20849A=Li, Na,
K, Rb, Cs, In, Tl, Cu, Ag, and Au /H20850as a function of A+radius
/H20849standard values /H20850.DENSITY-FUNCTIONAL CALCULATIONS OF THE ¼ PHYSICAL REVIEW B 73, 184101 /H208492006 /H20850
184101-5Cu close to EFis associated with completely filled dstates.
The almost empty sandpbands indicate that Cu has donated
its valence electron to the oxygen atoms. The two types of Cratoms have topologically different DOS profiles. The overallfeatures are similar to the findings for the alkali-metal mem-bers of the ACr
3O8/H20849Refs. 6 and 7 /H20850family. Cr1 has sharp
peaks close to EF/H20849from −1.5 to EF/H20850with more states in the
majority-spin channel than in the minority-spin channel. Onthe other hand, both the majority- and minority-spin channelsof Cr2 for CuCr
3O8are almost equally filled, resulting in
negligible exchange splitting and hence in a negligible mag-netic moment. From this observation we conclude that smallmagnetic moment at the Cr2 site is not due to small amountof electrons at the Cr2 state /H208496+ oxidation state has been
expected from simple ionic model /H20850. Moreover, the states at
the Cr2 site are more localized than those at the Cr1 site /H20851see
the −6 to −2 eV range in Fig. 4 /H20849b/H20850/H20852. If Cr2 had been in the
Cr
6+/H20849d0/H20850state, its dband should have been empty. However,
the appreciable DOS seen in the VB of Cr2 demonstrates that
the valence state of Cr2 is certainly not Cr6+as expected
experimentally. Similar to the findings for the other ACr3O8
compounds with KCr 3O8-type structure, the O atoms exhibit
somewhat different DOS profiles, even though they areenergetically degenerate with themselves and with the Cratoms. The result is distinct covalent hybridization.
Mixed-valence compounds have been receiving particular
attention as they exhibit interesting optical, electrical andmagnetic properties. Most of them are attractive candidatesfor technological applications owing to their CMR behavior.Even though occurrence of mixed-valence state is commonto all transition-metal ions, compounds with Mn have beenreceiving particular attention, may be due to their potentialapplications. In order to understand the exotic physical prop-erties exhibited by these interesting compounds, evaluationof the valence states of different ions is imminent. Quite afew empirical rules such as Pauling rules and Hume-Rotheryrule are used to ascribe valences for binary and ternary com-pounds. For compounds with transition-metal species semi-empirical Hund’s rules are useful to determine spin and va-lence configurations. The magnetic moment measurementsand concepts like bond-valence sums provide qualitative val-ues for valence states. The calculated magnetic moments,site-, and orbital-projected DOS at the transition-metal sitesin conjunction with crystal-field effects provide some cluesabout the valence states. Attempts to derive the formal va-lences from Born-effective charges for LiCr
3O8provided
1.36, 3.54, 4.57, and −1.61 for Li, Cr1, Cr2, and O,respectively.
7One of the reasons for the unexpected valence
FIG. 4. Total and site-projected density of states for CsCr 3O8/H20849prototype /H20850and CuCr 3O8/H20849KCr 3O8type /H20850. The Fermi level /H20849EF/H20850is indicated
by vertical dashed line.VIDYA et al. PHYSICAL REVIEW B 73, 184101 /H208492006 /H20850
184101-6states of the constituents is that, counting of valence states
basically starts from oxygen which is assumed to be ideallyin 2- /H20849pure ionic /H20850state. However, in reality valence electrons
participate in mixed ionocovalent bonding interactions whichresult in valence states different from ideally expected val-ues. Since the calculated Born-effective charge values can beconsidered as upper limits for the formal valence state of aconstituent, A, Cr1, Cr2, and O can be assigned 1+, 3+, 4+,
and 1.5− valence states, respectively.
D. Bonding characteristics
In spite of the fact that the CsCr 3O8structure possesses 48
atoms in the unit cell which certainly obscures the clarity ofthe picture, we have attempted to elucidate its bonding char-acteristics using charge-density and electron localizationfunction /H20849ELF /H20850plots /H20849see Refs. 6–8, 23, and 24 for back-
ground information and utilization of these tools for othermembers of the ACr
3O8family /H20850. ELF is a measure of the
probability of finding an electron near another electron withthe same spin. It is a ground state property which discrimi-nates between different kinds of bonding in a very sharp,quantitative way. The ELF is represented as a contour plot inreal space where different contour correspond to numericalvalues ranging from 0 to 1. A region where ELF is 1, there isno chance of finding two electrons with the same spin. Thisusually occurs in places where bonding pairs /H20849molecular or-
bitals /H20850or lone pairs /H20849atomic orbitals /H20850reside. An ELF 0 cor-
respond to the area where there is no electron density. For ahomogeneous electron gas like in metals ELF is 0.5.
25It
should be noted that ELF is a measure of the Pauli principleand not of electron density. It is seen from Fig. 5 /H20849a/H20850that the
covalent interaction between Cr2 and O is stronger than thatbetween Cr1 and O. The ionic nature of Cs at the interlayerposition is clearly evident from the more or less sphericallydistributed charge around Cs. The ELF is negligible at the Cr
sites whereas it attains local maximum values at the O sites;another manifestation of covalent interaction. The bondingsituation in CsCr
3O8is similar to that seen for other mem-
bers of ACr3O8family in the sense that chromium and oxy-
gen atoms form ionocovalent-bonded subunits whereas Cshas distinct ionic character. This may be one of the reasonsfor characteristic magnetic features observed in these com-pounds.
IV. CONCLUSION
The prediction of ground-state crystal structures for the
ACr3O8series have been extended to A=In, Tl, Cu, Ag, and
Au considering several potential structure types. The calcu-lated ground-state structure for CsCr
3O8/H20849space group Pnma /H20850
is found to be in good agreement with experimental data.Except NaCr
3O8all members of the ACr3O8family exhibit
antiferromagnetic ordering as the ground-state configurationsand the electronic structures show that all compounds /H20849ex-
cept InCr
3O8/H20850are insulators at 0 K; a pseudogaplike feature
is established for InCr 3O8. Different magnetic moments and
DOS profiles for the Cr atoms clearly confirm mixed-valencesituations. Bonding analysis undertaken using density-of-states, charge-density, and electron-localization plots indicateionic behavior for the Aatoms whereas the Cr and O atoms
mutually experience largely covalent interaction, however,with different degree of covalence for Cr1 and Cr2. In orderto have more insight into these compounds, more experimen-tal studies on electrical and magnetic properties are needed.
ACKNOWLEDGMENT
The authors are grateful to the Research Council of Nor-
way for financial support and computer time at the Norwe-gian supercomputer facilities.
FIG. 5. /H20849Color online /H20850Calculated /H20849a/H20850charge density and /H20849b/H20850electron-localization function for CsCr 3O8.DENSITY-FUNCTIONAL CALCULATIONS OF THE ¼ PHYSICAL REVIEW B 73, 184101 /H208492006 /H20850
184101-7*Electronic address: vidya.ravindran@kjemi.uio.no
1C. N. R. Rao and A. K. Raychauduri, in Colossal Magnetoresis-
tance, Charge Ordering and Related Properties of ManganeseOxides , edited by C. N. R. Rao and B. Raveau /H20849World Scientific,
Singapore, 1998 /H20850,p .1 .
2H. Smolinski, C. Gros, W. Weber, U. Peuchert, G. Roth, M.
Weiden, and C. Geibel, Phys. Rev. Lett. 80, 5164 /H208491998 /H20850.
3S. G. Bompadre, A. F. Hebard, V . N. Kotov, D. Hall, G. Maris, J.
Baas, and T. T. M. Palstra, Phys. Rev. B 61, R13321 /H208492000 /H20850.
4R. Melzer and C. H. Ruscher, Phase Transitions 58, 285 /H208491996 /H20850.
5J. A. Alonso, M. J. Martínez-Lope, M. T. Casais, J. L. García-
Muñoz, and M. T. Fernández-Díaz, Phys. Rev. B 61, 1756
/H208492000 /H20850.
6R. Vidya, P. Ravindran, P. Vajeeston, H. Fjellvåg, and A. Kjek-
shus, Phys. Rev. B 72, 014411 /H208492005 /H20850.
7R. Vidya, P. Ravindran, A. Kjekshus, and H. Fjellvåg, Phys. Rev.
B/H20849to be published /H20850.
8R. Vidya, P. Ravindran, A. Kjekshus, and H. Fjellvåg, J. Electro-
ceram. /H20849to be published /H20850.
9S. Okamoto, S. Ishihara, and S. Maekawa, Phys. Rev. B 63,
104401 /H208492001 /H20850.
10P. E. Bløchl, Phys. Rev. B 50, 17953 /H208491994 /H20850; G. Kresse and D.
Joubert, ibid. 59, 1758 /H208491999 /H20850.
11G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6,1 5 /H208491996 /H20850.
12J. P. Perdew, K. Burke, and Y . Wang, Phys. Rev. B 54, 16533/H208491996 /H20850; J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
Lett. 77, 3865 /H208491996 /H20850.
13P. Vinet, J. H. Rose, J. Ferrante, and J. R. Smith, J. Phys.: Con-
dens. Matter 1, 1941 /H208491989 /H20850.
14K.-A. Wilhelmi, Ark. Kemi 26, 141 /H208491966 /H20850; Chem. Commun.
/H20849London /H208501966 , 437 /H208491966 /H20850; Doctoral thesis, University of
Stockholm, Stockholm, 1966.
15U. Häussermann, H. Blomqvist, and D. Noréus, Inorg. Chem. 41,
3684 /H208492002 /H20850.
16M. J. Saavedra, C. Parada, and E. J. Baran, J. Phys. Chem. Solids
57, 1929 /H208491996 /H20850.
17H. Fjellvåg /H20849unpublished /H20850.
18Inorganic Crystal Structure Database, Version 2004.2.
19A. I. Liechtenstein, V . I. Anisimov, and J. Zaanen, Phys. Rev. B
52, R5467 /H208491995 /H20850.
20A. Svane and O. Gunnarsson, Phys. Rev. Lett. 65, 1148 /H208491990 /H20850.
21A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod.
Phys. 68,1 3 /H208491996 /H20850.
22R. Zimmermann, P. Steiner, R. Claessen, F. Reinert, S. Hüfner, P.
Blaha, and P. Dufek, J. Phys.: Condens. Matter 11, 1657 /H208491999 /H20850.
23P. Ravindran, P. Vajeeston, R. Vidya, A. Kjekshus, and H.
Fjellvåg, Phys. Rev. Lett. 89, 106403 /H208492002 /H20850.
24A. Savin, R. Nesper, S. Wengert, and T. Fässler, Angew. Chem.,
Int. Ed. Engl. 36, 1809 /H208491997 /H20850.
25N. Oai and J. B. Adams, J. Comput. Electron. 3,5 1 /H208492004 /H20850.VIDYA et al. PHYSICAL REVIEW B 73, 184101 /H208492006 /H20850
184101-8 |
PhysRevB.103.195129.pdf | PHYSICAL REVIEW B 103, 195129 (2021)
One-particle spectral functions of the one-dimensional Fermionic Hubbard
model with one fermion per site at zero and finite magnetic fields
José M. P. Carmelo,1,2,3Tilen ˇCadež ,4and Pedro D. Sacramento5
1Center of Physics of University of Minho and University of Porto, P-4169-007 Oporto, Portugal
2Department of Physics, University of Minho, Campus Gualtar, P-4710-057 Braga, Portugal
3Department of Physics, Boston University, 590 Commonwealth Ave., Boston, Massachusetts 02215, USA
4Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea
5CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, P-1049-001 Lisboa, Portugal
(Received 28 December 2020; accepted 29 April 2021; published 13 May 2021)
Although charge-spin separation has important consequences for the properties of one-dimensional (1D) Mott-
Hubbard insulators, matrix elements in some of its dynamical correlation functions involve the coupling of spinand charge degrees of freedom. The corresponding interplay of the 1D Mott-Hubbard insulator’s charge andspin degrees of freedom is an issue of both fundamental and technological interest, for instance, concerningtheir dynamics at subpicosecond timescales. On the other hand, the up- and down-spin one-particle spectralfunctions are the simplest dynamical correlation functions that involve excitation of both the charge and spindegrees of freedom. They are thus suitable to extract basic useful physical information on the above interplayat finite magnetic fields. Here the line shape of such functions at and in the ( k,ω) plane’s vicinity of their cusp
singularities is studied for the Mott-Hubbard insulator described by the 1D Hubbard model with one fermion persite at zero and finite magnetic fields. At zero field, they can be accessed in terms of electrons by photoemissionexperiments. At finite field, such functions and corresponding interplay of correlations and magnetic-field effectsrefer to an involved nonperturbative many-particle problem that is poorly understood. The Mott-Hubbard gap thatseparates the addition and removal spectral functions is calculated for all spin densities and interaction values.The qualitative differences in the one-particle properties of the Mott-Hubbard insulator and corresponding dopedinsulator are also investigated. The relation of our theoretical results and predictions to both condensed-matterand ultracold spin-1 /2 atom systems is discussed.
DOI: 10.1103/PhysRevB.103.195129
I. INTRODUCTION
Charge-spin separation has important physical conse-
quences for the properties of one-dimensional (1D) Mott-Hubbard insulators [ 1]. Their spin spectra are gapless in spite
of the finite charge Mott-Hubbard gap 2 /Delta1
MH. However, the
matrix elements in some of its dynamical correlation functions
involve the coupling of spin and charge degrees of freedom,which refers both to frequency and time dependencies. Forinstance, the coupling of the spin degrees of freedom to theFloquet sector with shifted transfer energies 2 /Delta1
MH±ωby
virtual absorption and emission of photons [ 2] of energy ω
plays a key role in the dynamics of 1D Mott-Hubbard insu-
lators at subpicosecond timescales. This is an issue of both
fundamental and technological interest [ 2,3]. The same ap-
plies to the more general problem of the 1D Mott-Hubbardinsulators’s interplay of charge and spin degrees of freedom.For instance, after a sudden quench, such insulators realizea nonthermal state that is an admixture of spin and chargedensity wave states [ 4].
The 1D Hubbard model with one fermion per lattice
site, on-site repulsion U, and first neighboring integral t
(usually called half-filled 1D Hubbard model) is exactly solv-
able by the Bethe ansatz [ 5–8]. It corresponds to phase Vin the phase diagram shown in Fig. 1 of Ref. [ 9], which
presents a recent study of the 1D Hubbard model in amagnetic field’s quantum critical behavior and thermodynam-ics. The half-filled 1D Hubbard model is the simplest andan emblematic example of a 1D Mott-Hubbard insulator.However, the interplay between correlation effects associ-ated with the gapped charge excitations and its spin degreesof freedom is an involved many-particle problem that waslittle studied, is not well understood, and deserves furtherinvestigations.
A very recent study on the spin dynamical properties of
that model in a magnetic field was on the spin longitudinaland transverse dynamical structure factors [ 10]. Beyond pre-
vious studies [ 11], it accounted for contributions from spin
n-strings. It was found that the momentum-dependent expo-
nents that control such factors line shape at and near the cuspsingularities depend very little on u=U/4t[10,11]. The main
effect of correlations was found to be on the energy band-width of the dynamical structure factors’ ( k,ω)-plane spectra,
which increases upon decreasing u, yet preserving the same
shape.
Such a study revealed that both the isotropic spin-1 /2
Heisenberg model and the half-filled 1D Hubbard model foranyfinite u=U/4tvalue and a suitable choice of units for
2469-9950/2021/103(19)/195129(24) 195129-1 ©2021 American Physical SocietyCARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
the spectra’s energy bandwidths describe the same spin dy-
namical properties. This suggests that the suitable values ofthe interaction for chain compounds and ultracold spin-1 /2
atom systems described by 1D Mott-Hubbard insulators couldbe settled by the agreement with experimental results on theup- and down-spin one-particle spectral functions, at energyscales above the Mott-Hubbard gap. In addition and, impor-tantly, they are the simplest dynamical correlation functionsthat involve excitation of both the charge and spin degrees
of freedom, being more sensitive to magnetic effects than thecharge dynamics [ 12].
For electrons, the one-particle spectral function de-
scribes at zero field spectral-weight distributions that canbe accessed by angle-resolved photoemission spectroscopy(ARPES) [ 13–15] in low-dimensional Mott-Hubbard insu-
lators and corresponding doped insulators [ 16]. 1D Mott-
Hubbard insulators can be studied within condensed matterby inelastic neutron scattering in spin chains whose chargedegrees of freedom are gapped, as well as a number of quasi-1D organic compounds [ 17].
On the other hand, in the presence of a magnetic field,
electrons are deflected and the up- and down-spin one-particlespectral functions are not accessible via ARPES. Such spec-tral functions and corresponding interplay of correlations andmagnetic-field effects, though, refers to a physically interest-ing and involved nonperturbative many-particle problem thatis poorly understood. It could, in principle, be experimentallyaccessible in systems of ultracold spin-1 /2 atoms on optical
lattices [ 18–24].
In the case of the 1D half-filled Hubbard model at chemical
potential μ=0, there is for u=U/4t>0 no one-particle
spectral weight in the excitation energy range within the Mott-Hubbard gap 2 /Delta1
MH[5,25]. The spectral-weight distributions
of the removal and addition one-particle spectral functionsthus occur in ( k,ω)-plane regions ω<−/Delta1
MHandω>/Delta1 MH,
respectively.
Most studies of the 1D half-filled Hubbard model’s
dynamical correlation functions [ 26–28] and related prop-
erties [ 29–32] refer to zero magnetic field. Few of them
focused specifically on the one-particle spectral functions.In the case of zero and low temperatures, the latter con-sidered interaction values in the u=U/4t∈[1,2] range
and relied on time-dependent density-matrix renormalization
group (tDMRG) computations [ 26], a combination of Bethe
ansatz results, Lanczos diagonalizations, and field theoreticalapproaches [ 27], and the dynamical density-matrix renormal-
ization group (DDMRG) method [ 28].
Other studies of the half-filled 1D Hubbard model at
zero magnetic field concerned, for instance, static andone-particle dynamical quantities [ 29], the gapped optical
conductivity [ 30], the dynamical density-density correlation
function [ 31], and the Mott-Hubbard insulator’s enhancement
of lattice dimerization by Coulomb correlations [ 32]. On the
other hand, studies on half-filled 1D Hubbard model at finitemagnetic field refer to the low-energy limit [ 33].
Our goal is the study of the line shape of the up- and down-
spin one-particle spectral functions of the 1D Hubbard modelwith one fermion per site both at zero and finite magneticfields at and in the ( k,ω) plane’s vicinity of such functions’s
cusp singularities by means of a suitable dynamical theory.The latter refer to peaks in the vicinity of which most one-
particle spectral weight is located.
The Mott-Hubbard gap that separates the addition and
removal spectral functions is calculated for all spin den-sities and interaction values. Accounting for the relationB
σ,+1(k,ω)=B¯σ,−1(π−k,−ω) between the one-particle
addition Bσ,+1(k,ω) and removal B¯σ,−1(k,ω) spectral func-
tions, where here and below ¯ σis the spin projection opposite
toσ, for simplicity, our study focuses explicitly on the latter
function.
As shortly reported in Sec. IIand Appendix A,t h ei r r e -
ducible representations of the η-spin SU(2) symmetry group
are for the 1D Hubbard model naturally described by con-figurations of rotated fermions whose relation to fermionshas been uniquely defined in terms of the corresponding uni-tary operator’s matrix elements between all Hilbert space’senergy eigenstates in the case of electrons [ 34]. Indeed, the
symmetries of the Hubbard model on any bipartite latticeinclude such an η-spin SU(2) symmetry, the corresponding
η-spins of projection −1/2 and +1/2 referring to SU(2) η-
spin symmetry degrees of freedom of sites doubly occupiedand unoccupied, respectively, by rotated fermions [ 34–36].
Before studying the Mott-Hubbard insulator’s spectral
function line shape near their cusp singularities, we ac-count for such a SU(2) symmetry irreducible representationsto show that the one-particle addition upper Hubbard band(UHB) stems from transitions to excited states of (i) theMott-Hubbard insulator and (ii) doped Mott-Hubbard with asmall yet finite deviation from fermionic density 1 that have adifferent η-spin character. Such states refer to creation of one
ηspin of projection −1/2 (i) associated with an η-spin doublet
and (ii) that emerges within an η-spin singlet pair. [Property
(ii) applies to all densities of the metallic phase [ 34].]
Theηspins of projection −1/2 and +1/2, whose one-
particle state configurations are shown in this paper to bedifferent for nondoped and doped Mott-Hubbard insulators,control important physical effects associated with other typesof excitations. Specifically, they control third-harmonic gen-eration spectroscopy, which plays a key role in the realizationof all-optical switching, modulating, and computing devicesof modern optical technology [ 3,37]. Indeed, excited states
containing a pair of ηspins with opposite projection −1/2
and+1/2 are behind the anomalously enhanced third-order
nonlinear optical susceptibility χ
(3), as observed in the 1D
Mott-Hubbard insulator Sr 2CuO 3[37]. Within a description
in terms of the 1D half-filled Hubbard model, this followsfrom odd- and even-parity states being nearly degeneratewith a large transition dipole moment between them. Thatdegeneracy is due to the spin-charge separation [ 1]. For large
u, rotated fermions become fermions and ηs p i n so fp r o j e c -
tion−1/2 and +1/2 refer to specific degrees of freedom of
sites doubly occupied and unoccupied by unrotated fermions,which have been called doublons and holons, respectively [ 4].
A simplified u/greatermuch1 holon-doublon model reproduces very
well the characteristic behaviors of the experimental χ
(3),
including data from the third-harmonic generation spec-troscopy [ 1].
Conversely, ηspins of projection −1/2 and +1/2a r e
a generalization of doublons and holons, respectively, forthe whole u>0 range. This involves the replacement of
195129-2ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
theu/greatermuch1 fermion’s doubly occupied and unoccupied sites
by rotated-fermion’s doubly occupied and unoccupied sites,respectively, for u>0. In the present case of one-particle
excited states, our studies account for the qualitative differentη-spin configurations of nondoped and doped Mott-Hubbard
insulators. They rely on a corresponding extension to thegapped subspace associated with the one-particle excited en-ergy eigenstates of the 1D Hubbard model with one fermionper site of the dynamical theory for the quantum metallicphases of that model introduced in Refs. [ 38,39]. That theory
was used in Ref. [ 34] to calculate expressions for the line
shape at and near the up- and down-spin one-particle spectralfunctions’ cusp singularities in the case of the 1D Hubbardmodel in a magnetic field’s metallic phase. For the up- anddown-spin one-particle spectral functions of the half-filled1D Hubbard model, whose ground state refers to the Mott-Hubbard insulating quantum phase, the dynamical theory usedin this paper refers to a limiting case of that used in Ref. [ 34].
On the other hand, in Refs. [ 10,11], the dynamical the-
ory was used for a subspace of the half-filled 1D Hubbardmodel spanned by spin excited energy eigenstates. The Mott-Hubbard gap is not seen by spin excitations, which are gaplessat half filling. The corresponding gapless spin quantum prob-lem is qualitatively different from the gapped one-particleproblem studied in this paper. Indeed, for it the Bethe-ansatzcharge cbranch of excitations does not contribute to the
dynamical theory. In contrast, for the one-particle problem,both the charge cand spin sbranch of excitations contribute
to the dynamical theory and this applies both to the metallic
and Mott-Hubbard insulator quantum phases.
In the case of integrable models, the general dynami-
cal theories of Refs. [ 10,11,34,38,39] are equivalent to and
account for the same microscopic processes [ 40]a st h e
mobile quantum impurity model scheme of Refs. [ 41,42].
Momentum-dependent exponents in the expressions of spec-tral functions have also been obtained in Refs. [ 43,44].
The dynamical theory of Refs. [ 34,38,39] is a general-
ization to the whole u=U/4t>0 range of the approach
used in the u→∞ limit in Refs. [ 45,46]. In that limit, the
Bethe-ansatz solution simplifies, which makes possible thederivation of several one-particle quantities [ 47,48].
The paper is organized as follows. In Sec. II, the model
and its up- and down-spin one-particle spectral functions areintroduced and the spin-density and udependencies of the
Mott-Hubbard gap are studied. The rotated fermions, corre-sponding fractionalized particles, and the differences relativeto the doped Mott-Hubbard insulator are the issues addressedin Sec. III. In Sec.
IV, the spectra and the general expressions
of the spectral functions at and near their cusp singularities areintroduced. The zero spin-density spectra and exponents arestudied in Sec. V. In Sec. VI, the specific line shapes at and
in the vicinity of the branch and boundary lines of the up- anddown-spin one-particle spectral functions are calculated forfinite spin density. The effects of the charge-spin separationand charge-spin recombination are the issues discussed inSec. VII. Finally, the discussion of the results and the con-
cluding remarks are presented in Sec. VIII. Some useful sideresults and derivations needed for our studies are presented in
three Appendices.
II. THE MODEL, THE SPECTRAL FUNCTIONS, AND
THE mDEPENDENCE OF THE MOTT-HUBBARD GAP
The Hubbard model with one fermion per site in a magnetic
field hunder periodic boundary conditions on a 1D lattice with
an even number Na→∞ of sites is given by
ˆH=ˆHH+2μBhˆSz
s,where ˆHH=tˆT+UˆVD,(1)
and
ˆT=−/summationdisplay
σ=↑,↓L/summationdisplay
j=1(c†
j,σcj+1,σ+c†
j+1,σcj,σ),
ˆVD=L/summationdisplay
j=1ˆρj,↑ˆρj,↓;ˆρj,σ=c†
j,σcj,σ−1/2(2)
are the kinetic-energy operator in units of tand the on-site
repulsion operator in units of U, respectively. The oper-
ator c†
j,σ(and cj,σ) creates (and annihilates) a σ=↑,↓
fermion (electron or atom) at lattice site j=1,...,Na.T h e
fermion number operators read ˆN=/summationtext
σ=↑,↓ˆNσand ˆNσ=/summationtextNa
j=1ˆnj,σ=/summationtextNa
j=1c†
j,σcj,σ. The model, Eq. ( 1), describes
N=N↑+N↓=Nainteracting spin-1 /2 fermions in a lat-
tice with Na→∞ sites for spin densities m=(N↑−N↓)/
Na∈[0,1[.
We use in general units of lattice constant and Planck’s
constant one, the number of lattice sites Na=N↑+N↓equal-
ing the lattice length L.I nE q .( 1),μBis the Bohr magneton,
for simplicity, in gμBwe have taken the Landé factor to
read g=2, and the operator ˆSz
s=−1
2/summationtextNa
j=1(ˆnj,↑−ˆnj,↓)i st h e
diagonal generator of the model global spin SU(2) symmetry,where ˆ n
j=/summationtext
σˆnj,σ. The model has also a global η-spin
SU(2) symmetry with diagonal generator ˆSz
η=−1
2/summationtextNa
j=1(1−
ˆnj). We denote by SsandSz
sthe energy eigenstates’s spin
and spin projection and by SηandSz
ηsuch states’s ηspin and
η-spin projection.
The global symmetry of ˆHHfor 1D and any bipar-
tite lattice [ 36]i sa t U/negationslash=0 given by [SO(4) ⊗U(1)]/Z2=
[SU(2) ⊗SU(2) ⊗U(1)]/Z2
2.H e r e1 /Z2
2refers to the num-
ber 4Naof its irreducible representations, which refer
to configurations of fractionalized particles considered inAppendix Aand correspond to the 4
Naenergy eigenstates
ofˆH, being four times smaller than the dimension 4Na+1
of the group SU(2) ⊗SU(2) ⊗U(1). The global clattice
U(1) symmetry beyond SO(4) =[SU(2) ⊗SU(2)] /Z2is as-
sociated with the lattice degrees of freedom and does notexist at U=0, emerging at any arbitrarily small u=U/4t
value [ 34–36].
The lowest weight states (LWSs) and highest weight states
(HWSs) of the ηspin ( α=η) and spin ( α=s)S U ( 2 )
symmetry algebras have numbers S
α=−Sz
αand Sα=Sz
α,
respectively.
195129-3CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
Our studies involve one-particle excited states of ground states with vanishing chemical potential μ=0, so their zero-energy
reference level lies in the middle of the Mott-Hubbard gap 2 /Delta1MH. The up- and down-spin one-particle spectral functions
Bσ,γ(k,ω) of the 1D half-filled Fermionic Hubbard model in a magnetic field hwhere γ=−1 (and γ=+1) for one-particle
removal (and addition) then read
Bσ,−1(k,ω)=/summationdisplay
ν−|/angbracketleftν−|ck,σ|GS/angbracketright|2δ(ω+/epsilon1σ,ν−(k)) for ω/lessorequalslant−/Delta1MH,
Bσ,+1(k,ω)=/summationdisplay
ν+|/angbracketleftν+|c†
k,σ|GS/angbracketright|2δ(ω−/epsilon1σ,ν+(k)) for ω/greaterorequalslant/Delta1MH,where σ=↑,↓. (3)
Here ck,σandc†
k,σare fermion annihilation and creation operators, respectively, of momentum k.|GS/angbracketrightdenotes the initial
N↑,N↓particle ground state of energy ENσ,N¯σ
GS.T h eν−andν+summations run over the Nσ−1 and Nσ+1-particle excited energy
eigenstates, respectively. ENσ−1,N¯σ
ν− andENσ+1,N¯σ
ν+ are in /epsilon1σ,ν−(k)=(ENσ−1,N¯σ
ν− −ENσ,N¯σ
GS) and/epsilon1σ,ν+(k)=(ENσ+1,N¯σ
ν+ −ENσ,N¯σ
GS)t h e
corresponding energies.
Since the chemical potential, μ=0, lies at the middle of the Mott-Hubbard gap, the following exact symmetry:
Bσ,+1(k,ω)=B¯σ,−1(π−k,−ω)f o r σ=↑,↓,with ¯↑=↓ and ¯↓=↑ (4)
holds. We also rely on the following symmetry that refers to
the spin density intervals m∈[−1,0] and m∈[0,1]:
Bσ,−1(k,ω)|m=B¯σ,−1(k,ω)|−mform∈[0,1].(5)
We thus only consider explicitly the removal function
Bσ,−1(k,ω) for the spin density interval m∈[0,1].
The Mott-Hubbard gap 2 /Delta1MHplays an important role in
the one-particle spectral properties. Its general expressionfor general spin densities, m∈[0,1], has not been given
explicitly in the literature. Relying on the Bethe-ansatz rep-resentation reported in Appendix B, we have derived it in
Appendix Cwith the result
2/Delta1
MH=U−4t+/integraldisplayB
−Bd/Lambda12tηs(/Lambda1)/Phi1(/Lambda1)+2μBh
=U−4t+/integraldisplay∞
−Bd/Lambda12tηs(/Lambda1)/Phi1(/Lambda1),where
/Phi1(/Lambda1)=2
πarctan/parenleftbigg/Lambda1
u/parenrightbigg
for/Lambda1< B
=1f o r /Lambda1> B. (6)
Here the distribution 2 tηs(/Lambda1) is solution of the coupled inte-
gral equations, Eqs. ( B6) of Appendix B, and the parameter
Bis defined in Eq. ( B8) of that Appendix. At m=0, its
expression simplifies and is well known [ 5,6,25]. At m=0
and in the m→1 limit, it reads
2/Delta1MH=U−4t+8t/integraldisplay∞
0dωJ1(ω)
ω(1+e2ωu)
=16t2
U/integraldisplay∞
1dω√
ω2−1
sinh/parenleftbig2πtω
U/parenrightbigatm=0 and
=/radicalbig
(4t)2+U2−4tform→1, (7)respectively, where J1(ω) is a Bessel function. This gives
2/Delta1MH≈8√
tU
πe−2π(t
U)foru/lessmuch1 and m=0,
≈U2
8tforu/lessmuch1 and m→1,
≈U−4tforu/greatermuch1 and m∈[0,1[. (8)
The Mott-Hubbard gap, Eqs. ( 6) and ( 7), is plotted in
Fig. 1(a) as a function of u=U/4tfor spin densities m=0
andm=1. Although it is very little udependent, note that
as shown in the inset of that figure, there is a small de-viation 2 /Delta1
MH(1)−2/Delta1MH(0) as a function of u=U/4tfor
spin densities m=0 and m=1. That deviation reaches a
maximum value at u=U/4t=u∗∼1. The Mott-Hubbard
gap 2/Delta1MH(m) is for all spin densities m∈[0,1] an increas-
ing function of u. To compare its dependence on the spin
density mconveniently, we present that gap deviation from
itsm=0 value in Fig. 1(b) for three representative uval-
ues. As the intersect of the lines for u=0.4 and u=5.0
reveals, for smaller mthe deviation 2 /Delta1MH(m)−2/Delta1MH(0)
is larger for u=5.0 than for u=0.4 and the opposite
FIG. 1. (a) The Mott-Hubbard gap 2 /Delta1MH,E q .( 6), in units of t
as a function of u=U/4tfor spin densities m=0a n d m=1a n d
(b) the small deviation 2 /Delta1MH(m)−2/Delta1MH(0) in units of twhere
2/Delta1MH(0) corresponds to m=0 as a function of the spin density m
foru=0.4,1.0,5.0. The inset in (a) shows 2 /Delta1MH(1)−2/Delta1MH(0) as
a function of u.
195129-4ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
occurs for mlarger than approximately 0.45. However, we
recall that 2 /Delta1MH(m)|u=5.0>2/Delta1MH(m)|u=1>2/Delta1MH(m)|u=0.4
for all spin densities m∈[0,1].
That the Mott-Hubbard small mdependence is more pro-
nounced at intermediate u=U/4t∼1 values is consistent
with that gap having the same values for all spin densitiesm∈[0,1] in the u→0 and u/greatermuch1 limits in which it reads
2/Delta1
MH=0 and 2 /Delta1MH=U−4t, respectively. 2 /Delta1MHis asso-
ciated with charge degrees of freedom, its little dependenceonmbeing consistent with the u>0 charge-spin separation
being strongest at half filling.
The interval m∈[0,1[ refers to fields in the range h∈
[0,h
c[, where hcis the critical field for m→1 fully polarized
ferromagnetism. The spin density curve h(m)∈[0,hc[f o r
m∈[0,1[ and hcare given by
h(m)=−ε0
s(kF↓)
2μB,hc=/radicalbig
(4t)2+U2−U
2μB, (9)
respectively. Here the bare s-band energy dispersion ε0
s(q)i s
defined in Eqs. ( C8) of Appendix C. In the thermodynamic
limit, one has that the momentum kF↓inε0
s(kF↓) and kF↑=
π−kF↓are for m∈[0,1[ given by
kF↓=π
2(1−m),kF↑=π
2(1+m). (10)
III. DIFFERENT ONE-PARTICLE PROCESSES OF THE
DOPED AND UNDOPED MOTT-HUBBARD INSULATOR
The dynamical theory used in our studies relies on a rep-
resentation in terms of fractionalized particles that naturallyemerge from the rotated-fermion degrees of freedom sepa-ration [ 10,11,34,38,39]. It is briefly described for the whole
Hilbert space in Appendix A. This is useful for the intro-
duction of its simplified form for the Mott-Hubbard insulatorsubspace of our study, which is spanned by energy eigen-states described only by real Bethe-ansatz rapidities. Theyare populated by cparticles, sparticles, unpaired spin 1 /2’s
of projection +1/2, and zero or one unpaired η-spin 1 /2o f
projection +1/2. Only the candsbands whose Bethe-ansatz
quantum numbers q
j=2π
LIc
jandqj=2π
LIs
j, respectively, are
given in Eqs. ( B2) and ( B3) of Appendix Bhave finite occu-
pancy.
In contrast to the doped Mott-Hubbard insulator and gen-
eral metallic phase, the states that span that subspace are notpopulated by the ηparticles considered in Appendix B, which
also refer to real Bethe-ansatz rapidities. The Bethe-ansatzequations and quantum numbers are given in Eqs. ( B1)–(B3)
of Appendix B. In the thermodynamic limit for u>0 and
m/greaterorequalslant0, the ground-state s-band Fermi points and limiting mo-
mentum values read ±k
F↓and±kF↑, respectively, Eq. ( 10).
Mott-Hubbard insulator ground states are for m∈[0,1[
not populated by η-spin 1 /2’s whose number is denoted by
Mη,±1/2for projections ±1/2, are populated by a number
Nc=N=Naofcparticles without internal degrees of free-
dom, a number Ns=/Pi1s=N↓ofsparticles whose internal
degrees of freedom refer to one unbound spin-singlet pairof spin 1 /2’s, and a number M
s,+1/2=N↑−N↓of unpaired
spin 1 /2’s of projection +1/2. The translational degrees of
freedom of such unpaired spin 1 /2s are described by the Nh
s=
2Ss=N↑−N↓s-band holes. The m=0 ground state forwhich Ms=Ms,+1/2+Ms,−1/2=2Ss=0 is not populated by
unpaired spins 1 /2 and thus has no s-band holes.
Ground states of a doped Mott-Hubbard insulator with
concentration δ=|1−nf|very small yet finite are populated
by a number Ncofcparticles and Nsofsparticles that for spin
LWSs for which m/greaterorequalslant0a r eg i v e nb y
Nc=Nand Ns=N↓fornf=N/Na∈[0,1],
Nc=Nhand Ns=Nh
↓fornf∈[1,2]. (11)
Here,
Nh=2Na−N,Nh
↑=Na−N↓,Nh
↓=Na−N↑(12)
are the numbers of fermionic holes that are more suitable
to describe the quantum problem for nf∈[1,2]. Indeed, the
corresponding fermionic hole density nh
f=Nh/L=Nh/Na
varies in the range nh
f∈[0,1]. (Each site has two orbitals,
Nh=2Na−Nbeing the number of orbitals that are not oc-
cupied by fermions and thus refer to fermionic holes.) Thespin densities of such ground states are m∈[0,n
f[f o r nf∈
[0,1] and m∈[0,nh
f[f o r nf∈[1,2] where nh
f=Nh/Na.T h e
present representation Ls=2Ss+2/Pi1sspins 1 /2a r ef o r nf∈
[1,2] those of the Nhrotated fermionic holes that single oc-
cupy sites. For nf∈[0,2], there are Ms=2Ss=N↑−N↓=
Nh
↑−Nh
↓unpaired spin 1 /2’s whose translational degrees of
freedom are described by Nh
s=2Ss=N↑−N↓=Nh
↑−Nh
↓s-
band holes. In the ground states of the doped case, thereareM
η=2Sη=Na−Nunpaired ηspins of projection +1/2
fornf∈[0,1] and Mη=2Sη=N−Naunpaired η-spins of
projection −1/2f o r nf∈[1,2].
To address the qualitative differences of the one-particle
problem for the Mott-Hubbard insulator and the correspond-ing doped insulator, respectively, we start by consideringfermionic densities n
f∈[0,2] and a Hamiltonian ˆH+2μˆSz
η.
Here ˆHis given in Eq. ( 1) andμis the chemical potential.
The ranges nf∈[0,1] and nf∈[1,2] refer to the η-spin
LWS and HWS Bethe ansatzes and thus to subspaces spannedby energy eigenstates that are S
z
η=−Sηη-spin LWSs and
Sz
η=Sηη-spin HWSs, respectively. An useful symmetry rela-
tion that connects the σ=↑,↓one-particle spectral functions
for fermionic densities nf<1 and n/prime
f=2−nf>1, respec-
tively, is
Bσ,γ(k,ω)|n<1=B¯σ,−γ(π−k,−ω)|n/prime=2−n>1. (13)
Fornf→1 and thus nh
f=2−nf→1, this general relation
gives that provided in Eq. ( 4).
Theσ=↑,↓one-particle spectral functions obey the fol-
lowing sum rules:
/summationdisplay
k/integraldisplay∞
−∞dωBσ,−1(k,ω)=Nσ,
/summationdisplay
k/integraldisplay∞
−∞dωBσ,+1(k,ω)=Na−Nσ,
(14)/summationdisplay
k/integraldisplay∞
−∞dωBLHB
σ,+1(k,ω)=Na−N,
/summationdisplay
k/integraldisplay∞
−∞dωBUHB
σ,+1(k,ω)=N−Nσ=N¯σ
195129-5CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
fornf∈[0,1] and
/summationdisplay
k/integraldisplay∞
−∞dωBσ,−1(k,ω)=Na−Nh
σ,
/summationdisplay
k/integraldisplay∞
−∞dωBσ,+1(k,ω)=Nh
σ,
(15)/summationdisplay
k/integraldisplay∞
−∞dωB−LHB
σ,−1(k,ω)=N−Na,
/summationdisplay
k/integraldisplay∞
−∞dωB−UHB
σ,−1(k,ω)=Nh−Nh
σ=Nh
¯σ
fornf∈[1,2]. Here the symbols’ lower Hubbard band (LHB)
and UHB (and −LHB and −UHB) refer for nf∈[0,1[ (and
nf∈]1,2]) to one-particle addition (and removal) for ω> 0
(andω< 0.)
The LHB (and −LHB) is generated by transitions to ad-
dition (and removal) excited energy eigenstates that are notpopulated by ηspins of projection −1/2 (and +1/2). The
UHB (and −UHB) is generated by transitions to such excited
states that are populated by ηspins of projection −1/2 (and
+1/2). Nearly all UHB (and −UHB) spectral weight stems
from transitions to addition (and removal) excited energyeigenstates that are populated by a single ηspin of projection
−1/2[34] (and +1/2.)
The first two sum rules in Eqs. ( 14) and ( 15)a r ew e l l
known and exact for all uvalues. The (i) B
LHB
σ,+1(k,ω) and
BUHB
σ,+1(k,ω) sum rules and (ii) B−LHB
σ,−1(k,ω) and B−UHB
σ,−1(k,ω)
sum rules are for u>0 found to be exact in the (i) nf→0
andnf→1 limits and (ii) nh
f→0 and nh
f→1 limits, re-
spectively. Both in the u/lessmuch1 and u/greatermuch1 limits, they are exact
for all fermionic densities nf∈[0,2] and all corresponding
spin densities. They are exact or a very good approxima-tion also for intermediate uvalues. Clarification of this issue
is not needed for our studies, since for the Mott-Hubbardinsulator and the doped Mott-Hubbard insulator for whichδ=|1−n
f|/lessmuch1 is finite but very small all such sum rules
apply.
For simplicity, in general, in this paper we consider
spin densities m/greaterorequalslant0 associated with spin LWSs such that
δMs,−1/2=0, yet similar results are obtained for m/lessorequalslant0. The
processes associated with one-particle removal (REM) forn
f/lessorequalslant1 and one-particle addition for nf/greaterorequalslant1 are similar for the
doped Mott-Hubbard insulator and the present nf=nh
f=1
Mott-Hubbard insulator. (See the REM fractionalized parti-cles numbers deviations in Table Iforn
f∈[0,1[. For how
they relate to those of addition for nf∈]1,2], see that table
caption.)
A first qualitative difference is that the one-particle addi-
tion LHB for nf∈[0,1[ and the one-particle removal −LHB
fornf∈]1,2] do not exist for the Mott-Hubbard insulator.
As justified in the following, the processes associated withthe UHB one-particle addition for n
f/lessorequalslant1 and −UHB one-
particle removal for nf/greaterorequalslant1 are also qualitatively different
for the doped Mott-Hubbard insulator and the Mott-Hubbardinsulator.
Let|ν
+,0,N/angbracketrightand|ν−,0,N/angbracketrightdenote energy eigenstates
of the Hamiltonian ˆH+2μˆSz
η, where the upper index 0TABLE I. Fractionalized particle number deviations for one-
particle types I and II UHB addition, LWS addition, and removal
fornf∈[0,1]. Similar values apply for types I and II −UHB re-
moval, −LWS removal, and addition for nf∈[1,2] provided that the
numbers δNσandδMη,∓1/2of rotated-fermion unoccupied ( +1/2)
and doubly occupied ( −1/2) sites are replaced by those for δNh
σand
δMη,±1/2, respectively. (For excited states of m=0 ground states, the
LHB deviations for δN↓=1 and the REM deviations for δN↑=−1
should be replaced by those for δN↑=1a n dδN↓=−1, respectively,
with the number δMs,−1/2of rotated-fermion singly occupied sites of
spin projection −1/2 replaced by δMs,1/2.)
δNcδNsδNηδMη,−1/2δMη,+1/2δMs,1/2
IUHBδN↑=1−1−10 1 0 1
IIUHBδN↑=1−1−11 0 −11
IUHBδN↓=1−10 0 1 0 −1
IIUHBδN↓=1−10 1 0 −1 −1
LHBδN↑=11 0 0 0 −11
LHBδN↓=11 1 0 0 −1 −1
REMδN↑=−1−10 0 0 1 −1
REMδN↓=−1−1−10 0 1 1
indicates they are η-spin LWSs and HWSs for upper indices
+and−, respectively. They thus refer to two correspond-
ing Bethe-ansatz solutions for nf∈[0,1[ and nf∈]1,2],
respectively, and u>0. Their label νrepresents u=U/4t
and the set of all u-independent quantum numbers other
than Nneeded to uniquely specify an energy eigenstate.
This refers to occupancy configurations of Bethe-ansatz
momentum quantum numbers qj=2π
LIβ
j.H e r e Iβ
jare succes-
sive integers, Iβ
j=0,±1,±2,..., or half-odd integers ,Iβ
j=
±1/2,±3/2,±5/2,..., according to well-defined boundary
conditions. Their allowed Pauli-like occupancies are zero
and one. The index βlabels the Bethe-ansatz band β=
c,s,η,η n,sn, where n>1a r e n-string lengths. We denote by
|GS+,0,N/angbracketrightand|GS−,0,N/angbracketrightthe ground states that are η-spin
LWSs and HWSs for nf∈[0,1[ and nf∈]1,2], respectively.
Most of the UHB and −UHB spectral weight results
from transitions from the ground states |GS+,0,N/angbracketrightand
|GS−,0,N/angbracketrightto energy eigenstates populated by a single η-spin
of projections −1/2 and +1/2, respectively. There are three
qualitatively different types of such states:
Type I: The single η-spin of projections −1/2 and +1/2
is unpaired and is not generated from an η-spin of projection
+1/2 and−1/2, respectively, by a η-spin flip process.
Type II: The single η-spin is paired within an η-spin singlet
pair.
Type III: The single η-spin of projection −1/2 and+1/2i s
unpaired and is generated from an η-spin of projection +1/2
and−1/2, respectively, by an η-spin flip process.
Our goal here is to confirm that the type-I and -II excited
states are those that play an active role in the cases of theMott-Hubbard insulator and doped Mott-Hubbard insulatorfor which δ=|1−n
f|/lessmuch1 is finite but very small, respec-
tively. The I UHB and II UHB fractionalized particle number
deviations are given in Table Ifor the UHB type-I and -II
excited states, respectively. That table caption reports howthey relate to those of the −UHB.
195129-6ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
In the case of a doped Mott-Hubbard insulator for which
δ=|1−nf|/lessmuch1 is finite but very small, creation of a single
unpaired η-spin of projection −1/2 and +1/2 under UHB
creation of one fermion for nf<1 and−UHB annihilation of
one fermion for nf>1, respectively, can only occur through
transitions from the ground state to type-II or type-III ex-cited states. Indeed, it turns out that type-I excited states arepopulated by a single η-spin 1 /2 whose projection is −1/2
and+1/2 for fermion creation and annihilation, respectively.
Type-I excited states thus do not exist in the subspace ofthe doped Mott-Hubbard insulator under consideration. Ourfollowing analysis thus involves matrix elements between thatinsulator’s ground state and type-II and type-III excited states,respectively.
We start by considering transitions to the latter type-III
UHB and −UHB excited energy eigenstates of the doped
Mott-Hubbard insulator, which we denote by |ν
+
UHB,N+1/angbracketright
and|ν−
UHB,N−1/angbracketright, respectively. The absence of the upper
label 0 means they are not η-spin LWSs and HWSs, respec-
tively. They can be written as
|ν+
UHB,N+1/angbracketright=1√Na−N+1ˆS+
η|ν+,0,N−1/angbracketright,
|ν−
UHB,N−1/angbracketright=1√N−Na+1ˆS−
η|ν−,0,N+1/angbracketright.(16)
These type-III excited states are η-spin non-LWSs ( +) [and
non-HWSs ( −)] of the doped Mott-Hubbard insulator that are
generated from corresponding η-spin LWSs ( +)|ν+,0,N−1/angbracketright
[and HWSs ( −)|ν−,0,N+1/angbracketright]. The latter state’s η-spin is
such that 2 Sη=Na−N+1>0 (and 2 Sη=N−Na+1>
0) and thus do not exist for the Sη=0 Mott-Hubbard insula-
tor. They are populated by N−1 (and N+1) fermions and
byMη=2Sη=|Na−N+1|unpaired η-spins of projection
+1/2 (and −1/2) and are not populated by ηspins of projec-
tion−1/2 (and +1/2.)
The off-diagonal generators of the η-spin SU(2) symmetry
algebra appearing in Eqs. ( 16)a r eg i v e nb y[ 34,35,49]
ˆS+
η=Na/summationdisplay
j=1(−1)jc†
j,↓c†
j,↑,ˆS−
η=Na/summationdisplay
j=1(−1)jcj,↑cj,↓ (17)
or, equivalently, in terms of momentum k:
ˆS+
η=/summationdisplay
kc†
π−k,↓c†
k,↑,ˆS−
η=/summationdisplay
kck,↑cπ−k,↓. (18)
The effect of ˆS±
ηin Eq. ( 16) is to produce an η-spin flip that
transforms one unpaired η-spin of projection ±1/2 into one
unpaired η-spin of projection ∓1/2.
We consider the following matrix elements associated with
transitions from the N-fermion ground states to the type-III
excited states |ν+
UHB,N+1/angbracketrightand|ν−
UHB,N−1/angbracketright:
/angbracketleftν+
UHB,N+1|c†
k,σ|GS+,0,N/angbracketright
=1√Na−N+1/angbracketleftν+,0,N−1|ˆS−
ηc†
k,σ|GS+,0,N/angbracketright
=1√Na−N+1/angbracketleftν+,0,N−1|[ˆS−
η,c†
k,σ]|GS+,0,N/angbracketright
=−γσ√Na−N+1/angbracketleftν+,0,N−1|cπ−k,¯σ|GS+,0,N/angbracketright(19)and
/angbracketleftν−
UHB,N−1|ck,σ|GS−,0,N/angbracketright
=1√N−Na+1/angbracketleftν−,0,N+1|ˆS+
ηck,σ|GS−,0,N/angbracketright
=1√N−Na+1/angbracketleftν−,0,N+1|[ˆS+
η,ck,σ]|GS−,0,N/angbracketright
=γσ√N−Na+1/angbracketleftν−,0,N+1|c†
π−k,¯σ|GS−,0,N/angbracketright,(20)
where the bra representation of the states, Eqs. ( 16), was used
and the coefficient γσis forσ=↑,↓given by
γ↑=+1 and γ↓=−1, (21)
respectively. From the use of the relations, Eqs. ( 19) and ( 20),
one finds for a doped Mott-Hubbard insulator with very smallyet finite concentration δ=|(1−n
f)|for which 1 /(Na−
N+1)≈1/Naδand 1/(N−Na+1)≈1/Naδthe following
matrix elements square ratios:
|/angbracketleftν+
UHB,N+1|c†
k,σ|GS+,0,N/angbracketright|2
|/angbracketleftν+,0,N−1|cπ−k,¯σ|GS+,0,N/angbracketright|2=1
Naδ,
|/angbracketleftν−
UHB,N−1|ck,σ|GS−,0,N/angbracketright|2
|/angbracketleftν−,0,N+1|c†
π−k,¯σ|GS−,0,N/angbracketright|2=1
Naδ. (22)
Such ratios then vanish in the thermodynamic limit.
On the other hand, the type-II UHB and −UHB excited
energy eigenstates |ν+,0
UHB,N+1/angbracketrightand|ν−,0
uhb,N−1/angbracketrightof such a
doped insulator are η-spin LWSs and HWSs, respectively. In
Eqs. ( 14), both the sum rules of B¯σ,−1(k,ω) (with σreplaced
by ¯σ) and BUHB
σ,+1(k,ω) give exactly the same number value
N¯σ.I nE q s .( 15), the sum rules of B¯σ,+1(k,ω) (again with σ
replaced by ¯ σ) and B−UHB
σ,−1(k,ω) also give the same number
value Nh
¯σ. One then finds that
|/angbracketleftν+,0,N−1|cπ−k,¯σ|GS+,0,N/angbracketright|2
|/angbracketleftν+,0
UHB,N+1|c†
k,σ|GS+,0,N/angbracketright|2≈1,
|/angbracketleftν−,0,N+1|c†
π−k,¯σ|GS−,0,N/angbracketright|2
|/angbracketleftν−,0
UHB,N−1|ck,σ|GS−,0,N/angbracketright|2≈1. (23)
That such k-dependent ratios are finite in the thermodynamic
limit is what matters for our analysis. On average, they aregiven approximately by one, as given here.
Combining the relations, Eqs. ( 23), with those given in
Eqs. ( 19) and ( 20), one then finds that
|/angbracketleftν
+
UHB,N+1|c†
k,σ|GS+,0,N/angbracketright|2
|/angbracketleftν+,0
UHB,N+1|c†
k,σ|GS+,0,N/angbracketright|2≈1
Naδ,
(24)
|/angbracketleftν−
UHB,N−1|ck,σ|GS−,0,N/angbracketright|2
|/angbracketleftν−,0
UHB,N−1|ck,σ|GS−,0,N/angbracketright|2≈1
Naδ
for very small yet finite concentration δ=|(1−nf)|. That
the ratios in this equation vanish in the thermodynamic limit,confirms that for the doped insulator the type-III UHB and−UHB excited states do not contribute to the one-particle
spectral-weight distributions.
Since no type-I excited states exist for the doped insulator,
the dominant processes that are associated with the UHB
195129-7CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
creation of one fermion for nf<1 and with the −UHB an-
nihilation of one fermion for nf>1 refer to type-II excited
states.
On the other hand, both type II and type III excited states
are not allowed for the Mott-Hubbard insulator whose groundstates are not populated by η-spins 1 /2. Its UHB and −UHB
are rather generated by transitions to type I excited statesthat are populated by one unpaired η-spin with projection
−1/2 and+1/2, respectively. They are not created by η-spin
flips, as in Eqs. ( 16) for the type-III excited states. For the
Mott-Hubbard insulator, there exist only the UHB and −UHB
that refer to the addition and removal spectral functions, re-spectively, which are related as given in Eq. ( 4).
Only for excited states of ground states for which |N
a−N|
is finite and δ=|(1−nf)|is of order 1 /Naand thus vanishes
in the thermodynamic limit, the UHB for N<Naand the
−UHB for N>Naresult from transitions to both excited
states of type II and type III.
The UHB one-particle processes reported here for a doped
Mott-Hubbard insulator with nf<1 are the same as those of
the metallic phase already studied in Ref. [ 34]. This is why,
in this paper, we limit our study to the up- and down-spinone-particle spectral functions, Eqs. ( 3), of the Mott-Hubbard
insulator.
An important qualitative difference of the dynamical the-
ory version suitable to the Mott-Hubbard insulator used in thestudies of this paper relative to that of the metallic phase usedin the studies of Ref. [ 34] is thus that the excited energy eigen-
states populated by ηparticles do not contribute to the line
shape near the ( k,ω)-plane cusp singularities under study. As
reported in the following, also the dynamical theory’s spectralparameters and candsparticle’s phase shifts have specific
values and uandmdependencies for the Mott-Hubbard insu-
lator, different from those of its version suitable to the metallicphase.
Due to the symmetry, Eq. ( 4), our study refers explicitly
to one-particle removal. As given in Table I, for both cases
of up- and down-spin one-particle removal, one cparticle is
removed and one unpaired η-spin 1 /2 of projection +1/2i s
created. Within removal of one down-spin fermion for m>0,
the unbound spin-singlet pair of one sparticle is broken along
with its annihilation and one unpaired spin 1 /2 of projec-
tion+1/2 is created. The spin 1 /2 of projection −1/2 that
also originates from the s-particle annihilation recombines
with the annihilated cparticle within the removed down-spin
fermion. For removal of one up-spin fermion for m>0,
the number of sparticles remains unchanged and one un-
paired spin 1 /2 of projection +1/2 is removed. It recombines
with the annihilated cparticle within the removed up-spin
fermion.
As reported in the caption of Table I, some of the numbers
provided in it are different for excited states of m=0 ground
states. Then the unbound spin-singlet pair of one sparticle
is broken along with its annihilation and one unpaired spin1/2 of projection +1/2 and−1/2 is created under removal of
one down-spin fermion and one up-spin fermion, respectively.The spin 1 /2 of projection −1/2 and+1/2 left over that also
originates from the sparticle annihilation recombines with the
annihilated cparticle within the removed down-spin fermion
and up-spin fermion, respectively.FIG. 2. The ( k,ω)-plane regions defined by the spectrum /epsilon1(k),
Eq. ( 38), where for spin density m=0a n d( a ) u=1.0, (b) u=1.75,
(c)u=1.935, and (d) u=2.0, there is in the thermodynamic limit
more spectral weight in the removal one-particle spectral function
of the 1D Hubbard model with one fermion per site. The c,c/prime,
andsbranch lines whose spectra are given in Eqs. ( 40) are repre-
sented by solid lines and the c−sboundary line by a dashed-dotted
line. The ( k,ω)-plane distributions presented here and in Figs. 3
and 6–12do not provide information on the relative amount of
spectral weight contained within each spectrum’s colored contin-
uum, most weight being actually located in the vicinity of thebranch lines and boundary line. For the above intermediate uval-
ues, the present one-particle spectra were studied by other methods
and other authors. They are to be compared with those plotted in( a )F i g .3 ( a )o fR e f .[ 26]f o r U/t=4.0( a n d u=U/4t=1.0) cal-
culated with tDMRG, (b) Figure 7(b) of Ref. [ 27]f o r U/t=7.0
(and u=1.75) derived combining Bethe-ansatz results, Lanczos
diagonalizations, and field theoretical approaches, (c) Fig. 10 of
Ref. [ 28]f o r U/t=7.74 (and u=1.935) calculated with DDMRG,
and (d) Fig. 1(a) of Ref. [ 26]f o r U/t=8.0( a n d u=2.0) obtained
with tDMRG.
IV . THE SPECTRA AND THE SPECTRAL FUNCTIONS
NEAR THEIR SINGULARITIES
For simplicity, we do not provide here the details of the
present dynamical theory that are common to those alreadygiven in Ref. [ 34] for the metallic case. The theory applies
to several dynamical correlation functions. In the case of thepresent quantum problem, it provides the line shape of the up-and down-spin one-particle spectral functions, Eqs. ( 3), at and
in the vicinity of two types of singular spectral features calledbranch lines and boundary lines, respectively. Those are themost important spectral features of such functions.
A. The spectra of the excitations behind most
one-particle spectral weight
For an initial insulator ground state, any transition under
which the deviation δNcis finite refers to a gapped excita-
tion spectrum, consistent with the term −/Delta1MHin the c-band
energy dispersion in Eq. ( B4) of Appendix B. This applies
to the removal up- and down-spin one-particle spectra forwhich δN
c=−1, as given in Table Ifor one-particle removal
(REM) excited energy eigenstates. Those are the states that
195129-8ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
span the present subspaces for which δMη,+1/2=δNh
c=1
andδMs,+1/2=δNh
s=1 and=−1f o rδN↓=−1 andδN↑=
−1, respectively. Due to the perturbative character of the
present quantum problem in terms of the number of ele-mentary fractionalized-particle processes [ 34], transitions to
one-particle removal excited energy eigenstates associatedwithδN
c<0 deviations such that |δNc|>1 lead to very littlespectral weight and do not contribute to the line shape at and
near the one-particle spectral function singularities studied inthis paper.
Within a kextended zone scheme, the −/epsilon1
↓(k)<0 and
−/epsilon1↑(k)<0 spectra of such excited states that generate
most of the down- and up-spin one-particle removal spectralweight, respectively, has the general form
/epsilon1↓(k)=−εc(q)−εs(q/prime),where k=ιπ−q−q/primeandι=±1f o r q∈[−π,π ] and q/prime∈[−kF↓,kF↓], (25)
and
/epsilon1↑(k)=−εc(q)+εs(q/prime),where k=−q+q/primeforq∈[−π,π ] and |q/prime|∈[kF↓,kF↑], (26)
respectively. The energy dispersions εc(q) andεs(q/prime) appear-
ing here are defined in Eq. ( B4) of Appendix B. The number
ofsband holes is, in the present subspace, given by Nh
s=
Nc−2Ns. This leads to δNh
s=−1 for up-spin one-particle
removal for which δNc=−1 andδNs=0, as given in Table I
(δN↑=−1 REM deviations.) The annihilation of one s-band
hole is behind the q/primeplus sign in the excitation momentum
k=−q+q/prime,E q s .( 26).
The ( k,ω)-plane continuum associated with the two-
parametric spectrum in Eqs. ( 25) that has two overlapping
ι=±1 branches is shown for several uvalues in Figs. 2
and3form=0 and in Figs. 7–9for a set of mvalues. That
associated with the two-parametric spectrum in Eqs. ( 26)i s
shown in Figs. 10–12, respectively, for a set of spin density
manduvalues. The zero energy of all such spectra is that ofthe deviation ω+/Delta1MHwhere ω/lessorequalslant−/Delta1MH. The energy level
ω=0 refers to the middle of the Mott-Hubbard gap 2 /Delta1MH,
Eqs. ( 6)–(8). The uandmdependencies of that gap amplitude
are illustrated in Figs. 1(a)and1(b).
The ( k,ω)-plane distributions shown in Figs. 2,3, and 7–
12do not provide information on the relative amount of
spectral weight contained within each spectrum’s colored con-tinuum. Most one-particle spectral weight is located in thevicinity of the branch lines and boundary line shown in suchfigures.
B. The spectral functions near their singularities
The following number and current number deviations un-
der transitions to excited states play an important role in thedynamical theory used in our studies:
δNF
c,ιandδNF
s,ιforι=1,−1 (right ,left) particles ,
δNF
c=/summationdisplay
ι=±1δNF
c,ιandδNF
s=/summationdisplay
ι=±1δNF
s,ι
δJF
c=1
2/summationdisplay
ι=±1ιδNF
c,ιandδJF
s=1
2/summationdisplay
ι=±1ιδNF
s,ι,
δNβ=δNF
β+δNNF
βforβ=c,s. (27)
Theβ=c,sdeviations δNF
β,ιrefer to changes in the number of βparticles at the ι=+1 right and ι=−1 left Fermi points
qι
Fβ, respectively. Except for 1 /Lcorrections, qι
Fβ≈ιπ
LNβ,s oqι
Fc≈ιπandqι
Fc≈ιkF↓. Rather than δNF
β,ι, one uses in general
β=c,snumber deviations δNF
βand number current deviations δJF
β,E q s .( 27), which contain exactly the same information.
Removal of β=c,sparticles at β-band momenta |q|<qFβaway from the Fermi points q±1
Fβand addition of sparticles at
s-band momenta kF↓<|q/prime|<kF↑are associated with number deviations denoted by δNNF
β,s oδNβ=δNF
β+δNNF
β,E q s .( 27).
The index ¯β=c,c/prime,slabels the branch lines that run within the ( k,ω)-plane continua associated with the spectra, Eqs. ( 25)
and ( 26). An index βc=c,c/primeis used only for the candc/primebranch lines. The branch-line processes lead to a one-parametric
(k,ω)-plane ¯β=c,c/prime,sbranch line spectrum of the general form
/epsilon1σ,¯β(k)=δ¯β,s/Delta1MH+c¯βε¯β(q)/greaterorequalslant0,where k=k0+c¯βqfor ¯β=c,c/prime,s. (28)
The convention is that when, as here, ¯βlabels an energy dispersion ε¯β(q), Eq. ( B4) of Appendix B, it reads ¯β=cboth for the c
andc/primebranch lines. In Eqs. ( 28),σ=↑,↓refers to the one-particle spectral function under consideration and the constants c¯β
are given by
cc=cc/prime=−1f o r q∈]−π,π [ and σ=↑,↓fermion c,c/primebranch lines ,
cs=−1f o r q=q/prime∈]−kF↓,kF↓[ and the ↓fermion sbranch line ,
=1f o r |q|=| q/prime|∈]kF↓,kF↑] and the ↑fermion sbranch line . (29)
195129-9CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
The momentum k0in Eqs. ( 28) is determined by the current
number deviations as follows:
k0=2πδJF
c+2kF↓δJF
s. (30)
Within the dynamical theory used in our studies, the up-
and down-spin spectral functions have for ω< 0 and small
values of the deviation ( ω+/epsilon1σ,¯β(k))/lessorequalslant0 at and near a ¯β=
c,c/prime,sbranch line the following general behavior:
Bσ,−1(k,ω)=Cσ,¯β(ω+/epsilon1σ,¯β(k))ξσ
¯β(k)
forsmall ( ω+/epsilon1σ,¯β(k))/lessorequalslant0. (31)
Here the constant Cσ,¯βis am- and u-dependent function that
has a fixed value for the kandωranges corresponding to small
values of the energy deviation ( ω+/epsilon1σ,¯β(k))/lessorequalslant0f o rw h i c h
this expression is valid. The momentum-dependent exponentsin it have the general form
ξ
σ
¯β(k)=−1+/summationdisplay
ι=±1/summationdisplay
β=c,s/Phi12
¯β,β,ι(q), (32)
where the spectral functionals /Phi1¯β,β,ι(q) are defined below.
The spectral function expression, Eq. ( 31), is exact when there
is no spectral weight just above the corresponding ¯βbranch
line. In the present case of one-particle spectral functions,either there is no weight above the branch lines or the amountof that weight is small. In the latter case, the very weakcoupling to it leads to a higher order contribution to the line-shape expression given in Eq. ( 31) that can be neglected in
the present thermodynamic limit. Indeed, such a contributiononly slightly changes the value of the momentum-dependentexponent, preserving its negativity or positivity.
The spectral functionals /Phi1
¯β,β,ι(q)i nE q .( 32) depend on
the¯β=c,c/prime,sbranch lines’ excitation momentum spectrum
k=k0±q,E q s .( 28). (The index ¯βthat labels such function-
als also reads ¯β=cboth for the candc/primebranch lines.) In
the present case of the Mott-Hubbard insulator, the phase-shiftrelated parameters ξ
β,β/primein the general expression of such spec-
tral functionals [ 34] have the specific form given in Eq. ( B22)
of Appendix B. Their use leads to the following expressions:
/Phi1¯β,c,ι(q)=ιδNF
c
2+δJF
c+ξscδJF
s
+c¯β/Phi1c,¯β(ιπ,q) (33)
and
/Phi1¯β,s,ι(q)=−ιξcs
2ξssδNF
c+ιδNF
s
2ξss+ξssδJF
s
+c¯β/Phi1s,¯β(ιkF↓,q). (34)
Here the ¯β-band momentum qis outside the ¯β=c,sFermi
points and c¯βare the constants, Eqs. ( 29). The β=c,spar-
ticle phase shifts /Phi1β,¯β(q,q/prime) in units of 2 πare defined by
Eqs. ( B15)–(B18) of Appendix B. Physically, ∓2π/Phi1β,¯β(q,q/prime)
is the phase shift acquired by a βparticle of momentum q
upon creation of one ¯βhole (−2π/Phi1β,¯β) and one ¯βparticle
(+2π/Phi1β,¯β) at a momentum q/prime.
The down- and up-spin one-particle removal excited states
may generate an overall c-band momentum shift and /or a
s-band momentum shift. Their possible values are 0 ,±π/L.Such shifts are behind the possibility of the β=c,sdevia-
tionsδNF
β,ιin Eq. ( 27) being half-odd integer numbers and are
implicitly accounted for by the functionals, Eqs. ( 33) and ( 34).
A coefficient bβcis used for βc=c,c/primein some of the
expressions given in the following that refer to the candc/prime
branch lines, respectively. It reads
bc=1,bc/prime=−1. (35)
There is a second type of ( k,ω)-plane feature near which
the present dynamical theory provides an analytical expres-sion of the one-particle spectral functions. It is generatedby processes where one cparticle is removed at a c-band
momentum value qand one sparticle is created or removed at
as-band momentum value q
/prime.
Those are one-parametric features called c−sboundary
lines. Indeed, they are part of the limiting boundary lines ofthe ( k,ω)-plane continua associated with the two-parametric
spectra, Eqs. ( 25) and ( 26). Ac−sboundary line ( k,ω)-plane
spectrum has the following general form:
/epsilon1
σ,c−s(k)=(−εc(q)+csεs(q/prime))δvc(q),vs(q/prime),where
k=k0−q+csq/prime. (36)
Here cs=−1o r cs=1, Eqs. ( 29), and several qandq/primelim-
iting values that obey the equality vc(q)=vs(q/prime) are defined
by the relations, Eqs. ( 62).
Near such a c−sboundary line, the up- and down-spin
one-particle spectral functions behave as [ 39]
Bσ,−1(k,ω)∝(ω+/epsilon1σ,c−s(k))−1/2
for small ( ω+/epsilon1σ,c−s(k))/lessorequalslant0. (37)
The branch- and boundary-line spectra studied in this paper
are defined in the momentum interval k∈[0,π].
V. T h e m=0 ZERO-MAGNETIZATION
SPECTRA AND EXPONENTS
For vanishing spin density and magnetic field of interest
for ARPES, there are previous studies on the present Mott-Hubbard insulator’s spectral function [ 26–28]. At h=0 and
m=0, the spectral function B
−1(k,ω) equals the spectral
functions B↓,−1(k,ω) and B↑,−1(k,ω) obtained in the m→
0 limit from m>0 and m<0 values, respectively. In the
m→0 limit from m>0 values, the up-spin one-particle
removal spectrum /epsilon1↑(k), Eqs. ( 26), becomes one-parametric
and coincides with the candc/primebranch lines of the down-
spin one-particle removal spectrum /epsilon1↓(k)=/epsilon1(k), Eqs. ( 25).
In that limit from m<0 values, the situation is the oppo-
site, the down-spin one-particle removal spectrum becomingone-parametric and the up-spin one-particle removal spectrumgiving /epsilon1(k).
Within a kextended zone scheme, the −/epsilon1(k)<0 spectrum
of the excitations that contain most of the one-particle removalspectral weight is given by
/epsilon1(k)=−ε
c(q)−εs(q/prime),
where k=ιπ−q−q/primeandι=±1
forq∈[−π,π ] and q/prime∈[−π/2,π/2]. (38)
195129-10ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
FIG. 3. The ( k,ω)-plane regions defined by the spectrum /epsilon1↓(k),
Eq. ( 38), where for spin density m=0a n d( a ) u=0.1, (b) u=0.4,
(c)u=5.0, and (d) u=15.0 there is in the thermodynamic limit
more spectral weight in the removal one-particle spectral function
of the 1D Hubbard model with one fermion per site. Most spectralweight is located in the vicinity of the c,c
/prime,a n d sbranch lines,
Eqs. ( 40), and of the c−sboundary line represented here by solid
lines and a dashed-dotted line, respectively.
The c- and s-band energy dispersions εc(q) and εs(q/prime) ap-
pearing here have at m=0 simplified expressions, defined by
Eqs. ( B12) of Appendix B.
The ( k,ω)-plane continuum shown in Figs. 2and3for
several uvalues in the range u∈[0.1,15.0] refers to the
two overlapping ι=±1 branches two-parametric spectrum,
Eq. ( 38). The continuum shown in Fig. 2is to be compared
with those plotted in (a) Fig. 3(a) of Ref. [ 26]f o r U/t=4.0
(and u=U/4t=1.0) calculated with tDMRG, (b) Fig. 7(b)
of Ref. [ 27]f o r U/t=7.0 (and u=1.75) derived combin-
ing Bethe-ansatz results, Lanczos diagonalizations, and fieldtheoretical approaches, (c) Fig. 10 of Ref. [ 28]f o rU/t=7.74
(and u=1.935) calculated with DDMRG, and (d) Fig. 1(a) of
Ref. [ 26]f o r U/t=8.0 (and
u=2.0) derived with tDMRG.
For excitation momentum k>0, the two-parametric spec-
trum, Eq. ( 38), contains the c,c/prime, and sbranch lines
represented by solid lines in Figs. 2and3and a c−sbound-
ary line represented by a dashed-dotted line.
The one-particle removal c,c/prime, and sbranch-line spectra
have the general form, Eq. ( 31), and are given by
B−1(k,ω)=C¯β(ω+/epsilon1¯β(k))ξ¯β(k)
for small ( ω+/epsilon1¯β(k))/lessorequalslant0. (39)
Here the βc=c,c/primeandsbranch-line spectra /epsilon1¯β(k) read
/epsilon1βc(k)=−εc(q)f o r ¯β=βc=c,c/primeand
/epsilon1s(k)=/Delta1MH−εs(q/prime)f o r ¯β=s, (40)
where /Delta1MHis half the m=0 Mott-Hubbard gap, Eqs. ( 7)
and ( 8), and the expressions of kare given below.
The functionals, Eqs. ( 33) and ( 34), simplify at m=0t o
/Phi1¯β,c,ι(q)=1
2/parenleftbig
ιδNF
c+2δJF
c+δJF
s/parenrightbig
+c¯β/Phi1c,¯β(ιπ,q)FIG. 4. The negative c-branch line exponent, Eq. ( 42), as a func-
tion of the excitation momentum kfor spin density m=0a n das e t
ofuvalues.
/Phi1¯β,s,ι(q)=1√
2/parenleftbigg
−ιδNF
c
2+ιδNF
s+δJF
s/parenrightbigg
+c¯β/Phi1s,¯β(ιπ/2,q). (41)
The use of the m=0 related phase-shift parameters and
phase-shift expressions, Eqs. ( B23) and ( B24) of Appendix B,
respectively, leads for u>0 to the following simplified ex-
pressions for the exponents in Eq. ( 39):
ξc(k)=−1
2+1
8/parenleftbigg
4/Psi1/parenleftbiggsinkc(q)
u/parenrightbigg
+1/parenrightbigg2
,where
k=−π
2−q∈/bracketleftBig
0,π
2/bracketrightBig
and q∈/bracketleftBig
−π,−π
2/bracketrightBig
,
k=3π
2−q∈/bracketleftBigπ
2,π/bracketrightBig
,
ξc/prime(k)=−1
2+1
8/parenleftbigg
4/Psi1/parenleftbiggsinkc(q)
u/parenrightbigg
−1/parenrightbigg2
,where
k=π
2−q∈[0,π]and q∈/bracketleftBig
−π
2,π
2/bracketrightBig
,
ξs(k)=−1
2+1
2π2/parenleftbigg
arctan/parenleftbigg
sinh/parenleftbiggπ
2/Lambda1s(q)
u/parenrightbigg/parenrightbigg/parenrightbigg2
,where
k=−q∈/bracketleftBig
0,π
2/bracketrightBig
and q∈/bracketleftBig
−π
2,0/bracketrightBig
. (42)
The rapidity functions kc(q) and/Lambda1s(q) in these expressions
are defined in terms of their inverse functions q=qc(k)f o r
k∈[−π,π ] and q=qs(/Lambda1)f o r/Lambda1∈[−∞,∞]i nE q s .( B13)
of Appendix B, respectively, and /Psi1(x) is the function given in
Eq. ( B25) of that Appendix.
The exponents in Eq. ( 42) are plotted as a function of
the momentum kfor several uvalues in Figs. 4–6.T h e ya r e
negative for their kintervals and for all u>0 values. Hence,
consistently with results obtained by other methods [ 26–28],
there are line-shape ( k,ω)-plane cusp singularities in the one-
particle removal spectral function, Eq. ( 39), at and in the
vicinity of the c,c/prime, and sbranch lines shown in Figs. 2and3.
The spectrum of the one-particle removal c−sboundary
line represented in these figures by a dashed-dotted line hasthe general form, Eq. ( 36). It reads /epsilon1
c−s(k)=−εc(q)−εs(q/prime)
195129-11CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
FIG. 5. The same as in Fig. 4for the c/prime-branch line exponent,
Eq. ( 42).
forvc(q)=vs(q/prime), where k=π−q−q/prime∈[π/2−kc−s,π].
Here kc−s=π/2f o r u/lessmuch1 and kc−s=π/4uforu/greatermuch1.
The spectrum −/epsilon1c−s(k) reaches a minimum value at k=π,
∂/epsilon1c−s(k)/∂k|k=π=0. Again, consistent with results obtained
by other methods [ 26–28], at and near this c−sboundary line
the spectral function displays singular behavior, B−1(k,ω)∝
(ω+/epsilon1c−s(k))−1/2
.
VI. THE BRANCH AND BOUNDARY LINES OF THE
ONE-PARTICLE SPECTRAL FUNCTIONS FOR m>0
In the ( k,ω)-plane distributions shown in Figs. 7–12,t h e
m>0 branch line kintervals for which the corresponding ex-
ponents, Eq. ( 32), are negative and positive are represented by
solid and dashed lines, respectively, and the boundary line bya dashed-dotted line. Expressions of the spectra and exponentsspecific to each branch line are provided in the following.
A. The down-spin ¯β=c,c/prime,sbranch lines
Thecandc/primebranch line one-parametric spectrum is con-
tained in that defined by Eq. ( 25) and reads
/epsilon1βc,↓(k)=−εc(q)f o r βc=c,c/prime, (43)
FIG. 6. The same as in Fig. 4for the s-branch line exponent,
Eq. ( 42).FIG. 7. The ( k,ω)-plane regions corresponding to the spectrum
/epsilon1↓(k), Eqs. ( 25), where for spin densities (a) m=0.01, (b) m=0.30,
(c)m=0.80, and (d) m=0.99 and u=0.4, there is in the ther-
modynamic limit more spectral weight in the removal down-spin
one-particle spectral function. Most such weight is located in thevicinity of the c,c
/prime,a n d sbranch lines and of the c−sboundary
line. Here and in Figs. 8–12, the branch line kintervals for which the
corresponding exponents are negative and positive are representedby solid and dashed lines, respectively, and the boundary line by a
dashed-dotted line.
where εc(q)i st h e c-band energy dispersion defined in
Eq. ( B4) of Appendix B. This and all branch-line spectra given
in the following have the general form, Eqs. ( 28), and all the
spectral weight near them has been transferred to the firstBrillouin zone for k>0.
Thecandc
/primebranch lines refer to excited states with devi-
ations δNF
c=0,δNF
s=−1,δNNF
c=−1,δJF
c=−bβcbcs/2,
andδJF
s=bβc/2. Here the coefficient bcs=±1 refers to two
spectral weight contributions from different extended-zone k
regions that in the first Brillouin zone refer to candc/primebranch
lines with the same energy spectrum but different weightdistributions. In the following, the value b
cs=1 is that used
because it refers to the processes that determine the line shape.
Thecbranch has for k>0 two subdomains,
k=−kF↑−q∈[0,kF↓]f o r q∈[−π,−kF↑] and
k=π+kF↓−q∈[kF↓,π]f o r q∈[kF↓,π], (44)
whereas the k>0 interval of the c/primebranch line is
k=kF↑−q∈[0,π]f o r q∈[−kF↓,kF↑]. (45)
Near the present branch lines, B↓,−1(k,ω) reads
B↓,−1(k,ω)=C↓,βc(ω+/epsilon1βc,↓(k))ξ↓
βc(k)
for (ω+/epsilon1βc,↓(k))/lessorequalslant0. (46)
Theβc=c,c/primeconstants C↓,βcand all multiplicative constants
in the branch-line spectral function expressions of the generalform, Eq. ( 31), provided in the following have a fixed value
for the kandωranges corresponding to small values of the
energy deviation for which such expressions are valid, whichhere is given by ( ω+/epsilon1
βc,↓(k)).
The use of the above deviations in the spectral function-
als, Eqs. ( 33) and ( 34)f o r ¯β=c,c/prime, leads to the following
195129-12ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
expression for the exponents in Eq. ( 46):
ξ↓
βc(k)=−1+/summationdisplay
ι/parenleftbigg
−bβc(1−ξcs)
2−/Phi1c,c(ιπ,q)/parenrightbigg2
+/summationdisplay
ι/parenleftbigg
−ι
2ξss+bβcξss
2−/Phi1s,c(ιkF↓,q)/parenrightbigg2
(47)
of general form, Eq. ( 32), where k=k(q)i sg i v e ni nE q s .( 44)
and ( 45)f o rt h e candc/primebranch lines, respectively.
The down-spin candc/primebranch-line exponents, Eq. ( 47),
are plotted in Figs. 13and14, respectively, as a function of
kfor a set of manduvalues. For the kintervals for which
they are negative, there are ( k,ω)-plane cusp singularities in
the down-spin spectral function, Eq. ( 46), at in the vicinity
of the corresponding branch lines. In contrast to their m=0
behavior, Figs. 4and5,f o rs o m e kintervals these exponents
are positive.
The spectrum of the down-spin sbranch line reads
/epsilon1s,↓(k)=/Delta1MH−εs(q/prime),where
k=−q/prime∈[0,kF↓]f o r q/prime∈[−kF↓,0].(48)
Here/Delta1MHis half the Mott-Hubbard gap, Eq. ( 7), and εs(q/prime)
is the s-band energy dispersion defined in Eq. ( B4)o f
Appendix B. This branch line corresponds to excited energy
eigenstates with deviations δNF
c=−1,δNF
s=1,δNNF
s=
−1, and δJF
c=δJF
s=0.
Near the sbranch line the expression of B↓,−1(k,ω)i s
B↓,−1(k,ω)=C↓,s(ω+/epsilon1s,↓(k))ξ↓
s(k)
for (ω+/epsilon1s,↓(k))/lessorequalslant0. (49)
The exponent obtained by the use in the functionals, Eqs. ( 33)
and ( 34), of the above deviations is given by
ξ↓
s(k)=−1+/summationdisplay
ι/parenleftBig
−ι
2−/Phi1c,s(ιπ,−k)/parenrightBig2
+/summationdisplay
ι/parenleftbiggιξcc
2ξss−/Phi1s,s(ιkF↓,−k)/parenrightbigg2
. (50)
The down-spin sbranch-line exponent, Eq. ( 50), is plotted
in Fig. 15as a function of the excitation momentum kfor a
set of spin density manduvalues. It is negative in all its k
intervals, so there are ( k,ω)-plane cusp singularities in the
down-spin spectral function, Eq. ( 49), at and in the vicinity of
this branch line.
B. The up-spin ¯β=c,c/prime,sbranch lines
Theβc=c,c/primebranch lines refer to excited energy eigen-
states with deviations δNF
c=δNF
s=0,δNNF
c=−1,δJF
c=
0, and δJF
s=−bβc/2. Their spectra read
/epsilon1βc,↑(k)=−εc(q),where
k=−kF↓−q∈[0,kF↑]f o r q∈[−π,−kF↓],
k=2π−kF↓−q∈[kF↑,π]f o r q∈[kF↑,π],
andβc=c,
k=kF↓−q∈[0,π]f o r q∈[−kF↑,kF↓],
andβc=c/prime. (51)Near the βc=c,c/primebranch lines, B↑,−1(k,ω) is given by
B↑,−1(k,ω)=C↑,βc(ω+/epsilon1↑
βc(k))ξ↑
βc(k)
for (ω+/epsilon1↑
βc(k))/lessorequalslant0. (52)
The exponent obtained from the use in the functionals,
Eqs. ( 33) and ( 34), of the above deviations reads
ξ↑
βc(k)=−1+/summationdisplay
ι/parenleftbigg
−bβcξcs
2−/Phi1c,c(ιπ,q)/parenrightbigg2
+/summationdisplay
ι/parenleftbigg
−bβcξss
2−/Phi1s,c(ιkF↓,q)/parenrightbigg2
. (53)
These candc/primebranch-line exponents are plotted in Figs. 16
and17, respectively, as a function of the excitation momentum
kfor a set of spin density manduvalues. They are mostly
negative, yet upon decreasing uand increasing mthey become
positive for some kintervals. For those for which such expo-
nents are negative, there are cusp singularities in the up-spinspectral function, Eq. ( 52), at and near the β
c=c,c/primebranch
lines.
The up-spin one-particle sbranch line refers to excited
states with deviations δNF
c=δNF
s=−1,δNNF
s=1,δJF
c=
1/2, and δJF
s=0. Its spectrum is given by
/epsilon1s,↑(k)=/Delta1MH+εs(q/prime),where k=π+q/prime∈[kF↓,kF↑]
forq/prime∈[−kF↑,−kF↓]. (54)
Near the present sbranch line, B↑,−1(k,ω) reads
B↑,−1(k,ω)C↑,s(ω+/epsilon1s,↑(k))ξ↑
s(k)
for (ω+/epsilon1s,↑(k))/lessorequalslant0. (55)
The exponent obtained from the use of the above deviations
in the functionals, Eqs. ( 33) and ( 34), reads
ξ↑
s(k)=−1+/summationdisplay
ι/parenleftbigg(1−ι)
2+/Phi1c,s(ιπ,q/prime)/parenrightbigg2
+/summationdisplay
ι/parenleftbigg
−ι(1−ξcs)
2ξss+/Phi1s,s(ιkF↓,q/prime)/parenrightbigg2
.(56)
This exponent is plotted in Fig. 18as a function of the
excitation momentum kfor a set of spin density mand u
values. It is negative for smaller spin densities and becomespositive upon increasing m.F o rt h e kintervals, spin densities,
anduvalues for which it is negative, there are ( k,ω)-plane
cusp singularities in the up-spin spectral function, Eq. ( 55), at
and near this branch line.
C. Branch lines for h=hcandm=1
For the fully polarized limit at h=hcandm=1, one has
thatB↓,−1(k,ω)=0. Indeed, all spins are up and thus there
are no spin-down electrons to be annihilated. In spite of theon-site interaction playing no role when all spins are up, anni-hilation of one up-spin electron generates energy eigenstateswith one rotated-electron unoccupied site and thus one ηspin
of projection 1 /2 (see Appendix A.) This is a high-energy
process that involves the many-electron interactions.
Similarly, B
↑,+1(k,ω)=0a t h=−hcand m=−1
and creation of one down-spin electron generates energy
195129-13CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
eigenstates with one rotated-electron doubly occupied site and
thus one η-spin of projection −1/2. Again, this is a high-
energy process that involves the electronic correlations.
In the present case of h=hcandm=1, the form of the
−/epsilon1↑(k)<0 spectrum, Eqs. ( 26), simplifies. Within a kex-
tended zone scheme, it reads
/epsilon1↑(k)=2tcosq+1
2/radicalbig
(4t)2+U2
−2t
π/integraldisplayπ
−πdq/prime/primesinq/prime/primearctan/parenleftbiggsinq/prime/prime−/Lambda1s(q/prime)
u/parenrightbigg
,
where k=−q+q/primefor q∈[−π,π ] and q/prime∈[−π,π ].
(57)
Here the s-band rapidity function /Lambda1s(q/prime) is defined by its
inverse function as
q/prime=1
π/integraldisplayπ
−πdq/prime/primearctan/parenleftbigg/Lambda1s(q/prime)−sinq/prime/prime
u/parenrightbigg
∈[−π,π ]
for/Lambda1∈[−∞,∞]. (58)
Them=1 spectrum, Eqs. ( 57), has nearly the same form as
that plotted in Fig. 10(d) form=0.99.
The up-spin one-particle removal spectral function has in
the vicinity of the ¯β=c,c/prime,sbranch lines the general form,
Eq. ( 31). It is given by
B↑,−1(k,ω)=C↑,¯β(ω+/epsilon1¯β,↑(k))ξ↑
¯β(k)
for small ( ω+/epsilon1¯β,↑(k))/lessorequalslant0. (59)
The form of the βc=c,c/primeand sbranch-line spectra
−/epsilon1¯β,↑(k)<0 appearing in this expression also simplifies. It
reads
/epsilon1c,↑(k)=/epsilon1c/prime,↑(k)=2tcosk+1
2/radicalbig
(4t)2+U2,
/epsilon1s,↑(k)=−2t
π/integraldisplayπ
−πdqsinqarctan/parenleftbiggsinq−/Lambda1s(π−k)
u/parenrightbigg
+/radicalbig
(4t)2+U2−4tboth for k∈[0,π].
(60)
Here and in the following, /Lambda1s(π−k)i st h e s-band rapidity
function defined by its inverse function in Eq. ( 58)f o r q/prime=
π−k∈[0,π]. The c,c/prime, and sbranch-line spectra, Eqs. ( 60),
have nearly the same form as those shown in Fig. 10(d) for
m=0.99. Note that at m=1, the candc/primebranch-line spectra
have exactly the same shape.
The form of the corresponding βc=c,c/primeandsexponents
in Eq. ( 59) also simplifies for h=hcandm=1:
ξ↑
βc(k)=−1
2−2bβc
πarctan/parenleftbiggsink
u/parenrightbigg
+2
π2/braceleftbigg
arctan/parenleftbiggsink
u/parenrightbigg/bracerightbigg2
,
ξ↑
s(k)=1
2−2
πarctan/parenleftbigg/Lambda1s(π−k)
2u/parenrightbiggFIG. 8. The same as Fig. 7for a larger uvalue, u=1.0.
+2
π2/parenleftBigg/braceleftbigg
arctan/parenleftbigg/Lambda1s(π−k)
u/parenrightbigg/bracerightbigg2
+/braceleftbigg
arctan/parenleftbigg/Lambda1s(π−k)
2u/parenrightbigg/bracerightbigg2/parenrightBigg
both for k∈[0,π].
(61)
Here the coefficient bβcused for βc=c,c/primeis defined in
Eqs. ( 35). These c,c/prime, and sexponents have nearly the same k
dependence as those shown in Figs. 16(f) –18(f) , respectively,
form=0.99.
When two branch lines have exactly the same spectrum,
as occurs here for /epsilon1c,↑(k)=/epsilon1c/prime,↑(k), the line shape in their
vicinity has the form shown in Eq. ( 59) with the exponent
being the smallest of the corresponding two exponents. Asconfirmed by comparison of the curves plotted in Figs. 16(f)
and17(f) , the smallest exponent is ξ
↑
c(k), Eqs. ( 61)f o rβc=c
andbβc=bc=1.
The values of the exponent curves plotted in Figs. 16(f)
and18(f) reveal that at m=1 the exponent ξ↑
c(k)i sn e g a t i v e
andξ↑
s(k) is positive. Hence there are spectral peaks associ-
ated with the cusp singularities near the cbranch line shown
in Fig. 10(d) .
D. The line shape near the c−sboundary lines
The limiting values of the kintervals of the down- and
up-spin c−sboundary line spectra, which have general
form, Eq. ( 36), involve the group velocities, Eqs. ( B5)o f
Appendix B.F o r u>0 and m>0, they are determined by
thec-band values q=±q0
candq=± ¯q0
csuch that q0
c<¯q0
c
and the s-band values q/prime=±kF↓,q/prime=± ¯qs,q/prime=± ˜qs, and
q/prime=kF↑such that kF↓<¯qs<˜qs<kF↑, which obey the rela-
tions
vc(0)=vs(kF↑),vc/parenleftbig
q0
c/parenrightbig
=vs(kF↓)=vs(˜qs),
vc/parenleftbig
¯q0
c/parenrightbig
=vs(¯qs),|vs(¯qs)|≡max|vs(q/prime)|. (62)
Fork>0 and finite uvalues, the spectra of the down-
and up-spin c−sboundary lines represented in Figs. 7,8,9
195129-14ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
FIG. 9. The same as Fig. 7for a larger uvalue, u=5.0.
and Figs. 10,11,12, respectively, by dashed-dotted lines are
given by
/epsilon1↓,c−s(k)=(−εc(q)−εs(q/prime))δvc(q),vs(q/prime),where
k=±π−q−q/prime∈/bracketleftbig
kF↑−q0
c,π/bracketrightbig
for
q∈/bracketleftbig
0,q0
c/bracketrightbig
and q/prime∈[0,kF↓] (63)
and
/epsilon1↑,c−s(k)=(−εc(q)+εs(q/prime))δvc(q),vs(q/prime),where
k=−q+q/prime∈/bracketleftbig/parenleftbig
¯qs−¯q0
c/parenrightbig
,kF↑/bracketrightbig
for
q∈/bracketleftbig
0,¯q0
c/bracketrightbig
and q/prime∈[¯qs,kF↑], (64)
respectively. These spectra have a minimum value at k=π
andk=kF↑, respectively, such that
∂/epsilon1↓,c−s(k)
∂k|k=π=0
−/epsilon1↓,c−s(π)=−/Delta1MH−4t+εs(0),
FIG. 10. The ( k,ω)-plane regions corresponding to the spectrum
/epsilon1↑(k), Eqs. ( 26), where for spin densities (a) m=0.01, (b) m=0.30,
(c)m=0.80, and (d) m=0.99 and u=0.4 there is in the thermody-
namic limit more spectral weight in the removal up-spin one-particlespectral function. Most such weight is located in the vicinity of the
c,c
/prime,a n d sbranch lines and of the c−sboundary line. The lines
notations are the same as in Fig. 7.∂/epsilon1↑,c−s(k)
∂k|k=kF↑=0
−/epsilon1↑,c−s(kF↑)=−/Delta1MH−4t−εs(kF↑). (65)
In the m→1 limit, the down-spin c−sboundary line
does not exist. At and near the c−sboundary lines, the
up- and down-spin spectral functions show cusp singularities,B
σ,−1(k,ω)∝(ω+/epsilon1σ,c−s(k))−1/2,a sg i v e ni nE q .( 37).
VII. EFFECTS OF THE CHARGE-SPIN SEPARATION
AND CHARGE-SPIN RECOMBINATION
In contrast to Fermi liquids, 1D interacting systems are
characterized by a breakdown of the basic quasiparticle pic-ture. Indeed, no quasiparticles occur when the electron’s rangeof motion is restricted to a single spatial dimension. In thequantum problem studied in this paper, correlated electronsrather split into basic fractionalized ccharge-only and sspin-
only particles. These particles can move with different speedsand even in different directions in the 1D many-electron sys-tem.
In the case of both the metallic and Mott-Hubbard insulat-
ing phases of the 1D Hubbard model, the usual quasiparticleFermi momenta are replaced by the c-particle Fermi points
±2k
Fands-particle Fermi points ±kF↓whose bands limits
are±πand±kF↑, respectively. Here ±2kFbecomes ±πfor
the Mott-Hubbard insulator. Importantly, and again in con-trast to the spin-1 /2 quasiparticle Fermi momenta, for it the
c-particle Fermi points read ±πand thus coincide with the
limits of the Brillouin zone for the whole spin density intervalm∈[0,1].
Within studies of doped high- T
Csuperconductors, a
shadow band has appeared in angle-resolved photoemission
spectra that corresponds in two-dimensional antiferromag-netic Fermi liquids to an extra line of peaks that at lowωoccurs at k
F+π[50,51]. The nonperturbative and non-
Fermi-liquid physics of the present 1D Hubbard model refersto a different quantum problem. However, at h=0 a line
of peaks that runs in the interval k∈[0,3k
F], which for the
Mott-Hubbard insulator refers to k∈[0,kF+π] with an ex-
treme at kF+πalso corresponding to low ω, occurs in the
one-particle removal spectral function. By analogy with thetwo-dimensional Fermi liquid, it has also been named shadow
band [45]. (See lower figure in Fig. 1 of Ref. [ 45]f o rt h e
metallic phase at quarter filling.)
The separation of the charge an spin degrees of freedom
that gives rise to independent charge candc
/primeand spin sbranch
line peaks in the one-particle removal spectral function occursboth in the metallic and Mott-Hubbard insulating phases of the1D Hubbard model for all densities. Near such branch lines,the spectral functions line shape is of the form B
σ,−1(k,ω)=
Cσ,¯β(ω+/epsilon1σ,¯β(k))ξσ
¯β(k),E q .( 31), where ¯β=c,c/prime,sand the k,
u, and density dependence of the exponents ξσ
¯β(k)s t e m sf r o m
the overlaps within the one-electron matrix elements.
In the general case of both the metallic and Mott-Hubbard
insulating phases, such a shadow band refers to the (i) down-spin and (ii) up-spin one-particle removal spectral functionsto a charge branch line of peaks that runs within an extendedscheme from k=0 at high negative values of ω, reaches
195129-15CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
FIG. 11. The same as Fig. 10for a larger uvalue, u=1.0.
a minimum ωof largest absolute value at (i) k=kF↑and
(ii)k=kF↓, and at low ωvalues runs until (i) k=2kF↑+
kF↓and (ii) k=2kF↓+kF↑, respectively. This shadow band
comes from effects on the charge degrees of freedom of spinfluctuations [ 45].
Hence in the specific case of the 1D Mott-Hubbard in-
sulator under study in this paper, the shadow band of the(i) down-spin and (ii) up-spin one-particle removal spectralfunctions runs within an extended zone scheme in the interval(i)k∈[0,k
F↑+π] and (ii) k∈[0,kF↓+π], respectively. In
(i), Figs. 7,8,9, and (ii), Figs. 10,11,12,f o r0 <h<hc,i t
is the c/primebranch line for k∈[0,π] and its interval (i) k∈
[π,kF↑+π] and (ii) k∈[π,kF↓+π] was brought to the first
Brillouin zone and corresponds to the part of the cbranch line
that runs in the interval (i) k∈[kF↓,π] and (ii) k∈[kF↑,π],
respectively.
As mentioned above, at h=0 the shadow band interval
becomes k∈[0,kF+π]. In Figs. 2and3forh=0, it is the c/prime
branch line for k∈[0,π] whereas its interval k∈[π,kF+π]
was brought again to the first Brillouin zone. It correspondsto the part of the cbranch line that runs in the interval k∈
[k
F,π].
FIG. 12. The same as Fig. 10for a larger uvalue, u=5.0.At spin density m=1, when the candsbands have the
same limiting momenta ±π,t h e candc/primebranch lines merge.
In the case of the down-spin one-particle removal spectralfunction, all weight actually vanishes at m=1. The line
shape near the up-spin one-particle removal spectral functionis controlled only by the cbranch line exponent ξ
↑
c(k). As
found in Sec. VI C ,ξ↑
c(k) is smaller than the exponent ξ↑
c/prime(k)
of the c/primebranch line, Eqs. ( 61)f o rβc=c,c/prime. Moreover, the
nonshadowed-band interval k∈[0,kF↑]o ft h e cbranch line
extends to k∈[0,π]. The corresponding lack of the shadow
band thus occurs only when the candsbands have the same
limiting momenta ±πand the sband is empty.
Interestingly, the physics behind the other type of singular
features, called boundary lines, is related to a charge-spinrecombination that occurs in the ( k,ω) plane only at and very
near such lines. As shown in Sec. 3.2 of Ref. [ 39], rather than
from one-electron matrix elements overlaps, the boundary-line power-law behavior B
σ,−1(k,ω)∝(ω+/epsilon1σ,c−s(k))−1/2,
Eq. ( 37), stems from singularities in the density of states of
the two-parametric spectra E(k)=−εc(q)+csεs(q/prime), where
k=k0−q+csq/prime.H e r e cs=−1o rcs=1, Eqs. ( 29).
The physical reason why such one-electron spectral-
function singularities are controlled by an exponent −1/2
that does not depend on the momentum k, interaction u, and
spin density mis indeed a phenomenon of charge-spin recom-
bination. As reported above, in the present nonperturbativemany-electron problem, there is a separation of the chargeand spin degrees of freedom such that the ccharge and sspin
excitations propagate in general with different group veloc-ities v
c(q) and vs(q/prime), respectively. Only at and very near a
(k,ω)-plane boundary line does one have that vc(q)=vs(q/prime).
This equality of the ccharge and sspin velocities is associated
with a partial recombination of the charge and spin degrees atand near such ( k,ω)-plane lines, consistently with the corre-
sponding Fermi-liquid-like exponent −1/2, in that it does not
depend on k,u, and m.
VIII. DISCUSSION OF THE RESULTS
AND CONCLUDING REMARKS
In this paper, we have studied the one-particle spectral
properties of the 1D Hubbard model with one fermion persite both at zero and finite magnetic field. Specifically, themomentum and energy dependence of the exponents and en-ergy spectra that control the line shape at and near the cuspsingularities of the up- and down-spin one-particle spectralfunctions, Eqs. ( 3), was derived.
An important qualitative difference of the Mott-Hubbard
insulator relative to the one-particle properties of the modelmetallic phase studied in Ref. [ 34] refers to the values of the
exponents ξ
σ
¯β(k)i nE q .( 31)a s u→0 when m<1. In the
metallic case, that for a given ¯β=c,sbranch line kinterval
the exponent reads ξσ
¯β(k)=−1i nt h e u→0 limit means
that the exact expression of the spectral function is not that
given in Eq. ( 31) because the four functionals /Phi1¯β,β,ι(q)i n
Eq. ( 32)f o rβ=c,sandι=±1 all exactly vanish. The cor-
responding one-electron spectral functions behavior is insteadδ-function-like. For the corresponding kmomentum intervals,
one recovers the exact U=0 up- and down-spin one-electron
195129-16ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
FIG. 13. The down-spin c-branch line exponent as a function of
the excitation momentum kfor spin densities (a) m=0.01, (b) m=
0.10, (c) m=0.30, (d) m=0.50, (e) m=0.80, and (f) m=0.99
and a set of uvalues. For small m, this exponent is negative. It
acquires kintervals, where it becomes positive whose momentum
width increases upon increasing m.
spectra [ 34]. Indeed, in the metallic phase the noninteracting
spectral functions are smoothly obtained upon continuouslydecreasing u.
In contrast, for the Mott-Hubbard insulator, the exponents
ξ
σ
¯β(k) of the branch lines whose spectra give rise to the metal-
lic half-filled noninteracting spectra do not read −1a su→0
when m<1. Inspection of Figs. 4-6and13-18reveals that
all branch-line exponents are typically larger than −1/2. This
is because under the quantum phase transition from the Mott-Hubbard insulator to the metallic half-filling phase occurringatU=0, there is a singular change of the spectral-weight line
shapes.
An exception is, though, the m→1 limit. In that case, the
up-spin cbranch-line exponent ξ
↑
c(k), Eq. ( 53)f o rβc=c,
plotted in Fig. 16indeed reads −1f o r k∈[0,π]i nt h e u→0
limit. This implies that the line shape is δ-function-like at
that line. Consistently, one recovers the exact U=0 up-spin
one-electron spectrum since in that limit that branch-line spec-trum, Eq. ( 51)f o rβ
c=c, is given by /epsilon1c,↑(k)=2t(1+cosk)
fork∈[0,π]. This behavior results from the proximity of
them=1 fully polarized quantum phase associated with the
m→1 limit.
Studies of the spin dynamical structure factors of the
half-filled 1D Hubbard model in magnetic field has shownthat the momentum-dependent exponents that control theirline shape at and near the cusp singularities depend verylittle on u=U/4t[10,11]. The main effect of correla-
tions was found to be on the energy bandwidth of thedynamical structure factors’ ( k,ω)-plane spectra, which in-
creases upon decreasing u, yet preserving the same shape.FIG. 14. The kdependence of the down-spin c/prime-branch line ex-
ponent for the same spin densities and uvalues as Fig. 13.F o rs m a l l
m, this exponent is negative. It becomes positive in kintervals whose
momentum width increases upon increasing m.
Hence, the half-filled 1D Hubbard model for anyfinite u=
U/4tvalue and a suitable choice of units for the spectra’s
energy bandwidths can describe the same spin dynamicalproperties.
In this paper, we have investigated whether suitable values
of the interaction for chain compounds at m=0 and for
ultracold atom systems for m>0 described by 1D Mott-
Hubbard insulators can be settled by the agreement with
experimental results on the up- and down-spin one-particlespectral functions. Analysis of the spectra plotted in Figs. 2,3,
and7–12reveals that again their energy bandwidth increases
upon decreasing u, yet preserves the same shape. For m=0
(Figs. 4–6) and low mvalues (Figs. 13–18), the momentum-
dependent exponents are little udependent, the same applying
to the sbranch-line exponents (Figs. 15and18) for all spin
densities.
In the down-spin case for m>0, the exponents that are
negative do not depend much on u.T h e candc
/primebranch-line
exponents, though, depend much on m.T h e sbranch-line
exponent depends less on mand stays negative. On the other
hand, the up-spin case is different: The exponents are lessdependent on m. Both for up and down spins, the candc
/prime
branch-line exponents (Figs. 13,14,16, and 17) are quite ude-
pendent upon increasing m. Indeed, changing uat some fixed
mvalues leads to corresponding changes of the kintervals for
which the exponents are positive and negative. This refers tocusp singularities not emerging and emerging in the spectralfunctions, respectively.
Hence, we conclude that at m=0, changing the interaction
values changes little the one-electron spectral properties ofchain compounds described by 1D Mott-Hubbard insulators,
195129-17CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
FIG. 15. The kdependence of the negative down-spin s-branch
line exponent for the same spin densities and uvalues as Fig. 13.
since a suitable choice of units for the spectra’s energy band-
widths can describe the same spectral properties for differentuvalues. This implies that physical quantities other than the
spin dynamical structure factors [ 10,11] and the one-particle
spectral functions studied in this paper should be used to findthe interaction values suitable to such compounds. Specifi-
FIG. 16. The kdependence of the up-spin c-branch line exponent
for the same spin densities and uvalues as Fig. 13. Except for a small
kinterval that emerges for small uvalues and intermediate mvalues,
this exponent is negative.FIG. 17. The kdependence of the up-spin c/prime-branch line expo-
nent for the same spin densities and uvalues as Fig. 13. Upon
increasing m, it acquires for small uvalues kintervals where it
becomes positive.
cally, and in spite of the lack of superconductivity in 1D,
the gapped ( k,ω)-plane distribution of the cusp singularities
at and just below the spectra’s thresholds of the spin singletand triplet pairs’ spectral functions could provide physicallyimportant information on the issue under consideration bothfor the Mott-Hubbard insulator and the corresponding dopedinsulator. Such a study could be interesting, for instance, forthe physics of chain cuprates [ 17].
An interesting related problem that could be studied else-
where is whether a description in terms of ηspins of
projection −1/2 and +1/2, which replace the u/greatermuch1 dou-
blons and holons [ 4], respectively, for u>0, could at m=0
and for values of unot necessarily large to reproduce the
characteristic behaviors of the experimental χ
(3)observed
in 1D Mott-Hubbard insulators [ 37]. This is a problem of
technological interest for third-harmonic generation spec-troscopy [ 3,37].
On the other hand, in the case of ultracold atom systems
on optical lattices described by 1D Mott-Hubbard insulators,upon increasing the spin density mone reaches a regime
where changing uat fixed mchanges the one-particle spectral
properties. Hence at large enough spin densities m, suitable
values of the interaction for such atomic systems could in-deed be settled by the agreement with experimental resultson the up- and down-spin one-particle spectral functions.This involves the specific u-dependent kintervals for which
the exponents plotted in Figs. 4–6and13–18as a function
of momentum kfor different mand uvalues are nega-
tive and positive, respectively. This predicts a u-dependent
(k,ω)-plane distribution of cusp singularities in the up- and
down-spin spectral functions, Eqs. ( 3), whose comparison
with experiments could settle the uvalue suitable to the
195129-18ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
FIG. 18. The kdependence of the up-spin s-branch line exponent
for the same spin densities and uvalues as Fig. 13. The general trend
of this exponent for all uvalues is that it is negative at small mand
becomes positive upon increasing m.
ultracold atoms. (Cusp singularities also emerge in the ( k,ω)-
plane regions at and near the c−sboundary lines.)
The interacting spin-1 /2 fermions described by the model
under study here can either be electrons or spin-1 /2a t o m s .
In condensed matter materials at zero magnetic field, ARPESdirectly measures the spectral function of the electrons [ 13],
which are removed via the photoelectric effect. On theother hand, the 1D Hubbard model with one fermion persite in a magnetic field can be implemented with ultra-cold atoms [ 18–24]. Momentum-resolved radio-frequency (rf)
spectroscopy [ 21,22] and Bragg spectroscopy [ 23,24]a r e
techniques to measure the spectral functions in ultracoldatomic systems. In particular, momentum-resolved rf is a toolto achieve an analog of ARPES for ultracold atomic systems.The spectroscopy takes advantage of the many spin states ofthe atoms in these cold systems.
Can the ( k,ω)-plane distribution of spectral peaks at fi-
nite magnetic field associated with the cusp singularities ofour theoretical predictions be accessed by ultracold spin-1 /2
atomic experiments? In a magnetic field, the degeneracy ofthe atoms’s spin states is split by the Zeeman interaction atmagnetic field strengths around the Feshbach resonance. ThisZeeman splitting is much larger than other energy scales in thesystem. Fortunately, all the spin-relaxation mechanisms avail-able to atoms in some of their spin states are either forbiddenor strongly suppressed, so a system of atoms in those spinstates stays that way without relaxing.
Momentum-resolved rf spectroscopy takes advantage of
the fact that the rf photon has a negligible momentumcompared to the momentum of the atom. As a result, thespin-flip transition does not change its momentum state. Inthe language of photoemission spectroscopy, this is a verticaltransition. The momentum of the spin-flipped atom, and thus
the momentum of the atom inside the interacting system, canbe measured in a time-of-flight experiment. Importantly, withthis information, one can indeed reconstruct the one-particle
spectral function and thus use the present results as a theoreti-cal prediction and check their relevance and consequences foractual physical systems [ 21,22].
Hence, our theoretical predictions are of experimental in-
terest for condensed-matter chain compounds at m=0 where
they could be tested by ARPES and for systems of ultracoldspin-1 /2 atoms on optical lattices for finite spin densities.
ACKNOWLEDGMENTS
We thank Karlo Penc for illuminating discussions.
J.M.P.C. would like to thank Boston University’s Con-densed Matter Theory Visitors Program for support andBoston University for hospitality during the initial pe-riod of this research. He acknowledges support fromFCT through the Grants No. PTDC /FIS-MAC /29291 /2017,
No. SFRH /BSAB /142925 /2018, and No. POCI-01-0145-
FEDER-028887. J.M.P.C. and T. ˇC. acknowledge support
from FCT through Grant No. UID /FIS/04650 /2013. T. ˇC.
gratefully acknowledges support by the Institute for Ba-sic Science in Korea (Grant No. IBS-R024-D1). P.D.S.acknowledges support from FCT through Grants No.UID/CTM/04540 /2013 and No. UID /CTM/04540 /2019.
APPENDIX A: ROTATED-FERMION RELATED
FRACTIONALIZED PARTICLES
In this Appendix, the general rotated-fermion represen-
tation of the 1D Hubbard model for its full Hilbert spaceis briefly described. It applies both when its operators c
†
j,σ
andcj,σin the expression, Eqs. ( 1) and ( 2), refer to elec-
trons and spin-1 /2 atoms. Rotated fermions are generated
from the fermions associated with such operators by a unitarytransformation. The unique definition of the correspondingfermion-rotated-fermion unitary operator in terms of its 4
Na×
4Namatrix elements between the model’s 4Naenergy eigen-
states is given in Ref. [ 34], which considers electrons.
As for fermions in the u→∞ limit, up- and down-
spin single-site occupancy, double-site occupancy, and siteno occupancy are for the whole u>0 range good quantum
numbers for rotated fermions. This means that in terms ofsuch fermions, the local occupancy configurations whose su-perposition generates any energy eigenstate have fixed valuesfor the numbers L
s,±1/2of sites singly occupied by rotated
fermions of spin projection ±1/2,Lη,+1/2of sites unoccu-
pied, and Lη,−1/2of sites doubly occupied by them. Hence
Ls=Ls,+1/2+Ls,−1/2gives the total number of sites singly
occupied by rotated fermions and Lη=Lη,+1/2+Lη,−1/2that
of sites doubly occupied and unoccupied by rotated fermions.
The rotated-fermion degrees of freedom naturally sepa-
rate for u>0 into occupancy configurations of three basic
fractionalized particles that generate the exact irreduciblerepresentations of the groups associated with the indepen-dent spin and η-spin SU(2) symmetries and the clattice
U(1) symmetry, respectively, in the model global [ SU(2)⊗
SU(2)⊗U(1)]/Z
2
2symmetry [ 34,35]. (See operator relations
195129-19CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
in Eq. (32) of Ref. [ 34].) For u>0, a number Lsof spin
1/2’s emerge from the rotated fermions that singly occupy
sites and describe their spin degrees of freedom, a numberL
ηofη-spin1 /2’s emerge from the rotated fermions that dou-
bly occupy sites and from the unoccupied sites and describetheη-spin degrees of freedom of such site occupancies, and
a number N
c=Ls=Na−Lηofcparticles and Nh
c=Lη=
Na−Lsofcholes also emerge and describe the translational
degrees of freedom associated with the relative positions ofspin 1/2’s and η-spin 1 /2’s, respectively.
Theη-spin 1 /2’s only see the set of L
η=Na−Lssites
unoccupied and doubly occupied by rotated fermions. Theη-spin 1 /2’s thus live in an η-spin squeezed effective lat-
tice [ 46,47,52] with L
η=Na−Lssites that corresponds to
anη-spin-1 /2XXX chain. The spin 1 /2’s only see the set of
Ls=Na−Lηsites singly occupied by rotated fermions. They
live in a spin-squeezed effective lattice with Ls=Na−Lη
sites that corresponds to a spin-1 /2XXX chain. These 1D
squeezed lattices are known in the u→∞ limit in which the
rotated fermions become fermions [ 46,47,52]. On the other
hand, the cparticles live on an effective lattice identical to the
original lattice.
The irreducible representations of the clattice U(1) sym-
metry group store full information on the relative positions inthe model’s original lattice of the spin and η-spin squeezed
effective lattices sites, respectively. The irreducible represen-tations of the groups associated with the η-spin and spin SU(2)
symmetries are generated by η-spin-1 /2 and spin-1 /2 occu-
pancy configurations, respectively, in their squeezed effectivelattices. They are similar to those of an η-spin-1 /2 and a spin-
1/2XXX chain on a lattice with L
ηandLssites, respectively.
Let us denote generally by SαandSz
αtheη-spin ( α=η)
and spin ( α=s) numbers. The spin 1 /2’s and η-spin 1 /2’s are
then generally named α-spin 1 /2’s. The α-spin squeezed ef-
fective lattice’s local configurations of each energy eigenstatecontain a set of M
α=2Sαα-spin 1 /2’s that participate in a α-
spin multiplet configuration and a complementary set of evennumber 2 /Pi1
α=Lα−2Sαofα-spins 1 /2 that participate in
Sz
α=Sα=0α-spin singlet configurations. The energy eigen-
states are superpositions of such configuration terms, eachbeing characterized by a different partition of L
αα-spin 1 /2’s
into Mα=2Sαα-spin 1 /2’s that participate in a 2 Sα+1α-
spin multiplet and a product of α-spin singlets involving the
remaining even number 2 /Pi1α=Lα−2Sαofα-spin 1 /2’s that
form a tensor product of α-spin singlet states.
Forα=s, the numbers of unpaired spin 1/2’s and paired
spin 1/2’s thus are Ms≡2Ssand 2/Pi1s≡Ls−2Ss, respec-
tively. For α=η, the numbers of unpaired η-spin 1/2’s
andpaired η-spin 1/2’s are Mη≡2Sηand 2/Pi1η=Lη−2Sη,
respectively. Hence, Ls=Ls,+1/2+Ls,−1/2=2/Pi1s+Msand
Lη=Lη,+1/2+Lη,−1/2=2/Pi1η+Mη, where Ls,±1/2=/Pi1s+
Ms,±1/2andLη,±1/2=/Pi1η+Mη,±1/2. Here, Ms,±1/2=Ss∓Sz
s
gives the number of unpaired spin 1 /2’s of projection ±1/2
andMη,±1/2=Sη∓Sz
ηthat of unpaired η-spin 1 /2’s of pro-
jection ±1/2, respectively. /Pi1s≡Ls/2−Ssis the number of
paired spins of both projections ±1/2 and /Pi1η=Lη/2−Sη
that of paired ηspins of both projections ±1/2.
The basic fractionalized particles are directly related to
the quantum numbers of the Bethe-ansatz solution whoseoccupancy configurations generate the exact energy eigen-states [ 34,35]. One has that /Pi1
s=/summationtext∞
n=1nNsnand/Pi1η=/summationtext∞
n=1nNηn.H e r e Nsnis for n=1, the number called here
Nsofsparticles (named s1 pseudoparticles in Ref. [ 34])
and of spin n-strings for n>1 and Nηnis for n=1t h e
number called here Nηofηparticles (named η1 pseudopar-
ticles in Ref. [ 34]) and of charge n-strings for n>1. Such
particles and strings correspond to Bethe-ansatz quantumnumbers, spin and charge n-strings being described by non-
real complex rapidities and c,s, andηparticles by only real
rapidities [ 34,35].
The internal degrees of freedom of one sparticle refers
to one unbound spin-singlet pair of spins 1 /2. The inter-
nal degrees of freedom of one spin n-string refers to n>1
spin-singlet pairs of spin 1 /2’s bound within it. The internal
degrees of freedom of one ηparticle refers to one unbound
η-spin-singlet pair ofη-spin 1 /2’s. The internal degrees of
freedom of one charge n-string refers to n>1η-spin-singlet
pairs of η-spin 1 /2’s bound within it [ 34,35].
The Bethe-ansatz quantum numbers are distributed by c
andαnbands whose number of occupied values are the
above numbers N
candNαn, respectively, where α=η,sand
n=1,2,...,∞. The translational degrees of freedom of the
Ms=2Ssunpaired spins 1 /2 (and Mη=2Sηunpaired η-spins
1/2) are described by the sn-band holes (and c-band and ηn-
band holes [ 35].) For the subspace of this paper, only the cand
sbands whose quantum numbers qj=2π
LIc
jandqj=2π
LIs
j,
respectively, are given in Eqs. ( B2) and ( B3) of Appendix B
have finite occupancy.
Finally, the usual holons and spinons refer to the c- and
s-band holes in excited energy eigenstates of Sη=0 and
Ss=0 ground states, respectively. They describe translational
degrees of freedom of the unpaired η-spin 1 /2’s and unpaired
spin 1/2’s, respectively.
APPENDIX B: SOME USEFUL AND NEEDED QUANTITIES
The 1D Hubbard model with one fermion per site in the
subspace of the quantum problem studied in this paper in-volves the following Bethe-ansatz equations:
q
j=2
LNa/summationdisplay
j/prime=1Nc(qj/prime)a r c t a n/parenleftbigg/Lambda1(qj)−sink(qj/prime)
u/parenrightbigg
−2
LN↑/summationdisplay
j/prime=1Ns(qj/prime)a r c t a n/parenleftbigg/Lambda1(qj)−/Lambda1(qj/prime)
2u/parenrightbigg
,
where j=1,...,Nasand Nas=N↑,
k(qj)=qj−2
LN↑/summationdisplay
j/prime=1Ns(qj/prime)a r c t a n/parenleftbiggsink(qj)−/Lambda1(qj/prime)
u/parenrightbigg
,
where j=1,...,Nacand Nac=N=Na. (B1)
Here,
qj=2π
LIβ
jforβ=c,s, (B2)
where the j=1,...,Naβquantum numbers Iβ
jare for β=c,s
either integers or half-odd integers according to the following
195129-20ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
boundary conditions [ 7]:
Ic
j=0,±1,±2,... forN↓even
=±1/2,±3/2,±5/2,... forN↓odd,
Is
j=0,±1,±2,... forN↑odd
=±1/2,±3/2,±5/2,... forN↑even.(B3)
The energy dispersions εc(q) andεs(q/prime) that appear in the
one-particle spectra are defined as
εc(q)=¯εc(k(q)) and εs(q/prime)=¯εs(/Lambda1(q/prime)),where
¯εc(k)=−/Delta1MH+/integraldisplayk
πdk/prime2tηc(k/prime),
¯εs(/Lambda1)=/integraldisplay/Lambda1
Bd/Lambda1/prime2tηs(/Lambda1/prime), (B4)
and the corresponding group velocities are given by
vc(q)=∂εc(q)
∂qand vs(q/prime)=∂εs(q/prime)
∂q/prime. (B5)
In Eq. ( B4),/Delta1MHrefers to the Mott-Hubbard gap 2 /Delta1MH,
Eq. ( 6), and the distributions 2 tηc(k) and 2 tηs(/Lambda1) are solu-
tions of the coupled integral equations,
2tηc(k)=2tsink+cosk
πu/integraldisplayB
−Bd/Lambda12tηs(/Lambda1)
1+/parenleftbigsink−/Lambda1
u/parenrightbig2,
2tηs(/Lambda1)=1
πu/integraldisplayπ
−πdk2tηc(k)
1+/parenleftbig/Lambda1−sink
u/parenrightbig2
−1
2πu/integraldisplayB
−Bd/Lambda1/prime2tηs(/Lambda1/prime)
1+/parenleftbig/Lambda1−/Lambda1/prime
2u/parenrightbig2. (B6)
The rapidity distribution functions k(q)f o r q∈[−π,π ]
and/Lambda1(q/prime)f o r q/prime∈[−kF↑,kF↑] in the arguments of the disper-
sions ¯εcand ¯εs,E q s .( B4), are defined in terms of their inverse
functions,
q(k)=k+1
π/integraldisplayB
−Bd/Lambda12πσ(/Lambda1)a r c t a n/parenleftbiggsink−/Lambda1
u/parenrightbigg
fork∈[−π,π ],
q/prime(/Lambda1)=1
π/integraldisplayπ
−πdk2πρ(k)a r c t a n/parenleftbigg/Lambda1−sink
u/parenrightbigg
−1
π/integraldisplayB
−Bd/Lambda1/prime2πσ(/Lambda1/prime)a r c t a n/parenleftbigg/Lambda1−/Lambda1/prime
2u/parenrightbigg
,
for/Lambda1∈[−∞,∞], (B7)
respectively. The parameter Bin Eqs. ( B4), (B6), and ( B7)i s
such that
B=/Lambda1(kF↓) with
lim
m→0B=∞ and lim
m→1B=0. (B8)
Furthermore, the distributions
2πρ(k)=dq(k)
dk,2πσ(/Lambda1)=dq/prime(/Lambda1)
d/Lambda1, (B9)in Eqs. ( B7) are solutions of the coupled equations,
2πρ(k)=1+cosk
πu/integraldisplayB
−Bd/Lambda12πσ(/Lambda1)
1+/parenleftbigsink−/Lambda1
u/parenrightbig2,
2πσ(/Lambda1)=1
πu/integraldisplayπ
−πdk2πρ(k)
1+/parenleftbig/Lambda1−sink
u/parenrightbig2
−1
2πu/integraldisplayB
−Bd/Lambda1/prime2πσ(/Lambda1/prime)
1+/parenleftbig/Lambda1−/Lambda1/prime
2u/parenrightbig2, (B10)
and obey the sum rules
1
π/integraldisplayπ
−πdk2πρ(k)=2,1
π/integraldisplayB
−Bd/Lambda12πσ(/Lambda1)=(1−m).
(B11)
Ath=0 and m=0t h e c- and s-band energy dispersions,
Eq. ( B4), can for u>0 be written as
εc(q)=¯εc(kc(q)) for q∈[−π,π ],where
¯εc(k)=U
2−2tcosk
−4t/integraldisplay∞
0dωcos(ωsink)
ω(1+e2ωu)J1(ω)
fork∈[−π,π ] and
εs(q)=¯εs(/Lambda1s(q)) for q∈/bracketleftBig
−π
2,π
2/bracketrightBig
,where
¯εs(/Lambda1)=−2t/integraldisplay∞
0dωcos(ω/Lambda1)
ωcosh(ωu)J1(ω)
for/Lambda1∈[−∞,∞], (B12)
respectively. At m=0, the inverse functions of kc(q) and
/Lambda1s(q), Eqs. ( B7), read
q=qc(k)=k+2/integraldisplay∞
0dωsin(ωsink)
ω(1+e2ωu)J0(ω),
q=qs(/Lambda1)=/integraldisplay∞
0dωsin(ω/Lambda1)
ωcosh(ωu)J0(ω). (B13)
Here and above J0(ω) and J1(ω) are Bessel functions.
Form=0 and u>0 the bare c- and s-band energy disper-
sions defined below in Appendix Care given by
¯ε0
c(k)=−U
2−2tcosk
−4t/integraldisplay∞
0dωcos(ωsink)
ω(1+e2ωu)J1(ω)
fork∈[−π,π ],
¯ε0
s(/Lambda1)=−2t/integraldisplay∞
0dωcos(ω/Lambda1)
ωcosh(ωu)J1(ω)
for/Lambda1∈[−∞,∞]. (B14)
Theβ=c,sphase shifts in Eqs. ( 33) and ( 34) read
2π/Phi1β,β/prime(q,q/prime)=2π¯/Phi1β,β/prime(r,r/prime),where β,β/prime=c,s
r=sink(q)
u|β=c,r=/Lambda1(q)
u|β=s,
r/prime=sink(q/prime)
u|β/prime=c,r/prime=/Lambda1(q/prime)
u|β/prime=s, (B15)
195129-21CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
and the rapidity phase shifts obey the integral equations
¯/Phi1s,s(r,r/prime)=1
πarctan/parenleftbiggr−r/prime
2/parenrightbigg
+/integraldisplayB/u
−B/udr/prime/primeG(r,r/prime/prime)¯/Phi1s,s(r/prime/prime,r/prime),
¯/Phi1s,c(r,r/prime)=−1
πarctan( r−r/prime)
+/integraldisplayB/u
−B/udr/prime/primeG(r,r/prime/prime)¯/Phi1s,c(r/prime/prime,r/prime),
¯/Phi1c,c(r,r/prime)=1
π/integraldisplayB/u
−B/udr/prime/prime¯/Phi1s,c(r/prime/prime,r/prime)
1+(r−r/prime/prime)2,
¯/Phi1c,s(r,r/prime)=−1
πarctan( r−r/prime)
+1
π/integraldisplayB/u
−B/udr/prime/prime¯/Phi1s,s(r/prime/prime,r/prime)
1+(r−r/prime/prime)2, (B16)
where kernel G(r,r/prime)i sf o r u>0 given by
G(r,r/prime)=−1
2π/parenleftbigg1
1+((r−r/prime)/2)2/parenrightbigg
. (B17)
The phase shifts appearing in Eqs. ( 33) and ( 34) read
/Phi1s,s(ιkF↓,q/prime)=¯/Phi1s,s/parenleftbigg
ιB
u,/Lambda1(q/prime)
u/parenrightbigg
,
/Phi1s,c(ιkF↓,q)=¯/Phi1s,c/parenleftbigg
ιB
u,sink(q)
u/parenrightbigg
,
/Phi1c,c(ιπ,q)=¯/Phi1c,c/parenleftbigg
0,sink(q)
u/parenrightbigg
,
/Phi1c,s(ιπ,q/prime)=¯/Phi1c,s/parenleftbigg
0,/Lambda1(q/prime)
u/parenrightbigg
forι=±1.(B18)
The phase-shift related spectral parameters are given by
ξββ/prime=δβ,β/prime+/summationdisplay
ι=±1(ι)/Phi1β,β/prime(qFβ,ιqFβ/prime),
where β,β/prime=c,s, (B19)
and, except for unimportant 1 /Lcontributions,
qFs=kF↓,qFc=π. (B20)
For the present Mott-Hubbard insulator one has that
ξsc=0f o r u>0a t nf=1,
ξcc=1f o r u>0a t nf=1. (B21)
The entries of the conformal-field theory dressed-charge ma-
trixZ, which are the parameters Eqs. ( B19)[53], read
Z=/bracketleftbigg
ξccξcs
ξscξss/bracketrightbigg
=/bracketleftbigg
1ξcs
0ξss/bracketrightbigg
. (B22)
(Here the dressed-charge matrix definition of Ref. [ 54] has
been used, which is the transposition of that of Ref. [ 55].) For
u>0, the parameter ξcscontinuously decreases upon increas-
ing the spin density mfromξcs=1/2a t m=0t oξcs=0
form=1. On the other hand, for u>0 the parameter ξsscontinuously increases from ξss=1/√
2a tm=0t oξss=1
form=1.
Form=0 and u>0 the matrix, Eq. ( B22), and the c- and
s-particle phase shifts, Eqs. ( B18), simplify to
Z=/bracketleftbigg
1ξcs
0ξss/bracketrightbigg
=/bracketleftbigg11 /2
01/√
2/bracketrightbigg
(B23)
and
/Phi1c,c(ιπ,q)=/Psi1/parenleftbiggsinkc(q)
u/parenrightbigg
,
/Phi1c,s(ιπ,q)=1
2πarctan/parenleftbigg
sinh/parenleftbiggπ
2/Lambda1s(q)
u/parenrightbigg/parenrightbigg
,
/Phi1s,c(ιπ/2,q)=−ι
2√
2,
/Phi1s,s(ιπ/2,q)=ι
2√
2forq/negationslash=ιπ/2
=ι/parenleftbigg3
2√
2−1/parenrightbigg
forq=ιπ/2,(B24)
respectively, where u>0,ι=±1, and the function /Psi1(x)i s
given by
/Psi1(x)=1
π/integraldisplay∞
0dzsin(zx)
z(1+e2z)
=i
2πln/parenleftbigg/Gamma1/parenleftbig1
2+ix
4/parenrightbig
/Gamma1/parenleftbig
1−ix
4/parenrightbig
/Gamma1/parenleftbig1
2−ix
4/parenrightbig
/Gamma1/parenleftbig
1+ix
4/parenrightbig/parenrightbigg
. (B25)
Here/Gamma1(x) is the usual gamma function.
APPENDIX C: DERIV ATION OF THE MOTT-HUBBARD
GAP FOR m∈[0,1]
The derivation of the Mott-Hubbard gap involves ground
states and excited states described by only real Bethe-ansatzrapidities. We start by considering fermionic densities n
f∈
[0,1] and spin densities m∈[0,nf] for the 1D Hubbard
model in the subspaces spanned by energy eigenstates de-scribed by only real rapidities. This involves addition to theHamiltonian, Eq. ( 1), of the term μˆN=μ/summationtext
σ=↑,↓ˆNσwhere
μis the chemical potential. For such subspaces, the Bethe-
ansatz equations have for nf∈[0,1] the form, Eqs. ( B1)o f
Appendix B, only differing from the nf=1 case in the occu-
pancies of the c-band momentum distribution Nc(qj/prime), which
are such that Nc=NandNh
c=Na−N.
The first step of our derivation involves the introduction
ofc- and s-band momentum distribution deviations for the
excited energy eigenstates under consideration,
δNβ(qj)=Nβ(qj)−N0
β(qj)f o r β=c,s,where
j=1,...,Naβ,Nac=N,Nas=N↑. (C1)
Here,
N0
c(qj)=θ(qj−q−
Fc)θ(q+
Fc−qj),
(C2)
N0
s(qj)=θ(qj−q−
Fs)θ(q+
Fs−qj),
are the corresponding ground-state c- and s-band momen-
tum distributions for densities nf∈[0,1] and m∈[0,nf].
The c- and s-band Fermi momentum values q±
Fβwhere β=
195129-22ONE-PARTICLE SPECTRAL FUNCTIONS OF THE … PHYSICAL REVIEW B 103, 195129 (2021)
c,sappearing here are provided in Eqs. (C.4)–(C.11) of
Ref. [ 56]. Ignoring O(1/L) corrections within the present
thermodynamic limit simplifies the ground-state distributions,Eqs. ( C2), toN
0
β(qj)=θ(qFβ−|qj|), where
qFc=2kF=πnf,qFs=kF↓=πnf,↓. (C3)
The energy spectrum of the present subspace’s eigenstates
is within the Bethe-ansatz solution given by
E=Na/summationdisplay
j=1Nc(qj)/parenleftBig
−U
2−2tcosk(qj)/parenrightBig
−2μBh/parenleftBigg
1
2Na/summationdisplay
j=1Nc(qj)−N↑/summationdisplay
j=1Ns(qj)/parenrightBigg
+μNa/summationdisplay
j=1Nc(qj), (C4)
where the rapidity function k(qj) specific to each state is
defined by the Bethe-ansatz equations.
Next, we derive an energy functional suitable to our goal by
using in the Bethe-ansatz equations, Eqs. ( B1) of Appendix B,
and energy spectrum, Eq. ( C4), the c- and s-band momentum
distribution functions Nβ(qj)=N0
β(qj)+δNβ(qj). The com-
bined and consistent solution of those equations and spectraup to second order in the deviations δN
β(qj), Eqs. ( C1), leads
to
E=E0+δE,where
δE=/summationdisplay
β=c,sLβ/summationdisplay
j=1εβ(qj)δNβ(qj)
+1
L/summationdisplay
β=c,s/summationdisplay
β/prime=c,sLβ/summationdisplay
j=1Lβ/prime/summationdisplay
j/prime=11
2fββ/prime(qj,qj/prime)
×δNβ(qj)δNβ/prime(qj/prime). (C5)
Here the zeroth-order term E0is the energy of the ground
state corresponding to given values of the densities nf∈[0,1]
andm∈[0,nf]. The ffunctions in the second-order terms
involve the candsgroup velocities and phase shifts defined
in Appendix B. Their expressions, though, are not needed for
the present derivation since the corresponding second-ordercontributions in the deviations lead to 1 /Lcorrections to the
Mott-Hubbard gap that vanish in the thermodynamic limit.
Only the first-order terms in the deviations contribute to
the quantities calculated here. The c- and s-band energy dis-
persions ε
β(qj) in such first-order terms are found to be
given by εc(q)=¯εc(k(q)) and εs(q/prime)=¯εs(/Lambda1(q/prime)), Eq. ( B4)
of Appendix B, where the rapidity functions k(q) and/Lambda1(q/prime)
are defined by the Bethe-ansatz equations. Importantly, therapidity-dependent c- and s-band energy dispersions are for
densities n
f∈[0,1] and m∈[0,nf] found to obey the fol-
lowing relations:
¯εc(k)=μ−μBh+¯ε0
c(k),
¯εs(/Lambda1)=2μBh+¯ε0
s(/Lambda1)=¯ε0
s(/Lambda1)−¯ε0
s(B). (C6)The zero-energy level of the bare rapidity-dependent c- and
s-band energy dispersions ¯ ε0
c(k) and ¯ε0
s(/Lambda1) in these relations
is shifted relative to that of the corresponding energy disper-sions ¯ε
c(k) and ¯ εs(/Lambda1), respectively. The former dispersions
are found to be given by
¯ε0
c(k)=−U
2−2tcosk
+1
π/integraldisplayB
−Bd/Lambda12tηs(/Lambda1)a r c t a n/parenleftbiggsink−/Lambda1
u/parenrightbigg
,
¯ε0
s(/Lambda1)=/integraldisplay/Lambda1
∞d/Lambda1/prime2tηs(/Lambda1/prime)
=1
π/integraldisplayQ
−Qdk2tηc(k)a r c t a n/parenleftbigg/Lambda1−sink
u/parenrightbigg
−1
π/integraldisplayB
−Bd/Lambda1/prime2tηs(/Lambda1/prime)a r c t a n/parenleftbigg/Lambda1−/Lambda1/prime
2u/parenrightbigg
,(C7)
where Q=k(2kF) and the distributions 2 tηc(k) and 2 tηs(/Lambda1)
are solutions of the coupled integral equations, Eqs. ( B6)o f
Appendix B, with the integrals/integraltextπ
−πdkreplaced by/integraltextQ
−Qdk.
Related bare c- and s-band energy dispersions that are a func-
tion of the corresponding c- and s-band momentum values are
given by
ε0
c(q)=¯ε0
c(k(q)),ε0
s(q/prime)=¯ε0
s(/Lambda1(q/prime)), (C8)
where k(q) and /Lambda1(q/prime) are rapidity functions defined by the
Bethe-ansatz equations.
Finally, we use the important relations given in Eqs. ( C6)
fornf=1. The zero-energy level of the present quantum
problem corresponds for that fermionic density to vanishingchemical potential, μ=0, at the middle of the Mott-Hubbard
gap. Consistently, from the use of the expression ¯ ε
c(k)=
−/Delta1MH+/integraltextk
πdk/prime2tηc(k/prime), Eq. ( B4) of Appendix B, one finds
that ¯εc(π)=−/Delta1MH. On the other hand, from the use of the
expression ¯ εs(/Lambda1)=/integraltext/Lambda1
Bd/Lambda1/prime2tηs(/Lambda1/prime) in that equation, one
finds that the zero-energy level of the s-band energy dis-
persion refers to ¯ εs(B)=0. In terms of the corresponding
momentum-dependent energy dispersions εc(q) and εs(q/prime),
this gives εc(qFc)=εc(π)=−/Delta1MHandεs(qFs)=εs(kF↓)=
0, respectively.
Taking μ=0 at the middle of the Mott-Hubbard gap and
using ¯εc(π)=−/Delta1MHand ¯εs(B)=0, the relations in Eq. ( C6)
give
¯εc(π)=−/Delta1MH=−μBh+¯ε0
c(π), (C9)
and ¯εs(B)=0=2μBh+¯ε0
s(B) for the specific values k=π
and/Lambda1=B. It then follows that the Mott-Hubbard gap is
given by
2/Delta1MH=−2¯ε0
c(π)−¯ε0
s(B)=−2ε0
c(π)−ε0
s(kF↓) (C10)
and the spin-density curve reads h(m)=− ¯ε0
s(B)/(2μB),
which can be rewritten as given in Eq. ( 9).
The use of the bare energy dispersions’ expressions,
Eqs. ( C7), in that for the Mott-Hubbard gap 2 /Delta1MHgiven in
Eq. ( C10) leads to that gap expression provided in Eq. ( 6). Its
derivation for the whole spin density range m∈[0,1] was the
main goal of this Appendix.
195129-23CARMELO, ˇCADEŽ, AND SACRAMENTO PHYSICAL REVIEW B 103, 195129 (2021)
[1] Y . Mizuno, K. Tsutsui, T. Tohyama, and S. Maekawa, Phys.
Rev. B 62, R4769(R) (2000) .
[2] J. H. Mentink, K. Balzer, and M. Eckstein, Nat. Commun. 6,
6708 (2015) .
[3] K. Shinjo and T. Tohyama, Phys. Rev. B 98, 165103 (2018) .
[4] K. Zawadzki and A. E. Feiguin, Phys. Rev. B 100, 195124
(2019) .
[ 5 ] E .H .L i e ba n dF .Y .W u , P h y s .R e v .L e t t . 20, 1445 (1968) .
[ 6 ] E .H .L i e ba n dF .Y .W u , Physica A 321, 1 (2003) .
[7] M. Takahashi, Prog. Theor. Phys. 47, 69 (1972) ;
[8] M. J. Martins and P. B. Ramos, Nucl. Phys. B 522, 413 (1998) .
[9] O. I. Pâtu, A. Klümper, and A. Foerster, Phys. Rev. B 101,
035149 (2020) .
[10] J. M. P. Carmelo and T. ˇCadež, Phys. Rev. B 103, 045118
(2021) .
[11] J. M. P. Carmelo and T. ˇCadež, Nucl. Phys. B 904, 39 (2016) ;
Corrigendum: 961, 115233 (2020) .
[12] R. G. Pereira, K. Penc, S. R. White, P. D. Sacramento, and
J. M. P. Carmelo, P h y s .R e v .B 85, 165132 (2012) .
[13] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys.
75, 473 (2003) .
[14] M. Sing, U. Schwingenschlögl, R. Claessen, P. Blaha, J. M. P.
Carmelo, L. M. Martelo, P. D. Sacramento, M. Dressel, andC. S. Jacobsen, P h y s .R e v .B 68, 125111 (2003)
[15] J. M. P. Carmelo, K. Penc, P. D. Sacramento, M. Sing, and R.
Claessen, J. Phys.: Condens. Matter 18, 5191 (2006) .
[16] B. J. Kim, H. Koh, E. Rotenberg, S.-J. Oh, H. Eisaki, N.
Motoyama, S. Uchida, T. Tohyama, S. Maekawa, Z.-X. Shen,and C. Kim, Nat. Phys. 2, 397 (2006)
[17] M. Raczkowski, F. F. Assaad, and L. Pollet, Phys. Rev. B 91,
045137 (2015) .
[18] M. T Batchelor and A. Foerster, J. Phys. A: Math. Theor. 49,
173001 (2016) .
[19] X.-W. Guan, M. T. Batchelor, and C. Lee, Rev. Mod. Phys. 85,
1633 (2013) .
[20] N. T. Zinner, EPJ Web Conf. 113, 01002 (2016) .
[21] T. L. Dao, A. Georges, J. Dalibard, C. Salomon, and I.
Carusotto, Phy. Rev. Lett. 98, 240402 (2007) .
[22] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature (London) 454,
744 (2008) .
[23] D. Clément, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio,
P h y s .R e v .L e t t . 102, 155301 (2009) .
[24] N. Fabbri, S. D. Huber, D. Clément, L. Fallani, C. Fort, M.
Inguscio, and E. Altman, Phys. Rev. Lett. 109, 055301 (2012) .
[25] A. A. Ovchinnikov, Sov. Phys. JETP 30, 1160 (1970).
[26] A. Nocera, F. H. L. Essler, and A. E. Feiguin, P h y s .R e v .B 97,
045146 (2018) .
[27] V . Lante and A. Parola, P h y s .R e v .B 80, 195113 (2009) .
[28] H. Benthien and E. Jeckelmann, Phys. Rev. B 75, 205128
(2007) .[29] M. Balzer, W. Hanke, and M. Potthoff, Phys. Rev. B 77, 045133
(2008) .
[30] J. M. P. Carmelo, N. M. R. Peres, D. K. Campbell, and A. W.
Sandvik, Z. Phys. B 103, 217 (1997).
[31] D. Controzzi and F. H. L. Essler, P h y s .R e v .B 66, 165112
(2002) .
[32] D. Baeriswyl, J. Carmelo, and K. Maki, Synth. Met. 21, 271
(1987) .
[33] H. Frahm and T. Vekua, J. Stat. Mech. (2008) P01007 .
[34] J. M. P. Carmelo and T. ˇCadež, Nucl. Phys. B 914, 461
(2017) .
[35] J. M. P. Carmelo, S. Nemati, and T. Prosen, Nucl. Phys. B 930,
418 (2018) .
[36] J. M. P. Carmelo, S. Östlund, and M. J. Sampaio, Ann. Phys.
325, 1550 (2010) .
[37] H. Kishida, H. Matsuzaki, H. Okamoto, T. Manabe, M.
Yamashita, Y . Taguchi, and Y . Tokura, Nature (London) 405,
929 (2000) .
[38] J. M. P. Carmelo, K. Penc, and D. Bozi, Nucl. Phys. B 725, 421
(2005) ;737, 351(E) (2006) .
[39] J. M. P. Carmelo, D. Bozi, and K. Penc, J. Phys.: Condens.
Matter 20, 415103 (2008) .
[40] J. M. P. Carmelo and P. D. Sacramento, Phys. Rep. 749,1
(2018) .
[41] A. Imambekov and L. I. Glazman, Science 323, 228
(2009) .
[42] A. Imambekov, T. L. Schmidt, and L. I. Glazman, Rev. Mod.
Phys. 84, 1253 (2012) .
[43] S. Sorella and A. Parola, P h y s .R e v .L e t t . 76, 4604 (1996) .
[44] S. Sorella and A. Parola, P h y s .R e v .B 57, 6444 (1998) .
[45] K. Penc, K. Hallberg, F. Mila, and H. Shiba, P h y s .R e v .L e t t .
77, 1390 (1996) .
[46] K. Penc, K. Hallberg, F. Mila, and H. Shiba, Phys. Rev. B 55,
15475 (1997) .
[47] M. Ogata and H. Shiba, P h y s .R e v .B 41, 2326 (1990) .
[48] S. Sorella and A. Parola, J. Phys: Condens. Matter 4, 3589
(1992)
[49] F. H. L. Essler, V . E. Korepin, and K. Schoutens, Phys. Rev.
Lett. 67, 3848 (1991) .
[50] A. P. Kampf and J. R. Schrieffer, Phys. Rev. B 42, 7967
(1990) .
[51] A. V . Chubukov, Phys. Rev. B 52, R3840(R) (1995) .
[52] H. V . Kruis, I. P. McCulloch, Z. Nussinov, and J. Zaanen, Phys.
Rev. B 70, 075109 (2004) .
[53] J. M. P. Carmelo and A. H. Castro Neto, Phys. Rev. Lett. 70,
1904 (1993) .
[54] F. Woynarovich, J. Phys.: Math. Gen. 22, 4243 (1989) .
[55] H. Frahm and V . E. Korepin, P h y s .R e v .B 42, 10553 (1990) .
[56] J. M. P. Carmelo, J. M. Román, and K. Penc, Nucl. Phys. B 683,
387 (2004) .
195129-24 |
PhysRevB.95.085416.pdf | PHYSICAL REVIEW B 95, 085416 (2017)
Spin-orbit and anisotropy effects in unoccupied bulk, surface, and image-potential states on W(110)
Henry Wortelen,1,*J¨urgen Henk,2and Markus Donath1
1Physikalisches Institut, Westf ¨alische Wilhelms-Universit ¨at M ¨unster, Wilhelm-Klemm-Strasse 10, 48149 M ¨unster, Germany
2Institut f ¨ur Physik, Martin-Luther-Universit ¨at Halle-Wittenberg, Von-Seckendorff-Platz 1, 06120 Halle (Saale), Germany
(Received 10 January 2017; revised manuscript received 24 January 2017; published 9 February 2017)
We report on joint experimental and theoretical investigations of the unoccupied surface electronic structure of
W(110). The spin-resolved inverse-photoemission experiments reveal a number of bands influenced by spin-orbitinteraction and an image-potential state. The bands disperse differently within the two nonequivalent mirror planesof the surface, which is explained by their origin and their localization within the surface region. Surprisingly,the image-potential state also exhibits anisotropic dispersion, although it is strongly located within the surfacebarrier. The experimental findings are confirmed by first-principles electronic-structure calculations.
DOI: 10.1103/PhysRevB.95.085416
I. INTRODUCTION
Tungsten is a 5 dtransition metal with a high atomic number
(Z=74) and, therefore, well known for a variety of properties
that are produced by spin-orbit coupling (SOC). For decades,it has been used for analyzing the spin polarization of electronsby spin-dependent low-energy electron diffraction from aW(001) surface [ 1,2]. Concerning W(110), detailed stud-
ies with spin- and angle-resolved photoemission (SARPES)revealed surface states with Rashba-type spin splitting onclean as well as on hydrogen- and alkali-covered surfaces[3–10]. And recently, the experimental identification of a spin-
polarized Dirac-cone-like surface state in a spin-orbit-inducedsymmetry gap [ 11–14] has called forth a number of theoretical
studies [ 15–18]. This particular state is highly anisotropic:
it exhibits linear dispersion along
/Gamma1H, while it is strongly
deformed along /Gamma1N, resulting in a flattened or squeezed Dirac
cone. On top of this, it is topologically protected by mirrorsymmetry [ 19]. Hence, W belongs to the class of topological
crystalline transition metals.
As the occupied bands, including their spin texture, have
been investigated in detail, there are only a few studies of theunoccupied bands [ 20–22]. All these inverse-photoemission
(IPE) measurements are restricted to the center of the Bril-louin zone
/Gamma1; an exception is Ref. [ 23], in which angle-
resolved spectra have been preliminary interpreted. A recentspin-resolved IPE study addresses spin signals from highlysymmetric unpolarized states at
/Gamma1(Ref. [ 24]). No information
on dispersing bands and their spin polarization is availableso far.
The intention of this paper is to shed more light on the
unoccupied surface electronic structure of W(110), which isexpected to be strongly influenced by SOC. Therefore, thissurface is studied experimentally with spin-resolved IPE andtheoretically by first-principles calculations. We focus on theenergy dispersions along the nonequivalent high-symmetrylines
/Gamma1N and /Gamma1H; their comparison reveals the anisotropy of
the dispersions of the unoccupied states.
While the occupied and unoccupied surface states are
derived from bulk states, it is not too surprising that theyexhibit anisotropic dispersions: they “inherit” the twofold
*Corresponding author: henry.wortelen@uni-muenster.desymmetry of the bulk states, which is corroborated by theirlocalization within the surface region. This picture may nothold for image-potential states because these are derivedfrom the surface barrier (rather than from bulk states) andare localized mostly within the surface barrier, that is, withtypically small penetration into the surface layers. Therefore,we pay special attention to an image-potential state andput its expected isotropic free-electron-like dispersion to thetest.
The paper is organized as follows. In Sec. IIwe present
details of the experiment (Sec. II A) and of the theory
(Sec. II B). Results are discussed in Sec. III: band dispersions
(Sec. III A ), spin polarizations (Sec. III B), and properties of
the image-potential state (Sec. III C).
II. METHODS
A. Experimental details
Spin- and angle-resolved IPE experiments have been
performed in a multifunctional ultrahigh-vacuum system [ 25].
The geometry of the experiment is schematically shown inFig. 1(a). An electron beam from a spin-polarized electron
source impinges on the sample at a defined angle θ, which
is varied by rotating the sample about the yaxis [ 26]. For
symmetry reasons, only the spin component in the surfaceplane and perpendicular to the momentum k
/bardbl, the so-called
Rashba component, is expected to result in spin asymmetries[15]. Therefore, we will only show measurements with
electrons being spin-polarized in this direction. The electronincidence angle θwas varied within mirror planes of the
sample. Thereby, the electron momentum k
/bardblwas varied along
the two high-symmetry directions /Gamma1N and /Gamma1H of the surface
Brillouin zone, as defined in Fig. 1(b). The particular mirror
plane was selected by azimuthal rotation of the sample.
The emitted photons with an energy of ¯ hω=9.9e V h a v e
been detected by three Geiger-M ¨uller counters, CA,CB, and
CC. The detection geometries of these counters are speci-
fied by ( α,β): for CA(70◦,180◦), for CB(35◦,90◦), and for
CC(35◦,69◦), as sketched in Fig. 4. The polar angle αand the
azimuth βare defined with respect to the Cartesian coordinate
system, as shown in Fig. 1(a).T h ezaxis coincides with the
electron incidence direction. The overall energy resolution ofour IPE experiment was about 350 meV (Ref. [ 25]).
2469-9950/2017/95(8)/085416(7) 085416-1 ©2017 American Physical SocietyHENRY WORTELEN, J ¨URGEN HENK, AND MARKUS DONATH PHYSICAL REVIEW B 95, 085416 (2017)
FIG. 1. (a) Experimental geometry of the spin- and angle-
resolved inverse-photoemission experiment. (b) LEED pattern of theW(110) surface with superimposed surface Brillouin zone (SBZ).
The W(110) surface was cleaned by repeated cycles
of heating in an oxygen atmosphere (from 6 ×10−8mbar
down to 1 ×10−8mbar) at 1500 K and subsequent flashing to
2300 K. During the final flash, the pressure did not exceed6×10
−9mbar. This cleaning procedure was successful in
removing contaminants, such as carbon and oxygen, from thesurface. The surface quality was confirmed by Auger electronspectra showing no intensity from contaminants and by a sharp(1×1) low-energy-electron diffraction (LEED) pattern with
low background intensity [Fig. 1(b)]. The W(110) sample was
at room temperature during the IPE measurements.
B. Theoretical details
The electronic structure of W(110) has been calculated
within density-functional theory, using generalized gradientexchange-correlation functionals [ 27,28]( s e ea l s oR e f .[ 24]).
We have applied relativistic multiple-scattering theory asformulated in the Korringa-Kohn-Rostoker (KKR) approach[29,30]. In this approach, spin-orbit coupling is accounted for
by solving the Dirac equation. The site-dependent potentialshave been initially obtained by a full-potential augmentedplane waves (FLAPW) calculation, performed with theFLEUR code [ 31]; these were then transferred to our KKR
code. The muffin-tin form of the potentials used in KKRreproduces very well the full-potential results obtained byFLAPW. The present computational setup follows closely thatof Ref. [ 32].
The surface electronic structure is described by the KKR
Green’s function Gof the semi-infinite system. The spectral
density
n
l(E,k/bardbl)=−1
πIm TrGll(E+iη,k/bardbl)
of layer lis computed for a small positive η(0.02 eV). It
is decomposed with respect to angular momentum and spinprojection. Spin textures are discussed by means of spindifferences n
l↑(E,k/bardbl)−nl↓(E,k/bardbl), in which ↑and↓refer to
a quantization axis specified by the experiments. For the /Gamma1H
and the /Gamma1N lines, the electron spin lies within the surface plane
and is perpendicular to the wave vector, which is commonlynamed the Rashba-type spin texture.
We have taken into account the surface relaxation that has
been determined from ab initio earlier [ 19]. The surface barrier
is modeled by a smooth function as proposed by Rundgren andMalmstr ¨om [33].θΓ
W1W2
IS
W0N
1°
-1°
-3°3°5°7°11°
-5°
-9° ytisnetnI(stinu .bra)
01 23 456 -1 01 23 456 -1
E- (eV)EF E- (eV)EF0°2°4°6°
-2°
-4°
-6°
-8°
-10°
-12°10°12°14°16°(a) (b)HΓ
θ
W0W2
ISW1
FIG. 2. Spin-integrated IPE spectra of W(110) for various angles
of electron incidence θalong (a) /Gamma1Na n d( b ) /Gamma1H. Photons were
detected by counter CC. Distinct intensity structures are indicated
by W0, W1, W2, and IS. The blue lines serve as guides to
the eye, indicating the dispersion of the spectral features. For
normal incidence, the gray solid line indicates the spectrum takenapproximately 3 h after the data indicated as circles.
III. RESULTS AND DISCUSSION
A. Dispersion
Figure 2presents spin-integrated IPE spectra, obtained by
counter CC, for various angles of electron incidence θalong
(a)/Gamma1N and (b) /Gamma1H. Four spectral features—W0, W1, W2,
and IS—with distinct energy dispersion, indicated by bluelines, are clearly observed. Figures 3(a) and3(e) summarize
the dispersions for W0, W1, and W2 in an E(k
/bardbl) diagram. For
a discussion of the spectral feature IS the reader is referred toSec. III C.
To determine accurately the energetic peak positions of the
spectral features, we utilized three methods: (i) a fitting routineas described in Refs. [ 34,35], (ii) determining the minima of the
085416-2SPIN-ORBIT AND ANISOTROPY EFFECTS IN . . . PHYSICAL REVIEW B 95, 085416 (2017)
4
3
2
1
0
-0.4 -0.2 0.0 0.2 0.4)Ve(
E-EF
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
k( Å)||-14
3
2
1
0)Ve(
E-EFΓ N N
Γ H HΓ N N
Γ H HΓ N N
Γ H H(b) (c) (d)
(f) (g) (h)W0W1W2
W0W1W2
W0W1W2
W1W2
W0W1W2W0W2
W1W2
W1
W0W2
W1
4
3
2
1
0
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4Γ H H Γ H H
k( Å)||-14
3
2
1
0Γ N N
Γ H HΓ N N
Γ H HΓ N N
Γ H HΓ N N Γ N N
(b) (c) (d)
(f) (g) (h)
W0W1W2
W1W2
W0W1W2W0W2
W1W2
W1
W0W2
W1(a)
(e)
FIG. 3. E(k/bardbl) diagrams for /Gamma1N( t o pr o w )a n d /Gamma1H (bottom row). (a), (e) Experimental results derived from spin-integrated IPE spectra
in Fig. 2via fitting procedure (black solid dots; error bars are smaller than the symbol size except for W0) and via second derivative (red
open diamonds). (b)–(d), (f)–(h) Computed spectral densities n(E,k/bardbl) for (b), (f) a bulk layer and for (c), (g) the topmost surface layer. The
panels share a linear gray scale (white: minimum; dark: maximum values). (d), (h) Difference of the spin-resolved surface spectral densities
n↑(E,k/bardbl)−n↓(E,k/bardbl), illustrated in red (blue) where spin-up (spin-down) intensity prevails (white denotes zero spin difference).
second derivatives d2I/dE2, and (iii) estimating the maximum
intensity by eye.
The determination of the positions of intensity maxima that
are closer to EFthan the experimental resolution (here: W0)
is a delicate issue [ 34]. To obtain a reasonable fit for W0, only
the energy position has been optimized, while the intensityand linewidth have been kept fixed. Hence, the fitting method(i) is the method of choice for W0, yet these results have tobe handled with care. For W1 and W2, the results of all threemethods agree within the symbol size of the data points inFigs. 3(a)and3(e).
The experimental data are compared with calculated bulk
and surface spectral densities in Figs. 3(b)and3(f)as well as in
Figs. 3(c) and3(g), respectively. Furthermore, the difference
between the spin-projected surface spectral densities is shownin Figs. 3(d) and3(h).
W0 around
/Gamma1, just above EF, is attributed to transitions into
a surface-derived state, appearing with high surface spectraldensity in Figs. 3(c) and3(g). With a maximum energy at
/Gamma1, it disperses downwards in energy for increasing |k/bardbl|and
eventually crosses EF. The Fermi-level crossings along /Gamma1N
and/Gamma1H differ; this anisotropic dispersion is consistent with
the twofold rotational symmetry of the W(110) surface. Thedispersion of W0 in the occupied regime (below the Fermilevel) has been investigated by ARPES experiments [ 12].
The IPE experiments show that the intensity for W0 is
higher for large photon-detection angles (counters C
AandCC)
than for smaller ones (counter CB). This suggests that W0
is composed of orbitals with a dipole axis orientated alongthe surface normal. Indeed, the orbital decomposition of thecomputed spectral density assigns a pronounced p
zcharacter
to W0. Recently, this surface state was identified to betopologically nontrivial [ 19]: it is one of two surface states
along
/Gamma1H that are topologically protected by mirror symmetry.
W1 at 2 .3 eV and W2 at 3 .2e V a t /Gamma1are assigned to
transitions into unoccupied dstates of tungsten [ 22]. They
consist of bulk and surface contributions; see Figs. 3(b),3(c),
3(f), and 3(g). Along /Gamma1N, W1 has strong negative dispersion
and disappears for |θ|>11◦, while along /Gamma1H it has positive
dispersion and merges with W2 into one feature for |θ|>16◦.
Hence, W1 forms a saddle point in E(k/bardbl)a t/Gamma1. W2 has positive
dispersion along /Gamma1N, while it shows nearly no dispersion
along /Gamma1H. Both W1 and W2 disperse differently in the
two high-symmetry directions, emphasizing again the twofoldsymmetry of W(110).
B. Spin information
We now discuss and interpret the spin dependence of the
IPE intensities. Figure 4presents spin-resolved spectra for W1
and W2 for various angles of electron incidence θalong (a)
/Gamma1N and (b) /Gamma1H, obtained with the three counters CA,CB, and
CC. The corresponding geometries are sketched in the top part
of the figure; red up-pointing triangles represent spin-up, whileblue down-pointing triangles represent spin-down intensities.
For W0 and W1, the calculations predict a tiny k
/bardbl-
dependent Rashba-type spin splitting into two oppositely spin-polarized states [see Figs. 3(d) and3(h)]. This splitting is too
small to be resolved by the experiment, which is corroboratedby the fact that we do not observe any spin dependence for W0
085416-3HENRY WORTELEN, J ¨URGEN HENK, AND MARKUS DONATH PHYSICAL REVIEW B 95, 085416 (2017)
FIG. 4. Spin-resolved IPE spectra of W(110) for various angles of electron incidence θalong (a) /Gamma1Na n d( b ) /Gamma1H. Photons were detected
by counter CA,CB,a n dCC. Spin-up (spin-down) intensities are marked as red up-pointing (blue down-pointing) triangles. Spin differences
between the two spin channels are marked as pale red (blue) areas where spin-up (spin-down) intensity exceeds.
(not shown) and for W1 in four of the six detection geometries
(shown in Fig. 4). However, in two geometries, i.e., CAand
CCfor/Gamma1N, we observe strong spin-dependent intensities for
W1, independent of k/bardbland with opposite sign for counters
CAandCC. This effect, which is observed even for normal
electron incidence, is a consequence of the symmetry breakingof the IPE experiment by the counter position; it is discussed indetail in Ref. [ 24]. It persists for off-normal electron incidence
and appears in addition to any k
/bardbl-dependent Rashba-type
spin effect. Via rotating the sample azimuthally by /Phi1these
spin-dependent intensities show a sinusoidal behavior witha periodicity of 180
◦which is consistent with the twofold
symmetry of the sample [ 36]. For W1 along /Gamma1H, we observe
a tiny trend that spin-up (spin-down) intensity prevails forhigher positive (negative) angles, which, however, appearsmore pronounced in theory.
For W2 along
/Gamma1N,k/bardbl-dependent spin asymmetries are
observed for all counters, even for small angles of elec-tron incidence. While for positive angles spin-up intensitiesdominate, spin-down is dominant for negative angles. The
spin signal is most pronounced for angles around 8
◦.F o r
normal electron incidence θ=0◦, the spin asymmetry is
zero. W2’s spin texture shows the signature of a Rashba-typewave-vector-dependent spin polarization. A spin-dependentenergy splitting is, however, not observed. Compared with
/Gamma1N, the spectra along /Gamma1H show only minor spin effects,
as predicted by theory [see Figs. 3(d) and 3(h)]. This is
a consequence of the orbital compositions of the involvedstates; it highlights once more the difference between the twohigh-symmetry lines also in view of the spin information.
C. Effective mass of the image-potential state
We now come back to the spectral feature IS, briefly
addressed in the description of Fig. 2. For normal electron
incidence, it appears at 4 .53 eV and disperses to higher
energies away from /Gamma1. In earlier studies, it was identified
as a transition into image-potential-induced surface states
085416-4SPIN-ORBIT AND ANISOTROPY EFFECTS IN . . . PHYSICAL REVIEW B 95, 085416 (2017)
Γ H H
k( Å)||-1)Ve(
E-EF
k( Å)||-1)Ve(
E-EFΓ N N )b( )a(
FIG. 5. Dispersion of the image potential state IS along the high-symmetry lines (a) /Gamma1Na n d( b ) /Gamma1H.E(k/bardbl) data derived from spin-integrated
IPE spectra in Fig. 2. To give a measure of uncertainty three different approaches have been used: (i) fitting procedure, (ii) second derivative,
and (iii) by eye. Effective mass m∗/m eof IS determined by fitting a quadratic function to the data (black line): (a) m∗/m e=0.93±0.06, (b)
m∗/m e=1.36±0.10. The free-electron-like parabola with m∗/m e=1 is indicated as red thin line.
[22,24,37]. More precisely, the states lie in a gap for states
with/Sigma11symmetry [ 37]. However, spin-orbit coupling allows
hybridization with states of /Sigma13and/Sigma14symmetry existing in
this energy range. Therefore, the image-potential states onW(110) are surface resonances with finite probability densityin the bulk. Image-potential states are pinned to the vacuumlevel and are expected to show free-electron-like dispersionparallel to the surface [ 38]. Indeed, experiments on ultrathin
Fe films on W(110) show that IS shifts to lower and higherenergies according to the thickness-dependent change of thework function of this overlayer system [ 39]. In addition,
IS appears with higher spectral intensity for larger photon-detection angles (not shown); this suggests a composition oforbitals with dipole axis along the surface normal, as expectedfor an image-potential state [ 40]. Spin-resolved measurements
(not shown) did not show any Rashba-type spin splitting of theimage-potential state, which is expected to be only a few meVand is therefore below the detection limit.
Image-potential states exhibit free-electron-like energy
dispersions with effective masses m
∗/meon the order of 1 ( me
electron mass). Detailed studies with IPE and two-photon-
photoemission on various materials and surfaces resulted inratios m
∗/mebetween 0.95 and 1.3 (Ref. [ 41]). Deviations
from 1 have been intensively discussed in the literature, e.g.,in the case of m
∗/me=1.3 for Ag(100) (Ref. [ 42]).Artificially
created anisotropic surfaces are reported to show anisotropicdispersion behavior of image-potential states: dispersion alongand perpendicular to atomic steps on Cu(100) (Ref. [ 43]) and
to indium chains on Si(111) (Ref. [ 44]). To the best of our
knowledge, there have been no reports of an experimentallyobserved anisotropic effective mass on a naturally anisotropic
surface. There is only a theoretical prediction for the specificcase of Be(10
10) (Ref. [ 45]). The dispersions of the image-
potential states along the high-symmetry lines /Gamma1M and /Gamma1A
differ significantly, which is caused by a large penetrationof the states’ wave functions into the bulk [ 46] and by the
anisotropic band gap edges of states with certain symmetries.The authors’ call for experiments remains unheard.From the preceding it is evident that W(110) is a good
candidate to test the hypothesis of an anisotropic effectivemass of the image-potential state for two reasons. First,all bulk and surface states show a strongly anisotropicbehavior; and second, the image-potential states on W(110)are expected to show a considerable bulk penetration depthdue to the lack of a total energy gap in the respective energyrange.
Figure 5depicts the dispersion of the image-potential state
along both high-symmetry lines
/Gamma1N and /Gamma1H. The energetic
positions of the spin-integrated data for IS (see Fig. 2)
were deduced by the three methods described above (seeSec. III A ). The black bars in Fig. 5reflect the uncertainty
of the energetic positions as obtained from the three methods.The effective masses m
∗/me, determined by fitting parabolas
to the data in Figs. 5(a) and5(b), read 0 .93±0.06 along /Gamma1N
and 1.36±0.10 along /Gamma1H, thereby revealing a substantial
anisotropy of the free-electron-like paraboloid.
Our spectral-density calculations confirm the trend that
m∗/mealong /Gamma1N( 0.72±0.05) is smaller than m∗/me
along /Gamma1H( 0.93±0.10). Recall that the surface barrier is
modeled by a static smooth function. An improved barriershape has to depend on both energy and wave vector [ 47];
such a dynamic barrier could provide a better descriptionof the image-potential state’s dispersion but relies on fittingparameters to the experimental dispersions.
The size of the effective mass is at variance with two-
photon-photoemission results, which report an anomalouslyand unexplained small effective mass of 0.5 along
/Gamma1H
(Ref. [ 37]). All attempts to explain this unexpected small
number on the basis of the band structure of W(110) resultedin an increased rather than a decreased effective mass.Speculations about an experimental artifact caused by a biasvoltage applied to the sample in order to extract the low-energy electrons and leading to a reduced angular distributioncould not be confirmed. Therefore, the unexpected loweffective mass observed by two-photon photoemission remainspuzzling.
085416-5HENRY WORTELEN, J ¨URGEN HENK, AND MARKUS DONATH PHYSICAL REVIEW B 95, 085416 (2017)
The above finding—different effective masses for the
nonequivalent high-symmetry directions—is an observationof an anisotropic dispersion of an image-potential state ona naturally anisotropic surface. It is in qualitative agreementwith the spectral-density calculations. This result proves thatthe image-potential-state electrons are not completely free;they reflect the twofold symmetry of the (110) surface andits electronic structure via a finite probability density in thebulk.
IV . CONCLUSIONS
The twofold rotational symmetry of W(110) manifests itself
in the anisotropic dispersions of the unoccupied spin-polarizedsurface electronic structure. As the occupied surface states arestrongly spin-polarized, due to the Rashba-Bychkov spin-orbitcoupling, their unoccupied pendants are spin-polarized as well.
On top of this, we find that an image-potential state exhibitsas well anisotropic dispersion; the latter is quantified by thesizable difference of the effective masses of the parabolicdispersions for the two nonequivalent mirror planes of thesurface. While anisotropic dispersions of image-potentialstates have been reported for stepped surfaces earlier, ourfindings prove this essential feature for a naturally anisotropicsurface.
ACKNOWLEDGMENTS
It is a pleasure to thank A. B. Schmidt and Th. Fauster
for fruitful discussions. Financial support by the DeutscheForschungsgemeinschaft (DFG) is gratefully acknowledged.
[1] J. Kirschner and R. Feder, Phys. Rev. Lett. 42,1008 (1979 ).
[2] D. Yu, C. Math, M. Meier, M. Escher, G. Rangelov, and
M. Donath, Surf. Sci. 601,5803 (2007 ).
[3] R. H. Gaylord and S. D. Kevan, Phys. Rev. B 36,9337 (1987 ).
[4] R. H. Gaylord and S. D. Kevan, Phys. Rev. B 37,8491 (1988 ).
[5] R. H. Gaylord, K. H. Jeong, and S. D. Kevan, Phys. Rev. Lett.
62,2036 (1989 ).
[6] E. Rotenberg and S. D. Kevan, Phys. Rev. Lett. 80,2905 (1998 ).
[7] E. Rotenberg, J. W. Chung, and S. D. Kevan, Phys. Rev. Lett.
82,4066 (1999 ).
[8] M. Hochstrasser, J. G. Tobin, E. Rotenberg, and S. D. Kevan,
Phys. Rev. Lett. 89,216802 (2002 ).
[9] A. M. Shikin, A. Varykhalov, G. V . Prudnikova, D. Usachov,
V . K. Adamchuk, Y . Yamada, J. D. Riley, and O. Rader,Phys. Rev. Lett. 100,057601 (2008 ).
[10] E. Rotenberg, O. Krupin, and S. D. Kevan, New J. Phys. 10,
023003 (2008 ).
[11] K. Miyamoto, A. Kimura, K. Kuroda, T. Okuda, K. Shimada,
H. Namatame, M. Taniguchi, and M. Donath, Phys. Rev. Lett.
108,066808 (2012 ).
[12] K. Miyamoto, A. Kimura, T. Okuda, K. Shimada, H. Iwasawa,
H. Hayashi, H. Namatame, M. Taniguchi, and M. Donath,Phys. Rev. B 86,161411 (2012 ).
[13] K. Miyamoto, A. Kimura, T. Okuda, and M. Donath, J. Electron.
Spectrosc. Relat. Phenom.
201,53(2015 ).
[14] K. Miyamoto, H. Wortelen, H. Mirhosseini, T. Okuda, A.
Kimura, H. Iwasawa, K. Shimada, J. Henk, and M. Donath,Phys. Rev. B 93,161403 (2016 ).
[15] H. Mirhosseini, M. Flieger, and J. Henk, New J. Phys. 15,
033019 (2013 ).
[16] H. Mirhosseini, F. Giebels, H. Gollisch, J. Henk, and R. Feder,
New J. Phys. 15,095017 (2013 ).
[17] J. Braun, K. Miyamoto, A. Kimura, T. Okuda, M. Donath, H.
Ebert, and J. Min ´ar,New J. Phys. 16,015005 (2014 ).
[18] G. Kirczenow, P h y s .R e v .B 94,205414 (2016 ).
[19] D. Thonig, T. Rauch, H. Mirhosseini, J. Henk, I. Mertig,
H. Wortelen, B. Engelkamp, A. B. Schmidt, and M. Donath,Phys. Rev. B 94,155132 (2016 ).
[20] D. Funnemann and H. Merz, J. Vac. Sci. Technol., A 5,657
(1987 ).[21] I. R. Collins, A. D. Laine, P. T. Andrews, and P. J. Durham,
J. Phys.: Condens. Matter 3,5307 (1991 ).
[22] D. Li, P. A. Dowben, J. E. Ortega, and F. J. Himpsel, Phys. Rev.
B47,12895 (1993 ).
[23] D. Funnemann, Winkelaufgel ¨oste inverse Photoemission an
Wolfram(110), Ph.D. thesis, M ¨unster University, 1986 .
[24] H. Wortelen, H. Mirhosseini, K. Miyamoto, A. B. Schmidt, J.
Henk, and M. Donath, P h y s .R e v .B 91,115420 (2015 ).
[25] M. Budke, T. Allmers, M. Donath, and G. Rangelov, Rev. Sci.
Instrum. 78
,113909 (2007 ).
[26] S. D. Stolwijk, H. Wortelen, A. B. Schmidt, and M. Donath,
Rev. Sci. Instrum. 85,013306 (2014 ).
[27] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 (1996 ).
[28] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 78,
1396 (1997 ).
[29] H. S. Nalwa (ed.), Handbook of Thin Film Materials (Academic
Press, San Diego, 2002).
[30] J. Zabloudil, R. Hammerling, L. Szunyogh, and P. Weinberger
(eds.), Electron Scattering in Solid Matter (Springer, Berlin,
2005).
[31] The J ¨ulich FLAPW code family: FLEUR. Forschungszentrum
J¨ulich, D-52425 J ¨ulich, Germany.
[32] F. Giebels, H. Gollisch, and R. Feder, Phys. Rev. B 87,035124
(2013 ).
[33] J. Rundgren and G. Malmstr ¨om,J. Phys. C 10,4671 (1977 ).
[34] M. Donath and V . Dose, Europhys. Lett. 9,821(1989 ).
[35] S. D. Stolwijk, A. B. Schmidt, and M. Donath, P h y s .R e v .B 82,
201412 (2010 ).
[36] In Ref. [ 24] all three counters show a periodicity of 180◦.B y
mistake, the periodicity of CBwas denoted as 90◦.
[37] U. Thomann, C. Reuß, T. Fauster, F. Passek, and M. Donath,
Phys. Rev. B 61,16163 (2000 ).
[38] P. M. Echenique and J. B. Pendry, J. Phys. C 11,2065 (1978 ).
[39] F. Passek, M. Donath, and K. Ertl, J. Magn. Magn. Mater. 159,
103(1996 ).
[40] V . Dose, W. Altmann, A. Goldmann, U. Kolac, and J. Rogozik,
Phys. Rev. Lett. 52,1919 (1984 ).
[41] W. Steinmann and T. Fauster, in Advanced Series in Physical
Chemistry, Vol. 5, Laser Spectroscopy and Photo-Chemistry on
085416-6SPIN-ORBIT AND ANISOTROPY EFFECTS IN . . . PHYSICAL REVIEW B 95, 085416 (2017)
Metal Surfaces ,P a r tI ,e d i t e db yI .H . - L .D aa n dW .H o( W o r l d
Scientific, Singapore, 1995), pp. 184–242.
[42] J. Pendry, C. Larsson, and P. Echenique, Surf. Sci. 166,57
(1986 ).
[43] J. E. Ortega, F. J. Himpsel, R. Haight, and D. R. Peale,
Phys. Rev. B 49,13859 (1994 ).[44] I. G. Hill and A. B. McLean, P h y s .R e v .L e t t . 82,2155
(1999 ).
[45] V . M. Silkin, E. V . Chulkov, and P. M. Echenique, Phys. Rev. B
60,7820 (1999 ).
[46] T. Fauster, Appl. Phys. A 59,479(1994 ).
[47] E. Tamura and R. Feder, Z. Phys. B 81,425(1990 ).
085416-7 |
PhysRevB.74.195125.pdf | Three-dimensional band structure of layered TiTe 2: Photoemission final-state effects
V. N. Strocov,1,*E. E. Krasovskii,2W. Schattke,2,3N. Barrett,4H. Berger,5D. Schrupp,6and R. Claessen6,7
1Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
2Institut für Theoretische Physik, Christian-Albrechts-Universität, D-24098 Kiel, Germany
3Donostia International Physics Center, 20018 San Sebastian, Basque Country, Spain
4CEA-DSM/DRECAM-SPCSI, CEA-Saclay, 91191 Gif-sur-Yvette, France
5Institut de Physique de la Matière Complexe, EPFL, CH-1015 Lausanne, Switzerland
6Experimentalphysik II, Universität Augsburg, D-86135 Augsburg, Germany
7Experimentelle Physik 4, Universität Würzburg, D-97074 Würzburg, Germany
/H20849Received 5 April 2006; published 28 November 2006 /H20850
Three-dimensional band structure of unoccupied and occupied states of the prototype layered material TiTe 2
is determined focusing on the /H9003Aline of the Brillouin zone. Dispersions and lifetimes of the unoccupied states,
acting as the final states in the photoemission process, are determined from a very-low-energy electron dif-fraction experiment supported by first-principles calculations based on a Bloch waves treatment of multiplescattering. The experimental unoccupied states of TiTe
2feature dramatic non-free-electron effects such as
multiband composition and nonparabolic dispersions. The valence band layer-perpendicular dispersions arethen determined from a photoemission experiment consistently interpreted on the basis of the experimentalfinal states to achieve control over the three-dimensional wave vector. The experimental results demonstrate theabsence of the Te 4 p
z*Fermi surface pocket at the /H9003point and significant self-energy renormalization of the
valence band dispersions. Photoemission calculations based on a Bloch waves formalism within the one-steptheory reveal limitations of understanding photoemission from layered materials such as TiTe
2in terms of
direct transitions.
DOI: 10.1103/PhysRevB.74.195125 PACS number /H20849s/H20850: 79.60. /H11002i, 71.15.Ap, 61.14. /H11002x, 73.20.At
I. INTRODUCTION
TiTe 2is a prototype material in a large family of layered
transition metal dichalcogenides /H20849TMDCs /H20850whose quasi-two-
dimensional /H20849quasi-2D /H20850structural and electronic properties
have been intensively studied during the past few decades/H20849for a recent review see, for example, Ref. 1/H20850. TiTe
2crystal-
lizes in the 1T-CdI 2structure, which is characterized by
tightly bound chalcogen-metal-chalcogen layers separatedby a van der Waals gap. Strong intralayer and weak inter-layer bonding in such a structure result in highly anisotropicquasi-2D properties of TiTe
2characterized, for example, by
a ratio of the out-of-plane to in-plane resistivity as much as35–40, a value typical of the TMDCs. The electronic struc-ture of TiTe
2is formed by partial overlap of the Te 5 p
derived valence states with the Ti 3 dderived conduction
states, which results in a semimetallic behavior of this mate-rial. Due to small electron-phonon coupling parameter /H20849/H9261
/H110110.22 /H20850it does not seem to exhibit superconductivity or
charge-density-wave instabilities typical of other quasi-2D
materials.
Extensive angle-resolved photoemission /H20849PE/H20850experiments
on TiTe
2have delivered a good knowledge of its band struc-
ture E/H20849k/H20850with resolution in the kspace /H20849for a few recent
entries see Refs. 2–5/H20850. The PE electron removal spectra are
relevant for the material properties because they reflect thehole spectral function A/H20849
/H9275,k/H20850weighted with the PE matrix
element. Of particular interest are such studies on the Fermi
surface /H20849FS/H20850immediately connected to the transport proper-
ties. For example, the Ti 3 dz2band forming an electron
pocket of the FS around the ML line of the Brillouin zone
/H20849BZ/H20850has been exploited as a playground to test the Fermiliquid theory.2–5However, such studies analyzed the PE data
mostly with respect to E/H20849k/H20850as a function of the layer-parallel
/H20849with the natural cleavage plane of TMDCs, surface-parallel /H20850
wave vector component k/H20648, remaining thus basically within a
2D view of the electronic structure.
Three-dimensional /H208493D/H20850effects in the electronic structure
of TiTe 2arise from the interlayer interactions. They are ex-
pressed by E/H20849k/H20850as a function of the layer-perpendicular
/H20849surface-perpendicular /H20850wave vector component k/H11036. Despite
the quasi-2D nature of TiTe 2, the 3D effects are significant.
First, PE spectra measured with variable photon energy h/H9263
suggest that the E/H20849k/H11036/H20850dispersion can reach a range of
/H110112.5 eV over the BZ extension.2,6The available band calcu-
lations yield similar figures. Furthermore, the very fact ofnonzero out-of-plane conductivity suggests existence ofbands having nonzero layer-perpendicular group velocity
/H11509E//H11509k/H11036at the FS. Recent high-resolution PE studies3,5have
found that even the model Ti 3 dz2derived FS pocket shows a
residual PE linewidth of 14–17 meV surviving in the limit of
negligible electron-phonon and electron-electron scattering.Extrapolated to the limit of negligible impurity scattering,
3
this figure suggests a residual 3D dispersion with /H11509E//H11509k/H11036of
the order of 0.12 eV Å.
k-resolved studies of the 3D effects in TiTe 2suffer from a
fundamental difficulty of the PE experiment: In a simplifiedpicture of the PE process, which includes photoelectron ex-citation within the crystal bulk and its escape into vacuum,k
/H11036is conserved at the excitation stage but gets distorted at
the escape stage. The information on the initial-state k/H11036can
however be recovered if the final-state surface-perpendiculardispersion E/H20849k
/H11036/H20850back into the crystal bulk is known. A com-
mon solution to this problem is the use of empirically ad-PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
1098-0121/2006/74 /H2084919/H20850/195125 /H2084914/H20850 ©2006 The American Physical Society 195125-1justed free-electron-like /H20849FE-like /H20850final states. However, for
TiTe 2such an approach fails. This is clear, for example, from
the results of PE band mapping of the valence band E/H20849k/H11036/H20850
from FE-like final states:2the resulting experimental points
are highly inconsistent. Another example is broadening of PEpeaks at low energies:
4under assumption of FE-like final
states it shows an energy dependence opposite to the trendpredicted by the mean free path “universal curve.” Thesefacts evidence that the final states in TiTe
2, similarly to other
quasi-2D materials,7–9feature dramatic non-free-electron
/H20849non-FE /H20850effects —deviations from the FE-like approximation
resulting from photoelectron multiple scattering by the crys-tal potential. Furthermore, the final states can experience sig-nificant energy shifts due to band- and k-dependent excited-
state self-energy corrections /H9004/H9018.
10
Another aspect of the PE experiment is that the finite
photoelectron lifetime damps the final-state wave function inthe surface-perpendicular direction towards the crystalinterior.
11,12Such a confinement results in intrinsic final-state
broadening in k/H11036. The PE peaks then reflect not exactly the
initial-state E/H20849k/H11036/H20850dispersion, but its average over the broad-
ening interval.
Interpretation of the PE data from TiTe 2with respect to
the valence band E/H20849k/H11036/H20850requires therefore knowledge of the
true final-state E/H20849k/H11036/H20850dispersions, including the non-FE and
self-energy effects, and the final-state lifetimes describing its
damping. This can be achieved with an independent very-low-energy electron diffraction /H20849VLEED /H20850experiment. The
relevance of VLEED to PE is based on the one-step PEtheory which, neglecting the electron-hole interactions, treatsthe final states as the time-reversed LEED states /H20849see, for
example, Ref. 11/H20850. In the VLEED spectra of elastic reflectiv-
ityR/H20849E/H20850, energies of the spectral structures reflect the char-
acteristic points in the final-state E/H20849k
/H11036/H20850such as the band gap
edges, and their broadening and relative amplitudes reflect
the corresponding lifetimes /H20849see Refs. 13and14and refer-
ences therein /H20850.
Here, we present a study of 3D effects in TiTe 2using a
combination of the VLEED and angle-resolved PE spec-troscopies. The study focuses on the /H9003Adirection of the BZ.
First, the final-state E/H20849k
/H11036/H20850dispersions and lifetimes are de-
termined from the VLEED experiment supported by first-
principles calculations of complex band structure. Thenon-FE effects in the final states such as nonparabolic dis-persions and multiband composition are analyzed in detail.Second, the valence band E/H20849k
/H11036/H20850is determined to a great
detail from extensive h/H9263-dependent PE experimental data in-
terpreted using the VLEED derived final states. The full con-trol over the 3D wave vector achieved with such a combina-tion of experimental techniques yields new findings aboutthe electronic structure of TiTe
2such as the absence of the
Te 4 pz*electron pocket of the FS.
II. UNOCCUPIED STATES
A. VLEED experiment and results
Our experimental technique is described in detail
elsewhere.15,16Briefly, we used a standard four-grid LEEDoptics operating in the retarding field mode: The electrons
are first accelerated in the gun to energies around 300 eV toform a well-focused beam, and then decelerated to the re-quired primary energy Ein a retarding field between the gun
and the sample, maintaining focusing down to the lowestenergies. For the angle-dependent measurements, distortionof the off-normal electron trajectories and thus incidentsurface-parallel wave vector K
/H20648due to the retarding field was
taken into account as described in Ref. 15. The VLEED
spectra were measured as the elastic electron transmissionspectra T/H20849E/H20850, which are related to the total elastic reflectivity
R/H20849E/H20850integrated over all diffracted beams as T/H20849E/H20850=1− R/H20849E/H20850.
The measurements were performed in the target current cir-
cuit, taking advantage of fairly structureless inelastic reflec-tivity contribution to the target current. The energy spread ofthe primary electrons was /H110110.25 eV HWHM. Atomically
clean surface of TiTe
2was obtained by usual in situ cleav-
age. The workfunction was determined to be 5.5±0.2 eV.
The experimental angle-dependent T/H20849E/H20850spectra are
shown in Fig. 1. They were measured under K/H20648variation
along the /H9003Kazimuth of the surface BZ /H20849/H9003AHK plane of the
bulk BZ /H20850. Corresponding energy dependences of the surface-
parallel incident wave vector K/H20648are shown in the inset. With
each spectrum taken at a fixed sample rotation angle, theretarding field increases the incident angle towards lower en-ergies, reducing energy variations of K
/H20648along the spectrum
compared to the field-free case.15
The experimental T/H20849E/H20850spectra show prominent structures
dispersing with K/H20648. In a series of previous works /H20849see, for
example, Refs. 7,14, and 16and the references therein /H20850it
has been established that the VLEED spectral structuresreflect unoccupied three-dimensional E/H20849k/H20850along the BZ
direction /H20849s/H20850defined by K
/H20648conservation. Of all bands avail-
able for given incident Eand K/H20648, only so-called coupling
/H20849orconducting /H20850bands —whose wave functions allow effec-
tive coupling to the incident plane wave and electron trans-port into the crystal—are involved in formation of theVLEED spectrum. Specifically, the T/H20849E/H20850sharp changes
/H20849identified as the dT/dEextremes /H20850reveal in the E/H20849k
/H11036/H20850dis-
persions of these bands the critical points such as the band
edges. In view of further implications of VLEED for analysisof the PE data, it is important to note that the same couplingbands effective in VLEED are effective as the final bands inthe PE process
14,17/H20849see Sec. II C 1 /H20850. Further, the coupling
bands in the PE context will be synonymously referred to asthe final bands.
K
/H20648dispersion of the experimental spectra is represented in
Fig. 2. Here, the energy intervals within the T/H20849E/H20850minima
/H20849identified by d2T/dE2/H110220/H20850in each spectrum are represented
by white areas in /H20849K/H20648,E/H20850coordinates, and the intervals within
theT/H20849E/H20850maxima /H20849d2T/dE2/H110210/H20850by shaded areas. With the
above physical meaning of the T/H20849E/H20850structures, the borders
between these areas directly represent E/H20849k/H20648/H20850surface pro-
jected dispersion of the critical points in final-state E/H20849k/H11036/H20850
dispersion for given K/H20648.
Apart from the gross T/H20849E/H20850structures due to the bulk
E/H20849k/H20850, weak narrow oscillations can be distinguished in the
K/H20648dispersion map. They are placed near the diffraction
thresholds E=/H92572/H20849K/H20648+g/H208502/2mmarked in Fig. 2by dashedSTROCOV et al. PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-2lines corresponding to different g. Such oscillations manifest
the surface resonance /H20849SR/H20850states formed by the pre-
emergent diffraction beam traveling along the surfacethrough multiple reflections between the surface barrier andthe crystal bulk /H20849physics of the SR states is discussed in
detail in Refs. 14,18, and 19/H20850. The multiple scattering
mechanism makes the SR states similar in origin to the im-age potential states, but the reflection phases on the crystalside are different because of different energies falling abovethe vacuum level and K
/H20648associated with the surface-parallel
movement of the beam. With the wave functions concen-trated outside the bulk crystal, the SR states experienceweaker potential corrugations and show almost free-electrondispersion. Their presence, highly sensitive to the surfacecontamination, indicates excellent surface quality. SR phe-nomena have also been observed for other quasi-2D materi-als such as TiS
2.20
B. Computational procedure and results
1. Complex band structure and VLEED spectra
Reference calculations of the VLEED spectra and corre-
sponding unoccupied E/H20849k/H20850are crucial for interpretation of
the experimental VLEED data in terms of band structure. We
focus on the normal-incidence data reflecting E/H20849k/H20850along the
/H9003Adirection of the BZ. The cornerstone of our computa-
tional approach is the Bloch wave formalism of the multiplescattering LEED theory /H20849see, for example, the seminal works
in Refs. 21–23/H20850. In this formalism, the electron scattering in
the VLEED experiment is described by matching the elec-tron wave function in the vacuum half-space /H9021
vac/H20849r/H20850/H20849=a
superposition of the plane waves corresponding to the inci-
dent and all diffracted beams /H20850to that in the crystal half-space
/H9021c/H20849r/H20850/H20851= a superposition /H20858kAk/H9278k/H20849r/H20850of the Bloch waves
/H9278k/H20849r/H20850excited in the crystal /H20852under conservation of Eandk/H20648.
The set of /H9278k/H20849r/H20850in the semi-infinite crystal is determined
from the Schrödinger equation
/H20875−/H60362
2m/H9004+V/H20849r/H20850−iVi−E/H20876/H9278k/H20849r/H20850=0 ,
where V/H20849r/H20850is the crystal potential and the inelastic scattering
is included through the spatially constant absorption poten-
tialViconnected to the electron lifetime as Vi=/H6036//H9270.I nt h e
elastic limit Vi=0, the /H9278k/H20849r/H20850solutions are either propagating
into the crystal interior /H20849bulk Bloch waves, having real k/H11036/H20850or
damped in this direction /H20849surface ones, having complex k/H11036
with Im k/H11036reflecting the damping rate /H20850. With Vi/HS110050, all
/H9278k/H20849r/H20850become damped into the crystal interior, and are de-
scribed on equal footing by complex k/H11036/H20851note however that
FIG. 1. Experimental VLEED angle-dependent electron trans-
mission spectra T/H20849E/H20850measured along the /H9003Kazimuth of the surface
BZ at the indicated sample rotation angles. The inset shows thecorresponding energy dependences of incidence K
/H20648. The spectra
show prominent structures, reflecting 3-dimensional E/H20849k/H20850band
structure of the PE final states.
FIG. 2. K/H20648dispersion map of the experimental spectra from Fig.
1. The shaded and white areas show, respectively, the T/H20849E/H20850maxima
and minima energy intervals between the dT/dEextrema reflecting
the critical points in 3D final states. Grayscale within the shadedareas characterizes d
2T/dE2in logarithmic scale. Dashed lines
show the diffraction thresholds E=/H60362/H20849K/H20648+g/H208502/2mnear which the
surface resonances are found.THREE-DIMENSIONAL BAND STRUCTURE OF LAYERED … PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-3by virtue of the surface-parallel invariance of the LEED pro-
cess/H9278k/H20849r/H20850are undamped along the surface and have real k/H20648/H20852.
The corresponding E/H20849k/H20850is the complex band structure in the
sense of real Edepending on complex k/H11036. Calculation of the
complex E/H20849k/H20850and corresponding /H9278k/H20849r/H20850is an inverse band
structure problem: Given Eand k/H20648, the secular equation is
solved for complex k/H11036values. Computationally, this is the
most demanding part of the Bloch waves formalism.
A unique feature of the present computational scheme is
incorporation of a realistic potential in the surface region,where it significantly deviates from the periodic potential inthe bulk. Two self-consistent calculations are performed: /H20849i/H20850
for the infinite TiTe
2crystal, and /H20849ii/H20850for the surface region,
which is a fragment of a periodic slab with the unit cellcontaining three Te-Ti-Te threelayers /H20849nine atomic layers /H20850
and a vacuum region to separate the threelayers from eachother. The former yields the self-consistent V/H20849r/H20850in the bulk,
and the latter that in the surface region matching the bulk.
Figure 3shows the obtained potential distribution in the sur-
face region. The slab is thick enough so that the interactionbetween the slabs is negligible. The V/H20849r/H20850profile in the
middle of the slab coincides with that in the bulk crystal, and
in the vacuum region it grows to reach a constant value of5.6 eV, which is in good agreement with our VLEED experi-mental workfunction.
The LEED states in the self-consistent V/H20849r/H20850are calculated
with the recently developed embedding method,
24within
which a fragment is cut out of the slab unit cell and embed-ded between the bulk crystal half-space on one side and thevacuum half-space with a constant potential on another side.In the embedded region and vacuum half-space V
iis set to
zero. In the bulk half-space the LEED function is a superpo-sition of the inverse band structure solutions
/H9278k/H20849r/H20850, and in
the embedded region it is expanded in terms of the eigen-
functions of the repeated slab Hamiltonian for a given k/H20648.I n
the vacuum region the LEED wave function is representedby the incoming plane wave and reflected plane waves /H20849in-
cluding decaying ones /H20850corresponding to all surface recipro-
cal vectors g. The function in the embedded region matches
/H20849with high but finite accuracy /H20850the function in the crystal
half-space and matches /H20849exactly /H20850the function in the vacuum
half-space. As the Schrödinger equation in the crystal half-space is satisfied by construction, the problem reduces toobtaining the solution of the Schrödinger equation in theembedded region and in the vacuum half-space. This is done
by minimizing the energy deviation /H20648/H20849Hˆ−E/H20850/H9023/H20648within these
two regions and simultaneously minimizing the function and
derivative mismatch at the matching plane between the bulkcrystal and the embedded region. A variational method usedfor this purpose is described in detail in Ref. 24.
Further details of our computational methodology have
been presented in Refs. 16,24, and 25. Briefly, the standard
density functional theory /H20849DFT /H20850formalism with the local
density approximation /H20849LDA /H20850exchange correlation is used.
Both the self-consistent and scattering calculations are per-formed with the extended linearized augmented plane waves/H20849ELAPW /H20850method. The radial basis sets for the lower angular
momenta were extended by additional functions to ensurehigh accuracy of the wave functions over the energy regionup to 50 eV above E
F. The inverse band structure problem
for complex k/H11036is solved using an exact k·pmethod using a
basis set of bulk band structure wave functions. This methodallows computationally efficient reduction of the inverseband structure problem to a linear algebra eigenvalue prob-lem. The energy dependence of V
iis determined empirically
by fitting the energy broadening and relative amplitudes ofthe experimental spectral structures.
16,26
The theoretical normal-incidence T/H20849E/H20850spectrum in com-
parison with the experimental one is shown in Fig. 4. The
excellent agreement with the experiment regarding the posi-tions, energy broadenings and relative amplitudes of all spec-tral structures proves the relevance of our theoretical frame-work of the VLEED process in application to TiTe
2. The
correct description of the potential distribution in the surfaceregion has turned out to be crucial to achieve accurate theo-retical T/H20849E/H20850: The approximation of a steplike surface barrier,
which was found sufficient for NbSe
2and graphite,16,26in-
troduced a considerable error for TiTe 2.
The inset in Fig. 4shows the Vi/H20849E/H20850energy dependence
used in the calculations. It was estimated by varying Vito fit
the energy broadenings and relative amplitudes of the spec-tral structures in the experimental normal-incidence T/H20849E/H20850
spectrum. With the actual sensitivity of the theoretical spec-
tra to the variations of V
i, it was sufficient to judge the qual-
ity of the fit by eye in order to estimate Vito within ±20%.
The Vi/H20849E/H20850dependence shows a sharp increase in the low-
energy region. Our supplementary ab initio calculation of the
dielectric function in the random phase approximation yieldthe bulk plasmon energy /H6036
/H9275paround 18 eV, in agreement
with the available experimental data.27This suggests that the
main contribution to the Vi/H20849E/H20850increase is due to excitation of
the bulk plasmon. The characteristic plasmon step26is how-
ever not resolved in our Vi/H20849E/H20850dependence, within the accu-
racy of our Vievaluation.
The theoretical unoccupied E/H20849k/H20850along/H9003A, underlying the
normal-incidence T/H20849E/H20850calculations, is shown in Figs. 5/H20849a/H20850
and5/H20849b/H20850. Reflecting the damped nature of the /H9278k/H20849r/H20850Bloch
FIG. 3. /H20849Color online /H20850Calculated potential distribution in the
surface region between the bulk crystal and vacuum region. The useof this potential in VLEED calculations considerably improvesagreement with experiment over the steplike surface barrier.STROCOV et al. PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-4waves involved in the VLEED process, this is the complex
band structure in the sense of real Edepending on complex
k/H11036=Re k/H11036+iImk/H11036.O fa l l /H9278k/H20849r/H20850generated by our calcula-
tions, the figure shows only those characterized by a smalldamping rate Im k/H11036/H11021/H20648/H9003A/H20648. Note that with Vi/HS110050 the com-
plex band structure is radically different from that in the Vi
=0 limit: The E/H20849Rek/H11036/H20850dispersions pass through the band
gaps continuously, with the critical points surviving only as
distinct changes in the dispersion slope.13The band gaps are
better distinguished in the E/H20849Imk/H11036/H20850plot as loop-like en-
hancements of Im k/H11036.
2. Partial absorbed currents
Identification of the coupling bands, dominating in the
VLEED and PE processes, employed a calculation of partial
absorbed current sTkcharacterizing the partial contributions
of each /H9278k/H20849r/H20850, constituting the total LEED state in the crystal
/H9021c/H20849r/H20850=/H20858kAk/H9278k*/H20849r/H20850,t ot h e T/H20849E/H20850total absorbed current. In the
Vi/H110050 elastic case, Tkare calculated with the usual expression
Tk=i/H20841Ak/H208412/H6036
2m/H20873/H9278k*·/H11509
/H11509r/H11036/H9278k−/H9278k/H11509
/H11509r/H11036/H9278k*/H20874
for the propagating /H9278k/H20849r/H20850, and Tk=0 for the damped ones.
In the Vi/HS110050 case all /H9278k/H20849r/H20850become damped, and the usual
concept of the currents associated with propagating wave
functions collapses. In Refs. 13and17it was shown that the
current absorbed in the crystal appears in this case due to theelectrons inelastically scattered away from the coherent wavefunction, and can be calculated by integrating the density ofthe LEED state over the crystal half-space as
FIG. 4. Theoretical VLEED normal-incidence T/H20849E/H20850spectrum
compared with the experimental one from Fig. 1/H20849linear background
subtracted /H20850. The inset shows the used Vi/H20849E/H20850dependence. Notable
energy shifts between the theoretical and experimental T/H20849E/H20850struc-
tures are attributed to the self-energy corrections.
FIG. 5. /H20849a,b/H20850Theoretical unoccupied complex E/H20849k/H20850along /H9003Aas a function of k/H11036=Re k/H11036+iImk/H11036. Shown are only the Bloch waves
characterized by small damping rate Im k/H11036/H11021/H20648/H9003A/H20648. Grayscale shows the Tkpartial absorbed current contribution of each band to total T/H20849E/H20850.
Significant Tkvalues identify, in the multitude of bands available for given Eandk/H20648, the coupling bands dominating in the VLEED and PE
processes. Availability of a few such bands /H20849numbered 1 to 3 /H20850and their nonparabolic dispersions are beyond the free-electronlike approxi-
mation; /H20849c/H20850The corresponding theoretical T/H20849E/H20850anddT/dEspectra.THREE-DIMENSIONAL BAND STRUCTURE OF LAYERED … PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-5Tk=2Vi
/H6036/H20885
/H9024/H20841/H9021c/H20849r/H20850/H208412dr
/H20851with this generalization, the current conservation theorem
was utilized in our calculations as a criterion of the compu-tational accuracy: the sum of the currents carried by the in-cident and diffracted beams, i.e., the current in the vacuumhalf-space, must be equal to the current T/H20849E/H20850absorbed in the
crystal half-space /H20852. The T
kpartial absorbed currents are then
defined as
Tk=2Vi
/H6036/H20885
/H9024/H20841Ak/H9278k/H20849r/H20850/H208412dr
In the Vi→0 limit this expression reduces to the usual elastic
current. Physically, large Tkvalues are characteristic of /H9278k/H20849r/H20850
which /H208491/H20850effectively couple to the incident plane wave and
thus receive large excitation amplitudes, and /H208492/H20850penetrate
deep into the crystal to enable effective electron transport. Inthe free-electron case one Bloch wave, identical to the in-coming plane wave, receives T
k=1 and all others Tk=0.
In contrast to the Vi=0 case, with Vi/HS110050 the surface re-
lated/H9278k/H20849r/H20850can in principle acquire nonzero Tkand contrib-
ute to the total current similarly to the bulk-related ones.
Owing to the interference terms /H20848/H9024Ak·Ak/H9278k·/H20849r/H20850/H9278k/H20849r/H20850drbe-
tween the /H9278k/H20849r/H20850constituents of total /H9021c/H20849r/H20850in the expression
forT/H20849E/H20850, the sum of Tkis not exactly equal to total T/H20849E/H2085017,26
andTkvalues can even exceed 1.
In Fig. 5/H20849a/H20850the calculated Tkvalues for each band are
shown in grayscale. The bands with strong damping/H20648Imk
/H11036/H20648/H11022/H20648/H9003A/H20648are omitted from the plot, because they any-
way have vanishing Tk.
C. Properties of the final states
1. Role of the T kpartial absorbed currents in VLEED and PE
The unoccupied E/H20849k/H20850, due to the progressive increase
with energy of the number of bands folding into the reduced
BZ, always appears as a multitude of bands. This general factis illustrated by our results for TiTe
2in Figs. 5/H20849a/H20850and5/H20849b/H20850.
Among all bands, however, only a few coupling bands iden-tified by sufficient T
kvalues /H20849these bands are labeled 1−3 )
make significant contributions to the T/H20849E/H20850total absorbed cur-
rent and thus form the T/H20849E/H20850spectrum. The majority of other
bands in the multitude available for given Eandk/H20648are char-
acterized by vanishing Tkvalues and are thus ineffective in
the VLEED process.
By virtue of the time-reversal relation between the
VLEED state and PE final state, the latter is composed of thesame
/H9278k/H20849r/H20850. In analogy with the VLEED process, the partial
contribution of each /H9278k/H20849r/H20850to the total photocurrent I/H20849E/H20850can
be characterized by partial photocurrents I kdefined in the
framework of the one-step PE theory as
Ik/H11008/H20841/H20855Ak*/H9278k*/H20849r/H20850/H20841A·P/H20841/H9023/H20849r/H20850/H20856/H208412,
where Ais the electromagnetic field vector potential, pthe
momentum operator, and /H9023/H20849r/H20850the initial-state wave func-
tion. In Ref. 17it was shown that Ikare in fact proportional
toTksuch thatIk/H11008/H20841Mfi/H20849k/H20850/H208412Tk,
where Mfiis the phototransition matrix element /H20851involving
the oscillating part of the final-state /H9278k/H20849r/H20850to characterize the
phototransition process decoupled from the photoelectron
escape11/H20852. Therefore, the coupling bands dominating in
VLEED also dominate as the final bands in PE /H20849if not im-
peded by vanishing Mfi/H20850. Physically, the states effective in
coupling to the incoming plane wave and electron transportinto the crystal in VLEED are equally effective in photoelec-tron transport out of the crystal and coupling to the outgoingplane wave in PE. The coupling bands in our unoccupiedE/H20849k/H20850in Fig. 5/H20849a/H20850are thus exactly the final bands in the PE
process.
2. Non-free-electron effects
The final-state bands in TiTe 2, represented by the
E/H20849Rek/H11036/H20850panel of Fig. 5/H20849b/H20850, demonstrate dramatic non-FE
effects:
/H20849i/H20850Within the FE-like approximation, implying spatially
constant crystal potential, the PE final state with energyE
fincludes only one coupling band /H20849whose surface-parallel
wave vector matches K/H20648in vacuum /H20850whereas Tkof all
other bands are strictly zero.17,28Figure 5/H20849a/H20850shows that for
TiTe 2this is not the case: through the whole energy range the
final state comprises multiple /H20849mostly two /H20850coupling bands
having comparable Tk—and thus Ikpartial photocurrent—
magnitudes. As we will illustrate in Sec. III C 1, multipleRek
/H11036available in such a multiband final state result in mul-
tiple PE peaks corresponding to direct transitions at differentRek
/H11036.
Multiband composition of the final state is in fact a typical
non-FE effect, in conventional terms of PE spectroscopy of-ten referred to as umklapp /H20849see, for example, Ref. 29/H20850or
secondary cone emission.
30,31Note that this effect must in-
clude band hybridization in the final state.
The multiband composition significantly complicates the
relation of the final states to the VLEED spectra. In the caseof one coupling band this relation is simple: the band gapregions manifest themselves as the T/H20849E/H20850minima and the re-
gions of smooth dispersion between them as the T/H20849E/H20850
maxima, with the critical points corresponding to the dT/dE
extremes /H20849see, for example, Refs. 8and14/H20850. To illustrate this
relation in our case, in Fig. 5/H20849c/H20850we reproduce the calculated
T/H20849E/H20850spectrum together with dT/dE. One interesting region
is around 15 eV, where one of the two bands forming the
spectrum, the band 1, passes through a band gap. Its T
kde-
creases here due to enhanced damping of the corresponding
/H9278k/H20849r/H20850. Surprisingly, the total T/H20849E/H20850shows here a maximum.
This occurs due to even stronger increase of Tkin another
coupling band, the band 2. Physically, this increase can beexplained by that the enhanced damping of the band 1 re-duces the strengths of its hybridization with the band 2 de-termined by overlap of the corresponding
/H9278k/H20849r/H20850; as a result,
the latter gets closer to the free-electron character and yields
larger Tk. This example illustrates that in the multiband case
the band gaps are not necessarily manifested by T/H20849E/H20850
minima. Another interesting region is around 31 eV. Despite
Tkof both bands 1 and 2 reach here a maximum, the totalSTROCOV et al. PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-6T/H20849E/H20850, surprisingly, shows a minimum. This fact is attributed
to interference between the two /H9278k/H20849r/H20850so that the sum of Tkis
not exactly equal to the total T/H20849E/H20850/H20849see Sec. II B 2 /H20850. To em-
brace such cases, the relation of the multiband states to
VLEED can be best understood with reference to the dT/dE
spectra: the critical points in the coupling bands, where theBloch waves composition undergo sharp changes, alwayscorrespond to the extremes /H20849minima or maxima /H20850indT/dE;
/H20849ii/H20850While the FE-like dispersion is parabolic, in TiTe
2
each of the final bands strongly deviates from such a behav-
ior in the regions where it experiences hybridization withother bands /H20849which would form the band gaps in the V
i=0
band structure /H20850. The FE-like approximation with the inner
potential V000and effective mass m0as adjustable parameters
will describe such dispersions only as a local fit with V000
andm0strongly depending on Eandk.
The non-FE effects are also apparent immediately in the
VLEED experimental surface projection of the final states inFig.2. The E/H20849k
/H20648/H20850dispersions remain parabolic only over lim-
ited Eand k/H20648intervals, with clear discontinuities between
them indicating discontinuities in the V000andm0parameters
of the FE-like fit. The FE-like approximation thus fails todescribe the final bands of TiTe
2.
3. Final-state k /c142broadening
Any final-state /H9278k/H20849r/H20850, a damped Bloch wave with com-
plex k/H11036, can be represented as a superposition of propagating
waves with real k/H11036. These k/H11036form a Lorentzian distribution
centered at Re k/H11036and having a full width of 2 Im k/H11036.12,32
Thus, /H9254k/H11036=2 Im k/H11036represents the final-state broadening in
k/H11036. This aspect of the final bands in TiTe 2is reflected in the
E/H20849Imk/H11036/H20850panel of Fig. 5/H20849a/H20850. The following properties of /H9254k/H11036
are observed:
/H20849i/H20850/H9254k/H11036is different for each band and undergoes signifi-
cant energy variations. In particular, it shows loop-like en-hancements corresponding to the band gaps in the V
i=0 band
structure, where the Bloch waves experience additionaldamping due to scattering off the crystal potential /H20849see, for
example, Refs. 13,23, and 26/H20850;
/H20849ii/H20850Outside the band gaps where
/H9254k/H11036is predominantly
due to inelastic scattering and connected with Vias/H9254k/H11036
=Vi/H11509Rek/H11036
/H11509E,23the energy dependence of /H9254k/H11036follows that of Vi
/H20849see inset in Fig. 4/H20850.
In conventional PE data analysis, /H9254k/H11036is estimated from
the final-state energy broadening /H9254Efof the PE peaks mea-
sured in the constant-initial-state mode with the initial statefixed at the Fermi level to reduce the initial-state lifetimecontribution; the experimental
/H9254Efis then translated into
/H9254k/H11036=/H9254Ef/H11509k/H11036
/H11509E. However, such analysis does not take into ac-
count the non-FE effects in the final states. In particular, forTiTe
2/H20849Ref.4/H20850it has yielded in a low Efregion around 13 eV
a/H9254k/H11036value of /H110110.27 Å−1, which is much too large com-
pared to the “universal curve” of the mean free path /H9261mir-
roring /H9254k/H11036as/H9261/H110051//H9254k/H11036. Furthermore, this /H9254k/H11036value was
found to decrease to /H110110.15 Å−1when going to higher Ef
around 21 eV, whereas the “universal curve” predicts an in-
crease of /H9254k/H11036in this energy region. The origin of these in-
consistencies is the multiband composition of the final statesin TiTe 2discussed above. Different final bands, see Fig. 5/H20849a/H20850,
unresolved in this PE experiment in separate peaks mergedin one peak with enhanced E
fbroadening, whose unwary
translation into /H9254k/H11036resulted in overestimated /H9254k/H11036values.
With increase of Eftowards 21 eV energy separation of the
two final bands somewhat decreases, explaining the apparentdecrease of overall
/H9254k/H11036. An additional drawback of such an
incautious analysis of the PE data is neglecting the enhance-ments of
/H9254k/H11036=2 Im k/H11036in the band gap regions discussed
above.
4. Experimental energy corrections
One-to-one correspondence of the spectral structures in
the theoretical and experimental normal-incidence T/H20849E/H20850in
Fig. 4allows a correction of the theoretical final states ac-
cording to the VLEED experiment.
The values of the energy shifts /H9004Ebetween the experi-
mental and theoretical dT/dEextremal points are shown in
Fig. 6. Assuming that the remnant computational inaccura-
cies in the matching procedure and underlying E/H20849k/H20850are neg-
ligible, these shifts are fundamentally due to the excited-state
self-energy corrections /H9004/H9018, which appear due to the differ-
ence of the excited-state exchange-correlation potential fromthe ground-state one implied by our DFT based calculations/H20849assuming that the LDA approximation is accurate /H20850. The ob-
served /H9004Eenergy dependence is nonmonotonous, with a
clear discontinuity near 10 eV where the dominant bandforming the VLEED structures hops from 1 to 2, an oscilla-tion near 20 eV, and prominent increase starting from 30 eV.Such peculiarities are expected in view of the non-monotonous band and energy dependence of /H9004/H9018.
10
The experimental /H9004Evalues, absorbing the self-energy
effects and partly computational inaccuracies, were used tocorrect the energy position of the theoretical final states fromFig. 4/H20849a/H20850according to the VLEED experiment. The energy
correction was taken as a smooth curve obtained by Gaussiansmoothing of the scattered /H9004Evalues to remove kinks in the
corrected E/H20849Rek
/H11036/H20850dispersions. In the low-energy region the
curves for the bands 1 and 2 were taken different, and above
15 eV the same curve characteristic of the band 2 was ap-plied to all bands 1−3. Our further PE analysis utilizes thecorrected final states.
III. V ALENCE BANDS
A. PE experiment
1. Experimental procedure and results
The PE experiment was performed at the SA73 bending
magnet beamline of the SuperACO storage ring at LURE,France. Synchrotron radiation polarized in the horizontalplane was incident at angle of 45° relative to the surfacenormal in the M /H9003M
/H11032azimuth. The spectra were measured at
normal emission in the EDC mode with photon energies hv
varying from 11.5 to 33 eV in steps of 0.5 eV. The combinedmonochromator and analyzer energy resolution varied from23 meV at the lowest hvlimit to 130 meV at the highest one.
The monochromator energy scale was calibrated with respectto the Fermi edge through all measured PE spectra assumingTHREE-DIMENSIONAL BAND STRUCTURE OF LAYERED … PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-7an analyzer workfunction of 4.3 eV. The spectra were ac-
quired with statistics between /H110113/H11003104and 2/H11003105counts
per energy window of /H1101133 meV. Control spectra, taken after
completing the measurement series /H1101150 h after the cleavage,
showed only insignificant background increase, indicatingnegligible surface contamination.
The raw EDC spectra I/H20849E/H20850are shown in Fig. 7/H20849left/H20850. The
achieved statistics allows clear resolution of finer spectraldetails such as merging peaks. The spectra /H20849minus the inten-
sity above E
Fdue to high-order light /H20850are normalized to the
same integral intensity.
In Fig. 8/H20849a,left/H20850the above normalized EDC spectra are
rendered into a PE intensity map as a function of the final-and initial-state energies E
fandEi=Ef−h/H9263/H20849relative to EF/H20850.
The individual EDCs taken at constant hvare seen as in-
clined lines.
Figure 8/H20849b,left/H20850shows a similar map obtained from the
negative second derivative − d2I/dE2of the EDCs, with the
negative values of − d2I/dE2set to zero. Such a representa-
tion enhances the spectral structures including the peaks andshoulders
14/H20851except for the structures whose dispersion in the
/H20849Ef,Ei/H20850coordinates is along the derivation direction, i.e.,
along the EDC lines /H20852. Due to noise enhancement in the sec-
ond derivative, the original EDCs were denoised here byGaussian smoothing using a half width linearly increasingfrom 60 meV at the high- E
iend to 240 meV at the low- Ei
end in accordance with increase of the spectral structures
energy width. Due to large intensity variations, a logarithmicintensity scale is used in our − d
2I/dE2map.
2. Overall picture of the PE spectral structures
The PE experiment was supported by calculations of the
valence band E/H20849k/H20850representing the bulk initial states of the
PE process. These states, in contrast to the damped final
states, are described by propagating Bloch waves and thus byrealk
/H11036. The calculations were performed within the standard
DFT-LDA formalism using the self-consistent ELAPWmethod /H20849see Sec. II B 1 /H20850. The scalar relativistic effects with
the spin-orbit coupling in the second variation were included.The theoretical valence band E/H20849k/H20850is shown in Fig. 9. Our
FIG. 6. Energy shifts /H9004Ebetween the dT/dEextremal points in
the experimental and theoretical normal-incidence VLEED spectraas a function of the theoretical dT/dEenergies. The smooth curves
represent correction to the theoretical final bands 1 and 2 used inour PE analysis.
FIG. 7. Experimental /H20849left
panel /H20850and theoretical /H20849right /H20850
normal-emission EDC spectra.Dispersion of the spectral peakswith hvreflects E/H20849k
/H11036/H20850of the indi-
cated valence bands along /H9003A.STROCOV et al. PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-8results appear in general agreement with the previous
calculations,2,4although there are some quantitative dis-
agreements /H20849see Sec. III E /H20850.
Our spectra measured at normal emission reflect the
E/H20849k/H11036/H20850layer-perpendicular valence band dispersions along
the/H9003Adirection. Comparison with the calculated valence
band identifies the following origin of the principal spectralstructures:
/H20849i/H20850The dominant peaks dispersing through the lower and
upper part of the valence band originate from the bonding
Te 5 p
zand antibonding Te 5 pz*states, respectively /H20849note that
in the Efregion near 18 eV the Te 5 pz*peaks disappear in the
−d2I/dE2plot because they disperse along the derivation di-
rection /H20850. As discussed in Sec. III C 1, multiple dispersion
branches of the Te 5 pzand Te 5 pz*peaks reflect multiband
composition of the final state. The PE dispersions in Efre-
flect strong dispersion of the Te 5 pzand 5 pz*states in k/H11036
which results from the orientation of their electron orbitals
allowing effective overlap in the layer-perpendicular direc-tion across the van der Waals gap;
/H20849ii/H20850The weak nondispersive peak at E
i/H11011−1.7 eV, vanish-
ing in the middle of our Efinterval, results from the bonding
Te 5 pxyband. Due to excellent statistics of our experiment,
in the − d2I/dE2plot we clearly resolve spin-orbit splitting of
this band. The absence of dispersion in Efreflects vanishingdispersion of the Te 5 pxystates in k/H11036which results from the
orientation of their orbitals minimizing overlap in the layer-perpendicular direction;
/H20849iii/H20850The weak narrow peak just below E
Fdoes not have
any direct counterpart in calculated E/H20849k/H20850along /H9003A. When
going away from the normal emission, it dramatically scales
up /H20849minimization of its amplitude was used in our experi-
ment to adjust the normal emission angle /H20850and disperses
down in Ei/H20849see the off-normal PE data in Ref. 2/H20850. With the
absence of dispersion in hv, such dispersion in K/H20648compared
with calculated E/H20849k/H20850suggest the origin of this peak as due to
the antibonding Te 5 px,y*band. In principle, according to all
available band calculations including ours, exactly at the /H9003A
line the Te 5 px,y*band comes slightly above EF. However,
already with small deviation in K/H20648/H110110.1 Å−1it disperses be-
low EF. In this case the Te 5 px,y*signal can mix into the
normal-emission spectra due to nonzero angular acceptanceof the analyzer /H20849±1° HWHM /H20850as well as certain planarity
errors over the sample surface /H20849of the order of ±1° HWHM,
as estimated from angular spread of reflected white lightbeam on the chamber wall /H20850. The resulting combined /H9004K
/H20648
spread varies from ±0.03 at the low-energy end of our Ef
interval to 0.07 Å−1at the high-energy end. Given the very
large magnitude of the Te 5 px,y*signal,2such/H9004K/H20648should be
sufficient to built up a sizeable contribution to the normal-emission spectrum;
FIG. 8. /H20849Color online /H20850Experimental /H20849left panels /H20850and theoretical /H20849right /H20850normal-emission PE data rendered from the EDCs in Fig. 7:/H20849a/H20850
Intensity map as a function of the initial-state and final-state energies /H20849relative to EF/H20850, and /H20849b/H20850Map of negative second derivative − d2I/dE2
of the EDCs /H20849negative values − d2I/dE2/H110210 set to zero /H20850in a logarithmic colorscale. Multiple dispersion branches within the Te 5 pzand
Te 5 pz*energy regions demonstrate a multiband composition of the final state, an effect beyond the FE-like approximation. Lines on top of
the theoretical maps show the direct transitions /H20849DT/H20850plot constructed with the theoretical initial bands from Fig. 9and final bands 1−3
having Tk/H110220.1 from Fig. 5/H20849b/H20850; the DT branches are indexed according to the individual final bands in Fig. 5/H20849b/H20850they originate from.
Theoretical PE peaks show notable intrinsic shifts from the DT positions.THREE-DIMENSIONAL BAND STRUCTURE OF LAYERED … PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-9/H20849iv/H20850The peak marked by the vertical dashed line runs
through the EDCs at constant Efindependent of hv. This fact
identifies its origin as due to secondary electron emission/H20849SEE /H20850excited by high-order light.
B. PE computations
The PE spectra were calculated within the framework of
one-step theory using a new Bloch waves based method de-veloped within the ELAPW formalism.
33Compared to the
KKR based methods, it provides the most direct link of thePE spectra to the initial and final state band structure.
The final state of the PE process is treated as complex
conjugate of the LEED state described in Sec. II B. The ini-tial state in the crystal half-space /H9023
c/H20849r/H20850is a standing wave
represented by a linear combination of Bloch waves /H9023c/H20849r/H20850
=/H20858kBk/H9023k/H20849r/H20850of semi-infinite crystal, including propagating
and decaying waves. In the bulk asymptote /H9023c/H20849r/H20850retains
only propagating waves incident from the crystal interior on
the surface, plus a number of reflected waves traveling in the
opposite direction /H20849in case of the Te 5 pzandpz*bands there
is only one reflected wave for each incident wave /H20850.
To calculate the initial states, a grid of Nequidistant k/H11036is
created along the A/H9003Aextent of the BZ /H20849in the present cal-
culation N=80 /H20850. For each k/H11036the direct band structure prob-
lem /H20849determination of the energy eigenvalues for given k/H11036/H20850is
solved. Of the NBloch waves /H9274k/H20849r/H20850yielded by this proce-
dure within each band, N/2 ones with positive/H11509E
/H11509k/H11036group
velocity are selected. They represent /H9274k/H20849r/H20850incident on the
surface from the crystal interior. Then for each of the corre-
sponding N/2 energies within each band the inverse complex
band structure problem is solved. This yields the partial
/H9274k/H20849r/H20850—including decaying ones with complex k/H11036—whose
linear combination forms the initial-state wave function in
the crystal half-space /H9023c/H20849r/H20850=/H20858kBk/H9274k/H20849r/H20850. To determine the
expansion coefficients Bk, the scattering problem is solved
with the embedding method of Ref. 24. The same procedure
as for the LEED states is used, only the incident wave nowcomes from the interior of the crystal, and there are nopropagating waves in the vacuum half-space. As the relativ-istic spin-orbit effects are not included in our scattering cal-culations, for further matrix element calculations the wave
functions are ascribed to the spin-orbit split states accordingto the expansion coefficients coming from the second-variation treatment of the spin-orbit coupling /H20849this approxi-
mation is plausible everywhere except the points near the
bottom of the Te 5 p
z*band where it strongly hybridizes with
the Te 5 pxyband /H20850.
To calculate the momentum matrix elements between the
initial and final states, the wave functions are expanded inplane waves using the gouging technique described in Ref.34: Each atom is surrounded by a small gauging sphere with
radius r
g/H20849in our case 0.5 a.u., with the muffin-tin spheres
radii of 2.57 a.u. /H20850within which the wave function is forced
to damp to reduce its oscillations. The resulting pseudo wavefunction has a rapidly convergent plane wave expansion,which facilitates the matrix element calculations. Althoughthe contribution from the small spheres is damped within thisapproximation, the introduced error reduces faster than thethird power of r
g. Convergence of the results with respect to
this parameter indicates negligible magnitude of the remnanterror.
Using these matrix elements, the EDCs were calculated.
With the initial-state energy broadening due to the hole life-time combined with the final-state k
/H11036broadening, the PE
intensity at given Eiappears as an integral over the k/H11036ex-
tension of the BZ. The integration was performed by sum-mation over the whole k
/H11036grid assuming linear E/H20849k/H11036/H20850disper-
sion and constant wave functions within each k/H11036interval.
The initial-state broadening was represented by Lorentzianwith a variable FWHM, which grew linearly from 0.05 eV atE
Fto 0.4 eV at −6 eV.
The results of our PE calculations are shown in Figs. 7
and8/H20849right /H20850represented in parallel with the experimental
data. Comparison with their experimental counterparts /H20849left/H20850
demonstrates remarkable agreement, in particular on disper-sions of the spectral structures. The systematic displacementsinE
fandEiare due to the /H9004/H9018renormalization of the final
state and valence band dispersions /H20849see Sec. III E /H20850. Notable
flattening of the Te 5 pz*bottom compared to the experiment
is caused by too much hybridization with the Te 5 pxyband,
whose calculated energy location is in the experiment pushed
by the /H9004/H9018corrections to higher Eiaway from the Te 5 pz*
bottom. To the best of our knowledge, the achieved level of
agreement with the experiment is the best among all one-stepPE calculations on layered materials reported so far. The keyelement of the calculations has been the use of accurate finalstates.
Certain disagreements between the calculated and experi-
mental spectral structures in intensity and shape may traceback to the basic approximations of the one-step PE theorysuch as using Kohn-Sham solutions for quasiparticle wavefunctions, neglecting the interaction of the photoelectronwith the hole left behind, and describing the inelastic effectswith damped coherent waves. At the present theoretical levelthe validity of this basically one-particle approach cannot bejudged a priori , and the comparison with the experiment is
the only way to feel its limitations. The achieved excellentdescription of the PE peak dispersions allows us to assumethat the quasiparticle band structure is treated by the presenttheory correctly /H20849to within the self-energy shift /H20850. We there-
FIG. 9. Theoretical valence band E/H20849k/H20850along representative
high-symmetry BZ directions.STROCOV et al. PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-10fore ascribe the failure to reproduce the exact intensities and
shapes of the spectral structures to deficiencies of the one-particle description of the photoemission process.
Our further analysis of relation between the PE spectra
and initial- and final-state band structure will employ a direct
transitions (DT) plot which shows the /H20849E
f,Ei/H20850positions of
the PE peaks dictated by momentum conservation betweenthe initial and final states. Such plot constructed for TiTe
2
with the theoretical initial bands from Fig. 9and final bands
from Fig. 5/H20849b/H20850/H20849including only the coupling bands 1−3
whose significant Tkvalues enable effective coupling to the
outgoing photoelectron plane wave /H20850is shown in Fig. 8/H20849a,
right /H20850superimposed on the calculated PE intensity. In gen-
eral the PE peaks follow the DT lines /H20851note that in some /H20849Ef,
Ei/H20850regions the predicted PE peaks can disappear due to van-
ishing Mfimatrix element /H20852. There are however notable de-
viations which trace back essentially to the final-state k/H11036
broadening /H20849see Sec. III C 2 /H20850. In this respect the PE calcula-
tions allow testing the limits of the DT model in applicationto 3D band dispersions.
C. Final-state effects
1. Signatures of non-FE final-state dispersions
With FE-like final states, including one single band cou-
pling to vacuum, any valence band dispersing in k/H11036mani-
fests itself as one single PE peak whose Eias a function of Ef
displays characteristic regular oscillations between the E/H20849k/H11036/H20850
extreme energies. This is illustrated in Fig. 10/H20849a/H20850which
shows the DT plot for TiTe 2derived from the theoretical
Te 5 pzand 5 pz*valence bands in Fig. 9as the initial states
and FE-like final states. The dispersion ranges of the valencebands were adjusted to fit those of the experimental PE peaksin order to incorporate the /H9004/H9018renormalization in the valence
band /H20849see Sec. III E /H20850. The final states employed free-electron
m
0andV000=14.5 eV, as empirically determined in Ref. 2
/H20849in close agreement with a value of 14.0 eV from Ref. 4/H20850.
Comparison with the experimental PE data shows that
these FE-like final states have some relevance in a limitedinterval of E
fbetween 17.5 and 27.5 eV, but severely fails
through the rest of the experimental Efrange. Interestingly,
the optimal description of our normal-emission data withfree-electron m
0is achieved with V000of/H110112.1 eV above the
vacuum level, the dashed line in Fig. 10/H20849a/H20850; such an anoma-
lous value demonstrates the purely empirical character of theFE-like approximation. Although such empirical adjustmentsofV
000andm0can reduce the discrepancies, the FE-like final
states inherently fail to explain the multiple dispersion
branches of the Te 5 pzand 5 pz*peaks. Furthermore, the ex-
perimental PE dispersions appear notably distorted comparedto those expected from the FE-like approximation.
Incorporation of the non-FE and self-energy effects into
the final states radically improves the description of the ex-perimental PE data. Figure 10/H20849b/H20850shows the DT plot for
TiTe
2derived from the same initial states, but with the final
states replaced by the VLEED derived ones, the final bands1−3 from theoretical unoccupied E/H20849k/H20850in Fig. 5/H20849b/H20850, with the
experimental energy corrections from Fig. 6.The DT plot constructed from the VLEED derived final
states reproduces most of the peculiarities of the experimen-tal data. In particular, the multiple dispersion branches of the
Te 5 p
zand 5 pz*peaks find their natural explanation as origi-
nating from the multiple bands 1−3 with different k/H11036com-
posing the final state /H20849energy position of the band 3 may be
less accurate because in the VLEED spectrum its signal wasobscured by the bands 1 and 2 having larger T
k/H20850. The experi-
mental peak dispersions are also well reproduced. Such aradical improvement over the FE-like final states conjectureproves that the non-FE behavior of the final states is crucialfor correct interpretation the of PE spectra of TiTe
2. Further-
more, the incorporated VLEED experimental corrections tothe final state energies /H20849Fig.6/H20850are seen to considerably im-
prove the agreement with the experiment in the low- andhigh- E
fregions; in the low- Efone the PE data clearly con-
firms the /H9004/H9018difference between the bands 1 and 2. These
FIG. 10. /H20849Color online /H20850/H20849a/H20850Direct transitions /H20849DT/H20850plot resulting
from the Te 5 pzand 5 pz*initial states /H20849the non-dispersive Te 5 pxy
band excluded for clarity /H20850and FE-like final states with /H20849solid lines /H20850
conventional V000=14.5 eV and /H20849dashed /H20850optimized V000=−2.1 eV
above the vacuum level, on top of the experimental − d2I/dE2data
from Fig. 8/H20849b/H20850. Limited relevance of this conjecture, in particular
failure to reproduce the multiple branches of PE peaks, evidencesnon-FE effects in the final states; /H20849b/H20850DT plot resulting from the
same initial states and VLEED derived final states, with thebranches indexed 1−3 according to the individual final bands withT
k/H110220.1 in Fig. 5/H20849b/H20850. Due to incorporating the non-FE and self-
energy effects in the final states, the plot reproduces most of thepeculiarities of the experimental PE data, in particular the multipledispersion branches.THREE-DIMENSIONAL BAND STRUCTURE OF LAYERED … PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-11facts demonstrate that the self-energy renormalization of the
final state dispersions is also important for interpretation ofthe PE data for TiTe
2.
Deviations of the experimental peaks from the DT plot
near the top of the Te 5 pz*band /H20849note that near its bottom
such deviations disappear /H20850indicate flattening of its E/H20849k/H11036/H20850
dispersion towards EF/H20849see Sec. III E /H20850.I nt h e Efregion near
17 eV within the Te 5 pzband the peaks from the branches 1
and2merge into one, and in an extended Efregion around
23 eV the peaks from all three branches merge. This resultsfrom large broadening of the peaks in E
idue to decrease of
hole lifetime when going towards deeper Ei. The remaining
minor discrepancies between the experimental data and DTplot are mostly the intrinsic shifts due to the final-state k
/H11036
broadening discussed in Sec. III C 2.
Our PE calculations in Fig. 8/H20849right /H20850incorporate the same
non-FE final states /H20849without the VLEED experimental cor-
rections /H20850. They reproduce therefore all peculiarities of the PE
data including the multiple dispersion branches.
Non-FE effects in the final states, including their multi-
band composition, are in fact prominent for the quasi-2Dmaterials due to strong modulation of the crystal potential inthe layer-perpendicular direction.
8,9,35However, in certain E
andk/H20648ranges such effects were observed for Bi,31GaAs,36
GaN,37and even Cu /H20849Refs. 14and17/H20850and Al /H20849Ref. 38/H20850
which are materials conventionally considered to have purelyFE-like final states. Interestingly, recent PE results on Alreported in Ref. 39evidence that non-FE effects can extend
to fairly high energies: the PE intensity map /H20849Fig.1of this
reference /H20850shows in the hvrange from 210 to 380 eV two
additional, although weaker, dispersion branches of PE peaksidentifying multiple final bands.
2. Signatures of the final-state k /c142broadening: Intrinsic shifts
and peaks due to one-dimensional density of states
On a qualitative level, the final-state k/H11036broadening /H20849see
Sec. II C 3 /H20850causes the PE signal to represent an average of
theMfiweighted initial-state E/H20849k/H11036/H20850dispersion over the /H9254k/H11036
interval. In this respect /H9254k/H11036appears as intrinsic k/H11036resolution
of the PE experiment in the sense of being limited by thephysics of the PE process rather than by the measurementaccuracy.
12The averaging over /H9254k/H11036can result in intrinsic
shifts of PE peaks from the energy positions dictated by strict
momentum conservation between the initial and final states.Such shifts can occur by different mechanisms, for example:
/H20849i/H20850Sharp asymmetric variations of M
fithrough the /H9254k/H11036
interval;
/H20849ii/H20850Averaging of nonlinear E/H20849k/H11036/H20850over/H9254k/H11036near the band
edges, which pushes the PE peaks towards the band
interior12,40/H20849in other words, the k/H11036broadening results in
compression of the dispersion range apparent in the PE spec-trum /H20850. In our experimental data such in-band shifts are evi-
dent, for example, by comparison of the two PE dispersion
extremes in the Te 5 p
z*bottom as reached with Ef/H1101116 eV
and /H1101128 eV: Although they correspond to the same initial
state in the Apoint of the BZ, the second extreme appears
/H110110.1 eV higher in Eidue to larger /H9254k/H11036value delivered by
the final state, see Fig. 5/H20849a/H20850. Our PE calculations in compari-
son with the DT lines in Fig. 8/H20849right /H20850deliver direct estimateof such phenomena: The PE peaks near all band extremes
show in-band shifts of the order of 0.2 eV.
Another effect of the /H9254k/H11036averaging is formation of dis-
persionless spectral structures resulting from singularities of
the one-dimensional density of states /H208491DOS /H20850/H11509k/H11036
/H11509Epiling up at
the band edges.11,12In our experimental data such 1DOS
peaks appear as weak structures visible in the logarithmically
scaled − d2I/dE2map, Fig. 8/H20849b,left/H20850, at the Te 5 pz*band up-
per edge in a wide Efregion around 15 eV. 1DOS peaks can
also be identified at the Te 5 pz*lower edge and Te 5 pzupper
edge near Efof 15 eV. As the hole lifetime decrease towards
deeper Eismears the 1DOS singularities at the band edges,
these peaks develop only small intensity normally hidden inthe slopes of the gross DT peaks, and become visible only inthose E
fregions where all DT peaks move to the opposite
band edge and their slopes flatten. The experimental 1DOS-peaks are well reproduced by the calculations, Fig. 8/H20849b,
right /H20850.
Beyond the simple averaging picture, intrinsic shifts can
also appear due to interference of different Bloch wave con-stituents of the final state and initial state. For detailed dis-cussion of the interference effects see Ref. 33. Such interfer-
ence spectral structures deviating from the DT positionsoccur primarily in the regions where DT branches corre-sponding to different final bands intersect, for example in the
E
fregion near 22 eV within the Te 5 pz*band, Fig. 8/H20849left/H20850.
They are again reproduced by our calculations, Fig. 8/H20849right /H20850.
The intrinsic shifts put certain limits on the accuracy of
mapping the 3D dispersions with PE spectroscopy. In gen-eral, their magnitude scales up with the ratio of
/H9254k/H11036to the
surface-perpendicular BZ extension k/H11036BZ.11,12These phenom-
ena are therefore of particular concern for layered materials,
characterized by small k/H11036BZ. In our case, for example, Fig.
5/H20849a/H20850shows that /H9254k/H11036=2 Im k/H11036exceeds 0.5 /H20648/H9003A/H20648, a significant
value compared to k/H11036BZ=2/H20648/H9003A/H20648. One-step PE calculations,
delivering reliable estimate of the intrinsic shifts, can be usedto account for them in 3D band mapping. The first analysisof the intrinsic shifts in the framework of one-step PE theoryin application to layered materials was reported in Ref. 35.
D. Band mapping
In the band mapping procedure one determines the va-
lence band E/H20849k/H11036/H20850by mapping Eiof the PE peaks against k/H11036
determined by Efof the peaks through the final-state E/H20849k/H11036/H20850
dispersion. We have employed here all experimental peaks
excluding the regions where the peaks overlapped with eachother /H20849and were thus susceptible to the interference effects
33/H20850
as well as the 1DOS peaks. The peak energies were evalu-ated in − d
2I/dE2. The VLEED derived final states used in
the DT plot of Fig. 10/H20849b/H20850were employed. The plot has al-
lowed us to identify each experimental PE peak with one ofthe band 1−3 in the final state /H20851note that in view of the
multitude of all unoccupied bands along /H9003A, see Fig. 5/H20849b/H20850,
the analysis of T
kto distinguish the dominant final bands was
crucial for such identification /H20852. The corresponding k/H11036values
were then extracted from the final-state E/H20849k/H11036/H20850dispersions.
Note that the use of the true VLEED derived final states
takes our band mapping procedure beyond the limitations ofSTROCOV et al. PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-12the FE-like as well as ground-state DFT approximation for
the final states. Neglected however are the slight intrinsicshifts of PE peaks from the DTs positions, see Fig. 8/H20849right /H20850.
The results of our band mapping are shown in Fig. 11
superimposed on the theoretical E/H20849k
/H11036/H20850. Amplitudes of the
−d2I/dE2peaks, representing amplitude and sharpness of the
spectral peaks, are shown in grayscale. The experimentalpoints show very consistent dispersions in both Te 5 p
zand
5pz*bands. Note that the band extremes exactly fit the sym-
metry points /H9003andA. The remaining minor disparities in the
experimental dispersions /H20849apart from the systematic energy
shift due to /H9004/H9018effects in the valence band, see below /H20850are
attributed to the intrinsic shifts of the PE peaks. Theachieved band mapping consistency should be contrasted tothe erratic results for TiTe
2returned by FE-like final bands
/H20849see, for example, Ref. 2/H20850as expected from the disagree-
ments in Fig. 10/H20849a/H20850. Radical improvements over the FE-like
approximation delivered by the use of VLEED derived finalbands were also found for VSe
2and TiS 2,7,9illustrating the
importance of non-FE effects in the final states of layeredmaterials, at least for low excitation energies. Furthermore,we have found that the VLEED derived /H9004/H9018renormalization
of the final-state dispersions was vital for the experimentalvalence band extremes to fit the symmetry points.
E. Properties of the experimental 3D valence band structure
Consistent control over k/H11036in our band mapping proce-
dure has enabled us to achieve qualitatively new informationabout the 3D states in the valence band of TiTe
2. Experimen-
talE/H20849k/H11036/H20850in Fig. 11shows the following peculiarities:
/H20849i/H20850At the top of the Te 5 pz*band the dispersion flattens.
The experimental points corresponding to the Te 5 pz*band
maximum in the /H9003point appear /H110110.35 eV below EF. Withthe intrinsic shifts here being of the order of 0.1 eV, see Fig.
8/H20849b,right /H20850, the E/H20849k/H11036/H20850maximum appears at /H110110.25 eV below
EF. The appearance of the 1DOS peak coming from the band
maximum confirms its position below EF. Therefore, con-
trary to the early calculations,2,6the Te 5 pz*band does not
cross EFand the corresponding FS pocket in the /H9003point does
not exist. This fact agrees however with the later calculationsin Ref. 4and in this work. Interestingly, the correct position
of the Te 5 p
z*band is achieved in our calculations only if the
spin-orbit interaction is included.
By virtue of the reliable control over k/H11036our analysis
gives, to the best of our knowledge, the first experimental
evidence of the absence of the Te 5 pz*derived 3D electron
pocket in the /H9003point. This fact has serious implications for
the transport properties of TiTe 2, because such a pocket
would have delivered a significant isotropic contribution tothe electron transport;
/H20849ii/H20850The Te 5 p
z*experimental band shows an energy shift
relative to the calculated one varying, with the intrinsic shiftsdeconvoluted, from −0.3 eV at the upper band edge to−0.6 eV at the lower one. For the Te 5 p
xyband the shift is
about +0.4 eV. Similarly to the unoccupied states /H20849see Sec.
I IC4 /H20850such shifts manifest mostly the /H9004/H9018self-energy cor-
rections to the DFT ground-state band structure. The differ-
ent and opposite /H9004/H9018values observed for the Te 5 pz*and 5 pxy
bands identify the band dependence of /H9004/H9018. Similar differ-
ences in /H9004/H9018between the valence /H9268and/H9266states were ob-
served for graphite.40
It should be noted that our calculations and the pseudopo-
tential Gaussian orbital calculations from Ref. 4both locate
the Te 5 pzand Te 5 pz*bands by /H110110.5 eV lower in energy
compared to the earlier LMTO calculations from Ref. 2,i n
closer agreement with the experiment. It remains to be seenwhether the improvement is mostly due to the use of fullpotential or the inclusion of spin-orbit interaction.
IV . CONCLUSION
We have investigated 3D effects in the electronic structure
of the TiTe 2prototype layered material, focusing on the
layer-perpendicular band dispersion along the /H9003Aline. The
VLEED experiment, supported by calculations of the com-plex band structure E/H20849Rek
/H11036+iImk/H11036/H20850within the full-
potential ELAPW formalism, has been used to determine the
dispersions and lifetimes of the final states engaged in the PEprocess. The PE experimental data, interpreted on the basisof the VLEED derived final states, has yielded consistentexperimental E/H20849k
/H11036/H20850dispersions in the valence band. The PE
experiment was supported by calculations based on a unique
Bloch waves formalism within the one-step PE theory, whichhas delivered most accurate description of PE spectra ofTiTe
2. Of specific results, we have found that:
/H20849i/H20850The final states of TiTe 2show strong non-free-electron
effects due to scattering off its highly modulated quasi-2Dcrystal potential. The final states feature multiband structure,each of the bands showing significantly nonparabolic disper-sion;
/H20849ii/H20850Consistent PE band mapping of the valence band
FIG. 11. /H20849Color online /H20850Experimental valence band E/H20849k/H11036/H20850along
/H9003Aachieved by band mapping with the VLEED derived final states.
Amplitude*sharpness of the PE peaks is shown in grayscale
/H20849maximum=black /H20850. The experimental points show high consis-
tency, contrasting to the results returned by FE-like final bands. Theexperimental points are superimposed on the LDA-DFT calculation.THREE-DIMENSIONAL BAND STRUCTURE OF LAYERED … PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-13k/H11036dispersions in TiTe 2critically depends on taking into ac-
count the non-free-electron and self-energy effects in the fi-nal states;
/H20849iii/H20850The FS of TiTe
2does not have any Te 5 pz*derived
sheet along the /H9003A line, which excludes the corresponding
isotropic component in the transport properties.ACKNOWLEDGMENTS
We acknowledge the support of Deutsche Forschungsge-
meinschaft /H20849CL124/5-2 and Forschergruppe FOR 353 /H20850and
the EC support for the experiments at LURE within the Ac-cess to Research Infrastructure program.
*Corresponding author. Email address: vladimir.strocov@psi.ch
1Electron spectroscopies applied to low-dimensional materials , ed-
ited by H. I. Starnberg and H. P. Hughes /H20849Kluwer, Netherlands,
2000 /H20850.
2R. Claessen, R. O. Anderson, G.-H. Gweon, J. W. Allen, W. P.
Ellis, C. Janowitz, C. G. Olson, Z. X. Shen, V. Eyert, M. Ski-bowski, K. Friemelt, E. Bucher, and S. Hüfner, Phys. Rev. B 54,
2453 /H208491996 /H20850.
3L. Perfetti, C. Rojas, A. Reginelli, L. Gavioli, H. Berger, G. Mar-
garitondo, M. Grioni, R. Gaál, L. Forró, and F. Rullier Albenque,Phys. Rev. B 64, 115102 /H208492001 /H20850.
4K. Rossnagel, L. Kipp, M. Skibowski, C. Solterbeck, T. Strasser,
W. Schattke, D. Voß, P. Krüger, A. Mazur, and J. Pollmann,Phys. Rev. B 63, 125104 /H208492001 /H20850.
5G. Nicolay, B. Eltner, S. Hüfner, F. Reinert, U. Probst, and E.
Bucher, Phys. Rev. B 73, 045116 /H208492006 /H20850.
6D. K. G. de Boer, C. F. van Bruggen, G. W. Bus, R. Coehoorn, C.
Haas, G. A. Sawatzky, H. W. Myron, D. Norman, and H. Pad-more, Phys. Rev. B 29, 6797 /H208491984 /H20850.
7V. N. Strocov, H. I. Starnberg, P. O. Nilsson, H. E. Brauer, and L.
J. Holleboom, Phys. Rev. Lett. 79, 467 /H208491997 /H20850; J. Phys.: Con-
dens. Matter 10, 5749 /H208491998 /H20850.
8V. N. Strocov, P. Blaha, H. I. Starnberg, M. Rohlfing, R. Claessen,
J.-M. Debever, and J.-M. Themlin, Phys. Rev. B 61, 4994
/H208492000 /H20850.
9V. N. Strocov, in Ref. 1.
10V. N. Strocov, R. Claessen, F. Aryasetiawan, P. Blaha, and P. O.
Nilsson, Phys. Rev. B 66, 195104 /H208492002 /H20850.
11P. J. Feibelman and D. E. Eastman, Phys. Rev. B 10, 4932 /H208491974 /H20850.
12V. N. Strocov, J. Electron Spectrosc. Relat. Phenom. 130,6 5
/H208492003 /H20850.
13V. N. Strocov, H. Starnberg, and P. O. Nilsson, J. Phys.: Condens.
Matter 8, 7539 /H208491996 /H20850.
14V. N. Strocov, R. Claessen, G. Nicolay, S. Hüfner, A. Kimura, A.
Harasawa, S. Shin, A. Kakizaki, P. O. Nilsson, H. I. Starnberg,and P. Blaha, Phys. Rev. Lett. 81, 4943 /H208491998 /H20850; Phys. Rev. B
63, 205108 /H208492001 /H20850.
15V. N. Strocov, Meas. Sci. Technol. 7, 1636 /H208491996 /H20850.
16E. E. Krasovskii, W. Schattke, V. N. Strocov, and R. Claessen,
Phys. Rev. B 66, 235403 /H208492002 /H20850.
17V. N. Strocov, H. I. Starnberg, and P. O. Nilsson, Phys. Rev. B56, 1717 /H208491997 /H20850.
18E. G. McRae, Rev. Mod. Phys. 51, 541 /H208491979 /H20850.
19R. O. Jones and P. J. Jennings, Surf. Sci. Rep. 9, 165 /H208491988 /H20850.
20V. N. Strocov, A. R. H. F Ettema, and H. I. Starnberg, Solid State
Commun. 96, 659 /H208491995 /H20850.
21G. Capart, Science 13, 361 /H208491969 /H20850.
22J. B. Pendry, J. Phys. C 2, 2273 /H208491969 /H20850.
23J. B. Pendry, Low Energy Electron Diffraction /H20849Academic Press,
London, 1974 /H20850.
24E. E. Krasovskii, Phys. Rev. B 70, 245322 /H208492004 /H20850; E. E. Kras-
ovskii and W. Schattke, Phys. Rev. Lett. 93, 027601 /H208492004 /H20850.
25E. E. Krasovskii and W. Schattke, Phys. Rev. B 56, 12874
/H208491997 /H20850.
26N. Barrett, E. E. Krasovskii, J.-M. Themlin, and V. N. Strocov,
Phys. Rev. B 71, 035427 /H208492005 /H20850.
27T. Buslaps /H20849unpublished /H20850.
28V. N. Strocov, Solid State Commun. 106, 101 /H208491998 /H20850.
29Y. Petroff and P. Thiry, Appl. Opt. 19, 3957 /H208491980 /H20850.
30G. D. Mahan, Phys. Rev. B 2, 4334 /H208491970 /H20850.
31C. R. Ast and H. Höchst, Phys. Rev. B 70, 245122 /H208492004 /H20850.
32R. Matzdorf, Appl. Phys. A 63, 549 /H208491996 /H20850; Surf. Sci. Rep. 30,
153 /H208491998 /H20850.
33E. E. Krasovskii, V. N. Strocov, N. Barrett, H. Berger, W. Schat-
tke, and R. Claessen /H20849unpublished /H20850.
34E. E. Krasovskii, F. Starrost, and W. Schattke, Phys. Rev. B 59,
10504 /H208491999 /H20850; E. E. Krasovskii and W. Schattke, ibid. 60,
R16251 /H208491999 /H20850.
35E. Pehlke and W. Schattke, Solid State Commun. 69, 419 /H208491989 /H20850.
36J. Henk, W. Schattke, H. Carstensen, R. Manzke, and M. Ski-
bowski, Phys. Rev. B 47, 2251 /H208491993 /H20850.
37T. Strasser, C. Solterbeck, F. Starrost, and W. Schattke, Phys. Rev.
B60, 11577 /H208491999 /H20850.
38A. Mugarza, A. Marini, T. Strasser, W. Schattke, A. Rubio, F. J.
García de Abajo, J. Lobo, E. G. Michel, J. Kuntze, and J. E.Ortega, Phys. Rev. B 69, 115422 /H208492004 /H20850.
39Ph. Hofmann, Ch. Søndergaard, S. Agergaard, S. V. Hoffmann, J.
E. Gayone, G. Zampieri, S. Lizzit, and A. Baraldi, Phys. Rev. B
66, 245422 /H208492002 /H20850.
40V. N. Strocov, A. Charrier, J.-M. Themlin, M. Rohlfing, R. Claes-
sen, N. Barrett, J. Avila, J. Sanchez, and M.-C. Asensio, Phys.Rev. B 64, 075105 /H208492001 /H20850.STROCOV et al. PHYSICAL REVIEW B 74, 195125 /H208492006 /H20850
195125-14 |
PhysRevB.82.155409.pdf | Structure, energy, and electronic states of vacancies in Ge nanocrystals
Kenneth Bayus
Department of Material Science and Engineering, Cornell University, Ithaca, New York 14850, USA
Oscar Paz *
Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA
S. P. Beckman†
Department of Material Science and Engineering, Iowa State University, Ames, Iowa 50014, USA
/H20849Received 1 June 2010; published 6 October 2010 /H20850
The atomic structure, energy of formation, and electronic states of vacancies in H-passivated Ge nanocrys-
tals are studied by density-functional theory methods. The competition between quantum self-purification andthe free surface relaxations is investigated. The free surfaces of crystals smaller than 2 nm distort the Jahn-Teller relaxation and enhance the reconstruction bonds. This increases the energy splitting of the quantumstates and reduces the energy of formation to as low as 1 eV per defect in the smallest nanocrystals. In crystalslarger than 2 nm the observed symmetry of the Jahn-Teller distortion matches the symmetry expected for bulkGe crystals. Near the nanocrystal’s surface the vacancy is found to have an energy of formation no larger than0.5–1.4 eV per defect, but a vacancy more than 0.7 nm inside the surface has an energy of formation that is thesame as in bulk Ge. No evidence of the self-purification effect is observed; the dominant effect is the freesurface relaxations, which allow for the enhanced reconstruction. From the evidence in this paper, it is pre-dicted that for moderate sized Ge nanocrystals a vacancy inside the crystal will behave bulklike and not interactstrongly with the surface, except when it is within 0.7 nm of the surface.
DOI: 10.1103/PhysRevB.82.155409 PACS number /H20849s/H20850: 66.30.Pa, 73.22. /H11002f, 61.46. /H11002w, 61.72.uf
I. INTRODUCTION
Germanium is a particularly attractive material for use in
semiconducting devices. The charge carriers have a high mo-bility due to their low effective mass and it is possible toachieve a high level of n- and p-type dopant activation.
1,2
The designers of microelectronics initially focused on Si in-
stead of Ge because Ge lacks a native oxide that can be usedas a dielectric. Fortunately this limitation can be overcomeby several techniques that have been developed within thelast decade: a thin Si overlayer can be grown over the Ge sothat SiO
2can be used as the dielectric,3a Ge-oxynitride di-
electric layer can be grown over the Ge /H20849Ref. 4/H20850or a high- /H9260
dielectric crystal, such as ZrO 2, can be used in the device.5
These techniques allow for the development of Si/Ge hetero-
structure devices such as metal-oxide-semiconductor field ef-fect transistors, MOSFETs. The heterostructure MOSFET isprimarily Si so existing fabrication technology can be used,but Ge is included as a buried channel between the sourceand drain to allow for high-speed conductivity.
6–9
Before Ge can become an industrially important material,
it must be possible to introduce and control a variety ofdopant species in the crystal.
10Due to the initial challenge of
finding a suitable dielectric material, there has been muchless effort in understanding Ge as compared to Si. Subse-quently much less is known about the control of dopants inGe than in Si. There exists a close relationship between im-purity diffusion and self-diffusion; therefore, it is fundamen-tally important to understand self-diffusion to understand thecontrol of impurity atoms.
Following Refs. 11and12, the self-diffusion coefficient
D/H20849T/H20850is written as a sum of the vacancy /H20849
v/H20850, interstitial /H20849i/H20850,
and direct-exchange /H20849ex/H20850diffusion mechanisms,D/H20849T/H20850=fv/H20849T/H20850Cveq/H20849T/H20850Dv/H20849T/H20850+fi/H20849T/H20850Cieq/H20849T/H20850Di/H20849T/H20850+Dex/H20849T/H20850,
/H208491/H20850
where f/H9257/H20849T/H20850are correlation factors, C/H9257eq/H20849T/H20850are the equilib-
rium concentrations of the intrinsic defects, and D/H9257/H20849T/H20850are
the diffusion coefficients corresponding to /H9257=v,i,ex. The
concentrations and diffusion coefficients are expressed interms of their thermodynamic quantities,
C
/H9257eq/H20849T/H20850= exp/H20875/H9004Sf/H9257
kB/H20876exp/H20875−/H9004Ef/H9257
kBT/H20876, /H208492/H20850
D/H9257/H20849T/H20850=Kmexp/H20875/H9004Sm/H9257
kB/H20876exp/H20875−/H9004Em/H9257
kBT/H20876, /H208493/H20850
where /H9004Sf/H9257and/H9004Ef/H9257are the entropy and energy of formation,
/H9004Sm/H9257and/H9004Em/H9257are the entropy and energy of migration, kBis
the Boltzmann constant, and Kmis a constant prefactor. The
constant Kmis independent of temperature and depends on
the lattice geometry and the vibrational frequencies. The en-tropy terms include both the configurational and the vibra-tional entropies. The energy of formation is determined bythe atomic bonding at the defect site and the energy of mi-gration is determined by the energy of the saddle-point con-figuration along the minimum-energy transition path betweenstable atomic configurations.
Direct exchange is the slowest diffusion mechanism.
Straining the lattice to allow the atoms to move past eachother requires a prohibitively large amount of energy. Theprincipal diffusion pathways involve either vacancy- orinterstitial-assisted migration. The work presented here fo-cuses on these intrinsic point defects. Based on the energiesPHYSICAL REVIEW B 82, 155409 /H208492010 /H20850
1098-0121/2010/82 /H2084915/H20850/155409 /H208497/H20850 ©2010 The American Physical Society 155409-1listed in Table Ithe self-diffusion coefficient in Si is con-
trolled by a self-interstitial kick-out mechanism at high tem-peratures /H20849T/H11022900 °C /H20850and vacancy-mediated diffusion at
lower temperatures.
13
In Ge, vacancy-assisted diffusion is the primary mode.30
Although the migration barrier for Ge vacancies and self-interstitials is roughly equivalent, the energy of formation forinterstitial Ge atoms is approximately 1 eV greater than va-cancies whereas in Si the energy to create vacancies andinterstitials is equivalent. In Ge self-interstitial atoms willonly be formed at very high temperatures or after highlyenergetic processes such as irradiation.
31Therefore, vacan-
cies in Ge are substantially more influential than interstitialatoms for assisting diffusion under thermal equilibrium, ascompared to Si.
The dominance of vacancies-assisted diffusion is ob-
served experimentally, both for self-diffusion
32and impurity-
atom diffusion.30,33,34The interaction between vacancies and
impurity atoms is complicated. It is believed that vacanciesand impurities form mobile defect pairs.
26,35,36This defect
pair can become pinned when a second impurity, such as C,joins the complex.
26In addition to forming complexes, the
vacancies and impurities often carry a charge.21,26,35It is
likely that an isolated vacancy in bulk Ge is charged −2.35In
the work presented here only isolated, charge neutral, impu-rities are investigated, which is consistent with the nanoscalecontext of this study.
To improve the engineering control of material properties
and increase device efficiency, it is desirable to move frombulk to nanoscale structures. There are many examples ofsituations where nanostructured Ge offers benefits. The useof Ge nanocrystals as the floating gate of MOS memory de-vices results in a dramatic shift in the threshold-voltage, im-proved switching characteristics, and decreased leakagecurrent.
37,38Ge films with nanostructured surfaces offer the
ability to tune the optical properties of thin films.39Ge nano-
wires are considered for use as MOSFETs.40,41
Ultimately the nanostructures used to create devices need
to be tailored by controlling their size, surfaces, and dopants.With respect to introducing dopants, the electronic propertiesof nanostructures are believed to be sensitive to the relativeposition of the impurities in the structure. The mean free pathof charge carriers within nanowires depends strongly on theradial dopant profile.
42This will influence the conductivity.
In addition to the challenge of selectively incorporating the
dopant atoms into the nanostructure, the impurity distribu-tion must be maintained for the lifetime of the device.
At the quantum scale the primary difference between a
bulk crystal and a nanocrystal is the interaction of the wavefunction with the surfaces. As the size of a structure de-creases, the crystal’s translational symmetry ceases to bemeaningful. The electronic band structure that is nominally afunction of the quantum number kis projected onto the /H9003
point in the center of the Brillouin zone. The crystal’s energybands become discrete quantum energy states. Whereas inbulk the wave function is distributed across the entire crystalas Bloch waves, u
k/H20849r/H20850eik·r, in nanocrystals the wave function
is confined by the surfaces. The size of the nanostructuredirectly impacts the energy states, analogous to the elemen-tary particle-in-a-box problem. Consider, for example, a/H20851110 /H20852Ge nanowire. When the wire diameter is sufficiently
small the crystal’s translational symmetry is only meaningfulin the /H20851110 /H20852direction and the bulk Ge states are projected
along the k=/H20851110 /H20852direction in kspace. This projection trans-
forms Ge from an indirect to direct band-gap material.
43The
confinement is predicted to distort the shape of the energydispersion for wires with diameters as large as 2 nm. Theenergy bands of nanowires with diameter greater than 2 nmare found to undergo a rigid shift, even for wires as large as5 nm.
43
Fundamentally, there are two effects that differentiate the
behavior of defects in nanostructures from bulk: quantumconfinement and free surfaces. Dalpian
44claims that the con-
finement of the defect’s wave function results in the so-called
self-purification effect that increases the defect’s /H9004Ef/H9257.I nt h e
case of dopant species, this increase hinders the incorpora-tion of dopant atoms into the nanostructures. This is a con-troversial subject and worthwhile investigating.
45–47In the
present calculations evidence of self-purification will besought.
The free surfaces allow the nanostructure to expand or
contract to reduce the strain energy surrounding the defect.From an energetics perspective the self-purification effectand the free surfaces compete with one another. The self-purification increases the energy and the free surfaces de-crease the energy. From a kinetics perspective they comple-TABLE I. The energies of formation and migration for vacancies and interstitial defects in Si and Ge
/H20849in eV /H20850.
Element /H9004Efv/H9004Efi/H9004Emv/H9004Emi
Si 3.1–3.6 /H20849Ref. 13/H20850 3.2 /H20849Ref. 14/H20850 0.4–1.40 /H20849Ref. 13/H20850 0.45 /H20849Ref. 15/H20850
3.7 /H20849Ref. 16/H20850 3.31–3.84 /H20849Ref. 17/H20850 0.43–0.49 /H20849Ref. 12/H20850 0.84 /H20849Ref. 12/H20850
3.49 /H20849Ref. 18/H20850 3.27 /H20849Ref. 19/H20850
3.53 /H20849Ref. 20/H20850
Ge 2.3 /H20849Ref. 21/H20850 2.29 /H20849Ref. 22/H20850 0.7 /H20849Ref. 23/H20850 0.5 /H20849Ref. 24/H20850
1.7–2.0 /H20849Ref. 25/H20850 2.3–4.1 /H20849Ref. 26/H20850 0.36–0.7 /H20849Ref. 27/H20850
2.4 /H20849Ref. 23/H20850 3.55 /H20849Ref. 28/H20850
2.6 /H20849Ref. 27/H20850 3.50 /H20849Ref. 29/H20850
2.56 /H20849Ref. 29/H20850BAYUS, PAZ, AND BECKMAN PHYSICAL REVIEW B 82, 155409 /H208492010 /H20850
155409-2ment one another because it is likely that the surfaces will
getter impurities out of the nanostructure. In the case of Sinanocrystals it is observed that the relative energy to intro-duce vacancies decreases as the nanocrystal’s size decreases.This indicates that energetically the free surfaces dominatethe self-purification effect.
48As the vacancy is moved toward
the surface the energy further decreases and when the va-cancy is within 0.6 Å of the surface it becomes unstable andis spontaneously moved to the surface of the crystal.
48,49
In this paper the structure and energies of vacancies in Ge
nanocrystals are examined as a function of the nanocrystal’ssize and the position of the vacancy in the crystal. Because
the energies, /H9004E
f/H9257, depend on the size of the crystal and the
position within the crystal, the concentrations and diffusivi-ties, from Eqs. /H208492/H20850and /H208493/H20850, also depend on the size and po-
sition. Self-diffusion within nanostructures is not a simplematter that can be easily described by a single coefficient.The work here is a first step toward building a comprehen-sive model.
Following this introduction, in Sec. II, the methods used
will be presented, including a discussion of the computa-tional approach and the details of the nanocrystals’ morphol-ogy. The results from these calculations will be presentedand discussed in Sec. III. A concluding summary will be
presented in Sec. IV.
II. METHODS
A. Computational approach
The calculations are performed within the framework of
the density-functional theory50/H20849DFT /H20850, using the local-
density approximation51/H20849LDA /H20850for the exchange-correlation
functional, as it is implemented in the SIESTA computational
package.52The electrons in the core atomic region are sub-
stituted by norm-conserving pseudopotentials of theTroullier-Martins type,
53and the valence charge is repre-
sented by a set of atom-centered basis functions. In SIESTA
these functions correspond to numerical atomic orbitals ofstrictly finite range, a particular choice that is specially suitedto treat isolated systems.
All calculations are carried out using a double-
/H9256plus po-
larization orbitals basis set. The cut-off radii of the basisfunctions are optimized for bulk germanium in the diamondstructure, following the method proposed in Ref. 54, where a
fictitious external pressure of 0.2 GPa is employed on thefree atom. This basis size and radius lengths are proven togive a good balance between the computational accuracy andcost. A theoretical lattice parameter of a
0=5.64 Å and a bulk
modulus of B0=80.0 GPa are obtained from the fitting to the
Murnaghan equation of state.55Both values are in good
agreement with the structural and elastic properties from
experiments56/H20849a˜0=5.66 Å and B˜0=75.8 GPa /H20850, given the
fact that the LDA tends to underestimate lattice constants bya 1–3 % but also to overestimate bulk moduli with errorsranging from 5% up to 20%.
57
The theoretical method employs periodic boundary condi-
tions. The nanocrystals are placed inside the supercell sur-rounded by a buffer of empty space. The size of the atomicclusters ranges from 44 to 244 Ge atoms and the vacuum
region is chosen to be large enough as to avoid any interac-tion between their periodic replicas. A kinetic-energy cutoffof 250 Rydberg is chosen for the real-space integrations in-volving the Hartree and the exchange-correlation contribu-tions to the self-consistent potential. In this respect, a strin-gent criterion is employed in the convergence of the densitymatrix and total energy. All atomic coordinates are then re-laxed according to a conjugate-gradient minimization algo-rithm, until the maximum residual forces are below0.02 eV /Å.
B. Nanocrystal morphology
The nanocrystal geometries used in these studies are
hydrogen-passivated, bond-centered crystals. Experimentalnanostructures frequently have amorphous, glassy, or poly-meric coatings that result from the method of crystal growth.It is possible to treat the surfaces to reduce or remove these,although it is uncommon experimentally to work with baresurfaces. The nanocrystals investigated here have their sur-faces passivated with an extremely “soft” H pseudopotential.Surface passivation removes surface states and allows thecompetition between the self-purification effect and the freesurfaces to be studied without considering the complicatedsurface chemistry.
Surface Ge atoms are identified and the dangling bonds of
the Ge atoms are capped with H. Any Ge atom that is foundto have three dangling bonds is replaced by a single H atom.The resulting nanocrystals are shown in Fig. 1. The surface
morphologies are examined and crystals that are highly fac-eted are excluded. Only nanocrystals with near spherical ge-ometry are studied.
III. RESULTS AND DISCUSSION
A. Atomic structure and defect states
The local atomic structure at the vacancy site is directly
related to the electronic states introduced to the gap from thebroken bonds. The atomic structure of the vacancy in the1.02 and 2.20 nm nanocrystals is given in Table II, using as
reference the ABCD indices shown in Fig. 2. The associated
electronic states are diagrammed in Fig. 3. The left column
of Fig. 3shows the states for a nanocrystal with no vacancy.
The band gap for the 1.02 nm crystal is 3.13 eV and the gapfor the 2.20 nm crystal is 2.0 eV .
An undistorted vacancy has T
dsymmetry. There are three
degenerate states in the gap, belonging to the t2representa-
tion, associated with this structure. These are shown in themiddle column of Fig. 3. There are two electrons localized at
the defect so the states are partially occupied. It is the partialoccupancy of the degenerate energy levels that allows thedefect to undergo a spontaneous symmetry breaking that re-duces the degeneracy and lowers the electronic energy. Inbulk crystals the Jahn-Teller distortion produces aD
2d-symmetrized structure with the fully occupied state be-
longing to the b2representation and the doubly degenerate,
empty state having the erepresentation.48,58,59STRUCTURE, ENERGY , AND ELECTRONIC STATES OF … PHYSICAL REVIEW B 82, 155409 /H208492010 /H20850
155409-3Here, as in the case of a vacancy in a Si nanocrystal,48the
symmetry of the structure approaches D2dbut due to the
surfaces there is additional distortion. In the case of the 1.02nm crystal, the symmetry of the vacancy structure is C
s. For
the 2.20 nm crystal the symmetry is essentially D2dbut a
minuscule 0.01 nm distortion of the bonds lowers the sym-metry, i.e., if
AC= AD= BC= BD then the symmetry would
beD2d. From Table IIit is determined that the Jahn-Teller
distortion in the 1.02 nm crystal is approximately 10% largerthan that in the 2.20 nm crystal and as a result the defectstates undergo a larger energy split, over 2.7 eV , which al-most pushes the states out of the gap. In the 2.20 nm crystalthe splitting is much smaller, around 0.75 eV .
B. Crystal size
The energy of the fully optimized vacancy structures in
the different sized nanocrystals is calculated. Subtracting thisenergy from the energy of the perfect nanocrystals yields theenergy of formation for a vacancy plus the chemical poten-tial for Ge, i.e., the energy to remove a Ge atom from thesystem. The chemical potential is variable and depends onthe local chemical environment. By assuming that all thenanocrystals are located in the same environment and havethe same chemical potential it is possible to compare therelative energy of formation for vacancies in different sized
nanocrystals. It is known that as a nanocrystal’s diameterapproaches infinity the energy of formation approaches thatfound in bulk Ge. Using this energy limit the calculated en-ergy of formation versus crystal diameter is plotted in Fig.4/H20849a/H20850. It is assumed that the size dependence goes as
E/H20849D/H20850=
/H9251
D/H9252+/H9253, /H208494/H20850
where /H9251,/H9252, and/H9253are fitting coefficients. In the limit that the
diameter, D, goes to infinity, the energy equals /H9253. The coef-
ficients are determined to be /H9251=−1.1395 eV and /H9252=6.2574.
The energy zero is shifted so that /H9253=2.0 eV, which is taken
from the energies reported in Table I. The quality of this fit
appears good. It is observed that the energy of formation isnear the bulk value for crystals as small as 2.0 nm. This issurprising because quantum confinement continues to
TABLE II. The local bond lengths /H20849in Å /H20850at a vacancy site in
crystals with diameters 1.02 and 2.20 nm. The segment labels ref-erence the tetrahedral structure in Fig. 2. Using the DFT-LDA the
theoretical bond length in bulk Ge is 2.44 Å, which corresponds tosegment lengths of 3.99 Å before atomic relaxation.
Segment D=1.02 nm D=2.20 nm
AB 2.55 2.90
AC 3.57 3.47
AD 3.57 3.47
BC 3.79 3.46
BD 3.79 3.46
CD 2.50 2.90
FIG. 1. /H20849Color online /H20850Ge nanocrystal morphology for /H20849a/H20850Ge44H42,/H20849b/H20850Ge130H98, and /H20849c/H20850Ge244H158. The corresponding sizes are 1.02,
1.70, and 2.20 nm, respectively. A central vacancy is depicted as a hatched atom surrounded by four missing bonds /H20849broken lines /H20850, all in red.
Ge atoms are colored in blue /H20849dark gray /H20850; saturating H atoms are in light gray.
[110]
AB
CD
FIG. 2. The reference geometry of the atomic structure at the
vacancy site. The dashed circle is the vacancy and the solid circlesare the nearest-neighbor Ge atoms that form a tetrahedron. Thetetrahedron, without atomic motion has T
dsymmetry. The calcu-
lated bond lengths for various sized nanocrystals are shown in TableII.BAYUS, PAZ, AND BECKMAN PHYSICAL REVIEW B 82, 155409 /H208492010 /H20850
155409-4strongly influence the band gap for crystals with a similar
size, as shown in Fig. 4/H20849b/H20850.
C. Distance from crystal center
To determine the influence of a vacancy’s position on its
energy a 2.20 nm crystal /H20849Ge244H158/H20850is examined with a
vacancy at various locations within it. The calculated ener-gies are shown in Fig. 5. The configuration where the va-
cancy is adjacent to the center of the crystal /H20851Fig. 1/H20849c/H20850/H20852is
defined as the zero. Near the center of the crystal there islittle change in the energy, but once the vacancy is within 0.7nm of the surface it begins to drop substantially. The laststable vacancy site is 0.3 nm from the surface. The energy ofa vacancy at this site is a full 1.2 eV less than a vacancy nearthe center. From the results in Sec. III B it is known that the
energy of formation in the center of the 2.20 nm crystal isalmost that observed in bulk or slightly smaller. Using theenergies in Table Iit is deduced that the energy of formation
for the vacancies near the surface can be no larger than 0.5–1.4 eV .
IV. CONCLUSION
A vacancy in a Ge nanocrystal undergoes a Jahn-Teller
distortion. The Td-symmetrized broken bonds located at the
vacancy introduce a set of threefold degenerate, partially oc-cupied, states in the gap. When these dangling bonds recon-struct the defect symmetry is lowered. This reduces the de-generacy of the defect states by splitting them into a lower-energy, fully occupied state and two higher-energy,degenerate empty states. In a bulk crystal it is known that thesymmetry of the vacancy site is D
2d,58but in the nanocrystals
the surfaces introduce additional distortion. For the smallestcrystal the surface influence is great; the defect symmetry isC
sand the energy splitting is approximately 2.7 eV . This
results in a dramatic reduction in the energy of formation. In-2-101234Energy (eV)
D=1.02 nmPerfect CrystalVacancy
No RelaxationVacancy
Jahn-Teller
-2-101234Energy (eV)
D=2.2 0nm
FIG. 3. /H20849Color online /H20850The top frame shows the energy levels for
a crystal with 1.02 nm diameter and the bottom frame shows a 2.20nm crystal. The red dashed line is the Fermi energy. The bandalignment between the diagrams is arbitrary. The crystals in the leftcolumn are perfect and contain no vacancy. The crystals in thecenter column have vacancies, but the structure has not been al-lowed to relax. The states at the Fermi level are threefold degener-ate. The right column shows the states after the Jahn-Teller distor-tion is completed, which allows the atomic structure to break thesymmetry and lift the degeneracy of the gap state.0.511.52Energy of formation (eV )
1 1.5 2 2.5 3
Diameter (nm)1.522.533.5Band gap (eV)(a)
(b)
FIG. 4. /H20849Color online /H20850The relative energy of formation for a
vacancy in different sized nanocrystals is plotted in frame /H20849a/H20850.I n
frame /H20849b/H20850band gap is plotted versus the crystal size. The solid lines
show Eq. /H208494/H20850with the coefficients fitted to the data.
02 4 68
Distance from center (Å)-1.2-0.8-0.40Relative energy (eV)Vacancy energy versus pos ition
22 Å Ge nanocrystal
FIG. 5. The relative energy of a vacancy in a 2.20 nm Ge nano-
crystal /H20849Ge244H158/H20850. The zero of the energy scale is defined to be the
innermost atomic site.STRUCTURE, ENERGY , AND ELECTRONIC STATES OF … PHYSICAL REVIEW B 82, 155409 /H208492010 /H20850
155409-5the 2.20 nm crystal the defect almost has the D2dsymmetry
that is found in bulk. The energy splitting is also smaller thanthat in the 1.02 nm crystal, only around 0.75 eV; therefore,the energy reduction due to the bond reconstruction is lowerand the energy of formation is larger in the 2.20 nm crystal.This is consistent with the calculated prediction that the en-ergy of formation will approach the bulk value for nanocrys-tals larger than 2.0 nm.
The band gap of the crystal continues to change greatly
even when the diameter is as large as 2 nm. It is deduced thatalthough quantum confinement continues to impact the en-ergy levels in the crystal, the primary influence on the va-
cancy is the surface’s ability to enhance the internal struc-tural relaxation. It is concluded that in this example thequantum self-purification effect plays a small role if any. Asimilar observation has been made for vacancies in Si.
48It is
hypothesized that this is due to the defect’s wave functionbeing highly localized at the reconstruction bonds.
Finally, it is determined that vacancies placed within 0.7
nm of the surface are spontaneously removed. Surprisinglyvacancies in the interior of the crystal are stable and do notappear to be drawn toward the exterior. An additional conse-quence is that if a surface were to act as a vacancy source,the vacancies produced from the surface are unlikely to pen-etrate deeply into the nanocrystal. The system studied herehas H-passivated surfaces, which allows for large relax-
ations. Experimental crystals that have surface reconstruc-tions or polymer coatings will have more rigidity and theinfluence of the surfaces will be further muted inside thecrystal.
The picture that emerges from this work is that moderate
sized crystals will have an interior where vacancies behavebulklike and a thin exterior surface region where the surfaceeffects will dominate. Assuming that the properties of theself-interstitial defect are not strongly modified by the sur-faces, then the evidence in this paper predicts that the self-diffusion in the interior of Ge nanocrystals will not be sub-stantially different from that observed in bulk. However,recent experiments indicate that Ge surfaces are not sinks forinterstitial atoms, but instead reflect the interstitial Ge backinto the crystal.
31If this observation holds within the nanore-
gime then it is possible that the large surface to volume ratioin nanostructures will magnify the impact of the interstitial-assisted diffusion.
ACKNOWLEDGMENT
The authors gratefully acknowledge support from the Na-
tional Science Foundation under Grant No. DMR-0755231.
*Present address: Institut de Ciència de Materials de Barcelona,
ICMAB-CSIC, Campus UAB, 08193 Bellaterra, Spain.
†sbeckman@iastate.edu
1P. Y . Yu and M. Cardona, Fundamentals of Semiconductors , 3rd
ed. /H20849Springer, New York, 2001 /H20850.
2C. O. Chui, K. Gopalakrishnan, P. B. Griffin, J. D. Plummer, and
K. C. Saraswat, Appl. Phys. Lett. 83, 3275 /H208492003 /H20850.
3M. L. Lee, C. W. Leitz, Z. Cheng, A. J. Pitera, T. Langdo, M. T.
Currie, G. Taraschi, E. A. Fitzgerald, and D. A. Antoniadis,Appl. Phys. Lett. 79, 3344 /H208492001 /H20850.
4H. L. Shang et al. , Tech. Dig. - Int. Electron Devices Meet.
2002, 441.
5C. O. Chui, S. Ramanathan, B. B. Triplett, P. C. McIntyre, and
K. C. Saraswat, IEEE Electron Device Lett. 23, 473 /H208492002 /H20850.
6M. L. Lee, E. A. Fitzgerald, M. T. Bulsara, M. T. Currie, and A.
Lochtefeld, J. Appl. Phys. 97, 011101 /H208492005 /H20850.
7H. Shang, J. O. Chu, X. Wang, P. M. Mooney, K. Lee, J. Ott, K.
Rim, K. Chan, K. Guarini, and M. Ieong, Dig. Tech. Pap. -Symp. VLSI Technol. 2004, 204.
8H. Shang, M. M. Frank, E. P. Gusev, J. O. Chu, S. W. Bedell, K.
W. Guarini, and M. Ieong, IBM J. Res. Dev. 50, 377 /H208492006 /H20850.
9Z. Y . Cheng, J. W. Jung, M. L. Lee, A. J. Pitera, J. L. Hoyt, D. A.
Antoniadis, and E. A. Fitzgerald, Semicond. Sci. Technol. 19,
L48 /H208492004 /H20850.
10H. J. Queisser and E. E. Haller, Science 281, 945 /H208491998 /H20850.
11H. Bracht, MRS Bull. 25,2 2 /H208492000 /H20850.
12P. Ganster, G. Treglia, and A. Saul, Phys. Rev. B 79, 115205
/H208492009 /H20850.
13Y . Shimizu, M. Uematsu, and K. M. Itoh, Phys. Rev. Lett. 98,
095901 /H208492007 /H20850, and references therein.14J. Zhu, T. Diaz dela Rubia, L. H. Yang, C. Mailhiot, and G. H.
Gilmer, Phys. Rev. B 54, 4741 /H208491996 /H20850.
15B. Sahli and W. Fichtner, Phys. Rev. B 72, 245210 /H208492005 /H20850.
16S. A. Centoni, B. Sadigh, G. H. Gilmer, T. J. Lenosky, T. D. de
la Rubia, and C. B. Musgrave, Phys. Rev. B 72, 195206 /H208492005 /H20850.
17W. K. Leung, R. J. Needs, G. Rajagopal, S. Itoh, and S. Ihara,
Phys. Rev. Lett. 83, 2351 /H208491999 /H20850.
18A. Antonelli, E. Kaxiras, and D. J. Chadi, Phys. Rev. Lett. 81,
2088 /H208491998 /H20850.
19C. Z. Wang, C. T. Chan, and K. M. Ho, Phys. Rev. Lett. 66, 189
/H208491991 /H20850.
20A. F. Wright, Phys. Rev. B 74, 165116 /H208492006 /H20850.
21A. Fazzio, A. Janotti, A. J. R. da Silva, and R. Mota, Phys. Rev.
B61, R2401 /H208492000 /H20850.
22A. J. R. da Silva, A. Janotti, A. Fazzio, R. J. Baierle, and R.
Mota, Phys. Rev. B 62, 9903 /H208492000 /H20850.
23B. P. Uberuaga, G. Henkelman, H. Jonsson, S. T. Dunham, W.
Windl, and R. Stumpf, Phys. Status Solidi B 233,2 4 /H208492002 /H20850.
24A. Carvalho, R. Jones, C. Janke, J. P. Goss, P. R. Briddon, J.
Coutinho, and S. Öberg, Phys. Rev. Lett. 99, 175502 /H208492007 /H20850.
25C. J. Hwang and L. A. K. Watt, Phys. Rev. 171, 958 /H208491968 /H20850.
26A. Chroneos, R. W. Grimes, B. P. Uberuaga, and H. Bracht,
Phys. Rev. B 77, 235208 /H208492008 /H20850, and references therein.
27H. Pinto, J. Coutinho, V . Torres, S. berg, and P. Briddon, Mater.
Sci. Semicond. Process. 9, 498 /H208492006 /H20850.
28M. D. Moreira, R. H. Miwa, and P. Venezuela, Phys. Rev. B 70,
115215 /H208492004 /H20850.
29J. Vanhellemont, P. Śpiewak, and K. Sueoka, J. Appl. Phys. 101,
036103 /H208492007 /H20850.
30H. H. Silvestri, H. Bracht, J. L. Hansen, A. N. Larsen, and E. E.BAYUS, PAZ, AND BECKMAN PHYSICAL REVIEW B 82, 155409 /H208492010 /H20850
155409-6Haller, Semicond. Sci. Technol. 21, 758 /H208492006 /H20850.
31H. Bracht, S. Schneider, J. N. Klug, C. Y . Liao, J. L. Hansen, E.
E. Haller, A. N. Larsen, D. Bougeard, M. Posselt, and C. Wün-disch, Phys. Rev. Lett. 103, 255501 /H208492009 /H20850.
32M. Werner, H. Mehrer, and H. D. Hochheimer, Phys. Rev. B 32,
3930 /H208491985 /H20850.
33H. Bracht, H. H. Silvestri, I. D. Sharp, and E. E. Haller, Phys.
Rev. B 75, 035211 /H208492007 /H20850.
34H. Bracht, N. A. Stolwijk, and H. Mehrer, Phys. Rev. B 43,
14465 /H208491991 /H20850.
35S. Brotzmann, H. Bracht, J. L. Hansen, A. N. Larsen, E. Simoen,
E. E. Haller, J. S. Christensen, and P. Werner, Phys. Rev. B 77,
235207 /H208492008 /H20850.
36A. Chroneos, H. Bracht, R. W. Grimes, and B. P. Uberuaga,
Appl. Phys. Lett. 92, 172103 /H208492008 /H20850.
37S. Das, R. K. Singha, K. Das, A. Dhar, and S. K. Ray, J.
Nanosci. Nanotechnol. 9, 5484 /H208492009 /H20850.
38X. Ma and C. Wang, Appl. Phys. B: Lasers Opt. 92, 589 /H208492008 /H20850.
39S. Sato, H. Morisaki, and M. Iwase, Appl. Phys. Lett. 66, 3176
/H208491995 /H20850.
40L. N. Zhang, J. He, J. Zhang, F. Liu, Y . Fu, Y . Song, and X.
Zhang, IEEE Trans. Electron Devices 55, 2907 /H208492008 /H20850.
41Y . Jiang et al. ,IEEE Electron Device Lett. 29, 595 /H208492008 /H20850.
42T. Markussen, R. Rurali, A. P. Jauho, and M. Brandbyge, Phys.
Rev. Lett. 99, 076803 /H208492007 /H20850.
43S. P. Beckman, J. X. Han, and J. R. Chelikowsky, Phys. Rev. B
74, 165314 /H208492006 /H20850.44G. M. Dalpian and J. R. Chelikowsky, Phys. Rev. Lett. 96,
226802 /H208492006 /H20850.
45M. H. Du, S. C. Erwin, A. L. Efros, and D. J. Norris, Phys. Rev.
Lett. 100, 179702 /H208492008 /H20850.
46G. M. Dalpian and J. R. Chelikowsky, Phys. Rev. Lett. 100,
179703 /H208492008 /H20850.
47J. B. Li, S. H. Wei, S. S. Li, and J. B. Xia, Phys. Rev. B 77,
113304 /H208492008 /H20850.
48S. P. Beckman and J. R. Chelikowsky, Physica B 401, 537
/H208492007 /H20850.
49S. P. Beckman and J. R. Chelikowsky /H20849unpublished /H20850.
50W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 /H208491965 /H20850.
51D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 /H208491980 /H20850.
52J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P.
Ordejón, and D. Sánchez-Portal, J. Phys.: Condens. Matter 14,
2745 /H208492002 /H20850.
53N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 /H208491991 /H20850.
54E. Anglada, J. M. Soler, J. Junquera, and E. Artacho, Phys. Rev.
B66, 205101 /H208492002 /H20850.
55F. D. Murnaghan, Proc. Natl. Acad. Sci. U.S.A. 30, 244 /H208491944 /H20850.
56O. Madelung, U. Rössler, and M. Schulz, Group IV Elements,
IV-IV and III-V Compounds Part a , Landolt-Börnstein Database
V ol. 41A1a /H20849Springer-Verlag, Berlin, 2001 /H20850.
57D. R. Hamann, Phys. Rev. Lett. 76, 660 /H208491996 /H20850.
58G. D. Watkins, Deep Centers in Semiconductors /H20849Gordon and
Breach Science, New York, 1986 /H20850, Chap. 3, p. 147.
59S. Ögüt and J. R. Chelikowsky, Phys. Rev. B 64, 245206 /H208492001 /H20850.STRUCTURE, ENERGY , AND ELECTRONIC STATES OF … PHYSICAL REVIEW B 82, 155409 /H208492010 /H20850
155409-7 |
PhysRevB.93.075301.pdf | PHYSICAL REVIEW B 93, 075301 (2016)
Long-distance entanglement of spin qubits via quantum Hall edge states
Guang Yang,1Chen-Hsuan Hsu,1Peter Stano,1Jelena Klinovaja,2and Daniel Loss1,2
1RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
2Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
(Received 6 October 2015; revised manuscript received 12 January 2016; published 1 February 2016)
The implementation of a functional quantum computer involves entangling and coherent manipulation of a
large number of qubits. For qubits based on electron spins confined in quantum dots, which are among the mostinvestigated solid-state qubits at present, architectural challenges are often encountered in the design of quantumcircuits attempting to assemble the qubits within the very limited space available. Here, we provide a solutionto such challenges based on an approach to realizing entanglement of spin qubits over long distances. We showthat long-range Ruderman-Kittel-Kasuya-Yosida interaction of confined electron spins can be established byquantum Hall edge states, leading to an exchange coupling of spin qubits. The coupling is anisotropic and canbe either Ising type or XY type, depending on the spin polarization of the edge state. Such a property, combinedwith the dependence of the electron spin susceptibility on the chirality of the edge state, can be utilized to gainvaluable insights into the topological nature of various quantum Hall states.
DOI: 10.1103/PhysRevB.93.075301
I. INTRODUCTION
Quantum computers, exploiting entanglement and super-
position of quantum mechanical states, promise much betterperformance than classical computers tackling a collectionof important mathematical problems [ 1]. Over the past few
decades, a variety of solid-state systems have been studied forthe implementation of qubits, the building blocks of a quantumcomputer. Among such systems, a very promising candidate[2] makes use of the spin of electrons confined in semicon-
ductor quantum dots (QDs). In that scheme, entanglementof qubits is achieved through the direct exchange interactionbetween confined electrons, and manipulation of individualqubits can be realized by magnetic or electrical means [ 3].
Recent advances in QD technology have established longcoherence times [ 4] exceeding 0 .2 ms and fast gate-operation
times [ 3] on the order of tens of nanoseconds for spin qubits
in QDs.
With the great progress in the development of quality
spin qubits, scalability becomes the next major challengetowards building a functional quantum computer capableof performing fault-tolerant quantum computing [ 5]. The
implementation of quantum-error-correction algorithms [ 6]
requires that the system reach a size of several thousands ofqubits. In practice, however, one faces tremendous difficultiesin assembling so many spin qubits, among which entanglementmust be selectively established and maintained. Indeed, thenearest-neighbor nature of the direct exchange interaction, theprimary source of entanglement, restricts drastically access ofeach qubit to the rest of the system and thus the space that canbe used for installing the quantum circuits. A natural way toovercome such difficulties is to employ means of entanglingspin qubits over long distances, which creates extra space forwiring the quantum circuits. In principle, this may be achievedby coupling the spin qubits to an electromagnetic cavity[7–10], a floating metallic gate [ 11], or a dipolar ferromagnet
[12]. Recently, it was shown that coupling of distant spin qubits
can also be realized via photon-assisted cotunneling [ 13].
In this paper, we propose a mechanism to achieve long-
distance entanglement of spin qubits. We make use of theRuderman-Kittel-Kasuya-Yosida (RKKY) interaction [ 14–16]
between confined electron spins in QDs, mediated by theconducting edge states of quantum Hall (QH) liquids [ 17],
to which the QDs are tunnel coupled [ 18]. The spin qubit
coupling obtained in such a way is particularly interesting.Depending on whether the edge state is spin polarized or not,the induced coupling between the spin qubits can be Ising typeand perpendicular to the plane of the system, or XY type and inplane. This offers great versatility in the design of large-scalequantum circuits. The advantage of using QH edge states istwofold. First, the edge states and the QDs can be formed inthe same material (by top gates) such as a two-dimensionalelectron gas (2DEG) in GaAs heterostructures. Second, theQH edge states are topologically stable and thus much morerobust against disorder effects compared to one-dimensional(1D) conduction channels in quantum wires. Moreover, we findthat the spin susceptibility of QH edge states manifests theinequivalence between the opposite directions, “clockwise”and “counterclockwise”, along the QH edge. In chiral edgestates, conduction electrons propagate in only one direction,leading to a “rectified” spin susceptibility in the propagationdirection of electrons. In nonchiral edge states, the spinsusceptibility is nonzero in both directions along the QH edge,but with different magnitudes. The spin susceptibility has thesame type of anisotropy as the coupling between qubits. Thus,measuring the spatial dependence of the spin susceptibility[19] can serve as a powerful probe of the chirality and spin
polarization of the edge state, and thus of the topological order[17] in a QH liquid.
II. MODEL
We now discuss the physics of RKKY interaction mediated
by QH edge states. The basic setup is shown in Fig. 1.T w o
QDs are placed adjacent to a QH liquid, separated by a distanceLand labeled by the site index i=1,2. Conduction electrons
in the QH edge state can tunnel into and out of the QDs[18] and thus can interact with the localized spins in them.
This establishes coupling between the QH edge and the QDs.
2469-9950/2016/93(7)/075301(15) 075301-1 ©2016 American Physical SocietyY ANG, HSU, STANO, KLINOV AJA, AND LOSS PHYSICAL REVIEW B 93, 075301 (2016)
FIG. 1. The basic setup consisting of two QDs (yellow disks)
tunnel coupled to the edge (white lines and arrows) of a QH liquid
(blue sheet) confined in the x-yplane. In general, the QH edge may
support multiple edge modes, propagating in the same or opposite
direction(s), which we do not depict explicitly. The QDs are separated
by a distance Lalong the QH edge. Each QD contains a single electron
(blue spheres), whose spin (red arrows) serves as a qubit. The coupling
strength between the QDs and the QH edge is controlled by gates
(not shown). We assume no direct interaction between the localizedelectron spins I
1andI2in the QDs.
For simplicity, we treat the QDs as two spatial points. The
Hamiltonian describing such a system has the form
H=Hedge+/summationdisplay
i=1,2/Gamma1iSi·Ii, (1)
where Hedge is the Hamiltonian of conduction electrons in
the edge state, Ii=(Ix
i,Iy
i,Iz
i) denotes the localized spin
in the ith QD, and Si=(Sx
i,Sy
i,Sz
i) denotes the spin of
conduction electrons coupled to Ii, with coupling strength /Gamma1i.
Experimentally, /Gamma1ican be tuned by gating. We define Sito be
the spin density in the edge state multiplied by the confinementlength of the QDs. For the setup, we assume Lis large so that
there is no direct interaction between the spins in the QDs.
In the weak tunnel coupling regime such that /Gamma1
i/lessmuchEF,
where EFis the Fermi energy of conduction electrons, the
dynamics of the spins in the QDs effectively decouples fromthat of the conduction electrons. In such a case, one canderive an effective Hamiltonian for the spins in QDs, validin the adiabatic regime, by performing a Schrieffer-Wolfftransformation [ 20,21]o fE q .( 1) followed by tracing out the
degrees of freedom of conduction electrons (see Appendix A
for the derivation of effective Hamiltonian and a discussion ofadiabaticity),
H
eff=/summationdisplay
ij,αβJαβ
ijIα
iIβ
j−/summationdisplay
iBi·Ii, (2)
where the spin-component indices α,β=x,y,z . The first
term is the RKKY interaction, with Jαβ
ij=/Gamma1i/Gamma1jχαβ
ij/2. Here,
χαβ
ijis the static spin susceptibility of conduction electrons,
χαβ
ij=−i/integraltext∞
0dt e−ηt/angbracketleft[Sα
i(t),Sβ
j(0)]/angbracketright, where η=0+and/angbracketleft.../angbracketright
denotes the average determined by Hedge. Physically, conduc-
tion electrons in the vicinity of a QD develop a spin-densityoscillation due to their interaction with the spin in the QD. This
spin-density response, determined by χαβ
ij, can be perceived by
the spins in other QDs coupled to the QH edge. In this way,the RKKY interaction is established. For spin-unpolarizedQH states, we assume /angbracketleftS
x
i/angbracketright=/angbracketleftSy
i/angbracketright=/angbracketleftSz
i/angbracketright=0, such that
χαβ
ij=δαβχαα
ij. On the other hand, the in-plane spin operators
Sx
i,Sy
iare less relevant (in the renormalization group sense)
than the out-of-plane ones Sz
iin a QH state with full spin
polarization, as we discuss in the following. In this case,
we set Sx
i=Sy
i=0 and hence χαβ
ij=δαzδβzχzz
ij. Thus, in
general we have Jαβ
ij=δαβJαα
ij. The RKKY interaction leads
to an effective exchange coupling Jα=Jαα
12+Jαα
21,a sa
function of the interdot distance L, between the localized
spinsIα
1andIα
2. The effective onsite Zeeman fields Biare a
direct consequence of time-reversal (TR) symmetry breaking
in QH systems, Bα
i=(/Gamma12
i/4)/integraltext∞
0dt e−ηt/epsilon1αβγ/angbracketleft{Sβ
i(t),Sγ
i(0)}/angbracketright.
We find that Bα
i=δαzBz
iin spin-unpolarized QH states and
Bi=0 in spin-polarized QH states (for more details and
estimates, we refer to Appendix A).
III. RKKY INTERACTION IN VARIOUS QH STATES
The RKKY interaction in Eq. ( 2) is by nature long ranged
and can be used as an approach to entangle spin qubits overlong distances. Thus, it is important to understand how theinteraction looks in various QH systems. To this end, it isconvenient to adopt a continuum description of the QH edgestates that is well approximated by the chiral Luttinger liquid(LL) model at low energy [ 17]. In general, the edge of a QH
liquid may support (electron-) density-fluctuation modes aswell as Majorana fermions (zero modes), with the action
S
edge=/integraldisplay
dxdt/bracketleftBigg/summationdisplay
IJ1
4π(KIJ∂tφI∂xφJ−VIJ∂xφI∂xφJ)
+/summationdisplay
KiλK(∂t−vK∂x)λK/bracketrightBigg
, (3)
written in the bosonization language [ 17] (throughout the paper
we set /planckover2pi1=1). The bosonic fields φIdescribe the density
modes, and λKdenote the Majorana fermions. The symmetric
matrix KIJencodes the topological properties of the QH state,
while the positive-definite symmetric matrix VIJspecifies the
velocities and interactions of φI. The parameter vKis the
velocity of λK:vK>0(vK<0) if the λKis left moving
(right moving).
Upon passing to the continuum limit, we replace the spin
operators Sα
i(t)/lwith spin-density operators Sα(xi,t), where l
is the confinement length of the QDs and xiis the position of the
ith QD. The nonvanishing components of the spin susceptibil-
ity are given by χαα
ij=−il2/integraltext∞
0dt e−ηt/angbracketleft[Sα(xi,t),Sα(xj,0)]/angbracketright.
Assuming translation invariance along the QH edge, whichis justified for clean samples, we may further write χ
αα
ij=
χαα(xi−xj)[22], where
χαα(x)=2l2/integraldisplay∞
0dt e−ηtIm/angbracketleftTSα(x,t)Sα(0,0)/angbracketright, (4)
withTthe time-ordering operator. The correlators are evalu-
ated in the zero-temperature limit. We define
Sα(x,t)=1
2/summationdisplay
σσ/primeψ†
σ(x,t)σα
σσ/primeψσ/prime(x,t), (5)
075301-2LONG-DISTANCE ENTANGLEMENT OF SPIN QUBITS VIA . . . PHYSICAL REVIEW B 93, 075301 (2016)
where ψσ=/summationtext
μψμ
σis the sum of the most relevant electron
operators ψμ
σwith spin σ=↑,↓on the QH edge.
The number of ψμ
↑operators is not necessarily equal to that
ofψμ
↓operators since TR symmetry is broken. For instance,
the most relevant electron operators have the same spin ina spin-polarized QH state, so that S
x=Sy=0. This is in
contrast to the situation in 1D systems where TR symmetry ispresent [ 23,24]. Using bosonization, we express S
αin terms
of the fields φIandλK, and compute the spin susceptibility.
We sketch the calculation of the spin susceptibility for
a generic QH edge state (for particular examples, seeAppendix B). First of all, we assume separation of charged
and neutral degrees of freedom in the QH edge state. Thisphenomenon, as has been demonstrated experimentally in anumber of QH systems [ 25,26], results from strong Coulomb
interaction among the elementary density modes φ
Iand
resembles “charge-spin separation” in a generic TR-invariant1D system [ 22]. As a result, the physical modes that propagate
on the QH edge are the charged and neutral collectivemodes as well as Majorana fermions. The physical parametersrelevant to experiment are the velocities and interactionsof these propagating modes, whose magnitudes are set bydifferent energy scales in the QH system. For instance, thecharged-mode velocity, determined by the dominant Coulombenergy scale, is much greater than the velocity of neutralmode and other parameters [ 25]. We make use of this fact
in our calculation. For a moment, we consider the case of twodensity modes in the edge theory [see Eq. ( 3)]. To compute
the correlators in Eq. ( 4), we define a new set of fields which
diagonalize the action of the density modes φ
I. The action
takes the form
Sdensity=/integraldisplay
dxdt1
4π[∂tφ+∂xφ++ε∂tφ−∂xφ−
−v+∂xφ+∂xφ+−v−∂xφ−∂xφ−]( 6 )
in the basis of new fields φ+andφ−. Here, ε=1(ε=− 1)
if the edge states are chiral (nonchiral) and v+,v−>0. New
velocities v+andv−are well approximated by the velocities
of the physical charged mode and neutral mode, respectively,so that v
+/greatermuchv−. Upon expressing the spin-density operators
in terms of the free fields φ+,φ−, andλK, it is straightforward
to compute the correlators
/angbracketleftTSα(x,t)Sα(0,0)/angbracketright∝ cos(/Delta1kx )/bracketleftbigg1
δ+i(t+x/v+)/bracketrightbigggα
+
×/bracketleftbigg1
δ+i(t+εx/v −)/bracketrightbigggα
−
, (7)
where δ> 0 is an infinitesimal and /Delta1kis the gauge-invariant
momentum difference between the edge modes. The case/Delta1k=0 corresponds to the scattering of an edge mode with
itself. Here, we have omitted the terms that are less relevant,and assumed |v
K|=v−as both of the velocities are determined
by less dominant energy scales in the system. The exponentsg
α
+,gα
−are functions of the matrices KIJandVIJand
as we show 0 <gα
+/lessmuch1 and gα
−>1 (see Appendix Bfor
the expressions in different QH states). Evaluating the timeintegral in Eq. ( 4), we obtain χ
αα(x), which in general may
contain multiple terms for different momentum differences.We keep only the most relevant terms.The various QH states can be divided into three types:
(i) those with a chiral edge state containing a single densitymode, such as the Laughlin states at filling factors ν=1/m,
where mis an odd integer; (ii) those with a chiral edge
state containing multiple interacting density modes, such asthe QH state at ν=2; (iii) those with a nonchiral edge
state, such as the particle-hole dual states [ 27] of Laughlin
states.
For QH states of type (i), we find χ
αα(x)=0, taking into
account the most relevant spin operators in the edge state.Thus, to the lowest order, the RKKY interaction cannot be es-tablished. Physically, the vanishing spin susceptibility reflectsthe homogeneous electronic structure in an independent QHedge mode, a property originating from the incompressibilityof the QH liquid which prevents the formation of electronicspin texture. In reality, however, a small nonzero spin suscep-tibility may still be measured, due to higher-order processesinvolving virtual transitions to edge states in higher Landaulevels.
In QH edge states of types (ii) and (iii), the spin sus-
ceptibility is nonzero to the lowest order. In these cases,the interedge interactions introduce inhomogeneous degreesof freedom (“noise”) to the stream of conduction electrons,allowing for the development of spin-density oscillations. Wefind
χ
αα(x)=cos(/Delta1kx )
|x|gα/Theta1(−x)Cα(gα,v)( 8 )
for left-moving type (ii) edge states, where gα=gα
++gα
−−1,
/Theta1(x) is the Heaviside step function, and Cα(gα,v)a r e
functions of gα=(gα
+,gα
−) andv=(v+,v−), whose explicit
definitions are given in Appendix B. If the edge state is right
moving, one replaces /Theta1(−x) with /Theta1(x), and sends v→−vin
Cα(gα,v). These findings suggest that the spin susceptibility
in type (ii) edge states is “rectified”, i.e., directed in thedownstream direction of the propagation of conduction elec-trons [see Fig. 2(a)], where left- and right-moving directions
are defined with respect to the lower edge of the QH liquid[the same in Fig. 2(b)]. This result is not surprising and can
be understood also intuitively. In a left-moving edge state,conduction electrons move in the −xdirection, leading to the
factor /Theta1(−x) in the expression of χ
αα(x). Formally, such an
interesting form of the spin susceptibility is a manifestation of
I1 I2xy
I1 I2xy
(a) (b)
FIG. 2. Spin susceptibility in QH edge states of (a) type (ii) and
(b) type (iii). For the type (ii) case, the spin susceptibility is directed
in the propagation direction of the edge modes. For the type (iii) case,
the spin susceptibility is nonzero in both directions along the QHedge.
075301-3Y ANG, HSU, STANO, KLINOV AJA, AND LOSS PHYSICAL REVIEW B 93, 075301 (2016)
the causality principle in 1D chiral systems, where information
is transported one way and novel physical rules can emerge,e.g., see Ref. [ 28] for fluctuation-dissipation relations in chiral
QH systems.
Lastly, we find
χ
αα(x)=cos(/Delta1kx )
|x|gα{/Theta1(x)Cα
>(gα,v)+/Theta1(−x)Cα
<(gα,v)}
(9)
for type (iii) edge states, where Cα
>(gα,v) andCα
<(gα,v)a r e
functions of gαandv, defined in Appendix B.T h es p i n
susceptibility in this case is “both-way”, as shown in Fig. 2(b),
with different magnitudes in the +xand−xdirections, i.e.,
Cα
>(gα,v)/negationslash=Cα
<(gα,v). This again reflects the inequivalence
between left moving and right-moving edge modes. Imaginingnow the chirality of all edge modes is reverted, e.g., by TRoperation, the profile of the spin susceptibility should also bereverted. Indeed, we find that C
α
>(gα,v) are related to Cα
<(gα,v)
by the exchange of arguments v+↔v−andgα
+↔gα
−, which
technically carries out the chirality-reverting procedure (seeAppendix B). In the above discussion, we have assumed that
spin excitations do not extend into the L
0−Lpart of the
QH edge, where L0is the total edge length. In practice, this
is realized by grounding the L0−Lpart or by choosing the
sample such that L0/greatermuchL.
The exponents gα, where α=x,y,z , determine how the
RKKY interaction scales with distance. In Table I, we list
them in different QH states. In general, gαdepend on both the
chirality and spin polarization of the QH edge state. For chiraledge states, i.e., those of type (ii), these exponents are integralinvariants depending on the topological order of the bulkliquid, whereas for nonchiral edge states they are nonuniversaland depend on the parameters in the Hamiltonian. In the lattercase, we write g
α=gα
0+δgα, where gα
0is the integer part of
gα. As shown in Appendix B,δgα/gα
0/lessmuch1 for all the nonchiral
edge states in the table, assuming “charge-neutral separation”on the edge. Moreover, we find that the in-plane componentsof the RKKY interaction vanish in a spin-polarized QHstate, leading to an Ising-type exchange coupling of spinqubits. On the other hand, the RKKY interaction has zeroout-of-plane component and equal in-plane components in aspin-unpolarized QH state, which is XY type. This suggeststhat a transformation of the anisotropy type of the RKKYinteraction may be observed in the QH liquid at ν=
2
3, which
was found to be spin unpolarized at low fields and spinpolarized at high fields [ 29].
The QH state at ν=
5
2is also of special interest. We
consider both Abelian and non-Abelian topological ordersproposed to describe this state. The former include the Halperin331 state [ 30] and 113 state [ 31], and the latter include
the Moore-Read (Pfaffian) state [ 32], the anti-Pfaffian state
[33,34], and the SU(2)
2state [ 35]. The 331 and 113 states can
be both spin polarized and unpolarized, just like the ν=2
3QH state. The Pfaffian state, like the Laughlin states, supports
a single density mode on the edge and thus has vanishingRKKY interaction. The particle-hole dual state of the Pfaffianstate, the anti-Pfaffian state, has a nonchiral edge state and anoninteger scaling exponent. For the SU(2)
2state, we assume
that the Majorana fermion and the neutral collective modepropagate at different velocities, as they should in reality,which is necessary to obtain a nonvanishing scaling exponent.
Such careful treatment is not essential for other ν=
5
2states.
We have assumed that the RKKY interaction is mediatedsolely by the fractional edge modes in the second Landaulevel, while the integer edge modes in the lowest Landaulevel do not play a role. Experimentally, this can be fulfilled,using the fact that edge modes in different Landau levels arespatially separated [ 36]. For instance, the QDs in Fig. 1can
be moved out of the plane of the QH liquid and formedin a second two- or quasi-one-dimensional electron gas inthe vertical direction [ 37–40], such that they are in tunnel
contact with the fractional edge modes but far away fromthe integer edge modes. The coupling between the integeredge and the QDs and the interaction between the integeredge and the fractional edge can be neglected to a goodapproximation.
IV . DISCUSSION
Let us estimate the coupling between the two spin qubits
in Fig. 1, given by Jα=/Gamma11/Gamma12{χαα(L)+χαα(−L)}/2. In
Appendix B, we obtain the dimensional part [ χαα(x)] of the
spin susceptibility
[χαα(x)]/similarequall2agα−1|x|−gα/v− (10)
for both type (ii) and type (iii) edge states, where ais the
lattice constant of the underlying material hosting the QHsystem. For example, let us consider the QH state at ν=2,
realized in GaAs heterostructures. We have a=0.565 nm
for GaAs, g
x=gy=1, and v−/similarequal104m/s[25]. Using /Gamma11=
/Gamma12=/Gamma1/similarequal0.1 meV and l=30 nm (see Appendix Cfor the
estimates), we find Jx=Jy/similarequal1μeV for L=1μm. This is
about one order of magnitude smaller than the direct exchangestrength J
direct/similarequal10–100 μeV in typical GaAs double QDs
[2] and is experimentally measurable. The RKKY interaction
established by QH edge states thus provides a way to realizeentangled quantum gates over mesoscopic distances. Theimplementation of two-qubit gates using Hamiltonians of theform of Eq. ( 2) is well known: see, e.g., Ref. [ 2] (footnote 13)
for Ising-type coupling and Ref. [ 7] for XY-type coupling. The
μeV exchange strength converts to gate-operation times of the
order of nanoseconds, which is well below the coherence times[3] of spin qubits.
It is interesting to compare the RKKY interaction in QH
edge states with that in semiconductor quantum wires. As-suming spin-rotation symmetry, the dimensional part [ χ
w(x)]
of the spin susceptibility in quantum wires can be found inRef. [ 23]. The ratio
r
α(x)=[χαα(x)]
[χw(x)]=vF
v−/parenleftbigga
|x|/parenrightbigggα−gw
(11)
characterizes the relative strength of the RKKY interaction in
the two sorts of systems, where vFis the Fermi velocity in the
quantum wire and gwdepends on the interaction of electrons.
In noninteracting case, gw=1. Consider the ν=2 QH edge
state and GaAs quantum wire. We find rx(L)=ry(L)/similarequal1.5
forL=1μm, using gw=0.75 and vF/similarequal105m/s[23]. In
principle, quantum wires can also be used to mediated RKKYinteraction between spin qubits. However, using QH edgestates offers more advantages. From technical aspect, the edge
075301-4LONG-DISTANCE ENTANGLEMENT OF SPIN QUBITS VIA . . . PHYSICAL REVIEW B 93, 075301 (2016)
TABLE I. Scaling exponents gαand anisotropy type of the RKKY interaction in various QH states. An overline is used to indicate a
spin-polarized state, e.g.,2
3denotes the spin-polarized QH state at ν=2
3. We consider several topological orders at ν=5
2, including both
Abelian ones (the 331 state and the 113 state, denoted as 331 /331 and 113 /113, respectively) and non-Abelian ones (the Pfaffian state, the
anti-Pfaffian state, and the SU(2) 2state, denoted as Pf,APf, and SU(2) 2, respectively). The 331 state and the 113 state both have spin-unpolarized
and spin-polarized versions. For chiral edge states, the exponents are integers and we add arrows to indicate that the spin susceptibility is
nonzero only in the downstream direction. For nonchiral edge states, the exponents are nonintegers and we enter the integer parts gα
0of the
exponents. We put “ −” in the entry if the corresponding component of the spin susceptibility (and thus that of the RKKY interaction) vanishes.
The RKKY interaction is XY type in spin-unpolarized states and Ising type in spin-polarized states.
QH state 1/m 22
32
3331 331 113 113 Pf APf SU(2) 2
gx–−→11 –−→3–3– – – –
gy–−→11 –−→3–3– – – –
gz–– –1 –−→3–3 – 1−→1
RKKY type XY XY Ising XY Ising XY Ising Ising Ising
states and the spin qubits can be realized in the same material,
for instance, in a 2DEG in GaAs heterostructures, which ismore experimentally accessible than a setup with quantumwires. More importantly, the topologically protected QH edgestates are more immune to disorder effects and perturbationsin the system than quantum wires. This guarantees a betterquality of the long-distance quantum gates.
Our discussions so far have focused on the RKKY in-
teraction between spin qubits. Interestingly, the treatmentscan also be applied to obtain the RKKY interaction betweennuclear spins embedded in the 1D QH edge state (see alsoRef. [ 41]). To this end, let /Gamma1
i→A/N andIi→˜Iiin Eq. ( 1)
and the following equations, where Ais the hyperfine coupling
constant, Nis the number of nuclear spins in a cross section
(labeled by i) of the QH edge, and ˜Iiis the total nuclear
spin operator in a given cross section. Given a nonchiraledge state with both spin-up and -down electrons, e.g., thespin-unpolarized state at ν=
2
3, the nuclear spins may form a
helical magnetic order [ 23] at low temperatures, induced by the
RKKY interaction. The nuclear magnetic order acts back onthe electronic system by gapping out conducting edge modes.Experimentally, such an order is evidenced by the reductionof the conductance at low temperatures [ 40].
By measuring the spatial dependence of RKKY interaction
[19,42], one can obtain information about the chirality and
spin polarization of the QH edge state, which in turn arerelated to the topological order of the bulk QH liquid [ 17].
In particular, this technique may be used to detect the natureof the QH liquid at ν=
5
2: One can distinguish between a
chiral edge state and a nonchiral edge state by confirmingwhether the spin susceptibility is unidirectional along theedge. One can rule out either a spin-polarized state or aspin-unpolarized state by comparing the in-plane and out-of-plane components of the RKKY interaction, by measuringthe spin states in the QDs. For this, one can make use ofexperimental techniques based on spin-to-charge conversion[2] developed for readout of spin qubits in QDs [ 43–45].
The numerical values of the scaling exponents also help toidentify the true ν=
5
2state. The advantages of measuring
the spin susceptibility are obvious, compared with otherapproaches detecting topological orders based on edge-bulkcorrespondence [ 17], such as measuring the temperature and
voltage dependence of quasiparticle tunneling [ 46]. First, it iseasier to vary the sampling point in space than in temperature
or voltage, e.g., one may use the setup in Fig. 1with an
array of QDs. Second, information encoded in spin degreesof freedom is more robust than that encoded in charge current,against unfavorable modification due to long-range Coulombinteraction in the device [ 47]. Compared with electronic Fabry-
P´erot [ 48,49] and Mach-Zehnder [ 50–52] interferometries, our
setup probes the non-Abelian topological orders at ν=
5
2with
a much simpler device geometry and more straightforwarddata.
The scenario becomes more complicated if one replaces
the QDs with quantum antidots [ 53]. In that case, tunneling
of quasiparticles, rather than electrons, defines the couplingbetween the QH edge and the antidots. It is still possible todefine an RKKY interaction mediated by quasiparticles in theedge state, whose spatial dependence can be used to distinguishdifferent Abelian QH states. For non-Abelian states, however,there are ambiguities in the scaling behavior of the RKKYinteraction, arising from the multiple fusion channels of non-Abelian quasiparticles.
To conclude, we have introduced an approach to achieving
long-distance entanglement of spin qubits confined in QDs,based on the RKKY interaction mediated by QH edge states.The approach allows for the implementation of quantumgates with long coupling ranges and fast operation times,which would greatly facilitate the development of large-scalequantum computers. From a fundamental point of view, theability to probe the chirality and the spin polarization of aQH edge state via measuring the spatial form of the RKKYinteraction opens up a new venue for studying electronic andspin physics in QH systems.
ACKNOWLEDGMENT
We acknowledge support from the Swiss NSF and NCCR
QSIT.
APPENDIX A: EFFECTIVE HAMILTONIAN
Our starting point is the Hamiltonian in Eq. ( 1). For weak
tunnel coupling between the QH edge and the QDs, wecan treat H
/Gamma1=/summationtext
i/Gamma1iSi·Iias a perturbation and make a
Schrieffer-Wolff transformation [ 20,21] to remove terms linear
075301-5Y ANG, HSU, STANO, KLINOV AJA, AND LOSS PHYSICAL REVIEW B 93, 075301 (2016)
in/Gamma1ifrom the Hamiltonian. The transformed Hamiltonian
reads as
¯H=eSHe−S=Hedge−1
2[[Hedge,S],S]+..., (A1)
where Ssatisfies [ Hedge,S]=H/Gamma1. Written in terms of the
Liouvillian superoperator L,S=L−1H/Gamma1. The leading-order
terms in /Gamma1iin¯Hare given by
¯H/Gamma1=−1
2[[Hedge,S],S]=1
2[L−1H/Gamma1,H/Gamma1]. (A2)
Using L−1=−i/integraltext∞
0dt e−ηteiLt, where η=0+, we find
¯H/Gamma1=−i
2/integraldisplay∞
0dt e−ηt[H/Gamma1(t),H/Gamma1]
=−i
2/summationdisplay
ij/Gamma1i/Gamma1j/integraldisplay∞
0dt e−ηt[Si(t)·Ii,Sj(0)·Ij]
=−1
2/integraldisplay∞
0dt e−ηt⎧
⎨
⎩/summationdisplay
iji/Gamma1i/Gamma1jIα
iIβ
j/bracketleftbig
Sα
i(t),Sβ
j(0)/bracketrightbig
+/summationdisplay
i/Gamma12
i/epsilon1αβγIα
iSβ
i(t)Sγ
i(0)/bracerightBigg
, (A3)
where we have defined ˆO(t)=eiHedgetˆOe−iHedgetfor an oper-
ator ˆOand used [ Iα
i,Iβ
j]=iδij/epsilon1αβγIγ
i, with /epsilon1αβγthe Levi-
Civita symbol. Summation over repeated spin-componentindices (Greek letters) is implied throughout this appendix.
Next, we take the expectation /angbracketleft.../angbracketrightover the electronic
degrees of freedom in the QH edge state. This gives an effectiveHamiltonian describing the dynamics of localized spins in theadiabatic limit,
H
eff=/angbracketleft ¯H/Gamma1/angbracketright=/summationdisplay
ij/Gamma1i/Gamma1j
2χαβ
ijIα
iIβ
j−/summationdisplay
iBα
iIα
i, (A4)
where we have identified the spin susceptibility of conduction
electrons
χαβ
ij=−i/integraldisplay∞
0dt e−ηt/angbracketleftbig/bracketleftbig
Sα
i(t),Sβ
j(0)/bracketrightbig/angbracketrightbig
, (A5)
and defined effective onsite Zeeman fields for the QDs
Bα
i=/Gamma12
i
2/integraldisplay∞
0dt e−ηt/epsilon1αβγ/angbracketleftbig
Sβ
i(t)Sγ
i(0)/angbracketrightbig
. (A6)
Hermiticity of the Hamiltonian ( A4) requires Bα
ibe real,
and thus the correlator /angbracketleftSβ
i(t)Sγ
i(0)/angbracketrightshould be replaced by
/angbracketleft{Sβ
i(t),Sγ
i(0)}/angbracketright/2. We are thus led to the Hamiltonian in
Eq. ( 2).
In deriving the above effective Hamiltonian, we have
neglected the external magnetic field Bextthat leads to the
formation of the QH liquid. We now estimate Bα
i(in units
of energy) and compare it with Bext. To this end, we take
the continuum limit of Eq. ( A6) assuming translation invari-
ance:Bα
i=Bα(x=a), where ais the natural short-distance
cutoff, taken as the lattice constant of the host material,and
Bα(x)=l2/Gamma12
i
4/integraldisplay∞
0dt e−ηt/epsilon1αβγ/angbracketleft{Sβ(x,t),Sγ(0,0)}/angbracketright,(A7)
which in turn can be written as a time-ordered product, such
that
Bα
i=lim
x→al2/Gamma12
i
2/integraldisplay∞
0dt e−ηt/epsilon1αβγRe/angbracketleftTSβ(x,t)Sγ(0,0)/angbracketright.
(A8)
In Appendix B, we have obtained the expressions of the
spin-density operators Sα(x,t) in various QH states. For
spin-unpolarized QH states we find Sx∝cos(φn+/Delta1kx ),
Sy∝sin(φn+/Delta1kx ), and Sz∝∂xφn, where φnis a neutral
edge mode (defined up to a multiplicative constant). Evaluatingthe correlators in Eq. ( A8), we find B
x
i=By
i=0 and Bz
i∼
(sin/Delta1ka )/Gamma12
i[χzz(a)], where [ χzz(x)] is given by Eq. ( 10).
The momentum difference /Delta1k depends on the transverse
distance between edge modes and can be taken as /Delta1k∼
1/lB, where lB∼10 nm is the magnetic length (the precise
evaluation of /Delta1kgives a similar result). Using a=0.565 nm
for GaAs, we perform an estimation similar to that for theeffective exchange coupling J
αand find Bz
i/similarequal0.06 meV .
This result is independent of the particular QH state inconsideration. On the other hand, B
ext/similarequal0.1 meV for typical
field strengths of several Tesla in QH liquids. Thus, Bz
iis in
general smaller than or comparable to Bextin spin-unpolarized
states.
For spin-polarized QH states, applying the assumption
Sx
i=Sy
i=0 yields Bx
i=By
i=Bz
i=0. In this case, we
consider fluctuations in the next order, associated with thenext-most-relevant spin operators δS
x
i,δSy
iin the edge theory.
We have /angbracketleftδSx
i/angbracketright=/angbracketleftδSy
i/angbracketright=0. The fluctuations give rise to
effective onsite Zeeman fields
δBα
i=/Gamma12
i
4/integraldisplay∞
0dt e−ηt/epsilon1αβγ/angbracketleftbig/braceleftbig
δSβ
i(t),δSγ
i(0)/bracerightbig/angbracketrightbig
, (A9)
which are fully out of plane δBα
i=δαzδBz
i. Simple dimen-
sional analysis shows that the order of magnitude of δSx
i,δSy
i
differs from those of the (nonvanishing) most relevant spin
operators by a factor of a/¯vτ, where ¯ vis the mean edge
velocity and τ∼1/EFis a typical time scale for the dynamics
of conduction electrons. Accordingly, the factor ( a/¯vτ)2
enters the relative strength of the effective onsite fields ( Bz
i)
in spin-unpolarized states to those ( δBz
i) in spin-polarized
states (where Bx
i=By
i=Bz
i=0). Our estimation shows that
a/¯vτ < 0.1, so that δBz
i/lessmuchBz
i<Bext. In the main text we
have neglected δBz
ifor simplicity.
In principle, the Zeeman terms HZ=−/summationtext
iBextIz
i(as-
suming Bext=Bextˆz) should be included in the unperturbed
Hamiltonian in the Schrieffer-Wolff procedure, i.e., Hedge→
Hedge+HZin Eqs. ( A1) and ( A2) and the definition of time
evolution. As a consequence, the first localized-spin operatorI
α
iappearing in the two terms in Eq. ( A3) acquires time
dependence, in addition to the time dependence in the firstconduction-spin operator S
α
i. The dynamics of Iα
i, set by
the Zeeman energy Bext, however decouples from that of
Sα
i, set by the Fermi energy EF, since EF/greatermuchBextaccording
to the estimation above. Thus, to a good approximation we
075301-6LONG-DISTANCE ENTANGLEMENT OF SPIN QUBITS VIA . . . PHYSICAL REVIEW B 93, 075301 (2016)
may neglect the time dependence in Iα
i. We do this for
spin-unpolarized states. For spin-polarized states, Eq. ( A3)
is exact: only the terms with α=β=zsurvive in the
equation and we have Iα
i(t)=Iα
i(0) since HZcommutes
withIz
i.
We note moreover that HZalso appears in Eq. ( A4)f o r
both spin-unpolarized and -polarized QH states. In the maintext, we have neglected this term for simplicity. However, H
Z
must be taken into account for the purpose of implementing
two-qubit quantum gates (see Ref. [ 11] for spin qubits working
in perpendicular Zeeman fields).
The effective Hamiltonian in Eq. ( A4) describes the system
in Fig. 1in equilibrium. Given a change in the spin state of one
of the qubits, the entire electronic system readjusts to achievenew equilibrium. The change of the qubit must be adiabaticin order for the other qubit to sense the change and respond.This means that the switching time t
swof the first qubit satisfies
tsw/greatermuchL/¯v. On the other hand, if the qubit state is changed very
fast (nonadiabatically), there will be no effect on the secondqubit within time L/¯v. In that case, the process is dynamic
and is described by the spin susceptibility at finite frequencies.ForL=1μm,L/¯v/similarequal10 ps, which is much shorter than the
ideal gate-operation time t
sw/similarequal1 ns. Thus, the requirement for
adiabaticity does not place much restriction on the operationof spin qubits.
APPENDIX B: SPIN SUSCEPTIBILITY
In this appendix, we calculate the spin susceptibility for the
QH states listed in Table I. The formula is given by Eq. ( 4).
First, we compute the correlators in the zero-temperaturelimit
G
α(x,t)=/angbracketleftTSα(x,t)Sα(0,0)/angbracketright, (B1)
where α=x,y,z . We focus on the scaling behaviors of these
correlators and neglect the proportionality constants. Next, weevaluate the time integral
χ
αα(x)=2l2/integraldisplay∞
0dt e−ηtImGα(x,t), (B2)
where η=0+. Restoring the proportionality constants, we
obtain the full expression of the spin susceptibility.
1. Correlators
a. Laughlin states at ν=1/m
The Lagrangian density that describes the edge state of the
ν=1/m(mis an odd integer) Laughlin state is
L=m
4π[∂tφ∂xφ−v(∂xφ)2], (B3)
where vis the velocity of the edge mode described by bosonic
fieldφ. We assume the edge state is left moving. Electrons
in the edge state are described by the vertex operator ψ=
1√
2πae−ikFxe−imφ, where ais the short-distance cutoff and kF
is the Fermi momentum. Here and throughout this appendix,
we omit the Klein factors in the electron operators, which willdrop out when evaluating the average. Since the edge stateis spin polarized, all the electrons have the same spin σ.L e t
us assume σ=↑.U s i n gE q .( 5) and neglecting transitions tohigher Landau levels, we find S
x=Sy=0, and
Sz=1
2ψ†ψ∝∂xφ. (B4)
The correlator of φcan be read from Eq. ( B3),
/angbracketleftTφ(x,t)φ(0,0)/angbracketright=−νln(x+vt−iδ)+const, where δis
defined as a positive infinitesimal throughout the appendix.This gives
G
z(x,t)∝ν
(x+vt−iδ)2, (B5)
whereas Gx=Gy=0. Substituting Eq. ( B5)i nE q .( B2)w e
obtain the spin susceptibility in Laughlin states.
b. QH state at ν=2
Theν=2 QH state has two bosonic edge modes φ↑,φ↓,
propagating in the same direction, where φ↑has spin up and
φ↓has spin down. The Lagrangian density is
L=1
4π⎧
⎨
⎩/summationdisplay
i=↑,↓[∂tφi∂xφi−vi(∂xφi)2]−2u∂xφ↑∂xφ↓⎫
⎬
⎭,
(B6)
where viis the velocity of φiandu> 0 is the repulsive
Coulomb interaction between φ↑andφ↓. We assume the edge
modes are left moving. The most relevant electron operatorsareψ
i=1√
2πae−ikF,ixe−iφi, where kF,iis the Fermi momentum
ofφi. The spin-density operators are
Sx=1
2(ψ†
↑ψ↓+H.c.)∝ei/Delta1kxei(φ↑−φ↓)+H.c.,
Sy=1
2(−iψ†
↑ψ↓+H.c.)∝−iei/Delta1kxei(φ↑−φ↓)+H.c.,
Sz=1
2(ψ†
↑ψ↑−ψ†
↓ψ↓)∝∂x(φ↑−φ↓), (B7)
where /Delta1k=kF,↑−kF,↓is the gauge-invariant momentum
difference, proportional to the magnetic flux penetratingbetween the two edge modes.
To compute the correlators, we define eigenmodes
φ
+=cosϕφ↑+sinϕφ↓,
(B8)
φ−=− sinϕφ↑+cosϕφ↓,
where tan 2 ϕ=2u
v↑−v↓, which diagonalize the edge theory
L=1
4π/summationdisplay
i=+,−[∂tφi∂xφi−vi(∂xφi)2], (B9)
where v±=1
2[v↑+v↓±/radicalbig
(v↑−v↓)2+4u2]. According to
the experiment [ 25],v+/greatermuchv−as a result of the strong
Coulomb interaction u. Expressing the spin-density operators
in eigenmodes, it is straightforward to obtain
Gx(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbiggc2
+/bracketleftbigg1
x+v−t−iδ/bracketrightbiggc2
−
,
Gy(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbiggc2
+/bracketleftbigg1
x+v−t−iδ/bracketrightbiggc2
−
,
Gz(x,t)∝c2
+
(x+v+t−iδ)2+c2
−
(x+v−t−iδ)2, (B10)
075301-7Y ANG, HSU, STANO, KLINOV AJA, AND LOSS PHYSICAL REVIEW B 93, 075301 (2016)
where the functions c±(ϕ)=cosϕ∓sinϕ. Notice that
c2
+(ϕ)+c2
−(ϕ)=2, i.e., the scaling exponents of the corre-
lators are integral invariant, independent of the angle ϕwhich
depends on the interedge interaction. This is a well-knownproperty of chiral QH edge states [ 17].
c. QH state at ν=2
3
Theν=2
3QH state can be spin unpolarized at low fields
and spin polarized at high fields [ 29]. We first consider the spin-
unpolarized state. It has two bosonic edge modes φ↑andφ↓,
where φ↑has spin up and φ↓has spin down. The Lagrangian
density is
L=1
4π/summationdisplay
i,j=↑,↓[Kij∂tφi∂xφj−Vij∂xφi∂xφj], (B11)
where
K=/parenleftbigg
12
21/parenrightbigg
andV=/parenleftbigg
v↑u
uv ↓/parenrightbigg
, (B12)
withvithe velocity of φianduthe interedge interaction. The
eigenvalues of the Kmatrix have opposite signs, so the edge
state is nonchiral.
Experiment [ 26] revealed that the ν=2
3edge state consists
of a charged mode and a neutral mode, moving in oppositedirections. To connect the parameters in the edge theorydescribed by Eq. ( B11) with experiment, we change to the
physical basis of charged mode φ
ρ=φ↑+φ↓and neutral
modeφn=φ↑−φ↓:
L=1
4π/bracketleftbigg3
2∂tφρ∂xφρ−1
2∂tφn∂xφn−3
2vρ(∂xφρ)2
−1
2vn(∂xφn)2−2vρn∂xφρ∂xφn/bracketrightbigg
, (B13)
where vρ=1
3(v↑
2+v↓
2+u),vn=v↑
2+v↓
2−u, and vρn=
1
4(v↑−v↓). In general, v↑/negationslash=v↓due to finite Zeeman splitting.
The charged-mode velocity vρ, determined by the large
Coulomb energy scale, is expected to be much greater inorder of magnitude than the neutral-mode velocity v
nand
the interaction vρn. We therefore assume vρ/greatermuchvn∼vρn.
In particular, we assume that the scaling dimensions ofquasiparticle operators in the real case do not deviate muchfrom those in the case v
ρn=0. With this assumption, we can
determine the most relevant electron operators in the edgetheory, which are ψ
↑∝e−i(2kF,↑+kF,↓)xe−i(2φ↑+φ↓), with spin
up, and ψ↓∝e−i(kF,↑+2kF,↓)xe−i(φ↑+2φ↓), with spin down, where
kF,↑andkF,↓are momentumlike constants related to the spatial
locations of the edge modes φ↑andφ↓. The spin-density
operators are obtained by computing the operator productexpansions (OPEs) of the electron operators and keeping themost singular terms. We find
S
x=1
2(ψ†
↑ψ↓+H.c.)∝ei/Delta1kxeiφn+H.c.,
Sy=1
2(−iψ†
↑ψ↓+H.c.)∝−iei/Delta1kxeiφn+H.c., (B14)
Sz=1
2(ψ†
↑ψ↑−ψ†
↓ψ↓)∝∂xφn,
where /Delta1k=kF,↑−kF,↓.In terms of eigenmodes
φ+=/radicalbigg
3
2coshθφρ+/radicalbigg
1
2sinhθφn,
(B15)
φ−=/radicalbigg
3
2sinhθφρ+/radicalbigg
1
2coshθφn,
where tanh 2 θ=4√
3vρn
vρ+vn, the edge theory is diagonalized,
L=1
4π/bracketleftBigg
∂tφ+∂xφ+−∂tφ−∂xφ−−/summationdisplay
i=+,−vi(∂xφi)2/bracketrightBigg
,
(B16)
where v+=1
cosh 2 θ(cosh2θvρ−sinh2θvn) and v−=
1
cosh 2 θ(cosh2θvn−sinh2θvρ). Since vρ/greatermuchvn∼vρn,w e
haveθ/lessmuch1 and thus v+/similarequalvρ,v−/similarequalvn, andv+/greatermuchv−.T h e
correlators are evaluated to be
Gx(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg˜c2
+/bracketleftbigg1
x−v−t+iδ/bracketrightbigg˜c2
−
,
Gy(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg˜c2
+/bracketleftbigg1
x−v−t+iδ/bracketrightbigg˜c2
−
,
Gz(x,t)∝˜c2
+
(x+v+t−iδ)2+˜c2
−
(x−v−t+iδ)2, (B17)
where the functions ˜c+(θ)=√
2s i n h θand ˜c−(θ)=√
2 cosh θ. Notice that ˜c2
+(θ)+˜c2
−(θ)=2(1+2s i n h2θ), i.e.,
the scaling exponents are nonuniversal and depend on theparameters in the Hamiltonian, through θ. This reflects the
nonchiral nature of the edge state.
Next, we discuss the spin-polarized state at ν=
2
3. It has
two bosonic edge modes φ1andφ2, having the same spin
polarization (assuming they are spin up). The Lagrangiandensity has the same form of Eq. ( B11), with
K=/parenleftbigg
10
0−3/parenrightbigg
andV=/parenleftbigg
v
1u/prime
u/prime3v2/parenrightbigg
. (B18)
This is also a nonchiral state. The charged mode and the neutral
mode in the edge theory are identified as φρ=φ1+φ2and
φn=φ1+3φ2, respectively, in terms of which the Lagrangian
density recovers the expression in Eq. ( B13), with vρ=3
2v1+
1
2v2−u/prime,vn=1
2v1+3
2v2−u/prime, andvρn=−3
4v1−3
4v2+u/prime.
Again, we assume vρ/greatermuchvn∼vρn. The most relevant elec-
tron operators are ψ1∝e−i(2kF,1+3kF,2)xe−i(2φ1+3φ2)andψ2∝
e−ikF,1xe−iφ1, both with spin up, where kF,1andkF,2are
constants. Using Eq. ( 5) and OPE, we find Sx=Sy=0 and
Sz=Sz
f+Sz
b, where
Sz
f=1
2(ψ†
1ψ1+ψ†
2ψ2)∝∂xφρ,
(B19)
Sz
b=1
2(ψ†
1ψ2+H.c.)∝ei/Delta1kxeiφn+H.c.,
where /Delta1k=kF,1+3kF,2is interpreted as the Fermi-
momentum difference between the elementary edge modesφ
1andφ2. The rest of the analysis resembles that for the
spin-unpolarized state. We diagonalize the edge theory usingthe free fields φ
+,φ−defined in Eq. ( B15) and evaluate the
075301-8LONG-DISTANCE ENTANGLEMENT OF SPIN QUBITS VIA . . . PHYSICAL REVIEW B 93, 075301 (2016)
correlators. We find Gx=Gy=0 and Gz=Gz
f+Gz
b, where
Gz
f(x,t)∝˜c2
−
(x+v+t−iδ)2+˜c2
+
(x−v−t+iδ)2,
Gz
b(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightBig˜c2
+/bracketleftBig1
x−v−t+iδ/bracketrightbigg˜c2
−
.
(B20)
d. 331 state at ν=5
2
We now turn to the QH state at ν=5
2. This QH state is
usually modeled by combining a ν=2 integer QH state in the
lowest Landau level, which is treated as an inert backgroundassuming no Landau level mixing, and a ν=
1
2fractional QH
state in the second Landau level, which is assumed to capturethe full topological order of the QH liquid. We study the RKKYinteraction mediated solely by the fractional edge state. In thefollowing, we consider several topological orders proposedfor the fractional edge state, including Halperin’s 331 and 113states [ 30,31], the Pfaffian state [ 32], the anti-Pfaffian state
[33,34] and the SU(2)
2state [ 35]. Motivated by the experiment
[26], we will always assume separation of charged and neutral
degrees of freedom in the edge state. Moreover, we assume thatthe charged-mode velocity is much greater than other physicalparameters, by a similar argument to that for the QH state atν=
2
3.
We start from Halperin’s 331 state, which has a spin-
unpolarized version and a spin-polarized version. TheLagrangian density for the edge of the spin-unpolarized 331state has the same form of Eq. ( B11), with
K=/parenleftbigg
31
13/parenrightbigg
andV=/parenleftbigg
v
↑u
uv ↓/parenrightbigg
, (B21)
where v↑andv↓are the velocities of edge modes φ↑and
φ↓, respectively, and uthe interedge interaction. Here, φ↑is
a spin-up mode and φ↓is a spin-down mode. The 331 state
is chiral. The physical charged mode and neutral mode aredefined as φ
ρ=φ↑+φ↓andφn=φ↑−φ↓, respectively, in
terms of which the Lagrangian density is
L=1
4π[2∂tφρ∂xφρ+∂tφn∂xφn−2vρ(∂xφρ)2
−vn(∂xφn)2−2vρn∂xφρ∂xφn], (B22)
where vρ=v↑
8+v↓
8+u
4,vn=v↑
4+v↓
4−u
2, and vρn=
1
4(v↑−v↓). Assuming vρ/greatermuchvn∼vρn, the most relevant
electron operators are ψ↑∝e−i(3kF,↑+kF,↓)xe−i(3φ↑+φ↓), with
spin up, and ψ↓∝e−i(kF,↑+3kF,↓)xe−i(φ↑+3φ↓), with spin down,
where kF,↑andkF,↓are constants. The spin-density operators
are
Sx=1
2(ψ†
↑ψ↓+H.c.)∝ei/Delta1kxei2φn+H.c.,
Sy=1
2(−iψ†
↑ψ↓+H.c.)∝−iei/Delta1kxei2φn+H.c., (B23)
Sz=1
2(ψ†
↑ψ↑−ψ†
↓ψ↓)∝∂xφn,where /Delta1k=2kF,↑−2kF,↓. To evaluate the correlators of Sα,
we define eigenmodes
φ+=√
2 cosθφρ+sinθφn,
(B24)
φ−=−√
2s i nθφρ+cosθφn,
where tan 2 θ=√
2vρn
vρ−vn.W eh a v e θ/lessmuch1. We find
Gx(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg¯c2
+/bracketleftbigg1
x+v−t−iδ/bracketrightbigg¯c2
−
,
Gy(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg¯c2
+/bracketleftbigg1
x+v−t−iδ/bracketrightbigg¯c2
−
,
Gz(x,t)∝¯c2
+
(x+v+t−iδ)2+¯c2
−
(x+v−t−iδ)2, (B25)
where ¯c+(θ)=2s i nθand ¯c−(θ)=2 cosθ. The parameters v+
andv−are the velocities of φ+andφ−, respectively. We have
v+/similarequalvρ,v−/similarequalvn, andv+/greatermuchv−.
The spin-polarized 331 state has two bosonic edge modes
φ1andφ2, having the same spin polarization (assuming they
are spin up). The Lagrangian density has the same form ofEq. ( B11), with
K=/parenleftbigg
3−2
−24/parenrightbigg
andV=/parenleftbigg
v
1u/prime
u/primev2/parenrightbigg
. (B26)
The physical charged and neutral modes are identified as φρ=
φ1andφn=−φ1+2φ2, respectively, in terms of which the
Lagrangian density recovers the form in Eq. ( B22), with vρ=
1
2v1+1
8v2+1
2u/prime,vn=1
4v2, andvρn=1
4v2+1
2u/prime. Assuming
vρ/greatermuchvn∼vρn, the most relevant electron operators are ψ1∝
e−i(kF,1+2kF,2)xe−i(φ1+2φ2)andψ2∝e−i(3kF,1−2kF,2)xe−i(3φ1−2φ2),
both with spin up, where kF,1andkF,2are constants. The spin-
density operators are Sx=Sy=0 andSz=Sz
f+Sz
b, where
Sz
f=1
2(ψ†
1ψ1+ψ†
2ψ2)∝∂xφρ,
(B27)
Sz
b=1
2(ψ†
1ψ2+H.c.)∝ei/Delta1kxei2φn+H.c.,
with/Delta1k=− 2kF,1+4kF,2. Using the definition of eigen-
modes in Eq. ( B24), we find Gx=Gy=0 and Gz=Gz
f+Gz
b,
where
Gz
f(x,t)∝¯c2
−
(x+v+t−iδ)2+¯c2
+
(x+v−t−iδ)2,
(B28)
Gz
b(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg¯c2
+/bracketleftbigg1
x+v−t−iδ/bracketrightbigg¯c2
−
.
e. 113 state at ν=5
2
The 113 state also has a spin-unpolarized version and a spin-
polarized version. The edge theory of the spin-unpolarized 113state is of the form of Eq. ( B11), with
K=/parenleftbigg
13
31/parenrightbigg
andV=/parenleftbigg
v
↑u
uv ↓/parenrightbigg
, (B29)
where v↑andv↓are the velocities of edge modes φ↑and
φ↓, respectively, and uthe interedge interaction. Here, φ↑
is a spin-up mode and φ↓is a spin-down mode. The 113
state is nonchiral. Switching to the physical basis of charged
075301-9Y ANG, HSU, STANO, KLINOV AJA, AND LOSS PHYSICAL REVIEW B 93, 075301 (2016)
mode φρ=φ↑+φ↓and neutral mode φn=φ↑−φ↓,t h e
Lagrangian density becomes
L=1
4π[2∂tφρ∂xφρ−∂tφn∂xφn−2vρ(∂xφρ)2
−vn(∂xφn)2−2vρn∂xφρ∂xφn], (B30)
where vρ=v↑
8+v↓
8+u
4,vn=v↑
4+v↓
4−u
2, and vρn=
1
4(v↑−v↓). Assuming vρ/greatermuchvn∼vρn, the most relevant elec-
tron operators are ψ↑∝e−i(3kF,↑+kF,↓)xe−i(3φ↑+φ↓), with spin
up, and ψ↓∝e−i(kF,↑+3kF,↓)xe−i(φ↑+3φ↓), with spin down, where
kF,↑andkF,↓are constants. The spin-density operators are
Sx=1
2(ψ†
↑ψ↓+H.c.)∝ei/Delta1kxei2φn+H.c.,
Sy=1
2(−iψ†
↑ψ↓+H.c.)∝−iei/Delta1kxei2φn+H.c., (B31)
Sz=1
2(ψ†
↑ψ↑−ψ†
↓ψ↓)∝∂xφn,
where /Delta1k=2kF,↑−2kF,↓. The eigenmodes are defined as
φ+=√
2 cosh θφρ+sinhθφn,
φ−=√
2s i n h θφρ+coshθφn, (B32)
where tanh 2 θ=√
2vρn
vρ+vn.W eh a v e θ/lessmuch1. We find
Gx(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg2˜c2
+/bracketleftbigg1
x−v−t+iδ/bracketrightbigg2˜c2
−
,
Gy(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg2˜c2
+/bracketleftbigg1
x−v−t+iδ/bracketrightbigg2˜c2
−
,
Gz(x,t)∝˜c2
+
(x+v+t−iδ)2+˜c2
−
(x−v−t+iδ)2, (B33)
where ˜c+(θ)=√
2s i n h θand ˜c−(θ)=√
2 cosh θ. The param-
etersv+andv−are the velocities of φ+andφ−, respectively.
We have v+/similarequalvρ,v−/similarequalvn, andv+/greatermuchv−.
The spin-polarized 113 state has two bosonic edge modes
φ1andφ2, having the same spin polarization (assume they
are spin up). The Lagrangian density has the same form ofEq. ( B11), with
K=/parenleftbigg
12
2−4/parenrightbigg
andV=/parenleftbigg
v
1u/prime
u/primev2/parenrightbigg
. (B34)
The charged and the neutral modes are φρ=φ1andφn=
−φ1+2φ2, respectively, in terms of which the Lagrangian
density recovers the form in Eq. ( B30), with vρ=1
2v1+
1
8v2+1
2u/prime,vn=1
4v2, andvρn=1
4v2+1
2u/prime. Assuming vρ/greatermuch
vn∼vρn, the most relevant electron operators are ψ1∝
e−i(kF,1+2kF,2)xe−i(φ1+2φ2)andψ2∝e−i(3kF,1−2kF,2)xe−i(3φ1−2φ2),
both with spin up, where kF,1andkF,2are constants. The
spin-density operators are Sx=Sy=0 and Sz=Sz
f+Sz
b,
where
Sz
f=1
2(ψ†
1ψ1+ψ†
2ψ2)∝∂xφρ,
(B35)
Sz
b=1
2(ψ†
1ψ2+H.c.)∝ei/Delta1kxei2φn+H.c.,with/Delta1k=− 2kF,1+4kF,2. Using the definition of eigen-
modes in Eq. ( B32), we find Gx=Gy=0 and Gz=Gz
f+Gz
b,
where
Gz
f(x,t)∝˜c2
−
(x+v+t−iδ)2+˜c2
+
(x−v−t+iδ)2,
Gz
b(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg2˜c2
+/bracketleftbigg1
x−v−t+iδ/bracketrightbigg2˜c2
−
.
(B36)
f. Pfaffian state at ν=5
2
The Pfaffian state is spin polarized and has a chiral edge
state. The Lagrangian density for the edge is
L=2
4π[∂tφ1∂xφ1−v1(∂xφ1)2]+iλ(∂t−vλ∂x)λ, (B37)
where φ1is a bosonic charged mode and λis a Majorana
fermion. We assume the edge state is left moving. The mostrelevant electron operator is ψ∝λe
−i2φ1. The spin-density
operators are Sx=Sy=0, and
Sz=1
2ψ†ψ∝∂xφ1, (B38)
w h e r ew eh a v eu s e d λ2=1. We find Gx=Gy=0, and
Gz(x,t)∝1
(x+v1t−iδ)2. (B39)
g. Anti-Pfaffian state at ν=5
2
The anti-Pfaffian state is the particle-hole dual of the Pfaf-
fian state. The state is spin polarized. We consider the situationof a clean sample where disorder effect can be neglectedand there is translation invariance on the edge. The edgeLagrangian density then takes the form
L=1
4π[2∂tφρ∂xφρ−∂tφn∂xφn−2vρ(∂xφρ)2
−vn(∂xφn)2−2vρn∂xφρ∂xφn]+iλ(∂t+vλ∂x)λ,
(B40)
where φρis a left-moving charged boson, φnis a right-moving
neutral boson, and λis a right-moving Majorana fermion. The
edge state is nonchiral. Assuming charge-neutral separation inthe edge state, i.e., v
ρ/greatermuchvn∼vρn∼vλ, we find three most
relevant electron operators: ψ1∝λe−i2φρ,ψ2∝e−iφne−i2φρ,
andψ3∝eiφne−i2φρ. The spin-density operators are Sx=
Sy=0 andSz=Sz
f+Sz
b1+Sz
b2, where
Sz
f=1
2(ψ†
1ψ1+ψ†
2ψ2+ψ†
3ψ3)∝∂xφρ,
Sz
b1=1
2(ψ†
1ψ2+ψ†
1ψ3+H.c.)∝ei/Delta1kxλeiφn+H.c., (B41)
Sz
b2=1
2(ψ†
2ψ3+H.c.)∝ei/Delta1k/primexei2φn+H.c.,
with /Delta1k,/Delta1k/primethe momentum differences between the
edge modes. Upon diagonalizing the edge theory inEq. ( B40), we find G
x=Gy=0 and Gz=Gz
f+Gz
b1+Gz
b2,
075301-10LONG-DISTANCE ENTANGLEMENT OF SPIN QUBITS VIA . . . PHYSICAL REVIEW B 93, 075301 (2016)
where
Gz
f(x,t)∝˜c2
−
(x+v+t−iδ)2+˜c2
+
(x−v−t+iδ)2,
Gz
b1(x,t)∝cos(/Delta1kx )/bracketleftbigg1
x+v+t−iδ/bracketrightbigg1
2˜c2
+/bracketleftbigg1
x−v−t+iδ/bracketrightbigg1
2˜c2
−
×1
x−vλt+iδ,
Gz
b2(x,t)∝cos(/Delta1k/primex)/bracketleftbigg1
x+v+t−iδ/bracketrightbigg2˜c2
+/bracketleftbigg1
x−v−t+iδ/bracketrightbigg2˜c2
−
,
(B42)
where ˜c+(θ)=√
2s i n h θand ˜c−(θ)=√
2 cosh θ. The param-
etersv+/similarequalvρ,v−/similarequalvn, and v+/greatermuchv−. Notice that Gz
f,Gz
b1
dominate over Gz
b2at long distances.
h. SU (2)2state at ν=5
2
This is a spin-polarized state. The edge Lagrangian density
is
L=1
4π[2∂tφρ∂xφρ+∂tφn∂xφn−2vρ(∂xφρ)2
−vn(∂xφn)2]+iλ(∂t−vλ∂x)λ, (B43)
where φρis a charged boson, φnis a neutral boson, and λ
is a Majorana fermion. The edge state is chiral. The mostrelevant electron operators and the spin-density operators havethe same forms as those in the anti-Pfaffian state. However,note that the fields φ
ρ,φnhere have different origins from those
in Eq. ( B40). The correlators are found to be Gx=Gy=0 and
Gz=Gz
f+Gz
b1+Gz
b2, where
Gz
f(x,t)∝1
(x+vρt−iδ)2,
Gz
b1(x,t)∝cos(/Delta1kx )1
(x+vnt−iδ)(x+vλt−iδ),
Gz
b2(x,t)∝cos(/Delta1k/primex)1
(x+vnt−iδ)4, (B44)
with/Delta1k,/Delta1k/primethe momentum differences between the edge
modes.
2. Time integral
The QH states we have discussed can be divided into three
types.
Type (i): The edge state is chiral and contains one bosonic
mode. Examples include Laughlin states at ν=1/mand the
Pfaffian state at ν=5
2. The in-plane correlators vanish, while
the out-of-plane correlator has the form
G(i)(x,t)=/bracketleftbigg1
δ+i(t+x/v)/bracketrightbiggn
, (B45)
neglecting the proportionality constant and assuming the edge
state is left moving, where n/greaterorequalslant2 is an even integer and v> 0
is the speed of the edge mode.
Type (ii): The edge state is chiral and contains multiple
interacting bosonic modes. Examples include the QH state atν=2 and the 331 state at ν=5
2. The correlators can have the
form of Eq. ( B45), or
G(ii)(x,t)=/bracketleftbigg1
δ+i(t+x/v+)/bracketrightbiggg+/bracketleftbigg1
δ+i(t+x/v−)/bracketrightbiggg−
,
(B46)
neglecting the proportionality constant and the modulating
factor, and assuming the edge state is left moving, where g+
andg−are nonintegers but g++g−is an even integer. From
previous calculations, we have 0 <g+/lessmuch1 and g−>1. To
a good approximation, v+andv−can be considered as the
speeds of the physical charged mode and neutral mode inthe edge state, respectively, so that v
+/greatermuchv−>0. We have
suppressed the spin-component index for simplicity.
Type (iii): The edge state is nonchiral. Examples include the
QH state at ν=2
3and the 113 state at ν=5
2. The correlators
can have the form of Eq. ( B45), or
G(iii)(x,t)=/bracketleftbigg1
δ+i(t+x/v+)/bracketrightbiggg+/bracketleftbigg1
δ+i(t−x/v−)/bracketrightbiggg−
,
(B47)
neglecting the proportionality constant and the modulating
factor, where g+,g−, andg++g−are all nonintegers. We
have 0 <g+/lessmuch1 andg−>1. The parameters v+andv−can
again be considered as the speeds of the physical charged modeand neutral mode, respectively, so that v
+/greatermuchv−>0.
In writing Eqs. ( B45)–(B47), we have assumed that there
are only two distinct velocities in the system: the charged-mode velocity and the neutral-mode velocity. In particular,for the anti-Pfaffian state we make the approximation thatthe Majorana fermion and the neutral boson propagate at thesame speed. For the SU(2)
2state, there is no need for such
an approximation and the correlators take the forms of eitherEq. ( B45)o r( B46).
In the following, we evaluate I=/integraltext
∞
0dt e−ηtImGa(x,t),
where η=0+anda=(i), (ii), (iii).
a. Type (i)
For type (i) edge states,
I=1
2i/braceleftbigg/integraldisplay∞
0dt e−ηt/bracketleftbigg1
δ+i(t+x/v)/bracketrightbiggn
−c.c./bracerightbigg
≡I1−I2. (B48)
The integrand of I1has an nth-order pole at t1=−x/v+iδ,
while the integrand of I2has an nth-order pole at t2=−x/v−
iδ. By residue theorem,
/integraldisplay∞
0dt1
(t−tk)n=− Res/parenleftbigglnt
(t−tk)n;tk/parenrightbigg
, (B49)
where k=1,2. This gives
I1=I2=1
2i(−1)n/2
n−1/parenleftbiggx
v/parenrightbigg1−n
, (B50)
so that I=0. This suggests that the spin susceptibility in type
(i) edge states vanishes to the lowest order (i.e., consideringonly the most relevant operators).
075301-11Y ANG, HSU, STANO, KLINOV AJA, AND LOSS PHYSICAL REVIEW B 93, 075301 (2016)
b. Type (ii)
For type (ii) edge states,
I=/integraldisplay∞
0dt e−ηtIm/bracketleftbigg1
δ+i(t+x/v+)/bracketrightbiggg+
×/bracketleftbigg1
δ+i(t+x/v−)/bracketrightbiggg−
. (B51)
The integrand has two branch points −x/v++iδand
−x/v−+iδ. Choosing the branch cut appropriately,
Im/bracketleftbigg1
δ+i(t+x/v+)/bracketrightbiggg+/bracketleftbigg1
δ+i(t+x/v−)/bracketrightbiggg−
=Im{e−iπ
2g+sgn(t+x
v+)e−iπ
2g−sgn(t+x
v−)}|G(ii)(x,t)|
=/Theta1/parenleftbigg
t+x
v+/parenrightbigg
/Theta1/parenleftbigg
−t−x
v−/parenrightbigg
sin/bracketleftbiggπ
2(g−−g+)/bracketrightbigg
|G(ii)(x,t)|,
(B52)
where /Theta1(x) is the Heaviside step function, sgn( x) is the signum
function, and
|G(ii)(x,t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsinglet+x
v+/vextendsingle/vextendsingle/vextendsingle/vextendsingle−g+/vextendsingle/vextendsingle/vextendsingle/vextendsinglet+x
v−/vextendsingle/vextendsingle/vextendsingle/vextendsingle−g−
. (B53)
We have used the fact that g++g−is an even integer, so that
Im{e−iπ
2(g++g−)}=0. Notice also that I=0i fw es e t v+=v−,
which is consistent with the previous result for type (i) edgestates. In our scenario, v
+/greatermuchv−>0. The integral is nonzero
only when x< 0. Explicitly,
I=/Theta1(−x)s i n/bracketleftbiggπ
2(g−−g+)/bracketrightbigg/integraldisplay−x/v−
−x/v+dt e−ηt|G(ii)(x,t)|
=/Theta1(−x)s i n/bracketleftbiggπ
2(g−−g+)/bracketrightbigg/parenleftbigg1
v+−1
v−/parenrightbigg−g
×B(1−g+,1−g−)|x|−g, (B54)
where B(x,y) is the Euler beta function and we define g=
g++g−−1.
The above calculation applies to left-moving edge states.
For right-moving edge states, one replaces /Theta1(−x) with /Theta1(x),
and sends v+,v−→−v+,−v−in Eq. ( B54). The exponent g
determines the scaling of the spin susceptibility with distance,and may take different values g
αfor different spin components
α=x,y,z . For type (ii) edge states, gαare integral invariants
depending on the topological order of the bulk QH liquid. Forinstance, g
x=gy=1 for the QH state at ν=2.
c. Type (iii)
For type (iii) edge states,
I=/integraldisplay∞
0dt e−ηtIm/bracketleftbigg1
δ+i(t+x/v+)/bracketrightbiggg+
×/bracketleftbigg1
δ+i(t−x/v−)/bracketrightbiggg−
. (B55)We have
Im/bracketleftbigg1
δ+i(t+x/v+)/bracketrightbiggg+/bracketleftbigg1
δ+i(t−x/v−)/bracketrightbiggg−
=Im{e−iπ
2g+sgn(t+x
v+)e−iπ
2g−sgn(t−x
v−)}|G(iii)(x,t)|
=/braceleftbigg/bracketleftbigg
/Theta1/parenleftbigg
−t−x
v+/parenrightbigg
−/Theta1/parenleftbigg
−t+x
v−/parenrightbigg/bracketrightbigg
sin/bracketleftbiggπ
2(g+−g−)/bracketrightbigg
−/Theta1/parenleftbigg
t+x
v+/parenrightbigg
/Theta1/parenleftbigg
t−x
v−/parenrightbigg
sin/bracketleftbiggπ
2(g+1)/bracketrightbigg/bracerightbigg
|G(iii)(x,t)|,
(B56)
where
|G(iii)(x,t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsinglet+x
v+/vextendsingle/vextendsingle/vextendsingle/vextendsingle−g+/vextendsingle/vextendsingle/vextendsingle/vextendsinglet−x
v−/vextendsingle/vextendsingle/vextendsingle/vextendsingle−g−
. (B57)
The integral is nonzero for both x> 0 and x< 0. We find
I=/Theta1(x)I>+/Theta1(−x)I<, where
I>=sin/bracketleftbiggπ
2(g−−g+)/bracketrightbigg/integraldisplayx/v−
0dt e−ηt|G(iii)(x,t)|
−sin/bracketleftbiggπ
2(g+1)/bracketrightbigg/integraldisplay∞
x/v−dt e−ηt|G(iii)(x,t)|
=/braceleftbigg
sin/bracketleftbiggπ
2(g−−g+)/bracketrightbiggvg+
+vg−−1
−
1−g−F/parenleftbigg
1,g+;2−g−;−v+
v−/parenrightbigg
−sin/bracketleftbiggπ
2(g+1)/bracketrightbigg/parenleftbigg1
v++1
v−/parenrightbigg−g
B(g,1−g−)/bracerightbigg
|x|−g,
(B58)
and
I<=sin/bracketleftbiggπ
2(g+−g−)/bracketrightbigg/integraldisplay−x/v+
0dt e−ηt|G(iii)(x,t)|
−sin/bracketleftbiggπ
2(g+1)/bracketrightbigg/integraldisplay∞
−x/v+dt e−ηt|G(iii)(x,t)|
=/braceleftbigg
sin/bracketleftbiggπ
2(g+−g−)/bracketrightbiggvg+−1
+vg−
−
1−g+F/parenleftbigg
1,g−;2−g+;−v−
v+/parenrightbigg
−sin/bracketleftbiggπ
2(g+1)/bracketrightbigg/parenleftbigg1
v++1
v−/parenrightbigg−g
B(g,1−g+)/bracerightbigg
|x|−g,
(B59)
where F(a,b;c;x) is the hypergeometric function. Notice that
I>andI<are related by the exchange of parameters
g+↔g−andv+↔v−, (B60)
which technically reverts the chirality of all the edge modes,
as seen from Eq. ( B47). For type (iii) edge states, g(i.e.,gα,
where α=x,y,z ) takes noninteger values. Let us write gα=
gα
0+δgα, where gα
0is the integer part of gα. We find δgα/lessmuchgα
0
for all the type (iii) edge states being discussed. For instance,
δgx=δgy=4s i n h2θ, where θ/lessmuch1, while gx
0=gy
0=1i n
the spin-unpolarized QH state at ν=2
3.
075301-12LONG-DISTANCE ENTANGLEMENT OF SPIN QUBITS VIA . . . PHYSICAL REVIEW B 93, 075301 (2016)
3. Full expression of spin susceptibility
Substituting the above results in Eq. ( B2), we obtain the
spin susceptibility in QH edge states
χαα(x)=cos(/Delta1kx )
4π2l2agα−1v−gα
+
+v−gα
−
−×I, (B61)
where we have restored the spin-component index and the pro-
portionality constant. The short-distance cutoff acan be taken
as the lattice constant of the host material of the QH system. Forleft-moving type (ii) edge states, Iis given by Eq. ( B54). Wehaveχ
αα(x)=cos(/Delta1kx )|x|−gα/Theta1(−x)Cα(gα,v), where gα=
(gα
+,gα
−),v=(v+,v−), and
Cα(gα,v)=l2agα−1
4π2sin/bracketleftbiggπ
2(gα
−−gα
+)/bracketrightbiggvgα
−−1
+vgα
+−1
−
(v−−v+)gα
×B(1−gα
+,1−gα
−). (B62)
For type (iii) edge states, Iis given by Eqs. ( B58) and
(B59). We have χαα(x)=cos(/Delta1kx )|x|−gα{/Theta1(x)Cα
>(gα,v)+
/Theta1(−x)Cα
<(gα,v)}, where
Cα
>(gα,v)=l2agα−1
4π2/braceleftBigg
sin/bracketleftbiggπ
2(gα
−−gα
+)/bracketrightbiggv−1
−
1−gα
−F/parenleftbigg
1,gα
+;2−gα
−;−v+
v−/parenrightbigg
−sin/bracketleftbiggπ
2(gα+1)/bracketrightbiggvgα
−−1
+vgα
+−1
−
(v++v−)gαB(gα,1−gα
−)/bracerightBigg
,
Cα
<(gα,v)=l2agα−1
4π2/braceleftBigg
sin/bracketleftbiggπ
2(gα
+−gα
−)/bracketrightbiggv−1
+
1−gα
+F/parenleftbigg
1,gα
−;2−gα
+;−v−
v+/parenrightbigg
−sin/bracketleftbiggπ
2(gα+1)/bracketrightbiggvgα
−−1
+vgα
+−1
−
(v++v−)gαB(gα,1−gα
+)/bracerightBigg
.
(B63)
We see that Cα
>(gα,v) and Cα
<(gα,v) are related by the
exchange of arguments: gα
+↔gα
−andv+↔v−.
Equations ( B62) and ( B63) show that the RKKY interaction
is ferromagnetic at short distances. To estimate the strengthof the RKKY interaction, we extract the dimensional part[χ
αα(x)] of the spin susceptibility. For type (ii) edge states,
[χαα(x)]=l2agα−1vgα
−−1
+vgα
+−1
−
(v+−v−)gα|x|−gα. (B64)
For type (iii) edge states, there are multiple terms in χαα(x),
with
[χαα(x)]=l2agα−1vgα
−−1
+vgα
+−1
−
(v++v−)gα|x|−gα;
l2agα−1v−1
−|x|−gα;l2agα−1v−1
+|x|−gα.(B65)
Using 0 <gα
+/lessmuch1 andv+/greatermuchv−, we find
[χαα(x)]/similarequall2agα−1v−1
−|x|−gα(B66)
for both type (ii) and type (iii) edge states.
APPENDIX C: EXCHANGE
Here, we estimate the strength of the exchange interaction
between the QD electron and the electrons in the edge modes.The textbook formula gives the exchange integral as
J=C/integraldisplay
dr
1dr2/Psi1∗
1(r1)/Psi1∗
2(r2)1
|r1−r2|/Psi11(r2)/Psi12(r1)(C1)
for two particles in single-particle orbitals /Psi11,/Psi12, interacting
through an unscreened Coulomb interaction parametrized by
C=e2
4π/epsilon10/epsilon1r, where eis the elementary charge, /epsilon10the vacuum
permittivity, and /epsilon1rthe relative permittivity of the medium.Providing a microscopic theory of the exchange for our
case is well beyond the scope of this paper. Instead, we areinterested only in the interaction strength scale. To get a roughestimate, let us assume that the exchange interaction is local
J=β/integraldisplay
dr
1dr2/Psi1∗
1(r1)/Psi1∗
2(r2)δ(r1−r2)/Psi11(r2)/Psi12(r1),
(C2)
which transforms the equation into a density-density interac-
tion
J=β/integraldisplay
drρ1(r)ρ2(r). (C3)
One can explicitly evaluate Eq. ( C1) for a tunnel-coupled
double dot modeled by a 2D harmonic confinement [ 54] and
then compare to the result given by Eq. ( C3). The calculated
energies scale the same with the interdot distance, and theoverall prefactors are related by β=Cl, withlthe confinement
length of the dot potential. We further guide ourselves byexperiments, which measured the exchange energy in few-electron QDs made in 2DEG in GaAs. The maximal scaleC/l, which evaluates to /similarequal3 meV for typical GaAs parameters
/epsilon1
r=12.9 and l=30 nm, is indeed approached in a single
dot where the densities overlap in Eq. ( C3)i so fo r d e r1
in dimensionless units ( l−2). A suppression of the interdot
tunneling (by increasing the interdot distance) leads to adecreasing exchange, which reaches J
DD/similarequal0.1–0.01 meV
in a tunnel-coupled double dot. Assuming that an analogoussuppression will result from the tunnel coupling of our dotcoupled to the edge finally gives J
DDas an order of magnitude
estimate for /Gamma1, which we used in the main text as the coupling
constant between an electron spin in a QD and a quasi-1D spindensity of the edge.
[ 1 ]M .A .N i e l s e na n dI .L .C h u a n g , Quantum Computation and
Quantum Information (Cambridge University Press, Cambridge,
2000).[2] D. Loss and D. DiVincenzo, Quantum computation
with quantum dots, Phys. Rev. A 57, 120
(1998 ).
075301-13Y ANG, HSU, STANO, KLINOV AJA, AND LOSS PHYSICAL REVIEW B 93, 075301 (2016)
[3] C. Kloeffel and D. Loss, Prospects for spin-based quantum
computing in quantum dots, Annu. Rev. Condens. Matter Phys.
4,51(2013 ).
[4] H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V .
Umansky, and A. Yacoby, Dephasing time of GaAs electron-spinqubits coupled to a nuclear bath exceeding 200 μs,Nat. Phys. 7,
109 (2011 ).
[5] K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, Local
fault-tolerant quantum computation, P h y s .R e v .A 72,022317
(2005 ).
[6] A. M. Childs, H. L. Haselgrove, and M. A. Nielsen, Lower
bounds on the complexity of simulating quantum gates, Phys.
Rev. A 68,052311 (2003 ).
[7] A. Imamog-lu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo,
D. Loss, M. Sherwin, and A. Small, Quantum InformationProcessing Using Quantum Dot Spins and Cavity QED, Phys.
Rev. Lett. 83,4204 (1999 ).
[8] L. Childress, A. S. Sørensen, and M. D. Lukin, Mesoscopic
cavity quantum electrodynamics with quantum dots, Phys. Rev.
A69,042302 (2004 ).
[9] G. Burkard and A. Imamoglu, Ultra-long-distance interaction
between spin qubits, P h y s .R e v .B 74,041307(R) (2006 ).
[10] M. Trif, V . N. Golovach, and D. Loss, Spin dynamics in InAs
nanowire quantum dots coupled to a transmission line, Phys.
Rev. B 77,045434 (2008 ).
[11] L. Trifunovic, O. Dial, M. Trif, J. R. Wootton, R. Abebe, A.
Yacoby, and D. Loss, Long-distance spin-spin coupling viafloating gates, P h y s .R e v .X 2,011006 (2012 ).
[12] L. Trifunovic, F. L. Pedrocchi, and D. Loss, Long-Distance
Entanglement of Spin Qubits via Ferromagnet, P h y s .R e v .X
3,041023 (2013 ).
[13] P. Stano, J. Klinovaja, F. R. Braakman, L. M. K. Vander-
sypen, and D. Loss, Fast long-distance control of spin qubits
by photon-assisted cotunneling, P h y s .R e v .B 92,075302
(2015 ).
[14] M. A. Ruderman and C. Kittel, Indirect exchange coupling of
nuclear magnetic moments by conduction electrons, Phys. Rev.
96,99(1954 ).
[15] T. Kasuya, A theory of metallic ferro- and antiferromagnetism
on zener’s model, Prog. Theor. Phys. 16,45(1956 ).
[16] K. Yosida, Magnetic properties of Cu-Mn alloys, Phys. Rev.
106,893 (1957 ).
[17] X.-G. Wen, Quantum Field Theory of Many-Body Systems: From
the Origin of Sound to an Origin of Light and Electrons (Oxford
University Press, New York, 2004).
[18] H. Kiyama, T. Nakajima, S. Teraoka, A. Oiwa, and S. Tarucha,
Spin-dependent current through a quantum dot from spin-polarized nonequilibrium quantum Hall edge channels, Phys.
Rev. B 91,155302 (2015 ).
[19] P. Stano, J. Klinovaja, A. Yacoby, and D. Loss, Local spin
susceptibilities of low-dimensional electron systems, Phys. Rev.
B88,045441 (2013 ).
[20] J. R. Schrieffer and P. A. Wolff, Relation between the anderson
and Kondo hamiltonians, Phys. Rev. 149,491 (1966 ).
[21] S. Bravyi, D. P. DiVincenzo, and D. Loss, Schrieffer-Wolff
Transformation for quantum many-body systems, Ann. Phys.
(Amsterdam) 326,2793 (2011 ).
[22] T. Giamarchi, Quantum Physics in One Dimension (Oxford
University Press, New York, 2003).[23] B. Braunecker, P. Simon, and D. Loss, Nuclear magnetism and
electron order in interacting one-dimensional conductors, Phys.
Rev. B 80,165119 (2009 ).
[24] Y .-W. Lee and Y .-L. Lee, Electrical control and interaction
effects of the RKKY interaction in helical liquids, Phys. Rev. B
91,214431 (2015 ).
[25] E. Bocquillon, V . Freulon, J-.M Berroir, P. Degiovanni, B.
Plac¸ a i s ,A .C a v a n n a ,Y .J i n ,a n dG .F `eve, Separation of neutral
and charge modes in one-dimensional chiral edge channels, Nat.
Commun. 4,1839 (2013 ).
[26] A. Bid, N. Ofek, H. Inoue, M. Heiblum, C. L. Kane, V .
Umansky, and D. Mahalu, Observation of neutral modes inthe fractional quantum Hall regime, Nature (London) 466,585
(2010 ).
[27] S. M. Girvin, Particle-Hole Symmetry in the anomalous quan-
tum Hall effect, P h y s .R e v .B 29,6012(R) (1984 ).
[28] C. Wang and D. E. Feldman, Chirality, Causality, and
Fluctuation-Dissipation Theorems in Nonequilibrium SteadyStates, Phys. Rev. Lett. 110,030602 (2013 ).
[29] J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, and K. W. West,
Evidence for a spin transition in the ν=2/3 fractional quantum
Hall effect, P h y s .R e v .B 41,7910(R) (1990 ).
[30] B. I. Halperin, Theory of the quantized Hall conductance, Helv.
Phys. Acta. 56, 75 (1983).
[31] G. Yang and D. E. Feldman, Experimental constraints and
a possible quantum Hall state at ν=5/2,P h y s .R e v .B 90,
161306(R) (2014 ).
[32] G. Moore and N. Read, Nonabelions in the fractional quantum
Hall effect, Nucl. Phys. B 360,362 (1991 ).
[33] M. Levin, B. I. Halperin, and B. Rosenow, Particle-Hole
Symmetry and the Pfaffian State, Phys. Rev. Lett. 99,236806
(2007 ).
[34] S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, Particle-Hole
Symmetry and the ν=
5
2Quantum Hall State, Phys. Rev. Lett.
99,236807 (2007 ).
[35] X.-G. Wen, Non-Abelian Statistics in the Fractional Quantum
Hall States, P h y s .R e v .L e t t . 66,802 (1991 ).
[36] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, Electro-
statics of edge channels, P h y s .R e v .B 46,4026 (1992 ).
[37] J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Independently
contacted two-dimensional electron systems in double quantumwells, Appl. Phys. Lett. 57,2324 (1990 ).
[38] O. M. Auslaender, A. Yacoby, R. de Picciotto, K. W. Baldwin,
L. N. Pfeiffer, and K. W. West, Experimental Evidence forResonant Tunneling in a Luttinger Liquid, Phys. Rev. Lett. 84,
1764 (2000 ).
[39] L. Pfeiffer, H. L. St ¨ormer, K. W. Baldwin, K. W. West, A.
R. Go ˜ni, A. Pinczuk, R. C. Ashoori, M. M. Dignam, and
W. Wegscheider, Cleaved edge overgrowth for quantum wirefabrication, J. Cryst. Growth 127,849 (1993 ).
[40] C. P. Scheller, T.-M. Liu, G. Barak, A. Yacoby, L. N. Pfeiffer,
K. W. West, and D. M. Zumb ¨uhl, Possible Evidence for Helical
Nuclear Spin Order in GaAs Quantum Wires, Phys. Rev. Lett.
112,066801 (2014 ).
[41] T. Meng, P. Stano, J. Klinovaja, and D. Loss, Helical nuclear
spin order in a strip of stripes in the quantum Hall regime, Eur.
P h y s .J .B 87,203 (2014 ).
[42] L. Zhou, J. Wiebe, S. Lounis, E. Vedmedenko, F. Meier,
S. Bl ¨ugel, P. H. Dederichs, and R. Wiesendanger, Strength
075301-14LONG-DISTANCE ENTANGLEMENT OF SPIN QUBITS VIA . . . PHYSICAL REVIEW B 93, 075301 (2016)
and directionality of surface Ruderman-Kittel-Kasuya-Yosida
interaction mapped on the atomic scale, Nat. Phys. 6,187
(2010 ).
[43] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A.
Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, andA. C. Gossard, Coherent manipulation of coupled electronspins in semiconductor quantum dots, Science 309,2180
(2005 ).
[44] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L.
M. K. Vandersypen, Spins in few-electron quantum dots, Rev.
Mod. Phys. 79,1217 (2007 ).
[45] C. Barthel, D. J. Reilly, C. M. Marcus, M. P. Hanson, and A. C.
Gossard, Rapid Single-Shot Measurement of a Singlet-TripletQubit, Phys. Rev. Lett. 103,160503 (2009 ).
[46] X.-G. Wen, Edge transport properties of the fractional quantum
Hall states and weak-impurity scattering of a one-dimensionalcharge-density wave, P h y s .R e v .B 44,5708 (1991 ).
[47] G. Yang and D. E. Feldman, Influence of device geometry on
tunneling in the ν=
5
2quantum Hall liquid, Phys. Rev. B 88,
085317 (2013 ).[48] A. Stern and B. I. Halperin, Proposed Experiments to Probe the
Non-Abelian ν=5/2 Quantum Hall State, Phys. Rev. Lett. 96,
016802 (2006 ).
[49] P. Bonderson, A. Kitaev, and K. Shtengel, Detecting Non-
Abelian Statistics in the ν=5/2 Fractional Quantum Hall State,
Phys. Rev. Lett. 96,016803 (2006 ).
[50] D. E. Feldman and A. Kitaev, Detecting Non-Abelian Statistics
with an Electronic Mach-Zehnder Interferometer, Phys. Rev.
Lett.97,186803 (2006 ).
[51] C. Wang and D. E. Feldman, Identification of 331 quantum
Hall states with Mach-Zehnder interferometry, P h y s .R e v .B 82,
165314 (2010 ).
[52] G. Yang, Probing the ν=5
2quantum Hall state with electronic
Mach-Zehnder interferometry, P h y s .R e v .B 91,115109 (2015 ).
[53] M. R. Geller and D. Loss, Aharonov-Bohm effect in the Chiral
Luttinger liquid, P h y s .R e v .B 56,9692 (1997 ).
[54] D. Stepanenko, M. Rudner, B. I. Halperin, and D. Loss,
Singlet-triplet splitting in double quantum dots due to spin-orbit and hyperfine interactions, Phys. Rev. B 85,075416
(2012 ).
075301-15 |
PhysRevB.76.205310.pdf | Spin coherence of a two-dimensional electron gas induced by resonant excitation of trions
and excitons in CdTe/ „Cd,Mg …Te quantum wells
E. A. Zhukov,1,2D. R. Yakovlev,1,3M. Bayer,1M. M. Glazov,3E. L. Ivchenko,3G. Karczewski,4T. Wojtowicz,4and
J. Kossut4
1Experimentelle Physik 2, Universität Dortmund, D-44221 Dortmund, Germany
2Faculty of Physics, M.V . Lomonosov Moscow State University, 119992 Moscow, Russia
3A.F . Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
4Institute of Physics, Polish Academy of Sciences, PL-02668 Warsaw, Poland
/H20849Received 27 July 2007; revised manuscript received 12 October 2007; published 8 November 2007 /H20850
The mechanisms for generation of long-lived spin coherence in a two-dimensional electron gas /H208492DEG /H20850have
been studied experimentally by means of a picosecond pump-probe Kerr rotation technique. CdTe/ /H20849Cd,Mg /H20850Te
quantum wells with a diluted 2DEG were investigated. The strong Coulomb interaction between electrons andholes, which results in large binding energies of neutral excitons and negatively charged excitons /H20849trions /H20850,
allows one to address selectively the exciton or trion states by resonant optical excitation. Different scenariosof spin coherence generation were analyzed theoretically, among them the direct trion photocreation, theformation of trions from photogenerated excitons, and the electron-exciton exchange scattering. Good agree-ment between experiment and theory is found.
DOI: 10.1103/PhysRevB.76.205310 PACS number /H20849s/H20850: 73.21.Fg, 75.75. /H11001a, 72.25.Dc, 78.47. /H11001p
I. INTRODUCTION
The spin coherence of electronic states is one of the key
features involved in numerous concepts for spintronics de-vices /H20849see, e.g., Refs. 1and2/H20850. It has been studied in semi-
conductor structures of different dimensionality, includingbulklike thin films, quantum wells /H20849QWs /H20850, and quantum dots.
Accordingly, the spin coherence time of an electron has beenfound to vary over a wide range from a few picoseconds upto a few microseconds.
An ensemble of electron spins subject to a magnetic field
is commonly characterized by three relaxation times:
3the
longitudinal spin relaxation time T1, which is related to the
relaxation of the spin component parallel to the field, and the
transverse relaxation times T2andT2*. The T2time describes
spin decoherence /H20849i.e., relaxation of the spin components
transverse to the field /H20850of asingle electron, while the T2*time
describes the decoherence of the spin ensemble taking intoaccount the inhomogeneous broadening of the electron g
factor.
4,5
The coherence time T2of a single spin is often few orders
of magnitude longer than the dephasing time T2*of a spin
ensemble. For example, in /H20849In,Ga /H20850As/GaAs quantum dots
these times are 3 /H9262s and 0.4 ns, respectively, at B=6 T.6,7A
T2*of 300 ns has been measured in bulk GaAs by means of
the Hanle effect on optically oriented electrons at liquidhelium temperature.
8For QWs the longest spin dephasing
times reported so far are 10 ns for GaAs/ /H20849Al,Ga /H20850As /H20849Ref.9/H20850
and 30 ns for CdTe/ /H20849Cd,Mg /H20850Te.11,12They have been
measured for structures with a very diluted two-dimensional
electron gas /H208492DEG /H20850. Electron spin coherence in
CdTe/ /H20849Cd,Mg /H20850Te QWs attracts recently an increasing
interest.10–17
Figure 1illustrates three typical situations realized in ex-
periments on coherent spin dynamics under resonant opticalexcitation of the QWs. These cases are different in the den-sity of resident 2D electrons, n
e. In the undoped samples/H20851panel /H20849a/H20850,ne=0/H20852spin oriented excitons are photogenerated.
Depending on experimental conditions the coherent spin dy-namics of either an exciton or an electron in the exciton canbe seen. However, this dynamics cannot be studied for timescales exceeding the exciton lifetime as the exciton recom-bination depopulates the photoexcited states. For the dense
2DEG /H20851panel /H20849c/H20850,n
eaB2/H110221, where aBis the exciton Bohr
radius /H20852, exciton formation is suppressed because of state-
filling and screening effects. After photogeneration a holelooses its spin and energy quickly and recombines with anelectron from a 2DEG. However, the spin oriented electronphotogenerated at the Fermi level has an infinite lifetimewhich allows one to study its long-lived spin coherence andspin relaxation. In this case a circularly polarized photon canincrease the spin polarization of the 2DEG by S=±1/2. In
FIG. 1. Schematic presentation of generation of carrier spin co-
herence by circularly polarized laser pulses. The three cases differin the density of 2DEG in the QW: /H20849a/H20850empty QW, only photoge-
nerated carriers, which form excitons; /H20849b/H20850low density 2DEG, trions
with a singlet ground state formed by a photogenerated exciton anda background electron. Interaction of the trion with the 2DEG isnegligible; /H20849c/H20850dense 2DEG with a Fermi energy exceeding the ex-
citon binding energy. Excitons and trions are suppressed by the2DEG.PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
1098-0121/2007/76 /H2084920/H20850/205310 /H2084916/H20850 ©2007 The American Physical Society 205310-1case of a diluted 2DEG /H20851panel /H20849b/H20850,neaB2/lessmuch1/H20852the mechanism
for generation of the electron spin coherence is not soobvious.
18,19Indeed the ground state is a singlet trion with an
antiparallel orientation of electron spins. Being resonantlyexcited, this state would not contribute to the spin polariza-tion because the hole undergoes fast decoherence and thetotal spin of the two electrons is S=0.
However, generation of electron spin coherence has been
observed experimentally under resonant excitation of trionsin QWs
11,20and quantum dots.6There are two equivalent
approaches to explain the generation mechanism in this case.The first one suggests that a coherent superposition of elec-tron and trion states is excited by a circularly polarized pho-ton when the system is subject to an external magneticfield.
6,20,21The second one considers the 2D electrons cap-
tured for trion formation: under circularly polarized excita-tion electrons with a specific spin orientation will be ex-tracted from the 2DEG and correspondingly spin polarizationwith the opposite sign will be induced.
11In order to avoid the
compensation of the induced spin polarization by the return-ing electrons after trion recombination either hole spin flip intrion or magnetic field is needed.
6
In this paper we report on a detailed study of the coherent
spin dynamics of electrons and electron-hole complexes inCdTe/Cd
0.78Mg0.22Te QWs with a low density 2DEG, per-
formed by a pump-probe Kerr rotation /H20849KR /H20850technique. A
theoretical analysis of the various mechanisms of spin coher-ence generation is carried out.
The paper is organized as follows. Section II is devoted to
/H20849i/H20850the theoretical analysis of the physical mechanisms lead-
ing to Kerr and Faraday rotations of the probe-pulse polar-ization, and /H20849ii/H20850the theoretical description of various sce-
narios of electron spin coherence generation. Theexperimental results are presented in Sec. III, which alsocontains a comparison with the model considerations of Sec.II. The paper is concluded by Sec. IV in which we alsocomment on the specifics of p-doped QWs.
II. THEORY: MODEL CONSIDERATIONS
Below we underline the two main aspects of the devel-
oped theory, namely, the nature of Kerr and Faraday rotationsignals and the models of spin dynamics of a 2DEG andelectron-hole complexes, both trions and excitons.
In short, the basic features of an experiment designed for
time-resolved measurements of the electron spin coherencecan be summarized as follows: the sample containing a2DEG is excited along the structure growth axis /H20849zaxis /H20850by
an intense pump pulse which induces resonant interbandtransitions. Then a much weaker, linearly polarized probepulse with the frequency either coinciding with or differentfrom the pump frequency arrives at the sample. The rotationof the polarization plane of the reflected or transmitted probepulse is analyzed as a function of the delay between thepump and probe pulses. An external magnetic field Bis ap-
plied in the QW plane, say, along the xaxis and leads to
precessions of the zandyelectron spin components with the
Larmor frequency /H9024/H11013/H9024
x=ge/H9262BB//H6036, where geis the elec-
tron in-plane gfactor along the xdirection and/H9262Bis theBohr magneton. For 2D heavy holes bound into excitons or
trions, the in-plane gfactor is very small and can be
ignored.22
A. Kerr and Faraday rotation in quantum wells
In order to describe the KR in the pump-probe experiment
under normal-incidence resonant excitation we first considerthe amplitude reflection coefficient of an axially symmetricsingle QW, which in the vicinity of the exciton or trion reso-nance is given by
23
rQW/H20849/H9275/H20850=i/H90030
/H92750−/H9275−i/H20849/H90030+/H9003/H20850, /H208491/H20850
where/H9275is the incident light frequency and /H92750,/H90030, and/H9003are
the exciton /H20849trion /H20850resonance frequency, and the radiative and
nonradiative damping rates, respectively. Taking into accountalso the cap layer as constituent of the heterostructure, thetotal amplitude reflection of the light incident on the struc-ture from vacuum reads
r=r
01+rQWe2i/H9278
1−r10rQWe2i/H9278. /H208492/H20850
Here r01=−r10=/H208491−nb/H20850//H208491+nb/H20850is the reflection coefficient at
the boundary between the cap layer and vacuum, nbis the
refractive index of the cap layer which for simplicity is as-sumed to coincide with the background refractive index ofthe well material, and
/H9278=kbb, where bis the cap-layer thick-
ness and kbis the light wave vector in the cap layer. The
above equation is valid for a spin-unpolarized system inwhich case the coefficient ris insensitive to the light polar-
ization. For the spin-polarized resident electrons the reflec-tion coefficient for right /H20849/H11001/H20850and left /H20849/H11002/H20850circularly polarized
light has the form of Eq. /H208491/H20850butr
QWis replaced by
rQW,±/H20849/H9275/H20850=i/H90030,±
/H92750,±−/H9275−i/H20849/H90030,±+/H9003±/H20850, /H208493/H20850
with the parameters /H92750,/H90030, and/H9003dependent on the light
helicity. These parameters are, in general, determined by theconcentration and spin polarization of the carriers and theircomplexes as well as by the delay between the pump andprobe pulses. When describing the Faraday rotation effect wehave to analyze the amplitude transmission coefficientt
QW,±/H20849/H9275/H20850=1+ rQW,±/H20849/H9275/H20850. Kerr rotation signal measured in the
reflection geometry is proportional to the difference
/H9018+−/H9018−=/H90180Im/H20853r+*r−/H20854, /H208494/H20850
where/H90180and/H9018±are the time-integrated intensities of the
incident and reflected probe pulses in time-resolved experi-ments /H20849or the stationary intensities under steady-state photo-
excitation /H20850. Let us introduce the symmetrized, r
¯, and anti-
symmetrized, r˜, combinations of r±and assume r˜to be small.
Then in first approximation one obtains
Im/H20853r+*r−/H20854=−2/H208491−r102/H20850
/H208411−r10r¯e2i/H9278/H208412Im/H20875r˜/H20849r01e2i/H9278+r¯*/H20850
1−r10r¯e2i/H9278/H20876. /H208495/H20850
Ifr¯is also small as compared with r01the right-hand side of
the above equation reduces to −2 r10/H208491−r102/H20850Im/H20853e2i/H9278r˜/H20854.ZHUKOV et al. PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-2The probe at the trion resonance frequency is described
by Eqs. /H208492/H20850–/H208494/H20850, where/H90030,±is the oscillator strength for reso-
nant trion excitation by /H9268±circularly polarized light.24For
heavy-hole optical transitions the values of /H90030,±are propor-
tional to the density of resident electrons with the spin com-ponent ±1/2, respectively.
Note that in the nonlinear regime of high excitation den-
sity the exciton-exciton interaction may play an importantrole in the formation of the Kerr and Faraday signals, see,e.g., Refs. 25and26.
In the case of a dense 2DEG the Kerr rotation angle is
proportional to
29
/H20873e/H6036/H20841pcv/H20841
m0EgQW/H208742
Re/H20877e2i/H9278/H20885
0/H11009
d/H9255D/H20851f−/H20849/H9255e/H20850−f+/H20849/H9255e/H20850/H20852
EgQW+/H9255−/H6036/H9275−i/H6036/H9003eh/H20878.
Here pcv=/H20855S/H20841pˆx/H20841X/H20856is the interband matrix element of the
momentum operator, EgQWis the band gap renormalized by
the quantum confinement of conduction electrons and heavyholes,/H9003
ehis the damping rate for an electron-hole pair, m0is
the free electron mass, /H9255is the sum of the 2D electron and
hole kinetic energies, /H9255e=/H20849/H9262/me/H20850/H9255;/H9262=memhh//H20849me+mhh/H20850,me
andmhhare the reduced, electron and heavy-hole effective
mass, respectively, f±/H20849/H9255e/H20850is the energy distribution function
for electrons with spin ±1/2, and Dis the reduced density of
states proportional to /H9262//H60362.
In the following we discuss, one after another, physical
mechanisms of the pump-probe signal Eq. /H208494/H20850for three cases
of interest: /H20849i/H20850a diluted 2DEG subject to resonant circularly
polarized photoexcitation in the singlet trion state, /H20849ii/H20850a di-
luted 2DEG with resonant generation of excitons at low tem-peratures favoring the binding of excitons and resident elec-trons into trions, /H20849iii/H20850exciton generation in a dense 2DEG,
and /H20849iv/H20850a diluted 2DEG with photogeneration of carriers
with kinetic energy considerably exceeding the exciton bind-ing energy.
B. Resonant excitation of trions
We start the analysis from the case of KR by a QW with
a diluted 2DEG and for resonant trion generation. In thiscase the radiative homogeneous broadenings /H9003
0,±/H20849coinciding
with the oscillator strength of the trion /H20850in Eq. /H208493/H20850are pro-
portional to the concentrations of the resident electrons withspin down and spin up, see Refs. 24and30. Qualitatively,
the physical picture looks as follows. According to the selec-tion rules, the absorption of a circularly polarized photonleads to the formation of an electron-hole pair with fixed spinprojections: /H20849e,−1/2; hh,+3/2 /H20850and /H20849e,+1/2; hh,−3/2 /H20850for
right /H20849
/H9268+/H20850and left /H20849/H9268−/H20850circularly polarized photons, respec-
tively. At weak and moderate magnetic fields the ground
state of negatively charged trions has a singlet electron con-figuration with antiparallel orientation of electron spins.Thus, for resonant excitation only resident electrons with ori-entation opposite to the photogenerated electrons can con-tribute to trion formation. This means that the 2DEG be-comes depleted of electrons with z-spin component S
z
=+1/2 under/H9268+pumping and of Sz=−1/2 electrons for /H9268−
pumping. An external magnetic field applied in the plane ofthe structure leads to precession of the spin polarization of
resident electrons and, therefore, to a modulation of /H90030,±and
oscillations of the Kerr signal. This process is shown sche-matically in Fig. 2.
Now we turn to an analytical description of this scenario.
The kinetic equations describing the spin dynamics of elec-trons and trions after resonant, pulsed excitation of trionshave the form
dS
z
dt=Sy/H9024−Sz
/H9270s+ST
/H92700T,
dSy
dt=−Sz/H9024−Sy
/H9270s,
dST
dt=−ST
/H9270T. /H208496/H20850
Here ST=/H20849T+−T−/H20850/2 is the effective trion spin density with
T±being the densities of negatively charged trions with the
heavy-hole spin ±3/2, SyandSzare the corresponding com-
ponents of the electron-gas spin density, /H9270Tis the lifetime of
the trion spin including the trion lifetime /H92700Tand the spin
relaxation time /H9270sT, i.e.,/H9270T=/H92700T/H9270sT//H20849/H92700T+/H9270sT/H20850, and/H9270sis the elec-
tron spin relaxation time. Under normal incidence of the /H9268+
polarized pump the initial conditions are Sy/H208490/H20850=0, ST/H208490/H20850=
−Sz/H208490/H20850=n0T/2 with n0Tbeing the initial density of photogener-
ated trions. Remember that the magnetic field is directed
along the xaxis; therefore, the xcomponent of electron spin
density is conserved.
Let us introduce the complex function S+/H20849t/H20850=Sz/H20849t/H20850
+iSy/H20849t/H20850. It satisfies the equationFIG. 2. Scheme of generation of 2DEG spin coherence in exter-
nal magnetic fields via resonant photogeneration of trions. /H20849a/H20850Initial
state of a 2DEG which polarization in the plane perpendicular to themagnetic field is zero. These spins are precessing around B./H20849b/H20850
/H9268−
polarized photon generates a /H20849e,+1/2; hh,−3/2 /H20850electron-hole pair,
which captures a −1/2 resident electron to form a trion. The 2DEGbecame polarized due to uncompensated +1/2 electron spin left. /H20849c/H20850
During trion lifetime,
/H92700T, the 2DEG polarization precesses around
the magnetic field. Trion state does not precess in magnetic field ason the one hand its electronic configuration is singlet and on theother hand in-plane hole gfactor is zero. /H20849d/H20850After trion recombi-
nation the −1/2 electron is returned to the 2DEG /H20849we neglect here
spin relaxation of the hole in the trion /H20850. Final state of the 2DEG
with induced polarization is shown.SPIN COHERENCE OF A TWO-DIMENSIONAL ELECTRON … PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-3dS+
dt=−/H208731
/H9270s+i/H9024/H20874S++ST
/H92700T, /H208497/H20850
with the initial condition S+/H208490/H20850=−n0T/2. The solution for
ST/H20849t/H20850is readily written as ST/H20849t/H20850=1/H114082n0Te−t//H9270T. Substituting
ST/H20849t/H20850into Eq. /H208497/H20850one finds
S+/H20849t/H20850=−n0T
2/H20851/H208491−/H9257/H20850e−/H20849/H9270s−1+i/H9024/H20850t+/H9257e−t//H9270T/H20852, /H208498/H20850
where/H9257=/H20849/H92700T/H20850−1//H20851/H20849/H9270T/H20850−1−/H9270s−1−i/H9024/H20852. The real part of S+/H20849t/H20850is
equal to Sz/H20849t/H20850and given by
Sz/H20849t/H20850=n0T
2/H20851/H208411−/H9257/H20841sin/H20849/H9024t−/H9021/H20850e−t//H9270s−/H9257/H11032e−t//H9270T/H20852, /H208499/H20850
where
/H9021= arctan /H20851/H208491−/H9257/H11032/H20850//H9257/H11033/H20852, /H2084910/H20850
/H9257/H11032and/H9257/H11033are the real and imaginary parts of /H9257.I nt h e
particular case when electron and trion spin relaxation can beignored, Eq. /H208499/H20850is reduced to
S
z/H20849t/H20850=n0T
2/H20851− cos2/H9021e−t//H9270T+ sin/H9021sin/H20849/H9024t−/H9021/H20850/H20852, /H2084911/H20850
where/H9021=arctan /H20849/H9024/H9270T/H20850.
In strong magnetic field such that /H9024T*/H112711/H20851with /H20849T*/H20850−1
=/H20849/H9270T/H20850−1−/H9270s−1/H20852the parameter /H9257→0 and sin/H9021→1, cos/H9021
→0 yielding/H9021→/H9266/2. The high field asymptotics reads
/H9021/H11015/H9266
2−1
/H9024/H92700T. /H2084912/H20850
For low magnetic fields /H9024T*/H112701 we have
/H9021/H11015/H9266
2−/H9024/H20849T*/H208502
/H92700T−T*. /H2084913/H20850
The dependence of the phase /H9021on magnetic field has a
minimum
/H9021min= arctan /H208492/H9024min/H92700T/H20850, /H2084914/H20850
reached at
/H9024min=1
T*/H208811−T*
/H92700T. /H2084915/H20850
Note that Eqs. /H2084914/H20850and /H2084915/H20850are valid provided that /H9270sis
magnetic field independent. In principle, this might not bethe case in the experiment, where the main contribution tothe ensemble dephasing time arises from the spread of
g-factor values, but for typical conditions
/H9270s/H11271/H92700Tso that T*
/H11015/H92700Tand is almost field independent.
At zero magnetic field, the electron spin exhibits no pre-
cession, Sy/H110130,S+/H20849t/H20850=Sz/H20849t/H20850, and one has
Sz/H20849t/H20850=−n0T
2/H20851/H92570e−t//H9270T+/H208491−/H92570/H20850e−t//H9270s/H20852, /H2084916/H20850
where/H92570=/H9257/H20849B=0/H20850. This can be understood as follows. At
the moment right after photoexcitation into the trion reso-nance by a pulsed, say, /H9268−polarized light, the system con-
tains n0Tsinglet trions with completely polarized holes −3/2,
andn0Telectrons with uncompensated spin +1/2, because the
same number of electrons with spin −1/2 were extractedfrom the 2DEG to form trions. This stage is illustrated byFig.2/H20849b/H20850. In the absence of spin relaxation, trions decay by
emitting
/H9268−photons and giving back lent electrons with spin
−1/2. As a result the initially generated electron spin polar-ization is compensated by the returned electrons and tends tozero as the trions vanish. It is important to note here that inthe absence of spin relaxation of either trions or electrons nolong-lived spin coherence of the resident electrons can begenerated.
The compensation takes place also for the limiting case
/H92700T/H11270/H9270sT,/H9270sand also for the special case of /H9270s=/H9270sT, see solid
curve in Fig. 3. If the spin relaxation times /H9270sand/H9270sTcoin-
cide, then/H92570=1 and the second contribution to Sz/H20849t/H20850propor-
tional to e−t//H9270sdisappears. In this case the electron polariza-
tion Sz/H20849t/H20850decays with the trion spin lifetime /H9270T.
Spin relaxation of any subsystem brings in an imbalance,
namely, the spins of the leftover and the returned electronscannot completely compensate each other, see dashed curvein Fig. 3. The two characteristic parts of the dashed line
correspond to the fast and slow negative contributions toS
z/H20849t/H20850in Eq. /H2084916/H20850governed by/H9270T=30 ps and/H9270s=2 ns, respec-
tively.
In the limit of/H9270s→/H9270T, the eigenfrequencies of the system
/H208496/H20850are degenerate and
Sz/H20849t/H20850=−n0T
2e−t//H9270s/H208731−t
/H92700T/H20874, /H2084917/H20850
so that the exponential electron spin decay is modified by
linear function of t.
In an in-plane external magnetic field the imbalance arises
even if spin relaxation is absent or /H9270s=/H9270sT. Indeed, as sche-
matically illustrated in Fig. 2/H20849c/H20850, both the heavy holes /H20849due toFIG. 3. Kerr signal calculated for pump and probe resonant with
the trion in absence of an external magnetic field, B=0. The elec-
tron spin relaxation time /H9270s=2 ns; the trion radiative lifetime /H92700T
=60 ps. The dashed curve corresponds to the trion spin relaxation
time/H9270sT=60 ps, while the solid curve corresponds to an unrealisti-
cally high/H9270sT=2 ns. The latter situation corresponds to the compen-
sation of the initial spin polarization by the returned electrons, /H92570
=1.ZHUKOV et al. PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-4their zero in-plane heavy hole gfactor /H20850and the singlet elec-
tron pairs bound in trions are not affected by the magneticfield, whereas the spins of the resident electrons precess. Thetrion recombines radiatively by emission of a
/H9268−photon and
the returned electron is spin-up polarized along the zaxis
/H20851Fig.2/H20849c/H20850/H20852. As a result, even after the trions have vanished
/H20849e−t//H9270T→0/H20850, the electron spin polarization is nonzero and it
oscillates with frequency /H9024.I nE q . /H208498/H20850, the/H9257-independent
term describes the spin precession of resident electrons leftafter the pump pulse ends while the terms proportional to
/H9257
take into account the electrons left after the trions have re-combined and decayed exponentially. At the moment of trionrecombination, these electrons become polarized antiparalleltoz, their spin is added to the total electron spin, and rotates
as well with frequency /H9024in the /H20849y,z/H20850plane.
Time-resolved KR can originate both from the electron
and the trion spin polarizations. Namely, Kerr rotation angleis given by
/H9008
K=C1Sz/H20849t/H20850+C2ST/H20849t/H20850, /H2084918/H20850
where C1andC2are constants. In particular, the constant C1
can be obtained by the expansion of Im /H20853r+*r−/H20854in terms of
/H90030,+−/H90030,−. According to Eq. /H208499/H20850the electron contribution to
the Kerr signal contains both exponential monotonous andoscillatory components, whereas the trion contribution shows
only a monotonous behavior, proportional to e
−t//H9270T. It should
be stressed that the initial number of photogenerated trions
under resonant excitation, n0T, cannot exceed ne/2, where ne
is the density of the 2DEG. Thus, the factor n0Tin Eq. /H208499/H20850
increases linearly with the pump intensity for small excita-tion density and then saturates with density increase at thevalue n
e/2. In the simplest model
n0T=ne
2G/H92700T//H208491+G/H92700T/H20850, /H2084919/H20850
where Gis the generation rate being proportional to the
pump power. Thus the initial spin polarization of the 2DEGshows a saturation behavior /H20849see Fig. 4/H20850which can be related
to saturation of trion absorbtion.C. Resonant excitation of excitons in a diluted
two-dimensional electron gas
If the pump photon energy is tuned to the exciton transi-
tion then, at low temperatures where kBT/H11021EBT/H20849with the Bolt-
zmann constant kB, and the trion binding energy EBT/H20850, the
photogenerated excitons tend to bind into trions as long asthey find resident electrons with proper spin orientation. Thetrion thermal dissociation, on the other hand, can be ne-glected. Experimentally, the exciton contribution to the Kerroscillating signal is determined by the Larmor spin preces-sion of the electrons in the excitons. In the pump-probe ex-periment the correlation between the electron and hole spinsvia the electron-hole exchange interaction is as a rule sup-pressed, see Refs. 31and32. Therefore, the spin s
Xof an
electron in an exciton precesses about an in-plane magneticfield with the same frequency as that of a resident electron.Note that if the heavy holes forming excitons are unpolarizedthe excitons can be labeled by the electron spin s
X. More-
over, the trions formed from these excitons are unpolarizedas well.
At zero magnetic field the rates of an electron and an
exciton binding into a trion are given by
/H20873dn±
dt/H20874
T=/H20873dn/H11007X
dt/H20874
T=−/H9253n±n/H11007X, /H2084920/H20850
where n±,n±Xare the densities of electrons and excitons with
electron spin ±1/2, and /H9253is a constant. In addition to this
constant the system is characterized by four times, namely,
the exciton and trion radiative lifetimes, /H92700Xand/H92700T, respec-
tively, and the spin relaxation times of resident electrons /H20849/H9270s/H20850
and excitons /H20849/H9270sX/H20850. In the presence of an in-plane magnetic
field B/H11036z, the spins of the resident electrons and the elec-
trons in excitons precess with the same frequency /H9024.I nt h e
coordinate frame which rotates around Bat a rate/H9024, one can
apply the simple equation /H2084920/H20850to describe the spin-
dependent decay of resident electrons and photoexcited ex-citons.
At low excitation intensities satisfying the condition n
0X
/H11270ne/2/H20849here n0Xis the number of photogenerated excitons /H20850
the total spin of the electron gas after the decay of all exci-tons can be estimated as
/H20841S
e/H20841=/H9270X
2/H9270bn0X, /H2084921/H20850
where/H9270Xis the total lifetime of the exciton spin, including
the radiative decay time, the time of exciton binding into atrion,
/H9270b/H11011/H20849/H9253ne/H20850−1, and the spin relaxation time /H20849/H9270X/H20850−1
=/H20849/H92700X/H20850−1+/H9270b−1+/H20849/H9270sX/H20850−1. In order to simplify the analysis we
assume the time of exciton binding into a trion, /H9270b
/H11011/H20849/H9253ne/H20850−1, to be shorter than the exciton radiative lifetime /H92700X
and the spin relaxation time of the electron in an exciton /H9270sX.
In this case, shortly after the pulsed optical excitation allexcitons are bound to trions and the QW structure contains
n
0Xtrions and ne−n0Xresident electrons with a total precessing
spinFIG. 4. Initial spin of the 2DEG as function of the pumping
power /H20849solid /H20850. The asymptotic lines correspond to linear growth
/H20849dashed /H20850and saturation /H20849dotted /H20850. Here it has been assumed that the
laser is circularly polarized and is resonant with the trion energy.The pump power is given in units of photocreated trions in thelinear regime.SPIN COHERENCE OF A TWO-DIMENSIONAL ELECTRON … PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-5/H20841Se/H20841=n0X/2, n0X/H11349ne/2. /H2084922/H20850
As a result, n0Xspins of resident electrons contribute to the
Kerr rotation oscillations.
At higher excitation intensity, n0X/H11350ne/2, the spin-
polarized excitons extract almost at once ne/2 electrons to
form trions. Therefore, in the absence of electron-in-excitonspin relaxation processes the trion density cannot exceedn
e/2, thus the total spin density of the electron gas is limited
byne/4. The electron-in-exciton spin relaxation allows to
convert the remaining n0X−/H20849ne/2/H20850excitons in trions. Obvi-
ously, the maximum number of formed trions cannot exceed
the concentration of background electrons, ne. The total spin
of resident electrons after the excitons and trions have re-combined can be estimated as /H20849provided that the holes are
unpolarized /H20850
/H20841S
e/H20841/H110151
4/H20902ne−2n0X−ne
1+ /H208492/H9270sX//H92700X/H20850,ne/H110222n0X−ne
1+ /H208492/H9270sX//H92700X/H20850
0 otherwise. /H20903/H2084923/H20850
This equation is valid both for B=0 and B/HS110050 when n0X
/H11350ne/2, otherwise Eq. /H2084922/H20850holds. At n0X=ne/2, the values of
/H20841Se/H20841given by Eqs. /H2084922/H20850and /H2084923/H20850coincide and are equal to
ne/4.
When deriving the above equation we have neglected the
spin relaxation of the resident electrons assuming /H9270sX/H11270/H9270s.I n
experiment the exciton radiative lifetime, /H92700X, may be compa-
rable with the trion formation time, /H9270b. In this case the above
estimation should be taken as a qualitative result predicting anonmonotonous dependence of the Kerr signal amplitude onpump intensity. This nonmonotonous behavior is illustratedin Fig. 5. An initial linear growth of /H20841S
e/H20841followed by a linear
decrease is seen. The decrease of initial electron spin as afunction of pump intensity is steeper for smaller values of
/H9270sX//H92700X, i.e., for shorter hole spin relaxation times. It is worth
to stress that in this regime the electron spin polarizationvanishes at very high pumping whereas, under resonant trion
excitation, /H20841Se/H20841monotonously increases with increasing
pump power and saturates at ne/4.
Detection aspects
Turning to the detection aspects, we find that selective
addressing of the exciton resonance also results in temporaloscillations of the probe-pulse Kerr rotation. The modulationcomes from the photoinduced difference in the resonancefrequencies
/H92750,±and/or the nonradiative damping rates /H9003±.
Both/H92750,+−/H92750,−and/H9003+−/H9003−become nonzero taking into ac-
count the exchange interaction between an electron in anexciton and the resident electrons of the 2DEG: the first dif-ference is related to the Hartree-Fock renormalization of theelectron energy in the spin-polarized electron gas, and thesecond one is related to the spin dependence of the electron-exciton scattering.
24As a result, the rotation of the total spin
of the electron ensemble leads to a modulation of the excitonresonance frequency and nonradiative broadening and, thus,to oscillations of the KR angle. We note that an in-planemagnetic field results also in spin precession of the electronin an exciton and the total Kerr signal will be a superpositionof 2DEG and exciton signals.
The situation for the probe tuned to the trion resonance is
qualitatively the same. The KR amplitude will contain com-ponents arising due to spin precession of the 2DEG and tothe electron-in-exciton spin precession. However, it is ex-pected that detection at trion resonance will be less sensitiveto the exciton spin dynamics as compared to detection at theexciton resonance, see Sec. III B. We note here that the am-plitude of the KR signal induced by the same number ofcoherent electron spins will be different for detection at thetrion or the exciton energies. The difference comes from thedifferent oscillator strengths of these resonances
24and from
the different efficiencies of signal modulation.
D. Resonant excitation of excitons in a dense
two-dimensional electron gas
An increase of the 2DEG density and its Fermi level leads
to dissociation of trions due to state-filling and screeningeffects. We consider an intermediate situation where the tri-
ons are suppressed, n
eaT2/H110221, but the excitons are not, neaB2
/H112701, where aBis the exciton Bohr radius and aTis the char-
acteristic trion radius. Therefore, the scenario of spin coher-ence generation for electrons involves two subsystems, the2DEG and the spin-polarized excitons resonantly excited bythe circularly polarized optical pulse. The electrons beinginitially unpolarized can gain spin polarization due to theelectron-electron exchange interaction in electron-excitonflip-flop scattering processes. At zero magnetic field, the flip-flop scattering rates for the resident electrons and thosebound in excitons are given by
/H20873dn±
dt/H20874
exch=/H20873dn/H11007X
dt/H20874
exch=−/H9253/H11032/H20849n±n/H11007X−n/H11007n±X/H20850, /H2084924/H20850
where/H9253/H11032is a constant different from /H9253in Eq. /H2084920/H20850. The total
number of electrons, ne=n++n−, is fixed while the exciton
density nX=n+X+n−Xdecays to zero.FIG. 5. Schematic plot of the spin of the 2DEG after exciton and
trion recombination as a function of the initial exciton density /H20849i.e.,
pumping power /H20850. The solid curve is plotted according to Eq. /H2084922/H20850
and dotted, dashed, and dash-dotted curves are plotted according toEq. /H2084923/H20850. These curves correspond to different values of
/H9270sX//H92700X
=0.1, 1, and 10, respectively. ne=1010cm−2. The crossover density
n0X=ne/2 is shown by the arrow.ZHUKOV et al. PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-6Similarly to the scenario in Sec. III C, we assume that the
holes in the excitons are unpolarized and the spins of boththe resident and bound electrons precess with the same fre-quency. The Kerr rotation signal is a result of the differencein shift between the Fermi levels for spin-up and spin-downelectrons and the concomitant renormalization of the reso-nance frequencies
/H92750,±due to the exchange interaction be-
tween the spin-polarized carriers.
At weak pump intensities so that
/H9253n0X/H112701
/H9270X=1
/H92700X+1
/H9270sX+/H9253ne,
and for negligible resident-electron spin relaxation, /H9270X/H11270/H9270s,
one can use, in agreement with Eq. /H2084921/H20850, the following esti-
mate for the amount of resident-electron spins oriented by
excitons: /H20841Se/H20841=1
2/H9253/H9270Xnen0X. With increasing pump intensity the
electron spin saturates at /H20841Se/H20841max/H110111
2ne/H208491−/H9270X//H9270sX/H20850when all
2DEG electrons have become spin polarized.
It should be noted that a similar description can be applied
for resonant photoexcitation of excitons in the high-
temperature regime where kBT/H11271EBT, but kBT/H11021EBX, so that the
trion states are thermally unstable but the exciton boundstates are still stable.
E. Nonresonant excitation of carriers
In the case of nonresonant pumping, the photogeneration
of free electrons and holes is followed by their separate,uncorrelated energy relaxation toward the bottoms of the cor-responding bands. The energy relaxation is accompanied bycarrier spin relaxation. The holes have lost their spin afterfew scattering events. The total spin of the electron ensembleafter energy relaxation can be estimated as
S
z/H11015/H9270*
s
/H9270*
s+/H9270/H9280n0e,
where n0eis the number of photogenerated electrons, /H9270/H9280is the
energy relaxation time, and /H9270*
sis the spin relaxation time of
hot electrons. We note that /H9270*
scan be much shorter than the
spin relaxation time /H9270sof the electron gas in quasiequilib-
rium, entering Eq. /H208496/H20850.
After the particles have reached the band bottoms they
can bind to form excitons and trions. In the diluted electrongas and for moderate pumping densities, the trion formationis more preferable and the subsequent spin dynamics can bedescribed by the model in Sec. II B. At very strong pumping,when the number of photogenerated electrons exceeds that ofthe resident electrons, or for a dense 2DEG, for which thetrions are suppressed, formation of excitons takes place. Thespin dynamics in this case can be described by the scenarioin Sec. II D.
F. Summary
We have developed a comprehensive model to describe
the Kerr rotation signal in QWs with varying 2DEG densitiesand for different excitation regimes. The processes of theoptical transition and spin precession are considered as sepa-rated ones.
27,28The signal dynamics reveals spin oscillations
of the resident electrons and/or of the electrons forming ex-citons. The pump power dependence of the Kerr rotationamplitude is as follows: at low pumping the amplitude growslinearly with the pump power, whereas in the limit of highpumping the amplitude behavior strongly depends on the ex-citation energy. For trion resonant excitation it saturates /H20849as
the number of generated trions is limited by half of the resi-dent electron concentration /H20850, while for resonant excitation of
excitons /H20849atk
BT/H11021EBTand in the diluted 2DEG /H20850the amplitude
exhibits a nonmonotonic behavior and eventually vanishes.
A theoretical model of excitation of electron spin polar-
ization by short laser pulses tuned to the trion resonance hasbeen also proposed in Refs. 6,20, and 21. Their approach is
based on analysis of the temporal dynamics of the coherentquantum beats between an electron and a trion localized by aquantum dot or by well width fluctuations in a QW plane.
Our approach and the approach in Refs. 6,20, and 21are
essentially equivalent. Indeed, let us consider a localized un-polarized electron. Its spin density matrix is diagonal with
/H92671/2,1/2 =/H9267−1/2,−1/2 =1/2. After optical excitation with a /H9268+cir-
cularly polarized pulse a superposition state of the localizedelectron and a trion is formed. After the trion radiative decaythe electron remains, but its density matrix is different fromthe initial one. For example, in the limiting case where thehole spin relaxation time is much shorter than the trion ra-diative lifetime which, in turn, is shorter than the electronspin relaxation time, the electron spin density matrix aftertrion recombination remains diagonal with
/H92671/2,1/2 =1
2/H208731−/H20841D/H208412
2/H20874,/H9267−1/2,−1/2 =1
2/H208731+/H20841D/H208412
2/H20874./H2084925/H20850
Here Dis a complex coefficient which depends on the opti-
cal pulse parameters. The resident electron acquires spin-down polarization under pulsed
/H9268+photoexcitation and a
train of such pulses leads to complete spin polarization. Thesame single electron spin density matrix, Eq. /H2084925/H20850, describes
an ensemble where the number of spin-up electrons issmaller than that of spin-down electrons. The constant Din
this case has a transparent physical sense: it characterizes theefficiency of singlet trion formation under polarized excita-tion of the unpolarized ensemble. One can readily check thatthis result is equivalent to Eq. /H2084916/H20850taken at t=0.
III. EXPERIMENTAL RESULTS
A. Experimentals
The studied CdTe/Cd 0.78Mg0.22Te QW heterostructure
/H20849031901C /H20850was grown by molecular-beam epitaxy on an
/H20849100 /H20850-oriented GaAs substrate followed by a 2 /H9262m CdTe
buffer layer. It has five periods, each of them consisting of a110 nm thick Cd
0.78Mg0.22Te barrier and a 20 nm thick CdTe
QW. An additional 110 nm thick barrier was grown on top ofthis layer sequence to reduce the contribution of surfacecharges. The barriers contain 15 nm layers doped by iodinedonors. Undoped 15 nm spacers separate the modulation-doped layers from the QWs. Electrons from the barrier do-nors, being collected into QWs, provide there a 2DEG with aSPIN COHERENCE OF A TWO-DIMENSIONAL ELECTRON … PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-7low density of about ne=1.1/H110031010cm−2. We have observed
very similar experimental results for structures with singleCdTe/Cd
0.7Mg0.3Te QWs of 12 and 8 nm widths. This con-
firms the general character of the reported data.
A time-resolved pump-probe Kerr rotation technique was
used to study the coherent spin dynamics of the electrons.1
We used a Ti:sapphire laser generating 1.5 ps pulses at arepetition frequency of 75.6 MHz. The laser beam was splitin pump and probe beams and the time delay between thepump and probe pulses was varied by a mechanical delayline. The pump beam was circularly polarized by means ofan elasto-optical modulator operated at 50 kHz. The probebeam was linearly polarized, and rotation of the polarizationplane was measured by a balanced photodetector. The time-resolved Kerr rotation signal allows us to follow the evolu-tion of the spin coherence of carriers and their complexesgenerated by the pump pulses. From an analysis of the decayof the Kerr rotation amplitude the spin dephasing time of the
electron ensemble T
2*can be extracted. The details of this
analysis through which the mechanisms of spin dephasing ofthe 2DEG can be understood have been reportedelsewhere.
11,17
The Kerr rotation technique has been used in two regimes.
For degenerate Kerr rotation the pump and probe beams havethe same photon energy, as they originate from the samelaser. This regime will be denoted as one-color experimenthere. We performed, however, also a two-color experimentwhere the pump and probe energy can be tuned indepen-dently. For that purpose two synchronized Ti:sapphire laserswere used. Experiments were performed in magnetic fields/H208490–7 T /H20850applied in the plane of the structure, i.e., in the
Voigt geometry. The sample temperature was tuned in a
range 1.9–100 K.
Furthermore, time-resolved photoluminescence was used
to study recombination dynamics of excitons and trions. Thesame laser as described above was used for excitation, andtime-resolved emission spectra have been recorded by a syn-chroscan streak camera connected to a 0.5 m spectrometer.The time resolution in this experiment was about 5 ps.
A typical photoluminescence /H20849PL/H20850spectrum of the QW
measured at a temperature T=1.9 K is shown in Fig. 6/H20849a/H20850.I t
shows the exciton and trion recombination lines separated by2 meV, corresponding to the trion binding energy. The fullwidth at half maximum of the exciton line is about 0.5 meVand is mainly due to exciton localization on the QW widthfluctuations.
The reflectivity spectrum of the same QW in the energy
range of the exciton and trion resonances is given in Fig.6/H20849b/H20850. Following the procedure described in Ref. 24we have
evaluated the oscillator strengths of the resonances andfound that the exciton oscillator strength is ten times largerthan that of the trion resonance. This fact should be takeninto account when the intensities of the Kerr rotation signalsmeasured at the exciton and trion energies are compared:Firstly, the probe response is proportional to the oscillatorstrength and, secondly, the number of photogenerated carri-ers is proportional to the oscillator strength.
For interpretation of the results of the spin dynamics ex-
periments, information on the recombination dynamics ofexcitons and trions under resonant excitation is necessary.We performed corresponding measurements under linearly
polarized excitation using the streak camera for detection.The results for pumping into the exciton and trion resonancesare very similar to each other. The typical recombinationkinetics for the trion is given in Fig. 6/H20849c/H20850. For resonant exci-
tation at the trion energy /H20849curve 2, detuning /H9004E=0/H20850about
80% of the PL intensity decays with a time of 30 ps and therest decays with a time of 100 ps. When the excitation en-ergy is detuned by 0.8 meV above the trion resonance a re-distribution of the two exponential decays with 30 and100 ps time occurs in favor of the longer decay component.Such a behavior is typical for QW emission.
17,35The shorter
decay time, 30 ps, can be attributed to the radiative recom-bination time of trions and excitons generated in the radiativecone where their wave vectors are transferred to the photons.For excitons scattered out of the radiative cone, a longer timeof about 100 ps is required to be returned to the cone viaemission of acoustical phonons. The exciton luminescencelifetime is slowed down to 100 ps in this case. The recom-bination of trions is not restricted to the radiative cone con-ditions as the electron left in the system can take the finitemomentum to satisfy the wave vector conservation. There-1.595 1.600
0 100 2001.595 1.600PL intensity
Energy (eV)T
X(a)PL intensity(c)
4
3
2
Time (ps)1 Laser pulse
2∆E=0m e V
3∆E=0 . 8m e V
4∆E=2 7m e V
1T=1 . 9KReflectivity
Energy (eV)T
X(b)
FIG. 6. /H20849a/H20850Photoluminescence spectrum of a 20 nm
CdTe/Cd 0.78Mg0.22Te QW measured under nonresonant cw excita-
tion with a photon energy of 2.33 eV. The exciton /H20849X/H20850and trion /H20849T/H20850
resonances are separated by 2 meV, which is the trion binding en-ergy. /H20849b/H20850Reflectivity spectrum of the same structure. The oscillator
strength of the exciton resonance is ten times larger than the trionone. /H20849c/H20850Kinetics of the photoluminescence measured by a streak
camera under 1.5 ps pulsed excitation resonant to the trion energy/H20849curve 2 /H20850and detuned from it by 0.8 and 27 meV to higher energies.
The dashed line shows the laser pulse.ZHUKOV et al. PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-8fore we may expect the fast decay of trion PL of about 30 ps
even for nonresonant photoexcitation. In most cases, how-ever, for these experimental conditions the trions are formedfrom photogenerated excitons, which dominate the absorp-tion due to their larger oscillator strength. As a result thedecay of the trion PL in this case is determined not by trionrecombination but by trion formation, and is contributed bythe exciton lifetime and trion formation time. For small de-tuning exemplified by curve 3, the fast 30 ps process coexistswith the longer /H20849100 ps /H20850one. When the excitation energy is
tuned to the band-to-band absorption /H20849curve 4,/H9004E
=27 meV /H20850, the PL decay is extended to 250 ps as additional
time is required for the free carriers to be bound to excitons.Note that the exciton binding energy in the studied QW is
E
BX=12 meV.
Below we present the experimental results for the two-
color and the degenerate pump-probe experiment. Further-more, we show the experimental data for the pump power,temperature, and magnetic field dependencies of the Kerrrotation signals.
B. Two-color pump probe
The two-color Kerr rotation technique enables indepen-
dent tuning of the energies of the pump and probe beams.This allows us to perform experiments with either constantexcitation or detection conditions, which simplifies identifi-cation of the studied relaxation processes.
Figure 7shows Kerr rotation signals detected at the trion
and exciton resonances for three pump energies: /H20849a/H20850resonant
with the trion, /H20849b/H20850resonant with the exciton, and /H20849c/H20850nonreso-
nantly excited 72 meV above the exciton energy. The sampleis subject to an in-plane magnetic field B=1 T. All signals
show damped oscillations with a frequency of 23 GHz,which is the Larmor precession frequency of the electronspins. This frequency corresponds to the Zeeman splitting ofthe conduction band electrons with a g-factor value of 1.64
which is in good agreement with literature data for the elec-tron gfactor in CdTe/ /H20849Cd,Mg /H20850Te QWs.
33Another common
feature of all signals shown in Fig. 7is the appearance of
long-living spin beats which are observed beyond delays of2.7 ns. The typical recombination times of excitons and tri-ons do not exceed 30–100 ps for resonant and quasiresonantexcitation and 250 ps for nonresonant excitation /H20851see Fig.
6/H20849c/H20850and Sec. III A /H20852; therefore, we identify the long-living
signal with the coherent spin precession of the resident elec-trons in the 2DEG. One can see that this coherence is excitedefficiently for all pump energies and can be detected by prob-ing both the trion and exciton resonances.
Some of the signals shown in Fig. 7contain a short-living
part right after the pump pulse with a typical decay time of50–70 ps. This part is especially pronounced for the “pumpX/probe X” condition /H20849i.e., pump and probe are degenerate
with the exciton resonance /H20850, see panel /H20849b/H20850. This fast compo-
nent is related to the exciton contribution to the Kerr rotationsignal, see Sec. II C and discussion below. It is in line withRef.9, where spin dynamics has been measured by the Hanle
effect under steady-state photoexcitation. To extract thetimes and relative amplitudes of the short- and long-livingcomponents in the spin beat signals each trace has been fitted
with a biexponential decay function
y/H20849t/H20850=/H20849Ae
−t//H92701+Be−t//H92702/H20850sin/H20849/H9275t+/H9272/H20850, /H2084926/H20850
where AandBare constants describing the amplitudes of the
fast /H20849/H92701/H20850and slow /H20849/H92702/H20850components, respectively, /H9275is the
Larmor frequency which is taken to be the same for both
components, and /H9272is the initial phase.
The parameters extracted from these fits are collected in
Table I. All signals, except the one for “pump T/probe T,” are
symmetric with respect to the abscissa. The pump T/probe Tsignal shows an initial relaxation of the center of gravity ofthe electron beats with a time constant of about 75 ps, whichcan be attributed to the spin relaxation time of the hole in thetrions, see the second term in Eq. /H208499/H20850and Ref. 11. A detailed
analysis of hole spin coherence in CdTe/ /H20849Cd,Mg /H20850Te QWs
will be reported elsewhere.
The decay times and relative amplitudes in Table Iare
given for B=1 T and T=1.9 K. It is important to note that
these parameters depend strongly on pump intensity, mag-netic field strength, and lattice temperature. These dependen-cies and the underlying physical mechanisms will be dis-FIG. 7. KR signals measured by a two-color technique at T
=1.9 K, B=1 T, and various pump excitation energies: /H20849a/H20850resonant
with trion at 1.5982 eV, /H20849b/H20850resonant with exciton at 1.6005 eV, and
/H20849c/H20850nonresonant at 1.6718 eV, which is 72 meV above the exciton
resonance. The resonant energies for X and T are shown by arrowsin Fig. 6/H20849b/H20850. Pump density 56 W/cm
2and probe density 8 W/cm2.SPIN COHERENCE OF A TWO-DIMENSIONAL ELECTRON … PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-9cussed in detail below. Here we focus our attention on the
pump energy dependence of these parameters. We first con-centrate on the relative amplitudes as they are a key to un-derstanding the generation of 2DEG spin coherence and therole of the trions in this process.
For the pump T/probe T regime only the long-living elec-
tron signal of a 2DEG is observed /H20851see panel /H20849a/H20850of Fig. 7/H20852.
This is in line with the model expectations as in this caseonly trions are photogenerated. As these trions are in thesinglet ground state with antiparallel electron spins, they donot contribute to the KR signal. Moving the pump energy inresonance with the exciton and further to a band-band tran-sition leads to appearance of the fast-decaying componentfor the signal probed at the trion energy. This can be attrib-uted to the spin dynamics of the exciton which is excitedeither resonantly or nonresonantly, see Secs. II C and II E.The probe has a finite spectral width /H20849about 1 meV /H20850and
tuned to the trion resonance slightly overlaps with the exci-ton resonance. The shortening of the electron spin dephasingtime from 5.7 down to 3.5–3.6 ns is attributed to heating ofthe 2DEG by photocarriers with excess kinetic energy /H20849for a
detailed discussion see Secs. III D and III E /H20850.
The Kerr rotation signal probed at the exciton energy has
two contributions: /H20849i/H20850given by the coherent precession of the
electrons in excitons and /H20849ii/H20850given by the spin precession of
the 2DEG. The former decays with the exciton recombina-tion time. This fast exciton component is clearly seen inpanels /H20849b/H20850and /H20849c/H20850and is absent for pumping at the trion
resonance. For nonresonant excitation its relative amplitudedoes not exceed 50%. In this case electrons and holes arephotogenerated 72 meV above the exciton resonance; there-fore, they have a high probability to scatter and relax sepa-rately to the bottoms of their bands, where they are boundinto trions and excitons. The relative amplitudes of the fastand long-lived signals reflect the probability of trion andexciton formation. It may be expected that trion formation ispreferable because the 2DEG density exceeds by at least anorder of magnitude the concentration of photocarriers whichis in line with the experimental findings.
A very different ratio of the relative amplitudes, 90% for
the fast decay and 10% for the long-living dynamics, is seenfor the signal when resonantly pumped at the exciton energyand detected at the exciton /H20851see panel /H20849b/H20850in Fig. 7/H20852. There are
at least two factors which favor an exciton population incomparison with a trion one under resonant pumping of theexcitons. First, the photogeneration leads to formation of ex-citons with very low kinetic energy; therefore, they remainwithin the radiative cone and quickly recombine /H20849during
30–50 ps /H20850prior to becoming bound to trions.
35Second, a
part of excitons is localized, so that they are not mobile andcannot reach the sites in the QW where the background elec-trons are localized. Consequently, the formation of trions outof this exciton reservoir is suppressed. Moreover, the ratiobetween the contributions of the excitons and the 2DEG tothe KR signal is spectrally dependent: detection at the exci-ton resonance is more sensitive to the spin precession of theelectron in the exciton, see Sec. II C.
The results of the two-color experiments presented in Fig.
7are in good agreement with the model expectations,
namely, /H20849i/H20850the signal oscillating with the electron Larmor
frequency is contributed by resident electrons of the 2DEGand by electrons precessing in the excitons, and /H20849ii/H20850trion
formation resulting either from resonant photoexcitation ofthe trions or from capture of excitons or free carriers is avery efficient mechanism for spin coherence generation in adiluted 2DEG.
C. Degenerate pump-probe Kerr rotation
The degenerate pump-probe technique is simpler in tech-
nical realization and therefore is a more common method toaddress spin coherence. In this case the pump and probepulses are generated by the same laser beam without addi-tional spectral selection. Examples of degenerate Kerr sig-nals can be found already in Fig. 7, for example, the pump
T/probe T and pump X/probe X traces measured for coincid-ing energies of the two lasers.
The degenerate pump-probe signal presented in Fig. 8
was measured with the use of one laser, as will be the casefor the most experiments presented in the rest of this paper. Itis consistent with spin dynamics studied by two-color tech-nique. Namely, only long-lived spin beats of resident elec-trons are observed when the laser is resonant with the trionand two component decay is seen for the laser hitting exci-ton. The experimental conditions were modified as comparedwith Fig. 7in order to achieve longer electron spin coherence
times. Namely, the pump density was reduced to 1.5 W/cm
2TABLE I. Decay times /H92701and/H92702and amplitude ratios A/B
extracted from biexponential fits to the experimental data using Eq./H2084926/H20850in Fig. 7.B=1 T and T=1.9 K.
Pump T Pump X Nonresonant
Probe T —/5.7 ns 40 ps/3.5 ns 56 ps/3.6 ns
0/1 0.5/0.5 0.2/0.8
Probe X —/2.6 ns 50 ps/2.0 ns 70 ps/2.8 ns
0/1 0.9/0.1 0.5/0.5
FIG. 8. Kerr rotation measured by degenerate pump-probe reso-
nant either with the trion or the exciton energies. T=1.9 K, pump
density is 1.5 W/cm2and probe density is 0.3 W/cm2.ZHUKOV et al. PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-10and a weaker magnetic field B=0.25 T was applied.11The
delay time range between pump and probe in Fig. 8covers
6 ns. Under these conditions the spin dephasing time of theresident electrons reaches 13.7 ns for pumping in the trionresonance and 4.2 ns for pumping in the exciton resonance.Moreover the spin coherence does not fully decay during thetime interval of 13.2 ns between the pump pulses as isclearly seen by the beats at negative delays in Fig. 8.
D. Dependence on pump density
In this part we investigate the modifications of Kerr rota-
tion signal occurring when the pump density is increased in awide range of pump powers Pfrom 1 to 320 W/cm
2. De-
generate pump-probe resonant either with the exciton or thetrion transition energies is used.
Results for excitation at the exciton transition energy are
collected in Fig. 9. With increase of the pump density a very
different behavior is observed for the fast-decaying excitoncomponent and the long-living 2DEG component. One canclearly see that the exciton part is rather weak at low powers,evidencing the high probability for an exciton to becomebound to a resident electron and form a trion. At higher ex-citation power the fast-decaying component becomes morepronounced and even dominates in the power range P
=20–180 W/cm
2where the concentration of photogenerated
excitons exceeds the 2DEG density. Moreover the absoluteamplitude of the 2DEG signal at longer delays is decreasedfor pump powers exceeding 20 W/cm
2. This result will be
discussed below together with the data plotted in Fig. 9.
Relaxation times obtained from the fit with Eq. /H2084926/H20850are
given in Fig. 10. The fast-decaying component falls in the
range of 25–40 ps and is independent of the pump density. Itis clearly determined by the radiative recombination of exci-tons resonantly excited in the light cone. The decay time ofthe long-living component related to the spin dephasing ofthe 2DEG decreases from 2.1 ns at 15 W/cm
2down to
1.4 ns at 240 W/cm2. A possible reason for this behavior
can be the heating of the resident electrons by the photoex-
citation, leading to a reduction of T2*.
Kerr rotation signals for resonant pumping into the trion
state are shown in Fig. 11. Their shape does not change
markedly for varying pump densities. Only the long-livingelectron spin dephasing is observed, which becomes shorterfor higher pump densities. The corresponding dephasingtimes are plotted in Fig. 12. Two characteristic regions are
seen in Fig. 12: a strong decrease from 14 ns down to 4 ns at
low densities, and a much slower decrease for pump densi-ties exceeding 100 W/cm
2. The decrease of the dephasing
time can be understood in terms of delocalization of the resi-dent electrons caused by their heating due to interaction withphotogenerated carriers. Thus, the electron localization in thestudied samples favors longer spin dephasing times. Thechange of the behavior of spin dephasing times at P
/H11011100 W/cm
2can be attributed to saturation of the trion
absorption: a further increase of the pumping intensity doesnot change strongly the number of photogenerated carriers.
Kerr rotation amplitudes of the 2DEG measured at the
trion and exciton resonances as functions of pump density atB=1 T are shown by the circles in Fig. 13. The signal has
been measured at a delay of 0.5 ns, where the contribution ofthe fast-decaying component is vanishingly small. At lowexcitation density both dependencies show to a good ap-proximation a linear behavior as is illustrated by the inset. Athigher excitation density the Kerr rotation amplitudes dem-FIG. 9. Kerr rotation signals for degenerate pump-probe reso-
nant with the exciton energy measured at different pump densities.T=1.9 K. For clarity of presentation upscaling factors have been
used for delays exceeding 0.5 ns. Also the initial parts of the signalsfor pump densities exceeding 20 W/cm
2have been shifted in time
to negative delays. For all signals a coherent signal at zero delayformed due to the temporary overlap of the probe with the pumphas been removed.FIG. 10. Pump density dependence of the 2DEG spin dephasing
time T2*=/H92702/H20849circles /H20850and the time of the fast decay /H92701/H20849triangles /H20850
related to the exciton recombination dynamics. T=1.9 K.SPIN COHERENCE OF A TWO-DIMENSIONAL ELECTRON … PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-11onstrate pronounced nonlinear behaviors: for the “pump/
probe T” configuration saturation is observed, while for the“pump/probe X” configuration the signal decreases with in-crease of the pump power. Both these results are in agree-ment with the theoretical predictions of Secs. II B and II C,respectively, as illustrated by Figs. 4and5. The amplitude of
the fast-decaying component related to the electron-in-exciton precession in given in Fig. 13by triangles. Note the
scaling factor 0.15, which shows that this component domi-nates over other signals at moderate and high pump densities/H20849see also Fig. 9/H20850.
In order to make a quantitative comparison we replot the
experimental points in Fig. 13using a different vertical scale,
see Fig. 14. Namely, we normalize the Kerr signals to theirmaximum values /H20849which, according to our theoretical predic-
tions, correspond to the electron spin density being equal ton
e/4/H20850. This allows us to compare the efficiency of spin co-
herence generation. One can see that in the low pumpingFIG. 11. Kerr rotation signals for degenerate pump-probe reso-
nant with the trion energy measured at different pump densities. T
=1.9 K.
FIG. 12. Pump density dependence of the 2DEG spin dephasing
time T2*.T=1.9 K.FIG. 13. Kerr rotation amplitude versus pump density for the
experimental data of Figs. 9and11. Results for the pump resonant
with the excitons and trions are shown, respectively. The amplitudeof the fast-decaying component evaluated for zero delay and mul-tiplied by a factor 0.15 is given by the triangles. The amplitudes ofthe long-lived 2DEG coherence measured at a delay of 0.5 ns areshown by open and closed circles for trions and excitons, respec-tively. The inset highlights the low density regime. T=1.9 K.
FIG. 14. Normalized long-lived amplitude of the 2DEG Kerr
rotation shown in Fig. 13measured at a delay of 0.5 ns under reso-
nant pumping of the excitons /H20849closed circles /H20850and the trions /H20849open
circles /H20850. Model calculations are shown by the lines /H20851the dashed
curve has been calculated for trion resonant excitation according toEq. /H2084919/H20850/H20852. The solid lines have been calculated for excitation at the
exciton frequency according to Eqs. /H2084922/H20850and /H2084923/H20850. In the latter case,
the fitting parameter was the ratio of
/H9270sX//H92700which is found to be 10.ZHUKOV et al. PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-12regime the spin coherence generation efficiency per absorbed
photon is practically the same for the laser tuned either to theexciton or the trion resonance. This is in a good agreementwith our theory: each absorbed photon creates a trion eitherdirectly or via an intermediate excitonic state, thus an elec-tron with a given spin orientation is removed from the2DEG. The strong pumping regime is different for excitonand trion excitations. In the case of the laser tuned to thetrion resonance the spin of the 2DEG saturates /H20849the small
decrease of the Kerr rotation amplitude can be attributed toheating of the 2DEG /H20850while for the laser tuned to the exciton
resonance a strong decrease of the spin coherence generationefficiency is seen.
The curves in Fig. 14are the results of theoretical calcu-
lations based on the models outlined in Secs. II B and II C.The dashed curve corresponds to trion resonant excitation,while the solid line is for exciton resonant excitation. For thedashed curve the only fitting parameter was the saturationlevel. For the solid line the only fitting parameter was theratio between the electron-in-exciton spin relaxation time
and the exciton radiative lifetime,
/H9270sX//H92700X. The best fit corre-
sponds to a/H9270sX//H92700X=10. The fact that the spin relaxation of an
electron in the exciton /H9270sX/H110110.5 ns is shorter than that for the
resident electrons might be due to its interaction with thehole in the exciton, which provides additional channel forspin relaxation.
E. Temperature dependence
The electron spin coherence in CdTe/ /H20849Cd,Mg /H20850Te QWs is
robust against temperature increase and can be clearly traced
up to 100 K. Kerr rotation signals measured at the trion andexciton resonances in a temperature range from 1.9 up to100 K are presented in Fig. 15. The signals are normalized to
their values at zero time delay in order to highlight the trendsin signal decay with increasing delay. One can see that thedecay of the spin beats is thermally accelerated both for reso-nant pumping in the trion and in the exciton resonance, butin the former case the decrease is slower. The decay timesare shown in Fig. 16. The Kerr rotation signals for the
“pump/probe trion” configuration were fitted by a single ex-ponential decay, and the resulting decay times are given in
the figure by the open circles. A relatively long T
2*time of
440 ps is measured at T=100 K at the trion resonance. The
signals for the “pump/probe exciton” configuration were fit-ted by double exponential decays, see Eq. /H2084926/H20850, for tempera-
tures below 60 K. Above T=60 K only the fast component
with a decay time in the 200–250 ps range is apparent. Thistime can be assigned to exciton recombination and we mayconclude that for high temperatures the exciton spin coher-ence dominates over the 2DEG signal. This can be explainedby a reduction of the trion formation from excitons when the2DEG electrons have elevated kinetic energies.
We turn now to the analysis of the Kerr rotation ampli-
tude. Its temperature dependence for trion resonant pumpingis plotted in Fig. 17. The amplitude of the first maxima after
the pump pulse, which closely coincides with the amplitudeobtained by extrapolation to zero delay, is plotted. In themain panel the Kerr rotation amplitude is shown as a func-tion of temperature on a linear scale. The signal rapidly
looses about 80% of its intensity by a temperature increasefrom 1.9 up to 20 K and then gradually decreases in intensityup to 100 K, which is the maximum temperature at whichsignal could be recorded in experiment. The insert shows thedata in a form which allows extraction of characteristic acti-vation energies in the temperature dependence. In the tem-perature range from 12 to 45 K we obtain an activation en-ergy of 2 meV, which equals to the trion binding energy.
The strong temperature dependence of the measured sig-
nals can be explained by two mechanisms: /H20849i/H20850delocalization
of electrons and /H20849ii/H20850dissociation of trions. At very low tem-
peratures k
BT/H113510.5–1 meV, both electrons and photogener-
ated trions are localized in the quantum well plane by alloyfluctuations and/or interface roughnesses. The Kerr rotationsignal at fixed pumping is proportional to the number ofphotogenerated trions. Increase of the temperature up toFIG. 15. Kerr rotation signal measured at different temperatures
by degenerate pump-probe resonant with the trion /H20849a/H20850and the exci-
ton /H20849b/H20850. The signals are normalized on their amplitudes at zero
delay. Pump density 5 W/cm2and probe density 1 W/cm2.SPIN COHERENCE OF A TWO-DIMENSIONAL ELECTRON … PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-13kBT=0.5–1 meV is accompanied by electron delocalization
which in turn leads to a decrease of the interaction betweenphotogenerated electron-hole pairs and electrons. Thus, thenumber of photogenerated trions decreases. Furthermore, theincrease of the temperature leads to thermal dissociation of
trions, whose number is proportional to exp /H20849−E
BT/kBT/H20850,
where EBT/H110152 meV. Thus, the trion contribution to the KR
signal becomes weaker. At temperatures strongly exceedingE
B/kBthe main channel for spin coherence formation is due
to exciton-electron spin-flip scattering, see Sec. II D.
F. Initial phase shift
The total spin of the electron ensemble precesses around
the external magnetic field. The decay of the total spin withincreasing delay is governed by the spin dephasing pro-
cesses. The spins of the electrons forming a trion are in asinglet state and therefore do not precess around the mag-netic field as the resident electrons do. Recombination oftrions leads to return of the partially z-polarized electrons to
the 2DEG. Their spin orientation differs from that of theprecessing electrons, which results in a shift of the initialphase of the measured KR signal, see Eq. /H208499/H20850. This effect is
illustrated in Fig. 18, where the full arrow shows the total
spin of the 2DEG, while the open arrow shows the spin ofthe returned electrons. One can see that after trion recombi-nation the total spin of the electron gas has been rotated by alarger angle as compared with the rotation of the residentelectrons spin. This induces a phase-shift of the oscillatingKerr signal.
We have analyzed the initial phase by extrapolation of the
spin beats at longer delays back toward zero delay. The ex-periments were performed at low excitation density withpump and probe at the trion resonance. The results are givenin Fig. 19. A pronounced minimum is seen at B=0.6–0.8 T
in qualitative agreement with the model calculations shownby the solid line, which was calculated with a trion lifetime
/H92700T=30 ps. We also assumed an electron spin dephasing time
/H9270s=T2*=2 ns /H20849which corresponds to the experimental value at
B=3 T /H20850. In experiment this time is longer at smaller fields.
However, the calculated curve is insensitive to the choice ofFIG. 16. Decay times determined from the Kerr rotation signals
measured at different temperatures in Fig. 15. The data evaluated
from a single exponential fit to the pump/probe trion data are shownby the open circles and a line as a guide to the eyes. The data for thepump/probe exciton configuration are given by the closed symbols:by the circles for
/H92702and by the triangles for /H92701.B=0.25 T, pump
density 5 W/cm2, and probe density 1 W/cm2.
FIG. 17. Temperature dependence of the Kerr rotation amplitude
of the 2DEG signals in Fig. 15/H20849a/H20850. The inset shows a logarithmic
plot of the amplitude on inverse temperature. The line correspondsto an activation energy of 2 meV.Spin of
returning electrons
FIG. 18. Diagram explaining the phase shift of the Kerr rotation
signal.
0.0 0.5 1.0 1.5 2. 00.81.01.21.41.6
60 psInitial phase (rad)
Magnetic field (T)pump / probe trion
τST=2 0p s
FIG. 19. Initial phase of the Kerr rotation spin beats versus
magnetic field: /H20849a/H20850experimental data measured for degenerate
pump-probe resonant with the trion. Pump density 0.64 W/cm2and
probe density 0.5 W/cm2. The model calculations have been per-
formed according to Eq. /H2084910/H20850.ZHUKOV et al. PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-14this value as far as /H9270sremains the longest time in the system.
The only free parameter in the calculations was the spin re-
laxation time of the hole in a trion /H9270sT. The two curves cor-
respond to/H9270sT=20 ps /H20849dashed /H20850and 60 ps /H20849solid /H20850. It is seen
that, in accordance with Eqs. /H2084914/H20850and /H2084915/H20850, the depth of the
mininum and its field position is controlled by /H9270sT. The model
results demonstrate qualitative agreement with the experi-mental data.
G. Excitons and trions probing different resident electrons
As we have already noted above, the studied sample con-
tains a diluted 2DEG with the Fermi energy smaller than thetypical localization potential caused by well width fluctua-tions. As a result, at low temperatures of both the lattice andthe electron gas, a major fraction of electrons is localized.We have concluded also from the pump density and tempera-
ture dependencies that the dephasing time T
2*is strongly sen-
sitive to electron localization, see Figs. 12and16.T2*can be
extremely long for the localized electrons and shortens withelectron delocalization.
To have a deeper insight into the effects of electron local-
ization on the spin coherence we have done two-color pump-probe measurements in the regime where the longest dephas-ing times have been achieved, i.e., using a low pump densityin a weak magnetic field of 0.25 T. Figure 20shows the
results of such an experiment, where the pump beam is reso-nant with the exciton transition and the Kerr rotation signalis detected at either the trion or the exciton energy. The spindephasing times were independent of probe beam intensitywhen it decreased by a factor of 2 from 0.4 to 0.2 W/cm
2.
Under these experimental conditions the fast-decaying com-ponent is very small for probing at the exciton energy. Theexcitation conditions are identical for both signals in Fig. 20,
therefore, one can expect to detect the same dephasing timesfor the 2DEG, irrespective of the probe energy. Therefore, itis rather surprising to observe that the dephasing times of the
electron coherence differ by a factor of 2: T
2*=10.8 and
5.3 ns for probing at the trion and the exciton resonance,respectively.To explain this difference we suggest that different frac-
tions of the resident electrons contribute to the Kerr rotationsignal measured at the trion or the exciton energies. This canbe understood if we take into account that different mecha-nisms lead to Kerr rotation signal at the exciton and trionenergies. For detection at the trion resonance the effect iscontributed by the variation of the trion oscillator strength,which is directly proportional to the concentration of elec-trons with a specific spin orientation, see Sec. II A. As thein-plane localization is important to achieve stability of thetrions in QWs one may expect that the trions are much moredependent on the localized electrons, which have a longerdephasing time. The Kerr rotation signal at free exciton en-ergy monitors 2DEG spin beats mostly due to the spin-dependent exciton-electron scattering, see Sec. II C and Ref.24. Possibility for this scattering to occur implies existence
of free electrons. Therefore, at the exciton energy we addressfree or quasifree resident electrons which have shorterdephasing times.
IV. CONCLUSIONS
We have demonstrated experimentally the possibility to
excite spin coherence of a 2DEG by resonant pumping intothe trion or the exciton states, as well as by nonresonantexcitation. It was shown that the time-resolved Kerr rotationsignal detected experimentally at the trion and excitonfrequencies contains two components: a fast contribution/H20849which vanishes /H1101130 ps after the pump pulse /H20850and a long-
living one /H20849with typical decay times of the order of nanosec-
onds /H20850. The fast component is related with electrons in exci-
tons, while the long one is due to the resident electrons of the2DEG. Experimentally we can clearly separate these contri-butions.
A theoretical model has been developed to describe vari-
ous scenarios of spin coherence generation in QWs with a2DEG. It is based on a classical approach to spins and ac-counts for resonant excitation in the trion and exciton reso-nances and also for nonresonant photoexcitation. A compre-hensive set of experimental results is consistently describedin the frame of this model.
The suggested model can be generalized to the description
of spin coherence generation in a 2D hole gas. In this casethe in-plane component of the heavy-hole gfactor is negli-
gible and, therefore, the regime of weak magnetic fields isrealized. It can be expected that due to the stronger spin orbitinteraction the free hole spin relaxation is faster than that offree electrons. On the other hand, in a low density hole gasmost of the carriers are localized, so that the role of the spinorbit interaction is diminished and the hole spin relaxationtimes can reach up to 600 ps in GaAs/ /H20849Al,Ga /H20850As QWs,
34for
example.
It is also worthwhile to note here, that the results of our
model consideration can be also applied for describing theexcitation of spin coherence in other low-dimensional semi-conductor structures such as in an ensemble of singlycharged quantum dots.FIG. 20. Kerr rotation measured by two-color pump probe. T
=1.9 K. Pump density 1 W/cm2and probe density 0.4 W/cm2.SPIN COHERENCE OF A TWO-DIMENSIONAL ELECTRON … PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-15ACKNOWLEDGMENTS
We acknowledge fruitful discussions with Al. L. Efros, A.
Shabaev, I. V. Ignatiev, and I. A. Yugova. This work wassupported by the BMBF “nanoquit” program, the DeutscheForschungsgemeinschaft /H20849Grant No. YA 65/5-1 /H20850, and by the
Russian Foundation for Basic Research. E.A.Z.’s stays inDortmund were financed by the Deutsche Forschungsge-
meinschaft via Grants Nos. 436RUS17/79/04, 436RUS17/93/05, and 436RUS17/77/06. An ELI research visit to Dort-mund was supported by the Gambrinus guest-professorprogram of the Universität Dortmund. M.M.G. was partiallysupported by the “Dynasty” foundation—ICFPM.
1Semiconductor Spintronics and Quantum Computation , edited by
D. D. Awschalom, D. Loss, and N. Samarth /H20849Springer-Verlag,
Berlin, 2002 /H20850.
2I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76, 323
/H208492004 /H20850.
3A. Abragam, The Principles of Nuclear Magnetism /H20849Oxford Uni-
versity Press, Oxford, 1961 /H20850,p .4 4 .
4X. R. Wang, Y. S. Zheng, and Sun Yin, Phys. Rev. B 72,
121303 /H20849R/H20850/H208492005 /H20850.
5A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. 88,
186802 /H208492002 /H20850.
6A. Greilich, R. Oulton, E. A. Zhukov, I. A. Yugova, D. R. Yak-
ovlev, M. Bayer, A. Shabaev, Al. L. Efros, I. A. Merkulov, V.Stavarache, D. Reuter, and A. Wieck, Phys. Rev. Lett. 96,
227401 /H208492006 /H20850.
7A. Greilich, D. R. Yakovlev, A. Shabaev, Al. L. Efros, I. A.
Yugova, R. Oulton, V. Stavarache, D. Reuter, A. Wieck, and M.Bayer, Science 313, 341 /H208492006 /H20850.
8R. I. Dzhioev, B. P. Zakharchenya, V. L. Korenev, D. Gammon,
and D. S. Katzer, JETP Lett. 74, 182 /H208492001 /H20850.
9R. I. Dzhioev, V. L. Korenev, B. P. Zakharchenya, D. Gammon,
A. S. Bracker, J. G. Tischler, and D. S. Katzer, Phys. Rev. B 66,
153409 /H208492002 /H20850.
10P. Gilliot, D. Brinkmann, J. Kudrna, O. Cregut, R. Levy, A. Ar-
noult, J. Cibert, and S. Tatarenko, Phys. Rev. B 60, 5797 /H208491999 /H20850.
11E. A. Zhukov, D. R. Yakovlev, M. Bayer, G. Karczewski, T. Woj-
towicz, and J. Kossut, Phys. Status Solidi B 243, 878 /H208492006 /H20850.
12H. Hoffmann, G. V. Astakhov, T. Kiessling, W. Ossau, G. Karc-
zewski, T. Wojtowicz, J. Kossut, and L. W. Molenkamp, Phys.Rev. B 74, 073407 /H208492006 /H20850.
13G. V. Astakhov, T. Kiessling, D. R. Yakovlev, E. A. Zhukov, M.
Bayer, W. Ossau, B. P. Zakharchenya, G. Karczewski, T. Woj-towicz, and J. Kossut, Phys. Status Solidi B 243, 858 /H208492006 /H20850.
14R. Bratschitsch, Z. Chen, S. T. Cundiff, E. A. Zhukov, D. R.
Yakovlev, M. Bayer, G. Karczewski, T. Wojtowicz, and J. Kos-sut, Appl. Phys. Lett. 89, 221113 /H208492006 /H20850.
15Z. Chen, R. Bratschitsch, S. G. Carter, S. T. Cundiff, D. R. Yak-
ovlev, G. Karczewski, T. Wojtowicz, and J. Kossut, Phys. Rev. B
75, 115320 /H208492007 /H20850.
16J. Tribollet, E. Aubry, G. Karczewski, B. Sermage, F. Bernardot,
C. Testelin, and M. Chamarro, Phys. Rev. B 75, 205304 /H208492007 /H20850.
17D. R. Yakovlev, E. A. Zhukov, M. Bayer, G. Karczewski, T.
Wojtowicz, and J. Kossut, Int. J. Mod. Phys. B 21, 1336 /H208492007 /H20850.
18R. I. Dzhioev, B. P. Zakharchenya, V. L. Korenev, P. E. Pak, D. A.Vinokurov, O. V. Kovalenkov, and I. S. Tarasov, Phys. Solid
State 40, 1587 /H208491998 /H20850/H20851Fiz. Tverd. Tela /H20849S.-Peterburg /H2085040, 1745
/H208491998 /H20850/H20852.
19A. S. Bracker, E. A. Stinaff, D. Gammon, M. E. Ware, J. G.
Tischler, A. Shabaev, Al. L. Efros, D. Park, D. Gershoni, V. L.Korenev, and I. A. Merkulov, Phys. Rev. Lett. 94, 047402
/H208492005 /H20850.
20T. A. Kennedy, A. Shabaev, M. Scheibner, Al. L. Efros, A. S.
Bracker, and D. Gammon, Phys. Rev. B 73, 045307 /H208492006 /H20850.
21A. Shabaev, Al. L. Efros, D. Gammon, and I. A. Merkulov, Phys.
Rev. B 68, 201305 /H20849R/H20850/H208492003 /H20850.
22X. Marie, T. Amand, P. Le Jeune, M. Paillard, P. Renucci, L. E.
Golub, V. D. Dymnikov, and E. L. Ivchenko, Phys. Rev. B 60,
5811 /H208491999 /H20850.
23E. L. Ivchenko, Optical Spectroscopy of Semiconductor Nano-
structures /H20849Alpha Science, Harrow, 2005 /H20850.
24G. V. Astakhov, V. P. Kochereshko, D. R. Yakovlev, W. Ossau, J.
Nürnberger, W. Faschinger, and G. Landwehr, Phys. Rev. B 62,
10345 /H208492000 /H20850.
25P. Palinginis and H. Wang, Phys. Rev. Lett. 92, 037402 /H208492004 /H20850.
26Y. Shen, A. M. Goebel, G. Khitrova, H. M. Gibbs, and H. Wang,
Phys. Rev. B 72, 233307 /H208492005 /H20850.
27Optical Orientation , edited by F. Meier and B. P. Zakharchenja
/H20849North-Holland, Amsterdam, 1984 /H20850.
28S. E. Economou, L. J. Sham, Y. Wu, and D. G. Steel, Phys. Rev.
B74, 205415 /H208492006 /H20850.
29A. G. Aronov and E. L. Ivchenko, Fiz. Tverd. Tela /H20849S.-Peterburg /H20850
15, 231 /H208491973 /H20850/H20851Sov. Phys. Solid State 15, 160 /H208491973 /H20850/H20852.
30G. V. Astakhov, V. P. Kochereshko, D. R. Yakovlev, W. Ossau, J.
Nürnberger, W. Faschinger, G. Landwehr, T. Wojtowicz, G.Karczewski, and J. Kossut, Phys. Rev. B 65, 115310 /H208492002 /H20850.
31T. Amand, X. Marie, P. Le Jeune, M. Brousseau, D. Robart, J.
Barrau, and R. Planel, Phys. Rev. Lett. 78, 1355 /H208491997 /H20850.
32M. Dyakonov, X. Marie, T. Amand, P. Le Jeune, D. Robart, M.
Brousseau, and J. Barrau, Phys. Rev. B 56, 10412 /H208491997 /H20850.
33A. A. Sirenko, T. Ruf, M. Cardona, D. R. Yakovlev, W. Ossau, A.
Waag, and G. Landwehr, Phys. Rev. B 56,2 1 1 4 /H208491997 /H20850.
34M. Syperek, D. R. Yakovlev, A. Greilich, J. Misiewicz, M. Bayer,
D. Reuter, and A. D. Wieck, Phys. Rev. Lett. 99, 187401
/H208492007 /H20850.
35V. Ciulin, P. Kossacki, S. Haacke, J.-D. Ganiere, B. Deveaud, A.
Esser, M. Kutrowski, and T. Wojtowicz, Phys. Rev. B 62,
R16310 /H208492000 /H20850.ZHUKOV et al. PHYSICAL REVIEW B 76, 205310 /H208492007 /H20850
205310-16 |
PhysRevB.83.224509.pdf | PHYSICAL REVIEW B 83, 224509 (2011)
Two pseudogaps with different energy scales at the antinode of the high-temperature Bi 2Sr2CuO 6
superconductor using angle-resolved photoemission spectroscopy
K. Nakayama,1T. Sato,1Y. - M . X u ,2,*Z.-H. Pan,2,†P. Richard,3,4H. Ding,2,4H.-H. Wen,4,‡K. Kudo,5,§T. Sasaki,5
N. Kobayashi,5and T. Takahashi1,3
1Department of Physics, Tohoku University, Sendai 980-8578, Japan
2Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
3World Premier International Research Center, Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
4Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
5Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
(Received 26 May 2011; published 23 June 2011)
We performed high-resolution angle-resolved photoemission spectroscopy on single-layered cuprate
Bi2Sr2CuO 6to clarify the origin of the pseudogap. By using various photon energies, we succeeded in directly
observing two different pseudogaps with two different energy scales which coexist in the antinodal region:one reflects the d
x2−y2-wave pairing strength while the other has a larger energy scale suggesting an origin
distinct from superconductivity. The observed two-pseudogap behavior provides a key to fully understanding thepseudogap phenomena in cuprates.
DOI: 10.1103/PhysRevB.83.224509 PACS number(s): 74 .25.Jb, 74.20.Mn, 74 .72.−h, 79.60.−i
I. INTRODUCTION
The pseudogap observed in the excitation spectrum as
a suppression of spectral weight in the normal state ofcuprate superconductors
1has attracted much attention since
it is closely related to the mechanism of high- Tc(tran-
sition temperature) superconductivity. The opening of thepseudogap has been interpreted either as a precursor ofCooper pairing above T
cwithout phase coherence2or as the
development of some sort of ordered state which competeswith superconductivity.
3–5However, in spite of intensive
studies, the origin of the pseudogap is still highly controversial.This is largely due to the lack of consensus on the energyscale of the pseudogap. Some experiments pointed out thatthe pseudogap has a different energy scale from that ofthe superconducting (SC) gap, indicative of the presenceof two energy scales (possibly two distinct energy gaps)in the SC state.
6–11This two-gap behavior suggests that
the pseudogap has a competing nature and is not directlyrelated to superconductivity. It has been reported that thetwo-gap behavior is pronounced in low- T
csystems such as
heavily underdoped Bi 2Sr2CaCu 2O8(Bi2212), single-layered
Bi2Sr2CuO 6(Bi2201), and La 2−xSrxCuO 4(LSCO).6–11On
the other hand, even in the low- Tcsystems, there are some
recent experimental studies reporting the presence of a singleenergy scale where the SC gap below T
cand the pseudogap
above Tcshow an identical energy scale with no evidence
for the two-gap behavior,12–16strongly supporting a pairing
origin of the pseudogap. The apparent contradiction requiresfurther experimental investigation on the energy scale of thepseudogap in low- T
ccuprates to elucidate the origin of the
pseudogap.
In this paper, we report high-resolution angle-resolved pho-
toemission spectroscopy (ARPES) results on single-layeredcuprate Bi2201. By comparing ARPES data obtained withtwo different photon energies (8.437 and 21.218 eV), weclearly found two energy scales at the antinode below T
c.W e
demonstrate that these energy scales persist even above Tc,
suggesting the presence of two different types of pseudogapscoexisting in the same momentum ( k) region. We discuss the
implications of the present experimental results in relation tothe existing models as well as the origin of the pseudogap.
II. EXPERIMENTS
High-quality single crystals of slightly overdoped (Bi,Pb) 2
Sr2CuO 6+δ(Pb-Bi2201; Tc∼21 K) and nearly optimally doped
Bi2Sr1.6La0.4CuO 6+δ(La-Bi2201; Tc∼32 K) were grown by
the floating-zone17,18and the traveling-solvent floating-zone
methods,19respectively. High-resolution ARPES measure-
ments were performed using VG-SCIENTA SES2002 andMBS A1 photoemission spectrometers with xenon (Xe) andhelium (He) plasma discharge lamps.
20We used one of the
Xe-I lines ( hν=8.437 eV) and the He-I αline (21.218 eV)
to excite photoelectrons. The energy resolution was set at2–4 and 6–12 meV for the measurements with the Xe andHe lamps, respectively. The angular resolution was set at0.2
◦. We cleaved samples under ultrahigh vacuum better than
4×10−11Torr to obtain a clean and fresh sample surface for
ARPES measurements. The Fermi level ( EF) of samples was
referenced to that of a gold film evaporated onto the sampleholder.
III. RESULTS AND DISCUSSION
First we present ARPES data in the SC state. Figures 1(a)
and1(b) show the ARPES intensity plot at EFof Pb-Bi2201
as a function of the two-dimensional wave vector measuredwith the Xe-I and He-I αlines, respectively. While the ARPES
intensity distribution in the kspace is different between
these plots likely due to matrix-element effects, we find anearly identical Fermi-surface shape (red curve) centered atthe (π,π) point as determined by tracing the Fermi wave
vector ( k
F) points. In the SC state, both the Xe-I and He-I α
spectra commonly show a holelike band crossing EFin the
nodal region [Figs. 1(c) and 1(d)] and a clear leading-edge
shift toward higher binding energy in the antinodal region[Figs. 1(e) and 1(f)]. Although these experimental results
224509-1 1098-0121/2011/83(22)/224509(4) ©2011 American Physical SocietyK. NAKAY AMA et al. PHYSICAL REVIEW B 83, 224509 (2011)
Binding Energy (meV)EF4080 120 EF40 80
Xe I
(8.437 eV)
(0, 0)(π, π)(a)
(π, π)
He Iα
(21.218 eV)
(0, 0)(b)
High Low
Intensity (arb. units)Pb-Bi2201
(c) (e)
(f)Xe I
He Iα (d)
FIG. 1. (Color online) (a) and (b) Plot of ARPES intensity at
EFfor Pb-Bi2201 ( Tc∼21 K) as a function of two-dimensional
wave vector measured at 10 K with the Xe-I ( hν=8.437 eV) and
the He-I α(21.218 eV) lines, respectively. The intensity at EFwas
obtained by integrating the spectra within ±15 meV with respect to
EF. Red curve represents the Fermi surface determined by smoothly
tracing the experimentally determined kFpoints. (c) and (d) ARPES
spectra measured at 10 K along the orange, left arrows shown in (a)and (b), respectively. (e) and (f) Same as (c) and (d), but measured
along the pink, right arrows.
suggest the similarity of the basic electronic structure between
the He-I αand Xe-I spectra, a closer look further reveals
marked differences in the gap behavior.
Figures 2(a) and2(b) display symmetrized ARPES spectra
of Pb-Bi2201 measured at kFpoints with various Fermi-surface
angles φat 10 K (below Tc) with the Xe-I and He-I αlines,
respectively. The gap size, defined by the energy separationbetween the peak position and E
F, monotonically increases on
going from the nodal (bottom in the panel) to the antinodal(top) regions, which is consistent with the anisotropic gapopening in the SC state. As also visible in Fig. 2(d), there is a
striking difference between the He- and Xe-spectra on the peakposition around the antinode, i.e., the peak in the He spectrumis located at much higher binding energy than that of the Xespectrum (18.5 and 11.5 meV , respectively). We use /Delta1
Xeand
/Delta1Heto note the gap size obtained from the Xe and He spectra,
respectively. In Fig. 2(e), we plot estimated /Delta1Xeand/Delta1Heat
10 K at various kFpoints. The kdependence of /Delta1Xeis well
fi t t e db yt h e dx2−y2-wave gap function with a small admixture
of a higher-order component, representing the energy scale ofthe SC gap.
16Although /Delta1Heshows a quantitative agreement
with/Delta1Xenear the node, it gradually deviates from /Delta1Xewith
approaching the antinode. A similar trend is also observedin La-Bi2201, whose T
cvalue (32 K) is much higher than
Pb-Bi2201 (21 K). As shown in Fig. 2(f), the difference
between /Delta1Xeand/Delta1He(arrow vs dashed line) exceeds 20 meV
at the antinode, whereas /Delta1Xeand/Delta1Heappear identical near the
node. As visible in Fig. 2(g), the observed significant deviation
of/Delta1Hefrom the ideal dx2−y2-wave gap function appears similar
to previous ARPES results which have been interpreted withtwo types of energy gaps in different kregions, i.e., (i) the SC
gap which dominates the gap symmetry near the node , and (ii)He Iα Xe I (a) (b)
φ
(π, 0)(0, π)
(0, 0)(π, π)(c)
Δ (meV)
(π, 0)20
040
30
10
(0, 0)(π, π)La-Bi2201
(Tc ~ 32 K)
(π, 0)10
020
15
5Δ (meV)
(0, 0)(π, π)Intens ity (ar b. units)
Energy relative to EF40 20 0 20 40He IαXe I(d)
φ
T = 10 KPb-Bi2201
(Tc ~ 21 K)02 0 4 02040
Energy relative to EF02 0 4 02040Intens ity (ar b. units) Intens ity (ar b. units)
Energy relative to EF80 40 0 40 80He IαXe I(f)
φ
T = 10 K(e)
(g)ΔHe
ΔXe
ΔHe
ΔXe
FIG. 2. (Color online) (a) and (b) kdependence of the Pb-Bi2201
(Tc∼21 K) ARPES spectra at 10 K, measured at various kFpoints
shown by circles in (c), using the Xe-I and He-I αlines, respectively.
The coloring of the spectra is the same as that of the circles in (c).
Each spectrum has been symmetrized with respect to EFto remove the
effect of the Fermi-Dirac distribution function. (c) Schematic Fermi
surface and definition of the Fermi-surface angle φ. (d) Comparison
of symmetrized spectra at the antinodal kFpoint measured with the
Xe-I and He-I αlines. The black arrow and the dashed line denote
the peak position for the Xe-I and He-I αspectrum, respectively.
(e)kdependence of the gap size at 10 K obtained with the Xe-I and
He-Iαlines ( /Delta1Xeand/Delta1He). The gap size was determined by fitting
the symmetrized spectra with the phenomenological gap functionconvoluted with the energy resolution.
21(f) and (g) Same as (d) and
(e) but measured in La-Bi2201 ( Tc∼32 K).
the large gap which develops near the antinode .6–8The good
agreement between /Delta1Heand/Delta1Xenear the node in the present
ARPES result is consistent with the pairing nature of the gaparound the node in the He spectra. In addition, the markeddifference between /Delta1
Heand/Delta1Xenear the antinode provides a
direct evidence for the presence of two energy gaps below Tc(a
small gap and a large gap) even in the samekregion, although
we cannot completely rule out a possible kzdependence
of the gap size to account for the difference between /Delta1He
and/Delta1Xe. This observation should be strictly distinguished
224509-2TWO PSEUDOGAPS WITH DIFFERENT ENERGY SCALES ... PHYSICAL REVIEW B 83, 224509 (2011)
(b)
24 KHe IαXe I
ΔXe (24 K)ΔXe (10 K)
ΔHe (24 K)ΔHe (10 K)Pb-Bi2201 ( Tc ~ 21 K)(a)
200
Energy (meV) Fermi surface angle φ (deg.)01 025Gap size Δ (meV)20Intens ity (a. u.) Intens ity (a. u.)(c)
40 20402004 0 204015
10
5
0
20 30 40
FIG. 3. (Color online) (a) and (b) Photon-energy dependence of
symmetrized kFspectra in Pb-Bi2201 ( Tc∼21 K) at 24 K measured
atφ∼30◦and∼0◦, respectively. (c) kdependence of the gap size
in the SC ( T=10 K) and pseudogap (24 K) states of Pb-Bi2201
measured with the Xe-I and He-I αlines.
from previous works reporting the “two gaps,”6–8in the
sense that two gaps appear simultaneously at the antinodalregion.
To clarify how these gaps evolve into the pseudogap above
T
c, we have performed ARPES measurements at 24 K (just
above Tc) on Pb-Bi2201 with the Xe-I and He-I αlines.
As seen in both sets of data in Fig. 3(a), the symmetrized
spectrum near the node shows a single peak at EF, while the
spectrum at the antinode exhibits spectral weight suppressionin the vicinity of E
F, a signature of the pseudogap opening
[Fig. 3(b)]. In the antinodal region, the characteristic energy
scales of the pseudogap are ∼12 and ∼20 meV for the Xe and
He spectrum, respectively, which are similar to the values of/Delta1
Xeand/Delta1Hebelow Tcin the antinodal region, as shown in
Fig. 3(c). It is thus inferred that there exist two pseudogaps
above Tcwith precursor-pairing and unknown origin which
smoothly evolve from the dx2−y2-wave SC gap and the larger
gap below Tc, respectively. It is emphasized that, although a
few previous ARPES results suggested a two-pseudogap-likebehavior,
22,23the present ARPES result directly demonstrates
for the first time the presence of two energy scales at the
antinode.
The present observation solves the contradiction among
recent ARPES experiments. While some studies supportedthe pairing origin of the pseudogap,
12–16others pointed out
that the pseudogap is not directly related to the pairing.6–9
Such difference is naturally understood by taking into account
the presence of two pseudogaps. Namely, the former studiesdetected only the small gap and the latter observed mostlythe large gap, essentially because of the difference in theexperimental conditions such as the photon energy. In fact,the previously reported pseudogap values of ∼15 meV
13,15
and∼35 meV7,9for La-Bi2201 (which differ among different
groups) agree well with the maximum values of /Delta1Xeand/Delta1He,
respectively. In addition, the difference of the gap anisotropyin the pseudogap phase of La
1.875Ba0.125CuO 422,24can also be
explained within the two-pseudogap picture.
Finally, we discuss the implication of the observed photon-
energy dependence. We revealed that the measurements usingthe Xe-I ( hν=8.437 eV) and the He-I α(21.218 eV) lines
are sensitive to the small gap and the large gap, respectively.One explanation of such behavior is that the two differentgaps suffer different matrix-element effects during the photo-excitation process and they can be selectively observed byspecific conditions of the photon energy. This explanation maybe valid if there are two different bands producing the smalland the large gaps, since the bilayer-split bands in Bi2212obey different matrix elements.
25On the other hand, Bi2201
is a single-layered system and there would be a single bandnearE
F. In this case, the appearance of two energy scales on
the single coherent quasiparticle band may be explained bythe idea that the large gap is not a complete gap but rathera soft gap,
9and the remaining density of states within the
large gap contributes to the formation of the small gap. It isalso possible to attribute the large and the small gaps to theincoherent and the coherent parts of the spectral function. Ineither case, the two gaps basically arise from a single-bandspectral function and their intensity ratio would not depend onthe photon energy. Hence we think that the present observationmay not be simply explained by the matrix-element effect.Another explanation is that the difference between the He andXe spectra originates from the surface and/or bulk sensitivity.In this case, it is inferred that the large gap, which seemsnot directly related to the superconductivity, is either (i) anextrinsic feature stabilized at the surface or (ii) an intrinsicfeature in bulk with much pronounced influence at the surface.On the other hand, the small gap, which is closely related tothe pairing, would reflect bulk properties because electronsexcited with the Xe-I line have a relatively long escape depth(20–40 ˚A) as compared to that excited with the He-I αline
(5–10 ˚A)
20. The bulk nature of the small gap is also supported
by a basic agreement between /Delta1Xein La-Bi2201 ( ∼14 meV)
and an energy scale observed in the B1gRaman spectrum
(∼17 meV).26While most of previous results on
Bi22017,9,15,16,23agree with the expectation that the spectral
weight related to the small gap feature is enhanced as thephoton energy is lowered (i.e., the photoelectron escape depthbecomes longer), there is one exceptional result which showsthe small gap feature at the antinode even with hν=22.5e Vi n
optimally doped La-Bi2201 with a zero residual resistivity.
13
Since the authors reported that the small gap disappearsin another optimally doped sample with a finite residualresistivity,
13the disorder effect may be an essential ingredient
in suppressing the small gap component and also in causing thedifference in the electronic states between surface and bulk.In any cases, the pairing interaction is essential in realizingthe origin of the pseudogap, and we conclude that the scenarioassuming the opening of a single competing pseudogap isinsufficient for the correct understanding of the pseudogapphenomena in cuprates.
IV . CONCLUSION
In conclusion, we performed a high-resolution ARPES
study of Bi2201 by using the Xe and He discharge lamps.The result clearly shows the presence of two energy scalesin the antinodal region below and above T
c, indicating the
existence of two different pseudogaps. We have concluded that
224509-3K. NAKAY AMA et al. PHYSICAL REVIEW B 83, 224509 (2011)
the smaller pseudogap originates from the precursor pairing
above Tc, while the larger pseudogap is not directly related
to the superconductivity. The present findings put a strongconstraint in modeling the pseudogap phenomena of cuprates.ACKNOWLEDGMENTS
This work was supported by grants from JST-CREST, JSPS,
MEXT of Japan, NSF of US, and MOST of China.
*Present address: Materials Sciences Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA.
†Present address: Condensed Matter Physics and Materials ScienceDepartment, Brookhaven National Lab, Upton, NY 11973, USA.
‡Present address: National Laboratory of Solid State Microstructuresand Department of Physics, Nanjing University, Nanjing 210093,China.
§Present address: Department of Physics, Okayama University,Okayama 700-8530, Japan.
1H. Ding, T. Yokoya, J. C. Campuzano, T. Takahashi, M. Randeria,M. R. Norman, T. Mochiku, K. Kadowaki, and J. Giapintzakis,Nature (London) 382, 51 (1996).
2V . J. Emery and S. A. Kivelson, Nature (London) 374, 434 (1995).
3C. M. Varma, P h y s .R e v .B 55, 14554 (1997).
4S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys.
Rev. B 63, 094503 (2001).
5T. Das, R. S. Markiewicz, and A. Bansil, P h y s .R e v .B 77, 134516
(2008).
6K. Tanaka, W. S. Lee, D. H. Lu, A. Fujimori, T. Fujii, Risdiana,I. Terasaki, D. J. Scalapino, T. P. Devereaux, Z. Hussain, and Z.-X.Shen, Science 314, 1910 (2006).
7T. Kondo, T. Takeuchi, A. Kaminski, S. Tsuda, and S. Shin, Phys.
Rev. Lett. 98, 267004 (2007).
8K. Terashima, H. Matsui, T. Sato, T. Takahashi, M. Kofu, and
K. Hirota, P h y s .R e v .L e t t . 99, 017003 (2007).
9J.-H. Ma, Z.-H. Pan, F. C. Niestemski, M. Neupane, Y .-M. Xu,
P. Richard, K. Nakayama, T. Sato, T. Takahashi, H.-Q. Luo,L. Fang, H.-H. Wen, Z. Wang, H. Ding, and V . Madhavan, Phys.
Rev. Lett. 101, 207002 (2008).
10M. L. Tacon, A. Sacuto, A. Georges, G. Kotliar, Y . Gallais,
D. Colson, and A. Forget, Nature Phys. 2, 537 (2006).
11M. C. Boyer, W. D. Wise, K. Chatterjee, M. Yi, T. Kondo,
T. Takeuchi, H. Ikuta, and E. W. Hudson, Nature Phys. 3, 802
(2007).
12U. Chatterjee, M. Shi, D. Ai, J. Zhao, A. Kanigel, S. Rosenkranz,H. Raffy, Z. Z. Li, K. Kadowaki, D. G. Hinks, Z. J. Xu, J. S. Wen,G .G u ,C .T .L i n ,H .C l a u s ,M .R .N o r m a n ,M .R a n d e r i a ,a n dJ .C .Campuzano, Nature Phys. 6, 99 (2010).13J. Wei, Y . Zhang, H. W. Ou, B. P. Xie, D. W. Shen, J. F. Zhao,
L. X. Yang, M. Arita, K. Shimada, H. Namatame, M. Taniguchi,Y . Yoshida, H. Eisaki, and D. L. Feng, Phys. Rev. Lett. 101, 097005
(2008).
14M. Shi, J. Chang, S. Pailh ´es, M. R. Norman, J. C. Campuzano,
M. M ˚ansson, T. Claesson, O. Tjernberg, A. Bendounan, L. Patthey,
N. Momono, M. Oda, M. Ido, C. Mudry, and J. Mesot, Phys. Rev.
Lett.101, 047002 (2008).
15J. Meng, W. Zhang, G. Liu, L. Zhao, H. Liu, X. Jia, W. Lu,
X. Dong, G. Wang, H. Zhang, Y . Zhou, Y . Zhu, X. Wang,Z. Zhao, Z. Xu, C. Chen, and X. J. Zhou, P h y s .R e v .B 79, 024514
(2009).
16K. Nakayama, T. Sato, Y . Sekiba, K. Terashima, P. Richard,T. Takahashi, K. Kudo, N. Okumura, T. Sasaki, and N. Kobayashi,Phys. Rev. Lett. 102, 227006 (2009).
17K. Kudo, T. Nishizaki, N. Okumura, and N. Kobayashi, Physica C
463-465 , 40 (2007).
18K. Kudo, Y . Miyoshi, T. Sasaki, T. Nishizaki, and N. Kobayashi,
J. Phys. Soc. Jpn. 75, 124710 (2006).
19H.-Q. Luo, P. Cheng, L. Fang, and H.-H. Wen, Supercond. Sci.
Technol. 21, 125024 (2008).
20S. Souma, T. Sato, T. Takahashi, and P. Baltzer, Rev. Sci. Instrum.
78, 123104 (2007).
21M. R. Norman, M. Randeria, H. Ding, and J. C. Campuzano, Phys.
Rev. B 57, R11093 (1998).
22R.-H. He, K. Tanaka, S.-K. Mo, T. Sasagawa, M. Fujita, T. Adachi,
N. Mannella, K. Yamada, Y . Koike, Z. Hussain, and Z.-X. Shen,Nature Phys. 5, 119 (2009).
23T. Kondo, Y . Hamaya, A. D. Palczewski, T. Takeuchi, J. S. Wen,
Z. J. Xu, G. D. Gu, J. Schmalian, and A. Kaminski, Nature Phys. 7,
21 (2010).
24T .V a l l a ,A .V .F e d o r o v ,J .L e e ,J .C .D a v i s ,a n dG .D .G u , Science
314, 1914 (2006).
25A. A. Kordyuk, S. V . Borisenko, T. K. Kim, K. A. Nenkov,
M. Knupfer, J. Fink, M. S. Golden, H. Berger, and R. Follath,Phys. Rev. Lett. 89, 077003 (2002).
26S. Sugai, H. Suzuki, Y . Takayanagi, T. Hosokawa, and N. Hayamizu,
Phys. Rev. B 68, 184504 (2003).
224509-4 |
PhysRevB.76.045301.pdf | Plasmon-cyclotron resonance in two-dimensional electron gas confined
at the GaN/Al xGa1−xN interface
Agnieszka Wolos
Institute of Semiconductor and Solid State Physics, Johannes Kepler Universität, Altenbergerstrasse 69, A-4040 Linz, Austria
and Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, 02-668 Warszawa, Poland
Wolfgang Jantsch
Institute of Semiconductor and Solid State Physics, Johannes Kepler Universität, Altenbergerstrasse 69, A-4040 Linz, Austria
Krzysztof Dybko and Zbyslaw Wilamowski
Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, 02-668 Warszawa, Poland
Czeslaw Skierbiszewski
Institute of High Pressure Physics, Polish Academy of Sciences, ulica Sokolowska 29/37, 01-142 Warszawa, Poland
/H20849Received 13 April 2007; revised manuscript received 23 May 2007; published 2 July 2007 /H20850
Edge magnetoplasma modes are studied in the two-dimensional electron gas confined at the
GaN/Al xGa1−xN interface using standard microwave resonance spectroscopy. The position and shape of the
resonance line are described by the theory for the dimension-dependent plasmon-cyclotron coupling and theDrude model of momentum relaxation. The analysis of the resonance line shape provides a contactless methodfor the determination of the sheet electron concentration and the mobility of the two-dimensional electron gas.In addition, we observe Shubnikov–de Haas oscillations using the same microwave resonance spectrometry.We compare values for the cyclotron and quantum mobilities obtained from the plasmon-cyclotron linewidthand the magnetic field dependence of the Shubnikov–de Haas signal, respectively.
DOI: 10.1103/PhysRevB.76.045301 PACS number /H20849s/H20850: 73.40.Kp, 76.30.Pk, 76.40. /H11001b
I. INTRODUCTION
Early work on the two-dimensional electron gas
/H208492DEG /H20850in silicon inversion layers1or in GaAs-based
heterostructures2–5showed that when the plasma frequency
of the 2D electrons approaches the cyclotron frequency, thetwo modes hybridize into a coupled excitation. Neglectingretardation effects
4for a disk-shaped sample, the plasmon-
cyclotron spectrum is characterized by two branches, whichat zero magnetic field converge at the frequency of theplasma oscillations. For high magnetic fields, the upper mag-netoplasma branch, a cyclotronlike mode, asymptotically ap-proaches the cyclotron frequency, while the lower branch,the so-called edge mode, approaches zero.
In order to correctly evaluate parameters from the cyclo-
tron resonance spectrum, particularly the value of the effec-tive mass, it is important to consider the plasmon-cyclotroncoupling. The latter becomes important when the resonancefrequency in the experiment is comparable to both theplasma and the cyclotron frequency.
In this paper, we present the results for the two-
dimensional electron gas confined at the GaN/AlGaN inter-face. Using a standard electron spin resonance /H20849ESR /H20850spec-
trometer, we have observed a broad line resembling acyclotron resonance, whose position corresponds, however,to an apparent cyclotron mass of 1.3 m
0. This value is far
from the effective mass of conduction electrons in GaN,0.2m
0, previously determined for GaN/AlGaN heterostruc-
tures in cyclotron resonance experiments.6–10The shift of the
observed line indicates plasmon-cyclotron coupling, and thisis further corroborated by the detailed analysis of the spectra.We show that the observed resonance is due to the edge
magnetoplasma mode.
Plasmon-cyclotron coupling has been extensively studied
in GaAs-based heterostructures, for which high sheet elec-tron mobilities of up to 10
6cm2/V s can be achieved.3–5The
high mobility of the 2D electrons is an essential prerequisitefor studying this type of excitations, as the electron scatter-ing time can be directly related to the linewidth of the reso-nance peak.
11The higher the mobility of the 2D electrons,
the narrower the linewidth.
Mobilities reaching 106cm2/V s have not been achieved
yet for GaN/AlGaN. However, thanks to the progress inGaN-based technology, due in particular to the use of nativeGaN substrates, it is possible to obtain samples with mobili-ties as high as 10
4–105cm2/V s.12,13Such samples have
been used for studying properties of the two-dimensionalmagnetoplasma in this paper.
II. SAMPLES AND EXPERIMENTAL DETAILS
Heterostructures were grown on semi-insulating GaN
bulk substrates14on the Ga-polarity /H208490001 /H20850surface by
plasma-assisted molecular beam epitaxy /H20849MBE /H20850. The surface
of the substrates was prepared for the growth by the standardmethod including mechanical polishing followed by mecha-nochemical polishing. The MBE-grown heterostructure con-sisted of a 0.9
/H9262m GaN layer followed by a 25-nm-thick
Al0.09Ga0.91N barrier and a 3-nm-thick GaN cap layer.12,13
The polarization-induced 2DEG at the GaN/AlGaN inter-
face exhibits high Hall mobility, of the order of /H9262H
=75 000–80 000 cm2/V s, measured at 4.2 K, owing to thePHYSICAL REVIEW B 76, 045301 /H208492007 /H20850
1098-0121/2007/76 /H208494/H20850/045301 /H208497/H20850 ©2007 The American Physical Society 045301-1high quality and low dislocation density of the substrate
used. The screw dislocation density is typically as low as10
4–105cm−2in these samples. The sheet carrier density
amounts to n2D=2/H110031012cm−2.
Properties of the 2DEG at the GaN/AlGaN interface were
studied by standard microwave resonance spectrometry us-ing a Bruker ELEXSYS E-580 ESR spectrometer. The spec-trometer was operating at a frequency of f=9.48 GHz with a
rectangular TE
102cavity, where the sample was placed in the
maximum of the microwave magnetic field. Coupling to thecyclotron-plasmon modes occurs due to inevitable in-planecomponents of the microwave electric field owing to the fi-
nite sample size and fringe fields in the perturbed cavity. Thetemperature in the experiment was lowered down to 2 Kusing a continuous-flow Oxford cryostat. The static magneticfieldBwas swept up to 1.5 T.
III. EXPERIMENTAL RESULTS
A. Shubnikov–de Haas oscillations
As shown in Fig. 1, Shubnikov–de Haas /H20849SdH /H20850oscilla-
tions are observed due to non-resonant microwave absorp-tion without any electrical contacts to the sample. Here, themagnetic field Bwas modulated with the amplitude of 5 G at
100 kHz. The modulation parameters were selected to avoiddistortion of the SdH signal. Due to the modulation of themagnetic field, the spectra recorded represent the first deriva-tive of the microwave absorption vs the magnetic field.
In Fig. 1, the measured Shubnikov–de Haas signal is plot-
ted as a function of the inverse magnetic field B
−1. The spec-
trum depends only on the perpendicular component of theapplied magnetic field /H20849not shown in the figure /H20850, confirming
the two-dimensional character of the observed spectrum. Theonset of the oscillations occurs at a magnetic field of about1 T, which corresponds to a Landau level filling factor ashigh as
/H9263=39. Fast Fourier transform /H20849FFT /H20850of the signal vs
the inverse magnetic field shows one dominant oscillationfrequency F/H20849see inset to Fig. 1/H20850due to a single occupiedelectrical subband in the GaN quantum well. From the FFT
peak position, we obtain a sheet electron concentrationofn
2D=1.95 /H110031012cm−2for this GaN/AlGaN sample
/H20849n2D=2eF/h/H20850, in good agreement with Hall data obtained on
a companion sample grown in the same process.
The amplitude of the SdH oscillations for low magnetic
fields /H20849/H6036/H9275C/H11270kBT/H20850measured by the dc resistivity /H9267is propor-
tional to15–17
/H9267/H20849B/H20850/H11011/H9252T
Bexp/H20875−/H9252/H20849T+TD/H20850
B/H20876cos/H208732/H9266hn2D
2e1
B/H20874, /H208491/H20850
where /H9252=2/H92662kBm*//H6036e. Here, n2Dis the sheet electron con-
centration, m*stands for the effective mass of the conduction
electrons, which is m*=0.2 m0for GaN,6,8,18and TDis the
Dingle temperature related to the quantum mobility by
/H9262q=/H9266//H9252TD.
The experimental data presented in Fig. 1show oscilla-
tions of the ac conductivity measured at a frequency of
/H9275=0.6/H110031011s−1. In Ref. 19it has been shown that Eq. /H208491/H20850is
also valid for the microwave-detected SdH effect /H20849at low
magnetic fields /H20850, resulting in values for the quantum mobility
that are consistent with those of dc resistivity measurements.
The first derivative of Eq. /H208491/H20850was fitted to the SdH spec-
trum recorded for a GaN/AlGaN sample, yielding a Dingletemperature of T
D=1.25 K, which corresponds to a quantum
mobility of /H9262q=8600 cm2/V s. This value is by a factor of 9
smaller than the Hall mobility obtained from magnetotrans-port measurements /H20849
/H9262H=75 000–80 000 cm2/V s /H20850and by a
factor of 18 smaller than the cyclotron mobility determined
in the resonance experiment /H20849/H9262CR=152 000 cm2/V s /H20850,a s
will be presented in Sec. III B.
The quantum mobility /H9262q, as determined from the mag-
netic field dependence of the amplitude of SdH oscillations,is usually lower than the transport mobility
/H9262Hdetermined
from Hall measurements. This is due to the fact that /H9262qis
related to the mean time a carrier remains in a particular statebefore being scattered to a different state, while the transportmobility is related to the mean time a carrier moves in aparticular direction. Thus, in calculating quantum lifetimesall scattering events are weighted equally, while in the caseof transport lifetimes weighting by a factor of /H208511−cos /H20849
/H9272/H20850/H20852
/H20849where /H9272is the scattering angle /H20850has to be included. In other
words, transport lifetimes emphasize the importance of large-over small-angle scattering events.
20
The low-field amplitude of SdH oscillations is very sen-
sitive to fluctuations of the Fermi level. It has been shownearlier for GaN/AlGaN heterostructures /H20849grown on GaN
templates prepared on sapphire substrates /H20850that in the pres-
ence of fluctuations of the sheet electron density by as littleas a few percent, the value of
/H9262qdetermined from the SdH
oscillations is substantially lowered with respect to the truequantum mobility. In these samples,
/H9262qdid not exceed
4000 cm2/V s.21The higher value of /H9262q=8600 cm2/Vs de-
termined for our homoepitaxial layers indicates higher ho-mogeneity of the 2D electron gas. The obtained value for
/H9262q
is also closer here to the actual quantum mobility.
The cyclotron mobility /H9262CRdeduced from a half-width at
half maximum of the cyclotron resonance line has been re-FIG. 1. Microwave-detected Shubnikov–de Haas oscillations
measured for a GaN/AlGaN sample /H20849B"c/H20850, solid line. Fit of Eq. /H208491/H20850
with TD=1.25 K, dotted line. The background originating from a
magnetoplasma resonance was subtracted. Inset: fast Fourier trans-form spectrum of the SdH signal. It shows one dominant oscillationfrequency, corresponding to a sheet electron concentration ofn
2D=1.95 /H110031012cm−2.WOLOS et al. PHYSICAL REVIEW B 76, 045301 /H208492007 /H20850
045301-2cently investigated for GaN/AlGaN samples having a sheet
electron concentration in the range of /H208491–4.5 /H20850/H110031012cm−2
and transport mobilities of 10 000–40 000 cm2/V s.21It has
been shown that /H9262CRcoincides with the transport mobility
/H9262Hfor lower electron densities, while for higher densities
/H9262CRbecomes much lower than the /H9262H. The cyclotron mobil-
ity has been identified with the quantum mobility, and thediscrepancy at higher electron densities has been attributedto dominant large-angle scattering in the samples investi-gated. It appears rather natural, however, to equate cyclotronand transport mobilities, as they both arise from the sameDrude formalism. Then, the discrepancy between
/H9262CRand
/H9262Hobserved at higher sheet electron densities should be ac-
counted for by other mechanisms broadening the cyclotronresonance line, e.g., electron-electron interaction and inho-mogeneities in the sample. In any case, n
2D=2/H110031012cm−2,
which is the concentration of our homoepitaxial samples, isstill in the range where
/H9262CRand/H9262Hgive the same values in
Ref. 21. In our experiment, however, we still observe a dis-
crepancy /H20849/H9262CRis higher than /H9262H/H20850, which will be discussed in
Sec. III B.
Sample illumination changes the carrier concentration,
which also leads to a change of the /H9262CRand/H9262q. Details are
described in Sec. III B.
B. Line shape of the coupled plasmon-cyclotron resonance
Figure 2shows a strong electric dipole-type resonance
/H20849first derivative of microwave absorption /H20850recorded in a
GaN/AlGaN sample. The spectra were measured for variousorientations of the magnetic field B, clearly showing a com-
mon dependence on the perpendicular component of the ap-
plied field, Bcos
/H9258, as is usual for two-dimensional electrons.
Here, /H9258is the angle between Band the GaN caxis, which is
perpendicular to the sample plane. The resonance position ofthe observed line is shifted, however, toward higher mag-netic fields, Bcos
/H9258=0.42 T, with respect to the pure cyclo-
tron resonance expected for the cyclotron mass of GaN elec-trons, Bcos
/H9258=0.07 T. Such a shift may originate from the
coupling of the cyclotron motion to plasma oscillations.1–5,22It has been observed, e.g., for GaAs-based 2D hetero-
structures, that the cyclotron motion of an electron and theplasma oscillations hybridize when the frequencies of thetwo modes approach each other. The two resonance frequen-cies for plasmon-cyclotron coupling, the upper cyclotronlikebranch and the lower edge mode, are then given by
2,22
/H9275res±=±/H9275c
2+/H20881/H9275p2+/H20873/H9275c
2/H208742
, /H208492/H20850
where /H9275Cstands for the cyclotron frequency and /H9275pis the
plasma frequency, which scales with the plasmon wave vec-tor and thus with the sample size. For a disk-shaped samplewith a radius R, this relation is given by
4,23
/H9275p2=n2De2
m*/H208491+/H9255/H20850/H92550R. /H208493/H20850
Here, n2Dis the sheet electron concentration, /H92550the vacuum
permittivity, and /H9255a static dielectric constant, which equals
10.4 for GaN.24The dependence of the coupled plasmon-
cyclotron resonance frequencies on the magnetic field isplotted in the inset in Fig. 3.
It may be worthwhile to note here that Eq. /H208492/H20850leads to a
1/cos
/H9258dependence of the resonance magnetic field, as it is a
case of a pure cyclotron resonance. It is thus not possible todistinguish pure cyclotron resonance and the plasmon-cyclotron coupling only from the angular analysis. Forplasmon-cyclotron coupling, the resonance position /H20849for the
edge mode /H20850is shifted toward higher magnetic fields, and the
shift depends on both the sample size and the sheet electronconcentration.
For a sample with a rectangular shape, two different
plasma frequencies exist, which are related to the samplelength and width. These modes are coupled in the presenceof a magnetic field. In our experiment, we had rectangularsamples. Due to the orientation of the sample inside the mi-crowave cavity, however, the applied linear polarized micro-FIG. 2. Plasma-cyclotron resonances in GaN/AlGaN, measured
at various angles /H9258between the external magnetic field Band the
direction normal to the sample plane, are plotted as a function ofBcos
/H9258. Inset: resonance magnetic field vs tilt angle /H9258/H20849points /H20850fol-
lowing a 1/cos /H9258dependence /H20849solid line /H20850.FIG. 3. Dots: First derivative of microwave absorption due to
the plasmon-cyclotron resonance measured in a GaN/AlGaNsample /H20849
/H9258=0, sample dimension 3.5 /H110034m m2/H20850. Some data points
have been omitted for the clarity of the picture. Solid line: leastsquares fit of Eq. /H208496/H20850with the best fit parameters listed in Table I.
The inset shows the upper and the lower branch of the coupledplasma-cyclotron resonance calculated according to Eq. /H208492/H20850. The
arrows mark plasma and microwave frequencies.PLASMON-CYCLOTRON RESONANCE IN TWO- … PHYSICAL REVIEW B 76, 045301 /H208492007 /H20850
045301-3wave electric field was always perpendicular to the longer
edge of a sample, so that only one plasma frequency relatedto the smaller sample dimension could be excited here. Toanalyze the recorded resonance, we have thus used a one-mode approximation, which, as shown below, gives a goodagreement with experimental results.
To describe the line shape of the ac electric absorption of
the coupled resonance recorded in our GaN/AlGaN hetero-structures, we assumed a Lorentzian dependence for the ab-sorption of circular polarized electromagnetic waves in thefrequency domain, with the two resonance frequencies givenby Eq. /H208492/H20850,
F
±/H20849/H9275,B/H20850=A/H9270C
/H92661
1+/H9270C2/H20851/H9275−/H9275res±/H20849B/H20850/H208522, /H208494/H20850
for/H9268+/H20849cyclotron resonance active /H20850and/H9268−/H20849cyclotron reso-
nance inactive /H20850polarization, respectively.22The function de-
cribed by Eq. /H208494/H20850reflects the ac Drude conductivity,6where
the scattering time /H9270Cis related to the mobility /H9262CRby the
relation
/H9262CR=e
m*/H9270C. /H208495/H20850
In order to fit the shape of the line recorded in our reso-
nance experiment, we need to take into account the linearpolarization of the absorbed microwaves and the fact that wemeasure the first derivative of the absorption vs magneticfield due to the use of modulation of Band lock-in detection.
We treat the linear polarization as a sum of
/H9268+and/H9268−polar-
ization. This leads to a final expression, which reproducesthe characteristic asymmetric line shape of the magneto-plasma resonance in the magnetic field domain,
f/H20849
/H9275,B/H20850=1
2/H11509
/H11509B/H20851F+/H20849/H9275,B/H20850+F−/H20849/H9275,B/H20850/H20852. /H208496/H20850
Figure 3shows a fit of Eq. /H208496/H20850to the spectrum recorded at
/H9258=0. The best fit parameters in this case are /H9275p
=/H208491.63±0.1 /H20850/H110031011s−1and /H9270C=/H208491.73±0.1 /H20850/H1100310−11s−1,
which correspond to /H9262CR=/H20849152 000±8000 /H20850cm2/V s. The
microwave frequency was set to the experimental value of
/H9275res=0.6/H110031011s−1.
The inset in Fig. 3shows two branches of magnetoplasma
modes calculated according to Eq. /H208492/H20850with parameters ob-
tained from the fit. As can be clearly seen, the fitted plasmafrequency is substantially higher than the microwave fre-quency, so the main contribution to the line shape of theobserved resonance originates from the edge magnetoplasmabranch. Due to the finite width of the resonance line, thecyclotronlike branch contributes slightly to the spectrum formagnetic fields close to B=0.
The obtained cyclotron mobility is by a factor of 2 higher
than the transport mobility determined from a Hall experi-ment. As argued in Sec. III A the cyclotron resonance in theGaN/AlGaN heterostructure should give a cyclotron mobil-ity close to that detemined from transport, at least forsamples with low sheet electron concentration.
21Thediscrepancy between the two mobilities values observed in
our experiment can be explained in terms of an ac characterof the cyclotron resonance measurement, which becomesmore pronounced for high-mobility samples. The main con-tribution of the electron gas /H20849oscillating at the resonance fre-
quency
/H9275res=0.6/H110031011s−1/H20850stems from electrons confined in
high-quality areas of the sample, where scattering potentialsoriginating from residual and remote impurities are wellscreened. The mobility measured here in a cyclotron reso-nance experiment can thus be closer to the value limited byintrinsic properties of the GaN host crystal, namely, byacoustic phonon scattering. Moreover, the interpretation ofdc conductivity measurements assumes a uniform currentdistribution. When the current is not uniform, e.g., due tolong range potential fluctuations, the resulting Hall mobilityis smaller than that corresponding to the scattering time.
C. Dependence of the plasmon-cyclotron resonance on the
sample size
The plasma frequency in a 2D electron gas depends on the
sample dimensions, as expected according to Eq. /H208493/H20850.T o
show that the observed resonance in GaN/AlGaN is reallysample size dependent, we have measured a set of samplescleaved from the same wafer, having equal sheet electronconcentrations and mobilities but different sizes. The threesamples measured had an approximately rectangular shape,with their bigger dimension equal to about 4 mm /H20849plasma
oscillations related to this dimension are not excited in ourexperiment /H20850and the smaller one equal to 3.5, 2.0, and
1.5 mm. The results are shown in Fig. 4. Indeed, with
decreasing sample size, the resonance is shifted towardhigher magnetic fields, reflecting an increase of theplasma frequency. All spectra can be fitted with verygood precision using Eq. /H208496/H20850, supporting the assignment to
the magnetoplasma resonance. The best fit parameters forall samples are listed in Table I. The obtained
/H9270C=const
=/H208491.8±0.1 /H20850/H1100310−11s, while /H9275pis size dependent.
From the fitted plasma frequencies, taking the sheet elec-
tron concentration equal to n2D=1.95 /H110031012cm−2,w ec a nFIG. 4. Plasmon-cyclotron resonance in GaN/AlGaN with dif-
ferent sample dimensions but the same sheet electron concentrationand mobility. Dots are experimental data, with some data pointsomitted for clarity. The solid lines represent least squares fits of Eq./H208496/H20850with best fit parameters listed in Table I.WOLOS et al. PHYSICAL REVIEW B 76, 045301 /H208492007 /H20850
045301-4calculate the effective diameter of the measured samples us-
ing Eq. /H208493/H20850. We obtain 2 Reff=2, 1.46, and 0.84 mm, which
are comparable to the actual widths of the sample /H208493.5, 2.0,
and 1.5 mm, respectively /H20850, but some discrepancies are
clearly visible. The observed discrepancy may be accountedfor considering the nonradial sample geometry and the factthat the 2DEG may not occupy the whole area of the mea-sured sample.
The frequency fof the magnetoplasma oscillations for a
millimeter-sized sample with a sheet electron density of 2
/H1100310
12cm−2is of the order of a few GHz. Reducing the
sample dimensions down to the micrometer regime, theplasma frequency is expected to fall into the THz region,which can be interesting for the generation or detection ofthe THz radiation.
25Making use of the plasmon-cyclotron
coupling, the resonance frequency can be further tuned to-ward both higher and lower frequencies by applying rela-tively low magnetic fields. The properties of GaN-based het-erostructures have already been successfully used in blueoptoelectronics as well as in high-power and high-temperature electronics. Here, we see that they can be alsoconsidered as a promising candidate for the THz technology.
D. Influence of illumination
It is widely recognized that a 2DEG is present at the
GaN/AlGaN interface even without any intentional dopingdue to the presence of strong polarization fields in the het-erostructure. Native defects from the surface of the AlGaNbarrier have been proposed as a source of the two-dimensional electrons in the GaN quantum well.
26There, it
has been calculated that the polarization fields are pushingnative defect levels above the Fermi level, promoting trans-fer of electrons from localized defect states at the surface tothe triangle well at the interface.
It has been also shown that illumination, which affects the
charge distribution in the GaN/AlGaN heterostructure, si-multaneously influences the electron concentration in the 2Dchannel and the surface barrier height.
27Under ultraviolet
illumination, the 2D electron concentration can be increasedand the surface barrier height is reduced. This effect wasexplained in terms of migration of holes, photogenerated atthe AlGaN barrier, toward the surface and the accumulationof photogenerated electrons at the GaN/AlGaN interface.
Photoionization of localized defect states in the AlGaN bar-rier can be also considered as a source of excess electrons inthe quantum well. Slow recombination of the photocarriershas been observed due to the charge separation by the AlGaNbarrier width and possibly due to capturing by impuritieslocalized in the barrier.
In order to vary the electron concentration in the 2D chan-
nel, the GaN/AlGaN sample /H208493.5/H110034m m
2/H20850was illuminated
in the spectrometer cavity and the resonance spectra were
recorded. The sheet electron concentration under illumina-tion was monitored by measuring the SdH oscillations. Thisapproach allowed us to test the applied model of theplasmon-cyclotron coupling without changing the sample di-mensions. Experimental data are shown in Fig. 5.
Illumination with a white halogen lamp light led to a slow
increase of the electron concentration. After 15 min ofillumination, the 2D concentration increased from1.95/H1100310
12to 2.2 /H110031012cm−2, and the effect did not satu-
rate. Illumination with infrared light had almost no influenceonn
2D, while illumination with UV laser light /H20849multiline 351
and 363 nm /H20850increased drastically the concentration up to
3.5/H110031012cm−2. After switching off the UV illumination, the
concentration slowly came back to its “dark” value, reachingn
2D=3/H110031012cm−2after 10 min. The observed response of
the sample is consistent with results described in Ref. 27.
Figure 5/H20849a/H20850shows the fast Fourier transform spectra of the
measured Shubnikov–de Haas oscillations in the dark andunder illumination. The shift of the FFT peak toward higherfrequencies under illumination is clearly visible, togetherwith the broadening of the FFT line, which may suggest adecrease of the mobility. Indeed, fitting Eq. /H208491/H20850to the SdH
spectra obtained under illumination reveals a drop of thequantum mobility from
/H9262q=8600 cm2/V s in the dark
down to /H9262q=2900 cm2/V s under the UV illumination
/H20849see Table I/H20850.
Plasmon-cyclotron resonances recorded simultaneously
with the SdH oscillations in the illumination experiment areshown in Fig. 5/H20849b/H20850. With increasing sheet electron concentra-
tion /H20849under illumination /H20850, the edge magnetoplasma resonance
moves toward higher magnetic fields, reflecting an increaseof the plasma frequency, as described by Eq. /H208492/H20850.
Figure 6shows all data collected in the illumination ex-
periment. The upper panel shows the square of the plasmaTABLE I. Transport parameters for GaN/AlGaN heterostructures obtained from the Shubnikov–de Haas
oscillations /H20849n2Dand/H9262q/H20850and from the analysis of the plasmon-cyclotron resonance line shape /H20849/H9275p,/H9270C, and
/H9262CR/H20850.
Sample size
/H20849mm2/H20850 Illuminationn2D
/H20849cm−2/H20850/H9262q
/H20849cm2/Vs /H20850
±500/H9275p
/H20849s−1/H20850
±0.1/H110031011/H9270C
/H20849s/H20850
±0.1/H1100310−11/H9262CR
/H20849cm2/Vs /H20850
±8000
3.5/H110034 Dark /H208491.95±0.05 /H20850/H1100310128600±500 1.63 /H1100310111.73/H1100310−11152000
2.0/H110034 Dark 1.98 /H1100310111.89/H1100310−11166000
1.5/H110034 Dark 2.58 /H1100310111.74/H1100310−11153000
3.5/H110034 White /H208492.2±0.1 /H20850/H1100310124300±1000 1.75 /H1100310111.53/H1100310−11135000
3.5/H110034U V /H208493.5±0.1 /H20850/H1100310122900±1000 2.45 /H1100310111.18/H1100310−11104000
3.5/H110034 Dark after UV 2.27 /H1100310111.34/H1100310−11118000PLASMON-CYCLOTRON RESONANCE IN TWO- … PHYSICAL REVIEW B 76, 045301 /H208492007 /H20850
045301-5frequency obtained from the fit of the magnetoplasma line
shape vs the sheet electron concentration obtained from theFourier transform of the SdH signal. A linear dependence isconsistent with Eq. /H208493/H20850. The slope of
/H9275p/H20849n2D/H20850corresponds to
the effective diameter of the sample equal to 2 Reff
=1.8±0.2 mm.
Figure 6/H20849b/H20850shows the cyclotron and the quantum mobili-
ties obtained from the magnetoplasma linewidth and from
the magnetic field dependence of the SdH oscillations, re-spectively, vs the sheet electron concentration. Both mobili-ties decrease with increasing n
2D. However, the drop of /H9262CR
is much smaller than that of /H9262q. The /H9262CR//H9262qratio equals 18
in the dark and 36 under the UV illumination.
As discussed above, illumination increases both the sheet
electron concentration in the GaN 2D channel and the con-centration of ionized defects in the AlGaN barrier. The in-creased number of remote scattering centers in the barriercan be responsible for the drop of both mobility values,
/H9262CR
and/H9262q, observed in the experiment. On the other hand, the
increase of the sheet electron concentration in the well isfollowed by the enlargement of the screening radius, whichleads to a longer characteristic length of potential fluctua-tions. This can explain the significant increase of the
/H9262CR//H9262q
ratio.IV. CONCLUSIONS
To summarize, we observed coupling of plasma oscilla-
tions to the cyclotron motion, an edge magnetoplasma mode,in the high-mobility 2D electron gas confined at theGaN/AlGaN interface using microwave resonance spec-trometry in a conventional ESR spectrometer. We haveshown that the analysis of the coupled plasmon-cyclotronresonance provides a contactless method to characterize
electronic properties of the 2D electron gas. Both the sheetelectron concentration and the mobility can be determinedwith high precision from the resonance position and the lineshape of coupled magnetoplasma modes, for which we havederived a simple formula based on the theory for dimen-sional resonances.
2,22
We also observed Shubnikov–de Haas oscillations in
GaN/AlGaN using the microwave resonance spectroscopy.SdH oscillations have been investigated earlier by thismethod, e.g., for InGaAs and AlGaAs quantum wells.
28–30
The observation failed, however, for high-mobility Si-based
heterostructures, in which large potential fluctuations arepresent.
31
The observation of the plasmon-cyclotron coupling was
possible, thanks to the high mobility of the electronsFIG. 5. Illumination experiment. /H20849a/H20850Fast Fourier transform of
the Shubnikov–de Haas signal in a GaN/AlGaN sample, measuredin the dark and under white halogen lamp or UV laser light illumi-nation. /H20849b/H20850Coupled plasma-cyclotron resonance measured simulta-
neously in the same experiment. Dots are experimental, with somedata points omitted for the clarity. Solid lines represent fits of Eq./H208496/H20850with best fit parameters listed in Table I.FIG. 6. /H20849a/H20850The square of the plasma frequency determined from
plasmon-cyclotron resonance versus the sheet electron concentra-tion obtained from the SdH oscillations for a GaN/AlGaN samplewith a width of 3.5 mm. The solid line represents a fit of Eq. /H208493/H20850,
yielding an effective diameter of the sample equal to 2 R
eff
=1.8 mm. /H20849b/H20850Quantum and cyclotron mobilities determined from
the SdH oscillations and from the magnetoplasma linewidth,respectively.WOLOS et al. PHYSICAL REVIEW B 76, 045301 /H208492007 /H20850
045301-6confined at the GaN/AlGaN heterojunction in the investi-
gated samples. High mobility was achieved here due to theoptimization of the MBE-growth technique, allowing us toobtain a high-quality interface between GaN and AlGaN, aswell as the use of a native GaN substrate, which resulted inlow dislocation density in the heterostructure.ACKNOWLEDGMENTS
This work has been supported by the Fonds zur Förderung
der Wissenschaftlichen Forschung, Austria, the ÖAD andGMe /H20849all Vienna /H20850, and by the funds for science 2007-2010 as
a research project PBZ/MNiSW/07/2006/39, Poland.
1T. N. Theis, J. P. Kotthaus, and P. J. Stiles, Solid State Commun.
24, 273 /H208491977 /H20850.
2S. J. Allen, Jr., H. L. Stormer, and J. C. M. Hwang, Phys. Rev. B
28, 4875 /H208491983 /H20850.
3I. V. Kukushkin, J. H. Smet, K. von Klitzing, and W. Wegsc-
heider, Nature /H20849London /H20850415, 409 /H208492002 /H20850.
4I. V. Kukushkin, J. H. Smet, S. A. Mikhailov, D. V. Kulakovskii,
K. von Klitzing, and W. Wegscheider, Phys. Rev. Lett. 90,
156801 /H208492003 /H20850.
5B. M. Ashkinadze and V. I. Yudson, Phys. Rev. Lett. 83, 812
/H208491999 /H20850.
6W. Knap, H. Alause, J. M. Bluet, J. Camassel, J. Young, M. Asif-
Khan, Q. Chen, S. Huant, and M. Shur, Solid State Commun.
99, 195 /H208491996 /H20850.
7Y. J. Wang, R. Kaplan, H. K. Ng, K. Doverspike, D. K. Gaskill, T.
Ikedo, I. Akasaki, and H. Amono, J. Appl. Phys. 79, 8007
/H208491996 /H20850.
8W. Knap, S. Contreras, H. Alause, C. Skierbiszewski, J. Ca-
massel, M. Dyakonov, J. L. Robert, J. Yang, Q. Chen, M. Asif-
Khan, M. L. Sadowski, S. Huant, F. H. Yang, M. Goiran, J.Leotin, and M. S. Shur, Appl. Phys. Lett. 70, 2123 /H208491997 /H20850.
9S. Syed, M. J. Manfra, Y. J. Wang, H. L. Stormer, and R. J.
Molnar, Phys. Rev. B 67, 241304 /H20849R/H20850/H208492003 /H20850.
10Z.-F. Li, W. Lu, S. C. Shen, S. Holland, C. M. Hu, D. Heitmann,
B. Shen, Y. D. Zheng, T. Someya, and Y. Arakawa, Appl. Phys.Lett. 80, 431 /H208492002 /H20850.
11G. Abstreiter, J. P. Kotthaus, and J. F. Koch, Phys. Rev. B 14,
2480 /H208491976 /H20850.
12C. Skierbiszewski, Z. Wasilewski, M. Siekacz, A. Feduniewicz,
B. Pastuszka, I. Grzegory, M. Leszczynski, and S. Porowski,Phys. Status Solidi A 201, 320 /H208492004 /H20850.
13C. Skierbiszewski, K. Dybko, W. Knap, M. Siekacz, J. Lusa-
kowski, W. Krupczynski, G. Nowak, M. Bockowski, Z. R.Wasilewski, D. Maude, T. Suski, and S. Porowski, Appl. Phys.Lett. 86, 102106 /H208492005 /H20850.
14S. Krukowski, M. Bockowski, B. Lucznik, I. Grzegory, S. Po-rowski, T. Suski, and Z. Romanowski, J. Phys.: Condens. Matter
13, 8881 /H208492001 /H20850.
15T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437
/H208491982 /H20850.
16S. Yamada and T. Makimoto, Appl. Phys. Lett. 57, 1022 /H208491990 /H20850.
17E. Skuras, R. Kumar, R. L. Williams, R. A. Stradling, J. E. Dmo-
chowski, E. A. Johnson, A. Mackinnon, J. J. Harris, R. B. Beall,C. Skierbiszewski, J. Singleton, P. J. van der Wel, and P. Wis-niewski, Semicond. Sci. Technol. 6, 535 /H208491991 /H20850.
18P. Perlin, E. Litwin-Staszewska, B. Suchanek, W. Knap, J. Ca-
massel, T. Suski, R. Piotrzkowski, I. Grzegory, S. Porowski, E.Kami ńska, and J. C. Chervin, Appl. Phys. Lett. 68, 1114 /H208491996 /H20850.
19H. Linke, P. Omling, P. Ramvall, B. Meyer, M. Drechsler, C.
Wetzel, R. Rudeloff, and F. Scholz, J. Appl. Phys. 73, 7533
/H208491993 /H20850.
20L. Hsu and W. Walukiewicz, Appl. Phys. Lett. 80, 2508 /H208492002 /H20850.
21S. Syed, M. J. Manfra, Y. J. Wang, R. J. Molnar, and H. L.
Stormer, Appl. Phys. Lett. 84, 1507 /H208492004 /H20850.
22E. D. Palik and J. K. Furdyna, Rep. Prog. Phys. 33, 1193 /H208491970 /H20850.
23A. L. Fetter, Phys. Rev. B 33, 5221 /H208491986 /H20850.
24A. S. Barker, Jr. and M. Ilegems, Phys. Rev. B 7, 743 /H208491973 /H20850.
25W. Knap, J. Lusakowski, T. Parenty, S. Bollaert, A. Cappy, V. V.
Popov, and M. S. Shur, Appl. Phys. Lett. 84, 2331 /H208492004 /H20850.
26L. Hsu and W. Walukiewicz, J. Appl. Phys. 89, 1783 /H208492001 /H20850.
27G. Koley, Ho-Young Cha, Jeonghyun Hwang, W. J. Schaff, L. F.
Eastman, and M. G. Spencer, J. Appl. Phys. 96, 4253 /H208492004 /H20850.
28G. Hendorfer, M. Seto, H. Ruckser, W. Jantsch, M. Helm, G.
Brunthaler, W. Jost, H. Obloh, K. Kohler, and D. J. As, Phys.Rev. B 48, 2328 /H208491993 /H20850.
29W. Timelthaler, W. Jantsch, and G. Weimann, Semicond. Sci.
Technol. 5, 686 /H208491990 /H20850.
30P. Omling, B. Meyer, and P. Emanuelsson, Appl. Phys. Lett. 58,
931 /H208491991 /H20850.
31Z. Wilamowski, N. Sandersfeld, W. Jantsch, D. Tobben, and F.
Schaffler, Phys. Rev. Lett. 87, 026401 /H208492001 /H20850.PLASMON-CYCLOTRON RESONANCE IN TWO- … PHYSICAL REVIEW B 76, 045301 /H208492007 /H20850
045301-7 |
PhysRevB.85.224506.pdf | PHYSICAL REVIEW B 85, 224506 (2012)
Impurity-induced electronic nematic state and C2-symmetric nanostructures
in iron pnictide superconductors
Yoshio Inoue, Youichi Yamakawa, and Hiroshi Kontani
Department of Physics, Nagoya University, and JST, TRIP , Furo-cho, Nagoya 464-8602, Japan
(Received 16 October 2011; revised manuscript received 12 April 2012; published 5 June 2012)
We propose that an impurity-induced electronic nematic state is realized above the orthorhombic structure
transition temperature TSin iron-pnictide superconductors. In the presence of strong orbital fluctuations near
TS, it is theoretically revealed that a single impurity induces local orbital order with C2symmetry, consistently
with recent Scanning Tunneling Microscopy/Spectroscopy (STM/STS) measurements. Each impurity-inducedC
2-symmetric nanostructure aligns along the aaxis by applying tiny uniaxial pressure along the ba x i s .I nt h i s
impurity-induced nematic phase, the resistivity shows sizable in-plane anisotropy ( ρb/ρa∼2) even above TS,
actually observed in various “detwinned” samples. The present study indicates the existence of strong orbitalfluctuations in iron-pnictide superconductors.
DOI: 10.1103/PhysRevB.85.224506 PACS number(s): 74 .70.Xa, 74 .20.Rp
I. INTRODUCTION
Since the discovery of iron-pnictide superconductors,1a
lot of effort has been devoted to understand the overall phase
diagram, including the superconducting (SC) state in the
tetragonal (T) phase and non-SC orthorhombic (O) phase.In Ba(Fe,Co)
2As2, the O structure transition at TSis of
second order,2and very large softening of shear modulus
CSsuggests the existence of strong ferro-quadrupole φS∝
ˆx2−ˆy2fluctuations above TS.3–6In many compounds, the
SC transition temperature Tctakes the highest value near the
endpoint of the O phase, suggesting a close relation between
the superconductivity and the orbital instability. The weakferro-quadrupole order in the O phase induces the spin-densitywave with Q=(π,0).
5
As for the SC mechanism, a spin-fluctuation-mediated
sign-reversing s-wave state ( s±-wave state) has been pro-
posed from an early stage because Coulomb interaction and
intraorbital nesting between hole and electron pockets has
been observed.7–9In iron pnictides, each Fermi pocket is
mainly composed of t2gorbitals of Fe atoms. An orbital-
fluctuation-mediated s-wave state without sign reversal ( s++-
wave state) had been investigated in Refs. 10–12: Strong
orbital fluctuations originate from the interorbital nestingbetween Fermi pockets in the presence of Coulomb and weak
electron-phonon ( e-ph) interactions. The latter scenario is
supported by the robustness of the SC state against impuritiesin many iron pnictides
13–16and by the orbital-independent
SC gap observed in Ba122 systems by laser angle-resolvedphotoelectron spectroscopy (APRES) measurement.
11,17Also,
an experimental “resonance-like” hump structure in the neu-tron inelastic scattering is well reproduced in terms of the
s
++-wave SC state, rather than the s±-wave SC state, by
taking the suppression in the inelastic scattering in the SCstate (dissipationless mechanism).
18
According to Ref. 5, the structure transition originates from
the ferro-charge quadrupole φS=Ox2−y2∝nxz−nyzinsta-
bility, realized by the bound-state formation of two orbitonswith opposite momenta. By this two-orbiton theory, we canfit the temperature dependence of C
Sin Ba(Fe 1−xCox)2As2
forx=0∼0.16 almost perfectly.19The spin nematic theory(or two-magnon process)3,20is another candidate. However,
incommensurate spin order is realized in Ba(Fe 1−xCox)2As2
forx/greaterorequalslant0.056,21although the latter theory requires commen-
surate fluctuations.
Furthermore, recent discovery of an “electronic nematic
transition” in the T phase, free from any lattice deformation,has attracted great attention. For example, in “detwinned”Ba(Fe
1−xCox)2As222,23under very small uniaxial pressure
(∼5 MPa), sizable in-plane anisotropy of resistivity emerges
atT∗, which is about 10–100 K higher than TS. The nematic
order is also observed in BaFe 2(As,P) 2by the magnetic
torque measurement.24Now, we need to find which degree
of freedom, spin or orbital, is more important for nematicity,orthorhombicity, and superconductivity.
In this paper, we discuss the impurity-induced electronic
nematic phase in iron pnictides, using the mean-field approxi-mation (MFA) in real space. When orbital fluctuations develop,we obtain various types of local orbital orders with lowersymmetries ( C
4,C2v,C2, etc.), actually reported by STM/STS
autocorrelation analyses.25,26The large cross section of the
local order gives giant residual resistivity, far beyond thes-wave unitary scattering value: ∼20μ/Omega1cm/%. When C
2
nanostructures are aligned along the aaxis, the in-plane
anisotropy of resistivity reaches 40%, which is consistent withexperiments.
22,23Such large anisotropy is not achieved when
isotropic impurity scattering is considered.27,28
In annealed Ba(Fe 1−xMx)2As2(M=Co, Ni), the difference
|ρb−ρa|is very small in the absence of Mimpurities ( x=
0),29while it increases in proportional to xforx/lessorequalslant4%.23,29
In contrast, both the magnetic moment and lattice deformation
monotonically decrease with x. These facts strongly support
the idea of impurity-induced nanostructures.
In strongly correlated electron systems, impurity potential
frequently causes drastic change in the electronic state. Forexample, in nearly antiferromagnetic metals, magnetic correla-tion is extremely enhanced near the nonmagnetic impurity site,giving rise to the local magnetic moment ( ∼1μ
B) and large
residual resistivity30that are indeed observed in optimally
and underdoped cuprates. In terms of the weak-couplingscheme, such phenomena originate from the Friedel oscillationsince the large local-density-of-states (LDOS) sites could
224506-1 1098-0121/2012/85(22)/224506(5) ©2012 American Physical SocietyYOSHIO INOUE, YOUICHI Y AMAKAW A, AND HIROSHI KONTANI PHYSICAL REVIEW B 85, 224506 (2012)
trigger the strong fluctuations around the impurity. As for
the iron-based superconductors, the system would be close tothe antiferro-orbital critical point. Thus, it is natural to expectthe occurrence of “impurity-induced local orbital order” iniron pnictides.
II. MODEL HAMILTONIAN AND METHOD
OF CALCULATION
Here, we study the single-impurity problem due to orbital-
diagonal impurity potential I10in a large cluster with 800 Fe
sites, based on the two-dimensional 10-orbital tight-bindingmodel for LaFeAsO in Refs. 7and 31.W es e t xandyaxes
parallel to the nearest Fe-Fe bonds. Then, the Fermi surfacesare mainly composed of t
2gorbitals ( xz,yz, andxy), although
egorbitals also play non-negligible roles. Here, we consider
the following quadrupole-quadrupole interaction:5,10–12
Hquad=−g/summationdisplay
i/braceleftbigˆOi
xzˆOi
xz+ˆOi
yzˆOi
yz+ˆOi
xyˆOi
xy/bracerightbig
,(1)
where ˆOi
/Gamma1is the quadrupole operator for channel /Gamma1at site i
introduced in Ref. 5,ˆOi
/Gamma1=/summationtext
l,m,σol,m
/Gamma1c†
i,lσci,mσ, where ol,m
/Gamma1
is the matrix element of the charge quadrupole operator. Note
that ˆOμν∝ˆlμˆlν+ˆlνˆlμ. The quadrupole coupling constant g
in Eq. (1)originates from both the e-ph interaction as well as
the Coulomb interaction for the charge sector, as discussed inRefs. 10and11. Since we are interested in the nonmagnetic
orbital order, we neglect the Coulomb interaction to simplifythe calculation. Then, strong orbital fluctuations for /Gamma1=xz,yz
channels are produced by relatively small g(∼0.2 eV).
10–12
Hereafter, the unit of energy is electon volts.
Here, we put T=0.02 and the electron filling n=6.0 per
Fe, which correspond to undoped compounds like BaFe 2As2.
In the absence of impurities, the bulk antiferro-orbital orderoccurs for g>g
c≡0.222. Below, we study the following
mean-field equation for g<g c:
Mi
l,m=/angbracketleftc†
i,lσci,mσ/angbracketrightI,g−/angbracketleftc†
i,lσci,mσ/angbracketrightI,0, (2)
where iis the Fe site, and l,m represent the dorbital. Mi
l,m
is impurity-induced mean field; ˆMi=0f o rI=0. Then, the
mean-field potential due to the Hartree term is
Si
l,m=/summationdisplay
l/prime,m/prime/Gamma1c
lm,l/primem/primeMi
l/prime,m/prime, (3)
where /Gamma1c
lm,l/primem/prime=− 2g/summationtextxz,yz,xy
/Gamma1 olm
/Gamma1ol/primem/prime
/Gamma1is the bare interaction
for charge sector5and the mean-field Hamiltonian is ˆHMF=
ˆH0+/summationtext
iˆSi+const. In the MFA, we solve Eqs. (2)and (3)
self-consistently.
III. NUMERICAL RESULTS AND DISCUSSIONS
In Figs. 1(a)–1(c), we show the obtained DOS at the Fermi
level ( EF) in real space, in which the center is the impurity
site with I=− 2. For g=0.200 (a), the impurity-induced
mean field is absent. The small modulation of the LDOSaround the impurity is caused by the Friedel oscillation.Forg> 0.207, impurity-induced local orbital order with
diagonal C
2vsymmetry appears, as shown in Fig. 1(b).
The suppression of the DOS is caused by the orbital order,-5
(d) (e) -4-3-2-1 0 1 2 3 4-4-3-2-1 0 1 2 3 4
0 0.5 1 1.5 2
-4-3-2-1 0 1 2 3 4-4-3-2-1 0 1 2 3 4
0 0.5 1 1.5 2
0-0.6 -0.4 -0.2 0 0.2 0.4 0.61234
Energy [eV](0,0)(2,0)(1.0)↓(4,4)LDOS
0.21 0.22 -2-1
g [eV]
]Ve[ygrenE0
no order
C2V
C2
0.20-10 -8 -6 -4 -2 0 2 4 6 8 10-6-4-2 0 2 4 6
0 0.2 0.4 0.6 0.8 1 1.2 1.4(c) g=0.218eV(a) I=-2eV, g=0.200eV (b) g=0.208eV
FIG. 1. (Color online) Obtained LDOS at EFforI=− 2a n d
(a)g=0.200: without orbital order, (b) g=0.208: orbital order with
diagonal C2vsymmetry, and (c) g=0.218: orbital with C2symmetry.
(d) Energy dependence of the LDOS for g=0.218. (e) gdependence
of the free energy.
consistent with a recent optical conductivity measurement.32
With increasing g, the orbital order changes to C2symmetry for
g> 0.212, shown in Fig. 1(c). The size of the nanostructure
is∼15aFe−Fe(∼7aFe−Fe) along the x(y) axis. Such a large
impurity-induced object is actually observed in Ba(Fe,Co) 2As2
by STM/STS.25,26When the impurity concentration nimpis
∼1%, the obtained C2order would be stabilized by the
weak overlap between neighbors against thermal fluctuationsomitted in the MFA. (Similar C
2order is also realized for
I=∞ .) Figure 1(d) shows the energy dependence of LDOS
forg=0.218 at r=(0,0) (impurity site), (1 ,0), (2,0), and
(4,4). Near (0 ,0), the LDOS is modified for a wide energy
range. Figure 1(e) presents the free energy as function of g.I n
the MFA, each transition at g≈0.207 and 0 .212 is of the first
order.
Since Mi
l,m=Mi
m,l, the present mean field has 15 com-
ponents at each site. They are represented as charge densityor monopole ( l=0), quadrupole ( l=2), and hexadecapole
(l=4) orders. The first two orders are give as ¯n
i=2/summationtext
l,lMi
l,l
and ¯Oi
/Gamma1=2/summationtext
l,mol,m
/Gamma1Mi
l,m, where /Gamma1=xz,yz,xy,z2, and
x2−y2. The hexadecapole order is negligibly small in the
present study. Figure 2shows the dominant four mean fields,
¯ni,¯Oi
xz,¯Oi
yz, and ¯Oi
xy,f o rg=0.218. We verified that the
224506-2IMPURITY-INDUCED ELECTRONIC NEMATIC STATE ... PHYSICAL REVIEW B 85, 224506 (2012)
-6 -4 -2 0 2 4 6-6-4-2 0 2 4 6
-2-1 0 1 2-6 -4 -2 0 2 4 6-6-4-2 0 2 4 6
-2-1 0 1 2
-6 -4 -2 0 2 4 6-6-4-2 0 2 4 6
-2-1 0 1 2
xyO-6 -4 -2 0 2 4 6-6-4-2 0 2 4 6
4 5 6 7 8n xzO
yzO
FIG. 2. (Color online) Obtained electron density ¯niand
quadrupole order ¯Oi
/Gamma1at Fe sites for I=− 2a n dg=0.218.
quadrupole interactions for /Gamma1=xz/yz channels in Eq. (1)are
indispensable for the C2order. The obtained quadrupole order
is very different from the uniform quadrupole ordered state(¯O
x2−y2∝nxz−nyz=const.) in the orthorhombic phase,5
and therefore the impurity-induced nematic order will exist
even below TS.
TheC2order in Fig. 1(c) can be aligned by the
strain-induced quadrupole potential; H/prime=/Delta1E/summationtext
iˆOi
x2−y2and
/Delta1E=ηS/epsilon1S·χQ
x2−y2(0)/χ(0)
x2−y2(0), where /epsilon1S∝a−bis the
strain and ηSis the strain-quadrupole coupling. χQ
x2−y2(0)i s
the ferroquadrupole susceptibility, which is strongly enhanced
nearTSdue to the two-orbiton process as discussed in Ref. 5.
This would be the reason why the nematically ordered state iseasily detwinned by small uniaxial pressure near T
S. In fact,
detwinning by uniaxial pressure is possible only when thestructure transition is of the second order.
33
Here, we assume x/bardblaaxis and y/bardblbaxis. In detwinned
compounds with a>b , ARPES measurements indicate /Delta1E <
0, that is, nxz>nyz.34,35For a single C2order, we obtain
the relation Fa−Fb≈2.5/Delta1E, where Fa(b)is the free energy
when the C2order is along the a(b) axis. Therefore, the nematic
order along the aaxis is realized by detwinning ( a>b ),
schematically shown in Fig. 3(a). Note that two kinds of C2
orders, the C2order in Fig. 1(c) and its inversion with respect to
thexaxis, still degenerate and coexist with equal probability.
Now, we calculate the in-plane resistivity in the nematic
state shown in Fig. 3(a).W eu s et h e T-matrix approximation,
which gives the exact result when nimp/lessmuch1 and localization
is negligible. The Tmatrix is given by solving the following
equation in the orbital-diagonal basis:
ˆTr,r/prime(ω)=(ˆI+ˆS)rδr,r/prime+/summationdisplay
r/prime/prime(ˆI+ˆS)rˆG(0)
r−r/prime/prime(ω)ˆTr/prime/prime,r/prime(ω),(4)
where ˆSris the impurity-induced mean-field potential, and
ˆG(0)
r(ω) is the Green’s function without impurities. ˆIr=Iˆ1δr,0(b)
0.2 0.21 0.22050100
ρa (without vertex)
ρb(without vertex)
ρbρa
g [eV]ρ [μΩcm]
ρaρb
c2c2vno orderCo
C2 orbital
order
b
a(a)
uniρ
I=-2eV
FIG. 3. (Color online) (a) Alignment of the impurity-induced C2
orders under uniaxial pressure ( a>b ). (b) Obtained ρa(b)fornimp=
1% and I=− 2:ρb>ρain the nematic phase.
is the impurity potential.10TheTmatrix is nonlocal when
ˆSr/negationslash=0. After the Fourier transformation, the self-energy in the
T-matrix approximation is ˆ/Sigma1(k,ω)=nimpˆTk,k(ω), and the full
Green’s function is ˆG(k,ω)=[ω+μ−ˆH0
k−ˆ/Sigma1(k,ω)]−1.
Then, the in-plane conductivity is given as
σν=e2
π1
N/summationdisplay
k,αvα
k,νJα
k,ν|Gα(k,iδ)|2, (5)
where ν=xory, andαrepresents the αth band. vα
k,νis the
group velocity and Gα(k,ω) is the full Green’s function in the
band-diagonal basis. Jα
k,νis the total current including the ver-
tex correction, which is given by solving the following Bethe-
Salpeter equation: Jα
k,ν=vα
k,ν+1
N/summationtext
p,βIα,β
k,p|Gβ(p,iδ)|2Jβ
p,ν
where Iα,β
k,k/prime=nimp|Tα,β
k,k/prime(iδ)|2is the irreducible vertex.
The obtained results for I=− 2 andnimp=1% are shown
in Fig. 3(b). Here, we assume the interlayer distance is 0.6 nm.
Without orbital order, the resistivity is 5.5 μ/Omega1cm, which
is about one-fourth of the maximum value without orbitalorder: ρ
uni∼20μ/Omega1cm for I≈+ 1. When diagonal C2vorder
appears, the resistivity exceeds ρuni, due to a large cross section
of the “effective impurity radius” as recognized in Fig. 1(b).
In the nematic phase with horizontal C2order, we obtain large
anisotropy ρb/ρa∼2: By including the vertex correction,
bothρaandρbare suppressed and the anisotropy ρb/ρais
enlarged, since the contribution of the forward scattering iscorrectly subtracted. The averaged resistivity ( ρ
a+ρb)/2 per
1% impurity reaches ∼50μ/Omega1cm, which is comparable to the
residual resistivity by 1% Co impurities observed in La111113
and Ba122.29
Now, we discuss the nematic transition at T∗in real
compounds. Beyond the MFA, the effective interaction ˜g(<g)
decreases with Tdue to the thermal fluctuation.12Then, one
possibility is that the phase transition from the diagonal C2v
to vertical C2occurs at T∗. (Then, ˜g≈0.212 at T∗.) Another
possibility is that C2order is realized even above T∗, while the
necessary condition for detwinning, χQ
x2−y2(0)/greatermuch1, is satisfied
only below T∗. In both cases, experimental results can be
explained.
We also study the impurity-induced local orbital order
forI=+ 1. Figure 4(a) shows the LDOS without orbital
224506-3YOSHIO INOUE, YOUICHI Y AMAKAW A, AND HIROSHI KONTANI PHYSICAL REVIEW B 85, 224506 (2012)
-4-3-2-1 0 1 2 3 4-4-3-2-1 0 1 2 3 4
0 0.5 1 1.5 2
(a) I=+1eV, g=0. 20eV
0.2 0.21 0.22050100
ρ (without vertex)
ρ
g [eV]ρ [μ Ω cm]
c4(c)
-4-3-2-1 0 1 2 3 4-4-3-2-1 0 1 2 3 4
0 0.5 1 1.5 2
(b) g=0.218eV
uniρ
I=+1eV
FIG. 4. (Color online) Obtained LDOS at EFin the case of
I=+ 1a n d( a ) g=0.200: without orbital order, and (b) g=0.218:
orbital order with C4symmetry. (c) Obtained resistivity for 1%
impurity with I=+ 1.
order: The realized Friedel oscillation pattern, different from
Fig. 1(a), would induce a new type of orbital order. In fact,
we obtain the orbital order with C4symmetry for g> 0.203:
Fig. 4(b) shows the LDOS for g=0.218. We also obtain a
metastable solution with C2symmetry similar to Fig. 1(c),
whose free energy is about 0 .1 eV higher than that for
theC4-symmetry solution. (When I=− 2, the C4-symmetry
solution is “unstable” with positive free energy.) Figure 4(c)
shows the resistivity ρ=ρa=ρbforI=+ 1 andnimp=1%:
It exceeds the unitary value as soon as C4order appears, and
it reaches ∼50μ/Omega1cm for g∼gc.
It is noteworthy that the obtained C4order looks similar to
Sn-impurity-induced “ring-shape object” in LiFeAs observedby Hanaguri
36very recently. The realized large reduction inthe DOS would result in the suppression of the s++-wave
state. In fact, in BaFe 1.89−2xZn2xCo0.11As2, the suppression in
Tcper 1% Zn impurity is −/Delta1T c/%∼3K/%:15Such small
suppression of Tcis consistent with the s++-wave state, since
−/Delta1T c/%∼20 K/% is expected in the s±-wave state when the
mass enhancement is m∗/mb∼3.16
Finally, we consider the impurities on other than Fe sites.
We expect that impurity-induced nematic phase is realized inBaFe
2(As,P) 2, since Psites give finite impurity potential on
the neighboring four Fe sites. In this case, we actually obtainimpurity-induced order with C
2orC1hsymmetry, which is
consistent with experiments.19In contrast, a nematic state is
not realized in (K,Ba)Fe 2As2,37possibly because K sites are
outside of FeAs planes.
IV . SUMMARY
In summary, we discussed an impurity-induced electronic
nematic state based on the orbital fluctuation theory. Theobtained local orbital orders with various symmetries ( C
2v,C2,
andC4) are consistent with recent STM/STS measurements.
In the case of C2order, the anisotropy of resistivity reaches
ρb/ρa∼2, which presents a natural explanation for the
nematic state in various “detwinned” iron pnictides. Thus,characteristic features of iron pnictides, nematic and structuretransitions as well as superconductivity, are well understoodbased on the orbital fluctuation theory.
ACKNOWLEDGMENTS
We thank Y . Matsuda, T. Shibauchi, S. Uchida, H. Eisaki,
M. Sato, M. Itoh, Y . Kobayashi, T. Hanaguri, D. Hirashima,S. Onari, and T. Saito for valuable discussions. This study hasbeen supported by grants-in-aid from MEXT of Japan and byJST, TRIP.
1Y . Kamihara et al. ,J. Am. Chem. Soc. 130, 3296 (2008).
2C. R. Rotundu and R. J. Birgeneau, P h y s .R e v .B 84, 092501
(2011).
3R. M. Fernandes, L. H. Van Bebber, S. Bhattacharya, P. Chandra,V . Keppens, D. Mandrus, M. A. McGuire, B. C. Sales, A. S. Sefat,and J. Schmalian, Phys. Rev. Lett. 105, 157003 (2010).
4M. Yoshizawa, R. Kamiya, R. Onodera, Y . Nakanishi, K. Kihou,
H. Eisaki, and C. H. Lee, J. Phys. Soc. Jpn. 81, 024604 (2012).
5H. Kontani, T. Saito, and S. Onari, Phys. Rev. B 84, 024528 (2011):
The structure transition due to two-orbiton process does not requirethe commensurability of antiferro-orbitons.
6T. Goto, R. Kurihara, K. Araki, K. Mitsumoto, M. Akatsu,Y . Nemoto, S. Tatematsu, and M. Sato, J. Phys. Soc. Jpn. 80, 073702
(2011).
7K. Kuroki, S. Onari, R. Arita, H. Usui, Y . Tanaka, H. Kontani, andH. Aoki, P h y s .R e v .L e t t . 101, 087004 (2008).
8I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev.
Lett.101, 057003 (2008).
9P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep. Prog. Phys.
74, 124508 (2011); S. Graser, G. R. Boyd, C. Cao, H.-P. Cheng,
P. J. Hirschfeld, and D. J. Scalapino, Phys. Rev. B 77, 180514(R)(2008); A. V . Chubukov, D. V . Efremov, and I. Eremin, ibid.78,
134512 (2008).
10H. Kontani and S. Onari, Phys. Rev. Lett. 104, 157001 (2010).
11T. Saito, S. Onari, and H. Kontani, Phys. Rev. B 82, 144510 (2010).
12S. Onari and H. Kontani, arXiv:1009.3882 (unpublished).
13M. Sato, Y . Kobayashi, S. C. Lee, H. Takahashi, E. Satomi, and
Y. M i u r a , J. Phys. Soc. Jpn. 79, 014710 (2009); S. C. Lee, E.
Satomi, Y . Kobayashi, and M. Sato, ibid.79, 023702 (2010).
14Y . Nakajima, T. Taen, Y . Tsuchiya, T. Tamegai, H. Kitamura, and
T. Murakami, Phys. Rev. B 82, 220504 (2010).
15J. Li, Y . Guo, S. Zhang, S. Yu, Y . Tsujimoto, H. Kontani,
K. Yamaura, and E. Takayama-Muromachi, Phys. Rev. B 84,
020513(R) (2011).
16S. Onari and H. Kontani, Phys. Rev. Lett. 103, 177001 (2009).
17T. Shimojima, F. Sakaguchi, K. Ishizaka, Y . Ishida, T. Kiss,
M. Okawa, T. Togashi, C.-T. Chen, S. Watanabe, M. Arita,
K. Shimada, H. Namatame, M. Taniguchi, K. Ohgushi, S. Kasahara,
T. Terashima, T. Shibauchi, Y . Matsuda, A. Chainani, and S. Shin,Science 332, 564 (2011).
18S. Onari, H. Kontani, and M. Sato, P h y s .R e v .B 81, 060504(R)
(2010); S. Onari and H. Kontani, ibid.84, 144518 (2011).
224506-4IMPURITY-INDUCED ELECTRONIC NEMATIC STATE ... PHYSICAL REVIEW B 85, 224506 (2012)
19H. Kontani (unpublished).
20C. Fang, H. Yao, W. F. Tsai, J. P. Hu, and S. A. Kivelson, Phys. Rev.
B77, 224509 (2008).
21D. K. Pratt, M. G. Kim, A. Kreyssig, Y . B. Lee, G. S. Tucker,
A .T h a l e r ,W .T i a n ,J .L .Z a r e s t k y ,S .L .B u d ’ k o ,P .C .C a n fi e l d ,B. N. Harmon, A. I. Goldman, and R. J. McQueeney, Phys. Rev.
Lett.106, 257001 (2011)
22J.-H. Chu, J. G. Analytis, K. D. Greve, P. L. McMahon, Z. Islam,
Y . Yamamoto, and I. R. Fisher, Science 329, 824 (2010); J. J. Ying,
X. F. Wang, T. Wu, Z. J. Xiang, R. H. Liu, Y . J. Yan, A. F. Wang,M. Zhang, G. J. Ye, P. Cheng, J. P. Hu, and X. H. Chen, Phys. Rev.
Lett.107, 067001 (2011).
23I. R. Fisher, L. Degiorgi, and Z. X. Shen, Rep. Prog. Phys. 74,
124506 (2011).
24Y . Matsuda (private communication).
25T.-M. Chuang, M. P. Allan, J. Lee, Y . Xie, N. Ni, S. L. Budko, G. S.Boebinger, P. C. Canfield, and J. C. Davis, Science 327, 181 (2010).
26C.-L. Song, Y .-L. Wang, P. Cheng, Y .-P. Jiang, W. Li, T. Zhang, Z.
Li, K. He, L. Wang, J.-F. Jia, H.-H. Hung, C. Wu, X. Ma, X. Chen,and Q.-K. Xue, Science 332, 1410 (2011).
27C.-C. Chen, J. Maciejko, A. P. Sorini, B. Moritz, R. R. P. Singh,
and T. P. Devereaux, Phys. Rev. B 82, 100504(R) (2010).
28R. M. Fernandes, E. Abrahams, and J. Schmalian, Phys. Rev. Lett.
107, 217002 (2011).29M. Nakajima, T. Liang, S. Ishida, Y . Tomioka, K. Kihou, C. H.
Lee, A. Iyo, H. Eisaki, T. Kakeshita, T. Ito, and S. Uchida,Proc. Natl. Acad. Sci. USA 108, 12238 (2011); S. Uchida
(unpublished).
30H. Kontani, Rep. Prog. Phys. 71, 026501 (2008); H. Kontani and
M. Ohno, Phys. Rev. B 74, 014406 (2006).
31T. Miyake, Nakamura, R. Arita, and M. Imada, J. Phys. Soc. Jpn.
79, 044705 (2010)
32L. Stojchevska, T. Mertelj, J.-H. Chu, I. R. Fisher, and
D. Mihailovic, arXiv:1107.5934 (unpublished).
33M. A. Tanatar, E. C. Blomberg, A. Kreyssig, M. G. Kim, N. Ni,
A. Thaler, S. L. Budko, P. C. Canfield, A. I. Goldman, I. I. Mazin,and R. Prozorov, arXiv:1002.3801 (unpublished).
34Q. Wang, Z. Sun, E. Rotenberg, F. Ronning, E. D. Bauer, H. Lin,
R. S. Markiewicz, M. Lindroos, B. Barbiellini, A. Bansil, and D. S.Dessau, arXiv:1009.0271 (unpublished).
35M .Y i ,D .H .L u ,J . - H .C h u ,J .G .A n a l y t i s ,A .P .S o r i n i ,A .F .
Kemper, S.-K. Mo, R. G. Moore, M. Hashimoto, W.-S. Lee,Z. Hussain, T. P. Devereaux, I. R. Fisher, and Z.-X. Shen, Proc.
Natl. Acad. Sci. USA 108, 6878 (2011).
36T. Hanaguri (unpublished).
37J. J. Ying, X. F. Wang, T. Wu, Z. J. Xiang, R. H. Liu, Y . J. Yan,
A. F. Wang, M. Zhang, G. J. Ye, P. Cheng, J. P. Hu, and X. H. Chen,Phys. Rev. Lett. 107, 067001 (2011)
224506-5 |
PhysRevB.80.165413.pdf | Electron-electron interactions and doping dependence of the two-phonon Raman intensity
in graphene
D. M. Basko,1,*S. Piscanec,2and A. C. Ferrari2
1Laboratoire de Physique et Modélisation des Milieux Condensés, Université Joseph Fourier and CNRS, 38042 Grenoble, France
2Department of Engineering, Cambridge University, 9 JJ Thomson Avenue, Cambridge CB3 OF A, United Kingdom
/H20849Received 8 June 2009; revised manuscript received 15 September 2009; published 16 October 2009 /H20850
Raman spectroscopy is a fast and nondestructive means to characterize graphene samples. In particular, the
Raman spectra are strongly affected by doping. While the resulting change in position and width of the Gpeak
can be explained by the nonadiabatic Kohn anomaly at /H9003, the significant doping dependence of the 2 Dpeak
intensity has not been understood yet. Here we show that this is due to a combination of electron-phonon andelectron-electron scattering. Under full resonance, the photogenerated electron-hole pairs can scatter not justwith phonons but also with doping-induced electrons or holes, and this changes the intensity. We explain thedoping dependence and show how it can be used to determine the corresponding electron-phonon coupling.This is higher than predicted by density-functional theory, as a consequence of renormalization by Coulombinteractions.
DOI: 10.1103/PhysRevB.80.165413 PACS number /H20849s/H20850: 78.30. /H11002j, 73.50.Bk
I. INTRODUCTION
Graphene is the latest carbon allotrope discovered and it
is now at the center of a significant research effort.1–6Near-
ballistic transport at room temperature and high mobility5–10
make it a potential material for nanoelectronics,11–14espe-
cially for high-frequency applications.15Furthermore, its
transparency and mechanical properties are ideal for micro-mechanical and nanomechanical systems, thin-film transis-tors, transparent and conductive composites and electrodes,and photonics.
16–20
Graphene layers can be readily identified in terms of num-
ber and orientation by inelastic and elastic light scatteringsuch as Raman
21and Rayleigh spectroscopies.22,23Raman
spectroscopy also allows monitoring of doping, defects,strain, disorder, chemical modifications, and edges.
21,24–38In-
deed, Raman spectroscopy is a fast and nondestructive char-acterization method for carbons.
39They show common fea-
tures in the 800–2000 cm−1region: the Gand Dpeaks,
around 1580 and 1350 cm−1, respectively. The Gpeak cor-
responds to the E2gphonon at the Brillouin-zone center /H20849/H9003
point /H20850. The Dpeak is due to the breathing modes of six-atom
rings and requires a defect for its activation.38,40,41It comes
from TO phonons around the Kpoint of the Brillouin
zone,38,41is active by double resonance /H20849DR /H20850,40and is
strongly dispersive with excitation energy due to a KohnAnomaly at K.
27The activation process for the Dpeak is
intervalley and is shown schematically in Fig. 1/H20849d/H20850:/H20849i/H20850a
laser-induced excitation of an electron/hole pair; /H20849ii/H20850
electron-phonon scattering with an exchanged momentumq/H11011K;/H20849iii/H20850defect scattering; /H20849iv/H20850electron-hole recombina-
tion. DR can also happen as intravalley process, i.e., con-necting two points belonging to the same cone around K/H20849or
K
/H11032/H20850, as shown in Fig. 1/H20849b/H20850. This gives the so-called D/H11032peak,
which is at about 1620 cm−1in defected graphite, when
measured at 514 nm excitation.
The 2 Dpeak is the second order of the Dpeak. This is a
single peak in single-layer graphene /H20849SLG /H20850whereas it splits
into four in bilayer graphene /H20849BLG /H20850, reflecting the evolutionof the band structure.21The 2 D/H11032peak is the second order of
theD/H11032peak. Since both 2 Dand 2 D/H11032originate from a process
where momentum conservation is satisfied by two phononswith opposite wave vectors /H20849qand − q/H20850, they do not require
the presence of defects for their activation and are thus al-
ways present. Indeed, high-quality graphene shows the G,
2D, and 2 D
/H11032peaks but not Dand D/H11032.21Also, under the
K (b) K (c) K (a)G peak 2D peak D peak
(e)(d)K KK K2D peak D peak
FIG. 1. /H20849Color online /H20850Role of the electron dispersion /H20849Dirac
cones, /H9280=/H11006vF/H20841p/H20841, shown by solid black lines /H20850in Raman scattering:
/H20849a/H20850intravalley one-phonon Gpeak, /H20849b/H20850defect-assisted intravalley
one-phonon D/H11032peak, /H20849c/H20850intravalley two-phonon 2 D/H11032peak, /H20849d/H20850
defect-assisted intervalley one-phonon Dpeak, and /H20849e/H20850intervalley
two-phonon 2 Dpeak. Vertical solid arrows represent interband tran-
sitions accompanied by photon absorption /H20849upward arrows /H20850or emis-
sion /H20849downward arrows /H20850/H20849the photon wave vector is neglected /H20850.
Dashed arrows represent phonon emission. Horizontal dotted ar-rows represent defect scattering.PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
1098-0121/2009/80 /H2084916/H20850/165413 /H2084910/H20850 ©2009 The American Physical Society 165413-1assumption of electron-hole symmetry, the two-phonon
peaks are fully resonant.42,43This means that energy and mo-
mentum conservation are satisfied in all elementary steps ofthe Raman process, as shown schematically in Figs. 1/H20849c/H20850and
1/H20849e/H20850. Then, all intermediate electronic states are real. As a
consequence, two-phonon Raman spectroscopy is sensitiveto the dynamics of the photoexcited electron-hole pair, inparticular, to the scattering processes it can undergo. This isof crucial importance for the present work.
Doping in graphene is commonly observed in as-
deposited samples, due to the presence of charges at the sur-face or interface.
24,44It can also be induced by applying a
voltage on an external gate electrode.26,32,33,36Substitutional
doping, either bulk or edge, is also possible,45,46however,
thus far, these samples are far from ideal, and the effects ofdoping and structural disorder overlap in their Ramanspectrum.
45In the present work we will thus focus on the
variation in the Raman spectra observed in samples where
the Fermi level moves as a result of charged impurities orapplied voltages, or both, as reported in Refs. 24,26,32,33,
and36. The Gpeak position, Pos /H20849G/H20850, increases and its full
width at half maximum, FWHM /H20849G/H20850, decreases for both elec-
tron and hole doping. The Gpeak stiffening is due to the
nonadiabatic removal of the Kohn anomaly at /H9003.
26,47The
FWHM /H20849G/H20850sharpening is due to Pauli blocking of phonon
decay into electron-hole pairs, when the electron-hole gap ishigher than the phonon energy,
26,48and saturates for a Fermi
shift bigger than half phonon energy.26,36,48A similar behav-
ior is observed for the LO- G−peak in metallic nanotubes,49
for the same reasons. In the case of BLG, the different band
structure renormalizes the phonon response to doping differ-ently from SLG.
33,50,51Also in this case the Raman Gpeak
stiffens and sharpens for both electron and hole doping, as aresult of the nonadiabatic Kohn anomaly at /H9003.
33However,
since BLG has two conduction and valence subbands, withsplitting dependent on the interlayer coupling, this changesthe slope in the variation in Pos /H20849G/H20850with doping, allowing a
direct measurement of the interlayer coupling strength.
33,51
Another significant result is that in SLG the ratio of the
heights of the 2 DandGpeaks, I/H208492D/H20850/I/H20849G/H20850, and their areas,
A/H208492D/H20850/A/H20849G/H20850, is maximum for zero doping,21,52and de-
creases for increasing doping. On the other hand, this showslittle dependence on doping for BLG.
32,33Figure 2plots the
combined data for SLG and BLG from Refs. 21,32,33,52,
and53. Note that Refs. 32and33reported height ratios,
while here, as discussed later, we analyze the area ratioA/H208492D/H20850/A/H20849G/H20850, which encompasses both trends of I/H208492D/H20850/I/H20849G/H20850
and FWHM /H208492D/H20850/FWHM /H20849G/H20850.
Due to residual disorder, the energy of the Dirac point can
fluctuate across the sample on a scale smaller than the laserspot, which leads to spatial inhomogeneity of the dopinglevel.
24,44We attribute the difference in the behavior of the
two SLG curves in Fig. 2to a different degree of residual
charge inhomogeneity in the polymeric electrolyte experi-ments of Refs. 32and33. On the other hand, the use of this
electrolyte enabled probing a very large doping range be-cause the nanometer-thick Debye layer gives a much highergate capacitance compared to the usual 300 nm SiO
2back
gate.26,32,33Note as well that A/H208492D/H20850/A/H20849G/H20850for the most in-
trinsic samples measured to date is about 12–17,21,52,53muchhigher than the zero gating values in Refs. 32and33,a s
shown in Fig. 2. This points again to sources of disorder in
the gated samples of Refs. 32and33, while the absence of a
significant Dpeak excludes large amounts of structural de-
fects. Finally, we stress that all data for varying EFin Fig. 2
are measured on samples on Si covered by the same SiO 2
thickness, thus the relative change in peaks’ intensities withdoping is not related to extrinsic interference effects.
22,54
Here, we show that the dependence on doping of the 2 D
peak intensity results from its sensitivity to the scattering ofthe photoexcited electron and hole. Assuming the dominantsources of scattering to be phonon emission and electron-electron collisions, we note that while the former is not sen-sitive to doping, the latter is. Then, the 2 Ddoping depen-
dence can be used to estimate the corresponding electron-phonon coupling /H20849EPC /H20850.FIG. 2. Experimental A/H208492D/H20850/A/H20849G/H20850, measured for 514.5 nm ex-
citation, as a function of EFfor SLG /H20849Refs. 21,32,33, and 52/H20850and
BLG /H20849Ref.33/H20850. The BLG data /H20849solid squares /H20850are divided by ten, to
make comparison easier. Note that the doping-dependent SLG dataare a combination of two experiments on two different samples,from Ref. 33 /H20849half-filled circles /H20850and Ref. 32 /H20849open circles /H20850, and a
data-point representative of intrinsic graphene from Refs. 21,52,
and53 /H20849solid star /H20850.BASKO, PISCANEC, AND FERRARI PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-2II. DOPING DEPENDENCE OF TWO-PHONON RAMAN
INTENSITY
A. Theoretical dependence
Raman scattering55is an electron-mediated process where
electromagnetic radiation exchanges vibrational quanta/H20849phonons /H20850with a crystal. A complete description requires the
detailed knowledge of /H20849i/H20850electronic structure, /H20849ii/H20850phonon
dispersions, and /H20849iii/H20850mutual interactions between electrons
and phonons /H20849i.e., electron-electron, electron-phonon, and
phonon-phonon scattering /H20850.
The Raman spectrum of graphene consists of a set of
distinct peaks. Each characterized by its position width,height, and area. The frequency-integrated area under eachpeak represents the probability of the whole process. It ismore robust with respect to various perturbations of the pho-non states than width and height. Indeed, for an ideal case ofdispersionless undamped phonons with frequency
/H9275phthe
shape of the n-phonon peak is a Dirac /H9254distribution
/H11008/H9254/H20849/H9275−n/H9275ph/H20850, with zero width, infinite height but well-
defined area. If the phonons decay /H20849e.g, into other phonons,
due to anharmonicity, or into electron-hole pairs, due toelectron-phonon coupling /H20850, the
/H9254peak broadens into a
Lorentzian, but the area is preserved, as the total number ofphonon states cannot be changed by such perturbations. Ifphonons have a weak dispersion, states with different mo-menta contribute at slightly different frequencies. This mayresult in an overall shift and a nontrivial peak shape butfrequency integration across the peak means counting allphonon states, as in the dispersionless case. Thus, the peakarea is preserved, as long as the Raman matrix element itselfis not changed significantly by the perturbation. The latterholds when the perturbation /H20849phonon broadening or disper-
sion /H20850is smaller than the typical energy scale determining the
matrix element. Converting this into a time scale using theuncertainty principle we obtain that, if the Raman process isfaster than the phonon decay, the total number of photonsemitted within a given peak /H20849i.e., integrated over frequency
across the peak /H20850, is not affected by phonon decay, although
their spectral distribution can be. Even if the graphenephonons giving rise to the DandD
/H11032peaks are dispersive due
to the Kohn anomalies at Kand/H9003,27their relative change
with respect to the average phonon energy is at most a fewpercent, thus we are in the weakly dispersive case discussedabove. The phonon decay in graphene is in the picosecondtime scale, while the Raman process is faster, in the femto-second time scale.
26,56,57Then, we will analyze the area ratio,
A/H208492D/H20850/A/H20849G/H20850, which encompasses both variations in height
ratio, I/H208492D/H20850/I/H20849G/H20850, and width FWHM /H208492D/H20850/FWHM /H20849G/H20850.
We first consider the Gpeak. For the one-phonon process,
allowed by momentum conservation, which gives rise to theGpeak, the picture is entirely different from the two-phonon
case. As shown in Fig. 1/H20849a/H20850, the process responsible for the G
peak is determined by virtual electrons and holes with energy/H11011E
L/2, where ELis the laser excitation energy /H20849for a typical
Raman measurement EL/2/H110111e V /H20850. If the Fermi energy, EF,
stays below EL/2, as in Refs. 32and33, these electronic
states are not strongly affected. Only the final phonon state isinfluenced by doping, which manifests itself in a change inPos /H20849G/H20850and FWHM /H20849G/H20850.
26,32,33,36However, the area of the
peak is determined by the total spectral weight of the phononstate, which is preserved. Thus, we do not expect any signifi-cant dependence of A/H20849G/H20850on doping, as long as the doping is
not too strong, /H20841E
F/H20841/H112701 eV. We can then take the measured
doping dependence of A/H208492D/H20850/A/H20849G/H20850as representative of the
A/H208492D/H20850trend. Note that A/H20849G/H20850can change as a function of
other external parameters such as the Raman excitationenergy.
21,38,58–60However, for fixed excitation, such as in the
experiments discussed here, the above argument holds.
In Ref. 43the following expressions for the 2 Dand 2 D/H11032
areas were obtained:
A/H208492D/H20850=8
3/H20873e2
c/H208742vF2
c2/H20873/H9253K
/H9253/H208742
, /H208491a/H20850
A/H208492D/H11032/H20850=4
3/H20873e2
c/H208742vF2
c2/H20873/H9253/H9003
/H9253/H208742
, /H208491b/H20850
where eis the electron charge, cis the speed of light,
e2/c/H110151/137 is the fine-structure constant, and vFis
the electron velocity /H20849its experimental value is
vF/H11015106m/s/H110156.6 eV Å /H20849Refs. 61–63/H20850.2/H9253is the scattering
rate of the photoexcited electron and hole. Note that we de-fine
/H9253as the imaginary part of the energy, so it determines
the decay of the amplitude, while the decay of the probabilityis determined by 2
/H9253. This includes all sources of inelastic
scattering. Assuming the two main mechanisms for electronscattering to be the emission of phonons and electron-electron collisions, we write
/H9253=/H9253e-ph+/H9253ee,/H9253e-ph=/H9253/H9003+/H9253K. /H208492/H20850
Here we include the phonons near /H9003andK, responsible for
DandD/H11032. The corresponding emission rates, 2 /H9253/H9003and 2/H9253K,
enter the numerators in Eqs. /H208491a/H20850and /H208491b/H20850.
Two points regarding Eqs. /H208491a/H20850and /H208491b/H20850should be empha-
sized. First, the scattering rates depend on the electron en-ergy,
/H9280, which is defined by half the laser energy, /H9280/H11015EL/2
/H20851see Eq. /H208499/H20850in the next section /H20852. Second, if impurity scatter-
ing is significant compared to other scattering mechanisms,the corresponding elastic-scattering rate cannot be simply in-cluded in
/H9253and Eqs. /H208491a/H20850and /H208491b/H20850. The whole Raman inten-
sity calculation should be done differently. Equations /H208491a/H20850
and /H208491b/H20850thus neglect impurity scattering. For short-range
impurities this assumption is justified by the absence of alarge Dpeak in the spectra of Refs. 32and33. Long-range
disorder is efficiently screened /H20849even though the vanishing
density of states at the Dirac point requires the screening tobe nonlinear
64–67/H20850; it is precisely this screening that gives rise
to the inhomogeneous concentration of electrons/holes andspatial fluctuations of the Dirac-point energy.
In principle, there are no reasons for a strong dependence
of
/H9253e-phon carrier density. However, /H9253eedoes exhibit such a
dependence. Indeed, in undoped graphene at low tempera-tures, the photoexcited electron finds itself in a state withsome momentum, p, measured from the Dirac point, in the
empty conduction band. To scatter into a state with a differ-ent momentum p
/H11032, it has to give away some energy and
momentum to another electron in the full valence band. ThisELECTRON-ELECTRON INTERACTIONS AND DOPING … PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-3second electron would have to be promoted to the conduc-
tion band /H20849as there are no available empty states in the va-
lence band /H20850into a state with momentum pe, leaving a hole in
the valence band with momentum ph. Momentum and energy
conservation require
p=p/H11032+pe+ph, /H208493a/H20850
/H9280/H20849p/H20850=/H9280/H20849p/H11032/H20850+/H9280/H20849pe/H20850+/H9280/H20849ph/H20850, /H208493b/H20850
where /H9280/H20849p/H20850is the quasiparticle dispersion, assumed the same
for electrons and holes. For Dirac particles /H9280/H20849p/H20850=vF/H20841p/H20841,s o
the only possibility to satisfy both conservation laws is tohave all four momenta parallel. If the spectrum is convex,d
2/H9280/H20849p/H20850/dp2/H110220, the two equations can be satisfied by a set of
momenta with nonzero measure, i.e., the phase space is fi-nite. If it is concave, d
2/H9280/H20849p/H20850/dp2/H110210, they are incompatible.
In SLG the spectrum is Dirac to a first approximation, result-ing in an uncertainty.
68This can be resolved by taking into
account corrections from electron-electron interactions,which make the spectrum concave
69,70and the interband pro-
cess forbidden.
As new carriers are added to the system, intraband
electron-electron collisions become allowed. The momentumand energy conservation become
p+p
e=p/H11032+pe/H11032, /H208494a/H20850
/H9280/H20849p/H20850+/H9280/H20849pe/H20850=/H9280/H20849p/H11032/H20850+/H9280/H20849pe/H11032/H20850, /H208494b/H20850
which can be satisfied for any quasiparticle dispersion. These
collisions give a contribution to /H9253eewhich increases with
carrier concentration. As a consequence, the total /H9253in Eq.
/H208491a/H20850increases, leading to an overall decrease in A/H208492D/H20850, con-
sistent with the experimental trend in Fig. 2.
The above arguments essentially use the nonconvexity of
the electronic spectrum in the conduction band, and thus ap-ply to SLG only. In BLG the spectrum is parabolic near theDirac point, so that d
2/H9280/dp2/H110220, and the phase-space restric-
tions are absent. Thus, electron-electron collisions are al-lowed even at zero doping and the collision rate has a muchweaker dependence on E
F, which, in first approximation, can
be neglected. Thus, A/H208492D/H20850is expected to have a weak de-
pendence on EF, as seen in Fig. 2, where the experimental
A/H208492D/H20850/A/H20849G/H20850for BLG shows a negligible variation with
doping.33
To quantify the doping effects on the SLG A/H208492D/H20850, we first
calculate the electron-electron scattering rate, 2 /H9253ee,i nt h e
random-phase approximation, analogously to Refs. 71and
72./H9253eeis given by the imaginary part of the on-shell elec-
tronic self-energy, Im /H9018ee/H20849p,/H9280/H20850for/H9280→vFp−0+, with /H9280andp
counted from the Dirac point.68Here we consider the limit-
ing case, when the energy of the photoexcited electron/H20849
/H9280=EL/2/H20850far exceeds EF. The carrier concentration is
n=EF2//H20849/H9266vF2/H20850. In this case, the collisions are dominated by
small momentum transfers, /H20841p−p/H11032/H20841/H11011/H20841EF/H20841/vF,s o/H9253eedoes not
depend on /H9280and is proportional to /H20841EF/H20841, the proportionality
coefficient depending only on the dimensionless Coulombcoupling constant r
s=e2//H20849/H9255vF/H20850/H20849/H9255being the dielectric con-
stant /H20850,/H9253ee=/H20841EF/H20841f/H20873e2
/H9255vF/H20874+O/H20849EF2//H9280/H20850. /H208495/H20850
The explicit form of the function fis given in the Appendix.
Figure 3plots its numerical values for rs/H110212.2, correspond-
ing to /H9255/H110221.FIG. 3. Numerical values of the function f/H20849rs/H20850, appearing in Eq.
/H208495/H20850.
FIG. 4. Fit of the experimental dependence /H20881A/H20849G/H20850/A/H208492D/H20850from
Ref. 32 /H20849open circles /H20850, Ref. 33 /H20849half-filled circles /H20850, and the data for
intrinsic graphene /H20849Refs. 21,52, and 53/H20850/H20849star /H20850using Eq. /H208498/H20850/H20849dashed
and solid lines, respectively /H20850.BASKO, PISCANEC, AND FERRARI PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-4Thus, we expect A/H208492D/H20850to change with EFas
A/H208492D/H20850=C
/H20851/H9253e-ph+/H20841EF/H20841f/H20849e2//H9255vF/H20850/H208522/H208496/H20850
with Ca constant. Note that a variation in the dielectric
constant /H9255will affect A/H208492D/H20850. Given the negligible depen-
dence of A/H20849G/H20850on doping, Eq. /H208496/H20850can be rewritten as
/H20881A/H20849G/H20850
A/H208492D/H20850=C/H11032/H20851/H9253e-ph+/H20841EF/H20841f/H20849e2//H9255vF/H20850/H20852, /H208497/H20850
where C/H11032is another constant.
B. Fit to experiments
Figure 4plots /H20881A/H20849G/H20850/A/H208492D/H20850as a function of EF. This
dependence, according to Eq. /H208497/H20850, should correspond to two
symmetric straight lines joining at EF=0. As noted in Sec. I,
close to EF=0 the data from the two polymer electrolyte
gating experiments do not converge to the same value. How-
ever, for both a linear rise of /H20881A/H20849G/H20850/A/H208492D/H20850is seen at higher
energies. Also, while the data represented by open circles inFig.4are almost symmetric, a significant asymmetry is seen
for electron doping in the set represented by the half-filledcircles, but the two sets are in good agreement for hole dop-ing.
A/H208492D/H20850/A/H20849G/H20850for intrinsic samples measured at 514.5 nm
excitation, the same as in Refs. 32and33, is in the range
12–17,
21,52represented by the star in Fig. 2at 14.5, corre-
sponding to /H20881A/H20849G/H20850/A/H208492D/H20850=0.26 at EF=0. This is in good
agreement with the ratio measured for carbon whiskers.53
These show a 2 Dpeak very similar to graphene, being com-
posed of misoriented graphene layers.53,73However, their
Raman spectra are much less susceptible to charged impuri-ties or surface doping, being bulk materials.
53We also need
to consider the dielectric constant of the polymer electrolyteused in the experiment of Ref. 32,/H9255=5, giving f/H20849e
2//H9255vF/H20850
/H110150.06. Thus, we fit the data with a one-parameter expression
/H20881A/H20849G/H20850
A/H208492D/H20850=0.26
/H9253e-ph/H20849/H9253e-ph+ 0.06 /H20841/H9280F/H20841/H20850. /H208498/H20850
We fit separately each branch of the two data sets, as shown
by solid and dotted lines in Fig. 4. As a result, we obtain
/H9253e-ph=18,21,29,65 meV, with an average /H9253e-ph=33 meV.74
III. RAMAN INTENSITIES AND ELECTRON-PHONON
COUPLING
A. Theoretical background and electron-phonon coupling
definitions
Even though graphite and other sp2-hybridized materials
have been investigated for more than 60 years,41,75all the
fundamental physical properties needed for the interpretationof the Raman spectra have undergone an intense debate,which seems to be just beginning to converge. Interestingly,several features of both phonon dispersions and band struc-ture of graphene are determined by the EPC. For example, inthe Kohn anomalies around /H9003orK/H20849Ref. 27/H20850the correctionto the phonon frequencies due to EPC results in a linear
slope of the optical phonon branches as the wave vector ap-proaches /H9003orK. The EPC and phonon-dispersions calcula-
tions of Ref. 27have been confirmed at the /H9003point by in-
elastic x-ray scattering
76and by the measurement of
FWHM /H20849G/H20850in graphite, graphene, and nanotubes,21,26,48,77
once anharmonic effects are taken into account.21,26,56For
theKpoint, the precise slope of the anomaly is still
debated.37,78,79Another EPC effect is the kink in the electron
dispersion, about 200 meV below EF, seen by angle-resolved
photoemission spectroscopy /H20849ARPES /H20850.63,80This is attributed
to a correction to the electron energy due to EPC,63,80,81al-
though alternative explanations also exist.82Thus, a correct
EPC determination is a fundamental step for an accurate de-scription of the physical properties of graphene and nano-tubes, these being rolled up graphene sheets.
To link the 2 Dintensity to the EPC we first consider the
rate of phonon emission by the photoexcited electron/hole,2
/H9253e-ph. This is obtained from the imaginary part of the elec-
tron self-energy, /H9253e-ph=Im/H9018e-ph/H20849/H9280/H20850. For EL/2/H11022EF+/H9275/H9003,a si n
the case of the Raman measurements at 2.41 eV excitation ofRefs. 32and33, we have
43
/H9253K=/H9261K
4/H20873EL
2−/H9275K/H20874,/H9253/H9003=/H9261/H9003
4/H20873EL
2−/H9275/H9003/H20874. /H208499/H20850
Then, from Eq. /H208492/H20850
/H9253e-ph=/H9261K
4/H20873EL
2−/H9275K/H20874+/H9261/H9003
4/H20873EL
2−/H9275/H9003/H20874. /H2084910/H20850
The dimensionless coupling constants /H9261/H9003,/H9261Kcorrespond to
phonons close to /H9003andK, respectively, and determine their
rate of emission. We define them as
/H9261/H9003,K=F/H9003,K2Au.c.
2M/H9275/H9003,KvF2. /H2084911/H20850
Here /H9275K=1210 cm−1=0.150 eV /H20849Ref. 79/H20850 and
/H9275/H9003=1580 cm−1=0.196 eV,21M/H110152.00/H1100310−23g=2.88
/H11003103/H20849eV Å2/H20850−1is the mass of the carbon atom and Au.c.
/H110155.24 Å2is the unit-cell area. F/H9003andFKhave the dimen-
sionality of a force and are the proportionality coefficientsbetween the change in effective Hamiltonian and the latticedisplacement along the corresponding phonon mode. Strictlyspeaking, the relevant phonon states are not exactly at /H9003and
K, as shown in Fig. 1. However, the corresponding devia-
tion, q/H11011E
L/vF, is small compared to the K−K/H11032distance and
is neglected. All observables depend on the dimensionlessEPCs, /H9261
/H9003and/H9261K.
Equation /H2084911/H20850follows the notation of Ref. 43. Since dif-
ferent EPC definitions are used in the literature, it is quiteuseful to give here matching rules for all of them, which willbe necessary when comparing the EPC values obtained herewith previous /H20849and future /H20850reports. The EPCs can be conve-
niently matched by either relating them to the nearest-neighbor tight-binding model, where they are expressed interms of a single parameter,
/H11509t0//H11509a, the derivative of the
nearest-neighbor electronic matrix element with respect tothe interatomic distance, or by comparing expressions forvarious observables. For example, doping leads to a GpeakELECTRON-ELECTRON INTERACTIONS AND DOPING … PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-5shift due to EPC. This is expressed in terms of EFas26,36,47,83
/H9254/H9275/H9003=/H9261/H9003
2/H9266/H20873/H20841EF/H20841+/H9275/H9003
4ln2EF−/H9275/H9003
2EF+/H9275/H9003/H20874. /H2084912/H20850
The corrections to the phonon dispersions as function of
wave vector q, measured from /H9003orK, are27,43,77
/H9254/H9275/H9003−LO=/H9261/H9003
8/H20881vF2q2−/H9275/H90032, /H2084913a /H20850
/H9254/H9275/H9003−TO=−/H9261/H9003
8/H9275/H90032
/H20881vF2q2−/H9275/H90032, /H2084913b /H20850
/H9254/H9275K=/H9261K
4/H20881vF2q2−/H9275K2. /H2084913c /H20850
Note that the E2gmode splits into longitudinal /H20849/H9003−LO /H20850and
transverse /H20849/H9003−TO /H20850at finite q. Note also that due to analyti-
cal properties of the logarithm and the square root, Eq. /H2084912/H20850
at/H20841EF/H20841/H11021/H9275/H9003/2 and Eqs. /H2084913a /H20850–/H2084913c /H20850atvFq/H11021/H9275K,/H9003acquire
imaginary parts, which correspond to the phonon decayinginto a continuum of electron-hole pairs.
48In this case
2I m/H9254/H9275gives the FWHM of the corresponding Lorentzian
profile. At vFq/H11271/H9275K,/H9003Eqs. /H2084913a /H20850and /H2084913c /H20850give the profile of
the Kohn anomalies.
In Refs. 26,27,78, and 84the EPCs are defined as the
matrix elements of the Kohn-Sham potential, differentiatedwith respect to the phonon displacements. What enters theobservables are their squares, averaged over the Fermi sur-face in the limit E
F→0. The matching rule is then
F/H90032=4 /H20855D/H90032/H20856F/H20849Refs. 26 and 78 /H20850=8M/H9275/H9003/H20855g/H90032/H20856F/H20849Refs. 27 and 84 /H20850,
/H2084914a /H20850
FK2=2 /H20855DK2/H20856F/H20849Refs. 26 and 78 /H20850=4M/H9275K/H20855gK2/H20856F/H20849Refs. 27 and 84 /H20850.
/H2084914b /H20850
In Refs. 36and83the dimensionless coupling constant /H9261is
defined as the proportionality coefficient in Eq. /H2084912/H20850. Thus,
/H9261/H20849Refs. 36 and 83 /H20850=/H9261/H9003
2/H9266. /H2084915/H20850
Note that the expression linking EPC to FWHM /H20849G/H20850in Ref.
36underestimates FWHM /H20849G/H20850by a factor 2, and should not
be used.
The dimensionless EPCs reported in the ARPES analysis
of Refs. 63,80,85, and 86and in the scanning tunneling
spectroscopy /H20849STS /H20850experiment of Ref. 87were measured
from the ratio of the electronic velocities below and abovethe kink in the electron dispersion. This ratio is determinedby the derivative of the real part of the electronic self-energyRe/H9018
e-ph/H20849/H9280/H20850due to the EPC. The latter can be calculated if
one takes the Dirac spectrum for electrons and a constantdispersion for phonons. For E
F/H110220 one has84/H9018e-ph/H20849/H9280/H20850=−/H9261K
4/H9266/H20849/H9280−/H9275K/H20850lnEM
/H20841/H9280−/H9275K−EF/H20841
−/H9261K
4/H9266/H20849/H9280+/H9275K/H20850lnEM/H20841/H9280+/H9275K−EF/H20841
/H20849/H9280+/H9275K/H208502
−/H9261/H9003
4/H9266/H20849/H9280−/H9275/H9003/H20850lnEM
/H20841/H9280−/H9275/H9003−EF/H20841
−/H9261/H9003
4/H9266/H20849/H9280+/H9275/H9003/H20850lnEM/H20841/H9280+/H9275/H9003−EF/H20841
/H20849/H9280+/H9275/H9003/H208502. /H2084916/H20850
Here EMis the ultraviolet cutoff, on the order of the elec-
tronic bandwidth. We then get the matching rule
/H9261/H20849kink /H20850=−/H20879/H11509Re/H9018e-ph
/H11509/H9280/H20879
/H9280=EF
=/H9261K
2/H9266/H20873EF−/H9275K
/H9275K+l nEM
/H9275K+EF/H20874
+/H9261/H9003
2/H9266/H20873EF−/H9275/H9003
/H9275/H9003+l nEM
/H9275/H9003+EF/H20874. /H2084917/H20850
However, we note that /H9261Kis subject to Coulomb
renormalizations.88This implies that /H9261Kdepends on the elec-
tronic energy scale, such as the electron energy /H9255, the Fermi
energy EF, or the temperature T, whichever is larger,
/H9261K=/H9261K/H20849max /H20853/H20841/H9280/H20841,/H20841EF/H20841,T/H20854/H20850. This dependence is shown in
Fig. 6 of Ref. 88. In a Raman measurement this scale is
given by the energy of the photoexcited electron, /H9280/H11015EL/2,
as long as EL/2/H11022/H20841EF/H20841. Thus, in Eq. /H2084910/H20850/H9261K=/H9261K/H20849EL/2/H20850.O n
the other hand, to estimate the EPC effects on the phonondispersions in intrinsic graphene, the relevant electron energyis on the order of the phonon energy. Thus, in Eq. /H2084913c /H20850
/H9261
K/H11011/H9261K/H20849/H9275K/H20850. From Fig. 6 of Ref. 88we estimate that
/H9261K/H20849/H9275K/H20850//H9261K/H20849EL/2/H20850/H110151.5 for /H9255=1 and 1.2 for /H9255=5 /H20849taking
EL/H110152 eV to represent Raman measurements in the visible
range /H20850.
The situation with Eq. /H2084917/H20850is more complicated since the
cutoff EMappears explicitly. The logarithmic term is deter-
mined by allenergy scales from EMdown to EF+/H9275K. Thus,
the proper expression is
/H9261/H20849kink /H20850=/H9261K/H20849EF/H20850
2/H9266EF−/H9275K
/H9275K+/H20885
EF+/H9275KEM/H9261K/H20849/H9280/H20850
2/H9266d/H9280
/H9280
+/H9261/H9003
2/H9266/H20873EF−/H9275/H9003
/H9275/H9003+l nEM
EF+/H9275/H9003/H20874. /H2084918/H20850
B. Experimental electron-phonon coupling
From Eq. /H2084912/H20850, our overall average /H9253e-ph=33 meV, de-
rived from a fit to all the data in Fig. 4, gives
/H9261/H9003+/H9261K/H110150.13. /H2084919/H20850
We also note that the hole doping side of Fig. 4shows two
data sets very consistent with each other. We can thus getanother estimate taken from the average
/H9253e-ph /H1101520 meV for
just the hole doping side. This would giveBASKO, PISCANEC, AND FERRARI PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-6/H9261/H9003+/H9261K/H110150.08. /H2084920/H20850
Based on measurements26,36and density-functional theory
/H20849DFT /H20850calculations,27the value of /H9261/H9003can be reliably taken
/H110150.03. Indeed, DFT gives27/H20855g/H90032/H20856F=0.0405 eV2and
vF=5.5 eV Å, corresponding, from Eqs. /H2084911/H20850and /H2084914a /H20850to
/H9261/H9003/H110150.028. Even though /H20855g/H90032/H20856FandvFare subject to Cou-
lomb renormalization, /H9261/H9003=4Au.c./H20855g/H90032/H20856F/vF2, which contains
their ratio, is not.88The experimental /H9261/H9003extracted from
FWHM /H20849G/H20850in graphene and graphite21,48according to Eq.
/H2084913a /H20850and from the dependence of Pos /H20849G/H20850on Fermi energy
according to Eq. /H2084912/H20850, are /H9261/H9003/H110150.034 /H20849Ref. 36/H20850and
/H9261/H9003/H110150.027.26
On the other hand, the value of /H9261Kis still debated.78,84,88
The calculated DFT /H20855gK2/H20856F=0.0994 eV2, together with the
DFT vF=5.5 eV·Å /H20849both taken from Ref. 27/H20850gives
/H9261K=0.034. However, Ref. 88suggested this should be en-
hanced by Coulomb renormalization by up to a factor 3,depending on the background dielectric constant. In order tocompare with our fits, we need consider that the correctionsto the phonon dispersion are determined by electronic stateswith energies lower than those contributing to the Ramansignal. As discussed in Sec. III A,/H9261
K/H20849/H9275K/H20850//H9261K/H20849EL/2/H20850/H110151.2
for/H9255=5. Our fit in Eq. /H2084919/H20850corresponds to /H9261K/H20849EL/2/H20850/H110150.1
while Eq. /H2084920/H20850gives /H9261K/H20849EL/2/H20850/H110150.05, resulting in
/H9261K/H20849/H9275K/H20850/H110150.12 and /H9261K/H20849/H9275K/H20850/H110150.06, respectively. These are
bigger than DFT by a factor of about 3.5 and 1.7, respec-tively.
A recent GW calculation gave /H20855D
K2/H20856F=193 eV2/Å2.78
Combining this with the GWvalue vF=6.6 eV Å,89we get
/H9261K/H20849/H9275K/H20850/H110150.054, a factor /H110111.6 greater than DFT, in good
agreement with our fitted average on the hole side.
Ref. 79reported inelastic x-ray scattering measurements
of the phonon dispersions near Kmore detailed than those
originally done in Ref. 76, now giving a phonon slope at K
of 73 meV Å. Using Eq. /H2084913c /H20850atq/H11271/H9275K/vFand taking the
experimental value vF=6.6 eV Å /H20849Ref. 63/H20850/H20849the bare elec-
tron velocity, i.e., below the phonon kink /H20850, we obtain
/H9261K/H20849/H9275K/H20850/H110150.044, a factor /H110111.3 higher than DFT, again in
good agreement with our fitted average on the hole side.
Another EPC estimate can be derived from the 2 Dand
2D/H11032area ratio. Combining Eqs. /H208491a/H20850,/H208491b/H20850,/H208499/H20850, and /H2084910/H20850we
get
A/H208492D/H20850
A/H208492D/H11032/H20850=2/H20873/H9261K
/H9261/H9003/H208742
. /H2084921/H20850
For intrinsic SLG and graphite whiskers, the experimental
ratio A/H208492D/H20850/A/H208492D/H11032/H20850is about 25–30,21,52,53which gives
/H9261K/H20849EL/2/H20850/H110150.11 and /H9261/H9003+/H9261K/H20849EL/2/H20850/H110150.13. Since in this case
/H9255=1, we obtain /H9261K/H20849/H9275K/H20850/H110150.16, 4.5 times higher than DFT, in
agreement with our upper estimate from Eq. /H2084919/H20850.
We finally consider the EPC derived from ARPES and
STS. For an estimate, we approximate the dependence /H9261K/H20849/H9280/H20850
as linear in ln /H9280. We take /H9261K/H20849EM/H20850=/H20849/H9275/H9003//H9275K/H20850/H9261/H9003, as given by
DFT /H20849assumed to be valid at high energies /H20850, and leave
/H9261K/H20849EL/2/H110151e V /H20850as the only free parameter determining this
linear dependence,/H9261K/H20849/H9280/H20850=/H9275/H9003
/H9275K/H9261/H9003−/H20875/H9275/H9003
/H9275K/H9261/H9003−/H9261K/H20849EL/2/H20850/H20876ln/H20849EM//H9280/H20850
ln/H20851EM//H20849EL/2/H20850/H20852.
/H2084922/H20850
Taking EF=0.4 eV,63,80,86,87EM=10 eV, and substituting
Eq. /H2084922/H20850in Eq. /H2084918/H20850,w eg e t
/H9261/H20849kink /H20850/H110150.7/H9261/H9003+ 0.6/H9261K/H20849EL/2/H20850. /H2084923/H20850
Note that the dependence on the precise value of EMis weak;
setting EM=5 eV changes the first coefficient to 0.5 and the
second /H20849more important as it multiplies the larger coupling
constant /H20850varies only by 2%. The measurements in Refs. 63,
80, and 85–87gave/H9261/H20849kink /H20850/H110150.4,0.3,0.26,0.2,0.14, respec-
tively. The smallest of these values, /H9261/H20849kink /H20850/H110150.14, from Eq.
/H2084923/H20850corresponds to /H9261/H9003+/H9261K/H20849EL/2/H20850/H110150.23 while the highest to
/H9261/H9003+/H9261K/H20849EL/2/H20850/H110150.66. Even the smallest is almost twice our
upper bound fit of Eq. /H2084919/H20850and would imply an EPC renor-
malization of almost 1 order of magnitude. Resolution ef-fects could play a role in this overestimation.
84
Thus, our fits to the doping-dependent Raman area ratios
point to a significant renormalization, by a factor 1.7–3.5, ofthe EPC for the TO mode close to K, responsible for the
Raman Dand 2 Dpeaks. Our lower bound estimate is con-
sistent with recent GW calculations and phonon measure-
ments, but our upper bound is much lower than the smallestestimate derived by ARPES, pointing to a problem in theway ARPES-based works have thus far extracted EPC fromtheir experimental data.
IV . CONCLUSIONS
We have shown that the 2 Dintensity dependence on dop-
ing can be explained considering the influence of electron-electron interactions on the total scattering rate of the photo-generated electrons /H20849holes /H20850. We have given a simple formula
linking 2 Dpeak area to the Fermi-level shift. Fitting this to
the available experimental data we got an estimate for theEPC value of the TO phonons close to K, responsible for the
Raman Dand 2 Dpeaks. This is larger than that from DFT
calculations, due to renormalization by Coulomb interac-tions. However, our fitted EPC is still significantly smallerthan those reported in ARPES or STS experiments.
ACKNOWLEDGMENTS
We acknowledge A. Das, S. Berciaud, A. Bonetti, and P.
H. Tan for useful discussions. A.C.F. acknowledges fundingfrom the Royal Society and the European Research Councilgrant NANOPOTS.
APPENDIX: THE FUNCTION f(rs)
The function f/H20849rs/H20850, appearing in Eq. /H208495/H20850, can be repre-
sented asELECTRON-ELECTRON INTERACTIONS AND DOPING … PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-7f/H20849rs/H20850=2
/H9266/H20885
0/H9266/2
d/H9272/H11003/H20877/H20885
02//H208491+cos /H9272/H20850dx x2sin/H9272R1
/H208512/H20849x/rs+4 /H20850xsin/H9272/H208522+R12
+/H20885
2//H208491+cos /H9272/H208502//H208491−cos /H9272/H20850dx x2sin/H9272R2
/H208512/H20849x/rs+4 /H20850xsin/H9272−R3/H208522+R22/H20849x,/H9272/H20850/H20878,
/H20849A1/H20850
where R1,R2, and R3are given by
R1/H20849x,/H9272/H20850=a+b+−a−b−−x2lna++b+
a−+b−, /H20849A2a /H20850R2/H20849x,/H9272/H20850=a+b+−x2lna++b+
x, /H20849A2b /H20850
R3/H20849x,/H9272/H20850=a−/H20881x2−a−2−x2arccosa−
x, /H20849A2c /H20850
a/H11006=2/H11006xcos/H9272,b/H11006=/H20881a/H110062−x2. /H20849A2d /H20850
Figure 3plots f/H20849rs/H20850, calculated numerically.
*denis.basko@grenoble.cnrs.fr
1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.
Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Sci-ence 306, 666 /H208492004 /H20850.
2A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 /H208492007 /H20850.
3A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,
and A. K. Geim, Rev. Mod. Phys. 81, 109 /H208492009 /H20850.
4J. C. Charlier, P. C. Eklund, J. Zhu, and A. C. Ferrari, in Carbon
nanotubes , edited by A. Jorio, G. Dresselhaus, and M. S.
Dresselhaus /H20849Springer, Berlin, 2008 /H20850.
5K. 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.
6Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature /H20849Lon-
don /H20850438, 201 /H208492005 /H20850.
7K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L.
Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A.K. Geim, Science 315, 1379 /H208492007 /H20850.
8S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin,
D. C. Elias, J. A. Jaszczak, and A. K. Geim, Phys. Rev. Lett.
100, 016602 /H208492008 /H20850.
9X. Du, I. Skachko, A. Barker, and E. Y. Andrei, Nat. Nanotech-
nol. 3, 491 /H208492008 /H20850.
10K. I. Bolotin, K. J. Sikes, J. Hone, H. L. Stormer, and P. Kim,
Phys. Rev. Lett. 101, 096802 /H208492008 /H20850; K. I. Bolotin, K. J. Sikes,
Z. Jiang, G. Fundenberg, J. Hone, P. Kim, and H. L. Stormer,Solid State Commun. 146, 351 /H208492008 /H20850.
11M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, Phys. Rev. Lett.
98, 206805 /H208492007 /H20850.
12Z. Chen, Y. M. Lin, M. Rooks, and P. Avouris, Physica E /H20849Am-
sterdam /H2085040, 228 /H208492007 /H20850.
13Y. Zhang, J. P. Small, W. V. Pontius, and P. Kim, Appl. Phys.
Lett. 86, 073104 /H208492005 /H20850.
14M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, IEEE
Electron Device Lett. 28, 282 /H208492007 /H20850.
15Y. M. Lin, K. A. Jenkins, A. Valdes-Garcia, J. P. Small, D. B.
Farmer, and P. Avouris, Nano Lett. 9, 422 /H208492009 /H20850.
16J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank,
D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L.McEuen, Science 315, 490 /H208492007 /H20850.
17P. Blake, P. D. Brimicombe, R. R. Nair, T. J. Booth, D. Jiang, F.
Schedin, L. A. Ponomarenko, S. V. Morozov, H. F. Gleeson, E.W. Hill, A. K. Geim, and K. S. Novoselov, Nano Lett. 8, 1704
/H208492008 /H20850.18Y. Hernandez, V. Nicolosi, M. Lotya, F. Blighe, Z. Sun, S. De, I.
T. McGovern, B. Holland, M. Byrne, Y. Gunko, J. Boland, P.Niraj, G. Duesberg, S. Krishnamurti, R. Goodhue, J. Hutchison,V. Scardaci, A. C. Ferrari, and J. N. Coleman, Nat. Nanotechnol.
3, 563 /H208492008 /H20850.
19G. Eda, G. Fanchini, and M. Chhowalla, Nat. Nanotechnol. 3,
270 /H208492008 /H20850.
20Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F.
Bonaccorso, D. M. Basko, and A. C. Ferrari, arXiv:0909.0457/H20849unpublished /H20850; T. Gokus, R. R. Nair, A. Bonetti, M. Bohmier, A.
Lombardo, N. S. Novoselov, A. K. Geim, A. C. Ferrari, and A.Hartschuh, arXiv:0909.3641 /H20849unpublished /H20850.
21A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri,
F. Mauri, S. Piscanec, K. S. Novoselov, Da Jiang, S. Roth, andA. K. Geim, Phys. Rev. Lett. 97, 187401 /H208492006 /H20850.
22C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyun-
yan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, Nano Lett. 7,
2711 /H208492007 /H20850.
23P. Blake, E. W. Hill, A. H. Castro Neto, K. S. Novoselov, D.
Jiang, R. Yang, T. J. Booth, and A. K. Geim, Appl. Phys. Lett.
91, 063124 /H208492007 /H20850.
24C. Casiraghi, S. Pisana, K. S. Novoselov, A. K. Geim, and A. C.
Ferrari, Appl. Phys. Lett. 91, 233108 /H208492007 /H20850.
25L. M. Malard, J. Nilsson, D. C. Elias, J. C. Brant, F. Plentz, E. S.
Alves, A. H. Castro Neto, and M. A. Pimenta, Phys. Rev. B 76,
201401 /H20849R/H20850/H208492007 /H20850.
26S. Pisana, M. Lazzeri, C. Casiraghi, K. S. Novoselov, A. K.
Geim, A. C. Ferrari, and F. Mauri, Nature Mater. 6, 198 /H208492007 /H20850.
27S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robert-
son, Phys. Rev. Lett. 93, 185503 /H208492004 /H20850.
28L. G. Cançado, R. Beams, and L. Novotny, arXiv:0802.3709
/H20849unpublished /H20850.
29C. Casiraghi, A. Hartschuh, H. Qian, S. Piscanec, C. Georgi, A.
Fasoli, K. S. Novoselov, D. M. Basko, and A. C. Ferrari, NanoLett. 9, 1433 /H208492009 /H20850.
30D. C. Elias, R. R. Nair, T. M. G. Mohiuddin, S. V. Morozov, P.
Blake, M. P. Halsall, A. C. Ferrari, D. W. Boukhvalov, M. I.Katsnelson, A. K. Geim, and K. S. Novoselov, Science 323, 610
/H208492009 /H20850.
31A. C. Ferrari, Solid State Commun. 143,4 7 /H208492007 /H20850.
32A. Das, S. Pisana, S. Piscanec, B. Chakraborty, S. K. Saha, U. V.
Waghmare, R. Yang, H. R. Krishnamurhthy, A. K. Geim, A. C.Ferrari, and A. K. Sood, Nat. Nanotechnol. 3, 210 /H208492008 /H20850.
33A. Das, B. Chakraborty, S. Piscanec, S. Pisana, A. K. Sood, andBASKO, PISCANEC, AND FERRARI PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-8A. C. Ferrari, Phys. Rev. B 79, 155417 /H208492009 /H20850.
34N. Ferralis, R. Maboudian, and C. Carraro, Phys. Rev. Lett. 101,
156801 /H208492008 /H20850.
35T. M. G. Mohiuddin, A. Lombardo, R. R. Nair, A. Bonetti, G.
Savini, R. Jalil, N. Bonini, D. M. Basko, C. Galiotis, N. Marzari,K. S. Novoselov, A. K. Geim, and A. C. Ferrari, Phys. Rev. B
79, 205433 /H208492009 /H20850.
36J. Yan, Y. Zhang, P. Kim, and A. Pinczuk, Phys. Rev. Lett. 98,
166802 /H208492007 /H20850.
37D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hi-
erold, and L. Wirtz, Nano Lett. 7, 238 /H208492007 /H20850.
38A. C. Ferrari and J. Robertson, Phys. Rev. B 61, 14095 /H208492000 /H20850;
Phys. Rev. B 64, 075414 /H208492001 /H20850.
39Raman spectroscopy in carbons: from nanotubes to diamond ,
edited by A. C. Ferrari and J. Robertson, theme issue of Philos.Trans. R. Soc. London, Ser. A 362, 2267 /H208492004 /H20850.
40C. Thomsen and S. Reich, Phys. Rev. Lett. 85, 5214 /H208492000 /H20850.
41F. Tuinstra and J. L. Koenig, J. Chem. Phys. 53, 1126 /H208491970 /H20850.
42D. M. Basko, Phys. Rev. B 76, 081405 /H20849R/H20850/H208492007 /H20850.
43D. M. Basko, Phys. Rev. B 78, 125418 /H208492008 /H20850.
44M. J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet,
K. Von Klitzing, and A. Yacoby, Nat. Phys. 4, 144 /H208492008 /H20850.
45X. Wang, X. Li, L. Zhang, Y. Yoon, P. K. Weber, H. Wang, J.
Guo, and H. Dai, Science 324, 768 /H208492009 /H20850.
46F. Cervantes-Sodi, G. Csanyi, S. Piscanec, and A. C. Ferrari,
Phys. Rev. B 77, 165427 /H208492008 /H20850.
47M. Lazzeri and F. Mauri, Phys. Rev. Lett. 97, 266407 /H208492006 /H20850.
48M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. Robert-
son, Phys. Rev. B 73, 155426 /H208492006 /H20850.
49A. Das, A. K. Sood, A. Govindaraj, A. M. Saitta, M. Lazzeri, F.
Mauri, and C. N. R. Rao, Phys. Rev. Lett. 99, 136803 /H208492007 /H20850.
50J. Yan, E. A. Henriksen, P. Kim, and A. Pinczuk, Phys. Rev. Lett.
101, 136804 /H208492008 /H20850.
51T. Ando, J. Phys. Soc. Jpn. 76, 104711 /H208492007 /H20850.
52S. Berciaud, S. Ryu, L. E. Brus, and T. F. Heinz, Nano Lett. 9,
346 /H208492009 /H20850; APS March Meeting Abstract, 2009, p. L26.00006.
53P. H. Tan, C. Y. Hu, J. Dong, W. C. Shen, and B. F. Zhang, Phys.
Rev. B 64, 214301 /H208492001 /H20850.
54Y. Y. Wang, Z. H. Ni, Z. X. Shena, H. M. Wang, and Y. H. Wu,
Appl. Phys. Lett. 92, 043121 /H208492008 /H20850; D. Yoon, H. Moon, Y.-W.
Son, J. S. Choi, B. H. Park, Y. H. Cha, Y. D. Kim, and H.Cheong, Phys. Rev. B 80, 125422 /H208492009 /H20850; E. Lidorikis and A. C.
Ferrari /H20849unpublished /H20850.
55G. Landsberg and L. Mandelshtam, Naturwiss. 16, 557 /H208491928 /H20850;
C. V. Raman and K. S. Krishnan, Nature /H20849London /H20850121, 501
/H208491928 /H20850;121, 619 /H208491928 /H20850.
56N. Bonini, M. Lazzeri, N. Marzari, and F. Mauri, Phys. Rev.
Lett. 99, 176802 /H208492007 /H20850.
57M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. Robert-
son, Phys. Rev. Lett. 95, 236802 /H208492005 /H20850.
58R. P. Vidano, D. B. Fischbach, L. J. Willis, and T. M. Loehr,
Solid State Commun. 39, 341 /H208491981 /H20850.
59I. Pócsik, M. Hundhausen, M. Koos, and L. Ley, J. Non-Cryst.
Solids 227-230 , 1083 /H208491998 /H20850.
60L. G. Cançado, A. Jorio, and M. A. Pimenta, Phys. Rev. B 76,
064304 /H208492007 /H20850.
61Z. Jiang, E. A. Henriksen, L. C. Tung, Y.-J. Wang, M. E.
Schwartz, M. Y. Han, P. Kim, and H. L. Stormer, Phys. Rev.Lett. 98, 197403 /H208492007 /H20850.
62A. Bostwick, Solid State Commun. 143,6 3 /H208492007 /H20850.63S. Y. Zhou, D. A. Siegel, A. V. Fedorov, and A. Lanzara, Phys.
Rev. B 78, 193404 /H208492008 /H20850.
64L. M. Zhang and M. M. Fogler, Phys. Rev. Lett. 100, 116804
/H208492008 /H20850.
65M. M. Fogler, D. S. Novikov, L. I. Glazman, and B. I. Shk-
lovskii, Phys. Rev. B 77, 075420 /H208492008 /H20850.
66E. Rossi and S. Das Sarma, Phys. Rev. Lett. 101, 166803 /H208492008 /H20850.
67M. Polini, A. Tomadin, R. Asgari, and A. H. MacDonald, Phys.
Rev. B 78, 115426 /H208492008 /H20850.
68J. González, F. Guinea, and M. A. H. Vozmediano, Phys. Rev.
Lett. 77, 3589 /H208491996 /H20850To be precise, the imaginary part of the
electronic self-energy has a discontinuity on the mass shell. Thisdiscontinuity is resolved by taking into account the real part aswell, which makes the electronic spectrum concave.
69A. A. Abrikosov and S. D. Beneslavskii, Zh. Eksp. Teor. Fiz 59,
1280 /H208491970 /H20850/H20851Sov. Phys. JETP 32, 699 /H208491971 /H20850/H20852.
70J. González, F. Guinea, and M. A. H. Vozmediano, Mod. Phys.
Lett. B 7, 1593 /H208491993 /H20850; Nucl. Phys. B 424, 595 /H208491994 /H20850;J .L o w
Temp. Phys. 99, 287 /H208491995 /H20850.
71E. H. Hwang, Ben Yu-Kuang Hu, and S. Das Sarma, Phys. Rev.
B76, 115434 /H208492007 /H20850.
72M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea, and
A. H. MacDonald, Phys. Rev. B 77, 081411 /H20849R/H20850/H208492008 /H20850.
73S. Latil, V. Meunier, and L. Henrard, Phys. Rev. B 76,
201402 /H20849R/H20850/H208492007 /H20850.
74Note that substrate interference effects /H20849Ref.54/H20850could mean that
the measured ratios A/H208492D/H20850/A/H20849G/H20850may differ from those intrinsic
to graphene by a constant factor, common for all the data points.Rescaling of the yaxis in Fig. 4by a constant factor, together
with the same rescaling of the coefficient 0.26 in Eq. /H208496/H20850, does
not affect the value of
/H9253e-phextracted from the fit. We note as
well that the calculated rescaling of A/H208492D/H20850/A/H20849G/H20850for SiO 2thick-
nesses of 0, 100, and 300 nm /H20849i.e., those most common in ex-
periments /H20850is anyway negligible,54thus again validating our fit-
ted/H9253e-ph.
75P. R. Wallace, Phys. Rev. 71, 622 /H208491947 /H20850.
76J. Maultzsch, S. Reich, C. Thomsen, H. Requardt, and P. Orde-
jón, Phys. Rev. Lett. 92, 075501 /H208492004 /H20850.
77S. Piscanec, M. Lazzeri, J. Robertson, A. C. Ferrari, and F.
Mauri, Phys. Rev. B 75, 035427 /H208492007 /H20850.
78M. Lazzeri, C. Attaccalite, L. Wirtz, and F. Mauri, Phys. Rev. B
78, 081406 /H20849R/H20850/H208492008 /H20850.
79A. Grüneis, J. Serrano, A. Bosak, M. Lazzeri, S. L. Molodtsov,
L. Wirtz, C. Attaccalite, M. Krisch, A. Rubio, F. Mauri, and T.Pichler, Phys. Rev. B 80, 085423 /H208492009 /H20850.
80A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg,
Nat. Phys. 3,3 6 /H208492007 /H20850.
81S. Y. Zhou, G. H. Gweon, and A. Lanzara, Ann. Phys. /H20849N.Y. /H20850
321, 1730 /H208492006 /H20850.
82P. E. Trevisanutto, C. Giorgetti, L. Reining, M. Ladisa, and V.
Olevano, Phys. Rev. Lett. 101, 226405 /H208492008 /H20850.
83T. Ando, J. Phys. Soc. Jpn. 75, 124701 /H208492006 /H20850.
84M. Calandra and F. Mauri, Phys. Rev. B 76, 205411 /H208492007 /H20850.
85C. S. Leem, B. J. Kim, Chul Kim, S. R. Park, T. Ohta, A.
Bostwick, E. Rotenberg, H.-D. Kim, M. K. Kim, H. J. Choi, andC. Kim, Phys. Rev. Lett. 100, 016802 /H208492008 /H20850.
86A. Grüneis, C. Attaccalite, A. Rubio, D. V. Vyalikh, S. L.
Molodtsov, J. Fink, R. Follath, W. Eberhardt, B. Büchner, and T.Pichler, Phys. Rev. B 79, 205106 /H208492009 /H20850.ELECTRON-ELECTRON INTERACTIONS AND DOPING … PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-987G. Li, A. Luican, and E. Y. Andrei, Phys. Rev. Lett. 102, 176804
/H208492009 /H20850.
88D. M. Basko and I. L. Aleiner, Phys. Rev. B 77, 041409 /H20849R/H20850
/H208492008 /H20850.89A. Grüneis, C. Attaccalite, T. Pichler, V. Zabolotnyy, H.
Shiozawa, S. L. Molodtsov, D. Inosov, A. Koitzsch, M. Kn-upfer, J. Schiessling, R. Follath, R. Weber, P. Rudolf, L. Wirtz,and A. Rubio, Phys. Rev. Lett. 100, 037601 /H208492008 /H20850.BASKO, PISCANEC, AND FERRARI PHYSICAL REVIEW B 80, 165413 /H208492009 /H20850
165413-10 |
PhysRevB.73.134107.pdf | First-principles prediction of the mechanical properties and electronic structure of ternary
aluminum carbide Zr 3Al3C5
Jingyang Wang,1,2Yanchun Zhou,1Zhijun Lin,1,3Ting Liao,1,3and Ling Feng He1,3
1Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences,
Shenyang 110016, China
2International Centre for Materials Physics, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China
3Graduate School of Chinese Academy of Sciences, Beijing 100039, China
/H20849Received 22 June 2005; revised manuscript received 8 February 2006; published 12 April 2006 /H20850
In this paper, we predicted the possible mechanical properties and presented the electronic structure of
Zr3Al3C5by means of first-principles pseudopotential total energy method. The equation of state, elastic
parameters /H20849including the full set of second order elastic coefficients, bulk and shear moduli, Young’s moduli,
and Poisson’s ratio /H20850, and ideal tensile and shear strengths are reported and compared with those of the binary
compound ZrC. Furthermore, the bond relaxation and bond breaking under tensile and shear deformation fromelasticity to structural instability are illustrated. Because shear induced bond breaking occurs inside the NaCl-type ZrC
xslabs, the ternary carbide is expected to have high hardness and strength, which are related to
structural instability under shear deformation, similar to the binary carbide. In addition, mechanical propertiesare interpreted by analyzing the electronic structure and chemical bonding characteristics accompanying de-formation paths. Based on the present results, Zr
3Al3C5is predicted to be useful as a hard ceramic for high
temperature applications.
DOI: 10.1103/PhysRevB.73.134107 PACS number /H20849s/H20850: 81.05.Je, 62.20.Dc, 71.20. /H11002b
I. INTRODUCTION
Refractory binary transition metal carbides /H20849TMC /H20850, such
as TiC, NbC, ZrC, and HfC, are characterized by high hard-ness, high strength, high melting point, good thermal shockand wear resistance, and chemical inertness. These carbidesare widely used in high temperature environments or as hardceramics.
1However, poor oxidation resistance and intrinsic
brittleness have restricted their applications. Recent develop-ments in synthesizing ternary aluminum carbides by incorpo-rating Al in binary carbides highlight a possible way to solvethese problems. It has been reported that ternary titaniumaluminum carbide, Ti
3AlC 2, platelets were found to form in
Al doped TiC.2Prepared under proper conditions, new ter-
nary aluminum carbides have been successfully synthesizedin Ti-Al-C, Nb-Al-C, Zr-Al-C, and Hf-Al-C systems, usingthe binary transition-metal carbides and aluminum as startingmaterials.
3
Ternary aluminum carbides in the Ti-Al-C and Nb-Al-C
systems, such as Ti 3AlC 2,T i 2AlC, and Nb 2AlC, have been
identified to crystallize with the space group P63/mmc.4,5
Microscopic investigations have shown that all of these ter-
nary carbides consist of two alternately stacked structuralunits. The crystal structures can be described as nanoscaleblocks of TiC
xor NbC xin a NaCl-type structure that is in-
tercalated and mirrored by close-packed Al atomic planes.These ternary aluminum carbides display outstanding prop-erties, such as high damage tolerance, good high temperatureoxidation resistance, and intrinsic toughness at room tem-perature due to their nanolaminated crystal structure.
6Fur-
thermore, these ternary carbides have high moduli andstrength that are desirable for structural materials.
Compounds in the Zr-Al-C system, on the other hand,
have different types of crystal structures. Three equilibriumphases, Zr
2Al3C5,Z r 3Al3C5/H20849previously reported with the
chemical formula of ZrAlC 2−xin Ref. 7 /H20850, and Zr 5Al3C, have
been reported in the Zr-Al-C system.5Among them,
Zr2Al3C5and Zr 3Al3C5were determined to have the hexago-
nal symmetry, and Zr 5Al3C was identified with the
Mo 5Si3-type crystal structure. Compared to the extensively
studied Ti-Al-C and Nb-Al-C based ceramics, the propertiesof ternary Zr-Al-C compounds are much less known. Thereasons are attributed to difficulties in synthesizing bulk Zr-Al-C materials.
Recently, the fracture strength and Vickers hardness were
primitively characterized for bulk Zr
2Al3C5and ZrAlC 2
/H20849Zr3Al3C5/H20850sintered from Zr 2Al3C5and ZrAlC 2powders pre-
pared by the solid-state reaction of ZrC, C, and Al.8,9The
authors have conducted theoretical studies of the electronicstructure, chemical bonding, and equations of state ofZr
2Al3C5by means of ab initio pseudopotential total energy
calculations.10Zr2Al3C5was predicted to have interesting
properties such as low hardness, easy machinability, damagetolerance, and oxidation resistance based on its nanolami-nated structure.
10Results also showed that Zr 2Al3C5might
have properties different from the binary carbide, ZrC. It wasshown that the intrinsic oxidation resistance of binary car-bide, such as TiC, could be greatly improved by making itinto ternary carbide that contains Al, such as Ti
3AlC 2.11The
excellent oxidation resistance resulted from the protectiveAl
2O3scale, which forms in high temperature environments.
For ternary Zr-Al-C based compounds, like Zr 3Al3C5, if con-
tinuous ZrO 2and/or Al 2O3scales can form, the material
would have good oxidation resistance at high temperatures.Recently, the authors have successfully synthesized Zr
3Al3C5
powders by means of a pressureless sintering using Zr-Alintermetallics and graphite as starting materials.
12Tetragonal
ZrO 2and/H9251-Al2O3were identified after Zr 3Al3C5powdersPHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
1098-0121/2006/73 /H2084913/H20850/134107 /H208499/H20850/$23.00 ©2006 The American Physical Society 134107-1were oxidized in air at temperatures up to 1500 °C. Unfor-
tunately, the oxidation kinetics of bulk Zr 3Al3C5is still not
fully clarified. Despite these new developments, intrinsic me-chanical, physical, and chemical properties are not yet avail-able for Zr
3Al3C5.
As can be seen from the crystal structures shown in Fig.
1, Zr 3Al3C5and Zr 2Al3C5have distinguishably different
crystal structures. The main difference centers on the atomicconfigurations linking the ZrC
xslabs. As described by Ges-
ing and Jeitschko,13Zr3Al3C5may be viewed as an inter-
grown structure consisting of two kinds of layers.13One is
the nonstoichiometric ZrC xslab in a NaCl-type structure and
the other consists of Al and C atoms in an arrangement simi-lar to that of the binary aluminum carbide, Al
4C3. The non-
stoichiometric ZrC xslabs in Zr 2Al3C5, however, are linked
by different Al-C units. Although both materials containZrC
xslabs, the mechanical properties of Zr 3Al3C5may be
different from those of Zr 2Al3C5, because of the different
interplanar adhesive strength linking the ZrC xslabs. There-
fore, it is necessary to clarify how the mechanical propertiesof such a material are determined by their intergrown struc-ture of ZrC
xslabs and Al 4C3-type Al-C layers. Because of
this crystallographic arrangement, it is of particular interestto investigate the mechanical properties and chemical bond-ing characteristics of Zr
3Al3C5. In this study, we computed
the equation of state /H20849EOS /H20850, elastic moduli, ideal strengths,
electronic structure, and chemical bonding characteristics ofZr
3Al3C5using the first-principles computational scheme.
The investigation aims to predict possible mechanical prop-erties, and further, to show the relationship between the elec-tronic structure and the mechanical properties.
The remainder of this paper is organized as follows. The
computational details are described in Sec. II. In Sec. III, wepresent our results for equilibrium geometry and electronicstructure characteristics. The EOS, elastic stiffness, the fullset of elastic coefficients, and mechanical parameters are re-ported in Sec. IV. The ideal stress-strain relationship, idealtensile and shear strength, together with the processes ofbond relaxation and bond breaking for material strained fromelasticity to structural instability, are illustrated in Sec. V.
Finally, the concluding remarks are given in Sec. VI.
II. COMPUTATIONAL DETAILS
The CASTEP code was used in the present calculations,14
wherein the Vanderbilt-type ultrasoft pseudopotential15and
generalized gradient approximation16/H20849GGA-PW91 /H20850were
employed. The plane-wave basis set cutoff was 450 eV forall calculations. The special points sampling integration overthe Brillouin zone was employed by using the Monkhorst-Pack method with a 10 /H1100310/H110032 special k-points mesh.
17To
investigate the ground state electronic structure and EOS, theequilibrium crystal structures were optimized at various iso-tropic hydrostatic pressures ranging from 0 to 50 GPa. Lat-tice parameters, including lattice constants and internalatomic coordinates, were modified independently to mini-mize the enthalpy and interatomic forces. The Broyden-Fletcher-Goldfarb-Shanno /H20849BFGS /H20850minimization scheme
18
was used in geometry optimization. The tolerances for geom-
etry optimization are difference on total energy within 5/H1100310
−6eV/atom, maximum ionic Hellmann-Feynman force
within 0.01 eV/Å, maximum ionic displacement within 5/H1100310
−4Å and maximum stress within 0.02 GPa. We have
shown that the present first-principles calculation scheme isreliable on predicting crystal structure, elastic stiffness, andinteratomic force constants of ternary transition metalcarbides.
19–21
The calculations of projected density of states were per-
formed using a projection of the plane-wave electronic statesonto a localized linear combination of atomic orbitals/H20849LCAO /H20850basis set. In the present calculation, the LCAO basis
set was the atomic pseudo-orbitals corresponding to theclosed valence shell containing the valence electrons. Thenumbers of pseudo-orbitals were chosen as 4 for C, 4 for Al,and 9 for Zr. The s,pvalence orbitals of C, as well as the s,
porbitals of Al and the s,p, and dorbitals of Zr were
included in the calculation of partial density of states/H20849PDOS /H20850. Since Zr
3Al3C5is a metallic system, partial occu-
pancies were introduced to eliminate discontinuous changesin the total energy, which were created when energy bandscrossed the Fermi level during self-consistent electronicminimization. Twelve additional empty bands were includedin the electronic minimization, and we used the Gaussiansmearing scheme with a smearing width of 0.1 eV.
The elastic coefficients were determined from a first-
principles calculation by applying a set of given homoge-neous deformations with a finite value and calculating theresulting stress with respect to optimizing the internal de-grees of freedoms, as implemented by Milman et al.
22The
criteria for convergence in optimizing atomic internal free-doms were selected as follows: difference on total energywithin 1 /H1100310
−6eV/atom, ionic Hellmann-Feynman forces
within 0.002 eV/Å and maximum ionic displacement within1/H1100310
−4Å. Two strain patterns, one with nonzero /H925511and/H925523
components and the other with a nonzero /H925533, generated
stresses related to all five independent elastic coefficients fora unit cell with a hexagonal symmetry. Three positive andthree negative amplitudes were applied for each strain com-
FIG. 1. /H20849Color online /H20850Crystal structures of Zr 2Al3C5and
Zr3Al3C5. The Zr, Al, and C atoms are indexed with numbers ac-
cording to various coordination environments.WANG et al. PHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
134107-2ponent with the maximum strain value of 0.5%. We deter-
mined the elastic stiffness from a linear fit of the calculatedstress as a function of strain. The compliance tensor Swas
calculated as the inverse of the stiffness tensor, S=C
−1. Other
mechanical parameters, such as the bulk modulus, Young’smoduli, and Poisson’s ratio were calculated from the compli-ance tensor. The shear modulus was calculated according tothe Voigt approximation.
23
Deformation and failure modes of Zr 3Al3C5lattice are
studied. To obtain the ideal stress-strain relationships, weemployed a method widely used to study the ideal strengthand lattice stability of metals and ceramics.
24–27By deform-
ing the crystal from its elastic state to structural instability,the stress-strain curves are computed for /H208550001 /H20856/H208490001 /H20850
uniaxial tension and /H2085512¯10/H20856/H208490001 /H20850shear deformation /H20849a pos-
sible easy slip system for hexagonal compound with large
c/aratio /H20850. For the tensile deformation, a series of incremen-
tal tensile strains were applied to the crystal. To ensure thatthe material was under an uniaxial stress state, relaxation of
the structure perpendicular to the applied strain direction wasperformed by holding the applied strain fixed and adjustingthe other two normal strain components independently untilthe calculated conjugate Hellmann-Feynman stresses were
both less than 0.2 GPa. For the /H2085512
¯10/H20856/H208490001 /H20850shear deforma-
tion path, the crystal was relaxed until all of the stresses
orthogonal to the applied stress are reduced to less than0.2 GPa. These computations produce the stress versus straincurves for both the tensile and shear deformations. The first-reached maximum in these curves is the ideal strength of thematerial under a particular strain path, provided that no otherinstability occurs before it. For comparison, the stress-straincurves for ZrC were also computed for the /H20849001 /H20850/H20855001 /H20856ten-
sion and /H20849110 /H20850/H208551
¯10/H20856shear deformation /H20849the primary slip sys-
tem for the NaCl-type transition metal carbide at low
temperature /H20850.28III. EQUILIBRIUM GEOMETRY AND
ELECTRONIC STRUCTURE
To compute the equilibrium crystal structure of Zr 3Al3C5,
we optimized the previously reported crystal structure13by
relaxing the cell degrees of freedom with constrained spacegroup P6
3mc. Then, we analyzed the space group of a fully
relaxed unit cell. It shows that Zr 3Al3C5belongs to the space
group P63/mmc within a precision of 0.005 Å by performing
symmetry operations. For the initial structure with spacegroup P6
3mc, the Al /H208492/H20850atoms were located off the center of
the trigonal bipyramid formed by C /H208491/H20850and C /H208492/H20850atoms. In
the present computation, the Al /H208492/H20850atoms locate on a mirror
plane on the center of the trigonal bipyramid, which yields toa higher symmetry space group P6
3/mmc. Very recently, Lin
et al. have confirmed the space group P63/mmc for Zr 3Al3C5
in experiments by means of selected area electron diffraction
and convergent beam electron diffraction methods.29Theo-
retical lattice parameters are listed in Table I, together withthe experimental values for comparison. The computed lat-tice constants aandcare consistent with experimental data
within 1% deviation. Furthermore, the two sets of internalatomic parameters, z, also have close values. The theoretical
density is 5.280 g/cm
3and it agrees well with the experi-
mental value found in the JCPDS data file /H20849card no. 32-
0030 /H20850, 5.282 g/cm3. Therefore, the present first-principles
computation is able to reliably reproduce the equilibriumcrystal structure of Zr
3Al3C5.
To better understand the nature of the interatomic bond-
ing, the electronic structure was examined. Figure 2 showsthe total and projected density of states /H20849PDOS /H20850of Zr
3Al3C5.
The lowest lying states from −14.39 to −8.78 eV originatemainly from the C 2 sorbitals with slight contributions from
Al-pand Zr- dorbitals. The p-pand p-dcovalent bonding
dominate the states ranging from −7.38 eV to the Fermilevel. From −7.38 to −3.77 eV, the p-pand s-pbonding
states come from C-Al interatomic bonds. We compared theTABLE I. Theoretical and experimental crystal symmetry, lattice constants aandc/H20849in Å /H20850,c/aratio, free
internal atomic parameters z/H20849atom /H20850of Zr 3Al3C5, as well as the bulk modulus B/H20849in GPa /H20850and its pressure
derivative B/H11032derived from the equation of state.
Method Symmetry ac c /az /H20849atom /H20850 BB /H11032
Calc. P63/mmc 3.316 27.387 8.259 z/H20849C1/H20850=0.0502 205 3.8
z/H20849C2/H20850=0.2498
z/H20849C3/H20850=0.1535
z/H20849Al1 /H20850=0.1777
z/H20849Al2 /H20850=0.2497
z/H20849Zr1 /H20850=0.5960
Expt.aP63mc 3.343 27.609 8.259 z/H20849C1/H20850=0.0510
z/H20849C2/H20850=0.2511
z/H20849C3/H20850=0.1508
z/H20849Al1 /H20850=0.1776
z/H20849Al2 /H20850=0.2448
z/H20849Zr1 /H20850=0.5955
aReference 13.FIRST-PRINCIPLES PREDICTION OF THE ¼ PHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
134107-3Zrdt2g−Cp/H9268-like bonds in ZrC and Zr 3Al3C5, and found
that the characteristics of Zr-C bonds are rather similar in thetwo carbides. The C 2 p−Zr 4 dbonding peak is located at
around −2.88 eV away from the Fermi level in Zr
3Al3C5,
which is similar to the corresponding peak at around−2.40 eV in ZrC. In addition, the bond lengths of Zr-C are2.289 Å, 2.360 Å, and 2.480 Å in Zr
3Al3C5, and 2.345 Å in
ZrC. Therefore, the strong Zr-C bonds are well preserved inthe ternary carbide. It should be noted that the bond length ofZr2-C2 /H208492.480 Å /H20850is about 8.34% and 5.08% longer than that
of Zr1-C3 /H208492.360 Å /H20850and Zr2-C3 /H208492.289 Å /H20850, respectively.
This implies that the Zr2-C2 bond may be weaker than other
Zr-C bonds in Zr
3Al3C5.
The bonding peak located at around −1.83 eV corre-
sponds to the Al 3 p−C 2 pcovalent bonds. These hybridiza-
tion states are located at a higher energy range than the Zr4d−C 2 pbonding states. Therefore, the chemical strength of
certain Al-C bond is weaker than that of the Zr-C covalentbonds. There are two types of Al-C bonds in Zr
3Al3C5ac-cording to different atomic arrangements. Identifying the
character of these Al-C bonds is very important to predict themechanical properties of Zr
3Al3C5, because variations of
these bonding strengths will lead to distinctly different per-formances. If the Al1-C2 is weaker than the Al1-C1 bond,Zr
3Al3C5could be regarded as a combination of ZrC xslabs
and Al-C blocks with a Al 4C3structure. On the other hand, if
the Al1-C2 is stronger than the Al1-C1 bond, Zr 3Al3C5could
be described as covalently bonded Zr-C-Al blocks being in-terleaved by close-packed Al-C atomic planes. Unfortu-nately, strengths of the Al1-C1 and Al1-C2 bonds could notbe obtained from the PDOS figures, due to an extensive en-ergy range of the p-derived states. To quantitatively analyze
the adhesions of Al-C bonds, the adhesive energies were cal-culated by cleaving the Al1-C2 and Al1-C1 bonds parallel tothe basal plane.
The energy of breaking the Al1-C2 bonds E
adhesi veAl1-C2was
computed by
Eadhesi veAl1-C2=EtotalZr3C4/H20002+Etotal/H20002Al3C−EtotalZr3Al3C5, /H208491/H20850
where EtotalZr3C4/H20002is the total energy of Zr 3C4/H20002blocks after
eliminating Al 3C blocks from the unit cell, Etotal/H20002Al3Cis the total
energy of Al 3C blocks after eliminating the Zr 3C4/H20002blocks
from the unit cell and EtotalZr3Al3C5is the total energy of
Zr3Al3C5. In the same way, the adhesive energy of breaking
the Al1-C1 bonds Eadhesi veAl1-C1was computed by
Eadhesi veAl1-C1=EtotalZr3C4Al2/H20002+Etotal/H20002AlC−EtotalZr3Al3C5, /H208492/H20850
where the EtotalZr3C4Al2/H20002is the total energy of Zr 3C4Al2/H20002blocks
after eliminating AlC atomic planes from the unit cell, Etotal/H20002AlC
is the total energy of Al-C atomic plane after eliminating
Zr3C4Al2/H20002blocks from the unit cell and EtotalZr3Al3C5is the total
energy of Zr 3Al3C5.
The computed adhesive energies yield 1.18 eV/atom and
0.41 eV/atom by breaking the Al1-C2 and Al1-C1 bonds,respectively. Therefore, the adhesion strength of the Al1-C1bond is much weaker than that of the Al1-C2 bond. We fur-ther calculated the adhesive energy of Zr2-C2 bond, which isthe longest Zr-C bond in Zr
3Al3C5. The result yields
0.92 eV/atom, which is located between that of the Al1-C1and Al1-C2 bonds. The bonding strength results suggest that,at the equilibrium crystal structure, Zr
3Al3C5could be de-
scribed as strong covalently bonded Al-C-Zr-C-Zr-C-Zr-C-Al blocks interleaved by close-packed Al-C atomic planes.In addition, the Al-C adhesion between those covalent-bonded atomic chains and Al-C planes are relatively weak.
To illustrate the bonding characteristics, Fig. 3 shows the
charge density distribution for a slice of the /H20849112
¯0/H20850plane in
a2/H110032/H110031 supercell. Figure 3 obviously shows strong cova-
lent bonding within the Al-C-Zr-C-Zr-C-Zr-C-Al atomicblocks /H20849abbreviated as Zr
3C4Al2/H20002/H20850. The adjacent Zr 3C4Al2
/H20002blocks are interleaved and mirrored by Al-C atomic
planes. The electronic density is relatively low in the regionbetween Zr
3C4Al2/H20002and Al-C units, which leads to weaker
interplanar adhesion.
FIG. 2. /H20849Color online /H20850Total and projected electronic density of
states of ZrC and Zr 3Al3C5.WANG et al. PHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
134107-4IV. EQUATION OF STATE AND ELASTIC STIFFNESS
Figure 4 plots the relative unit cell volume, V/V0,a sa
function of external pressure. By fitting the data with theBirch-Murnaghan equation,
30bulk modulus B0and its pres-
sure derivative B0/H11032are obtained to be 205 GPa and 3.8, re-
spectively. The bulk modulus of Zr 3Al3C5is comparable to
that of ZrC /H20849229 GPa /H20850, computed within the same first-
principles scheme. The B0of Zr 3Al3C5is definitely higher
than that of Zr 2Al3C5, which has a value of only 160 GPa.10
In Fig. 5, we present the pressure dependence of lattice con-stants aandcand display changes of the axial ratio, c/a,i n
the inset. The c/aratio decreases almost linearly with in-
creasing hydrostatic pressure continuously to /H1101125 GPa, and
then it decreases less rapidly at higher pressures. The trend inc/aversus pressure demonstrates that ccontracts more dra-
matically than ain the pressure range examined. Therefore,the material is stiffer in the basal plane than in cwhen
Zr
3Al3C5is under isotropic pressure. This elastic anisotropy
suggests that interplanar bonding along cis weaker than in-
traplanar bonding along the basal plane in Zr 3Al3C5.
Anisotropic elasticity has been investigated for Ti 3SiC 2at
various pressures both by experiment31and by first-
principles calculation.32In these studies, the strength of in-
teratomic bonding was characterized by its resistance against
external pressure. The Ti-Si and Ti-C bonds shrank by 8%and 5%, respectively, when hydrostatic pressure increasedfrom 0 to 50 GPa. Combined with electronic structure analy-sis, the interplanar Ti-Si covalent bond was found to beweaker than the Ti-C bonds in TiC
0.67blocks. This difference
in bond strength led to the mechanical anisotropy of Ti 3SiC 2.
Following the same method to evaluate the strengths of in-teratomic bonding against pressure for Zr
3Al3C5, we exam-
ine the degree of bond length contraction under various pres-sures, and illustrate the results in Fig. 6. The lowest lyingcurve is seen to be associated with the Al1-C1 bond, which isthe most compressible. Above it are the curves for Zr2-C2,Al1-C2, Zr2-C3, and Al2-C1 covalent bonds, and thesebonds show similar compressibility with increasing appliedpressure. The least compressible among these is the Zr2-C3bond. Figure 6 indicates that the Al1-C1 bonding is softerthan other interatomic bonds, such as Zr2-C2 and Al1-C2bonds, in Zr
3Al3C5under hydrostatic pressure.
The elastic stiffness of a crystal determines its response to
an applied strain /H20849or stress /H20850near equilibrium and provides
FIG. 3. /H20849Color online /H20850Valence electron density of a slice of the
/H20849112¯0/H20850plane in a 2 /H110032/H110031 supercell. The white contour lines range
from 0.08 to 0.27 electrons/Å3.
FIG. 4. Relative unit cell volume V/V0as a function of external
hydrostatic pressure. The bulk modulus B0and its pressure deriva-
tive B0/H11032are determined to be 205 GPa and 3.8, respectively, by
fitting the data with the Birch-Murnaghan equation.
FIG. 5. /H20849Color online /H20850Pressure dependence of lattice constants
aandc. The inset shows the c/aratio versus pressure.
FIG. 6. /H20849Color online /H20850Relative bond-length contractions at vari-
ous pressures.FIRST-PRINCIPLES PREDICTION OF THE ¼ PHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
134107-5information about bonding characteristics. To our best
knowledge, the elastic moduli of Zr 3Al3C5have not been
reported. In Table II, we include the computed full set ofsecond order elastic coefficients of Zr
3Al3C5, together with
the theoretical and experimental values of ZrC forcomparison.
33The theoretical data agree well with experi-
mental values of ZrC. It is also noted that Zr 3Al3C5and ZrC
have similar elastic coefficients. For example, the modulirepresenting stiffness against uniaxial strains, c
11andc33of
Zr3Al3C5, are about 94% and 83%, respectively, of c11of
ZrC. The c44andc66of Zr 3Al3C5, which correspond to the
resistance against /H20853100 /H20854/H20855110 /H20856and /H20853010 /H20854/H20855001 /H20856shear deforma-
tions, are about 1.18 and 1.05 times, respectively, of c44of
ZrC.
Table III presents the computed mechanical parameters of
Zr3Al3C5, such as bulk modulus B, shear modulus G, aniso-
tropic Young’s moduli E, and Poisson’s ratio, together with
theoretical and experimental values of ZrC.33The magni-
tudes of the mechanical parameters of Zr 3Al3C5are rather
similar to those of ZrC, varying within 10%. This impliesthat the complex ternary aluminum carbide Zr
3Al3C5has
comparable elastic stiffness to the binary carbide ZrC.
V. BOND-BREAKING AND IDEAL STRENGTHS
Material deformation is strain dependent and elastic pa-
rameters may not always give accurate account for all mac-roscopic mechanical properties, such as hardness. The reasoncan be attributed to the fact that these elastic parameters arecomputed under equilibrium conditions; while material de-formation associated with experimentally obtained strengthsoccurs at a specific range of strains where bonding charac-teristics change significantly. Therefore, studies of the stress-strain relationships following a material strained from elas-ticity to the limit of its structural stability and the underlyingbond-responding processes are useful in understanding its
experimental strengths and hardness at ambient conditions.
In Fig. 7, we present the calculated stress-strain curves for
Zr
3Al3C5. Several interesting features are noticed for the
/H2085512¯10/H20856/H208490001 /H20850shear path: /H20849i/H20850stress increases with strain al-
most linearly before the material reaches structural instabil-
ity; /H20849ii/H20850shear stress drops abruptly after a critical strain; /H20849iii/H20850
there is no obvious “yielding” when the compound passesfrom elasticity to structural instability. On the other hand, forthe /H208490001 /H20850/H208550001 /H20856tension: /H20849i/H20850breakdown of elasticity occurs
long before the strain reaches a critical bond-breaking point;/H20849ii/H20850there is an obvious “yielding” that follows a gradual re-
duction of the tensile stress after its maximum value. In thefollowing discussions, we will show that these features origi-nate from the softening and breaking of Al-C and Zr-C bondsthat are involved in different deformation modes.
To understand the trend in the stress-strain curve for shear
deformation, we examined the bond-relaxation processes atvarious strains. Figure 8 shows the valence charge density
distribution on the /H20849112
¯0/H20850atomic plane in Zr 3Al3C5at vari-
ous shear strains. It shows that all bonds accommodate strain
almost homogeneously, and no local bond softening occursbefore the critical strain, as seen in Fig. 8 /H20849a/H20850for/H9255=0.05.
Both the Al-C and Zr-C bonds remain strong up to the bond-breaking point. Of the most interest, breaking of the Zr2-C2bond, instead of the weakest Al1-C1 bond, is responsible forthe structural instability of Zr
3Al3C5under shear deformation
as shown in Fig. 8 /H20849b/H20850for/H9255=0.175. Because failure of this
material occurs inside the NaCl-type ZrC xslabs, it is likely
that the binary and ternary carbides will show similar me-TABLE II. Computed second order elastic coefficients cij/H20849in
GPa /H20850of ZrC and Zr 3Al3C5, together with experimental values of
ZrC for comparison.
Method c11 c12 c44 c33 c13 c66
ZrC Calc. 455 116 152
Expt.a472 99 159
Zr3Al3C5Calc. 429 110 179 378 93 160
aReference 33.
TABLE III. Computed bulk modulus B, shear modulus G, anisotropic Young’s moduli E, and Poisson’s
ratio/H9271of ZrC and Zr 3Al3C5, together with experimental values of ZrC polycrystalline for comparison.
Method B/H20849GPa /H20850 G/H20849GPa /H20850 E/H20849GPa /H20850 /H9271
ZrC. Cal. 229 170 408 vxy=0.21
Expt.a223 170 407 vxy=0.19
Zr3Al3C5 Calc. 202 166 Ex=388 vxy=0.22
Ez=346 vxz=0.19
vzx=0.17
aReference 33.
FIG. 7. /H20849Color online /H20850Ideal stress-strain curves of tensile and
shear deformation for Zr 3Al3C5.WANG et al. PHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
134107-6chanical properties that are determined by shear-induced
structural instability.
When we focus on the valence charge density distributing
on the /H20849112¯0/H20850atomic plane in Zr 3Al3C5under tension, a
different bond-relaxation mechanism is observed. As shown
in Fig. 9, the softening and breaking of the weakest Al1-C1bonds is now responsible for the failure of Zr
3Al3C5. In con-
trast to Al-C bonds being stable under shear deformation, theAl1-C1 bond softens considerably under tensile strains long
FIG. 9. /H20849Color online /H20850Valence electron density of a slice of the
/H20849112¯0/H20850plane in a 2 /H110032/H110031 supercell under tensile deformations of
/H20849a/H20850/H9255=0.05, and /H20849b/H20850/H9255=0.25. Zr 3Al3C5is seen to cleave by breaking
the Al1-C1 bonds.
FIG. 8. /H20849Color online /H20850Valence electron density of a slice of the
/H20849112¯0/H20850plane in a 2 /H110032/H110031 supercell under shear deformations of
/H20849a/H20850/H9255=0.05, and /H20849b/H20850/H9255=0.175. Obviously, the shear-induced struc-
tural instability occurs by breaking the Zr2-C2 bonds in the ZrC x
slab.FIRST-PRINCIPLES PREDICTION OF THE ¼ PHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
134107-7before bond breaking. The bond length of Al1-C1 elongates
6%, 17%, 32%, 57%, and 78% under tensile strains of 0.05,0.10, 0.15, 0.20, and 0.25, respectively. This produces thenotable “yielding” in the stress-strain curve shown in Fig. 7.When the tensile stress increases, the Al1-C1 bond does notabruptly experience a bond-breaking event, because of thedelocalized nature and spatial extension of its p-phybridiza-
tion. Therefore, “yielding” occurs, which follows a gradualreduction of the tensile stress under increased tensile strains.On the other hand, the Zr-C bonds in the NaCl-type ZrC
x
slabs remain stable during tensile deformation along the c
direction.
From the computed stress-strain curves, the ideal shear
and tensile strengths are 26.5 GPa and 26.8 GPa, respec-tively. These values lead to a shear-to-tensile strength ratio ofabout 1. For the binary carbide, ZrC, we compounded theideal shear and tensile strengths, which yield 29 GPa and31 GPa, respectively. The presented ideal shear strength ofZrC, 29 GPa, is comparable to some transition metal car-bides TiC, TiN, and HfC, which were reported as 35, 31, and30 GPa, respectively, by Jhi et al.
24The strengths of
Zr3Al3C5are, therefore, comparable with those of ZrC, and
they represent the upper limit of stresses attainable prior tofailure. Again, comparison on these ideal strengths indicatesthat Zr
3Al3C5may have similar mechanical properties to
ZrC.
Generally speaking, hardness reflects the resistance of a
material against permanent plastic deformation. The mecha-nism for plastic deformation involves the nucleation andmovement of dislocations. The mobility of dislocation couldbe estimated from the Peierls stress or the stress requiredmoving a dislocation one atomic Burgers vector. For stronglycovalent materials, like binary transition metal carbides, thevery low mobility of dislocation kinks is the rate-determining factor for dislocation motion and is in turn de-termined by the very high bond-breaking energy under alarge shear strain.
34–36With the help of calculated mechani-
cal parameters, we estimate the maximum value of Peierlsshear stress to initiate the movement of a dislocation in itsglide plane for Zr
3Al3C5by37
/H9268S=2G
1−/H9263exp/H20873−2/H9266/H9261
b/H20874 /H208493/H20850
where /H9261=3−2 /H9263/4/H208491−/H9263/H20850d,Gis the shear modulus, bthe
Burgers vector, dthe spacing between atomic slip plans, and
/H9263the Poisson’s ratio.Following Krenn et al. ,38we use dby 1/6 /H20855111 /H20856andbby
1/2 /H20855110 /H20856for ZrC with NaCl-type structure. Computed from
the lattice constant /H208494.691 Å /H20850of ZrC, dis 1.354 Å and bis
3.317 Å. Because slip in Zr 3Al3C5occurs by breaking the
Zr2-C2 bonds, the interplanar distance between Zr2-C2atomic planes is used as dto calculate its Peierls stress.
Therefore, dis set as 1.576 Å and bis the lattice constant
along the basal plane /H208493.316 Å /H20850;
/H9263for both carbides is taken
as 0.19. Results from Eq. /H208493/H20850can be used to compare the
relative maximum magnitude of Peierls stress between bi-nary and ternary carbides. The calculated
/H9268Sfor Zr 3Al3C5
reaches 78% of the binary carbide. Therefore, similar to ZrC,the ternary carbide, Zr
3Al3C5, is expected to show very low
dislocation mobility. Suggested by the present investigations,experimental hardness of Zr
3Al3C5is determined by the re-
sponse of strong Zr-C covalent bonds to shear strain, and isexpected to display a high magnitude.
VI. CONCLUDING REMARKS
In summary, we investigated the equilibrium crystal struc-
ture, EOS, elastic moduli, ideal strengths, bond-breakingprocesses, and electronic structure of Zr
3Al3C5by first-
principles calculations. The results show how interatomicbonds in this complex ternary carbide respond to appliedstrains, especially under strains near a critical point of struc-tural instability. We show that the Al-C and Zr-C bondsdominate tensile and shear failures of Zr
3Al3C5, respectively.
The material cleaves by breaking the weakest Al-C bond,while slip plane movement occurs by abruptly breaking theZr-C bonds under shear strains. Shear induced bond-breakingoccurs inside the NaCl-type ZrC
xslabs, which provides
Zr3Al3C5to have high shear strength and moduli similar to
ZrC. Obtained ideal tensile and shear strengths of this ter-nary carbide are comparable to its binary counterpart, ZrC.Furthermore, Peierls stresses of the two carbides are com-pared, and the result shows that dislocation mobility will below in the two compounds. Results led to a suggestion thatZr
3Al3C5has a great potential to be used as a hard ceramic
for high temperature applications and should be extensivelystudied in the future.
ACKNOWLEDGMENT
This work was supported by the National Outstanding
Young Scientist Foundation for Y. C. Zhou under Grant No.59925208, Natural Sciences Foundation of China underGrant Nos. 50232040, 90403027, and 50302011.
1H. Nowotny and S. Windisch, Annu. Rev. Mater. Sci. 3, 171
/H208491973 /H20850.
2R. Yu, L. L. He, and H. Q. Ye, Acta Mater. 51, 2477 /H208492003 /H20850.
3J. C. Schuster, H. Nowotny, and C. Vaccaro, J. Solid State Chem.
32, 213 /H208491980 /H20850.
4M. A. Pietzka and J. C. Schuster, J. Phase Equilib. 15, 392
/H208491994 /H20850.5J. C. Schuster and H. Nowotny, Z. Metallkd. 71, 341 /H208491980 /H20850.
6M. W. Barsoum, Prog. Solid State Chem. 28, 201 /H208492000 /H20850.
7S. I. Mikhalenko, Y. B. Kuz’ma, V. E. Popov, V. N. Gurin, and A.
P. Nechitailov, Izv. Akad. Nauk SSSR, Neorg. Mater. 15, 1948
/H208491979 /H20850.
8U. Leela-adisorn, S. M. Choi, S. Hashimoto, S. Honda, and H.
Awaji, in Proceedings of the 21st International Korea-JapanWANG et al. PHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
134107-8Seminar on Ceramics /H208492004 /H20850, p. 339.
9U. Leela-Adisorn, S. M. Choi, N. Tera, T. Takeuchi, S. Hash-
imoto, S. Honda, H. Awaji, K. Hayakawa, and A. Yamaguchi, J.Ceram. Soc. Jpn. 113, 188 /H208492005 /H20850.
10J. Y. Wang, Y. C. Zhou, Z. J. Lin, and T. Liao, Phys. Rev. B 72,
052102 /H208492005 /H20850.
11X. H. Wang and Y. C. Zhou, Corros. Sci. 45, 891 /H208492003 /H20850.
12L. F. He, Y. C. Zhou, Y. W. Bao, J. Y. Wang, and M. S. Li
/H20849unpublished /H20850.
13T. M. Gesing and W. Jeitschko, J. Solid State Chem. 140, 396
/H208491998 /H20850.
14M. D. Segall, P. L. D. Lindan, M. J. Probert, C. J. Pickard, P. J.
Hasnip, S. J. Clark, and M. C. Payne, J. Phys.: Condens. Matter
14, 2717 /H208492002 /H20850.
15D. Vanderbilt, Phys. Rev. B 41, R7892 /H208491990 /H20850.
16J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R.
Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671
/H208491992 /H20850.
17H. J. Monkhorst and J. D. Pack, Phys. Rev. B 16, 1748 /H208491977 /H20850.
18B. G. Pfrommer, M. Côté, S. G. Louie, and M. L. Cohen, J.
Comp. Physiol. 131, 233 /H208491997 /H20850.
19J. Y. Wang and Y. C. Zhou, Phys. Rev. B 69, 144108 /H208492004 /H20850.
20J. Y. Wang and Y. C. Zhou, Phys. Rev. B 69, 214111 /H208492004 /H20850.
21J. Y. Wang, Y. C. Zhou, Z. J. Lin, F. L. Meng, and F. Li, Appl.
Phys. Lett. 86, 101902 /H208492005 /H20850.
22V. Milman and M. C. Warren, J. Phys.: Condens. Matter 13, 241
/H208492001 /H20850.
23P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, J. Wills,
and O. Eriksson, J. Appl. Phys. 84, 4891 /H208491998 /H20850.
24S.-H. Jhi, S. G. Louie, M. L. Cohen, and J. W. Morris, Jr., Phys.Rev. Lett. 87, 075503 /H208492001 /H20850.
25R. H. Telling, C. J. Pickard, M. C. Payne, and J. E. Field, Phys.
Rev. Lett. 84, 5160 /H208492000 /H20850.
26S. Ogata, J. Li, and S. Yip, Science 298, 807 /H208492002 /H20850.
27Y. Zhang, H. Sun, and C. F. Chen, Phys. Rev. Lett. 93, 195504
/H208492004 /H20850.
28L. E. Toth, in Transition Metal Carbides and Nitrides /H20849Academic
Press, New York, 1971 /H20850.
29Z. J. Lin, M. J. Zhuo, L. F. He, Y. C. Zhou, M. S. Li, and J. Y.
Wang, Acta Mater. /H20849to be published /H20850.
30F. Birch, J. Geophys. Res. 83, 1257 /H208491978 /H20850.
31A. Onondera, H. Hirano, T. Yuasa, N. F. Gao, and Y. Miyamoto,
Appl. Phys. Lett. 74, 3782 /H208491999 /H20850.
32J. Y. Wang and Y. C. Zhou, J. Phys.: Condens. Matter 15, 1983
/H208492003 /H20850.
33D. J. Green, in An Introduction to the Mechanical Properties of
Ceramics /H20849Press Syndicate of the University of Cambridge,
Cambridge, 1998 /H20850.
34S.-H. Jhi, J. Ihm, S. G. Louie, and M. L. Cohen, Nature /H20849London /H20850
399, 132 /H208491999 /H20850.
35J. J. Gilman, Science 261, 1436 /H208491993 /H20850.
36J. J. Gilman, Mater. Sci. Eng. A 209,7 4 /H208491996 /H20850.
37A. M. Kosevich, “Crystal dislocations and the theory of elastic-
ity,” in Dislocations in Solids , edited by F. R. N. Nabarro
/H20849North-Holland Publishing Company, Amsterdam, 1979 /H20850.
38C. R. Krenn, J. W. Morris, Jr., S.-H. Jhi, and J. Ihm, Relationships
between atomistic bonding and intrinsic macroscopic hardness,”inHard Coatings Based on Borides, Carbides & Nitrides , edited
by A. Kumar, Y.-W. Chung, and R. W. J. Chia /H20849TMS, 1998 /H20850.FIRST-PRINCIPLES PREDICTION OF THE ¼ PHYSICAL REVIEW B 73, 134107 /H208492006 /H20850
134107-9 |
PhysRevB.92.195156.pdf | PHYSICAL REVIEW B 92, 195156 (2015)
Unusual Landau level pinning and correlated ν=1 quantum Hall effect in hole systems
confined to wide GaAs quantum wells
Yang Liu, S. Hasdemir, M. Shayegan, L. N. Pfeiffer, K. W. West, and K. W. Baldwin
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
(Received 2 July 2015; revised manuscript received 29 October 2015; published 30 November 2015)
In two-dimensional hole systems confined to wide GaAs quantum wells, where the heavy- and light-hole states
are close in energy, we observe a very unusual crossing of the lowest two Landau levels as the sample is tiltedin a magnetic field. At a magic tilt angle θ/similarequal34
◦, which surprisingly is independent of the well width or hole
density, in a large filling factor range near ν=1, the lowest two levels are nearly degenerate as evidenced by the
presence of two-component quantum Hall states. Remarkably, a quantum Hall state is seen at ν=1, consistent
with a correlated /Psi1111state.
DOI: 10.1103/PhysRevB.92.195156 PACS number(s): 73 .43.Qt,73.63.Hs
I. INTRODUCTION
Among the most fascinating phases of two-dimensional
electron systems (2DESs) in a strong perpendicular magnetic
field ( B⊥) are the quantum Hall states (QHSs). These are
incompressible phases signaled by vanishing longitudinalresistance ( R
xx) and quantized Hall resistance ( Rxy), and
are observed at integral or certain fractional Landau level(LL) filling factors ( ν)[1–3]. Adding a layer (or subband)
degree of freedom leads to exciting twists. A bilayer electronsystem with nearly degenerate LLs from different subbands
and comparable inter- and intralayer interaction can support
new, two-component (2C) QHSs that have no counterpartin standard single-layer (or one-component, 1C) 2DESs.An example is the /Psi1
331state, a QHS formed at the even-
denominator filling ν=1/2[4–7]. The correlated /Psi1111QHS,
stabilized at ν=1, is another example [ 5,8–12]. This state
is generally considered to be an excitonic superfluid, which
can support Josephson-like interlayer tunneling and superfluid
transport.
Recent experimental studies of 2D hole systems (2DHSs)
confined to wide GaAs quantum wells (QWs) have unraveledunique phenomena, arising from the nontrivial spin-orbitcoupling of the heavy and light holes. Graninger et al. reported
a reentrant behavior of the ν=1 QHS as a function of parallel
magnetic field B
||in symmetric, wide QWs [ 13]. Later, Liu
et al. observed an unusual crossing of the two lowest-energy
LLs at B||=0 as a function of B⊥[14]. For a given density
pand well-width W, the crossing occurs at a particular filling
[Fig. 1(a)]; it destroys or weakens the odd-denominator QHSs
near this filling, and stabilizes a unique even-denominator QHSwhen it happens at ν=1/2[14].
Here we present low-temperature transport data for 2DHSs
confined to symmetric, wide GaAs QWs, as we change thetilt angle θbetween the sample normal and the magnetic
field direction. We find that at low and high θ,i fWandp
are sufficiently large, LLs from different subbands are wellseparated from each other and the 2DHSs exhibit normalQHSs at the standard fillings ν=2/3, 1, 4 /3, 7/5, 8/5,
and 5/3. However, near an intermediate θ, the 2DHSs show
2C QHSs similar to those reported in bilayer 2DESs withvanishing subband separation [ 15]. This observation indicates
that the two lowest-energy LLs are nearly degenerate and isconsistent with a B
||-induced LL crossing [ 16]. Remarkably,
as schematically shown in Fig. 1(b), this near degeneracy
persists in a large magnetic field range nearν=1 when
θ/similarequal34◦, a magic angle, which does not depend on Wor
p. Moreover, when the two LLs are degenerate, the 2DHS
is compressible at ν=1i fpandWare large so that
d/lB/greaterorsimilar1.3, but exhibits a QHS when d/lB/lessorsimilar1.3, consis-
tent with the development of a correlated 2C ( /Psi1111) state
(dis the interlayer separation and lBis the magnetic length)
[8,12,17].
II. METHOD
Our samples, grown by molecular beam epitaxy on GaAs
(001) wafers, consist of GaAs QWs flanked by undopedAl
0.3Ga0.7As spacer and carbon δ-doped layers. The 2DHSs
have as-grown densities ranging from 0.98 to 2.12, in unitsof 10
11cm−2, which we use throughout this report, and very
high low-temperature mobilities μ/greaterorequalslant100 m2/Vs. We made
samples in a van der Pauw geometry, 4 ×4m m2, and alloyed
In : Zn contacts at their four corners. Each sample is fittedwith an evaporated Ti/Au front gate and an In back gateto control the 2DHS density and QW symmetry. The datapresented here were taken in symmetric QWs. The transportmeasurements were carried out in a dilution refrigerator with abase temperature of T≈30 mK and a superconducting magnet
up to 18 T. We changed θwith an in situ rotator, and used the
low-frequency ( ∼30 Hz) lock-in technique. Here, we focus
primarily on R
xxtraces; the Rxydata corroborate Rxxand
show the corresponding plateaus.
III. EXPERIMENTAL RESULTS AT θ=0
We first describe data taken in a 2DHS confined to a 40-nm-
wide QW as a function of density at θ=0◦.I nF i g . 2, the QHS
transitions (marked by solid circles), which appear when twoLLs are nearly degenerate, can be seen moving from low tohighνas we increase p.A tp=0.76, we observe QHSs at the
standard fillings, similar to what is seen in systems where LLsfrom different subbands are well-separated [ 3]. The ν=2/3
QHS becomes weak at p=0.82 but is restored at higher p.
The weakening of the ν=1 QHS at p/similarequal1.01 is evidenced by
a profound narrowing of its R
xxplateau, and serves as direct
1098-0121/2015/92(19)/195156(5) 195156-1 ©2015 American Physical SocietyY ANG LIU et al. PHYSICAL REVIEW B 92, 195156 (2015)Energy
1/ν(a) θ = 0° (b) θ ≈ 34° (c) θ ≈ 50°
1/ν 1/νν = 1ν = 1 ν = 1
FIG. 1. (Color online) Schematic diagram of the lowest two LLs
at different tilt angles ( θ).
evidence that the two lowest-energy LLs are crossing at ν=1
[18–22]. Atp=1.20, a strong ν=1 QHS is restored, and a
2C QHS develops at an unusual filling ν=19/15 [15,23]. The
transition continues moving to higher νatp=1.31. The ν=
5/3 QHS disappears and another 2C QHS develops at ν=3/2,
which is the particle-hole counterpart of the 2C ν=1/2(/Psi1331)
QHS [ 6]. In the top trace ( p=1.59), the 2DHS reverts back
to 1C for ν< 2, exhibiting QHSs at standard fillings. The
above evolution of the QHSs, which implies a LL crossingthat moves from low νto high νas the density is increased,
is consistent with previous observations and theoreticalcalculations [ 14].
The above LL crossing can be qualitatively understood in
a simplified picture (see the right panels in Fig. 2). When
confined to QWs, because of their heavier mass in the
zdirection, the heavy-hole (HH) subband is lower in energy
0.761.31
1.20
1.01
0123 Rxx (kΩ)ν = 1
1/ν1.0 1.5 0.52/3
4/3
3/25/3
19/15W = 40 nm
p
(1011 cm-2)8/57/54/5
1.59
0.82÷4 4/35/3
8/57/5(d)
ν = 1ν = 5/3
ν = 2/3ν > 2HH-S0
LH-S0(d)
(c)(c)
(b)(b)
(a)(a)
1/νEnergy
FIG. 2. (Color online) Rxxtraces measured in a 2DHS confined
to a 40-nm-QW at θ=0 and different densities. Right panels show
schematically the two lowest-energy LLs’ energy vs 1 /νat different
densities, corresponding to the traces as marked.than the light-hole (LH) subband. But the HHs have a smaller
effective mass in the xyplane than the LHs, so the ground-state
(N=0) LL of the HH symmetric subband, which we refer to
as HH-S0 for simplicity, increases faster in energy than theLH-S0 LL as we sufficiently increase B
⊥, leading to a LL
crossing. In a more quantitative picture, the spin-orbit couplingmixes the HH and LH subbands and LLs, and results in a morecomplex, nonlinear LL fan diagram. However, the crossingbetween the two lowest-energy LLs is preserved in symmetricQWs [ 14]. In our wide QW samples, the HH and LH subbands
are close in energy, so the two levels cross at moderateB
⊥.
IV . EXPERIMENTAL RESULTS OF FINITE θ
Data presented in Fig. 3reveal that QHS transitions can also
be induced at a fixed density by varying θ, but the behavior
is dramatically different. In Fig. 3(c),w es h o w RxxvsB⊥
traces measured at p=2.05 and different θ. The density is
high so that the LH-S0 LL is well below the HH-S0 LLatθ=0
◦in the range ν< 2 [see Fig. 2(d)], and the 2DHS
exhibits 1C QHSs at standard fillings. At θ/similarequal34◦, the 2DHS
becomes 2C in a large range of fillings 2 /3<ν< 2. This is
evidenced by the development of insulating phases aroundν=2/3 (i.e., around ν=1/3 for each component [ 24]),
the complete disappearance of the QHSs at ν=5/3 and 1,
as well as the stabilization of QHSs at twice the standard
fillings ν=4/3, 6/5, 6/7, 2/3, and at unusual fillings such
asν=19/15 and 29 /35 [15,23]. At larger θ,t h eν=1 and
5/3 QHSs reappear while many 2C QHSs remain, suggesting
the two lowest-energy LLs are separated by a small but finiteenergy [ 25].
Figure 3(d) data taken at p=1.59 exhibit a more complete
and revealing evolution. The system is essentially 1C for θ/lessorsimilar
20
◦andθ/greaterorsimilar44◦, showing strong QHSs at standard fillings
[25]. It becomes 2C for ν< 2 when 25◦/lessorsimilarθ/lessorsimilar44◦, exhibiting
insulating phases flanking ν=2/3 and 2C QHSs at ν=19/5,
6/5, 29/35, etc., while QHSs at ν=1 and 5 /3 become weak
and essentially disappear as θapproaches 34◦.
Figure 3(e) shows traces taken at p=1.28 where, at θ=0,
the LL crossing occurs near ν=3/2, as evidenced by the
stabilization of the correlated, 2C QHS at ν=3/2, and the
absence of a QHS at ν=5/3. Similar to the data of Figs. 3(c)
and3(d), the system becomes 2C near θ/similarequal34◦and 1C when
θ/greaterorsimilar49◦. However, in contrast to Figs. 3(c) and3(d) data, the
ν=1 QHS becomes weak at θ=34◦but never disappears.
The fact that the system is 2C near ν=1 suggests that the
ν=1 QHS seen at θ/similarequal34◦in Fig. 3(d) is also a 2C QHS; we
will return to this later.
V . DISCUSSION
The transition from 1C to 2C as a function of increasing
B||has been reported previously for electrons confined to
wide GaAs QWs [ 15,26]. In such systems, the coupling of
B||to the orbital (out-of-plane) motion of electrons renders
the system progressively more bilayer-like at higher B||
and quenches the energy separation between the N=0
LLs of the symmetric and antisymmetric subbands, makingthem essentially degenerate [ 15,26]. Further increasing B
||
195156-2UNUSUAL LANDAU LEVEL PINNING AND CORRELATED . . . PHYSICAL REVIEW B 92, 195156 (2015)
4 6 8
B┴ (T)ν = 14/3
5/3
4/5
θ = 0°25°27°34°44°49°59°
12/3 4/35/3
6/519/154/529/35
6/7
8/58/57/5
22/33/5(e) p = 1.28 x 1011 cm-2
2/3 3/5
48 61 0θ = 0°25°30°34°39°44°49°66°
0246
B┴ (T)Rxx (kΩ)(d) p = 1.59 x 1011 cm-2
(c) p = 2.05 x 1011 cm-2
4 81 2
B┴ (T)ν = 14/3
5/3
24/52/3
3/5
19/1529/354/5
6/5
4/35/3
2ν = 1
19/15
6/529/354/5
6/72/3
θ = 0°25°44°54°59°
34°(a)
W = 40 nmd
4/31
4/5
4/33/24/3
6/519/15
8/5(b) B
B||B┴
θ
Sample
61 0
FIG. 3. (Color online) (a) Self-consistently calculated charge distribution of the 2DHS confined to the 40-nm-wide QW at densities
p=2.05, 1.59, and 1.28. (b) Experimental geometry. (c)–(e) RxxvsB⊥traces measured at different θ. In all panels, the QHS at ν=1 is strong
atθ=0, disappears or weakens at θ/similarequal34◦, and becomes strong again at larger θ.
does not lift this degeneracy and the system remains 2C
at the highest B||. This is very different from our data
shown in Figs. 3(d) and 3(e), where the 2DHS near ν=1
becomes 2C only near θ/similarequal34◦, but is 1C at smaller and
higher B||.
We attribute the evolution in Fig. 3data to a B||-induced LL
crossing [ 13,27,28]. Unfortunately, no accurate calculations
of LLs in the presence of both B⊥andB||are available,
particularly for 2DHSs with multiband structure. The tilted-field geometry implies complicated couplings between Landauharmonic oscillators from different subbands, and makesnumerical calculations extremely demanding. Qualitatively,we can explain the crossing as follows. The densities of Fig. 3
data are sufficiently large so that the LH-S0 level is lower thanthe HH-S0 level near ν=1a tB
||=0 [Figs. 2(c) and2(d)].
Finite B||introduces additional confinement of the 2DHS in
thezdirection, raises the LH-S0 LL relative to the HH-S0 LL,
and causes a crossing of these levels at intermediate θ[see
Fig. 4(d)].
The most remarkable feature of Fig. 3data, however, is
not the LL crossing at an intermediate θ. Rather, it is the
behavior of the 2DHS near the crossing angle, suggesting avery unusual “pinning” (or near-pinning) of the LLs in a verylarge range of ν[Fig. 1(b)]. Note in Fig. 3that at a given
density the system exhibits 2C behavior in the entire rangeofν< 4/3a tθ/similarequal34
◦. This is very different from the θ=0
data of Fig. 2where the LL crossing features for any given
density appear near a specific ν, which moves from low to
high values as the density is increased. Moreover, in Fig. 3,
the angle θ/similarequal34◦at which the 2DHS becomes 2C appears to
be independent of the 2DHS density. In other 2DHS samples,confined to QWs with Wranging from 35 to 50 nm, we have
observed similar phenomena as in Fig. 3at the same θ/similarequal34
◦.
This independence of the 2C behavior on ν,p, and Wat
this critical angle is astonishing, and demands a theoreticalexplanation.
The evolution of the QHS at ν=1 is also very intriguing.
As seen in Fig. 3, it disappears completely at θ/similarequal34
◦when
p=2.05 but only becomes weak at p=1.28. In Fig. 4(a),
we summarize our results for many 2DHSs, illustrating theconditions for the stability of the ν=1 QHS. Data are shown
as a function of θandd/l
B, which compares the interlayer
(e2/4π/epsilon1d) and intralayer ( e2/4π/epsilon1lB) correlations and is
widely used to characterize bilayer QHSs [ 8–12,17,29,30].
Figure 4(a) shows that no LL crossing at ν=1 can be induced
via tilting if d/lB/lessorsimilar1.0, and the ν=1 QHS is always strong.
195156-3Y ANG LIU et al. PHYSICAL REVIEW B 92, 195156 (2015)
02 04 0 601.01.8
0.81.41.6
1.2d/lB
θ (degrees)40 nm
45 nm
50 nm35 nmd/lB
Δ
HH-S0LH-S0d
Δ
θEnergyW = 45 nm
2.05
1.59
1.28
0.981.10
0.951.39
2.12
1.710.78p (1011 cm-2)ν = 1
No crossingNo QHS
QHSQHSNo QHS
p = 1.39 × 1011 cm-2
(a)(b)(c)
(d) ≈ 34°
30 05 01
FIG. 4. (Color online) (a) Phase diagram for the stability of QHS
atν=1 as a function of the tilting angle θandd/lB. The solid
(open) symbols mark the presence (absence) of a QHS at ν=1. In
narrow QWs and at low density, no crossing is seen as a function
ofθ, shown as the blue region. Once d/lB/greaterorsimilar1.0, a crossing occurs
nearθ/similarequal34◦. At the crossing, a QHS appears at ν=1i fd/lB/lessorsimilar1.3,
consistent with the /Psi1111state. (b) Calculated charge distribution for
a 45-nm-wide QW with p=1.39 showing the interlayer distance d.
(c) and (d) Schematic phase and LL diagrams at ν=1 showing how
the LL separation /Delta1increases as θdeviates from /similarequal34◦.
When d/lB/greaterorsimilar1.0, atν=1, the LH-S0 level is lower than the
HH-S0 level at θ=0, and the two levels cross at θ/similarequal34◦;
see Fig. 4(d). At the crossing, we observe a QHS at ν=1
ifd/lB/lessorsimilar1.3, and the ground state becomes compressible
ifd/lB/greaterorsimilar1.3. The d/lB/lessorsimilar1.3 condition for the stability of
theν=1 QHS at the crossing, and the fact that the 2DHSis 2C at nearby fillings, suggest that it is a 2C QHS with
strong interlayer correlations, likely the /Psi1111state reported
in GaAs bilayer electron [ 8–10,12] or hole [ 11,17]s y s t e m s
confined to double QWs. In those systems, when the lowestLLs from different subbands are degenerate, the ν=1 QHS
is stable at d/l
B/lessorsimilar2, and turns into a compressible state if
d/lBbecomes large [ 8,12]. Also note that in our experiments
the energy separation between the two crossing LLs increases
asθdeviates from /similarequal34◦[see Fig. 4(d)]. We show in Fig. 4(c) a
schematic “phase diagram” for the stability of the ν=1 QHS
as functions of /Delta1andd/lB. The resemblance of Fig. 4(c)
and the phase diagram of ν=1 QHS in double QWs [ 8,12]
is striking. We emphasize that in our experiments, we areessentially tuning /Delta1through zero as we tilt the sample near
θ/similarequal34
◦; see Fig. 4(d).
In conclusion, 2DHSs confined to wide GaAs QWs and
with sufficiently high density, reveal an unusual crossingof the two lowest-energy LLs near ν=1a sw et i l tt h e
sample in magnetic field. It appears at a magic angle θ/similarequal34
◦,
essentially independent of the QW width, density, or B⊥
(filling), suggesting a pinning of the LLs near the crossing.
The crossing and the pinning likely stems from the complexinterplay of the heavy- and light-hole LLs in B
||, and should
stimulate further theoretical investigation. Near this angle, the2DHS becomes 2C at ν< 2 and, if d/l
Bis small, exhibits a
ν=1 QHS, consistent with a correlated, 2C, /Psi1111state.
ACKNOWLEDGMENTS
We acknowledge the DOE BES (DE-FG02-00-ER45841)
grant for measurements, and the NSF (Grants DMR-1305691and MRSEC DMR-1420541), the Gordon and Betty MooreFoundation (Grant GBMF4420), and Keck Foundation forsample fabrication and characterization. A portion of thiswork was performed at the NHMFL, which is supportedby the NSF Cooperative Agreement No. DMR-1157490,the State of Florida, and the DOE. We thank S. Hannahs,G. E. Jones, T. P. Murphy, E. Palm, A. Suslov, and J. H. Parkfor technical assistance. We are indebted to R. Winkler forilluminating discussions and providing the self-consistentlycalculated charge distribution.
[1] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45,
494(1980 ).
[ 2 ]D .C .T s u i ,H .L .S t o r m e r ,a n dA .C .G o s s a r d , Phys. Rev. Lett.
48,1559 (1982 ).
[3] J. K. Jain, Composite Fermions (Cambridge University Press,
Cambridge, UK, 2007).
[4] Y . W. Suen, L. W. Engel, M. B. Santos, M. Shayegan, and D. C.
Tsui, P h y s .R e v .L e t t . 68,1379 (1992 ).
[5] J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, K. W. West, and
S. He, Phys. Rev. Lett. 68,1383 (1992 ).
[6] Y . W. Suen, H. C. Manoharan, X. Ying, M. B. Santos, and
M. Shayegan, P h y s .R e v .L e t t . 72,3405 (1994 ).
[7] J. Shabani, Y . Liu, M. Shayegan, L. N. Pfeiffer, K. W. West, and
K. W. Baldwin, Phys. Rev. B 88,245413 (2013 ).
[8] S. Q. Murphy, J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer,
a n dK .W .W e s t , Phys. Rev. Lett. 72,728(1994 ).[9] I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West,
Phys. Rev. Lett. 84,5808 (2000 ).
[10] M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West,
Phys. Rev. Lett. 93,036801 (2004 ).
[11] E. Tutuc, M. Shayegan, and D. A. Huse, Phys. Rev. Lett. 93,
036802 (2004 ).
[12] J. Eisenstein, Annu. Rev. Condens. Matter Phys. 5,159
(2014 ).
[13] A. L. Graninger, D. Kamburov, M. Shayegan, L. N. Pfeiffer,
K. W. West, K. W. Baldwin, and R. Winkler, Phys. Rev. Lett.
107,176810 (2011 ).
[14] Y . Liu, S. Hasdemir, D. Kamburov, A. L. Graninger, M.
Shayegan, L. N. Pfeiffer, K. W. West, K. W. Baldwin, andR. Winkler, P h y s .R e v .B 89,165313 (2014 ).
[15] H. C. Manoharan, Y . W. Suen, T. S. Lay, M. B. Santos, and
M. Shayegan, P h y s .R e v .L e t t . 79,2722 (1997 ).
195156-4UNUSUAL LANDAU LEVEL PINNING AND CORRELATED . . . PHYSICAL REVIEW B 92, 195156 (2015)
[16] Spin-orbit coupling and B||introduce LL mixing, which may
lead to an anticrossing rather than crossing. This does not changeour conclusions.
[17] E. Tutuc and M. Shayegan, P h y s .R e v .B 72,081307 (2005 ).
[18] Note that, because of QHS ferromagnetism, certain integer
QHSs and fractional QHSs do not collapse when two LLs crossat the Fermi energy [ 19–22].
[19] D. K. Maude, M. Potemski, J. C. Portal, M. Henini, L. Eaves,
G. Hill, and M. A. Pate, P h y s .R e v .L e t t . 77,4604 (1996 ).
[20] K. Muraki, T. Saku, and Y . Hirayama, P h y s .R e v .L e t t . 87,
196801 (2001 ).
[21] M. Padmanabhan, T. Gokmen, and M. Shayegan, Phys. Rev.
Lett. 104,016805 (2010 ).
[22] Y . Liu, J. Shabani, and M. Shayegan, Phys. Rev. B 84,195303
(2011 ).
[23] The ν=19/15 QHS can be thought of as a 2C state where
the fillings for the two “components” are 2 /3a n d3 /5.
Other examples are QHSs at ν=11/15=(1/3+2/5) and
ν=29/35=(2/5+3/7). Such states have been documented
for bilayer electron systems confined to wide GaAs QWs inRef. [ 15].[24] M. B. Santos, Y . W. Suen, M. Shayegan, Y . P. Li, L. W. Engel,
a n dD .C .T s u i , Phys. Rev. Lett. 68,1188 (1992 ).
[25] At the highest angle that we can reach,
ν=1 in our experiments,
θ/similarequal59◦in Fig. 3(c) andθ/similarequal66◦in Fig. 3(d), the QHS at ν=1
weakens again ( Rxxplateau becomes narrow), signaling that the
2DHS is becoming 2C (bilayer) again because of the very largeB
||.
[26] S. Hasdemir, Y . Liu, H. Deng, M. Shayegan, L. N. Pfeiffer,
K. W. West, K. W. Baldwin, and R. Winkler, Phys. Rev. B 91,
045113 (2015 ).
[27] The crossing of the two lowest LLs in tilted fields is also seen
in geometric resonance data [ 28].
[28] M. A. Mueed, D. Kamburov, S. Hasdemir, M. Shayegan, L. N.
Pfeiffer, K. W. West, and K. W. Baldwin, Phys. Rev. Lett. 114,
236406 (2015 ).
[29] T. S. Lay, Y . W. Suen, H. C. Manoharan, X. Ying, M. B. Santos,
and M. Shayegan, Phys. Rev. B 50,17725 (1994 ).
[30] In Fig. 4,w eu s e dfrom self-consistent calculations that
were performed for B=0. It is likely that in the presence of
largeB||, as the 2DHS becomes more bilayerlike, dslightly
increases.
195156-5 |
PhysRevB.79.184418.pdf | Spin-polarized tunneling spectroscopy of fully epitaxial magnetic tunnel junctions
using Co 2FeAl 0.5Si0.5Heusler alloy electrodes
Hiroaki Sukegawa,1,*Wenhong Wang,1Rong Shan,1Tomoya Nakatani,2Koichiro Inomata,1and Kazuhiro Hono1,2
1Magnetic Materials Center, National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba 305-0047, Japan
2Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan
/H20849Received 16 June 2008; revised manuscript received 3 March 2009; published 18 May 2009 /H20850
Spin-dependent tunneling spectroscopy has been studied in fully epitaxial magnetic tunnel junctions with
full-Heusler Co 2FeAl 0.5Si0.5 /H20849CFAS /H20850alloys. We fabricated CFAS/MgO/CFAS structures with L21- and
B2-ordered CFAS layers and measured the bias voltage dependence of differential conductance G. We found
forL21-CFAS /MgO /L21-CFAS structure symmetrical conductance curves with respect to polarity of the bias
voltage for parallel /H20849P/H20850and antiparallel /H20849AP/H20850magnetization configurations and two characteristic crossovers in
Gbetween P and AP accompanied with a flat feature within /H110060.6 V in G/H20849P/H20850. On the other hand, only one
crossover was observed at a negative-bias voltage for L21-CFAS /MgO /B2-CFAS structure. The direct tun-
neling that reflects the specific spin-dependent density of states of the half-metallic L21-CFAS is proposed as
a possible transport mechanism leading to the notable crossovers.
DOI: 10.1103/PhysRevB.79.184418 PACS number /H20849s/H20850: 75.47. /H11002m, 85.75. /H11002d, 73.20.At, 75.50.Cc
I. INTRODUCTION
Half-metallic compounds which are fully spin polarized
near the Fermi level /H20849EF/H20850due to an energy gap in the
minority-spin band1have attracted great attention as key ma-
terials for creating spintronics devices such as future ultra-high density nonvolatile memory devices and spin metal-oxide-semiconductor field-effect transistor.
2Especially, Co-
based full-Heusler alloys have been intensively studied sincethe half metallicity is expected even at room temperature/H20849RT/H20850due to their high Curie temperature around 1000 K.
3–16
Recently, a very large tunnel magnetoresistance /H20849TMR /H20850ratio
of 570% was achieved in Co 2MnSi /H20849CMS /H20850/AlO x/CMS mag-
netic tunneling junctions /H20849MTJs /H20850at 2 K.11However, the
TMR ratio, which is represented as 2 P2//H208491−P2/H20850,17where P
is a tunneling spin polarization, significantly decreases withincreasing temperature and becomes 67% at RT. This behav-ior was attributed to a low-energy separation between Fermilevel and conduction-band edge,
11an inelastic tunneling pro-
cess including spin flip due to magnon-assisted tunneling7
through the formation of interface states near EFin the
minority-spin gap18or the formation of nonquasiparticle
states above EFin the minority-spin gap,19resulting in the
low TMR ratio at RT. Recently, we have reported a largeTMR ratio at RT and a relatively small temperature depen-dence of TMR ratio /H20849220% at RT and 390% at 5 K /H20850in epi-
taxial Co
2FeAl 0.5Si0.5 /H20849CFAS /H20850/MgO/CFAS MTJs.14The
TMR ratio of 390% corresponds to P=0.81 for CFAS. Ab
initio calculations on quarternary Co 2FeAl 1−xSixalloys
which were performed after this observation demonstratedthe half metallicity of these alloys with the L2
1structure and
predicted EFto be adjustable by controlling the composition
x.20–22Forx=0.5 /H20849CFAS /H20850theEFlies at the middle of the
minority-spin gap of about 1 eV, which will enhance thetemperature robustness of TMR ratio. This is consistent withthe large TMR ratio at RT obtained in the experiment.
14The
low density of states /H20849DOS /H20850of the majority-spin band near
EFin CFAS may also contribute to the relatively small tem-
perature dependence of TMR ratio due to the decreasing ofthe magnon excitation.In this paper we report two crossovers in differential
tunneling conductance G=dI/dVmeasured in epitaxial
L2
1-CFAS /MgO /L21-CFAS MTJs to discuss the origin of
the large TMR ratios obtained in the MTJs. The direct tun-neling that reflects the specific spin-dependent DOS of thehalf-metallic CFAS obtained from the first-principlecalculations
20–22is proposed as a possible transport mecha-
nism leading to the notable crossovers.
II. EXPERIMENTAL DETAILS
Multilayers were deposited on single-crystal MgO /H20849001 /H20850
substrates using an ultrahigh vacuum magnetron sputteringsystem with the base pressure below 8 /H1100310
−8Pa. We used
MgO buffer layer in place of Cr buffer layer used in Ref. 14
because Cr buffer impedes a highly L21-ordered CFAS struc-
ture due to the Cr atoms interdiffusion by postannealing athigher temperatures above 723 K and thus limits to achievehigher spin polarization. Typical MTJ structure fabricated isa top-spin-valve type consisting of MgO /H2084920/H20850/CFAS /H2084930/H20850/Mg
/H208490.3/H20850/MgO /H208491.1 or 1.4 /H20850/CFAS /H208495/H20850/Co
75Fe25 /H208492/H20850/Ir20Mn80
/H2084915/H20850/Ru /H208497/H20850/H20849thickness in nm /H20850on MgO /H20849001 /H20850substrates. MgO
layers were sputter deposited from a sintered MgO targetdirectly using rf magnetron, whereas other metallic layerswere deposited using dc magnetron. We inserted the thin Mglayer under the MgO barrier layer in order to avoid the oxi-dation of the bottom-CFAS layer. CFAS layers were depos-ited at RT from a stoichiometric target /H20849Co 50.0%, Fe 25.0%,
Al 12.5%, and Si 12.5% /H20850. After the deposition of the bottom-
CFAS layer we postannealed for 15 min in situ atT
bottom
=813 K. Postannealing was also performed at Ttop=RT or
813 K after the deposition of the top-CFAS layer. Finally weannealed the multilayers at 633 K in a furnace under a mag-netic field of 5 kOe fo r1hi n order to exchange bias the top
electrode. Both MTJs /H20849junction area: 5 /H110032–10/H1100310
/H9262m2/H20850
and electrodes were patterned using a typical microfabrica-tion technique, i.e., electron-beam lithography, photolithog-raphy, and Ar ion-beam etching. The crystal structure ofCFAS films was investigated by x-ray diffraction /H20849XRD /H20850.PHYSICAL REVIEW B 79, 184418 /H208492009 /H20850
1098-0121/2009/79 /H2084918/H20850/184418 /H208496/H20850 ©2009 The American Physical Society 184418-1MTJs were characterized by measuring R-H/H20849TMR /H20850and
current-voltage /H20849I-V/H20850curves with a dc four-probe method in
a temperature range of 7–290 K, where a magnetic field wasapplied along CFAS /H20851110 /H20852/H20849
/H20648MgO /H20851100 /H20852/H20850. In this study, the
positive current is defined as that electrons flow from thetop-CFAS layer to the bottom-CFAS layer. The compositionof deposited CFAS films was confirmed to be nearly stoichi-ometric /H20849Co 51.0%, Fe 23.2%, Al 14.1%, and Si 11.7% /H20850by
inductively coupled plasma analysis.
III. RESULTS
A. Structural characterization
Cross-sectional high-resolution transmission electron mi-
croscopy /H20849HRTEM /H20850was used to clarify the detailed crystal
structure of the multilayer /H20851CFAS /H20849Ta=873 K /H20850/MgO /
CFAS /H20849Ta=813 K /H20850/H20852. In the HRTEM lattice image, as shown
in Fig. 1, very smooth and abrupt interfaces are realized in
the CFAS/MgO/CFAS structure. Both bottom- and top-CFAS layers are grown epitaxially and single crystalline. Thelattice spacing of the CFAS layers are resolved with d
=0.20 nm, corresponding to body-centered cubic /H20853110 /H20854
planes. The MgO layer shows a /H20851001 /H20852growth orientation,
nevertheless a large number of structural imperfections areobserved. Interestingly, the defects are introduced not only atthe CFAS/MgO interfaces but also inside the MgO layer,which is shown by circles in an MgO barrier to relax thelattice mismatch between MgO and CFAS /H20849approximately
4.7% /H20850. In the reports of the epitaxial MTJs with MgO layers
prepared using an electron-beam evaporation /H20851Fe/MgO/Fe
/H20849Ref. 23/H20850and CMS/MgO/CoFe /H20849Refs. 12and16/H20850/H20852, in which
TMR enhancement due to the coherent tunneling has beenobserved, almost all lattice dislocations form at the interfacesand the desired epitaxial /H20849001 /H20850growth for MgO layer is suc-
cessfully achieved, unlike in our MgO barrier. Therefore, thecoherent tunneling effect would be weakened in our MTJs aswell as in the MTJ with the polycrystalline MgO barrier.
24
B. TMR properties
We prepared MTJs with different ordering of CFAS films
by changing the postannealing temperature, in whichL21-CFAS /MgO /L21-CFAS structure was achieved by post-
annealing at 813 K for both bottom- and top-CFAS layers/H20849T
bottom =813 K and Ttop=813 K /H20850, accompanied with almost
perfect /H20849001 /H20850orientation and very flat surface with an aver-
age surface roughness of 0.10 nm.25The L21-CFAS /
MgO /B2-CFAS structure was formed when annealed at 813
K for only the bottom-CFAS layer /H20849Tbottom =813 K and Ttop
=RT /H20850. Thus, we can investigate the L21and B2 ordering
dependences on spin-dependent tunneling properties throughthe two different structures for the top-CFAS layer.
25TMR
curves at both 7 and 290 K /H20849RT/H20850are shown for a
L21/MgO /H208491.1 nm /H20850/L21structure in Fig. 2/H20849a/H20850and a
L21/MgO /H208491.4 nm /H20850/B2 structure in Fig. 2/H20849b/H20850. Bias voltages
o f1m V /H208497K /H20850a n d5m V /H20849RT/H20850were applied for the measure-
ment. TMR ratios at RT for both structures are approxi-mately equal, 130–140 %. However a large difference was
observed at 7 K, 308% and 227% for the L2
1-CFAS /
MgO /L21-CFAS and L21-CFAS /MgO /B2-CFAS structures,
respectively, which implies higher PinL21than B2a t7K
/H20851P/H20849L21/H20850=0.78 and P/H20849B2/H20850=0.68 determined by Julliere’s
formula /H20852.17
The lower tunneling spin polarization of 0.78 than that
expected from the half metallicity of the L21-CFAS suggests
that the TMR enhancement due to the coherent tunnelingmay be negligibly small and the inelastic tunneling processstill contributes in our MTJs due to the low-quality MgObarrier as seen in Fig. 1. In fact, TMR ratios of
L2
1-CFAS /MgO /H208491.4 nm /H20850/CoFe MTJs on MgO-buffered
MgO /H20849001 /H20850substrates, which were fabricated using the same
process as described in this study and have a degraded MgObarrier as demonstrated in in situ reflective high-energy elec-
FIG. 1. Cross-sectional high-resolution transmission electron
microscopy image of a L21-CFAS /MgO /L21-CFAS magnetic tun-
nel junction along the /H20851110 /H20852direction of the CFAS layers. Structural
imperfections are circled.
FIG. 2. /H20849Color online /H20850TMR curves and schematic illustrations
of magnetic tunnel junctions with /H20849a/H20850L21-CFAS /MgO /H208491.1 nm /H20850/
L21-CFAS and /H20849b/H20850L21-CFAS /MgO /H208491.4 nm /H20850/B2-CFAS structures.
Bias voltages of 1 mV /H208497K /H20850a n d5m V /H20849RT/H20850were applied for the
measurement.SUKEGAWA et al. PHYSICAL REVIEW B 79, 184418 /H208492009 /H20850
184418-2tron diffraction, showed TMR of 130% and 95% at 7 K and
RT, respectively.26These TMR values correspond to tunnel-
ing spin polarizations PCoFe=0.505 /H208497K /H20850and 0.495 /H20849RT/H20850for
CoFe if we assume P/H20849L21-CFAS /H20850=0.78 and 0.65 /H20849RT/H20850as
estimated above. PCoFe=0.505 is close to the value of 0.506
derived from CoFe /AlO x/CoFe tunnel structures,27which
also suggests negligibly small TMR enhancement byMgO /H20849001 /H20850structure in our MTJs.
Figure 3/H20849a/H20850shows the bias voltage dependences of TMR
ratio at 7 and 290 K /H20849RT/H20850and Figs. 3/H20849b/H20850and3/H20849c/H20850show the
resistance for antiparallel /H20849AP/H20850and parallel /H20849P/H20850states as a
function of bias voltage a t 7 K and RT, respectively. The
TMR ratio a t 7 K decreases rapidly at a low-bias voltage and
gradually in a higher voltage region; however, such sharp-bias voltage dependence of TMR ratios does not appear atRT. Note that the TMR becomes negative over 1 eV asshown in the inset in Fig. 3/H20849a/H20850. The resistance change in the
low-bias voltage region a t7Ki s small for P but is remark-
able for AP similar to the TMR change, indicating that theTMR reduction by the bias voltage is originated from theresistance reduction in the AP state.
C. Spin-polarized tunneling spectroscopy
We obtained the bias voltage dependences of dI/dVby
the numerical calculation of I-Vcurves in order to investi-
gate the DOS of the CFAS films. Figure 4/H20849a/H20850shows the result
at 7 K for the L21-CFAS /MgO /L21-CFAS structure. Sym-
metrical curves are observed with respect to polarity of thebias voltage for both P /H20849dotted curve /H20850and AP /H20849solid curve /H20850states, indicating similar interface structures for both bottom
CFAS/MgO and MgO/top CFAS. A clear dip appears aroundzero-bias voltage for G
APcurve, which is characteristic of
the magnon excitation contribution to spin-dependent tunnel-ing as commonly observed in typical MTJs.
28However, this
dip disappears at RT /H20849data not shown /H20850. The GPcurve of
L21-CFAS /MgO /L21-CFAS exhibits a small change within
/H110060.6 V range as seen in Fig. 4/H20849a/H20850. Such a feature has al-
ready been reported in the MTJs with half-metallic CMSlayers.
11Additionally, these curves demonstrate a distin-
guishing feature; two obvious crossovers between GPand
GAPat + /H20849−/H208500.6 and + /H20849−/H208501.2 V and a distinct peak in GAPat
+/H20849−/H208500.9 V. The two crossovers and the peak in GAPwere
observed even at RT /H20849data not shown /H20850. On the other hand, as
shown in Fig. 4/H20849b/H20850, the conductance characteristics of the
L21-CFAS /MgO /B2-CFAS structure differ considerably
from those of the L21-CFAS /MgO /L21-CFAS structure,
which exhibits the pronounced asymmetry regarding the po-larity and a crossover between G
PandGAPin the negative-
bias voltage region only /H20849electrons flow: L21-CFAS
→B2-CFAS /H20850.
IV. DISCUSSION
We propose the direct tunneling that reflects the specific
spin-dependent DOS of the half-metallic bulk CFAS as apossible transport mechanism leading to the notable cross-
FIG. 3. /H20849Color online /H20850Bias voltage dependences of /H20849a/H20850TMR
ratio, /H20849b/H20850resistance area /H20849RA /H20850product of antiparallel state, and /H20849c/H20850
RA of parallel state for L21-CFAS /MgO /H208491.1 nm /H20850/L21-CFAS
structure. Inset in /H20849a/H20850indicates closeup near 1 V for the TMR curve
measured at 7 K.
FIG. 4. /H20849Color online /H20850dI/dVspectra of MTJs with /H20849a/H20850
L21-CFAS /MgO /H208491.1 nm /H20850/L21-CFAS and /H20849b/H20850L21-CFAS /
MgO /H208491.4 nm /H20850/B2-CFAS structures at 7 K. The dotted lines /H20849solid
lines /H20850represent the parallel /H20849antiparallel /H20850magnetization configura-
tion. The arrows represent the points where the parallel and theantiparallel lines cross each other.SPIN-POLARIZED TUNNELING SPECTROSCOPY OF … PHYSICAL REVIEW B 79, 184418 /H208492009 /H20850
184418-3overs. In order to clarify the direct tunneling effect, we mod-
eled the calculated bulk band structure of CFAS /H20849Ref. 22/H20850
and estimated the tunneling conductance. For simplicity, weassume that the conductance is determined only by DOS ofboth electrodes and neglect parameters for the tunnel barrierand the possible spin-filtering effect of MgO. Additionally,we assumed no spin-flip process during tunneling for sim-plicity. This assumption, however, does not change the es-sential interpretation of the tunneling spectra. If spins of theboth electrodes are denoted by
/H9268/H20849=↑,↓/H20850and/H9268/H11032/H20849=↑,↓/H20850, re-
spectively, these assumptions provide tunneling current I/H9268,/H9268/H11032
due to /H9268to/H9268/H11032spin channel under the application of finite
bias voltage Vas written in Eq. /H208491/H20850/H20849Ref. 29/H20850:
I/H9268,/H9268/H11032/H20849V/H20850=2/H9266e
/H6036/H20885/H20855/H20841Tˆ/H208412/H20856D/H9268/H20849E/H20850D/H9268/H11032/H20849E+eV/H20850/H20851f/H9268/H20849E/H20850
−f/H9268/H11032/H20849E+eV/H20850/H20852dE, /H208491/H20850
where /H20855/H20841Tˆ/H208412/H20856represents an averaged tunneling probability and
fis the Fermi distribution function. D/H9268and D/H9268/H11032represent
DOSs for /H9268and/H9268/H11032spin bands of both electrodes, respec-
tively. If we consider at T=0 K and assume that /H20855/H20841Tˆ/H208412/H20856is
independent of spin, Eq. /H208491/H20850is simplified as
I/H9268,/H9268/H11032/H20849V/H20850=G/H9268,/H9268/H11032V=2/H9266e
/H6036/H20855/H20841Tˆ/H208412/H20856/H20885
EF−eVEF
D/H9268/H20849E/H20850D/H9268/H11032/H20849E+eV/H20850dE,
/H208492/H20850
where G/H9268,/H9268/H11032represents the tunneling conductance for /H9268to/H9268/H11032
spin channel. The conductances for the P and AP states are
given by GP=G↑,↑+G↓,↓and GAP=G↑,↓+G↓,↑, respectively.
In order to calculate Eq. /H208492/H20850, we refer the DOS of L21-CFAS
as shown in Fig. 5/H20849a/H20850. The calculated bias voltage depen-
dence of GPandGAPusing Eq. /H208492/H20850and DOS in Fig. 5/H20849a/H20850are
shown in Fig. 5/H20849b/H20850in a positive voltage range /H20849the same for
the negative voltage /H20850forL21-CFAS /MgO /L21-CFAS struc-
ture. The calculated curves can reproduce the experimentalresult in Fig. 4/H20849a/H20850on the whole; two crossovers between G
P
andGAP/H20849corresponding to VBandVE/H20850and a local maximum
/H20849corresponding to VD/H20850at around 1.0 V for GAP. The appear-
ance of the maximum around 1.8 eV in GPin Fig. 5/H20849b/H20850is
attributed to the assumed infinite barrier height in the calcu-lation. In fact the effective barrier height of our MgO barrieris determined to be approximately 2 eV by Simmons’ fittingfor the measured I-Vcurves.
30The GPis almost constantuntil V=VC/H110111.2 V due to the gap Egin the minority-spin
band of L21-CFAS /H20851Fig.5/H20849a/H20850/H20852and no magnon contribution
assumed. For further increasing the bias voltage the GPin-
creases monotonically up to 1.8 V in Fig. 5/H20849b/H20850. On the other
hand, the GAPis zero until V=VA/H110110.5 V, the conduction-
band edge /H20849or the valence-band edge /H20850of the minority-spin
band over which the minority-spin electrons start to contrib-ute to the tunneling and G
APincreases monotonically in the
range of 0.5 V /H20849VA/H20850/H11021V/H110211.1 V /H20849VD/H20850with increasing Vand
gives a inflection point at V/H110111.1 V /H20849VD/H20850. The values of the
bias voltage VBandVEat the first and second crossovers,
respectively, are nearly the same as those in Fig. 4/H20849a/H20850, respec-
tively, indicating that the plausible band structure ofL2
1-CFAS calculated. The flat feature within /H110060.6 V ob-
served in GPofL21-CFAS /MgO /L21-CFAS /H20851Fig.4/H20849b/H20850/H20852may
support the existence of a wide gap in the vicinity of EFin
the minority band since the total conductance within /H20841V/H20841
/H11021Egmust be dominated mainly by majority-majority spin
channel for a half-metal/insulator/half-metal MTJ structure.10
Here, the crossover at a smaller bias voltage in the experi-ments is not E
gitself but the crossover voltage will be influ-
enced by the existence of the dip, namely, magnon contribu-tion, which changes the shape of the G
APcurve. However,
the influence is not significant for CFAS with a wide bandgap because the magnon contribution is limited to near smallbias voltage. The second crossover at V
Emainly reflects the
shape of the DOS around −1 eV. In the case of CFAS theDOS of majority-spin band is low and nearly constant downto −1.5 eV, while the DOS of minority-spin band around−/H208491–2 /H20850eV has a deep minimum. This feature provides a
maximum around 1.1 V for G
AP/H20849VD/H20850. On the other hand, the
DOSs of both majority- and minority-spin bands of the CMSincrease greatly below −0.5 eV and there is no deep mini-mum in the minority band.
4,5,8Consequently, the second
crossover and the peak in GAPmay not appear in a MTJ
using CMS electrodes. In fact, Ishikawa et al.12reported only
one crossover of the conductance curves in a CMS/MgO/CoFe structure.
The only one crossover accompanied with the asymmetric
dI/dVcurve observed in L2
1-CFAS /MgO /B2-CFAS struc-
ture suggests that the band structure of the B2-CFAS differs
from that of the L21-CFAS. The /H20849002 /H20850peak intensity in XRD
pattern for the B2-CFAS annealed at 633 K was weak,25
suggesting the existing of significant A2o rD O 3-type disor-
dering structure in the B2-CFAS film, which destroys half
metallicity.21
FIG. 5. /H20849Color online /H20850/H20849a/H20850
Spin-resolved DOS ofL2
1-Co 2FeAl 0.5Si0.5obtained from
the generalized gradient approxi-mation /H20849GGA /H20850+Utheorem. /H20849b/H20850
Calculated dI/dVspectra based on
Fig. 5/H20849a/H20850and Eq. /H208492/H20850for parallel
/H20849dotted line /H20850and antiparallel /H20849solid
line /H20850configurations.SUKEGAWA et al. PHYSICAL REVIEW B 79, 184418 /H208492009 /H20850
184418-4A crossover in tunneling conductance curves is also re-
ported in the Fe/MgO/Fe structure due to the carbon /H20849C/H20850
contamination of the bottom Fe layer.31The residual C atoms
on a substrate diffuse into bottom Fe/MgO interface and thetransport is predicted to be dominated by the Fe-C/MgOelectronic structure.
31In this paper, however, we grew the
CFAS/MgO/CFAS structure on a 20-nm-thick MgO bufferlayer, which acts as antidiffusion barrier which traps the re-sidual C impurities and prevents their diffusion within thelayers during subsequent annealing stages. Therefore, C con-tamination effect for the crossovers is ruled out in our junc-tions. In fact, when a 10-nm-thick MgO underlayer wasused, the crossover was observed in Fe/MgO/Fe.
31In addi-
tion, we have observed two crossovers accompanied with apeak in the AP curve as seen in Fig. 4/H20849a/H20850, which is very
different from the C contamination effect.
31
Recently, we have fabricated MTJs on MgO substrates
using an AlO xbarrier consisting of CFAS /AlO x/CoFe and
found 162% TMR ratio at 26 K, which will be reportedelsewhere. From this TMR ratio, we obtained P=0.9 for
CFAS by assuming Julliere’s model and P=0.5 for CoFe,
which implies the half metallicity for CFAS. The smaller P
=0.78 in the present L2
1-CFAS /MgO /L21-CFAS structure
may be attributed to the magnon-assisted inelastic tunnelingin a low-bias voltage region.
18If we assume the interface
states near EFin the minority-spin band gap as shown sche-
matically in Fig. 6, magnon-assisted inelastic tunneling is
possible in a low-bias voltage region, which reduces TMRsignificantly as observed in Fig. 3. The interface states, how-
ever, do not affect the spin-dependent tunneling at highervoltages since they lie only near E
F. Thus the two crossovers
are maintained even if the interface states assumed near EF
inL21-CFAS /MgO /L21-CFAS structure. In the case of theL21-CFAS /MgO /B2-CFAS structure, the top B2-CFAS may
not be half metallic as mentioned above due to the lowerannealing temperature, which leads to the asymmetric con-ductance without two crossovers as shown in Fig. 4/H20849b/H20850.
The other possible reasons of the discrepancy between the
experiment and the calculation would be the lower L2
1or-
dering parameter, which reduces the gap. In fact the gapobtained in this study is about 0.6 eV as seen in Fig. 4/H20849a/H20850,
while it is 1.2 eV in the calculation. Note that we obtain nodistinct proof of the TMR enhancement by the coherent tun-neling effect in the present MTJs as mentioned in Sec. III B.
In this study we have used the sputtering method for the
MgO barrier formation, which gives a poor-quality barrier asmentioned in Sec. III A. In order to achieve a higher TMR
reduction in inelastic tunneling process by realizing cleanbarrier/CFAS interfaces is required. For this purpose, wewould need a high-quality MgO barrier using electron-beamdeposition method, which has been performed in the previ-ous experiments of Heusler/MgO/Heusler MTJs.
12–15
V. CONCLUSIONS
In this work, we measured TMR and tunneling spectros-
copy in fully epitaxial MTJs using full-Heusler CFAS/MgO/CFAS structures with L2
1and B2 for CFAS layers, fabri-
cated on MgO-buffered MgO /H20849001 /H20850substrates using
magnetron sputtering. Two distinct crossovers in differentialconductance curves G
PandGAPwere observed at different
voltages for both positive and negative bias voltages in theL2
1-CFAS /MgO /L21-CFAS structure, while only one cross-
over was observed at a negative voltage in theL2
1-CFAS /MgO /B2-CFAS structure. We proposed the di-
rect tunneling as a possible transport mechanism leading tothe notable crossovers, which reflects the specific spin-dependent density of states of the half-metallic L2
1-CFAS.
Based on this idea, we calculated GPandGAPby the DOS
for half-metallic L21-CFAS predicted from first-principle
calculations and successfully reproduced the conductancecurves as seen in the experiment.
ACKNOWLEDGMENTS
This work was partly supported by the NEDO, the
CREST program of the JST, the Grant-in-Aid for Young Sci-entists /H20849B/H20850under Grant No. 20760478 from the Ministry
of Education, Culture, Sports, Science and Technology andJST-SFG project.
*sukegawa.hiroaki@nims.go.jp
1R. A. deGroot, F. M. Mueller, P. G. vanEngen, and K. H. J.
Buschow, Phys. Rev. Lett. 50, 2024 /H208491983 /H20850.
2S. Sugahara and M. Tanaka, Appl. Phys. Lett. 84, 2307 /H208492004 /H20850.
3S. Ishida, S. Fujii, S. Kashiwagi, and S. Asano, J. Phys. Soc. Jpn.
64, 2152 /H208491995 /H20850.
4I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B
66, 174429 /H208492002 /H20850.5S. Picozzi, A. Continenza, and A. J. Freeman, Phys. Rev. B 66,
094421 /H208492002 /H20850.
6K. Inomata, S. Okamura, R. Goto, and N. Tezuka, Jpn. J. Appl.
Phys., Part 2 42, L419 /H208492003 /H20850.
7J. Schmalhorst, S. Kämmerer, G. Reiss, and A. Hütten, Appl.
Phys. Lett. 86, 052501 /H208492005 /H20850.
8H. C. Kandpal, G. H. Fecher, C. Felser, and G. Schonhense,
Phys. Rev. B 73, 094422 /H208492006 /H20850.
FIG. 6. /H20849Color online /H20850Schematic illustrations of the DOS of the
CFAS electrodes.SPIN-POLARIZED TUNNELING SPECTROSCOPY OF … PHYSICAL REVIEW B 79, 184418 /H208492009 /H20850
184418-59N. Tezuka, N. Ikeda, A. Miyazaki, S. Sugimoto, M. Kikuchi, and
K. Inomata, Appl. Phys. Lett. 89, 112514 /H208492006 /H20850.
10Y. Sakuraba, T. Miyakoshi, M. Oogane, Y. Ando, A. Sakuma, T.
Miyazaki, and H. Kubota, Appl. Phys. Lett. 89, 052508 /H208492006 /H20850.
11Y. Sakuraba, M. Hattori, M. Oogane, Y. Ando, H. Kato, A. Sa-
kuma, T. Miyazaki, and H. Kubota, Appl. Phys. Lett. 88,
192508 /H208492006 /H20850.
12T. Ishikawa, T. Marukame, H. Kijima, K.-I. Matsuda, T. Uemura,
M. Arita, and M. Yamamoto, Appl. Phys. Lett. 89, 192505
/H208492006 /H20850.
13T. Marukame, T. Ishikawa, K.-I. Matsuda, T. Uemura, and M.
Yamamoto, Appl. Phys. Lett. 88, 262503 /H208492006 /H20850.
14N. Tezuka, N. Ikeda, S. Sugimoto, and K. Inomata, Jpn. J. Appl.
Phys., Part 2 46, L454 /H208492007 /H20850.
15T. Ishikawa, S. Hakamata, K. Matsuda, T. Uemura, and M.
Yamamoto, J. Appl. Phys. 103, 07A919 /H208492008 /H20850.
16S. Tsunegi, Y. Sakuraba, M. Oogane, K. Takanashi, and Y. Ando,
Appl. Phys. Lett. 93, 112506 /H208492008 /H20850.
17M. Julliere, Phys. Lett. 54A, 225 /H208491975 /H20850.
18P. Mavropoulos, M. Lezaic, and S. Blugel, Phys. Rev. B 72,
174428 /H208492005 /H20850.
19L. Chioncel, Y. Sakuraba, E. Arrigoni, M. I. Katsnelson, M. Oo-
gane, Y. Ando, E. Burzo, T. Miyazaki, and A. I. Lichtenstein,Phys. Rev. Lett. 100, 086402 /H208492008 /H20850.20G. H. Fecher and C. Felser, J. Phys. D 40, 1582 /H208492007 /H20850.
21Z. Gercsi and K. Hono, J. Phys.: Condens. Matter 19, 326216
/H208492007 /H20850.
22T. M. Nakatani, A. Rajanikanth, Z. Gercsi, Y. K. Takahashi, K.
Inomata, and K. Hono, J. Appl. Phys. 102, 033916 /H208492007 /H20850.
23S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando,
Nature Mater. 3, 868 /H208492004 /H20850.
24T. Dimopoulos, G. Gieres, J. Wecker, N. Wiese, Y. Luo, and K.
Samwer, J. Appl. Phys. 98, 073705 /H208492005 /H20850.
25W. H. Wang, H. Sukegawa, R. Shan, T. Furubayashi, and K.
Inomata, Appl. Phys. Lett. 92, 221912 /H208492008 /H20850.
26W. H. Wang, H. Sukegawa, R. Shan, and K. Inomata, Appl.
Phys. Lett. 93, 122506 /H208492008 /H20850.
27X. F. Han, T. Daibou, M. Kamijo, K. Yaoita, H. Kubota, Y.
Ando, and T. Miyazaki, Jpn. J. Appl. Phys., Part 2 39, L439
/H208492000 /H20850.
28J. S. Moodera and J. Nowak, R. J. M. vandeVeerdonk, Phys.
Rev. Lett. 80, 2941 /H208491998 /H20850.
29Spin Dependent Transport in Magnetic Nanostructures , edited
by S. Maekawa and T. Shinjo /H20849Taylor & Francis, London, 2002 /H20850.
30J. G. Simmons, J. Appl. Phys. 34, 238 /H208491963 /H20850.
31C. Tiusan, M. Sicot, J. F.-Vincent, M. Hehn, C. Bellouard, F.
Montaigne, S. Andrieu, and A. Schuhl, J. Phys.: Condens. Mat-ter18, 941 /H208492006 /H20850.SUKEGAWA et al. PHYSICAL REVIEW B 79, 184418 /H208492009 /H20850
184418-6 |
PhysRevB.96.195422.pdf | PHYSICAL REVIEW B 96, 195422 (2017)
Second quantization of Leinaas-Myrheim anyons in one dimension and their relation
to the Lieb-Liniger model
Thore Posske,1Björn Trauzettel,2and Michael Thorwart1
1I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
2Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
(Received 20 September 2016; revised manuscript received 24 January 2017; published 15 November 2017)
In one spatial dimension, anyons in the original description of Leinaas and Myrheim are formally equivalent
to locally interacting bosons described by the Lieb-Liniger model. This allows an interesting reinterpretation ofinteracting bosons in the context of anyons. We elaborate on this parallel, particularly including the many-bodybound states from the attractive Lieb-Liniger model. In the anyonic context these bound states are created solelyby quantum-statistical attraction and coined the quantum-statistical condensate, which is shown to be more robustthan the Bose-Einstein condensate. We introduce the second quantization formalism for the present anyons andconstruct the generalized Jordan-Wigner transformation that connects them to the bosons of the Lieb-Linigermodel.
DOI: 10.1103/PhysRevB.96.195422
I. INTRODUCTION
In modern mesoscopic systems, electronic excitations
are effectively confined to a lower-dimensional world. Anunexpected consequence of such a reduced spatial dimension
is the occurrence of particles that neither obey Fermi nor Bose
statistics. These are known as anyons [ 1–6]. Especially in
two dimensions, anyons have been theoretically extensively
studied [ 7–9] and indicated to exist in several experimental
systems [ 10–15]. The spatial exchange of two-dimensional
anyons and the accompanied, fixed unitary transformation of
the anyonic wave function could pave the way to topological
quantum computing [ 16]. Exchangeability is also apparent
in special one-dimensional systems, e.g, ringlike ones or
Tstructures [ 17,18]. Sparked by this idea, the interest in
lower-than-two-dimensional anyons has recently increased,especially in conjunction with the possible detection of
Majorana bound states in quantum wires [ 19–23]. Those are
expected to be non-Abelian anyons, potentially applicable intopological quantum computing [ 18,24]a sw e l l .
In this paper, we describe a first-principles theory of
one-dimensional anyons whose exchange is prohibited by thegeometry of the system, i.e., anyons on a line and in a box. Tothis end, we employ the concepts introduced in the fundamen-tal work of Leinaas and Myrheim [ 1] extended to many-particle
systems. Their approach follows one fundamental idea: to setup the proper classical theory of indistinguishable particlesand subsequently quantize it. In two dimensions, this resultsin “standard” Chern-Simons anyons, that can be interpretedas bosons with an attached flux acquiring an Aharonov-Bohmphase when physically exchanged [ 6,9]. This renders the work
of Leinaas and Myrheim [ 1] one of the standard references
in the field. Their theory is less frequently applied to onespatial dimension. There, a manifold of different theoriesexists [ 5,19,25–40], which also describe locally interacting
anyons [ 41–44]; see Appendix Af o rab r i e fs u m m a r y .I n
particular, Leinaas-Myrheim anyons have to be contrasted tothe emergent excitations of the Calogero-Sutherland model,the Haldane-Shastry chain, and the fractional quasiparticlesin Tomonaga-Luttinger liquids [ 19,35–39] that are as well
called anyons [ 37]. The defining property of these kinds ofanyons is that the wave function acquires a fixed phase κ
when the coordinates of two anyons get permuted, in completeanalogy to the statistical angle in two spatial dimensions.While this behavior is seen as natural, it conflicts with theimpenetrability of anyons, an essential ingredient to derivingthe two-dimensional theory of anyons in the framework ofLeinaas and Myrheim. The question arises how anyonic wavefunctions can acquire a phase upon the exchange of coordinatesif the particles themselves cannot be exchanged. Interestingly,corresponding theories are still described by the approach ofLeinaas and Myrheim with a proper continuation procedure[37]. To strengthen the plausibility of Leinaas’ and Myrheim’s
approach, we furthermore want to stress its fundamental depth.First, it does not require an underlying theory of additional,constituting particles. Only the indistinguishability of theconsidered particles, the validity of canonical quantizationfor flat spaces, and the Hermiticity of the Hamiltonian isassumed. Second, Leinaas-Myrheim anyons appear naturally
when two-dimensional Chern-Simons anyons are confined
to one dimension by a potential. In the process of thedimensional crossover, the complete statistical angle getsgradually absorbed and encoded into the scattering behaviorof the anyons [ 32], which eliminates the need for an additional
statistical phase in one spatial dimension. As a concretephysical example, we can imagine a fractional quantum Hallinsulator [ 11] where anyonic bulk excitations are confined to
one spatial dimension by an electric potential.
As a twist of history, Leinaas’ and Myrheim’s theory
of one-dimensional anyons (1977) turns out to be formallyequivalent to the Lieb-Liniger model, describing locallyinteracting bosons in one spatial dimension (1963) [ 45]. With
“formally equivalent” we mean that the equations appearingin both theories are identical. However, the calculation ofphysical observables is different. This leads to both systemsbeing described by the same equations but exhibiting differentphenomenologies (see Sec. IIbelow). Lieb and Liniger derived
the solutions of their model by first rewriting it employing
boundary conditions, not knowing that these equations wouldseveral years later be employed by Leinaas and Myrheimto describe one-dimensional anyons. While the Lieb-Linigermodel has substantially advanced since 1963, the results
2469-9950/2017/96(19)/195422(9) 195422-1 ©2017 American Physical SocietyPOSSKE, TRAUZETTEL, AND THORWART PHYSICAL REVIEW B 96, 195422 (2017)
have, to the best of our knowledge, not been carried over
to the theory of Leinaas-Myrheim anyons. In this work, weclose this gap. This includes the calculation of observables[19,20,46–48] for confined anyons: the energy spectrum,
momentum density, and finite-size density oscillations. Ourresults are applicable to quasiparticle excitations in quasi-one-dimensional systems, such as interacting cold-atom/ion chainsand edge liquids of topological insulators, that potentiallycarry anyonic excitations [ 19,20,40,49,50]. Additionally, the
interpretation of interacting bosons as anyons introduces afresh perspective into the established field of the Lieb-Linigermodel. For instance, the results for confined anyons alsodescribe Lieb-Liniger bosons in a box, where the Dirichletboundary conditions result in modified Bethe ansatz equations[51,52]. Motivated by the anyonic interpretation, we develop
the second quantization formalism for Leinaas-Myrheimanyons and, hence, an alternative second quantization forthe Lieb-Liniger model. While presenting the formalism, wetake particular care of including complex momenta, which, asusual, describe spatially bound states. In the exact many-bodysolutions, they build up a stable quantum phase, which we callthe quantum-statistical condensate. At first sight, the existenceof bound states is surprising in the anyonic interpretation, butimmediately becomes clear when interpreted as attractivelyinteracting bosons in the Lieb-Liniger model. Lieb and Linigerhave first disregarded this regime as unphysical and unstable
[45]. More recent work has albeit revealed its soundness
[53–58]. Furthermore, it has been pointed out that, within the
attractive regime, additional gaslike phases may exist [ 59,60].
The structure of the paper is as follows. In Sec. II,w e
concisely review both relevant models, i.e., the Lieb-Linigermodel and the model of Leinaas and Myrheim for one-dimensional anyons and state their formal equivalence. InSec. III, we construct the anyonic wave functions, which are
the basis of the second quantization formalism that we derivein Sec. IV. We also provide the generalized Jordan-Wigner
transformation from Lieb-Liniger bosons to Leinaas-Myrheimanyons. The Bethe ansatz equations for systems of finite sizeare discussed in Sec. V, which are consequently, in Sec. VI,
applied to derive some properties of anyons in a box. Thecase of a negative statistical parameter is covered in Sec. VII,
where we introduce the quantum-statistical condensate and theinterpretation of the clusters as individual anyons themselves.We conclude our work in Sec. VIII.
II. MODEL
We start by reviewing the models of Lieb-Liniger and
Leinaas and Myrheim and highlight how their formal equiv-alence still results in different phenomenology. The Lieb-Liniger model [ 45] describes a number of nlocally interacting
bosons in one dimension. In real space, the system isrepresented by its totally symmetric wave function /Psi1, which
mapsnreal numbers to a complex one, and is governed by the
Hamiltonian
H
LL=−¯h2
2mn/summationdisplay
j=1∂2
xj+2c/summationdisplay
i/negationslash=jδ(xi−xj). (1)Here, mdenotes the mass of the particles and cis the
real-valued interaction strength that has the dimension ofmomentum. The δfunctions can be directly implemented into
the wave function by demanding boundary conditions, which,because of the symmetry of the wave function, turn out to bethe so-called Robin boundary conditions
(∂
xj+1−∂xj)/Psi1(x)|xj→xj+1=c/Psi1(x)|xj→xj+1, (2)
for each jbetween 1 and n−1, and we restrict ourselves to the
region R={x|x1<x 2<···<xn}of the parameter space
[45]. In exchange for the boundary conditions, the Hamiltonian
onRbecomes the one of free particles, i.e.,
HLL|R=−¯h2
2mn/summationdisplay
j=1∂2
xj. (3)
Let us now recapitulate and slightly extend the theory of
Leinaas and Myrheim [ 1] for indistinguishable quantum
particles. First, consider nclassical particles on a line. The
spatial configurations of a system of distinguishable particleswould be described by tuples of positions x=(x
1,..., x n).
Because the particles are indistinguishable, however, usingtuples is ambiguous: For n=2, (x
1,x2) and ( x2,x1) label the
same configuration. Instead, we employ the sets {x1,..., x n}
ofndistinct positions. The family of all these sets is called
configuration space Rand inherits various properties by local
equivalence to Rn. Here, the notational correspondence of the
configuration space to the parameter region of the Lieb-Linigermodel is on purpose, since the real-space variables x
1<···<
xnparametrize R. To obtain the quantum-mechanical theory,
space and momentum variables get promoted to operatorsacting on the wave functions /Psi1:R→C. We consider, for
concreteness, the particles to obey the free Hamiltonian ofEq. ( 3) as well. However, electromagnetic potentials and
particle interactions can be added without changing the generalformalism. Finally, we demand the Hamiltonian Hto be
Hermitian. Interestingly, Hermiticity is granted if and onlyif/Psi1fulfills the Robin boundary conditions of Eq. ( 2)[32]. In
this context, the interaction strength cis called the statistical
parameter η≡c.
In conclusion, both theories use the same differential
equation and boundary conditions, which constitutes a formalequivalence between them. The phenomenology of bothmodels, however, can differ significantly. The reason for this isthe differing calculation of physical observables. For the modelof Leinaas and Myrheim, given an operator A, its expectation
value is calculated by an integral over the configuration spaceRonly, according to
/angbracketleftA/angbracketright
LM=/integraldisplay∞
xn−1dxn···/integraldisplayx3
x2dx2/integraldisplayx2
−∞dx1/Psi1∗(x)A/Psi1(x),(4)
while for the Lieb-Liniger model, the integration region is the
full real space, such that
/angbracketleftA/angbracketrightLL=/integraldisplay∞
−∞dnx/Psi1∗(x)A/Psi1(x). (5)
In the latter equation, /Psi1(x) has been symmetrically continued
by/Psi1(x):=/Psi1(π(x)), where πis a permutation such that π(x)
is inR. The fact that Eqs. ( 4) and ( 5) can differ is known in
the context of impenetrable bosons, i.e., the Tonks-Girardeau
195422-2SECOND QUANTIZATION OF LEINAAS-MYRHEIM ANYONS . . . PHYSICAL REVIEW B 96, 195422 (2017)
gas, and free fermions. While the former is the limit of the
Lieb-Liniger theory at infinite repulsion c→∞ , the latter
is the limit of the Leinaas-Myrheim theory for η→∞ .
However, impenetrable one-dimensional bosons differ fromfree fermions, for instance, by their momentum distribution[61].
The twofold interpretation of Eq. ( 2) in combination with
Eq. (3) provides simple explanations of seemingly complicated
facts. For instance, if the bosons attract each other, i.e., c<0,
they form clusters [ 58]. This intuitive property of the Lieb-
Liniger model seems surprising for Leinaas-Myrheim anyons,where the attraction would be mediated by the statistics only[62].
There is indication that the coincidentally looking formal
equivalence between locally interacting bosons and Leinaas-Myrheim anyons is in fact not coincidental. To explain this,we refer to the connection between bosons and anyons in twospatial dimensions. There, anyons are equivalent to bosons thatacquire an Aharonov-Bohm flux when circling around eachother. Hence, also in two dimensions, anyons are equivalent toparticularly interacting bosons. The reason for this is that theconfiguration space for two-dimensional anyons has holes, thepoints where the position of a pair of particles would coincide.The holes themselves are irrelevant concerning scattering sinceparticles can move around them by infinitesimally alteringtheir path. However, the holes potentially induce a holonomy
in the wave function, the Aharonov-Bohm phase. In one spatial
dimension, the holes no longer induce a holonomy becauseparticles cannot be exchanged. However, the holes themselvesbecome relevant as, by normal propagation, particles canscatter off each other at some time. Then, the holes inducethe boundary condition of Eq. ( 2) and thereby again serve as
the origin of the bosonic interaction that connects bosons andanyons.
In the remaining course of the paper, we present and
interpret the solutions to the equivalent models from the morerarely employed anyonic point of view.
III. CONSTRUCTION OF THE WA VE FUNCTIONS
We next construct all wave functions that fulfill Eq. ( 2),
thereby combining the solutions of the attractive and therepulsive Lieb-Liniger model [ 45,58], with the final aim to pro-
vide the second quantization formalism for Leinaas-Myrheimanyons. To this end, we employ the ansatz [ 45]/Psi1(x)=/integraltext
k∈Cndnkα(k)eikx. Complex momenta kare explicitly in-
cluded. These are needed to describe anyonic bound statesthat form for a negative statistical parameter. In momentumspace, the boundary conditions translate to
α(k)=e
−iφη(kj+1−kj)α/parenleftbig
σjk/parenrightbig
ifkj+1−kj/negationslash=iη, (6)
α(k)=0i fkj+1−kj=iη. (7)
Here,σjdenotes the elementary permutation which permutes
thejth and ( j+1)th element of a tuple and
φη(kj+1−kj)=2a r c t a n/bracketleftbig
η/(kj+1−kj)/bracketrightbig
(8)
is the statistical phase. By iteration, these conditions con-
nect coefficients of relatively permuted momenta α(k)=
FIG. 1. Examples of composite anyons described by clusters μ,
called strings in the context of the Lieb-Liniger model. These are
the fundamental building blocks of the anyonic wave functions andcan be conceived as individual particles. For clusters of more than
one anyon, η< 0 is implicit. (a) Single anyon; (b) two-anyon bound
state; (c) maximally bound cluster of nanyons: the quantum-statistical
condensate.
eiφP
η(k)α(Pk). Here, P=σj1···σjris a general permuta-
tion written with an ras small as possible and φP
η(k)=/summationtextr
i=1φη[(σj1···σjik)ji−(σj1···σjik)ji+1]. The basis func-
tions are therefore of the form /Psi1k(x)∝/summationtext
P∈SneiφP
η(k)ei(Pk)x.
On physical grounds, divergent elements of this set need tobe excluded. This is done by only permitting special valuesofk. These are build up by tuples μof complex momenta
where the difference between adjacent momenta μ
j+1−μj
is−iη. These tuples are called strings in the context of
the Lieb-Liniger model [ 58]. Within the anyonic context,
we call them clusters. Examples of clusters are sketched inFig.1.
Physically, clusters with more than one element represent
composite anyons whose constituents move collectively, sep-arated by a characteristic length scale of 1 /η. They can be
conceived as individual particles themselves. For positive η,
clusters only consist of single particles, and hence describefree (unbound) anyons. In order to uniquely label the basisfunctions, we introduce the cluster ordering O. This is done
in direct analogy to the ordering that needs to be introduced tolabel fermionic basis states in standard quantum many-bodytheory [ 63]. To apply Oto a tuple Dof clusters, first take
the union of the clusters’ momenta. Then sort all momentaby their real parts (smaller values first). If there are momentawith equal real parts, sort them by their imaginary parts (again,smaller values first).
In conclusion, the basis functions are given by momenta
that describe composite and free anyons in momentum space.Given an ordered tuple O(D) of clusters, the corresponding
basis function obtains the form
/Psi1
(k=O(D))(x)=Nk/summationdisplay
P∈SneiφP
η(k)ei(Pk)x, (9)
where Nkis the normalization [ 64] andeiφP
η(k)plays the role
of a generalized Slater determinant.
IV . SECOND QUANTIZATION
An advantage of the anyonic interpretation of Eqs. ( 2)
and ( 3) is that a second quantization of the solution is
reasonably motivated. In contrast, this endeavor seems tobe discouraged in the bosonic picture of the Lieb-Linigermodel, where the bosons are already given in their secondquantized form. The formalism can facilitate the calculation ofvarious properties, similar to the original second quantization
195422-3POSSKE, TRAUZETTEL, AND THORWART PHYSICAL REVIEW B 96, 195422 (2017)
of bosons and fermions. Details on the formalism are presented
in Appendix B. Given the basis wave functions of Eq. ( 9),
second quantization amounts to defining creation operators toconstruct all basis states from a vacuum state [ 65]. For a cluster
μ, we define its creation operator by
a
†
μ/Psi1O(D)=/radicalbig
M(μ)+1ei/Phi1μ
η(D)/Psi1O({μ}∪D) (10)
and linear continuation to all states. Here, M(μ)i st h e
number of clusters μinD. The phase /Phi1μ
η(D)=/summationtext
˜μ<μϕ˜μ,μ
η
is composed of the cluster-cluster exchange phases ϕ˜μ,μ
η=/summationtextN(μ)
i=1/summationtextN(˜μ)
i=jφη(˜μj−μi). Here, ˜μ<μif˜μis ordered to the
left of μby cluster ordering and N(μ) denotes the number of
anyons in μ. Employing Eq. ( 10), the algebra of the cluster
creation operators is
a†
μ1a†
μ2=eiϕμ1,μ2ηa†
μ2a†
μ1. (11)
To be concrete, we consider the case of unbound anyons,
described by clusters with exactly one element. Here,
a†
pa†
q=eiφη(p−q)a†
qa†
p,
apa†
q=e−iφη(p−q)a†
qap+δ(p−q), (12)
where the annihilation operator apis the Hermitian conjugate
ofa†
p, which is the shorthand notation for a†
(p).
It is striking that the one-dimensional anyonic algebra
depends on the relative momentum instead of providing a fixedstatistical phase as familiar from two-dimensional anyons [ 1]
and the different types of one-dimensional anyons mentionedin the Introduction [ 19,35–39].
Employing the anyonic second quantization, the Hamilto-
nian of Eq. ( 3) becomes
H=/summationdisplay
μ/epsilon1μa†
μaμ, (13)
and describes free anyonic clusters. In Eq. ( 13), the sum runs
over all possible clusters μ, which have the energy
/epsilon1μ=¯h2
2m/parenleftbigg
nμK2
μ−1
12η2(nμ−1)nμ(nμ+1)/parenrightbigg
. (14)
The latter equation is derived in Appendix C. Here, Kμis the
real-valued center-of-mass momentum (see Fig. 1) andnμis
the number of bare anyons forming the cluster. One can seethat the contribution of clusters to the energy separates, whichfurther motivates their interpretation as individual particlesthemselves. Furthermore, each cluster contributes by its kineticenergy and its internal binding energy. This is reflected in thefirst and second summand of Eq. ( 14), respectively.
Given the momentum-space operator algebra of Eq. ( 12),
we can address the algebra of the real-space operators /Psi1
†(x)=/integraltext∞
−∞dpeipx√
2πa†
p. We obtain
{/Psi1(x),/Psi1†(y)}=δ(x−y)
+/integraldisplay∞
0dz2e−z
|η|
|η|/Psi1†(y−z)/Psi1(x−z),
{/Psi1†(x),/Psi1†(y)}=/integraldisplay∞
0dz2e−z
|η|
|η|/Psi1†(y−z)/Psi1†(x+z),(15)where {...,... }denotes the anticommutator. Here,
limη→0/integraltext∞
0dz1
|η|e−z/|η|f(z)=f(0) yields the bosonic com-
mutation algebra, while the fermionic anticommutation re-
lations for η→∞ are trivially contained. If we set x=y,
we arrive at a smeared anyonic Pauli principle in real spacerepresented by
[/Psi1
†(x)]2=/integraldisplay∞
0dz1
|η|e−z/|η|/Psi1†(y−z)/Psi1†(x+z).(16)
In one dimension, there exist ways to transform between
different statistics regarding bosons, fermions, and spins, e.g.,by bosonization, refermionization, and the Jordan-Wignertransformation [ 19,66]. Likewise here, there is a generalized
Jordan-Wigner transformation from the present anyons to the
bosons of the Lieb-Liniger model. Ultimately, this reflects thefact that the Fock space of anyons is naturally isomorphic tothe one of the Lieb-Liniger model (if η/negationslash=± ∞ ). To this end,
consider the bosonic operators b
lwith the algebra [ bk,b†
l]=
δ(k−l) and [bk,bl]=0 with k,l∈R.F o rη/negationslash=± ∞ , we define
the generalized Jordan-Wigner transformation,
˜a(j)=lim
/epsilon1→0+ei/integraltextj−/epsilon1
−∞dkb†
kbkφη(k−j)b(j). (17)
Calculating the algebra of ˜a, we find ˜aj˜ak=˜ak˜ajeiφη(k−j)
and ˜aj˜a†
k=˜a†
k˜aje−iφη(j−k)+δ(j−k), which is exactly the
anyonic algebra described in Eq. ( 12). As an apparent
peculiarity, we have ˜a†
k˜ak=b†
kbk, which results, for η> 0,
in the same free Hamiltonian, Eq. ( 3), using either the bosonic
or the anyonic description. One would naively expect that thetransformation should generate the interacting Hamiltonianof Eq. ( 1) instead. However, since the theory of Leinaas
and Myrheim is intrinsically constrained to the region R,
defined at the beginning of Sec. II, it makes sense that the
transformation yields Eq. ( 3). The information about the
interactions remains encoded in the boundary conditions ratherthan in the Hamiltonian.
V . SYSTEMS OF FINITE SIZE
When anyons are confined to the length L, one would
expect the Dirichlet boundary conditions /Psi1(0,x2,..., x n)=
/Psi1(x1,..., x n−1,L)=0 to quantize the allowed momenta,
similar to the particle-in-a-box problem. In fact, the conditionstranslate to
α(−k
1,..., k n)=−α(k),
α(k1,...,−kn)=−e2iknLα(k). (18)
These constraints of Eq. ( 18) are only consistent with Eqs. ( 6)
and ( 7) if the system of transcendental equations
Lkj+/summationdisplay
1/lessorequalslant(i/negationslash=j)/lessorequalslantn[φη(ki−kj)−φη(ki+kj)]/2=πzj(19)
is fulfilled for jbetween 1 and n. Here, the zjare positive
integers. The momenta that solve Eq. ( 19) are discrete and
readily numerically obtainable. In the context of the Lieb-Liniger model, these equations are very similar to the so-calledlogarithmic Bethe ansatz equations [ 59,67]. Note, however,
that the Lieb-Liniger Bethe ansatz equations originally de-scribe particles that are confined to a ring, while Eq. ( 19)
195422-4SECOND QUANTIZATION OF LEINAAS-MYRHEIM ANYONS . . . PHYSICAL REVIEW B 96, 195422 (2017)
is adjusted to the case of particles in a box. Although no
differences in the thermodynamic limit are to be expected,these Bethe ansatz equations should give more reasonablefinite-size results for confined particles (see Refs. [ 51,52]
for the discussion in the context of the Lieb-Linigermodel).
VI. APPLICATION
Equipped with the developed formalism, we next consider
observables of experimental interest. First, we calculate thespectrum of two confined anyons numerically by solvingEq. ( 19). The result is depicted in Fig. 2(a). The anyonic
spectra interpolate between the familiar bosonic and fermionicparticle-in-a-box spectra for positive η. For instance, setting
E
0=¯h2π2/(2mL2), the bosonic level with an energy of
2E0continuously evolves to the fermionic level with 5 E0.
At negative η, a two-anyon bound state forms with an
energy proportional to −η2in the infinite-size limit L→
∞. Energetically higher anyonic bound states correspond to
kinetic excitations of this composite anyon in analogy tothe behavior of a single particle in a box. Some anyoniclevels refuse to form bound states and instead converge tofermionic energies as η→− ∞ . These levels ensure that
the finite-size spectrum coherently converges to the infinite-size spectrum. Energy spectra could be a viable observ-able in systems with few anyons, such as interacting cold-atom chains [ 19,20,40], and are detectable by spectroscopic
techniques.
Turning to systems containing many anyons, as possibly
being the case in solid state systems, unavoidable levelbroadening renders an accurate measurement of the discretespectrum unfeasible. Yet, the momentum distribution coulduncover the character of the anyons [ 19]. We depict the
momentum density n
kat zero temperature in Fig. 2(b).T h i s
function gives the number of anyons with momentum between
k1andk2by/integraltextk2
k1dkn k. For bosons and fermions, it is pro-
portional to the Bose-Einstein and Fermi-Dirac distribution,respectively. Anyons with a positive statistical parametertransform these distributions into each other, still preserving asharply defined chemical potential reflected by a discontinuityinn
k. This has to be seen in contrast to the behavior of a
Tomonaga-Luttinger liquid. The depicted momentum densityis well known from the Lieb-Liniger model [ 45]. Because
of the difference between Eqs. ( 4) and ( 5), however, the
momentum density is not the physically measurable one ofthe interacting bosons [ 61].
If the spectral properties of a system are inaccessible,
the statistics is still inferable via local properties, e.g., thefinite-size density fluctuations [ 20,48]. While bosons condense
to the middle of the system, fermions distribute equally spaced(by Pauli repulsion), resulting in oscillations of the particledensity. Figure 2(c) depicts the scenario for four anyons in
the ground state. Unbound anyons suppress the fermionicpeaks and broaden the bosonic one, which is characteristicof intermediate statistics [ 20,48].
VII. THE QUANTUM-STATISTICAL CONDENSATE
Forη< 0, the anyonic ground state is a cluster of the
formμj=iη
2(nμ−2j+1) as depicted in Fig. 1. We call this
cluster the quantum-statistical condensate since, in the anyonicpicture, its origin is solely based on the quantum statistics. Itcan be conceived as a single composite anyon and correspondsto the bound state of bosons that forms in the Lieb-Linigermodel [ 54,55] for an attractive interaction. Therefore, its local
density is similar to the one of a single quantum particle,which, in turn, is the same as the one of the Bose-Einsteincondensate [cf. Fig. 2(c)]. In fact, the Bose-Einstein conden-
sate can be interpreted as the limit of the quantum-statisticalcondensate as η→0
−. Besides this, both condensates differ
profoundly: Bosons condense into their single-particle groundstate, but anyons into an inseparable many-body groundstate.
Let us derive further characteristics of the quantum-
statistical condensate. First, we obtain its ground-stateenergy
/epsilon1
GS=−¯h2
24mη2(n−1)n(n+1) (20)
by Eq. ( 14). The proportionality to n3reveals an exceptional
stability of the condensate [ 68]. Let us for a moment regard
charged anyons exhibiting Hubbard repulsion, which has anassociated energy proportional to n
2. Then, providing a suffi-
ciently large number of anyons, the negative statistical energyoutperforms the positive one created by charge repulsion.
FIG. 2. Observables for confined anyons. The anyonic properties for 0 <η< ∞continuously interpolate between bosons ( η=0) and
fermions ( η=∞ ). The statistical condensate forms at η< 0. (a) Discrete energy spectrum of two anyons. To depict the full range of η, we plot
against φη(1/L). The two-anyon bound state and its excitations emerge for negative η. (b) Momentum density at zero temperature in the limit
of infinitely many particles (numerical calculations for n=512 anyons, where the curves almost converge to the limit n→∞ ). (c) Finite-size
oscillations of the particle density ρfor four anyons.
195422-5POSSKE, TRAUZETTEL, AND THORWART PHYSICAL REVIEW B 96, 195422 (2017)
The quantum-statistical condensate is hence stable against the
introduction of charge. We conjecture that this property couldlead to anyon superconductivity [ 69–71]. In fact, in the context
of the Lieb-Liniger model, a phase of pairwisely bound bosonshas been predicted [ 59].
A cluster behaves as an individual anyon, the energy of
which separates into a kinetic and an internal part. Addition-ally, by Eq. ( 11), clusters acquire different statistical phases
than their constituents. For instance, clusters of two anyonsbehave as anyons with the statistical phase 2 φ
η+φ2η(see
Appendix Dfor details). In the vocabulary of topological field
theories for two-dimensional anyons [ 9,16,72], the formation
of clusters is linked to anyon fusion.
VIII. CONCLUSIONS
On the basis of the general assumptions of Leinaas
and Myrheim [ 1], we derive an exact quantum many-body
formalism for one-dimensional anyons including the exactwave functions, the second quantization, and the momentumdiscretizing equations for anyons in a box. The formalism isbased on the equivalence to the Lieb-Liniger model of locallyinteracting bosons for which an interpretation in the anyoniccontext is established. We numerically calculate characteristicobservables, namely, the energy spectrum, the momentumstatistics, and the finite-size density fluctuations. For a negativestatistical parameter, anyons attract each other with a forcepurely induced by their quantum statistics and form thequantum-statistical condensate. This genuine quantum many-body phase is shown to be more robust than the Bose-Einsteincondensate. In particular, the statistical condensate is stableagainst the introduction of charged anyons. The clustersthemselves should be conceived as individual anyons andobtain a different statistical phase than their constituents. Ourwork shows that one-dimensional anyons exhibit original andinteresting physics even in the absence of spatial exchange.Furthermore, it emphasizes the link between anyons andinteracting bosons and thereby opens possibilities of syn-thesizing either physical system by its formally equivalentpartner.
ACKNOWLEDGMENTS
We would like to thank T. Hans Hansson and J. Magne
Leinaas for their clarifying remarks. Additionally, we ac-knowledge interesting discussions with P. Burset, FrançoisCrépin, D. Hetterich, A. Pelster, and N. Rosehr. B.T.thanks the DFG for financial support through the SFB 1170(“ToCoTronics”).
APPENDIX A: NOTIONS OF INTERMEDIATE STATISTICS
IN ONE SPATIAL DIMENSION
There exists a variety of formalisms describing particles of
intermediate statistics in one dimension, which are expectedto be applicable to different physical situations. Although theydiffer in their phenomenology, these particles are all calledanyons. For clarity, we briefly introduce some prominenttheories that are applicable to one spatial dimension.If the occupation number of a single-particle quantum
state is restricted to maximally assume a given integer,the particles can be described as parafermions, which areclosely related to Potts and clock models [ 26,27] and Gentile
statistics [ 25]. Such particles are, among others, expected
to exist as magnetic excitations [ 28,29]. Another kind of
intermediate statistics considers the representations of thelocal current algebra (the commutation relations betweenthe particle density and the particle currents in all spatialdimensions) [ 5,30] or the quantization of the algebra of
allowed observables of indistinguishable particles. The latterhas been applied to superconducting vortices [ 31] and two-
dimensional anyons effectively confined to one dimension bya strong magnetic field [ 32]. Yet another notion of anyons in
one dimension can be derived from Haldane’s generalizationof the Pauli principle [ 33], which is, for instance, applicable
to spinon excitations in spin chains. In this approach, thesingle-particle Hilbert space dimension depends on the totalnumber of particles in the system. Finally, the term anyons isused in one dimension to describe low-energy quasiparticleexcitations of interacting fermionic systems [ 34] linked to
the Calogero-Sutherland model [ 35,39], the Haldane-Shastry
chain, and the fractional excitations in Tomonaga-Luttingerliquids [ 19,36–38]. It is known that these particles (considering
each channel separately in the case of a Tomonaga-Luttingerliquid) break time-reversal symmetry on the fundamental
level of their operator algebra, reflected by an asymmetric
momentum distribution [ 19,40].
APPENDIX B: DETAILS ON THE CONSTRUCTION OF
THE SECOND QUANTIZATION
In this Appendix, we give details on the construction of
the second quantization formalism in the main text. First,we define the Fock space based on the valid wave functionsin momentum space, given by linear combinations of thebasis functions in Eq. ( 9), together with an auxiliary vacuum
state, and construct the creation and annihilation operators inmomentum space.
Then-particle anyonic Hilbert space H
nin a real-space
representation is formed by the wave functions described inEq. ( 9)a c t i n go nt h e ndimensional configuration space R
n=
{x∈Rn|x1<x 2<···<xn}.
These functions form an orthonormal basis. It is worth
noting in this regard, that although we generally wouldtake an overcomplete set of basis functions into accountby this approach, the constraints of Eq. ( 6), however, filter
out a complete set of mutually orthogonal functions if onlyproperly ordered vectors kare considered. For most of the
pairings, the orthogonality can be deduced by noticing thateigenfunctions of the Hermitian Hamiltonian with differenteigenvalues are orthogonal. It is worth mentioning here thatthe scalar product in a real-space representation is an integralrestricted to the configuration space as opposed to runningoverR
n. For two wave functions /Psi11,/Psi12:Rn→Ctheir scalar
product is
(/Psi11,/Psi12)=/integraldisplay
x∈Rn/Psi1∗
1(x)/Psi12(x). (B1)
195422-6SECOND QUANTIZATION OF LEINAAS-MYRHEIM ANYONS . . . PHYSICAL REVIEW B 96, 195422 (2017)
The Fock space is constructed by F=/circleplustext
n∈N0Hn. Here, H0
is the auxiliary vacuum space, isomorphic to C. The scalar
product of Eq. ( B1) is extended to act on Fby demanding
(/Psi1,/Psi1/prime)=0i f/Psi1and/Psi1/primeare states with a different particle
number. We now define the action of the linear creationoperators a†
μ:Hn→Hn+1for a cluster μas in Eq. ( 10)o f
the main text by its action on the basis function and linearcontinuation to all states, i.e.,
a
†
μ/Psi1O(D)=/radicalbig
M(μ)+1ei/Phi1μ
η(D)/Psi1O({μ}∪D), (B2)
where M(μ) is the number of clusters μinD. It is straightfor-
ward to show by virtue of Eq. ( 9) thata†
μis well defined, i.e.,
Eq. (9) results in a unique representation of a†
μin the given basis
functions. Its adjoint counterpart, the annihilation operator, iscorrespondingly defined employing the scalar product
/parenleftbig
/Psi1
1,a†
μ/Psi12/parenrightbig
=/parenleftbig
aμ/Psi11,/Psi12/parenrightbig
. (B3)
These definitions result in the operator algebra described in
t h em a i nt e x t[ s e eE q s .( 11) and ( 12)], by standard algebraic
manipulations, i.e.,
a†
μ1a†
μ2=eiϕμ1,μ2ηa†
μ2a†
μ1. (B4)
Applied to clusters of single particles, we have
a†
pa†
q=eiφη(p−q)a†
qa†
p,
apa†
q=e−iφη(p−q)a†
qap+δ(p−q). (B5)
Finally, we want to mention that the anyonic algebra for any
values of the statistical parameter but η=± ∞ results in
bosonic relations for equal momentum: [ ap,ap]=0, and, if
p→q,[ap,a†
q]→δ(p−q). However, for η=± ∞ , we, by
notation, strictly set e−iφ±∞(p−q)=−1, which results in the
familiar fermionic anticommutation algebra.
APPENDIX C: DERIVATION OF EQ. ( 14)—CLUSTER
ENERGY
The eigenvalues of Eq. ( 3) determine, as usual, the energy
of an eigenstate. Inserting an eigenstate [Eq. ( 9)] defined by
a vector kinto Eq. ( 3), the eigenenergy assumes the familiar
form/epsilon1=¯h2
2m/summationtextnk
j=1k2
j, where nkis the number of elements of
k. To derive the energy of a single cluster, Eq. ( 14), we insert
the general form of a momentum vector describing a singlecluster,
μ=/parenleftbigg
K
μ+iη
2(nμ−2j+1)/parenrightbigg
|nμ
j=1, (C1)
into the mentioned equation to obtain
/epsilon1μ=¯h2
2m⎛
⎝nμK2
μ+iηK μnμ/summationdisplay
j=1(nμ−2j+1)
−η2
4nμ/summationdisplay
j=1(nμ−2j+1)2⎞
⎠, (C2)
where nμis the number of elements of μ.U s i n g/summationtextnμ
j=1j=
nμ(nμ+1)
2and/summationtextnμ
j=1j2=1
6nμ(nμ+1)(2nμ+1), we see thatthe imaginary part vanishes and obtain Eq. ( 14) with
/epsilon1μ=¯h2
2m/parenleftbigg
nμK2
μ−1
12η2(nμ−1)nμ(nμ+1)/parenrightbigg
. (C3)
APPENDIX D: INTERPRETATION OF ANYON CLUSTERS
AS INDIVIDUAL ANYONS
We want to show how the exchange phase of clusters can
be interpreted as the statistical phase of a composite species ofanyons reaching further than the interpretation supported byEq. ( 11). To this end, we consider two clusters of anyons μ
1=
(K1+iη/2,K1−iη/2) and μ2=(K2+iη/2,K2−iη/2),
the cluster structures of which are depicted in Fig. 1(b).W e
introduce the center-of-mass coordinates X1=(x1+x2)/2
andX2=(x3+x4)/2, as well as the relative coordinates Z1=
(x2−x1)/2 andZ2=(x4−x3)/2. Under the assumption that
the two clusters are sufficiently far away from each other, i.e.,X
2−X1→∞ andZ1,Z2finite, we obtain
/Psi1(μ1,μ2)(X1,X2,Z1,Z2)
∝[e2i(K1X1+K2X2)+eiϕμ1,μ2ηe2i(K2X1+K1X2)]eη(Z1+Z2).
(D1)
This wave function resembles a wave function of two compos-
ite anyons with an altered statistical phase of ϕμ1,μ2η , especially
if we recall that Z1andZ2are of the order of 1 /η. This can be
physically interpreted as the fusion of anyons to clusters whichthemselves behave as a composite anyon species. Interestingly,the combined statistical phase is
ϕ
μ1,μ2
η=2φη(K2−K1)+φ2η(K2−K1), (D2)
where φηis the statistical phase defined in Eq. ( 8). This has an
appealing geometric interpretation, which we depict in Fig. 3.
FIG. 3. Geometric interpretation of the statistical phase of two
clusters, each consisting of two anyons. The radius of the circle
denotes the relative momentum between clusters K2−K1.T h e
statistical angle is obtained by adding up three summands. Two ofthese summands are the normal statistical angle φ
η, and the third
summand is the statistical angle of the doubled statistical parameter
φ2η. The statistical parameter ηappears in the lengths of the drawn
tangential segments.
195422-7POSSKE, TRAUZETTEL, AND THORWART PHYSICAL REVIEW B 96, 195422 (2017)
[1] J. Leinaas and J. Myrheim, Nuovo Cimento Soc. Ital. Fis. 37B,
1(1977 ).
[2] M. Kretzschmar, Z. Phys. 185,73(1965 ).
[3] M. Laidlaw and C. DeWitt, P h y s .R e v .D 3,1375 (1971 ).
[4] J. S. Dowker, J. Phys. A 5,936(1972 ).
[5] G. A. Goldin, R. Menikoff, and D. H. Sharp, J. Math. Phys. 21,
650(1980 ).
[6] F. Wilczek, P h y s .R e v .L e t t . 49,957(1982 ).
[7] L. Biedenharn, E. Lieb, B. Simon, and F. Wilczek, Phys. Today
43(8),90(1990 ).
[8] A. Khare, Fractional Statistics and Quantum Theory (World
Scientific, Singapore, 2005).
[9] A. Kitaev, Ann. Phys. (NY) 321,2(2006 ).
[10] G. Moore and N. Read, Nucl. Phys. B 360,362(1991 ).
[11] R. B. Laughlin, Phys. Rev. Lett. 50,1395 (1983 ).
[12] D. A. Ivanov, Phys. Rev. Lett. 86,268(2001 ).
[13] F. E. Camino, W. Zhou, and V . J. Goldman, Phys. Rev. B 72,
075342 (2005 ).
[14] R. L. Willett, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 82,
205301 (2010 ).
[15] Y . P. Zhong, D. Xu, P. Wang, C. Song, Q. J. Guo, W. X. Liu, K.
X u ,B .X .X i a ,C . - Y .L u ,S .H a n ,J . - W .P a n ,a n dH .W a n g , Phys.
Rev. Lett. 117,110501 (2016 ).
[16] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das
Sarma, Rev. Mod. Phys. 80,1083 (2008 ).
[17] S. Park and P. Recher, Phys. Rev. Lett. 115,246403
(2015 ).
[18] J. Alicea, Y . Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher,
Nat. Phys. 7,412(2011 ).
[19] G. Tang, S. Eggert, and A. Pelster, New J. Phys. 17,123016
(2015 ).
[20] T. Keilmann, S. Lanzmich, I. McCulloch, and M. Roncaglia,
Nat. Commun. 2,361(2011 ).
[21] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo,
A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346,
602(2014 ).
[22] V . Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A.
M. Bakkers, and L. P. Kouwenhoven, Science 336,1003
(2012 ).
[23] A. Y . Kitaev, Phys. Usp. 44,131(2001 ).
[24] B. I. Halperin, Y . Oreg, A. Stern, G. Refael, J. Alicea, and F. von
Oppen, P h y s .R e v .B 85,144501 (2012 ).
[25] G. Gentile j., Nuovo Cimento Soc. Ital. Fis. 17,493(1940 ).
[26] P. Fendley, J. Stat. Mech. (2012 )P11020 .
[27] J. Alicea and P. Fendley, Annu. Rev. Condens. Matter Phys. 7,
119(2016 ).
[28] W.-S. Dai and M. Xie, J. Stat. Mech. (2009 )P04021 .
[29] A. Hutter, J. R. Wootton, and D. Loss, P h y s .R e v .X 5,041040
(2015 ).
[30] G. A. Goldin, R. Menikoff, and D. H. Sharp, J. Math. Phys. 22,
1664 (1981 ).
[31] J. M. Leinaas and J. Myrheim, Phys. Rev. B 37,9286
(1988 ).
[32] T. Hansson, J. Leinaas, and J. Myrheim, Nucl. Phys. B 384,559
(1992 ).
[33] F. D. M. Haldane, P h y s .R e v .L e t t . 67,937(1991 ).
[34] Z. Ha, Nucl. Phys. B 435,604(1995 ).
[35] V . Pasquier, in Integrable Models and Strings , Lecture Notes in
Physics V ol. 436 (Springer, Berlin, 1994), pp. 36–48.[36] M. P. A. Fisher and L. I. Glazman, Transport in a one-
dimensional Luttinger liquid, in Mesoscopic Electron Transport ,
edited by L. L. Sohn, L. P. Kouwenhoven, and G. Schön(Springer, Dordrecht, 1997), pp. 331–373.
[37] M. T. Batchelor, X.-W. Guan, and N. Oelkers, Phys. Rev. Lett.
96,210402 (2006 ).
[38] A. Kundu, P h y s .R e v .L e t t . 83,1275 (1999 ).
[39] L. Brink, T. Hansson, and M. Vasiliev,
Phys. Lett. B 286,109
(1992 ).
[40] Y . Hao, Y . Zhang, and S. Chen, Phys. Rev. A 79,043633
(2009 ).
[41] O. I. Pu, V . E. Korepin, and D. V . Averin, J. Phys. A 41,145006
(2008 ).
[42] O. I. Pu, V . E. Korepin, and D. V . Averin, J. Phys. A 41,255205
(2008 ).
[43] O. I. Pu, V . E. Korepin, and D. V . Averin, J. Phys. A 42,275207
(2009 ).
[44] O. I. Pu, V . E. Korepin, and D. V . Averin, J. Phys. A 43,115204
(2010 ).
[45] E. H. Lieb and W. Liniger, Phys. Rev. 130,1605 (1963 ).
[46] A. Haim, E. Berg, F. von Oppen, and Y . Oreg, P h y s .R e v .B 92,
245112 (2015 ).
[47] B. Rosenow, I. P. Levkivskyi, and B. I. Halperin, Phys. Rev.
Lett.116,156802 (2016 ).
[48] C. Sträter, S. C. L. Srivastava, and A. Eckardt, Phys. Rev. Lett.
117,205303 (2016 ).
[49] L. Fu and C. L. Kane, P h y s .R e v .L e t t . 100,096407 (2008 ).
[50] M. Levin and A. Stern, P h y s .R e v .L e t t . 103,196803
(2009 ).
[51] M. Gaudin, Phys. Rev. A 4,386(1971 ).
[52] Y . Hao, Y . Zhang, J. Q. Liang, and S. Chen, P h y s .R e v .A 73,
063617 (2006 ).
[53] J. G. Muga and R. F. Snider, Phys. Rev. A 57,3317
(1998 ).
[54] A. Montina and F. T. Arecchi, Phys. Rev. A 71,063615
(2005 ).
[55] R. Kanamoto, H. Saito, and M. Ueda, P h y s .R e v .A 67,013608
(2003 ).
[56] K. Sakmann, A. I. Streltsov, O. E. Alon, and L. S. Cederbaum,
Phys. Rev. A 72,033613 (2005 ).
[57] A. G. Sykes, P. D. Drummond, and M. J. Davis, P h y s .R e v .A
76,063620 (2007 ).
[58] P. Calabrese and J.-S. Caux, P h y s .R e v .L e t t . 98,150403
(2007 ).
[59] M. T. Batchelor, M. Bortz, X. W. Guan, and N. Oelkers, J. Stat.
Mech. (2005 )L10001 .
[60] G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini,
Phys. Rev. Lett. 95,190407 (2005 ).
[61] M. Olshanii, Phys. Rev. Lett. 81,938(1998 ).
[62] There also is a vivid explanation for why anyons can bind
that does not leave the anyonic point of view. To this end, notethat a scattering event of two anyons with a negative statisticalparameter results in a negative time shift of the correspondingmatter wave [ 32]. Thus, anyons effectively scatter backwards in
time, have the chance to scatter again, and form a bound stateby infinite repetition.
[63] A. Altland and B. Simons, Condensed Matter Field Theory
(Cambridge University Press, Cambridge, UK, 2006).
[64] For real vectors k,w eh a v e N
k=√1/(2πnn!).
195422-8SECOND QUANTIZATION OF LEINAAS-MYRHEIM ANYONS . . . PHYSICAL REVIEW B 96, 195422 (2017)
[65] Technically, we define the vacuum state as /Psi1{}=1.
[66] J. von Delft and H. Schoeller, Ann. Phys. (Berlin) 7,225
(1998 ).
[67] M. T. Batchelor, X.-W. Guan, and J.-S. He, J. Stat. Mech. (2007 )
P03007 .
[68] In the context of the Lieb-Liniger model this has first been
wrongly described in Ref. [ 45] but corrected in Ref. [ 73].
[69] R. B. Laughlin, Phys. Rev. Lett. 60,2677 (1988 ).[70] F. Wilczek, Fractional Statistics and Anyon Superconductivity ,
Series on Directions in Condensed Matter Physics V ol. 10(World Scientific, Singapore, 1990).
[71] W. Bishara and C. Nayak, Phys. Rev. Lett. 99,066401
(2007 ).
[72] S. Trebst, M. Troyer, Z. Wang, and A. W. W. Ludwig, Prog.
Theor. Phys. Suppl. 176,384(2008 ).
[73] J. B. McGuire, J. Math. Phys. 5,622(1964 ).
195422-9 |
PhysRevB.78.233404.pdf | Electron transport through single phthalocyanine molecules studied using scanning
tunneling microscopy
A. F. Takács,1,2F. Witt,1S. Schmaus,1T. Balashov,1M. Bowen,3E. Beaurepaire,3and W. Wulfhekel1,2
1Physikalisches Institut, Universität Karlsruhe (TH), Wolfgang-Gaede-Strasse 1, 76131 Karlsruhe, Germany
2DFG-Center for Functional Nanostructures, Universität Karlsruhe (TH), Wolfgang-Gaede-Strasse 1, 76131 Karlsruhe, Germany
3Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS-ULP, UMR 7504,
23 Rue du Loess B.P . 43, 67034 Strasbourg, France
/H20849Received 18 September 2008; published 8 December 2008 /H20850
Using low-temperature scanning tunneling microscopy, electron transport across single H 2-o rC o -
phthalocyanine molecules was studied. The molecules were adsorbed onto clean Cu /H20849111/H20850surface and Co
nanostructures on Cu /H20849111/H20850and were contacted through a controlled approach of the tip. Soft phonons from the
molecular side groups promote a discrete jump to contact followed by bond formation. Molecular conductancesstrongly depend on the substrate but not on the central atom of the molecules. This is explained by chargetransfer and hybridization of the molecular orbitals with the substrate states as seen by scanning tunnelingspectroscopy.
DOI: 10.1103/PhysRevB.78.233404 PACS number /H20849s/H20850: 68.37.Ef, 85.65. /H11001h, 34.35. /H11001a, 82.37.Gk
In the last few decades, remarkable progress has been
made in the semiconductor industry towards the miniaturiza-tion of electronic devices following Moore’s law. The physi-cal limits of this top-down approach will be reached in theforeseeable future as devices of the size of single atoms ormolecules will be required. A new approach in miniaturiza-tion of electronic devices is the study of molecular electron-ics, i.e., the fabrication of circuits at the molecular scale.Electron transport through single molecules is at the heart ofthis approach and has as such become a question of generalinterest in view of its possible applications.
1,2
The most important characteristics of a molecular junc-
tion are the electron transfer rates between the metal elec-trodes contacting the molecule and the molecule itself, aswell as the transmission of electrons through the molecularorbitals. Yet, surprisingly, very few quantitative studies di-rectly address these parameters.
3This discrepancy reflects an
insufficient control of the molecular adsorption geometry,compounded by the lack of control of the electrode geometryin the common break junction experiments.
4An alternate,
more precise method to contact single molecules is the use ofscanning tunneling microscopy /H20849STM /H20850.
3,5–8Indeed, STM en-
ables the imaging of the substrate, which plays the role ofone of the electrodes, and of the molecules on this substratebefore contacting a single molecule at a well defined posi-tion. The family of phthalocyanine /H20849Pc/H20850molecules has re-
ceived much attention due to their thermal stability, as wellas the ability to tune their structural, chemical, and transportproperties by substituting the metal cation /H20849Me/H20850within the Pc
molecular cage or by grafting atoms or radicals onto the sidegroups /H20851see Fig. 1/H20849a/H20850/H20852. For Pc molecules adsorbed on a metal
substrate, recent reports on controlling the Kondo effect
9or
studying their vibrational states during single molecule elec-tron transport
10have been published.
Determining the precise experimental geometry of a mo-
lecular junction is of considerable importance towards a con-vergence with ab initio theoretical calculations. In the con-
text of molecular spintronics, it is also essential tounderstand the nature of hybridization between a moleculeand a magnetic surface. The behavior of Pc molecules onmagnetic /H20849Co/H20850and nonmagnetic /H20849Cu/H20850surfaces is the aim of
our work. In this Brief Report, we present an experimental
study of the contact and electrical transport between CoPc orH
2Pc molecules and a Cu /H20849111/H20850or Co/Cu /H20849111/H20850surface using
STM. At first, distance curves were recorded, in which thetunneling current Iwas measured as a function of the dis-
tance between tip and sample until a molecular junction wasformed. We used inelastic scanning tunneling spectroscopy/H20849ISTS /H20850to show how soft phonon modes within the molecular
side groups promote the observed abrupt formation of themolecular junctions. Finally, scanning tunneling spectros-copy /H20849STS /H20850was performed on the molecules to explain the
difference in conductance of the molecules when placed onthe Co or Cu surface.
The experiments were carried out with a home-built ultra-
FIG. 1. /H20849Color online /H20850/H20849a/H20850Chemical structure of a Me-
phthalocyanine molecule consisting of four benzene pyrole groupsconnected over four nitrogen bonds. The nitrogen sites of the pyrolecan form a complex compound with a metal cation /H20849Me/H20850. Here
Me=H
2,Co. /H20849b/H20850–/H20849d/H20850Topographic scans at T=4.2 K of CoPc /H20849b/H20850on
Cu/H20849111/H20850and /H20849c/H20850on Co/Cu /H20849111/H20850, and H 2Pc/H20849d/H20850on Cu /H20849111/H20850and /H20849e/H20850on
Co/Cu /H20849111/H20850.PHYSICAL REVIEW B 78, 233404 /H208492008 /H20850
1098-0121/2008/78 /H2084923/H20850/233404 /H208494/H20850 ©2008 The American Physical Society 233404-1high vacuum STM at 4.2 K. The tungsten STM tips were
cleaned in situ by Ar+sputtering and annealing. An atomi-
cally clean and flat Cu /H20849111/H20850single crystal was used as a
substrate. Auger-electron spectroscopy showed no contami-nation on the Cu surface and low energy electron diffractionrevealed sharp spots and low background, indicating a highcrystal quality. STM scans revealed atomically clean terracesof widths exceeding 50 nm. A submonolayer amount of Cowas deposited onto Cu /H20849111/H20850by electron-beam evaporation,
resulting in the formation of pseudomorphic double-layer is-lands of Co.
11Finally, a small amount of purified H 2Pc or
CoPc was deposited from a Knudsen cell, followed by STMexperiments at 4.2 K. During the deposition of the mol-ecules, a sample temperature of around 270 K was main-tained to avoid molecular diffusion on the bare Cu towardthe step edges of the Co islands. Topographic measurementsreveal that both molecules were adsorbed on the Co islandsas well as on the bare Cu. Figures 1/H20849b/H20850–1/H20849e/H20850show a small
area scan for the two different molecules on bare Cu /H20849111/H20850
and Co/Cu /H20849111/H20850. Note that, in these scans, the CoPc mol-
ecules display a bright spot in the center of the molecule—corresponding to the Co site—while H
2Pc displays a depres-
sion.
After choosing a single molecule from topographical
scans, current versus distance curves were obtained in thefollowing way: the STM tip was positioned above a mol-ecule at a low bias voltage of typically 10 mV . Then thefeedback loop of the STM was opened, and by ramping thevoltage of the zpiezo, the tip was moved toward the sample
surface in a controlled manner. During the tip approach, thetunneling current Iwas recorded. Afterward, the tip was re-
tracted back to its initial position. The approach was repeated25 times over the same molecule.
We present in Fig. 2/H20849a/H20850experiments on the formation of a
CoPc molecular junction using a STM tip. The initial tipapproach on bare Cu /H20849111/H20850reveals an exponential increase of
the current that reflects the reduction of the tunneling barrierwidth as expected. The extracted work function of4.3 eV on Cu agree well with previous work functions mea-sured in the STM geometry.
12The tip approach toward a
CoPc molecule that is adsorbed onto Cu initially shows thesame behavior, yet deviates from the expected exponentialdependence below a certain distance. The current rises super-exponentially, which can be explained by a partial lifting ofthe molecule due to the close proximity of the tip. Ulti-mately, the molecule jumps into contact with the tip, therebybridging the tunnel junction formed by the STM tip and theCu/H20849111/H20850surface. All subsequent approach curves reveal an
almost constant conductance of the molecular junction, i.e.,the bond that is initially formed remains intact after returningto the initial tip-sample distance. In this bridged position ofthe molecule, a relatively high conductance of 0.1 G
0was
observed, which most likely reflects the conductance via thedelocalized
/H9266orbitals of the molecule.
After measuring several distance curves in the bridged
position, but before proceeding with the topographical scan,the feedback loop is closed again. In order to reach the origi-nal set-point current, the tip is retracted until the currentdrops, i.e., until the molecular bridge is broken. When thetopographic scan was continued after taking the distancecurves, the molecule disappeared from the image /H20851see Fig.
2/H20849c/H20850/H20852. This indicates that, in the process of breaking the
bridge, the CoPc molecule was transferred to the tip.
13Ob-
viously, the Pc forms a more stable bond to the W tip than tothe Cu surface, as is expected from the low chemical reac-tivity of Cu compared to W. Measurements of H
2Pc on
Cu/H20849111/H20850show similar results /H20851see Figs. 2/H20849b/H20850and2/H20849d/H20850/H20852. The
same behavior of the tunneling current, including the jump tocontact, was observed and similar values for the conduc-tance, /H110110.1G
0, were found. Apparently, neither the contact
behavior nor the conductance is influenced by the centralmetal cation. This further proves that the conductance is notrelated to the central Pc metal atom but to the delocalized
/H9266
electron system of the organic ligands.
Measurements of H 2Pc and CoPc on the Co/Cu /H20849111/H20850is-
lands /H20849Fig. 3/H20850show a similar “jump-to-contact” behavior as
on bare Cu /H20849111/H20850. However, in contrast to the experiments on
Cu/H20849111/H20850, the subsequent distance curves measured on Co
start essentially at the original current set point. This meansthat the molecule-tip bond that occurs during the jump tocontact is broken again when the tip is retracted. In agree-ment with this finding, the molecule can also be seen in atopographic scan taken after the tip retraction /H20849see the insets
of Fig. 3/H20850. Thus the Pc molecules form a stronger bond to Co
than to Cu. Furthermore, we observe a conductance of
FIG. 2. /H20849Color online /H20850Tip approach experiments on /H20849a/H20850CoPc
and /H20849b/H20850H2Pc on Cu /H20849111/H20850show how distance curves measured atop
the molecule /H20849solid red/gray line /H20850deviate over a 0.2 nm distance
range from those measured on bare Cu /H20849dotted black line /H20850and upon
a subsequent approach /H20849dashed blue/dark gray line /H20850remain essen-
tially constant, reflecting the molecular contact. Here, the offsetstands for the displacement of the tip toward the surface. /H20849c/H20850CoPc
/H208495/H110035nm
2/H20850and /H20849d/H20850H2Pc/H208497/H110037nm2/H20850topographical scans showing
the abrupt transfer of the molecules to the tip after recording dis-tance curves.BRIEF REPORTS PHYSICAL REVIEW B 78, 233404 /H208492008 /H20850
233404-2/H110110.3G0across both CoPc or H 2Pc molecular junctions on a
Co surface, which is approximately a factor of 3–4 higherthan on a Cu surface. These findings suggest that the resis-tance of the molecular junction is not only given by themolecule itself, but also by the details of the contact betweenthe molecule and the electrodes.
Our extensive studies on assembling CoPc and H
2Pc mo-
lecular junctions indicate that the molecular jump is morelikely to occur if the tip approaches the benzene side groupsrather than the center of the molecule. In the sudden jump tocontact, the molecule geometry is changed in a conforma-tional transition between two local minima of the total en-ergy: the flat lying and the bridging configuration. As thistransition is due to a mechanical soft mode, it should berelated to low energy phonons of the flat molecule. ISTSmeasurements were carried out by measuring the second de-rivative of the tunneling current, which can reveal vibronicexcitations.
14Indeed, when the energy of the tunneling elec-
trons is sufficient to create an inelastic excitation in the formof a phonon, a peak in d
2I/dU2appears.15Spatial resolution
was achieved by recording inelastic spectra at different posi-tions of the Pc molecule. Figure 4shows the d
2I/dU2spec-
trum of a CoPc molecule on Co/Cu /H20849111/H20850. An inelastic exci-
tation at around /H1100620 meV appears in the spectrum as an
antisymmetric combination of a peak and a dip. This excita-tion falls within the same energy range as the excitation en-ergy of benzene-substrate phonons.
16To locate the vibronic
excitation within the molecule, we plotted the value of theinelastic spectrum at /H1100620 meV as a function of tip position
atop the molecule /H20849see insets of Fig. 4/H20850. A high inelastic
excitation probability is indicated by a combined dark fea-ture at negative and a bright feature at positive bias. We thussee that the excitations are located on the side groups ofCoPc and can therefore be associated with a vibrationalmode of the molecular side group. These low energy vibra-tions represent the molecular degrees of freedom of the mo-
lecular jump to contact when the tip is close enough to themolecule /H20849see Figs. 2and3/H20850.
To gain more insight into the transport properties of the
molecular junctions, spatial STS measurements were per-formed by opening the feedback loop for each tip position.The spatially resolved differential conductance dI/dUwas in
turn measured using lock-in techniques. Since the differentialconductance is proportional to the local density of states/H20849LDOS /H20850at the tip position,
17this allows both to measure the
LDOS of the molecule as function of energy and to laterallymap the LDOS at a specific energy, i.e., to image the mo-lecular orbitals. The experimental results for CoPc are shownin Fig. 5. The energy-dependent LDOS, which was averaged
over all individual spectra recorded above different positionsof the molecule, exhibits peaks reflecting the molecular-orbital energy levels.
18The spectra for CoPc on Co and on
Cu have a similar overall shape, but the spectrum of CoPc onCo appears to be shifted by approximately +500 meV rela-tive to that of CoPc on Cu. This shift is confirmed by thesimilar shape of the molecular orbitals when acquired on therelevant molecular-orbital energy levels /H20851see within Fig. 5
the scan pairs /H20849c/H20850-/H20849f/H20850and /H20849d/H20850-/H20849g/H20850and their corresponding en-
ergy positions on panel /H20849a/H20850/H20852. A similar behavior was found
regarding H
2Pc molecules, where a shift of 400 mV is ob-
served. We note that the molecular orbitals, which arestrongly hybridized with the metal states of the substrate,differ from the orbitals in vacuum.
19This energy shift indi-
cates an electron transfer from the molecule to the less nobleCo surface, suggesting a stronger bond as evidenced above.As a consequence, one molecular orbital is shifted to theFermi energy, leading to a high LDOS at the Fermi edge.This in turn accounts for the higher molecular junction con-ductance when the molecules are placed on Co, since in thiscase electrons can be transferred efficiently from the elec-trodes to the molecule. Given the similar work functions
/H9278=4.94 eV for Cu /H20849111/H20850/H20849Ref. 20/H20850and/H9278=5.03 eV for
Co/H20849111/H20850,21this 0.5 eV energy shift cannot be explained
within the interface dipole model /H20849see, e.g., Ref. 22/H20850, but
FIG. 3. /H20849Color online /H20850Averaged distance curves measured atop
CoPc /H20849solid red/gray line /H20850and H 2Pc/H20849dotted blue/dark gray line /H20850
molecules adsorbed onto Co/Cu /H20849111/H20850reveal a reversible jump-to-
contact behavior. For CoPc above approximately 0.13 nm there aretwo curves: the upper one is the average after a jump, the otherwithout a jump. The offset between the two data sets reflects thedifferent current set points for these measurements. Insets: topo-graphical scans of H
2Pc/H20851upper left; /H208493/H110033n m2/H20850/H20852and CoPc /H20851lower
right; /H208498/H110038n m2/H20850/H20852during taking distance curves.
FIG. 4. /H20849Color online /H20850Energy dependence of ISTS for CoPc on
Co/Cu /H20849111/H20850. Insets: d2I/dU2/H20849/H1100620 meV /H20850spatial maps, with high
inelastic excitation probability on the molecular side groups. Themolecular overlay serves as a guide to the eyes.BRIEF REPORTS PHYSICAL REVIEW B 78, 233404 /H208492008 /H20850
233404-3likely reflects fundamentally different adsorption mecha-
nisms /H20849physisorption vs chemisorption /H20850.4
In conclusion, we have presented systematic studies on
assembling Pc molecular junctions with a STM tip. We haveclarified the “jump-to-contact” mechanism that relates junc-tion formation to a bending of the Pc molecular side groupsdue to soft phonons. We find that neither the contact behaviornor the conductance is influenced by the Pc’s central metalcation. The measured conductance across our molecularjunctions depends strongly on the metallic surface throughthe parameters of hybridization and charge transfer. On theCo substrate, the Pc molecule exhibits metallic behavior,while on Cu it is semiconducting with a gap at the Fermi
energy. Our results, which nicely illustrate the interplay be-tween the mechanical or adhesive properties of, and thecharge transfer or electrical conduction across, a molecularjunction, can help account for the large scatter of the con-ductance in molecular break junction experiments reportedthus far.
4
This work was supported by the Deutsche Forschungs-
gemeinschaft /H20849DFG /H20850, the Center for Functional Nano-
structures /H20849CFN /H20850of the DFG, and Agence National de la
Recherche /H20849ANR /H20850contract “Spinorga.”
1C. Joachim et al. , Nature /H20849London /H20850408, 541 /H208492000 /H20850.
2A. Nitzan and M. A. Ratner, Science 300, 1384 /H208492003 /H20850.
3W. Haiss et al. , Nature Mater. 5, 995 /H208492006 /H20850.
4H. B. Akkerman and B. de Boer, J. Phys.: Condens. Matter 20,
013001 /H208492008 /H20850.
5C. Joachim et al. , Phys. Rev. Lett. 74, 2102 /H208491995 /H20850.
6F. Moresco, Phys. Rep. 399, 175 /H208492004 /H20850.
7N. Néel et al. , Phys. Rev. Lett. 98, 065502 /H208492007 /H20850.
8R. Temirov et al. , Nanotechnology 19, 065401 /H208492008 /H20850.
9A. Zhao et al. , Science 309, 1542 /H208492005 /H20850.
10X. H. Qiu et al. , Phys. Rev. Lett. 92, 206102 /H208492004 /H20850.
11M. T. Kief and W. F. Egelhoff, Phys. Rev. B 47, 10785 /H208491993 /H20850.
12L. Olesen et al. , Phys. Rev. Lett. 76, 1485 /H208491996 /H20850.
13The molecule on the tip can be detected by recording approach
curves on bare Cu which deviate from the initial curves.14B. C. Stipe et al. , Science 280, 1732 /H208491998 /H20850.
15E. L. Wolf, Principles of Electron Tunneling Spectroscopy /H20849Ox-
ford University Press, New York, 1985 /H20850.
16J. I. Pascual et al. , Phys. Rev. Lett. 86, 1050 /H208492001 /H20850.
17J. Tersoff and D. R. Hamann, Phys. Rev. Lett. 50, 1998 /H208491983 /H20850.
18The molecular states are broadened significantly with respect to
those of free molecules due to hybridization with the metallicstates of the substrate. The latter’s contribution to the LDOS wasmitigated by subtracting the substrate’s spectrum from the spec-trum recorded on the molecules.
19C. Chavy et al. , Chem. Phys. Lett. 214, 569 /H208491993 /H20850.
20H. Kawano, Prog. Surf. Sci. 83,1/H208492008 /H20850.
21S. Pick, Surf. Sci. 601, 5571 /H208492007 /H20850.
22C. Shen and A. Kahn, J. Appl. Phys. 90, 4549 /H208492001 /H20850.
()
()FIG. 5. /H20849Color online /H20850/H20849a/H20850En-
ergy dependence of the LDOS forCoPc on Cu /H20849111/H20850/H20849dotted black
line/H20850and Co/Cu /H20849111/H20850/H20849solid red/
gray line /H20850. Topographical scans of
CoPc on /H20849b/H20850Cu and /H20849e/H20850Co. Spa-
tial maps of the molecular orbitalsat/H20849c/H20850600 meV on Cu and /H20849f/H20850100
meV on Co and at /H20849d/H20850400 meV on
Cu and /H20849g/H20850800 meV on Co. The
scans all span 2.5 /H110032.5 nm
2./H20849h/H20850
Energy dependence of the LDOSfor H
2Pc on Cu /H20849111/H20850/H20849dotted black
line/H20850and Co/Cu /H20849111/H20850/H20849solid red/
gray line /H20850.BRIEF REPORTS PHYSICAL REVIEW B 78, 233404 /H208492008 /H20850
233404-4 |
PhysRevB.71.235416.pdf | Local-barrier-height images of TiO 2„110 …surfaces
D. Ostermann,1G. Walther,1and K. D. Schierbaum1,*
1Institut für Physik der kondensierten Materie, Abteilung Materialwissenschaft, Heinrich-Heine-Universität Düsseldorf,
Universitätsstrasse 1, Gebäude 25.23, 40225 Düsseldorf, Germany
/H20849Received 29 October 2004; revised manuscript received 11 January 2005; published 27 June 2005 /H20850
Local barrier height /H20849LBH /H20850and constant current mode /H20849CCM /H20850images of nonreconstructed /H20849110 /H20850single-
crystalline titanium dioxide surfaces are determined under ultrahigh vacuum conditions. The barrier heightmeasurements are performed with the tip-oscillation technique in the constant current mode /H20849CCM /H20850of a
beetle-type scanning tunneling microscope with tunneling currents between 1 and 0.1 nA. The mean apparentheight
/H9278of the surface is derived as a function of the tunneling gap width. The data /H9278are in accordance with
the results from I−zspectroscopy. A simple electrostatic space-charge layer model explains the observed
decrease of the apparent height for small tunneling gaps. The LBH technique is applied to investigate thestructures of point defects on TiO
2/H20849110 /H20850and their effects on the work function on an atomic scale. We find
small concentrations of vacancies of the protruding bridging oxygen atoms, giving rise to bright atomic-scalespots between the Ti rows. The barrier height images indicate a localized and reduced apparent barrier height
/H9278at these sites, in correspondence to the overall decrease in the work function /H9021that has been observed earlier
in ultraviolet photoelectron spectra of ion-sputtered and reduced TiO 2/H20849110 /H20850surfaces. Besides the oxygen
vacancies, the surface reveals the presence of distinct other types of atomic-scale features, both in topographicand barrier height images. Along the Ti rows, depressions with a concentration of approximately 0.02 mono-layers can be identified in the constant current as well as in the barrier height images extending over one and/H20849to a significantly lower extent /H20850two atomic distances along /H20851001 /H20852. They are attributed to the so-called type-B
defects and they exhibit a local increase of the work function /H9021.
DOI: 10.1103/PhysRevB.71.235416 PACS number /H20849s/H20850: 68.37.Ef, 68.47.Gh
I. INTRODUCTION
Atomic-scale studies of surface defects of TiO 2/H20849110 /H20850
single crystals and the local variations of the electronic struc-
ture and work function, which are connected with the surfacelattice disorder, are of great interest to an understanding ofchemisorption and interface formation with metal overlayers.It is well known from a variety of experimental investiga-tions that oxygen vacancies are formed preferentially at the/H20849110 /H20850surface in ultrahigh vacuum due to the entropy-driven
loss of lattice oxygen, thereby producing electronic donorstates.
1Many studies on chemisorption show that these sites
are involved in elementary steps of the solid-gas interaction.An example is the chemisorption of TiO
2/H20849110 /H20850with water
molecules which takes place at oxygen vacancy sites of the
surface.2,3In addition to these point defects, which are ex-
pected at T/H110220 K for thermodynamic reasons, other types of
surface lattice disorder including step edges, extended de-fects and reconstructions occur at TiO
2/H20849110 /H20850.4–6They have
been identified primarily by means of atomically resolved
scanning tunneling microscopy /H20849STM /H20850. This technique, in
particular, has entailed considerable progress in the develop-ment of widely accepted structural models.
Until now, only a few atomic-scale investigations have
been reported in which tunneling spectroscopy and apparentheight measurements complement STM data of TiO
2/H20849110 /H20850
surfaces. Such techniques can resolve the local variations of
the electronic structure and work functions that are inducedby defects. The work function is a fundamental property of asurface and its overall sensitivity towards adsorption and de-fect formation is well known. A recent tunneling spectros-copy study /H20849STS /H20850on defective TiO
2/H20849110 /H20850has shown an in-
creased differential conductance at approximately −1 V
which has been associated with the defect band of such sur-faces which lies /H110111 eV below the Fermi energy.
7In this
case, 1 /H110032 reconstructed strands and step edges along /H20851001 /H20852
is the predominant type of surface disorder. Moreover,charged subsurface impurities have been found that are asso-ciated with characteristic band bending effects that affectboth STM images and STS data. Maeda et al. have reported
a local barrier height /H20849LBH /H20850study, carried out with the
distance-oscillation technique on the clean TiO
2/H20849110 /H208501/H110031
and the 1 /H110032 reconstructed surface and the effect of the
deposition of Au nanoparticles.8
Although the technique of local barrier imaging has been
reported for a few metal and semiconductor surfaces9–11
since the pioneering work by Binning, Rohrer andco-workers,
12the interpretation of the data is not at all
straightforward because of the appearance of tip-sample in-teraction at low tunneling widths.
13,14This technique has not
yet attained importance like STM. The sensitivity of the tun-neling current Ito variations of the tunneling gap zdefines
the apparent barrier height
/H9278=/H60362
8m/H20873dlnI
dz/H208742
. /H208491/H20850
In the simplest quantum mechanic treatment of electron tun-
neling through a rectangular potential barrier between the tipand the sample,
/H9278represents the mean work function
1
2/H20849/H9021tip+/H9021sample /H20850/H20849for small tunneling voltages V/H11270/H9278/H20850and is
related with the inverse decay lengthPHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
1098-0121/2005/71 /H2084923/H20850/235416 /H2084910/H20850/$23.00 ©2005 The American Physical Society 235416-1/H9260=/H208812m/H9278
/H6036/H208492/H20850
of the electron wave function outside the solid. According to
Yoon et al. , the atomic resolution of LBH images may be
explained in the idealized Tersoff-Hamann picture if one re-lates the current modulation to the zvariation of the local
density of states
/H9267at the Fermi level EF, evaluated at the
center of the curvature rtof the tip,15
/H11509I
/H11509z/H11008/H11509
/H11509z/H20875/H20858
/H9263/H20841/H9274/H9263/H20849rt/H20850/H208412/H9254/H20849E/H9263−EF/H20850/H20876=/H11509
/H11509z/H9267/H20849rt,EF/H20850; /H208493/H20850
/H9274/H9263denotes a wave function of the sample unperturbed by the
tip. The LBH images would then map the lateral variation ofthe tunneling current due to the decay lengths of the varioussurface wave functions of the sample which depend on thelocation in the surface Brillouin zone.
16
A second effect, responsible for the contrast in LBH im-
ages, is the gap-width dependence of the apparent height.Theory predicts that the actual barrier drops down with de-creasing gap distance due to image force and exchange-correlation effects.
17,18The effect of barrier lowering was
first shown experimentally by Binnig et al. , but other studies
indicate contradictory results. Recently, Olesen et al. have
presented a comprehensive experimental study for Au, Ni,and Pt single crystal surfaces in which they showed that
/H9278is
constant until point contact is reached if one takes into ac-count the finite input impedance of the current preamplifierand evaluates
/H9278from measurements of both the tunneling
voltage and current.19The distance dependence of /H9278is con-
nected with the presence of strong adhesive forces at smallgap widths which lead to significant displacements of tip andsample atoms and decrease the actual tunneling width. Chenshowed in a study on clean Si /H20849111 /H208507/H110037 with a clean W tip
that the experimental
/H9278−zdependence can be fully under-
stood by taking into account the effect of attractive and re-pulsive forces which follow a Morse curve and which modi-fies the actual displacement of the tunneling gap belowapproximately 3 Å.
20In the context of the present work, we
will show that the /H9278−zdependence is the major contrast
mechanism in local-barrier images of “geometrically” idealTiO
2/H20849110 /H20850surfaces, that are obtained in the constant current
mode. Hence, the /H9278−zdependence must be determined to
achieve a quantitative evaluation of the LBH data of thissurface.
In this report, we use weakly reduced TiO
2/H20849110 /H20850single
crystals which exhibit a small concentration of intrinsic de-
fects. Typical bulk donor concentrations of the samples arearound 10
19cm3, as determined from electrical conductivi-
ties.
In the first part we describe the variation of LBH data as
a function of the tunneling current. The data are used toderive the mean apparent height of the surface as a functionof the tunneling gap width. This is compared with I−zspec-
troscopy. In the second section, we will present a simpleelectrostatic space-charge layer model, applicable to TiO
2,
which explains the observed decrease of the apparent heightfor small tunneling gaps. The third part of this report ex-plores the applicability of this method to investigate the
structures of point defects on TiO
2/H20849110 /H20850and their effects on
the work function on an atomic scale.
II. INSTRUMENTATION AND SAMPLE PREPARATION
The experiments were carried out in an ultrahigh vacuum
system which consists of a preparation and an STM chamber.It is equipped with a homemade beetle-type scanning tunnel-ing microscope, operated with a commercial scan unit /H20849RHK,
SPM 100 /H20850. We used electrochemically etched W tips. Tun-
neling voltages are given with respect to the tip. TheTiO
2/H20849110 /H20850single crystals were prepared by several Ar+sput-
tering and annealing cycles at 1000 K. The electrical resis-
tance /H20849R=14.8 /H9024/H20850was determined in a four point arrange-
ment with stainless steel contacts that can be pressed in situ
to the sample’s sides. We used a constant current source/H20849Burster, Digistant 4405 /H20850and a voltage meter /H20849Prema, 5017 /H20850.
The bulk conductivity is
/H9268=0.15 /H9024−1cm−1at room tempera-
ture, as calculated with the van der Pauw equation. Here weassume flat-band conditions without significant surface con-ductance. From this value we determine the bulk density ofelectrons, n
b=1.5/H110031018cm−3, according to nb=/H9268//H20849e/H9262/H20850.W e
assume that /H9262=10−6/H20849T/K/H20850−2.5cm2/H20849Vs/H20850−1holds at a tempera-
ture of T=300 K.21
III. EXPERIMENTAL RESULTS AND DISCUSSION
A. Evaluation of LBH images and point spectroscopy
In a first set of experiments, we have determined 400
/H11003400 Å2wide CCM and LBH images of the clean
TiO 2/H20849110 /H20850surface at different tunneling currents Ibetween
0.1 and 0.8 nA and a constant bias voltage of V=1.4 V. The
resolution of the images is 512 /H11003512 pixels and the scan
velocity is 100 ms per line. The root-mean-square amplitudeand the frequency of the tip modulation are
/H9254z=0.238 Å and
/H9263=7/H11003103s−1, respectively. The current is step-wisely
changed six times per image /H20849with the sequence 0.1 →0.25
→0.4→0.6→0.8→0.1 nA /H20850while both CCM and LBH data
are recorded simultaneously.26In addition, the tunneling cur-
rent is recorded. /H20849See Fig. 1. /H20850
Thus, the LBH image consists of six regimes from which
we then determine the spatially averaged lock-in voltage.The latter is transformed into the root-mean-square ampli-tude of the harmonic component of I, denoted as i
rms.I ti s
related to the peak value by iˆ=/H208812irms. Similarly, we obtain
the corresponding tunneling current Ifrom the current image.
These values are used to determine the apparent barrierheights as described in the following.
For an oscillating tip, we assume the applicability of the
relationship /H208491/H20850and write for the time-dependent tunneling
current,
I/H20849t/H20850=VBexp /H20851−A/H20881/H9278z/H20849t/H20850/H20852, /H208494/H20850
with Vas voltage and Bas a coefficient. The latter is propor-
tional to the local density of states at the Fermi energy EF
and hence may show pronounced differences at the Ti and O
sites of the TiO 2surface. The value of Abecomes 1.025 if /H9278OSTERMANN, WALTHER, AND SCHIERBAUM PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-2andzare given in the units electron volt and Ångstrom. The
function z/H20849t/H20850is given by
z/H20849t/H20850=z0+/H208812/H9254zsin/H9275t, /H208495/H20850
in which z0is the static displacement of the tip, /H9254zthe root
mean square amplitude and /H9275=2/H9266/H9263the angular frequency.
Hence, one can write
I/H20849t/H20850=VBexp /H20851−A/H20881/H9278z0/H20852exp /H20851−A/H20881/H9278/H208812/H9254zsin/H9275t/H20852. /H208496/H20850
We assume that /H9278is constant during the tip oscillation. As it
is shown in the Appendix A, the evaluation of this equationyields the ratio of the the time-averaged value Iand the
amplitude of the first harmonic component iˆ,
I
iˆ=1+1
4b2+1
64b4
b+1
8b3, /H208497/H20850
with b=−A/H20881/H9278/H208812/H9254z.
Figure 2 displays the calculated ratio I/iˆas a function of
/H9278. An approximate solution of the polynomial /H208497/H20850is
/H20881/H9278=2
A/H9254z/H20875I
irms−/H20881/H20873I
irms/H208742
−1/H20876, /H208498/H20850
if one neglects the third- and fourth-order term. Note that b
refers to a situation in which /H9278does not alter during the tip
oscillation, i.e., the amplitudes are sufficiently small. In mostcases, large amplitudes are necessary to reduce the noise-to-signal ratio or to obtain spatially resolved information. This
gives rise to a systematic error in the determination of I/iˆandhence of
/H9278if its value depends on the gap width.
We will assign the experimental I/iˆdata with the index
“la” that signifies that large amplitudes have been applied.Corresponding barrier heights deviate therefore from the ac-tual values and are denoted with
/H9278la. We will demonstrate in
the following that one can take /H9278laas the first estimated
values to obtain more accurate data of /H9278. By using the func-
tion I/iˆ/H20849z0/H20850, we determine the barrier heights from the ratio
of the spatially averaged values of /H20855iˆ/H20856=/H208812/H20855irms/H20856. The results
are given in Table I. The value of /H20855/H9278/H20856larefers to the mean
value of the surface. It can be concluded from the decreasing
slope of the function I/iˆ/H20849z0/H20850/H20849compare with Fig. 2 /H20850that the
statistical error of /H20855/H9278/H20856labecomes larger at lower tunneling
currents if one assumes a constant absolute uncertainty in the
determination of Iand /H20855iˆ/H20856. The statistical errors in /H20855/H9278/H20856lahave
been computed for the lowest and highest values from the
standard deviations in the images of the currents Iandiˆ.
TABLE I. Spatially averaged values of the root-mean-square
amplitudes /H20855irms/H20856at different tunneling currents I. Barrier heights
/H20855/H9278/H20856laare determined from the plot I/iˆvs/H9278in Fig. 2. The values of
/H20855/H9278/H20856are determined from Eq. /H2084911/H20850.
I/nA /H20855irms/H20856/nA /H20849I
/H20855iˆ/H20856/H20850
la/H20855/H9278/H20856la/eV /H20855/H9278/H20856/eV
0.118±0.005 0.0923±0.005 0.90±0.09 15.61±4 5.48±0.7
0.255 0.1180 1.53 4.04 3.10
0.411 0.1594 1.82 2.74 2.33
0.603 0.2186 1.95 2.37 1.91
0.812±0.005 0.2216±0.005 2.59±0.09 1.30±0.08 1.60±0.1
FIG. 1. Experimental values of the topographic data /H20849i.e.,
height /H20850at a tunneling current of I=0.6 nA from the CCM image
of the TiO 2/H20849110 /H20850sample 1 /H20849solid circles /H20850and corresponding values
of the root-mean-square amplitude of the first harmonic componentof the tunneling current i
rms/H20849open squares /H20850in the LBH image along
/H2085111¯0/H20852. The second scale on the left hand side corresponds to the z0
scale which is used throughout this paper. Top: Schematic cross
section of the TiO 2/H20849110 /H20850surface indicating the bridging and in-
plane O atoms and the Ti atoms in fivefold /H20849in-plane /H20850and sixfold
coordination. Further explanations are given in the text.
FIG. 2. Ratio I/iˆas a function of barrier height /H9278for/H9254z
=0.238 Å. Vertical lines correspond to experimental values of
/H20849I//H20855iˆ/H20856/H20850la. The inset shows I/H20849t/H20850, its Talyor series expansion ITaylor , the
time average I, the peak value iˆ, and the root-mean-square value irms
for/H9278=15.61 eV.LOCAL-BARRIER-HEIGHT IMAGES OF TiO 2/H20849110 /H20850SURFACES PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-3The next step in the refinement of our determination of
/H20855/H9278/H20856concerns the calculation of the current-distance curves.
We use the relationship
I=Irefexp /H20851−A/H20881/H20855/H9278/H20856laz0/H20852. /H208499/H20850
Here, we refer Ito a value of Iref=1 nA that we attribute
toz0=0 /H20849at a certain gap width or topographic height, see
Fig. 1 /H20850.
The iteration is based on a fitting procedure in which five
individual functions I/Iref/H20849z0/H20850are calculated that reflect the
behavior of Iin the intervals I to V /H20849Fig. 3 /H20850. The procedure is
explained in Appendix B. The first function I/Iref/H20849z0/H20850at
/H20855/H9278/H20856la=1.30 eV holds between 1 nA and 0.603 nA /H20849dashed
line, used as the start function of the iteration /H20850. The dotted
curve depicts the function I/Iref/H20849z0/H20850and corresponds to the
last step of the iteration at /H20855/H9278/H20856la=15.61 eV. The other curves
are not shown in Fig. 3 for a simplification.
In order to obtain a common I/Irefcurve for the entire
experimentally adjusted range of z0, we then calculate a
fourth-order polynomial function I/Irefthat fits the individual
I/Irefcurves in the different regimes /H20849solid line labeled “fit” /H20850,
I/Iref= 1.017 − 1.220 z0+ 0.530 z02− 0.1 z03+ 0.007 z04.
/H2084910/H20850
z0is given in the unit Ångstrom. From this function, one
obtains the mean apparent barrier height of the TiO 2/H20849110 /H20850
surface from
/H20855/H9278/H20856= 1.025/H20875dln/H20849I/Iref/H20850
dz0/H208762
. /H2084911/H20850
The numeric determination yields the /H20855/H9278/H20856-z0relationship
which is revealed by the dot-dashed curve in Fig. 3. As the
minimum experimental value of Iis 0.118 nA, an upper limit
of the determination of /H20855/H9278/H20856appears, being 5.48 eV. A physi-
cally more realistic model would give a progressive transi-tion from the polynomial /H2084910/H20850into this limit which has not
been considered in Fig. 3. The values /H20855/H9278/H20856are given in Table
I for the experimentally chosen tunneling currents. We esti-
mate the error in /H20855/H9278/H20856in such a way that we repeat the fitting
procedure described above and take into account the varia-
tion of /H20855/H9278/H20856la.
One expects barrier heights close to /H20855/H9278/H20856=/H20849/H9021W+/H9021TiO2−eV /H20850/2/H11015/H208494.55+5.2−1.4 /H20850/2=4.2 eV for low tunneling cur-
rents and hence larger tip-sample spacing. In the next part of
this report we will show that one must take into account thata significant portion of the applied bias voltage of 1.4 Vdrops down within a space-charge region of TiO
2. Hence, eV
is smaller /H208490.18 eV /H20850and one obtains /H20855/H9278/H20856=4.8 eV which is
closer to our low-current value of /H20855/H9278/H20856and in the range of the
estimated error. Moreover, one may also attribute the differ-
ence between these values to a higher work function of theW tip at the apex. Maeda et al. have reported a local work
function of 4.95 eV of the “ideal” surface from their LBHdata.
8They employed very large tip oscillation amplitudes of
2.5 to 5 Å and did not comment on the gap-width depen-dence of
/H9278.
It is evident that the large barrier heights /H20855/H9278/H20856laat low
tunneling currents differ from the more consistent values of
/H20855/H9278/H20856. The difference leads to the significant deviation between
the fitted overall I/Iref/H20849z0/H20850-function and the I/Iref/H20849z0/H20850curve
for /H20855/H9278/H20856la=15.61 eV in Fig 3. A detailed analysis of this prob-
lem is given in Appendix C.
We have also determined conventional I−zcurves /H20849“point
spectroscopy” /H20850. Figure 4 shows a typical result for a
clean TiO 2/H20849110 /H20850surface, together with the overall
lnI/Iref/H20849z0/H20850-function of Fig. 3 /H20849solid line /H20850.
Here, the zscale has been adapted to the z0scale. The
standard deviation of Ihas been computed from ten I−z
FIG. 3. Relative tunneling current I/Iref/H20849solid line /H20850and barrier
height /H20855/H9278/H20856/H20849dot-dashed line /H20850as a function of distance z0. The value
z0=1.4 Å corresponds to the lowest tunneling current. Further ex-
planations are given in the text.
FIG. 4. Experimental ln I−zdata from point spectroscopy /H20849dots /H20850
and calculated ln /H20849I/Iref/H20850values /H20849solid curve /H20850as a function of z0. The
dot-dashed line corresponds to a hypothetical z0dependence of the
current ln /H20849I*/Iref/H20850if one assumes a constant apparent height of
/H20855/H9278/H20856=5.48 eV. The z0data are referenced to z0at which Iis 1 nA.
The dotted lines indicate the low and high tunneling currents in the
tip-oscillation measurements of /H20855/H9278/H20856.OSTERMANN, WALTHER, AND SCHIERBAUM PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-4curves. We find a good agreement of the point spectroscopy
data with the fitted curve. It is evident that point spectros-copy exhibits a large statistical error when compared withthe tip-oscillation technique. We have also computed the val-ues of ln I/I
reffor 0 Å /H33355z0/H333551.4 Å for the case that the ap-
parent height /H20855/H9278/H20856=5.48 eV /H20849which corresponds to the tunnel-
ing current of 0.118 nA /H20850is constant upon variation of the
gap width. The result is depicted by the dot-dashed line andshows a significant deviation from the point spectroscopydata at lower values of z
0. By comparison we find that the
I−zdata reveal a decreasing slope with a decreasing gap
width. Moreover, the I−zdata give evidence of an upper
limit of the apparent height as we have assumed in Fig. 3.We will now deal with the effect of the gap width on theapparent height in more detail. It was previously pointed outby Bonnell that tips and voltage induced band bending ef-fects occur in STM experiments on transition metal oxidesurfaces.
22
B. Gap-width dependence of the apparent height
Figure 5 depicts a schematic energy band diagram of the
tunneling gap and the important energy levels associatedwith the W tip and the TiO
2. We assume the work function of
the W tip is equal to /H9021W=4.55 eV. From ultraviolet photo-
electron spectra, /H9021TiO2=5.2 eV has been determined for the
clean and geometrically ideal TiO 2surface.23We determine
the bulk position of the Fermi energy EFrelative to the con-
duction band minimum ECin the Boltzmann approximation,
EC−EF=kTlnnb
NC= − 0.2 eV, /H2084912/H20850
where NCdenotes the effective density of states, calculated
with an effective electron mass of 20 mefor a temperature of
T=300 K. Hence, the electron affinity /H9273TiO2of TiO 2is5.0 eV. Furthermore, we assume that the surface density of
electronic states in the band gap of TiO 2/H20849110 /H20850is negligible
so that the conduction and valence bands are not pinned with
respect to EF. These band gap states are associated with in-
trinsic defects whereas the Ti and O-derived “danglingbonds” of the undercoordinated ions at the surface are ener-getically localized in the region of the conduction and va-lence bands and are therefore completely empty or occupied.The depicted situation bears analogy with an ideal metal-oxide-semiconductor capacitor. In this case, the band bend-ing at the semiconductor is determined by the difference inthe work functions at zero applied bias voltage. The bandbending can be removed and flat band condition is reachedby applying a compensating external bias given by
V
fb=/H9021M−/H9273S−EC+EF=/H9021M−/H9021S, /H2084913/H20850
where /H9021Mand/H9021Sare the metal and semiconductor work
functions and /H9273Sis the semiconductor electron affinity. On
the other hand, an ideal Schottky barrier may be formed ifthe vacuum gap separating tip and sample finally disappearsupon reducing the gap width. Then the Schottky-Mott rule,
/H9278SB=/H9021M−/H9273S, /H2084914/H20850
holds.
Here/H9278SBdenotes the barrier height. It is obvious that the
difference Vbetween the electrostatic potentials outside the
surfaces of the metal and the semiconductor tend to zeroupon reducing the gap width and disappears altogether whenan intimate contact is formed. In order to evaluate quantita-tively the effect of variations of the vacuum-gap width s, one
may treat the system as to consist of two capacitors that areconnected in series. For the gap we write
C
gap/H11008s−1/H2084915/H20850
as capacitance per unit area, and for the depleted region in
the TiO 2,
Csc/H11008/H9255w−1, /H2084916/H20850
as the differential capacitance. Here, wdenotes the width of
the space charge layer. Its value can be calculated in thedepletion approximation,
24
w=lD/H208812eVsc
kT. /H2084917/H20850
Here,
lD=/H20881/H9255/H92550kT
e2ND/H2084918/H20850
is the Debye length for a semiconductor with fully ionized
donors. For TiO 2with a dielectric coefficient /H9255of approxi-
mately 100, we calculate lD=139 Å and 438 /H33355w/H333551028 Å
for 0.1 /H33355s/H3335510 Å at T=300 K. We have assumed nD
=0.5 nb, i.e., doubly ionized donor states, for the sake of sim-
plicity. Even for typical tunneling gaps, the value of the dif-ferential capacitance becomes comparable with C
gapdue to
the large value of the dielectric coefficient of TiO 2.
The voltage applied between the tungsten tip and the TiO 2
drops down over the tunneling gap and the depletion region.
FIG. 5. Energy band diagram of the system W/TiO 2separated
by a narrow vacuum gap of the width sunder an applied external
bias voltage V. Potentials at the gap and the TiO 2depletion layer
areVgapandVsc, respectively. The donor states which are associated
with the oxygen vacancies in the TiO 2are not shown. For a descrip-
tion, see the text.LOCAL-BARRIER-HEIGHT IMAGES OF TiO 2/H20849110 /H20850SURFACES PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-5Only for larger distances, Vgap=Vholds. One can determine
the voltage drop in the TiO 2from the total capacitance,
1
C=1
Cgap+1
Csc/H2084919/H20850
and
Vsc=VCgap/Csc
1+Cgap/Csc. /H2084920/H20850
Correspondingly, the gap potential is
Vgap=V−Vsc, /H2084921/H20850
which yields 0.0097 V /H33355Vgap/H333550.69 V for an applied bias
voltage of 1.4 V /H208490.1/H33355s/H3335510 Å /H20850. Evidently, the voltage
drop over the tunneling gap is not at all constant when the
tip-sample distance is changed. As shown in Fig. 6, the dif-ference ln /H20849I/I
ref/H20850−ln /H20849I*/Iref/H20850becomes larger when z0de-
creases. This behavior is consistent with a corresponding de-
crease of the gap voltage. We will now investigate whetherthis effect is related to the observed current-distance behav-ior and the decrease of the apparent height. We have includedthe ln /H20849I/I
ref/H20850vsz0graph in the same diagram, together with
the straight line ln /H20849I*/Iref/H20850vsz0at a constant apparent height
of/H20855/H9278/H20856=5.48 eV. For a quantitative comparison, one must
align the scales sandz0/H20849Fig. 6 /H20850. The criterion for the align-
ment is whether the difference ln /H20849I/Iref/H20850−ln /H20849Vgap/H20850vsz0
/H20849which is associated with the tunneling conductance g/H20850ex-
hibits the same slope than the “ideal” ln /H20849I*/Iref/H20850vsz0curve
/H20849Vgapis given in volts /H20850. In this case, ln /H20849I/Iref/H20850vs ln /H20849Vgap/H20850fol-
lows a function of the type g/H11008exp /H20849−1.025 /H20881/H20855/H9278/H20856z0/H20850with a
constant value of /H9278.
Figure 6 shows that such a situation occurs if the ln /H20849Vgap/H20850
curve is shifted by −0.6 Å with respect to the z0scale /H20849dotted
line /H20850. Here, the ln /H20849Vgap/H20850values which are marked as opensquares have been used to calculate the difference ln /H20849I/Iref/H20850
−ln /H20849Vgap/H20850/H20849solid squares /H20850. We conclude that the tunneling
conductance would indicate a constant apparent height over
the entire range of widths from 0.18 to 1.4 Å if the bias volt-age is V
gapinstead of V. Similar to the interpretation given by
Olesen et al. , our findings suggest that the tunneling voltage
depends on the gap width . This dependence leads to the re-
duction of /H20855/H9278/H20856. In contrast to their work, however, we cannot
measure the gap voltage since the bias voltage drops down
within the TiO 2crystal. In spite of simplified treatment of the
tip’s geometry, our electrostatic space charge model explainsthe observed decrease of the apparent height quantitatively.We will now analyze the effects of intrinsic defects onTiO
2/H20849110 /H20850on the apparent barrier height which are observed
in the LBH images.
C. Point defects
Point defects can be easily created at TiO 2/H20849110 /H20850surfaces
by heating crystals in ultrahigh vacuum to high temperatures
of about 1000 K.1This process is usually kinetically con-
trolled as point defects exhibit high diffusion coefficients inthe bulk, thus generating nonequilibrium defect distributions.Recent STM studies have shown that oxygen vacancies V
o
can be identified on TiO 2/H20849110 /H20850surfaces25and can be distin-
guished from hydroxyl groups.3The vacancies are formed by
the desorption of molecular oxygen and result from missingoxygen atoms at two-fold coordinated bridging sites. Theyare associated with bright spots which appear in STM imagesbetween the /H20855001 /H20856rows of the fivefold-coordinated surface
Ti atoms.
Figure 7 /H20849a/H20850shows the CCM map of a /H20849110 /H20850terrace on
which such oxygen vacancies are visible among other typesof pointlike features. They are marked by V
o. One estimates
a surface concentration of approximately 5 /H110031012cm−2from
the image which corresponds to the model of statisticallydistributed and noninteracting point defects. The value islower than the reported defect density of TiO
2/H20849110 /H20850after a
high-temperature treatment at 1100 K and a subsequent O 2
dosing of 100 L which might result from a lower heating
temperature in our study. The LBH map /H20851Fig. 7 /H20849b/H20850/H20852indicates
a reduced value of irmsat the positions of the oxygen vacan-
cies.
Quantitative data can be derived from the line profile
along line Awhich is depicted in Fig. 8. We determine 30 pA
atVoand approximately 75 pA at the regular Ti sites. These
values correspond to 1.81 and 0.73 of /H20849I/iˆ/H20850la, respectively,
using iˆ=/H208812JirmsandI=85 pA. /H20851Table I provides a relation-
ship between the large-amplitude values of the ratio /H20849I//H20855iˆ/H20856/H20850la
and the spatially averaged apparent height /H20855/H9278/H20856for ratios in
the range between 2.59 and 0.90, which almost cover the
range of /H20849I/iˆ/H20850lavalues in the LBH image /H20852. The polynomial fit
of the values of the Table I yields
/H20855/H9278/H20856= 8.29 − 1.82 /H20873I
/H20855iˆ/H20856/H20874
la− 2.06/H20873I
/H20855iˆ/H20856/H20874
la2
+ 0.68/H20873I
/H20855iˆ/H20856/H20874
la3
.
/H2084922/H20850
FIG. 6. Tunneling current I/Iref/H20849thick solid line /H20850and gap volt-
ageVgap/H20849in volts /H20850as a function of the gap width z0/H20849dotted line /H20850.
Further explanations are given in the text.OSTERMANN, WALTHER, AND SCHIERBAUM PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-6This function is also applicable to calculate the spatially
resolved apparent heights /H9278/H20849x11¯0/H20850from the data iˆ/H20849x11¯0/H20850
=/H208812irms/H20849x11¯0/H20850.
The result is shown in Fig. 9 by the dotted line, labeled
“/H9278/H20849x11¯0/H20850.” As outlined in Appendix B, /H9278depends on the gap
width which alters along the x/H2085111¯0/H20852direction due to the cor-
rugation of /H110110.5 Å. This was explained by the distance de-
pendence of the tunneling voltage V. Consequently, the value
/H9278/H20849x11¯0/H20850is the apparent barrier at a certain gap width or, by
referring to our z0scale, at a certain value of z0. In order to
compensate the distance dependence, which appears in
/H9278/H20849x11¯0/H20850, one may compute the spatially averaged apparent
height /H20855/H9278/H20856of the surface at the same gap width /H20849i.e., thesame value of z0/H20850as the function of x11¯0. We can apply the
/H20855/H9278/H20856/H20849z0/H20850polynomial, which is given in Appendix C, to obtain
the function /H20855/H9278/H20856/H20849x11¯0/H20850. To do so, one must convert the height
scale of Fig. 8 into a z0scale by introducing a /H20849constant /H20850
displacement /H9004zbetween both scales. The value of /H9004zis
varied until the mean values /H20855/H9278/H20856/H20849x11¯0/H20850and/H9278/H20849x11¯0/H20850become
equal. /H20849The averaging is performed in the region outside the
vacancy. /H20850The curve /H20855/H9278/H20856/H20849x11¯0/H20850is depicted by the solid line in
Fig. 9. The displacement is /H9004z=−0.55 Å and the z0scale
agrees reasonably well with the z0scale of Fig. 1. It is, how-
ever, evident that the maximum values of /H20855/H9278/H20856/H20849x11¯0/H20850are larger
than 5.48 eV and beyond the range for which the applicabil-
ity of the polynomial /H20849C1/H20850has been evidenced /H20849compare with
Fig. 3 /H20850. Taking also into account the statistical errors of /H20855/H9278/H20856
FIG. 7. /H20849a/H20850CCM and /H20849b/H20850LBH image of TiO 2/H20849110 /H20850/H20849sample 2 /H20850.
Scan range: 110 /H11003150 Å2; bias voltage V=1.4 V; tunneling current
I=0.085 nA; tip oscillation: root-mean-square amplitude /H9254z
=0.238 Å, frequency 7000 s−1. Further explanations are given in
the text.
FIG. 8. Line scan along A of the CCM /H20849solid curve /H20850and LBH
images /H20849dotted curve /H20850of Fig. 7. Vertical dotted lines indicate in-
plane Ti sites; the solid line refers to the position of an oxygenvacancy.
FIG. 9. Line scan of the topographic data of Fig. 7 in a plot z0vs
x11¯0. The apparent barrier height /H9278is obtained from the irmsdata
/H20849dotted curve /H20850. The apparent barrier height /H20855/H9278/H20856as calculated from
thez0values /H20849solid curve /H20850./H9004/H9278shows the difference between the
solid and dotted curves.LOCAL-BARRIER-HEIGHT IMAGES OF TiO 2/H20849110 /H20850SURFACES PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-7that are given in Table I, one concludes that there is no sig-
nificant difference between both curves at the in-plane Ti/H20849maxima /H20850and bridging O sites /H20849minima /H20850whereas the differ-
ence is significant at the vacancy site. The difference,
/H9004
/H9278=/H9278/H20849x11¯0/H20850−/H20855/H9278/H20856/H20849x11¯0/H20850, /H2084923/H20850
is shown by the thin solid curve in Fig. 9, together with an
error bar. It represents the deviation of the apparent heightfrom the spatially averaged apparent height at an /H20849approxi-
mately /H20850identical gap width. Ideally, /H9004
/H9278does not exhibit a z0
dependence, in contrast to the /H9278/H20849x11¯0/H20850curve. This demon-
strates that LBH images and line scans can only be evaluated
by taking quantitatively into account their gap width depen-dence. There is one exception, of course, with respect toLBH data that are taken at different sites of the surface but atidentical topographic heights which can be directly evalu-ated. This situation is almost fulfilled for the in-plane Ti siteand the oxygen vacancy. We find a strong decrease with anegative value of /H9004
/H9278/H11015−3.8 eV at the oxygen vacancy site.
This may be attributed to a reduced work function /H9021at the
oxygen vacancy of about 1.4 eV. A local work function of1.9 eV has been reported by Maeda et al. for strong reduced
TiO
2/H20849110 /H20850surfaces, exhibiting a /H208491/H110032/H20850reconstruction.8
Such a reduction of /H9021is in correspondence with the overall
decrease in the work function /H9278to about 3 eV that has been
observed earlier in ultraviolet photoelectron spectra of ion-sputtered and reduced TiO
2/H20849110 /H20850surfaces, although the de-
gree of quantitative agreement remains a subject for further
experimental improvements of the measurements.
The second type of pointlike defects appears on the Ti
rows as depressions in the CCM image, which are labeledhere as a “type-B” defect in accordance with Diebold’snotation.
4They extend over one and /H20849to a significantly lower
extent, see also Fig. 7 /H20850two atomic distances along /H20851001 /H20852.
Although the existence of these defects is well known forTiO
2/H20849110 /H20850surfaces, their nature is still under debate. Subsur-
face defects have been proposed which possibly could con-
sist of missing oxygen atoms under the in-plane Ti ions,thereby producing a reconstruction with minima in the to-pography.
Later, Fig. 10 reveals the line scan along B in a z
0vsx/H20851001 /H20852
plot that is calculated from the topographic height according
to the procedure described above. We also include a 6 Ålong arrow /H20849i.e., twice the surface lattice unit in the /H20851001 /H20852
direction, 2 /H110032.96 Å /H20850, indicating the lateral dimension of
this defect. Figure 10 shows also the effect on the apparentheight
/H9278and on /H9004/H9278. It is obvious that this defect is associ-
ated with a local increase of /H9278. The CCM images show that
the majority of the B-type defects appear brighter on bothsides that neighbor the depression in the /H20851001 /H20852direction. On
a few spots we notice point defects that appear such as theyare composed of two adjacent B-type defects on neighboredTi rows. Although the LBH variations are similar, the natureof these features remains as an open problem like the type-Bdefect itself. The same holds for the adsorbates at the sur-face, which appear in the CCM images as bright spots be-tween the Ti rows. Interestingly, they lead to different “fin-gerprints” in the LBH images. The majority of them areassociated with depressions, but they may also appear as
protrusions. The topographic heights at these spots exceedthe surface plane by about 2.3 Å. This is likely related tomore pronounced changes of the local density of states andbeyond the applicability of our evaluation method.
IV. SUMMARY AND CONCLUSIONS
We have employed the tip-oscillation technique, comple-
mented by conventional constant-current-mode STM, to im-age intrinsic defects on the TiO
2/H20849110 /H20850surface and to inves-
tigate how they influence the work function on an atomic
scale. As an important step towards an accurate measurementof apparent heights and their dependence on the tunnelingwidths, we have demonstrated and discussed an evaluationprocedure for LBH data and compared it with point spectros-copy. A simple electrostatic space charge model, based onwell known semiconductor physics approaches, has beenproposed and applied to TiO
2that explains quantitatively the
observed gap-width dependence of the apparent height. It isapplicable to weakly reduced /H20849110 /H20850surfaces on which the
concentration of band gap states, that are attributed to Ti3 d
states and associated with oxygen vacancies, is small andhence a pinning does not occur. This model has interestingimplications for scanning tunneling spectroscopy, employedto such surfaces, as this model may provide information onthe actual gap voltage as a function of the bias.
In the third part we have demonstrated that this LBH
method provides an access to local work function changeswhich are induced by surface oxygen vacancies. Other typesof pointlike defects have been found, including type-B de-fects.
ACKNOWLEDGMENTS
This work is funded by the European Community under
Project No. 505895-1. Stimulating discussions with G.Thornton /H20849University College London /H20850are gratefully ac-
knowledged.
FIG. 10. Line scan along B of the CCM and LBH images of
Fig. 7.OSTERMANN, WALTHER, AND SCHIERBAUM PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-8APPENDIX A: CALCULATION OF I/iˆ
If one applies a Taylor series expansion to
exp /H20851−A/H20881/H9278/H208812/H9254zsin/H9275t/H20852=exp /H20851bsin/H9275t/H20852with b=−A/H20881/H9278/H208812/H9254z
and uses the trigonometric expressions sin2/H9275t=1
2/H208491
−cos 2 /H9275t/H20850, cos2/H9275t=1
2/H208491+sin 2 /H9275t/H20850, and sin3/H9275t=−1
4/H20849sin 3/H9275t
−3 sin /H9275t/H20850one obtains for the tunneling current
I/H20849t/H20850=VBexp /H20851−A/H20881/H9278z0/H20852/H208731+bsin/H9275t+1
4b2−1
4b2cos 2/H9275t
−1
24b3sin 3/H9275t+1
8b3sin/H9275t+1
64b4
−1
64b4cos 2/H9275t±¯/H20874. /H20849A1/H20850
Truncation after the fourth-degree term yields the ampli-
tude of the first harmonic component of I,
iˆ=/H20879VBexp /H20851−A/H20881/H9278z0/H20852/H20873b+1
8b3/H20874/H20879, /H20849A2/H20850
for which we take positive values only. In a similar fashion,
one can write for the time-averaged value of I,
I/H20849t/H20850=VBexp /H20851−A/H20881/H9278z0/H20852/H208731+1
4b2+1
64b4/H20874. /H20849A3/H20850
Note that the term /H208491+1
4b2+1
64b4/H20850is associated with an in-
crease of the tunneling current when the tip is oscillating.27
Equations /H20849A2/H20850and /H20849A3/H20850suggest that the term VBexp /H20851/H20881/H9278/H20852is
cancelled out by division. It is trivial that this applies to
exp /H20849z0/H20850as LBH and CCM images are recorded simulta-
neously, i.e., at the same value of z0.
It is, however, not as simple for the expression VBexp/H20881/H9278.
As shown in Fig. 1, irmsand hence iˆvary laterally while Iis
constant. The variation of along /H2085111¯0/H20852reveals the periodicity
of the surface lattice unit /H208496.5 Å /H20850like the topographic data.
As the consequence, the ratio I/iˆshould be determined from
theiˆdata at identical sites /H20849e.g., at the minima or maxima /H20850.
Since a sufficient atomic resolution is not obtained at all
values of I, we compute the spatial average of /H20855iˆ/H20856from the
LBH image. /H20849In the following, we use /H20855/H20856when spatially
averaged data must be indicated. /H20850
The procedure of averaging /H20855iˆ/H20856is suitable since the ratio
of the bridging O and in-plane Ti sites, at which the largest
differences of /H20855iˆ/H20856occur, is close to 1 and the number of
defects and adsorbates is small. Similarly, we determine /H20855I/H20856
from the current image to increase the accuracy for the val-
ues of I. From the Eqs. /H20849A1/H20850and /H20849A2/H20850, we then obtain the
ratio
I
iˆ=1+1
4b2+1
64b4
b+1
8b3. /H20849A4/H20850APPENDIX B: DETERMINATION OF I/Iref„z0…
We first calculate the function I/Iref/H20849z0/H20850from Eq. /H208499/H20850, us-
ing /H20855/H9278/H20856la=1.30 eV that passes through the value 0.812 in the
interval I. At the low-current value of I/Irefat the border
between the intervals I and II /H20849and, hence, at a certain value
z0/H11032/H20850we calculate a new function I/Iref=exp /H20851−A/H208812.37 eV /H20849z0
+/H9004z/H20850/H20852in such a way that both curves have the same value of
I/Irefatz0/H11032=z0+/H9004z. Graphically, /H9004zcorresponds to a horizon-
tal shift of the curve I/Iref. If we repeat this procedure, five
I/Irefcurves are generated which individually reflect the tun-
neling current- versus -distance relationships in a small range
ofz0. These ranges correspond to the intervals I to V in Fig.
3.
APPENDIX C: EFFECT OF OSCILLATING APPARENT
HEIGHT
We obtained the barrier height /H20855/H9278/H20856laby evaluating Iand
/H20855iˆ/H20856. The relationship with Eq. /H208494/H20850is illustrated by Fig. 11. It
indicates the oscillation of exp /H20851−A/H20881/H20855/H9278/H20856z/H20849t/H20850/H20852 /H20849to which the
tunneling current is proportional /H20850for two different cases. The
image /H20849a/H20850shows the situation if one assumes a constant tun-
neling barrier height in the evaluation of the experimentaldata. We choose /H20855
/H9278/H20856la=15.61 eV and z0=1.4 Å in line with
the values at a tunneling current of I=0.118 nA.
The image /H20849b/H20850demonstrates the case of an oscillating tip
/H20849/H9263=7/H11003103s−1,/H9254z=0.238 Å, and z/H20849t/H20850=z0+/H208812/H9254zsin/H9275twith
z0=1.4 Å /H20850and an oscillating barrier height /H20855/H9278/H20856/H20849t/H20850. We have
computed /H20855/H9278/H20856/H20849t/H20850from the /H20855/H9278/H20856−z0curve of Fig. 1 by means
of a fourth-order polynomial fit,
/H20855/H9278/H20856= 1.431 + 0.073 z0+ 3.959 z02− 4.532 z03+ 2.346 z04
/H20849C1/H20850
/H20849/H20855/H9278/H20856andz0are given in the units eV and Ångstrom, respec-
tively /H20850. Here, the values of exp /H20851−A/H2088115.61 eV z/H20849t/H20850/H20852are multi-
FIG. 11. exp /H20851−A/H20881/H9278z/H20849t/H20850/H20852,/H9278/H20849t/H20850andz/H20849t/H20850as a function of time tfor
an oscillating tip. Further explanations are given in the text.LOCAL-BARRIER-HEIGHT IMAGES OF TiO 2/H20849110 /H20850SURFACES PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-9plied by a factor of 11, to adjust the scales. It can be easily
shown by Fourier analysis /H20849which is not illustrated here /H20850that
the same ratios I//H20855iˆ/H20856are obtained as in /H20849a/H20850. This comparison
demonstrates that the finite value of /H9254zrepresents a majorsource for systematic errors in the determination of /H20855/H9278/H20856.
Consequently, it appears to be necessary to determine LBH
data at different tunneling currents and hence different tun-neling widths for a quantitative analysis of images.
*Corresponding author. Electronic address: schierb@uni-
duesseldorf.de
1W. Göpel, G. Rocker, and R. Feierabend, Phys. Rev. B 28, 3427
/H208491983 /H20850.
2I. M. Brookes, C. A. Muryn, and G. Thornton, Phys. Rev. Lett.
87, 266103 /H208492001 /H20850.
3R. Schaub, P. Thostrup, N. Lopez, E. Lægsgaard, I. Stensgaard, J.
K. Nørksov, and F. Besenbacher, Phys. Rev. Lett. 87, 266104
/H208492001 /H20850.
4U. Diebold, Surf. Sci. Rep. 48,5 3 /H208492003 /H20850.
5M. Sander and T. Engel, Surf. Sci. Lett. 302, L263–L268 /H208491994 /H20850.
6R. A. Bennett, P. Stone, N. J. Price, and M. Bowker, Phys. Rev.
Lett. 82, 3831 /H208491999 /H20850.
7M. Batzill, K. Katsiev, D. J. Gaspar, and U. Diebold, Phys. Rev.
B66, 235401 /H208492002 /H20850.
8Y . Maeda, M. Okumura, S. Tsubota, M. Kohyama, and M.
Haruta, Appl. Surf. Sci. 222, 409 /H208492004 /H20850.
9B. Marchon, P. Bernhardt, M. E. Bussell, G. A. Somorjai, M.
Salmeron, and W. Siekhaus, Phys. Rev. Lett. 60, 1166 /H208491988 /H20850.
10R. Wiesendanger, L. Eng, H. R. Hidber, P. Oelhafen, L. Rosentha-
ler, U. Staufer, and H.-J. Güntherodt, Surf. Sci. 189/190 ,2 4
/H208491987 /H20850.
11L. Olesen, M. Brandbyge, M. R. Sørensen, K. W. Jacobsen, E.
Lægsgaard, I. Stensgaard, and F. Besenbacher, Phys. Rev. Lett.
76, 1485 /H208491996 /H20850.
12G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Appl. Phys.
Lett. 40, 178 /H208491982 /H20850.
13J. B. Pethica and A. P. Sutton, J. Vac. Sci. Technol. A 6, 2490
/H208491988 /H20850.
14C. J. Chen and R. J. Hamers, J. Vac. Sci. Technol. B 9, 503/H208491991 /H20850.
15M. Yoon, H. Mai, and R. F. Willis, Europhys. Lett. 54, 626
/H208492001 /H20850.
16W. A. Hofer, Prog. Surf. Sci. 71, 147 /H208492003 /H20850.
17G. Binnig, N. Garcia, H. Rohrer, J. M. Soler, and F. Flores, Phys.
Rev. B 30, 4816 /H208491984 /H20850.
18N. D. Lang, Phys. Rev. B 37, R10 395 /H208491988 /H20850.
19L. Olesen, M. Brandbyge, M. R. Sørensen, K. W. Jacobsen, E.
Lægsgaard, I. Stensgaard, and F. Besenbacher, Phys. Rev. Lett.
76, 1485 /H208491996 /H20850.
20C. J. Chen, in Scanning Tunneling Microscopy III: Theory of STM
and Related Scanning Probe Methods , edited by R. Wiesendan-
ger and H. J. Güntherodt, Springer Series in Surface ScienceV ol. 29 /H20849Springer, New York, 1996 /H20850, p. 141.
21V . N. Bogomolov and V . P. Zhuse, Fiz. Tverd. Tela /H20849S.-Peterburg /H20850
10, 100 /H208491962 /H20850/H20851Sov. Phys. Solid State 5, 2404 /H208491963 /H20850/H20852.
22D. A. Bonnell, Prog. Surf. Sci. 57, 187 /H208491998 /H20850.
23K. D. Schierbaum, S. Fischer, M. C. Torquemada, J. L. de Seg-
ovia, E. Rom, and J. A. Mart-Gago, Surf. Sci. 345, 261 /H208491996 /H20850.
24E. H. Rhoderick and R. H. Williams, Metal-Semiconductor Con-
tacts /H20849Oxford Science Publications, Clarendon Press, Oxford,
1988 /H20850.
25U. Diebold, J. Lehman, T. Mohmoud, M. Kuhn, G. Leonardelli,
W. Hebenstreit, M. Schmid, and P. Varga, Surf. Sci. 41, 137
/H208491998 /H20850.
26As the result of the integration times of the amplifiers, small shifts
below 0.2 Å occur between the different images.
27This change of the tunneling current should carry information of
the apparent height but has not been evalutated here.OSTERMANN, WALTHER, AND SCHIERBAUM PHYSICAL REVIEW B 71, 235416 /H208492005 /H20850
235416-10 |
PhysRevB.81.085301.pdf | Atomistic tight-binding theory of multiexciton complexes in a self-assembled InAs quantum dot
M. Zieli ński,*M. Korkusi ński, and P. Hawrylak
Institute for Microstructural Sciences, National Research Council, Ottawa, Canada K1A 0R6
/H20849Received 20 August 2009; revised manuscript received 8 December 2009; published 1 February 2010 /H20850
We present atomistic tight-binding theory of electronic structure and optical properties of InAs/GaAs self-
assembled quantum dots. The tight-binding model includes zincblende symmetry, faceting, and sp3d5s/H11569atomic
orbitals accounting for interband and intervalley couplings. The equilibrium positions of atoms are calculatedusing valence force field method and modification of the tight-binding Hamiltonian due to strain is accountedfor using Harrison’s law. The electronic and optical properties of multiexciton complexes are then determinedby diagonalizing the many-body Hamiltonian for interacting electrons and holes using the configuration-interaction approach. The calculations of strain distribution approach 10
8atoms while the electron and valence
hole single-particle states are calculated by diagonalization of the Hamiltonian matrix with size on the order of10
7. The dependence of predicted electronic and optical properties on InAs/GaAs valence-band offset and InAs
absolute valence-band deformation potentials are described. The reliability of the atomistic calculations isassessed by comparison with results obtained from the effective bond orbital model and empirical pseudopo-tentials method.
DOI: 10.1103/PhysRevB.81.085301 PACS number /H20849s/H20850: 73.21.La, 73.22. /H11002f, 78.67.Hc, 71.15. /H11002m
I. INTRODUCTION
Self-assembled quantum dots /H20849SADs /H20850/H20849Refs. 1–3/H20850involve
millions of atoms and their electronic properties cannot atpresent be computed using ab initio methods, such as, e.g.,
GW-BSE approach.
4Approximate methods, capturing atom-
istic structure of quantum dots and their matrix, includetight-binding
5–12and pseudopotential13–16approaches. One
of the approximate methods, valence force field-tight-binding-configuration interaction /H20849VFF-TB-CI /H20850discussed
here, involves three steps:
5,6,11,17/H20849a/H20850calculation of equilib-
rium position of constituent atoms using VFF model, /H20849b/H20850
calculation of quasielectron and quasihole states /H20849equivalent
to the GW step /H20850using a linear combination of atomic orbitals
/H20849LCAO /H20850approach in a TB approximation, and /H20849c/H20850inclusion
of final-state interactions by defining an effective Hamil-tonian of interacting excited quasiparticles, diagonalized us-ing the CI method. The approximate nature of such an ap-proach requires careful analysis of results, and in particularan analysis of sensitivity of results to approximations made.Benefits include predictive capability allowing us to under-stand the dependence on size, geometry, composition, andexternal electric and magnetic fields of the electronic andoptical properties of SADs.
In this paper we present results of the VFF-TB-CI meth-
odology applied to InAs/GaAs SADs, with convergent straindistribution computed using the VFF approach for hundredsof millions of atoms, the electron and hole single-particle/H20849SP/H20850states computed using the 20-band sp
3d5s/H11569tight-binding
model for millions of atoms, and energies, states, and emis-sion spectra from up to ten exciton complexes obtained inthe configuration-interaction method. In the VFF calculationwe use the Keating model with material parameters derivedfrom bulk elastic constants c
ij.18The TB parameters for un-
strained InAs and GaAs are obtained by fitting of the TBbulk band edges and effective masses to those obtained inexperiment or by ab initio calculations, with the valence-
band offset /H20849VBO /H20850built into the parameter set.
17The depen-dence of band edges on lattice deformation computed using
density-functional theory /H20849DFT /H20850/H20849Ref. 19/H20850is used to find
strain corrections to TB parameters. We generate two sets ofparameters corresponding to two DFT results predicting op-posite behavior of the valence-band edge. The Coulomb ma-trix elements needed for CI are obtained with TB wave func-tions involving /H1101110
8orbitals, with on-site terms computed
by approximating the TB basis with Slater orbitals. The in-teractions are screened by a distance-dependent dielectricfunction. In the CI step, typically /H1101110
4configurations are
used as a basis for each multiexciton system, while emissionspectra are calculated from Fermi’s golden rule.
We illustrate the method by computing the electronic and
optical properties of a lens-shaped SAD. We find that in bothcases the quasielectron states are organized in degenerateshells, a result independent of the VBO and strain param-eters. The quasihole states are sensitive to these constantsand do not reveal a shell structure for the lens-shaped dot.We study the signature of this sensitivity in multiexcitonemission spectra. The reliability of the atomistic calculationsis assessed by comparison with results obtained from theeffective bond orbital model /H20849EBOM /H20850rooted in the k·pap-
proximation and empirical pseudopotentials method ofZunger and co-workers.
The paper is organized as follows. Section IIcontains the
definition of the geometry of the system. Next we discuss thedetails of the model, starting with the calculation of strain/H20849Sec. III/H20850, the tight-binding model for electronic-structure
calculation /H20849Sec. IV/H20850and the coupling of these two elements
/H20849Sec. V/H20850. Sections VIandVIIdiscuss the resulting evolution
of the bulk bands as a function of strain, and their sensitivityto the valence-band offset, respectively. In Sec. VIII we out-
line the calculations of the Coulomb and dipole matrix ele-ments, while in Sec. IXwe describe the computational pro-
cedure used to diagonalize the many-body Hamiltonian.Section Xpresents a detailed discussion of all aspects of our
VFF-TB-CI computation on the example of a lens-shapedquantum dot. Finally, in Sec. XIwe summarize the paper.PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
1098-0121/2010/81 /H208498/H20850/085301 /H2084912/H20850 ©2010 The American Physical Society 085301-1II. GEOMETRY DEFINITION
We start with InAs quantum dots embedded in GaAs. We
define the size and shape of InAs quantum dot and embed thedot in a box of barrier GaAs material. In, As, and Ga atomsare placed on the sites of GaAs zincblende lattice. The di-mension of the surrounding GaAs box defines the size of thecomputational domain.
The theory will be illustrated by calculations carried out
for a lens-shaped quantum dot, for which results of EBOMand pseudopotential calculations are available. The height ofthe dot is h=3.5 nm and the base diameter is D=25 nm.
The center of the base of the quantum dot is placed on theanion /H20849As/H20850atom. The dot is placed on one lattice constant
/H20849/H110110.6 nm /H20850thick wetting layer.
III. ATOMISTIC CALCULATION OF STRAIN
There is a /H110117% lattice mismatch between InAs and
GaAs. The resulting strain is the driving force for the growthof self-assembled quantum dots. It also plays an importantrole in determining their electronic and optical properties.We use the approach in which the strain calculation processis equivalent to finding atomic positions that minimize thetotal elastic energy. At the same time the knowledge ofatomic positions is a prerequisite for the atomistic calcula-tion of the electronic structure.
In the continuous elasticity theory
20the elastic energy is
defined as a sum of local distortions of a continuous mediumdiscretized on a computational grid. Unfortunately, this ap-proach neglects atomistic details of interfaces and the lack ofinversion symmetry of the zincblende lattice. Here, we usethe atomistic approach of Keating,
21in which the total elastic
energy ETOTcontains the stretching and bending terms from
each atomic bond
ETOT=1
2/H20858
i=1N
/H20858
j=1nn
Aij/H20851/H20849R/H6023i−R/H6023j/H208502−dij2/H208522
+/H20858
i=1N
/H20858
j=1nn
/H20858
k=j+1nn
Bijk/H20875/H20849R/H6023j−R/H6023i/H20850/H20849R/H6023k−R/H6023i/H20850−1
3dijdik/H208762
.
/H208491/H20850
Here, R/H6023idenotes the position of the ith atom, dijis the bulk
bond length between the ith and jth atoms, and AijandBijk
are material-dependent elastic parameters. The summations
go over Natoms and nearest neighbors /H20849nn/H20850. We start from a
uniform lattice with GaAs lattice constant and to obtainstrain field, we minimize total elastic energy with respect tothe atomic positions using the conjugate gradient algorithm.The equilibrium positions of atoms are displaced from thosein the bulk, and these displacements, i.e., the lengths anddirections of atomic bonds, vary across the sample. As ameasure of the displacement field, distribution of strain ten-sor elements across the sample is then computed by compar-ing the deformed zincblende unit cells with their unstrainedbulk counterparts.
20
Because strain is long ranged, the GaAs buffer /H20849computa-
tional domain /H20850used in the strain-energy minimization mustbe large enough to ensure that the strain fields vanish at the
GaAs buffer boundaries. Lee et al.22investigated in detail the
vertical size of the GaAs buffer needed to obtain vanishinghydrostatic strain at the box boundaries.
IV. ATOMISTIC TIGHT-BINDING ELECTRONIC-
STRUCTURE CALCULATION
With the equilibrium atomic positions known, we can at-
tempt to calculate the single-particle electronic structure ofthe system. The single-particle spectrum describes a quasi-particle moving in a field of atoms and dressed by interactionwith all other electrons. The quasiparticle Hamiltonian inGW approximation reads
4
HˆQP=p/H60232/2m+Vatomic /H20849r/H6023/H20850+VHartree /H20849r/H6023/H20850+/H9018/H20849E,r/H6023/H20850, /H208492/H20850
where Vatomic /H20849r/H6023/H20850is the sum of atomic potentials, VHartree /H20849r/H6023/H20850is
the Hartree potential produced by all electrons, and /H9018/H20849E,r/H6023/H20850is
the energy-dependent self-energy due to exchange and cor-relation. In the density-functional approximation the self-energy is replaced by the exchange-correlation potential. Ifwe were able to carry out fully self-consistent density-functional calculations, the Kohn-Sham Hamiltonian wouldhave been expressed in terms of atomic, Hartree, andexchange-correlation potentials, themselves functionals ofelectronic density. Since we do not know the Hamiltonian,we parametrize it in a tight-binding form by first expandingthe wave function in a basis of atomic orbitals
/H9278=/H20858
R/H6023,/H9251cR/H6023/H9251/H20841R/H6023/H9251/H20856, /H208493/H20850
and next forming the Hamiltonian matrix in the atomic basis.
By comparison, in a pseudopotential approach13–16the
atomic, Hartree, and self-energy potentials are replaced by asum of effective atomic potentials. These atomic potentialsare next used to generate one-electron potential of the nano-structure.
In our tight-binding approach the wave function on each
atom is described by ten valence orbitals for each spin: oneof type s, three of type p, five of type d, and an additional s
/H11569
orbital that accounts for higher lying states. Each orbital is
doubly spin degenerate, thus resulting in a total of 20 bands.The resulting Hamiltonian of quasiparticle in an N-atom
quantum dot, written in the language of second quantization,reads
Hˆ
TB=/H20858
i=1N
/H20858
/H9251=120
/H9280i/H9251ci/H9251+ci/H9251+/H20858
i=1N
/H20858
/H9251=1,/H9252=120
/H9261i/H9251,/H9252ci/H9251+ci/H9252
+/H20858
i=1N
/H20858
j=14
/H20858
/H9251,/H9252=120
ti/H9251,j/H9252ci/H9251+cj/H9252, /H208494/H20850
where ci/H9251+/H20849ci/H9251/H20850is the creation /H20849annihilation /H20850operator of a car-
rier on the orbital /H9251localized on the site i,/H9280i/H9251is the corre-
sponding on-site energy, and ti/H9251,j/H9252describes the hopping of
the particle between orbitals on neighboring sites. Couplingto farther neighbors is neglected. Finally, /H9261
i/H9251,/H9252accounts for
the spin-orbit interaction by introducing finite matrix ele-ZIELI ŃSKI, KORKUSI ŃSKI, AND HAWRYLAK PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-2ments /H9004connecting porbitals of opposite spin, residing on
the same atom, following the description given by Chadi.23
For example, /H20855py,↑/H20841H/H20841↓,pz/H20856=−i/H9004. Spin-orbit-type coupling
between d-type orbitals is neglected. Here we assume that
each site holds 20 orbitals and is surrounded by four neigh-bors.
Hopping, i.e., off-diagonal matrix elements of our Hamil-
tonian are calculated according to the recipe given by Slaterand Koster.
24In this approach the hopping matrix elements
ti/H9251,j/H9252are expressed as geometric functions of two-center in-
tegrals and depend only on the relative positions of the atomsiandj. Contributions from three-center and higher integrals
are neglected. For example, if the two atoms are connectedby a bond along the xaxis then orbitals sandp
zcreate a /H9266
bond and the matrix element ts,pz=Vs,pz/H9266=0 vanishes because
of the symmetry. On the other hand, if the direction of the
bond is along yaxis, i.e., vertical, then the bond is of a /H9268
type and ts,pz=Vs,pz/H9268is finite. In the general case the
nearest neighbors are connected by bonds of any direction
d/H6023=/H20841d/H20841/H20849lxˆ+myˆ+nzˆ/H20850, with dbeing the bond length and l,m,
n—the direction cosines. Then the tunneling ts,pzelement can
be expressed in terms of projecting the pzorbital onto the
bond and in the direction perpendicular to it. Since the per-pendicular projections give
/H9266-type bonds, their contribution
is zero. The Hamiltonian matrix element is thust
i,s,j,pz=nVsp/H9268. Similar sets of rules are defined for all other
t-matrix elements.24
This approach reduces the number of unknown matrix
elements as they can be related via Slater-Koster rules to arelatively small subset of two-center integrals V
/H9251/H9252,/H9253. This is
particularly useful within the framework of empirical tightbinding, where E
/H9251,/H9261/H9251,/H9252, and V/H9251/H9252,/H9253parameters are not di-
rectly calculated, but rather obtained by fitting25the TB bulk
model results to experimentally known band gaps and effec-tive masses at high-symmetry points of the Brillouin zone.We want to stress here that we are fitting TB model not onlyto bulk properties at /H9003point, but also at X and L points to
account for multivalley couplings.
The most frequent parameterizations used so far are given
in Refs. 17,18, and 26. These previous works demonstrated
that the inclusion of dorbitals in the basis allows to obtain
much better fits of the masses and energy gaps to the targetmaterial values. In particular, the treatment of theconduction-band edge is significantly improved, which is im-portant for small nanostructures.
27In this work we use our
own parameterization, analogous to work by Klimecket al. ,
17but giving better agreement with target bulk proper-
ties. More details will be presented in our future work.
In order to address the treatment of the interface between
InAs and GaAs we note that this two materials share thesame anion /H20849arsenic /H20850. Thus during the fitting procedure diag-
onal matrix elements on arsenic are kept the same in bothmaterials. This approach removes the necessity of averagingon-site matrix elements for interface atoms. Additionally, toaccount for the BO between the materials forming the inter-face, fit for InAs is performed in such a way that it includesband offset, i.e., the top of the valence band of InAs is set tobe equal to the band offset value. This removes the necessityof shifting values of diagonal matrix element for interfaceatoms, which would result in two different sets of parameters
for arsenic: one for InAs and another for GaAs. We analyzethe importance of the choice of the VBO in Sec. X.
Finally, there is a second type of interface that arises on
the edges of the computational box. There, the appearance offree surfaces leads to the existence of dangling bonds. Theirpresence results in spurious surface states, with energy insideof the gap of the barrier material, making it difficult to dis-tinguish spurious states from the single-particle states of thequantum dot. A dangling-bond-energy shift that mimics thepassivation procedure, described in Ref. 21is performed in
order to shift the energies of surface-localized states awayfrom the energies corresponding to confined QD orbitals.
Finally, a parallel Lanczos diagonalizer with Kramers
symmetry is used to resolve the Kramers-degenerate dou-blets. We performed a systematic study /H20849similar to the one in
Ref. 22/H20850of the effect of the size of the tight-binding compu-
tational domain on the convergence of energies and wavefunctions of states confined in a quantum dot. This allowedus to ensure that the TB domain is large enough so thatfurther extension of the computational box would change thesingle-particle energies by less then a small fraction of mil-lielectron volt. Because the computational domain necessaryfor the converged tight-binding calculation involves /H20849case-
dependent /H20850/H110151 million atoms, resulting tight-binding matri-
ces are very large, i.e., /H20849/H1101520 million by 20 millon /H20850. This
presents a significant numerical problem, but utilizing matrixsparsity, parallel computer and the fact that we need onlyseveral lowest electron and hole states, and not the entireHamiltonian eigenspectrum, we achieved linear scaling as afunction of the number of atoms.
V. INCLUSION OF STRAIN INTO TIGHT-BINDING
HAMILTONIAN
As mentioned above, tight-binding parameters are ob-
tained by fitting the bulk TB band structure to experimentallymeasured band structure of the unstrained bulk semiconduc-tors. Since strain changes bond angles and lengths, theHamiltonian matrix elements change as well. The Slater-Koster approach is particularly convenient in introducing thestrain effects into the model since changes in bond angles aretaken into account by the set of rules involving directioncosines.
To account for changes in bond lengths we use a general-
ized version of Harrison law:
28V/H9251/H9252,/H9253=V/H9251/H9252,/H92530/H20849dij/d0/H20850/H9257, where
V/H9251/H9252,/H92530is two-center integral for the unstrained case, dij/d0is
the ratio of the new to old /H20849ideal /H20850bond length dscaled by the
exponent /H9257, value of which will be discussed later. Modified
V/H9251/H9252/H9253are used to build tight-binding Hamiltonian for the
strained system.
Boykin et al.18argued that because tight-binding orbitals
are not true atomic orbitals, but rather they are the orthogo-nalized Löwdin orbitals, the diagonal matrix elements might,in principle, also vary in response to displacements of neigh-boring atoms. Similarly, Jancu et al.
26introduced uniaxial
strain-induced splitting of otherwise degenerate dxy,dyx, and
dzxlevels. These authors claim that uniaxial strain induces a
tetragonal crystal field which lifts the degeneracy of the d
atomic levels.ATOMISTIC TIGHT-BINDING THEORY OF … PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-3To summarize, both models include modification of on-
site /H20849diagonal /H20850matrix elements due to strain, and we follow
the model developed by Klimeck as it is more general andnot limited to the case of the uniaxial strain. In this approachwe have
/H9280R/H6023/H9251=/H9280R/H6023/H92510+/H20858
R/H6023/H33528nn/H20858
/H9252CR/H6023/H9251,R/H11032/H6023/H9252tR/H6023/H9251,R/H11032/H6023/H92520−tR/H6023/H9251,R/H11032/H6023/H9252/H11032
/H9280R/H6023/H92510+/H9280R/H11032/H6023/H92520, /H208495/H20850
where Care empirical material parameters, yet to be deter-
mined.
VI. BAND EVOLUTION AS A FUNCTION OF STRAIN
In the original work by Harrison,28/H9257was assumed to be
the same for all integrals V/H9251/H9252,/H9253. It was also determined to be
equal to 2.0 by comparison of the TB model with the nearlyfree-electron model. In this work we fit the parameters
/H9257and
Cto match evolution of bulk band edges given by the Bir-
Pikus /H20849BP/H20850model29with experimentally measured deforma-
tion potentials for the case of hydrostatic strain. However, wenote that there is ongoing discussion
30–32in literature regard-
ing the sign of the absolute hole band deformation potentialsa
v, both for InAs and GaAs. Henceforth we compare the
results obtained using positive and negative values of av.
Figures 1/H20849a/H20850and1/H20849b/H20850show evolution of InAs band edges
as a function of hydrostatic deformation given as Tr /H20849/H9280/H20850,where /H9280is the hydrostatic strain tensor as calculated with our
tight-binding model fitted to reproduce the Bir-Pikus model29
for two different values of absolute valence-band deforma-tion potential, /H20849a/H20850a
v=1.0 and /H20849b/H20850av=−1.0. In this model, for
purely hydrostatic strain, the light and heavy holes remaindegenerate and the evolution of the top of the valence band isgiven as
E
v=Ev0+avTr/H20849/H9280/H20850, /H208496/H20850
where Ev0is unstrained bulk top of the valence band.
Arrows denote different trend of the heavy/light hole band
evolution as a function of compressive strain. The avparam-
eter is important for confined system as it determineswhether the confining potential for holes becomes deeper orshallower under hydrostatic strain. We note that other authorsused both positive a
v/H20849Ref. 20/H20850and negative av/H20849Ref. 15/H20850in
their calculation.
The strain in self-assembled quantum dots is not purely of
hydrostatic kind, there is also a significant biaxial strain con-tribution. For the more complicated case of the biaxial strainwe fit the tight-binding model to reproduce results obtainedby both the Bir-Pikus model, and more elaborate DFT.
19
Additionally, after the VFF strain relaxation, the in-plane
lattice distances /H20849“constants” /H20850in the wetting layer and in the
quantum dot itself are strongly distorted from InAs bulkequilibrium value, almost matching that of GaAs. Because ofthis it is important to fit TB Hamiltonian to reproduce notonly the evolution of band edges for small deformation /H20849de-
formation potentials /H20850but also band trends in a wide range of
deformation.
Figures 1/H20849c/H20850and1/H20849d/H20850show the evolution of InAs band
edges under biaxial strain as calculated with our model fittedto reproduce /H20849c/H20850Bir-Pikus model /H20849with a
v=−1.0 /H20850and /H20849d/H20850
DFT calculation33/H20849av=−0.88 /H20850. We were able to achieve a
much better fit to DFT than to the simple Bir-Pikus case, asBir-Pikus model, compared to DFT, overestimates the energyevolution of the bottom of conduction band by almost 0.2 eV/H20849dashed line /H20850. Such a difference will result in a similar, sig-
nificant differences of energies of confined quantum dotstates as described later on.
In order to combine both hydrostatic and biaxial strain in
one set of parameters we developed a genetic algorithm thatperforms simultaneous fit to hydrostatic and biaxial straincases.
34In the later part of the work we will analyze the
properties of electron and hole states as a function of differ-ent strain coupling methods.
VII. VALENCE-BAND OFFSET
Apart from the uncertainty of the sign of the absolute
InAs /H20849GaAs /H20850valence-band deformation potential there is
also a considerable spread in the values reported for theInAs/GaAs valence-band offset.
35,36The reported values
vary from 60 to 330 meV. Although specific choice of VBOwill not affect the effective QD gap significantly,
10large dif-
ferences between extreme VBO values, combined with un-certainty of a
v, may correspond to a very different confining
potential profile for the hole states depending on the choiceof parameters. In this work we will study these problems by1.0
Bir Pikus-0.09 -0.06 -0.03 0.0 0-0.6-0.30.00.30.60.91.2Bir Pikus av=-1.0 (eV)
Our fitEnergy (eV)
Tr(ε)-0.09 -0.06 -0.03 0.00-0.6-0.30.00.30.60.91.2Bir Pikus av=1.0 (eV)
Our fitEnergy (eV)
Tr(ε)
1.0
DFTa) b)
c) d)HH/LH HH/LHCB
SOCB
SO
5.7 5.8 5.9 6.0-0.6-0.4-0.20.00.20.40.60.8Our fitEnergy (eV)
In-plane lattice constance (Å)5.7 5.8 5.9 6.0-0.6-0.4-0.20.00.20.40.60.8DFT
Our fitEnergy (eV)
In-plane lattice constance (Å)c) d)
HHCB
SOLHHHCB
SOLH
FIG. 1. /H20849Color online /H20850InAs band edges as a function of the
hydrostatic deformation Tr /H20849/H9280/H20850as calculated with our tight-binding
model fitted to reproduce Bir-Pikus model for two different valuesof valence band absolute deformation potential /H20849a/H20850a
v=1.0 and /H20849b/H20850
av=−1.0. InAs band edges under biaxial strain as calculated with
our model fitted to reproduce /H20849c/H20850Bir-Pikus model /H20849av=−1.0 eV /H20850
and /H20849d/H20850DFT calculation /H20849av=−0.88 eV /H20850.ZIELI ŃSKI, KORKUSI ŃSKI, AND HAWRYLAK PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-4using two different band offset values: VBO=210 meV as
“recommended” by Ref. 35and consistent with Ref. 36and
VBO=60 meV obtained by Wei and Zunger30and used by
Zunger and co-workers in their empirical pseudopotentialcalculations.
15,16
VIII. COULOMB AND DIPOLE MATRIX ELEMENTS
Once single-particle energy states are found, the next step
is the calculation of many-body states of excited electronsand holes populating single-particle levels. The interaction ofelectrons and holes and their interaction with light requirescalculation of Coulomb and dipole matrix elements.
In a GW approach one calculates the effective interaction
Wself-consistently. Not being able to carry out this calcula-
tion, we assume a statically screened Coulomb interaction.Hence the Coulomb matrix elements V
ijklare given by
Vijkl=/H20885/H20885/H9278i/H11569/H20849r1/H6023/H20850/H9278j/H11569/H20849r2/H6023/H20850e2
/H9280/H20849r1/H6023,r2/H6023/H20850/H20841r1/H6023−r2/H6023/H20841/H9278k/H20849r2/H6023/H20850/H9278l/H20849rl/H6023/H20850,
/H208497/H20850
where /H9280/H20849r1/H6023,r2/H6023/H20850is the position-dependent dielectric function,
/H9278are single-particle wave functions obtained by diagonaliza-
tion of the tight-binding Hamiltonian and given as LCAO
/H9278i=/H20858
R/H6023,/H9251cR/H6023/H9251i/H20841R/H6023/H9251/H20856. /H208498/H20850
If we substitute /H9278iin this LCAO form into the formula /H208497/H20850,
we get
Vijkl=/H20858
R1/H6023/H92511/H20858
R2/H6023/H92512/H20858
R3/H6023/H92513/H20858
R4/H6023/H92514cR1/H6023/H92511i/H11569cR2/H6023/H92512j/H11569cR3/H6023/H92513kcR4/H6023/H92514l
/H11003/H20855R1/H6023/H92511,R2/H6023/H92512/H20841e2
/H9280/H20849r1/H6023,r2/H6023/H20850/H20841r1/H6023−r2/H6023/H20841/H20841R3/H6023/H92513,R4/H6023/H92514/H20856./H208499/H20850
In the sums we separate out the onsite terms
/H20849R1/H6023=R2/H6023=R3/H6023=R4/H6023/H20850and use for them the unscreened Coulomb
interaction with dielectric constant /H9280=1. In the remaining
terms we take the Coulomb interaction screened by the bulkdielectric constant. In the derivation we take into account
only two-center contributions /H20849i.e.,R
1/H6023=R4/H6023andR2/H6023=R3/H6023/H20850and
assume that for sites which are far apart the exact structure ofthe localized orbitals is not important
7/H20849i.e., in the integral we
setr1/H6023=R1/H6023andr2/H6023=R2/H6023/H20850. As a result we obtain an approximate
form of Coulomb matrix elements
Vijkl=/H20858
R1/H6023/H20858
R2/H6023/HS11005R1/H6023/H20875/H20858
/H92511cR1/H6023/H92511i/H11569cR1/H6023/H92511l/H20876/H20875/H20858
/H92512cR2/H6023/H92512j/H11569cR2/H6023/H92512k/H20876e2
/H9280/H20841R1/H6023−R2/H6023/H20841
+/H20858
R1/H6023/H20858
/H92511/H92512/H92513/H92514cR1/H6023/H92511i/H11569cR1/H6023/H92512j/H11569cR1/H6023/H92513kcR1/H6023/H92514l
/H11003/H20855R1/H6023/H92511,R1/H6023/H92512/H20841e2
/H20841r1/H6023−r2/H6023/H20841/H20841R1/H6023/H92513,R1/H6023/H92514/H20856. /H2084910/H20850
The first term is the long-range contribution to the two-center
integral built from the monopole interaction of two chargedensities localized at different atomic sites. The second termis the on-site unscreened part, calculated by direct integration
using atomic orbitals. As a first step, following Refs. 8and9
we model the tight-binding orbitals with atomic Slater orbit-als. This approximation does not account for the monopole-dipole and dipole-dipole contributions,
37which will be in-
vestigated in the future in relation with electron-holeexchange interactions.
In the calculation of Coulomb matrix elements we ap-
proximate LCAO basis orbitals by Slater orbitals. Lee et al.
8
investigated different orbitals, including nonorthogonal
Slater and both nonorthogonal and orthogonalized Gaussian-type orbitals. The authors found the dependence of the re-sults on the choice of basis orbitals decreasing quickly withthe increasing dot size. They found that for the radius largerthan /H110151.5–2 nm, and much smaller then dot investigated in
this work, Coulomb interactions can be calculated reliablyusing the simple approximate orbitals as basis.
Dipole matrix element for light polarized along xdirec-
tion are calculated by the following formulas:
7,38
/H20855/H9278e/H20841x/H20841/H9278h/H20856=/H20858
R/H6023/H9251RxcR/H6023/H9251e/H11569cR/H6023/H9251h+/H20858
R/H6023/H9251/H20858
/H9252/HS11005/H9251cR/H6023/H9251e/H11569cR/H6023/H9252h/H20855/H9251/H20841x/H20841/H9252/H20856
+/H20858
R1/H6023/H9251/H20858
R2/H6023/HS11005R1/H6023/H9252cR1/H6023/H9251e/H11569cR2/H6023/H9252h/H20855R1/H6023/H9251/H20841x/H20841R2/H6023/H9252/H20856, /H2084911/H20850
where the first sum gives the “volume” contribution built
from the atomic position dipole moments determined by the
position of atom R/H6023=/H20849Rx,Ry,Rz/H20850. The second term is built
from intra-atomic dipole moments for atomic transitions be-tween orbitals on the same atom and the last term collectscontributions coming from orbitals centered on different at-oms. As in the case of Coulomb matrix element we useSlater orbitals
27to calculate intra-atomic and interatomic di-
pole elements.
IX. MULTIEXCITON HAMILTONIAN
Once the single-particle eigenstates /H9278i, their energies Ei,
and Coulomb matrix elements Vijklare found, the Hamil-
tonian for the interacting electrons and holes can be writtenin second quantization as
1
Hˆex=/H20858
iEieci†ci+/H20858
iEihhi†hi+1
2/H20858
ijklVijkleeci†cj†ckcl
+1
2/H20858
ijklVijklhhhi†hj†hkhl−/H20858
ijklVijkleh,dirci†hj†hkcl
+/H20858
ijklVijkleh,exchgci†hj†ckhl. /H2084912/H20850
We note that this Hamiltonian does include vertex correc-
tions in the form of electron-hole interaction, but self-energycorrections are included indirectly in the electron and holeenergies fitted to experimental transitions of bulk material.The investigation of self-energy correction will be carriedout in the future.
Multiexciton configurations are built from several elec-
tron and hole single-particle states. We take into account sixlowest electron and six hole levels, but each level corre-ATOMISTIC TIGHT-BINDING THEORY OF … PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-5sponds to doublet of two Kramers degenerate states giving a
total of 12 electron and 12 holes states. The multiexcitonwave function is expanded in terms of these configurations,the Hamiltonian matrix constructed in configuration spaceand diagonalized giving energies of ground and excitedstates of multiexciton complexes. As number of possiblemany-body configurations grows factorially with number ofparticles, in exact diagonalization approach we introduce acutoff in number of configurations used in calculation. How-ever, we make sure that analyzed features both in energy andoptical spectra are converged with respect to the number ofconfigurations and single-particle states used in building con-figurations.
Finally, the optical spectra are found by calculating the
intensity of photoluminescence from the recombination ofone electron-hole pair in a N-exciton state using Fermi’s
golden rule
I/H20849
/H9275/H20850=/H20858
f/H20841/H20855f,N−1/H20841P−/H20841i,N/H20856/H208412/H9254/H20849Ei−Ef−/H6036/H9275/H20850, /H2084913/H20850
where /H20841i,N/H20856isith state of the N-exciton system. The operator
P−describes all the possible electron-hole recombination
channels
P−=/H20858
lm/H20855le/H20841/H9280/H6023·r/H6023/H20841mh/H20856clhm, /H2084914/H20850
where /H20855le/H20841/H9280/H6023·r/H6023/H20841mh/H20856is a dipole matrix element calculated from
single-particle tight-binding wave functions for a given po-larization of light
/H9280/H6023.
X. RESULTS
A. Single-particle levels
We present here the results of calculations for a lens-
shaped InAs dot, shown in Fig. 2/H20849a/H20850. The height of the lens is
h=3.5 nm and the base diameter is D=25 nm. We chose
this particular size and shape to be able to compare our re-sults with results of a different atomistic methodology,namely, empirical pseudopotentials /H20849Refs. 15and16/H20850.
Figure 2/H20849b/H20850shows charge distributions and energies cor-
responding to several lowest electron and hole levels. In theatomistic calculation, due to the underlying zincblende lat-tice, the rotational symmetry C
/H11009is reduced to C2v, despite
the cylindrical shape of the dot. As a result, the electronicstates can no longer be labeled as eigenstates of angular mo-mentum L
z, which is not a good quantum number. In prin-
ciple, one should label states by different irreducible repre-sentations of the “crystal+dot” symmetry group, but forclarity and simplicity one can still label states approximatelyas of s,p,o rdcharacter by analysis of their nodal structure
in real space. This approach works very well for electronstates, which can be well described by single-band modeland have well-defined directional and nodal properties.Therefore the ground electron state e
1is of ssymmetry with
no nodal plane /H20849Fig. 2/H20850, while the first and second excited
states /H20849e2ande3/H20850are of psymmetry, with one nodal plane
each. First of the pstates is localized along /H20851110/H20852crystal
direction while the second pis localized perpendicularlyalong the /H2085111/H60180/H20852crystal direction. This localization and elon-
gation, also visible for other states, is due to underlying lat-
tice symmetry and is enhanced further by strain effects.
The next three excited electron states have a more com-
plicated nodal structure. Two of them, e4and e5, can be
denoted as dx2−y2anddxyas they have two nodal planes on
thexyplane. Above these states of dsymmetry there is an e6
state of approximate d/H20849or 2s/H20850symmetry. This state has one
node along radial coordinate, thus index 2 in contrast to thenodeless 1 sstate. In the effective-mass model with parabolic
confinement the state corresponding to e
6would be acci-
dently degenerate with the dlevels, forming a shell, while in
atomistic calculation for self-assembled quantum dots all de-generacies /H20849apart from Kramers /H20850are removed.
The effective gap between electron and hole ground levels
is 0.7797 eV, which is a reasonable value for pure /H20849nonal-
loyed /H20850InAs dot. Spacing between ground /H20849s-type /H20850and first
excited /H20849p-type /H20850electron level is 52.7 meV, while the split-
ting of first and second excited /H20849p-type /H20850states is 1.28 meV.
Spacing between the higher of two p-states and the lowest of
1.201.251.301.35Energy (eV)da)
b)25 nm
3.5 nm
0.6 nm
0.400.420.44
sp
6 8 10 12 14772774776778780Egap=E1-H1(meV)
Buffer hei ght(nm)c)
FIG. 2. /H20849Color online /H20850/H20849a/H20850Lens-shaped quantum dot embedded
in GaAs /H20849only indium and arsenic atoms are shown /H20850./H20849b/H20850Electron
/H20849right /H20850and hole /H20849left/H20850probability density isosurfaces and energies
for dot /H20849a/H20850./H20849c/H20850Single-particle effective gap Egap=E1−H1as a func-
tion tight-binding computational domain height for dot /H20849a/H20850.ZIELI ŃSKI, KORKUSI ŃSKI, AND HAWRYLAK PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-6dstates is 51.7 meV, and is very similar to that between
ground and first excited states, thus very close to the predic-tion given by harmonic oscillator model, with the overallshell structure being very well preserved.
However, the structure of hole levels is different then
those of electrons, more complicated due to the mixing ofangular momenta and anisotropic effect of strain. Surpris-ingly, the hole energy levels cannot be grouped into quaside-generate shells. Yet, the structure of lower hole levels chargedensities is similar to those of electrons, with the ground holeh
1state of ssymmetry and the two first excited states of
approximate psymmetry. Note that the first excited hole
state h2is elongated along /H2085111/H60180/H20852, perpendicular to the elon-
gation axis of the first excited electron level e2.
The energy difference between the ground and excited
hole states is 14.4 meV, approximately one third of the cor-responding value for electrons. Smaller spacings betweenhole levels in self-assembled quantum dots can be attributedto higher effective mass of holes. On the contrary, the split-ting between p-type excited hole states h
2andh3is 9.8 meV,
is much larger then for electrons and comparable with energydifference between sandp. Large splitting of the plevels
can be understood in terms of different biaxial strain contri-bution along /H20851110/H20852and /H2085111/H60180/H20852axes. It affects holes only as
electrons, built predominantly from atomic sorbitals, are vir-
tually immune, at least in the simplest Bir-Pikus model, tobiaxial strain.
The higher lying state h
4does not show a clear symmetry
character, with spacing between h3andh4equal only 5.5
meV. However, two higher lying states h5andh6are well
separated /H2084913.5 meV /H20850from h4and form a well-defined dou-
blet with small, 1.8 meV, splitting. While the shell structureof holes is not visible in lens-shaped dots, Indium flushtechnique,
39which creates thinner, more disklike dots leads
to a hole shell structure11observed experimentally.40
B. Dependence of electron levels on valence-band offset and
deformation potential
Figure 3shows energies corresponding to several lowest
electron /H20849blue /H20850and hole /H20849red/H20850levels obtained for different
values of InAs/GaAs valence-band offset /H20849VBO /H20850, different
absolute InAs /H20849GaAs /H20850valence-band deformation potential av,
and different models, BP or DFT, used in fitting tight-bindingHamiltonian to reproduce strained bulk band edges.
Different choice of VBO results in a simultaneous shift of
the depth of both strained and unstrained electron and holeconfining potentials. The bigger the VBO the deeper the wellfor holes and the shallower the well for electrons. However,the energy difference between the top of the valence bandand the bottom of the conduction band remains unchangedand equals bulk InAs band gap in the unstrained case. Thisresults in a very small change, /H110152%, of the effective QD
e
1−h1gap when going from VBO=210 to 60 meV. Simi-
larly, the different choice of avdoes not lead to significant
change of the effective gap as overall InAs band-gap defor-mation potential a
gap=ac−av=−6.08 eV is kept fixed for
different values of av. The only notable difference in the gap
prediction between Bir-Pikus and DFT models can be attrib-uted to overestimation of conduction-band energies under bi-
axial strain, as mentioned previously. This difference empha-sizes the importance of the proper modeling of strain inatomistic calculations and leaves open door for future re-search.
As mentioned above, a
gapdoes not depend on the choice
ofav, however, different choice of avresults in different
value of absolute conduction-band deformation potential be-cause a
c=agap+av, i.e., ac=−5.08 or −7.08 eV, resulting in
different energies of the ground electron level for differenta
c. However, we note that general structure of electron levels
remains fairly unchanged with well-pronounced shell struc-ture and ground-first-excited-states spacing /H1101550 meV for
different choice of input parameters.
We also note here that structure of probability densities of
confined electron states is very stable with respect to thechoice of VBO and a
vparameters. For all cases considered
in this work their charge density looks exactly the same as inFig.2, with the same nodal structure, elongation directions,
and the spatial extent.
C. Dependence of valence hole levels on valence-band offset,
deformation potential, and methodology
The detailed structure of hole states differs significantly
with a different choice of input parameters. Figure 4shows
detailed probability density isosurfaces and energies forholes, calculated for different values of InAs/GaAs VBO,different absolute valence-band deformation potential a
vand
different fitting targets, BP or DFT. The ground hole statesav(eV) 1.0 -1.0 -1.0 -0.88 -0.88StrainBP BP BP DFT DFT
Fit
1.201.251.301.35gy (eV)1.351.401.45
1.201.251.30
1.251.301.35
1.101.151.20VBO 210 210 60 210 60
(meV)
0.260.281.15Energy
0.400.420.441.30
0.260.281.15
0.420.440.461.20
0.260.280.301.05
E1-H1(eV) 0.9136 0.9073 0.9169 0.7797 0.7969
0.00.20.40.60.8Eg=E1-H1 (eV)
FIG. 3. /H20849Color online /H20850Electron /H20849blue/upper/dark gray /H20850and hole
/H20849red/lower/light gray /H20850single-particle energies calculated for differ-
ent values of InAs/GaAs VBO, different absolute /H20849InAs /H20850valence-
band deformation potential avand different models, BP or DFT,
used in fitting tight-binding Hamiltonian to reproduce strained bulkband edges.ATOMISTIC TIGHT-BINDING THEORY OF … PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-7remain of approximately stype, however, with different level
of elongation depending on the model. Also, the first excitedstate is of psymmetry in all cases, localized along /H2085111/H60180/H20852.
Higher lying states change their character and energy
spacing depending on the choice of parameters, making italmost impossible to address the details of hole level struc-ture without a better understanding of hole bulk properties:VBO and a
v. As these parameters are input to any atomistic
calculation, we believe that similar uncertainties are presentin results obtained by other authors.
To illustrate the problem, we show in Fig. 5results of
calculation for the same dot obtained with effective bondorbital method /H20849EBOM /H20850,
41tight-binding /H20849TB/H20850model with in-
put parameters chosen similar to the ones used in empiricalpseudopotential calculation /H20849EMP1 /H20850from Ref. 15. Another
empirical pseudopotential calculation /H20849EMP2 /H20850is shown for
comparison.
16
As expected, the structure of electron states is similar in
all cases, however, hole states differ significantly. EBOMmaintains approximately a shell-like structure of holelevels. This differs from both TB and EMP and canbe attributed to the replacement of zincblende with cubicsymmetry in EBOM. There is a good agreement betweenTB and EMP1 calculation, with a characteristic
“large/large/small” level spacing between subsequent holelevels h
1−h2/H11015h2−h3/H11271h3−h4.
Surprisingly, two pseudopotential /H20849EMP1/EMP2 /H20850calcula-
tions predict different details of hole levels, most likely dueto slightly different choice of fitting parameters or fittingresults in pseudopotential fitting procedure. Figure 5illus-
trates our point on importance of the choice of proper param-etrization for all empirical atomistic calculations.
D. Single-particle optical properties
Symmetries of single-particle states directly influence
quantum dot optical properties via the dipole moment matrixelements. Figure 6shows joint optical density of states
/H20849single-particle absorption spectrum /H20850calculated for light po-
larized along xaxis /H20849/H20851100/H20852crystal direction /H20850, i.e., /H20855
/H9274el/H20841x/H20841/H9278ho/H20856.
We observe three main groups of peaks corresponding to
transitions between states from shells of similar symmetry,av 1.0 eV -1.0 eV -1.0 eV -0.88 eV -0.88 eVS t r a i n B PB P B PD F TD F T
h1
h2
h4
h3
VBO 0.21 eV 0.21 eV 0.06 eV 0.21 eV 0.06 eV
0.260.280.30
0.420.440.46
0.240.260.28
0.400.420.44
0.260.280.30
h6
h5
h1 h1 h1 h1 h1Energies (eV)
FIG. 4. /H20849Color online /H20850Hole probability density isosurfaces and
energies calculated for different values of InAs/GaAs VBO, differ-ent absolute valence-band deformation potential a
vand different
models, BP or DFT, used in fitting tight-binding Hamiltonian toreproduce strained bulk band edges.1.251.301.35(eV)EBOM TB EMP1 EMP2
12 01.251.301.35
12 51.301.351.40
1.201.251.30
0.240.260.281.201.25Energy
0.180.200.221.20
0.160.180.201.25
0.260.281.15
FIG. 5. /H20849Color online /H20850Electron /H20849blue/upper/dark gray /H20850and hole
/H20849red/lower/light gray /H20850single-particle energies calculated for EBOM,
TB model with parameters chosen similar to EMP1 from Ref. 15.
Another EMP2 is shown for comparison from Ref. 16.
Se(arb. units )X polarization
Z polarization (x5)
P
0.80 0.84 0.88 0.92DAmplitud e
Energy (eV)
FIG. 6. /H20849Color online /H20850Joint optical density of states /H20849single-
particle absorption spectra /H20850calculated for two light polarize along x
/H20849red/light gray /H20850andz/H20849blue/dark gray /H20850axes.ZIELI ŃSKI, KORKUSI ŃSKI, AND HAWRYLAK PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-8i.e.,s-s,p-p, and d-d, with spacings among groups deter-
mined by spacing between s,p, and delectronic shells. How-
ever, as single-particle hole states are not purely s,p,o rd,
but rather of mixed angular momentum character, we alsoobserve additional lines, not present in predictions of single-band effective-mass approximation. Figure 6shows also that
contribution from light polarized along zaxis /H20849/H20851001/H20852/H20850, i.e.,
growth direction, is negligible comparing to in-plane contri-bution, reflecting dominant confinement in zdirection.
The atomistic character of underlying lattice, which re-
sults in inequivalency of crystal directions /H20851110/H20852and /H2085111/H60180/H20852
can also be observed in the optical spectrum. To quantify thiseffect we calculate the polarization ratio defined as
/H9261=P
/H20851110/H20852
P/H2085111/H60180/H20852=/H20855/H9274e1/H20841r/H20851110/H20852/H20841/H9274h1/H20856
/H20855/H9274e1/H20841r/H2085111/H60180/H20852/H20841/H9274h1/H20856. /H2084915/H20850
This ratio measures the difference in optical matrix elements
for light polarization along different optical axis. For thesystem analyzed here, this ratio is equal 0.9 for the transitionbetween ground electron and hole states. This is differentfrom 1.0 for a system with full cylindrical symmetry.
As mentioned above, well-pronounced character of elec-
tron states and their stability with respect to different param-eterizations enables us to label/classify hole states accordingto optical transitions to different electron states. Electronstates e
2ande3are of well-defined pcharacter and we label
them as p1andp2correspondingly. Similarly we label e4,e5,
ande6asd1,d2, and d3, respectively.Figure 7shows the comparison between joint optical den-
sity of states calculated for two different values of InAs/GaAs VBO: /H20849a/H20850VBO=210 meV and /H20849b/H20850VBO=60 meV. In
both cases DFT model was used as a fitting target to describethe biaxial strain evolution with a
v=−0.88 as derived from
DFT.
The overall blueshift in the VBO=60 meV case corre-
sponds to larger effective gap. In both cases the characteristicsplitting within the pshell reflects the splitting of the hole p
shell /H20849h
3−h2/H1101510–12 meV /H20850.
For VBO=210 meV allowed transitions occur only
within a given shell /H20849stos,ptop, etc /H20850, which again allows
us to classify h2andh3states as of pcharacter and higher
lying states as of mixed dcharacter.
However, for VBO=60 meV, apart from the ptoptran-
sitions there are noticeable p1-d1andp2-d1lines correspond-
ing to transitions from electronic pshell to hole d1/H20849h4/H20850state
and from electronic dshell to hole p2/H20849h3/H20850state revealing
mixed p/dcharacter of h3/h4states. This hybridization of
states can also be expected from energy spectra, as these twostates are very close in energy /H20849Fig. 4/H20850. Notice that mixed
character of states, well pronounced in JDOS, is not clearlyvisible in the charge-density distribution on Fig. 7, which
emphasizes the usefulness of JDOS as a way of labeling QDstates.
E. Many-body properties
Next we turn to calculating the many-body spectrum of
the quantum dot.
1. Excitonic absorption spectrum
Figure 8shows exciton absorption spectra calculated
for two different values of InAs/GaAs VBO, /H20849a/H20850
VBO=210 meV and /H20849b/H20850VBO=60 meV, using different
levels of approximation in the many-body calculation.
First, in the single-particle /H20849SP/H20850picture all interactions
between electron and hole states are neglected, SP being thusthe joint optical density of states. Then, in the Hartree-Fock/H20849HF/H20850approximation only diagonal matrix elements of
electron-hole Hamiltonian are included, corresponding effec-tively to a perturbative treatment of many-body effects. Fi-nally, mixing between different configurations by Coulombscattering is taken into account in a full configuration-interaction /H20849CI/H20850treatment.
First noticeable difference between different approxima-
tions is overall shift toward lower energies when going from/H20849a/H20850single particle to /H20849b/H20850“Hartree-Fock” case. This redshift is
due to the attractive electron-hole Coulomb interaction equal33.6 and 33.3 meV for VBO=210 and 60 meV correspond-ingly. Note that despite the 2% difference in the single-particle energy gap between two VBO cases the electron-hole Coulomb integral is almost identical.
The energy corresponding to the ground state of
exciton is 746.1 meV /H20849VBO=210 meV /H20850and 763.5 meV
/H20849VBO=60 meV /H20850and it is redshifted from the single-particle
gap by electron-hole Coulomb interaction 33.6 and 33.3meV for VBO=210 and 60 meV, respectively. In the CI case/H20849c/H20850the ground-state energy is further redshifted by the cor-0.80 0.84 0.88 0.92 0.9 6p2-p1
d1-d2d2-d2d1-d1p2-p1s-s
p1-p2
p1-p1p2-p2DSAmplitude (arb. units)
Energy (eV)P
d2-d1VBO=210meV a)
p2-p1
0.80 0.84 0.88 0.92 0.96s-d3d1-d1
d1-p2d2-d3
d3-sp2-p1s-s
p1-p2
p2-p2p1-p1 D-DSAmplitude (arb. units)
Energy (eV)P-P
d2-d1p1-d1
p2-d1VBO=60meV b)
p2-p1
FIG. 7. /H20849Color online /H20850Joint optical density of states /H20849single-
particle absorption spectra /H20850calculated for two different values of
InAs/GaAs VBO /H20849a/H20850VBO=210 meV and /H20849b/H20850VBO=60 meV.ATOMISTIC TIGHT-BINDING THEORY OF … PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-9rection due to correlation effects /H20849/H110151 meV /H20850.
Figure 8also shows a quite significant difference in rela-
tive heights of absorption lines, when going from /H20849a/H20850to/H20849c/H20850.
Surprisingly, some of the noticeable differences betweenVBO=210 and 60 meV visible in initial single-particle prop-erties are smeared out by interactions in the final, fully cor-related calculation. These two observations stress out the ne-cessity of full many-body treatment of excitonic effects inquantum dots. Interestingly, the excitonic absorption spectracomprising of a single “ s” shell and dominant “ p” shell ab-
sorption peaks seem to compare very well with absorptionspectra predicted from the effective-mass model.
42
2. Multiexciton complexes
Finally, we analyze the spectrum of multiexciton com-
plexes. As already mentioned, the lens-type quantum dotshows only quasidegenerate electronic shells for the electronstates with hole states being more complicated, without aclear shell structure. However, as we could see above in theexciton absorption spectra, the optical properties of QDs in-cluding many-body effects are still dominated by the single-
particle spacings of the electron states s,p, etc.
Figure 9shows the emission spectra from the multiexci-
ton complexes as a function of the number of excitons,showing filling of quasidegenerate shells. In other words,Fig.9presents recombination spectra of an electron-hole pair
in the presence of several other electron-hole pairs. Namely,for the 3 Xcomplex we show emission spectra of the
electron-hole pair recombing in the presence of two otherelectron-hole pairs, etc. We see well-defined groups of peaksbelonging to s,p, and dapproximate shells. We see also that
the emission from the s,p, and dshells is not a sensitive
function of the number of excitons as postulated by “hiddensymmetry” arguments.
43–45The small structure of the pshell
emission /H20849for 3 Xand 4 X/H20850is related to the total spin and
scattering to higher shells, while more complicated pshell
emission for the 5 Xand 6 Xcomplexes is related to the split-
ting of the hole pstates. The associated emission structure in
thesshell energy range is not related to hidden symmetries
and is a sensitive function of the filling of the shell, in agree-ment with results obtained using the effective-mass harmonicoscillator states.
44
Figure 9/H20849b/H20850shows the exciton charging spectrum, defined
as the energy needed to add one exciton to a system already0.760 .80 0 .840.88 0 .92 0 .96DSAmplitude (arb .units)
Energy (e V)P
Sarb.units)0.760 .80 0 .840.88 0 .92 0 .96DSAmplitude (arb .units)
Energy (e V)P
Sarb.units)VBO=210 meV
SP SP
HF HFa) VBO=60 meVb)
0.760 .80 0 .840.88 0 .92 0 .96DAmplitude ( a
Energy (e V)P
0.760 .80 0 .840.88 0 .92 0 .96DAmplitude ( a
Energy (e V)P
CI CI
0.760 .80 0 .840.88 0 .92 0 .96DSAmplitude (arb .units)
Energy (e V)P
0.760 .80 0 .840.88 0 .92 0 .96DSAmplitude(arb.units)
Energy (e V)P
FIG. 8. Exciton absorption spectra calculated for two different
values of InAs/GaAs VBO: /H20849a/H20850VBO=210 meV and /H20849b/H20850
VBO=60 meV using different level of approximation in many-body calculation. First in SP picture all interactions between elec-tron and hole are neglected, next in HF picture, only diagonal ma-trix elements of electron-hole Hamiltonian are included, finallymixing between different configurations /H20849off diagonal, i.e., Cou-
lomb scattering terms /H20850is taken into account in full CI picture.
a)
MULTIEXCITO/CID49SPECTRA
BLACKVBO=210meV,REDVBO=60meVIntensity( arb.units)
b)
123456789 1 00.750.800.850.90TB (VBO = 210 meV)
EBOM (shifted -0.2eV)ds h e l l
ps h e l lChargingenergy(eV )
/CID49umberofexcitonsss h e l lEnergy(eV )
FIG. 9. /H20849Color online /H20850/H20849a/H20850Multiexciton emission spectra calcu-
lated for two different values of InAs/GaAs VBO:VBO=210 meV /H20849black /H20850and VBO=60 meV /H20849red, light gray /H20850,/H20849b/H20850
exciton charging energy as function of number of excitons.ZIELI ŃSKI, KORKUSI ŃSKI, AND HAWRYLAK PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-10containing Nexcitons. Despite the complicated structure of
hole levels, the charging spectra also reveal well-definedsteps corresponding to filling of subsequent shells. However,a comparison with EBOM /H20849Ref. 5/H20850/H20849shifted by −0.2 eV to
match the energy of ground the exciton states /H20850shows a small
step for the case of 5 Xand 9 Xdue to the splitting of the hole
panddstates.
When the fifth /H20849ninth /H20850hole is added to the system it has to
occupy the second of the pstates /H20849third of the dstates /H20850due to
the Pauli exclusion principle. Due to the splitting in holespectrum this state has a different single-particle energy re-sulting in a step in charging spectrum.
XI. CONCLUSIONS
We presented here an atomistic sp3d5s/H11569tight-binding
theory of electronic structure and optical properties of InAs/GaAs self-assembled quantum dots. The atomistic theory in-cludes zincblende symmetry, faceting, and atomic orbitalsaccounting for interband and intervalley couplings. The equi-librium position of atoms is calculated using the valenceforce field /H20849VFF /H20850method and modification of the tight-
binding Hamiltonian due to strain is accounted for usingHarrison’s law. The electronic and optical properties of mul-tiexciton complexes are determined by solving the many-body Hamiltonian for interacting electrons and holes usingthe configuration-interaction approach. The methodology isapplied to an InAs/GaAs lens-shaped quantum dot. The de-pendence of calculated electron and hole electronic states onthe InAs/GaAs valence-band offsets and InAs absolutevalence-band deformation potentials is described. It is shown
that the electron levels are well described by the effective-mass harmonic oscillator model. The hole levels are foundnot to have a well-defined shell structure, and their levels arefound to be sensitive to the choice of the valence-band offsetand valence-band dependence on strain, i.e., parameterswhich are not well known. Given the set of bulk parameters,the reliability of the atomistic calculations was positivelyassessed by comparison with results of the empirical pseudo-potentials method and effective bond orbital method. Futurework will address the structure of valence holes by compari-son with available experiments. In the end, this will establishtight-binding methodology as a useful tool in designingsemiconductor nanostructures starting with their atomic con-stituents and ending with their electronic and optical proper-ties.
ACKNOWLEDGMENTS
The authors acknowledge useful discussions with G. Bry-
ant, H. Guo, and G. Klimeck and support from the CanadianInstitute for Advanced Research, QuantumWorks, NRC-NSERC-BDC Nanotechnology Projects and NSERC. M.Zielinski acknowledges the financial support of the Futureand Emerging Technologies /H20849FET /H20850programme within the
Seventh Framework Programme for Research of the Euro-pean Commission, under the FET-Open grant agreementCORNER No. FP7–ICT-213681. We thank W. Sheng forproviding the results of EBOM calculations and E. Kadant-sev for DFT simulations.
*Present address: Instytut Fizyki UMK, Grudzi ądzka 5, 87-100 To-
ruń, Poland.
1P. Hawrylak and M. Korkusinski, in Single Quantum Dots: Fun-
damentals, Applications, and New Concepts , Topics in Applied
Physics Vol. 90, edited by P. Michler /H20849Springer, New York, 2003.
2L. Jacak, P. Hawrylak, and A. Wojs, Quantum Dots /H20849Springer,
Berlin, 1998 /H20850.
3D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot
Heterostructures /H20849Wiley, New York, 1998 /H20850.
4C. D. Spataru, S. Ismail-Beigi, L. X. Benedict, and S. G. Louie,
Phys. Rev. Lett. 92, 077402 /H208492004 /H20850.
5W. Sheng, S.-J. Cheng, and P. Hawrylak, Phys. Rev. B 71,
035316 /H208492005 /H20850.
6W. Sheng and P. Hawrylak, Phys. Rev. B 72, 035326 /H208492005 /H20850.
7S. Schulz, S. Schumacher, and G. Czycholl, Phys. Rev. B 73,
245327 /H208492006 /H20850.
8S. Lee, L. Jönsson, J. W. Wilkins, G. W. Bryant, and G. Klimeck,
Phys. Rev. B 63, 195318 /H208492001 /H20850.
9K. Leung and K. B. Whaley, Phys. Rev. B 56, 7455 /H208491997 /H20850.
10R. Santoprete, Belita Koiller, R. B. Capaz, P. Kratzer, Q. K. K.
Liu, and M. Scheffler Phys. Rev. B 68, 235311 /H208492003 /H20850.
11M. Korkusinski, M. Zielinski, and P. Hawrylak, J. Appl. Phys.
105, 122406 /H208492009 /H20850.
12W. Jaskólski, M. Zieli ński, G. W. Bryant, and J. Aizpurua, Phys.
Rev. B 74, 195339 /H208492006 /H20850.13A. Canning, L. W. Wang, A. Williamson, and A. Zunger, J. Com-
put. Phys. 160,2 9 /H208492000 /H20850.
14G. Bester and A. Zunger, Phys. Rev. B 71, 045318 /H208492005 /H20850.
15L. He and A. Zunger, Phys. Rev. B 73, 115324 /H208492006 /H20850.
16A. J. Williamson, L. W. Wang, and A. Zunger, Phys. Rev. B 62,
12963 /H208492000 /H20850.
17G. Klimeck, R. C. Bowen, T. B. Boykin, and T. A. Cwik, Super-
lattices Microstruct. 27, 519 /H208492000 /H20850.
18T. B. Boykin, G. Klimeck, R. C. Bowen, and F. Oyafuso, Phys.
Rev. B 66, 125207 /H208492002 /H20850.
19P. R. C. Kent, G. L. W. Hart, and A. Zunger, Appl. Phys. Lett.
81, 4377 /H208492002 /H20850.
20C. Pryor, J. Kim, L. W. Wang, A. J. Williamson, and A. Zunger,
J. Appl. Phys. 83, 2548 /H208491998 /H20850; M. Tadi ć, F. M. Peeters, K. L.
Janssens, M. Korkusinski, and P. Hawrylak, ibid. 92, 5819
/H208492002 /H20850.
21P. N. Keating, Phys. Rev. 145, 637 /H208491966 /H20850; R. M. Martin, Phys.
Rev. B 1, 4005 /H208491970 /H20850.
22S. Lee, F. Oyafuso, P. von Allmen, and G. Klimeck, Phys. Rev. B
69, 045316 /H208492004 /H20850.
23D. J. Chadi, Phys. Rev. B 16, 790 /H208491977 /H20850.
24J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 /H208491954 /H20850.
25P. Vogl, H. P. Hjalmarson, and J. D. Dow, J. Phys. Chem. Solids
44, 365 /H208491983 /H20850.
26J. M. Jancu, R. Scholz, F. Beltram, and F. Bassani, Phys. Rev. BATOMISTIC TIGHT-BINDING THEORY OF … PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-1157, 6493 /H208491998 /H20850.
27J. G. Díaz and G. W. Bryant, Phys. Rev. B 73, 075329 /H208492006 /H20850.
28W. A. Harrison, Electronic Structure and the Properties of Solids
/H20849Freeman, New York, 1980 /H20850.
29G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects
in Semiconductors /H20849Wiley, New York, 1974 /H20850.
30S. H. Wei and A. Zunger, Phys. Rev. B 60, 5404 /H208491999 /H20850.
31Y. H. Li, X. G. Gong, and S. H. Wei, Phys. Rev. B 73, 245206
/H208492006 /H20850.
32Chris G. Van de Walle, Phys. Rev. B 39, 1871 /H208491989 /H20850.
33E. Kadantsev /H20849unpublished /H20850.
34M. Korkusinski and M. Zielinski /H20849unpublished /H20850.
35I. Vurgaftman, J. R. Meyer, and W. L. R. Ram-Mohan, Appl.
Phys. Rev. 89, 5815 /H208492001 /H20850.
36C. G. Van de Walle and J. Neugebauer, Nature /H20849London /H20850423,
626 /H208492003 /H20850.37S. V. Goupalov and E. L. Ivchenko, Phys. Solid State 43, 1867
/H208492001 /H20850.
38G. W. Bryant and W. Jaskólski, Phys. Rev. B 67, 205320 /H208492003 /H20850.
39Z. R. Wasilewski, S. Fafard, and J. P. McCaffrey, J. Cryst.
Growth 201-202 , 1131 /H208491999 /H20850.
40S. Raymond, S. Studenikin, A. Sachrajda, Z. Wasilewski, S. J.
Cheng, W. Sheng, P. Hawrylak, A. Babinski, M. Potemski, G.Ortner, and M. Bayer, Phys. Rev. Lett. 92, 187402 /H208492004 /H20850.
41W. Sheng /H20849unpublished /H20850.
42P. Hawrylak, G. A. Narvaez, M. Bayer, and A. Forchel, Phys.
Rev. Lett. 85, 389 /H208492000 /H20850.
43A. Wojs and P. Hawrylak, Solid State Commun. 100, 487
/H208491996 /H20850.
44P. Hawrylak, Phys. Rev. B 60, 5597 /H208491999 /H20850.
45P. Hawrylak, Solid State Commun. 127, 753 /H208492003 /H20850.ZIELI ŃSKI, KORKUSI ŃSKI, AND HAWRYLAK PHYSICAL REVIEW B 81, 085301 /H208492010 /H20850
085301-12 |
PhysRevB.74.115103.pdf | Excited and ground state properties of LaSrMnO 4: A combined x-ray spectroscopic study
K. Kuepper,1,2,*R. Klingeler,3P. Reutler,3B. Büchner,3and M. Neumann2
1Forschungszentrum Rossendorf, Institute of Ion Beam Physics and Materials Research, P .O. Box 51 01 19, D-01314 Dresden, Germany
2Department of Physics, University of Osnabrück, Barbarastrasse 7, D-49069 Osnabrück, Germany
3Institute for Solid State Research IFW Dresden, P .O. Box 27 01 16, D-01171 Dresden, Germany
/H20849Received 18 February 2006; revised manuscript received 21 June 2006; published 5 September 2006 /H20850
The electronic properties of the parent compound of the single layered manganites La 1−xSr1+xMnO 4, namely
LaSrMnO 4have been investigated in a detailed spectroscopic study. We apply different complementary x-ray
spectroscopic techniques in the soft x-ray regime. X-ray photoelectron spectroscopy and normal-like x-rayemission spectroscopy were used to reveal a detailed picture of the total and partial densities of states in thiscompound. Furthermore we apply resonant x-ray emission spectroscopy to the Mn L
2,3edges. The spectra
exhibit a rich multiplet structure. Resonant x-ray emission spectroscopy at the O Kedge is used to study the
local partial densities of states of the in plane and out of plane oxygen atoms. The results are discussed alongavailable band structure calculations as well as charge transfer multiplet calculations.
DOI: 10.1103/PhysRevB.74.115103 PACS number /H20849s/H20850: 71.20. /H11002b, 75.47.Lx, 78.70.En, 79.60. /H11002i
I. INTRODUCTION
The perovskite manganites /H20849La,Sr /H20850n+1Mn nO3n+1display a
remarkably rich phase diagram as a function of temperature,
magnetic field, and doping that is due to the intricate inter-play of charge, spin, orbital, and lattice degrees of freedom.This competition of different phases on the atomic scale hasbeen the subject of many studies during the last years.
1–4In
particular, the cubic manganites La 1−xSrxMnO 3and the bilay-
ered compound La 2−2xSr1+2xMn 2O7/H20849n=2/H20850show colossal
magnetoresistance /H20849CMR /H20850.1,2,5,6In the last few years the
discussion of the orbital degree of freedom is reopened forboth three dimensional compounds such as La
7/8Sr1/8MnO 3
and the single layered compound La 0.5Sr1.5MnO 4.7–11
However, the parent compound of the single layered man-
ganites, namely LaSrMnO 4has attracted much less attention
up to now. LaSrMnO 4crystallizes in the K 2NiF 4structure.
The oxygen octahedra are elongated along the caxis, result-
ing in a strongly anisotropic crystal field /H20849CF/H20850. Thus, in
LaSrMnO 4, the CF favors a ferro-orbital ordering of d3z2−r2
orbitals, which was recently studied theoretically,12,13and
also experimentally by means of x-ray linear dichroism at theMnLedge.
9Compared to the cubic manganites LaSrMnO 4
also shows different magnetic properties. LaSrMnO 4is an
antiferromagnetic /H20849AFM /H20850insulator below TN=130 K. Ac-
cording to the Goodenough-Kanamori-Anderson /H20849GKA /H20850
rules, C-type AFM spin order is also realized.11Up to now
experimental investigations of the thermal expansion coeffi-cient, the magnetic and the crystal structure by means ofdifferent diffraction, dilatometry and resonance techniqueshave been reported.
10,11,14,15
As to the electronic structure of LaSrMnO 4, Park carried
outfirst-principles electronic structure calculations, and Wu
et al. performed x-ray absorption spectroscopy.12,13Very re-
cently results of a comprehensive study by means of ex-tended x-ray absorption fine structure /H20849EXAFS /H20850at the Mn K
edge have been published.
16Besides XANES /H20849x-ray absorp-
tion near edge structure /H20850and EXAFS, the methods of x-ray
emission spectroscopy /H20849XES /H20850and x-ray photoelectron spec-
troscopy /H20849XPS /H20850provide a tool of unique precision for theinvestigation of the spatial distribution of the electron den-
sity. Very recently, Kuepper et al. have performed an elec-
tronic structure study by means of XPS and XES /H20849Ref. 17/H20850.
Element and site specific resonant x-ray emission spec-
troscopy /H20849RXES /H2085018,19is another complementary powerful
tool to investigate the electronic structure of transition metalcompounds. The possibilities in the study of correlated sys-tems by means of RXES go from Coulomb interactions onhigh energy scales over charge transfer excitations to lowerexcitation energies, especially regarding optical weakly ac-cessible bands, such as ddtransitions.
20–22RXE spectra,
which are obtained by tuning the excitation energy of theincoming photons across a corresponding x-ray absorptionspectroscopy /H20849XAS /H20850spectrum often comprise two different
components. On the one hand the so-called resonant inelasticx-ray scattering /H20849RIXS /H20850features often reflect the local elec-
tronic structure as ddtransitions or charge transfer excita-
tions in correlated electron systems. In RIXS, an incidentphoton is inelastically scattered and a photon of lower en-
ergy is detected with respect to that of elastically scattered
photons. On the other hand ordinary fluorescence is also de-tected at constant photon energies. This process is oftennormal x-ray emission spectroscopy /H20849NXES /H20850and represents
to a large extent the partial densities of states /H20849PDOS /H20850.A
number of RXES studies have been carried out regarding thecubic CMR manganites. Kurmaev et al. investigated
Pr
0.5Sr0.5MnO 3by means of RXES at the Mn Ledge. Later
Butorin et al. correlated the multiplet structure of the Mn3+
sublattice of La 0.5Ca0.5MnO 3with charge transfer multiplet
calculations.23More systematic investigations over a series
of samples with different dopant concentrations have beenperformed on La
1−xNaxMnO 3and La 1−xBaxMn 1−yTM yO3
/H20849TM=Co,Ni /H20850.24,25As to the layered manganites only the
double layered La 1.2Sr1.8Mn 2O7has been probed by resonant
x-ray emission at the Mn Ledge.26On the other hand RXES
on the O Kedge is an appropriate tool to study the local
partial densities of states in anisotropic materials likeNaV
2O5or Sr 2RuO 4/H20849Refs. 27and28/H20850. In the present case
we study the site specific contributions of the inequivalent inplane /H20849O1/H20850and out of plane /H20849O2/H20850oxygen atoms to the O KPHYSICAL REVIEW B 74, 115103 /H208492006 /H20850
1098-0121/2006/74 /H2084911/H20850/115103 /H208497/H20850 ©2006 The American Physical Society 115103-1RXE spectra in dependence of the excitation energy along
the O Kabsorption edge.
We present here a detailed experimental picture of the
electronic properties of LaSrMnO 4, obtained by applying a
number of complementary x-ray spectroscopic techniques,namely XPS, /H20849R/H20850XES, and XAS in the soft x-ray regime. 3 d
transition metal compounds show strong interactions be-tween the 2 pand the 3 delectrons, and hence 2 pcore level
spectra are dominated by the interactions between the 2 p
hole with the valence electrons, i.e., multiplet effects. This isthe case for the Mn L
2,3XAS, and also to a large extent, for
the Mn 2 p→3dRXES process. Whereas transition metal
2p→3dRXES spectra with excitation energies below or
close to the TM 2 p3/2threshold largely reflect the local elec-
tronic structure of the TMO 6cluster, bandlike features be-
come more important if the excitation energy is set to justabove the TM L
3absorption maximum since the continuum
excited normal fluorescence becomes more dominant. Theinterested reader is, e.g., referred to the review of de Groot.
29
In contrast the core hole interaction between O 1 sand O 2 p
is negligible compared to the core level spin orbit couplingand consequently multiplet effects have no significant influ-ence on the oxygen spectra. These can be described in theframework of electronic structure calculations based uponthe so-called one-electron approximation. In the presentwork we compare our experimental x-ray spectroscopic stud-ies with both available ab initio band structure calculations
as well as full multiplet calculations.
13,23,30
II. EXPERIMENTAL DETAILS
A high quality LaSrMnO 4single crystal was grown by the
floating zone method; the details are described by Reutler et
al.31
The XAS, XES, and RXES data experiments on
LaSrMnO 4were performed at room temperature at beamline
8.0.1 at the Advanced Light Source, Berkeley, CaliforniaUSA, using the x-ray fluorescence end station of the Univer-sity of Tennessee at Knoxville.
32Linearly polarized light
with polarization in the horizontal plane was incident on thesample whose surface was in the vertical plane. Emissionwas measured along the electric vector of the incident lightin the horizontal plane, that is, at a scattering angle of 90°.This geometry minimizes diffuse elastic scattering from thesurface, since the Brewster angle in the soft x-ray range is
usually very close to 45° so that the reflectivity for p
/H6023light is
very close to zero. The light was incident at 30° to thesample normal. Photons with an energy of 500–700 eV areprovided to the end station via a spherical 925 lines/mmgrating monochromator. The Mn 3 d→2pand O 2 p→1s
RXE spectra were obtained with a 1500 lines/mm, 10 meterradius grating. The excitation energies for the NXE spectrawere set to 641.8 eV for the Mn L
3edge and to 550 eV for
the O Kedge. The overall resolution /H20849beamline plus spec-
trometer /H20850was set to around 0.6 eV, which can be obtained
from the width of the Mn L2,3elastic recombination peak at
full width at half maximum /H20849FWHM /H20850. This is an essentially
better resolution than in most of the work which has beendone on manganites so far. The spectra were calibrated usinga reference sample of pure Mn metal and MgO, respectively.
The Mn 2 pand O 1 sx-ray absorption spectra were mea-
sured under the same experimental conditions in total elec-tron yield mode /H20849TEY /H20850. For the XAS experiments the instru-
mental resolution was set to 0.3 eV. The LaSrMnO
4sample
was rinsed with isopropanol in order to reduce surface con-tamination just before mounting it into the transfer chamber.
The XPS valence band was recorded at the Department of
Physics, University of Osnabrück, Germany, using a PHI5600CI multitechnique spectrometer with monochromaticAlK
/H9251/H20849h/H9263=1486.6 eV /H20850radiation of 0.3 eV at FWHM. The
overall resolution of the spectrometer is 1.5% of the pass
energy of the analyzer, 0.35 eV in the present case. The spec-trometer was calibrated using an Au foil as a referencesample /H20849the binding energy of the Au f
7/2core level is
84.0 eV /H20850. The measurements were performed at room tem-
perature. To get a surface free of contamination, the samplewas fractured in situ .
III. RESULTS AND DISCUSSION
A. RXES on the Mn Ledge
The top panel of Fig. 1displays the Mn L2,3edge XA
spectrum of LaSrMnO 4. The Mn L2,3XA spectrum, which is
dominated by transitions to Mn 3 dstates, consists of two
broad multiplets due to the spin-orbit splitting of the Mn 2 p
core hole. The Mn L2,3XAS can be compared with ligand-
field multiplet calculations for Mn3+inD4hcrystal
symmetry.33However, quite good agreement is achieved
with recently reported model calculations based upon a/H20849MnO
6/H2085010−charge transfer cluster model in D4hsymmetry,30
assuming a ground state configuration comprising 73.6% 3 d4
states and 26.4% 3 d5Lcharge transfer states. This calcula
tion has been fitted to experiments on LaMnO 3/H20849Ref. 34/H20850, and
one can conclude that the Mn L2,3XAS of LaMnO 3and
LaSrMnO 4are quite similar.
In the bottom panel we present the resonant Mn L2,3edge
x-ray emission spectra excited across the Mn L2,3absorption.
Labels A–I correspond to the excitation energies as markedin the XA spectrum. The RXE spectra depend strongly on theincident photon energy. The emission spectra correspondingto excitation energies A–E consist of three main resonantfeatures, namely the elastic recombination peak and RIXS-like loss features, which appear at constant energy /H20849about
2.2 eV and 6–9 eV /H20850below to the elastic recombination
peak. At higher excitation energies above the Mn L
3thresh-
old /H20849from excitation energy E /H20850the emission spectra contain
more nonresonant features which contribute to the spectra asdispersing features compared to spectra taken at incidentphoton energies which are below and at the Mn L
3XAS
edge. If the excitation energy is increased to the Mn L2edge,
the elastic peak almost disappears while the loss feature lo-cated 2.2 eV below the recombination peak shows a strongresonance. Also the charge transfer excitations appear, whileone can investigate normal fluorescence due to transitionsfrom the Mn 3 dstates to the Mn 2 p
3/2level about 12–14 eV
below the elastic peak.
In Fig. 2we compare the Mn L2,3RXE spectra with avail-
able Mn3+charge transfer multiplet calculations,23the valuesKUEPPER et al. PHYSICAL REVIEW B 74, 115103 /H208492006 /H20850
115103-2of the lifetime broadening of the intermediate
state were estimated 0.4 eV at the Mn L2edge and 0.6 eV at
the Mn L3edge, respectively. This comparison is some-
what limited by the fact that the multiplet calculations shownhere have been performed for an almost cubic manganiteapplying very small values for Dsand Dt,
23whereas
LaSrMnO 4is strongly distorted along the zaxis. On the other
hand the same calculations have been used to analyze theMnL
2,3RXES of another layered manganite, namely
La1.2Sr1.8Mn 2O7/H20849Ref. 26/H20850, and the simulations reproduce the
RXES spectra of LaSrMnO 4rather well.
According to the calculation and earlier experimental re-
sults obtained on perovskite manganites24,25the multiplet
structures located around 2.2 eV and 6–9 eV below the elas-tic recombination peak can be described as local ddtransi-
tions and O 2 p→Mn3 dcharge transfer excitations, respec-
tively. We want to point out that the present Mn L
2,3RXE
spectra do not have such a high resolution as very recentlypublished data on MnO /H20849Ref. 35/H20850. However, the signal to
noise ratio is somewhat better in this work and it is of par-ticular importance to extract spectral features at lower exci-tation energies /H20849A–C /H20850. In the spectra C–E presented here also
two weaker features located around 4 eV and 5.5 below theelastic recombination peak are nicely visible, with an overallspectral resolution essentially below 1 eV and a good signalto noise ratio. The 2.2 eV loss peak is also present at excita-tion energy E, corresponding to the maximum of the Mn L
3
absorption. For this energy and those recorded at higher ex-
citation energies the excitonic states begin to overlap withthe ordinary Mn 3 d→2pemission, representing at least in
part the occupied density of states. In addition, a furtherweak peak around 1.2 eV occurs if the excitation energy isset close to or to the Mn L
3XAS intensity maximum
/H20849marked with an arrow in Fig. 2/H20850. This feature is only very
weak or even not present in the cluster calculations and weassociate this feature with more bandlike transitions, mainlyarising from the Mn e
gelectrons /H20849see Sec. III B for a more
detailed discussion /H20850.TEY / I0
665 660 655 650 645 640 635 630
Photon Energy (eV)ABCD E
F
GH
ILaSrMnO4 XAS on Mn L edge
Mn 2p XAS theor.Counts (arb. units)
660 655 650 645 640 635 630 625
Det. Photon Energy (eV)ABCDEFGHILaSrMnO4: RXES on Mn L edge
FIG. 1. Upper panel: Mn Ledge XAS of LaSrMnO 4in com-
parison with charge transfer cluster calculations from Taguchi andAltarelli /H20849Ref. 30/H20850. Lower panel: Mn L
2,3RXE spectra recorded at
different excitation energies across the Mn L2,3absorption edges.
The excitation energies are those labeled A–I in the XAS /H20849upper
panel /H20850: A: 639.3 eV, B: 639.5 eV, C: 639.8 eV, D: 641.2 eV, E:
641.8 eV, F: 643.1 eV, G: 650.7 eV, H: 652.4 eV, I: 653.4 eV. Theexcitation energies are marked by arrows, the ddtransitions by dark
gray bars, and the charge transfer features by bright gray bars,respectively.
Counts (arb. units)
-15 -10 -5 0 5
Energy Loss (eV)C exp.
C theor.E exp.
E theor.F theor.F exp.G theor.G exp.H theor.H exp.LaSrMnO4
FIG. 2. The Mn L2,3RXE spectra, brought to an energy loss
scale and in comparison with corresponding Mn3+charge transfer
multiplet calculations which have been adapted from Butorin et al.
/H20849Ref. 23/H20850.EXCITED AND GROUND STATE PROPERTIES OF ¼ PHYSICAL REVIEW B 74, 115103 /H208492006 /H20850
115103-3There are some other differences in detail. The elastic
peak is overestimated in the calculations. The elastic recom-bination peak comprises several components. As indicated inSec. II there is some scattering which comes from the surfaceroughness. Moreover, for excitation energies close to reso-nance, the dielectric constants of the sample change rapidly,what might, e.g., have influence on the Brewster angle andsubsequently to the elastic peak. Finally, a number of otherphenomena may occur at resonancelike quasielastic scatter-ing. In spectrum F the intensity of the charge transfer showsa resonance. This is partly reproducible by the multiplet cal-culations, but also here quite strong bandlike features areoverlapping the multiplet structure due to the continuum ex-cited normal fluorescence which becomes much more impor-tant from excitation energy E on. Since we investigate asystem with a strong anisotropic crystal field around theMn
3+ions the scattering geometry can also have some influ-
ence on the weight of the spectral features. This has beenvery recently shown on NaV
2O5/H20849Ref. 36/H20850. The above men-
tioned phenomena can be investigated by angle resolvedmeasurements or the use of variable polarization beamlines,which is beyond the scope of this work.
B. XPS valence band and NXES
In Fig. 3we show the XPS valence band of LaSrMnO 4
/H20849top panel /H20850, the Mn L3/H20849middle panel /H20850, and O KXES /H20849bottom
panel /H20850results along with results of available LSDA
+/H20849U/H20850/H20849Ueff=2 eV /H20850electronic structure calculations.13In a
very recent work we compared the valence band of
LaSrMnO 4with a number of available electronic structure
calculations,12,13and found the quite moderate U=2 eV
value to be in the best agreement with the experiment.17
Such moderate values for the onsite correlation potentialhave been reported also for LaMnO
3, e.g., U=2 eV by
Sawada et al.37Ravindran et al. performed calculations using
the full-potential linearized augmented plane-wave methodwithout invoking any special treatment of the intra-atomiccorrelation beyond the conventional generalized gradient ap-proximation /H20849GGA /H20850.
38On the other side much larger values
were under discussion such as U=8 eV in the work of
Medvedeva et al.39According to earlier findings we compare
the results presented here with the mentioned U=2 eV
calculation.13,17
The XPS valence band consists of four distinct peaks la-
beled /H20849i/H20850–/H20849iv/H20850. A quite weak band /H20849i/H20850is located just below
Fermi level. A shoulder with increasing intensity /H20849ii/H20850is lo-
cated at a binding energy of around 2 eV, followed by anabsolute maximum in intensity at 4 eV /H20849iii/H20850, and finally a
local maximum between 6–7 eV /H20849iv/H20850on the binding energy
scale.
The Mn L
3and O KXE spectra, which have been con-
verted to the binding energy scale by using the correspond-ing XPS core level binding energies, are an experimentalsupport for the identification of the XPS features. Besides theelastic recombination peak /H20849as described in Sec. III A /H20850, the
MnL
3/H20849R/H20850XES is dominated by an intense spectral feature at
2 eV, furthermore shoulders at 0.7 eV and 4–5 eV are vis-ible, followed by a second broad peak /H20849/H110158–10 eV /H20850. TheOKx-ray emission spectrum spans the energy range from
1 eV to about 7 eV, with a rather sharp intensity maximumat 3 eV. With the help of the experimental x-ray emissionand the corresponding calculated Mn 3 dand O 2 ppartial
densities of states we now can identify the features in theXPS valence band. According to the band structure calcula-tion band /H20849i/H20850can be assigned to Mn e
gstates, also the Mn L3
/H20849R/H20850XES shows a corresponding feature.12,13According to the
theory and the Mn RXE spectrum the shoulder at /H110152e V
comprises the main part of the Mn t2gorbitals, also O 2 p
states are present, likely energetically overlapping the Mn 3 d
contributions. Finally bands /H20849iii/H20850and /H20849iv/H20850are built up out of
Mn 3 dand O 2 pstates are present, which are hybridized via
charge transfer.
In the lower part of the valence band /H20851bands /H20849iii/H20850and /H20849iv/H20850/H20852
there are also a few likely La 4 dand Sr 4 pstates present,
such as, e.g., in LaMnO 3or Sr 2FeMoO 6/H20849Refs. 25,38, and
40/H20850having some influence on the XPS valence band due to
their large photoionization cross section. The remaining dif-ferences between the Mn L
3/H20849R/H20850XES and the calculated Mn
PDOS can be explained by the remaining multiplet effects inthe experimental spectrum.1 2 1 0 86420 - 21 2 1 0 86420 - 21 2 1 0 86420 - 2LDA+U
(Ueff = 2eV)XPS VB
Mn L3 RXES
Mn pDOS:
LDA+U (Ueff = 2eV)
O K XES
O pDOS:
LDA+U
(Ueff = 2eV)Intensity (arb. units)
Binding Energy (eV)i)ii)iv) iii)
α
FIG. 3. The XPS valence band and XE spectra taken at the
MnL3/H20849spectrum Ein Fig. 1/H20850and the O Kedges of LaSrMnO 4. The
calculated densities of states have been extracted from Park /H20849Ref.
13/H20850.KUEPPER et al. PHYSICAL REVIEW B 74, 115103 /H208492006 /H20850
115103-4RXES on the O Kedge
In the following we want to discuss the O KRXES which
also show rather strong dependence on the excitation energy/H20849Fig. 4/H20850. RXE spectra have been recorded at several excita-
tion energies across the corresponding O Kspectrum, la-
beled with a-f. In the lower panel of Fig. 4the excitation
energies are indicated by arrows. At lowest excitation energyathe XES spectrum looks like an approximate 6 eV broad
peak with an asymmetric tail to lower photon energies /H20849see
also Fig. 5/H20850. At slightly higher excitation energies /H20849bandc/H20850
the shape of the RXE spectra changes significantly. Thesespectra show a four peak structure, a shoulder located 1.5 eVbelow the small elastic recombination peak, followed by anintense and rather narrow main peak. Another small shoulderis present at lower photon energies, finally a second peakaround 5 eV below the elastic recombination peak appears.
The excitation energy dependence can be associated with
site specific contributions from the different oxygen sites inLaSrMnO
4, namely the in-plane, or equatorial O1 site, andthe out of plane, or apical O2 site. Similar effects have been
observed in a few previous works dealing with other highlyanisotropic materials as Sr
2RuO 4or NaV 2O5.27,28On the
other side no such energy dependence of the O KRXES in
materials showing cubic symmetry has been found.28,41In
order to reveal more detailed information from the resonantOK
/H9251RXE spectra we compare our experimental results
with the calculated local partial densities of states /H20849LPDOS /H20850
of the in-plane /H20849O1/H20850and out of plane /H20849O2/H20850oxygen atoms.13
Figure 5displays a comparison of selected experimental
RXE spectra with the O1 LPDOS, the O2 LPDOS, and fi-nally the summarized /H20849O1+O2 /H20850oxygen PDOS. In the top
panel a comparison of the corresponding O KXAS with the
calculated unoccupied densities of states is shown.
At higher excitation energies /H20849eandf/H20850both, the O1 and
the O2 oxygen sites contribute equally to the emission spec-tra and we note a quite good coincidence between the calcu-TEY / I0
550 545 540 535 530 525
Photon Energy (eV)abdcefLaSrMnO4: XAS on O K edgeCounts (arb. units)
535 530 525 520 515
Det. Photon Energy (eV)acdef
bLaSrMnO4: RXES on O K edge
FIG. 4. Resonant x-ray emission spectra of LaSrMnO 4at the
OKedge. The excitation energies are those labeled a-fin the cor-
responding XA spectrum /H20849upper panel /H20850:a: 528.3 eV, b: 528.6 eV,
c: 528.8 eV, d: 529.8 eV, e: 533.6 eV, f: 535.8 eV. The four peak
structure /H20849see text /H20850is marked with bars.
Intensity (arb. units)
10 8 6 4 2 0 -2
Ener gy Loss (eV) a exp. (x5)
pDOS O1 c exp.
c pDOS O2 f exp.
pDOS O1 + O2LaSrMnO4 Intensity (arb. units)
-4 -3 -2 -1 0
Energy (eV) pDOS O2 (x2) pDOS O1 pDOS O XAS on O K edge
abc d
FIG. 5. Upper panel: The O KXAS of LaSrMnO 4has been
brought to a common energy scale with the electronic structurecalculations of Park for comparison Ref. 13. Lower panel: Selected
OKRXE spectra are compared with the calculated occupied oxy-
gen densities of states. Spectrum ais compared with the in plane
/H20849O1/H20850PDOS, spectrum cwith the out of plane /H20849O2/H20850PDOS, and
spectrum fwith the summarized /H20849O1+O2 /H20850PDOS.EXCITED AND GROUND STATE PROPERTIES OF ¼ PHYSICAL REVIEW B 74, 115103 /H208492006 /H20850
115103-5lated overall O PDOS and the experimental spectra /H20849spec-
trum fand the corresponding calculation in Fig. 5/H20850. Above
energy bthe incoming x-ray photons are also excited into
the unoccupied states, the out of plane or apical O2 sites,hence one can expect a strong resonance in the RXE spectrataken at these energies. Therefore the strong enhancementof the narrow peak located at 3 eV can be observed. As toRXE-spectrum c, both the shoulder at around 1.5 eV binding
energy as well as the narrow but intense main peak located at3 eV are well reproduced by theory. The shoulder at 5–6 eVcan be associated with remaining transitions from O1 unoc-cupied sites since at this energy both oxygen sites are ex-cited. According to theory the apical O2 atoms do not con-tribute to this region of the occupied DOS whereas for theO1 atoms a strong local intensity maximum is predicted.
13
As in a lot of layered perovskites as cuprates and other man-ganites this can be understood as being due to strong chargetransfer effects arising from Mn 3 dand O 2 pstates coming
from the in-plane O1 atoms. At the lower excitation energy a
only excitations from the O 1 sstates into unoccupied O1
states occur. Nevertheless the agreement between the calcu-lated O1 LPDOS and the RXES spectrum corresponding toexcitation energy ais much less satisfactory than for the O2
LPDOS or the oxygen PDOS /H20849bottom spectrum in Fig. 5/H20850.
The shoulder around 5–6 eV is pronounced rather intenselybut not as much as suggested by theory indicating that thereare already significant O2 contributions in the emission spec-trum. A similar behavior has been investigated before.
24,28
An interesting onset for an explanation has been given by
Woods et al.27who found that under these resonant condi-
tions different spatial symmetries can be enhanced and sup-pressed not only on excitation energy dependence but also onthe orientation of the polarization vector of the incomingx-ray photons. From an experimental point of view furtherinvestigation making use of polarization dependence on anoriented piece of single crystal would be desirable. From atheoretical point of view the analysis by means of a calcula-tion taking explicitly into account the framework of the sec-ond order Kramers-Heisenberg relation, and the intra- andinteratomic correlation effect has been proposed.
19With such
a calculation in combination with the above proposed exten-sion of the experiment presented here perhaps one could getmore detailed information on this yet not fully understood
RIXS process at excitation energies very close to or belowthe O 1 sabsorption threshold in 3 dtransition metal com-
pounds.IV. CONCLUSIONS
In summary we have presented a combined x-ray spectro-
scopic study of the single layer manganite LaSrMnO 4lead-
ing to a number of results. RXE spectra recorded at theMnL
2,3edges with a quite high resolution of 0.6 eV com-
pared to some previous work exhibits a rich multiplet struc-ture. The spectra consist of three main features, which can beassociated with the elastic recombination peak and two lossfeatures. The peak located around 2.2 eV below the elasticpeak can be associated with a local Mn ddtransition in
agreement with multiplet calculations,
23the second main fea-
ture around 6–9 eV below the elastic peak can be assignedto charge transfer transitions. Further fine structures pre-dicted by theory are nicely resolved in the present experi-ment. If the excitation energy is tuned just above the Mn L
3
XAS maximum or to higher energies the spectra represent
parts of the joint DOS, with the states projected on the Mnsite. Regarding the valence band, by comparing the XPSvalence band and complementary XES spectra reflecting theMn 3 dand O 2 ppartial densities of states with electronic
structure calculations we find the features close to the Fermienergy and at 2 eV to be due to Mn e
gand rather localized
Mn 3 dt2gstates. The lower lying features are due to O 2 p,
strongly overlapping with Mn 3 dand La 4 dstates. Regard-
ing the valence band, the experimental results are in goodagreement with a full-potential linearized augmented plane-wave method–local density approximation /H20849FLAPW-LDA /H20850
+Ucalculation invoking a quite moderate effective Coulomb
potential U=2 eV for the Mn 3 dstates.
13As to the RXES
performed at the O Kedge we were able to resolve excita-
tion energy dependent features which we could attribute tosome extent to the site specific LPDOS of the inequivalent inplane /H20849O1/H20850and out of plane /H20849O2/H20850oxygen atoms. Whereas a
very good coincidence between experiment and theory is
achieved in the case of the oxygen /H20849O1+O2 /H20850PDOS and the
contributions of the out of plane /H20849O2/H20850LPDOS, less in agree-
ment for the in-plane /H20849O1/H20850oxygen atoms is found. Regard-
ing this point further experimental and theoretical efforts are
necessary in the future.
ACKNOWLEDGMENTS
Financial support of the Ph.D. program of the federal state
of Lower Saxony, Germany is gratefully acknowledged.Most of the present work has been performed at the Ad-vanced Light Source /H20849A.L.S. /H20850which is supported by the U.S.
Department of Energy under Contract No. DE-AC03-76SF00098.
*Electronic address: k.kuepper@fz-rossendorf.de
1A. J. Millis, Nature /H20849London /H20850392, 147 /H208491998 /H20850.
2Y . Tokura and N. Nagaosa, Science 288, 462 /H208492000 /H20850.
3M. B. Salamon and M. Jaime, Rev. Mod. Phys. 73, 583 /H208492001 /H20850.
4Y . D. Chuang, A. D. Gromko, D. S. Dessau, T. Kimura, and Y .
Tokura, Science 292, 1509 /H208492001 /H20850.5R. von Helmolt, J. Wecker, B. Holzapfel, L. Schultz, and K. Sam-
wer, Phys. Rev. Lett. 71, 2331 /H208491993 /H20850.
6Y . Moritomo, A. Asamitsu, H. Kuwahara, and Y . Tokura, Nature
/H20849London /H20850380, 141 /H208491996 /H20850.
7K. Kuepper, F. Bondino, K. C. Prince, M. Zangrando, M. Zacchi-
gna, A. F. Takács, T. Crainic, M. Matteucci, F. Parmigiani, V . R.KUEPPER et al. PHYSICAL REVIEW B 74, 115103 /H208492006 /H20850
115103-6Galakhov, Y . M. Mukovskii et al. , J. Phys. Chem. B 109, 15667
/H208492005 /H20850.
8J. Geck, P. Wochner, D. Bruns, B. Büchner, U. Gebhardt, S.
Kiele, P. Reutler, and A. Revcolevschi, Phys. Rev. B 69, 104413
/H208492004 /H20850.
9D. Huang, W. Wu, G. Guo, H.-J. Lin, T. Hou, C. F. Chang, C.
Chen, A. Fujimori, T. Kimura, H. Huang et al. , Phys. Rev. Lett.
92, 087202 /H208492004 /H20850.
10R. Klingeler, D. Bruns, C. Baumann, P. Reutler, A. Revcolevschi,
and B. Büchner, J. Magn. Magn. Mater. 290-291 , 944 /H208492005 /H20850.
11D. Senff, P. Reutler, M. Braden, O. Friedt, D. Bruns, A. Cousson,
F. Bourée, M. Merz, B. Büchner, and A. Revcolevschi, Phys.Rev. B 71, 024425 /H208492005 /H20850.
12W. Wu, D. J. Huang, G. Guo, H.-J. Lin, T. Hou, C. F. Chang, C.
Chen, A. Fujimori, T. Kimura, H. Huang et al. , J. Electron Spec-
trosc. Relat. Phenom. 137-140 , 641 /H208492004 /H20850.
13K. T. Park, J. Phys.: Condens. Matter 13, 9231 /H208492001 /H20850.
14S. Larochelle, A. Mehta, L. Lu, P. K. Mang, O. P. Vajk, N.
Kaneko, J. W. Lynn, L. Zhou, and M. Greven, Phys. Rev. B 71,
024435 /H208492005 /H20850.
15C. Baumann, G. Allodi, A. Amato, B. Büchner, D. Cattani, R. de
Renzi, R. Klingeler, P. Reutler, and A. Revcolevschi, Physica B
374,8 3 /H208492006 /H20850.
16J. Herrero-Martín, J. García, G. Subías, J. Blasco, and M. C.
Sánchez, Phys. Rev. B 72, 085106 /H208492005 /H20850.
17K. Kuepper, R. Klingeler, P. Reutler, B. Büchner, and M. Neu-
mann, J. Appl. Phys. 99, 08Q308 /H208492006 /H20850.
18J. A. Carlisle, E. L. Shirley, E. A. Hudson, L. J. Terminello, T. A.
Callcott, J. J. Jia, D. L. Ederer, R. C. C. Perera, and F. J.Himpsel, Phys. Rev. Lett. 74, 1234 /H208491995 /H20850.
19A. Kotani and S. Shin, Rev. Mod. Phys. 73, 203 /H208492001 /H20850.
20S. M. Butorin, J.-H. Guo, M. Magnuson, P. Kuiper, and J. Nor-
dgren, Phys. Rev. B 54, 4405 /H208491996 /H20850.
21L. C. Duda, J. Nordgren, G. Dräger, S. Bocharov, and T. Kirch-
ner, J. Electron Spectrosc. Relat. Phenom. 110-111 , 275 /H208492000 /H20850.
22G. P. Zhang, T. A. Callcott, G. T. Woods, L. Lin, B. Sales, D.
Mandrus, and J. He, Phys. Rev. Lett. 88, 077401 /H208492002 /H20850.
23S. M. Butorin, C. Såthe, F. Saalem, J. Nordgren, and X.-M. Zhu,
Surf. Rev. Lett. 9, 989 /H208492002 /H20850.
24F. Bondino, M. Platé, M. Zangrando, D. Cocco, A. Comin, I.Alessandri, L. Malavasi, and F. Parmigiani, J. Phys. Chem. B
108, 4018 /H208492004 /H20850.
25K. Kuepper, M. C. Falub, K. C. Prince, V . R. Galakhov, I. O.
Troyanchuk, S. G. Chiuzb ăian, M. Matteucci, D. Wett, R.
Szargan, N. A. Ovechkina et al. , J. Phys. Chem. B 109, 9354
/H208492000 /H20850.
26A. Agui, S. M. Butorin, T. Käämbre, C. Sathe, T. Saitoh, Y . Mori-
tomo, and J. Nordgren, J. Phys. Soc. Jpn. 74, 1772 /H208492005 /H20850.
27G. T. Woods, G. P. Zhang, T. A. Callcott, L. Lin, G. S. Chang, B.
Sales, D. Mandrus, and J. He, Phys. Rev. B 65, 165108 /H208492002 /H20850.
28E. Z. Kurmaev, S. Stadler, D. L. Ederer, Y . Harada, S. Shin, M.
M. Grush, T. A. Callcott, R. C. C. Pereira, D. A. Zatsepin, N.Ovechkina et al. , Phys. Rev. B 57, 1558 /H208491998 /H20850.
29F. M. F. de Groot, Chem. Rev. /H20849Washington, D.C. /H20850101, 1779
/H208492001 /H20850.
30M. Taguchi and M. Altarelli, Surf. Rev. Lett. 9, 1167 /H208492002 /H20850.
31P. Reutler, O. Friedt, B. Büchner, M. Braden, and A. Revcolevs-
chi, J. Cryst. Growth 249, 222 /H208492003 /H20850.
32J. J. Jia, T. A. Callcott, J. Yurkas, A. W. Ellis, F. J. Himpsel, M. G.
Samant, G. Stöhr, D. L. Ederer, J. A. Carlisle, E. A. Hudson et
al., Rev. Sci. Instrum. 66, 1394 /H208491995 /H20850.
33C. W. M. Castleton and M. Altarelli, Phys. Rev. B 62, 1033
/H208492000 /H20850.
34J.-H. Park, S.-W. Cheong, and C. T. Chen, Phys. Rev. B 55,
11072 /H208491997 /H20850.
35G. Ghiringhelli, M. Matsubara, C. Dallera, F. Fracassi, A. Taglia-
ferri, N. B. Brookes, A. Kotani, and L. Braicovich, Phys. Rev. B
73, 035111 /H208492006 /H20850.
36G. P. Zhang and T. A. Callcott, Phys. Rev. B 73, 125102 /H208492006 /H20850.
37H. Sawada, Y . Morikawa, K. Terakura, and N. Hamada, Phys.
Rev. B 56, 12154 /H208491997 /H20850.
38P. Ravindran, A. Kjekshus, H. Fjellvåg, A. Delin, and O. Eriks-
son, Phys. Rev. B 65, 064445 /H208492002 /H20850.
39J. E. Medvedeva, M. A. Korotin, V . I. Anisimov, and A. J. Free-
man, Phys. Rev. B 65, 172413 /H208492002 /H20850.
40K. Kuepper, M. Kadirog ¯lu, A. V . Postnikov, K. C. Prince, M.
Matteucci, V . R. Galakhov, H. Hesse, G. Borstel, and M. Neu-mann, J. Phys.: Condens. Matter 17, 4309 /H208492005 /H20850.
41K. C. Prince, M. Matteucci, K. Kuepper, S. G. Chiuzba ˇian, S.
Bartkowski, and M. Neumann, Phys. Rev. B 71, 085102 /H208492005 /H20850.EXCITED AND GROUND STATE PROPERTIES OF ¼ PHYSICAL REVIEW B 74, 115103 /H208492006 /H20850
115103-7 |
PhysRevB.103.125161.pdf | PHYSICAL REVIEW B 103, 125161 (2021)
Finite and infinite matrix product states for Gutzwiller projected mean-field wave functions
Gabriel Petrica ,1,*Bo-Xiao Zheng,2,3Garnet Kin-Lic Chan,2and Bryan K. Clark1,†
1Institute for Condensed Matter Theory and IQUIST and Department of Physics, University of Illinois at Urbana-Champaign,
Urbana, Illinois 61801, USA
2Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
3AxiomQuant Investment Management LLC, Shanghai 200120, China
(Received 18 January 2021; revised 15 March 2021; accepted 17 March 2021; published 31 March 2021)
Matrix product states (MPS) and “dressed” ground states of quadratic mean fields (e.g., Gutzwiller projected
Slater determinants) are both important classes of variational wave functions. This latter class has played impor-tant roles in understanding superconductivity and quantum spin liquids. We present a method to obtain both thefinite and infinite MPS (iMPS) representation of the ground state of an arbitrary fermionic quadratic mean-fieldHamiltonian (which in the simplest case is a Slater determinant and in the most general case is a Pfaffian). Wealso show how to represent products of such states (e.g., determinants times Pfaffians). From this representationone can project to single occupancy and evaluate the entanglement spectra after Gutzwiller projection. We thenobtain the MPS and iMPS representation of Gutzwiller projected mean-field states that arise from the variationalslave-fermion approach to the S=1 bilinear-biquadratic quantum spin chain. To accomplish this, we develop an
approach to orthogonalize degenerate iMPS to find all the states in the degenerate ground-state manifold. We findthe energies of the MPS and iMPS states match the variational energies closely, indicating the method is accurateand there is minimal loss due to truncation error. We then present an exploration of the entanglement spectra ofprojected slave-fermion states, exploring their qualitative features and finding good qualitative agreement withthe respective exact ground-state spectra found from density matrix renormalization group.
DOI: 10.1103/PhysRevB.103.125161
I. INTRODUCTION
Variational wave functions are frequently used to under-
stand quantum many-body systems. Two important classesof variational wave functions are dressed Slater determinantsand tensor networks. Dressed Slater determinants introducea correlation on top of a mean-field ground state. On theother hand, a tensor network is represented as a network ofconnected tensors providing a natural framework in whichto understand and represent low-entangled quantum states(see Fig. 1).
Slater determinants (SDs) [and other generalized quadratic
ground states such as Bogoliubov–de Gennes (BdG) [ 1] and
Pfaffian states [ 2]] have played a key role in the understanding
of physical systems ranging from their use as the Hartree-Focksolution in quantum chemistry to being applied as a startingmean-field Ansatz for strongly correlated systems. These latter
Ansatzë are then dressed in various ways: Slater-Jastrow wave
functions are the de facto standard for simulating material
systems in quantum Monte Carlo; many prototypical quantumHall states are represented as powers or products of Slaterdeterminants and Pfaffians; and projected mean-field statesare an important starting point for probing the physics ofhigh-temperature superconductivity as well as quantum spinliquids.
*petrica2@illinois.edu
†bkclark@illinois.eduWhile dressed mean-field states are often easy to represent
in variational Monte Carlo (VMC), it is also often difficult toextract certain properties from VMC. Foremost among theseare the entanglement spectra which are an important metricused for understanding topological phases of matter. Evenproperties which can be extracted easily, such as the energy,can be statistically noisy, making aspects such as optimizationdifficult. Moreover, evaluating dressed mean-field states inMonte Carlo scales cubically with the system size, makingthe approach to the thermodynamic limit costly. Matrix prod-uct states avoid many of these problems in one-dimensionalsystems and ladders: They are ideally suited for extractingentanglement spectra, computing observables exactly withoutany statistical noise, and directly representing (gapped) phys-ical systems in the thermodynamic limit.
Our main contribution in this paper is to describe a series of
efficient and highly parallel algorithms which take (projected)mean-field (i.e., quadratic) eigenstates and generate both finite(fMPS) and infinite (iMPS) matrix product states (MPS) fromthem. We will also show how to generate fMPS and iMPSfor products of mean-field wave functions. We will then applyour approach to compute the MPS and entanglement spectraof a series of projected slave-fermion wave functions of thebilinear-biquadratic model. This example will bring to light anumber of interesting aspects of generating multiple degen-erate ground states from Gutzwiller projected slave-fermionsystems in iMPS.
Beyond this particular application, being able to generate a
MPS from a projected SD is generically useful. It allows for
2469-9950/2021/103(12)/125161(15) 125161-1 ©2021 American Physical SocietyPETRICA, ZHENG, CHAN, AND CLARK PHYSICAL REVIEW B 103, 125161 (2021)
FIG. 1. (a) Graphical representation of the left-boundary, bulk,
and right-boundary Atensors forming an open-boundary matrix
product state. The α’s are the virtual indices; the σ’s are the physical
indices. (b) Graphical representation of an eight-site matrix product
state.
more faithful comparisons between slave-fermion and density
matrix renormalization group (DMRG) results which oftendisagree on the underlying phase of spin liquids. It could beused to initialize DMRG with a good initial mean-field guessfor certain Hamiltonians. This can be useful both for calcula-tions on discrete lattices as well as DMRG in the continuum.Because there exist algorithms which build MPS on quantumcomputers, it immediately gives an additional approach togenerate a dressed quadratic mean-field state on a quantumdevice.
We are aware of two other algorithms which convert Slater
determinants to MPS [ 3–5]. Both of these are based on the
idea of applying quantum gates or matrix product operators toa simpler quantum state to generate the MPS. Our approachdiffers from these techniques in two key ways: (1) We cangenerate the infinite MPS for a family of Slater determinantsand (2) we generate our (i)MPS by directly generating theMPS coefficients without the application of any operators tothe system. We also note that Ref. [ 6] represents Slater de-
terminants in a MPS-like framework in a Gaussian fermionicrepresentation.
In Sec. II, we will describe our key algorithm for turning a
SD into an (i)MPS. In Sec. III, we will show a series of exam-
ples for how to use this basic procedure for generating morecomplicated mean-field states (i.e., Pfaffians) as well as stateswhich are products of mean-field states. Finally, in Sec. IV
we focus on computing (i)MPS for the slave-fermion statesof the bilinear-biquadratic model showing their entanglementspectra and energy.
II. SLATER DETERMINANTS TO MPS
In this section we are going to show how to generate either
a finite matrix product state (fMPStoSD) or an infinite matrixproduct state (iMPStoSD) from a Slater determinant (SD).This will not only be useful in its own right but will be thekey operation used in the rest of this work to produce MPSfor both more complicated quadratic mean-field states as wellas dressed versions of these states.
The fMPStoSD generates the matrix product state site by
site in an approach that is highly reminiscent of the site-decimation canonical technique to convert a generic wave
function (i.e., a multisite tensor) into a matrix product state[7]. The typical site-decimation procedure involves perform-
ing singular value decompositions (SVDs) over matricesgenerated by collecting different subsets of indices into thetwo matrix dimensions. This general approach will become ef-ficient to use with Slater determinants because SVDs of Slaterdeterminants are efficient and generate sums of products ofSlater determinants.
In fMPStoSD we perform a series of Schmidt decom-
positions over all bipartitions of our system. Each Schmidtdecomposition generates a set of Schmidt vectors; each suchSchmidt vector is a Slater determinant. The MPS is then gen-erated by taking overlaps of these Slater determinant Schmidtvectors with each other in the correct way.
In iMPStoSD we can easily generate the bulk uniform
iMPS tensor from justtwo Schmidt decompositions: one for
each of two ground-state Slater determinants of the sameHamiltonian defined on sufficiently large systems that differin size by one unit cell. Again, these Schmidt decompositionswill have Slater determinant Schmidt eigenvectors. After weappropriately fix the gauge of the two Schmidt decomposi-tions, the uniform bulk MPS tensor will be generated fromappropriate overlaps of these Schmidt eigenvectors.
A. Slater determinant →finite MPS
In this section, we show in detail how to convert a Slater
determinant into a finite matrix product state. The Schmidtdecomposition of a Slater determinant |SD/angbracketrightonNsites bipar-
titioned into two regions cut between sites iandi+1 will be
notated as
|SD/angbracketright=/summationdisplay
αλi;N
α/vextendsingle/vextendsingleLi;N
α/angbracketrightbig/vextendsingle/vextendsingleRi;N
α/angbracketrightbig
, (1)
where |Lα/angbracketrightand|Rα/angbracketrightare the αth left and right Schmidt
vectors, respectively (with support in their respective subsys-tem), and λ
αis the αth Schmidt eigenvalue. Note that, for
a Slater determinant, each of the individual left and rightSchmidt vectors are also Slater determinants and efficientlycomputable [ 8–11] [see also Supplemental Material (SM) 3
[12] for more details regarding the Schmidt decomposition of
Slater determinants]. Slater determinants are specified by aset of single-particle orbitals and all the Slater determinantsin the set of right Schmidt vectors {|R
α/angbracketright}are specified by
subsets of single-particle orbitals from a set of (at most) N
single-particle orbitals {φR
1···φR
N}defined on the (inclusive)
sites [( i+1),..., N]. There are, at most, 2Nsuch subsets.
Analogous statements hold for the left Schmidt vectors.
A general matrix product state can be written as
|MPS/angbracketright=/summationdisplay
{σ},{a}A[1]σ1
1,a1···A[N]σN
aN−1,1|σ1···σN/angbracketright, (2)
where A[k]is the kth three-tensor specified by the physical
index σk(e.g., occupancy or spin) and the virtual indices
(αk−1,αk)[7]. To generate the MPS of a Slater determinant,
we compute each three-tensor A[i+1]as
A[i+1]σi+1
αkαk+1=/parenleftbig
/angbracketleftσi+1|⊗/angbracketleft Ri+1;N
αk+1|/parenrightbig
|Ri;N
αk/angbracketright, (3)
125161-2FINITE AND INFINITE MATRIX PRODUCT STATES FOR … PHYSICAL REVIEW B 103, 125161 (2021)
giving a matrix which is in right canonical form, i.e.,/summationtext
σA[i+1]σ(A[i+1]σ)†=I. Note that this procedure is very
similar to the one which transforms a vector into a MPS[7] and works for the same reason: The sets {|R
i;N
α/angbracketright}and
{|σi/angbracketright⊗| Ri+1;N
β/angbracketright}span the same space and therefore there is a
transformation Awhich rotates between them. In practice, we
keep the bond dimension of Acontrolled by only computing
the Schmidt vectors whose Schmidt values are above a certainthreshold /epsilon1. This can be done without computing any Schmidt
eigenvector with an eigenvalue less than /epsilon1. Here, we have
focused on the bulk tensors and slight modifications need tobe made for the boundary tensors A
[1]σ1andA[N]σN(see SM
1[12] for the mathematical expression of the boundary ten-
sors). We now describe how to efficiently evaluate the matrixelements of each A. We start by noting that |σ
i+1/angbracketright⊗| Ri+1;N/angbracketrightis
also a Slater determinant. It is specified by the single-particleorbitals
{[0φ
a],[0φb],..., [0φc],φi+1}, (4)
where [0 φa] is the single-particle orbital with coefficients
in the lattice basis [0 ,φa(i+2),φa(i+3),...,φ a(N)] and
φi+1is the single-particle orbital in analogous notation,
[1,0,..., 0]. Equation ( 3) then reduces to the overlap of two
Slater determinants of size ( N−i)×(N−i) which can be
computed in O(N3) time.
While naively each element of Arequires such a com-
putation, there is a significant overlap in these differentcomputations which reduces the naive computational com-plexity of the tensor computation. There are two steps incomputing the overlap of two Slater determinants: evaluatingthe overlap matrix between all pairs of single-particle orbitalsthat make up the two determinants and computing the de-terminant of this overlap matrix. All the Slater determinantsused in the ket (respectively bra) of Eq. ( 3) (over different
terms in A) come from a subset of single-particle orbitals of
theN-orbital set {φ
R
1,...,φR
N}. We can compute the overlap
matrix of all these respective single-particle orbitals once perthree-tensor Aat a cost of O(N
3).The entries of Aare then
determinants of submatrices of this overlap matrix. Whilenaively each determinant also costs O(N
3) to compute, the
submatrices differ only in the bottom log2Dcolumns and
right log2Drows where Dis the bond dimension of A; de-
terminant update formulas can then be used to accelerate thiscomputation, letting each determinant be computed in timeO(N
2log2D) after an initial O(N3) operation to evaluate the
inverse of the upper-left ( N−log2D)×(N−log2D) block
of the overlap matrix. The whole evaluation of each tensorAcan be done in O(N
3)+O(D2N2log2D) time. This can
be further attenuated somewhat by more aggressive use ofdeterminant update formulas [ 13].
Notice that there are significant parts of this algorithm that
can be run in parallel. Each three-tensor Acan be computed
separately. Within each A, the Schmidt decomposition can be
partially parallelized; each element of the overlap matrix canbe computed in parallel; and, after the initial evaluation ofthe inverse of the upper-left block of the overlap matrix, eachdeterminant can then be computed in parallel. See Fig. 2.
FIG. 2. Top: Graphical representation of the Schmidt decom-
position of the wave function |/Psi18/angbracketright=/summationtext
βλ4;8
β|L4;8
β/angbracketright|R4;8
β/angbracketrightover a
bipartition [1 ,..., 4]×[5,..., 8] of an eight-site system. Middle
and bottom: Two additional ways of representing the quantum state
|/Psi18/angbracketright. The tensor A[4]is constructed by having the overlap of the right
five sites of the bottom two figures equal one.
B. Gapped Slater determinant →infinite MPS
The above procedure generates a finite MPS approximation
(the accuracy of the representation is given as an input tothe algorithm) of any Slater determinant. In this section, wedescribe how to generate an infinite MPS from the Slaterdeterminant ground state of a gapped mean-field Hamiltonian.This infinite MPS can be described by left Land right R
boundary tensors which sandwich the bulk tensor A,g i v i n g
us an infinite matrix product state of the form
|iMPS/angbracketright=/summationdisplay
σLσL···Aσn−1AσnAσn+1···RσR
×|σL···σn−1σnσn+1···σR/angbracketright, (5)
with an arbitrary number of bulk tensors A.Land Rare
tensors which span a fixed number kof sites. Note that any
thermodynamic observable can be computed directly in thethermodynamic limit of the Slater determinant using only thebulk tensor A. In addition, we can compute the amplitude for
the Slater determinant on any (large enough) system size, byinserting the corresponding number of bulk tensors betweenthe boundary tensors LandR(i.e., to generate the MPS for an
N-site Slater determinant from the infinite MPS, we therefore
useN−2kbulk tensors A); see Fig. 3.
To generate the iMPS, we start off by producing two Slater
determinants defined on 2 Nand 2 N+1 sites (see Fig. 4),
where Nis sufficiently large such that the entanglement spec-
FIG. 3. We obtain the finite MPS |ψ2N+n/angbracketrightby inserting n(in the
figure n=3)AiMPS bulk tensors between the left |LN/2;N/angbracketrightand right
|RN/2;N/angbracketrightSchmidt vectors obtained from |ψ2N/angbracketright.
125161-3PETRICA, ZHENG, CHAN, AND CLARK PHYSICAL REVIEW B 103, 125161 (2021)
FIG. 4. Illustration of the algorithm for generating the infinite
MPS representation of a Slater determinant. Lines 1 and 2 correspondto the standard Schmidt decomposition after site Nof wave functions
defined on 2 Nand 2 N+1 sites. For line 3, we use our gauge freedom
to replace the left Schmidt eigenvalues λ
N;2N+1and eigenvectors
|LN;2N+1/angbracketrightwith eigenvalues λN;2Nand eigenvectors |LN;2N/angbracketrightrotated by
Cwhere Cis defined in Eq. ( 9). Finally, the tensor Ais constructed
by having the overlap of the right N+1 sites (including C)o ft h e
bottom two lines equal one.
trum is constant over cuts in the “bulk” of the wave functions.
For gapped systems, we generically expect the entanglementspectrum over the bulk to be constant; see Fig. S3 in SM 5[12] for an example of this and for entanglement spectrum
data in the context of the Su-Schrieffer-Heeger (SSH) model.We then generate the Schmidt decompositions
|/Psi1
2N/angbracketright=/summationdisplay
αλN;2N
α/vextendsingle/vextendsingleLN;2N
α/angbracketrightbig/vextendsingle/vextendsingleRN;2N
α/angbracketrightbig
, (6)
|/Psi12N+1/angbracketright=/summationdisplay
αλN;2N+1
α/vextendsingle/vextendsingleLN;2N+1
α/angbracketrightbig/vextendsingle/vextendsingleRN;2N+1
α/angbracketrightbig
. (7)
Both|LN;2N
α/angbracketrightand|LN;2N+1
α /angbracketrightare going to be the same up to
a gauge freedom. We fix this gauge freedom by defining aunitary
C
N;2N+1
αβ=0i fλα/negationslash=λβ
=/angbracketleftLN;2N|α|LN;2N+1/angbracketrightβotherwise, (8)
which rotates between Schmidt eigenvectors with the same
Schmidt eigenvalue allowing the state on 2 N+1 sites to be
defined as
|/Psi12N+1/angbracketright=/summationdisplay
αγ|LN;2N/angbracketrightαλN;2N+1
α CN;2N+1
αγ |RN;2N+1/angbracketrightγ.(9)
Then the tensor Afor the iMPS is
AσN+1
αβ=/summationdisplay
γC[2N+1]
αγ /angbracketleftσN+1|/angbracketleftRN;2N|β||RN,2N+1/angbracketrightγ. (10)
As in the finite MPS case, we have that the single-particle
orbitals of the Slater determinant |σN+1/angbracketright|RN;2N/angbracketrightare shifted to
the right with an initial zero as their first element. The overlapof this tensor can be computed in exactly the same way as forthe finite MPS case. Here, though, we only need to evaluate
one tensor Ainstead of a tensor per site, with the assumption
that we are using an iMPS defined by a single tensor (i.e.,single-site unit cell) A. This process can be generalized to
multisite unit cells as well (see SM 2 [ 12] for the mathematical
derivation). Note that by directly applying the finite MPSalgorithm to large systems to try to find the bulk tensor Awill
fail because the gauge freedom available in the tensors willprevent a single identical bulk tensor from being produced ateach step.
C. Numerical validation
We numerically validate our algorithms by applying
fMPStoSD and iMPStoSD on the ground state of the Su-Schrieffer-Heeger (SSH) model [ 14],
H
SSH=v/summationdisplay
n(c†
n,1cn,2+H.c.)
+w/summationdisplay
n(c†
n+1,1cn,2+H.c.). (11)
The model describes spinless fermions on a one-dimensional
(1D) lattice, with a two-site unit cell made up of A,Bsites,
with different (real) parameters for intracell hopping ( v)
and intercell hopping ( w). It admits two different quantum
ground states, distinct in their topological properties: a triv-ially gapped phase for v>wand a (symmetry-protected)
topological gapped ground state, characterized by the pres-ence of two zero-energy edge modes inside the gap, forv<w, separated by a quantum critical point at v=w.
We will discuss here the trivial ground state. A small sub-
tlety related to choosing the same gapless boundary mode inthe Slater determinant wave functions used for generating theuniform tensor in the iMPS procedure is delegated to SM 4[12] (where we show how we deal with gapless boundary
modes in the context of the iMPS procedure). For the finiteSlater determinant, we compare the MPS we generate usingtwo different truncation values against the exact Slater de-terminant by comparing all of the amplitudes (see the firstcolumn in Fig. 5). For the infinite case, we generate the
iMPS and then use the bulk tensor we have computed alongwith the boundary tensors to compute amplitudes for a muchlarger system and again compare amplitudes against the exactsolution for that much larger system (see the second columnin Fig. 5). In both cases, we find that the amplitudes are in
very good agreement for all amplitudes down to the Schmidteigenvalue cutoff.
III. GENERAL (DRESSED) QUADRATIC MEAN FIELDS
In Sec. IIwe showed how to generate a matrix product state
from a Slater determinant. In this section, we show that thismachinery gives us the means to generate the matrix productstate representations of ground states of arbitrary quadraticmean-field Hamiltonians.
All quadratic Hamiltonians can be easily diagonalized
using a canonical transformation [ 15]. Without loss of gen-
erality, in our derivations, we will use translation invariantsystems for ease of presentation. We will first go throughtwo canonical examples. In Sec. III A we will show how to
125161-4FINITE AND INFINITE MATRIX PRODUCT STATES FOR … PHYSICAL REVIEW B 103, 125161 (2021)
FIG. 5. Comparison of amplitudes (normalized by the largest amplitude seen) between our fMPS /iMPS wave functions and the exact
SD. The largest (normalized) amplitude is at the “origin” of the graphs with smaller amplitudes toward both edges. Orange triangles arevalues at which the fMPS /iMPS gives zero amplitude; for these the “ y” coordinate is arbitrarily set to 1. The amplitudes for the top row
are less accurate as they are generated with larger MPS thresholds /epsilon1. (a) and (c) compare all amplitudes for N=8o n[ v=1.0;w=0.6;
Eq. ( 11)] and [ t=1;μ=3;/Delta1=1; Eq. ( 16)], respectively. (b) We compare 459 428 random configurations (top and bottom are different
configurations) between the N=24 Slater determinant with ( v=1.0;w=0.6) of Eq. ( 11) and a MPS generated from eight uniform iMPS
bulk tensors (generated from SD on N=16,17 sites) sandwiched between the eight left and eight right tensors from the 16-site Slater
determinant. (d) We compare 49 972 (top) and 34 933 (bottom) random configuration between the N=32 Pfaffian ground state of the Kitaev
chain with ( t=1.0;/Delta1=−1μ=−2.2) of Eq. ( 17) and a MPS generated from eight uniform iMPS bulk tensors (generated from SD on
N=24,25 sites) sandwiched between the 12 left and 12 right tensors from the 24-site Pfaffian.
produce the MPS representation of the ground states of BdG
Hamiltonians which are Slater determinants in disguise. InSec. III B, we show how to compute the MPS representa-
tion of the p-wave pairing ground state of the Kitaev chain
[16]. We then generalize this result to general Pfaffian wave
functions which are the most general quadratic mean-fieldground states. Finally, we show how to take products (orpowers) of quadratic mean-field Hamiltonians and turn theminto (i)MPS.
A. BdG →MPS
The key trick to convert a BdG wave function into a
MPS will be to (1) convert it to a Slater determinant througha particle-hole transformation, (2) convert this Slater de-terminant to a MPS, and (3) then undo the particle-holetransformation in the MPS language.
Consider a BdG Hamiltonian,
H
BdG=−/summationdisplay
/angbracketleftij/angbracketright,σtij(c†
iσcjσ+H.c.)
−/summationdisplay
/angbracketleftij/angbracketright/Delta1ij(c†
i↑c†
j↓+H.c.)−μ/summationdisplay
iσc†
iσciσ.(12)Under a canonical particle-hole transformation in the ↓sector,
f†
2i−1=c†
i↑,
f†
2i=ci↓,(13)
the BdG Hamiltonian becomes
Hph
BdG=−/summationdisplay
/angbracketleftij/angbracketrighttij(f†
2i−1f2j−1−f†
2if2j+H.c.)
−/summationdisplay
/angbracketleftij/angbracketright/Delta1ij(f†
2i−1f2j+H.c.)
−μ/summationdisplay
iσ(f†
2i−1f2i−1−f†
2if2i), (14)
and the new vacuum is |0ph/angbracketright=c†
1↓c†
2↓···c†
N↓|0/angbracketright, where |0/angbracketrightis
the vacuum of the original theory. The ground state of HBdGis
thus the Slater determinant ground state of Hph
BdGon top of the
new vacuum |0ph/angbracketright.
Using the results from Sec. II, we convert the Slater
determinant ground state of Hph
BdGinto a MPS |/Psi1MPS/angbracketright=/summationtext
{σ}A[1]σ1···A[2N]σ2N|σ1···σ2N/angbracketrightwhere σ2i−1={0,1}(i∈
[1,N]) indicates the absence /presence of a ↑particle and
σ2i={0,1}indicates the absence /presence of a hole on top
of the filled ↓Fermi sea at site i.
125161-5PETRICA, ZHENG, CHAN, AND CLARK PHYSICAL REVIEW B 103, 125161 (2021)
To “undo” the particle-hole transformation, we need to deal
with the fact that the fiact on the false vacuum |0ph/angbracketright(and not
the real vacuum) by swapping, for all i, the matrices A[2i]1and
A[2i]0. Moreover, by ordering the fermionic operators by site,
and then spin, the matrices A[2i−1]1,A[2i]1will pick up factors
of (−1)i−1. We can now combine these transformations giv-
ing us our final MPS for the BdG ground state of the form|/Phi1
GS/angbracketright=/summationtext
{σ=0,↑,↓,↑↓}B[1]σ1···B[N]σN|σ1···σN/angbracketrightwhere
B[i]0=(−1)i−1×A[2i−1]0A[2i]1,
B[i]↑=(−1)2i−2×A[2i−1]1A[2i]1,
B[i]↓=(−1)0×A[2i−1]0A[2i]0,
B[i]↑↓=(−1)i−1×A[2i−1]1A[2i]0. (15)
This approach works both for the finite and infinite MPS as
we just used our (i)MPS →Slater determinant approach as a
subroutine. For the infinite MPS it produces a unit cell of size2 as every other Bdiffers by a sign.
As a check of our algorithm, we consider the ground state
of the BdG Hamiltonian of the form
H
BdG=−/summationdisplay
/angbracketleftij/angbracketright,σtij(c†
iσcjσ+H.c.)
−/summationdisplay
/angbracketleftij/angbracketright/Delta1ij(c†
i↑c†
j↓+c†
j↑c†
i↓+H.c.)−μ/summationdisplay
iσc†
iσciσ,
(16)
and compare the amplitudes of the exact ground state with the
MPS generated (see the third column in Fig. 5).
B. Pfaffian →MPS
We will show how to generate a MPS representation of the
Pfaffian ground state of the Kitaev p-wave chain:
Hc=−t/summationdisplay
n(c†
ncn+1+H.c.)+/Delta1/summationdisplay
n(c†
nc†
n+1+H.c.)
+μ/summationdisplay
nc†
ncn. (17)
We first consider Hext=Hc/circleplustextHdwhose ground state is given
by the tensor product of two identical Pfaffians,
|GS/angbracketright=/summationdisplay
σc,σdPf(Mσc)|σc/angbracketright/circlemultiplydisplay
Pf(Mσd)|σd/angbracketright, (18)
where Mis an N×Nmatrix built from parameters of the
model and Mσcis a submatrix of Mobtained by selecting
indices as given by the σcconfiguration.
Using the local canonical transformation of fermions,
c†
n=(¯c†
n,↑+¯c†
n,↓)
√
2,
d†
n=i(¯c†
n,↑−¯c†
n,↓)
√
2,(19)converts Hextto
HBdG=−t/summationdisplay
n,σ(¯c†
n,σ¯cn+1,σ+H.c.)
+/Delta1/summationdisplay
n(¯c†
n↑¯c†
n+1↓+¯c†
n+1,↓¯c†
n,↑+H.c.)
+μ/summationdisplay
nσ¯c†
n,σ¯cn,σ. (20)
The transformation leaves the vacuum unchanged. Given a
BdG Hamiltonian, we can obtain the ground state as a MPSas done in Sec. III A ,
|GS/angbracketright=/summationdisplay
σ···A[2i−1]σ2i−1A[2i]σ2i···|σ1σ2···σ2N/angbracketright,(21)
where A[2i−1]σ2i−1is the tensor on site ifor the ↑local physical
sector and A[2i]σ2iis the tensor on site ifor the ↓local physical
sector.
Notice that the canonical transformation given in Eq. ( 19)
mixes the ↑,↓physical sectors on site i. Hence we can obtain
the MPS for the GS in the c,dspace by choosing the on-site
tensor in the following way,
|GS/angbracketright=/summationdisplay
σC[1]σ1C[2]σ2···C[N]σN|σ1σ2···σN/angbracketright, (22)
withσ={0,c,d,cd}where
C[n],0=A[2n−1],0A[2n],0,
C[n],c=A[2n−1],1A[2n],0+A[2n−1],1A[2n],0
√
2,
C[n],d=iA[2n−1],1A[2n],0−A[2n−1],1A[2n],0
√
2,
C[n],cd=iA[2n−1],1A[2n],1. (23)
By projecting out the dparticles in the |GS/angbracketrightwave func-
tion, we obtain a Pfaffian wave function in the c-particle
sector: |GS/angbracketright=const×/summationtext
σcPf(Mσc)|σc/angbracketright. At the level of the
MPS this projection is realized by eliminating the sectorsσ={d,cd},
|Pf/angbracketright=/summationdisplay
σ={0,c}C[1]σ1C[2]σ2···C[N]σN|σ1σ2···σN/angbracketright. (24)
C. Pfaffian →MPS generalization
While we focused in the previous section on a specific
example, here we consider a generic quadratic HamiltonianH=/summationtext
n,mC†
nhn,mCmwith gspecies of fermions per unit cell
where the vector C†
n=(c†
n,1,cn,1,c†
n,2,cn,2,..., c†
n,g,cn,g).
We form an extended Hamiltonian Hextwhich is a sum of
two copies of H,
Hext=/summationdisplay
hhop
iα,jβ(c†
i,αcj,β+d†
i,αdj,β+H.c.)
+/summationdisplay
hpair
iα,jβ(c†
i,αc†
j,β+d†
i,αd†
j,β+H.c.),(25)
where α,β∈(1,..., g). As before, its ground state |GS/angbracketrightext=/summationtext
σc,σdPf(Mσc)|σc/angbracketright/circlemultiplytextPf(Mσd)|σd/angbracketrightis a tensor product of two
identical Pfaffian wave functions. We then obtain the Pfaffian
125161-6FINITE AND INFINITE MATRIX PRODUCT STATES FOR … PHYSICAL REVIEW B 103, 125161 (2021)
ground state of Hby projecting out all the dsectors. Under
the following linear canonical transformation
¯c†
n,α,↑=c†
n,α+id†
n,α√
2,
¯c†
n,α,↓=c†
n,α−id†
n,α√
2,(26)
Hextbecomes a BdG-like Hamiltonian when expressed in
terms of c↑andc↓:
Hext
BdG=/summationdisplay
hhop
iα,jβ(¯c†
i,α,↑¯cj,β,↑+¯c†
i,α,↓¯cj,β,↓+H.c.)
+/summationdisplay
hpair
iα,jβ(¯c†
i,α,↑¯c†
j,β,↓+¯c†
i,α,↓¯c†
j,β,↑+H.c.).(27)
We can then solve for the MPS representation of the ground
state of the above BdG-like Hamiltonian using the methodsdescribed in Sec. III A .
We obtain the Slater determinant ground state |/Psi1
ext/angbracketrightby
diagonalizing the particle-hole transformed Hext
BdGand then
computing its MPS representation. Each unit cell Mis de-
scribed by 2 gtensors A[M,p]σwithσ=0,1 signifying the
absence /presence of a particle of type p∈[0,1,..., 2g−1].
A particle of type p=2kcorresponds to the flavor k,↑;a
particle of type p=2k+1 corresponds to the flavor k,↓.
From the above MPS (which is in ¯ c↑,¯c↓local physical space)
we construct the MPS tensors in c,dspace. In particular, the
matrices describing the absence /presence of a particle of type
cion site Mare given by
B[M,i]0=A[M,2i−1][0]A[M,2i][1],
B[M,i]1=(−1)g(M−1)+i−1
√
2
×[A[M,2i−1][1]A[M,2i][1]+A[M,2i−1]0A[M,2i][0]],(28)
where ( −1)g(M−1)+i−1takes care of fermionic ordering. and
the MPS representation of |/Psi1GS/angbracketrightdefined on Ngsites is
given by
|/Psi1GS/angbracketright=/summationdisplay
{σ}/parenleftbig
B[1,1]σ11B[1,2]σ12···B[1,g]σ1g/parenrightbig
···
×/parenleftbig
B[N,1]σN1B[N,2]σN2···B[N,g]σNg/parenrightbig
×|/parenleftbig
σ11σ12···σ1g/parenrightbig
···/parenleftbig
σN1σN2···σNg/parenrightbig
/angbracketright.(29)
By suitably contracting tensors we can obtain an N-tensor
MPS representation with a physical dimension 2g:|Pf/angbracketright=/summationtext
σC1σ1C2σ2···CNσN|σ1σ2···σN/angbracketright. For instance, the tensor
corresponding to the presence of particles of type t1,t2,···,ts
on site Mis
C[M](t1,t2,...,ts)=B[M,1][0]···
×B[M,t1][1]B[M,t1+1][0]···B[M,ts][1]···B[M,g][0].
(30)
D. Power of Slater determinants
In this section we describe how to obtain the MPS repre-
sentation of a wave function
|ψ1/n/angbracketright=/summationdisplay
r1,r2,...,rn/angbracketleftr1,r2,..., rn|ψ/angbracketrightn|r1,r2,..., rn/angbracketright,(31)where |ψ/angbracketrightis a Slater determinant. We will use as an example
n=3. Products of other mean-field wave functions can be
obtained similarly.
We extend our N-site system to a 3 N-site system for
which we label the sites as {111213212223···N1N2N3}.W e
then write a single Slater determinant (by padding and inter-lacing the orbitals to keep the above ordering) of the form|ψ/angbracketright/circlemultiplytext|ψ/angbracketright/circlemultiplytext|ψ/angbracketrightfor which we then convert into a MPS
given by
|MPS/angbracketright=/summationdisplay
ip(B[11]i11B[12]i12B[13]i13)···
×(B[N1]iN1B[N2]iN2B[N3]iN3)
×|i11i12i13···iN1iN2iN3/angbracketright. (32)
Projecting on the sector in1=in2=in3gives us the desired
results of
|ψ1/3/angbracketright=A[1]i1A[2]i2···A[N]iN|i1i2···iN/angbracketright, (33)
where we define
A[n]in=B[n1]in1B[n2]in2B[n3]in3. (34)
IV . BILINEAR-BIQUADRATIC S=1M O D E L
In this section, we use our approach to compute the MPS
representation and entanglement spectra of the Gutzwillerprojected slave-fermion mean-field states [ 17] of the bilinear-
biquadratic (BLBQ) S=1 model,
H=/radicalBig
J2
1+J2
2/summationdisplay
/angbracketlefti,j/angbracketright[cosθSi·Sj+sinθ(Si·Sj)2],(35)
The physics of the 1D quantum Heisenberg spin chain
is qualitatively different for different spin representations[18]; half-integer spins have a gapless ground state and
power-law spin correlations; integer spins have a gappedground state with exponentially decaying correlations, theHaldane /Affleck-Kennedy-Lieb-Tasaki (AKLT) phase [ 19].
This latter phase is robust due to a combination of symme-tries which protect its topological properties [ 20,21]. This
symmetry protection can be understood in terms of “fraction-alization”: A S=1 spin effectively splits into two S=1/2
edge modes that transform under nontrivial projective repre-sentations of the symmetries (the product of the symmetryrepresentations differs from the representation of the product).These features are reflected by nontrivial degeneracies in theentanglement spectrum [ 22,23], i.e., the eigenvalues of H
ent
inρA=e−Hentwhere ρA=TrB|ψ/angbracketright/angbracketleftψ|is the reduced density
matrix on an Asubsystem [ 24–26]. The BLBQ model has four
phases as shown in Fig. 6. This includes the Haldane phase
(at the Heisenberg point), as well as a dimerized and criticalphase.
One can derive the relevant projected mean-field state from
the slave-fermion construction by fractionalizing the spin op-erators ˆSin terms of fermionic parton operators,
ˆS
i=f†
i;αSαβfi;β, (36)
where f†
i;αis the α-flavor fermionic parton creation operator
at site iandSαβare the matrix elements of the spin operators
125161-7PETRICA, ZHENG, CHAN, AND CLARK PHYSICAL REVIEW B 103, 125161 (2021)
FIG. 6. Phase diagram of the bilinear-biquadratic S=1 model.
Figure reproduced from Ref. [ 17].
in a given representation S[27]. Substituting these expressions
into the original Hamiltonian gives a quartic fermionic Hamil-tonian H
fwhich can be decoupled through a mean field. The
resulting mean-field ground state must then be projected backinto the original Hilbert space, essentially “gluing” togetherthe fractionalized degrees of freedom.
The slave-fermion construction of the bilinear-biquadratic
model was studied in Ref. [ 17] where the authors used VMC
to optimize the ( χ,δ,μ ) parameters of the projected wave
function and studied the energy of this slave-fermion statecompared to the exact energy achieved by the time-evolvingblock decimation (TEBD) algorithm. In this section, we willtake the same points as studied in ref. [ 17,28], convert the
slave-fermion states to MPS, and compute the entanglementspectra and the energies.
At the level of the Gutzwiller projected ground states, the
two gapped phases, the dimer phase and the AKLT phase, aredistinguished by their “fingerprint” in the low-lying structureof the entanglement spectrum: The lowest level of the entan-glement spectrum of the dimerized topologically trivial phaseis singly degenerate (for between dimer cuts); the lowest levelof the entanglement spectrum of the Haldane phase groundstate is doubly degenerate, corresponding to the presence oftwo boundary S=1/2 edge modes.
A. Generating the MPS
The relevant mean-field Hamiltonian that arises from the
parton construction of the bilinear-biquadratic S=1 model
[27,28]i s
Hmf=−Jχ/summationdisplay
i,α=−1,0,1[c†
i,αci,α+H.c.]
+(J−K)/Delta1/summationdisplay
i,j[c†
i,−1c†
j,1−c†
0ic†
0j+c†
1ic†
−1j+H.c.]
+λ/summationdisplay
i,αc†
iαciα, (37)
where the c†
−1,c†
0, and c†
1are the on-site fermion parton flavors
corresponding to Sz=−1,0,1.This could be converted into a MPS by treating it as a
general Pfaffian and then applying the techniques in Sec. III B.
In models such as this, though, where the mean-field Hamilto-nian in the parton basis has a tensor sum structure where oneor more of the Hilbert subspaces can be treated with a simplermean field (i.e., with a SD or BdG ground state) it makescomputational sense to obtain the MPS representation in eachsector and then “glue” the two MPS together; we exemplifythis approach here.
Introducing a Nambu spinor, in the kbasis the Hamiltonian
is block-diagonal,
H
k
mf=1
2/bracketleftbig
c†
k,1c−k,−1c†
k,0c−k,0/bracketrightbig
×⎡
⎢⎣χk/Delta1k 00
/Delta1∗
k−χk 00
00 χk−/Delta1k
00 −/Delta1∗
k−χk⎤
⎥⎦⎡
⎢⎢⎣ck,1
c†
−k,−1
ck,0
c†
−k,0⎤
⎥⎥⎦.(38)
The one-body Hamiltonian is a tensor sum of BdG-like
Hamiltonian HBdGand a p-wave Hamiltonian Hp. The mean-
field ground state is (we consider antiperiodic boundaryconditions and an even number of sites)
|/Psi1
GS/angbracketright=/Pi10<k<2π(uk+vkc†
k,1c†
−k,−1)
×/Pi10<q<π(uq−vqc†
q,0c†
−q,0)|0/angbracketright, (39)
where ukandvkare given in terms of the parameters of the
Hamiltonian.
By performing a particle-hole transformation in the Sz=
{↑,↓}sector (see Sec. III A ) and the Pfaffian artificial exten-
sion in the Sz=0 sector (see Sec. III B),
f†
1,k=c†
k,1,
f†
2,k=c−k,−1,
f†
3,k=c†
k,0,
f†
4,k=c−k,0, (40)
where {fk,α,fq,β}=δαβδkq,g i v i n gu s
/vextendsingle/vextendsingle/Psi1ext
GS/angbracketrightbig
=/Pi1k(ukf†
k,2+vkf†
k,1)
×/Pi10<q</Pi1(uqf†
q,4−vqf†
k,3)|vac/angbracketright, (41)
with|vac/angbracketright=/Pi10<k<2πck↓|0/angbracketright/Pi10<q<πck,0|0/angbracketright. Since |/Psi1ext
GS/angbracketrightis a
Slater determinant, we can obtain the MPS representation us-ing the methods in Sec. II. We now “undo” the transformation
(see again Secs. III A andIII B) and write the MPS in the
following form,
|MPS/angbracketright=/summationdisplay
{i}(C[1↑]i1↑C[1↓]i1↓C[1,→]i1,→)
× ··· (C[N↑]iN↑C[N↓]iN↓C[N,→]iN→)
×|(i1↑i1↓i1→)···(iN↑iN↓iN→)/angbracketright, (42)
where in,α∈{0,1}indicates the absence /presence of a parti-
cle of type αon site n.
To obtain the MPS with on-site tensors A[n]σn,σn∈{ ↑,↓
,→,↑→,↓→,↑↓,↑↓→} , we “glue” together appropriate
125161-8FINITE AND INFINITE MATRIX PRODUCT STATES FOR … PHYSICAL REVIEW B 103, 125161 (2021)
sectors. For example,
A[n]↑=C[n↑]1C[n↓]0C[n→]0, (43)
and the Gutzwiller projection is realized by summing only
over the one-particle per site physical indices σn∈{ ↑,↓,→}:
PG|/Psi1GS/angbracketright=/summationdisplay
σn∈{↑,↓,→}A[1]σ1···A[N]σN|σ1···σN/angbracketright.(44)
B. iMPS orthogonalization
In this section, we will discuss orthogonalizing our iMPS
states. This includes a brief overview of the standard iMPSorthogonalization as well as a detailed description of howwe address the degeneracies that appear when Gutzwillerprojecting slave-fermion mean-field states onto degenerateground-state manifolds.
The orthogonalization procedure for a typical iMPS is stan-
dard (see Ref. [ 29] and SM 6 [ 12] for an intuitive derivation
and for more details). The method relies on obtaining theleading right /left eigenvectors of the transfer matrix operator
E=/summationtext
σAσ⊗Aσ∗where σruns over the on-site physical
index. Eadmits the following decomposition,
E=/summationdisplay
iλi|R/angbracketrighti/angbracketleftL|i, (45)
where |L/angbracketrightiand|R/angbracketrightiare left /right eigenvectors of Eand
/angbracketleftLi|Ri/angbracketright=0f o rλinondegenerate. In the infinite limit only the
leading left /right eigenvectors of Esurvive. If the dominant
eigenvalue is nondegenerate, the transfer matrix is given by
lim
N→∞EN=|R/angbracketright/angbracketleftL|, (46)
where |R(L)/angbracketrightare by definition the eigenvectors correspond-
ing to the dominant eigenvalue. Thus, the dominant leftand right eigenvectors correspond to a pure state. The im-plicitly restarted Arnoldi method can be efficiently usedfor this purpose by noting that (/summationtext
σAσ⊗Aσ∗)vec(v)=/summationtext
σvec(Aσ∗vAT), where the vec( v) operation takes the square
matrix vand stacks the columns together. The entanglement
spectrum and observables are then easily obtained.
When the leading eigenvalues of the transfer matrix are
degenerate in magnitude,
lim
N→∞EN=|R1/angbracketright/angbracketleftL1|+|R2/angbracketright/angbracketleftL2|, (47)
ENis in mixed form. In general the output of the Arnoldi
method gives /angbracketleftLi|Ri/angbracketright/negationslash=0. Thus, additional steps are required
to obtain the canonical form of the iMPS. The degeneracyof the leading eigenvalues signals the presence of degeneratestates. This is indeed what happens for the twofold degeneratedimer phase and the fourfold degenerate Haldane phase (inthe thermodynamic limit). In order to access all the states inthe ground-state manifold, we need to obtain the proper setof pure iMPS states. The transfer matrix of a pure iMPS hasunique left /right leading eigenvectors.
Here, we consider the case of twofold degeneracy present
in the dimer phase states (other states and higher degeneraciescan be dealt with using a similar procedure). For a twofolddegenerate iMPS, we need to find two pure iMPS generatedby bulk tensors A
1andA2. Any iMPS within the degenerate
manifold is then able to be written as a linear superposition
FIG. 7. Illustration of decomposition of a mixed transfer matrix
into pure components. In step 1 (first line), we find the dominantleft eigenvector of the transfer matrix; In step 4 (second line), we
findU
Land ˜S1and ˜S2. In step 5 (third, fourth, and fifth lines), we
form two new tensors ˜Biand find their right leading eigenvectors.
In step 6 (other lines), we find the entanglement spectrum√/Lambda1iand
the right /left canonical matrices BR
iandBL
i, corresponding to the two
pure states.
of these pure states, |ψB/angbracketright=α1|ψ(A1)/angbracketright+α2|ψ(A2)/angbracketright, where
the notation |ψ(B)/angbracketrightindicates the iMPS generated by bulk
tensor B. Note that the entanglement spectrum of the reduced
density matrix ρB=|α1|2ρA1+|α2|2ρA2is given by the com-
bined spectra of α1ρ1andα2ρ2.
To generate these pure iMPS, we start from a noncanonical
bulk tensor AiMPS with a single-site unit cell (typically gen-
erated by projection). In the case of the dimer phase, this bulktensor has two leading right (respectively left) eigenvectors,v
1andv2, with equal magnitude eigenvalues |η1|=|η2|,b u t
different signs (i.e., η1=−η2).
125161-9PETRICA, ZHENG, CHAN, AND CLARK PHYSICAL REVIEW B 103, 125161 (2021)
TABLE I. Comparison of energies per site between the exact ground state (iTEBD /DMRG), variational Monte Carlo (VMC), and the
fMPS and iMPS generated from the projected slave-fermion states. fMPS are computed with N=64 or N=96 and 10−4/lessorequalslant/epsilon1/lessorequalslant3×10−4.
The bulk tensors used in iMPS have bond dimension D≈1000. Column headings correspond to ( J,K).
(1,1)ULS (1,1
3)AKLT(1,0)Heisenberg (1,−1)TB (1,−2) (1 ,−3) (0 ,−1) ( −1,−3) ( −1,−2)
iTEBD [ 17] 0.2971 −2
3−1.4015 −4 −6.7531 −9.5330 −2.7969 −7.3518 −4.5939
DMRG 0.2978 −2
3−1.4015 −3.9999 −6.7526 −9.5314 −2.7969 −7.3516 −4.5939
VMC [ 17] 0.2997 −2
3−1.4001 −3.9917 −6.7372 −9.5103 −2.7953 −7.2901 −4.4946
±0.0004 ±7×10−15±0.0004 ±0.0012 ±0.0023 ±0.0034 ±0.0005 ±0.0038 ±0.0028
fMPS 0.2995 −2
3−1.3999 −3.9895 −6.7369 −9.5073 −2.7948 −7.2877 −4.4935
iMPS −2
3−1.3999 −6.7368 −9.5071 −2.7947 −7.2877 −4.4934
χ 1 1 1 1 1100 0
/Delta1 03
20.98 1.11 1.15 1.79 1 1 1
λ 1 0 1.78 2.00 2.07 2.22 0.14 0.21 0.12
According to Theorem 5 in Ref. [ 30] and Theorem 11 in
Ref. [ 31], there is a unitary that transforms each of the matri-
cesBσσ/prime=AσAσ/primeinto block-diagonal form, with two blocks;
the two blocks are the two-site uniform tensors correspondingto the two pure states.
Based on the mathematical theorems in Refs. [ 30,31], we
use the following procedure to compute the pure states (seeFig.7):
(1) Start with the D×D(Dis the bond dimension of
the bulk tensors A) left leading eigenvectors (with the same
eigenvalue), V
L
1andVL
2, of the completely positive map E2,
i.e.,/summationtext
σ,σ/primeB†σσ/primeVL
iBσ/prime=VL
i.
(2)VL
iare transformed into Hermitian matrices: VL
i:=
1/2[VL
i+(VL
i)†]; this is possible because if VL
iis an eigen-
vector of E2, then ( VL
i)†is also an eigenvector and so their
sum is Hermitian. If VL
i=UiDiU†
i, then we can write VL
i=
Y†
iYiwith Yi=√DiU†
i.
(3) Diagonalize VL
1andVL
2together; this can be done since
[VL
1,VL
2]=0 so that U†VL
iU=Si, with Sibeing a diagonal
matrix.
(4) Form two linear combinations˜VL
i=VL
1−αiVL
2
where αiis one of the two nonzero values obtained
by the elementwise division of S1and S2. Then
U†
L˜VL
1ULwill be a diagonal matrix ˜S1with entries
(˜d1
1,˜d2
1,..., ˜dp
1,0,0,..., 0) and U†
L˜VL
2ULa diagonal matrix
˜S2with entries (0 ,0,..., 0,˜dD−k
2,˜dD−k−1
2,..., ˜dD
2,) and
D−k/greaterorequalslantp; in fact, it will almost always be the case that
D−k>p, since the bond dimension of the canonical bulk
tensor decreases after projection; this decomposition isguaranteed by Theorem 5 in Ref. [ 30].
(5) Form two new two-site bulk tensors ˜B
i=/radicalbig˜SiU†
LBUL/radicalbig˜Si−1and obtain their transfer matrix
right-leading eigenvectors; they will each have a uniqueleading Hermitian semipositive definite diagonal eigenvector,
V
R
i=UR,i/Lambda1i(UR,i)†; we can write VR
i=XiX†
iwith
Xi=UR,i√
/Lambda1; then√/Lambda1iis the entanglement spectrum of
the corresponding pure state.
(6)BR
i=√/Lambda1i−1(UR,i)†˜BiUR,i√/Lambda1iare the right canoni-
cal tensors and BL
i=(UR,i)†˜BiUR,iare the left canonical
tensors; since BL
i√/Lambda1i=√/Lambda1iBR
ithe uniform two-site trans-lationally invariant bulk tensors can be written as Ai=/radicalbig√/Lambda1iBR
i/radicalbig√/Lambda1i−1=/radicalbig√/Lambda1i−1BL
i/radicalbig√/Lambda1i.
C. Energy of BLBQ slave-fermion wave functions
We compute both the MPS and iMPS (except at the critical
points) for the variational Gutzwiller projected wave functionscorresponding (as found in Ref. [ 17] by minimizing the vari-
ational energy) to the points in Fig. 6. We directly compare
the energy for all of these points (see Table I) and find that
the energies are all within the error bars reported for the VMCcalculation [ 17].
D. Entanglement spectra of BLBQ slave-fermion wave functions
1. Dimer phase
In this section, we will consider entanglement spectra of
the dimerized phase of the BLBQ model. The ground stateof the dimerized phase is twofold degenerate depending onwhether the dimer covering spans even or odd bonds; the en-tanglement spectra also depends on whether the entanglementcut is made through or between dimers. For the fMPS, we canobtain both the even and odd cut entanglement spectra of thedimerized states by choosing two consecutive cuts whereasfor the iMPS we use the procedure described in Sec. IV B to
find the two pure states which correspond respectively to theeven and odd cuts.
We start by considering a generic slave-fermion point in
the BLBQ model; see Fig. 8for the entanglement spectrum
(ES). The iMPS and fMPS slave-fermion point agrees wellboth with each other and the exact ES from DMRG.
The low-lying level of the entanglement spectrum cycles
between a singlet and a triplet as we move the locationof the entanglement cut within the chain. This is indicativeof translation invariance breaking and the dimerized structureof the ground state: Namely the low-level singlet is associatedwith a cut between dimers, whereas the low-level triplet isassociated with a cut inside dimers.
We can also further understand the higher states in the
entanglement spectra. A generic point in the dimer phase ofthe BLBQ model is SU(2) symmetric. Consequently, the en-tanglement levels transform under SU(2) representation and
125161-10FINITE AND INFINITE MATRIX PRODUCT STATES FOR … PHYSICAL REVIEW B 103, 125161 (2021)
FIG. 8. Entanglement spectra of the projected mean-field state
from iMPS and fMPS at ( J,K)=(1,−2) showing (a) the compar-
ison with DMRG between dimers, and the fMPS spin-resolved ES
(b) between dimers and (c) within dimers. Note that the single and
triple degeneracies seen in (b) and (c) are the expected low-levelstructures seen in a pure VBS state. The lowest three entanglement
levels of (b) are representations of SU(2): singlet, triplet, and quintet.
therefore we expect that degeneracies should go as the dimen-
sion of SU(2) representations (i.e., 2 n+1 for non-negative
integer n); this can be seen in the multiplet structure of both
the entanglement spectra in Fig. 8(bottom left), where the
lowest three degeneracies between dimers form the singlet(S=0), triplet ( S=1), and quintuplet ( S=2).
Beyond considering a generic point within the dimer
phase, we now consider the exactly solvable point where
(J,K)=(0,−1) [the so-called Klümper-Barber-Batchelor
(KBB) point [ 32,33]], which is invariant under a larger
symmetry group, SU(3) [as opposed to SU(2)]. This largersymmetry group forces the triplet and quintet to form anoctet [the adjoint representation of SU(3)]. Variationally, thevanishing of the hopping parameter in the slave-fermionmean-field Hamiltonian forces this larger symmetry group atthe level of the variational Gutzwiller projected wave function.See Fig. 9.
The points ( J,K)=(−1,−2) and ( J,K)=(−1,−3)
which are found at a variational minima with t=0i nt h e
parent Hamiltonian by Ref. [ 17] also have SU(3) symmetry.
This symmetry is not present in the true DMRG ground statewhich transforms only under SU(2) symmetry; therefore, abetter agreement is obtained by perturbing slightly away fromthis point (see SM 7 [ 12] for entanglement spectrum data of
the slightly perturbed points).
2. Haldane phase
In this section we compute the entanglement spectra of
the AKLT and Heisenberg points belonging to the Haldane
FIG. 9. Spin-resolved entanglement spectrum (between dimers)
for the ( J,K)=(0,−1) variational point generated from fMPS. The
SU(3) symmetry forces the S=1a n d S=2 into a degenerate octet.
phase of BLBQ. The slave-fermion mean field for this model
has a fourfold degeneracy at the Fermi level that allows forchoosing six orthogonal preprojected mean-field states (at halffilling). Projecting each of these states generates (postpro-jection) a space of MPS which span four degenerate groundstates which correspond to the representations of the sum ofthe two fractionalized S=1/2 edge modes of the Haldane
phase.
Here, we start by considering the AKLT point which has
an exact analytic solution. The AKLT point ( J,K)=(1,1/3)
is exactly mapped under the above slave-fermion projectiveconstruction to ( χ,δ,μ )=(1,3/2,0). To find the AKLT state
which (for example) has the edge modes ↑↓we can either
search in the fourfold projected degenerate space or choosethe correct orbitals at the Fermi-level preprojection. We findthe entanglement spectra for each of the four fMPS whichcorrespond to ↓↓,↓↑,↑↓,↑↑edge spin configurations is
equal to ln(2).
For the iMPS, unlike the dimer phase, where there was
twofold degeneracy in the leading eigenvalues of the trans-fer matrix operator for points in the Haldane phase, we findfourfold degeneracy. We obtain two-negative and two-positive(equal in magnitude) leading eigenvalues. Taking the spacespanned by the two eigenvectors with positive eigenvalues, weapply the iMPS orthogonalization procedure from Sec. IV B .
From this process, for the AKLT slave-fermion point we findafter orthonormalization the iMPS
A
0=/parenleftbigg
−10
01/parenrightbigg
,
A↓=√
2/parenleftbigg
0−e−iθ
00/parenrightbigg
,
A↑=√
2/parenleftbigg00
eiθ0/parenrightbigg
,(48)
associated with two pure states (i.e., |↑↓/angbracketright and|↓↑/angbracketright of the edge
modes). In SM 8 [ 12], we also obtain the iMPS representation
of the S=1/2 VBS ground state of the Majumdar-Ghosh
(MG) chain [ 34,35].
125161-11PETRICA, ZHENG, CHAN, AND CLARK PHYSICAL REVIEW B 103, 125161 (2021)
FIG. 10. Top: Comparison of entanglement spectrum obtained
from fMPS, iMPS, and DMRG at the ( J,K)=(1,0) Heisenberg
point. Bottom: Spin-resolved entanglement spectrum from fMPS at
the (J,K)=(1,0) Heisenberg point in the ( S=1,Sz=1) sector.
In Fig. 10we also present the entanglement spectrum of the
Heisenberg point ( J,K)=(1,0) obtained using both fMPS
and iMPS. We see that the lower levels match well the en-tanglement spectrum levels obtained from DMRG of the trueHeisenberg ground state. Discrepancies occur naturally higherup in the spectrum as the variational wave function is not theexact ground state. However, the entanglement spectrum of
the variational ground state (qualitatively) captures the sym-metries and degeneracies of the true entanglement spectrum.
Note the fact that every level has even degeneracy comes fromthe topological nature of the phase. Moreover, notice that thelowest parts of the entanglement spectra match the AKLT statein the same sector.
3. Critical points
We compute the two critical points at ( J,K)=(1,−1) [the
Takhtajan-Babujian (TB) point [ 36,37]] and at ( J,K)=(1,1)
[the Uimin-Lai-Sutherland (ULS) [ 38–40]]; they are gapless
and hence we analyze them only in the framework of ourfMPS method. The TB ground state is unique; the associatedeffective conformal field theory is SU(2) |
k=2.
The ULS ground state is also unique. However, it has
an enlarged SU(3) symmetry group; the associated effec-tive conformal field theory is SU(3) |
k=1. In particular it
can be mapped to the SU(3) nearest-neighbor Heisenbergmodel [ 41]. This enforces an equal number of “quark”
FIG. 11. (a) Comparison of entanglement spectra at the ULS
point for fMPS and DMRG. (b) Comparison of entanglement spectra
at the TB point for fMPS and DMRG. (c) Spin-resolved entangle-
ment spectrum for the (left) TB point and (right) ULS point.
particle constraints (and hence a global equal number of
spins 1 ,−1,0). At the mean-field level, the pairing parameter
vanishes since J−K=0. The Hamiltonian is then a tensor
sum of 3 identical hopping Hamiltonians acting indepen-dently on the fermions of flavor “up,” “down,” and “zero.”The particle number constraint of c
1,c−1,c0is naturally en-
forced at the mean-field level if the number of sites Nis a
multiple of 3.
The SU(3) symmetry of the ULS point is reflected in the
degeneracies of the entanglement spectrum where the S=1
andS=2 levels combine together to form SU(3) octets as
can be seen in Fig. 11. For the TB point the S=1 and S=2
entanglement levels remain separated.
The central charge of SU( N)|kconformal field theories
(CFTs) is given by c=k(N2−1)/(N+k). Hence, ana-
lytically c=1.5 for the TB point and c=2 for ULS.
Calabrese and Cardy [ 42] obtained the following expression
for the entanglement entropy scaling for a 1D critical gap-less point of finite size Lwith open-boundary conditions and
partition size x,
S(x,L)=c
6ln2L
πsin/parenleftBigπx
L/parenrightBig
+lng+s1/2, (49)
125161-12FINITE AND INFINITE MATRIX PRODUCT STATES FOR … PHYSICAL REVIEW B 103, 125161 (2021)
FIG. 12. The von Neumann entanglement entropy scaling of the
slave-fermion wave functions describing the ULS critical point (top)
for a system of size L=120 and the TB (bottom) critical point for
a system of size L=300 computed with fMPS. Shown also is the
central charge obtained from least squares fitting.
where ln gis a boundary entropy term and S(x,L)i st h ev o n
Neumann entanglement entropy.
It was found in Ref. [ 43] that there is an additional
alternating term in S(x,L) which decays away from the
boundaries. In Fig. 12we plot the entanglement entropy
againstc
6ln2L
πsin (πx
L) for both TB and ULS models. We
work on a system of size L=300 and plot the region x∈
[75,110] for TB and a system of size L=120 and plot the re-
gion x∈[21,44] for ULS. We picked the lower bounds to be
far enough from the boundary and the upper bounds to workin the region of the sine curve with xaway from L/2 where
the curve becomes very flat. For the SU(3) ULS point, whenxis a multiple of 3, the highest eigenvalue Schmidt vector
contains equal numbers of “quarks” and hence is dominant.For cuts at x=3k+1,3k+2 (for integer k), the Schmidt
vectors cannot satisfy the particle conservation constraint andhence the highest eigenvalue Schmidt vectors are degenerate.A similar situation occurs at the SU(2) TB point where the2-periodicity is easily explained in the dimer picture: For evencuts we cut between dimers, whereas for odd cuts we breakdimers and hence split the singlet apart. The alternating termis still significant for the parameters we chose. Hence, it isdifficult to reliably extract the central charge. However, wemanage to obtain results for central charges at both pointswhich are remarkably close to their theoretical values: 2.013as compared to 2.0 for the ULS point and 1.505 as comparedto 1.5 for the TB point. We overlay the lines obtained fromleast-squares fitting for both models.V . DISCUSSION AND FUTURE WORK
We have developed a series of efficient and highly parallel
algorithms to obtain the finite and infinite (for gapped states)MPS representation of fermionic mean-field states. Gutzwillerprojection is easily implemented by eliminating the doublyoccupied and unoccupied physical sectors of the mean-fieldslave-fermion MPS tensors. We have used these methodsto obtain the (i)MPS representation of Gutzwiller projectedmean-field states that arise from the variational slave-fermionapproach to the S=1 bilinear-biquadratic (BLBQ) quantum
spin chain introduced in Ref. [ 17]. We first verify that the
energies we obtain via both finite MPS and infinite MPS (notapplicable to the critical points) for the points considered arewithin the error bars of their VMC calculations [ 17].
Additionally, we obtain the entanglement spectra at two
critical points (ULS and TB) and several generic points in thedimer and Haldane phases of the BLBQ model. We find goodqualitative (and quantitative) agreement with results obtaineddirectly from DMRG. We briefly discuss the salient structuralfeatures of the entanglement spectrum in all the phases (butsee Ref. [ 41] for a more detailed analysis). Extracting the
central charges of the conformal field theories describing thetwo gapless critical points from a numerical computation ofthe entanglement spectrum on finite open-boundary systemsis made difficult by a slowly decaying oscillatory term inthe entanglement entropy. However, we do obtain very goodagreement for the central charge at both the ULS and TBpoint: 2.013 as compared to the exact analytical value of2.0 for the ULS point and 1.505 as compared to the exactanalytical value of 1.5 for TB.
We also introduce an algorithmic procedure that orthogo-
nalizes an iMPS by breaking it down into its pure states. Thisis essential when dealing with degenerate ground states thatappear upon Gutzwiller projection as is the case with pointsin the dimer phase of the BLBQ model. Having obtainedthe pure states, we can compute the entanglement spectrumfor any state in the ground-state manifold. We check thatthe entanglement spectrum obtained from iMPS matches theone we obtain from the finite MPS procedure. Discrepanciesnaturally appear as we approach values close to the thresholdsused to generate the finite MPS and infinite MPS.
The methods can be easily adapted to the study of systems
on 2D ladders (infinite in length but with finite width). TheiMPS unit cell is now formed by the tensors sitting on thewidth of the cylinder. We will explore the applications ofGutzwiller projected variational wave functions to the study of2D quantum spin liquids in future publications. This methodmay also be applicable to topological states such as quantumHall and fractional Chern insulators that are represented asproducts of mean-field wave functions.
ACKNOWLEDGMENTS
B.K.C. acknowledges support from the Department of
Energy Grant No. DOE de-sc0020165. This project is part ofthe Blue Waters sustained petascale computing project, whichis supported by the National Science Foundation (AwardsNo. OCI-0725070 and No. ACI-1238993) and the State ofIllinois. Blue Waters is a joint effort of the University of
125161-13PETRICA, ZHENG, CHAN, AND CLARK PHYSICAL REVIEW B 103, 125161 (2021)
Illinois at Urbana-Champaign and its National Center for
Supercomputing Applications. G.K.-L.C. was supported bythe US National Science Foundation via Grant No. 1839204.G.K.-L.C. also acknowledges support from the Simons Foun-
dation via the Investigator Award and the Many-ElectronCollaboration.
[1] A. J. Leggett, Quantum Liquids: Bose Condensation and Cooper
Pairing in Condensed-Matter Systems (Oxford University Press,
Oxford, U.K., 2006).
[2] N. Read and D. Green, Paired states of fermions in two di-
mensions with breaking of parity and time-reversal symmetriesand the fractional quantum Hall effect, P h y s .R e v .B 61, 10267
(2000) .
[3] M. T. Fishman and S. R. White, Compression of correlation
matrices and an efficient method for forming matrix productstates of fermionic Gaussian states, Phys. Rev. B 92, 075132
(2015) .
[4] Y .-H. Wu, L. Wang, and H.-H. Tu, Tensor Network Represen-
tations of Parton Wave Functions, Phys. Rev. Lett. 124, 246401
(2020) .
[5] H.-K. Jin, H.-H. Tu, and Y . Zhou, Efficient tensor network rep-
resentation for Gutzwiller projected states of paired fermions,P h y s .R e v .B 101, 165135 (2020) .
[6] N. Schuch and B. Bauer, Matrix product state algorithms
for Gaussian fermionic states, Phys. Rev. B 100, 245121
(2019) .
[7] U. Schollwöck, The density-matrix renormalization group in
the age of matrix product states, Ann. Phys. 326, 96 (2011) .
[8] S.-A. Cheong and C. L. Henley, Many-body density matrices
for free fermions, Phys. Rev. B 69, 075111 (2004) .
[9] I. Klich, Lower entropy bounds and particle number fluctuations
in a Fermi sea, J. Phys. A: Math. Gen. 39, L85 (2006) .
[10] G. Knizia and G. K.-L. Chan, Density Matrix Embedding: A
Simple Alternative to Dynamical Mean-Field Theory, Phys.
Rev. Lett. 109, 186404 (2012) .
[11] I. Peschel, Special review: Entanglement in solvable many-
particle models, Braz. J. Phys. 42, 267 (2012) .
[12] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.103.125161 for more details regarding the
Schmidt decomposition of Slater determinants, the mathemati-cal expression of the boundary tensors, entanglement spectrumdata in the context of the SSH model, a mathematical derivation,how we deal with gapless boundary modes in the context ofthe iMPS procedure, an intuitive derivation, entanglement spec-trum data of the slightly perturbed points, and derivations andnumerical data.
[13] D. A. Harville, Matrix Algebra from a Statistician’s Perspective
(Springer, New York, 1997).
[14] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in Poly-
acetylene, P h y s .R e v .L e t t . 42, 1698 (1979) .
[15] S. Ripka, J. Blaizot, and G. Ripka, Quantum Theory of Finite
Systems (MIT Press, Cambridge, MA, 1986).
[16] A. Y . Kitaev, Unpaired Majorana fermions in quantum wires,
Phys. Usp. 44, 131 (2001) .
[17] Z.-X. Liu, Y . Zhou, H.-H. Tu, X.-G. Wen, and T.-K. Ng,
Gutzwiller projected wave functions in the fermionic theory ofS=1s p i nc h a i n s , P h y s .R e v .B 85, 195144 (2012) .
[18] F. D. M. Haldane, Continuum dynamics of the 1-D Heisenberg
antiferromagnet: Identification with the O(3) nonlinear sigmamodel, Phys. Lett. A 93, 464 (1983) .[19] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Valence
bond ground states in isotropic quantum antiferromagnets, inCondensed Matter Physics and Exactly Soluble Models(Springer, Berlin, 1988), pp. 253–304.
[20] Z.-C. Gu and X.-G. Wen, Tensor-entanglement-filtering renor-
malization approach and symmetry-protected topological order,Phys. Rev. B 80, 155131 (2009) .
[21] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Sym-
metry protection of topological phases in one-dimensionalquantum spin systems, Phys. Rev. B 85, 075125 (2012) .
[22] A. M. Turner, F. Pollmann, and E. Berg, Topological phases
of one-dimensional fermions: An entanglement point of view,Phys. Rev. B 83, 075102 (2011) .
[23] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, En-
tanglement spectrum of a topological phase in one dimension,Phys. Rev. B 81, 064439 (2010) .
[24] H. Li and F. D. M. Haldane, Entanglement Spectrum as
a Generalization of Entanglement Entropy: Identification ofTopological Order in Non-Abelian Fractional Quantum HallEffect States, P h y s .R e v .L e t t . 101, 010504 (2008) .
[25] M. Levin and X.-G. Wen, Detecting Topological Order in a
Ground State Wave Function, Phys. Rev. Lett. 96, 110405
(2006) .
[26] A. Kitaev and J. Preskill, Topological Entanglement Entropy,
Phys. Rev. Lett. 96, 110404 (2006) .
[27] Z.-X. Liu, Y . Zhou, and T.-K. Ng, Fermionic theory for quantum
antiferromagnets with spin S>
1
2,P h y s .R e v .B 82, 144422
(2010) .
[28] Z.-X. Liu, Y . Zhou, and T.-K. Ng, Gutzwiller approach for
elementary excitations in S=1 antiferromagnetic chains, New
J. Phys. 16, 083031 (2014) .
[29] R. Orus and G. Vidal, Infinite time-evolving block decimation
algorithm beyond unitary evolution, Phys. Rev. B 78, 155117
(2008) .
[30] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac,
Matrix product state representations, Quantum Inf. Comput. 7,
401 (2007).
[31] M. S. Ruiz, Tensor networks in condensed matter, Ph.D. thesis,
Technische Universität München, 2011.
[32] A. Klümper, New results for q-state vertex models and the pure
biquadratic spin-1 Hamiltonian, Europhys. Lett. 9, 815 (1989) .
[33] M. N. Barber and M. T. Batchelor, Spectrum of the biquadratic
spin-1 antiferromagnetic chain, P h y s .R e v .B 40, 4621 (1989) .
[34] C. K. Majumdar and D. K. Ghosh, On next-nearest-neighbor
interaction in linear chain. I, J. Math. Phys. 10, 1388 (1969) .
[35] V . Karimipour and L. Memarzadeh, Matrix product represen-
tations for all valence bond states, Phys. Rev. B 77, 094416
(2008) .
[36] L. Takhtajan, The picture of low-lying excitations in the
isotropic Heisenberg chain of arbitrary spins, Phys. Lett. A 87,
479 (1982) .
[37] H. Babujian, Exact solution of the one-dimensional isotropic
Heisenberg chain with arbitrary spins S,Phys. Lett. A 90, 479
(1982) .
125161-14FINITE AND INFINITE MATRIX PRODUCT STATES FOR … PHYSICAL REVIEW B 103, 125161 (2021)
[38] G. Uimin, One-dimensional problem for S=1 with modified
antiferromagnetic Hamiltonian, JETP Lett. 12, 225 (1970).
[39] C. Lai, Lattice gas with nearest-neighbor interaction in one
dimension with arbitrary statistics, J. Math. Phys. 15, 1675
(1974) .
[40] B. Sutherland, Model for a multicomponent quantum system,
inExactly Solvable Models Of Strongly Correlated Electrons
(World Scientific, Singapore, 1994), pp. 287–297.[41] R. Thomale, S. Rachel, B. A. Bernevig, and D. P. Arovas,
Entanglement analysis of isotropic spin-1 chains, J. Stat. Mech.:
Theory Exp. (2015) P07017 .
[42] P. Calabrese and J. Cardy, Entanglement entropy and quantum
field theory, J. Stat. Mech.: Theory Exp. (2004) P06002 .
[43] N. Laflorencie, E. S. Sørensen, M.-S. Chang, and I. Affleck,
Boundary Effects in the Critical Scaling of Entanglement En-tropy in 1D Systems, P h y s .R e v .L e t t . 96, 100603 (2006) .
125161-15 |
PhysRevB.76.045418.pdf | Selection rule for the optical absorption of graphene nanoribbons
Han Hsu and L. E. Reichl
Center for Complex Quantum Systems and Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
/H20849Received 7 March 2007; published 19 July 2007 /H20850
We demonstrate that the optical absorption of zigzag-edge graphene nanoribbons is qualitatively different
from that of armchair nanotubes. Unlike the selection rule for nanotubes, when the incident beam ispolarized along the longitudinal direction, the interband transitions at direct gaps are forbidden for graphenenanoribbons. This selection rule is due to the finite width of graphene nanoribbons. We also demonstrate thatthe edge states of graphene nanoribbons play an important role in the optical absorption. They are involved inmany of the absorption peaks within optical range /H20849/H6036
/H9275/H110210.12 a.u. /H20850and have no contribution to the absorption
peaks beyond optical range.
DOI: 10.1103/PhysRevB.76.045418 PACS number /H20849s/H20850: 78.40.Ri, 78.67. /H11002n, 73.22. /H11002f
I. INTRODUCTION
Carbon-based nanoscale low-dimensional materials, such
as carbon nanotubes, single- or few-layer graphene, andgraphene nanoribbons /H20849finite-width strips made of graphene /H20850,
have attracted much attention because of their properties andtheir potential in future application.
1–25Recently, the study
for graphene nanoribbons /H20849GNRs /H20850is a rapidly growing
field.3–15The quasi-one-dimensional atomic structure of
GNRs is similar to that of single-walled carbon nanotubes/H20849SWNTs /H20850, but the electronic structures
3–9and transport
properties10–15of GNRs can be very different from that of
SWNTs. As to optical properties, only a few graphene-basedsystems, such as few-layer graphene
16and triangular
graphene fragments,17have been studied, while SWNTs have
been intensively studied in the past decade.18–23
In this paper, we study the optical response of GNRs in
the perturbative regime, and we demonstrate that the opticalabsorption of GNRs is qualitatively different from that ofSWNTs. For SWNTs, when the polarization of the incidentbeam is longitudinal, namely, parallel to the nanotube axis/H20849xdirection /H20850, the optical absorption peaks are mainly caused
by the interband transitions at direct gaps, as explicitly dis-cussed in Sec. III and some of the references.
20,21Such a
selection rule, however, does not apply to GNR. In the caseof GNR, due to its finite width, the eigenstates are eithersymmetric or antisymmetric along the transverse direction /H20849y
direction /H20850, and the transitions at direct gaps are thus forbid-
den.
Another important feature of GNR is the existence of
edge states, which do not occur in SWNTs and infinitegraphene sheets. Edge states are electronic states that local-ize at /H20849and extend along /H20850the edge and decay exponentially
into the center of GNR. The properties of edge states havebeen widely discussed in many theoretical works.
3–15As
shall be discussed later, edge states play an important role inoptical absorption. They are involved in many of the absorp-tion peaks in the optical range /H20849/H6036
/H9275/H110210.12 a.u. /H20850. Beyond op-
tical range, they have no contribution to absorption peaks.
The atomic structure of GNR merits a brief review before
we proceed to any further discussion. While SWNTs are gen-erally categorized by the arrangement of carbon atoms alongthe cross section of the tube,
1,2GNRs are categorized by thearrangement of carbon atoms on the side edges.3Two main
types of GNR are zigzag-edge GNR /H20849ZGNR /H20850and armchair-
edge GNR. Both types of GNRs can be further specified bytheir width, in other words, by the number of longitudinalatomic lines. For example, ZGNR with 12 zigzag lines isreferred to as 12-ZGNR.
7,15By comparing 12-ZGNR and
/H208496, 6 /H20850SWNT, we will notice that for these two quasi-one-
dimensional objects, the number of carbon atoms per unitstrip is the same /H2084924 atoms /H20850. If we roll up a 12-ZGNR and
allow covalent bonds to connect the carbon atoms on oppo-site edges, a /H208496, 6 /H20850SWNT can be constructed.
Two cases will be considered in this paper: The case for
/H208496, 6/H20850SWNT and 12-ZGNR and the case for /H2084910, 10 /H20850SWNT
and 20-ZNGR. The relevant theoretical tools will be re-viewed in Sec. II, and the results of calculations will bediscussed in Sec. III. In Sec. IV, concluding remarks will begiven.
II. THEORETICAL FORMULATION
It is well known that the electronic and optical properties
of nanotubes and graphene are mainly determined by the /H9266
electrons of carbon atoms.1,2To model those /H9266electrons, the
tight-binding /H20849TB/H20850approximation has been widely used. The
TB Hamiltonian His written as
H=/H9253/H20858
/H20855i,j/H20856/H20849/H20841i/H20856/H20855j/H20841+/H20841j/H20856/H20855i/H20841/H20850, /H208491/H20850
where /H20841i/H20856is the /H9266state at site i,/H20855i,j/H20856represents pairs of
nearest neighbor sites iand j, and /H9253=−0.1115 a.u. is the
transfer integral. With this Hamiltonian, the band structureE
n,kcan be calculated, and the spatial symmetry of the eigen-
states /H20841n,k/H20856can be determined by inspecting the eigenvec-
tors, where nandkrepresent the nth energy level for a par-
ticular Bloch wave vector k.
In this paper, we compare the effect of external laser
fields polarized longitudinal /H20849referred to as xdirection
later /H20850to SWNTs and GNRs. To study the response of
SWNTs and GNRs to such fields, two quantities are useful:The joint density of states D
j/H20849/H9275/H20850and conductance /H9268/H20849/H9275/H20850
=/H92681/H20849/H9275/H20850+i/H92682/H20849/H9275/H20850. The joint density of states /H20849JDOS /H20850can be
determined from the band structure using the formulaPHYSICAL REVIEW B 76, 045418 /H208492007 /H20850
1098-0121/2007/76 /H208494/H20850/045418 /H208495/H20850 ©2007 The American Physical Society 045418-1Dj/H20849/H9275/H20850=2
L/H20858
n,n/H11032,k/H20851f/H20849En,k/H20850−f/H20849En/H11032,k/H20850/H20852/H9254/H20849En/H11032,k−En,k−/H6036/H9275/H20850,
/H208492/H20850
and the real part of conductance, /H92681/H20849/H9275/H20850, which is directly
related to the absorption of incident energy, can be calculated
using perturbation theory26for the weak fields
/H92681/H20849/H9275/H20850=2/H9266e2
me2/H9275L/H20858
n,n/H11032,k/H20851f/H20849En,k/H20850−f/H20849En/H11032,k/H20850/H20852/H20841/H20855n,k/H20841Px/H20841n/H11032,k/H20856/H208412
/H11003/H9254/H20849En/H11032,k−En,k−/H6036/H9275/H20850, /H208493/H20850
where Lis the length of the nanoribbon /H20849or nanotube /H20850,f/H20849En,k/H20850
is the Fermi-Dirac distribution function, Pxis the xcompo-
nent of the momentum operator, and the factor of 2 is forspin degeneracy. Here, we assume f/H20849E
n,k/H20850=1 for En,k/H11021EF
andf/H20849En,k/H20850=0 for En,k/H11022EF. Since the on-site energy for the
/H9266electrons is chosen to be zero, the Fermi level EFis zero
for both ZGNR and SWNT.
In the presence of an incident laser beam with photon
energy /H6036/H9275, a peak in /H92681/H20849/H9275/H20850indicates an absorption peak for
the photons with energy /H6036/H9275, and vanishing /H92681/H20849/H9275/H20850indicates
no absorption for /H6036/H9275. The imaginary part of conductance,
/H92682/H20849/H9275/H20850, can be calculated from /H92681/H20849/H9275/H20850by using Kramers-
Kronig relations. The dielectric function /H9255/H20849/H9275/H20850can also be
calculated from /H9268/H20849/H9275/H20850by using26
/H9255/H20849/H9275/H20850=1+4/H9266i
/H9275/H9268/H20849/H9275/H20850. /H208494/H20850
In other words, starting from /H92681/H20849/H9275/H20850, we can calculate pretty
much all the optical properties of graphene nanoribbons, in-
cluding the refraction index and reflectivity.
When calculating the matrix element /H20855n,k/H20841Px/H20841n/H11032,k/H20856in Eq.
/H208493/H20850, we only consider the effect of nearest neighbors, and we
use /H20855i/H20841/H11612/H20841j/H20856=Meˆij, where M=0.206 and eˆijis the unit vector
connecting from site ito site j.24It should be mentioned that
the value of Mdoes not affect the shape of /H92681/H20849/H9275/H20850; therefore,
the shape of the optical absorption spectrum is independent
of the numerical value of M.
III. RESULTS AND DISCUSSION
First, we compare the optical response of /H208496, 6 /H20850SWNT
and 12-ZGNR. As discussed in Sec. I, these two systemshave similar atomic structure. However, the eigenstates /H20841n,k/H20856
of ZGNR have different symmetric properties from theeigenstates of SWNT. In 12-ZGNR, due to the finite widthand the potential being symmetric along the transverse direc-tion /H20849ydirection /H20850, the eigenstates are either symmetric or
antisymmetric along the ydirection, namely, /H20855−y/H20841n,k/H20856
=±/H20855y/H20841n,k/H20856. This symmetry property of ZGNR makes its se-
lection rule for interband transitions qualitatively different
from that of SWNT, as shall be discussed later.
In Fig. 1, the band structure of /H208496, 6 /H20850SWNT and 12-
ZGNR is plotted in panels /H20849a/H20850and /H20849b/H20850, respectively. In Fig.
1/H20849a/H20850, we use different colors and formats to represent differ-
ent subband indices for SWNT.
1,21,25In Fig. 1/H20849b/H20850,w eu s eblack dashed lines to represent the transversely symmetric
states and red solid lines to represent transversely antisym-metric states. For 12-ZGNR, an eigenstate /H20841n,k/H20856is symmetric
ifnis odd /H20849n=1 being the state with lowest energy for a
given k/H20850and antisymmetric if nis even. When the incident
beam is polarized longitudinal to SWNT, only the transitionsbetween states with the same subband index are allowed. Inthe case of nanoribbons, only transitions between states withthe same parity are allowed. In other words, only states withthe same color and format in Fig. 1can have interband tran-
sition. Next, we will use JDOS D
j/H20849/H9275/H20850and real conductance
/H92681/H20849/H9275/H20850to demonstrate this selection rule.
In Figs. 2and3, JDOS Dj/H20849/H9275/H20850and real conductance /H92681/H20849/H9275/H20850
are plotted. In each figure, panel /H20849a/H20850is for /H208496, 6 /H20850SWNT and
panel /H20849b/H20850is for 12-ZGNR. Let us start by discussing the case
of/H208496, 6 /H20850SWNT. In Fig. 2/H20849a/H20850, we see peaks at different /H9275
known as van Hove singularities. Consider the peaks at /H9275
=0.1114, 0.1573, and 0.1929 a.u., for example. From Fig.0 1−0.15−0.1−0.0500.050.10.15
ka/πEnergy (a.u.)(a)
0 1−0.15−0.1−0.0500.050.10.15
ka/πn=1 2n=1 3(b)
FIG. 1. /H20849Color online /H20850Band structure of /H20849a/H20850/H208496, 6 /H20850SWNT and
/H20849b/H2085012-ZGNR. For the case of /H208496, 6 /H20850SWNT, different colors and
formats represent different subband indices /H20849Refs. 1,21, and 25/H20850.
For the case of 12-ZGNR, black dashed lines /H20849odd n/H20850represent
transversely symmetric states, and red solid lines /H20849even n/H20850represent
transversely antisymmetric states. In both cases, only the transitionsbetween the states with the same color and format are possiblewhen the polarization of the incident beam is along the longitudinaldirection.
0 0.05 0.1 0.15 0.2 0.25 0.30300Dj(ω)(a)
0 0.05 0.1 0.15 0.2 0.25 0.30300
ω(a.u. )Dj(ω)(b)
FIG. 2. Joint density of states Dj/H20849/H9275/H20850for/H20849a/H20850/H208496, 6/H20850SWNT and /H20849b/H20850
12-ZGNR. Peaks in Dj/H20849/H9275/H20850are known as van Hove singularities.
The effects of dipole matrix elements and selection rules are not
considered here.HAN HSU AND L. E. REICHL PHYSICAL REVIEW B 76, 045418 /H208492007 /H20850
045418-21/H20849a/H20850, we see that the green subbands /H20849thick dotted lines /H20850at
k=0.715 /H9266/aand the black subbands /H20849thick dashed lines /H20850at
k=0.829 /H9266/ahave vanishing derivatives, /H11509E//H11509k, which cause
the peaks of Dj/H20849/H9275/H20850in Fig. 2/H20849a/H20850to occur at /H9275=0.1114 and
0.1929 a.u., respectively. At k/H110150.762/H9266/ain Fig. 1/H20849a/H20850, the
green and black subbands below EFhave the same derivative
with the black and green subbands above EF, respectively.
This is the reason that Dj/H20849/H9275/H20850in Fig. 2/H20849a/H20850has a peak at /H9275
=0.1573 a.u. However, the dipole matrix element /H20855green,
0.762/H9266/a/H20841Px/H20841black, 0.762 /H9266/a/H20856vanishes; in other words,
such a transition is forbidden. Therefore, as shown in Fig.3/H20849a/H20850,
/H92681/H20849/H9275/H20850does not have a peak at /H9275=0.1573 a.u., but it
does have peaks at /H9275=0.1114 and 0.1929 a.u. Another peak
in Fig. 3/H20849a/H20850occurs at /H9275=0.2227 a.u. This peak results from
the transition between the red subbands /H20849thick dot-dashed
lines /H20850in Fig. 1/H20849a/H20850atk/H11015/H9266/a. Notice that /H92681/H20849/H9275/H20850=0 for
0/H11021/H9275/H110210.1114 a.u., which appears to contradict the fact that
/H208496, 6 /H20850SWNT is metallic and the blue subbands /H20849thick solid
lines /H20850in Fig. 1/H20849a/H20850have energy gaps between 0 and
0.1114 a.u. The interband transition between the blue sub-bands does not occur because those states are also the eigen-states of momentum P
x.21,25Therefore, the off diagonal ma-
trix elements in Eq. /H208493/H20850are zero, and the transition cannot
occur.
As discussed in the previous paragraph, the interband
transitions of SWNT occur at the direct gaps at k
=0.715 /H9266/a,k=0.829 /H9266/a, and k/H11015/H9266/afor green, black, and
red subbands, respectively. For GNR, however, direct-gaptransitions cannot occur when the incident beam is polarizedalong the longitudinal direction. In Fig. 1/H20849b/H20850, we see that the
subband E
n,kwith n=11 /H20849black dashed line /H20850has a maximum
atk=0.714 /H9266/a, and the subband with n=14 /H20849red solid line /H20850
has a minimum at the same k. In other words, there is a direct
gap between n=11 and 14 at k=0.714 /H9266/a. This direct gap
makes Dj/H20849/H9275/H20850, as shown in Fig. 2/H20849b/H20850, peak at /H9275=0.0772 a.u.,
which is the energy difference of these two states. However,
the parity of these two states is different, and the transition isforbidden. Therefore, there is no peak at
/H9275=0.0772 a.u. in
/H92681/H20849/H9275/H20850, as shown in Fig. 3/H20849b/H20850. We also see a peak at /H9275
=0.0639 a.u. in Dj/H20849/H9275/H20850, which results from the energy gap atk=0.750 /H9266/abetween n=12 and 15. Again, due to parity,
there is no optical absorption at /H9275=0.0639 a.u., as shown in
Fig.3/H20849b/H20850. Although 12-ZGNR is metallic, its /H92681/H20849/H9275/H20850vanishes
at low frequency, just like the case for /H208496, 6 /H20850SWNT. The
reason is that for 12-ZGNR, states with n=12 and n=13
have different parity, which forbids the transition.
In ZGNRs, there exist edge states: The electronic states
that are localized on the edge and decay exponentially intothe center. For 12-ZGNR, the edge states are /H20841n,k/H20856, with n
=12 and 13 for k/H110220.74. The edge states play an important
role in the interband transitions in optical range. In Fig. 3/H20849b/H20850,
the peaks at
/H9275=0.0423, 0.0834, and 0.1081 a.u. are due to
edge states. The initial states /H20841n,k/H20856and final states /H20841n/H11032,k/H20856of
these transitions are listed in Table I. The absorption peaks
beyond /H9275=0.1081 a.u. have nothing to do with the edge
states, which can be seen from the band structure shown inFig.1/H20849b/H20850. Note that both /H208496, 6/H20850SWNT and 12-ZGNR have an
absorption peak at
/H9275/H110150.221 a.u., which results from the al-
lowed transition at k/H11015/H9266/a,En,k/H11015±0.111 a.u.
We make a comparison of /H2084910, 10 /H20850SWNT and 20-ZGNR
in Figs. 4and5. The optical response is similar to the case
for/H208496, 6 /H20850SWNT and 12-ZGNR because the selection rules
do not change with the diameter of SWNT or the width ofZGNR. The only difference is that for the SWNT withTABLE I. Absorption peaks /H20849shown in Fig. 3/H20850that involve the
transition from /H20849to/H20850the edge states /H20841n,k/H20856/H20849 /H20841n/H11032,k/H20856/H20850, with n=12 /H20849n/H11032
=13 /H20850.
/H9275
/H20849a.u./H20850 k/H20849/H9266/a/H20850/H20849 n,n/H11032/H20850/H20849 n,n/H11032/H20850
0.0423 0.741 /H2084912, 14 /H20850/H20849 11, 13 /H20850
0.0834 0.801 /H2084912, 16 /H20850/H20849 9, 13 /H20850
0.1081 0.929 /H2084912, 18 /H20850/H20849 7, 13 /H208500 0.05 0.1 0.15 0.2 0.25 0.300.1σ1(ω)(a)
0 0.05 0.1 0.15 0.2 0.25 0.300.1
ω(a.u.)σ1(ω)(b)
FIG. 3. Real part of the conductance /H92681/H20849/H9275/H20850for/H20849a/H20850/H208496, 6/H20850SWNT
and /H20849b/H2085012-ZGNR. Peaks in /H92681/H20849/H9275/H20850indicate strong absorption for
photons with energy /H6036/H9275, and vanishing /H92681/H20849/H9275/H20850indicates no absorp-
tion. Notice how the peaks correspond to the allowed transitionstates shown in Fig. 1.
0 1−0.15−0.1−0.0500.050.10.15
ka/πEnergy (a.u.)(a)
0 1−0.15−0.1−0.0500.050.10.15
ka/πn=2 0n=2 1(b)
FIG. 4. /H20849Color online /H20850Band structure of /H20849a/H20850/H2084910, 10 /H20850SWNT and
/H20849b/H2085020-ZGNR. For the case of /H2084910, 10 /H20850SWNT, different colors and
formats represent different subband indices /H20849Refs. 1,21, and 25/H20850.
For the case of 20-ZGNR, black dashed lines /H20849odd n/H20850represent
transversely symmetric states, and red solid lines /H20849even n/H20850represent
transversely antisymmetric states. In both cases, only the transitionsbetween the states with the same color and format are possiblewhen the polarization of the incident beam is along the longitudinaldirection.SELECTION RULE FOR THE OPTICAL ABSORPTION OF … PHYSICAL REVIEW B 76, 045418 /H208492007 /H20850
045418-3greater diameter and ZGNR with greater width, there are
more absorption peaks, as can be seen by comparing Figs. 3
and 5. This is because they have a greater number of
states per unit length, which allows more transitions,as can be seen by comparing the band structures shownin Figs. 1and 4. Again, edge states are involved in
many of the absorption peaks for ZGNR, including
/H9275=0.0265, 0.0559, 0.0816, 0.1000, and 0.1102 a.u. in Fig.
5/H20849b/H20850. The peak at /H9275=0.1102 a.u. is related to the interband
transition from /H20849to/H20850the edge states n=20 /H20849n/H11032=21 /H20850atk
/H11015/H9266/ashown in Fig. 4/H20849b/H20850. Beyond /H9275=0.1102 a.u., no
absorption peak is related to the edge states. From the previ-ous discussion for 12- and 20-ZGNR, we see that
/H9275
/H110150.11 a.u. is the threshold beyond which no absorption
peak is contributed by edge states, and this threshold isbarely dependent on the width of nanoribbons. Similar toFig.3, we can see absorption peaks at
/H9275/H110150.222 a.u. for both
/H2084910, 10 /H20850SWNT and 20-ZGNR. Like /H208496, 6 /H20850SWNT and
12-ZGNR, this peak also corresponds to the transition thatoccurs at k/H11015
/H9266/a,En,k/H11015±0.111 a.u.
IV. CONCLUSION
In this paper, we have studied the optical response of
zigzag-edge graphene nanoribbons within the tight-bindingapproximation, and we have compared it to the case of arm-chair single-walled carbon nanotubes. Although these twomaterials have somewhat similar atomic structure, their elec-tronic structure and the symmetry of eigenstates are quitedifferent. For zigzag-edge nanoribbons, the eigenstates areeither transversely symmetric or antisymmetric, whichmakes their optical response qualitatively different from thatof SWNTs. In the presence of laser beams polarized longitu-dinal to nanotubes or nanoribbons, the interband transitionsin nanotubes can occur at direct gaps, while the direct-gaptransition in nanoribbons is forbidden.
We have also demonstrated that the edge states play an
important role in the optical absorption of nanoribbons. Theyare involved in many of the absorption peaks in the opticalrange. The greatest photon energy with which edge states cancause an absorption peak is around 0.11 a.u., and we canalways see an absorption peak around 0.11 a.u. for zigzag-edge nanoribbons with even number of zigzag lines. Beyondthis threshold, which happens to be close to the boundary ofoptical range, edge states have no contribution to absorptionpeaks.
ACKNOWLEDGMENT
The authors would like to thank the Robert A. Welch
Foundation /H20849Grant No. F-1051 /H20850for their support.
1R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Prop-
erties of Carbon Nanotubes /H20849Imperial College Press, London,
1998 /H20850, and references therein.
2Steven G. Louie, Top. Appl. Phys. 80,1 1 3 /H208492001 /H20850, and references
therein.
3Kyoko Nakada, Mitsutaka Fujita, G. Dresselhaus, and M. S.
Dresselhaus, Phys. Rev. B 54, 17954 /H208491996 /H20850.
4Yoshiyuki Miyamoto, Kyoko Nakada, and Mitsutaka Fujita, Phys.
Rev. B 59, 9858 /H208491999 /H20850.
5Takazumi Kawai, Yoshiyuki Miyamoto, Osamu Sugino, and
Yoshinori Koga, Phys. Rev. B 62, R16349 /H208492000 /H20850.
6Motohiko Ezawa, Phys. Rev. B 73, 045432 /H208492006 /H20850.
7Young-Woo Son, Marvin L. Cohen, and Steven G. Louie, Phys.
Rev. Lett. 97, 216803 /H208492006 /H20850.
8L. Brey and H. A. Fertig, Phys. Rev. B 73, 235411 /H208492006 /H20850.
9Katsunori Wakabayashi, Mitsutaka Fujita, Hiroshi Ajiki, and
Manfred Sigrist, Phys. Rev. B 59, 8271 /H208491999 /H20850.
10A. H. Castro Neto, F. Guinea, and N. M. R. Peres, Phys. Rev. B
73, 205408 /H208492006 /H20850.
11Dmitry A. Abanin, Patrick A. Lee, and Leonid S. Levitov, Phys.Rev. Lett. 96, 176803 /H208492006 /H20850.
12N. M. R. Peres, A. H. Castro Neto, and F. Guinea, Phys. Rev. B
73, 195411 /H208492006 /H20850.
13Denis A. Areshkin, Daniel Gunlycke, and Carter T. White, Nano
Lett. 7, 204 /H208492006 /H20850.
14D. Gunlycke, H. M. Lawler, and C. T. White, Phys. Rev. B 75,
085418 /H208492007 /H20850.
15Young-Woo Son, Marvin L. Cohen, and Steven G. Louie, Nature
/H20849London /H20850444, 347 /H208492006 /H20850.
16C. L. Lu, C. P. Chang, Y. C. Huang, R. B. Chen, and M. L. Lin,
Phys. Rev. B 73, 144427 /H208492006 /H20850.
17Takahiro Yamamoto, Tomoyuki Noguchi, and Kazuyuki Wa-
tanabe, Phys. Rev. B 74, 121409 /H20849R/H20850/H208492006 /H20850.
18Hiroshi Ajiki and Tsuneya Ando, Physica B 201, 349 /H208491994 /H20850.
19M. F. Lin and Kenneth W.-K. Shung, Phys. Rev. B 50, 17744
/H208491994 /H20850.
20Shuichi Tasaki, Koji Maekawa, and Tokio Yamabe, Phys. Rev. B
57, 9301 /H208491998 /H20850.
21A. Gruneis, R. Saito, Ge. G. Samsonidze, T. Kimura, M. A. Pi-
menta, A. Jorio, A. G. Souza Filho, G. Dresselhaus, and M. S.0 0.05 0.1 0.15 0.2 0.25 0.300.1σ1(ω)(a)
0 0.05 0.1 0.15 0.2 0.25 0.300.1
ω(a.u.)σ1(ω)(b)
FIG. 5. Real part of the conductance /H92681/H20849/H9275/H20850for /H20849a/H20850/H2084910, 10 /H20850
SWNT and /H20849b/H2085020-ZGNR. Peaks in /H92681/H20849/H9275/H20850indicate strong absorp-
tion for photons with energy /H6036/H9275, and vanishing /H92681/H20849/H9275/H20850indicates no
absorption. Notice how the peaks correspond to the allowed transi-tion states shown in Fig. 4.HAN HSU AND L. E. REICHL PHYSICAL REVIEW B 76, 045418 /H208492007 /H20850
045418-4Dresselhaus, Phys. Rev. B 67, 165402 /H208492003 /H20850.
22M. Machon, S. Reich, C. Thomsen, D. Sanchez-Portal, and P.
Ordejon, Phys. Rev. B 66, 155410 /H208492003 /H20850.
23G. Y. Guo, K. C. Chu, Ding-sheng Wang, and Chun-gang Duan,
Phys. Rev. B 69, 205416 /H208492004 /H20850.24Ashish Kumar Gupta, Ofir E. Alon, and Nimrod Moiseyev, Phys.
Rev. B 68, 205101 /H208492003 /H20850.
25Han Hsu and L. E. Reichl, Phys. Rev. B 74, 115406 /H208492006 /H20850.
26Michael P. Marder, Condensed Matter Physics /H20849Wiley, New York,
2000 /H20850.SELECTION RULE FOR THE OPTICAL ABSORPTION OF … PHYSICAL REVIEW B 76, 045418 /H208492007 /H20850
045418-5 |
PhysRevB.91.094301.pdf | PHYSICAL REVIEW B 91, 094301 (2015)
Non-Hermitian Hamiltonian approach to quantum transport in disordered networks with sinks:
Validity and effectiveness
Giulio G. Giusteri,1 ’2’3’* Francesco Mattiotti,1 and G. Luca Celardo1 ,2
1 Dipartimento di Matematica e Fisica and Interdisciplinary Laboratories fo r Advanced Materials Physics,
Universita Cattolica del Sacro Cuore, via Musei 41,1-25121 Brescia, Italy
2 Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via Bassi 6,1-27100, Pavia, Italy
3International Research Center on Mathematics & Mechanics of Complex Systems, via XIX marzo 1,1-04012 Cisterna di Latina, Italy
(Received 23 December 2014; revised manuscript received 18 February 2015; published 9 March 2015)
We investigate the validity of the non-Hermitian Hamiltonian approach in describing quantum transport in
disordered tight-binding networks connected to external environments, acting as sinks. Usually, non-Hermitian
terms are added, on a phenomenological basis, to such networks to summarize the effects of the coupling to the
sinks. Here, we consider a paradigmatic model of open quantum network for which we derive a non-Hermitian
effective model, discussing its limit of validity by a comparison with the analysis of the full Hermitian model.
Specifically, we consider a ring of sites connected to a central one-dimensional lead. The lead acts as a sink that
absorbs the excitation initially present in the ring. The coupling strength to the lead controls the opening of the
ring system. This model has been widely discussed in literature in the context of light-harvesting systems. We
analyze the effectiveness of the non-Hermitian description both in absence and in presence of static disorder on
the ring. In both cases, the non-Hermitian model is valid when the energy range determined by the eigenvalues
of the ring Hamiltonian is smaller than the energy band in the lead. Under such condition, we show that results
about the interplay of opening and disorder, previously obtained within the non-Hermitian Hamiltonian approach,
remain valid when the full Hermitian model in presence of disorder is considered. The results of our analysis can
be extended to generic networks with sinks, leading to the conclusion that the non-Hermitian approach is valid
when the energy dependence of the coupling to the external environments is sufficiently smooth in the energy
range spanned by the eigenstates of the network.
DOI: 10.1103/PhysRevB.91.094301 PACS number(s): 05.60.Gg, 71.35.- y , 72.15.Rn
I. INTRODUCTION
Open quantum systems are nowadays at the center of many
research fields in physics, ranging from quantum computing to
transport in nano- and mesoscale solid state systems as well as
biological aggregates. In particular, charge/excitation transport
in the quantum coherent regime can be considered one of the
central subjects in modern solid state physics [1,2]. Transport
properties depend strongly on the degree of openness of the
system. In important applications, the effect of the opening
is large, and cannot be treated perturbatively. The analysis
of open quantum systems beyond the perturbative regime
is often difficult due to the presence of infinitely many degrees
of freedom. Thus a consistent way to take the effect of the
opening into account for arbitrary coupling strength between
the system and the external world is highly desirable.
In a typical situation, we have a discrete quantum sys
tem coupled to an external environment characterized by
a continuum of states. Elimination of the continuum leads
to an effective non-Hermitian Hamiltonian. This approach
to open quantum systems has been shown to be a very
effective tool in dealing also with the strong coupling regime
[3-8]. The non-Hermitian Hamiltonian approach offers several
advantages: (i) it reduces an infinite dimensional problem to a
finite dimensional one; (ii) it allows to compute conductance
and the whole time evolution of the relevant subsystem;
(iii) the effects of interference between discrete states and
’Corresponding author: giulio.giusteri@unicatt.itthe continuum, such as superradiance or Fano resonances can
be easily analyzed [9].
Tight-binding networks are often considered in literature
to model quantum transport and decay, and their coupling
with external environments, acting as sinks, is taken into
account by adding non-Hermitian terms to the Hamiltonian
[10,11]. Indeed, non-Hermitian models are more and more
used to describe trapping or loss of excitation into transport
channels of complex biological aggregates [12-14], but a
proper justification of the employed non-Hermitian model is
often overlooked.
Together with the coupling to a sink, such networks
are usually coupled to other environments, which induce
different kinds of disorder: static disorder (space-dependent)
and dynamical disorder (time-dependent). When disorder is
added to the system to take into account the effect of other
environments, the strength of the coupling to the sink is usually
assumed to be unaffected by the disorder itself.
This assumption has been used both when dealing with
dynamical disorder [15,16] and with static disorder [14,17,18].
Specifically, some of the authors of this paper have pre
viously analyzed the interplay of opening and static disor
der in paradigmatic models of quantum transport, such as
one-dimensional and three-dimensional tight-binding mod
els [17,18], Within the framework of the non-Hermitian
Hamiltonian approach, a novel cooperative regime was found
characterized by the presence of subradiant hybrid states.
Moreover, cooperative robustness to disorder has been shown
[17] to play an important role in the dynamics of quantum
systems with sinks. As a matter of fact, all of those results were
obtained assuming the coupling to the sinks to be independent
1098-0121 12015/91 (9)/094301(18) 094301-1 ©2015 American Physical Society
GIUSTERI. MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91, 094301 (2015)
FIG. 1. (Color online) Tight-binding model described by the
Hermitian Hamiltonian H given in Eq. (22): a ring of NR sites,
connected with nearest-neighbor coupling S 3 , and a lead of NL sites,
connected with nearest-neighbor coupling SlL. The ring sites are
equally coupled to the first lead site with tunneling amplitude Q .RL.
of the disorder strength, even if we expect this assumption to
fail for large disorder.
In this paper, we consider a tight-binding network com
posed by a ringlike structure coupled to a semi-infinite lead
(Fig. 1). This model has been discussed in several publications
in literature due to its relevance to light-harvesting complexes
and to proposals of bio-engineered devices for photon sensing
[14,15,19-22]. Here, we derive a non-Hermitian Hamiltonian
able to describe the transport properties of the model. By
comparing the results of the full Hermitian model with the
results obtained with the non-Hermitian model, we want to
assess the limit of validity for the use of a non-Hermitian
Hamiltonian to model the decay properties in the presence of
a sink. We will analyze in detail the case with no disorder,
while for the case in the presence of disorder our main goal
is to give a qualitative discussion of the limit of validity of
the non-Hermitian approach and to ascertain its reliability in
reproducing the physics of the full Hermitian system, focusing
on the existence of subradiant hybrid states and cooperative
robustness to static disorder.
In Sec. II, we introduce the non-Hermitian Hamiltonian
approach to open quantum systems; in Sec. Ill, we present
our Hermitian model and we derive the corresponding non-
Hermitian Hamiltonian, showing, in Sec. IV, the effects of
superradiance in such a system. We then analyze the validity
and effectiveness of the non-Hermitian model in reproducing
the dynamics of the Hermitian system, in both absence (Sec. V)
and presence (Sec. VI) of diagonal disorder. In Sec. VII, we
show how our results generalize to generic networks with
sinks. A summary of the results and their implications for the
modeling of quantum sinks is given in the concluding section.
II. DERIVATION OF THE NON-HERMITIAN
HAMILTONIAN
In this section, we present a standard derivation of the non-
Hermitian effective Hamiltonian. Alternative derivations can
also be found in Refs. [3,8,23].
Let us consider a discrete quantum system A, interacting
with another system B, which represents the environment.
We assume that the subspace A is spanned by NA discretestates |i), while the subsystem B represents the environment
with states \c,E), where c — 1,... ,M is a discrete quantum
number, labeling the decay channels, and E is another discrete
quantum number, representing the energy (we will take the
continuum limit of this quantum number later).
In order to derive the effective non-Hermitian Hamiltonian,
which describes the intrinsic system A, let us consider the
projectors, within the Hilbert space of the total system A + B ,
on the two subsystems:
Na M N b
Pa = J > X i | , Pb = X ] E l c’£ > < c’£ l -
i= 1 e = 1 E = 1
Under the orthogonality conditions (i\j) = <$,j, {c,E\c' ,E') =
Sc.c’ Se-e1 , (i\c,E) = 0, we can rewrite the total Hamiltonian
of the system as
H = H0 + V =
where
HA A = P AHPA, H ab = P aHPb, (3)
and similarly for the other terms.
We can now define the unperturbed propagator Gq(x) =
(x - Ho)-1 and the total propagator G(x) — (x — H)~l, re
lated by the Dyson equation
G(x) = G0 (x) + G0 (x)VG(x),
which gives rise to the following coupled equations for GA A =
PaGPa and GB A = PB GPA:
GAa — G ° aa + G °a aHa bGba,
GBa = G ° bbHbaGaa.
By substitution we obtain
Gaa — G ° a a + G ° aaHabG °bbHbaGa a
and, taking into account that G ° B B = (x - Hbb)~x , we have
x Haa — HAB j^ - bHb a
From this expression, we obtain an effective Hamiltonian,
which defines the propagator over the subspace A and takes
the form
He ff(x) = Haa + H ab-----l —— Hba . (4)
X ~ nBB
The effective Hamiltonian, Eq. (4), can be written in an
explicit form taking into account the orthogonality conditions
of the states in the subsystems A and B . Without loss of
generality, we assume that the total Hamiltonian is diagonal
on the subsystem B :
(c,E \H \ c',E ,) = E8 c,c,8e_e,
Using the projectors, Eq. (1), we have
U j
+ \i)(i\H\c,E)-^(c,E\H\j)(j\.0
Hb b0
Hb aHa b
0(2)
094301-2
NON-HERMITIAN HAMILTONIAN APPROACH TO QUANTUM ... PHYSICAL REVIEW B 91, 094301 (2015)
Let us now define the transition amplitudes between the
intrinsic states and the states of the environment:
A'(E)=(i\H\c,E). (5)
Taking the continuum limit
E-E/ p ^ dE
c,E
and using the identity
1= Pv1
± in8(x — x q),
X — Xo \ X — Xo,
the non-Hermitian Hamiltonian can be written as
H e % (x) = Ha a + A(x) l - Q (x), (6)
where
Qij(x) = 2 n ? / Ac i(E) (Ac j(E))* p(E)8(x — E)dE
= 2nY ^ A-(x) (. Ac j(x))* p(x)(7)
and
A ij(x)
= *>/A'(E)(AC j(E))* p(E)
x — EdE, (8)
with p(E) the density of states in the environment B .
The ambiguity in the sign of the last term in Eq. (6),
producing two distinct forms of the effective Hamiltonian,
comes from the fact that the propagator G ° B B , which appears
in Eq. (4), can be associated with either the forward or the
backward evolution: the minus sign gives the forward-in-time
evolution, i.e.,
9(t — to )U (ti r +0°
To) = - r - r /2ni J _ o oe x p [ - ix ( t - t0)]
dx, (9)
* - W )
while the plus sign gives the backward-in-time evolution, i.e.,
0(fo - t)U (ti r+°°
To) = — /2m J.ooe x p [ - ^ x ( f - t0)]
dx. (10)
Note that H(t,t0 ) is the projection through P A of the full
evolution operator of the system A + B . Thus, if the initial
state of the total system has components only on the intrinsic
system A, its evolution under the operator U{t,0) in Eq. (11)
gives the projection of the wave function of the total system
over the intrinsic system.
From now on we will use the notation H eff(x) for H + n(x),
referring to H e s(x) as the effective Hamiltonian. To actually
compute the evolution of an initial state, it is convenient to
make use of the (x-dependent) basis of eigenstates of H eff(x).
Since H eff(x) is in general non-Hermitian, due to the presence
of the decay operator Q(x), its eigenvalues
£r(x) = Er(x) - l - Tr(x), r — l , , N a ,
are complex, and it has left and right eigenstates
tfeffM I r,x) = £r(x)\r,x), (r,x\H f,ff(x) = (r,x \£ r{x),which are biorthogonal, i.e., the identity operator is given by
1 = \r,x)(r,x\.
r
Note that when //eff(*) is sym m etric, (r ,x | equals the transpose
of |r,x), and not its Hermitian conjugate, as it happens in the
case of Hermitian Hamiltonian operators.
We observe that the decomposition of the identity given
above is correct only when the eigenstates of the Hamiltonian
form a complete set. This will be always true in the systems
considered in this paper, but it is not always the case for
non-Hermitian Hamiltonians. Indeed, the breakdown of such
a condition for parameter-dependent non-Hermitian operators
defines the so-called exceptional points in parameter space,
whose study is relevant to many physical systems [24].
Assuming now t > 0, the evolution operator on states of
the intrinsic system A can be written as
U(t+°°e h xt\r,x){r ,x|
-OO X - £V(x)+ i r r(x)■ dx. (ID
Due to the coupling between the intrinsic system and the
environment, the total probability for an initially intrinsic state
to remain in A may not be conserved in time, this is why the
evolution operator U is, in general, nonunitary. This property
can be already gathered form Eq. (11), but it will become more
evident in the next section.
In the case [//eff(^).77eff(x )] = 0 (which is always true in
the case NA = 1) we can write the evolution operator in a
particularly useful form. Indeed, we can define, for any r —
1
G f(x) =\r,x){r,x |
x — Er(x) ± |Tr(x)
and express the propagator in the form
GW = £(G+ w - g;w)
- E- t T r(x)|r,x)(f,x|(12)
r [ x -E r{x)? + \Tr{x)2'
so that the evolution operator on states of the intrinsic system
A reads
1 ^ r +°° e - s * T r(x)|r,x)(f,x|
2x ^ J - o o [x - Er{x )}2 + iT r(x)2dx. (13)
This form of the evolution operator will be used in the next
sections.
A. Energy-independent non-Hermitian Hamiltonian
The effective non-Hermitian Hamiltonian, Eqs. (6,7,8) can
be greatly simplified if the x dependence can be neglected. In
the case of a single quantum level of energy Eq, the effective
Hamiltonian becomes a complex number and we have
Heff(x) = E0 + A (x) -
In order to see under which conditions the x dependence can
be neglected, we can analyze the expression for the evolution
094301-3
GIUSTERI, MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91, 094301 (2015)
operator given in Eq. (13), where £j(x) = Eq + A(x) and
ri(x ) = Q(x). If F | (x) and A(x) are smooth and slowly
varying function around j c = E o , the propagator, Eq. (12), can
be well approximated around Eq by setting
A(x) = A (£0),
r,(x) = r l(£0).(14)
With this approximation the evolution operator becomes
U(t,0) e-x (£'o+A (£o )V e-2kr,(£:o)!i
Clearly, the range of times in which the latter will be a
good approximation for the actual evolution will depend on
how well the propagator is approximated by a Lorentzian
function even for energies distant from Eq. Note that the
approximated propagator implies an exponential decay of the
unstable quantum state with a decay width
r I(£0 ) = 2 7 r ^ |A c(£ 0)|2p(£o),
see Eq. (7), which corresponds to the Fermi golden rule. Hence
the problem of the validity of the energy-independent effective
Hamiltonian in the case of a single state is formally equivalent
to the problem of the validity of the exponential decay given
by the Fermi golden rule of an unstable quantum state [25-27],
When we have many levels coupled to the same continuum,
the evolution operator in a generic situation is given by
Eq. (11). The problem of obtaining an energy-independent
non-Hermitian Hamiltonian able to describe such evolution is
now more delicate since, in this case, we may have different
energies associated with the different levels. This general
problem will be treated in a subsequent publication.
On the other side, when AI;(x) and Q,j{x) are smooth
and slowly varying functions of x in the whole physically
relevant energy range, determined by the eigenvalues of the
Hamiltonian of the intrinsic system, we can completely neglect
the dependence on the energy of the initial state. Under these
conditions, we can obtain an energy-independent effective
Hamiltonian by setting
A (x)ij = A (E0 )ij,
(15)
Q(x)ij = Q(E0 )ij,
where Eq is any energy lying in the relevant range. We
can thus treat also the case of many levels coupled to the
same continuum with an energy-independent non-Hermitian
Hamiltonian, which reads
He f f = / /a a + A - -Q. (16)
This approximation will be used in Sec. VI to analyze a system
in presence of disorder.
The energy-independent non-Hermitian Hamiltonian be
haves as an effective Hamiltonian and allows us to compute
the time evolution of the projection of the total wave function
on the intrinsic system, see Eq. (11). Indeed, we can expand
any initial state of the intrinsic system, over the eigenstates
of the effective Hamiltonian and its time evolution can be
determined as follows:Note that, for the case of a single particle/excitation in the
intrinsic system and zero temperature in the external envi
ronment, the energy-independent non-Hermitian Hamiltonian
description is equivalent to the standard master equation in
Lindblad form obtained under the Born-Markov secular ap
proximation [9]. Nevertheless, the non-Hermitian Hamiltonian
approach is computationally much more efficient because one
has to integrate only NA equations while, with the master
equation approach, one has 0(N\) equations to deal with.
B. Superradiance
A very important phenomenon that can be easily analyzed
by means of the energy-independent effective Hamiltonian, see
Eq. (16), is superradiance. It is the cooperative effect which
produces a strong inhomogeneity in the decay rates of the states
of the intrinsic subsystem: some states, named superradiant,
display large decay rates, while the decay of some other states
is very slow, sometimes even negligible. We note that such an
effect is also known as “resonance trapping” and is present in
many physical systems such as nuclei, microwave resonators,
and optical resonators (see, e.g., Ref. [28]).
The roots of this effect lie in the interference due to the
competition of different states of the intrinsic subsystem to
decay in the same channel in the continuum. Considering
a generic situation, let us assume that we can obtain an
x-independent form of the terms Q and A of Eqs. (7) and
(8), respectively. Necessarily, the effective Q possesses a
factorized structure, since it is derived by the tensor product of
the (rectangular) transition matrices . It thus can have only as
many nonzero eigenvalues as the number M of decay channels.
In the energy-independent approximation, A usually displays
the same factorized structure of Q , and thus [A ,(7] = 0.
We must now distinguish two situations: (1) when
WaA'Q] = 0, the eigenvalues £ , of H c f f are given by the sum
of those of Ha a [Eq. (3)], A, and — ( i/2)Q , so that we can have
at most M nonvanishing decay widths, while NA — M eigen
states of Htff are not decaying at all. And (2) when [HA A ,Q\ ^
0, we encounter an additional effect, named superradiance
transition. Indeed, the relative energy scale of the opening term
A — (i/2 )(7 with respect to that of the intrinsic Hamiltonian
Ha a becomes important in determining how the decay width
is distributed among the eigenstates of H^: when the opening
is weak, the eigenstates will be close to the eigenstates of
Ha a and all of them will have a similar decay width; on
the other hand, when the opening is strong, the eigenstates
will approach those of A — ( i/2)Q , and only M of them will
have a significant decay width. We then see a transition from
a nonsuperradiant regime (weak opening) to a superradiant
regime (strong opening). This transition is not present in case 1
above, where we are in a superradiant regime for any opening
strength. One could also consider the case [A,<2] ^ 0, but
under this condition it is not possible to predict the behavior of
the system regarding superradiance on general grounds, and
we need to look at the eigenvalues of the specific Heff at hand.
III. THE MODEL
\\jf(t)) = e~iHtBt^h \i{f(0)) — y ^ e _,£r'/ ,i|r)(r|i/f(0)). (17) We consider here a simple model with Nr two-level
r systems arranged in a ring structure and coupled to a
094301-4
NON-HERMITIAN HAMILTONIAN APPROACH TO QUANTUM ... PHYSICAL REVIEW B 91, 094301 (2015)
common decay channel, the sink, which we model with a
one-dimensional lead. Such a ringlike structure has been
considered in several papers [14,15,19-22] as a paradigmatic
model to describe different systems, such as molecular J
aggregates [29], bioinspired devices for photon sensing [20]
and efficient light-harvesting systems [21]. In particular, it has
been often considered in the frame of exciton transport in
natural photosynthetic systems, where chlorophyll molecules
aggregate in ringlike structures around a reaction center,
representing a central core absorber, where the excitation can
be trapped [30]. Chlorophyll molecules are able to absorb
photons and can be modeled as two-level systems. Under
low light intensity, only one excitation is considered and the
molecular aggregate becomes equivalent to a tight-binding
model where one particle can hop from site to site.
We first introduce a Hermitian model to describe the decay
of excitation from the ringlike structure to the central core
absorber represented by a lead, as described in Fig. 1. Note that
also in Ref. [20] the central core absorber of the photon-sensing
device was modeled by a lead. Specifically, we consider a ring
with Nr sites, connected with nearest-neighbor coupling Q ,
described by the tight-binding Hamiltonian
//« = Q ^ ( |r ) ( r ,| + |r')(r|), (18)
{ r , r ' )
where the sum runs over the pairs of neighboring sites. In what
follows, we will measure energies in units of cm-1 and times in
ps. This choice, common in models for molecular aggregates,
corresponds to setting \/h = 0.06 ;r cm/ps.
Each site of the ring is connected, through the tunneling
amplitude £ 2rl, to the first site of a lead, described by a linear
chain of NL resonant sites with nearest-neighbor coupling U 2 /.
The Hamiltonian for the lead isthe subsystem represented by the lead. In this derivation, we
will follow the procedure described in Sec. II.
The eigenvalues of the lead Hamiltonian are given by
Eq = -2QLcoskq a, q - l , . . . , N L, (23)
where a is the distance between adjacent sites and the wave
number is
k =
“ a(NL + l)'
The components on the site | tj) of the lead eigenstates read
Wjltq) = y N^ + ] sin kqja. (24)
We perform now the continuum limit taking NL — ► oo. The
discrete wave number kq becomes a continuous parameter
k e (0,7t/a). We obtain from Eq. (23),
E(k) = — 2Ql cos ka (25)
and, recalling that
. f E ( k ysin ka = / 1 -------- —,
V
we can derive the density of states p(E) = dk/dE as
2rrVL J\-(E /2Q .l )2'
Moreover, the eigenstates in the continuum are given by
{lj\i/s(E)) = V2sin kja,
nl - t
Hi = Sh £ ( I W ; + i l + 1^+iX ^I), (19)
7=1
and the interaction between the ring and the lead is described
bywith E given by Eq. (25), and their components on the first
lead site (j — 1) read
(hmE)) = V 2 -E2
4
Nr
Vr l = ^ r l^ 2 ( I'-W tl + I W I ) . (20)
r= 1
so that the total Hamiltonian of the system, written on the site
basis
{\r)M j),r=\,...,NR,j = \,...,N L} , (21)
readsWe can now compute the matrix elements connecting
the site r of the ring with the eigenstates of the lead with
energy E:
Ar(E)=(r\H\ll)(ll\x lr (E ))
= S lR Lj2y/\ - (E/2£2L)2 .(26)
Computing now the coupling terms
H = H r + V rl + Hl . (22)
One can imagine that, when NL is large enough, the lead
represents a good sink, in that it absorbs most of the excitation
present in the system.
A. The non-Hermitian Hamiltonian
Since our main focus is on the decay of the excitation from
the ring and not on its dynamics in the lead, we will now
derive an effective Hamiltonian for the subsystem formed by
the ring, summarizing into non-Hermitian terms the effects of^2
Ar(E)Ar .(E)*p(E) = - £ V l - (E/2Vl )2 ,
n£2L
we define the transition matrix Q(x), see Eq. (7), by
Qr r ,(x) - \ r p % . for x e [— 2^ ,2^ ] ,
(0 otherwise,
where we introduced the opening strength(27)
(28)
(29)
094301-5
GIUSTERI, MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91, 094301 (2015)
By using Eq. (27), we can also derive the expression for the
matrix A(x), see Eq. (8), as
A r r ’ My f 2 Q= — Pv /
2 tr ,/_ 20.L y/\ - (E/2S2J2
- 2 Q .l X — E
thus obtaining the effective HamiltoniandE, (30)
He ff(x) = Hr + A(x) - l -Q(x). (31)
Note that the matrix elements Qr r '(x ) and Ar r '(x) do
not depend on r or r'. This fact implies that they commute
and they both have only one nonzero eigenvalue. The state
corresponding to that eigenvalue is the fully symmetric state
|S> (32)
which is also an eigenstate of HR , corresponding to the
maximum energy 2f2. The remaining Nr — 1 eigenstates of
<200 and A(x) are degenerate and can always (i.e., for any
x) be chosen to match those of Hr orthogonal to | S). For this
reason, Q(x) and A(x) commute with Hr.
The only nonzero eigenvalue of Q(x) is given by
r srCO= ( NR Y y/T for r e [-2£2t ,2ni ]
otherwise,
(33)
and the only nonzero eigenvalue of A(x) is
Asr(x) = yNR P wL2nL ^ 1 - £- 2/4^ 2
2 Ql x - EdE. (34)
From the foregoing facts, we obtain the important con
sequence that we can diagonalize the effective Hamiltonian
HeS (x) on the x-independent basis of eigenstates of Hr. The
only eigenvalue of the intrinsic system (the ring) which is
modified by the opening is
£sr = 2ft + Asr(x) - ^T sr(x),
while the others are
_ 2 rtrE r — Er — 2S2cos -----, for r = 1,... ,Nr — 1,Nr
which coincide with the eigenvalues of Hr [17].
Remarkably, we are in the peculiar situation in which
only one ring state [IS), Eq. (32)] is coupled to the lead,
and the number of relevant degrees of freedom, as far as
decay properties are concerned, may look already dramatically
reduced to 1. Nevertheless, the dependency on x of r sr and Asr,
keeps the actual number of degrees of freedom infinite.
The time-evolution operator for the sole ring state, |S),
which is coupled to the lead is given by
Us(t, 0)J _ f + 2 C l L e ~ ^ T sr(x) ^
2?r J-2 aL [ x - 2 f i - A s r (x)]2 + iT sr(x)2 *'
(35)
while the ring states which are orthogonal to \S ) are effectively
decoupled from the lead. For those states, the opening termFIG. 2. (Color online) Tight-binding model described by the
effective Hamiltonian //e ff given in Eq. (38): the NR resonant sites
are coupled with nearest-neighbor tunneling amplitude £2. Moreover,
they are equally open towards a common decay channel with opening
strength y.
in the effective Hamiltonian vanishes, and it is trivially, but
exactly, x-independent: those states will never decay.
To complete the wanted dimensional reduction, we then
need to derive an x-independent (or energy-independent)
approximation of Heff(x), Eq. (31). Now, if we let — * ■ oo
and Qr l — > oo keeping y fixed (wide-band limit), we clearly
obtain an exact energy-independence with
Asr(x) -» 0 and Tsr(x) yNR . (36)
With those assumptions we get
,, , / 2 Q i YN r \Usd, 0) = exp 1— — f - -—-t | .2 h )(37)
and the effective energy-independent non-Hermitian Hamilto
nian describing the evolution of the intrinsic system, the ring,
becomes
He ff = Hr — i ^-0, (38)
where O is a full matrix with all entries equal to 1, and the
components of He ff on the ring-site basis read
(He ff)r r . = (HR ) r r , - i^.
Accordingly, the evolution operator on the whole intrinsic
subspace is given by
U {t, 0) = e~iH c n t/h . (39)
In summary, the effective non-Hermitian model, depicted
in Fig. 2, is given by an open ring of Nr resonant sites
equally coupled, with strength y, to a common decay channel,
in which the excitation can be lost. We will analyze in
Sec. V below the limit of validity of the energy-independent
non-Hermitian Hamiltonian of Eq. (38). Note that the non-
Hermitian Hamiltonian just derived contains, together with
the Hamiltonian of the closed ring Hr, another term O,
representing the decay matrix. Since the matrix O is a full
matrix, it represents a long-range hopping between the sites of
the ring, mediated by the coupling of the sites of the ring to the
common decay channel in the lead. This long-range hopping
will be relevant to understand the interplay of opening and
disorder discussed in Sec. VI.
IV. SUPERRADIANCE IN TRANSPORT
The ring subsystem is, for any y ^ 0, in a superradiant
regime, with a single superradiant state |S),Eq. (32), absorbing
094301-6
NON-HERMITIAN HAMILTONIAN APPROACH TO QUANTUM ... PHYSICAL REVIEW B 91, 094301 (2015)
all the decay width yNR , and NR — 1 subradiant states with
vanishing decay widths. Thus our tight-binding model offers a
paradigmatic realization of case 1 of Sec. IIB, showing that the
symmetry of the coupling between each ring site and the sink
is responsible for the effective perfect segregation of the decay
widths. Moreover, the superradiance transition introduced in
case 2 of Sec. IIB plays a fundamental role in determining
the dynamics of our system when static disorder is added, and
Sec. VI is devoted to the analysis of such effect in both the
Hermitian and the non-Hermitian models introduced above.
Here we consider the effects of superradiance on the decay
of states which are initially excited on the ring, showing that
the non-Hermitian model correctly reproduces the dynamics
of the full Hermitian system. Given an initial ring state |^o),
we consider the survival probability
Nr
P(t) — l(r IV r(0>|2> (40)
r = l
computing the time evolution in both the Hermitian model,
Eq. (22),
I f H(0) = e~iH '/nW o), (41)
and the non-Hermitian one, Eq. (38),
\i/r//' < r( t) ) = e - iH“ " /nli/r0). (42)
In Fig. 3, we compare the P(t) obtained with the two models
for the superradiant initial state of Eq. (32), varying the system
size as Nr = 1,2,10 (black, red, and blue data, respectively).
The agreement between the Hermitian model (circles) and the
non-Hermitian one (curves) is excellent. Moreover, the decay
FIG. 3. (Color online) Survival probability P(t) vs time t. Re
sults obtained with the Hermitian model, Eq. (22), (symbols), are
compared with the results obtained with the non-Hermitian model,
Eq. (38), (curves), for different values of NR . Circles represents data
obtained starting from the fully symmetric superradiant state |S) of
Eq. (32), while crosses refer to the antisymmetric state of Eq. (43),
which is subradiant. Values of the parameters are C ! = 1, SlR L = 10,
Q :l = 100, y = 2, and NL = 1000.width, given by
Nr2Q.2 r,
~ n f k = NrY'
increases well above the single-site decay rate y as NR is
increased, signaling the presence of cooperative effects in the
dynamics.
The same excellent agreement between the Hermitian
model (green crosses) and the non-Hermitian one (green curve)
is found in the P(t) computed for the initial state
|AS) = - ^ ^ ( - i y » , (43)
which is remarkably different from the one computed for
| t^o) = 1 5). Indeed, as anticipated above by analyzing the
non-Hermitian model, we are in a superradiant regime. The
state | A S) is a subradiant eigenstate of He f f with vanishing
decay width, and then its survival probability is constantly
equal to 1. This suppression of decay, due to interference
effects, is somehow surprising if one consider that all the
sites are coupled to a semi-infinite lead and nevertheless the
excitation never leaves the system.
It is important to stress that the super/subradiant dynamics,
predicted on the basis of the reduced non-Hermitian system,
faithfully reproduces the Hermitian evolution, at least up to
the times shown in the figure. For larger times or for very short
times the behavior of the two models will depart from each
other, due to the fact that, in our simulations, both G/ and NL
are finite. In Sec. V, we will analyze in detail the critical times
up to which the agreement persists.
V. LIMIT OF VALIDITY OF THE NON-HERMITIAN
MODEL: NONEXPONENTIAL DECAY
The non-Hermitian Hamiltonian, Eq. (38), constitutes a
great simplification of the full Hermitian problem, since it
eliminates the infinite number of degrees of freedom of
the lead. As it was shown in the previous sections, the
non-Hermitian Hamiltonian becomes exact for infinite length
and infinite energy band in the lead.
In this section, we want to clarify the effect of £ lL and Ni
being finite on the validity of the non-Hermitian approach.
We will restrict our attention to the survival probability P(t)
computed for the superradiant initial state |S) of Eq. (32), for
which the non-Hermitian Hamiltonian predicts an exponential
decay with a decay width given by NR y. For all the other
initial states, orthogonal to the state | S), the non-Hermitian
Hamiltonian predicts that they are subradiant and do not
decay at all. Since we have only one level, |5), coupled to
the lead, the problem of the validity of the non-Hermitian
Hamiltonian approach is formally equivalent to the validity of
the exponential decay, given by the Fermi golden rule, of the
survival probability of a single unstable quantum state coupled
with a continuum of states [25-27,31], Also in our case we
will show that the exponential behavior is typically valid for
intermediate times, while for both short and long times the
decay is not exponential.
094301-7
GIUSTERI, MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91. 094301 (2015)
A. Finite-size effects
For finite lead length the decay of the excitation from the
ring will not be irreversible, since the excitation can bounce
back at the end of the lead, inducing a revival of the survival
probability. Thus, we can expect deviations of the Hermitian
evolution from the exponential decay due to the reflection of
the wave packet at the end of the lead. This bouncing effect
is always present in any finite-size sink and it is known in
literature as mesoscopic echo [27].
We can analytically estimate the bouncing time tB assuming
that the excitation, after leaving the ring, goes through the
lead with a certain group velocity vg, and bounces back once
reached the end of the lead.
We can describe the eigenstates of the lead as plane waves
[see Eq. (24)]t
{Zjlfq) oc sin kqja,
with wave numbers
Knq
a(NL + l)’j = 1 (44)
kq e (0,7 x/a). (45)
From this point of view, a superposition of lead eigenstates
forms a wave packet. The group velocity of this wave packet
is given by
_ d c o (k ) _ 1 dE(k)
V s ~ dk * ~ ~ h dk
where k is the mean wave number of the waves that form
the wave packet. Using the wave numbers of the eigenstates
defined above, we can write the energies of the lead, Eq. (23),
as(46)
E(kq) = — 2 cos kq a,
so that the group velocity becomes
2QL a . -v „ =------- sin A m .
* H(47)
(48)
The excitation has to go through the entire lead twice before
reaching again the starting point. From Eq. (48), we see that the
maximum velocity of a wave packet is vg — 2£ 2 L a/h. Hence,
we can expect the agreement between the Hermitian evolution
and the non-Hermitian one to persist up to the time
2a Nl _ H N l
Vg S2 l(49)
In order to check our estimate for tB , in Fig. 4 we compare
the superradiant decay exp(— NB yt), produced by the non-
Hermitian model (green curve), with the Hermitian evolution
computed for different values of the lead size NL. As NL
increases, the agreement between the Hermitian and the non-
Hermitian evolution persists up to a critical time after which
we have deviations from the exponential decay and a revival of
the survival probability. For small values of Nl, the agreement
time increases linearly with NL and it is well estimated by
the values of tB , see vertical arrows. On the other side, for
larger values of NL, the agreement time becomes independent
of the length of the lead. This suggests that a different effect,
see discussion below, causes the departure of P(t) from the
exponential decay.FIG. 4. (Color online) The survival probability P (t) computed
starting from the superradiant state of Eq. (32) is plotted vs time.
The analytic decay exp(— NR yt) (green curve), obtained from the
non-Hermitian model with y given by Eq. (38), is compared with
the data obtained from the Hermitian model with NR = 10, £ 2 = 1 ,
S iR L = 10, and QL = 100 for different values of NL . Vertical arrows
mark the values of tB given by Eq. (49) for the corresponding values
of AT.
B. Finite-bandwidth effects: S I < & S lL
From Fig. 4, we notice that when we increase NL above
a certain value, the agreement time does not improve, even if
the bouncing time increases. Indeed, the large-size regime is
characterized by an Nl-independent agreement time, marking
the transition from the superradiant decay to a much slower
one. The origin of this brake in the decay is very general and
it is due to the presence of a finite energy band in the lead,
whose bandwidth equals 4 S 1 L, see discussion in Ref. [25].
Indeed from Eq. (35) we see that the time evolution of
the superradiant state is given by the Fourier transform of the
function
£(*) = -r sr(x)/2
n [ x - 2 S l- Asr(x)]2 + 4 r sr(x)2(50)
In the limit S lB -> oo, Tsr, and Asr do not depend on x (see
discussion in Sec. III). Moreover, the limits of integration in
Eq. (35) go to infinity. Thus, in this limit, the time evolution of
the superradiant state is the Fourier transform of a Lorentzian
function, which gives an exponential decay. On the other side,
for finite bandwidth in the lead, we can expect deviations from
the exponential decay due to two reasons: (i) the Lorentzian
function is now distorted due the fact that both r s r and As r
depend on x and (ii) the limits of integration do not go to
infinity anymore.
In this section, we will consider the situation in which the
energy range of the ring is much smaller than the energy band in
the lead, so that the transition amplitude Ar(E) and the density
of states p(E), see Eq. (27), are very smooth and slowly varying
function of the energy in the whole energy range of the ring.
We are thus allowed to set the width Fsr(x) = r sr(0) = NBy,
see Eq. (33), and Asr(x) = Asr(0) = 0, see Eq. (34). For this
reason, our results will not depend on the energy 2 S T 2 of the
initial state |S) and we can use the approximation £ 2 % 0. This
094301-8
NON-HERMITIAN HAMILTONIAN APPROACH TO QUANTUM ... PHYSICAL REVIEW B 91, 094301 (2015)
regime is characterized by the following conditions:
£ 2 < $ ; f2£, Tsr = Nry < £ . 4 ( 5 1 )
In Appendix A, using the conditions in Eq. (51), we
show that the function £(x) is well approximated by a
Lorentzian function. In such a case, the main deviations from
the exponential decay are due solely to the truncation of the
function £(x) outside the energy band of the lead.
Specifically, in Appendix A, we show that the decay of P(t)
is exponential between two time-scales t o and t$ and we have:
Pit) %1
const./?3for t < t0 ,
for t 0 < t < ts,
for t > ts.(52)
From Eq. (52), we see that the decay is initially quadratic in
time, as predicted by perturbation theory, then it is exponential
with a decay width Tsr = NR y given by the Fermi golden rule,
and eventually it decays as a power law.
The transition from quadratic to exponential decay occurs
at a time t 0 given by
(53)
which has been derived in Appendix A and, with a more
heuristic approach, in Appendix B.
The quadratic initial decay given by the full Hermitian
model is shown in Fig. 5 (symbols) for different values of S lL .
In the same figure, the short-time anlytic estimate (curves)
given in Eq. (52) is shown to be a good estimate of the initial
behavior of the Hermitian system up to the time t o , marked
with arrows in Fig. 5.
The power-law decay P(t) o t t~3 for t > ts is in agreement
with numerical results, see dashed line in Fig. 4. The critical
time ts for which we have the transition from the exponential
FIG. 5. (Color online) Evolution of the survival probability P(t),
computed starting form the state | S), Eq. (32), is plotted vs time t
for different values of Q ,L. The evolution of the Hermitian model
(symbols) is well approximated by the parabolic decay (full curves)
of Eq. (52) for times shorter than t0 = h/2£lL (vertical arrows).
Parameters are £ 2 = l, NR = 10, NL = 100, and £ 2 R L = so
that we keep the opening y = 2 fixed.4fVrs r
FIG. 6. (Color online) The ratio between the time ts at which the
decay of the survival probability P{t), computed with the Hermitian
evolution of |S), departs from the exponential decay predicted by
the non-Hermitian model and the characteristic decay time rsr =
h/ r sr is plotted against the ratio 4ftL/ T sr. Data are obtained keeping
£lL = O r R L and NR = 10, so that the decay rate y = 2 is fixed. We
considered a fixed value of £ 2 = 1 (circles). The logarithmic scaling
predicted in Eq. (54) is apparent and it has been highlighted by means
of the dotted curve.
decay to the power-law decay has also been derived in
Appendix A and we have
ts o c4£h
Tsr '(54)
For the exponential decay to be a good approximation on
a significant time range, we need ts to be several times
the mean life-time rsr = hf rs r . This can be achieved only
if the logarithmic term in Eq. (54) is large enough. In Fig. 6,
the logarithmic dependence of ts on is shown to agree with
numerical results. We also observe that both the finite-size and
finite-bandwidth effects disappear in the thermodynamic limit
(Nl — > oo, £li — ► oo, p(E) — 1 /2 t t) considered in Sec. Ill
during the derivation of H tg , since both tB and ts grow to
infinity, while t o goes to zero.
C. Finite-bandwidth effects: ft ~ ftt
Here, we analyze the situation in which the energy range of
the ring can be comparable with the energy band of the lead.
Specifically, we analyze what happens if £ 2 and Tsr are not
small compared to QL , so that the conditions in Eq. (51) are
no longer satisfied.
In Fig. 7, we show the survival probability starting from
the state |S) for different values of the ratio £l/£2L and fixed
r sr. We compare the exact results with the results given by the
non-Hermitian model under the conditions given in Eq. (51).
In the range £l/£lL 1/4 there is a very good agreement
(compare dashed line in Fig. 7) with numerical results, red
solid curve). As we increase the ratio £2 /£ 2 L , the exponential
decay is still valid in a significant time range, but the decay
width is different. According to the discussion in Sec. II A,
094301-9
GIUSTERI, MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91, 094301 (2015)
FIG. 7. (Color online) Survival probability P(t), computed start
ing from the superradiant state |S) Eq. (32), is plotted vs time
t, for different ratios £2/£2f.. The results obtained with the full
Hermitian model (solid curves) are compared with the exponential
decay obtained setting r sr(0) = NR y (dashed curve). Dotted lines
indicate the exponential decays with decay width given by Eq. (55).
Data shown refer to the case NR = 10, NL = 1000, £ 1 L = 200, y = 2,
and different values of £2.
since £ 2 is not small if compared to £2^, we should build our
non-Hermitian Hamiltonian by evaluating Asr(x) and r sr(x)
at the energy Eq = 2£2, so that decay width of the survival
probability can be predicted by evaluating r sr(x) at the energy
2£2 of the initial state:
r.<2a) = ^ i - ( | f ) , (« )
which is deduced by the actual form of Fsr(x) given in
Eq. (33). We plotted such exponential decays as dotted lines
in Fig. 7. Note that the non-Hermitian Hamiltonian derived in
the previous Section was obtained by evaluating Asr(x) and
Fsr(x) at the energy Eq = 0.
When 2£2 + r sr(2£2)/2 > 2£21 we have deviations also
from the exponential decay obtained by employing the width
in Eq. (55). Notice also that as soon as the energy of the initial
state is outside the energy band of the lead the decay is strongly
suppressed (see data corresponding to £2/£2 L = 1.05). We will
see, in the next section, that the strong suppression of decay
when the energy of the initial state lies outside the energy band
of the lead will be crucial in understanding the limit of validity
of our effective Hamiltonian approximation in presence of
disorder.
In our model, we have only one state, with energy 2£2,
coupled to the continuum of states in the lead. Even if the
non-Hermitian Hamiltonian model obtained by evaluating
Asr(x) and Fsr(x) at the energy Eq = 2£2 would be valid
in a larger range of parameters, that approximation is not
readily extendable to a situation in presence of disorder, which
will be considered in the following sections. Indeed, in this
case, we do not have a single level coupled to the lead, but
many levels, each with its own energy. For this reason, we
are mainly interested in comparing the full dynamics with
the non-Hermitian Hamiltonian model obtained by evaluating
Asr(x) and Fsr(x) at the energy Eq = 0, which is valid when theFIG. 8. (Color online) Survival probability P(t), computed start
ing from the superradiant state |S) Eq. (32), is plotted vs the rescaled
time t* = rs T t/h = NR yt/h time, for different values of the ratio
Tsr over the energy bandwidth in the lead 4£2f.. The results obtained
with the full Hermitian model (solid curves) are compared with the
result predicted by the non-Hermitian model (dashed curve). Data
shown refer to the case NR = 10, £ 2 = 1, NL = 1000, QL = 100,
and different values of QR L .
dependence on the energy of the initial state can be neglected
and thus it can be easily used also in presence of disorder.
Note that all of the exponential decays leave place to a
power-law decay above a critical time, which we estimated in
the previous section only in the case £ 2 / £2L < 5 £ 1. Here, we will
not discuss how this transition time is modified as we increase
the ratio £2/£2/., since in this case the deviation from non-
Hermitian Hamiltonian model obtained by evaluating Asr(x)
and r sr (x) at the energy Eq — 0, occurs for all times.
In Fig. 8, we show the survival probability starting from
the state \S ) for different values of the ratio r sr /4£2L and for
fixed energy of the initial state in the regime £2/£2/, < £ £ 1. In
this case, deviations from the exponential decay predicted by
the non-Hermitian Hamiltonian start already when r s r /4£2t -
0.1, thus showing that the agreement between the Hermitian
and the non-Hermitian model is very sensitive to the decay
width of the initial state. The strong oscillations that can be
seen in Fig. 8 are due to the fact that, for large F sr, the coupling
£2fl£ between the ring and the first lead site is large. Our results
show that, for the non-Hermitian Hamiltonian approach to be
effective, the coupling £ 2 ^ between the ring and the lead does
not need to be small with respect to the characteristic energy
scale £ 2 of the ring, but only with respect to the characteristic
energy scale £2 L of the lead.
VI. THE EFFECTS OF STATIC DISORDER
In this section, we aim at studying the effectiveness of
the non-Hermitian Hamiltonian approach in describing the
effects of static diagonal disorder on the transport properties
of the system under consideration. Note that we add disorder
only in the ring, leaving the lead unchanged. Such a disorder
is modeled by position-dependent, but time-independent,
fluctuations of the ring site energies, that is, we added to the
094301-10
NON-HERMITIAN HAMILTONIAN APPROACH TO QUANTUM ... PHYSICAL REVIEW B 91, 094301 (2015)
ring Hamiltonian HR , Eq. (18), the term
Nr
D = ^ € r |r)(r|, (56)
r= 1
where er are independent random variables uniformly dis
tributed on [— W /2, W /2], and W represents the disorder
strength.
It is well known that, in one-dimensional systems with
short-range interactions [32], static diagonal disorder induces
Anderson localization: the eigenstates of the system become
exponentially localized. The critical disorder strength in one
dimensional aggregates for such a localization to occur is given
by
100
Wlo c % — , (57)
VN
where N is the system size [17].
To understand the effects of disorder on the excitation
transport from the ring into the lead, we can make the following
considerations: since disorder destroys the perfect symmetry
of the ring, which produces zero decay widths of the subradiant
states, it will decrease the decay width of the superradiant state,
while it will increase the decay widths of the subradiant states.
Thus, in the presence of disorder, we do not have only one
state coupled to the lead as in the previous section, but we
have a genuine many-level problem. Note that opening and
disorder have competing effects, since the opening induces a
long-range hopping among the sites of the ring, as it is clearly
seen from the structure of the full matrix O in Eq. (38), which
can be expected to contrast the localization effects of disorder.
The nontrivial competition between opening and disorder
has been analyzed in Refs. [17,18], within the framework of
the non-Hermitian Hamiltonian approach to open quantum
systems. It was there shown that, upon increasing the disorder
strength, the decay widths of the subradiant and superradiant
states become the same, and equal to y for W > lV sr, where
Wsr represents the critical disorder strength above which
superradiance is quenched.
The analysis was performed assuming the coupling y to the
continuum to be independent of the disorder strength. Such an
assumption is often used in literature and greatly simplifies the
calculations. On the other side, one can expect the presence
of diagonal disorder to affect the outcome of the reduction
procedure leading to He g, Eq. (38). For instance, the coupling
to the continuum will in general depend on the disorder
strength. In order to understand this point, one can consider
only one site coupled to a lead with an energy bandwidth of
4£2/,. If we assume the opening strength to the lead to be
independent of disorder, the non-Hermitian Hamiltonian of
this system reads = E q + eo — iy/2, which implies that
the survival probability decays exponentially as e~Y ‘ ^ h for any
value of the disorder strength W. Clearly, this cannot be true
when W J > > A £lL, since in that case the probability of the initial
state to be outside the energy band of the lead will be large
and the decay will be consequently suppressed, as discussed
in Sec. V C.
Even in the presence of disorder, the effective non-
Hermitian Hamiltonian can be built following the procedurepresented in Sec. Ill A, and reads
He f [ (x) = Hr + D + A( x) - Q(x), (58)
with Q(x) and A(x) given by Eq. (28) and Eq. (30), respec
tively. It is clear from that expression that, in the wide-band
limit Ql — > oo (with y fixed), the non-Hermitian Hamiltonian
for the disordered ring becomes energy-independent and can
be written as
He f f = HR + D-i^O , (59)
where O is a full matrix with all entries equal to 1. Note that
this expression coincides with the value for x = 0 of H t f f in
Eq. (58).
Clearly, the foregoing energy-independent /7eff is exact
solely in the infinite-bandwidth limit, while, for any finite
bandwidth in the lead, it will be a good approximation of the
true dynamics only for a disorder strength W sufficiently small
if compared to the lead bandwidth, and even in that case only
in a certain time window. The problem of determining such
ranges of validity is very complicated, since we are dealing
with a many-level system and the considerations used in the
previous sections cannot be used blindly. A discussion of this
problem will be given in the next section.
The main puipose of this section is to see whether, for
a sufficiently large (but finite) bandwidth in the lead, the
important effects, found in Refs. [17,18], are indeed present
in the full Hermitian model considered in this paper. The
two main findings of Refs. [17,18] can be summarized as
follows, (i) Cooperative robustness to disorder. For large
enough opening strength, the critical disorder Wsr needed
to quench superradiance increases linearly with the system
size, (ii) Subradiant hybrid regime. In the superradiant regime,
the response of the superradiant and subradiant subspaces to
disorder is very different. While superradiant states display
robustness to disorder by remaining extended up to Wsr, sub
radiant states show strong signatures of localization. Indeed,
they have hybrid features displaying both an exponentially
localized peak and a uniform delocalized plateau.
A. Comparison between Hermitian and non-Hermitian
models in presence of disorder
To assess the effectiveness of the non-Hermitian descrip
tion, under the assumption that the coupling to the continuum
is independent of disorder, we will first study the survival
probability P(t) of finding the excitation on the ring at time
t, comparing both the results given by the Hermitian and the
non-Hermitian model in presence of disorder.
We considered a generic initial state generated as a random
superposition of the ring eigenstates. Since most of the
eigenstates are, for sufficiently small disorder, subradiant,
a random initial state will have mainly components on the
subradiant subspace, so that we can expect that disorder
will initially increase the transport efficiency. For very large
disorder, the non-Hermitian model predicts an exponential
decay of the survival probability P(t) = e~Y 'in, while we can
expect a much slower decay from the full Hermitian model.
The average P(t) computed for different values of W/4
is shown in Fig. 9. We observe that, in agreement with the
094301-11
GIUSTERI, MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91, 094301 (2015)
FIG. 9. (Color online) Average survival probability P(/), starting
from a randomly generated ring state, computed for different values of
the disorder strength W. A perfect agreement between the Hermitian
model (symbols) and the non-Hermitian one (curves) appears, except
for the largest considered value, W / A Q .L = 2.5 (blue), in which case
the prediction of the non-Hermitian model (curve) is remarkably
different from the Hermitian one (circles). As a dashed curve, we
plotted the exponential decay e~r'/h corresponding to the independent
sites limit of the non-Hermitian model. Parameters are NR = 10,
£ 2 = 1, nR L = 10, nL = 100, and NL = 250.
foregoing discussion, when W/4£2i, is large, the behavior of
the non-Hermitian model departs from that of the Hermitian
system for all times. Indeed, while the decay of P(t) in the
non-Hermitian case is faster and faster as disorder increases,
approaching the limiting decay rate y/h, for the Hermitian
case the decay has a nonmonotone behavior with the disorder
strength, since it increases for small disorder and it is strongly
suppressed for large disorder.
On the other hand, we can see that, for W/4£2/, small, our
non-Hermitian model reproduces the Hermitian dynamics in
the time window shown in the figure. For any finite bandwidth,
we expect a departure from the non-Hermitian description for
very small times and very large times. Specifically, the time
t o up to which we have a quadratic decay, can be estimated
also in the many-level case, see Appendix B, where we show
that t o is again given by Eq. (53). On the other side, the timets above which we have a departure from the non-Hermitian
Hamiltonian decay is much more difficult to estimate in the
many-level case. Indeed, the departure from the exponential
decay has been interpreted in Ref. [27] as a consequence of the
fact that, for finite bandwidth in the lead, there is a finite return
probability from the lead to the initial state: the transition
from exponential to power-law occurs when the probability to
be in the inital state and the return probability are comparable.
In presence of many levels, the return probability will not
only repopulate the inital state, but also all the other states
connected to the lead. For this reason, the estimation of ts in
the many-level case is a delicate issue.
The rigorous analysis of this problem will be the subject of
a future publication, here we just stress that, as the bandwidth
in the lead goes to infinity, we have that t o goes to zero and
ts grows to infinity. To illustrate this point we analyzed the
survival probability P(t) starting from the exact eigenstates
of the effective Hamiltonian of Eq. (59). The dynamics
determined by the non-Hermitian model gives an exponential
decay of P(t) with a decay width determined by the imaginary
part of the complex eigenvalue corresponding to the initial
state. In Fig. 10, we compare the non-Hermitian evolution
with the Hermitian one obtained from the same initial states
as we vary the bandwidth in the lead. In the left panel, we
show the P{t) computed starting from the state with the largest
width (superradiant), while in the right panel we show the P(t)
computed starting from the state with the second-largest width
(subradiant). In both cases, the agreement time ts between the
two models increases as we increase the bandwidth in the lead.
Most importantly, as we decrease the ratio W/4Q l this
time window goes to infinity (see Fig. 10) independently of
the strength of the disorder with respect to the energy scale of
the intrinsic system (measured in our case by the ratio W/4£2).
FIG. 10. (Color online) Survival probability P(t) computed
starting from the state with the largest width (left panel) and starting
from the state with the second-largest width (right panel) for different
values of the coupling £2t within the lead. A good agreement between
the Hermitian model (symbols) and the non-Hermitian one (curves)
is present up to a critical time ts which increases upon increasing the
energy bandwidth (4£2L) in the lead. Parameters are NR = 4, £ 2 = 1,
y = 2, Nl = 4000, and the disorder strength is given by W = 1.
094301-12
NON-HERMITIAN HAMILTONIAN APPROACH TO QUANTUM ... PHYSICAL REVIEW B 91, 094301 (2015)
On the contrary, for any finite bandwidth, as we increase W
approaching £2^, the evolution given by two models differs
for all times. This means that for any given disorder strength
the non-Hermitian description will be a good approximation
provided that the energy bandwidth of the lead is sufficiently
large.
To further illustrate this point, we now analyze the
agreement between the two models looking at the transport
efficiency p(r), commonly used in literature, defined as
m = 1 - P(t). (60)
Note that r j(t) is the probability that the excitation has escaped
into the lead within the time t. In our simulations, we set
t = h/y.lf P(t) decays with a rate y/h, corresponding to that
of noninteracting decaying sites, r](h/y) assumes the value
1 — l/e. Hence, a value of rj(h/y) grater than 1 — l/e signals
a superradiant cooperative decay, while a value of r](h/y)
smaller than that threshold signals a subradiant decay. In what
follows, we will denote simply by ? ? the value rj(h/y).
In Fig. 11, we show the efficiency r ] varying the coupling £2L
in the lead (and accordingly modifying £2fii and NL to keep
the decay rate y fixed and remove the bouncing effect). The
results of the non-Hermitian model are shown as full curves,
while those of the Hermitian model are shown as symbols. In
the upper panel, we consider the fully symmetric initial state
| S ) of Eq. (32), while in the lower panel, we consider the fully
antisymmetric state |A5) of Eq. (43). For zero disorder, the
state | S ) is superradiant and the state |A 2>) is subradiant with
zero decay width.
For the non-Hermitian case, the behavior of r j is inde
pendent of £2/,, since we kept y fixed: the efficiency of
W
FIG. 11. (Color online) The efficiency, Eq. (60), computed start
ing from the symmetric state of Eq. (32) ( r js, upper panel) and
computed starting from the antisymmetric state of Eq. (43) ( t)as,
lower panel) is plotted versus the disorder strength W . By varying
£2l, we see that the non-Hermitian prediction (full curves) agrees with
the Hermitian evolution (symbols) up to a disorder strength (vertical
dashed lines) proportional to £2L. The dotted horizontal lines indicate
the noninteracting sites efficiency 1 — l/e. The parameters used are
Nr = 4, £ 2 = 1, y = 2, Nl = £2L (to avoid bouncing effects).the symmetric state (Fig. 11, upper panel) decreases with
the disorder strength, asymptotically approaching the value
1 - l/e (dotted line), which would be the efficiency of
noninteracting decaying sites, while the efficiency of the
antisymmetric state (Fig. 11, lower panel) increases with the
disorder up to the same limiting value.
As for the Hermitian model, it is in perfect agreement with
the non-Hermitian one for small disorder strength, while, for
strong disorder, the efficiency goes to zero. This is due to the
fact that, when W > 4£2L , some of the energy levels in the
ring lie outside the energy band in the lead, thus producing
a suppression of decay. Such a suppression is completely
neglected in the non-Hermitian model which is derived by
assuming an infinite energy band in the lead, as explained at
the beginning of this section. Most importantly, we notice that
the agreement between the Hermitian and the non-Hermitian
model increases proportionally to £2i; see vertical dashed lines
in Fig. 11. In the following sections, we will analyze whether
the interesting effects found in the non-Hermitian model
and described at the beginning of this section (cooperative
robustness and subradiant hybrid states) can be found also in
the Hermitian model for W < $ £ £2L.
B. Cooperative robustness to disorder
As already mentioned, disorder will quench superradiance
and the critical disorder W sr at which this occurs has been
computed in Ref. [ 17, Eq. (11)], for the non-Hermitian model,
assuming a disorder-independent opening strength. For the
sake of clarity, we report below that result:
Wsr =
NE,48£22(A« - 1)
n7-i i
<?=1(61)
(cos^ - U 2+ ^ £
For the parameter range, NRy 4£2, Eq. (61) reduces to
W sr = V3NRy. (62)
We stress that the linear growth of Wsr with the ring size
Nr, Eq. (62), is a manifestation of cooperative robustness to
disorder.
To illustrate this effect, we plotted in Fig. 12 the transport
efficiency p versus disorder computed taking as initial state the
symmetric state | S), for different ring sizes NR . The results for
the non-Hermitian model (full curves) are compared with the
results for the Hermitian model (symbols). The agreement
between the two models persists up to a certain value of W
indicated by the vertical arrow in Fig. 12. The fact that above
this value of disorder the agreement between the two models
becomes poor is due to the finite energy bandwidth in the lead.
As it has been explained in the previous section, the value of
W up to which the to models agree, depends only on W/4£2i,
which is kept fixed for the data shown in Fig. 12.
Figure 12 clearly shows that, even in the full Hermitian
model, upon increasing the ring size, the disorder needed
to quench the superradiant transport increases. That disorder
strength is well estimated by W sr given in Eq. (62) (see
vertical dashed lines in Fig. 12). We also checked that the
full expression for Wsr, Eq. (61), gives a good estimate of
094301-13
GIUSTERI, MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91, 094301 (2015)
FIG. 12. (Color online) The efficiency r / s, Eq. (60), computed
starting from the symmetric state of Eq. (32), is plotted vs the disorder
strength W . The size NR of the ring has been varied, keeping fixed
£ 2 = 1, S iR L = 10, Ql = 100, and NL = 100. Symbols are obtained
with the Hermitian model, while solid curves with the non-Hermitian
one. The dashed horizontal line indicates the noninteracting sites
efficiency 1 — 1 /e, asymptotically approached by the non-Hermitian
evolution. The vertical dashed lines mark the superradiance transition
W sr, Eq. (62), and the arrow roughly indicates the value of disorder
up to which the two models agree.
the critical disorder quenching superradiance in any param eter
range for the Herm itian model provided that W < 3 C £2R.
C. Hybrid subradiant states
In Refs. [17,18], it was shown that the superradiant state
does not localize at the finite-size Anderson transition, W|0 C ,
Eq. (57), but it starts to localize only above the superradiant
transition, Ws r . On the other side, subradiant states feel the
Anderson transition in a way similar to that of the states
of the closed system. Specifically, it was shown that, for
W[0 C < W < Wsr, subradiant states display a hybrid nature,
with an exponentially localized peak and an extended plateau.
The persistence o f signatures of Anderson localization in
the subradiant regime is somehow surprising; since in this
regim e, the opening is large, one could expect that the
long-range coupling induced by the opening would destroy
localization. This regim e was named subradiant hybrid regime
in Ref. [18],
To show that this regime is present also in the Hermitian
model, we cannot follow the same procedure that was followed
in Refs. [17,18], where the structure of the eigenstates of the
effective Hamiltonian was analyzed. On the other side, we can
analyze the long-term dynamics of a state initially localized
on a single site of the ring. This state has a small overlap with
the superradiant state. That com ponent will decay fast, and
the dynam ics will bring the system in the subradiant subspace
with a much slower decay. Thus we can expect that the hybrid
structure of the subradiant states will reveal itself in the long
time form of the wave function.
In order to show this point, in Fig. 13, we plot the probability
of being on the ring site r, obtained by the long-tim e evolution
of an excitation initially localized on site 1. We chose theFIG. 13. (Color online) Probability of being on a ring site at
distance d from site 1, obtained by the long-time evolution of an
excitation initially localized on site 1, for different values of the
ring size NR . The wave function i/r* is normalized by setting to
1 the probability of being on the ring. We construct the long
time shape of the probability by letting the system evolve until
a steady configuration is reached. We chose the disorder strength
W = 10 in a regime where Anderson localization should be achieved,
while superradiance is not yet destroyed, that is, Wlo c < W < W s t .
Parameters are £ 2 = 1, £2/^ = 10, QL = 100, y = 2, and NL = 100.
The agreement between the Hermitian model H + D (circles) and
the non-Hermitian one Heff + D (solid curves) is very good. The
exponential peak on the initially excited site corresponds to the one
obtained for a closed ring (£2fii = y = 0), indicated by the black
dashed curve. Dotted horizontal lines mark the values 0.38 /NR and
have been drawn to highlight the scaling of the plateau with the ring
size.
disorder strength V P in a regime where Anderson localization
should be achieved, while superradiance is not yet destroyed,
that is, Wioc < W < Ws r . O f course, we chose a value of £2R
for which the agreement between the Hermitian and the non-
Hermitian model is good in the relevant disorder range. The
probability plotted in Fig. 13 is normalized by setting to 1 the
probability of being on the ring.
In the localized regime and in absence of the coupling
with the lead, the diffusion of an excitation initially placed
on one site would be suppressed, resulting in a long-time
probability distribution exponentially localized on the initial
site (see dashed curve in Fig. 13).
On the other side, in the full model, we obtain a hybrid
state, characterized by an exponential peak on the initial site
and a fully extended plateau on the other sites. The important
features of this hybrid structure are (i) the exponential peak
coincides with the one obtained in a closed ring (for which
= y = 0) and (ii) the probability on the extended plateau
decreases as \/N R as we increase the ring size. Again we
observe that the non-Herm itian model (solid curves) is in very
good agreement with the Hermitian one (circles), thus proving
that the presence of hybrid subradiant states, described in
Ref. [ 18], is a genuine feature of the full Hermitian model from
which / / eff is deduced. Importantly, in the limit NR ->• oo, the
subradiant states become fully localized. For a more detailed
discussion of the origin of this regime see Ref. [18],
094301-14
NON-HERMITIAN HAMILTONIAN APPROACH TO QUANTUM ... PHYSICAL REVIEW B 91, 094301 (2015)
VII. TRANSPORT IN A GENERIC NETWORK
In order to discuss the range of applicability of the
results discussed so far, we consider here a generic example
of network with a sink represented by an external one
dimensional lead. From our previous results, we expect that
the energy-independent non-Hermitian Hamiltonian approach
will be valid under the condition that the energy band in the
lead is much larger than the energy range of the network.
We consider a fully-connected disordered network, de
picted in the upper panel of Fig. 14. The system is described
by a tight-binding Hamiltonian with site energies chosen
randomly in the interval [-W /2,W /2]. Moreover, each site
is coupled to all of the other sites with a tunneling coupling £ 2
randomly distributed in the interval [-1,1]. One of the sites
is coupled to an external one-dimensional lead with coupling
£ 2rl. The external lead is an ordered chain of sites connected
with tunneling coupling QL.
The non-Hermitian Hamiltonian for this model can be
obtained, following Sec. Ill, by adding to the energy of the
site connected to the lead the imaginary term — iy/2, with
Y given in Eq. (29). The diameter in the complex plane of
the eigenvalues of the non-Hermitian Hamiltonian defines the
energy range AE of the disordered network, while the energy
tc1.0,
1 Y °°O n A A. ° o 0 oooooc0 0 AE/4nL = 0 .3
o o AE/4£lL = 1 .4
° 0 0 0 0 o o 0 0 0 °0 0 0 0 0 0 0 '
v v v v v
0 . 0 1 - - - - - - - - - - - - ‘- - - - - - - - - - - - - - - - - - ■ --------------------------. ------------------- I -------------------. _ _ _ _ _ _ _ . ___________I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
0 100 2 0 0 300 4 0 0 500 600 700 800
t
FIG. 14. (Color online) In this figure, we consider a fully con
nected disordered network, depicted in the upper panel and described
in Sec. VII. One of the sites (red bottom-right site) is coupled to an
external one-dimensional lead with NL = 4000. In the lower panel,
we compare the results of the Hermitian model (symbols) with those
of the non-Hermitian one (curves). We analyze the probability P(t) of
being in the network vs time, computed for an initial state localized
on one site (green upper-left site, upper panel) that is not directly
connected to the lead. Data shown refer to a single realization of the
disordered network, and they have been obtained by changing the
ratio AE/4QL between the energy range of the disordered network
and the energy band in the lead (see legend). Note that we kept the
value y = 2 constant and we chose W = 1.bandwidth in the lead is given by 4£2L. We thus expect the
non-Hermitian approximation to be effective when AE/4£iL
is sufficiently small.
To show this point, we analyze the probability of being
in the network versus time, computed for an initial state
localized on one site that is not directly connected to the
lead. The typical result is shown in Fig. 14, lower panel: when
AE/4£lL < 1, the Hermitian and non-Hermitian models agree
(compare diamonds with green curve) in a large time window;
on the other side, when AE/4£lL > 1, the non-Hermitian
model looses its validity (compare circles with red curve).
VIII. CONCLUSIONS
We analyze the problem of describing the transport prop
erties of quantum networks coupled to external environments
acting as sinks, in the sense that they absorb the excitation
from the network in an irreversible way. To this end, we
analyze a paradigmatic model for quantum transport and
decay. Our tight-binding model consists of a network of
sites arranged in a ring and connected to a central lead. We
derive an energy-independent non-Hermitian model, which
greatly simplifies the analysis of its transport properties. This
non-Hermitian model retains only the degrees of freedom of
the ring, summarizing the coupling with the infinite degrees
of freedom of the lead into non-Hermitian opening terms,
which induce a decay of the probability to be on the ring.
Such non-Hermitian terms can be obtained from the same
quantities, which are used in the Fermi golden rule: the
transition amplitudes from the discrete states of the quantum
network to the continuum of states in the external sinks, and
the density of states in the sinks.
Such a kind of non-Hermitian models are widely used in
literature, but the problem of their validity is often overlooked.
Here, by comparing the results of the full Hermitian model with
those given by the non-Hermitian one, we demonstrate that
the energy-independent non-Hermitian Hamiltonian approach
is valid in the regime of large energy band in the lead. Under
that condition, we show that the interesting effects usually
described with the non-Hermitian model, such as super- and
subradiance in transport, are present also in the full Hermitian
model.
We also consider the decay from the ring in presence of
static disorder. We discuss the validity of the assumption
that the opening strength to the continuum is independent
of disorder, which is often used in literature since it greatly
simplifies the problem. We show that the non-Hermitian
Hamiltonian, with opening terms independent of disorder, is
able to describe the decay in the full Hermitian model for a
range of disorder for which the energy range in the ring is
much smaller than the energy band in the lead. In this regime,
we were able to confirm the existence of the interesting effects
predicted within the non-Hermitian Hamiltonian approach also
in the full Hermitian model. Indeed, superradiant states are
cooperatively robust to disorder, while subradiant states show
a different behavior, displaying a hybrid nature, due to the
interplay of disorder and opening.
Our results have a wide range of applicability: if we con
sider a generic quantum network of sites coupled to an external
lead, the energy-independent non-Hermitian Hamiltonian
094301-15
GIUSTERI, MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91, 094301 (2015)
approach is valid under the condition that the energy band in
the lead is much larger than the energy range of the network.
Specifically, in this limit, it can give an accurate description of
any observable of the network. In the case of generic external
environments acting as sinks, the same approach is effective
when the transition amplitudes from the network states to
the sink states are smooth and slowly varying functions
of the energy, in the range determined by the eigenvalues
of the disordered network. Moreover, we want to stress that, for
this approach to be valid, the coupling between the system and
the environment does not need to be small with respect to the
characteristic energy scale of the system, but only with respect
to the characteristic energy scale of the external environment.
ACKNOWLEDGMENTS
We would like to thank F. Borgonovi for his valuable
comments on earlier drafts of the present paper. We also
acknowledge useful discussions with A. Biella, R. Fazio, L.
Kaplan, F. Izrailev, S. Pascazio, and H. M. Pastawski.
APPENDIX A
In order to estimate the effect of the finite bandwidth
on the decay, we consider a different approximation of the
time-evolution operator of Eq. (35), slightly more refined
than the one leading to He ff and Eq. (37). We assume the
bandwidth to be finite, but large enough to justify the following
approximation: we consider the transition amplitude Ar(E)
and the density of states p(E), see Eq. (27), to be constant in
the finite energy band [— 2£2 l,2Ql ]. Specifically, we assume
that Ar(E)(Ar '(E))*p(E) — y/2n inside the energy band of
the lead and zero outside. We can now substitute in Eq. (35) the
limiting values given by Eq. (36), that corresponds to choosing
r srGOYN r, forx e [— 2£lL,2Q .L]
0, otherwise.(Al)
Moreover, we also assume Asr(x) = 0. The evolution operator
of Eq. (35) becomes then
Ws(r,0)1 e~Tixl y Nr------ / ----------------------------- (lx2?rZ J _ 2 n L [X _ 2fi]2 + i / 2A | ’(A2)
which is the Fourier transform of a truncated Lorentzian
suitably normalized by means of the factor Z to ensure that
U ${0,0) = 1. The evolution will be well approximated by an
exponential only for intermediate times and we will have
deviations both for small and large times, due to the truncation
at the edges of the energy band of the lead.
Some remarks on the accuracy of the approximation leading
to Eq. (A2) are in order. Given the choice of Tsr(x) in Eq. (A l),
it is possible to explicitly compute the energy shift
A srM = y In|x + 2G/J
|x — 2Q,i\
The latter function is odd, with derivative(A3)
A'(0)Y
2:xQ.i ’
and slowly divergent as x approaches the edges of the band.
We then see that, by setting Asr(x) = 0, we obtain an integrandin Eq. (A2) significantly distorted if compared to the exact one
[Eq. (35)] only in a neighborhood of the edges of the band.
To minimize the effects of such a distortion, it is then crucial
that the maximum point of the exact integrand function lies
far enough from the edges of the energy band of the lead.
Moreover, we need the decay width of the superradiant state
to be much smaller than the energy band in the lead. Since the
position of the maximum point is determined (to leading order)
by the average energy of the superradiant state (S| H IS) =
2fi, we obtain the conditions:
G « Q l, NR y<^AQ.L. (A4)
These conditions are necessary for the approximation leading
to Eq. (A2) to be accurate.
Starting from the evolution operator obtained in Eq. (A2),
we can now give an estimate of the times r0 and ts at which
the decay of the survival probability P(t), computed with
the Hermitian evolution of |S), changes from the quadratic
behavior to the exponential decay predicted by the non-
Hermitian model (to ) and from the exponential decay to a
power-law decay (ts).
The evolution operator Us(t ,0) of Eq. (A2) is given by the
Fourier transform of a Lorentzian function multiplied by a
rectangular function with support on [— 2Gi ,2G i ], Recalling
that the Fourier transform of that rectangular function is given
by
T C t
and that the Fourier transform of a product is the convolution
of the Fourier transforms, Eq. (A2) becomes
/•+0° sin (2Et t) 2S2i,, s yNr u IU s(t, 0 ) = / ----\ ^ - L e—K«-r)e- 2^ (A5)
J -oo Zttt
Now, since
sin(cur) lim ----------w -> oo 7 T r= S(T)
for any c o in the sense of distributions, if we consider in
Eq. (A5) the wide-band limit Q/ -> oo, we immediately
recover the evolution given by Eq. (37) for any time t. On the
other hand, the effects of a finite bandwidth strongly modify
the decay at both small and large times.
Under the assumption of Eq. (A4), we can set G 0 and
neglect the oscillating term e so that Us(t, 0) reduces
to the convolution of the exponential decay with the
kernel
K(t) =Zt c t(A6)
The normalization factor Z, needed to compensate the approx
imation A(x) ~ 0 already introduced in Eq. (A2), can be easily
found by applying the normalization condition Us(0,0) — 1.
The fact that the small-time decay is quadratic can be easily
seen by considering the parity of and K(r): since they
are both even functions, their derivatives are odd and
Zttt dt1=0dr = 0.
094301-16
NON-HERMITIAN HAMILTONIAN APPROACH TO QUANTUM ... PHYSICAL REVIEW B 91, 094301 (2015)
2.0
1.5
1.0
0.5
0.0
-0.5
0.0 0.2 0.4 0.6 0.8
2.0
1.5
1.0
0.5
0.0
-0.5
-0.5 0.0 0.5 1.0 1.5 2.0
TT
FIG. 15. (Color online) To illustrate the argument presented in
this section, we plotted the exponential function e~2 7t?|r| (dashed
red curve) and the kernel K(z — t) of Eq. (A6) (solid curves) for
different times t. The shaded regions are those providing the dominant
contribution to the convolution product of Eq. (A5) at each time.
Consequently, the derivative of Us(t,0) vanishes for t = 0 and
the decay is quadratic. This is true for any finite value of Q/ ,
but we see a sharp transition to a linear short-time decay in the
limit Ql — > - oo.
An intuitive explanation of that transition can be given with
the aid of Fig. 15. In the first panel we plotted and
K(r — t) for t = 0. The evolution at each time t is given by the
integral of the product of the exponential and the kernel K and
the dominant contribution for t = 0 comes from the shaded
region in Fig. 15 first panel. The amplitude of this region is
twice the inverse of the oscillation frequency 2 £ 2 L /h. As we
increase t of a small amount dt, since the shaded region lies on
both sides of the peak of the exponential function, the variation
in the integral is of order 0(dt2 ) , producing a quadratic decay.
This is no longer true in the limit £ 2 L — ► oo, since K tends
to a Dirac function whose support can lie only on one side of
the peak, so that the variation of the convolution integral is of
order O(dt), entailing a linear small-time decay.
From analogous considerations, we can estimate t o as the
time at which the relevant region (shaded region in Fig. 15,
second panel) lies only on one side of the peak of the
exponential. Since the peak of the kernel K(t — r) is at z — t,
we have
(A7)
We can then see that, for t > t o the decay is exponential, since
the main contribution to the convolution integral (see shadedregions in Fig. 15, second panel) is proportional to
r i * Re - hId
For even larger times, together with the previously de
scribed exponential term (right shaded region in Fig. 15, third
panel), a second term contributes to the convolution integral
(left shaded region in Fig. 15, third panel). The first involves
the central part of the kernel K and the tail of the exponential
function, while the second involves the tail of the kernel K
and the central part of the exponential function. The first
contribution is again proportional to and the second
one to h/{2ElR t). When the term involving the tail of K is
dominant, we have a power-law decay. Hence we can estimate
the transition time ts as the time at which the two contributions
are comparable by setting
o 2 h lS
2 Q it$
which leads to the equation
YNr 4£2 l yNR
— — 1 $ = In -------- h I n------- ts2 h yNR 2 h
and, neglecting the last logarithmic term, to the estimate
ts oc2 h AQ.iIn-yNR yNR(A8)
The exponent of the power-law decay, being determined
by the long-time behavior of the convolution kernel K, is
strongly dependent on how the Lorentzian density of Eq. (A2)
is deformed to be zero outside the energy band of the lead.
Indeed, the sharp truncation considered above, given by the
definition of rs r in Eq. (A l), corresponds to multiplying the
Lorentzian with a rectangular function, that produces a 1/t
decay due to the form of the kernel K of Eq. (A6).
Nevertheless, we can easily understand the effect of a
different deformation: if we multiply the Lorentzian function
in Eq. (A2) by a compactly supported function, which goes to
zero as (x - Eedge)p in proximity of the edges of the energy
band, by a well-known result in Fourier analysis [25,33], we
will obtain a convolution kernel K, which decays as l/tp + l
for large times. Such a modification does not affect any of
the foregoing results, but produces a long-time decay of the
probability amplitude proportional to 1 /tp+ l. Consequently,
the survival probability P(t) will decay as 1 /t2 (p + ^.
If we consider now the detailed structure of Fsr in the
finite-bandwidth case, Eq. (33), we see that it goes to zero
in proximity of the edges of the energy band with exponent
p — 1/2. This implies a decay 1 /? 3/2 of the convolution kernel
and the decay 1 /r3 of the survival probability P(t), which was
indeed found in the numerical results shown in Fig. 4.
APPENDIX B
To determine the behavior for very short times, we will
follow now a different approach, more heuristic than the one
used in Appendix A. If we consider an initial state on the ring,
then it “becomes aware” of the presence of the lead only after
some time. In particular, we can expect the initial dynamics
to be determined by the interaction of the ring with the first
site of the lead. If we had = 0, the fully symmetric ring
094301-17
GIUSTERI, MATTIOTTI, AND CELARDO PHYSICAL REVIEW B 91, 094301 (2015)
state | S) would be only coupled to the first lead site, and its
dynamics would be determined by the 2 x 2 Hamiltonian
( \/N rQrl\
0 ) ’
which has eigenvalues
Li,2 = £ 2 ± + NrQ,2 r l .
Consequently, we obtain the following estimate for the short-
time decay of the survival probability of the superradiant state:
Nr£2~ d, n
P(t)*l- t ■ (Bl)fr
According to the foregoing argument, the time to up to
which Eq. (Bl) can be a good approximation of the dynamicsshould decrease upon increasing the coupling QL within the
lead. Indeed, the value to = h/2Q,i, presented in Eq. (A7),
gives a good estimate of this threshold. Clearly, the non-
Hermitian model cannot reproduce the true dynamics of the
system up to to , since that model is obtained considering the
effect of a lead with an infinite coupling C/ in the lead.
Let us now consider the case of a disordered ring described
by the Hamiltonian H + D, Eqs. (22, 56). Also in this case,
for short times the true evolution will be different from the
evolution given by the non-Hermitian model. Indeed, the
short-time dynamics is well approximated by the evolution
under H + D, where H describes the subsystem formed by
the ring and the first lead site. We can estimate with the same
to given above the time up to which the system does not feel
the presence of the other lead sites and, consequently, the
non-Hermitian model is not applicable.
[1] C. W. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
[2] P. A. Lee and T. V . Ramakrishnan, Rev. Mod. Phys. 57, 287
(1985).
[3] V. V . Sokolov and V . G. Zelevinsky, Nucl. Phys. A 504, 562
(1989); Phys. Lett. B 202, 10 (1988); I. Rotter, Rep. Prog. Phys.
54, 635 (1991); V. V . Sokolov and V. G. Zelevinsky, Ann. Phys.
(N. Y.) 216, 323 (1992).
[4] H. Feshbach, Ann. Phys. 5, 357 (1958); 19, 287 (1962); 43,
410 (1967); C. Mahaux and H. A. Weidenmiiller, Shell Model
Approach to Nuclear Reactions (North Holland, Amsterdam,
1969).
[5] A. F. Sadreev and I. Rotter, J. Phys. A 36, 11413 (2003);
M. Weiss, J. A. Mendez-Bermudez, and T. Kottos, Phys. Rev. B
73, 045103 (2006).
[61 F. M. Dittes, Phys. Rep. 339, 215 (2000).
[7] A. Amir, Y . Oreg, and Y . Imry. Phys. Rev. A 77, 050101(R)
(2008).
[8] N. Auerbach and V. Zelevinsky, Rep. Prog. Phys. 74, 106301
(2011 ).
[9] G. L. Celardo (unpublished).
[10] G. L. Celardo and L. Kaplan, Phys. Rev. B 79, 155108 (2009);
G. L. Celardo, A. M. Smith, S. Sorathia, V. G. Zelevinsky,
R. A. Sen’kov, andL. Kaplan, ibid. 82, 165437 (2010).
[11] A. Ziletti, F. Borgonovi, G. L. Celardo, F. M. Izrailev, L. Kaplan,
and V. G. Zelevinsky, Phys. Rev. B 85, 052201 (2012); S.
Sorathia, F. M. Izrailev, V . G. Zelevinsky, and G. L. Celardo,
Phys. Rev. E 86, 011142 (2012); I. F. Herrera-Gonzalez,
J. A. Mendez-Bermudez, and F. M. Izrailev, ibid. 90. 042115
(2014).
[12] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik,
J. Chem. Phys. 129. 174106 (2008); M. B. Plenio and S. F.
Huelga, New J. Phys. 10, 113019 (2008); F. Caruso, A. W.
Chin, A. Datta, S. F. Huelga, and M. B. Plenio, J. Chem. Phys.
131, 105106 (2009).
[13] G. L. Celardo, F. Borgonovi, V. I. Tsifrinovich, M. Merkli, and
G. P . Berman, J. Phys. Chem. C 116, 22105 (2012); D. Ferrari,
G. L. Celardo, G. P. Berman, R. T. Sayre, and F. Borgonovi,
ibid. 118, 20 (2014).
[14] A. Olaya-Castro, C. F. Lee, F. Fassioli Olsen, and N. F. Johnson,
Phys. Rev. B 78, 085115 (2008).[15] J. Grad, G. Hernandez, and S. Mukamel, Phys. Rev. A 37, 3835
(1988); F. C. Spano, J. R. Kuklinski, and S. Mukamel, J. Chem.
Phys. 94, 7534 (1991); M. J. Stephen, ibid. 40, 669 (1964);
R. H. Lehmberg, Phys. Rev. A 2, 883 (1970).
[16] G. L. Celardo, P. Poli, L. Lussardi, and F. Borgonovi, Phys. Rev.
B 90, 085142 (2014).
[17] G. L. Celardo, G. G. Giusteri, and F. Borgonovi, Phys. Rev. B
90. 075113 (2014).
[18] G. L. Celardo, A. Biella, L. Kaplan, and F. Borgonovi, Fortschr.
Phys. 61, 250 (2013); A. Biella, F. Borgonovi, R. Kaiser, and
G. L. Celardo, Europhys. Lett. 103, 57009 (2013).
[19] J. M. Moix, M. Khasin, and J. Cao, New J. Phys. 15, 085010
(2013).
[20] K. D. B. Higgins, S. C. Benjamin, T. M. Stace, G. J. Milbum,
B. W. Lovett, and E. M. Gauger, Nat. Commun. 5,4705 (2014).
[21] M. Sarovar and K. B. Whaley, New J. Phys. 15, 013030 (2013).
[22] F. C. Spano and S. Mukamel, J. Chem. Phys. 91, 683 (1989).
[23] A. Messiah, Quantum Mechanics (North Holland Publishing
Company, Amsterdam, 1962), Vol. II, Chap. XXI. 13.
[24] C. Dembowski, H.-D. Graf, H. L. Harney, A. Heine, W. D. Heiss,
H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86, 787 (2001);
W. D. Heiss, J. Phys. A: Math. Theor. 45, 444016 (2012).
[25] P . Facchi and S. Pascazio, La Regola d'oro di Fermi (Quademi
di Fisica Teorica, Bibliopolis, 1999).
[26] A. Peres, Ann. Phys. 129, 33 (1980).
[27] H. M. Pastawski, Physica B 398, 278 (2007); E. Rufeil Fiori,
H. M. Pastawski, Chem. Phys. Lett. 420, 35 (2006).
[28] E. Persson, T. Gorin, and I. Rotter, Phys. Rev. E 58,1334 (1998);
J. Wiersig, Phys. Rev. Lett. 97, 253901 (2006).
[29] H. Fidder, J. Knoester, and D. A. Wiersma, J. Chem. Phys.
95, 7880 (1991); J. Moll, S. Daehne, J. R. Durrant, and D. A.
Wiersma, ibid. 102, 6362 (1995).
[30] X. Hu, T. Ritz, A. Damjanovic, and K. Schulten, J. Phys. Chem.
B 101, 3854(1997).
[31] M. Peshkin, A. Volya, and V. Zelevinsky, Europhys. Lett. 107,
40001 (2014).
[32] P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[33] S. Bochner and K. Chandrasekharan, Fourier Transforms , Issue
19 of Annals of Mathematics Studies (Princeton University
Press, NJ, 1950).
094301-18
Copyright
of
Physical
Review
B:
Condensed
Matter
&
Materials
Physics
is
the
property
of
American
Physical
Society
and
its
content
may
not
be
copied
or
emailed
to
multiple
sites
or
posted
to
a
listserv
without
the
copyright
holder's
express
written
permission.
However,
users
may
print,
download,
or
email
articles
for
individual
use.
|
PhysRevB.98.195403.pdf | PHYSICAL REVIEW B 98, 195403 (2018)
Effect of illumination on quantum lifetime in GaAs quantum wells
X. Fu,1A. Riedl,1M. Borisov,1M. A. Zudov,1,*J. D. Watson,2,†G. Gardner,2M. J. Manfra,2,3,4K. W. Baldwin,5
L. N. Pfeiffer,5and K. W. West5
1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
2Department of Physics and Astronomy and Birck Nanotechnology Center , Purdue University, West Lafayette, Indiana 47907, USA
3Station Q Purdue, Purdue University, West Lafayette, Indiana 47907, USA
4School of Materials Engineering and School of Electrical and Computer Engineering, Purdue University,
West Lafayette, Indiana 47907, USA
5Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
(Received 16 August 2018; revised manuscript received 7 October 2018; published 5 November 2018)
Low-temperature illumination of a two-dimensional electron gas in GaAs quantum wells is known to greatly
improve the quality of high-field magnetotransport. The improvement is known to occur even when the carrierdensity and mobility remain unchanged, but what exactly causes it remains unclear. Here, we investigate theeffect of illumination on microwave photoresistance in low magnetic fields. We find that the amplitude ofmicrowave-induced resistance oscillations grows dramatically after illumination. Dingle analysis reveals thatthis growth reflects a substantial increase in the single-particle (quantum) lifetime, which likely originates fromthe light-induced redistribution of charge enhancing the screening capability of the doping layers.
DOI: 10.1103/PhysRevB.98.195403
Even though low-temperature illumination of a two-
dimensional electron gas (2DEG) in GaAs quantum wells isknown to improve the quality of high-field magnetotransport[1–3], systematic investigations of this effect remain limited.
One study [ 2] has investigated the effect of illumination
on a 2DEG residing in a 30-nm-wide GaAs /Al
0.34Ga0.66As
quantum well with Si δ-doping layers placed directly in
Al0.34Ga0.66As barriers on both sides at setback distances of
100 nm (above the well) and 120 nm (below the well). Theinitial effect of illumination is a considerable increase of boththe density n
eand the mobility μof the 2DEG [ 4] which,
predictably, resulted in better developed fractional quantumHall (FQH) states. However, additional, higher-intensity il-lumination left n
eandμessentially unchanged, while the
transport features, e.g., the fragile FQH states in the N=1
Landau level, were further improved. This improvement wasattributed to the enhanced screening of ionized impurities byan increased number of polarized neutral shallow donors [ 5].
Another study [ 3] investigated the effect of illumination
in a 2DEG hosted by a 30-nm-wide GaAs /Al
0.24Ga0.76As
quantum well utilizing a “modern” doping scheme. This het-erostructure was also remotely doped on both sides, but Siatoms were placed inside very narrow GaAs “doping” wellssandwiched between thin AlAs layers [ 6–12]. Such a doping
scheme avoids formation of deep donor states; all Si atomsare ionized, but a significant fraction of donated electronspopulate the X band in surrounding AlAs layers. Interestingly,illumination of such a structure can also lead to improvementof high-field transport characteristics even though it does
*Corresponding author: zudov001@umn.edu
†Present address: Microsoft Station-Q at Delft University of Tech-
nology, 2600 GA Delft, The Netherlands.not appreciably change neandμ. For example, Ref. [ 3]
has shown that illumination can significantly enhance themeasured energy gap of the FQH state at filling factor ν=5/2
and better development of other fragile quantum Hall states.The enhancement of transport quality was linked to improvedhomogeneity of the 2DEG achieved after illumination.
In this paper we (i) examine the effect of illumination on
the quality of the low-field magnetotransport under microwave
irradiation and (ii) quantitatively assess the effect of illumi-nation on total (quantum) lifetime τ
q, which is a measure
of electron-remote impurity scattering. To measure τqwe
employ microwave-induced resistance oscillations (MIRO)[13,14] which, in contrast to Shubnikov-de Haas oscillations
[15], are believed to be largely immune to macroscopic
density fluctuations. We find that after illumination MIRObecome more pronounced while extending to lower magneticfields. The Dingle analysis reveals that the observed im-provement is a result of significant enhancement of quantumlifetime which increases by a factor of about two. This en-hancement presents strong evidence that illumination resultsin reduced scattering from remote impurities, presumablydue to light-induced redistribution of charge improving thescreening capability of the doping layers. Whether or notthe increase of τ
qalso contributes to the improvement of
high-field transport [ 1–3] remains an open question [ 9,15].
While we have investigated several samples with similar
outcomes, here we present the results from two samples whichexhibited almost no change in mobility due to illumination.The 2DEG in sample A (B) resides in a GaAs quantum wellof width 30 nm (24.9 nm) surrounded by Al
xGa1−xAs barriers
withx=0.24 (x=0.28). Sample A (B) utilized Si doping in
narrow GaAs doping wells surrounded by thin AlAs layersand positioned at a setback distance of 75 nm (80 nm) onboth sides of the GaAs well hosting the 2DEG [ 16]. Both
2469-9950/2018/98(19)/195403(4) 195403-1 ©2018 American Physical SocietyX. FU et al. PHYSICAL REVIEW B 98, 195403 (2018)
samples were 4 ×4 mm squares with eight indium contacts
fabricated at the corners and the midsides. When cooled in thedark, sample A (B) had the density n
e≈2.57×1011cm−2
(ne≈3.33×1011cm−2). Low-temperature mobility was es-
timated to be μ≈1.5×107cm2V−1s−1in sample A and
μ≈1.6×107cm2V−1s−1in sample B. Measurements were
performed in Faraday configuration; microwave radiation wasdelivered to the sample immersed in liquid
3He inside a super-
conducting solenoid via a rectangular (WR-28) stainless steelwaveguide with the magnetic field was applied perpendicularto the 2DEG. The longitudinal resistance Rin sample A
(B) was recorded using a standard low-frequency (a few Hz)four-terminal lock-in technique under continuous irradiationby microwaves of f=34 GHz ( f=64 GHz) at a constant
coolant temperature T≈0.3K(T≈1.8K ) .
Both sample A and sample B were illuminated by visible
light (either green or white light-emitting diode) via the mi-crowave waveguide at zero magnetic field for 10 minutes. Forsample A, we followed a procedure outlined in Ref. [ 3]; illu-
mination at base temperature ( T≈0.3 K in our case) follow-
ing up by an annealing step at T≈2.5 K for 15 minutes. For
sample B, we used “conventional” illumination temperatureofT≈5 K after which the sample was cooled down in the
dark. After illumination procedure, the density of sample A(B) increased only by ≈4×10
9cm−2(≈9×109cm−2) while
the mobilities remained essentially unchanged. However, aswe show next, both illumination protocols yielded substantialimprovement of the quality of low-field magnetotransport,manifested by more pronounced MIRO, which we link to theenhancement of the quantum lifetime.
Before presenting our experimental results, we recall that
the oscillatory microwave photoresistance δR, i.e., the change
of resistance caused by microwave radiation, can be written as[17–20]
δR(/epsilon1)
R0∝− 2π/epsilon1λ2Psin 2π/epsilon1. (1)
Here, R0is the resistance at B=0,/epsilon1=2πf/ω c,ωc=
eB/m⋆is the cyclotron frequency, m⋆≈0.06m0is the elec-
tron effective mass [ 21–23],λ=exp(−π/ω cτq) is the Dingle
factor, and P(/epsilon1) is the effective microwave power which, for
linearly polarized microwaves, is given by [ 24,25]
P(/epsilon1)=P0
2/summationdisplay
±1
(1±/epsilon1−1)2+β2ω,P0=e2E2
acv2
F
/epsilon1eff¯h2ω4, (2)
where βω≡(ωτem)−1+(ωτ)−1,τ=(m⋆/e)μis the momen-
tum relaxation time, τ−1
em=nee2/2√/epsilon1eff/epsilon10m⋆c[24,26]i st h e
radiative decay rate, 2√/epsilon1eff=√ε+1 defines the effective
dielectric constant /epsilon1eff,ε=12.8 is the dielectric constant of
GaAs, vFis the Fermi velocity, and Eacis the microwave
electric field.
The effect of low-temperature illumination on MIRO in
sample A is illustrated in Fig. 1(a) which shows the re-
sistance Rnormalized to its zero-field value R0measured
before (dotted line) and after (solid line) illumination undermicrowave irradiation of frequency f=34 GHz at temper-
atureT≈0.3 K. Vertical lines are drawn at integer /epsilon1,a s
marked. The data clearly reveal that after illumination MIRObecome more pronounced and extend to higher orders. Similar = 3
= 3
FIG. 1. Resistance in units of the zero-field resistance R/R 0as
a function of Bmeasured before (dotted line) and after (solid line)
illumination in (a) sample A at T≈0.3Ka n d f=34 GHz and
(b) sample B at T≈1.8Ka n d f=68 GHz. Vertical lines are drawn
at integer /epsilon1,a sm a r k e d .
measurements in sample B, though employing different illu-
mination procedure, yielded qualitatively identical results, asillustrated in Fig. 1(b) showing the data at f=68 GHz and
T≈1.8K .
The results in Fig. 1reveal that the enhancement of MIRO
after illumination is significantly more pronounced at higher/epsilon1, signaling an increase in quantum lifetime. To quantify
this increase, we performed Dingle analysis of MIRO. Fol-lowing Eq. ( 1), we introduce a reduced MIRO amplitude
A=|δR|
maxP0/2π/epsilon1PR0[27], where |δR|maxis the measured
MIRO amplitude. The results for sample A and sample Bare presented in Figs. 2(a) and 2(b), respectively, which
show Aas a function of /epsilon1extracted from the data acquired
before (•) and after ( /squaresolid) illumination. Fitting the data with
A=A0exp(−/epsilon1/fτ q) (solid lines) reveals that illumination
enhances τqfrom 23 ps to 44 ps in sample A and from 16 ps
to 32 ps in sample B.
It is known that in contrast to Shubnikov-de Haas oscilla-
tions [ 28,29], MIRO yield the quantum lifetime which is re-
duced by electron-electron scattering [ 30]. More specifically
[18,31–34], one can write:
τ−1
q=τ−1
q,0+τ−1
ee, (3)
195403-2EFFECT OF ILLUMINATION ON QUANTUM LIFETIME IN … PHYSICAL REVIEW B 98, 195403 (2018)
FIG. 2. Reduced MIRO amplitude A=|δR|max(P0/P)/2π/epsilon1R 0
[27] as a function of /epsilon1for (a) sample A and (b) sample B be-
fore (•)a n da f t e r( /squaresolid) illumination. Fitting the data with A=
A0exp(−/epsilon1/fτ q) (solid lines) reveals that illumination enhances τq
f r o m2 3p st o4 4p si ns a m p l eAa n df r o m1 6p st o3 2p si n
sample B.
where τ−1
q,0represents the electron-impurity contribution and
the electron-electron contribution τ−1
eecan be written as
[17,31,32]
¯h
τee=πk2
BT2
4EFln2¯hvF/aB
πkBT. (4)
Here,EFis the Fermi energy and aB≈11 nm is the Bohr ra-
dius in GaAs. It is clear that subtracting the electron-electroncontribution will only increase the change in impurity-limitedquantum lifetime caused by illumination. Because measure-ments on sample A were performed at low temperature ( T≈
0.3 K), electron-electron scattering rate is much smaller than
τ
−1
q≈τ−1
q,0. In sample B, however, Eq. ( 4) yields τee≈80 ps
and using Eq. ( 3) we can estimate that τq,0increases from
τq,0≈20 ps to τq,0≈53 ps upon illumination.
While the observed increase of quantum lifetime after
illumination in both samples is quite significant, the momen-tum relaxation time τremained virtually unchanged. This
observation allows us to establish the source of disorder whichis affected by illumination. Since the quantum scattering rate,in general, is much more sensitive to remote impurities thanthe transport scattering rate, we can conclude that illuminationprimarily affects scattering from remote impurities rather than
from those in the vicinity of the GaAs quantum well. Insen-sitivity of τto illumination then suggests that contribution
of the remote impurities to the momentum relaxation rate isnegligible even before the illumination, i.e., that τis limited
by scattering from unintentional background impurities withinthe GaAs quantum well and in the AlGaAs barriers [ 12,35].
The quantum scattering rate, on the other hand, can stillcontain a sizable or even dominant contribution from theremote impurities, e.g., Si ions in the doping layers, beforethe sample has been illuminated.
Recent theoretical examination [ 12,35] of the doping layers
has shown that excess electrons which occupy the X bandsof the AlAs miniwells form compact dipoles with donorsof their choice (to minimize their energy) which reside inGaAs miniwells. These X electrons can effectively screenthe random potential from the remaining unpaired ionized Siatoms and the screening effectiveness grows rapidly with theirnumber. Because of this fast growth, the doping layer whichhas fewer X electrons will contribute much more strongly toscattering than the other one. In typical samples, such as ours,this would be the top doping layer which donates a significantnumber of electrons to compensate surface states. If theillumination can increase the number of X electrons in thetop doping layer, e.g., by returning electrons from the surface[36], one can expect a significant reduction of the quantum
scattering rate. Assuming that after illumination the numberof X electrons in the top doping layer becomes similar to thatin the bottom doping layer, theoretical estimates [ 12,35]s h o w
that the remote impurity-limited quantum lifetime should beseveral times higher than observed in our experiment. Ourfindings thus suggest that the quantum scattering rate afterillumination is limited by scattering off background impuritiesresiding in the main GaAs quantum well and in surroundingAlGaAs barriers.
While we clearly established that illumination significantly
reduces the quantum scattering rate, whether the observedreduction is the sole cause for the concurrent improvementin high-field transport characteristics [ 1–3] can be debated
[9,15]. Indeed, as mentioned in Ref. [ 3], low-temperature
illumination can also lead to improved density homogeneityof the 2DEG under study which must lead to improved devel-opment of FQH states, e.g., the increase of the excitation gapatν=5/2. MIRO, on the other hand, are nearly immune to
macroscopic density fluctuations and therefore their enhance-ment can be linked directly to the increase of the quantumlifetime. Indeed, the enhancement of MIRO accompaniedby an increase in τ
qhas been observed even when samples
became less homogeneous after illumination.
In summary, we have investigated the effect of low-
temperature illumination on low-field magnetotransport char-acteristics of two-dimensional electrons in GaAs quantumwells subjected to microwave radiation. We have found thatmicrowave-induced resistance oscillations become signifi-cantly enhanced after illumination and that this enhance-ment is due to the increase of the quantum lifetime of the2D electrons. We believe that the observed increase likelyoriginates from the light-induced redistribution of chargewhich increases the number of X electrons in the top dopinglayer. Insensitivity of transport scattering rate to illumination
195403-3X. FU et al. PHYSICAL REVIEW B 98, 195403 (2018)
confirms that electron mobility is limited by background
impurities in the vicinity of GaAs quantum well hosting the2DEG even before illumination.
We thank M. Sammon and B. Shklovskii for discussions.
The work at Minnesota was supported by the NSF AwardNo. DMR-1309578 (measurements on sample A) and by theU.S. Department of Energy, Office of Science, Basic EnergySciences, under Award No. ER 46640-SC0002567 (measure-
ments on sample B). Sample growth at Purdue was supportedby the U.S. Department of Energy, Office of Science, BasicEnergy Sciences, under Award No. DE-SC0006671. L.N.P.and K.W.W. of Princeton University acknowledge the Gordonand Betty Moore Foundation Grant No. GBMF 4420 andthe National Science Foundation MRSEC Grant No. DMR-1420541.
[1] K. B. Cooper, M. P. Lilly, J. P. Eisenstein, T. Jungwirth,
L. N. Pfeiffer, and K. W. West, Solid State Commun. 119,89
(2001 ).
[2] G. Gamez and K. Muraki, P h y s .R e v .B 88,075308 (2013 ).
[3] M. Samani, A. V . Rossokhaty, E. Sajadi, S. Lüscher, J. A. Folk,
J. D. Watson, G. C. Gardner, and M. J. Manfra, P h y s .R e v .B
90,121405 (2014 ).
[4] L. Pfeiffer, K. W. West, H. L. Stormer, and K. W. Baldwin,
Appl. Phys. Lett. 55,1888 (1989 ).
[5] Light-induced conversion of negatively-charged DX centers to
neutral shallow donors has been demonstrated much earlier inRef. [ 37] which systematically investigated the effect of illumi-
nation on time-dependent density and mobility in modulation-doped GaAs /Al
xGa1−xAs single heterojunctions. This work
also showed that illumination can increase mobility withoutaffecting the carrier density.
[6] T. Baba, T. Mizutani, and M. Ogawa, Jpn. J. Appl. Phys. 22,
L627 (1983 ).
[7] K.-J. Friedland, R. Hey, H. Kostial, R. Klann, and K. Ploog,
P h y s .R e v .L e t t . 77,4616 (1996 ).
[8] L. Pfeiffer and K. W. West, Physica E 20,57(2003 ).
[9] V . Umansky, M. Heiblum, Y . Levinson, J. Smet, J. Nübler, and
M. Dolev, J. Cryst. Growth 311,1658 (2009 ).
[10] M. J. Manfra, Annu. Rev. Condens. Matter Phys. 5,347(2014 ).
[11] G. C. Gardner, S. Fallahi, J. D. Watson, and M. J. Manfra,
J. Cryst. Growth 441,71(2016 ).
[12] M. Sammon, M. A. Zudov, and B. I. Shklovskii, Phys. Rev.
Materials 2,064604 (2018 ).
[13] M. A. Zudov, R. R. Du, J. A. Simmons, and J. L. Reno, Phys.
Rev. B 64,201311(R) (2001 ).
[14] P. D. Ye, L. W. Engel, D. C. Tsui, J. A. Simmons, J. R. Wendt,
G. A. Vawter, and J. L. Reno, Appl. Phys. Lett. 79,2193
(2001 ).
[15] Q. Qian, J. Nakamura, S. Fallahi, G. C. Gardner, J. D. Watson,
S. Lüscher, J. A. Folk, G. A. Csáthy, and M. J. Manfra, Phys.
Rev. B 96,035309 (2017 ).
[16] For a band diagram and a schematic of sample structure such
as ours, see, e.g., Fig. 1 in Ref. [ 9] or Fig. 7 in Ref. [ 10]. The
schematic and discussion of the doping layer can also be foundin Ref. [ 12].[17] I. A. Dmitriev, M. G. Vavilov, I. L. Aleiner, A. D. Mirlin, and
D. G. Polyakov, Phys. Rev. B 71
,115316 (2005 ).
[18] I. A. Dmitriev, M. Khodas, A. D. Mirlin, D. G. Polyakov, and
M. G. Vavilov, Phys. Rev. B 80,165327 (2009 ).
[19] I. A. Dmitriev, A. D. Mirlin, D. G. Polyakov, and M. A. Zudov,
Rev. Mod. Phys. 84,1709 (2012 ).
[20] Equation ( 1) was obtained assuming 2 πkBT/greatermuch¯hωand is ac-
curate away from the cyclotron resonance (2 π/epsilon1/greatermuch1), when the
microwave power is not too high ( P/lessmuch1) and when Landau
levels are overlapping ( λ/lessmuch1). All these conditions are satis-
fied reasonably well in our experiment.
[21] A. T. Hatke, M. A. Zudov, J. D. Watson, M. J. Manfra, L. N.
Pfeiffer, and K. W. West, P h y s .R e v .B 87,161307(R) (2013 ).
[22] A. V . Shchepetilnikov, D. D. Frolov, Y . A. Nefyodov, I. V .
Kukushkin, and S. Schmult, Phys. Rev. B 95,161305 (2017 ).
[23] X. Fu, Q. A. Ebner, Q. Shi, M. A. Zudov, Q. Qian, J. D. Watson,
a n dM .J .M a n f r a , P h y s .R e v .B 95,235415 (2017 ).
[ 2 4 ] K .W .C h i u ,T .K .L e e ,a n dJ .J .Q u i n n , Surf. Sci. 58,182(1976 ).
[25] M. Khodas and M. G. Vavilov, Phys. Rev. B 78,245319 (2008 ).
[26] Q. Zhang, T. Arikawa, E. Kato, J. L. Reno, W. Pan, J. D. Wat-
son, M. J. Manfra, M. A. Zudov, M. Tokman, M. Erukhimovaet al. ,P h y s .R e v .L e t t . 113,047601 (2014 ).
[27] P/P
0was calculated using m⋆=0.067m0entering /epsilon1andτem.
[28] G. W. Martin, D. L. Maslov, and M. Y . Reizer, Phys. Rev. B 68,
241309 (2003 ).
[29] Y . Adamov, I. V . Gornyi, and A. D. Mirlin, Phys. Rev. B 73,
045426 (2006 ).
[30] A. T. Hatke, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Phys.
Rev. Lett. 102,066804 (2009 ).
[31] A. V . Chaplik, Zh. Eksp. Teor. Fiz. 60, 1845 (1971) [Sov. Phys.
JETP 33, 997 (1971)].
[32] G. F. Giuliani and J. J. Quinn, P h y s .R e v .B 26,4421 (1982 ).
[33] V . Ryzhii and R. Suris, J. Phys.: Condens. Matter 15,6855
(2003 ).
[34] V . Ryzhii, A. Chaplik, and R. Suris, JETP Lett. 80,363(2004 ).
[35] M. Sammon, T. Chen, and B. I. Shklovskii, Phys. Rev. Materials
2,104001 (2018 ).
[36] M. Sammon and B. I. Shklovskii (unpublished).[37] M. Hayne, A. Usher, J. J. Harris, V . V . Moshchalkov, and C. T.
Foxon, P h y s .R e v .B 57,14813 (1998 ).
195403-4 |
PhysRevB.80.241412.pdf | Breakdown of the N=0 quantum Hall state in graphene: Two insulating regimes
L. Zhang,1J. Camacho,1H. Cao,2Y. P. Chen,2M. Khodas,1,3D. E. Kharzeev,3A. M. Tsvelik,1T. Valla,1and
I. A. Zaliznyak1,*
1CMPMSD, Brookhaven National Laboratory, Upton, New York 11973, USA
2Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA
3Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA
/H20849Received 13 April 2009; revised manuscript received 20 November 2009; published 23 December 2009 /H20850
We studied the unusual quantum Hall effect /H20849QHE /H20850near the charge neutrality point in high-mobility
graphene sample for magnetic fields up to 18 T. We observe breakdown of the delocalized QHE transport andstrong increase in resistivities
/H9267xx,/H20841/H9267xy/H20841with decreasing Landau-level filling for /H9263/H110212, where we identify two
insulating regimes. First, /H9267xx,xyincreases nearly exponentially within the range of several resistance quanta RK,
while the Hall effect gradually disappears and the off-diagonal resistivity /H9267xyeventually becomes independent
of the direction of magnetic field, consistent with the Hall insulator with local transport. Then, at a filling /H9263
/H110151/2, there is a cusp in /H9267xx/H20849/H9263/H20850and an onset of even faster growth with the decreasing /H9263, indicating transition
to a collective insulator state. A likely candidate for this state is a pinned Wigner crystal.
DOI: 10.1103/PhysRevB.80.241412 PACS number /H20849s/H20850: 73.43. /H11002f, 71.70.Di, 73.63. /H11002b
Graphene is a honeycomb monolayer of C atoms, which
forms the most common carbon allotrope graphite, but hasonly recently been isolated in the two-dimensional /H208492D/H20850
form.
1A bipartite honeycomb sp2-bonded lattice gives
graphene its unique electronic structure. Two dispersionsheets associated with the electrons belonging to two differ-ent sublattices form the filled
/H9266and the empty /H9266/H11569bands,
which meet at two distinct isolated points /H20849valleys KandK/H11032/H20850,
yielding pointlike Fermi surfaces. Low-energy electronicstates in each valley have a linear 2D conical dispersion/H9255/H20849p/H20850=
vFpand, in addition to 2D momentum, have a 2D
pseudospin quantum number accounting for the two-sublattice structure. Such quasiparticles are formally de-
scribed by the Dirac equation for chiral massless fermionsand have peculiar transport properties.
2,3
In a magnetic field Hperpendicular to the graphene layer,
the spectrum of Dirac quasiparticles is quantized into Landau
levels /H20849LLs/H20850with energies EN=/H11006/H6036/H9275c/H20881N. Plus and minus
signs correspond to electrons and holes, respectively, /H9275c
=vF/H208812eH //H20849/H6036c/H20850is the “cyclotron frequency” for Dirac fermi-
ons, and N=n/H11032+1 /2/H110061/2, where n/H11032=0,1,2,... enumerates
orbital wave functions and /H110061/2 are pseudospin eigenval-
ues. For each EN, there are four states, corresponding to dif-
ferent spin and valley indices. In the presence of Zeemaninteraction and intervalley scattering, these states might splitfurther, as illustrated in Fig. 4/H20849d/H20850. Unlike the case of Landau
quantization for nonrelativistic massive electrons, where E
N
=/H6036/H9275c/H20849N+1 /2/H20850and/H9275c=eH //H20849mc/H20850, in graphene there is a
field-independent level at E=0 for N=0. At the charge neu-
trality point /H20849CNP /H20850/H20849in undoped graphene /H20850, this level is half-
filled, being equally shared between particles and holes. Ex-perimentally, such peculiar LL structure is manifested in theunusual QHE observed in graphene,
4,5where Hall conduc-
tivity is quantized as /H9268xy=4/H20849l+1 /2/H20850/RK,lis an integer, and
RK=h/e2is the resistance quantum. The nature of electronic
states on the N=0 level remains unclear and has recently
become the focus of considerable attention.3,6–10
Here we report an experimental investigation of charge
transport in a high-mobility graphene sample for low carrierdensity nnear the CNP and in magnetic fields up to 18 T,
that is, in the QHE regime near the N=0 Landau level, where
previous studies have yielded conflicting results.6–10In par-
ticular, some studies have found finite longitudinal resistance
/H9267xx/H11011RKnear the CNP even for high magnetic fields, where
the plateau at /H9267xy=RK/2 around /H9263=n/H90210/H=/H110062/H20849/H90210is the
flux quantum /H20850corresponding to either particle or hole filling
of the two lowest N=0 LL is well developed.6–8Others, re-
ported /H9267xx/H11271RK, in the M /H9024range, indicating an insulating
N=0 state at high fields.9,10In Ref. 10, the/H9267xx/H20849H/H20850divergence
with Hwas analyzed as an ad hoc Kosterlitz-Thouless tran-
sition and associated with a critical field Hc/H20849rather than a
filling /H9263c/H20850, which was found to be sample dependent. The
discrepancies between different measurements could besomewhat reconciled by the fact that cleaner samples showstronger divergence of
/H9267xx/H20849H/H20850with increasing Hin the N
=0 state.10Hence, sample quality appears crucial for under-
standing the physics of the N=0 LL state in graphene.
We have studied a monolayer graphene sample prepared
by mechanical exfoliation of ZYA grade highly oriented py-rolytic graphite on a Si /SiO
2substrate using the standard
procedure described in Ref. 1. Transport properties summa-
rized in Figs. 1/H20849a/H20850and1/H20849b/H20850reveal high quality of our sample.
Magnetoresistance measurements were performed using thelow-frequency /H2084917.777 Hz /H20850lock-in technique with a driving
current I=10 nA and the 18 T superconducting magnet at
the National High Magnetic Field Laboratory /H20849NHMFL /H20850.
The follow-up measurements in fields up to 6 T were per-formed at Purdue University. The sample mobility betweenthe two measurements remained unchanged within the error
/H9262=2.82 /H208496/H20850/H11003104cm2/V/s, while the CNP has shifted by
/H110152V .A t T=150 K, /H9262decreased by less than 10%, to
/H9262/H20849150K/H20850/H110152.6/H11003104cm2/V/s. A slight electron-hole asym-
metry of the resistance in Fig. 1is typical of devices with
invasive contacts /H20851see inset in Fig. 1/H20849b/H20850/H20852and is explained by
the work-function difference between graphene and theAu/Cr electrodes in our device.
12
Figure 2summarizes the QHE in our sample for three
charge-carrier densities near the CNP, n=1/H110031011, 3.2PHYSICAL REVIEW B 80, 241412 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
1098-0121/2009/80 /H2084924/H20850/241412 /H208494/H20850 ©2009 The American Physical Society 241412-1/H110031011, and 4.7 /H110031011cm−2. The latter was refined by fitting
the low-field linear part of the Hall resistivity to /H9267xy/H20849H/H20850
=H//H20849nec/H20850/H20851Fig. 2/H20849a/H20850/H20852. This yields dn /dVg=−8.07 /H208499/H20850
/H110031010cm−2/V. Clear plateaux corresponding to half the re-sistance quantum in /H9267xyand zero /H9267xxdevelop for all three
carrier densities upon approaching the LL filling /H20841/H9263/H20841=2/H20851Figs.
2/H20849a/H20850–2/H20849c/H20850/H20852. This is a hallmark of the QHE in graphene, re-
sulting from only two of the total four N=0 LL being avail-
able to either electrons or holes /H20851Fig.4/H20849d/H20850/H20852. Hence, two con-
ducting channels and /H9267xy=/H11006RK/2 plateaux. The developed
QHE regime is also manifested by the plateau at /H9266/2 in the
Hall angle /H9258Hall=atan /H20849/H9267xy//H9267xx/H20850/H20851Fig. 2/H20849d/H20850/H20852. Fitting the low-
field linear part of /H9258Hallto the result of the Boltzman trans-
port theory, tan /H20849/H9258Hall/H20850=/H20849/H9262/c/H20850H, allows for an alternative re-
finement of the sample mobility /H9262/H20851dash-dotted lines in Fig.
2/H20849d/H20850/H20852. The obtained Hall mobility agrees very well with the
Drude field mobility /H20851Fig.1/H20849b/H20850/H20852.
When LL filling decreases far enough past /H20841/H9263/H20841=2 with the
increasing magnetic field, the QHE resistance quantizationbreaks down and the system enters a resistive state, where
/H9267xx/H20849H/H20850/H110220 and fluctuates widely with H. This fluctuating be-
havior of /H9267xx/H20849H/H20850and/H9267xy/H20849H/H20850in Figs. 2/H20849a/H20850and2/H20849b/H20850is repro-
ducible. In fact, the H/H110220 parts of the curves for n=4.7
/H110031011cm−2overlay results of three different field sweep
measurements, which essentially coincide. While for this car-rier density LL filling in our field range is
/H9263/H114071 and/H9267xx,xy/H20849H/H20850
increase only moderately, much lower N=0 LL fillings /H9263
/H112701 are achievable for n=1011cm−2. Here, we observe a
dramatic increase in /H9267xx,xy/H20849H/H20850, which is detailed in Fig. 3.I n
terms of the Hall conductivity, it appears as a “zero plateau”state with
/H9268xy=0 around /H9263=0/H20851Fig.2/H20849c/H20850/H20852. However, the mag-
netic field reversal /H20849Onsager /H20850symmetry /H9267xy/H20849H/H20850=−/H9267xy/H20849−H/H20850is
violated in this state, indicating a breakdown of the dissipa-tionless QHE transport via delocalized states, typical of a
FIG. 1. /H20849Color online /H20850Longitudinal resistivity /H9267xxof our sample
in zero magnetic field as a function of the gate voltage Vg./H20849a/H20850Solid
lines are /H9267xxatT=5 K for increasing /H20849thick /H20850and decreasing /H20849thin/H20850
Vg, as indicated by arrows. Small hysteresis results from mobile
charges in Si /SiO 2substrate. The inset shows the same data to-
gether with the results of similar follow-up measurements at T
=2.5 K and T=150 K. Dotted lines are fits to /H9267xx
=1 //H20881/H9268min2+/H20849e/H9262n/H208502./H20849b/H20850Drude mobility /H9262=1 //H20849ne/H9267xx/H20850/H20849Ref.11/H20850. The
dashed line shows /H9262=2.82 /H208496/H20850/H11003104cm2/V/s, resulting from the
fit. Filled and open symbols correspond to different Vgsweep di-
rections shown by arrows. Triangles are Hall mobilities obtainedfrom fits to the data in Fig. 2/H20849d/H20850. The inset is the optical image of
our sample.
-15-10-5051015ρxy(kΩ)
Vg-VD=-1.27V
Vg-VD=-3.92V
Vg-VD=-5.55Vdn−dVg= −8.07 x1010cm-2V-1 (a)
-18 -12 -6 0 6 12 18
H(T)024681012ρxx(kΩ)
ν=-2
ν=2
1.0x1011cm-2
3.2x1011cm-2
4.7x1011cm-2(b)-1.0-0.50.00.51.0
σxy(kΩ-1)(c)
-12 -6 0 612 18
H(T)-1.0-0.50.00.51.0
θHall/π(d)
FIG. 2. /H20849Color online /H20850QHE in our sample for different gate
offsets Vg−VDfrom the Dirac point VD, i. e. for different carrier
densities n./H20849a/H20850/H9267xy,/H20849b/H20850/H9267xx,/H20849c/H20850/H9268xy, and /H20849d/H20850Hall angle /H9258Hall./H9263
=/H110062 filling for each nis shown by the corresponding arrow. Dash-
dotted lines in /H20849a/H20850are linear fits of the low-field part of /H9267xy/H20849H/H20850used
to refine the Hall constant and obtain n=H//H20851ec/H9267xy/H20849H/H20850/H20852. Curves in
/H20849b/H20850are shifted for clarity; the zero level for each is given by the
broken horizontal line. Horizontal dotted lines in /H20849a/H20850and /H20849c/H20850show
the resistance and the conductivity quantization in graphene.05001000ρxx,xy(kΩ)-18 -12 -6 0 6 12 18H(T)
n= 1.0x1011cm2
ρxx
ρxyνK,K’≈-1/4
νK,K’≈1/4νK,K’≈-1/2
νK,K’≈1/2
HI HI CI CI(a)
-400-2000200400dρxx/dH (k Ω/T)
-4 -3 -2 -1 0 1 2 3 4
1/ν[= H/n Φ0](b)-10 -5 0 5 10H(T)
-20020406080100
ρxx,xy(kΩ)(c)
-3 -2 -1 0 1 2 3
1/ν[= H/n Φ0]-20246
[ρxy(H)+_ρxy(-H)]/RKνK,K’≈-1/2
νK,K’≈1/2
HI HI CI CI(d)
FIG. 3. /H20849Color online /H20850Breakdown of QHE in graphene for n
/H110151011cm−2andT=0.25 K. Top scale shows magnetic field H
corresponding to the inverse LL filling 1 //H9263on the bottom. Filling
per valley is /H9263K,K/H11032/H11015/H9263/2./H20849a/H20850/H9267xx/H20849H/H20850/H20849dark /H20850and/H9267xy/H20849H/H20850/H20849light /H20850;/H20849b/H20850
d/H9267xx/H20849H/H20850/dH. Dramatic increase in /H9267xx/H20849H/H20850beyond a cusp at /H110154RK
and a steplike anomaly in d/H9267xx/H20849H/H20850/dHindicate transition to a col-
lective insulator state at /H9263/H11015/H110061/2./H20849c/H20850Blowup of the initial growth
in/H9267xx,xy/H20849H/H20850and /H20849d/H20850the sum /H20849dark /H20850and the difference /H20849light /H20850of/H9267xy
for two opposite directions of magnetic field show the disappear-
ance of the Hall component following the breakdown of /H20841/H9263/H20841=2 QHE
state with increasing H/H20851see also Figs. 2/H20849a/H20850and2/H20849b/H20850/H20852. Dash-dotted
curves in /H20849c/H20850,/H20849d/H20850nearly coincident with the data in the HI phase at
1/H11407/H20841/H9263/H20841/H114071/2 are fits to an exponential increase /H9267a/H20849H/H20850=aeb/H20849/H20841H/H20841−Hc/H20850
/H20849Ref. 15/H20850. Dotted curve in /H20849c/H20850is the difference /H9267xx/H20849H/H20850−/H9267a/H20849H/H20850.ZHANG et al. PHYSICAL REVIEW B 80, 241412 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
241412-2Hall insulator /H20849HI/H20850.13This behavior is further emphasized by
the Hall angle dependence in Fig. 2/H20849d/H20850, which deviates from
the/H20841/H9258Hall/H20849H/H20850/H20841=/H9266/2 QHE plateau and tends toward zero in
high magnetic field.
The breakdown of the Hall effect and the development of
an insulating state induced by magnetic field in our sampleforn=10
11cm−2are further quantified in Fig. 3. Panel /H20849a/H20850
shows the full range of /H9267xx/H20849H/H20850and/H9267xy/H20849H/H20850variation, whose
initial parts are also included in Fig. 2. Careful examination
of the data allows identifying distinct transport regimes.First, for some
/H9263in the range 1 /H11351/H9263/H113512, the QHE resistance
quantization breaks down and a resistive state forms. TheHall angle becomes /H20841
/H9258Hall/H20841/H11021/H9266/2, but/H9267xy/H20849H/H20850=−/H9267xy/H20849−H/H20850rela-
tion still holds, as shown by the sum of Hall resistivities fortwo opposite magnetic field directions
/H9267xy/H20849H/H20850+/H9267xy/H20849−H/H20850/H110150
in Fig. 3/H20849d/H20850. Such behavior could be a sign of the full split-
ting of N=0 level shown in Fig. 4/H20849d/H20850and of a developing
/H9263=1 plateau or of the unusual resistive Hall metal state at
N=0 LL considered in Refs. 6and14.
In the 1 /2/H11351/H9263/H113511 regime, the Hall effect gradually disap-
pears. A field-symmetric component breaking the magneticfield reversal symmetry appears in the transverse resistivity
/H9267xy, such that /H9267xy/H20849H/H20850/HS11005−/H9267xy/H20849−H/H20850. The antisymmetric Hall
component /H9267xy/H20849H/H20850−/H9267xy/H20849−H/H20850decreases, while /H9267xx/H20849H/H20850and
/H9267xy/H20849H/H20850steadily increase, reaching several resistance quanta
/H113514RK. Both /H9267xx/H20849H/H20850and the symmetrized /H9267xy/H20849H/H20850+/H9267xy/H20849−H/H20850
follow the same exponential dependence /H9267a/H20849H/H20850=aeb/H20849/H20841H/H20841−Hc/H20850
shown by dash-dotted lines in Figs. 4/H20849c/H20850and4/H20849d/H20850. Indepen-
dent fits in panels /H20849c/H20850and /H20849d/H20850give consistent Hc=4.6 /H208492/H20850T,
corresponding to /H9263/H110151, or a filling /H9263K,K/H11032/H11015/H9263/2/H110150.5 per val-
ley.Such behavior can be understood as a HI /H20849Refs. 6and13/H20850,
resulting from the Zeeman splitting of the N=0 LL /H20851Fig.
4/H20849d/H20850/H20852. In this case, the delocalized quantum Hall state and the
related N=0 mobility edge shift to a finite energy E0
/H11015g/H9262BH/H20849gis the Lande factor and /H9262Bis the Bohr’s magne-
ton/H20850. For filling factors below /H9263/H110151, the chemical potential
falls below E0and the electrons are localized. Thermally
activated Hall transport via delocalized states is possible forT/H110220, but it vanishes with the increasing field-induced spin
splitting of the N=0 level. At T/H110150, the system is insulating,
the transport occurs via hopping between localized states,and is dominated by the mesoscopic conductance fluctua-tions in the sample. The transverse voltage drop results fromthe sample and the leads average asymmetry. The I−Vcurve
is expected to be strongly nonlinear, as it was indeed ob-served in Ref. 10, although authors there have interpreted
this nonlinearity as resulting from sample heating. A similarN=0 insulating regime in a lower mobility sample was re-
cently reported in Ref. 15.
Probably the most surprising and intriguing is the
/H9263
/H113511/2 regime, which is characterized by marked changes in
behavior of /H9267xx/H20849H/H20850and/H9267xy/H20849H/H20850. Both /H9267xx/H20849H/H20850and the symme-
trized/H9267xy/H20849H/H20850+/H9267xy/H20849−H/H20850deviate from the exponential growth
describing the HI phase. The deviation from the fit /H9267xx/H20849H/H20850
−/H9267a/H20849H/H20850is shown by the dotted line in Fig. 3/H20849c/H20850. It empha-
sizes cusp in /H9267xx/H20849H/H20850, which is also visible in Fig. 3/H20849a/H20850, and
which is followed by an abrupt increase in resistance beyond
/H9267xx/H110114RK. The cusplike singularity in /H9267xx/H20849H/H20850is also identi-
fied by the prominent jump in the derivative d/H9267xx/dHat/H9263
/H110151/2/H20851Fig.3/H20849b/H20850/H20852. The Hall component of the transverse re-
sistivity given by the difference /H9267xy/H20849H/H20850−/H9267xy/H20849−H/H20850also shows
singular behavior, abruptly approaching zero at the same fill-ing/H20851Fig.3/H20849d/H20850/H20852, so that
/H9267xy/H20849H/H20850/H11015/H9267xy/H20849−H/H20850for/H9263/H113511/2.
Simultaneous singular behavior of /H9267xy/H20849H/H20850−/H9267xy/H20849−H/H20850and
d/H9267xx/dHat/H9263/H110151/2 indicate transition to a different, more
insulating state, which we identify as a collective insulator/H20849CI/H20850. A likely candidate for such a CI state is a pinned
Wigner crystal /H20849WC/H20850, where electrons are collectively rather
than individually localized. Our interpretation is based uponthe analogy with QHE in high-mobility 2D electron gases/H208492DEG /H20850in semiconductor heterostructures, where an onset of
strongly insulating behavior at low filling factors
/H9263/H113511/4 has
also been reported.16–18In 2DEG, this CI behavior is under-
stood as a 2D WC, which can be pinned by an arbitrarilysmall disorder at T=0 K.
19,20In our case of graphene, tran-
sition at /H9263/H110151/2 corresponds to a filling /H9263K,K/H11032/H110151/4 per val-
ley, roughly consistent with 2DEG results.
Finally, we have also investigated the breakdown of the
N=0 quantum Hall state and the appearance of the insulating
behavior in our graphene sample by varying the gate voltageV
gin magnetic field H=18 T. The results shown in Fig. 4
are in agreement with those in Fig. 3, corroborating the pic-
ture presented above. There is a well-developed QHE plateauaround filling /H20841
/H9263/H20841=2. The quantization breaks down in the
1/H11351/H20841/H9263/H20841/H113512 range and there is a hint of a plateau at /H20841/H9267xy/H20841=RK
developing around /H20841/H9263/H20841/H110151/H20851Figs. 4/H20849b/H20850and4/H20849c/H20850/H20852. An onset of
strong increase in /H9267xxand /H20841/H9267xy/H20841beyond /H110114RKis seen at /H20841/H9263/H20841
/H110151/2. It can also be traced in the behavior of the derivative
d/H9267xy/dVg. The fact that such an abrupt onset occurs here and
in Fig. 3at very different /H20841n/H20841and /H20841B/H20841but similar /H20841/H9263/H20841-500050010001500ρxx,xy(kΩ)
-5 0 5 10 0
Vg(V)-2 -1 0 1nΦ0/H(=ν)
4RKH=1 8T
ρxx
ρxy(a)-10 -5 0 5Vg-VD(V, @H=18T)
-202
ρxx,xy/RK(b)
1 0 -1 -2
nΦ0/H (=ν)-4-2024
σxyRK18 T
6T(c)
E/--hωcDOS↑↓↑↓ ↑↓ ↑↓ ↑↓ ↑↓
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5K’K(d)
FIG. 4. /H20849Color online /H20850Breakdown of the quantum Hall state in
graphene at H=18 T as a function of gate voltage /H20849carrier concen-
tration /H20850atT=0.25 K. /H20849a/H20850Onset of a strong increase in /H9267xxand /H20841/H9267xy/H20841
beyond /H110154RKnear filling /H9263=/H110061/2/H20849shown by arrows /H20850. Blowup of
/H9267xyin/H20849b/H20850and/H9268xyin/H20849c/H20850show a well-developed QHE plateau around
/H9263=−2 and a hint of plateaux developing at /H20841/H9263/H20841=1 at RKand 1 /RK,
respectively. The apparent /H9263=0 plateau in /H20849c/H20850corresponds to a bulk
insulator with zero Hall angle and no Hall effect. /H20849d/H20850Schematics of
Landau levels in graphene, including Zeeman splitting /H20849up/down
arrows /H20850andK,K/H11032valley splitting /H20849bars/H20850.BREAKDOWN OF THE N=0 QUANTUM HALL STATE IN … PHYSICAL REVIEW B 80, 241412 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
241412-3=/H20841nh /eH/H20841/H110151/2 suggests that it is driven by the interaction
between electrons in graphene, again consistent with a tran-sition to a bulk collective insulator at /H20841
/H9263/H20841/H113511/2. Due to the
contact-induced electron-hole asymmetry, there is a slightshift of the curves in Fig. 4with respect to the nominal n
=0, making such measurements by sweeping V
gless favor-
able compared to the sweep of magnetic field at a constantV
g.
In summary, we have investigated the breakdown of the
quantum Hall effect and the emergence of an insulating be-havior in the N=0 Landau Level in a high-mobility single-
layer graphene sample. The LL filling in the range /H20841
/H9263/H20841/H113512i s
achieved by either increasing the magnetic field at a constantcarrier density nor by varying n/H20849V
g/H20850atH=18 T. Careful
analysis of our data leads us to identify two different insu-lating regimes as a function of the decreasing LL filling
/H9263=n/H90210/H. First, the well-developed resistance quantization
on the /H20841/H9263/H20841=2 plateau breaks down and a dissipative state
develops near the LL filling /H20841/H9263/H20841/H110151. This can be understood
as a HI resulting from the Zeeman splitting of the N=0 level,
similar to that observed for higher Nstates.7,9,15ForN=0,
however, there remain no delocalized states occupied byelectrons /H20849holes /H20850at sufficiently low filling /H20841
/H9263/H20841/H113511, as a result
of this splitting /H20851Fig. 4/H20849d/H20850/H20852, and the transport is local at
T→0. This observation agrees with recent findings reported
in Ref. 15and largely rules out the picture of a Hall metalwith spin-polarized chiral currents,6although not its refined
version in Ref. 14, which effectively leads to a HI.
Our most striking finding, though, is a well-defined onset
of the marked resistance increase with decreasing filling at
/H9263/H110151/2. It is clearly revealed by the anomalies in the field
dependencies /H9267xx/H20849H/H20850and/H9267xy/H20849H/H20850and can be understood as a
transition from the local HI to a bulk collective insulatorstate. Collective bulk insulating states have been previouslyobserved in 2DEG systems at low filling
/H9263and are com-
monly associated with pinned Wigner crystals.16–20We sug-
gest that a pinned WC is also a likely candidate for thestrongly insulating state, which we have identified in ourgraphene sample for /H20841
/H9263/H20841/H113511/2.
We thank I. Childres, J.-H. Park, and E. Palm for help
with the measurements and S. Suchalkin, Z. Jiang, M. Stron-gin, D. Abanin, and A. Shytov for discussions. We also thankF. Camino, A. Stein, and D. Nykypanchuk for help at theBrookhaven Center for Functional Nanomaterials, where oursamples were prepared. This work was supported by the U.S.
DOE under the Contract No. DE-AC02-98CH10886. Partialsupport by Purdue University and Miller Family Endowmentis gratefully acknowledged. Work at the NHMFL is also sup-ported by the NSF through Grant No. DMR-0084173 and bythe State of Florida.
*Corresponding author; zaliznyak@bnl.gov
1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.
Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Sci-ence 306, 666 /H208492004 /H20850.
2C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 /H208492008 /H20850.
3A. H. Castro Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and
A. K. Geim, Rev. Mod. Phys. 81, 109 /H208492009 /H20850.
4K. 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.
5Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature /H20849Lon-
don/H20850438, 201 /H208492005 /H20850.
6D. A. Abanin, K. S. Novoselov, U. Zeitler, P. A. Lee, A. K.
Geim, and L. S. Levitov, Phys. Rev. Lett. 98, 196806 /H208492007 /H20850.
7A. J. M. Giesbers, U. Zeitler, M. I. Katsnelson, L. A. Ponomar-
enko, T. M. Mohiuddin, and J. C. Maan, Phys. Rev. Lett. 99,
206803 /H208492007 /H20850.
8Z. Jiang, Y. Zhang, H. L. Stormer, and P. Kim, Phys. Rev. Lett.
99, 106802 /H208492007 /H20850.
9Y. Zhang, Z. Jiang, J. P. Small, M. S. Purewal, Y.-W. Tan, M.
Fazlollahi, J. D. Chudow, J. A. Jaszczak, H. L. Stormer, and P.Kim, Phys. Rev. Lett. 96, 136806 /H208492006 /H20850.
10J. G. Checkelsky, L. Li, and N. P. Ong, Phys. Rev. Lett. 100,206801 /H208492008 /H20850; Phys. Rev. B 79, 115434 /H208492009 /H20850.
11Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H.
Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, Phys. Rev.Lett. 99, 246803 /H208492007 /H20850.
12B. Huard, N. Stander, J. A. Sulpizio, and D. Goldhaber-Gordon,
Phys. Rev. B 78, 121402 /H20849R/H20850/H208492008 /H20850.
13M. Hilke, D. Shahar, S. H. Song, D. C. Tsui, Y. H. Xie, and D.
Monroe, Nature /H20849London /H20850395, 675 /H208491998 /H20850.
14E. Shimshoni, H. A. Fertig, and G. V. Pai, Phys. Rev. Lett. 102,
206408 /H208492009 /H20850.
15A. J. M. Giesbers, L. A. Ponomarenko, K. S. Novoselov, A. K.
Geim, M. I. Katsnelson, J. C. Maan, and U. Zeitler, Phys. Rev. B
80, 201403 /H20849R/H20850/H208492009 /H20850.
16H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui, L. N.
Pfeiffer, and K. W. West, Phys. Rev. Lett. 65, 633 /H208491990 /H20850.
17S. D. Suchalkin, Yu. B. Vasil’ev, M. Zundel, G. Nachtwei, K.
von Klitzing, and K. Eberl, JETP Lett. 74, 564 /H208492001 /H20850.
18Y. P. Chen, G. Sambandamurthy, Z. H. Wang, R. M. Lewis, L.
W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and K. W. West,Nat. Phys. 2, 452 /H208492006 /H20850.
19U. Merkt, Phys. Rev. Lett. 76, 1134 /H208491996 /H20850.
20H. A. Fertig, Phys. Rev. B 59, 2120 /H208491999 /H20850.ZHANG et al. PHYSICAL REVIEW B 80, 241412 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
241412-4 |
PhysRevB.94.134101.pdf | PHYSICAL REVIEW B 94, 134101 (2016)
Hexagonal structure of phase III of solid hydrogen
Bartomeu Monserrat,1,2,*Richard J. Needs,2Eugene Gregoryanz,3,4and Chris J. Pickard5,6
1Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA
2TCM Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
3Centre for Science at Extreme Conditions and School of Physics and Astronomy, University of Edinburgh,
Edinburgh EH9 3JZ, United Kingdom
4Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China
5Department of Materials Science & Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom
6Advanced Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba, Sendai 980-8577, Japan
(Received 21 June 2016; published 3 October 2016)
A hexagonal structure of solid molecular hydrogen with P6122 symmetry is calculated to be more stable
below about 200 GPa than the monoclinic C2/cstructure identified previously as the best candidate for phase
III. We find that the effects of nuclear quantum and thermal vibrations play a central role in the stabilization ofP6
122. The P6122 and C2/cstructures are very similar and their Raman and infrared data are in good agreement
with experiment. However, our calculations show that the hexagonal P6122 structure provides better agreement
with the available x-ray diffraction data than the C2/cstructure at pressures below about 200 GPa. We suggest
that two phase-III-like structures may be formed at high pressures: hexagonal P6122 below about 200 GPa and
monoclinic C2/cat higher pressures.
DOI: 10.1103/PhysRevB.94.134101
I. INTRODUCTION
Experimental and theoretical studies of hydrogen at high
pressures have progressed rapidly in recent years. On theexperimental front, improvements in diamond anvil celltechniques have enabled the exploration of static pressuresabove 300 GPa in hydrogen [ 1–3] and even higher pressures
in other materials [ 4]. The solid molecular crystalline phases
I, II, III, and IV /IV
/primeof hydrogen have been extensively
studied experimentally. However, only the structure of the low-
temperature and pressure phase I is known with precision, and
it is found to be a quantum solid consisting of molecules withangular momentum L=0i nam i x t u r eo f ortho andpara states
and arranged on a hexagonal close-packed lattice [ 1–3]. At low
temperatures and pressures above about 100 GPa (25 GPa fordeuterium), hydrogen enters phase II, in which the molecularrotations are restricted [ 5]. The detailed structure of phase
II is unknown, although infrared (IR) and Raman vibrational
data [ 6,7] and x-ray and neutron diffraction data [ 8–11] impose
constraints on it. At about 160 GPa, phase II transforms into theordered molecular phase III [ 12]. Phase III has a single strong
IR active vibron peak and an IR activity much larger than thatof phase II [ 13]. The phases IV and IV
/prime, which are similar to
each other, become stable at high pressures and temperaturesabove about 300 K [ 14–16]. They exhibit a high-frequency
vibron peak that is weakly dependent on pressure and a
strong Raman vibron peak at lower frequencies which softensrapidly with applied pressure. More recently, a new phase Vof hydrogen has been observed in Raman experiments aroundroom temperature reaching pressures of 388 GPa [ 17], but the
structure of this new phase also remains unknown.
On the theoretical front, high-pressure structures of
hydrogen have been investigated extensively using ab
initio molecular dynamics [ 18–22], path-integral molec-
ular dynamics [ 23,24], quantum Monte Carlo methods
*bm418@cam.ac.uk[3,22,25–30], first-principles density functional theory (DFT)
methods [ 31–35], and many-body methods [ 26,36]. The recent
widespread adoption of DFT structure searching techniqueshas led to the discovery of high-pressure hydrogen structuresthat are consistent with many of the experimental observations.Using the ab initio random structure searching (AIRSS)
method, we found a hydrogen structure of P2
1/csymmetry
that is a plausible model for phase II [ 37]. We also discovered
a monoclinic structure of C2/csymmetry and 24 atoms per
primitive cell (henceforth called C2/c-24), which provides a
good match to the experimental vibrational data for phase IIIand is the lowest-enthalpy phase found over the pressure rangein which phase III is observed, of 160 to above 300 GPa [ 32].
Energetically competitive “mixed structures” of C2,Pbcn ,
andIbam symmetries have also been found [ 32] that consist
of alternate layers of strongly and weakly bonded molecules,which provide simple models for phases IV /IV
/prime.W eh a v e
developed improved models for these phases, in particular thePcstructure with 48 atoms per primitive unit cell [ 33,34],
and also a slightly better structure with 96 atoms per cell (seethe Supplemental Material of Ref. [ 33]). Structure searching
methods have found widespread application beyond hydrogen,in the discovery of many new structures that were subsequentlysynthesized. For example, AIRSS has been used to determinestructures of silane [ 38], aluminum hydride [ 39], ammo-
nia [ 40,41], ammonia hydrates [ 42], and xenon oxides [ 43]
that were subsequently verified by experiments.
Candidate structures for phases II [ 37], III [ 32], and
IV/IV
/prime[33] have been determined by structure searching using
AIRSS. These searches did not use experimental input, but theresulting structures provide Raman and IR vibrational data inreasonable agreement with experiment. Despite this success,there are still discrepancies between theory and experiment.In particular, there remains an outstanding question about thestructure of phase III. Although the vibrational signaturesof the monoclinic C2/c-24 structure agree well with the
experimental data for phase III, there is an inconsistency
2469-9950/2016/94(13)/134101(7) 134101-1 ©2016 American Physical SocietyMONSERRAT, NEEDS, GREGORY ANZ, AND PICKARD PHYSICAL REVIEW B 94, 134101 (2016)
between the experimental x-ray diffraction data for phase
III, reported in Ref. [ 11], and the simulated x-ray data for
C2/c-24. The experimental x-ray data are consistent with a
hexagonal space group, but notwith the monoclinic space
group of C2/c-24 [ 44].
In this work we investigate this discrepancy using DFT
methods [ 45]. We find a new hexagonal structure of high-
pressure hydrogen of P6122 symmetry, which is calculated
to be more stable than the C2/c-24 structure below pressures
of about 200 GPa, once the effects of quantum and thermalmotion are incorporated. The Raman and IR spectra of P6
122
are in good agreement with those observed experimentallyfor phase III, and the hexagonal symmetry leads to the bestagreement of any known candidate structure with the x-raydiffraction data available. We propose P6
122 as a candidate
structure for phase III of solid hydrogen.
The rest of the paper is organized as follows. In Sec. II
we describe the structure searches and in Sec. IIIwe calculate
the relative free energies of the most competitive candidatestructures. We then characterize the new P6
122 structure in
Sec. IVand propose it as the candidate structure of phase III of
solid hydrogen in Sec. V. We draw our conclusions in Sec. VI.
II. STRUCTURE SEARCHES
We used AIRSS to search for low-enthalpy static-lattice
structures of solid hydrogen at high pressures. In contrast
to previous searches, we focused on structures containing anumber of atoms or molecules equal to a highly compositenumber. Highly composite numbers are positive integers thathave more divisors than any smaller positive integer, andsearches over structures containing a highly composite numberof atoms or molecules explore structures containing severaldifferent numbers of formula units during each search. Foreach structure, a physically reasonable volume and set ofatomic positions were selected at random. Although somesearches were performed without symmetry constraints, formost searches we imposed common space group symmetriesof molecular crystals and in particular space groups P2
1/c,
P212121,Pca 21,Pna 21, andC2/c. We constrained the min-
imum initial atomic separations using data from preliminaryshort AIRSS runs, with different minimum separations at eachpressure. This helps to space out the atoms appropriately whileretaining a high degree of randomness. The structures werethen relaxed until the forces on the atoms were small and thepressure took the required value. This procedure was repeatedmany times, and a total of 85 424 structures were generated.
The searches were performed using the
CASTEP [46]D F T
plane-wave pseudopotential code with “on the fly” ultrasoftpseudopotentials [ 47] and the Becke-Lee-Yang-Parr (BLYP)
density functional [ 48,49], which has been shown to provide a
good description of molecular hydrogen at high pressures [ 50].
We employed an energy cutoff of 230 eV , and k-point grids of
spacings 2 π×0.07˚A
−1and 2π×0.05˚A−1.
AIRSS found a previously unknown hydrogen structure of
hexagonal P6122 symmetry which is energetically competi-
tive with monoclinic C2/c-24. A primitive unit cell of P6122
contains 36 atoms, which is a highly composite number. Itappeared that this system size had not been explored previouslyin hydrogen.100 200 300-10123456
-2Relative Gibbs free energy (meV/proton)Static
100 200 300
Pressure (GPa)T=0K
100 200 300T=300K
C2/c-24P6122
C2/c-12P21/cP63/m
FIG. 1. Relative stability of P6122 (blue), C2/c-12 (green),
P21/c(orange), and P63/m(violet) with respect to C2/c-24 (red) at
the static lattice level, T=0K ,a n d T=300 K.
III. FREE ENERGY CALCULATIONS
We next evaluate the relative enthalpies and free energies
of the most competitive structures of high-pressure hydrogenin the pressure range 100–350 GPa, as shown in Fig. 1.W e
include the best-known candidate structures for phase II, ofP2
1/candP63/msymmetry; the best-known candidates for
phase III, the C2/c-24 structure and a 12-atom variant (referred
to asC2/c-12); and the newly discovered P6122 structure.
For both static lattice and vibrational energy calculations,
we used an energy cutoff of 1000 eV and k-point grids of
spacing 2 π×0.025 ˚A−1. These parameters provide energy
differences between frozen-phonon structures that are con-verged to better than 10
−4eV/atom, forces to better than
10−4eV/˚A, and stresses to better than 10−3GPa.
The low mass of hydrogen leads to large vibrational
energies and amplitudes and to significant anharmonic nuclearmotion, which must be accounted for if accurate energiesare to be calculated. We evaluated the free energies usingthe method proposed in Ref. [ 51]. The low-energy part
of the Born-Oppenheimer energy surface was mapped well
beyond the harmonic region in a finite-displacement approach.
We took advantage of the recently introduced nondiagonalsupercells method [ 52] to reach unprecedented levels of
convergence with respect to the size of the simulation cell. Asan example, the results reported here for the P6
122 structure
were obtained using nondiagonal supercells containing amaximum of 72 atoms, but these results are the same as
those that would be obtained using the standard supercell
approach and a supercell containing 288 atoms. Tests withlarger nondiagonal supercells containing a maximum of 108atoms (equivalent to standard supercells containing 972 atoms)show that the final vibrational energies are converged to betterthan 0.2 meV/proton. After construction of the anharmonicpotential, the resulting Schr ¨odinger equation was solved
using a vibrational self-consistent-field approach, in which
134101-2HEXAGONAL STRUCTURE OF PHASE III OF SOLID . . . PHYSICAL REVIEW B 94, 134101 (2016)
the vibrational wave function was represented in a basis
of simple-harmonic-oscillator functions for each degree of
freedom, and converged results were achieved by includingup to 50 basis functions per mode.
The relative enthalpies and free energies reported in Fig. 1
correspond to static lattices, T=0 K, and T=300 K. At
300 K, P6
122 is thermodynamically stable at pressures below
about 180 GPa when the vibrational energy is included. The
energy difference between C2/c-24 and P6122 is small,
but it is clear that the new P6122 structure is energetically
competitive in the pressure range where phase III is observedexperimentally. The structural similarities between P6
122
andC2/c-24 suggest that errors in the total free energies
arising, for example, from the choice of exchange-correlationfunctional, should largely cancel when evaluating their relative
free energies. The C2/c-12 structure has a static lattice
energy higher than that of the C2/c-24 structure, and the
inclusion of quantum and thermal vibrations destabilizes itfurther. This demonstrates the importance of the stacking oflayers in determining the relative stability of these otherwisevery similar structures. The candidate phase II structures aresignificantly destabilized by the inclusion of quantum nuclear
motion, but it has recently been shown that a quantum Monte
Carlo description of the electronic energy is necessary toaccurately describe the relative energy of these structurescompared to C2/c-24 [ 29]. Finally, we note that even at 300 K,
the vibrational energy is dominated by the quantum zero-pointmotion.
We have also calculated the relative Gibbs free energy of
P6
122 with respect to C2/c-24 for the heavier deuterium
isotope. As atomic vibrations drive the thermodynamic sta-bility of P6
122 compared to C2/c-24, the heavier deuterium
compound has a narrower stability range. For example, at300 K the transition into the C2/c-24 structure is predicted
to occur at about 160 GPa.
IV . PROPERTIES OF THE P6122 STRUCTURE
The data reported in Fig. 1suggest that P6122 is a
competitive candidate structure for phase III. Therefore, inthis section we characterize the P6
122 structure and compare
its spectroscopic signatures with experiment and with those of
C2/c-24, which is the best candidate for phase III known at
present.
A. Structure
BothP6122 and C2/c-24 are layered molecular structures,
with two extra layers in the hexagonal primitive cell of P6122,
as shown in Fig. 2. Their structural details at 200 GPa
are provided in Table I, which shows that their primitive
cells are similar, differing mainly in the length of the c
axis (about 50% longer in P6122 as a consequence of the
two extra layers) and in the slight monoclinic distortioninC2/c-24. Two slightly different molecular bond lengths
(BL) appear in these structures, and they differ by less than0.001 ˚A between the two structures. The volume per proton
ofP6
122 is only 0.1% larger than that of C2/c-24. We
include a structure file of the P6122 structure as Supplemental
Material [ 53].
FIG. 2. The P6122 and C2/c-24 layered molecular structures.
B. Band structure and phonon dispersion
In Fig. 3(a) we show the band structure and density of
states of P6122 at a pressure of 200 GPa. In Fig. 3(b) we show
the corresponding phonon dispersion and associated density ofvibrational states. The absence of imaginary frequencies in thephonon dispersion shows that P6
122 is a dynamically stable
structure.
C. Raman and IR
The vast majority of experiments on pressurized hydrogen
report Raman and/or IR spectra. In Fig. 4we show the theoret-
ical Raman and IR spectra of C2/c-24 and P6122 at 200 GPa.
As the C2/c-24 and P6122 structures are almost identical,
the frequencies of the active modes are indistinguishable andagree well with those observed experimentally [ 13]. The main
difference between the two signals is the stronger IR vibronpeak for P6
122, which is consistent with the observation that
in phase III the IR activity is much larger than that in phaseII [13]. Overall, the IR and Raman spectra of C2/c-24 and
P6
122 agree well with the corresponding spectra observed for
phase III, and therefore we cannot unambiguously identify thestructure of phase III based purely on its vibrational response.
We note that the Raman and IR spectra were obtained using
the Perdew-Burke-Ernzerhof (PBE) functional [ 54] instead of
the BLYP functional. The latter is not implemented in
CASTEP
within the density functional perturbation theory formalismneeded to evaluate these spectra. The main difference betweenthe spectra obtained using PBE and the one that would beobtained using a different functional is the position of the
134101-3MONSERRAT, NEEDS, GREGORY ANZ, AND PICKARD PHYSICAL REVIEW B 94, 134101 (2016)
TABLE I. Static lattice structural details of C2/c-24 and P6122 at 200 GPa.
abc α β γ V olume per proton BL1 BL2
C2/c-24 3.025 ˚A 3.025 ˚A 5.408 ˚A9 0 .1◦90.1◦119.9◦1.787 ˚A30.719 ˚A 0.716 ˚A
P6122 3.022 ˚A 3.022 ˚A 8.143 ˚A9 0 .0◦90.0◦120.0◦1.789 ˚A30.719 ˚A 0.715 ˚A
peaks, caused by the slightly different bond lengths predicted
by the various functionals.
D. X-ray diffraction
Hydrogen having the lowest atomic number Z is a very poor
scatterer of x rays. This, combined with the restrictive access toa diamond anvil cell, makes the structural studies of hydrogenat high pressures notoriously difficult: even the structure ofphase II, which appears above 25 GPa for deuterium, is stillnot known. In a remarkable work, Akahama and co-workersrecently published x-ray data for phase III of hydrogen up to apressure of 183 GPa [ 11]. Figure 5shows x-ray diffraction
HK ΓAL M Γ-20-1001020Band structure (eV)
DOS(a)
HK ΓAL M Γ0.00.10.20.30.40.50.6Phonon dispersion (eV)
DOS(b)
FIG. 3. (a) Band structure along a high-symmetry path (left) and
electronic density of states (right) of P6122 at 200 GPa. The dashed
black line represents the Fermi level. (b) Phonon dispersion along a
high-symmetry path (left) and vibrational density of states (right) ofP6
122 at 200 GPa.data from Ref. [ 11], together with our data for C2/c-24
andP6122, simulated using the wavelength λ=0.4122 ˚A
and the lattice parameters corresponding to a pressure of174 GPa. The experimental data for phase III shows two strongreflections at about 15 .5
◦and 17 .7◦. These data might show
the continuation of the 100 and 101 strong reflections observedin hexagonal phase I. Experimentally, a weaker peak is alsoobserved at about 17 .1
◦at some pressures, which could be
the continuation of the 002 peak of phase I. The 002 peak isnot easily observed because the crystallites tend to grow withtheircaxis perpendicular to the diamond culets, and due to the
geometrical constraints of the diamond anvil cell this preventsaccess to the 001 planes. The available experimental data fromRef. [ 11] suggests that the x-ray diffraction pattern of a good
candidate structure for phase III should (i) exhibit two peaksof similar intensity at 15 .5
◦and at 17 .7◦and (ii) could exhibit
a weaker peak at 17 .1◦.
The x-ray diffraction pattern of C2/c-24 has a double peak
around 15 .5◦(see double arrow in Fig. 5), resulting from the
monoclinic distortion of the structure, and is inconsistent withthe number of peaks observed experimentally. This, therefore,rules out C2/c-24 as a possible structural candidate for phase
III below 200 GPa. P6
122 shows a single peak at 15 .5◦,
which is consistent with its hexagonal symmetry and withexperiment. However, the relative peak intensities of P6
122
are not in good agreement with those observed experimentally.The strongest peaks in P6
122 are those at 17 .1◦and 17 .7◦,
while the peak at 15 .5◦is weaker. The intensities of the
calculated peaks would match experiment better if the peaksat 15.5
◦and 17 .1◦were interchanged.
0123 4
Frequency (103cm-1)Intensity (arb. units)
0123 4C2/c-24
P6122C2/c-24
P6122Raman IR
FIG. 4. Raman and IR spectra of C2/c-24 and P6122 at
P=200 GPa.
134101-4HEXAGONAL STRUCTURE OF PHASE III OF SOLID . . . PHYSICAL REVIEW B 94, 134101 (2016)
10 15 20 25
2θ(degree)Intensity (arb. units)C2/c-24
P6122Experiment100
101002
FIG. 5. Experimental x-ray diffraction data for phase III [ 11],
and simulated x-ray diffraction data for the C2/c-24 and P6122
structures at P=174 GPa and wavelength λ=0.4122 ˚A. The black
arrows in the experimental data indicate the H 2peaks, while the other
peaks correspond to the rhenium metal gasket. The two arrows in thecalculated spectrum of C2/c-24 indicate the double peak at around
15.5
◦caused by the monoclinic distortion, and the arrow in the P6122
spectrum shows that in this hexagonal structure there is a single peakaround 15 .5
◦.
V . PHASE III OF SOLID HYDROGEN
The structural and vibrational characteristics of P6122,
together with the energies reported in Fig. 1, suggest that it
is the best candidate for phase III of solid hydrogen of allknown structures. It is the first hexagonal candidate for thisphase, and its x-ray spectrum exhibits the correct number ofpeaks at the appropriate locations. The remaining challengeis to explain the discrepancy in the intensities of these peaksbetween theory and experiment.
Based on the data shown in Fig. 1, we speculate that phase
III might in reality be two distinct phases, with the P6
122 phase
being stable below about 200 GPa and C2/c-24 being stable athigher pressures. The almost identical Raman and IR spectra of
these two phases would make it difficult to distinguish betweenthem using spectroscopic techniques, and x-ray data, whichmight distinguish between the hexagonal and monoclinic sym-metries, is only available up to 183 GPa, where the hexagonalstructure is predicted to be stable. It would be very useful ifexperimental x-ray data could be collected above 200 GPa.
VI. CONCLUSIONS
In conclusion, we have discovered a new candidate structure
for phase III of high-pressure hydrogen with hexagonalsymmetry and space group P6
122. Our calculations suggest
thatP6122 is the most stable structure at pressures up to about
200 GPa and that its structural and vibrational properties arein better agreement with experiment than any other knowncandidate structures. Furthermore, the C2/c-24 structure is
predicted to be stable at pressures above 200 GPa, whichsuggests that phase III might be two distinct phases: ahexagonal phase below 200 GPa and a monoclinic phase athigher pressures.
ACKNOWLEDGMENTS
We thank Nicholas Worth for help with the x-ray diffraction
pattern calculations. B.M. acknowledges Robinson College,Cambridge, and the Cambridge Philosophical Society fora Henslow Research Fellowship. R.J.N., E.G., and C.J.P.acknowledge financial support from the Engineering andPhysical Sciences Research Council (EPSRC) of the UnitedKingdom (Grants No. EP/J017639/1, No. EP/J003999/1,and No. EP/K013688/1, respectively). C.J.P. is also sup-ported by the Royal Society through a Royal SocietyWolfson Research Merit award. The calculations were per-formed on the Darwin Supercomputer of the University ofCambridge High Performance Computing Service facility(http://www.hpc.cam.ac.uk/ ) and the Archer facility of the
UK national high performance computing service, for whichaccess was obtained via the UKCP consortium and funded byEPSRC Grant No. EP/K014560/1.
All the data supporting the findings of this work are
available within the article and its Supplemental Material.
[1] I. F. Silvera, The solid molecular hydrogens in the condensed
phase: Fundamentals and static properties, Rev. Mod. Phys. 52,
393(1980 ).
[2] H.-k. Mao and R. J. Hemley, Ultrahigh-pressure transitions in
solid hydrogen, Rev. Mod. Phys. 66,671(1994 ).
[3] J. M. McMahon, M. A. Morales, C. Pierleoni, and D. M.
Ceperley, The properties of hydrogen and helium under extremeconditions, Rev. Mod. Phys. 84,1607 (2012 ).
[4] L. Dubrovinsky, N. Dubrovinskaia, E. Bykova, M. Bykov,
V . Prakapenka, C. Prescher, K. Glazyrin, H.-P. Liermann,M. Hanfland, M. Ekholm, Q. Feng, L. V . Pourovskii, M.I. Katsnelson, J. M. Wills, and I. A. Abrikosov, The mostincompressible metal osmium at static pressures above 750gigapascals, Nature (London) 525,226(2015 ).[5] I. F. Silvera and R. J. Wijngaarden, New Low-Temperature Phase
of Molecular Deuterium at Ultrahigh Pressure, Phys. Rev. Lett.
47,39(1981 ).
[6] I. I. Mazin, Russell J. Hemley, A. F. Goncharov, M. Hanfland,
and H.-k. Mao, Quantum and Classical Orientational Orderingin Solid Hydrogen, P h y s .R e v .L e t t . 78,1066 (1997 ).
[7] A. F. Goncharov, E. Gregoryanz, R. J. Hemley, and H.-K. Mao,
Spectroscopic studies of the vibrational and electronic propertiesof solid hydrogen to 285 GPa, Proc. Natl. Acad. Sci. U.S.A. 98,
14234 (2001 ).
[8] P. Loubeyre, R. LeToullec, D. Hausermann, M. Hanfland, R. J.
Hemley, H. K. Mao, and L. W. Finger, X-ray diffraction andequation of state of hydrogen at megabar pressures, Nature
(London) 383,702(1996 ).
134101-5MONSERRAT, NEEDS, GREGORY ANZ, AND PICKARD PHYSICAL REVIEW B 94, 134101 (2016)
[9] H. Kawamura, Y . Akahama, S. Umemoto, K. Takemura, Y .
Ohishi, and O. Shimomura, X-ray powder diffraction from soliddeuterium, J. Phys.: Condens. Matter 14,10407 (2002 ).
[10] I. Goncharenko and P. Loubeyre, Neutron and x-ray diffraction
study of the broken symmetry phase transition in solid deu-terium, Nature (London) 435,1206 (2005 ).
[11] Y . Akahama, M. Nishimura, H. Kawamura, N. Hirao, Y .
Ohishi, and K. Takemura, Evidence from x-ray diffraction oforientational ordering in phase III of solid hydrogen at pressuresup to 183 GPa, Phys. Rev. B 82,060101 (2010 ).
[12] R. J. Hemley and H. K. Mao, Phase Transition in Solid Molecular
Hydrogen at Ultrahigh Pressures, Phys. Rev. Lett. 61,857
(1988 ).
[13] R. J. Hemley, Z. G. Soos, M. Hanfland, and H.-k. Mao, Charge-
transfer states in dense hydrogen, Nature (London) 369,384
(1994 ).
[14] M. I. Eremets and I. A. Troyan, Conductive dense hydrogen,
Nat. Mater. 10,927(2011 ).
[15] R. T. Howie, C. L. Guillaume, T. Scheler, A. F. Goncharov, and
E. Gregoryanz, Mixed Molecular and Atomic Phase of DenseHydrogen, P h y s .R e v .L e t t . 108,125501 (2012 ).
[16] R. T. Howie, T. Scheler, C. L. Guillaume, and E. Gregoryanz,
Proton tunneling in phase IV of hydrogen and deuterium, Phys.
Rev. B 86,214104 (2012 ).
[17] P. Dalladay-Simpson, R. T. Howie, and E. Gregoryanz, Evidence
for a new phase of dense hydrogen above 325 gigapascals,
Nature (London) 529,63(2016 ).
[18] J. Kohanoff, S. Scandolo, G. L. Chiarotti, and E. Tosatti, Solid
Molecular Hydrogen: The Broken Symmetry Phase, Phys. Rev.
Lett.78,2783 (1997 ).
[19] J. Kohanoff, S. Scandolo, S. de Gironcoli, and E. Tosatti,
Dipole-Quadrupole Interactions and the Nature of Phase III ofCompressed Hydrogen, Phys. Rev. Lett. 83,4097 (
1999 ).
[20] I. B. Magd ˘au and G. J. Ackland, Identification of high-pressure
phases III and IV in hydrogen: Simulating Raman spectra usingmolecular dynamics, Phys. Rev. B 87,174110 (2013 ).
[21] R. Singh, S. Azadi, and T. D. K ¨uhne, Anharmonicity and finite-
temperature effects on the structure, stability, and vibrationalspectrum of phase III of solid molecular hydrogen, Phys. Rev.
B90,014110 (2014 ).
[ 2 2 ] J .C h e n ,X .R e n ,X . - Z .L i ,D .A l f `e, and E. Wang, On the room-
temperature phase diagram of high pressure hydrogen: An ab
initio molecular dynamics perspective and a diffusion Monte
Carlo study, J. Chem. Phys. 141,024501 (2014 ).
[23] H. Kitamura, S. Tsuneyuki, T. Ogitsu, and T. Miyake, Quantum
distribution of protons in solid molecular hydrogen at megabarpressures, Nature (London) 404,259(2000 ).
[24] J. Chen, X.-Z. Li, Q. Zhang, M. I. J. Probert, C. J. Pickard, R.
J. Needs, A. Michaelides, and E. Wang, Quantum simulationof low-temperature metallic liquid hydrogen, Nat. Commun. 4,
2064 (2013 ).
[25] D. M. Ceperley and B. J. Alder, Ground state of solid hydrogen
at high pressures, Phys. Rev. B 36,2092 (1987 ).
[26] S. Azadi, W. M. C. Foulkes, and T. D. K ¨uhne, Quantum Monte
Carlo study of high pressure solid molecular hydrogen, New J.
Phys. 15,113005 (2013 ).
[27] S. Azadi, B. Monserrat, W. M. C. Foulkes, and R. J. Needs,
Dissociation of High-Pressure Solid Molecular Hydrogen: AQuantum Monte Carlo and Anharmonic Vibrational Study,Phys. Rev. Lett. 112,165501 (2014 ).[28] J. McMinis, R. C. Clay, D. Lee, and M. A. Morales, Molecular
to Atomic Phase Transition in Hydrogen under High Pressure,Phys. Rev. Lett. 114,105305 (2015 ).
[29] N. D. Drummond, B. Monserrat, J. H. Lloyd-Williams, P. L ´opez
R´ıos, C. J. Pickard, and R. J. Needs, Quantum Monte Carlo study
of the phase diagram of solid molecular hydrogen at extreme
pressures, Nat. Commun. 6,7794 (2015 ).
[30] S. Azadi, N. D. Drummond, and W. M. C. Foulkes, Nature of the
metallization transition in solid hydrogen, arXiv:1608.00754 .
[31] K. A. Johnson and N. W. Ashcroft, Structure and bandgap
closure in dense hydrogen, Nature (London) 403,632(2000 ).
[32] C. J. Pickard and R. J. Needs, Structure of phase III of solid
hydrogen, Nat. Phys. 3,473(2007 ).
[33] C. J. Pickard, M. Martinez-Canales, and R. J. Needs, Density
functional theory study of phase IV of solid hydrogen, Phys.
Rev. B 85,214114 (2012 ).
[34] C. J. Pickard, M. Martinez-Canales, and R. J. Needs, Erratum:
Density functional theory study of phase IV of solid hydrogen[Phys. Rev. B 85, 214114 (2012)], Phys. Rev. B 86,059902(E)
(2012 ).
[35] S. Azadi and Th. D. K ¨uhne, Absence of metallization in solid
molecular hydrogen, JETP Lett. 95,449(2012 ).
[36] M. Dvorak, X.-J. Chen, and Z. Wu, Quasiparticle energies and
excitonic effects in dense solid hydrogen near metallization,Phys. Rev. B 90,035103 (2014 ).
[37] C. J. Pickard and R. J. Needs, Structures at high pressure from
random searching, Phys. Status Solidi B 246,536(2009 ).
[38] C. J. Pickard and R. J. Needs, High-Pressure Phases of Silane,
Phys. Rev. Lett. 97,045504 (2006 ).
[39] C. J. Pickard and R. J. Needs, Metallization of aluminum hydride
at high pressures: A first-principles study, Phys. Rev. B 76,
144114 (2007 ).
[40] C. J. Pickard and R. J. Needs, Highly compressed ammonia
forms an ionic crystal, Nat. Mater. 7
,775(2008 ).
[41] S. Ninet, F. Datchi, P. Dumas, M. Mezouar, G. Garbarino,
A. Mafety, C. J. Pickard, R. J. Needs, and A. M. Saitta,Experimental and theoretical evidence for an ionic crystal ofammonia at high pressure, P h y s .R e v .B 89,174103 (2014 ).
[42] A. Dominic Fortes, E. Suard, M.-H. Lem ´ee-Cailleau, C. J.
Pickard, and R. J. Needs, Crystal structure of ammonia mono-hydrate phase II, J. Am. Chem. Soc. 131,13508 (2009 ).
[43] A. Dewaele, N. Worth, C. J. Pickard, R. J. Needs, S. Pascarelli,
O. Mathon, M. Mezouar, and T. Irifune, Synthesis and stabilityof xenon oxides Xe
2O5and Xe 3O2under pressure, Nat. Chem.
8,784(2016 ).
[44] A. F. Goncharov, R. T. Howie, and E. Gregoryanz, Hydrogen at
extreme pressures, Low Temp. Phys. 39,402(2013 ).
[45] R. O. Jones, Density functional theory: Its origins, rise to
prominence, and future, Rev. Mod. Phys. 87,897(2015 ).
[46] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. I. J.
Probert, K. Refson, and M. C. Payne, First principles methodsusing
CASTEP ,Z. Kristallogr. 220,567(2005 ).
[47] D. Vanderbilt, Soft self-consistent pseudopotentials in a gener-
alized eigenvalue formalism, Phys. Rev. B 41,7892 (1990 ).
[48] A. D. Becke, Density-functional exchange-energy approxima-
tion with correct asymptotic behavior, Phys. Rev. A 38,3098
(1988 ).
[49] C. Lee, W. Yang, and R. G. Parr, Development of the Colle-
Salvetti correlation-energy formula into a functional of theelectron density, P h y s .R e v .B 37,785(1988 ).
134101-6HEXAGONAL STRUCTURE OF PHASE III OF SOLID . . . PHYSICAL REVIEW B 94, 134101 (2016)
[50] R. C. Clay, III, J. Mcminis, Jeffrey M. McMahon, C. Pierleoni,
D. M. Ceperley, and M. A. Morales, Benchmarking exchange-correlation functionals for hydrogen at high pressures usingquantum Monte Carlo, P h y s .R e v .B 89,184106 (2014 ).
[51] B. Monserrat, N. D. Drummond, and R. J. Needs, An-
harmonic vibrational properties in periodic systems: Energy,electron-phonon coupling, and stress, Phys. Rev. B 87,144302
(2013 ).[52] J. H. Lloyd-Williams and B. Monserrat, Lattice dynamics
and electron-phonon coupling calculations using nondiagonalsupercells, Phys. Rev. B 92,184301 (2015 ).
[53] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.94.134101 for the file reporting the details
of the P6
122 structure.
[54] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient
Approximation Made Simple, Phys. Rev. Lett. 77,3865 (1996 ).
134101-7 |
PhysRevB.101.155118.pdf | PHYSICAL REVIEW B 101, 155118 (2020)
Multipolar magnetism in d-orbital systems: Crystal field levels,
octupolar order, and orbital loop currents
Sreekar V oleti ,1D. D. Maharaj ,2B. D. Gaulin,2,3,4Graeme Luke,2,3,5and A. Paramekanti1,*
1Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7
2Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1
3Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada L8S 4M1
4Canadian Institute for Advanced Research, 661 University Ave., Toronto, Ontario, Canada M5G 1M1
5TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A3
(Received 14 February 2020; revised manuscript received 30 March 2020; accepted 30 March 2020;
published 15 April 2020)
Quantum magnets with spin J=2, which arise in spin-orbit coupled Mott insulators, can potentially display
multipolar orders. Motivated by gaining a better microscopic understanding of the local physics of such d-orbital
quantum magnets, we carry out an exact diagonalization study of a simple octahedral crystal field Hamiltonianfor two electrons, incorporating spin-orbit coupling (SOC) and interactions. While the rotationally invariantKanamori interaction in the t
2gsector leads to a fivefold degenerate J=2 manifold, we find that either explicitly
including the egorbitals, or going beyond the rotationally invariant Coulomb interaction within the t2gsector,
causes a degeneracy breaking of the J=2 levels. This can lead to a low-lying non-Kramers doublet carrying
quadrupolar and octupolar moments and an excited triplet which supports magnetic dipole moments, bolsteringour previous phenomenological proposal for the stabilization of ferro-octupolar order in heavy transition metaloxides. We show that the spontaneous time-reversal symmetry breaking due to ferro-octupolar ordering withinthe non-Kramers doublet leads to electronic orbital loop currents. The resulting internal magnetic fields canpotentially explain the small fields inferred from muon-spin relaxation ( μSR) experiments on cubic 5 d
2osmate
double perovskites Ba 2ZnOsO 6,B a 2CaOsO 6,a n dB a 2MgOsO6, which were previously attributed to weak
dipolar magnetism. We make further predictions for oxygen NMR experiments on these materials. We alsostudy the reversed level scheme, where the J=2 multiplet splits into a low-lying magnetic triplet and excited
non-Kramers doublet, presenting single-ion results for the magnetic susceptibility in this case, and pointingout its possible relevance for the rhenate Ba
2YReO 6. Our work highlights the intimate connection between the
physics of heavy transition metal oxides and that of f-electron based heavy fermion compounds.
DOI: 10.1103/PhysRevB.101.155118
Multipolar orders have been proposed and discussed exten-
sively in f-orbital based heavy fermion compounds [ 1–14].
Such multipolar orders have also been proposed to occur ind-orbital metals with large spin-orbit coupling (SOC), such
as LiOsO
3and Cd 2Re2O7, via Pomeranchuk instabilities of
the Fermi liquid [ 15]. Optical second-harmonic generation
experiments on Cd 2Re2O7have found evidence for such an
inversion broken quadrupolar ordered state below Tc∼200 K
[16]. Other candidates for multipolar orders include proposed
quadrupolar order in A 2OsO 4(with A =K, Rb, Cs) [ 17].
In recent work we have studied d-orbital Mott insulators
with large SOC and a d2configuration in a local octahedral
environment, and proposed these systems as candidates forrealizing ferro-octupolar order [ 18,19]. Previous studies of
such d
2quantum magnets [ 20–22] have argued that the com-
bination of crystal field and interaction effects leads to thestabilization of a state with total L=1 and S=1, which are
locked by SOC into a J=2 spin. Motivated by experiments
*arunp@physics.utoronto.ca[18,23–26] on certain cubic double perovskite (DP) Mott in-
sulators, Ba 2ZnOsO 6,B a 2CaOsO 6, and Ba 2MgOsO6, which
host a 5 d2configuration on Os, we have instead proposed
[19] that their observed nontrivial phenomenology, such as
entropy and a spin gap, could be captured by assuming that thefivefold J=2 multiplet is weakly split, resulting in a ground
state non-Kramers doublet carrying quadrupolar and octupo-lar moments. The lack of any observed crystal distortionsin x-ray and neutron diffraction experiments appears to ruleout quadrupolar order [ 18]. Uniform ferro-octupolar ordering
in the low lying doublet manifold then provides the mostviable route to further reconciling the cubic symmetry, theobservation of time-reversal symmetry breaking seen via μSR
oscillations [ 23], the apparent lack of any magnetic Bragg
peaks in elastic neutron diffraction experiments [ 18], and the
spin gap observed in inelastic neutron scattering experiments[18,19].
In this paper we provide further theoretical calculations
in favor of the above scenario. We first present exact diago-nalization results on a simple local crystal field Hamiltoniankeeping the t
2gandeglevels in an octahedral environment,
showing that the combination of SOC and interactions does
2469-9950/2020/101(15)/155118(8) 155118-1 ©2020 American Physical SocietySREEKAR VOLETI et al. PHYSICAL REVIEW B 101, 155118 (2020)
favor a non-Kramers ground state doublet. We show how the
splitting between this doublet and the excited magnetic tripletdepends on SOC and the Hund’s coupling and results fromperturbative t
2g-egmixing. Such t2g-egmixing was discussed
previously but its importance for the low energy physicsappears not to have been properly recognized [ 21,27]. We
also examine a model of just t
2gelectronic states, and show
that deviations of the Coulomb interaction from sphericalsymmetry, perhaps engendered by hybridization with oxygenorbitals [ 28], can lead to a similar non-Kramers doublet state.
This doublet-triplet splitting may be too small to be resolvedusing resonant inelastic x-ray scattering experiments [ 29,30],
but it is crucial for the low energy symmetry-breaking orders.We study the impact of ferro-octupolar order within this lowenergy non-Kramers doublet, and show that this leads toorbital electronic currents, generating internal magnetic fieldsand semiquantitatively explain the μSR oscillations seen in
Ba
2ZnOsO 6,B a 2CaOsO 6, and Ba 2MgOsO6. The nonspher-
ical Coulomb interaction mechanism for splitting the J=2
multiplet discussed above also permits for the possibility forthe level ordering to be reversed, with a magnetic tripletground state and an excited non-Kramers doublet. We presentsingle ion results for the magnetic susceptibility in this case,arguing that this reversed level scheme is likely to be relevantto the 5 d
2rhenate [ 31]B a 2YReO 6.
Our theory strengthens the case for multipolar orders in
ac l a s so f d-orbital Mott insulators, pointing to a smooth
conceptual link between the physics of heavy d-orbital oxides
andf-electron based heavy fermion materials. Such octupolar
order with a high transition temperature may provide a newtemplate to store information.
I. LOCAL MODEL
We use the following Hamiltonian for two electrons in a
d-orbital placed in an octahedral environment:
H=HCEF+HSOC+Hint, (1)
where we include the octahedral crystal field splitting,
SOC, and Kanamori interactions, written in the orbital basis({yz,xz,xy},{x
2−y2,3z2−r2})↔({1,2,3},{4,5}) where
α≡{1,2,3}label t2gorbitals and α≡{4,5}label egorbitals.
The CEF term is given by
HCEF=VC/summationdisplay
α=4,5/summationdisplay
snα,s, (2)
where sis the spin. The SOC term is
HSOC=λ
2/summationdisplay
α,β/summationdisplay
s,s/prime/angbracketleftα|L|β/angbracketright·/angbracketlefts|σ|s/prime/angbracketrightc†
α,scβ,s/prime, (3)
where σrefers to the vector of Pauli matrices, and Lis the
orbital angular momentum. Its components in the orbital basisare given in Appendix A. We assume a Kanamori interaction
for all five dorbitals given by
H
int=U/summationdisplay
αnα↑nα↓+U/prime/summationdisplay
α>βnαnβ−JH/summationdisplay
α/negationslash=β/vectorSα·/vectorSβ
+JH/summationdisplay
α/negationslash=βc†
α↑c†
α↓cβ↓cβ↑, (4)
FIG. 1. Low energy spectrum (15 lowest eigenvalues) of the
Hamiltonian in Eq. ( 1) with two electrons, corresponding to states
where both electrons predominantly occupy the t2gorbitals. The
numbers at the end of the curves, and in the zoomed-in insetswhich show weak splittings, indicate the degeneracies of the different
energy levels.
where /vectorSα=(1/2)c†
αs/vectorσs,s/primecαs/prime. This simple form, where we use
the same interaction parameters for all t2gandegorbitals,
is used to avoid a proliferation of interaction parameters.Assuming spherical symmetry of the Coulomb interaction, wehave U
/prime=U−2JH(see, e.g., Ref. [ 32]).
For electronic configurations with partially filled t2gor-
bitals, the most commonly used approach is to simply ignorethee
gorbitals and restrict attention to the low energy t2gstates.
We find that the ground state manifold in this approximationconsists of a fivefold degenerate J=2 state. However, we
show below that this degeneracy is further split due to twopossible microscopic mechanisms: t
2g-egmixing and devia-
tions of the Coulomb interaction from spherical symmetry.
A.t2g-egmixing: Exact results, perturbation theory
We consider two electrons in the full d-orbital manifold
including t2gandegstates, and study this using numerical
exact diagonalization in the 45 basis states. For couplingconstants we use values typical for 5 dtransition metal oxides:
V
C=3e V , U=2.5e V ,λ=0.4 eV, and JH=0.25 eV. Fig-
ure1plots the evolution with JHof the lowest 15 energy levels
which correspond to eigenstates where the two electrons arepredominantly both in the t
2gsector. The indicated numbers
mark the degeneracies of these multiplets. For JH=0, there
are just three energy levels, which, in increasing order ofenergy, correspond to having (i) both electrons in j=1/2,
(ii) one electron in j=1/2 and one electron in j=3/2
(energy cost 3 λ/2), and (iii) both electrons in j=3/2 (energy
cost 3 λ). We see that the lowest energy set of five states
evolves adiabatically out of the first sector as we increaseJ
H; this set of five states corresponds to the J=2 moment.
However, a zoom-in of this multiplet, as well as of one ofthe higher energy multiplets, shows that the apparent fivefolddegeneracy of these states is actually weakly broken as 2 ⊕3
due to weak t
2g-egmixing. In particular, the naively expected
fivefold degenerate J=2 ground state is split into a non-
Kramers doublet ground state and an excited magnetic triplet;for the typical values listed above, this splitting is ∼8m e V .
Figure 2shows the dependence of this lowest energy
doublet-triplet energy splitting (blue solid line) on V
C. We find
155118-2MULTIPOLAR MAGNETISM IN d-ORBITAL SYSTEMS: … PHYSICAL REVIEW B 101, 155118 (2020)
FIG. 2. Energy difference between the lower energy non-
Kramers doublet ( Ed) and the excited triplet ( Et), given by /Delta1=
Et−Ed, obtained via exact diagonalization of the Hamiltonian in
Eq. ( 1) (blue, solid line) plotted as a function of the dominant t2g-eg
splitting VC. We compare this with the third-order perturbation theory
result (red, dashed line) induced by small ( JH/VC,λ/VC) which leads
to weak t2g-egmixing.
that this splitting can be semiquantitatively captured within
third-order perturbation theory, as discussed in Appendix B,
where we first eliminate the egstates, to find an effective
t2gmodel, and then diagonalize this reduced Hamiltonian.
The relevant terms arise at O(λ2JH/V2
C), from the following
sequence: (i) SOC λpromoting one electron from the t2g
manifold into the egsector, (ii) intermediate state t2g-eginter-
actions driven by Hund’s coupling set by JH, and finally (iii)
de-exciting back via SOC λto end up with both electrons in
thet2gmanifold. Diagonalizing this third-order perturbative
Hamiltonian, in conjunction with the bare t2gHund’s cou-
pling, leads to the non-negligible splitting shown (red dashedline) in Fig. 2, which agrees well with the full numerical cal-
culation in the regime of large V
C. Our result is in contrast with
a previous conjecture that the splitting would appear at fourthorder in perturbation theory [ 21], which would have indeed
rendered this effect negligible. This highlights a nontrivialeffect of t
2g-egmixing, showing that it can be important for
nucleating multipolar order in 5 dMott insulators. However,
this effect by itself may be too small to account for the spingap observed in neutron scattering experiments [ 18,19]o n
Ba
2ZnOsO 6,B a 2CaOsO 6, and Ba 2MgOsO6.W en e x tt u r nt o
an additional mechanism, which can cooperate to enhance thissplitting, or even reverse the level ordering which we argue isimportant in certain other materials.
B. Nonspherical Coulomb interactions in t2gmodel
The second important physical effect we consider is that
projecting the Coulomb interaction to the t2gWannier or-
bitals can lead to deviations from the spherical symmetryassumption, so that U
/prime/negationslash=U−2JH. This is expected to be
more important for 5 dorbitals which have more significant
overlap with the oxygen cage, as has been previously noted inanab initio study [ 28]. We therefore numerically diagonalize
FIG. 3. Energy difference /Delta1=Et−Edbetween the magnetic
triplet and the non-Kramers doublet obtained via exact diagonaliza-tion of the t
2g-only model, shown as a function of the normalized de-
viation δU/prime/U/primeof the Coulomb interaction from spherical symmetry.
ForδU/prime>0, the non-Kramers doublet has lower energy so /Delta1> 0.
the above model Hamiltonian, restricting ourselves to the
Hilbert space where both electrons occupy the t2gorbitals,
and varying δU/prime=U/prime−(U−2JH) to simulate the deviation
from spherical symmetry. Figure 3shows how the low energy
degeneracy gets split as we go away from δU/prime=0. We see
from here that even a small deviation δU/prime/U/prime∼0.1 leads
to a substantial splitting ∼20 meV. For δU/prime>0, we find
that the non-Kramers doublet is lower in energy than themagnetic triplet, which we argue is relevant to osmates such asBa
2ZnOsO 6,B a 2CaOsO 6, and Ba 2MgOsO6. The case where
theδU/prime<0, so that the magnetic triplet lies lower in energy
than the doublet, may be important to understand aspects ofthe unusual magnetism of the rhenate [ 31]B a
2YReO 6;t h i s
will be discussed in Sec. III.
II. MAGNETIC FIELDS FROM OCTUPOLAR ORDER
On phenomenological grounds, and the above microscopic
calculations, 5 d2oxides are candidates for a low-lying non-
Kramers doublet. As shown previously [ 19], this doublet may
be described using the wave functions of the J=2 manifold
in terms of |Jz/angbracketrighteigenstates written as pseudospin-1 /2 states:
|ψg,↑/angbracketright=| 0/angbracketright;|ψg,↓/angbracketright=1√
2(|2/angbracketright+|− 2/angbracketright). (5)
Each of these two states is individually time-reversal in-
variant. The angular momentum operators ( J2
x−J2
y) and
(3J2
z−J2), restricted to this basis, act as pseudospin-1 /2
operators ( τx,τz), forming the two components of an XY-
like quadrupolar order parameter, while JxJyJz(with overline
denoting symmetrization) behaves as τy, and serves as the
Ising-like octupolar order parameter. The mean field ferro-octupolar ordered ground state is described by each site beingin the superposition state |ψ
oct
±/angbracketright=|ψg,↑/angbracketright±i|ψg,↓/angbracketright. Here the
signs reflect the Z2nature of the Ising order, and ireflects the
breaking of time-reversal symmetry.
155118-3SREEKAR VOLETI et al. PHYSICAL REVIEW B 101, 155118 (2020)
The broken time-reversal symmetry of the octupolar
ground state would lead to internal magnetic fields in thecrystal. Using exact diagonalization, we obtain |ψ
oct
±/angbracketrightas
the two-electron wave function obtained by superposing thetwo degenerate time-reversal invariant ground eigenstates asabove, and compute the electronic currents in these stateswhich generate the internal magnetic fields. In the single-sitepicture, the orbital currents responsible for the internal fieldslive on the d
2ion. We thus define the orbital current density
operator as
J(r)=ie¯h
2m/summationdisplay
s[/Psi1†
s(∇/Psi1s)−(∇/Psi1†
s)/Psi1s], (6)
where ssums over the physical electron spin. We expand the
operator /Psi1in the orbital basis as
/Psi1†
s=/summationdisplay
αψn/lscriptα(r,θ,φ )c†
α,s, (7)
where r≡(r,θ,φ ),ψn/lscriptαrefers to the real hydrogenlike wave
function, with n=5 and/lscript=2f o rt h e5 dwave functions, and
αdenotes the orbital. We thus arrive at the spatially varying
expectation value of the current density operator:
/angbracketleftJ(r)/angbracketright±=ie¯h
2m/summationdisplay
s/summationdisplay
αβ/angbracketleftψoct
±|c†
α,scβ,s|ψoct
±/angbracketrightξαβ, (8)
ξαβ=R2
n/lscript(r)/parenleftbig
Y/lscriptα∇Y/lscriptβ−Y/lscriptβ∇Y/lscriptα/parenrightbig
, (9)
where the two Ising states have /angbracketleftJ(r)/angbracketright−=− /angbracketleft J(r)/angbracketright+.H e r e
Y/lscriptα(θ,φ) are real Tesseral harmonics, and Rn/lscript(r) is the radial
wave function. To compute the current density, we use avariational ansatz for the radial wave function, which takeson a hydrogenic form, but with an effective nuclear chargewhich decreases with r, from a bare nuclear charge Z
0for
r→0 to the screened effective charge Z∞forr→∞ , over
a length scale r0.F o rt h eO s6+ion relevant to Ba 2ZnOsO 6,
Ba2CaOsO 6, and Ba 2MgOsO6,w eu s e Z0=76 and Z∞=
7, and consider different values of r0; details are given in
Appendix B.
Using this expectation value for the current density, we
compute the magnetic field via
B±(r)=μ0
4π/integraldisplay
d3r/prime/angbracketleftJ(r/prime)/angbracketright±×(r−r/prime)
|r−r/prime|3, (10)
where the integral is carried out over primed variables. The
two Ising time-reversed partner states have opposite magneticfields B
−(r)=−B+(r).
The orbital current pattern which creates this field is shown
schematically in Fig. 4(left panel), highlighting that it is
analogous to loop current orders proposed in certain cuprateand heavy fermion materials [ 33,34]. In a more realistic calcu-
lation, which retains hybridization with oxygen, the octupolarorder we have uncovered may in fact be identical to plaquetteloop current order in the OsO
6cage. We find that the magnetic
field has a pattern which, appropriately, might be expectedfrom a set of eight alternating “magnetic monopoles” arrangedon a cube, as shown in Fig. 4(right panel), to form an octupole
centered on the Os
6+ion. Figure 5shows the magnetic field
expected from these orbital currents as a function of distancefrom the Os
6+ion along the [111] direction, where the fieldFIG. 4. Left: Schematic plot of the orbital current pattern on the
5d2Os ion (indicated by the ball), showing that it has the same
symmetry as plaquette loop current order residing on the OsO 6octa-
hedral cage. Right: Configuration of fictitious “magnetic monopoles”forming an octupole, which would produce the octupolar current
loop pattern shown in the left panel.
strength is the largest, for two different choices of r0as in-
dicated. Figure 6shows the same calculation, but normalizing
the field by that generated by a 1 μBdipole located at the Os6+
site.
While we have discussed above the magnetic field due to
octupolar order as a function of distance from Os, in orderto make a comparison with μSR experiments, we have to
estimate the fields produced by the octupolar order at possiblemuon stopping sites. We thus next estimate the magneticfield distribution over the surface of a sphere of radius 1 Åcentered around the oxygen site, which is where the muonis expected to be bound [ 35,36]. Figure 7shows a plot of
the field distribution, where we find the maximum field tobe present at points on this sphere located near the Os
6+
ion. (This calculation retained nine Os6+ions closest to the
oxygen ion, beyond which the contribution was negligible.)We note that these maxima lie between the /angbracketleft111/angbracketrightand/angbracketleft100/angbracketright
directions. The presence of four symmetric maxima of the
FIG. 5. Magnetic field generated within the crystal in the pres-
ence of ferro-octupolar order, plotted as a function of distance from
the 5 d2Os ion along the [111] direction. The two curves correspond
to different choices of the screening parameter r0, which impacts the
field only at short distances. The wiggles reflect the structure of the
radial wave function.
155118-4MULTIPOLAR MAGNETISM IN d-ORBITAL SYSTEMS: … PHYSICAL REVIEW B 101, 155118 (2020)
FIG. 6. Magnetic field in the presence of ferro-octupolar order,
plotted as a function of distance from the 5 d2Os ion along the [111]
direction. The data are the same as in Fig. 5, but normalized by Bdip
which denotes the magnetic field at the same location generated by
a1μBdipole moment located at the origin and pointing along the
[111] direction.
field strength is consistent with the residual symmetry in
the ferro-octupolar state of C4rotations about the Os-O axis
followed by time reversal. The computed maximum field isfound to be ∼30 G, within a factor of 2 of the ∼50 G mag-
netic field inferred from μSR experiments on Ba
2ZnOsO 6,
Ba2CaOsO 6, and Ba 2MgOsO6below a transition tempera-
ture T∗. A quantitative computation with the μSR results
would need to retain the Os-O hybridization and ab initio
calculations for the optimal muon stopping sites [ 35,36]. The
magnetic field inferred from μSR experiments was previously
attributed to possible weak magnetic dipolar order, with a tinyordered moment /lessorsimilar0.02μ
B. Such a tiny ordered moment is
difficult to explain given the typical ∼1μBlocal moments
FIG. 7. Color plot of the ferro-octupolar magnetic field distribu-
tion over a sphere of radius 1 Å around the oxygen site where themuon is expected to be bound. The oxygen site is located half-way
between Os and the B-site ion (Mg, Zn, Ca). The largest field strength
(in red) appears near the Os
6+ion.
FIG. 8. Temperature dependence of the inverse magnetic suscep-
tibility (normalized to its value at T=300 K) in the single-site prob-
lem with a low lying magnetic triplet and an excited non-Kramers
doublet; see text for details of the model and parameters. At high
temperature, we find a Curie-Weiss-like linear form χ−1(T)∝(T+
Ts), as indicated by the dashed line, with Ts∼275 K for the chosen
parameters. At low temperature, we find the Curie law χ−1(T)∝T.
The temperature where the low Tand high Tlines meet denotes a
crossover temperature scale Tcr≈30 K. Varying the doublet-triplet
splitting, we find that kBTcr≈0.07|/Delta1|andkBTs≈0.35|/Delta1|.
expected in such Mott insulators, unless one is fine tuned to
be near a quantum critical point. Our work instead naturallyrules out dipolar order, and instead explains this weak field asarising from loop currents in a phase which supports octupolarorder.
III. REVERSED LEVEL SCHEME: MAGNETIC
TRIPLET GROUND STATE
In previous work and in the above sections, we have ex-
tensively explored the case where the J=2 multiplet is split
into a low-energy non-Kramers doublet and a spin-gappedmagnetic triplet. In this section we explore the single-ionphysics of the reversed level scheme which has also not beenstudied in the oxides literature. As an illustrative example ofa model which leads to this level ordering, we explore theHamiltonian in Eq. ( 1), but with δU
/prime=U/prime−(U−2JH)<
0, and projecting onto just the t2gorbitals. We note that
this deviation is not necessarily the only way in which theCoulomb interaction can deviate from spherical symmetry—indeed, imposing only the octahedral point group symmetrywill allow for a broader set of interactions.
Figure 8shows the inverse magnetic susceptibility χ
−1(T)
in this single-ion case, normalized by its value at T=300 K,
for a choice of parameters VC=3e V , U=2.5e V , λ=
0.4 eV, and JH=0.25 eV (as used in the previous sections),
but with δU/prime=− 0.5 eV. (This choice of an admittedly large
δU/primeis only used for the simplest model to illustrate the impact
of splitting the lowest energy J=2 multiplet; it is not meant
to capture the full spectrum of higher energy excitations.) Thisleads to a triplet ground state, with an excited non-Kramersdoublet at an energy |/Delta1|∼37 meV. Interestingly, we find
155118-5SREEKAR VOLETI et al. PHYSICAL REVIEW B 101, 155118 (2020)
thatχ−1(T)∝(T+Ts) in this case, exhibiting an apparent
“Curie-Weiss”-like form with Ts≈275 K, over a wide range
of temperatures /greaterorsimilar150 K. Based on this, one might mislead-
ingly infer a Curie-Weiss temperature ∼− 275 K. Only upon
going to lower temperatures, do we observe a change ofslope and the correct χ
−1(T)∝TCurie law associated with
the single-ion low energy magnetic triplet. We find a verysimilar result in an even simpler model where we split theJ=2 multiplet using symmetry-allowed Stevens operators,
viaH
eff=−Veff(O40+5O44), with Veff<0, where
O40=35J4
z−[30J(J+1)−25]J2
z+3J2(J+1)2
−6J(J+1), (11)
O44=1
2(J4
++J4
−), (12)
suggesting that it is a robust consequence of triplet-doublet
splitting, with Tsreflecting single-ion physics; in this model,
|/Delta1|=120|Veff|. Varying the doublet-triplet splitting, we
find kBTs≈0.35|/Delta1|, while the crossover from the high
temperature Curie-Weiss-like form to the low temperaturebehavior occurs at a temperature T
crgiven by kBTcr≈
0.07|/Delta1|.
Remarkably, precisely such a behavior, with a Curie-
Weiss-like form for χ−1(T) and a break in slope on going
below/lessorsimilar150 K has been observed [ 31]i nB a 2YReO 6, leading
us to suspect that the experimentally reported large Curie-Weiss temperature ∼− 600 K may in fact be misleading,
and could partly reflect this modified single-ion physics. Thetrue Curie-Weiss temperature in this material may thus well
be much smaller, and likely closer to that seen in the d
2
osmates discussed above. Our exploration thus serves to partly
rationalize the widely diverging Curie-Weiss temperatures re-ported in this class of materials as arising from the differencesin the single-ion physics of different 5 dions. The nature
and strength of exchange interactions between such magneticions will be discussed elsewhere, in the context of ongoingexperiments on Ba
2YReO 6.
IV . DISCUSSION
We have shown that the physics of spin-orbit coupled J=
2 magnets can exhibit unconventional multipolar orders whichemerge from a low energy non-Kramers doublet. This doubletarises from crystal field splitting of the J=2 multiplet due
to multiple physical effects: weak t
2g-egmixing as well as
deviation of the Coulomb interaction from spherical symme-try. Ferro-octupolar ordering within this doublet, which canresult from the interplay of magnetic exchange and orbitalrepulsion [ 19], provides the most viable explanation for the
huge body of experimental data, including the μSR oscilla-
tions which we have shown results from orbital electroniccurrents. As a further test of our theory, we propose thatnuclear magnetic resonance (NMR) studies on the oxygen siteshould show no sign of any internal fields below T
∗due to
its octupolar structure, which is evident from the schematicplot in Fig. 4; specifically, the octupolar configuration in a
cubic system is invariant under C
4rotations about the Os-Oaxis followed by time reversal. This vanishing of the field
in oxygen NMR would serve to further distinguish octupolarorder from possible dipolar order for which we do expect tosee an internal field in the NMR spectrum. Applying uniaxialpressure along the /angbracketleft111/angbracketrightor/angbracketleft110/angbracketrightdirections would break
thisC
4symmetry, leading to a nonzero field at the oxygen
site which may be detectable by NMR. In previous work[19] we have also shown how Raman scattering in a /angbracketleft111/angbracketright
magnetic field can uncover octupolar order via the appearanceof new modes below T
∗. Our work makes a compelling
case for octupolar order in a d-orbital Mott insulator. Future
experimental studies using pressure or doping, to suppressthe octupolar transition temperature and induce metallicity,may allow one to study possible non-Fermi liquid statesassociated with fluctuating multipolar orders [ 37]. Our work
emphasizes the need for additional ab initio studies of 5 d
oxides at various filling factors to construct the appropriateWannier functions in order to extract the local interactionHamiltonian. In light of our work, it is also imperative torevisit the entire body of experiments on other 5 d
2materials,
such as Ba 2YReO 6,a sw e l la s5 doxides at other filling
factors.
ACKNOWLEDGMENT
This work was supported by the Natural Sciences and
Engineering Research Council of Canada.
APPENDIX A: ORBITAL WA VE FUNCTIONS
AND L MATRICES
Thedorbital basis is constructed out of the lzeigenstates
of the angular momentum l=2 manifold as
|yz/angbracketright=i√
2(|−1/angbracketright+| 1/angbracketright)≡|1/angbracketrightα,
|xz/angbracketright=1√
2(|−1/angbracketright−| 1/angbracketright)≡|2/angbracketrightα,
|xy/angbracketright=i√
2(|−2/angbracketright−| 2/angbracketright)≡|3/angbracketrightα,
|x2−y2/angbracketright=1√
2(|−2/angbracketright+| 2/angbracketright)≡|4/angbracketrightα,
|3z2−r2/angbracketright=| 0/angbracketright≡| 5/angbracketrightα,(A1)
where the states |m/angbracketrightrefer to |l=2,m/angbracketrightand states with the
subscript αindicate the orbital basis. Since this is the basis
we will be working with in this paper, the αindex will be
dropped. The |m/angbracketrightstates in position space can be represented
using spherical harmonics (employing the Condon-Shortleyphase), and the particular linear combinations above ensurethat the orbital wave functions are real, giving the so-calledtesseral harmonics. In this basis, the angular momentum ma-
155118-6MULTIPOLAR MAGNETISM IN d-ORBITAL SYSTEMS: … PHYSICAL REVIEW B 101, 155118 (2020)
trices can be constructed as
Lx=⎛
⎜⎜⎜⎜⎜⎝00 0
−i−i√
3
00 i 00
0−i0 00
i 00 00
i√
300 00⎞
⎟⎟⎟⎟⎟⎠,
L
y=⎛
⎜⎜⎜⎜⎜⎝00 −i
00
00 0 −ii√
3
i 00 00
0 i 0 00
0−i√
30 00⎞
⎟⎟⎟⎟⎟⎠,
L
z=⎛
⎜⎜⎜⎜⎜⎝0 i 0
00
−i00 00
00 0 2i0
00 −2i00
00 0 00⎞
⎟⎟⎟⎟⎟⎠.(A2)
The top left blocks in the above matrices show the t
2gsub-
space, and it is clear that the angular momentum is completelyquenched in the e
gsubspace.
APPENDIX B: PERTURBATION THEORY
We carry out a perturbation theory study, using HCEF
[Eq. ( 2)] as the unperturbed Hamiltonian and treating JH
(interactions) and λ(SOC) as perturbations. Working in the
two-electron basis |α1,s1;α2,s2/angbracketright≡c†
α1,s1c†
α2,s2|0/angbracketright, where the
α’s are orbital indices, and the s’s are spin indices, the unper-
turbed eigenspace consists of three energy levels {0,VC,2VC},
with degeneracies {15,24,6}. These correspond to double
occupancy within the t2glevel, shared occupancy between
thet2gandeglevels, and double occupancy in the eglevel,
respectively. The perturbations couple these different sectors.For instance, SOC can excite an electron from a t
2glevel into
aneglevel, across the gap VC. Similarly, pair hopping can hop
a pair of electrons from a t2glevel into an eglevel, across
an energy gap 2 VC. Treating such terms within perturbation
theory we find that in order to project out the egsubspace,
we treat all such mixing terms adding second-order and third-order perturbation effects, which leads to an effective t
2g
subspace Hamiltonian. At second order, we find that U/prime−U,
pair hopping, and magnetic Hund’s coupling are renormalizeddifferently, but in a way that does not break spherical sym-metry, i.e., the renormalized Kanamori couplings obey [ 32]
U
/prime−U=JP+JH(where JPandJHdenote, respectively, the
strength of the interorbital pair hopping and magnetic Hund’scoupling). Diagonalizing the resulting effective Hamiltonian,which sums the full Hamiltonian projected to t
2glevels with
the above perturbed interactions, we find that the ground stateremains a fivefold degenerate J=2 multiplet. However, at
third order, we find new interactions that arise in the effectiveHamiltonian in the t
2gmanifold which cannot be described
as renormalizations of existing interactions; specifically, thereare terms schematically given by
/Delta1H(3)
L,L/prime=/summationdisplay
H,H/prime(HSOC)L,H(HHund)H,H/prime(HSOC)H/prime,L/prime
V2
C,
where L,Hrefer to low and high energy states with Lhaving
both electrons in the t2gorbitals, and Hhaving one electron
int2gand the other in eg. This term leads to a splitting of the
J=2 manifold into a low energy non-Kramers doublet and
a high energy magnetic triplet, with the splitting emerging atO(λ
2JH/V2
C) at large VC.
APPENDIX C: ORBITAL CURRENTS AND
MAGNETIC FIELDS
In order to study the impact of ferro-octupolar order in gen-
erating time-reversal breaking electronic currents and mag-netic fields, we explicitly write out the orbital wave functionsin position space which enter the angular momentum states.For this, we multiply the radial part of the hydrogenlike wavefunction with the tesseral harmonic of the orbital. We use thefollowing form for the radial wave function:
R
nl(r)=Nnlρl(r)e−ρ(r)/2L2l+1
n−l−1[ρ(r)], (C1)
where nis the principal quantum number, lis the angular mo-
mentum quantum number, and ρ(r)=2r/na(r).L2l+1
n−l−1is the
generalized Laguerre polynomial, and Nnlis a normalization
constant. a(r) is a function which captures the screening by
the inner electrons, which we call the “effective” Bohr radius.The function must be chosen such that
lim
r→0a(r)=a0/Z0; lim
r→∞a(r)=a0/Z∞, (C2)
where a0is the (hydrogen) Bohr radius, Z0is the bare charge
of the nucleus, and Z∞is the effective charge that an electron
sees at large distances. We propose the following simple form:
a(r)=a0
Z(r);Z(r)=Z∞+(Z0−Z∞)e−r/r0, (C3)
with r0being a tuning parameter which determines how the
effective charge falls off with distance. For instance, for anOs
6+ion,Z0=76 and Z∞=7 (since all electrons except the
one 5 delectron we focus on will contribute to screening at
large distances). A reasonable value for r0is that it is smaller
than the ionic radius ∼70 pm; we thus consider r0=10–
20 pm. If we are interested in 5 delectrons, the radial wave
function is of the form
R52(r)=N52/parenleftbigg2r
5a(r)/parenrightbigg2
e−r/5a(r)L5
2/parenleftbigg2r
5a(r)/parenrightbigg
. (C4)
The normalization constant N52depends on r0. Hence the full
wave function is given by ψnlα=Rnl(r)Ylα(θ,φ), where Ylα
is the tesseral harmonic associated with the αorbital. The
current operator thus becomes
J=ie¯h
2m/summationdisplay
α,β/summationdisplay
s(ψnlα∇ψnlβ−ψnlβ∇ψnlα)c†
α,scβ,s.(C5)
All the spatial dependence of the current is encoded in
the factor ψnlα∇ψnlβ−ψnlβ∇ψnlα≡ξαβ(r,θ,φ ). Since the
wave functions can be separated into the radial and angular
155118-7SREEKAR VOLETI et al. PHYSICAL REVIEW B 101, 155118 (2020)
components, i.e., ψnlα=Rnl(r)Ylα(θ,φ), this factor becomes
ξαβ=R2
nl(Ylα∇Ylβ−Ylβ∇Ylα). (C6)
From the exact diagonalization, we can obtain the ground state
of the system as some linear combination of our basis states.Let us call this ground state |ψ
g/angbracketright:
|ψg/angbracketright=/summationdisplay
/Omega1a/Omega1|/Omega1/angbracketright, (C7)
where |/Omega1/angbracketrightrefers to our basis states of the form |α1,s1;α2,s2/angbracketright.
Since we are interested in the matrix elements of the currentin Eq. ( C5) in this state, we can recast the problem as
/angbracketleftJ/angbracketright=ie¯h
2m/summationdisplay
α,βwαβξαβ, (C8)
where each factor ξαβis associated with a “weight” wαβ,g i v e n
by
wαβ=/summationdisplay
/Omega1,/Omega1/primea∗
/Omega1a/Omega1/prime/summationdisplay
s/angbracketleft/Omega1|c†
α,scβ,s|/Omega1/prime/angbracketright. (C9)
It can be seen that wαβ=w∗
βα. The Hermiticity of Jconstrains
the weights to be purely imaginary. Once the expectationvalue of the current density is obtained, we use Eq. ( 10)t o
compute the magnetic field.
[1] P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani,
and G. H. Lander, Rev. Mod. Phys. 81,807(2009 ).
[2] K. Haule and G. Kotliar, Nat. Phys. 5,796(2009 ).
[3] P. Santini and G. Amoretti, Phys. Rev. Lett. 85,2188 (2000 ).
[4] J. A. Paixão, C. Detlefs, M. J. Longfield, R. Caciuffo, P. Santini,
N. Bernhoeft, J. Rebizant, and G. H. Lander, Phys. Rev. Lett.
89,187202 (2002 ).
[5] A. Kiss and P. Fazekas, P h y s .R e v .B 68,174425 (2003 ).
[6] Y . Tokunaga, D. Aoki, Y . Homma, S. Kambe, H. Sakai, S.
Ikeda, T. Fujimoto, R. E. Walstedt, H. Yasuoka, E. Yamamoto,A. Nakamura, and Y . Shiokawa, Phys. Rev. Lett. 97,257601
(2006 ).
[7] T.-h. Arima, J. Phys. Soc. Jpn. 82,013705 (2013 ).
[8] A. Sakai and S. Nakatsuji, J. Phys. Soc. Jpn. 80,063701 (2011 ).
[9] T. J. Sato, S. Ibuka, Y . Nambu, T. Yamazaki, T. Hong, A. Sakai,
and S. Nakatsuji, Phys. Rev. B 86,184419 (2012 ).
[10] M. Tsujimoto, Y . Matsumoto, T. Tomita, A. Sakai, and S.
Nakatsuji, Phys. Rev. Lett. 113,267001 (2014 ).
[11] K. Hattori and H. Tsunetsugu, J. Phys. Soc. Jpn. 85,094001
(2016 ).
[12] F. Freyer, J. Attig, S. Lee, A. Paramekanti, S. Trebst, and Y . B.
Kim, P h y s .R e v .B 97,115111 (2018
).
[13] S. Lee, S. Trebst, Y . B. Kim, and A. Paramekanti, P h y s .R e v .B
98,134447 (2018 ).
[14] A. S. Patri, A. Sakai, S. Lee, A. Paramekanti, S. Nakatsuji, and
Y. B . K i m , Nat. Commun. 10,4092 (2019 ).
[15] L. Fu, P h y s .R e v .L e t t . 115,026401 (2015 ).
[16] J. W. Harter, Z. Y . Zhao, J.-Q. Yan, D. G. Mandrus, and D.
Hsieh, Science 356,295(2017 ).
[17] S. Hayami, H. Kusunose, and Y . Motome, Phys. Rev. B 97,
024414 (2018 ).
[18] D. D. Maharaj, G. Sala, M. B. Stone, E. Kermarrec, C. Ritter,
F. Fauth, C. A. Marjerrison, J. E. Greedan, A. Paramekanti, andB. D. Gaulin, P h y s .R e v .L e t t . 124,087206 (2020 ).
[19] A. Paramekanti, D. D. Maharaj, and B. D. Gaulin, Phys. Rev. B
101,054439 (2020 ).
[20] G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82,174440
(2010 ).
[21] G. Chen and L. Balents, P h y s .R e v .B 84,094420 (2011 ).
[22] C. Svoboda, M. Randeria, and N. Trivedi, Phys. Rev. B 95,
014409 (2017 ).
[23] C. M. Thompson, J. P. Carlo, R. Flacau, T. Aharen, I. A. Leahy,
J. R. Pollichemi, T. J. S. Munsie, T. Medina, G. M. Luke,J. Munevar, S. Cheung, T. Goko, Y . J. Uemura, and J. E.
Greedan, J. Phys.: Condens. Matter 26,306003 (2014 ).
[24] E. Kermarrec, C. A. Marjerrison, C. M. Thompson, D. D.
Maharaj, K. Levin, S. Kroeker, G. E. Granroth, R. Flacau, Z.Yamani, J. E. Greedan, and B. D. Gaulin, Phys. Rev. B 91,
075133
(2015 ).
[25] C. M. Thompson, C. A. Marjerrison, A. Z. Sharma, C. R.
Wiebe, D. D. Maharaj, G. Sala, R. Flacau, A. M. Hallas, Y . Cai,B. D. Gaulin, G. M. Luke, and J. E. Greedan, P h y s .R e v .B 93,
014431 (2016 ).
[26] C. A. Marjerrison, C. M. Thompson, A. Z. Sharma, A. M.
Hallas, M. N. Wilson, T. J. S. Munsie, R. Flacau, C. R. Wiebe,B. D. Gaulin, G. M. Luke, and J. E. Greedan, P h y s .R e v .B 94,
134429 (2016 ).
[27] G. L. Stamokostas and G. A. Fiete, Phys. Rev. B 97,085150
(2018 ).
[28] T. Ribic, E. Assmann, A. Tóth, and K. Held, Phys. Rev. B 90,
165105 (2014 ).
[29] B. Yuan, J. P. Clancy, A. M. Cook, C. M. Thompson, J.
Greedan, G. Cao, B. C. Jeon, T. W. Noh, M. H. Upton, D. Casa,T. Gog, A. Paramekanti, and Y .-J. Kim, P h y s .R e v .B 95,235114
(2017 ).
[30] A. Paramekanti, D. J. Singh, B. Yuan, D. Casa, A. Said,
Y .-J. Kim, and A. D. Christianson, P h y s .R e v .B 97,235119
(2018 ).
[31] T. Aharen, J. E. Greedan, C. A. Bridges, A. A. Aczel, J.
Rodriguez, G. MacDougall, G. M. Luke, V . K. Michaelis, S.Kroeker, C. R. Wiebe, H. Zhou, and L. M. D. Cranswick, Phys.
Rev. B 81,064436 (2010 ).
[32] A. Georges, L. d. Medici, and J. Mravlje, Annu. Rev. Condens.
Matter Phys. 4,137(2013 ).
[33] M. E. Simon and C. M. Varma, Phys. Rev. Lett. 89,247003
(2002 ).
[34] P. Chandra, P. Coleman, J. A. Mydosh, and V . Tripathi, Nature
(London) 417,831(2002 ).
[35] W. K. Dawson, K. Tibbs, S. P. Weathersby, C. Boekema, and
K. B. Chan, J. Appl. Phys. 64,5809 (1988 ).
[36] F. R. Foronda, F. Lang, J. S. Möller, T. Lancaster, A. T.
Boothroyd, F. L. Pratt, S. R. Giblin, D. Prabhakaran,and S. J. Blundell, Phys. Rev. Lett. 114,017602
(2015 ).
[37] A. S. Patri, I. Khait, and Y . B. Kim, Phys. Rev. Res. 2,013257
(2020 ).
155118-8 |
PhysRevB.42.1332.pdf | PHYSICAL REVIEW B VOLUME 42,NUMBER 2 15JULY1990-I
Effectofelectron andholeaccumulation onmagneto-optical spectra
ofanundoped GaAs/Ga Al,Asquantum well
A.S.Plaut*
TheClarendon Laboratory, University ofOxford,ParksRoad,OxfordOX13PU,UnitedICingdom
R.T.Harley
TheClarendon Laboratory, University ofOxford,ParksRoad,OxfordOX13PU,UnitedKingdom
andGEC,HirstResearch Centre,EastLane,Wembley, Middlesex HA94PP,UnitedKingdom
S.R.Andrews andT.M.Kerr
GEC,HirstResearch Centre,EastLane,8'embley, Middlesex HA94PP,UnitedKingdom
(Received 17January 1990;revisedmanuscript received 19March1990)
Wehaveperformed magnetophotoluminescence andmagnetophotoluminescence-excitation spec-
troscopy at1.8KonaGaAs/Ga„All „Asquantum-well structure inwhichthecarrierdensityina
0
single50Aquantum wellcanbevariedfromzerouptoabout2X10"carriers cmbyuseofa
Schottky gate.Thisresultsfromtransferofeitherelectrons orholesphotoexcited inthickerGaAs
layersinthestructure andthereby allowstheinvestigation ofcarrier-density-dependent effectsina
singlesample. Thedatafortheemptywellareconsistent withprevious studiesofmagneto-optics of
atomicexcitons. WithaFermiseaofheavyholesorelectrons present, thespectrashowevidenceof
band-gap renormalization andphase-space filling.Alsothelowestinter-Landau-level transition was
observed tofollowalinearfielddependence forfieldslowerthanfillingfactorv=2buttohavea
particularly weakmagnetic fielddependence athigherfields.Thisisconsistent withcrossover from
freecarriertoexcitonic behavior atv=2andiscompared torecenttheoretical calculations.
I.INTRODUCTION
Optical studiesofsemiconductor quantum wellshave
mainlyconcentrated ontheproperties ofundoped struc-
tures.Inthesesystems thetwo-dimensional confinement
isseentoenhance theexcitonic natureoftheinterband
transitions, resulting inincreased oscillator strengths and
binding energies compared tothevaluesforbulkGaAs.
Consequently, noveleffectshavebeenobserved suchas
well-defined excitonic behavior evenatroomtemperature
andthequantum confined Starkeffect.'Spectroscopic
measurements, inwhichphotoluminsecence techniques
haveplayedamajorrole,havebeenundertaken withand
without applied magnetic fieldandhaveyielded the
valuesof,forinstance, theexciton binding energyand
heavy-hole effective massasafunctionofwellwidth.
Recently aninteresthasbeentakenintheopticalprop-
ertiesofquantum wellscontaining freecarriers whether n
orptype.'Inthesemodulation-doped structures the
carriers arespatially separated fromthedopants, which
resultsinalargereduction ofscattering bytheCoulomb
potentials oftheionized impurities andtherefore inhigh
mobilities. Thepresence ofthefreecarriers altersthe
electronic properties ofthequantum wellsconsiderably.
Forinstance, itcausesarenormalization ofthebandgap
duetomany-body, exchange andcorrelation effects.
Phase-space fillingandshort-range exchange interactions
between carriers intheFermiseaalsoleadtoquenching
oftheatomicexcitonoscillator strength andtoareduc-
tioninitsbinding energy. Insteadofanatomicexciton,atlowtemperatures, thedynamical responseoftheFermi
seaproduces anexcitonlike peak,whichisaboundstate
withrespecttotheFermilevel.Itiscalledthe"Mahan
exciton"' andissimilartotheFermi-edge singularity ob-
servedinx-rayabsorption andemission spectraofmet-
als15,16
Inmodulation-doped structures thetypeandtosome
extentthenumberofcarriers present inthewellsisfixed
duringthegrowthprocess. Various attempts havebeen
madetoalterthecarrierconcentration, eitherbypho-
toexcitation''orbyapplication ofanelectrical
bias.''Theformermethod resultsinsimultaneous
injectionofbothelectrons andholesproducing aneutral
plasma, whilethelatterinjectseitherelectrons orholes
producing acharged plasma. Theoretical descriptions of
thesedifferent situations havebeenreviewed byBauer:
Themeasurements described herewereperformed ona
sample(Fig.l)consisting offoursingleundo@ed quan-
tumwellsofdifferent widthsranging from50Ato800A
inwhichthecarrierconcentration couldbevariedbythe
quasiresonant injectionofeitherelectrons orholesonap-
plication ofelectrical bias.Thustheadvantages ofa
largecarrier-dopant separation arecombined withacon-
tinuously variable carrierdensity, enabling studiesofthe
electronic structure asafunctionofcarrierconcentration
andwellwidthwithout theunavoidable fluctuations of
parameters inherent incomparison ofmeasurements
madeondifferent samples. Inthispaperweconcentrate
uponthechanges inthemagnetophotoluminescence
(MPL}andmagnetophotoluminescence excitation
421332 1990TheAmerican Physical Society
42 EFFECT OFELECTRON ANDHOLEACCUMULATION ON... 1333
34AlO3Ga07As barriers
ITOn'GaAs
Lz5102080nm
GaAswells
0MPLE)spectraofthe50Awellasthecarrierconcentra-
tionisvariedfrom10"heavyholescmthroht rougempty
wellconditions upto2X10"electrons cm
II.EXPERIMENTALFIG.l..Schematic ofsamplestructure andpotential profile
forzeroapplied bias.ITOindicates theindium tinoxidioxie
chottkygate.Dimensions ofthebarriers andbufferlayersare
giveninthetext.biasthebandsareeffectively flatbecause thereisanequal
andopposite internal voltage inthesampleduetothen+
substrate. Thespectraaretypicalofanundoped quan-
tumwellwithnocarriers inthewell.Atzerofield,sharp
excitonic features areobserved forboththelight-hole
N=1subband toelectron N=1subband (lhl-el)transi-
tionandheavy-hole N=1toelectron N=1(hh1-el)
transition at1.6145and1.5940eV,respectively. The
lh1-e1freecarrieredgeisalsoobservable at1.6265eV.
The2sesstateoftheexciton isjustdiscernible asapeakat
thiscontinuum onset.Theequivalent hh1-e1continuum
onsetismasked bythelh1-elexciton. Inanexternal
magnetic field8thecontinuum isseentosplitinto
iscretelineswhichareinterpreted asexcitonic inter-
Landau leveltransitions involving mainlyhhl-el sub-
bandswiththeselection ruleAn=0.
TheMPLEspectra with+1.0Vappliedacrossthe
sample (seeFig.3)aresignificantly modified fromthe
emptywelldata.Theexcitonic peaksatzerofieldare
broader andreduced intotalintegrated arearelativeto
thatofthecontinuum athigherenergy, indicating ade-
creaseinoscillator strength byafactorof2or3.The
arealsoblueshiftedcompared tothetransitions observed
Thesamplestudiedconsisted ofintrinsic (background
impurity concentration -5X10' cm type)GaAs
wellsofthickness 50,100,200,and800Aseparated b
340AAlGp3Gap7Asbarriers grownbymolecular-beamseparate y
epitaxy(MBE)inthesequence depicted inFig.1,onan
Si-doped (ND—10'cm)GaAssubstrate withaninter-
vening0.5-pm-wide undoped GaAsbu6'erlayer.Electri-
calbiaseswereappliedperpendicular tothelayersusinga
00-pm-thick, 90%transparent indium tinoxide(ITO)
Schottky contact onthetopsurfaceandanindium-alloy
contact tothen+substrate. Photoluminescence was
excitedusingaPyridine 2(LC7300)dyelayer.Thelight,
polarized withanEvectorintheplaneofthesurface,
wasincident perpendicular totheplaneofthequantum
wellsandfocusedtoa-200-pm spotonthesample. The
incident powerdensities were200mWcmorless.The
sample wasimmersed inpumped liquidheliumat1.8K
andmagnetic fieldsoriented alongtheincident beam
wereprovided byoneoftwosuperconducting magnets
withmaximum attainable fieldsof6and10T.Theback-
wardemitted luminescence wasanalyzed usinga0.5-m
monochromator witharesolution of0.05meVandhav-
ingacooledphotomultiplier asadetector. PLwasexam-
inedusingafixeddyelaserenergywellabovetheabsorp-
tionedgeandscanning thespectrometer through there-
gionofinterest. PLEspectra wereobtained bysetting
thespectrometer todetectatthemaximum ofthefirst
conduction subband tofirstheavy-hole subband recom-
bination peakandscanning thedyelasertohigherener-
gy.Inasampleofthistype,forphoton energies below
thebandgapofthebarriers, weexpectthePLEintensity
toapproximately reflectabsorption.
A.Magnetophotoluminescence excitationOC6)O
lJl
Q)C1
03
160 165
Energy(eV)
Figure2showsthePLEspectraforvarious magnetic
fieldswith+0.375Vappliedacrossthesample. AtthisFIG.2.PLEspectraofthe50Aquantum wellatvarious
magnetic fieldsat+0.375Vappliedbias,corresponding tothe
flatbandcondition.
1334 PLAUT HARLEY,ANDREWS, ANDKERR 42
atanappliedbiasof+0.375V.hh1-e1isnow
eVandlh1-elat1.6160V.
diff'erence("Stokesshift"e.Fromthead~''dditional energy
oessift")between theemission andab-
sorption peakscompared tothatfor+0.375Vbia
ctronscmpresent
eweunderanapplied biasof+10V
orp
einsitpartlyoffsetbyband-ar
associated 'thfill'ewiingofthee1subbyan-gaprenormalization
sionofthhsubband. Further discus-
foundelse~hereoecangesincurred intheezero-field datacanbe
Inafinitemagnetic fieldthespectraaainsho
Landau-level transit' Th 'ti
spreading otu,asinteemmetwellc'tions,insteadof
py
eaanenergywellabovetheexciton eak
eriescoincident withthem
Similarbehbehavior isobserved (seeFig.4)when
pieissubjected to—0.5Vbiasriasproducing apopulation of
cavyolescminthe50Awell.A
o-fildk bod areroadened andtheiroscillatorstrength reduced andinfieldthewholesep p
au-evetransitions. Inthiscaseanext
peaktolowerenergyisoserveaove3.25T.
Thedifferences between theemta
wellbecome evneemptyandthen-orp-type
Landau levelsappeartoigs.—.Intheemptywell(Fi.5ig.)the
ppeartoconverge atzeromane
hhl-elcontinuum edge.Withcar'exciton eatureandcorrespondin to'gothe
and7}theinter-Landau-level trans''ege.stcarriers present(Figs.6
atzerofieldransitions extrapolate back
zeroetoamuchlowerenergy,whichwetaketobe
rmaizeandedge.Itisclea
creasesareaconsequence ofhase-seein-
p-p g
oransitions involving lowerLandau lev
theypassupwards throhhF'.i
larlyremarkable thatthlugteermilevel.Iti
sitionsforthewellfilled'h~~aeowestinter-Landau-1 -eveltran-
eeeitherwithelectrons orholes,
I I I I
10
OT OT
2.0T1.0T
Vl
thC
CU
C
OCI
Vl
CU
C:30T
4.0Tnfl
C
U
tflC
C
0)O
O
th0)C2.0T
3.0T—
3.25T
3.5T
4.0T
1.60 1.65
Energy {eV}I
1.60
Energy(eV}I
1.65
FIG.3.PLEspectraatvarious manetic
d cm(+1.0Vappliedbias).FIG.4.PLEsespectraatvariousmanetic
h1d'
yo—cm(—0.5Vappliedbias)
42 EFFECT OFELECTRON ANDHOLEACCUMULATION ON... 1335
1.70 1.70
)1.65
1.60—
ooooooo1.60
I I I I I
2 468
Magnetic Field(T)I ) I I I I I I
2 468
Magnetic Field(T)
FIG.5.PLE(solidcircles}andPL(opencircles)transition
energies asafunctionofmagnetic fieldatBatbands(+0.375V
bias).Thesolidcurvesarecalculations fortwodifferent values
ofp,asdescribed inthetext.FIG.7.PLE(solidcircles) andPL(opencircles)transition
energies asafunction ofmagnetic fieldataholedensityof—10"cm'.Thesolidlinesarecalculations asdescribed inthe
text.
whichemerges onlyathighmagnetic fields,doesnot
seemtoshowanyfielddependence (seeFig.8).Thelines
plottedinFigs.5-8willbediscussed indetailinSec.III.
B.Magnetophotoluminsecence
ThePLpeakpositions areplotted asopencirclesin
Figs.5—8.Withfiatbands(Fig.5)asinglenarrowPL
1.70
~~~~linewasobserved (at1.5914eVat8=0),whichinmag-
neticfieldfollowed thediamagnetic shiftoftheheavy-
holePLEwithaconstant "Stokesshift"ofabout3meV.
Thissmallshiftisinterpreted asduetotheefficient trap-
pingofthedelocalized excitons atregionswherethewell
isonetotwomonolayers widerthanaverage. Withelec-
tronsinthewell(Figs.6and8)thePLlineatzeromag-
neticfieldwassignificantly broadened andredshifted
(1.5871eV)andinmagnetic fieldwasseentosplitinto
twoduetoformation ofLandau levels:Asthefieldin-
creased, thelowerpeakgrewinintensity whilethatat
higherenergydiminished anddisappeared atthemagnet-
1.65
OJ
~1.59
Ch
CIJC
LU0 ——~-~~--—~—~-~- ~L0~~
0 0-0—-0-
)~o
t)0~0
1.60
I l ) I I I I
2468
Magnetic Field(T)1.580I I I
2 46
Magnetic Field(T)
FIG.6.PLE(solidcircles) andPL{opencircles)transition
energies asafunctionofmagnetic fieldatanelectron densityof-2X10"cm'.Thesolidlinesarecalculations asdescribed in
thetext.FIG.8.Expanded viewofthelowenergyPLE(solidcircles}
andPL(opencircles)transition energies asafunctionofmag-
neticfieldatanelectron densityof-2X10" cm.Thesolid
lineisacalculation asdescribed inthetext.
1336 PLAUT, HARLEY, ANDRE%S, ANDKERR 42
icfie1dvalueatwhichanewtransition wasfirstobserved
inthePLEspectra (3T).Above4.25T,whichwecan
identify asfillingfactorv=2,thePLpeakshowsthe
samelackofmagnetic fielddependence asthelowest
PLEpeak(seeFig.8).Withholespresent(Fig.7)no
splittingofthePLlinewasseen,consistent withtherebe-
ingarelatively smalldensityofholes.Againabove3.25
T(i.e.,belowv=2)theenergypositionofthelumines-
cencelineisveryinsensitive tothemagnetic fieldand
parallels thatofthelowestPLEline.
III.DISCUSSION
Thedatafortheemptywell(Fig.5)closelyresemble
thatfrommagnetoreflectivity onsimilarsamples. Asin
thatcase,thedatahavebeentakeninthelow-magnetic-
fieldregimewhereexcitonic effectsandnonparabolicity
oftheheavy-hole Landau levelshaveastronginfluence
onthefielddependence oftheinterband transitions and
conversely electron nonparabolicity issmallenoughtobe
ignored. Themodelcalculation superimposed onthe
datainFig.5isbasedonthenumerical resultsofAkimo-
toandHasegawa foratwo-dimensional (2D)exciton in
asmallmagnetic field.Theadaptation ofthismodel
fromtheexact2Dtothequasi-two-dimensional caseis
achieved byscalingtheirsystemindependent numerical
resultsbyboththeobserved zero-field exciton binding en-
ergyandthereduced massoftheelectron-hole pair
p=(1/m,'+I/mz")'(m,'andmz'beingtheelectron
andheavy-hole in-plane effective masses,respectively). It
wasfoundpreviously thatpdecreased withdecreasing
magnetic fieldreflecting asimilardecrease intheheavy-
holeeffective mass.ForthecurvesshowninFig.5are-
ducedmassofp=0.0500mo wasusedfor0.0T&B&2.5
Tandp=0.0605mo for2.0T&B&6.0T.Anexciton
binding energyof11meVwasusedinbothmagnetic field
ranges.
Foroccupied n-andp-typewells(Figs.6—8)thedata
confirm thatthecarriercontentofthewellsisessentially
independent ofmagnetic field.Fromthemagnetic field
valuecorresponding inthePLEdatatov=2weobtain
anelectron densityof2.05X10"cmandaholedensity
of1.57X10"cm,compared with2X10" cmand
1X10"cm,respectively, usingthemeasured addition-
al"Stokes shift."Theagreement between thesevalues
isquitesatisfactory considering thattheformermethod
willtendtooverestimate thecarrierconcentration be-
causetherehastobeasignificant numberofemptystates
intheLandau levelbeforetheabsorption intensity be-
comesobservable.
Thedataalsoreadilyyieldvaluesfortheband-gap re-
normalization. Weobtainan11.8-meVrenormalization
with1.8X10"cmelectrons presentandavalueof11.4
meVwith0.75X10" cmholesinthewell.Asde-
scribedinmoredetailelsewhere thistakesintoaccount
asmallcontribution tothelineshiftsduetoboththe
electric fieldandthebandbending effectsofthespace
charge. Thesevaluesareingeneral agreement with
theoretical expectations,'however, itisinteresting to
notethattherenormalization inthecaseofholesis
significantly greaterthanforelectrons atagivenconcen-tration.
Forthecalculations showninFigs.6—8,wehaveas-
sumedthattheCoulomb effectsarescreened givingthe
freecarrierLandau levelformula foraninterband transi-
tion:
E(8)=Eo+(n+—,')p
whereEoisthezero-field transition energyandnisthe
Landau-level index.p=0.063mowastakenforthesefits
whichisthevaluemeasured inhighfieldsinsimilarsam-
plesunderwhichconditions thereisstrongheavy-and
light-hole mixing. Phase-space fillingcausestheinter-
bandtransitions tooccuratfinitewavevectorwhereeven
atzeromagnetic fieldthereissignificant valence subband
mixing,'thusjustifying theadoption ofthisvalue.
Thecalculated curvesseemtogiveareasonable represen-
tationofthebehavior ofthehigher-energy transitions in
bothcases(Figs.6and7),indicating thatwhenthereare
carriers inthewelltheexciton binding energybecomes
verysmall,aspredicted,'eventhough itsoptical
strength isnotsodrastically effected. Thebinding energy
ofa2D"Mahan exciton" is,moreover, expected tobe
lessthan1meV(Refs.14and32)whichisalsoconsistent
withourfindings.
Onthebasisofthesimplecalculation plotted inFigs.
6—8,onewouldexpectthelowestLandau leveltoexhibit
afree-electron-like magnetic fielddependence whichit
seemstodoatlowfieldsabovev=2witheitherelectrons
orholesinthewell.However itsslopeforhigherfields,
i.e.,belowv=2,ismuchless,beingalmostzero(seeFig.
8);thereisachangeover atv=2fromfreeelectron to
field-independent behavior.
Various typesof"anomalous" behavior havebeenob-
servedpreviously ininterband andcyclotron resonance
spectra. Anoscillatory dependence oftheluminescence
energyinperpendicular magnetic fieldhasbeenobserved
byPerryetal.'inawiden-typemodulation doped
GaAs/Al„Ga, „Asquantum well.Similarbehavior has
0alsobeenseeninan100An-typeIn„Ga,„As/InP
quantum well.Thesamples studied werequitestrongly
doped(4X10" cmand9.2X10"cm,respectively)
andthedeviations fromlinearbehavior wereobserved at
evenfillingfactors,i.e.,lowermagnetic fieldthanv=2,in
thelowerLandau levels.Withintheexperimental uncer-
taintyimposed bythePLlinewidths, ourdata(opencir-
clesinFigs.6—8)donotshowtheseanomalies which
havebeeninterpreted intermsofoscillatory screening
effectsontheself-energies.
Disorder maybeexpected toperturb thefreeelectron-
likebehavior ofLandau levels.Firstly,itwillresultina
finitewidthforthelevelsandconsequently maycause
shiftsinabsorption andluminescence peakpositions with
respecttothecenteroftheLandau levelasthefillingfac-
torchanges. Boththesmallmeasured widthofthe
lowestPLEpeaks(Figs.3and4),andthefactthatthe
component ofthe"Stokes shift"notassociated with
phase-space fillingisrelatively smallandfieldindepen-
dent(Figs.5—8)suggestthatthisisnotalargeeffecthere.
Secondly, measurements ofcyclotron resonance inquan-
tumwellsandheterostructures haverevealed anomalies
42 EFFECT OFELECTRON ANDHOLEACCUMULATION ON... 1337
inLandau levelsforv~2(seeRefs.36and37andpapers
citedtherein). Theeffectcanbedescribed empirically by
an=0to1cyclotron transition energy givenby
(fico,+b,)',where b,isan"offset" whichisdisorder
relatedandmayrepresent anelectron binding energy. b
variesstrongly withfieldincreasing rapidlynearv=2.
Theoriginof6isnotfullyunderstood anditisnotobvi-
ousthatasimilaroffsetshouldbeexpected ininterband
transitions between Landau levelsofthesameindex.
Nevertheless, avalueof6increasing withfieldupto10
meVat8Twouldbeneededtoexplainthedeviation of
ourmeasured peakpositions fromthefreeelectron values
(Fig.8).Ontheotherhand,inourmeasurements anesti-
mateofthebinding energyassociated withdisorder is
provided bythepartofthe"Stokesshift"notcausedby
phase-space filling.Fortheemptywell(Fig.5)thisis-3
meVforallfields,whereas forthewellwith2X10"cm
(Fig.8)itisconstant at—1,4meVforv&2.Thus,while
disorder relatedeffectsmaybepresent inthedata,it
seemsthattheyarenotlargeenoughtoaccount fullyfor
theobserved behavior forv&2.
Achangeover fromfreeelectron toapproximately
field-independent behavior hasbeenobserved inmagneto-
luminescence froma107-A-wide quantum wellwithelec-
tronconcentrations between 1.5X10"cm and
5.3X10" cm.Thisoccured atfieldscorresponding
roughly tov=3andwasassigned asatransition from
freeelectrontoexcitonic behavior.
Suchachangeover hasbeenpredicted byBauerfora
two-dimensional plasma butatv=2ratherthanv=3.
Heemphasizes theimportance oftherelative magnitudes
ofquantum wellwidthL,andThomas-Fermi (TF)
screening length A,T„ininterpretations ofmagneto-
opticalspectraofquasi-two-dimensional charged plas-
mas.Inourmeasurements onarelatively weaklypopu-
lated50Awellwehavethesituation A,TF&L„whereas
fortheluminescence measurements citedabove
A,TF&(L,.Thusthepresent experimental situation pro-videsarelatively goodapproximation toa2Dcharged
plasma. Bauer hascalculated theeffectofexcitonic
correlations onthemagnetoluminescence andabsorption
of2Dplasma inamean-field (Hartree-Fock) approxima-
tion.Hepredicts thatindeedforaonecomponent plas-
maatlowmagnetic fieldtheoccupied low-energy states
arenotmodified bythevertexcorrection (theeffective
electron-hole interaction) butthatclosetov=2the
luminescence undergoes atransition fromfreeparticleto
excitonic behavior. Thechangeoccursbecause themag-
neticfieldcausestheelectron andholewavefunctions to
besqueezed together andhenceincreases theexciton
binding energy. Although thetheoryusedbyBaueris
strictly2Dandtherfore ignoresthefinitethickness ofthe
quantum wellthereisexcellent qualitative agreement
withourobservations (seeFig.4ofRef.23).
IV.SUMMARY
%ehavestudied thechanges intheMPLandMPLE
spectraofasingle50-Aquantum wellcausedbyresonant
injection andaccumulation ofbothelectrons andholes
separately. %ehavetherefore studied theeffectsof
phase-spacing fillingonboththebandgapandalsoon
thebinding energyoftheexcitonoverarangeofcarrier
concentrations from-2X10" cmelectrons through
emptyto—10"cmholes.Theapplication ofmagnetic
fieldalsoallowsverification ofthecarrierdensityitself
andthevaluessoobtained areingoodagreement with
thezero-field measurements. Atfieldscorresponding to
v=2whenthewellisporntypewehaveobserved devia-
tionsfromalinearvariation ofthetransition energyof
thelowestLandau levelwhichweinterpret asduetoa
changeover fromfreetoexcitonic behavior aspredicted
thoeretically. Forthiswellwidthandrangeofcarrier
concentrations wedonotobserve oscillatory behavior of
luminescence energies inamagnetic fieldwhichmaybe
associated withfield-dependent screening effectsonthe
electron andholeself-energies.
'Present address: Max-Planck-Institut furFestkorper-
forschung, Heisenbergstrasse 1,7000Stuttgart 80,WestGer-
many.
Present address: Department ofPhysics, TheUniversity,
Southampton SO95NH,UnitedKingdom.
'D.A.B.Miller,D.S.Chemla,T.C.Damem, A.C.Gossard,
W.Wiegmann, andC.A.Burrus, Phys.Rev.Lett.53,2173
(1984).
2D.A.B.Miller,D.S.Chemla,D.J.Eilenberger, P.W.Smith,
A.C.Gossard, andW.T.Tsang,Appl.Phys.Lett.41,679
(1982).
R.C.Miller,D.A.Kleinman, W.T.Tsang,andA.C.Gos-
sard,Phys.Rev.B24,1134(1982).
4J.C.Maan,G.Belle,A.Fasolino, M.Altarelli, andK.Ploog,
Phys.Rev.B30,2253(1984).
5D.C.Rogers,J.Singleton, R.J.Nicholas, C.T.Foxon,andK.
Woodbridge, Phys.Rev.B34,4002(1986).
P.Dawson,K.J.Moore,G.Duggan, H.I.Ralph,andC.T.B.
Foxon,Phys.Rev.B34,6007(1986).
~A.S.Plaut,J.Singleton, R.J.Nicholas, R.T.Harley,S.R.Andrews, andC.T.B.Foxon,Phys.Rev.B38,1323(1988).
8A.Pinczuk,J.Shah,H.L.Stormer,R.C.Miller,A.C.Gos-
sard,andW.Wiegmann, Surf.Sci.142,492(1984).
R.C.MillerandD.A.Kleinman,J.Lumin.30,520(1985).
'C.H.Perry,J.M.Worlock, M.C.Smith,andA.Petrou, High
Magnetic FieldsinSemiconductor Physics,Vol.71ofSpringer
SeriesinSolidStateSciences, edited byG.Landwehr
(Springer-Verlag, Berlin,1987),p.202."Y.Iwasa,J.S.Lee,andN.Miura,SolidStateCommun. 64,
597(1987).
'M.S.Skolnick,J.M.Rorison,K.J.Nash,D.J.Mowbray, P.
R.Tapster,S.J.Bass,andA.D.Pitt,Phys.Rev.Lett.58,
2130(1987).
C.Delalande, G.Bastard,J.Orgonasi,J.A.Brum,H.W.Lui,
andM.Voos,Phys.Rev.Lett.59,2690(1987).
G.D.Mahan, Phys.Rev.153,882(1967).
G.D.Mahan, Phys.Rev.163,612(1967).
'P.Nozieres andC.T.deDominicis, Phys.Rev.178,1097
(1968).
'C.Delalande, J.Orgonasi, M.H.Meynadier, J.A.Brum,G.
1338 PLAUT, HARLEY, ANDREWS, ANDKERR 42
Bastard,G.Weimann, andW.Schlapp, SolidStateCommun.
59,613(1986).
C.Weber,C.Klingshirn, D.S.Chemla,D.A.B.Miller,J.E.
Cunningham, andC.Ell,Phys.Rev.B38,12748(1988).
M.K.Saker,M.S.Skolnick, P.A.Claxton,J.S.Roberts, and
M.J.Kane,Semicond. Sci.Technol. 3,691(1988).
I.BarJoseph,J.M.Kuo,C.Klingshirn, G.Livescu,T.Y.
Chang,D.A.B.Miller,andD.S.Chemla, Phys.Rev.Lett.
59,1357{1987).
'H.Yoshimura, G.E.W.Bauer,andH.Sakaki,Phys.Rev.B
38,10791(1988).
H.Yoshimura andH.Sakaki,Phys.Rev.B39,13024{1989).
G.E.W.Bauer, inProceedings ofthe8thInternational
Conference onElectronic Properties of2DSystems, Greno-
ble,1989[Surf.Sci.(tobepublished)].
S.R.Andrews, A.S.Plaut,R.T.Harley, andT.M.Kerr,
Phys.Rev.B41,5040(1990).
E.Burstein, Phys.Rev.93,632(1954).
O.Akimoto andH.Hasegawa, J.Phys.Soc.Jpn.22,181
(1967).
S.Schmitt-Rink, D.S.Chemla, andD.A.B.Miller,Phys.Rev.B32,6601{1985).
F.Ancilotto, A.Faslino, andJ.C.Maan,Superlatt. Micros-
truct.3,187{1987).
M.Altarelli, U.Ekenberg, andA.Fasolino, Phys.Rev.B32,
5138(1985).
G.D.Sanders andY.C.Chang,Phys.Rev.B31,6893(1985).
D.A.Kleinman, Phys.Rev.B32,3766{1985).
J.Gavoret, P.Nozieres, B.Roulet, andM.Combescot, J.
Phys.30,987(1969).
M.S.Skolnick, K.J.Nash,S.J.Bass,P.E.Simmonds, and
M.J.Kane,SolidStateCommun. 67,637(1988).
34S.Katayama, andT.Ando,SolidStateCommun. 70,97
(1989).
T.Uenoyama andL.J.Sham,Phys.Rev.B39,11044(1989).
G.Wiggins,R.J.Nicholas,J.J.Harris,andC.T.Foxon,in
Proceedings ofthe8thInternational Conference onElectron-
icProperties of2DSystems, Grenoble, 1989[Surf.Sci.(tobe
published)].
R.J.Nicholas, M.A.Hopkins,D.J.Barnes,M.A.Brummel,
H.Sigg,D.Heitmann, K.Ensslin,J.J.Harris,C.T.Foxon,
andG.Weimann, Phys.Rev.B39,10955{1989).
|
PhysRevB.79.220517.pdf | Nanoscale superconducting-gap variations and lack of phase separation in optimally doped
BaFe 1.86Co0.14As2
F. Massee, *Y. Huang, R. Huisman, S. de Jong, J. B. Goedkoop, and M. S. Golden
Van der Waals–Zeeman Institute, University of Amsterdam, 1018XE Amsterdam, The Netherlands
/H20849Received 8 April 2009; revised manuscript received 20 May 2009; published 30 June 2009 /H20850
We present tunneling data from superconducting BaFe 1.86Co0.14As2and its parent compound, BaFe 2As2.I n
the superconductor, clear coherencelike peaks are seen across the whole field of view, and their analysis revealsnanoscale variations in the superconducting gap value, /H9004. The average peak-to-peak separation gives a 2 /H9004of
/H110117.4k
BTc, which exceeds the BCS weak coupling value for either s-o r d-wave superconductivity. The char-
acteristic length scales of the deviations from the average gap value and of anticorrelations observed betweenthe gap magnitude and both the zero-bias conductance and coherence peak strength match well with theaverage separation between the Co dopant ions in the superconducting FeAs planes.
DOI: 10.1103/PhysRevB.79.220517 PACS number /H20849s/H20850: 74.70. /H11002b, 68.37.Ef, 74.25.Jb
The pnictide high- Tcsuperconductors,1with maximal Tc’s
currently exceeding 55 K,2are the subject of global scrutiny
on a level at par with that seen for the cuprates. One of themost debated issues are their similarities and differences withrespect to the cuprates.
3The pnictides display many features
that we recognize from the cuprate repertoire, yet there arearguments that they are essentially different.
4In the last few
years, scanning tunneling microscopy and spectroscopy/H20849STM/STS /H20850have played a major role in investigating the
electronic structure of the cuprates on length scales down tothose of the atom.
5–9This effort has brought the role of in-
trinsic disorder introduced by dopant atoms into sharp focus.Consequently, BaFe
2−xCoxAs2is of great interest not only as
an electron-doped pnictide but also because the electroni-cally active dopants in this system are situated in the super-conducting layers themselves.
Single crystals of superconducting BaFe
1.86Co0.14As2and
the nonsuperconducting parent compound BaFe 2As2/H20849Ba122 /H20850
were grown using a self-flux method. Typical crystal sizesare 1/H110031/H110030.1 mm
3/H20851see Fig. 1/H20849b/H20850/H20852. The high quality of our
crystals is illustrated in Fig. 1/H20849a/H20850, with the parent compound
displaying the well-known resistivity kink at 130 K, whichhas been associated with a spin-density wave and accompa-nying orthorhombic phase transition.
10–13Co doping erases
any sign of these transitions in the resistivity but brings withit a very sharp transition into the superconducting state,which takes place at 22 /H110060.25 K in this case.
For the STM investigation, crystals were cleaved at a
pressure of 5 /H1100310
−10mbar at room temperature in a prepa-
ration chamber and were immediately transferred into theSTM chamber /H20849base pressure /H110211.5/H1100310
−11mbar /H20850, where
they were cooled to 4.2 K. The experiments were carried outusing electrochemically etched W and cut Pt/Ir tips, whichwere conditioned before each measurement on a Au /H20849778/H20850
single crystal and yielded identical results. In all cases, thespectral shapes obtained were independent of the tip tosample distance.
14
Low-energy electron diffraction /H20849LEED /H20850was performed
in situ , directly after the STM/STS measurements. For all
measured samples, only the tetragonal unit cell spots wereseen in LEED /H20851see Fig. 2/H20849a/H20850/H20852, with no sign of extra spots /H20849or
extinctions /H20850as would occur as a result of a significant andstructurally coherent reconstruction of the atomic positions at
the surface.
We begin our discussion with the Co-doped crystals. A
constant current image with atomic resolution is shown inFig.2/H20849a/H20850. In general, over areas of up to 150 /H11003150 nm
2,w e
saw no sign of steps on the surface, with the maximal appar-ent corrugation being less than 1.5 Å on all the data shownhere.
From the zooms shown in Figs. 2/H20849b/H20850and2/H20849c/H20850, one can
immediately see that the surface atoms lie arranged along thedirection of the clear /H208491/H110031/H20850tetragonal unit cell we measure
using LEED but that the inter-atomic spacing is quite irregu-lar. The most frequent separation seen is /H110118 Å, twice the
tetragonal unit cell. Occasionally a row of atoms with a sepa-ration of 3.9 Å occurs, as seen in panel /H20849c/H20850, and sometimes,
the/H20849bright /H20850atoms sit on a black background. The irregularity
in interatomic distances is reflected in the Fourier transformshown in Fig. 2/H20849d/H20850in which smeared-out features predomi-
nate, corresponding to a real-space separation of /H110118Å
/H20849marked with an ’X’ on the FFT /H20850. The tetragonal lattice is
barely visible in the form of weak spots, highlighted in Fig.2/H20849d/H20850with a yellow arrow.
Inspection of the crystal structure of Ba122 leads to the
supposition that cleavage occurs at the As-Ba interface. Inorder to avoid creation of a polar surface, the charge of the
0.16
0.12
0.08
0.04
0.00ρ(mΩcm)
30 20 10 0
T(K)0.6
0.4
0.2
0.0ρ(mΩcm)
3002001000
T( K )0.16
0.12
0.08
0.04
0.00ρ(mΩcm)
30 20 10 0
T(K)0.6
0.4
0.2
0.0
3002001000
T( K )
500/CID5maa b
FIG. 1. /H20849Color online /H20850/H20849a/H20850Co-doped Ba122 shows a sharp su-
perconducting transition /H20849/H9004Tc/H110110.5 K /H20850at 22 K /H20849red curves /H20850. The
inset displays the full temperature range, with data from the parentcompound /H20849top trace /H20850./H20849b/H20850Optical micrograph of the very flat cleav-
age surface typically obtained.PHYSICAL REVIEW B 79, 220517 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
1098-0121/2009/79 /H2084922/H20850/220517 /H208494/H20850 ©2009 The American Physical Society 220517-1termination layer of a system such as the pnictides should be
−1/2 of that of the layer beneath,4and this condition can be
met in Ba122 if exactly half of the Ba atoms remains on eachof the surfaces created by cleavage. For a room-temperaturecleave, the Ba ions may reorder to minimize their mutualCoulomb repulsion, resulting in interatomic distances largerthat the in-plane tetragonal lattice constant, as seen in Fig. 2.
Although the LEED patterns from all studied surfaces
after room-temperature cleavage show a nonreconstructed/H208491/H110031/H20850tetragonal pattern, on an STM length scale, the de-
tails of the topography vary from cleave to cleave and occa-sionally also for different locations on the same cleave. A
commonly encountered variation is shown in Fig. 3/H20849a/H20850from
a different Co-doped crystal of the same doping level inwhich the atomic contrast is absent, and a two-dimensional/H208492D/H20850, mazelike network is seen, oriented along the tetragonal
axes with typical period of /H1101112 Å. This image
resembles previously reported one-dimensional stripelikestructures,
15,16whereby our “stripes” appear cut into shorter
and more disordered segments, probably as a result of thehigher cleavage temperature. This is in keeping with a recentreport of a temperature dependence of the surface topologyin a related system.
17In Fig. 3/H20849b/H20850, we show an image from
pristine Ba122, which displays a very similar surface topol-ogy as in panel /H20849a/H20850, as can also be seen from the line scans
through both images shown in panel /H20849c/H20850.We now move on to the tunneling spectra. Differential
conductance spectra /H20849dI/dV/H20850of both the Co-doped and pris-
tine Ba122 systems were recorded /H20849modulation amplitude of
2mV at a lock-in frequency of 427.3 Hz /H20850on a square
64/H1100364 pixel grid at 4.2 K. The spectra for the Co-doped
system vary significantly between different locations withina single field of view /H20849FOV /H20850.
In Fig. 4/H20849a/H20850we plot representative STS spectra for differ-
ing gap values /H20849each spectrum is an average of four adjacent
pixels /H20850. The spectra show not only a clear variation in peak-
to-peak separation, hereafter taken as a measure of the super-conducting gap /H208492/H9004/H20850, but also in the value of the conduc-
tance at zero bias /H20849ZBC /H20850and in the strength of the coherence
peaks. They also vary in their asymmetry and in the form ofthe higher-energy structures. While these latter variations aremore pronounced than those reported in Ref. 16, they are
less dramatic than observed in a study of the relatedSr
1−xKxFe2As2compound.15Naturally, we are interested in
the real-space 2 /H9004distribution and thus plot the peak-to-peak
separation from all 4096 spectra as a gap map in Fig. 4/H20849b/H20850.
The map indicates that the major part of the FOV supports agap,/H9004, of 7 meV, with smaller patches of dimension between
5 and 10 Å possessing significantly smaller /H20849darker /H20850and
larger /H20849brighter /H20850gaps. From this FOV, only a handful of
spectra did not exhibit coherencelike peaks, and thus if thesepeaked structures give the superconducting gap, we can ex-clude phase separation between superconducting and nonsu-perconducting /H20849magnetic /H20850regions from these data.
From Figs. 4/H20849a/H20850and4/H20849b/H20850it is clear that BaFe
1.86Co0.14As2
supports 2 /H9004/kBTcvalues between 5.3 for the smaller gaps,
through 7.4 for the modal gap value /H208497 meV /H20850to 10.6 for the
largest gaps. Our modal gap is close to that seen recently inangle resolved photoemission spectroscopy /H20849ARPES /H20850from
optimally Co-doped Ba122 for the outermost /H9003centered /H20849/H9003
2
or/H9252/H20850Fermi surface18and is in keeping with data from an-
50 Å450Åa
d
1.2{1/nm}xdd
189 Åb
c
1.6
1.2
0.8
2520151050
distance (Å)
z(Å)0 3
intensity (arb. units )
FIG. 2. /H20849Color online /H20850/H20849a/H20850Constant current image /H20849Vsample =
−35 mV, Isetup=40 pA /H20850of Co-doped Ba122. The inset shows
LEED from the same surface with E0=114 eV. /H20849b/H20850Zoom of panel
/H20849a/H20850: the surface atoms can be seen as bright dots. /H20849c/H20850Further zoom
of/H20849b/H20850showing a row of four surface atoms separated by the tetrag-
onal cell dimension of 3.9 Å. /H20849d/H20850Fourier transform of panel /H20849a/H20850,
whereby the yellow X and arrow indicate real-space distances of 8and 3.9 Å, respectively. All data /H20849with exception of the LEED /H20850
were recorded at 4.2 K.
189 Å
189 Å
0.7
0.6
0.5a b
cc dd
1.0
0.8
0.6
60 40 20 0
distance (Å)corrugation ( Å)
30
25
20
15
10
dI/dV (arb. units )
-20 020
Vsample(mV)
FIG. 3. /H20849Color online /H20850/H20849a/H20850Constant current image from a differ-
ent sample of BaFe 1.86Co0.14As2/H20849Vsample =−88 mV, Isetup=40 pA /H20850.
/H20849b/H20850Analogous topography from BaFe 2As2 /H20849Vsample =
−100 mV, Isetup=33 pA /H20850./H20849c/H20850Line scans from the superconducting
/H20849top/H20850and parent /H20849bottom /H20850compounds taken along the arrows. Typi-
cal distances between the bright rows in the images is 12 Å indi-cated by gray ticks in the line scans. /H20849d/H20850Overlay of 400 dI/dV
conductance spectra from pristine Ba122, together with their aver-age /H20849bold line /H20850. All data were recorded at 4.2 K.MASSEE et al. PHYSICAL REVIEW B 79, 220517 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
220517-2other STS experiment on Co-doped Ba122.16We also note
that our normalized gap values are in line with ARPES datafrom the outer hole pocket Fermi surface in hole-doped/H20849Ba,K /H20850-122 system.
19At present it is unclear whether the
STS tunneling matrix elements favor the hole or electronpocket Fermi surfaces, but in any case, all of the normalizedgap values we observe are greater than those expected for a
weakly coupled BCS s-o rd-wave superconductor.
Closer inspection of Fig. 4/H20849a/H20850reveals a relation among the
ZBC, 2 /H9004, and the coherence peak strength: when 2 /H9004is small
/H20849large /H20850, the ZBC is large /H20849small /H20850, and the coherence peaks are
strong /H20849weak /H20850. To see whether these relations hold for the
whole data set, we conduct three qualitative tests. First, weplot in Fig. 4/H20849c/H20850the results of binning all the STS spectra into
five groups, depending on their peak-to-peak separation.
20Second, we show in Fig. 4/H20849d/H20850a map of the ZBC value, with
an inverted color scale to ease comparison with panel /H20849a/H20850.
Third, we map the spatial distribution of the coherence peakstrength for this sample /H20851Fig.4/H20849f/H20850/H20852and note that this gives a
nearly identical image to the ZBC map in panel /H20849d/H20850. The fact
that the inverse relation between 2 /H9004and ZBC /H20849and between
2/H9004and the coherence peak strength /H20850is also clearly visible in
the binned /H20849i.e., highly averaged /H20850STS spectra and the fact
that the gap, ZBC and peak strength maps resemble eachother closely /H20849providing the color scale in the latter two maps
is inverted /H20850indeed indicate that the relation among 2 /H9004, ZBC,
and peak strength holds for the data set as a whole and notjust for the STS spectra shown in Fig. 4/H20849a/H20850.
In a next step, we formalize the relations between the
2/H9004⇔ZBC and peak-strength maps by comparing their
azimuthal-integrated cross correlation traces. Figure 4/H20849e/H20850
shows the results, indicating a clear anticorrelation that diesoff over 8–10 Å. The question arises as to the origin of the
observed spatial disorder in the 2 /H9004values. As a first suspect,
we consider the nontrivial surface topography illustrated inFigs. 2and3. Here, pristine Ba122 offers the best test of this
point as it possesses the same complex Ba termination topog-raphy and yet is void of substitutional disorder within theFeAs structural unit. Figure 3/H20849d/H20850shows a compilation of one-
tenth of a complete STS data set /H20849400 of 4096 STS spectra /H20850
and their average—taken from pristine Ba122 across thesame FOV as the topograph shown in Fig. 3/H20849b/H20850. The near- E
F
electronic states in the parent compound are fairly spatially
homogeneous in terms of both the shape of the spectra andthe absolute dI/dVvalue.
To compare the variation in absolute conductance with
that seen in the superconductor, we show a ZBC map frompristine Ba122 in Fig. 4/H20849g/H20850, using the same color scale as Fig.
4/H20849d/H20850, after correction for the difference in junction resistance.
Although the two ZBC maps are from different cleaves ofdiffering materials, it is still obvious that the pristine materialhas much less variation and does not possess structure on thelength scales of the superconductor. Thus, the gap disorderseen in Fig. 4is a property of the superconducting system,
and the finger is quickly pointed in the direction of the Codopants as the culprit. For our doping level, 1 in 14 Fe atomsis replaced by a Co dopant, and at a zeroth level this gives aCo-Co separation of a little over 10 Å. The first length scaleover which Co disorder would be visible is therefore 10 Å,as observed here. In Fig. 4/H20849e/H20850we show autocorrelation traces
for the gap map in Fig. 4/H20849b/H20850and the analogous trace for a gap
map recorded from an identical FOV as Fig. 3/H20849a/H20850, i.e., a
superconductor cleave with a clearly different topographyyet similar STS spectra. Both these /H20849positive /H20850correlation
traces show that 8 Å is the characteristic length scale of thesignificant gap variations, which is very close to the Co-Colength scale. The fact that the atomically resolved cleaves/H20851Fig. 2/H20849a/H20850/H20852and those with the 2D mazelike structure /H20849Fig.
3/H20849a/H20850/H20850both give the same length scales for the deviations from
2/H9004is a further indication that in these cases the topographic
details—which most likely track the particulars of the sur-face Ba /H20849dis/H20850order—do not have much direct effect on the
superconducting system. We note that—albeit on a some-what coarser energy scale—similar conclusions have beendrawn from a recent photoemission study of the Ba122
c>9 8 7 6
189 Åd40
30
20
10dI/dV (arb. units)-20 0 20Vsample(mV)corr. coeff.0.4
0.0
-0.4
20151050
189 Åe40
30
20
10dI/dV (arb. units)a
30
25
20
15
10
5
distance (Å)g f
189 Åb∆(meV)
<5
30
25
20
15
10
5
FIG. 4. /H20849Color online /H20850/H20849a/H20850STS spectra from BaFe 1.86Co0.14As2
/H20849averages of four adjacent pixels /H20850displaying different values of /H9004,
taken from the region marked in panel /H20849b/H20850./H20849b/H20850Gap map with iden-
tical FOV as Fig. 2/H20849b/H20850, taken with the same setup conditions. /H20849c/H20850
Binned spectra for five ranges of /H9004/H20849Ref. 20/H20850, plotted in colors
matching those in the gap map. /H20849d/H20850Map of the zero-bias conduc-
tance on an inverted color scale from the same data set as panel /H20849b/H20850.
/H20849e/H20850Correlation functions. From top to bottom: autocorrelation of the
gap map shown in panel /H20849b/H20850/H20849red squares /H20850and of an analogous data
set recorded from the FOV shown in Fig. 3/H20849a/H20850/H20849black triangles /H20850.
Yellow dots: cross correlation of the gap map of panel /H20849b/H20850with the
corresponding topograph /H20851Fig.2/H20849b/H20850/H20852. Blue /H20849green /H20850diamonds: cross
correlations between the gap and ZBC /H20849peak-strength /H20850maps. /H20849f/H20850
Map of the coherence peak strength /H20851for half of the FOV of panels
/H20849b/H20850and /H20849d/H20850, and on an inverted color scale /H20852, extracted from a com-
parison of the coherence peak weight to the high energy back-
ground. /H20849g/H20850ZBC map of pristine Ba122 recorded from half the FOV
shown in Fig. 3/H20849b/H20850. The conductance scale /H20849shown on the right /H20850is
the same as that used in panel /H20849d/H20850.NANOSCALE SUPERCONDUCTING-GAP VARIATIONS … PHYSICAL REVIEW B 79, 220517 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
220517-3system.21Finally, we remark that the cross correlation traces
between the gap map and the topography are zero for bothtypes of topography /H20851see Fig. 4/H20849e/H20850/H20852.
How do the spatial gap variations found here in an elec-
tron doped pnictide compare with those from STS studies ofthe cuprate high T
c’s at analogous doping levels? Optimally
doped Bi2212 yields a similar total spread of a factor of 2 innormalized gap values but upshifted with respect to thosehere to lie between 6 and 13 k
BTc.7Recently, the emphasis
has come to lie on the role played by the pseudogap in theobserved large apparent superconducting gap disorder seenin the cuprates.
22In the pnictide STS data presented here, it
would be natural to take the modal gap value as that repre-senting areas with Co doping occurring in the FeAs plane atthe nominal level. Consequently, the small gaps could eitherbe under- or overdoped regions formed due to clustering ofCo since both would—in principle—lead to a lower T
c/H20849and
presumably /H9004/H20850. This would leave only the larger peak-to-
peak separations unaccounted for. To decide whether, as inthe cuprates, these large-gap regions are related to the pres-ence of a pseudogap or not will require detailed temperature-dependent measurements.
In summary, we present detailed STM and STS investiga-
tions of pristine Ba122 and samples of the electron dopedsuperconductor BaFe
1.86Co0.14As2. In the first part of the
Rapid Communication we describe the complex topographyof the surfaces of these single crystals, which is probably a
result of partial liftoff of the Ba ions upon cleavage. We goon to demonstrate that the termination-plane topographic dis-order encountered here has little effect on the low-lying elec-tronic states of these systems.
The STS data from the superconducting samples display
clear coherence-peak-like features defining an energy gap ofon average 7.4 k
BTc. There exist, however, significant spatial
deviations from this modal gap value, with the gap distribu-tion ranging from 5–10 k
BTc. If these gaps are indeed super-
conducting gaps, we can clearly rule out nanoscopic phaseseparation in these samples. There is a robust anticorrelationbetween the peak-to-peak separation and the zero-bias con-ductance, and coherence peak strength, operating over lengthscales of /H110118 Å. The spatial correlation of the low and high
gap deviations from the modal gap value also displays thesame length scale, one which is very close to the averageseparation of the Co atoms in the FeAs superconductingblocks, highlighting their importance as local dopants.
We thank H. Luigjes, H. Schlatter, and J. S. Agema for
valuable technical support, and J. van den Brink and A. deVisser for useful discussions. This work is part of the re-search program of FOM, which is financially supported bythe NWO.
*f.massee@uva.nl
1Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.
Chem. Soc. 130, 3296 /H208492008 /H20850.
2Z.-A. Ren, G.-C. Che, X.-L. Dong, J. Yang, W. Lu, W. Yi, X.-L.
Shen, Z.-C. Li, L.-L. Sun, F. Zhou, and Z.-X. Zhao, Chin. Phys.Lett. 25, 2215 /H208492008 /H20850.
3M. R. Norman, Physics 1,2 1 /H208492008 /H20850.
4G. Sawatzky, I. Elfimov, J. van den Brink, and J. Zaanen,
arXiv:0808.1390 /H20849unpublished /H20850.
5Ø. Fischer, M. Kugler, I. Maggio-Aprile, and C. Berthod, Rev.
Mod. Phys. 79, 353 /H208492007 /H20850.
6S. H. Pan, E. W. Hudson, K. M. Lang, H. Eisaki, S. Uchida, and
J. C. Davis, Nature /H20849London /H20850403, 746 /H208492000 /H20850.
7K. McElroy, R. W. Simmonds, J. E. Hoffman, D.-H. Lee, J.
Orenstein, H. Eisaki, S. Uchida, and J. C. Davis, Nature /H20849Lon-
don/H20850422, 592 /H208492003 /H20850.
8J. E. Hoffman, K. McElroy, D.-H. Lee, K. M. Lang, H. Eisaki, S.
Uchida, and J. C. Davis, Science 297, 1148 /H208492002 /H20850.
9K. K. Gomes, A. N. Pasupathy, A. Pushp, S. Ono, Y. Ando, and
A. Yazdani, Nature /H20849London /H20850447, 569 /H208492007 /H20850.
10M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes,
and R. Pöttgen, Phys. Rev. B 78, 020503 /H20849R/H20850/H208492008 /H20850.
11J. Dong, H. J. Zhang, G. Xu, Z. Li, G. Li, W. Z. Hu, D. Wu, G.
F. Chen, X. Dai, J. L. Luo, Z. Fang, and N. L. Wang, EPL 83,
27006 /H208492008 /H20850.
12T. Nomura, S. W. Kim, Y. Kamihara, M. Hirano, P. V. Sushko, K.
Kato, M. Takata, A. L. Shluger, and H. Hosono, Supercond. Sci.Technol. 21, 125028 /H208492008 /H20850.
13G. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong, P. Zheng, J.
L. Luo, and N. L. Wang, Phys. Rev. Lett. 100, 247002 /H208492008 /H20850.
14C. Renner and Ø. Fischer, Phys. Rev. B 51, 9208 /H208491995 /H20850.
15M. Boyer, K. Chatterjee, W. Wise, G. Chen, J. Luo, N. Wang,
and E. Hudson, arXiv:0806.4400 /H20849unpublished /H20850.
16Y. Yin, M. Zech, T. L. Williams, X. F. Wang, G. Wu, X. H. Chen,
and J. E. Hoffman, Phys. Rev. Lett. 102, 097002 /H208492009 /H20850.
17D. Hsieh, Y. Xia, L. Wray, D. Qian, K. Gomes, A. Yazdani, G.
Chen, J. Luo, N. Wang, and M. Hasan, arXiv:0812.2289 /H20849unpub-
lished /H20850.
18K. Terashima, Y. Sekiba, J. H. Bowen, K. Nakayama, T. Kawa-
hara, T. Sato, P. Richard, Y.-M. Xu, L. J. Li, G. H. Cao, Z.-A.Xu, H. Ding, and T. Takahashi, Proc. Natl. Acad. Sci. U.S.A.
106, 7330 /H208492009 /H20850.
19L. Wray, D. Qian, D. Hsieh, Y. Xia, L. Li, J. Checkelsky, A.
Pasupathy, K. Gomes, A. Fedorov, G. Chen, J. Luo, A. Yazdani,N. Ong, N. Wang, and M. Hasan, Phys. Rev. B 78, 184508
/H208492008 /H20850.
20/H9004values have been binned as follows: 0–5.5, 5.5–6.5, 6.5–7.5,
7.5–8.5, 8.5- /H9004maxmeV.
21S. de Jong, Y. Huang, R. Huisman, F. Massee, S. Thirupathaiah,
M. Gorgoi, F. Schaefers, R. Follath, J. B. Goedkoop, and M. S.Golden, Phys. Rev. B 79, 115125 /H208492009 /H20850.
22M. C. Boyer, W. D. Wise, K. Chatterjee, M. Yi, T. Kondo, T.
Takeuchi, H. Ikuta, and E. W. Hudson, Nat. Phys. 3, 802 /H208492007 /H20850.MASSEE et al. PHYSICAL REVIEW B 79, 220517 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS
220517-4 |
PhysRevB.69.214301.pdf | Spectral response of crystalline acetanilide and N-methylacetamide: Vibrational self-trapping
in hydrogen-bonded crystals
Julian Edler *and Peter Hamm†
Universität Zürich, Physikalisch Chemisches Institut, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
(Received 12 February 2004; published 28 June 2004 )
Femtosecond pump-probe and Fourier transform infrared spectroscopy is applied to compare the spectral
response of the amide I band and the NH-stretching band of acetanilide (ACN )andN-methylacetamide
(NMA ), as well as their deuterated derivatives. Both molecules form hydrogen-bonded molecular crystals that
are regarded to be model systems for polypeptides and proteins. The amide I bands of both ACN and NMAshow a temperature-dependent sideband, while the NH bands are accompanied by a sequence of equidistantlyspaced satellite peaks. These spectral anomalies are interpreted as a signature of vibrational self-trapping. Twodifferent types of states can be identified in both crystals in the pump-probe signal: a delocalized free-excitonstate and a set of localized self-trapped states. The phonons that mediate self-trapping in ACN and deuteratedACN are identified by their temperature dependence, confirming our previous results. The study shows that thesubstructure of the NH band in NMA (amideAand amide Bbands )originates, at least partly, from vibrational
self-trapping and not, as often assumed, from a Fermi resonance.
DOI: 10.1103/PhysRevB.69.214301 PACS number (s): 63.20.Pw, 63.20.Ry
I. INTRODUCTION
The ahelix is one of the most common secondary struc-
ture motifs in proteins. The three-dimensional (3D)configu-
ration of the ahelix is stabilized by three quasi-one-
dimensional chains of hydrogen bonds, which run throughthe helix and connect the C vO and N-H groups of the
peptide units. It had been proposed many years ago that vi-brational excitations of the amide groups in such a systemcould self-localize and play an important role in the energytransport of proteins.
1
Self-localization or self-trapping of vibrational excitations
is described by the polaron Hamiltonian, which combinestwo different coupling mechanisms:
2,3(a)exciton coupling
and(b)phonon coupling. Exciton coupling leads to delocal-
ization of a vibrational excitation, caused by electrostatic in-teraction between the individual peptide units.
4,5The vibra-
tional excitation can either be the N-H or the C vO
stretching vibration of the peptide unit (the latter is often
referred to as the amide I mode ). Analogous to electronic
molecular excitons, the delocalized states are also called vi-brational excitons or vibrons.
The vibrational excitons, in turn, are coupled to lattice
phonons through an anharmonic (nonlinear )coupling term,
which is mediated through the hydrogen bonds that stabilizethe three-dimensional (3D)structure of the system. The
exciton-phonon coupling has been rationalized as follows.
6It
is well known that the length of the hydrogen bond modu-lates the transition frequency of either the amide I or NHvibration mode of the peptide unit.
7This interaction is re-
sponsible for a contraction of the hydrogen bond once one ofthe amide modes is vibrationally excited. As a consequence,the initially delocalized vibrational exciton collapses to forma self-localized state.
Hydrogen-bonded crystals, such as acetanilide
(CH
3-CONH-C 6H5, ACN )and N-methylacetamide
(CH3-CONH-CH 3, NMA ), are commonly regarded as modelsystems for regular secondary protein structures. It had been
recognized some time ago that the infrared absorption spec-tra of ACN exhibits anomalies in the regions of the amide Iand NH-stretching modes.
8,9The amide I mode consists of a
single peak at 1666 cm−1at room temperature. With decreas-
ing temperatures, an additional peak appears at 1650 cm−1.
The NH mode, on the other hand, is weakly temperaturedependent and consists of a main peak at 3295 cm
−1, accom-
panied by an almost regular series of satellite peaks towardslower frequencies. The origin of the unconventional amide Iband has been the topic of many theoretical and experimentalstudies, including infrared spectroscopy, Raman scattering,x-ray scattering, and neutron scattering.
3,6,8,10–19After care-
fully excluding more conventional explanations, Careri andco-workers have proposed that the spectral anomaly is re-lated to vibrational self-trapping.
18
In theoretical discussions of vibrational self-trapping, the
exciton coupling is often neglected in a first step, since it issmaller than the exciton-phonon coupling, and included onlyafterwards in a perturbative manner.
3,6The problem simpli-
fies significantly when one further takes into account the oneorder of magnitude difference between the frequencies of thephonon modes and the high frequency vibrational mode, al-lowing an adiabatic “Born-Oppenheimer”-like separation oftime scales. In this approximation, the high frequency vibra-tion adopts adiabatically to the position of the phonon coor-dinates. As a result, the potential energy surface for eachexcitation level of high frequency mode is expressed as afunction of phonon coordinates, as shown in Fig. 1. Thenonlinear exciton-phonon coupling gives rise to a displace-ment of the potential energy surface of the self-trapped state.The minimum of that potential energy surface is shifted to-wards smaller intermolecular distances, giving rise to astronger hydrogen bond in the excited sate. Consequently, a“Franck-Condon-like” progression is obtained for the ab-sorption spectrum, consisting of one vibrational excitationplus several quanta of phonon excitation.PHYSICAL REVIEW B 69, 214301 (2004 )
0163-1829/2004/69 (21)/214301 (8)/$22.50 ©2004 The American Physical Society 69214301-1When the displacement of ground- and excited-state po-
tential energy surface is small (e.g., for the C vO band in
ACN ), only the zero-phonon transition (i.e., the 1650 cm−1
band )carries a noticeable oscillator strength at zero tempera-
ture and all other transitions are weak. With increasing tem-peratures, phonons in the C vO ground state are thermally
excited, and the intensity of the zero-phonon line diminishes.The intensity is expected to decrease with temperature ac-
cording to a e
−gT2law, in perfect agreement with the experi-
mentally observed dependence of the 1650 cm−1band in
ACN.6,18,20This agreement is presently considered to be the
strongest evidence for self-trapping of the amide I band ofACN. In the case of the N-H mode, exciton-phonon couplingis significantly larger and the binding energy E
BDamounts to
several phonon quanta. Therefore, one can observe a pro-gression of self-trapped states in the NH band (see Fig. 1 ).
Recently, we performed femtosecond infrared pump-
probe experiments, providing strong evidence for the self-trapping theory.
21–23A study of the NH band revealed the
phonons that couple to the NH mode and mediateself-trapping.
21Moreover, we showed that the two amide I
substates ofACN exhibit a distinctively different response onselective excitations, a result which has been interpreted interms of the degree of delocalization of both states.
22In ad-
dition, we were able to definitely exclude the possibility of aFermi resonance and conformational substates as alternativeexplanations for the temperature dependence of the amide Imode,
23confirming previous studies.12,24
Compelling evidence for vibrational self-trapping has so
far only been gathered for crystalline ACN. However, themechanism is expected to be generic, and should occur inany crystal with comparable structure. In particular, one ex-
pects to observe it in NMA, which is often regarded to be the
model compound for peptides and proteins. Both crystals,ACN and NMA, have an orthorhombic structure and consistof quasi-one-dimensional chains of hydrogen-bonded peptideunits (-CO-NH- )with structural properties that are similar to
those of
ahelices.25,26Nevertheless, no convincing experi-
mental evidence for self-trapping in NMAhas been found sofar. In the present study, we follow the strategy to transfer theknowledge we have gathered forACN, for which vibrationalself-trapping is clear and by now, well established, to NMA.To this end we compare the infrared absorption spectra andthe pump-probe spectra of the amide I and NH mode of fourdifferent hydrogen-bonded crystals: ACN, deuterated ACN(CD
3-CONH-C 6D5, ACN-D 8), NMA, and deuterated NMA
(CD3-CONH-CD 3, NMA-D 6). We will see that certain non-
linear spectroscopic fingerprints appear in comparable waysfor both molecules, from which we conclude that the mecha-nism giving rise to the complex band shape of the NH andCO mode of both materials has the same origin.
II. MATERIALS AND METHODS
Femtosecond IR pump-probe experiments were per-
formed with pulses with a bandwidth of 200 cm−1full width
at half maximum (FWHM ), a pulse energy of 1–1.5 mJ, and
a pulse duration of 150 fs at a 1 kHz repetition rate.27A
small fraction of the infrared pulses was split off to obtainbroadband probe and reference pulses, which were spectrallydispersed after interaction with the sample and detected witha 31-channel HgCdTe detector array (8c m
−1resolution ).The
main part of the infrared pulses was used as pump pulses thatwere spectrally filtered for some of the experiments using anadjustable Fabry-Perot filter (bandwidth of 30 cm
−1), yield-
ing a pump-pulse duration of 250 fs (FHWM ). Infrared ab-
sorption spectra were measured on a BioRad FTS 175Cspectrometer equipped with a highly sensitive mercury cad-mium telluride (MCT )detector.
ACN (zone refined, purity 99.95% )and NMA (purity
99+% )were obtained from Aldrich, ACN-D
8s98%dfrom
Cambridge Isotope Laboratories, and NMA-D 6s98.3% d
from C/D/N isotopes. Monocrystalline ACN, NMA, and
NMA-D 6were prepared by cooling a thin layer of the molten
substance between two CaF 2windows. In the case of NMA,
the sample was heated in a dry box to about 80°C before-hand in order to remove any possible water contaminationsof the highly hygroscopic substance. MonocrystallineACN-D
8was grown out of a concentrated solution of
ethanol/chloroform s20/80 don a CaF 2window. For the ex-
periments we carefully selected samples that showed only
one crystalline orientation. The spectra were obtained withtheEvector parallel and perpendicular to the hydrogen bond
chain (in ACN the bandcaxes, and in NMA the aandc
axes). The comparison between spectra obtained with differ-
ent polarizations confirmed that our samples were in amonocrystalline phase. In the cases of ACN and NMA-D
6
the crystals in the sample were perfectly aligned in one di-rection, while the ACN-D
8and NMA samples showed some
slight disorder. The samples were placed in a cryostat and
FIG. 1. (a)Absorption spectrum of the CO band of crystalline
ACN, showing a temperature-dependent doublet. The peak at1650 cm
−1corresponds to a self-trapped state and the one at
1666 cm−1to a free exciton. (b)Absorption spectrum of the NH
band, consisting of a main peak at 3295 cm−1(free exciton )and a
sequence of satellite peaks (self-trapped states ). The proposed
schemes of potential energy surfaces are depicted on the right-handside.E
BDrepresents the binding energy of the self-trapped state.JULIAN EDLER AND PETER HAMM PHYSICAL REVIEW B 69, 214301 (2004 )
214301-2experiments were performed at temperatures between 18 and
293 K. All spectra of NMA were measured at temperaturesbelow the solid phase transition, which occurs between 274and 283 K.
26,28
III. EXPERIMENTAL RESULTS
A. Absorption spectra of the NH and the C BO bands
Figures 2 (a)–2(h)show the absorption spectra in the spec-
tral range of the NH mode of crystalline ACN, ACN-D 8,
NMA, and NMA-D 6at high (250 and 220 K )and low tem-
peratures s<30 K d. Since the spectra are congested we will
first discuss the contributions from other bands than the NH
stretching band. We can identify CH stretching modes bycomparing the C deuterated with the nondeuterated com-pounds. Figure 2 shows that the CH modes appear around3000 cm
−1(ACN 3005, 3042, 3058, and 3118 cm−1; NMA
2995 and 2993 cm−1).Another peak, which is not part of the
NH band, is labeled with “A” in theACN spectrum [see Fig.
2(e)]and has been assigned previously to an overtone of the
amide I mode.10All spectra show a strong polarization de-
pendence and most bands are only observed when the E
vector is oriented parallel to the NH groups.9,29Only the B
andB8bands in the spectra of ACN and ACN-D 8(ACN
2860 and 2925 cm−1; ACN-D 82860 and 2913 cm−1)show
hardly any polarization dependence, and hence are attributedto overtone/combination modes of unknown origin.
If one combines the above observations and takes only the
peaks that can be assigned to the NH band into account, onerealizes that all four molecules exhibit an NH band whichconsists of a main band accompanied by a series of almostequidistant satellite peaks towards lower frequencies. Thespacing between the four satellite peaks is <60 cm
−1for
ACN and <70 cm−1for ACN-D 8. In NMA, the separation
between the satellite peaks is about three times larger. Thesatellite peaks in NMA and NMA-D 6further split into dou-
blets at low temperatures [Figs. 2 (g)and 2 (h)]. This splitting
has been observed before and is of unclear origin.30–32In
ACN the NH main peak at 3295 cm−1is observed both in the
parallel and perpendicular measurements, however, with arelative shift of 11 cm
−1[Fig. 2 (a)]. The shift corresponds to
the “Davydov” splitting and shows that the main peak inACN is a delocalized state, i.e., a free-exciton state.
9
Previous works have started from the assumption that the
NH spectrum ofACN contains nine satellite peaks.6,14,18,20,21
However, given the polarization dependence and the com-parison withACN-D
8, one must conclude that only the high-
est four satellite peaks are “real” (i.e., belong to the NH
band ), while the lower frequency part of the spectrum is
perturbed by CH vibrations and weak overtones and/or com-bination modes.
Figures 3 (a)and 3 (b)show the absorption spectra of the
amide I mode of ACN and NMA at different temperatures.As described before, an “anomalous” band appears at1650 cm
−1in ACN with decreasing temperatures. The “nor-
FIG. 2. IR spectrum of the NH
band of ACN, ACN-D 8, NMA,
and NMA-D 6at high temperatures
(a)–(d)and at low temperatures
(e)–(h). All four crystals show
similar band shapes, consisting ofa main peak and a sequence ofthree to four satellite peaks. TheCH modes, the amide I overtone(A)and theBmodes are not part
of the NH mode. The thick linesare for parallel polarization of theIR light with respect to thehydrogen-bonded chain and thethin lines for perpendicular polar-ization. For NMA-D
6the perpen-
dicular spectra are magnified by afactor of 10, for ACN by a factorof 4. The self-trapped states aremarked by black bars and the free-exciton peak by FE.
FIG. 3. IR spectrum of the amide I band of crystalline (a)ACN
and (b)NMA at three different temperatures. The vertical line
marks the position of the temperature-dependent sideband. The IRlight was polarized parallel to the hydrogen-bonded chain.SPECTRAL RESPONSE OF CRYSTALLINE PHYSICAL REVIEW B 69, 214301 (2004 )
214301-3mal” amide I band, on the other hand, splits into three sub-
bands and decreases in intensity at low temperatures.18,33In-
terestingly, we also observe a temperature-dependentsideband in the amide I spectrum of NMA at 1634 cm
−1,
whose separation from the main band, however, is aboutthree times smaller than in ACN.
B. Impulsive excitation of the NH band
Crystalline ACN . In this section we use broadband pump
pulses (bandwidth 200 cm−1)to impulsively excite a section
of the absorption spectrum, which includes the NH mainpeak and the first three satellite peaks. As a representativeexample, Fig. 4 shows the transient response for probe fre-quencies of 3200 cm
−1. The striking feature is strong oscil-
lations, which at 77 K persist up to 14 ps. With increasingtemperature, the oscillations decay faster and their amplitudedecreases. A spectral analysis of the beating structure is ob-tained from the Fourier transformation
DAs
v,vprd=E
0‘
dtpu,preiwtpu,prDAstpu,pr,vprd, s1d
where DAstpu,pr,vprdis the pump-probe signal (i.e., the tran-
sient response change plotted in Fig. 4 )for delay times tpu,prbetween pump and probe pulse at probe frequency vpr. The
absolute value spectrum uDAsv,vprduis shown in Figs. 5 (d)
and 5 (e)as a two-dimensional (2D)plot for 293 and 77 K. In
this representation, the xaxis corresponds to the probe fre-
quency vprand theyaxis to the frequency v, which is re-
vealed by the Fourier transformation. The 2D plot at 77 Kshows clearly two major frequency components at 54 and83 cm
−1. The oscillations reach their maxima at probe fre-
quencies which correspond to the peaks of the NH absorp-tion band. Vertical cuts through the spectra at the position ofthe satellite peaks are shown in Figs. 5 (c)and 5 (f). A com-
parison between low and high temperature data revealsclearly that the frequencies of the two major oscillations de-crease to 48 and 76 cm
−1at 293 K. In Fig. 6, the frequencies
of the two major modes are plotted against temperature andcompared with the frequencies of the phonon modes in thelow frequency Raman spectrum (the latter taken from
Johnston et al.
34).
The impulsive excitation with an ultrafast laser pulse may
result in the creation of vibrational wave packets, which giverise to oscillations in the pump-probe signal. Principallyspeaking, two different generation mechanisms of such wavepackets are possible: (i)impulsive absorption, yielding a
wave packet in the excited state, in which case the beatingfrequency should represent the energy separation between(coupled )states in the NH manifold of states, and (ii)a
Raman-like process resulting in a ground-state wave packet,giving rise to phonon excitation. Impulsive absorption andRaman-like excitation are expected to occur simultaneouslywith probabilities that are given by the Franck-Condon fac-tors of the individual transitions.
In Ref. 21, we have used two observations as an argument
to assign the beatings in the pump-probe signal of ACN to aground-state wave packet: (i)The lifetime of the excited
state [400 fs, see Fig. 7 (a)]is too short to support an excited
state wave packet that persists for many picoseconds. Thisdiscrepancy becomes even more pronounced at low tempera-tures, where the coherence decay time increases significantly,while the excited state lifetime stays about the same [600 fs,
FIG. 4. The transient response of ACN after impulsive excita-
tion for a probe frequency s3200 cm−1dwhich coincides with one of
the self-trapped states. The temperature is varied from 77 to 293 K.
FIG.5. (a)and(b)AbsorptionspectraofcrystallineACN,showingtheNHbandfor293and77 K. (d)and(e)The2Dplotoftheabsolute
value of the Fourier transformation of the transient pump-probe signal after impulsive excitation. vpris the frequency of the probe pulse and
vresults from the Fourier transformation of the pump-probe signal with respect to delay time tpu,prbetween pump and probe pulses. (c)and
(f)Vertical cuts through the 2D spectrum at frequencies corresponding to the satellite peaks.JULIAN EDLER AND PETER HAMM PHYSICAL REVIEW B 69, 214301 (2004 )
214301-4see Fig. 7 (a)].(ii)At room temperature, the frequency of the
quantum beats does not fit the frequency spacings in theabsorption spectrum. At lower temperatures, however, thebeating frequency increases slightly and the absorption spec-trum becomes more structured, so that the 54 cm
−1beating
frequency in fact matches the frequency spacing between the3198 (the second satellite peak )and the 3252 cm
−1bands
(the first overtone of the amide I )at 77 K.
Nevertheless, our previous interpretation of a ground-state
wave packet is validated by the temperature dependence ofthe beating frequency of the two major components (Fig. 6 ).
In the case of a ground-state wave packet, the temperaturedependence of the beating frequencies should match that ofcertain Raman modes. The low-frequency Raman spectrumof ACN has been the subject of several studies
18,34–36andcontains a total of 24 Raman active modes. In Fig. 6, we
compare the temperature dependence of the beating fre-quency with that of the Raman-active phonons of the crystaland find an excellent agreement. Furthermore, the linewidthof the phonon modes narrows,
34in agreement with the Fou-
rier transform spectra in Figs. 5 (c)and 5 (f). Hence, we con-
clude that the beatings in the pump-probe signal reflect acoherent excitation of two distinct phonons in the NH groundstate.
Interestingly, the pump-probe experiment selectively ex-
cites just 2 out of 24 Raman-active phonons. In the presentexperiment, the phonons are excited by a stimulated impul-sive Raman process, which is resonantly enhanced by theNH absorption band. Such a process is only possible if therespective Raman modes are coupled to the NH excitation.In the representation of Fig. 1, such a coupling gives rise toa shift of the potential energy surfaces, i.e., it renders thephonon to be “Franck-Condon” active. In other words, thecoupling observed in the pump-probe experiment is preciselythe same as the one that is responsible for polaron formation.Combining these arguments, we conclude that the phononswe observe in the pump-probe experiment are indeed thosephonons that mediate self-trapping.
Crystalline ACN-D
8. The transient signal in ACN-D 8
shows also a striking beating pattern (Fig. 8 )similar toACN.
The Fourier transform spectrum contains clearly two fre-quency components: a sharp peak at 47 cm
−1and a broad
peak at about 80 cm−1. Since the molecular weight and crys-
tal structure ofACN-D 8are comparable toACN, one indeed
expects to excite the same phonons with similar frequencies.The comparison with the splittings in the absorption spec-trum rules out an excited-state wave packet. Raman spectraof fully deuterated ACN show that the spectra below60 cm
−1do not change much upon deuteration and contain a
distinct mode at about 50 cm−1and a broad band between 60
and 100 cm−1.37Consequently, the oscillations are assigned
to a ground-state wave packet, using the same arguments asin the previous paragraph. This is direct proof that the NHmode in ACN-D
8is coupled to low frequency phonons and
indicates that the multiplicity of the NH band in ACN-D 8is
caused by self-trapping, just as in ACN.
FIG. 6. Temperature dependence of the low-frequency Raman
modes s+din ACN, taken from Johnston et al. (Ref. 34 ), and of the
beating frequencies observed in the pump-probe experiment s•d.
The temperature dependence of the beating frequencies perfectlymatches that of the Raman modes.
FIG. 7. The transient signal of ACN and NMA after selective
excitation of the free-exciton peak. In the case of ACN the signalrecovers on a 400 fs time scale at 293 K and on a 600 fs time scaleat 77 K. In NMA the signal relaxes on a 1 ps time scale.
FIG. 8. (a)The transient signal of ACN-D 8after impulsive ex-
citation for a probe frequency which coincides with the free exci-ton, showing pronounced oscillations. (b)The Fourier-transform
spectra show a sharp band at 47 cm
−1and a broadband at 80 cm−1.SPECTRAL RESPONSE OF CRYSTALLINE PHYSICAL REVIEW B 69, 214301 (2004 )
214301-5In the case of NMA and NMA-D 6, we were unable to
observe beatings in the pump-probe signal. Since the split-ting between the satellite states is about three times larger inNMAthan inACN [Figs. 2 (c)and 2 (d)], one expects that the
frequency of the corresponding phonon modes is larger byabout the same factor. Part of this difference might be ex-plained by the smaller mass of NMA. Such a high frequencymode would be at the detection limit of our setup, which isdetermined by the IR pulses duration. Technical improve-ments, i.e., pulse compression techniques, might make it pos-sible to uncover these modes in the future. It could be alsopossible that the lifetime of the phonons in NMAare signifi-cantly shorter than in ACN, as suggested by the broader Ra-man lines,
32and that they are thus much harder to detect.
C. Frequency-selective excitation of the NH band
Crystalline ACN . In this section we selectively excite the
main peak and the satellite peaks in the NH spectrum ofACN and NMA using narrow-band, tunable pump pulses(see Fig. 9 ). For ACN we have discussed this response in
detail in a previous publication.
21In brief, the response of the
main peak, which is, in the case of ACN, the peak with thestrongest intensity and the highest frequency, differs distinc-tively from that of the satellite peaks. Immediately afterpumping the main peak [Fig. 9 (d)], a strong bleach and
stimulated emission signal is observed at the position of thatpeak. The signal recovers on a 400 fs±100 fs time scale,representing the lifetime of this state [see Fig. 7 (a)]. Simul-
taneously, the population is transferred into lower-lyingstates, giving rise to negative signals at the position of thesatellite bands. On the other hand, when exciting one of thesatellite states directly [Fig. 9 (g)], we observe negative sig-
nals at the position of all satellite peaks, but not at the posi-tion of the main band. In Ref. 21, we have interpreted thisresponse as a direct manifestation of self-trapping. Pumpingthe main peak transfers population on an ultrafast 400 fstime scale into the satellite peaks, i.e., into the self-trappedstates. The main peak shows a small Davydov splitting [Fig.
2(e)], which indicates that the band corresponds to a delocal-
ized state, i.e., a free-exciton state. On the other hand, whenpumping the self-trapped states directly, no back populationof the free exciton state is observed.
Crystalline NMA and NMA -D
6. Interestingly, a similar re-sponse is also found in the pump-probe response of NMA
and NMA-D 6, except that the free exciton no longer relates
to the peak with the highest intensity, but to a shoulder on thehigh-frequency side (marked with “FE” in Figs. 2 and 9 ).
Also, this band shows a Davydov splitting between the par-allel and perpendicular spectrum [Fig. 2 (h)]. Both in NMA
and in NMA-D
6, a bleach of all bands is observed when the
pump pulse is resonant with that shoulder, and the signal atthe shoulder band decays on an ultrafast timescale compa-rable to the free exciton band in ACN (Fig. 7 ). A direct
excitation of one of the satellite peaks, on the other hand,reveals negative signals for all bands except for that shoul-der, just as an excitation of a self-trapped state in ACN. Inconclusion, we observed also in NMAtwo different types ofstates, whose nonlinear response matches the free-excitonand the self-trapped state in ACN.
IV. DISCUSSION
In the present paper, we compare the spectral response of
two commonly used model systems for peptides: crystallineACN and crystalline NMA, as well as their carbon-deuterated derivates. While the absorption spectra of thesecompounds, as well as their pump-probe signals, differ indetail, they exhibit striking similarities, which shall be sum-marized again.
(1)The amide I bands of ACN and NMA both contain a
temperature-dependent “anomalous” band; however, theseparation from the main peak is about three times smaller inNMA (Fig. 3 ).
(2)The NH bands of ACN and NMA are both accompa-
nied by a sequence of equidistant satellite peaks towardslower frequencies (Fig. 2 ). The spacing between these satel-
lite peaks is about three times larger in NMA.
(3)The narrow-band pump-probe signals of the NH band
of ACN and NMA reveal similar responses (Fig. 9 ). In par-
ticular, two different types of states can be identified in bothcases, one being the free-exciton (the highest frequency
band )and the other being the self-trapped states (the satellite
peaks ).
(4)Quantum beats are observed in the broadband pump-
probe signal of both ACN and ACN-D
8. The temperature
dependence of the quantum beats substantiated our previous
FIG. 9. Sections of the NH
band of (a)ACN, (b)NMA-D 6,
and (c)NMA. Pump-probe re-
sponse after selective excitation ofthe high frequency band (d)–(f)
and of the second satellite peaks(g)–(i)for 400 fs s•dand 4 ps s+d
delay times between the pump andthe probe pulse. The arrows indi-cate the position of the narrow-band pump pulse. All three mol-ecules show qualitatively similarresponses.JULIAN EDLER AND PETER HAMM PHYSICAL REVIEW B 69, 214301 (2004 )
214301-6result that they relate to the phonons which mediate
self-trapping.
In the case of crystalline ACN, it is now well established
that the signatures in the CO and NH spectra are a manifes-tation of vibrational self-trapping.The spectral appearance ofACN is very special, with probably the clearest manifesta-tion of vibrational self-trapping. Nevertheless, we find strik-ing similarities in the spectral response of crystalline NMA,from which we conclude that the mechanism behind thecomplicated band shapes are the same in both cases. Hence,one might speculate that self-trapping is not a unique prop-erty of crystalline ACN, but a general phenomenon inhydrogen-bonded crystals. This is indeed expected since thetheory behind vibrational self-trapping is rather general. Itessentially only depends on two coupling parameters, the
exciton coupling and the exciton-phonon coupling, which areexpected to be similar in crystals of comparable structure.
Also the NH absorption band of polypetides and proteins
contains sidebands, which have been discussed in many earlypapers.
29,38–42In 1951, Ambrose and Elliott already sug-
gested that the sidebands of the NH stretching frequencies inpolyamides, polypeptides, proteins, andACN are caused by asystem of coupled hydrogen-bonded NH oscillators.
38Brad-
bury and Elliott proposed that the NH band in NMA is acombination between a genuine NH stretching vibration anda low-frequency vibration of the hydrogen bond (i.e., the
same mechanism as the one discussed here, although theauthors did not use the term “vibrational self-trapping” ).
29
Nevertheless, it is now considered to be textbook knowl-
edge that the doublet found for the NH band in polypetidesand proteins (i.e., the main peak at 3300 cm
−1and the first
satellite peak at 3100 cm−1)is the result of a Fermi reso-
nance between the NH mode and an overtone of the amide IIvibration.
39,40,42,43One generally refers to these two bands as
the amide Aand amide Bband. The Fermi resonance assign-
ment predominantly goes back to work by Krimm et al., who
have performed normal mode calculations for variouspolypeptides based on empirical force fields.
44–47These
works tried to assign in detail the overtone and/or combina-tion mode which is supposed to be involved in the Fermiresonance. As a criterion for this assignment, the frequencyand symmetry species of the modes have been used, but, ofcourse, little was known about anharmonic couplings, i.e.,the third ingredient needed to obtain a Fermi resonance. Inthe case of a
bsheet, two different symmetry species of
amide II must couple to form the amide Bband, while in
case of an ahelix, an overtone of only one amide II species
is necessary.44,45In different bsheets, such as
b-poly (L-glycine )andb-poly (L-alanylglycine ), the amide B
mode originates from different pairs of symmetry species.45
Furthermore, studies on N-deuterated bpolypeptides46used
a three-level Fermi resonance to explain the experimentallyobserved amide AandBbands. All these calculations agree
well with the experimental data and explain temperature-dependent shifts in the spectra. Nevertheless, it seems strik-ing thatdifferent sets of modes are required to explain the
uniformappearance of the amide Bmode in various second-
ary structure motifs. It is not surprising that one can alwaysfind a combination of modes that fits the Fermi resonance
picture, given the huge density of states in a molecule aslarge as a polypeptide.
Also in the case of NMA, the two intense peaks in the NH
band (3100 and 3250 cm
−1)have been previously assigned to
a Fermi resonance and were referred to as amide AandB
modes.39,40The remaining bands in the sequence of satellite
peaks (2650 and 2900 cm−1)have been interpreted as Fermi
resonances with the overtone of the amide III band and acombination mode of amide II + amide III, respectively.
41
Nevertheless, the Fermi resonance interpretation for the NHband in NMA has been questioned frequently in theliterature.
29,32,48,49Temperature-dependent infrared and Ra-
man studies on NMA and various isotopic species showedthat a simple Fermi interpretation cannot account for the twointense peaks in NMA. The present work provides strongevidence that vibrational self-trapping at least contributes tothe peculiar line shape of NMA.
The Fermi resonance picture completely ignores the effect
of hydrogen bonding on the NH mode. It is well known thatthe NH mode in hydrogen-bonded systems strongly couplesto low frequency vibrations of the hydrogen bond itself.
50,51
This coupling, which is a result of the highly anharmonichydrogen-bond potential, leads to a redsshift of the absorp-tion band, an intensity increase, and a band broadening ac-companied by a peculiar band shape with a rich substructure.Both coupling mechanisms, Fermi resonance and coupling tolow frequency modes, can coexist in a hydrogen-bonded sys-tem. Recent theoretical studies have taken both couplingmechanisms into account to describe the spectra ofhydrogen-bonded vibrational modes.
52,53The simulations re-
sult in band shapes that depend strongly on the relativestrength of the two couplings and range from a typical Fermiresonance doublet to complex substructures with regularlyspaced sidebands.
V. CONCLUSION
NMAis considered to be the prototype molecule to study
vibrational spectra of peptides, with an obvious extension topolypeptides and proteins. Our study suggests that the previ-ous assignment of the amide Aand amide Bbands in NMA
solely to a Fermi resonance is clearly not sufficient, and thatexciton-phonon coupling, leading to vibrational self-trapping, plays an important additional role. However, whenthe coupling mechanism in the prototype molecule needs tobe reevaluated, then the interpretation of the amide AandB
modes in polypeptides and proteins may be questioned aswell. Clearly a careful analysis, which considers hydrogenbonding coupling to low frequency modes and Fermi reso-nances, is needed to understand the complex NH band shapein polypeptides. We are currently performing similar experi-ments on a real
ahelix which also yield strong evidence for
vibrational self-trapping of the NH band.54
ACKNOWLEDGMENT
The work was supported by the Swiss National Science
Foundation under Contract No. 2100-067573/1.SPECTRAL RESPONSE OF CRYSTALLINE PHYSICAL REVIEW B 69, 214301 (2004 )
214301-7*Email address: jedler@pci.unizh.ch
†Email address: phamm@pci.unizh.ch
1A. S. Davydov, J. Theor. Biol. 66, 379 (1977 ).
2H. Fröhlich, Adv. Phys. 3, 325 (1954 ).
3A. C. Scott, Phys. Rep. 217,1(1992 ).
4H. Torii and M. Tasumi J. Chem. Phys. 96, 3379 (1992 ).
5S. Krimm and J. Bandekar, Adv. Protein Chem. 38, 181 (1986 ).
6D. M. Alexander and J. A. Krumhansel, Phys. Rev. B 33, 7172
(1986 ).
7H. Torii, T. Tatsumi, and M. Tasumi, Mikrochim. Acta 14, 531
(1997 ).
8G. Careri, U. Buontempo, F. Carta, E. Gratton, and A. C. Scott,
Phys. Rev. Lett. 51, 304 (1983 ).
9N. B. Abbot and A. Eliott, Proc. R. Soc. London, Ser. A 234A,
247(1956 ).
10A. Scott, E. Gratton, E. Shyamsunder, and G. Careri, Phys. Rev.
B32, 5551 (1985 ).
11C. B. Eilbeck, P. S. Lomdahl, and A. C. Scott, Phys. Rev. B 30,
4703 (1984 ).
12J. Sauvajol, G. D. Nunzio, R. Almairac, J. Moret, and M. Barthés,
Solid State Commun. 77, 199 (1991 ).
13C. T. Johnston and B. I. Swanson, Chem. Phys. Lett. 114, 547
(1985 ).
14G. B. Blanchet and C. R. Fincher, Jr., Phys. Rev. Lett. 54, 1310
(1985 ).
15A. Spire, M. Barthes, H. Kellouai, and G. D. Nunzio, Physica D
137, 392 (2000 ).
16M. Barthes, H. Kellouai, G. Page, J. Moret, S. W. Johnson, and J.
Eckert, Physica D 68,4 5 (1993 ).
17M. Barthes, R. Almairac, J.-L. Sauvajol, and J. Moret, Phys. Rev.
B43, 5223 (1991 ).
18G. Careri, U. Buontempo, F. Galluzzi, A. C. Scott, E. Gratton,
and E. Shyamsunder, Phys. Rev. B 30, 4689 (1984 ).
19H. J. Wassermann, R. R. Ryan, and S. P. Layne, Acta Crystallogr.,
Sect. C: Cryst. Struct. Commun. C41, 783 (1985 ).
20D. M. Alexander, Phys. Rev. Lett. 54, 138 (1985 ).
21J. Edler, P. Hamm, and A. C. Scott, Phys. Rev. Lett. 88, 067403
(2002 ).
22J. Edler and P. Hamm, J. Chem. Phys. 117, 2415 (2002 ).
23J. Edler and P. Hamm, J. Chem. Phys. 119, 2709 (2003 ).
24M. Barthes, J. Mol. Liq. 41, 143 (1989 ).
25C. J. Brown and D. E. C. Corbridge, Acta Crystallogr. 7,7 1 1
(1954 ).
26J. L. Katz and B. Post, Acta Crystallogr. 13, 624 (1960 ).
27P. Hamm, R. A. Kaindel, and J. Stenger, Opt. Lett. 25, 1798
(2000 ).
28J. Eckert, M. Barthes, W. T. Klooster, A. Albinati, R. Aznar, andT. F. Koetzle, J. Phys. Chem. B 105,1 9 (2001 ).
29E. M. Bradbury and A. Elliott, Spectrochim.Acta 19, 995 (1963 ).
30F. Fillaux, Chem. Phys. 62, 287 (1981 ).
31M. Barthes, H. N. Bordallo, J. Eckert, O. Maurus, G. de Nunzio,
and J. Léon, J. Phys. Chem. B 102, 6177 (1998 ).
32W. A. Herrebout, K. Clou, and H. O. Desseyn, J. Phys. Chem. A
105, 4865 (2001 ).
33G. Careri, E. Gratton, and E. Shyamsunder Phys. Rev.A 37, 4048
(1988 ).
34C. T. Johnston, S. F. Agnew, J. Eckert, L. H. Jones, B. I. Swan-
son, and C. J. Unkefer, J. Phys. Chem. 95, 5281 (1991 ).
35V. P. Gerasimov, Opt. Spektrosk. 43, 705 (1977 ).
36M. Sakai, N. Kuroda, and Y. Nishina, Phys. Rev. B 47, 150
(1993 ).
37J. Sauvajol, R. Almairac, J. Moret, M. Barthes, and J. Ribet, J.
Raman Spectrosc. 20, 517 (1989 ).
38E. J. Ambrose and A. Elliott, Proc. R. Soc. London, Ser. A 205,
47(1951 ).
39T. Miyazawa, J. Mol. Spectrosc. 4, 168 (1960 ).
40M. Beer, H. B. Kessler, and G. B. B. Sutherland, J. Chem. Phys.
29, 1097 (1958 ).
41H. Pivcová, B. Schneider, and J. Sˆtokr, Collect. Czech. Chem.
Commun. 30, 2215 (1965 ).
42T. Miyazawa, in Poly aAmino Acids , edited by G. D. Fasman
(Marcel Dekker, New York, 1967 ), pp. 69–103.
43R. M. Badger and A. D. E. Oullin, J. Chem. Phys. 22, 1142
(1954 ).
44W. H. Moore and S. Krimm, Biopolymers 15, 2439 (1976 ).
45W. H. Moore and S. Krimm, Biopolymers 15, 2465 (1976 ).
46S. Krimm and A. M. Dwivedi, J. Raman Spectrosc. 12, 133
(1982 ).
47S.-H. Lee, N. G. Mirkin, and S. Krimm, Biopolymers 49, 195
(1999 ).
48G. Dellepiane, S. Abbate, P. Bosi, and G. Zerbi, J. Chem. Phys.
73, 1040 (1980 ).
49F. Fillaux and M. Baron, Chem. Phys. 62, 275 (1981 ).
50Theoretical Treatment of Hydrogen Bonding , edited by D. Hadži
(Wiley, New York, 1997 ).
51O. Henri-Rousseau and P. Blaise Adv. Chem. Phys. 103,1
(1998 ).
52H. Ratajczak and A. M. Yaremko, Chem. Phys. Lett. 314, 122
(1999 ).
53K. Belhayara, D. Chamma, and O. Henri-Rousseau, J. Mol.
Struct.648,9 3 (2003 ).
54J. Edler, R. Pfister, V. Pouthier, C. Falvo, and P. Hamm (unpub-
lished ).JULIAN EDLER AND PETER HAMM PHYSICAL REVIEW B 69, 214301 (2004 )
214301-8 |
PhysRevB.82.035303.pdf | Enhanced binding energy of manganese acceptors close to the GaAs(110) surface
J. K. Garleff,1,*A. P. Wijnheijmer,1A. Yu. Silov,1J. van Bree,1W. Van Roy,2J.-M. Tang,3
M. E. Flatté,4and P. M. Koenraad1
1COBRA Inter-University Research Institute, Department of Applied Physics, Eindhoven University of Technology,
P .O. Box 513, NL-5600 MB Eindhoven, The Netherlands
2IMEC, Kapeldreef 75, B-3001 Leuven, Belgium
3Department of Physics, University of New Hampshire, Durham, New Hampshire 03824, USA
4Optical Science and Technology Center and Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA
/H20849Received 26 March 2010; published 2 July 2010 /H20850
Scanning tunneling spectroscopy was performed at low temperature on buried manganese /H20849Mn/H20850acceptors
below the /H20849110/H20850surface of gallium arsenide. The main Mn-induced features consisted of a number of dI/dV
peaks in the band gap of the host material. The peaks in the band gap are followed by negative differentialconductivity, which can be understood in terms of an energy-filter mechanism. The spectroscopic featuresdetected on the Mn atoms clearly depend on the depth of the addressed acceptor below the surface. Combiningthe depth dependence of the positions of the Mn-induced peaks and using the energy-filter model to explain thenegative resistance qualitatively proves that the binding energy of the hole bound to the Mn atom increases forMn acceptors closer to the surface.
DOI: 10.1103/PhysRevB.82.035303 PACS number /H20849s/H20850: 68.37.Ef, 68.47.Fg, 73.20.Hb, 71.55. /H11002i
I. INTRODUCTION
Following Moore’s law of increasing computer perfor-
mance, electronic semiconductor devices have been scaleddown in size since the invention of the transistor.
1Nowadays
the dimensions reached a level where individual dopant at-oms, interfaces, and the distances in between them start tobecome important.
2Mn acceptors in GaAs have attracted a
lot of research interest especially since they have been foundas a promising option to make semiconductors magnetic.
3,4
Their properties in scanning tunneling microscopy /H20849STM /H20850
measurements are also well known.5–7The bow-tie-shaped
contrast found with constant current mode STM around+1.5 V is generally interpreted as the wave function of thehole bound to the Mn acceptor.
6–8Within the approach of
Tersoff and Hamann,9the STM always maps wave functions
but it is not straightforward to determine whether one indi-vidual wave function is addressed that belongs to one spe-cific Mn state. STM data measured in the constant currentmode represent the integrated local density of states /H20849LDOS /H20850
between the Fermi level E
Fand the applied voltage
/H20849EF+Vbias/H20850.9Based on scanning tunneling spectroscopy we
now show that the well-known contrast around 1.5 V splitsinto three contributions.
Strong effects of the surface have been shown: the sym-
metry of the Mn contrast is broken due to strain and theelectric field but also due to the broken symmetry in thebuckled relaxation at the GaAs /H20849110/H20850surface.
10–13Also the
presence of other structures such as quantum dots14and ad-
sorbed atoms13on the surface close to the Mn atom have
been shown to disturb the electronic structure of a Mn atom.Mn acceptors located very close to the GaAs /H20849110/H20850surface
show a rather intense contrast which is restricted laterally toa few lattice cells. In contrast, the wave functions of deeplyembedded Mn atoms are smoothly smeared out over severalnanometers. According to the basic model of a particle in abox, spatial restriction of a state is correlated with higherbinding energies. Enhanced binding energy has theoreticallybeen predicted
15for Mn atoms closely below GaAs /H20849110/H20850.W estudied the electronic structure of Mn acceptors located at
different depths below this surface experimentally. Due tolocal fluctuations of the Mn concentration on the scale oftypical frame sizes mapped in STM we also detected thedistortion of the electronic properties of Mn atoms by accep-tors in close vicinity.
II. EXPERIMENTAL SETUP
We used an Omicron low-temperature STM /H20849LT-STM /H20850
setup operated a t 5 K and a base pressure below 10−11mbar.
The tips were electrochemically etched from polycrystallineW wire, further preparation in UHV guaranteed tips ofatomic resolution and stability over several hours in scanningtunneling spectroscopy /H20849STS/H20850mode.
16The measurements
were carried out on hetero structures of Mn-doped GaAsgrown by molecular beam epitaxy on p
+-doped substrates in
a cross-sectional geometry /H20849X-STM /H20850. The exact structure of
the samples as well as the procedure applied to approach theheterostructures are described in Ref. 13.
On the cleaved surface we performed constant current to-
pography in order to find a suitable area that contains Mnatoms buried in different depths below the surface and that isfree of step edges and other unwanted defects, which com-plicate the interpretation of the data. On ensembles of Mnacceptors residing at different depths below the surface, asshown in Fig. 1,I/H20849V/H20850spectra were acquired at every pixel of
the topographic image /H20851current imaging tunneling spectros-
copy /H20849CITS /H20850, see e.g., Ref. 17/H20852. The depth of individual Mn
acceptors below the GaAs /H20849110/H20850surface is obtained from the
STM topographies.
13Figuring out the absolute depth can be
difficult for Mn acceptors in deeper layers but the relativedepth of the acceptors imaged in one frame can be identifiedunambiguously. The Mn atoms will be referred to as Mn
2to
Mn 11, with the depth in atomic layers as an index. Focusing
on the ensemble of Mn atoms we adjusted the stabilizationvoltage /H20849V
bias/H20850applied to the sample and the setpoint current
in order to minimize the topographic contrast of the embed-PHYSICAL REVIEW B 82, 035303 /H208492010 /H20850
1098-0121/2010/82 /H208493/H20850/035303 /H208496/H20850 ©2010 The American Physical Society 035303-1ded Mn atoms. Possible artifacts in the tunneling spectra
resulting from a changing tip-sample distance are avoided inthis manner. Typical parameters are V
bias/H11015+2.3 V or
Vbias/H11015−1.5 V with a feedback current of up to 3 nA. At
every pixel on a grid of up to 2562we performed I/H20849V/H20850spec-
troscopy measurements. Each I/H20849V/H20850curve consisted of typical
330 voltage steps between −2 V and +2 V. After smoothingthe spectra with Gauss filters and cubic spline fits, and nu-merical differentiation, the spectroscopic resolution was/H1101150 mV on the external voltage scale. In the resulting data
matrix dI/dV/H20849x,y,V/H20850
z0we have the full information to study
lateral properties of the LDOS as a function of the energy.
III. EXPERIMENTAL RESULTS
Within the spectroscopic information we focus on the
voltage range where the Mn-induced peaks appear. In thefollowing we will discuss the three main aspects observedaround the Mn atoms in the dI/dVmaps shown in Fig. 2.
First of all, the CITS maps show the characteristic bow-tie-shaped contrast which is well known from constant currenttopography images of buried Mn atoms in GaAs at positivepolarity around +1.5 V.
6,8,12,13The actual spectroscopic po-
sition depends on the depth of the addressed Mn atom belowthe surface as is shown in Fig. 2. The Mn atoms deep below
the surface show the bow tie already around +0.8 V, seeFigs. 2/H20849a/H20850and2/H20849b/H20850whereas the voltage has to be increased to
+1.5 V in order to find the contrast stemming from Mn
2
much closer to the surface, see Figs. 2/H20849e/H20850and2/H20849f/H20850. Second,
the Mn state consists of three peaks as will be discussed later.However, the CITS data show that all of them are character-ized by a very similar bow-tie-shaped pattern, and cannot bedecomposed into different lateral contributions appearing atdifferent sample voltages. The third observation concerns theLDOS at voltages slightly below the voltage at which theMn-induced contrast in the band gap is addressed. At /H110111V
/H20851see Fig. 2/H20849b/H20850/H20852large blurry rectangular features are observed
around the Mn atoms. With increasing voltage the lateralextension of the rectangles shrinks whereas their contrast
amplitude increases, see Figs. 2/H20849c/H20850and2/H20849d/H20850. In this process,
the rectangular shape deforms, and the depth-specific Mn-induced patterns evolve; bow ties for Mn in deeper layersand more asymmetric shapes for Mn close to the surface,
13
see Figs. 2/H20849c/H20850–2/H20849f/H20850. The decreasing extension of the rimlike
rectangular feature with increasing voltage shows the samevoltage dependence as the rings of ionization for Mn accep-tors in InAs.
18
The decreasing diameter of the rings surrounding Mn ac-
ceptors in InAs with increasing voltage was explained by thetip-induced band bending /H20849TIBB /H20850/H20849Ref. 19/H20850that decreases
with increasing distance from the tip and with decreasingvoltage. The distance-dependent nature of this electrostaticeffect results in the circular symmetry of the rings. The non-circular symmetry of the rectangular rim surrounding the Mnatoms in GaAs proves that mapping the Mn wave function isfully entangled with the mechanism behind this feature andthat a description purely based on electrostatics is not suffi-cient. A second deviation of the observed rectangles from therings of ionization can be seen where the rims of neighboringMn atoms overlap. In that case the size of the rim around Mnatoms in GaAs does not significantly change whereas therings of ionization around Si in GaAs and Mn in InAs arereduced in diameter where neighboring rings overlap.
18,20We
conclude that the Coulomb effect leading to a ring of ioniza-tion and mapping genuine properties of the Mn wave func-tion are strongly entangled resulting in the rectangular rimsaround Mn atoms below GaAs /H20849110/H20850.
The remainder of the paper will focus on the qualitative
and quantitative interpretation of the voltages needed to ad-dress Mn atoms that clearly depend on the depth of the ad-dressed Mn atom below the surface. Finally we will address
AL2
AL5AL7
AL8AL11
5n m110
001
FIG. 1. /H20849Color online /H20850STM topography image of an ensemble
of Mn acceptors in GaAs /H20849110/H20850measured at +2 V and 30 pA. The
depth of the acceptors below the surface /H20849Ref. 13/H20850is given in the
image /H20849in units of atomic layers counting the surface layer as 1 /H20850.
(b) +0.94 V (a) +0.73 V
(d) +1.25 V (c) +1.03 V
(f) +1.64 V (e) +1.48 VAL13
AL14
dI/dV
FIG. 2. /H20849Color online /H20850dI/dVmaps of the Mn ensemble shown
in Fig. 1. The voltage is indicated in each image.GARLEFF et al. PHYSICAL REVIEW B 82, 035303 /H208492010 /H20850
035303-2the threefold splitting of the Mn acceptor peak.
Laterally averaging over the spectra on top of individual
Mn acceptors in different atomic layers is used to study thedepth dependence of the spectroscopic signature of the Mnacceptors. Around each of the five specific Mn atoms identi-fied in Fig. 1, the spectra were averaged in a box of
/H110112/H110032.5 nm
2/H20849/H11011900 individual spectra /H20850. The resulting av-
eraged spectra are plotted in Fig. 3/H20849a/H20850. We compare the spec-
tra taken on the five acceptors with an average spectrum onthe bare GaAs surface in the same data set as a reference. Inthe upper left and lower left corner of Fig. 2/H20849c/H20850two addi-
tional Mn atoms are buried deeper below the surface than 11atomic layers /H20849AL/H20850. We include them in the further discus-
sion. Based on the relative strength of their contrasts, theirdepth is roughly estimated to be 13AL and 14AL below thesurface.
The main spectroscopic signature of the Mn acceptors is
found in the band gap below the onset of the conductionband of GaAs, which is in good agreement with earlier pub-lications for spectroscopy on Mn in InAs,
21and for Mn in the
first layer of GaAs.7It also fits to the reported position of the
Mn-induced contrast in constant current topographies onMn-doped GaAs.
6,8,12,13Figure 3/H20849a/H20850highlights this voltage
range. The Mn signature consists of three peaks that are fol-lowed by negative differential conductivity /H20849NDC /H20850.
22,23
The Mn-induced peaks are quantitatively characterized by
Gaussian fits. The fits to the peaks in the band gap are shownin the insets in Fig. 3. Correct assignment of the first, second,
and third peak is crucial because comparing wrong peakswith each other would severely distort the derived depth de-pendence of the peak positions. The spectroscopic positionof the NDC was used as a landmark in order to identify thecorresponding peaks for the Mn atoms in different depth.The resulting peak positions from the Gauss fits are summa-rized in Table Iand plotted versus the depth of the corre-
sponding Mn atom in Fig. 4. This quantitatively confirms the
trend that the peaks in the band-gap shift to lower voltagesfor acceptors deeper below the surface. The uncertainty of
the resulting peak positions is /H110115 mV. The amplitude and
width of the peaks, however, do not show unambiguoustrends. In general, features stemming from acceptors close tothe surface are stronger than those induced by deeply buriedMn atoms. In most cases the second peak is the strongest ofthe threefold peaks in the band gap.
A closer look at the spectroscopic signature of Mn
11in
Fig.3, Table I, and Fig. 4reveals that the lowest energy peak
of this Mn atom appears at significantly lower voltage thanexpected from a linear fit. The peak stemming from Mn
11lies
even below the corresponding peak belonging to Mn 13and
Mn 14. Figures 1and2/H20849c/H20850show that Mn 11is located close to
Mn 7and to Mn 13. The lateral distance to both neighbors is
around /H110112.7 nm. Peaks at reduced voltages are systemati-
cally found for Mn atoms situated in close vicinity to neigh-boring acceptors in all our measurements. This supports theconclusion that the interactions between neighboring Mn at-oms lower the spectroscopic position of the first peak in theband gap, consistent with Ref. 7. The critical interaction
length of /H110113 nm is much larger than the average distance
between the defects at the metal-insulator transition.
24Mn
acceptors deeper than /H110112.5 nm below the surface are hardly
visible in topographic images; this depth is very similar tothe critical interaction length. Therefore the local neighbor-hood of the individual Mn atoms results in shifted peak po-sitions similar to the variations between individual emittersresulting in inhomogeneous broadening in photo lumines-cence experiments.TABLE I. Results of fitting three Gaussians to the dI/dVspectra
measured at the Mn atoms in different atomic layers./H11569peak 1 of
Mn 11is shifted due to other Mn atoms close by.
Mn 2Mn 5Mn 7Mn 8Mn 11Mn 13Mn 14
Peak 1 /H20849mV/H208501338 1329 1222 1227 874/H11569996 959
Peak 2 /H20849mV/H208501439 1440 1367 1248 1207 1139 1121
Peak 3 /H20849mV/H208501669 1562 1498 1449 1328 1300 1293
0,8 1,0 1,2 1,4 1,6 1,8-20020406080100120140Mn AL2
Mn AL5
Mn AL7
Mn AL8
Mn AL11
Mn AL13
Mn AL14
GaAsdI/dV [pA/V]
sample voltage [V]0,8 1,0 1,2 1,4 1,60246810121416C
A(b) AL2
0,8 1,0 1,2 1,4 1,60246810C
A
0,8 1,0 1,2 1,4 1,601234567C
A
0,8 1,0 1,2 1,4 1,60,00,51,01,52,02,53,0D
0,8 1,0 1,2 1,4 1,60,00,51,0H(f) AL11
0,8 1,0 1,2 1,4 1,6-0,50,00,51,01,52,02,5(g) AL13
0,8 1,0 1,2 1,4 1,6-0,50,00,51,01,5Mn0g(h) AL140.8 1.2 1.6
Sample voltage [V]1.0 1.4 1.8-20 dI/dV [nA /V]
020406080100120140
0.8 1.2 1.6 0.8 1.2 1.6 0.8 1.2 1.6 0.8 1.2 1.6(d) AL7
(e) AL8(a)
(c) AL5
FIG. 3. /H20849Color online /H20850/H20849a/H20850Differential conductivity spectra on
Mn acceptors in different depths below GaAs /H20849110/H20850./H20849b/H20850–/H20849h/H20850Gauss-
ian fits to the Mn-induced peaks in the gap: black dots=raw data,red full lines=individual Gaussians, black dashed line=sum of theGaussians; depth of the Mn atom as indicated.
FIG. 4. /H20849Color online /H20850Peak positions of the Mn-induced fea-
tures in the band gap for Mn atoms in different depth from theGaussian fits shown in Fig. 3. The symbols red dots, green stars,
and black squares depict the peaks 1, 2, and 3 in the gap.ENHANCED BINDING ENERGY OF MANGANESE … PHYSICAL REVIEW B 82, 035303 /H208492010 /H20850
035303-3Splitting of the acceptor levels due to interaction with
neighboring Mn atoms has been predicted theoretically forbulk Mn-GaAs.
25The reduced energetic position of the low-
est state depends on the relative position of both Mn atoms.Our observation of a peak at significantly lower voltage forMn atoms located close to each other confirms the theoreticalprediction qualitatively. The limited visibility of Mn atoms indeeper layers, and their specific properties close to the sur-face hinder a quantitative comparison with Ref. 25. This
splitting was measured experimentally between atoms in thesurface layer
7but until now had not been demonstrated for
subsurface magnetic dopant pairs.
IV. ENHANCED BINDING ENERGY
It is tempting to interpret the shifted spectroscopic posi-
tion of the Mn-induced peaks in the band gap directly as anenhanced binding energy for Mn atoms closer to the surface.A similar argument was used for Mn atoms in the first atomiclayer.
7,15However, the situation is more complex since the
Fermi level EFis not pinned on GaAs /H20849110/H20850, and TIBB /H20849Ref.
19/H20850cannot be neglected. Quantitative calculations of the
TIBB are challenging because several crucial parameters areonly known by estimation, e.g., geometry and work functionof the tip, and distance between tip and sample. We thereforedraw qualitative conclusions on the effect of the depth de-pendence of the Mn-induced features before we discuss theestimated binding energy of Mn atoms in different layersbelow the surface.
A. Qualitative analysis
Addressing Mn atoms closer to the surface at increased
voltage rises the question on the mechanism behind the Mn-induced peaks. In general, a peak in dI/dVstems from an
additional tunneling channel. In the case of Mn in GaAs, thishappens when the addressed acceptor state close to the sur-
face /H20849E
Asurf/H20850lines up with the Fermi level deep in the sample,
EFbulk, as schematically depicted in Fig. 5. A second ingredi-
ent for interpretation is the negative differential conductancefollowing the Mn-induced peaks in the band gap. NDC canbe explained in a straightforward manner by the energy-filtermechanism which is briefly introduced here. The tunnelingcurrent flows from the tip first into the empty acceptor stateof the addressed Mn atom close to the surface. Then in asecond step, the electrons have to reach an empty state in the
partially filled acceptor band in the bulk at the same energy.
The decaying TIBB toward the bulk allows to align E
Asurf
with EFbulk. Increasing the external voltage lifts EAsurfabove
EFbulk, and the tunneling channel through the Mn state close to
the surface is closed because the electrons can no longer bedrained into the impurity band in the bulk. Blocking the tun-neling channel thus decreases the current for increased volt-age and explains the observed NDC. Details can be found inRef. 23.
Assuming the binding energy of the acceptor to remain
unchanged close to the surface, E
Asurf=EAbulk, means that it
lines up with EFbulkwhen the bands are flat. In this situation,
the TIBB and its decay toward the bulk both vanish. There-
fore all Mn atoms in different depths would cross EFbulkat
exactly the same external voltage, and no depth dependencewould be observed for the Mn-induced peaks. This is clearlynot the case in our experiments as shown in Fig. 4. We con-
clude that E
Asurfindeed depends on the depth of the Mn atom
below the surface. In case of upward /H20849downward /H20850TIBB
needed to achieve EAsurf=EAbulk, the Mn binding energy will be
decreased /H20849increased /H20850close to the surface.
We measured the flat-band voltage independently by z/H20849IT/H20850
/H20849Ref. 23/H20850spectroscopy, resulting in a flat band voltage of
2.5/H110060.5 V, clearly above the Mn-induced peaks. In agree-
ment with Ref. 22the Mn-induced peaks and the NDC are
detected at downward TIBB, proving the enhanced bindingenergy of Mn atoms close to the surface. Figures 5/H20849a/H20850and
5/H20849b/H20850qualitatively compare the band bending needed to ad-
dress an acceptor close to the surface /H208515/H20849a/H20850/H20852and deeper in the
crystal /H208515/H20849b/H20850/H20852. The higher voltage needed to align the accep-
tor close to the surface causes a smaller total TIBB than the
lower voltage at which the deeper acceptor aligns with theimpurity band. Due to the rapid decay of the TIBB towardthe bulk, the local TIBB at the acceptor site is still higher for
the acceptor close to the surface. This shows that bindingenergy of Mn acceptors increases monotonously for decreas-ing depth below the surface.
The depth dependence of the Mn-induced peaks is repro-
duced in all other STS measurements even though the abso-lute peak positions differ. The deviations most probably stemfrom different tips resulting in modified band bending con-figurations. All our spectroscopy measurements with suffi-cient resolution reproduced the threefold splitting of the Mn-induced peak in the band gap. We conclude that all peaksshow an enhanced binding energy as predicted by Ref. 15.
B. Quantitative analysis
In order to extract the binding energy as a function of the
depth of the Mn acceptor below the GaAs /H20849110/H20850surface quan-
titatively, the TIBB has to be calculated for the voltages atwhich the Mn-induced features are detected in Fig. 4.A s
discussed in the previous paragraph and shown in Fig. 5, the
Mn peaks in the gap originate from lining up E
Asurfwith EFbulk.
The binding energy is then given by the equation
EA=/H20849EFbulk−EVB/H20850+/H20841TIBB /H20841. The decay of the TIBB into the
material is crucial because the addressed Mn atoms arelocated in different depths between 2AL and 14AL in thesurface.bulk bulk surf deep
Mn-GaAs tip tip Mn-GaAs(a) (b)
EF EAeVeVEA EFEF,tip
EF,tip
FIG. 5. /H20849Color online /H20850Schematic band bending of the
Mn:GaAs /H20849110/H20850surface in case of lining up a Mn atom /H20849a/H20850close to
or/H20849b/H20850deep below the surface with EFbulk.GARLEFF et al. PHYSICAL REVIEW B 82, 035303 /H208492010 /H20850
035303-4We calculated the TIBB using the code published in Ref.
26. The obtained binding energy depends on the flat band
voltage and on the tip radius Rtip. As mentioned before, the
flat band voltage was independently determined by z/H20849IT/H20850
spectroscopy to FB=2.5/H110060.5 V. Furthermore we assume
similar Rtipas we characterized earlier Rtip/H1134910 nm /H20849Refs.
20and27/H20850because all our tips are prepared using the same
procedure. Although these values seem to have small errors,the resulting variation in E
Ais still large. Therefore we need
further restrictions to narrow down the range of parametersin the TIBB calculations. It is close at hand to assume thebinding energy to approach the bulk value asymptotically foracceptors at increasing depth below the surface. Thereforewe demand that the calculated binding energies of Mn
13and
Mn 14are very similar to each other. EAAL13/H11015EAAL14is
achieved for a flat band condition FB/H110152.3 V, and a tip
radius Rtip=2 nm, both in good agreement with the expected
parameters, FB=2.5/H110060.5 V, and Rtip/H1102110 nm.
The resulting depth-dependent binding energy is fitted by
an exponential decay, depicted by the dashed-dotted red linein Fig. 6. The aymptotic limit of this fit deeply below the
surface is 148 meV , 25 meV higher than the bulk bindingenergy of 113 meV . This deviation most probably stems froma different position of E
Fin the bulk. We assumed it to be
113 meV above the VB. However, the position of EFbulkde-
pends on the doping level. Shifting EFbulkwith respect to EVB
in the equation we used, EA=/H20849EFbulk−EVB/H20850+/H20841TIBB /H20841, results in
a constant offset to the extracted binding energies. We re-moved the offset such that the extracted binding energy as-ymptotically approaches 113 meV for very deep layers. Theresults for the investigated Mn atoms are plotted in Fig. 6
relative to the top of the VB defined as VB=0 meV. The red
dots, green stars, and black squares depict the binding ener-gies of the first, second, and third peak. All energetic posi-tions significantly depend on the depth of the addressed Mnatom below the surface. For Mn atoms closer to the surface,the Mn-induced states in the band-gap shift to higher energy.The extracted binding energy equals /H11011117 meV for Mn
14,
which is close to EAbulk, and increases up to /H11011170 meV forMn 2. The extracted absolute binding energies depend on the
choice of the parameters in order to calculate the TIBB.Therefore it is difficult to give the error bars. The errorwithin one choice of parameters is smaller than /H110065 meV but
there is a possible scaling of /H1100630% by choosing different
parameters. The binding energies that we extract from theexperiment agree satisfyingly well with a recent theoreticalstudy of Mn atoms closely below GaAs /H20849110/H20850
15that is plotted
as black triangles. Theory predicts a much stronger increase
inEAsurfin AL1 and AL2, which is not found in the experi-
ment whereas the furtherly predicted decay toward the bulkvalue on a larger length scale fits much better with our ex-perimental results.
Now we focus on the Mn-induced feature consisting of
three peaks. The Mn 3 d
5electrons with a total spin of
S=5 /2, the hole in the acceptor state with an orbital
momentum L=1 and a spin of s=1 /2 allow in total 36
angular-momentum states. We define the total momentumJ=S+s+L, and the 36 states correspond to one J=1, two
J=2, two J=3 and one J=4 multiplet. The crystal field splits
all multiplets with J/H110221, so there are many energetic levels
possible. The ground state is given by J=1, and if we assume
the other two levels visible are two of the excited levels/H20849e.g., J=2,3 /H20850, then the splitting between the ground state and
the exited states can be assumed to originate either fromexchange coupling or from spin-orbit interaction. Exchangecoupling would produce progressive splitting betweenJ=1,2,3. T able IIlists the splitting between the peaks in our
experiment for comparison. /H9004
i−jdenotes the spectroscopic
distance between neighboring peaks iandjcorresponding to
the same Mn atom. Our experimental data clearly do notshow progressive splitting so we exclude exchange couplingas the source of the splitting into three levels.
For spin-orbit interaction the states with J=1,2,3 are
evenly split by /H1101140 meV,
25which appears superficially
similar to the results in Table II. The spin-orbit interaction
was also proposed as an explanation for the multiple levelsmeasured in Ref. 21. The calculation of the spin-orbit split-
ting for Mn in bulk GaAs,
25however, has to be compared to
the peaks associated with the Mn atoms in the deepest layers,which are split by 3–4 meV in the experiment. This is oneorder of magnitude smaller than predicted in Ref. 25,s ow e
also exclude spin-orbit interaction as the source of the ob-served splitting.
We conclude that the threefold splitting in our experimen-
tal data does not originate from energy splitting between theJ=1 and other, higher energy angular-momentum states. In-
stead we propose that the threefold degeneracy of J=1 is
lifted close to the surface as a result of symmetry lower thanthe bulk crystal. The symmetry is broken because the relax-TABLE II. Splitting between the Mn-induced peaks
/H9004i−j=EAi−EAj./H11569EAof Mn 11is shifted due to the interaction with
neighboring acceptors.
Mn 2Mn 5Mn 7Mn 8Mn 11 Mn 13 Mn 14
/H90041–2/H20849meV /H20850 9 771 1 3/H1156934
/H90042–3/H20849meV /H20850 2 0 758 34 4
FIG. 6. /H20849Color online /H20850Energetic positions relative to the top of
the VB of Mn acceptors in different layers below the GaAs /H20849110/H20850
surface extracted from the experiment. The red dots, green stars,and black squares depict the three peaks in the band gap of GaAs.The open black triangles give the theoretical prediction /H20849Ref. 15/H20850.ENHANCED BINDING ENERGY OF MANGANESE … PHYSICAL REVIEW B 82, 035303 /H208492010 /H20850
035303-5ation at the GaAs /H20849110/H20850surface introduces strain into the lat-
tice, and because the voltage applied between the sample andthe STM tip results in an electric field at the position of theacceptor. Both effects break the symmetry of acceptor wavefunctions.
12,14,28Under these conditions mJ=+1,0,−1, are no
longer the proper energy eigenstates. However, we assumethat three linear combinations will be formed, as described inRefs. 14and28, and lead to the threefold splitting of the Mn
state in our spectra. Strain or electric fields directed along the/H20851110/H20852direction will produce evenly split states whereas strain
or electric fields along the /H20851111/H20852direction will produce two
degenerate states and one split-off one. We conclude fromTable IIthat deep within the crystal the strain or electric field
is directed along the surface normal, corresponding to evenlysplit states whereas at the surface there is a strong componentof the strain which is along the /H20851111/H20852direction and produces
an uneven energy splitting. This effect of strain along the/H20851111/H20852direction is consistent with the results of Ref. 12. Since
the states connected to J=1 are fully degenerate in the bulk,
a small splitting of only 3 meV in our experiment 14 layersdeep is not surprising.
V. CONCLUSIONS
In summary, the spectroscopic signature of Mn atoms in
different layers below GaAs /H20849110/H20850was measured by scanningtunneling spectroscopy. The observed features, a threefold
split peak in the band gap, shift to higher voltage for Mnatoms closer to the surface. The peaks in the band gap arefollowed by NDC which is explained by an energy-filtermodel at negative TIBB. This allows the qualitative conclu-sion that the binding energy of Mn atoms is enhanced closeto the surface. Quantitatively the binding energy was ex-tracted from the measured peak positions by calculating theTIBB in the respective depth below the surface. The result-ing binding energy of /H11011170 meV for Mn atoms close the
GaAs /H20849110/H20850surface is in satisfactory agreement with a recent
theoretical prediction.
15The threefold splitting of the Mn
state is not identified with the theoretical prediction of athreefold state split by the spin-orbit interaction,
25or as due
to exchange coupling. Instead, we assign it to a splitting oftheJ=1 ground state by the strain and electric field present at
the surface.
ACKNOWLEDGMENTS
We thank M. Bozkurt, C. Çelebi, S. Loth, and M. Wen-
deroth for valuable discussions, and the STW-VICI underGrant No. 6631, NAMASTE, and COBRA for financialsupport.
*j.k.garleff@tue.nl
1J. Bardeen and H. W. Brattain, Phys. Rev. 74, 230 /H208491948 /H20850.
2S. Roy and A. Asenov, Science 309, 388 /H208492005 /H20850.
3T. Dietl, F. Matsukura, and H. Ohno, Phys. Rev. B 66, 033203
/H208492002 /H20850.
4T. Dietl, Semicond. Sci. Technol. 17, 377 /H208492002 /H20850.
5P. I. Arseev, N. S. Maslova, V . I. Panov, S. V . Savinov, and C. v.
Haesendock, JETP Lett. 77, 172 /H208492003 /H20850.
6A. M. Yakunin, A. Yu. Silov, P. M. Koenraad, J. H. Wolter,
W. Van Roy, J. De Boeck, J.-M. Tang, and M. E. Flatté, Phys.
Rev. Lett. 92, 216806 /H208492004 /H20850.
7D. Kitchen, A. Richardella, J.-M. Tang, M. E. Flatté, and
A. Yazdani, Nature /H20849London /H20850442, 436 /H208492006 /H20850.
8C. Çelebi, P. M. Koenraad, A. Yu. Silov, W. Van Roy, A. M.
Monakhov, J.-M. Tang, and M. E. Flatté, Phys. Rev. B 77,
075328 /H208492008 /H20850.
9J. Tersoff and D. R. Hamann, Phys. Rev. Lett. 50, 1998 /H208491983 /H20850.
10S. Loth, M. Wenderoth, and R. G. Ulbrich, Phys. Rev. B 77,
115344 /H208492008 /H20850.
11J.-M. Jancu, J.-C. Girard, M. O. Nestoklon, A. Lemaître, F. Glas,
Z. Z. Wang, and P. V oisin, Phys. Rev. Lett. 101, 196801 /H208492008 /H20850.
12C. Çelebi, J. K. Garleff, A. Yu. Silov, A. M. Yakunin, P. M.
Koenraad, W. Van Roy, J.-M. Tang, and M. E. Flatté, Phys. Rev.
Lett. 104, 086404 /H208492010 /H20850.
13J. K. Garleff, C. Çelebi, W. Van Roy, J.-M. Tang, M. E. Flatté,
and P. M. Koenraad, Phys. Rev. B 78, 075313 /H208492008 /H20850.
14A. M. Yakunin, A. Y . Silov, P. M. Koenraad, J.-M. Tang, M. E.
Flatté, J.-L. Primus, W. van Roy, J. de Boeck, A. M. Monakhov,K. S. Romanov, I. E. Panaiotti, and N. S. Averkiev, Nature
Mater. 6, 512 /H208492007 /H20850.
15T. O. Strandberg, C. M. Canali, and A. H. MacDonald, Phys.Rev. B 80, 024425 /H208492009 /H20850.
16J. K. Garleff, M. Wenderoth, K. Sauthoff, R. G. Ulbrich, and
M. Rohlfing, Phys. Rev. B 70, 245424 /H208492004 /H20850.
17R. J. Hamers and D. F. Padowitz, in Methods of Tunneling Spec-
troscopy with the STM in Scanning Tunneling Microscopy andSpectroscopy, Theory, Techniques and Applications , edited by
D. Bonnell /H20849VCH, New York, 1993 /H20850.
18F. Marczinowski, J. Wiebe, F. Meier, K. Hashimoto, and R. Wie-
sendanger, Phys. Rev. B 77, 115318 /H208492008 /H20850.
19R. M. Feenstra and J. A. Stroscio, J. Vac. Sci. Technol. B 5, 923
/H208491987 /H20850.
20A. P. Wijnheijmer, J. K. Garleff, K. Teichmann, M. Wenderoth,
S. Loth, R. G. Ulbrich, P. A. Maksym, M. Roy, and P. M. Koen-raad, Phys. Rev. Lett. 102, 166101 /H208492009 /H20850.
21F. Marczinowski, J. Wiebe, J.-M. Tang, M. E. Flatté, F. Meier,
M. Morgenstern, and R. Wiesendanger, Phys. Rev. Lett. 99,
157202 /H208492007 /H20850.
22S. Loth, M. Wenderoth, L. Winking, R. G. Ulbrich, S. Malzer,
and G. H. Döhler, Phys. Rev. Lett. 96, 066403 /H208492006 /H20850.
23A. P. Wijnheijmer, J. K. Garleff, M. A. v. d. Heijden, and P. M.
Koenraad /H20849unpublished /H20850.
24A. Richardella, P. Roushan, S. Mack, B. Zhou, D. A. Huse, D. D.
Awschalom, and A. Yazdani, Science 327, 665 /H208492010 /H20850.
25J.-M. Tang and M. E. Flatté, Phys. Rev. Lett. 92, 047201 /H208492004 /H20850.
26R. M. Feenstra, J. Vac. Sci. Technol. B 21, 2080 /H208492003 /H20850.
27K. Teichmann, M. Wenderoth, S. Loth, R. G. Ulbrich, J. K. Gar-
leff, A. P. Wijnheijmer, and P. M. Koenraad, Phys. Rev. Lett.
101, 076103 /H208492008 /H20850.
28J.-M. Tang, J. Levy, and M. E. Flatté, Phys. Rev. Lett. 97,
106803 /H208492006 /H20850.GARLEFF et al. PHYSICAL REVIEW B 82, 035303 /H208492010 /H20850
035303-6 |
PhysRevB.103.035111.pdf | PHYSICAL REVIEW B 103, 035111 (2021)
Electronic mechanism for nanoscale skyrmions and topological metals
Deepak S. Kathyat , Arnob Mukherjee , and Sanjeev Kumar
Department of Physical Sciences, Indian Institute of Science Education and Research Mohali,
Sector 81, S.A.S. Nagar, Manauli PO 140306, India
(Received 22 July 2020; revised 26 November 2020; accepted 9 December 2020; published 8 January 2021)
We report a microscopic electronic mechanism for nanoscale skyrmion formation and topological metalicity
originating from Rashba and double-exchange physics. The results are based on hybrid simulations in a modelthat explicitly retains itinerant electronic degrees of freedom. A simple physical picture is provided via aneffective short-range spin model. We identify hexagonal and square lattice arrangements of skyrmions in twodifferent regimes of the parameter space. Sparse skyrmions emerge at finite temperatures as excitations of theferromagnetic phase. The skyrmion states are characterized as topological metals via explicit calculations of theBott index and the Hall conductivity. Oscillations in local density of states are shown to arise from a combinationof confinement effects and emergent gauge-fields. We also emphasize the importance of a consistent treatmentof spin-orbit coupling for calculating electronic properties of metals hosting unconventional magnetic texturessuch as skyrmions.
DOI: 10.1103/PhysRevB.103.035111
I. INTRODUCTION
Magnetic skyrmions are being envisioned as building
blocks of next-generation data storage and processing de-vices [ 1–6]. This has led to a surge in research activity
geared towards identifying candidate materials [ 7–19]. Such
textures in metals are particularly important since they canbe manipulated by ultralow electrical currents [ 10,11,20,21].
The appearance of skyrmions has been reported in bulk aswell as in thin films of a variety of chiral metallic mag-nets [ 12–15,22–27]. However, the current understanding of
skyrmion formation in magnets is via spin Hamiltonians thateither include Dzyaloshinskii-Moriya (DM) interactions orgeometrical frustration [ 28–32]. Such studies have also shown
the formation of three-dimensional lattices of skyrmions,relevant for skyrmions in bulk [ 33,34]. This approach is
inconsistent for metals as the aforementioned terms areusually understood as arising from the effect of spin-orbitcoupling in Mott insulators [ 35]. Therefore, the importance
of electronic Hamiltonian-based understanding of skyrmionformation in metals has been recognized and a mechanismbased on Ruderman-Kittel-Kasuya-Yosida interactions hasrecently been put forward [ 36,37]. Furthermore, most theo-
retical studies describe states that are periodic arrangementsof skyrmions, whereas experiments on certain thin films oron constricted samples also support a phase with sparseskyrmions [ 9,27,38,39].
The introduction of the double-exchange (DE) mechanism
by Zenger represents a milestone in our understanding offerromagnetic metals [ 40–42]. The mechanism has played
a key role in the description of magnetic and magneto-transport phenomena across families of materials, such asperovskite manganites, dilute magnetic semiconductors, andHeusler metals [ 43–46]. Surprisingly, the role of DE physics
in skyrmion formation has largely remained unexplored. Onthe other hand, the DE mechanism is commonly invoked whenstudying the effect of magnetic textures, including skyrmions,
on transport properties in metals. The implication of spin-orbit-modified DE physics on transport properties has recentlybeen discussed [ 47].
In this work, we show that the Rashba DE (RDE) model
in the presence of Zeeman field leads to states hostingnanoskyrmions. We explicitly demonstrate the appearanceof skyrmions using the state-of-the-art hybrid Monte Carlo(HMC) simulations. An effective spin Hamiltonian is studiedfor a comprehensive understanding of the origin as well asthe stability of these spin textures. A filamentary domain wall(fDW) phase is identified as the parent of sparse skyrmions(sSk), which are found to be stable only at finite temperaturesand metastable in the ground state, and a single-Q (SQ) spi-ral state leads to packed skyrmions (pSk). Our findings areconsistent with small-angle neutron scattering (SANS) andLorentz transmission electron microscopy (LTEM) data onthin films of Co-Zn-Mn alloys, FeGe, MnSi, and transitionmetal multilayers [ 10–13,18,26,27,48,49]. Furthermore, we
explicitly demonstrate by calculating the Bott index and thetopological Hall conductivity that the skyrmion phases are anatural realization of amorphous topological metals. This isparticularly important in view of recent attempts to engineertight-binding models for the realization of amorphous topo-logical phases [ 50–52]. We also present local density of states
(LDOS) calculations to show the importance of consistenttreatment of spin-orbit coupling for the skyrmion formationand for electronic transport aspects. A combination of dI/dV
measurements and LDOS analysis can be a useful alternateto the existing methods for estimating the strength of Rashbacoupling in real materials [ 53].
II. SKYRMIONS IN THE RDE MODEL
We start with the ferromagnetic Kondo lattice model
(FKLM) in the presence of Rashba SOC, described by the
2469-9950/2021/103(3)/035111(9) 035111-1 ©2021 American Physical SocietyKATHYAT, MUKHERJEE, AND KUMAR PHYSICAL REVIEW B 103, 035111 (2021)
following Hamiltonian:
H=−t/summationdisplay
/angbracketleftij/angbracketright,σ(c†
iσcjσ+H.c.)+λ/summationdisplay
i[(c†
i↓ci+x↑−c†
i↑ci+x↓)
+i(c†
i↓ci+y↑+c†
i↑ci+y↓)+H.c.]−JH/summationdisplay
iSi·si.(1)
Here, ciσ(c†
iσ) annihilates (creates) an electron at site iwith
spinσ, and/angbracketleftij/angbracketrightimplies that iandjare nearest-neighbor (nn)
sites.λandJHdenote the strengths of Rashba and Hund’s
coupling, respectively. siis the electronic spin operator at site
i, and Si, with |Si|=1, denotes the localized spin at that
site. We parametrize t=(1−α)t0andλ=αt0and set t0=1
as the reference energy scale. Assuming large JHand tak-ing the double-exchange approximation, we obtain the RDE
Hamiltonian [ 54],
HRDE=/summationdisplay
/angbracketleftij/angbracketright,γ/bracketleftbig
gγ
ijd†
idj+H.c./bracketrightbig
−hz/summationdisplay
iSz
i, (2)
where di(d†
i) annihilates (creates) an electron at site iwith
spin parallel to the localized spin. The second term representsthe Zeeman coupling of local moments to external magneticfield of strength h
z. Site j=i+γis the nn of site ialong the
spatial direction γ=xandy. The projected hopping gγ
ij=
tγ
ij+λγ
ijdepends on the orientations of the local moments Si
andSj[54]:
tγ
ij=−t/bracketleftbigg
cos/parenleftbiggθi
2/parenrightbigg
cos/parenleftbiggθj
2/parenrightbigg
+sin/parenleftbiggθi
2/parenrightbigg
sin/parenleftbiggθj
2/parenrightbigg
e−i(φi−φj)/bracketrightbigg
,
λx
ij=λ/bracketleftbigg
sin/parenleftbiggθi
2/parenrightbigg
cos/parenleftbiggθj
2/parenrightbigg
e−iφi−cos/parenleftbiggθi
2/parenrightbigg
sin/parenleftbiggθj
2/parenrightbigg
eiφj/bracketrightbigg
, (3)
λy
ij=iλ/bracketleftbigg
sin/parenleftbiggθi
2/parenrightbigg
cos/parenleftbiggθj
2/parenrightbigg
e−iφi+cos/parenleftbiggθi
2/parenrightbigg
sin/parenleftbiggθj
2/parenrightbigg
eiφj/bracketrightbigg
,
where θi(φi) denotes the polar (azimuthal) angle for the local-
ized spin Si.
We study the RDE Hamiltonian using numerically exact
HMC simulations [ 55](see also Refs. [ 56,57] and references
therein). The presence of skyrmions is inferred via the localskyrmion density [ 29],
χ
i=1
8π[Si·(Si+x×Si+y)+Si·(Si−x×Si−y)].(4)
The total skyrmion density is defined as χ=/summationtext
iχi.W e
also compute the spin structure factor (SSF) as
Sf(q)=1
N2/summationdisplay
ijSi·Sje−iq·(ri−rj), (5)
and we compute the relevant component of the vector chirality
ηas
η=1
N/summationdisplay
i(Si×Si+x)·ˆy−(Si×Si+y)·ˆx. (6)
The averaging of all quantities over MC steps is implicitly
assumed, unless stated otherwise.
Results obtained via HMC simulations for two represen-
tative values of αare shown in Fig. 1. Upon increasing hz,
the magnetization, Mz=1
N/summationtext
iSz
i, increases and ηdecreases.
The magnitude of χinitially increases with the applied field
and then decreases on the approach to the saturated ferro-magnetic (sFM) state [see circles in Figs. 1(a)and1(d)]. The
qualitative behavior appears to be similar between α=0.25
andα=0.45. The negative sign of χreveals that the polarity
of skyrmions is opposite to the orientation of the backgroundmagnetization.
The existence of skyrmions in the RDE Hamiltonian is
explicitly demonstrated via the spin configurations as well asskyrmion density maps in the ground state. We find that smallvalues of αlead to sparse skyrmions within the zero-field-
cooled (ZFC) protocol [see Fig. 1(b)] and the packing (size)
of skyrmions increases (decreases) with increasing α[see
Fig.1(e)]. The negative polarity is consistent with the fact that
the central spin in the skyrmion texture is oriented opposite tothe magnetization direction [see Figs. 1(c)and1(f)]. We also
note that the skyrmions obtained here are of the Neel typewith negative effective magnetic monopole charge. In order tounderstand the origin and stability of sSk and pSk, we presentthe results of an effective spin model derived from the RDEHamiltonian.
III. ORIGIN AND STABILITY OF SPARSE AND PACKED
SKYRMIONS
Including the Zeeman coupling term in the recently derived
effective spin model for HRDE[54], we obtain
Heff=−/summationdisplay
/angbracketleftij/angbracketright,γDγ
ijfγ
ij−hz/summationdisplay
iSz
i,
√
2fγ
ij={t2(1+Si·Sj)+2tλˆγ/prime·(Si×Sj)
+λ2[1−Si·Sj+2(ˆγ/prime·Si)(ˆγ/prime·Sj)]}1/2,
Dγ
ij=/angbracketleftbig/bracketleftbig
eihγ
ijd†
idj+H.c./bracketrightbig/angbracketrightbig
gs. (7)
In the above, ˆγ/prime=ˆz׈γ,fγ
ij(hγ
ij) is the modulus (ar-
gument) of the complex number gγ
ij, and /angbracketleftˆO/angbracketrightgsdenotes
expectation values of operator ˆOin the ground state. It has
been shown that using a constant value of Dγ
ijcaptures the
essential physics of the Hamiltonian Eq. ( 7); therefore, we set
Dγ
ij≡D0=1 in our simulations [ 54]. Note that a derivation
starting with a simple two-site picture also leads to an identi-cal functional form for the effective spin model [ 58].
035111-2ELECTRONIC MECHANISM FOR NANOSCALE SKYRMIONS … PHYSICAL REVIEW B 103, 035111 (2021)
FIG. 1. Magnetization Mz(triangles), total skyrmion density χ
(circles), and vector chirality η(squares) as a function of the applied
Zeeman field for (a) α=0.25 and (d) α=0.45. Snapshots of spin
configurations[panels (b) and (e)] and the local skyrmion density[panels (c) and (f)] at T=0.01 for representative values of αand
h
z: (b) and (c) α=0.25 and hz=0.03; (e) and (f) α=0.45 and
hz=0.09.
We simulate Heffusing the conventional classical MC
scheme [ 55]. We find that the field dependence of magneti-
zation, ηandχforHeff, is similar to that obtained via HMC
[compare Figs. 1(a)and1(d) and Fig. 2]. For small values of
α, the magnetization increases linearly for small hz, followed
by a slower than linear rise. This change to nonlinear behavioris accompanied by a sharp increase in the magnitude of χ
[see Figs. 2(a) and2(b)]. A simple understanding is that the
emergence of skyrmions arrests the ease with which spinsalign along the direction of the external magnetic field. Afinite value of ηin the absence of a magnetic field originates
from the DM-like terms present in the effective Hamiltonian.The variation of ηis anticorrelated with that of magnetiza-
tion and the former shows a sharp decrease accompanyingthe increase in magnitude of χ[see Figs. 2(a) and2(b)].
Finally, for still larger values of the applied field, the systemapproaches the sFM state, with both χandηvanishing. For
α=0.5, the change in χnear h
z=0.25 is sharper and is
accompanied by a weak discontinuity in both magnetizationandη[see Fig. 2(c)]. This qualitatively different behavior is
an indicator of the pSk state, as is illustrated below with thehelp of real-space spin configurations. For α=0.6,χis finite
even at h
z=0. This is consistent with our results reported
for Rashba FKLM [ 54]. Interestingly, the magnitude of χ0.0 0.4 0.8
hz−85.00.0
χ(c)α=0.5
0.0 0.6 1.2
hz−75.00.0
χ(d)α=0.60.0 0.03 0.06−1.50.0
χ(a)α=0.12
0.0 0.15 0.3−20.00.0
χ(b)α=0.3
0.01.0 η,M z
0.01.0 η,M z0.01.0 η,M z
0.01.0 η,M z
FIG. 2. (a)–(d) Magnetization (triangles), total skyrmion density
(circles), and vector chirality (squares) as a function of hzfor differ-
ent values of α.T h el e f t y-axis scale is for χ.
reduces with increasing hzand then again increases before
finally vanishing on approach to the sFM state [see Fig. 2(d)].
The re-entrant behavior of χshows that the skyrmion crystal
(SkX) state does not directly lead to the sFM state via isolatedskyrmions, instead a SQ spiral phase is stabilized at interme-diate h
zbefore the sFM state appears in the strong field limit.
This suggests that in contrast to the pSk phase, which can beviewed as a packed arrangement of isolated skyrmions, theSkX phase should be interpreted as a fully cooperative orderedarrangement of spins stable only in the low-field regime. Notethat square lattice of skyrmions has also been reported inexperiments [ 59,60]
We find that, within the ZFC protocol at finite tempera-
tures, the domain junctions in the fDW states for small α[see
Fig. 3(a)] become nucleation centers for skyrmions when a
magnetic field is applied [see Fig. 3(b)]. For larger values of α,
the SQ spiral state gives way to the pSk phase [see Fig. 3(c)].
For a given α, increasing h
zleads, initially, to a reduction of
the size [compare Figs. 3(c)and3(d)] and then to a reduction
of the number of skyrmions [ 55]. We have also confirmed
that the skyrmion formation in the model is not an artifactof the ZFC protocol by verifying their existence using thefield-cooled protocol [ 55]. However, an important question is
whether the sSk phase is a thermodynamically stable groundstate phase. By comparing energies between increasing- anddecreasing- h
zsimulations at low temperatures we find that a
saturated ferromagnet has lower energy compared to that ofthe sSk phase. Therefore, the sSk phase is not a stable groundstate phase. We perform additional simulations starting at lowtemperatures with a sFM configuration and find that isolatedskyrmions spontaneously form by simply increasing the tem-perature in simulations (see Fig. 4). This suggests that the
sSk phase is entropically favored over the sFM phase and
035111-3KATHYAT, MUKHERJEE, AND KUMAR PHYSICAL REVIEW B 103, 035111 (2021)
FIG. 3. Low-temperature snapshots of spin configurations for
representative values of αandhz. (a) fDW state at α=0.16 and
hz=0, (b) sparse skyrmions at α=0.16 and hz=0.036, (c) pSk
atα=0.32 and hz=0.13, and (d) pSk at α=0.32 and hz=0.21.
hence it should be relevant to real systems. We show how the
skyrmion count nSkfirst increases and then decreases upon
increasing T[see inset in Fig. 4(b)]. A possible interpretation
is that isolated skyrmions exist as thermal excitations in theferromagnetic background. However, a confirmation of thisrequires more systematic exploration of the model at finitetemperatures which will be taken up in a separate study. Thereduction of n
Skwith increasing temperature correlates with
the transition of the parent ferromagnetic state into a param-agnetic state.
We now summarize the results discussed above in the form
of a phase diagram in Fig. 5(a). We identify, in addition to
the trivial sFM state, (i) a fDW state, (ii) a SQ spiral withpeaks in the spin structure factor at (0 ,Q)o r( Q,0), (iii) a
pSk state, and (iv) a SkX with square geometry. Furthermore,a metastable sSk region is also indicated in the ground statephase diagram. The boundary between fDW /SQ and sSk /pSk
is determined from the sharp increase in the magnitude of χ
with increasing h
z[see dashed black lines in Figs. 2(a)–2(d)].
Similarly, the boundary between SkX and SQ is inferred fromthe variation in χ[see dashed red line in Fig. 2(d)]. Note
that the sharp change in χis accompanied by a weak but
noticeable change in h
zdependence in magnetization and
chirality. The sFM boundary is defined by the saturation ofmagnetization together with a complete vanishing of chiralityand skyrmion density. It is important to mention that thefinite- TsSk phase can only be characterized by real-space im-
ages showing the presence of isolated skyrmions. Since theseisolated skyrmions exist in the ferromagnetic background anybulk indicators, such as the SSF, will identify this phase asa ferromagnet. Since the definition of the sSk state as a truethermodynamic phase is not possible, we indicate this as ametastable region just below the sFM phase boundary. Thisis to be understood as the region where isolated skyrmions
FIG. 4. Snapshots of typical spin configurations (left column)
and corresponding skyrmion density map (right column) taken from
Monte Carlo simulations with increasing temperature starting from
t h es F Ms t a t ea t T=0.001. (a) and (b): T=0.044, (c) and (d):
T=0.052, and (e) and (f): T=0.058. The inset in panel (b) shows
the variation in the skyrmion count nSkwith the temperature con-
firming the existence of isolated skyrmions in the ferromagneticbackground as finite- Texcitations. The calculations were performed
on a 60 ×60 lattice at α=0.25 and h
z=0.1.
will emerge at finite temperatures. The boundary between pSk
and sSk states is obtained from the hzdependence of the
explicit skyrmion count nSk.In the pSk phase, the number
of skyrmions does not change upon changing the externalmagnetic field [see the inset in Fig. 5(a)]. Strictly at T=0,
the skyrmion count should exhibit a steplike jump to zero.However, at finite Tthere is a narrow region in h
zdisplaying
a steep decrease in the skyrmion count. A similar gradualdecrease in the skyrmion count is expected upon increasingthe temperature close to the paramagnetic phase boundary.This can be interpreted as a melting of the skyrmion latticevia the sSk state [ 61].
The SSF for the fDW, pSk, and SkX states are displayed
in Figs. 5(b)–5(d), in that order. The circular diffuse pattern
for small α[see Figs. 5(b)and5(c)] matches well with SANS
experiments and Fourier transform of LTEM images on MnSiand Co-Zn-Mn alloys [ 11,15,18]. We also characterize the
pSk state by plotting the number of skyrmions n
Skas a func-
tion of the applied field. A plateau in nSkis an indicator of the
pSk state [see the inset in Fig. 5(a)].
The SSF in the pSk phase seems to have a hexagonal
symmetry. This is expected as close packing of disk-shapedparticles will naturally lead to the formation of a triangular
035111-4ELECTRONIC MECHANISM FOR NANOSCALE SKYRMIONS … PHYSICAL REVIEW B 103, 035111 (2021)
FIG. 5. (a) Low-temperature phase diagram in the α-hzplane.
SSFs for (b) fDW at α=0.22 and hz=0, (c) pSk at α=0.4a n d
hz=0.16, and (d) SkX at α=0.6a n d hz=0.3. The inset in panel
(a) shows an explicit count of the skyrmion centers nSkas a function
ofhzalong the vertical dashed line at α=0.5. The sSk phase is
metastable and the sFM state is the true ground state in that parameterregime.
lattice. However, on a closer look one finds that the points on
thekyaxis are more intense than those located close to the
diagonals. We have identified the origin of this asymmetry inthe SSF of the h
z=0 state. As mentioned earlier, the spirals in
thehz=0 SQ phase are doubly degenerate with the (0 ,Q) and
(Q,0) spirals having identical energies. In simulations, one of
these spirals is spontaneously stablized at low temperatures.By selecting two such simulations where different SQ stateswere stabilized, and by increasing h
zin the ZFC protocol,
we obtain pSk phases characterized by an SSF pattern thathas a relative 90
◦rotation [compare Figs. 6(b) and6(d)]. In
either case, the peaks located on the kxorkyaxis are relatively
intense. Observation of these asymmetries suggest that forma-tion of an emergent lattice of extended particles residing on,and defined from, the sites of a square lattice cannot form aperfect hexagonal structure. Clearly, in continuum a triangularlattice can spontaneously form with arbitrary orientations ofthe three defining axes. On a square lattice, however, the xor
ydirection becomes a natural choice for one of the axes of
the triangular lattice, leading to a stronger intensity in the SSFalong that direction.
In Sec. II, we have shown that the RDE model can account
for the formation of skyrmions within an electronic Hamilto-nian without the need to write a spin-only model. In Sec. III,
we explicitly verified that the connection of the electronicHamiltonian approach to the standard DM-interaction-basedapproach to skyrmion formation is understood via the ef-fective spin Hamiltonian derived in our earlier work [ 54].
While it is well known that itinerant electrons stronglycoupled to a magnetic skyrmion background generate an
FIG. 6. SSF for hz=0 [panels (a) and (c)] and hz=0.16 [panels
(b) and (d)] in two independent simulations. The ground state in the
absence of the magnetic field is doubly degenerate with (0 ,Q)a n d
(Q,0) spirals having equal energy. The hexagonal pattern in the SSF
at a finite magnetic field displaying a slight asymmetry related to the
hz=0 spiral state.
anomalous response in transport [ 31,62], the influence of
Rashba coupling on transport properties has been pointed outonly recently [ 47]. For consistency and completeness, in the
next section we discuss the effect of magnetic skyrmion stateson the electronic properties. This is important to emphasizeas in the existing literature it is common practice to use thestandard DE model for studying the response of itinerantelectrons to unconventional spin textures [ 30,62–64]. The fact
that spin-orbit interactions play a crucial role in stabilizingskyrmion textures is commonly ignored when analyzing theresponse of itinerant electrons to skyrmions and the resultinganomalous Hall physics.
IV. BOTT INDEX AND TOPOLOGICAL METALICITY
The possibility of finding amorphous analogs of transla-
tionally invariant topological insulators has attracted muchattention in recent years. Models for disordered topologicalmetallic or insulating states have been proposed [ 50,65]. A
crucial feature of these models is a spatially dependent patternof hopping parameters which may not be easy to realize. It iswell known that electrons coupled to noncoplanar magneticpatterns experience an effective magnetic field and generatean anomalous Hall effect [ 31,62]. We study how the presence
of Rashba coupling in the DE model affects this anomalousresponse. We present topological characterization of the sSkand pSk states by computing the Bott index Band the Hall
conductivity σ
xy[55] (see also Refs. [ 51,52] therein). Loring
and Hastings first introduced the concept of Bott index incondensed matter systems [ 66,67]. It is a measure of the inca-
pability of the system to form localized Wannier orbitals fromthe occupied states [ 67]. Our motivation to compute the Bott
index here is to mathematically confirm the topological aspectof the band structure of electrons in the presence of magnetic
035111-5KATHYAT, MUKHERJEE, AND KUMAR PHYSICAL REVIEW B 103, 035111 (2021)
FIG. 7. (a) Bott index Band Hall conductivity σxy(in units of e2/h) as functions of the Fermi level EF, (b) low-temperature magnetic
configuration obtained via simulations with open boundary conditions, (c) LDOS at skyrmion cores with (red lines) and without (blue lines)
local gauge fields, and (d) real-space map of LDOS at E=−3.38 in the absence of local gauge fields ( hγ
ij≡0). Panels (a)–(d) display results
forα=0.15. Panels (e)–(h) show the same quantities as shown in panels (a)–(d), in that order, for α=0.30. LDOS in panel (h) is shown for
E=−2.87. The results are obtained on 100 ×100 lattice using open boundary conditions.
skyrmion states. The implementation details of the Bott index
calculation can be found in the literature [ 50–52,68]. Never-
theless, for completeness we outline the key steps below. Thefirst step is to construct a projection operator out of all theoccupied states:
P=
Nel/summationdisplay
k=1|ψk/angbracketright/angbracketleftψk|, (8)
where |ψk/angbracketrightis the occupied eigenstate corresponding to the
kth eigenvalue Ek, and Nelis the number of electrons in the
system. The position coordinates ( xi,yi) of any lattice site i
can be mapped into the spherical coordinates ( /Theta1i,/Phi1i)o na
torus where 0 /lessorequalslant/Theta1i<2πand 0/lessorequalslant/Phi1i<2π. The next step is
to define the projected position operators,
U=Pei/Theta1P,V=Pei/Phi1P, (9)
where /Theta1and/Phi1are the diagonal matrices with /Theta1iand/Phi1ias
diagonal elements, respectively. The Bott index is given by
B=1
2πIm{tr[log( VUV†U†)]}, (10)
For numerical stability of the algorithm, we perform sin-
gular value decomposition of the projected position operators
UandVfollowing Huang and Liu [ 51,52]. The Hall conduc-
tivityσxyis computed using the Kubo formula [ 55]. Both sSk
and pSk states support finite values of σxyas well as B[see
Figs. 7(a)and7(e)]. Both quantities display a change of sign
as the Fermi level crosses zero. Since the Bott index is ananalog of the Chern index for inhomogeneous systems, theaforementioned correspondence between the Hall conductiv-ity and the Bott index confirms the topological aspect of theskyrmion phases in the RDE model in the same manner as thefinite Chern index confirms the topological nature of states intranslationally invariant systems. Therefore, metallic systemswith few or many skyrmions can be classified as topological
metals in close analogy with recently proposed hopping-pattern-engineered tight-binding models [ 50,65]. Further, the
larger magnitude of σ
xyin Fig. 7(e) compared to that in
Fig.7(a)is due to the increase in the number of skyrmions. In
the sparse skyrmion regime, the Hall conductivity increaseslinearly with the number of skyrmions. The dependence be-comes sublinear on approach to a pSk phase. On the otherhand, in the pSk phase increasing the number of skyrmionsnecessarily leads to a decrease in the size of skyrmions. Theconsequence is larger gauge fields and hence a smaller andquantized Hall conductivity (see Fig. S1 in the SupplementalMaterial [ 55]).
In order to further differentiate between the Rashba-
modified DE mechanism from the standard DE physics, wecalculate the LDOS, ρ
i(E)=1/N/summationtext
k|ψk
i|2δ(E−Ek), where
ψk
iis the amplitude on site iof the single-particle eigenstate
|ψk/angbracketrightcorresponding to eigenvalue Ekof the RDE Hamiltonian
Eq. ( 2). A Lorentzian with broadening parameter 0.01 is used
to approximate the Dirac delta function. We find that theskyrmion textures in magnetization have strong implicationsfor the electronic wave functions in this unusual metallicphase. We use open boundary conditions for LDOS calcu-lations in order to illustrate the presence of edge modes dueto skyrmion-induced gauge fields [ 55]. Note that the open
boundary condition results lead to visible textures in mag-netization along the edges. These are a simple consequenceof competition between DM-like and ferromagnetic terms inthe spin Hamiltonian along the edges and are unrelated to thepresence of skyrmions in the bulk. We focus on the LDOSfor sites located in skyrmion cores. In the sparse skyrmioncase, there is a weak enhancement in LDOS near the bandedge [see Fig. 7(c)]. The effect becomes much pronounced
for the packed skyrmion state. Furthermore, periodic modu-lations as a function of energy become clear [see Fig. 7(g)].
035111-6ELECTRONIC MECHANISM FOR NANOSCALE SKYRMIONS … PHYSICAL REVIEW B 103, 035111 (2021)
(c)
(d)
(a)
(b)
0.00.040.080.12
0.00.20.40.6
FIG. 8. For the sSk configuration obtained at α=0.15 and hZ=
0.04 [shown in Fig. 7(b)]: LDOS map for energy near the band edge
(a) in an inconsistent electronic model with α=0.0 and (b) for the
consistent calculation with α=0.15 in the itinerant model. For the
pSk configuration shown in Fig. 7(f): LDOS maps within (c) the in-
consistent calculation with α=0.0 and (d) the consistent calculation
withα=0.30. Note the qualitative difference between the left and
right columns: while the skyrmion cores behave as repulsive centers
for electrons in α=0 calculations, they become attractive centers in
the consistent calculations.
The inset in Fig. 7(g) shows the energy difference of two
consecutive peaks, /Delta1En, as a function of peak index. There
are two possible interpretations of the spikes in LDOS. Theycan appear either due to the confinement effect, similar tothose reported in metallic nanoislands and carbon nanotubeswith defect [ 69,70] or due to effective magnetic flux hidden
in the gauge fields. We find a clear approach to disentanglethese two effects. Ignoring the phases in the complex hop-ping parameters g
γ
ijin the RDE Hamiltonian sets the gauge
fields to zero, and the resulting model with real hoppingparameters contains pure confinement effects. The results ofLDOS calculations using h
γ
ij≡0i nE q .( 2) [blue lines in
Figs. 7(c) and7(g)] show that the periodic modulations van-
ish and only a single peak near the band edge survives. InFigs. 7(d) and7(h), we plot lattice maps of LDOS for the
energy fixed at peak locations. The resulting maps displayinhomogeneities and a clear localization of electronic wavefunctions at skyrmion cores for the pSk state [see Fig. 7(h)].
Note that the depletion of LDOS along the edges visible inFig.7(d) is related to the magnetic texture along the edges in
Fig.7(b) and is not a topological feature. The above analysis
proves that, although the confinement effects are present dueto changes in the magnitude of g
γ
ij, the oscillations can only
be explained by Landau level physics arising from effectivemagnetic flux hidden in complex gγ
ij. This is further confirmed
by performing LDOS calculations on ideal skyrmion latticeswhere we explicitly show quantization of σ
xy[55]( s e ea l s o
Ref. [ 71] and references therein).In order to clearly emphasize the importance of a consis-
tent treatment of Rashba coupling in the DE mechanism ofskyrmion formation, we demonstrate the qualitative differencebetween LDOS maps obtained without and with the Rashbaterm. Once again we take the typical configurations from sSkand pSk phases for this demonstration. For both sSk and pSkstates, LDOS maps calculated by setting α=0 display a de-
pletion of electron density near skyrmion cores [see Figs. 8(a)
and8(c)]. When the consistent calculations are performed by
setting the value of αequal to that used for obtaining the
skyrmion textures, an opposite qualitative picture emerges.The skyrmion cores tend to behave as attraction centers forthe electronic charge [see Figs. 8(b)and8(d)]. This change of
qualitative behavior is a clear sign of caution for the calcula-tions performed within the conventional DE approach.
V. CONCLUSION
The double-exchange mechanism provides a basis for un-
derstanding ferromagnetism in a variety of metallic magnets.We have uncovered an aspect associated with the classic DEmechanism by including the effect of Rashba SOC and theZeeman field in the DE model. An explicit demonstration ofthe existence of nanoscale skyrmions in an electronic modelwith no direct spin-spin interactions is presented. In the pres-ence of a magnetic field, at finite temperatures, phases withsparse as well as packed skyrmions are stabilized. While thepSk states are shown to be true ground states of the model,the sparse skyrmions are metastable in the ground state butoccur at finite temperature as excitations of the ferromag-net. The circular patterns in the SSF are remarkably similarto those reported in the SANS experiments on Co-Zn-Mnalloys and MnSi [ 15,18]. The corresponding real-space im-
ages, representative of fDW states, are also in agreementwith the LTEM images on FeGe, Co-Zn-Mn, and transitionmetal multilayers [[ 8–10,12,13,15,27],[48,49]]. The origin of
the skyrmion states lies in the anisotropy terms of the DMinteraction and the pseudodipolar form that become apparentin the effective Hamiltonian derived from the RDE model. Inaddition to a hexagonal packed lattice of skyrmions, we alsofind a qualitatively different square lattice skyrmion crystalstable for larger values of α. Hall conductivity and Bott index
calculations are presented to explicitly demonstrate that theskyrmion states are examples of amorphous topological met-als. Analogy with recently proposed tight-binding models foramorphous topological metals and insulators is also discussed.The LDOS calculations are presented in order to emphasizethe importance of Rashba coupling in the DE mechanism thatis commonly used for analyzing the influence of magnetictextures on electronic transport. The characteristic oscillationsin LDOS as a function of energy can be directly measured inexperiments via dI/dVspectra. Such dI/dVmeasurements
can also be used to estimate the strength of Rashba couplingin a metal with the help of a careful modeling of the data.
ACKNOWLEDGMENTS
We thank Goutam Sheet and Yogesh Singh for valuable
discussions. We acknowledge the use of the computing facilityat IISER Mohali.
035111-7KATHYAT, MUKHERJEE, AND KUMAR PHYSICAL REVIEW B 103, 035111 (2021)
[1] A. Fert, N. Reyren, and V . Cros, Nat. Rev. Mater. 2, 17031
(2017) .
[2] R. Wiesendanger, Nat. Rev. Mater. 1, 16044 (2016) .
[3] A. Fert, V . Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152
(2013) .
[4] N. Nagaosa and Y . Tokura, Nat. Nanotechnol. 8, 899 (2013) .
[5] B. Göbel, A. Mook, J. Henk, and I. Mertig, Phys. Rev. B 99,
020405(R) (2019) .
[6] A. N. Bogdanov and C. Panagopoulos, Phys. Today 73(3), 44
(2020) .
[7] B. Dupé, M. Hoffmann, C. Paillard, and S. Heinze, Nat.
Commun. 5, 4030 (2014) .
[ 8 ]S .D .P o l l a r d ,J .A .G a r l o w ,J .Y u ,Z .W a n g ,Y .Z h u ,a n dH .
Yang, Nat. Commun. 8, 14761 (2017) .
[9] A. Soumyanarayanan, M. Raju, A. L. Gonzalez Oyarce,
A. K. C. Tan, M.-Y . Im, A. Petrovi ´c, P. Ho, K. H. Khoo, M.
Tran, C. K. Gan, F. Ernult, and C. Panagopoulos, Nat. Mater.
16, 898 (2017) .
[10] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter,
K. von Bergmann, A. Kubetzka, and R. Wiesendanger, Science
341, 636 (2013) .
[11] X. Z. Yu, N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara, K.
Kimoto, Y . Matsui, Y . Onose, and Y . Tokura, Nat. Commun. 3,
988 (2012) .
[12] X. Z. Yu, N. Kanazawa, Y . Onose, K. Kimoto, W. Z. Zhang, S.
Ishiwata, Y . Matsui, and Y . Tokura, Nat. Mater. 10, 106 (2011) .
[13] X. Zhao, C. Jin, C. Wang, H. Du, J. Zang, M. Tian, R. Che, and
Y . Zhang, Proc. Natl. Acad. Sci. U.S.A. 113, 4918 (2016) .
[14] S. Meyer, M. Perini, S. von Malottki, A. Kubetzka, R.
Wiesendanger, K. von Bergmann, and S. Heinze, Nat.
Commun. 10, 3823 (2019) .
[15] A. Tonomura, X. Yu, K. Yanagisawa, T. Matsuda, Y . Onose,
N. Kanazawa, H. S. Park, and Y . Tokura, Nano Lett. 12, 1673
(2012) .
[16] M. Hirschberger, T. Nakajima, S. Gao, L. Peng, A. Kikkawa, T.
Kurumaji, M. Kriener, Y . Yamasaki, H. Sagayama, H. Nakao,K. Ohishi, K. Kakurai, Y . Taguchi, X. Yu, T.-h. Arima, and Y .Tokura, Nat. Commun. 10, 5831 (2019) .
[17] C. Jin, Z.-A. Li, A. Kovács, J. Caron, F. Zheng, F. N. Rybakov,
N. S. Kiselev, H. Du, S. Blügel, M. Tian, Y . Zhang, M. Farle,and R. E. Dunin-Borkowski, Nat. Commun. 8, 15569 (2017) .
[18] K. Karube, J. S. White, N. Reynolds, J. L. Gavilano, H. Oike, A.
Kikkawa, F. Kagawa, Y . Tokunaga, H. M. Rønnow, Y . Tokura,and Y . Taguchi, Nat. Mater. 15, 1237 (2016) .
[19] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A.
Neubauer, R. Georgii, and P. Böni, Science 323, 915 (2009) .
[20] K. M. Song, J.-S. Jeong, B. Pan, X. Zhang, J. Xia, S. Cha, T.-E.
Park, K. Kim, S. Finizio, J. Raabe, J. Chang, Y . Zhou, W. Zhao,W. Kang, H. Ju, and S. Woo, Nat. Electron. 3, 148 (2020) .
[21] J. Sampaio, V . Cros, S. Rohart, A. Thiaville, and A. Fert, Nat.
Nanotechnol. 8, 839 (2013) .
[22] A. K. Nayak, V . Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo,
F. Damay, U. K. Rößler, C. Felser, and S. S. P. Parkin, Nature
(London) 548, 561 (2017) .
[23] C. Pfleiderer, D. Reznik, L. Pintschovius, H. v. Löhneysen, M.
Garst, and A. Rosch, Nature (London) 427, 227 (2004) .
[24] J. Jena, B. Göbel, T. Ma, V . Kumar, R. Saha, I. Mertig, C. Felser,
a n dS .S .P .P a r k i n , Nat. Commun. 11, 1115 (2020) .
[25] P.-J. Hsu, L. Rózsa, A. Finco, L. Schmidt, K. Palotás, E.
Vedmedenko, L. Udvardi, L. Szunyogh, A. Kubetzka, K. vonBergmann, and R. Wiesendanger, Nat. Commun. 9, 1571
(2018) .
[26] X. Z. Yu, W. Koshibae, Y . Tokunaga, K. Shibata, Y . Taguchi, N.
Nagaosa, and Y . Tokura, Nature (London) 564, 95 (2018) .
[27] T. Nagase, M. Komatsu, Y . G. So, T. Ishida, H. Yoshida, Y .
Kawaguchi, Y . Tanaka, K. Saitoh, N. Ikarashi, M. Kuwahara,and M. Nagao, Phys. Rev. Lett. 123, 137203 (2019) .
[28] U. K. Rößler, A. N. Bogdanov, and C. Pfleiderer, Nature
(London) 442, 797 (2006) .
[29] J. P. Chen, D.-W. Zhang, and J. M. Liu, Sci. Rep. 6, 29126
(2016) .
[30] N. Mohanta, E. Dagotto, and S. Okamoto, Phys. Rev. B 100,
064429 (2019) .
[31] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys. Rev.
Lett.107, 136804 (2011) .
[32] J. Iwasaki, A. J. Beekman, and N. Nagaosa, P h y s .R e v .B 89,
064412 (2014) .
[33] S. G. Yang, Y . H. Liu, and J. H. Han, P h y s .R e v .B 94
, 054420
(2016) .
[34] X. X. Zhang, A. S. Mishchenko, G. De Filippis, and N.
Nagaosa, P h y s .R e v .B 94, 174428 (2016) .
[35] A. Farrell and T. Pereg-Barnea, Phys. Rev. B 89, 035112
(2014) .
[36] R. Ozawa, S. Hayami, and Y . Motome, Phys. Rev. Lett. 118,
147205 (2017) .
[37] Z. Wang, Y . Su, S.-Z. Lin, and C. D. Batista, Phys. Rev. Lett.
124, 207201 (2020) .
[38] W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch,
F. Y . Fradin, J. E. Pearson, Y . Tserkovnyak, K. L. Wang, O.Heinonen, S. G. Te Velthuis, and A. Hoffmann, Science 349,
283 (2015) .
[39] N. Mathur, M. J. Stolt, and S. Jin, APL Mater. 7, 120703 (2019) .
[40] C. Zener, Phys. Rev. 82, 403 (1951) .
[41] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 (1955) .
[42] P. G. de Gennes, Phys. Rev. 118, 141 (1960) .
[43] E. Dagotto, Nanoscale Phase Separation and Collosal Magne-
toresistance (Springer, Berlin, 2002).
[44] K. Pradhan and S. K. Das, Sci. Rep. 7, 9603 (2017) .
[45] A. Yaouanc, P. Dalmas de Réotier, B. Roessli, A. Maisuradze,
A. Amato, D. Andreica, and G. Lapertot, P h y s .R e v .R e s . 2,
013029 (2020) .
[46] D. Bombor, C. G. F. Blum, O. V olkonskiy, S. Rodan, S.
Wurmehl, C. Hess, and B. Büchner, Phys. Rev. Lett. 110,
066601 (2013) .
[47] S.-S. Zhang, H. Ishizuka, H. Zhang, G. B. Halász, and C. D.
Batista, Phys. Rev. B 101, 024420 (2020) .
[48] K. Karube, J. S. White, D. Morikawa, C. D. Dewhurst, R.
Cubitt, A. Kikkawa, X. Yu, Y . Tokunaga, T.-h. Arima, H. M.Rønnow, Y . Tokura, and Y . Taguchi, Sci. Adv. 4, eaar7043
(2018) .
[49] Y . Tokunaga, X. Z. Yu, J. S. White, H. M. Rønnow, D.
Morikawa, Y . Taguchi, and Y . Tokura, Nat. Commun. 6, 7638
(2015) .
[50] A. Agarwala and V . B. Shenoy, Phys. Rev. Lett. 118, 236402
(2017)
.
[51] H. Huang and F. Liu, P h y s .R e v .L e t t . 121, 126401 (2018) .
[52] H. Huang and F. Liu, P h y s .R e v .B 98, 125130 (2018) .
[53] D. C. Vaz, F. Trier, A. Dyrdał, A. Johansson, K. Garcia, A.
Barthélémy, I. Mertig, J. Barna ´s, A. Fert, and M. Bibes, Phys.
Rev. Mater. 4, 071001(R) (2020) .
035111-8ELECTRONIC MECHANISM FOR NANOSCALE SKYRMIONS … PHYSICAL REVIEW B 103, 035111 (2021)
[54] D. S. Kathyat, A. Mukherjee, and S. Kumar, P h y s .R e v .B 102,
075106 (2020) .
[55] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.103.035111 for further details.
[56] S. Kumar and P. Majumdar, E u r .P h y s .J .B 50, 571 (2006) .
[57] A. Mukherjee, N. D. Patel, C. Bishop, and E. Dagotto, Phys.
Rev. E 91, 063303 (2015) .
[58] S. Banerjee, J. Rowland, O. Erten, and M. Randeria, Phys. Rev.
X4, 031045 (2014) .
[59] N. D. Khanh, T. Nakajima, X. Yu, S. Gao, K. Shibata, M.
Hirschberger, Y . Yamasaki, H. Sagayama, H. Nakao, L. Peng,K. Nakajima, R. Takagi, T. hisa Arima, Y . Tokura, and S. Seki,Nat. Nanotechnol. 15, 444 (2020) .
[60] T. Tanigaki, K. Shibata, N. Kanazawa, X. Yu, Y . Onose, H. S.
Park, D. Shindo, and Y . Tokura, Nano Lett. 15, 5438 (2015) .
[61] Y . Nishikawa, K. Hukushima, and W. Krauth, P h y s .R e v .B 99,
064435 (2019) .
[62] B. Göbel, A. Mook, J. Henk, and I. Mertig, E u r .P h y s .J .B 91,
179 (2018) .[63] K. Hamamoto, M. Ezawa, and N. Nagaosa, P h y s .R e v .B 92,
115417 (2015) .
[64] B. Göbel, A. Mook, J. Henk, and I. Mertig, Phys. Rev. B 95,
094413 (2017) .
[65] Y .-B. Yang, T. Qin, D.-L. Deng, L.-M. Duan, and Y . Xu, Phys.
Rev. Lett. 123, 076401 (2019) .
[66] T. A. Loring and M. B. Hastings, Europhys. Lett. 92, 67004
(2010) .
[67] M. B. Hastings and T. A. Loring, Ann. Phys. (New York) 326,
1699 (2011) .
[68] D. Toniolo, Phys. Rev. B 98, 235425 (2018) .
[69] H.-T. Yang, J.-W. Chen, L.-F. Yang, and J. Dong, Phys. Rev. B
71, 085402 (2005) .
[70] M. V . Rastei, B. Heinrich, L. Limot, P. A. Ignatiev, V . S.
Stepanyuk, P. Bruno, and J. P. Bucher, Phys. Rev. Lett. 99,
246102 (2007) .
[71] F. Tejo, A. Riveros, J. Escrig, K. Y . Guslienko,
and O. Chubykalo-Fesenko, Sci. Rep. 8
, 6280
(2018) .
035111-9 |
PhysRevB.87.075407.pdf | PHYSICAL REVIEW B 87, 075407 (2013)
Electronic structure of CoPc adsorbed on Ag(100): Evidence for molecule-substrate interaction
mediated by Co 3 dorbitals
E. Salomon,1,*P. Amsalem,2N. Marom,3M. V ondracek,4L. Kronik,5N. Koch,2,6and T. Angot1
1Aix-Marseille Universit ´e, CNRS, PIIM UMR 7345, 13397 Marseille, France
2Humboldt Universit ¨at zu Berlin, D-12489 Berlin, Germany
3Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, Texas 78712, USA
4Institute of Physics AS CR, Na slovance 2, CZ-182 21 Praha 8, Czech Republic
5Department of Materials and Interfaces, Weizmann Institute of Science, Rehovoth 76100, Israel
6Helmholtz-Zentrum Berlin f ¨ur Materialien und Energie GmbH, BESSY II, Berlin, Germany
(Received 21 November 2012; published 6 February 2013)
The electronic structure of cobalt-phthalocyanine (CoPc) molecules adsorbed on Ag(100) is investigated by
photoemission spectroscopy. The results are compared to first-principles electronic structure calculations, basedon many-body perturbation theory in the GW approximation. The photoemission data, obtained from bothmultilayer and monolayer films of CoPc, show that charge transfer occurs between the first molecular layerand the metal surface. Varying the photon energy, to tune the photoionization cross sections, reveals that thecharge-transfer-related interface states mainly involve the Co 3 datomic orbitals of the Co central atom. GW
calculations for the neutral CoPc molecule and its anion compare well with the experimental observations for amultilayer and a monolayer CoPc film, respectively. They confirm the major role played by the Co atom in thecharge-transfer process and elucidate the complex energy rearrangement of the molecular electronic levels uponmetal adsorption.
DOI: 10.1103/PhysRevB.87.075407 PACS number(s): 73 .20.−r, 73.22.−f
I. INTRODUCTION
The performance of organic-molecule-based electronic
devices depends critically on the energy level alignment atelectrode-organic interfaces. When contacting π-conjugated
organic materials with (electrode) metal surfaces, large mod-ifications of the properties of the pristine materials are oftenobserved, which therefore strongly affect the electronic behav-ior of (opto-)electronic devices.
1–6Despite considerable effort
devoted to characterization of the electronic properties of suchinterfaces, our understanding of the fundamental processesgoverning the metal-molecule interactions is still incomplete.This is because the interaction is affected by the interplayof many complex mechanisms, e.g., dispersive interactions,Pauli repulsion, interface states, covalent and/or ionic bonds,charge transfer, and more.
2,7,8Metal-phthalocyanines (MPc)
are among the most widely studied organic molecular materialsas they can be used in a broad range of applications,including electronic, optical, or magnetic devices.
9–11The
variety of possible central metal atoms offers the opportu-nity to tune their optoelectronic properties as well as theirreactivity with metal electrodes. MPcs are known to formwell-ordered films on various metal surfaces, which serveas model systems for both device-relevant and theoreticalinvestigations. Depending on the substrate and the centralmetal atom used, the balance between molecule-substrate
and intermolecular interactions (and possibly the competition
between kinetics and thermodynamics) can stabilize severalmolecular superstructures with different lattice parameters,commensurate or not with the metal substrate.
12–15This feature
allows for tuning the intermolecular distance and thereforestudying collective electronic properties. Additionally, theability to use magnetic central metal atoms (e.g., Co, Fe, andMn) in these molecules is promising for the development oforganic-based spintronic devices. Fundamental understandingof magnetism and many-body effects at the molecular level is
essential for this purpose.
16–23
In recent studies of the electronic properties of well-
ordered MPc films adsorbed on metal single crystal surfaces,
charge-transfer states, Kondo resonances, and related interfaceplasmons were identified.
17,21,24–27These effects result from
electron donation from the metal to the (delocalized) lowest
unoccupied molecular orbital (LUMO) and from the important
role played by many-body effects in these systems.12,13,25–29
FePc and CoPc films exhibit a different behavior, because
the specific electronic configuration of the Fe and Co 3 d
atomic orbitals gives rise to a molecule-substrate interaction,
which occurs mainly via the central metal atom.17,22,30–39Most
experiments on FePc and CoPc molecules to date have beenperformed on Au single crystals, whose surface is relativelyinert toward the MPc molecules. The investigation of CoPc
films adsorbed on more reactive surfaces, such as Ag, is of
interest because the intimate nature of the interaction betweenCoPc and Ag may involve stronger bonds or different orbitalsthan in the case of Au surfaces.
First-principles electronic structure calculations may
substantially help the interpretation of experimentaldata.
20,22,37,40–47Typically, density functional theory (DFT) is
the method of choice for such calculations. For MPcs DFTshould be used with great care.
41,42,46,48–52In calculations
employing semilocal functionals, significant distortions tothe photoemission spectra have been found, arising fromsevere self-interaction errors (SIE, the spurious Coulombinteraction of an electron with itself
53), involving orbitals
strongly localized on the metal atom. The addition of a fractionof exact (Fock) exchange, as in hybrid functionals, was foundto mitigate the SIE effect considerably and improve the agree-ment with photoemission spectroscopy (PES) experiments.However, DFT eigenvalues, including those obtained fromhybrid functionals, are not formally equivalent to quasiparticle
075407-1 1098-0121/2013/87(7)/075407(9) ©2013 American Physical SocietyE. SALOMON et al. PHYSICAL REVIEW B 87, 075407 (2013)
(QP) excitation energies.54,55Therefore, further improvement
may be achieved by calculating the QP energies directly withinthe framework of many-body perturbation theory, typicallyemployed within the GW approximation.
56,57Here, G is the
one-particle Green function and W is the dynamically screenedCoulomb interaction.
Unlike DFT eigenvalues, the QP energies obtained from
GW calculations can be directly comparable to the electronremoval energies measured in PES. Another advantage of theGW approach for metal-adsorbed molecules is that it treatsinterface polarization effects correctly, whereas DFT does notwith either semilocal or conventional hybrid functionals.
58–61
Partly owing to the prohibitive computational cost of fully
self-consistent GW calculations and partly because they donot always offer improvement over non-self-consistent GWcalculations,
62–67GW calculations are typically performed
non-self-consistently. Within the G 0W0approximation, the QP
energies are obtained as a perturbative first-order correction tothe DFT eigenvalues, using the DFT orbitals and eigenvalues tocalculate the self-energy. Due to the non-self-consistent natureof the G
0W0approach, the results depend on the quality of the
underlying DFT calculation. It has been shown that thanks tothe mitigation of SIE in hybrid functionals they often serve asa better G
0W0starting point than semilocal functionals.62,68–70
Indeed, such calculations, performed recently for CuPc, have
yielded excellent agreement with experiment and have furtherimproved upon DFT results, particularly with respect to theposition of highly localized Cu-derived orbitals.
68
Here, we study the electronic properties of CoPc films
adsorbed on a Ag(100) surface by means of PES. By tuning thephoton energy, the photoionization cross sections are changingand thus, we are able to demonstrate that for one monolayer(ML) of CoPc, only the molecular orbitals (MO) in which thereis a substantial contribution of the Co 3 datomic orbitals (AO)
are strongly modified upon adsorption. This indicates that the
molecules mainly interact with the substrate via their centralCo atom, toward which charge transfer occurs. We comparethe experimental spectra with DFT and GW calculations for aCoPc molecule. The GW results for the neutral molecule arein excellent agreement with the multilayer PES, in terms ofboth peak positions and their AO character. GW calculationsperformed for a CoPc anion confirm the reorganization of theCo 3dorbitals observed in the PES data for the monolayer
film.
II. EXPERIMENTAL DETAILS
Direct PES experiments were carried out at the Material
Science Beamline of the Elettra synchrotron (Trieste, Italy) andrecorded in angle-integrated mode. The experimental chamberis equipped with a high luminosity electron energy analyzer(Specs Phoibos, 150 mm mean radius, with nine channels)and a low-energy electron diffraction (LEED) apparatus. Theresolution of the PES measurements, determined from thewidth of the Fermi edge on the clean Ag(100), was 0.15 eV .Ag(100) single crystals (misorientation /lessorequalslant0.1
◦) from Surface
Preparation Laboratory were prepared by repeated cyclesof Ar-ion sputtering (800 eV) and annealing (700 K). Thisprocedure produces a clean and well-ordered Ag surfaceexhibiting a sharp LEED pattern of a square (1 ×1) unit cell,with a low-elastic background. CoPc powder, bought from
Sigma-Aldrich, was purified by outgassing under ultrahighvacuum conditions at temperatures up to 600 K. They weresubsequently thermally evaporated onto the clean Ag(100)surface, which was kept at room temperature.
III. COMPUTATIONAL DETAILS
G0W0calculations were performed using the all-electron
numerical atom-centered orbital (NAO) code FHI-AIMS .71,72
The NAO basis sets are grouped into a minimal basis, con-
taining only basis functions for the core and valence electronsof the free atom, followed by four hierarchically constructedsets of additional basis functions, denoted as tiers 1–4.
71A
detailed account of the all-electron implementation of G 0W0in
FHI-AIMS i sg i v e ni nR e f . 69. Briefly, the implementation makes
use of the resolution-of-identity (RI) technique, in which a setof auxiliary basis functions is introduced to represent both theCoulomb potential and the noninteracting response function.This allows for efficient GW calculations with NAO basisfunctions. The self-energy is first calculated on the imaginaryfrequency axis and then analytically continued to the realfrequency axis using a two-pole fitting procedure. The G
0W0
calculations presented here were performed with the highlyconverged tier 4 NAO basis set.
Due to the non-self-consistent nature of the G
0W0ap-
proach, the results depend on the quality of the underlyingDFT calculation. Indeed, for CuPc it has been shown that aG
0W0calculation based on a hybrid functional yields excellent
agreement with experiment, whereas a G 0W0calculation based
on a semilocal functional does not, due to the propagation ofSIE from the DFT level to the G
0W0level.68Owing to the
resemblance of CoPc to CuPc in terms of SIE at the DFTlevel,
48,49one may expect similar behavior at the G 0W0level.
We therefore examined two starting points for G 0W0calcu-
lations: (i) the generalized gradient approximation of Perdew,Burke, and Ernzerhof (PBE)
73,74and (ii) the one-parameter
hybrid functional, known as PBEh (or PBE0),75–77in which
25% of exact (Fock) exchange are added to PBE. We denotethese calculations as G
0W0@PBE and G 0W0@PBEh, respec-
tively. The detailed comparison between the two starting pointsis reported in the Appendix. As expected, the G
0W0@PBEh
results are in better agreement with the experimental datapresented here than the G
0W0@PBE ones. Therefore the
results reported throughout this paper are those obtained withthe PBEh starting point. Finally, a Gaussian broadening of0.3 eV was applied to the computed energy levels to simulatethe effective experimental broadening. Cross-section effectsare not taken into account in the computed spectra andthe comparison of theory to experiment is focused on peakpositions.
IV . RESULTS AND DISCUSSION
A. CoPc multilayer film
Figure 1presents the valence band (VB) region of a
multilayer CoPc film, measured using two different photonenergies. These spectra are, as expected, in good agreementwith the experimental data presented by Aristov et al. for the
case of a 70 ˚A multilayer film of CoPc adsorbed on Au(001).
40
075407-2ELECTRONIC STRUCTURE OF CoPc ADSORBED ON ... PHYSICAL REVIEW B 87, 075407 (2013)
7 6 5 4 3 2 1 0Binding energy with respect to Ef (eV)
12 11 10 9 8 7 6 5
Energy (eV)Intensity (arb. unit) hν = 126 eV
hν = 25 eV
GW@PBEh
ABRegion 2Region 1
FIG. 1. (Color online) Photoemission spectra of the valence band
region of a CoPc multilayer film, obtained using photon energies
of 126 and 25 eV , compared with the simulated valence bandphotoemission spectrum of an isolated CoPc molecule, obtained using
the G
0W0@PBEh approximation.
The lowest binding energy (BE) peak ( A) is attributed to
the highest occupied molecular orbital (HOMO) of CoPc, ascommon to most of the metallophthalocyanine molecules sinceit is independent of the central atom.
49,50,78The second lowest
BE peak ( B), centered around 2.2 eV , is more specific to CoPc.
To establish the physical origin of this latter peak, we variedthe photon energy and compared the VBs recorded at a 25 anda 126 eV photon energy. In Fig. 1, one can clearly see that
the intensity ratio between the two lowest BE peaks stronglydepends on the photon energy: At the 25 eV photon energy,theBtoAratio (B/A ) is close to one, but it is roughly three
times larger at the 126 eV photon energy. Moreover, it isclear that region 1 of the density of states (DOS), denoted inFig. 1, is also strongly enhanced at 126 eV , as compared to the
HOMO and to region 2. As the photoelectron inelastic meanfree path does not change drastically in this energy range,the observed modification of the DOS intensity cannot beattributed to different probing depths. In addition, according tothe calculations of Yeh and Lindau, the atomic photoionizationcross sections (PICS) of the N 2 pand C 2 pAO decrease
dramatically with increasing photon energy, as compared to amuch more moderate decrease for the Co 3 dAO (cf. Table I).
79
TABLE I. Photoionization cross-section values for selected
atomic orbitals for the two photon energies used in this study, from
Ref. 79.
PICS (Mbarn)
Atomic orbital hν=25 eV hν=126 eV
Co 3d 5.40 4.03
C2p 4.80 0.09
N2p 8.34 0.287 6 5 4 3 2 1 0
12 11 10 9 8 7 6 5Energy (eV)Intensity (arb. unit)
a1u eg
b2g
hν = 25 eV
GW@PBEh
PBEh
AB
Region 2Region 1
Binding energy with respect to Ef (eV)
FIG. 2. (Color online) Experimentally measured valence band
photoemission spectrum of a CoPc multilayer film, obtained using
a photon energy of 25 eV , compared with simulated valence band
photoemission spectra of an isolated CoPc molecule based on DFTusing the PBEh functional and many-body perturbation theory within
the G
0W0@PBEh approximation. Illustrations of selected orbitals are
also shown.
Therefore, the modification in the intensity of the VB
suggests that there is a substantial contribution from the Co 3 d
AO to the DOS in the region around 2, 4, and 5.2 eV BE.
For further corroboration and interpretation of the ex-
perimental findings we turn to the GW calculations. TheG
0W0@PBEh spectrum, as well as the PBEh spectrum used as
its starting point, of the valence spectrum of an isolated CoPcmolecule, are compared in Fig. 2to the above multilayer-film
experimental data. We note that both the GW and the DFT datawere rigidly shifted so as to align the position of the HOMOwith the leading photoemission peak. Unlike in the PBEh data,the QP energies obtained from a GW calculation for an isolatedmolecule are directly comparable to the electron removalenergies obtained from gas-phase PES (see the Appendix andRef. 68). However a rigid shift is still necessary here to account
for polarization effects in the multilayer film.
Considering both Figs. 1and 2it appears that the main
features of the experimental spectrum are well reproducedin the G
0W0@PBEh spectrum, but not with the PBEh
spectrum. In the latter, the peak labeled B, which exhibits
a strong dependence on the photon energy, is clearly absent.Furthermore, in region 1, around BEs of 4 and 5.2 eV , theexperimental data show a strong enhancement of the DOS.According to the PICS this means that there is a significantcontribution of the Co 3 dAO in these subregions. Looking at
the molecular orbitals, illustrated in Fig. 2, it appears that their
075407-3E. SALOMON et al. PHYSICAL REVIEW B 87, 075407 (2013)
energy distribution obtained in the G 0W0@PBEh calculation
more accurately represents the electronic structure than theone resulting from the PBEh calculation. Peak Ais associated
with the HOMO, an a
1uorbital delocalized over the whole
macrocycle, similarly to other MPcs. The second peak ( B)
is composed of three distinct frontier orbitals: eg↓,eg↑, and
b2g↓. These orbitals have significant contributions from the Co
atom, which accounts for ∼40% of their electron density. In
the PBEh spectrum these orbitals are too high in binding energywith respect to the HOMO, such that they are merged with thethird PES peak in region 1 (cf. Figs. 2and7). G
0W0@PBEh
places these orbitals at the correct distance from the HOMO, inexcellent agreement with PES. This is reminiscent of the caseof the Cu-derived b
1g↑orbital in CuPc.68At higher energy
in region 1, there are also several MOs with significant Co3dcontributions. Because of the drastic change in the PICS
of the latter AO as compared to the C 2 pand N 2 p, these
MOs are enhanced at 126 eV . This induces, as observedexperimentally, a strong modification of the DOS intensitywith the photon energy, and confirms the above mentionedhypothesis regarding the contribution of the Co 3 dAO to the
DOS around 2, 4, and 5.2 eV .
B. CoPc monolayer
At ML coverage, the molecules arrange on the surface
following a (5 ×5) reconstruction. A representative LEED
pattern of this molecular superstructure in presented in Fig. 3.
As in the above case of the multilayer film, the VB of the
ordered ML has been investigated with both 25 and 126 eVphoton energies. The corresponding spectra are presentedin Fig. 4. At a 25 eV photon energy the VB exhibits two
peaks located at 1.4 and 1.0 eV from the Fermi level. Similarfeatures have been recently reported after adsorption of CoPcmolecules on the Ag(111) surface.
38,39Changing the photon
energy from 25 to 126 eV greatly increases the VB signal, andmore particularly of the lowest BE peak [cf. Fig. 4(a)]. As in
the multilayer film discussed above, this strong modificationof the VB intensity with the photon energy is indicative of
FIG. 3. (Color online) LEED pattern of a CoPc monolayer
adsorbed on the Ag(100) surface. The primary beam energy was
set to 45 eV . The long and short arrows represent the Ag and CoPc
superstructure basis vectors, respectively.Intensity (arb. unit)
3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5
Binding energy Ef (eV)Clean Ag(100) hν = 25eV
hν = 126eV
1ML CoPc / Ag(100) hν = 25eV
hν = 126eVa
Intensity (arb. unit)
3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5AB Clean Ag(100)
1ML CoPc / Ag(100)
TF CoPc / Ag(100) b
with respect to Binding energy Ef (eV) with respect to
FIG. 4. (a) Valence band photoemission spectra measured for
both a clean and a Ag(100) surface with an adsorbed CoPc monolayer,
using different photon energies. (b) Valence band photoemissionspectra of the clean, as well as the CoPc monolayer and multilayer
adsorbed surfaces, recorded at a photon energy of 25 eV .
a substantial contribution of the Co 3 dAO to the pertinent
MOs. As the HOMO of the neutral (isolated) CoPc is mostlycomprised of C 2 pand N 2 pAO, we attribute the presence
of a Co-derived peak at 1.0 eV to an interface state resultingfrom the interaction of the molecule with the metal surface.The DOS at 1.4 eV BE cannot be strictly related to the formerHOMO of the isolated molecule as it too is enhanced at 126 eVphoton energy, again demonstrating a participation of the Co3dAO. This suggests a nontrivial charge redistribution upon
charge transfer. In addition, Fig. 4(b) provides a comparison
of the low-energy region of the multilayer and monolayerspectra taken at a 25 eV photon energy. Markedly, the 2.2 eVBE feature observed for the multilayer film, in which the CoAO are strongly involved, completely vanishes at monolayercoverage. Still, both the 2.2 eV (multilayer) and the 1.0 eV(ML) states are enhanced by increasing the photon energyfrom 25 to 126 eV , i.e., are related to Co-derived states.
It is known that upon adsorption on metal surfaces,
molecular adsorbates often undergo charge transfer (CT).
2
This interaction mechanism usually involves electron transferfrom the metal to the LUMO of the neutral molecules. Becausein the case of the MPc molecules, the central metal atomtypically has a small contribution to the LUMO (3% for theCoPc), a strong modification of the metal AO is unlikely ifonly the LUMO is involved in the CT mechanism. In fact,because of the specific occupancy of the dorbitals of the
central Co atom of the CoPc molecules, it is possible thatthe unpaired electrons of the d
z2orbital of the Co form a
partial bond with the underlying Ag substrate. Therefore,we suggest that the metal-organic CT occurs mainly viathe central Co atom. Such interaction was already proposedby Gargiani and co-workers and more recently by Mugarzaet al.
30,80Explicit G 0W0calculations of the metal-adsorbed
ML raise very serious formal and computational difficulties.59
Fortunately, they can be avoided here. Under the assumptionthat the rearrangement of the Co 3 dorbitals upon transfer of
approximately one electron from the substrate (as shown inRef. 80) is similar to the rearrangement upon one electron
075407-4ELECTRONIC STRUCTURE OF CoPc ADSORBED ON ... PHYSICAL REVIEW B 87, 075407 (2013)
IIntensity (arb. unit)10 8 6 4 2 0Energy (eV)
eg a1g eg a1u eg eg b2g
FIG. 5. (Color online) Simulated electronic structure obtained
using G 0W0@PBEh for a neutral CoPc molecule (bottom), a CoPc
anion in an open-shell singlet configuration (middle), and a CoPcanion in a closed-shell configuration (top). The lowest binding energy
a
1uMO of the anion has been shifted and aligned with the HOMO of
the neutral CoPc. Illustrations of selected orbitals are also shown.
addition, it suffices to compare the experimental data to a
G0W0@PBEh calculation for the CoPc anion.
Figure 5shows a comparison between the G 0W0@PBEh
spectra of a neutral CoPc molecule and a CoPc anion.Similarly to other MPcs,
50the CoPc anion has two electronic
configurations that are close in energy, an “open-shell singlet”configuration and a “closed-shell” configuration, both ofwhich are shown. As different chemical environments maystabilize different electronic configurations, we examine bothconfigurations with respect to the experimentally observeddifferences between the multilayer and ML spectra. In thefigure, the lowest binding energy a
1uMO of the anion has
been shifted and aligned with the HOMO of the neutralCoPc. The main differences between the spectra of the neutralCoPc and its anion in the open-shell singlet configurationare the presence of an additional peak at a lower BE thanthe former HOMO, followed by a double peak, spread overabout 2 eV . While the former difference is consistent withthe experimental observation, the latter is not. Apart from thelowest BE peak, only one peak is observed experimentallyrather than two (see Fig. 4). Furthermore, the broad double
peak is inconsistent with the quenching of peak B, which is
clearly evident in the experimental data. Finally, the intensityratio between the lowest BE peak and the double peakis inverted, as compared to experiment. We thus concludethat the open-shell singlet configuration does not sufficientlyagree with experiment and turn to examine the closed-shellconfiguration. The main differences between the spectra ofthe neutral molecule and the closed-shell anion are in betteragreement with the experimental observations, notably, (i) thepresence of an additional (single) peak at a lower BE thanthe former HOMO and (ii) the quenching of the peak B,centered around 7.5 eV with respect to the vacuum, and 2.2 eV
with respect to the Fermi level. In addition, the intensity ratiobetween the two lowest BE peaks agrees with experiment.We thus conclude that at monolayer coverage, the closed-shellconfiguration is more favorable and we may now use it tointerpret the experimental data, in terms of orbital assignment.
The (closed-shell) G
0W0@PBEh calculations for the anion
reveal that the lowest BE peak is not the egorbital, which is
the LUMO of neutral CoPc. Rather, the extra electron inducessignificant charge redistribution, such that the anion HOMO isthea
1gorbital, which is the LUMO +1 of neutral CoPc. This
agrees with a similar finding by Gargiani and co-workers inthe context of doping CoPc films using alkali-metal atoms.
81
Unlike the a1uHOMO of the neutral, the a1gHOMO of
the anion is highly localized on the Co central atom. Thisparticipation of the Co 3 dAO in the lowest MO is consistent
with the assignment of the experimentally observed interfacestate at 1.0 eV to a Co 3 d-derived orbital. This interface state
arises from electron transfer from the metal surface to themolecule, via its Co atom. In the context of a surface-adsorbedmolecule, the preferential filling of the a
1gorbital may be
explained from spatial considerations. This orbital, derivedfrom the Co d
z2AO, sticks out from the plane of the macrocycle
and forms a partial bond with the substrate.
In addition to the a1gHOMO, G 0W0@PBEh predicts an
egHOMO-1, an a1uHOMO-2, and a b2gHOMO-3. This
orbital assignment is again consistent with the experimentallyobserved change in the intensity of the peaks when goingfrom a photon energy of 25 eV to 126 eV (see Fig. 4and
additional analysis in the Appendix), which indicate that theHOMO and HOMO-3 possess a significant Co 3 dcontribution,
whereas the HOMO-1 and HOMO-2 do not. We note that
22.0 1.5 1.0 0.5 0.001
2
301
23
Intensity (arb. unit)
Binding energy (eV)
FIG. 6. (Color online) Experimental VB (open circle) of the CoPc
monolayer recorded at 25 eV (bottom) and 126 eV (top) together with
its fit (solid line) using four Gaussian functions.
075407-5E. SALOMON et al. PHYSICAL REVIEW B 87, 075407 (2013)
TABLE II. Full width at half maximum (FWHM) and relative
area of the Gaussian functions used in the decomposition procedure
depicted in Fig. 6.
hν=25 eV hν=126 eV
BE FWHM Rel. FWHM Rel.
Peak (eV) (eV) area (%) (eV) area (%)
0 0.7 0.3 7.5 0.4 281 1.0 0.3 61.5 0.4 51.52 1.4 0.3 29 0.4 16.5
3 1.8 0.3 1 0.4 4
the quenching of peak Band the strong Co 3 dcontribution
to the anion HOMO are not captured at the DFT level (seefurther discussion in the Appendix), underscoring again theimportance of many-body perturbation theory here. The factthat the experimental valence features are reproduced bycharge addition while completely neglecting the underlyingsubstrate surface suggests that the interaction mechanism doesnot include formation of significant covalent bonds with themetal states, but rather that the bonding has a significant ioniccharacter. Furthermore, the orbital rearrangement shown inFig. 5reveals that the DOS of the CoPc anion cannot be
addressed naively, i.e., in terms of an additional peak and anotherwise rigid shift with respect to the neutral. Rather, it isthe result of a complex energy redistribution of the molecularorbitals occurring upon electron addition.
To gain further insight into the nature of the orbitals, in
terms of the Co 3 dcontributions, we perform a Gaussian
decomposition of the lowest BE part of the experimental datafor the CoPc monolayer, taken at both 25 and 126 eV . Figure 6
shows the results of this fitting procedure performed, in bothcases, using four Gaussian functions, whose parameters aresummarized in Table II.
Based on the modification of the relative area of each peak
upon changing the photon energy from 25 to 126 eV , it is clearthat peaks 0 and 3 are strongly enhanced (roughly by a factor of4 for both peaks), while peaks 1 and 2 are attenuated. Becausethe Co 3 dPICS is enhanced at 126 eV , we attribute peaks
0 and 3 to the Co-derived a
1gandb2gorbitals, respectively,
whereas peaks 1 and 2 may be attributed to the eganda1u
orbitals, respectively. This assignment once again agrees with
the G 0W0@PBEh calculation in the closed-shell configuration.
V . CONCLUSION
In summary, valence band photoemission spectroscopy
employing different excitation energies was used to studythe DOS of a multilayer and a monolayer film of Ag(100)-adsorbed CoPc. For the monolayer film, the presence ofan interface state, involving the Co 3 dAO, was revealed.
This indicates that in the case of the CoPc molecule, and incontrast with most other MPcs, the molecule-substrate inter-action mainly occurs via the metal central atom. Many-bodyperturbation theory calculations, performed in the frameworkof the GW approximation for the neutral molecule and theCoPc anion, yield for good agreement with experimental datafor both CoPc multi- and monolayer films, respectively. Thecombination of theory and experiment enables the assignmentof the origin of the interface state to charge transfer into the a
1g
molecular orbital, associated with the Co dz2atomic orbital,
rather than into the egmolecular orbital, which corresponds to
the LUMO of the neutral molecule. Furthermore, the theoryallows for a detailed assignment of other spectral featuresand reveals a complex energy rearrangement of the electroniclevels, associated with the Co 3 dorbitals of Ag(100)-adsorbed
CoPc, owing to charge transfer from the metal surface.
ACKNOWLEDGMENTS
Work at the Weizmann Institute was supported by the Israel
Science Foundation and the Lise Meitner Minerva Centerfor Computational Chemistry. Computational resources wereprovided by the National Energy Research Scientific Comput-ing Center (NERSC), the Texas Advanced Computing Center(TACC), and the Argonne Leadership Computing Facility(ALCF). P.A. and N.K. acknowledge financial support bythe DGF (SPP1355 and SFB951). The Materials ScienceBeamline is supported by the Ministry of Education of theCzech Republic under Grants No. LA08022, No. LD11047,and No. LG12003. We thank Pr. C. Ziegler for kindly providingpermission to use their gas-phase photoemission data of CoPc.
APPENDIX
In light of the significant starting point dependence of G 0W0
calculations for CuPc68and the similarity of CoPc to CuPc, in
terms of SIE at the DFT level,48,49we compare the performance
of G 0W0based on the PBE and PBEh functionals for CoPc.
Figure 7shows the DFT and G 0W0spectra obtained with
PBE and PBEh, compared to the recently published gas-phase
FIG. 7. (Color online) Simulated electronic structure of the CoPc
molecule, obtained from DFT and G 0W0based on the PBE and
PBEh functionals, compared to the gas-phase photoemission data of
V ogel et al. (Ref. 42). In the figure the a1uorbital corresponds to the
HOMO of the CoPc molecule. The DFT spectra are shifted to align
the HOMO level with the ionization potential obtained from the total
energy difference between neutral CoPc and its cation.
075407-6ELECTRONIC STRUCTURE OF CoPc ADSORBED ON ... PHYSICAL REVIEW B 87, 075407 (2013)
photoemission data of Ref. 42, which were not available at the
time of publication of Ref. 49. The DFT spectra were shifted
to align the HOMO level with the ionization potential obtainedfrom the total energy difference between the CoPc neutral andcation. No shift was applied to the G
0W0spectra because the
calculated QP energies should be directly comparable to theelectron detachment energies measured in gas-phase PES.
At the DFT level, the main differences between the PBE
and PBEh spectra are in the positions of the occupied e
g
andb2gorbitals and the half-filled a1gorbitals, all of which
are localized on the Co atom and contain significant Co3dcontributions. PBE underbinds the occupied e
gandb2g
orbitals due to the repulsion caused by SIE. This leads to a
wide peak, which is in disagreement with the narrow HOMOpeak obtained in experiment. The addition of a fraction ofexact exchange as in PBEh mitigates the effect of SIE andresults in significantly improved agreement with experiment,as suggested earlier in Ref. 49. However, the spacing between
the HOMO and the HOMO-1 is now too large and theHOMO-1 peak, which is clearly visible in the photoemissiondata, is missing from the PBEh spectrum.
Turning to the a
1gorbital, PBE underestimates its spin
splitting, considerably owing to SIE, such that the unoccupieda
1gorbital is predicted to be the LUMO and is shifted to a much
lower binding energy, resulting in an unreasonably narrow gapof less than 0.5 eV .
48PBEh yields a larger spin splitting for the
half-filled a1gorbitals, such that the unoccupied a1gorbital
is predicted to be the LUMO +1 and the HOMO-LUMO
gap increases to 2.4 eV . Although the HOMO-LUMO gapsobtained from PBE and PBEh are not expected to equalthe fundamental gap,
82,83their magnitude is still important
because the overscreening due to a severely underestimatedgap may be detrimental for the G
0W0calculations.
The differences between the PBE and PBEh spectra, caused
by SIE at the DFT level, are manifested at the G 0W0level as
well. G 0W0@PBE improves on the PBE result by increasing
the spin splitting of the egandb2gorbitals and shifting the
b2gorbitals to a higher binding energy with respect to the eg
orbitals. However, this still results in a wide feature, which
disagrees with the narrow HOMO peak in the PES, andthere is still no peak that matches the PES HOMO-1 peak.Qualitatively, the G
0W0@PBEh corrections are of similar
nature. However, in this case, owing to the better PBEh startingpoint, the positions of the e
gorbitals are in excellent agreement
with the PES HOMO-1 and HOMO-2 peaks. Similarly, thespin splitting of the half-filled a
1gorbital obtained from
G0W0@PBE is still considerably smaller than that obtained
from G 0W0@PBEh. As a result, the a1gorbital is predicted
to be the LUMO with G 0W0@PBE but the LUMO +1 with
G0W0@PBEh. This is also manifested in the HOMO-LUMO
gap. While G 0W0with either starting point increases the
gap significantly with respect to DFT, the HOMO-LUMOgap with G
0W0@PBE is 3.3 eV , compared to 4.0 eV with
G0W0@PBEh. For lack of experimental data we cannot
confirm our predictions for the ordering of the unoccupiedorbitals of CoPc. However, based on its performance for theoccupied states and on the precedent of CuPc, we considerG
0W0@PBEh to be more reliable than G 0W0@PBE.68
The case of CoPc exemplifies, yet again, the propagation
of SIE from the DFT level to the G 0W0level. Owing to the
FIG. 8. (Color online) Simulated valence band photoemission
spectra of the CoPc anion, obtained from DFT and G 0W0using
the PBE and PBEh functionals. For PBEh, both the closed-shell and
the open-shell singlet electronic configurations are shown. The DFT
spectra are shifted to align the anion HOMO level with the electronaffinity obtained from the total energy difference between neutral
CoPc and its anion.
localization of metal-derived states, metal-organic molecules
are particularly prone to severe SIE, which manifests as strongG
0W0starting point dependence. Therefore, choosing an
appropriate starting point is essential for obtaining meaningfulresults for such systems.
The starting point sensitivity of the G
0W0calculations is
also apparent for the CoPc anion. Figure 8shows the DFT
and G 0W0spectra obtained with PBE and PBEh. For PBEh,
both the closed-shell and the open-shell singlet electronicconfigurations are shown. The open-shell configuration isslightly more energetically stable than the closed-shell con-figuration. However, as suggested for other MPcs,
50the latter
may be stabilized by the interaction with the Ag surface.As in Fig. 7, the DFT spectra are shifted to align the anion
HOMO level with its ionization potential obtained from thetotal energy difference between the neutral CoPc and itsanion. For the CoPc anion, PBE and PBEh (in both electronicconfigurations) predict that the additional electron occupiesthea
1gorbital. Other than that, the two functionals predict
rather different electronic configurations. This is apparent inthe orbital ordering and in the resulting line shape of thepredicted spectra (cf. Fig. 7).
We are not aware of gas-phase PES data for the CoPc
anion. Therefore, we can only compare the anion calculatedspectra to the measured ML spectra of the HOMO region,as done in Fig. 4of the main text. The main differences
between the multilayer and ML spectra, which we interpret ascorresponding to the difference between neutral and anionicCoPc, respectively, are (i) two frontier peaks in the MLspectrum, as opposed to one HOMO peak in the multilayer
075407-7E. SALOMON et al. PHYSICAL REVIEW B 87, 075407 (2013)
spectrum and (ii) peak B in the multilayer spectrum is
quenched in the ML spectrum. None of the DFT spectracorrectly capture these differences. For the open-shell singletconfiguration, neither does GW. However, in the closed-shellsinglet configuration, both G
0W0@PBE and G 0W0@PBEh
reproduce the main features of the ML spectrum, in thesense that both possess two frontier peaks followed by a gap.The main difference between these two G
0W0spectra lies
in the HOMO-2 and HOMO-3 levels, where the b2ganda1u
change their order. Based on a comparison to the Gaussian
decomposition of the monolayer spectrum, shown in Fig. 6,w econclude that the orbital ordering obtained from G 0W0@PBEh
is in better agreement with experiment.
To summarize, the CoPc neutral and anion exhibit sig-
nificant self-interaction errors at the DFT level, associatedwith the strong localization of states derived from the Coatom. These errors are “inherited” at the G
0W0level and
are manifested by an unusually strong G 0W0starting point
dependence. Therefore, choosing an appropriate starting pointis essential for obtaining meaningful results for such systems.Our study indicates that for the CoPc system, the PBEhfunctional provides such an appropriate starting point.
*eric.salomon@univ-amu.fr
1J. Blochwitz, M. Pfeiffer, T. Fritz, and K. Leo, Appl. Phys. Lett. 73,
729 (1998).
2N. Koch, ChemPhysChem. 8, 1438 (2007).
3X. Zhou, M. Pfeiffer, J. Blochwitz, A. Werner, A. Nollau, T. Fritz,
and K. Leo, Appl. Phys. Lett. 78, 410 (2001).
4M. A. Baldo and S. R. Forrest, Phys. Rev. B 64, 085201 (2001).
5G. Horowitz, R. Hajlaoui, H. Bouchriha, R. Bourguiga, and
M. Hajlaoui, Adv. Mater. 10, 923 (1998).
6D. Cahen, A. Kahn, and E. Umbach, Mater. Today 8, 32 (2005).
7H. Ishii, K. Sugiyama, E. Ito, and K. Seki, Adv. Mater. 11, 605
(1999).
8L. Kronik and N. Koch, MRS Bull. 35, 417 (2010).
9J. Jiang, Functional Phthalocyanine Molecular Materials
(Springer-Verlag, Berlin, Heidelberg, 2010).
10G. Chaidogiannos, F. Petraki, N. Glezos, S. Kennou, andS. Nespurek, Appl. Phys. A: Mater. Sci. Process. 96, 763 (2009).
11D. Hohnholz, S. Steinbrecher, and M. Hanack, J. Mol. Struct. 521,
231 (2000).
12B. Stadtmuller, I. Kroger, F. Reinert, and C. Kumpf, P h y s .R e v .B
83, 085416 (2011).
13I. Kroger, B. Stadtmuller, C. Stadler, J. Ziroff, M. Kochler, A. Stahl,
F. Pollinger, T.-L. Lee, J. Zegenhagen, F. Reinert et al. ,New J. Phys.
12, 083038 (2010).
14C. Stadler, S. Hansen, I. Kroger, C. Kumpf, and E. Umbach, Nat.
Phys. 5, 153 (2009).
15S. C. Bobaru, E. Salomon, J.-M. Layet, and T. Angot, J. Phys.
Chem. C 115, 5875 (2011).
16K. J. Franke, G. Schulze, and J. Pascual, Science 332, 940 (2011).
17A. Zhao, Q. Li, L. Chen, H. Xiang, W. Wang, S. Pan, B. Wang,
X. Xiao, J. Yang, J. G. Hou et al. ,Science 309, 1542 (2005).
18X. Chen, Y .-S. Fu, S.-H. Ji, T. Zhang, P. Cheng, X.-C. Ma, X.-L.
Zou, W.-H. Duan, J.-F. Jia, and Q.-K. Xue, Phys. Rev. Lett. 101,
197208 (2008).
19N. Ishikawa, Struct. Bond. 135, 211 (2010).
20H. Wende, M. Bernien, J. Luo, C. Sorg, N. Ponpandian, J. Kurde,
J. Miguel, M. Piantek, X. Xu, P. Eckhold et al. ,Nature Mater. 6,
516 (2007).
21U .G .E .P e r e r a ,H .J .K u l i k ,V .I a n c u ,L .G .G .V .D i a sd aS i l v a ,S. E. Ulloa, N. Marzari, and S.-W. Hla, Phys. Rev. Lett. 105, 106601
(2010).
22J. Brede, N. Atodiresei, S. Kuck, P. Lazic, V . Caciuc, Y . Morikawa,G. Hoffmann, S. Blugel, and R. Wiesendanger, Phys. Rev. Lett.
105, 047204 (2010).23N. Atodiresei, J. Brede, P. Lazic, V . Caciuc, G. Hoffmann,
R. Wiesendanger, and S. Blugel, P h y s .R e v .L e t t . 105, 066601
(2010).
24N. Tsukahara, S. Shiraki, S. Itou, N. Ohta, N. Takagi, and M. Kawai,Phys. Rev. Lett. 106, 187201 (2011).
25L. Giovanelli, P. Amsalem, T. Angot, L. Petaccia, S. Gorovikov,
L. Porte, A. Goldoni, and J.-M. Themlin, P h y s .R e v .B 82, 125431
(2010).
26P. Amsalem, L. Giovanelli, J. M. Themlin, and T. Angot, Phys. Rev.
B79, 235426 (2009).
27E. Salomon, J.-M. Layet, and T. Angot, P h y s .R e v .B 85, 125420
(2012).
28W. Dou, Y . Tang, C. Lee, S. Bao, and S. Lee, J. Chem. Phys. 133,
144704 (2010).
29J. Ziroff, S. Hame, M. Kochler, A. Bendounan, A. Scholl, andF. Reinert, P h y s .R e v .B 85, 161404 (2012).
30P. Gargiani, M. Angelucci, C. Mariani, and M. G. Betti, Phys. Rev.
B81, 085412 (2010).
31M. G. Betti, P. Gargiani, R. Frisenda, R. Biagi, A. Cossaro,
A. Verdini, L. Floreano, and C. Mariani, J. Phys. Chem. C 114,
21638 (2010).
32J. Xiao and P. A. Dowben, J. Phys.: Condens. Matter 21, 052001
(2009).
33T. R. Ellis, K. T. Park, M. D. Ulrich, S. L. Hulbert, and J. E. Rowe,J. Appl. Phys. 100, 093515 (2006).
34H. Peisert, I. Biswas, U. Aygul, A. V ollmer, and T. Chasse, Chem.
Phys. Lett. 493, 126 (2010).
35F. Petraki, H. Peisert, F. Latteyer, U. Aygul, A. V ollmer, and
T. Chasse, J. Phys. Chem. C 115, 21334 (2011).
36F. Petraki, H. Peisert, I. Biswas, and T. Chasse, J. Phys. Chem. C
114, 17638 (2010).
37J. D. Baran, J. A. Larsson, R. A. J. Woolley, Y . Cong, P. J. Moriarty,
A. A. Cafolla, K. Schulte, and V . R. Dhanak, P h y s .R e v .B 81,
075413 (2010).
38M. Schmid, A. Kaftan, H.-P. Steinruuck, and J. M. Gottfried, Surf.
Sci.606, 945 (2012).
39M. Toader, P. Shukrynau, M. Knupfer, D. R. T. Zahn, and
M. Hietschold, Langmuir 28, 13325 (2012).
40V . Maslyuk, V . Aristov, O. Molodtsova, D. Vyalikh, V . Zhilin,
Y . Ossipyan, T. Bredow, I. Mertig, and M. Knupfer, Appl. Phys. A
94, 485 (2009).
41B. Brena, C. Puglia, M. de Simone, M. Coreno, K. Tarafder,
V . Feyer, R. Banerjee, E. Gothelid, B. Sanyal, P. M. Oppeneeret al. ,J. Chem. Phys. 134, 074312 (2011).
075407-8ELECTRONIC STRUCTURE OF CoPc ADSORBED ON ... PHYSICAL REVIEW B 87, 075407 (2013)
42M. V ogel, F. Schmitt, J. Sauther, B. Baumann, A. Altenhof, S. Lach,
and C. Ziegler, Anal. Bioanal. Chem. 400, 673 (2011).
43M. Dauth, T. Korzdorfer, S. Kummel, J. Ziroff, M. Wiessner,
A. Scholl, F. Reinert, M. Arita, and K. Shimada, Phys. Rev. Lett.
107, 193002 (2011).
44F. Petraki, H. Peisert, P. Hoffmann, J. Uihlein, M. Knupfer, and
T. Chasse, J. Phys. Chem. C 116, 5121 (2012).
45X. Sun, B. Wang, and Y . Yamauchi, J. Phys. Chem. C 116, 18752
(2012).
46L. Kronik and Y . Morikawa, in The Molecule-Metal Interface ,
edited by N. Koch, N. Ueno, and A. T. S. Wee (Wiley-VCH,Weinheim, 2013).
47P. Puschnig, E. M. Reinisch, T. Ules, G. Koller, S. Soubatch,M. Ostler, L. Romaner, F. S. Tautz, C. Ambrosch-Draxl, and M. G.Ramsey, Phys. Rev. B 84, 235427 (2011).
48N. Marom, O. Hod, G. E. Scuseria, and L. Kronik, J. Chem. Phys.
128, 164107 (2008).
49N. Marom and L. Kronik, Appl. Phys. A 95, 159 (2009).
50N. Marom and L. Kronik, Appl. Phys. A 95, 165 (2009).
51D. Stradi, C. Diaz, F. Martin, and M. Alcami, Theor. Chem. Acc.
128, 497 (2011).
52C. J. Cramer and D. G. Truhlar, Phys. Chem. Chem. Phys. 11, 10757
(2009).
53J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
54M. Jain, J. R. Chelikowsky, and S. G. Louie, Phys. Rev. Lett. 107,
216806 (2011).
55R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689
(1989).
56L. Hedin, Phys. Rev. 139, A796 (1965).
57M. S. Hybertsen and S. G. Louie, P h y s .R e v .B 34, 5390 (1986).
58J. B. Neaton, M. S. Hybertsen, and S. G. Louie, P h y s .R e v .L e t t . 97,
216405 (2006).
59I. Tamblyn, P. Darancet, S. Y . Quek, S. A. Bonev, and J. B. Neaton,P h y s .R e v .B 84, 201402(R) (2011).
60A. Biller, I. Tamblyn, J. B. Neaton, and L. Kronik, J. Chem. Phys.
135, 164706 (2011).
61J. M. Garcia-Lastra, C. Rostgaard, A. Rubio, and K. S. Thygesen,
P h y s .R e v .B 80, 245427 (2009).62N. Marom, F. Caruso, X. Ren, O. T. Hofmann, T. Korzdorfer, J. R.
Chelikowsky, A. Rubio, M. Scheffler, and P. Rinke, Phys. Rev. B
86, 245127 (2012).
63K. Delaney, P. Garcia-Gonzalez, A. Rubio, P. Rinke, and R. W.
Godby, Phys. Rev. Lett. 93, 249701 (2004).
64B. Holm and U. von Barth, Phys. Rev. B 57, 2108 (1998).
65W.-D. Schone and A. G. Eguiluz, Phys. Rev. Lett. 81, 1662 (1998).
66K. Kaasbjerg and K. S. Thygesen, Phys. Rev. B 81, 085102 (2010).
67A. Schindlmayr, T. J. Pollehn, and R. W. Godby, Phys. Rev. B 58,
12684 (1998).
68N. Marom, X. G. Ren, J. E. Moussa, J. R. Chelikowsky, andL. Kronik, P h y s .R e v .B 84, 195143 (2011).
69X. Ren, A. Sanfilipo, P. Rinke, J. Wieferink, A. Tkatchenko,
K. Reuter, V . Blum, and M. Scheffler, New J. Phys. 14, 053020
(2012).
70T. Korzdorfer and N. Marom, Phys. Rev. B 86, 041110(R) (2012).
71V . Blum, R. Gehrke, F. Hanke, P. Havu, V . Havu, X. Ren, K. Reuter,
and M. Scheffler, Comput. Phys. Commun. 180, 2175 (2009).
72V . Havu, V . Blum, P. Havu, and M. Scheffler, J. Comput. Phys. 228,
8367 (2009).
73J. P. Perdew, K. Burke, and M. Ernzerhof, P h y s .R e v .L e t t . 77, 3865
(1996).
74J. P. Perdew, K. Burke, and M. Ernzerhof, P h y s .R e v .L e t t . 78, 1396
(1997).
75C. Adamo and V . Barone, J. Chem. Phys. 110, 6158 (1999).
76J. P. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982
(1996).
77M. Ernzerhof and G. E. Scuseria, J. Chem. Phys. 110, 5029 (1999).
78N. Papageorgiou, Y . Ferro, E. Salomon, A. Allouche, J. M. Layet,
L. Giovanelli, and G. Le Lay, Phys. Rev. B 68, 235105 (2003).
79J. J. Yeh and I. Lindau, At. Data Nucl. Data Tables 32, 1 (1985).
80A. Mugarza, R. Robles, C. Krull, R. Korytar, N. Lorente, and
P. Gambardella, P h y s .R e v .B 85, 155437 (2012).
81P. Gargiani, A. Calabrese, C. Mariani, and M. G. Betti, J. Phys.
Chem. C 114, 12258 (2010).
82L. Kronik, T. Stein, S. Refaely-Abramson, and R. Baer, J. Chem.
Theory Comput. 8, 1515 (2012).
83S. Kummel and L. Kronik, Rev. Mod. Phys. 80, 3 (2008).
075407-9 |
PhysRevB.88.174506.pdf | PHYSICAL REVIEW B 88, 174506 (2013)
Doping and critical-temperature dependence of the energy gaps in Ba(Fe 1−xCo x)2As2thin films
P. Pecchio, D. Daghero, G. A. Ummarino, and R. S. Gonnelli
Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, 10129 Torino, Italy
F. Kurth, B. Holzapfel, and K. Iida
Leibniz-Institut f ¨ur Festk ¨orper-und Werkstoffforschung (IFW) Dresden, P .O. Box 270116, 01171 Dresden, Germany
(Received 1 August 2013; revised manuscript received 22 October 2013; published 11 November 2013)
The dependence of the superconducting gaps in epitaxial Ba(Fe 1−xCox)2As2thin films on the nominal doping
x(0.04/lessorequalslantx/lessorequalslant0.15) was studied by means of point-contact Andreev-reflection spectroscopy. The normalized
conductance curves were well fitted by using the two-dimensional Blonder-Tinkham-Klapwijk model with twonodeless, isotropic gaps—although the possible presence of gap anisotropies cannot be completely excluded.The amplitudes of the two gaps /Delta1
Sand/Delta1Lshow similar monotonic trends as a function of the local critical
temperature TA
c(measured in the same point contacts) from 25 K down to 8 K. The dependence of the gaps
onxis well correlated to the trend of the critical temperature, i.e., to the shape of the superconducting region
in the phase diagram. When analyzed within a simple three-band Eliashberg model, this trend turns out to becompatible with a mechanism of superconducting coupling mediated by spin fluctuations, whose characteristicenergy scales with T
caccording to the empirical law /Omega10=4.65kBTc, and with a total electron-boson coupling
strength λtot=2.22 for x/lessorequalslant0.10 (i.e., up to optimal doping) that slightly decreases to λtot=1.82 in the overdoped
samples ( x=0.15).
DOI: 10.1103/PhysRevB.88.174506 PACS number(s): 74 .45.+c, 74.70.Xa, 74 .78.−w
I. INTRODUCTION
The research on Fe-based superconductors has been re-
cently boosted by the progress in the techniques of film depo-sition. Films of very high quality are necessary for applicationsin superconducting electronics, i.e., for the fabrication ofJosephson junctions,
1superconducting quantum interference
devices,2and so on. However, they can be fruitfully used also
to investigate fundamental properties of these compounds.For instance, they are the perfect playground for transport,optical, and spectroscopic measurements of various kinds;thanks to strain/stress effects that can be induced by thesubstrate,
3thin films offer an additional way to tune the
critical temperature; finally, they are necessary to realize someproposed phase-sensitive experiments
4to determine the order
parameter symmetry ( s++ ors±).
So far, thin films of 122 Fe-based compounds have been
used to investigate, for example, the gap amplitude andstructure, which are probably the most intriguing open issuesof these superconductors. As a matter of fact, the emergenceof zeros or nodes in the gap has been predicted theoreticallywithin the s±symmetry
5–8as a result of the strong sensitivity
of the Fermi surface (FS) to the details of the lattice structure. In10% Co-doped BaFe
2As2(Ba-122) thin films, measurements
of the complex dynamical conductivity9have shown a small
isotropic gap of about 3 meV and a larger, highly anisotropicgap of about 8 meV—possibly featuring vertical node lines—located on the electronlike FS sheet. A superconducting gapof 2.8 meV has been measured also by terahertz conductivityspectroscopy in thin films of the same compound with T
c=
19 K, but has been associated with the electronlike FS.10
Optical reflectivity and complex transmittivity measurementsin Co-doped Ba-122 films (with nominal x=0.10) have
given instead an isotropic gap of 1 .85±0.15 meV ,
11but
have also shown a low-frequency absorption much strongerthan expected for an s-wave gap. Further measurements of
optical conductivity and permittivity in similar films alloweddiscriminating a small gap /Delta1
S=1.85 meV on the electron-
like FS and a larger gap /Delta1L=3.95 meV on the holelike
FS.12Recent transmittance and reflectance measurements at
terahertz frequencies in ultrathin films with x=0.08 and
Tc=17.5 K have given even smaller gaps, i.e., /Delta1S/similarequal1.0m e V
and/Delta1L/similarequal2.1m e V .13
Clearly, the results collected up to now do not give a
consistent picture, either about the presence and location ofthe nodal lines, or about the amplitude of the gaps. To try toaddress this point, we have performed point-contact Andreev-reflection spectroscopy (PCARS) measurements in epitaxialBa(Fe
1−xCox)2As2thin films with nominal Co content x
ranging from 0.04 to 0.15, i.e., from the underdoped to theoverdoped region of the phase diagram. The PCARS spectrado not show any clear hint of the emergence of extendednode lines, and can be well fitted by the two-band two-
dimensional (2D) Blonder-Tinkham-Klapwijk (BTK) model
using isotropic gaps—although the shape of the spectra doesnot allow excluding some degree of gap anisotropy. Thedependence of the gap amplitudes /Delta1
Sand/Delta1Lon the local
critical temperature TA
cis discussed. In underdoped and
optimally doped films, the gap ratios are 2 /Delta1S/kBTc/similarequal3.7 and
2/Delta1L/kBTc/similarequal9, but decrease to 2.6 and 6.5, respectively, in
the overdoped region. When analyzed within a three-bandEliashberg model, these results turn out to be perfectlycompatible with s±superconductivity mediated by spin fluc-
tuations, whose characteristic energy is /Omega1
sf
0=4.65kBTc(as
found experimentally by neutron scattering experiments14). A
reduction of the electron-boson coupling strength is observed
in the overdoped regime, which can be rationalized as being
related to the suppression of spin fluctuations in this region ofthe phase diagram.
174506-1 1098-0121/2013/88(17)/174506(9) ©2013 American Physical SocietyP. PECCHIO et al. PHYSICAL REVIEW B 88, 174506 (2013)
II. EXPERIMENTAL DETAILS
The Ba(Fe 1−xCox)2As2thin films with a thickness of
50 nm were deposited on (001)CaF 2substrates by pulsed
laser deposition15using a polycrystalline target with high
phase purity.15,16The surface smoothness was confirmed by
insitu reflection high energy electron diffraction during the
deposition; only streaky patterns were observed for all filmsindicative of smooth surfaces. The details of the structuralcharacterization and of the microstructure of these high-quality, epitaxial thin films can be found in Ref. 15. Standard
four-probe resistance measurements were performed in a
4He
cryostat to determine the transport critical temperature and thetransition widths, reported in Table I. With respect to most
phase diagrams of Ba(Fe
1−xCox)2As2single crystals,17–19
where the optimal doping corresponds to x/similarequal0.065, the
highest T90
cof our films (i.e., the temperature at which the
resistance is 90% of its normal-state value immediately beforethe transition) is attained for x=0.10, and in the x=0.15
sample T
90
cis still about 22 K. This wide doping range of
highTcis presumably due to a combination of epitaxial strain
from the substrate and of reduced Co content in the filmwith respect to the nominal one. Detailed investigation isunder way. In the following of this paper we will thereforealways refer to the doping content of the target. This doesnot hamper our discussion, since we will refer all the resultsto the critical temperature of the contact, which is a localproperty directly correlated to the gap amplitudes (as we havealready demonstrated in many different cases
20) and is thus
well defined irrespectively of the actual Co content.21
PCARS measurements have been performed by using the
“soft” technique we introduced many years ago,22in which
at h i nA uw i r e( ∅=18μm) is kept in contact with the
film surface by means of a small drop ( ∅/lessorequalslant100μm) of
Ag conducting paste. The effective size of the point contact(PC) is of course much smaller than the area covered bythe Ag paste: Parallel nanoscopic contacts are likely to beformed here and there, between individual Ag grains and thesample surface. This technique has some advantages over theconventional “needle-anvil” one and has indeed been adoptedalso by other groups.
23,24In the specific case of the Co-doped
Ba-122 films we studied here, conventional point-contactmeasurements gave either featureless spectra, or spectra farfrom ideality, with a small Andreev signal superimposed toa background strongly decreasing with bias voltage,
25while
TABLE I. Critical temperatures of our films determined from
electric transport measurements. T90
candT10
care the temperatures at
which the resistance (the resistivity) is 90% and 10% of the normal-
state value immediately before the transition. /Delta1T cis defined here as
(T90
c−T10
c).
xT90
c(K) T10
c(K) /Delta1T c(K)
0.04 9.5 7.0 2.5
0.08 25.6 24.2 1.4
0.08 25.4 23.9 1.50.08 25.4 23.6 1.8
0.10 26.6 24.6 2.0
0.15 22.0 20.6 1.4the soft technique provides very often good spectra with a
clear spectroscopic signal, as we will show in the following.Analyses of the surface of these films carried out by means ofatomic force microscopy, x-ray photoemission spectroscopy,and scanning spreading resistance microscopy
25have provided
an explanation for this difference. They have shown thepresence of a thin, inhomogeneous, poorly conducting layer,due to reaction with air, that makes the local conductivity of thesurface highly position dependent. In these conditions, sincethe conventional technique probes a very small portion of thesurface, the probability of making the contact in a clean regionis rather low. In contrast, the drop of Ag paste used in the softtechnique covers a larger surface and allows a natural selectionof the more conducting channels within a micrometric region.
The PCARS spectra simply consist of the differential
conductance dI/dV of the N-S contact, as a function of the
voltage. In principle, a point contact can provide spectroscopicinformation only if the conduction is ballistic, i.e., electronsdo not scatter in the contact region. This is achieved if thecontact radius ais smaller than the electronic mean free path,
/lscript.
26According to Sharvin’s27or Wexler’s28equations, ais
also related to the normal-state contact resistance RN.T h e
fact that in these films most of the contacts, irrespective oftheir resistance, do show clear Andreev signals is again relatedto the particular nature of the film surface and to the factthat each contact is actually the parallel of many nanoscopiccontacts. As a matter of fact, the high residual resistivity(120μ/Omega1cm for x=0.10) of the films implies a small mean
free path /lscript, reasonably of the order of a few nanometers (even
though its precise determination from the resistivity is notstraightforward and would at least require the calculation of theplasma frequencies of the different bands). In these conditions[analogous to those discussed in the case of PCARS on thinfilms of SmFeAs(O,F) and LaFeAs(O,F)
29] the ballistic—or,
at least, the diffusive26—regime can only be achieved when
the (microscopic) PC is the parallel of several nanoscopiccontacts that fulfill the ballistic or diffusive conditions, andwhose individual resistance is thus much greater than that ofthe microscopic contact as a whole. This occurs rather naturallyin our films thanks to the surface characteristics mentionedabove.
Owing to the epitaxial structure of the films and to the
smoothness of their surface, the current that flows throughthe point contact is mainly parallel to the crystallographiccaxis. By placing the contacts in different regions of the
sample surface, we were able to check the homogeneity ofthe superconducting properties and to obtain some informa-tion about their distribution. To allow a comparison of theexperimental dI/dV vsVcurves to the theoretical models,
the former must be first normalized, i.e., divided by thenormal-state conductance curve ( dI/dV )
NvsV(in principle,
recorded at the same temperature). This curve is inaccessibleto experiments because of the very high upper critical field,and the normal-state conductance measured just above T
cis
unusable because of an anomalous shift of the conductancecurves across the superconducting transition, clearly visible inFig. 1. This effect is typical of very thin films and is related
to a temperature- and current-dependent spreading resistancecontribution arising from the portion of the film between thepoint contact and the second voltage electrode.
30A quantitative
174506-2DOPING AND CRITICAL-TEMPERATURE DEPENDENCE OF ... PHYSICAL REVIEW B 88, 174506 (2013)
-75 -50 -25 0 25 50 750.030.040.05 T (K)
4.22
8.46
12.38
16.05
18.01
20.23
21.72
22.37
22.84
23.12
23.36
23.75
24.07
24.29
24.67
24.96
25.20
25.67
25.78
26.17
27.31Conductance ( Ω-1)
Voltage (mV)x=0.08
-50 -25 0 25 500.0450.0500.055Conductance ( Ω-1)
Voltage (mV)22 23 24 25 26 270246810 Resistance ( Ω)
T (K)
FIG. 1. (Color online) Temperature dependence of the spectrum
of a point contact on an 8% Co-doped film, up to the critical
temperature and above. The shift of the spectra is clearly seen. The
left inset shows the low-temperature spectrum and the polynomialcurve that fits its tails used for normalization. The right inset reports
the temperature dependence of the resistance of the film. The shift of
the spectra correlates with the onset of resistivity in the film. Here,T
A
cis between 25.20 and 25.67 K, i.e., TA
c/similarequal25.4±0.3.
model has been recently proposed in Ref. 31. For these reasons,
as shown in detail elsewhere,32the normalization can be rather
critical in Fe-based compounds; here we chose to divide thelow-temperature conductance curves by a polynomial fit oftheir own high-voltage tails, as shown in the left inset ofFig. 1. For the same reason, we will here concentrate on the
low-temperature spectra.
The normalized curves were fitted with the BTK model
generalized by Kashiwaya and Tanaka
33,34(later on called
the “2D-BTK model”) in order to extract the gap values,as discussed in the following section. This model is basedon the assumption of spherical FSs in both the normalmetal and the superconductor, which is clearly a strongsimplification in the case of Fe-based compounds. A morerefined (and complicated) three-dimensional-BTK model
32,35
could be used to calculate the conductance curves accounting
for the real shape of the FS but, as shown elsewhere,36,37this
would not significantly change the resulting gap amplitudes.The local critical temperature (in the following indicated byT
A
c) can be determined by simply looking at the temperature
dependence of the raw conductance curves; TA
cis identified
with the temperature at which the features related to Andreevreflection disappear, and its uncertainty is determined by thewidth of the temperature steps between the acquired spectra.TheT
A
cvalues were generally in very good agreement with the
critical temperature of the whole film determined by resistancemeasurements and/or susceptibility or magnetization measure-ments since in most cases they fall between T
10
candT90
c.
A noticeable exception is the underdoped sample ( x=0.04)
in which some spectra show a zero-bias peak that becomesclearer on increasing temperature and persists in the normalstate. This effect has also been observed by other groups
24,38,39
and might be related to some magnetic scattering rather than
to superconductivity. This issue is still under debate24and willbe addressed in a forthcoming paper. Here, however, we will
only report PCARS spectra that do not show this anomaly.
III. RESULTS AND DISCUSSION
Figure 2shows some representative examples of the
many PCARS spectra recorded in films at different doping(symbols), from x=0.04 (top panel) to x=0.15 (bottom
panel). For x/greaterorequalslant0.08 the shape of all the curves is clearly
incompatible with a single gap. These spectra show twosymmetric maxima at low energy (or a small flat region aroundzero bias, as in the bottom panel) which are the hallmark of thesmall gap /Delta1
S, plus additional shoulders or changes in slope at
higher energy that are due to the second, larger gap /Delta1L.T h e
case of x=0.04, where the double-gap structure is not evident,
is in some sense anomalous and will be discussed in moredetail later. The additional structures often visible at higherenergy (about 20 meV for x/greaterorequalslant0.08) are related to the strong
coupling between electrons and spin fluctuations, as shown indetail elsewhere.
32,40Here, we are mainly interested in the gap
amplitudes, and we will thus disregard these structures (note
1.001.051.101.151.20
1.001.051.101.15
1.001.051.10
-40 -20 0 20 401.041.121.20RN = 11 ΩRN = 24 Ω(a)
ΔS=3.95 + 0.35 meV
ΔL=10.4 + 0.6 meVΔS=1.55 + 0.51 meV
ΔL=3.20 + 0.78 meVNormalized ConductanceTcA=7.3 + 0.9 K
x = 0.04
T = 2.0 K
(b)
T = 4.2 KTcA= 25.1 + 0.5 K
x = 0.08
RN = 2.7 ΩRN = 4.5 Ω(c)
T= 2.0 KΔS=3.1 + 0.1 meV
ΔL=9.3 + 1.5 meV TcA= 27.4 + 0.5 K
x = 0.10
(d)
T= 1.9 KΔS=2.51 + 0.24 meV
ΔL=6.95 + 0.55 meVTcA=21.6 + 0.6 K
x = 0.15
Voltage (mV)
FIG. 2. (Color online) Low-temperature normalized PCARS
spectra (symbols) in films with different Co content: x=0.04 (a),
x=0.08 (b), x=0.10 (c), and x=0.15 (d), together with their
two-band 2D-BTK fit (lines). The corresponding fitting parameters
are listed in Table II. The gap values /Delta1Sand/Delta1Lindicated in the
panels are instead the average over different possible fits of the same
curve, as explained in the text. The normal-state resistance of the
contacts is also indicated, as well as the local critical temperature TA
c.
174506-3P. PECCHIO et al. PHYSICAL REVIEW B 88, 174506 (2013)
TABLE II. Fitting parameters of the spectra shown in Fig. 2. Each set of parameters is relevant to the individual BTK curve shown in the
corresponding panel of Fig. 2. The gaps /Delta1Sand/Delta1Land the broadening parameters /Gamma1Sand/Gamma1Lare expressed in meV . The uncertainty on the
gap ratio is due to the uncertainty on the critical temperature (see Fig. 2).
x/Delta1 S /Gamma1S ZS /Delta1L /Gamma1L ZL wS 2/Delta1S/kBTA
c 2/Delta1L/kBTA
c
0.04 1.80 1.52 0.16 3.60 2.59 0.17 0.48 5.73 ±0.71 11.47 ±1.41
0.08 3.90 2.70 0.23 10.60 7.00 0.39 0.40 3.61 ±0.07 9.82 ±0.20
0.10 3.20 2.40 0.25 8.80 7.30 0.32 0.50 2.72 ±0.05 7.47 ±0.14
0.15 2.75 2.03 0.20 7.00 2.90 0.20 0.50 2.96 ±0.08 7.54 ±0.21
that their presence does not affect in any way the values of the
gaps extracted from the fit, as shown in Refs. 40and32).
Solid lines superimposed to the experimental data of Fig. 2
represent their best fit within the two-band 2D-BTK model.
This model assumes that the total conductance is simply the
sum of the partial contributions from two sets of equivalent
bands, i.e., holelike and electronlike, and each contributioncan be calculated by using the 2D-BTK model. The model thus
contains seven adjustable parameters: the two gap amplitudes
/Delta1
Sand/Delta1L, the broadening parameters /Gamma1Sand/Gamma1L, the barrier
parameters ZSandZL, and the relative weight of the two bands
that contribute to the conductance ( wSandwL=1−wS).41
The values of these parameters for the curves shown in Fig. 2
are reported in Table II. Because of the number of parameters,
the set of their best-fitting values for a given spectrum is not
univocal, especially when the signal is not very high as in Fe-
based compounds. To account for this, we always determinedthe maximum possible range of /Delta1
Sand/Delta1Lvalues compatible
with a given curve, when all the other parameters are changed
as well. Based on the results obtained in Ba(Fe 1−xCox)2As2
single crystals at optimal doping,40we initially assumed the
two gaps to be isotropic. This assumption works well inthewhole doping range analyzed here, thus indicating that
there are no clear signs of a change in the gap symmetry
and structure on increasing the doping content. In this respect
it should be noted that the 2D-BTK model is not the most
sensitive to the subtle details of the gap structure, so this resultdoes not exclude gap anisotropies either in the plane or in the
cdirection whose existence has been claimed or predicted in
Co-doped Ba-122 (Ref. 42) and more generally in the 122
systems.
6,8It must be noted, however, that if extended node
lines (predicted in particular conditions in 122 compounds7)
were present, they would give rise to quasiparticle excitations
with very small energy that can be detected by PCARS, as
shown in the case of Ca(Fe ,Co) 2As2.43
Figure 3shows two examples of the many (almost 20)
conductance curves measured in three different films withx=0.08. The curves have different shapes but the values of
the gaps extracted from their fit (indicated in the legend) arecompatible with one another. The other fitting parameters arelisted in the caption. As shown in panel (c) of the same figure,there is no correlation between the gap values extracted fromthe fit of different spectra and the resistance of the contact.This fact supports the spectroscopic nature of the contacts
26
and excludes the presence of spreading-resistance effects30
in our measurements at low temperature. More generally, theconsistency of the gap values obtained in different regions ofthesame film is good proof of the macroscopic homogeneity
of the superconducting properties, while the consistency of thevalues obtained in different films with the same doping is proof
of the reproducibility of the deposition process.
-50 -40 -30 -20 -10 0 10 20 30 40 501.001.051.101.15
-50 -40 -30 -20 -10 0 10 20 30 40 501.001.051.101.151.201.25(a)
RN = 1 Ωtwo-gap fit
ΔS= 4.00 + 0.25 meV
ΔL= 9.3 + 0.6 meVNormalized ConductanceVoltage (mV)
RN = 34 Ω(b)
two-gap fit
ΔS= 3.85 + 0.15 meV
ΔL= 9.2 + 0.3 meV
Voltage (mV)
0 20 40 60 80 100 120 140 160 1800246810121416
(c) (film 1)
(film 2)
(film 3)Energy Gaps (meV)
Contact Resistance RN (Ω)
FIG. 3. (Color online) (a), (b) Two examples of PCARS spectra
taken in different films of Ba(Fe 0.92Co0.08)2As2. Despite the different
shape of the spectra, the gap values obtained from the fit are consistent.
The fit shown in (a) was obtained with /Delta1S=4.25 meV , /Gamma1S=
3.60 meV , ZS=0.25,/Delta1L=9.90 meV , /Gamma1L=4.70 meV , ZL=0.34,
wL=0.60. The fit in (b) was obtained with /Delta1S=3.70 meV , /Gamma1S=
2.35 meV , ZS=0.18,/Delta1L=9.00 meV , /Gamma1L=3.25 meV , ZL=0.30,
wL=0.40. (c) Gap amplitudes as a function of the resistance of the
contacts, which shows the absence of any correlation between these
quantities and demonstrates the spectroscopic nature of the contacts.
This panel includes data taken in three different films.
174506-4DOPING AND CRITICAL-TEMPERATURE DEPENDENCE OF ... PHYSICAL REVIEW B 88, 174506 (2013)
-60 -40 -20 0 20 40 601.01.11.21.3
- 2 0 - 1 5 - 1 0- 5 0 5 1 01 52 01.001.051.101.151.20
0241.051.101.15two-gap fit
ΔL=2.75 + 0.05 meV
Δ*=9.5 + 0.3 meV
Voltage (mV)RN = 58 Ω(a)
(b)
one-band fit
Δ=1.92 meVtwo-band fit
ΔS=1.35 + 0.15 meV
ΔL=2.45 + 0.15 meVNormalized conductance
Voltage (mV)RN = 4.5 Ω
0 1 02 03 04 05 06 00246810 (c)
ΔS
ΔL
Δ*Energy gaps: ΔS, ΔL, Δ*
Contact resistance RN (Ω)Voltage (mV)
FIG. 4. (Color online) (a), (b) Two examples of PCARS spec-
tra taken in different points of the same film (5 ×5m m2)o f
Ba(Fe 0.96Co0.04)2As2. The spectrum in (a) shows clear shoulders
around 7 meV and conductance maxima at lower energy. The solid
line represents the best fit of the curve obtained within the 2D-BTK
model assuming that the shoulders are due to a superconducting
gap/Delta1∗. The fitting parameters are /Delta1L=2.8m e V , /Gamma1L=1.15 meV ,
ZL=0.24,/Delta1∗=9.8m e V , /Gamma1∗=4.75 meV , Z∗=0.25,wS=0.2.
The spectrum in (b) instead does not show shoulders but a single
maximum at zero bias, and the FWHM of the whole structure is of
the order of 3 meV . The solid line is the two-band BTK fit, obtainedwith parameters /Delta1
S=1.35 meV , /Gamma1S=0.84 meV , ZS=0.18,/Delta1L=
2.6m e V , /Gamma1L=1.8m e V , ZL=0.4,wS=0.6. The dashed line is
the single-band BTK fit, obtained with parameters /Delta1=1.92 meV ,
/Gamma1=1.74 meV , Z=0.12. A magnification of the low-energy region
(inset) shows that the two-gap fit is better. (c) Amplitudes of the
“gap” /Delta1∗and of the gaps /Delta1Sand/Delta1Las a function of the contact
resistance RN.
Forx=0.04 the spectra often show very clear shoulders
at energies of the order of 7 meV in addition to conductancemaxima at about 3 meV , as shown in Fig. 4(a). The shoulders
are fast suppressed on increasing temperature or upon appli-cation of a magnetic field (they look completely washed outalready at 5 K, or in a field of 1 T). If one assumes that theyare due to a superconducting gap and fits the spectra with thetwo-band 2D-BTK model, the relevant amplitude /Delta1
∗turns out
to be of the order of 6–9 meV . Since the measured film showedaT
90
cof less than 10 K, these values are clearly unphysical
for a superconducting gap. The smaller gap obtained from thesame 2D-BTK fit turns out to range between 1.1 and 3.2 meVand its Andreev signal shows a conventional dependence ontemperature and magnetic field. In a small number of spectra,of which an example is shown in Fig. 4(b), the structures
at about 7 meV are not present at all and a single, muchnarrower structure is observed, whose width is of the orderof 3 meV . These spectra admit a fit with the single-gap BTKmodel, giving a gap of the order of 2 meV , but the two-bandBTK model still works better [see the inset to Fig. 4(b)] and
gives a small gap /Delta1
Sof the order of 1.5 meV and a larger
gap/Delta1Lof about 2.5–3.0 meV . Although this fact alone does
not allow concluding that in this compound two gaps survive,we will refer from now on to the results of the two-gap fiton the basis of plausibility arguments. Indeed, even at 4%Co doping, the Ba-122 system retains a multiband electronicstructure and there is no reason to believe that the two gapsobserved at higher doping should “merge” into one. This effectis theoretically predicted in the presence of strong disorder
44
but would be accompanied by a strong suppression of thecritical temperature, while the T
cof our 4% Co-doped film is
identical to that of single crystals with the same Co content,where two gaps have been measured by specific heat.
45
Figure 4(c) shows a summary of the values of /Delta1S,/Delta1L,
and/Delta1∗obtained from the two-band fit of spectra of the first
and second type, plotted as a function of the resistance of thecontacts R
N. Clearly, the larger “energy scale” /Delta1∗depends
on the contact resistance, which (together with the anomalousdependence on temperature and magnetic field) indicates thatthe structures around 7 meV are not due to a superconductinggap. Further confirmation comes from the weight of /Delta1
∗in
the fit, which depends on RN(unlike for superconducting
gaps), decreasing from 0.8 to 0.6 when RNgoes from 58 /Omega1
to 17 /Omega1. This reflects the fact that the amplitude of the
relevant structures decreases on decreasing RN; consequently,
their position is less easily identifiable (which may partlyaccount for the dependence of /Delta1
∗fromRN). On this basis,
understanding the origin of these structures is a difficult task.The 4% doped sample falls well in the region of the phasediagram where superconductivity and magnetism coexist, andwhich is still poorly understood. One might speculate that thesestructures arise from a magnetic phase probed by a subsetof nanocontacts; the amplitude of the relevant signal in thespectrum, as well as the resistance of the contact as a whole,may thus be related to the fraction of conduction channels inthis phase.
Going back to Fig. 4(c), the smaller gaps do not show any
dependence on the contact resistance and seem to cluster in twogroups indicated by squares and circles for clarity. Althoughthe two energy ranges are very close to each other, they donot overlap (even taking into account the error bars), furthersupporting the picture of twogaps/Delta1
Sand/Delta1L.
Figure 5reports the (average) gap amplitudes /Delta1Sand/Delta1L
obtained in the various films as a function of the (average)
TA
cof the contacts. In other words, the values of /Delta1Sand/Delta1L
reported here are the midpoints of the corresponding range of
gap amplitudes obtained in the fit of different curves. The width
174506-5P. PECCHIO et al. PHYSICAL REVIEW B 88, 174506 (2013)
6 8 10 12 14 16 18 20 22 24 26024681012
024681012Energy Gaps (meV)
Local Critical Temperature, TA
c (K) Samuely (2009) PCARS - crystals
Nakamura (2011) Terahertz conductivity - films
Yin (2009) STS - crystals PCARS, this work - films
Tortello (2010) PCARS - crystals
Arham (2013) PCARS - crystals
Tu (2010) Optical measurements - crystals
Maksimov (2011) Optical measurements - films
Fischer (2010) Optical conductivity - films
Van Heumen (2010) Optical measurements - crystals
Perucchi (2013) Optical measurements - films
Terashima (2009) ARPES - crystals
Hardy (2010) Specific heat - crystals
FIG. 5. (Color online) Average gap amplitudes in
Ba(Fe 1−xCox)2As2thin films with different Co content obtained
by PCARS measurements (filled circles), plotted as a function of
TA
c. The other data points are taken from literature, and specifically
squares from Ref. 40,u pt r i a n g l e sf r o mR e f . 24, diamonds from
Ref. 47, down triangles from Ref. 12, left triangles from Ref. 9,s t a r s
from Ref. 65,a s t e r i s k sf r o mR e f . 13, right triangles from Ref. 49,
hexagons from Ref. 48, half-filled squares from Ref. 46, half-filled
circles from Ref. 10, and half-filled triangles from Ref. 50.T h e
techniques used for these measurements are indicated in the legend.The upper and lower dashed lines correspond to a gap ratio 2 /Delta1/k
BTc
equal to 3.52 and 9.0, respectively.
of the range is represented by the vertical error bars, while the
horizontal error bars indicate the range of TA
cvalues in all the
point contacts made on that film. Figure 5also shows the results
of PCARS in single crystals24,40,46as well as the gap ampli-
tudes determined either in films or single crystals by meansof other techniques, namely, optical measurements,
9,10,12,47
specific heat,48angle-resolved photoemission spectroscopy
(ARPES)49and scanning tunneling spectroscopy.50
At the highest Tcvalues, corresponding to x=0.08 and
x=0.10, the gap values agree rather well with those given
by PCARS in single crystals.24,40The large spread of /Delta1L
values given by PCARS has already been noticed in various
Fe-based compounds32and its origin may be either intrinsic
(e.g., anisotropy of /Delta1L) or extrinsic (uncertainty due to
the normalization). Finally, the values given by PCARS(especially for /Delta1
L) are systematically larger than those given
by optical measurements and specific-heat measurements. Thismay be due to the approximations on which the fit of the curvesis based, but may also hide some more fundamental property ofFe-based compounds. The small gap /Delta1
Sappears much better
defined; the values provided by different techniques are wellconsistent with one another. Concerning the gap values awayfrom optimal doping, it should be borne in mind that Fig. 5
reports in the same plot the data for underdoped and overdopedsamples; in particular, the points at T
c=20.2 K refer to the
x=0.15 film. If these points are temporarily excluded from
the analysis, a roughly linear trend of the gaps as a function ofT
ccan be observed. The dashed lines in Fig. 5have equations
2/Delta1/k BTc=3.52 and 2 /Delta1/k BTc=9.0; it can be clearly seen
that the small gap is approximately BCS for any xbetween2 4 6 8 10 12 14 16024681012
050100150200Energy gaps (meV)
Nominal Co content (%)ΔS (PCARS)
ΔL (PCARS)
|Δ2| (theory)
|Δ3| ~ Δ1 (theory)Tc (K)T90
c
T 10
c
Tctheor4 6 8 1 01 21 41 6012
λ13λ12
Nominal Co content (%)Coupling strengthλtot
FIG. 6. (Color online) Doping dependence of the gaps measured
by PCARS (circles, left vertical scale) and of the critical temperature
from resistivity measurements (lines, right vertical scale). Triangles
and squares indicate the values of the gaps and of the criticaltemperature calculated within the three-band Eliashberg model
described in the text. The inset shows the dependence of the coupling
strengths λ
12andλ13on the Co content, together with the total
electron-boson coupling constant.
0.04 and 0.10. Even though /Delta1Lis affected by a much larger
uncertainty, it can be said that 2 /Delta1L/kBTcranges between
7 and 10 in the same doping range. The points at x=0.15 are
instead outside this trend since the gap values here correspondto reduced gap ratios. This point can be clarified by plottingthe gap amplitudes as a function of the nominal doping, as inFig. 6.
As expected, the trend of the gaps mimics the trend of the
critical temperature, showing a maximum at x=0.08–0.10.
However, the trend is not symmetric in the sense that in theoverdoped region the gaps decrease “more” than the criticaltemperature, i.e., the gap ratios decrease. The theoreticalanalysis of these results is presented in the following section.
IV . INTERPRETATION OF THE RESULTS WITHIN
ELIASHBERG THEORY
We have shown elsewhere51,52that a simple three-band
Eliashberg model with a very small number of free parameterscan account surprisingly well for the phenomenology of Fe-based superconductors and allows explaining a large varietyof their properties. Here we use the same model to try torationalize the experimental trend of the gaps as a functionofT
cor of the doping content x. The first assumption of the
model is that the electronic structure of Ba(Fe 1−xCox)2As2
can be approximately described by one hole band (indicated
in the following as band 1) and two electron bands (2 and3).
40,52The gap symmetry is assumed to be s±(Ref. 53)
so that the sign of /Delta11(here assumed positive) is opposite
to that of /Delta12and/Delta13. Although PCARS, as well as many
other spectroscopic techniques, provides at most two gapamplitudes and does not allow associating them to a particularFS sheet, the use of (at least) three effective bands and thusthree gaps is necessary for the Eliashberg model to be able toreproduce the experimental results. However, ARPES results
174506-6DOPING AND CRITICAL-TEMPERATURE DEPENDENCE OF ... PHYSICAL REVIEW B 88, 174506 (2013)
in optimally Co-doped Ba-122 single crystals indicated that
the larger gap belongs to the holelike FS sheet.49With this
in mind, we will assume /Delta11/similarequal|/Delta13|and|/Delta12|to be the large
and the small gaps measured by PCARS, respectively. Thisassumption is consistent with the fact that the experimentalresults do not resolve the two larger gaps. To obtain the gapsand the critical temperature within the s ±wave three-band
Eliashberg model
54one has to solve six coupled equations for
the gaps /Delta1i(iωn) and the renormalization functions Zi(iωn),
where iis a band index ( i=1.3). The equations have
been reported elsewhere;51their solution requires a large
number of input parameters (18 functions and 9 constants);however, some of these parameters are correlated, some can beextracted from experiments, and some can be fixed by suitableapproximations. For example, the coupling constant matrixλ
ijcan be greatly simplified. In general, one should consider
that each matrix element has one contribution from phononsand one from antiferromagnetic (AFM) spin fluctuations (SF),
i.e.λ
ij=λph
ij+λsf
ij. However, the coupling between the two
electron bands is small, and we thus take λ23=λ32=0;
the total electron-phonon coupling in pnictides is generallysmall
55and phonons mainly provide intraband coupling, so
that we assume λph
ij=0; spin fluctuations mainly provide
interband coupling between the two quasi-nested FS sheets,53
and thus we assume λsf
ii=0. Finally, the electron-boson
coupling-constant matrix λijtakes the following form:40,51,56
λij=⎛
⎜⎝λph
11λsf12λsf13
λsf21λph220
λsf
31 0λph
33⎞
⎟⎠, (1)
where λsf
21=λsf
12ν12andλsf
31=λsf
13ν13, withνij=Ni(0)/Nj(0)
andNi(0) is the normal density of states at the Fermi
level for the ith band. Another fundamental ingredient is
the electron-boson spectral function α2F(/Omega1) of the boson
responsible for the pairing. The shape of the electron-phononspectral function is taken from literature
57and we assume
α2
11Fph(/Omega1)=α2
22Fph(/Omega1)=α2
33Fph(/Omega1) with λph
ii=0.2.58As
for spin fluctuations, we assume their spectrum to have aLorentzian shape:
51,59–61
α2
ijFsf(/Omega1)=Cij{L(/Omega1+/Omega1ij,Yij)−L(/Omega1−/Omega1ij,Yij)},(2)
where L(/Omega1±/Omega1ij,Yij)=1
(/Omega1±/Omega1ij)2+Y2
ij.Cijare normalization
constants, necessary to obtain the proper values of λijwhile
/Omega1ijandYijare the peak energies and half-widths of the
Lorentzian functions, respectively.51In all the calculations
we set /Omega1ij=/Omega1sf
0andYij=Ysf
ij=/Omega1sf
0/2.14Here,/Omega1sf
0is the
characteristic energy of the AFM SF, assumed to be equal tothe spin-resonance energy, as verified experimentally by us inoptimally Co-doped Ba-122 single crystals.
32,40Its value is de-
termined according to the empirical relation /Omega1sf
0=4.65kBTc
(proposed in Ref. 62). Band-structure calculations provide
information about the factors νijthat enter the definition
ofλij. In the case of optimally doped Ba(Fe 1−xCox)2As2,
ν12=1.12 and ν13=4.50.63As a first approximation, these
values have been used here for all Co contents. Moreover,based on the fact that the Coulomb pseudopotential is probablysmall in these compounds
8we assume all the elements of thepseudopotential matrix to be identically zero ( μ∗
ii=μ∗
ij=0);
finally, we neglect the effect of disorder, owing to the highquality of the films.
Finally, only two free parameters remain, i.e., the coupling
constants λsf
12andλsf
13. These parameters can be tuned in such
a way to reproduce the experimental values of the small gap/Delta1
Sand of the critical temperature, which are the best-defined
experimental data; the values of the large gap /Delta1Lare indeed
affected by a larger relative uncertainty, and moreover theymight actually be a sort of weighted “average” of the two gaps/Delta1
1and|/Delta13|. The larger gaps are therefore calculated with the
values of λsf
12andλsf
13that allow reproducing /Delta1SandTc.
The result of these calculations is that (i) the trend of the
experimental gaps /Delta1Sand/Delta1Las a function of Tcand of x
in the samples with nominal Co content x=0.04,0.08, and
0.10 can be reproduced by using λsf
12=0.8 and λsf
13=1.33,
and only changing the value of the characteristic SF energy/Omega1
0according to the change in Tc; (ii) to reproduce the values
of the gaps and of Tcin the overdoped sample ( x=0.15)
it is instead also necessary to reduce the values of the two
coupling constants: λsf
12=0.5 and λsf
13=1.21. The values of
these two parameters are shown as a function of xin the inset
of Fig. 6. Note that the total coupling is λtot=2.22 for x=
0.04,0.08, and 0 .10 and decreases to λtot=1.82 atx=0.15.
These values are in agreement with those found in previousworks,
52,58and indicate that Co-doped Ba-122 is a strong-
coupling superconductor at all the doping contents analyzedhere. The main panel of Fig. 6also reports the calculated values
of the gaps as a function of x. The agreement between the
theoretical and experimental values of T
cand of the small gap
is very good; the large gap is underestimated around optimaldoping, but the trend is qualitatively correct. The agreementmight be improved if the feedback effect of the condensate onthe bosonic excitations
60,61was taken into account, which was
not done in this paper for simplicity.
V . CONCLUSIONS
In conclusion, we have determined the energy gaps of
Ba(Fe 1−xCox)2As2in a wide range of nominal doping (0 .04/lessorequalslant
x/lessorequalslant0.15) by means of soft PCARS measurements in epitaxial
thin films. Several PCARS spectra were acquired on eachsample, with the probe current injected perpendicular to thefilm surface and thus mainly along the caxis. For any x/greaterorequalslant0.08
the PCARS spectra admit a fit with the two-band 2D-BTKmodel using two isotropic gaps, and their shape does notsuggest the presence of node lines on the FS; in the stronglyunderdoped sample ( x=0.04) a fit with a single isotropic
gap is also possible, though a little worse than the two-gapone. Altogether, these results show no clear hints of changesin the gap symmetry or structure in the doping range ofour films—although the shape of the spectra does not allowexcluding some degree of gap anisotropy.
The small gap turns out to be approximately BCS, with a
ratio 2 /Delta1
S/kBTc=3.7±0.8 (the uncertainty arises from the
statistical spread of gap values) for x/lessorequalslant0.10, and smaller
(2.6±0.3) atx=0.15. The second gap is much larger, with
a ratio 2 /Delta1L/kBTcof the order of 9 for x/lessorequalslant0.10 and 6.5 for
x=0.15.
174506-7P. PECCHIO et al. PHYSICAL REVIEW B 88, 174506 (2013)
The trend of the gaps and of Tcas a function of the Co
content can be reproduced by a simple s±Eliashberg model
in which the spectrum of the mediating boson is that of spinfluctuations, and its characteristic energy coincides with theenergy of the spin resonance. The decrease of the gap ratios inthe overdoped samples is reflected in the values of the couplingstrengths that are constant for x/lessorequalslant0.10 and slightly decrease
atx=0.15. This result finds a natural explanation within the
picture of s±superconductivity mediated by spin fluctuations:
In the overdoped regime, far from the AFM region of the phasediagram, superconductivity may suffer from a suppression ofthe spin fluctuations and the loss of nesting,
64which could lead
to a decrease in the superconducting interband coupling that,in turn, produces a larger decrease of the gaps in comparison
with the reduction of the critical temperature.
ACKNOWLEDGMENTS
D.D. and P.P. wish to thank the Leibniz Institute for Solid
State and Materials Research (IFW) in Dresden, Germany, andin particular, the Department of Superconducting Materials,where many of the PCARS measurements were performed.Particular thanks to V . Grinenko, J. H ¨anisch, and K. Nenkov
for valuable discussions and technical support. This work wasdone under the Collaborative EU-Japan Project “IRON SEA”(NMP3-SL-2011-283141).
1P. Seidel, Supercond. Sci. Technol. 24, 043001 (2011).
2T. Katase, Y . Ishimaru, A. Tsukamoto, H. Hiramatsu, T. Kamiya,
K. Tanabe, and H. Hosono, Supercond. Sci. Technol. 23, 082001
(2010).
3K .I i d a ,J .H ¨anisch, R. H ¨uhne, F. Kurth, M. Kidszun, S. Haindl,
J. Werner, L. Schultz, and B. Holzapfel, Appl. Phys. Lett. 95,
192501 (2009).
4A. A. Golubov and I. I. Mazin, Appl. Phys. Lett. 102, 032601
(2013).
5K. Kuroki, H. Usui, S. Onari, R. Arita, and H. Aoki, P h y s .R e v .B
79, 224511 (2009).
6S. Graser, A. F. Kemper, T. A. Maier, H.-P. Cheng, P. J. Hirschfeld,
and D. J. Scalapino, P h y s .R e v .B 81, 214503 (2010).
7K .S u z u k i ,H .U s u i ,a n dK .K u r o k i , J. Phys. Soc. Jpn. 80, 013710
(2011).
8P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep. Prog. Phys.
74, 124508 (2011).
9T. Fischer, A. V . Pronin, J. Wosnitza, K. Iida, F. Kurth, S. Haindl,
L. Schultz, B. Holzapfel, and E. Schachinger, P h y s .R e v .B 82,
224507 (2010).
10D. Nakamura, Y . Imai, A. Maeda, T. Katase, H. Hiramatsu, andH. Hosono, Physica C 471, 634 (2011).
11B. Gorshunov, D. Wu, A. A. V oronkov, P. Kallina, K. Iida,
S .H a i n d l ,F .K u r t h ,L .S c h u l t z ,B .H o l z a p f e l ,a n dM .D r e s s e l , Phys.
Rev. B 81, 060509 (2010).
12E. G. Maksimov, A. E. Karakozov, B. P. Gorshunov, A. Prokhorov,
A. A. V oronkov, E. S. Zhukova, V . S. Nozdrin, S. S. Zhukov,D. Wu, M. Dressel et al. ,Phys. Rev. B 83, 140502(R) (2011).
13A. Perucchi, L. Baldassarre, B. Joseph, S. Lupi, S. Lee, C. B. Eom,
J. Jiang, J. D. Weiss, E. E. Hellstrom, and P. Dore, Eur. Phys. J. B
86, 274 (2013).
14D. S. Inosov, J. T. Park, P. Bourges, D. L. Sun, Y . Sidis,
A. Schneidewind, K. Hradil, D. Haug, C. T. Lin, B. Keimer et al. ,
Nat. Phys. 6, 178 (2010).
15F. Kurth, E. Reich, J. H ¨anisch, A. Ichinose, I. Tsukada, R. H ¨uhne,
S. Trommler, J. Engelmann, L. Schultz, B. Holzapfel et al. ,Appl.
Phys. Lett. 102, 142601 (2013).
16F. Kurth, K. Iida, S. Trommler, J. H ¨anisch, K. Nenkov,
J. Engelmann, S. Oswald, J. Werner, L. Schultz, B. Holzapfel et al. ,
Supercond. Sci. Technol. 26, 025014 (2013).
17N. Ni, M. E. Tillman, J. Q. Yan, A. Kracher, S. T. Hannahs, S. L.
Budko, and P. C. Canfield, Phys. Rev. B 78, 214515 (2008).18J.-H. Chu, J. G. Analytis, C. Kucharczyk, and I. R. Fisher, Phys.
Rev. B 79, 014506 (2009).
19F. L. Ning, K. Ahilan, T. Imai, A. S. Sefat, M. A. McGuire, B. C.
Sales, D. Mandrus, P. Cheng, B. Shen, and H.-H. Wen, Phys. Rev.
Lett. 104, 037001 (2010).
20D. Daghero and R. Gonnelli, Supercond. Sci. Technol. 23, 043001
(2010).
21Please note that resistivity measurements unambiguously prove thatthe film with x=0.15 is in the overdoped region, since (i) its
ρ(T) curve does not show the low-temperature upturn typical of
underdoped samples (Ref. 18), observed instead in the films with
x=0.04 and 0.08 and (ii) its critical temperature is smaller than in
the optimally doped film ( x=0.10).
22R. S. Gonnelli, D. Daghero, G. A. Ummarino, V . A. Stepanov,
J. Jun, S. M. Kazakov, and J. Karpinski, Phys. Rev. Lett. 89, 247004
(2002).
23H. Z. Arham, C. R. Hunt, W. K. Park, J. Gillett, S. D. Das, S. E.S e b a s t i a n ,Z .J .X u ,J .S .W e n ,Z .W .L i n ,Q .L i et al. ,P h y s .R e v .B
85, 214515 (2012).
24H. Z. Arham, C. R. Hunt, J. Gillett, S. D. Das, S. E. Sebastian,
D. Y . Chung, M. G. Kanatzidis, and L. H. Greene, arXiv: 1307.1908 .
25T. Plecenik, M. Gregor, R. Sobota, M. Truchly, L. Satrapinskyy,
F. Kurth, B. Holzapfel, K. Iida, P. Kus, and A. Plecenik, Appl. Phys.
Lett. 103, 052601 (2013).
26Y . G. Naidyuk and I. K. Yanson, Point-Contact Spectroscopy ,
Springer Series in Solid-State Sciences V ol. 145 (Springer, NewYork, 2004).
27Y . V . Sharvin, Zh. Eksp. Teor. Fiz. 48, 984 (1965) [Sov. Phys. JETP
21, 655 (1965)].
28G. Wexler, Proc. Phys. Soc. London 89, 927 (1966).
29Y . G. Naidyuk, O. E. Kvitnitskaya, I. K. Yanson, G. Fuchs,
S. Haindl, M. Kidszun, L. Schultz, and B. Holzapfel, Supercond.
Sci. Technol. 24, 065010 (2011).
30T .Y .C h e n ,S .X .H u a n g ,a n dC .L .C h i e n , P h y s .R e v .B 81, 214444
(2010).
31S. D ¨oring, S. Schmidt, S. Gottwals, S. Schmidl, V . Tympel,
I. M ¨onch, F. Kurth, K. Iida, B. Holzapfel et al. ,a r X i v : 1309.1641 .
32D. Daghero, M. Tortello, G. Ummarino, and R. S. Gonnelli, Rep.
Prog. Phys. 74, 124509 (2011).
33S. Kashiwaya, Y . Tanaka, M. Koyanagi, and K. Kajimura, Phys.
Rev. B 53, 2667 (1996).
34S. Kashiwaya and Y . Tanaka, Rep. Prog. Phys. 63, 1641 (2000).
174506-8DOPING AND CRITICAL-TEMPERATURE DEPENDENCE OF ... PHYSICAL REVIEW B 88, 174506 (2013)
35D. Daghero, M. Tortello, P. Pecchio, V . A. Stepanov, and R. S.
Gonnelli, Low Temp. Phys. 39, 199 (2013).
36R. S. Gonnelli, M. Tortello, D. Daghero, P. Pecchio, S. Galasso,
V . A. Stepanov, Z. Bukovski, N. D. Zhigadlo, J. Karpinski, K. Iidaet al. ,J. Supercond. Nov. Magn. 26, 1331 (2013).
37R. S. Gonnelli, D. Daghero, and M. Tortello, Curr. Opin. Solid State
Mater. Sci. 17, 72 (2013).
38G .S h e e t ,M .M e h t a ,D .A .D i k i n ,S .L e e ,C .B a r k ,J .J i a n g ,J .D .
Weiss, E. E. Hellstrom, M. S. Rzchowski, C. B. Eom et al. ,Phys.
Rev. Lett. 105, 167003 (2010).
39W .K .P a r k ,C .R .H u n t ,H .Z .A r h a m ,Z .J .X u ,J .S .W e n ,Z .W .
Lin, Q. Li, G. D. Gu, and L. H. Greene, arXiv: 1005.0190 .
40M. Tortello, D. Daghero, G. A. Ummarino, V . A. Stepanov, J. Jiang,
J. D. Weiss, E. E. Hellstrom, and R. S. Gonnelli, Phys. Rev. Lett.
105, 237002 (2010).
41We used different barrier parameters for the two bands because of
the different Fermi velocities (Ref. 20)s i n c e Z=U0/(¯hvF)b e i n g
U0the height of the potential barrier at the interface. Interestingly,
in the x=0.08 case where we have the larger statistics, we found
thatZL/ZS=vF,S/vF,L=1.55±0.40. This is an indirect way to
estimate the ratio of Fermi velocities in the two equivalent bandsthat host the two gaps /Delta1
Land/Delta1S.
42I. I. Mazin, T. P. Devereaux, J. G. Analytis, J.-H. Chu, I. R. Fisher,
B. Muschler, and R. Hackl, P h y s .R e v .B 82, 180502(R) (2010).
43R. S. Gonnelli, M. Tortello, D. Daghero, R. K. Kremer, Z. Bukovski,
N. D. Zhigadlo, and J. Karpinski, Supercond. Sci. Technol. 25,
065007 (2012).
44D. V . Efremov, M. M. Korshunov, O. V . Dolgov, A. A. Golubov,a n dP .J .H i r s c h f e l d , P h y s .R e v .B 84, 180512(R) (2011).
45F. Hardy, P. Burger, T. Wolf, R. A. Fisher, P. Schweiss, P. Adelmann,
R. Heid, R. Fromknecht, R. Eder, D. Ernst et al. ,Europhys. Lett.
91, 47008 (2010).
46P. Samuely, Z. Pribulov ´a, P. Szab ´o, G. Prist ´aˇs, S. L. Bud’ko, and
P. C. Canfield, Physica C 469, 507 (2009).
47J. J. Tu, J. Li, W. Liu, A. Punnoose, Y . Gong, Y . H. Ren, L. J. Li,
G. H. Cao, Z. A. Xu, and C. C. Homes, P h y s .R e v .B 82, 174509
(2010).48F. Hardy, T. Wolf, R. A. Fisher, R. Eder, P. Schweiss, P. Adelmann,H. v. L ¨ohneysen, and C. Meingast, P h y s .R e v .B 81, 060501(R)
(2010).
49K. Terashima, Y . Sekiba, J. H. Bowen, K. Nakayama, T. Kawahara,T. Sato, P. Richard, Y .-M. Xu, L. J. Li, G. H. Cao et al. ,Proc. Natl.
Acad. Sci. U.S.A. 106, 7330 (2009).
50Y . Yin, M. Zech, T. L. Williams, X. F. Wang, G. Wu, X. H. Chen,
a n dJ .E .H o f f m a n , Phys. Rev. Lett. 102, 097002 (2009).
51G. A. Ummarino, M. Tortello, D. Daghero, and R. S. Gonnelli,
Phys. Rev. B 80, 172503 (2009).
52G. A. Ummarino, Phys. Rev. B 83, 092508 (2011).
53I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev.
Lett. 101, 057003 (2008).
54G. M. Eliashberg, Sov. Phys. JETP 11, 696 (1963).
55L. Boeri, M. Calandra, I. I. Mazin, O. V . Dolgov, and F. Mauri,
Phys. Rev. B 82, 020506 (2010).
56I. I. Mazin and J. Schmalian, Physica C 469, 614 (2009).
57R. Mittal, Y . Su, S. Rols, T. Chatterji, S. L. Chaplot, H. Schober,
M. Rotter, D. Johrendt, and T. Brueckel, P h y s .R e v .B 78, 104514
(2008).
58P. Popovich, A. V . Boris, O. V . Dolgov, A. A. Golubov, D. L. Sun,C. T. Lin, R. K. Kremer, and B. Keimer, P h y s .R e v .L e t t . 105,
027003 (2010).
59G. A. Ummarino, D. Daghero, M. Tortello, and R. S. Gonnelli, J.
Supercond. Novel Magn. 24, 247 (2011).
60G. A. Ummarino, J. Supercond. Novel Magn. 25, 1333
(2012).
61D. Daghero, M. Tortello, G. Ummarino, V . A. Stepanov,F. Bernardini, M. Tropeano, M. Putti, and R. S. Gonnelli,Supercond. Sci. Technol. 25, 084012 (2012).
62J. Paglione and R. L. Greene, Nat. Phys. 6, 645 (2010).
63I. I. Mazin (private communication).
64L. Fang, H. Luo, P. Cheng, Z. Wang, Y . Jia, G. Mu, B. Shen, I. I.
Mazin, L. Shan, C. Ren et al. ,P h y s .R e v .B 80, 140508(R) (2009).
65E. van Heumen, Y . Huang, S. de Jong, A. B. Kuzmenko,
M. S. Golden, and D. van der Marel, Eur. Phys. Lett. 90, 37005
(2010).
174506-9 |
PhysRevB.78.205427.pdf | Electronic transport in carbon nanotubes: Diffusive and localized regimes
P. A. Sundqvist, F. J. Garcia-Vidal, *and F. Flores
Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
/H20849Received 22 July 2008; revised manuscript received 7 September 2008; published 20 November 2008 /H20850
A fully self-consistent analysis of Boltzmann’s equations for electrons and phonons is used to study how the
resistance of single-walled carbon nanotubes evolves as a function of its length. We demonstrate that thepopulation of hot optical phonons controls the electronic transport of short nanotubes, whereas acousticalphonons take the leading role when the nanotube is very long. In this limit of long tubes, we also analyze theinterplay between the diffusive and localized transport regimes when the electron mean-free path and thelocalization length due to impurities are comparable.
DOI: 10.1103/PhysRevB.78.205427 PACS number /H20849s/H20850: 73.63.Fg, 73.23. /H11002b
Carbon nanotubes are ideal one-dimensional systems
where both basic science and nanodevice applications natu-rally merge.
1–7Their electronic properties are strongly de-
pendent on small structural variations, defects, and also onintrinsic properties as the electron-phonon interaction. In par-ticular, their metallic or insulating character is determined bythe chirality of the carbon atoms forming the nanotube. Inthis paper we analyze the transport properties of metallicsingle-walled carbon nanotubes /H20849SWNT /H20850.
The electronic transport properties of these nanotubes
roughly present three distinct regimes: /H20849a/H20850ballistic,
8/H20849b/H20850
diffusive,9–12and /H20849c/H20850localized.13,14Three different lengths
define the appearance of these regimes: Lis the nanotube
length, L0is the localization length, and /H9261is the electron
mean-free path. In SWNTs this mean-free path is mainlycontrolled by the electron-phonon interaction. If L/H11270L
0,/H9261,
electrons propagate ballistically between the two electrodes,and because of the two channels involved, the electrical re-sistance, R, can be simply written as 2 R=R
0, with R0being
the quantum of resistance. If /H9261/H11270L,L0, the transport process
is controlled by the diffusive propagation of electrons andthe resistance exhibits a linear dependence with L,2R
=R
0/H208491+L//H9261/H20850. Finally, the strong Anderson localization re-
gime in SWNTs emerges when L0/H11270L/H11021/H9261 and is character-
ized by an exponential law for R,2R=R0exp /H20849L/L0/H20850.
In this paper we will discuss theoretically the transport
properties of SWNTs in both the diffusive and localized re-gimes. Regarding the diffusive regime, we present a fullyself-consistent analysis of the propagation of electrons alonga/H2084910, 10 /H20850SWNT taking into account their interaction with
both acoustical and optical phonons. In particular, we discussin detail how the population of hot optical phonons modifiesthe electron mean-free path
15and demonstrate that optical
phonons get decoupled from electrons for long enoughSWNTs. In this limit, electron scattering with acousticalphonons determines the SWNT resistance. An interestingcase appears above this limit /H20849very long tubes, L/H1102250
/H9262m/H20850
where even a very low density of defects can modify thetransport properties of SWNTs. We will show how, if thethree characteristic lengths are such that L
0/H11021/H9261/H11021 L, the elec-
tronic transport in SWNTs enters into an intermediate regimein which the resistance depends on the length as 2 R=R
0/H208491
+L//H9261/H20850exp /H20849/H9261/L0/H20850, in good agreement with very recent experi-
mental evidence.16
In the diffusive regime, the nanotube resistance dependscritically on the applied bias, V, since optical phonons of
energy /H9280opt/H20849/H9280opt=0.18 eV /H20850can only be excited if eV/H11022/H9280opt.
ForeV/H11021/H9280opt, only acoustical phonons are operative and then
2R=R0/H208491+L//H9261ac/H20850,/H9261acbeing the electron mean-free path as-
sociated with acoustical phonons. In the opposite case /H20849eV
/H11022/H9280opt/H20850, both types of phonons play a competing role in the
diffusive regime. Notice that at room temperature, /H9261acis
around 1 /H9262m whereas its optical counterpart, /H9261opt,i so ft h e
order of 50 nm. Therefore, for small Land high biases, R
presents a linear dependence with Land the slope is deter-
mined by /H9261opt. For large enough L, however, the applied bias
varies very smoothly along the nanotube and optical phononsare excited less effectively. This competing effect betweenacoustical and optical phonons was analyzed in Ref. 17by
solving self-consistently Boltzmann’s equation for electronsalong the nanotube. In this analysis, optical phonons werenot treated at the same level of consistency and the bias-dependent /H9261
opt/H20849V/H20850was fitted to experimental data. This defi-
ciency is important because the population of hot opticalphonons determines the scattering length of the electron-phonon interaction.
15
In our study of the diffusive regime, we have solved self-
consistently the two semiclassical Boltzmann’s equations18
either for electrons /H20849two channels /H20850or for optical phonons
/H20849which are assumed to have a constant energy, /H9280opt/H20850. Acous-
tical phonons contribute to the electron-phonon scatteringbut they are assumed to be in thermal equilibrium with theenvironment. In our semiclassical model for electrons, for agiven voltage, the energy window between E
max=/H9280optand
Emin=−eV−/H9280opt, is discretized into Nlevels /H20849around 80 to
achieve convergence /H20850. We analyze, using Boltzmann’s equa-
tion for each energy, E, the local distribution functions,
n+/H20849x,E/H20850andn−/H20849x,E/H20850, representing electrons traveling at con-
stant Fermi velocity, vF, along the xpositive and negative
directions of the one-dimensional nanotube /H20849xmeasures the
distance along the SWNT /H20850. The corresponding hole distribu-
tions are p/H11006/H20849x,E/H20850=1− n/H11006/H20849x,E/H20850. Along their path, electrons
will be scattered by optical and acoustical phonons /H20849see Fig.
1/H20850. We quantify the interaction with the optical phonons by
means of the functions Hopt/H11006andGopt/H11006, where Hopt/H11006/H20849x,E/H20850de-
scribes the probability for an electron of energy E/H20849within a
length segment dxlocated at x/H20850of jumping into another en-
ergy level after the absorption or emission of an optical pho-non. To account for processes in which electrons of differentPHYSICAL REVIEW B 78, 205427 /H208492008 /H20850
1098-0121/2008/78 /H2084920/H20850/205427 /H208495/H20850 ©2008 The American Physical Society 205427-1energies jump into the level of energy E, we use the function
Gopt/H11006. In a similar way, electrons can also be scattered by
acoustical phonons, this process characterized by Hac/H11006and
Gac/H11006. Boltzmann’s equations for the electron distributions,
n+/H20849x,E/H20850andn−/H20849x,E/H20850, are
/H11006dn/H11006
dx=−n/H11006/H20849Hopt/H11006+Hac/H11006/H20850+p/H11006/H20849Gopt/H11006+Gac/H11006/H20850, /H208491/H20850
where we have omitted the xandEdependence for the sake
of brevity. Equation /H208491/H20850describes how n+andn−evolve as a
function of x, for constant energy, E, and constant velocity,
vF/H20849see Fig. 1/H20850; as the energy is constant along the trajectory,
no contribution from/H11509n/H11006
/H11509Eappears in that equation. Moreover,
we assume local charge neutrality conditions; this determinesthe local chemical potential,
/H9262/H20849x/H20850, and the electron momen-
tum by the equation E−/H9262/H20849x/H20850=/H11006kvF. The mathematical ex-
pressions for Hopt/H11006andGopt/H11006are
Hopt/H11006/H20849E/H20850=1
/H9261opt/H20853/H208491+nBopt/H20850/H20851/H9251¯p/H11007/H20849E−/H20850+/H9251p/H11006/H20849E−/H20850/H20852
+nBopt/H20851/H9251¯p/H11007/H20849E+/H20850+/H9251p/H11006/H20849E+/H20850/H20852/H20854, /H208492/H20850
Gopt/H11006/H20849E/H20850=Hopt/H11006/H20849E;p/H11006→n/H11007,E/H11006→E/H11007/H20850, /H208493/H20850
where /H9251/H20849/H9251¯=1−/H9251/H20850describes the forward /H20849backward /H20850scatter-
ing rate associated with the interaction between electrons andoptical phonons. According to theoretical calculations in Ref.18,
/H9251is close to 0.25. In Eqs. /H208492/H20850and /H208493/H20850, the population of
optical phonons is characterized by the function nBopt/H20849x/H20850.I n
Eq. /H208492/H20850, the first term with the factor /H208491+nBopt/H20850accounts for
the emission of an optical phonon /H20849final energy, E−=E−/H9280opt/H20850,
while the second one with the factor nBoptdescribes the ab-
sorption of an optical phonon /H20849final energy, E+=E+/H9280opt/H20850. The
different terms appearing in Eq. /H208493/H20850have similar interpreta-
tions. In Eq. /H208492/H20850,/H9261optincludes all the optical phonons, longi-
tudinal and transversal,18contributing to the electron-phonon
scattering rate. From Ref. 18, for a /H2084910, 10 /H20850carbon nanotube,
/H9261opt/H1101585 nm. In our case, the best fitting to the experimental
data is obtained, however, with /H9261opt/H1101550 nm /H20849see below /H20850.
As expressed in Eq. /H208491/H20850, longitudinal acoustical phononsalso contribute to the electron-phonon scattering rate, al-
though, being this process a quasielastic one, in our calcula-tions we only consider backscattering. Accordingly, we take
/H9251=0 in the corresponding equations for Hac/H11006andGac/H11006, analo-
gous to Eqs. /H208492/H20850and /H208493/H20850,
Hac/H11006/H20849E/H20850=1
/H9261ac/H20851/H208491+nBac/H20850p/H11007/H20849E−/H20850+nBacp/H11007/H20849E+/H20850/H20852 /H20849 4/H20850
Gac/H11006/H20849E/H20850=Hac/H11006/H20849E;p/H11006→n/H11007,E/H11006→E/H11007/H20850, /H208495/H20850
The inset of Fig. 2shows how electrons above /H9262/H20849x/H20850are
backscattered by acoustical phonons; momentum and energyconservation yields the phonon energy, h
/H9263ac, as a function of
the electron energy measured with respect to the local Fermienergy, /H6036
/H9275/H20851/H6036/H9275=E−/H9262/H20849x/H20850/H20852:h/H9263ac=2/H6036vs/H9275/vF, with vsandvF
being the acoustical phonon and electron velocities, respec-
tively. It is known that 1 //H9261acis proportional to q2//H9263ac,19with
qbeing the phonon momentum and /H9263ac=qvs. For the quasi-
elastic acoustical phonons, p/H11007/H20849E−/H20850/H11015p/H11007/H20849E+/H20850, and Eqs. /H208494/H20850
and /H208495/H20850can be well approximated by
Hac/H11006/H20849E/H20850=1
/H9261ac/H208491+2 nBac/H20850p/H11007/H20849E−/H20850, /H208496/H20850
Gac/H11006/H20849E/H20850=1
/H9261ac/H208491+2 nBac/H20850n/H11006/H20849E+/H20850, /H208497/H20850
in such a way that an effective /H9261accan be introduced as
1
/H9261aceff=1
/H9261ac/H208491+2 nBac/H20850/H20849 8/H20850
where
nBac=1
exp/H20873h/H9263ac
kBT/H20874−1. /H208499/H20850
Now we have to distinguish between two limits depend-
ing on the ratio between kBTandh/H9263ac. In the low- Tlimit
/H20849h/H9263ac/H11271kBT/H20850,a s nBac→0, 1 //H9261aceff/H110151//H9261acis proportional to
h/H9263ac, i.e., to /H6036/H9275. In the opposite limit /H20849h/H9263ac/H11270kBT/H20850,nBacEF1
h/CID1opt
h/CID1ac
EF2
FIG. 1. /H20849Color online /H20850Schematic picture showing the dis-
cretized electron levels in the nanotube. In our model, electrons canemit /H20849or absorb /H20850optical phonons of energy h
/H9263opt, and emit acousti-
cal phonons of energy h/H9263ac. All these processes yield, in a self-
consistent solution, the electron distribution function, n/H20849x,E/H20850and
the optical phonon distribution function, nBopt/H20849x/H20850. For a nanotube, all
electrons move with a constant velocity, vF, even if their energy
w.r.t. the local chemical potential, E−/H9262/H20849x/H20850, changes.eff
ac/CID2
/CID4/CID41/CID1
/CID4C/CID4/CID1acq/CID1
ach/CID3
FIG. 2. /H20849Color online /H20850Qualitative behavior of /H9261aceffas a function
of the electron energy, /H6036/H9275=E−/H9262/H20849x/H20850. The inset shows one electron
of energy /H6036/H9275being backscattered by acoustical phonons of energy
h/H9263acand momentum /H6036qac.SUNDQVIST, GARCIA-VIDAL, AND FLORES PHYSICAL REVIEW B 78, 205427 /H208492008 /H20850
205427-2/H11015kBT/h/H9263ac, and 1 //H9261aceff=2kBT//H20849h/H9263ac/H9261ac/H20850, independent of h/H9263ac
and proportional to kBT. Figure 2shows schematically /H9261aceffas
a function of /H6036/H9275, the energy of the incoming electron. For
/H6036/H9275C=vF
2vsh/H9263ac=vF
2vskBT,/H9261aceffhas a crossover from a constant
value /H20849proportional to kBT/H20850t oa1 //H9275dependence. At room
temperature, /H6036/H9275Cis of the order of 0.75 eV , and in our cal-
culations at that temperature we can then take /H9261aceffas a con-
stant value that will be fitted to the behavior of the resistanceat very low voltage. In the second part of this paper, we aregoing to consider a low- Tcase, and this approximation will
be reconsidered. In addition, as the relevant acoustical pho-non energies are between 0 and k
BT, we take, for the sake of
simplicity, an energy close to 25 meV as the mean energy ofthe excited acoustical phonons.
Boltzmann’s equation for the optical phonon distribution,
n
Bopt/H20849x/H20850, can be written as
/H11509nBopt
/H11509t+vopt/H11509nBopt
/H11509x=1
/H9270opt/H20851/H208491+nBopt/H20850S↓−nBoptS↑/H20852−nBopt
/H9270th,
/H2084910/H20850
where /H9270threpresents the thermalization time for optical
phonons, which contains contributions from nonlinear decay-ing processes to acoustical phonons and from heat exchangewith the substrate. On the other hand,
/H9270optis the lifetime of
the optical phonons in their interaction with electrons, /H9261opt
=vF/H9270opt. The magnitudes S↓andS↑account for the generation
and absorption rates of the optical phonons, respectively.These quantities depend on the electronic distribution func-tions, n
+andn−, as follows:
S↓/H20849x/H20850=/H20858
E/H20853/H9251¯/H20851n+/H20849E/H20850p−/H20849E−/H20850+n−/H20849E/H20850p+/H20849E−/H20850/H20852+/H9251/H20851n+/H20849E/H20850p+/H20849E−/H20850
+n−/H20849E/H20850p−/H20849E−/H20850/H20852/H20854, /H2084911/H20850
where the sum extends over all the energy levels. S↑/H20849x/H20850can
be obtained by just replacing E−byE+. In our approach we
assume to have optical phonons of constant energy whichimplies zero group velocity /H20849
vopt=0/H20850. Then, under stationary
conditions /H20849/H11509t→0/H20850, Eq. /H2084910/H20850simplifies to
nBopt/H20849x/H20850=/H9264S↓
1+/H9264/H20851S↑/H20849x/H20850−S↓/H20849x/H20850/H20852, /H2084912/H20850
where /H9264=/H9270th//H9270opt. In our calculations we will use /H9264as a fit-
ting parameter to available experimental data. Then, ourelectronic transport problem /H20849for a fixed nanotube length and
applied bias /H20850is just to solve self-consistently Eq. /H208491/H20850for
n
/H11006/H20849x,E/H20850and Eq. /H2084912/H20850fornBopt/H20849x/H20850. In our calculations, we take
room temperature /H20849300 K /H20850and assume /H9261opt,/H9261aceff, and/H9264to be
known. Once the self-consistent solution is reached, the elec-tronic current-voltage relation, I/H20849V/H20850, is obtained from the
equation I/H20849V/H20850=4e/h/H20848dE/H20851n
+/H20849L,E/H20850−n−/H20849L,E/H20850/H20852.
Figure 3renders the evolution of both the /H20849a/H20850resistance
and /H20849b/H20850differential resistance as a function of the length for
a/H2084910, 10 /H20850SWNT for four voltages /H20849V=0.4, 0.7, 1.0, and 1.5
V/H20850. Dots correspond to the experimental values as reported
in Ref. 17, whereas full curves show the numerical results
emerging from our self-consistent approach. The values forthe fitting parameters that lead to this excellent agreementbetween theory and experiment are /H9261
opt=50 nm, /H9261aceff
=650 nm, and /H9264=28. Notice that this value for /H9261optis a bit
smaller than the one obtained from ab initio calculations.18
The dependence of the population of the hot optical
phonons with the applied voltage and length of the nanotubeis analyzed in Fig. 4/H20849c/H20850. This panel shows how the mean
value of n
B/H20849x/H20850increases with voltage, explaining why in Ref.
17, a decaying function of /H9261optversus Vwas needed in order
to fit the experimental data. The origin of this behavior is thatmore optical phonons are excited when the voltage is high,resulting in a shorter effective mean-free path. It is alsoworth noticing that, for a fixed voltage, /H20855n
B/H20849x/H20850/H20856decreases as
Lis increased. This is due to the fact that when the length of
the nanotube is increased, due to the small voltage gradientalong the SWNT, electrons have less and less energy to ex-cite optical phonons. This results in a change in behavior forthe resistance; for short SWNTs, electron-phonon scatteringis dominated by optical phonons, whereas for long tubes,acoustical phonons play the dominant role. This change inbehavior is nicely visualized when looking at the evolutionofn
+/H20849x,E/H20850atV=1.0 V for two different lengths of the nano-
tube L=0.5/H9262m in Fig. 4/H20849a/H20850andL=4/H9262m in Fig. 4/H20849b/H20850. For
L=0.5/H9262m, optical phonons control the distribution function
and Rwhile for L=4/H9262m, acoustical phonons are much
more operative. We should mention that the results presentedµΩ
FIG. 3. Resistance /H20849a/H20850and differential resistance, dV /dI,i n
panel /H20849b/H20850versus L/H20849in microns /H20850. Four different values of Vare
studied: V=0.4 V /H20849full lines /H20850,V=0.7 V /H20849dashed lines /H20850,V=1.0 V
/H20849dash-dotted lines /H20850andV=1.5 V /H20849dotted lines /H20850. Dots corresponds
to experimental data as reported in Ref. 17.ELECTRONIC TRANSPORT IN CARBON NANOTUBES: … PHYSICAL REVIEW B 78, 205427 /H208492008 /H20850
205427-3in this paper show some small quantitative differences with
those of Ref. 17; this is due to the effects of the hot phonon
populations that make /H9261optto change with L/H20849in Ref. 17,/H9261opt
was taken constant and different for each bias /H20850.
It is convenient to comment also at this point that, appar-
ently, the calculations presented above depend on many dif-
ferent parameters: /H9261aceff,/H9261opt,/H9251, and/H9264. However, /H9251is taken
from LDA calculations; /H9261acefffrom the resistance of clean
nanotubes at low bias; and /H9261optfrom the resistance at V
=0.4 V /H20849in this case, there are very few hot phonons and the
resistance at small Ldetermines /H9261opt/H20850. Therefore, in our cal-
culations there is only one free parameter /H20849/H9264/H20850that we have
chosen to give the best fitting to the experimental data. Thequality of this fitting shows the good quality of our results.From the previous analysis, we conclude that for longenough SWNTs, the resistance is controlled by acousticalphonons when electrons do not have enough energy to excite
optical phonons. /H9261
aceffis around 1 /H9262m at room temperature,
but at very low temperatures /H20849T=1–5 K/H20850,/H9261aceffcan be of the
order of 100 /H9262m. In this low-temperature limit and for very
long tubes, impurities may start to play a role in the elec-tronic transport. In what follows, we will show how this isexactly the case analyzed very recently in the experimentsreported in Ref. 16, in which an interplay between the diffu-
sive and localized regimes is taking place.
As mentioned above, three lengths characterize the nano-
tube resistance behavior: L,/H9261
aceff, and L0. Apart from the three
canonical regimes already discussed, an unexplored regime
emerges when L0/H11021/H9261aceff/H11021L. This is a regime in which we can
expect to have localized states in the sample; but because ofthe electron-phonon interaction, we cannot expect a 2 R
=R
0exp /H20849L/L0/H20850law. The crucial point to realize is that if
/H9261aceff/H11022L, the localized wave functions extend coherently to
the whole system, with peaks in the density of states ran-domly distributed and having a linewidth proportional toexp /H20849−L/L
0/H20850,20which is responsible at the end of the expo-
nential behavior of the resistance. However, when the elec-
tron phonon is operative and /H9261aceff/H11021L, the localized states
only extend coherently to a region of length /H9261aceff. This indi-
cates that the sharp peaks in the density of states in this case
would have a linewidth proportional to exp /H20849−/H9261aceff/L0/H20850. At the
same time, the electron-phonon interaction introduces a dif-fusive process as electrons move incoherently between local-ized states. This suggests the introduction of the factor /H208491
+L//H9261
aceff/H20850contributing to the nanotube resistance. All these
arguments imply that for L0/H11021/H9261aceff/H11270L, the nanotube resis-
tance should go as 2 R=R0/H208491+L//H9261aceff/H20850exp /H20849/H9261aceff/L0/H20850. This equa-
tion can be easily generalized to include the case /H9261aceff/H11271Lby
replacing in the exponent 1 //H9261aceffby 1 //H9261aceff+1 /L. This yields
the equation
2R=R0/H208731+L
/H9261aceff/H20874exp/H20875/H9261aceffL
/H20849/H9261aceff+L/H20850L0/H20876 /H2084913/H20850
valid for any values of /H9261aceff,L, and L0, In particular, for L
→0, Eq. /H2084913/H20850yields 2 R=R0/H208511+L/H208491//H9261aceff+1 /L0/H20850/H20852, showing
that in this limit there is an effective mean-free path, /H9261eff,
given by 1 //H9261eff=1 //H9261aceff+1 /L0.
Before comparing the results emerging from Eq. /H2084913/H20850with
the experimental data reported in Ref. 16, it is convenient to
reconsider our approximation for /H9261aceffat low T. As shown in
Fig.2,/H9261aceffis proportional to 1 /kBTfor small /H6036/H9275; in the other
limit,/H9261aceffdecreases as 1 //H9275. Now if the applied voltage, eV,
is much higher than /H6036/H9275C,/H9261aceffshould decrease with that volt-
age as 1 /eV. This argument indicates that /H9261aceffsaturates for
very short tubes and high bias, and it suggests a length-
dependent /H9261aceff,
/H9261aceff/H20849T/H20850=/H9261aceff/H20849T0/H20850
T/T0+b/H9261eff/L, /H2084914/H20850
where T0is the room temperature and ba constant. In Eq.
/H2084914/H20850, we have introduced the factor /H9261eff/Lthat takes into
account how the effective bias is reduced for long nanotubesif/H9261
eff/H11021L; otherwise /H9261eff/Lshould be replaced by 1. Note
that Eq. /H2084914/H20850is just an interpolation between the two limits
of low and high biases. For very low bias, /H9261aceff/H20849T/H20850/H110081/T,
whereas for a high one, /H9261acefftakes a constant value. The pa-
rameter bin Eq. /H2084914/H20850determines the relative weights of the
two limiting values for low and high biases in the final
/H9261aceff/H20849T/H20850. Therefore, in principle, this parameter bdepends on
the applied voltage and will be used in our calculations as afitting parameter to the experiment.
1.0
0.8
0.6
0.4
0.2
0.0
X(/CID80m) X(/CID80/CID80m)0.0
-0.5
-1.0
0 0.50 4Energy(eV)(a) (b)
(c)
FIG. 4. /H20849Color /H20850Contour plots rendering n+/H20849x,E/H20850atV=1.0 V for
two different L’s/H20851L=0.5/H9262mi n /H20849a/H20850andL=4/H9262mi n /H20849b/H20850/H20852./H20849c/H20850Mean
value of nB/H20849x/H20850for the four voltages analyzed V=0.4 V /H20849blue full
line /H20850,V=0.7 V /H20849black dashed line /H20850,V=1.0 V /H20849red dotted line /H20850, and
V=1.5 V /H20849green dash-dotted line /H20850as a function of the length of the
nanotube. Inset shows nB/H20849x/H20850for two voltages /H20849V=0.4 V: blue
curves and V=1.0 V: red curves /H20850for three different L’s/H20849L
=0.1/H9262m: full lines; L=0.5/H9262m: dashed lines; and L=4.0/H9262m:
dash-dotted lines.SUNDQVIST, GARCIA-VIDAL, AND FLORES PHYSICAL REVIEW B 78, 205427 /H208492008 /H20850
205427-4Figure 5shows a comparison between Eq. /H2084913/H20850and the
experimental data of Ref. 16. We have taken /H9261aceff/H20849T0/H20850
=0.8/H9262m and /H9261eff/H20849L→0/H20850=7/H9262m as in the experiments.16As
stated above, in our analysis there is only one fitting param-eter, b=0.076. In this regards, it is worth stressing that the
applied bias is 6.4 meV /H2084975 K /H20850, this value explaining the
importance of the bias correction for /H9261
aceffas introduced in Eq.
/H2084914/H20850. The different behaviors for R/H20849L/H20850shown in Fig. 5for the
three temperatures is related to the change in /H9261aceffwith T/H20849see
inset of Fig. 5/H20850. In our calculations, we find L0=21/H9262m andfor long tubes /H20849L=370 /H9262m/H20850,/H9261aceff=89, 41, and 22 /H9262m for
T=1.65, 5, and 10 K, respectively. It is the high value of
/H9261aceff/L0forT=1.65 K found for long tubes, the cause of the
rapid increase in the resistance as a function of Lfor this
case. Note that in this limit, /H9261aceff/H110081/T, and then Eq. /H2084913/H20850
leads to R/H11008exp /H208491/T/H20850when L/H11271/H9261aceff. This is exactly the char-
acteristic Tdependence for the resistance within the so-
called thermally activated electron conduction in the local-ized regime, as described in Ref. 21. It is also worth
mentioning that due to the applied bias, /H9261
aceff/H20849L→0/H20850tends to a
constant value, independent on T/H20849see inset in Fig. 5/H20850. This
explains why in the experiments reported in Ref. 16, there is
a saturation value for /H9261effat very low temperatures.
In conclusion, we have shown how the electron transport
in SWNTs depends on the nanotube length and applied bias.
For high biases, hot optical phonons control the diffusiveregime settled in the nanotube. We have demonstrated that,however, for long enough tubes, acoustical phonons start todominate the nanotube resistance. A different transport re-gime appears if L
0, the localization length, is smaller than the
electron mean-free path, /H9261. We have shown that if /H9261/H11021L,
electron conduction enters to a new transport regime, inwhich electrons propagate through localized states assistedby phonons. This finding helps completing the landscape ofthe electronic transport regimes in single-walled carbonnanotubes.
We acknowledge financial support from the Spanish
CICYT under Projects No. MAT2005-01298 and No. NAN-2004-09183-C10-07, and the Comunidad de Madrid/FederProject under Contract No. 07N/0050/2001. We also ac-knowledge helpful discussions with M. Moreno-Moreno andJ. Gómez-Herrero.
*fj.garcia@uam.es
1S. Iijima, Nature /H20849London /H20850354,5 6 /H208491991 /H20850.
2S. J. Tans, A. R. Verschueren, and C. Dekker, Nature /H20849London /H20850
393,4 9 /H208491998 /H20850.
3P. M. Ajayan and T. W. Ebbesen, Rep. Prog. Phys. 60, 1025
/H208491997 /H20850.
4R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, Science
297, 787 /H208492002 /H20850.
5A. Bachtold, P. Hadley, T. Nakanishi, and C. Dekker, Science
294, 1317 /H208492001 /H20850.
6A. Javey, J. Guo, Q. Wang, M. Lundstrom, and H. J. Dai, Nature
/H20849London /H20850424, 654 /H208492003 /H20850.
7J.-C. Charlier, X. Blase, and S. Roche, Rev. Mod. Phys. 79, 677
/H208492007 /H20850.
8C. T. White and T. N. Todorov, Nature /H20849London /H20850393, 240
/H208491998 /H20850.
9Z. Yao, C. L. Kane, and C. Dekker, Phys. Rev. Lett. 84, 2941
/H208492000 /H20850.
10J.-Y . Park, S. Rosenblatt, Y . Yaish, V . Sazonova, H. Ustenel, S.
Braig, T. A. Arias, P. W. Brouwer, and P. L. McEuen, Nano Lett.
4, 517 /H208492004 /H20850.
11A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M. Lund-strom, and H. Dai, Phys. Rev. Lett. 92, 106804 /H208492004 /H20850.
12E. Pop, D. Mann, J. Cao, Q. Wang, K. Goodson, and H. J. Dai,
Phys. Rev. Lett. 95, 155505 /H208492005 /H20850.
13C. Gomez-Navarro, P. J. de Pablo, J. Gomez-Herrero, B. Biel, F.
J. Garcia-Vidal, A. Rubio, and F. Flores, Nature Mater. 4, 534
/H208492005 /H20850.
14B. Biel, F. J. Garcia-Vidal, A. Rubio, and F. Flores, Phys. Rev.
Lett. 95, 266801 /H208492005 /H20850.
15M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. Robert-
son, Phys. Rev. Lett. 95, 236802 /H208492005 /H20850.
16M. S. Purewal, B. H. Hong, A. Ravi, B. Chandra, J. Hone, and P.
Kim, Phys. Rev. Lett. 98, 186808 /H208492007 /H20850.
17P. Sundqvist, F. J. Garcia-Vidal, F. Flores, M. Moreno-Moreno,
C. Gomez-Navarro, J. S. Bunch, and J. Gomez-Herrero, NanoLett. 7, 2568 /H208492007 /H20850.
18M. Lazzeri and F. Mauri, Phys. Rev. B 73, 165419 /H208492006 /H20850.
19J. Ziman, Electrons and Phonons /H20849Oxford University Press, New
York, 1967 /H20850.
20J. B. Pendry, Adv. Phys. 43, 461 /H208491994 /H20850.
21Y . Imry, Introduction to Mesoscopic Physics /H20849Oxford University
Press, New York, 1998 /H20850.0 100 200 300 400020406080100T=1.65K
T=5K
T=10K
L(/CID80m)
0 100 200 300 40 003006009001200Resistance (k/CID58)
Length of the nanotube ( /CID80m)eff
ac/CID79
FIG. 5. Theoretical results for Ras obtained from Eq. /H2084913/H20850for
three different T’s:T=1.65 K /H20849full line /H20850,T=5 K /H20849dashed line /H20850, and
T=10 K /H20849dotted line /H20850. Dots correspond to the experimental data as
reported in Ref. 16. Inset displays the behavior of /H9261aceffas a function
ofLfor the three T’s analyzed in the main panel.ELECTRONIC TRANSPORT IN CARBON NANOTUBES: … PHYSICAL REVIEW B 78, 205427 /H208492008 /H20850
205427-5 |
PhysRevB.92.165110.pdf | PHYSICAL REVIEW B 92, 165110 (2015)
Edge structure of graphene monolayers in the ν=0 quantum Hall state
Angelika Knothe1,2and Thierry Jolicoeur1
1Laboratoire de Physique Th ´eorique et Mod `eles statistiques, Universit ´e Paris-Sud, 91405 Orsay, France
2Physikalisches Institut, Albert-Ludwigs-Universit ¨at Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
(Received 24 July 2015; revised manuscript received 17 September 2015; published 9 October 2015)
Monolayer graphene at neutrality in the quantum Hall regime has many competing ground states with various
types of ordering. The outcome of this competition is modified by the presence of the sample boundaries. In thispaper we use a Hartree-Fock treatment of the electronic correlations allowing for space-dependent ordering. Thearmchair edge influence is modeled by a simple perturbative effective magnetic field in valley space. We findthat all phases found in the bulk of the sample, ferromagnetic, canted antiferromagnetic, charge-density wave,and Kekul ´e distortion, are smoothly connected to a Kekul ´e-distorted edge. The single-particle excitations are
computed taking into account the spatial variation of the order parameters. An eventual metal-insulator transitionas a function of the Zeeman energy is not simply related to the type of bulk order.
DOI: 10.1103/PhysRevB.92.165110 PACS number(s): 73 .43.−f,73.22.Pr,73.20.−r
I. INTRODUCTION
When subjected to a strong perpendicular magnetic field,
the electrons confined to the two-dimensional (2D) carbonlattice of graphene form a unique quantum Hall (QH) sys-tem. Notably graphene at neutrality is an example of QHferromagnetism with many competing ground states [ 1–9].
In such systems we expect generically complex physics at theedge. Early work on graphene edge states [ 10] has shown
that when taking into account the electron spin degree offreedom, the edge states should be of helical nature, i.e.,exhibiting counterpropagating modes carrying opposite spinpolarizations. Graphene was thus proposed to be a modelcandidate for a quantum spin Hall (QSH) system. Furthermore,
graphene has unique features so that it may be an ideal
probe material to study QH edge physics experimentally: insemiconductor-based 2D electron gas systems the confiningpotential is soft at the scale of the magnetic length sothat there may be edge reconstruction [ 11–13], spoiling the
ideal theoretical description. Therefore, the understanding ofedge phenomena in these 2D systems is difficult [ 14,15]
and still controversial. In contrast, the boundaries of thegraphene lattice are directly defined at the atomic level.Therefore, they naturally represent atomically sharp QH edges.This should allow observation of QH edge state physicswithout complications from edge reconstruction [ 16,17]. The
fabrication, design, and control of graphene edge structures
with atomic level precision is a field of ongoing research; see,
e.g., Refs. [ 18–20]. Among the theoretical approaches, also
a mean-field treatment of a Hubbard-type model has beenapplied to the edge physics [ 21]. The QSH nature of the
graphene edge has been the subject of recent experimentalinvestigations [ 22]. At Landau level filling factor ν=0, upon
tilting the applied magnetic field with respect to the graphenesheet, there is a metal-insulator transition for some criticalangle. This suggests a change of the bulk state as a functionof tilting. Indeed, extensive theoretical studies of the ν=0
ground state (GS) structure of bulk graphene [ 6,7,23,24]h a v e
shown the existence of various competing phases with distinctsymmetry-breaking properties. While the graphene GS at neu-trality is a highly degenerate SU(4) ferromagnetic multiplet,
small symmetry-breaking terms due to short-distance physicslift this degeneracy and the system may form various different
phases. Among these phases are, e.g., the ferromagnet (F)
state [ 1,2] or the antiferromagnet (AF) state, where the latter
may form a canted antiferromagnetic (CAF) state as has beenpointed out by Kharitonov in Ref. [ 7]. Further possible phases
include a charge-density wave [ 3,6,25,26] (CDW) or a Kekul ´e
distorted state [ 27,28] (KD). Transitions between these states
may be induced by varying the Zeeman energy which is doneby tilting the field.
In this paper we study the edge structure of the ν=0Q H
state in the presence of SU(4)-symmetry-breaking interactions.We use a simple model of the edge potential in the basis of then=0 Landau level (LL) orbitals appropriate to an armchair
termination of the graphene lattice and treat interactions bya Hartree-Fock (HF) approximation. Our variational ansatz isorbital-dependent so it captures the spatial variations of the
ordering from the bulk to the edge (an effect which is absent
in previous HF studies [ 29]). We find that there is always a
crossover towards a Kekul ´e distorted region close to the edge.
There is appearance also of spin/valley nontrivial entanglementwhich is limited to the transition region towards Kekul ´
e order
and does not take place either in the bulk or close to the edge.We propose a quantitative measurement of the entanglementby computing the concurrence as a function of edge distance.Within HF we also compute the single-particle properties ofthe particle-hole excitation spectrum. We discuss how thisspectrum varies with the edge distance and also how it isinfluenced by the nature of the bulk order. Our main findingis that there should be always a metal-insulator transition as afunction of the Zeeman field whatever the nature of the orderedphase. So strictly speaking the experimental observations of
Ref. [ 22] do not imply that their graphene bulk is a CAF
state. It should be noted that our HF treatment has someshortcomings. Notably the Coulomb exchange interactionin the n=0 LL leads to a coupling between charge and
spin/valley degrees of freedom. So in general charge motion isdone through spin/valley textures as proposed in Refs. [ 30–33].
As in previous HF works [ 29] we will not try to model these
effects. While the edge effects will overcome exchange energyclose enough to boundary, they may change the nature ofexcitations right in the transition region. More work is neededto understand this point.
1098-0121/2015/92(16)/165110(14) 165110-1 ©2015 American Physical SocietyANGELIKA KNOTHE AND THIERRY JOLICOEUR PHYSICAL REVIEW B 92, 165110 (2015)
The paper is structured as follows: In Sec. II, we define
the theoretical framework describing the ν=0 QH state of
graphene in the presence of a boundary and the correspondingmodel Hamiltonian. We introduce the parametrization of theHartree-Fock (HF) GS wave function in Sec. III. In Sec. IV,
we present results for the GS and its properties obtainedfrom minimizing the HF energy functional. We describe theevolution of the different possible bulk phases when movingtowards to the edge. We find that the presence of a boundarygives rise to novel spin/isospin configurations which do notexist in the bulk. Furthermore, we show the existence ofan intermediate region between the bulk and the edge withnontrivial entanglement of spin and valley isospin. In Sec. V,
we extend our HF treatment to compute the spatial evolutionof the single-particle (SP) energy levels and correspondingeigenstates from the bulk to the boundary. As pointed out byprevious work [ 29], the SP spectra can either exhibit nonzero
gaps or support gapless edge states, depending on the systemparameters determining the bulk phase. We describe how thespatial variation of the spin/isospin texture influences the shapeof the SP energy levels. This leads to an understanding ofthe edge gap, the number of conducting channels, as well aspossible conclusions for the bulk symmetry properties drawnfrom the conductance behavior of the edges. In a final part, wecompute the spin and isospin properties of the correspondingSP eigenstates to directly prove that the edge states of the ν=0
QH state in graphene indeed exhibit the helical properties of a
QSH state. Section VIfinally discusses our results in relation
to experimental findings and contains our conclusions.
II. MODEL OF THE GRAPHENE EDGE
We first recall basic facts about the electronic structure
of graphene [ 34,35]. The hexagonal structure admits two
triangular Bravais sublattices AandBthat form the basis for a
tight-binding Hamiltonian. In the Brillouin zone there are twospecial degeneracy points: at these Dirac points KandK
/prime,t h e
valence and the conductance band form Dirac cones and touchat the Fermi level for neutral graphene. In the vicinity of theDirac points, for each orientation of the spin σ=↑,↓,the
electronic wave functions /Psi1
Aand/Psi1Bon the two sublattices
can be written as
/Psi1A,σ(r)=eiK·rψA,σ+e−iK/prime·rψ/prime
A,σ, (1a)
/Psi1B,σ(r)=eiK·rψB,σ+e−iK/prime·rψ/prime
B,σ. (1b)
Assembling the envelope functions in a four-spinor notation
we write for the electronic state
/Psi1σ=⎛
⎜⎜⎝ψA,σ
ψB,σ
ψ/prime
A,σ
ψ/prime
B,σ⎞
⎟⎟⎠
HK,K/prime⊗HA,B, (2)
where the subindex HK,K/prime⊗HA,Bindicates that the state /Psi1σ
lives in the Hilbert space formed as the direct product between
Dirac valley space HK,K/primeand the A,B sublattice space HA,B.
An applied magnetic field leads to the formation of Landaulevels (LL) with energies
En=vF
/lscriptB/radicalbig
2|n|sgn(n), (3)
where n∈Zis the LL index, /lscriptB=√/planckover2pi1c
eB⊥is the magnetic
length for perpendicular magnetic field strength B⊥, andvF
denotes the Fermi velocity. The corresponding eigenstates can
be written as
/Psi1n/negationslash=0,σ=1√
2⎛
⎜⎝|n−1/angbracketright
|n/angbracketright
−|n−1/angbracketright
−|n/angbracketright⎞
⎟⎠
HK,K/prime⊗HA,B,
/Psi1n=0,σ=⎛
⎜⎝0
|0/angbracketright
0
−|0/angbracketright⎞
⎟⎠
HK,K/prime⊗HA,B. (4)
The filling factor νof the Landau levels is defined by
ν=ne
2π/lscript2
B, (5)
where nedenotes the electronic density. The configuration of
neutral graphene, i.e., ν=0, is peculiar. Indeed this particle-
hole symmetric situation corresponds to the case in whichall the LLs with n< 0 are filled and all the LLs with n> 0
are empty, but the n=0 LL is exactly half filled with two
electrons per orbital. The form of the n=0w a v ef u n c t i o na s
given in Eq. ( 4)i m p l i e st h a t n=0 LL electrons reside on one
of the sublattices only. In the following, we consider the case ofneutral graphene and therefore study the properties of electronsin the n=0 LL. We simplify the notation by collecting only
the nonzero entries of the n=0 spinor in Eq. ( 4)a s
/Psi1
0=⎛
⎜⎝|↑+ /angbracketright
|↑− /angbracketright|↓+ /angbracketright|↓− /angbracketright⎞
⎟⎠
H, (6)
identifying the valley and the sublattice indices in a common
valley-isospin τasτ=+ ˆ=K↔Aandτ=− ˆ=K/prime↔B.
In the four-dimensional Hilbert space H=Hspin⊗Hvalley we
use the indices μ,ν to label the four possible configurations of
spin and isospin in this space: μ,ν∈{ ↑+ ,↑−,↓+,↓− } .
Due to the fourfold degeneracy in spin space ( σ=↑,↓) and
in valley space ( τ=+,−), the integer QH effect in graphene
is expected at values of νthat change in steps of four.
The total Hamiltonian we use is given by
H=Hkin+HZ+HCoul+Haniso, (7)
where we have a space-dependent kinetic energy induced by
the presence of the boundary:
Hkin=−/summationdisplay
iEkin(ri)τi
x, (8)
where the index ilabels the positions of the electron orbits. The
hexagonal graphene lattice can be terminated in many differentways, yielding several possible edge structures. Every differentatomic configuration leads to different boundary conditions forthe wave function [ 36]. Two extreme cases are the so-called
165110-2EDGE STRUCTURE OF GRAPHENE MONOLAYERS IN THE . . . PHYSICAL REVIEW B 92, 165110 (2015)
1 2 3 4 5 6R0.20.40.60.81Ekin
100 EZ
Edge Bulk
Armchair
Zigzag
FIG. 1. (Color online) Shape of the kinetic energy edge potential
EkininHkinof Eq. ( 8). The kinetic energy rises from Ekin=0
in the bulk (equivalent to the case of infinitely extended, transla-
tionally invariant graphene in the n=0 LL) to the energy of the
n=1 LL exactly at the edge. The curve was obtained from solving
the problem of free electrons on a graphene lattice in the presence
of a magnetic field in the presence of boundary, i.e., by applying
appropriate boundary conditions for an armchair edge (see inset).Zigzag edges have similar nonzero energy states but also additional
zero-energy modes that are beyond the scope of our work.
zigzag andarmchair edges [ 30]. A finite piece of graphene
terminated by a zigzag edge and an armchair edge is shown inthe inset of Fig. 1. The kinetic energy and the corresponding
eigenstates can be obtained analytically [ 10,31,37,38]. This is
equivalent to turning the level index into a space-dependentquantity n(R) where Rrelates to the distance to the edge
rasr=R/lscript
B/√
2. In Fig. 1we show the spatial shape
of the kinetic energy Ekinobtained by this procedure, as
we will use it in the subsequent calculations. We write thekinetic energy as a space-dependent potential proportional toτ
x(a comparable treatment can be found in Refs. [ 29,33]).
This corresponds to a perturbative treatment as it assumesan expansion of the perturbed edge states in terms of theunperturbed bulk basis states. It restricts our description tothe case of “armchair-like” boundaries: one can always inferthe number of branches in the SP edge spectrum as beingequal to the number of degenerate SP levels in the bulk andhence apply a perturbative expansion as implied by Eq. ( 8).
A derivation of such a Hamiltonian describing the kineticpotential of a graphene edge using arguments of perturbationtheory can be found in Ref. [ 29]. The edges terminated by
a zigzag boundary condition support dispersionless surfacestates [ 30,31] that break the simple bulk/edge correspondence.
They are beyond our simple treatment. The form of the kineticenergy in Eq. ( 8) is valid only in the regime E
kin/lessmuch/planckover2pi1ωc, i.e.,
spatially not too close to the edge. As can be seen from Fig. 1,
this condition is very well met if we restrict the subsequentdiscussion to the regime R> 3. Hence the restriction R> 3
corresponds to a minimal distance r
min≈2.12/lscriptB, which at
realistic experimental values corresponds to rmin≈120a0,
where a0denotes the lattice constant of graphene. The Zeeman
energy can be written as
HZ=−EZ/summationdisplay
iσi
z. (9)In the case of graphene the spacing between kinetic LL energy
levels, /Delta1E kin, easily exceeds the Zeeman energy by two orders
of magnitude. The Coulomb interaction is given by
HCoul=1
2/summationdisplay
i/negationslash=je2
ε1
|ri−rj|, (10)
where εis an effective dielectric constant which depends upon
the substrate [ 39]. It has full SU(4) symmetry. At neutrality
we have an example of quantum Hall ferromagnetism [ 1]
with this large symmetry: the ground state is thus highlydegenerate and forms an irreducible representation of SU(4).However this symmetry is only approximate. In fact it isweakly broken by lattice-scale effects that include short-rangeCoulomb interactions and electron-phonon couplings. It isdifficult to obtain precise estimates of these effects but theirsymmetry-breaking properties can be encoded in the followingHamiltonian:
H
aniso=1
2/summationdisplay
i/negationslash=j/bracketleftbig
gxτi
xτj
x+gyτi
yτj
y+gzτi
zτj
z/bracketrightbig
δ2(ri−rj),
(11)
withδdenoting the Dirac delta function. This Hamiltonian
Haniso has been proposed by Aleiner et al. [40] and its effects
have been analyzed at the mean-field level by Kharitonov [ 7].
Its symmetry properties and phase diagram have been studied
by exact diagonalization [ 41]. It is parametrized by the
coupling constants gx,y,z whose values are not known with
precision. It is likely that the ratio of the energy scales betweenCoulomb interaction and these anisotropies is of the order of10
2. It is thus best to explore the complete phase diagram
taking these parameters as unknowns. For monolayer andbilayer graphene at neutrality, there is a rich phase diagram asa function of these couplings [ 7,42]. The fractional quantum
Hall states are also sensitive to these effects [ 9].
We now perform a HF study of this Hamiltonian by
including the edge potential. We note that in this approachwe neglect all possible spatial dependence of the couplingconstants, which is justified as long as we analyze a spatialdomain not too close to the edge.
III. HARTREE-FOCK TREATMENT
The neutral ν=0 state corresponds to the half-filled case
where two of the four available states per orbital are occupied.We look for the ground state within the family of Slaterdeterminants of the form
|G/angbracketright=/productdisplay
p/parenleftBigg/summationdisplay
μ,νgμνc†
μ(p)c†
ν(p)/parenrightBigg
|0/angbracketright, (12)
where pdenotes the Landau-gauge momentum component
along the edge which labels the electron orbitals. The vacuum|0/angbracketrightconsists of the completely occupied set of states for all
n< 0 and completely empty states for all n> 0. In Eq. ( 12)
gis a 4×4 antisymmetric matrix, i.e., g
μν=−gνμ, in order to
describe a valid fermionic state and Tr[ gg†]=2 to ensure
normalization of the two-particle state. We minimize theenergy of the Slater determinant by varying g. To capture
the effect of the edge potential we take the gmatrix to
165110-3ANGELIKA KNOTHE AND THIERRY JOLICOEUR PHYSICAL REVIEW B 92, 165110 (2015)
be momentum-dependent, i.e., g≡g(p)i nE q .( 12). Due
to the duality between the longitudinal momentum pand
the transverse coordinate rp=p/lscript2
Bthis is equivalent to a
space-dependent description of the problem. The most generalantisymmetric matrix ghas 12 real parameters. By exploiting
the symmetry properties of the state |G/angbracketrightand the Hamiltonian,
one can reduce the number of free parameters. We use thesame strategy as in Ref. [ 43] where an equivalent problem
was studied in the context of electronic bilayer systems. It isconvenient to rewrite the problem in terms of the followingsimple expectation values:
S
α=1
2/angbracketleftG|c†(p)σαc(p)|G/angbracketright=1
2Tr[σαgg†], (13a)
Tα=1
2/angbracketleftG|c†(p)ταc(p)|G/angbracketright=1
2Tr[ταgg†], (13b)
Rαβ=1
2/angbracketleftG|c†(p)σατβc(p)|G/angbracketright=1
2Tr[σατβgg†].(13c)
The expressions of Eqs. ( 13a) and ( 13b) yield the compo-
nents of the total spin Sαand isospin Tαper orbital p.T h e
energy contribution from the symmetry-breaking interactionHamiltonian of Eq. ( 11)i sn o wg i v e nb y
/angbracketleftG|H
aniso|G/angbracketright=1
2/summationdisplay
αuα(Tr[ταgg†]2−Tr[ταgg†ταgg†]),
(14)
where the anisotropy energies uαare given by uα=gα
2π/lscript2
B.
Isotropy of the interaction potential in the x-yplane implies
thatux=uy=:u⊥. Using Eqs. ( 13) and ( 14), we obtain the
following expression for the functional of the total energyE
tot=/angbracketleftG|H|G/angbracketright:
Etot=− 2EkinTx−2EZSz+/summationdisplay
αuα/parenleftBigg
T2
α−/summationdisplay
iR2
iα−S2/parenrightBigg
.
(15)
The 12 −2=10 free parameters of the problem (dropping
the overall phase and normalization constant) are now encodedin the 6 components of the total spin Sand the total isospin
T, together with 4 out of 9 components of R
αβwhich can be
chosen independently. The invariance of Etotin Eq. ( 15) under
rotations of Sin spin space and rotations of Taround the zaxis
in isospin space allows us to choose Sy=Sx=Ty=0 with no
loss of generality, yielding seven variables to be determined.The dimension of parameter space can be further reduced bycareful consideration of all the symmetries of the problem.As demonstrated by Ezawa et al. [43] in a situation of an
equivalent symmetry class, reduction is possible to a totalnumber of three free parameters. For the present system, thisleads us to a minimization problem for the total energy E
tot
with respect to the variational parameters −1/lessorequalslantα/lessorequalslant1,−1/lessorequalslant
β/lessorequalslant1, and χ∈R, which are related to observables of Eq. ( 13)
by
Sz=1/radicalbig
1+χ2/radicalbig
1−α2,T x=χ/radicalbig
1+χ2α/radicalbig
1−β2,
Tz=χ/radicalbig
1+χ2αβ, (16)and
/summationdisplay
iR2
ix=T2
z
χ2,/summationdisplay
iR2
iy=χ2S2,/summationdisplay
iR2
iz=T2
x
χ2,(17)
where the index iruns over the spatial components {x,y,z}.
The density matrix ρg=gg†is connected to these quantities
as (summation convention implied)
ρg=1
21+1
2(σiSi+τiTi+σiτjRij). (18)
IV . GROUND-STATE PROPERTIES
A. Evolution of the spin-isospin texture close to the edge
The bulk GS of graphene within the model we use has
several different phases depending on the anisotropy energiesu
⊥anduzcompared to the Zeeman energy EZ[7,41]. The
anisotropies u⊥,uzand the Zeeman term EZselect some
subset of the manifold of SU(4) ferromagnetic ground states.These phases have distinct spin and isospin configurations,i.e., by different spin textures. The mean-field GS diagramis shown in Fig. 2. It is correct beyond mean-field as shown
by exact diagonalization techniques [ 41]. Notably all these
bulk phases do not involve spin/valley entanglement. We nowgeneralize the description of these phases by treating theeffect of a boundary of the lattice. We proceed as follows:we minimize the energy E
tot(α,β,χ )o fE q .( 15), including a
space-dependent edge potential of the shape shown in Fig. 1
by varying the parameters α,β,χ . Then from the knowledge of
the parameters α(R),β(R),χ(R), we can compute the values of
the observables Sz(R),Tx(R),Tz(R)o ft h eG S |G/angbracketrightvia Eq. ( 16).
-3 -2 -1 1 2 3
-3-2-112345
CAF
F KD
CDW5
2
-2u⊥/EZu/EZ
FIG. 2. (Color online) Bulk GS phase diagram as a function
of the coupling energies u⊥anduzof the ν=0 QH state for a
Hamiltonian as in Eq. ( 7). Different colors distinguish between the
different possible GS phases. The white, dotted lines indicate the
parameters we choose throughout this paper to perform cuts throughthe phase diagram in order to explore the behavior of all possible GS
phases when starting from the bulk and moving towards an edge of
the graphene lattice. The phase diagram for the different GS phasesin the bulk of graphene was first presented in Ref. [ 7].
165110-4EDGE STRUCTURE OF GRAPHENE MONOLAYERS IN THE . . . PHYSICAL REVIEW B 92, 165110 (2015)
It is also possible to construct the entire density matrix ρg
characterizing the GS via Eq. ( 18).
In order to obtain a full picture capturing the edge behavior
of all possible bulk phases shown in Fig. 2, our choice of system
parameters is guided by the following idea: for fixed Zeemanenergy E
Z, we vary the coupling energies u⊥anduzbecause
this can be realized experimentally by tilting the magnetic field.We choose three values of the perpendicular coupling energy
uz:uz=5EZ,uz=2EZ, anduz=− 2EZ, and we vary the
perpendicular coupling in the range −3EZ/lessorequalslantu⊥/lessorequalslant3EZ.T h i s
leads to horizontal cuts through the ν=0 GS phase diagram
in the u⊥,uzplane, shown by white dotted lines in the phase
diagram of Fig. 1.F o ruz=5EZanduz=2EZby varying u⊥
we meet the KD, CAF, and F phases:
u⊥=−∞KD
u⊥>−1
2uz+/radicalbig
2E2
Z+u2z/parenrightbigCAF
u⊥>−EZ
2Fu⊥
Foruz=− 2EZand varying again u⊥we find the KD, CDW, and F phases:
u⊥=−∞KD
u⊥>uzCDW
u⊥>−(EZ+uz)Fu⊥
The corresponding bulk phase transitions are indicated by
white arrows in the phase diagram in Fig. 1.
We first investigate the influence of the edge potential
on the spin and isospin observables SandT.M o r ep r e -
cisely, we discuss the spatial evolution of the componentsS
z(R),Tx(R),Tz(R) for different choices of the anisotropy
energies u⊥anduzcompared to the Zeeman energy EZ.
Figure 3shows the results for uz=5EZ, corresponding to
a cut through the upper half plane (we plot observables withcolored lines). The four different panels depict the situationfor different values of u
⊥:f o ru⊥=−EZ, the bulk system
is in the CAF phase with canting angle cos θ=Sz=1/2
(upper left panel), whereas for u⊥=− 0.2EZ,u⊥=0.5EZ,
andu⊥=1.5EZthe bulk system establishes a F phase in
00.20.40.60.81Sz, Tx, Tz, Cuz= 5EZ
u⊥= -EZ
CAFuz= 5EZ
u⊥= -0.2EZ
F
3 4 5 6 R 00.20.40.60.81
CSz, Tx, Tz, Cu⊥= 0.5EZ
F
3 4 5 6 R u⊥= 1.5EZ
SzTzTxF
FIG. 3. (Color online) Evolution of the spin and isospin as well as
the concurrence Cas functions of R=√
2r//lscriptB, with rthe distance
from the edge. We fix uz=5EZand vary u⊥. (Colored lines: Sz,
solid;Tz, dashed; Tx, dotted. Black, thin, solid line: Concurrence C.)
There is a transition between the bulk phase on the right-hand side
[CAF for u⊥=−EZin the upper left panel (green colors) and F
for all other choices of u⊥shown (blue colors)] to a KD phase at
the edge. In an intermediate regime spin and isospin are canted with
0<Sz<1a n d0 <Tx<0 at the same time and nonzero values of
the concurrence. With growing u⊥, this domain wall grows narrower
in space and moves closer to the boundary.which the spins are fully polarized. The colored curves shown
in Fig. 4correspond to the observables for a cut through the
lower half plane of the phase diagram at uz=− 2EZ. Here,
the perpendicular couplings are chosen so that the left panel atu
⊥=− 0.6EZcorresponds to a CDW phase whereas the right
panel at u⊥=1.2EZagain corresponds t oaFb u l k phase, as
predicted by the GS phase diagram of Fig. 2. Curves for values
of the anisotropy energies favoring a KD phase in the bulk arenot shown since in this case the system does not undergo anytransition whatsoever but remains in the bulk KD phase all theway to the edge.
In general, one can distinguish three different regimes for
the behavior of the observables as a function of the distancer=
/lscriptB√
2Rto the edge. For sufficiently large values of R, i.e.,
deep enough in the bulk, we recover the results of mean-field
theory [ 7]. Close enough to the edge, the system is driven into a
KD phase with Nx=1 andSz=Nz=0, independently of the
bulk phase it adopts. This behavior is due to the edge potentialin the kinetic energy Hamiltonian H
kinin Eq. ( 8): this term is
proportional to τx, so it acts as a Zeeman effect in isospin space,
polarizing the isospin along the xdirection as soon as Ekin(R)
is large enough. This behavior is also consistent with previousworks [ 32,33]. In an intermediate regime we find a finite
interval in space in which S
z/negationslash=1,Tx/negationslash=1 andTz/negationslash=0,Nx/negationslash=0;
i.e., the spin and the isospin are canted simultaneously with
3 4 5 6 R00.20.40.60.81Sz, Tx, Tz, Cuz= -2EZ
u⊥= -0.6EZ Tz
TxSz
CDW
3 4 5 6 RCuz= -2EZ
u⊥= 1.2EZ
F
FIG. 4. (Color online) Same as Fig. 3for transverse coupling
energy uz=− 2EZand different perpendicular coupling energies u⊥,
such that the phases of the lower half plane of the GS phase diagram
are established in the bulk. Left panel: u⊥=0.6 ,l e a d i n gt oaC D W
in the bulk (red colors). Right panel: u⊥=1.2EZ, at which the bulk
is in a F phase (blue colors).
165110-5ANGELIKA KNOTHE AND THIERRY JOLICOEUR PHYSICAL REVIEW B 92, 165110 (2015)
respect to their bulk values. There is thus a domain wall at a
small finite distance from the edge. For the CAF configuration,this domain wall connects smoothly to the bulk configuration.For a system in a F phase in the bulk, however, the changein spin and isospin is abrupt and the domain wall is narrowerwith increasing u
⊥. Hence, for larger values of u⊥, the F phase
of the bulk proves to be more resistant against the increasinginfluence of the edge.
From our analysis and the results shown in Figs. 3and 4
we therefore draw the following conclusions: The phasesin the bulk of a finite sample of graphene do not remainunaffected close enough to the edges. Indeed the effective edgepotential causes the bulk state to undergo a transition in whichthe polarizations of spin and isospin change simultaneously.Sufficiently close to the edge, the GS is driven into a KD phaseindependently of the nature of the bulk phase.
B. Spin-valley entanglement of edge states
The parametrization of S,T,and R αβin terms of the three
parameters α,β, andχin Eqs. ( 16) and ( 17) allows for
complete reconstruction of the density matrix ρgvia Eq. ( 18)
using the values of the parameters obtained by minimizingE
totfrom Eq. ( 15). Therefore, we have a full description
of the GS and its spatial evolution from the bulk towardsthe edge. In this paragraph we investigate the entanglement
between spin and isospin degrees of freedom in the system.For the infinite bulk case, product states of the form |s/angbracketright⊗|n/angbracketright,
where |s/angbracketrightdenotes the (single particle) spin state and |n/angbracketrightthe
(single particle) isospin state, have been used as an ansatz tominimize the GS energy [ 7,29,42]. Existing studies of edge
states using a variational trial wave function approach havesuggested [ 33], however, that for a nonzero edge potential,
spin and isospin might not remain independent, separableobservables, but become entangled . In order to quantify the
amount of entanglement in the bipartite two-level systemH=H
spin⊗Hvalley, we calculate the concurrence Caccording
to the definition [ 44]
C=max(λ1−λ2−λ3−λ4,0), (19)
where the λiare the eigenvalues of the matrix
R=√ρg(σy⊗σy)ρ∗
g(σy⊗σy)√ρg, (20)
in decreasing order λ2
i/greaterorequalslantλ2
i+1∀i.I nE q .( 20),σydenotes the
2×2 Pauli matrix. The quantity Cranges from 0 to 1 with
C=0 meaning no entanglement and C=1 for maximally
entangled states.
In order to study the entanglement of all phases, we
perform the same cuts through the phase diagram as in theprevious paragraph, fixing the transverse coupling at u
z=5EZ
anduz=2EZto investigate the upper half plane and at
uz=− 2EZto study the lower half plane, and we vary the
perpendicular coupling u⊥at these fixed values. Examples of
the spatial behavior of the concurrence C(R)a saf u n c t i o no f
the distance from the edge is depicted by the black solid linesin Figs. 3and4. The curves reveal several characteristics of
the behavior of the concurrence. It goes to zero deep enoughin the bulk for all values of the anisotropies lim
R→∞C(R)=
0∀uz,u⊥. Close enough to the edge, the concurrence is also
equal to zero for all possible bulk phases, as can be seen inFigs. 3and 4forC(R≈3)≡0∀uz,u⊥. In an intermediate
r e g i m ef o rw h i c ht h es y s t e mi si naFo rC A F phase in the
bulk, we find that the concurrence develops a sharp peak. Thispeak appears precisely within the domain wall separating itsbulk phase to the KD phase near the edge. The peak is sharperand higher with rising u
⊥, as the domain wall becomes more
and more narrow in space. Another behavior is observed whenthe bulk is CDW (left panel of Fig. 4). Here, the concurrence
remains zero independently of the distance from the edge:C(R)≡0∀R.
Our findings are summarized in Fig. 5,w h e r ew ep l o t
the maximum concurrence C
maxas a function of u⊥.T h e
resulting curves characterize the behavior of the spin-valleyentanglement of the edge states. Nonzero values of the con-currence are found only for anisotropies favoring a CAF phase(green squares) or a F phase (blue circles) in the bulk. In theseregimes, the maximum concurrence C
maxis a monotonically
rising function of the perpendicular coupling u⊥, with no
discontinuity at u⊥=−EZ/2, which would correspond to the
CAF/F transition in the bulk. Discontinuous jumps appearat values of u
⊥corresponding to the transitions KD/CAF
or CDW/F. Combining the information from Figs. 3,4
and Fig. 5, we draw the following conclusions: unlike the
states in an infinite system, the GS in the presence of aboundary may exhibit nonzero spin-valley entanglement. Asdemonstrated in Figs. 3and 4, the concurrence is exactly
zero in all configurations where either the spin or the isospin
is strictly zero. Nonzero values of the concurrence appear
-2 -1 1 20.20.40.60.81F
CAF
CDW
KD
u⊥/ EZCmax
uz = 5Ezuz= 2Ez
uz = -2Ez
FIG. 5. (Color online) Maximum concurrence Cmaxin the differ-
ent regimes from the bulk to the edge. For three different values
of the coupling energy uz,w ev a r y u⊥. Different symbols represent
different phases in the bulk at the respective value of u⊥: F (blue
circles), CAF (green squares), CDW (red diamonds), and KD (gray
triangles). Empty, shaded, or filled symbols distinguish between the
cuts at different uz. The dashed lines connect the data points as a guide
to the eye. Vertical, gray lines mark the values of {uz,u⊥},a tw h i c h ,i n
the bulk, transitions between the respective phases occur. We observe
that nonzero values of Chappen only if the system in the bulk is in
a CAF or in a F phase. In this case, the CAF/F transition is smooth.
If the bulk is CDW or KD, the concurrence remains equal to zero all
the way from the bulk to the edge. This leads to a discontinuous jumpin the curve of C
maxat the point of the CDW/F transition.
165110-6EDGE STRUCTURE OF GRAPHENE MONOLAYERS IN THE . . . PHYSICAL REVIEW B 92, 165110 (2015)
for configurations in which both spin and isospin are canted
simultaneously .
Compared to the bulk case, we find that the lattice boundary
gives rise to novel ground state phases close to the edge, withsimultaneous canting of spin and isospin, 0 <S<1,0<T<
1, and nonzero spin-valley entanglement. These phases cannotbe described using trial wave functions in the form of separableproduct states [ 7,29].
V . MEAN-FIELD SPECTRUM OF EXCITED STATES
A. Single-particle energy levels
We now diagonalize the single-particle HF Hamiltonian.
The spectrum of the excited states is of particular interest,since the conduction properties of real graphene samplesare governed by the edge modes in the QH regime. Recentconductance experiments have shown [ 22] that upon tilting the
applied magnetic field there is a transition from an insulatingregime to a phase where presumably edge states carry anonzero current. Tilting the magnetic field corresponds tovarying the parameter u
⊥/EZin the system. The experimental
observations therefore suggest that the gap to excited states inthe edge spectrum varies as a function of u
⊥/EZand closes,
eventually, giving rise to a metal-insulator transition. We writethe one-body HF Hamiltonian h
HFcorresponding to the full
Hamiltonian Hof Eq. ( 7). It consists of four terms:
hHF
μν(p)=−Ekin(p)[τx]μν−EZ[σz]μν+C/Delta1μν+A/Delta1μν,
(21)
which we obtain via standard HF decoupling. For the mean-
field potential from the Coulomb interaction Hamiltonian ofEq. ( 10) we find
C/Delta1μν=−u0[gg†]μν, (22)
where u0describes the exchange term of the Coulomb
interaction Hamiltonian of Eq. ( 10). This formula is valid
provided we neglect the spatial dependence of g. It means thatwe do not capture the spin texture effects of the exchange.
For completeness, in the following analytical calculationsand expressions, the Coulomb contribution of Eq. ( 22) will
be written explicitly. The mean-field potential due to theinteractions breaking SU(4) symmetry is given by
A/Delta1μν=/summationdisplay
αuα([τα]μνTr[gg†τα]−[ταgg†τα]μν).(23)
Within HF mean-field theory, diagonalizing hHFprovides
access to SP energies εi, satisfying hHF|i/angbracketright=εi|i/angbracketright, where
the state labeled by istands for the ith SP HF eigenstate.
In the following, we assume the eigenvalues to be orderedε
1/lessorequalslantε2/lessorequalslantε3/lessorequalslantε4. In earlier work [ 29], this Hamiltonian has
been studied with the assumption of constant order parameterto the edge.
However, the explicit effective valley field due to the edge
certainly invalidates this simple assumption. It should be notedalso that in the intermediate regime between the bulk state andthe edge regime the HF ground state is no longer a simpletensor product state even within the HF approximation andthere is some nontrivial spin/valley entanglement. To facilitatesubsequent discussion we summarize the results of Ref. [ 29]
in Table I. The analytical expressions for the mean-field
levels were presented in Ref. [ 29] for the F/CAF cases. A
straightforward calculation allows direct extension to the KDand the CDW configuration. Examples for the SP energy levelsε
±±obtained within this approximation for the different phases
are plotted in Fig. 6for various choices of the couplings.
The ansatz that we introduced in Secs. IIandIIIis able to
describe spatial dependence of the order and also to capturespin/valley entanglement.
We obtain the set of parameters {α(R),β(R),χ(R)}char-
acterizing the GS |g/angbracketright, by minimizing the total energy E
tot
of Eq. ( 15). From these space-dependent parameters, we
construct the corresponding density matrix ρg=gg†via
Eq. ( 18), which in turn allows us to reconstruct and diagonalize
hHFof Eq. ( 21). The analysis is repeated for every point in
TABLE I. Analytical expressions for the mean-field potential of the symmetry-breaking terms,A/Delta1, the eigenvalues of the full mean-field
Hamiltonian, ε±±, and the minimum gaps in the bulk and at the edge, /Delta1ε bulk/edge. We denote by sandtthe SP spin and isospin configuration
of the two electrons per orbital and θdescribes the canting angle between the two spins. In all the formulas for the mean-field potentials, we
dropped a constant term −1
2(u0+2u⊥+uz)1⊗1. These analytical results have been obtained within the approximation that the bulk phase
does not change as a function of space when approaching the edge.
CAF/F phase:
A/Delta1=A/Delta10z1⊗σz+A/Delta1zxσz⊗σx
withA/Delta10z=−1
2(u0+uz+2u⊥)c o sθandA/Delta1zx=−1
2(u0+uz−2u⊥)s i nθ,
ε±±=±/radicalBig
[Ekin(p)±(EZ−A/Delta10z)]2+(A/Delta1zx)2,
/Delta1ε edge=2|A/Delta1zx|,/Delta1εCAF
bulk=u0+uz−2u⊥,/Delta1εF
bulk=2|EZ−A/Delta10z|.
CDW/KD phase:
A/Delta1=A/Delta1x0σx⊗1+A/Delta1y0σy⊗1+A/Delta1z0σz⊗1
withA/Delta1x,y0=−1
2(u0tx,y−uztx,y−4u⊥tx,y)a n dA/Delta1z0=−1
2(u0tz−3uztz−2u⊥tz),
which impliesA/Delta1CDW
z0=−1
2(u0−3uz−2u⊥)a n dA/Delta1KD
x0=−1
2(u0−uz−4u⊥),
εCDW
±±=±EZ±/radicalBig
Ekin(p)2+(A/Delta1CDW
z0)2,andεKD
±±=±EZ±[Ekin(p)−A/Delta1KD
x0],
/Delta1εbulk=2|EZ−|A/Delta1z0/x0||.
165110-7ANGELIKA KNOTHE AND THIERRY JOLICOEUR PHYSICAL REVIEW B 92, 165110 (2015)
Δ = 2CDW
Δ = 0.3EZEZ
-4-2024ε±± / EzCAF/F
Δzx= 0EZ
Δzx ≈ 0.884EZ
3.5 4 4.5 5 5.5 R -4-2024 Δ = 0.3ε±± / EzKD
Δ = -0.7EZEZ
3.5 4 4.5 5 5.5 RΔ = 1.5 KD
Δ = -2EZEZ
FIG. 6. (Color online) SP energy levels ε±±for the different
phase regimes as they are obtained from the analytical formulas in
Table I. Upper left panel: CAF (solid, green lines), F (blue, dashed
lines). Upper right panel: CDW for |/Delta1|>E Z(solid, red lines) and
|/Delta1|<E Z(dashed, black lines). Lower left panel: KD for |/Delta1|<E Z
with/Delta1> 0 (solid, orange lines) or /Delta1< 0 (dashed, black lines).
Lower right panel: KD for |/Delta1|>E Zwith/Delta1> 0 (solid, orange
lines) or /Delta1< 0 (dashed, black lines). Depending on the sign and
the magnitude of /Delta1, the different cases for the SP spectra differ in
the number of level crossings, even within one and the same phase.
space, thereby yielding the spatial behavior as a function of
the distance from the edge. The results for the spectra εi
established near the edge for different phases in the bulk
are shown in Figs. 7and 8. We have chosen the same
parameters as in the plots of Figs. 3and 5.F o ruz>0i n
Fig. 7, we explore the upper half plane, whereas for uz<0
(Fig. 8), the evolution of the bulk phases of the lower half
plane is displayed. In all figures the thick colorful lines showthe outcome of our numerical studies. The thin black linesrepresent the analytical results for the SP energy levels forthe phase established in the bulk at the particular systemparameters shown, as given in Table I. Figure 7illustrates
the evolution of the edge SP levels for a transverse couplingenergy of u
z=5EZand different values of the perpendicular
coupling u⊥, for which the bulk phase configuration passes
from a CAF phase at u⊥=−EZ(green lines, upper left panel)
to a F phase at u⊥=− 0.2EZ,u⊥=0.5EZ, andu⊥=1.5EZ,
respectively (blue spectra in the upper right and the two lowerpanels). In general, the spectra in all four panels show thefollowing behavior: two flat energy levels, separated by thegap/Delta1ε
bulk, are present in the bulk, both twofold degenerate,
and they split into four branches when approaching the edge.The two intermediate levels, being labeled ε
2andε3,fi r s t
bend towards each other, establishing the minimum energygap/Delta1ε
edge</Delta1 ε bulk, before, even closer to the edge, the two
lowest and the two highest levels ε1,ε2andε3,ε4are driven
apart in two parallel pairs, respectively.
In Fig. 8, we show the spectra corresponding to the phases
of the lower half plane: at uz=− 2, we display the energy
levels at u⊥=− 0.6EZ(red lines in the left panel), for which
the bulk is CDW as well as for u⊥=1.2EZ(blue lines in
the right panel), where the bulk is F. The behavior of the SPenergy levels in a CDW bulk phase qualitatively differs fromthe situation of the CAF/F phase described above. In the left-10-50510 εi / Ezuz= 5EZ
1234
u⊥= -EZCAFΔεedge
u⊥= -0.2EZuz= 5EZ
F
3 4 5 6 R-10-50510uz= 5EZεi / Ez
u⊥= 0.5EZF
3 4 5 6 Ru⊥= 1.5EZΔεbulkuz= 5EZ F
FIG. 7. (Color online) Spatial behavior of the energetic SP spec-
tra in the presence of a boundary for positive transverse couplingu
z=5EZand different values of the perpendicular coupling u⊥:
CAF bulk phase at u⊥=−EZin the upper left panel (green lines),
F bulk phase at u⊥=− 0.2EZ,u⊥=0.5EZ,a n d u⊥=1.5EZin
the remaining panels, respectively (blue lines). Thick, colorful lines
show our numerical results. Here, different line shapes distinguish
between the different single-particle energy levels ε1/lessorequalslantε2/lessorequalslantε3/lessorequalslantε4.
Thin, black lines compare to the analytical formulas for ε±±(R)
listed in Table Ifor the different phases, respectively, in which no
modulation of the underlying spin/isospin texture towards the edge istaken into account. We see two bulk levels, being twofold degenerate
and separated by a gap of width /Delta1ε
bulkin the bulk, split into four
branches when approaching the edge. The branches exhibit kinksand regimes of different behavior corresponding to the transitions
between different spin and isospin phases during the evolution from
the bulk to the edge (see Fig. 3). The edge gap between the two
intermediate levels ε
2andε3(dotted and dashed lines, respectively)
/Delta1ε edge=min(|ε3−ε2|] remains finite over a certain range of u⊥,
reducing gradually as u⊥increases until it finally closes completely.
The lower right panel shows a configuration were the levels cross and
form gapless edge states. Special attention should be paid to the upper
right panel and the lower left panel in which the numerical resultsshow configurations with a F phase in the bulk were finite edge gaps
/Delta1ε
edge/negationslash=0 remain, whereas the analytical curves cross as soon as the
bulk passes into an F phase. The behavior of the /Delta1ε edgeas a function
ofu⊥is studied further in Fig. 9.
panel of Fig. 8there are four nondegenerate levels in the bulk.
In contrast to the levels of the CAF/F case, they do not bendtowards each other and there is no minimum energy induced bythe edge behavior. Hence, we find that the minimum edge gap isequal to the bulk gap: /Delta1ε
edge=/Delta1εbulk. Sufficiently close to the
edge, the levels again form two parallel pairs. The spectra forthe KD phase in the bulk are not shown because, as mentionedin Sec. IV, this state does not undergo any significant evolution
when approaching the edge. The spectra do not differ from theanalytical prediction for ε
±±shown forA/Delta1KD
x0in the lower
right panel of Fig. 6.
When comparing this behavior to the analytical results as
given in Table I(black lines), deep in the bulk we find that
all curves coincide as they should. Furthermore, from thediscussion in Sec. IV B , we know that there is no spin-valley
entanglement in the bulk; i.e., the bulk states indeed are ofseparable product form. Yet, significant deviations betweenthe numerical results capturing the full GS properties and the
165110-8EDGE STRUCTURE OF GRAPHENE MONOLAYERS IN THE . . . PHYSICAL REVIEW B 92, 165110 (2015)
3 4 5 6 R-10-50510uz= -2EZ
u⊥= -0.6EZεi / Ez
Δεedge= ΔεbulkCDW
3 4 5 6 Ru⊥= 1.2EZuz= -2EZ F
FIG. 8. (Color online) Same as Fig. 7foruz=− 2EZand differ-
entu⊥, hence displaying the SP spectra for the bulk phase regimes
of the lower half plane of the GS phase diagram. Left panel: CDW
bulk phase at u⊥=− 0.6EZ(red lines). Right panel: F bulk phase at
u⊥=− 1.3EZ(blue lines). Thin, black lines compare to the analytical
formulas given in Table Ifor the different phases, respectively,
in which the evolution of the bulk’s phase towards the edge is
neglected. For the CDW wave phase in the left panel we observe
four nondegenerate SP levels in the bulk; furthermore, as these levelsdo not bend towards each other but disperse when approaching the
edge, the minimum energy gap to SP excitations is given by the bulk
gap/Delta1ε
edge=/Delta1εbulk.
analytical curves from the simplified treatment are observed
when moving closer to the edges, where the GS spin/isospinconfiguration starts to deviate from the bulk phase; cf. Fig. 3.
The SP energies ε
ihave kinks whenever the underlying
spin/isospin texture changes and exhibit qualitatively differentbehavior in the different texture regimes. Thus, the emergenceof different spin/isospin configurations due to the edgepotential when approaching the edges directly translates intothe SP spectra leading to a complex energy structure as afunction of space.
Furthermore, in Ref. [ 29], it is claimed that for a system
being in a CAF phase, the edge spectra always exhibit a gap,which closes when approachin g a F phase, such that a system in
the F phase always supports gapless edge states. Due to the fact,however, that the system does not remain in its bulk phase whenapproaching the edge, we see from Fig. 7that configurations
can be found, in which the bulk indeed is i n a F phase, but
the edge states still exhibit a finite gap /Delta1ε
edge/negationslash=0. In Fig. 7,
this is the case for the anisotropy energies u⊥=− 0.2EZand
u⊥=0.5EZ(Blue lines in the upper right and lower left panel,
respectively). As the value of u⊥rises, the gap /Delta1ε edgebecomes
smaller, until it finally does close, as to be seen in the lowerright panel of Fig. 7foru
⊥=1.5EZ. We elucidate further
on the closing of edge gaps as consequences of the symmetrystructure of the underlying phases in the following Sec. VB.
B. Single-particle level crossings in the different texture regimes
The energy levels of the SP ground and excited states show
a complex structure as a function of space depending on thespatial changes of the spin and isospin texture when approach-ing the graphene edge. In particular, in some configurations,the SP spectra exhibit a finite gap, whereas for other systemparameters, the edge states are gapless when the SP levelscross. In this section, we investigate the crossings betweenSP energy levels which leads to gapless edge excitations. Wefirst discuss the properties of the edge gap and its behaviorwhen approaching the critical values where it closes. Then, weexplain the number of allowed crossing points between energylevels and the connection to the symmetry properties of the
underlying spin/isospin texture phases. The spatial variationof the order parameters has a direct impact on the overallshape of the dispersion of edge modes. This is readily seenin Figs. 7and 8, where we plot the dispersions from our
calculation including edge effects and results from a similarcalculation using only bulk values without spatial variation.In order to investigate the closure of the edge gap /Delta1ε
edgeas
a function of the ratio u⊥/EZ, we evaluate the size of the
minimum gap in the SP spectra for various system parameters.We choose the same values for the anisotropy energies as inSec. IVby fixing u
z=5EZ,uz=2EZ, anduz=− 2EZand
varying u⊥so that we access all bulk phases in the GS phase
diagram. The resulting curves /Delta1ε edge(u⊥/EZ) are shown in
Fig. 9. We find that the size of the edge gap /Delta1ε edgeis a strictly
monotonic decreasing function of u⊥/EZfor all values of uz.
When the bulk is KD or CDW the flat bulk SP levels splitfurther apart when approaching the edge so that the minimumgap in the spectrum is equal to the bulk gap /Delta1ε
edge=/Delta1εbulk.
In these two cases we find that the bulk gap is a linear function
of the perpendicular coupling energy: /Delta1εKD/CDW
bulk ∝u⊥/EZ.
The numerical results in Fig. 9follow exactly the analytical
prediction given in Table I:A tuz=− 2EZ, we find /Delta1εKD
bulk=
−4u⊥and/Delta1εCDW
bulk=− 2u⊥+4EZ, whereas the KD edge
gap at uz=2EZbehaves as /Delta1εKD
bulk=− 4u⊥−4EZ. These
analytical curves are plotted in Fig. 9as dotted lines for
comparison (they are shifted by a constant offset with respect to
-3 -2 -1 1 22468
F
CAF
CDW
KD
u⊥/ EZΔεedge/ EZ
uz= 2Ezuz= 5Ezuz= -2Ez
(-2x+4)+0.5(-4x)+0.5
(-2x+3.3)+0.2
(-1.85x+0.75)+0.5-0.5 0 0.5 1 1.500.511.5
(-4x-4)+0.5
FIG. 9. (Color online) Behavior of the gap /Delta1ε edge in the SP
spectra close to the edge. We use three couplings: uz=− 2EZ,uz=
2EZ,a n duz=5EZ(filled, shaded, and empty symbols, respectively).
Different colors and symbols stand for different bulk phases: KD
(gray triangles), CDW (red diamonds), CAF (green squares), and Fphase (blue circles). Dashed, colored lines connect the data points as
a guide to the eye. The dotted, black lines represent the behavior of
the data in the linear regimes (they are shifted by a constant offsetwith respect to the curves for better visibility). Gray vertical lines
indicate the critical values for bulk phase transitions. For all phases
the gap /Delta1ε
edgemonotonically decreases as u⊥grows until it finally
closes in the regime where the bulk is in an F phase. The inset is
a close-up on how the blue lines smoothly approach /Delta1ε=0. The
transitions from /Delta1ε edge/negationslash=0t o/Delta1ε edge=0 take place at u⊥≈0.3EZ,
u⊥≈EZ,u⊥≈1.625EZ.
165110-9ANGELIKA KNOTHE AND THIERRY JOLICOEUR PHYSICAL REVIEW B 92, 165110 (2015)
the numerical results for better visibility). As a consequence
of this linear behavior in u⊥, for couplings favoring KD or
CDW in the bulk, at the system parameters chosen in Fig. 9,
there is always a nonzero gap to SP excitations. When the GSin the bulk is a CAF or a F phase, the SP spectra now bendtowards each other and therefore exhibit a minimum energygap/Delta1ε
edgenear the edge which is smaller than the bulk gap.
Foru⊥/lessorequalslant−EZ/2 where the bulk is a CAF phase (green
squares), the spectrum always exhibits an nonzero edge gapwhich is almost linear as a function of the perpendicularcoupling. At values u
⊥/greaterorequalslant−EZ/2, hence for a F bulk, the
shape of the spectrum changes qualitatively and the bulk gapcloses in a nonlinear way, asymptotically approaching zero atsufficiently large values of u
⊥. For the transverse couplings
chosen in Fig. 9,uz=5EZ,uz=2EZ, and uz=− 2EZ,
the edge gap /Delta1ε edgecloses at u⊥≈EZ,u⊥≈0.3EZ, and
u⊥≈1.6EZ, respectively. All these values leads to a bulk
which is ferromagnetic. So the prediction for the gap closurepoint clearly differs from the value u
⊥=−EZ
2, which can be
read from the bulk phase diagram in Fig. 2. This is due to the
changes of the spin/isospin configuration of the GS inducedby the effective edge potential as we approach the boundary;cf. Fig. 3. Indeed the system does not remain in a F phase
configuration all the way from the bulk to the edge. During itstransition into a KD phase close to the boundary, there is anintermediate regime with nonzero spin-valley entanglement
and simultaneous canting of both spin and isospin. Hence,
in this transition regime there is no ap r i o r i justification for
/Delta1
CAF/F
edge of Table Ito yield a correct description of the edge
gap.
From this analysis of the gaps of the SP spectra we can
draw the following conclusion: when the bulk is CDW, KD, orCAF phase, the SP energy levels always have nonzero gaps.However for a bulk F phase one may have gapped or gaplessspectra. We note that ignoring the spatial variation of the trialHF state leads to qualitatively different results [ 29].
C. Number of level crossings
We now discuss in more detail the number of crossings of
the HF single-particle states. In the F phase with dispersionε
F
±±in Table I, the intersecting levels εF
+−andεF
−−cross
exactly once as their slope is given by the slope of thekinetic energy term and they are monotonic functions of thespatial coordinate. For several system parameters, such asin the spectrum in the lower left panel of Fig. 7atu
z=
5EZ,u⊥=1.5EZas well as in close-ups shown in Fig. 10,
we observe multiple crossings. We first discuss the occurrenceof multiple crossings and the relation with the underlyingspin/valley texture. The number of crossings is governed by thesymmetries of the HF Hamiltonian and the magnitude of theHF self-consistent potentials. After discussing the differentphases separately, we apply the insights to the edge-statestructure described in Sec. IV, where the GS phase changes as
a function of space when approaching the edge from the bulk.We first discuss the case of the CAF/F transition. We rewritethe mean-field Hamiltonian of Eq. ( 21) involving the CAF/F
mean-field potential given in Table Iby decomposing it into
εi / Ezεi / Ez
3.6 3.7 3.8 R-6-3036
u⊥= 1.2EZuz= 2EZ
3.2 3.3 3.4 3.5 R -2-1012
u⊥= 3EZεi / Ezuz= 2EZ33.5 44.5 55.5 R -8-4048
KD F
33.5 44.5 55.5 R -10-50510 εi / EzF KD
FIG. 10. (Color online) Close-up on the SP spectra in the regime
with multiple crossings for uz=2EZ. Left side: u⊥=1.2EZ, right
side:u⊥=3EZ. The blue lines show the SP energy levels εi.T h e
different background colors mark the spatial regimes in which the
GS establishes different spin/isospin textures: there is a F phase (blue
region). In the intermediate (white) region, the system undergoes atransition before it finally ends up in a KD phase (yellow region). We
observe the crossings of the SP levels to occur in regions of different
spin/isospin phases—their origins lie in the different symmetries of
the F and KD phases.
four 2×2 matrices as
hHF(p)=/bracketleftbigg
γ1γ2
γ3γ4/bracketrightbigg
, (24)
where the respective entries for the CAF/F phase are given by
γ1=A/Delta1zxσx−(EZ−A/Delta10z)σz,
γ2=γ3=−Ekin(p)1,
γ3=−A/Delta1zxσx−(EZ−A/Delta10z)σz, (25)
withA/Delta1zxandA/Delta10zdefined for the CAF/F phase in Table I.
The size of the gap is therefore governed by the first off-
diagonal coupling matrix elementsA/Delta1CAF/F
zx .I fA/Delta1CAF/Fzx/negationslash=
0, as is the case for any nonzero canting angle θ/negationslash=0, the
eigenvalues of the Hamiltonian hHF
CAF/F exhibit the character-
istic behavior of avoided crossings. The SP levels are allowed
to cross only forA/Delta1CAF/F
zx =0a tθ=0, i.e., in the F phase. In
the bulk, i.e., at Ekin≡0, all values of the coupling energies
uzandu⊥allowed for the F phase yield the same ordering
of the SP energy levels εF,0
±±=εF,0
±±(Ekin≡0), independently
of the sign or the modulus of /Delta1CAF/F
0z :εF,0
+−=εF,0
−+<0<
εF,0
−−=εF,0
++. Hence, there is only one possible scenario of level
crossings when approaching the boundary as the increasingedge potential is driving the SP levels away from their bulkvalues. This leads to exactly one crossing of the levels ε
F
+−and
εF
−−, shown by the blue, dashed lines in the upper left panel
of Fig. 6. We can perform the same analysis for CDW or KD
phases. Again, we rewrite the corresponding HF Hamiltonians
of Eq. ( 21) with the potentialsA/Delta1CDW/KD
z0/x0 from Table Iand
we find the respective entries for the CDW:
γ1=−EZσz+A/Delta1z01,
γ2=γ3=−Ekin(p)1, (26)
γ4=−EZσz−A/Delta1z01,
165110-10EDGE STRUCTURE OF GRAPHENE MONOLAYERS IN THE . . . PHYSICAL REVIEW B 92, 165110 (2015)
whereas for the KD phase we find
γ1=γ4=EZσz,
γ2=γ3=/parenleftbigA/Delta1KD
x0−Ekin(p)/parenrightbig
1. (27)
The Hamiltonians for the CDW phase and the KD phase thus
turn out to have higher symmetry than in the CAF phase: Inh
HF
CDW and hHFKD, all entries of the two first off-diagonals as
well as of the antidiagonal are zero. Pairwise degeneracy ofthe corresponding eigenvalues, i.e., crossings between the SPenergy levels, is now allowed. Note that, unlike the transitionfrom a CAF to a F phase, all other transitions do not correspondto smooth transitions. In these cases, a transition betweenphases goes along with an abrupt change of the symmetryproperties of the spin/isospin configuration of the GS and thecorresponding Hamiltonian.
We now discuss the different possible scenarios of SP level
crossings in CDW and KD phases. The SP energy levelsof the CDW ε
CDW
±± in Table Iare independent of the sign
ofA/Delta1CDW
z0. Different orderings of the bulk levels εCDW, 0
±± at
Ekin≡0 may, however, appear depending on the modulus
ofA/Delta1CDW
z0:f o r|A/Delta1CDW
z0|>E Z, the bulk states are ordered
asεCDW, 0
−− <εCDW, 0
+− <εCDW, 0
−+ <εCDW, 0
++ . In this case, when
approaching the boundary, the kinetic energy potential drivesthe positive and the negative energy states farther apart fromeach other such that they do not cross. In the case where|
A/Delta1CDW
z0|<E Z, however, the bulk states rather follow the
hierarchy εCDW, 0
−− <εCDW, 0
−+ <0<εCDW, 0
+− <εCDW, 0
++ .I nt h i s
case, turning on the effective edge potential drives the levels
εCDW, 0
−+ andεCDW, 0
+− towards each other and they cross at zero
energy. These two different scenarios are depicted in the upperright panel of Fig. 6, where the red, solid lines show the levels
ε
CDW
±± from Table IatA/Delta1CDW
z0=2EZ>E Zand the black,
dashed lines show the spectrum forA/Delta1CDW
z0=0.3EZ<E Z.
The latter case, |A/Delta1CDW
z0|<E Z, however, is prohibited by the
conditions imposed on the couplings uzandu⊥in order for
the system to establish a CDW phase in the bulk. Requiringu
z<u⊥anduz<−EZ−u⊥will always force |A/Delta1CDW
z0|>
EZ. Therefore, treating the system as having a stable CDW
phase in the bulk and all the way to the edge will neverlead to any crossings of the SP edge levels. Turning to themore important case of the KD phase, the situation becomeseven richer. Here, depending on the sign and the modulus of
A/Delta1KD
x0, four different SP level orderings in the bulk and four
resulting crossing scenarios may appear. For |A/Delta1KD
x0|<E Z,i f
A/Delta1KD
x0>0, there is one level crossing at zero energy and two
additional crossings above and below the zero-energy line,respectively, whereas for negative
A/Delta1KD
x0, only one crossing
at zero energy is present. The case |A/Delta1KD
x0|>E Zcan lead to
four crossings, two at zero energy plus one above and onebelow, respectively, if /Delta1
KD
x0>0, whereas forA/Delta1KD
x0<0, the
four levels do not cross.
Examples of the four different cases are shown in the lower
panels of Fig. 6, where the lower left panel shows the possible
situations for |A/Delta1KD
x0|<E Z(solid, orange lines forA/Delta1KD
x0=
−0.3EZ<0 and black dashed lines forA/Delta1KD
x0=0.7EZ>0),
whereas the right panel displays the corresponding spectra for|
A/Delta1KD
x0|>E Z(here, the solid, orange lines are forA/Delta1KD
x0=
−1.5EZ<0 and black dashed lines forA/Delta1KD
x0=2EZ>0).Note that, just as in the case of the CDW phase, not all
these cases are allowed by the restrictions on the parameterrange for a KD bulk: requiring the couplings u
zandu⊥to
fulfill the relations u⊥<uzanduz<E2
Z
2u⊥−u⊥always implies
A/Delta1KD
x0>E Z. Therefore, again, all cases including possible
crossings between edge levels are ruled out for a system witha KD phase in the bulk.
This simple picture drawn for constant order parameters
changes when considering the electronic GS structure de-scribed in Sec. IV. Indeed the GS spin/isospin texture deviates
from the bulk phase when moving towards the edge as aconsequence of the growing edge potential. Close enoughto the edge the system is always driven into a KD phase.Hence when moving sufficiently close to the edge the GS doesbecomes KD-ordered, even though the system parameters u
z
andu⊥do not allow KD order in the bulk. Two examples
are shown in the close-ups in Fig. 10. Parameters in both
panels are chosen such that the bulk system at Ekin≡0i s
in a F phase. When moving towards the edge the energylevels evolve according to ε
F
±±of Table I(corresponding to
the evolution within the blue region). A first crossing betweenthe intermediate levels occurs, as predicted by the analysisof the F phase energy levels. After the transition region (leftwhite), the GS becomes KD (marked by the yellow shading).However, the system parameters do not force
A/Delta1KD
x0>E Z:
in the left panel of Fig. 10, we findA/Delta1KD
x0=− 3.4EZand in
the right panel we haveA/Delta1KD
x0=− 7EZ. Therefore the energy
levels now evolve according to εKD
±±of Table Iin the case
A/Delta1KD
x0<0,|A/Delta1KD
x0|<E Z. As a consequence, in this regime,
one more level crossing may occur. Hence, the appearanceof several crossings of the SP energy levels in the numericalspectra as in the lower right panel of Fig. 7and in Fig. 10can be
explained combining the insight of Sec. IVthat any bulk phase
by the edge potential always is driven into a KD phase close tothe boundary, with the understanding of the possible behaviorofε
KD
±±depending on the value ofA/Delta1KD
x0as a function o the
coupling energies uzandu⊥. The SP energy levels describing
the numerical results of Figs. 7,8, and 10can be summarized
as
ε±±(R)=⎧
⎪⎨
⎪⎩εbulk
±±(R)f o r R>R 2,
unknown for R1<R<R 2,
εKD
±±(R)f o r R<R 1,(28)
where εbulk
±±(R) denotes the level spectra for the bulk phase
established at a given choice of system parameters and R2and
R1label the inner and outer limits in space of the domain
wall, for which there is no simple analytic expression. Theevolution of ε
KD
±±(R) is no longer limited to the noncrossing
behavior imposed for a bulk KD phase but it can exhibit any ofthe shapes drawn in the two lower panels of Fig. 6. Which of
these curves describes the KD-like evolution of the edge statescorrectly is determined by the system parameters u
⊥anduz
that govern the bulk texture phase. From the analysis of the
number of SP level crossings we hence learn that, in principle,by choosing appropriate values of u
zandu⊥, SP energy levels
can have zero, one, two, or even three crossings at zero energy.Among these crossings, one is due to the symmetry propertiesof the bulk F phase. The remaining crossings appear in the
165110-11ANGELIKA KNOTHE AND THIERRY JOLICOEUR PHYSICAL REVIEW B 92, 165110 (2015)
-0.5-0.2500.250.5 sz(i)uz= 5EZ 1
2
43
CAFu⊥= -EZ
3 4 5 6 R-0.5-0.2500.250.5 1
23
4tx(i)uz= 5EZ1
2
3
4Fu⊥= -0.2EZ
3 4 5 6 R1
23
4
FIG. 11. (Color online) Evolution of the single-electron spin and
isospin components sz(i)a n dtx(i) of the SP eigenstates from the bulk
towards the edge for the two anisotropy energies u⊥=−EZ(CAF
bulk GS, left panels with green lines) and u⊥=− 0.2EZ(F bulk
GS, right panels with blue lines). Different line shapes distinguish
between the four SP energy levels ε1/lessorequalslantε2/lessorequalslantε3/lessorequalslantε4. Green/blue
lines correspond to the observables of the two lowest-lying states
which are occupied orbitals in the HF GS. Gray lines indicate the
behavior of the higher-lying SP states. Arrows show the behavior ofthe spin and isospin polarizations. The second and third levels |2/angbracketright
and|3/angbracketrightare oppositely polarized in spin and isospin at the edge. In all
plots we set u
z=5EZ.
KD phase close to the boundary, which in this regime shows
novel properties not present for a KD phase in the bulk. Wenote that these crossings occur at different distances from theedge—at the distance where the corresponding KD phase SPlevels for a certain
A/Delta1KD
x0cross, it is necessary for the systemalready to have evolved from the bulk phase into the KD edge
phase in order for the additional SP level crossings to occur.This is the reason why we do not see any crossings in theSP spectrum shown in the lower left panel of Fig. 8where
the bulk is in a CDW phase: at the distance R
cross where the
KD-like levels near the edge would cross, the system stillbehaves according to its bulk CDW phase. In this case, thecrossing is thus prevented by the fact that the crossing point liesoutside the KD region: R
cross>R 2. Nevertheless, a situation
in which the bulk is a CDW but the SP edge states are gaplessdue to crossings of the KD-like levels close to the boundaryis not forbidden by the underlying symmetry principles asthe restrictions for the coupling energies of the CDW bulkphase allow negative values of
A/Delta1KD
x0. The exact distances
from the edge R1,R2,o rRcross which define the points of
crossing involve the explicit form of the kinetic energy Ekin(R)
as they are determined by the eventual dominance of thekinetic energy. Numerical values for R
1,R2,o rRcrosstherefore
strongly depend on the model potential chosen for Ekin(R).
This is not true, however, for the answer to the questionof whether crossings are allowed or not since the values of
A/Delta1are determined generically by the system parameters u⊥
anduz.
D. Properties of the underlying SP states
In order to obtain a better understanding of the nature of the
excited states we analyze the properties of the single-electronstates. We compute the spin and isospin components s
z(i)=
1
2/angbracketlefti|σz|i/angbracketrightandtx(i)=1
2/angbracketlefti|τx|i/angbracketrightand display the results as a
function of space in Fig. 11.
The evolution of the observables can be summarized as
follows:
sz(i) edge intermediate bulk tx(i) edge intermediate bulk
sz(1) : ↑/arrownortheast↑ tx(1) : ↑/arrownortheast↑
sz(2) : ↓/arrowsoutheast − → /arrownortheast↑ tx(2) : ↑/arrownortheast − → /arrowsoutheast↓
sz(3) : ↑/arrownortheast − → /arrowsoutheast↓ tx(3) : ↓/arrowsoutheast − → /arrownortheast↑
sz(4) : ↓/arrowsouthwest↓ tx(4) : ↓/arrowsouthwest↓(29)
where arrows represent schematically the spin and isospin
vectors.
The HF GS |G/angbracketrightis built from Slater determinants of |1/angbracketright
and|2/angbracketrightas the eigenstates of the two lowest lying branches ε1
andε2in Fig. 7. The lowest energy excitations correspond
to single-particle excitation from the second to the thirdlevelε
2→ε3. These two states have oppositely polarized
spin and isospin components. The closing of the gap /Delta1ε edge
between the second and the third SP level in Fig. 11hence
is a transition from insulating to conducting behavior withthe counterpropagating current-carrying edge states exhibitingopposite spin and isospin polarizations. This is the behavior ofgapless helical edge states of a QSH state [ 45].
VI. CONCLUSION
The SU(4) QH ferromagnetism leads to a highly degen-
erate manifold of ground states for neutral graphene. Thisdegeneracy is lifted by small lattice-scale anisotropies and
there is a competition between phases with several types oforder. This competition is affected notably by the substratesupporting the graphene sample. We have used a simple modelof these anisotropies to study the edge properties of neutralgraphene by means of a HF approach. The sharp atomicedge is then described by an effective field in valley spacewhich modifies the competition between phases. Ultimately,whatever the bulk order, the system is in a KD phase closeenough to the edge. In the transition region between thisedge order and the bulk order, we have obtained evidencefor an intermediate regime with spin/valley entanglement. Inthis regime there is a nontrivial change of the single-particlespectrum. We find that the number of levels crossing the Fermienergy can be varied by changing the parameter u
⊥/EZ.T h i s
means that there are metal-insulator transitions when tiltingthe magnetic field. This is consistent with the experimentalfindings of Young et al. [22]. If we adopt the estimates
165110-12EDGE STRUCTURE OF GRAPHENE MONOLAYERS IN THE . . . PHYSICAL REVIEW B 92, 165110 (2015)
for the approximate magnetic field dependencies of EZand
u⊥stated in Refs. [ 7] and [ 29]a sEZ(B)≈0.7B[T]K and
u⊥(B⊥)≈1–10B⊥[T]K, where Bdenotes the total magnetic
field and B⊥its component perpendicular to the device plane,
the values for the parameters stated in Ref. [ 22] suggest that
the authors were able to experimentally tune the ratio u⊥/EZ
roughly in a range from −13 to−0.5. The picture we obtain
is more complex than that obtained by assuming that the orderdoes not persist up to the edge [ 29]. Notably the occurrence
of the metal-insulator transition, while it sets constraints onthe microscopic parameters, does not imply that the bulk isCAF ordered. The observation of a conductance G≈2e
2/h
which corresponds to two conducting channels, i.e., to onesingle-level crossing, has two possible explanations: eitherthe bulk is in a F phase leading to one crossing unaffectedby the KD edge regime, or the bulk has noncrossing SPlevels, but the crossing occurs in the KD regime close to theedge. Furthermore, our results suggest that the observation ofexactly one crossing only corresponds to a limited parameterrange. Varying the anisotropy parameters may lead to theobservation of conductance values of higher multiples of two,corresponding to several crossings in the SP edge spectrum.
Of course there are obvious limitations of our theoretical
approach: we apply a perturbative treatment of the edge,whose validity is limited to a certain range [cf. the discussionfollowing Eq. ( 8)]. The appearance of a KD phase in the
vicinity of the edge is a direct consequence of treating the
effective edge potential perturbatively. Furthermore, in ourcalculations we assume the anisotropy energies u
⊥anduzto remain constant at their bulk values [as we discuss after
introducing Eq. ( 11)]. This approximation certainly becomes
less exact as we approach the boundary. Also we haveneglected the exchange energy effects that will create texturesin the charge-carrying states.
In conclusion, in this paper we have studied the influence
of an edge on the ν=0 QH state in monolayer graphene.
We found that the effective edge potential induces a changeof the GS spin/isospin texture. During this evolution, novelphases are observed, involving simultaneous canting of spinand isospin as well as nonzero spin/valley entanglement.Phases of this kind are not present in the bulk. Furthermore,we analyzed how this spatial evolution changes the SP excitedstates. Here we have shown that, as a consequence of thespatial modulation of the underlying spin/isospin texture, thedirect correspondence between the conductance propertiesof the edge states and the bulk phase is lost. The transportproperties are governed by either zero, one, or multiple SPlevel crossings. The analysis of the SP eigenstates shows thatthe lowest SP excitation describes counterpropagating helicaledge states carrying opposite spin and isospin polarizations.
ACKNOWLEDGMENTS
We acknowledge discussions with Allan H. MacDonald,
Inti Sodemann, Feng-Cheng Wu, and Ren ´eC o t ´e. A.K.
gratefully acknowledges support from the German Aca-
demic Exchange Service and the German National AcademicFoundation.
[1] K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96,256602
(2006 ).
[2] K. Yang, S. Das Sarma, and A. H. MacDonald, P h y s .R e v .B 74,
075423 (2006 ).
[ 3 ] J .A l i c e aa n dM .P .A .F i s h e r , Phys. Rev. B 74,075422
(2006 ).
[4] I. F. Herbut, P h y s .R e v .B 75,165411 (2007 ).
[5] I. F. Herbut, P h y s .R e v .B 76,085432 (2007 ).
[6] J. Jung and A. H. MacDonald, P h y s .R e v .B 80,235417
(2009 ).
[7] M. Kharitonov, Phys. Rev. B 85,155439 (2012 ).
[8] D. A. Abanin, B. E. Feldman, A. Yacoby, and B. I. Halperin,
Phys. Rev. B 88,115407 (2013 ).
[9] I. Sodemann and A. H. MacDonald, P h y s .R e v .L e t t . 112,126804
(2014 ).
[10] D. A. Abanin, P. A. Lee, and L. S. Levitov, Phys. Rev. Lett. 96,
176803 (2006 ).
[11] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, Phys. Rev.
B46,4026 (1992 ).
[12] C. de C. Chamon and X. G. Wen, Phys. Rev. B 49,8227 (1994 ).
[13] K. Yang, Phys. Rev. Lett. 91,036802 (2003 ).
[14] A. M. Chang, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett.
77,2538 (1996 ).
[15] M. Grayson, D. C. Tsui, L. N. Pfeiffer, K. W. West, and A. M.
Chang, P h y s .R e v .L e t t . 80,1062 (1998 ).
[16] Z.-X. Hu, R. N. Bhatt, X. Wan, and K. Yang, Phys. Rev. Lett.
107,236806 (2011 ).[17] G. Li, A. Luican-Mayer, D. Abanin, L. Levitov, and E. Y . Andrei,
Nat. Commun. 4,1744 (2013 ).
[18] M. S. Fuhrer, Nat. Mater. 9,611(2010 ).
[19] J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg,
M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng et al. ,Nature
(London) 466,470(2010 ).
[20] H. Huang, D. Wei, J. Sun, S. L. Wong, Y . P. Feng, A. H. Castro
Neto, and A. T. S. Wee, Sci. Rep. 2,983(2012 ).
[21] J. L. Lado and J. Fern ´andez-Rossier, Phys. Rev. B 90,165429
(2014 ).
[22] A. F. Young, J. D. Sanchez-Yamagishi, B. Hunt, S. H. Choi, K.
Watanabe, T. Taniguchi, R. C. Ashoori, and P. Jarillo-Herrero,Nature (London) 505,528(2014 ).
[23] B. Roy, M. P. Kennett, and S. Das Sarma, Phys. Rev. B 90,
201409 (2014 ).
[24] B. Roy, P h y s .R e v .B 89,201401 (2014 ).
[25] J. Alicea and M. P. A. Fisher,
Solid State Commun. 143,504
(2007 ).
[26] J.-N. Fuchs and P. Lederer, P h y s .R e v .L e t t . 98,016803 (2007 ).
[27] K. Nomura, S. Ryu, and D.-H. Lee, Phys. Rev. Lett. 103,216801
(2009 ).
[28] C.-Yu Hou, C. Chamon, and C. Mudry, Phys. Rev. B 81,075427
(2010 ).
[29] M. Kharitonov, Phys. Rev. B 86,075450 (2012 ).
[30] L. Brey and H. A. Fertig, P h y s .R e v .B 73,235411 (2006 ).
[31] L. Brey and H. A. Fertig, P h y s .R e v .B 73,195408 (2006 ).
[32] H. A. Fertig and L. Brey, P h y s .R e v .L e t t . 97,116805 (2006 ).
165110-13ANGELIKA KNOTHE AND THIERRY JOLICOEUR PHYSICAL REVIEW B 92, 165110 (2015)
[33] G. Murthy, E. Shimshoni, and H. A. Fertig, Phys. Rev. B 90,
241410 (2014 ).
[34] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,
and A. K. Geim, Rev. Mod. Phys. 81,109(2009 ).
[35] M. O. Goerbig, Rev. Mod. Phys. 83,1193 (2011 ).
[36] A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B 77,
085423 (2008 ).
[37] W. N. Mei and Y . C. Lee, J. Phys. A: Math. Gen. 16,1623
(1983 ).
[38] M. Janssen, O. Viehweger, U. Fastenrath, and J. Hajdu,
Introduction to the Theory of the Integer Quantum Hall Effect(Wiley-VCH, Weinheim, 1994).[39] E. J. G. Santos and E. Kaxiras, Nano Lett. 13,898(2013 ).
[40] I. L. Aleiner, D. E. Kharzeev, and A. M. Tsvelik, Phys. Rev. B
76,195415 (2007 ).
[41] F. Wu, I. Sodemann, Y . Araki, A. H. MacDonald, and
Th. Jolicoeur, P h y s .R e v .B 90,235432 (2014 ).
[42] M. Kharitonov, Phys. Rev. Lett. 109,046803 (2012 ).
[43] Z. F. Ezawa, M. Eliashvili, and G. Tsitsishvili, Phys. Rev. B 71,
125318 (2005 ).
[44] F. Mintert, A. R. R. Carvalho, M. Ku, and A. Buchleitner, Phys.
Rep.415,207(2005 ).
[45] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82,3045
(2010 ).
165110-14 |
PhysRevB.100.144507.pdf | PHYSICAL REVIEW B 100, 144507 (2019)
Vortex bound states of charge and magnetic fluctuations induced topological
superconductors in heterostructures
Hossein Hosseinabadi and Mehdi Kargarian*
Department of Physics, Sharif University of Technology, Tehran 14588-89694, Iran
(Received 27 April 2019; revised manuscript received 10 September 2019; published 16 October 2019)
The helical electron states on the surface of topological insulators or elemental bismuth become unstable
toward superconducting pairing formation when coupled to the charge or magnetic fluctuations. The lattergives rise to pairing instability in chiral channels d
xy±idx2−y2, as has been observed recently in the epitaxial
Bi/Ni bilayer system at relatively high temperature, while the former favors a pairing with zero total angular
momentum. Motivated by this observation we study the vortex bound states in these superconducting states. Weconsider a minimal model describing the superconductivity in the presence of a vortex in the superconductingorder parameter. We show that zero-energy states appear in the spectrum of the vortex core for all pairingsymmetries. Our findings may facilitate the observation of Majorana modes bounded to the vortices inheterostructures with no need for a proximity-induced superconductivity and relatively large value of /Delta1/E
F.
DOI: 10.1103/PhysRevB.100.144507
I. INTRODUCTION
For many years it was a common belief that quantum-
mechanical particles are either bosons or fermions whosewave functions take a plus or minus sign, respectively, uponthe exchange of two identical particles. But in the pastforty years it has been realized that this picture for point-like particles is correct only for a space with dimensionsequal to or greater than three. There are other possibilitiesin two-dimensional (2D) space [ 1]. In 2D systems, parti-
cles’ wave functions can in general acquire a complex phaseupon turning one particle around another one, the so-calledAbelian anyons [ 2,3]. The first physical realization of Abelian
anyons occurred in systems exhibiting the fractional quantumHall effect [ 4,5]. For the non-Abelian anyons, on the other
hand, the multicomponent wave functions live in a degeneratesubspace [ 6], and the interchange of particles amounts to a
unitary evolution matrix within the degenerate subspace. Thenoncommutative structure of matrices promises a platform forfault-tolerant topological quantum computations [ 7].
Majorana fermions, a special class of non-Abelian anyons,
discovered by theoretical particle physicist Ettore Majo-rana [ 8] in 1937, have the property that they are their own
antiparticles. If γ
iandγ†
iare donated as the annihilation and
creation operators for a Majorana fermion in a quantum state|i/angbracketrightthenγ
i=γ†
iand{γi,γj}=2δij. Majorana fermions were
not observed in elementary particle physics. But the developedconcepts were traced in condensed-matter physics years afterthe theoretical discovery [ 9–11]. In the seminal work by Reed
and Green [ 11], it is shown that a boundary between a 2D
topological p+ipsuperconductor and a trivial one hosts
a single Majorana mode, and in a vortex core a Majoranabound state (MBS) is formed. A prime example of a systemhosting a MBS was introduced by Kitaev [ 12]. The model is a
*kargarian@physics.sharif.educhain of spinless fermions with p-wave superconducting order
parameter. Under certain conditions, where the bulk of thesystem is topologically nontrivial, two unpaired MBSs appearat the ends of an open chain.
The discovery of topological insulators (TIs) [ 13,14]p r o -
vided a boost to the realization of Majorana fermions insolid state systems. In a celebrated work by Fu and Kane inRef. [ 15] it is shown that the MBSs appear in the vortex cores
of a conventional s-wave superconductor proximitized to the
surface of TIs arising from the spin-momentum locked struc-ture of the surface states of the TIs and the underlying Berryphase, which makes the s-wave pairing formally like a p-wave
pairing upon projection onto the surface states. Viewing thesurface states as massive Dirac electrons, the vortex boundstates of the corresponding fermion-vortex problem have beenstudied [ 16,17]. The MBSs can also be realized in doped
TIs [ 18,19]. Moreover, the surface of TIs can be replaced by
more conventional semiconductors with strong Rashba spin-orbit coupling. The latter lifts the spin degeneracy and createsmultiple spin-momentum locked Fermi surfaces. An externalmagnetic or Zeeman field is required to tune a quantum phasetransition from an induced s-wave superconductor to a topo-
logical superconductor with Majorana fermions dispersingalong the edges of 2D systems [ 20–22] or bounded to the end
points of semiconductor nanowires [ 23,24].
At the heart of these settings for generating a topological
phase with MBSs lies the existence of three conventional in-
gredients: a semiconductor quantum well with strong Rashbacoupling, an s-wave superconductor, and a ferromagnetic
insulator. The Zeeman field produced by the ferromagneticlayer should be strong enough to remove the extra Fermisurfaces, and simultaneously should be weak for the inducedsuperconducting pairing amplitude to survive, a conditionwhich severely restricts the choice of materials. Moreover,the induced superconducting gap is rather small to allow forresolving MBSs [ 25]. Therefore, it is highly demanding to
look for heterostructures with less impeding ingredients.
2469-9950/2019/100(14)/144507(7) 144507-1 ©2019 American Physical SocietyHOSSEIN HOSSEINABADI AND MEHDI KARGARIAN PHYSICAL REVIEW B 100, 144507 (2019)
TI(Bi)
FM(Ni)
FIG. 1. A schematic representation of epitaxial bilayer. A thin
film of bismuth (Bi) or topological insulator (TI) is deposited on a
ferromagnetic layer such as nickel (Ni), an experimentally realized
bilayer [ 26]. The Dirac cone and the black arrows indicate the helical
electron states near the Fermi surface. The wavy red line showsthe pairing states between electron states living on opposite sides
of the Fermi surface, induced by the magnetic fluctuations on the
in-plane magnetic moments (thick blue arrow) of nickel. Verticalarrows demonstrate a schematic view of a typical vortex.
The recently discovered superconductivity in epitaxial
bismuth /nickel (Bi /Ni) bilayer heterostructure with relatively
large transition temperature Tc≈4.2 K may provide an ex-
ample of an intrinsic topological superconductor with chirald
xy±idx2−y2order parameter [ 26]. The nodeless structure of
the proposed gap function is consistent with measurementsof frequency-dependent optical conductivity in time-domainTHz spectroscopy [ 27]. It is also argued that the superconduc-
tivity could result from the bulk alloys near the interface [ 28],
but it turns out that the surface superconductivity is consistentwith thickness dependent of transition temperature [ 26]. A
schematic representation of the bilayer system is shown inFig. 1. The main advantage of the latter system over the
quantum well structures discussed above is that here thesuperconductivity is intrinsically driven by the ferromagneticfluctuations circumventing the proximity to an extra s-wave
superconductor layer. Now the important question is what arethe vortex bound states in an intrinsic d
xy±idx2−y2topological
superconductor in the Bi /Ni bilayer system? The aim of this
paper is to theoretically answer this question. We show thatthe Berry phase effects change the angular momentum ofthe order parameter by one giving rise to odd parity withzero-energy states. Furthermore, contrary to conventional su-perconductors, large ratios of superconducting gap to Fermienergy /Delta1/E
F≈10−2–10−1in Bi/Ni allow for MBS to re-
main well-separated from the low-lying excited states.
For nonchiral d-wave superconductor dx2−y2the existence
of extended [ 29] and localized core states [ 30,31]h a v e
been discussed. The analog of discrete Caroli–de Gennes–Matricon core states [ 32] for spin-rotationally symmetric
chiral d-wave superconductors has been studied [ 29,33], but
no zero modes are reported. The heterostructure of topolog-ical insulator Bi
2Se3film on a nodal d-wave superconductor
Bi2Sr2CaCu 2O8+δhas been experimentally studied recently
where a rather large induced and isotropic superconductinggap was reported [ 34]. The isotropically nodeless gap wasattributed to an emergent s-wave component on the surface
of TI due to broken fourfold symmetry with possible MBSs invortex cores [ 35]. Note that our system is also distinct from the
case of d
x2−y2superconductor proximitized to an electron gas
studied in Ref. [ 36], where an external Zeeman field is applied
to the system while in our work the time-reversal symmetry isalready broken by the nature of chiral pairing.
The paper is organized as follows. In Sec. IIwe derive
the single-particle Hamiltonian describing the helical electronstates near the Fermi surface, then in Sec. IIIwe derive
the pairing correlations driven by magnetic fluctuations. InSec. IVwe study the vortex bound states for various cases
and the existence of zero-energy states, and finally Sec. V
concludes.
II. REAL-SPACE PROJECTED
NONINTERACTING HAMILTONIAN
We begin with the electronic structure of the surface of
Bi exposed to the vacuum as shown in Fig. 1. Most likely,
the interface adjacent to Ni does not contribute to super-conductivity due to strong pair breaking effects, which isconsistent with the thickness dependency of T
c[26,37]. The
strong Rashba coupling splits the surface electron states tospin-momentum locked states with the largest pocket centeredaround the center of the surface Brillouin zone. For simplicitywe only consider this pocket and, hence, the Bi thin film ino u rs e t u pi nF i g . 1can be replaced with a surface of TI as
well. Therefore the helical electron states are described by thefollowing action in Euclidean-time formalism:
S
e=/integraldisplay
dτdr/Psi1†(∂τ+˜H0)/Psi1. (1)
The noninteracting Hamiltonian ˜H0reads as
˜H0=/integraldisplay
dr/Psi1†(r)(vF[σ×p]z−μ)/Psi1(r), (2)
where /Psi1=(ψ↑,ψ↓)Twithψ/arrowbothvas electron annihilation oper-
ators, vFis the magnitude of spin-orbit interaction or equiva-
lently the Fermi velocity of electrons at the surface of TI, μ
is the chemical potential, σis a vector of Pauli matrices, and
p=−i∇.W es e t¯ h=kB=1 throughout.
Since we are interested in electron states near the Fermi
surface, we use helical eigenstates |k,±/angbracketright = (1,±ie−iφk)T/√
2
in momentum space, where ˜H0(k)|k,±/angbracketright = ± εk|k,±/angbracketrightwith
εk=vF|k|. We assume that the Fermi level crosses the “ +”
band and is located well above the node. We first write theannihilation operators in the spin basis in terms of annihilationoperators in the helical “ ±” basis:
ψ
↑k=1√
2(ψk++ψk−),ψ ↓k=ie−iφk
√
2(ψk+−ψk−).(3)
Then in real space they become
ψ↑(r)=1√
2[ψ+(r)+ψ−(r)], (4)
ψ↓(r)=i√
2/summationdisplay
ke−iφk(ψk+−ψk−)eik·r. (5)
144507-2VORTEX BOUND STATES OF CHARGE AND MAGNETIC … PHYSICAL REVIEW B 100, 144507 (2019)
To perform the momentum sum in Eq. ( 5) we approximate the
angular exponential term using
e−iφk≈kx
kF−iky
kF, (6)
which is justified so long as the electron states near the Fermi
surface are involved in the physical processes of interestsuch as pairing instabilities. Here k
F=μ/vFis the Fermi
momentum. The field operator ψ↓(r) then reads as
ψ↓(r)=1√
2kF(∂x−i∂y)[ψ+(r)−ψ−(r)]. (7)
By projection to the Fermi surface we obtain
ψ↑(r)≈1√
2ψ+(r),ψ ↓(r)≈1√
2kF(∂x−i∂y)ψ+(r).(8)
Rewriting Eq. ( 2) by use of these expressions, the projected
form of the noninteracting Hamiltonian reads as follows:
H0=−/integraldisplay
drψ†
+(r)/parenleftbiggvF
kF∇2+μ/parenrightbigg
ψ+(r), (9)
which is not dissimilar to the Hamiltonian of a 2D Fermi gas
via the identification vF/kF=1/2m. Hereafter we drop the
subindex in the field and write ψ+(r)≡ψ(r).
III. MODEL OF MAGNETIC FLUCTUATIONS
AND PAIRING HAMILTONIAN
To make the structure of the paper self-contained, in this
section we present the details of a minimal model describingthe superconductivity in the bilayer structure shown in Fig. 1
which is largely based on Ref. [ 26]. In the regime of interest
the superconducting T
cis much lower than the Curie temper-
ature of ferromagnetic Ni layer, and hence the ferromagnet isdeep inside the ordered phase. We assume that the in-planemoments are aligned along the ydirection (see Fig. 1) and
the low-energy fluctuations of the magnetic moments, the spinwaves, are described by the vector l(τ,r) in which l·ˆy=0.
The magnetic fluctuations and their coupling to electrons aredescribed, respectively, by the following actions [ 38]:
S
M=ρs
2/integraldisplay
dτdr[−i(l×∂τl)+κ(∇l)2], (10)
and
SeM=g/integraldisplay
dτdr/Psi1†(l·σ)/Psi1, (11)
where ρsis the density of magnetic moments, κis the char-
acteristic of spin waves, and gis the strength of interaction
between electron spins and magnetic moments. For simplicityand in the interest of formation of Cooper pairs with zerocenter-of-mass momentum we only consider the out-of-planefluctuations denoted by l
z≡b.
By taking Fourier transform to momentum space, Eqs. ( 10)
and ( 11) become
SM=T
2/summationdisplay
qD−1(q)b†
qbq, (12)and
SeM=gT
2/summationdisplay
k,q,αβ/parenleftbig
bqψ†
kασz
αβψk+q,β+H.c./parenrightbig
, (13)
where Tis the temperature, q=(q,2nπT),k=(k,(2n+
1)πT)with nas an integer, and D(q)=1/(κρs|q|2+ζ)i st h e
magnon propagator with a small gap ζdue to anisotropy [ 39].
Upon integrating out the bosonic field band subsequent pro-
jection of electron fields ψkαto the Fermi surface described by
effective spinless fermion operators ψkin Eq. ( 3), we obtain
the following effective interaction between electrons in theCooper channel [ 26]:
S
c=T
2A/summationdisplay
k,k/primeU(k,k/prime)e−i(φk−φk/prime)ψ†
kψ†
−kψ−k/primeψk/prime, (14)
whereAis the area of the system. Here U(k,k/prime)i sa ne v e n
function of its arguments and can be expanded in angularharmonics as
U(k,k
/prime)=/summationdisplay
l=evenUleil(φk−φk/prime). (15)
Therefore the even angular momentum components of
the interaction matrix contribute to the odd component ofthe condensate f=/angbracketleftψ
−kψk/angbracketright. This is a direct result of the
nontrivial topology of Dirac Fermions. The effective angularmomentum of the condensate fis decreased by 1 due to the
Berry phase. Inserting Eq. ( 15)i nE q .( 14) and decoupling the
interaction in the Cooper channels f, we obtain the mean-field
BCS Hamiltonian as follows:
H
/Delta1=/summationdisplay
k/Delta1(|k|)ei(l−1)φkψ†
kψ†
−k+H.c. (16)
In this work we only consider the channels with the lowest
angular momenta l=0,±2. The superconducting instability
in channels l=±2 is driven by the magnetic fluctuations be-
ing relevant to the Bi /Ni bilayer system, while the instability
with l=0 arises from phonons or charge fluctuations [ 40],
i.e.,σz→1in Eq. ( 11). In our formalism below we study all
cases.
IV . SPECTRUM OF VORTEX BOUND STATES
The formulation and arguments presented in the preced-
ing sections provide a minimal superconducting Hamiltonianusing Eqs. ( 9) and ( 16), i.e., H=H
0+H/Delta1. In the following
subsections we first derive the corresponding Bogoliubov–deGennes (BdG) equations for each channel land then study the
spectrum of vortex bound states.
A. Cases with l=2a n d l=0
By inspection we see that for both cases the phases of the
pairing in Eq. ( 16) are simply complex conjugates of each
other, and thus, they can be treated within the same formalism.We present the details of calculations for l=2 and will briefly
discuss the l=0 case at the end of this subsection. For the
former case the pairing term H
/Delta1in Eq. ( 16) is written as
H/Delta1=/summationdisplay
k/Delta1(|k|)
kF(kx+iky)ψ†
kψ†
−k+H.c., (17)
144507-3HOSSEIN HOSSEINABADI AND MEHDI KARGARIAN PHYSICAL REVIEW B 100, 144507 (2019)
where we used Eq. ( 6), assuming pairing occurs near the
Fermi surface. To introduce a vortex in the order parameter,we assume that the space profile of pairing gap in the polarcoordinate is /Delta1(r)=/Delta1(r)e
inθ, where ris measured from the
center of vortex and ndenotes the winding of the vortex, the
degree of vorticity. Thus, the full mean-field Hamiltonian ofthis system in real space can be recast as
H=/integraldisplay
dr/bracketleftbigg
−ψ
†/parenleftbiggvF
kF∇2+μ/parenrightbigg
ψ+i/Delta1(r)
2kFψ†{einθ,∂x
+i∂y}ψ†+i/Delta1(r)
2kFψ{einθ,∂x−i∂y}ψ/bracketrightbigg
, (18)
where {A,B}=(AB+BA)/2 is a symmetric operator. We
define γ†
ias a creation operator for the ith eigenstate of themean-field Hamiltonian satisfying
[HMF,γ†
i]=Eiγ†
i. (19)
The operator γ†is used to diagonalize the Hamiltonian and
hence can be written as a linear combination of ψ’s:
γ†
i=/integraldisplay
dr[ui(r)ψ†(r)+vi(r)ψ(r)]. (20)
From now on we drop the index ifor simplicity. Using
Eqs. ( 18) and ( 20)i nE q .( 19), we get a system of dif-
ferential equations of the form HBdGϕ(r)=Eϕ(r), where
ϕ(r)=(u(r),v(r))TandHBdGis the BdG Hamiltonian,
HBdG=/parenleftBigg
−vF
kF∇2−μ i/Delta1(r)
2kF{einθ,∂x+i∂y}
i/Delta1(r)
2kF{e−inθ,∂x−i∂y}vF
kF∇2+μ/parenrightBigg
. (21)
Rewriting the differential operators in polar coordinates, Eq. ( 21) assumes the following form:
HBdG=/parenleftBigg
−vF
kF/bracketleftbig1
r∂r(r∂r)+1
r2∂2
θ/bracketrightbig
−μ i/Delta1(r)
kFein/primeθ/parenleftbig
∂r+i1
r∂θ−n
2r/parenrightbig
i/Delta1(r)
kFe−in/primeθ/parenleftbig
∂r−i1
r∂θ−n
2r/parenrightbigvF
kF/bracketleftbig1
r∂r(r∂r)+1
r2∂2
θ/bracketrightbig
+μ/parenrightBigg
, (22)
where n/prime=n+1. We use a pseudorotation operator defined
by the unitary transformation U(θ)=e−i[m+(n/prime/2)τz]θ, where
τzis the Pauli matrix acting in particle-hole space, to re-
move the phase dependency of the pairing gap [ 21]. That
is, we write the wave function ϕ(r)i nt h ef o r m ϕ(r,θ)=
ei[m+(n/prime/2)τz]θϕ(r). The possible values for mare determined
by the single-valued condition of wave function implyingthat mis an integer (half integer) for even (odd) n
/prime.U s i n g
this transformation the eigenvalue problem turns into a set ofdifferential equations for v(r) and u(r)a s
∂
2
ru+1
r∂ru−m2
+
r2u+kFμ
vFu−i/Delta1(r)
vF/parenleftbigg
∂rv−2m−1
2rv/parenrightbigg
=−kFE
vFu, (23)
∂2
rv+1
r∂rv−m2
−
r2v+kFμ
vFv+i/Delta1(r)
vF/parenleftbigg
∂ru+2m+1
2ru/parenrightbigg
=kFE
vFv, (24)
where m±=(2m±n/prime)/2. We see that the equations are not
symmetric under n→− n. Note that the shift in n/primerelative
to winding nby 1 comes from the π-Berry phase of the
electron states on the Fermi surface. The latter phase shiftsthe relative angular momentum of pairs by 1 [ 41]. Therefore
the bound states of cores with opposite phase winding aroundthe vortex would have different energy spectra. A set ofequations similar to those quoted in Eqs. ( 23) and ( 24)i s
presented for a p-wave superconductor [ 11,42], where the
kinetic terms are usually absent in the long-wavelength limitand it is assumed that the chemical potential is negative in thevortex core (the strong-coupling phase) and positive outside(the weak-coupling phase) with /Delta1(r) as a constant. In our case
however we assume μto be constant and take a space-varying
order parameter like conventional superconductors.
D u et ot h e p-wave structure of the Hamiltonian ( 22)t h e
vortex core hosts a Majorana fermion. In Appendix Awe
employ an approach similar to Ref. [ 21] and explicitly show
that a zero-energy state exists in the vortex core. Furthermore,in order to get more insight into the spectrum of the boundstates we use a long-wavelength approximation [ 9,32]f o rt h e
wave function (see Appendix Bfor details) and show that the
spectrum reads as
˜E=m/integraltext
∞
0/parenleftbig˜/Delta1(x/prime)
x/prime+n/prime
x/prime2/parenrightbig
e−2χ(x/prime)dx/prime
/integraltext∞
0e−2χ(x/prime)dx/prime, (25)
where χ(x)=/integraltextx
0˜/Delta1(x/prime)
2dx/prime, and we have used dimensionless
parameters x=kFrand ˜/Delta1=/Delta1/μ and ˜E=E/μ. One has
to note that we are a little cavalier in using the semiclassicalapproach, since the value of /Delta1/E
Fin our system, as we dis-
cuss more in Sec. V, is rather large compared to conventional
superconductors. Within the approximations used it turns outthat the vortices with n
/prime/negationslash=0 would have very large energy
ifm/negationslash=0 simultaneously. Let us consider a vortex with the
lowest value of vorticity n/prime=0 corresponding to n=−1
as shown schematically in Fig. 1. The condition U(2π)=1
implies that the mhas to be an integer number with m=0
corresponding to a zero-energy state.
As we mentioned at the beginning of this subsection the
cases with l=2 and l=0 can be treated on equal footing,
since the corresponding equations are the same. The lattercase, l=0, yields n
/prime=n−1 and a vortex with lowest wind-
ing number will have n=1. Again the vortex can host a
zero-energy state.
144507-4VORTEX BOUND STATES OF CHARGE AND MAGNETIC … PHYSICAL REVIEW B 100, 144507 (2019)
B. Case with l=−2
In this case the BdG equations become third order and an
analytical solution for them is a formidable task if not impos-sible. To circumvent this problem, we use the semiclassicalapproximation used in the analysis of the Andreev boundstates in superconductors [ 43] and closely follow Refs. [ 44]
and [ 45]. The BdG equation in this case is
H
BdG=(h0−μ)τ3+i/Delta1(r)
2k3
F{einθ,(∂x+i∂y)3}τ+
+i/Delta1(r)
2k3
F{e−inθ,(∂x−i∂y)3}τ−, (26)
where h0=−vF
kF∇2is the kinetic energy. For solving BdG
equations we use an ansatz for the wave function as /Psi1=
ϕ(r)eiq·rwith an approximation that the momentum qis re-
stricted to the Fermi surface, i.e., q=kF(cosφ,sinφ) known
as the momentum of a quasiparticle in the Andreev approxi-mation. Using /Psi1in Eq. ( 26), we obtain
H=−iv·∇τ
3+/Delta1(r) cos(θ/prime)τ1+/Delta1(r)s i n (θ/prime)τ2,(27)
which acts only on ϕ(r). Here we defined v=(2vF/kF)qand
θ/prime=nθ+3φ. We rotate the coordinates such that the new x
axis becomes parallel to q:
H=−iv∂xτ3+/Delta1(r) cos[ nθ+(3−n)φ]τ1
+/Delta1(r)s i n [ nθ+(3−n)φ]τ2. (28)
Then the φdependence in the Hamiltonian can be removed
using the transformation ϕ→ei(3−n)φτ3/2ϕ:
H=−iv∂xτ3+/Delta1(r) cos( nθ)τ1+/Delta1(r)s i n( nθ)τ2.(29)
This is a quasi one-dimensional problem derived in Ref. [ 44]
[see Eq. (3.10) in the latter reference with replacement θ→
−nθ]. To proceed we define an impact parameter for quasipar-
ticles as b=rsinθwhich measures the minimum distance of
the quasiparticle trajectory from the origin of the vortex core.For the pairing gap we use a profile as /Delta1(r)=/Delta1/Theta1(r−R),
where /Theta1(x) is the usual step function and Ris the radius of
the vortex. The latter is of order of R/similarequalv
F//Delta1. With these
assumptions the quasi-one-dimensional model Eq. ( 29) can be
solved to obtain the energy spectrum of the bound states. Forsmall values of bk
F/lessmuch1, corresponding to trajectories passing
near the origin, the spectrum reads as
Em=ω0/bracketleftbig
−nπ+2π/parenleftbig
m+1
2/parenrightbig/bracketrightbig
, (30)
where ω0=vF/2Ris the angular velocity of the super-
fluid [ 44]. Now it is clearly seen that for n=1 the spectrum
becomes Em=2πω 0mand a zero mode corresponds to m=
0. Therefore the vortex bound states for the l=−2 case also
contain a zero-energy mode.
V . CONCLUSIONS
This work is mainly motivated by the efforts put forward
in recent years to find Majorana bound states in the vortexcore of superconducting states. We proposed the supercon-ducting epitaxial Bi /Ni bilayer as a platform to create and
manipulate the Majorana states. The system has an advantageover the heterostructures proposed in the literatures in that thechiral superconducting states are created intrinsically due tothe magnetic fluctuations of the ferromagnetic layer circum-
venting the need for a proximitized superconducting layer.The heterostructure here can be replaced by other materialscombinations, e.g., a thin film of topological insulator Bi
2Te3
deposited on the magnetic insulator layer FeTe [ 46,47], or
superconducting states in oxide interfaces [ 48], making our
proposal for creating and manipulating zero modes experi-mentally feasible.
The chiral superconducting states in Bi /Ni bilayer are
characterized by total angular momentum l=±2 correspond-
ing to d
xy±idx2−y2, which break the time-reversal symmetry.
We showed that the underlying strong spin-orbit couplingalters the bound-state spectrum in the vortex core. In particularwe demonstrated that a zero-energy state corresponding toa Majorana bound state appears at the vortex core for bothcases of the pairing wave functions. We also showed that thecase with total angular momentum l=0, which intrinsically
does not break the time-reversal symmetry and may be in-duced by charge fluctuations, can also support a zero-energystate. The setup studied in our work, as shown in Fig. 1,
should be contrasted with proposals in the literature wherethe superconductivity is induced by proximity. Our finingsmay motivate the search for Majorana zero modes in vorticesin heterostructures with no need for proximity to an extrasuperconducting layer.
Another peculiar aspect of vortex bound states in Bi /Ni is
that the Majorana bound state remains well separated fromthe low-lying excited states due to a relatively large valueof/Delta1/E
F. The superconducting gap is estimated to be about
/Delta1≈0.7m e V[ 27] and the Fermi energy EFis about 27 meV
for hole pockets and 10 meV for electron pockets [ 49] yielding
a ratio of about 10−2–10−1. The ratio is by an order of mag-
nitude larger than the corresponding values for conventionalsuperconductors. A clear observation of discrete bound statesin iron-based superconductor FeTe
0.55Se0.45[50] is reported
due to the large value of /Delta1/EF. The surface of the latter
compound was shown to be a topological superconductor [ 51]
hosting a well-resolved Majorana bound state [ 52]. Therefore
the same sort of well-resolved bound states and Majorana zeromode should be observed in epitaxial Bi /Ni bilayer.
In summary, our work offers the chiral superconductor in
epitaxial Bi /Ni bilayer as a platform to explore the vortex
states with (i) a robust Majorana bound state due to the largeratio of /Delta1/E
F, (ii) less complexity in the heterostructure, and
(iii) relatively high transition temperature.
ACKNOWLEDGMENT
The authors would like to acknowledge the support
from the Sharif University of Technology under Grant No.G960208.
APPENDIX A: EXPLICIT CALCULATION OF ZERO
ENERGY STATE
Using the dimensionless parameters x=kFrand ˜/Delta1=
/Delta1/μ and ˜E=E/μ,E q s .( 23) and ( 24) can be rewritten as
∂2
xu+1
x∂xu−m2
+
x2u+u−i˜/Delta1(x)/parenleftbigg
∂xv−2m−1
2xv/parenrightbigg
=− ˜Eu, (A1)
144507-5HOSSEIN HOSSEINABADI AND MEHDI KARGARIAN PHYSICAL REVIEW B 100, 144507 (2019)
∂2
xv+1
x∂xv−m2
−
x2v+v+i˜/Delta1(x)/parenleftbigg
∂xu+2m+1
2xu/parenrightbigg
=˜Ev. (A2)
Here we consider ( A1) and ( A2) and follow Ref. [ 21]f o r
˜E=0. We assume that the vortex boundary is at x=x0and
take ˜/Delta1(x) to be zero for x<x0and a constant value for x>
x0. For the region inside the vortex the equations become a
set of decoupled Bessel equations with the general analyticalsolution:/parenleftbigg
u(x)
v(x)/parenrightbigg
=/parenleftbigg
A
1Jm+(x)
A2Jm−(x)/parenrightbigg
, (A3)
where Jm(x) is the Bessel function of the first kind. A closed
solution of equations for x>x0is not tractable. Instead we try
to find the asymptotic solution of equations in the limit x/greatermuch1.
In this limit ( A1) and ( A2) become
∂2
xu+u−i˜/Delta1∂xv=0, (A4)
∂2
xv+v+i˜/Delta1∂xu=0. (A5)
Looking for a decaying solution of the form
/parenleftbigg
u
v/parenrightbigg
=/parenleftbigg
u0
v0/parenrightbigg
eκx
one gets
/parenleftbigg
u(x)
v(x)/parenrightbigg
=A3/parenleftbigg
1
−i/parenrightbigg
f+(x)+A4/parenleftbigg
1
−i/parenrightbigg
f−(x), (A6)
where f±(x) are decaying functions with the asymptotic
form f±(x)→e−κ±xandκ±=|˜/Delta1|
2±√
(˜/Delta1
2)2−1. We have to
match the solutions for inside and outside of the vortex at thevortex boundary. The condition ϕ(x
−
0)=ϕ(x+
0)g i v e s
A1Jm+(x0)=A3f+(x0)+A4f−(x0), (A7)
A2Jm−(x0)=−i[A3f+(x0)+A4f−(x0)], (A8)
ϕ/prime(x−
0)=ϕ/prime(x+
0) yields
A1J/prime
m+(x0)=A3f/prime
+(x0)+A4f/prime
−(x0), (A9)
A2J/prime
m−(x0)=−i[A3f/prime
+(x0)+A4f/prime
−(x0)]. (A10)
These equations along with the normalization condition/integraltext
[u2(r)+v2(r)]dr=1 should be satisfied in order to have
a solution. In general, it is not possible to satisfy all of theseconditions by adjusting only four unknowns ( A
1-A4) and the
problem is overconstrained and a zero mode solution doesnot exist. Nevertheless, in the special case of n
/prime=0 where
m+=m−from ( A7) and ( A8)w eh a v e A2=−iA1whichmakes ( A9) and ( A10) identical and therefore there are only
three independent boundary conditions which along with thenormalization condition assign a unique value to A
1-A4.T h i s
is compatible with ( 25) in which a zero energy state exists
only if n/prime=0.
APPENDIX B: ENERGY SPECTRUM OF BOUND STATES
Following Refs. [ 9] and [ 32], we assume that the wave
functions in ( A1) and ( A2) take the following form:
/parenleftbigg
u
v/parenrightbigg
=/parenleftbigg
f+
g+/parenrightbigg
H1
q(x)+/parenleftbigg
f−
g−/parenrightbigg
H2
q(x), (B1)
where H1
qandH2
qare the Hankel functions of the first and
second kind, respectively, and fand gare slowly varying
functions. We insert the above ansatz in Eqs. ( A1) and ( A2)
and neglect the second derivatives of f±and g±.U s i n g
the asymptotic behavior of Hankel functions, we obtain thefollowing differential equations governing f
±andg±:
df±
dx−i˜/Delta1
2g±=±i/parenleftbigg˜E
2−n/primem
2x2/parenrightbigg
f±∓˜/Delta1m
2xg±, (B2)
dg±
dx+i˜/Delta1
2f±=∓i/parenleftbigg˜E
2−n/primem
2x2/parenrightbigg
g±∓˜/Delta1m
2xf±. (B3)
The low-energy spectrum of bound states in the vortex
lies within the superconducting bulk gap. Therefore a naturalassumption is to assume ˜E<˜/Delta1in the equations above,
otherwise there would be no bound states at the vortex core.Physically the bound states result from the Andreev reflectionsof quasiparticles in the vortex core [ 44]. We also assume that
x/greatermuch1 which means that we are considering the long-distance
behavior of the system. Then by treating the expressions onthe right-hand side as perturbations, we obtain the followingexpressions for fandgup to first order:
/parenleftbigg
f
1
g1/parenrightbigg
=A1/braceleftbigg/parenleftbigg
1
i/parenrightbigg
e−χ(x)−/parenleftbigg
i
1/parenrightbigg
eχ(x)
×/integraldisplay∞
x/parenleftbigg˜/Delta1m
2x/prime+n/primem
2x/prime2−˜E
2/parenrightbigg
e−2χ(x/prime)dx/prime/bracerightbigg
,(B4)
/parenleftbigg
f2
g2/parenrightbigg
=A2/braceleftbigg/parenleftbigg
1
i/parenrightbigg
e−χ(x)+/parenleftbigg
i
1/parenrightbigg
eχ(x)
×/integraldisplay∞
x/parenleftbigg˜/Delta1m
2x/prime+n/primem
2x/prime2−˜E
2/parenrightbigg
e−2χ(x/prime)dx/prime/bracerightbigg
,(B5)
where χ(x)=/integraltextx
0˜/Delta1(x/prime)
2dx/prime. In order to avoid the singularity of
Hankel functions at the origin we should take A1=A2and
the second terms should vanish as x→0. These boundary
conditions eventually lead to the expression for the energyspectrum of vortex bound states ˜Ein (25).
[1] F. Wilczek, P h y s .R e v .L e t t . 48,1144 (1982 ).
[2] J. M. Leinaas and J. Myrheim, Nuovo Cimento B 37,1
(1977 ).
[3] F. Wilczek, P h y s .R e v .L e t t . 49,957(1982 ).
[4] B. I. Halperin, Phys. Rev. Lett. 52,1583 (1984 ).[5] D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53,
722(1984 ).
[6] J. Fröhlich, in Nonperturbative Quantum Field Theory ,e d i t e d
by G. ’t Hooft, A. Jaffe, G. Mack, P. K. Mitter, and R. Stora(Springer, Boston, 1988), pp. 71–100.
144507-6VORTEX BOUND STATES OF CHARGE AND MAGNETIC … PHYSICAL REVIEW B 100, 144507 (2019)
[7] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das
Sarma, Rev. Mod. Phys. 80,1083 (2008 ).
[8] E. Majorana, Nuovo Cimento 14,171(2008 ).
[9] N. B. Kopnin and M. M. Salomaa, Phys. Rev. B 44,9667
(1991 ).
[10] G. Moore and N. Read, Nucl. Phys. B 360,362(1991 ).
[11] N. Read and D. Green, Phys. Rev. B 61,10267 (2000 ).
[12] A. Y . Kitaev, Phys. Usp. 44,131(2001 ).
[13] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82,3045
(2010 ).
[14] M. Z. Hasan and J. E. Moore, Annu. Rev. Condens. Matter
Phys. 2,55(2011 ).
[15] L. Fu and C. L. Kane, P h y s .R e v .L e t t . 100,096407 (2008 ).
[16] R. Jackiw and P. Rossi, Nucl. Phys. B 190,681(1981 ).
[17] B. Seradjeh, Nucl. Phys. B 805,182(2008 ).
[18] P. Hosur, P. Ghaemi, R. S. K. Mong, and A. Vishwanath,
P h y s .R e v .L e t t . 107,097001 (2011 ).
[19] C.-K. Chiu, P. Ghaemi, and T. L. Hughes, P h y s .R e v .L e t t . 109,
237009 (2012 ).
[20] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys.
Rev. Lett. 104,040502 (2010 ).
[21] J. D. Sau, S. Tewari, R. M. Lutchyn, T. D. Stanescu, and S. Das
Sarma, Phys. Rev. B 82,214509 (2010 ).
[22] J. Alicea, P h y s .R e v .B 81,125318 (2010 ).
[23] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, P h y s .R e v .L e t t .
105,077001 (2010 ).
[24] Y . Oreg, G. Refael, and F. von Oppen, P h y s .R e v .L e t t . 105,
177002 (2010 ).
[25] M.-X. Wang, C. Liu, J.-P. Xu, F. Yang, L. Miao, M.-Y . Yao,
C. L. Gao, C. Shen, X. Ma, X. Chen, Z.-A. Xu, Y . Liu, S.-C.Zhang, D. Qian, J.-F. Jia, and Q.-K. Xue, Science 336,52
(2012 ).
[26] X. Gong, M. Kargarian, A. Stern, D. Yue, H. Zhou, X. Jin, V . M.
Galitski, V . M. Yakovenko, and J. Xia, Sci. Adv. 3,e1602579
(2017 ).
[27] P. Chauhan, F. Mahmood, D. Yue, P.-C. Xu, X. Jin, and N. P.
Armitage, P h y s .R e v .L e t t . 122,017002 (2019 ).
[28] S.-P. Chao, P h y s .R e v .B 99,064504 (2019 ).
[29] M. Franz and Z. Tešanovi ´c,P h y s .R e v .L e t t . 80,4763 (1998 ).
[30] I. Knezevic and Z. Radovic, Phys. C (Amsterdam, Neth.) 317-
318,640(1999 ).
[31] M. Kato and K. Maki, Europhys. Lett. 54,800(2001 ).[32] C. Caroli, P. D. Gennes, and J. Matricon, Phys. Lett. 9,307
(1964 ).
[33] B. Rosenstein, I. Shapiro, and B. Y . Shapiro, J. Low Temp.
Phys. 173,289(2013 ).
[34] E. Wang, H. Ding, A. V . Fedorov, W. Yao, Z. Li, Y .-F. Lv, K.
Zhao, L.-G. Zhang, Z. Xu, J. Schneeloch, R. Zhong, S.-H. Ji, L.Wang, K. He, X. Ma, G. Gu, H. Yao, Q.-K. Xue, X. Chen, andS. Zhou, Nat. Phys. 9,621(2013 ).
[35] Z.-X. Li, C. Chan, and H. Yao, P h y s .R e v .B 91,235143 (2015 ).
[36] L. Ortiz, S. Varona, O. Viyuela, and M. A. Martin-Delgado,
Phys. Rev. B 97,064501 (2018 ).
[37] X.-X. Gong, H.-X. Zhou, P.-C. Xu, D. Yue, K. Zhu, X.-F. Jin, H.
Tian, G.-J. Zhao, and T.-Y . Chen, Chin. Phys. Lett. 32,067402
(2015 ).
[38] M. Kargarian, D. K. Efimkin, and V . Galitski, P h y s .R e v .L e t t .
117,076806 (2016 ).
[39] Y . Tserkovnyak and D. Loss, P h y s .R e v .L e t t . 108,187201
(2012 ).
[40] L. P. Gor’kov and E. I. Rashba, P h y s .R e v .L e t t . 87,037004
(2001 ).
[41] S. Zeytino ˘glu, A. Imamo ˘glu, and S. Huber, arXiv:1810.05737 .
[42] M. Matsumoto and R. Heeb, P h y s .R e v .B 65,014504 (2001 ).
[43] A. F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964)[
J. Exp.
Theor. Phys. 19, 1228 (1964)].
[44] M. Stone, P h y s .R e v .B 54,13222 (1996 ).
[45] G. E. V olovik, J. Exp. Theor. Phys. Lett. 70,609(1999 ).
[46] S. Manna, A. Kamlapure, L. Cornils, T. Hänke, E. M. J.
Hedegaard, M. Bremholm, B. B. Iversen, P. Hofmann, J. Wiebe,and R. Wiesendanger, Nat. Commun. 8,14074 (2017 ).
[47] Q. L. He, H. Liu, M. He, Y . H. Lai, H. He, G. Wang, K. T. Law,
R. Lortz, J. Wang, and I. K. Sou, Nat. Commun. 5,4247 (2014 ).
[48] M. S. Scheurer and J. Schmalian, Nat. Commun. 6,6005
(2015 ).
[49] P. Hofmann, Prog. Surf. Sci. 81,191(2006 ).
[50] M. Chen, X. Chen, H. Yang, Z. Du, X. Zhu, E. Wang, and H.-H.
Wen, Nat. Commun. 9,970(2018 ).
[51] P. Zhang, K. Yaji, T. Hashimoto, Y . Ota, T. Kondo, K. Okazaki,
Z. Wang, J. Wen, G. D. Gu, H. Ding, and S. Shin, Science 360,
182(2018 ).
[52] D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y .
Sun, S. Du, J. Schneeloch, R. Zhong, G. Gu, L. Fu, H. Ding,and H.-J. Gao, Science 362,333(2018 ).
144507-7 |
PhysRevB.97.100503.pdf | PHYSICAL REVIEW B 97, 100503(R) (2018)
Rapid Communications
Tunneling probe of fluctuating superconductivity in disordered thin films
David Dentelski,1,2Aviad Frydman,1Efrat Shimshoni,1and Emanuele G. Dalla Torre1,2
1Department of Physics, Bar-Ilan University, 52900 Ramat Gan, Israel
2Center for Quantum Entanglement Science and Technology, Bar-Ilan University, 52900 Ramat Gan, Israel
(Received 26 June 2017; revised manuscript received 20 February 2018; published 9 March 2018)
Disordered thin films close to the superconductor-insulator phase transition (SIT) hold the key to understanding
quantum phase transition in strongly correlated materials. The SIT is governed by superconducting quantumfluctuations, which can be revealed, for example, by tunneling measurements. These experiments detect a spectralgap, accompanied by suppressed coherence peaks, on both sides of the transition. Here we describe the insulatingside in terms of a fluctuating superconducting field with finite-range correlations. We perform a controlleddiagrammatic resummation and derive analytic expressions for the tunneling differential conductance. We findthat short-range superconducting fluctuations suppress the coherence peaks even in the presence of long-rangecorrelations. Our approach offers a quantitative description of existing measurements on disordered thin filmsand accounts for tunneling spectra with suppressed coherence peaks.
DOI: 10.1103/PhysRevB.97.100503
Introduction. Superconducting thin films have attracted
recently a lot of attention due to the possibility of observinga direct superconductor-to-insulator transition (SIT) [ 1–3].
The SIT is considered as an excellent example of a quantumphase transition [ 4]: It occurs at temperature T=0 and is
driven by a nonthermal tuning parameter g. Experimentally,
the SIT can be driven by a wide variety of g’s, such as
thickness [ 5–16], magnetic [ 11,12,17–26] or electric fields
[27], chemical composition [ 28,29], and disorder [ 25,30]. Near
the quantum critical point g=g
c, the system is governed by
quantum fluctuations [ 31–35] and cannot be described in terms
of classical Ginzburg-Landau theories [ 36–39].
In the vicinity of the SIT, experiments show an intriguing
behavior of the superconducting energy gap /Delta1. Traditionally,
/Delta1is determined by fitting the tunneling conductivity to a
phenomenological extension of the BCS theory, which takesinto account an effective energy broadening /Gamma1and is known as
the Dynes formula [ 40],
dI
dV(V)=Re/bracketleftBigg
V−i/Gamma1/radicalbig
(V−i/Gamma1)2−/Delta12/bracketrightBigg
. (1)
This procedure is very useful in extracting values of /Delta1as a
function of temperature for a relatively clean superconductor.In these materials, /Delta1shrinks to zero as Tapproaches the critical
temperature T
c, in agreement with the predictions of the BCS
theory [ 41]. In contrast, in disordered thin films, tunneling
experiments have revealed that /Delta1smoothly evolves across the
transition [ 42,43] and through Tc[44].
In addition to this nonconventional behavior of /Delta1,s u -
perconducting thin films show a significant deviation of theexperimentally measured density of states (DOS) from Eq. ( 1)
due to a considerable suppression of the coherence peaks at thegap edges [ 43,44]. A similar disagreement was observed in
high-temperature superconductors [ 45,46]. This discrepancy
can be accounted for by assuming the two /Gamma1’s that appear in
the numerator and in the denominator of Eq. ( 1) are different[47–49], but this approach lacks physical insight (see the
Supplemental Material [ 50]).
In this Rapid Communication we suggest an alternative
approach which relates the experimental findings to a well-defined theoretical model. Instead of considering an effectiveenergy broadening, we include superconducting fluctuationsby postulating a bosonic field /Delta1(r,t) with a finite correlation
length. By summing the contributions of short-range (SR) and
long-range (LR) fluctuations, we obtain an excellent agreementwith experimental curves on the insulating side of the SIT. Ourresults demonstrate that there are two important length scales:One is the effective size of a superconducting island ξ
sc, and
the other is the typical size of quantum fluctuations ξfluc, which
diverges at the SIT.
Superconducting fluctuations. We begin our analysis by
introducing the Hamiltonian H=H0+H/Delta1, where H0=/summationtext
k,σεkc†
k,σck,σdescribes free electrons (quasiparticles) with
a Fermi surface at εk=0. Superconducting fluctuations are
represented by a randomly fluctuating bosonic field /Delta1(r,t)
coupled to the fermions by
H/Delta1=/Delta1(r,t)c†
↑(r,t)c†
↓(r,t)+H.c. (2)
The effects of small fluctuations of /Delta1on the superconducting
side of the transition were analyzed self-consistently in Refs.[51–53]. In this Rapid Communication, we instead focus on
the insulating side of the transition where the superconductingorder parameter averages to zero /angbracketleft/Delta1(r,t)/angbracketright=0. We model
finite-range superconducting fluctuations by a free field withtwo-point correlations,
C(r−r
/prime,t−t/prime)=/angbracketleft/Delta1(r,t)/Delta1∗(r/prime,t/prime)/angbracketright. (3)
The function Cdescribes the decay of the superconducting
correlations and tends to zero at long distances and longtimes. Our phenomenological model can be justified by thenumerical solution of the attractive Hubbard model with on-sitedisorder [ 32,54,55]. These earlier studies showed that, on the
insulating side of the transition, the interplay between disorder
2469-9950/2018/97(10)/100503(6) 100503-1 ©2018 American Physical SocietyDENTELSKI, FRYDMAN, SHIMSHONI, AND DALLA TORRE PHYSICAL REVIEW B 97, 100503(R) (2018)
FIG. 1. One-loop diagram—second-order perturbation in /Delta1(r,t).
The black thick arrows represent charge (i.e., the right arrow for a
particle and the left arrow for a hole), whereas the thin red ones
represent momentum and energy.
and interactions gives rise to a superconducting gap with
short-range correlations.
We now derive the relation between C(r−r/prime,t−t/prime) and
tunneling measurements. The tunneling differential conduc-tivity is proportional to [ 56]
dI
dV(V)∝/integraldisplay∞
−∞dωρ (V+ω)f/prime(ω). (4)
HereVis the voltage bias, f/prime(ω)=df/dω is the derivative
of the Fermi-Dirac distribution function, and ρ(ω)i st h e
DOS of the sample. Equation ( 4) assumes a constant DOS
of the tip and at T=0 simply reduces to dI/dV ∝ρ(V).
Within the Green’s function formalism in Nambu space [ 57],
ρ(ω)=− (1/π)/angbracketleftIm{Tr[/summationtext
kGret(k,ω)]}/angbracketright, where Gret(k,ω)i s
the retarded Green’s function, Tr is the trace in Nambu space,Im is the imaginary part, and /angbracketleft ···/angbracketright implies an average over the
fluctuations of the superconducting field /Delta1(r,t).
Our first step involves a Dyson resummation of the one-loop
contributions shown in Fig. 1, whose two vertices represent the
coupling term ( 2). By performing a trace over particle and hole
contributions, we find (see the Supplemental Material [ 58])
Tr[G
ret(k,ω)]=ω+i0+
(ω+i0+)2−ε2
k−D2(k,ω). (5)
Here we defined the pairing-fluctuations’ function Das
D2(k,ω)=/integraldisplay
d2q/integraldisplay
d/Omega1ω+i0++εk
i/Omega1+ω+i0++εk−qC(q,/Omega1),
(6)
withC(q,/Omega1)=/integraltext
d2rdt C (r,t)eiq·r−i/Omega1t.
Because the Green’s function in Eq. ( 5) is strongly peaked
atk=kF, we can approximate the density of states as
ρ(ω)=Re/bracketleftBigg
ω/radicalbig
ω2−D2(ω)/bracketrightBigg
, (7)
where we defined D(ω)≡D(kF,ω).
Equation ( 7) is analogous to ( 1), but involves the frequency-
dependent D(ω) instead of the quasiparticles’ lifetime /Gamma1.N o t e
that, if the correlation function C(r,t) does not decay in
space and time (i.e., the BCS limit), its Fourier transform isC(q,/Omega1)=/Delta1
2
0δ(q)δ(/Omega1). In this case, Eq. ( 6) yields a frequency-
independent D2(ω)=/Delta12
0, and one recovers the well-known
result.Our approach has some similarities to Refs. [ 36–38] where
superconducting fluctuations with a finite lifetime were con-sidered as well. These authors were interested in the thermalregime T> T
cwhere superconducting fluctuations are weak
and lead to small deviations of the density of states. As aconsequence, their approximation scheme does not recoverthe diverging density of states predicted by BCS. The Dysonresummation employed in the present Rapid Communicationallows us to consider strong superconducting fluctuations andget closer to the SIT. See also Ref. [ 59] for a microscopic model
describing the effect of superconducting fluctuations close tothe SIT and their effects on the DOS.
In what follows, for simplicity, we will generically assume
that the correlations are time independent, i.e.,
C(q,/Omega1)=C(q)δ(/Omega1). (8)
This assumption is justified if the collective-mode velocity vis
much smaller than the Fermi velocity v
F(see the Supplemental
Material [ 60]). Assuming a quadratic dispersion relation εk=
(k2−k2
F)vF/2kFand assuming that C(q) decays to zero at
q∼kF, we obtain
D2(ω)=/integraldisplay
d2qω
ω−vFqxC(q). (9)
Equations ( 7) and ( 9) are the key theoretical results of our
analysis, and we will now use them to model the density ofstates of disordered superconductors under various assump-tions in the form of C(q). In the following, the films are
assumed to be thin enough such that both kand qcan be
treated as two dimensional.
Short range vs long range. In order to understand the
effects of superconducting fluctuations on the density of states,we consider correlation functions decaying over a typicalinverse length scale q
0. This quantity can be associated with
the average size of superconducting islands in the granularmaterials and with an emergent electronic granularity ofamorphous materials [ 31–35]. In the specific case of C(q)=
/Delta12
0
π3/2v2
Fq2
0exp(−q2/q2
0), we can analytically solve the integral in
Eq. ( 9) to find
D2(ω)=/Delta12
01
vFq0exp/parenleftbigg
−ω2
v2
Fq2
0/parenrightbigg
ω/bracketleftbigg
erfi/parenleftbiggω
vFq0/parenrightbigg
−i/bracketrightbigg
.
(10)
Here erfi is the imaginary error function, which is a real
function. The real and imaginary parts of Eq. ( 10) are shown
in the upper panels of Figs. 2(a)–2(c) and the corresponding
DOS in the lower panels. Note that the real part of D2(ω)
closely resembles the local density of states ρ(ω), but these
two quantities have a different physical meaning: The formeractually needs to be substituted in Eq. ( 7) to deliver the latter.
We observe that both Re[ D
2] and Im[ D2] are peaked at a typical
energy scale vFq0. The effect of superconducting fluctuations
on the DOS changes dramatically, depending on the ratiobetween v
Fq0and/Delta10.
Let us consider two extreme cases, which we denote by
long range (LR) and short range (SR), respectively. Theformer occurs for v
Fq0∼vFξ−1
fluc/lessmuch/Delta10. In this case, D2(ω)
is approximately constant, and we recover the BCS limit [seeFig. 2(a)]. On the other hand, when v
Fq0∼vFξ−1
sc/greatermuch/Delta10,t h e
100503-2TUNNELING PROBE OF FLUCTUATING … PHYSICAL REVIEW B 97, 100503(R) (2018)
FIG. 2. Upper panel: (a)–(c) Real and imaginary parts of Eq. ( 10) for different values of vFq0//Delta1 0. (d) Real and imaginary parts of Eq. ( 12).
Lower panel: the corresponding density of states, Eq. ( 7).
fluctuations are short ranged. In this regime,
D2(ω)≈−iωγ (11)
is purely imaginary, and the DOS shows a deep without
coherence peaks [see Fig. 2(c)]. The distinction between
LR and SR fluctuations does not depend on the specificchoice of C(q) and can be related to the Anderson limit
of superconductivity [ 55,61]. The crossover between these
two regimes occurs when /Delta1
0is on the order of the typical
energy level spacing of a superconducting island of size 1 /q0
(the superconducting correlation length), i.e., /Delta10∼vFq0.A s
q0increases, the coherence peaks become less pronounced,
and their positions move to higher energies [ 62,63] (see the
Supplemental Material [ 64]).
In the vicinity of the SIT, superconducting fluctuations
are described by a universal critical theory [ 65], which in its
simplest form is given by C(q)=/Delta12
0
π31
q2+q2
0, where q0=
1/ξfluctends to zero at the transition. In this case, which we
denote as QLR, we can again solve analytically Eq. ( 9)t o
10-1100101
r0/vF00.511.5C(r)/2
0
SC flucLR
SR
SR+LR
-2 -1 0 1 2
/000.511.52()(b) (a)
FIG. 3. (a) Normalized spatial correlations of the superconduct-
ing fluctuations C(r). (b) Density of states ρ(ω) for different types
of superconducting correlations. Note that adding SR fluctuations
(vFq0//Delta1 0=3) suppresses the peaks even in the presence of LR
superconducting correlations ( vFq0//Delta1 0=0.1).obtain
D2(ω)=/Delta12
0
π2ω/radicalBig
v2
Fq2
0+ω2⎡
⎣ln⎛
⎝/radicalBig
v2
Fq2
0+ω2+ω
/radicalBig
v2
Fq2
0+ω2−ω⎞
⎠−iπ⎤
⎦.
(12)
As shown in Fig. 2(d),f o rq0→0, the real part of Eq. ( 12)
diverges logarithmically, whereas the imaginary part is propor-tional to sgn( ω). The resulting DOS resembles the LR situation
and is very weakly dependent on the infrared cutoff q
0.A sw e
will see below, QLR superconducting fluctuations give a betterdescription of the experiment than true LR correlations.
In actual materials, one generically expects to find a
combination of superconducting fluctuations with long-rangeand short-range correlations. The former are universal anddetermine the emergent properties of the material, whereasthe latter depend on the microscopical details and are oftenneglected. In contrast to this common practice, we find thatshort-range correlations strongly affect the density of states(see Fig. 3): Although the correlation functions denoted by LR
and SR +LR have the same asymptotic behavior [subplot (a)],
the corresponding DOS are very different [subplot (b)].
TABLE I. Comparison between different correlation functions,
i.e., Dynes, SR +LR, and SR +QLR.
DOS Best fit (meV) χ2
Dynes Equation ( 1) /Delta10=0.73 0.046
/Gamma1=0.16
SR+LR Equation ( 7) with /Delta10=0.67
D2(ω)=Eqs.(11)+(10) γ=0.22 0.022
vFq0=0.22
SR+QLR Equation ( 7) with /Delta10=0.69
D2(ω)=Eqs.(11)+(12) γ=0.11 0.011
vFq0=0.022
100503-3DENTELSKI, FRYDMAN, SHIMSHONI, AND DALLA TORRE PHYSICAL REVIEW B 97, 100503(R) (2018)
-5 -4 -3 -2 -1 0 1 2 3 4 5
[meV]00.20.40.60.811.21.4dI/dV
Experiment
SR+LR
SR+QLR
Dynes10-2100102
r0/vF00.511.5C(r) [meV2]
FIG. 4. Comparison between theoretical predictions and actual
measurements performed on an insulating thin film close to the SIT.The best theoretical fits for each curve are presented in Table I.T h e
best fit to the experiment is given by SR +QLR. The inset shows the
corresponding superconducting correlation functions ( ξ
SC=1
3q0).
Comparison with experiments. We now compare our
theoretical calculations with the tunneling measurement ofRef. [ 43], performed on an InO film at T=1K o n t h e
insulating side of the SIT. Note that, in order to isolate thesuperconducting contribution to the DOS, the experimentalraw data were normalized by the tunneling spectra at a highmagnetic field. The results of our analysis are summarized inTable Iand Fig. 4where we show the best-fitting parameters
and the minimal normalized χ
2distribution between theory
and experiment. We find that the sum of short-range and long-range superconducting fluctuations is required to obtain a goodfit. Furthermore, a detailed analysis reveals that the long-rangepart is best described by Eq. ( 12)(χ
2=0.011) rather than
Eq. ( 10)(χ2=0.022), in agreement with the expected critical
behavior of the SIT [ 65].
Discussion. In this Rapid Communication we studied the
effects of superconducting fluctuations on the tunneling con-ductivity of disordered thin films, focusing on the insulatingside of the SIT. The common approach, known as the Dynesformula ( 1), relies on a phenomenological parameter /Gamma1that
describes the inverse lifetime of the quasiparticles. In thisRapid Communication, we showed that the experiments arebetter fit by a theory of free electrons, coupled to supercon-ducting fluctuations with finite-range correlations. By using
a controlled diagrammatic approach, we derived a simple ex-pression that connects the correlations of the superconductingfluctuations to the tunneling spectra Eqs. ( 6) and ( 7). This
result has potential applications that go beyond the presentRapid Communication, including quantum as well as classicalsuperconducting phase transitions. Our analytical results showthat, generically, short-range fluctuations lead to tunnelingspectra with reduced or absent coherence peaks even in thepresence of long-range superconducting correlations.
By comparing our analytic expressions to experimental
measurements, we find that, in disordered thin films, thesuperconducting fluctuations are given by the sum of twocomponents. The long-range component is associated withuniversal fluctuations close to the SIT quantum critical point,characterized by a diverging length scale ξ
fluc. Accordingly,
the experimental data are best fit by a critical theory with q0∼
1/ξfluc/lessmuchkF(see the last row of Table I, taking into account
thatvFkF∼1 eV). In contrast, the short-range component
is determined by the microscopic details of the material.Specifically, short-range correlations are expected to play apredominant role in amorphous materials where Cooper pairsare localized by disorder. In granular materials, on the otherhand, the superconducting correlations decay over a muchlonger range, set by the typical scale of the grains. This distinc-tion can explain why the Dynes formula fits well experimentson granular Pb films [ 9] but does not fit amorphous InO
films [ 43]. The distinction between short-range and long-range
fluctuations can bridge the long-standing controversy betweenthe fermionic and the bosonic approach to the SIT [ 55]. On
a broader prospective, our approach contributes to the under-standing of puzzling spectrometric experiments in unconven-tional superconductors. Specifically, we find that, although SRfluctuations contribute to the local superconducting gap, theygenerically lead to a tunneling spectra with suppressed coher-ence peaks in analogy to the experimental observations in thepseudogap regime of underdoped cuprates (see, for example,Refs. [ 47,48,66,67]).
Acknowledgments . We thank A. Auerbach, I. Burmistrov,
T. Baturina, O. Eizenberg, and E. Demler for useful discus-sions. This Rapid Communication was supported by the IsraelScience Foundation, Grants No. 1452/14 (E.G.D.T. and D.D.)and No. 231/14 (E.S.). A.F. acknowledges support from theGIF Foundation Grant No. I-1250-303.10/2014. D.D. thanksthe Klein family for a fellowship in memory of Prof. MichaelKlein.
[1] Y . H. Lin, J. Nelson, and A. Goldman, Superconductivity of very
thin films: The superconductor–insulator transition, Physica C
514,130(2015 ).
[2] A. M. Goldman and N. Markovic, Superconductor-insulator
transitions in the two-dimensional limit, Phys. Today 51(11),
39(1998 ).
[3] N. Markovi ć, C. Christiansen, and A. M. Goldman, Thickness-
Magnetic Field Phase Diagram at the Superconductor-InsulatorTransition in 2D, P h y s .R e v .L e t t . 81,5217 (1998 ).[4] S. Sachdev, Quantum Phase Transitions (Cambridge University
Press, Cambridge, UK, 2001).
[5] M. Strongin, R. Thompson, O. Kammerer, and J. Crow, Destruc-
tion of superconductivity in disordered near-monolayer films,Phys. Rev. B 1,1078 (1970 ).
[6] R. C. Dynes, A. E. White, J. M. Graybeal, and J. P. Garno,
Breakdown of Eliashberg Theory for Two-Dimensional Super-
conductivity in the Presence of Disorder, Phys. Rev. Lett. 57,
2195 (1986 ).
100503-4TUNNELING PROBE OF FLUCTUATING … PHYSICAL REVIEW B 97, 100503(R) (2018)
[7] D. B. Haviland, Y . Liu, and A. M. Goldman, Onset of Super-
conductivity in the Two-Dimensional Limit, Phys. Rev. Lett. 62,
2180 (1989 ).
[8] J. M. Valles, Jr., R. C. Dynes, and J. P. Garno, Electron
Tunneling Determination of the Order-Parameter Amplitude atthe Superconductor-Insulator Transition in 2D, Phys. Rev. Lett.
69,3567 (1992 ).
[9] L. Merchant, J. Ostrick, R. P. Barber, Jr., and R. C. Dynes,
Crossover from phase fluctuation to amplitude-dominated su-perconductivity: A model system, Phys. Rev. B 63,134508
(2001 ).
[10] A. Frydman, O. Naaman, and R. C. Dynes, Universal transport
in two-dimensional granular superconductors, Phys. Rev. B 66,
052509 (2002 ).
[11] N. Hadacek, M. Sanquer, and J.-C. Villégier, Double reentrant
superconductor-insulator transition in thin TiN films, Phys. Rev.
B69,024505 (2004 ).
[12] M. D. Stewart, A. Yin, J. Xu, and J. M. Valles, Superconducting
pair correlations in an amorphous insulating nanohoneycombfilm, Science 318,1273 (2007 ).
[13] B. Sacépé, C. Chapelier, T. I. Baturina, V . M. Vinokur, M. R.
Baklanov, and M. Sanquer, Disorder-Induced Inhomogeneitiesof the Superconducting State Close to the Superconductor-Insulator Transition, Phys. Rev. Lett. 101,157006 (2008 ).
[14] S. M. Hollen, H. Q Nguyen, E. Rudisaile, M. D. Stewart, Jr.,
J. Shainline, J. M Xu, and J. M. Valles, Jr., Cooper-pair insu-lator phase in superconducting amorphous bi films induced by
nanometer-scale thickness variations, P h y s .R e v .B 84,064528
(2011 ).
[15] T. I. Baturina, V . M. Vinokur, A. Y . Mironov, N. M.
Chtchelkatchev, D. A. Nasimov, and A. V . Latyshev,Nanopattern-stimulated superconductor-insulator transition inthin TiN films, Europhys. Lett. 93,47002 (2011 ).
[16] S. Poran, T. Nguyen-Duc, A. Auerbach, N. Dupuis, A. Frydman,
and O. Bourgeois, Quantum criticality at the superconductor-insulator transition revealed by specific heat measurements,Nat. Commun. 8,14464 (2017 ).
[17] M. A. Paalanen, A. F. Hebard, and R. R. Ruel, Low-Temperature
Insulating Phases of Uniformly Disordered Two-DimensionalSuperconductors, Phys. Rev. Lett. 69,1604 (1992 ).
[18] A. Yazdani and A. Kapitulnik, Superconducting-Insulating
Transition in Two-Dimensional a-Moge Thin Films, Phys. Rev.
Lett. 74,3037 (1995 ).
[19] V . Gantmakher, M. Golubkov, V . Dolgopolov, A. Shashkin, and
G. Tsydynzhapov, Observation of the parallel-magnetic-field-induced superconductor-insulator transition in thin amorphousino films, JETP Lett. 71,473(2000 ).
[20] G. Sambandamurthy, L. W. Engel, A. Johansson, and D. Shahar,
Superconductivity-Related Insulating Behavior, Phys. Rev. Lett.
92,107005 (2004 ).
[21] G. Sambandamurthy, L. W. Engel, A. Johansson, E. Peled, and
D. Shahar, Experimental Evidence for a Collective InsulatingState in Two-Dimensional Superconductors, Phys. Rev. Lett. 94,
017003 (2005 ).
[22] M. A. Steiner, G. Boebinger, and A. Kapitulnik, Possible
Field-Tuned Superconductor-Insulator Transition in High- T
c
Superconductors: Implications for Pairing at High Magnetic
Fields, Phys. Rev. Lett. 94,107008 (2005 ).
[23] T. I. Baturina, J. Bentner, C. Strunk, M. R. Baklanov, and
A. Satta, From quantum corrections to magnetic-field-tunedsuperconductor–insulator quantum phase transition in TiN films,
Physica B 359,500(2005 ).
[24] T. I. Baturina, C. Strunk, M. R. Baklanov, and A. Satta, Quantum
Metallicity on the High-Field Side of the Superconductor-Insulator Transition, P h y s .R e v .L e t t . 98,127003 (2007 ).
[25] R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy,
D. Shahar, and G. Grüner, Fluctuations, dissipation, and nonuni-versal superfluid jumps in two-dimensional superconductors,Phys. Rev. B 75,094506 (2007 ).
[26] V . M. Vinokur, T. I. Baturina, M. V . Fistul, A. Y . Mironov,
M. R. Baklanov, and C. Strunk, Superinsulator and quantumsynchronization, Nature (London) 452,613(2008 ).
[27] K. A. Parendo, K. H. S. B. Tan, A. Bhattacharya, M. Eblen-
Zayas, N. E. Staley, and A. M. Goldman, Electrostatic Tuningof the Superconductor-Insulator Transition in Two Dimensions,Phys. Rev. Lett. 94,197004 (2005 ).
[28] M. Mondal, A. Kamlapure, M. Chand, G. Saraswat, S. Kumar, J.
Jesudasan, L. Benfatto, V . Tripathi, and P. Raychaudhuri, PhaseFluctuations in a Strongly Disordered s-Wave nbn Superconduc-tor Close to the Metal-Insulator Transition, Phys. Rev. Lett. 106,
047001 (2011 ).
[29] D. Sherman et al. , The higgs mode in disordered superconduc-
tors close to a quantum phase transition, Nat. Phys. 11,188
(2015 ).
[30] S. Poran, E. Shimshoni, and A. Frydman, Disorder-induced
superconducting ratchet effect in nanowires, Phys. Rev. B 84,
014529 (2011 ).
[31] E. Shimshoni, A. Auerbach, and A. Kapitulnik, Transport
through Quantum Melts, Phys. Rev. Lett. 80,3352 (1998 ).
[32] K. Bouadim, Y . L. Loh, M. Randeria, and N. Trivedi,
Single-and two-particle energy gaps across the disorder-drivensuperconductor-insulator transition, Nat. Phys. 7,884(2011 ).
[33] Y . Dubi, Y . Meir, and Y . Avishai, Nature of the superconductor–
insulator transition in disordered superconductors, Nature (Lon-
don) 449,876(2007 ).
[34] A. Erez and Y . Meir, Thermal phase transition in two-
dimensional disordered superconductors, Europhys. Lett. 91,
47003 (2010 ).
[35] A. Erez and Y . Meir, Proposed Measurement of Spatial Cor-
relations at the Berezinski-Kosterlitz-Thouless Transition ofSuperconducting Thin Films, Phys. Rev. Lett. 111,187002
(2013 ).
[36] R. W. Cohen, B. Abeles, and C. Fuselier, Effect of Fluctuations in
the Superconducting Order Parameter on the Tunneling Densityof States, Phys. Rev. Lett. 23,377(1969 ).
[37] E. Abrahams, M. Redi, and J. W. Woo, Effect of fluctuations
on electronic properties above the superconducting transition,Phys. Rev. B 1,208(1970 ).
[38] C. Di Castro, R. Raimondi, C. Castellani, and A. A. Varlamov,
Superconductive fluctuations in the density of states and tun-neling resistance in high- T
csuperconductors, Phys. Rev. B 42,
10211 (1990 ).
[39] A. Varlamov, G. Balestrino, E. Milani, and D. Livanov, The role
of density of states fluctuations in the normal state properties ofhighT
csuperconductors, Adv. Phys. 48,655(1999 ).
[40] R. C. Dynes, V . Narayanamurti, and J. P. Garno, Direct Measure-
ment of Quasiparticle-Lifetime Broadening in a Strong-CoupledSuperconductor, P h y s .R e v .L e t t . 41,1509 (1978 ).
[41] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of
superconductivity, Phys. Rev. 108,1175 (1957 ).
100503-5DENTELSKI, FRYDMAN, SHIMSHONI, AND DALLA TORRE PHYSICAL REVIEW B 97, 100503(R) (2018)
[42] R. P. Barber, Jr., L. M. Merchant, A. La Porta, and R. C. Dynes,
Tunneling into granular pb films in the superconducting andinsulating regimes, Phys. Rev. B 49,3409 (1994 ).
[43] D. Sherman, G. Kopnov, D. Shahar, and A. Frydman, Measure-
ment of a Superconducting Energy Gap in a HomogeneouslyAmorphous Insulator, P h y s .R e v .L e t t . 108,177006 (2012 ).
[44] B. Sacépé, T. Dubouchet, C. Chapelier, M. Sanquer, M. Ovadia,
D. Shahar, M. Feigel’man, and L. Ioffe, Localization of pre-formed cooper pairs in disordered superconductors, Nat. Phys.
7,239(2011 ).
[45] J. Zasadzinski, N. Tralshawala, P. Romano, Q. Huang, Jun. Chen,
and K. E. Gray, Tunneling density of states in cuprate andbismuthate superconductors, J. Phys. Chem. Solids 53,1635
(1992 ).
[46] R. Dynes, The order parameter of high T
csuperconductors;
experimental probes, Solid State Commun. 92,53(1994 ).
[47] M. R. Norman, M. Randeria, H. Ding, and J. C. Campuzano,
Phenomenology of the low-energy spectral function in high- Tc
superconductors, Phys. Rev. B 57,R11093(R) (1998 ).
[48] M. R. Norman, A. Kaminski, J. Mesot, and J. Campuzano,
Temperature evolution of the spectral peak in high-temperaturesuperconductors, Phys. Rev. B 63,140508 (2001 ).
[49] M. R. Norman, A. Kanigel, M. Randeria, U. Chatterjee, and J.
C. Campuzano, Modeling the fermi arc in underdoped cuprates,Phys. Rev. B 76,174501 (2007 ).
[50] See Supplemental Material (SM-1) at http://link.aps.org/
supplemental/10.1103/PhysRevB.97.100503 for a comparison
between this method (“two lifetimes”), which was used to fit
spectral functions in cuprates, and our approach.
[51] A. I. Larkin and Yu. N. Ovchinnikov, Density of states in
inhomogeneous superconductors, Zh. Eksp. Teor. Fiz. 61, 2147
(1972) [ Sov. Phys. JETP 34, 1144 (1972) ].
[52] J. S. Meyer and B. D. Simons, Gap fluctuations in inhomoge-
neous superconductors, Phys. Rev. B 64,134516 (2001 ).
[53] M. V . Feigel’man and M. A. Skvortsov, Universal Broadening
of the Bardeen-Cooper-Schrieffer Coherence Peak of Disor-dered Superconducting Films, Phys. Rev. Lett. 109,147002
(2012 ).
[54] A. Ghosal, M. Randeria, and N. Trivedi, Inhomogeneous pairing
in highly disordered s-wave superconductors. P h y s .R e v .B 65,
014501 (2001 ).[55] G. Seibold, L. Benfatto, C. Castellani, and J. Lorenzana, Am-
plitude, density, and current correlations of strongly disorderedsuperconductors. P h y s .R e v .B 92,064512 (2015 ).
[56] G. D. Mahan, Many-Particle Physics , 3rd ed., Physics of Solids
and Liquids (Kluwer Academic-Plenum, New York, 2000).
[57] A. Altland and B. Simons, Condensed Matter Field Theory
(Cambridge University Press, Cambridge, UK, 2006).
[58] See Supplemental Material (SM-4) at http://link.aps.org/
supplemental/10.1103/PhysRevB.97.100503 for the full deriva-
tion of Eqs. ( 5)a n d( 6).
[59] I. S. Burmistrov, I. V . Gornyi, and A. D. Mirlin, Local density
of states and its mesoscopic fluctuations near the transition to asuperconducting state in disordered systems, Phys. Rev. B 93,
205432 (2016 ).
[60] See Supplemental Material (SM-2) at http://link.aps.org/
supplemental/10.1103/PhysRevB.97.100503 for more details
about the validity of this approximation.
[61] P. W. Anderson, Theory of dirty superconductors, J. Phys. Chem.
Solids 11,26(1959 ).
[62] B. Cheng, L. Wu, N. J. Laurita, H. Singh, M. Chand, P. Ray-
chaudhuri, and N. P. Armitage, Anomalous gap-edge dissipationin disordered superconductors on the brink of localization,Phys. Rev. B 93,180511 (2016 ).
[63] G. Seibold, L. Benfatto, and C. Castellani, Application of the
Mattis-Bardeen theory in strongly disordered superconductors,Phys. Rev. B 96,144507 (2017 ).
[64] See Supplemental Material (SM-3) at http://link.aps.org/
supplemental/10.1103/PhysRevB.97.100503 for a detailed anal-
ysis of the effect of finite-range correlations on the coherencepeaks.
[65] S. Sondhi, S. Girvin, J. Carini, and D. Shahar, Continuous
quantum phase transitions, Rev. Mod. Phys. 69,315(1997 ).
[66] M. C. Boyer, W. D. Wise, K. Chatterjee, M. Yi, T. Kondo, T.
Takeuchi, H. Ikuta, and E. W. Hudson, Imaging the two gaps ofthe high-temperature superconductor Bi
2Sr2CuO 6+x,Nat. Phys.
3,802(2007 ).
[67] K. Fujita, A. R. Schmidt, E. Kim, M. J. Lawler, D. H. Lee,
J. C. Davis, E. Hiroshi, and S. Uchida, Spectroscopic imagingscanning tunneling microscopy studies of electronic structure inthe superconducting and pseudogap phases of cuprate high- T
c
superconductors, J. Phys. Soc. Jpn. 81,011005 (2011 ).
100503-6 |
PhysRevB.84.035318.pdf | PHYSICAL REVIEW B 84, 035318 (2011)
Spin dynamics in the strong spin-orbit coupling regime
Xin Liu,1Xiong-Jun Liu,1and Jairo Sinova1,2
1Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA
2Institute of Physics ASCR, Cukrovarnick ´a 10, 162 53 Praha 6, Czech Republic
(Received 15 February 2011; published 26 July 2011)
We study the spin dynamics in a high-mobility two-dimensional electron gas with generic spin-orbit interactions
(SOI’s). We derive a set of spin-dynamics equations that capture purely exponentially the damped oscillatoryspin evolution modes observed in different regimes of SOI strength. Hence we provide a full treatment of theD’yakonov-Perel mechanism by using the microscopic linear-response theory from the weak to the strong SOIlimit. We show that the damped oscillatory modes appear when the electron scattering time is larger than half ofthe spin precession time due to the SOI, in agreement with recent observations. We propose a way to measurethe scattering time and the relative strength of Rashba and linear Dresselhaus SOI’s based on these modes andoptical grating experiments. We discuss the physical interpretation of each of these modes in the context of Rabioscillation.
DOI: 10.1103/PhysRevB.84.035318 PACS number(s): 73 .21.Fg, 72 .25.Dc, 72 .25.Rb, 72 .10.−d
I. INTRODUCTION
In recent years, research in semiconductor-based devices
has incorporated the spin degree of freedom as a new statevariable in novel electronic devices with potential for futureapplications. The spin-orbit interaction (SOI) is a key toolto electrically manipulate the spin and realize such devices.However, the SOI is a double-edged sword because it will alsoinduce random spin precession through an angle /Omega1
soτbetween
collisions with impurities, where τis the electron lifetime.
This is known as the D’yakonov-Perel (DP) mechanism1–3
and dominates the spin relaxation in the technologically
important III-V semiconductors.4Therefore, it is very impor-
tant to understand fully the DP mechanism for the possibleapplication and further development of spintronics devices.Although study of the DP mechanism in semiconductors in thepresence of SOI was initiated long ago, most of the theoreticalresearch
5–10was focused on the weak spin-orbit coupling
(SOC) regime in which /Omega1soτ/lessmuch1. However, as high-mobility
two-dimensional electron gas (2DEG) systems are created, it isnow not difficult to reach the strong SOI regime experimentallywhere /Omega1
soτ> 1 at low temperatures as long as the mobility
is approximately larger than 1 .2×105cm2/Vs.11The spin
evolution in this regime is observed to be damped oscillationsin the uniform
11,12and nonuniform spin-polarized system,13–15
which cannot be described by spin-charge drift-diffusion
equations derived for the weak SOC regime and lacks a cleartheoretical explanation.
Here, we study the spin dynamics theoretically from the
weak to strong SOC regime. The method we use is linear-response theory.
5,7,16We derive a set of spin-dynamics equa-
tions in the uniform spin-polarized 2DEG with different SOI’sin the presence of the short-ranged impurity scattering. Forthe experiments we consider, even in the strong SOC regime,it is dominated by neutral impurity or interface roughnessscattering, which are short-ranged impurity scattering.
12The
weak localization effect on the spin relaxation17is neglected
in our work because we consider the spin relaxation in themetallic regime, where the weak localization effect is small.
We show analytically that for /Omega1
soτ>1
2, the damped oscil-
lations appear. The decay rate in this case is proportional to1
τinstead of τas in the weak SOC regime. The cubic Dresselhaus
term is shown to reduce the oscillatory frequency and increasethe decay rate in the strong SOC regime. The spin dynamicsfor nonuniform spin polarization with spatial frequency qin
the strong SOC regime is obtained by solving the equationsnumerically. We discuss these dynamics by using the analogywith Rabi oscillations between two momentum states that aregapped by the SOI. Our results agree quantitatively with theexperimental observations. We also show how to exploit ouranalysis to create an accurate measurement of the strengthof Rashba and linear Dresselhaus SOI’s in a 2DEG, henceallowing a full characterization of different device samplesthat will lead to a more accurate modeling and predictabilityof the optimal operating physical regimes.
II. MODEL HAMILTONIAN AND DENSITY-MATRIX
RESPONSE FUNCTION
Normally in the 2D semiconductor heterostructures, we
have three kinds of SOI’s, namely the linear Rashba18,19term
and the linear and cubic Dresselhaus20terms. The Hamiltonian
takes the form
H=k2
2m+h(k)·ˆσ, (1)
where h(k) is the effective magnetic and contains Rashba,
linear, and cubic Dresselhaus terms, which are
hR(k)=α(−ky,kx), (2)
hD1(k)=β1(ky,kx), (3)
hD3(k)=−2β3cos 2θ(−ky,kx), (4)
where kfis the Fermi wave vector. Here we take θas the
angle between the wave vector kand the [110] direction,
which is the xaxis in our coordinates. The above SOI’s split
the spin-degenerate bands and dominate the spin dynamics inthe 2DEG. The corresponding SOC Hamiltonian and the spinprecession frequency /Omega1
sotake the form:
Hso=(λ1−2β3cos 2θ)kxσy+(λ2+2β3cos 2θ)kyσx,(5)
where λ1=α+β1andλ2=β1−α.
035318-1 1098-0121/2011/84(3)/035318(8) ©2011 American Physical SocietyXIN LIU, XIONG-JUN LIU, AND JAIRO SINOV A PHYSICAL REVIEW B 84, 035318 (2011)
We derive the spin-dynamics equations from the density-
matrix response function.16The spin diffusion is dominated by
the pole of the spin-charge diffusion propagator or “diffuson:”5
D=[1−ˆI]−1(6)
and
ˆIσ1σ2,σ3σ4=1
2mτ/integraldisplayd2k
(2π)2GA
σ3σ1(k,0)GR
σ2σ4(k+q,/Omega1), (7)
where σiis just a number that can be 1 or 2.5It is more
convenient to write Eq. ( 7) in a classical charge-spin space,
Iαβ=Tr(σαˆIσβ), (8)
where α,β=c,x,y,z.5
If one calculates the response function by expanding in
term of /Omega1soτto the first order, the spin-relaxation behavior
obtained by this approximate response function is only validin the weak SOC regime, such as in Ref. 7. However, if one
calculates the response function exactly without any expansionin the parameter /Omega1
soτ, this response function can give the
spin relaxation in both the weak and strong SOC regime. Inthe appendix of Ref. 5,B u r k o v et al. give the expression
of the spin-charge diffuson in the presence of the Rashbaspin-orbit interaction. The authors in Ref. 5are interested in
finding a spin-charge drift diffusion equation, only applicablein the weak SOC regime, and therefore they expanded theexpressions in terms of /Omega1
soτto first order. However, they claimthat the expression should be useful in the strong SOC regime.
Here, we will calculate the diffuson matrix exactly with thegenetic SOI’s and find the poles of this exact expression.
III. UNIFORM SPIN POLARIZATION
In the case of a uniform spin-polarized 2DEG system, i.e.,
q=0, because the effective magnetic field due to the SOI has
inversion symmetry in momentum space, only the diagonalelements of the diffuson matrix are nonzero, which means thespinx,y,zand charge are not coupled to each other. Therefore,
when considering the uniform spin polarization along the z
direction, only I
zzneeds to be calculated. First, we neglect the
cubic Dresselhaus term, which is normally much smaller thanthe linear Dresselhaus term. We find the pole of the diffusion
matrix by solving the equation
1−I
zz=1−1−i/Omega1τ/radicalbig
[(1−i/Omega1τ)2+(/Omega1soτ)2]2−γ2(/Omega1soτ)4=0,
(9)
where /Omega1is the frequency of the spin evolution, /Omega1so=
2/radicalBig
α2+β2
1kf,γ=2αβ1
α2+β2
1=λ21−λ2
2
λ21+λ2
2,kfis the Fermi wave
vector, and Izzis obtained from the exact angular integration
of Eq. ( 7). The details of calculating Izzare shown in the
Appendix. There are four solutions of Eq. ( 9) ,w h i c ht a k et h e
form
/Omega1τ=−i/parenleftbigg
1±√
2
2/radicalBig
1−2(/Omega1soτ)2±/radicalbig
1−4(/Omega1soτ)2+4(/Omega1soτ)4γ2/parenrightbigg
. (10)
However, note that not all of these solutions give the spin evolution mode observed by the experiments.11,12To find the right one,
we explore the values of the above four solutions in the limit of the weak spin-orbit coupling regime, say /Omega1soτ=0, and write
them as
/Omega11τ=−i/parenleftbigg
1−√
2
2/radicalBig
1−2(/Omega1soτ)2+/radicalbig
1−4(/Omega1soτ)2+4(/Omega1soτ)4γ2/parenrightbigg
=0,
/Omega12τ=−i/parenleftbigg
1−√
2
2/radicalBig
1−2(/Omega1soτ)2−/radicalbig
1−4(/Omega1soτ)2+4(/Omega1soτ)4γ2/parenrightbigg
=−i,
(11)
/Omega13τ=−i/parenleftbigg
1+√
2
2/radicalBig
1−2(/Omega1soτ)2+/radicalbig
1−4(/Omega1soτ)2+4(/Omega1soτ)4γ2/parenrightbigg
=−2i,
/Omega14τ=−i/parenleftbigg
1+√
2
2/radicalBig
1−2(/Omega1soτ)2−/radicalbig
1−4(/Omega1soτ)2+4(/Omega1soτ)4γ2/parenrightbigg
=−i.
We know that for the spin relaxation dominated by the DP mechanism, /Omega1τ→0 when /Omega1soτ→0, which indicates that only the
first mode, /Omega11,i nE q .( 11) gives the right behavior of the spin relaxation, say /Omega1∝τ,5,7in the weak spin-orbit coupling regime. On
the other hand, in the strong spin-orbit coupling regime, there is only one mode observed in the uniform spin-polarized case.11,12
Therefore, we can conclude that only the first mode in Eq. ( 11) contributes to the spin relaxation. Therefore, the eigenmode of
the spin dynamic evolution takes the form
i/Omega1τ=1
2(2−√
2/radicalBig
1−2(/Omega1soτ)2+/radicalbig
1−4(/Omega1soτ)2+4(/Omega1soτ)4γ2). (12)
Note that γ/lessorequalslant1 and
[1−4(/Omega1soτ)2+4(/Omega1soτ)4γ2]/lessorequalslant/parenleftbig
1−2/Omega12
soτ2/parenrightbig
.
Therefore, as long as 1 −4(/Omega1soτ)2+4(/Omega1soτ)4γ2<0, a nonequilibrium spin polarization will exhibit damped oscillation with
respect to time (see Fig. 1).
035318-2SPIN DYNAMICS IN THE STRONG SPIN-ORBIT ... PHYSICAL REVIEW B 84, 035318 (2011)
2α kfτ2β1 kfτNormalized damped decay rate
−2 0 2 4−202
00.51
(a)
2α kfτ2β1 kfτNormalized oscillatory frequency
−2 0 2 4−202
00.51(b)
FIG. 1. (Color online) The uniform spin dynamics from the weak
to the strong spin-orbit coupling regime in the presence of both
Rashba and linear Dresselhaus terms. (a) The normalized exponential
decay rate, Im( /Omega1τ), is shown as a function of normalized Rashba
and linear Dresselhaus SOI. (b) The nonzero normalized oscillatoryfrequency,
Re(/Omega1)
/Omega1so, is nonzero whenever 2 αkfτ/greaterorequalslant1
2or 2β1kfτ/greaterorequalslant1
2.
In the case of α=0o rβ1=0, the eigenmode takes the
form
i/Omega1τ=1
2−1
2/radicalbig
1−4/Omega12soτ2. (13)
When /Omega1soτ> 1/2, the decay rate changes from the expo-
nential decay mode to the damped oscillation mode. Theoscillatory frequency in the clean limit, τ→∞ ,i s/Omega1
so.
Several experiments11–14observe the damped oscillation mode
of spin evolution at low temperature. However, their analysisdid not explain quantitatively when this kind of mode appearsbut just qualitatively argued that it appears in the regimewhere /Omega1
soτ> 1. Our theory agrees with a recent experiment12
in which the authors observe that when the temperature is
above 5 K, the oscillation will disappear. In their system,this corresponds to /Omega1
soτ∗
p≈0.48, which is close to our
result 1 /2. Here τ∗
pis different from the transport scattering
timeτpobtained from the mobility; this difference is due
to the Coulomb interaction effect on spin-currents and spindephasing.
8,13Thise-einteraction treatment is beyond the
scope of our paper and will not be discussed in this work. The τ
here corresponds to τ∗
p. When the oscillatory mode appears, the
damped decay rate is always equal to1
2τwhen either α=0o r
β1=0. This result agrees with a recent experiment11in which
the authors found that the decay rates for several different2DEG’s always equal
1
1.9τwhen the damped oscillatory mode
appears, in agreement with our theoretical result.
As the linear and cubic Dresselhaus terms always coexist,
we have to consider the effect of the cubic Dresselhaus termon Eq. ( 13). We do this in the simplest case, when the Rashba
coefficient is zero. In this case, the diffuson matrix element I
zztakes the form
Izz=1−i/Omega1τ/radicalBig
(1−i/Omega1τ)2+/Omega12soτ2/bracketleftbig
1+2/parenleftbigβ3
β1/parenrightbig2−2β3
β1/bracketrightbig
×1/radicalbig
(1−i/Omega1τ)2+/Omega12soτ2, (14)
where /Omega1so=2β1kfandδ=2β3
β1(1−β3
β1). The corresponding
spin decay rate is
i/Omega1τ=1
−/radicalBig/parenleftbig
1+/radicalbig
1−4/Omega12soτ2+2/Omega12soτ2δ+/Omega14soτ4δ2/parenrightbig2−/Omega14soτ4δ2
2.
(15)
Equations ( 13) and ( 15) show that the cubic term will
increase the exponential decay rate and decrease the oscillatoryfrequency. To show the effect of the cubic Dresselhaus term,the real (imaginary) value of the damped oscillatory frequencywhenβ
3/negationslash=0 is divided by the value when β3=0. This ratio
is plotted in Fig. 2with respect to β3/β1and 2β1τ. When
β3
β1<0.2, the effect of the cubic term is very small and can
be neglected. In this case, the damped decay rate is always
equal to1
2τas long as /Omega1so>1
2and the oscillatory frequency
/Omega1approaches /Omega1sowhen /Omega1soτ/greatermuch1. This provides a reliable
way to measure the momentum scattering time τ. Further, the
strength of the linear Dresselhaus SOI can be obtained fromEq. ( 13) once we know τand the oscillatory frequency from
the measurements. These will be discussed in a later section.
Now, let us choose α=β
1, which is a more unique case and
gives us the persistent spin helix for special qvalues.14,15,21
Ωsoτβ3/β1Normalized damped decay rate
1 1.5 2 2.5 300.10.20.30.4
11.21.4(a)
Ωsoτβ3/β1Normalized oscillatory frequency
1 1.5 2 2.5 300.50.20.30.40.5
0.40.60.8(b)
FIG. 2. (Color online) The uniform spin dynamics from the weak
to the strong spin-orbit coupling regime in the presence of linearβ
1and cubic β3SOI. (a) The normalized exponential decay rate,
Re(i/Omega1τ), is constant when β3is zero and slightly larger than1
2when
β3is nonzero. (b) The nonzero normalized oscillatory frequency,
Im(i/Omega1τ), appears when /Omega1soτ>1
2.
035318-3XIN LIU, XIONG-JUN LIU, AND JAIRO SINOV A PHYSICAL REVIEW B 84, 035318 (2011)
For the uniform spin polarization, the decay rate of the spin
satisfies
i/Omega1τ=1−/radicalbig
1−2(/Omega1soτ)2, (16)
where /Omega1so=2√
α2+β2
1kf. The damped oscillation mode
will happen when /Omega1soτ=2√
2αkfτ>√
2/2, say 2 αkfτ>
1/2, which is the same as the pure Rashba or Dresselhaus case.
The oscillating frequency in the clean limit is√
2/Omega1so=4αkf,
which is the twofold frequency for the pure Rashba orDresselhaus case. On the other hand, as the real part of i/Omega1τ
is equal to 1 when the damped oscillation mode appears, thedamped decay rate is also the twofold case of the pure Rashbaor Dresselhaus case.
IV . SPIN DYNAMICS AND RABI OSCILLATION
Before we discuss the spin dynamics for the nonuniform
spin-polarization system, let us give a physical explanation ofthe result we have obtained. We can construct a simple physicspicture to describe the spin-polarized wave theoretically.Taking the Rashba SOI, for example, we define the eigenstates|φ
a
k/angbracketrightto denote the majority band and the |φb
k/angbracketrightto denote the
minority band. The spin of the eigenstate of the SOC 2DEGlies in the x-yplane. The majority electron has opposite
spin to the minority electron when they have the same wavevector k.
As a result, the spin polarization along the zdirection can
be obtained by the superposition of the majority and minority
bands as
ψ
↑,q=A/bracketleftBigg/summationdisplay
ke(/epsilon1−/epsilon1f)2/4σ21√
2/parenleftbig/vextendsingle/vextendsingleφa
k/angbracketrightbig
+/vextendsingle/vextendsingleφb
k+q/angbracketrightbig/parenrightbig/bracketrightBigg
+A/bracketleftBigg/summationdisplay
ke(/epsilon1−/epsilon1f)2/4σ21√
2/parenleftbig/vextendsingle/vextendsingleφb
k/angbracketrightbig
+/vextendsingle/vextendsingleφa
k+q/angbracketrightbig/parenrightbig/bracketrightBigg
,(17)
where Ais the normalization coefficient, ψ↑,qis the wave
function of the system with positive spin polarization along the
zdirection with wave vector q, and the function e(/epsilon1−/epsilon1f)2/4σ2re-
stricts the spin-polarization electrons only in the narrow range
1
2σ/lessmuch/epsilon1faround the Fermi energy /epsilon1f. The expectation value
/angbracketleftψ↑,q|σzcosq/primex|ψ↑,q/angbracketrightis nonzero only when q/prime=q, which
confirms that ψ↑,qcan describe the spin-polarized wave. The
energy difference of these two electrons in the first (second)term on the right-hand side of Eq. ( 17)i s/Delta1
1(2),a ss h o w ni n
Fig.3. Therefore, |ψ/angbracketrightcan be treated as a collective two-level
system with two Rabi frequencies /Omega11(2)=/Delta11(2)
¯h. The uniform
spin polarization means q=0 and there is only one Rabi
frequency /Omega10=/Delta10
¯h,F i g . 3. When the system is very clean,
our results, Eqs. ( 13) and ( 16), show that the spin evolution is
damped oscillation and the oscillatory frequency is the Rabifrequency. It is a little surprising that when α=β, although
the SOC gap /Delta1
0is not a constant, the oscillatory frequency
corresponds to the maximum splitting energy 4 αkfinstead of
the average splitting energy 2√
2αkf. In the weak SOC regime,
the disorder is so strong that the splitting energy due to theSOI is completely submerged in the broadening of the band
¯h
τ. Therefore, the spin polarization just decays exponentially.
For the nonuniform spin-polarization case, since there are twoRabi oscillation frequencies /Omega1
1and/Omega12, we expect to have two
FIG. 3. (Color online) The dispersion relation due to the linear
Dresselhaus SOI. The SOI induces the energy gap /Delta10=2β1k,w h i c h
is the spin precession frequency for the single-electron spin. However,
when the system is excited to be a spin-polarization wave with wavevector q, the spin polarization along the zdirection is constructed by
the superposition of the two electrons with wave vectors kandk+q.
In this case, the spin precession frequency will be /Delta1
1(2)/similarequal/Delta10(1±q
Q),
where Q=2mβ 1.
damped oscillatory modes in the clean system corresponding
to energy differences /Delta11and/Delta12, respectively, in Fig. 3.
V . NONUNIFORM SPIN POLARIZATION
In the case of the nonuniform spin-polarized 2DEG, the
initial state is a spin wave with wave vector q, and the
momentum kis coupled to k+q, which makes the center
of the Fermi sea shift to near q. The average magnetic field
is nonzero and the of-diagonal elements of the diffusionmatrix appear to couple the different spin components. Whenonly considering the Rashba or linear Dresselhaus SOI, ournumerical calculation does have two kinds of spin dynamicalmodes, which are shown in Figs. 4and5.
The two damped oscillatory modes and their oscillatory
frequency satisfy our expectation based on the Rabi oscillationviewpoint. When qincreases, the Rabi frequency of the faster
mode always increases, which makes the damped oscillatorymode appear even when /Omega1
soτ<1
2. This means we can expect
to observe the oscillation for the nonzero spin polarization athigher temperature than for the uniform spin polarization. InRef. 12, where the spin polarization is uniform, the damped
oscillatory mode appears below 5 K. On the other hand, inRef.13, where the spin polarization is nonuniform, the damped
oscillatory mode appears below 50 K. The material, Fermienergy, and mobility in these two papers are similar. Thisseems support our Rabi oscillation viewpoint. For the slowoscillatory mode, when qis around Q, the corresponding Rabi
frequency /Omega1
2is around 0, which means the spin precession
is very slow. Because the Rabi frequencies are much smallerthan
1
τ, the spin polarization just decays exponentially and the
exponential decay rate has its minimum in this regime when q
is around Q.
A particular case is when α=β1andβ3=0. The analytical
solutions of these two modes can be obtained by finding thepoles of Eq. (20) of Ref. 21, and they have the form
i/Omega1τ=1−/radicalBigg
1−(/Omega1soτ)2/parenleftbigg
1±q
Q/parenrightbigg2
, (18)
where Q=4mα.A tq=Q, the Rabi frequency of the slower
mode is zero for all of the electron momentum k.O nt h e
035318-4SPIN DYNAMICS IN THE STRONG SPIN-ORBIT ... PHYSICAL REVIEW B 84, 035318 (2011)
0 0.5 1 1.5 20.20.40.60.81
q/Q(iRe Im Ωτ) Ωsoτ=0.4
Ωsoτ=0.5
Ωsoτ=0.6
Ωsoτ=0.7
Ωsoτ=1
Ωsoτ=1.5(a)
0 0.5 1 1.5 20246
q/Q(iΩτ)Ωsoτ=0.4
Ωsoτ=0.5
Ωsoτ=0.6
Ωsoτ=0.7
Ωsoτ=1
Ωsoτ=1.5(b)
FIG. 4. (Color online) The fast oscillatory mode of the nonuni-
form spin dynamics in the strong SOC regime when the system only
has bulk inversion asymmetry. (a) The normalized exponential decayrate, Re( i/Omega1τ), increases with increasing qand approaches 1 at large q.
(b) The nonzero normalized oscillatory frequency, Im( i/Omega1τ), increases
linearly at large q, the slope is close to /Omega1
soτ, and its value approaches
/Omega1so(1+q
Q), where Q=2mβ 1.
0.5 1 1.5 20.20.40.60.811.2
q/Q(iΩτ)Ωsoτ=1.5
Ωsoτ=1.7
Ωsoτ=2(a)
0.5 1 1.5 200.511.52
q/Q(iΩτ)Ωsoτ=1.5
Ωsoτ=1.7
Ωsoτ=2.0(b)Re Im
FIG. 5. (Color online) The slow oscillatory mode of the nonuni-
form spin dynamics in the strong SOC regime when the system only
has bulk inversion asymmetry. (a) The normalized exponential decay
rate, Re( i/Omega1τ), has a minimum around q=Qand approaches 1 at
largeq. (b) The nonzero normalized oscillatory frequency, Im( i/Omega1τ),
is always zero when qis around Qand increases linearly at large q.
The slope is close to /Omega1soτand the value approaches /Omega1so(1−q
Q)a t
largeq,w h e r e Q=2mβ 1.other hand, the spin yis a good quantum number for all the
electron states, which means the spin-independent disorderwill never couple the two electrons in different bands withdifferent spin directions. Therefore, the Rabi frequency of theslower mode is still exactly zero even in the presence of thespin-independent disorder no matter how strong it is. As aresult, the spin along the zdirection will never precess and
has an infinitely long lifetime. This provides another way tounderstand the persistent spin helix.
14,15However, the cubic
Dresselhaus SOI induces a band transition in the presenceof spin-independent impurities and makes the spin lifetimefinite.
7When α/negationslash=β1,e v e na t q=Q, the gap of the two
electrons with momentum kandk+qin different spin bands
is dependent on kand fluctuates around the average value of
the gap. The average value of the Rabi frequency of the slowermode is small but not zero. Therefore, the spin relaxationcannot be exactly suppressed. However, if the average valueof the gap is much larger than the fluctuation, normally whenq/greatermuchQ, the spin relaxation can be well described by Eq. ( 18)
for an arbitrary combination of αandβ
1.
VI. PROPOSED EXPERIMENTS
The spin dynamics in the strong SOC regime have several
special characters that can be used in experimental measure-ments.
Momentum scattering time τ
∗
p. In the spin dynamics, the
Coulomb interaction plays an important role in determining themomentum scattering time τ
∗
p.22,23This is quite different from
the charge-transport case, in which electron-electron ( e-e)
interaction will not change the ensemble momentum scatteringτ
p, which determines the electron mobility. This difference is
called spin Coulomb drag (SCD). In previous experimentalwork, SCD was observed through the spin diffusion coefficientD
s=1
2v2
fτ∗
pby fitting the spin decay rate in the weak SOC
regime. Here, we provide a way to observe SCD in the strongSOC regime by directly measuring the momentum scatteringtimeτ
∗
p. Based on Eqs. ( 13) and ( 15), when only Dresselhaus
SOI is presented, the damped decay rate is always almostequal to 1 /2 as long as
β3
β1<0.2, which is easily realized in
experiments.11,15
The strength of SOI’s . Here, we would like to emphasize
that 2β1kfτ=1
2is a very important case and corresponds
to the transition point between the pure exponential decaymode and the damped oscillatory mode. The decay rate at thispoint is not only equal to
1
2τbut also equal to1
2β1kfwhen
α=0. This means that at this point we can obtain the strength
of linear Dresselhaus SOI from the spin-polarization decayrate. When 2 β
1kfτ=1
2, we can increase the Rashba SOI by
adding a gate voltage. As long as 0 <α<β 1, according to
Eq. ( 12), the spin evolution still decays exponentially and
the decay rate is [1 −√
2
2/radicalbig
1−2(/Omega1soτ)2]/τ, where /Omega1so=
2√
α2+β2
1kf, which gives us the strength of Rashba SOI.
VII. CONCLUSION
We have discussed the spin dynamics in the strong spin-
orbit coupling regime. We describe quantitatively the specialcharacters of the damped oscillatory mode in this regime.We also compare our results to the previous experimental
035318-5XIN LIU, XIONG-JUN LIU, AND JAIRO SINOV A PHYSICAL REVIEW B 84, 035318 (2011)
data and find they match very well. Based on our theoretical
results, a reliable way is proposed to measure the Rashba andDresselhaus coefficients and electron momentum scattering
time, which does not correspond to the mobility due to
the Coulomb interaction. Furthermore, we find that the spindynamics in the 2DEG can be treated as a collective two-level
system. This helps us to understand semiquantitatively thespin dynamics in the strong spin-orbit coupling regime. Forthe nonzero spin-polarization case, we predict that there existdouble damped oscillatory modes at large q, and we explain
the persistent spin helix mode from the Rabi oscillation pointof view.ACKNOWLEDGMENTS
We acknowledge Chia-Ren Hu, Ar. Abanov, Yang Liu, and
Victor Galitski for very helpful discussion, and support fromDMR-0547875, NSF-MRSEC DMR-0820414, and SWAN-NRI. J.S. is a Cottrell Scholar of Research Corporation.
APPENDIX: SPIN DYNAMIC MATRIX FOR THE UNIFORM
SPIN POLARIZATION
In this section, we derive the spin evolution mode of the
uniform spin polarization. According to Eq. ( 5), the strength
of SOI is angle-dependent and can be written as
hso=/radicalBig
α2+β2
1kf/radicalBigg
1+cos 2ψcos 2θ+/bracketleftbigg
2/parenleftbiggβ3
λ/prime/parenrightbigg2
−2β3
λ/primesin(ψ+π/4)/bracketrightbigg
(1+cos 4θ)−4β3
λ/primecos(ψ+π/4) cos 2 θ, (A1)
where cos ψ=λ1/√
λ2
1+λ2
2.
First we consider the case for β3=0. The Hamiltonian is written as
H=k2
2m+(α+β)kxσy−(α−β)kyσx=k2
2m+λ1kxσy+λ2kyσx, (A2)
where kxis along the [110] direction, λ1=α+β, andλ2=−(α−β). The Green’s function for this Hamiltonian takes the form
GR(A)=E−k2
2m±i
2τ+(α+β)kxσy−(α−β)kyσx/parenleftbig
E−k2
2m±i
2τ/parenrightbig2−(α2+β2)k2/parenleftbig
1+2αβ
α2+β2cos 2θ/parenrightbig=E−k2
2m±i
2τ+λ1kxσy+λ2kyσx
/parenleftbig
E−k2
2m±i
2τ/parenrightbig2−(λ2
1+λ2
2)
2k2(1+γcos 2θ), (A3)
where τis the momentum scattering time, and γ=2αβ
α2+β2=λ2
1−λ2
2
λ21+λ2
2. It is more convenient to write down the element of the 2 ×2
Green’s function, Eq. ( A3), as
G11=G22=1
2/parenleftbigg1
E−k2
2m−λk√1+γcos 2θ±i
τ+1
E−k2
2m+λk√1+γcos 2θ±i
τ/parenrightbigg
,
G12=1
2/parenleftbigg1
E−k2
2m−λk√1+γcos 2θ±i
τ−1
E−k2
2m+λk√1+γcos 2θ±i
τ/parenrightbigg√
2(−icosψcosθ+sinψsinθ)√1+γcos 2θ, (A4)
G21=1
2/parenleftbigg1
E−k2
2m−λk√1+γcos 2θ±i
τ−1
E−k2
2m+λk√1+γcos 2θ±i
τ/parenrightbigg√
2(icosψcosθ+sinψsinθ)√1+γcos 2θ,
where
λ=/radicalBig/parenleftbig
λ2
1+λ2
2/parenrightbig/slashbig
2=/radicalbig
α2+β2,cosψ=λ1//radicalBig
λ2
1+λ2
2,andγ=cos 2ψ.
According to Eq. ( 8), the diagonal element of the spin polarization along the zdirection has the form
Izz=I11,11−I11,22−I22,11+I22,22=1
2mτ/integraldisplayd2k
(2π)2/parenleftbig
GA
11GR11−GA
21GR12−GA
12GR21+GA
22GR22/parenrightbig
. (A5)
The first term and the fourth term in Eq. ( A5) are equal to each other and have the form
1
2mτ/integraldisplayd2k
(2π)2GA
11GR11=1
2m/integraldisplayd2k
(2π)21
4/parenleftbigg1
E−/epsilon1+(k)−i
2τ+1
E−/epsilon1−(k)−i
2τ/parenrightbigg
×/parenleftbigg1
E+/Omega1−/epsilon1−(k)+i
2τ+1
E+/Omega1−/epsilon1−(k)+i
2τ/parenrightbigg
=1
16mπ/integraldisplay2π
0dθ
vf/parenleftbiggk+
1−i/Omega1τ+k−
1−i/Omega1τ+2iλk√1+γcos 2θ+k+
1−i/Omega1τ−2iλk√1+γcos 2θ+k−
1−i/Omega1τ/parenrightbigg
/similarequal1
16π/integraldisplay2π
0dθ/parenleftbigg1
1−i/Omega1τ+1
1−i/Omega1τ+2iλk√1+γcos 2θ+1
1−i/Omega1τ−2iλk√1+γcos 2θ+1
1−i/Omega1τ/parenrightbigg
, (A6)
035318-6SPIN DYNAMICS IN THE STRONG SPIN-ORBIT ... PHYSICAL REVIEW B 84, 035318 (2011)
where vf=∂Ef
∂k. In the polar coordinate,/integraltext
d2k=/integraltext
d(k2/2)dθ. As we assume that λkf/lessmuchEf,d(k2/2)/similarequalmdE , where mis the
effective mass.
The other two terms are also equal to each other and can be written as
1
2mτ/integraldisplayd2k
(2π)2GA
21GR12=/integraldisplayd2k
(2π)21
4/parenleftbigg1
E−/epsilon1+(k)−i
2τe−1
E−/epsilon1−(k)−i
2τe/parenrightbigg/parenleftbigg1
E+/Omega1−/epsilon1+(k)+i
2τe−1
E+/Omega1−/epsilon1−(k)+i
2τe/parenrightbigg
=1
16mπ/integraldisplay2π
0dθ
vf/parenleftbiggk+
1−i/Omega1τ−k−
1−i/Omega1τ+2iλk√1+γcos 2θ−k+
1−i/Omega1τ−2iλk√1+γcos 2θ+k−
1−i/Omega1τ/parenrightbigg
.
/similarequal1
16π/integraldisplay2π
0dθ/parenleftbigg1
1−i/Omega1τ−1
1−i/Omega1τ+2iλk√1+γcos 2θ−1
1−i/Omega1τ−2iλk√1+γcos 2θ+1
1−i/Omega1τ/parenrightbigg
. (A7)
Substituting Eqs. ( A6) and ( A7) into Eq. ( A5), we have
Izz=1
4π/integraldisplay2π
0dθ/parenleftbigg1
1−i/Omega1τ+2iλk√1+γcos 2θ+1
1−i/Omega1τ−2iλk√1+γcos 2θ/parenrightbigg
=1
2π/integraldisplay2π
0dθ1−i/Omega1τ
(1−i/Omega1τ)2+(/Omega1soτ)2(1+γcos 2θ)=1−i/Omega1τ
2π[(1−i/Omega1τ)2+(/Omega1soτ)2]/integraldisplayπ
0dx2
[1+acos(x)], (A8)
where x=2θanda=γ(/Omega1soτ)2/[(1−i/Omega1τ)2+(/Omega1soτ)2]. The indefinite integral/integraltext
dx1
1+acos(x)=2 arc tanh/bracketleftbig(−1+a)t a n [x
2]√
−1+a2/bracketrightbig
√
−1+a2 . Therefore,
we have
Izz=1−i/Omega1τ
2π[(1−i/Omega1τ)2+(/Omega1soτ)2]/integraldisplayπ
0dx2
[1+acos(x)]
=1−i/Omega1τ
2π[(1−i/Omega1τ)2+(/Omega1soτ)2]2⎛
⎝2 arc tanh/bracketleftbig(−1+a) tan[π
2]√
−1+a2/bracketrightbig
√
−1+a2−2 arc tanh/bracketleftbig(−1+a) tan[0
2]√
−1+a2/bracketrightbig
√
−1+a2⎞
⎠
=1−i/Omega1τ
2π[(1−i/Omega1τ)2+(/Omega1soτ)2]2πi√
−1+a2=1−i/Omega1τ/radicalbig
[(1−i/Omega1τ)2+(/Omega1soτ)2]2−γ2(/Omega1soτ)4. (A9)
When the Rashba SOI is zero, the strength of the SOI’s takes the form
hso=β1kf/radicalBigg
1+/parenleftbigg
2β2
3
β2
1−2β3
β1/parenrightbigg
(1+cos 4θ). (A10)
To obtain the spin diffusive matrix element Izz, it is easy to prove that we only need to replace the term λk√1+γcos 2θin
Eq. ( A9) with hsoin Eq. ( A10). Therefore, we have
Izz=1
4π/integraldisplay/parenleftbigg2(1−i/Omega1τ)
(1−i/Omega1τ)2+(2hsoτ)2/parenrightbigg
dθ=1−i/Omega1τ
/radicalbig
(1−i/Omega1τ)2+/Omega12soτ2/radicalBig
(1−i/Omega1τ)2+/Omega12soτ2/bracketleftbig
1+2/parenleftbigβ3
β1/parenrightbig2−2β3
β1/bracketrightbig.(A11)
1M. I. Dyakonov and V . I. Perel, JETP Lett. 13, 467 (1971).
2M. I. Dyakonov and V . I. Perel, Zh. Eksp. Teor. Fiz. 60, 1954 (1971)
[Sov. Phys. JETP 33, 1053 (1971)].
3M. I. Dyakonov and V . Yu. Kachorovskii, Fiz. Tekh. Poluprovodn.
20, 178 (1986) [Sov. Phys. Semicond. 20, 110 (1986)].
4F. Meier and B. P. Zakharchenya, Optical Orientation, Mod-
ern Problems in Condensed Matter Sciences (North-Holland,
Amsterdam, 1984).
5A. A. Burkov, A. S. Nunez, and A. H. MacDonald, P h y s .R e v .B
70, 155308 (2004).
6E. G. Mishchenko, A. V . Shytov, and B. I. Halperin, Phys. Rev.
Lett.93, 226602 (2004).
7T. D. Stanescu and V . Galitski, P h y s .R e v .B 75, 125307
(2007).8A. Punnoose and A. M. Finkel’stein, Phys. Rev. Lett. 96, 057202
(2006).
9J. H. Jiang and M. W. Wu, Phys. Rev. B 79, 125206 (2009).
10L. Yang, J. Orenstein, and D.-H. Lee, Phys. Rev. B 82, 155324
(2010).
11W. J. H. Leyland, R. T. Harley, M. Henini, A. J. Shields, I. Farrer,and D. A. Ritchie, Phys. Rev. B 76, 195305 (2007).
12M. A. Brand, A. Malinowski, O. Z. Karimov, P. A. Marsden,
R. T. Harley, A. J. Shields, D. Sanvitto, D. A. Ritchie, and M.Y . Simmons, Phys. Rev. Lett. 89, 236601 (2002).
13C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein, J. Stephens, and
D. D. Awshalom, Nature (London) 437, 1330 (2005).
14C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, J. Stephens,
and D. D. Awshalom, Phys. Rev. Lett. 98, 076604 (2007).
035318-7XIN LIU, XIONG-JUN LIU, AND JAIRO SINOV A PHYSICAL REVIEW B 84, 035318 (2011)
15J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C.
Zhang, S. Mack, and D. D. Awschalom, Nature (London) 458, 610
(2009).
16E. Akkermans and G. Montambaux, Mesoscopic Physics of Elec-
trons and Photons (Cambridge University Press, Cambridge, 2007).
17A. G. Mal’shukov, K. A. Chao, and M. Willander, P h y s .R e v .L e t t .
76, 3794 (1996); I. S. Lyubinskiy and V . Yu. Kachorovskii, Phys.
Rev. B 70, 205335 (2004).18E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).
19Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6093
(1984).
20G. Dresselhaus, Phys. Rev. 100, 580 (1955).
21B. A. Bernevig and J. Hu, P h y s .R e v .B 78, 245123 (2008).
22M. M. Glazov and E. L. Ivchenko, JETP Lett. 75, 476
(2002).
23I. DAmico and G. Vignale, Phys. Rev. B 68, 045307 (2003).
035318-8 |
PhysRevB.92.165420.pdf | PHYSICAL REVIEW B 92, 165420 (2015)
Atomic resolution imaging of the two-component Dirac-Landau levels
in a gapped graphene monolayer
Wen-Xiao Wang,1,2Long-Jing Yin,1,2Jia-Bin Qiao,1,2Tuocheng Cai,3,4Si-Yu Li,1,2Rui-Fen Dou,1Jia-Cai Nie,1
Xiaosong Wu,3,4and Lin He1,2,*
1Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China
2Center for Advanced Quantum Studies, Beijing Normal University, Beijing 100875, People’s Republic of China
3State Key Laboratory for Artificial Microsctructure and Mesoscopic Physics, Peking University, Beijing 100871, People’s Republic of China
4Collaborative Innovation Center of Quantum Matter, Beijing 100871, People’s Republic of China
(Received 12 June 2015; published 16 October 2015)
The wave function of Dirac fermions is a two-component spinor. In graphene, a one-atom-thick film showing
two-dimensional Dirac-like electronic excitations, the two-component representation, reflects the amplitude ofthe electron wave function on the AandBsublattices. This unique property provides unprecedented opportunities
to image the two components of Dirac fermions spatially. Here, we report atomic resolution imaging oftwo-component Dirac-Landau levels in gapped graphene monolayers by scanning tunneling microscopy andspectroscopy. A gap of about 20 meV , driven by inversion symmetry breaking by the substrate potential, isobserved in the graphene sheets on both SiC and graphite substrates. Such a gap splits the n=0 Landau level
(LL) into two levels, 0
+and 0 −. We demonstrate that the amplitude of the wave function of the 0 +LL is mainly
on the Asites and that of the 0 −LL is mainly on the Bsites of graphene, characterizing the internal structure of
the spinor of the n=0 LL. This provides direct evidence of the two-component nature of Dirac fermions.
DOI: 10.1103/PhysRevB.92.165420 PACS number(s): 73 .22.Pr
I. INTRODUCTION
Because of the bipartite honeycomb lattice [ 1–3], which
has two distinct sublattices (denoted by AandB), the wave
functions describing the low-energy excitations near the Diracpoints, commonly called KandK
/prime, in graphene monolayers
are two-component spinors. The Dirac spinors of the two conesin graphene have the form
|K/angbracketright=/parenleftbigg
ψ
KA
ψKB/parenrightbigg
=1√
2/parenleftbigg1
±ie−iθτ/parenrightbigg
,
(1)
|K/prime/angbracketright=/parenleftbigg
ψK/primeB
ψK/primeA/parenrightbigg
=1√
2/parenleftbigg1
∓ie−iθτ/parenrightbigg
.
Here,θτ=arctan(qτ,y
qτ,x) is defined as the angle of wave vector
qτ≡(qτ,x,qτ,y) in momentum space. The two-component rep-
resentation, which resembles that of a spin in a mathematicallyway [ 1–4], corresponds to the projection of the electron wave
function on the AandBsublattices.
A site energy difference 2 /Delta1between the sublattices,
generated by substrates, could break the inversion symmetryof graphene and lift the energy degeneracy of the AandB
sublattices [ 5,6], as shown in Fig. 1(a). This effect generates
ag a p /Delta1E=2/Delta1at the Dirac points [as shown in Fig. 1(b)],
which was previously observed for a graphene monolayer ontop of SiC [ 7], graphite [ 8–11], and hexagonal boron nitride
[12,13]. The gap, usually ranging from 10 meV to several
tens meV , can result in a valley contrasting Hall transportin graphene monolayers [ 5,13]. In the quantum Hall regime,
the broken symmetry of the graphene sublattices shifts theenergies of the n=0 Landau level (LL) in the KandK
/prime
valleys in opposite directions and therefore splits the n=0
*helin@bnu.edu.cnLL into the 0 +and 0 −LLs (here, λ=+,−denote the K/prime
andKvalleys, respectively) [ 8–11], as schematically shown
in Fig. 1(c). Generally, the wave functions |nλ/angbracketrightof the LLs in
graphene are given by [ 2,14,15]
|n−/angbracketright=/parenleftbiggψn−
A
ψn−
B/parenrightbigg
=/parenleftbigg
sinαn−φ|n|−1
cosαn−φ|n|/parenrightbigg
,
(2)
|n+/angbracketright=/parenleftbiggψn+
B
ψn+
A/parenrightbigg
=/parenleftbigg
cosαn+φ|n|−1
sinαn+φ|n|/parenrightbigg
,
where tan α±=(hωB√|n|sgn(n))/(√
h2ω2
B|n|+/Delta12−
/Delta1sgn(n))(n/negationslash=0) and ωB=√
2ev2
FB//planckover2pi1, with vFthe Fermi
velocity and Bthe magnetic field (here, φnis the usual
Landau-level wave function). For n=0,α0−=0 and
α0+=π/2, hence only the second components of the spinors
are nonzero and we have |0−/angbracketright=(ψ0−
A=0
ψ0−
B=φ0) and|0+/angbracketright=(ψ0+
B=0
ψ0+
A=φ0).
It indicates that we can detect the 0 −LL, i.e., the spinor |K0/angbracketright,
only on the Bsites and detect the 0 +LL, i.e., the spinor
|K/prime0/angbracketright, only on the Asites (see the Supplemental Material for
details of the analysis [ 16]). With the help of high energy and
spatial resolution of scanning tunneling microscopy (STM)and spectroscopy (STS), it is therefore possible to image thetwo components of the spinors in atomic resolution.
II. EXPERIMENTAL METHOD
The STM system was an ultrahigh vacuum single-probe
scanning probe microscope (USM-1500) from UNISOKU. AllSTM and STS measurements were performed at liquid-heliumtemperature and the images were taken in a constant-currentscanning mode. The STM tips were obtained by chemicaletching from a wire of Pt(80%)-Ir(20%) alloys. Lateraldimensions observed in the STM images were calibrated usinga standard graphene lattice, a Si (111)-(7 ×7) lattice, and
1098-0121/2015/92(16)/165420(5) 165420-1 ©2015 American Physical SocietyW ANG, YIN, QIAO, CAI, LI, DOU, NIE, WU, AND HE PHYSICAL REVIEW B 92, 165420 (2015)
FIG. 1. (Color online) Electronic band structure and Landau
quantization in a gapped graphene monolayer. (a) Schematic diagramof a graphene monolayer with a staggered sublattice potential
breaking the inversion symmetry. The AandBsites are denoted by
blue and red balls, respectively. (b) Energy spectrum of a graphenemonolayer with broken inversion symmetry. (c) Schematic Landau
levels and DOS of a gapped graphene monolayer in the quantum
Hall regime. Peaks in the DOS correspond to the Landau Levels n
λ
(λ=+,−denote the K/primeandKvalleys).
Ag (111) surface. The STS spectrum, i.e., the dI/dV-Vcurve,
was carried out with a standard lock-in technique using an873 Hz alternating current modulation with an amplitude of7 mV . Our experiments were carried on both SiC and a highlyoriented pyrolytic graphite (HOPG) surface. We prepared ahigh-quality graphene film on a silicon carbide crystal bythe thermal decomposition process, and the HOPG substratewas cleaved with adhesive tape prior to experiments (see the
Supplemental Material for details [ 16]).
III. EXPERIMENTAL RESULTS AND ANALYSIS
Figure 2(a) shows a representative STM image of a
graphene multilayer grown on a SiC substrate [ 17–20]. An
intensity imbalance between the AandBsublattices, as shown
in the atomic image of Fig. 2(b), indicates the inversion
symmetry breaking by the substrate potential [ 7–11]. Our
STS, as shown in Fig. 2(c), and Raman measurements (see
Supplemental Fig. S1 [ 16]) show decoupling behavior of the
topmost graphene monolayer and the underlying graphenemultilayer, which is in agreement with earlier transport [ 17,21]
and spectroscopy measurements [ 7,22,23]. Figure 2(d) shows
a typical STM image of a graphite surface. The topmostgraphene sheet usually decouples from the underlying graphitesubstrate [ 8–11,24–26]. However, the electrostatic potential of
the second commensurate layer still can break the inversionsymmetry of the topmost graphene sheet [ 11], as demonstrated
in our experiment by the triangular lattice shown in Fig. 2(e)
[usually, the main part of graphite is the Bernal ( AB-stacked)
graphite].
The spectra of the graphene sheets, recorded in the magnetic
field of 8 T [Figs. 2(c) and2(f)], on both the SiC and graphite
substrates, exhibit Landau quantization of Dirac fermions, asexpected to be observed in a gapped graphene monolayer[7–11,24]. The slight electron-hole asymmetry, as shown in
Figs. 2(c) and 2(f), may arise from a finite next-nearest-
FIG. 2. (Color online) STM images and STS spectra of gapped graphene sheets. (a) and (d) show STM images of graphene on a SiC (000 ¯1)
terrace and on highly oriented pyrolytic graphite (HOPG), respectively. The periodic protuberances in (a) are attributed to the moir ´ep a t t e r n
arising from a stacking misorientation between adjacent graphene layers. (b) and (e) show zoom-in atomic-resolution topographies obtained in
the black frame in (a) and (d), respectively. The honeycomb structures of graphene are overlaid onto the STM images. The STM images show
a triangular lattice, indicating the broken sublattice symmetry. Here, we define the bright spots as the A-site atoms and the dark spots as the
B-site atoms. (c) and (f) show dI/dVspectra obtained at different positions, as marked in different colors in (b) and (e), respectively. The black
arrows in both panels denote the position of the charge neutrality point of the topmost graphene sheets in zero magnetic field. In the magnetic
field of 8 T, the spectra exhibit Landau quantization of Dirac fermions, as expected in a gapped graphene monolayer. LL indices are marked.For the graphene on both substrates, the n=0 LL splits into two peaks and the intensity of the two peaks depends sensitively on the recorded
positions.
165420-2ATOMIC RESOLUTION IMAGING OF THE TWO- . . . PHYSICAL REVIEW B 92, 165420 (2015)
neighbor hopping in the topmost graphene sheets [ 27]. The
Fermi velocities for the graphene sheets on the SiC andgraphite substrates are estimated to be v
F=(0.79±0.03)×
106m/s andvF=(0.84±0.03)×106m/s, respectively (see
Supplemental Figs. S2–S4 for more experimental data andmethods to measure the Fermi velocities [ 16]). A notable
feature of the tunneling spectra recorded at 8 T is the splitting ofthen=0 peak and its sensitive dependency on the recorded
positions, as shown in Figs. 2(c) and 2(f). The split of the
n=0 peak, ∼20 mV , is attributed to a gap caused by the
inversion symmetry breaking by the substrate potential [ 8–11].
Here, we should point out that similar results have beenobserved in several different graphene sheets on both theSiC and graphite substrates and the value of the gap rangesfrom about 15 to 22 meV . (In our experiment, the inversionsymmetry breaking of the topmost graphene is generated bythe underlying graphene sheet on both the SiC and graphitesubstrates. Interestingly, we observe almost the same gap forgraphene sheets on the two substrates.) The tunneling spectrumgives direct access to the local density of states (DOS) of thesurface beneath the STM tip. The spectra recorded at differentpositions, as shown in Figs. 2(c) and 2(f), indicate that the
0
−LL is significant only on the Bsites and the 0 +LL is
pronounced only on the Asites. At the center of the hexagons of
the graphene sheets, for example, at the green dots in Figs. 2(b)
and2(e), the observed intensities of the 0 +and the 0 −LLs are
almost identical, as shown in Figs. 2(c) and2(f). Such a feature
reminds us of the characteristics of the internal structure of thetwo-component spinors of the 0
−and 0 +LLs [ 2,14]. We will
demonstrate below that the splitting of the n=0 LL is a direct
consequence of its two-component nature.
Figures 3(a) and3(b) show differential conductance maps
for the graphene on SiC substrate at 8 T at the bias voltagesof the 0
−and 0 +LLs, respectively. The maps reflect spatial
distribution of the local DOS at the bias voltages. Both themaps exhibit triangular contrasting, indicating a pronouncedasymmetry of the 0
+and 0 −LLs on the sublattices. However,
there is a very important difference between the two maps. Thebright spots in the conductance map of the 0
−LL correspond
to the dark spots of the triangular lattice, i.e., the Bsites, in
the STM image, whereas the bright spots in the map of the0
+LL correspond to the bright spots of the triangular lattice,
i.e., the Asites, in the STM image (similar conductance maps
and results are also observed in the gapped graphene sheet ongraphite surface, see Supplemental Figs. S2 and S5 [ 16]). At
a fixed energy, the local DOS at position ris determined by
the wave functions according to ρ(r)∝|ψ(r)|
2. Therefore, the
maps in Figs. 3(a) and3(b) reflect atomic resolution images
of the two-component Dirac-Landau levels. Theoretically, thespinor of the 0
−(0+) LL only has a nonzero component on the
B(A) sites, which is qualitatively consistent with the observed
large asymmetry of the 0 −and 0 +LLs on the sublattices.
Atn=0 LL, the broken inversion symmetry lifts the
degeneracies of both the sublattices and the valleys, as shownin Fig. 1(c).A tn/negationslash=0, the KandK
/primevalleys are doubly
degenerate in energy, whereas there is still a difference betweenthe amplitudes of the two components in the spinors of the LLs,as described by Eq. ( 2). Forn> 0(n< 0), the amplitude of the
A-site (B-site) component of the spinors in both the KandK
/prime
valleys is predicted to be slightly larger than that of the B-site
FIG. 3. (Color online) Conductance maps of the gapped
graphene monolayer on SiC at different energies. (a) The conductancemap recorded at the bias voltage of the 0
+LL (Vsample=65.5m V ) .
(b) The conductance map recorded at the bias voltage of the 0 −LL
(Vsample=45 mV). (c) The conductance map recorded at the bias
voltage of the +1L L( Vsample=108 mV). (d) The conductance map
recorded at the bias voltage of the −1L L( Vsample=− 22 mV). The
honeycomb structure of graphene and the atomic resolution STMimage are overlaid onto the maps. The amplitude of the 0
+and+1
LLs on the Asites is much stronger than that on the Bsites, whereas
the amplitude of the 0 −and−1 LLs on the Bsites is much stronger
than that on the Asites.
(A-site) component. Such a feature has also been demonstrated
explicitly in our experiment for the gapped graphene sampleson both the SiC and graphite substrates. Figures 3(c) and3(d)
show conductance maps for the graphene on SiC substrateat 8 T at the bias voltages of the n=+ 1 and n=− 1 LLs,
respectively. Both the maps exhibit triangular contrasting and,obviously, the amplitude of the n=+ 1(n=− 1) LL on the
A(B) sites is stronger than that on the B(A) sites (similar
conductance maps and results are also observed in the gappedgraphene sheet on the graphite surface, see SupplementalFigs. S2 and S6 [ 16]). However, the asymmetry between the
amplitudes of the A-site component and B-site component of
the spinors for the n=+ 1 and n=− 1 LLs is much weaker
than that for the 0
+and 0 −LLs, as demonstrated in Fig. 3.
Such an asymmetry will further decrease with increasing thevalue of |n|, according to Eq. ( 2). Therefore, we obtain almost
honeycomb contrasting in the conductance map recorded at thebias voltage of 170 mV (the value of nat 170 mV is estimated
to be 3), as shown in Fig. 4(a). Theoretically, the value of
sin
2(an/2)/cos2(an/2), which reflects the asymmetry between
the amplitudes of the A-site component and B-site component
of the spinors, is estimated to be about 1.12 for n=3.
To further compare our experimental results with the theory,
we plot vertical line cuts of the conductance maps of the0
+,0−,−1,+1, and +3 LLs, as shown in Figs. 3and4(a),
along the AandBatoms in Fig. 4(b). The theoretical result
showing the amplitudes of these LLs on the sublattices is alsoshown in Fig. 4(c), which reproduces the overall features of
our experimental results quite well. However, there is stillan obvious difference if we compare the experimental result
165420-3W ANG, YIN, QIAO, CAI, LI, DOU, NIE, WU, AND HE PHYSICAL REVIEW B 92, 165420 (2015)
FIG. 4. (Color online) Internal structures of different LLs. (a) The conductance map recorded at the bias voltage of the +3 LL for the
graphene on the SiC substrate ( Vsample=170 mV). (b) Vertical line cuts of the conductance maps of the 0 +,0−,−1,+1, and +3 LLs along the
AandBatoms. The curves are offset vertically for clarity, and the zero-line for these curves is denoted by dashed lines. (c) The theoretical
local DOS (LDOS) showing amplitudes of these LLs on the AandBatoms. The amplitudes of the two components are calculated according
to Eq. ( 2), and we use Gaussian peaks to reflect the linewidth of the DOS at the AandBatoms.
with the theoretical one quantitatively. For example, the A-site
component of the 0 −LL is predicted to be zero, however,
we always observe a nonzero value on the Asites in the
conductance map of the 0 −LL. Such a discrepancy is mainly
attributed to the overlap of the 0 +and 0 −LLs. There is a finite
linewidth of the 0 +and 0 −LLs and they are separated by
only about 20 mV , as shown in Figs. 2(c) and2(f). Therefore,
when we carry out conductance mapping at the bias voltageof the 0
−LL, there is a slightly contribution from the 0 +LL,
which results in the nonzero value of intensity on the Asites.
A concomitant result of this effect is that there is always aweak peak of the 0
+LL (or the 0 −LL) when we measure
the tunneling spectra on the Bsites (or Asites), as shown in
Figs. 2(c) and2(f).
The validity of our experimental result is further confirmed
by performing similar measurements on a graphene monolayerwithout breaking the inversion symmetry, i.e., /Delta1=0, on the
graphite surface. The stacking order of this region may beAA-stacked graphite, therefore, the underlying graphene will
not break the sublattice symmetry of the topmost decoupledgraphene sheet (see Supplemental Fig. S7 for experimentaldata [ 16]). For this zero-gap graphene, we always observe a
honeycomb lattice in the STM image and obtain honeycombcontrasting in the conductance maps recorded at 8 T at differentbias voltages, which is consistent with the prediction of Eq. ( 2)
by taking the limit /Delta1→0.IV . CONCLUSIONS
In summary, our experimental result demonstrates that the
splitting of the n=0 LL is a direct consequence of its two-
component nature, and we realize atomic resolution imaging ofthe two-component Dirac-Landau levels in gapped graphenesheets on both the SiC and graphite substrates. The abilityto tune the amplitude of the two components of the Diracspinors may pave the way for manipulating spins in otherDirac systems, such as topological insulators [ 4,28,29].
ACKNOWLEDGMENTS
This work was supported by the National Basic Re-
search Program of China (Grants No. 2014CB920903,No. 2013CBA01603, No. 2013CB921701, and No.2012CB933022), the National Natural Science Foundation ofChina (Grants No. 11422430, No. 11374035, No. 11474022,No. 51172029, No. 91121012, and No. 11222436), the pro-gram for New Century Excellent Talents in University of theMinistry of Education of China (Grant No. NCET-13-0054),and the Beijing Higher Education Young Elite Teacher Project(Grant No. YETP0238). L.H. also acknowledges supportfrom the National Program for Support of Topnotch YoungProfessionals.
W.-X.W., L.-J.Y ., and J.-B.Q. contributed equally to this
work.
[1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,
and A. K. Geim, The electronic properties of graphene, Rev.
Mod. Phys. 81,109(2009 ).
[2] M. O. Goerbig, Electronic properties of graphene in a strong
magnetic field, Rev. Mod. Phys. 83,1193 (2011 ).
[3] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Chiral
tunneling and the Klein paradox in graphene, Nat. Phys. 2,620
(2006 ).
[4] Y .-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hannaguri,
and T. Sasagawa, Imaging the two-component nature of Dirac-Landau levels in the topological surface state of Bi
2Se3,Nat.
Phys. 10,815(2014 ).
[5] D. Xiao, W. Yao, and Q. Niu, Valley-Contrasting Physics in
Graphene: Magnetic Moment and Topological Transport, Phys.
Rev. Lett. 99,236809 (2007 ).[6] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L.
Kleinman, and A. H. MacDonald, Intrinsic and Rashba spin-orbit interactions in graphene sheets, P h y s .R e v .B 74,165310
(2006 ).
[7] S. Y . Zhou, G.-H. Gweon, A. V . Fedorov, P. N. First, W. A. de
Heer, D.-H. Lee, F. Guinea, A. H. Castro Neto, and A. Lanzara,Substrate-induced bandgap opening in epitaxial graphene, Nat.
Mater .6,770(2007 ).
[8] G. Li, A. Luican, and E. Y . Andrei, Scanning Tunneling
Spectroscopy of Graphene on Graphite, Phys. Rev. Lett. 102,
176804 (2009 ).
[9] X. Du, I. Skachko, F. Duerr, A. Luican, and E. Y .
Andrei, Fractional quantum Hall effect and insulating phaseof Dirac electrons in graphene, Nature (London) 462,192
(2009 ).
165420-4ATOMIC RESOLUTION IMAGING OF THE TWO- . . . PHYSICAL REVIEW B 92, 165420 (2015)
[10] D. L. Miller, K. D. Kubista, G. M. Rutter, M. Ruan,
W. A. de Heer, P. N. First, and J. A. Stroscio, Observing thequantization of zero mass carriers in graphene, Science 324,924
(2009 ).
[11] G. Li, A. Luican-Mayer, D. Abanin, L. Levitov, and E. Y . Andrei,
Evolution of Landau levels into edge states in graphene, Nat.
Commun .4,1744 (2013 ).
[12] B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz,
B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino,P. Jarillo-Herrero, and R. C. Ashoori, Massive Dirac fermionsand Hofstadter butterfly in a van der Waals heterostructure,Science 340,1427 (2013 ).
[13] R. V . Gorbachev, J. C. W. Song, G. L. Yu, A. V . Kretinin,
F. Withers, Y . Cao, A. Mishchenko, I. V . Grigorieva,K. S. Novoselov, L. S. Levitov, and A. K. Geim, Detectingtopological currents in graphene superlattices, Science 346,448
(2014 ).
[14] M. Koshino and T. Ando, Anomalous orbital magnetism in
Dirac-electron systems: Role of pseudospin paramagnetism,Phys. Rev. B 81,195431 (2010 ).
[15] J. L. Lado, J. W. Gonzalez, and J. Fernandez-Rossier, Quantum
Hall effect in gapped graphene heterojunctions, Phys. Rev. B
88,035448 (2013 ).
[16] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.92.165420 for Raman spectra, more STM
images, STS spectra, and details of the analysis.
[17] C. Berger, Z. Song, T. Li, X. Li, A. Y . Ogbazghi, R. Feng, Z. Dai,
A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer,Ultrathin epitaxial graphite: 2D electron gas properties and aroute toward graphene-based nanoelectronics, J. Phys. Chem. B
108,19912 (2004 ).
[18] X. Wu, X. Li, Z. Song, C. Berger, and W. A. de Heer, Weak
Antilocalization in Epitaxial Graphene: Evidence for ChiralElectrons, Phys. Rev. Lett. 98,136801 (2007 ).
[19] T. Cai, Z. Jia, B. Yan, D. Yu, and X. Wu, Hydrogen assisted
growth of high quality epitaxial graphene on the C-face of4H-SiC, Appl. Phys. Lett. 106,013106 (2015 ).[20] R. Zhang, Y . Dong, W. Kong, W. Han, P. Tan, Z. Liao, X. Wu,
and D. Yu, Growth of large domain epitaxial graphene on theC-face of SiC, J. Appl. Phys. 112,104307 (2012 ).
[21] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud,
D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad,P. N. First, and W. A. de Heer, Electronic confinement andcoherence in patterned epitaxial graphene, Science
312,1191
(2006 ).
[22] J. Hicks, M. Sprinkle, K. Shepperd, F. Wang, A. Tejeda, A.
Taleb-Ibrahimi, F. Bertran, P. Le Fevre, W. A. de Heer, C.Berger, and E. H. Conrad, Symmetry breaking in commensurategraphene rotational stacking: Comparison of theory and experi-ment, P h y s .R e v .B 83,205403 (2011 ).
[23] J. Hass, F. Varchon, J. E. Millan-Otoya, M. Sprinkle, N. Sharma,
W. A. de Heer, C. Berger, P. N. First, L. Magaud, and E. H.Conrad, Why Multilayer Graphene on 4H-SiC(0001) BehavesLike a Single Sheet of Graphene, Phys. Rev. Lett. 100,125504
(2008 ).
[24] L.-J. Yin, S.-Y . Li, J.-B. Qiao, J.-C. Nie, and L. He, Landau
quantization in graphene monolayer, Bernal bilayer, and Bernaltrilayer on graphite surface, P h y s .R e v .B 91,115405 (2015 ).
[25] L.-J. Yin, J.-B. Qiao, W.-X. Wang, Z.-D. Chu, K. F. Zhang,
R.-F. Dou, C. L. Gao, J.-F. Jia, J.-C. Nie, and L. He, Tuningstructures and electronic spectra of graphene layers with tiltgrain boundaries, Phys. Rev. B 89,205410 (2014 ).
[26] L.-J. Yin, J.-B. Qiao, W.-J. Zuo, W.-T. Li, and L. He, Exper-
imental evidence for non-Abelian gauge potentials in twisted
graphene bilayers, P h y s .R e v .B 92,081406(R) (2015 ).
[27] K.-K. Bai, Y .-C. Wei, J.-B. Qiao, S.-Y . Li, L.-J. Yin, W. Yan,
J.-C. Nie, and L. He, Detecting giant electron-hole asymmetryin a graphene monolayer generated by strain and charged-defectscattering via Landau level spectroscopy, Phys. Rev. B 92,
121405(R) (2015 ).
[28] M. Z. Hasan and C. L. Kane, Colloquium: Topological insula-
tors, Rev. Mod. Phys. 82,3045 (2010 ).
[29] X.-L. Qi and S.-C. Zhang, Topological insulators and supercon-
ductors, Rev. Mod. Phys. 83,1057 (2011 ).
165420-5 |
PhysRevB.97.174517.pdf | PHYSICAL REVIEW B 97, 174517 (2018)
Pressure-temperature phase diagrams of CaK(Fe 1−xNix)4As4superconductors
Li Xiang,*William R. Meier, Mingyu Xu, Udhara S. Kaluarachchi, Sergey L. Bud’ko, and Paul C. Canfield†
Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA
and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
(Received 28 March 2018; published 22 May 2018)
The pressure dependence of the magnetic and superconducting transitions and that of the superconducting
upper critical field are reported for CaK(Fe 1−xNix)4As4, the first example of an Fe-based superconductor with
spin-vortex-crystal-type magnetic ordering. Resistance measurements were performed on single crystals with twosubstitution levels ( x=0.033,0.050) under hydrostatic pressures up to 5.12 GPa and in magnetic fields up to 9 T.
Our results show that, for both compositions, magnetic transition temperatures T
Nare suppressed upon applying
pressure; the superconducting transition temperatures Tcare suppressed by pressure as well, except for x=0.050
in the pressure region where TNandTccross. Furthermore, the pressure associated with the crossing of the TN
andTclines also coincides with a minimum in the normalized slope of the superconducting upper critical field,
consistent with a likely Fermi-surface reconstruction associated with the loss of magnetic ordering. Finally, atp∼4 GPa, both Ni-substituted CaK(Fe
1−xNix)4As4samples likely go through a half-collapsed-tetragonal phase
transition, similar to the parent compound CaKFe 4As4.
DOI: 10.1103/PhysRevB.97.174517
I. INTRODUCTION
Since the discovery of Fe-based superconductors (FeSC)
[1–4], many studies have been done on them, and they have
expanded into a large family. Among them, the AeFe2As2
compounds ( Ae=Ca, Sr, Ba, Eu) have received significant
attention because large, high-quality single crystals can beobtained with a variety of chemical substitution [ 5,6]. Studies
have revealed that members of this family share a globalphase diagram upon tuning by substitution or pressure [ 5,7].
At ambient pressure, the parent compounds undergo a struc-tural/magnetic transition upon cooling; substitution or pressureinduces superconductivity after sufficiently suppressing thestructural/magnetic transitions [ 5,6,8–11]. This suggests a
competition between the magnetism and superconductivity andthat magnetic fluctuations play an important role in formingsuperconductivity in this system [ 7,12–16].
Recently, a new FeSC AeA Fe
4As4(A=K, Rb, Cs)
structural type ( P4/mmm ) was discovered by Iyo et al. [17].
This is not a homogeneous substitution as in ( Ae0.5A0.5)Fe 2As2
where AeandAshare the same crystallographic site. Each Ae
andAin theAeA Fe4As4structure has a unique, well-defined,
crystallographic site, forming alternating AeandAplanes
along the caxis [ 17,18]. Among them, single crystals of
CaKFe 4As4were synthesized and found to be superconducting
at∼35 K, with no other phase transition from 1.8 to 300 K at
ambient pressure [ 18,19]. A pressure study up to 6 GPa shows
that the superconducting transition temperature Tcis sup-
pressed to about 28.5 K before it undergoes a half-collapsed-tetragonal (hct) phase transition at ∼4 GPa and loses bulk
superconductivity [ 20]. The hct phase transition occurs due
*ives@iastate.edu
†canfield@ameslab.govto the As-As bonding across the Ca layer under pressure, like
the collapsed-tetragonal transition in CaFe 2As2at∼0.35 GPa
[21–23].
From the perspective of electron count, CaKFe 4As4is
analogous to (Ba 0.5K0.5)Fe 2As2, and their many properties
are similar with each other [ 18]. In the latter compound, the
stripe-type spin density wave associated with BaFe 2As2is
suppressed by hole doping [ 7] (substituting K for Ba). A recent
study revealed that adding electrons to CaKFe 4As4via Ni or
Co substitution drives the system back towards a magnetic
phase. In contrast to the stripe-type antiferromagnetism in
the “122” systems, the order in the Ni- or Co-substituted
CaKFe 4As4is experimentally identified as a new hedgehog
spin-vortex-crystal (SVC) magnetism that has no structuralphase transition associated with it [ 24]. This type of magnetic
order had been theoretically predicted but, until the discovery
of Ni- or Co-substituted CaKFe
4As4, was considered to be a
“missing link” [ 25–27]. Increasing the substitution level of Ni
or Co in CaK(Fe 1−xTx)4As4leads to the suppression of the
superconducting transition temperature Tcand stabilizing the
SVC magnetism and increasing TN[24].
The application of pressure to Ba(Fe 1−xCox)2As2sup-
presses antiferromagnetism ( TNfalls) and increases Tc[28].
This has been taken as an indication that pressure, like doping,can tune T
Nand the associated antiferromagnetic (AFM)
fluctuations to favor the superconducting state when TN>T c.
Therefore, it is natural to study how the SVC magnetic orderbehaves under pressure, specifically, how the magnetism andsuperconductivity interact in this system and whether thisinteraction is similar to Ba(Fe
1−xCox)2As2.
In this work, we present the first pressure study on Ni-
substituted CaK(Fe 1−xNix)4As4(x=0.033 and 0.050) up to
5.12 GPa. The pressure-temperature ( p-T) phase diagrams
inferred from resistance measurements allow comparison ofT
N(p) andTc(p). Specifically, p-Tphase diagrams reveal that
2469-9950/2018/97(17)/174517(8) 174517-1 ©2018 American Physical SocietyLI XIANG et al. PHYSICAL REVIEW B 97, 174517 (2018)
0 100 200 3000.10.2
0
18 20 220.04
020 40 60 80R(Ω)
T(K)0G P a
0.070.43
0.82
1.341.54
1.83CaK(Fe1-xNix)4As4
x=0.033
Sample#1
TN
(a)
Tonset
c
(b)R(Ω)
T(K)Toffset
c (c)
dR/dT(a.u.)
T(K)TN
FIG. 1. (a) Evolution of the in-plane resistance with hy-
drostatic pressures up to 1.83 GPa measured in a PCC forCaK(Fe
0.967Ni0.033)4As4, sample 1. (b) Blowup of the low-
temperature region. Criteria for Tonset
c andToffset
c are indicated.
(c) Temperature derivative dR/dT , showing the evolution of the
magnetic transition TNwith offset criteria as shown.
TNis suppressed with pressure for both substitution levels.
In contrast to Ba(Fe 1−xCox)2As2,Tcis suppressed as well,
although more slowly. For x=0.050, it exhibits an anomaly
at the pressure where TcandTNcross. At ∼4 GPa both com-
positions appear to undergo the hct transition, as was observedin the undoped CaKFe
4As4. Furthermore, superconducting
upper critical fields studied up to 9 T suggest a Fermi-surfacereconstruction when T
N(p) crosses Tc(p).
II. EXPERIMENTAL DETAILS
Single crystals of CaK(Fe 1−xNix)4As4(x=0.033 and
0.050) with sharp superconducting transitions at ambient pres-sure [see Figs 1(b)–3(b)] were grown using high-temperature
solution growth [ 18,19]. The substitution level xwas de-
termined by performing wavelength-dispersive x-ray spec-troscopy as described in Ref. [ 24].
The in-plane abresistance was measured using the standard
four-probe configuration. The 25- μm Pt wires were soldered
to the samples using a Sn:Pb 60:40 alloy. For x=0.033, two
samples, 1 and 2, were cut from one single crystal. Theywere then measured in a piston-cylinder cell (PCC) [ 29] and a
modified Bridgman anvil cell (MBAC) [ 30], respectively. For
x=0.050, a single sample was prepared and measured in the
MBAC. Pressure values for both cells, at low temperature, wereinferred from the T
c(p) of lead [ 31]. For the PCC, a 4:6 mixture
of light mineral oil: n-pentane was used as the pressure medium,0 100 200 3000.51.01.5
0
01 0 2 0 3 00.4
020 40 60 803.71
4.014.254.634.86
5.120G P a
1.942.332.572.712.993.28
(a)CaK(Fe1-xNix)4As4
x=0.033
Sample#2R(Ω)
T(K)TN
Tonset
c
(b)R(Ω)
T(K)Toffset
c (c)
dR/dT(a.u.)
T(K)TN
FIG. 2. (a) Evolution of the in-plane resistance with hy-
drostatic pressures up to 5.12 GPa measured in a MBAC
for CaK(Fe 0.967Ni0.033)4As4, sample 2. (b) Blowup of the low-
temperature region. (c) Temperature derivative dR/dT , showing the
evolution of the magnetic transition TN.
which solidifies, at room temperature, in the range of 3–4 GPa.
For the MBAC, a 1:1 mixture of isopentane: n-pentane was used
as the pressure medium, which solidifies, at room temperature,in the range of 6–7 GPa. Both of the solidification pressuresare well above the maximum pressures achieved in the pressurecells, which suggests good hydrostatic conditions [ 29,32,33].
The ac resistance measurements were performed in a Quan-
tum Design physical property measurement system (PPMS)usingI=1 mA; f=17 Hz excitation, on cooling with a rate
of 0.25 K /min, and the magnetic field was applied along the c
axis.
III. RESULTS AND DISCUSSION
Figures 1(a) and 2(a) show the pressure dependence of
the temperature-dependent resistance for CaK(Fe 1−xNix)4As4,
x=0.033. Sample 1 was measured in the PCC for pressures
up to 1.83 GPa. Sample 2 was measured in the MBACfor pressures up to 5.12 GPa. For both samples, the 0-GParesistance was corrected for geometric changes to the samplevia normalization. (Details of the normalization are describedin the Appendix.) Figure 3(a)shows the pressure dependence
of the temperature-dependent resistance for the x=0.050
sample that was measured in the MBAC for pressures up to5.12 GPa. In general, for all samples, the resistance decreasesunder applied pressure.
For both compositions, the magnetic phase transition T
N
appears as a kinklike anomaly in the lower-temperature data
and is more pronounced in the x=0.050 compound. This
174517-2PRESSURE-TEMPERATURE PHASE DIAGRAMS OF CaK(Fe … PHYSICAL REVIEW B 97, 174517 (2018)
0 100 200 3000.10.20.3
0
0 5 10 15 200.1
020 40 60 80(a)CaK(Fe1-xNix)4As4
x=0.05R(Ω)
T(K)0G P a
1.582.112.412.923.11
3.31
3.533.864.045.12
TN
Tonset
c
(b)R(Ω)
T(K)Toffset
c(c)
dR/dT(a.u.)
T(K)TN
FIG. 3. (a) Evolution of the in-plane resistance with hydro-
static pressures up to 5.12 GPa measured in a MBAC forCaK(Fe
0.95Ni0.05)4As4. (b) Blowup of the low-temperature region.
(c) Temperature derivative dR/dT , showing the evolution of the
magnetic transition TN.
feature is more clearly revealed as a steplike anomaly in the
temperature derivative dR/dT [Figs. 1(c),2(c) and 3(c)].
These plots demonstrate that TNis suppressed by increasing
pressure before it disappears at higher pressures.
The blowups of the low-temperature resistance [Figs. 1(b),
2(b)and3(b)]s h o wh o w Tcchanges under increasing pressure.
Forx=0.033,Tcmonotonically decreases in the studied
pressure range. In contrast, for x=0.050, after 2.41 GPa there
is a slight enhancement of Tcbefore it is suppressed again at
higher pressures.
Upon increasing pressure above ∼4 GPa, the sharp super-
conducting transition at lower pressures becomes broadened athigher pressures. A similar behavior was also observed in theparent compound CaKFe
4As4and has been associated with the
hct phase transition at p/greaterorsimilar4G P a[ 20]. In order to understand
the nature of the broadening in the substituted system, ananalysis similar to that in Ref. [ 20] was carried out.
Figure 4presents the temperature dependence of the resis-
tance under magnetic field up to 9 T for selected pressures.The superconducting transition width, /Delta1T=T
onset
c−Toffset
c,
is broadened with increasing pressure, with the criteria forT
onset
c andToffset
c shown in Figs. 1(b),2(b) and3(b). In order to
determine whether the broadening is associated with any sort ofphase transition or is simply due to pressure inhomogeneitiesin the pressure medium when larger loads are applied, the fielddependence of the superconducting transition width /Delta1T(H)
was studied [ 20]. Specifically, the transition widths at magnetic
fields of 0 and 3 T (indicated by thicker lines in Figs. 4)10 15 200.250.50
010 15 20 05 1 0 1 5 2 0
05 1 00.060.12
0
05 1 0 05 1 0 1 5CaK(Fe1-xNix)4As4
x=0.033R(Ω)
T(K)0G P a
(a) (b)
T(K)1.94 GPa
0T9T
(c)
T(K)4.86 GPa
(d)R(Ω)
T(K)0G P ax=0.05
(e)
T(K)2.11 GPa
(f)
T(K)4.04 GPa
FIG. 4. Temperature dependence of resistance under magnetic
field up to 9 T for selected pressures for CaK(Fe 1−xNix)4As4, with
(a)–(c) x=0.033 and (d)–(f) x=0.050. The superconducting tran-
sition becomes broader as pressure is increased for both compounds;
to explore the nature of the broadening, transition widths at 0 and
3 T (indicated by thick lines) were analyzed and ae described in detailin the text.
were determined, and then the difference between them,
/Delta1T(3 T)−/Delta1T(0), was calculated. Any broadening due to the
pressure inhomogeneities is expected to be equally presentin the H=0 and 3 T data. Figures 5(a) and 5(c) present
the pressure dependence of the transition width difference.As is clearly seen, for both compositions, /Delta1T(3 T)−/Delta1T(0)
increases dramatically as pressure goes above p
∗∼4G P a
[indicated by arrows in Figs. 5(a)and5(c)]. Note that for x=
0.050, at 5.12 GPa, the transition width difference was taken
between H=0 and 1 T because Toffset
c is not clearly defined at
H=3 T. But we would expect the transition width difference
between H=0 and 3 T to be even larger at this pressure.
Furthermore, the pressure dependence of the resistance R(p)a t
fixed temperatures for both compositions [Figs. 5(b) and5(d)]
shows an anomaly at the same pressure at 40 K (indicatedby arrows), although it is subtle for x=0.033. Based on
the analogy with the parent compound CaKFe
4As4[20], we
identify this anomaly as an indication of the hct phase transitionthat exists from base temperature up to at least 40 K. As was thecase for pure CaKFe
4As4, we believe that superconductivity is
not bulk for p/greaterorsimilar4 GPa (i.e., in the hct phase).
The upper superconducting critical field Hc2can be evalu-
ated from Fig. 4at pressures lower than p∗, where supercon-
ductivity is considered bulk, using the offset criteria definedin Figs. 1–3. The temperature dependence of H
c2at various
pressures is presented in Figs. 6and7for CaK(Fe 1−xNix)4As4,
withx=0.033 and 0.050, respectively. For x=0.033, both
samples 1 and 2 are analyzed and plotted in Fig 6. Note that at
ambient pressure, Toffset
c values for the two samples differ by
∼0.5 K, possibly due to a small difference in the substitution
level at the different positions of the crystal they were cut from.As shown in Figs. 6and7,f o rx=0.033,H
c2is systematically
suppressed by increasing pressure, whereas, for x=0.050, the
evolution of the temperature-dependent Hc2is nonmonotonic.
174517-3LI XIANG et al. PHYSICAL REVIEW B 97, 174517 (2018)
0123
0123
01234560.511.5
012345670.10.20.3Sample#1
Sample#2CaK(Fe1-xNix)4As4
x=0.033T(3T)- T(0) (K)
(a) p*
T(3T)- T(0) (K)x=0.050
(c)
(b)Sample#2R(Ω)
p(GPa)406080100120160200300K
250
(d)
R(Ω)
p(GPa)406080100120160200300K
250
FIG. 5. (a), (c) Pressure dependence of the superconducting
transition width difference for CaK(Fe 1−xNix)4As4, with x=0.033
and 0.050, respectively. The superconducting transition width is
/Delta1T=Tonset
c−Toffset
c, and the width difference is taken between zero
field and 3 T. The open symbol in (c) is the width difference taken
between zero field and 1 T because of the lack of clear definition of
Toffset
c at 3 T for 5.12 GPa. (b), (d) Pressure dependence of resistance
atR(p) fixed temperatures for CaK(Fe 1−xNix)4As4, with x=0.033
and 0.050, respectively. The critical pressure p∗(arrows) which is
associated with the hct phase is described in detail in the text.
For both compositions, Hc2is linear in temperature except for
magnetic fields below 1 T. The curvature at low fields has beenobserved in other FeSC and can be explained by the nature
0 5 10 15 20 250246810
Sample #1
0G P a
1.34
1.83Sample #2
0G P a
1.94
2.33
2.71
3.28
3.71
4.01
CaK(Fe1-xNix)4As4
x=0.0330Hc2(T)
T(K)
FIG. 6. Temperature dependence of the upper superconducting
critical field Hc2(T) under selected pressures for CaK(Fe 1−xNix)4As4,
withx=0.033.Toffset
c is used. Half-filled and solid symbols are two
samples measured in the PCC and MBAC, respectively.02468 1 0 1 20246810
0G P a
1.58
2.41
2.92
3.31
3.53
3.86
CaK(Fe1-xNix)4As4
x=0.050Hc2(T)
T(K)
FIG. 7. Temperature dependence of the upper superconducting
critical field Hc2(T) under selected pressures for CaK(Fe 1−xNix)4As4,
withx=0.050.Toffset
c is used.
of superconductivity [ 34–36], which is also the case for the
parent compound CaKFe 4As4[37].
Figures 8(a) and9(a) present the p-Tphase diagrams for
CaK(Fe 1−xNix)4As4, with x=0.033 and 0.050, respectively,
withToffset
c andTNvalues obtained using the criteria shown
in Figs. 1–3and the indication of nonbulk superconductivity
above p∗. For both compositions, TNis suppressed by pressure;
specifically, TNis suppressed from 43 to 25 K at 2.71 GPa for
x=0.033 and suppressed from 51 to 13.8 K at 3.31 GPa for
x=0.050.
In terms of superconductivity, for x=0.033,Toffset
c is
monotonically suppressed with increasing pressure. It dropsfrom 20.5 to 15.1 K at 4.01 GPa before superconductivitybecomes nonbulk. A closer examination reveals that T
offset
c is
initially linearly suppressed by pressure up to 2.71 GPa; thena small, but clear, deviation from the linear suppression wasobserved above 2.99 GPa. An extrapolation of T
Nshows that
the deviation happens near the crossing of the TNandToffset
c
lines. For x=0.050, the behavior of Toffset
c(p) is distinctly
nonmonotonic. Toffset
c is initially linearly suppressed from 11 K
to a local minimum of 8.7 K at 2.41 GPa. Then it rises to amaximum of 10 K at 3.31 GPa, exhibiting a dome shape. Thisdome of enhanced T
offset
c coincides with the disappearance of
TN. After the local maximum in Toffset
c there is a much more
rapid suppression of Toffset
c with increasing puntil the hct
transition at p∗. For both compositions, a change in Toffset
c(p)
happens at the pressure where the TNandToffset
c lines cross.
Both compositions show signatures of nonbulk supercon-
ductivity above p∗∼4 GPa [blue symbols in Figs. 8(a) and
9(a)] similar to that of the parent compound CaKFe 4As4[20],
suggesting the same hct phase transition. Pressure-dependentresistance data in Fig. 5demonstrate that the hct phase
transition is discernible up to at least 40 K for the substitutedcompounds. The transition pressure does not appear to changewith Ni substitution. This is not too surprising given the factthat the hct transition does not involve the Fe plane but is,instead, As-As bonding across the Ca plane.
To better understand the superconducting properties of
CaK(Fe
1−xNix)4As4, the superconducting upper critical field
Hc2was analyzed following Refs. [ 35,36,38]. Generally
174517-4PRESSURE-TEMPERATURE PHASE DIAGRAMS OF CaK(Fe … PHYSICAL REVIEW B 97, 174517 (2018)
FIG. 8. (a) Temperature-pressure phase diagram of
CaK(Fe 1−xNix)4As4, with x=0.033, as determined from resistance
measurement. The squares and circles represent the superconducting
Toffset
c and magnetic TNphase transition. Half-filled and solid symbols
are two samples measured in the PCC and the MBAC, respectively.
Blue symbols represent Toffset
c for filamentary superconductivity.
Dashed lines are guides to the eye. The blue dotted line indicatesthe half-collapsed-tetragonal phase transition up to 40 K, inferred
from the pressure-dependent resistance R(p) data in Fig. 5.
(b) Pressure dependence of the normalized upper critical fieldslope−(1/T
c)(dμoHc2/dT)|Tc. A local minimum in the slope at pc
(indicated by the arrow) is observed near the pressure where the
Toffset
c andTNlines cross.
speaking, the slope of the upper critical field normalized by
Tcis related to the Fermi velocity and superconducting gap of
the system [ 34]. In the clean limit, for a single band,
−(1/Tc)(dμoHc2/dT)|Tc∝1/v2
F, (1)
where vFis the Fermi velocity. Even though the superconduc-
tivity in CaKFe 4As4compounds is multiband, Eq. ( 1) can give
qualitative insight into changes induced by pressure.
As shown in Figs. 8(b) and 9(b), the normalized slope
of the upper critical field −(1/Tc)(dμoHc2/dT)|Tc(the slope
dμoHc2/dT|Tcis calculated by linearly fitting the data from 1
to 5 T in Figs. 6and7) exhibits a similar pressure dependence
forx=0.033 and 0.050. It initially decreases upon increasing
pressure and then begins to increase above pressure pc, result-
ing in a minimum of −(1/Tc)(dμoHc2/dT)|Tcin the studied
pressure range. In both compositions, pccoincides with the
crossing of the TNandToffset
c lines, suggesting a common origin
of this feature.
In Fe-based superconductors, especially the 122 system,
Fermi-surface nesting can lead to a partial opening of agap at the Fermi surface below T
N. By tuning with dop-
ing or applying pressure, a Fermi-surface reconstructionFIG. 9. (a) Temperature-pressure phase diagram of
CaK(Fe 1−xNix)4As4, with x=0.050, as determined from resistance
measurement. The squares and circles represent the superconducting
Toffset
c and magnetic TNphase transition. Blue symbols represent
Toffset
c for filamentary superconductivity. Dashed lines are guides to
the eye. The blue dotted line indicates the half-collapsed-tetragonal
phase transition up to 40 K, inferred from the pressure-dependent
resistance R(p) data in Fig. 5. (b) Pressure dependence of the
normalized upper critical field slope −(1/Tc)(dμoHc2/dT)|Tc.A
local minimum in the slope at pc(indicated by the arrow) is observed
near the pressure where the Toffset
c andTNlines cross.
could happen due to the disappearance of magnetism
[39–47]. For CaK(Fe 1−xNix)4As4(x=0.033 and 0.050), a
clear change in the pressure dependence of the normalizedslope−(1/T
c)(dμoHc2/dT)|Tcis observed at pc, indicating
a possible Fermi-surface reconstruction near pc. Note that
forx=0.050, there appears to be a discontinuous change
in the normalized slope −(1/Tc)(dμoHc2/dT)|Tcand a subtle
anomaly in Tc(p) from 2.41 to 2.92 GP, suggesting there may
be a Liftshiz transition near this pressure. Such features are notobserved for x=0.033.
Figures 8and 9, then, combine surprising and not un-
expected features. The hct phase transition pressure appearsinsensitive to Ni substitution. This is reasonable because thistransition involves bonding of As atoms across the Ca plane.The clear feature at p
cin−(1/Tc)(dμoHc2/dT)|Tc,a sw e l l
as the subtler features in Tc(p), is again not too surprising
and can be associated with the change (with increasing p)
fromTN>T ctoTN<T c, i.e., from Tcoccurring in an AFM
ordered state to Tcoccurring in a state lacking the AFM
order and associated additional periodicities. The surprisingfeature shown in Figs. 8and9is the weak suppression of T
c
concurrent with the strong suppression of TN. This is contrary
to what is seen in a Co substitution and pressure study ofBaFe
2As2(where Tcincreases as TNis suppressed) [ 5,6,11,28]
174517-5LI XIANG et al. PHYSICAL REVIEW B 97, 174517 (2018)
0.10.20.30.4
0
0 100 200 3000.51.01.5
00G P a
0.070.43
0.82
1.341.541.83CaK(Fe1-xNix)4As4
x=0.033
Sample#1R(Ω)
(a)
(b)R(Ω)Sample#2
T(K)0G P a
1.942.332.57
2.71
2.993.28
3.71
4.01 4.25
4.63
4.86
5.12
FIG. 10. Evolution of the in-plane resistance with the hydrostatic
pressure of (a) sample 1 measured in a PCC and (b) sample 2 measured
in a MBAC for CaK(Fe 1−xNix)4As4, with x=0.033. Solid lines are
the actual resistance data measured; dashed lines are the normalized
resistance for 0 GPa. Notice that the 0-GPa resistance is measured on
a PPMS puck outside of either pressure cell (i.e., ambient pressure); inboth cases there is a sudden change between the resistance measured
at ambient pressure and inside the pressure cell. Possible reasons for
the sudden change and details of normalization are explained in detailin the text.
and brings into question the exact effects suppression of TNhas
on the magnetic fluctuations that the superconducting state isnominally built out of.
IV . CONCLUSION
In conclusion, the resistance of the Ni-substituted iron-
based superconductor CaK(Fe 1−xNix)4As4(x=0.033 and
0.050) has been studied under pressures up to 5.12 GPaand in magnetic fields up to 9 T. For both substitutionlevels, the hedgehog spin-vortex-crystal magnetic transitiontemperature T
Nis suppressed with increasing pressure. In both
compositions, Tcis initially suppressed as well and exhibits
a weak anomaly near the crossing of the TNandTclines. As
pressure exceeds ∼4 GPa, both compositions likely go through
the half-collapsed-tetragonal phase transition, similar to theone observed in the parent compound. This demonstrates theinsensitivity of the hct transition pressure to Ni substitution.The minimum observed in the normalized slope of the uppercritical field, −(1/T
c)(dμoHc2/dT)|Tc, at the pressure where
theTNandTclines cross indicates a possible Fermi-surface
01234560.050.10
0Sample#2
Sample#1CaK(Fe1-xNix)4As4
x=0.033
T=6 0K
p(GPa)R(Ω)
0.20.40.6
0
R(Ω)
FIG. 11. Pressure dependence of resistance at 60 K for
CaK(Fe 1−xNix)4As4,w i t h x=0.033; black solid squares are data
from sample 1 measured in the PCC, and red solid circles are data
from sample 2 measured in the MBAC. Dashed lines are linear fittingof the data before 4 GPa (not including 0 GPa); notice the clear
deviation from the linear fitting for the 0-GPa data. Open symbols
are the corresponding normalized 0-GPa resistance for samples 1 and2a t6 0K .
reconstruction associated with the disappearance of antiferro-
magnetism.
ACKNOWLEDGMENTS
We would like to thank A. Kreyssig for useful discussions.
This work is supported by the U.S. DOE, Basic EnergySciences, Materials Science and Engineering Division underContract No. DE-AC02-07CH11358. L.X. was supported, inpart, by the W. M. Keck Foundation. W.R.M. was supportedby the Gordon and Betty Moore Foundation’s EPiQS Initiativethrough Grant No. GBMF4411.
APPENDIX
Figure 10presents the evolution of the in-plane resistance
with hydrostatic pressure for CaK(Fe 1−xNix)4As4,x=0.033;
solid lines are the actual measured resistance data, and dashedlines are the resistance after normalization. Sample 1 wasmeasured in a PCC for pressures up to 1.83 GPa, and sample2 was measured in a MBAC for pressures up to 5.12 GPa.Note that the 0-GPa resistance data were measured on a PPMSpuck outside of either pressure cell (i.e., ambient pressure);a sudden change in resistance between the ambient pressureand inside the pressure cell was observed in both samples. Forsample 1, when the sample was moved from the PPMS puckand mounted onto the PCC, one contact of the voltage channelbecame detached from the sample and that contact had to be re-attached. As a result, the changed position of the contact led tochanges in the resistance before and after the move. For sample2, nothing was intentionally done to the sample before andafter it was put into the MBAC, and the sudden change in theresistance is most likely due to the exfoliation or cracking of thesample when pressure was first applied as the pressure cell wasclosed. Despite the abrupt change in resistance from ambientpressure to the first finite pressures inside the pressure cell,the resistance of CaK(Fe
1−xNix)4As4(x=0.033 and 0.05)
continuously and systematically decreases upon increasing
174517-6PRESSURE-TEMPERATURE PHASE DIAGRAMS OF CaK(Fe … PHYSICAL REVIEW B 97, 174517 (2018)
pressure, consistent with the behavior that is observed in the
parent compound CaKFe 4As4[24] and many 122 systems
[38,48,49].
To better evaluate the resistance evolution with pressure,
especially the pressure dependence of resistance at varioustemperatures [Figs. 5(b) and5(d)], the ambient-pressure re-
sistance is shifted via normalization (assuming in each casethat the shift was due to geometric changes). Figure 11presents
the pressure dependence of the resistance at T=60 K for
samples 1 and 2 (solid symbols). Note that T=60 K was
chosen because the pressure values are determined from theT
c(p) of lead [ 31]a t∼7 K, and the pressure cells are known
to have pressure changes with temperature. With the pressurecells and liquid medium we used in this study, the pressurechange from room temperature to 7 K can be 0.2–0.3 GPa[30,50]. The value of 60 K was chosen based on the idea that
at this temperature, the pressure medium has already solidified[33], the temperature dependence of the thermal expansion
of cell materials flattens at low temperature, and the pressuredifference between 60 and 7 K should be small [ 50]. The fact
that 60 K is still above the magnetic transition temperatureT
Nguarantees that the pressure dependence of the resistance
at this temperature has no feature related to magnetism. Ass h o w ni nF i g . 11, except for the ambient-pressure data, the 60 K
resistance for both samples is linearly suppressed by pressurebefore 4 GPa, so it is assumed that the ambient-pressureresistance should also follow this pressure dependence (opensymbols in Fig. 11). To do that, the ambient-pressure resistance
curves for the two samples are multiplied by two correspondingfactors and moved to the dashed lines shown in Fig. 10.
[1] Y . Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.
Chem. Soc. 130,3296 (2008 ).
[2] Z. A. Ren, G. C. Che, X. L. Dong, J. Yang, W. Lu, W. Yi, X. L.
Shen, Z. C. Li, L. L. Sun, F. Zhou, and Z. X. Zhao, Europhys.
Lett.83,17002 (2008 ).
[3] M. Rotter, M. Tegel, and D. Johrendt, P h y s .R e v .L e t t . 101,
107006 (2008 ).
[4] H. Takahashi, K. Igawa, K. Arii, Y . Kamihara, M. Hirano, and
H. Hosono, Nature (London) 453,376(2008 ).
[ 5 ]P .C .C a n fi e l da n dS .L .B u d ’ k o , Annu. Rev. Condens. Matter
Phys. 1,27(2010 ).
[6] N. Ni and S. L. Bud’ko, MRS Bull. 36,620(2011 ).
[7] J. Paglione and R. L. Greene, Nat. Phys. 6,645(2010 ).
[8] M. S. Torikachvili, S. L. Bud’ko, N. Ni, and P. C. Canfield, Phys.
Rev. B 78,104527 (2008 ).
[9] P. L. Alireza, Y . T. C. Ko, J. Gillett, C. M. Petrone, J. M. Cole,
G. G. Lonzarich, and S. E. Sebastian, J. Phys.: Condens. Matter
21,012208 (2009 ).
[10] S. A. J. Kimber, A. Kreyssig, Y .-Z. Zhang, H. O. Jeschke, R.
Valentí, F. Yokaichiya, E. Colombier, J. Yan, T. C. Hansen, T.Chatterji, R. J. McQueeney, P. C. Canfield, A. I. Goldman, andD. N. Argyriou, Nat. Mater. 8,471(2009 ).
[11] E. Colombier, S. L. Bud’ko, N. Ni, and P. C. Canfield, Phys.
Rev. B 79,224518 (2009 ).
[12] D. K. Pratt, W. Tian, A. Kreyssig, J. L. Zarestky, S. Nandi,
N .N i ,S .L .B u d ’ k o ,P .C .C a n fi e l d ,A .I .G o l d m a n ,a n dR .J .McQueeney, P h y s .R e v .L e t t . 103
,087001 (2009 ).
[13] A. D. Christianson, M. D. Lumsden, S. E. Nagler, G. J. Mac-
Dougall, M. A. McGuire, A. S. Sefat, R. Jin, B. C. Sales, and D.Mandrus, Phys. Rev. Lett. 103,087002 (2009 ).
[14] R. M. Fernandes, D. K. Pratt, W. Tian, J. Zarestky, A. Kreyssig,
S .N a n d i ,M .G .K i m ,A .T h a l e r ,N .N i ,P .C .C a n fi e l d ,R .J .McQueeney, J. Schmalian, and A. I. Goldman, P h y s .R e v .B 81,
140501 (2010 ).
[15] A. D. Christianson, E. A. Goremychkin, R. Osborn, S.
Rosenkranz, M. D. Lumsden, C. D. Malliakas, I. S. Todorov,H. Claus, D. Y . Chung, M. G. Kanatzidis, R. I. Bewley, and T.Guidi, Nature (London) 456,930(2008 ).
[16] G. Yu, Y . Li, E. M. Motoyama, and M. Greven, Nat. Phys. 5,873
(2009 ).
[17] A. Iyo, K. Kawashima, T. Kinjo, T. Nishio, S. Ishida, H. Fujihisa,
Y . Gotoh, K. Kihou, H. Eisaki, and Y . Yoshida, J. Am. Chem.
Soc.138,3410 (2016 ).[18] W. R. Meier, T. Kong, U. S. Kaluarachchi, V . Taufour, N. H.
Jo, G. Drachuck, A. E. Böhmer, S. M. Saunders, A. Sapkota,A. Kreyssig, M. A. Tanatar, R. Prozorov, A. I. Goldman, F. F.Balakirev, A. Gurevich, S. L. Bud’ko, and P. C. Canfield, Phys.
Rev. B 94,064501 (2016 ).
[19] W. R. Meier, T. Kong, S. L. Bud’ko, and P. C. Canfield, Phys.
Rev. Mater. 1,013401 (2017 ).
[20] U. S. Kaluarachchi, V . Taufour, A. Sapkota, V . Borisov, T. Kong,
W. R. Meier, K. Kothapalli, B. G. Ueland, A. Kreyssig, R.Valentí, R. J. McQueeney, A. I. Goldman, S. L. Bud’ko, andP. C. Canfield, P h y s .R e v .B 96,140501 (2017 ).
[21] M. S. Torikachvili, S. L. Bud’ko, N. Ni, and P. C. Canfield, Phys.
Rev. Lett. 101,057006 (2008 ).
[22] W. Yu, A. A. Aczel, T. J. Williams, S. L. Bud’ko, N. Ni, P. C.
Canfield, and G. M. Luke, Phys. Rev. B 79,020511 (2009 ).
[23] A. Kreyssig, M. A. Green, Y . Lee, G. D. Samolyuk, P. Zajdel,
J. W. Lynn, S. L. Bud’ko, M. S. Torikachvili, N. Ni, S. Nandi,J. B. Leão, S. J. Poulton, D. N. Argyriou, B. N. Harmon, R. J.McQueeney, P. C. Canfield, and A. I. Goldman, Phys. Rev. B
78,
184517 (2008 ).
[24] W. R. Meier, Q.-P. Ding, A. Kreyssig, S. L. Bud’ko, A. Sapkota,
K. Kothapalli, V . Borisov, R. Valentí, C. D. Batista, P. P. Orth,R. M. Fernandes, A. I. Goldman, Y . Furukawa, A. E. Böhmer,a n dP .C .C a n fi e l d , npj Quantum Mater. 3,5(2018 ).
[25] R. M. Fernandes, S. A. Kivelson, and E. Berg, P h y s .R e v .B 93,
014511 (2016 ).
[26] V . Cvetkovic and O. Vafek, Phys. Rev. B 88,134510 (2013 ).
[27] J. O’Halloran, D. F. Agterberg, M. X. Chen, and M. Weinert,
Phys. Rev. B 95,075104 (2017 ).
[28] E. Colombier, M. S. Torikachvili, N. Ni, A. Thaler, S. L.
Bud’ko, and P. C. Canfield, Supercond. Sci. Technol. 23,054003
(2010 ).
[29] S. L. Bud’ko, A. N. V oronovskii, A. G. Gapotchenko, and
E. S. ltskevich, Zh. Eksp. Teor. Fiz. 86, 778 (1984) [Sov. Phys.
JETP 59, 454 (1984)].
[30] E. Colombier and D. Braithwaite, Rev. Sci. Instrum. 78,093903
(2007 ).
[31] B. Bireckoven and J. Wittig, J. Phys. E 21,841(1988 ).
[32] S. K. Kim, M. S. Torikachvili, E. Colombier, A. Thaler,
S. L. Bud’ko, and P. C. Canfield, P h y s .R e v .B 84,134525
(2011 ).
[33] M. S. Torikachvili, S. K. Kim, E. Colombier, S. L. Bud’ko, and
P. C. Canfield, Rev. Sci. Instrum. 86,123904 (2015 ).
174517-7LI XIANG et al. PHYSICAL REVIEW B 97, 174517 (2018)
[34] V . G. Kogan and R. Prozorov, Rep. Prog. Phys. 75,114502
(2012 ).
[35] U. S. Kaluarachchi, V . Taufour, A. E. Böhmer, M. A. Tanatar,
S .L .B u d ’ k o ,V .G .K o g a n ,R .P r o z o r o v ,a n dP .C .C a n fi e l d , Phys.
Rev. B 93,064503 (2016 ).
[36] L. Xiang, U. S. Kaluarachchi, A. E. Böhmer, V . Taufour, M. A.
Tanatar, R. Prozorov, S. L. Bud’ko, and P. C. Canfield, Phys.
Rev. B 96,024511 (2017 ).
[37] D. Mou, T. Kong, W. R. Meier, F. Lochner, L.-L. Wang, Q. Lin,
Y . Wu, S. L. Bud’ko, I. Eremin, D. D. Johnson, P. C. Canfield,and A. Kaminski, Phys. Rev. Lett. 117,277001 (2016 ).
[38] V . Taufour, N. Foroozani, M. A. Tanatar, J. Lim, U. Kalu-
arachchi, S. K. Kim, Y . Liu, T. A. Lograsso, V . G. Kogan, R.Prozorov, S. L. Bud’ko, J. S. Schilling, and P. C. Canfield, Phys.
Rev. B 89,220509 (2014 ).
[39] S. Jiang, H. Xing, G. Xuan, C. Wang, Z. Ren, C. Feng, J. Dai, Z.
Xu, and G. Cao, J. Phys. Condens. Matter 21,382203 (2009 ).
[40] J. Dai, Q. Si, J.-X. Zhu, and E. Abrahams, Proc. Natl. Acad. Sci.
USA 106,4118 (2009 ).
[41] M. Gooch, B. Lv, B. Lorenz, A. M. Guloy, and C.-W. Chu, Phys.
Rev. B 79,104504 (2009 ).
[42] M. Gooch, B. Lv, B. Lorenz, A. M. Guloy, and C. W. Chu,
J. Appl. Phys. 107,09E145 (2010 ).[43] J. Maiwald, H. S. Jeevan, and P. Gegenwart, Phys. Rev. B 85,
024511 (2012 ).
[44] S. Arsenijevi ć, H. Hodovanets, R. Gaál, L. Forró, S. L. Bud’ko,
a n dP .C .C a n fi e l d , Phys. Rev. B 87,224508 (2013 ).
[45] C. Liu, T. Kondo, N. Ni, A. D. Palczewski, A. Bostwick, G. D.
Samolyuk, R. Khasanov, M. Shi, E. Rotenberg, S. L. Bud’ko,P. C. Canfield, and A. Kaminski, P h y s .R e v .L e t t . 102,167004
(2009 ).
[46] G. Liu, H. Liu, L. Zhao, W. Zhang, X. Jia, J. Meng, X. Dong, J.
Zhang, G. F. Chen, G. Wang, Y . Zhou, Y . Zhu, X. Wang, Z. Xu,C. Chen, and X. J. Zhou, Phys. Rev. B 80,134519 (2009 ).
[47] R. S. Dhaka, S. E. Hahn, E. Razzoli, R. Jiang, M. Shi, B. N.
Harmon, A. Thaler, S. L. Bud’ko, P. C. Canfield, and A.Kaminski, P h y s .R e v .L e t t . 110,067002 (2013 ).
[48] E. Hassinger, G. Gredat, F. Valade, S. R. de Cotret, A. Juneau-
Fecteau, J.-P. Reid, H. Kim, M. A. Tanatar, R. Prozorov, B. Shen,H.-H. Wen, N. Doiron-Leyraud, and L. Taillefer, Phys. Rev. B
86,140502 (2012 ).
[49] E. Hassinger, G. Gredat, F. Valade, S. R. de Cotret, O. Cyr-
Choinière, A. Juneau-Fecteau, J.-P. Reid, H. Kim, M. A. Tanatar,R. Prozorov, B. Shen, H.-H. Wen, N. Doiron-Leyraud, and L.Taillefer, P h y s .R e v .B 93,144401 (2016 ).
[50] J. D. Thompson, Rev. Sci. Instrum. 55,231(1984 ).
174517-8 |
PhysRevB.99.245404.pdf | PHYSICAL REVIEW B 99, 245404 (2019)
Molecular spintronics using single-molecule magnets under irradiation
Kieran Hymas and Alessandro Soncini*
School of Chemistry, University of Melbourne, Parkville, Victoria 3010, Australia
(Received 21 May 2018; revised manuscript received 6 May 2019; published 10 June 2019)
We theoretically investigate a single-molecule magnet (SMM) grafted to a quantum dot in contact with
metallic leads and interacting with a resonant electromagnetic radiation. We explore both the explicit time-dependent behavior and the steady-state current-voltage characteristics of the device when the source lead isferromagnetic. At zero-bias voltage, a net current is pumped through the device with the source spin currentbeing reversed and amplified in the drain lead; this effect also persists for nonzero bias. We explain this effectin terms of spin transitions in the nanomagnet induced by the resonant radiation followed by their subsequentrelaxation via spin-asymmetric charge-transfer processes. We demonstrate that the same effects are recovered inthe time-averaged current when the device interacts with pulsed resonant radiation. Moreover, within the pulsedirradiation regime, appropriate choices of pulse length and wait times are shown here to allow the detection ofcoherent Rabi oscillations of the SMM spin states, via time-averaged spin current measurements.
DOI: 10.1103/PhysRevB.99.245404
I. INTRODUCTION
Single-molecule magnets (SMMs) are magnetically
anisotropic inorganic complexes with large spin momentsthat display a slow relaxation of the magnetization below agiven blocking temperature [ 1]. When grafted to graphene
quantum point contacts or carbon nanotubes, single-moleculemagnets have been shown to impart highly anisotropicmagnetoconductance hysteresis fingerprints on local electriccurrents, providing compelling evidence for the existenceof an exchange interaction between the giant spin of theSMM and the spin of conduction electrons of the carbonnanostructure [ 2,3] or phthalocyaninato quantum dots in the
case of TbPc
2break junction devices [ 4,5]. SMMs have been
studied in the context of molecular spintronics [ 6] and show
potential as molecular memory units [ 7] and spin valves
[3,8] that may eventually form the foundations of complex
spintronic technologies or even more ambitiously, quantumcomputers.
Recent spin-polarized scanning tunneling microscopy
(STM) studies of quantum magnets on surfaces have demon-strated that polarized spin currents can influence and evenflip the nanomagnet’s spin moment via a spin-transfer torqueeffect [ 9,10]. This effect could be used to read or write bits
of information to single nanomagnets in spintronics devices.A crucial challenge in the development of molecular quantumspintronics consists of injecting a spin current into a SMM-based device. To date, a spintronics experiment with thisformat has not yet been realized. A feasible strategy to achievecoupling between a spin current and the quantum spin states ofa single-molecule magnet is to graft SMMs onto the surfaceof a graphene quantum point contact since (i) efficient spininjection in graphene has already been achieved [ 11,12] and
(ii) coupling between SMMs and a graphene quantum dotdevice has been demonstrated [ 2].
*asoncini@unimelb.edu.auIn this paper we propose and theoretically study a molecu-
lar spintronics setup based on a SMM device under resonant
irradiation. The aim is to perturb the populations of SMM spinstates by inducing simple coherent spin dynamics behavior in
the SMM and assess its influence on the spin current flowing
through a device via the aforementioned exchange interaction,so that the spin current effectively measures the dynamics
of the SMM spin states under irradiation. In SMM-based
transport experiments, a sweeping magnetic field is often usedin this spirit to probe the incoherent dynamics related to theslow relaxation of the nanomagnet [ 2,3,13] but here, by using
resonant electromagnetic radiation, we are able to study also
the coherent oscillatory dynamics of the magnetic subsystemand its interplay with the dissipative dynamics of the leads.
While the spectroscopy of nanomagnets in the bulk phase is
relatively commonplace, addressing single (or few) moleculesin a spintronic device with radiation is not at all trivial.Recently, STM tips have been employed in this vein to in-duce atomically localized time-dependent modulations to thecrystal field of magnetic atoms adsorbed to a MgO /Ag(001)
substrate [ 14]. Another approach to achieve coherent transi-
tions within a SMM device was demonstrated by Thiele et al.
[5] whereby the nuclear spin states of a single TbPc
2molecule
in a molecular break junction were coupled to resonant mi-crowave signals via the hyperfine Stark effect. While theexperimental details of inducing resonant coherent transitionsin a nanomagnet spintronics device are intricate and systemspecific, a radiation-magnetic dipole coupling is archetypal ofmore general coupling schemes (discussed in Appendix A)
that may be utilized in an experimental nanomagnet spintron-ics setup. In this paper we focus on this simple regime ofradiation-dipole coupling in order to illustrate the interestingphenomena that can arise from a nanomagnet spintronic de-vice subject to a resonant, time-dependent perturbation.
We contribute to the already extensive nanomagnet-based
spintronics literature [ 15–22] by considering a SMM con-
figuration with the potential to work as a spin pump and
2469-9950/2019/99(24)/245404(9) 245404-1 ©2019 American Physical SocietyKIERAN HYMAS AND ALESSANDRO SONCINI PHYSICAL REVIEW B 99, 245404 (2019)
FIG. 1. A schematic representation of electron transport from
a ferromagnetic lead through a quantum dot that is antiferromag-netically coupled to a SMM subject to resonant radiation. Energy
is supplied to the system to tilt the giant spin of the SMM (thick,
red) allowing a spin-majority electron to charge the device from theferromagnetic source. On relaxation, the SMM aligns against the
longitudinal field reversing the spin of the conduction electron as it
is emitted to the nonmagnetic drain.
spin switch. Although noncollinear magnetic molecules have
been previously presented as efficient spin-switching devices[23,24], the possibility of inducing spin current switching is
presented here via a more general SMM system (i.e., withoutinvoking specific noncollinear spin configurations). Finally,we discuss the possibility of reading out Rabi oscillationsbetween spin states via time-averaged spin-current measure-ments, a result already observed in experiment between thenuclear spin states of a TbPc
2molecule, in which case,
however, the device also required a sweeping magnetic field[5].
In Sec. IIwe present a model describing the operation of
our SMM-based spintronic device under irradiation utilizingthe density matrix formalism. In Sec. IIIwe show results
from our model when both continuous and pulsed radiationare applied and discuss the underlying mechanism that leadsto pumping, switching, and amplification of the spin current.Finally, in Sec. IVwe recapitulate and make concluding
remarks.
II. THEORETICAL MODEL
A. Model Hamiltonian
We consider a device (Fig. 1) consisting of a SMM grafted
to a quantum dot that is weakly coupled to two metallic leads.We include an interaction with a static longitudinal magneticfield and a gate electrode. At sufficiently low temperatures,we assume that the device operates in the Coulomb blockaderegime such that charging and discharging to and from thedot occurs sequentially. We suppose that the onsite Coulombrepulsion between electrons on the dot is large enough toexclude doubly charged states from participating in transportthrough the device. We also include a coupling between thetotal spin of the device and the magnetic component of anapplied radiation.The total Hamiltonian for the device reads as
H(t)=H
L+HS+V(t)+HT, (1)
where
HL=/summationdisplay
αkσ(/epsilon1αkσ−μα)a†
αkσaαkσ (2)
is the isolated source and drain Hamiltonian, in which a(†)
αkσ
destroys (creates) an electron in lead αwith wave vector
k,s p i n σ, and energy /epsilon1αkσ. Here, μαcorresponds to the
chemical potential of electrons in the Fermi level of lead α
which is often modulated in experiment by the applicationof an antisymmetric bias voltage V
bsuch that μL=Vb/2 and
μR=−Vb/2. The system Hamiltonian is
HS=−DS2
z+/summationdisplay
σ(/epsilon1−eVg)c†
σcσ
+μBBz(g1Sz+g2sz)−JS·s, (3)
where S=(Sx,Sy,Sz) is the SMM spin operator, c(†)
σan-
nihilates (creates) an electron on the dot with spin σ, and
s=(sx,sy,sz) is the spin operator for the aforementioned
radical. Dis the uniaxial anisotropy characterizing the zero-
field splitting of the SMM spin states, g1and g2are the
gfactors for the SMM and the dot, respectively, μBis the
Bohr magneton, Bzis the amplitude of a static longitudinal
magnetic field, /epsilon1is the one-electron dot-orbital energy, Vg
is the magnitude of an applied gate voltage, and Jis the
exchange coupling between the SMM and an electron on thedot. The tunneling Hamiltonian is simply
H
T=/summationdisplay
αkσT∗
αa†
αkσcσ+Tαc†
σaαkσ, (4)
where Tαare the tunneling amplitudes for charging and dis-
charging events between lead αand the dot; we neglect the
possibility of direct tunneling between source and drain leads.
We discuss here the simplest radiation-dipole coupling
regime that can induce magnetic dipole-allowed resonanttransitions in the ground spin multiplet of the nanomagnet.We approximate the magnetic component of radiation prop-agating along the easy axis of the nanomagnet as a rotatingtransverse magnetic field that couples to the giant spin of theSMM by a Zeeman interaction. We take the field to be rotatingclockwise with a frequency ωin the plane perpendicular to the
easy axis of the SMM so that
V(t)=g
1μBB⊥[Sxcos(ωt)−Sysin(ωt)], (5)
where B⊥is the amplitude of the magnetic component of the
radiation.
After noting that the zcomponent of the total spin operator
(defined by St=S+s) commutes with HS, it is convenient to
enumerate the energy eigenstates of HSwith the eigenvalues
ofSt
z. We use a notation where |n,m/angbracketrightdenotes an electronic
state of the SMM-dot hybrid with a total spin projectionmand with nelectrons occupying the lowest unoccupied
molecular orbital (LUMO) of the dot. The energy eigenstatesof the neutral and charged systems are |0,m/angbracketright≡| m/angbracketright⊗|vac/angbracketright
and|1,m/angbracketright
±≡A±
m|m+1/2/angbracketright⊗| ↓ /angbracketright+ B±
m|m−1/2/angbracketright⊗| ↑ /angbracketright ,r e -
spectively; the fully polarized states are simply |1,s+1/2/angbracketright≡
245404-2MOLECULAR SPINTRONICS USING SINGLE-MOLECULE … PHYSICAL REVIEW B 99, 245404 (2019)
|s/angbracketright⊗| ↑ /angbracketright and|1,−s−1/2/angbracketright≡| − s/angbracketright⊗| ↓ /angbracketright . The coefficients
A±
mandB±
mare of the form
A±
m=±|J|
J√2/Delta1/epsilon1(m)∓[(2D−J)m−μBBzδg]
2√/Delta1/epsilon1(m),
B±
m=|J|/radicalbig
s(s+1)−m2+1/4
2√/Delta1/epsilon1(m)√2/Delta1/epsilon1(m)∓[(2D−J)m−μBBzδg](6)
with /Delta1/epsilon1(m)=[(μBBzδg/2)2+μBBzδg(2D−J)m/2+
D(D−J)m2+(J/4)2(2s+1)2]1/2and δg=g1−g2.
The energies of the electronic states of the SMM-dothybrid are E(0,m)=−Dm
2−g1μBmB zand E(1,m)±=
/epsilon1−Vg+J/4−D(m2+1/4)−g1μBmB z±/Delta1/epsilon1(m). The
energies of the fully polarized charged states are given byE(1,±s±1/2)
+when 2 D−J/greaterorequalslant0 and E(1,±s±1/2)−
otherwise.
From here we shall be concerned with the 2 D−J>0
regime in which the charged ground states are the antiferro-magnetic |1,±s∓1/2/angbracketright
−states. Note that the exchange part
of the Hamiltonian in Eq. ( 3) mixes states of the SMM-dot
hybrid that conserve the axial projection of the total spin ofthe device. Thus, in the antiferromagnetic coupling regime thecharged ground states are linear combinations of SMM spinstates; this is a crucial condition for the operation of the deviceas discussed later. We choose B
z<0 to lift the degeneracy
of both neutral and charged spectra but are careful not tochoose |B
z|so large that the ferromagnetic |1,s+1/2/angbracketrightstate
becomes the new ground state of the charged system. Finally,we impose a level degeneracy condition between the |0,s/angbracketrightand
|1,s−1/2/angbracketright
−states by choosing a suitable gate voltage Vgso
that|E(0,s)−E(1,s−1/2)−|=0.
B. Master equation in a time-dependent resonant field
and stationary current
The reduced density matrix describing the electronic
spin states of the SMM-dot hybrid is defined by ρ(t)=
TrL{ρtot(t)}where ρtot(t) is the density matrix for the entire
device and Tr L{...}denotes a trace over states in the leads. A
system of differential equations for ρ(t) is obtained within the
Born-Markov approximation by making standard manipula-tions [ 25] to the V on Neumann equation, however, neglecting
the effect of V(t) in the unperturbed propagators used to
transform the equations of motion of the density matrix intothe interaction picture. It is self-consistent to neglect the effectof the radiation in the definition of the interaction pictureprovided that the transitions caused by V(t) are much slower
than the decay of correlations in the leads induced by H
T[26].
After retaining only the secular terms in the resultant masterequation (the validity of which is investigated in Appendix B),
the evolution of a reduced density matrix element is governedby
˙ρ
mm/prime=−i
¯h[HS+V(t),ρ]mm/prime+δmm/prime/summationdisplay
lWl→mρl−γmm/primeρmm/prime,
(7)
where ρmm/prime=/angbracketleftn,m|ρ(t)|n,m/prime/angbracketrightis a matrix element between
eigenstates of HS(we do not consider coherences between
states from different charge spaces and so unambiguouslydrop the index ninρmm/prime),γmm/prime=1
2/summationtext
lWm→l+Wm/prime→lis the
total decoherence rate, and Wl→m=/summationtext
ασWl→m
ασ are rates of
charging /discharging (summed over leads and spin) from a
state|n,l/angbracketrightto a state |n/prime,m/angbracketrightgiven by [ 18]
Wl→m
ασ=/Gamma1α(1+2σPα)
2¯h⎧
⎨
⎩/vextendsingle/vextendsinglecn→n+1
σ,ml/vextendsingle/vextendsingle2fα(/Delta1ml),
/vextendsingle/vextendsinglecn→n−1
σ,ml/vextendsingle/vextendsingle2[1−fα(/Delta1lm)],(8)
where the upper case applies for charging transitions ( n/prime=
n+1) and the lower case applies for discharging transi-
tions ( n/prime=n−1). In the expression above, fα(/Delta1)={1+
exp[β(/Delta1∓Vb/2)]}−1is the Fermi-Dirac distribution for elec-
trons in lead α, the argument /Delta1is the energy difference
between the relevant charged and neutral states, −(+)Vbcor-
responds to the applied bias voltage at the source (drain) lead,β=1/k
BTwhere Tis temperature and kBis Boltzmann’s
constant, /Gamma1αis the coupling strength between lead αand the
SMM-dot hybrid, and Pαis the spin polarization inherent
to lead α. Finally, cn→n+1
σ,ml=/angbracketleftn/prime,m|c†
σ|n,l/angbracketrightand cn→n−1
σ,ml=
/angbracketleftn/prime,m|cσ|n,l/angbracketrightare the charging and discharging transition am-
plitudes, respectively. Note that Wl→mis only nonzero when
the number of conduction electrons is changed by one and thetotal spin of the SMM-dot hybrid is changed by one-half, i.e.,|n
/prime−n|=1 and|l−m|=1/2.
In the Coulomb blockade regime, at low temperatures and
bias voltages, only the |0,s/angbracketright,|0,s−1/angbracketright, and |1,s−1/2/angbracketright−
states make significant contributions to the current flowing
through the device and so we focus only on the evolution ofthese states. Since the |1,s−1/2/angbracketright
+state will not participate
in transport, we will from now on unambiguously refer to|1,s−1/2/angbracketright
−as|1,s−1/2/angbracketrightin order to ease notation. Due
to the presence of V(t) inside the commutator in Eq. ( 7)
we obtain rate equations with an explicit time dependencein the coefficients of the density matrix elements. This ex-plicit time dependence can be eliminated [ 26] by changing
to the rotating reference frame |n,m/angbracketright
R=eimωt|n,m/angbracketrightso that
ρmm/prime=eiω(m−m/prime)tρR
mm/prime. In the rotating frame, the relevant rate
equations take the form
˙ρR
s=√
2sg1μBB⊥
¯hIm/braceleftbig
ρR
s−1,s/bracerightbig
+Ws−1/2→sρR
s−1/2−Ws→s−1/2ρR
s,
˙ρR
s−1=√
2sg1μBB⊥
¯hIm/braceleftbig
ρR
s,s−1/bracerightbig
+Ws−1/2→s−1ρR
s−1/2−Ws−1→s−1/2ρR
s−1,
˙ρR
s−1/2=Ws→s−1/2ρR
s+Ws−1→s−1/2ρR
s−1
−(Ws−1/2→s+Ws−1/2→s−1)ρR
s−1/2,
˙ρR
s−1,s=i√
2sg1μBB⊥
2¯h/parenleftbig
ρR
s−1−ρR
s/parenrightbig
−i(/Delta1s−1,s−ω)ρR
s−1,s−γs−1,sρR
s−1,s,
˙ρR
s,s−1=i√
2sg1μBB⊥
2¯h/parenleftbig
ρR
s−ρR
s−1/parenrightbig
−i(/Delta1s,s−1+ω)ρR
s,s−1−γs,s−1ρR
s,s−1, (9)
245404-3KIERAN HYMAS AND ALESSANDRO SONCINI PHYSICAL REVIEW B 99, 245404 (2019)
where /Delta1s−1,s=[E(0,s−1)−E(0,s)]/¯h. We approximate
the coherences in the rotating frame by setting ˙ ρR
s−1,s=
˙ρR
s,s−1=0 so that by inverting the last two equations in Eq. ( 9)
we obtain expressions for ρR
s−1,sandρR
s,s−1. The imaginary
parts of the coherences are Lorentzian line shapes multipliedby the difference in the nonequilibrium populations of the twostates involved in the coherent superposition, and are given by
Im/braceleftbig
ρ
R
s−1,s/bracerightbig
=√
2sg1μBB⊥
2¯hγs−1,s/parenleftbig
ρR
s−1−ρR
s/parenrightbig
(/Delta1s−1,s−ω)2+γ2
s−1,s(10)
with Im {ρR
s,s−1}=− Im{ρR
s−1,s}. Note that the Lorentzian line
shapes appearing in Eq. ( 10) are broadened by the total
decoherence rate γs−1,s, and peaked at ω=/Delta1s−1,s, thus defin-
ing the resonance condition for the dissipative system. Afterinserting the imaginary part of the coherences into the top twoexpressions in Eq. ( 9) we obtain a 3 ×3 system of differential
equations containing only the diagonal components of the
reduced density matrix in the rotating reference frame. To ex-plore the stationary current limit, we invoke a further steady-state approximation and solve for the long-time behavior ofthe diagonal components of the density matrix. The solutionsmay be transformed back into the rest frame trivially as thediagonal components of the density matrix do not pick up anexplicit time dependence when shifting between frames.
We calculate the total current and the spin current at lead α
with
I
(α)
t=±e(Iα↑+Iα↓),
(11)I(α)
s=±e(Iα↑−Iα↓),
respectively, where the plus (minus) sign is used for the source
(drain), eis the elementary charge,
Iασ=Ws→s−1/2
ασ ρs+Ws−1→s−1/2
ασ ρs−1
−/parenleftbig
Ws−1/2→s
ασ +Ws−1/2→s−1
ασ/parenrightbig
ρs−1/2, (12)
andρs,ρs−1,ρs−1/2are the rest frame reduced density matrix
elements obtained above.
III. RESULTS AND DISCUSSION
For the purpose of our calculations we have chosen some
reasonable parameters describing an easy-axis spin systemcontaining all the necessary properties to behave as a SMMwith s=6,D=0.02 meV , and J=− 0.06 meV . We further
choose B
z=− 0.2T , B⊥=2×10−3T,/Gamma1S=/Gamma1D=10−3
meV , T=10 mK, and ω=/Delta1s−1,s=3.5×1011s−1.Vgis
always chosen to impose a level degeneracy condition be-tween the ground states of the neutral and charged manifoldsrendering /epsilon1an arbitrary parameter. We consider a system with
g
1=g2=2 but note that the implications of our model are
not restricted by this choice. Variation of g1will change the
position of the level degeneracy; this can be compensatedfor by adjusting V
g. With this choice of parameters, the
resulting energy levels of the neutral and singly charged statesof the device have the structure presented in Fig. 2.I ti s
particularly important to note that due to the antiferromagneticcoupling assumed here, the lowest-lying exchange coupledstate is |1,s−1/2/angbracketrightwhile the ferromagnetic state is thermally
inaccessible for charge transport. We consider the case of an
FIG. 2. Energy levels of the SMM-dot hybrid described by the
Hamiltonian given in Eq. ( 3) calculated using parameters chosen
above. The neutral states are represented by black dots and the plus
(minus) charged states by upward-facing (red) [downward-facing(blue)] triangles.
idealized spintronics experiment in which the source lead is
ferromagnetic and spin injection is 100% effective ( PS=1)
while the drain remains nonmagnetic ( PD=0).
A. Continuous radiation
In order to investigate the time-dependent coherent dynam-
ics of the magnetic system induced by the resonant radia-tion, we first performed brute force numerical integration ofEq. ( 7). In addition, numerical integration of Eq. ( 7) provides
a means to assess the robustness of the approximations leadingto the analytic steady-state solutions obtained for Eq. ( 9).
Figure 3shows the time evolution of the relevant diagonal
elements of ρ(t) obtained at V
b=0 when the SMM hybrid
is initially prepared in the |0,s/angbracketrightstate. The radiation induces
damped Rabi oscillations between |0,s/angbracketrightand|0,s−1/angbracketrightthat
quickly decay to a steady state due to decoherence introducedby the incoherent charge-transfer process between the theleads and the open quantum system. We find that the long-timebehavior of these solutions agrees with the analytical solutions
FIG. 3. Time evolution of the ρs,ρs−1/2,a n dρs−1density matrix
elements obtained by numerical integration of Eq. ( 7)a tVb=0 with
a ferromagnetic source.
245404-4MOLECULAR SPINTRONICS USING SINGLE-MOLECULE … PHYSICAL REVIEW B 99, 245404 (2019)
FIG. 4. The stationary charge current (top) and spin currents at
source and drain (bottom) flowing through the device as a function
of applied bias voltage.
obtained from our treatment of the master equation above,
therefore corroborating the steady-state approximations lead-ing to Eq. ( 10). The rate of population transfer between |0,s/angbracketright
and|0,s−1/angbracketrightat steady state is related to the imaginary part of
the off-diagonal matrix element given in Eq. ( 10) and is thus
maximal when ω=/Delta1
s−1,s. The energy supplied to the device
via continuous irradiation drives a population imbalance inthe neutral manifold leading to the manifestation of severalinteresting steady-state transport effects.
Figure 4shows the stationary charge and spin currents
as a function of applied bias voltage flowing through thedevice. A net current is pumped through the device at zero-bias voltage with the majority-spin current injected from theferromagnetic source being completely reversed and ampli-fied at the drain. When the SMM is prepared in the |0,s/angbracketright
ground state via an external magnetic field along the easyaxis but is not irradiated, then charging from the sourcecan not occur as the ferromagnetic reduced state |1,s+1/2/angbracketright
of the device is thermally inaccessible for transport (seeFig.2). One may view this configuration as the high-resistance
state of a molecular spin valve where the single-moleculemagnet acts as a spin analyzer. When energy is supplied tothe system by resonant electromagnetic radiation (see Fig. 1
for a schematic), then the giant spin of the SMM is tiltedvia transfer of population to the excited |0,s−1/angbracketrightstate. A
spin-majority electron may now charge the device owingto the nonzero amplitude /angbracketleft1,s−1/2|c
†
↑|0,s−1/angbracketrightbetweenthe|0,s−1/angbracketrightand|1,s−1/2/angbracketright=A−
s−1/2|s/angbracketright⊗| ↓ /angbracketright+ B−
s−1/2⊗
|s−1/angbracketright|↑/angbracketright states. The only nonzero discharging process that
can take place from |1,s−1/2/angbracketrightis one in which the SMM is
returned to its maximal spin ground state |s/angbracketright, and therefore
only discharging of spin-minority electrons is possible, due tothe coherent superposition structure of |1,s−1/2/angbracketright; crucially,
this can occur only at the drain owing to the fully spin-
polarized character of the ferromagnetic source lead. Thus,even at zero-bias voltage, a spin-switched current is pumpedthrough the device due to energy supplied via the resonantradiation and the spin-asymmetric charge-transfer processesat the ferromagnetic source and nonmagnetic drain. We notethat at low temperatures the |0,s−1/angbracketrightstate lies outside of
the conduction window provided that V
b<2/Delta1s−1,s−1/2=
D(2s−1)−g1μBBz. As a consequence, when the |0,s−1/angbracketright
is populated as a result of the resonant electromagnetic radi-ation, the device may also be charged by electrons from thedrain that also undergo a spin reversal before being emittedback to the drain. Although this process does not contributeto the net charge current flowing through the device, it doesprovide an additional contribution to the negative spin cur-rent at the drain resulting in an amplification of the drainspin current. These effects persist for nonzero bias voltageprovided that the bias is not so large as to activate the ferro-magnetic |1,s+1/2/angbracketrightcharged state or to include |0,s−1/angbracketrightin
the conduction window. While the charge pumping describedhere is reminiscent of the photon-assisted tunneling alreadyobserved in quantum dots [ 27,28], we stress that in this setup
it is the SMM that absorbs the radiation in order to overcomethe current blockade rather than the conduction electron.
B. Pulsed radiation
The continuous irradiation model described in the previous
section may present practical challenges in attaining constanttemperature of the system due to heat dissipation involvedby the absorption process. Thus, we also explore a perhapsmore easily realizable experimental setup, investigating thespintronics problem under pulsed radiation. Accordingly, wedefine a timescale t
p+w=tp+twcorresponding to a single
pulse-wait sequence. During the interval t∈[0,tp] the radi-
ation is switched on and V(t)i sg i v e nb yE q .( 5), whereas
in the interval t∈[tp,tp+w] the radiation is switched off and
V(t)=0; this sequence is repeated for multiples of tp+w.W e
calculate the average current through the device by numericalintegration of the master equation followed by averaging ofthe time-dependent current over an arbitrary number of pulse-wait sequences occurring after the initial pulse. For clarity,we define the time average of a function f(t) over the time
domain T={t∈R|t
a/lessorequalslantt/lessorequalslanttb}by
/angbracketleftf/angbracketrightT=1
tb−ta/integraldisplaytb
taf(t)dt. (13)
We focus on the case when Vb=0 and investigate the depen-
dence of the time-averaged current on the pulse and wait times
tpandtw, respectively.
Figure 5shows the time-averaged current flowing through
the device at zero bias for values of tpandtw. Even here
we obtain a finite time-averaged charge current for all valuesoft
p/negationslash=0 which tends toward saturation as tp→∞ and
245404-5KIERAN HYMAS AND ALESSANDRO SONCINI PHYSICAL REVIEW B 99, 245404 (2019)
FIG. 5. The time-averaged charge current flowing through the
device at Vb=0 as a function of various pulse times tpand wait
times tw.
manifests an oscillatory behavior as tp→0. By increasing
the wait time in-between pulses, we see that the averagecharge current per t
p+wcycle diminishes and tends to zero for
tw→∞ .
As noted previously, the resonant radiation causes damped
Rabi oscillations between elements of the density matrix (seeFig. 3) which is consequently reflected in the time-dependent
current. When t
pis shorter than the decay of the damped
oscillations, /angbracketleftItot/angbracketrightTprovides piecewise measurements of the
time evolution of the Rabi oscillations between |0,s/angbracketrightand
|0,s−1/angbracketright. Conversely, when tpis longer than the decay of the
damped Rabi oscillations, the system is able to reach a quasi -
steady-state limit (as in the continuous irradiation model)within the pulse phase of each t
p+wcycle and, therefore, the
oscillations are averaged out in /angbracketleftItot/angbracketrightT. During wait sequences
[where V(t)=0] the coherences in Eq. ( 7) become com-
pletely decoupled from the diagonal elements of the densitymatrix and the master equation becomes completely solubleup until the next pulse. Specializing to the 3 ×3s y s t e m
discussed above, we solve ˙ ρ=Mρover t
p/lessorequalslantt/lessorequalslanttp+wwhere
ρ=(ρs,ρs−1,ρs−1/2)TandMis the time-independent rate
matrix describing charging and discharging processes be-tween the dot and the leads. A great deal of simplification canbe made when V
b=0a s Ws−1/2→s−1≈0 and Ws→s−1/2=
Ws−1/2→s, leading one to discover the eigenvalues of M
as{0,−2Ws→s−1/2,−Ws−1→s−1/2}. Recalling that the long-
time limit of the system in the absence of resonant radiationleads to a blockage of current we see that, regardless of t
p,
fortw>max(−2Ws→s−1/2,−Ws−1→s−1/2) no current flows
through the device resulting in a diminishing value of /angbracketleftItot/angbracketrightT
astw→∞ .
Although in this section we have focused only on the
time-averaged charge current in the pulsed radiation regime,we note that the time-averaged spin currents (not shown) arealso switched and amplified at the drain as in the continuousradiation model. The behavior of the time-averaged spin cur-rents as functions of t
pandtwis mirrored in the discussion
above and so we omit it here. For the device to functionoptimally, t
pandtwshould be chosen such that the |0,s−1/angbracketright
state is sufficiently populated on each tp+wcycle and also such
that heat acquired from the resonant radiation diffuses awayfrom the SMM before the next pulse. We do not investigatethe added complexity of heat diffusion in this paper.C. Candidate nanomagnets for the device
In the model presented above, we have made no mention
of the specific SMM that should be used in the junctionas we predict the pumping, switching, and amplification ef-fects described above to be achievable with any nanomagnetthat is well described by the Hamiltonian given in Eq. ( 3).
In a practical setting, however, the choice of SMM is farfrom arbitrary as the frequency of radiation required for them=s→s−1 transition may also couple to vibrational
modes in the molecule or contribute to other undesirable in-teractions. Fe
4-based nanomagnets could be prime candidates
for the device proposed above as their magnetic propertiesare retained following surface deposition [ 29] and have been
shown to be robust under successive oxidation and reductionin three terminal devices [ 30,31]. A first-principles theoretical
study of an Fe
4nanomagnet attached to metallic leads has
furthermore indicated that the magnetic properties of Fe 4
are likely to be preserved in such a junction and may enjoya modest increase in uniaxial anisotropy on reduction [ 32].
The aforementioned theoretical work by Nossa et al. partially
corroborates the assumption made by Burzuri et al. [31]i n
that Fe
4acquires a S=9
2ground state on reduction, implying
an antiferromagnetic coupling between the giant spin of themagnet and the radical. In addition, the gap between theground and excited states on graphene has been reported∼1c m
−1which could be probed with microwave radiation
[33,34]. Octanuclear Fe(III) nanomagnets are also good can-
didates for the device since the gap /Delta1s−1,s∼4c m−1is also
amenable to microwave radiation. In fact, the m=s→s−1
transition in Fe 8SMM crystals has already been probed with
pulsed microwave radiation in previous studies [ 35–39]. The
required radiation-induced transition in Mn 12could also be
achieved with microwave radiation as it has been reported topossess a /Delta1
s−1,sof∼9c m−1[40]. Cr 7M(M=Cd, Mn, Ni)
molecular wheels may also be excellent candidates for ourdevice given their stability on surface deposition [ 41,42] and
under microwave radiation [ 43,44].
IV . CONCLUSION
We have proposed a model for electron transport through a
SMM nanostructure under irradiation in the Coulomb block-ade regime. We demonstrated that a spin current is pumpedthrough the device at zero-bias voltage when coupled to aferromagnetic source as a result of radiation-induced transi-tions in the SMM followed by spin-asymmetric dischargingat the source and drain leads. In addition to this spin-pumpingeffect, we find that the spin-polarized current pumped fromthe source is reversed and amplified at the drain even when
V
b/negationslash=0. We also investigated the behavior of the device under
pulsed irradiation and discussed the time-averaged current asa function of pulse length and wait time. We find that forlong enough pulse lengths and short wait times the station-ary current results are recovered. Interestingly, however, forshort pulse lengths and long wait times we also find thatthe proposed device can be used to measure coherent Rabioscillations between the SMM spin states, which could offeran as yet unexplored strategy to integrate SMM-based spinqubits into spintronics circuits.
245404-6MOLECULAR SPINTRONICS USING SINGLE-MOLECULE … PHYSICAL REVIEW B 99, 245404 (2019)
ACKNOWLEDGMENTS
K.H. acknowledges the support from the Australian Gov-
ernment Research Training Program Scholarship. A.S. ac-knowledges support from the Australian Research Council,Future Fellowship No. FT180100519. We thank A. Candiniand M. Affronte for their suggestions regarding the pulsedradiation model.
APPENDIX A: ALTERNATE RESONANT PERTURBATION
COUPLING SCHEMES
Throughout this paper we have discussed the spin-transport
dynamics imparted to a SMM-dot hybrid device with a spe-cific radiation-dipole coupling scheme [see Eq. ( 5)]. A more
general coupling between the magnetic states of the SMMand a resonant coherent perturbation can be achieved with theHamiltonian
V
N(t)=ν[SN
+eiωt+SN
−e−iωt], (A1)
where N∈Nandνis some constant specific to the applied
resonant perturbation in a given experimental setup. Notethat Eq. ( 5) is recovered from Eq. ( A1) when N=1 and
ν=g
1μBB⊥. To investigate the consequences on the spin-
transport dynamics of the device when the ground spin state|0,s/angbracketrightis coherently and resonantly coupled to an excited spin
state|0,s−N/angbracketrightwith N>1, we proceeded as in the main
text and developed a master equation for the reduced densitymatrix elements akin to Eq. ( 7), but with the substitution
V(t)/mapsto→V
N(t). Using the parameters from Sec. IIIwithω=
/Delta1s−N,s, we computed the zero-bias steady-state spin currents
at the ferromagnetic source and nonmagnetic drain elec-trodes for several values of Nand nanomagnet spin quantum
numbers s.
IfN>s, the resonant perturbation will induce popula-
tion transfer over the nanomagnet anisotropy barrier thusquenching the spin pumping, switching, and amplificationeffects described above. Multiple excitations caused by V
N(t),
while occurring on the slowest timescales can, in the steady-state limit, also result in population transfer over the barriereven when N<s; these processes are suppressed, however,
with increasing sorD. Provided that Nis not too large
with respect to the barrier height, Fig. 6demonstrates that
coupling the ground state |0,s/angbracketrightto excited states with axial
spin projections less than s−1 with a resonant perturbation
can still give rise to the current pumping, switching, andamplification effects described in the main text. As Nis
increased, these effects are augmented owing to the multistep
FIG. 6. Zero-bias steady-state spin currents at the source and
drain electrodes for SMM devices with various spin quantum num-bers sand resonant perturbations V
N(t).
charging /discharging cascade that is required to relax the
nanomagnet from the excited state |0,s−N/angbracketrightto the ground
state|0,s/angbracketright. For example, consider a resonant perturbation
that couples |0,s/angbracketrightand|0,s−2/angbracketright: after each excitation from
the ground state to |0,s−2/angbracketrightrelaxation proceeds via the
charging /discharging cascade |0,s−2/angbracketright→| 1,s−3/2/angbracketright−→
|0,s−1/angbracketright→| 1,s−1/2/angbracketright−→|0,s/angbracketright. In this example, two
electrons much sequentially charge and discharge from the de-vice both with their individual spin moments switched henceleading to a larger spin current than observed in the N=1
case. Finally, we note that for the nanomagnet spintronicssetup outlined here with a resonant perturbation V
N>1(t),
calculation of the steady-state spin currents can be reducedto that of the effective three-state model described in themain text with only the lead-dot coupling /Gamma1
αrenormalized to
account for the multistep charging /discharging cascade.
APPENDIX B: NONSECULAR RATE EQUATION
In order to confirm the validity of the secular approxima-
tion leading to Eq. ( 7), we performed a numerical integration
of the full nonsecular quantum master equation. The evolutionof a reduced density matrix element is
˙ρ
mm/prime=−i
¯h[HS+V(t),ρ]mm/prime+(Rρ)mm/prime, (B1)
where Rcontrols the full nonsecular dynamics of the device
owing to the dissipative effects from coupling to the leads andis given explicitly by
(Rρ)mm/prime=S,D/summationdisplay
α↑↓/summationdisplay
σ/Gamma1α(1+2σPα)
4¯h/braceleftBiggn+1/summationdisplay
klcn+1→n
σ,mlρlkcn→n+1
σ,km/prime[2−fα(/Delta1lm)−fα(/Delta1km/prime)]
+n−1/summationdisplay
klcn−1→n
σ,mlρlkcn→n−1
σ,km/prime[fα(/Delta1ml)+fα(/Delta1m/primek)]−n/summationdisplay
kn+1/summationdisplay
lfα(/Delta1lk)/parenleftbig
cn+1→n
σ,mlcn→n+1
σ,lkρkm/prime+ρmkcn+1→n
σ,klcn→n+1
σ,lm/prime/parenrightbig
−n/summationdisplay
kn−1/summationdisplay
l[1−fα(/Delta1kl)]/parenleftbig
cn−1→n
σ,mlcn→n−1
σ,lkρkm/prime+ρmkcn−1→n
σ,klcn→n−1
σ,lm/prime/parenrightbig/bracerightBigg
. (B2)
245404-7KIERAN HYMAS AND ALESSANDRO SONCINI PHYSICAL REVIEW B 99, 245404 (2019)
We found the numerical solutions to Eq. ( B1) to be indistin-
guishable from those that result from a quantum rate equationcontaining only secular terms (Fig. 3). The secular approx-
imation is thus justified for this system as the nonseculardynamics of the coherences induced by the dissipative effect
of the leads occurs on a much faster timescale than the timeevolution of the populations ρ
mand as such are canceled out
upon integration.
[1] G. Christou, D. Gatteschi, D. N. Hendrickson, and R. Sessoli,
Single-molecule magnets, MRS Bull. 25,66(2000 ).
[2] A. Candini, S. Klyatskaya, M. Ruben, W. Wernsdorfer, and
M. Affronte, Graphene spintronic devices with molecular nano-magnets, Nano Lett. 11,2634 (2011 ).
[3] M. Urdampilleta, S. Klyatskaya, J.-P. Cleuziou, M. Ruben, and
W. Wernsdorfer, Supramolecular spin valves, Nat. Mater. 10,
502(2011 ).
[4] R. Vincent, S. Klyatskaya, M. Ruben, W. Wernsdorfer, and
F. Balestro, Electronic read-out of a single nuclear spin us-ing a molecular spin transistor, Nature (London) 488,357
(2012 ).
[5] S. Thiele, F. Balestro, R. Ballou, S. Klyatskaya, M. Ruben, and
W. Wernsdorfer, Electrically driven nuclear spin resonance insingle-molecule magnets, Science 344,1135 (2014 ).
[6] L. Bogani and W. Wernsdorfer, Molecular spintronics using
single-molecule magnets, Nat. Mater. 7,179(2008 ).
[7] M. Mannini, F. Pineider, P. Sainctavit, C. Danieli, E. Otero,
C. Sciancalepore, A. M. Talarico, M.-A. Arrio, A. Cornia,D. Gatteschi et al. , Magnetic memory of a single-molecule
quantum magnet wired to a gold surface, Nat. Mater. 8,194
(2009 ).
[8] M. Urdampilleta, S. Klayatskaya, M. Ruben, and W.
Wernsdorfer, Magnetic interaction between a radical spin anda single-molecule magnet in a molecular spin-valve, ACS Nano
9,4458 (2015 ).
[9] A. A. Khajetoorians, B. Baxevanis, C. Hübner, T. Schlenk,
S. Krause, T. O. Wehling, S. Lounis, A. Lichtenstein, D.Pfannkuche, J. Wiebe et al. , Current-driven spin dynamics
of artificially constructed quantum magnets, Science 339,55
(2013 ).
[10] S. Krause, L. Berbil-Bautista, G. Herzog, M. Bode, and R.
Wiesendanger, Current-induced magnetization switching witha spin-polarized scanning tunneling microscope, Science 317,
1537 (2007 ).
[11] W. Han, K. Pi, K. M. McCreary, Y . Li, J. J. I. Wong,
A. G. Swartz, and R. K. Kawakami, Tunneling Spin Injectioninto Single Layer Graphene, P h y s .R e v .L e t t . 105,
167202
(2010 ).
[12] M. Ohishi, M. Shiraishi, R. Nouchi, T. Nozaki, T. Shinjo, and
Y . Suzuki, Spin injection into a graphene thin film at roomtemperature, Jpn. J. Appl. Phys. 46,L605 (2007 ).
[13] F. Troiani, C. Godfrin, S. Thiele, F. Balestro, W. Wernsdorfer,
S. Klyatskaya, M. Ruben, and M. Affronte, Landau-ZenerTransition in a Continuously Measured Single-Molecule SpinTransistor, P h y s .R e v .L e t t . 118,257701 (2017 ).
[14] S. Baumann, W. Paul, T. Choi, C. P. Lutz, A. Ardavan, and A. J.
Heinrich, Electron paramagnetic resonance of individual atomson a surface, Science 350,417(2015 ).
[15] C. Timm and F. Elste, Spin amplification, reading, and writing
in transport through anisotropic magnetic molecules, Phys. Rev.
B73,235304 (2006 ).[16] C. Romeike, M. R. Wegewijs, W. Hofstetter, and H. Schoeller,
Quantum-Tunneling-Induced Kondo Effect in Single MolecularMagnets, P h y s .R e v .L e t t . 96,196601 (2006 ).
[17] M. R. Wegewijs, C. Romeike, H. Schoeller, and W. Hofstetter,
Magneto-transport through single-molecule magnets: Kondo-peaks, zero-bias dips, molecular symmetry and Berry’s phase,New J. Phys. 9,344(2007 ).
[18] M. Misiorny and J. Barna ´s, Spin polarized transport through a
single-molecule magnet: Current-induced magnetic switching,Phys. Rev. B 76,054448 (2007 ).
[19] F. Delgado, J. J. Palacios, and J. Fernández-Rossier, Spin-
Transfer Torque on a Single Magnetic Adatom, Phys. Rev. Lett.
104,026601 (2010 ).
[20] J. W. González, F. Delgado, and J. Fernández-Rossier,
Graphene single-electron transistor as a spin sensor for mag-netic adsorbates, Phys. Rev. B 87,085433 (2013 ).
[21] A. Hurley, N. Baadji, and S. Sanvito, Spin-flip inelastic electron
tunneling spectroscopy in atomic chains, Phys. Rev. B 84,
035427 (2011 ).
[22] H. Xie, Q. Wang, B. Chang, H. Jiao, and J.-Q. Liang, Spin
current and polarization reversal through a single-moleculemagnet with ferromagnetic electrodes, J. Appl. Phys. 111,
063707
(2012 ).
[23] J. Crabtree and A. Soncini, Toroidal quantum states in molecu-
lar spin-frustrated triangular nanomagnets with weak spin-orbitcoupling: Applications to molecular spintronics, P h y s .R e v .B
98,094417 (2018 ).
[24] A. Soncini and L. F. Chibotaru, Molecular spintronics using
noncollinear magnetic molecules, P h y s .R e v .B 81,132403
(2010 ).
[25] K. Blum, Density Matrix Theory and Applications (Springer,
Berlin, 2012).
[26] H.-A. Engel and D. Loss, Single-spin dynamics and decoher-
ence in a quantum dot via charge transport, P h y s .R e v .B 65,
195321 (2002 ).
[27] L. P. Kouwenhoven, S. Jauhar, K. McCormick, D. Dixon, P. L.
McEuen, Y . V . Nazarov, N. C. vanderVaart, and C. T. Foxon,Photon-assisted tunneling through a quantum dot, Phys. Rev. B
50,2019 (1994 ).
[28] L. P. Kouwenhoven, S. Jauhar, J. Orenstein, P. L. McEuen, Y .
Nagamune, J. Motohisa, and H. Sakaki, Observation of Photon-Assisted Tunneling through a Quantum Dot, Phys. Rev. Lett.
73,3443 (1994 ).
[29] A. Cini, M. Mannini, F. Totti, M. Fittipaldi, G. Spina, A.
Chumakov, R. Rüffer, A. Cornia, and R. Sessoli, Mössbauerspectroscopy of a monolayer of single molecule magnets, Nat.
Commun. 9,480(2018 ).
[30] A. S. Zyazin, J. W. G. van den Berg, E. A. Osorio, H. S. J.
van der Zant, N. P. Konstantinidis, M. Leijnse, M. R. Wegewijs,F. May, W. Hofstetter, and C. Danieli, Electric field controlledmagnetic anisotropy in a single molecule, Nano Lett. 10,3307
(2010 ).
245404-8MOLECULAR SPINTRONICS USING SINGLE-MOLECULE … PHYSICAL REVIEW B 99, 245404 (2019)
[31] E. Burzurí, A. S. Zyazin, A. Cornia, and H. S. J. Van der Zant,
Direct Observation of Magnetic Anisotropy in an Individual Fe 4
Single-Molecule Magnet, Phys. Rev. Lett. 109,147203 (2012 ).
[32] J. F. Nossa, M. F. Islam, C. M. Canali, and M. R. Pederson,
Electric control of a Fe 4single-molecule magnet in a single-
electron transistor, P h y s .R e v .B 88,224423 (2013 ).
[33] C. Cervetti, A. Rettori, M. G. Pini, A. Cornia, A. Repolles,
F. Luis, M. Dressel, S. Rauschenbach, K. Kern, M. Burghardet al. , The classical and quantum dynamics of molecular spins
on graphene, Nat. Mater. 15,164(2016 ).
[34] C. Schlegel, J. Van Slageren, M. Manoli, E. K. Brechin, and M.
Dressel, Direct Observation of Quantum Coherence in Single-Molecule Magnets, P h y s .R e v .L e t t . 101,147203 (2008 ).
[35] M. Bal, J. R. Friedman, W. Chen, M. Tuominen, C. Beedle,
E. Rumberger, and D. Hendrickson, Radiation-and phonon-bottleneck–induced tunneling in the Fe
8single-molecule mag-
net,Europhys. Lett. 82,17005 (2008 ).
[36] M. Bal, J. R. Friedman, E. Rumberger, S. Shah, D.
Hendrickson, N. Avraham, Y . Myasoedov, H. Shtrikman, andE. Zeldov, Photon-induced magnetization changes in single-molecule magnets, J. Appl. Phys. 99,08D103 (2006 ).
[37] S. Bahr, K. Petukhov, V . Mosser, and W. Wernsdorfer, Pump-
Probe Experiments on the Single-Molecule Magnet Fe
8: Mea-
surement of Excited Level Lifetimes, Phys. Rev. Lett. 99,
147205 (2007 ).
[38] K. Petukhov, S. Bahr, W. Wernsdorfer, A. L. Barra, and V .
Mosser, Magnetization dynamics in the single-molecule magnetFe8under pulsed microwave irradiation, P h y s .R e v .B 75,
064408 (2007 ).
[39] L. Sorace, W. Wernsdorfer, C. Thirion, A.-L. Barra, M.
Pacchioni, D. Mailly, and B. Barbara, Photon-assisted tunnelingin a Fe
8single-molecule magnet, P h y s .R e v .B 68,220407(R)
(2003 ).
[40] A. L. Barra, D. Gatteschi, and R. Sessoli, High-frequency epr
spectra of a molecular nanomagnet: Understanding quantumtunneling of the magnetization, P h y s .R e v .B 56,8192 (1997 ).
[41] V . Corradini, R. Biagi, U. Del Pennino, V . De Renzi, A.
Gambardella, M. Affronte, C. Muryn, G. Timco, and R.Winpenny, Isolated heterometallic Cr
7Ni rings grafted on Au
(111) surface, Inorg. Chem. 46,4937 (2007 ).
[42] A. Ghirri, V . Corradini, C. Cervetti, A. Candini, U. Del Pennino,
G. Timco, R. J. Pritchard, C. Muryn, R. Winpenny, and M.Affronte, Deposition of functionalized Cr
7Ni molecular rings
on graphite from the liquid phase, Adv. Funct. Mater. 20,1552
(2010 ).
[43] A. Ardavan, O. Rival, J. Morton, S. J. Blundell, A. M.
Tyryshkin, G. A. Timco, and R. Winpenny, Will Spin-Relaxation Times in Molecular Magnets Permit Quantum In-formation Processing? P h y s .R e v .L e t t . 98,057201 (2007 ).
[44] S. Piligkos, H. Weihe, E. Bill, F. Neese, H. El Mkami, G. M.
Smith, D. Collison, G. Rajaraman, G. A. Timco, R. Winpenny et
al., EPR spectroscopy of a family of Cr
III7MII(M=Cd, Zn, Mn,
Ni)“wheels”: Studies of isostructural compounds with differentspin ground states, C h e m .E u r .J . 15,3152 (2009 ).
245404-9 |
PhysRevB.72.205333.pdf | Electrical properties of epitaxial junctions between Nb:SrTiO 3and optimally doped,
underdoped, and Zn-doped YBa 2Cu3O7−/H9254
W. Ramadan,1S. B. Ogale,1S. Dhar,1L. F. Fu,2,3S. R. Shinde,1Darshan C. Kundaliya,1M. S. R. Rao,1N. D. Browning,2,3
and T. Venkatesan1
1Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
2Department of Chemical Engineering and Materials Science, University of California Davis, One Shields Ave., Davis, California 95616,
USA
3The National Center for Electron Microscopy, Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley,
California 94720, USA
/H20849Received 7 April 2005; revised manuscript received 2 September 2005; published 22 November 2005 /H20850
Epitaxial thin films of optimally doped, underdoped, and Zn-doped YBa 2Cu3O7−/H9254/H20849YBCO /H20850were grown on
single crystal /H20849001 /H20850Nb:SrTiO 3substrates by pulsed laser deposition /H20849PLD /H20850and the electrical properties of the
corresponding interface junctions were examined. The growth conditions were optimized in each case to getthe appropriate crystalline quality of the films as well as the desired normal state and superconducting prop-erties. The films or heterointerfaces were characterized by x-ray diffraction, Rutherford backscattering /H20849RBS /H20850
ion channeling spectrometry in normal and oxygen resonance modes, magnetic susceptibility, four probein-plane resistivity, and the temperature dependent current-voltage /H20849I-V/H20850characteristics. Nonlinear I-Vcurves
/H20849forward and reverse /H20850were obtained in all the cases, revealing some characteristic differences and interesting
temperature evolution. These data, when analyzed within the framework of a standard description of transportacross the metal-semiconductor /H20849Schottky /H20850interface, suggest lateral intrinsic nanoscale electrical inhomogene-
ity in the system. Also as compared to the case of optimally doped YBCO a small but definitive lowering of theeffective Schottky barrier height /H9021
Bis observed for junctions based on oxygen underdoped and Zn-doped
YBCO.
DOI: 10.1103/PhysRevB.72.205333 PACS number /H20849s/H20850: 73.30. /H11001y, 73.40. /H11002c, 73.61.Ga, 74.78 /H11002w
I. INTRODUCTION
Interfaces between two dissimilar materials have always
been a subject of great scientific curiosity and technologicalinterest.
1–10With the significant advances in thin film growth
and characterization techniques over the past decade, the in-terest in the formation and properties of high quality, atomi-cally flat and well characterized interfaces in epitaxial het-erostructures has grown dramatically.
9–18Indeed, such
interfaces are now being looked at as new materials withunique properties,
13,15,16capable of supporting novel device
functions. Amongst the various interface systems, thesemiconductor-semiconductor /H20849p-njunction /H20850and metal-
semiconductor /H20849Schottky /H20850interfaces are perhaps the most
widely studied in view of their widespread applicability inmicroelectronics technology. Not surprisingly, most theoret-ical considerations have also been directed to these interfacesand the same have matured over the years especially in thecontext of the electrical transport across interfaces.
19–27
However, with the ever shrinking dimensions of devices andthe corresponding physics as well as processing problemsfaced by conventional semiconductor microelectronics,
28–30
attention is now being directed to new materials with highcarrier concentration and the related interface systems. To-wards this end, metal oxides such as the high temperaturesuperconductors, colossal magnetoresistive managanites andtransparent conducting oxides are being activelyinvestigated.
31–36The ongoing research on oxide-oxide and
semiconductor-oxide interfaces is envisaged to lay the foun-dations of a potentially novel oxide electronics. Most effortsin this domain have been primarily directed towards the
growth and microstructural characterization of epitaxial in-terfaces, and the measurements of the electrical properties ofthe interfaces are now beginning to grow.
37–47
In this work we have examined the growth and properties
/H20849structural and electrical /H20850of the interface between an oxide
superconductor YBa 2Cu3O7−/H9254/H20849YBCO /H20850and an n-type oxide
semiconductor 0.5 wt. % Nb-doped SrTiO 3/H20849NSTO /H20850.W e
have considered three different cases of optimally doped,oxygen underdoped and Zn-doped YBCO grown epitaxiallyon NSTO. The normal state and superconducting propertiesfor all the three cases are known to be intriguing and con-tinue to interest the scientific community for over adecade.
48–56In addition to the basic characterizations of ma-
terials properties, we have focused on the temperature depen-dence of the I-Vcharacteristics. We have analyzed the data in
terms of the standard physical picture of a metal-semiconductor /H20849Schottky /H20850interface
20and have extracted po-
tentially useful information regarding the barrier properties.
II. EXPERIMENT
In our experiments commercially procured /H20849Crystek /H20850high
quality single crystalline /H20849001 /H208500.5 wt. % Nb-doped SrTiO 3
single crystal substrates were used for film growth. The
YBCO films were grown by pulsed laser deposition /H20849PLD /H20850
technique. A pulsed KrF Excimer laser was used for ablation.The corresponding energy density and pulse repetition ratewere 1.8 J/cm
2and 10 Hz, respectively. For optimally doped
YBCO, the films were grown at 200 mTorr oxygen pressurePHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
1098-0121/2005/72 /H2084920/H20850/205333 /H208499/H20850/$23.00 ©2005 The American Physical Society 205333-1at the substrate temperature of 800 °C and then cooled to
room temperature in oxygen pressure of 400 Torr. Thisgrowth condition is known to yield films with good crystal-linity and superconducting properties.
57–59Some films were
also grown at 750 °C, to explore the effect of growth tem-perature on the electrical properties of the interface. One setof the optimally doped /H20849T
c/H1101191 K /H20850films were annealed at
about 300 °C for several hours to achieve the desired degree
of oxygen deficiency and underdoping. A separate set of Zn-doped YBCO films /H20849YBa
2Cu3−xZnxO7−/H9254,x=0.2 /H20850were grown
at 750 °C at 130 mTorr oxygen and cooled under 400 Torroxygen. This low temperature growth procedure was used tocontrol the Zn loss at high temperature as suggested byOgale et al.
60The films were characterized by x-ray diffrac-
tion /H20849structural quality, lattice parameter /H20850, Rutherford back-
scattering /H20849RBS /H20850channeling /H20849crystalline quality, composi-
tion, thickness /H20850, oxygen resonance RBS technique /H20849oxygen
stoichiometry /H20850, and four probe electrical measurements. In-
dium contacts were made to NSTO while In-Ag contactswere made to the YBCO layer. This ensured ohmic contactcharacteristics. The approximate contact area in all cases wasabout 0.1 cm
2. The resistivity values for the superconductor
films were measured on a pure SrTiO 3/H20849001 /H20850substrate, on
which films were grown concurrently with those on NSTO,to avoid the interference of the conductivity of NSTO in themeasurement. For the standard RBS and ion channeling mea-surement 2 MeV He ions were used. For the RBS oxygenresonance study the He ion energy was set to 3.05 MeVwhich enhanced the oxygen signal dramatically allowingoxygen detection in films with an accuracy of a few percent.We should like to point out that the data reported and ana-lyzed in this paper were reproduced at least in three samplesin each case.
III. RESULTS AND DISCUSSION
The details regarding the structural and compositional
properties of films grown under the described conditionshave been discussed previously. Specifically, the x-ray dif-fraction /H20849XRD /H20850patterns for all the three samples can be in-
dexed to the /H2084900/H5129/H20850family of the 123 superconducting phase
up to /H5129=11 indicating their c-axis orientation. In Fig. 1 we
summarize the results of RBS, ion channeling and oxygendetermination studies. Figure 1 /H20849a/H20850shows the representative
RBS angular scan for the optimally doped sample. A goodminimum channeling yield /H20849
/H9273min/H20850of 8% is realized, reflect-
ing the high crystalline quality and epitaxial nature of the
film. The minimum channeling /H20849/H9273min/H20850yields for the oxygen
deficient and Zn-doped films were 13% and 22%, respec-
tively. The higher /H9273minvalues in these samples reflect defect-
strain related local distortions of the matrix due to oxygenvacancies or Zn substitution, and are indeed expected to berevealed by the highly site distortion sensitive RBS ionChanneling technique.
53We employed the oxygen resonance
detection to explore the changes in the oxygen stoichiometry,both in the normal and channeling modes. From Fig. 1 /H20849b/H20850
/H20849upper panel /H20850it can be seen that the oxygen stoichiometry
for the Zn-doped case is almost the same as that for theoptimally doped sample. This is again consistent with thefact that Zn dopant substitutes for the Cu ions in the CuO
2
plane sites with the same valence state and thus the oxygen
concentration is expected to remain undisturbed.51,55As
shown in Fig. 1 /H20849c/H20850/H20849upper panel /H20850, the oxygen deficient
sample has about 5% /H20849±2% /H20850less oxygen as compared to the
optimally doped sample, when the integrated areas under the
resonance peak are compared for the two cases. If we assumethe oxygen stoichiometry /H208497−
/H9254YBa 2Cu3O7−/H9254/H20850for the opti-
mally doped sample to be about 6.93 /H20849/H9254=0.07 /H20850based on the
observed transition temperature of 91 K, we obtain the 7
−/H9254value of about 6.6±0.1 /H20849/H9254=0.4 /H20850for the oxygen deficient
sample. As expected, this corroborates well52with the ob-
served transition temperature of about 58 K. Finally, it isuseful to point out an interesting fact about oxygen under-doping which has not been revealed by previous studies. Ascan be noted from the channeling portions of Fig. 1 /H20849c/H20850/H20849lower
panel /H20850, the “channeled” oxygen contribution in the under-
doped case is actually higher than that in the optimally dopedsample. This implies that the process of introducing oxygendeficiency in the sample also causes lattice distortions lead-ing to enhanced dechanneling in the measurement. Thesedistortions may have a contribution to the suppression of thetransition temperature in addition to the other factors.
The data for in-plane resistivity /H20849
/H9267ab/H20850as function of tem-
perature /H20849T/H20850for four samples /H20849two optimally doped samples
grown at 800 °C and 750 °C, Zn-doped and oxygen defi-
cient samples /H20850are shown in Fig. 2. It is evident that the
transition temperature, TO/H20849onset /H20850,i s9 2K ,5 8K ,a n d4 5K
for the optimum-doped, oxygen-deficient and Zn-doped/H20849YBa
2Cu3−xZnxO7−/H9254,x=0.2 /H20850samples, respectively. The
properties of the optimally doped sample are as expected interms of the residual resistively ratio and the sharpness of
FIG. 1. Results of RBS, ion channeling, and oxygen determina-
tion studies: /H20849a/H20850The channeling angular scans for the optimally
doped, underdoped, and Zn-doped samples grown on Nb:SrTiO 3;
/H20849b/H20850comparison of oxygen yield in the resonance mode for the three
cases in the normal and channeling modes.RAMADAN et al. PHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
205333-2transition. The shift of the transition to lower temperature
and its broadening as observed for the underdoped and Zn-doped samples are also consistent with the previous
results.
48,51,52The normal state resistivity values for the
optimum-doped, oxygen-deficient and Zn-doped sampleswere found to be 300±100
/H9262/H9024cm, 1200±200 /H9262/H9024cm, and
540±100 /H9262/H9024cm, respectively, and are in a reasonable
agreement with the literature.
The primary effects of Zn doping48and related scattering
is to add a nominally temperature independent contributionto the transport scattering rate. However, a small increase inthe average slope of d
/H9267ab/dTcan be noted and the same can
be attributed either to a decrease in carrier concentration orsome temperature dependent scattering contribution. EachZn ion is suggested to represent a scattering cross section ofa diameter of about 4.2 Å /H20849encompassing four oxygen neigh-
bors /H20850and this strong scattering may well be one of the causes
of the rapid suppression of T
cwith Zn incorporation.48Zn is
a nonmagnetic impurity, but it induces a magnetic moment inCuO
2plane when substituted at the Cu site, the precise value
of the moment being dependent upon oxygen content.
The peculiar Sshape characteristic of the resistivity curve
for the oxygen deficient film has been noted and discussed inthe literature
52for oxygen under-doped YBCO crystals, thin
films, as well as for cobalt doped YBCO crystals, and isdistinctly different as compared to a uniform upward shift inresistivity and decrease in the transition temperature seen forZn doped films. Indeed the initial considerably higher slope/H20849d
/H9267/dT/H20850from 300 K down to about 125 K as compared to
the case of optimally doped film, followed by the tapering
off of the slope observed in our film is completely consistentwith the reports in the literature.
52The pronounced and sys-
tematic nonlinear behavior in oxygen deficient films hasbeen attributed to reduced hole densities /H20849electron doping /H20850as
analyzed by Hall measurements. The change in the slope oftheTdependence of resistivity at an independent tempera-
ture corresponds to the change in the nature of dependence ofHall carrier density on temperature.
Figure 3 compares the room temperature I-Vcharacteris-
tics for the three cases of junctions based on optimallydoped, oxygen deficient and Zn-doped YBCO. The general
features in all the three cases appear to reflect the basic char-acteristics expected for a metal-semiconductor /H20849M-S /H20850junc-
tion, although there are clear differences between the opti-mally doped case and the two other cases of underdoped andZn-doped films.
Figures 4 /H20849a/H20850,4/H20849c/H20850, and 4 /H20849e/H20850, show the current-voltage
characteristics /H20849I-V/H20850for the three junctions /H20849optimally doped,
underdoped, and Zn-doped samples grown on Nb:STO /H20850un-
der forward biased conditions at some temperatures, withnegative /H20849positive /H20850bias applied to NSTO /H20849YBCO /H20850. The cor-
responding reverse biased characteristics are shown in Figs.4/H20849b/H20850,4/H20849d/H20850, and 4 /H20849f/H20850, respectively. No superconductivity re-
lated features /H20849expected in the meV range /H20850could be seen
possibly due to the specific nature of interfacial layer forma-tion and the role of the corresponding electronic states insuppressing the features. Also, hysteretic features are notedin the characteristics, which have been observed in previousworks
61–64on perovskite matrices and attributed to complex
effects involving interface trap states, oxygen vacancy de-fects and their slow dynamics under current and field drivenby electrochemistry.
65–68Such effects can lead to distribution
of charge in the interfacial region affecting the band line-upand tunneling probability. Clearly, the degree of hystereticcharacteristic should differ in the three cases being discussedherein. In the underdoped and Zn-doped cases the leakagecurrent density on reverse bias is also comparatively higher.For comparison, the room temperature values of reverse cur-rent density /H20849J
rev/H20850at −1 V bias for optimally doped, under-
doped, and Zn-doped films are /H110116/H1100310−4A/cm2,1 8
/H1100310−4A/cm2,2 7/H1100310−3A/cm2, respectively. The relatively
large values of reverse current densities are expected for thesmall barrier widths arising from a large carrier concentra-tion in NSTO due to a high Nb concentration. With muchlower Nb concentration such as 0.01 wt. % one should beable to reduce J
revsubstantially. It may further be noted that
for this oxygen deficient case, at intermediate and low tem-peratures one observes large excess current in the reverseregion, exhibiting a more symmetrical behavior at small volt-ages. Similar symmetric nonlinearity was also seen in the
FIG. 2. The in-plane resistivity /H20849/H9267ab/H20850as a function of tempera-
ture /H20849T/H20850for the optimally doped /H20849grown at 800 °C and 750 °C /H20850,
underdoped, and Zn-doped films on SrTiO 3.
FIG. 3. Comparison of the room temperature I-Vcharacteristics
for the junctions based on optimally doped, oxygen deficient, andZn-doped YBCO.ELECTRICAL PROPERTIES OF EPITAXIAL … PHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
205333-3optimally doped case but at temperatures below about 60 K
and in Zn-doped case even at relatively higher temperatures.The origin of this contribution is not clear at this time.
Before we proceed to discuss the observed transport data,
it is important to mention that we ensured the high quality ofour interfaces by high resolution scanning transmission elec-tron microscopy /H20849STEM /H20850; the representative data for an op-
tically doped film being shown in Fig. 5. The interface isclearly seen to be abrupt and epitaxial. The stacking fault
type features running parallel to the interface are well knownin the highest quality films.
69Scattered nanoscale inclusions
which are also known in high quality films were noted, buttheir being mostly away from the interface should not influ-ence the intrinsic features of the interface transport problemat hand. We have also performed chemical analysis of thefilms across the interface using electron loss spectroscopy
FIG. 4. Current-voltage /H20849I-V/H20850characteristics for YBCO/Nb:SrTiO 3junctions: /H20849a/H20850,/H20849c/H20850, and /H20849e/H20850correspond to the forward bias /H20849lower
current curve in the hysteresis representing the positive going trace /H20850, and /H20849b/H20850,/H20849d/H20850, and /H20849f/H20850to reverse bias /H20849higher current curve in the
hysteresis representing the negative going trace /H20850, for junctions with optimally doped, Zn-doped, and oxygen underdoped YBCO films,
respectively.RAMADAN et al. PHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
205333-4/H20849EELS /H20850and found that the interfaces are also chemicaly
sharp within subnanometer scale. The details of these studieswill be reported separately.
70As discussed earlier, the high
epitaxial quality of the films was separately establishedthrough RBS channeling results.
Over a decade ago, Hasegawa et al.
71had studied the
properties of the contact between the high- Tcsuperconductor
Er-Ba-Cu-O and Nb:STO for two cases of Nb doping/H208490.05 wt. % and 0.5 wt. % /H20850in STO. The T
c/H20849zero resistance /H20850
of their optimally doped film was 69 K much lower than theexpected 92 K.
72It may be noted that the Tcin our optimally
doped films was /H1101191 K. As in our case, these authors had
also not observed any superconductivity related structure intheI-Vcharacteristics, which is expected in the meV regime.
As pointed out by Hasegawa et al. this nonobservance can be
attributed to the presence of some interfacial layer or stateson the surface of the semiconductor Nb:STO. These authorshad found that with decreasing temperature the forward cur-rent decreases while the reverse current increases. In theirsample with higher Nb content /H208490.5% resembling our case /H20850
they observed that both the forward and reverse currentswere relatively higher as compared to those for lower Nb/H208490.05 wt. % /H20850doped sample, and the I-Vcharacteristic was
less temperature dependent. The latter suggested that tunnel-
ing current is dominant in this case. Defining the reversebreakdown voltage as the voltage at which the reverse cur-rent density becomes 1 /H1100310
−4A/cm2, these authors exam-
ined the temperature dependence of the breakdown voltage.In the sample with 0.5 wt. % /H20849heavy /H20850Nb doping, which is
closer to our case, the breakdown voltage was lower as com-pared to the lower Nb doped /H208490.05 wt. % /H20850sample, and it
showed only weak temperature dependence. In the heavily
Nb doped sample the Schottky barrier is thin causing tunnel-ing current to increase. This in turn causes an increase in theZener breakdown contribution, leading to a relatively lowerbreakdown voltage and its weak temperature dependence.
It is now useful to comment on the applicability of spe-
cific models for analyzing transport across a metal-semiconductor junction to the situation at hand, which rep-resents the case of a heavily doped wide band-gap oxidesemiconductor in contact with an oxide metal having highcarrier concentration. The transport across a Schottky contactcan be analyzed within the frameworks of the thermionicemission theory, diffusion theory or a hybrid theory account-ing for both the effects, depending on the specifics of the
problem, because of the different assumptions of themodels.
20
For the case moderately doped high mobility semiconduc-
tors the thermionic model is generally applicable. In this casetheI-Vrelation for the current flow through a Schottky con-
tact is given by
I=I
sT/H20851exp /H20849qV/nkT /H20850−1/H20852, /H208491/H20850
where qis the electronic charge, kis the Boltzmann constant,
Vis the bias voltage and nis the ideality factor. The reverse
saturation current IsTis given by
IsT=AA**T2exp /H20849−q/H9272B/kT/H20850, /H208492/H20850
where Ais the diode area, A**is the effective Richardson
constant, and /H9272Bis the apparent barrier height.
For the diffusion theory, which is generally applicable for
low mobility semiconductors, the current density expressionis similar to the thermionic emission theory:
I=I
sD/H20851exp /H20849qV/nkT /H20850−1/H20852. /H208493/H20850
However, the saturation current density has a different form
given by
IsD=q2DnNC
kT/H20875q/H20849Vbi−V/H208502ND
/H9255s/H208761/2
exp /H20849−q/H9272B/kT/H20850./H208494/H20850
Here Dnis the diffusion coefficient, NCis the effective
density of states in the conduction band, Vbiis the built-in
potential, Vis the applied potential, /H9255sis the semiconductor
permittivity, and NDis the donor impurity density. A few
things can be noted from the comparison of Eqs. /H208492/H20850and /H208494/H20850.
First, IsDvaries more rapidly with the voltage but is less
sensitive to the temperature as compared to IsT. Second, IsD
depends on ND, which is not the case for pure thermionic
theory. Third, over low voltage range /H20849V/H11270Vbi/H20850the extra volt-
age dependence of Eq. /H208494/H20850will weaken yielding voltage de-
pendence only from the exponential term in Eqs. /H208491/H20850and /H208493/H20850.
Although SrTiO 3has fairly high carrier mobility for an
oxide, it is certainly not high enough to eliminate the pos-sible contribution of diffusion model to our case. Moreover,we did observe that the I-Vcharacteristics depend on Nb
concentration /H20849data not shown /H20850, further implying a role of
carrier diffusion. It is also clear however that even within theframework of diffusion model a more realistic estimate ofthe effective Schottky barrier height should be possible byrestricting to low voltage regime /H20849V/H11270V
bi/H20850. Finally, it is im-
portant to recognize that in our current case of fairly heavily
doped semiconductor, the tunneling current also makes animportant contribution. The modified equations then lead tothe definition of ideality factor given by the relation
n=1
2.3026 /H11003kT/q/H11003d/H20849logI/H20850/dV. /H208495/H20850
As Sze20has pointed out, the ideality factors can depart sub-
stantially from unity in the case of heavily doped semicon-ductors /H20849as in our case /H20850, the departure increasing with low-
ering temperature.
FIG. 5. High resolution scanning transmission electron micro-
scopy /H20849STEM /H20850data for optimally doped YBCO films grown on
Nb:SrTiO 3.ELECTRICAL PROPERTIES OF EPITAXIAL … PHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
205333-5In Fig. 6 /H20849a/H20850we show the dependence of the effective
Schottky barrier height /H20849SBH, denoted by /H9021B/H20850, calculated
using the rising portions of the forward biased I-Vcurves.
The observed decrease in SBH with temperature reflects thedegree of non-ideality of the junctions which we addresslater. The rates of decrease of /H9021
Bwith temperature for the
optimally doped, Zn-doped and oxygen deficient cases are/H20849a/H20850optimally doped film: 2.6 meV/K, /H20849b/H20850Zn-doped film:
2.2 meV/K, and /H20849c/H20850oxygen deficient film: 2.15 meV/K. The
room temperature values of the ideality factor nare /H20849a/H20850op-
timally doped film: 4.4, /H20849b/H20850Zn-doped film: 7.7, and /H20849c/H20850oxy-
gen deficient film: 7.27. The significant departure of nfrom
unity can be attributed to the heavily doped nature of thesemiconductor.
20Interestingly the values for the Zn-doped
and oxygen deficient films are considerably higher than theoptimally doped case. The rate of increase of nwith decreas-
ing temperature /H20849not shown /H20850is also much smaller in the op-
timally doped case as compared to the other two cases. Thevalue of SBH /H20849/H9021
B/H20850at room temperature is found to be
/H110110.73±0.02 eV for the case of the optimally doped YBCO
films, /H110110.63±0.02 eV for the Zn-doped and oxygen defi-
cient cases. Here the indicated deviation around the meanvalue reflects the sample to sample variation. The smallnessof this deviation reflects the robustness of the numbers, mak-ing them worthy of analyses. It also speaks for the intrinsicnature of the changes observed between different cases con-sidered. It is interesting that the SBH values for the oxygen
underdoped and Zn-doped YBCO films are lower than thosefor optimally doped YBCO cases but close to each other; thetwo cases representing considerably suppressed but closelyplaced superconducting transitions. We found that the SBHvalue for the optimally doped film grown at 750 °C is alsoclose to 0.73±0.02 eV, which establishes that the decrease inSBH for the junction based on Zn-doped film is not due togrowth condition related change, but is intrinsic to Zn dop-ing. Given the connection between the transition temperatureand the density of states at the Fermi energy /H20849
/H9267EF/H20850, this result
reflects the role of /H9267EFin defining /H9021Band possibly the related
changes in the work function. Clearly, these differences aresmall because they should appear via shifts in the weight ofa distribution. In Fig. 6 /H20849b/H20850we show the SBH dependence on
temperature for the three cases, estimated by restricting onlyto the low current regimes /H20849/H333555m A /H20850to see whether any
unexpected contributions at higher current influence the cal-
culated SBH values significantly. The changes are rathersmall, but we do see an interesting feature resolved in theoxygen deficient case, which relates to the correspondingS-shaped resistivity curve shown in Fig. 2. For this case we
observe a distinct downward jump of about 40–50 meV near/H11011125 K, which also corresponds to the temperature at which
the resistivity dependence on temperature /H20849Fig. 2 /H20850shows a
change in the slope. As pointed out earlier, over low voltage/H20849and therefore current /H20850range /H20849V/H11270V
bi/H20850the extra voltage de-
pendence of Eq. /H208494/H20850corresponding to carrier diffusion effects
is weak yielding a voltage dependence only from the expo-nential term in Eqs. /H208491/H20850and /H208493/H20850. The SBH values obtained
from the form of Eq. /H208491/H20850employed over low voltage /H20849cur-
rent /H20850range, presented in Fig. 6 /H20849b/H20850, may thus reveal the in-
trinsic features more clearly, such as the feature near 125 Kfor oxygen underdoped case. When the same equation is em-ployed over a broader voltage range /H20851as in Fig. 6 /H20849a/H20850/H20852some
such fine features could be wiped out because of the partialinapplicability of Eq. /H208491/H20850.
As discussed by Singh et al.
37the mechanisms contribut-
ing to the effective Schottky barrier height /H20849SBH /H20850and ideal-
ity factor nof metal-semiconductor /H20849MS /H20850junctions and their
interplay leading to a specific nature of temperature /H20849T/H20850de-
pendence of SBH and nhave been the subjects of many
interesting scientific investigations over the past fewdecades.
19–26Many theoretical models developed and dis-
cussed in this respect hinge upon the notion of Fermi level/H20849E
F/H20850pinning by the electronic midgap states or the interface
defect states the corresponding Tdependence being sugges-
tive of the mechanism of Fermi level pinning. For an EF
pinned by midgap states, the Tdependence of the barrier
height should mimic that of the band gap. However, if EFis
pinned by the interface defects, the Tdependence of the SBH
is controlled by the ionization entropy of such defects. Thedifferences between the Tdependences in the two cases
should, in principle, allow one to discern the operative pin-ning mechanism in the specific case. Card and Rhoderick
19
have shown that the ideality factor for a Schottky diode isrelated to the density of interface states between the metaland semiconductor by the equation
n=1+/H20849
/H9254//H9255i/H20850/H20849/H9255s/W+qDsb/H20850
1+ /H20849/H9254//H9255i/H20850qDsa.
FIG. 6. Dependence of the effective Schottky barrier height
/H20849SBH /H20850on temperature for optimally doped, underdoped, and Zn-
doped films on SrTiO 3obtained from Eq. /H208491/H20850/H20849a/H20850over larger current
range /H208495–20 mA /H20850, and /H20849b/H20850over low current regime /H208490–5 mA /H20850.RAMADAN et al. PHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
205333-6Here/H9254is the thickness of the insulating surface layer, W
is the width of the depletion region in the semiconductor, and/H9255
iand/H9255sare the permittivities of the insulating layer and the
semiconductor, respectively. Also, DsaandDsbrepresent the
density of states that are in equilibrium with the metal andthe semiconductor, respectively. Clearly, a higher density ofthe interface states in equilibrium with the semiconductor ascontrolled by the corresponding Fermi energy /H20849D
sb/H20850increase
the ideality factor n. As Dharmadasa et al.22have discussed,
the value of Dsband the energy positions of the correspond-
ing states in the band have implications for the value of SBHas well as the precise nature of the I-Vcharacteristics. How-
ever, in all these theoretical models a laterally homogeneousSBH is presumed and the role of SBH inhomogeneity, whichcan occur under most realistic experimental conditions, is notaddressed. Indeed, experimental investigations have estab-lished the existence of such nanoscale local non-uniformitiesof SBH. Tung has modeled and analyzed the implications ofsuch SBH inhomogeneity for the properties such as the inte-gral SBH.
8,23Assuming nanoscale lateral variations in local
SBH he has been able to offer satisfactory explanation of thetemperature evolution of the apparent barrier height /H20849/H9021
B/H20850
and ideality factor /H20849n/H20850. It is very important to mention here
that the effects discussed by Tung occur when the length
scale of lateral fluctuations is smaller than the width of thedepletion layer width W. On the other hand, if the fluctua-
tions are on a longer length scale than W, regions with dif-
ferent local SBHs are essentially electrically independent andthe total junction current is a sum of currents through differ-ent areas.
73
In our case, the observed, rather strong temperature de-
pendence of SBH /H20849/H110112.2±0.1 meV/K /H20850cannot be explained
by midgap or interface state pinning mechanisms. This is
because the value of temperature coefficient of SBH for thecase of midgap state pinning, which mimics the band-gapchange, cannot be stronger than a very small fraction of theabove values. Also, the ionization entropy of the interfacedefects depends only very weakly on temperature.
37Thus, as
in the case of many real Schottky junctions, in our case thelateral nanoscale SBH inhomogeneity as proposed and ana-lyzed by Tung
23and Werner and Guttler24appear to be re-
sponsible for the observed dependences of /H9021BandnonT.
Assuming a Gaussian type distribution function for SBH in-homogeneity, Dobrocka and Osvald
26have shown that this
dependence is strongest for the largest standard deviation ofthe distribution function.
In real junctions lateral inhomogeneity could result from
factors such as strain, chemical composition or interface to-pological fluctuations, defect distributions etc. There can alsobe intrinsic electronic inhomogeneity in the two layers form-ing the junction which would manifest in the I-Vresponse.
The fact that the temperature variation of /H9021
Bcorrelates with
theS-shaped resistivity dependence on temperature in the
case of the junction based on oxygen deficient film is impor-tant to note in this context. Also, a higher degree of nonide-ality in oxygen deficient and Zn-doped case could possiblybe due to the sensitivity of local electronic states to Zn oroxygen vacancy concentration and the natural lateral fluctua-tions in these. One may also need to consider the fluctuationsin the concentration of Niobium in the substrate /H20849NSTO /H20850which would cause changes in the local carrier concentration
and thereby fluctuations in SBH. Given the use of heavilydoped /H208490.5 wt. % /H20850Nb:SrTiO
3semiconductor in our experi-
ments implying small depletion width W, our observations
suggest that nanoscale electrical property fluctuations arepresent in the optimally doped, underdoped and Zn-dopedYBCO film even in the normal state. Presence of nanoscaleelectrical property fluctuations in oxide superconductorshave indeed been addressed in the literature.
54–56,58,59,74Pan
et al. have shown that such inhomogeneity manifests in the
form of spatial variations in both the local density of statesspectrum and the superconducting energy gap. These occurover surprisingly short length scale of the order of 1–2 nm.Lang et al. have shown an apparent segregation of the elec-
tronic structure into domains of /H110113 nm reflecting the hole
redistribution. Since SBH is basically the difference betweenthe metal work function and the electron affinity of thesemiconductor,
20the basic manifestation of the inhomogene-
ity in the effective SBH must emanate from the inhomoge-neity in the superconductor work function. Indeed suchnanoscale work function inhomogeneity has been reported inoxide superconductors by direct STM studies.
75,54–56In the
case of superconductors /H20849optimally doped, oxygen under-
doped, and Zn-doped /H20850there could be a close connection of
the work function fluctuations to the modulations of carrierdensity.
It is interesting to point out here that inhomogeneity over
different length scales has been reported in different mixed-valent colossal magnetoresistance manganite systems.
76Yet,
in specific cases involving manganite electrode/H20849La
0.67Sr0.33MnO 3/H20850in contact with Nb:SrTiO 3nearly ideal
Schottky characteristics have been recently realized.77The
reason for this can be attributed to /H20849a/H20850much larger length
scale of electronic phase separation /H20849and the corresponding
electrical property fluctuation /H20850in the more metallic LSMO
system /H20849as compared to cases such as /H20849La0.67Ca0.33MnO 3or
PrxLa0.67− xCa0.33MnO 3/H20850,76/H20849b/H20850major differences in the elec-
trical properties of the two components in the phase sepa-rated state /H20849ferromagnetic metal and charge ordered
insulator /H20850,
76and /H20849c/H20850use of 0.01 wt. % and 0.1 wt. %
Nb:SrTiO 3substrates yielding larger value of depletion
width W, reducing tunneling contribution /H20849as against the case
of 0.5 wt. % Nb:SrTiO 3in the present work /H20850.
IV . CONCLUSIONS
The properties of Schottky junctions formed by pulsed
laser deposited films of optimally doped, underdoped, andZn-doped YBa
2Cu3O7−/H9254on single crystal /H20849001 /H208500.5 wt.%
Nb:SrTiO 3substrates were examined. The growth condi-
tions were optimized in each case to get the appropriate crys-talline quality of the films as well as the desired normal stateand superconducting properties. The high quality of the filmsas well as the interfaces was established using different ap-propriate techniques. Nonlinear I-Vcurves /H20849forward and re-
verse /H20850were obtained in all cases. Analyses of these data
brought out the intrinsic differences between the parametricvalues in the three cases, as well as the temperature variationof the Schottky Barrier Height /H20849SBH, /H9021/H20850and the idealityELECTRICAL PROPERTIES OF EPITAXIAL … PHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
205333-7factor /H20849n/H20850in each case. The small but definitive lowering of
/H9021in the case of oxygen underdoped and Zn-doped YBCO
reflects the role of /H9267EFin defining /H9021Band possibly the related
changes in the work function. These differences are smallbecause they should appear via shifts in the weight of a dis-tribution. The systematics of temperature dependence of /H9021
andncan be accounted for within the model of nanoscale
electrical property inhomogeneity. Given the high quality ofinterfaces, we suggest that this inhomogeneity is intrinsic innature. Interesting correlation is noted between the S-shaped
resistivity dependence on Tin the oxygen deficient sample
and the temperature variation of SBH estimated over lowcurrent regime, bringing out an energy shift of about 40 meVat the inflection point.ACKNOWLEDGMENTS
One of the authors /H20849W.R. /H20850will like to thank the Fulbright
commission for support. The authors would like to thankRichard Greene for fruitful discussions. The work was per-formed using the shared experimental facilities /H20849SEFs /H20850of
pulsed laser deposition and Pelletron accelerator supportedby the Center for Superconductivity Research and UMDNSF-MRSEC Grant No. DMR 00-80008. The microscopywork /H20849N.D.B. /H20850was supported by NSF Grant No. DMR-
0335364 and performed in the National Center for ElectronMicroscopy at Lawrence Berkeley National Laboratory sup-ported by the Department of Energy under Contract No. DE-AC03-73SF00098.
1T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437
/H208491982 /H20850.
2V. A. Markov and S. Stoyanov, Contemp. Phys. 28, 267 /H208491987 /H20850.
3P. Batra, E. Tekman, and S. Ciraci, Prog. Surf. Sci. 36, 289
/H208491991 /H20850.
4G. P. Das, Pramana, J. Phys. 38, 545 /H208491992 /H20850.
5A. Franciosi and C. G. VandeWalle, Surf. Sci. Rep. 25,1/H208491996 /H20850.
6M. Peressi, N. Binggeli, and A. Baldereschi, J. Phys. D 31, 1273
/H208491998 /H20850.
7G. Margaritondo, Rep. Prog. Phys. 62, 765 /H208491999 /H20850.
8R. T. Tung, Mater. Sci. Eng., R. 35,1 /H208492001 /H20850.
9S. P. Wilks, J. Phys. D 35, R77 /H208492002 /H20850.
10B. T. Jonker, Proc. IEEE 91, 727 /H208492003 /H20850.
11J. M. Triscone and O. Fischer, Rep. Prog. Phys. 60, 1673 /H208491997 /H20850.
12D. P. Norton, Annu. Rev. Mater. Sci. 28, 299 /H208491998 /H20850.
13K. Ueda, H. Tabata, and T. Kawai, Science 280, 1064 /H208491998 /H20850.
14G. D. Wilk, R. M. Wallace, and J. M. Anthony, J. Appl. Phys. 89,
5243 /H208492001 /H20850.
15A. Ohtomo, D. A. Muller, J. L. Grazul, and H. Y. Hwang, Nature
/H20849London /H20850419, 378 /H208492002 /H20850.
16A. Ohmoto and H. Y. Hwang, Nature /H20849London /H20850427, 423 /H208492004 /H20850.
17Y. Ijiri, J. Phys.: Condens. Matter 14, R947 /H208492002 /H20850.
18D. P. Norton, Mater. Sci. Eng., R. 43, 139 /H208492004 /H20850.
19H. C. Card and E. H. Rhodreick, J. Phys. D 4, 1589 /H208491971 /H20850.
20S. M. Sze, Physics of Semiconductor Devices , 2nd ed. /H20849Wiley
Interscience, New York, 1981 /H20850.
21J. L. Freeouf, T. N. Jackson, S. E. Laux, and J. M. Woodall, Appl.
Phys. Lett. 40, 634 /H208491982 /H20850.
22I. M. Dharmadasa, G. G. Roberts, and M. C. Petty, J. Phys. D 15,
901 /H208491982 /H20850.
23R. T. Tung, Appl. Phys. Lett. 58, 2821 /H208491991 /H20850.
24J. H. Werner and H. H. Guttler, Appl. Phys. Lett. 56, 1113
/H208491990 /H20850; J. Appl. Phys. 69, 1522 /H208491991 /H20850; J. Appl. Phys. 73, 1315
/H208491993 /H20850.
25J. P. Sullivan, R. T. Tung, and M. R. Pinto, J. Appl. Phys. 70,
7403 /H208491991 /H20850.
26E. Dobrocka and J. Osvald, Appl. Phys. Lett. 65, 575 /H208491994 /H20850.
27V. G. Bozhkov and D. Ju. Kuzyakov, J. Appl. Phys. 92, 4502
/H208492002 /H20850.
28L. L. Chang and C. M. Hu, Superlattices Microstruct. 28, 351
/H208492000 /H20850.29H. Morkoc and Y. Taur, J. Korean Phys. Soc. 42, S 555 /H208492003 /H20850.
30S. E. Thompson, R. S. Chou, T. Ghani, K. Mistry, S. Tyagi, and
M. T. Bohr, IEEE Trans. Semicond. Manuf. 18,2 6 /H208492005 /H20850.
31S. B. Ogale, V. Talyansky, C. H. Chen, R. Ramesh, R. L. Greene,
and T. Venkatesan, Phys. Rev. Lett. 77, 1159 /H208491996 /H20850.
32D. M. News, J. M. Misewich, C. C. Tsuei, A. Gupta, B. A. Scott,
and A. Schrott, Appl. Phys. Lett. 73, 780 /H208491998 /H20850.
33T. Venkatesan, M. Rajeswari, Z. W. Dong, S. B. Ogale, and R.
Ramesh, Philos. Trans. R. Soc. London, Ser. A 356, 1661
/H208491998 /H20850.
34Y. Tokura and Y. Tomioka, J. Magn. Magn. Mater. 200,1/H208491999 /H20850.
35T. Wu, S. B. Ogale, J. E. Garrison, B. Nagaraj, A. Biswas, Z.
Chen, R. L. Greene, R. Ramesh, T. Venkatesan, and A. J. Millis,Phys. Rev. Lett. 86, 5998 /H208492001 /H20850.
36T. Minami, MRS Bull. 25,3 8 /H208492000 /H20850.
37R. Singh, S. K. Arora, R. Tyagi, S. K. Agarwal, and D. Kanjilal,
Bull. Mater. Sci. 23, 471 /H208492000 /H20850.
38H. Tanaka, J. Zhang, and T. Kawai, Phys. Rev. Lett. 88, 027204
/H208492002 /H20850.
39F. X. Hu, J. Gao, J. R. Sun, and B. G. Shen, Appl. Phys. Lett. 83,
1869 /H208492003 /H20850.
40A. Tiwari, C. Jin, D. Kumar, and J. Narayan, Appl. Phys. Lett.
83, 1773 /H208492003 /H20850.
41J. R. Sun, C. M. Xiong, T. Y. Zhao, S. Y. Zhang, Y. F. Chen, and
B. G. Shen, Appl. Phys. Lett. 84, 1528 /H208492004 /H20850.
42J. R. Sun, C. M. Xiong, B. G. Shen, P. Y. Wang, and Y. X. Weng,
Appl. Phys. Lett. 84, 2611 /H208492004 /H20850.
43J. R. Sun, C. M. Xiong, Y. F. Chen, B. G. Shen, and L. Kang,
Europhys. Lett. 66, 868 /H208492004 /H20850.
44J. R. Sun, C. H. Lai, and H. K. Wong, Appl. Phys. Lett. 85,7 3
/H208492004 /H20850.
45T. Muramatsu, Y. Muraoka, and Z. Hiroi, Solid State Commun.
132, 351 /H208492004 /H20850.
46M. Ziese, U. Kohler, R. Hohne, A. Bollero, and P. Esquinazi, J.
Magn. Magn. Mater. 290, 1116 /H208492005 /H20850.
47C. Ren, J. Trbovic, P. Xiong, and S. Von Molnar, Appl. Phys.
Lett. 86, 012501 /H208492005 /H20850.
48T. R. Chien, Z. Z. Wang, and N. P. Ong, Phys. Rev. Lett. 67, 2088
/H208491991 /H20850.
49J. M. Harris, Y. F. Yan, and N. P. Ong, Phys. Rev. B 46, R14293
/H208491992 /H20850.RAMADAN et al. PHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
205333-850T. Ito, K. Takenaka, and S. Uchida, Phys. Rev. Lett. 70, 3995
/H208491993 /H20850.
51Y. Fukuzumi, K. Mizuhashi, K. Takenaka, and S. Uchida, Phys.
Rev. Lett. 76, 684 /H208491996 /H20850.
52B. Wuyts, V. V. Moshchalkov, and Y. Bruynseraede, Phys. Rev. B
53, 9418 /H208491996 /H20850and references therein.
53R. P. Sharma, S. B. Ogale, Z. H. Zhang, J. R. Liu, W. K. Chu, B.
Veal, A. Paulikas, H. Zheng, and T. Venkatesan, Nature/H20849London /H20850404, 737 /H208492000 /H20850.
54S. H. Pan, J. P. O’Neal, R. L. Badzey, C. Chamon, H. Ding, J. R.
Engelbrecht, Z. Wang, H. Eisaki, S. Uchida, A. K. Gupta, K.-W.Ng, E. W. Hudson, K. M. Lang, and J. C. Davis, Nature/H20849London /H20850413, 282 /H208492001 /H20850.
55D. N. Shi, Z. D. Wang, and J. X. Li, Physica C 371,6 9 /H208492002 /H20850.
56K. M. Lang, V. Madhavan, J. E. Hoffman, E. W. Hudson, H.
Eisaki, S. Uchida, and J. C. Davis, Nature /H20849London /H20850415, 412
/H208492002 /H20850.
57Z. Y. Chen, A. Biswas, I. Zutic, T. Wu, S. B. Ogale, R. L. Greene,
and T. Venkatesan, Phys. Rev. B 63, 212508 /H208492001 /H20850.
58E. Li, R. P. Sharma, S. B. Ogale, Y. G. Zhao, T. Venkatesan, J. J.
Li, W. L. Cao, and C. H. Lee, Phys. Rev. B 65, 184519 /H208492002 /H20850.
59E. Li, S. B. Ogale, R. P. Sharma, T. Venkatesan, J. J. Li, W. L.
Cao, and C. H. Lee, Phys. Rev. B 69, 134520 /H208492004 /H20850.
60S. B. Ogale, M. Rajeswari, R. P. Sharma, and T. Venkatesan,
Appl. Phys. Lett. 68, 421 /H208491996 /H20850.
61A. Yoshida, H. Tamura, K. Gotoh, H. Takauchi, and S. Hasuo, J.
Appl. Phys. 70, 4976 /H208491991 /H20850.
62A. Baikalov, Y. Wang, B. Shen, B. Lorenz, S. Tsui, Y. Y. Sun, Y.
Y. Xue, and C. W. Chu, Appl. Phys. Lett. 83, 957 /H208492003 /H20850.
63A. Sawa, T. Fujii, M. Kawasaki, and Y. Tokura, Appl. Phys. Lett.
85, 4073 /H208492004 /H20850.
64T. Fujii, M. Kawasaki, A. Sawa, H. Akoh, Y. Kawazoe, and Y.Tokura, Appl. Phys. Lett. 86, 012107 /H208492005 /H20850.
65B. W. Veal, H. You, A. P. Paulikas, H. Shi, Y. Fang, and J. W.
Downey, Phys. Rev. B 42, 4770 /H208491990 /H20850.
66B. H. Moeckly, D. K. Lathrop, and R. A. Buhrman, Phys. Rev. B
47, 400 /H208491993 /H20850.
67N. Chandrasekhar, O. T. Valls, and A. M. Goldman, Phys. Rev.
Lett. 71, 1079 /H208491993 /H20850.
68S. H. Huerth, M. P. Taylor, H. D. Hallen, and B. H. Moeckly,
Appl. Phys. Lett. 77, 2127 /H208492000 /H20850.
69R. Ramesh, D. M. Hwang, T. Venkatesan, T. S. Ravi, L. Nazar, A.
Inam, X. D. Wu, B. Dutta, G. Thomas, A. F. Marshall, and T. H.Gaballe, Science 247,5 7 /H208491990 /H20850.
70L. F. Fu et al. /H20849unpublished /H20850.
71H. Hasegawa, T. Fukazawa, and T. Aida, Jpn. J. Appl. Phys., Part
228, L2210 /H208491989 /H20850.
72J. H. Takemoto, C. M. Jackson, H. M. Manasevit, D. C. St. John,
J. F. Burch, K. P. Daly, and R. W. Simon, Appl. Phys. Lett. 58,
1109 /H208491991 /H20850.
73I. Ohdomari and K. N. Tu, J. Appl. Phys. 51, 3735 /H208491980 /H20850.
74H. A. Mook, P. C. Dai, F. Dogan, and R. D. Hunt, Nature
/H20849London /H20850404, 729 /H208492000 /H20850.
75J. F. Jia, Y. Hasegawa, T. Sakurai, and H. Zhang, Solid State
Commun. 105, 533 /H208491998 /H20850.
76Colossal Magnetoresistive Oxides , edited by Y. Tokura /H20849Gordon
and Breach, New York, 2000 /H20850; M. Uehara, S. Mori, C. H. Chen,
and S.-W. Cheong, Nature /H20849London /H20850399, 560 /H208491999 /H20850; J. Burgy,
A. Moreo, and E. Dagotto, Phys. Rev. Lett. 92097202 /H208492004 /H20850,
and references therein.
77F. M. Postma, R. Ramaneti, T. Banerjee, H. Gokcan, D. H. A.
Blank, R. Jansen, and J. C. Lodder, J. Appl. Phys. 95, 7324
/H208492004 /H20850.ELECTRICAL PROPERTIES OF EPITAXIAL … PHYSICAL REVIEW B 72, 205333 /H208492005 /H20850
205333-9 |
PhysRevB.97.024515.pdf | PHYSICAL REVIEW B 97, 024515 (2018)
p-wave superconductivity in weakly repulsive 2D Hubbard model with Zeeman splitting
and weak Rashba spin-orbit coupling
Henning G. Hugdal*and Asle Sudbø†
Department of Physics, NTNU, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, and Center for Quantum
Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
(Received 7 November 2017; revised manuscript received 11 January 2018; published 24 January 2018)
We study the superconducting order in a two-dimensional square lattice Hubbard model with weak repulsive
interactions, subject to a Zeeman field and weak Rashba spin-orbit interactions. Diagonalizing the noninteractingHamiltonian leads to two separate bands, and by deriving an effective low-energy interaction we find the meanfield gap equations for the superconducting order parameter on the bands. Solving the gap equations just below thecritical temperature, we find that superconductivity is caused by Kohn-Luttinger-type interaction, while the pairingsymmetry of the bands is indirectly affected by the spin-orbit coupling. The dominating attractive momentumchannel of the Kohn-Luttinger term depends on the filling fraction nof the system, and it is therefore possible to
change the momentum dependence of the order parameter by tuning n. Moreover, nalso determines which band
has the highest critical temperature. Rotating the magnetic field changes the momentum dependence from statesthat for small momenta reduce to a chiral p
x±ipytype state for out-of-plane fields, to a nodal p-wave-type state
for purely in-plane fields.
DOI: 10.1103/PhysRevB.97.024515
I. INTRODUCTION
Much attention has been paid to the possibility of un-
conventional superconductivity due to weak repulsive inter-actions, as first pointed out by Kohn and Luttinger in 1967[1]. They found that because of oscillations in long-range
interactions, a p-wave superconducting state could be formed
in a three-dimensional electron gas at O(U
2) in the inter-
action strength U. In two dimensions, no such state can be
formed at O(U2)[2]; it is only present at O(U3) and zero
temperature [ 3]. However, by applying a magnetic field the
effect is present on the majority band also at second orderinU[2,4].
In systems with broken inversion symmetry, either due
to crystal structure or an applied electric field, one has toinclude the effects of spin-orbit interactions by including aRashba spin-orbit coupling (SOC) term [ 5,6]. A Rashba term
in the system Hamiltonian will lead to a coupling between thespin-up and spin-down Fermi surfaces, and hence opens upthe possibility of proximity-induced superconductivity on theminority band [ 4,7]. Reference [ 8] provides a recent review on
superconductivity in systems with broken inversion symmetry.The effects of magnetic fields and spin-orbit coupling intwo-dimensional systems has been studied in various cases,in limiting cases of, e.g., the strength of the SOC or thedirection of the magnetic field [ 4,9–14]. Recently Lake et al. [7]
studied a weakly spin-orbit-coupled two-dimensional electrongas (2DEG) with a magnetic field which could be rotated inand out of the plane. They reported that topological p+ip
superconductivity is realized when the field is perpendicular
*henning.g.hugdal@ntnu.no
†asle.sudbo@ntnu.noto the plane, while an in-plane magnetic field in the xdirection
leads to a pymomentum dependence of the order parameter.
In either case, only the majority band was found to besuperconducting.
In this paper, we perform an analysis similar to that of
Ref. [ 7] to study the superconducting order in a weakly repul-
sive, spin-polarized Hubbard model on a two-dimensional (2D)square lattice with weak SOC. Such systems can be realized,e.g., at the interface between LaAlO
3and SrTiO 3, which has
been shown to exhibit a 2D superconducting state [ 15–17], a
magnetic state [ 18], and coexistence of superconductivity and
magnetism [ 19–22]. Moreover, it has been shown that the SOC
at the interface can be tuned by a gate voltage or an appliedelectric field [ 23,24].
By finding the superconducting state that emerges at the
critical temperature T
c, we study the dominating pairing
symmetries on the two bands for different filling fractions andmagnetic field orientations. We find that superconductivity canbe induced on both bands , depending on the filling fraction.
We also find that two different pairing symmetries are realized,one for nearly empty or nearly filled bands, and one close tohalf filling. However, the small-momentum limit of the orderparameters are the same in both regions, a chiral p
x±ipy
symmetry for purely out-of-plane fields and p-wave state state
for purely in-plane fields. We also find that the Cooper pairshave a finite center-of-mass momentum [ 7,14,25], i.e., a Fulde-
Ferrell-Larkin-Ovchinnikov state (FFLO) [ 26,27], whenever
the magnetic field has an in-plane component.
The remainder of the paper is organized as follows: The
model system is presented in Sec. IItogether with the derivation
of the effective Hamiltonian and self-consistent equations forthe mean field superconducting gap. The numerical solutionstrategy is discussed in Sec. III, the results of which are pre-
sented in Sec. IV. Finally, we summarize our results in Sec. V.
2469-9950/2018/97(2)/024515(10) 024515-1 ©2018 American Physical SocietyHENNING G. HUGDAL AND ASLE SUDBØ PHYSICAL REVIEW B 97, 024515 (2018)
xyz
B
δ
θ
FIG. 1. Sketch of system geometry, where the 2D lattice is located
in the xyplane and the magnetic field Bcan point in any direction.
II. MODEL
Our starting point is a two-dimensional lattice in the pres-
ence of an external magnetic field, and with broken inversionsymmetry such that SOC is present. A sketch of the geometry iss h o w ni nF i g . 1. We use the Hubbard model augmented by SOC
to describe the fermions on the lattice, with a spin-diagonalhopping integral between nearest-neighbor lattice sites givenbyt, and the electrons interact via a onsite repulsion Un
i↑ni↓,
U> 0. We will assume that the interaction is weak, i.e., the
energy scale of the Hubbard interaction is small comparedto the kinetic energy, U/t/lessmuch1. Time-reversal symmetry is
broken by applying an external magnetic field B, which couples
to the electrons via the Zeeman coupling −gμ
BB·σ/2, where
gis thegfactor and μBis the Bohr magneton. This lifts the
degeneracy between the spin directions. The effect of SOCis included via a Rashba term with spin-orbit axis normalto the lattice plane, α
R(p×σ)·ˆz, where αRis the strength
of the spin-orbit coupling. We thus obtain the total systemHamiltonian H=H
t+HB+HR+HI=H0+HI, with
Ht=/summationdisplay
σ,k/epsilon1kc†
kσckσ, (1a)
HR=αR/summationdisplay
k/summationdisplay
σ,σ/prime/parenleftbig
σy
σσ/primesinkx−σx
σσ/primesinky/parenrightbig
c†
kσckσ/prime,(1b)HB=−H·/summationdisplay
k/summationdisplay
σ,σ/primeσσσ/primec†
kσckσ/prime, (1c)
HI=U
V/summationdisplay
k1,k2,k3c†
k1↑c†
k2↓ck3↓ck1+k2−k3,↑, (1d)
where /epsilon1k≡−2t(coskx+cosky)−μis the square-lattice
tight-binding dispersion relative the chemical potential μ,
σ=↑,↓denotes spin-up and spin-down electrons re-
spectively, H=gμBB/2=h(cosθsinδˆx+sinθsinδˆy+
cosδˆz), and Vis the volume of the system. For notational
simplicity, we have set ¯ hand the lattice constant ato 1
throughout the paper.
A. Diagonalization of noninteracting Hamiltonian
Following Ref. [ 7], we will treat the SOC as a perturbation,
assuming that αR/h/lessmuch1. Hence, we expect that when diago-
nalizing the noninteracting Hamiltonian H0, the lowest order
expression will simply be that of a tight-binding system withspins polarized along the direction of H. We therefore rotate
the spin quantization axis to point along the magnetic fieldusing the unitary rotation operator R
n(α)=exp(−iασ·ˆn/2),
where αis the angle of rotation about an axis ˆn:W efi r s tr o t a t e
an angle θabout ˆn=ˆz, and then an angle δabout ˆn=ˆy.T h i s
yields
H0=/summationdisplay
k/summationdisplay
σ,σ/primeEσσ/prime(k)c†
kσckσ/prime, (2)
where
E(k)=/epsilon1kσ0−hσz+αR[(sinkxsinθ−sinkycosθ) cosδσx
+(sinkxcosθ+sinkysinθ)σy
+(sinkysinθ−sinkycosθ)s i nδσz]. (3)
Diagonalizing H0leads to two bands with eigenenergies
/epsilon1λ(k)=/epsilon1k−ζλ/radicalBig
h2−2hαR(sinkxsinθ−sinkycosθ)s i nδ+α2
R(sin2kx+sin2ky)
≈/epsilon1k−ζλ/bracketleftbigg
h−αR(sinkxsinθ−sinkycosθ)s i nδ+α2
R
2h(sinkxcosθ+sinkysinθ)2
+α2
R
2h(sinkxsinθ−sinkycosθ)2cos2δ/bracketrightbigg
, (4)
where ζλ=1(2)=+(−)1 for the majority (minority) band. In
the last line, we have kept terms only up to first order inα
R/h. In the limit |k|/lessmuch1 and θ=0, this result agrees with
Ref. [ 7]. When the magnetic field has an in-plane component,
the momentum qcorresponding the minima of the band
dispersions will shift away from the origin according to
qx≈−ζλαR
2tsinδsinθ, (5a)
qy≈+ζλαR
2tsinδcosθ. (5b)This shift is illustrated in Fig. 2.
Using the eigenvalues in Eq. ( 4), we also find relations
between the spin and band creation and annihilation operators,which to second order in α
R/hare given by
ck↑=/bracketleftbigg
1−α2
R
8h2|γ(k,δ,θ)|2/bracketrightbigg
ak1+αR
2hγ(k,δ,θ)ak2,(6a)
ck↓=−αR
2hγ†(k,δ,θ)ak1+/bracketleftbigg
1−α2
R
8h2|γ(k,δ,θ)|2/bracketrightbigg
ak2,
(6b)
024515-2p-WA VE SUPERCONDUCTIVITY IN WEAKLY … PHYSICAL REVIEW B 97, 024515 (2018)
−1.0 −0.5 0.0 0.5 1.0
kx/π−1.0−0.50.00.51.0ky/πλ=1
λ=2
FIG. 2. Plot of the Fermi levels for an in-plane magnetic field ( δ=
π/2) with angle θ=π/4 relative to the xaxis, for filling fraction n=
0.3, magnetic field strength h/t=1, and SOC strength αR/t=0.2.
The momenta corresponding to the minima of the band dispersions
are shifted away from the origin according to Eq. ( 5). Note that the
shift is exaggerated compared to what will be considered throughout
the paper.
where we have defined the function
γ(k,δ,θ)=(sinkxsinθ−sinkycosθ) cosδ
−i(sinkxcosθ+sinkysinθ)
anda†
kλandakλare the creation and annihilation operators
for band λrespectively. Using these relations, we find that
the expectation value of the zcomponent of the spin is
1/2 for the majority λ=1 band and −1/2 for the minority
λ=2 band, with the corrections being second order in αR/h.
Hence, to lowest order, the majority and minority bands consistof spin-up and spin-down particles, respectively. This hasconsequences for the momentum dependence of any intrabandinteraction which could lead to superconductivity. In the nextsection, we will transform the interaction Hamiltonian usingthe above operator relations and obtain an effective low-energytheory using a Schrieffer-Wolff transformation [ 28].
B. Transformation of the interaction Hamiltonian
Since the interaction Hamiltonian HIis proportional to U,
where we have assumed that the interaction is weak, U/t/lessmuch1,
we have to consider what powers of UandαR/hto keep when
transforming the Hamiltonian to the eigenbasis according toEq. ( 6). Following Ref. [ 7], we keep terms of O(U
2/t2) and
O(Uα2
R/th2), while disregarding terms of O(U2αR/t2h); i.e.,
we assume αR/h/greatermuchU/t.
Transforming the creation and annihilation operators in
HI, we get four main types of terms: intraband and pair-
hopping terms a†
k1λa†
k2λak3μak4μofO(Uα2
R/h2); interband
terms a†
k1λa†
k2¯λak3¯λak4λofO(U); and mixed terms such as
a†
k1λa†
k2λak3λak4¯λofO(UαR/h) and higher. The notation ¯λ
denotes the opposite band of λ. We collect the intraband andpair-hopping terms in H1and the remaining terms in H2:
H1=/summationdisplay
k,k/prime,q/summationdisplay
λ,μUα2
R
4Vh2/Gamma1λ/parenleftBig
k+q
2/parenrightBig
/Gamma1†
μ/parenleftBig
k/prime+q
2/parenrightBig
×a†
−k+q
2,λa†
k+q
2,λak/prime+q
2,μa−k/prime+q
2,μ, (7)
where
/Gamma1λ(k)=ζλ[sinkxcosθ+sinkysinθ
+iζλ(sinkxsinθ−sinkycosθ) cosδ]( 8 )
and
H2=U
2V/summationdisplay
k1,k2,k3/summationdisplay
λa†
k1λa†
k2¯λak3¯λak1+k2−k3,λ+O/parenleftbiggUαR
h/parenrightbigg
.
(9)
The terms in H2correspond to processes where the resulting
quasiparticles are on different bands, and including suchinteractions in a mean-field treatment would require orderparameters with mixed-band indices. In order to get a formof the interaction suitable for analysis within a mean-fieldtheory, we perform a Schrieffer-Wolff transformation; see, e.g.,Ref. [ 29] for a review. This enables us to get rid of the lowest
order processes in H
2while still including the effects of H2to
higher order, such as an intraband process at O(U2). This is
obtained by the unitary transformation
H/prime=e−SHeS=H0+H1+H2+[H0+H1+H2,S]
+1
2[[H0+H1+H2,S],S]+··· , (10)
where Sis an antiunitary operator chosen such that
[H0,S]=−H2. The lowest order term in Sis necessar-
ily of O(U/t), and this is the only contributing term
to the order we are working. Using as an ansatz S=/summationtext
k1,k2,k3/summationtext
λCλ(k1,k2,k3,k4)a†
k1λa†
k2¯λak3¯λak4λ, where k4=
k1+k2−k3, we find
S=U
2V/summationdisplay
k1,k2,k3,k4/summationdisplay
λa†
k1λa†
k2¯λak3¯λak4λδ(k1+k2−k3−k4)
/epsilon1λ(k4)+/epsilon1¯λ(k3)−/epsilon1¯λ(k2)−/epsilon1λ(k1).
(11)
Since Scomes with a factor U, we can neglect most of the
terms in the transformed Hamiltonian, leaving us with H/prime=
H0+H1+[H2,S]/2. Hence, the contributing higher order
processes due to H2are found by calculating the commutator
between H2andS.
The commutator leads to two kinds of terms of relevant
order: a four-operator interband term proportional to a†
λa†
¯λa¯λaλ
and six-operator terms a†
λa†
λaλaλa†
¯λa¯λ, both of O(U2/t2).
However, since the interactions must conserve momentum,and the interacting particles are close to the Fermi level, thephase space of the interband interaction is severely limited,as illustrated in Fig. 3. Although the figure does not include
the shifts in the minima of the dispersions away from theorigin, Eq. ( 5), these shifts are small when α
R/h/lessmuch1, and the
argument should still hold. Hence we will neglect this term,and include only the six-operator terms.
An effective intraband process on band λis obtained
from the six-operator terms a†
λa†
λaλaλa†
¯λa¯λby projecting the
024515-3HENNING G. HUGDAL AND ASLE SUDBØ PHYSICAL REVIEW B 97, 024515 (2018)
−1 −0.50 0 .−1−0.500.51
k+q
2−k+q
2
q
k+q
2−k+q
2
λ=1λ=2ky/π
−1 −0.50 0 .51
51−1−0.500.51
k+q
2−k+q
2
qk+q
2
−k+q
2
kx/πky/π
FIG. 3. The figures illustrate that the interband scattering from
k+q/2a n d−k+q/2t ok/prime+q/2a n d−k/prime+q/2 has a very limited
phase space for both low (top) and high (bottom) filling fractions n.
Here we have not included the shifts in center-of-mass momenta, since
the shifts are small when αR/h/lessmuch1.
operators a†
¯λa¯λto the noninteracting ¯λband, which results
in a replacement a†
¯λka¯λk/prime→δ(k−k/prime)f(/epsilon1¯λ(k)) [4,7]. Here,
f(/epsilon1) is the Fermi-Dirac distribution function. Since the shifts
in center-of-mass momenta are small, including them in theinteraction terms leads to a correction of higher order thanwe are considering. We therefore specialize to the case wherethe total momentum of the particles interacting is zero, whichyields the result for the commutator
1
2[H2,S]=U2
2V/summationdisplay
k,k/prime/summationdisplay
λχ¯λ(k−k/prime)a†
−k/prime,λa†
k/prime,λak,λa−k,λ,
(12)
where we have defined the susceptibility
χλ(q)=1
V/summationdisplay
pf(/epsilon1λ(p+q))−f(/epsilon1λ(p))
/epsilon1λ(p+q)−/epsilon1λ(p). (13)
In contrast to the 2DEG case [ 4,7], we have not been able to
calculate the susceptibility analytically for the lattice model.However, a numerical calculation is possible, the results ofwhich will be discussed in Sec. III A.
Setting the total momentum of an interacting pair of parti-
cles to zero also in H
1, and collecting all terms, we arrive atthe effective low-energy Hamiltonian
H/prime=H0+/summationdisplay
k,k/prime/summationdisplay
λ,μgλμ(k,k/prime)a†
−k,λa†
k,λak/prime,μa−k/prime,μ,(14)
where we have defined the interaction matrix
gλμ(k,k/prime)=U2
2Vδλμχ¯λ(k/prime−k)+Uα2
R
4Vh2/Gamma1λ(k)/Gamma1†
μ(k/prime),(15)
where /Gamma1λ(k) is defined in Eq. ( 8). The first term in Eq. ( 15)i s
an intraband interaction due to the Kohn-Luttinger mechanism.The second term, which is caused by the SOC, contains bothintraband and pair-hopping terms, with opposite signs due tothe factors ζ
λζμ. We thus expect the two terms in Eq. ( 15)t og i v e
rise to different superconducting states. The first term gives riseto uncoupled ordered states on the two bands with different T
c,
while the second term couples the order parameters and shouldlead to simultaneous superconductivity on both bands.
C. Mean-field treatment
Defining the mean-field order parameters (gap functions)
/Delta1λ(k)=−/summationdisplay
k/prime,μ2gλμ(k,k/prime)/angbracketleftak/primeμa−k/primeμ/angbracketright, (16)
/Delta1†
μ(k)=−/summationdisplay
k/prime,λ2gλμ(k/prime,k)/angbracketlefta†
−k/primeλa†
k/primeλ/angbracketright, (17)
we rewrite the Hamiltonian in the standard way
H/prime=/summationdisplay
k,λ1
2{[/epsilon1λ(−k)−μ]+/Delta1λ(k)/angbracketlefta†
−kλa†
kλ/angbracketright}
+1
2/summationdisplay
k,λψ†
kλEλ(k)ψkλ. (18)
Here, we have defined the Nambu spinors ψkλ=(akλa†
−kλ)T
and the matrix
Eλ(k)=/parenleftbigg/epsilon1λ(k)−μ/Delta1 λ(k)
/Delta1†
λ(k)−/epsilon1λ(−k)+μ/parenrightbigg
. (19)
Performing a Bogoliuobov transformation yields
H/prime=E0+/summationdisplay
k,λ/bracketleftbigg/epsilon1λ(k)−/epsilon1λ(−k)
2+Eλ(k)/bracketrightbigg
nkλ, (20)
where nkλis the number operator of the Bogoliubov quasipar-
tices in the rotated basis and
Eλ(k)=/radicalBig
ξ2
λ(k)+|/Delta1λ(k)|2 (21)
is the approximate quasiparticle dispersion with ξλ(k)≡
[/epsilon1λ(k)+/epsilon1λ(−k)]/2. Moreover,
E0=1
2/summationdisplay
k,λ[ξλ(k)−Eλ(k)+/Delta1λ(k)/angbracketlefta†
−kλa†
kλ/angbracketright]. (22)
Since the SOC term in the system Hamiltonian is the only
term which breaks inversion symmetry, we have [ /epsilon1λ(k)−
/epsilon1λ(−k)]/2∼αR.T h e[ /epsilon1λ(k)−/epsilon1λ(−k)] term in the diagonal-
ized Hamiltonian thus leads to higher order corrections andwill therefore be neglected. Minimizing the free energy with
024515-4p-WA VE SUPERCONDUCTIVITY IN WEAKLY … PHYSICAL REVIEW B 97, 024515 (2018)
respect to /Delta1†
λ(k) yields the gap equations
/Delta1λ(k)=−/summationdisplay
k/prime,μgλμ(k,k/prime)/Delta1μ(k/prime)
Eμ(k/prime)tanh/parenleftbiggβEμ(k/prime)
2/parenrightbigg
,(23)
where β=1/kBT. Note that we have set αR=0i nEλ(k),
since including the effects of the SOC in the dispersion giverise to terms of higher order than what we are considering.
III. NUMERICAL SOLUTION STRATEGY
A. Calculation of the susceptibility
The susceptibility is obtained numerically from Eq. ( 13)
in the zero-temperature limit. Since the susceptibility entersthe gap equations Eq. ( 23) with a prefactor proportional to
U
2, we can neglect the effects of SOC and thus set αR=0
in the calculations. The results for the majority band for threedifferent nare shown in Fig. 4, together with the analytical
result for the 2DEG with Zeeman splitting treated in Refs. [ 4,7].
For low n, the susceptibility is isotropic and resembles the
2DEG result. Closer to half-filling, the susceptibility becomesmore anisotropic due to the anisotropy of the dispersion.
In order to find the dominating attractive pairing channels
due to the Kohn-Luttinger term in the gap equations, we expandthe results for the susceptibility in square lattice harmonics;see the Appendix for details. Considering only the dominantattractive pairing channels, we find that the susceptibility togood approximation can be written
χ
λ(k−k/prime)
=χ1
λ[gx+iy(k)gx−iy(k/prime)+gx−iy(k)gx+iy(k/prime)]
+χ2
λ[gx(k)gx(k/prime)+gy(k)gy(k/prime)]
+χ3
λ[gx(kx,2ky)gx(k/prime
x,2k/prime
y)+gy(kx,2ky)gy(k/prime
x,2k/prime
y)
+gx(2kx,ky)gx(2k/prime
x,k/prime
y)+gy(2kx,ky)gy(2k/prime
x,k/prime
y)],(24)
where we have defined the functions
2πgx+iy(k)=sinkx+isinky, (25a)
2πgx−iy(k)=sinkx−isinky, (25b)
2πgx(k)=2πgx(kx,ky)=2s i nkxcosky,(25c)
2πgy(k)=2πgy(kx,ky)=2 coskxsinky.(25d)
These functions are orthonormal, i.e.,/integraltext
1BZdkgi(k)g†
j(k)=
δij.
The values for the expansion coefficients χi
λfor different
filling fractions nare shown in Fig. 5forh=0.2tat zero
temperature. Notice that χi
λ(n)=χi
¯λ(1−n). We will in the
following focus on filling fractions where the first two termsin Eq. ( 24) suffice to describe the most attractive pairing
channel, i.e., the channel with the most negative coeffi-cientχ
i
λ. Regions where this does not simultaneously hold
for both susceptibilities, because of significant or dominant
FIG. 4. Plot of numerically calculated susceptibilities for the majority band at filling fraction (a) n=0.02, (b) n=0.2, and (c) n=0.45,
which is close to half-filling of the band. The spikes at q=0 are numerical divergences that do not contribute to the results when expanding in
square lattice harmonics. The susceptibility in the 2DEG case with Zeeman splitting treated in Refs. [ 4,7],χ(q)∝−1+Re/radicalbig
q2−(2kF)2/q,
is shown in panel (d) with kF/π=0.2 for comparison.
024515-5HENNING G. HUGDAL AND ASLE SUDBØ PHYSICAL REVIEW B 97, 024515 (2018)
0.0 0.2 0.4 0.6 0.8 1.0
n−1.5−1.0−0.50.00.5χi
λ[t−1]
n=0.10
n=0.45χ1
1
χ2
1
χ3
1χ1
2
χ2
2
χ3
2
FIG. 5. Plot of expansion coefficients χi
λin Eq. ( 24) as a function
of filling fraction nforh=0.2tat zero temperature. Filling fraction
n=0 corresponds to a completely empty system, and n=1t ot w o
completely filled bands. The gray regions indicate where keeping only
the first two terms in the expansion in Eq. ( 24) is not sufficient due to
dominant contributions to attractive pairing from other square latticeharmonics, such as the χ
3
λterm in Eq. ( 24).
contributions from other channels, are indicated by the gray
regions in the figure. The coefficients χi
λshould, strictly
speaking, be calculated at the temperature of the system, butwe expect the superconducting transition temperature to besufficiently low for this to be a good approximation.
The plots of the coefficients χ
i
λin Fig. 5illustrate two
important points. First, the dominant attractive Kohn-Luttingerpairing channel depends strongly on the filling fraction. Forinstance, the dominant attractive channel for intermediatefilling fractions differs from low and high filling fractions. Thisis related to the shape of the Fermi surfaces in these regions andcould lead to significantly different kdependences of the order
parameters in these regions. Second, the plots also show that themajority and minority bands have the most negative expansioncoefficient in different filling fraction intervals. Therefore,there exists a possibility that there can be a switching betweenbands with the highest T
c.
B. Momentum dependence of the order parameter
From the preceding subsection, we found that the poten-
tially dominating momentum dependence of the supercon-ducting gap due to the Kohn-Luttinger term in the interactionEq. ( 15) could be any of the four functions in Eq. ( 25). If,
however, the solution were to be determined by the secondterm in Eq. ( 15), the solution should be proportional to /Gamma1
λ(k)
in Eq. ( 8), which can be rewritten in terms of gx±iy(k),
/Gamma1λ(k)=πgx+iy(k)(ξλ−cosδ)(cosθ−isinθ)
+πgx−iy(k)(ξλ+cosδ)(cosθ+isinθ).(26)
Therefore, keeping only the dominant terms, the superconduct-
ing gap can be expanded using the four functions in Eq. ( 25),
/Delta1λ(k)=/Delta1x+iy
λgx+iy(k)+/Delta1x−iy
λgx−iy(k)+/Delta1x
λgx(k)
+/Delta1y
λgy(k). (27)C. Solutions close to the critical temperature Tc
The physically realizable solution of the gap equations
is the solution which corresponds to a global minimum ofthe free energy. However, when solving the gap equationsnumerically using, e.g., a root solver, the solution might just aswell correspond to a local minimum of the free energy. Thesesolutions will have a lower T
cand will therefore not be realized
when cooling down the system. In order to circumvent thisproblem, we instead calculate T
cand find the corresponding
solution.
Close to and below Tc, we linearize the gap equations,
/Delta1λ(k,T−
c)=−/summationdisplay
k/prime,μgλμ(k,k/prime)/Delta1μ(k/prime,T−
c)
|ξμ(k/prime)|tanh/parenleftbiggβc|ξμ(k/prime)|
2/parenrightbigg
.
(28)
By multiplying this equation by (2 π)2g†
j(k)/V, where j=
{x+iy,x−iy,x,y }, and summing over the first Brillouin
zone, we get a system of linear equations
/Delta1i
λ=/summationdisplay
j/summationdisplay
μMij
λμ(Tc)/Delta1j
μ, (29)
where
Mij
λμ(Tc)=−(2π)2
V/summationdisplay
k/prime/bracketleftBigg/parenleftBigg/summationdisplay
kgλμ(k,k/prime)g†
j(k)/parenrightBigg
×gi(k/prime)
|ξμ(k/prime)|tanh/parenleftbiggβc|ξμ(k/prime)|
2/parenrightbigg/bracketrightbigg
, (30)
which may conveniently be written in the form
/vector/Delta1=M(Tc)/vector/Delta1. (31)
Here, /vector/Delta1=(/Delta1x+iy
1/Delta1x−iy
1 ... /Delta1y2)T. Thus, for a nontrivial
solution to exist we require that det( M(Tc))=0, which allows
for a computation of Tc. In cases where this holds for multiple
temperatures, the highest Tccorresponds to the channel where
superconductivity actually occurs. When Tcis determined, /vector/Delta1is
found by calculating the eigenvector of M(Tc) corresponding
to eigenvalue 1. The eigenvector only gives information aboutthe relative size of the coefficients in Eq. ( 27), not the absolute
scale. This is nonetheless enough information to determinethe dominant momentum dependence of the order parameterclose to T
c, and hence in which channel superconductivity first
appears upon cooling.
IV . RESULTS AND DISCUSSION
Using the procedure described in the previous section, we
have calculated the eigenvector of M(Tc), focusing on filling
fractions n=0.1 andn=0.45. These values are indicated in
Fig. 5. All results are obtained with h=0.2t.F o rn=0.1,
the results as a function of tilt angle δatθ=0i ss h o w ni n
Fig.6. We see that for a pure out-of-plane field, δ=0,/Delta11(k)∝
sinkx+isinky, which for small momenta corresponds to a
chiral kx+ikyorder parameter. For a pure in-plane field in
thexdirection, /Delta11(k)∝sinky, which corresponds to a ky
dependence in the low- |k|limit. This is in agreement with
the results of Lake et al. [7]. It is important to note that when
calculating the eigenvectors at Tc, we do not get information
024515-6p-WA VE SUPERCONDUCTIVITY IN WEAKLY … PHYSICAL REVIEW B 97, 024515 (2018)
0.00 0.25 0.50
δ/π0.00.51.0|Δ|[a.u](a)
Δx+iy
1
Δx−iy
1
Δx
1
Δy
1
0.00 0.25 0.50
δ/π−0.50.00.51.0ReΔ[a.u](b)
0.00 0.25 0.50
δ/π0.000.020.040.06ImΔ[a.u](c)
FIG. 6. Plot of the (a) absolute value, (b) real part, and (c) imaginary part of the dominant elements of the eigenvector of M(Tc) corresponding
to eigenvalue 1 as a function of δforn=0.1a n d θ=0. The terms proportional to the function gx±iy(k) are the dominant terms in /Delta11(k).
/Delta12(k)=0, not shown in the plot.
about the absolute value of the gaps, nor the relative size of the
gap coefficients between, e.g., δ=0 andδ=π.
Rotating the magnetic field in the xyplane, the kde-
pendence of the gap also changes accordingly, from a puresink
ydependence for θ=0, to a pure sin kxdependence
forθ=π/2, as seen from the values of the coefficients in
Fig.7(a). This change coincides with the rotation of the center
momentum qin Eq. ( 5). The reason for this might be that the
superconducting state is of FFLO kind whenever there is anin-plane component of the field. In the above calculations, we
0.0 0.1 0.2 0.3 0.4 0.5
θ/π−1.0−0.50.00.51.0Δ[a.u.](a)
ReΔx+iy
1
ReΔx−iy
1ImΔx+iy
1
ImΔx−iy
1
0.0 0.1 0.2 0.3 0.4 0.5
θ/π−1.0−0.50.00.51.0Δ[a.u.](b)
ReΔx
2
ReΔy
2ImΔx
2
ImΔy
2
FIG. 7. Plot of dominating terms of the eigenvector as a function
ofθfor pure in-plane magnetic field and filling fraction (a) n=0.1
and (b) n=0.45. In the small- |k|limit, the kdependence is changed
from pure kyto a pure kxas the field is rotated. The overall phase is
chosen such that the dominating contribution at θ=0 is real.neglected the shift in the center momentum of the Fermi levels,
Eq. ( 5), since they lead to higher order corrections. However,
since the Fermi levels in fact are shifted, the Cooper pairs havea finite center momentum 2 qand thus are FFLO Cooper pairs.
This is in agreement with Ref. [ 7].
Though the majority band has the highest T
chere, we see
from Fig. 5that also the minority band is attractive in the
gx±iychannel for low filling fractions, in contrast to what
has been found in other studies with quadratic dispersions[4,7]. Instead of being completely flat for |k−k
/prime|<2|kFλ|,
as in the quadratic case, the susceptibility develops a domein this region when increasing the filling fraction. In thisway, the susceptibility on the majority band also becomesk-dependent for interactions between particles close to the
Fermi surface on the minority band, leading to the possibilityof attractive interactions. We therefore expect that the minorityband becomes superconducting at some finite temperaturelower than T
con the majority band.
Performing similar calculations close to half-filling, with
n=0.45, we find that the momentum dependence for the
order parameter is dominated by the functions gx(k) andgy(k).
Moreover, superconductivity is now induced on the minorityband at T
c, as shown in Fig. 8.T h ev a l u e n=0.45 is close to
the filling fraction for which the majority band is half-filled,which corresponds to a van Hove singularity in the densityof states of the majority band. The fact that the minorityband has the highest T
ccan thus be explained by the vast
number of particles on the majority band which can mediatean effective intraband interaction. Again, the functional formof the gap is changed by rotating the magnetic field: Whenδ=0,/Delta1
2(k)∝sinkxcosky−icoskxsinky, which in the
small- |k|limit corresponds to kx−iky, and thus has the
opposite chirality compared to the n=0.1 case. For a pure
in-plane field, we get /Delta12(k)∝coskxsinky, which for small
momenta corresponds to a pure kydependence. As for n=0.1,
rotating the field in-plane changes the kdependence, as shown
in Fig. 7(b).
Since the coefficients χi
λhave the symmetry χi
λ(n)=
χi
¯λ(1−n), we have also performed the above analysis for
n=0.9 and n=0.55. In both cases, superconductivity is
now present on the opposite band compared to the n=
0.1 and n=0.45 cases, again with helicity kx+ikyfor
λ=1 and kx−ikyforλ=2. Therefore, it appears that a
superconducting state with the same helicity as the band isfavored [ 7].
024515-7HENNING G. HUGDAL AND ASLE SUDBØ PHYSICAL REVIEW B 97, 024515 (2018)
0.00 0.25 0.50
δ/π0.00.51.0|Δ|[a.u](a)
0.00 0.25 0.50
δ/π0.00.51.0ReΔ[a.u](b)
0.00 0.25 0.50
δ/π0.000.350.70ImΔ[a.u](c)
Δx+iy
2
Δx−iy
2
Δx
2
Δy
2
FIG. 8. Plot of the (a) absolute value, (b) real part, and (c) imaginary part of the dominant elements of the eigenvector of M(Tc) corresponding
to eigenvalue 1 as a function of δforn=0.45 and θ=0. The terms proportional to the functions gx(k)a n dgy(k) are the dominant terms in
/Delta12(k), which differs from the n=0.1 case in Fig. 6./Delta11(k)=0, not shown in the plot.
In both previous cases, only one band is superconducting
atTc. This indicates that the second term in Eq. ( 15) does
not contribute significantly to the superconducting pairing, asthis would lead to simultaneous superconductivity on bothbands. Moreover, from the form of /Gamma1
λ(k), we see that this
term should lead to superconductivity of opposite chiralityof what was found here, k
x∓ikyon the majority/minority
band [ 7]. Notice also that it is in principle possible to read
off the dominating functional form of the superconducting gapdirectly from Fig. 5.
Finally, we have found that the value of α
R/hhas no
impact on Tc, while it depends strongly on the value of
U/t. These are indications that the Kohn-Luttinger term in
the interaction is responsible for the physically realizablesuperconducting order, and thus due to pure intraband in-teractions. This allows to make some predictions regardingparts of the gray regions in Fig. 5, where the χ
3
λterm is
the dominating attractive term. From the above results, itis reasonable to assume that the solution in these regions
is of the form /Delta1
λ(k)=/Delta1x,2y
λgx(kx,2ky)+/Delta1y,2y
λgy(kx,2ky)+
/Delta1x,2x
λgx(2kx,ky)+/Delta1y,2x
λgy(2kx,ky), with the same small- |k|
functional form as found above. This, however, has not beenchecked explicitly.
The fact that superconductivity is not proximity induced on
the opposite band by the second term in Eq. ( 15), requires that
/Delta1
λ(k) satisfies
/summationdisplay
k/Gamma1†
λ(k)/Delta1λ(k)
|ξλ(k)|tanh/parenleftbiggβ|ξλ(k)|
2/parenrightbigg
=0. (32)
Using this requirement, we derive an ansatz for the functional
form of the superconducting gaps,
/Delta11(k)=/Delta11
1/bracketleftbigg1+cosδ/radicalbig
2(1+cos2δ)(cosθ−isinθ)gx+iy(k)
−1−cosδ/radicalbig
2(1+cos2δ)(cosθ+isinθ)gx−iy(k)/bracketrightbigg
+/Delta12
1/bracketleftbiggcosθ−icosδsinθ√
1+cos2δgy(k)
−sinθ+icosδcosθ√
1+cos2δgx(k)/bracketrightbigg
, (33a)/Delta12(k)=/Delta11
2/bracketleftbigg1+cosδ/radicalbig
2(1+cos2δ)(cosθ+isinθ)gx−iy(k)
−1−cosδ/radicalbig
2(1+cos2δ)(cosθ−isinθ)gx+iy(k)/bracketrightbigg
+/Delta12
2/bracketleftbiggcosθ+icosδsinθ√
1+cos2δgy(k)
−sinθ−icosδcosθ√
1+cos2δgx(k)/bracketrightbigg
, (33b)
where /Delta1i
λin general can depend on the field alignment angle.
Using this ansatz to find Tcand the solution eigenvectors,
we find the same results as presented above. Hence, we seethat even though the results do not depend directly on theSOC strength, the fact that SOC is present affects the realizedpairing symmetry [ 7]. The results of Ref. [ 14] indicate that
this conclusion might not hold for all values of α
R/h, and
an interesting development would therefore be to study thissystem for general SOC strengths.
TheT
cquickly decreases with decreasing U/t, and for
values in the regime set by the derivation of the gap equations,a numerical solution is impossible. Hence, we have performedthe above analysis for a range of values of U/t andα
R/h, and
found that the results were qualitatively unchanged. The factsthat the results agree with Ref. [ 7] for small filling fractions and
thatT
cdepends only on U/t indicate that the results presented
above should be valid also for realistic values of U/tandαR/h.
There could in principle exist a transition to a magnetic state,
such as the antiferromagnetic phase found for the 2D repulsiveHubbard model at half-filling in the weak-coupling limit [ 30].
However, applying a Zeeman field splits the degenerate spinbands, and we therefore expect that no antiferromagneticordering can exist as long as h>U . Though the application of
a Zeeman field could favor a ferromagnetic phase, other studieshave indicated that ferromagnetic ordering does not appear inthe weak-coupling limit of the 2D Hubbard model [ 30,31], a
result we expect to hold also in the present case.
V . CONCLUSION
We have investigated the role of a weak spin-orbit coupling
on a spin-polarized weakly repulsive Hubbard system on asquare lattice. Performing an analysis along the same lines asdone by Lake et al. [7] for the 2D electron gas, we found that
024515-8p-WA VE SUPERCONDUCTIVITY IN WEAKLY … PHYSICAL REVIEW B 97, 024515 (2018)
the superconducting order was caused by the SOC-independent
Kohn-Luttinger term in the interaction. The pairing symmetrywas, however, indirectly determined by the SOC: The realizedsuperconducting gap has the same chirality as the band. We alsofound that the momentum dependence of the superconductinggap could be tuned by rotating the magnetic field and changingthe filling fraction. The filling fraction also determines whichband has the highest T
c.
ACKNOWLEDGMENTS
H.G.H. would like to thank F. N. Krohg for useful dis-
cussions. This work was supported by the Research Coun-cil of Norway through Grant No. 250985, “Fundamentalsof Low-Dissipative Topological Matter“, and through GrantNo. 262633 Center of Excellence, “Center for QuantumSpintronics“.
APPENDIX: EXPANSION OF SUSCEPTIBILITY
IN SQUARE LATTICE HARMONICS
To the order we are working, we can set αR=0 when
calculating the susceptibility, Eq. ( 13). In this case, the dis-
persion in Eq. ( 4) has the symmetries of the C4vgroup and is
invariant under spatial inverision, k→− k, 4-fold rotations,
(kx,ky)→(ky,−kx), mirror operations, ( kx,ky)→(−kx,ky),
etc. Since we in Eq. ( 13) sum over the 1 BZ, it can be shown
that the susceptibility has the same symmetries. The expansionof the susceptibility thus has to be invariant under the sameoperations, which greatly reduces the possible terms in theexpansion. Since the susceptibility is even under inversions(only the SOC term breaks inversion symmetry, which isneglected here), the expansion must contain only even terms,which we write in a general form [ 32]a s
χ(q)=/summationdisplay
m,namncos(mqx+nqy), (A1)
where mandnare integers, and the band index has been
dropped for notational simplicity. From the requirementχ(q
x,qy)=χ(−qx,qy), we find amn=am,−n=a−m,n, and
similarly from χ(qx,qy)=χ(qy,−qx) we find amn=a−n,m=
anm. Using these relations, we simplify the above equation:
χ(q)=a00+/summationdisplay
(m,n)>02amn[cos(mqx+nqy)
+cos(mqx−nqy)]. (A2)
Separating the terms according to if m=nor not, we get
χ(q)=D00G00+/summationdisplay
m>n> 0DmnGmn(q)
+/summationdisplay
m>0[D0mG0m(q)+DmmGmm(q)], (A3)where we have redefined the expansion coefficients amnand
defined the orthonormal functions
G00=1
2π, (A4a)
G0m(q)=cosmqx+cosmqy
2π, (A4b)
Gmm(q)=cosmqxcosmqy
π, (A4c)
Gmn(q)=cosmqxcosnqy+cosnqxcosmqy√
2π.(A4d)
We now insert q=k−k/primeand rewrite the above functions in
terms of products of functions of kork/primeseparately,
4πG 0m(k−k/prime)=[(sinmkx+isinmky)(sinmk/prime
x−isinmk/prime
y)
+H.c.]+[sin→cos],
πGmm(k−k/prime)=[(cosmkxcosmky)(cosmk/prime
xcosmk/prime
y)
+(cosmkxsinmky)(cosmk/prime
xsinmk/prime
y)]
+[sin↔cos],
√
2πGmn(k−k/prime)=[(cosmkxcosnky)(cosmk/prime
xcosnk/prime
y)
+(cosmkxsinnky)(cosmk/prime
xsinnk/prime
y)
+(sinmkxcosnky)(sinmk/prime
xcosnk/prime
y)
+(sinmkxsinnky)(sinmk/prime
xsinnk/prime
y)]
+[n↔m].
Since the SOC is weak, the interaction can be regarded to be
between particles of equal spin to the order we are working.Hence, the interaction must be odd in kandk
/prime. In this way,
we can neglect most of the above terms and are left with anexpansion of the form
χ(k−k
/prime)=/summationdisplay
mχ0m[gx+iy(mk)gx−iy(mk/prime)+H.c.]
+/summationdisplay
mχmm[gx(mk)gx(mk/prime)+gy(mk)gy(mk/prime)]
+/summationdisplay
m>nχmn[gx(mkx,nky)gx(mk/prime
x,nk/prime
y)
+gy(mkx,nky)gy(mk/prime
x,nk/prime
y)+m↔n],(A5)
where m,n > 0 and we have used the functions defined
in Eq. ( 25). The leading-order terms included in Eq. ( 24)
correspond to the χ01,χ11, andχ21terms in the above equation.
[1] W. Kohn and J. M. Luttinger, P h y s .R e v .L e t t . 15,524(1965 ).
[2] M. Yu. Kagan and A. V . Chubukov, JETP Lett. 50, 517 (1989).
[3] A. V . Chubukov, P h y s .R e v .B 48,1097 (1993 ).
[4] S. Raghu and S. A. Kivelson, Phys. Rev. B 83,094518
(2011 ).[5] Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984).
[6] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine,
Nat. Mater. 14,871(2015 ).
[7] E. Lake, C. Webb, D. A. Pesin, and O. A. Starykh, Phys. Rev. B
93,214516 (2016 ).
024515-9HENNING G. HUGDAL AND ASLE SUDBØ PHYSICAL REVIEW B 97, 024515 (2018)
[8] M. Smidman, M. B. Salamon, H. Q. Yuan, and D. F. Agterberg,
Rep. Prog. Phys. 80,36501 (2017 ).
[ 9 ]L .P .G o r ’ k o va n dE .I .R a s h b a , Phys. Rev. Lett. 87,037004
(2001 ).
[10] V . Barzykin and L. P. Gor’kov, P h y s .R e v .L e t t . 89,227002
(2002 ).
[11] D. F. Agterberg and R. P. Kaur, Phys. Rev. B 75,064511 (2007 ).
[12] O. Vafek and L. Wang, Phys. Rev. B 84,172501 (2011 ).
[13] S. Raghu, S. A. Kivelson, and D. J. Scalapino, Phys. Rev. B 81,
224505 (2010 ).
[14] F. Loder, A. P. Kampf, and T. Kopp, J. Phys.: Condens. Matter
25,362201 (2013 ).
[15] N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis, G.
Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S. Rüetschi,D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, and J.Mannhart, Science 317,1196 (2007 ).
[16] A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider,
M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M.Triscone, Nature (London) 456,624(2008 ).
[17] S. Gariglio, N. Reyren, A. D. Caviglia, and J. Triscone, J. Phys.
Condens. Matter 21,164213 (2009 ).
[18] A. Brinkman, M. Huijben, M. van Zalk, J. Huijben, U. Zeitler,
J .C .M a a n ,W .G .v a nd e rW i e l ,G .R i j n d e r s ,D .H .A .B l a n k ,
and H. Hilgenkamp, Nat. Mater. 6,493(2007 ).[19] J. A. Bert, B. Kalisky, C. Bell, M. Kim, Y . Hikita, H. Y . Hwang,
and K. A. Moler, Nat. Phys. 7,767(2011
).
[20] L. Li, C. Richter, J. Mannhart, and R. C. Ashoori, Nat. Phys. 7,
762(2011 ).
[21] M. Sachs, D. Rakhmilevitch, M. B. Shalom, S. Shefler, A.
Palevski, and Y . Dagan, Phys. C 470,S746 (2010 ).
[22] D. A. Dikin, M. Mehta, C. W. Bark, C. M. Folkman, C. B.
Eom, and V . Chandrasekhar, Phys. Rev. Lett. 107,056802
(2011 ).
[23] M. B. Shalom, M. Sachs, D. Rakhmilevitch, A. Palevski, and Y .
Dagan, P h y s .R e v .L e t t . 104,126802 (2010 ).
[24] A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cancellieri,
and J. M. Triscone, P h y s .R e v .L e t t . 104,126803 (2010 ).
[25] K. Michaeli, A. C. Potter, and P. A. Lee, Phys. Rev. Lett. 108,
117003 (2012 ).
[26] P. Fulde and R. A. Ferrell, Phys. Rev. 135,A550 (1964 ).
[27] A. Larkin and Y . Ovchinnikov, Sov. Phys. JETP 20, 762 (1965).
[28] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149,491(1966 ).
[29] S. Bravyi, D. P. Di Vincenzo, and D. Loss, Ann. Phys. (NY) 326,
2793 (2011 ).
[30] J. E. Hirsch, P h y s .R e v .B 31,4403 (1985 ).
[31] H. Q. Lin and J. E. Hirsch, Phys. Rev. B 35,3359 (1987 ).
[32] E. Otnes, M.Sc. thesis, Norwegian University of Science and
Technology, Trondheim, Norway, 1997.
024515-10 |
PhysRevB.95.174504.pdf | PHYSICAL REVIEW B 95, 174504 (2017)
Orbital selective pairing and gap structures of iron-based superconductors
Andreas Kreisel,1,2Brian M. Andersen,1P. O. Sprau,3,4A. Kostin,3,4J. C. Séamus Davis,3,4and P. J. Hirschfeld5
1Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK 2100 Copenhagen, Denmark
2Institut für Theoretische Physik, Universität Leipzig, D-04103 Leipzig, Germany
3LASSP , Department of Physics, Cornell University, Ithaca, New York 14853, USA
4CMPMS Department, Brookhaven National Laboratory, Upton, New York 11973, USA
5Department of Physics, University of Florida, Gainesville, Florida 32611, USA
(Received 8 November 2016; revised manuscript received 5 April 2017; published 8 May 2017)
We discuss the influence on spin-fluctuation pairing theory of orbital selective strong correlation effects
in Fe-based superconductors, particularly Fe chalcogenide systems. We propose that a key ingredient for animproved itinerant pairing theory is orbital selectivity, i.e., incorporating the reduced coherence of quasiparticlesoccupying specific orbital states. This modifies the usual spin-fluctuation theory via suppression of pair scatteringprocesses involving those less coherent states and results in orbital selective Cooper pairing of electrons in theremaining states. We show that this paradigm yields remarkably good agreement with the experimentally observedanisotropic gap structures in both bulk and monolayer FeSe, as well as LiFeAs, indicating that orbital selectiveCooper pairing plays a key role in the more strongly correlated iron-based superconductors.
DOI: 10.1103/PhysRevB.95.174504
I. INTRODUCTION
In both copper-based and iron-based high-temperature
superconductors, fundamental issues include the degree ofelectron correlation and its consequences for enhancing super-conductivity. In both archetypes, there are multiple active or-bitals (two O porbitals and one Cu dorbital in the former, and
five Fe dorbitals in the latter). This implies the possibility of
orbital selective physics, where states dominated by electronsof one orbital type may be weakly correlated and others muchmore strongly correlated, leading to substantial differencesin quasiparticle spectral weights, interactions, magnetism, andorbital ordering [ 1–7]. Cooper pairing itself could then become
orbital selective [ 8,9], with the electrons of a specific orbital
character binding to form the Cooper pairs of the supercon-ductor. The superconducting energy gaps of such a materialwould therefore generically be highly anisotropic [ 8,9], i.e.,
large only for those Fermi surface regions where a specificorbital character dominates. Such phenomena, although longthe focus of theoretical research on higher-temperature super-conductivity in correlated multiorbital superconductors, haveremained largely unexplored because orbital selective Cooperpairing has not been experimentally accessible.
Spin fluctuations are proposed as the dominant mecha-
nism driving Cooper pairing in a wide variety of uncon-
ventional superconductors: heavy-fermion systems, cuprates,two-dimensional (2D) organic charge transfer salts, and iron-based superconductors (FeSC) [ 13–16]. There is currently
no version of spin-fluctuation-based pairing theory that en-joys either the well-controlled derivation from fundamentalinteractions or the consensual success explaining observedproperties of the BCS-Migdal-Eliashberg theory of conven-
tional superconductivity. On the other hand, the calculational
scheme referred to as random phase approximation (RPA)in the case of one-band systems [ 17,18], or matrix RPA in
the case of multiband systems [ 19,20], has achieved consider-
able qualitative progress for unconventional systems.
While material-specific calculations of the critical temper-
atureT
cwithin spin-fluctuation theory appear distant, consid-
erable success has been achieved understanding qualitative as-pects of pairing, particularly in Fe-pnictide systems [ 15,21,22].
In the 122 materials, which were the subject of the mostintensive early study, itinerant spin-fluctuation theory providedconvincing, material-specific understanding of the variation ofgap anisotropy with doping within the dominant sign-changings-wave channel, particularly the existence or nonexistence of
nodes; the interplay with d-wave pairing; the rough size of T
c;
and the origin of particle-hole asymmetry in the phase diagram.
In retrospect, such agreement was somewhat fortuitous, possi-bly because the 122 systems have large Fermi surface pocketsof both hole and electron type, and are relatively weaklycorrelated. In other pnictides like 111 [ 4,23,24], and in 11 Fe-
chalcogenide systems [ 3,25], correlation effects are consider-
ably more significant. In LiFeAs, for example, angle-resolvedphotoemission spectroscopy (ARPES) measurements [ 26,27]
show that the /Gamma1-centered d
xz/dyzhole pockets are considerably
smaller than predicted by density functional theory (DFT),while the d
xypocket is larger. Taking these effects into account
via a set of renormalized energy bands is insufficient, however,to account for the accurate gap structure of LiFeAs within spin-fluctuation theory [ 12] (see Ref. [ 15] and references therein).
The consequences of correlations for the band structure of
FeSC are more profound than simple Fermi surface shifts,however. If one examines compounds where the dbands are
closer to half-filling (5 electrons/Fe), the effect of electron-electron interactions are enhanced in a way distinctly differentfrom one-band systems: different d-orbital effective masses
are enhanced by different factors. This “orbital selectivity”
predicted by theory [ 1–3,28–30] has been confirmed by
ARPES experiments. While most Fe-based systems have moreelectrons/Fe, closer to 6, the effects are still nontrivial in theFe-chalcogenides. For example, the electrons in bands withd
xy-orbital character have been claimed to exhibit single-
particle masses up to 10–20 times the band mass, while in
dxz/dzystates the renormalization is closer to 3–4 [ 31,32].
In Fermi-liquid theory, excitations in a system of interacting
fermions are described by quasiparticles that have the samequantum numbers but deviate from the free particles in prop-erties such as the quasiparticle mass, which renormalizes the
2469-9950/2017/95(17)/174504(12) 174504-1 ©2017 American Physical SocietyANDREAS KREISEL et al. PHYSICAL REVIEW B 95, 174504 (2017)
dxy=dxz
dxy
dyz=dxy
dyz
dxz=dyz
dxz 00
CRLH0.51
0.5 1
FIG. 1. Fermi surfaces together with orbital character of the models considered in this work obtained from tight-binding models fit to
ARPES and quantum oscillation experiments. The individual sheets are labeled as indicated: (a) model for FeSe (bulk) [ 10] including orbital
order, (b) 2D model for FeSe monolayer derived from the previous one where maps of ARPES intensities obtained from measurements withhorizontally polarized (LH) and circular polarized (CR) initial photons have been overlayed to show agreement to experimental results [ 11]
and (c) model for LiFeAs [ 12]. Plots as a function of the angle ϕaround the Fermi surface sheets are done with the angle measured from the
k
xaxis as indicated in (b).
Fermi velocity. Generally, interactions in electronic systems
also lead to reduced quasiparticle weights, corresponding toreduced values of the residue at the pole of the Green’sfunction describing those dressed electrons. Even in one-band
systems where orbital selectivity does not play a role, pairing in
superfluid systems with reduced Landau quasiparticle weightis an important unsolved theoretical problem. While onegenerally expects pairing interactions to be reduced as the
quasiparticle weight is suppressed as other aspects of pairing
are held fixed, pairing in completely incoherent non-Fermiliquids is not impossible, as discussed recently in Ref. [ 33]. The
effect of orbital selective quasiparticle weights on pairing inFeSC has been discussed elsewhere in various approximations
[8,9], with differing conclusions.
In this work, we implement a simple scheme to incorporate
aspects of renormalization of the electronic band structure,including reduced quasiparticle coherence that is orbitalselective into spin-fluctuation pairing theory, and apply it
to several FeSC. This orbital selective approach to pairing
provides an excellent description for the superconducting gapdeduced from quasiparticle interference measurements on thenematic Fermi surface pockets of bulk FeSe, as shown alreadyin Ref. [ 10]. Here, we discuss the generality of this approach,
and show how it explains the exotic gap structures of FeSe,
FeSe monolayers, and in the LiFeAs system as well. Thesefindings encourage us to believe that the proposed paradigm isthe correct way to understand the physics in these materials,but we cannot rule out completely that other effects affecting
the gap such as spin-orbit coupling or orbital fluctuations
[34] may contribute. While the microscopic origin of the
phenomenology remains an open challenge, we believe that itprovides a major step towards a quantitative, material-specifictheory of superconductivity in strongly correlated FeSC.
II. MODEL
The starting point of any uncorrelated multiband system
is the electronic structure described by a tight-binding model[12,34–36]
H=/summationdisplay
kσ/lscript/lscript/primet/lscript/lscript/prime
kc†
/lscriptσ(k)c/lscript/primeσ(k), (1)where c†
/lscriptσ(k) is the Fourier amplitude of an operator that
creates an electron in Wannier orbital /lscriptwith spin σand
t/lscript/lscript/prime
kis the Fourier transform of the hoppings. By a unitary
transformation from orbital to band space, Hbecomes
diagonal H=/summationtext
kσμξμ(k)c†
μσ(k)cμσ(k), with eigenenergies
ξμ(k) andc†
μσ(k) creating an electron in Bloch state μ,k.
There is no way to determine empirically the electronic
structure ξμ(k) of the uncorrelated reference system corre-
sponding to a given real material. However, experimentalprobes like ARPES and quantum oscillations provide infor-mation on the real single-particle spectrum, which we willcall ˜E
μ(k). Since we do not have access to ξμ(k), we will
henceforth use the term “uncorrelated” to mean a modelfor an electronic structure where the quasiparticles have unitweight; in this work we only work with such models where theeigenenergies ˜E
μ(k) have been obtained by fit to experiment.
In Fig. 1, we show examples of Fermi surfaces derived from
the eigenenergies ˜Eμ(k). For three-dimensional (3D) models
considered in this work, the zero-energy surfaces, i.e., the setofkvectors with ˜E
μ(k)=0, are corrugated tubes identified
asα, δ, andεsheets in Fig. 1(a) (FeSe, bulk) or the βandγ
sheets in Fig. 1(c) (LiFeAs), but can also be closed surfaces
as the αpocket in Fig. 1(c). For a two-dimensional model as
shown in Fig. 1(b), the Fermi surface is given by elliptical lines
such that it is convenient to plot quantities as a function of theangleϕ.
In the orbital basis, the “uncorrelated” Green’s function is
given by
G
/lscript/lscript/prime(k,ωn)=/summationdisplay
μa/lscript
μ(k)a/lscript/prime∗
μ(k)
iωn−˜Eμ(k), (2)
where a/lscript
μ(k) are the matrix elements of the unitary transfor-
mation mentioned above. The orbital weight |a/lscript
μ(k)|2becomes
important when discussing low-energy (Fermi-surface driven)properties and is therefore visualized color coded for theimportant Fe dorbitals /lscript={d
xy,dxz,dyz}in Fig. 1as well.
In order to include the full effects of correlations,
we further make the orbital selective ansatz that theoperators c†
/lscript(k) create quasiparticles with weight√Z/lscriptin
174504-2ORBITAL SELECTIVE PAIRING AND GAP STRUCTURES . . . PHYSICAL REVIEW B 95, 174504 (2017)
orbital /lscript,c†
/lscript(k)→√Z/lscriptc†
/lscript(k). Note that /lscriptruns over the Fe
3dorbitals ( dxy,dx2-y2,dxz,dyz,d3z2-r2).
The associated Green’s function becomes
˜G/lscript/lscript/prime(k,ωn)=/radicalbig
Z/lscriptZ/lscript/prime/summationdisplay
μa/lscript
μ(k)a/lscript/prime∗
μ(k)
iωn−˜Eμ(k), (3)
where ˜Eμ(k) are the renormalized band energies. A similar ap-
proach has been used recently when parametrizing the normal-state Green’s functions in a Fermi-liquid picture [ 37], with the
formal difference that we explicitly employ the renormalizedquasiparticle energies ˜E
μ(k), which include the static real
part of the self-energy, and retain the quasiparticle weights inthe numerator. Following state-of-the-art pairing calculationsfrom spin-fluctuation theory [ 12,38–40] (see Appendix C),
important effects of the√
Z/lscriptfactors enter in two places:
(1) the calculation of the susceptibility includes the renormal-ized quasiparticle Green’s function, and (2) when projectingthe pairing interaction from orbital to band space, one needs
to account for the replacement of c
†
/lscript(k)→√Z/lscriptc†
/lscript(k). In
cases where the Hamiltonian already correctly describes thequasiparticle energies of a correlated system ξ
μ(k)→˜Eμ(k)
(as obtained, e.g., from fits to measured quasiparticle energiesfrom spectroscopic experiments), the bare susceptibility inorbital space needs to be simply multiplied by the quasiparticleweights
˜χ
0
/lscript1/lscript2/lscript3/lscript4(q)=/radicalbig
Z/lscript1Z/lscript2Z/lscript3Z/lscript4χ0
/lscript1/lscript2/lscript3/lscript4(q), (4)
in order to obtain the corresponding quantity (with tilde) in
the correlated system. Our models as shown in Fig. 1already
match the true quasiparticle energies ˜Eμ(k), such that we
can use Eq. ( 4) to examine the effect of the quasiparticle
weights on the susceptibility. In Fig. 2(a), the diagonal
components of the orbitally resolved susceptibilities where/lscript
1=/lscript2=/lscript3=/lscript4are plotted as obtained from our model
of FeSe (bulk). For all orbitals, the overall magnitude issimilar (except for /lscript=d
z2that does not play any role for
the subsequent discussion), but the momentum structure isdistinct: the d
xycomponent has a maximum at q=(π,π),
whereas the components for dyz(dxz) have maxima at
q=(π,0) [q=(0,π)]. Introducing quasiparticle weights as
indicated in Fig. 2(b), it is obvious that some components are
suppressed more than others such that for the present choiceof{√
Zl}=[0.2715,0.9717,0.4048,0.9236,0.5916], the dyz
contribution dominates.1In a similar way, the pairing inter-
action gets modified by prefactors from quasiparticle weights
(see Appendix C). Physically, this means that orbital selective
pairing occurs because pairing from certain quasiparticle statesis suppressed more than others because the states themselvesare less coherent.
To visualize this effect, we have plotted the spectral function
A(k,ω)=−1/πIm TrG(k,ω)f o rk
z=0 at zero energy in
1Note that the dx2-y2component is still large because of the choice
of a quasiparticle weight close to 1. It therefore contributes to the
physical susceptibility, but has little influence on the superconductingorder parameter since the orbital weight for states at low energies is
small (see Fig. 1).00.51
(0,0) (π,0) ( π,π) (0, π) (0,0)00.511.5
FIG. 2. Comparison of the orbitally diagonal components of the
susceptibility of the uncorrelated model for bulk FeSe (a) and the
same quantities including the quasiparticle weights that suppress
contributions from orbitals with small weight factors according to
Eq. ( 4)( b ) .
Fig. 3(a) for the uncorrelated system and in Fig. 3(b) with
the same choice of quasiparticle weights as discussed above.We use the bulk FeSe Fermi surface discussed below asan illustration of the idea, but details of the bands arenot important for this purpose. The superconducting orderparameter is now determined by the strength of the pair
FIG. 3. Plot of the spectral function at zero energy in the first
Brillouin zone. (a) Spectral function A(k,0)=−1/πIm TrG(k,0)
of the uncorrelated model for FeSe (bulk) at kz=0 with the Green’s
function as in Eq. ( 2). (b) Spectral function ˜A(k,0) of the model
including quasiparticle weights inducing orbital selective reduced
coherence. For the pair scattering of Cooper pairs at momenta ktok/prime
on the Fermi surface (arrows) two quantities determine the scattering
strength: (i) the susceptibility ˜ χ(q) to which the pairing vertex /Gamma1k,k/prime
is proportional and (ii) the quasiparticle weight at initial and final
momentum. In summary, some processes get largely suppressed (thin
red and blue arrows) such that other processes (thick green arrow)dominate the Cooper pairing.
174504-3ANDREAS KREISEL et al. PHYSICAL REVIEW B 95, 174504 (2017)
05101520
(0,0) (π,0) ( π,π) (0, π) (0,0)χ(q)
FIG. 4. Results for FeSe (bulk): (a) calculated susceptibility with quasiparticle weights ( ˜ χ, thick lines) compared to the susceptibility
without quasiparticle weights ( χ, thin dashed lines), (b) gap symmetry function as obtained from conventional spin-fluctuation pairing, and
(c) the same quantity when taking into account orbital-dependent quasiparticle weights. For both calculations, the dominant pair scattering
processes leading to a large order parameter are symbolized with a double arrow. The calculations are done for a fixed ratio J=U/6, but with
an overall scale Uas indicated.
scattering /Gamma1k,k/primeof a Cooper pair at ktok/primewhich is proportional
to the susceptibility within the spin-fluctuation approach. In theuncorrelated case, scattering processes involving three pairs ofkvectors as depicted by the arrows in Fig. 3are comparable in
magnitude (with the process in blue involving d
xystates being
slightly larger). Taking into account the quasiparticle weights,the spectral function and thus the pair scattering is suppressedon parts of the Fermi surface. Consequently, the processesinvolving d
yzstates (green, thick arrow) dominate over those
involving dxystates (blue) and dxzstates (red), making the
pairing orbital selective.
III. BULK FeSe
Early thermodynamic and transport studies of bulk FeSe
as well as STM supported a state with gap nodes [ 41,42].
However, more recent measurements of low-temperaturespecific heat [ 43,44], STM [ 44], thermal conductivity [ 45,46],
and penetration depth [ 47,48] have found a tiny spectral gap,
indicating that the gap function is highly anisotropic but maynot change sign on any given sheet. The only experiments thatprovide information on the location of these deep minima arean ARPES measurement on the related Fe(Se,S) material [ 49]
and a recent quasiparticle interference (QPI) experiment [ 10],
both of which find deep minima on the tips of the hole ellipseat the center of the Brillouin zone. The latter also distinguishesdeep minima on the tips of the εelectron pocket “ellipse”.
To test the mechanism of orbital selective pairing deter-
mined by reduced coherence of some quasiparticles, we showfirst how this mechanism modifies results for the susceptibilityand the superconducting gap for bulk FeSe. Our startingpoint is a tight-binding model with hoppings adapted suchthat the spectral positions of the quasiparticle energies fitrecent findings using ARPES, quantum oscillations, and STMexperiments [ 10,50–53]. As the band energies are “measured”
in this case, these can be identified with the renormalized bandenergies ˜E
μ(k) in the presence of correlations, yielding the
Fermi surface in Fig. 1(a).
To construct a proper approximation of the quasipar-
ticle Green’s function [Eq. ( 3)], we need to additionally
include quasiparticle weights. Next, we fix the ratio J=
U/6 as found in cRPA calculations [ 54,55] and optimizethe weights in the orbital basis. The result is {√Zl}=
[0.2715,0.9717,0.4048,0.9236,0.5916] such that the gap
function yields a nodeless order parameter with a largeanisotropic gap on the αpocket, as seen from Fig. 4(c). These
values for Z
lare in reasonable agreement with general trends in
FeSC: the dxyorbital exhibits strongest correlations (smallest
weight) [ 31], while the dx2-y2orbital is the most weakly
correlated [ 1–3]. We note that the resulting gap structure
is very different from the one obtained from conventionalspin-fluctuation calculations (which also show a distortionfrom tetragonal symmetry as expected) [ 56], a result of the
very different momentum structure of the pairing interaction
[compare Figs. 4(b) and 4(c)]: The largest gap magnitude
is on the tip electron pocket ( ε) centered at the Xpoint
for the conventional calculation because the largest pairscattering /Gamma1
k,k/primeconnects this area of the Fermi surface with
the corresponding one on the Y-centered pocket [blue arrow in
Figs. 3(a)and4(b)]. It appears on the αpocket when using the
orbital selective pairing ansatz because the dressed electrons
mediate the strongest Cooper pair scattering from the flat area
of the αpocket to the flat area of the εpocket, where also
the gap is maximal [green arrow in Figs. 3(b) and4(c)]. The
physical origin of this can be attributed to the strong splittingof weights of the d
xzanddyzorbitals where states of the dxz
orbital are very incoherent.
We observe that the susceptibility ˜ χ, originally strongly
dominated by ( π,π), now shows dominant stripe fluctuations
withq=(π,0) [see Fig. 4(a)]. This result is in agreement with
findings from neutron scattering experiments [ 57,58] which
find strong stripe fluctuations at low energies. Taking intoaccount the results of a recent ARPES experiment [ 59] with the
conclusion that the electronic structure of FeSe evolves in sucha way that it becomes less correlated as temperature increases,
we can conclude that weight of the spin fluctuations should
shift from ( π,0) towards ( π,π) as temperature increases. This
can be understood directly from Eq. ( 4), where the different
orbital components of the susceptibility are weighted accord-ing to the quasiparticle weights; the d
xycomponents which are
peaked at ( π,π) get suppressed. The dxzcomponents, peaked
at (0,π), are suppressed as well (see Fig. 2). On individual
pockets, the gap function then follows the orbital content of
the orbital with strongest contribution (in this case, the dyz
orbital) [compare Fig. 1(a)].
174504-4ORBITAL SELECTIVE PAIRING AND GAP STRUCTURES . . . PHYSICAL REVIEW B 95, 174504 (2017)
10123
-180 -90 0 90 18010123
FIG. 5. Results for FeSe (bulk): plot of the gap function around
the Fermi surface pockets for (a) the conventional spin-fluctuation
calculation and (b) a calculation using the spin-fluctuation pairingin presence of quasiparticle weights. For direct comparison, the
data from a Bogoliubov QPI analysis from Ref. [ 10]a n daA R P E S
investigation on a related compound FeSe(S) [ 49] are displayed as
well.
Consequently, the pairing is changed by two mechanisms:
First, it is modified directly by the quasiparticle weights asdiscussed earlier and, second, the peak shifts in qin the (RPA)
susceptibility. Both of these effects make the pair scattering inthed
yzorbital more important [green thick arrow in Fig. 3(b)]
yielding the gap structure as shown in Fig. 4(c). To make the
agreement to experiment evident, we plot in Fig. 5the gapfunction at a cut of the Fermi surface at kz=πcomparing
to results from two different spectroscopic methods. Whilethe conventional calculation [Fig. 5(a)] does not show any
similarities, the correspondence in Fig. 5(b)is evident. Finally,
we note that this picture is different than that ascribed toorbital selective physics in the “strong-coupling” t-Jmodel
approach, where the d
xypairing channel is enhanced rather
than suppressed [ 9].
IV . MONOLAYER FeSe ON SrTiO 3
Despite considerable excitement over the high critical
temperature in the FeSe/STO monolayer system, limitedinformation is available regarding the structure of the su-perconducting gap. Early ARPES measurements suggestedan isotropic gap on electron pockets [ 60,61]. Theoretical
possibilities for pairing states in the presence of missing/Gamma1-centered hole band were discussed in Ref. [ 15]. Quite
recently, a new ARPES study identified significant and unusualanisotropy on a single unhybridized elliptical electron pocket[11], whereby the gap acquired global maxima at the ellipse
tips and additional local maxima at the ellipse sides. Theseauthors showed that the structure cannot be explained usingany of the low-order Brillouin zone harmonics expected fromso-called “strong-coupling” electronic pairing theories.
Within the model for the electronic structure of bulk FeSe,
we perform a calculation with a few modifications to accountfor differences in the monolayer from the bulk: (1) We ignoreall hoppings out of the plane, yielding a strictly 2D system.(2) We neglect orbital order, which has never been observedin the monolayer. (3) Experimentally, only electronlike Fermipockets have been detected, suggesting that the monolayeris actually electron doped. Possible reasons for this dopingare charge transfers from the substrate or surface defects.We therefore apply a rigid band shift by δμ=60 meV,
which removes the /Gamma1-centered hole pocket and leaves electron
pockets that have the size and shape of measured spectralfunctions in ARPES [ 11], with n=6.12 electrons/Fe [see
Figs. 1(b) and6(a) for a plot of the orbital character]. The
quasiparticle weights in the monolayer may be different fromthe bulk for two reasons: (1) The absence of the orbital order,i.e., the tetragonal crystal structure dictates that the weights
00.51
dxydxzdyz
180 90 0 90 18 0(a)
180 90 0 90 18051015(b)
180 90 0 90 18051015(c)
FIG. 6. Results for monolayer FeSe: (a) orbital weight at the Fermi surface, (b) superconducting gap obtained from conventional spin-
fluctuation theory, and (c) the same quantity including orbital-dependent quasiparticle weights compared to measured gap functions in
ARPES [ 11]. Symmetry operations of the tetragonal system have been applied to the measured data. All calculations were done for a fixed
ratioJ=U/10, with overall scale Uas indicated.
174504-5ANDREAS KREISEL et al. PHYSICAL REVIEW B 95, 174504 (2017)
αβγ(a)
g(k)U=3 eV
180 90 0 90 180864202468(b)
180 90 0 90 180864202468(c)
FIG. 7. Results for LiFeAs: (a) 3D plot of the gap function as obtained from spin-fluctuation calculation including quasiparticle weights.
(b) Cut at kz=πof the result of the s-wave gap function from conventional spin-fluctuation theory (solid lines) plotted as a function of angle
ϕ( a sd e fi n e di nF i g . 1) around the pockets [ /Gamma1-centered hole pocket ( α, magenta), M-centered hole pocket ( γ, cyan), and X-centered electron
pocket ( β, black)] together with experimental results. The measured magnitudes of the gap from an ARPES experiment [ 27] are symmetrized
and displayed as crosses, and those from a Bogoliubov QPI experiment [ 62] as filled dots. (c) The same quantity for the gap function as shown
in (a) also compared to experimental data. All calculations are done for a fixed ratio J=0.37U[12], but with overall scale Uas indicated.
fordxzanddyzorbitals are degenerate (unlike bulk FeSe).
(2) Correlations may be different in the monolayer where atendency towards weaker correlations was found recently [ 6],
such that we fix the ratio J=U/10 in this case.
At this point, we note that the states on the Fermi surface
have only tiny orbital weight of d
z2anddx2-y2character,
and additionally there are no pair scattering processes fromktok
/primewith q=(π,0) [or q=(0,π)] such that a fit
procedure with all quasiparticle weights will be underdeter-mined. In the optimization procedure, we therefore fix theweights to/radicalbig
Zx2-y2=0.8>/radicalbig
Zz2=0.7 and obtain {√Zl}=
[0.4273,0.8000,0.9826,0.9826,0.700] for the best agreement
to the gap measured in ARPES [ 11]. This result does change
the susceptibility slightly, but keeps the ( π,π) fluctuations
dominant; for details we refer to Fig. 8in the Appendix. These
fluctuations drive an overall (nodeless) d-symmetry ground
state as expected, but with an unusual structure modifiedstrongly by orbital correlations, with the result as shownin Figs. 6(b) and 6(c). Evidently the gap function for the
standard spin-fluctuation calculation [Fig. 6(b)] mostly follows
the orbital content of the d
xyorbital [compare Fig. 6(a) for a
plot of the orbital weights as a function of angle ϕaround
theX-centered pocket2]. For the current Fermi surface, this
is expected because the pairing interaction is dominated byintraorbital processes, and the d
xyorbital has large weight at
positions kandk/primeon the Fermi surface which are separated
roughly by ( π,π) and can take advantage of the strong peak
in the susceptibility at that qvector. The other two orbitals
play a negligible role in the pairing process. This situation ismodified once the pairing interaction is renormalized by thequasiparticle weights and therefore reduces the contributionof the d
xyorbital. The main effect is that a second maximum
in the gap function appears at a position in momentum spacewhere the d
xzordyzorbital is dominant [see Fig. 6(c)].
In the pairing process, intraorbital, interpocket contribu-
tions dominate, whereby one pair on the Xpocket of dyz
2TheY-centered pocket is symmetry related and will not be
discussed further at this point.character scatters into another pair on the Ypocket with the
same orbital character, meaning that the latter pair must belocated on the tip of the Ypocket where the gap has largest
magnitude. Because the total weight of this orbital is smallerthere, the order parameter for kstates dominated by this orbital
is enhanced. In summary, one gets a gap structure with a largemaximum at the tip of the ellipse and a small maximum at theflat part of the ellipse, remarkably similar to that detected byexperiment [ 11].
V . LiFeAs
LiFeAs is another Fe-based superconductor that is known
to have a Fermi surface quite different from that predicted fromDFT. Several theoretical attempts [ 12,34,36,63] to understand
the ARPES-determined gap structure [ 26,27,62,64]w e r e
reviewed recently in Ref. [ 15]. All were based on an “engi-
neered” tight-binding band structure consistent with ARPESdata [ 12], i.e., containing the correct spectral positions of the
bands (including the orbital content). Despite some success inexplaining certain features of the gap structure, others werenot reproduced properly in all approaches, although Ref. [ 34]
claimed a good overall fit to experiment.
To reveal how and whether the standard spin-fluctuation
theory result changes upon inclusion of quasiparticle weights,we use the same method as described above for a bandstructure relevant to LiFeAs [ 12]. The corresponding Fermi
surface is shown in Fig. 1(c). First, we note that moderate
changes in the quasiparticle weights which we set to {√
Zl}=
[0.5493,0.969,0.5952,0.5952,0.9267] do change the gap
structure, but largely preserve the structure of the susceptibility(see Appendix D). The gap functions, however, undergo a
remarkable change relative to unrenormalized spin-fluctuationtheory. These include first a stronger tendency towards s
±sym-
metry, even with small values of J. Note that the conventional
spin-fluctuation scenario, dandswave solutions are nearly
degenerate, a consequence of the poor ( π,0) nesting properties
of LiFeAs [ 24,26]. Second, orbital selectivity enhances the
gap on the small /Gamma1-centered hole pocket ( αpocket) [see
Fig. 7(a)]. This appears to correct the crucial discrepancy in
174504-6ORBITAL SELECTIVE PAIRING AND GAP STRUCTURES . . . PHYSICAL REVIEW B 95, 174504 (2017)
the calculation of Wang et al. [12] relative to experiment [see
Figs. 7(b) and7(c)]. Finally, the procedure leads to weaker
anisotropy of the gap on the large dxydominated pocket, also
in better agreement with experiment, whereas small deviationsbetween the ARPES data [ 26] and our calculation on the
electron pockets persist which could be due to hybridization ofthe corresponding bands. We did not investigate effects of spin-orbit coupling in this case since these are supposed to be small[12]. Note further that the (angular) position of the maximum
gap on the electron pockets changes from 0
◦to slightly
off 90◦, opening the possibility of two maxima (and two
minima). Unlike the models for FeSe (bulk) and monolayerFeSe, all three orbitals ( d
xy,dxz,dyz) play an important role
in determining the gap anisotropy on the βpockets, making it
more sensitive to changes in the electronic structure.
VI. DISCUSSION
The above results are extremely encouraging, suggest-
ing that the orbital selective correlation effects are indeedrequired when applying spin-fluctuation pairing theory toFe-chalcogenide and more strongly correlated Fe-based su-perconductors. We caution, however, that we have not derivedthe renormalizations entering the pair vertex self-consistentlyfrom a microscopic theory. Efforts along these lines are inprogress. Second, by construction the quasiparticle renormal-izations describe only the states near the Fermi level. Compari-son with ARPES measurements should be performed carefully,as these analyses tend to emphasize renormalizations on muchlarger energy scales, which may be quite different. Possible im-prints of the orbital selectivity could be visible in the penetra-tion depth [ 47] if calculated within the same theoretical frame-
work, or Friedel oscillations close to impurities in the caseof bulk FeSe which are rotating in direction as a function ofenergy [ 42]. Calculations along these lines are also in progress.
VII. CONCLUSIONS
In the absence of a fully controlled many-body treatment
of electronically paired superconductivity, it may be veryvaluable to have a simple phenomenological yet microscopicapproach that includes aspects of the low-energy quasiparticlerenormalizations that affect pairing most strongly. We havepresented a paradigm that allows for suppressed quasiparticleweight within the framework of conventional spin-fluctuationpairing theory, and argued that it provides accurate descriptionsfor the previously inexplicable superconducting energy gapstructures of the most strongly correlated FeSC. We havegiven results of explicit calculations in three cases wherecorrelations are known to play an important role: bulk FeSe,monolayer FeSe on STO, and LiFeAs. These results revealan immediate challenge to determine if our approach canbe combined with microscopic calculations of quasiparticleweights to yield a material-specific theory with predictivepower for strongly correlated FeSC.
ACKNOWLEDGMENTS
We would like to thank A. V . Chubukov, D. J. Scalapino,
and D. D. Scherer for useful discussions. A.Kr. and B.M.A.acknowledge support from a Lundbeckfond fellowship (Grant
No. A9318). P.J.H. acknowledges support through Departmentof Energy Grant No. DE-FG02-05ER46236. J.C.S.D.acknowledges gratefully support from the Moore Foundation’sEPiQS Initiative through Grant No. GBMF4544, and thehospitality and support of the Tyndall National Institute,University College Cork, Cork, Ireland. P.O.S. and A.Ko.acknowledge support from the Center for Emergent Supercon-ductivity, an Energy Frontier Research Center, headquarteredat Brookhaven National Laboratory and funded by the US De-partment of Energy under Grant No. DE-2009-BNL-PM015.
APPENDIX A: HAMILTONIAN AND CONSTRUCTION
OF GREEN’S FUNCTION
Considering the tight-binding Hamiltonian ( 1) together
with its diagonalization to band basis, one can constructthe Green’s function in the band basis G
μ(k,ωn)=[iωn−
ξμ(k)]−1. The unitary transformation that takes one from
the band basis (Greek indices) to the orbital basis (Romanindices) is
c
/lscriptσ(k)=/summationdisplay
νa/lscript
ν(k)cνσ(k). (A1)
Unitarity implies
/summationdisplay
/lscripta/lscript
ν(k)a/lscript
μ(k)∗=δμν (A2)
so we can invert ( A1) to find the orbital basis Green’s function
as stated in the main text:
G/lscript/lscript/prime(k,ωn)=/summationdisplay
μa/lscript
μ(k)a/lscript/prime∗
μ(k)Gμ(k,ωn)=/summationdisplay
μa/lscript
μ(k)a/lscript/prime∗
μ(k)
iωn−ξμ(k).
(A3)
APPENDIX B: QUASIPARTICLE DESCRIPTION
IN BAND SPACE
At this point, we make a short remark about the implications
of quasiparticles in band representation. Starting from Eq. ( 3),
we can transform back to the band basis and obtain thequasiparticle Green’s function
˜G
ν(k,ωn)=/summationdisplay
s,pas
ν∗(k)ap
ν(k)˜Gsp(k,ωn)
=/parenleftBigg/summationdisplay
s,p/vextendsingle/vextendsingleas
ν(k)/vextendsingle/vextendsingle2/vextendsingle/vextendsingleap
ν(k)/vextendsingle/vextendsingle2/radicalbig
Zs/radicalbig
Zp/parenrightBigg
Gν(k,ωn)
=˜Zν(k)Gν(k,ωn)≡˜Gν(k,ωn), (B1)
where ˜Zν(k)≡[/summationtext
s|as
ν(k)|2√Zs]2are the quasiparticle band
weights near the Fermi surface. If the point kon the Fermi
surface sheet νis dominated by a particular orbital weight
|as
ν(k)|2, the quasiparticle weight for that band will be given
predominantly by Zs. Calculating the spectral function from
such a Green’s function and plotting versus katω=0,
one directly sees that part of the Fermi surface is stronglysuppressed in intensity whenever an orbital dominates that hassmall quasiparticle weight, i.e., is strongly correlated. In Fig. 3,
174504-7ANDREAS KREISEL et al. PHYSICAL REVIEW B 95, 174504 (2017)
we show this effect of the spectral function on the example of
our model for FeSe (bulk).
We stress that the approach applied in this paper is
phenomenological in the sense that the band renormalizationsand the quasiparticle weights are not obtained self-consistentlyfrom the same bare interaction parameters. Thus, we donot address the problem of how to quantitatively capturenontrivial self-energy effects and the eventual transition tonon-Fermi-liquid behavior with increasing correlations or holedoping [ 4], but simply rely on a wealth of previous theoretical
studies showing the existence of orbital selectivity, and studytheir influence on the superconducting pairing structure.
APPENDIX C: SPIN-FLUCTUATION PAIRING:
UNCORRELATED MODEL
Here, we remind the reader of the approach to calculating
the gap function in the usual spin-fluctuation pairing model[38,40]. First, local interactions are included via the five-orbital
Hubbard-Hund Hamiltionan
H=H
0+U/summationdisplay
i,/lscriptni/lscript↑ni/lscript↓+U/prime/summationdisplay
i,/lscript/prime</lscriptni/lscriptni/lscript/prime
+J/summationdisplay
i,/lscript/prime</lscript/summationdisplay
σ,σ/primec†
i/lscriptσc†
i/lscript/primeσ/primeci/lscriptσ/primeci/lscript/primeσ
+J/prime/summationdisplay
i,/lscript/prime/negationslash=/lscriptc†
i/lscript↑c†
i/lscript↓ci/lscript/prime↓ci/lscript/prime↑, (C1)
where the interaction parameters U,U/prime,J,J/primeare given in
the notation of Kuroki et al. [65] with the choice U/prime=
U−2J,J=J/prime, leaving only UandJ/U to specify the
interactions. Here, /lscriptis an orbital index with /lscript∈(1,..., 5)
corresponding to the Fe 3 dorbitals ( dxy,dx2-y2,dxz,dyz,d3z2-r2).
The orbital susceptibility tensor in the normal state is nowgiven as
χ
0
/lscript1/lscript2/lscript3/lscript4(q)=−/summationdisplay
k,μνMμν
/lscript1/lscript2/lscript3/lscript4(k,q)Gμ(k+q)Gν(k),(C2)
where we have adopted the shorthand k≡(k,ωn), and defined
Mμν
/lscript1/lscript2/lscript3/lscript4(k,q)=a/lscript4
ν(k)a/lscript2,∗
ν(k)a/lscript1
μ(k+q)a/lscript3,∗
μ(k+q).(C3)
The Matsubara sum in Eq. ( C2) is performed analytically,
and we then evaluate χ0
/lscript1/lscript2/lscript3/lscript4by integrating over the full
Brillouin zone. As noted earlier [ 56], the Fermi surface nesting
condition gives significant contributions to the susceptibility,but finite-energy nesting also contributes. The spin- ( χ
RPA
1)
and charge-fluctuation ( χRPA
0) parts of the RPA susceptibility
forq=(q,ωn=0) are now defined within the random phase
approximation as
χRPA
1/lscript1/lscript2/lscript3/lscript4(q)={χ0(q)[1−¯Usχ0(q)]−1}/lscript1/lscript2/lscript3/lscript4,(C4a)
χRPA
0/lscript1/lscript2/lscript3/lscript4(q)={χ0(q)[1+¯Ucχ0(q)]−1}/lscript1/lscript2/lscript3/lscript4.(C4b)
The total spin susceptibility at ω=0 is then given by the sum
χ(q)=1
2/summationdisplay
/lscript/lscript/primeχRPA
1/lscript/lscript/lscript/prime/lscript/prime(q). (C5)
The interaction matrices ¯Usand ¯Ucin orbital space are
composed of linear combinations of U,U/prime,J,J/primeand their formsare given, e.g., in Ref. [ 39]. We focus here on the spin-singlet
vertex for pair scattering between bands νandμ,
/Gamma1νμ(k,k/prime)=Re/summationdisplay
/lscript1/lscript2/lscript3/lscript4a/lscript1,∗
ν(k)a/lscript4,∗
ν(−k)
×/Gamma1/lscript1/lscript2/lscript3/lscript4(k,k/prime)a/lscript2
μ(k/prime)a/lscript3
μ(−k/prime),(C6)
where kandk/primeare quasiparticle momenta restricted to the
pockets k∈Cνandk/prime∈Cμ, and is defined in terms of the the
orbital space vertex function
/Gamma1/lscript1/lscript2/lscript3/lscript4(k,k/prime)=/bracketleftbig3
2¯UsχRPA
1(k−k/prime)¯Us+1
2¯Us
−1
2¯UcχRPA
0(k−k/prime)¯Uc+1
2¯Uc/bracketrightbig
/lscript1/lscript2/lscript3/lscript4.
(C7)
Using this approximation to the vertex, we now consider the
linearized gap equation
−1
VG/summationdisplay
μ/integraldisplay
FSμdS/prime/Gamma1νμ(k,k/prime)gi(k/prime)
|vFμ(k/prime)|=λigi(k)( C 8 )
and solve for the leading eigenvalue λand corresponding
eigenfunction g(k). Here, vFμ(k/prime) is the Fermi velocity of
bandμand the integration is over the Fermi surface FS μ.T h e
eigenfunction gi(k) for the leading eigenvalue then determines
the symmetry and structure of the leading pairing gap /Delta1(k)∝
g(k) close to Tc. Finally, the area of the Fermi surface sheets
is discretized using a Delaunay triangulation algorithm thattransforms the integral equation ( C8) into an algebraic matrix
equation which is solved numerically. Typically, we use a k
mesh of 80 ×80×30 points for the kintegration and totally
≈1200 points on all Fermi sheets for a 3D calculation, while
for a 2D calculation the kmesh is on the order of 100 ×100 and
≈200 points on all Fermi sheets are required for reasonably
converged results.
APPENDIX D: SPIN-FLUCTUATION PAIRING
INCLUDING QUASIPARTICLE WEIGHTS
In this Appendix, we show the modified equations for
the pairing calculation as outlined above, but includingquasiparticle weights from dressed electrons. Taking the ansatz
05101520
(0,0) (π,0) ( π,π) (0, π) (0,0)χ(q)
FIG. 8. Susceptibility ˜ χfor our model for the monolayer FeSe
as calculated from the orbital selective ansatz using the quasi-
particle Green’s functions with {√Zl}=[0.4273,0.8000,0.9826,
0.9826,0.700] compared to the conventional calculation ( χ), where
the interactions have been scaled down.
174504-8ORBITAL SELECTIVE PAIRING AND GAP STRUCTURES . . . PHYSICAL REVIEW B 95, 174504 (2017)
0246
(0,0) (π,0) ( π,π) (0, π) (0,0)χ(q)
FIG. 9. Total susceptibility χfor LiFeAs as calculated from the
electronic structure using a 3D model and same quantity ˜ χ,b u t
calculated using the quasiparticle Green’s functions with {√Zl}=
[0.5493,0.969,0.5952,0.5952,0.9267].
for the dressed Green’s function ( 3), it is obvious that from
Eq. ( C2) immediately follows Eq. ( 4) which is then used in
Eqs. ( C4) instead of χ0
/lscript1/lscript2/lscript3/lscript4(q) for the dressed quantities. The
total susceptibility then reads as
˜χ(q)=1
2/summationdisplay
/lscript/lscript/prime˜χRPA
1/lscript/lscript/lscript/prime/lscript/prime(q). (D1)
For the FeSe (bulk) model, the total susceptibility is displayed
and discussed in the main text because the quasiparticleweights have a strong effect on the qualitative behavior. Atthis point, it is worth mentioning that this is not the casefor the model of monolayer FeSe, where the quasiparticleweights are chosen closer to unity (accounting for smallercorrelation effects in this material). In Fig. 8, it can be seenthat the total susceptibility is practically unchanged. Similar
conclusions can also be drawn from the comparison of the totalsusceptibilities for LiFeAs in the uncorrelated and correlatedmodel (see Fig. 9). Note that the quasiparticle weights Z
lare
consistent with DMFT results where it is found that t2gorbitals
are strongly correlated with dxystrongest, and components of
the susceptibility get suppressed ( dxystrongest) [ 66].
The equation
˜/Gamma1/lscript1/lscript2/lscript3/lscript4(k,k/prime)=/bracketleftbig3
2¯Us˜χRPA
1(k−k/prime)¯Us+1
2¯Us
−1
2¯Uc˜χRPA
0(k−k/prime)¯Uc+1
2¯Uc/bracketrightbig
/lscript1/lscript2/lscript3/lscript4
(D2)
for the orbital space vertex function is basically unchanged
except for the addition of the tilde. In the construction of thepair scattering vertex, additional quasiparticle weights enter
from the replacement c†
/lscript(k)→√Z/lscriptc†
/lscript(k) such that it reads as
˜/Gamma1νμ(k,k/prime)=Re/summationdisplay
/lscript1/lscript2/lscript3/lscript4/radicalbig
Z/lscript1/radicalbig
Z/lscript4a/lscript1,∗
ν(k)a/lscript4,∗
ν(−k)
ט/Gamma1/lscript1/lscript2/lscript3/lscript4(k,k/prime)/radicalbig
Z/lscript2/radicalbig
Z/lscript3a/lscript2
μ(k/prime)a/lscript3
μ(−k/prime)
(D3)
and enters Eq. ( C8) instead of /Gamma1νμ(k,k/prime).
APPENDIX E: COMPARISON OF 2D CALCULATIONS
AND 3D CALCULATIONS
In this paper, we discuss three different physical systems,
two of them parametrized using a band structure including a kz
dispersion as well. As noted already earlier, the susceptibility
as calculated from a 3D model (with weak dispersion in
180 90 0 180 18010123
angle (degrees)Δk
180 90 0 180 18010123
angle (degrees)Δk
180 90 0 180 18010123
angle (degrees)Δk
180 90 0 180 18010123
angle (degrees)Δk
180 90 0 180 18010123
angle (degrees)Δk
180 90 0 180 18010123
angle (degrees)Δk
180 90 0 180 180210123
angle (degrees)Δk
180 90 0 180 180210123
angle (degrees)Δk
FIG. 10. Comparison of the calculated gap function for FeSe (bulk) to experimental data from Refs. [ 10,49]. Calculated gap function
from the two-dimensional model at kz=0 with conventional spin-fluctuation pairing and interaction parameters U=0.33 eV,J=U/6
(a), a calculation with the orbitally selective pairing ansatz as described in the main text (b). Since the quasiparticle weights reduce the
susceptibility in general, a slightly larger interaction of U=0.54 eV was chosen, while the ratio J=U/6 is kept constant. Cuts of the
results as shown in the main text for a 3D calculation: (c) kz=0 cut from the conventional spin-fluctuation calculation, (d) the same cut
from the orbitally selective ansatz, cuts for kz=πare shown in Fig. 4. Variations of the fits for the 2D model, where the ratio of the
quasiparticle weights of the dyzanddxzorbital is constrained to the value as indicated on the figure [(g) and (h)]. The resulting values are then
{√Zl}=[0.2264,0.9717,0.4658,0.9317,0.6916] (g) and {√Zl}=[0.2633,0.9000,0.5998,0.8997,0.3630] (h).
174504-9ANDREAS KREISEL et al. PHYSICAL REVIEW B 95, 174504 (2017)
kzdirection) shows only very small dependence on kz[12].
Conclusions similar to the ones in the main text can also bedrawn in a two-dimensional calculation, where the initial bandstructure is just the one at k
z=0. Taking the same interaction
parameters and quasiparticle weights, one obtains qualitativelysimilar results as for the 3D calculation. This is expected sincethe electronic structure is found to be quasi-two-dimensional,and especially since the susceptibility and thus the pairinginteraction have little dependence on q
z. Differences in the
relative magnitudes of the gap functions on the individualpockets can, however, arise due to the variation of the Fermivelocities as a function of k
z, e.g., the weight at kz=0a s
included in a 2D calculation is not just the average of thepartial contributions to the density of states from different k
z
[12]. In the solution of the linearized gap equation, this can
increase the gap on individual pockets [ 12] or reduce the gapas seen on the αpocket for the 3D calculation in Fig. 10(d) .
Overall, the variation of the results is small and mostly ofquantitative nature rather than qualitative. We note that theFermi surface properties can still strongly influence the actualsuperconducting order parameter in such a calculation even ifthe pairing interaction itself has negligible variation in q
z.T h i s
will occur in a 2D calculation for the LiFeAs model where theFermi surface is different at cuts in k
z=0 and πbecause of
the closed αpocket. Because of this, we have not considered
any results of a 2D calculation for this model further. Finally,we present results for the gap structure obtained from a fitwhere the relative magnitudes of the quasiparticle weights ofthed
xzanddyzorbitals are kept fixed. Even when lowering the
ratio between those, the agreement is still good [see Figs. 10(g)
and10(h) ], but not allowing a larger quasiparticle weight in
thedyzorbital does not yield an agreement (not shown).
[1] L. de Medici, in Iron-Based Superconductivity, Weak and Strong
Correlations in Fe Superconductors , edited by P. D. Johnson,
G. Xu, and W.-G. Yin, Springer Series in Materials Science(Springer, Berlin, 2015).
[2] E. Bascones, B. Valenzuela, and M. Calderón, Magnetic inter-
actions in iron superconductors: A review, C. R. Phys. 17,36
(2016 ).
[3] A. van Roekeghem, P. Richard, H. Ding, and S. Biermann, Spec-
tral properties of transition metal pnictides and chalcogenides:Angle-resolved photoemission spectroscopy and dynamicalmean-field theory, C. R. Phys. 17,140(2016 ).
[4] Z. P. Yin, K. Haule, and G. Kotliar, Kinetic frustration and the
nature of the magnetic and paramagnetic states in iron pnictidesand iron chalcogenides, Nat. Mater. 10,932(2011 ).
[5] Y . Li, Z. Yin, X. Wang, D. W. Tam, D. L. Abernathy, A.
Podlesnyak, C. Zhang, M. Wang, L. Xing, C. Jin, K. Haule,G. Kotliar, Thomas A. Maier, and P. Dai, Orbital Selective SpinExcitations and Their Impact on Superconductivity of LiFe
1-x
CoxAs,Phys. Rev. Lett. 116,247001 (2016 ).
[6] S. Mandal, P. Zhang, and K. Haule, How correlated is the
FeSe/SrTiO 3system? arXiv:1609.06815 .
[7] Z. R. Ye, Y . Zhang, F. Chen, M. Xu, J. Jiang, X. H. Niu, C. H. P.
Wen, L. Y . Xing, X. C. Wang, C. Q. Jin, B. P. Xie, and D. L.Feng, Extraordinary Doping Effects on Quasiparticle Scatteringand Bandwidth in Iron-Based Superconductors, Phys. Rev. X 4,
031041 (2014 ).
[8] N. Arakawa and M. Ogata, Orbital-selective superconductivity
and the effect of lattice distortion in iron-based superconductors,J. Phys. Soc. Jpn. 80,074704 (2011 ).
[9] R. Yu, J.-X. Zhu, and Q. Si, Orbital-selective superconductivity,
gap anisotropy, and spin resonance excitations in a multiorbitalt-J
1-J2model for iron pnictides, P h y s .R e v .B 89,024509
(2014 ).
[10] P. O. Sprau, A. Kostin, A. Kreisel, A. E. Böhmer, V . Taufour,
P. C. Canfield, S. Mukherjee, P. J. Hirschfeld, B. M. Andersen,and J. C. Séamus Davis, Discovery of Orbital-Selective CooperPairing in FeSe, arXiv:1611.02134 .
[11] Y . Zhang, J. J. Lee, R. G. Moore, W. Li, M. Yi, M. Hashimoto,
D. H. Lu, T. P. Devereaux, D.-H. Lee, and Z.-X. Shen,Superconducting Gap Anisotropy in Monolayer FeSe thin Film,Phys. Rev. Lett. 117,117001 (2016 ).[12] Y . Wang, A. Kreisel, V . B. Zabolotnyy, S. V . Borisenko, B.
Büchner, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino,Superconducting gap in LiFeAs from three-dimensional spin-fluctuation pairing calculations, P h y s .R e v .B 88,174516 (2013 ).
[13] D. J. Scalapino, A common thread: The pairing interaction
for unconventional superconductors, Rev. Mod. Phys. 84,1383
(2012 ).
[14] A. Ardavan, S. Brown, S. Kagoshima, K. Kanoda, K. Kuroki,
H. Mori, M. Ogata, S. Uji, and J. Wosnitza, Recent topicsof organic superconductors, J. Phys. Soc. Jpn. 81,011004
(2012 ).
[15] P. J. Hirschfeld, Using gap symmetry and structure to reveal the
pairing mechanism in Fe-based superconductors, C. R. Phys.
17,197(2016 ).
[16] A. Chubukov, in Iron-Based Superconductivity, Itinerant Elec-
tron Scenario for Fe-based Superconductors ,e d i t e db yP .D .
Johnson, G. Xu, and W.-G. Yin, Springer Series in MaterialsScience (Springer, Berlin, 2015).
[17] N. F. Berk and J. R. Schrieffer, Effect of Ferromagnetic Spin
Correlations on Superconductivity, Phys. Rev. Lett. 17,433
(1966 ).
[18] D. J. Scalapino, E. Loh, and J. E. Hirsch, d-wave pairing near a
spin-density-wave instability, Phys. Rev. B 34,8190 (1986 ).
[19] T. Takimoto, T. Hotta, and K. Ueda, Strong-coupling theory of
superconductivity in a degenerate Hubbard model, Phys. Rev. B
69,104504 (2004 ).
[20] M. Altmeyer, D. Guterding, P. J. Hirschfeld, T. A. Maier, R.
Valentí, and D. J. Scalapino, Role of vertex corrections in thematrix formulation of the random phase approximation for themultiorbital hubbard model, Phys. Rev. B 94,214515 (2016 ).
[21] P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Gap
symmetry and structure of Fe-based superconductors,
Rep. Prog.
Phys. 74,124508 (2011 ).
[22] C. Platt, W. Hanke, and R. Thomale, Functional renormalization
group for multi-orbital Fermi surface instabilities, Adv. Phys. 62,
453(2013 ).
[23] J. Ferber, K. Foyevtsova, R. Valentí, and H. O. Jeschke, LDA +
DMFT study of the effects of correlation in LiFeAs, Phys. Rev.
B85,094505 (2012 ).
[24] G. Lee, H. S. Ji, Y . Kim, C. Kim, K. Haule, G. Kotliar, B.
L e e ,S .K h i m ,K .H .K i m ,K .S .K i m ,K . - S .K i m ,a n dJ .H .
174504-10ORBITAL SELECTIVE PAIRING AND GAP STRUCTURES . . . PHYSICAL REVIEW B 95, 174504 (2017)
Shim, Orbital Selective Fermi Surface Shifts and Mechanism of
HighTcSuperconductivity in Correlated aFeAs ( a=Li, Na),
Phys. Rev. Lett. 109,177001 (2012 ).
[25] R. Yu, Q. Si, P. Goswami, and E. Abrahams, Electron correlation
and spin dynamics in iron pnictides and chalcogenides, J. Phys.:
Conf. Ser. 449,012025 (2013 ).
[26] S. V . Borisenko, V . B. Zabolotnyy, D. V . Evtushinsky, T. K.
Kim, I. V . Morozov, A. N. Yaresko, A. A. Kordyuk, G. Behr, A.Vasiliev, R. Follath, and B. Büchner, Superconductivity WithoutNesting in LiFeAs, P h y s .R e v .L e t t . 105,067002 (2010 ).
[27] S. V . Borisenko, V . B. Zabolotnyy, A. A. Kordyuk, D.
V . Evtushinsky, T. K. Kim, I. V . Morozov, R. Follath,and B. Büchner, One-sign order parameter in iron basedsuperconductor, Symmetry 4,251(2012 ).
[28] R. Yu and Q. Si, Mott transition in multiorbital models for iron
pnictides, Phys. Rev. B 84,235115 (2011 ).
[29] M. Yi, D. H. Lu, R. Yu, S. C. Riggs, J.-H. Chu, B. Lv, Z. K.
Liu, M. Lu, Y .-T. Cui, M. Hashimoto, S.-K. Mo, Z. Hussain,C. W. Chu, I. R. Fisher, Q. Si, and Z.-X. Shen, Observationof Temperature-Induced Crossover to an Orbital-Selective MottPhase in A
xFe2-ySe2(a=K, Rb) Superconductors, Phys. Rev.
Lett.110,067003 (2013 ).
[30] R. Yu and Q. Si, Orbital-Selective Mott Phase in Multiorbital
Models for Alkaline Iron Selenides K 1-xFe2-ySe2,Phys. Rev.
Lett.110,146402 (2013 ).
[31] M. Yi, Z.-K. Liu, Y . Zhang, R. Yu, J.-X. Zhu, J. J. Lee, R. G.
Moore, F. T. Schmitt, W. Li, S. C. Riggs, J.-H. Chu, B. Lv, J.H u ,M .H a s h i m o t o ,S . - K .M o ,Z .H u s s a i n ,Z .Q .M a o ,C .W .Chu, I. R. Fisher, Q. Si, Z.-X. Shen, and D. H. Lu, Observationof universal strong orbital-dependent correlation effects in ironchalcogenides, Nat. Commun. 6,7777 (2015 ).
[32] Z. K. Liu, M. Yi, Y . Zhang, J. Hu, R. Yu, J.-X. Zhu, R.-H.
He, Y . L. Chen, M. Hashimoto, R. G. Moore, S.-K. Mo,Z. Hussain, Q. Si, Z. Q. Mao, D. H. Lu, and Z.-X. Shen,Experimental observation of incoherent-coherent crossover andorbital-dependent band renormalization in iron chalcogenidesuperconductors, Phys. Rev. B 92,235138 (2015 ).
[33] M. A. Metlitski, D. F. Mross, S. Sachdev, and T. Senthil, Cooper
pairing in non-Fermi liquids, Phys. Rev. B 91,115111 (2015 ).
[34] T. Saito, S. Onari, Y . Yamakawa, H. Kontani, S. V . Borisenko,
and V . B. Zabolotnyy, Reproduction of experimental gapstructure in lifeas based on orbital-spin fluctuation theory:s
++-wave, s±-wave, and hole- s±-wave states, Phys. Rev. B 90,
035104 (2014 ).
[35] H. Eschrig and K. Koepernik, Tight-binding models for
the iron-based superconductors, Phys. Rev. B 80,104503
(2009 ).
[36] F. Ahn, I. Eremin, J. Knolle, V . B. Zabolotnyy, S. V . Borisenko,
B. Büchner, and A. V . Chubukov, Superconductivity fromrepulsion in LiFeAs: Novel s-wave symmetry and potential
time-reversal symmetry breaking, Phys. Rev. B 89,144513
(2014 ).
[37] Y . Yamakawa, S. Onari, and H. Kontani, Nematicity and Mag-
netism in FeSe and Other Families of Fe-based Superconductors,Phys. Rev. X 6,021032 (2016 ).
[38] S. Graser, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino, Near-
degeneracy of several pairing channels in multiorbital modelsfor the Fe pnictides, New J. Phys. 11,025016 (2009 ).
[39] A. F. Kemper, T. A. Maier, S. Graser, H.-P. Cheng, P. J.
Hirschfeld, and D. J. Scalapino, Sensitivity of the superconduct-ing state and magnetic susceptibility to key aspects of electronic
structure in ferropnictides, New J. Phys. 12,073030 (2010 ).
[40] A. Kreisel, Y . Wang, T. A. Maier, P. J. Hirschfeld, and
D. J. Scalapino, Spin fluctuations and superconductivity inK
xFe2-ySe2,Phys. Rev. B 88,094522 (2013 ).
[41] C.-L. Song, Y .-L. Wang, P. Cheng, Y .-P. Jiang, W. Li, T. Zhang,
Z. Li, K. He, L. Wang, J.-F. Jia, H.-H. Hung, C. Wu, X. Ma, X.Chen, and Q.-K. Xue, Direct observation of nodes and twofoldsymmetry in FeSe superconductor, Science 332,1410 (2011 ).
[42] S. Kasahara, T. Watashige, T. Hanaguri, Y . Kohsaka, T.
Yamashita, Y . Shimoyama, Y . Mizukami, R. Endo, H. Ikeda,K. Aoyama, T. Terashima, S. Uji, T. Wolf, H. von Löhneysen,T. Shibauchi, and Y . Matsuda, Field-induced superconductingphase of FeSe in the BCS-BEC cross-over, Proc. Natl. Acad.
Sci. USA 111,16309 (2014 ).
[ 4 3 ] J . - Y .L i n ,Y .S .H s i e h ,D .A .C h a r e e v ,A .N .V a s i l i e v ,Y .
Parsons, and H. D. Yang, Coexistence of isotropic and extendeds-wave order parameters in FeSe as revealed by low-temperature
specific heat, P h y s .R e v .B 84,220507 (2011 ).
[44] L. Jiao, C.-L. Huang, S. Rößler, C. Koz, U. K. Rößler, U.
Schwarz, and S. Wirth, Superconducting gap structure of fese,Sci. Rep. 7,44024 (2017 ).
[45] J. K. Dong, T. Y . Guan, S. Y . Zhou, X. Qiu, L. Ding, C.
Zhang, U. Patel, Z. L. Xiao, and S. Y . Li, Multigap nodelesssuperconductivity in FeSe
x: Evidence from quasiparticle heat
transport, P h y s .R e v .B 80,024518 (2009 ).
[46] P. Bourgeois-Hope, S. Chi, D. A. Bonn, R. Liang, W. N. Hardy,
T. Wolf, C. Meingast, N. Doiron-Leyraud, and L. Taillefer,Thermal Conductivity of the Iron-Based Superconductor FeSe:Nodeless Gap with a Strong Two-Band Character, Phys. Rev.
Lett.117,097003 (2016 ).
[47] M. Li, N. R. Lee-Hone, S. Chi, R. Liang, W. N. Hardy,
D. A. Bonn, E. Girt, and D. M. Broun, Superfluid density andmicrowave conductivity of FeSe superconductor: ultra-long-lived quasiparticles and extended s-wave energy gap, New J.
Phys. 18,082001 (2016 ).
[48] S. Teknowijoyo, K. Cho, M. A. Tanatar, J. Gonzales,
A. E. Böhmer, O. Cavani, V . Mishra, P. J. Hirschfeld, S. L.Bud’ko, P. C. Canfield, and R. Prozorov, Enhancement ofsuperconducting transition temperature by pointlike disorderand anisotropic energy gap in FeSe single crystals, Phys. Rev.
B94,064521 (2016 ).
[49] H. C. Xu, X. H. Niu, D. F. Xu, J. Jiang, Q. Yao, Q. Y . Chen,
Q. Song, M. Abdel-Hafiez, D. A. Chareev, A. N. Vasiliev,Q. S. Wang, H. L. Wo, J. Zhao, R. Peng, and D. L. Feng,Highly Anisotropic and Twofold Symmetric SuperconductingGap in Nematically Ordered FeSe
0.93S0.07,P h y s .R e v .L e t t . 117,
157003 (2016 ).
[50] T. Terashima, N. Kikugawa, A. Kiswandhi, E.-S. Choi, J. S.
Brooks, S. Kasahara, T. Watashige, H. Ikeda, T. Shibauchi, Y .Matsuda, T. Wolf, A. E. Böhmer, F. Hardy, C. Meingast, H.v. Löhneysen, M.-T. Suzuki, R. Arita, and S. Uji, AnomalousFermi surface in FeSe seen by Shubnikov–de Haas oscillationmeasurements, P h y s .R e v .B 90,144517 (2014 ).
[51] A. Audouard, F. Duc, L. Drigo, P. Toulemonde, S. Karlsson,
P. Strobel, and A. Sulpice, Quantum oscillations and uppercritical magnetic field of the iron-based superconductor FeSe,Europhys. Lett. 109,27003 (2015 ).
[52] M. D. Watson, T. K. Kim, A. A. Haghighirad, N. R. Davies,
A. McCollam, A. Narayanan, S. F. Blake, Y . L. Chen, S.
174504-11ANDREAS KREISEL et al. PHYSICAL REVIEW B 95, 174504 (2017)
Ghannadzadeh, A. J. Schofield, M. Hoesch, C. Meingast, T.
Wolf, and A. I. Coldea, Emergence of the nematic electronicstate in FeSe, Phys. Rev. B 91,155106 (2015 ).
[53] M. D. Watson, T. K. Kim, L. C. Rhodes, M. Eschrig, M. Hoesch,
A. A. Haghighirad, and A. I. Coldea, Evidence for unidirectionalnematic bond ordering in fese, Phys. Rev. B 94,201107
(2016 ).
[54] T. Miyake, K. Nakamura, R. Arita, and M. Imada, Comparison of
ab initio low-energy models for LaFePO, LaFeAsO, BaFe
2As2,
LiFeAs, FeSe, and FeTe: Electron correlation and covalency,J. Phys. Soc. Jpn. 79,044705 (2010 ).
[55] D. D. Scherer, A. C. Jacko, C. Friedrich, E. S ¸as¸ıo˘glu, S. Blügel,
R. Valentí, and B. M. Andersen, Interplay of nematic andmagnetic orders in FeSe under pressure, P h y s .R e v .B 95,094504
(2017 ).
[56] A. Kreisel, S. Mukherjee, P. J. Hirschfeld, and B. M. Andersen,
Spin excitations in a model of FeSe with orbital ordering,Phys. Rev. B 92,224515 (2015 ).
[57] Q. Wang, Y . Shen, B. Pan, X. Zhang, K. Ikeuchi, K. Iida, A. D.
Christianson, H. C. Walker, D. T. Adroja, M. Abdel-Hafiez, X.Chen, D. A. Chareev, A. N. Vasiliev, and J. Zhao, Magneticground state of fese, Nat. Commun. 7,12182 (2016 ).
[58] Q. Wang, Y . Shen, B. Pan, Y . Hao, M. Ma, F. Zhou, P. Steffens,
K. Schmalzl, T. R. Forrest, M. Abdel-Hafiez, X. Chen, D.A. Chareev, A. N. Vasiliev, P. Bourges, Y . Sidis, H. Cao,and J. Zhao, Strong interplay between stripe spin fluctuations,nematicity and superconductivity in FeSe, Nat. Mater. 15,159
(2016 ).
[59] Y . Kushnirenko, A. A. Kordyuk, A. Fedorov, E. Haubold,
T. Wolf, B. Büchner, and S. V . Borisenko, Anomaloustemperature evolution of the electronic structure of FeSe,arXiv:1702.02088 .[60] D. Liu, W. Zhang, D. Mou, J. He, Y .-B. Ou, Q.-Y . Wang, Z. Li,
L. Wang, L. Zhao, S. He, Y . Peng, X. Liu, C. Chen, L Yu, G.Liu, X. Dong, J. Zhang, C. Chen, Z. Xu, J. Hu, X. Chen, X. Ma,Q. Xue, and X. J. Zhou, Electronic origin of high-temperaturesuperconductivity in single-layer FeSe superconductor,Nat. Commun. 3,931(2012 ).
[61] S. Tan, Y . Zhang, M. Xia, Z. Ye, F. Chen, X. Xie, R. Peng, D.
Xu, Q. Fan, H. Xu, J. Jiang, T. Zhang, X. Lai, T. Xiang, J. Hu,B. Xie, and D. Feng, Interface-induced superconductivity andstrain-dependent spin density waves in FeSe/SrTiO3 thin films,Nat. Mater. 12,634(2013 ).
[62] M. P. Allan, A. W. Rost, A. P. Mackenzie, Y . Xie, J. C. Davis,
K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, and T.-M. Chuang,Anisotropic energy gaps of iron-based superconductivity fromintraband quasiparticle interference in LiFeAs, Science 336,563
(2012 ).
[63] Z. P. Yin, K. Haule, and G. Kotliar, Spin dynamics and orbital-
antiphase pairing symmetry in iron-based superconductors,Nat. Phys. 10,845(2014 ).
[64] K. Umezawa, Y . Li, H. Miao, K. Nakayama, Z.-H. Liu, P.
Richard, T. Sato, J. B. He, D.-M. Wang, G. F. Chen, H. Ding,T. Takahashi, and S.-C. Wang, Unconventional Anisotropics-Wave Superconducting Gaps of the Lifeas Iron-Pnictide
Superconductor, P h y s .R e v .L e t t . 108,037002 (2012 ).
[65] K. Kuroki, S. Onari, R. Arita, H. Usui, Y . Tanaka, H. Kontani,
and H. Aoki, Unconventional Pairing Originating from theDisconnected Fermi Surfaces of Superconducting LaFeAsO
1-x
Fx,P h y s .R e v .L e t t . 101,087004 (2008 ).
[66] R. Nourafkan, G. Kotliar, and A.-M. S. Tremblay, Correlation-
Enhanced Odd-Parity Interorbital Dinglet Pairing in the Iron-Pnictide Superconductor LiFeAs, Phys. Rev. Lett. 117,137001
(2016 ).
174504-12 |
PhysRevB.83.205128.pdf | PHYSICAL REVIEW B 83, 205128 (2011)
Screened hybrid and self-consistent GW calculations of cadmium/magnesium
indium sulfide materials
Melissa J. Lucero,1Irene Aguilera,2Cristian V . Diaconu,1Pablo Palacios,2,3Perla Wahn ´on,2and Gustavo E. Scuseria1,4
1Department of Chemistry, Rice University, Houston, Texas 77005-1892, USA
2Instituto de Energ ´ıa Solar and Departmento Tecnolog ´ıas Especiales, ETSI Telecomunicati ´on, UPM,
Ciudad Universitaria, Madrid E-28040, Spain
3F´ısica y Qu ´ımica Aplicadas a la T ´ecnica Aeron ´autica, E. de Ingenier ´ıa Aeron ´autica y del Espacio, UPM,
Ciudad Universitaria, Madrid E-28040, Spain
4Department of Physics and Astronomy, Rice University, Houston, Texas 77005-1827, USA
(Received 14 February 2011; revised manuscript received 4 April 2011; published 25 May 2011)
The cadmium and magnesium indium sulfides are medium-gap semiconductors demonstrating a propensity to
form intermediate band materials when doped with transition metals. The inherent structural diversity exhibitedbyM
+2In2S4thiospinels and related AB 2X4compounds often precludes definitive experimental determination of
the band-gap width and type of transition. Employing a series of traditional semilocal functionals (e.g., the localspin density approximation; the Perdew, Burke, and Enzerhof functional; and the Tao, Perdew, Staroverov, andScuseria functional) the screened hybrid of Heyd, Scuseria, and Ernzerhof (HSE), band gaps, projected densitiesof states, and band structures are calculated for the normal, full inverse, and intermediate configurations of[Cd/Mg]
8In16S32. Band structures and band gaps are also obtained via self-consistent many-body methods, using
the static Coulomb-hole and screened exchange approximation to GW as a starting point for perturbative G0W0
calculations. Comparison to experiment indicates that HSE provides an accurate, computationally efficient, and
relatively rapid means for predicting band-gap properties in spinel-type photovoltaic materials.
DOI: 10.1103/PhysRevB.83.205128 PACS number(s): 71 .15.Mb, 71 .20.Nr, 78 .20.Bh
I. INTRODUCTION
Photovoltaic cells containing intermediate band (IB) ma-
terials are capable of efficiently absorbing photons over abroad range of the solar spectrum. An IB optimally situatedbetween the valence and conduction bands can result inelectron promotion using two photons with a combined energyexpenditure smaller than the typical one-photon electronic
excitation across the analogous semiconductor gap, boosting
ideal efficiencies from 40.7%
1to 63.1%.2Rapid prescreening
of semiconductors with ∼2–3 eV band gaps will facilitate
selection, suggest modification, and expedite fabrication ofIB-forming, doped semiconductors.
To date, the majority of solid-state computational studies
employ semilocal
3exchange correlation functionals [e.g., lo-
cal spin density approximation (LSDA), generalized-gradientapproximation (GGA), or meta-GGA variants] that consis-tently fail to reproduce experimental band gaps despite thedevelopment of more sophisticated approximations.
4Interme-
diate band photovoltaics are formally metallic, yet still closelyresemble the undoped parent semiconductors that tend to haveunderestimated band gaps on the order of 1 eV .
5–9
Fortunately, more accurate results, comparable to those of
full, self-consistent (sc) GW calculations,10are accessible,
at reduced computational cost, by employing the Coulomb-
hole and screened exchange (COHSEX)11,12approximation.
The COHSEX approximation accounts for statically screenedexchange and correlation in the form of the classical interactionbetween and additional point charge in the system and thesurrounding polarization cloud that this additional charge
induces. The static COHSEX result is then augmented with
dynamic effects through a perturbative G
0W0calculation. The
main effects neglected by this scheme are the excitonic andpolaronic effects.The scCOHSEX +G
0W0(scGW) scheme has been suc-
cessfully applied to a wide range of materials,11,13–16yielding
fundamental band gaps (as opposed to the smaller optical gaps)
and band structures, often in good agreement with experiment.Nevertheless, when applied to IB materials doped with highconcentrations of transition metals, even these many-bodycorrections become prohibitively expensive for all but thesmallest systems.
17
Other well-established, less CPU-intensive corrections are
also unsuitable for modeling transition-metal-doped IB mate-rials: perturbative G
0W0after LSDA cannot accurately address
the influence of populated dorbitals on band gaps,11,18while
density functional theory plus Hubbard U(DFT+U) methods
require system-dependent parameters that are unknown fornovel materials. It is worth noting that the screened short-rangeHartree-Fock exchange interactions in the hybrid functionalof Heyd, Scuseria, and Ernzerhof
19–21(HSE) are reminiscent
of the role that the Hubbard on-site repulsion Uplays in
DFT+U. However, unlike +Umethods, HSE can allocate
a unique effective “ U” to different orbital interactions. In
fact, a recent paper22advocates the use of HSE to determine
Uwhen experimental information is lacking and the need
to reduce computational effort surmounts the desire for in-creased prediction quality. Significantly, HSE alone produces
semiconductor band gaps and lattice parameters in excellentagreement with experiment,
19,23without requiring multiple
calculations, perturbative adjustments or material-dependentparameters—and at significantly reduced cost relative tomany-body corrections.
13,14,24–26
Recently, the M+2In2S4semiconductors containing Mg and
Cd, garnered considerable attention due to their potential ap-plication in high-efficiency solar cells.
6,27Spinel-type chalco-
genides are capable of adopting a variety of related crystallineforms,
28a consequence of the AB 2X4lattice affording the
205128-1 1098-0121/2011/83(20)/205128(12) ©2011 American Physical SocietyMELISSA J. LUCERO et al. PHYSICAL REVIEW B 83, 205128 (2011)
anions freedom to expand or contract around their fractional
coordinates, thus allowing facile accommodation of a widerange of cation sizes, while maintaining overall symmetry. Theliterally hundreds of known spinels are classified accordingto the 24 occupied interstices of the fcc lattice (defined byX). By convention, the eight smaller, usually divalent cations
occupying T
dholes are designated A, while the remaining 16,
typically higher-valent Bcations reside in Ohholes29yielding
crystallographic unit cells of composition A8B16X32, where
X=O, S, Se, or Te.
The limiting designations30for cation occupancy in spinel
structures are (a) normal , with all Acations filling Tdsites
and all Bcations in Ohholes, or (b) full inverse , in which
A=Bfor occupied Ohsites, forcing half of the Bcations
intoTdholes. The term partial inverse describes the spectrum
of intermediate spinel structures, x=A1−xBx[AxB2−x]X4,
where brackets denote Ohsites. Thus defined, the degree of
inversion, x, ranges from 0 (normal) to 1 (full inverse), with
x=2
3representing a fully stochastic system.
II. COMPUTATIONAL METHODS
A. Density functional calculations
Electronic-structure calculations were performed using the
periodic boundary-condition code31–33within the GAUSSIAN
suite of programs.34Data analysis and visualization were
performed using GaussView35and VMD.36
Gaussian basis sets modified for solids are provided in the
supplementary material37and are of the following quality: Mg:
8-511G (all-electron); S: 6-311G*(all-electron); Cd: 6-311G(all-electron); In: 4 s4p2d(ECP, modified). Unless otherwise
noted, initial geometries are the conventional, crystallographicunit cells ( Fd¯3m, 227), downloaded as CIF files from the
ICSD,
38The 56-atom crystallographic unit cells are optimized
in redundant internal coordinates39with 36 kpoints on a 4 ×
4×4 mesh for the reciprocal space integration. The 14-atom
primitive cells are optimized similarly, but employ 112 kpoints
on a 6 ×6×6 mesh.
Reported band gaps and related properties for fully relaxed
(lattice parameters and geometries) periodic systems wereobtained using three semilocal and one screened hybridfunctional to create a series of increasingly sophisticatedexchange-correlation approximations. Specifically, we com-pare the local spin density approximation (LSDA)
40(with
SVWN541), the GGA corrected functional of Perdew, Burke,
and Enzerhof42,43(PBE), the meta-GGA functional of Tao,
Perdew, Staroverov, and Scuseria44(TPSS), and the nonlocal
Heyd-Scuseria-Enzerhof45screened hybrid functional (HSE).
B. Many-body calculations
All many-body calculations were performed on 14-atom
cells using the plane-wave based code ABINIT .46For the
COHSEX and G0W0calculations,47a basis set of ∼25 000
plane waves was required for convergence. A Monkhorst-Packk-point mesh of 3 ×3×3 was used to sample the Brillouin
zone. Norm-conserving pseudopotentials
48were generated
with the fh i98PP code,49accounting for semicore states
(e.g., the 4s4p4dof In) explicitly in the valence, follow-
ing Hybertsen.50Details of the pseudopotential generationprocedure can be found in the supplementary material.37
The plasmon-pole model51is used for G0W0calculations
and COHSEX wave functions are represented on a restrictedLSDA basis set as proposed by Bruneval.
11All scGW calcu-
lations start from relaxed LSDA-optimized structures: normalCd
2In4S8,ao=10.775 ˚A; normal Mg 2In4S8,ao=10.682 ˚A;
full inverse Mg 2In4S8,ao=10.634 ˚A.
III. CADMIUM INDIUM SULFIDE
Numerous applications, particularly in photovoltaics and
light-emitting diodes (LED’s),52–54render cadmium indium
sulfide an extremely well-studied thiospinel, generally ac-cepted to crystallize in a normal structure, with x≈1,
although studies involving partial inverse structures andmixed crystals have been reported.
55,56DFT calculations of
normal CdIn 2S4were performed on the conventional 56-atom
crystallographic unit cells ( Fd¯3m, 227), as well as the 14-atom
primitives. The inverse ordering is modeled using only theprimitive cells.
A. Cd normal spinel structure
Measured band gaps are on the order of 2.1–2.7 eV .52,57–66
A rather broad range of band gaps is also observed
for the related spinel-type transparent conducting oxides(TCO’s) CdIn
2O4,Eg=2.67–3.24,67and Cd 2SnO 4,Eg=
2.06–3.00.67,68Comparison of bulk and thin-film specimens
of the Cd 1+xIn2−2xSnxO4solid solution demonstrates that the
optical gaps for thin films are significantly larger than for bulksamples,
69this difference most likely arising from a Burstein-
Moss shift.70Furthermore, the gap narrows as the cation
ordering becomes more inverted,71a consequence of the order-
disorder phenomena discussed in Sec. IV. Note that there are
many documented larger lattice constants than the commonly
citedao=10.797. Lee et al.60report that the CdIn 2S4ao
varies according to the method of crystal growth, ranging
from 10.838 to 10.860 ˚A, rather larger than the 10.797 ˚A
reported by Hahn.72Thus, the band-gap widths for these
systems are affected by dimensionality and degree of inversion,which is dependent upon method of synthesis: films havelarger gaps, and any reaction condition that facilitates inversionresults in lower gaps.
As indicated in Table I, the three semilocal functionals
underestimate the gap for the normal spinel by at least 1 eV ,as expected, while the screened hybrid HSE provides bandgaps close to that of experiment at 2.33 eV , tending toward thebottom of the reported experimental band gaps. There is alsoan experimental lack of consensus (see Ref. 73and references
therein) regarding the nature of the transition. However, all
four functionals predict an indirect transition that is ∼10 meV
lower in energy than that for the direct path, perhaps indicatingthat this minor energy difference is somehow related to thegeneral disagreement regarding the type of band gap. (Thecommon practice of reporting only one decimal place inducesa coalescence of theoretical gaps, thus forcing inference of adirect gap.) This vanishingly small /Delta1Eis not unique: β-In
2S3
also has experimental band gaps ranging from 2 to 3 eV in
magnitude with disputed indirect-direct transitions typicallyvarying by ∼10 meV .
74Note that the β-In2S3structure can
205128-2SCREENED HYBRID AND SELF-CONSISTENT G W ... PHYSICAL REVIEW B 83, 205128 (2011)
TABLE I. Normal and full inverse CdIn 2S4: Functional depen-
dence of band gap (eV) and lattice parameters ( ˚A).
Functional LSDA PBE TPSS HSE
Nature of gap EiEdEiEdEiEdEiEd
Normal Cd8In16S32 56-atom cell
Experiment 2.2–2.7aao=10.797b
ao 10.840 11.106 11.073 11.000
Band gapc1.34 1.44 1.21 1.28 1.52 1.60 2.33 2.41
Rlx→HSE spd2.14 2.25 2.13 2.21 2.19 2.28
Full inverse Cd2In4S8 14-atom cell
a 7.671 7.861 7.835 7.776
b 7.722 7.909 7.883 7.841
c 7.656 7.842 7.813 7.763
Band gapc0.21 0.22 0.13 0.14 0.37 0.40 1.19 1.23
Rlx→HSE spd1.39 1.42 1.02 1.06 1.08 1.11
aReferences 52,57–66.
bICSD ID 300725.72
cFully relaxed geometry and forces.
dRelaxed using LSDA/PBE/TPSS and then HSE energy calculation.
be considered a parent of the many indium thiospinels, and
it is often described as a quasiquaternary defect spinel withcationic vacancies in the T
dsites ordered along the caxis.75
The 0 K lattice parameters predicted by LSDA most closely
resemble measured values, yet the HSE-relaxed geometry,with a slightly larger volume, has a band gap in much betteragreement with experiment. In fact, while LSDA geometriesare generally considered to be better for semiconductors, theUV photoemission spectra of CdIn
2S4and related spinels
exhibit little sensitivity to small crystallographic deviations.76
Full relaxation using each of the semilocal functionals, fol-lowed by HSE single point energy calculations, is summarizedin the “Rlx →HSE
sp” row of Table I. All relaxed lattices,
withaovarying from experiment by 4–30 pm, result in
gaps close to—or within—the experimental range, clearlyillustrating the profound effect that the introduction of nonlocalHartree-Fock-type exchange has on bandwidth.
Moreover, the HSE single point energy of the LSDA-
relaxed structure, with a gap of 2.14 eV , and the HSE-relaxedgap of 2.33 eV are at the bottom of the experimental range,which correctly corresponds to bulk
58,59rather than thin-
film52,53band gaps. Indeed, a recent study of hierarchical
nanostructured CdIn 2S4produced at low temperatures using
different methods resulted in multiple morphologies, yet theband gaps were constrained to a narrow range of 2.23–2.27 eV .
77
Admittedly, evaluation of Hartree-Fock exchange is com-
putationally expensive for anyhybrid functional. This potential
bottleneck may be surmounted by first performing a fullrelaxation with a less expensive functional, followed bya single point energy calculation using HSE. This simpleshortcut yields more accurate band gaps and would workequally well for anyfunctional considered to produce superior
lattice parameters, whether traditionally semilocal or next-generation, designed specifically for solids, e.g., HSEsol.
78
This procedure should prove quite useful, particularly forstudies of formation energies of interstitial defects,
79defect
transition levels80(HSE performs particularly well for both),or any investigation requiring large supercells. Moreover,
this shortcut is possible in any software package with animplementation of HSE.
B. Cd full inverse spinel structure
The experimentally unobserved full inverse structure (bot-
tom section, Table I), like the normal spinel, also has a
marginally indirect band gap, predicted to have a width of0.1–04 eV by all functionals except HSE, which produces asomewhat larger, 1.2 eV gap. The HSE single point energiesof structures relaxed using semilocal functionals also show anincreased in gap, with the indirect transition favored, again, byonly a milli-electron volt. Notably, the analogous spinel oxide,CdIn
2O4, was also calculated to have a smaller band gap in
the inverse spinel structure.81HSE thus provides an interesting
prediction of a 1.2–1.4 eV band gap should such a structure beisolated.
C. Densities of states
The Cd 8In16S32normal spinel projected density of states
(PDOS) is plotted for each functional in Fig. 1.T h eI n5 s
(blue) and S 3 porbitals (yellow) dominate the conduction
band, while the primary contribution to the valence band isalmost exclusively S 3 porbital. This pattern is strikingly
similar to that observed for β-In
2S3, which has a gap of around
2.1 eV .82,83T h eC d5 scontribution is minimal in both the top of
the valence and bottom of the conduction bands, demonstratingthat metal insertion into the β-In
2S3manifold produces more
significant structural consequences (ordered defect spinel →
normal spinel) than for electronic properties relevant tothe band gap. As the exchange correlation approximationsimprove, LSDA →TPSS, a clear blueshift is observed
for the conduction band, which dramatically increases uponintroduction of nonlocal Hartree-Fock exchange (HSE). Incontrast, very little of interest transpires in the valenceband, which is somewhat wider for HSE than the semilocalfunctionals. The HSE band resembles that for LSDA, buthas more structure and a slightly extended ( ∼0.2–0.3 eV)
low-energy tail.
The transition from normal to full inverse spinel structures
results in a marked decrease in the predicted band gaps—from 2.3 to 1.2 eV—and both the valence and conductionbands broaden and change morphology, as is illustrated inFig. 2. Further, a small band on the edge of the low-energy
tail of the conduction band appears in both spinels, which isdiscernible in the bottom of Fig. 2, between 1 and 2 eV (2
and 3 eV in the normal spinel). Enlargements of these regionsare depicted in Fig. 3to facilitate comparison of the HSE
PDOS. While the In 5 sorbitals predominate in both cases, the
enlargements indicate that this small, almost isolated, featureclosely resembles the larger section of the conduction band, yetthe relative contributions of the sulfur and cadmium orbitalschange. In the normal spinel, Figs. 3(a)and3(c), the sulfur 3 s,
3p, and Cd 5 sorbital contributions are nearly identical, while
in the full inverse structure, Figs. 3(b) and3(d), the sulfur 3 s
contribution increases as does that of the In 4 dorbitals, which
were not present in the tail of the normal spinel (c) at all.
205128-3MELISSA J. LUCERO et al. PHYSICAL REVIEW B 83, 205128 (2011)
FIG. 1. (Color online) Projected density of states for Cd 8In16S32
as anormal spinel structure, calculated with LSDA, PBE, TPSS, and
HSE. The Fermi level is indicated by the dashed black line at E=0.
The top of the valence band does not terminate exactly at zero due toa 10 meV Gaussian line broadening.
IV . MAGNESIUM INDIUM THIOSPINELS
While observed in a natural spinel, MgAl 2O4, the normal
structure is not adopted by many synthetic Mg-containingspinel oxides.
84,85As already discussed, this dependence of
configuration upon the method of formation and synthesisis also observed for many thiospinels, including those withA=Mg,
86and is consequence of the oxygen or chalcogenide
anions forming a highly adaptable fcc structure, allowinga wide range of cations to not only occupy, but move inbetween the T
dandOhholes. This structural mobility is
influenced by the chemical composition, but is more sensitiveto the ordering of occupied holes, which, in turn, variesaccording to cation size, electrostatic interactions, structuredefects, and temperature.
87,88Experimental determination
of the ground-state cation distributions is thus nontrivial,
particularly since the high temperatures requisite for mostolder spinel syntheses mimic the formation conditions ofthe natural minerals known to form metastable crystallinestates,
89and consequent adherence to Ostwald’s rule.90At
lower temperatures, thermal equilibrium is also difficult toobtain due to very low diffusion rates.
91
FIG. 2. (Color online) The HSE projected density of states for
Cd8In16S32in normal (top) and full inverse (bottom) spinel structures.
The Fermi level is indicated by the dashed black line at E=0. The
top of the valence band does not terminate exactly at zero due to a10 meV Gaussian line broadening.
Not surprisingly, several order-disorder phenomena92have
been recognized in spinels. The normal spinels generallyexhibit long-range, nonconvergent order-disorder behavior, inwhich the extent of inversion changes continuously withouta phase transition, while inverse spinels exhibit two types oforder-disorder behavior: (1) an ordered inverse →disordered
inverse first-order transition stabilized by configurationalentropy-associated cation exchange in O
hsites, or (2) a
nonconvergent disordered inverse to another disordered statestabilized entropically by cation exchange in both T
dandOh
sites.93
The stability of the normal versus inverse structures, for
Cd/MgIn 2S4, assuming low-temperature thermal equilibrium
is presented in Table II. From a 0 K perspective, the Cd system
makes sense thermodynamically, implying that a normalstructure should predominate, assuming thermal equilibriumis achieved. Recall from Sec. IIIthat experimentally, normal
(or close to normal) structures are observed and a full inverseanalog has not been isolated.
For the Mg thiospinels, the energy preference between
either inverse ordering and the normal structure is significantlyreduced. This is not surprising, as both MgIn
2S4and its
oxide equivalent, MgIn 2O4, are observed to adopt some
form of inverse structure.86The partial inverse configuration
is calculated to be less stable than the full inverse, yet
experimentally, a fully inverse structure has not been isolated.
205128-4SCREENED HYBRID AND SELF-CONSISTENT G W ... PHYSICAL REVIEW B 83, 205128 (2011)
FIG. 3. (Color online) Enlargement of the low-energy region
in the conduction bands of the HSE projected density of states
for Cd 8In16S32adopting normal (left) and full inverse (right) spinel
structures. The top figures (a) and (b) highlight the similar bandshapes, but somewhat different population densities, while the
increased zoom in the bottom plots (c) and (d) further illustrate the
disparate contributions from the relevant S, Cd, and In orbitals asthe spinel structure is inverted.
Nevertheless, MgGa 2O4, a spinel with an experimental degree
of inversion similar to our partial inverse structure ( ∼0.84),94
was shown via finite-temperature MC calculations,91to prefer
an inverse-type structure at RT, strongly implying that
synthetic MgIn 2S4is subject to some form of order-disorder
TABLE II. M2+In2S4relative energies by type (kcal /mol).
Functional
Spinel type LSDA PBE TPSS HSE
MgIn 2S4
Normal 0 .00 0 .00 0 .00 0 .00
Partial inverse 4 .60 3 .27 3 .02 3 .37
Full inverse 4 .13 2 .67 2 .42 2 .15
CdIn 2S4
Normal 0 .00 0 .00 0 .00 0 .00
Full inverse 17 .01 17 .00 17 .49 16 .57behavior. Recent specialized models95and finite-temperature
simulations96demonstrate that predicting whether a normal or
inverse-type ordering scheme will predominate in the 0–278 Krange and identifying the relative stability of the threedisordered states possible for inverse structures is complexand labor-intensive.
97
Fortunately, there is abundant experimental data for
Mg 8In16S32, so we simulate cation distributions by using
the 56-atom crystallographic unit cell as a template toconstruct the normal, partial, and full inverse orderings, shownas (a), (b), and (c), respectively, in Fig. 4. All structures
started with an approximate Fd¯3msymmetry prior to full
relaxation.
A. Mg normal spinel structure
The heretofore unobserved normal -type Mg 8In16S32is the
structure available from crystallographic databases and it isalso the easiest to benchmark computationally. The results ofseveral theoretical studies
61,73,98,99also provide an alternate
means for comparison in the absence of experimental data.The DFT predictions are summarized in Table III.
Paralleling the Cd system, all functionals produce fully
relaxed Mg
8In16S32cells with expanded lattice parameters,
LSDA deviating the least. The band gap is observed to increaseas the cation ordering approaches the normal extreme for theanalogous oxide,
81,100and the Cd thiospinel also followed this
pattern, so it is reasonable to expect the Mg normal spinel
will also have a larger gap. The experimentally observed bandgaps for the Mg system correspond to what is known to be apartial inverse configuration (see Table V), implying that the
normal band gap should be larger than 2.1–2.3 eV . In fact,
HSE predicts a band gap of 2.83 eV—similar in magnitudeto the high end 2.7 eV of reported Cd thiospinel gaps. Thethree semilocal functionals all produce smaller gaps, thusHSE>TPSS >LSDA >PBE. Whether one references the
smaller partial inverse measured gaps or trusts the larger HSEprediction paralleling the Cd system (Table I), errors are on
the order of 20%–30%.
These data indicate again that the presence of nonlocal
Hartree-Fock exchange in the calculation far outweighs anylattice differences: indeed, LSDA predicts the smallest relaxedvolume as well as the narrowest gap, whereas TPSS has a muchlarger a
o, yet still fails to produce a band gap of the magnitude
predicted by HSE, and PBE has the largest cell but the smallest
TABLE III. Normal Mg 8In16S32: Functional dependence of band
gap (eV) and lattice parameters ( ˚A).
Functional LSDA PBE TPSS HSE
Nature of gap EiEdEiEdEiEdEiEd
Mg 8In16S32
Experiment ao=10.687a
ao 10.715 10.935 10.898 10.840
Band gapb1.73 1.69 2.01 2.83
Rlx→HSE spc2.88 2.63 2.71
aICSD ID 59551.72
bFully relaxed geometry and forces.
cHSE energy calculation of fully-relaxed structure.
205128-5MELISSA J. LUCERO et al. PHYSICAL REVIEW B 83, 205128 (2011)
FIG. 4. (Color online) The three 56-atom conventional, crystallographic unit cells addressing cation ordering for Mg 8In16S32in this study:
the (a) normal, (b) partial inverse, and (c) full inverse spinel structures. The Mg+2cation is green, In+3is brown, and the S−2anion is yellow.
Cations in Tdholes are surrounded by yellow tetrahedra.
band gap. The HSE single point calculations on the structures
relaxed using semilocal functionals provide larger gaps, allwithin ∼0.2 eV of the HSE prediction.
Comparison of the normal spinel PDOS for the four
functionals is presented in the column to the far left ofFig.5. Again paralleling the Cd thiospinel, the valence band is
dominated by the sulfur 3 porbitals, with minor contributions
from the indium 5 pand 4dorbitals. The conduction bandis also dominated by indium 5 sand sulfur 3 porbitals in
nearly equal amounts. The population patterns remain moreor less the same, and a blueshift is again evident. Unlike the
Cd system, there is no extra structure observed at the edgeof the low-energy tail of the conduction band, and all fourfunctionals predict that the band gap is direct . Nevertheless, a
normal spinel structure for Mg
8In16S32has yet to be isolated,
so the HSE band gap of 2.83 is purely predictive.
FIG. 5. (Color online) Projected density of states for Mg 8In16S32for the normal, partial inverse, and full inverse spinel structures as
calculated using LSDA, PBE, TPSS, and HSE. The Fermi level is indicated by the dark red line at E=0. The top of the valence band does not
terminate exactly at zero due to a 10 meV Gaussian line broadening.
205128-6SCREENED HYBRID AND SELF-CONSISTENT G W ... PHYSICAL REVIEW B 83, 205128 (2011)
TABLE IV . Full inverse Mg 8In16S32: Functional dependence of
band gap (eV) and lattice parameters ( ˚A).
Functional LSDA PBE TPSS HSE
Nature of gap EiEdEiEdEiEdEiEd
Mg 8In16S32
Experiment 2.1–2.3aao=10.687a
a 10.674 10.904 10.867 10.802
b 10.680 10.910 10.873 10.809
c 10.677 10.906 10.869 10.804
Band gapb0.98 1.04 0.87 0.91 1.13 1.18 1.98 2.04
aICSD ID 59551,72with 8 Mg and 8 In exchanged.
bFully relaxed (geometry and forces) 56-atom cells.
B. Mg full inverse spinel structure
A fully inverse structure is also unobserved in nature or syn-
thetically, but serves as a close approximation to experiment(x=1 versus x=0.84) facilitating direct comparison. As is
evident from Table IV, all functionals predict an indirect band
gap; the semilocal functions severely underestimate the gap,while HSE yields 1.98 eV , slightly below the experimentalrange of 2.1–2.3 eV , which is reasonable given that thepartial inverse gap should be larger. LSDA predicts a slightlycontracted lattice, analogous to what is observed for the Cdanalog, while all other functionals predict an expansion. TheHSE lattice parameters again show the smallest increase involume.
Cursory visual inspection reveals several striking contrasts
in both the shape and populations of the calculated PDOS inFig. 5for the normal (far left) and full inverse (far right)
thiospinels. In the normal configuration, the valence bandhas considerable structure, which is drastically attenuatedin the full inverse motif. As with the Cd compounds, thenormal spinel conduction band has a slowly diminishingtail and a maximum near the high-energy edge of theconduction band, whereas the full inverse conduction bandhas a more symmetrical population density and an overallsmoother “band shape.” In general, both structure types exhibitsimilar contributions from the sulfur 3 p(yellow) and indium
4dorbitals (blue). However, the indium 5 porbitals (cyan),
observed primarily in the valence band of the normal spinel,also show a non-negligible presence in the conduction band of
the full inverse spinel. This increased In 5 pcontribution can be
considered a migration from the high-energy band beginning at∼4 eV (LSDA) in the normal structure, into the lower-energy
conduction band of the full inverse structure. Finally, themagnesium 3 sorbitals (magenta) are seen to contribute—
albeit marginally—to both the valence and conduction bandsfor full-inverse ordering, while not at all, at least in the bandsof relevance to the gap, for normal ordering. These dramaticchanges in population densities are evident for HSE as wellas the semilocal functionals—the main distinction being theincreasingly wider band gaps.
C. Partial inverse spinel structure
Most characterizations of synthetic Mg 8In16S32report a
partial inverse structure.58,60,101–105An intermediate degree
of inversion for the 56-atom, full fcc conventional unit cellTABLE V . Partial inverse Mg In thiospinels: Functional
dependence of band gap (eV) and lattice parameters ( ˚A).
Functional LSDA PBE TPSS HSE
Nature of gap EiEdEiEdEiEdEiEd
Mg 8In16S32a
Experiment 2.1–2.3bao=10.687c
a 10.698 10.927 10.889 10.827
b 10.689 10.919 10.888 10.817
c 10.694 10.922 10.878 10.822
Band gap 1.06 1.08 0.94 0.95 1.18 2.04 2.06Full inverse
c0.98 1.04 0.87 0.91 1.13 1.18 1.98 2.04
aStarted with Ref. 72, see text for ordering description, x≈
ICSD ID 59551.
bReferences 58,61,102,103,105and106.
cMg 8In16S32from Table IV.
Mg 8In16S32was obtained by taking the structure of Hahn72
and swapping six cations. Specifically, two In3+are moved
fromOhtoTdholes, with four Ohholes swapped with
Mg2+“randomly” according to their order in the input
file to produced a normality of x=0.84.102,106Table V
summarizes relevant data for the partial inverse MgIn 2S4
structure. Predictions for the full inverse structure are alsoincluded for ease of reference.
As expected, the gaps for full and partial inverse structures
are of similar magnitude and the gap is indirect. The semilocalresults underestimate the gap by at least 1 eV , whereas theHSE prediction is within 6 meV of the lower bound for theRT experimental values of 2.14 eV .
103Interestingly, low-
temperature (4 K) experiments indicate an indirect transitionacross a gap of 2.26 eV , which is 14 meV lower than thedirect transition.
104,107A gap of ∼2.1 eV suggests that a
partial inverse structure should be a dark red color, which is,
in fact, what is observed.58,106This vanishingly small energy
difference is also observed in the cubic tin indium thiopinel,108
several zinc spinels [e.g., ZnRh 2O4(Ref. 109)o rZ n G a 2O4
(Ref. 110)], and the parent β-indium sulfide structure:74in all
cases, the band gaps are ∼2–3 eV with a disputed band-gap
type.
Comparison of both inverse structure PDOS’s in Fig. 5
(center and right columns) demonstrates the degree of simi-larity between the partial and full inverse Mg thiospinels. Thepopulation densities are similar for both inverse structures,and HSE exhibits patterns resembling those produced by thesemilocal functionals. Alas, the systems are not identical.Closer examination of the conduction band (HSE) reveals thatMgsorbitals contribute slightly more in the peak of the tail
for the partial inverse structure, Fig. 6(top), which is also
slightly blueshifted with respect to the full inverse structure,Fig.6(bottom), where Mg porbitals begin to contribute. This
enlargement demonstrates that there is also generally more Mgs-andp-orbital contribution in the conduction band for the
full inverse structure. At higher energies, the partial inversestructure shows redshifted indium p, d, and sulfur sorbitals.
and noticeable orbital-ordering-by-contribution differencesoccur at 2.5, 2.9, and 3.0 eV in the conduction band. Note thatthere is no such contribution in the normal spinel configurationas there is no additional structure at the bottom of the
205128-7MELISSA J. LUCERO et al. PHYSICAL REVIEW B 83, 205128 (2011)
FIG. 6. (Color online) HSE projected density of states for
Mg 8In16S32for the (a) partial inverse and (b) full inverse spinel
structures. The bottom of the conduction band is enlarged to illustratedissimilar population patterns. Each plot uses 10 meV Gaussian line
broadening.
conduction band, and the Mg contribution (left column, Fig. 5)
is primarily at the higher end of the conduction band, not nearthe band gap.
The spinel systems are clearly disordered systems.
29,93
When the shape of the optical absorption edge is exponential,
producing “Urbach tails,”111information about the degree
of disorder can be inferred. In recent investigations of thedisorder in α-silicon,
112the calculated DOS and fitting for
tails bear a striking resemblance to what is observed inspinels.
58While the valence band in all Cd/Mg thiospinels
is sharply terminated, the conduction band has exponentialtails. The inverse structures are necessarily more disorderedwhile the normal Cd thiospinel would be more disordered thanthe unknown Mg analog (which shows no extra band) becauseof the larger cation.
V . MANY-BODY CALCULATIONS OF THIOSPINELS
The 14-atom M+2In2S4primitives for Cd and Mg were
relaxed using LSDA. Corrected band gaps for the normalCd, normal Mg, and full inverse spinel structures were ob-tained using a scCOHSEX +G
0W0many-body treatment (see
Sec. II). Symmetry considerations require a 56-atom unit cell
for the partial inverse structure, which is not computationallyfeasible for sc GW, but as was shown previously, properties of
the intermediate structure can be inferred from the behaviorof the limiting structures, particularly that of the full inversestructure. The Fermi level is taken to be zero in all plots. For
comparison, the same primitives were optimized using HSE.
The resulting band structures are presented in Fig. 7, with
the scGW and HSE bands on top and bottom, respectively. It is
immediately evident that the sc GW and HSE band structures
are very similar. Further, all calculated band structures,regardless of ordering, exhibit a flat, nondisperse characterin the valence band and exhibit a high degree of structure inthe conduction band—a pattern typically observed for spineloxides.
113–115
In the conduction band, the Cd and Mg cations adopting
the normal configuration (left and right columns of Fig. 7)
display slightly different structure than those of the Mgfull inverse ordering (center column), particularly along theW→Kpath, where the bands show less curvature, notably
around L. It is interesting that the experimentally observed
Cd normal (right) and Mg inverse (center) structures manifestsimilar curvature at the bottom of the conduction band, witha clear separation of the In 5 sand slightly higher in energy S
3sorbitals. In contrast, the same bands of the unobserved Mg
normal compound [Figs. 7(c)and7(e)] overlap.
In terms of band gaps, both sc GW and HSE predict an
indirect transition for the normal Cd compound [Figs. 7(a)and
7(d), respectively]. The sc GW correction locates the valence-
band maximum along the K→/Gamma1path yielding an indirect
gap of 2.98 eV , while HSE predicts a gap of 2.33 eV along thesame path.
For the full inverse Mg compound, sc GW predicts an
indirect transition, spanning a 3.04 eV gap that originates
from the valence-band maximum in the K→/Gamma1direction. The
scGW gap is overestimated by ∼1 eV relative to experiment
(Table VI) and the flat top of the valence band conceals the
fact that the indirect gap is only 10 meV lower than thedirect transition. On the other hand, the smaller HSE band gapunderestimates experiment by only ∼0.3 eV . While the HSE
prediction is somewhat lower than the experimental range foraknown partial inverse structure, this is to be expected—the
TABLE VI. Normal and full inverse Mg 2In4S48: scGW band gaps
(eV) compared to four functionals.
Functional LSDA PBE TPSS HSE
Nature of gap EiEdEiEdEiEdEiEd
Cd normal
Expt. 2.2–2.7a
Band gapb1.34 1.43 1.21 1.30 1.52 1.59 2.33 2.41
scGWc2.98 3.10
Mg normal
Band gapd1.75 1.69 2.01 2.84
scGWc3.85
Mg full inverse
Expt. 2.1–2.2e
Band gapd0.96 0.85 1.12 1.93
scGWc3.04 3.05
aFully relaxed 14-atom cells. ICSD ID 300725.72
bReferences 52,57–66.
c14-atom normal spinel cell, initially relaxed using LSDA.
dICSD ID 59551.72
eUsing partial inverse structure data: Refs. 58,61,102,103,105,106.
205128-8SCREENED HYBRID AND SELF-CONSISTENT G W ... PHYSICAL REVIEW B 83, 205128 (2011)
FIG. 7. (Color online) Comparison of sc GW (top) and HSE (bottom) band structures for the Cd/Mg indium thiospinels along the L/Gamma1XW
K/Gamma1path. Normal Cd 2In4S8spinel (a) and (d); full inverse Mg 2In4S8spinel (b) and (e), and the predicted normal spinel ordering for Mg 2In4S8
(c) and (f).
scGW error would also be expected to decrease slightly if a
true partial inverse, not a full inverse structure, was examined.The small energetic distinction between indirect and directgaps does not exist for HSE, nor any of the DFT calculations,when using the smaller, 14-atom primitives, as it did for the56-atom conventional cells (Tables IIIandIV), which is not
surprising considering the magnitude of /Delta1E
dir−Eindand the
reduction of information inherent to using a smaller systemwith fewer electrons.
In the last case, the purely theoretical Mg normal structure,
scGW predicts a direct band gap with a magnitude of 3.85 eV .
HSE also predicts a direct transition, but the band gap isnarrower at 2.84 eV . Nevertheless, the sc GW and HSE bands
strongly resemble each other (right column of Fig. 7). As there
are no experimental data for comparison, these gaps remainexclusively predictive, but the HSE band gap is, as was pointedout earlier (see Sec. IV A ), very reasonable.
A. Discussion
For the known Cd and Mg compounds, the difference
between experiment and sc GW is opposite in sign, but nearly
equal in magnitude to the error that LSDA and GGA typicallyshow for these systems.
73,99While the sc GW scheme used
in this work is known to overestimate indirect semiconductorband gaps,
116the∼1 eV disparity is somewhat larger than the
expected 0.1–0.3 eV .11There are, however, numerous factors
(beyond the precision of the method itself) with the potentialto create this large disparity between sc GW predictions and
experimental measurements.
The most likely issue is probably the neglect of excitonic
effects: in medium-gap materials, screening is lower and theelectron-hole interaction becomes stronger.
117Although there
is no clear experimental evidence supporting the presence ofan excitonic effect, the absorption spectra of Ruiz-Fuerteset al.
58might support this hypothesis. Polaronic effects, which
are also absent in sc GW14methods and also show some
dependence on the system,118may also be relevant. While
again unverified experimentally, significant polaronic effectsare expected from the large /epsilon1
0−/epsilon1∞that these spinels present
[/epsilon10=18.8–20.74, /epsilon1infty=5.5–5.8 for MgIn 2S4(Refs. 103,
119) and /epsilon10=18.71,/epsilon1infty=6.49 for CdIn 2S4(Ref. 120)].
Recently, Vidal121showed that neglecting polaronic effects in
many-body approaches can lead to band-gap overestimationsof up to 1 eV . To a lesser extent, the differences between theLSDA and the experimental structural parameters (not only thelattice parameter abut also the ratio c/aand the internal anion
distortion u), the finite temperature of the experiments, and the
presence of other types of defects in the experimental samples(such as silica
103or Mg vacancies119) may also contribute.
The combined contributions of these otherwise small effectsmay explain why sc GW consistently overestimates Cd /Mg
indium thiospinels by ∼1 eV . Nevertheless, recent reports
of the successful application of HSE +G
0W0for band
structures122,123suggest an interesting alternative for future
exploration.
VI. CONCLUSION
The indium thiopinels of Mg and Cd were examined
by a theoretical treatment consisting of DFT and sc GW
many-body corrections to LSDA. Investigation into the relativeperformance of LSDA, PBE, TPSS, and HSE reaffirms earlierobservations that semilocal functionals underestimate the band
205128-9MELISSA J. LUCERO et al. PHYSICAL REVIEW B 83, 205128 (2011)
gaps of these semiconductors, regardless of cation ordering,
while demonstrating that the screened hybrid HSE providesband gaps and lattice parameters consistently in excellentagreement with experiment. It is also evident that the predictivepower of HSE extends beyond the idealized extrema ofnormal and full inverse spinel occupancies through successfulpredictions for an experimentally observed partial inversespinel structure.
The DFT calculations also indicate that while LSDA
geometries are generally considered to be better, spinel-typeband gaps are far more sensitive overall to the amount ofnonlocal Hartree-Fock exchange than they are to pm scaledeviations in the lattice parameters. The projected DOSillustrates that the presence of Hartree-Fock exchange inducesa significant blueshift in the location of the bottom conductionband—regardless of the M
+2metal present. For all functionals,
the conduction band also exhibits distinctive morphologicalchanges as the degree of inversion increases from normal to fullinverse, indicating population redistribution to lower states. Inthe valence band, the sulfur 3 porbitals provide the dominant
contribution, while the conduction band consists primarily ofthe In 5 sorbitals, followed closely by the sulfur 3 porbitals—a
pattern strikingly similar to that of β-In
2S4.
The sc GW analysis of band structures reveals that the
method overestimates thiospinel band gaps, relative to bothexperiment and HSE, yet the structure and dispersion patternsof the sc GW bands resemble those for other experimen-
tally characterized spinel systems, as well as parallelingthe predictions from the more expedient and accurate HSEcalculations. Both sc GW and HSE predict a minute, meV-scale
energetic distinction between indirect and direct transitionsthat is observed for isolable spinel compounds, regardlessof configuration type and irrespective of the identity of theM
+2cation. Additionally, the strong agreement between the
many-body and screened hybrid band structures implies thatthe details of reliable spinel band structure might serve asa useful adjunct to experimental determination of cationordering because the normal and inverse spinels manifestdissimilar band patterns.
Thus, this combined DFT /scGW study confirms that the
screened hybrid HSE provides an accurate, computationallyefficient means for predicting band gaps for the structurallycomplex Cd /Mg indium sulfide semiconductors.
ACKNOWLEDGMENTS
The work at Rice was funded by the Department of
Energy (Grant No. DE-FG02-09ER16053) and The WelchFoundation (C-0036). The Spanish effort was supported bythe Ministerio de Ciencia y Innovacion: Consolider Ingenio2010 GENESIS-FV (Grant No. CSD2006-04), FOTOMAT(Grant No. MAT2009-14625-C03-01), and the Comunidadde Madrid NUMACIA 2 project (Grant No. S-2009ENE-1477). The authors thankfully acknowledge the computerresources and technical expertise provided by the Centro deSupercomputaci ´on y Visualizaci ´on de Madrid (CeSViMa) and
the Spanish Supercomputing Network.
1W. Shockley and H. J. Queisser, J. Appl. Phys. 32, 510
(1961).
2A. Luque and A. Mart ´ı,P h y s .R e v .L e t t . 78, 5014 (1997).
3J. P. Perdew and K. Schmidt, Jacob’s Ladder of Density Functional
Approximations for the Exchange-Correlation Energy, in DensityFunctional Theory and Its Applications to Materials ,e d i t e db yV .
E. Van Doren, K. Van Alsenoy, and P. Geerlings (AIP, Melville,NY , 2001), pp. 1–20.
4G. E. Scuseria and V . N. Staroverov, Theory and Applications of
Computational Chemistry: The First 40 Years (Elsevier, Amster-
dam, 2005).
5I. Aguilera, P. Palacios, and P. Wahn ´on,Thin Solid Films 516, 7055
(2008).
6R. Lucena, I. Aguilera, P. Palacios, P. Wahn ´on, and J. C. Conesa,
Chem. Mater. 20, 5125 (2008).
7P. Palacios, K. S ´anchez, J. C. Conesa, J. J. Fern ´andez, and
P. Wahn ´on,Thin Solid Films 515, 6280 (2007) see also U.S. Patent
6444897.
8P. Palacios, P. Wahn ´on, S. Pizzinato, and J. C. Conesa, J. Chem.
Phys. 124, 014711 (2006).
9P. Wahn ´on and C. Tablero, Phys. Rev. B 65, 165115 (2002).
10S. V . Faleev, M. van Schilfgaarde, and T. Kotani, Phys. Rev. Lett.
93, 126406 (2004).
11F. Bruneval, N. Vast, and L. Reining, Phys. Rev. B 74, 045102
(2006).
12L. Hedin, Phys. Rev. 139, A796 (1965).13J. Vidal, S. Botti, P. Olsson, J.-F. Guillemoles, and L. Reining,
P h y s .R e v .L e t t . 104, 056401 (2010).
14J. Vidal, F. Trani, F. Bruneval, M. A. L. Marques, and S. Botti,
P h y s .R e v .L e t t . 104, 136401 (2010).
15M. Gatti, F. Bruneval, V . Olevano, and L. Reining, Phys. Rev. Lett.
99, 266402 (2007).
16F. Bruneval, N. Vast, L. Reining, M. Izquierdo, F. Sirotti, and
N. Barrett, P h y s .R e v .L e t t . 97, 267601 (2006).
17I. Aguilera, Ph.D. thesis, Universidad Polit ´ecnica de Madrid (2010)
[http://oa.upm.es/3669/ ].
18M. van Schilfgaarde, T. Kotani, and S. Faleev, Phys. Rev. Lett. 96,
226402 (2006).
19T. M. Henderson, A. F. Izmaylov, G. Scalmani, and G. E. Scuseria,J. Chem. Phys. 131, 044108 (2009).
20A. V . Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria,
J. Chem. Phys. 125, 224106 (2006).
21J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118,
8207 (2003).
22A. Stroppa, G. Kresse, and A. Continenza, Phys. Rev. B 83, 085201
(2011).
23A. F. Izmaylov, E. N. Brothers, and G. E. Scuseria, J. Chem. Phys.
125, 224105 (2006); J. Heyd, J. E. Peralta, G. E. Scuseria, and R.
L. Martin, ibid. 123, 174101 (2005); J. Heyd and G. E. Scuseria,
ibid. 120, 7274 (2004); 121, 1187 (2004).
24J .P o h la n dK .A l b e , J. Appl. Phys. 108, 023509
(2010).
205128-10SCREENED HYBRID AND SELF-CONSISTENT G W ... PHYSICAL REVIEW B 83, 205128 (2011)
25B. G. Janesko, T. M. Henderson, and G. E. Scuseria, Phys. Chem.
Chem. Phys. 11, 443 (2009); T. M. Henderson, J. Paier, and G. E.
Scuseria, Phys. Status Solidi B 248, 767 (2011).
26E. N. Brothers, A. F. Izmaylov, J. O. Normand, V . Barone, and
G. E. Scuseria, J. Chem. Phys. 129, 011102 (2008).
272008 Featured Articles: (a) July 31, Solar Cell Material Can
Soak Up More Sun, [ http://www.newscientist.com ]; (b) August 9,
A Step Closer to a Next Generation of Solar Cells,[http://technology4life.wordpress.com ]; (c) September 30, Tech-
nical Insights Report, Intermediate-Band Materials for Solar Cells,Profile of the work done on indium sulfide-based materials,[www.frost.com ].
28T. F. W. Barth and E. Posnjak, Z. Krist. 82, 325 (1932).
29K. E. Sickafus and J. M. Wills, J. Am. Ceram. Soc. 82, 3279 (1999).
30E. J. W. Verwey and E. L. Heilman, J. Chem. Phys. 15, 174 (1947).
31K. N. Kudin and G. E. Scuseria, P h y s .R e v .B 61, 16440 (2000).
32K. N. Kudin and G. E. Scuseria, Chem. Phys. Lett. 289, 611 (1998).
33K. N. Kudin and G. E. Scuseria, Chem. Phys. Lett. 283, 61 (1998).
34M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A.
Robb, J. R. Cheeseman, G. Scalmani, V . Barone, B. Mennucci,G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian,A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada,M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Naka-jima, Y . Honda, O. Kitao, H. Nakai, T. Vreven, J. A. MontgomeryJr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers,K. N. Kudin, V . N. Staroverov, R. Kobayashi, J. Normand,K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi,M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross,V . Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann,O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski,R. L. Martin, K. Morokuma, V . G. Zakrzewski, G. A. V oth,P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels,O. Farkas, J. B. Foresman, J. V . Ortiz, J. Cioslowski, and D. J.Fox, Gaussian 09 Revision A.1 (Gaussian Inc., Wallingford, CT,
2009).
35R. Dennington II, T. Keith, and J. Millam, GaussView 5.0
(Semichem, Inc., Wallingford, CT, 2000).
36W. Humphrey, A. Dalke, and K. Schulten, J. Mol. Graph. 14,3 3
(1996), VMD—Visual Molecular Dynamics, version 1.8.7.
37See supplemental material at [ http://link.aps.org/supplemental/
10.1103/PhysRevB.83.205128 ] for modified basis sets, informa-
tion about pseudopotential generation, and high resolution graphicsof the 56-atom unit cells.
38ICSD, Inorganic Crystallographic Structural Database[www.fiz-karlsruhe.de/icsd_web.html ] (2010).
39K. N. Kudin, G. E. Scuseria, and H. B. Schlegel, J. Chem. Phys.
114, 2919 (2001).
40J. P. Perdew and Y . Wang, P h y s .R e v .B 45, 13244 (1992).
41S. H. V osko, L. Wilk, and M. Nusair, Can. J. Phys 58, 1200
(1980).
42J. P. Perdew, K. Burke, and M. Ernzerhof, P h y s .R e v .L e t t . 77,
3865 (1996).
43J. P. Perdew, K. Burke, and M. Ernzerhof, P h y s .R e v .L e t t . 78,
1396 (1997).
44J .T a o ,J .P .P e r d e w ,V .N .S t a r o v e r o v ,a n dG .E .S c u s e r i a , Phys.
Rev. Lett. 91, 146401 (2003).
45Specifically, we use the HSEh parametrization (Ref. 19) of
HSE06 (Ref. 20) and (Ref. 21), called by the GAUSSIAN keyword
HSEh1PBE.46X. Gonze, B. Amadon, P.-M. Anglade, J.-M. Beuken, F. Bottin,P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Cote,T. Deutsch, L. Genovese, P. Ghosez, M. Giantomassi,S. Goedecker, D. R. Hamann, P. Hermet, F. Jollet, G. Jomard,S. Leroux, M. Mancini, S. Mazevet, M. J. T. Oliveira, G. Onida,Y . Pouillon, T. Rangel, G.-M. Rignanese, D. Sangalli, R. Shaltaf,M. Torrent, M. J. Verstraete, G. Zerah, and J. W. Zwanziger,Comput. Phys. Commun. 180, 2582 (2009),
ABINIT .
47M. Torrent, F. Jollet, F. Bottin, G. Zerah, and X. Gonze, Comput.
Mater. Sci. 42, 337 (2008).
48G. B. Bachelet, D. R. Hamann, and M. Schluter, P h y s .R e v .B 26,
4199 (1982).
49M. Fuchs and M. Scheffler, Comput. Phys. Commun. 119,6 7
(1999).
50M. S. Hybertsen and S. G. Louie, P h y s .R e v .B 34, 5390 (1986).
51R. W. Godby and R. J. Needs, P h y s .R e v .L e t t . 62, 1169 (1989).
52S. N. Baek, T. S. Jeong, C. J. Youn, K. J. Hong, J. S. Park, D. C.
Shin, and Y . T. Yoo, J. Cryst. Growth 262, 259 (2004).
53H. Nakanishi, Jpn. J. Appl. Phys. 19, 103 (1980).
54S. I. Radautsan, V . F. Zhitar, I. G. Kosnichan, M. I. Shmiglyuk, Fiz.
Tekh. Poluprovodn. 5(1971) 2240; Sov. Phys. Semicond. (English
Transl.) 5(1971) 1959.
55W. Czaja and L. Krautsbauer, Phys. Status Solidi 33, 191 (1969).
56M. R. Brown, M. D. Martin, and W. A. Shand, J. Phys. C 3, 1329
(1970).
57Y . Li, R. Dillert, and D. Bahnemann, Thin Solid Films 516, 4988
(2008).
58J. Ruiz-Fuertes, D. Errandonea, F. J. Manjon, D. Martinez-Garcia,A. Segura, V . V . Ursaki, and I. M. Tiginyanu, J. Appl. Phys. 103,
063710 (2008).
59L. Betancourt, V . Sagredo, C. Rincon, and G. E. Delgado, Rev.Mex. Fis. 52, 164 (2006).
60S.-J. Lee, J.-E. Kim, and H.-Y . Park, J. Mater. Res. 18, 733 (2003).
61M. Marinelli, S. Baroni, and F. Meloni, Phys. Rev. B 38, 8258
(1988).
62M. Bohm, G. Huber, A. MacKinnon, and O. Madelung, Semicon-
ductors Subvolume H: Physics of Ternary Compounds , Landolt-
Bornstein Numerical Data and Functional Relationships in Scienceand Technology, New Series, Group III, V ol. 17 (Springer-Verlag,New York, 1985), and references therein. See p. 137.
63G. van Veen and R. D. Beer, J. Phys. Chem. Solids 38, 217 (1977).
64H. Nakanishi, S. Endo, and T. Irie, J. Phys. (Paris) Colloq. 36,C 3
(1975); see also Jpn. J. Appl. Phys. 19, 103 (1980).
65J. A. Beun, R. Nitsche, and M. Lichtensteiger, Physica 26, 647
(1960).
66H. Koelmans and H. G. Grimmeiss, Physica 25, 1287 (1959).
67T. Pisarkiewicz, K. Zakrzewska, and E. Leja, Thin Solid Films
153, 479 (1987).
68A. J. Nozik, P h y s .R e v .B 6, 453 (1972).
69D. R. Kammler, T. O. Mason, D. L. Young, T. J. Coutts, D. Ko,
K. R. Poeppelmeier, and D. L. Williamson, J. Appl. Phys. 90, 5979
(2001).
70E. Burstein, Phys. Rev. 93, 632 (1954); T. S. Moss, Proc. Phys.
Soc. London, B 67, 775 (1954).
71L. N. Brewer, D. R. Kammler, T. O. Mason, and V . P. Dravid,
J. Appl. Phys. 89, 951 (2001).
72H. Hahn and W. Klinger, Z. Anorg. Allg. Chem. 263, 177 (1950).
73I. Aguilera, P. Palacios, K. S ´anchez, and P. Wahn ´on,Phys. Rev. B
81, 075206 (2010).
205128-11MELISSA J. LUCERO et al. PHYSICAL REVIEW B 83, 205128 (2011)
74N. Barreau, A. Mokrani, F. Couzini ´e-Devy, and J. Kessler, Thin
Solid Films 517, 2316 (2009).
75N. Barreau, Solar Energy 83, 363 (2009).
76F. Cerrina, C. Quaresima, I. Abbati, L. Braicovich, P. Picco, and
G. Margaritondo, Solid State Commun. 33, 429 (1980).
77S. K. Apte, S. N. Garaje, R. D. Bolade, J. D. Ambekar, M. V .
Kulkarni, S. D. Naik, S. W. Gosavi, J. O. Baeg, and B. B. Kale,J. Mater. Chem. 20, 6095 (2010).
78L. Schmicka, J. Harl, and G. Kresse, J. Chem. Phys. 134, 024116
(2011).
79E. R. Batista, J. Heyd, R. G. Hennig, B. P. Uberuaga, R. L. Martin,G. E. Scuseria, C. J. Umrigar, and J. W. Wilkins, Phys. Rev. B 74,
121102 (2006).
80D. O. Scanlon, B. J. Morgan, G. W. Watson, and A. Walsh, Phys.
Rev. Lett. 103, 096405 (2009).
81S . - H .W e ia n dS .B .Z h a n g , Phys. Rev. B 63, 045112 (2001).
82M. Womes, J. C. Jumas, and J. Olivier-Fourcade, Solid State
Commun. 131, 257 (2004).
83N. Barreau, J. C. Bern `ede, S. Marsillac, and A. Mokrani, J. Cryst.
Growth 235, 439 (2002).
84G. D. Price, S. L. Price, and J. K. Burdett, Phys. Chem. Miner. 8,
69 (1982).
85U. Schmocker, H. R. Boesch, and F. Waldner, Phys. Lett. 40A, 237
(1972).
86R. J. Hill, J. R. Craig, and G. V . Gibbs, Phys. Chem. Miner. 4, 317
(1979).
87G. D. Price and N. L. Ross, The Stability of Minerals , Mineralogical
Society Series V ol. 3 (Chapman & Hall, New York, 1992).
88M. O’Keefe and B. G. Hyde, Cation Ordering and Electron Transfer
Structure and Bonding 61, 77 (1985).
89J. C. Deelman, Neues Jb. Miner. Monat. 7, 289 (1999).
90T. Threlfall, Org. Proc. Res. Dev. 7, 1017 (2003).
91A. Seko, K. Yuge, F. Oba, A. Kuwabara, and I. Tanaka, Phys. Rev.
B73, 184117 (2006).
92A. Putnis, Introduction to Mineral Sciences (Cambridge University
Press, NY , NY 40 w. 20th Street 1992).
93A. Seko, F. Oba, and I. Tanaka, Phys. Rev. B 81, 054114 (2010).
94H. Schmalzried, Z. Phys. Chem. 28, 203 (1961); J. E. Wieden-
borner, N. Stemple, and Y . Okaya, Acta Crystallogr. 20, 761 (1966).
95V. S t eva n ov i ´c, M. d’Avezac, and A. Zunger, P h y s .R e v .L e t t . 105,
075501 (2010).
96A. Seko, K. Yuge, F. Oba, A. Kuwabara, I. Tanaka, andT. Yamamoto, P h y s .R e v .B 73, 094116 (2006), and references
therein.
97A. Zunger, in Statistics and Dynamics of Alloy Phase Transfor-
mations , Rhodes, 1993, edited by P. E. A. Turchi and A. Gonis
(Plenum, New York, 1994).
98F. Semari, R. Khenata, M. Rabah, A. Bouhemadou, S. B. Omran,A. H. Reshak, and D. Rached, J. Solid State Chem. 183, 2818
(2010).99P. Palacios, I. Aguilera, K. S ´anchez, J. C. Conesa, and P. Wahn ´on,
P h y s .R e v .L e t t . 101, 046403 (2008).
100B. Li, L. Zeng, and F. Zhang, Phys. Status Solidi A 201, 960
(2004).
101V. V. U r s a k i , F. M a n j ´on, I. Tiginyanu, and V . E. Tezlevan, J. Phys.
Condens. Matter 14, 6801 (2002).
102S. Katsuki, Solid State Commun. 39, 767 (1981).
103M. Wakaki, O. Shintani, T. Ogawa, and T. Arai, Jpn. J. Appl. Phys.
19S3 , 255 (1980).
104D. Fiorani and S. Viticoli, Solid State Commun. 32, 889 (1979).
105L. Gastaldi and A. Lapiccirella, J. Solid State Chem. 30, 223
(1979).
106P. M. Sirimanne, N. Sonoyama, and T. Sakata, J. Solid State Chem.
154, 476 (2000).
107E. Fortin, S. Fafard, A. Anedda, F. Ledda, and A. Charlebois, Solid
State Commun. 77, 165 (1991).
108R. Dedryvere, P. E. Lippens, J. C. Jumas, I. Lefebvre-Devos, and
C. P. Vicente, Solid State Sci. 3, 267 (2001).
109D. J. Singh, R. C. Rai, J. L. Musfeldt, S. Auluck, N. Singh,
P. Khalifah, S. McClure, D. G. Mandrus, W. Shockley, andH. J. Queisser, Chem. Mater. 18, 2696 (2006).
110L. Pisani, T. Maitra, and R. Valenti, Phys. Rev. B 73, 205204
(2006).
111F. Urbach, Phys. Rev. 92, 1324 (1953).
112D. A. Drabold, Y . Li, B. Cai, and M. Zhang, Phys. Rev. B 83,
045201 (2011), and references, therein.
113W. Setywan and S. Curtarolo, Comput. Mater. Sci. 49, 299
(2010).
114N. Mansourian-Hadavi, S. Wansom, N. H. Perry, A. R. Nagaraja,T. O. Mason, L. H. Ye, and A. J. Freeman, Phys. Rev. B 81, 075112
(2010).
115S. Lopez, A. H. Romero, P. Rodriguez-Hernandez, and A. Munoz,P h y s .R e v .B 79, 214103 (2009).
116F. Trani, J. Vidal, S. Botti, and M. A. L. Marques, Phys. Rev. B 82,
085115 (2010); 83, 039901(E) (2011).
117S. Botti, F. Sottile, N. Vast, V . Olevano, L. Reining, H. C. Weissker,
A. Rubio, G. Onida, R. DelSole, and R. W. Godby, Phys. Rev. B
69, 155112 (2004).
118G. D. Mahan, Many Particle Physics (Physics of Solids and
Liquids) 3rd ed. (Springer, 2000).
119M. Wakaki, O. Shintani, T. Ogawa, and T. Arai, Jpn. J. Appl. Phys.
21, 958 (1982).
120N. Syrbu, M. Bogdanash, and N. A. Moldovyan, Infrared Phys.
Technol. 37, 763 (1996).
121J. Vidal, F. Trani, F. Bruneval, M. A. L. Marques, and S. Botti,
P h y s .R e v .L e t t . 104, 136401 (2010).
122A. Stroppa, M. Marsman, G. Kresse, and S. Picozzi, New J. Phys.
12, 13 (2010).
123Y .-S. Kim, K. Hummer, and G. Kresse, P h y s .R e v .B 80, 035203
(2009).
205128-12 |
PhysRevB.94.081410.pdf | RAPID COMMUNICATIONS
PHYSICAL REVIEW B 94, 081410(R) (2016)
Density functional theory of the Seebeck coefficient in the Coulomb blockade regime
Kaike Yang,1Enrico Perfetto,2,3Stefan Kurth,1,4Gianluca Stefanucci,2,3and Roberto D’Agosta1,4
1Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Departamento de F ´ısica de Materiales, Universidad
del Pa ´ıs Vasco UPV/EHU, Avenida de Tolosa 72, E-20018 San Sebasti ´an, Spain
2Dipartimento di Fisica and European Theoretical Spectroscopy Facility (ETSF), Universit `a di Roma Tor Vergata, Via della Ricerca
Scientifica 1, 00133 Rome, Italy
3INFN, Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati, Italy
4IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain
(Received 23 December 2015; revised manuscript received 22 June 2016; published 23 August 2016)
The Seebeck coefficient plays a fundamental role in identifying the efficiency of a thermoelectric device. Its
theoretical evaluation for atomistic models is routinely based on density functional theory calculations combinedwith the Landauer-B ¨uttiker approach to quantum transport. This combination, however, suffers from serious
drawbacks for devices in the Coulomb blockade regime. We show how to cure the theory through a simplecorrection in terms of the temperature derivative of the exchange correlation potential. Our results compare well
with both rate equations and experimental findings on carbon nanotubes.
DOI: 10.1103/PhysRevB.94.081410
The quest for increasingly energy-efficient technologies
has recently led to significant scientific and technologicalinterest in thermoelectricity [ 1–4]. Indeed, thermoelectric
devices convert waste heat to electric power: The basicworking principle at their heart is the Seebeck effect [ 5]. The
corresponding Seebeck coefficient is an important ingredientin the thermoelectric figure of merit (an efficiency measureof a thermoelectric device). At present, the method of choicefor an atomistic modeling of the Seebeck and other transportcoefficients is density functional theory (DFT) combined withthe Landauer-B ¨uttiker formalism (LB-DFT) [ 6,7]. However,
an incautious use of LB-DFT as guide to material and systemselection may point in the wrong direction. In fact, LB-DFTis unable to capture the ubiquitous Coulomb blockade (CB)phenomenon of quantum devices weakly coupled to leads,thereby overestimating the conductance and, as we shall see,underestimating the Seebeck coefficient. In Refs. [ 8–10]i t
was shown that the erroneous high conductance predictedby LB-DFT stems from neglecting exchange-correlation (xc)corrections to the bias [ 11–15]. According to a recently
proposed DFT framework for thermal transport (and thusfor the calculation of the Seebeck coefficient) [ 16,17]x c
corrections to the temperature gradient are also expected tooccur.
In this Rapid Communication we propose an alternative
DFT approach to the Seebeck coefficient well suited forquantum devices in the CB regime. Following a recent ideaon the construction of xc corrections to the conductance [ 8],
we find a very simple xc correction to the LB-DFT Seebeckcoefficient in terms of static DFT quantities. To illustrate thetheory we first consider the Anderson impurity model (AIM), a
paradigm for the CB effect [ 18], and subsequently extend the
analysis to multiple-level systems. The proposed equationsare validated by benchmarking the results against those of therate equations [ 19–21] (RE), demonstrating the crucial role of
the xc correction. Finally, we apply the theory to single-wallcarbon nanotubes and find good qualitative agreement withexperiment.
The Seebeck coefficient Sis defined as the ratio S=
(/Delta1V//Delta1T )
I=0, where /Delta1V is the voltage that must be appliedto cancel the current Igenerated by a small temperature
difference /Delta1Tbetween the left and right leads. This definition
corresponds to the phenomenological Seebeck coefficientof Refs. [ 22,23]. For an AIM symmetrically coupled to
featureless leads the Seebeck coefficient takes the form [ 24]
(atomic units are used throughout)
S=−1
T/integraltext
ωf/prime(ω)A(ω)/integraltext
f/prime(ω)A(ω),/integraldisplay
≡/integraldisplay∞
−∞dω
2π, (1)
withA(ω) being the interacting spectral function, f(ω)=
1/(1+eβ(ω−μ)) being the Fermi function at temperature T=
1/βand chemical potential μ, and f/prime≡df/dω .W en o w
show how to rewrite Sin terms of quantities which are all
accessible by DFT. The starting point is the many-body (MB)equation N=2/integraltext
f(ω)A(ω) for the electron occupation at the
impurity. For an uncoupled impurity Adepends on Tandμ
exclusively through N. This exact property is also a very good
approximation at weak coupling and in the CB regime [ 25].
Hence the MB equation can be converted into a self-consistentequation for N, and at the self-consistent solution we can write
dA/dT =(dA/dN )(dN/dT ). After some algebra we find
dN
dT=−2
T/integraltext
ωf/prime(ω)A(ω)
1+R, (2)
where we have defined R=−2/integraltext
f(ω)dA(ω)/dN . With
similar steps we can also express the compressibility as
dN
dμ=−2/integraltext
f/prime(ω)A(ω)
1+R. (3)
The combination of Eqs. ( 2) and ( 3) then gives
S=−dN/dT
dN/dμ. (4)
This rewriting of the Seebeck coefficient is extremely in-
teresting from a DFT perspective since it involves exclu-sively the occupation Nof the equilibrium system. We then
calculate the derivatives in Eq. ( 4) using the Kohn-Sham
(KS) expression N=2/integraltext
f(ω)A
s(ω), where Asis the KS
spectral function. For a gated impurity with energy vwe have
2469-9950/2016/94(8)/081410(5) 081410-1 ©2016 American Physical SocietyRAPID COMMUNICATIONS
YANG, PERFETTO, KURTH, STEFANUCCI, AND D’AGOSTA PHYSICAL REVIEW B 94, 081410(R) (2016)
As(ω)=/lscript(ω−v−vHxc), with /lscript(ω)=γ/(ω2+γ2/4) being
a Lorentzian of width determined by the dot-lead tunnelingrateγ/2. The Hartree-xc (Hxc) potential v
Hxcdepends on N
andT. Therefore, at self-consistency Asdepends implicitly
(through N)o nμand both implicitly and explicitly on T.B y
calculating the required density derivatives and taking into ac-count that
dvHxc
dT=(∂vHxc
∂N)TdN
dT+(∂vHxc
∂T)N, we obtain the exact
relation
S=Ss+/parenleftbigg∂vHxc
∂T/parenrightbigg
N. (5)
HereSsis the KS Seebeck coefficient obtained from Eq. ( 1)b y
replacing A(ω) with As(ω), and it is precisely the coefficient
predicted by the LB-DFT approach. We further observe thatfor the derivation of Eq. ( 5) it is not necessary that A
sis a
Lorentzian; the only requirement is that Asis a function of
(ω−v−vHxc).
Equation ( 5), the central result of this Rapid Communica-
tion, provides a rigorous route to cure LB-DFT through theinclusion of the xc correction ∂v
Hxc/∂T while still remaining
in a pure DFT framework. As we shall see, Eq. ( 5) also suggests
how to correct Ssin larger systems.
At temperatures T/greatermuchγ, but still T/lessmuchUwhere Uis the
onsite repulsion, the CB phenomenon leaves clear fingerprintson the Seebeck coefficient. Nevertheless, these are onlypartially captured by S
s, even when the exact vHxcis used
(see below). The Anderson model is particularly instructivesince it allows us to disentangle the coordinated actions of theCB effect on S
sand of the xc correction in reproducing the
interacting S.
In the following we assume that γis the smallest energy
scale and we approximate vHxcby the exact Hxc potential of
the isolated ( γ=0) impurity [ 26,27],
vHxc[N]≈vimp
Hxc[N]=U
2+gU(N−1), (6)
where gU(x)=U
2+1
βln (x+√
x2+exp(−βU)(1−x2)
1+x). At low tem-
peratures, the Hxc potential exhibits a sharp (but continuous)
step of size Uat occupation N=1[26,28,29]. With an
analytic expression for vHxcwe can evaluate both terms on the
right-hand side of Eq. ( 5). In Fig. 1we show Scalculated from
our DFT equation (black) versus the gate v. To demonstrate
the accuracy of the result we also show the Seebeck coefficientcalculated from Eq. ( 1) using the MB spectral function A(ω)=
N
2/lscript(ω−v−U)+(1−N
2)/lscript(ω−v)[25] (blue) as well as the
one calculated using the RE approach of Ref. [ 20] (red),
exact in the limit γ→0. All three approaches give the same
Seebeck coefficient and densities (see inset). Let us nowdiscuss how the two terms in Eq. ( 5) contribute. The KS
Seebeck S
s(green) accounts for the correct linear behavior
(with slope proportional to T−1) at large values of |v|.I n
fact, for γ→0 the KS spectral function becomes As(ω)=
2πδ(ω−v−vHxc) and consequently Ss=−(v+vHxc)/T.
The linear behavior at large |v|is not surprising since the
noninteracting Seebeck coefficient behaves in the same way.Noteworthy is instead the plateau of S
sforv∈(−U,0). This
is a direct consequence of the step in vHxcwhich pins the
KS level to the chemical potential, thereby blocking electronswith energy below v+Ufrom entering the impurity site
FIG. 1. Seebeck coefficient Sand density N(inset) for the
Anderson model vs gate vfor our corrected DFT (black), MB (blue),
and RE (red). The Ss(KS, green) and the xc correction ∂vHxc/∂T
(cyan) are also displayed. The parameters are T=0.1a n dγ=0.01
(energies in units of U).
(see inset). The CB-induced plateau in Ssopens a gap in
the noninteracting straight line −v/T, shifting it leftward
byUforv<−Uand generating the correct behavior at
large negative values of v. However, Ssmisses entirely the
oscillation of SforN≈1, thus severely underestimating the
true Seebeck coefficient. Remarkably, this deficiency is exactlycured by the xc correction ∂v
Hxc/∂T (cyan). The temperature
variation of vHxcis the key ingredient for the nonvanish-
ing Seebeck coefficient in the CB regime [ 20,30–32]. We
emphasize that the LDA potential misses both the plateau(no step in v
Hxc) and the oscillation induced by ∂vHxc/∂T
[33].
We now extend the DFT approach to junctions with more
than one level. For T/greatermuchγthe Seebeck coefficient exhibits
a sawtooth behavior as a function of v, with rapid (but
continuous) jumps occurring when the number Nof electrons
crosses an integer. Furthermore, if the level spacing /Delta1εis much
larger than T, a superimposed fine structure of wiggles spaced
by/Delta1εemerges [ 20]. The wiggles originate from excitations
that bring the system with ( N−1) particles in the ground state
to some excited state with Nparticles.
The physics of the Seebeck coefficient in a multiple-
level junctions is well captured by the constant interactionmodel (CIM). The CIM Hamiltonian reads ˆH=/summationtext
iσεiˆniσ+
1
2/summationtext
iσ/negationslash=jσ/primeUijˆniσˆnjσ/prime, where ˆniσis the occupation operator
of the ith level with spin σ. The indices i,jrun over M
levels and for M=1 we are back to the Anderson model.
For simplicity we assume that each level is equally coupled tothe left and right leads with tunneling rate γ/2. In this case the
derivation of Eq. ( 4) can be repeated step by step by replacing
the spectral function Awith its trace Tr[ A]. Consequently, we
can again express Sin a pure DFT framework by calculating
the derivatives of the total number of electrons from the KSexpression N=2/integraltext
f(ω)Tr[A
s(ω)]. The KS spectral function
[As]ij=δijAs,iis diagonal in the level basis and reads
As,i(ω)=/lscript(ω−εi−vHxc,i), where the Hxc potential of level
idepends on the occupations {n}of all the levels. It is
081410-2RAPID COMMUNICATIONS
DENSITY FUNCTIONAL THEORY OF THE SEEBECK . . . PHYSICAL REVIEW B 94, 081410(R) (2016)
straightforward to show that
S=Ss+/summationdisplay
j/integraltext
f/prime(ω)As,j(ω)/integraltext
f/prime(ω)Tr[As(ω)]/parenleftbigg∂vHxc,j
∂T/parenrightbigg
N. (7)
In Ref. [ 34] we proved that at zero temperature the Hxc
potential of the isolated ( γ=0) CIM Hamiltonian is uniform
and depends only on N, i.e., vHxc,i[{n}]=vHxc[N]. The
zeroth-order approximation at finite temperatures and weakcoupling to the leads therefore consists in neglecting thenonuniformity and the local dependence on the {n}.I nt h i s
approximation Eq. ( 7) reduces to Eq. ( 5). The interacting See-
beck coefficient then follows once we specify the functionalform of v
Hxc. Following Ref. [ 8] we construct vHxc[N]a st h e
sum of single-impurity Hxc potentials according to
vHxc[N]=2M−1/summationdisplay
K=1/bracketleftbiggUK
2+gext
UK(N−K)/bracketrightbigg
, (8)
where UKis the charging energy needed for adding one
electron to the system with Kelectrons, and the extended
gext
Ufunction is defined according to
gext
U(N−1)=⎧
⎪⎨
⎪⎩−U/2 N< 0
gU(N−1) 0 /lessorequalslantN/lessorequalslant2,
U/2 N> 2(9)
withgUgiven below Eq. ( 6). The Hxc potential in Eq. ( 8)
has a staircase behavior with steps of width UKbetween two
consecutive integers.
To assess the quality of our approximate Hxc potential we
first consider a two-level CIM with Uij=U.I nF i g . 2we
display results at temperature T=0.03, coupling γ=0.001,
andεi=ε0
i+v, where ε0
1=0 andε0
2=0.3 (energies in units
ofU). The left panel shows the total occupation Nas well
as the occupation n2=/summationtext
σn2σof the highest level calculated
using both DFT and RE. Although a perfect agreement is foundforN, exponentially small discrepancies are seen for the local
occupation. In fact, the uniformity (i.e., level independence)of our zeroth-order approximation v
Hxcof Eq. ( 8) neglects
thermal excitations, which corresponds to mixing only ground
FIG. 2. Density (left) and Seebeck coefficient (right) of CIM with
two spin-degenerate levels computed from RE and DFT using the
approximate functional of Eq. ( 8). The KS Seebeck coefficient is also
shown.
FIG. 3. Seeebeck coefficient for the Anderson model with non-
degenerate single-particle levels (left) and for three spin-degenerate
levels (right). The parameters are ε0
↑=0,ε0
↓=0.3 (left panel) and
ε0
1=0,ε0
2=0.3,ε0
3=0.6 (right panel). In both panels T=0.03 and
γ=0.001 (all energies in units of U).
states of different N. Accordingly, the DFT Seebeck coefficient
is expected to exhibit only those wiggles associated withthe addition of one electron in the lowest available level.This is confirmed by the right panel of Fig. 2, where the
wiggles associated to the addition energies of excited statesare captured by the RE (red) but missed by DFT (black). Forimproving the agreement between DFT and RE one shouldabandon the zeroth-order uniform approximation and considera level-dependent Hxc potential which correctly reproducesthe level occupations. To further support this analysis onthe relation between the nonuniformity of v
Hxcand physical
excitations we show in Fig. 3the Seebeck coefficient for
the Anderson model with broken spin degeneracy (left) andfor a three-level CIM (right). In the first case DFT agreeswith RE since there exists only one addition energy, whereasin the second case DFT misses the wiggles of excited-stateaddition energies. We emphasize that the wiggles stem fromS
s(green line) and are not due to the xc correction. The latter
is responsible for the large sawtooth oscillations and, as Figs. 2
and3clearly show, it is the dominant contribution to S.
Recently experimental measurements of the Seebeck coef-
ficient and the electrical conductance in the CB regime havebeen reported for an individual single-wall carbon nanotube[35] as well as for quantum dots [ 31,32]. For the transport
properties of nanotubes, we can extract both single-particleenergies and charging energies from the experimental resultsand then use them to calculate GandSwithin our DFT scheme.
We again consider the Hxc potential of Eq. ( 8) but, in contrast
to the model calculations described previously, the chargingenergies U
Kdepend on the charging state K.
In Fig. 4we plot the conductance Gcalculated using
the formalism of Ref. [ 8] (upper panel) and the Seebeck
coefficient Scalculated from Eq. ( 7) (lower panel) versus the
gate voltage vfor the parameters of Table Iand for temperature
T=4.5 K and coupling γ=0.02 meV. For comparison
we also report the Seebeck coefficient as calculated fromthe LB-DFT formalism with the same parameters. LB-DFTfails in reproducing the characteristic sawtooth behaviourof the experimental results. Instead, the Seebeck coefficient
081410-3RAPID COMMUNICATIONS
YANG, PERFETTO, KURTH, STEFANUCCI, AND D’AGOSTA PHYSICAL REVIEW B 94, 081410(R) (2016)
FIG. 4. Conductance (upper panel) and Seebeck coefficient
(lower panel) of a single-wall carbon nanotube from DFT (black)
and experiment (red, data from Ref. [ 35]). Also shown is the KS
Seebeck coefficient (dashed green). The single particle and chargingenergies are given in Table I. The other parameters are T=4.5Ka n d
γ=0.02 meV.
calculated with our DFT scheme clearly shows the peak and
valley structures observed in experiment, confirming again thecrucial role of the xc correction. Also, all the fine structurewiggles (kinks in some cases) are correctly captured.
In conclusion, we have proposed a DFT scheme for the
calculation of the Seebeck coefficient which corrects thedeficiencies of the canonical Landauer-B ¨uttiker approach inTABLE I. Single-particle energies ε0and charging energies UK
(in meV), for modeling the calculation of Fig. 4.
ε0−6.0 −3.75 −3.75 −3.75 −1.5 0.75
2, 5.0 4, 2.25 6, 4.5 8, 4.5 10.5.25
K,UK 1, 3.75
3, 6.25 5,5.75 7,2.0 9,6.5 11,6.75
the Coulomb blockade regime. We found that two ingredients
in the Hxc potential are essential: (i) the step feature atinteger electron number opens a gap in the linear dependenceon gate voltage and (ii) the temperature derivative generatesthe sawtooth behavior in this gap region. Remarkably, thexc correction represents the dominant contribution to Sjust
as the xc correction to the conductance dominates in theCoulomb blockade regime [ 8]. We have compared our theory
with both rate equations and experimental results on a carbonnanotube and found good quantitative agreement in all cases.The present approach is valid in the linear response regime,where the applied thermal gradient is small. Going beyond thelinear response would pave the way for a deeper understandingof the thermoelectric effect and allow to study materials forextreme applications. The recently proposed DFT frameworkfor thermal transport by Eich et al. [16] appears to be a
promising starting point for this purpose.
We would like to acknowledge useful discussions with
F. Eich at the early stage of this project. K.Y ., S.K.,and R.D’A. acknowledge financial support from DYN-XC-TRANS (Grant No. FIS2013-43130-P) and NANOTherm(Grant No. CSD2010-00044) of the Ministerio de Economiay Competitividad (MINECO) and the Grupo ConsolidadoUPV/EHU del Gobierno Vasco - Eusko Jaurlaritza (Grant No.IT578-13). E.P. and G.S. acknowledge funding by MIUR FIRBGrant No. RBFR12SW0 and EC funding through the RISECoExAN (GA644076).
[ 1 ] F .J .D i S a l v o , Science 285,703(1999 ).
[2] C. B. Vining, Nat. Mater. 7,765(2008 ).
[3] C. B. Vining, Nat. Mater. 8,83(2009 ).
[4] Y . Dubi and M. Di Ventra, Rev. Mod. Phys. 83,131(2011 ).
[5] H. J. Goldsmid, Introduction to Thermoelectricity in Springer
Ser. Mater. Sci. (Springer-Verlag, Berlin, 2010).
[6] M. Di Ventra, Electrical Transport in Nanoscale Systems
(Cambridge University Press, New York, 2008).
[7] R. D’Agosta, Phys. Chem. Chem. Phys. 15,1758 (2013 ).
[8] S. Kurth and G. Stefanucci, P h y s .R e v .L e t t . 111,030601 (2013 ).
[9] Z.-F. Liu and K. Burke, Phys. Rev. B 91,245158 (2015 ).
[10] G. Stefanucci and S. Kurth, Nano Lett. 15,8020 (2015 ).
[11] G. Stefanucci and C.-O. Almbladh, Phys. Rev. B 69,195318
(2004 ).
[12] G. Stefanucci and C.-O. Almbladh, Europhys. Lett. 67,14
(2004 ).
[13] N. Sai, M. Zwolak, G. Vignale, and M. Di Ventra, Phys. Rev.
Lett. 94,186810 (2005 ).[14] M. Koentopp, K. Burke, and F. Evers, Phys. Rev. B 73,121403
(2006 ).
[15] G. Vignale and M. Di Ventra, Phys. Rev. B 79,014201
(2009 ).
[16] F. G. Eich, M. Di Ventra, and G. Vignale, P h y s .R e v .L e t t . 112,
196401 (2014 ).
[17] F. G. Eich, A. Principi, M. Di Ventra, and G. Vignale, Phys. Rev.
B90,115116 (2014 ).
[18] T. A. Costi and V . Zlati ´c,P h y s .R e v .B 81,235127 (2010 ).
[19] C. W. J. Beenakker, P h y s .R e v .B 44,1646 (1991 ).
[20] C. W. J. Beenakker and A. A. M. Staring, Phys. Rev. B 46,9667
(1992 ).
[21] X. Zianni, P h y s .R e v .B 78,165327 (2008 ).
[22] J. Cai and G. D. Mahan, P h y s .R e v .B 74,075201 (2006 ).
[23] Y . Apertet, H. Ouerdane, C. Goupil, and P. Lecoeur, Eur. Phys.
J. Plus 131,76(2016 ).
[24] B. Dong and X. Lei, J. Phys.: Condens. Matter 14,11747
(2002 ).
081410-4RAPID COMMUNICATIONS
DENSITY FUNCTIONAL THEORY OF THE SEEBECK . . . PHYSICAL REVIEW B 94, 081410(R) (2016)
[25] H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and
Optics of Semiconductors (Springer, New York, 2008).
[26] G. Stefanucci and S. Kurth, P h y s .R e v .L e t t . 107,216401 (2011 ).
[27] E. Perfetto and G. Stefanucci, P h y s .R e v .B 86,081409 (2012 ).
[28] P. Tr ¨oster, P. Schmitteckert, and F. Evers, Phys. Rev. B 85,
115409 (2012 ).
[29] J. P. Bergfield, Z.-F. Liu, K. Burke, and C. A. Stafford, Phys.
Rev. Lett. 108,066801 (2012 ).
[30] A. A. M. Staring, L. W. Molenkamp, B. W. Alphenaar, H. van
Houten, O. J. A. Buyk, M. A. A. Mabesoone, C. W. J. Beenakker,and C. T. Foxon, Europhys. Lett. 22,57(1993 ).[31] A. S. Dzurak, C. G. Smith, M. Pepper, D. A. Ritchie, J. E. F.
Frost, G. A. C. Jones, and D. G. Hasko, Solid State Commun.
87,1145 (1993 ).
[32] A. S. Dzurak, C. G. Smith, C. H. W. Barnes, M. Pepper, L.
Mart ´ın-Moreno, C. T. Liang, D. A. Ritchie, and G. A. C. Jones,
Phys. Rev. B 55,R10197 (1997 ).
[33] U. Gupta and A. K. Rajagopal, Phys. Rep. 87,259(1982 ).
[34] G. Stefanucci and S. Kurth, Phys. Status Solidi B 250,2378
(2013 ).
[35] J. P. Small, K. M. Perez, and P. Kim, P h y s .R e v .L e t t . 91,256801
(2003 ).
081410-5 |
PhysRevB.72.113402.pdf | High coverage of hydrogen on a (10,0) single-walled boron nitride nanotube
Sang Soo Han, Sung Hoon Lee, Jeung Ku Kang, and Hyuck Mo Lee *
Department of Materials Science and Engineering, Korea Advanced Institute of Science and Technology,
Kusung-dong 373-1, Yusung-gu, Daejon 305-701, Korea
/H20849Received 11 May 2005; revised manuscript received 13 July 2005; published 2 September 2005 /H20850
The binding energy of hydrogen atoms to a /H2084910,0 /H20850single-walled boron nitride nanotube /H20849SWBNNT /H20850is
calculated at 25%, 50%, 75%, and 100% coverage using the density functional theory. The average bindingenergy is highest at 50% coverage when the H atoms are adsorbed on the adjacent B and N atoms along thetube axis and the value is −53.93 kcal/mol, which is similar to half of the H uH binding energy. Also, the
band gap /H20849−4.29 eV /H20850of the pristine /H2084910,0 /H20850SWBNNT is decreased up to −2.01 eV for the H-adsorbed BNNT
with 50% coverage.
DOI: 10.1103/PhysRevB.72.113402 PACS number /H20849s/H20850: 68.43. /H11002h, 31.10. /H11001z, 73.22. /H11002f, 82.65. /H11001r
Due to low dimensionality and high surface-area-to-
volume ratio, the physical properties of nanotubes can bedramatically influenced by the surface addition of selectedatomic or molecular species. Such functionalization can leadto significant enhancement of properties relevant to techno-logical applications. For instance, it has been recently dem-onstrated that carbon nanotubes /H20849CNTs /H20850functionalized with
hydrogen represent a new type of a nanoscale electronicdevice.
1
In contrast to CNTs, boron nitride nanotubes /H20849BNNTs /H20850
have a uniform electronic band gap independent of the diam-
eter and chirality of the tube and their native state is electri-cally insulting.
2,3Because of large ionicity of the B uN
bond in BNNTs, their properties are different from CNTs. Itwas reported that BNNTs prefer a nonhelical or zigzag ori-entation during growth,
4,5which prefigures the advantage of
BNNTs in the potential application of nanoscale electronicdevices similar to CNTs. The doping behavior of BNNTs isalso important since it may create acceptors or donors for theuse in the electronic device such as the p-njunction. How-
ever, studies of BNNTs have been focused on growth andproperties of clean BNNTs so far. The doping behavior orfunctionalization of BNNT is very rare compared with thestudies of CNTs.
In this work, we study the chemisorption behavior of hy-
drogen on the sidewall of the /H2084910,0 /H20850single-walled BNNT
/H20849SWBNNT /H20850with 25%, 50%, 75%, and 100% coverages by
the density functional theory /H20849DFT /H20850. We will show that the
binding energies of hydrogen on the BNNT wall are depen-dent on its coverage and the electronic structures of the tubecan be modified by the chemisorption process of hydrogen.
The binding energies of hydrogen were calculated using
the local-orbital DFT method implemented with the
DMOL3
package.6All the electron calculations were used together
with the double numerical plus polarization /H20849DNP /H208507basis set
and the generalized gradient approximation /H20849GGA /H20850employ-
ing the Perdew-Wang scheme.8The DNP basis set is compa-
rable in quality to the commonly used Gaussian analyticalbasis set, 6-31G
**.9To model the interaction between hydro-
gen atoms and a /H2084910,0 /H20850SWBNNT, we used a tetragonal cell
of the size 25 Å /H1100325 Å/H110034.38 Å with the length of cequal
to the periodicity of the /H2084910,0 /H20850SWBNNT, assuming a B uNbond length of 1.46 Å.10The supercell includes 20 B and 20
N atoms. The experimentally synthesized SWBNNTs wereobserved to have a range of diameters ranging from/H110110.5 to 1.2 nm.
11,12It was also reported that the synthesis of
the zigzag-type BNNT is more favorable to that of the am-chair type,
4,5thus the /H2084910,0 /H20850SWBNNT is chosen in this
study since the zigzag /H2084910,0 /H20850BNNT has a diameter of
0.81 nm. In calculating with a DMOL3code, all the calcula-
tions were performed only at the gamma point /H20849k/H6023=0/H20850.
Densities of states /H20849DOSs /H20850for both the pristine /H2084910,0 /H20850
SWBNNT and the hydrogenated one were also investigatedby the
CASTEP software.13The exchange-correlation of elec-
trons was handled with the PW91 functional8of GGA level
and the norm-conserving pseudopotentials generated by us-ing the Troullier-Martins scheme were adopted to describethe electron-ion interaction.
14We set a kinetic energy cutoff
of 700 eV for the BNNTs and used the Monkhorst-Packscheme
15to generate the k-space grid. With the optimized
structures of the nanotubes predicted in the above chemi-sorption energy studies of hydrogen with coverage, we per-formed single point calculations for the structures to calcu-late DOS because the plane-wave type DFT calculationgenerally requires high simulation time cost. The cell size isconsidered to be same as that in the chemisorption energystudies performed by the
DMOL3package. We also found out
that a k-point sampling of 2 /H110032/H110034 is sufficient for energy
convergence. Integration over a three-dimensional Brillouinzone was carried out using the total of eight kpoints.
According to Bauschlicher,
16it was proved that hydrogen
is more strongly bonded to the single-walled carbon nano-tube /H20849SWCNT /H20850with hydrogen coverage of a high symmetry
than the random structure. As a result, the SWBNNTs withhydrogen coverage of only high symmetry were considered
in this study. The /H2084910,0 /H20850SWBNNT plus H models considered
here are shown in Fig. 1 in which the structures are opti-mized with all the atoms free. To show the adsorption struc-tures clearly, we repeat the unit cell in the cdirection up to
double periodicity. The hydrogen binding energies are sum-marized in Table I. The adsorption energy /H9004E
adsis defined as
follows:PHYSICAL REVIEW B 72, 113402 /H208492005 /H20850
1098-0121/2005/72 /H2084911/H20850/113402 /H208494/H20850/$23.00 ©2005 The American Physical Society 113402-1/H9004Eads=Etot/H20849BNNT + nH/H20850−Etot/H20849BNNT /H20850−n/H11003Etot/H20849H/H20850
/H208491/H20850
where Etot/H20849BNNT+ nH/H20850,Etot/H20849BNNT /H20850, and Etot/H20851H/H20852are the to-
tal energies of the fully optimized /H2084910,0 /H20850SWBNNT-H struc-
ture, the nanotube alone and the hydrogen atom, respectively,andnis the number of hydrogen. The minus value of /H9004E
ads
implies that adsorption is exothermic.
In Table I, it is noticeable that the binding energies of
hydrogen are higher for all the high hydrogen coverages thanthe binding energies of one B uH bond or one N uH bond.
Especially, the N uH binding energy has a positive value
when one H atom is adsorbed on the top site of one N atomin the /H2084910,0 /H20850SWBNNT; however, it changed to a negative
value if the hydrogen coverage is increased /H20849see 25cov /H60182i n
Table I /H20850. The result for the adsorption of a single H shows
that the adsorption is site selective; in other words , a H atom
prefers to be adsorbed on the top site of the B atom of thepristine SWBNNT, which agrees well with the DFT result ofWuet al.
17Moreover, the B uH binding energy in the 25%
/H2084925cov /H60181/H20850or 50% /H2084950cov /H60182/H20850coverage is significantly higher
than that in the single B uH bond state. It is because only
one BuH bond state leaves an open-shell radical state /H20849dou-
blet spin state /H20850on the nanotube, while the BNNTs with high
hydrogen coverage, such as 25cov /H60181 and 50cov /H60182, have a
close-shell structure /H20849singlet spin state /H20850.
In addition, the highest stability of the BNNT structure
with high hydrogen coverage is observable when hydrogen ischemisorbed on adjacent B and N atoms along the tube axis,
which probably results from two reasons. Hydrogen bondingto adjacent B and N atoms would imply a loss of one B uN
/H9266bond, while that with two B atoms would have a greater
disruption of the B uN/H9266bond system. Another reason can
be discovered by atomic charges of hydrogen. For example,in the case of 25% hydrogen coverage, the atomic chargeshave negative values as hydrogen is bonded with only Batoms /H20849see 25cov /H60181 in Fig. 1 /H20850, and vice versa for N uH
bonds. Thus the repulsive interaction is affected between twohydrogen atoms chemisorbed on the nanotube wall and two
hydrogen atoms tend to repel each other. On the other hand,if hydrogen is chemically adsorbed with adjacent B and Natoms, as in the case of 25cov /H60183, the charges for hydrogen
atoms are negative for B uH bonds and positive for N uH
bonds, indicating attractive interaction between two hydro-gen atoms. The attractive force causes two H atoms to pulleach other. Additionally, one can consider the attraction be-tween nonbonded H and N atoms in the case of 25cov /H60183
since H atoms bonded with N atoms have positive atomiccharges and other N atoms do not. In the case of 25cov /H60181,
this attraction is not present because all H atoms bonded withB atoms have negative charges. Therefore, the most favor-able hydrogen configuration at a constant coverage has Hatoms upon adjacent B and N atoms.
FIG. 1. /H20849Color online /H20850Hydrogen decorations considered in this
study to investigate the binding energies of H on the exterior wall of/H2084910,0 /H20850SWBNNT as a function of H coverage. In DFT calculations,
only four layers of the BNNT /H20849half of the present tube structures
along the tube axis /H20850were considered due to computational limita-
tion. Here, the pink, blue, and white atoms mean boron, nitrogen,and hydrogen, respectively. In the case of 25% H coverage, the25cov /H60181 indicates all boron pattern, the 25cov /H60182 all nitrogen pattern,
the 25cov /H60183 the line pattern, and the 25cov /H60184 the zigzag pattern. In
50% coverage, the 50cov /H60181 means the pairs of rings pattern, the
50cov /H60182 all boron pattern, the 50cov /H60183 the line pattern, the 50cov /H60184
the spiral pattern, and the 50cov /H60185 the pairs of lines pattern. Also,
the 75cov /H60181 is the pattern of 75% H coverage having the 20 B uH
and 10 N uH in unit cell of the BNNT and the 75cov /H60182i st h e
pattern having 10 B uH and 20 N uH.TABLE I. Summary of H binding energies /H20849kcal/mol /H20850on the
/H2084910,0 /H20850SWBNNT exterior wall as a function of H coverage. The
calculations were performed at the level of DFT/PW91 with a DNPbasis set. The case that all H atoms adsorb on nitrogen atoms incoverage of 50% is not shown here because the calculation is notconverged.
Coverage /H9004E
adsa/H9004Eaveb
25%
25cov /H60181 −178.98 −17.90 /H2084910 BuH/H20850
25cov /H60182 −274.29 −27.43 /H2084910 NuH/H20850
25cov /H60183 −538.15 −53.81 /H208495BuH&5NuH/H20850
25cov /H60184 −518.71 −51.87 /H208495BuH&5NuH/H20850
50%
50cov /H60181 −1045.76 −52.29 /H2084910 BuH&10 NuH/H20850
50cov /H60182 −415.48 −20.77 /H2084920 BuH/H20850
50cov /H60183 −1007.41 −50.37 /H2084910 BuH&10 NuH/H20850
50cov /H60184 −1068.48 −53.42 /H2084910 BuH&10 NuH/H20850
50cov /H60185 −1078.57 −53.93 /H2084910 BuH&10 NuH/H20850
75%
75cov /H60181 −1210.78 −40.36 /H2084920 BuH&10 NuH/H20850
75cov /H60182 −1132.28 −37.74 /H2084910 BuH&20 NuH/H20850
100%
100cov −1835.44 −45.89 /H2084920 BuH&20 NuH/H20850
2.5%
One BuH on BNNT −8.10
One NuH on BNNT +7.25
a/H9004Eads=Etot/H20851BNNT+ nH/H20852−Etot/H20851BNNT /H20852−n/H11003Etot/H20851H/H20852/H20849nis the num-
ber of hydrogen atom /H20850
bAverage H binding energy calculated by the following equation:
/H9004Eave=/H9004Eads/nBRIEF REPORTS PHYSICAL REVIEW B 72, 113402 /H208492005 /H20850
113402-2In Table I, we can also find that the most favorable ad-
sorption configuration is that the numbers of H-adsorbing Band N atoms are equal. However, the hydrogen binding en-ergies at 100% coverage are lower than at 25% and 50%coverage with the equal number of B uH and NuH bonds.
To form B uH and NuH bonds, the B and N atoms must
occupy sp
3hybridization, which results in the B and N bulg-
ing out the tube. For both 100% coverages, the B and Natoms still form a good tube structure since any deformationthat improves one B uHo rNuH bond will weaken an-
other, which is similar to the case of CNT.
16This means that
for the 100% coverage, the B and N atoms cannot changetheir hybridization to enhance the B uH and NuH bond,
which results in weaker adsorption than the 50% coveragewhere half of the B and N atoms can bulge out of the tube tomaximize the B uH and N uH bonding. The reason why
hydrogen binding energy at 100% coverage is higher than at75% is that the electrostatic attraction between nonbondedBuH and N uH is maximized when numbers of the
formed B uH and NuH bonds are same. If the number is
different, the additional electrostatic repulsive forces be-tween two H atoms bonded with B atoms /H2084975cov /H60181/H20850or with
N atoms /H2084975cov /H60182/H20850are generated. Despite any limitation in
the calculation methods used, it is clear that the formation of75% and 100% coverage will be a very endothermic processbecause the H uH bond energy is 105.27 kcal/mol at the
present DFT calculation level. Because the highest bindingenergies /H20849−53.93 kcal/mol in the case of 50cov /H60185/H20850computed
for the 50% coverage are sufficiently close to one-half/H20849−52.64 kcal/mol /H20850of the HuH bond energy, it might be
possible to achieve the coverage level in a thermodynamic
process starting with H
2and BNNT. The formation of sig-
nificantly higher than 50% coverage in a thermodynamicprocess seems unlikely since a higher coverage would re-quire deformation of the B and N atoms on the tube, whichwould weaken some of the existing B uH and NuH bonds
and hence result in a smaller average H binding energy.
By calculating the electronic DOS, we investigated the
electronic structures of the /H2084910,0 /H20850SWBNNT with high hy-
drogen coverage. Figure 2 shows the total DOS /H20849TDOS /H20850for
electrons in pristine and hydrogen adsorbed /H2084910,0 /H20850
SWBNNT where the hydrogen coverage corresponds to50cov /H60185, the most stable state in Fig. 1. We found that the
pristine /H2084910,0 /H20850SWBNNT has a band gap of 4.29 eV from the
gap of the highest occupied molecular orbital /H20849HOMO /H20850and
the lowest unoccupied molecular orbital /H20849LUMO /H20850in Fig.
2/H20849a/H20850, which is similar to the published DFT result of
4.03 eV.
18In the SWBNNT with 50% hydrogen coverage,
the band gap /H20849HOMO-LUMO gap /H20850is decreased to 2.01 eV,
indicating a semiconductor with a wide band gap. From thisfact, we know that the SWBNNT can be modulated throughhydrogen adsorption from an insulator /H20849for the pristine state /H20850
to a semiconductor /H20849for the 50% coverage state /H20850. The reason
why the band structure can be changed by hydrogen adsorp-tion is easily understood from the partial DOS /H20849PDOS /H20850for B
and N atoms in the nanotube, although the result is notshown here. In the case of the pristine /H2084910,0 /H20850SWBNNT, the
PDOS demonstrates that the structure of the valence band isalmost completely determined by nitrogen. The contributionof boron is small but its effect on the formation of states atthe edges of the conduction band is greater. In the BNNT
with 50% hydrogen coverage, a new peak is found near0.0 eV in the band gap, indicated by an arrow in Fig. 2 /H20849b/H20850.
The peak is mainly caused by the sorbital of H atoms
bonded with B atoms.
As already mentioned, it is believed that the BNNT is an
insulator with a 4–5 eV band gap independent of itshelicity.
2,3To modulate the band gap of the nanotube, there
are two ways: carbon doping19and radial deformation20of
the BNNT. According to the local density functional calcu-lation of Blase et al. ,
19the calculated band gap is decreased
to 2.0 and 0.5 eV for BC 2N and BC 3nanotubes, respectively,
as the carbon composition is increasesd in B xCyNznanotube.
Thus, the synthesis of the composite B xCyNznanotube would
make possible its application for electronic and photonic de-vices with a variety of electronic properties. Also, Kim et
al.
20reported that in the zigzag BNNTs the radial deforma-
tion that gives rise to transverse pressure of about 10 GPadecreases the gap from 5 to 2 eV, allowing for optical appli-
FIG. 2. TDOS for /H20849a/H20850pristine /H2084910,0 /H20850SWBNNT and /H20849b/H20850/H2084910,0 /H20850
SWBNNT with 50% H coverage, called 50cov /H60185 in Fig. 1. The
dotted lines in each figure refer to the Fermi energy level. Thearrow in /H20849b/H20850indicates one band in the band gap, caused by hydro-
gen chemisorption.BRIEF REPORTS PHYSICAL REVIEW B 72, 113402 /H208492005 /H20850
113402-3cation in the visible range. We suggest a new method, hydro-
gen adsorption, to modulate a band gap of the BNNT. Theband gap /H208494.29 eV /H20850of the insulating /H2084910,0 /H20850SWBNNT can be
decreased to 2.01 eV, indicating a semiconductor with a
wide band gap.
In summary, it is possible for hydrogen to be chemically
adsorbed on the exterior surface of BNNT up to 50% cover-age. Since the average hydrogen binding energy /H20849per H
atom /H20850at 50% coverage is very close to half of a H
2bond
energy, it might be possible to achieve this level of coveragein a thermodynamic process starting with H
2and pristine
SWBNNT. The band structure of the SWBNNT can bemodulated through hydrogen chemisorption. In detail, if hy-
drogen is chemisorbed on the exterior wall of BNNT up tothe coverage of 50%, the pristine insulating BNNT ischanged to a semiconductor with a wide band gap. It is cer-tain that the present results for the electronic properties ofBNNT activate further works on the electronic and photonicdevices using BNNTs.
This research was supported by Grant No. 04K1501-
02210 from the Center for Nanostructured Materials Tech-nology under the 21st Century Frontier R&D Programs ofthe Ministry of Science and Technology, Korea.
*Corresponding author: Prof. Hyuck Mo Lee: Email:
hmlee@kaist.ac.kr TEL: /H1100182-42-869-3334: FAX: /H1100182-42-869-
3310
1K. S. Kim, D. J. Bae, J. R. Kim, K. A. Park, S. C. Lim, J.-J. Kim,
W. B. Choi, C. Y . Park, and Y . H. Lee, Adv. Mater. /H20849Weinheim,
Ger. /H2085014, 1818 /H208492002 /H20850.
2X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, Europhys.
Lett. 28, 335 /H208491994 /H20850.
3A. Rubio, J. L. Corkill, and M. L. Cohen, Phys. Rev. B 49, 5081
/H208491994 /H20850.
4M. Terauchi, M. Tanaka, K. Suzuki, A. Ogino, and K. Kimura,
Chem. Phys. Lett. 324, 359 /H208492000 /H20850.
5H. J. Xiang, J. Yang, J. G. Hou, and Q. Zhu, Phys. Rev. B 68,
035427 /H208492003 /H20850.
6B. Delley, J. Chem. Phys. 113, 7756 /H208492000 /H20850;DMOL3is a regis-
tered software product of Molecular Simulation Inc.
7B. Delley, J. Chem. Phys. 92, 508 /H208491990 /H20850.
8J. P. Perdew and Y . Wang, Phys. Rev. B 45, 13244 /H208491992 /H20850.
9B. Mårlid, K. Larsson, and J.-O. Carlsson, J. Phys. Chem. B 103,7637 /H208491999 /H20850.
10S. W. Yang, H. Zhang, J. M. Soon, C. W. Lim, P. Wu, and K. P.
Loh, Diamond Relat. Mater. 12, 1194 /H208492003 /H20850.
11A. Loiseau, F. Willaime, N. Demoncy, G. Hug, and H. Pascard,
Phys. Rev. Lett. 76, 4737 /H208491996 /H20850.
12E. Bengu and L. D. Marks, Phys. Rev. Lett. 86, 2385 /H208492001 /H20850.
13M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D.
Joannopoulous, Rev. Mod. Phys. 64, 1045 /H208491992 /H20850.
14N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 /H208491991 /H20850.
15H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 /H208491976 /H20850.
16C. W. Bauschlicher, Jr., Nano Lett. 1, 223 /H208492001 /H20850.
17X. Wu, J. Yang, J. G. Hou, and Q. Zhu, Phys. Rev. B 69, 153411
/H208492004 /H20850.
18T. M. Schmidt, R. J. Baierle, P. Piquini, and A. Fazzio, Phys. Rev.
B67, 113407 /H208492003 /H20850.
19X. Blase, J.-C. Charlier, A. De Vita, and R. Car, Appl. Phys. Lett.
70, 197 /H208491997 /H20850.
20Y .-H. Kim, K. J. Chang, and S. G. Louie, Phys. Rev. B 63,
205408 /H208492001 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 72, 113402 /H208492005 /H20850
113402-4 |
PhysRevB.94.165155.pdf | PHYSICAL REVIEW B 94, 165155 (2016)
Kernel-corrected random-phase approximation for the uniform
electron gas and jellium surface energy
Adrienn Ruzsinszky,1Lucian A. Constantin,2and J. M. Pitarke3,4
1Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
2Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy
3CIC nanoGUNE, Tolosa Hiribidea 76, E-20018 Donostia, Basque Country
4Materia Kondentsatuaren Fisika Saila, DIPC, and Centro F ´ısica Materiales CSIC-UPV/EHU,
644 Posta kutxatila, E-48080 Bilbo, Basque Country
(Received 11 July 2016; published 21 October 2016)
We introduce and test a nonlocal energy-optimized model kernel (NEO) within the adiabatic connection
fluctuation-dissipation (ACFD) density-functional theory for the jellium surface and uniform electron gas, asbenchmarks for simple metallic systems. Our model kernel is short ranged for the uniform electron gas paradigmsystem and one-electron self-correlation free. One-electron self-interaction freedom is provided by an iso-orbitalindicator. We show how several versions of the NEO kernel perform for the uniform electron gas and jelliumsurface energies, and in addition we explain the underlying physics of self-interaction-free exchange-only kernelsfor exponentially decaying surface densities.
DOI: 10.1103/PhysRevB.94.165155
I. INTRODUCTION
Nonempirical density-functional theory (DFT) relies on
the knowledge of paradigms [ 1]. One of these paradigms is
theuniform electron gas [ 2–4], which plays a key role in the
construction of many density-functional approximations [ 1].
The uniform electron gas provides relevant information aboutcorrelation in materials and serves as a model for metallicsystems [ 5]. The surface of a bounded electron gas, which is
different from the bulk, delivers additional information aboutthe ground-state correlation.
As the uniform electron gas has done in the past, the jellium
surface can also guide the construction of density functionals.In the jellium model of a simple metal surface, the ions arereplaced by a semi-infinite uniform positive background ofdensity n, which is neutralized by a valence electron density
n(z) allowed to leak out into the vacuum side of the surface.
The accuracy of [ 6–8] local and semilocal density func-
tionals is limited by the approximating form of the exchange-correlation (xc) energy or its corresponding potential [ 9].
Density-functional approximations are usually benchmarkedagainst correlated wave-function-based methods [ 10–15].
Approximations that rely on the concept of the slowlyvarying limit of the perturbed uniform electron gas deliveraccurate lattice properties [ 16]. The simplest approximation,
the local density approximation (LDA) [ 3], is reasonably
accurate for periodic solids [ 17]. Generalized gradient ap-
proximations (GGAs) utilize information about slowly varyingdensities [ 18–22]. These GGAs have proven accurate for both
bulk solids and surfaces. Meta-GGAs beyond the GGA leveladd the positive kinetic-energy density as a new ingredientto the existing electron density and density gradient inGGAs [ 23–29]. With all these ingredients, the meta-GGA
density functionals are the potentially most accurate semilocalapproximations, with the flexibility to describe bulk solids,surfaces, and molecules at the same time [ 29,30]. Adiabatic-
connection fluctuation-dissipation (ACFD) approximations[the random-phase approximation (RPA) in particular] standon the fifth and highest rung of a ladder [ 31] of density-functional approximations, employing the unoccupied as well
as the occupied Kohn-Sham (KS) orbitals in a fully nonlocalway that can potentially solve problems such as capturing weakvan-der-Waals interactions [ 32] and static correlation for the
H
2molecule in a spin-restricted formalism [ 33].
Jellium surface energies were thoroughly investigated
within an ACFD approach [ 34–37], and these results were
compared to Fermi-hypernetted chain (FHNC) [ 13] and
diffusion-Monte-Carlo (DMC) surface energies [ 14,15]. Later,
Yan et al. made an assessment of several density-functional
approximations to the jellium surface energy [ 38]. Many
refinements of DFT (including ACFD), as summarized inRef. [ 39], produced jellium surface energies close to those
obtained in the LDA and thus much lower than those of FHNCand early DMC [ 14]. A more recent DMC calculation [ 40]
agrees well with the DFT values.
By combining an adiabatic-connection (AC) formula with
the fluctuation-dissipation (FD) theorem, one obtains an exact
expression for the xc energy of an arbitrary many-electronsystem (unless stated otherwise, atomic units are used through-out) [ 41]:
E
xc=1
2/integraldisplay1
0dλ/integraldisplay
dr/integraldisplay
dr/primev(r,r/prime)/braceleftbigg/bracketleftbigg
−1
π/integraldisplay∞
0dω
×χλ(r,r/prime,iω)/bracketrightbigg
−n(r)δ(r−r/prime)/bracerightbigg
, (1)
where v=1/|r−r/prime|is the bare Coulomb interaction and
χλrepresents the interacting density-response function of
afictitious many-electron system with the electron-electron
interaction strength λe2. In the framework of time-dependent
DFT (TDDFT), the interacting density-response function χλ
obeys a Dyson-like integral equation [ 42]:
χλ=/bracketleftbig
1−χ0/parenleftbig
λv+fλ
xc/parenrightbig/bracketrightbig−1χ0, (2)
where χ0is the density-response function of noninteracting
KS electrons. The RPA sets the xc kernel fλ
xc(r,r/prime,ω) to zero.
In much of this work, we will need only the exchange-only
2469-9950/2016/94(16)/165155(7) 165155-1 ©2016 American Physical SocietyRUZSINSZKY , CONSTANTIN, AND PITARKE PHYSICAL REVIEW B 94, 165155 (2016)
(linear in λ) kernel, but we will leave λgeneral in the notation.
I nt h eR P A ,E q s .( 1) and ( 2) are combined, bootstrapping a
crude approximation for χλto a more sophisticated one for
Exc[43]. For the uniform electron gas, it was shown [ 44] that
an adiabatic (static) local kernel overshoots the correlationenergy by about as much ( ∼0.5 eV) as the RPA undershoots
it; a static nonlocal kernel [ 45,46] was found to reduce the error
to∼0.1 eV , and a dynamic nonlocal kernel [ 47] was found to
reduce the error down to ∼0.02 eV .
There are other routes to correct for the missing short-range
correlation. The quantum chemistry community often uses thesecond-order screened exchange (SOSEX) contribution (com-ing from the wave-function antisymmetry) [ 48]. The SOSEX
correction was found to perform somewhat controversially inquantum chemistry.
The first implementation of the ACFD scheme for a
nonuniform system was reported in Ref. [ 34] for the jellium
surface. Although RPA delivers too deep correlation energyfor the short range, jellium surface energies are surprisinglyaccurate. The better performance of RPA for the jelliumsurface energy was explained by the relevance of the long-range correlation for surfaces. Even if RPA does not providean accurate short-range correlation, the error tends to cancelout of the surface energy. Therefore exchange-correlationkernels, which can correct the deep RPA correlation forshort range in bulks or inhomogeneous systems, do not give
much contribution for the jellium surface energy. The same
conclusion can be drawn when the correction to the RPAcalled RPA +is applied to the jellium surface. For energy
differences in processes that conserve the electron number,it was argued [ 38] that the correction from RPA +, although
large for the total energy (about +0.5 eV per electron), tends
to cancel out almost completely. Beyond the jellium model,recent calculations showed a remarkably accuracy of the RPAfor various properties (including the surface energy) of realmaterials [ 49–55].
To account for this missing short-range part of the RPA
correlation energy in the uniform electron gas and inhomoge-neous systems, we rely on (for the jellium surface) a nonlocalenergy-optimized (NEO) model kernel [ 56,57] which has been
introduced and tested recently [ 58]. This kernel is designed to
satisfy exact constraints utilizing the iso-orbital Zindicator, a
meta-GGA ingredient. The original construction of NEO wasdesigned to produce a correctly long-ranged ( ∼1/u) exchange
kernel for one- and two-electron systems, where Z=1. A
problem arises in that it also produces a long-ranged exchangekernel in the tail of the density of a jellium surface, sinceZ→1 andk
F→0 there. (Here kF=(3π2n)1/3is the Fermi
wave vector.) In this paper we present the NEO kernel and testit for the jellium surface. Along with the original NEO kernel,we also introduce a modification of the original expressionto restore the correct decay of the density tail for the jelliumsurface. With this construction the NEO-kernel-corrected RPAshould be correct at long range, as pointed out by Ref. [ 34].
II. COMPUTATIONAL FRAMEWORK
Consider a many-electron system that is neutralized by a
uniform positive background (jellium) of density ¯ncut off
sharply at a planar surface (at z=z0). The xc surface energyis obtained as follows [ 35]
σxc=N
A/braceleftbig
εxc[n]−εunif
xc(¯n)/bracerightbig
, (3)
where εxc[n] andεunif
xc(¯n) represent, respectively, the xc energy
per particle of the actual semi-infinite many-electron system[of density n(z)] and a uniform electron gas (of density ¯n)
cut off sharply at z=z
0; here, ¯n=k3
F/(3π2),kFbeing the
magnitude of the bulk Fermi wave vector. Using Eq. ( 1), one
finds:
εxc[n]=/integraldisplay∞
0d(q/kF)εxc,q[n], (4)
where
εxc,q[n]=kF
4π/integraldisplay
dz/integraldisplay
dz/primen(z)vq(|z−z/prime|)/bracketleftbigg
−1
πn(z)
×/integraldisplay1
0dλ/integraldisplay∞
0dωχ λ(z,z/prime;q,iω )−δ(z−z/prime)/bracketrightbigg
.
(5)
Here, qrepresents the magnitude of a two-dimensional (2D)
wave vector parallel to the surface, vq(|z−z/prime|)i st h e2 D
Fourier transform of the bare Coulomb interaction v, and
χλ(z,z/prime;q,iω ) is the 2D Fourier transform of the interacting
density-response function χλof Eq. ( 2). If the interacting
density-response function χλis replaced for all λby the
noninteracting density response function χ0, then Eq. ( 5)
reduces to the exact exchange energy per particle εx,q[n]. We
define the correlation energy per particle εc,q[n]=εxc,q[n]−
εx,q[n].
Forεunif
xc(¯n), one simply needs to replace (i) the interacting
density-response function χλ(z,z/prime;q,iω ) entering Eq. ( 5)b y
that of a uniform electron gas and (ii) the electron density n(z)
[also entering Eq. ( 5)] by the step function ¯nθ(z0−z). This
yields a 2D wave-vector analysis [Eq. ( 4)] of the uniform-gas
xc energy per particle εunif
xc(¯n), which will be needed below
for a 2D wave-vector analysis of the xc surface energy.Alternatively, one can use Eq. ( 1) to reach the following
three-dimensional (3D) wave-vector analysis:
ε
unif
xc(¯n)=/integraldisplay∞
0d(Q/k F)εunif
xc,Q(¯n), (6)
where
εunif
xc,Q(¯n)=kF
4π2vQ/bracketleftbigg
−1
π¯n/integraldisplay1
0dλ/integraldisplay∞
0dωχ λ(Q,iω )−1/bracketrightbigg
.
(7)
Here,Qrepresents the magnitude of a 3D wave vector, vQis
the 3D Fourier transform of the bare Coulomb interaction v,
andχλ(Q,iω ) is the 3D Fourier transform of the interacting
density-response function χλ[see Eq. ( 2)] of a uniform
electron gas of density ¯n.
For the numerical calculations reported here, we start with
a jellium slab of finite width along the zdirection, and we
then take the limit of large thickness. In this paper we used thecode described in Refs. [ 34,35,59] that computes numerically
Eqs. ( 3)–(5) using accurate (occupied and unoccupied) LDA
orbitals [ 35]. Instead of considering the double-cosine Fourier
representation of the density-response function (as done in
165155-2KERNEL-CORRECTED RANDOM-PHASE APPROXIMATION . . . PHYSICAL REVIEW B 94, 165155 (2016)
Ref. [ 35]), we worked directly in the zspace. This approach
simplifies considerably the computational implementation ofthe kernels, but needs a large number of grid points in the z
direction, in order to obtain converged results (up to 1100 z
points on a Gaussian grid).
III. METHODOLOGY
A. The NEO-I kernel
Here we invoke NEO: a nonlocal energy-optimized model
kernel [ 56,57] which has been introduced and tested recently
for the uniform electron gas [ 58]. This kernel (which can
be applied to an arbitrary nonuniform system) is based onexact constraints and on the concept of the uniform electrongas [ 44]; the most widely-applicable approximations of DFT
and TDDFT are well known to rely on this paradigm. TheNEO kernel, as introduced in Ref. [ 58] (NEO-I), is:
f
λ,NEO
xc ([n],r,r/prime)=−λv(r,r/prime)/summationdisplay
σ/parenleftBignσ
n/parenrightBig2
×erfc(aNEO−I|r−r/prime|), (8)
where
aNEO−I=/radicalBig
˜c/parenleftbig
1−Z2σ/parenrightbig
kFσ, (9)
Zσ=τW
σ/τσbeing a meta-GGA ingredient [ 24,25],τσbeing
the KS kinetic-energy density
τσ=1
2occup/summationdisplay
α|∇φασ|2, (10)
andτW
σ=| ∇nσ|2/(8nσ) being the von Weizs ¨acker kinetic-
energy density [ 60] (which equals τσfor one- and two-
electron ground states). The decaying function erfc is thecomplementary error function, n
σandnare the σspin and
the total electron density, respectively, and kFσ=(6π2nσ)1/3.
These quantities are all evaluated at ( r+r/prime)/2.
For one-electron densities, one finds the expected result
fλ,NEO
xc =−λv. For two electrons in a spin singlet, one finds
the exact-exchange form fλ,NEO
xc =−λv/2, which is exact
in the high-density limit. In the short-range limit, the NEO-Ikernel becomes −λv/summationtext
σ(nσ/n)2as does the PGG kernel [ 61];
in the spin-polarized case, this further simplifies to −λv,
while in the spin-unpolarized case it becomes −λv/2. In the
long-range limit, fλ,NEO
xc vanishes rapidly, except in the one-
and two-electron regions. The exact xc kernel of the uniformelectron gas is known to be nonlocal but short ranged, and theNEO-I kernel has these features.
The ˜cparameter entering Eq. ( 9) is taken to fit the
exact second-order exchange contribution to the uniform-gascorrelation energy, which can be evaluated from explicitexpressions given by Langreth and Perdew [ 41] and by von
Barth and Hedin [ 62]; one finds ˜c=0.264, which makes a
large improvement over the RPA ( ˜c→∞ ).
B. NEO-II kernel
The NEO-I kernel of Eqs. ( 8) and ( 9) is simply the bare
Coulomb interaction λvmultiplied by a decaying function
of the variable aNEO−I|r−r/prime|. This is designed in order to0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4 -0.2 0 0.2 0.4
z/λFaNEO-I
aNEO-II
FIG. 1. The coefficients aNEO−IandaNEO−IIdefining the ker-
nels NEO-I and NEO-II, for a jellium surface with the electron-density parameter r
s=6. The surface is at z=0, the bulk is at
z< 0, and the vacuum is at z> 0.
(i) produce a correctly long-ranged exchange kernel for one-
and two-electron systems, where Zσ=1, and (ii) produce
no second-order gradient correction to the RPA correlationenergy in a slowly-varying high electron density, where Z
σ→
0+O(∇2). The problem is that this kernel (as introduced in
Ref. [ 58]) produces an unwanted long-ranged exchange kernel
in the tail of the electron density of a jellium surface ( Zσ→1
andkFσ→0) where the RPA should be recovered as discussed
in Ref. [ 35]. Hence, here we replace Eq. ( 9)b y
aNEO−II=/radicalBig
˜c/parenleftbig
3ασ−3α2σ+α3σ/parenrightbig
kFσ, (11)
where ασ=(τσ−τW
σ)/τunif
σ,τunif
σ being the Thomas-Fermi
kinetic-energy density [ 63,64]:
τunif
σ=(1/2)(3/10)(3π2)2/3(2nσ)5/3. (12)
This new approach (NEO-II) represents a clear improve-
ment, as it leaves unchanged the correct behavior of the NEO-I
kernel for one- and two-electron densities (at ασ=0) and also
for slowly-varying densities [at ασ=1+O(∇2)] and kills, at
the same time, the unwanted kernel in the tail of the electrondensity far away from the surface (at α
σ→∞ ). As one moves
from NEO-I to NEO-II, the parameter ˜cdoes not need to be
refitted.
By construction, for αvalues that are close to 1 (slowly-
varying densities) the coefficient 3 ασ−3α2
σ+α3
σentering
Eq. ( 11) deviates little from unity, so the range of our NEO-II
kernel is of the order of k−1
F. Hence, over this range of αvalues
the kernel correction to RPA is nearly local, as in the semilocalcorrection to RPA of Yan, Kurth, and Perdew [ 38].
In Fig. 1, we show a comparison between the coefficients
a
NEO−IandaNEO−IIdefining the kernels NEO-I and NEO-II,
for a jellium surface with the electron-density parameterr
s=6. By construction, these functions agree well in the bulk
and even at the surface, while in the vacuum side far fromthe surface one finds: a
NEO−I→0 and aNEO−II→∞ ,a s
expected. The quantity αwe are introducing here represents
a kinetic-energy dependent ingredient that plays a key rolein the construction of meta-GGA functionals [ 27,28,65–68]
165155-3RUZSINSZKY , CONSTANTIN, AND PITARKE PHYSICAL REVIEW B 94, 165155 (2016)
TABLE I. Correlation energy of the uniform electron gas (in
mHa), as obtained from Eq. ( 1) with the use of the NEO-I and NEO-III
kernels, for various values of the electron-density parameter rs.F o r
the uniform electron gas, the kernels NEO-I and NEO-II coincide. ThePW92 correlation energy [ 73] is given for comparison. The values
in parenthesis are relative deviations from the exact (PW92) values.
The last line reports the root mean square (RMS).
rs NEO-I NEO-III exact (PW92)
1 −65.12 ( −0.0895) −60.29 ( −0.0087) −59.77
2 −50.56 ( −0.1296) −44.08 (0.0152) −44.76
3 −42.92 ( −0.1619) −36.34 (0.0162) −36.94
4 −37.93 ( −0.1901) −31.76 (0.0035) −31.87
5 −34.33 ( −0.2165) −28.65 ( −0.0152) −28.22
6 −31.55 ( −0.2407) −26.37 ( −0.0369) −25.43
10 −24.60 ( −0.3247) −20.90 ( −0.1255) −18.57
100 −6.50 (−1.0376) −6.12 (−0.9185) −3.19
1000 −1.31 (−2.3590) −1.29 (−2.3077) −0.39
RMS 5.351 1.374
and is also relevant for a correct asymptotic description
of the electron density [ 69,70]. Note, for example, that
1/[1+α(r)2] is the electron-localization function often used
in the characterization of chemical bonds [ 71,72].
C. NEO-III kernel
One of the ingredients of the NEO-I kernel [the parameter
˜centering Eq. ( 9)] is constructed to fit the exact second-
order exchange contribution to the uniform-gas correlationenergy. Now we construct NEO-III by replacing the coefficienta
NEO−Iby the new coefficient
aNEO -III=/radicalBigg
˜c
1+brcs/parenleftbig
1−Z2σ/parenrightbig
kFσ, (13)
where the energy-optimization coefficient ˜centering Eq. ( 9)
has been replaced by the new electron-density dependentcoefficient ˜c/(1+br
c
s), with the parameters b=1.1 and
c=1.35 taken to fit the PW92 [ 73] correlation energy for
electron densities down to rs=1000.
We have not tried to construct the analog of Eq. ( 13)u s i n g
αinstead of Z.Zandαin NEO-I and NEO-II present different
physics in the density tail, while the replacement of ˜cby/radicalBig
˜c
1+brcsin NEO-III was designed to test the applicability of
the kernel for density regions which are different from high
densities.
In Table I, the uniform-gas correlation energy is given for
various values of rs, as obtained with the use of the NEO-I
(or NEO-II) and NEO-III kernels. NEO-III correlation energyshows reduced deviations for each r
scompared to NEO-I.
This fact indicates that NEO-III is performing better for theintegrated correlation energies of individual r
svalues. This is
not surprising since the parameters in NEO-III were obtainedby fitting to the uniform electron gas over a wide range ofdensities. Notice that the better correlation energies in NEO-IIIshow up in the wave-vector analysis of Fig. 2only in the
correlation energy, as an integrated area under the curve.-0.045-0.04-0.035-0.03-0.025-0.02-0.015-0.01-0.005 0
0 0.5 1 1.5 2 2.5 3εc,Q
Qrs=2
Exact
RPA
NEO-I
NEO-III
-0.07-0.06-0.05-0.04-0.03-0.02-0.01 0 0.01
0 0.2 0.4 0.6 0.8 1 1.2 1.4εc,Q
Qrs=5
Exact
RPA
NEO-I
NEO-III
FIG. 2. 3D wave-vector analysis of the correlation energy per
particle εunif
c,Q(¯n) of a uniform electron gas with rs=2a n d rs=5,
as obtained from Eq. ( 7) in the RPA and with the use of the
NEO-I and NEO-III kernels. The same quantity, as obtained from the
Perdew-Wang parametrization of the uniform-gas correlation-holedensity [ 74] (exact) is given for comparison. For the uniform electron
gas, the kernels NEO-I and NEO-II coincide.
IV . RESULTS AND DISCUSSION
Figure 2exhibits the Q-dependent correlation energy
per particle εunif
c,Q(¯n) (3D wave-vector analysis) of a uniform
electron gas with rs=2 andrs=5, as obtained from Eq. ( 7).
In the long-wavelength ( Q→0) limit, where the RPA is
exact, all calculations converge, as expected. At moderatevalues of Q, NEO-I agrees very well with the Perdew-Wang
parametrization [ 74] in a region where RPA starts to deviate
significantly. At the shortest wavelengths, all kernel-correctedcalculations improve considerably over the RPA, although theygiveQ-dependent correlation energies that are still below the
exact calculation.
Since both NEO-I and NEO-II kernels were fitted against
the second-order exchange energy, they both have the samephysics built in for the high-density limit. In Ref. [ 58] one of
the authors has shown that for r
sbetween 1 and 20, the fitted
parameter produces a distribution of errors compared to PW92between approximately 3 and 5 mHa. Small displacementbelow and above this fitted parameter delivers a balance ofsmall absolute errors and small distribution of error overa large range of densities. Therefore this particular fitting
165155-4KERNEL-CORRECTED RANDOM-PHASE APPROXIMATION . . . PHYSICAL REVIEW B 94, 165155 (2016)
-0.16-0.14-0.12-0.1-0.08-0.06-0.04-0.02 0
0 0.5 1 1.5 2εc,q
qISTLS
RPA
NEO-I
NEO-II
NEO-III
FIG. 3. 2D wave-vector analysis of the correlation energy per
particle εc,q[n] of a jellium slab of width a=2.23λFandrs=2.07,
as obtained from Eq. ( 5) in the RPA and with the use of the NEO-I,
NEO-II, and NEO-III kernels. ISTLS calculations (see Ref. [ 39]) are
given for comparison.
provides some flexibility for the kernel to be accurate in both
high-density and lower density regions as well. The balanceof good absolute errors and error distribution transfers tothe wave-vector analysis by tuning the agreement for smallto intermediate Qwithout changing ˜c. NEO-III is worse
than NEO-I for intermediate wave vectors, but its integratedcorrelation energies are better.
Figure 3shows the q-dependent correlation energy per
particle ε
c,q[n] (2D wave-vector analysis) of a jellium slab
of width a=2.23λF(λF=2π/kFis the Fermi wavelength)
andrs=2.07 (the electron-density parameter corresponding
to valence electrons in Al), as obtained from Eq. ( 5). Here
we compare our RPA and beyond-RPA calculations to theinhomogeneous Singwi-Tosi-Land-Sj ¨olander (ISTLS) calcu-
lations reported in Ref. [ 39]. As in the case of the uniform
electron gas, all calculations converge, as expected, in thelong-wavelength ( q→0) limit. At moderate values of q, both
NEO-I and NEO-II agree well with our reference calculation(ISTLS) in a region where RPA starts to deviate significantly.At the shortest wavelengths, all kernel-corrected calculationsare slightly below our reference calculation (ISTLS), as occursin the case of the uniform electron gas. The best results here(compared to the ISTLS reference) are obtained by using theNEO-I and NEO-II kernels.
In Fig. 4, we show (for r
s=2.07) our wave-vector analysis
of the jellium surface correlation energy
γc,q=(N/A )/braceleftbig
εc,q[n]−εunif
c,q(¯n)/bracerightbig
, (14)
where Nis the total number of electrons and Arepresents a
normalization area. The area under each curve represents thecorrelation surface energy σ
c, which we give in Table II.A l l
NEO kernels yield accurate jellium surface energies, whichare (i) close to the ISTLS calculation and (ii) within the errorbar of DMC calculations.
Figure 4shows that in the long-wavelength ( q→0) limit
all calculations coincide with the RPA calculation, which isexact in this limit. In the large- qlimit, where the RPA fails
badly, all kernel-corrected calculations agree with each other 0 100 200 300 400 500 600 700 800 900 1000 1100
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2γc,q
q/kFISTLS
RPA
NEO-I
NEO-II
NEO-III
FIG. 4. 2D wave-vector analysis γc,qof the correlation surface
energy of a jellium slab of width a=2.23λFandrs=2.07. The area
under each curve represents the corresponding correlation surface en-ergyσ
c. Units are erg /cm2(1 hartee /bohr2=1.557×106erg/cm2).
and with our reference calculation (ISTLS); this is an expected
result, since all calculations yielding accurate uniform-gascorrelation energies are expected to also yield an accurateγ
qin this region [ 36]. Differences among our kernel-corrected
calculations and between these calculations and our referencecalculation (ISTLS) arise at intermediate values of q.T h e
NEO-II kernel yields correlation energies γ
qthat are very
close to our reference calculation (ISTLS) for wave vectors uptoq≈0.4. This is an expected result, since this is the only
kernel that is free from an unrealistic long-ranged behaviorin the tail of the electron density into the vacuum side ofthe surface. At larger values of qthe NEO-II kernel yields
correlation energies that are slightly below the ISTLS result,thus leading to a total NEO-II surface energy that is slightlysmaller than the ISTLS surface energy.
We close this paper by looking at the position-dependent
xc energy per particle ε
xc([n],z), which for a many-electron
system that is invariant in two directions we define asfollows [ 59]
E
xc=A/integraldisplay
dzn(z)εxc([n],z), (15)
Excbeing the xc energy of Eq. ( 1). Equation ( 15) itself does
not define εxc([n],z) uniquely [ 75–77], but here we use the
choice made in Eq. ( 1).
Figure 5exhibits the correlation energy per particle
εc([n],z). Only the NEO-II kernel is found to capture both
the correct εc([n],z) in the bulk (which in the case of the RPA
TABLE II. NEO-I, NEO-II, and NEO-III correlation surface ener-
giesσcof a jellium surface with rs=2.07. LDA, PBEsol, ISTLS [ 39],
and DMC [ 40] correlation energies are given for comparison. NEO
and ISTLS calculations represent the surface energy of a semi-infinite
jellium, which has been obtained from finite-slab calculations by
following the extrapolation procedure described in Ref. [ 35]. Units
are erg /cm2(1 hartee /bohr2=1.557×106erg/cm2).
rs LDA PBEsol NEO-I NEO-II NEO-III ISTLS DMC
2.07 287 645 702 692 714 730 697 ±45
165155-5RUZSINSZKY , CONSTANTIN, AND PITARKE PHYSICAL REVIEW B 94, 165155 (2016)
-0.07-0.06-0.05-0.04-0.03-0.02-0.01 0 0.01
-6 -4 -2 0 2 4 6 8 10 12εc(z)
z (a.u.)RPA
NEO-I
NEO-II
NEO-III
FIG. 5. Correlation energy per particle ( εcversus z, for a jellium
slab of width a=2.23λFandrs=2.07). The surface is at z=0, the
bulk is at z< 0, and the vacuum is at z> 0.
is too negative) and the correct imagelike εc([n],z) far away
from the surface (where the RPA is exact [ 59] and NEO-I and
NEO-III are all wrong). This is an expected result, since theNEO-I and NEO-III kernels produce an unwanted long-rangedbehavior in the tail of the electron density due to the inabilityof the Zingredient entering Eq. ( 9) to distinguish between the
surface tail and one- or two-electron regions. The relevanceof the αparameter versus Zwas mentioned in the context of
orbital overlap in closed-shell species [ 27,29,78].V . CONCLUSIONS
We have constructed a nonlocal energy-optimized model
kernel [ 56,57] with various inhomogeneity parameters, which
we have tested for the jellium-surface problem. Our workreveals the role and significance of α, a dimensionless
deviation from the single orbital shape, as an ingredient forexponentially decaying surface densities.
A kernel-corrected RPA calculation of the xc jellium
surface energy was reported in Ref. [ 36]. In Ref. [ 36], the
xc kernel f
λ
xc(r,r/prime,ω) was taken to be (by assuming that the
electron density variation is small within the short range ofthe kernel) equal to the xc kernel of a uniform electrongas of density [ n(r)+n(r
/prime)]/2. Krotscheck and Kohn [ 13],
however, had argued that this local-density approximation forthe particle-hole interaction might be inadequate to calculatethe surface energy of simple metals. The present work bringsus to the conclusion that the use of an appropriate kernel (likeNEO-II), which does not only depend on the electron densityat (r+r
/prime)/2 but also on its gradient as well as the kinetic
energy density, does not change the jellium surface energysignificantly compared to the RPA value and leaves it close toour reference ISTLS and DMC calculations.
ACKNOWLEDGMENTS
A.R. acknowledges the support of the National Science
Foundation under Grant No.DMR-1553022. We thank Prof. J.P. Perdew for many stimulating and enjoyable discussions.
[1] J. P. Perdew, A. Ruzsinszky, J. Tao, V . N. Staroverov, G.
E. Scuseria, and G. I. Csonka, J. Chem. Phys. 123,062201
(2005 ).
[2] P. Hohenberg and W. Kohn, Phys. Rev. 136,B864 (1964 ).
[3] W. Kohn and L. J. Sham, Phys. Rev. 140,A1133 (1965 ).
[4] S. Ichimaru, Rev. Mod. Phys. 54,1017 (1982 ).
[5] C. Kittel, Introduction to Solid State Physics (Wiley, New York,
2005).
[6] N. D. Lang and W. Kohn, Phys. Rev. B 1,4555 (1970 ).
[7] N. D. Lang and W. Kohn, Phys. Rev. B 3,1215 (1971 ).
[8] N. D. Lang and W. Kohn, Phys. Rev. B 7,3541 (1973 ).
[9] J. P. Perdew and K. Schmidt, in Density Functional Theory and
its Application to Materials ,e d i t e db yV .V a nD o r e n ,C .V a n
Alsenoy, and P. Geerlings, AIP Conf. Proc. No. 577 (AIP, NewYork, 2001), p. 1.
[10] Z. Y . Zhang, D. C. Langreth, and J. P. Perdew, Phys. Rev. B 41,
5674 (1990 ).
[11] D. C. Langreth and M. J. Mehl, P h y s .R e v .L e t t . 47,446(1981 ).
[12] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 (1996 ).
[13] E. Krotscheck, W. Kohn, and G.-X. Qian, Phys. Rev. B 32,5693
(1985 ).
[14] X.-P. Li, R. J. Needs, R. M. Martin, and D. M. Ceperley, Phys.
Rev. B
45,6124 (1992 ).
[15] P. H. Acioli and D. M. Ceperley, P h y s .R e v .B 54,17199 (1996 ).
[16] J. P. Perdew, L. A. Constantin, E. Sagvolden, and K. Burke,
Phys. Rev. Lett. 97,223002 (2006 ).[17] F. Tran, J. Stelzl, and P. Blaha, J. Chem. Phys. 144,204120
(2016 ).
[18] R. Armiento and A. E. Mattsson, Phys. Rev. B 72,085108
(2005 ).
[19] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E.
Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev.
Lett.100,136406 (2008 ).
[20] Z. Wu and R. E. Cohen, Phys. Rev. B 73,235116 (2006 ).
[21] Y . Zhao and D. G. Truhlar, J. Chem. Phys. 128,184109 (2008 ).
[22] L. A. Constantin, A. Terentjevs, F. Della Sala, P. Cortona, and
E. Fabiano, Phys. Rev. B 93,045126 (2016 ).
[23] T. Van V oorhis and G. E. Scuseria, J. Chem. Phys. 109,400
(1998 ).
[24] J. Tao, J. P. Perdew, V . N. Staroverov, and G. E. Scuseria, Phys.
Rev. Lett. 91,146401 (2003 ).
[25] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin, and
J. Sun, Phys. Rev. Lett. 103,026403 (2009 ).
[26] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin, and
J. Sun, Phys. Rev. Lett. 106,179902(E) (2011 ).
[27] J. Sun, B. Xiao, and A. Ruzsinszky, J. Chem. Phys. 137,051101
(2012 ).
[28] J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115,
036402 (2015 ).
[29] Y . Zhao and D. G. Truhlar, J. Chem. Phys. 125,194101 (2006 ).
[30] J. Sun, B. Xiao, Y . Fang, R. Haunschild, P. Hao, A. Ruzsinszky,
G. I. Csonka, G. E. Scuseria, and J. P. Perdew, P h y s .R e v .L e t t .
111,106401 (2013 ).
165155-6KERNEL-CORRECTED RANDOM-PHASE APPROXIMATION . . . PHYSICAL REVIEW B 94, 165155 (2016)
[31] F. Furche and T. Van V oorhis, J. Chem. Phys. 122,164106
(2005 ).
[32] J. F. Dobson, J. Wang, B. P. Dinte, K. McLennan, and H. M. Le,
Int. J. Quantum Chem. 101,579 (2005 ).
[33] A. Heßelmann and A. G ¨orling, Phys. Rev. Lett. 106,093001
(2011 ).
[34] J. M. Pitarke and A. G. Eguiluz, Phys. Rev. B 57,6329 (1998 ).
[35] J. M. Pitarke and A. G. Eguiluz, P h y s .R e v .B 63,045116 (2001 ).
[36] J. M. Pitarke and J. P. Perdew, P h y s .R e v .B 67,045101 (2003 ).
[37] J. M. Pitarke, P h y s .R e v .B 70,087401 (2004 ).
[38] Z. Yan, J. P. Perdew, S. Kurth, C. Fiolhais, and L. Almeida,
Phys. Rev. B 61,2595 (2000 ).
[39] L. A. Constantin, J. M. Pitarke, J. F. Dobson, A. Garcia-Lekue,
and J. P. Perdew, Phys. Rev. Lett. 100,036401 (2008 ).
[40] B. Wood, N. D. M. Hine, W. M. C. Foulkes, and P. Garc ´ıa-
Gonz ´alez, Phys. Rev. B 76,035403 (2007 ).
[41] D. C. Langreth and J. P. Perdew, Sol. State Comm. 17,1425
(1975 ).
[42] E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52,997 (1984 ).
[43] D. C. Langreth and J. P. Perdew, P h y s .R e v .B 21,5469 (1980 ).
[ 4 4 ]M .L e i n ,E .K .U .G r o s s ,a n dJ .P .P e r d e w , P h y s .R e v .B 61,
13431 (2000 ).
[45] M. Corradini, R. Del Sole, G. Onida, and M. Palummo, Phys.
Rev. B 57,14569 (1998 ).
[46] L. A. Constantin and J. M. Pitarke, Phys. Rev. B 75,245127
(2007 ).
[47] C. F. Richardson and N. W. Ashcroft, Phys. Rev. B 50,8170
(1994 ).
[48] J. Paier, X. Ren, P. Rinke, G. E. Scuseria, A. Gr ¨u n e i s ,G .K r e s s e ,
and M. Scheffler, New J. Phys. 14,043002 (2012 ).
[49] J. Harl and G. Kresse, P h y s .R e v .L e t t . 103,056401 (2009 ).
[50] C. E. Patrick and K. S. Thygesen, J. Chem. Phys. 143,102802
(2015 ).
[51] S. Lebegue, J. Harl, T. Gould, J. G. ´Angy ´an, G. Kresse, and
J. F. Dobson, P h y s .R e v .L e t t . 105,196401 (2010 ).
[52] L. Schimka, J. Harl, A. Stroppa, A. Gr ¨uneis, M. Marsman, F.
Mittendorfer, and G. Kresse, Nat. Mater. 9,741 (2010 ).
[53] P. D. Mezei, G. I. Csonka, and A. Ruzsinszky, J. Chem. Theory
Comput. 11,3961 (2015 ).
[54] B. Grabowski, S. Wippermann, A. Glensk, T. Hickel, and J.
Neugebauer, P h y s .R e v .B 91,201103 (2015 ).
[55] H. Eshuis, J. Yarkony, and F. Furche, J. Chem. Phys. 132,234114
(2010 ).[56] J. Jung, P. Garcia-Gonzalez, J. F. Dobson, and R. W. Godby,
Phys. Rev. B 70,205107 (2004 ).
[57] J. F. Dobson and J. Wang, Phys. Rev. B 62,10038
(2000 ).
[58] J. E. Bates, S. Laricchia, and A. Ruzsinszky, Phys. Rev. B 93,
045119 (2016 ).
[59] L. A. Constantin and J. M. Pitarke, Phys. Rev. B 83,075116
(2011 ).
[60] C. F. v. Weizs ¨acker, Z. Physik 96,431 (1935 ).
[61] M. Petersilka, U. J. Gossmann, and E. K. U. Gross, Phys. Rev.
Lett.76,1212 (1996 ).
[62] U. von Barth and L. Hedin, J. Phys. C: Sol. State Phys. 5,1629
(1972 ).
[63] L. H. Thomas, in Mathematical Proceedings of the Cambridge
Philosophical Society , V ol. 23 (Cambridge University Press,
Cambridge, 1927), pp. 542–548.
[64] E. Fermi, Rend. Accad. Naz. Lincei 6, 602 (1927).
[65] A. Ruzsinszky, J. Sun, B. Xiao, and G. I. Csonka, J. Chem.
Theory Comput. 8,2078 (2012 ).
[66] J. Sun, J. P. Perdew, and A. Ruzsinszky, Proc. Natl. Acad. Sci.
112,685 (2015 ).
[67] J. Wellendorff, K. T. Lundgaard, K. W. Jacobsen, and T.
Bligaard, J. Chem. Phys. 140,144107 (2014 ).
[68] K. T. Lundgaard, J. Wellendorff, J. V oss, K. W. Jacobsen, and
T. Bligaard, P h y s .R e v .B 93,235162 (2016 ).
[69] F. Della Sala, E. Fabiano, and L. A. Constantin, P h y s .R e v .B
91,035126 (2015 ).
[70] L. A. Constantin, E. Fabiano, J. M. Pitarke, and F. Della Sala,
Phys. Rev. B 93,115127 (2016 ).
[71] B. Silvi and A. Savin, Nature (London) 371,683 (1994 ).
[72] A. Savin, R. Nesper, S. Wengert, and T. F. F ¨assler, Angew.
Chem., Int. Ed. Engl. 36,1808 (1997 ).
[73] J. P. Perdew and Y . Wang, P h y s .R e v .B 45,13244
(1992 ).
[74] J. P. Perdew and Y . Wang, Phys. Rev. B 46,12947 (1992 ).
[75] J. Tao, V . N. Staroverov, G. E. Scuseria, and J. P. Perdew, Phys.
Rev. A 77,012509 (2008 ).
[76] K. Burke, F. G. Cruz, and K.-C. Lam, J. Chem. Phys. 109,8161
(1998 ).
[77] K. Burke, F. G. Cruz, and K.-C. Lam, Int. J. Quantum Chem.
70,583 (1998 ).
[78] G. K. Madsen, L. Ferrighi, and B. Hammer, J. Phys. Chem. Lett.
1,515 (2009 ).
165155-7 |
PhysRevB.100.155421.pdf | PHYSICAL REVIEW B 100, 155421 (2019)
Double flat bands in kagome twisted bilayers
F. Crasto de Lima*and R. H. Miwa†
Instituto de Física, Universidade Federal de Uberlândia, Caixa Postale 593, 38400-902, Uberlândia, Minas Gerais, Brazil
E. Suárez Morell‡
Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile
(Received 17 July 2019; revised manuscript received 16 September 2019; published 21 October 2019)
We have studied how a generic bilayer kagome lattice behaves upon layer rotation. We employed a tight-
binding model with one orbital per site and found: (i) for low rotational angles and at low energies, the same flatbands structure, such as in twisted bilayer graphene; although, for a larger value of the magic angle. Moreover,(ii) at high energies due to the superstructure symmetry regions, we found the characteristic three-band dispersionof the kagome lattice. In the latter, its bandwidth decreases for lower angles confining them within a few meV .Therefore, we found, in the twisted kagome lattice, the coexistence of two sets of flat bands in different energiesand lying in different spatial regions of the bilayer system.
DOI: 10.1103/PhysRevB.100.155421
The unexpected discovery of superconductivity and corre-
lated insulating behavior in twisted bilayer graphene (TBG)[1] at a certain “magic” angle for which the band structures
present very flat bands near the Fermi level [ 2–5] has turned
TBG into a puzzle the community is trying to understand[6,7]. The temperature at which the material turns supercon-
ductor ( T
C) is low, but the fact that the electrons density
is a thousand times smaller than in other superconductingmaterials is a promising avenue to explore. Moreover, recentlyin a tetralayer structure composed of two-stacked AB bilayergraphene with a rotational angle between them, a supercon-ducting and correlated insulating behavior was also foundby two independent groups [ 8,9]. Additionally, in a trilayer
ABC structure over hexagonal boron nitride a Mott insulatorstate can be tuned by an external electric field [ 10]. In each
case, the correlated behavior was related with the flat bands ofthe system. Furthermore, flat bands have also been found inother non-carbon-based materials with interesting properties[11,12].
Another aspect to consider is the role that interlayer cou-
pling plays to tune the magic angle value. For instance,uniaxial pressure increases the coupling between layers, thus,augmenting the values of the magic angle [ 13,14]. This is
not a trivial factor; it has been suggested that, in this kindof system, with flat bands in their electronic structure, thecritical temperature depends linearly on an attractive electron-electron interaction and on the area of the flat band [ 15,16].
An increase in the value of the magic angle would increasethe size of the Brilloiun zone and extend the area of the flatbands.
Aside from the graphene honeycomb structure, thousands
of other two-dimensional (2D) structures have been proposed
*felipe.lima@ufu.br
†hiroki@ufu.br
‡eric.suarez@usm.cl[17,18]. The kagome lattice is a network composed with the
vertices and edges of the trihexagonal Archimedean tiling,
and it gained theoretical relevance as a realization of per-fect flat bands in its electronic structure if you consider asimple nearest-neighbor hopping Hamiltonian. Indeed, a sim-ple three-orbital tight-binding (TB) model of the monolayerkagome lattice shows a flat band at high energies plus a Dirac-like linear band touching at the Kpoints similar to graphene
bands [see the dashed lines in Fig. 1(b1) ]. Several proposals
of exciting new phenomena that could be found in this latticehave come out, e.g., spin liquid phases [ 19], the fractional
quantum Hall effect [ 20], and the quantum anomalous and
spin Hall effects [ 21,22].
In the realm of 2D materials, the kagome lattice shows up
in different metal-organic framework (MOF) systems. Suchsystems display different properties by designing combina-tions of metals and organic ligands [ 23]. Recently, it has
been shown that Fe
3Se2forms a kagome bilayer structure
[24] where the system is a soft ferromagnet with intrinsic
anomalous Hall conductivity and a Dirac cone gapped by30 meV as a consequence of the spin-orbit coupling andclose to the Fermi energy ∼−125 meV. The TB fitting of the
experimental results shows an interlayer coupling of ∼0.3t,
almost three times the value in TBG.
In principle, in the MOFs, the distance between metallic
atoms within the layer might be larger than the interlayerdistance, which would result in relatively larger values of theinterlayer effective hopping. This property, together with arelative rotation between layers, might introduce new ingredi-ents to the formation of flat bands in kagome twisted bilayers(KTBs).
In this paper, we explore the consequences that twisting
a bilayer kagome lattice have to its electronic properties.We employ a TB model with one orbital per metallic atomand only three parameters, which has been shown to captureprevious density functional theory studies of kagome bilayersystems [ 25]. The intralayer hopping tand interlayer hopping
2469-9950/2019/100(15)/155421(4) 155421-1 ©2019 American Physical SocietyCRASTO DE LIMA, MIWA, AND SUÁREZ MORELL PHYSICAL REVIEW B 100, 155421 (2019)
Γ Γ Γ Γ Γ Γ
FIG. 1. Different stackings of kagome bilayer (a1) C6,( a 2 ) C3,
and (a3) C2symmetry; (b1)–(b3) the band structures for the corre-
sponding stacking (red lines) C6,C3,a n d C2, respectively, the dashed
lines are the monolayer kagome band structure; (c) identification of
symmetry regions of a 3 .89◦twisted kagome bilayer lattice; the blue
line in the left panel shows the unit cell.
tzstrength decays accordingly to the distance between the sites
∝exp[−β(d−di)] with dias the nearest-neighbor distance
and interlayer distance in each case. Therefore, the threeparameters t,t
z, and decaying factor βcontrol our model.
The calculations are performed for a generic kagome lattice;nonetheless, we used a t
z/tratio between inter- and intralayer
nearest hopping of 0.3 and β=20/di, similar to the value
obtained in Ref. [ 24] to fit experimental results. Succinctly, we
found KTB exhibits a larger value of the magic angle (2 .28◦)
in comparison with its counterpart TBG. The low-energyelectrons at this angle are spatially localized in the AA regionsandC
6symmetry sites of the moiré pattern. Concomitantly,
the superlattice structure composed by C2symmetry sites,
which conforms to a (new) C2-kagome lattice, lead to a new
set of kagome bands at higher energies. Those findings showthat twisted kagome lattices are characterized by two sets offlat bands lying at different energy and spatial regions.
The electronic structure of the bilayer kagome lattice dras-
tically depends on the stacking order. We study first three pos-sible stacking orders of the bilayer structure. We label themaccording to the symmetry of the structure, which will helpus to explain below the properties of different regions of thetwisted structure. In Figs. 1(a1) –1(a3) , we show the stacked
AA (eclipsed) bilayer with C
6symmetry, the AB stacking with
C3symmetry, and another stacking obtained by shifting one
of the layers half a lattice vector along the horizontal axis;this stacking has a lower C
2symmetry. We have calculated
the electronic structure of these three structures using our TBmodel with distance-dependent hoppings within and betweenlayers.
The electronic structure of a bilayer kagome lattice in the
AA stacking, Fig. 1(b1) , is similar to the structure of bilayer
stacked AA graphene, and we see a split of the monolayerdispersion (dashed lines) forming two identical sets of bands(red lines) shifted in energy. In fact, such behavior is observedΓ Γ Γ Γ
Γ Γ Γ Γ
FIG. 2. Twisted kagome band structure close to Dirac bands for
(a) 21.78◦,( b )6.01◦,( c )3.14◦,a n d( d )2 .28◦. The bands close to the
Dirac dispersion are highlighted in red, whereas blue dashed linesare the monolayer dispersion.
in the bilayer MOF (NiC 4S4)3with AA stackings [ 26]. The
AB stacking with C3symmetry, Fig. 1(b2) ,s h o w s ,a tl o w
energy, two sets of bonding-antibonding parabolic bands, sim-ilar to Bernal bilayer graphene. In the C
2symmetry stacking,
Fig. 1(b3) , the degeneracy is broken at the Kpoint with no
electron-hole symmetry and linear crossings of the bands intheK-/Gamma1direction.
We can build a rotated commensurate unit cell from
two stacked AA layers following the same procedure as inTBG [ 27]; the rotation axis passes through the center of the
hexagon. We obtain a superlattice unit cell where we canidentify three regions with the symmetries C
6,C3, and C2.O n
the right side of Fig. 1(c), we show a schematic of how these
regions are distributed over the KTB cell. Below, we associatethe low-energy states with their spatial localizations.
We study now how the band structure changes when we
rotate one of the kagome layers. We will concentrate first onthe Dirac-like bands and then on the kagome flat band. Weshow, in Fig. 2, the band structure for four rotational angles. A
gap is clearly visible in the 21 .78
◦structure, for lower angles,
the gap disappears, and the fermions velocity is renormalizeduntil the bands are flat, such as in TBG. It is worth noting thatthe symmetry group is preserved upon twisting, and for lowertwist angles, the energy bands along the M-K-/Gamma1directions
are nearly degenerated, whereas along the /Gamma1-Mdirection, the
degeneracy is visibly lifted.
We can further understand the evolution of the Dirac bands
by looking at the projection in the band structures of the dif-ferent symmetry regions, Fig. 3(a), and the spatial localization
of the Dirac states, Fig. 3(b). For the high twisted angle,
Fig. 3(a1) , the Dirac bands comprise a hybridization among
theC
6,C3, and C2regions with most contributions coming
from the C6symmetry region. With the lowering of the angle,
the contribution of the C3andC2regions diminish where for
2.28◦[Fig. 3(a4) ], it has vanished. In that sense, a simplified
approach tells us that as the twisting angle is lowered and the
C6regions become apart from each other [Figs. 3(b1) –3(b4) ],
its hybridization diminishes forming the almost flat band in
155421-2DOUBLE FLAT BANDS IN KAGOME TWISTED BILAYERS PHYSICAL REVIEW B 100, 155421 (2019)
ΓΓ
ΓΓ ΓΓΓΓ
FIG. 3. Projected band structure in the symmetric regions for
twisted angles of (a1) 21 .78◦,( a 2 )6 .01◦,( a 3 )3 .14◦, and (a4) 2 .28◦.
Real-space distribution of the Dirac states for twisted angles of (b1)
21.78◦, (b2) 6 .01◦, (b3) 3 .14◦, and (b4) 2 .28◦.
the previous Dirac point, which is associated with the presence
of van Hove singularities (VHSs). In Fig. 3(a), we show the
angular dependence of the VHS [ 28] taking place along the
/Gamma1-Mpoint. On the other hand, increasing the angle brings
theC6sites closer where its interaction with each other leads
to dispersive bands. Indeed, the role the long-range moirépotential plays on the electronic properties is, currently, anarea of intense research [ 15,29–33].
It is interestingly to note that, in contrast with the trian-
gular lattice composed by the C
6symmetry sites, we found
that the C2sites form a new kagome lattice embedded in
the twisted system, Fig. 1(c). Those C2sites promote the
emergence of a new set of kagome bands above the Fermilevel, namely, at ∼1.5t[shaded region in Fig. 1(b1) ] where
the energy dispersions of those kagome bands are ruled bythe interlayer interactions of the intrinsic flat bands of eachkagome monolayer. In Fig. 4, we show the band structures
and the orbital localization of those new kagome bands asa function of the twist angle. The evolution with the angleshows that the energy width of these three bands decreaseswith the angle, note also that the high-energy band becomesless dispersive, strengthening the localization on the C
2sites
as the angle gets lower. Thus, similar with what we found fortheC
6sites, the decrease in the bandwidth and the flatness
of these high-energy bands for low angles is related with areduction in the effective hopping between the C
2regions as
the distance between them increases.
Finally, by tracing the flatness of the twisted Dirac bands
[Fig. 3], we will discuss how the interlayer coupling affects
the value of the magic angle. Increasing the interlayer hop-ping will increase the angle for which the flat bands appear.Experimentally, that can be achieved by exerting a uniaxialperpendicular pressure over the structure [ 14]. However, if we
normalize the interlayer hopping in terms of the intralayer
Γ Γ Γ Γ
Γ Γ Γ Γ
FIG. 4. Twisted kagome symmetric site projected band struc-
tures close to flat bands [shaded regions in Fig. 1(b1) –1(b3) ]f o r
(a1) 21 .78◦,( a 2 )1 3 .17◦,( a 3 )9 .43◦, and (a4) 7 .34◦. Real-space
distribution of the states shown in (a) for (b1) 21 .78◦, (b2) 13 .17◦,
(b3) 9.43◦, and (b4) 7 .34◦.
hopping, the sequence for the magic angle will follow the
same pattern. It is possible to define a dimensionless pa-rameter [ 6]α=κt
z/tsin(θ/2), where κ=2√
3/4πfor the
kagome lattice, tzandtare the inter- and intralayer nearest-
neighbor hoppings, and θis the rotational angle. We plot how
the bandwidth at the Mpoint depends on 1 /αfor different
angles and obtain a sequence for the first four magic angleswith basically the same pattern for all twisted structures (seeFig. 5). We are considering the width at the Mpoint, which
is a good indicator of the flatness of low-energy bands as itbetter describes its VHS positions [ 34,35].
With the help of the αparameter, it is clear that to increase
the value of the magic angle, we need larger ratios between
α×
FIG. 5. Bandwidth evolution as a function of the 1 /αparameter
for different angles.
155421-3CRASTO DE LIMA, MIWA, AND SUÁREZ MORELL PHYSICAL REVIEW B 100, 155421 (2019)
the inter- ( tz) and intra- ( t) layer hoppings. We have employed
throughout this article a relation between tz/t’s of 0.3, and it
gave us a magic angle of 2 .28◦, but it is straightforward to
obtain that, for tz/t=1, the magic angle will be around 8◦.
Moreover, if we assume a linear dependence of the TCon
the size of the flat bands as proposed in Ref. [ 15] with the
same coupling strength ( λ) in TBG and in twisted Fe 3Se2,a
simple calculation relating the size of the two Brillouin zonesresults in an increase of 4.7 in the value of the T
C, and going a
little bit further, if we are able to find a material with a magicangle around 8
◦, we will get an astonishing increase of around
50 in the critical temperature.
In conclusion, we have studied a twisted bilayer kagome
lattice which can be obtained in different metal-organic frame-works and found that two sets of flat bands are generatedin their electronic spectra. The origin of the low-energy flatbands is the same as in twisted bilayer graphene with acharacteristic triangular spatial localization; the other set of
flat bands arises at the same energy of the monolayer kagomeflat band, and it has the typical three-band dispersion ofthe kagome lattice; these states are localized in a differentregion of the superlattice unit cell, and its origin is relatedwith an emergent kagome-like lattice in the generated moirésuperstructure. We showed also that the value of the magicangle increases in this kind of structure due to a largerinterlayer /intralayer hopping ratio and this eventually might
result in an structure where the insulating correlated statesappear at higher T
C.
The authors acknowledge financial support from the
Brazilian agencies CNPq, FAPEMIG, the CENAPAD-SP,and LNCC-ScafMat for computer time. E.S.M. acknowl-edges financial support from FONDECYT Regular Grant No.1170921 (Chile).
[1] Y . Cao, V . Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y .
Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E.
Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Nature (London)
556,80(2018 ).
[2] E. Suárez Morell, J. D. Correa, P. Vargas, M. Pacheco, and Z.
Barticevic, Phys. Rev. B 82,121407(R) (2010 ).
[3] R. Bistritzer and A. H. Macdonald, Proc. Natl. Acad. Sci. USA
108,12233 (2011 ).
[4] G. Li, A. Luican, J. M. B. Lopes dos Santos, A. H. Castro Neto,
A. Reina, J. Kong, and E. Y . Andrei, Nat. Phys. 6,109(2009 ).
[5] G. Trambly de Laissardière, D. Mayou, and L. Magaud, Nano
Lett. 10,804(2010 ).
[6] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Phys. Rev.
Lett. 122,106405 (2019 ).
[7] J. González and T. Stauber, P h y s .R e v .L e t t . 122,026801
(2019 ).
[8] X. Liu, Z. Hao, E. Khalaf, J. Yeon Lee, K. Watanabe, T.
Taniguchi, A. Vishwanath, and P. Kim, arXiv:1903.08130 .
[9] Y . Cao, D. Rodan-Legrain, O. Rubies-Bigordà, J. M.
Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero,arXiv:1903.08596 .
[10] G. Chen, L. Jiang, S. Wu, B. Lyu, H. Li, B. L. Chittari, K.
Watanabe, T. Taniguchi, Z. Shi, J. Jung, Y . Zhang, and F. Wang,Nat. Phys. 15,237(2019 ).
[11] F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald,
P h y s .R e v .L e t t . 122,086402 (2019 ).
[12] D. Ospina, C. Duque, J. Correa, and E. S. Morell, Superlattices
Microstruct. 97,562(2016 ).
[13] S. Carr, S. Fang, P. Jarillo-Herrero, and E. Kaxiras, Phys. Rev.
B98,085144 (
2018 ).
[14] M. Yankowitz, J. Jung, E. Laksono, N. Leconte, B. L. Chittari,
K. Watanabe, T. Taniguchi, S. Adam, D. Graf, and C. R. Dean,Nature (London) 557,404(2018 ).
[15] T. J. Peltonen, R. Ojajärvi, and T. T. Heikkilä, P h y s .R e v .B 98,
220504(R) (2018 ).
[16] T. T. Heikkila, N. B. Kopnin, and G. E. V olovik, JETP Lett. 94,
233(2011 ).
[17] K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro
Neto, Science 353,aac9439 (2016 ).
[18] S. Haastrup, M. Strange, M. Pandey, T. Deilmann, P. S.
Schmidt, N. F. Hinsche, M. N. Gjerding, D. Torelli, P. M.Larsen, A. C. Riis-Jensen, J. Gath, K. W. Jacobsen, J. J.
Mortensen, T. Olsen, and K. S. Thygesen, 2D Mater. 5,042002
(2018 ).
[19] T. H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-
Rivera, C. Broholm, and Y . S. Lee, Nature (London) 492,406
(2012 ).
[20] E. Tang, J.-W. Mei, and X.-G. Wen, Phys. Rev. Lett. 106,
236802 (2011 ).
[21] Z. F. Wang, Z. Liu, and F. Liu, P h y s .R e v .L e t t . 110,196801
(2013 ).
[22] B. Zhao, J. Zhang, W. Feng, Y . Yao, and Z. Yang, P h y s .R e v .B
90,201403(R) (2014 ).
[23] D. Rodríguez-San-Miguel, P. Amo-Ochoa, and F. Zamora,
Chem. Commun. 52,4113 (2016 ).
[24] M. Kang, D. C. Bell, R. Comin, A. Bostwick, F. von Cube, J.
Liu, L. Ye, J. G. Checkelsky, L. Fu, C. Jozwiak, E. Rotenberg,C. R. Wicker, and T. Suzuki, Nature (London) 555,638(2018 ).
[25] F. Crasto de Lima, G. J. Ferreira, and R. H. Miwa, J. Chem.
Phys. 150,
234701 (2019 ).
[26] F. C. de Lima, G. J. Ferreira, and R. H. Miwa, Phys. Rev. B 96,
115426 (2017 ).
[27] E. Suárez Morell, P. Vargas, L. Chico, and L. Brey, Phys. Rev.
B84,195421 (2011 ).
[28] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro
Neto, Phys. Rev. Lett. 99,256802 (2007 ).
[29] B. Lian, Z. Wang, and B. A. Bernevig, P h y s .R e v .L e t t . 122,
257002 (2019 ).
[30] L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, P h y s .R e v .B
98,085435 (2018 ).
[31] B. Roy and V . Juri ˇci´c,P h y s .R e v .B 99,121407(R) (2019 ).
[32] F. Wu, A. H. MacDonald, and I. Martin, P h y s .R e v .L e t t . 121,
257001 (2018 ).
[33] L. Rademaker and P. Mellado, P h y s .R e v .B 98,235158 (2018 ).
[34] Y . Choi, J. Kemmer, Y . Peng, A. Thomson, H. Arora, R. Polski,
Y . Zhang, H. Ren, J. Alicea, G. Refael, F. von Oppen, K.Watanabe, T. Taniguchi, and S. Nadj-Perge, Nat. Phys. (2019 ),
doi:10.1038/s41567-019-0606-5 .
[35] A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M.
Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C.Dean, A. Rubio, and A. N. Pasupathy, Nature (London) 572,95
(2019 ).
155421-4 |
PhysRevB.97.075432.pdf | PHYSICAL REVIEW B 97, 075432 (2018)
Third-order optical conductivity of an electron fluid
Zhiyuan Sun,1D. N. Basov,1,2and M. M. Fogler1
1Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
2Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027
(Received 23 November 2017; published 20 February 2018)
We derive the nonlinear optical conductivity of an isotropic electron fluid at frequencies below the interparticle
collision rate. In this regime, governed by hydrodynamics, the conductivity acquires a universal form at anytemperature, chemical potential, and spatial dimension. We show that the nonlinear response of the fluid toa uniform field is dominated by the third-order conductivity tensor σ
(3)whose magnitude and temperature
dependence differ qualitatively from those in the conventional kinetic regime of higher frequencies. We obtainexplicit formulas for σ
(3)for Dirac materials such as graphene and Weyl semimetals. We make predictions for the
third-harmonic generation, renormalization of the collective-mode spectrum, and the third-order circular magneticbirefringence experiments.
DOI: 10.1103/PhysRevB.97.075432
I. INTRODUCTION
In typical metals and semiconductors, electrons experi-
ence frequent collisions with impurities and phonons. Thecombined rate /Gamma1
d=/Gamma1dis+/Gamma1phof these collisions far ex-
ceeds the rate /Gamma1eeof electron-electron scattering. However,
in several pure materials, the opposite case /Gamma1d/lessmuch/Gamma1eehas
recently shown to be possible in a range of temperatures. Undersuch conditions [ 1,2] electrons behave as a fluid that obeys
hydrodynamic equations [ 3]. Evidence for the hydrodynamic
behavior has been obtained from dc transport experiments withtwo-dimensional (2D) electron gases in GaAs [ 4], graphene
[5–7], and a quasi-2D metal PdCoO
2[8]. These discoveries
stimulated many theoretical studies [ 9–21]. The conceptual
simplicity of hydrodynamics arises from dealing with only afew degrees of freedom: the local temperature T(r,t), chemical
potential μ(r,t), and the flow velocity u(r,t). The complicated
many-body collisions need not be considered explicitly. In ourprevious paper [ 20], we used this hydrodynamic formalism
to calculate the electrodynamic response of an electron fluid,in particular, its linear and second-order nonlinear opticalconductivities. The magnitude and functional form of thesequantities in the hydrodynamic regime of frequencies ω/lessmuch/Gamma1
ee
were shown to differ qualitatively from their counterparts in the
conventional kinetic regime ω/greatermuch/Gamma1ee. Here we continue this
line of investigation by addressing the third-order nonlinearity,which controls, e.g., the third-harmonic generation (Fig. 1), the
Kerr effect, and four-wave mixing.
One reason the third-order nonlinearity warrants attention
is dictated by the symmetry. In general, the electrodynamicresponse of a conductor is characterized by the tensors σ
(n)
describing the components of the induced current proportional
to the nth power of the electric field. (The definition of these
tensors is given in Sec. II). Unless the field is very strong,
the nonlinear response of a material lacking the inversionsymmetry is dominated by the second-order conductivity σ
(2).
However, in centrosymmetric systems, such as graphene, σ(2)
must vanish if the electric field is uniform, in which case the
third-order conductivity σ(3)becomes more important.As we show below in this paper, the derivation of the
nonlinear conductivities is straightforward within a certainmodel that we call the Dirac fluid. This model is simpleyet flexible enough to describe several types of solid-state
materials. The model assumes that the quasiparticles of the
system behave as Dirac fermions with the energy-momentumdispersion ε
2(p)=m2v4+p2v2. The massless case m=0
corresponds to electrons in graphene; the massive case m> 0
is a reasonable approximation for narrow-gap semiconduc-tors. Neglecting fermion-fermion interactions, one can readilycompute the equilibrium thermodynamic parameters of thissystem [ 9,10,13,17,20], such as the pressure P=P(μ,T) and
the energy density n
E=nE(μ,T). The crucial simplification
of the Dirac model is that the energy-momentum tensor ofthe moving fluid can be derived from that of the static oneby a Lorentz transformation with vin lieu of the speed of
lightc. (In the noninteracting case, the moving fluid is defined
as the Fermi distribution of quasiparticles with the Doppler-
shifted energies ε(p)−pu.) The Lorentz invariance ensures
that the hydrodynamic equations of a Dirac fluid have a simple“relativistic” form [ 3,9,10,13,14,17,20]. Precisely because v/negationslash=
c, the solid-state systems with real Coulomb interactions are
not truly Lorentz invariant. However, the Dirac fluid shouldbe a reasonable approximation if the Coulomb interactionsare not too strong, so that Pandn
Eare dominated by the
kinetic energy. Besides graphene, examples of such Diracfluids may include the surface states of topological insulatorsand three-dimensional Dirac/Weyl semimetals.
Note that the hydrodynamic regime probed by recent dc
transport experiments [ 5–7] is less than 1 THz wide. If one
wants to expand it toward higher ω, it is necessary to increase
/Gamma1
ee, which can be done by raising electron temperature T
(Fig. 2). This must be done without heavily increasing the
electron-phonon scattering rate /Gamma1ph, which is also temperature-
dependent. One possible solution [ 20] is to work with (steady
or transient) states where electrons are “hot” but lattice stays“cold.” Such nonequilibrium states can be created by opticalpumping or electric-current heating.
2469-9950/2018/97(7)/075432(9) 075432-1 ©2018 American Physical SocietyZHIYUAN SUN, D. N. BASOV , AND M. M. FOGLER PHYSICAL REVIEW B 97, 075432 (2018)
FIG. 1. Illustration of the third-order optical nonlinearity in
graphene.
At frequencies ω>/Gamma1 eewhere the hydrodynamic theory
fails, the response of the system is better described bymore conventional approaches, e.g., the Boltzmann kineticequation neglecting electron-electron collisions. Among theDirac materials, graphene has been the most common targetof such calculations. Nonlinear conductivity of graphene hasbeen addressed in many theoretical studies [ 20,22–33]. As
discussed in our previous paper [ 20], the differences between
the hydrodynamic and kinetic regimes (Fig. 2) become con-
spicuous at temperatures Texceeding the chemical potential
μ, where graphene contains two types of carriers, electrons,
and thermally excited holes. In the kinetic regime, electronsand holes tend to move in opposite directions when driven bythe electric field. Their contributions to the electric current addup. In the hydrodynamic regime, due to frequent interparticlecollisions, all the carriers tend to move together. Hence, theelectron and hole currents partially cancel. As a consequence,there is an increased effective mass per unit charge, resultingin reduced linear and second-order nonlinear conductivities[20]. In this paper, we show that the third-order electrodynamic
response also exhibits distinct behaviors in the two regimes, inaccord with this physical picture.
The remainder of the paper is organized is follows.
Section IIgives a summary of our main results such as the
analytical formula for the third-order nonlinear conductivity ofan isotropic Dirac fluid. This formula is simple and universal.
FIG. 2. Schematic: Kinetic and hydrodynamic domains in the
frequency-temperature diagram of graphene for the carrier density
n=1012cm−2, corresponding to the zero-temperature chemical po-
tential μ(n,0)=0.12 eV. The (upper) dashed line separating the two
regimes is ω=/Gamma1ee(n,T). The lower dashed line is the momentum
relaxation rate due to electron scattering by acoustic and A/prime
1zone-
boundary phonons.It is valid for any mass m, chemical potential μ, temperature T,
and space dimension dif momentum nonconserving processes
can be neglected, /Gamma1d→0. We also discuss the general form
of the higher nonlinear conductivities tensors σ(n)of odd order
n. In Sec. III, we introduce the relativistic hydrodynamic
equations and apply them to the massless case, such asgraphene. In Sec. IV, we give the derivation of σ
(3), including
the case of an external applied magnetic field. In Sec. V,w e
apply our results for σ(3)to computing the third harmonic
generation. In Sec. VI, we discuss the Kerr effect and its
influence on the hydrodynamic collective modes of the fluid. InSec. VII, we compute the magnetic-field-induced third-order
circular birefringence. The concluding remarks are given inSec. VIII. Appendix Aprovides a summary of the analytical
expressions for the thermodynamic quantities of a masslessDirac fluid. Appendix Boutlines the derivation of σ
(3)for a
more realistic case of a finite scattering rate /Gamma1d.
II. MAIN RESULTS
TheNth-order nonlinear ac conductivity σ(N)is defined as
a rank (1 ,N) tensor, which maps electric fields to the Nth order
electrical current:
j(N)
i(q,ω)
=/summationdisplay
ν1...νN/summationdisplay
(q1,ω1),(q2,ω2)...(qN,ωN)δ/parenleftBiggN/summationdisplay
a(qa,ωa)−(q,ω)/parenrightBigg
×σ(N)
iν1...νN(q1,ω1,q2,ω2...,qN,ωN)
×Eν1(q1,ω1)Eν2(q2,ω2)...EνN(qN,ωN). (1)
The even, e.g., second-order conductivities vanish at zero
momentum in centrosymmetric systems. We therefore focuson odd-order (e.g., N=3) conductivities and disregard O(q
2)
nonlocal corrections.
For a general d-dimensional charged ideal fluid with O(d)
(rotation and reflection) symmetry, the Nth-order nonlinear
optical conductivity is found to be
σ(N)
iν1ν2...νN=iD(N)
h
ω1ω2...ω N/Delta1iν1ν2...νN, (2)
where
/Delta1iν1ν2...νN=δiν1δν2ν3...δνN−1νN+permutations (3)
is the totally symmetric rank N+1 tensor, which is the sum
of the N!! isotropic tensors. Note that O(d) symmetry only
requires that σ(N)is a linear combination of the isotropic
tensors. As we will show later, due to the additional conditionof thermal equilibrium in the hydrodynamic regime, σ
(N)can
only be proportional to the totally symmetric rank N+1 tensor
/Delta1iν1ν2...νN. The hydrodynamic Nth-order optical weight D(N)
h
should be understood as a thermodynamic quantity, which is
generally unknown.
Applied to the Dirac fluid, which has (quasi) Lorentz
symmetry, the linear optical conductivity is recovered asσ
ij=iDh/π
ωδijwhere the hydrodynamic Drude weight is Dh=
πne2/m∗(see, e.g., Supplemental Material of Ref. [ 20]). The
most important result of this paper is that the third-order
075432-2THIRD-ORDER OPTICAL CONDUCTIVITY OF AN … PHYSICAL REVIEW B 97, 075432 (2018)
nonlinear optical conductivity is given by
σ(3)
ilmn=iD(3)
h
ω1ω2ω3/Delta1ilmn
=iD(3)
h
ω1ω2ω3(δilδmn+δimδln+δinδlm). (4)
Here the third-order optical weight,
D(3)
h=1−Cise
3!e4n
m∗3v2=1−Cise
3!W
ρ4v4/parenleftbiggDh
π/parenrightbigg4
,(5)
is expressed in terms of thermodynamic quantities and the
asymptotic velocity v. These quantities are defined in Sec. III
below.
III. HYDRODYNAMICS
A. Hydrodynamic equations
The hydrodynamic equations for an ideal relativistic
charged fluid are [ 3]
∂μTμν=JμFνμ,∂μJμ=0, (6)
where Tνμis the energy-momentum tensor, Fνμis the elec-
tromagnetic field tensor, and
Jμ=(ρv,j),j=(jx,jy,jz), (7)
is the four-current and its spatial part, respectively. By ideal we
mean a fluid with vanishing viscosity and thermal conductivity.Although a “covariant” notation is implemented, Eqs. ( 6)
hold even for systems without Lorentz symmetry because theconservation of the stress tensor requires only the translationalsymmetry. The form of stress tensor for a general interactingfluid is unknown; however, for Lorentz-invariant systems, i.e.,Dirac fluids, the stress tensor is related to thermodynamicquantities [ 3]:
T
μν=Wuμuν−Pgμν. (8)
The four-current is related to the proper charge den-
sity through Jμ=ρ0uμ=ρ0γ(v,u)=ρ(v,u) where γ=
1//radicalbig
1−u2/v2. In solid-state systems, the vis the asymp-
totic velocity such that electrons have a Dirac-like energy-momentum dispersion ε
2
p=(pv)2+(mv2)2. The thermody-
namic quantities, the proper charge density ρ0=en0,t h e
enthalpy density W, and the pressure Pare all defined in the
fictitious proper frame moving with the local liquid. Note thatn≡ρ/e is defined as the effective charge-carrier density and
is, in general, not the same as particle density. For example,in graphene at high temperature, there are both electronsand holes, and nwill be the number of electrons minus the
number of holes. We define the hydrodynamic effective massasm
∗≡W/(nv2). And we define
Cise=n
W/parenleftbigg∂W
∂n/parenrightbigg
ise−1=n
W/parenleftbigg∂P
∂n/parenrightbigg
ise=1
m∗v2/parenleftbigg∂P
∂n/parenrightbigg
ise
(9)
as the dimensionless bulk isentropic modulus of the electron
fluid [ 20]. For example, it has the value 1 /dfor massless Dirac
particles in space dimension d. Out of the three thermodynamic
quantities W,P , andρ0, only two are independent. Thus theindependent variables are any two thermodynamic quantities
and the local flow velocity u. This set of Eqs. ( 6)i sc l o s e d .
Alternatively, these hydrodynamic equations can be derived
from the Boltzmann kinetic equation with the interparticlecollision integral but neglecting the many-body interactioncorrection to the thermodynamic quantities, as shown inRef. [ 13] and also the covariant version in Appendix C.
From Eq. ( 8), the first part of Eqs. ( 6) could be written in
another form:
Wu
μ∂μuν−∂νP+uνuμ∂μP=JμFνμ. (10)
Separating the time and spatial components in a proper way,
Eqs. ( 6) have another form:
(∂t+uk∂k)ui=1
γ2W/parenleftbigg
−∂iP−ui∂tP+ρEi
+ρ/epsilon1iklukv
cBl−uij·E/parenrightbigg
, (11)
∂t(nE)+∇(γ2Wu)=j·E, (12)
∂tρ+∇· j=0, (13)
where nE=γ2W−Pis the energy density of the electrons
(relative to zero doping and temperature case), and nE0=W−
Pis the same quantity but in the proper frame. Equation ( 11)
is the relativistic version of the Euler equation, Eq. ( 12)i s
the conservation of the energy current, and Eq. ( 13)i st h e
conservation of the charge current. Terms due to viscosity anddissipative thermal conductivity are neglected because theyaffect the conductivities only through O(q
2) terms. To simplify
the notations, the asymptotic velocity has been taken to bev=1 except for the Lorentz force term due to the magnetic
field B. Note that the magnetic field is related to the electric
one by ∇× E=−c
−1∂tB. Since we neglect finite- qeffects in
this paper, we must set ∇× E=0, so that the magnetic field
is considered time independent.
Starting from Eqs. ( 11)–(13), we can compute the linear and,
in principle, any higher-order nonlinear optical conductivitiesσ
(N)by expanding all the dynamic variables in powers of the
electric field E. This procedure and its results are presented in
the following sections.
B. Hydrodynamic regime of graphene
Hydrodynamic regime is not a phase of matter but a domain
of frequency-momentum diagram (see, e.g., Fig. 1 of Ref. [ 20])
where the hydrodynamic equations work well as an effectivetheory. This regime is defined by inequalities /Gamma1
d,ω/lessmuch/Gamma1ee
andq/lessmuchl−1
ee, which can be satisfied in some pure solid-state
systems. The electron-electron collision rate /Gamma1ee(n,T) that
sets the upper bound on the hydrodynamic regime dependson temperature and doping level of the electron system. Forexample, in graphene, /Gamma1
eescales as ∼ln(2μ/T )(T2/μ)a tl o w
Tand as ∼α2TatT/greatermuchμwithα∼1. For a rough estimate,
we connect these two formulas by a naive interpolation withthe relative weights μ/(T+μ) andT/(T+μ), respectively.
The corresponding boundary of the hydrodynamic domain(blue region) is shown by the upper dashed line in Fig. 2.T h e
other important scattering rate shown in the same Figure is
075432-3ZHIYUAN SUN, D. N. BASOV , AND M. M. FOGLER PHYSICAL REVIEW B 97, 075432 (2018)
/Gamma1d. For ultraclean graphene encapsulated in hexagonal boron
nitride (hBN), the major contribution to /Gamma1dis the electron-
phonon scattering /Gamma1ph. The electron-phonon scattering sup-
presses the hydrodynamic behavior if /Gamma1ph>/Gamma1 ee. However,
our theoretical estimation of /Gamma1phand/Gamma1eeindicates that the
hydrodynamic regime of graphene is fairly large, as shownin Fig. 2. Furthermore, in a nonequlibrum situation where
the electron system is at high temperature Twhile the lattice
temperature T
lremains low, /Gamma1eeis enhanced while the /Gamma1ph=
/Gamma1ph(Tl,T,n ) remains small, and so the hydrodynamic window
could be wider. Such a nonequlibrum situation can be realizedeither through optical pumping [ 34] or by Joule heating due to
an electric current [ 35].
Recent experiments [ 5–8] explored the dc transport in
several pure 2D conductors in the regime /Gamma1
d/lessmuch/Gamma1ee, i.e.,
near the horizontal axis ω=0o fF i g . 2. These measure-
ments revealed signatures of viscous electron flow, whichmanifest themselves through a combination of high viscosityν=v
2//Gamma1 eeand geometrical restriction on the flow. Because
of the particular focus of these studies, the lower viscosity,higher temperature regime was labeled as nonhydrodynamic.From our point of view, viscosity is just one of very many
hydrodynamic phenomena rather than its essential element.
Actually, in the high-temperature regime, where the electron-hole plasma becomes a more perfect fluid [ 10], hydrodynamic
effects should be of crucial importance. They are predicted togive dramatically different optical responses compared to thenoninteracting kinetic theory [ 20].
IV . THIRD-ORDER NONLINEAR OPTICAL
CONDUCTIVITY
A. The general charged fluid
Since we neglect the O(q2) nonlocal corrections, σ(N)could
be derived by simply considering the fluid driven by a uniformelectric field. (The dynamic magnetic field Bis zero in this
case.) The hydrodynamic Eqs. ( 6) simplify to
∂
tp=ρE,∂tρ=0,∂tSn=0, (14)
where pis the momentum density and Sn=S/n is the
entropy per unit charge. The last relation in Eqs. ( 14) comes
from the fact that the hydrodynamic flow is isentropic. The
second relation in Eqs. ( 14) comes from the charge continuity
equation. It entails that the charge density ρstays constant.
In turn, the first relation in Eqs. ( 14) implies that pis strictly
linear in E. If the electric field in the system is composed of
Fourier harmonic with amplitudes Eaand frequencies ωa, then
the momentum density is
p=N/summationdisplay
a=1i
ωaρEae−iωat. (15)
The current density can be treated as a thermodynamic function
ofn,Sn, and p:
j=j(n,Sn,p). (16)
Since particle density and entropy are conserved, the Nth-
order current where N=2m+1, is just the Nth-order Taylor
expansion of jwith respect to p. Due to the isotropy of the fluid,
current density must be parallel to the momentum: j=jˆp.Therefore,
j(N)
i=1
N!/parenleftbig
∂N
pj/parenrightbig
(p2)mpi. (17)
Using Eq. ( 15), we find the Nth-order current of frequency
ω=/summationtextN
aωato be
j(N)
i=1
N!/parenleftbig
∂N
pj/parenrightbigi(−1)m
ω1ω2...ω Nδiν1δν2ν3...δνN−1νN
×E1,ν1E2,ν2...E N,νN+perm(1 ,2,..., N ).(18)
Here and below, “perm” stands for permutations. Therefore,
Eq. ( 2) is proven with
D(N)
h=(−2)mm!
(N!)2/parenleftbig
∂N
pj/parenrightbig
. (19)
B. The Lorentz invariant fluid (Dirac fluid)
As we mentioned above, Eqs. ( 14) imply that the charge
density ρis a constant. From Lorentz invariance, the current
isji=ρuiand therefore j(3)
i=ρu(3)
i. The flow velocity u
can be found from its nonlinear relation to the momentump
i=Wγ2ui. The left-hand side is linear in electric field, thus
the third-order terms on the right-hand side must vanish:
0=Wu(3)
i+W(2)u(1)
i+Wu(1)2u(1)
i. (20)
It follows that
u(3)
i=−W(2)
Wu(1)
i−u(1)2u(1)
i=−/parenleftBigg/parenleftbig∂W
∂n0/parenrightbig
Snn(2)
0
W+u(1)2/parenrightBigg
u(1)
i.
(21)
From the relativistic relation n=γn0,w eh a v e
n0=n(1−u2/2+O(u4)), (22)
Thus, n(2)
0=nu(1)2/2 and
u(3)
i=1
2(Cise−1)u(1)2u(1)
i. (23)
Therefore, we arrive at
j(3)=ρu(3)=1
2(Cise−1)(p/W )3, (24)
which renders the third-order optical weight Eq. ( 5).
C. 2D Dirac fluid with a static magnetic field
The uniform hydrodynamic equations with a static magnetic
field are
∂tpi=ρEi+ρ1
c/epsilon1ijkujBk,∂tρ=0,∂tSn=0.(25)
The momentum density pis no longer strictly linear in E.
Below we focus on the 2D case, so the Dirac fluid is on x-y
plane and the static magnetic field is in the zdirection. The
linear conductivity is modified to [ 9]
σij=Dh/π
ω2−ω2c(iωδij−ωc/epsilon1ij), (26)
where /epsilon1ijis the antisymmetric tensor in 2D, and ωc=
eB/(m∗c) is the hydrodynamic cyclotron frequency. Note that
ωcis, in general, not equal to the usual cyclotron frequency
because the hydrodynamic effective mass m∗≡W/(nv2)i s
075432-4THIRD-ORDER OPTICAL CONDUCTIVITY OF AN … PHYSICAL REVIEW B 97, 075432 (2018)
not exactly the same as the quasiparticle effective mass in
a Fermi liquid. From the Euler equation, the third-ordermomentum is related to the third-order flow velocity
−iω
sp(3)
i=ρ1
c/epsilon1ijku(3)
jBk, (27)
where ωsis the frequency of the third-order current (the sum
frequency). Together with the relation
p(3)
i=Wu(3)
i+1
2(1−Cise)W(u(1))2u(1)
i, (28)
we get the equation for the third-order current
Mijj(3)
j=1
2(1−Cise)W
ρ3(j(1))2j(1)
i, (29)
where ˆM=−i
ωsρˆσ−1, therefore
j(3)
i=1
2(1−Cise)W
ρ4iωsσil(ωs)(j(1))2j(1)
l
=1
2(1−Cise)W
ρ4iωsσiα(ωs)σαl(ω1)σkm(ω2)σkn(ω3)
×E1lE2mE3n+perm(1 ,2,3). (30)
The symmetrized third-order conductivity reads
σ(3)
ilmn=1
3!·2(1−Cise)W
ρ4iωsσiα(ωs)σαl(ω1)σkm(ω2)σkn(ω3)
+perm(1 ,2,3)(l,m,n ), (31)
where perm(1 ,2,3)(l,m,n ) denotes the 3! =6 permutations of
the indices ( l,m,n ) together with (1 ,2,3). Therefore, we have
proven that the third-order conductivity σ(3)is determined by
the linear one σ.
For moderate magnetic field ωc/lessmuchω, and the case of a
single frequency ω1=ω2=ω3=ω, we can expand σ(3)to
linear order in B:
σ(3)
ilmn=iD(3)
h
ω3/bracketleftbigg
/Delta1ilmn+i4ωc
3ω/Xi1ilmn/bracketrightbigg
, (32)
where /Xi1ilmn=δlm/epsilon1in+δln/epsilon1im+δmn/epsilon1il.
D. Analysis of σ(3)
The simple tensorial structure of Eq. ( 4)i sar e s u l to f
rotational symmetry and local equilibrium nature of an idealcharged fluid. For comparison, in the high-frequency ki-netic/quantum regime of graphene, the tensorial structure ofσ
(3)is more complicated due to contributions from interband
transitions and disorder scattering effects [ 25,27,28]. In the
hydrodynamic regime, the interband transitions are suppressedby fast e-e scattering, resulting in the simple expression ofEq. ( 4).
The magnitude of hydrodynamic σ
(3)is also different from
that of the kinetic theory. Applied to graphene at T=0, our
result for D(3)
his
D(3)
h(T=0)=g
48πe4vF
¯h3kF=2D(3)
k, (33)
which is twice the third-order spectral weight D(3)
kfrom the
collisionless Boltzmann transport theory [ 22,25,28]. This dif-ference could be measured by, e.g., third harmonic generation
to be discussed in Sec. V.
Comparing the third-order nonlinear response with the
usual linear one, we notice that the third-order current issuppressed by the parameter
ξ=/parenleftbigg−eE/ω
m∗v/parenrightbigg2
/lessmuch1. (34)
This factor is different in the nonrelativistic and the ultrarela-
tivistic regimes because m∗depends on the Fermi momentum
pF. As a result, in the nonrelativisitic case, parameter ξis
smaller by the factor of ( vF/v)2/lessmuch1 than the ultrarelativistic
case. This factor vanishes for a system with the parabolicdispersion, which corresponds to v→∞ . Indeed, for such
a system, all nonlinearities at zero qshould be absent because
of the Galilean invariance (Kohn’s theorem). On the otherhand, in graphene at zero temperature, which is an example ofthe ultrarelativistic system, m
∗v=pF, so that ξ=(δp/p F)2.
Hereδp=−eE/ω has the physical meaning of the amplitude
of electron momentum oscillation caused by the electric field.
The third-order conductivity Eq. ( 4) diverges at the zero
frequency limit, which is unphysical. In reality, the divergenceis curbed by the momentum relaxation rate /Gamma1
d, similar to the
first-order conductivity. A simple but crude way to includethe effect of the momentum relaxation is to change all thefrequencies ω
atoω+
a=ωa+i/Gamma1d. However, this approach
neglects the increase of entropy density due to momentumrelaxation. Thus, special care needs to be taken to computethe true nonlinear dc response, as shown in Appendix B.
V . THIRD HARMONIC GENERATION
One quantity we can derive from σ(3)[Eq. ( 4)] is the third
harmonic generation (THG) [ 28,36]. Assume the ac electric
field of the incident light is in the xdirection:
E(t)=ˆxE(ω)e−iωt+c.c. (35)
The current, which determines the observable THG signal is
j(3)(t)=ˆxσ(3)
xxxx(ω,ω,ω )E(ω)3e−i3ωt+c.c. (36)
Therefore, σ(3)
xxxx(ω,ω,ω ) represents the magnitude of the
THG. This quantity is plotted in Fig. 3as a function of T.
FIG. 3. The THG signal as a function of temperature at fixed n.
The frequency of incident light is ¯ hω=εF/20 with εFbeing the zero
temperature Fermi energy. The blue curve is the prediction of the
hydrodynamic theory Eq. ( 4), the green curve is from the RPA [ 27].
The observable signal should behave as sketched by the dashed curve.
075432-5ZHIYUAN SUN, D. N. BASOV , AND M. M. FOGLER PHYSICAL REVIEW B 97, 075432 (2018)
Also shown in Fig. 3is the prediction of the conventional
theory based on random-phase approximation (RPA), [ 27,28],
which is valid in the kinetic regime. The two curves exhibitdifferent behavior. At zero temperature, the hydrodynamictheory predicts the THG signal, which is twice that of thekinetic theory. However, since /Gamma1
ee=0a tT=0, the electron
system much be in the kinetic regime (Fig. 2), and so the actual
σ(3)
xxxx(ω,ω,ω ) should be close to the kinetic theory value, as
sketched by the dashed line. As temperature increases, /Gamma1ee
grows, and the system will experience a crossover from the
kinetic to hydrodynamic regime at certain T∗. This crossover
temperature is determined by /Gamma1ee(n,T∗)=ω. As temperature
increases further, the hydrodynamic third-order optical weight
drops as D(3)
h∝(m∗)−3∝T−9due to the thermal enhance-
ment of the hydrodynamic effective mass m∗(n,T). Therefore,
the THG drops much faster than what the conventional kinetictheory would predict.
VI. THE KERR EFFECT AND THE DEMONS
The Kerr effect refers to the change of the effective
permittivity of a medium due to the third-order nonlinearity[37,38]. For a 2D charged Dirac fluid, this effect is more
conveniently described as the shift of the effective conductivity.One manifestation of the Kerr effect is the renormalization ofthe frequency of the collective modes in a strong optical field.In the kinetic regimes, these modes are the familiar plasmons[36,39]. In the hydrodynamic regime, they are the demons
[17,20].
In general, to describe the collective modes, we need to
study response at a finite q. For small enough q, we can
approximate the result using q=0 quantities, as follows. The
charge density fluctuation could be represented by means ofthe Fourier amplitude ρ
ω,q:
ρ=ρω,qei(q·r−ωt)+c.c. (37)
Assuming qis in the ˆxdirection, the corresponding Fourier
amplitude of the electric field is Ex=(−iq)vqρω,q, where vq
is the Coulomb potential. In 2D, it is given by vq=2π/κq .
The electric field induces the current
jx(ω)=σ(ω)Ex+3σ(3)
xxxx(ω,ω,−ω)ExExE∗
x. (38)
Using the charge continuity equation ∂tρ+∇j=0, we obtain
−iωρ+q2vqσ(ω)ρ+3q4v3
qσ(3)
xxxx(ω,ω,−ω)ρρρ∗=0.
(39)
The weak-field dispersion can be obtained from this equation
by dropping the last term. When this term is retained, thedispersion acquires the frequency shift proportional to σ
(3):
δω=−3i
2q4v3
qσ(3)
xxxx(ω,ω,−ω)|ρω,q|2
=−3i
2q2vqσ(3)
xxxx(ω,ω,−ω)|Eω,q|2. (40)
[To obtain this relation, we also assumed that the linear
conductivity has the Drude form σ(ω)∝ω−1.] Applied to
Eq. ( 4), we obtain the fractional shift of the frequency of the
FIG. 4. The dispersion of the demons in the hydrodynamic regime
of graphene. The black curve is for the weak field limit while the
red dashed curve includes the Kerr-effect-induced shift in a strong
field (E=104V/cm). The carrier density and temperature are n=
1012cm−2andT=300 K.
demon:
δω
ω=−9
2q2vqD(3)
h
ω4|Eω,q|2=−3
4(1−Cise)ξ, (41)
where ξis defined by Eq. ( 34). The negative sign of the shift
means the collective mode is softened by the strong field. Thereason for this is that the third-order conductivity is oppositein sign compared to the linear one, which is due to the current j
being a concave function of the momentum density pi naD i r a c
fluid. The results for the original and shifted demon dispersionin graphene is illustrated by Fig. 4. It is remarkable that an
appreciable shift occurs already at a relatively low field ofE=10
4V/cm.
VII. THIRD-ORDER CIRCULAR BIREFRINGENCE
In the presence of an applied magnetic field, there is a finite
second term ∝ωcin Eq. ( 32), which causes the third-order
circular birefringence. As in Sec. IV C , let us consider a 2D
system subject to a normally incident monochromatic light offrequency ωwith the electric field polarized in the xdirection.
The generated third-order current has frequency ω
s=3ωand
has a nonzero y-component
jx(3ω)=σ(3)
xxxxE3
x(ω)=iD(3)
h3
ω3,
jy(3ω)=σ(3)
yxxxE3
x(ω)=iD(3)
h3
ω3/parenleftbigg4iωc
3ω/parenrightbigg
. (42)
In the dissipationless limit, ωis real, and jyhas aπ/2 phase
difference relative to jx. Therefore, the third harmonic light
will be elliptically polarized with the principal axis along x.
Its ellipticity, conventionally denoted by tan θ, is given by
tanθ=/vextendsingle/vextendsingle/vextendsingle/vextendsinglejy
jx/vextendsingle/vextendsingle/vextendsingle/vextendsingle=4
3ωc
ω. (43)
Therefore, the ellipticity scales as ωc=eB/m∗c, which de-
cays with temperature if the carrier density nis fixed. This is
illustrated by Fig. 5for the case of graphene. From Eq. ( 26),
there is also circular birefringence in the linear response, withtanθ=ω
c/ω, which differs only by the constant numerical
factor 4 /3. It is also plotted in Fig. 5, for an easy comparison.
075432-6THIRD-ORDER OPTICAL CONDUCTIVITY OF AN … PHYSICAL REVIEW B 97, 075432 (2018)
FIG. 5. Circular birefrigence in graphene according to the hydro-
dynamic theory. The blue line is the ellipticity of the third-harmonic
light (|Ey/Ex|) as a function of temperature at n=1012cm−2.T h e
black line is the ellipticity of the first-harmonic reflected light. Thefrequency is ω=1.41 THz, the magnetic field is B=0.1T . I n s e t :
illustration of the elliptical polarization.
VIII. DISCUSSION
We showed that the third-order nonlinear conductivity σ(3)
of a Dirac fluid has a universal functional form for any mass,
chemical potential, temperature, and space dimension. It isremarkable that the third-order and the linear conductivities aresimply related through Eqs. ( 5) and ( 31). Although we have
used graphene as an example in the numerical calculations,our formulas, e.g., Eqs. ( 4) and ( 5), hold for any Lorentz-
invariant Dirac fluid. As such, these formulas should be a goodapproximation to surface states of topological insulators and
Dirac/Weyl semimetals, provided they are in the hydrodynamicregime. We also studied the field-induced renormalization ofthe dispersion of the collective modes (demons) and the third-order circular birefringence in the presence of a static magneticfield.
In the future, it would be interesting to investigate hydro-
dynamics of non-Dirac fluids, that is, systems without theLorentz symmetry. This will be important for more realisticmodeling of ultrapure solid-state systems where hydrodynamicregime has been reported (GaAs, graphene, and PdCoO
2). In
the above systems, although the quasiparticle band dispersionis approximately Dirac-like, the Coulomb interaction tendsto break this quasi Lorentz symmetry because it propagateswith the speed of light crather than v. Moreover, for ultrathin
slabs of Weyl semimetals, the contribution from the FermiArc surface states might be appreciable. The latter forms anelectron fluid that breaks the rotational symmetry and thereforeneeds special treatment. It would also be interesting to studynonlinear thermal transport in the Dirac fluid. The case ofphonon fluids has been studied half a century ago [ 40].
ACKNOWLEDGMENTS
This paper is supported by the Office of Naval Research
under Grant No. N00014-15-1-2671. D.N.B. is an investigatorin quantum materials funded by the Gordon and Betty MooreFoundation’s EPiQS Initiative through Grant No. GBMF4533.We thank B. I. Shklovskii for helpful discussions.
APPENDIX A: THERMODYNAMIC QUANTITIES IN GRAPHENE
For convenience, below we list the expressions for the thermodynamic quantities of graphene in the noninteracting limit. The
charge density n(μ,T)[17]i s
n=/integraldisplay∞
−∞[f(μ,T,/epsilon1 )−f(0,0,/epsilon1)]g(/epsilon1)d/epsilon1=1
πμ2
¯h2v2
F/bracketleftbigg
1+π2
3T2
μ2+4T2
μ2Li2(−e−μ/T)/bracketrightbigg
, (A1)
where Li s(x) is the polylogarithm function. The energy density nEis defined relative to the ( μ,T)=(0,0) case:
nE=/integraldisplay∞
−∞[f(μ,T,/epsilon1 )−f(0,0,/epsilon1)]/epsilon1g(/epsilon1)d/epsilon1=2
πT3
¯h2v2
F/bracketleftbiggπ2
3μ
T+1
3μ3
T3−4L i 3(−e−μ/T)/bracketrightbigg
=2
3πμ3
¯h2v2
F/bracketleftbigg
1+π2T2
μ2−12T3
μ3Li3(−e−μ/T)/bracketrightbigg
. (A2)
The enthalpy density is W=3
2nE, the pressure is P=1
2nE, and the entropy density is
s=/parenleftbigg∂P
∂T/parenrightbigg
μ=1
3πμ2
¯h2v2
F/bracketleftbigg
2π2T
μ−12T
μLi2(−e−μ/T)−36T2
μ2Li3(−e−μ/T)/bracketrightbigg
. (A3)
The hydrodynamic effective mass m∗(μ,T)i s
m∗(μ,T)=1
v2W(μ,T)
n(μ,T). (A4)
The dimensionless bulk isentropic modulus is
Cise=1
m∗v2/parenleftbigg∂P
∂n/parenrightbigg
ise=1
d=1
2. (A5)
075432-7ZHIYUAN SUN, D. N. BASOV , AND M. M. FOGLER PHYSICAL REVIEW B 97, 075432 (2018)
APPENDIX B: DERIV ATION OF σ(3)WITH MOMENTUM AND ENERGY RELAXATION
In the homogeneous case, the hydrodynamic equations are
(∂t+/Gamma1d)pi=ρEi,∂tnE=ρujEj−/Gamma1EδnE−/Gamma1kWu2,∂tρ=0, (B1)
where /Gamma1dis the phenomenological momentum relaxation rate, /Gamma1Ecan be called the cooling rate, δnE=nE−nEeqis the fluctuation
of energy density with respect to its steady-state value, and /Gamma1kis the relaxation rate of the center-of-mass kinetic energy of a
moving fluid. The last equation entails ρis constant. Therefore j(3)
i=ρu(3)
i, and the momentum piis strictly linear in electric
field, same as the dissipationless case. The third-order velocity can be found from Eq. ( 20):
u(3)
i=−/parenleftbig
W(2)/W+u(1)2/parenrightbig
u(1)
i. (B2)
By rotational symmetry, the leading-order perturbation to the scalar quantities are second order in the electric field [ 20]:
n(2)
0=−1
2n0u(1)2,n(2)
E=ω+
2−i/Gamma1k
ω1+ω2+i/Gamma1EWu(1)
1iu(1)
2i+perm,n(2)
E0=n(2)
E−Wu(1)2, (B3)
where “perm” stands for permutations among subscripts 1 ,2, and 3, corresponding to frequencies ω1,ω2, andω3, respectively.
Note the difference between density n0and energy density nE0in proper frame and their counterparts n,nEin laboratory frame.
With the second-order expansion of the enthalpy
W(2)=/parenleftbigg∂W
∂n0/parenrightbigg
nE0n(2)
0+/parenleftbigg∂W
∂nE0/parenrightbigg
n0n(2)
E0, (B4)
we are ready to write down the third-order flow velocity
u(3)
i=−/bracketleftBigg
1
W/parenleftbigg∂W
∂n0/parenrightbigg
nE0/parenleftbigg
−1
2n0/parenrightbigg
u(1)2+/parenleftbigg∂W
∂nE0/parenrightbigg
n0/parenleftbiggω+
2−i/Gamma1k
ω1+ω2+i/Gamma1Eu(1)
1ju(1)
2j+perm/parenrightbigg
−/parenleftbigg∂W
∂nE0/parenrightbigg
n0u(1)2+u(1)2/bracketrightBigg
u(1)
i,(B5)
where we defined ω+
a≡ωa+i/Gamma1d. The equation for the Fourier amplitude u(3)(ωs) of the combined frequency ωs=ω1+ω2+ω3
becomes
u(3)
i=−/bracketleftBigg
1
W/parenleftbigg∂W
∂n0/parenrightbigg
nE0/parenleftbigg
−1
2n0/parenrightbigg
+/parenleftbigg∂W
∂nE0/parenrightbigg
n0/parenleftbiggω+
2−i/Gamma1k
ω1+ω2+i/Gamma1E/parenrightbigg
−/parenleftbigg∂W
∂nE0/parenrightbigg
n0+1/bracketrightBigg
u(1)
1ju(1)
2ju(1)
3i+perm
=/bracketleftBigg
1
2n0
W/parenleftbigg∂W
∂n0/parenrightbigg
nE0+/parenleftbigg∂W
∂nE0/parenrightbigg
n0−1−1
2/parenleftbigg∂W
∂nE0/parenrightbigg
n0/parenleftbiggω+
1+ω+
2−2i/Gamma1k
ω1+ω2+i/Gamma1E/parenrightbigg/bracketrightBigg
u(1)
1ju(1)
2ju(1)
3i+perm
=1
2/bracketleftBigg
Cise−1−/parenleftbigg∂W
∂nE0/parenrightbigg
n0/parenleftbigg2i/Gamma1d−i/Gamma1E−2i/Gamma1k
ω1+ω2+i/Gamma1E/parenrightbigg/bracketrightBigg
u(1)
1ju(1)
2ju(1)
3i+perm. (B6)
After the standard symmetrization procedure, Eq. ( B6) renders σ(3)with dissipation. The cooling rate /Gamma1Ecould arise due to
electron-phonon coupling and is crucial for eliminating the divergence of σ(3)in the dc limit. In this dc limit, due to work done
by the electric field, the electron-hole fluid would be heated up by order E2//Gamma1E, thus inducing a large correction to the current
at the third order. This is a physical reason why setting /Gamma1E→0 would lead to a diverging σ(3).
In the dissipationless limit, Eq. ( B6) becomes Eq. ( 23). Moreover, it can be readily checked that if /Gamma1E+2/Gamma1k=2/Gamma1d,E q .( B6)
becomes identical to Eq. ( 23) as well. Under this condition, the fluid dynamics becomes isentropic again: the kinetic energy is
lost to the environment due to momentum relaxation instead of converted into heat of the fluid.
APPENDIX C: RELATIVISTIC BOLTZMANN EQUATION
The relativistic Boltzmann equation is
/parenleftbig
Pμ∂μ+FμνPμ∂Pν/parenrightbig
fR(X,P )=I[fR], (C1)
where fR(X,P ) is the relativistic distribution function and I[fR] is the collision integral due to interactions. Note that the
space-time coordinate Xμand the momentum Pμ=muμare covariant ones. For a given Xμ, the distribution function fR(X,P )
can be defined as the density of world lines whose local tangent is Pμ. Mathematically, fR(X,P ) is a scalar function defined on
the tangent bundle of the d+1 dimensional space-time. If we focus on one species of particle with a fixed mass m, thenfR(X,P )
is related to the ordinary distribution function through
fR(X,P )=f(t,r,p)δ(E2−p2−m2). (C2)
075432-8THIRD-ORDER OPTICAL CONDUCTIVITY OF AN … PHYSICAL REVIEW B 97, 075432 (2018)
We integrate Eq. ( C1) over Pto get the charge continuity equation:
∂μJμ=0,
Jμ=/integraldisplay
dPPμfR(X,P )=/integraldisplay
dp(1,v)f(t,r,p), (C3)
which is the second equation in Eqs. ( 6).
Next, we multiply Eq. ( C1)b yPαand again integrate it over P. We get the continuity equation for the energy-momentum
tensor,
∂μTμν=Fν
μJμ,
Tμν=/integraldisplay
dPPμPνfR(X,P )=/integraldisplay
dpPμPν1
Ef(t,r,p), (C4)
which is the first equation in Eqs. ( 6).
[1] R. N. Gurzhi, Sov. Phys. Usp. 11,255(1968 ).
[2] A. V . Andreev, S. A. Kivelson, and B. Spivak, Phys. Rev. Lett.
106,256804 (2011 ).
[3] L. D. Landau and E. M. Lifshitz, Fluid Mechanics , 2nd ed.
(Pergamon Press, Oxford, 1987).
[4] M. J. M. de Jong and L. W. Molenkamp, P h y s .R e v .B 51,13389
(1995 ).
[5] D. A. Bandurin, I. Torre, R. K. Kumar, M. Ben Shalom, A.
Tomadin, A. Principi, G. H. Auton, E. Khestanova, K. S.Novoselov, I. V . Grigorieva, L. A. Ponomarenko, A. K. Geim,and M. Polini, Science 351,1055 (2016 ).
[6] J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim, A. Lucas,
S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe, T. A. Ohki, andK. C. Fong, Science 351,1058 (2016 ).
[7] R. Krishna Kumar, D. A. Bandurin, F. M. D. Pellegrino, Y .
Cao, A. Principi, H. Guo, G. H. Auton, M. Ben Shalom, L. A.Ponomarenko, G. Falkovich, K. Watanabe, T. Taniguchi, I. V .Grigorieva, L. S. Levitov, M. Polini, and A. K. Geim, Nat. Phys.
13,1182 (2017 ).
[8] P. J. W. Moll, P. Kushwaha, N. Nandi, B. Schmidt, and A. P.
Mackenzie, Science 351,1061 (2016 ).
[9] M. Müller, L. Fritz, and S. Sachdev, Phys. Rev. B 78,115406
(2008 ).
[10] M. Müller, J. Schmalian, and L. Fritz, P h y s .R e v .L e t t . 103,
025301 (2009 ).
[11] T. V . Phan, J. C. W. Song, and L. S. Levitov, arXiv:1306.4972 .
[12] D. Forcella, J. Zaanen, D. Valentinis, and D. van der Marel,
Phys. Rev. B 90,035143 (2014 ).
[13] U. Briskot, M. Schütt, I. V . Gornyi, M. Titov, B. N. Narozhny,
and A. D. Mirlin, P h y s .R e v .B 92,115426 (2015 ).
[14] B. N. Narozhny, I. V . Gornyi, M. Titov, M. Schütt, and A. D.
Mirlin, P h y s .R e v .B 91,035414 (2015 ).
[15] A. Principi and G. Vignale, P h y s .R e v .B 91,205423 (2015 ).
[16] A. Principi and G. Vignale, P h y s .R e v .L e t t . 115,056603 (2015 ).
[17] Z. Sun, D. N. Basov, and M. M. Fogler, P h y s .R e v .L e t t . 117,
076805 (2016 ).
[18] A. Lucas, J. Crossno, K. C. Fong, P. Kim, and S. Sachdev,
Phys. Rev. B 93,075426 (2016 ).
[19] A. Lucas, Phys. Rev. B 95,115425 (2017 ).
[20] Z. Sun, D. N. Basov, and M. M. Fogler, arXiv:1704.07334 .
[21] H. Guo, E. Ilseven, G. Falkovich, and L. S. Levitov, Proc. Natl.
Acad. Sci. USA 114,3068 (2017 ).[22] S. A. Mikhailov, Europhys. Lett. 79,27002 (2007 ).
[23] S. A. Mikhailov, Phys. Rev. B 84,045432 (2011 ).
[24] M. Gullans, D. E. Chang, F. H. L. Koppens, F. J. Garcia
de Abajo, and M. D. Lukin, P h y s .R e v .L e t t . 111,247401
(2013 ).
[25] J. L. Cheng, N. Vermeulen, and J. E. Sipe, New J. Phys. 16,
053014 (2014 ).
[26] X. Yao, M. Tokman, and A. Belyanin, Phys. Rev. Lett. 112,
055501 (2014 ).
[27] J. L. Cheng, N. Vermeulen, and J. E. Sipe, P h y s .R e v .B 91,
235320 (2015 ).
[28] S. A. Mikhailov, Phys. Rev. B 93,085403 (2016 ).
[29] M. Tokman, Y . Wang, I. Oladyshkin, A. R. Kutayiah, and A.
Belyanin, P h y s .R e v .B 93,235422 (2016 ).
[30] J. L. Cheng, N. Vermeulen, and J. E. Sipe, Sci. Rep. 7,43843
(2017 ).
[31] M. T. Manzoni, I. Silveiro, F. J. G. D. Abajo, and D. E. Chang,
New J. Phys. 17,83031 .
[32] Y . Wang, M. Tokman, and A. Belyanin, Phys. Rev. B 94,195442
(2016 ).
[33] H. Rostami, M. I. Katsnelson, and M. Polini, Phys. Rev. B 95,
035416 (2017 ).
[34] G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S.
McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. CastroNeto, J. Hone, M. M. Fogler, and D. N. Basov, Nat. Photonics
10,244(2016 ).
[35] S.-k. Son, M. Šiškins, C. Mullan, J. Yin, V . G. Kravets, A.
Kozikov, S. Ozdemir, M. Alhazmi, M. Holwill, K. Watanabe,T. Taniguchi, D. Ghazaryan, K. S. Novoselov, V . I. Fal’ko, andA. Mishchenko, 2D Mater. 5,011006 (2018 ).
[36] F. Giorgianni, E. Chiadroni, A. Rovere, M. Cestelli-Guidi,
A. Perucchi, M. Bellaveglia, M. Castellano, D. Di Giovenale,G. Di Pirro, M. Ferrario, R. Pompili, C. Vaccarezza, F. Villa, A.Cianchi, A. Mostacci, M. Petrarca, M. Brahlek, N. Koirala, S.Oh, and S. Lupi, Nat. Commun. 7,11421 (2016 ).
[37] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous
Media (Pergamon, Oxford, 1984).
[38] R. W. Boyd, Nonlinear Optics , 3rd ed. (Academic Press, Cam-
bridge, MA, 2008).
[39] S. A. Mikhailov, ACS Photonics 4,3018 (2017
).
[40] H. Nielsen and B. I. Shklovskii, Zh. Eksp. Teor. Fiz. 56, 709
(1969) [JETP 20, 386 (1969)].
075432-9 |
PhysRevB.76.045329.pdf | Kondo effect in coupled quantum dots with RKKY interaction:
Effects of finite temperature and magnetic field
Chung-Hou Chung1,3and Walter Hofstetter2
1Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany
2Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, 60438 Frankfurt/Main, Germany
3Electrophysics Department, National Chiao-Tung University, Hsinchu, Taiwan 300, Republic of China
/H20849Received 22 May 2007; published 24 July 2007 /H20850
We study transport through two quantum dots coupled by a Ruderman-Kittel-Kasuya-Yoshida interaction as
a function of temperature and magnetic field. By applying the numerical renormalization group method, weobtain the transmission and the linear conductance. At zero temperature and magnetic field, we observe aquantum phase transition between the Kondo screened state and a local spin singlet as the RKKY interactionis tuned. Above the critical RKKY coupling, the Kondo peak is split. However, we find that both finitetemperature and magnetic field restore the Kondo resonance. Our results agree well with recent transportexperiments on gold grain quantum dots in the presence of magnetic impurities /H20851H. B. Heersche et al. , Phys.
Rev. Lett. 96, 017205 /H208492006 /H20850/H20852.
DOI: 10.1103/PhysRevB.76.045329 PACS number /H20849s/H20850: 73.23.Hk, 73.63.Kv, 75.30.Hx, 71.27. /H11001a
I. INTRODUCTION
In recent years, the Kondo effect1in semiconductor quan-
tum dots has gained significant interest both theoretically andexperimentally.
2–4Electronic transport in quantum dots is
strongly influenced by the Coulomb blockade5due to their
small size. In these systems, a single unpaired spin can in-teract with conduction electrons, leading to screening of thespin and an enhanced conductance at low bias and low tem-peratures. More recently, due to rapid progress in spintronicsand quantum information, it becomes desirable to gain moretunable spin control in double quantum dots where an effec-tive spin-spin interaction known as the Ruderman-Kittel-Kasuya-Yoshida /H20849RKKY /H20850
6coupling is generated between the
two dots via conduction electrons in the lead. The RKKYcoupling competes with the Kondo effect in these systems,leading to a quantum phase transition between the Kondoscreened phase at weak RKKY coupling and a local spin-singlet state at strong coupling. Part of this rich physics hasbeen studied previously in the two-impurity Kondo
7and
Anderson8problems.
Recently, indications for a competition between Kondo
screening and the RKKY interaction have been observedexperimentally.
9This experiment stimulated theoretical and
experimental efforts on the above-mentioned quantum phasetransition in coupled dots. Very recent measurements havebeen reported to observe the restoring of the Kondo reso-nance in a gold quantum dot with magnetic impurities atfinite temperatures and magnetic fields where an effectiveRKKY coupling is generated between the dot and theimpurities.
10Though similar systems have been studied theo-
retically via several approaches,11–13little is known about the
transport properties at finite temperatures and magneticfields. Motivated by these recent experiments, here we sys-tematically study transport properties of double quantumdots coupled by a RKKY interaction at finite temperaturesand magnetic fields via the numerical renormalization group/H20849NRG /H20850method,
14a nonperturbative approach to quantum im-
purity systems. In contrast to other techniques, this methoddoes not rely on any assumptions regarding the ground state
or the leading divergent couplings, which is crucial in ouranalysis.
II. MODEL
We consider a two-impurity Anderson model, as shown in
Fig. 1, describing two quantum dots which are separately
coupled to two-channel leads and subject to an antiferromag-netic RKKY spin-spin interaction and a local magnetic field.This setup is general enough to describe both experimentsmentioned above. The Hamiltonian of the system is given by
H=H
D+Hl+Ht+HJ+HB,
HD=/H20858
is/H20873/H9280di+U
2/H20874di/H9268†di/H9268+U
2/H20858
i/H20849Ni−N/H208502,
Hl=/H20858
/H9251ik/H9268/H9280/H9251ik/H9268c/H9251ik/H9268†c/H9251ik/H9268,
Ht=/H20858
/H9251ik/H9268Vi/H9251c/H9251ik/H9268†di/H9268+ H.c.,
HJ=JS1S2,HB=−B/H20849S1z+S2z/H20850. /H208491/H20850
Here, /H9251=L/Rdenotes the left/right lead, i=1,2 denotes
the two dots, Ni=/H20858/H9268di/H9268†di/H9268is the number of electrons occu-
µR µL
21
J
V VV V11
2 2
FIG. 1. Two quantum dots /H20849denoted by the two circles /H20850with
two-channel lead /H20849chemical potentials /H9262Land/H9262R/H20850.V1andV2denote
the tunneling matrix element between the dots and the lead. Thereexists a RKKY coupling Jbetween the two dots.PHYSICAL REVIEW B 76, 045329 /H208492007 /H20850
1098-0121/2007/76 /H208494/H20850/045329 /H208496/H20850 ©2007 The American Physical Society 045329-1pying dot i, and Si=/H208491/2 /H20850/H20858/H9268/H9268/H11032di/H9268†/H9268/H9268/H9268/H11032di/H9268/H11032are the spins of
the two levels. Each dot is subject to a charging energy U
andJis the RKKY exchange coupling between the two dots
which is assumed to be antiferromagnetic /H20849J/H110220/H20850. The mag-
netic field Binduces a Zeeman splitting, where we have set
g=1. Here, we consider the model with particle-hole sym-
metry, i.e., /H9280di=−U
2, and single occupation for each dot, i.e.,
N=1. The energies /H9280diof the two dots and their precise po-
sition in the Coulomb blockade valley can be tuned experi-mentally by an external magnetic field and the gate voltage.Here, we neglect the energy dependence of the tunnelingmatrix elements V
i/H9251. Without loss of generality, we assume
they have left-right symmetry, i.e., V1L=V1R=V1,V2L=V2R
=V2, but V1/HS11005V2, in general. The intrinsic linewidth of the
dot levels due to tunnel coupling to the leads is /H9003=/H9003L+/H9003R
with/H9003L/R=2/H9266/H20841V/H208412NL/R, where NL/Ris the density of states in
the leads.
Since the correct choice of the effective model and the
number of leads per dot are important, they deserve somediscussion at this point. In both experiments, we refer toRefs. 9and10, there are two effective quantum impurities,
which interact with conduction electrons in the lead via ex-change processes. In Ref. 9, these are two separate quantum
dots, while the experiments of Ref. 10have been interpreted
in the sense that a single quantum dot /H20849the gold grain /H20850and a
magnetic impurity both interact with conduction electrons. Inreality, the antiferromagnetic Kondo coupling and the RKKYcoupling between the dots are both generated by interactionwith conduction electrons in the lead, the latter due tosecond-order exchange processes. Essentially, the picture isthe following: The two dots couple to different points in themetallic lead, i.e., to different conduction electron modes.These modes are not completely orthogonal /H20849otherwise there
would be no RKKY /H20850but they are also not identical. As a
result, one obtains a model Hamiltonian where two dotscouple to two separate /H20849weakly nonorthogonal /H20850electronic
modes. The RKKY coupling depends both on the distancebetween the dots and on the Fermi wavelength. But both ofthese quantities are not known accurately enough in the ex-periments in order to calculate the RKKY coupling micro-scopically. In that sense, it is a free parameter which has tobe inferred from the transport results.
A good approximation to this model—which is much
easier to treat computationally—is the situation considered inour work: Each dot couples to its own electronic mode,
where the overlap between the modes is neglected. TheRKKY coupling is then treated as an additional free param-eter, which can be chosen independently of the Kondo ex-change couplings. We emphasize that the number of freeparameters in this approach is no greater than in the detailed“microscopic” one described above. Moreover, the param-eters in our model—the effective Kondo exchange couplingsand the RKKY coupling—are much easier extracted fromexperimental results than, e.g., the distance between the twodots. One, therefore, concludes that for both experiments, thetwo-channel situation is generic, except for the case wherethe two dots sit exactly at the same position. In fact, it ismuch more difficult to realize a double dot coupled to thesingle-channel lead /H20849one example is a setup very different
from the ones considered here: a multilevel dot with single-mode lead close to pinch-off
15/H20850.For J=0, and Vir=0, three triplet configurations
/H208411,1 /H20856=d1↑†d2↑†/H208410/H20856,/H208411,0 /H20856=/H208491//H208812/H20850/H20849d1↑†d2↓†+d1↓†d2↑†/H20850/H208410/H20856,/H208411,−1 /H20856
=d1↓†d2↓†/H208410/H20856and the singlet /H208410,0 /H20856=/H208491//H208812/H20850/H20849d1↑†d2↓†−d1↓†d2↑†/H20850/H208410/H20856
are degenerate. Finite tunneling Virleads to independent
spin-1/2 Kondo screening in each of the two dots. The cor-
responding Kondo temperatures TK1andTK2are generally dif-
ferent, and given by TK1/H208492/H20850/H11008Dexp /H20849−1/2/H9267cJ1/H208492/H20850/H20850, where Dis
the bandwidth, /H9267cthe density of states of the lead, and J1/H208492/H20850
=4V1/H208492/H208502/Uthe effective Kondo coupling.1
The above degeneracy is lifted at finite RKKY coupling
J/H110220: Three triplet states are shifted to energy Et=J/4 and
the singlet state to energy Es=−3J/4. There exists competi-
tion between Kondo screening and a local spin-singletground state: the former is expected to be the ground statefor small J, the latter for large J. A quantum phase transition
at zero temperature between these two phases occurs as theRKKY coupling is tuned.
7,8Note that a related singlet-triplet
transition in two-level quantum dots was studied both fortwo-mode
16and for single-mode leads.15,17,18In the latter
case, a Kosterlitz-Thouless transition from a local singletstate to a single-channel S=1 underscreened Kondo model
was observed.
15
An interesting aspect of the Kondo-to-singlet transition in
our setup is the tunability between these two phases sinceone can get good control over the various parameters in ex-periments. In particular, as observed in the experiment,
10the
Kondo resonance is restored at finite temperatures and mag-netic fields close to the singlet-triplet transition. The goal ofour work is to describe this behavior theoretically.
The NRG method is applied here to extract ground state
properties of the Anderson impurity model. The key ideaintroduced by Wilson
14is the logarithmic discretization of
the conduction band via the parameter /H9011/H110221. After perform-
ing a Lanczos transformation, the conduction band can bemapped onto a linear chain of fermions.
14The transformed
model can be solved by iterative diagonalization, keeping ineach step only the lowest levels. Here, we iterate 44 times,keep the lowest 600 states, and set /H9011=4.
III. TRANSPORT PROPERTIES
We are interested in calculating electronic transport
through the dot close to the transition. To this end, we use thegeneralized Landauer formula
19
I=2e
h/H20885d/H9275„f/H20849/H9275−/H9262L/H20850−f/H20849/H9275−/H9262R/H20850…T/H20849/H9275/H20850, /H208492/H20850
with the Fermi function f/H20849/H9275/H20850and the transmission
coefficient15
T/H20849/H9275/H20850=−/H20858
i,/H9268/H9003L/H9003R
/H9003L+/H9003RImGii/H9268/H20849/H9275/H20850. /H208493/H20850
Here, we have introduced the retarded dot Green’s functions
Gii/H11032/H9268/H20849t/H20850=−i/H9258/H20849t/H20850/H20855/H20853di/H9268/H20849t/H20850,di/H11032/H9268†/H20854/H20856. In the following, we focus on
the low bias regime, where T/H20849/H9275/H20850can be evaluated in equi-
librium, using the NRG. Using the current formula /H208492/H20850,w e
determine the behavior of the linear conductance /H20841G/H20849T/H20850CHUNG-HOU CHUNG AND WALTER HOFSTETTER PHYSICAL REVIEW B 76, 045329 /H208492007 /H20850
045329-2=dI
dV/H20841V=0at finite temperatures. Note that the equilibrium
transmission T/H20849/H9275/H20850also yields a good approximation to the
differential conductance dI/dVat finite bias measured in ex-
periments.
IV . QUANTUM PHASE TRANSITION AT ZERO
TEMPERATURE FOR ZERO FIELD
In Fig. 2/H20849a/H20850, we plot the transmission coefficient T/H20849w/H20850as a
function of the RKKY coupling Jfor zero temperature and
zero field. It shows a quantum phase transition between theKondo screened phase /H20849J/H11021J
c/H20850and the local spin-singlet
phase /H20849forJ/H11022Jc/H20850, where TK1/H11021Jc/H11021TK2. The value of Jcis
consistent with the previous results by Sakai et al.8For
J/H11021Jc, a Kondo peak at /H9275=0 is observed, where the trans-
mission T/H20849/H9275→0/H20850tends to reach its unitary limit of 2 /H20849each
dot acts as an unitary Kondo channel /H20850. Due to systematic
numerical errors in the NRG calculation, this limit is under-estimated by about 10%. As Jis increased, the Kondo peak
becomes narrower, indicating a vanishing low-energy scaleas the system approaches the critical point. As J→J
c, the
transmission reaches the unstable fixed point valueT/H20849
/H9275→0/H20850=1. For J/H11022Jc, the two dots form a local spin-
singlet state where the Kondo peak is split and T/H20849/H9275→0/H20850=0.
The splitting of the Kondo peak increases as Jis increased.
In our analysis, the RKKY coupling Jis varied indepen-
dently, while the two-stage Kondo temperatures TK1andTK2
are essentially kept fixed, so that the system remains in the
regime TK1/H11021Jc/H11021TK2. Although complete control over the
coupling parameters in the experiment is a challenge, oneshould keep in mind that in Ref. 10—which we mostly refer
to—a large number of different samples with different dI/dV
traces was considered. Our statement is that among thesesamples, there are some which have dI/dVcharacteristics
that are well described by our model and parameter choices.
In that sense, the regime T
K1/H11021Jc/H11021TK2is not generic, but has
been observed experimentally.
Nevertheless, we want to point out that it is still possible
to tune the RKKY coupling independently while keeping TK1andTK2approximately fixed. In fact, the experiment9is an
example, where the RKKY coupling is tuned by varying thecouplings between the central big conducting island and eachof the two dots on the left and right. While doing so, the
Kondo temperatures T
K1andTK2can still roughly stay con-
stant, as they are determined mostly by the stronger lead-dotcouplings, not the couplings between each dot and the cen-tral conducting island.
V . TRANSPORT AT FINITE TEMPERATURES
We now determine the behavior of the linear conductance
G/H20849T/H20850/G0at finite temperatures where G0=2e2/his the con-
ductance unit. Results are shown in Fig. 2/H20849b/H20850. The Kondo
screened and local spin-singlet phases are characterized bystable fixed points with G/H20849T→0/H20850/H110152G
0or 0, respectively. In
the Kondo regime, upon lowering the temperature, the con-
ductance rises up to the unitary limit with two steplike struc-
tures indicating the two crossover Kondo temperatures Tk1
andTk2.
In the local spin-singlet regime, we find a nonmonotonic
behavior of the conductance when Tis lowered: After an
initial rise due to the Kondo effect, G/H20849T/H20850decreases to zero as
the temperature is lowered. The broad peak of G/H20849T/H20850at finite
temperatures is, in fact, a signature of the reappearance of the
Kondo resonance which is also seen in the experiment.10
To gain more insight into the finite temperature behavior,
we present the plot of transmission T/H20849/H9275/H20850at different tem-
peratures /H20849see Fig. 3/H20850when RKKY is strong enough
/H20849J/H11022Jc/H20850so that the Kondo peak is split at zero temperature.
We find that as temperature increases, T/H20849/H9275/H20850inside the dip
first increases due to thermal broadening of the split peaks
around /H9275/H11015±Juntil the Kondo resonance reappears. Then
the peak height decreases again as Tis further increased,
similar to the Kondo effect at finite temperatures withoutRKKY interaction. The transmission T/H20849
/H9275=0/H20850reaches its
maximum value at a temperature Tmax/H11008J−Jc. Note that the
NRG data for T/H20849/H9275/H20850at/H9275/H33355Tare estimated by extrapolating
T/H20849/H9275/H20850from higher frequencies. The qualitative behavior of−0.002 −0.001 0 0.001 0.00200.511.52
J=0
J = 0.00015
J = 0.00041
J = 0.00043
J = 0.00075
J = 0.00125
J = 0.0025T()
ωω
10−810−710−610−510−410−310−2
T00.511.52
J=0
J = 0.00035
J = 0.0005
J = 0.00125
0 G/G(b) (a)
FIG. 2. /H20849Color online /H20850/H20849a/H20850Transmission coefficient at zero temperature for different RKKY interactions. The unit of energy is the half
bandwidth D=1 of the conduction electrons. Here, U=1,/H9280d1=/H9280d2=−0.5, /H90031L=/H90031R=0.05, /H90032L=/H90032R=0.1, /H9011=4,Tk1/H110150.002, Tk2/H110150.000 02
/H20849forJ=0/H20850. The critical RKKY coupling is Jc/H110150.000 42. /H20849b/H20850Linear conductance G/H20849T/H20850for different RKKY couplings. Parameters are the same
as in /H20849a/H20850.KONDO EFFECT IN COUPLED QUANTUM DOTS WITH … PHYSICAL REVIEW B 76, 045329 /H208492007 /H20850
045329-3the transmission—reappearance of the Kondo peak at finite
T—agrees well with recent experiments.10
VI. EFFECT OF A MAGNETIC FIELD
In the presence of a magnetic field, a singlet-triplet cross-
over occurs. Here, we discuss the effect of a Zeeman split-ting at zero temperature in the presence of a large RKKY
coupling Jsuch that the Kondo peak is split in the absence of
magnetic fields /H20849see Fig. 4/H20850.
For finite B/H11021J, we find an increase of the transmission
T/H20849/H9275/H20850at/H9275=0. When Bis comparable to the value of the
RKKY interaction, B/H11015J, we observe the reappearance of the
Kondo peak where T/H20849/H9275/H20850reaches the unitary limit of 1, cor-
responding to a single-channel S=1/2 Kondo effect. When
the magnetic field increases further, the Kondo peak splitsagain. What happens is that due to the Zeeman splitting, onecomponent of the triplet /H20849/H208411,1/H20856/H20850is “pulled down” and even-
tually becomes degenerate with the singlet /H208410,0/H20856/H20851see inset /H20849b/H20850
of Fig. 4/H20852. At this point, a Kondo effect—analogous to S
=1/2 Kondo—arises between these two states, which has
been discussed previously for a two-level quantum dotwithin a perturbative scaling approach.
20Our nonperturba-
tive NRG results confirm this scenario: The transmissionT/H20849
/H9275=0/H20850indeed reaches the unitary limit of the S=1/2
Kondo effect. Note that due to tunneling into the lead, de-
generacy of the singlet and triplet states occurs at a renor-malized value B/HS11005J. Close to the degeneracy point, a four-
peak structure emerges in the transmission /H20851see inset /H20849a/H20850of
Fig.4/H20852, corresponding to the splitting between the singlet and
the lowest /H20849second-lowest /H20850triplet state. Due to NRG broad-
ening effects, the third triplet state is not visible.
In Fig. 5, we present results for the singlet-triplet cross-
over at smaller RKKY coupling, for different magnetic fieldsand finite temperature. Compared to the sharp transition in−0.04 −0.02 0 0.02 0.0400.20.40.60.8
T=0
T=0.00025
T=0.001
T=0.00125
T=0.002
T=0.004
ω(ω)T
FIG. 3. /H20849Color online /H20850Transmission coefficient T/H20849/H9275/H20850for
a fixed RKKY coupling J=0.002 at finite temperatures. Here,
/H90031L=/H90031R=0.04, /H90032L=/H90032R=0.08. The remaining parameters are the
same as in Fig. 2.
−0.01 −0.005 0 0.005 0.0100.050.10.150.20.25
ωT( )ω
−0.006 −0.004 −0.002 0 0.002 0.004 0.00 600.20.40.60.81
B=0
B=0.0016
B=0.00184
B=0.00186
B=0.003
ω(ω)T(a)(b)
S=1
S=0
B1/4 J − B−3/4 J1/4 J1/4 J + B
J
FIG. 4. /H20849Color online /H20850Transmission coefficient T/H20849/H9275/H20850of the coupled dots at different magnetic fields. Here, /H90031L=/H90031R=/H90032L=/H90032R=0.06,
J=0.002. The remaining parameters are the same as in Fig. 2. Inset /H20849a/H20850: the transmission T/H20849/H9275/H20850forB=0.0016, which clearly shows four peaks
in pairs around ± /H20849B±J/H20850. Inset /H20849b/H20850: The energy levels of the triplet and singlet states in the presence of a finite RKKY coupling Jand a
magnetic field B. The three triplet states are split into E1,1=1/4 J−B,E1,0=1/4 J, and E1,−1=1/4 J+B. The singlet /H208410,0/H20856and one of the triplet
states /H208411,1/H20856become degenerate at B=J.CHUNG-HOU CHUNG AND WALTER HOFSTETTER PHYSICAL REVIEW B 76, 045329 /H208492007 /H20850
045329-4Fig.4where Jis much larger than Tk, the crossover here is
smoother and closer to the experimental observation in Ref.10. Note that due to thermal broadening at a finite tempera-
ture, T/H20849
/H9275=0/H20850/H110220 even in the absence of a magnetic field. For
antiferromagnetic RKKY coupling /H20851see Fig. 5/H20849a/H20850/H20852,T/H20849/H9275/H20850
shows the dip-peak-dip structure with increasing magnetic
field. For ferromagnetic RKKY /H20851see Fig. 5/H20849b/H20850/H20852, the Kondo
peak at B=0, which is due to complete screening of the
triplet, splits monotonically with increasing field due to Zee-man splitting of the triplet levels, as shown in Ref. 10. Both
results are qualitatively in good agreement with experiment
10
/H20849for more recent measurements on the magnetic field depen-
dence, see Ref. 21/H20850.
VII. CONCLUSIONS
We have studied transport properties of a double quantum
dot system with RKKY interaction. Using the numericalrenormalization group, we have calculated the transmissionat finite frequency as a function of temperature and magnetic
field. A quantum phase transition between the Kondoscreened phase and a local spin-singlet state is observed.Moreover, we have shown that both finite temperature and amagnetic field can restore the Kondo resonance in the pres-ence of a RKKY coupling. This crossover back into a Kondoscreened state is in good agreement with recentmeasurements.
10For the differential conductance in a finite
magnetic field, we predict a multiple peak structure which isyet to be observed in future experiments.
ACKNOWLEDGMENTS
The authors would like to thank M. Wegewijs, M. Vojta,
and H. van der Zant for useful discussions and feedback.This work was supported by the Center for Functional Nano-structures /H20849CFN /H20850Karlsruhe /H20849C.H.C. /H20850. C.H.C. acknowledges
support from the National Science Council of Taiwan/H20849NSCT /H20850and the MOE ATU Program of Taiwan, R.O.C.
1A. C. Hewson, The Kondo Problem to Heavy Fermions /H20849Cam-
bridge University Press, Cambridge, 1997 /H20850.
2L. Kouwenhoven and L. Glazman, Phys. World 14,3 3 /H208492001 /H20850.
3D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-
Magder, U. Meirav, and M. A. Kastner, Nature /H20849London /H20850391,
156 /H208491998 /H20850; W. G. van der Wiel, S. De Franceschi, T. Fujisawa,
J. M. Elzerman, S. Tarucha, and L. P. Kouwenhoven, Science
289, 2105 /H208492000 /H20850.
4L. I. Glazman and M. E. Raikh, Sov. Phys. JETP 47, 452 /H208491988 /H20850;
T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 /H208491988 /H20850.
5L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R.
M. Westervelt, and N. S. Wingreen, in Mesoscopic Electron
Transport , NATO ASI Series E: Applied Science, edited by L. L.
Sohn, L. P. Kouwenhoven, and G. Schön /H20849Kluwer, Dordrecht,
1997 /H20850, Vol. 345, pp. 105–214.
6M. A. Ruderman and C. Kittel, Phys. Rev. 96,9 9 /H208491954 /H20850;T .
Kasuya, Prog. Theor. Phys. 16,4 5 /H208491956 /H20850; K. Yosida, Phys. Rev.106, 893 /H208491957 /H20850.
7C. Jayaprakash, H. R. Krishna-murthy, and J. W. Wilkins, Phys.
Rev. Lett. 47, 737 /H208491981 /H20850; B. A. Jones and C. M. Varma, ibid.
58, 843 /H208491987 /H20850; B. A. Jones, C. M. Varma, and J. W. Wilkins,
ibid. 61, 125 /H208491988 /H20850; B. A. Jones and C. M. Varma, Phys. Rev.
B40, 324 /H208491989 /H20850; I. Affleck, A. W. W. Ludwig, and B. A. Jones,
ibid. 52, 9528 /H208491995 /H20850.
8O. Sakai and Y. Shimizu, J. Phys. Soc. Jpn. 61, 2333 /H208491992 /H20850;61,
2348 /H208491992 /H20850.
9N. J. Craig, J. M. Taylor, E. A. Lester, C. M. Marcus, M. P.
Hanson, and A. C. Gossard, Science 304, 565 /H208492004 /H20850.
10H. B. Heersche, Z. de Groot, J. A. Folk, L. P. Kouwenhoven, H.
S. J. van der Zant, A. A. Houck, J. Labaziewicz, and I. L.Chuang, Phys. Rev. Lett. 96, 017205 /H208492006 /H20850.
11P. Simon, R. López, and Y. Oreg, Phys. Rev. Lett. 94, 086602
/H208492005 /H20850.
12R. López, R. Aguado, and G. Platero, Phys. Rev. Lett. 89, 136802−0.02 −0.01 −0.01 −0.005 00 . 0 1 0 0.02 0.005 0.0101
023
0.54
11.52
B=0B=0.008
T()ω
ωB=0
B=0.001
B=0.002
B=0.003
B=0.004
B=0.005
T( )ω
ω(a) (b)
FIG. 5. /H20849Color online /H20850/H20849a/H20850Transmission coefficient T/H20849/H9275/H20850of the coupled quantum dots for different magnetic fields. Here, U=1,
/H9280d1=/H9280d2=−0.5, /H90031L=/H90031R=/H90032L=/H90032R=0.1, /H9011=4,J=0.007, Tk/H110150.0025 /H20849forJ=0/H20850,Jc/H110150.005, T=0.000 01. The values of Bare in steps of
0.001, and the traces of T/H20849/H9275/H20850are shifted in steps of 400 B./H20849b/H20850Transmission vs frequency for ferromagnetic RKKY coupling J=−0.005 and
different magnetic fields. The remaining parameters are the same as in /H20849a/H20850.KONDO EFFECT IN COUPLED QUANTUM DOTS WITH … PHYSICAL REVIEW B 76, 045329 /H208492007 /H20850
045329-5/H208492002 /H20850.
13M. G. Vavilov and L. I. Glazman, Phys. Rev. Lett. 94, 086805
/H208492005 /H20850.
14K. G. Wilson, Rev. Mod. Phys. 47, 773 /H208491975 /H20850; T. A. Costi, A. C.
Hewson, and V. Zlati ć, J. Phys.: Condens. Matter 6, 2519
/H208491994 /H20850; W. Hofstetter, Phys. Rev. Lett. 85, 1508 /H208492000 /H20850.
15W. Hofstetter and H. Schoeller, Phys. Rev. Lett. 88, 016803
/H208492002 /H20850; M. Vojta, R. Bulla, and W. Hofstetter, Phys. Rev. B 65,
140405 /H20849R/H20850/H208492002 /H20850.
16W. Izumida, O. Sakai, and S. Tarucha, Phys. Rev. Lett. 87,
216803 /H208492001 /H20850.17M. Pustilnik and L. I. Glazman, Phys. Rev. Lett. 87, 216601
/H208492001 /H20850.
18W. Hofstetter and G. Zarand, Phys. Rev. B 69, 235301 /H208492004 /H20850;
M. Pustilnik, L. I. Glazman, and W. Hofstetter, ibid. 68,
161303 /H20849R/H20850/H208492003 /H20850.
19Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. 70, 2601
/H208491993 /H20850.
20M. Pustilnik, Y. Avishai, and K. Kikoin, Phys. Rev. Lett. 84, 1756
/H208492000 /H20850; M. Pustilnik and L. I. Glazman, Phys. Rev. B 64,
045328 /H208492001 /H20850.
21E. Osorio et al. /H20849unpublished /H20850.CHUNG-HOU CHUNG AND WALTER HOFSTETTER PHYSICAL REVIEW B 76, 045329 /H208492007 /H20850
045329-6 |
PhysRevB.84.195404.pdf | PHYSICAL REVIEW B 84, 195404 (2011)
Transport properties of graphene across strain-induced nonuniform velocity profiles
F. M. D. Pellegrino,1,2G. G. N. Angilella,1,2,3,4and R. Pucci1,2
1Dipartimento di Fisica e Astronomia, Universit `a di Catania, Via S. Sofia, 64, I-95123 Catania, Italy
2CNISM, UdR Catania, I-95123 Catania, Italy
3Scuola Superiore di Catania, Universit `a di Catania, Via V aldisavoia, 9, I-95123 Catania, Italy
4INFN, Sezione di Catania, I-95123 Catania, Italy
(Received 8 July 2011; revised manuscript received 11 September 2011; published 2 November 2011)
We consider the effect of uniaxial strain on ballistic transport in graphene, across single and multiple tunneling
barriers. Specifically, we show that applied strain not only shifts the position of the Dirac points in reciprocalspace, but also induces a deformation of the Dirac cones, and that both effects are of the same order on the appliedstrain intensity. We therefore study the deviations thereby induced on the angular dependence of the tunnelingtransmission across a single barrier, as well as on the conductivity and Fano factor across a single barrier anda superstructure of several, periodically repeated, such sharp barriers. Our model is generalized to the case ofnonuniform barriers, where either the strain or the gate potential profiles may depend continuously on position.This should afford a more accurate description of realistic “origami” nanodevices based on graphene, where“foldings” are expected to involve several lattice spacings.
DOI: 10.1103/PhysRevB.84.195404 PACS number(s): 73 .20.Mf, 62 .20.−x, 81.05.ue
I. INTRODUCTION
Graphene is an atomically thin, two-dimensional layer of
carbon atoms arranged according to a honeycomb lattice.After having being speculated since long ago as the ideal
building block of graphite and other sp
2carbon compounds,
it has been recently obtained in the laboratory,1thereby
kindling an extraordinary outburst of experimental as wellas theoretical research activity.
2,3Reduced dimensionality
and its peculiar structure conspire toward the formation of
low-energy quasiparticles, which can be described as massless
Dirac fermions with a cone dispersion relation in reciprocalspace around the so-called Dirac points K,K
/prime, and a linearly
vanishing density of states (DOS) at the Fermi level. This
is reflected in several electronic properties already in the
noninteracting limit, e.g., Klein tunneling,4–8the reflectivity,9
the optical conductivity,10–14and the plasmon dispersion
relation.15–18
Graphene is also remarkable for its exceptional mechanical
properties, as is generic for most carbon compounds. For in-stance, notwithstanding its reduced dimensionality, grapheneis characterized by a relatively large tensile strength andstiffness,
19with graphene sheets being capable to sustain
elastic deformations as large as ≈20%.20–24Larger strains
would then induce a semimetal-to-semiconductor transition,with the opening of an energy gap,
25–28and it has been
demonstrated that such an effect critically depends on the
direction of applied strain.14,29The effect of uniaxial strain
on the linear-response electronic properties of graphene hasbeen studied on quite general grounds.
30
Recently, it has been suggested that graphene-based elec-
tronic devices might be designed by suitably tailoring theelectronic structure of a graphene sheet under applied strain.
31
Indeed, a considerable amount of work has been devotedto the study of the transport properties in graphene acrossstrain-induced single and multiple barriers.
32,33There, the
main effect of strain has usually been considered to be thatof shifting the position of the Dirac points in reciprocal space.However, it has been demonstrated that a nonuniform spacevariation of the underlying gate potential would result in a
modulation of the Fermi velocity.
32,34,35
Here, we show that both effects are of the same order on
the applied strain intensity, and should therefore be considered
on the same ground, when studying the transport properties of
strained graphene. We shall therefore explicitly consider notonly the strain-induced displacement of the Dirac points inreciprocal space, but also a strain-induced deformation of theDirac cones, resulting in a strain-dependent anisotropic Fermi
velocity. Specifically, we will consider tunneling through a
single strain-induced sharp barrier, possibly subjected to agate potential, and through a superstructure made of severalsuch barriers, periodically repeated. More interestingly, we
will generalize our results to the problem of transport through
a tunneling structure, characterized by a nonuniform variation
of both the Fermi velocity and of the gate potential, as can,e.g., be brought about by a continuous deformation or applieduniaxial strain.
The paper is organized as follows. After introducing our
model in Sec. II, we discuss the effect of a strain-induced
modulation of the Fermi velocity on the angular dependence ofthe transmission across a single sharp barrier, as well as on theconductivity and Fano factor for ballistic transport (Sec. III).
We then consider the case of several such barriers, arrangedin a periodic fashion (Sec. IV). In Sec. V, we generalize our
results to the case of nonuniform strain across a smooth barrier.
Finally, in Sec. VIwe summarize and give directions for future
investigation.
II. MODEL
In unstrained graphene, low-energy quasiparticles can be
described by the linear Hamiltonian in momentum space,
H(0)=¯hvFIσ·p, (1)
where vFis the Fermi velocity, σ=(σ1,σ2), with σiandτi
(i=1,2,3) Pauli matrices and Ithe identity matrix associated
with the two-dimensional spaces of the sublattices ( AandB,
195404-1 1098-0121/2011/84(19)/195404(11) ©2011 American Physical SocietyF. M. D. PELLEGRINO, G. G. N. ANGILELLA, AND R. PUCCI PHYSICAL REVIEW B 84, 195404 (2011)
say), and of the two valleys around the Dirac points ( Kand
K/prime), respectively. Equation ( 1) acts on the four-component
spinors,36,37
/Psi1p=(/Psi1A,K(p),/Psi1B,K(p),/Psi1B,K/prime(p),−/Psi1A,K/prime(p))/latticetop,(2)
where pis measured from the Dirac point one is referring
to. Here and below, a superscript zero denotes absence ofstrain. The effect of uniaxial strain in real space is that
of modifying the lattice vectors as δ
/lscript=(I+ε)·δ(0)
/lscript(/lscript=
1,2,3), where δ(0)
1=a(√
3,1)/2,δ(0)
2=a(−√
3,1)/2,δ(0)
3=
a(0,−1) are the relaxed (unstrained) vectors connecting two
nearest-neighbor (NN) carbon sites, with a=1.42˚A, the
equilibrium C-C distance in a graphene sheet,2andεis the
strain tensor,26
ε=1
2ε[(1−ν)I+(1+ν)A(θ)], (3)
where
A(θ)=cos(2θ)σz+sin(2θ)σx, (4)
where the Pauli matrices now are understood to act on vectors
of the two-dimensional direct or reciprocal lattice. In Eq. ( 3),θ
denotes the angle along which the strain is applied, with respectto the xaxis in the lattice coordinate system, εis the strainmodulus, and νis Poisson’s ratio. While in the hydrostatic limit
ν=−1 and ε=εI, in the case of graphene one has ν=0.14,
as determined from ab initio calculations,
38to be compared
with the known experimental value ν=0.165 for graphite.39
The special values θ=0 andθ=π/2 refer to strain along the
zigzag and armchair directions, respectively.
The possibility of describing the effects of strain through
Eq. ( 3), i.e., elastically, implies that applied strain does not
induce any irreversible process or mechanical failure of thegraphene sheet, such as dislocations, grain boundaries, orcracks. In fact, such dramatic effects are not expected forstrain below ∼20%, as is predicted by calculations within
density-functional theory,
21,40and confirmed experimentally
by means of atomic force microscopy (AFM).41
In momentum space, the effect of uniaxial strain on
the Hamiltonian Eq. ( 1) is likewise accounted for by the
strain tensor, Eq. ( 3). This is usually described as a shift in
momentum space of the location of the Dirac points. However,starting from the more general, tight-binding Hamiltonian,
2
expanding to first order in the strain modulus, and to secondorder in the impulses, one may show that applied strain alsoinduces a deformation of the Dirac cones, at the same (first)order in ε. Explicitly, one finds
H=¯hvFσ1/braceleftbig/bracketleftbig
1+/parenleftbig1
2−κ0/parenrightbig
ε(1−ν)+/parenleftbig1
2−1
2κ0/parenrightbig
ε(1+ν) cos(2 θ)/bracketrightbig
px+/parenleftbig1
2−1
2κ0/parenrightbig
ε(1+ν)s i n ( 2 θ)py/bracerightbig
+¯hvFσ2/braceleftbig/bracketleftbig
1+/parenleftbig1
2−κ0/parenrightbig
ε(1−ν)−/parenleftbig1
2−1
2κ0/parenrightbig
ε(1+ν) cos(2 θ)/bracketrightbig
py+/parenleftbig1
2−1
2κ0/parenrightbig
ε(1+ν)s i n ( 2 θ)px/bracerightbig
−1
4¯hvFτ3/bracketleftbig
σ1/parenleftbig
p2
x−p2
y/parenrightbig
−2σ2pxpy/bracketrightbig
−¯hvFτ3σ1ε(1+ν) cos(2 θ)+¯hvFτ3σ2ε(1+ν)s i n ( 2 θ), (5)
where κ0=(a/2t)|∂t/∂ a|≈1.6 is related to the logarithmic
derivative of the nearest-neighbor hopping tatε=0.
Our model is based on the tight-binding approximation for
the band structure, including only nearest-neighbor hopping.To this level of approximation, one does not observe anystrain-induced modification of the work function /Phi1. In order
to include also such effects, one needs to consider alsonext-nearest-neighbor hopping.
2Making use of the expression
for the hopping function between two neighboring carbon p
orbitals involved in a πbond, as a function of the bond length
/lscript,Vppπ(/lscript)=t0e−3.37(/lscript/a−1), with t0=−2.7e V ,26one finds
/Phi1=3
2(1−ν)√
3adVppπ(/lscript)
d/lscript/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/lscript=√
3aε≈1.7e V×ε,(6)
viz. a scalar term, going linear with the strain modulus ε, whose
order of magnitude agrees with the ab initio results of Ref. 23.
At any rate, the work function, Eq. ( 6), can be absorbed in an
effective scalar potential U, which we conventionally refer to
as a gate potential below.
Another effect that is not explicitly considered in our model
is the deformation of the πorbitals due to off-plane bending,
as would be, e.g., generated by an AFM tip. However, achange in the hopping parameters due to the bending ofthe graphene sheet can be described as an effective in-planestrain.
42Specifically, one may expect that strain induced by
an AFM tip would be characterized by cylindrical symmetry,which is beyond the scope of the present work, whereonly linear barriers are considered. We note in passing that
other efficient ways to realize controllable strain consists indepositing graphene on top of deformable substrates.
43,44
The spectrum of the strained Hamiltonian, Eq. ( 5), is
still linear, but now around the shifted Dirac points qDa=
±(κ0ε(1+ν) cos(2 θ),−κ0ε(1+ν)s i n ( 2 θ))/latticetop. To first order
in the wave-vector displacement q=p∓qDfrom such shifted
Dirac points, one finds
H=¯hvFσ·q/prime, (7)
where
q/prime={[1−κε(1−ν)]I−κε(1+ν)A(θ)}q, (8)
andκ=κ0−1
2. However, it is convenient to work in the
reference frame with the xaxis along the direction of applied
strain. This is accomplished by a rotation in the sublattice AB
space, described by the unitary matrix
U(θ)=/parenleftbigg10
0e−iθ/parenrightbigg
, (9)
so that
H=¯hvFU†(θ)[σ1(1−λxε)qx+σ2(1−λyε)qy]U(θ),
(10)
195404-2TRANSPORT PROPERTIES OF GRAPHENE ACROSS ... PHYSICAL REVIEW B 84, 195404 (2011)
where λx=2κ,λy=−2κν. After the rotation, Eq. ( 9), the
location of the Dirac points is given by
qDa=±(κ0ε(1+ν) cos(3 θ),−κ0ε(1+ν)s i n ( 3 θ))/latticetop.(11)
The density operator can be expressed as
ρ(r)=/Psi1†(r)/Psi1(r), (12)
where /Psi1(r)=(2π)−2/integraltext
d2ke−ik·r/Psi1(k). Correspondingly, the
current density operator can be derived as45J=−ie
¯h[H,r],
yielding
Ji(r)=−evF/Psi1†(r)(1−λiε)U†(θ)σiU(θ)/Psi1(r).(13)
In the following, for the sake of definitiveness, we shall restrict
to the valley Konly, thus having q=p−qD.
III. TUNNELING ACROSS A SINGLE BARRIER
Potential barriers for single quasiparticle tunneling in
graphene are conventionally designed by suitably changingthe underlying gate voltage. Recently, it has been suggestedthat an equivalent effect may be induced by local uniaxialstrain.
31,42Therefore we start by considering a strain-induced
one-dimensional steplike barrier, characterized by uniaxialstrain applied along the direction θ, with respect to the x
axis, Eq. ( 3), with strain modulus εfor 0/lessorequalslantx/lessorequalslantD, and
zero otherwise. Correspondingly, the Hamiltonian and currentdensity vector are given by Eqs. ( 10) and ( 13), respectively.
In addition, for the sake of generality, we may also consider anonzero gate potential V
gwithin the barrier (Fig. 1).
Since we are interested in stationary solutions and the strain
barrier is uniform along the ydirection, the energy Eand the
component kyof the wave vector of an incoming wave are
conserved. We look therefore for solutions of the stationaryDirac equation of the form
ψ(x,y)=⎧
⎪⎨
⎪⎩U
†(θ)ψI(x)eikyy,x < 0,
U†(θ)ψII(x)eikyy, 0/lessorequalslantx/lessorequalslantD,
U†(θ)ψIII(x)eikyy,x > D ,(14)
where
ψI(x)=/bracketleftbigg1√
2/parenleftbigg1
seiϕ/parenrightbigg
eikxx+r√
2/parenleftbigg1
−se−iϕ/parenrightbigg
e−ikxx/bracketrightbigg
,(15a)
xk qk
I II III0Dε , θ , Vg
FIG. 1. One-dimensional single tunneling barrier along the x
direction. Region II (0 /lessorequalslantx/lessorequalslantD) is characterized by applied strain ε
along the θdirection, as well as by a gate voltage Vg.ψII(x)=/bracketleftbigga√
2/parenleftbigg1
s/primeeiα/parenrightbigg
ei(qx+qD)x
+b√
2/parenleftbigg1
−s/primee−iα/parenrightbigg
e−i(qx−qD)x/bracketrightbigg
, (15b)
ψIII(x)=t/parenleftbigg1
seiϕ/parenrightbigg
eikxx. (15c)
In Eqs. ( 15),ϕdenotes the angle of incidence with respect
to the barrier, kx=(|E|/¯hvF) cosϕ,ky=(|E|/¯hvF)s i nϕ,
(E−Ug)2=¯h2v2
F[(1−λxε)2q2
x+(1−λyε)2(ky−qDy)2],s/prime=
sgn (E−Ug), with Ug=−eVg. Propagating waves corre-
spond to real values of qx, while evanescent waves correspond
to having qxpurely imaginary.
Given the stationary character of the solution, the continuity
equation implies that ∇·J=0 everywhere. In particular,
/angbracketleftJ/angbracketright≡/angbracketleftψ|J|ψ/angbracketrightmay only depend on x, therefore /angbracketleftJx/angbracketrightis con-
stant. The latter condition implies, at the barrier boundaries,
ψI(0−)=(1−λxε)−1/2ψII(0+), (16a)
(1−λxε)−1/2ψII(D−)=ψIII(D+). (16b)
Enforcing the above conditions in Eqs. ( 15), one eventually
finds for the tunneling transmission, T=|t|2,
T=C2cos2ϕ
C2cos2ϕcos2(qxD)+(1−ss/primeSsinϕ)2sin2(qxD),
(17)
where qy=ky−qDy,qx=(1−λxε)−1|(E−Ug)2/¯h2v2
F−
(1−λyε)2q2
y|1/2,C=(1−λxε)¯hvFqx/|E−Ug|,S=(1−
λyε)¯hvFqy/|E−Ug|.
A. Angular dependence
In order to discuss the dependence of the tunneling trans-
mission on the incidence angle ϕ, we preliminarily observe
that propagation within the barrier is allowed whenever
¯h2v2
F(1−λyε)2(ky−qDy)2/lessorequalslant(E−Ug)2, (18)
where ky=(E/¯hvF)s i nϕ. Within such a range, one has
moreover total transmission ( T=1) whenever
qxD=nπ, (19)
nbeing an integer. Equation ( 18) differs from the usual
condition for propagation across strain-induced barriers31in
that we are not only considering a shift of the Dirac point qD,
but also a strain-induced deformation of the Dirac cone, hereexemplified by the substitution v
F/mapsto→vF(1−λyε).
Figures 2and3show our results for the tunneling transmis-
sionT=T(ϕ), Eq. ( 17), as a function of the incidence angle
ϕ,f o rE=80 meV , D=100 nm (Fig. 2) andE=150 meV ,
D=100 nm (Fig. 3). In both figures, the left (respectively,
right) panel refers to uniaxial strain applied along the zigzag(θ=0; respectively, armchair, θ=π/2) direction.
In the case of strain applied along the zigzag direction ( θ=
0, Figs. 2and3, left panels), curves (b) neglect a strain-induced
deformation of the Dirac cone. Comparison with curves (c),where such a deformation is fully included, shows that theeffect of a strain-induced anisotropy of the Fermi velocity
195404-3F. M. D. PELLEGRINO, G. G. N. ANGILELLA, AND R. PUCCI PHYSICAL REVIEW B 84, 195404 (2011)
-1-0.5 0 0.5 1
0 0.5 1θ = 0
(a)
(b)
(c)
-1-0.5 0 0.5 1
0 0.5 1θ = π / 2
FIG. 2. (Color online) Dependence on the incidence angle ϕof
the tunneling transmission T,E q .( 17). Left panel refers to strain
applied along the zigzag direction ( θ=0), and (a) ε=0.03,Ug=
0m e V ;( b ) ε=0.03,Ug=−20 meV (the strain-induced deformation
of the Dirac cone is neglected); (c) ε=0.03,Ug=−20 meV . Right
panel refers to strain applied along the armchair direction ( θ=π/2),
and (a) ε=0.01,Ug=0m e V ;( b ) ε=0.01,Ug=0 meV (the strain-
induced deformation of the Dirac cone is neglected); (c) ε=0.01,
Ug=−20 meV .
is that of shifting the angular location of the maxima [ T=1,
Eq. (19)] of the tunneling transmission. Such an effect becomes
more important with increasing energy (from Fig. 2to Fig. 3),
while the number of peaks increases, Eq. ( 19), and the angular
range in which the propagating regime is allowed widens. Theeffect of a strain-induced deformation of the Dirac cone is evenmore dramatic in the absence of a gate potential [ U
g=0m e V ,
curve (a)]. Indeed, in such a case, neglecting the Fermi velocityanisotropy for strain applied along the zigzag direction wouldyield a uniform tunneling transmission T=1, for all incidence
angles ϕ, whereas we find that transmission via propagating
waves is allowed only for |ϕ|/lessorequalslantarcsin[(1 −λ
yε)−1], with
small oscillations below T=1 within, and evanescent waves
beyond that range. A similar analysis applies to the case ofstrain applied along the armchair direction ( θ=π/2, Figs. 2
-1-0.5 0 0.5 1
0 0.5 1θ = 0
(a)
(b)
(c)
-1-0.5 0 0.5 1
0 0.5 1θ = π / 2
FIG. 3. (Color online) Same as Fig. 2,b u tf o r E=150 meV and
D=100 nm.gLL
WV Vy
x
V
LD
FIG. 4. Schematic top view of a graphene layer contacted by
metallic leads, as considered in Sec. III B.
and3, right panels), which is characterized by an asymmetric
transmission T=T(ϕ), with pronounced oscillations for ϕ>
0 close to the propagating edge.
The origin of such an asymmetry of the ϕdependence
of the transmission can be traced back to the particularDirac cone vertex q
D, whose shift is here considered. Global
symmetry would be restored upon inclusion of the other Diraccone. In that case, one would obtain the same picture, butwithϕ/mapsto→−ϕ. It should be emphasized that the stationarity
condition, Eq. ( 19), characterizes the occurrence of peaks in
the transmission T(ϕ) in any case. In addition, for a potential
barrier, in the absence of strain, one also recovers completetransmission ( T=1) atϕ=0 (Klein tunneling).
Summarizing, at variance with previous studies,
31from
Eq. ( 17) one obtains that the overall effect of a strain-
induced deformation of the Dirac cones is that of shifting thetransmission peaks, and of reducing the range in ϕat which
transmission takes place.
B. Ballistic transport
We now consider a more realistic device, viz. a graphene
strip of length Land width W, subjected to two leads at a
distance D(Fig. 4).33,34,46,47Following Ref. 46, we assume
thatW/L/greatermuch1, and that the gate potential within the strip
is much less than the potential of the leads, |Vg|/lessmuch|VL|.
Moreover, we assume the graphene strip to be characterizedby uniaxial strain, with modulus εand strain direction θ, and
explicitly consider the deformation of the Dirac cones inducedby the applied strain. The energy levels of Ref. 46are therefore
modified into
E=U
L+s¯hvF/radicalBig
k2x+k2y,x < 0,x > D , (20a)
=Ug+s/prime¯hvF/radicalBig
(1−λxε)2q2x+(1−λyε)2q2y,
0<x<D , (20b)
195404-4TRANSPORT PROPERTIES OF GRAPHENE ACROSS ... PHYSICAL REVIEW B 84, 195404 (2011)
where again qy=ky−qDy, andUL=−eVLandUg=−eVg.
The limit |VL|→∞ is equivalent to the limit ϕ→0, and the
transmission, Eq. ( 17), reduces to
Tprop
α(ky)=1
cos2(qxD)+η(ky)s i n2(qxD), (21)
for propagating waves in the valley α=K, where
η(ky)=(E−Ug)2
(E−Ug)2−¯h2v2
F(1−λyε)2(ky±qDy)2,(22)
and the minus (respectively, plus) sign applies to the valley
α=K(respectively, α=K/prime). Analogous expressions hold
for the transmission Tevan
α(ky) in the evanescent case, with
η(ky)/mapsto→−η(ky), cos( qxD)/mapsto→cosh(qxD), and sin( qxD)/mapsto→
sinh(qxD). The transmission for a general (propagating or
evanescent) wave therefore reads
Tα(ky)=/Theta1[η(ky)]Tprop
α(ky)+{1−/Theta1[η(ky)]}Tevan
α(ky),
(23)
where /Theta1(t) is the Heaviside (step) function. Integrating over ky
and summing over both valleys, one obtains the conductance
across the barrier (Landauer formula),48,49
G=2e2
hW/summationdisplay
α/integraldisplay∞
−∞dky
2πTα(ky), (24)
where the factor of 2 takes into account for the spin degeneracy,
the conductivity
σ=D
WG, (25)
and the Fano factor50,51
F=1−/summationtext
α/integraltext∞
−∞dky
2πT2
α(ky)
/summationtext
α/integraltext∞
−∞dky
2πTα(ky). (26)
In Eq. ( 25) for the conductivity, the summation over the valleys
contributes with an additional factor of 2, whereas this factorcancels in the definition of the Fano factor, Eq. ( 26).
Before discussing our results, let us observe that the
inclusion of a strain-induced deformation of the Dirac conein the expressions of the conductivity, Eq. ( 25), and of the
Fano factor, Eq. ( 26), amounts to the replacements
D/mapsto→D
eff≡ξD, (27a)
E/mapsto→Eeff≡ζE, (27b)
for the strip width and incident energy, respectively, in the
corresponding expressions, σ(0)andF(0), say, without cone
deformation, with
ξ=1−λyε
1−λxε, (28a)
ζ=1
1−λyε. (28b)
In particular, one explicitly finds
σ(D,E )=ξ−1σ(0)(Deff,Eeff). (29)As a consequence, while lim E→0σ(0)(D,E )=4e2/πh,a
universal constant,52in the presence of applied uniaxial strain
one finds
lim
E→0σ(D,E )=1
ξ4e2
πh. (30)
Only in the case of hydrostatic strain ( ν=−1,λx=λy,ξ=1)
does one recover the universal limit, regardless of the strainmodulus.
34On the other hand, one finds lim E→0F(D,E )=1
3,
corresponding to strongly sub-Poissonian noise,46regardless
of applied strain.
In the opposite limit, the conductivity across a single
barrier in the absence of strain is linear in energy, σ(0)≈
(e2/h)D|E|/¯hvFforE→∞ , with damped oscillations char-
acterized by a pseudoperiod /Delta1Esuch that47D/Delta1E/ ¯hvF=π.
In the presence of strain, such results are modified by Eqs. ( 28),
so that σ(E)≈σ∞(E)f o rE→∞ , with
σ∞(E)=4e2
hD|E|
4ζ, (31)
with damped oscillations characterized by a pseudoperiod
given by
ξζD/Delta1E
¯hvF=π. (32)
In view of the fact that |λx|>|λy|, one may conclude that
applied strain induces a slight change in the slope of σvs|E|,
while it modifies the pseudoperiod of the oscillations moresubstantially.
Figure 5shows our results for the scaled conductivity in the
presence of uniaxial strain ( ε=0.03−0.15) applied along the
armchair direction ( θ=π/2). When the conductivity σ(E)
is normalized with respect to its asymptotic limit, Eq. ( 31),
and plotted against energy Escaled with the strain-dependent
pseudoperiod /Delta1E,E q .( 32), results corresponding to different
values of the strain modulus collapse into a single curve,displaying damped oscillations, as prescribed by Eq. ( 32).
Similarly, Fig. 6reports our results for the Fano factor as
a function of scaled energy. Again, the results for all thestrain moduli here considered ( ε=0.03−0.15) collapse into
0.96 0.98 1 1.02 1.04
0 1 2 3 4 5 6 7 8σ / σ∞
E / Δ E
FIG. 5. Conductivity across a graphene strip ( D=100 nm)
normalized to asymptotic large-energy behavior, Eq. ( 31), vs energy
scaled to the pseudoperiod, Eq. ( 32). Actually shown are four curves,
all collapsing into a single one, corresponding to strain applied along
the armchair direction ( θ=π/2), with ε=0.03, 0.05, 0.10, 0.15.
195404-5F. M. D. PELLEGRINO, G. G. N. ANGILELLA, AND R. PUCCI PHYSICAL REVIEW B 84, 195404 (2011)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0 1 2 3 4 5 6 7 8F
E / Δ E
FIG. 6. Fano factor for ballistic transport across a graphene strip.
All parameters are as in Fig. 5. Dashed lines represent the universal
low- and large-energy asymptotic values, F(0)=1
3andF∞=1
8,
respectively.
a single, oscillating curve. Note that the universal limits
F(E=0)=1
3andF∞≡limE→∞F(E)=1
8are recovered
in all cases, regardless of applied strain. Such results do notdepend on the direction θof applied strain.
IV . TRANSMISSION ACROSS MULTIPLE BARRIERS
We next consider quasiparticle tunneling across Nidentical
barriers, each of width /lscript, two nearest-neighbor (NN) barriers
being separated by the distance /lscript, such that 2 N/lscript=D(Fig. 7).
We assume a position-dependent strain modulus ε(x) and gate
potential energy U(x), with
ε(x)=ε−,(m−1)/lscript/lessorequalslantx/lessorequalslantm/lscript, (33a)
=ε+,m /lscript/lessorequalslantx/lessorequalslant(m+1)/lscript, (33b)
and
U(x)=U−,(m−1)/lscript/lessorequalslantx/lessorequalslantm/lscript, (34a)
=U+,m /lscript/lessorequalslantx/lessorequalslant(m+1)/lscript, (34b)
withm=1,...2N−1. We further consider the possibility
of contacting the two extrema of the chain of barriers with leadsat the potential V
L. Equations ( 16) then suggest to look for a
solution of the Dirac equation in the form
ψ(x,y)=U†(θ)φ(x)√1−λxε(x)eikyy(35)
ε
Dl l lε−+
V−V+
FIG. 7. Schematic plot of the multiple barrier, as considered in
Sec. IV.so that φ(x) is a continuous function at the barriers’ edges.
The stationary Dirac equation for φ(x) can then be casted
in the form of an evolution equation,47so that φ(x)=
T(N)(x,x 0)φ(x0), where the evolution matrix T(N)(x,x 0)i n
turn obeys the equation
d
dxT(N)(x,x 0)=/bracketleftbigg
iq(0)
Dxε(x)τzI+i
¯hvFE−U(x)
1−λxε(x)σx
+1−λyε(x)
1−λxε(x)/parenleftbig
ky−q(0)
Dyε(x)τz/parenrightbig
σz/bracketrightbigg
×T(N)(x,x 0), (36)
withT(N)(x0,x0)=I. For a single barrier, the evolution matrix
is related to the transfer matrix by52
M(1)(x,x 0)=Q−1
s(ϕ)T(1)(x,x 0)Qs(ϕ), (37)
where
Qs(ϕ)=1√
2/parenleftbigg11
seiϕ−se−iϕ/parenrightbigg
(38)
includes the incidence angle ϕof the incoming spinor,
Eq. ( 15a), and s=sgn (E). In the limit of metallic
leads ( |VL|→∞ ), one has ϕ→0, with Q+(0)=1√
2(σz+
σx),Q−1
+(0)=Q+(0), and Q−(0)=Q+(0)σx,Q−1
−(0)=
σxQ+(0). The elements of the transfer matrix can be further-
more related to the elements of the scattering matrix across thebarrier,
S=/parenleftbiggrt
/prime
tr/prime/parenrightbigg
, (39)
where r,t(respectively, r/prime,t/prime) are the amplitudes of the
reflected and transmitted waves in region I (respectively, III),cf. Fig. 1. Indeed, one explicitly finds
52,53
M(1)=/parenleftBigg
(t†)−1r/prime(t/prime)−1
−(t/prime)−1r(t/prime)−1/parenrightBigg
. (40)
Therefore, for the conductance across a single barrier, one
finds
G=2e2
hTr (t†t)=2e2
hTr/parenleftbig/parenleftbig
M(1)†
11M(1)
11/parenrightbig−1/parenrightbig, (41)
where Tr ≡W/summationtext
α/integraltext∞
−∞dky/2π. Correspondingly, the trans-
mission for an incoming quasiparticle with energy Eand trans-
verse wave vector kyin valley αisTα(ky)=(M(1)†
11M(1)
11)−1,
and the expressions for the conductivity, Eq. ( 25), and Fano
factor, Eq. ( 26), follow straightforwardly.
The solution of Eq. ( 36) for the transfer matrix is derived
analytically in the Appendix, both for a single and for a mul-tiple barrier, in the presence of strain-induced deformation ofthe Dirac cone. Making use of Eqs. ( A10) for the transmission
T
α(ky) in Landauer’s formula for the conductivity, Eq. ( 25),
and in the definition for the Fano factor, Eq. ( 26), one again
finds that the conductivity in strained graphene and strainedgraphene where the strain-induced velocity anisotropy has
195404-6TRANSPORT PROPERTIES OF GRAPHENE ACROSS ... PHYSICAL REVIEW B 84, 195404 (2011)
been neglected are related by means of Eqs. (28) and ( 29),
but now with D=2N/lscript, and
ξ=1
2(ξ++ξ−), (42a)
ζ=1
2(ζ++ζ−), (42b)
ξ±=1−λyε±
1−λxε±, (42c)
ζ±=1
1−λyε±. (42d)
Equation ( 30) in the limit E→0 then follows straight-
forwardly, with ξgiven now by Eq. ( 42a). Moreover, the
conductivity at large energies is characterized by an overalllinear behavior, interrupted by dips with decreasing depth,which result from the coherent superposition of the dampedoscillations produced by scattering off the edges of the singlebarriers. The energies E
nat which such dips occur are
asymptotically given by (cf. the Appendix)
En
¯hvFD
N1
2(ξ+ζ++ξ−ζ−)=nπ, (43)
withnan integer.
Figure 8shows our numerical results for the conductivity in
strained graphene, with strain applied nonuniformly along the
armchair direction, across a superlattice of N=10 barriers.
At variance with Fig. 5, we have not scaled σwith its
asymptotic behavior at large energies, Eq. ( 31). As expected,
the overall linear behavior of σ(E) is interrupted by dips,
whose approximate energy location is given by Eq. ( 43).
While such dips get damped as energy increases, they arenonetheless enhanced with respect to the case in which thestrain-induced deformation of the Dirac cones is neglected,
33
especially those corresponding to even integer values of n
in Eq. ( 43). Correspondingly, the Fano factor (Fig. 9)i s
0 10 20 30 40 50 60 70
0 1 2 3 4 5 6 7 8 9σ/σ0
E / E1(a)
(b)
(c)
(d)
(e)
FIG. 8. (Color online) Conductivity σ(E) in units of σ0=4e2/h,
vs energy E, scaled with respect to the approximate location of the
first dip, E1,a sg i v e nb yE q .( 43). Subsequent dips then occur close to
integer values of the ratio E/E 1. Uniaxial strain is applied along the
armchair direction ( θ=π/2) in the case of a multibarrier superlattice,
withN=10 barriers, /lscript=25 nm ( D=500 nm). Different curves
refer to nonuniform strain moduli within and outside NN barriers(cf. Fig. 7), with (a) ε
+=0.004,ε−=0; (b) ε+=0.003,ε−=0;
(c)ε+=0.002,ε−=−0.001; (d) ε+=0.002,ε−=0.001; (e) ε+=
0.0005,ε−=0. In all cases, we set U±=0, for the sake of simplicity. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 1 2 3 4 5 6 7 8 9F
E / E1 (a)
(b)
(c)
(d)
(e)
0 0.1 0.2 0.3 0.4 0.5
0 0.5 1 1.5 2 2.5
E [meV]
FIG. 9. (Color online) Fano factor Fvs scaled energy E/E 1,f o r
transport across a multibarrier superlattice, with nonuniform uniaxial
strain applied along the armchair direction ( θ=π/2). All parameters
are as in Fig. 8. Inset shows the universal low-energy asymptotic
behavior in the various cases. In the limit E→0, the universal
asymptotic value, F(0)=1
3, is recovered.
characterized by essentially analogous features, with bumps
occurring at approximately En,E q .( 43). In particular, the
universal limit at low energy, F(0)=1
3, is recovered as in the
single-barrier case, regardless of applied strain.
Figure 10shows our numerical results for the conductivity
in strained graphene, but now for nonuniform strain appliedalong the zigzag direction. At variance with the armchaircase (Fig. 8), for strain applied along the zigzag direction
the conductivity seems not to be characterized by prominentdips as a function of energy. This may explained by a reducedcoherent superposition of the effects due to each single barrier.However, if the trailing linear dependence on energy is dividedout (Fig. 10, inset), one may again recognize “oscillations,”
with extrema approximatively occurring at E
n, as given by
0 10 20 30 40 50 60 70
0 1 2 3 4 5 6 7 8σ(E)/σ0
E / E1(a)
(b)
(c)
(d)
(e)
0.97 0.98 0.99 1 1.01
1 2 3 4 5 6 7 8
σ(E)/( σ0 E D/4 )
FIG. 10. (Color online) Conductivity σ(E) in units of σ0=
4e2/h,v se n e r g y E, scaled with respect to E1,a sg i v e nb yE q .( 43).
Uniaxial strain is applied along the zigzag direction ( θ=0) in the
case of a multibarrier superlattice, with N=10 barriers, /lscript=25 nm
(D=500 nm). Different curves refer to nonuniform strain moduli
within and outside NN barriers (cf. Fig. 7), with (a) ε+=0,ε−=0;
(b)ε+=0.03,ε−=0; (c)ε+=0.05,ε−=0; (d)ε+=0.07,ε−=0;
(e)ε+=0.10,ε−=0. In all cases, we set U±=0, for the sake of
simplicity. Inset shows the conductivity scaled with respect to its
large-energy asymptotic limit, σ/σ∞, as a function of scaled energy,
E/E 1.
195404-7F. M. D. PELLEGRINO, G. G. N. ANGILELLA, AND R. PUCCI PHYSICAL REVIEW B 84, 195404 (2011)
0.105 0.115 0.125 0.135
0 1 2 3 4 5F
E / E1(a)
(b)
(c)
(d)
(e) 0.2 0.25 0.3 0.35
0 0.5 1 1.5 2 2.5
E [meV]
FIG. 11. (Color online) Fano factor Fvs scaled energy E/E 1,f o r
transport across a multibarrier superlattice, with nonuniform uniaxial
strain applied along the zigzag direction ( θ=0). All parameters are
as in Fig. 10. Note the deviations from the large-energy asymptotic
limit for the unstrained case, F∞=1
8(dashed line). The low-energy
universal limit, F(0)=1
3(inset, dashed line), is recovered, regardless
of strain.
Eq. ( 43). At variance with the armchair case, the Fano factor
exhibits a strain-dependent asymptotic limit, for large energies(Fig. 11), with increasing deviations from the unstrained
behavior F
∞=1
8, with increasing strain modulus ε(at least
within the strain range that has been numerically investigated).On the other hand, both the oscillations as a function ofscaled energy E/E
1and the low-energy limit F(0)=1
3(Fig. 11, inset) are recovered.
V . TRASMISSION ACROSS A SMOOTH BARRIER:
EFFECT OF CONTINUOUS STRAIN
Although considerable insight is afforded by analytical
solutions to the problem of tunneling across single or multiplesharp barriers, there is sufficient evidence, both experimental
54
and theoretical,32that barrier edge effects are also important to
determine the transport properties across corrugated graphene.Here, we therefore consider the case in which uniaxial strain isapplied in a nonuniform but continuous fashion to a graphenesheet, which can be modeled by a single barrier with smooth
strain and gate potential profiles, ε=ε(x) and U=U(x),
respectively. Such a description includes and generalizes,in particular, a continuous Fermi wave-vector profile, asconsidered in Ref. 32.
On quite general grounds, one may expect that a smooth
potential profile (whether induced by strain or by gating)introduces a new length scale, asay [as in Eq. ( 48) below],
which is the linear size over which the potential strain variesappreciably. Such a new length scale has then to be comparedwith the atomic scale, measured by the lattice step a,o n
one hand, and with the Fermi wavelength λ
F=¯hvF/(2πE)
corresponding to the incident energy E, on the other. The
approximation of a sharp barrier (no smoothing) then holdswhenever a/lessmucha/lessmuchλ
F, i.e., at sufficiently large incident
energies. On the other hand, the detailed structure of the barrierneeds to be considered when a∼λ
F. In both cases, we are
interested to the more general and realistic cases where a/lessmucha,
where one may additionally neglect the occurrence of K-K/primecoupling. Indeed, truly sharp electrostatic barriers on the order
of the electron wavelength are quite difficult to be realized,as is, e.g., demonstrated by the occurrence of Fabry-P ´erot
oscillations of the conductance in graphene heterostructuresas narrow as ∼20 nm, where a resonant cavity is formed
between two electrostatically created bipolar junctions.
55Such
oscillations are more accurately described when the smoothstructure of these potential barriers is taken into account,whereas intervalley scattering can be safely neglected (seeSupplementary Information in Ref. 55). Another instance of
nonuniform barrier, where smoothing effects are important, isthe strain-induced ripples superlattice experimentally realizedin Ref. 43, in which smoothing is essential on a length scale
of∼100 nm, whereas intervalley processes are negligible.
The kinetic part of the Hamiltonian for graphene subjected
to uniform strain εalong the direction θis
H=U
†(θ)σi¯hvi/parenleftbigg1
i∇i−qDi/parenrightbigg
U(θ), (44)
where vi=vF(1−λiε), and summation over the repeated
index i=1,2 is understood. In order to generalize Eq. ( 44)
to the case of a nonuniform, but continuous strain profileε=ε(x), one may be tempted to perform the replacements
v
i/mapsto→vi(r)≡vF[1−λiε(x)] and qD/mapsto→qD(r), Eq. ( 11), with
ε=ε(x). However, the resulting Hamiltonian must be sym-
metrized, in order to preserve hermiticity, thus leading to themodel Hamiltonian for a nonuniform strain profile:
H=U
†(θ)σi1
2/bracketleftbigg
¯hvi(r)/parenleftbigg1
i∇i−qDi(r)/parenrightbigg
+/parenleftbigg1
i∇i−qDi(r)/parenrightbigg
¯hvi(r)/bracketrightbigg
U(θ). (45)
Equation ( 45) includes the effect of nonuniform, continuous
strain both as a shift in the position of the Dirac points,and as a deformation of the Dirac cones (nonuniform andanisotropic Fermi velocity), at variance e.g., with Ref. 35,
where a nonuniform velocity is considered, but an isotropicprofile is assumed. As in the case of a single, sharp barrier(Sec. III), continuity of the current density, Eqs. ( 16), suggests
to seek for a solution of the stationary Dirac equation in a formanalogous to Eq. ( 35), viz.
ψ(x,y)=U
†(θ)φ(x)√vx(x)eikyy. (46)
One explicitly finds [cf. Eq. ( 36)]
dφ(x)
dx=/bracketleftbigg1−λyε(x)
1−λxε(x)/parenleftbig
ky−q(0)
Dyε(x)/parenrightbig
σz
+iE−U(x)
[1−λxε(x)]¯hvFσx+iq(0)
Dxε(x)I/bracketrightbigg
φ(x). (47)
We have solved Eq. ( 47) numerically, for the nonuniform,
smooth strain profile
ε(x)=ε0
tanh(D/4a)/parenleftbigg1
1+e−x/a−1
1+e−(x−D)/a/parenrightbigg
,(48)
as shown in Fig. 12. Such a strain profile is essentially flat
for|x−D/2|/lessmucha, where ε(x)≈ε0, and for |x−D/2|/greatermucha,
where ε(x)≈0. In the limit a/D→0, Eq. ( 48) tends to the
sharp barrier considered in Sec. III. Therefore asymptotically
195404-8TRANSPORT PROPERTIES OF GRAPHENE ACROSS ... PHYSICAL REVIEW B 84, 195404 (2011)
0 D2 a
FIG. 12. Schematic single tunneling barrier, with smooth strain
profile, Eq. ( 48). Dashed line depicts a sharp barrier, corresponding
to the limit a→0.
for|x|→∞ , the solutions of Eq. ( 47) must merge into
Eqs. ( 15), in regions I and III. We have therefore taken an
initial value φ(x=x0) in the form of Eq. ( 15c), forx0=5D,
and integrated Eq. ( 47) backward for x/lessmuch0. Comparing
the numerical solution with Eq. ( 15a), one may extract the
reflection coefficient r, relative to an incident wave with
unit amplitude incoming from x> 0, as the Fourier weight
with respect to its negative frequency component, whence thetransmission T(φ) follows straightforwardly. As a cross check
of our procedure, we have also verified that the continuityequation, Eq. ( 16), holds true, within the numerical error.
Figures 13and 14show our numerical results for the
tunneling transmission T(ϕ) across the smooth strain barrier,
Eq. ( 48), with D=100 nm and different values of the
smoothing parameter, a/D. Figure 13refers to an incidence
energy E=80 meV , corresponding to an incident wavelength
λ
F=¯hvF/(2πE)≈1.3 nm. One finds that transmission of
-1-0.5 0 0.5 1
0 0.5 1θ = 0
(a)
(b)
(c)
-1-0.5 0 0.5 1
0 0.5 1θ = π / 2
FIG. 13. (Color online) Tunneling transmission vs incidence
angleϕacross a smooth strain barrier, Eq. ( 48), with D=100 nm,
and incidence energy E=80 meV ( λF=¯hvF/(2πE)≈1.3 nm).
Left panel refers to strain applied along the zigzag direction ( θ=0),
withε0=0.1. Right panel refers to strain applied along the armchair
direction ( θ=0), with ε0=0.01. In both cases, the different lines
correspond to different values of the smoothing parameter, viz.
(a)a=0 (sharp barrier); (b) a=10−2D=1 nm; (c) a=10−1D=
10 nm. In all cases, U(x)=0, for the sake of simplicity.-1-0.5 0 0.5 1
0 0.5 1θ = 0
(a)
(b)
(c)
-1-0.5 0 0.5 1
0 0.5 1θ = π / 2
FIG. 14. (Color online) Same as Fig. 13, but with E=150 meV
(λF≈0.7 nm).
propagating waves is allowed for incidence angles ϕsuch that
ϕcr−/lessorequalslantϕ/lessorequalslantϕcr+, with
ϕcr±=± arcsin/parenleftbigg1
1−λyε0/parenrightbigg
, (49)
in the zigzag case ( θ=0), and ϕ>ϕ cr, with
arcsin/parenleftbigg
−1
1−λyε0+¯hvF
|E|ε0κ(1−ν)/parenrightbigg
, (50)
in the armchair case ( θ=π/2), independent of the smoothing
parameter a/D. Outside that window, transmission takes place
via evanescent waves only, and T(ϕ)≈0. For strain applied
along the zigzag direction ( θ=0, Fig. 13, left panel), Eq. ( 49)
predicts the existence of critical angles |ϕcr±|<π / 2. This
is a direct consequence of the strain-induced deformationof the Dirac cones [ λ
y/negationslash=0i nE q .( 49)]. Both in case of
strain applied along the zigzag and armchair directions,increasing the smoothness parameter a/D away from the limit
of a sharp barrier ( a/D=0) suppresses the oscillations in
T(ϕ) within the propagating window, until a>λ
F, in which
case transmission is almost undisturbed by the presence ofthe barrier. These results are confirmed by Fig. 14, where
we consider quasiparticles with larger incident energy E=
150 meV , corresponding to a smaller Fermi wavelength λ
F≈
0.7 nm. While the transmission window widens and the num-
ber of oscillations increases, smoothening the strain profileimmediately washes out the deviations of the tunneling trans-mission from unity. In ending this section, we note that the pro-cedure applied to extracting the tunneling transmission fromthe numerical solution of Eq. ( 47) can be generalized, in princi-
ple, to the case of an arbitrary nonuniform strain potential, such
as a superlattice of several smooth barriers, such as Eq. ( 48).
VI. CONCLUSIONS
We have studied the effect of a strain-induced modulation of
the Fermi velocity on several transport properties of graphene,such as the angular dependence of the tunneling transmission,the conductivity, and the Fano factor. After considering thecases of a single sharp tunneling barrier, and of a superstructureof several, periodically repeated, such sharp barriers, we have
195404-9F. M. D. PELLEGRINO, G. G. N. ANGILELLA, AND R. PUCCI PHYSICAL REVIEW B 84, 195404 (2011)
specifically studied the case in which both the modulus of
applied uniaxial strain, and possibly an applied gate potential,depend continuously on position. This is expected to afford amore accurate description of real “origami” device,
31in which
“foldings” of a graphene sheet would conceivably involveseveral lattice spacings. In the case of sharp tunneling barriers,we have demonstrated that the effect of a strain-induceddeformation of the Dirac cone is of the same order of thestrain-induced shift of the Dirac points, and should thereforebe taken into account on the same basis. In particular, wehave found that strain modifies the quasiperiod in energythat regulates the occurrence of dips in the conductivityacross a superstructure of several sharp barriers, due tocoherent scattering off their edges. Such effect is, however,less dramatic in the energy dependence of the Fano factor.Finally, we have generalized our results to embrace the caseof a generic nonuniform strain, and possibly a gate potential,
profile. Besides allowing a more accurate analysis of tunnelingtransmission across smooth barriers, especially at low incidentenergies, which are expected to be more sensitive to localdeviations from uniformity, such an approach can be appliedto describe arbitrary strain superstructures, albeit numerically.
Among the already available experimental results, which
could be described in terms of a strain-induced deformationof the Dirac cones, we mention Raman spectroscopy
56and the
strain dependence of both the longitudinal and the recently pre-dicted transverse plasmon mode.
30,57Moreover, transmittance
measurements with polarized light between the near-infraredand the ultraviolet on uniaxially strained graphene may provideinformation on the Dirac cone deformation.
9,30
Note added in proof . Recently, we became aware of Ref. 59,
where the effect of anisotropic mechanical strain on sometransport properties of graphene is studied.
ACKNOWLEDGMENTS
F. M. D. P. acknowledges D. M. Basko for discussions and
correspondence over the general area embraced by the presentwork.
APPENDIX: TRANSFER MATRIX ACROSS A MULTIPLE
BARRIER
In the case of a single barrier [( N=1, 2/lscript=D), Fig. 7],
Eq. ( 36) for the transfer matrix admits the analytical solution
M(1)(D,0)=exp/parenleftbig
iq(0)
DxεD/parenrightbig
exp/parenleftbiggi
¯hvF(E−Ug)D
1−λxεσz
+1−λyε
1−λxε/parenleftbig
ky−q(0)
Dy/parenrightbig
Dσx/parenrightbigg
, (A1)
corresponding to the initial condition M(1)(0,0)=I, and to
a uniform strain εand to a gate potential energy Ugacross
the barrier. The second matrix exponential in Eq. ( A1) can be
made more explicit, by making use of the following identityfor a linear combination of the Pauli matrices,
exp(a·σ)=sinha
aa·σ+Icosha, (A2)
where a=(/summationtext
ia2
i)1/2, andai∈C(i=1,2,3).We next consider a single barrier, but now with nonuniform
strain modulus and gate potential energy, i.e., ε(x)=ε−and
U(x)=U−within the barrier (0 <x</lscript ), andε(x)=ε+and
U(x)=U+beyond the barrier’s second edge ( /lscript<x< 2/lscript;c f .
Fig. 7). In this case, one finds M(1)(2/lscript,0)=M+(/lscript)M−(/lscript),
where M±(/lscript) are given by Eq. ( A1), with D/mapsto→/lscript,ε/mapsto→ε±,
andU/mapsto→U±. One finds
M(1)(2/lscript,0)=eiq(0)
Dx(ε++ε−)/lscript˜M1, (A3)
where ˜M1is a unimodular matrix, det ˜M1=1. Specifically,
one finds
(˜M1)11=λ+iη, (A4)
where
λ=sinh(q−/lscript)
q−sinh(q+/lscript)
q+(κ−κ+−u−u+)
+cosh(q−/lscript) cosh( q+/lscript), (A5a)
η=u−
q−sinh(q−/lscript) cosh( q+/lscript)
+u+
q+sinh(q+/lscript) cosh( q−/lscript), (A5b)
with
κ±=1−λyε±
1−λxε±/parenleftbig
ky−q(0)
Dyε±/parenrightbig
, (A6a)
u±=E−U±
¯hvF(1−λxε±), (A6b)
q±=/radicalBig
κ2
±−u2
±, (A6c)
whence Eq. ( 41) follows straightforwardly.
Finally, in the case of Nbarriers ( D=2N/lscript,F i g . 7),
iterating Eq. ( A3)Ntimes, one has
M(N)(D,0)=eiq(0)
Dx(ε++ε−)N/lscript˜MN
1, (A7)
where for the Nth power of the unimodular matrix ˜M1one may
use an identity due to Chebyshev,58and specifically obtain
/parenleftbig˜MN
1/parenrightbig
11=sinh(Nz)
sinhz(˜M1)11−sinh[(N−1)z]
sinhz.(A8)
Here, we have denoted the eigenvalues of ˜M1bye±z, with
z∈C. Finally, one finds for the transmission
T(ky)=[(M(N)(D,0))∗
11(M(N)(D,0))11]−1
=/bracketleftbigg
cosh2(Nz)+η2
λ2−1sinh2(Nz)/bracketrightbigg−1
.(A9)
Sinceλ=1
2Tr˜M1=coshz, one finds explicitly
Tα(ky)≡Tprop
α(ky)
=/bracketleftbigg
cos2(Ny)+η2
λ2−1sin2(Ny)/bracketrightbigg−1
, (A10a)
withy=arccos λ,i f|λ|<1,
≡Tevan
α(ky)
=/bracketleftbigg
cosh2(Nx)+η2
λ2−1sinh2(Nx)/bracketrightbigg−1
, (A10b)
withx=ln|λ+√
λ2−1|,i f|λ|>1,
=[1+η2N2]−1, (A10c)
if|λ|=1. In particular, one finds λ∼cos(u+/lscript+u−/lscript), for
E→∞ , whence Eq. ( 43) follows.
195404-10TRANSPORT PROPERTIES OF GRAPHENE ACROSS ... PHYSICAL REVIEW B 84, 195404 (2011)
1K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V . V . Khotkevich,
S. V . Morozov, and A. K. Geim, Proc. Nat. Acad. Sci. 102, 10451
(2005).
2A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, andA. K. Geim, Rev. Mod. Phys. 81, 000109 (2009).
3D. S. L. Abergel, V . Apalkov, J. Berashevich, K. Ziegler, and
T. Chakraborty, Adv. Phys. 59, 261 (2010).
4J. M. Pereira, V . Mlinar, F. M. Peeters, and P. Vasilopoulos, Phys.
Rev. B 74, 045424 (2006).
5M. Barbier, P. Vasilopoulos, and F. M. Peeters, P h y s .R e v .B 80,
205415 (2009).
6M. Barbier, P. Vasilopoulos, and F. M. Peeters, P h y s .R e v .B 81,
075438 (2010).
7M. Barbier, P. Vasilopoulos, and F. M. Peeters, P h y s .R e v .B 82,
235408 (2010).
8N. M. R. Peres, J. Phys.: Condens. Matter 21, 323201 (2009).
9R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J.
Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, Science 320,
1308 (2008).
10A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel,P h y s .R e v .L e t t . 100, 117401 (2008).
11F. Wang, Y . Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and
Y. R . S h e n , Science 320, 206 (2008).
12K .F .M a k ,M .Y .S f e i r ,Y .W u ,C .H .L u i ,J .A .M i s e w i c h ,a n dT .F .
Heinz, P h y s .R e v .L e t t . 101, 196405 (2008).
13T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78,
085432 (2008).
14F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, P h y s .R e v .B
81, 035411 (2010).
15E. H. Hwang and S. Das Sarma, P h y s .R e v .B 75, 205418
(2007).
16S. H. Abendipour, G. Vignale, A. Principi, M. Polini, W.-K. Tse,and A. H. MacDonald, P h y s .R e v .B 84, 045429 (2011).
17F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, P h y s .R e v .B
82, 115434 (2010).
18F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, High Press.
Res. 31, 98 (2011).
19T. J. Booth et al. ,Nano Lett. 8, 2442 (2008).
20K. S. Kim, Y . Zhao, H. Jang, S. Y . Lee, J. M. Kim, K. S. Kim, J. H.
Ahn, P. Kim, J. Choi, and B. H. Hong, Nature (London) 457, 706
(2009).
21F. Liu, P. Ming, and J. Li, P h y s .R e v .B 76, 064120 (2007).
22E. Cadelano, P. L. Palla, S. Giordano, and L. Colombo, Phys. Rev.
Lett. 102, 235502 (2009).
23S.-M. Choi, S.-H. Jhi, and Y .-W. Son, Phys. Rev. B 81, 081407(R)
(2010).
24J.-W. Jiang, J.-S. Wang, and B. Li, P h y s .R e v .B 81, 073405
(2010).
25G. Gui, J. Li, and J. Zhong, P h y s .R e v .B 78, 075435 (2008).
26V . M. Pereira, A. H. Castro Neto, and N. M. R. Peres, Phys. Rev. B
80, 045401 (2009).27R. M. Ribeiro, V . M. Pereira, N. M. R. Peres, P. R. Briddon, and
A. H. C. Neto, New J. Phys. 11, 115002 (2009).
28G. Cocco, E. Cadelano, and L. Colombo, Phys. Rev. B 81, 241412
(2010).
29F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, High Press.
Res. 29, 569 (2009).
30F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, e-print
arXiv: 1110.4036 (accepted to Phys. Rev. B).
31V . M. Pereira and A. H. Castro Neto, Phys. Rev. Lett. 103, 046801
(2009).
32J. Cayssol, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. B 79,
075428 (2009).
33S. Gattenl ¨ohner, W. Belzig, and M. Titov, Phys. Rev. B 82, 155417
(2010).
34A. Concha and Z. Te ˇsanovi ´c,Phys. Rev. B 82, 033413 (2010).
35A. Raoux, M. Polini, R. Asgari, A. R. Hamilton, R. Fazio, and
A. H. MacDonald, P h y s .R e v .B 81, 073407 (2010).
36I. L. Aleiner and K. B. Efetov, Phys. Rev. Lett. 97, 236801 (2006).
37D. M. Basko, Phys. Rev. B 78, 125418 (2008); 79, 129902(E)
(2009).
38M. Farjam and H. Rafii-Tabar, Phys. Rev. B 80, 167401 (2009).
39O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and
T. Weng, J. Appl. Phys. 41, 3373 (1970).
40C. A. Marianetti and H. G. Yevick, Phys. Rev. Lett. 105, 245502
(2010).
41C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321, 385 (2008).
42E.-A. Kim and A. H. C. Neto, Europhys. Lett. 84, 57007 (2008).
43W. Bao, F. Miao, Z. Chen, H. Zhang, W. Jang, C. Dames, and C. N.
Lau, Nat. Nanotechnol. 5, 562 (2009).
44T. M. G. Mohiuddin et al. ,P h y s .R e v .B 79, 205433 (2009).
45I. Paul and G. Kotliar, P h y s .R e v .B 67, 115131 (2003).
46J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J.
Beenakker, P h y s .R e v .L e t t . 96, 246802 (2006); see also Appendix
in the preprint version e-print arXiv: cond-mat/0603315 .
47W.-R. Hannes and M. Titov, Europhys. Lett. 89, 47007 (2010).
48R. Landauer, IBM J. Res. Dev. 1, 223 (1957).
49M. B ¨uttiker, Phys. Rev. Lett. 57, 1761 (1986).
50U. Fano, Rev. Mod. Phys. 29, 74 (1957).
51Ya. M. Blanter and M. B ¨uttiker, Phys. Rep. 336, 1 (2000).
52M. Titov, Eur. Phys. Lett. 79, 17004 (2007).
53H. Bruus and K. Flensberg, Many-Body Quantum Theory in
Condensed Matter Physics: An Introduction (Oxford University
Press, Oxford, 2004).
54E. J. H. Lee, K. Balasubramanian, R. T. Weitz, M. Burghard, andK. Kern, Nat. Nanotechnol. 3, 486 (2008).
55A. F. Young and P. Kim, Nat. Phys. 5, 222 (2009).
56M. Huang, H. Yan, T. F. Heinz, and J. Hone, Nano Lett. 10, 4074
(2010).
57S. A. Mikhailov and K. Ziegler, Phys. Rev. Lett. 99, 016803 (2007).
58P. Yeh, A. Yariv, and C. Hong, J. Opt. Soc. Am. 67, 423 (1977).
59M. Alidoust and J. Linder, P h y s .R e v .B 84, 035407 (2011).
195404-11 |
PhysRevB.99.035435.pdf | PHYSICAL REVIEW B 99, 035435 (2019)
Pump-probe spectroscopy study of ultrafast temperature dynamics in nanoporous gold
Michele Ortolani,1Andrea Mancini,1Arne Budweg,2Denis Garoli,3Daniele Brida,2,4and Francesco de Angelis3
1Department of Physics, Sapienza University of Rome, 00185 Rome, Italy
2Department of Physics and Center for Applied Photonics, University of Konstanz, 78457 Konstanz, Germany
3Plasmon Nanotechnology Department, Istituto Italiano di Tecnologia (IIT), 16163 Genoa, Italy
4Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
(Received 9 August 2018; revised manuscript received 4 January 2019; published 24 January 2019)
We explore the influence of the nanoporous structure on the thermal relaxation of electrons and holes excited
by ultrashort laser pulses ( ∼7 fs) in thin gold films. Plasmon decay into hot electron-hole pairs results in the
generation of a Fermi-Dirac distribution thermalized at a temperature Tehigher than the lattice temperature
Tl. The relaxation times of the energy exchange between electrons and lattice, here measured by pump-probe
spectroscopy, is slowed down by the nanoporous structure, resulting in much higher peak Tethan for bulk gold
films. The electron-phonon coupling constant and the Debye temperature are found to scale with the metal fillingfactor fand a two-temperature model reproduces the data. The results open the way for electron temperature
control in metals by engineering of the nanoporous geometry.
DOI: 10.1103/PhysRevB.99.035435
I. INTRODUCTION
The optical excitation of electrons and holes at high en-
ergy levels in metal nanostructures has been the subject ofconsiderable attention in the last decade [ 1–6], with the aim
of enabling chemical reactions and charge transfer from themetal to adjacent materials at ambient temperature for energyharvesting and storage [ 1,4], most notably H
2production
by water splitting [ 7–10]. In particular, gold nanostructures
have been investigated because of the relative ease of ob-taining plasmonic field enhancement at their surfaces [ 11].
The absorption of optical energy by free carriers in a metalimplies collective oscillation of electron currents (plasmons)[12–14]. Such coherent plasmons rapidly decay into non-
thermalized electron-hole (e-h) pairs occupying high kineticenergy states. The e-h pairs decay via electron-electron scat-tering on the femtosecond timescale into hot carriers, whichcan be represented by a Fermi-Dirac distribution at an electrontemperature T
e, much higher than the lattice temperature Tl.
Subsequently, electron-phonon interaction leads to equilib-rium defined as T
e≈Tlon the picosecond timescale [ 15,16].
Very recently, ab initio calculations of all electron and
phonon states of gold have been employed to confirm theabove interpretation of ultrafast pump-probe spectroscopy inthe case of spherical nanoparticles of 60 nm diameter inaqueous solution [ 17]. For such a simple geometry, electron
and phonon distributions may be taken as constant in space,and the introduction of statistical thermal baths for electronsatT
eand phonons at Tlmay not be conceptually necessary
any more. The present paper, however, explores the oppositelimit of an extended nanoscale filament network, also callednanoporous gold (NPG). In NPG, geometrical parameterssuch as gold filling factor and filament diameter play a keyrole in determining the electron-phonon thermalization timedue to spatially inhomogeneous excitation intensity at thenanoscale, therefore the previous simplified approach of twocoupled statistical thermal baths (so called two-temperature(TT) model [ 16,18,19]) will be followed in this paper so as
to effectively include the geometrical parameters of the NPGstructure in the model.
Hot electron plasmonics experiments have been mostly
conducted on nanoparticles dispersed in solutions [ 2,7–10,20–
22], and the ultrafast temperature dynamics are poorly un-
derstood due to an extremely varied experimental landscape[4,23,24].
NPG [ 26–29] represents an interesting system for appli-
cations, as it allows liquid and gas samples to fill the emptyspaces among gold ligaments [ 7–10] where the radiation field
is strongly enhanced by cusplike geometries of the fractalstructure [see Figs. 1(a)–1(b)][28–30]. Nanoporous materials
of different kinds (e.g., glass [ 31], silicon [ 32,33], and poly-
mers [ 31]) are also well known for their thermal and acoustic
insulation properties. The nanoporous structure should thenimpact the ultrafast electron temperature dynamics followingthe absorption of optical energy by plasmons in NPG. Ifcompared to bulk gold, the decrease in the thermal conduc-tivity at the interior of the effective material constituted bythe nanoporous metal should then lead to higher maximumtemperatures and slower local energy relaxation, in a waysimilar to that observed in gold nanoparticles [ 17] and clus-
ters [ 21]. In this paper, we present an ultrafast pump-probe
spectroscopy study and related thermal modeling of plasmonenergy relaxation in NPG. Interestingly, relevant fundamentalquantities of the TT model such as the speed of sound, the De-bye temperature, and the electron-phonon coupling constantare found to follow a simple power-scaling law with the metalfilling fraction fin NPG, which quantitatively explains both
the longer timescales and the higher electron temperaturesobserved in our experiments.
II. EXPERIMENT
NPG samples were prepared by chemical dealloying from
an Ag0.67Au0.33thin film following the procedure reported in
2469-9950/2019/99(3)/035435(6) 035435-1 ©2019 American Physical SocietyMICHELE ORTOLANI et al. PHYSICAL REVIEW B 99, 035435 (2019)
FIG. 1. (a), (b) Scanning electron micrographs (SEM) of the two
NPG samples characterized by different fanddwire. (c) Scheme of
the solid thin-film samples with optical beams. (d), (e) Reflectance
and transmittance spectra of the NPG films at equilibrium. The
transmission dip around 0.3 eV in panel (e) is due to multiphononabsorption in the diamond substrate. In panel (e), the skin depth of
gold taken from Ref. [ 25] is also reported to highlight the dielectric
resonance of gold at 2 .5 eV, mainly due to 5 d-6spinterband transi-
tion at the L-point of the first Brillouin zone.
Ref. [ 30]. The two films studied in this work are characterized
by different dealloying times (3 h for NPG3 and 9 h for NPG9)and have a similar f(mainly related to the composition of
the initial alloy). Different dealloying times lead to differentaverage diameter of the gold ligaments d
wire[30]. In particular,
by numerical analysis of the SEM images of Figs. 1(a) and
1(b) [30], we found f=0.39 and dwire∼50 nm for NPG3,
f=0.37 and dwire∼80,nm for NPG9. In Figs. 1(c) and1(d)
the optical reflectance Rand transmittance trof the two NPG
films in the infrared and visible ranges are reported. A redshiftof the plasma edge is observed from 0.5 eV in NPG3 to 0.2 eVin NPG9 [ 28–30]. The dielectric resonance of gold at 2.5 eV
is clearly visible in all samples. The broad peak barely seenin the spectra of NPG9 around 1.8 eV is due to an effectivemedium resonance [ 30].
In a simplified model of optical excitations of gold, the
lowest-energy interband transition is the 5 d-6sptransition
at the L-point, which leads to the lowest-energy resonance
in the dielectric function of gold. The spectral lineshapeof this resonance is a Lorentz function centered at 2.5 eV[16,17,25]. In this paper, to focus on the geometrical effect of
the nanoporous structure rather than on the details of electro-magnetic interactions, we will make use of a correspondingsimplified model for ultrafast pump-probe spectroscopy ofgold: the infrared pump pulse spectrum, being located atphoton energies well below the L-point transitions at 2.5 eV
[see Fig. 2(a)], mainly excites the intraband transitions within
the 6spband. As a 6 spintraband transition of gold can be seen
as a pure free-electron excitation, it can also be interpretedas a plasmon excitation. The plasmon then decays into a6spe-h pair that subsequently thermalizes in a hot carrier
population in the 6 spband, which we model with a simpleFermi-Dirac distribution thermalized at T
e. The white-light
probe pulse, instead, encompasses a broader spectral rangeincluding the dielectric function resonance at 2.5 eV , here usedas a qualitative probe of T
eas a function of pump-probe delay.
Figure 2(b) is a sketch that summarizes the simplified model
for ultrafast pump-probe spectroscopy of gold. However, ithas been recently established, both theoretically [ 34,35] and
experimentally [ 36], that 5 d-6spinterband transitions at the
X-point can actually be excited by pump photons with energy
higher than a threshold approximately set at 1.8 eV . The effectofX-point transitions is to depress plasmon excitation in the
6spband taking place at pump photon energies higher than
1.8 eV , therefore the simplified picture described above andsketched in Fig. 2(b), which implies pure plasmonic excitation
in gold for all pump photon energies below the dielectric func-tion resonance at 2.5 eV , has to be rigorously rejected [ 36].
At odds with the L-point transitions, however, the weaker
X-point transition oscillator does not produce a true resonance
in the dielectric function of gold at 1.8 eV [ 25], so our probe
pulse will not be sensitive to hot holes in the 5 dband at
that energy. Also, the pump-pulse spectrum in our experimentextends between 1.4 eV to 1.9 eV as shown in Fig. 2(a),s o
it overlaps only marginally with the X-point transitions at
1.8 eV . Therefore, the simplified model of Fig. 2(b) can be
fairly employed for the scopes of the present paper henceallowing us to describe the electron system, after e-h pairthermalization, with the single parameter T
e.
Transient absorption experiments were performed with
an ultrafast laser system based on a Yb:KGW regenerativeamplifier operating at a repetition time of 20 μs. A home-built
noncollinear optical parametric amplifier delivers excitationpulses with a bandwidth of 0.53 eV at a central energy of E
p∼
1.65 eV as reported in Fig. 2(a), hence excluding the 5 d-6sp
transition [see Fig. 2(b)]. Dielectric chirped mirrors compress
the pulses to a duration of 7 fs. In Fig. 2(c), the evolution of the
Fermi-Dirac distribution following the excitation of the pumppulse is sketched. At t=0 the pulse excites a nonequilibrium
distribution whose shape is determined by the pulse-energyspectrum in Fig. 2(a), which can be roughly approximated
by a multiple-step function [black dashed curve in Fig. 2(c)]
[13,16]. The nonequilibrium e-h pair distribution generated
by the pump pulse thermalizes to a Fermi-Dirac distributionatT
eon a timescale of the order of hundreds of fs, mainly
through electron-electron interactions. At this stage, Teis still
much higher than Tl[red curve in Fig. 2(c)]. On a longer
timescale on the order of ps, the carriers cool down throughelectron-phonon interactions to a new lattice temperature T
l=
Te[orange curve in Fig. 2(c)] higher than the environment
temperature Tenv/similarequal300 K.
The pump-induced optical transmission change tr(t)i s
probed by a synchronous white light pulse obtained fromsupercontinuum generation in a 2-mm thick sapphire crystal[37]. Probe pulses cover a spectral range between 1.55 and
2.64 eV including the 5 d-6sptransition. Spectra of sub-
sequent probe pulses are used to calculate the differentialtransmission signal /Delta1tr(t)/tr=[tr(t)−tr(t/lessorsimilar0)]/tr(t/lessorsimilar
0) with a modulation of the excitation pulses at half therepetition rate. In Figs. 2(d)–2(f), color plots of /Delta1tr(t)/tras
a function of pump-probe time delay tand probe wavelength
λare shown for a reference bulk gold thin film and for the
035435-2PUMP-PROBE SPECTROSCOPY STUDY OF ULTRAFAST … PHYSICAL REVIEW B 99, 035435 (2019)
FIG. 2. (a) Spectrum of the pump pulse used in the experiments (duration is 7 fs). (b) Simplified sketch of the density of states (DOS) of
gold at the L-point employed in this paper for interpretation of the pump-probe data. (c) Simplified sketch of the evolution of the Fermi-Dirac
distribution following the pump pulse excitation. The shift of the chemical potential with temperature is neglected for clarity. (d)–(f) /Delta1tr(t)/tr
maps for a reference bulk gold thin film (thickness 30 nm) (d) and for the two NPG samples (e), (f). Inset of panel (d), green curve: Cut of the
map in (d) at λ=600 nm; red curve: the Te(t) obtained from the extended TT model.
NPG3 and NPG9 samples. By comparing the three plots of
Figs. 2(d)–2(f), one immediately sees a strongly increased
transmittance around λ=560 nm in both NPG samples which
is almost absent in the bulk gold film [ 28,29], accompanied by
a decay of /Delta1tr(t)/trslower than that of the gold film at all
wavelengths. For probe wavelengths shorter than ∼550 nm,
the sign of /Delta1tr(t)/trchanges to negative because of pump-
induced interband absorption [ 15,38–41]. High-energy non-
thermalized carriers impact the transmittance of gold filmsand nanostructures only for t/lessmuch0.5p s[ 16,40]. The transmit-
tance dynamics for probe delays above 0.5 ps, instead, canbe almost entirely attributed to thermalized carriers and tochanges in their T
e, displaying a relaxation timescale inde-
pendent on the probe wavelength [ 42]. In this perspective, the
strongly increased transmittance observed in NPG [positiveareas in Figs. 2(e) and 2(f)] indicates a much higher value
ofT
Max
e if compared to that reached in bulk gold [Fig. 2(d)].
These facts demonstrate that NPG is a very promising candi-date for hot-electron plasmonics applications.
III. MODEL
Numerical evaluation of Te(t) andTl(t) dynamics is per-
formed within the TT model, in which energy relaxation tothe lattice from the free carriers, heated by e-h pair thermal-ization via the fast electron-electron interaction, is mediatedby the relatively slow electron-phonon interaction [ 18]. In an
improved version of the TT model [ 19], e-h pairs produced
by plasmon decay act as the external heat source for boththe Fermi-Dirac free carrier distribution and the lattice viaelectron-electron and electron-phonon scattering processes,respectively, resulting in the following coupled equations:
C
edTe
dt=−g(Te−Tl)−e−(τ−1
e,relax+τ−1
p,relax)t
t2
×[t+τe,relax(1−et/τ e,relax)]·Pa,
ClTl
dt=g(Te−Tl)−e−(τ−1
e,relax+τ−1
p,relax)t
tτp,relax
×[τe,relax(1−et/τ e,relax)]·Pa, (1)
where CeandClare the electronic and lattice heat capacities
per unit volume, gis the electron-phonon coupling con-
stant, τe,relax andτp,relax are characteristic times related to
the electron-electron and electron-phonon energy relaxation[19]. The pump-pulse power in the instantaneous pump-pulse
approximation is P
a=Fa/d, with dthe film thickness, Fa=
(1−R−tr)F, andF=180μJ/cm2the pump fluence. For
bulk gold thin films, the values of the parameters used in theextended TT model are C
e=γTe,γ=68 J m−3K−2,Cl=
2.5·106Jm−3K−1,g=2.2·1016Wm−3K−1,EF=7.3e V ,
τe,relax=136 fs, τp,relax=1650 fs, EP=1.65 eV [ 16]. In the
inset of Fig. 2(c),t h eTecurve obtained from Eqs. ( 1)fi t s
to the/Delta1tr(t)/tr(0) data for bulk gold, provided that the
delay scale is normalized by the relative change factor ξ=
ln(/Delta1trMax/tr)/ln(/Delta1TMax
e/Te(t/lessorsimilar0))/similarequal3.
To analyze the ultrafast temperature dynamics of NPG
within the extended TT model, we scale all quantities ofEqs. ( 1)b yf
β, where βis the corresponding scaling expo-
nent, as summarized in Table I.F o rCeandCl, the scaling ex-
ponent is a trivial βC=1 as they scale linearly with the mass
density. For the thermal conductance, the problem is consid-erably more complex due to the NPG network connectivity.Previous works have employed the Asymmetric Bruggeman
035435-3MICHELE ORTOLANI et al. PHYSICAL REVIEW B 99, 035435 (2019)
TABLE I. Geometrical scaling of the TT model parameters.
Quantity Ce,C l vs g /Theta1D τp,relax
scaling ( f< 1) f1f3/2f3f3/2f−3/2
Theory (ABT) [ 43] to calculate the electron thermal conduc-
tivity in NPG [ 44,45] and the lattice thermal conductivity of
nanoporous glass [ 31]. In both cases, the results point toward
an experimental value of βk=3/2 for thermal conductivities
of nanoporous solids. The lattice thermal conductivity is writ-ten as k
l=1/3Clvslph, where Clis the lattice specific heat, vs
is the speed of sound, and lphis the phonon mean free path.
Since lphandClare microscopic quantities that should not
depend on the geometry, vsshould scale with the exponent
βv=βk=+ 3/2a sw e l l[ 31]. There are two quantities in the
TT model of Eqs. ( 1) that depend on vs. The first quantity
isg[46],
g=π2menev2
s
6Teτ(Te,Tl), (2)
where neis the microscopic electron density, meis the elec-
tron mass, and τ(Te,Tl) is the total electron scattering time in-
cluding electron-electron τeeand electron phonon τepscatter-
ings. Following Matthiessen’s rule and assuming momentum-independent scattering, the effect of electron scattering atphysical boundaries in NPG ligaments can be included inthe model by considering an additional scattering time τ
B=
vF/dwire, where vF=1.40·106m/s is the Fermi velocity ingold [ 44]:
1
τ(Te,Tl)=1
τee+1
τep+1
τB=AT2
e+BT l+vF
dwire.(3)
In Eq. ( 3)AandBare temperature-independent coefficients
that in gold can be taken equal to A=1.2·107K−2s−1,B=
1.23·1011K−1s−1[47]. In bulk gold, dwire→∞ and the
contribution of τBis negligible. The case of gold nanoparticles
can also be obtained by using f=1 and dwiresimilar to the
value of the nanoparticle diameter [ 48]. In Eq. ( 2), the only
quantity that scales with fis the speed of sound vs, therefore
forgwe obtain a scaling exponent βg=2βv=+ 3.
The second quantity of the TT model proportional to vsis
the Debye temperature /Theta1D:
/Theta1D=¯hkDvs
kB, (4)
where kD=(6πNa)1/3(Nais the atomic density) and kBis
the Boltzmann constant. kB/Theta1Drepresents the average phonon
energy and, as such, enters in the definition of the electron-phonon energy relaxation time as τ
p,relax=τepEP/kB/Theta1D.
Therefore, from β/Theta1=βv=+ 3/2 we obtain βτ=−β/Theta1=
−3/2f o rτp,relax.
IV . DISCUSSION
Using the scaling exponents of Table I, we can describe the
ultrafast electron dynamics of NPG by solving the extendedTT model of Eqs. ( 1) as a function of f. It is important
to notice that the scaled quantities are effective quantities
1.01.035 nm
FIG. 3. (a) Effect of varying fon the Te(t) dynamics ( f=1 corresponds to bulk gold). (b) Same curves as (a) normalized at TMax
e to
highlight the different temperature dynamics. (c) Effect of dwireon the Te(t) decay for f=0.4. (d)–(i) Comparison of the spectra obtained from
the/Delta1tr(t)/trcolor plots of Fig. 2atλ=600 nm (d)–(f) and at λ=500 nm (g)–(i) with the electron temperatures obtained from the extended
TT model (dark blue curves). (d), (g): bulk gold; (e), (h): NPG3; (f), (i): NPG9. Inset of panels (b) and (c) show the full undershoot at short
delay due to the generation of nonthermalized carriers in NPG. Note that /Delta1tr(t)/trin panels (g)–(i) is reported with negative multiplication
factors.
035435-4PUMP-PROBE SPECTROSCOPY STUDY OF ULTRAFAST … PHYSICAL REVIEW B 99, 035435 (2019)
purposely defined for the nanoporous solid, and do not cor-
respond to an actual variation of the microscopic quantities ofbulk gold. In Figs. 3(a)–3(c) the model results are reported,
highlighting the effect of fandd
wireonTe. In the model,
the temperature dynamics is clearly slowed down for low f
andTMax
e is considerably increased. Electron scattering at
physical boundaries, which is almost absent in bulk gold,becomes relevant only when the electron mean free path ingold/lscript∼40 nm [ 44,49] is of the same order of the mean
ligament diameter d
wire(as it is in our samples NPG3 and
NPG9 with dwireof 50 nm and 80 nm, respectively).
In Figs. 3(d)–3(i), we compare cuts of the experimental
data of Figs. 2(d)–2(f) at fixed λ=600 nm and λ=500 nm
with the prediction of the TT model scaled by f=0.39 for
NPG3 and f=0.37 for NPG9. The relaxation dynamics for
t>0.5 ps is fairly reproduced by the TT model in all plots
of Fig. 3. The much higher /Delta1tr(t)/trfor NPG if compared
to bulk gold at λ=600 nm is indicative of the much higher
TMax
e reached in NPG. The TT model accounts only for the
dynamics of thermalized electrons and therefore it cannotreproduce the ultrafast variations of /Delta1tr(t)/trat very short
t/greaterorequalslant0. Especially at λ=600 nm, a strong induced absorption
signal can be seen for t<100 fs [see insets of Figs. 3(e)
and 3(f)] and it can be attributed to the excitation of non-
thermalized high-energy carriers [ 15,38–41]. Hot carriers are
almost absent in bulk gold for the same excitation conditionsas for NPG, as expected due to the high density of field-enhancement hotspots in NPG and to the high surface/volumeratio [ 13] of the NPG fractal structure [ 30]. Atλ=500 nmthe contribution of non-thermalized carriers to /Delta1tr(t)/tris
much smaller [ 16,17] and it does not impact the fitting of
the model to the data as seen in Figs. 3(h) and 3(i). It has
been observed [ 22] that surface functionalization of gold
nanostructures leads to similar slowdown of the tempera-ture dynamics. Further studies of functionalized NPG forfuture hot-electron chemistry applications will be required tounderstand the combination of the two different slowdowneffects.
V . CONCLUSION
In conclusion, the predictions of a geometrical scaling
theory of NPG concerning the reduced thermal capacitance,the weaker thermal link between electrons and phonons,and the longer electron-phonon energy relaxation time ifcompared to bulk gold, could quantitatively account for theultrafast temperature dynamics experimentally observed bypump-probe spectroscopy. On the basis of these results, higherelectron temperatures and longer plasmon decay times can beengineered in gold nanostructures for future applications ofhot-electron plasmonics.
ACKNOWLEDGMENTS
This work was funded by the Italian Ministry of
Research through the program PRIN2015 (MIUR GrantNo. 2015FSHNCB) and by Sapienza University of Romethrough the program Ricerca d’Ateneo 2017 (Grant No.PH11715C7E435F41).
[1] H. A. Atwater and A. Polman, Nat. Mater. 9,205(2010 ).
[2] A. Manjavacas, J. G. Liu, V . Kulkarni, and P. Nordlander, ACS
Nano 8,7630 (2014 ).
[3] M. L. Brongersma, N. J. Halas, and P. Nordlander, Nat. Nan-
otechnol. 10,25(2015 ).
[ 4 ]K .W u ,J .C h e n ,J .R .M c B r i d e ,a n dT .L i a n , Science 349,632
(2015 ).
[5] F. Benz, M. K. Schmidt, A. Dreismann, R. Chikkaraddy,
Y . Zhang, A. Demetriadou, C. Carnegie, H. Ohadi, B. de Nijs,R. Esteban et al. ,Science 354,726(2016 ).
[6] E. Cortés, W. Xie, J. Cambiasso, A. S. Jermyn, R.
Sundararaman, P. Narang, S. Schlücker, and S. A. Maier, Nat.
Commun. 8,14880 (2017 ).
[7] J. Lee, S. Mubeen, X. Ji, G. D. Stucky, and M. Moskovits, Nano
Lett.12,5014 (2012 ).
[8] S. Mukherjee, F. Libisch, N. Large, O. Neumann, L. V . Brown,
J. Cheng, J. B. Lassiter, E. A. Carter, P. Nordlander, and N. J.Halas, Nano Lett. 13,240(2012 ).
[9] Y . Zhang, S. He, W. Guo, Y . Hu, J. Huang, J. R. Mulcahy, and
W. D. Wei, Chem. Rev. 118,2927 (2017 ).
[10] Y . Zhang, T. R. Nelson, S. Tretiak, H. Guo, and G. C. Schatz,
ACS Nano 12,8415 (2018 ).
[11] G. V . Naik, V . M. Shalaev, and A. Boltasseva, Adv. Mater. 25,
3264 (2013 ).
[12] P. Ruello, A. Ayouch, G. Vaudel, T. Pezeril, N. Delorme, S.
Sato, K. Kimura, and V . E. Gusev, P h y s .R e v .B
92,174304
(2015 ).[13] L. V . Besteiro, X.-T. Kong, Z. Wang, G. Hartland, and A. O.
Govorov, ACS Photonics 4,2759 (2017 ).
[14] W. M. Deacon, A. Lombardi, F. Benz, Y . del Valle-Inclan
Redondo, R. Chikkaraddy, B. de Nijs, M.-E. Kleemann, J.Mertens, and J. J. Baumberg, P h y s .R e v .L e t t . 119,023901
(2017 ).
[15] H. Baida, D. Mongin, D. Christofilos, G. Bachelier, A. Crut, P.
Maioli, N. Del Fatti, and F. Vallée, P h y s .R e v .L e t t . 107,057402
(2011 ).
[16] G. Della Valle, M. Conforti, S. Longhi, G. Cerullo, and D.
Brida, Phys. Rev. B 86,155139 (2012 ).
[17] A. M. Brown, R. Sundararaman, P. Narang, A. M.
Schwartzberg, W. A. Goddard III, and H. A. Atwater,Phys. Rev. Lett. 118,087401 (2017 ).
[18] S. Anisimov, B. Kapeliovich, and T. Perelman, Zh. Eksp. Teor.
Fiz.66, 776 (1974) [Sov. Phys. JETP 39, 375 (1974)].
[19] E. Carpene, P h y s .R e v .B 74,024301 (2006 ).
[20] J. H. Hodak, I. Martini, and G. V . Hartland, J. Phys. Chem. B
102,6958 (1998 ).
[21] C. Hrelescu, J. Stehr, M. Ringler, R. A. Sperling, W. J. Parak, T.
A. Klar, and J. Feldmann, J. Phys. Chem. C 114,7401 (2010 ).
[22] K. O. Aruda, M. Tagliazucchi, C. M. Sweeney, D. C. Hannah,
G. C. Schatz, and E. A. Weiss, Proc. Natl. Acad. Sci. USA 110,
4212 (2013 ).
[23] E. L. Keller and R. R. Frontiera, ACS Nano 12,5848 (2018 ).
[24] Y . Yu, V . Sundaresan, and K. A. Willets, J. Phys. Chem. C 122
,
5040 (2018 ).
035435-5MICHELE ORTOLANI et al. PHYSICAL REVIEW B 99, 035435 (2019)
[25] R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh,
G. D. Boreman, and M. B. Raschke, P h y s .R e v .B 86,235147
(2012 ).
[26] J. Biener, G. W. Nyce, A. M. Hodge, M. M. Biener, A. V .
Hamza, and S. A. Maier, Adv. Mater. 20,1211 (2008 ).
[27] Y . Ding and M. Chen, MRS Bull. 34,569(2009 ).
[28] X. Lang, L. Qian, P. Guan, J. Zi, and M. Chen, Appl. Phys. Lett.
98,093701 (2011 ).
[29] E. Detsi, M. Salverda, P. R. Onck, and J. T. M. De Hosson,
J. Appl. Phys. 115,044308 (2014 ).
[30] D. Garoli, E. Calandrini, A. Bozzola, A. Toma, S. Cattarin, M.
Ortolani, and F. De Angelis, ACS Photonics 5,3408 (2018 ).
[31] R. M. Costescu, A. J. Bullen, G. Matamis, K. E. O’Hara, and
D. G. Cahill, Phys. Rev. B 65,094205 (2002 ).
[32] P. E. Hopkins, C. M. Reinke, M. F. Su, R. H. Olsson III, E.
A. Shaner, Z. C. Leseman, J. R. Serrano, L. M. Phinney, andI. El-Kady, Nano Lett. 11,107(2010 ).
[33] Z. Wang, J. E. Alaniz, W. Jang, J. E. Garay, and C. Dames,
Nano Lett. 11,2206 (2011 ).
[34] R. Sundararaman, P. Narang, A. S. Jermyn, W. A. Goddard III,
and H. A. Atwater, Nat. Commun. 5,5788 (2014 ).
[35] A. M. Brown, R. Sundararaman, P. Narang, W. A. Goddard III,
and H. A. Atwater, ACS Nano 10,957(2015 ).
[36] G. Tagliabue, A. S. Jermyn, R. Sundararaman, A. J. Welch,
J. S. DuChene, R. Pala, A. R. Davoyan, P. Narang, and H. A.Atwater, Nat. Commun. 9,3394
(2018 ).
[37] A. Grupp, A. Budweg, M. P. Fischer, J. Allerbeck, G. Soavi,
A. Leitenstorfer, and D. Brida, J. Opt. 20,014005 (2017 ).[38] N. Rotenberg, J. N. Caspers, and H. M. van Driel, P h y s .R e v .B
80,245420 (2009 ).
[39] N. Rotenberg, M. Betz, and H. M. van Driel, Phys. Rev. Lett.
105,017402 (2010 ).
[40] G. Della Valle, D. Polli, P. Biagioni, C. Martella, M. C.
Giordano, M. Finazzi, S. Longhi, L. Duo, G. Cerullo, andBuatier de Mongeot, P h y s .R e v .B 91,235440 (2015 ).
[41] L. Di Mario, T. O. Otomalo, D. Catone, P. O’Keeffe, L. Tian, S.
Turchini, B. Palpant, and F. Martelli, Phys. Rev. B 97,115448
(2018 ).
[42] T. Qiu and C. Tien, Int. J. Heat Mass Transf. 35,719
(1992 ).
[43] D. S. McLachlan, M. Blaszkiewicz, and R. E. Newnham, J. Am.
Ceram. Soc. 73,2187 (1990 ).
[44] P. E. Hopkins, P. M. Norris, L. M. Phinney, S. A. Policastro,
a n dR .G .K e l l y , J. Nanomater. 2008 ,22(2008 ).
[45] R. Makinson, in Mathematical Proceedings of the Cambridge
Philosophical Society (Cambridge University Press, Cam-
bridge, 1938), V ol. 34, pp. 474–497.
[46] M. Kaganov, I. Lifshitz, and L. Tanatarov, Zh. Eksp. Teor. Fiz.
31, 232 (1957) [Sov. Phys. JETP 4, 173 (1957)].
[47] X. Y . Wang, D. M. Riffe, Y .-S. Lee, and M. C. Downer, Phys.
Rev. B 50,8016 (1994 ).
[48] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.99.035435 for full calculations of g in dif-
ferent cases and how the different limits of nanoporous filmsand nanoparticles can be retrieved within our scaling model.
[49] D. Gall, J. Appl. Phys. 119,085101
(2016 ).
035435-6 |
PhysRevB.102.081108.pdf | PHYSICAL REVIEW B 102, 081108(R) (2020)
Rapid Communications
Doping effects on electronic states in electron-doped FeSe:
Impact of self-energy and vertex corrections
Youichi Yamakawa, Seiichiro Onari, and Hiroshi Kontani
Department of Physics, Nagoya University, Nagoya 464-8602, Japan
(Received 18 November 2019; revised 15 July 2020; accepted 15 July 2020; published 11 August 2020)
The pairing glue of high- Tcsuperconductivity in heavily electron-doped (e-doped) FeSe, in which hole pockets
are absent, has been an important unsolved problem. Here, we focus on a heavily e-doped bulk superconductorLi
1−xFexOHFeSe ( Tc∼40 K). We construct a multiorbital model beyond the rigid band approximation
and analyze the spin and orbital fluctuations by taking both vertex corrections (VCs) and self-energy intoconsideration. Without e-doping ( x=0), the ferro-orbital order without magnetism in FeSe is reproduced by
the VCs. The orbital order quickly disappears when the hole pocket vanishes at x∼0.03. With increasing x
further, the spin fluctuations remain small, whereas orbital fluctuations gradually increase with xdue to the VCs.
The negative feedback due to the self-energy is crucial to explain experimental phase diagram. Thanks to bothvertex and self-energy corrections, the orbital-fluctuation-mediated s
++-wave state appears for a wide doping
range, consistent with experiments.
DOI: 10.1103/PhysRevB.102.081108
The high- Tcsuperconducting (SC) state in heavily
electron-doped (e-doped) FeSe systems attracts great atten-tion, but its pairing mechanism is still an open question. Oneof the characteristics of e-doped FeSe is the lack of magneticorder. Bulk FeSe exhibits spontaneous orbital polarizationn
xz/negationslash=nyzatTS=90 K, whereas no magnetic order occurs
down to the SC transition temperature Tc=9K[ 1]. The SC
state has been studied intensively [ 1–7]. On the other hand,
the orbital order is suppressed by only a few percent e-doping,and instead, a high- T
cSC phase with Tc/greaterorequalslant40 K appears for a
wide doping range in various e-doped FeSe compounds, suchas an ultrathin FeSe layer on SrTiO
3(Tc=40–100 K) [ 8–12],
K-dosed FeSe ( Tc∼40 K) [ 13,14], and intercalated supercon-
ductors ( Tc∼40 K) [ 15–21]. Angle-resolved photoemission
spectroscopy (ARPES) and scanning tunneling microscopy(STM) measurements have revealed that the SC gaps on theelectron Fermi surfaces (FSs) are fully gapped [ 9–11,17–21].
In usual Fe-based superconductors with electron-FSs
(eFSs) and hole-FSs (hFSs), strong spin orbital fluctuationscoexist in many compounds. This fact means that two kindsofs-wave SC states, the s
±-wave state with sign reversal
and the s++-wave state without sign reversal, can be medi-
ated by spin and orbital fluctuations, respectively [ 22–28].
Up to now, much experimental effort has been devoted todetecting the presence or absence of sign reversal [ 2,4,29–32].
The recently reported impurity-induced s
±→s++crossover
in Ba(Fe ,Rh) 2As2[33,34] has clarified the coexistence of
sizable repulsive and attractive pairing glues in Fe-basedcompounds.
In e-doped FeSe compounds, in contrast, the top of the
hFSs completely sinks below the Fermi level [ 9,13,19]. In
spite of its high T
c, NMR studies have revealed that the spin
fluctuations at Tcin e-doped FeSe are considerably weaker
than those in undoped FeSe [ 35]. It is still a significantmystery that the high- Tcstate ( Tc>40 K) is realized in
e-doped FeSe in spite of its weak spin fluctuation. Therefore,the pairing glue for the high- T
cstate in e-doped FeSe is
still controversial. Up to now, d-wave state [ 36–39] and the
incipient s±-wave state [ 37,40,41] have been proposed based
on the spin fluctuation theory, while Tcwill not be high. In
FeSe/SrTiO 3, it is expected that strong interfacial electron-
phonon coupling increases Tcup to∼60 K [ 8,9,42,43]. How-
ever, Tc∼40 K is realized in (Li,Fe)OHFeSe even in the
absence of strong interfacial electron-phonon interaction [ 16],
indicating that the main pairing glue originates from electroncorrelations.
The present authors investigated the pairing mechanism in
e-doped FeSe by focusing on the vertex corrections (VCs) thatinduce the orbital order and fluctuations [ 44]. It was found that
orbital-fluctuation-meditated s
++-wave SC can occur even
in the absence of hFSs. This result is consistent with therecent quasiparticle interference (QPI) measurement reportedin Ref. [ 10], while another QPI study indicates sign reversal
between inner- and outer-electron FSs [ 17,21]. However, the
reason that a high- T
cstate is realized in various e-doped FeSe
families for a very wide doping range ( x=0.05–0.20) was not
explained [ 12,14]. Therefore, further progress on the theory of
pairing mechanisms is still necessary.
In this Rapid Communication, we discuss the mechanism
of high- Tcsuperconductivity in bulk heavily e-doped com-
pound Li 1−xFexOHFeSe ( Tc∼40 K). To understand the x-T
phase diagram with wide superconducting region, we analyzethe model using the self-consistent-vertex correction (SC-VC) theory [ 44–46], by incorporating the x-dependent self-
energy into the theory [ 47]. At x=0, the ferro-orbital order
without magnetism in FeSe is reproduced. With increasingx, the orbital order quickly disappears, and spin fluctua-
tions remain small for 0 .05<x<0.20. Interestingly, orbital
2469-9950/2020/102(8)/081108(6) 081108-1 ©2020 American Physical SocietyYAMAKAWA, ONARI, AND KONTANI PHYSICAL REVIEW B 102, 081108(R) (2020)
(b)
Q'sQsorbitalspin
spin(a)
C
χsχs
χsXc(q) =χs
q=Qs+Q's +
FIG. 1. (a) Interference process between spin and orbital fluc-
tuations. Cis the three-boson coupling made by electron Green’s
functions. (b) Charge channel χ-VC Xc.
fluctuations start to increase gradually for x>0.06. There-
fore, the orbital-fluctuation-mediated s++-wave state appears
for a wide doping range, in agreement with experiments. Thex-dependent self-energy is crucial to explain the appropriate
e-doping phase diagrams of FeSe compounds.
The key ingredient of the present orbital /spin fluctuation
theory is the interference process between spin and orbitalfluctuations shown in Fig. 1(a)[48]. The three-boson coupling
Cis given by three electron Green’s functions. The correction
for the orbital susceptibility in Fig. 1(b), which we call the
χ-VC, is composed of the process in Fig. 1(a) twice. Note
that the χ-VC at q=0is proportional to/summationtext
q{χs(q)}2∝
max qχs(q) in two-dimensional systems [ 45,49]. This “pos-
itive feedback” through χ-VC is the physical reason why
orbital fluctuations are enlarged by spin fluctuations [ 45]. This
bottom-up approach towards strongly correlated systems byintroducing the VCs enables us to explain the nematicity andanomalous transport phenomena in Refs. [ 45,50].
Here, we construct the model Hamiltonian for
Li
1−xFexOHFeSe. We first perform the WIEN 2Kband
calculation for general xusing the virtual crystal
approximation (VCA). Next, we derive the eight-orbitaltight-binding model ˆH
VCA
x using the WANNIER 90 package.
Then, the unfolded eight-orbital d-pHubbard model
for Li 1−xFexOHFeSe is given as ˆH=ˆH0
x+rˆHU, where
ˆH0
x=ˆHVCA
x+ˆ/Sigma1exp(k). Here, the static self-energy ˆ/Sigma1exp(k)
is introduced to reproduce the experimental FSs of FeSeatx=0[51–55], by following Ref. [ 46]. The microscopic
derivation of ˆ/Sigma1
exp(k)[56,57] is an important future issue.
ˆHUis the first-principles screened Coulomb interaction for
dorbitals in FeSe (averaged Coulomb interaction is 7.2 eV)
[58], and ris the reduction factor. In the present study, we fix
r=0.355 to reproduce weakly developed spin fluctuations
forx=0–0.2 above TS[59,60]. A more detailed explanation
is summarized in Supplemental Material (SM) A [ 61].
Figures 2(a) and2(b) show the folded FSs in the two-Fe
Brillouin zone (BZ) derived from ˆHxforx=0 and 0.15,
respectively. Here, we introduced the spin-orbit interaction(SOI) η
SOIl·swithηSOI=50 meV . At x=xc∼0.03, the
hFS around the /Gamma1point disappears. The d-orbital density of
states (DOS) at the Fermi level is shown in Fig. 2(c).A s
xincreases from x=0, both the total DOS and xzorbital
DOS decrease for x/lessorequalslantxc.F o r x>xc,t h e xy-orbital DOS is
dominant and it increases gradually with doping. The DOSin the FeSe rigid-band approximation ( ˆH
0
x=0)i ss m a l l e rt h a n
the DOS in the present VCA-based model, consistently witha previous study on K-dosed FeSe [ 62]. This non-rigid-band
increment of the DOS magnifies T
cforx∼0.2.0 ππ
0π
0
kyky
kxxDOS (eV−1)(a)
(b)(c)
xzyz xy
hFSeFS1
eFS2
eFS2eFS1x=0
x=0.15present model
RB approximation
xy orbital
xz orbital
xcq~(π,π/2)
0 0.1 0.2 0.300.51
θxzyz
xy
FIG. 2. (a), (b) FSs for (a) x=0 and (b) 0.15 in the folded BZ
withηSOI=50 meV . The green, red, and blue lines correspond to
thexz,yz,a n d xyorbitals, respectively. eFS1 and eFS2 are inner and
outer eFSs, respectively. q=(π,π/ 2) is the nesting vector; eFS1
touches eFS2 by the translational wave vector q.( c )xdependence of
d-orbital DOS at the Fermi level. The hFS around /Gamma1disappears at
xc∼0.03.
The self-energy in the fluctuation exchange (FLEX) ap-
proximation [ 63] in the absence of the SOI is given as
/Sigma1l,m(k)=T
N/summationdisplay
k/prime,l/prime,m/primeV/Sigma1
l,l/prime;m,m/prime(k−k/prime)Gl/prime,m/prime(k/prime), (1)
where ˆV/Sigma1=3
2ˆVs+1
2ˆVc,k=[k,/epsilon1n=πT(2n+1)], and
l,l/prime,m,m/primerepresent the d-orbital indices ( l=1,2,3,4,5
corresponds to 3 z2−r2,xz,yz,xy,x2−y2orbitals). The
electron Green’s function is ˆG(k)=[{ˆG0(k)}−1−ˆ/Sigma1(k)]−1,
where ˆG0(k) is the bare Green’s function.
The spin (charge) susceptibility is ˆ χs(c)=ˆχ0[ˆ1−
ˆUs(c)ˆχ0]−1, where χ0
l,l/prime;m,m/prime(q)=−T
N/summationtext
kGl,m(k+q)Gm/primel/prime(k)
is the irreducible susceptibility. The spin (charge) channelinteraction in Eq. ( 1)i sˆV
s(c)=ˆUs(c)+ˆUs(c)ˆχs(c)ˆUs(c), where
ˆUs(c)is the spin (charge) channel Coulomb interaction.
The self-energy represents the mass-enhancement andquasiparticle damping due to spin fluctuations. Here, to keepthe shape of FSs, we drop the static Hermite part in theself-energy [ ˆ/Sigma1(k,+iδ)+ˆ/Sigma1(k,−iδ)]/2[47,64].
Figure 3(a) shows total spin susceptibility χ
s(q)≡/summationtext
l,mχs
l,l;m,m(q)f o r x=0 and 0.2 with r=0.355. Here,
we calculate χsat low T(=1 meV) in order to clarify its
peak structure. At x=0,χs(q) has commensurate peaks at
q=(π,0) and (0 ,π) due to the nesting between hFSs and
eFSs. In addition, a broad peak appears around q=(π,π ).
Atx=0.2,χs(q) has incommensurate peaks at the nesting
vector between eFSs shown in Fig. 2(b), which is observed
experimentally [ 60]. The importance of the latter nesting
has been stressed in various FeSe and FeAs compounds inliterature [ 22,36,37,47].
Figure 3(b) shows the xdependence of the spin Stoner
factor α
s, which is the maximum eigenvalues of ˆUsˆχ0(q);
αs=1 is the magnetic critical point. The obtained spin
081108-2DOPING EFFECTS ON ELECTRONIC STATES IN … PHYSICAL REVIEW B 102, 081108(R) (2020)
(a)
xαs
m*/m
xxz orbitalxy orbital)c( )b(
FLEX
RPA
xc xc00 . 1 0 . 224(π,π) (0,0) (π,0) (π,π)x=0.0
qx=0.2
χs
00 . 1 0 . 20.80.910510
FIG. 3. (a)–(c) Total spin susceptibility χs(q)f o r( a ) x=0a n d
(b)x=0.2f o r T=1m e V .( c ) xdependence of spin Stoner factors
αs. (d) Mass-enhancement factors Z=m∗/mforxzandxyorbitals.
Here, r=0.355 and 0.218 are used in the FLEX and RPA calcula-
tions, respectively.
fluctuations remain small even for the heavily e-doped case,
which is consistent with NMR results [ 35]. A similar result is
also obtained by the random-phase approximation (RPA) forr=0.209. Figure 3(c) shows the mass-enhancement factor
Z=m
∗/mgiven by the self-energy for xz(yz) and xyorbitals.
Zxyis approximately 3–4 and is larger than Zxzforx>xc,
due to the strong spin fluctuations and large DOS on the xy
orbital. The obtained values agree with previous dynamicalmean-field theory (DMFT) studies [ 62,65].
Next, we study the orbital susceptibility by including the
VC for the susceptibility χ-VC ( ˆX
c) and /Sigma1[47]. Here,
we consider the Aslamazov-Larkin (AL) processes shown inFig. 1(b) and analytically shown in SM B [ 61], because its
significance in Fe-based superconductors has been discoveredin previous studies [ 5,44–47,49]. At x=0, strong orbital
fluctuations with respect to O≡n
xz−nyzappears due to the
AL term [ 45,46]. In this case, χc
l;m(q)≡χc
l,l;m,m(q)s h o w s
a large positive (negative) value for l=m=2 and 3 ( l=
2,m=3) at q=0, and they diverge when orbital order
nxz/negationslash=nyxappears.
Hereafter, we set T=20 meV . Figure 4(a) shows the
numerical results for x=0. Due to the χ-VC on xz,yz
orbitals, χc
2;2(0) develops divergently due to the χ-VC in
spite of the weak spin fluctuations in FeSe [ 46]. Since
χc
2;3(0)≈−χc
2;2(0), strong ferro-orbital fluctuations with re-
spect to Ox2−y2≡nxz−nyzin undoped FeSe is satisfactorily
explained. The nonmagnetic nematic order can be explainedin the present theory as discussed in Ref. [ 46] and SM C [ 61].
In contrast, χ
c
4;4(4=xyorbital) remains small because the
χ-VC on the xyorbital is unimportant due to the smallness of
xyorbital spin fluctuations.04812
04812(0,0)
(π,0)(π,π)
(0,0)
(π,0)(π,π)qxqy
qxqyαc
x(a)x=0
(b)x=0.2(c)χc
2;2
χc
4;4
without VCwith χ-VC
xc00 . 1 0 . 20.60.81
χc
4;4χc
2;2
FIG. 4. (a) Orbital susceptibilities χc
l;l(q)f o r l=2=xz(green)
andl=4=xy(blue) for x=0. (b)χc
l;l(q)f o r x=0.2. (c) xdepen-
dence of αcwith and without χ-VC. Here, we set r=0.355.
In contrast, for x=0.2, strong ferro-orbital fluctuations
with respect to Oz2≡nxy−(nxz+nyz)/2 appears. The ob-
tained χc
4;4(q) is shown in Fig. 4(b), and the relations χc
2;2≈
χc
4;4/4,χc
2;4≈−χc
4;4/2, and χc
2;3≈χc
4;4/4 hold. As we dis-
cussed in Ref. [ 47],Oz2orbital fluctuations develop when the
χ-VC develop for all 2–4 orbitals. The crossover of domi-
nant orbital fluctuations from Ox2−y2toOz2occurs at x∼xc,
reflecting the increment of xy-orbital spin fluctuations due to
inter-eFS nesting. Both Ox2−y2- and Oz2-orbital fluctuations at
q∼0enlarge Tcirrespective of the gap symmetry [ 25,66].
Figure 4(c) shows the xdependence of the charge Stoner
factor αc, which is the maximum eigenvalue of ˆUc(ˆχ0+ˆXc).
Thus, αcis strongly enlarged by the χ-VC. Here, αc∼1
forx=0 corresponds to the ferro-orbital order ( nxz/negationslash=nyz)i n
undoped FeSe. Through doping, αcdrops quickly since hFS
disappears at x=xc. Interestingly, αcgradually increases for
x/greaterorsimilarxc, indicating the orbital-fluctuation-meditated supercon-
ductivity in the absence of the hFS.
To confirm the numerical results in Fig. 4, we also analyze
the same model based on the density-wave (DW) equationdeveloped in Refs. [ 67–70]i nS MD[ 61]. By solving the DW
equation, the higher-order AL processes are automaticallygenerated. Also, the criteria of the conserving approximation[71] are satisfied by including the FLEX self-energy. At x=0
andx=0.25, strong ferro-orbital fluctuations are obtained,
consistently with Figs. 4(a) and 4(b). In addition, strong
incommensurate charge channel fluctuations at q≈(π,π/ 2)
develop at x=0.25. They are overlooked in the SC-VC
theory since the strong kdependence of the form factor, which
represents the fluctuation of hopping integrals, is essential.These incommensurate “bond fluctuations” will be importantfor the pairing mechanism in heavily e-doped FeSe.
Next, we study the SC state in e-doped FeSe by following
the theoretical procedure reported in Refs. [ 44,72] The lin-
earized gap equation is given by
λ
SCZα(k,/epsilon1n)/Delta1α(k,/epsilon1n)
=−πT
(2π)2/summationdisplay
m,β/contintegraldisplay
βdp
vβ(p)VSC
α,β(k,/epsilon1n;p,/epsilon1m)/Delta1β(p,/epsilon1m)
|/epsilon1m|,
(2)
081108-3YAMAKAWA, ONARI, AND KONTANI PHYSICAL REVIEW B 102, 081108(R) (2020)
repulsive
attractivee-FS2e-FS1θ=0π/2
π
3π/2(d)s++ (x=0.15)
xλSC
nodal-ds++(c)
fullgap-d
(without SOI)
00 . 1 0 . 200.20.4
xc(a)
Vα,βave
xxcintra-FS
inter-FS(e)
unfolded BZ(f) intra-FS
inter-FS
0 0.1 0.2−6−4−202=1 ^+++Λc
l,l';m,m'(k,p)
ll 'mm'
kpC C
MT term AL temrs U-VCλSCΔk = +
k− kp− pVΛSC(k,p) VSC
cross(k,p)
(b)
FIG. 5. (a) Beyond-ME gap equation in the present analysis.
Both the first term VSC
/Lambda1and and the second term VSC
crossare significant.
(b) Charge channel U-VC/Lambda1c.( c ) xdependence of eigenvalues
of linearized gap equation λSC. Here, the SOI ηSOI=50 meV is
considered except for the dashed lines. x<xcis the orbital order
region. (d) Angle θdependence of the s++-wave gap function on FSs
forx=0.15. (e) Averaged intra-FS ( α=β) and inter-FS ( α/negationslash=β)
interactions Vave
α,βon FSs in (f) unfolded BZ without SOI.
where λSCis the eigenvalue, which is roughly proportional
toTc, and the relation λSC=1 is satisfied at T=Tc. Further,
/Delta1α(k) is the gap function, vα(k)≡∂/epsilon1α(k)
∂kis the Fermi velocity,
andZα(k) is the mass-enhancement factor. Here, αandβare
the indices of the folded FSs with the SOI shown in Figs. 1(a)
and1(b), considering the significance of the SOI on the SC
gap. We omit the SOI in calculating the pairing interactionsince its influence on the fluctuations is small [ 72].
The pairing interaction V
SCin the present beyond the
Migdal-Eliashberg (ME) formalism is shown in Fig. 5(a).T h e
first term is the single-fluctuation exchange process with theVC for the electron-boson coupling, which we call the U-VC.
TheU-VC/Lambda1
ν(ν=s,c) is composed of the AL processes and
Maki-Thompson (MT) term expressed in Fig. 5(b). The first
term is symbolically expressed as ˆVSC
/Lambda1=/summationtexts,c
νbνˆ/Lambda1μˆVνˆ/Lambda1ν,
where bs=3/2 and bc=−1/2. In the presence of moderate
spin fluctuations, |/Lambda1c|2/greatermuch1 for the low-energy region nearthe Fermi momentum [ 73–75]. Therefore, U-VC is significant
for the pairing mechanism.
The second crossing term ˆVSC
crossin Fig. 5(a) is one of the
lowest beyond-ME processes that are absent in the first term.In Ref. [ 44], we revealed that ˆV
SC
crossgives a large attractive
interaction between eFS1 and eFS2. The existence of thisinterpocket attractive pairing in heavily e-doped FeSe is con-firmed based on the DW equation study as we explain in SM E[61]. It is found that ˆV
SC
crossrepresents the interpocket attractive
interaction due to the bond fluctuations.
Figure 5(c)shows the doping dependence of λSC. The fully
gapped s++-wave state has the largest eigenvalue throughout
the entire doping region. The resulting λSCfor the s++-wave
state is enhanced due to the synergy between U-VC and VSC
cross,
as we discuss in Ref. [ 44] in detail. The resulting fully gapped
state with moderate anisotropy is shown in Fig. 5(d), which is
consistent with experimental reports in Refs. [ 9–11,19]. The
relation with the SC state in bulk FeSe is briefly discussed in
SM F [ 61]. In contrast, λSCfor the spin-fluctuation-mediated
d-wave state is small, since the spin fluctuations are weak
and the gap is suppressed by the SOI-induced band mixing[42]. Note that a fully gapped d-wave state is realized if
/Delta1/greaterorsimilarη
SOI[39].
To clarify why the s++-wave state is realized,
we show in Fig. 5(e) the xdependence of the
averaged intra- and interpocket interactions: Vave
α,β≡1
(2π)2/summationtext
/epsilon1n,/epsilon1m=±πT/contintegraltext
αdk
vα(k)/contintegraltext
βdp
vβ(p)VSC
α,β(k,/epsilon1n;p,/epsilon1m). Here, we con-
sider two electron pockets in unfolded BZ without the SOI
shown in Fig. 5(f). The intra-FS ( α=β) interaction changes
from positive to negative with e-doping because of thedevelopment of the orbital fluctuations shown in Fig. 4(c).
In addition, sizable attractive inter-FS ( α/negationslash=β) interaction is
mainly given by V
SC
cross,a sw ed i s c u s s e di nR e f .[ 44] and in
SM E [ 61].
It is interesting to discuss the similarities between e-doped
FeSe and heavily e-doped RFeAsO ( R=rare earth), both of
which exhibit high- Tcphases in spite of weak spin fluctuations
[35,76]. These high- Tccompounds have very similar FSs:
Large eFSs and tiny or absent hFSs. The FSs in heavilye-doped RFeAsO are shown in Fig. 1(c) in Ref. [ 47]. As
discussed in Ref. [ 47], the xy-orbital DOS is dominant, and
weak spin fluctuations on the xyorbital efficiently induce
strong orbital fluctuations on the xyorbital, χ
c
xy, due to the
AL-VCs. The present analysis indicates that a similar pairingmechanism is realized in these compounds.
The improvement of the self-energy beyond the FLEX
approximation is one of the important future issues. We notethat the enhancement of s-wave T
cin the Hund’s metal state
was discussed in a recent DMFT study [ 77].
In summary, we discussed the pairing mechanism in the
heavily e-doped bulk compound Li 1−xFexOHFeSe. At x=0,
the ferro-orbital order without magnetism in FeSe is repro-duced. With increasing x, the orbital order quickly disappears,
and the spin fluctuations remain weak for 0 <x<0.20. In-
terestingly, small spin fluctuations cause large orbital fluctua-tions due to the AL-VC X
ALfor a wide doping range. There-
fore, the orbital-fluctuation-mediated s++-wave state appears
for 0.05<x<0.2, consistent with experiments. Therefore,
the orbital fluctuations will be the main pairing glue in bothe-doped FeSe and H-doped 1111 systems.
081108-4DOPING EFFECTS ON ELECTRONIC STATES IN … PHYSICAL REVIEW B 102, 081108(R) (2020)
We are grateful to Y . Nomura for fruitful discussion on the
doping effect on the band structure in FeSe beyond the rigid-band approximation. This work was supported by the Grants-in-Aid for Scientific Research from MEXT, Japan (Grants No.
JP19H05825, No. JP18H01175, No. JP17K05543, and No.JP17K14338).
[1] A. E. Böhmer and A. Kreisel, J. Phys.: Condens. Matter 30,
023001 (2017) .
[2] Q. Wang, Y . Shen, B. Pan, Y . Hao, M. Ma, F. Zhou, P. Steffens,
K. Schmalzl, T. R. Forrest, M. Abdel-Hafiez, X. Chen, D. A.Chareev, A. N. Vasiliev, P. Bourges, Y . Sidis, H. Cao, and J.Zhao, Nat. Mater. 15, 159 (2016) .
[3] P. O. Sprau, A. Kostin, A. Kreisel, A. E. Böhmer, V . Taufour,
P. C. Canfield, S. Mukherjee, P. J. Hirschfeld, B. M. Andersen,a n dJ .C .S .D a v i s , Science 357, 75 (2017) .
[4] Y . Sun, A. Park, S. Pyon, T. Tamegai, and H. Kitamura, Phys.
Rev. B 96, 140505(R) (2017) .
[5] Y . Yamakawa and H. Kontani, Phys. Rev. B 96, 144509 (2017) .
[ 6 ] A .K r e i s e l ,B .M .A n d e r s e n ,P .O .S p r a u ,A .K o s t i n ,
J. C. Seamus Davis, and P. J. Hirschfeld, Phys. Rev. B 95,
174504 (2017) .
[7] S.-H. Baek, J. M. Ok, J. S. Kim, S. Aswartham, I. Morozov,
D. Chareev, T. Urata, K. Tanigaki, Y . Tanabe, B. Büchner, andD. V . Efremov, npj Quantum Mater. 5, 8 (2020) .
[8] Q.-Y . Wang, Z. Li, W.-H. Zhang, Z.-C. Zhang, J.-S. Zhang, W.
Li, H. Ding, Y .-B. Ou, P. Deng, K. Chang, J. Wen, C.-L. Song,K. He, J.-F. Jia, S.-H. Ji, Y . Wang, L. Wang, X. Chen, X. Ma,and Q.-K. Xue, Chin. Phys. Lett. 29, 037402 (2012) .
[9] J. J. Lee, F. T. Schmitt, R. G. Moore, S. Johnston, Y .-T. Cui, W.
Li, M. Yi, Z. K. Liu, M. Hashimoto, Y . Zhang, D. H. Lu, T. P.Devereaux, D.-H. Lee, and Z.-X. Shen, Nature (London) 515,
245 (2014) .
[10] Q. Fan, W. H. Zhang, X. Liu, Y . J. Yan, M. Q. Ren, R. Peng,
H .C .X u ,B .P .X i e ,J .P .H u ,T .Z h a n g ,a n dD .L .F e n g , Nat.
Phys. 11, 946 (2015) .
[11] Y . Zhang, J. J. Lee, R. G. Moore, W. Li, M. Yi, M. Hashimoto,
D. H. Lu, T. P. Devereaux, D.-H. Lee, and Z.-X. Shen, Phys.
Rev. Lett. 117, 117001 (2016) .
[12] X. Shi, Z.-Q. Han, X.-L. Peng, P. Richard, T. Qian, X.-X. Wu,
M.-W. Qiu, S. C. Wang, J. P. Hu, Y .-J. Sun, and H. Ding, Nat.
Commun. 8, 14988 (2017) .
[13] Y . Miyata, K. Nakayama, K. Sugawara, T. Sato, and T.
Takahashi, Nat. Mater. 14, 775 (2015) .
[14] C. H. P. Wen, H. C. Xu, C. Chen, Z. C. Huang, X. Lou, Y . J.
Pu, Q. Song, B. P. Xie, M. Abdel-Hafiez, D. A. Chareev, A. N.Vasiliev, R. Peng, and D. L. Feng, Nat. Commun. 7, 10840
(2016) .
[15] T. Noji, T. Hatakeda, S. Hosono, T. Kawamata, M. Kato, and Y .
Koike, Physica C 504, 8 (2014) .
[ 1 6 ]X .F .L u ,N .Z .W a n g ,H .W u ,Y .P .W u ,D .Z h a o ,X .Z .Z e n g ,
X. G. Luo, T. Wu, W. Bao, G. H. Zhang, F. Q. Huang, Q. Z.Huang, and X. H. Chen, Nat. Mater. 14, 325 (2015) .
[17] Z. Du, X. Yang, H. Lin, D. Fang, G. Du, J. Xing, H. Yang, X.
Zhu, and H.-H. Wen, Nat. Commun. 7, 10565 (2016) .
[18] Y . J. Yan, W. H. Zhang, M. Q. Ren, X. Liu, X. F. Lu, N. Z. Wang,
X. H. Niu, Q. Fan, J. Miao, R. Tao, B. P. Xie, X. H. Chen, T.Zhang, and D. L. Feng, P h y s .R e v .B 94, 134502 (2016) .
[19] X. H. Niu, R. Peng, H. C. Xu, Y . J. Yan, J. Jiang, D. F. Xu, T. L.
Yu, Q. Song, Z. C. Huang, Y . X. Wang, B. P. Xie, X. F. Lu,N. Z. Wang, X. H. Chen, Z. Sun, and D. L. Feng, Phys. Rev. B
92, 060504(R) (2015) .
[20] M. Ren, Y . Yan, X. Niu, R. Tao, D. Hu, R. Peng, B. Xie, J. Zhao,
T. Zhang, and D.-L. Feng, Sci. Adv. 3, e1603238 (2017) .
[21] Q. Gu, Q. Tang, S. Wan, Z. Du, X. Yang, H. Yang, Q.-H.
Wang, H. Lin, X. Zhu, and H.-H. Wen, Phys. Rev. B 98, 134503
(2018) .
[22] K. Kuroki, S. Onari, R. Arita, H. Usui, Y . Tanaka, H. Kontani,
and H. Aoki, Phys. Rev. Lett. 101, 087004 (2008) .
[23] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys.
Rev. Lett. 101, 057003 (2008) .
[24] S. Graser, G. R. Boyd, C. Cao, H.-P. Cheng, P. J. Hirschfeld,
and D. J. Scalapino, Phys. Rev. B 77, 180514(R) (2008) .
[25] H. Kontani and S. Onari, Phys. Rev. Lett. 104, 157001 (2010) .
[26] A. V . Chubukov, D. V . Efremov, and I. Eremin, Phys. Rev. B
78, 134512 (2008) .
[27] P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep. Prog.
Phys. 74, 124508 (2011) .
[28] Z. P. Yin, K. Haule, and G. Kotliar, Nat. Phys. 10, 845 (2014) .
[29] M. Sato, Y . Kobayashi, S. Chul Lee, H. Takahashi, E. Satomi,
and Y . Miura, J. Phys. Soc. Jpn. 79, 014710 (2010) ;S .C .L e e ,
E. Satomi, Y . Kobayashi, and M. Sato, ibid. 79, 023702 (2010) .
[30] J. Li, Y . F. Guo, S. B. Zhang, J. Yuan, Y . Tsujimoto, X. Wang,
C. I. Sathish, Y . Sun, S. Yu, W. Yi, K. Yamaura, E. Takayama-Muromachiu, Y . Shirako, M. Akaogi, and H. Kontani, Phys.
Rev. B 85, 214509 (2012) .
[31] D. S. Inosov, J. T. Park, P. Bourges, D. L. Sun, Y . Sidis, A.
Schneidewind, K. Hradil, D. Haug, C. T. Lin, B. Keimer, and V .Hinkov, Nat. Phys. 6, 178 (2010) .
[32] C. Zhang, H.-F. Li, Y . Song, Y . Su, G. Tan, T. Netherton,
C. Redding, S. V . Carr, O. Sobolev, A. Schneidewind, E.Faulhaber, L. W. Harriger, S. Li, X. Lu, D.-X. Yao, T. Das, A. V .Balatsky, Th. Brückel, J. W. Lynn, and P. Dai, Phys. Rev. B 88,
064504 (2013)
.
[33] G. Ghigo, D. Torsello, G. A. Ummarino, L. Gozzelino, M. A.
Tanatar, R. Prozorov, and P. C. Canfield, P h y s .R e v .L e t t . 121,
107001 (2018) .
[34] M. B. Schilling, A. Baumgartner, B. Gorshunov, E. S. Zhukova,
V . A. Dravin, K. V . Mitsen, D. V . Efremov, O. V . Dolgov, K.Iida, M. Dressel, and S. Zapf, Phys. Rev. B 93, 174515 (2016) .
[35] M. M. Hrovat, P. Jegli ˇc, M. Klanjšek, T. Hatakeda, T. Noji, Y .
Tanabe, T. Urata, K. K. Huynh, Y . Koike, K. Tanigaki, and D.Arˇcon, Phys. Rev. B 92, 094513 (2015) .
[36] T. A. Maier, S. Graser, P. J. Hirschfeld, and D. J. Scalapino,
Phys. Rev. B 83, 100515(R) (2011) .
[37] T. Saito, S. Onari, and H. Kontani, P h y s .R e v .B 83, 140512(R)
(2011) .
[38] Z.-X. Li, F. Wang, H. Yao, and D.-H. Lee, Sci. Bull. 61, 925
(2016) .
[39] D. F. Agterberg, T. Shishidou, J. O’Halloran, P. M. R. Brydon,
and M. Weinert, Phys. Rev. Lett. 119, 267001 (2017) .
[40] X. Chen, S. Maiti, A. Linscheid, and P. J. Hirschfeld, Phys. Rev.
B92, 224514 (2015) .
081108-5YAMAKAWA, ONARI, AND KONTANI PHYSICAL REVIEW B 102, 081108(R) (2020)
[41] V . Mishra, D. J. Scalapino, and T. A. Maier, Sci. Rep. 6, 32078
(2016) .
[42] D.-H. Lee, Chin. Phys. B 24, 117405 (2015) .
[43] L. Rademaker, Y . Wang, T. Berlijn, and S. Johnston, New J.
Phys. 18, 022001 (2016) .
[44] Y . Yamakawa and H. Kontani, Phys. Rev. B 96, 045130 (2017) .
[45] S. Onari and H. Kontani, P h y s .R e v .L e t t . 109, 137001 (2012) .
[46] Y . Yamakawa, S. Onari, and H. Kontani, Phys. Rev. X 6, 021032
(2016) .
[47] S. Onari, Y . Yamakawa, and H. Kontani, Phys. Rev. Lett. 112,
187001 (2014) .
[48] R. Tazai and H. Kontani, P h y s .R e v .B 100, 241103(R) (2019) .
[49] H. Kontani and Y . Yamakawa, Phys. Rev. Lett. 113, 047001
(2014) .
[50] H. Kontani, Rep. Prog. Phys. 71, 026501 (2008) .
[51] J. Maletz, V . B. Zabolotnyy, D. V . Evtushinsky, S.
Thirupathaiah, A. U. B. Wolter, L. Harnagea, A. N. Yaresko,A. N. Vasiliev, D. A. Chareev, A. E. Böhmer, F. Hardy, T. Wolf,C .M e i n g a s t ,E .D .L .R i e n k s ,B .B ü c h n e r ,a n dS .V .B o r i s e n k o ,P h y s .R e v .B 89, 220506(R) (2014) .
[52] K. Nakayama, Y . Miyata, G. N. Phan, T. Sato, Y . Tanabe, T.
Urata, K. Tanigaki, and T. Takahashi, Phys. Rev. Lett. 113,
237001 (2014) .
[53] T. Shimojima, Y . Suzuki, T. Sonobe, A. Nakamura, M. Sakano,
J. Omachi, K. Yoshioka, M. Kuwata-Gonokami, K. Ono, H.Kumigashira, A. E. Böhmer, F. Hardy, T. Wolf, C. Meingast,H. v. Löhneysen, H. Ikeda, and K. Ishizaka, Phys. Rev. B 90,
121111(R) (2014) .
[54] Y . Suzuki, T. Shimojima, T. Sonobe, A. Nakamura, M. Sakano,
H. Tsuji, J. Omachi, K. Yoshioka, M. Kuwata-Gonokami,T. Watashige, R. Kobayashi, S. Kasahara, T. Shibauchi, Y .Matsuda, Y . Yamakawa, H. Kontani, and K. Ishizaka, Phys. Rev.
B92, 205117 (2015) .
[55] T. Terashima, N. Kikugawa, A. Kiswandhi, E.-S. Choi, J. S.
Brooks, S. Kasahara, T. Watashige, H. Ikeda, T. Shibauchi, Y .Matsuda, T. Wolf, A. E. Böhmer, F. Hardy, C. Meingast, H. v.Löhneysen, M.-T. Suzuki, R. Arita, and S. Uji, Phys. Rev. B 90,
144517 (2014) .
[56] M. Hirayama, T. Miyake, and M. Imada, Phys. Rev. B 87,
195144 (2013) .
[57] J. M. Tomczak, M. van Schilfgaarde, and G. Kotliar, Phys. Rev.
Lett. 109, 237010 (2012) .[58] T. Miyake, K. Nakamura, R. Arita, and M. Imada,
J. Phys. Soc.
Jpn. 79, 044705 (2010) .
[59] Q. Wang, Y . Shen, B. Pan, X. Zhang, K. Ikeuchi, K. Iida, A. D.
Christianson, H. C. Walker, D. T. Adroja, M. Abdel-Hafiez,X. Chen, D. A. Chareev, A. N. Vasiliev, and J. Zhao, Nat.
Commun. 7, 12182 (2016) .
[60] B. Pan, Y . Shen, D. Hu, Y . Feng, J. T. Park, A. D. Christianson,
Q. Wang, Y . Hao, H. Wo, Z. Yin, T. A. Maier, and J. Zhao, Nat.
Commun. 8, 123 (2017) .
[61] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.102.081108 for the details of the tight-
binding model, the formulation of vertex corrections, the orbitalorder without magnetism in bulk and low-doped FeSe, theresults of the DW equation with self-energy, which satisfies theconserving approximation, the origin of the pairing interactionsbeyond Migdal-Eliashberg formalism, and the superconductiv-ity in bulk and low-doped FeSe.
[62] Y . W. Choi and H. J. Choi, Phys. Rev. Lett. 122, 046401 (2019) .
[63] N. E. Bickers and S. R. White, P h y s .R e v .B 43, 8044 (1991) .
[64] H. Ikeda, R. Arita, and J. Kuneš, P h y s .R e v .B 81, 054502
(2010) .
[65] Z. P. Yin, K. Haule, and G. Kotliar, Nat. Mater. 10, 932
(2011) .
[66] J. Kang and R. M. Fernandes, Phys. Rev. Lett. 117, 217003
(2016) .
[67] S. Onari, Y . Yamakawa, and H. Kontani, Phys. Rev. Lett. 116,
227001 (2016) .
[68] S. Onari and H. Kontani, Phys. Rev. B 100, 020507(R) (2019) .
[69] S. Onari and H. Kontani, arXiv:1911.10689 .
[70] K. Kawaguchi, M. Tsuchiizu, Y . Yamakawa, and H. Kontani,
J. Phys. Soc. Jpn. 86, 063707 (2017) .
[71] G. Baym and P. Kadanoff, Phys. Rev. 124, 287 (1961) .
[72] T. Saito, Y . Yamakawa, S. Onari, and H. Kontani, Phys. Rev. B
92, 134522 (2015) .
[73] R. Tazai, Y . Yamakawa, M. Tsuchiizu, and H. Kontani, Phys.
Rev. B 94, 115155 (2016) .
[74] R. Tazai, Y . Yamakawa, M. Tsuchiizu, and H. Kontani, J. Phys.
Soc. Jpn. 86, 073703 (2017) .
[75] R. Tazai and H. Kontani, Phys. Rev. B 98, 205107 (2018) .
[76] N. Fujiwara, N. Kawaguchi, S. IImura, S. Matsuishi, and H.
Hosono, P h y s .R e v .B
96, 140507(R) (2017) .
[77] L. Fanfarillo, A. Valli, and M. Capone, arXiv:1908.10901 .
081108-6 |
PhysRevB.70.085203.pdf | Hysteretic electroluminescence in organic light-emitting diodes for spin injection
G. Salis, S. F. Alvarado, M. Tschudy, T. Brunschwiler, and R. Allenspach
IBM Research, Zurich Research Laboratory, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
(Received 31 March 2004; revised manuscript received 11 May 2004; published 12 August 2004 )
Organic light-emitting diodes with ferromagnetic contacts are fabricated, and their emission intensity is
studied at room temperature for parallel and antiparallel magnetization configuration of anode and cathode.Sweeping the magnetic field applied parallel to the electrode allows the magnetization of the two electrodes tobe switched independently. The electroluminescence intensity for the antiparallel magnetic configuration isfound to be enhanced as compared to the parallel one. We show that this increase is not evidence of spininjection but is a consequence of the magnetic-field dependence of the electroluminescence intensity combinedwith magnetic stray fields from the electrodes.
DOI: 10.1103/PhysRevB.70.085203 PACS number (s): 85.75. 2d, 72.25.Hg, 78.60.Fi, 78.66.Qn
Spin lifetimes in organic materials are expected to be on
the order of microseconds,1making these materials poten-
tially useful for spin-based information-processing devices.An essential requirement for “spintronics” devices is the ef-ficient injection of spin-polarized carriers from a ferromag-netic contact into the active material. In the case of semicon-ductors, such spin injection has been demonstrated bymeasuring the polarization of light generated by recombiningcarriers in a quantum well.
2In III-V semiconductors, the
degree of circular polarization of the emitted photons iscoupled to the spin polarization of the recombining carriersdue to optical selection rules for transitions between the spin-orbit split valence and the conduction-band.
3This scheme is
not applicable to organic light-emitting diodes (OLEDs ),
where the optical selection rules for electric dipole transi-tions are not determined by spin-orbit interaction and there-fore the polarization of emitted photons is not directly relatedto the spin of the electron. However, simultaneous polariza-tion of both electron and hole spins of recombining carriersmay influence the OLED electroluminescence (EL)intensity
because of the nonradiative nature of triplet excitons:
4In the
case of antiparallel configuration of electron and hole spins,
the formation of singlet states is enhanced, increasing theluminescence intensity, whereas triplet states should domi-nate in parallel configuration, thus suppressing light emis-sion. Spin-injection into OLEDs is the topic of increasingresearch activities.
4–7
An important role is attributed to the interface between
the ferromagnetic metal (FM)and the organic material. As
has been shown for FM/semiconductor layers with Ohmiccontacts, large differences in the conductivities of the layersprevent efficient spin injection.
8For inorganic semiconduc-
tors it has been shown theoretically and experimentally that atunnel barrier
9or hot-carrier injection10circumvent the
conductivity-mismatch problem and potentially allow largespin-injection efficiencies. This should also hold for our FM/organic interfaces (see experimental details below ).
In this paper we investigate the EL intensity Iof OLEDs
with magnetic electrodes and find a dependence on the rela-tive magnetization orientation of anode and cathode. We ob-serve an enhancement of Ifor antiparallel magnetic configu-
ration compared with the value for a parallel configuration.This could be an indication that the electron and hole spin-polarizations of the excitons are correlated with the magne-
tization of the corresponding electrodes. We compare thedata with those of samples containing only one magneticelectrode, where no spin-dependent modulation of Iis ex-
pected. In combination with an analysis of the magnitude ofthe effect as a function of the voltage applied across theOLED, we find evidence that in the samples discussed in thiswork the EL modulation originates from the magnetic strayfields emanating from the electrodes. We deduce an upperlimit of 5 310
−5for the product of the spin polarization of
electrons and holes attributable to spin injection.
We present results obtained with OLEDs that consist
of an organic stack of tris (8-hydroxyquinolinato )aluminum
sAlq3dand 2,2 8,7,78-tetrakis (diphenylamino )-9,98-
spirobifluorene (STAD ). Alq 3serves as an electron-
transporting and emitting material, whereas STAD is thehole-transporting material [Fig. 1 (a)]. For the magnetic elec-
trodes, we choose Ni and Ni
0.81Fe0.19(Py)because of the
high degree of spin polarization at the Fermi level in thesemetals.
11,12In addition, their coercive fields are very differ-
ent, allowing us to establish parallel and antiparallel magne-tization orientations of the electrodes. Several samples withthe following anode/cathode combinations were fabricated:Al/Py, Ni/Py, and Ni/Ca. The OLEDs are made on glasssubstrates with prepatterned anodes prepared in a separatemetal-deposition chamber. The thickness of the anode metalthin films is in the range of 50–70 nm.Athin Al
2O3layer is
obtained by oxidizing a 1-nm-thick film of Al deposited ontop of the anode. The final step in the preparation of theanode is the deposition of a thin film of fluorocarbon sCF
xd
in a reactive sputter apparatus. The freshly prepared anode
substrates are inserted into an Ar-filled glovebox. Next, thesubstrates are introduced into an appended evaporationchamber for organic-material and cathode deposition. Theorganic layers, deposited by thermal sublimation, consist of a55-nm-thick film of STAD followed by 50 nm of Alq
3.A
semitransparent cathode is made by depositing a thin film ofLiF(approx. 0.5 nm thick )and a metal thin film of nominal
thickness in the range of 6–9 nm and capped with a2-nm-thick Al layer. The base pressure of the evaporationsystem is in the mid-10
−7mbar range.At the initial stages of
the metal evaporation, the pressure increases up toPHYSICAL REVIEW B 70, 085203 (2004 )
1098-0121/2004/70 (8)/085203 (6)/$22.50 ©2004 The American Physical Society 70085203-110−6mbar. Anode and cathode overlap on an area of 2
33m m2, which is the size of the light-emissive region. Re-
garding the properties of the cathode, we have already shownthat the intercalation of a LiF thin film between the organicand the metal electrode leads to a reduction of the barrierheight at the interface from 1.1 to 0.4 eV.
13In the present
devices we have added a thin aluminium oxide buffer layerbetween the cathode and the LiF layer. Therefore we con-clude that in these devices the cathodes are not Ohmic. Re-garding the anodes, we performed a comparative study be-tween devices having either FM or indium-thin oxide (ITO)
anodes
14and found that significantly lower operating volt-
ages can be achieved when using ITO. This indicates a non-zero potential drop at the FM-organic interface, i.e., the an-odes used in the devices here reported are not Ohmic.Therefore, we do not expect the conductivity mismatch
8be-
tween FM electrodes and organic layers to prevent efficientspin injection in our devices. Further details on anode andcathode preparation will be published elsewhere. TheOLEDs are encapsulated in Ar gas and extracted from theglovebox to perform the magnetic-field-dependent EL mea-
surements. The modified anode and cathode configurationsused in the OLEDs allow us to achieve voltage thresholds of2–2.2 V for the onset of ELand current densities higher than10
−2A/cm2for bias voltages below 6 V. This represents a
considerable improvement when compared to typical OLEDshaving transition-metal electrodes, where typical operatingvoltages are much higher.
4–6The realization of low EL
threshold and high current densities at low bias voltages isnecessary for efficient spin injection into and detection inorganic materials.
The ELintensity Iof the OLEDs is measured at a constant
biasUat room temperature using an unbiased Si photodiode
as a detector.Amagnetic field His applied in the plane of the
sample [Fig. 1 (a)]monitored by an in situHall probe. The
presented data of IsHdis obtained by averaging over several
hysteresis loops, whereby a slow drift of Iwith time has been
corrected by using a linear approximation.
Figure 1 (b)shows the magnetic hysteresis loop obtained
by a longitudinal magnetooptical Kerr-effect measurement ofa Ni/Py sample. It reveals two distinct transitions per sweepdirection, corresponding to the magnetization reversal of thePy and Ni layers occurring at ,2.5 and 17 Oe, respectively.
By appropriate sweeps of Hit is therefore possible to switch
the magnetization of electrode and cathode independentlyinto parallel and antiparallel configuration, as indicated bythe arrows in Fig. 1 (b).
In Fig. 1 (c), data ofI/I
0atU=5.0 V are shown, where I0
denotes the minimum value of I. At this bias, the current
density through the OLED is 2.8 310−3A/cm2.Ahysteretic
behavior in Iis observed with an apparent increase in I/I0of
<0.15% for the antiparallel magnetic configuration as com-
pared with the value for the parallel one. Superimposed onthe hysteresis loop is a monotonic increase of Iwith the
modulus of H.Amagnetic-field dependence of luminescence
is known for crystalline organic materials
15,16and a positive
field dependence has been recently observed for amorphousfilms.
17,18An explanation of these effects is based on a
magnetic-field-dependent conversion rate between singletand triplet states.
19We note that we see the same EL inten-
sity modulations as shown in Fig. 1 (c)if we measure at
constant OLED current. We therefore exclude that thesemodulations are directly induced by changes in the injectioncurrent. As in Ref. 17, we find a negative magnetoresistiveeffect in the injection current at constant voltage. Themagnetic-field-dependent modulation of the device currenthas a similar shape as the ELintensity shown in Fig. 1 (c); its
magnitude, however, is about two orders of magnitude lower.
The EL efficiency modulation induced by the electrode
magnetization reversal could be interpreted as an indicationof spin-polarized charge-carrier injection effects. However,we find a higher EL efficiency for the antiparallel magneti-zation configuration, which seems to contradict the expecta-tion of a higher singlet-to-triplet exciton population ratio forthe parallel configuration.
4This expectation takes into ac-
count that the hole spin is opposite to that of an electron, butneglects the possibility that the two electrodes might emitspins of either majority or minority type. In fact, a higher ELefficiency for the antiparallel magnetic configuration is ex-pected if one of the electrodes acts as a source of predomi-
FIG. 1. (a)Schematic view of the thin-film OLED structure used
in these studies. The magnetic field His applied in the sample
plane. (b)Magnetic hysteresis loop of the OLED with a Ni anode
and Py cathode, as measured with the longitudinal magneto-opticalKerr effect. The two magnetic layers display different switchingfields, enabling the parallel and antiparallel configuration of themagnetization (arrows ).(c)Corresponding OLED intensity IvsH
for up (triangles )and down (squares )sweeps shows intensity en-
hancement for the antiparallel magnetic configuration sU=5.0 V d.
Dotted lines are Lorentz fits to up and down sweeps in the magneti-cally saturated regions, defining two curves that are simply dis-placed in the direction of the field. The height hdefines the relative
intensity change at magnetization reversal of the Ni layer. Area A(shaded )is the difference between I/I
0and the Lorentz fit, inte-
grated over the region with antiparallel magnetization.G. SALIS et al. PHYSICAL REVIEW B 70, 085203 (2004 )
085203-2nantly minority spins while the other is of the majority type.
Thiscouldbethecase,forexample,forNiandFeelectrodes,where the former has a predominant density of minoritystates near the Fermi level whereas the latter exhibits pre-dominant majority spin one.
20,21For the Ni and Py electrodes
used here we cannot be certain of the sign of the spin polar-ization of the injected carriers for a number of reasons: (a)
the electrodes are polycrystalline, and thus the density ofstates is a superposition of contributions from several crystalorientations, (b)the density of states of the FM surface can
be affected by the presence of the oxide layer, and (c), the
relative weighting of the states contributing to the current isdetermined by the injection voltage bias.
In order to understand the hysteretic EL intensity, we in-
vestigate samples with only one magnetic electrode. In suchsamples, no spin-dependent intensity modulation is expectedbecause the charge carrier species injected by the nonmag-netic electrode are not spin-polarized, thus eliminating thedependence of the exciton singlet-triplet ratio on the spinpolarization of the opposite carrier species. In the following,we show results from a Ni/Ca sample and an Al/Py sample.
Figure 2 (a)shows hysteresis loops of the magnetization of
the Ni anode of the Ni/Ca OLED, as obtained by longitudi-nal Kerr-effect measurements. The coercive field of the Nilayer is ,20 Oe. The EL traces are affected by the magnetic
hysteresis [Fig. 2 (b)]: A minimum in Ioccurs at different
fieldsH=H
0,uporH0,downfor up or down sweeps, respec-
tively. Starting from this minimum, Iinitially increases with
identical shape IsH−H0,idfori=upori=down [dotted lines
in Fig. 2 (b)]. At magnetization switching, Iabruptly jumps
from one curve to the other. This behavior can be understoodby assuming that (i)the total magnetic field H
totat the emis-
sive region of the organic layer is composed of the appliedfield,H, and an additional stray field, H
st, originating from
the magnetic electrode layer, and that (ii)the EL intensity is
monotonically increasing with uHtotu. Assumption (ii)is vali-dated in measurements of IsHdin OLEDs with nonmagnetic
electrodes, where the minimum of Iis reached at H=Htot
=0(not shown ). The minimum in IatH=H0,upandH
=H0,downthus indicates that Htotis at a minimum value at
those field positions. Assuming that Hstis collinear with H,
the jump in Iat magnetization reversal is induced by a
change from Hst=−H0,uptoHst=−H0,down. From the data in
Fig. 2 (b)one obtains H0,up=6 Oe and H0,down=−6 Oe. This
suggests that the average stray field is of magnitude 6 Oeand oriented antiparallel to H. This stray field is averaged
over the volume of the emissive region of the OLED, whichis located within a layer that starts at the STAD/Al q
3inter-
face and extends about 10 nm into the Alq 3material. We
note that the size of Hstis about one order of magnitude
larger than expected for a demagnetizing field at a distanceof 40 nm from an ideal, homogenously magnetized Ni filmof thickness 50 nm and lateral extension of 2–3 mm. Pos-sible explanations of this discrepancy are magnetic imperfec-tions and surface roughness of the Ni film (Néel orange peel
coupling
22). This is supported by our observation that the
value ofHstdepends on the conditions of growth of the FM
electrodes. Another indication that the observed Hstis not
simply a demagnetizing field is our observation of a signifi-cant increase in H
stif the thickness of the organic layers is
reduced by a factor of 2.
Figure 3 (a)shows magnetic hysteresis loops of the sample
with a Py cathode and a nonmagnetic anode. The magneti-zation switches at a coercive field of ,6 Oe with a more
gradual transition than in the Ni/Ca sample. In Fig. 3 (b),
measurements of I/I
0vsHare displayed for U=6 V. Similar
to the Ni/Ca sample, magnetization switching affects I, but
whereas in the Ni/Ca sample Idecreases within a few Oe at
magnetization reversal, here Iincreases before magnetization
reversal, and then asymptotically approaches, within,10 Oe, the values of the opposite sweep direction. From
the positions of the two minima of the up and down sweeps,one obtains a stray field of ,1.5 Oe oriented against the
magnetization. If the stray field were simply antiparallel tothe magnetization, no increase in Iwould be expected at
FIG. 2. (a)Longitudinal magneto-optical Kerr measurement of
the OLED sample with a magnetic Ni anode and Ca cathode revealsthe magnetic hysteresis of the Ni layer. (b)Corresponding OLED
electroluminescence intensity Imeasured at U=6.0 V, with the up
(triangles )and down sweeps (squares )falling on two different
curves offset by 2 H
st. At magnetization reversal, Idrops towards
the other curve.
FIG. 3. (a)Longitudinal magneto-optical Kerr measurement of
OLED sample with an Al anode and permalloy cathode. (b)The
hysteretic OLED intensity IatU=6.0 V increases at magnetization
reversal, followed by a slow decrease towards saturation.HYSTERETIC ELECTROLUMINESCENCE IN ORGANIC PHYSICAL REVIEW B 70, 085203 (2004 )
085203-3magnetization reversal, since it would switch from an orien-
tation parallel to Hto an antiparallel one, thus reducing Htot.
An increase of Ican only be explained by the presence of an
additional stray-field contribution that increases Htot, which
could be induced by a stray-field component perpendicular tothe applied field that builds up around magnetization rever-sal. A possible source for such a perpendicular stray field isthe formation of domain walls around magnetization reversalleading to a correspondingly large stray field. An averageperpendicular component of ,7 Oe would be needed to ex-
plain the maximum increase in Iof 0.3% at H=7 Oe. Note
that because Idepends on the modulus of the magnetic field,
the perpendicular stray field associated with domain walls ofdifferent sign does not cancel out. These experiments on thesingle magnetic layers imply that the hysteretic EL depen-dence can be qualitatively explained by magnetic stray fields.However, the observed differences between the Ni and thePy film show that these effects are related to the micromag-netic behavior of these films. We have performed magneto-optical Kerr microscopy studies on these films and find a
different mode of magnetization reversal in Ni than in Py.Whereas in Ni the reversal is driven by domain-wall propa-gation, in Py it is mainly dominated by nucleation, similar toobservations in ultrathin magnetic layers.
23Correspondingly,
the Py domains are much smaller, resulting in a much largeroverall domain-wall length. Hence, the perpendicular strayfield emanating from the domain walls is expected to belarger in Py than in Ni, in line with the EL observations. Wesuspect that these differences in magnetic behavior originatein the different film morphology. Scanning tunneling micros-copy reveals a large roughness of the Py film with a peak-to-peak amplitude of 4 nm, and a rather flat Ni film with aroughness of less than 0.5 nm.
As will be shown in the following, also the hysteretic
increase of Iat the antiparallel magnetization of the Ni/Py
sample [Fig. 1 (c)]can be related to magnetic stray-fields. For
a quantitative analysis, IsHdhas been measured on the Ni/Py
sample for different Uand fit with a Lorentz function I
=I
0f1+DI/s1+H1/22/4sH−H0,id2dg, where DIis a constant,
H0,iis the field position of minimum I, andH1/2is the full
width at half maximum. Up and down sweeps are fitted sepa-rately using data points between −50 and −5 Oe for upsweeps (5 and 50 Oe for down sweeps ). These fits describe
the field-dependence of Iin the region of parallel magneti-
zation very well, as can be seen in Fig. 1 (c)showing fits for
U=5.0 V (dotted lines ). Figure 4 summarizes the parameters
obtained. For this sample, values of H
0,ido not depend on U
and differ by ,20 Oe for up and down sweeps [Fig. 4 (a)].
As both spin-injection efficiency as well as spin transport areexpected to depend on U, the constant field positions H
0,ican
not reflect spin-injection effects, but rather must be explainedby the total stray field, H
st, being the sum of the stray fields
of the Py and Ni layers. The fitted H1/2values [Fig. 4 (b)]
depend neither on Unor on the sweep direction. On the other
hand, DI— which indicates the magnitude of the overall
magnetic-field-dependent EL—monotonically decreases withU[solid line in Fig. 4 (c)]. This tendency was also observed
in Ref. 17.
We define two quantities, handA, as a measure for the
hysteretic behavior of IsHd, and compare their dependenceonUwith the dependence of DIonU. Because DIis ob-
tained by fitting data outside of the hysteretic range, it doesnot depend on the relative orientation of magnetization di-
rections (and thus spin-injection effects ). We define has the
relative change in I/I
0at magnetization reversal of Ni [Fig.
1(c)], andAas the area obtained from the integrated differ-
ence between IsHd/I0and the extrapolated fit from the satu-
rated magnetization region [shaded area in Fig. 1 (c)].Ais a
measure of the change in intensity when Py is magnetizedantiparallel rather than parallel to Ni.Any spin-injection sig-nal would be included in A. The measured values for h
(crosses ),A(open squares ), and DI(line)are compared in
Fig. 4 (c). BothhandAare found to follow the same depen-
dence on UasDI. While it is unlikely that the spin-injection
efficiency is proportional to DI, stray-field effects automati-
cally provide this dependence. This is because a stray-fielddisplaces the curve IsHdin field-direction by an amount that
depends on the magnetic configuration, but not on U, leading
to hysteretic differences in IsHdthat are proportional to DI.
The observed proportionality between Aand DIfor
U.2.7 V indicates that no significant spin injection has
been achieved for these voltages. For lower U, the EL inten-
sity becomes very faint, making it difficult to compare thedependence of AandDIonU.
One can quantify an upper limit for spin polarization by
comparing normalized traces of IsHdmeasured at different
U. In the case of injection of charge carriers without spin
correlation, IsHd/I
0only depends on Uthrough DI. This as-
sumption is valid if H1/2andH0are not affected by U. Fur-
FIG. 4. Parameters of Lorentz-curve fits to measured IvsHfor
the Ni/permalloy OLED. Panel (a)shows the different values of H0
for up (triangles )and down sweeps (squares ), and (b)the full width
at half maximum, H1/2. These two parameters do not depend on U.
(c)The total electroluminescence increase DI(solid line )decreases
withU, with same functional form as the area A(squares )and the
heighth(crosses )of the intensity change at Ni magnetization
reversal.G. SALIS et al. PHYSICAL REVIEW B 70, 085203 (2004 )
085203-4thermore, we have to assume that the stray-field components
of the two magnetic layers do not individually depend on U.
Because of its Lorentz shape, IsHd/I0−1 is directly propor-
tional to DI, and different curves can be scaled by multiply-
ing with a constant factor a. Figure 5 (a)shows two curves
forU=2.7 and 5.0 V measured on the Ni/Py sample, where
the curve at 5.0 V is multiplied by a=2.00. The scaling fac-
torais given by the ratio of DIat 2.7 and 5.0 V, as taken
from the data in Fig. 4 (c). The scaled curve at 5.0 V falls on
the curve at 2.7 V. A detailed analysis of the differences inthe two scaled curves gives an upper limit on the spin polar-ization of the charge carriers. Figures 5 (b)and 5 (c)show the
difference between the scaled intensity data at 2.7 and 5.0 Vfor up and down sweeps of the magnetic field. The fluctua-tions in the data points are mainly due to the experimentalnoise in the EL intensity measurement at 2.7 V. The valuesin Figs. 5 (b)and 5 (c)are averaged separately for regions
with antiparallel and parallel magnetic configuration (solid
lines ). The averaged values deviate by less than 1 310
−4in
intensity between the two magnetic configurations. Thisnumber can be turned into an upper limit for spin polariza-tion. We obtain an EL intensity proportional to 1− p
ephby
assuming that singlet and (nonradiating )triplet excitons form
with equal probability for antiparallel electron and hole spinconfiguration, whereas parallel spins form nonradiating trip-let excitons. Here, p
eis the electron and phthe hole spin
polarization. If at 5.0 V the spin injection is inefficient andthe hysteretic ELentirely due to stray-field effects, the matchof the two curves to within 1 310
−4gives an upper limit for
pephof 5310−5atU=2.7 V. We note that this upper limit
for spin polarization is given by the noise level of the ELintensity measurements at low U.
In conclusion, we have shown that the EL of OLEDs sig-
nificantly depends on magnetic fields, including stray fieldsfrom FM layers. In the search for spin injection into organicmaterials, the possibility that such stray fields mimic the spineffects has to be considered. We observe intensity increasesfor antiparallel configuration of magnetic electrodes of up to0.3%, which would correspond to a spin polarization of bothelectrons and holes of ,4%.After subtracting the stray-field
effect, we find an upper limit for spin polarization of
p
eph.5310−5forUø2.7 V in the investigated samples.
Further experiments will have to explore EL at lower tem-peratures and lower bias ranges where higher values of p
e
andphare expected. In addition, the important role of the
interface layers between the magnetic electrode and the or-ganic material and the possibility of magnetically dead layerswill have to be considered. In order to understand the detailsof the stray-field-induced intensity changes, spatially re-solved intensity measurements will allow the differentiationto be made between averaged and local stray fields.
We acknowledge E. Lörtscher and W. Riess for allowing
us to use their experimental setup, R. F. Mahrt for valuablediscussions, A. Bischof and P. Müller for technical support.
1V. Dediu, M. Murgia, F. C. Matacotta, C. Taliani, and S. Barban-
era, Solid State Commun. 122, 181 (2002 ).
2R. Fieldering, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A.
Waag, and L. W. Molenkamp, Nature (London )402, 787
(1999 ); Y. Ohno, D. K. Young, B. Beschoten, F. Matsukura, H.
Ohno, and D. D. Awschalom, ibid.402, 790 (1999 ).
3Optical Orientation , edited by F. Meier and B. P. Zakharchenya
(Elsevier, Amsterdam, 1984 ).
4A. H. Davis and K. Bussmann, J. Appl. Phys. 93, 7358 (2003 ).
5E. Arisi, I. Bergenti, V. Dediu, M. A. Loi, M. Muccini, M.Murgia, G. Ruani, C. Taliani, and R. Zamboni, J. Appl. Phys.
93, 7682 (2003 ).
6E. Shikoh, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater.
272, 1921 (2004 ).
7Z. H. Xiong, Di Wu, Z. Valy Vardeny, and Jing Shi, Nature
(London )427, 821 (2004 ).
8G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J.
van Wees, Phys. Rev. B 62, R4790 (2000 ).
9E. I. Rashba, Phys. Rev. B 62, R16 267 (2000 ).
10X. Jiang, R. Wang, S. van Dijken, R. Shelby, R. Macfarlane, G. S.
FIG. 5. (a)Relative electroluminescence intensity sI−I0d/I0vs
Hfor the Ni/permalloy OLED collected at 2.7 (circles )and 5.0 V
(solid line ). The data for 5.0 V are scaled by a factor of 2 to show
that the curve matches that of the 2.7 V data. The difference be-tween two datasets is shown for up sweeps in (b)and for down
sweeps in (c)of the magnetic field. Averaging the values over an-
tiparallel and parallel regions (solid lines )indicates a maximum
change of ,10
−4due to spin injection.HYSTERETIC ELECTROLUMINESCENCE IN ORGANIC PHYSICAL REVIEW B 70, 085203 (2004 )
085203-5Solomon, J. Harris, and S. S. P. Parkin, Phys. Rev. Lett. 90,
256603 (2003 ).
11R. J. Soulen, Jr., J. M. Byers, M. S. Osofsky, B. Nadgorny, T.
Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka, J. Nowak,J. S. Moodera, A. Barry, and J. M. D. Coey, Science 282,8 5
(1998 ).
12D. J. Monsma and S. S. P. Parkin, Appl. Phys. Lett. 77, 720
(2000 ).
13S. F. Alvarado and W. Riess, Mater. Res. Soc. Symp. Proc. 665,
121(2001 ).
14S. F. Alvarado (unpublished ).
15R. C. Johnson, R. E. Merrifield, P. Avakian, and R. B. Flippen,
Phys. Rev. Lett. 19, 285 (1967 ).
16N. E. Geacintov, M. Pope, and F. Vogel, Phys. Rev. Lett. 22, 593
(1969 ).17J. Kalinowski, M. Cocchi, D. Virgili, P. Di Marco, and V. Fattori,
Chem. Phys. Lett. 380, 710 (2003 ).
18E. Lörtscher, diploma thesis, ETH Zürich 2003 (unpublished ).
19M. Pope and C. E. Swenberg, Electronic Processes in Organic
Crystals and Polymers , 2nd ed. (Oxford University Press, New
York, 1999 ).
20V. L. Moruzzi, J. F. Janak, and A. R. W. Williams, in Calculated
Electronic Properties of Metals (Pergamon, Oxford, 1978 ).
21Polarized Electrons in Surface Physics , Advanced series in sur-
face science, edited by R. Feder (Word Scientific, Singapore,
1985 ).
22L. Néel, Comptes Rendus Acad. Sci. 255, 1676 (1962 ).
23J. Pommier, P. Meyer, G. Pénissard, J. Ferré, P. Bruno, and D.
Renard, Phys. Rev. Lett. 65, 2054 (1990 ).G. SALIS et al. PHYSICAL REVIEW B 70, 085203 (2004 )
085203-6 |